University of Southern California Dissertations and Theses

All-optical devices based on carrier nonlinearities for optical filtering and spectral equalization

(USC Thesis Other)

All-optical devices based on carrier nonlinearities for optical filtering and spectral equalization

PDF

Download

Open document

Flip pages

Download a page range

Download transcript

Copy asset link

Request this asset

Transcript (if available)

Content NOTE TO USERS This reproduction is the best copy available. __ ® UMI Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ALL-OPTICAL DEVICES BASED ON CARRIER NONLINEARITIES FOR OPTICAL FILTERING AND SPECTRAL EQUALIZATION by Johan Petrus Burger A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2001 Copyright 2001 Johan Petrus Burger Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3027697 UMI UMI Microform 3027697 Copyright 2001 by Bell & Howell Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. Bell & Howell Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UNIVERSITY OF SOUTHERN CALIFORNIA The Graduate School University Park LOS ANGELES, CALIFORNIA 90089^1695 This dissertation, w ritten b y a n p . & Under th e direction o f h .lfk. D issertation Com m ittee, and approved b y a ll its m em bers, has been p resen ted to an d accepted b y The Graduate School, in p a rtia l fulfillm ent o f requirem ents fo r th e degree o f DOCTOR OF PHILOSOPHY Dean o f Graduate Studies D ate May 11, 2001 DISSER TA TIO N CO M M ITTEE ....____________________ _ Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgments I would like to thank Dr. William H. Steier, my advisor, for willingly taking me on as a student in his research group nearly five years ago, for financially supporting me through his research grants, and for seeing me through to grad uation. I thank him for being an accessible and fine teacher, and for tutoring me to think and work according to the scientific method. I sincerely appreciate his integrity and friendly attitude towards me. I am grateful for all the help and personal advice I have received from Dr. Serge Dubovitsky, who contributed in one way or another to most of the work presented in this dissertation. He also provided helpful tips on technical writing. I am thankful to Dr. Dan Dapkus and his group, who supplied us with certain devices for part of our research. The excellent quality of the devices was instrumental in getting good results in the wave-mixing experiments. I would also like to acknowledge him for the epi-wafer that he donated, even when we were not working on a joint project. His good will is appreciated. Dr.Mehrdad Ziari from SDL also supplied us with devices, on more than one occasion, which were successfully used in experiments (mine as well as those of ii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. others), and helped me in creating another chapter in the dissertation. I would, therefore, like to extend my sincere gratitude to Dr. Ziari in this regard. I would like to thank members of my qualifying and defense committees, Professors William Steier, Martin Gundersen, Dan Dapkus, Robert Hellwarth, and Keith Jenkins, for their patient reading, listening and feedback. Several of my fellow colleagues were also very helpful. I would like to thank Dr. Denis Tishinin for his general help in clean room activities, and especially for passing along the knowledge of coating semiconductor active devices and device processing. I guess I wanted to be more of a device processing guy, so Denis fulfilled my dream. Dr. Vadim Chuyanov also helped me on more than occasion, and taught me how to dice with a wafer saw. I thank Dr. In Kim for telling me a few things about MOCVD growth and teaching me how to do photoluminescence measurements. Dr. S. Kalluri also showed me how things worked when he was around. It was a pleasure doing experiments with Dr. Xiaoxing (Daniel) Zhu. All the other ‘polymer dudes’ in our group were also great companions. I would also like to thank Betty Madrid for the excellent and friendly ad ministrative support. Of course my wife, Elvira, had to jointly endure the rigors of Ph. D. study, and I thank her for putting up with it. I am thankful for her unfaltering support iii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. during this time. I am also grateful to her for her help in typing some of references in this dissertation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Contents Acknowledgments ii List of Tables ix List of Figures x Abstract xxvii Preface xxix 1 Introduction 1 1.1 Digital Switching for Optical Communications Applications . . . 2 1.2 Analog Applications ................. 7 1.3 Scope of this Work: All-Optical Devices Based on Carrier Nonlinearities....................................................................... 8 1.4 Summary ......................................................................................... 12 Reference List for Chapter 1 ........................................................... 13 2 An Optical Filter Based on Carrier Nonlinearities for Optical RF Channelizing and Spectrum Analysis 21 2.1 Introduction...................................................... 22 2.2 Broad Area SOA Device Fabrication and Characterization . . . 28 2.3 Experimental P ro ced u re................................................................ 37 2.4 Experimental Characterization of the Filter .............................. 39 2.5 Spectral Analysis with the XWave F i l t e r .................................... 44 2.6 RF Channelizing of Optical S ig n als.............................................. 47 2.7 Time Domain Filtering................................................................... 54 2.8 Noise S ources.............. 59 2.9 Conclusion......................................................................................... 65 Reference List for Chapter 2 ........................................................... 66 v Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 Kerr-Like Nonlinear Mode Converters for Inte grated Optic Device Applications 71 3.1 In tro d u ctio n ...................................................................................... 72 3.2 C o n c ep t............................................................................................. 74 3.2.1 Basic I d e a .............................................................................. 74 3.2.2 Interaction R e g io n ............................................................... 78 3.2.3 Mode Sorting Region ......................................... 78 3.3 Overlap of Nonlinear Material with the Modes of the Device: Placement of Nonlinearity . ........................... . 81 3.3.1 Classic Kerr M e d ia ............................................................... 81 3.3.2 Semiconductor M edia............................................................ 83 3.4 Coherent Interaction........................................................................ 87 3.4.1 Analytic Calculation of Four-Wave Mixing Efficiency . . 87 3.4.1.1 Copropagating p ro b e...................... 100 3.4.1.2 Pump a and probe in the same m ode........... 101 3.4.1.3 Pump b and the probe in the same mode .... 104 3.4.1.4 Counterpropagating p ro b e .............................. 105 3.4.2 Analytical Analysis of Two-Wave Interaction..................... 107 3.4.2.1 Phase shift of gratings in two-wave interaction . 107 3.4.2.2 Two-wave mixing equations.......................... 110 3.4.2.3 Kerr m e d ia ...................................................... I l l 3.4.2.4 Effect of gain g ra tin g s.................................... 112 3.4.2.5 Solution of two-wave mixing equations..........113 3.4.2.6 Efficiency of pump suppression.................... 115 3.4.3 Design Example and S im ulations..................................... 117 3.5 Incoherent In te ra c tio n ..................................................................... 125 3.6 Conclusion.......................................................................................... 125 Reference List for Chapter 3 ........................................................... 126 4 A Theoretical Study of Integrated Optic Limiter Architectures 132 4.1 Introduction....................................................................................... 133 4.2 Limiting Using Kerr N onlinearities............................................... 138 4.3 Numerical Simulation of the Structures by a Mod ified Finite Difference Beam Propagation M e th o d ...................... 140 4.3.1 General Methodology........................................................... 140 4.3.2 Special Modifications of the BPM for Anal ysis of Propagating Fields in Semiconductors................... 145 4.4 The Use of Strained Quantum Well M aterial................................ 151 4.5 The Asymmetric Adiabatic Y-branch with Kerr Nonlinearities . 162 vi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5.1 Mechanims for Limiting in Asymmetric Nonlinear Y-branches.......................................................... 162 4.5.1.1 Introduction ....................................................... 162 4.5.1.2 Conceptual framework: The dig ital optical switch and the local normal mode formalism . ...................................163 4.5.1.3 Limiting mechanism in the Y- branch with a self-focusing Kerr- like nonlinearity ................................................. 168 4.5.1.4 Limiting mechanism in the Y- branch with a defocusing Kerr-like nonlinearity.......................................................... 172 4.5.2 Beam Propagation Studies of Selected Structures .... 173 4.5.2.1 BPM simulation of structures with defocusing Kerr nonlinearities.................. 173 4.5.2.2 A numerical BPM study of Y- branch structures with self-focusing Kerr-like nonlinearities........................................ 178 4.6 Limiting in a Power Unbalanced Interferom eter..............................182 4.7 Methods for the Selective Placing of Semiconductor Kerr-like Nonlinearities in Optical Integrated Structures .... 189 4.8 Conclusion.......................................................................................... 195 Reference List for Chapter 4 ........................................................... 197 5 Optical Limiting in an Optical Fiber Sagnac In terferometer with an Intraloop Saturable Ab sorber 207 5.1 Introduction....................................................................................... 207 5.2 C o n c ep t............................................................................................. 209 5.3 Implementation of the C o n c e p t...................... 210 5.4 Analysis .............................................................................. 215 5.5 Experimental M ethod........................................................................ 222 5.6 Experimental Limiting Results.................................. 228 5.7 Two-Photon A bsorption ........................................................... 234 5.8 Polarization-dependent Phase S h i f t............................................... 237 5.9 Phase Shift Efficiency........................................................................ 241 5.10 Discussion.................................. 242 5.11 Figure of Merit for Lim iter...................................................................246 5.12 Conclusion...................... 248 Reference List for Chapter 5 ........................................................... 250 6 Future Work 254 vii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6.1 Suggestions for Improving Devices based on Four- wave Mixing M edium............... 254 6.1.1 Quantum Dot Gain M edium .................................................. 254 6.1.2 Spin G ratings....................................................................... 259 6.1.3 Sub-Bandgap Wave-Mixing .............................................. 260 6.2 Suggestions for Further Work on Limiters........................................261 Reference List for Chapter 4 .............................. 263 Appendix A P-side Down Mounting of Broad Area Laser B a r s ...................................267 Reference List for Appendix A ........................................................... 271 Appendix B Noise Figure for the Four-Wave Mixing Process in an SO A Due to Amplified Spontaneous E m issio n ................................272 B.l Noise Figure for the XWave F ilter................................................ 272 B.2 Noise Figure in Terms of Measured V a lu es.....................................274 Reference List for Appendix B ....................................................... 279 Appendix C Overlap Factor Corrections in Semiconductors Due to Spatial Carrier Hole Burning in Presence of Diffu sion ................................................................................................... 280 C.l Statement of Problem ................................................................... 280 C.2 First Order Correction of Overlap F a c to r....................................... 284 C.2.1 Weak E x c ita tio n .............................. 286 C.2.2 Mathematical Analysis of Higher Order Corrections . . . 290 C.2.3 Strong Excitation ............................................................292 Reference List for Appendix C ........................................................ 294 Appendix D Derivation of Two-Photon Absorption Terms .............................. 295 Reference List for Appendix D ................................................. 297 viii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Tables Table 3.1 VALUES USED FOR PARAMETERS FOR COM PARING THE ANALYSIS WITH NUMERICAL CALCULATIONS (SEE Fig. 3.5). .............................................. 103 3.2 VALUES OF DEVICE PARAMETERS USED IN THE DESIGN EXAM PLE.............................................................. 120 4.1 THE PARAMETERS USED FOR THE SIMULA TION SHOWN IN FIG. 4.20........................................................... 186 5.1 VALUES OF DEVICE PARAMETERS USED IN THE FIT OF THE WAVEGUIDE ABSORPTION ...................... 237 ix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. List of Figures Figure 1.1 The predicted nonlinear refractive index changes due to the gain change (KK or Kramers-Kronig contribution) and the plasma effect in bulk In- GaAsP which is lattice-matched to InP, when the carrier density changes from 1 x 101 8 cm~3 to 2 x 101 8 m-3. The gain curve for a carrier density of 2 x 101 8 m~3 is also shown. All Coulomb effects, except bandgap shrinkage, were neglected in the calculation of these curves........................................ 2.1 A functional diagram of the XWave filter as it was implemented in this experiment.............................. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2 This figure shows the possible application of a nar row band optical filter to the filtering of RF-waves in the optical domain. In the electronics warfare community this is known as channelizing. In the scenario shown in the picture, a wideband RF band (like the W-band) is picked up by an antenna. This band may contain many different signals of many formats. One might like to convert one of the (wide band) signals to baseband or IF (intermediate fre quency) for further analysis. Normal heterodyning might not work to separate the signals because of their close spacing. This requires some kind of RF filtering (before heterodyning) to separate the sig nals. This can be done by mixing the RF with an optical carrier (the laser spectrum is shown in the figure) and then using the optical filter to separate the signals. Heterodyning by photomixing in a p-n junction diode can then give one the desired base band or IF signal.......................................................... 2.3 A simplified diagram of the broad area semiconduc tor amplifier after device processing (WG=waveguide and QW=quantum well)............................................. 2.4 The light current curve (pulsed) for device #SQW3_C2_8 used in the wave mixing experi ments. The two curves are for the uncoated device and its characteristics after coating with antireflec tion coatings on both facets........................................ 2.5 The electroluminesence measured from the edge of device #SQW3_C2_8 after the application of the antireflection coating. A resolution bandwidth of O.lnm was used. This measurement was taken at a peak current of 1A. The device clearly has strong luminescence in both polarizations, and peaks are also well matched.......................................................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.6 The beam profile of the beam coupled to the broad area SOA. Fig. 2.6(a) is the beam profile in the horizontal direction (parallel to optical table). The vertical control was adjusted for the maximum peak power into the fiber. Fig. 2.6(b) is the beam profile along the vertical direction after the power has been peaked along the horizontal axis. .............................. 33 2.7 The absolute chip gain of the broad area SOA in the TM polarization, as seen by a weak probe. On the same graph is a plot of the relative magnitude of the amplified spontaneous emission, which is pro portional to the gain at higher current............................................ 34 2.8 A diagram of the setup used to characterize gain ripple from the SOA. A grating monochromator (~ 4nm bandwidth) in the Monk-Gillieson configura tion was used so that only light near the laser line of the tunable laser can reach the detector..................................... 34 2.9 The variation in the chip gain for a TM probe, as a function of the probe wavelength. The measure ment was done at an injected current of 1.5A................................ 36 2.10 The measured variation of the gain for the TM po larization as a function of the output power in the TM polarization. ............... 36 2.11 The experimental setup..................... 37 2.12 Definition of the input angle............................................................ 40 2.13 The filter shape for the probe obtained at a total input angle 9 = 7° and an injected current of 1.5A....................... 41 xii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.14 The filter FWHM as a function of the injected cur rent into the device................................................... 41 2.15 The FWHM off the filter is directly proportional to the square of the grating’ s wavenumber, (2t t /A )2, as was measured here, and fitted by using linear regression. The fit enables one to extract some of the material parameters.................................................................... 43 2.16 The FWM (four-wave mixing) efficiency as a func tion of the total injected power in the device. The straight line fit shows a slope of nearly 3, which shows that the efficiency grows as the cube of the total input power................................................................................ 44 2.17 This diagram shows how the XWave filter was used to measure the complex spectrum, that was gener ated when a single frequency laser was modulated with a RF sine wave. The spectra that is shown, is that of the laser before and after modulation................................. 45 2.18 The spectrum of an amplitude-modulated optical carrier as measured by the XWave filter. The inset shows the spectrum as measured by a Fabry-Perot etalon with 7.9GHZ FSR................................................................... 46 2.19 The spectrum of the modulated optical carrier, as measured by the XWave filter, when the pump and signal are TM-polarized..................................................................... 47 2.20 This diagram shows how the XWave filter was used to demonstrate channelizing. Extra 400Hz side bands are added onto the optical carrier and the 3GHz sidebands due to amplitude modulation of the RF signals (compare with Fig. 2.17). Note that the spacing and amplitude of the spectral lines are not to scale.................. 48 xiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.21 The amplitude of the modulation detected at 400Hz for different detunings between the pump/probe and signal when the RF is modulated at 400Hz. The inset shows the (normalized) theoretical pre diction obtained by doing a time domain-based sim ulation of the experiment.................................................................. 49 2.22 An example of the spectrum of light modulated by a Mach-Zehnder optical modulator. Only the positive frequencies of the Fourier transform of Eq. 2.4 is plotted. The following values were used for the parameters u j0 =1000 rad/s, u RF=23 rad/s, ua— 5 rad/s, rai=0.9, ra2=0.25, and /3 m=1. The latter three values correspond to the experimen tal conditions. The spectrum was convolved with g(u> ) = e~u . The upper half of the plot shows the amplitude spectrum and the bottom half the phase spectrum. The 5 rad/s beats in this spectrum is analogous to that of the 400Hz beats in the spec trum obtained in the experiment..................................................... 53 2.23 The amplitude of the modulation detected at 400Hz for different detunings between the pump/probe and signal when the RF is modulated at 400Hz. A TM-polarized pump and signal were used com pared to TE in Fig. 2.21................................................................... 54 2.24 This diagram shows how a single signal was selected by the XWave filter. The probe picks up the mod ulation through the pulsating grating and is routed to the detector.................................................................................... 55 xiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.25 An eye-diagram of the light output from the SOA. The current pulse duration is ~ 500ns. The ob served light mostly originates from the diffracted probe. The modulator is DC biased for maxi mum transmission in this picture. Data bursts are aligned with the ’ ’quiet” part of the pulse. When a data pulse enters, it switches off the light. The kind of trace showed in this picture aids in the lineup of the data, but after alignment the modulator is DC biased for minimum transmission, so that a data pulse switches the light on....................................... 2.26 Two oscilloscope photographs of the input (a) and output pulses (b) for a 30Mb/s pulse train mea sured on 125MHz bandwidth photoreceiver. The time per division is 20ns. The XWave filter is tuned to the carrier frequency of the modulated light. The output pulses were measured in real time (i.e., sin gle shot) using a digital storage oscilloscope. . . . 2.27 The first oscilloscope photograph (a) shows the eye- diagram of a 30Mb/s pseudo-random bit pattern transmitted by the filter. By using the limiting action, due to photoreceiver saturation, the eye- diagram on the left (b) is obtained. This particular eye-diagram (b) was recorded over a period of an hour, but it still shows clear eyes, which translate into error free transmission. The bottom rails of the eye-diagrams are much brighter than the up per rails due to the low duty cycle of the data bursts. An infinite persistence setting was used in the recording of both eye-diagrams......................... Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.28 The output spectrum of the signal filtered with XWave device as measured with an optical spec trum analyzer set to a resolution bandwidth of 0.2nm. The signal is the big peak. It is situated on top of another peak formed by the filtering of the amplified spontaneous emission (ASE) by the pass- band of the Monk-Gillieson monochromator. From this measurement one can deduce a signal-to-ASE background ratio of 35dB.................................................................. 59 2.29 The effect of the finite beam isolation on the out put powers measured at the different output ports is illustrated in this drawing. A beam isolation of 28 dB when TM-polarized pump and signal beams were used at input angle of 9 = 7°. Better isola tion can be obtained for larger angles, but this also adversely affects the four-wave mixing efficiency......................... 60 2.30 This diagram shows how two frequency components of the signal entering the SOA can be mixed by a four-wave process. Apart from generating new harmonics, the mixing also results in the trading of power between frequency components themselves and all their harmonics because of the non-zero linewidth enhancement factor. ........................... 62 3.1 The classical type of nonlinear interaction by opti cal beams in a bulk medium.............................. 75 3.2 The concept of the nonlinear modeconverter device. The upper half of the figure shows the physical lay out of the device. Light is coupled into and out of the interaction region by using adiabatic mode sorters. The lower half of the figure shows the cor responding k-vector diagram............................................................. 77 xvi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 To prevent diffusion in semiconductor media, which washes out nonlinear gratings written by interfering modes, the gain media has to be placed assymetri- cally in the waveguide formed by the lateral index steps. This can be done by placing the quantum well gain either assymetrically (a), or symmetri cally with a central gap in the optical confinement region(b).............................................................................................. 86 3.4 The different fields involved in the nonlinear interaction............... 89 3.5 A comparison of the mixing efficiencies obtained by analytical calculation with the exact results. The diamonds represent the exact results, the solid line is calculated using Eq. 3.52, and the dashed line with Eq. 3.53. The gain of the amplifier is also shown in the dash-dotted line. The parameters used in the calculations are shown in Table 3.1 102 3.6 The analytically calculated (Eqs. 3.80 and 3.84) power transfer between pumps a and b. The pa rameters used in the calculations are shown in table 3.1. r 0 i = 0.022 in this calculation. It is assumed the powers in field a and b are both 85 / j,W at the start of interaction region. Also shown is the phase shift between the grating and interference pattern............................ 115 3.7 The cross-section of our design example. The de vice has a ridge waveguide structure with a sepa rate confinement heterostructure gain region. Gain is provided by six compressively quantum wells In- GaAs embedded in InGaAsP waveguide. The wells are etched out in the central region to prevent the washing out of a carrier grating by diffusion...................................... 117 3.8 A contour plot of the electric field magnitude of the modal distribution of the lowest order mode of the waveguide in Fig. 3.7............................................................................. 118 xvii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.9 A contour plot of the electric field magnitude of the modal distribution of the first order mode of the waveguide in Fig. 3.7......................................................................119 3.10 The effective gain seen by the two pumps A and B after propagation through the nonlinear diffraction region. Their respective gains critically depend on the detuning between the pumps.................................................... 121 3.11 This figure shows how light is scattered from the probe (C) into a field D by a grating formed by pumps A as B. The efficiency of the interaction is critically dependent on the detuning between A and B.................... 122 3.12 One possible design for a mode sorter based on an asymmetric Y-junction. The quantum well con taining layer has been partially etched out under one arm of the gain medium to reduce the effective index in this arm.................................................................................... 123 3.13 The extinction ratio of the local normal modes at the end of an asymmetric Y-branch as a function on the length of the length of the Y-branch, assuming a design similar to that shown in Fig. 3.12. The two branches are separated by 5.Qfim at the end of the branch. The characteristics of two types of tapers is shown in this graph: a straight taper and a shaped taper defined in the text................. 124 4.1 The transfer characteristics of the ideal power lim iter, which can either take the form of an optical hard limiter (solid line) (which requires a certain threshold to switch on) or an optical clamp (dashed line) (which has a linear transfer function up to the limiting threshold). Either characteristic may be preferable depending on the precise application.................................133 xviii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.2 The layout of the wavelength division multiplexed optical analog-to-digital conversion (ADC) system. A single ultrashort pulse with a very wide spec trum is generated by a mode-locked erbium doped fiber laser (EDFL), followed by spectral broaden ing in a nonlinear fiber. This pulse is then spec trally sliced and the slices differentially delayed to generate a string of pulses, by using an arrayed waveguide grating (the first wavelength demulti plexer or WDM). These pulses sample the analog signal via mixing in an amplitude modulator. Dif ferent sampling pulses of different wavelengths are then routed to different quantizers (photodetectors and electronic ADC’s)........................................................................... 135 4.3 The limiting characteristics of semiconductor waveguides with various types of nonlinearities. The solid curve is for a semiconductor optical am plifier with only gain saturation present (no ultra fast effects). The dashed curve is for a passive semi conductor waveguide exhibiting only two-photon absorption (TPA) 137 4.4 The apertured nonlinear waveguide optical limiter (all-optical cutoff modulator). The core contains a defocusing Kerr-type nonlinearity.......................... 139 4.5 This graph shows the relationships between the ma terial band-edge wavelengths for a compressively strained quantum well device and the input light spectrum of the light that is being is switched, by a device based on this material. ......................... 152 4.6 The gain for TE-polarized fields in a 7.5nm wide 1% compressively strained InGaAsP quantum well embedded in lattice-matched InGaAsP with a bandgap of 1.06/rm for carrier densities from 1V=1 x 101 7 cm~3 to N=5 x 101 8 cm~3 in ten equal steps. . 154 xix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.7 The gain for TM-polarized fields in a 7.5nm wide 1% compressively strained InGaAsP quan tum well embedded in lattice-matched InGaAsP with a bandgap of 1.06/un for carrier densities from iV=l x 101 7 crn~3 to N = 5 x 101 8 cm~3 in ten equal steps..................155 4.8 The nonlinear index change for the TM polariza tion due to the Kramers-Kronig transformation of the gain changes in a 7.5nm wide 1% compres sively strained InGaAsP quantum well embedded in lattice-matched InGaAsP with a bandgap of 1.06/wn for carrier densities from N = l x 101 7 cra-3 to N=5 x 101 8 cm~3 in ten equal steps..................................... 156 4.9 The total nonlinear index changes in a 6nm In- GaAs quantum well embedded in InP-based ma terial. The solid line represents the index change for the TM polarization and the dashed line the in dex change for the TE polarization. The curves are for carrier densities of 5 x 101 7 cmT3 to 4 x 101 S cm-3 in 5 x 101 7 cm~3 steps. The semiconductor with a carrier density of N=1 x 101 6 cm~3 is the reference....................... 161 4.10 A diagram of the generic structure discussed in Paragraph 4.5. In order to ensure adiabatic mode evolution the angle between the branches is typi cally very small. One of the arms contains a non linear material which can exhibit a nonlinear index change, that can be either be switched by a voltage (in the case of the digital optical switch), or by all- optical means in the case of the limiter. In this way asymmetry in the effective indices of the branches can be switched. Light propagation can either be from one of the branches to the stem or vice versa............................163 4.11 An illustration of the operation of the adiabatic nonlinear y-branch limiter, with a self-focusing nonlinearity in the one branch (left).................................................... 169 xx Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.12 The nonlinear Y-branch with a defocusing nonlin earity in one arm. It is assumed that the dual po larization scheme (as discussed in Paragraph 4.4) is used.................................................................................................. 172 4.13 The approximate structure used in evaluating the adiabatic Y-branch with defocusing nonlinearity. The branch marked by — n2 contains the defocusing nonlinearity....................................................................................... 174 4.14 The limiting characteristics of the structure shown in Fig. 4 .1 3 ........................... 177 4.15 The power-dependent transmission of the structure shown in Fig. 4.16..................................................................................179 4.16 The structure with a self-focusing type of nonlin earity that can be used to obtain the limiting char acteristics shown in Fig. 4.15 180 4.17 The propagation of the field in the structure shown in Fig. 4.16 for an input power of P=100mW. The solid lines are the outlines of the waveguides.................................. 181 4.18 The power unbalanced interferometer. This is a top-down view of a Mach-Zehnder interferometer. ................ 183 4.19 This diagram shows how a nonlinear phase shift and the interferometer transmission can be com bined to obtain an ideal limiting characteristic. This method works the best when the phase shift saturates at 7 r as shown here.................................................................184 xxi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.20 The limiting characteristics of a device similar to that shown in Fig. 4.18. Pumped quantum wells are used as the Kerr-like material. The device pa rameters used in this calculation are shown in Table 4.1..................185 4.21 The limiting characteristics of a device similar to that shown in Fig. 4.18. Saturable absorptive quantum well material is used as the Kerr-like medium in the dual polarization scheme.............................................188 4.22 The effect of impurity-induced disordering (IID) on the shape of energy band profile of the conduction band of a quantum well. Note the shift in the en ergy of the allowed states of the well. The same thing happens in the valence band, which results in a net blue shift of the bandgap.................................................... 190 4.23 The shift in room temperature photoluminescence (PL) brought about by vacancy-induced disorder ing in samples of InP-based material (See Fig. 4.24), which contain strained InGaAsP quantum wells. The samples were selectively capped with a thin layer of S 1O2 and then rapidly thermally an nealed (RTA). Three curves are shown for the PL for coming from a small area on the wafer. The first curve is for the original material which is not ther mally annealed, the other for material which was put through RTA but not capped and the third for an area that was capped and annealed................................................192 4.24 The layer structure of the sample used in the in termixing experiments. Q(1.2) in the annotation means InGaAsP lattice-matched to InP and with a bandgap of 1.2fim................... 194 xxii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.1 A polarization-sensitive saturable device placed as- symetrically in a Sagnac loop can be used to create a nonreciprocal phase shift for a probe. TE light is absorbed by the waveguide and generates a phase shift by carrier nonlinearities while the TM light is transmitted and only reads the nonlinear phase shift....................... 212 5.2 A nonlinear intensity dependent phase shift ele ment in a Sagnac interferometer can be used to produce a limiter at the transmitting port. The upper right hand graph shows a nonlinear nonre ciprocal phase shift generated in a Sagnac interfer ometer. When this characteristic is combined with the transmission characteristic of the transmitting characteristic of the unbalanced Sagnac interferom eter (lower left hand graph), one obtains the limit ing characteristic shown on the right hand graph..............................213 5.3 The layout of the experiment. ................................................... 223 5.4 The autocorrelation trace obtained at connector #1 (See Fig. 5.3) when equal power is launched along both eigenaxes of the PM fiber. The autocorrela tion is based on second harmonic generation and is highly polarization-sensitive. In order to detect two polarizations, a waveplate was used to rotate the in put light going into the correlator (which contains a TE and TM pulse) by 45 degrees, with respect to the sensitive axis. This figure shows that greater than 4ps of delay is obtained in the 2m of PM fiber......................... 224 5.5 The electroluminescence in both the TE and TM polarization measured from the edge of the com mercial semiconductor optical amplifier (SOA) de vice, at a current of 20mA. This is well below the material gain transparency current of ~ 45mA.............................. 225 xxiii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.6 The magnitude of the modulation in the voltage across the p-n junction of the SOA as a function of the current into the junction, caused by a modula tion in the input light. The minimum shifts as a function of the average input power of light, which is unexplained at this time. The legends show that the polarization is also varied and also results in a shift in the transparency current. (TE:TM is the TE to TM polarization ratio). .................................................... 227 5.7 Limiting characteristics measured in experim ent..............................229 5.8 The spectrum of the mode-locked laser before and after the limiter (measured minutes apart). Take into account that the time averaged spectrum is also slowly fluctuating in time.......................................................... 230 5.9 The interferometer intraloop transmission as a function of energy (crosses), as well as the transmis sion of the pulses through the nonlinear waveguide for the clockwise direction (circles), and counter clockwise direction (diamonds).............................................................233 5.10 By fitting the interferometer transmission shown in fig5.9, one obtains a nonreciprocal phase shift which seems to saturate as a function of e n e rg y .............................234 5.11 Absorption characteristics of the waveguide when TM light only is launched. The energies are mea sured inside the device. The transmission is mea sured between the device facets and only includes the estimated on-chip losses..................................................................238 xxiv Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5.12 Change in the relative interferometer transmission as a function of the input polarization state. The ratios shown are the ratio of TE:TM light of the input pulse. The total input power is kept the same for every polarization ratio................................... . 240 5.13 The change in the transmission of the Sagnac inter ferometer due to only the TE (pump) pulse because the counterpropagating TM waves has been equal ized by adjusting the coupler................................... 242 5.14 The output power characteristics for a generic lim iter, and the definitions of parameters associated with the figure of merit of such a device. The de vice exhibits hmiting between points A and B. The noise reduction is graphically illustrated by the re duction in the size of sinusoid in the hmiting region..................... 248 5.15 An energy limiting characteristic obtained with the Sagnac loop limiter, which is used in the figure of merit calculation................................................................................. 249 6.1 Illustration of lateral quantum dot to barrier diffu sion for four different quantum dot states (by Kim 6.2 A diagram of a single-mode waveguide containing quantum dots. Counterpropagating fields write a carrier grating which can be used for contradirec- tional coupling of a probe beam. This reflective carrier grating can exist because of the quantum confinement in the quantum dots, preventing dif fusion. (Note: this drawing does not show actual relative scales).........................................................................................257 et al., ch.6,ref.[2]) 256 XXV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A.l A scanning electron microscope photograph of a series of broad area lasers mounted on a copper heatsink...................................................................................................267 A.2 A top view of the heatsink on which lasers were mounted and diced with a wafer saw. The lasers were peeled off by hand to reveal voids in the solder.......................269 A. 3 A photograph of a laser bar mounted p-side down on a copper heatsink using PbSn and a water sol uble soldering flux. The solder only sticks to the gold-coated p-side contacts, but not the SiN x in the other areas........................................................................................270 xxvi Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract InGaAsP-based quantum wells can display nonlinear refractive index changes of ~ 0 .1 near the band-edge for intrawell carrier density changes of 1 x 1 0 18cm~3, due to effects like bandfilhng and the plasma effect, which make these materials promising for the realization of all-optical signal processing de vices, as demonstrated here. A novel single passband filter with sub-gigahertz bandwidth and greater than 40nm of tunability was experimentally demonstrated. The filter uses the detuning characteristics of nearly degenerate four-wave mixing in a broadarea semiconductor optical amplifier to obtain frequency selectivity. The key to this demonstration was the spatial separation of the filtered signal from the input signal, based on their different propagation directions. An analysis of an analogous integrated optic dual-order mode nonlinear mode-converter, with integrated mode sorters which separate the signal from the interacting modes, was also undertaken. This device is promising as a filter, a wavelength con verter, notch filter, and a wavelength recognizing switch. Novel ways to prevent carrier diffusion, which washes out the nonlinear grating, were suggested. It is xxvii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. important to have a large mutual overlap to modal overlap ratio of the two in teracting modes on the nonlinear medium, because the mixing efficiency scales as the fourth power of this number. Three types of integrated optic limiters (based on Kerr-like nonlinearities) namely an all-optical cutoff modulator, a nonlinear Y-branch and an interfer ometer with an internal Kerr element, were theoretically investigated. A beam propagation program, which can solve the propagation of an optical field in a semiconductor in the presence of carrier diffusion, was developed for the numer ical analysis of these structures. A negative feedback mechanism was identified in the Y-branch devices and a new limiting configuration was discovered in a Y- branch with a selectively placed defocusing nonlinearity. Dichroic materials like compressively strained quantum wells are a promising material for the limiters. A novel wavelength insensitive pulse limiter based on a polarization-maintaining fiber Sagnac loop with such an intraloop dichroic waveguide was demonstrated, and displays superior noise reduction characteristics compared to a saturating optical amplifier/ two-photon absorption waveguide. xxviii Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Preface Im achten Manne ist ein Kind versteckt; das will spielen. - F. Nietzche xxix Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 1 Introduction The work presented in this dissertation is concerned with novel devices for all- optical processing of signals transmitted by optical fiber. The meaning of all- optical is basically ‘light switched by light’ in analogy with the (all) electronic transistor. These types of devices are of great importance in the fields of optical communications, signal analysis, radio frequency signal processing, and optically assisted analog-to-digital conversion. A small survey of the current state-of-the- art devices and applications is shown in this chapter (paragraphs 1 .1 and 1 .2 ), illustrating the pressing need for these devices and the significant progress made in the last few years in the field of optical processing. These types of devices are becoming more practical and are quite likely to be introduced into ‘real world’ applications, even at commercial levels. The material properties which make these devices possible are briefly discussed in Paragraph 1.3. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.1 Digital Switching for Optical Communications Applications The largest driver for optical signal processing research has been the field of optical communications, in which applications like wavelength conversion [29, 36, 31, 64], simple boolean operations [45, 22, 41, 23], all-optical regenera tion [30, 46, 63, 47], clock recovery [48, 59], and demultiplexing [19, 20], all via all-optical means, have been investigated. At the time of this writing, so-called dense wavelength division multiplexing (DWDM) has become the prevalent and, in fact, ubiquitous technique to increase composite data rates on a single op tical fiber beyond 10 Gb/s. Currently there is a push to enable DWDM to move from being a pure transmission technology into a state where it can be applied in so-called transparent all-optical networks [11]. In other words, in current networks which employ EDFA’ s (erbium doped fiber amplifiers) optical data transmission is a largely analog endeavor, with signal processing functions such as demultiplexing of data tributaries and signal verification restricted to electronic equipment at link endpoints or nodes [24]. On the other hand, in all-optical networks one strives to keep the signals in the optical domain, even when routing or switching at some network node. This has the possible bene fits of scalability, modularity, and survivability of the optical network [1 0 ]. A particularly good example of an all-optical processing function that can provide significant advantages in terms of these attributes is wavelength conversion [1 0 ]. 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Wavelength conversion is the technique, witch which the carrier or center fre quency of a signal is shifted to another wavelength. Wavelength conversion can be used to interface different networks, or it can be utilized to route and switch wavelengths while ensuring added functionalities, such as contention resolution and blockage removal, which are vitally needed under dynamic traffic patterns. Wavelength conversion can be accomplished by detecting a signal on a detector and remodulating the electrical signal onto another laser. To avoid the signal degradation associated with the unneeded optoelectronic and electrooptic tech niques, techniques that rely on all-optical conversion have been investigated [1 0 ]. The all-optical wavelength conversion techniques include cross gain modulation (XGM) in an optical amplifier (usually a semiconductor optical amplifier-SOA), interferometric converters that utilize a Kerr-like element’ s (usually an SOA) nonlinear cross phase modulation (XPM) to switch, or wave-mixing (four-wave mixing in SOA or difference frequency generation in a material like LiNbO 3). Recently it has been recognized that the XPM type of wavelength converters have nonlinear transfer functions that make them suitable for noise suppression and, therefore, regeneration as well [61, 62], These interferometric devices offer 2R regeneration (re-amplification and re-shaping) with full bit-rate transparency or 3R regeneration (i.e., re-timing is also included) [30]. The regenerative capa bilities of the wavelength translators also ensure good cascadability, with up to 75 cascaded wavelength conversions experimentally demonstrated at 2.5 Gb/s 3 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. with a 2dB penalty [36], which was 8 times more than could be obtained with optoelectronic translators with comparable small signal bandwidth. Wavelength converters based on four-wave mixing have the interesting property of conjugat ing the input (spectral inversion), which can be used for mid-span dispersion compensation in high bit rate systems [34]. It must be stressed that in many of these devices (like the wavelength con verters based on XPM and XGM) the processing function’s character is not really different from electronic processing in the sense that optical transparency is broken. Therefore, the choice of processing technique in a particular applica tion context is likely to depend on practical factors such as performance, cost, size, and power consumption. In this context all-optical processing may have advantages because the receiver/ transmitter interfaces required for an electronic device may be to some degree be ‘ integrated’ / economized in an all-optical im plementation. The relative economy and performance of electronic solutions also drop rapidly as the bit rate increases. Newer modulation formats and proposed bitrates seem to necessitate an increased channel bandwidth in DWDM systems, which will stretch switching electronics to its limits. A common DWDM channel bit rate utilized in present systems is lOGb/s and uses the NRZ (non-return-to-zero) ASK (amplitude shift keying) format, so that the total required system bandwidth is just greater than 5GHz. The current trend is to move the channel bit rate to 40Gb/s [67], and 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. replace the NRZ format with RZ modulation (return to zero), which has better transmission characteristics [40]. RZ replaces the NRZ logical ON databit with a pulse. Even better transmission performance can be obtained when the pulses are pre-chirped [14]. When the pulses get very short and have high enough power, so-called solitons may be formed. The newest incarnation of systems based on soliton transmission goes by the name of DMS (dispersion managed solitons) [7]. CRZ (chirped RZ) and DMS formats are in practice actually pretty close to each other in character [40]. These RZ, CRZ, and DMS data formats have many Fourier components and for a lOGb/s system the system bandwidth is at least 10GHz. The better performance of RZ modulation compared to NRZ modulation can be mainly attributed to the enhanced power tolerance of single pulses, for which distortions due to the limiting nonlinear fiber effect (XPM) can be balanced with fiber dispersion [37]. It was calculated that changing the transmission system modulation format from NRZ to RZ gives the same benefit as replacing the link’s SMF (standard single-mode) fibers with NZDF (Non-zero dispersion shifted fibers) [13]. When SMF is used, the RZ format enables one to increase the transmission span length with 2dB at lOGb/s or 6dB at 40Gb/s [13]. The higher power tolerance of RZ leads to enhanced power budget, which is of particular significance for island hopping to avoid remotely pumped amplifiers or expensive installation of (undersea) amplifiers [37]. RZ and especially chirped RZ are also more robust against effects like polarization 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. mode dispersion [28]. Experimental results seem to confirm the theory and show that error free transmission can be achieved over 432km for RZ at 40Gb/s and only 218km for NRZ using the same bit rate and fiber [33]. Higher bit rates up to lOOGb/s over 100km have been demonstrated using DMS transmission [27]. The high bandwidth characteristics of these newer modulation formats place more stringent requirements on any signal processing component, and it is not yet clear if current semiconductor electronic technology will be sufficient to handle all of the needed functions. Semiconductors based devices are also typically used for all-optical signal processing, and it may also be asked if these devices have sufficient bandwidth [39], since the switching speed is seemingly determined by the carrier lifetime, which is of the order of Ins. However, traveling-wave effects have to be taken into account to understand the frequency response of wavelength conversion based on XGM and XPM in SOA’s [25]. The wavelength converted signals may be high frequency-filtered due to saturation effects which increase bandwidth beyond what one would normally expect. Optimized SOA’s are expected to have nonlinear bandwidths exceeding 90GHz [25]. Ultrafast effects in the devices can extend the bandwidth further [39], when very short and high power input pulses are used. Structural methods, like differential delay techniques in interferometers, are also used to get rid of slow carrier density recovery effects [24]. In a very simple SOA delayed-interference loop device up to lOOGb/s signals have been wavelength-converted [32]. 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.2 Analog Applications Photonics can also be used in the transmission and processing of analog signals. Optically assisted analog-to-digital conversion of high speed electronic signals [4, 2] can also be considered an analog application of photonics. The main im petus for this work seems to be the wide bandwidth and low loss capabilities of photonics. Linearity of transmission and all signal processing functions, like wavelength conversion, are quite important for certain analog systems. Cross gain modulation (XGM) or four-wave mixing(FWM) [56] in a SOA can be used to do the wavelength translation of analog signals. XGM results in more 2nd harmonic distortion than FWM [56]. Four-wave mixing preserves phase and amplitude, and can be used with more complicated general modulation schemes than AM (amplitude modulation). Nevertheless, by using XGM in SOA’s, in teresting functionalities, like all-optical microwave mixing [51] and an all-optical microwave notch filtering [5], have been demonstrated. XPM switching in an interferometer can also be used to generate a wide fractional bandwidth ASK (amphtude shift keying) digital RF signal [15]. Besides the modulation format transparency and superior linearity attributes of FWM, it also offers the oppor tunity for ultra-broadband switching by using the ultrafast effects in the SOA [66]. Diez et al. [6 ] demonstrated the sampling of a 160Gb/s data signals with 1.7ps temporal resolution and high linearity, based on FWM in an SOA. 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.3 Scope of this Work: All-Optical Devices Based on Carrier Nonlinearities The work summarized in this dissertation is concerned with the realization of novel all-optically switched waveguide devices based on the quaternary mate rial InGaAsP, which is grown epitaxially on InP substrates. The emphasis is on engineering, fabrication and characterization of novel types of devices, based on carrier nonlinearities, near the bandgap of these materials. In order to under stand the inner workings of these devices, a short explanation of the nonlinear materials properties is called for. Semiconductor materials display a rich variety of nonlinear optical effects that can be utilized for optical signal processing applications. Illumination of the semiconductor can lead to interband transitions (either absorption or stimulated emission, depending on the carrier density), which result in a change in the car rier density and carrier-dependent dielectric permittivity of the material. Vari ous effects are responsible for this change of the material properties, depending on the time duration and intensity of illumination, and the material’s prevailing background carrier density. The nonlinear effects include, but are not limited to, spectral hole burning, carrier heating or cooling [16, 65, 38, 60, 35], bandgap renormalization [43], screening of the excitons or the continuum Coulomb en hancement [43], bandfilling [43], the plasma effect [18, 57, 55], intervalence band absorption [52, 1, 12, 26], thermal nonlinearities [21] and two-photon absorption 8 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [53, 16]. Because of the causality principle, the gain changes due to these effects have to be accompanied by a refractive index change. The relationships between the real and imaginary parts of the medium response are known as the Hilbert transforms [44] or, in the physics community, as the Kramers-Kronig transforms [3]. Given the spectral dependence of the optically induced gain change, Ag(co), over the whole spectrum, one can calculate the corresponding index change [3], An(uO = r dfi, (1.1) 7T«o J _ o o f t 2 ~ ^ where u is optical angular frequency, c the speed of light, nQ the background refractive index, and Ag the spectral dependence of the material gain change. The devices discussed here are based on direct bandgap semiconductors like InGaAsP, with bandgap energies between 0.67eV and 0.8eV. The materials are illuminated with light just above the bandgap energies. The background carrier density is such (by carrier injection) that the exciton effects are already screened out. The ultrafast effects (carrier heating, spectral heating, and two-photon absorption) also play a small role when the input light is continuous and has relatively low intensity (~10 M W /cm 2 or less). The largest contributions to the changes in the materials properties due to the optically induced carrier density changes come from the effects of bandfilling and the plasma effect [18, 57]. Bandfilling is the change in occupancy of the bands, and results in gain changes Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. for interband transitions. Electronic carriers in the device also move in the optical field according to Newton’s Second Law, which produces a polarization proportional to the electrical fields, which reduces the refractive index. By using the effective mass approximation [54] in semiconductors and assuming the classical treatment of the interaction is correct, the refractive index change due to the electric field interaction with carriers can be shown to be [50] Na2 A n = - ■ ■ ■ ■ »... , (1.2) 2 urm*n0 where q is the electron charge, N the carrier density, c o the optical angular frequency, n0 the background refractive index, and m* the effective mass of the carriers. A numerical calculation in MATLAB for bulk InGaAsP gives the reader a good feel for the relative magnitude of the nonlinearities (See Fig. 1.1). The magnitude of the nonlinear index changes shown in Fig. 1.1 agrees quite well with experimentally measured values [49]. The index changes in quan tum well materials with the same bandgap are about an order of magnitude larger [9, 8]. There are, therefore, quite big Kerr-like nonlinearities in these types of materials, which can result in big changes in the phase and shape of a waveguide mode as a function of the injected optical power. The self-phase modulation coefficient of multiple-quantum-well optical amplifiers, for exam ple, varies between 16 x 104W ~1 m~1 and 2.7 x 104W ~1 m~1, which is more 10 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. than seven orders of magnitude larger than the number for single-mode opti cal fibers, which is 1.8 x 10~zW ~lm~l [58]. Due to the asymmetric shape of x 10 \ —y 0 3 O w o o o C M 0 3 -10 .2 TOTAL INDEX CHANGE PLASMA EFFECT KK- CONTRIBUTION -14 -14 1.1 0.8 1 0.6 0.7 0.9 Energy (eV) Figure 1.1: The predicted nonlinear refractive index changes due to the gain change (KK or Kramers-Kronig contribution) and the plasma effect in bulk InGaAsP which is lattice-matched to InP, when the carrier density changes from 1 x 101 8 cra“3 to 2 x 101 8 ra-3. The gain curve for a carrier density of 2 x 101 8 m-3 is also shown. All Coulomb effects, except bandgap shrinkage, were neglected in the calculation of these curves. the gain/absorption (and carrier-dependent gain/absorption changes) in semi conductors and the nearly constant negative nonlinear refractive index change close to the bandgap, due to the plasma effect, one always finds a big negative index change (for increasing carrier density) close to the bandgap. The change 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in refractive index is commonly described by the so-called Henry linewidth en hancement factor, which relates the real and imaginary parts of the differential carrier density (N) dependent refractive index (n) [42]: = d[He{n}]/dN d[Im{n}\/dN' { ] a is always positive close to the band-edge and has a value of approximately 5-6 in InP/InGaAsP lasers [42]. The finite value of a at the gain peak helped Henry [17] to explain the reason for the larger than expected (from conventional laser theories) linewidth of semiconductor lasers, which he showed to be propor tional to (1 + a 2). This factor also plays an important role in describing some of the optical signal processing devices in this dissertation, and a large a is, for example, desirable in devices based on four-wave mixing in semiconductor waveguides. 1.4 Summary All-optical signal processing is important for optical digital communications and analog systems. The research summarized in this dissertation focuses on the use of waveguide devices based on InGaAsP, to create novel, all-optical signal processing functionalities. The InGaAsP semiconductor displays large carrier 12 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. nonlinearities around the bandgap. The main contributions to these carrier nonlinearities come from bandfilling and the plasma effect. 13 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1 ] G. P. Agrawal and N. K. Dutta, Semiconductor lasers, 2nd ed., pp. 139- 142, Van Nostrand Reinhold, New York, 1993. [2] A. S. Bhushan, F. Coppinger, S. Yegnanarayanan, and B. Jalali, Nondisper- sive wavelength-division sampling, Opt. Lett. 24 (1999), no. 11, 738-740. [3] S. L. Chuang, Physics of optoelectronic devices, pp. 671-674, John Wiley and Sons, New York, 1995. [4] F. Coppinger, A. S. Bhushan, and B. Jalali, Time magnification of electrical signals using chirped optical pulses, Electron. Lett. 34 (1998), 399-400. [5] F. Coppinger, S. Yegnanarayanan, P.D. Trinh, and B.Jalali, All-optical RF filter using amplitude inversion in a semiconductor optical amplifier, IEEE Trans. Microwave Theory Tech. 45 (1997), no. 8, 1473-1477. [6 ] S. Diez, R. Ludwig, C. Schmidt, U. Feiste, and H. G. Weber, 160 Gbit/s op tical sampling by a novel ultra-broadband switch based on four-wave mixing in a semiconductor optical amplifier, Tech. Digest OFC 1999, 1999, Paper PD38, pp. 381-383. [7] N. J. Doran, Dispersion managed soliton systems, Proceedings of the Euro pean Conference on Optical Communication (ECOC ’98) (Madrid, Spain), September 1998, pp. 97-99. [8 ] J. E. Ehrlich, D. T. Neilson, and A. C. Walker, Carrier-dependent nonlin earities and modulation in an InGaAs SQW waveguide, IEEE J. Quantum Electron. 29 (1993), no. 8, 2319-2324. [9 ] _____ , Guided-wave measurements of real-excitation optical nonlinearities in a tensile strained InGaAs on InP quantum well at 1.5 micron, Opt. Comm. 102 (1993), 473-477. [10] J. M. H. Elmirghani and H. T. Mouftah, All-optical wavelength conversion: Technologies and applications inDWDM networks, IEEE Communications Magazine (2000), 86-92. 14 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [11] ---------, Technologies and architectures for scalable dynamic dense WDM networks, IEEE Communications Magazine (2000), 58-66. [12] G. Fuchs, J. Horer, A. Hangleiter, V. Harle, F. Scholz, R. W. Glew, and L. Goldstein, Intervalence band absorption in strained and unstrained In GaAs multiple quantum well structures, Appl. Phys. Lett. 60 (1992), no. 2, 231-233. [13] C. Furst, G. Mohs, H. Geiger, and G. Fischer, Performance limits of non linear RZ and NRZ coded transmissions at 10 and fO Gb/s on different fibers, Tech. Digest OFC 2000, 2000, Paper WM31-1, pp. 302-304. [14] E. A. Golovchenko, A. N. Pilipetskii, and N. S. Bergano, Transmission properties of chirped retum-to-zero pulses and nonlinear intersymbol in terference in lOGb/s WDM transmission, Tech. Digest OFC 2000, 2000, Paper FC3-1, pp. 38-40. [15] K. L. Hall, S. R. Chinn, D. M. Boroson, and K. A. Rauschenbach, Wide band RF signal generation using all-optical switching, Proceedings of CLEO 1997, 1997, Paper CWOl, pp. 296-298. [16] K. L. Hall, G. Lenz, and A. M. Darwish, Subpicosecond gain and index nonlinearities in InGaAsP diode lasers, Opt. Comm. I l l (1994), 589-612. [17] C. H. Henry, Theory of the linewidth of semiconductor lasers, IEEE J. Quantum Electron. QE-18 (1982), no. 2, 259-264. [18] C. H. Henry, R. A. Logan, and K. A. Bertness, Spectral dependence of the change in reflective index due to carrier injection and GaAs lasers, J. Appl. Phys. 52 (1981), no. 7, 4457-4461. [19] R. Hess, M. Caraccia-Gross, W. Vogt, E. Gamper, P.A. Besse, M. Duelk, E. Gini, H. Melchior, B. Mikkelsen, M. Vaa, K. S. Jepsen, K. E. Stubk- jaer, and S. Bouchoule, All-optical demultiplexing of 80 to 10 Gb/s signals with monolithic integrated high-performance Mach-Zehnder interferometer, IEEE Photon. Technol. Lett. 10 (1998), no. 1, 165-167. [20] R. Hess, M. Duelk, W. Vogt, E. Gamper, E. Gini, P. A. Besse, H. Mel chior, K.S. Jepsen, B. Mikkelsen, M. Vaa, H. N. Poulsen, A. T. Clausen, K. E. Stubkjaer, S. Bouchoule, and F. Devaux, Simultaneous all-optical add and drop multiplexing of f.0Gbit/s OTDM signals using monolithically integrated Mach-Zehnder interferometer, Electron. Lett. 34 (1998), no. 6, 579-580. 15 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [21] H.Haug (ed.), Optical nonlinearities and instabilities in semiconductors. Academic Press, San Diego, 1988. [22] T. Houbavlis, K. Zoiros, A. Hatziefremidis, H. Avramopoulos, L. Occhi, S. Hansmann, H. Burkhard, and R. Dall’ Ara, lOGbit/s all-optical boolean XOR with SOA fibre Sagnac gate, Electron. Lett. 35 (1999), no. 19, 1650- 1652. [23] T. Houbavlis, K. Zoiros, K. Vlachos, T. Papakyriakopoulos, H. Avramopoulos, F. Girardin, G. Guekos, R. Dall’Ara, S. Hans mann, and H. Burkhard, All-optical XOR in a semiconductor optical amplifier-assisted fibre Sagnac gate, IEEE Photon. Technol. Lett. 11 (1999), no. 3, 334-336. [24] C. Janz, All-optical signal processing with photonic integrated circuits, Tech. Digest OFC 2000, 2000, Paper ThF6, pp. 90-92. [25] C. Joergensen, S. L. Danielsen, K. E. Stubkjaer, M. Schilling, K. Daub, P. Dousierre, F. Pommerau, P. B. Hansen, H. N. Poulsen, A. Kloch, M. Vaa, B. Mikkelsen, E. Lach, G. Laube, W.Idler, and K. Wunstel, All-optical wavelength conversion at rates above lOGb/s using semiconductor optical amplifiers, IEEE J. Select. Top. Quantum. Electron. 3 (1997), no. 5, 1168- 1180. [26] I. Joindot and J. L. Beylat, Intervalence band absorption coefficient and measurements in bulk layer, strained and unstrained multiquantumwell 1.55 pm semiconductor lasers, Electron. Lett. 29 (1993), no. 7, 604-606. [27] W. I. Kaechele, M. L. Dennis, R. B. Jenkins, T. F. Carruthers, and I. N. Duling, Dispersion-managed transmission of 100 Gb/s time-division- multiplexed retum-to-zero format data, Tech. Digest OFC 2000, 2000, Pa per TuP3-l, pp. 232-234. [28] R. Khosravani and A. E. Willner, Comparison of different modulation for mats in terrestrial systems with high polarization mode dispersion, Tech. Digest OFC 2000, 2000, Paper W15-1, pp. 201-203. [29] T. L. Koch and X. Pan, Semiconductor inferometric optical wavelength conversion device, United States Patent Number 6,069,732, 2000. [30] B. Lavigne, D. Chiaroni, P. Guerber, L. Hamon, and A. Jourdan, Improve ment of regeneration capabilities in a semiconductor optical amplifier-based 3R regenerator, Tech. Digest OFC 1999, 1999, Paper TuJ3-l, pp. 128-130. 16 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [31] J. Leuthold, C. H. Joyner, B. Mikkelsen, G. Raybon, J. L. Pleumeekers, B. I. Miller, K. Dreyer, and C. A. Burr us, Compact and fully packaged wavelength converter with integrated delay loop for 40 Gbit/s RZ signals, Tech. Digest OFC 2000, 2000, Paper PD17-1, pp. 218-220. [32] -------- , 100 Gbit/s all-optical wavelength conversion with an integrated SOA delayed interference configuration, Electron. Lett. 36 (2000), no. 13, 1129-1130. [33] R. Ludwig, U. Feiste, E. Dietrich, H.G. Weber, D. Breuer, M. Martin, and F. Kuppers, Experimental comparison of 40 Gbit/s RZ and NRZ trans mission over standard singlemode fibre, Electron. Lett. 35 (1999), no. 25, 2216-2218. [34] D. D. Marcenac, D. Nesset, A. E. Kelly, M. Brierley, A.D. Ellis, D. G. Moodie, and C. W. Ford, 40Gbit/s transmission over 406km of NDSF us ing mid-span spectral inversion by four-wave-mixing in a 2mm long semi conductor optical amplifier, Electron. Lett. 33 (1997), no. 10, 879-880. [35] J. Mark and J. Mork, Subpicosecond gain dynamics in InGaAsP optical amplifiers, Appl. Phys. Lett. 61 (1992), no. 19, 2281-2283. [36] B. Mikkelsen, G. Raybon, T. N. Nielsen, U. Koren, B. I. Miller, and K. Dreyer, Opto-electronic and all-optical wavelength translators and their cascadabillity, Tech. Digest OFC 1999, 1999, Paper FJ1-1, pp. 146-148. [37] G. Mohs, C. Furst, H. Geiger, and G. Fisher, Advantages of nonlinear RZ over NRZ on 10 Gb/s single-span links, Tech. Digest OFC 2000, 2000, Paper FC2-1, pp. 35-37. [38] J. Mork and A. Mecozzi, Response function for gain and refractive index dynamics in active semiconductor waveguides, Appl. Phys. Lett. 14 (1994), no. 3, 1736-1738. [39] --------- , Semiconductor devices for all-optical signal processing: Just how fast can they go?, Proceedings of LEOS 1999,1999, Paper ThCCl, pp. 900- 901. [40] R. M. Mu, T. Yu, V. S. Grigoryan, and C. R. Menyuk, Convergence of the CRZ and DMS formats in WDM systems using dispersion management, Tech. Digest OFC 2000, 2000, Paper FCl-1, pp. 32-34. [41] C. Nintjas, M. Kalyvas, G. Theophilopoulos, T. Stathopoulos, A. Avramopoulos, L. Occhi, L. Schares, G. Guekos, S. Hansmann, and 17 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. R. DalPAra, 20Gb/s all-optical XOR with UNI gate, IEEE Photon. Tech- nol. Lett. 12 (2000), no. 7, 834-836. [42] M. Osinski and J.Buus, Linewidth enhancement factor in semiconductor lasers - an overview, IEEE J. Quantum Electron. QE-23 (1987), no. 1, 9-29. [43] N. Peyghambarian and S. W. Koch, Semiconductor nonlinear materials, Nonlinear Photonics (H. M. Gibbs, G. Khitrova, and N. Peyghambarian, eds.), Springer-Verlag, Berlin, 1990. [44] A. D. Poularikas and S. Seely, Signals and systems, 2nd ed., ch. 4, pp. 300- 312, PWS-Kent Publishing Company, Boston, 1991. [45] A. J. Poustie, K. J. Blow, A. E. Kelly, and R. J. Manning, All-optical parity checker, Tech. Digest OFC 1999, 1999, Paper TuJ6-l, pp. 137-139. [46] G. Raybon, B. Mikkelsen, U. Koren, B. I. Miller, K. Dreyer, L. Boivin, S. Chandrasekhar, and C. A. Burr us, 20 Gbit/s all optical regeneration and wavelength conversion using SO A based interferometers, Tech. Digest OFC 1999, 1999, Paper FB2-1, pp. 27-29. [47] _____ , 20 Gbit/s all-optical regeneration and wavelength conversion using SO A based interferometers, Tech. Digest OFC 1999, 1999, Paper FB2-1, pp. 27-29. [48] B. Sartorius, C. Bornholdt, S. Bauer, M. Mohrle, P. Brindel, and O. Leclerc, System application of f0 GHz all-optical clock in a f.0 Gbit/s optical 3R regenerator, Tech. Digest OFC 2000, 2000, Paper PD11-1, pp. 199-201. [49] G. Schraud, G. Muller, L. Stoll, and U. Wolff, Simple measurement of carrier enduced refractive-index change in InGaAsP pin rigde waveguide structures, Electron. Lett. 27 (1991), no. 4, 297-298. [50] Karlheinz Seeger, Semiconductor physics, pp. 348-360, Springer Verlag, Berlin, 1991. [51] W. Shieh, S. X. Yao, G. Lutes, and L. Maleki, An all-optical microwave mixer with gain, Tech. Digest OFC 1997, 1997, Paper ThGl, pp. 263-264. [52] J. Shim, M. Yamaguchi, P. Delansay, and M. Kitamura, Refractive in dex and loss changes produced by current injection in InGaAs(P)-InGaAsP multiple quantum well (MQW) waveguides, IEEE J. Select. Top. Quantum. Electron. 1 (1995), no. 2, 408-415. 18 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [53] P. M. V. Skovgaard, R. J. Mullane, D. N. Nikogosyan, and J. G. Mclnerney, Two-photon photoconductivity in semiconductor waveguide autocorrelators, Opt. Comm. 153 (1998), 78-82. [54] S.L.Chuang, Physics of optoelectronic devices, pp. 651-661, John Wiley and Sons, New York, 1995. [55] H. N. Spector, Free-carrier absorption in quasi-two-dimensional semicon ducting structures, Phys. Rev. B. 28 (1983), 971-976. [56] H. J. Thiele, I. Brener, T. Nielsen, P. Freeman, H. Presbey, and U. Ko- ren, Analog performance of semiconductor optical amplifier wavelength converter: cross-gain modulation versus four-wave mixing, Proceedings of CLEO ’97, 1997, Paper CWN5, p. 296. [57] L. F. Tiemeijer, P. J. A. Thijs, J. J. M. Binsma, and T. van Don- gen, Effect of free carriers on the linewidth enhancements factor of InGaAs/InP(strained-layer) multiple quantum well lasers, Appl. Phys. Lett. 60 (1992), no. 20, 2466-2468. [58] L. F. Tiemeijer, P. J. A. Thijs, T. van Dongen, J. J. M. Binsma, and E. J. Jansen, Self-phase modulation coefficient of multiple-quantum-well optical amplifiers, IEEE Photon. Technol. Lett. 8 (1996), no. 7, 876-878. [59] K. Vlachos, G. Theophilopoulos, A. Hatziefremidis, and A. Avramopoulos, 30 Gb/s all-optical clock recovery circuit, IEEE Photon. Technol. Lett. 12 (2000), no. 6, 705-707. [60] M. Willatzen, J. Mark, and J. Mork, Carrier temperature and spectral holebuming dynamics in InGaAsP quantum well laser amplifiers, Appl. Phys. Lett. 64 (1994), no. 2, 143-145. [61] D. Wolfson, S. L. Danielsen, H. N. Poulsen, P. B. Hansen, and K. E. Stubk- jaer, Experimental and theoretical investigation of the regenerative capabil ities of electrooptic and all-optical interferometric wavelength converters, IEEE Photon. Technol. Lett. 10 (1998), 1413-1415. [62] D. Wolfson, T. Fjelde, A. Kloch, C. Janz, A. Coquelin, I. Guillemot, F. Ga- borit, F. Poingt, and M. Renaud, Experimental investigation at lOGb/s of the noise suppression capabilities in a pass-through configuration in SOA- based interferometric structures, IEEE Photon. Technol. Lett. 12 (2000), 837-839. 19 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [63] D. Wolfson, P. B. Hansen, A. Kloch, and T. Fjelde, All-optical 2R regen eration at 40 Gbit/s in an SOA-based Mach-Zehnder interferometer, Tech. Digest OFC 1999, 1999, Paper PD36, pp. 361-363. [64] D. Wolfson, A. Kloch, T. Fjelde, C. Janz, B. Dagens, and M. Renaud, Jd- Gb/s all-optical wavelength conversion, regeneration, and demultiplexing in an S OA - based- al I- active Mach-Zehnder interferometer, IEEE Photon. Technol. Lett. 12 (2000), no. 3, 332-334. [65] J. Zhou, N. Park, J. W. Dawson, and K. J. Vahala, Terahertz four-wave mixing spectroscopy for study of ultrafast dynamics in a semiconductor op tical amplifier, Appl. Phys. Lett. 63 (1993), no. 9, 1179-1181. [66] J. Zhou, N. Park, J. W. Dawson, K. J. Vahala, M. A. Newkirk, and B. I. Miller, Efficiency of broadband four-wave mixing wavelength conversion us ing semiconductor traveling-wave amplifiers, IEEE Photon. Technol. Lett. 6 (1994), no. 1, 50-52. [67] Y. Zhu, W. S. Lee, G. Pettitt, M. Jones, and A. Hjifotiou, Eight-channel fOGb/s RZ transmission over four 80km spans (328km) of NDSF with a net dispersion tolerance in excess of 180 ps/nm, Tech. Digest OFC 2000, 2000, pp. 51-53. 20 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 2 An Optical Filter Based on Carrier Nonlinearities for Optical RF Channelizing and Spectrum Analysis A novel filter, the so-called XWave (Wave miXing) filter, which is based on the finite time response of the interband carrier nonlinearities in a direct bandgap semiconductor, was experimentally demonstrated. The filter was implemented in a mixed strain polarization insensitive multiple quantum well, broad area semiconductor optical amplifier (SOA). The key to the filter implementation is the spatial separation of a four-wave mixing-generated sideband from the linearly amplified inputs to the SOA. The fabrication of the traveling-wave broad area SOA and the characterization of this device as a filter is discussed. The results of some systems-level experiments, are shown. Both spectrum analysis and channelizing of RF modulated optical carriers were demonstrated. The time domain response was also investigated by looking at the eye-diagram of a filtered signal. Sources of noise, which degrade the signal-to-noise ratio, is also investigated. 21 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.1 Introduction In this chapter an experimental demonstration of the XWave (Wave miXing) filter and its use in optical channelization of RF signals and in optical spec trum analysis are discussed [7, 8]. The XWave filter is tunable over the FWM efficiency bandwidth of the SOA (~40 nm FWHM) and has a passband of ~ 1 GHz which is determined by the time response of the carrier density in the SOA. There are no additional passbands, so that the filter has infinite effective finesse. It is virtually impossible to achieve all of these properties in a conventional reso nant filter such as a Fabry-Perot etalon or a microring resonator. An FSR (free spectral range) of a few tens of nanometers implies a resonator with dimensions of the order of a few wavelengths. The Q (quality factor) of the resonator scales as the volume of the resonator (which is very small for these tiny cavities), and it is inversely proportional to the energy losses from the resonator [3] (which there fore needs to be very low). Much research has recently gone into the realization of a practical microring resonator device [28, 21]. Typically, the finesse in these devices is of the order of ~ 250. It is hard to obtain much better numbers due to various energy loss mechanisms like sidewall scattering loss, leakage loss to the substrate, scattering, mode mismatch at resonator-to-waveguide coupling junctions, and curvature-induced bending loss. In the XWave filter, though, loss and the Q are essentially decoupled, and calculations show that increasing 22 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. internal losses may even improve the nonlinear mixing efficiency (given a fixed device gain). The device described in this chapter uses the response bandwidth, Au, of the nonlinearity of the medium in which the optical fields propagate, to construct a filter with the same bandwidth, Au. Hong and Chang [16] described this principle and demonstrated an RF notch filter based on photorefractive two- beam coupling. This principle can be extended to four-wave interactions in other Kerr-like media, and the tunable narrowband optical filter, based on four- wave mixing (FWM) in a semiconductor optical amplifier (SOA), was described and analyzed by Dubovitsky and Steier [11], The work described in this chapter is a direct outflow of the latter proposal. The XWave filter is based on nearly degenerate FWM in a Kerr-like medium which has a finite response time, as shown in Fig. 2.1. Two laser beams, specifically a local oscillator pump beam (monochromatic) and a signal beam (to be analyzed or filtered), are injected into the device at an angle with respect to each other. The signal beam is a single frequency laser, modulated with some information, which broadens the spectrum as shown in Fig. 2.1. The interacting beams only write nonlinear grating(s) if the frequency detuning between the beams falls within the response bandwidth of the medium, which is determined by the dominant lifetime, r, of the medium. In the XWave filter the nonlinear grating is of the form of a carrier grating written in the SOA, through the process of stimulated emission 23 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (interband transitions). Only a transmission grating is effectively written in the SOA, due to carrier diffusion considerations, as is shown in Fig. 2.1. In general, this grating will be pulsating when the frequency components of the signal and pump are detuned. These pulsations are limited to a repetition rate of 2irr, where r is the dominant nonlinearity’s characteristic response time (and is actually just the effective carrier lifetime of medium). Therefore, the grating pulsations can only contain frequency components of the signal which are within 1/ 27t t , of the local oscillator pump beam frequency. In other words, only signal frequencies close to the pump frequency contribute to the pulsations. A probe beam (single frequency) can read this grating. Some part of the probe S IG N A L .F il te r AMPLIFIED local oscillator " « ■ * 2/t SOA = 1 = = - « * ) Nonlinear FILTERED transmission SIGNAL "to LOCAL OSCILLATOR (PUMP) grating (DIFFRACTED PROBE) Figure 2.1: A functional diagram of the XWave filter as it was implemented in this experiment. is diffracted by the grating and, therefore, picks up the pulsations because the 24 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. probe is essentially mixed with grating. The diffracted part of the probe beam is then a replica of the spectral part of the signal that falls within the nonlinear filter’s passband around the local oscillator pump frequency. The XWave filter is tuned by changing the pump laser’s frequency. Phase-matched conditions exist, which ensure the suppression of undesired sidebands, that can cause spurious beat tones at the detector. The key to the filter’s implementation is that the filtered output beams should be spatially separable from the pump or the signal. This is because these different beams are very closely spaced in frequency (of the order of ~ 1GHz), which makes it very hard to separate them using a spectral filter. In the implementation of the XWave filter, shown in this chapter, the separation is accomplished by using the nondegenerate beam directions, but it can also be done by using polarization or, in an integrated device, by using different transverse waveguide modes [11]. The XWave filter has many possible applications. Because the XWave filter has no other passbands, it can excel at separating a single narrowband (com pared to the optical frequencies) information signal from a wavelength band that contains many closely spaced signals. The situation that was just described is encountered in DWDM (dense wavelength division multiplexed) optical com munication systems, in which a single signal carried on a specific wavelength needs to be demultiplexed [20] at some user’s terminal. Another possible ap plication of the XWave filter is in channelizing [18, 2], which can be considered 25 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. as the filtering and separation of many signals which comprise an RF band. One such scenario is shown in Fig. 2.2. The emphasis in RF photonics is, in general, to modulate a very high frequency RF signal onto an optical carrier and distribute it by fiber instead of by very lossy coaxial cables or metal waveg uides [26]. Therefore, it can be advantageous to channelize this signal coming from an antenna (or some other source) while it is in the optical domain. The channelization process requires an optical filter to separate out ~lGHz slices from the modulated spectrum which can then be processed further. The XWave filter is well suited for this application since it has broad tunability, which is very hard to accomplish with RF filters, and has a very high effective Q (a Q of nearly 4 x 105 is calculated). There are also possible weight, size, and power consumption advantages in using the optical approach [36]. An application closely related to the channelizing just described, is subcarrier multiplexing [19] (onto an optical carrier) for optical communications. Usually all the subcarrier multiplexed signals are detected on a single photodetector and then electrically filtered. If one has the availability of a very high speed external modulator (like a high speed electro-optic Mach-Zehnder modulator), many subcarriers may be multiplexed over a span of 20GHz or more. Optical prefiltering before photodetection with a device like the XWave filter may then be practical to 26 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. extract a single subcarrier. This prefiltering serves the purpose of demultiplex ing and reduces the bandwidth requirements on the detector and, therefore, the transimpedance gain and detection sensitivity are enhanced. Broadband RF (like whole W-band) Laser Optical modulator Modulated spectrum Digital Other AM PM/FM Signal B V Signal A Separate single signal1 for further processing Carrier Figure 2.2: This figure shows the possible application of a narrow band opti cal filter to the filtering of RF-waves in the optical domain. In the electronics warfare community this is known as channelizing. In the scenario shown in the picture, a wideband RF band (like the W-band) is picked up by an antenna. This band may contain many different signals of many formats. One might like to convert one of the (wideband) signals to baseband or IF (intermediate fre quency) for further analysis. Normal heterodyning might not work to separate the signals because of their close spacing. This requires some kind of RF filtering (before heterodyning) to separate the signals. This can be done by mixing the RF with an optical carrier (the laser spectrum is shown in the figure) and then using the optical filter to separate the signals. Heterodyning by photomixing in a p-n junction diode can then give one the desired baseband or IF signal. 27 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.2 Broad Area SOA Device Fabrication and Characterization Used in this work were 70 nm wide by 800 fj,m long ridge waveguide devices with mixed strain InGaAsP quantum wells [35]. The approximate structure is shown in Fig. 2.3. The structure was grown on a (100) substrate by the MOCVD (metal organic chemical vapor deposition) technique at a temperature of 640 o °C and a pressure of 76 Torr. The active region consists of three 60 A thick, o 0.9% compressively strained I no.g Gao. i Aso.52Pqas quantum wells and three 1 0 0 A thick, 1 .1 % tensile strained Ino.51Gao.49Aso.78Po.22 quantum wells embedded in an InGaAsP waveguide, which is lattice-matched to InP. The waveguide mate rial has a 1.05[xm bandgap. The compressively strained wells provide gain to TE-polarized optical fields, and the tensile-strained wells provide gain mostly to TM-polarized fields but also some gain to TE [9]. This structure is designed so that an optical amplifier, based on these materials has a peak gain at approx imately 1.32 fmn. With careful crystal growth the peak gains for TE and TM polarization can be equalized in amplitude and wavelength. The finite rate of carrier escape from the quantum wells also provides for carrier-induced coupling between the populations of the two well types and, therefore, couples TE and TM gains so that the gain saturation is also polarization insensitive [10, 12]. The wafer was thinned to ~ 125jim before cleaving it into ~ 800jim x lew bars containing not more than 20 lasers. The bars were soldered onto copper 28 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. buffer: n-lnP, 900nm wafer: n-lnP — Figure 2.3: A simplified diagram of the broad area semiconductor amplifier after device processing (WG=waveguide and QW=quantum well). heatsinks with top dimensions nearly the same as those of the bars. In general, the n-side was soldered to the copper using either indium or PbSn solder, with the simultaneous use of water soluble flux. It should be noted that p-side down soldered devices have some thermal advantages. Some results in this regard are shown in Appendix A. The flux residue was subsequently removed in an ultrasonically agitated bath (filled with water) to obtain clean endfaces on the laser bars. In order to obtain a traveling-wave SOA from the laser bar, the endface Fres- nel reflectance of the devices has to be reduced from ~ 0.3 to around 1 x 10~4. A single layer antireflection coating with index na ~ ne// [14] (where neff is the laser’s modal effective index) was deposited on both facets of the device 29 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 350 Coated Uncoated 300 250 — 200 a. 150 T 3 © 100 -50 0.5 2.5 Peak pulsed current (A) Figure 2.4: The light current curve (pulsed) for device #SQW3_C2_8 used in the wave mixing experiments. The two curves are for the uncoated device and its characteristics after coating with antireflection coatings on both facets. using an e-beam evaporation system. The general methodology is similar to that done at USC in the past few years [40, 34]. SiOx is one material which has the right refractive index to fulfill the requirement on the refractive index. The composition of SiOx can be varied by changing the process conditions. By o using a starting pressure of 5 x 10~7Torr, a deposition rate of 4A/s, and dry air bleed-in to raise the pressure to 5 x 10~5 Torr, the right index (as measured by an ellipsometer) can be obtained. SiOx is not environmentally stable, and it was observed to peel off the laser endfaces after a couple of months. There fore, a stochiometric material, in the form of 1 2 0 3 , was also tried as alternative material. But it has a melting point of 2410 °C compared to 1702 °C for SiO 30 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -90 TE TM -95 -100 -a “ -110 -120 1.2 1.25 1.3 1.35 1.4 Wavelength (micron) Figure 2.5: The electroluminesence measured from the edge of device #SQW3_C2_8 after the application of the antireflection coating. A resolution bandwidth of O.lnm was used. This measurement was taken at a peak current of 1A. The device clearly has strong luminescence in both polarizations, and peaks are also well matched. [17], and it was experimentally found that it required a factor of about 3x more power to evaporate than SiO. This heats up the vacuum chamber due to in frared radiation to the point that liquid cooling is probably needed on the chuck where the lasers were mounted. The layer facets were covered with a single layer of SiOx, using in situ opti cal reflectometry to determine a A/4 optical thickness at 1.32/ i m on a separate dummy wafer. The threshold current (pulsed) of the laser could be increased from 500mA (precoated) to over 1.6A. But the field exiting the laser device has 31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a curved wave front due to the small beam size of the confined mode and, there fore, the thickness of the thin film on the dummy wafer (A/4 for a plane wave) may not be the most optimal for the laser [40]. Furthermore, the deposition conditions on the thin edge of the laser bar may not be the same as the dummy wafer’s. It is better to determine the right thickness for each specific laser on the device itself. Therefore, a different type of in situ technique was used, where the rear facet laser diode power of single device kept at some current bias is monitored as a function of antireflection thickness on the front facet [29]. The monitored power is a minimum at the lowest effective reflectance. By using this method the threshold current of a laser diode could be increased to ~3A on the best devices. The light-current curves for the device used in the experiments are shown in Fig. 2.4, before and after application of its anti-reflection coat ings. The electroluminesence (See Fig. 2.5) also shows a very low ripple, which indicates low reflections from the endfaces. In order to couple light efficiently into the whole gain region of the device, one has to use highly elliptical input beams. The collimated light coming from fibers has circular beam profiles. Therefore, beam compressors were used to convert these circular beams into elliptical profiles. A lensed fiber tip mounted on a piezoelectric actuated stage was used as diagnostic tool for characterizing the beams. The beam profile at the beam waist was measured, and is shown in Fig. 2.6. From photocurrent measurements in the device (and assuming a 32 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.30 « Intensity — Gaussian fit 0.25 3 0.20 nS % 0.15 | °-1 0 § 0.05 -40 -20 0 20 40 0.30 f Gaussian fit 0.25 I °'2° > 0.15 E 2 °-1 0 ^ 0.05 -4 -2 0 2 4 Position (micron) Position (micron) Figure 2.6: The beam profile of the beam coupled to the broad area SOA. Fig. 2.6(a) is the beam profile in the horizontal direction (parallel to optical table). The vertical control was adjusted for the maximum peak power into the fiber. Fig. 2.6(b) is the beam profile along the vertical direction after the power has been peaked along the horizontal axis. quantum efficiency of 100% in the photodetection process), one can estimate a coupling efficiency of over 50% into the device gain region. The chip gain was also measured for both polarizations. At an injected current of 1.5A, gains of 20dB and 19.2dB were measured for TM and TE polarizations, respectively. The variation in TM gain as a function of injected current is shown in Fig. 2.7. The chip gain is estimated by assuming a 3dB coupling loss at each facet and directly measuring the light input and output to and from the SOA, respectively. This gain was also double checked in another way. The chip gain is known at the original threshold current of the uncoated device, if one assumes mirror reflectivities of ~30% (Fresnel reflection between uncoated semiconductor and air). By using the measured relative gain vs. current variation, one knows the 33 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 ■ a 15 "c 10 - ® - Gain Relative ASE power 0.25 0.75 1.25 1.75 2.25 2.75 Current (A) Figure 2.7: The absolute chip gain of the broad area SOA in the TM polar ization, as seen by a weak probe. On the same graph is a plot of the relative magnitude of the amplified spontaneous emission, which is proportional to the gain at higher current. Optical Fiber Optical spectrum analyzer Laser beam Lens Lens Lens Lens Grating Tunable external cavity laser SOA Beamsplitter Aperture Detector Figure 2.8: A diagram of the setup used to characterize gain ripple from the SOA. A grating monochromator (~ 4nm bandwidth) in the Monk-Gillieson configuration was used so that only light near the laser line of the tunable laser can reach the detector. 34 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. absolute gain at all other currents. The gain numbers obtained in this way agree with the ’ ’directly” measured gain. It is also important to directly measure the gain ripple caused by the resid ual reflectivity from the device facets to ensure that the device operates as a traveling-wave SOA. This was measured using the setup in Fig. 2.8. An ex ternal cavity diode laser (ECDL) was frequency-tuned over a certain range and the gain measured as a function of the laser wavelength. The peak in the in put spectrum was measured using a spectrum analyzer, and the output was filtered so that amplified spontaneous emission (ASE) from both the SOA and the ECDL is rejected. It is very important to reject the ASE using a filter be cause it may significantly alter the measured value of the ripple (making it look smaller than it really is). The measured ripple for the TM polarization is shown in Fig. 2.9. For the TE polarization the ripple was even smaller than the value shown in Fig. 2.9 (0.2dB peak-to-peak variation in gain). These measurements indicate very low effective reflectivities, and show that the device operates as a traveling-wave SOA and is well suited for wave mixing experiments. The saturation output power of the SOA was also measured. Such a mea surement is shown in Fig. 2.10. This indicates that the saturation output power of the device is around 21.5 dBm at a current of 1.5A. 35 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20.5 19.5 1321.5 1322 1320.5 1321 Wavelength (nm) Figure 2.9: The variation in the chip gain for a TM probe, as a function of the probe wavelength. The measurement was done at an injected current of 1.5A. (3 2 0 1 19.5 - 19 - r '1 8 .5 ■ 18 - 17.5 - 17 - 5 10 15 Output power (dBm) 20 Figure 2.10: The measured variation of the gain for the TM polarization as a function of the output power in the TM polarization. 36 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LWE 122 Optical modulator. Laser source RF source - © TOP VIEW amplified TE (pump) +TM probe TE (signal Broad Far field image Beam BS compressor r a B Beam ' compressor1 Aperture TE (pump) +TM (probe) amplified TE (signal) + diffracted TM probe LWE 126 Laser source CL=Cylindrical Lens BS=Beam Splitter MO=Microscope Objective PD=Photodetector PD Polarizer blocks TE Figure 2.11: The experimental setup. 2.3 Experimental Procedure In the experimental demonstration of the filter, a broad area SOA (device # SQW3-C2-8) was used, and the different interacting beams were injected at an angle with respect to the facets and each other in order to set up a transmis sion grating with Bragg match conditions. The experimental setup used in the wave mixing experiments, is shown in Fig. 2.11. The amplifier was pulsed at a repetition rate of 10 kHz with 0.5% duty factor, i.e., with 500ns pulses. A Peltier cooler (bonded to the device heatsink) was set at 20 °C to reduce possible 37 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. temperature variation. The output was either detected with a large 5mm diam eter germanium (which was easy to align) or a high speed preamplified InGaAs detector (New Focus 1801). In the case of the large detector the 10kHz modula tion was removed using lock-in detection and with the fast detector a box-cart integrator was used to average over a number of pulses. The light sources used were diode-pumped 1319 nm Nd-YAG lasers (Lightwave Electronics models 122 and 126). The light was delivered to fiber-to-freespace collimators via PM op tical fiber (polarization-maintaining fiber). The signal laser was TE-polarized in free space. An integrated optic Mach-Zehnder intensity modulator, with a 3dB bandwidth of 3GHz and a halfwave voltage of ~ 3V, was used to create the desired optical spectra by modulating the signal laser. The modulator has an extra bias input which enables one to adjust the operating point very easily with a DC voltage source. A second laser was used as the pump and the probe but with orthogonal polarizations (TE for the pump and TM for the probe). The amplified probe and the nonlinearly diffracted probe can be separated by using suitably placed apertures due to their different propagation directions. The diffracted probe is separated from the amplified signal and the scattered pump by using a free-space polarizer with an extinction ratio greater than 40dB. Beam shaping optics were used to transform the circular beams of the fibers to elliptical beams for injection into the amplifier. The pump/probe laser could be tuned over a 80GHz bandwidth by adjusting the temperature of the Nd.YAG 38 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. crystals. Both beams were steered into the optical amplifiers. Two clearly sep arated output beams subsequently showed up in the far field of the output, as verified by a vidicon camera. ASE noise was suppressed by using apertures (spatial filtering) and by using the same monochromator (spectral filtering) as shown in Fig. 2.8. The onset of nonlinear interaction can be most easily verified by looking at the power transfer between the pump and signal as a function of detuning, which results in gain to the beam on the low frequency side of the other (for small detunings). The latter is just a manifestation of the well-known Bogatov gain asymmetry [4], which is actually just two-wave mixing between two beams in a Kerr-like medium characterized by a finite time constant [5]. After two-wave mixing is observed, both the linearly amplified pump and signal can be blocked by a combination of spatial filtering and polarization optics, so that only the (weak) diffracted probe reaches the detector and can be observed. 2.4 Experimental Characterization of the Filter Theoretically, the grating formation efficiency can be expressed as a function of the frequency detuning, Au, between the pump and the signal [22, 11]: T = „ I'D . 0 1 + P0 + ( 4 7 T — sm - VhsAvTc -l (2 .1) 39 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where rc is the carrier lifetime, La = \/D arc is the carrier diffusion length (Da is the ambipolar diffusion coefficient), Pq is the total output power normalized to the saturation power of the amplifier, A is the free space optical wavelength, and 6 is the input angle between the beams as shown in Fig. 2.12. The four- wave mixing (FWM) efficiency, 77(A^, 9), is just proportional to |T|2. The FWM efficiency is defined as (assuming single frequency signal, pump and probe lasers) the following: From Eq. 2.1 one can see that the filter shape (oc rj) has a Lorentzian type dependence on the detuning between the local oscillator and the signal, with a FWHM of 7 r/reff, where re// is the effective medium lifetime given by Power output in diffracted probe (2.2) Power input in signal SOA Gratinq V A =A /(2sin(0/2)) Figure 2.12: Definition of the input angle. (2.3) 4 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Experimental Lorentzian fit 35 Av, FWHM' •4 ■ 3 -2 -1 Detuning (GHz) o 2 3 1 4 Figure 2.13: The filter shape for the probe obtained at a total input angle 0 = 7° and an injected current of 1.5A. £ 0.6 1.5 2 2.5 Injected current (A) Figure 2.14: The filter FWHM as a function of the injected current into the device. 41 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. By using the following injected powers measured inside the amplifier: P(pump) = 2.9 dBm, P(probe) = -2.4 dBm, and P(signal) = -7.6 dBm, one can measure a filter shape like that shown in Fig. 2.13. The filter’s full width at half maximum (FWHM) was found to increase nearly linearly with the injected current in the device (See Fig. 2.14). This is due to the decrease in the values of the carrier lifetime and diffusion coefficient as the carrier density increases [38] which, in turn, decreases the effective carrier lifetime (See Eq. 2.3). The injection angle was optimized in order to maximize the mixing effi ciency, while simultaneously trying to obtain good spatial separation between the output beams, so that the best signal-to-noise ratio could be obtained. The isolation obtained between the beams at the output was determined to be 20dB for a full angle of 6° between the beams. Under these conditions a FWM effi ciency of 7.1dB is also measured. The filter FWHM also depends on the material parameters (See Eq. 2.3) and, by varying the injection angle, one can, in fact, extract their values. The result of such a measurement and fit is shown in Fig. 2.15. At a current of 1.5A, it was determined that rc=2.5ns, Da— 5.7cm2/s, and La— 3.8 /ma. These values should probably be seen as a composite value of the materials in both the compressive and tensile-strained quantum wells because of the carrier cross coupling between these two types of wells [10, 12, 27], due to the small detunings used in this experiment. The device was also tested un der different conditions, during which the signal and pump were TM-polarized 42 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1.1 0.8 0.7 0.6 0.5 ° '40 0.05 0.1 0.15 0.2 0.25 0.3 0.35 ( 2 id A )2 (u m ~2) Figure 2.15: The FWHM off the filter is directly proportional to the square of the grating’ s wavenumber, (27r/A)2, as was measured here, and fitted by using linear regression. The fit enables one to extract some of the material parameters. and the probe was TE-polarized. The powers used (measured inside device) were P (pump)=1.76dBm, P (signal)=0dBm, and P (probe)=0dBm. A current of 1.75A was injected into the device. At an angle of 9 « 7°, an isolation be tween the beam ports of -28dB was obtained. Under these same conditions the FWM efficiency was measured to be 9.2dB. The total input power was also re duced by using a neutral density filter wheel and the FWM efficiency measured at these reduced powers (See Fig. 2.16). From the measurements shown in Fig. 2.16, it can be deduced that the efficiency is still showing an upward trend as a function of power up to the highest powers. Mukai et al. [25] have studied four-wave mixing in SOA’s and found theoretically and experimentally that the efficiency increases as the cube of the total power (in all the interacting beams), 43 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. until it reaches a plateau and then tapers off and decreases at very high powers. The optimum condition (from the standpoint of efficiency) is to operate at or near the plateau, which was seemingly not reached in this experiment. Calcula tions show that the position of this plateau as a function of the amount of gain saturation, depends, among other things, on the internal loss in the device. Slope=2.85 CO • M easured — Linear fit u. Total Input Power (dBm) Figure 2.16: The FWM (four-wave mixing) efficiency as a function of the total injected power in the device. The straight line fit shows a slope of nearly 3, which shows that the efficiency grows as the cube of the total input power. 2.5 Spectral Analysis w ith the XWave Filter This paragraph highlights the use of the XWave device as a tunable filtering component for a spectrum analyzer. The excellent performance of the XWave filter as a spectrum analyzer was shown by doing the experiment outlined in Fig. 2.17. In this experiment a single RF source and microwave amplifier were 44 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3GHz t T une p u m p freq o d u lato r Noumea* uency Direct detection Filtered signal 3GHz sine wave Figure 2.17: This diagram shows how the XWave filter was used to measure the complex spectrum, that was generated when a single frequency laser was modulated with a RF sine wave. The spectra that is shown, is that of the laser before and after modulation. used to modulate the signal laser at 3GHz via a Mach-Zehnder modulator. The modulator was biased near quadrature and driven over a phase range of 0.97T (peak-to-peak). The resultant optical domain power spectral density before the XWave filter is shown as the inset to Fig. 2.18. It was measured with a Fabry- Perot etalon (FPE) with a FSR of 7.9GHz and a finesse of approximately 100. The small peaks seen approximately 2 GHz from the carrier, are the second sidebands of the modulation which actually occur 6 GHz from the carrier and are folded into this spectra because of the limited FSR of the FPE. To measure the spectra with the XWave filter, one can slowly tune the central frequency of the pump, as shown in Fig. 2.17, and measure the diffracted probe using a relatively slow detector and lock-in detection at 10kHz (the repetition rate of the amplifier). The results are shown in Fig. 2.18. The widths of the peaks 45 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. give the bandwidth of the XWave filter because the laser is narrow line. The powers used were P(pump) = 2.9 dBm, P(probe) = -2.4 dBm, and P(signal) = -7.6 dBm, with the pump and signal TE-polarized and probe TM-polarized. Figure 2.18: The spectrum of an amplitude-modulated optical carrier as mea sured by the XWave filter. The inset shows the spectrum as measured by a Fabry-Perot etalon with 7.9GHZ FSR. The experiment outlined in this paragraph was also repeated for the TM- polarized pump and signal and TE-polarized probe with the following powers: P(pump)=1.76dBm, P(signal)=-1.54dBm, and P(probe)=0dBm. The device current was 1.75A. The result of this experiment, is shown in Fig. 2.19. The filter FWHM was bigger in this case (Fig. 2.19) than for the results shown in Fig. 2.18, but a better diffracted probe power to background power ratio was 7: 0 -6 -5 -4 -3 -2 -1 0 1 2 3 6 7 Detuning Frequency (GHz) 46 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Detuning Frequency (GHz) Figure 2.19: The spectrum of the modulated optical carrier, as measured by the XWave filter, when the pump and signal are TM-polarized. obtained due to better beam isolation and larger signal power. The data in Fig. 2.19, show the actual background measured in this case (i.e., no subtraction was done). 2.6 RF Channelizing of Optical Signals In order to demonstrate channelizing, an additional coherent signal was added into the spectrum by amplitude modulating the 3 GHz source at 400 Hz with a modulation index of 25%, as shown in Fig. 2.20. The optical spectrum is now complicated and consists of the optical carrier with 400Hz sidebands and the 3GHz sidebands each with its 400Hz sidebands. The phases of the harmonics are such that the beat between the carrier and its 400Hz sidebands 47 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. i Tune pump frequency SG-z [Ncn'inea*. Direct detection + „ 1 Synchronous detection r^rt c at 400Hz . ^Modulator 3GHz s i n e Q w ave modulated at 400Hz (25%) Figure 2.20: This diagram shows how the XWave filter was used to demonstrate channelizing. Extra 400Hz sidebands are added onto the optical carrier and the 3GHz sidebands due to amplitude modulation of the RF signals (compare with Fig. 2.17). Note that the spacing and amplitude of the spectral lines are not to scale. produces a 400Hz signal, which is out of phase with the 400Hz signal produced by the beat between the first and second 3GHz sidebands and their 400Hz sidebands. The signal was synchronously detected at 400Hz. The resultant output is shown in Fig. 2.21 as the signal beam is tuned. Notice that the signal shifts phase when the channelizer filters out the carrier and its 400Hz sidebands as compared to when it filters out the 3GHz sidebands and their respective 400Hz sidebands. This is consistent with the calculated modulated spectrum (which is discussed below) and demonstrates that the XWave filter preserves both the amplitude and the phase of the modulation within the filter bandwidth. In other words, this experiment shows that the filter can separate closely spaced signals, 3GHz apart, where the different signals are the 3GHz sidebands and the optical carrier with their respective 400Hz sidebands. The 48 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. experimental result does not approach zero away from the peaks (as it should) due to the combination of cross gain modulation at 400Hz on the amplified probe and the fact that there is non-ideal separation between the beams. Therefore, some of the diffracted probe, as well as the amplified probe (which is modulated), reaches the detector. THEORY: EXPERIMENT TS Q- Mode Hop ' Detuning frequency (GHz) Figure 2.21: The amplitude of the modulation detected at 400Hz for different detunings between the pump/probe and signal when the RF is modulated at 400Hz. The inset shows the (normalized) theoretical prediction obtained by doing a time domain-based simulation of the experiment. To understand the results shown in Fig. 2.21, one should look at the precise mathematical form of the optical field after modulation. The electrical field 49 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of the optical beam going into the filter can be written as (neglecting other constant scale factors) _|_ ejn/2{l3m+inicos(u}RFt)[l+m2smuiAt]}\ e jui0t c c where o > o , vrf, uja are the optical carrier, RF and audio angular frequencies, respectively, mi and m2 are the modulation indices on the optical carrier and RF carrier, respectively, and j3 m a biasing coefficient, c.c. stands for complex conjugate. In this experiment s = e 0.9 and m2 = 0.25. Let us now concentrate on the expansion of the term T — exp (j-^m i cos (ujRFt) [1 + m 2 sin (uAt)]^ x exp (juQ t) x exp , (2.5) by using the Bessel function expansion identity [30]: OO (2.6) 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Jn(£) is the Bessel function of the first kind of order n (n is an integer) and argument £. Now let Pi = |m i and p2 = \m im 2. Then T (Eq. 2.5) can be written as follows and then be expanded using Eq. 2.6: 2 rp _ gj'/3i sin (f-u > RFt) e jfo sin ([uA+WRF]t) g jf e sin ([uA~uRF]t) oo oo = ^ Jm(&)e*m< W A + a ,J U ^ x i=- oo E •*.(&)< n = — oo OO OO OO = E E E WJmWnifh) X Z=— oo m = — oo n= — oo £jl% e jm(u}A+uRp ) te -jlu)RFt ejm.(uA-+uRF)t 2 l=—oo )ejn(uA-uRF)t Therefore, the Fourier transform of T (Eq. 2.7) is T= ^ P m Y, E E Wl)J m {P 2 )J n {p 2 )e^ OO OO OO ^ \ -r / ^ V o 7 2 L l=—oo m ——O Q n = — oo (5 { w — cjq + I tO R p t — m { u ) A + ^ r f ) — w (w .a — u z f tp ) } , ( 2 . 8 ) 51 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where S(u) is the Dirac delta function, and u is the angular frequency variable. Eq. 2.8, therefore, points the way to writing the full Fourier transform of Eq. 2.4, O O O O o o S = ir[5(u - u > o ) + eP^m x l~ — oo m —— oo l= — oo 5 {co — u j q + lujRpt — m(u>A + wrf) — u(uja ~ ^ f ) } ] + *F(c.c.), (2.9) where F{c.c.) is the Fourier transform of the complex conjugate part of Eq. 2.4. T{c.c) represents the Fourier components of the optical field situated at nega tive frequencies. Eq. 2.9 shows that a large number of Bessel expansion terms needs to be considered when /3X and are large, i.e., at high modulation indices mx and m2. Therefore, Eq. 2.9 is more conveniently evaluated by computer. To make the spectrum more plotable on a computer, the spectrum can be con volved with some other function like a Gaussian, as shown in the example in Fig. 2.22. The actual theoretical 400Hz spectrum shown in the inset to Fig. 2.4, was calculated using a direct time domain calculation, where the effects of the XWave and instrumentation filters were calculated using convolution integrals. The experiment outlined in this paragraph, was also repeated for the TM- polarized pump and signal and TE-polarized probe with the following powers: 52 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. P(pump)=1.76dBm, P(signal) =-1. 54dBm, and P(probe)=OdBm. As was men tioned before, much better beam isolation was achieved in this case, as compared to the situation where the pump and signal were TE-polarized. Therefore, the reader sees no background signal in the results shown in Fig. 2.23. There is quite good agreement with the theoretically predicted behaviour. The main discrep ancy can be attributed to the fact that the experimentally obtained detuning does not seem entirely linear. 1100 -2 950 1000 1050 1100 900 Angular frequency (rad/s) Figure 2.22: An example of the spectrum of light modulated by a Mach-Zehnder optical modulator. Only the positive frequencies of the Fourier transform of Eq. 2.4 is plotted. The following values were used for the parameters q j0 =1000 rad/s, ujrf=2Z rad/s, u > a— 5 rad/s, rai=0.9, 7712=0.25, and /3 m= l. The latter three values correspond to the experimental conditions. The spectrum was convolved with g(u> ) = . The upper half of the plot shows the amplitude spectrum and the bottom half the phase spectrum. The 5 rad/s beats in this spectrum is analogous to that of the 400Hz beats in the spectrum obtained in the experiment. 53 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Measured Theory o Detuning Fi (GHz) requency Figure 2.23: The amplitude of the modulation detected at 400Hz for different detunings between the pump/probe and signal when the RF is modulated at 400Hz. A TM-polarized pump and signal were used compared to TE in Fig. 2 .21. 2.7 Time Domain Filtering In the previous three paragraphs the frequency domain characteristics of the XWave filter have been discussed. These characteristics were, however, mea sured under conditions where a lot of signal averaging takes place (i.e., slow scans). The intended use of the filter is to extract actual signals in certain applications. Therefore, it is useful to consider the signal transmission in the time domain, to make sure that there are no transient effects (like, for example, filamentation [23]) that would adversely affect the transmission of parts of the signal. 54 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Patterr Geners «*) Optical .Modulator AMPLIFIED Polarizer PR? BE SIGNAL 2/xj A 11 ___ «b 0 ) (DIFFRACTED PROBE) LOCAL OSCILLATOR (PUMP) Figure 2.24: This diagram shows how a single signal was selected by the XWave filter. The probe picks up the modulation through the pulsating grating and is routed to the detector. The transmission of a single modulated laser through the filter was studied, lated by a Mach-Zehnder modulator. When the XWave filter was tuned to the signal laser frequency, the amplitude modulation could be observed, and when the XWave filter was detuned from the signal frequency, the modulation rapidly dropped as a function of the detuning, which is consistent with the behaviour of the XWave device as a filter. Simple ON-OFF modulation was chosen as the signal format for testing the XWave filter. This type of modulation has high and low frequency harmonics, which can fill the whole passband of the filter. The eye-diagram and bit error as is shown in Fig. 2.24. The signal consists of a laser beam amplitude modu- 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. US Current pulse duratiop L i m - Data bursts Figure 2.25: An eye-diagram of the light output from the SOA. The current pulse duration is ~ 500ns. The observed light mostly originates from the diffracted probe. The modulator is DC biased for maximum transmission in t his picture. Data bursts are aligned with the ” quiet” part of the pulse. When a data pulse enters, it switches off the light. The kind of trace showed in this picture aids in the lineup of the data, but after alignment the modulator is DC biased for minimum transmission, so that a data pulse switches the light on. rate analysis also provide widely accepted and objective criteria for the evalu ation of the filter. ON-OFF keying is also used in digital RF communications. Most importantly from an experimental standpoint however, ON-OFF keying allows the signal to be transmitted, only when the SOA is switched on, without the introduction of current switching harmonics into the output signal. It was observed (See Fig. 2.25) that the current pulses entering SOA cause significant ringing in the light output. This ringing can adversely affect data transmission. Therefore, bursts of data were electronically aligned with current pulses, so that the signal was effectively only transmitted after the oscillations died down, but before the SOA was switched off again. Therefore, the duty 56 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • i J •! I^v Lm , C *-£ T 7 , Z O y & r t f * ; M It I v j •, u (a) (b) Figure 2.26: Two oscilloscope photographs of the input (a) and output pulses (b) for a 30Mb/s pulse train measured on 125MHz bandwidth photoreceiver. The time per division is 20ns. The XWave filter is tuned to the carrier frequency of the modulated light. The output pulses were measured in real time (i.e., single shot) using a digital storage oscilloscope. cycle for the data (~ 0.15%) is even lower than that of the current pulses (0.5%). The results shown in Fig. 2.26 indicate that a 30Mb/s pulse train is clearly transmitted. There did not seem to be a drop-out of transmission on any of the pulses. It also seems that the decay time constant of the grating (~ 2.5ns) measured in the time domain, is inconsistent with the frequency domain data shown in the previous paragraphs. Furthermore, the grating decay data is not well described by a single exponential because of a very slow decrease versus time in the tail parts of the decaying pulses, which is unexplained at this time. The ‘hair’ on top of the output pulses is due to some of the undiffracted probe which leaks through and ‘beats’ with the diffracted probe. The level of beat noise is consistent with the independently measured probe leakage. 57 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a j t e - ' j u s . Figure 2.27: The first oscilloscope photograph (a) shows the eye-diagram of a 30Mb/s pseudo-random bit pattern transmitted by the filter. By using the limiting action, due to photoreceiver saturation, the eye-diagram on the left (b) is obtained. This particular eye-diagram (b) was recorded over a period of an hour, but it still shows clear eyes, which translate into error free transmission. The bottom rails of the eye-diagrams are much brighter than the upper rails due to the low duty cycle of the data bursts. An infinite persistence setting was used in the recording of both eye-diagrams. The beat noise affects the transmission of the data detrimentally. This is obvious in an eye-diagram. The pulse train was substituted by a pseudo-random bit pattern of length 223 — 1. The beatnoise can be seen to have a broadening effect on the upper rail of the eye-pattern (Fig. 2.27(a)). The eye-diagram can be slightly enhanced by using the limiting action (due to receiver saturation) to clamp the highest transmitted power to a certain level (Fig. 2.27(b)). The clamping causes a narrowing in the upper rail. The eye-diagram seems to be affected by the slow decay of the grating. This can be seen to pinch off the eye at the bottom half, preventing the measurement of clear eyes at >50Mb/s. 58 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. There also seemed to be pattern-dependent jitter in the downgoing edge, which is unexplained at this time. 2.8 Noise Sources -4 0 -a < - 5 0 -6 0 8-70 a. 1.325 1.32 Wavelength (micron) Figure 2.28: The output spectrum of the signal filtered with XWave device as measured with an optical spectrum analyzer set to a resolution bandwidth of 0 .2 nm. The signal is the big peak. It is situated on top of another peak formed by the filtering of the amplified spontaneous emission (ASE) by the passband of the Monk-Gillieson monochromator. From this measurement one can deduce a signal-to-ASE background ratio of 35dB. There are different mechanisms that degrade the performance of the XWave filter. The additive noise due to the amplified spontaneous emission (ASE) in an optical amplifier is one such effect. The SNR degradation due to this mechanism is usually quantified through the amplifier noise figure, NF, in analogy with 59 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. O0 -20dBm A . ®/V — -j-28dBm OdBm f Figure 2.29: The effect of the finite beam isolation on the output powers mea sured at the different output ports is illustrated in this drawing. A beam isola tion of 28 dB when TM-polarized pump and signal beams were used at input angle of 6 = 7°. Better isolation can be obtained for larger angles, but this also adversely affects the four-wave mixing efficiency electronic amplifiers. Such a number can also be defined for the nonlinear wave mixing in the SOA and can be calculated to be (See Appendix B): 2W NF = — -2 , (2.10) rjhv where hu is the photon energy, rj the mixing efficiency, and Wsp the power spectral density of the ASE noise. A measurement of the signal-to-background ratio (measured with an optical spectrum analyzer) enables one to calculate the noise figure (See Appendix B). Such a measurement is shown in Fig. 2.28. The ratio of the signal to spontaneous emission background (SBR) was measured in a 0.2nm bandwidth and was found to be 35dB. The filter passband could be accurately fitted with a Gaussian profile, which means that the calculations in Appendix B are directly 60 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. applicable. The corresponding noise figure is 18.6dB [33]. The powers used were P(pump) = 2.9 dBm, P(probe) = -2.4 dBm, and P(signal) = -3.3 dBm, and the pump and signal TE-polarized and probe TM-polarized. Perhaps the biggest problem encountered with the XWave filter was the finite beam isolation at the output terminals, as shown in Fig. 2.29. This basically means some of input the probe light lands on the detector together with the diffracted probe. If the input signal is single frequency and detuned from the pump by Au, it leads to a detuning of A v between the probe and diffracted probe as well. There is therefore an extra beat signal on top of the diffracted probe signal (as measured on a photodiode). This is what leads to the noise on top of the signal shown in Fig. 2.26(b). In the general case, where the signal has a complicated continuous frequency spectrum spanning a bandwidth greater than the passband of the XWave filter, the diffracted probe and input probe cannot be separated by spectral filtering at all because the spectra of these two fields will overlap. Under the best experimental conditions, when a TM-polarized pump and signal and a TE-polarized signal were used, a beam isolation of 28dB was obtained (direct measurement). In another indirect measurement, a peak-to-peak fluctuation as low as 1 0 % was measured on top of the signal when the pump/probe and signal are nearly at the same frequency, which means that the leaking probe power is about 26dB below the diffracted probe level. The gain for TM was estimated to be around 18dB under the 61 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Av<=1/jw Figure 2.30: This diagram shows how two frequency components of the signal entering the SOA can be mixed by a four-wave process. Apart from generating new harmonics, the mixing also results in the trading of power between fre quency components themselves and all their harmonics because of the non-zero linewidth enhancement factor. conditions of the measurements and 77 ~ 9dB. These numbers translates into an effective beam isolation of 33dB. The nonlinear effects in the amplifier can also distort the channelized signal and this must be minimized. The beat between different frequency components of the input signal can produce carrier density pulsations, if the frequency dif ference is ~ 1/ 27t t or less. The distortion process is illustrated in Fig. 2.30. The distortion is actually due to the non-phase-matched, four-wave mixing between different frequency components over a distance much shorter than the coherence length. This type of interaction has been studied extensively in the literature [25, 1, 37, 24]. More importantly, nonlinear sidebands are also generated on the diffracted probe. Let us assume that there are two discrete signal components, with equal power Ps(z — 0), detuned by a frequency much smaller than the response bandwidth of the carrier nonlinearity (carrier density pulsations). As sume as before that a dual polarization pump/probe is also entering the device. 62 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The first nonlinear sidebands generated on the diffracted probe due to the beat between the two signal components, have a power level of [37] P u l * P ? ( z = ^ P p r o U z = L ) C ( x (3)) ( 2 .1 1 ) where Ps(z — L) is the power in the signal frequency components at the output of the device (assuming small power transfer between the signal components), Pprobe(z = L) is the power in the probe at the output, and C (x^) is a nonlinear gain term depending on the degree of device saturation, the material gain, device length, linewidth enhancement factor, carrier lifetime, and the device’s saturation power. The phase mismatch is very small due to the near degeneracy of the signal frequency components. The power in one of the desired signal components at the output of the XWave filter is Psignal ~ Ps(,Z = L)Pprobe{z = L)Ppump{z = I /)C '(X (3)) , ( 2 .1 2 ) where P pUmP ( z = L) is the output power in the pump. Note that C < C due to the effect of diffusion on the transmission grating. Therefore, the ratio 63 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. between a filtered signal component and an undesired nonlinear sideband at the photodetector is obtained by dividing Eq. 2.12 by Eq. 2.11 and is just Desired signal component power C(x^)Ppump(z = L) Undesired nonlinear sideband power assuming that ~ C and a small degree of two-wave mixing between the pump and signal. From the ratio shown in Eq. 2.13, one can see that the power in the signal should be made substantially lower than that of the pump to ensure the linear transmission of a two-tone signal with intertone frequency separation much less than the filter bandwidth. In general, the lowest possible signal peak spectral density (compared to the pump’ s) is preferable, so that the filtered sig nal spectrum does not become distorted. For a given total power in the signal, a wider spectrum (and lower spectral density) signal, would result in smaller signal induced carrier pulsation frequency components, spread over a wide fre quency range, resulting in a smaller nonlinear polarization at a single frequency (compared to a two-tone discrete component signal). The linearity of the trans mitted signal is very important when the XWave device is applied to the filtering of analog signal formats like some forms of FM and AM, which might have dis crete signal components [31]. The filter is, therefore, probably more suitable for digital formats pulse code modulation formats like FSK (frequency shift 64 c (x(3 )) a ( z = l ) P p u m p {z = 0 ) rri 1 Ps(z = 0 ) ’ {2Ai} Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. keying), PSK (phase shift keying), or ASK (Amplitude shift keying), because these formats typically reduce the peak spectral density by spreading the signal energy more uniformly over a wide band. The NRZ (non-return to zero) format of these signals does not have any discrete spectral components when random data are transmitted [32]. 2.9 Conclusion The first experimental demonstration of a novel class of optical filters was pre sented in this chapter. The filter have a passband of less than one GHz and can be tuned over a 40 nm 3dB-bandwidth (efficiency) [39]. Its use in the optical channelization of an RF signal was demonstrated, and it was shown that the channelized signal preserves both the phase and amplitude of the RF signal. No phase noise originating from the pump or probe is added to the conjugate (filtered) signal, because the pump and probe are derived from the same laser. This is because the phases of the pump and the probe are subtracted to obtain the FWM product (filtered signal). This is a very important consideration for some classes of RF signals. ~9dB mixing efficiency was achieved (with a TM-polarized pump and signal) for less than 3dB gain saturation. It is expected that significantly better noise figures (~ 1 0 dB) can be achieved for total input powers of the order of lOOmW, 65 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. due to a bigger FWM signal [25] as well as suppressed ASE. A properly opti mized device will have a higher gain overlap factor x device length product by either increasing the number of quantum wells or increasing the length of the device, resulting in higher efficiency and simultaneously a lower noise figure. The beneficial effects of increasing device length (and therefore gain) have been experimentally shown by others [13]. An integrated version of the XWave filter based on nonlinear coupling in an over-moded waveguide coupler [15, 6 ] has also been proposed. The latter configuration has possible advantages compared to the current demonstration due to a longer possible interaction length, ease of fiber pigtailing and reduced saturation power, and would not be affected by carrier diffusion. The integrated optic version of the XWave device is the topic of discussion in the next chapter. 66 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1] G. P. Agrawal, Population pulsation and nondegenerate four-wave mixing in semiconductor lasers and amplifiers, J. Opt. Soc. Am., B 5 (1988), no. 1, 147-159. [2] G. W. Anderson, D. C. Webb, A. E. Spezio, and J. N. Lee, Advanced chan nelization technolgy for RF, microwave and millimeterwave applications, Proc. IEEE 79 (1991), 355-388. [3] C. A. Balanis, Advanced engineeering electromagnetics, pp. 390-391, John Wiley and Sons, New York, 1993. [4] A. P. Bogatov, P. G. Eliseev, and B. N. Sverdlov, Anomalous interaction of the spectral modes in a semiconductor laser, IEEE J. Quantum Electron. QE-11 (1975), 510-515. [5] R. W. Boyd, Nonlinear optics, pp. 269-274, Academic Press, San Diego, 1992. [ 6 ] J. P. Burger, S. Dubovitsky, and W. H. Steier, Nonlinear optics in a dual-moded waveguide semiconductor optical amplifier, Proc. LEOS An nual Meeting, vol. 1, 1998, pp. 202-203. [7 ] J. P. Burger, W. H. Steier, S. Dubovitsky, D. Tishinin, K. Uppal, and P. D. Dapkus, A tunable optical domain microwave filter suitable for wide band channelizing, Proc. LEOS Annual Meeting, vol. 1, 2000, Paper ThX3, pp. 100-102. [ 8 ] ______, An optical filter based on carrier nonlinearities for optical RF channelizing and spectrum analysis, IEEE Photon. Technol. Lett. 13 (2001), no. 3, 224-226. [9 ] S. L. Chuang, Physics of optoelectronic devices, John Wiley and Sons, New York, 1995. [10] S. Dubovitsky, P. D. Dapkus, A. Mathur, and W. H. Steier, Wavelength conversion in a quantum well polarization insensitive amplifier, IEEE Pho ton. Technol. Lett. 6 (1994), no. 7, 804-807. 67 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [11] S. Dubovitsky and W. H. Steier, Tunable wavelength filters based on nonlin ear optical interactions in semiconductor amplifiers, J. Lightwave Teehnol. 14 (1996), 1020-1026. [12] S. Dubovitsky, W. H. Steier, A. Mathur, and P. D. Dapkus, Gain saturation properties of a semiconductor gain medium with tensile and compressive strain quantum wells, IEEE J. Quantum Electron. 30 (1994), no. 2, 380- 391. [13] F. Girardin, J. Eckner, G. Guekos, R. Dall’Ara, A. Mecozzi, A. D’Ottawi, F. Martelli, S. Scotti, and P. Spano, Low-noise and very high-efficiency four-wave mixing in 1.5-mm-long semiconductor optical amplifiers, IEEE Photon. Teehnol. Lett. 9 (1997), 746-748. [14] E. Hecht, Optics, Second ed., ch. 9, pp. 375-376, Addison-Wesley, Reading, Massachusetts, 1990. [15] B. Hoanca, S. Dubovitsky, D. X. Zhu, A. A. Sawchuk, W. H. Steier, and P. D. Dapkus, All-optical routing using wavelength recognizing switches, J. Lightwave Teehnol. 16 (1998), 2243-2254. [16] J. H. Hong and T. Y. Chang, Frequency-agile RF notch filter that used photorefractive two-beam coupling, Opt. Lett. 18 (1993), no. 2, 164-166. [17] Johnson Matthey Catalog Company, Research chemicals, metals and ma terials, 1999-2000, Company Catalog, 2000. [18] T. Jung, J. Shin, D. T. K. Jong, S. Murthy, M. C. Wu, T. Tanbun-Ek, W. Wang, R. Lodenkamper, R. Davis, and J. C. Brock, CW injection locking of a mode-locked semiconductor laser as a local oscillator comb for channelizing broad-band RF signals, IEEE Trans. Microwave Theory Tech. 47 (1999), 1225-1233. [19] L. Kazovsky, S. Benedetto, and A.Willner, Optical fiber communication systems, ch. 7, pp. 529-602, Artech House, Boston, 1996. [20] L. Kazovsky, S. Benedetto, and A. Willner, Optical fiber communication systems, ch. 7, pp. 529-602, Artech House, Boston, 1996. [21] B. E. Little, J. S. Foresi, G. Steinmeyer, E. R. Thoen, S. T. Chu, H. A. Haus, E. P. Ippen, L. C. Kimberling, and W. Greene, Ultra-compact Si — SiO < 2 microring resonator optical channel dropping filters, IEEE Photon. Teehnol. Lett. 10 (1998), no. 4, 549-551. 68 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [22] M. Lucente, J. G. Fujimoto, and G. M. Carter, Spatial and frequency de pendence of four-wave mixing in broad-area diode lasers, Appl. Phys. Lett. 53 (1988), 1897-1899. [23] J. R. Marciante and G. P. Agrawal, Spatio-temporal characteristics offila- mentation in broad-area semiconductor lasers, IEEE J. Quantum Electron. 33 (1997), no. 7, 1174-1179. [24] A. Mecozzi, S. Scotti, A. D’Ottavi, E. Iannone, and P. Spano, Four-wave mixing in traveling-wave semiconductor amplifiers, IEEE J. Quantum Elec tron. 31 (1995), no. 4, 689-699. [25] T. Mukai and T. Saitoh, Detuning characteristics and conversion efficiency of nearly degenerate four-wave mixing in a 1.5 -pm travelling-wave semi conductor laser amplifier, IEEE J. Quantum Electron. 26 (1990), 865-875. [26] G. F. Nelson, R.F. optical links, IEEE AES Maganize (1992), 12-15. [27] R. Paiella, G. Hunziker, U. Koren, and K. J. Vahala, Polarization- dependent optical nonlinearities of multiquantum-well laser amplifier stud ied by four-wave mixing, IEEE J. Select. Top. Quantum. Electron. 3 (1997), no. 2, 529-540. [28] D. Rafizadeh, J. P. Zhang, R. C. Tiberio, and S. T. Ho, Propagation loss measurements in semiconductor microcavity ring and disk resonators, J. Lightwave Teehnol. 16 (1998), no. 7, 1308-1314. [29] A. Somani, P. A. Goud, and C. G. Englefield, Real-time monitoring of laser diode reflectivity while being coated with SiOx, Appl. Optics 27 (1988), 1391-1393. [30] F. G. Stremler, Introduction to communication systems, 3rd ed., ch. 6, pp. 310-312, Addison-Wesley, Reading, Massachusetts, 1990. [31] _____ , Introduction to communication systems, 3rd ed., pp. 219-369, Addison-Wesley, Reading, Massachusetts, 1990. [32] _____ , Introduction to communication systems, 3rd ed., ch. 9, pp. 567- 571, Addison-Wesley, Reading, Massachusetts, 1990. [33] M. A. Summerfield and R. S. Tucker, Noise figure and conversion efficiency of four-wave mixing in semiconductor optical amplifiers, Electron. Lett. 31 (1995), 1159-1160. [34] D. V. Tishinin, Novel elements for dense wavelength division multiplexing systems, Ph.D. thesis, University of Southern California, May 2000. 69 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [35] K. Uppal, D. Tishinin, and P. D. Dapkus, Characterization of mixed strain quantum well structures, 81 (1997), 390-393. [36] Tim van Eck, May 2000, Personal communication. [37] J. Zhou, N. Park, J. W. Dawson, K. J. Vahala, M. A. Newkirk, and B. I. Miller, Efficiency of broadband four-wave mixing wavelength conversion us ing semiconductor traveling-wave amplifiers, IEEE Photon. Teehnol. Lett. 6 (1994), no. 1, 50-52. [38] D. X. Zhu, S. Dubovitsky, W. H. Steier, J.Burger, D.Tishinin, K.Uppal, and P.D.Dapkus, Ambipolar diffusion coefficient and carrier lifetime in a compressively strained ingaasp multiple quantum well device, Appl. Phys. Lett. 71 (1997), no. 5, 647-649. [39] D. X. Zhu, S. Dubovitsky, W. H. Steier, K. Uppal, D. Tishinin, J. P. Burger, and P. D. Dapkus, Noncollinear four-wave mixing in a broad area semiconductor optical amplifier, Appl. Phys. Lett. 70 (1997), 2082-2084. [40] X. Zhu, Novel all-optical switches based on travelling-wave semiconductor optical amplifiers, Ph.D. thesis, University of Southern California, August 1997. 70 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 3 Kerr-Like Nonlinear M ode Converters for Integrated Optic Device Applications In this chapter, an integrated optic structure is proposed in order to make Kerr-like nonlinear interactions practical for optical signal processing applica tions. The interacting electric fields propagate as different modes of a multimode waveguide and couple to each other or cause a nonlinear cross gain or cross phase modulation through the nonlinear dielectric permittivity perturbation caused by the propagating fields. Integrated optic mode combiners/mode splitters provide a way to spatially separate/ combine the interacting fields, negating the need for spectral filtering in nonlinear interactions like four-wave mixing. This concept is applicable to all types of Kerr-like materials but it is particularly attractive for realizing novel devices, based on the large carrier nonlinearities in semicon ductors. The device discussed in this chapter can be considered as an integrated optic version of the XWave device discussed in the previous chapter. Instead of sepa rating the interacting waves based on their propagation direction, the integrated 71 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. optic structure enables one to separate the waves based on the magnitude of the wavenumbers of the modes. In this way, free-space optics is replaced by a planar device monolithically integrated on small chip area, which allows one to butt-couple optical fiber directly to the input and output ports and, therefore, allows the compact packaging of the device. The main features and new concepts of the device are pointed out in para graphs 3.1 and 3.2. The placement of the nonlinearity is important if there is some diffusion process present and is discussed in Paragraph 3.3. An analysis of coherent interactions in this device in Paragraph 3.4 quantifies the dependence of the efficiency of the nonlinear interactions on the device parameters. The device can also be used in incoherent interactions (Paragraph 3.5). 3.1 Introduction Optical nonlinearities play a key role in present day photonics devices. In par ticular, second order nonlinearities form the basis for the implementation of the commercially available phase and amplitude modulators, optical switches, and frequency doublers [41]. The third-order nonlinearities, which are needed to implement all-optical signal processing devices, have not found a widespread practical use so far. These nonlinearities are less efficient and are harder to implement in a practical, i.e., integrated optics, configuration. An integrated 72 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. optics implementation allows one to reduce the required optical power by in creasing intensities and interaction lengths in guided-wave interactions, but it also creates a serious problem of separating inputs from generated outputs. In a laboratory one can use high-quality tunable wavelength filters to suppress the amplified pump, but in practice that is not feasible because tunable optical filters themselves are very difficult devices to stabilize. An example of a wave length converter based on four-wave mixing in semiconductor optical amplifiers [31] illustrates this point well. One can use the highly efficient but narrowband nearly-degenerate interaction [14] in such a device, but then it requires pro hibitively narrow-band filters to isolate the desired product. To avoid the filter problem one is forced to use a highly-degenerate interaction in which the mixing efficiency is significantly reduced [44]. This problem does not exist in a classical bulk-optics four-wave-mixing geometry because the pump and the probe come out at different angles, i.e., k-vectors, and therefore can be easily isolated based on the direction of propagation. The key, therefore, is to find an integrated optics configuration which preserves the benefits of a guided-wave interaction and yet enables the separation of the desired interaction-generated signals from the input signals. In this paper an integrated optics device configuration suit able for implementation of high efficiency all-optical nonlinear interactions with easily separable outputs is described and analyzed. 73 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The device concept described in this chapter is, in general, applicable to all devices implemented in a material with a third order linearity, but it is particularly interesting for implementation in semiconductors due to the large effective nonlinear indices in this material due to carrier nonlinearities like band- filling and plasma effects [15, 12, 11]. Therefore, the nonlinear interactions in a pumped semiconductor are discussed as an example. The peculiarities of semi conductors, for example the diffusion of carriers, in fact enable us to keep the discussion more general on certain aspects of device operation since ’ classical’ electronic Kerr nonlinearities are a simpler subcategory of more general class of Kerr-like nonlinearities (as in semiconductors) in a phenomological sense. 3.2 Concept 3.2.1 Basic Idea To implement an efficient nonlinear optics interaction with easily separable out puts, we need to design a configuration in which the interacting optical fields have a significant overlap and are easily distinguished from each other. For ex ample, in a typical bulk-optics four-wave mixing interaction geometry, shown in Fig. 3.1, the two inputs have significant spatial overlap needed for an efficient interaction, and yet the outputs are easily separated based on their direction of propagation. By analogy, the key to implementation of a similar interaction 74 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. configuration in integrated optics is to find an electric field property suitable for distinguishing the fields. In general, any electric field has four quantities describing it: frequency, polarization, spatial location, and wave-vector. The two interacting beams must have an overlapping spatial location and, there fore, spatial location cannot be used to distinguish the beams. Frequency is not a good distinguishing mark because optical frequency filters with required resolution and stability are not easy to implement and therefore, this leaves polarization and wave-vector as possible distinguishing properties. The new concept, shown in Fig. 3.2(a), uses the wave-vector as a distinguishing quantity to isolate the interacting beams. The device basically consists of two functional building blocks, namely a nonlinear (mode converting) interaction region and integrated optic mode combiner/ splitters at the input/output terminals to the nonlinear region. c (ft free-space Nonlinear Interaction free.S pace propagation propagation Figure 3.1: The classical type of nonlinear interaction by optical beams in a bulk medium. In Fig. 3.2(a) the two inputs enter two completely isolated waveguides A and B. The isolated single-mode waveguides have propagation constants (3 a and (3 b - 75 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The waveguides are then adiabatically brought together in a transition region in a single dual-moded waveguide which supports modes 0 and 1 with propagation constants /3 q and (3\. In the output transition region the dual-moded waveguide adiabatically separates into two isolated waveguides identical to those at the in put. The relative magnitudes of the four propagation constants, or equivalently effective indices, are shown in Fig. 3.2(b). Assuming an adiabatic transition, the field of the waveguide A, which has a higher effective index, couples ex clusively to the lowest order mode of the dual-moded waveguide, whereas the field of waveguide B with a lower effective index couples only to the first order mode of the dual-moded waveguide. At the output the process reverses itself and, therefore, in the absence of any refractive index perturbation, the input into port A comes out of the output port A. Similarly, input into the waveguide B comes out from the waveguide B. On the other hand, a nonlinear refractive index perturbation can couple the two orthogonal modes of the dual-moded waveguide and consequently transfer input from port A to the output port B. An efficient phase-matched interaction can be realized by placing a Kerr-type nonlinearity into the dual-moded waveguide. In this case, the two modes, Eq and E % write a grating in the nonlinear medium with a k-vector equal to the difference between their propagation constants, Ak = /?0 — /A, which creates an automatically phase-matched coupling between the two modes. The two modes can write a strong grating because they have high spatial overlap, and yet the 76 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. two modes can be separated by the adiabatic transition sections based on the magnitude of their k-vectors. The integrated optics compatible configuration shown in Fig. 3.2(a) is conceptually identical to the one shown in Fig. 3.1 ex cept that, instead of the direction of propagation, the beams are isolated based on the magnitude of their k-vector by adiabatic asymmetric Y-j unctions at the input and output. p P o AP Adiabatic following Adiabatic following Nonlinear Interaction (b) Figure 3.2: The concept of the nonlinear modeconverter device. The upper half of the figure shows the physical layout of the device. Light is coupled into and out of the interaction region by using adiabatic mode sorters. The lower half of the figure shows the corresponding k-vector diagram. Certain design criteria for the device are discussed now. The justification for some of these surfaces in some of our analytical and numerical calculations which subsequently follow. 77 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.2.2 Interaction Region The purpose of this region in the device is to spatially overlap the interacting fields but still be able to separate them. The strength of interaction and separa bility are NOT tradeoffs, at least to first order. We, therefore, need to have an interaction region such that there is maximum spatial overlap between the two beams since the ’interaction strength’ is an monotonically increasing function of the overlap. Low linear coupling between the two interacting fields is also required. Coupling decreases rapidly with an increase in the difference of the propagation constants of the modes [7]. We, therefore, need a high phase mis match so that the fields do not couple to each other except through nonlinear interaction. This has the extra benefit that only certain nonlinear products are phase-matched. This is described by the well known nonlinear mixing phase mismatch factor [40]: e = (where L is the length if the interaction region and A/3 the difference in the propagation constants of fields). 3.2.3 M ode Sorting Region Yajima [39] was the first to show that each mode from a two-moded waveguide section incident on an asymmetric planar-dielectric branching waveguide couples power into just one arm of the branch. In particular, the mode chooses the arm with an effective index closest to the effective index that characterized its 78 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. propagation in the two-moded section. For larger taper angles the device just operates as a power divider. Burns and Milton devised a criterion on the taper angle [5], which describes the transition boundary between a mode splitter and a power divider, and obtained analytical solutions of the local normal mode equations for structures whose shape has a specific functional form [6]. This specific functional form corresponds to a taper where the ratio of the local coupling coefficient, Co,i, which appears in local normal mode equations, is made proportional to the difference in the propagation constants of the local normal modes (/?o and Pi) along every point of the position taper [26]: where C 1 is a constant number. With only one mode incident on the taper, the extinction ratio (between the local normal modes) will oscillates sinusoidally along the branch between zero and (. The authors in [26] showed that this taper shape is the optimum (under certain assumptions), in the sense that it gives the shortest position taper with maximum extinction ratio £ for given input and output cross-sections. On the other hand, Song and Tomlinson [36] showed that the extinction ratio can be written as a Fourier Transform of the taper function T0 > l and subsequently showed that electrical filter theory can be used to find even better taper shapes. In particular one can try, for example, the Hamming, 79 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Hanning, or Blackmann filters that give different behaviour of the extinction ratio versus taper length. For all of these synthesis methods one has to calculate the local normal modes, their corresponding propagation constants, and the coupling coefficient for a large number of different waveguide separations. This proves to be quite arduous, especially in the case where the problem can not be reduced to slab waveguides, i.e., the general three-dimensional waveguide taper. In this case, Love et al. [26] pointed out that it is more expedient to try to approximate the optimum filter shape with, for example, a cosinusiodal or parabolic shape since the extinction ratio does not appear to be critically dependent on the precise splitting shape used, as long as there is reasonable asymmetry between the waveguide widths (between 6:4 and 8:2). It should be pointed out that the adiabatic asymmetric is not the only way to do mode separation, but that multimode interference [23], or a three-guide coupler [25] separator, may also be used to do the sorting operation. 80 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.3 Overlap of Nonlinear x ^ M aterial with the M odes of the Device: Placem ent of Nonlinearity 3.3.1 Classic Kerr M edia It is fairly intuitive that the nonlinear material should be positioned in such a way as to maximize the overlap between the guided waves and the nonlinear material in order to make the most efficient use of the input power and to achieve the biggest nonlinear effects. It is later shown in Paragraph 3.4 that the greatest switching contrast in coherent interactions can be achieved when the ratio of the crossmodal overlap integral to the overlap integrals of the individual respective modes is maximized. An overlap factor with physical entity situated in the waveguide, with a dimensionless distribution G(x,y,z), is defined as / OO P O O / G(x,y, z)Ui(x,y)Uj(x,y) dx dy , (3.2) - o o J — OO where Uij{x,y) are the mode profiles of modes/fields i ,j normalized so that I Z fZ U iU jd x d y = 8ijm2 (5ij is the Kronecker delta function), x and y are the spatial coordinates perpendicular to the propagation direction, and z the spatial coordinate parallel to the propagation direction. If % — j, Py is called the auto overlap factor (or just overlap factor) of mode j. When i ^ j, it is called the crossmodal overlap factor (between modes/fields i and j). 81 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In a nonlinear device where the x (3 ) nonlinearity is due to bound electrons (the classic Kerr effect), it is advantageous to place the nonlinear material over the whole waveguide area (the core and/or the cladding) in order to maxi mize the overlap factors. When two optical pumps, with real (scalar) electric fields Sa(x ,y ,z ,t) and £b(x,y, z,t), are propagating in the lowest and the first mode, respectively, the refractive index of the waveguide area is changed from n(x, y, z, t) = nb(x, y, z) to n(x, y, z, t) — nb(x, y, z) + An(x, y, z, t) where [34] A n(x, y, z, t) = n2 E(x, y, z) (£a(x, y, z, t) + Sb(x, y, z, t ) f , (3.3) and where u2 e is the nonlinear Kerr coefficient distribution, given by y ,z ) = n2E-^(x, y, z) , (3.4) where r P 2 E is the magnitude of the Kerr nonlinear coefficient (in m 2/V 2), and N is equal to unity where the nonlinear material is present and zero elsewhere (assuming that the nonlinear coefficient is uniform over the area where it is distributed). The spatial distribution of the nonlinear index change is important since this gives a nonzero crossmodal overlap factor with the index change. By assuming that the fields are harmonically time varying, the spatially varying part of the 82 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. time averaged nonlinear index change due to intermodal interference can be written as Ana b = n2(x, y, z)Da(z)Db(z)Ua(x, y)Ub(x, y) , (3.5) where Da and Db are the electric field amplitudes and Ua and Ub the spatial field profiles of the interacting fields £a and £b, respectively. The crossmodal overlap factor with Ana b can then be found by substituting Eq. 3.5 in Eq. 3.2: / + 0 O / • + OO / n ° 2Da(z)Db(z)N{x, y, z)U„(x, y)Ub{x, y) dx dy . (3.6) ■ oo J —O O This overlap factor is clearly nonzero even though the modes are orthogonal. Therefore, the Kerr nonlinearity writes an index distribution that essentially breaks the orthogonality between the linear modes of the waveguides and cou ples them. This is similar to using a slanted grating in a passive dual-mode waveguide to couple the supermodes [19]. 3.3.2 Semiconductor M edia In semiconductors in which carrier nonlinearities are utilized, the nonlinear carrier grating can be affected by carrier diffusion. This is especially true for the carrier distribution in the lateral direction (perpendicular to the propagation direction) since the dimensions of semiconductor waveguides are of the order of 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2jj,m micron while the diffusion length is of the order of 5 fxm in InGaAsP based quantum well materials [47]. Therefore, carrier density variations brought about by stimulated interband transitions can be easily washed out by carrier d iffu s io n . The carrier concentration can be assumed to be nearly constant over the lateral dimension of the gain region. Only at very high intensities is the light able to burn significant variations into this carrier distribution. On the other hand, phase mismatch between the modes can be designed so that carrier density variations in the propagation direction can be made large enough (ps KXffim), which is sufficiently larger than the diffusion length to be undisturbed by carrier diffusion. Assume that gain is provided by quantum wells in a buried heterostructure confinement structure. Assume there are n quantum wells of individual thick ness t. The buried heterostructure optical waveguide is centered around x = 0 and the center of the \th quantum well is situated at a depth y — yi from the top of the chip. The material gain distribution is approximately constant across the wells situated between xa and X b in the lateral dimension, x : n g{x, y) = ^ 2 9°rect ( “ ~ c) rect ( f - Vi) ’ (3.7) 84 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where w = \{xa — Xb)| and c = (xa + Xb)/2. The rect-function has its usual definition: rect(a:) = < 1 when |ar| < | (3.8) 0 when |cc| > | , and g0 is a constant. The crossmodal overlap factor can be obtained by substi tuting Eq. 3.7 into Eq. 3.2: / -f-oo p-roo / g{x, y)Ua{x, y)Ub{x, y) dx dy , -oo J — oo “ 9»e/i r « / — oo */ — 0 0 X Ua(x,y)Ub(x,y)dxdy . (3.9) If the optical modes are symmetrical around x = 0 (the midpoint of the waveg uide) , and the quantum well active region is placed uniformly symmetrical around this point in the waveguide (i.e., xa < 0 and xb > 0 with xa = \xb\), Eq. 3.9 predicts that the Toi is always zero due to the mode orthogonality theorem. This means that under very weak or no optical excitation there is no nonlinear coupling possible between the modes, if the gain is placed over the whole core of the waveguide region. Note that the crossmodal overlap factor Toi is never exactly zero and may become appreciable under very strong optical excitation (which may burn depletion holes in the carrier distribution). This is analytically investigated in Appendix C. 85 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. = Quantum Well gain medium Waveguide core Waveguide core (a) (b) Figure 3.3: To prevent diffusion in semiconductor media, which washes out non linear gratings written by interfering modes, the gain media has to be placed assymetrically in the waveguide formed by the lateral index steps. This can be done by placing the quantum well gain either assymetrically (a), or symmetri cally with a central gap in the optical confinement region(b). In order to get a nonzero T^, one has to place the gain assymetrically with respect to the waveguide’s symmetry axis (usually the middle of the waveguide). Such a gain placement is illustrated in Fig. 3.3(a). This asymmetric gain place ment only works if there is strong index guiding and the gain area can be treated as perturbation of the index guiding. Another way around the problem is to include to a central heterostructure barrier in the gain area in order to prevent diffusion. This is also shown in Fig. 3.3(b). This would amount to manufactur ing the quantum wells into laterally separated buried heterostructures. 86 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4 Coherent Interaction In this paragraph coherent interaction in a two-moded waveguide is discussed by using the special case of interaction in a device with a direct bandgap semicon ductor nonlinear material, with the associated carrier nonlinearities. Analyses of four-wave mixing and two-wave mixing are presented in order to show the important factors in these interactions. A design example shows the expected performance of these kind of devices. 3.4.1 Analytic Calculation of Four-Wave M ixing Efficiency An analytical formula is derived to show the dependence of four-wave mixing efficiency in a two-moded waveguide on the overlap factors, the saturation pow ers, and the degree of saturation. This enables one to set up design rules for this class of device. Consider electric fields propagating in a two-moded nonlinear waveguide. A certain field can propagate in one of two modes which is denoted by mode 0 and mode 1. The real fields in the medium can be described by equations of the form Sm(x, y, z, t) = A n [Drn(z)e~i(u ’ mt- km Z' > + c.c.) . (3.10) 87 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where c.c. denotes the complex conjugate. The frequencies of these fields are arbitrary. It is assumed that the fields are transverse to the propagation direc tion. Um(x, y) is the dimensionless transverse mode profile of field m, with the normalization condition Um(x, y)Un(x, y) dx dy = 5mn x lm 2, where 5mn is the Kronecker delta function. Furthermore, em is the normahzed polarization vector, uim the angular frequency of the field, km the propagation constant, and Dm(z) the electric field variation in the propagation direction (z). The power in field m can then be described by nm ffE ~ Pm = = (3.11) where * signifies the complex conjugation operation. r]0 is the impedance of free space and the effective index of the field m and S = lm 2 is a dimensionality constant. r/e// = r j o / n is the effective impedance seen by field m. In practice, the effective indices of the two modes of a weakly guided dual-mode waveguide differ only by a small amount and, therefore, for the purpose of power calcula tions, one can just assume that is the average of effective modal impedances of the lowest and first order mode, i.e., = (Veff + vlff)/2- The five fields depicted in Fig. 3.4 are involved in the nonlinear interaction. Field a (£a) is assumed to be the local oscillator (a local pump beam) and 8b is a control pump beam which can nonlinearly interact with pump a. It is assumed 88 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. that fields a and b are detuned by an angular frequency of 0. These two fields will, in general, write a refractive and gain (or loss) grating. There is assumed to be a relatively large difference in the propagation constants of these fields, Ak = \ka — kb\, which is largely determined by the geometry of the structure in which the waves propagate. It is, therefore, reasonable to assume that energy and phase-matching requires that these pumps only scatter into each other and do not generate any sidebands on each other. T3 C L Frequency Figure 3.4: The different fields involved in the nonlinear interaction. It is assumed that there is a third probe field £c incident on the structure. The detuning of £c from £a and £ & is much larger than the 3-dB bandwidth of the nonlinear response due to the dominant nonlinear process. In the case of a semiconductor optical amplifier this process would be the carrier density pulsations. In fact, in this paper the smaller effects of the ultrafast processes 89 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. like carrier heating and spectral hole burning [43] are ignored. Therefore, Ec would approximately not interact nonlinearly with £a and £ & , except through the grating written by the two pumps. Due to energy considerations and the phase-matching condition, it is clear that £c can therefore be scattered into only one of two sidebands: £d or £e, depending on the sign of Q or the propagation direction. At this point, field a can be in any of the two modes (a= l or 0), but b must propagate as the other mode once the propagation mode of a has been chosen. Field c can again propagate as mode 0 or 1, and the scattered products d and e must then be both of the opposite mode compared to c. When all of the fields are assumed to be of the same polarization (in this case TE), and the active region is small enough that transverse carrier variations wash out, it is possible to show that one arrives at the following familiar four- wave mixing equations similar to those derived in earlier works [1, 28, 32, 27], which describe the evolution of the fields in the medium: ^ = 5 ( ^ ( 1 L _ E l i _ x T e f f a , 2.T, 2rjeffA 1 -f~ i£lTej j 90 £ A, (3.12) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where and where lh U )a dg(uC ,N) -/ yy-N + ( 1 - iUac)— dN 1 ° }DcD *eDbei{- k'> ~ k«+k°-k^ z fkua d y ~ / 7\r\ + (1 - iaac)— } DdD*Dbei(- kb- k“ +kd- kJz, (3.13) dDb 1 /r, ^ ^ ( 1 — iotaa)g(u)a, N) — £ > & cfe 2 1 r oi v re// X 2E 2rjeffA 1 — zOr( e// -£ , (3.14) S = (1 - ^ A Ifl. I 2 + (1 - ia » ) - - S ' ’5^ c’ ^ DaD'cD j J T j O J q + (1 - zaac)- - - ^ ^ l ± l f l D aD * dDcei (^-^+fcc-fcd )Z ) (3.i5) 91 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and where dDc = i ( r c c dz 2d 2rjef fA. 1 + iOre// , J _ _ I L _ X P f f n . 2H 2rjeffA 1 — iQreff (3.16) Ce = (1 - i a c a ) - ^ L ^ 9 ^ a , N } D a D * D e e i ( k a ~ k h + k ° - k c ) z hua 0 dg{ujc,N) / ^rvs + ( l - f a ^ ) ’ o c io .l2 d g (u > c,N) - / a V \ + (1 - c, )DeDlDdei{-k*~M *-kJz, (3.17) as well as, C d = (l - iaca) - - W lg^ a’ Dd D * aDbe^kd- k^ - k^ z huja + (1 _ De \Dd f> tlU Jc + ( l _ fa + (1 Xfe) ^ DeD*D,iel(kd kc+k* f c c )2j (3.18) 92 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and where and where dDa dz = \ ( ~ ( l - i a ) g ( u c, N ) - ^ j Dd x Teff -r> 2c. 2 tjeffA 1 + iQ,reff V = (1 - ia c c ) - ^ L ^ c’ ^ Dd |Dc |2 + (1 - ioica) — DaDZDce1 (kc~kd+ka~kb )z fkoa + (1 - iacc) - ^ DdD2ei (kc~kd+kc~ke)z hu)P c ’ dDe 1 / F e e L (1 -ia)g{uc,N) £ > e 1 r oi x _ _ M _ £ 2 H 2t]qA 1 — iflTeff d g (u > a,N) - / s r \ £ = (1 — ZQcc)— D |J) |2 + ( 1 - z a c a ) l ^ 0’ ^ -D *aDbDcei ( fcc - f c . + f c i . - f c a )2 + (1 - m cc) — ^ D^D^e1 (k-~k-+k- T iu Jq Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.19) (3.20) (3.21) (3.22) 93 In writing the coupled four-wave mixing equations, it was assumed that the u)a ~ oJb and o jc « u > d & oje. A is the cross-sectional area of the active region, t the thickness of the active area, and £ the waveguide scattering and absorp tion loss which is assumed to be approximately the same for both modes. The eraged carrier density N(z), and the appropriate frequencies um. reff is the effective carrier lifetime which is in general intensity-dependent and is given by the following formula: q is the charge of a single electron and J the current density at a certain position z along the waveguide’ s longitudinal axis. The remainder of the discussion as sumes that J is constant over the device. rs(N) is the carrier density-dependent lifetime. Furthermore, there are dimensionless overlap factors appearing in the above equations, which are given by Eq. 3.2. G(x, y, z) — G(x, y) in Eq. 3.2 equals 1 in areas where the active medium is present and zero elsewhere. In the four-wave mixing equations there also appears a couple of crosswave length Henry linewidth enhancement factors, which can be defined as gain g(oum,N) and the differential gain — are evaluated at the time av- dg(um, N) r mm DmD\ aij(N) — —2k, dn/dN(u>i, N) 'j0dg/dN(u)j,N) (3.24) 94 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where kjo is the free space propagation constant at Uj and n the real refractive index of the active region. Eqs. 3.12 through 3.21 are, in general, only numer ically solvable. A first order estimate of the mixing efficiency can be made by making various approximations. The first assumption is that pump a is much stronger than all the other fields and is, therefore, responsible for the gain sat uration effects. In fact, it is desirable to gain clamp the amplifying medium with a strong pump in order to minimize cross gain modulation in the case of multiple probes at multiple wavelengths. The second assumption is that the propagating loss can be neglected, com pared to the high material gain. This greatly simplifies the mathematics. Also assume that the linewidth enhancement factor, gain, and differential gain are approximately constant (i.e., no gain dispersion) over the wavelength range that the interacting fields span. The gain is linearized in the following way S W = ^ ( X - N C ), (3.25) where N0 is the transparency carrier density. A logarithmic type function may be more appropriate for quantum well devices [8], but in this case the FWM equations can only be numerically solved. Solving for N from the steady-state 95 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. rate equation for the semiconductor and substituting in Eq. 3.25, one obtains the following expression for the linearized (in carrier density) gain: It was assumed that field a is the dominant saturating field. In Eq. 3.26 the saturation field for field a is given by The carrier lifetime is assumed to be independent of carrier density. Consider the scenario where all the fields are co-propagating in the direction of increasing z-coordinates. Furthermore, assume that pump a is undepleted by nonlinear interactions with the other fields. Therefore, the differential equation for the modulus squared field a can be written as Eq. 3.28 is a separable differential equation and can be solved to give (3.26) (3.27) d\Da\2 _ Taag0\Da\2 1 (3.28) ■sat (3.29) 96 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where S(z) = \Da(z)\2 and the saturated integrated gain G at z is defined by G(z) = S(z)/S(0). Eq. 3.29 is a simple transcendental equation which can be solved numerically to find the relation between the saturated integrated gain and the input power. Let’ s now look at the differential equation describing the evolution of DaD'l. This term describes the ’ ’nonlinear grating” from which the probe c scatters. Using Eqs. 3.12 and 3.14 one can obtain (neglecting the effects of the probe and its nonlinear sidebands) « £ J S L „ + dz dz dz = ^ [ T n ( l+ ia ) + r „ ( l - ~ d :d ‘ 1 ig r 01(l — m)go ^ r, 2S dN 2rjE ffAtkj[l + SjA2 ^ ] " “ "[l + S /A U 1 dg r 01(l + ia)gq , ,2 „ n,* rs /o oq\ 2E dN 2rkffAto[l + S/A2 sat] W [1 + S/A**] ^ The second term of Eq. 3.30 will be neglected because it is proportional to the cube of Db, which in turn is much smaller than Da. It has been assumed that there is zero detuning between the pumps. Now do the following definitions: T(z) = DaDl , (3.31) x _ i [r**(1 + iod) + r aa(i — ioL)\ ( ^ dab = ---------------- ----------------- • (3.32) £ J - an. 97 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The solution of Eq. 3.30 can then be found to be T = T(Q)GSabHlc , (3.33) where 1 + S(z)/Alat H = 1 + S ( 0 P L ’ (3-34) and 7c = (3.35) & A aa In order to simplify the calculations, one can assume that the amplifier is weakly to moderately saturated so that S{z)/A2 sat is small and, therefore, H « 1. Now also assume that the probe (field c) is not depleted by the nonlinear interactions. Then Eq. 3.16 can be approximated as dDc 1 dz 2 r cc(l-ify ): 90 Dc . (3.36) i + s M L j Eq. 3.36 is a separable differential equation and can be integrated to give Dc(z) = Dc(0)GSc, (3.37) where 8c = — ^ ’• (3.38) aa 98 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Now we are in a position to solve the equation for the scattered products. It can be verified that field e will be phase-matched in the case where all of the fields are co-propagating and when a and c are in the same mode. In writing the simplified nonlinear equation for the nonlinearly scattered field the nonlinear gain term proportional to D* A, is assumed to be the dominant nonlinear gain term dDe ~ 1 Fee(1 ~ ia)9°De{z) - (3.39) where dz 2 1 + S/A2 m ' (1 + S / A ^ f 7oi ^ (3.40) and a crossmodal saturation electric field has been defined as The first term on the right hand side of Eq. 3.39 is just the linear gain term and the second the nonlinear gain term (nonconstant driving term). Eq. 3.39 has the boundary condition that J9e(0) = 0. When fields b and c are in the same mode, field d is the phase-matched wave and an equation describing the evolution of field d, similar to Eq. 3.39, is obtained except that is substituted for De and T is not conjugated. 99 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4.1.1 Copropagating probe Let’s concentrate on the solution of Eq. 3.39, which is of the form = -p(S(z))De - q(S(z)). (3.42) If one defines that P{S(z)) = f ‘ p(S(z))dz , (3.43) Jo it can be shown that the solution of Eq. 3.42 is given by [3 ] De(z) = De(0)e~p{s(z» + e~pWz» f ep^ q ( S ( t ) ) d t . (3.44) Jo In our case we have that p(S(z)) = -- re e (l - ia)' . 1 + S /A * , . (3.45) Eq. 3.45 can be integrated to give Jp{S(z))dz = Jp(S) dS = In . (3.46) 100 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Now the following integral can be solved: I = J ep^ q ( S ( t) ) d t (S)"is 7oiT(Q)*Dc(0) f s^ dS ^sat^aa90 L S(0) X X ^ ( f S ) i \S (0 )J (1 + S(z)/Al.t)S ' where Sab = Ifab 1 1 ^6 6 (l d- ^o) + ra a (l icx) 9 0 ^ aa 2 4 = ~ ( 1 -«*)• * aa 3.4.1.2 Pump a and probe in the same mode When a and c are in the same mode it can be verified that k = — Se + S * }, + 5C — 1 = 0 , and Eq. 3.47 reduces to nS(L) dS K Js(o) 1 5 (o ) i + S(z)/A2 sat Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (3.47) (3.48) (3.49) (3.50) 101 as 10 0 5 , = 10 forrQ i=Q.0Q5 •2 ■ 3 ■ 4 10 10 10 Total input power (Watt) Figure 3.5: A comparison of the mixing efficiencies obtained by analytical cal culation with the exact results. The diamonds represent the exact results, the solid line is calculated using Eq. 3.52, and the dashed line with Eq. 3.53. The gain of the amplifier is also shown in the dash-dotted line. The parameters used in the calculations are shown in Table 3.1 where K T(0) Dc{0)jOlg ^ - 6 c-S*b + 8e ^aa9oCg( (3.51) sat Eq. 3.50 can be integrated and the result substituted in Eq. 3.44, enabling one to write the following formula for the absolute squared field of the scattered signal at the end of the amplifier (z — L): 45(0) r4 A aa 1 + G(L)S(0)/AI sat 1 + S(0)/Al sat . (3.52) 102 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.1: VALUES USED FOR PARAMETERS FOR COMPARING THE ANALYSIS WITH NUMERICAL CALCULATIONS (SEE Fig. 3.5). Device property Value Device Length 0.8mm Gain overlap factor for mode 0 0.02185 m2 Gain overlap factor for mode 1 0.02351 m2 Material gain 3000 cm~l Material saturation intensity lA2MW/cm2 Linewidth enhancement factor 8 Photon energy 0.8e V Carrier lifetime 200ps Average Effective index of modes 3.23 Eq. 3.52 is only in terms of the saturated gain, which can be easily obtained from Eq. 3.29, and the other input fields. It has been assumed that a2 1. Clearly it is very important to maximize the ratio of crossmodal overlap factor (Poi) to the overlap factor of the pump and probe (Paa). A big a-factor is also beneficial. Under light saturation, that is S(z)/A2 sat < C 1 for 2 = 0 to L, the integrand in Eq. 3.50 can be simplified, resulting in the following familiar nearly cubed (if Paa « P^) dependence of the scattered signal on the gain of the device which is valid as long as G{L) ;$ > 1. The mixing efficiency is defined as |_ D e(L)|2/|D c(Q)|2 and can be calculated using either Eq. 3.52 or Eq. 3.53. The analytically calculated efficiency can 103 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. be compared with the exact number, which is obtained by numerically solving the four-wave mixing equations, and is shown in Fig. 3.5. In this graph the mixing efficiency is plotted as a function of the total input power. The ratio of input powers was chosen as P(a) : P(b) : P(c) = 100 : 10 : 2. The device parameters used in this calculation are shown in table 3.1. It can be seen that even a small change in T0i can lead to a large change in the mixing efficiency due to a 4th power dependence on this factor. The analytical results show very good agreement with the exact results, except for very high input powers when the amplifier is substantially saturated. In the latter regime Eq. 3.52 gives better results than Eq. 3.53. The value of the analytical results is that they directly show the dependence of the efficiency on certain factors. 3.4.1.3 Pump b and the probe in the same mode When pump b and the probe are in the same mode, field d becomes the phase- matched sideband. The following formula can be obtained for field d, based on the same kind of assumptions as before Dd{L) = K'S{L)s* j (3.54) where (3.55) 104 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and (3.56) Eq. 3.54 is difficult to integrate exactly, but one can look at the limiting case where the amplifier is lightly saturated (i.e., S(z) < C A2 sat for all z) and a » 1. The following is then obtained: From the power coefficient of G in Eq. 3.57, it can be seen that it can be advantageous to make Yb b > r aa. This basically implies that the weaker pump and probe are amplified more than the strong pump. 3.4.1.4 Counterpropagating probe One may want to inject a counterpropagating probe. This section derives formu las for the signal power under this condition. When the probe is counterpropa gating, the scattered signal is zero at the end of amplifier, i.e., De(L)/Dd{L) = 0. From Eq. 3.44 one can obtain the field of the counterpropagating scattered field at the output end of amplifier (z = 0) : [ ep^ q ( t ) d t Jo (3.58) 105 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (An equation of the same form can be written if d is the phase-matched field). If e is the phase-matched field one obtains that 1 r ee(l 2 1 + S(z)/A2 sat (3.59) The amplitude of the probe is described by (3.60) When the probe is launched in the same mode as the pump field a (but opposite propagation direction), field d is the phase-matched field. Similarly, if field c is launched in the same mode as b, field e is the phase-matched field. In the latter case, the same formula for the backward propagating power is obtained, that would be obtained if the probe is launched in the forward direction, but in the same mode as a, i.e., If the probe is launched in the same mode as a, field e is described by the same formula as Eq. 3.54, except that the input probe field becomes DC (L) instead of Dc(0). There is, therefore, this interesting symmetry which tells us that backward propagation of the probe in a certain mode gives us the same output l+G (L)S{0)/Alt- 1 + S(0)M L, . ’ 106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. power for the scattered signal as when the probe is forward propagating, but in the other mode. 3.4.2 Analytical Analysis of Two-Wave Interaction The possibility of power coupling between the two pumps a and b (See Fig. 3.4) is important in determining the efficiency of scattering a 3rd field from the grating written by the pumps. This is basically because the grating’s amplitude is determined by the product of the intensities of the two pumps. A large power transfer from the weaker pump (at the input) to a stronger pump diminishes this product and the resulting grating is weaker. Efficient two-wave interaction opens up other functionalities when considered on its own. The important factor governing the transfer of power between two pumps is the phase shift between the interference pattern of the two waves and the resultant index grating, as will be shown below. 3.4.2.1 Phase shift of gratings in two-wave interaction The phase shift between the light interference pattern and the nonlinear grating generated by the two pumps is determined by the time response of the relevant nonlinearity of the medium in which the light propagates. In a semiconductor the nonlinear response is characterized by an effective lifetime, reff (See Eq. 3.23), which is dependent on the carrier lifetime and the degree of material 107 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. saturation. The finite lifetime leads to detuning dependent interaction strength between two optical pumps. Therefore, when two pumps a and b interact, the magnitude of the nonlinear gratings (the gain and index modulations of the medium) declines with increasing detuning. This is because the interference pattern between a and b starts moving through the medium for nonzero detuning and the medium with its finite response cannot follow the rapid modulation. But there is also an associated phase shift of the grating with respect to the light intensity pattern written by a and b. The interference between the pumps sets up a modulation in the carrier density, N = N + i (ANe-im + c.c.) , (3.62) where AN = l/(Ar,e„)Teff(I^g(^N)D'aDte ^ - ^ M ^ + <A2 T*J j In Eq. 3.63 the phase shift of the grating (< p ) with respect to the light interfer ence pattern is given by < f) = atan(Qreff) . (3.64) In Eqs. 3.62, 3.63 and 3.64 the symbols have the same meanings as in the previous paragraph (derivation of analytic expressions for FWM signal power). From Eq. 3.64, one can see that < f goes to 90 degrees for large positive values of 108 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C l and to -90 degrees for large negative values of C l. The gain and index gratings are phase shifted with respect to the interference pattern. As is shown, the gain grating does not result in any transfer of power between the two fields even when phase shifted. On the other hand, the index grating does not couple the two beams when they are at the same frequency (no detuning) but, with detuning, the associated nonzero phase shift makes it possible to couple the power from one beam to another, with the direction of power transfer determined by the sign of detuning. In the case of two-wave mixing in Kerr media based on bound electron nonlinearities, however, where the nonlinearity is ’ ’fast” enough to fol low rapid fluctuations in the fields, there is no phase shift between the index grating and the interference pattern. It was rigorously shown by Silberberg et al. [35] that light in a two-moded waveguide with an electronic Kerr nonlinear ity can interact only by phase shifting each other without mixing which only occurs near some (high) critical power. The need for a phase shift between the grating and interference pattern is somewhat similar to the two-wave mixing in photorefractives, where the 90-degree spatial shift between the interference pattern and the index grating results in efficient power transfer between the waves (See for example reference [42]). 109 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4.2.2 Two-wave mixing equations It is possible to derive analytic results substantiating the abovementioned state ments. Under the conditions that a semiconductor gain (or loss) medium is only lightly saturated by the lightwaves, the two-wave mixing equations for pumps a and b can be written in the following form (using Eqs. 3.12 and 3.14) (3.65) (3.66) where the a and b symbols stand for the absolute squared fields Yz = Pad - Qab[q - r] , ^ = Pbb- Qab[q + r] , a = P a l2 , (3.67) b = \Db\2. (3.68) Furthermore the gain coefficients are T'aa& dO /r, r-n\ Pa = — =— , (3.69) ft = (3.70) 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. a n d t h e s y m b o l s q a n d r a r e d e f i n e d b y q — crcos{4> ) , (3-71) r = asin{4>) , (3.72) where a is again the Henry linewidth enhancement factor and 0 the phase shift of the index and gain gratings with respect to the light interference pattern, and which is given by Eq. 3.64 for semiconductors. A symbol a is used in Eqs. 3.69 through 3.72 as ‘marker’ for gain processes, and has a value of 1 for semiconductors. The factor Q is the coupling constant and takes on the value where the symbols have their usual meanings. With the proper redefinition of Q and by making pa = Pb, two-wave mixing in bulk media can also be described by exactly the same equations (See [42]). 3.4.2.3 Kerr media Eqs. 3.65 and 3.66 are valid for two-wave mixing in Kerr media with electronic nonlinearities (in the slowly varying envelope approximation), as long as q is zero and the coupling constant Q is changed to take into account the different mechanism and geometry and a is given the proper value. In Kerr media one 111 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. finds that ( j> — 0 (no phase shift between the index grating and interference pattern) and < 7 = 0 (no gain) resulting in zero nonlinear gain terms in Eqs. 3.65 and 3.66. Therefore, there cannot be an exchange of power between the fields in this case, and the fields can only phase shift each other. 3.4.2.4 Effect of gain gratings It is interesting to look at the roles that the gain and index grating respectively play in the transfer of power. If there is no index grating present (q = 0), one can find from Eqs. 3.65 and 3.66 that d(a — b) , . . ~ ~ = paa ~ pbb. (3.74) Eq. 3.74 implies that the change in the power difference between fields a and b as a function of propagation distance comes about only due to the difference in the linear gains of the two fields, irrespective of the phase of the gain grating. Therefore, one can generalize and say that the gain grating by itself scatters as much from a to b, as it does from b to a and does not by itself result in a net transfer of power. The total energy that can be extracted by the two beams is, however, affected by the nonlinear gain grating, as also shown in the seminal paper by Agrawal [1]. 1 1 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4.2.5 Solution of two-wave mixing equations Eqs. 3.65 and 3.66 can be simplified by the transformation a — uepaZ 6 = vepbZ , (3.75) (3.76) which allows the differential equations to be written in the form du dz dv dz —QuvepbZ[q - r], — QuvepaZ[q + r]. (3.77) (3.78) Under the simplification that pa ~ pb, one can find a simple solution to Eqs. 3.77 and 3.78. Dividing Eq. 3.77 by Eq. 3.78 and integrating one finds v(z) « 6(0) + q — r q + r {u(z) + a(0)} , (3.79) where a(0) and 6(0) are the absolute squared fields at the start of the interaction region (z = 0). By substituting Eq. 3.79 in Eq. 3.77, one can solve the resulting differential equation by the separation of variables method. Following 113 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. some algebra, one then ends up with the following equation which describes the propagation of field a through the medium: a(z) = - - M - NeR(z)' a p a Z (3.80) where M = q — r q + r HO) + a(0) , (3.81) and JS(*) = _ £ M ( g + r ) (eP b Z — 1) Pb (3.82) and N = a(0) M a{ 0) (3.83) For field b the result is given by substituting Eq. 3.79 in Eq. 3.76: b(z) = ep b Z 6(0) + q — r q + r a (0 ) - M 1 - NeRW (3.84) The analytically calculated (Eqs. 3.80 and 3.84) power transfer between pumps a and b is shown in Fig. 3.6. 114 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 100 26 24 -25 -75 -100 -10 £1 x (dimensionless) Figure 3.6: The analytically calculated (Eqs. 3.80 and 3.84) power transfer between pumps a and 6. The parameters used in the calculations are shown in table 3.1. F0i = 0.022 in this calculation. It is assumed the powers in field a and b are both 85 fiW at the start of interaction region. Also shown is the phase shift between the grating and interference pattern. 3.4.2.6 Efficiency of pump suppression By using the analytically derived formulas for the evolution of pump power, it is possible to determine which parameters are important in the two-wave interaction. Let’s look at the case when the pump powers are nearly equal, i.e., a(0) 6(0). Assume that the detuning is such that < f > m 90°. This implies that q 0 (and positive) and r « a. Under these conditions one also finds that M — > 0, R — ► 0 and N — * 1. The efficiency of suppression of pump a at the 115 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. end of the amplifier (z = L), normalized to the linear unsaturated gain, is then given by lim - M (3.85) 1 - M (3.86) l i m ---------------------- a(0) M — *o 1 — exp(R(L)) a (0 ) m ->o p r (l ) 2 E M K J e d M 1 1 (3.87) a(0)MF> (3.88) (3.89) a r 01Gba(0)(l + n ^ ) 1/2' It was assumed that Ga — exp(paL) 1 and Gb = exp(pbL) 1, and L’Hospital’s rule was used to get Eq. 3.87 from Eq. 3.86. Eq. 3.89 shows that in order to suppress pump a , the ratio of the crossmodal to the modal overlap factor for field b should be made as large as possible. Furthermore, the ratio between the power in pump a and the crossmodal saturation power should be maximized. A large Henry linewidth enhancement factor and a large gain (Gb) are also desirable. It can be seen that by proper design of the device, and using sufficient input power, it is possible to obtain suppression of a of the order of l/Gb, which can be bigger than 20dB for typical SOA’s. Simulation shows that even larger suppression can be obtained if the input powers are un equal. The two-wave mixing in a phase-matched configuration shows promise for optical carrier suppression and optical narrowband notch filtering. 116 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.4.3 Design Example and Simulations p+ doped InGaAs cap ---------- IriP etch stop lays lantum wells InGaAsP waveguide i-doped InP -4 -3 -2 -1 0 1 2 3 4 Horizontal direction (pm) Figure 3.7: The cross-section of our design example. The device has a ridge waveguide structure with a separate confinement heterostructure gain region. Gain is provided by six compressively quantum wells InGaAs embedded in In GaAsP waveguide. The wells are etched out in the central region to prevent the washing out of a carrier grating by diffusion. In order to estimate the actual device performance, a realistic design was drawn up and the nonlinear interaction and wave propagation in the device numerically simulated. The relevant device structure is shown in Fig. 3.7. This is a cross-section of the waveguide in the interaction region. It consists of a dual mode ridge waveguide structure with a separate confinement heterostructure gain region. Gain is provided by six 1% compressively strained quantum wells (/n 0. 68G ! ao,32As) embedded in a lattice-matched InGaAsP waveguide with a 1.08 fim bandgap wavelength. The well containing core layer is etched out in the central region to prevent the washing out of a carrier grating by diffusion. 117 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. -0.5 x coordinate (pm) Figure 3.8: A contour plot of the electric field magnitude of the modal distri bution of the lowest order mode of the waveguide in Fig. 3.7. It is preferable to have of the order of ~ 10 grating periods along the lon gitudinal dimension of the interaction region. This is so that only the desired nonlinear wavemixing product is phase-matched. One also wants the shortest possible device, but with lateral feature dimensions which can still be easily processed by standard microfabrication procedures. The intermodal longitudi nal beatlength in a dual-mode waveguide is strongly correlated with the lateral confinement of the modes; the higher the confinement, the shorter the beat length. Higher lateral confinement can be obtained by increasing the lateral index step in a waveguide, but it also implies a smaller lateral dimension so that it stays dual-mode. The device dimensions of the designed structure are 118 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p -0.5 x coordinate (jxm) Figure 3.9: A contour plot of the electric field magnitude of the modal distri bution of the first order mode of the waveguide in Fig. 3.7. compatible with standard lithography, but the beatlength (320 /loti) is still suf ficiently small to ensure a short device. The modes of this device were calculated by the correlation method [13], using a commercial beam propagation package. The zero order and first order mode of the device is shown Fig. 3.8 and Fig. 3.9. The four-wave mixing was simulated by using a coupled wave approach based on the Eqs. 3.12, 3.14, 3.16, 3.19 and 3.21. The parameters used for the device is shown in table 3.2. The effect of amplified spontaneous emission was neglected; it was assumed that the signals saturated the device sufficiently so that ASE was suppressed, and plays a small role in the device saturation. In the simulation, there are two pumps, called A and B, of equal power. A is launched into mode 0 of the waveguide and B into mode 1 of the waveguide. 119 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 3.2: VALUES OF DEVICE PARAMETERS USED IN THE DESIGN EXAMPLE Device property Value Device Length 2.2mm Gain overlap factor for mode 0 0.02185 to2 Gain overlap factor for mode 1 0.02351 m2 Crossmodal overlap factor 0.02163 m2 Material gain 3750 cm~l Material saturation intensity \A 2M W j cm? Linewidth enhancement factor 3 Photon energy 0.8e V Carrier lifetime 200ps Average effective index of modes 3.23 Scattering and free carrier absorption losses 50cm-1 Both had an equal input power of 150 pW . A probe field C is injected into the same mode as A, which can then be scattered into a new field D (mode 1), by the grating written by pumps A and B. The power in field C is assumed to 5 fiW. The beat between fields C and D is, therefore, very weak compared to the one between A and B. The pumps can transfer power by two-wave mixing. The gain of the pumps versus detuning is shown in Fig. 3.10. For A and B the gain is defined as ~ Power output m field X Gam(X) = —............. x= A or B. (3.90) Power input m field X It can be seen that one of the pumps (A) is suppressed -16dB (with reference to the gain of field A at large detunings) when it is detuned from pump B 120 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 20 S ' jj- H -i'3 c ffl O10 5 0 Figure 3.10: The effective gain seen by the two pumps A and B after propagation through the nonlinear diffraction region. Their respective gains critically depend on the detuning between the pumps. over the range 1 to 2GHz. This kind of interaction shows promise for optical notch filtering. One application of this could be all-optical carrier suppression for optically transmitted RF waves. Due to the phase-matched conditions the pumps only transfer power and no spurious nonlinear sidebands are generated. One can also look at the transmission of C and D. The gain for C is defined in the same way as for A and B. The gain (four-wave mixing efficiency) for field D is defined as = Power output in field D Power input m field C The ’gain’ of D, critically depends on the detuning between A and B. In fact, these detuning characteristics may serve as a filter for one of the pumps. This has recently been demonstrated in experiments in a broad-area semiconductor 121 16dEp I suppression -20 -10 10 0 20 Detuning between A and B (GHz) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C Q T S ■ — c ■ f f l O 20r 15 10 5- 0 -5 T — C - D -1 IS o -20 0 20 40 Detuning between A and B (GHz) Figure 3.11: This figure shows how light is scattered from the probe (C) into a field D by a grating formed by pumps A as B. The efficiency of the interaction is critically dependent on the detuning between A and B. optical amplifier [4]. If there is information modulated on one of the pumps, the modulation will be transferred to D. This can be used to very efficiently wavelength convert data signals over tens of nanometers, as has been shown in a experiment in a broad-area amplifier [46]. The significance of four-wave mixing in the proposed structure is that although C and D are closely spaced in frequency, they are spatially separated by the modeconverter. This is in stark contrast with wavelength conversion schemes that use four-wave mixing based on the ultrafast nonlinearities [37], which are much more inefficient. In order to realistically evaluate the device design, one also has to look at the design of the mode sorter. One possible type of design is the asymmetric y-branch shown in Fig. 3.12. In this design the 4fim wide interaction region 1 2 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. QW’s etched out Low n8 f f Figure 3.12: One possible design for a mode sorter based on an asymmetric Y-junction. The quantum well containing layer has been partially etched out under one arm of the gain medium to reduce the effective index in this arm. of the device is split into two 2.5fim wide ridges. Under one of the ridges the quantum wells has been etched out. This waveguide has a lower effective index than the upper waveguide, due to the absence of high index material under a section of the ridge. The zero order mode of the interaction region, therefore, evolves into the mode of the upper waveguide, and the first order mode into the mode of the lower waveguide. For this design of the mode sorter the waveguides have to be separated by 5.6 fj,m to achieve an extinction ratio (defined as the power overlap of the higher order mode on the upper waveguide) of -40dB. The separation characteristics of two types of adiabatic mode sorters were investigated: a straight taper and a curved taper. The curved taper is defined by the fact that the ratio of the local normal Co,i mode coupling constant to the difference in propagation constants (of the local normal modes) is made constant along the taper. These two types of tapers were compared to each other 123 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ihaped Straight -10 -20 -30 -35 -40 1000 2000 3000 4000 5000 6000 Length (>im) Figure 3.13: The extinction ratio of the local normal modes at the end of an asymmetric Y-branch as a function on the length of the length of the Y-branch, assuming a design similar to that shown in Fig. 3.12. The two branches are separated by 5.6fj,m at the end of the branch. The characteristics of two types of tapers is shown in this graph: a straight taper and a shaped taper defined in the text. in terms of their modal extinction ratio versus taper length characteristics. At the end of the taper the waveguides were separated by 5.6jum. The results of a local normal mode analysis is shown in Fig. 3.13. It can be seen that a length of ~ 2500/ttm is sufficient to achieve an extinction of -30dB between the fields. Therefore,quite a compact overall device length can be achieved. The Y-junction can be made passive at the wavelengths of the interacting fields even though it contains quantum wells. This can be accomplished by using techniques of selective area growth [29, 18, 30, 2, 16] or quantum well intermixing [33, 24]. 124 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.5 Incoherent Interaction When there is no mechanism to prevent lateral carrier diffusion inside the inter action region (in the case when a semiconductor is used as the active region), any carrier gratings are washed out. In this instance two fields can only incoherently interact through cross gain modulation (XGM) [17, 9] or cross phase modulation (XPM) [10]. The interacting fields can then again be separated by the adiabatic mode sorter in our proposed device. XPM and XGM in a dual-mode waveguide have recently being exploited for wavelength conversion in wavelength division multiplexed optical communications. The so-called DOMO (dual-order mode) converter uses a dual-mode waveguide interaction region together with an MMI mode separator [22, 20, 21] inside a Mach-Zehnder interferometer to obtain up to 25dB separation between the interacting fields [38]. This has been exploited to obtain error-free wavelength conversion up to 20 Gb/s. 3.6 Conclusion A new type of nonlinear device to exploit the Kerr-type nonlinearities in an inte grated optics structure has been proposed. This device concept seems especially promising when implemented in a direct bandgap semiconductor with carrier nonlinearities. The simulations presented in this chapter show that highly ef ficient nonlinear mode conversion can be achieved placed in a compact device 125 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. implemented in a multiple quantum well semiconductor medium. It was shown that by the careful selective placement of the gain regions it is possible to negate the effects of carrier diffusion, which inhibits this interaction. Instead of separating the interacting waves based on their propagation di rection (like in the usual bulk nonlinear optics configuration), the integrated optic structure enables one to separate the waves based on the magnitude of the wavenumbers of the modes. In this way free-space optics is replaced by a planar device monolithically integrated on a small chip area, which allows one to butt-couple optical fiber directly to the input and output ports, and allows the compact packaging of the device. This should make the device attractive for applications like a tunable filter, wavelength conversion, and wavelength recognizing switch [45] in the case of four-wave mixing, and for notch filter ing and optical carrier suppression in the case of two-wave mixing. Incoherent interactions in a dual-mode structure have already proven useful. 126 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1 ] G. P. Agrawal, Population pulsation and nondegenerate four-wave mixing in semiconductor lasers and amplifiers, J. Opt. Soc. Am., B 5 (1988), no. 1, 147-159. [2] M. Aoki, T. Taniwatari, M. Suzuki, and T. Tsutsui, Detuning adjustable multiwavelength MQW-DFB laser array grown by effective index/quantum energy control selective area MOVPE, IEEE Photon. Technol. Lett. 6 (1994), no. 7, 789-791. [3] G. Birkhoff and G. Rota, Ordinary differential equations, 4th ed., John Wiley and Sons, New York, 1989. [4] J. P. Burger, W. H. Steier, S. Dubovitsky, D. Tishinin, K. Uppal, and P. D. Dapkus, An optical filter based on carrier nonlinearities for optical RF channelizing and spectrum analysis, IEEE Photon. Technol. Lett. 13 (2001), no. 5, 224-226. [5] William K. Burns and A. Fenner Milton, Mode conversion in planar- dielectric separating waveguides, IEEE J. Quantum Electron. QE-11 (1975), no. 1, 32-39. [6] ______ , An analytical solution for mode coupling in optical branch waveg uides, IEEE J. Quantum Electron. QE-16 (1980), no. 4, 446-454. [7] S. L. Chuang, Physics of optoelectronic devices, pp. 294-315, John Wiley and Sons, New York, 1995. [8] L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, pp. 111-184, John Wiley and Sons, New York, 1995. [9] S. Dubovitsky, P. D. Dapkus, A. Mathur, and W. H. Steier, Wavelength conversion in a quantum well polarization insensitive amplifier, IEEE Pho ton. Technol. Lett. 6 (1994), no. 7, 804-807. [10] T. Durhuus, B. Mikkelsen, C. Joergensen, S. L. Danielsen, and K. E. Stubk- jaer, All-optical wavelength conversion by semiconductor optical amplifiers, J. Lightwave Technol. 14 (1996), 942-954. 127 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [11] J. E. Ehrlich, D. T. Neilson, and A. C. Walker, Carrier-dependent nonlin earities and modulation in an InGaAs SQW waveguide, IEEE J. Quantum Electron. 29 (1993), no. 8, 2319-2324. [12] ______ , Guided-wave measurements of real-excitation optical nonlinearities in a tensile strained InGaAs on InP quantum well at 1.5 pm, Opt. Comm. 102 (1993), 473-477. [13] M. D. Felt and J. A. Fleck, Computation of mode eigenfuctions in graded- index optical fibers by the propaqatinq beam method, Applied Optics 19 (1980), no. 13, 2240-2246. [14] G. Grosskopf, R. Ludwig, R. Schnabel, N. Schunk, and H. G. Weber, Fre quency conversion by four-wave-mixing in semiconductor laser amplifiers: How to relax the filter problem?, Proc. Fourth Photon. Switching OSA Topical Meeting (Salt Lake City, Utah), 1989, Paper WB3-1. [15] H. Haug (ed.), Optical nonlinearities and instabilities in semiconductors, Academic Press, San Diego, 1988. [16] R. Iga, T. Yamada, and H. Sugiura, Lateral band-gap control of InGaAsP multiple quantum wells by laser-assisted metaorganic molecular beam epi taxy for a multiwavelength laser array, Appl. Phys. Lett. 64 (1994), no. 8, 983-985. [17] C. Joergensen, T. Duurhuus, C. Braagaard, B. Mikkelsen, and K. E. Stubk- jaer, 4 Gb/s optical wavelength conversion using semiconductor optical am plifiers, IEEE Photon. Technol. Lett. 5 (1993), no. 6. [18] Y. Katoh, T. Kunii, Y. Matsui, and T. Kamijoh, Four-wavelength DBR laser array with waveguide couplers fabricated using selective MOVPE growth, Opt. Quantum Electronics 28 (1996), 533-540. [19] A. S. Kewitsch, G. A. Rakuljic, P. A. Willems, and A. Yariv, All-fiber zero- insertion-loss add-drop filter for wavelength-division multiplexing, Opt. Lett. 23 (1998), 106-108. [20] J. Leuthold, P. Besse, and M. Bachman, Compact optical-optical switches and wavelength converters by means of multimode interference mode con verters, U.S. Patent number 5,933,554, August 3, 1999. [21] J. Leuthold, P. Besse, E. Gamper, M. Dulk, S. Fischer, G. Guekos, and H. Melchior, All-optical Mach-Zender interferometer wavelength converters and switches with integrated data- and control-signal separation scheme, J. Lightwave Technol. 17 (1999), no. 6, 1056-1066. 128 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [22] J. Leuthold, P.A. Besse, E. Gamper, M. Dulk, S. Fischer, and H. Mel chior, Cascadable dual-order mode all-optical switch with intergrated data- and control-signal separators, Electron. Lett. 34 (1998), no. 16, 1598-1600. [23] J. Leuthold, J. Eckner, E. Gamper, P. A. Besse, and H. Melchior, Multi- mode interference couplers for the conversion and combining of zero- and first-order modes, J. Lightwave Technol. 16 (1998), no. 7, 1228-1239. [24] E. H. Li (ed.), Semiconductor quantum wells intermixing, Optoelectronic Properties of Semiconductors and Superlattices, vol. 8, Gordon and Breach Science Publishers, Australia, 2000. [25] M. J. Lin, M. A. A. Neil, and E. G. S. Page, Integrated optic mode order converting and manipulating structures in strip waveguides, Opt. Comm. 110 (1994), 218-227. [26] J. D. Love, R. W. C. Vance, and A. Joblin, Asymmetric adiabatic multi pronged planar splitters, Opt. Quantum Electronics 28 (1996), 353-369. [27] A. Mecozzi, S. Scotti, A. D'Ottavi, E. Iannone, and P. Spano, Four-wave mixing in traveling-wave semiconductor amplifiers, IEEE J. Quantum Elec tron. 31 (1995), no. 4, 689-699. [28] T. Mukai and T. Saitoh, Detuning caracteristics and conversion efficiency of nearly degenerate four-wave mixing in a 1.5-pm traveling-wave semi conductor laser amplifier, IEEE J. Quantum Electron. 26 (1990), no. 5, 865-875. [29] T. Sasaki, M. Kitamura, and I. Mito, Selective metalorganic vapor phase epitaxial growth of InGaAsP/InP layers with bandgap energy control in In- GaAs/InGaAsP multiple-quantum well structures, J. Crystal Growth 132 (1993), 435-443. [30] T. Sasaki, M. Yamaguchi, and M. Kitamura, Monolithically integrated multi-wavelength MQ W-DBRlaser diodes fabricated by selective metalor ganic vapor phase epitaxy, J. Crystal Growth 145 (1994), 846-851. [31] N. Schunk, All-optical frequency conversion in a traveling-wave semicon ductor laser amplifier, IEEE J. Quantum Electron. 27 (1991), no. 6, 1271. [32] N. Schunk, G. Grosskopf, R. Ludwig, R. Schnabel, and H. G. Weber, Fre quency conversion by nearly-degenerate four-wave mixing in travelling-wave semiconductor laser amplifiers, IEE Proc. Pt.J 137 (1990), no. 4, 209-214. 129 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [33] S. K. Si, D. H. Yeo, K. H. Yoon, and S. J. Kim, Area selectivity of InGaAsP-InP multiquantum-well intermixing by impurity-free vacancy dif fusion, IEEE J. Select. Top. Quantum. Electron. 4 (1998), no. 4, 619-623. [34] A. E. Siegman (ed.), Lasers, pp. 379-380, University Science Books, Sausal- ito: California, 1986. [35] Y. Silberberg and G. I. Stegeman, Nonlinear coupling of waveguide modes, Appl. Phys. Lett. 50 (1987), no. 30, 801-803. [36] G. Hugh Song and W. J. Tomlinson, Fourier analysis and synthesis of adiabatic tapers in integrated optics, J. Opt. Soc. Am., A 9 (1992), no. 8, 1289-1300. [37] M. A. Summerfield and R. S. Tucker, Optimization of pump and signal powers for wavelength converters based on FWM in semiconductor optical amplifiers, IEEE Photon. Technol. Lett. 8 (1996), no. 10, 1316-1318. [38] D. Wolfson, T. Fjelde, A. Kloch, C. Janz, F. Poingt, F. Pommerau, I. Guillemot, F. Gaborit, and M. Renaud, Detailed experimental inves tigation of all-active dual-order mode Mach-Zehnder wavelength converter, Electron. Lett. 36 (2000), no. 15, 1296-1297. [39] H. Yajima, Dielectric thin film optical branching waveguide, Appl. Phys. Lett. 22 (1973), 647-649. [40] A. Yariv and P. Yeh, Optical waves in crystals: Propagation and control of laser radiation, pp. 521-524, John Wiley and Sons, 1984. [41] ______ , Optical waves in crystals, John Wiley and Sons, New York, 1990. [42] Pochi Yeh, Introduction to photorefractive nonlinear optics, John Wiley and Sons, New York, 1993. [43] J. Zhou, N. Park, J. W. Dawson, and K. J. Vahala, Terahertz four-wave mixing spectroscopy for study of ultrafast dynamics in a semiconductor op tical amplifier, Appl. Phys. Lett. 63 (1993), no. 9, 1179-1181. [44] J. Zhou, N. Park, J. W. Dawson, K. J. Vahala, M. A. Newkirk, and B. I. Miller, Efficiency of broadband four-wave mixing wavelength conversion us ing semiconductor traveling-wave amplifiers, IEEE Photon. Technol. Lett. 6 (1994), no. 1, 50-52. [45] D. X. Zhu, S. Dubovitsky, W. H. Steier, K. Uppal, D. Tishinin, J. P. Burger, P. D. Dapkus, and S. Garner, A novel photonic packet switch with 130 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. all-optical routing control, OSA TOPS Optical Amplifiers and Their Ap plications (M.N.Zervas, A.E.Willner, and S.Sasaki, eds.), vol. 16, 1997, pp. 278-283. [46] D. X. Zhu, D. Tishinin, K. Uppal, S. Dubovitsky, J. P. Burger, W. H. Steier, and P. D. Dapkus, Filter-free four-wave mixing wavelength conversion in semiconductor optical amplifiers, Electron. Lett. 34 (1998), no. 1, 87-88. [47] Daniel X. Zhu, Serge Dubovitsky, William H. Steier, Johan Burger, Denis Tishinin, Kushant Uppal, and P. Daniel Dapkus, Ambipolar diffusion co efficient and carrier lifetime in a compressively strained InGaAsP multiple quantum well device, Appl. Phys. Lett. 71 (1997), no. 5, 647-649. 131 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 4 A Theoretical Study of Integrated Optic Limiter Architectures In this chapter a theoretical study of integrated optics limiters based on Kerr- like nonlinearities, which can either be of the self-focusing or defocusing nature, is presented. The limiting is based either on nonlinear guiding or the self-phase modulation that can be achieved in devices which contain Kerr-like materials in some areas. By using these effects, one can obtain a transmission in a device which is inversely dependent on the input power, as is required in a limiter. The nonlinear optical properties of strained and unstrained compressive quan tum wells in direct bandgap materials are discussed, because these types of materials show interesting properties, which make them attractive for the real ization of Kerr-like regions in the limiters. Some of these devices were analyzed with special beam propagation software because of the analytical complexity of the problems. Therefore, theoretical basis and assumptions made in these cal culations are also shown. Possible manufacturing methods are also considered 132 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. since these are important in the practical implementation of the limiters. Specif ically, one needs to solve the problem of obtaining integrated optic structures with Kerr-like nonlinearities in only some selected areas of the device. 4.1 Introduction INPUT POWER Figure 4.1: The transfer characteristics of the ideal power limiter, which can either take the form of an optical hard limiter (solid line) (which requires a certain threshold to switch on) or an optical clamp (dashed line) (which has a linear transfer function up to the limiting threshold). Either characteristic may be preferable depending on the precise application. There is a need for a guided wave limiter for spectral equalization of long chirped pulses or a serial sequence of short transform limited pulses with dif ferent wavelengths in the new optical analog-to-digital converters (ADC). In one of these systems, chirped pulses with a time duration of the order of 10 nanoseconds are mixed with electrical signals in an optical modulator to achieve time to wavelength mapping of the information carried on the electrical signal 133 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [95, 44, 21, 43], which enables ultrahigh speed ADC. The chirped pulses are de rived from mode-locked laser sources, which do not necessarily generate Fourier limited pulses. The spectral bandwidths may be tens of nanometers. In some of the newer wavelength-division ADC sampling schemes [27, 8], the chirped pulses are replaced by a serial string of near-transform limited pulses at sepa rate wavelengths. A diagram of a wavelength-division ADC system is shown in Fig. 4.2. It has been observed that the short pulse sources, from which this string of pulses are derived, may exhibit pulse-to-pulse spectral variation, which cannot be easily compensated for in the ADC systems, showing up as noise in the sampled signal [42]. In order to get rid of this noise, the optical amplitude has to be clamped at a certain level using a transfer function similar to that shown in Fig. 4.1. This calls for an optical limiter, which is preferably a guided wave design, for compatibility with the rest of the system. Guided-wave limiters are very crucial components in other types of photonic systems as well. Optical code-division-multiple-access (CDMA) optical commu nications systems [67] is one such architecture. Systems engineers have proposed using optical hard limiters, before and after the sender CDMA bit pattern is correlated with the receiver bit pattern. The use of optical hard limiters in CDMA systems has been shown to improve the transmission penalty dramati cally in theory for the case where channel interference is the ultimate limit in link performance [63]. The nonlinear transmission function, which is used in 134 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Spectral slicing and delay lines WDM=Wavelength demultiplexer True Time Delay Analog Signal Quantizer Dispersion-free sampling pulses EDFL + Spectrum Broadening Modulator Quantizer Sampled Analog Signal Quantizei Figure 4.2: The layout of the wavelength division multiplexed optical analog- to-digital conversion (ADC) system. A single ultrashort pulse with a very wide spectrum is generated by a mode-locked erbium doped fiber laser (EDFL), fol lowed by spectral broadening in a nonlinear fiber. This pulse is then spectrally sliced and the slices differentially delayed to generate a string of pulses, by us ing an arrayed waveguide grating (the first wavelength demultiplexer or WDM). These pulses sample the analog signal via mixing in an amplitude modulator. Different sampling pulses of different wavelengths are then routed to different quantizers (photodetectors and electronic ADC’ s). optical limiting also has general applicability to the transformation of the noise distributions present in ON-OFF keyed optical communication systems [93] and can be used to clean up the noisy rails visible in the eye-diagrams of transmitted signals [94]. The limiting properties can therefore be used to all-optically regen erate optical signals degraded by transmission over long spans [26]. In addition to classical noise reduction, one may even achieve quantum noise suppression in structures similar to those used in limiting. In particular, amplitude squeezing 135 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. has been proposed [50] and realized [61, 51] by essentially using optical limiting in nonlinear interferometers. One might use some material properties to obtain the desired optical power transfer function of a limiter like the saturating properties of a semiconductor optical amplifier or the two-photon absorption (TPA) characteristics of a passive semiconductor waveguide. In order to evaluate the performance of the saturable amplifier or TPA waveguide, some calculations were performed for these types of devices using commonly accepted material parameters (See Fig.4.3). Presume a small signal gain of 30dB and a saturation energy of 5 pJ for the amplifier (which is typical for a InGaAsP-based semiconductor optical amplifier [92]). Assume bandfilling is the only nonlinearity in the amplifier and TPA the only effect in the passive waveguide. A TPA coefficient of (3 = 70cm /G W [90], a device length of L=lm m , and an effective modal area of 1.35 x 1Q ~8cm2 were used in the TPA waveguide calculations. The pulse width was taken to be 5ps (square pulse). In the amplifier calculation one can use the theory developed by Agrawal and Ollson [3]. In the case of the two-photon absorber, the output energy is given by a simple analytic result [68, 80, 36] - jmrrr ( 4 - 1 ) 136 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Ein is the input pulse energy, L the device length, and J0 the input intensity, which is related to the input energy by lo Ein A m o d X P V V (4.2) PW is the input pulse width, and Am od is the effective modal area. 100 03 Saturable amplifier Output TPA Output x 10 Input energy (pJ) Figure 4.3: The limiting characteristics of semiconductor waveguides with var ious types of nonlinearities. The solid curve is for a semiconductor optical am plifier with only gain saturation present (no ultrafast effects). The dashed curve is for a passive semiconductor waveguide exhibiting only two-photon absorption (TPA). It can be seen from Fig. 4.3 that neither the saturable amplifier nor the two- photon absorber output power characteristics display a slope of zero, as required 137 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in an ideal limiter. Furthermore, the two-photon absorber only effectively limits at very high energies. 4.2 Limiting Using Kerr Nonlinearities In certain applications the use of the Kerr effect as the nonlinear switching mechanism in the limiter may be superior to the methods suggested in the previous paragraph. Specifically, combining the Kerr effect with an optical guiding mechanism can prove to give better results than by using material properties like saturable amplification or TPA. The Kerr effect be used in two regimes. First, a comparitively weak Kerr effect can be present in a strongly confined waveguide, which results in an intensity-dependent nonlinear phase shift for the propagating mode. When an optical mode is weakly guided, a big nonlinear index change via the Kerr effect may play a substantial role in modifying the path and shape of the advancing wavefront of the optical field. In this chapter three structures based on combining the Kerr-effect with external guiding structures are considered: • The power-dependent guiding-waveguide or all-optical cutoff modulator [62] (S ee Fig. 4.4). This device uses a defocusing Kerr-like nonlinearity in the core of a waveguide. With increasing optical power the guided optical 138 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4.4: The apertured nonlinear waveguide optical limiter (all-optical cutoff modulator). The core contains a defocusing Kerr-type nonlinearity. mode becomes bigger, or even unguided due to the defocusing effect. Light falling outside the core of the waveguide is then blocked with an aperture. • The adiabatic asymmetric Y-junction with localized Kerr-nonlinearity, which is discussed in Paragraph 4.5. • The optical interferometer with localized Kerr nonlinearity. The Kerr nonlinearity is placed on one arm of the waveguide, which results in a nonlinear phase shift for the beam traversing this arm. This unbalances the interferometer and may lead to a power-dependent transfer function for the device. The power unbalanced interferometer is discussed in Para graph 4.6. 139 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.3 Numerical Simulation of the Structures by a Modified Finite Difference Beam Propagation M ethod 4.3.1 General M ethodology The limiting structures based on nonlinear guiding were studied using numerical methods rather than doing an analytical computation because of the complex ity of analytically analyzing these problems. Even in some special cases where analytical results can be obtained [7, 4, 71], it is still difficult to obtain physical insight into the device operation, due to the complicated nature of the ex pressions which still requires substantial numerical evaluation. The previously mentioned analytical approaches’ calculations also consider ‘stationary’ modes of nonlinear structures. In the real world some field is launched into a structure and may evolve as a function of the propagation axis coordinate. The problem becomes even more complicated when gain or loss is added in the structure. The method of choice for analysis of propagating fields was the so-called finite difference beam propagation method (FDBPM). As the name implies, this is a method of solving Maxwell’s equations for a propagating field by doing a stepwise calculation of the advancing wavefront of the field. The FDBPM is an especially numerically efficient beam propagation of a family of beam propagation methods [74, 37, 25]. Let’ s look at the scalar wave equation for a 140 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. monochromatic wave in one transverse (x) and one longitudinal dimension(z). By assuming a reference index h for the medium in which the beam propagates, one can factor out much of the rapidly varying phase. Therefore, assume the propagating wave can be written in the form U{x, z, t) = H(x, z)eikze~™ t , (4.3) where k = k^n and k$ = to/c, and where c and ui are the vacuum speed and angular velocity of the light, respectively. When Eq. 4.3 is inserted into the well known scalar wave equation, one obtains the so-called scalar Helmholtz equation [34]: dH _ j_&H_ _ iPH_ , dz 2k dz2 2k where the operator P is defined by P = kl + V i, (4.5) where e is the permittivity of the medium, and eo the permittivity of free space. Also Vj_ = d2/d x 2. Often the parabolic approximation is made, which involves neglecting the term involving d /d z2 in Eq. 4.4. This gives several restrictions on the applicability of Eq. 4.4, namely [74]: • The gradient of k along z must be small. 141 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • It requires that ~ n. • The angular spectrum must be limited to k±/k\\ < C 1. Some of these requirements may be relaxed when the so-called Pade approxi- mant operators are used. To see how these work, let us write Eq. 4.4 formally in the following form: fiij iE . = 2 J ^ r H . (4.6) dz i - s f e Eq. 4.6 suggests a recurrence relation can be used to obtain higher order derivates in z from lower order derivatives in z: 8 — _ | - _________ 2k _ ( 4 7 ' j 1 2feazl(r e - 1) In other words, Eq. 4.7 can be used to replace the z-derivative in the denom inator of Eq. 4.6 with an expression containing only the operator P, in the following way: where N and D are polynomials in P. In fact, Eq. 4.8 contains in effect an (n,d) Pade approximant for the exact Helmholtz operator [34, 32, 6], i(s/P + k2 — k). 142 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. n and d are the highest degrees of P in the polynomials, N and D, respectively. The (1,0) Pade approximant in P is P/2k and the (1,1) approximant is [34, 32]: N — D = i f t (4'9) ^ 1 \ AU2 Various higher order approximants are also possible. In this work the (1,1) approximant is used (Eq. 4.9). The so-called Crank Nicholson scheme [65, 81, 16] is used to discretize Eq. 4.8. In discretizing H (x,z), the notation HI = H (x„ zr) is used, where the transverse grid is [xs = Xq + sAx; s = 0, N; A x — (x0 — X j v ) / N ] and the longitudinal grid is [zr = rA z, r = 0,1,....]. The z-differencing is done by centering with respect to z as follows: D(Hr+ 1 - H T ) = + Hr+1). (4.10) Furthermore, centered spatial differencing results in the following form for the operator P for the two-dimensional case: P H \S - 1/(Ax f{ v sHs + HS + 1 + F s_x), (4.11) where = k2 0{ A x f ( - n2 ) - 2 (4.12) 143 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. By applying Eq. 4.11 to Eq. 4.10, one obtains [(a - 6)1 /, + l]//;+ 1 + (a - + (a - b)Hr+ } = H l + (a + b)[v,H: + H U + H ; +l] The symbols a and b stand for the following: - 1 1 a (Ax)2 4k2 ’ _ iA z (Ax)2 = 4k ' One can see that Eq. 4.13 can be written in a matrix form CH r+ 1 = D, (4.16) where Hr+ 1 is a field (column vector) to be determined and C and D are known matrices. D is in terms of the currently known field H r. Eq. 4.16 is a tridi agonal system and can be solved by a simple form of matrix inversion [66]. This algorithm was implemented using the mathematical programming pack age MATLAB, which was translated to C ++ by using MATLAB compiler, and subsequently compiled to give fast executables. Dirichlet boundary conditions, which make the fields zero at the edges of the computational window, were implemented. Numerically this actually gives reflection of the waves at the 144 (4.14) (4.15) (4.13) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. boundary. Simple transparent boundary conditions [33], were also available as an option, and were used in most of the calculations. The refractive indexes was also adapted in the vicinity of interfaces [38] to obtain a more accurate and effective beam propagation method (BPM). The accuracy of the BPM calcula tions was compared with those obtained by a commercial BPM software package (RSoft Beamprop). Rsoft Beamprop did not have the necessary flexibility for the more complicated problems that needed to be analyzed. Beamprop has been benchmarked against analytical solutions and is widely used in industry. The custom software gave excellent agreement with the commercial package for sim ple problems where classical Kerr nonlinearities were involved. This provided confidence that the software would work correctly for certain problems that the commercial software could not handle. 4.3.2 Special M odifications of the B P M for Analysis of Propagating Fields in Semiconductors Compared to RSoft Beamprop, the custom software was modified in three significant ways . This is so that Kerr-like nonlinearities that arise in semicon ductor media can be analyzed. In semiconductors one has to be able to analyze strong nonlinear guiding. In order to do this, iterative beam propagation steps have to be taken. This is to ensure that the index distribution, which includes the nonlinear index change, is self-consistent with the light distribution at that 145 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. position. This is accomplished by starting with the previously obtained field at z — zr. The index distribution is then calculated at z — zr + A z/2 by using the background index distribution as well as the nonlinear index change (brought about by H r, the field a zr). This enables one to calculate H r+1. The field Hr and the field Hr+ 1 can then be averaged to obtain a better estimate of the actual field and subsequently the nonlinear refractive index distribution at the halfstep, z = zr + A z/2. Then Hr+ 1 can again be recalculated. In the same way the index distribution at the halfstep can again be calculated. This process can be repeated to obtain better and better estimates of the index distribution at the halfstep and the field at the full step. Another way to deal with strong nonlinear guiding is just to take smaller steps in the z-direction than one would when analyzing linear structures. The commercial package was also modified upon the urging of the author to include the possibility to iterate the field at a single step. The custom BPM package was also modified to handle Kerr-like nonlinear ities that saturate like those found in saturating amplifiers and absorbers. The vendor of the commercial package did the same modifications upon our rec ommendation. Lastly the custom software was modified to include the effects of carrier dynamics in order to directly calculate the propagation of lightwaves in semiconductors, without making assumptions about the nature of the car rier nonlinearities. Specifically, the effects of carrier injection, carrier diffusion, 146 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. stimulated and spontaneous emission, as well as, absorption were added. The transverse rate equations for carriers are basically solved at every propagation step. The rate equation is given by [20] The symbols in Eq. 4.17 have the following meanings: • J is the injected current density. • q is the electron charge. • N is carrier density. ® No and Ntr are fitting parameters with the same units as carrier density. They are used to accurately describe the gain in quantum well amplifiers [20]. Ntr is essentially the gain-transparency carrier density. • The carrier lifetime is defined as dN T N = vga(N - Ntr)Nph + D V 2N. (4.17) 1 , where A,B and C are constants. (4.18) a + b n + c m • The differential gain of the medium, a, is defined as N+N, N + N0- N tr' (4.19) 1 4 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The type of gain variation shown in Eq. 4.19 is appropriate for quantum well devices [20]. The differential gain is defined in this way, so that the stimulated emission term is a linear function of N. • D is the diffusion coefficient. • Nph is the photon density. ® vg is the group velocity of the optical field in the device. The following approximations were also made: ® W ~ 0 ? i-e-i steady-state conditions. • = 0 , i.e., slow variation in N along the z-axis. We are interested in only two-dimensional problems (for the sake of sim plicity and computational speed). The approximations enable one to write Eq. 4.17 in the form where S is a ” stimulus” function dependent on the photon density, NP h, and the injected current density, J. The stimulus contains no explicit carrier density terms, and are only implicitly related to carrier density through the carrier density dependent lifetime,r, and the differential gain a. The zero of Eq. 4.20 (4.20) 148 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. must be found at every beam propagation step in order to find the carrier distribution at that point. Eq. 4.20 can again be discretized with a centered spatial differencing formula similar to Eq. 4.11. In other words, by using finite differencing, one can write Eq. 4.20 in the the matrix form A(N)N = B(N), (4.21) where N is column vector describing the carrier distribution, and A(N) and B(N ) are matrices dependent on N. Simple Dirichlet boundary conditions seem to suffice for this problem. Eq. 4.21 is solved by using a slight variation to the fixed point iteration algorithm [28]: 1 . One has to first obtain an approximate initial distribution of N to start the calculation. When the very first beam propagation step has to be taken, one can make a guess for r and a, given J at that point, and then solve the diffusion equation (Eq. 4.21) for N (neglecting stimulated emission or absorption). Otherwise, if a beam propagation step has already been taken, one may just use the carrier distribution at that previous step as the initial condition. Let us call the initial value N=Nj. 2. Now calculate the matrices Aj — A(Nj) and Bj — B(Nj). 149 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3. Solve the matrix equation AjNj+i = Bj for the new carrier distribution Nj+i = A~lBj. A j 1 is the inverse of A ,-. 4. The updated carrier distribution is given as the average of the new carrier distribution and the old carrier distribution, Nj+\ = |(iVj + Nj+i). This makes the algorithm stable. Now go back to step 1 and make Nj = Nj+\. Repeat the whole process until a desired accuracy is reached. This algorithm can be used to solve the propagation of optical waves in strongly saturated amplifiers or saturable absorbers with weak index guiding waveguides. A full carrier distribution calculation is necessary for the realistic evaluation of limiting devices based on nonlinear guiding in semiconductors. By using the carrier density distribution calculation in conjunction with the FDBPM, it was demonstrated that the absorption in quantum well devices is in fact too high to make the all-optical cutoff modulator (shown in Fig. 4.4) viable. It was found that all light is absorbed before nonlinear guiding can start to play a role in the transmission characteristics. One needs a material with a lower absorption to make the device concept shown in Fig. 4.4 viable like, for example sihcon as in the original work by Normandin [62]. Silicon essentially has a very high linewidth enhancement factor, compared to direct bandgap semiconductors. This is due to the low absorption near the bandgap of silicon, which is an implication of the indirect bandgap. This also makes the 150 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. differential gain lower. The differential index change dn/dN (n is the refractive index, and N the carrier density) may still be high due to the plasma effect. 4.4 The Use of Strained Quantum Well Material Semiconductors are desirable materials for use in limiters because of the large achievable refractive index changes in these materials [30, 35, 24, 23, 10]. In order to bring about the index changes, semiconductors need to be utilized in spectral regions where they have interband gain or absorption so that light can change the material carrier density by stimulated absorption or emission, which leads to an index change. The gain or absorption, or the changes in gain/absorption brought about by interband transitions may be undesirable under certain circumstances, as it may be more difficult to engineer certain power-dependent transmission characteristics. This is because not only refrac tive index changes have to be taken into account, but now also simultaneously gain effects. Furthermore, high absorption and spontaneous emission are always undesirable properties of saturable absorbers or amplifiers, respectively. Quantum confinement using heterostructures, and compressive strain in semiconductors give certain interesting properties to semiconductors. These are essentially ways to achieve a transparent material in which very big index 151 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LH1-C1 HH1-C1 Bandedge Bandedge J 1 TE Absorption o TM Absorption' ^ Input spectrum W avelength Figure 4.5: This graph shows the relationships between the material band-edge wavelengths for a compressively strained quantum well device and the input light spectrum of the light that is being is switched, by a device based on this material. changes can still be achieved [19, 20]. The effect of quantum confinement is to lift the degeneracy between the heavy-hole (HH) and light-hole (LH) valence bands for electron waves in the material with crystal wavenumber k = 0. The splitting is such that the HH-levels are closer to the conduction bands than the LH-levels. Compressive strain leads to a perturbation in the Schrodinger equa tion which leads to an extra and therefore even larger splitting between the HH- and LH-levels. The matrix elements of the first conduction band level (Cl) to the first light-hole valence band level (LH1-C1) and the first conduction band level Cl to first heavy-hole valence band level (HH1-C1) interband transitions are such that the C1-HH1 transition provides absorption (or gain in a pumped semiconductor) exclusively to the TE-polarized optical fields in a device, while the LH1-C1 transition provides mostly TM absorption/gain as well as some 152 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TE-absorption/gain. A diagram showing the relative position of the bandedges in a saturable absorber containing a compressively strained quantum well ma terial is shown in Fig. 4.5. The main idea is to use the compressively strained quantum well material as the nonlinear medium in a device and use a dual polarization input beam with a wavelength that falls between the LH1-C1 and HH1-C1 bandedges. The TE-polarized part of the light can serve as a ‘pump’ beam which brings about interband transitions (either absorption or stimulated emission). The ’ probe’, which is the TM-polarized field, can read the refractive index changes brought about by the TE-polarized light. If the material is trans parent enough and does not respond to the TM fight at all (the ideal case), the ‘probe’ beam’s power can be made much larger than that of the ‘pump’ beam, and therefore the amount of power that actuates the nonlinear device is essen tially much smaller than the power that is being switched. Therefore, optical ‘gain’ can be achieved in a similar way to that in an electronic transistor. This idea has been exploited in the so-called SAND-switch (saturable absorber non linear dichroic switch), which is a waveguide Fabry-Perot interferometer device containing strained quantum wells [31]. A device gain of 23 was predicted by the authors of the latter paper based on their experiments. An accurate calculation of the properties of these materials is also shown here (See Figs. 4.6, 4.7 and 4.8) to give the reader a quantitative feel for the num bers involved in typical devices. This specific numerical calculation was done 153 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4000 2000 E < j C- -2000 e '& Ui I -4000 -6000 -8000 - 1 0 0 0 0 '_______ L ..............1...______ 8 ..............I , , 1 ------------1......... .....I— - I - ..... ,1____ .___ 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5 1.55 1.6 1.65 Wavelength (um) Figure 4.6: The gain for TE-polarized fields in a 7.5nm wide 1% compressively strained InGaAsP quantum well embedded in lattice-matched InGaAsP with a bandgap of 1.06/rm for carrier densities from N = 1 x 1017cm~ 3 to N = 5 x 1 0 18cm~ 3 in ten equal steps. using the commercial program LASTIP. The author’ s more approximate calcula tions, which neglect intervalence band mixing, give similar results. The calcula tions are for a 7.5 nm thick In i-xGaxAsyPi-y quantum well, where x— 0.26 and y=0.85 (~ 1% compressive strain), embedded in quaternary In i-xGaxAsyPi-.y (lattice-matched to InP), with a bandgap of 1.06/rm. For the barrier material one obtains that x=0.1116 and y=0.2446. Broadening of spectral features was modeled using a Landsberg distribution, and it was assumed that the intraband scattering time is rscat=0.5ps. The Coulomb interaction between the electrons and holes was neglected. The TE-gain is shown in Fig. 4.8. The family of 154 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4000 2000 £ -2000 | -4000 I. | -6000 -8000 -10000 -12000 1.65 1.25 1.3 1.35 1.4 1.45 1.5 1.55 Wavelength (um) 1.15 Figure 4.7: The gain for TM-polarized fields in a 7.5nm wide 1% compressively strained InGaAsP quantum well embedded in lattice-matched InGaAsP with a bandgap of 1.06/xm for carrier densities from N —l x 1017cra~ 3 to 77=5 x 1 0 18cm~ 3 in ten equal steps. curves shown are for carrier densities from iV=lxl0 17cm~ 3 to 5xl018cra~ 3 in ten equal steps. The plot in Fig. 4.8 shows the TM-index change due to gain changes, i.e., the Kramers-Kronig transformation (KK-transformation) of the gain changes (0.8eV corresponds to 1.55 um free space wavelength). Clearly there are substantial index changes for the TM light below the LH1-C1 band edge, where TM light has very little absorption. 155 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.75 0.8 0.85 0.9 0.95 1 1.05 Energy (eV) Figure 4.8: The nonlinear index change for the TM polarization due to the Kramers-Kronig transformation of the gain changes in a 7.5nm wide 1% com pressively strained InGaAsP quantum well embedded in lattice-matched In GaAsP with a bandgap of 1.06/xm for carrier densities from N = 1 x 1017cm" 3 to N = 5 x 1018cm~ 3 in ten equal steps. The plasma effect also leads to a negative index change for the TM-polarized light for increasing carrier density, with nearly the same magnitude below the LH1-C1 band-edge as that produced by the KK-transformation [24, 23], but simultaneously a small gain/absorption change. The large plasma effect-induced index change, with the simultaneous absence of large absorption effects in this wavelength range, is due to the large KK-transformed, free-carrier losses in the far infrared region of the spectrum. The plasma effect refractive index contribution is, therefore, a proverbial ‘free lunch’. 156 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A quantitative example of the free-carrier loss is calculated here to confirm the relatively low loss of this mechanism. The quantum theory of this mecha nism is quite complicated [75] and it involves the calculation of phonon-assisted interband transitions. Subtler effects like optical deformation potential scatter ing and ionized impurity scattering may also play a role [75], Therefore, the simpler classical approach is shown here. In the Drude-Zener theory [79], the imaginary part of the refractive index for the plasma effect is given by to is the angular frequency of the light, 7 is the damping constant of the material, and up is the plasma frequency of the material: where N is the carrier density of the material, q the electron charge, and e0 the permittivity of free space, m* is just the effective mass of the electron in the conduction band, which is 0.0427 multiplied by the rest mass of an electron (for InGaAs lattice-matched to InP [19]). The holes have little effect on the free carrier losses due to their much bigger effective masses. The damping constant can (by a quantum mechanical calculation [75]) be shown to be equal to the inverse of the interband scattering time, which in turn is around O.lps [20] in (4.23) 157 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. these materials. For a carrier density of iV = 1 x 1019cm~3, the absorption coefficient can be calculated to be about 48cm-1. Even at the high carrier density of N = 1 x 1019cm~3, the loss coefficient is more than two orders of magnitude smaller than the interband absorption numbers visible in Figs.4.6 and 4.7. The linewidth enhancement factor for the plasma effect, at the NIR (near infrared) wavelengths in direct bandgap semiconductors (like InGaAsP) , can in the classical theory be shown to be otplasma — x j (4.24) 2 fW 7 where nmat is the refractive index of the material. Eq. 4.24 gives a linewidth enhancement factor ~17.4 at 1.5pm in materials like InGaAsP, depending on the precise value of 7 . This can be compared to the linewidth enhancement factor due to the KK-contribution at the gain peak for a quantum well material, which is typically around 1-2 [ 8 8 ]. This shows that the plasma effect is much more effective to create nonlinear index changes in the N IR than the KK- contribution for a given gain change. The nonlinear refractive index cross- sections, a, are also still about the same at the gain peak of an amplifier or saturable absorber for both mechanisms [ 8 8 , 24, 23]. (The nonlinear index change is usually in the form An — oN, where N is the carrier density). 158 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. According to common wisdom [ 8 8 , 70], the plasma effect can only be seen for the TE polarization due to the ‘fact’ that electron motion is limited to in-the- plane motion in the quantum well. Experimentally the latter statement does not seem to bear out since large TE and TM nonlinear index changes can be measured far below the respective bandgaps, which cannot be explained by KK- transformations of gain changes only [23]. This is also confirmed by Tomita’s theoretical work [89] which showed that the Drude model provides a good ap proximation of the plasma effect for the TE mode in a quantum well device. He also showed that the susceptibility for the TM mode is almost the same as that for the TE mode, in spite of the quantization of carrier motion, as long as the photon energy is much larger than the intersubband energy. Spector [ 8 6 ] ex tended the quantum theory of free carrier absorption to quasi-two-dimensional structures and showed that the free carrier absorption may be a function of the thickness of the layer due to the quantization. This theory predicts an enhance ment factor (compared to bulk material) of the refractive index cross-section due free carrier absorption of the order of ~3 for a 6 nm InGaAs quantum well with InP barriers [23]. It should be pointed out that intervalence band absorp tion (I VBA) [76] may contribute to carrier density losses above and below the bandgap and should be included in gain and refractive index cross-section calcu lations if higher accuracy is desired. The IVBA mechanism is due to transitions 159 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. between the heavy-hole and split-off valence bands. It is, however, quite com plicated to calculate since it requires the solution of the 8 x 8 Luttinger-Kohn Hamiltonian. The literature is also unclear about the correctness of theoreti cal and experimental values of the intervalence band contribution to the total carrier-dependent absorption [2]. Values of IVBA which were ‘directly mea sured’ seem to range between ~ 12cm- 1 and ~ 25cm.-1 in p-type InGaAsP at 1.3fim and 1.6/xm, respectively, at a carrier density of 1018cm-3. Experiments in which measurements of the carrier-dependent losses below the bandgaps in strained and unstrained quantum wells were done are also unclear on this sub ject since the IVBA and the plasma effect are not taken into account separately [45, 29]. The latter experiments also seem to indicate that IVBA may be turned off in highly strained quantum wells. The predicted index changes in a 6 nm wide Ino.53Gao.47As well embedded in quaternary material with a bandgap of 1.06 fxm for carrier densities from 1 x 1016cm- 3 to 4 x 1018cm~ 3 in 5 x 1017cm~ 3 steps, are shown in Fig. 4.9. The calculated index changes include both the KK-transform contributions as well as the plasma effect, but ignores IVBA. The data shown in Fig. 4.9 were calculated using a custom program in MATLAB and neglect valence band intermixing, i.e., the parabolic band approximation was used. A free carrier absorption enhancement factor of 3 compared to bulk material was assumed [23]. Excitonic 160 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. - 0.02 -0.06 $ - 0.1 £ - 0.12 -0.14 -0.16 1.3 1.4 1.5 1.6 Wavelength (p m) Figure 4.9: The total nonlinear index changes in a 6 nm InGaAs quantum well embedded in InP-based material. The solid line represents the index change for the TM polarization and the dashed line the index change for the TE polariza tion. The curves are for carrier densities of 5 x 1017cm~ 3 to 4 x 1018cm~ 3 in 5 x 1017cm~ 3 steps. The semiconductor with a carrier density of N=1 x 1016cm- 3 is the reference. effects were neglected. The HH1-C1 band-edge is at about 1.5/im and the LH1- C1 band-edge at 1.415/rm. In between the bandedges where TM absorption is low, there are substantial and nearly wavelength independent index changes for the TM polarization. The difference in the LH1-C1 and HH1-C1 bandedges is smaller than for the previously shown 7.5nm well due to the absence of the compressive strain. It should be pointed out that the index changes obtained from these theoretical calculations seem to agree quite well with experimental measurements [24, 23] in similar material systems. 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4.5 The Asym metric Adiabatic Y-branch with Kerr Nonlinearities 4.5.1 M echanims for Limiting in Asym m etric Nonlinear Y-branches 4.5.1.1 Introduction It was found that the asymmetric adiabatic Y-branch with selectively placed Kerr-like nonlinearities, like shown in Fig. 4.10, can display interesting trans mission characteristics. In particular, two specific configurations (See Fig. 4.11 and Fig. 4.12) show transmission characteristics suitable for power limiting. Beam propagation simulations seem to agree with a negative feedback mecha nism identified in these structures. A literature search shows that this kind of structure has, in fact, attracted previous attention from the all-optical switching community. Murata [59] showed analytically that a coupled waveguide system, with a nonlinearity in one of the waveguides (like that in a Y-branch with one nonlinear branch), may show bistability in the dispersion (as function of power) of the effective indices of the supermodes. By using these analytical results he was able to show that an adiabatic Y-branch with a focusing Kerr-nonlinearity in one of branches may exhibit bistability when light is propagating from the stem of the branch to the branches, in the case of very small angles in the branch. Chen and Najafi [17, 18] earlier analyzed the same kind of structure but also looked 162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Branch #1 Branch #2 Small angle to ensure adiabacity Index change by external means Stem (usually single mode) Figure 4.10: A diagram of the generic structure discussed in Paragraph 4.5. In order to ensure adiabatic mode evolution the angle between the branches is typically very small. One of the arms contains a nonlinear material which can exhibit a nonlinear index change, that can be either be switched by a voltage (in the case of the digital optical switch), or by all-optical means in the case of the limiter. In this way asymmetry in the effective indices of the branches can be switched. Light propagation can either be from one of the branches to the stem or vice versa. at transmission from the branches to the stem. Their calculations showed that the structure actually behaves in a nonreciprocal way, i.e., the transmission as a function of power looks different depending on the direction of propagation of light. They also found that the transmission of light from the linear branch to the stem of Y-branch showed an interesting power limiting characteristic. 4.5.1.2 Conceptual framework: The digital optical switch and the local normal mode formalism The type of devices discussed here are in fact all-optically actuated digital optical switches (DOS) [78, 82, 83]. The DOS can be considered as adiabatic 163 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. mode evolution switch (or switchable postion taper) that can be switched by external means into one of two states. The usual definition of a taper is a device which changes the shape of mode through a ’ ’slow” transition in the waveguide without loss to higher modes and radiation modes. This class of devices can be conveniently described in terms of the coupled normal mode formalism [84, 85, 54, 9, 11, 12]. In the coupled local normal mode formalism coupled waveguides, or arbitrary guiding structures, are considered as a single waveguide and the mode evolution described in terms of the supermodes of the structure. A short summary of this theory is given here since it aids in the understanding how the structures being considered in this chapter operate. Propagation of the light is described in terms of the evolution of the so-called local normal modes. The supermodes or local normal modes are calculated at every position along the propagation direction, as if the structure at that point is unchanged as one goes backward or forward along the propagation direction. Assume that the transverse electric fields in polarization p and mode order 7 at position z can be written as £jp(x, y, z, t) = ^ Up(x, y, z)Dp{z)el^ ( - z)z~u lt) + c.c. , (4.25) where c.c. is the complex conjugate, (x , y, z) the spatial coordinates, t the time variable, and (3P is the propagation constant. Up is the mode profile and Dp the 164 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. field amplitude. Let’ s first consider only power transfer between modes 7 = i and j. Using the continuity of the tangential E and H-fields across a small step; neglecting curvature, radiation losses, and reflections; and using mode orthogonality and power relationships and dropping the polarization index, one obtains a set of local normal mode equations [56]: qJ = CjiDi — ifijDj on = CijDj — ipiDi , (4.26) where A flji = — Aflij = (3j — $ and Cji = — C q- is the coupling constant given by CH = k - 2 m h M m d A (4 2 7 ) Here n is the average effective refractive index at position 2 , and n the refractive index at the coordinates (x,y,z). Usually power is only launched into either mode i or j. Assuming that all the power is launched into mode i the extinction £ (i.e., the ratio of the power in mode j to the power in mode i is given by) £ = | r Cjie^o A^ dz"dz'\2 . (4.28) Jo Eq. 4.26 shows that totally adiabatic conversion of modal shapes is not possible in a structure with z-dependent features. In fact, undesired mode conversion is 165 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. predicted by Eq. 4.28. The structures discussed here are nearly adiabatic and, therefore, £ is small. The conceptual ideas of the local normal mode theory are, however, still highly useful in this discussion. The previously mentioned DOS is a special type of Y-branch which acts as a mode sorter if the Y-Branch angle is small enough and if there is sufficient asymmetry in the waveguide branches [14]. The lowest order mode incident on the y-branch from the stem would ‘choose’ to go the higher effective index branch. In other words, the lowest order local normal mode of the stem evolves into the symmetric local normal of the Y-branch, which adiabatically couples only to the mode of the higher effective index branch. If the stem was mul- timoded, the first order mode incident on the branch would ‘choose’ to go to the lower index waveguide because the higher order mode of the stem evolves into the assymetric mode of the y-branch, which adiabatically transforms into the mode of the lower effective index branch. In a sense the higher mode of the stem is phase-matched to the mode of the lower effective index branch and the lowest order mode to the mode of the high effective index branch. In other words, a mode would choose the arm in which it could propagate with an effective index closest to the effective index that characterized its propagation before the waveguide divided. If the angle of the branch is ‘big’, or if the index asymmetry is small, the power is just split between the waveguides. Burns and Milton [14] 166 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. studied these structures extensively and found that mode sorting is obtained, if the branching angle satisfies 9 < A / ? / 7 , (4.29) where A/3 is the average difference between propagation constants of the two normal modes and 7 is their transverse propagation constant in the cladding region. Eq. 4.29 usually implies branching angles of at least smaller than 1 degree in typical photonic structures implemented in semiconductors. In the digital optical switch the stem is single-mode. Therefore, only the lowest order mode is excited before the branch (in the stem) and will go to the branch with the higher index. In the digital optical switch one can transform a symmetric Y-branch into an asymmetric branch by changing the index of one of the branches through the electro-optic effect by applying a voltage across one of arms [78, 13]. Of course,the digital optical switch can be implemented not only in electro-optic materials like LiNbOz [78, 13], but also in semiconductor materials. In semi conductors the effective index of an arm of the Y-branch can be changed by either current injection [15, 60] or the Franz-Keldysh (or quantum confined Stark effect in quantum wells) [49, 82, 83]). Of course, the electro-optic effect also still contributes in semiconductors. The work referenced here shows that the nonlinear effects in semiconductors are big enough to change light guiding 167 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. in these structures. Patrick LiKamWa’s research also demostrated, that the carrier nonlinearities and Stark effect in the DOS can be switched on by an optical beam, instead of electrically activating these nonlinearities in the semi conductor, which shows that a limiter based on these materials and structure is also viable[46, 53]. 4.5.1.3 Limiting mechanism in the Y-branch with a self-focusing Kerr-like nonlinearity The first configuration that seems promising as a limiter is shown Fig. 4.11. Light propagates from the linear branch to the stem in the structure shown in Fig. 4.11. The stem and individual branches are assumed to be single-moded so that the stem rejects the asymmetric normal mode incident upon it from the branches. The nonlinear branch contains a self-focusing type of Kerr-like nonlinearity. It is assumed that the structure is manufactured in such a way that it is asymmetric (at low input power). In particular, the linear arm’s effective index is assumed to be higher than that of the nonlinear arm. A diagram of the propagation constants of the guided waves is also shown next to the Y-branches in Fig. 4.11. Under normal operation the light would only be launched into the linear arm. The simultaneous propagation of a hypothetical probe light beam is also considered for explanatory purposes. The probe beam is assumed to be very weak, so that so that it does not affect the propagation constants. It is launched along the nonlinear arm and denoted by dashed lines. The propagation constant 168 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. mode cutoff 1st 1st mode cutoff 1st' + n g I & I O J c Figure 4.11: An illustration of the operation of the adiabatic nonlinear y-branch limiter, with a self-focusing nonlinearity in the one branch (left). 169 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. diagram, therefore, denotes both the propagation of the so-called normal beam (solid lines) as well as the hypothetical probe beam (dashed lines). The limiting phenomenon in this structure can now be explained as follows: • When a small amount of power is launched into the normal beam it does not affect the propagation constants of the local normal modes in the Y-branch (See Fig. 4.11a). The power launched into the normal mode in the linear branch evolves into the lowest order mode of the Y-branch, which adiabatically couples to the mode supported by the stem. The hypothetical probe excites the asymmetric mode of the Y-branch, which has a smaller effective index, than the lowest order symmetric mode sup ported in the branching section (See the propagation constant diagram). The assymetric mode cannot couple to the mode in the stem, where it is cutoff, and therefore it would radiate to the substrate at the point where the branches meet. • At intermediate powers in the normal beam, the effective index of the non linear arm is raised through Kerr-like effects. In Fig. 4.11b, the situation is shown where the effective indices of the arms are about equal, so that the normal beam excites equal amounts of the symmetric and antisym metric modes in the Y-branch region. But only the symmetric mode can couple to the mode of the stem, so half of the light is cutoff and radiates into the cladding regions. 170 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • At very high power in the normal beam, the index of the nonlinear arm has increased to such an extent that the effective index of the nonlinear arm, exceeds that of the linear branch (Fig. 4.11c). Now the normal beam couples almost exclusively to the antisymmetric mode of the Y-branch, which radiates into the cladding at the point where the branches meet. The primary limiting mechanism is, therefore, the monotonically increasing fraction of input power coupled to the cladding as a function of power. One needs some extra mechanism in order to obtain very flat power transfer charac teristics for the limiter over an extended power range. Such a mechanism exists in the form of negative optical feedback, which is now explained. Assume the amount of power in the normal beam is such that the index of the nonlinear arm is increased to equal that of the linear branch. At the point where the effec tive indices of the branches are equal, the Y-branch operates as a 3dB splitter and the normal beam launched into the linear branch excites equal parts of the symmetric and asymmetric modes. Consider the situation where the normal beam power is increased by a small amount, SP. The power increase leads to a small positive change in the index of the nonlinear arm, Sn, which increases asymmetry of the Y-branch, so that the normal beam now excites more of the asymmetric mode, which in turn has a larger overlap on the linear branch than the nonlinear branch. A smaller fraction of power then overlaps the nonlinear branch, which causes the effective index of the arm to decrease, so that the 171 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. branches are pulled closer to symmetry again. A negative feedback mechanism, therefore, exists in the form of the fact that the nonlinear arm wants to keep the power incident on it constant. POLARIZER— * ^ I TMonlyA OUTPUT A j \A . “ ,nUT 7 # An=-n2l (only TE) | TM (probe) * — "► T P /P n m n l Figure 4.12: The nonlinear Y-branch with a defocusing nonlinearity in one arm. It is assumed that the dual polarization scheme (as discussed in Paragraph 4.4) is used. 4.5.1.4 Limiting mechanism in the Y-branch with a defocusing Kerr- like nonlinearity Another version of the asymmetric Y-branch with a selectively placed Kerr nonlinearity is depicted in Fig. 4.12. In the structure shown in Fig. 4.12, the light propagates from the stem to the nonlinear branch. This type of device works in a somewhat similar way to the Y-branch with the self-focusing non- linearity. The device in Fig. 4.12 is assumed to be manufactured in such a way 172 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. that the nonlinear branch has a higher effective index than the linear branch (at low intensities). Therefore, the mode of stem, that has to couple to the symmetric mode of the Y-branch, adiabatically evolves into the mode of the nonlinear branch. At higher powers the effective index of the nonlinear branch is diminished so that the structure is nearly symmetric. Both the symmetric and asymmetric modes of the structure are excited when the structure is sym metric and only half of the light goes to the end of the nonlinear branch. At even higher intensities, the index of the nonlinear arm can be reduced so much that the stem-mode couples to the symmetric mode. The symmetric mode is now falling mostly across the linear arm and evolves into the output mode of the linear branch. Most of the light is, therefore, now routed to the linear branch. A negative feedback mechanism can clearly also be identified for this structure. The structure in Fig. 4.12 is novel and is presented for a first time in this dissertation. 4.5.2 Beam Propagation Studies of Selected Structures 4.5.2.1 BPM simulation of structures w ith defocusing Kerr nonlinearities Beam propagation numerical simulations were performed in order to find the best shaped structures for limiting. The Kerr-like nonlinearities were assumed to be of the same magnitude as those that can be obtained in direct bandgap semiconducting materials (like InGaAsP) at the optical fiber communication 173 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Input Figure 4.13: The approximate structure used in evaluating the adiabatic Y- branch with defocusing nonlinearity. The branch marked by — n^ contains the defocusing nonlinearity. wavelengths of 1.3 and 1.55/xm. The first structure that was analyzed is shown in Fig. 4.13 and consists of a defocusing nonlinearity in the one arm of the y-branch. The nonlinear material is in the form of unpumped quantum wells (QW). The refractive index of the QW’s can be changed by the absorption of TE-polarized light. The rest of the structure is assumed to be pumped up to transparency, which decreases the refractive index compared to areas with no current injection. The effective refractive index step for the nonlinear arm is 0.0045 and for the rest of the structure it is 0.002 (due to the current injection). One would like to get an idea how this structure might operate in the dual polarization scheme (See Paragraph 4.4). At the time this simulation was done, only the RSoft BPM was available. RSoft BPM had only the ability to calculate the effects of a classic Kerr nonlinearity. Therefore, the structure was assumed 174 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. to be totally transparent, and the classical Kerr nonlinearity coefficient n2 (in cm2/W ) was related to the semiconductor nonlinearity. For the classical Kerr- nonlinearities the intensity dependent refractive index is defined as n = n0 + n2I, (4.30) where I is the time averaged intensity and no is the background refractive index. In an unpumped semiconductor, where it is assumed that the material absorption and refractive index are directly proportional to the carrier density, one finds from the steady-state rate equation that , crIN0/Is ^ " = "0 + 1 T T J T . ■ ( 5 Is = E/(ar) is the material saturation intensity (a is the differential gain, E the photon energy and r the carrier lifetime), a the refractive index cross- section (in cm3) and fV 0 the optical transparency carrier density. From the work of Erlich et al. [24, 23], who considered the nonlinearities in an unstrained 6nm wide InGaAs quantum well embedded in InP, a has been estimated to be around — 1 x 101 9 cm-3 close to the bandgap. For this type of quantum well, one also finds that the transparency carrier density is approximately 2.3 x 101 8 cm3 [20]. Therefore, the maximum index change is A™ ax = a x N0 = 0.23. Now also assume Is = 1.7M W /cm 2, and that the vertical overlap factor (the 175 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. fraction of the optical energy that is contained in the quantum well) of the three-dimensional wave is r 2 /=0.0108. It has been assumed that we are dealing shape U(x, y) — X(x)Y(y), where x and y are the transverse coordinates in the three-dimensional case, which is reduced to only the lateral coordinate, x, in the two-dimensional calculation, through the use of the effective index technique. Therefore, the modal overlap with a quantum well with width w situated at y = yo (i.e., the gain distribution q(x, y) = rect ( ^ f a)) is with a (laterally) weakly index-guided wave with a separable type of mode — OO j — OG (4.32) where f + oo r „ = / X 2(x) dx and (4.33) J —o o (4.34) 1 7 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. If one assumes the semiconductor nonlinear index change can not be satu rated, one can relate n2 to the other parameters using Eqs. 4.30 and 4.31: n 2 — -y^- = 0.1353cm2 / M W (4.35) l s Of course,the number in Eq. 4.35 has to be multiplied by the vertical overlap r„=0.0108 to find the correct n2 for a two-dimensional beam propagation cal culation. The results of calculating the power-dependent transmission for the 1 00 ^ -3 < 2 % 0.9 variation 0.8 0.7 0.3 150 200 250 300 Input power (mW) 50% power i fluctuation i Figure 4.14: The limiting characteristics of the structure shown in Fig. 4.13 structure in Fig. 4.13 are shown in Fig. 4.14. The power is launched into the mode of the stem, and the output is defined to be the amount of output power in the mode at the output of the nonlinear waveguide. Clearly, excellent power 177 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. limiting can be obtained over a fairly large range. As shown in the Fig. 4.14, an input power fluctuation of ±50% can be converted to a small power variation of only 2%, i.e., classical noise reduction of ~14dB. The structure in Fig. 4.13 was also simulated with the custom BPM pack age, which can accurately simulate the propagation of both polarizations in the presence of absorption and saturation effects. The calculation showed that the TE polarization gets absorbed before it can properly switch the structure (be cause of the high interband absorption associated with the nonlinearity). The TM-polarized light, which is defined as the output, is then not limited at all. 4.5.2.2 A numerical BPM study of Y-branch structures with self- focusing Kerr-like nonlinearities The power-dependent transmission of structures, similar to that shown in Fig. 4.11, with carrier nonlinearities, was also simulated using the custom BPM software. The semiconductor must exhibit optical gain, in order for the nonlin earities to be of the self-focusing nature. When the dual polarization scheme is used (Paragraph 4.4), one polarization experiences gain (TE) while the other (TM) sees a transparent medium. The TE and TM polarizations also do not ex perience the same carrier-induced index change (TE sees a bigger index change than TM near the HH1-C1 bandgap; see Fig. 4.9). It was, therefore, found that 178 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.2 0.05 0.15 Input power (Watt) Figure 4.15: The power-dependent transmission of the structure shown in Fig. 4.16. the amount and threshold of limiting for the two polarizations were desynchro nized. In fact, it was impossible to obtain good limiting characteristics for the TM field. One can, therefore, resort to using only one polarization in the device, namely the polarization that causes the interband transitions (TE). The use of the single polarization results in very good performance as shown in Fig. 4.15. The relevant structure for this simulation is shown in Fig. 4.16. It was assumed that the nonlinear arm contained quantum well material (6nm InGaAs quantum well in InP). To make the simulation more realistic, a nonsaturable loss of 5cm-1 was assumed for the passive parts of the structure. If the gain 179 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Output Figure 4.16: The structure with a self-focusing type of nonlinearity that can be used to obtain the limiting characteristics shown in Fig. 4.15 is linearized in carrier density, one obtains the following expression for the gain (from the steady-state carrier rate equation) 9o i > (4.36) where g0 m 1500cm”1 at a carrier density of 4.4 x 101 8 cm“3 for this type of quantum well. The other symbols have the same meanings as in Paragraph 4.5.2.1. Therefore, the imaginary part of the refractive index in the 2-D simu lation scenario is 9^y 2ko (4.37) where k0 is the optical free-space wavenumber, r y=0.005 is the vertical confine ment factor of the 3-D field, and g is given by Eq. 4.36. The same magnitude 180 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. of nonlinear real index change is assumed as in Paragraph 4.5.2.1. This means that the real part of the refractive index is given by (the 2-D case) n = n 0 + 0.23F„I (4.38) where no is a background index. The linear branches are assumed to have a lateral effective index step of An=0.004, and the nonlinear branch an index step of An=0.0Q55. The refractive index of the nonlinear branch has to be reduced with 0.45P,, due to the carrier injection. Is is taken to be 1.7 M W j cm?. 5000 4000 3000 2000 1000 1 m -20 -10 0 10 20 30 40 X(jtim) 1.0 I 0.0 Figure 4.17: The propagation of the field in the structure shown in Fig. 4.16 for an input power of P=100mW. The solid lines are the outlines of the waveguides. 181 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 4.15 shows that the slope of the power transfer characteristics of the limiter is quite sensitive to the separation (angle) between the arms of the lim iter in the same way that a directional coupler coupling ratio is sensitive to waveguide parameters. A small vertical overlap factor seems essential to make the device operate properly. For larger Ty, the power transfer characteristics exhibit a bow, which seems to be caused by the saturation of the nonlinearities. When the material saturates, the normal mode extinction ratio of the device does not change any more, which means the device stops responding and con sequently lets through more light with increasing input power. Fig. 4.17 shows a snapshot of the propagating field amplitude obtained by the beam propaga tion software (the original image is in color for better visualization). Limiting is taking place, and mostly the asymmetric mode is excited in the branching region. The asymmetric mode is cutoff in the stem region and propagates to the substrate. 4.6 Limiting in a Power Unbalanced Interferometer The nonlinear phase shift due to Kerr-type nonlinearities in semiconductor waveguides like the semiconductor optical amplifier can also be utilized in in terferometer structures to do all-optical switching, like all-optical wavelength 182 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. OUTPUT INPUT \ — rr7 /? rT / N ..Jm ____ / Kerr-like nonlinear material Figure 4.18: The power unbalanced interferometer. This is a top-down view of a Mach-Zehnder interferometer. conversion [93]. Recently, the noise suppression capabilities of the interferomet- ric wavelength converters systems were recognized [94]. It is also well known now that by using the nonlinear index effects in conjunction with the saturation effects one obtains much more efficient switching [22] than one could by only using the gain saturation effects. To achieve a zero slope in the output power characteristics (and therefore effective limiting), one has to produce a transmission with an inverse dependence on the input intensity over a certain range. A Mach-Zehnder or Michelson interferometer has a decreasing transmission as a function of phase shift in one of its arms, if the interferometer was originally biased near maximum transmission. This phase shift can be brought about by placing a Kerr nonlinear element in one arm like shown in Fig. 4.18. The Kerr-like element generates a nonlinear 183 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Power limiting region ! » £ Input Power Slope 1/Power Input Power Phase Figure 4.19: This diagram shows how a nonlinear phase shift and the interfer ometer transmission can be combined to obtain an ideal limiting characteristic. This method works the best when the phase shift saturates at 7 r as shown here. phase shift roughly proportional to power injected into the device as shown in Fig. 4.19. The structure in Fig. 4.18 was analyzed by again assuming the use of quan tum well material in the device in a dual polarization scheme like previously discussed in Paragraph 4.4. The result of such an analysis is shown in Fig. 4.20 in which it was assumed that the quantum wells are electrically/optically pumped to exhibit gain in the TE polarization. The input light was assumed to be in the form of short pulses of FWHM A t which is much shorter than the carrier lifetime r. The pulses contain TE- and TM-polarized light, but the 184 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.9 0.8 30.7 « ^0.6 o > fco.5 ...... 0.4 0.3 0.1 0.5 1 1.5 2 TE Input Energy (pj) 2.5 Figure 4.20: The limiting characteristics of a device similar to that shown in Fig. 4.18. Pumped quantum wells are used as the Kerr-like material. The device parameters used in this calculation are shown in Table 4.1. TE-polarized light is blocked by a polarizer at the output. The relevant de vice parameters off the nonlinear Kerr-like section are shown in Table 4.1. The cross-polarization linewidth enhancement factor is defined as d\Re(xTM)] dN d[Im{xTE)} dN (4.39) X t m and x t e are the carrier-dependent susceptibilities for the TM-polarized and TE-polarized fields, respectively, and N the carrier density. Re(*) and Im(*) are operators that take the real and imaginary parts of a number. The form of the linewidth enhancement factor shown in Eq. 4.39 takes into account the fact that the phase shift on the TM-polarized light is brought about by the TE-polarized field. Agrawal and Olsson’s theory [3], was used to calculate the 185 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 4.1: THE PARAMETERS USED FOR THE SIMULATION SHOWN IN FIG. 4.20 Material parameter Value Cross-polarization linewidth enhancement factor 1.3 Amplifier length 1mm Gain overlap factor 0.02 Peak material gain 1500cm-1 Saturation energy 3pJ Small signal integrated gain 20 dB nonlinear phase shift. In the limit that A t « r , and in a reference frame that moves with the pulse, one finds that the nonlinear phase shift imparted on the TM part of the pulse is ^ N L i U f n E ^ ) ) = - O i T E - * T M h i (4.40) where GqE is the unsaturated single-pass gain of the amplifier for the T E po larization and UfnE{t) = f P™ (t')dt’ , (4.41) J — o o where P?E(t) is the power in the TE-polarized part of the input pulse. There fore, U in(t)TE represents the fraction of the pulse energy in the TE polarization contained in the leading part of the pulse up to t‘ < t. When the TM part of the input pulse is slightly delayed with respect to the TE part of the pulse by a 186 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. time which is longer than the pulse (as assumed in this calculation) but which is still much smaller than the carrier lifetime, U ff (t) just becomes the input energy in the TE portion of the pulse (E lf), which is time-independent. By us ing the delay between the TE and TM polarizations, a (nearly) constant phase shift is obtained across the TM pulse, which ensures a good extinction ratio of switching. There is also nearly no chirp imparted onto the TM field, which could be important in applications connected with optical communications. To obtain the maximum flatness in the limiting characteristics, the interferometer can either be unbalanced or biased to obtain the best possible power transfer characteristics. To obtain the characteristics shown in Fig. 4.20, the device was assumed to be slightly imbalanced, and the output in the TM light was taken to be proportional to: TM Output o c 1.08 + cos (0jvl(0) - < j ) NL(E ff)) ; (4.42) The form of Eq. 4.42 assumes that the interferometer is biased for maximum transmission (at zero input power). Essentially the same theory can be used when the interferometer contains a defocusing Kerr nonlinearity in the form of a quantum well saturable absorber. If one uses the same parameters as in Table 4.1 for this device, except that the small signal gain (absorption), GEE, of the device is now -20dB, one can obtain 187 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 ©1.5 a 0.5 TE input energy (p j) Figure 4.21: The limiting characteristics of a device similar to that shown in Fig. 4.18. Saturable absorptive quantum well material is used as the Kerr-like medium in the dual polarization scheme. the limiting characteristics shown in Fig. 4.21. In this case, it is assumed the device is unbalanced so that the output is of the form TM Output o c 1.96 + cos (4 > n l {0 ) - 4 > n l {E J® )) . (4.43) 4 > nl is again given by Eq. 4.40. The result of an energy transmission calculation is shown in Fig. 4.21. Essentially the same kind of results as those shown in Figs.4.20 and 4.21 can be achieved, when the light is assumed to be continuous wave (CW), (by solving the traveling wave equations for CW fields in a saturable amplifier/ absorber). In the instance of CW light, the relevant saturation parameter is the saturation power Psat — Esat/ r (r is the carrier lifetime). Typically, the energy gain (or 188 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. absorption) saturates slower as a function of Ein/E sat than the CW power gain as a function of P ^w / Psat (P ^w is the CW input power) [3]. In the CW regime, the best looking limiting characteristics may also be obtained by unbalancing or biasing the interferometer. In general, the best characteristics may be obtained when one has the ability to tune both of these latter parameters. 4.7 M ethods for the Selective Placing of Semiconductor Kerr-like Nonlinearities in Optical Integrated Structures One has to be able to selectively create areas with Kerr-like nonlinearities in the previously discussed optic integrated devices. The focus in this chapter has primarily been on integrated optic devices implemented in semiconductor materials that contain active semiconductor quantum wells. Quantum wells are deposited in a planar process and special steps have to be taken to obtain both passive and active areas on a wafer. One way to make a certain region of the chip passive is by selective current junction in certain areas up to the point where material gain transparency is reached. This approach has three problems. The ’’passive” material still generates amplified spontaneous emission (ASE), which can mask the signal. Secondly a lot of heat can be generated, if large areas of the device need to be pumped, creating secondary problems. Lastly material transparency can only be reached at only one single wavelength. Wavelengths below the transparency wavelength experience gain, and above 189 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. experience absorption. This might be a problem if the device needs to operate over a wide wavelength range. Another approach is to selectively etch out the quantum wells in certain areas [58] and replace the active waveguides with passively regrown waveguides. This typically requires at least one crystal regrowth step after the etching, which is sometimes problematical/complicated. Good waveguide mode matching can be achieved (~95%) [58], which probably makes this method good enough for the monolithic integration of the nonlinear interferometer with the Kerr-like section in one arm. But since it is hard to achieve complete mode matching, and reflections may occur at the butt-coupling interface, it is probably not the best method for the integration of the adiabatic Y-junctions limiters. This is because the latter devices rely on nonlinear mode conversion and modal pu rity is therefore required. A butt-coupled joint may cause undesirable mode conversion, which can adversely affect the operation of the device. no E2 E1 Figure 4.22: The effect of impurity-induced disordering (IID) on the shape of energy band profile of the conduction band of a quantum well. Note the shift in the energy of the allowed states of the well. The same thing happens in the valence band, which results in a net blue shift of the bandgap. 190 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. well layers during the process since the disordering is strongly depth-dependent [1], Disordered regions seem to extend up to 1.7pm in InGaAs/InP type of structures [55]. Spatial resolution up to 2pm can be achieved if the quantum wells are buried about lOOnm from the surface of the epitaxial structure [87]. 191 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Without cap, RTA Under cap, RTi 1100 n r 1000 .Original 600 400 300 1420 1430 1440 1450 1460 1470 1480 1490 1500 1510 1520 1530 Wavelength (nm) Figure 4.23: The shift in room temperature photoluminescence (PL) brought about by vacancy-induced disordering in samples of InP-based material (See Fig. 4.24), which contain strained InGaAsP quantum wells. The samples were selectively capped with a thin layer of S i0 2 and then rapidly thermally annealed (RTA). Three curves are shown for the PL for coming from a small area on the wafer. The first curve is for the original material which is not thermally annealed, the other for material which was put through RTA but not capped and the third for an area that was capped and annealed. The vacancy disordering method was experimentally tested on an InP-based sample (See Figs.4.23 and 4.24) with compressively quantum well layers that was epitaxially grown by the MOCVD (metal organic chemical vapor deposition) method for 1.55/xm optical amplifiers. The sample was topped with 0.5mm wide and 200nm thick S i0 2 strips (evaporated with an electron beam gun). The photoluminescence (PL) of the material was measured at a specific spot after patterning the S i0 2. The material was then pressed together with an GaAs 192 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. wafer (to prevent the decomposition of InGaAs at the top of the InP wafer) and rapidly thermally annealed at a temperature of 725 degrees centigrade for 2 minutes. The PL was then measured at the same spot as before. Two different PL curves are obtained if the laser spot is moved slightly to fall either on a capped or uncapped region of the InP-wafer (See Fig. 4.23). A differential PL shift of greater than 30nm is obtained even though the quantum wells are buried at a distance of more than 1/xra from the top of the wafer. Note that there seems to be no degradation in the intensity of the PL. Ideally one would stop the initial growth before the offset layer (See Fig. 4.24) is reached and then do intermixing on the quantum wells contained in the optical waveguide first, after which one can then regrow the top layers like the offset layer and the ridge layer. The fact that the structure already contains zinc as a dopant in the structure is also detrimental because zinc also diffuses under the application of heat. Zinc indiffusion enhances the diffusion of cations, but leaves anions unaffected [40]. The quantum well structure may be maintained with different characteristics, or even ordering may occur [91]. This may lead to a competing red shift in the PL spectrum [77]. The diffusion of the sulphur is not a problem since it gives a redshift [64]. 193 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. p+-doped InGaAs con- tanct layer T J Q ) C L O ■ o O c K 5 0.8 jim thick InP layer (for ridge) 5nm thick Q (1.2) Offset aynr '(In?)': ?6bnni thick ' etchstop layer ■ ■ ■ I llilM l w 0.6 pm thick waveguide (Q(1.2)) containing two 9nm wide compressively strained InGaAsP quantum wells Sulfur doped In'3 substrate Figure 4.24: The layer structure of the sample used in the intermixing exper iments. Q(1.2) in the annotation means InGaAsP lattice-matched to InP and with a bandgap of 1.2jim. It is believed that the quantum well disordering technique is a highly promis ing technique for active/passive integration in all-optical devices like the lim iters. The utility of the disordering technique has, in fact, already been demon strated for nonlinear Mach-Zehnder interferometers [47]. Differential bandgap shifts of around 170meP have been demonstrated [77], which are sufficient for wideband applications. Extremely high spatial resolution of around 90nm has also been shown in a special intermixing technique [69]. One can also envision using special growth techniques like the selective area growth method in order to spatially define the bandgap [72, 48, 73, 5, 41]. The serious drawback of the selective area growth method, when it is applied to the type of devices discussed in this chapter, is that the spatial resolution is limited 194 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. at least to the surface migration length of the species during growth (typically > 1 — 10 /mi). This might be insufficient for realizing the nonlinear Y-branch limiters and possibly also for the nonlinear Mach-Zehnder interferometers. 4.8 Conclusion Kerr-like nonlinearities, and specifically the carrier nonlinearities in direct bandgap semiconductors, can be effectively used in certain structures to show characteristics suitable for power limiting. The LH1-C1 and HH1-C1 bandgap offsets in unstrained and compressively strained quantum well semiconductor material seem to exhibit some unique properties suitable for integrated optic limiters. The simulation of the strongly nonlinear guided but absorptive de vices (i.e., defocusing effect-devices) in this chapter seems to suggest that the absorption in of the HH1-C1 transition is too high to make these devices work. This is because these devices need to be typically very long. The light is ab sorbed before it can significantly change the index over the whole interaction length. The absorption problem is the reason why the waveguide limiter with the (absorptive) defocusing nonlinearity (See Fig. 4.4) does not work. High absorption is not a problem in the nonlinear interferometer limiter, which still generates big nonlinear phase shifts for the TM polarization, even though all the 195 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TE-polarized light is absorbed. It was shown that when the material is exter nally pumped (i.e., exhibits gain), the nonlinear Y-branch can effectively operate as a limiter, if only one polarization (TE) is used. The nonlinear interferometer and (amplifying) nonlinear Y-branch seem to be very suitable candidates for practical guide wave limiters. Quantum well intermixing seems to be the most promising method to fabri cate the devices because it gives good spatial resolution for bandgap patterning and gives no reflection/ mode conversion at the interfaces of active and passive material. Theoretically there is some absorption of TM-polarized light with a wave length falling between the HH1-C1 and LH1-C1 bandedges of a compressively strained quantum well (See Fig. 4.7). An interesting question is whether the ab sorption for the TM polarization between the HH1-C1 and LH1-C1 bandedges is large enough so that impinging TM-polarized light can create enough carriers to influence the guiding of the TM light through the (big) plasma effect, but be still low enough to let enough TM light through to the output of the device. In this scenario only TM light with a wavelength falling between the HH1-C1 and LH1-C1 bandedges may be used in the switches. The switching power (the power needed to create a certain amount of carriers needed for actuating the device) can be kept low using a large number of quantum wells in the waveg uide region, thereby ensuring a large overlap of the propagating field with the 196 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. nonlinear material. A great unknown is the precise value of the TM absorption between the LH1-C1 and HH1-C1 bandedges. 197 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1] Depth-dependent native-defect-induced layer disordering in AlxG a\-xAs — GaAs quantum well heterostructures, Appl. Phys. Lett. 54 (1989), no. 3, 262-264. [2 ] G. P. Agrawal and N. K. Dutta, Semiconductor lasers, 2nd ed., pp. 139- 142, Van Nostrand Reinhold, New York, 1993. [3] G. P. Agrawal and N. A. Olsson, Self-phase modulation and spectral broad ening of optical pulses in semiconductor laser amplifiers, IEEE J. Quantum Electron. 25 (1989), 2297-2306. [4] S. J. Al-Bader and H. A. Jamid, Nonlinear waves in saturable self-focusing thin films bounded by linear media, IEEE J. Quantum Electron. 24 (1988), no. 10, 2052-2058. [5] M. Aoki, T. Taniwatari, M. Suzuki, and T. Tsutsui, Detuning adjustable multiwavelength MQW-DFB laser array grown by effective index/quantum energy control selective area MOVPE, IEEE Photon. Technol. Lett. 6 (1994), no. 7, 789-791. [6 ] G. A. Baker, Essentials of Pade approximants, Academic Press, San Diego, 1975. [7] J. D. Begin and M. Cada, Exact analytic solutions to the nonlinear wave equation for a saturable Kerr-like medium: Modelling of nonlinear opti cal waveguides and couplers, IEEE Trans. Quantum Electron. 30 (1994), no. 12, 3006-3016. [8 ] A. S. Bhushan, F. Coppinger, S. Yegnanarayanan, and B. Jalali, Nondisper- sive wavelength-division sampling, Opt. Lett. 24 (1999), no. 11, 738-740. [9 ] R. J. Black, S. Lacroix, F. Gonthier, and J. D. Love, Tapered single-mode fibres and devices. Part 2: Experimental and theoretical quantification, IEE Proc. Part J 138 (1991), no. 5, 355-364. [10] R. W. Boyd, Nonlinear optics, pp. 159-164, Academic Press, San Diego, 1992. 198 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [11] W. K. Burns, Normal mode analysis of waveguide devices. Part 1: Theory, J. Lightwave Technol. 6 (1988), no. 6, 1051-1057. [12] _____ , Normal mode analysis of waveguide devices. Part II: Device output and crosstalk, J. Lightwave Technol. 6 (1988), no. 6, 1058-1068. [13] --------, Voltage-length product for modal evolution-type digital switches, J. Lightwave Technol. 8 (1990), no. 6, 990-997. [14] W. K. Burns and A. F. Milton, Mode conversion in planar-dielectric sepa rating waveguides, IEEE J. Quantum Electron. 11 (1975), no. 1, 32-39. [15] J. A. Cavailles, M. Renaud, J. F. Vinchant, and M. Erman, First digital optical switch based on InP/GalnAsP double heterostructure waveguides, Electron. Lett. 27 (1991), no. 9, 699-700. [16] M. A. Celia and W. G. Gray, Numerical methods for differential equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1992. [17] J. Y. Chen and I. Najafi, Adiabatic polymer-glass-waveguide all-optical switch, Appl.Opt. 33 (1994), no. 16, 3375-3383. [18] J. Y. Chen and S. I. Najafi, Nonlinear waves and bistabilities in waveguides and all-optical switches, Proceedings of SPIE, vol. 1852, pp. 90-105. [19] S. L. Chuang, Physics of optoelectronic devices, John Wiley and Sons, New York, 1995. [20] L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, John Wiley and Sons, New York, 1995. [21] F. Coppinger, A. S. Bhushan, and B. Jalali, Time magnification of electrical signals using chirped optical pulses, Electron. Lett. 34 (1998), 399-400. [22] T. Durhuus, B. Mikkelsen, C. Joergensen, S. L. Danielsen, and K. E. Stubk- jaer, All-optical wavelength conversion by semiconductor optical amplifiers, J. Lightwave Technol. 14 (1996), 942-954. [23] J. E. Ehrlich, D. T. Neilson, and A. C. Walker, Carrier-dependent nonlin earities and modulation in an InGaAs SQW waveguide, IEEE J. Quantum Electron. 29 (1993), no. 8, 2319-2324. [24] _____ , Guided-wave measurements of real-excitation optical nonlinearities in a tensile strained InGaAs on InP quantum well at 1.5 micron, Opt. Comm. 102 (1993), 473-477. 199 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [25 [26 [27 [28 [29 [30 [31 [32 [33 [34 [35 [36 [37 M. D. Feit and J. A. Fleck, Computation of mode eigenfunctions in graded- index optical fibers by the propagating beam method, Appl.Opt. 19 (1980), no. 13, 2240-2246. S. Fischer, M. Diilk, E. Gamper, W. Vogt, E. Gini, H. Melchior, W.Hunziker, D. Nesset, and A. D. Ellis, Optical 3R regenerator for fOGb/s networks, Electron. Lett. 35 (1999), no. 23, 2047-2049. M. Y. Frankel, J. U. Kang, and R. D. Esman, High-performance hybrid- photonic analog-digital converter based on time-wavelength mapping, Pro ceedings of LEOS ’98, 1998, Paper PD2.2. Carl-Erik Froberg, Numerical mathematics, pp. 178-184, The Ben jamin/ Cummings Publishing Company, Menlo Park, California, 1985. G. Fuchs, J. Horer, A. Hangleiter, V. Harle, F. Scholz, R. W. Glew, and L. Goldstein, Interualence band absorption in strained and unstrained In GaAs multiple quantum well structures, Appl. Phys. Lett. 60 (1992), no. 2, 231-233. H. M. Gibbs, G. Khirova, and N. Peyghambarian (eds.), Nonlinear pho tonics, ch. 2, Springer-Verlag, Berlin, 1990. I. Gontijo, D. T. Neilson, A. C. Walker, G. T. Kennedy, and W. Sibbett, Nonlinear InGaAs-InGaAsP single-quantum-well all-optical switch - theory and experiments, IEEE J. Quantum Electron. 32 (1996), 2112-2121. G. R. Hadley, Multistep method for wide-anqle beam propagation, Opt. Lett. 17 (1992), no. 24, 1743-1745. , Transparent boundary condition for the beam propagation metod, IEEE J. Quantum Electron. 28 (1992), no. 1, 363-370. , Wide-angle beam propagation using Pade approximant operators, Opt. Lett. 17 (1992), no. 20, 1426-1428. H. Haug (ed.), Optical nonlinearities and instabilities in semiconductors, Academic Press, San Diego, 1988. G. S. He, C. Weder, P. Smith, and P. N. Prasad, Optical power limiting and stabilization based on a novel polymer compound, IEEE J. Quantum Electron. 34 (1998), no. 12, 2279-2285. H. J. W. M. Hoekstra, On beam propagation methods for modelling in in tegrated optics, Opt. Quantum Electronics 29 (1997), 157-171. 2 0 0 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [38] H. J. W. M. Hoekstra, G. J. M. Krijnen, and P. V. Lambeck, Efficient interface conditions for the finite difference beam propagation method, J. Lightwave Technol. 1 0 (1992), no. 10, 1352-1355. [39] N. Holonyak, Impurity-induced layer disordering of quantum-well het erostructures: Discovery and prospects, IEEE J. Select. Top. Quantum. Electron. 4 (1998), no. 4, 584-594. [40] D. M. HWang, S. A. Schwarz, R. Bhat, C. Y. Chen, and T. S. Ravi, Cation diffusion in In P j Jno.53Gao.47As superlattices: strain build-up and relax ation, Opt. Quantum Electronics 23 (1991), S829-S846. [41] R. Iga, T. Yamada, and H. Sugiura, Lateral band-gap control of InGaAsP multiple quantum wells by laser-assisted metaorganic molecular beam epi taxy for a multiwavelength laser array, Appl. Phys. Lett. 64 (1994), no. 8 , 983-985. [42] B. Jalali, Personal communication. [43] B. Jalali, A. S. Bhushan, and F. Coppinger, Photonic time-stretch: A poten tial solution for ultrafast A /D conversions, Proceedings of IEEE Microwave Photonic Conference, MWP’98, 1998. [44] Z. Jiang and X. C. Zhang, Single-shot spatiotemporal terahertz field imag ing, Opt. Lett. 23 (1998), no. 14, 1114-1116. [45] I. Joindot and J. L. Beylat, Intervalence band absorption coefficient and measurements in bulk layer, strained and unstrained multiquantumwell 1.55 pm semiconductor lasers, Electron. Lett. 29 (1993), no. 7, 604-606. [46] A. M. Kan’an, P. LiKamWa, M. Dutta, and J. Pamulapati, 1.7-ps consec utive switching in an integrated multiple-quantum-well Y-junction optical switch, IEEE Photon. Technol. Lett. 8 (1996), no. 12, 1641-1643. [47] A. M. Kan’an, P. LiKamWa, Mitra-Dutta, and J. Pamulapati, Area- selective disordering of multiple quantum well structures and its applica tions to all-optical devices, J. Appl. Phys. 80 (1996), no. 6 . [48] Y. Katoh, T. Kunii, Y. Matsui, and T. Kamijoh, Four-wavelength DBR laser array with waveguide couplers fabricated using selective MOVPE- growth, Opt. Quantum Electronics 28 (1996), 533-540. [49] M. N. Khan, J. E. Zucker, T. Y. Chang, N. J. Sauer, M. D. Divino, T. L. Koch, C. A. Burrus, and H. M. Presby, Design and demonstration of weighted-coupling digital Y-Branch optical switches inlnGaAs/InGaAlAs 201 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. electron transfer waveguides, J. Lightwave Technol. 12 (1994), no. 11, 2032- 2039. [50] M. Kitagawa and Y. Yamamoto, Number-phase minimum-uncertainty state with reduced number uncertainty in a Kerr nonlinear interferometer, Phys. Rev. A 34 (1986), no. 5, 3974-3988. [51] D. Krylov and K. Bergman, Amplitude-squeezed solitons from an asymmet ric fiber interferometer, Opt. Lett. 23 (1998), no. 17, 1390-1392. [52] M. Kuzuhara, T. Nosaki, and T. Kameija, Characterization of Ga-out dif fusion from GaAs into SiOxNy films during rapid thermal annealing, J. Appl. Phys. 66 (1989), no. 12, 5833-5836. [53] P. LiKamWa and A. M. Kan’an, Ultrafast all-optical switching in multiple- quantum-well Y-junction waveguides at the band gap resonance, IEEE J. Select. Top. Quantum. Electron. 2 (1996), no. 3, 655-660. [54] J. D. Love, W. M. Henry, W. J. Streward, R. J. Black, S. Lacroix, and F. Gonthier, Tapered single-mode fibres and devices. Part 1: Adiabaticity criteria, IEE Proc. Part J 138 (1991), no. 5, 343-354. [55] T. Miayazawa, H. Iwamura, O. Mikami, and M. Naganuma, Compositional disodering of Ino.5 3Gao.4 7A s/In P mulitquantum well structures by repeti tive rapid thermal annealing, Jpn. J. Appl. Phys. 28 (1989), no. 6. [56] A.Fenner Milton and William K. Burns, Mode coupling in optical waveguide horns, IEEE J. Quantum Electron. 13 (1977), no. 10, 828-835. [57] T. Miyazawa, H. Iwamura, and M. Naganuma, Integrated external-cavity InGaAs/InP lasers using cap-annealing disordering, IEEE Photon. Tech nol. Lett. 3 (1991), no. 5, 421-423. [58] T. Mukaihara, T. Kurobe, N. Yamanaka, N. Iwai, and A. Kasukawa, High performance mode field converter GalnAsP laser integrated with improved butt-coupled passive waveguide, International Conference on Indium Phos phide and Related Materials, 2000. Conference Proceedings, 2000, pp. 29- 32. [59] H. Murata, M. Izutsu, and T. Sueta, Optical bistability and all-optical switching in novel waveguide junctions with localized optical nonlinearity, J. Lightwave Technol. 16 (1998), no. 5, 833-840. [60] W. H. Nelson, A. N. M. Masum Choudhury, M. Abdalla, R. Bryant, E. Me- land, and W. Niland, Wavelength- and polarization-independent large angle 202 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. InP/InGaAsP digital optical switches with extinction ratios exceeding 2 0 dB, IEEE Photon. Technol. Lett. 6 (1994), no. 11, 1332-1334. [61] N. Nishizawa, M. Hashiura, T. Horio, and M. Mori, Generation and detec tion of squeezed ligth with a phase tunable fiber loop mirror using polariza tion beam splitter, J. Appl. Phys. 37 (1998), no. part 2, No. 7A, 792-794. [62] R. Normandin, D. C. Houghton, M. Simard-Normandin, and Y . Zhang, All-optical silicon-based integrated fiber-optic modulator, optical gate, and self-limiter, Can. J. Phys. 66 (1988), 833-839. [63] T. Ohtsuki, Performance analysis of direct-detection optical asynchronous CDMA systems with double optical hard-limiters, J. Lightwave Technol. 15 (1997), no. 3, 452-457. [64] I. J. Pape, P. LiKamWa, J. P. R. David, P. A. Claxton, and P. N. Robson, Disordering of Gao.4 7Ino.5zA s/InP multiple quantum well layers by sulfur diffusion, Electron. Lett. 24 (1988), no. 19, 1217-1218. [65] W. H. Press, S. A. Teukolsky, and W. T. Vetter ling, Numerical recipes in C, ch. 19, Cambridge University Press, Cambridge, 1996. [66] _____ , Numerical recipes in C, Cambridge University Press, Cambridge, 1996. [67] P. R. Prucnal, M. A. Santoro, and T. R. Fan, Spread spectrum fiber-optic local area network using optical processing, J. Lightwave Technol. LT-4 (1986), no. 5, 547-554. [68] J. Reintjies and R. C. Eckardt, Two-photon absorption in ADP and KD*P at 266.1 nm, IEEE J. Quantum Electron. 13 (1977), 791-795. [69] J. P. Reithmaier and A. Forschel, Focused ion-beam implantation induced thermal quantum-well intermixing for monolithic optoelectronic device inte gration, IEEE J. Select. Top. Quantum. Electron. 4 (1998), no. 4, 595-605. [70] S. Ryu, I. Kim, W. G. Jeong, B. Choe, and S. Park, Theoretical investiga tion of gain and linewidth enhancement factor for 1.55-p,m tensile strained quantum-well lasers, J. Lightwave Technol. 15 (1997), no. 4, 711-716. [71] M. Saini and E. K. Sharma, Analysis of nonlinear MQW waveguides: A simple numerical approach, IEEE Photon. Technol. Lett. 8 (1996), no. 3, 384-386. 203 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [72] T. Sasaki, M. Kitamura, and I. Mito, Selective metalorganic vapor phase epitaxial growth of InGaAsP/InP layers with bandgap energy control in In- GaAs/InGaAsP multiple-quantum well structures, J. Crystal Growth 132 (1993), 435-443. [73] T. Sasaki, M. Yamaguchi, and M. Kitamura, Monolithically integrated multi-wavelength MQW-DBR laser diodes fabricated by selective metalor ganic vapor phase epitaxy, J. Crystal Growth 145 (1994), 846-851. [74] R. Scarmozzino and R. M. Osgood, Comparison of finite-difference and Fourier-transform solutions of the parabolic wave equation with emphasis on integrated-optics applications, J. Opt. Soc. Am., A 8 (1991), no. 5, 724- 731. [75] Karlheinz Seeger, Semiconductor physics, pp. 348-360, Springer Verlag, Berlin, 1991. [76] J. Shim, M. Yamaguchi, P. Delansay, and M. Kitamura, Refractive in dex and loss changes produced by current injection in InGaAs(P)-InGaAsP multiple quantum well (MQW) waveguides, IEEE J. Select. Top. Quantum. Electron. 1 (1995), no. 2, 408-415. [77] S. K. Si, D. H. Yeo, K. H. Yoon, and S. J. Kim, Area selectivity of InGaAsP-InP multiquantum-wel I intermixing by impurity-free vacancy dif fusion, IEEE J. Select. Top. Quantum. Electron. 4 (1998), no. 4, 619-623. [78] Y. Silberberg, P. Perlmutter, and J. E. Baran, Digital optical switch, Appl. Phys. Lett. 51 (1987), no. 16, 1230-1232. [79] J. Singh, Physics of semiconductors and their heterostructures, pp. 564- 566, McGraw-Hill, New York, 1993. [80] P. M. V. Skovgaard, R. J. Mullane, D. N. Nikogosyan, and J. G. Mclnerney, Two-photon photoconductivity in semiconductor waveguide autocorrelators, Opt. Comm. 153 (1998), 78-82. [81] G. D. Smith, Numerical solution of partial differential equations: Finite difference methods, Clarendon Press, Oxford, 1985. [82] A. Sneh, J. E. Zucker, and B. I. Miller, Compact, low-crosstalk, and low- propagation-loss quantum well y-branch switches, IEEE Photon. Technol. Lett. 8 (1997), no. 12, 1644-1646. [83] A. Sneh, J. E. Zucker, B. I. Miller, and L. W. Stulz, Polarization insensi tive InP-based MQW digital optical switch, IEEE Photon. Technol. Lett. 9 (1997), no. 12, 1589-1591. 204 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [84] A. W. Snyder, Coupling of modes on a tapered dielectric cylinder, IEEE Trans. Microwave Theory Tech. 18 (1970), no. 7, 383-392. [85] _____ , Mode propagation in a nonuniform cylindrical medium, IEEE Trans. Microwave Theory Tech. (1971), 402-403. [86] H. N. Spector, Free-carrier absorption in quasi-two-dimensional semicon ducting structures, Phys. Rev. B. 28 (1983), 971-976. [87] S. Sudo, H. Onishi, Y. Nakano, Y. Shimogaki, K. Tada, M. J. Mondry, and L. A. Coldren, Impurity-free disordering of InGaAs/InGaAlAs quantum wells on InP by dielectric thin cap films and characterization of its in plane spatial resolution, Jpn. J. Appl. Phys., Part 1 35 (1996), no. 2B, 1276-1279. [88] L. F. Tiemeijer, P. J. A. Thijs, J. J. M. Binsma, and T.van Dongen, Effect of free carriers on the linewidth enhancements factor of InGaAs/InP (strained- layer) multiple quantum well lasers, Appl. Phys. Lett. 60 (1992), no. 20, 2466-2468. [89] A. Tomita, Free carrier effect on the refractive index change in quantum well structures, IEEE J. Quantum Electron. 30 (1994), no. 12, 2798-2801. [90] H. K. Tsang, R. V. Penty, I. H. White, R. S. Grant, W. Sibbett, J. B. D. Soole, H. P. Le Blanc, N. C. Andreakis, R. Bhat, and M. A. Khoza, Two-photon absorption and self-phase modulation in InGaAsP/InP multi- quantum-well waveguides, J. Appl. Phys. 70 (1991), 3992-3994. [91] G. J. van Gurp, W. M. van de Wijgert, G. M. Fontijn, and P. J. A. Thijs, Zn diffusion-enchanced disordering and ordering of InGaAsP/InP quantum well structures, J. Appl. Phys. 67 (1990), no. 5, 2919-2926. [92] J. M. Wiesenfeld, G. Eisenstein, R. S. Tucker, G. Raybon, and P. B. Hansen, Distortionless picosecond pulse amplification and gain compres sion in a travelling-wave InGaAsP optical amplifier, Appl. Phys. Lett. 53 (1998), 1239-1241. [93] D. Wolfson, S. L. Danielsen, H. N. Poulsen, P. B. Hansen, and K. E. Stubk- jaer, Experimental and theoretical investigation of the regenerative capabil ities of electrooptic and all-optical interferometric wavelength converters, IEEE Photon. Technol. Lett. 10 (1998), 1413-1415. [94] D. Wolfson, T. Fjelde, A. Kloch, C. Janz, A. Coquelin, I. Guillemot, F. Ga- borit, F. Poingt, and M. Renaud, Experimental investigation at lOGb/s of 205 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the noise suppression capabilities in a pass-through configuration in SOA- based interferometric structures, IEEE Photon. Technol. Lett. 12 (2000), 837-839. [95] X. Yang, Implementation of time lenses and optical temporal processors, Opt. Comm. 116 (1995), 193-207. 206 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 5 Optical Limiting in an Optical Fiber Sagnac Interferometer with an Intraloop Saturable Absorber An experimental demonstration of an energy limiter for short pulses is reported in this chapter. The limiter is based on an optical fiber Sagnac loop with an intraloop dichroic saturable absorber based on an InGaAsP-based waveguide with a compressively strained multiple quantum well absorptive region. The saturable absorber is placed assymetrically in the loop and is used in a Kerr- like nonlinear phase shifter. The device is operated in a novel dual polarization scheme, which offers some tantalizing advantages. Excellent limiting and noise reduction characteristics are demonstrated with the device. 5.1 Introduction The device that is discussed in this chapter builds on the theoretical work of the previous chapter. The device is a special case of the interferometers with an internal Kerr-like element discussed there. The nonlinear interferometer is 207 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. built out of discrete components consisting of optical fiber components and a saturable amplifier. This simple setup is just to prove the concept of the limiter based on the nonlinear phase shift in an interferometer, and it is totally possible, and in fact desirable, to monolithically integrate this system using one of the methods discussed in the last chapter. Because the interferometer is built out of fiber, a Sagnac configuration was utilized to ensure stability against bias drift caused by environmental variables like temperature, pressure, sound etc. This is an experimental necessity, but it does not diminish the generality of the results achieved, here which can be extended to integrated optic Mach-Zehnder or Michelson interferometers. The type of device discussed here has been previously studied in connection with optically controlled switching for optical communications. These devices go by names of TOAD’ s (Terahertz Optical Asymmetric Demultiplexer) [21], SLALOM’ s (Semiconductor Laser Amplifier in a Loop Mirror) [14] or TWSLA- NOLM’s (Traveling Wave Semiconductor Laser Amplifier placed within a Non linear Optical Loop Mirror) [8], and basically consist of a semiconductor optical amplifier assymetrically placed in a fiber optic Sagnac loop. The control light may enter either of the input ports of the Sagnac loop, or it can be injected us ing an extra intraloop coupler [21, 8]. The work described in this chapter builds 208 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. on this. A novel application (optical limiting) is demonstrated by using a dif ferent approach (dual polarization scheme and a dichroic saturable absorber) of switching the Sagnac interferometer. 5.2 Concept It has been pointed out in the previous chapter that good power limiting char acteristics can be achieved in saturable devices by utilizing the Kerr-like effects in these guided wave devices. One can use the Kerr-like element in the interfer ometer, to sharpen the transmission of a saturable absorber/amplifier in order to obtain a power-dependent transmission that is inversely proportional to the input power. One needs the largest possible index change with the smallest possible change in absorption in the saturable Kerr-like device in order to properly engineer the power-dependent transmission of the device. This may be achieved by using a dichroic device and by splitting the input light into a so-called pump polarization to which the device is opaque and a transparent probe polarization, which serves as the output of the device. The pump is absorbed and changes the refractive index of the dichroic device, which phase shifts the transparent polarization’s field (probe). Increasing pump power results in an unbalanced interferometer for the probe, and a transmission inversely proportional to the input pump 209 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. intensity and, thereforem the device input power. The dichroic characteristics can be achieved in an unstrained/compressively strained quantum well device (See Chapter 4). If the light is the form of pulses and a Sagnac interferometer is utilized like in our case, one can achieve the largest differential nonlinear phase shift by assymetrically placing the dichroic device inside the Sagnac loop and by using an extra intraloop polarizer for the pump polarization. The input probe is slightly delayed with respect to the pump, which also helps eliminate/reduce chirp on the probe, which is very important in, for example, long haul optical communications systems. 5.3 Im plementation of the Concept The implementation of the aforementioned concept is based on a Sagnac in terferometer because of the requirement for zero wavelength sensitivity. It is important that the limiter works for the spectral equalization of a string of pulses of different wavelengths, spanning tens of nanometers in wavelength. Wavelength insensitivity can only be achieved in a zero-pathlength difference interferometer. The reason for this is that the bias point of an unbalanced inter ferometer is different for every wavelength. Even small pathlength differences can translate into big shifts of bias point as a function of wavelength when the 210 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. relevant wavelength band of the input light spans many tens of nanometers. The change in the bias point (in radians) as a function of the wavelength spread is i = 21 a (; 2 £ ) aa . ( a n A (nL) is the difference in optical path length of the arms, A is the free space wavelength, and A A is the wavelength spread. Eq. 5.1 predicts that a wave length sweep of 50nm can result in a bias shift of 0.2 radians if the optical path length difference is equal to the free space wavelength, given by 1.55 gm. In theory, an integrated optical Mach-Zehnder interferometer can have zero path- length difference, but in practice this is hard to realize because of inevitable manufacturing deviations. Therefore,a fiber Sagnac interferometer is utilized in this experiment. The implementation of the limiter is shown in Fig. 5.1. It consists of an optical fiber Sagnac loop with an intraloop absorbing semiconductor waveguide and an adjustable coupler. The waveguide is placed assymetrically inside the loop at a distance AL from the center of the loop. An input pulse is split into two orthogonally polarized pulses which are slightly delayed, with respect to each other, before injection into the Sagnac loop system. The polarizations are referenced with respect to the epitaxial growth planes of the semiconduc tor waveguides. The nonlinear waveguide strongly absorbs TE-polarized light 211 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (pump), but it is mostly transparent to TM-polarized light (probe). Conse quently only TE light creates carriers and generates a nonlinear index change through the bandfilling and plasma effects. Both the TE and TM polarizations can see some of this index change. The transmitting port is used as the output terminal in order to obtain good isolation between the input and output light signals. In this case one can only achieve limiting over the nonlinear phase shift range of 7 r to 2n as shown in Fig. 5.2. Nonlinear phase shift in waveguide due to absorbed TE -Vx Center of Sagnac Loop PM fiber x=Carrier Lifetime TIME TE absorbed in device ■■■■ Nonlinear INPUT LIGHT: < — Semiconductor TM probe") Waveguide (1-C) . time Reflecting port Coupler 7 Polarizer / c 'J L block TE INPUT / ppump") Path of TE light Transmitting port OUTPUT Figure 5.1: A polarization-sensitive saturable device placed assymetrically in a Sagnac loop can be used to create a nonreciprocal phase shift for a probe. TE light is absorbed by the waveguide and generates a phase shift by carrier nonlinearities while the TM light is transmitted and only reads the nonlinear phase shift. 2 1 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Input Power Power limiting region Slope ,1/Power Input Power Nonreciprocal Phase shift Figure 5.2: A nonlinear intensity dependent phase shift element in a Sagnac in terferometer can be used to produce a limiter at the transmitting port. The up per right hand graph shows a nonlinear nonreciprocal phase shift generated in a Sagnac interferometer. When this characteristic is combined with the transmis sion characteristic of the transmitting characteristic of the unbalanced Sagnac interferometer (lower left hand graph), one obtains the limiting characteristic shown on the right hand graph. Asymmetric polarization-dependent loss is the primary mechanism which is utilized together with the intraloop polarization-sensitive saturable device to create a nonreciprocal combination inside a Sagnac interferometer. The setup in Fig. 5.1 shows that we are using an intraloop polarizer to keep counterclockwise propagating light from reaching the nonlinear waveguide. This blocks the light for the polarization (TE) that has the biggest absorption and which generates a big index change by carrier nonlinearities. The other polarization (TM) affects the device to a lesser degree and passes through the device without significant absorption, but it does see the index change caused by the other polarization 213 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (TE). Therefore, only the clockwise propagating TE pulse generates an index change via the carrier nonlinearities. The clockwise TM pulse closely follows the TE pulse and sees this index change, giving a big phase shift to the pulse. It is assumed AL is chosen so that the medium in the waveguide has relaxed by the time the counterclockwise TM-polarized pulse reaches the device, and therefore only sees a small index change. Consequently, these two counterpropagating TM pulses see a large difference in phase shifts when they propagate through the nonlinear waveguide. The limiter architecture shows some similarity to the all-optical loop mirror switch employing an asymmetric semiconductor optical amplifier/attenuator combination [20]. In the latter case, a nonreciprocal phase shift (caused by carrier nonlinearities) is induced in a saturable device due to different amounts of power incident on the device facets from opposite directions. In our case the difference in incident powers is brought about by the polarizer (of course the Kerr-like nonlinear waveguide is dichroic also). As pointed out before, the dichroic properties needed in the intraloop Kerr- like nonlinear waveguide can be achieved in a semiconductor waveguide with a compressively strained quantum well. Device parameters like the transmis sion and nonlinear index change of the nonlinear phase shifting device fortu nately also appear to be nearly wavelength-independent over the wavelength range where limiting is desired. The TM absorption between the HH1-C1 (first heavy-hole valence subband to first conduction subband interband transition) 214 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and LH1-C1 (first light-hole valence subband to first conduction subband in terband transition) bandedges of a compressively strained quantum well has a low and slowly varying absorption (according to calculations shown in Chapter 4). Photons in this energy band can also experience a substantial nonlinear refractive index change via the plasma effect due to any carrier density changes in the quantum well (See chapter 4). Other researchers have measured these index changes and they are nearly constant, between the HH1-C1 and LH1-C1 bandedges, for the TM polarization [7]. 5.4 Analysis In this section, a simple phenomological model to describe the operation of the limiter is developed. In general, both the TE and TM pulses may experience a whole range of ultrafast (as well as slower) effects as they propagate through the device. The aim is to include the major effects in a descriptive way, which will be adequate for describing the experimentally measured results. It is assumed that the input pulses are sufficiently short, so that the bandfilfing effect has an integrating nature, i.e., the pulsewidths are much shorter than the carrier lifetime. When the TE pulse arrives at the device, it is absorbed via interband tran sitions and generates an index change via the bandfilfing and plasma effects. 215 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This translates into nonlinear phase shift for the TM-polarized probe pulses. It is assumed that the probe pulses trail the pump pulses and do not overlap it in any way. The phase change seen by the probe pulses decays due to recombina tion. From the standpoint of the succeeding TM pulses, it can be approximated as < t > — ®max exp (5.2) This assumes the pump pulse arrived at t 0 at the device and generates a maxi mum phase shift $ max, which decays at a rate determined by the carrier lifetime effect utilized to switch the Sagnac interferometer. $ max is directly proportional to the energy in the TE pulse. Consider the propagation of the two counterpropagating TM inputs from the reflective port of the Sagnac interferometer to the transmitting port. As is shown in Fig. 5.1, the TM probe pulse is assumed to lag the pump pulse by a time At. Therefore,the clockwise propagating TM pulse arrives at the nonlinear waveguide at time t\ = to+At. Since the waveguide is not placed in the center of the Sagnac loop, the counterclockwise pulse arrives at the nonlinear waveguide at time r of the medium. u(t) is the unity step function. This phase shift is the main 2 rax AL (5.3) ^ 2 — t\ T c 216 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where c is the speed of light in vacuum, neff the effective index of the fiber mode, and AL the distance from the nonlinear waveguide to the center of the loop. In general, the probe pulses will in general be able to generate time- varying gain, and a time varying phase shift across the profile of the pulse, as it traverses the nonlinear waveguide, due to effects like spectral hole burning, carrier heating or cooling, two-photon absorption, and bandfilling effects [19, 16, 6, 18, 11, 9, 26, 15, 1, 13]. Assume for this part of the analysis that these effects are small enough to be neglected, since the interband matrix element for the TM polarization is small below the LH1-C1 bandgap energy. Therefore, the main nonlinear effect is the phase shift caused by the TE pump. Assume that this phase change is decaying slowly enough so that a TM probe pulse essentially sees a constant phase shift over the pulse profile. C is the portion of the power that gets coupled across the coupler, and (1 — C) is the fraction coupled through. The clockwise propagating TM field’ s amplitude just before the detector at the transmitting port of the interferometer can be written as E c w ^ e n d i t) = y/K(l - C ) e ^ D [Einput(0, t)} e* , (5.4) where K is an intensity transmission factor and etd a constant phase factor. The propagation coordinate varies from z = 0 at the input to z = zend at the 217 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. output. Einput is the electric field of the input TM pulse, and D[Einput] is the dispersed version of the input pulse, i.e., D is the dispersion operator for the whole Sagnac loop for the TM polarization defined by [23] / OO Einputio, t')h(t - t')d£ , (5.5) -OO where h(t) is the impulse-response function for the optical fiber Sagnac loop from the reflecting port to the transmitting port. The precise form of h(t) is not important for the sake of the discussion. Assume that the nonlinearities in the fiber itself are too small to have any noticeable effect, and that the fiber is lossless. Therefore,the energies in the input and dispersed version of the input pulses are equal. The counterclockwise pulse gets a 180-degree phase shift with respect to the clockwise pulse since it crosses the coupler twice and can be expressed as Eccw(t) = - V K C ^ D [Einput(t)] eiS . (5.6) By combining the Eqs. 5.4 and 5.6, the output intensity at the transmitting port can then be written as = | E ^ t ) + Eccw (t) I 2 = K ((1 - C )2 + C 2 - 2(1 — C )C cos (A < 3> )) i g , (5.7) 218 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where A# = < 3?(ti) - $(£2) = ®max ( 6Xp '- A t — exp T -At — 2 n AL/c T (5.8) is the energy-dependent differential nonlinear phase shift, and IT = I D[Einput}(zend,t)\2. From the experiment, it is known that one- and two-photon absorption for exists for the TM probe. One has to account for the reduction in the TM transmission due to these effects. The absorption of the TM light is also accom panied by an instantaneous index change, and by slower effects caused by the bandfilling effect. Assuming that the pulses arriving at the device are squared shaped and neglecting the phase effects for the time being, it is possible to show that the amplitude of the clockwise pulse (Eq. 5.4) has to be multiplied by the following expression [22, 24]: exp ( -a L dev) l + {(3I^v/a ) ( l- e x p ( - a L dev)) ’ (5.9) where Laev is the length of the semiconductor waveguide, (3 is the two-photon absorption coefficient, and a is the one-photon (interband) absorption coeffi cient. I ^ v is the intensity of the TM pulse at the input facet of the nonlinear 219 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. waveguide for the clockwise propagating direction. This quantity is related to the input intensity 7jn = \Ei n p u t \2 by a constant factor Ca which describes the optical losses and reduction in pulse intensity due to dispersion between the input port of the Sagnac loop and the device facet l t V = CJin- (5.10) An electric field transmission factor similar to Eq. 5.9 can be used for the counterclockwise propagating TM pulse: rp _ I e X p ( a L d e y ) _________________ / r Iccw U 1 + 1 - exp(-aL d e,)) ‘ j I^w is the intensity of the TM pulse at input facet of the nonlinear waveguide for the counterclockwise propagating direction. Note that I t z = W in, (5.12) where C & is a constant. The phase changes accompanying the carrier index changes brought about the probe pulses are in general nonuniform across the pulses due to for exam ple the integrating nature of the bandfilling effect for ~ 10 picosecond pulses. This extra nonlinear phase shift is taken into account by adding a timevarying 220 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. self-phase modulation phase shift 4>spm{t) of unknown functional form to the existing differential phase shift. In other words, the total differential phase shift, which replaces A < 3 > in Eq. 5.7, is defined as 4 > n l{t) — -f 4>sPM(t)- (5.13) Assume (for simplicity) that the pulses are still square shaped at the out put. Then the average output can still be described by equations of the same functional form, given that the nonlinear phase shift, 4>nli is replaced with the quantity < fiNL = A < 3 ? + (j)SpM where < P s p m is defined by — 1 f to u t+ P W / 2 C O S (A$ + (t> S P M) = ~p\xr / cos (A$ + (f>SPM {t)) (5.14) r w Jtout-PW/2 and where P W is the pulse width of the output pulse centered at time t = tout. Therefore effect of the time varying phase is averaged over the width of the output pulse. 2 2 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Therefore, by combining Eqs. 5.7, 5.9, 5.11 and 5.14, one can write the following equation for the output intensity (averaged over the output pulse width time interval) as T , (1 - C f C 2 lout — K x (-- + ' 1 + IinCa/3 1 + linCbfl 2C^ cos ($NL))lP , (5.15) \ / ( l + hnCb(3){l + IinCa 0 ) The time averaged transmission of the system as a function of the input intensity is then simply T = I ^ / l P . This is, in fact, also the energy transmission of the Sagnac loop because the output energy is proportional to the average measured power or intensity. 5.5 Experimental M ethod The experimental setup is shown in Fig. 5.3. Pulses are generated by a Clark- MXR erbium-doped fiber laser which generates pulses with a temporal width of 300fs. The pulses have a spectral width of 60nm centered at 1560nm. The pulse repetition rate is 40MHz. The input pulse energy can be controlled by using a free space liquid crystal polarization rotator in conjunction with a polarizer. The input pulse energy is measured by splitting light off and filling the aperture of a 5mm diameter germanium photodiode and detecting the photocurrent. 2 2 2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Lensed tip fibers -5.5 d B ^ SOA § / connectoi PZ fiber, L=3.5 meter -1.75 dB PM fiber TM ("probe") ^connector #2 connectoi time PM tunable coupler Detector connector IE ("pump") Figure 5.3: The layout of the experiment. The pulses are injected into the polarization-maintaining (PM) fiber Sagnac interferometer, at a specific angle with respect to eigenaxes of the PM fiber by using a A/2-waveplate in conjunction with a polarizer. This causes the pulse to break up due to the different group velocities of the polarizations along the eigenaxes. TE polarization is launched along the fast axis of the fiber and the TM polarization along the slow axis. The polarization modal dispersion is around 2ps/m and was measured by utilizing second harmonic autocorrelation (See Fig. 5.4). There is estimated to be a 15ps delay between the pulses of different polarization reaching the SOA from side A (See Fig. 5.3) and that pulse width (full width at half maximum) is 6ps (assuming Gaussian type pulses). 223 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. r0.8 0.4 o0.2 /iv v * * “ Delay (picosecond) Figure 5.4: The autocorrelation trace obtained at connector #1 (See Fig. 5.3) when equal power is launched along both eigenaxes of the PM fiber. The auto correlation is based on second harmonic generation and is highly polarization- sensitive. In order to detect two polarizations, a waveplate was used to rotate the input light going into the correlator (which contains a TE and TM pulse) by 45 degrees, with respect to the sensitive axis. This figure shows that greater than 4ps of delay is obtained in the 2m of PM fiber. A commercial compressively strained multiple quantum well semiconductor optical amplifier (SOA) with an active length of ~ 1 mm is used as the non linear element inside the Sagnac loop. The device is angle facetted and has integrated spot-size converters for efficient fiber coupling. This device is not pumped under normal operation of the limiter and, therefore, is operated as a saturable absorber. This was found to be the most stable mode operation due to polarization extinction considerations. It is hard to preserve a pure state of polarization due to the many connections between the components. By using 224 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TM TE Sensitivity=-81dBm Analyzer Resolutions Onm Current=20mA Temperature=20 degree Centigrade 1.30 1.35 1.40 1.45 1.50 1.55 1.60 1.65 1.70 Wavelength (micron) Figure 5.5: The electroluminescence in both the TE and TM polarization mea sured from the edge of the commercial semiconductor optical amplifier (SOA) device, at a current of 20mA. This is well below the material gain transparency current of 45mA. the device as an absorber it cleans up the polarization, by acting as a polar izer for the TE polarization. When the device was used as an amplifier, TE light could not be fully extinguished by the fiber polarizer, and fluctuation in the output light was seen due to the effects of temperature and vibration on polarization modal dispersion, which can result in dynamic polarization varia tions. Better limiting characteristics were also displayed when the device was operated as a saturable absorber. By measuring the polarization-dependent electroluminesence from the SOA, the first heavy-hole and light-hole subband edges can be estimated to be situated at, respectively, 1566nm and 1485nm (See Fig. 5.5). When a current of 200mA is injected into the SOA, it yields a gain of 30dB for the TE polarization with negligible gain ripple, which means the 225 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. reflectances of the facets are insignificant. The material gain transparency point was measured to be ~45mA. This was done by measuring the amplitude and phase of the low frequency modulation of the voltage across the p-n junction of the SOA, caused by the low frequency chopping of the mode-locked laser input to the device. This is because bandfilling due to the injected light leads to a change in the position of the Fermi levels. At gain transparency this modulation in the voltage is zero, and the phase of the modulation inverts. Such a mea surement is shown in Fig. 5.6. From experimental measurements on the Sagnac interferometer, with the intraloop saturable absorber, it also possible to deter mine that the smallest nonlinear phase shifts occurs when the current applied to the device is 45mA. This agrees with the gain transparency measurements. Light is directly coupled from the fiber to the nonlinear waveguide device by using fiber pigtails with conical tips. Both fibers are aligned for the maximum coupling efficiency. At port A of the device (See Fig. 5.3), an efficiency of -1.75dB for fiber pigtail to device coupling was achieved and at the other port (B) an efficiency of -5.5dB. Therefore, there is an inherent coupling asymmetry in the ring. Connector loss and transmission loss in each part of the system were also carefully measured. The directional coupler, which forms part of the loop, is mechanically tunable over the range 0 to 100%. The eigenaxes of the PM fiber in the setup are aligned with the growth plane of the device. Both Fujikura PANDA and elliptical cladding (3M) fiber are used in the setup. A PZ 226 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 ,3 > 10' o > ,2 10' ,1 Q 10 ,0 10 0 20 40 60 80 100 Bias current(mA) Figure 5.6: The magnitude of the modulation in the voltage across the p-n junction of the SOA as a function of the current into the junction, caused by a modulation in the input light. The minimum shifts as a function of the average input power of light, which is unexplained at this time. The legends show that the polarization is also varied and also results in a shift in the transparency current. (TE:TM is the TE to TM polarization ratio). (Polarizing fiber) of length 3.5m, which blocks TE polarization, is also placed inside the loop. The inclusion of the PZ fiber also makes the position of the semiconductor waveguide asymmetric in the loop. The TE pump pulse generates carriers via interband absorption in the de vice. The index change generated by these carriers is read a few picoseconds later by the TM pulse entering from side A, i.e., the clockwise traveling pulse. The TM pulse which travels counterclockwise, reaches the device an estimated 1.75ns after the clockwise pulses. By this time most of the carriers have recom bined and, therefore, the counterclockwise pulse experiences a smaller phase 227 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. shift than the clockwise propagating pulse. The lifetime for InGaAsP alloys with a bandgap of 0.8eV at room temperature are of the order of Ins or less [2]. The device was reversed biased up to 2V to generate of field of approximately 100kV/cm over the active region, but resulted in no visible changes in transmis sion characteristics. Reverse biasing sweeps out carriers, reducing the lifetime [5]. The fact that no changes could be detected probably means that the life time was sufficiently short, so as not to influence the operation of the device. In other words, the device could recover fast enough to ensure the maximum phase shift. The output energy is measured as the average photocurrent on a slow 5mm diameter germanium photodetector. 5.6 Experimental Limiting Results By carefully trimming the coupling ratio C of the directional coupler, adjusting the TE:TM polarization ratio launched into the fiber and biasing the saturable absorber with a current of 5mA (the transparency current is 45mA), we were able to measure the limiting characteristics shown in Fig. 5.7. The shape of the limiting characteristics is nearly ideal since the change in slope is abrupt and then nearly zero over an octave. 228 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.25 Q . 0.15 P > Q. 60 80 100 120 140 160 Input energy before coupler (pj) Figure 5.7: Limiting characteristics measured in experiment The limiter appears to be fairly wavelength-insensitive and broad band since the input spectrum and spectrum of optically limited light have nearly the same spectral bandwidth (See Fig. 5.8). Only some of the input light on the short wavelength side of the spectrum seems to have been absorbed in the saturable absorber. By using the theory from Paragraph 5.4, the effects of the nonlinear phase shift and an absorption increase due to TPA can be shown to contribute to the excellent limiting characteristics shown in Fig. 5.7. It is possible to figure out what the role of each of the nonlinear mechanisms was, by extracting the exact nonlinear phase shift from the experiment. This can also be done by 229 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. — Before limiter ....... Aft8r B fm Ster 1JO as Wavelength (micron) Figure 5.8: The spectrum of the mode-locked laser before and after the limiter (measured minutes apart). Take into account that the time averaged spectrum is also slowly fluctuating in time. measuring the intraloop powers counterpropagating in the Sagnac interferome ter after the pulses passed through the nonlinear waveguide and by measuring the interferometer transmission. Therefore, after the measurement of the in terferometer transmission, the transmission from the input through to point A (See Fig. 5.3) was measured (i.e., the intraloop counterclockwise transmission), and also the transmission from the input to point B (i.e the intraloop clockwise device transmission). The shape of the transmission characteristics measured at the various points are shown in Fig. 5.9. The two curves measured for the waveguide transmission are dissimilar in shape due to the fact that the contra- directional powers coupled into the SOA differ a lot in magnitude due to the value of the coupling ratio and the unequal coupling efficiencies, leading to a 230 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. large difference in the resultant two-photon absorption for the two directions of propagation. The clockwise propagating TM pulse’ s transmission can be fitted with a function of the form T - A ” l + BEln where Ein is the average input energy before the interferometer. A and B are fitting constants. This equation is the squared form of Eq. 5.9. Energy was substituted for the average intensity since this two quantities are just propor tional: B in — P ^ V in X I in X A mode • (5-17) PWin is the pulsewidth of the input pulse (assuming a square pulse), and Amode is the effective modal area of the propagating mode. As can be seen from Fig. 5.9, the counterclockwise propagating TM pulse experiences nearly constant transmission up the highest average input power due to the small value of the coupling ratio and the coupling efficiency to the nonlinear waveguide. (5.16) 2 3 1 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The nonlinear phase shift can now be extracted by fitting the following interferometer transmission characteristic (similar to Eq. 5.15) to the data T. ^ (1 - C,)2 , r* 2 C ( l- C 7 ) c o B f e \ mt \ l + BEin y/T+ B E in ) (5.18) where C is coupling ratio of the tunable coupler, ( f > NL, is the nonlinear nonre ciprocal phase experienced by the TM pulses, if is a proportionality constant, and Ein represents the input energy in a pulse. This equation is actually the same as Eq. 5.15, except that intensity has been substituted by energy by using Eq. 5.17. It was found that the nonreciprocal nonlinear phase shift, could, be fitted by the following type of dependence on the injected intensity: (5.19) 1 + \ P \l Es is a saturation parameter (with units of pj) and nc a nonlinear phase shift proportionality constant (with units of rad/pj). The extracted phase shift is shown in Fig. 5.10 and approaches nearly 2ir. It was found that the waveguide absorbed all the impinging TE absorption even up to the highest injection energies. The injected TM light also experienced one-photon absorption in the waveguide, nearly uniformly across the spectrum 232 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2.5 % 0 < fi O O O O O /> O Q O O O Q O O O O O tf i 00 INTERFEROM ETER FIT STRAIGHT THROUGHfA T O B) STRAIGHT THROU GH (B TO A) 0.5, 50 100 150 Input energy before coupler (pj) 200 Figure 5.9: The interferometer intraloop transmission as a function of energy (crosses), as well as the transmission of the pulses through the nonlinear waveg uide for the clockwise direction (circles), and counterclockwise direction (dia monds). of the pulses, to yield a transmission loss for the TM-polarized field of -6.6dB. The measured photocurrent for a given injected TM power, was consistent with the measured TM polarization propagation losses. Not only is this detrimental for the transmission of the limiter, but also because TM light, therefore, may phase shift itself. The magnitude of the self-phase shift is in general nonuniform over the pulse, and this may for example lead to pulse distortion and chirp. 233 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 * ; Q.' 2 1 0 50 100 150 200 Input energy before coupler (pJ) Figure 5.10: By fitting the interferometer transmission shown in fig5.9, one obtains a nonreciprocal phase shift which seems to saturate as a function of energy 5.7 Two-Photon Absorption In this paragraph, a direct measurement of the transmission of light through the SOA seems to indicate that TPA is a reasonable explanation for the absorption increase for TM polarization as a function of increasing TM power. The TM light experiences two-photon absorption at high intensities, manifesting itself as increasing absorption of a passive waveguide at higher intensities. This hypoth esis was partially proven correct by the fact that a model which includes both the linear absorption and the two-photon absorption could successfully match the absorption data. 234 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A simple model similar to the one used by Agrawal and Olsson [4 ] is used. Two-photon effects are added in perturbational way [12]. The pulse propaga tion can be modeled in a time frame moving with the pulse by making the transformation t = t — z/vg (5.20) where t is the time in the stationary observer time frame, vg is the group velocity of the mode in the waveguide, and z is the length coordinate of the waveguide. The amplitude A and phase of the pulse, 0, can be separated using the trans formation, A = VPexp (i(p) , (5.21) where P is the power of the propagating pulse. This enables one to write the following two equations which govern the amplitude of the wave (See Appendix D) for the derivation of the two-photon absorbing terms): dP BP2 oT = ( j - <*nt)P - y — (5.22) U Z - tH n o d e dg _ (go - g) gP a T(3P2 S t tc Ent + 2 A l^ tu u ( ' > Dispersion of material parameters and the effects of group velocity dispersion are neglected. /? is the two-photon absorption coefficient, Am o £ z e is the effective modal area, g0 is the small signal gain or loss coefficient of the waveguide, rc 235 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. is the carrier lifetime, g is the saturated gain (or loss), aint is scattering loss, r is the overlap factor of the optical mode over the gain region, and a is the differential gain of the semiconductor waveguide. The bandfilling due to two- photon absorption is taken into account by the second term in Eq. 5.23 and plays a negligible role, given reahstic device parameters. The equations are solved using a finite difference formula based on Eq. 5.22 in conjunction with the Runge Kutta integration of Eq. 5.23 at every discrete z-step. By using the device parameters stated in Table 5.1, one can do a successful fit of the waveguide transmission, which is shown in Fig. 5.11. Note that the parameters apply to the TM polarization and, therefore, the saturation energy used is quite large. This is because the saturation energy scales as the inverse of the differential gain (actually differential absorption in our case). According to calculations, differential gain may be factor ~ 10 smaller for the TM polarization than for TE (See chapter 4). The absorption characteristics of the saturable absorber can also be fitted with a formula of the same form as Eq. 5.16. In fact, the energy transmission of the waveguide for the TM input polarization can be numerically expressed as T® m = -------------- , (5.24) ™ 1 + 0.0616 x Ein ’ K J 236 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table 5.1: VALUES OF DEVICE PARAMETERS USED IN THE FIT OF THE WAVEGUIDE ABSORPTION Device property Symbol Value TPA coefficient P 70 cm/GW Small signal interband absorption coefficient 9o -15 cmT1 Device Length L 0.1 cm Saturation energy Esat 100 pJ Modal cross-section 1.35 x 10~8 cm2 Differential gain a 4 x 10-16 cm2 Photon energy h u > 0.8eV where Ein is the input energy (in units of pJ) of the TM-polarized pulse launched into the waveguide. This is just the same relationship that was used earlier (See Eqs. 5.9 and 5.11). 5.8 Polarization-dependent Phase Shift In this paragraph, the usefulness of the dual polarization switching technique for optical limiting is shown by additional measurements. These measurements show that the TE (pump) polarization is more effective than the TM (probe) light in creating a nonlinear phase shift. By using the TE light in conjunction with the TM light, one can to reach the ir — ► 2ix nonlinear phase shift range where the slope of the output energy vs. input energy is flat. This is very critical in order to achieve good limiting characteristics. 237 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.25 * Measured — Calculation .<20.15 0.05 Input energy in TM (pJ) Figure 5.11: Absorption characteristics of the waveguide when TM light only is launched. The energies are measured inside the device. The transmission is measured between the device facets and only includes the estimated on-chip losses. By measuring the nonlinear transmission characteristics for various launched polarization ratios, one can figure out the amount of nonreciprocal phase shift induced by each polarization. The transmission characteristics of the Sagnac in terferometer were measured for a number of different polarization ratios, while at the same time keeping the total injected power into the limiter constant. Some of the total relative transmissions (including both polarizations) for the interferometer are shown in Fig. 5.12. These transmission characteristics are for a directional coupler coupling ratio of C=0.55. These transmission char acteristics can again be fitted by using an equation similar to Eq. 5.15 and 238 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. utilizing the transmission characteristics shown in Fig. 5.11. In particular, the transmission characteristics of the interferometer are modeled by , (1 - C f C 2 K X ( - --------------— xr + 1 + EiCa/3 1 + EtCb/3 2(7(1 — C) ^ (1 + ElCb p)(l + ElC j ) cos ( < f > NL)) . (5.25) Energy was again substituted for intensity. Ei is the total energy launched into the interferometer and includes both polarizations. The coefficients Ca and Cb are the fractions of the total input energy launched into the interferometer, which is coupled into the TM-polarized mode of the semiconductor waveguide. Ca is the fraction coupled into side A and Cb the fraction coupled into side B. /3=Q.061 p J -1 is the two-photon transmission reduction factor (the same factor that appears in the denominator of Eq. 5.24). The nonreciprocal phase shift, ( f) NL, is caused by both the asymmetry in TM power and, to a larger degree, by the fact that the TE light is launched from only one side. Therefore,this phase shift has two components dependent on the amount of energy launched in the TE (E fE) and the TM (EfM) polarizations: 4 > . nl = n™ x E ™ + nT cE x E fE (5.26) 239 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The total launched energy Ei = E fM + E fE stays the same during the exper iment. nEM and nEE are the phase shift coefficients with the units of radians per picojoule. 0.35 0.3 0.25 1:14 0.2 2:1 0.15 0.1 0.05 40 Total energy launched Into coupler (pJ) 80 20 60 100 Figure 5.12: Change in the relative interferometer transmission as a function of the input polarization state. The ratios shown are the ratio of TE:TM light of the input pulse. The total input power is kept the same for every polarization ratio. Therefore,one obtains the maximum phase shift as a function of polarization ratio, enabling one to deduce that the TE polarization causes a 2 ± 0.2 larger nonreciprocal phase shift than TM polarization, for a fixed total input power, i.e., nEE/n EM = 2±0.2. This shows that the TE polarization is more efficient at generating a nonlin ear phase shift than the TM polarization. There are three reasons for this. The first is that TE is totally absorbed to generate the maximum amount of carriers 240 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. per unit energy whereas TM is not fully absorbed. The self-phase shift gen erated by TM is also highly nonuniform across the TM pulse profile, resulting only in the partial switching of the probe. Third, both the clockwise and an ticlockwise TM pulses generate self-phase shifts, but the differential self-phase shift may still be small if these phase shifts are nearly equal. In other words, a large part of the TM-induced phase shifts is common mode for the clockwise and counterclockwise pulses. 5.9 Phase Shift Efficiency It was shown that the dual polarization idea is useful for the practical implemen tation of a limiter. The power needed to actuate the limiter can be determined if one knows the material parameters of the nonlinear waveguide. Therefore, it is very useful to know how many radians of phase are generated for every pico- joule of optical energy that is absorbed. The latter number can be considered to be the phase shifting efficiency of the nonlinear device. In fact, it is possible to extract the phase shifting efficiency for absorbed TE light. This was achieved by first tuning the coupler to the point where the contra-directional TM-polarized optical powers injected into the device are exactly equal. Therefore, the self-phase modulation experienced by the coun- terpropagating TM pulses should be the same (to the first order). The only 241 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. change in transmission, therefore, occurs due to the nonreciprocal phase shift brought about by the TE light and two-photon absorption effects. The two- photon effects can be divided out in this case. The resultant transmission is again fit with a simple interferometer transmission function which then yields the phase shift. The interferometer transmission is shown in Fig. 5.13. This yields a phase shift efficiency of 0.5 rad/pJ for low energies. p > Q_ Measured Fitted data Input energy in TE (pJ) Figure 5.13: The change in the transmission of the Sagnac interferometer due to only the TE (pump) pulse because the counterpropagating TM waves has been equalized by adjusting the coupler. 5.10 Discussion The experimental measurements shown earlier prove that the Sagnac loop device has very good limiting characteristics and is a promising serial processor of 242 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. strings of pulses. This device can optimized in various ways so that limiting can be achieved on very high repetition rate pulsetrains. There are also ways to improve the transmission, and the switching power where limiting sets in. The recovery time of the intraloop nonlinear waveguide is estimated to be of the order of Ins. This was sufficient for the experiment since the repetition rate of the mode-locked laser is only 40MHz. For pulses spaced less than one nanosecond, the device has to recover faster, in order to still obtain a big phase shift for every pulse. In particular, one wants r T, where r is the carrier lifetime and T the repetition period of the source. A passive device, like used in our experiment, has a significant advantage here since the recovery time can be sped up quite easily by reducing the carrier lifetime by either creating recombination centers or traps in the material [25, 17], or by sweeping out the carriers by reverse biasing the device [5]. This enables one to lower the lifetime to around 10 to 20ps. There is, therefore, an added degree of freedom compared to active devices like a semiconductor optical amplifier in the sense that one can engineer the material itself to a larger degree to suit the application. This also does not significantly change the switching energy for pulses of the order of the order of a few picoseconds, as long as the lifetime is a few times bigger than the pulsewidth, i.e., as long as the integrating properties of the bandfilling effect is not compromised. 243 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The spatial asymmetry of the semiconductor waveguide in the Sagnac loop should be such that it does not affect the limiting of a single pulse. Therefore, the time difference for the clockwise and anticlockwise pulses (TM) to reach the device should be less than the input pulse spacing. In other words, the offset of the nonlinear waveguide from the center of the loop should be AL < (5.27) 2 nef f where T is the input pulse spacing, c the speed of light and ne// the effective index of the fiber mode. Since the position of the nonlinear waveguide in the loop and the material lifetime can be adjusted, one has the freedom to engineer the system to meet the requirements. We expect the limiter to be able to be scaled up in order to achieve the limiting of pulses at ~ 10GHz rates. The concept also shows promise for optical integration either in a all-semiconductor or hybrid technology. Further flexibility in shaping the transmission characteristics is obtained by adding a tunable coupler inside the structure. This provides a knob whereby the limiting characteristics can be fine tuned in order to obtain a flat slope. The polarization ratio can also be adjusted so that the amount of energy absorbed is adequate to produce a certain phase shift. 2 4 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The limiter described in this article is totally passive. This means the device gain is smaller than 1, but this also implies that virtually no spontaneous emis sion is generated (compared to a semiconductor optical amplifier). Subsequent optical amplification may be needed to boost the output, but in this case an Erbium-doped fiber amplifier with a low noise figure may, for example, be used. Currently the TM pulse is partially absorbed into the waveguide. This leads to low transmission and self-phase modulation effects on the TM part of the pulse. These may be undesirable in a optical ADC (Analog-to-digital converter) or optical communication systems. It should be pointed out that the current device is not optimized for the limiting application. The device is engineered for amplification, and the quantum well-overlap x device length may be too high. We believe a shorter device may still absorb all or most of the TE (” pump”) light, but have significantly less absorption of the TM polarization by either one or two-photon absorption. The SOA structure also includes doped layers which have some free carrier loss, especially the intervalence absorption in the p-doped layers [3]. Efficient coupling to the waveguide and low system losses are also necessary. The current interferometer can be used in the reflecting mode, i.e., the lim ited light exits the same port as the input port. This would require a device like an optical circulator in order to separate the outgoing light from the incom ing light. A nonlinear phase shift of 7 r would be required to achieve complete 245 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. turn-off of the light at the reflecting port. This is in comparison with the cur rent > 2 7 r phase shift necessary for limiting at the transmitting port. There is also some saturation of phase shift for bigger phase shifts (See Eq. 5.19). One can expect more than two times reduction in the energy necessary for limiting when one uses the reflecting port, compared to the current situation where the transmitting port is being utilized. 5.11 Figure of M erit for Limiter In order to quantitatively compare different kinds of limiters, one has to have some number, to indicate how effectively the input noise is reduced. The trans mission characteristics of a generic limiter are described by a function, iQ = /(**), where z * is the input intensity and i0 is the output intensity, as shown in Fig. 5.14. One can define a figure of merit for the limiter, and the following was chosen: FO M = r T . (5.28) h a h + Some of these variables are defined in Fig. 5.14, which shows the transmission characteristics of a generic limiter. Limiting occurs between point A and B (arbitrarily chosen). The highest possible transmission, /<,//*, is preferable up to the point where the device start limiting. The widest fractional limiting range Ah/{Ii + A/*) is also advantageous. Finally, the average slope across the 246 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. switching region, a, has to be as close as possible to zero in order to get the maximum noise reduction. We define a as the average of the absolute slope over the limiting region. In other words the limiting region is divided in N section with equal width Sii — A/j/IV. Then, Um + (5.29) N -*o o N Sii 3 = 0 1 Furthermore, a good limiter for noise reduction has a well defined point where it starts limiting, i.e., the output characteristic has to change slope within a short intensity span. An additional multiplicative transmission factor for the limiter (low intensity transmission) has been left out of the figure of merit expression (Eq. 5.28) because in it possible to use a low noise amplifier at the output port (like an erbium-doped fiber amplifier) to boost the intensity level again in the case of lower transmission devices. This is true as long as the signal power is high enough, so that additive noise in the form of amplified spontaneous emission does not degrade the signal to noise ratio appreciably. To compare various types of limiters for noise reduction, it is necessary to specify the amount of fluctuation of the source. The appropriate fractional limiting range, A / j / ( J j + A /*), can then be chosen. Given this number, one can 247 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. then choose the most favorable part of the output characteristics to maximize the figure of merit. Input intensity Figure 5.14: The output power characteristics for a generic limiter, and the definitions of parameters associated with the figure of merit of such a device. The device exhibits limiting between points A and B. The noise reduction is graphically illustrated by the reduction in the size of sinusoid in the limiting region. 5.12 Conclusion It was demonstrated that limiting can be achieved in a Sagnac loop with an a polarization-sensitive semiconductor saturable absorber placed assymetrically in the loop. The process was aided by two-photon absorption. Significant flexibility to change the transfer function for the system is obtained by having a tunable coupler amd the ability to change the input polarization. This enables one to trim the system to obtain a slope in the output characteristics which 248 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3.5 ,0OOOO<JOOOOOOOOOOO Limiting range - 2.5 > Q. 1.5 0.5 In p u t e n e r g y (A .U .) Figure 5.15: An energy limiting characteristic obtained with the Sagnac loop limiter, which is used in the figure of merit calculation. approaches zero. In order to quantify the ‘goodness’ of the limiter, a figure of merit is introduced was introduced in Paragraph 5.11. If we choose the inverse of the fractional limiting range, A A /(1 * + Alj), (See Eq. 5 .28 and Fig. 5.14) to be 1.25 (i.e., if we are expecting up to ~ 25% variation from a certain base value), we can quantitatively compare different approaches with our limiter. For our system we were able to achieve a figure of merit of up to 58. This is by using the particularly good looking data set in Fig. 5.15. In recent years two-photon absorption has also been investigated as a promis ing mechanism for limiting of free space laser beams [10]. The Sagnac loop limiter can be compared with the two-photon limiting action obtained in the nonlinear waveguide (See Fig. 5.11). For a limiting range of input energies 249 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. (inside the device) between 131 and 175pJ, the figure of merit is calculated to be 1.13. For a saturating optical amplifier with a chip gain of 20dB, which is operated as a limiter in an range of input energies close to its saturation energy (optimum), the figure of merit is calculated to be 0.73 by using the theory of Agrawal and Ollson [4]. Therefore, by utilizing the nonlinear phase shift in the semiconductor device, one is able to achieve a superior figure of merit, com pared to limiting based only on the saturation and absorption characteristics of semiconductor waveguides. 250 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1 ] M. Y. Hong A. Dienes, J. P. Heritage and Y. H. Chang, Time- and spectral domain evolution of subpicosecond pulses in semiconductor optical ampli fiers, Opt. Lett. 17 (1992), no. 22, 1602-1604. [2 ] G. P. Agrawal and N. K. Dutta, Semiconductor lasers, 2nd ed., pp. 120- 126, Van Norstrand Reinhold, New York, 1993. [3 ] _____ , Semiconductor lasers, 2nd ed., pp. 139-142, Van Nostrand Rein hold, New York, 1993. [4] G. P. Agrawal and N. A. Olsson, Self-phase modulation and spectral broad ening of optical pulses in semiconductor laser amplifiers, IEEE J. Quantum Electron. 25 (1989), 2297-2306. [5] I. E. Day, P. A. Snow, R. V. Penty, I. H. White, R. S. Grant, G. T. Kennedy, W. Sibbett, D. A. 0. Davies, M. A. Fisher, and M. J. Adams, Bias dependent recovery time of all-optical resonant nonlinearity in an InGaAsP/InGaAsP multiquantum well waveguide, Appl. Phys. Lett. 65 (1994). [6] A. Dienes, J. P. Heritage, C. Jasti, and M. Y. Hong, Femtosecond optical pulse amplification in saturated media, J. Opt. Soc. Am., A 13 (1996), no. 4, 725-734. [7] J. E. Ehrlich, D. T. Neilson, and A. C. Walker, Carrier-dependent nonlin earities and modulation in an InGaAs SQW waveguide, IEEE J. Quantum Electron. 29 (1993), 2319-2324. [8] A. D. Ellis, D. A. O. Davies, A. Kelly, and W. A. Pender, Data driven operation of semiconductor amplifier loop mirror at f0Gb/s, Electron. Lett. 31 (1995), no. 15, 1245-1247. [9] K. L. Hall, G. Lenz, and A. M. Darwish, Subpicosecond gain and index nonlinearities in InGaAsP diode lasers, Opt. Comm. I l l (1994), 589-612. [10] G. S. He, C. Weder, P. Smith, and P. N. Prasad, Optical power limiting and stabilization based on a novel polymer compound, IEEE J. Quantum Electron. 34 (1998), no. 12, 2279-2285. 251 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 11] M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, and P. J. Delfyett, Subpicosecond pulse amplification in semiconductor laser amplifiers: theory and experiment, IEEE J. Quantum Electron. 30 (1994), no. 4, 1122-1131. 12] M. Y. Hong, Y. H. Chang, A. Dienes, J. P. Heritage, P. J. Delfyett, Sol Dijaili, and F. G. Patterson, Femtosecond self- and cross-phase modulation in semiconductor laser amplifiers, IEEE J. Select. Top. Quantum. Electron. 2 (1996), 523-539. 13] A. Mecozzi J. Mork and C. Hultgren, Spectral effects in short pump probe measurements, Appl. Phys. Lett. 68 (1996), no. 4, 449-451. 14] W. Pieper M. Eiselt and H. G. Weber, SLALOM: Semiconductor laser amplifier in a loop mirror, J. Lightwave Technol. 13 (1995), no. 10, 2009- 2112. 15] J. Mark and J. Mork, Subpicosecond gain dynamics in InGaAsP optical amplifiers, Appl. Phys. Lett. 61 (1992), no. 19, 2281-2283. 16] A. Mecozzi and J. Mork, Saturation induced by picosecond pulses in semi conductor optical amplifiers, J. Opt. Soc. Am., A 14 (1997), no. 4, 761-770. 17] D. B. Mitchell, B. J. Robinson, D. A. Thompson, Q. Li, S. D. Benjamin, and P. W. E. Smith, He-plasma assisted epitaxy for highly resistive, optically fast InP-based materials, Appl. Phys. Lett. 69 (1996), 509-511. 18] J. Mork and A. Mecozzi, Response function for gain and refractive index dynamics in active semiconductor waveguides, Appl. Phys. Lett. 14 (1994), no. 3, 1736-1738. 19] _____ , Theory of the ultrafast optical response of active semiconductor optical waveguides, J. Opt. Soc. Am., A 13 (1996), no. 8, 1803-1816. 20] A. W. O’Neill and R. P. Webb, All-optical loop mirror switch employing an asymmetric amplifier/attenuator combination, Electron. Lett. 26 (1990), 2008-2009. 21] P. R. Prucnal and J. P. Sokoloff, Terahertz optical asymmetric demulti plexer, United States Patent Number 5,493,433, 1996. 22] J. Reintjies and R. C. Eckardt, Two-photon absorption in ADP and KD*P at 266.1 nm, IEEE J. Quantum Electron. 13 (1977), 791-795. 23] B. E. A. Saleh and M. C. Teich, Fudamentals of photonics, pp. 176-191, John Wiley and Sons, New York, 1991. 252 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [24] P. M. V. Skovgaard, R. J. Mullane, D. N. Nikogosyan, and J. G. Mclnerney, Two-photon photoconductivity in semiconductor waveguide autocorrelators, Opt. Comm. 153 (1998), 78-82. [25] E. R. Thoen, J. P. Donnelly, S. H. Groves, K. L. Hall, and E. P. Ippen, Pro ton bombardment for enhanced four-wave mixing in InGaAsP-InP waveg uides, IEEE Photon. Technol. Lett. 12 (2000), 311-313. [26] M. Willatzen, J. Mark, and J. Mork, Carrier temperature and spectral holebuming dynamics in InGaAsP quantum well laser amplifiers, Appl. Phys. Lett. 64 (1994), no. 2, 143-145. 253 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Chapter 6 Future Work The devices that were studied in this dissertation are at the proof-of-concept level and are not optimized for practical use. Several suggestions to improve the performance of these experimentally demonstrated devices were mentioned in Chapter 2 and Chapter 5. The author is of the opinion that the dissertation work points to some other new research directions. 6.1 Suggestions for Improving Devices based on Four-wave M ixing M edium 6.1.1 Quantum D ot Gain M edium One of the important issues uncovered in the experiments and simulations of four-wave mixing in semiconductor optical amplifiers was the ability to obtain good isolation between the amplified and diffracted probe fields. In order to obtain better isolation in bulk four-wave mixing setup (shown in Chapter 2), one can increase the angle between the beams, reducing the four-wave mixing 254 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. efficiency due to the effects of diffusion (and decreasing field overlap). Diffusion affects efficiency because the diffusion length is of the order of the grating period. If one could find a way to reduce the carrier diffusion length (without affecting the carrier lifetime), one would be able to obtain better isolation by using a larger injection angle without a reduction in mixing efficiency. Therefore,some kind of lateral carrier confinement, which would keep the carriers from diffusing, would be desirable. Highly efficient lateral confinement has recently been shown in submicron stripe lasers using quantum dot (QD) gain regions [20, 21]. QD’s are discon nected laterally and cladded with a higher energy material and carriers injected into the dots are confined by a lateral energy barrier, as illustrated in Fig. 6.1. Carriers in the lower QD states experience a higher barrier and should be better confined laterally. In the same way, a deeper-ground state is desirable, which means the dots should be large and rich in indium [21]. The work of Kim et al. [21] showed a factor of ~ 9 reduction in diffusion coefficient compared to a quantum well of similar material (for a device operating on the ground state at room temperature). As Kim et al. [21] pointed out, self-organized dots with lateral and ver tical size distributions work perfectly well for carrier confinement. QD lasers themselves have fallen short in terms of other criteria like high characteristic 255 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. e * ei. .e* e 160meV l 45rneV Figure 6.1: Illustration of lateral quantum dot to barrier diffusion for four dif ferent quantum dot states (by Kim et al., ch.6,ref.[2]) . temperature, threshold, and high speed, partly due to the size distribution prob lem. On the other hand, the three-dimensional confinement in QD’s is a natural and promising way of obtaining carrier confinement in lasers and nonlinear op tical devices, as long as the size distribution is small enough to provide sufficient gain at the lower QD states. Recent experimental work indicates that QD gain material may be practical for lasers and nonlinear devices as proposed here [9, 3, 27]. QD lasers with an energy offset greater than lOOMeV between the ground and first excited state have been demonstrated, enabling ground state operation (continuous wave) for temperatures up to ~50° C [24]. The linewidth enhancement factor is lower at the gain peak in QD’ s (~0.5) [3] than for quantum wells (~ 1 — > 3) due to a more symmetric gain shape. The linewidth enhancement may however still be appre ciable on the long wavelength side of the gain peak. The linewidth enhancement factor are also usually higher for an SOA (semiconductor optical amplifier, than 256 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. / N c n f n s a r Q 'Et'ng Waveguide core containing quantum dots. Figure 6.2: A diagram of a single-mode waveguide containing quantum dots. Counterpropagating fields write a carrier grating which can be used for con- tradirectional coupling of a probe beam. This reflective carrier grating can exist because of the quantum confinement in the quantum dots, preventing diffusion. (Note: this drawing does not show actual relative scales). for a laser) [26]. Pulsed four-wave mixing in InAs/InGaAs QD amplifiers (under electrical injection) has also been reported [5]. Much progress has been made in realizing QD’ s with long wavelength emission, like at the important 1.3 fim optical fiber communications wavelength [3, 22, 27] where a large number of optical components and sources like 1.319 urn Nd:YAG lasers are also available. Ground state lasing of 40°C at 1.31 fj,m has, for example been, demonstrated in columnar shaped InGaAs dots [27]. Technological improvements in QD gain regions are likely, which could result in high modal gain and short devices (the latter is not really an important requirement for nonlinear devices though). One might also be able to think up even simpler device structures for four- wave mixing based on the properties of the QD’s. One such idea is illustrated in 257 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Fig. 6.2, which shows how a counterpropagating wave in a single-mode waveg uide with a QD gain media can set up a reflective grating for contradirectional coupling. This grating could exist due to the shorter expected diffusion length in this media. Therefore, probe beam could be therefore reflected by this grat ing, which could result in the simple separation of the reflected probe from the incident probe. The proposed wave-mixing configuration represents a very easily implemented experiment (neglecting crystal growth) due to easy device processing and the fact that the optical fibers can be directly butt-coupled to the device. A question is to how much light will be linearly scattered by the dense distribution (2 — 6 x 10locmT3) of quantum dots of size of 13 nm height by 17nm diameter (for columnar shaped dots). Mie’s theory [4] cannot be used to estimate the effect due to the dense subwavelength spacing of the dots. Scat tering from randomly spaced (and sized) dielectric spheres with subwavelength spacing is in fact, still one of the great unsolved problems for theorists [18, 17], and in our case it is probably more easily calculated using a numerical tech nique like the finite difference time domain method [7]. The scattering effect is expected to be small, however, because of the small overlap factor of the optical modes with the dots and the nonresonant nature of the backscattering due to the random dense placement of the dots with sizes much smaller than 258 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the optical wavelength. This needs to be investigated numerically and exper imentally. Wave-mixing may also be a valuable probe to study the electronic characteristics of the quantum dots. 6.1.2 Spin G ratings Electron spin may provide an extra degree of freedom for realizing new op toelectronic devices[11, 2, 19, 28, 25, 23, 1], and can also be used to set up spin-polarized gratings[6] in the case of four-wave mixing interactions. Spin- polarized carrier populations are typically set up by the interband absorption (or stimulated emssion) of a circularly polarized laser beam propagating per pendicular to a wafer containing semiconductor quantum wells. Light has to propagate in this way, due to the pinning of the (heavy-hole valence band to conduction band) interband matrix element in the plane of the well. Due to the thin width of the quantum well, the interaction length is very small (except if the quantum well is placed in resonant cavity). Phase-matched conditions are also nonexistent. It is therefore desirable to have light propagating along the plane containing the active semiconductor material to realize efficient phase-matched nonlinear interactions over a long interaction length. For in-plane propagation one would require some ‘matrix element engineering’ which is in fact possible by using properly shaped quantum wires/dots [8]. It is desirable to have a nearly isotropic matrix element for in-plane electromagnetic field propagation. The 259 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. confinement of carriers in quantum dots may prove to be a highly desirable property because of the high mobility[6] of (spin-polarized) electrons. The opti cal waveguide should also be carefully engineered to avoid birefringence, which would destroy phase-matching. Of course splitting of the heavy-hole and light- hole valence bands in semiconductors can not only accomplished by quantum confinement, but also by strain[2, 28] and it should be possible to demonstrate four-wave mixing by spin-polarization in a suitably strained bulk crystal as well. 6.1.3 Sub-Bandgap W ave-M ixing In wave-mixing it might be advantageous to inject probe light below the bandgap in order to read the grating written by beams with wavelengths shorter than the bandgap. In Chapter 4, it was seen that the plasma effect can result in big (and increasing with longer wavelength) refractive index changes below the bandgap. This has been demonstrated in the wave-mixing experiments of Diez et al. [10]. This approach reduces the mixing efficiency due to the device gain but it is not expected to affect the diffracted fraction, i.e., the switching contrast, of the probe drastically. The probe power can be made quite high because it does not affect the gain of an SOA. The latter property may be important in wavelength conversion where the gain transparency at the probe wavelength enables one to avoid pattern dependence (for digital signals) of the conversion efficiency. In this approach there is also no ASE present in the output signal, 260 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. so that the signal-to-noise ratio can be arbitrarily high (in the absence of other noise sources). 6.2 Suggestions for Further Work on Limiters A few ideas for integrated optic limiters were discussed in chapter 4. Because of the relative simplicity of fabrication, the all-optical cutoff modulator seemed to be one of the more elegant ideas, to realize an optical limiter. It is desirable to have a defocusing Kerr-like medium in the core. This means the semiconductor has to be used in the absorptive regime. A material with a big linewidth en hancement factor and low absorption is needed for this, and it was pointed out in chapter 4 that the compressively strained quantum used with TM-polarized light with a wavelength falling between the HH1-C1 and LH1-C1 bandedges might be ideal for this. Software has already been written to use the relevant material parameters in simulations (See chapter 4). Experimental values for the material parameters in this wavelength range and polarization are not very reliably calculated theoretically, and there does not seem to be any detailed experimental results as well. One needs to know the interband absorption, as well as the free carrier absorption and refraction for TM light for a known structure, in order to properly design and simulate such a structure. Other studies have shown the big refractive effects for TM-polarized 261 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. light, but not the simultaneous gain/absorption effects [15, 14, 16]. The ex perimental setup shown in chapter 5 can, in fact, be used for the spectroscopy of carrier-dependent gain and index effects for TM light. This is an analogy to others who have shown that the nonlinear properties like the linewidth en hancement factor can be measured by putting the nonlinear waveguide in an interferometer [13, 12]. This is basically a direct measurement of the nonlinear ities and does not require an acousto-optic modulator like in the selfheterodyne method [26]. Because the Sagnac interferometer is a differential interferometer, either CW input light or the current in the device can be modulated at one of eigenfrequencies of the loop to get the largest (nonlinear refraction) phase excursion, which the interferometer converts into amplitude modulation (AM) for a probe beam, which can be measured directly on an oscilloscope. The am plitude modulation of the waveguide transmission due to the light or current modulation can also be measured directly after the nonlinear waveguide to get the carrier-dependent gain/absorption. The differential carrier density, dN/dP (where N is the carrier density and P optical power), is also important for nonlinear applications and can, for example, be determined by measuring the current modulation across a pn-junction containing the quantum wells caused by a modulation in input light. With a tunable laser (tunable between the LH1-C1 and HH1-C1 bandedges), the dispersion of the carrier-dependent (light 262 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. and/or current induced) refractive index changes and absorption/ gain can be measured. This can be done with existing laboratory equipment. 263 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1] D. D. Awschalom, Electron spin and optical coherence in semiconductors, Physics Today (1999), 33-38. [2] V. L. Berkovits, V. I. Safarov, and A. N. Titkov, Effects of uniaxial defor mation on the polarized luminescence of isotropic semiconductors, Seriya Fizicheskaya 40 (1976), no. 11, 48-51. [3] D. Bimberg, N. Kirstaedter, N.N. Ledentsov, I. Alferov, P.S. Kop’ev, and V. M. Ustinov, InGaAs-GaAs quantum-dot lasers, IEEE J. Select. Top. Quantum. Electron. 3 (1997), no. 2, 196-205. [4] M. Bom and E. Wolf, Principles of optics, ch. 13, pp. 633-664, Cambridge University Press, Cambridge, UK, 1997. [5 ] P. Borri, W. Langbein, J.M. Hvam, F. Heinrichsdorff, M. H. Mao, and D. Bimberg, Time-resolved four-wave mixing in InAs/InGaAs quantum-dot amplifiers under electrical injection, Appl. Phys. Lett. 76 (2000), no. 11, 1380-1382. [6] A. R. Cameron, P. Riblet, and A. Miller, Spin gratings and the measurement of electron drift mobility in multiple quantum well semiconductors, Phys. Rev. Lett. 76 (1996), no. 25, 4793-4796. [7] S. Chu and S. K. Chaudhuri, A finite-difference time-domain method for the design and analysis of guided-wave optical structures, J. Lightwave Technol. 7 (1989), no. 12, 2033-2038. [8] L. A. Coldren and S. W. Corzine, Diode lasers and photonic integrated circuits, pp. 518-524, John Wiley and Sons, New York, 1995. [9 ] D. G. Deppe, D. L. Huffaker, G. Park, and O. B. Shchekin, Quantum dots: A new generation of semiconductors lasers?, LEOS NewsLetter (2000), 3- 6. [10] S. Diez, R. Ludwig, C. Schmidt, U. Feiste, and H. G. Weber, 160 Gbit/s op tical sampling by a novel ultra-broadband switch based on four-wave mixing in a semiconductor optical amplifier, Tech. Digest OFC 1999, 1999, Paper PD38, pp. 381-383. 264 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [11] M. I. D’ Yakonov and V. I. Perel, Spin orientation of electrons associated with the interband absorption of light in semiconductors, Soviet Physics JETP 33 (1971), no. 5, 1053-1059. [12] J. Ehrhardt, A. Villeneuve, G. Assanto, B. Mersadi, A. Accard, G. Geliy, and B. Fernier, Interferometric measurement of the linewidth enhancement factor of a 1.55pm Gain As P optical amplifier, Appl. Phys. Lett. 58 (1991), no. 8, 816-818. [13] J. Ehrhardt, A. Villeneuve, G. I. Stegeman, H. Nakajima, J. Landreau, and A. Ougazzaden, Interferometric measurement of the linewidth enhancement factor of a 1.55pm strained multiquantum-welI InGaAs/InGaAsP ampli fier, IEEE Photon. Technol. Lett. 4 (1992), no. 12, 1335-1338. [14] J. E. Ehrlich, D. T. Neilson, and A. C. Walker, Carrier-dependent nonlin earities and modulation in an InGaAs SQW waveguide, IEEE J. Quantum Electron. 29 (1993), no. 8, 2319-2324. [15] ______ , Guided-wave measurements of real-excitation optical nonlinearities in a tensile strained InGaAs on InP quantum well at 1.5 micron, Opt. Comm. 102 (1993), 473-477. [16] I. Gontijo, D. T. Neilson, A. C. Walker, G. T. Kennedy, and W. Sibbett, Nonlinear InGaAs-InGaAsP single-quantum-well all-optical switch - theory and experiments, IEEE J. Quantum Electron. 32 (1996), 2112-2121. [17] A. K. Hamid, I. R. Ciric, and M. Hamid, Iterative solution of the scatter ing by an arbitrary configuration of conducting or dielectric spheres, Proc. IEEE 138 (1991), no. 6, 565-572. [18] A. Ishimaru, Wave propagation and scattering in random media and rough surfaces, Proc. IEEE 79 (1991), no. 10, 1359-1366. [19] T. Kawazoe, T. Tomobumi, and Y. Masamuto, Highly repetitive pisosec- ond poarization switching in type-II AlGaAs/ AlAs multiple quantum well structures, Jpn. J. Appl. Phys. 32 p a rt 2 (1993), no. 12A, 1756-1759. [20] J. K. Kim and L. A. Coldren, Lateral carrier confinement for ultralow threshold quantum dot VCSELs, Proc. CLEO/Pacific-Rim ’99, 1999, Paper ThD5, pp. 620—621. [21] J. K. Kim, R. L. Naone, and L.A. Coldren, Lateral carrier confinement in miniature lasers using quantum dots, IEEE J. Select. Top. Quantum. Electron. 6 (2000), no. 3, 504-510. 265 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. [22] M. V. Maximov, Y. M. Shernyakov, I. N. Kaiander, D. A. Bedarev, E. Y. Kondrat’ eva, P. S. Kop’ev, A. R. Kovsh, N. A. Maleev, S. S. Mikhrin, A. F. Tsatsul’nikov, V. M. Ustinov, B. V. Volovik, A.E. Zhukov, Z. J. Alferov, N. N. Ledentsov, and D. Bimberg, Single transverse mode operation of long wavelength (~ 1.3pm) InAs/GaAs quantum dot laser, Electron. Lett. 35 (1999), no. 23, 2038-2039. [23] Y. Nishikawa, A. Tackeuchi, S. Nakamura, S. Muto, and N. Yokoyama, All-optical picosecond switching of a quantum well etalon using spin- polarization relaxation, Appl. Phys. Lett. 66 (1995), no. 7, 839-841. [24] O. B. Shchekin, G. Park, D. L. Huffaker, Q. Mo, and D.G. Deppe, Low- threshold continuous-wave two-stack quantum-dot laser with reduced tem perature sensitivity, IEEE Photon. Technol. Lett. 12 (2000), no. 9, 1120- 1122. [25] M. J. Snelling, P. Perozzo, D. C. Hutchings, I. Galbraith, and A. Miller, Investigation of excitonic saturation by time-resolved circular dichroism in GaAs — AlxGa\-xAs multiple quantum wells, Phys. Rev. B 49 (1994), no. 24, 17160-17169. [26] N. Storkfelt, B. Mikkelsen, D. S. Oleson, M. Yamaguchi, and K. E. Stubk- jaer, Measurement of carrier lifetime and linewidth enhancement factor for 1.5pm ridge-waveguide laser amplifier, IEEE Photon. Technol. Lett. 3 (1991), no. 7, 632-634. [27] M. Sugawara, K. Mukai, Y. Nakata, K. Otsubu, and H. Ishilkawa, Per formance and physics of quantum-dot lasers with self-assembled columnar shaped and 1.3-pm emitting InGaAs quantum dots, IEEE J. Select. Top. Quantum. Electron. 6 (2000), no. 3, 462-474. [28] L. R. Tessler, C. Hermann, G. Lampel, Y. Lassailly, C. Fontaine, E. Daran, and A. Munoz-Yague, Highly polarized photoluminescence from 2 — pm- thick strained GaAs grown on CaF2, Appl. Phys. Lett. 64 (1994), no. 7, 895-897. 266 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix A P-side Down M ounting of Broad Area Laser Bars Figure A.l: A scanning electron microscope photograph of a series of broad area lasers mounted on a copper heatsink. In industry, p-side down mounting is used to obtain good thermal character istics for laser diodes [1]. P-side down mounting is preferable to n-side mounting because the p-n junction is of the order of less than a micron away from the p-side contact (for devices grown on a n-doped substrate), and therefore the 267 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. thermal resistance between the p-n junction and the highly thermally conduc tive mounting material is very small. This enables one to run high current, broad area devices at room temperature with a continuous current [1], P-side down mounting is harder to do than n-side mounting because of the proximity of the laser junction to the mounting interface. Mounting and soldering the laser onto its heatsink is a dirty process. The facets of the laser chip have to be kept clean near the laser junction to obtain good reflectivity (in the case of a laser), or high transmission (no scattering or absorption) of laser light in the case of a SOA. An attempt was made to realize p-side down mounting for the broad area devices used in the four-wave mixing experiments. A bar of lasers (a cleaved processed chip, containing many lasers side-by-side), was mounted on a copper heatsink and then cut into separate devices using a wafer saw (See Fig. A.l), so that contact to an individual laser could be made. Alignment of the cuts is done by first measuring the distances of the metal contacts of the devices from the edge of the bar, when looking at the p-side. When the bar is turned around to show the n-side and mounted p-side down, the proper cutting positions can be determined and the blade positioned at the correct distance from the edge of the whole bar. In initial attempts In or PbSn solder was used to mount the laser bar in the absence of a flux. Clean endfaces could be obtained and the devices which 268 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure A.2: A top view of the heatsink on which lasers were mounted and diced with a wafer saw. The lasers were peeled off by hand to reveal voids in the solder. were cut, lased up to higher pulsed current duty cycle than n-side down devices. Some of the device peeled off during cutting or when they were pulled on with a tweezer. The solder beneath these lasers revealed voids (See Fig. A.2), which are partly responsible for the poor adhesion of the devices to the solder. The voids also increase thermal resistance. Therefore, the decision was made to use a water soluble flux in the soldering process. This flux is washed off in an ultrasonic water bath. The result is shown in Fig. A.3. Clean endfaces could be obtained, and there also seems to be good adhesion between the metal contact of the lasers and the solder. The cuts on this specific sample were unfortunately misaligned with the air voids and destroyed the devices. No further material was available at that point to 269 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. continue the work. This experiments show that the methodology used seem to be promising for producing p-side down devices mounted next to each other on a common heatsink. The cutting process on top of the heatsink produces minimal chipping, and working devices can be obtained. AIR GAPS ENDFACE OF CHIP (LASER MIRROR) Figure A.3: A photograph of a laser bar mounted p-side down on a copper heatsink using PbSn and a water soluble soldering flux. The solder only sticks to the gold-coated p-side contacts, but not the SiNx in the other areas. 270 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1 ] R. Parke, D. F. Welch, A. Hardy, R. Lang, D. Mehuys, S. O’Brien, K. Dzurko, and D. Scifres, 2.0 W diffraction limited operation of a monolith- ically integrated master oscillator power amplifier, ’ ’IEEE Photon. Technol. Lett.” 5 (1993), 297-300. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. A ppendix B N oise Figure for the Four-Wave M ixing Process in an SOA Due to Amplified Spontaneous Emission B .l Noise Figure for the XWave Filter All amplifiers, and in our case mixers, based on amplifiers, degrade the signal- to-noise ratio of a transmitted signal because of additive spontaneous emission during the amplification process. This signal degradation can be described by the noise figure, which is defined as where SNR is the signal-toratio of the electrical signal generated when the optical signal is converted to electrical current in the photodiode. One can obtain a simple expression for the NF when one considers an ideal detector 272 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. limited by shot noise only [1]. The SNR of the input signal (with power Pin) is given by (SNR)*, where {I)=RPin is the average photocurrent, hv the photon energy (u is the optical frequency and h is Planck’s constant), q the electron charge, R = q/hv is the responsivity of the photodetector with unit quantum efficiency, and a 2 = 2q(RPin)A f (B.3) is the shot noise in the detector bandwidth, A/ [2]. Now the dominant contribution to the receiver noise is the beat between the spontaneous emission and the signal [1]. The variance of the photocurrent after amplification of the signal by a factor G in the amplifier is, therefore, a 2 = 2q(RGPin)Af+4{GRPin)(RSsp)Af, (B.4) where Sp is the spectral density of the spontaneous emission noise. The first term in Eq. B.4 is the shot noise, and the second term results from the heterodyning 273 { i f (RP inf 2q (RPm)2 A/ d b ’ (R 2 ) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. between the signal and spontaneous emission. Now let r/ be the four-wave mixing efficiency. Then the SNR of the resultant nonlinear output product is (I)2 (SNR) out = H - . (B.5) By substituting Eq. B.4 in Eq. B.5, and dividing the result with Eq. B.2, one obtains the noise figure Np = Ghu + 2GSP Tjhv 2GSP r)hv - ^ ■ (R 6 ) where Wap = GSP is the spectral density of the amplified spontaneous emission (ASE) noise. The approximation made to obtain the last relation is valid for G 1 [1] and was also numerically checked to be valid under the experimental conditions. Eq. B.6 is similar to the result shown in [3]. B.2 Noise Figure in Terms of Measured Values In order to determine the noise figure from experiment, the variables in Eq. B.6 have to be related to directly measured parameters. The signal-to-background 274 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ratio (SBR) is the ratio of the signal peak to the background ASE noise measured in the wavelength domain with an optical spectrum analyzer. It is therefore convenient to express the noise figure in terms of this easily obtainable number. Let’s assume the SBR is measured with a spectrum analyzer with a Gaussian passband given by F(v,n) = Foexp ( ~ ('" 2 ~2/‘)2) . (B.7) where v is the frequency variable, fx is the center frequency of the passband, Fq is a constant, and a is related to the FWHM of the filter, Aufwhm, (the resolution bandwidth) by A f w h m / 1 /T3 C T = 2 V 2 1 ^ 2 ( R 8 ) If the input power in the signal (single frequency) before the SOA is given by Ps(z — 0) and the XWave filter is tuned to the signal, the spectral domain representation of the output signal will just be given by Ps(u, z = L) = r]Ps(z = 0)S(u - us) , (B.9) 275 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where S is the Dirac delta function and ua the signal frequency. When the spectrum analyzer’ s filter is tuned to the signal frequency, i.e., = us, the power detected at frequency v is, therefore, just given by / OO (.Pa(u, z = L) + Wsp) F(u, 11 — us) du (B.10) -OO If one now assumes that the integrated signal power within the spectrum ana lyzer filter bandwidth is much larger than the integrated ASE noise power, the detected signal power (Eq. B.10) becomes PS ig{v = Va) ~ rjPs(0)F0 (B .ll) When the spectrum analyzer filter is detuned from the output signal frequency, the detected power in the filter passband is Pase = J WsPF (i/) dv fts WspFdI where J = J exp ( ^ 3 . 2 ^ ) ’ (B.12) assuming that the spectral density of the ASE is constant within the spectrum analyzer’ s filter passband. The integral X can be evaluated by making the transformation x — (u — h)/V2< t: X = r = {B 13) 2Vln2 J - 00 2\/]n2 276 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Eq. B .ll can now be divided by Eq. B.12 (with Eq. B.13 substituted in it) to yield the SBR: SBR = n (B.14) W rr sp •J t c AvpwHM 2\/lii2 But the SBR is measured in practice, and one would like to know the ASE power spectral density Wsp. Therefore,by using Eq. B.14, one obtains Wsp (in terms of wavelength variables): = 2 A g Vln2 v Ps(z = 0) , sp SBR c s/n AXfwhm ’ ^ ' where A o is the free space wavelength at which the SBR was measured, c is the speed of light and A A fwhm is the FWHM of the spectrum analyzer in units of meter. The SBR, 77, and Pa{z = 0) are all easily experimentally measured. Eq. B.15 can be substituted in Eq. B .6 to yield the noise figure in terms of experimentally measured values: 4 Ag V E2 P ,( z = 0) SB R h c2 A X f w h m ' At the wavelength of 1.32fim, where the experiments described in Chapter 2 were done, one obtains that Eq. B.16 simplifies to (when S.I. units are used): _ 1 „ „ second A Ps(z = 0) _ NF « 0.1 -T ....■ ■ x - —A—— I— (B.17) Newton J SBR A A fwhm 277 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In Eq. B.17, Ps(z = 0) has to be in the units of Watt, SBR dimensionless and AXfwhm in units of meters. 278 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1] G. P. Agrawal and N. K. Dutta, Semiconductor lasers, 2nd. ed., ch. 10, pp. 491-493, Van Nostrand Reinhold, New York, 1993. [2] B. E. A. Saleh and M. C. Teich, Fundamentals of photonics, ch. 17, pp. 673- 690, John Wiley and Sons, New York, 1991. [3] M. A. Summerfield and R. S. Tucker, Noise figure and conversion efficiency of four-wave mixing in semiconductor optical amplifiers, Electron. Lett. 31 (1995), 1159-1160. 279 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix C Overlap Factor Corrections in Semiconductors D ue to Spatial Carrier Hole Burning in Presence of Diffusion In this Appendix the crossmodal overlap factor in a direct bandgap semicon ductor (above the bandgap) is studied further. This builds on the work shown in Chapter 3. The crossmodal overlap factor between the orthogonal modes of a multimode waveguide, which contains a single semiconductor nonlinear sec tion, are never entirely zero and may be significant in the case of high optical intensities when the photons can burn significant variations into the carrier dis tribution which is mostly determined by carrier injection and diffusion. In this Appendix this phenomenon is analytically investigated for a quantum well gain medium in a buried heterostructure. C .l Statem ent of Problem In order to find the carrier density, one has to solve the carrier continuity equa tion at each point along the propagation direction. Assuming that there is zero 280 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. detuning between the pump fields a and b (propagating in different modes), which responsible for the nonlinear grating which couples these beams, the par tial derivative with respect to time of the carrier density can be written as [3 ] 8N(xg^ = D V 2N + ^ - vgg(N)(Nph(x, y, z))T - ~ = 0 (C.l) The meanings of the symbols in the above equation are the following: • N — carrier density • D— ambipolar diffusion coefficient • J— carrier density • q= charge of single electron • t= thickness of single quantum well • vg— group velocity of optical fields (assume it to be approximately equal for both modes) • g(N)— the carrier density dependent material gain • Nph=th.e photon density • r=carrier lifetime 281 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. • n=number of quantum wells • {-)r=time average over one optical cycle It was assumed that the electron-hole plasma is neutral and, therefore, that the electron density is everywhere equal to the hole density, resulting in a zero am- due to the difference in diffusion coefficients for electrons and holes. In the case of a quantum well active region, the variation of the optical field along the axis perpendicular to the quantum well plane (the y-axis) can be assumed to be small. Furthermore, the variations along the propagation direction (z) can be assumed to be much smaller than the variations across the lateral axis x. This is because the intermodal beat length is of the order of 100ymn and the modes are slowly growing (at least on the lym scale). This enables one to simplify Eq. C.l to a one-dimensional equation at a certain point z — z0 along the propagation direction. Assuming that the quantum wells are situated at a mean depth y « y0, one can then rewrite Eq. C.l as: bipolar mobility. It was also assumed that there are no internal fields generated d2N(x,y0 ,z0) dx2 vgg(N)Nph(x,yo,z0) 282 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This is a linear second order inhomogeneous ordinary differential equation with the boundary conditions that N(x = ± 0 0 ) = 0 and can, in general, only be numerically solved, given the electric fields and injection current density. Of course there is also lateral confinement of the carriers due to the het- erojunction interfaces, and this might possibly be handled by solving for the electron wavefunction and eigenenergy and looking at the transmission proba bilities across the lateral potential step, given the temperature of the carriers. A possible way to incorporate this result in the carrier diffusion equation (Eq. C.2) is to make the diffusion a rapidly decaying position-dependent function outside the active region, i.e., D = D{x, Zq), s o that Eq. C.2 describes the carrier distribution of carriers within the active region and with energy equal to the quantized level energy plus the appropriate kinetic energy. Although this conjecture will not be proven here, it does make physical sense and allows one to match the boundary conditions of Eq. C.2 to obtain a unique solution. A suitable choice for the diffusion constant for an active gain region distributed over xa < x < xb is D = DQ f(x), where f(x) = u(x, Xa, Xb) + s(xa - x) exp j + s(x - xb ) exp (C.3) 283 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where 1 when xa < x < Xb u(x,xa,xb)=<j (C.4) 0 when x > xb or x < xa, and s is the unit step function: 1 when x > 0 s(x) = { (C.5) 0 when x < 0 and C is some characteristic decay length (electron wave penetration depth), with a size smaller than fh 5nm, and D0 is the ambipolar diffusion coefficient in the quantum well region. The form of Eq. C.3, ensures that the carriers are physically confined to the active region, while making D continuous, which is required in order to match boundary conditions. C.2 First Order Correction of Overlap Factor It is now possible to derive qualitative and quantitative results for the cross- modal overlap factor if the assumption is made that the gain can be linearized in the carrier density, i.e.,, g(N) « a(N - N0) (C.6) 284 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where a = dg/dN (the differential gain) and N 0 is the transparency carrier density. It is useful to use the transformation N = N — N0 to write Eq. C.2 in the form f( x) ^ - 7 T ~ K p ( x ) N = - J ( x ) + ^ (C.7) (J b jU D D In this equation: « Ld — y/DQ T (the diffusion length when no optical fields are present) _ tr a ( £ > q + r > ^ ) — 2-q h u ) • rj =effective impedance of the waveguide material • fio»=photon energy (u is the angular frequency of the waves) • P(x) = Dffef (Dl uaix> y) + Dbub(x > y) + 2DaDbUa(x, y)Ub{x, y)) (The normalized spatial power distribution function for the incident pho tons) • J(x) = J(x)/nqtD0 Also assume that the input current distribution is given by { Jo when xa < x < xb ( C .8 ) 0 when x > xb or x < xa i.e., the current density is constant over the active area and zero elsewhere, like in a typical buried heterostructure. 285 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. It is basically important to see that diffusion and stimulated emission are in competition, and the “stronger” process determines the strength of the nonlinear grating and, as will be shown, the crossmodal overlap factor (with the gain). C.2.1 Weak Excitation It can be assumed that since the current injection is uniform and because the carriers are confined, the carrier distribution is uniform in the quantum well regions if there is no or very weak optical signal present. In this case the carrier distribution is approximately (using Eq. C.7): using Eqs. C.4,C.6,C.8 and C.9. Clearly if the optical modes are symmetrical is placed uniformly symmetrical around this point (i.e., xa < 0 and xb > 0 with J0r/nqt — Nq when xa < x < xb when x > xb ox x < xa (C.9) The crossmodal overlap with the gain can then be calculated to be (See Eq. 3.2) : / +oo /*+oo / g(x, Vo, z0)Ua{x, y)Ub(x, y) dx dy -OC J — OO — OO J — OO u(x, xa, xb)Ua(x, y0)Ub(x, y0) dx (C.10) around x=0 (the midpoint of the waveguide) and the quantum well active region 286 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xa = |xb|), Eq. C.10 predicts that the Tat will always be zero due to the mode orthogonality theorem. This says that under very weak or no excitation there is no nonlinear coupling possible between the modes, if the gain is placed over the whole core of the waveguide region. Eq. C.10 is the same relation as shown in Chapter 3. A more rigorous and accurate calculation can be performed by using regular perturbation theory. Under “weak” illumination the optical excitation can be seen as a perturbation of the carrier distribution determined by the carrier injection, diffusion, and spontaneous relaxation. The zero order approximation is then determined by solving Eq. C.7 under the requirement that e = K, — > 0. For xa < x < Xb the general solution can be found to be where c and d are constant coefficients. In the regions outside the active region (i.e x > xa and x < x & ) the following differential equation must be solved P = c sinh(L£>:r) + d cosh(LDx) + — N0 nqt (C .ll) cPN( °) f[(°) (C.12) 287 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. where Xi = \x — Xi\, i = a or b. This equation is just a transformed Bessel differential equation with a solution of [2 ] iV(0 ) = e j0 exp(xi/2£)^j + foKo ex p (x i/2 £ ) Nn (C.13) where e* and fi are constant coefficients and / 0 and K0 are modified Bessel Functions of the first and second kind, respectively. Because one requires that N(°\ — > N0 if Xi — > oo, one finds that e, = 0 because I0 is a monotonically increasing and K0 a monotonically decaying function for large values of x. By imposing the continuity of carrier density and diffusion current at x a and Xb, one ends up with a matrix equation A X = B with X consisting of the four unknown coefficients: X = d f a h (C.14) which can be solved through simple matrix inversion, i.e., X — A lB. Now that N is known, it is possible to search for a first order correction for the case of weak optical field. If the carrier distribution is written as N = 288 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iV(°) + e N ^ + e2N ^ +... and substituted in Eq. C.7 one can find the following differential equation for the first order correction dNW jvd) s ( x ) ~ d ^ ~ 1 j r = N p {x ) (C U 5) It is possible to solve this equation by the Green’s function method, as shown in Paragraph C.2.2. The solution of Eq. C.15 and higher order perturbation anal ysis yields correction functions, which show up in a modified power-dependent crossmodal overlap factor as follows: „ _ aJr < ■ + “ ab / +oo />+oo r _ / (x) + e N ^ (x) + e2 (x) + ... • O O J — O O ? J —oo J — o o x U0 (x,yQ )Ui(x,y0)dx (C.16) n J T r+OO r+OO ~ -----/ / N ^(x ) U0{x,y)U1(x,y)dxdy (C.17) Q J— o o J — o o This last approximation can be made if the gain is placed symmetrically in the wave guide and the only the first order correction is assumed to be significant. The main result from Eq. C.16 is that the spatial distributions of the higher order corrections may be such as that they break the orthogonality of the eigen- modes in the overlap factor integral (Eq. C.16). Therefore,even when \xa\ = \xb\ the overlap factor can be nonzero, but even then it will usually be small since 289 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. the optical fields only perturb the distribution the carrier distribution mostly determined by diffusion. C .2.2 M athem atical Analysis of Higher Order Corrections It is possible to solve Eq. C.15 by the Green’s function method (See, for exam ple, Powers [1]). On the active region (between xa and Xb) f(x) — 1. Let us re quire that Eq. C.15 has the boundary conditions (xa) = A and (x^ = B where A and B are constants to be determined from the overall boundary con ditions. The two linearly independent solutions of the homogeneous part of this Eq. C.15 is then N {l\x ) = < sinh(x/Ld) and (C.18) cosh(x/Ld)- We have to find linearly independent linear combinations (x) and of the above solutions which now satisfy iV ^(xa) = 0 and = 0. Such solutions are (C.19) (C.20) 290 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The Wronskian of these two functions is W (x ) = det N ^(x ) N^l)(x) (i) / dN dN . dx dx 1 ( x b- x a sinn --------- L d V L d (C.21) where use was made of some hyperbolic identities. Therefore,a Green’ s function is known for this problem G(x,t) = j xa < x < t, W(t) t < X < Xh (C.22) and the solution can directly be written down as n[1\x ) + —^ ----N ^ \x ) + -rB----- N[l\x ) N2 (xa) N ? \ x a) f xb / G{x,t)N{0\t)P(t)dt J X a sinh[(xb- xa)/LD] \ xb — X ( A sinh - r / % i LD JX a - - f Ld Jx X a X b Ld x - xa L sinh sinh D xb — x Ld t - x a Ld xb- t Ld + B sinh x — x„ L D N^{t)P{t)dt N^(t)P{t)dt) (C.23) The modes of the waveguide, and therefore the normalized power distribution P, are usually numerically calculated, and the integral in Eq. C.23 would, therefore, be typically numerically evaluated for every required x-value. 291 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. In order to find the values for A and B, one has to solve Eq. C.15 also outside the active region. The Green’ s function method can again be used in the most general case but, if it assumed that Eq. C.9 is a good enough approximation to the zero order carrier distribution, one only has to solve an equation similar to Eq. C.12. Therefore,for xi > 0 (where again Xi = \x — Xj|, where i = a,b), one finds that the carrier solution is given by N-1^ — hiKo ^j^exp(xi/2£j . where i — a,b (C.24) Now ha, hb, A, and B are calculated by matching the solutions and their derivatives (the latter probably has to be evaluated numerically) at xa and Xb- This again results in a matrix equation for the unknown coefficients (ha, hb, A and B) which can then be solved by simple matrix inversion. Higher order corrections can also easily calculated in the same way, since the differential equations for all higher order corrections are of the same form as Eq. C.15. C .2.3 Strong Excitation Under very strong optical excitation the modes may burn significant carrier variations into the normal distribution. A rough criterion for this is when the effective diffusion length is nearly equal to the active area dimension, i.e., Leff ~ 292 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. \xa — £j|. Here the effective diffusion length is defined as Letf = , 1 The \ f TeffD0 effective carrier lifetime is approximately given by Teft = -----_ T " . . (C.25) ff i + E i P'/Pl where Pl is the incident power in mode i at z = z0 and Pl s is the saturating power for mode i. Let 7 = E i Pl/P l s (the degree of saturation). Therefore,the effective diffusion length becomes Le// = L p /y T + ~ 7 where Lp has the same meaning as before. In the materials of interest (InGaAsP based strained quantum wells), Lp was measured to be approximately 4/xra [4]. The typical dimensions of the active areas are, however, ~ 1 — > 2fim. Therefore,to reduce Leff by a factor of 1 — > 2 so that Leff Lp, one finds that 7 fa 3 — » ■ 15. This also implies that the local material gain will be compressed by the same factor. Such high powers are rarely encountered in typical semiconductor optical amplifiers utilized for nonlinear mixing. The formulas derived in the previous section are probably sufficient for practical cases, except in highly saturated amplifiers, in which case a lot of corrections will need to be determined. In the latter case it is probably more convenient to solve Eq. C.l numerically with for example the finite difference method (See Chapter 4). 293 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1] David L. Powers, Elementary differential equatiosn with boundary value problems, Prindle,Weber and Schmidt, Boston, 1981. [2] Lennart Rade and Bertil Westergren, Mathematics handbook for science and engineering, Birkhauser Boston, Cambridge, MA, 1996. [3] Michael Shu, Physics of semiconductor devices, Prentice Hall, New Jersey, 1990. [4] Daniel X. Zhu, Serge Dubovitsky, William H.Steier, Johan Burger, Denis Tishinin, Kushant Uppal, and P.Daniel Dapkus, Ambipolar diffusion co efficient and carrier lifetime in a compressively strained ingaasp multiple quantum well device, Appl. Phys. Lett. 71 (1997), no. 5, 647-649. 294 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Appendix D Derivation of Two-Photon Absorption Terms Extra two-photon absorption terms for the traveling wave equations of Agrawal and Ollson [1 ] are derived here. These terms are added to these equations in a perturbational way. Assume that the mode of the waveguide can be approxi mated by a square profile. The power in the mode P is just simply P = AmodeI where I is the intensity in the wave. Normally the evolution of a wave in the presence of two-photon absorption is specified in terms of intensity tionship between power and energy, the power absorption coefficient due to two-photon absorption is then (D.l) where (3 is the two-photon absorption coefficient. Using the simplified rela- ■mode T P A (D.2) 2 9 5 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. This also assumes j3 is constant across the whole modal cross-section. Clearly Eq. D.2 is just an approximation for the real situation. It is also possible to calculate the time change of rate in the gain due to the two-photon absorption. The number of photons absorbed within an infinitesimal length segment Sz and during a time duration St is i m p 2 1 Sn = ^ ~ — S zS t~ , (D.3) 2 Amode Hu where hu is the photon energy, and T the fraction of the light overlapping the active quantum well region. It is assumed that the pulses are short enough so that there is not enough time for the carriers generated outside the gain region to get captured in the quantum wells during the presence of the light. The rate of change of the number density can then be obtained by dividing with (T AjnodeS zSt'j (8 N \ _ /» » \ dt ) TpA 2 hu Then the change in gain due to two-photon absorption is just: I L = H ^ ) t p a Ta(3P2 where a is the differential gain of the quantum wells. (D.5) 296 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reference List [1] G. P. Agrawal and N. A. Olsson, Self-phase modulation and spectral broad ening of optical pulses in semiconductor laser amplifiers, IEEE J. Quantum Electron. 25 (1989), 2297-2306. 297 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Linked assets

University of Southern California Dissertations and Theses

Conceptually similar

PDF

Dispersive properties in photonic crystals applications to beam-steering and wavelength demultiplexing

PDF

Dispersive and nonlinear effects in high-speed reconfigurable WDM optical fiber communication systems

$Carrier transport in photorefractive multiple -quantum -well spatial light modulators$

PDF

Carrier transport in photorefractive multiple -quantum -well spatial light modulators

PDF

Active microdisk resonant devices and semiconductor optical equalizers as building blocks for future photonic circuitry

PDF

Investigation of degrading effects and performance optimization in long -haul WDM transmission systems and reconfigurable networks

PDF

Electro-optic and thermo -optic polymer micro-ring resonators and their applications

PDF

High performance components of free -space optical and fiber -optic communications systems

PDF

Characterization and compensation of polarization mode dispersion and chromatic dispersion slope mismatch for high bit -rate data transmission

PDF

Experimental demonstrations of all -optical networking functions for WDM optical networks

PDF

Growth control, structural characterization, and electronic structure of Stranski-Krastanow indium arsenide/gallium arsenide(001) quantum dots

PDF

Design, synthesis and characterization of novel nonlinear optical chromophores for electro-optical applications

PDF

Electro -optic microdisk RF -wireless receiver

PDF

Geometrical theory of aberrations for classical offset reflector antennas and telescopes

PDF

Experimental demonstration of optical router and signal processing functions in dynamically reconfigurable wavelength-division-multiplexed fiber -optic networks

PDF

III-V compound semiconductor nanostructures by selective area growth using block copolymer lithography

PDF

High speed low voltage semiconductor microresonator modulators

PDF

Applications of all-optical wavelength shifting using semiconductor optical amplifiers for switching and routing functions in a dynamically reconfigurable wavelength-division-multiplexed fiber-op...

PDF

Fading channel equalization and video traffic classification using nonlinear signal processing techniques

PDF

Structural and optical properties of crystalline organic thin films grown by organic molecular beam deposition

PDF

All-dielectric polymer electro-optic receivers and sensors

Asset Metadata

Creator Burger, Johan Petrus (author)

Core Title All-optical devices based on carrier nonlinearities for optical filtering and spectral equalization

Contributor Digitized by ProQuest (provenance)

School Graduate School

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag engineering, electronics and electrical,OAI-PMH Harvest,physics, condensed matter,physics, optics

Language English

Advisor Steier, William H. (committee chair), Dapkus, P. Daniel (committee member), Gundersen, Martin A. (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-92771

Unique identifier UC11327944

Identifier 3027697.pdf (filename),usctheses-c16-92771 (legacy record id)

Legacy Identifier 3027697.pdf

Dmrecord 92771

Document Type Dissertation

Rights Burger, Johan Petrus

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Access Conditions The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the au...

Repository Name University of Southern California Digital Library

Repository Location USC Digital Library, University of Southern California, University Park Campus, Los Angeles, California 90089, USA