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High capacity optical and RF/mm-wave communications utilizing orbital angular momentum multiplexing

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High capacity optical and RF/mm-wave communications utilizing orbital angular momentum multiplexing

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Content HIGH CAPACITY OPTICAL AND RF/MM-WAVE COMMUNICATIONS UTILIZING ORBITAL ANGULAR MOMENTUM MULTIPLEXING by Yan Yan A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2016 Copyright 2016 Yan Yan ii Dedication This dissertation is dedicated to my loving wife Yun Zhang for her whole-hearted support, to my loving parents Jine Yu and Jianguo Yan for their unconditional love and teaching me the virtues, and to my respectful Prof. Alan E. Willner for being a great mentor during my Ph.D. career. iii Acknowledgements I am greatly thankful to my dissertation advisor, Prof. Alan E. Willner, for giving me the opportunities to conduct research in one of the most advanced labs in the world, teaching me all the work ethics, virtues, and all the valuable guidance during my Ph.D. career. To my loving wife, Yun Zhang, for firmly standing beside me with love, belief in me, and support all these years, encouraging and helping me proceed during my down-turns, and sharing and appreciating my joy and success during the journey. To my dear parents, Jianguo Yan and Jine Yu, for giving me their lifetime unconditional love and setting the best examples for me to be honest, upright, hardworking and kind-hearted. To Dr. Lin Zhang, who introduced me to Optical Communications Lab that led to my fruitful academic research. To all professors, colleagues, family members and friends who provided instructions, guidance and encouragement to me. You make me who I am today and may all share the credit of my achievements, my happiness and my peace. iv Table of Contents Dedication ii Acknowledgements……………………………………………………………...iii List of Figures…………………………………………………………………...vi Abstract xiv Chapter 1 Introduction………………………………………………………1 1.1 Orbital Angular Momentum of EM waves ..................................... 1 1.2 Applications of OAM for Optical Communications ....................... 3 1.3 Thesis Outline ................................................................................. 5 1.4 References ....................................................................................... 5 Chapter 2 High Capacity 28-GHz Mm-wave Communications with Orbital Angular Momentum Multiplexing………………………………7 2.1 Introduction ..................................................................................... 7 2.2 Generation and multiplexing of eight co-propagating polarization- multiplexed OAM channels ............................................................ 9 2.3 Demultiplexing of eight polarization-multiplexed OAM channels and link capacity measurement ..................................................... 14 2.4 Discussion ..................................................................................... 18 2.5 Reference ....................................................................................... 19 Chapter 3 High Capacity 60-GHz Mm-wave Communications with Orbital Angular Momentum Multiplexing……………………………..23 3.1 Introduction ................................................................................... 23 3.2 Generation of 60-GHz millimeter OAM Beams ........................... 24 3.3 Multiplexing and Demultiplexing OAM Beams ........................... 28 3.4 Demonstration of 32-Gbit/s Link using Two OAM Channels on each of the Two Polarization ......................................................... 31 3.5 Reference ....................................................................................... 36 Chapter 4 Multipath Effects in Millimeter-Wave Wireless Communication using Orbital Angular Momentum Multiplexing……………..37 4.1 Introduction ................................................................................... 37 4.2 Concept of Multipath-Effect-Induced Intra- and Inter-Channel Crosstalk of OAM Channels ......................................................... 38 4.3 Multipath Effect on a Single OAM Channel and Intra-Channel Crosstalk ........................................................................................ 43 4.4 Multipath Effect of Multiplexed OAM Channels and Inter-Channel Crosstalk ........................................................................................ 49 v 4.5 Discussion ..................................................................................... 52 4.6 Reference ....................................................................................... 53 Chapter 5 Multicasting in Spatial Division-Multiplexing System based on Optical Orbital Angular Momentum…………………………..54 5.1 Introduction ................................................................................... 54 5.2 Design and results ......................................................................... 55 5.3 Reference ....................................................................................... 60 Chapter 6 Fiber Coupler for Generating Orbital Angular Momentum Modes…………………………………………………………….62 6.1 Introduction ................................................................................... 62 6.2 Design and results ......................................................................... 63 6.3 Reference ....................................................................................... 69 Chapter 7 A Novel Fiber Structure to Convert a Gaussian Beam to Higher- Order Optical Orbital Angular Momentum Modes…………..71 7.1 Introduction ................................................................................... 71 7.2 Design and results ......................................................................... 72 7.3 Reference ....................................................................................... 79 Chapter 8 The Efficient Generation and Multiplexing of Optical Orbital Angular Momentum Modes in a Ring Fiber by Using Multiple Coherent Inputs…………………………………………………82 8.1 Introduction ................................................................................... 82 8.2 Design and Results ........................................................................ 83 8.3 Reference……………………………………………….........…..91 vi List of Figures Figure 1.1 (a) Circularly polarized beam carrying SAM; (b) Phase structure of a light beam carrying OAM; (c) Intensity and phase of the wavefront of OAM beams with different values of ℓ. Figure 1.2. Utilizing OAM and polarization multiplexing in a free space mm-wave communication link. Figure 2.1. Concept of utilizing OAM and polarization multiplexing in a free space mm-wave communications link to significantly increase the data capacity over short distances. Such technique could have perspective applications in places, such as data centers where require large bandwidth links between computer clusters. Figure 2.2 (a) Schematic diagram of a millimeter-wave link based on OAM and polarization multiplexing. (b) Polarization-multiplexed (pol-muxed) OAM beams (ℓ= −3, −1, +1 and +3) are generated by dual-polarization lensed horn antennae followed by spiral phase plates. These beams are multiplexed by a 1×4 combiner (i.e., three beamsplitters in our experiment). (c1) SPP based OAM demultiplexer. The received OAM beams are split into four copies by a 1×4 splitter (three beamsplitters), for each of which (ℓ) is demultiplexed and converted into a Gaussian beam by a reverse spiral phase plate of −ℓ. (c2) OAM mode demultiplexer. The received OAM beams are demultiplexed by an OAM mode demultiplexer, which can spatially separate the four OAM beams without power-splitting loss. Pol: polarization, SPP: spiral phase plate, MUX: multiplexing, DeMUX: demultiplexing, Ch: channel, BS: beamsplitter. Figure 2.3. Normalised intensity and interferogram of mm-wave OAM beams. (a) Top row: Normalised measured intensity of four mm-wave OAM beams of charge ℓ =±1 and ℓ =±3, respectively. Bottom row: Interferogram images of a Gaussian beam and vii OAM beams combined by a beam splitter. (b) Normalised measured and simulated intensity distribution of the multiplexed OAM beams after beam splitters. Figure 2.4. Results of 32 Gbit/s data transmission using 8 pol-muxed mm-wave OAM channels. (a) The crosstalk of each OAM channel, measured at 28 GHz (CW) when only transmitting the four Y-pol channels (single Y-pol) and transmitting all of the X- and Y-pol (dual-pol) channels. (b) The received constellations of 1-Gbaud 16-QAM signals for OAM channel ℓ =+3 under single Y-pol and dual-pol conditions, respectively (SNR =19 dB). (c) Measured BER curves of 1-Gbaud 16-QAM signals for (i) single OAM channel (no crosstalk), (ii) four OAM channels on Y-pol (with crosstalk, Y-pol) and (iii) eight OAM channels on both X-and Y-pol (with crosstalk, X-pol and Y-pol). Figure 3.1. Concept of 60-GHz millimeter-wave wireless communication using OAM and polarization multiplexing techniques. The multiplexed OAM beams propagate through a single pair of apertures and could be demultiplexed with low crosstalk and recovered without the need for further signal processing to cancel channel interference. Figure 3.2 Power loss of Gaussian beam, OAM beams with l = +1 and l = +3 for 28 GHz and 60 GHz as a function of transmission distance. The diameter of the Gaussian beam at the transmitter is 15 cm. The diameter of the transmitter and the receiver is 30 cm. The OAM beam is generated by propagating the Gaussian beam through the SPP. Figure 3.3. Transmitter/receiver aperture size as a function of power loss for the 28- GHz and 60-GHz OAM beams, under the assumption that the transmission distance is 10 m. Figure 3.4. (a) Lensed-horn antenna, the SPP used to generate OAM beams and the intensity of the emitted Gaussian-like beam from the lens horn antenna. (b) Measured intensity and interferogram of the OAM beam with l = +1. (c) Measured intensity and interferogram of the OAM beam with l = +3. viii Figure 3.5. Four OAM channels (l = +1 and l = +3 on both X- and Y- polarizations) are multiplexed by using two dual-polarization antennae, two SPPs, and a 50/50 BS. On the receiver side, a demultiplexing SPP is used to demultiplex one of the multiplexed OAM channels one at a time, following which a receiver lensed horn antenna is used to receive the Gaussian-like beam for detection. The distance between the BS and the demultiplexing SPP (Demux-SPP) is 2.5 m. Figure 3.6. Mode purity of the transmitted OAM beams with (a) l = +1 and (b) l = +3. Figure 3.7. Transmitter and receiver of the 2-Gbaud 16-QAM signal used in the OAM communication link. Transmitter: AWG: arbitrary waveform generator. Receiver: detection of 2-Gbaud/s 16-QAM signal by using an LNA, down-conversion mixer, and an 80-GSample/s real-time oscilloscope followed by offline processing. Figure 3.8. BER measurement of 2-Gbaud/s 16-QAM signal on a single Gaussian beam (B2B only l = 0), a single OAM channel on a single OAM channel, two OAM channels on X-polarizations or Y-polarizations, and 4 OAM channels on both X-and Y- polarizations. Figure 3.9. Constellation of the OAM channel with l = +3 when the SNR is 20 dB. Figure 4.1. (a) Intensity and wavefront phase of OAM beams with ℓ = +1 and ℓ = +3. (b) Generation and detection of an OAM beam. (c) 1-D beam intensity of OAM beams at distances of 1 m and 2.5 m. Figure 4.2. Multipath effects of an OAM channel caused by specular reflection from a parallel ideal reflector. Owing to beam divergence, a part of the OAM beam will be reflected. The reflected OAM beam can be observed as an off-axis OAM beam from the imaging antenna Tx' and the imaging SPP' with an opposite OAM number. Figure 4.3 Simulation results showing the intensity, phase, and OAM spectrum of the direct path OAM beam, reflected path OAM beam, and the actual beam at the receiver. The actual beam is the superposition of the direct OAM and the reflected OAM beams. ix Because the reflected OAM beam is along the off-axis, it is no longer orthogonal to the OAM beams in the direct path and is likely to cause both inter-channel and intra- channel crosstalk at the receiver. The white solid line represents the reflector. Figure 4.4. (a) Experimental setup for investigating the multipath effects of OAM channels. A movable aluminium sheet is used as an ideal reflector, which is placed parallel to the propagation path. (b) Intensity of the OAM beam with ℓ = +3 without and with the multipath effects. Figure 4.5. (a) Setup to study intra-channel crosstalk of a single OAM channel. (b) Experimental results of the received power OAM beams with different values of ℓ as functions of the reflector distance h. (c) Experimental results of the reflected-to-direct power ratio for OAM beams with different values of ℓ. Figure 4.6. Simulation results of intra-channel crosstalk for OAM beams with ℓ = 0, ℓ = +1, and ℓ = +3 under different (a) reflector distance h and (b) propagation distance. Blue-squared curves: received power of OAM channel ℓ from the direct path in the receiver aperture. Red-circled line: total power collected by the receiver aperture from the reflected path. Black triangle line: received power of OAM channel ℓ from the reflected path. The difference between the black triangle curve and the blue-squared curve denotes the intra-crosstalk level. Figure 4.7. Measured BER and SNR as functions of the reflector distance h for (a) ℓ = +1 and (b) ℓ = +3. Stronger fluctuations of BER and SNR are observed for OAM with ℓ = +3 because of the stronger intra-channel crosstalk induced by multipath effects. Figure 4.8. (a) Experimental setup to study the inter-channel crosstalk of OAM channels. (b) Power received from different OAM channels when the transmitted OAM number is ℓ =+1 (c) Power received from different OAM channels when the transmitted OAM number is ℓ =+3. Figure 4.9. (a) Multipath effects in an OAM multiplexing communication link suffering from both the intra- and the inter-channel crosstalk induced by multipath effects. (b) x Measured BER as a function of the reflector distance for the ℓ = +1 channel. (c) Measured BER as a function of the reflector distance for the ℓ = +3. (d) Measured BER as a function of the SNR for the ℓ = +1 channel. (e) Measured BER as a function of the SNR for the ℓ = +3. Figure 5.1. Concept of multicasting function in a OAM multiplexing system. Figure 5.2. Approach of OAM multicasting using all-phase pattern with combination of two amplitude and phase sliced pattern. Left: input OAM state spectrum. Middle: Amplitude and phase patterns for multicasting. Right: OAM spectrum after multicasting. Figure 5.3. Experiment setup of OAM mode multicasting of 100 Gbit/s QPSK. SLM: Spatial Light Modulator. Col: Collimator. PC: polarization controller. The OAM multicasting block includes: OAM mode generation, multicasting input single mode to multiple OAM mode and OAM mode de-multiplexing. Figure 5.4. Data multicasting from one (OAM+15) to 5 OAM channels (OAM+6, OAM+9, OAM+12, OAM+15, OAM+18). (a), Left: intensity of input OAM+15 beam; middle, phase pattern used for multicasting; right: beam intensity after OAM multicasting. (b) The standard deviation over mean of the power of 5 multicast OAM channels as a function of " # . (c) Experimental results of OAM power spectrum before OAM multicasting. (d) Theoretical and experimental results of OAM power spectrum after 5-OAM multicasting. (e), BER curves of data channels after 5-OAM multicasting. Figure 5.5 Data multicasting from one to multiple OAM beams. a, Intensity input OAM channel ($ = 15), phase pattern, and intensity of beam after multicasting. b2, OAM charge spectrum before multicasting. b. Beam intensity after de-multiplexing by SLM- 3 c, Theoretical and experimental results of OAM charge spectrum after multicasting using phase pattern from b1. b4, BER curve of seven OAM multicast channels with <3- xi dB power variation. e, The standard deviation over mean of the power of 7 multicast OAM channels as a function of " ( and " ) which are shown in Fig.5.5b1. Figure 6.1. The structure of fiber coupler and the phase and polarization state of input lights. The red electric field components have a phase difference of 90 degrees relative to the blue ones. Figure 6.2. Blue line: Effective refractive index of the fundamental mode in the external cores as a function of the core radius. Red line: Effective refractive index of HEn,1 mode in the central ring. Figure 6.3. (left) Neff of the symmetric and asymmetric (even and odd) modes composed of HE3,1 in the ring and HE1,1 in the cores. White arrows illustrate the electric field vector direction. (right) Group velocity difference of asymmetric and symmetric modes ∆β1= βasym-βsym= 1/vg,asym-1/vg,asym. Figure 6.4. The intensity and phase of the azimuthal components of the generated OAM modes with odd charge numbers of +1, -3, +5, -7, and +9 in the central ring, respectively. Figure 6.5. Each column shows the OAM weight spectra of the generated OAM modes in the unit of dB with different charge number of n . The top is spectra of HE modes and the bottom is spectra of EH modes. The OAM weight determines the purity of the desired generated OAM mode. The purity of each OAM mode is >0.99 and the crosstalk among these modes is lower than -15dB. Figure 6.6. Dependence of OAM generation performance on the offset of external cores. (a) The light power in the external core as a function of the propagation distance for three offsets. (b) Coupling length as a function of the offset. Figure 6.7. Dependence of OAM generation on the input polarization state. For some charge number with appropriate polarization, the weight could exceed 99% and the extinction ratio could exceed 20 dB (i.e., the generation efficiency is >0.99). xii Figure 6.8. Dependence of OAM generation on the input wavelength. The higher-order OAM mode is much sensitive to the wavelength change because of the large waveguide dispersion. Figure 7.1. (a) Cross section of the designed fiber coupler with a central square and a ring structure. (b) Refractive index of different regions of the designed fiber. Figure 7.2 (a) The effective refractive index of the eigenmodes supported by the designed fiber. (b) Symmetric and asymmetric modes of HE1,1 (core) and EH3,1 (ring) in the designed fiber. The arrows show the direction of the electric field. Figure 7.3. The intensity and phase of the azimuthal electric field (HE polarized) and radial component (EH polarized) of the electric field of the generated OAM modes. Figure 7.4. Optical power in the square core as the function of fiber length when generating the HE5,1 OAM mode in the ring. Figure 7.5. Each column shows the OAM charge power weight spectra of the generated OAM modes in units of dB of the different charge number l. The top is the spectra of the HE modes and the bottom is the spectra of the EH modes. Figure 7.6. The dependence of the (a) mode purity and (b) extinction ratio of the generated OAM modes on the wavelength. Figure 7.7. (a) Dependence of coupling length L and wavelength λc on the fillet radius rf of the square core. (b) Dependence of the OAM mode purity (HE l=5) on the input offset d and tilt angle θ. Figure 8.1. (a) The generation of OAM modes of charge number l in a ring fiber with N coherent Gaussian inputs. (b) The refractive index of the ring fiber, which is designed to support OAMl,1 modes with l=-3~3. xiii Figure 8.2 (a). Left: Intensity of 8 Gaussian input modes. Right: The output intensity and phase patterns of the generated OAMl,1 modes with l=0~3 and LP4,1 mode. (b) The normalized optical power of the generated OAM modes with l=0~3 and LP4,1 mode in a 2 cm long ring fiber. (c) Comparison of the mode purity and crosstalk of the generated OAM modes in a ring fiber or a multi-mode fiber with a step refractive index. Figure 8.3. The power weight of the OAM state l at (a) the ring fiber input z=0cm and (b) ring fiber output z=2cm. Figure 8.4 (a) The dependence of the generated OAM mode (l= 0~3) purity and crosstalk on the (a) offset between the centers of the ring fiber and the multiple inputs and (b) the phase error of the inputs. Figure 8.5. The system diagram of an OAM mode-based MDM transmitter using multiple coherent inputs that generate and multiplex multiple OAM modes, in which each one carries independent data Dl. D: the transmitted independent data channel from different OAM modes; M: modulator; d: data from each coherent input. xiv Abstract One property of electromagnetic (EM) waves that has recently been explored is the ability to multiplex multiple beams carrying orbital angular momentum (OAM) such that each beam has a unique helical phase front. Such OAM-based multiplexing can potentially increase the system capacity and spectral efficiency of RF wireless and optical fiber communication links by transmitting multiple coaxial data streams. My dissertation will introduce the basic concept of OAM and the principle of using OAM for spatial multiplexing in RF/optical communications and my research in this area, including: high capacity mm-wave communications using OAM multiplexing, multipath effects of wireless OAM communications, multicasting network function in optical OAM communications, and different designs to generate and multiplexing OAM channels in optical fibers. 1 Chapter 1 Introduction This chapter will introduce the the basic concept of orbital angular momentum, followed by the motivations of using OAM for optical and RF/mm-wave communications. 1.1 Orbital Angular Momentum of EM waves Although EM waves have been studied for well over a century, one property of EM waves, OAM, was only discovered in the 1990s. In 1992, Les Allen and his colleagues discovered that the OAM of EM waves was associated with the helical transverse phase structure exp (/ℓθ), in which ℓ is the transverse azimuthal angle and ℓ is an unbounded integer [1]. The amount of phase front “twisting” indicates the OAM number, and beams with different OAM are spatially orthogonal. Note that OAM relates to the spatial phase profile rather than to the state of polarization of the beam, which is associated with the spin angular momentum (SAM). An EM beam carries SAM if the electrical field rotates along the beam axis, in other words circularly polarized EM waves. It carries OAM if the wave vector spirals around the beam axis, leading to a helical phase front, as shown in Fig. 1.1 (a) and (b) [2]. In its analytic expression, this helical phase front is usually related to a phase term of exp (iℓθ) in the transverse plane, where θ refers to the angular coordinate and ℓ is an integer indicating the number of intertwined helices; in other words, the number of 2π phase shifts along the circle around the beam axis. ℓ is an integer that can take a positive, negative, or even a zero value, corresponding to clockwise phase helices, counterclockwise phase helices, or no helix, respectively [3]. Although the SAM and OAM of EM waves can be coupled to each other under certain scenarios [4], they can be clearly distinguished 2 for a paraxial EM beam. Therefore, in the paraxial limit, OAM and polarization can be considered as two independent properties of EM beams. Fig. 1.1(c) shows the intensity and phase of the wavefront of OAM beams with different ℓ. When ℓ ≠ 0, the beam wavefront has a spiral phase distribution and intensity null at the center due to the phase singularity. In general, the OAM beams diverge faster as |ℓ| increases [5]. Figure 1.1 (a) Circularly polarized beam carrying SAM; (b) Phase structure of a light beam carrying OAM; (c) Intensity and phase of the wavefront of OAM beams with different values of ℓ. In general, an OAM-carrying beam could refer to any helically phased light beam irrespective of its radial distribution. Laguerre–Gaussian (LG) beams are a special subset among all OAM-carrying beams; their radical distribution is characterized by the fact that they are paraxial eigensolutions of the wave equation in cylindrical coordinates and in homogeneous media, such as free space. For an LG beam, both azimuthal and radial wavefront distributions are well defined and indicated by two indices, ℓ and p, in which ℓ has the same meaning as that of a general OAM beam – azimuthal phase dependence – and p refers to the radial nodes in the intensity 3 distribution. LG beams form an orthogonal and complete mode in the spatial domain. In contrast, a general OAM beam may be expanded into a group of LG beams, each with the same ℓ but a different p index, due to the absence of radial definition. Henceforth, the term “OAM beam” refers to all helically phased beams and is to be distinguished from LG beams. 1.2 Applications of OAM for Optical Communications The demand for capacity and spectral efficiency continues to grow exponentially due to the multitude of wireless applications, including indoor communications, data centers, and front-haul and back-haul connections. Such a multitude of applications produces a great interest in devising advanced multiplexing approaches in RF, one of which is OAM multiplexing. OAM beams with different ℓ values are mutually orthogonal in space, allowing them to be multiplexed together along the same beam axis and demultiplexed with low crosstalk [6,7,8,9]. Utilization of OAM for communications is based on the fact that coaxially propagating EM beams with different OAM states can be efficiently separated. For instance, consider two OAM beams U 1 and U 2 having an azimuthal index of ℓ 1 and ℓ 2 , respectively. Relying only on the azimuthal phase, the two OAM beams can be expressed as: 2 ( (3,5,6 )= 7 ( (3,6 )exp (/ℓ ( 5) (1.1) 2 ( (3,5,6 )= 7 ( (3,6 )exp (/ℓ ( 5) (1.2) where r and z refer to the radial position and propagation distance, respectively. From the above expressions, one can conclude that these two beams are spatially orthogonal in the following sense: 4 (1.3) Consequently, one can establish a well-defined line-of-sight (LoS) link, for which each OAM beam at the same carrier frequency can carry an independent data stream, thereby increasing the capacity and spectral efficiency by a factor that is equal to the number of OAM states. Fig. 1.2 illustrates a prospective application scenario, using OAM multiplexing as well as polarization multiplexing for short-range, high- speed wireless information exchange between a transmitter and a receiver. As mentioned above, in the paraxial limit, OAM and polarization can be considered as two independent properties of an EM wave. Therefore, polarization multiplexing is compatible with OAM multiplexing and can be used to produce a twofold increase in the capacity and spectral efficiency of a transmission link. OAM 3 OAM 1 OAM 2 y-polarization OAM 4 OAM 1 OAM 2 OAM 3 OAM 4 Multiple OAM and dual-polarization transmitter Multiple OAM and dual-polarization receiver OAM and polarization multiplexed beam Y X Figure 1.2. Utilizing OAM and polarization multiplexing in a free space mm-wave communication link. 5 1.3 Thesis Outline This dissertation is organized with the following structure: Chapter 2 presents a 32 Gbit/s mm-wave communication link using OAM multiplexing and polarization multiplexing (8 channels) at 28 GHz RF frequency. Chapter 3 presents a 32 Gbit/s mm-wave communication link using OAM multiplexing and polarization multiplexing (4 channels) at 60 GHz RF frequency. Chapter 4 presents the study of multipath effect of OAM communication links, including both intra- and inter-channel crosstalk. Chapter 5 presents a multicasting network function to distribute the information from one OAM channel to multiple OAM channels. Chapters 6, 7 and 8 present three optical fiber design to generate and multiplex optical OAM modes. 1.4 References [1]Allen, L., Beijersbergen, M.W., Spreeuw, R.J.C., and Woerdman, J.P. (1992) Orbital angular-momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys. Rev. A 45, 8185–8189 [2] Barnett, S. and Allen, L. (1994) Orbital angular momentum and non paraxial light beams. Opt. Commun., 110, 670–678. [3] Allen, L., Padgett, M., and Babiker, M. (1999) The orbital angular momentum of light. Prog. Opt. 39, 291–372. [4] Zhao, Y., Edgar, J.S., Jeffries, G.D.M., McGloin, D., and Chiu, D.T. (2007) Spin- to-orbital angular momentum conversion in a strongly focused optical beam. Phys. Rev. Lett., 99, 073901. [5] Padgett, M.J., Miatto, F.M., M. P.J. Lavery, Zeilinger, A., and Boyd, R.W. (2015) Divergence of an orbital-angular-momentum-carrying beam upon propagation. New J. Phys., 17, 023011. [6] Gibson, G., Courtial, J., Padgett, M., Vasnetsov, M., Pas'ko, V., Barnett, S., and Franke-Arnold, S. (2004) Free-space information transfer using light beams carrying orbital angular momentum. Opt. Express, 12, 5448–5456. 6 [7] Yao, A.M., and Padgett, M.J. (2011) Orbital angular momentum: origins, behavior and applications. Adv. Opt. Photonics, 3,161–204. [8]Molina-Terriza, G., Torres, J.P., and Torner, L. (2007) Twisted photons. Nat. Phys., 3, 305–310. [9] Yan, Y., Xie, G., Lavery, M.P.J., Huang, H., Ahmed, N., Bao, C., Ren, Y., Cao, Y., Li, L., Zhao, Z., Molisch, A.F., Tur, M., Padgett, M.J. and Willner, A.E. (2014) High-capacity millimetre-wave communications with orbital angular momentum multiplexing. Nat. Commun., 5, 4876. 7 Chapter 2 High Capacity 28-GHz Mm-wave Communications with Orbital Angular Momentum Multiplexing 2.1 Introduction Although electromagnetic (EM) waves have been studied for well over a century, a property of EM waves, namely orbital angular momentum (OAM), was discovered in the 1990’s which, with accompanying studies that followed, promised to enable the transmission of many independent data streams over the same spatial wireless medium [1,2]. The EM wave carrying OAM has a helical transverse phase structure of exp(/ℓ8), in which 8 is the transverse azimuthal angle and ℓ is an unbounded integer (the OAM state number). Note that OAM relates to the spatial phase profile rather than to the state of polarization of the beam which is associated with spin angular momentum [3]. OAM beams with different ℓ values are mutually orthogonal, allowing them to be multiplexed together along the same beam axis and demultiplexed with low crosstalk [2,4,5]. Consequently, one can establish a well- defined line-of-sight link, for which each OAM beam at the same carrier frequency can carry an independent data stream, thereby increasing the capacity and spectral efficiency by a factor that is equal to the number of OAM states. Indeed, OAM has seen exciting progress in the optical domain. Systems results have been reported for Tbit/s transmission over both free-space and special vortex fibre for which all OAM beams are multiplexed together and propagate along the same spatial axis [6,7]. Importantly, given that the demand for capacity and spectral efficiency continues growing exponentially due to the multitude of wireless 8 applications, including data center and back-haul connections [8-12], there is great interest to also show advanced multiplexing approaches at radio frequency (RF) and millimeter-wave (mm-wave) frequencies. However, given the different frequency range from optics, such RF and mm-wave systems would require different technologies to achieve the hoped-for results. Specifically, RF and mm-waves transmitted and received using a single aperture pair (i.e., the data passing through one aperture at the transmitter and received by one aperture at the receiver) have been used for free-space wireless communications since the days of Marconi [13-15]. Although there have been many advances since that time, such as advanced modulation format and polarization multiplexing, the basic system architecture has remained fairly unchanged. However, OAM mode multiplexing of multiple orthogonal data-carrying beams multiplexed at the transmitter and demultiplexed at the receiver using a single transmitter/receiver aperture pair would represent a significant architectural change. Recent reports have shown that OAM beams can be used for data transmission in RF links [16,17], including the transmission of one Gaussian and two OAM beams (ℓ= 0,±1) each carrying an 11 Mbit/s signal at ~17 GHz carrier [18-20]. However, different OAM beams in these demonstrations propagate along different spatial axes and are not transmitted through the same single aperture. Note that the spatial orthogonality coaxially propagating OAM beams could enable advantages such as: (a) inherent low crosstalk among OAM channels, (b) reduced need of further signal processing to cancel OAM channel interference after demultiplexing, and (c) scalability to increase the number of OAM channels 6 . In order to fully exploit the advantages of OAM mode-division multiplexing, OAM channels should be multiplexed and transmitted along the same spatial axis though a single aperture. In this chapter, we demonstrate a high-capacity millimeter-wave communication link by transmitting 8 multiplexed OAM beams (4 OAM beams on each of the 2 9 orthogonal polarizations), each carrying a 1-Gbaud 16-QAM signal, thereby achieving a capacity of 32 Gbit/s (1 Gbaud × 4 bit/symbol × 4 OAM beams × 2 polarizations) and a spectral efficiency of 16 bit/s/Hz at a single carrier frequency of 28 GHz [21]. Four different OAM beams of state numbers -3, -1, +1, and +3 on each of two polarizations are generated and multiplexed using spiral phase plates (SPPs) and specially-designed beamsplitters (BSs) at the transmitter, such that all beams are co- propagating from a single transmitter aperture. After propagating through 2.5 meters, the OAM channels are demultiplexed at the receiver using two different techniques: (a) SPPs [22], which can convert an OAM beam back into an approximate Gaussian beam, and (b) an mm-wave OAM mode demultiplexer for simultaneous demultiplexing of multiple OAM beams. The on-axis propagation of all OAM channels and the orthogonality among them allow their demultiplexing to be accomplished using physical microwave components with low crosstalk, reducing the need for further signal processing to cancel channel interference [23]. 2.2 Generation and multiplexing of eight co-propagating polarization-multiplexed OAM channels Figure 2.1. Concept of utilizing OAM and polarization multiplexing in a free space mm-wave communications link to significantly increase the data capacity over short distances. Such technique 10 could have perspective applications in places, such as data centers where require large bandwidth links between computer clusters. Figure 2.1 illustrates a prospective application scenario of using OAM multiplexing for short range, high speed wireless information exchange in a data center. In this paper, we demonstrate a proof -of-concept experiment under laboratory condition. A schematic overview of an OAM-multiplexed mm-wave communication link is shown in Fig. 2.2a. We generate each mm-wave OAM beam of state ℓ by passing polarization-multiplexed (pol-muxed) Gaussian beams, emitted by a collimated dual-polarization, lensed horn antenna, through an SPP (see Fig. 2b). The OAM beams can be generated using different components, such as holograms and SPPs. Because SPPs are polarization independent (unlike holograms), two pol-muxed OAM beams with the same ℓ can be generated by using a single SPP and a dual-pol antenna. High-density polyethylene (HDPE) is chosen for the material of the SPPs due to its transparency at 28 GHz. The SPP is defined by its thickness, which varies azimuthally according to ℎ(8)= ( < )= )ℓ>/(@−1), acquiring a maximum thickness difference of Δℎ = ℓ>/(@−1) (8 is the azimuthal angle varying from 0 to 2π, n is the refractive index of the plate material, and > is the wavelength of the mm-wave) 4 . Normally, one surface of an SPP is flat, so the required thickness can be obtained simply by controlling the height of the other surface. The SPPs used to generate the OAM beams have circular apertures with a diameter of 30 cm. These SPPs are manufactured through the computer numerical control (CNC) milling of a solid block of HDPE, which has a refractive index of n = 1.52 at 28 GHz. For the four OAM states used in this experiment, the SPPs corresponding to ℓ= ±1 and ℓ= ±3 are designed to have height differences of Δh 1 = ±2.07cm and Δh 3 = ±6.21cm, respectively. 11 Figure 2.2 (a) Schematic diagram of a millimeter-wave link based on OAM and polarization multiplexing. (b) Polarization-multiplexed (pol-muxed) OAM beams (ℓ= −3, −1, +1 and +3) are generated by dual-polarization lensed horn antennae followed by spiral phase plates. These beams are multiplexed by a 1×4 combiner (i.e., three beamsplitters in our experiment). (c1) SPP based OAM demultiplexer. The received OAM beams are split into four copies by a 1×4 splitter (three beamsplitters), for each of which (ℓ) is demultiplexed and converted into a Gaussian beam by a reverse spiral phase plate of −ℓ. (c2) OAM mode demultiplexer. The received OAM beams are demultiplexed by an OAM mode demultiplexer, which can spatially separate the four OAM beams without power- splitting loss. Pol: polarization, SPP: spiral phase plate, MUX: multiplexing, DeMUX: demultiplexing, Ch: channel, BS: beamsplitter. 12 To verify the characteristics of the generated OAM beams, their spatial normalised intensity and phase distributions are captured (through interferometry) by a probe antenna with a small aperture diameter of 0.7 cm, whose output is recorded by an RF spectrum analyser. The probe antenna is attached to a two-dimensional linear translation stage with a scanning resolution of 1 cm and a transversal coverage area of 60 cm × 60 cm. Figure 2.3a clearly depicts the ring-shaped normalised intensity profile 24 of the generated OAM beams. The state number of the OAM beams can be deduced from the number of rotating arms in their interferograms, which are generated by interfering the different OAM beams with a Gaussian beam (ℓ= 0) through a coherent superposition, using a beamsplitter, as described below. The field distributions of the SPP-generated OAM beams are similar to the Laguerre-Gaussian (LG) modes 1 . The OAM states of such beams are preserved as the beams propagate in free-space. We note that, in general, the SPP-generated OAM beams can be decomposed to be a set of LG modes with the same OAM state ℓ but different radial indices p [4]. Figure 2.3b shows the normalised measured intensity distributions of the coherent superposition of different OAM beams, which are in good agreement with the corresponding simulation results. 13 Figure 2.3 Normalised intensity and interferogram of mm-wave OAM beams. (a) Top row: Normalised measured intensity of four mm-wave OAM beams of charge ℓ =±1 and ℓ =±3, respectively. Bottom row: Interferogram images of a Gaussian beam and OAM beams combined by a beam splitter. (b) Normalised measured and simulated intensity distribution of the multiplexed OAM beams after beam splitters. Four pol-muxed OAM beams, generated by 4 lensed horn antennae, each followed by an SPP of a different ℓ value, are spatially multiplexed by a 1×4 combiner (comprising 3 cascaded 50:50 planar beamsplitter, as shown in Fig. 2b). Our beamsplitters are similar in form to the polka-dot beamsplitters used in the optical regime. These beamsplitters are implemented by spatially varying the reflectivity of the surface. This is achieved by adding sub-wavelength sized square array of reflective material to obtain 50% transmission and 50% reflection at 28 GHz frequency. In our experiment, the beamsplitters are fabricated by patterning a standard printed circuit board (PCB) with the designed structure (see Supplementary Fig. 2). The multiplexed 14 8 OAM channels (4 OAM states on each of two orthogonal polarizations) are obtained by combining OAM beams of ℓ= ±1 and ℓ= ±3, using a 1×4 beam combiner. The divergence of OAM beams are related to the OAM state number ℓ. For example, the beam sizes of propagating LG beams are proportional to the square root of ℓ given they have the same Rayleigh range [25]. To resolve this ℓ-dependent divergence, positive RF lenses made of HDPE with focal lengths of 2 meters are fabricated and used to slightly focus the ℓ= ±3 modes, so that their beam sizes become similar to that of the OAM beams of ℓ= ±1 at the receiver over the designed link length of 2.5 meters. To generate OAM beams of ℓ= ±1, the SPPs are placed right after the antennae. To generate OAM of ℓ= ±3, the SPPs and positive lenses are placed 2 meters after the antennae. The diameters of the SPPs and lenses are larger than the diameters of the lensed horn antennae in order to cover the diverging Gaussian beams after propagating 2 meters. The resulting 8 co-axial channels, each carrying an independent 1-Gbaud 16-QAM data stream, co-propagate toward the receiver, transmitting a total of 32-Gbit/s on a single RF carrier of 28 GHz. 2.3 Demultiplexing of eight polarization-multiplexed OAM channels and link capacity measurement We recover 8 multiplexed OAM channels one at a time by using a corresponding SPP and a receiver antenna [26]. To recover an OAM beam (ℓ) of interest, an inverse SPP with a specified state (−ℓ) is used to remove the azimuthal phase term exp(/ℓ∅) of the OAM beam, which is then converted back into a beam with a planar phase front of ℓ= 0. This beam then has a bright high-intensity spot at the center, which is separable from other OAM beams with ‘doughnut’ intensity profiles through spatial filtering. In principle, the position of the inverse SPP should be at the center of the multiplexed OAM beams to optimise the conversion from the specific OAM beam to the beam with ℓ= 0 and minimize the crosstalk from other OAM beams. Rotating the 15 SPP should not affect the conversion or crosstalk. For example, in order to obtain the pol-muxed data streams of the ℓ=-3 OAM beam (created by an SPP with a thickness profile of ( < )= )(−3)>/(@−1) ), an inverse SPP with the opposite thickness profile of ( < )= )(+3)>/(@−1) is employed (see Fig. 2c). Note that transmitter and receiver lensed horn antennae are matched to the Gaussian beams (ℓ= 0). Consequently, the helical phase of the ℓ= −3 OAM beam is removed, and the emerging wave could be collected with a dual polarization lensed horn antenna. All other beams, while also ℓ= +6 ), maintain their ring-shaped profiles and helical phase, and therefore negligible signals can be coupled into the Gaussian-matched antenna due to the mode mismatch. The output from this dual polarization receiver antenna is recorded and digitized with an 80 Gsample/s real-time oscilloscope with an analogue bandwidth of 32 GHz, which is wide enough to faithfully capture the modulated 28 GHz mm-wave waveforms. The recorded signals are then processed offline to recover the 16-QAM constellations and calculate the bit-error rate (BER). In the experiment, the transmission distance is 2.5 meters and the diameter of the SPP on the receiver side is 45 cm. The transmission distance of OAM channel depends on different factors including the receiver aperture size and the divergence of OAM beams. Our calculation indicates that when the diameter of the receiver aperture is 45 cm and transmission distance is 2.5 meters, ~60% of the transmitted power could be recovered by the receiver aperture for OAM beams with ℓ =±1, and ~40% of the transmitted power could be recovered for OAM beams with ℓ =±3. Instead, if the transmission distance is 5 meters, ~10% of the transmitted power could be covered by the same receiver aperture for OAM beams with ℓ =±1, and ~1% of the transmitted power could be covered for OAM beams with ℓ =±3. The transmitted signal power and link budget are characterized as follows. The generated signal power of 8 dBm is fed into the lensed horn antenna. The power loss of a 2.5-meter link having only two lensed horn antennae without SPP is ~22 dB. After 16 two SPPs are included in the link, the power loss of the link is ~33 dB, indicating an extra 11 dB power loss from the generation of OAM beam and the back-conversion from the OAM beam to an approximate Gaussian beam. Each OAM beam passes through two beamsplitters for multiplexing, resulting in an additional 6 dB power loss. Therefore, the total power loss for the link is ~40 dB. Figure 2.4. Results of 32 Gbit/s data transmission using 8 pol-muxed mm-wave OAM channels. (a) The crosstalk of each OAM channel, measured at 28 GHz (CW) when only transmitting the four Y-pol channels (single Y-pol) and transmitting all of the X-and Y-pol (dual-pol) channels. (b) The received constellations of 1-Gbaud 16-QAM signals for OAM channel ℓ =+3 under single Y-pol and dual-pol conditions, respectively (SNR =19 dB). (c) Measured BER curves of 1-Gbaud 16-QAM signals for (i) single OAM channel (no crosstalk), (ii) four OAM channels on Y-pol (with crosstalk, Y-pol) and (iii) eight OAM channels on both X-and Y-pol (with crosstalk, X-pol and Y-pol). It is expected that the power from other channels would leak into the channel under detection, due to the imperfections of OAM generation, multiplexing and setup misalignment, which would essentially result in channel crosstalk when a specific channel is recovered at the receiver. The crosstalk for a specific OAM channel ℓ ( can be measured by G ℓHℓ I G ℓJℓ I , where G ℓHℓ I is the received power of channel ℓ ( when all channels except channel ℓ ( are transmitted, and G ℓJℓ I is the received power of channel ℓ ( when only channel ℓ ( is transmitted. Given that our RF-modulated signal 17 spectrally lies within 28±1GHz, and the fact that our SPPs and beamsplitters are frequency-dependent, for simplification we only measure the crosstalk at the frequency of 27.5 GHz, 28 GHz and 28.5 GHz for each channel by transmitting continuous wave (CW) signal, rather than 1-Gbaud 16-QAM modulated signals. For example, to obtain the crosstalk for channel ℓ= −1, we measure the parasitic powers of channels ℓ= +1,±3, and divide them by the signal power coming from channel ℓ= −1. Figure 2.4a presents the crosstalk of each OAM channel measured at 28 GHz when (i) only transmitting the four channels on Y-pol (single Y-pol case) and (ii) transmitting all eight channels on X-pol and Y-pol (dual-pol case). We observe that the crosstalk deteriorates significantly when two polarizations are included (see Fig. 4a), mainly due to the birefringence of the SPPs. We thus conjecture that during the machining process, some stress-induced birefringence is induced in the plates, giving rise to the observed polarization crosstalk. We also observed that the crosstalk value of channel ℓ= −1 is −23 dB, which is higher than those of other OAM channels for the single Y-polarization case. This value increases to −19 dB and −20 dB when the transmitted CWs are at 27.5 GHz and 28.5 GHz, respectively. This deterioration in crosstalk is mainly caused by (i) the material dispersion of our SPPs, and (ii) the deviations from the design frequency of 28 GHz of SPPs. This led to the deviation of the phase fronts of the generated OAM beams from that of the required azimuthal maximum variation of 2ℓL. Figure 2.4b shows the received constellations and error vector magnitudes (EVMs) 27 of 1-Gbaud 16-QAM signals at a signal-to-noise ratio (SNR) of 19 dB for channel ℓ= +3, in both single-polarization (single-pol) and dual-polarization (dual- pol) cases. We can observe that the constellations for channels ℓ= +3 of both X-pol and Y-pol become much worse in the dual-pol case, due to the fact that they experience higher crosstalk. For similar reasons, ℓ= +3 on X-pol has a more blurred constellation than ℓ= +3 on Y-pol. The measured BERs for each channel as functions of SNR under both single Y-pol and dual-pol cases are shown in Fig. 2. 4c. 18 For comparison, the BER curves when only OAM channel ℓ= −1,+1,−3 or +3 is transmitted on Y-pol (in the absence of crosstalk from other channels, black square) are also shown. We see that the power penalty of each OAM state (ℓ= −1,+1,−3 or +3) in the single-pol case is lower than that of either polarization in the dual-pol case. This is due to the fact that each channel in the single-pol case experiences lower crosstalk from the other channels. We also observe that the OAM channel with higher crosstalk will consequently have worse BER performance, as expected. It is clear that each channel is able to achieve a raw BER below 3.8×10 -3 , which is a level that allows to achieve extremely low block error rates through the application of efficient forward error correction (FEC) codes [28] . No pulse shaping or pre-filtering technique has been used at transmitter and the spectral efficiency in the experiment is 16 bit/s/Hz. In general, the number of the OAM states that can be accommodated in a system is limited by different factors, including receiver aperture size and inter-modal crosstalk. In terms of aperture size, the OAM beam of larger ℓ has a larger size at the receiver such that reduces the possible recovered power for limited aperture size thereby increasing the BER. In addition, higher inter-modal crosstalk leads to an increased power penalty and a limited number of OAM channels. Higher crosstalk can arise due to non-ideal performance of the SPPs, as well as from a decreased OAM state number separation between neighboring OAM channels. 2.4 Discussion The experiment is demonstrated over a 2.5-meter link, which is considered as a near-field experiment because the distance is shorter than the Fraunhofer distance. It seems likely that the approach could be expanded to longer distances in the far-field, provided that the aperture size is large enough to capture sufficient power and phase change of each of the OAM beams. Moreover, atmospheric turbulence would likely not present a critical challenge for this frequency range. The system could be 19 potentially scaled to include more OAM channels and to support higher-order modulation format of the signal if the inter-modal crosstalk of the OAM beams could be reduced by enhancing the techniques for OAM beam generation and demultiplexing. The number of OAM channels and transmission distance could be further increased if larger apertures are used. We emphasize that our implementation of OAM mode-multiplexing is different from traditional RF spatial multiplexing. The latter employs multiple spatially separated transmitter and receiver aperture pairs for the transmission of multiple data streams. Since each of the antenna elements receives a different superposition of the different transmitted signals, each of the original channels can be demultiplexed through the use of electronic digital signal processing. In our implementation, the multiplexed beams are completely co-axial throughout the transmission medium and use only one transmitter and receiver aperture (though with certain minimum aperture sizes), employing OAM beam orthogonality to achieve efficient demultiplexing without the need for further digital signal post-processing to cancel channel interference. There are thus significant implementation differences between the two approaches. 2.5 Reference 1. Allen, L., Beijersbergen, M. W., Spreeuw, R. J. C. & Woerdman, J. P. Orbital angular-momentum of light and the transformation of Laguerre-Gaussian laser modes. Phys. Rev. A 45, 8185–8189 (1992). 2. Gibson, G. et al. Free-space information transfer using light beams carrying orbital angular momentum. Opt. Express 12, 5448–5456 (2004). 20 3. Poynting, J. H. The wave motion of a revolving shaft, and a suggestion as to the angular momentum in a beam of circularly polarised light. in Proceedings of the Royal Society of London. 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Lightwave Technol. 30, 504–511 (2012). 23 Chapter 3 High Capacity 60-GHz Mm-wave Communications with Orbital Angular Momentum Multiplexing 3.1 Introduction 60-GHz frequency band has recently attracted attention in the field of wireless communication owing to its unlicensed bandwidth [1,2], low atmospheric absorption [3,4], and potentially reduced beam divergence (high carrier frequency). In particular, low divergence is preferred for OAM communication because one of the challenges associated with OAM communication is that OAM beams diverge faster than conventional Gaussian beams. and has less power at the very center. Operation at a higher carrier frequency will help to increase the propagation distance, the signal-to- noise ratio (SNR) of the channels, and the number of multiplexed OAM channels [5]. It may also decrease the size of the components and lead to more compact transmitter and receiver [6]. In this chapter, we demonstrate an mm-wave communication link using OAM multiplexing at 60 GHz. We multiplexed four channels with two different OAM numbers (l = +1 and l = +3) and two polarizations (X- and Y-polarizations) to achieve a capacity of 32 Gbit/s. Each channel carries a 2-Gbaud 16-QAM signal. We used spiral phase plates (SPPs) made of high-density polyethylene (HDPE) to generate and demultiplex the OAM beams. A 50/50 beam splitter (BS) is used to multiplex the four OAM channels. The transmission distance was 2.5 m in the laboratory. From the results of our experiment, we showed that crosstalk was low, allowing a decoding of the spatial 24 streams without the need for further processing by interference cancellers or maximum- likelihood detectors. 3.2 Generation of 60-GHz millimeter OAM Beams Figure 3.1. Concept of 60-GHz millimeter-wave wireless communication using OAM and polarization multiplexing techniques. The multiplexed OAM beams propagate through a single pair of apertures and could be demultiplexed with low crosstalk and recovered without the need for further signal processing to cancel channel interference. Figure 3.1 shows the concept of the 60-GHz wireless communication using four OAM channels with OAM numbers of l=+1 and l=+3 on both polarizations. We start with the Gaussian-like beams with l=0 emitted by standard mm-wave dual-polarization lens horn antennae and have them propagate through SPPs to generate OAM beams. The aperture diameter of the lensed horn antennae used in our experiment is 15 cm. 25 Figure 3.2 Power loss of Gaussian beam, OAM beams with l = +1 and l = +3 for 28 GHz and 60 GHz as a function of transmission distance. The diameter of the Gaussian beam at the transmitter is 15 cm. The diameter of the transmitter and the receiver is 30 cm. The OAM beam is generated by propagating the Gaussian beam through the SPP. Figure 3.2 shows the simulation results, which compare the power loss for carrier frequencies of 28 GHz and 60 GHz for different OAM beams. From this figure, it can be observed that as the OAM number increases, the difference between the carrier frequencies of 28 GHz and 60 GHz also increases. This observation indicates that a higher carrier frequency can help reduce the propagation loss. Particularly for OAM beams of higher l values, there is more loss reduction due to the use of a higher carrier frequency. 26 Figure 3.3. Transmitter/receiver aperture size as a function of power loss for the 28-GHz and 60-GHz OAM beams, under the assumption that the transmission distance is 10 m. Figure 3.3 shows the aperture size of the transmitter/receiver as a function of the transmission loss for a transmission distance of 10 m. From this figure, it can be observed that a higher carrier frequency also helps to reduce the aperture size under the same transmission loss. Again, we observe that for OAM beams of higher l values, there is more aperture size reduction due to the use of a higher carrier frequency. The lensed horn antenna and the intensity of the emitted 60-GHz millimeter-wave beam are shown in Fig. 3.4(a). An SPP is placed in front of the antenna to convert the Gaussian-like beam to an OAM beam. The spiral surface induces different phase shifts in different parts of the input Gaussian beam. When the height difference of the SPP Δh = lλ/(n - 1), where n is the refractive index of the plate material, λ is the wavelength of the millimeter wave, and l is the OAM charge number, the transmitted beam is converted to an OAM beam with charge number l [5]. Conversely, an SPP that has an inverse spiral surface can convert an OAM beam back to a beam with l = 0. These SPPs are made of HDPE, which has a refractive index of 1.52 at 60 GHz. Thus, to generate 27 the l = ±1 and l = ±3 OAM beams, the designed height differences are Δh 1 = ±9.52 mm and Δh 3 = 28.62 mm, respectively. The diameter of the SPPs in our experiment is 30 cm, which is greater than that of the aperture of the horn antenna that we used to generate a Gaussian beam. Further, these SPPs were fabricated by a precise Computer Numeric Control (CNC) machine process. The approach to using an SPP may impose limitations on the bandwidth because the OAM value is related to the wavelength of the carrier wave. Figures 3.4 (b) and (c) show the measured intensity of the OAM beams, which are generated by the Gaussian beam propagating through the SPPs, and the interferogram of the OAM beams generated by interfere them with a Gaussian-like beam with l = 0. The image is recorded using a probe antenna with a small aperture diameter of 0.2 cm that is connected to an electrical spectrum analyzer. The probe antenna is attached to a two-dimensional (X-Y) linear translation stage with a scanning resolution of 1 cm and a transversal coverage of 60 × 60 cm. Figures 2.4 (b) and (c) confirm the ring-shaped intensity profile of the generated OAM beams. The OAM values of the beams were deduced from the interferogram, and the numbers of the rotating arms indicate that the OAM values are l = +1 and l = +3. 28 Figure 3.4. (a) Lensed-horn antenna, the SPP used to generate OAM beams and the intensity of the emitted Gaussian-like beam from the lens horn antenna. (b) Measured intensity and interferogram of the OAM beam with l = +1. (c) Measured intensity and interferogram of the OAM beam with l = +3. 3.3 Multiplexing and Demultiplexing OAM Beams The dual-pol lensed horn antenna emits two Gaussian-like beams on both X- and Y- polarizations. The generated 60-GHz OAM beams are combined by a designed beamsplitter [6]. The beam splitter is a 2-dimenional array of metal rectangles on a dielectric substrate, which can be fabricated by using standard PCB techniques. The 29 size of the rectangular unit is 1 mm × 0.5 mm, and the period of the array is 1.25 mm in both the X- and the Y-directions. These parameters are designed to exhibit a 50/50 transmission-reflection ratio in the case when the incoming beam is incident on the surface with an angle of 45°. The designed parameters are based on the simulation performed by using a finite-difference time-domain method (FDTD) [7]. Figure 3.5 shows the diagram and a photograph of the experimental setup used for multiplexing and demultiplexing OAM beams. On the transmitter side, two lensed- horn antennae and two SPPs with l = +1 and l = +3 are used to generate four OAM beams with two OAM numbers for both polarization states. A BS is used to combine these four beams. Because OAM and polarization are the orthogonal domains of multiplexing, we can combine them to achieve a high transmission capacity and spectral efficiency. On the receiver side, one of the OAM beams was demultiplexed at one time. An SPP with l = -1 (or l = -3) is used to convert the OAM beams with l = +1 (or l = +3) on two polarizations back to Gaussian-like beams with l = 0, while the other ones remained as OAM beams with l=+2 (or -2). Another lensed horn antenna is used to receive the signal carried by the converted Gaussian beam, and the antenna receives little energy from the other OAM beams. Figure 3.5 (b) shows a photograph of the experimental setup. The link distance between the BS and the demultiplexing SPP is 2.5 m. 30 Figure 3.5. Four OAM channels (l = +1 and l = +3 on both X- and Y- polarizations) are multiplexed by using two dual-polarization antennae, two SPPs, and a 50/50 BS. On the receiver side, a demultiplexing SPP is used to demultiplex one of the multiplexed OAM channels one at a time, following which a receiver lensed horn antenna is used to receive the Gaussian-like beam for detection. The distance between the BS and the demultiplexing SPP (Demux-SPP) is 2.5 m. Figure 3.6 shows the mode purity of the generated OAM beams. The normalized received power is measured by fixing the transmitted OAM value and by changing the OAM value of the Demux-SPP. The measurement characterizes the amount of power that goes from the transmitted OAM channel to the other OAM channels. From this figure, it can be observed that for both l = +1 and l = +3, crosstalk with the adjacent channels is approximately -10 dB. Further, crosstalk with the channels next to the adjacent channels is approximately -20 dB. 31 (a) (b) Figure 3.6. Mode purity of the transmitted OAM beams with (a) l = +1 and (b) l = +3. 3.4 Demonstration of 32-Gbit/s Link using Two OAM Channels on each of the Two Polarization Figure 3.7 shows the transmitter and the receiver that are used to generate and detect a 2-Gbaud 16-QAM signal. A 15-GHz clock signal is generated from an RF oscillator as the input of an X4 frequency multiplier to generate a 60-GHz clock signal. An IQ mixer takes the 60-GHz clock signal as the LO and two independent IF signal waveforms from the arbitrary waveform generator to generate the 2-Gbaud 16-QAM signal at a carrier frequency of 60 GHz. The length of the PRBS signal is 2 15 -1. The output 16 QAM signal is amplified and then split into four paths by using a four-way power divider. Four cables of different lengths connect the four outputs of the power divider to four dual-pol antenna inputs. The lengths of the cables and the positions of 32 the antennae ensure that the signals on 4 different channels have time delays among them and the signal are decorrelated. On the receiver end, the signal is received by a horn-lensed antenna and is then amplified by a low noise amplifier (LNA). The resulting signal is then transferred to a down-conversion mixer such that the 60-GHz signal is mixed with a 50-GHz clock signal. The output is a 10-GHz signal. Then, an 80-Gsample/s real-time oscilloscope with an analog bandwidth of 32 GHz is used to capture the waveform of the 10-GHz signal. Finally, the recorded signal (2 × 10 6 sampled points corresponding to 4 × 10 5 bits for the 2-Gbaud/s 16-QAM signal) is processed offline to recover the 16-QAM signal and to calculate the SNR and the BER. Attenuators are placed before the real- time oscilloscope to vary the SNR. Figure 3.7. Transmitter and receiver of the 2-Gbaud 16-QAM signal used in the OAM communication link. Transmitter: AWG: arbitrary waveform generator. Receiver: detection of 2-Gbaud/s 16-QAM signal by using an LNA, down-conversion mixer, and an 80-GSample/s real-time oscilloscope followed by offline processing. Ideally, there should be no crosstalk from one OAM channel to others. However, in the implementation of OAM communication, the imperfections associated with OAM generation, multiplexing, demultiplexing, and setup misalignment may result in 33 a certain level of channel crosstalk. Crosstalk for a specific OAM channel l 1 can be measured by P l≠l1 /P l=l1 , where P l≠l1 is the received power when all other channels are transmitted on except channel ℓ ( , and P l=l1 is the received power on channel ℓ ( when only channel l 1 is transmitted. Table 3.1 Crosstalk of OAM channels measured at f = 60 GHz. Single-pol case: two channels with two different OAM numbers on X-polarization. Dual-pol case: four channels with two different OAM numbers on both polarizations. The first row of Table 3.1 shows the measured crosstalk of the two OAM channels on the X-polarization without the other two channels on the Y-polarization. The second row shows the measured crosstalk of the two OAM channels on the Y- polarization without the other two channels on the X-polarization. The third and fourth rows show the crosstalk from all the other three OAM channels on both polarizations. From this table, it is found that the crosstalk for all the channels is less than -18 dB. 34 Figure 3.8. BER measurement of 2-Gbaud/s 16-QAM signal on a single Gaussian beam (B2B only l = 0), a single OAM channel on a single OAM channel, two OAM channels on X-polarizations or Y- polarizations, and 4 OAM channels on both X-and Y-polarizations. 35 Figure 3.8 shows the BER measurement of the OAM channels. For each OAM charge number, we measured the BER in the absence of crosstalk from the other channels (blue star), in the presence of crosstalk from the other OAM channels on the same polarization (red circle), and in the presence of crosstalk from the other three OAM channels on both polarizations (black square). We also measured the BER for the baseline case when only the Gaussian beam was used in the link (B2B only l = 0). From this figure, it is observed that the power penalty of each OAM number in the single-pol case is lower than that of either polarization in the dual-pol case owing to the fact that each channel in the single-pol case experiences lower crosstalk. It is thus clear that each channel is able to achieve a raw BER below the 3.8 × 10 -3 forward error correction (FEC) limit, which is a level that enables very low packet error rates to be achieved by the use of appropriate FEC codes (e.g., concatenated RS-convolution code [17]). As an example, in Fig. 9, we show the constellations obtained for the OAM charge number l = +1 when the SNR is 20 dB. Figure 3.9. Constellation of the OAM channel with l = +3 when the SNR is 20 dB. -1.5 -1 -0.5 0 0.5 1 1.5 -1.5 -1 -0.5 0 0.5 1 1.5 36 3.5 Reference [1] J. Wang, et al., Terabit free-space data transmission employing orbital angular momentum multiplexing. Nat. Photon 6, 488–496 (2012). [2] H. Huang, et al., 100 Tbit/s free-space data link enabled by three-dimensional multiplexing of orbital angular momentum, polarization, and wavelength, Opt. Lett. 39, 197-200 (2014) [3] Z. Hao and T. A. Gulliver, On the capacity of 60 GHz wireless communications, in Proc. 2009 Canadian Conf. Electrical Computer Eng., pp. 936–939. [4] A. F. Molisch, “Wireless Communications, 2nd edition, IEEE Press Wiley, 2011, Chapter 20 [5] F. Tamburini, et al. Encoding many channels on the same frequency through radio vorticity: first experimental test. New J. Phys. vol.14, pp. 033001 (2012) [6] X. Hui, S. Zheng, Y. Chen, Y. Hu, X. Jin, H. Chi and X. Zhang, Multiplexed Millimeter Wave Communication with Dual Orbital Angular Momentum (OAM) Mode Antennas, Scientific Reports 5, 10148 (2015) [7] G.A. Turnbull, D.A. Robertson, G.M. Smith, L. Allen, M.J. Padget, The generation of free-space Laguerre-Gaussian modes at millimeter-wave frequencies by use of a spiral phase plate, Optics Communications vol. 127 pp.183- 188 (1996) 37 Chapter 4 Multipath Effects in Millimeter-Wave Wireless Communication using Orbital Angular Momentum Multiplexing 4.1 Introduction As in the case of all wireless communication systems, multipath effects [1-5] are likely to have significant effects on OAM multiplexing systems, particularly when the divergence of OAM beams with ℓ ≠ 0 is greater than that of conventional beams with ℓ = 0 (e.g. Gaussian beams). Several unique factors associated with an OAM- multiplexed link present the following interesting technical challenges: (1) Intra- and inter-channel crosstalk: The reflected energy can be coupled not only into the same data channel of the same OAM value (i.e. as in the case of a conventional single beam link) but also into another data channel with a different OAM value. Therefore, both intra- and inter-channel crosstalk can occur. (2) Beam intensity and phase: An OAM beam has a doughnut-shaped intensity profile showing low power in the center and a peak showing maximum power around a ring; further, the beam exhibits an azimuthal phase change with a value of 2πℓ. In addition, the detection of a specific OAM beam requires a spatial filter to filter out energy on the other beams, which may reduce the received power from the reflected beam. In this chapter, we analyse the multipath effects of OAM channels caused by specular reflection from a surface parallel to the communication link. We show that such reflection causes the distortion of the helical wavefront of the received OAM beams, leading to multipath-induced intra-channel and inter-channel crosstalk. The OAM beam divergence and the spatial filtering effect of the OAM receiver are found to determine the crosstalk. Results of the analysis and measurement show that OAM 38 channels with high values of ℓ tend to suffer from both strong intra-channel crosstalk and strong inter-channel crosstalk with other channels. 4.2 Concept of Multipath-Effect-Induced Intra- and Inter- Channel Crosstalk of OAM Channels One approach to generate and detect OAM millimeter-wave beams involves the use of spiral surface plates (SPPs), as shown in Fig. 4.1(a). At the transmitter end, a millimeter-wave OAM beam can be generated by propagating a beam with ℓ = 0 through an SPP with an OAM number ℓ [6]. At the receiver end, another SPP with -ℓ can be used to convert the OAM beam back to be a beam with ℓ = 0 for detection. Figure 4.1(b) shows the ring-shaped intensity and spiral phase of the OAM beam wavefront with ℓ = +1 and ℓ = +3. It should be noted that given the same size of the initial beam of ℓ = 0, OAM beams passing thought the SPPs with a higher value of ℓ will diverge faster. Figure 4.1(c) shows the simulation result of the one-dimensional (1-D) intensity profile of OAM beams with different values of ℓ at propagation distances of 1 m and 2.5 m. The frequency of the millimeter wave is 28 GHz, and the beam width of the initial beam with ℓ = 0 is 7.5 cm. From this figure, it is observed that OAM beams with a higher value of ℓ diverge more as they propagate and their intensity profile shows low power in the center. 39 Figure 4.1. (a) Intensity and wavefront phase of OAM beams with ℓ = +1 and ℓ = +3. (b) Generation and detection of an OAM beam. (c) 1-D beam intensity of OAM beams at distances of 1 m and 2.5 m. Figure 4.2 shows the concept of the multipath effects of an OAM beam caused by the specular reflection from a reflector parallel to the link. An OAM channel with an OAM number of ℓ 1 is transmitted along the link. At the receiver end, an SPP with an OAM number of ℓ 2 and an antenna is used to receive the OAM channel with an OAM 40 number of ℓ 2 . Ideally, power can only be recovered only when ℓ 1 = ℓ 2 owing to the orthogonality of OAM beams in a line-of-sight link. However, the orthogonality no longer holds when the receiver receives the reflected beam. As shown in Fig. 4.2, a reflector is placed at a distance of h away from the beam center. Assuming that the reflector has a reflection coefficient of 100%, the reflected beam can be observed as an OAM beam from an imaging antenna Tx' and an imaging SPP with an OAM number of -ℓ 1 [7] (reflection changes the sign of the OAM value). Therefore, the receiver will receive an OAM beam with an OAM number of ℓ 1 from the original link as well as a reflected beam with an OAM number of -ℓ 1 from an offset link, which is placed at a distance of 2h. Figure 4.2. Multipath effects of an OAM channel caused by specular reflection from a parallel ideal reflector. Owing to beam divergence, a part of the OAM beam will be reflected. The reflected OAM beam can be observed as an off-axis OAM beam from the imaging antenna Tx' and the imaging SPP' with an opposite OAM number. Orthogonality of the OAM beams depends on the spiral wavefront. Reflection is likely to distort the wavefront phase and induce both intra-channel and inter-channel crosstalk. Further, reflection causes the distortion of the OAM beam wavefront, giving rise to both intra-channel and inter-channel crosstalk. To illustrate this phenomenon, 41 we use an OAM beam with ℓ 1 = +3 as an example. In Fig. 4.3, the left-hand-side column shows the intensity, phase, and OAM spectrum of the OAM beam in the direct path. The entire power is in the OAM state of ℓ 1 = +3. The middle column shows the reflected OAM beam. The reflected OAM beam exhibits an OAM number of ℓ 1 ' = -3, and it is offset to the direct link. As a result, when the reflected OAM beam is decomposed with respect to the OAM basis along the direct path axis, power diverges onto a wide range of OAM states, leading to intra-channel crosstalk with an OAM channel with ℓ 1 = +3 and inter-channel crosstalk with the other OAM channels with ℓ 1 ≠ +3 [17]. The right- hand-side column in Fig. 4.3 shows the actual beam at the receiver, which is the superposition of the direct and the reflected beams. The intensity exhibits a fringing pattern owing to the interference between the direct and the reflected beams. The wavefront phase is also distorted owing to the multipath effect. The power of the actual received OAM with ℓ 1 = +3 differs from the one of the directed path because of the intra-channel crosstalk from the reflected beam, and the received power of the other OAM beams with values of ℓ 1 ≠ +3 is nothing but the inter-channel crosstalk from the reflected beam. 42 Figure 4.3 Simulation results showing the intensity, phase, and OAM spectrum of the direct path OAM beam, reflected path OAM beam, and the actual beam at the receiver. The actual beam is the superposition of the direct OAM and the reflected OAM beams. Because the reflected OAM beam is along the off-axis, it is no longer orthogonal to the OAM beams in the direct path and is likely to cause both inter-channel and intra-channel crosstalk at the receiver. The white solid line represents the reflector. The experimental setup for investigating the multipath effects of OAM channels is shown in Fig.4.4 (a). An aluminium sheet with an area of 2.5 m × 1.5 m is mounted on a cart and placed parallel to the link. The distance between the path and the reflector can be varied by moving the cart. It is observed that the multipath effect becomes stronger with a decrease in the distance between the path and the reflector. As an example, Fig. 4.4(b) shows the intensity of an OAM beam with ℓ = +3 in the absence 43 of a reflector, as well as the intensity of the OAM beam when the reflector is placed close to the link. From this figure, it is observed that the measured fringing intensity pattern caused by the interference of the direct and the reflected beams resembles the simulation result shown in Fig. 4.3. Figure 4.4 (a) Experimental setup for investigating the multipath effects of OAM channels. A movable aluminium sheet is used as an ideal reflector, which is placed parallel to the propagation path. (b) Intensity of the OAM beam with ℓ = +3 without and with the multipath effects. 4.3 Multipath Effect on a Single OAM Channel and Intra- Channel Crosstalk We first studied the intra-channel crosstalk induced by the multipath effect. This intra- channel crosstalk is shown in Fig. 4.5(a), where only an OAM channel with an OAM number of ℓ 1 is transmitted and received. A vector network analyser (VNA) is used to generate and detect a 28-GHz millimeter-wave signal. Two sets of lens horn antennae and SPPs with an OAM number of ℓ 1 are used on both sides to generate and detect OAM beams. According to the abovementioned multipath model, the received signal N can be analysed in terms of the addition of signal N d from the direct path and N r from the reflected path; the relationship among these signals is expressed as N= N r +N d . Figure 44 4.5(b) shows the experimental results of the normalised received power as a function of the reflector distance h for OAM beams with different values of ℓ. As shown in this figure, the following two effects are observed. First, for a high value of ℓ, the received signal power |N| 2 is low, which can be explained on the basis of the divergence property of OAM beams [18]. Second, when the reflector distance h is small, the received power |N| 2 fluctuates with a change in the reflector distance, which is explained in terms of the constructive or destructive interference between the signals from the direct and reflected paths [3]. Moreover, the higher the value of ℓ, the larger is the fluctuation in power, indicating stronger multipath effects for OAM channels with higher values of ℓ. To verify the abovementioned conjecture, we define |N r | 2 /|N d | 2 as the reflected-to-direct power ratio of the OAM channels, and we calculate this value by measuring N and N d with the VNA. N d is measured when the reflector is removed from the link. Further, N is measured when the reflector is included in the setup. Then, N r can be obtained from N r =N-N d . Figure 4.5(c) shows the measured reflected-to-direct power ratios as functions of the reflector distance for OAM beams with different values of ℓ ranging from 0 to 3. The result shows that an OAM channel with a higher value of ℓ exhibits a higher reflected-to-direct power ratio. 45 Figure 4.5. (a) Setup to study intra-channel crosstalk of a single OAM channel. (b) Experimental results of the received power OAM beams with different values of ℓ as functions of the reflector distance h. (c) Experimental results of the reflected-to-direct power ratio for OAM beams with different values of ℓ. The simulation results shown in Fig.4.6 (a) and (b) provide a more detailed analysis for OAM intra-channel crosstalk. In Fig. 4.6 (a) we show the received power of directed path and reflected path as functions of the reflector distance h. The blue lines show the received power of an OAM channel where ℓ = ℓ 1 from the direct path, which is collected by the receiver aperture. The power from the direct path is found to decrease with an increase in the OAM number ℓ 1 ; this change is explained on the basis of the divergence characteristics of OAM beams. The red lines show the total power collected by the receiver aperture from the reflection path. The total power from the reflected path increases with the OAM number, because for a high value of ℓ, the OAM beam spreads faster so that more energy gets reflected into the receiving aperture. The 46 black line shows the received power from the reflection path that only belongs to the OAM channel with ℓ = ℓ 1 , which indicates the absolute inter-channel crosstalk from the reflected path. It should be noted that although the total collected power (red curves) from the reflected path increases significantly with the OAM number ℓ, the received power of the specific OAM channel (black curves) for different OAM numbers is not that much different. This effect can be qualitatively explained as follows. Figure 4.6. Simulation results of intra-channel crosstalk for OAM beams with ℓ = 0, ℓ = +1, and ℓ = +3 under different (a) reflector distance h and (b) propagation distance. Blue-squared curves: received power of OAM channel ℓ from the direct path in the receiver aperture. Red-circled line: total power 47 collected by the receiver aperture from the reflected path. Black triangle line: received power of OAM channel ℓ from the reflected path. The difference between the black triangle curve and the blue-squared curve denotes the intra-crosstalk level. Only a part of the reflected power belongs to the OAM channel with an OAM number of ℓ 1 . To determine this amount of energy, we calculate the overlap integral between the fields of the received reflected beam and the OAM beam of ℓ 1 . This calculation method is known as OAM decomposition [8]. C ℓ = | ∅ x,y φ ℓ ∗ (x,y) | 2 (4.1) In the above equation, ∅ x,y is the complex amplitude of the reflected beam in the receiver aperture, and φ ℓ (x,y) denotes the complex amplitude of the OAM beam in the receiver aperture. Given that a higher value of ℓ results in a more rapid change in the azimuthal phase of φ ℓ (x,y) and that the phase of ∅ x,y is relatively uniform (because the reflected beam is off-centerd, see Fig. 4.3.), the value of the overlap integral C ℓ is expected to be low owing to the averaging effect in the overlapped integral. This observation explains the reason for an increase in the distance between the red lines and the black lines with an increase in the OAM value. On the other hand, it is observed that owing to beam divergence, power from the direct path decreases and the total collected reflected power increases with ℓ. Moreover, the phase structure of the receiver SPP helps reduce the intra-channel crosstalk from the reflected channel. The overall intra-channel crosstalk is, in fact, the difference between the black and the blue curves, which increases with the value of ℓ, indicating strong intra-channel crosstalk for OAM beams with higher values of ℓ. In Fig. 4.6 (b), we show simulation results of the received power from direct and reflected paths as functions of propagation distance z when the reflector distance h=25 cm. The blue curves shows the received power from the direct path drops more rapidly for a higher ℓ value due to the divergence property that there is less power in the center. 48 Another effect is that for a higher ℓ value, the received reflected power is higher at a shorter distance. That can also be explained by the divergence property of OAM beams that the beam spread more rapidly and get reflected at a shorter distance. For ℓ =+1 and ℓ =+3, it is observed that the reflected power starts decreasing after certain z=2.5 m. As illustrated in Fig. 4.1 and Fig. 4.2, since the reflected OAM beam's peak intensity changes at different distance, when the receiver is moved along the propagation distance it will experience the intensity change and see the highest power at certain distance where the intensity peak of the reflected beam locates. The overall effect, which is shown by the the difference between the black curve and blue curve, is that for a higher ℓ value, the same level of intra-channel crosstalk happens at a shorter distance. Next, we investigate the manner in which multipath-induced intra-channel crosstalk affects the performance of a single OAM channel. A 16-QAM signal is used with a data rate of 1 Gbaud. As shown in Figs. 4.5 and 4.6, significant multipath effects are observed when the reflector distance is within 35 cm, corresponding to a distance difference of 1.7 cm~5 cm (a time delay of 55 ps~166 ps) between the direct and the reflected paths, which is estimated from the geometry shown in Fig. 2. Given that the wavelength is 1.07 cm for the 28-GHz carrier frequency, the signal-to-noise ratio (SNR) variation with the reflection distance h is expected owing to the constructive or destructive interference. The signal symbol is 30 cm in length (1 ns in time), which is much longer than the multipath difference; therefore, in this experiment, the inter- symbol interference effect is considered to be weak. Figure 4.7 shows the SNR and the bit error rate (BER) as functions of the reflector distance h for OAM channels with ℓ = +1 and ℓ = +3, respectively. When ℓ = +1, given that the intra-channel crosstalk is relatively low, moderate SNR and BER variations are observed. When ℓ = +3, significantly stronger fluctuations of the SNR and BER are observed owing to the 49 stronger inter-channel crosstalk. The results show that the performance of an OAM channel with a higher value of ℓ is more affected by the intra-channel crosstalk. Figure 4.7. Measured BER and SNR as functions of the reflector distance h for (a) ℓ = +1 and (b) ℓ = +3. Stronger fluctuations of BER and SNR are observed for OAM with ℓ = +3 because of the stronger intra-channel crosstalk induced by multipath effects. 4.4 Multipath Effect of Multiplexed OAM Channels and Inter-Channel Crosstalk Inter-channel crosstalk is measured when the transmitter SPP OAM number ℓ 1 is fixed and the receiver SPP OAM number ℓ 2 takes different values. Figure 4.8(a) shows the experimental setup. A spectrum analyser is used to measure the amount of power that is received from the OAM channel ℓ 1 when the SPP with OAM number ℓ 2 is used at the receiver. Figures 4.8(b) and (c) show the results of the measurement of the received power of the OAM channels with OAM number ℓ 2 as functions of the reflector distance h when ℓ 1 = +1 and ℓ 1 = +3, respectively. From these figures, it is observed that with an increase in h, the power received from the other OAM channels increases owing to the strong multipath effects. When ℓ 1 = +1, the received power difference between ℓ 2 = +1 and the other channels is >20 dB, indicating that the inter- 50 channel crosstalk from the channel with ℓ = +1 to the other channels is less than 20 dB. In the case of the OAM channel with ℓ 1 = +3, the inter-channel crosstalk from the channel with ℓ = +3 to the other channels is 2~9 dB. Figure 4.8. (a) Experimental setup to study the inter-channel crosstalk of OAM channels. (b) Power received from different OAM channels when the transmitted OAM number is ℓ =+1 (c) Power received from different OAM channels when the transmitted OAM number is ℓ =+3. Next we study the channel performance when two OAM channels with ℓ = +1 and ℓ = +3 are multiplexed as shown in Fig. 4.9 (a). Two OAM channels are combined at the transmitter by using a beam combiner. At the receiver, one of the two OAM channels is detected by using the corresponding SPP. In this case, both the intra-channel and the inter-channel crosstalk exist because of the multipath effects. The signal on each OAM channel is a 1-Gbaud 16-QAM signal. 51 Figure 4.9. (a) Multipath effects in an OAM multiplexing communication link suffering from both the intra- and the inter-channel crosstalk induced by multipath effects. (b) Measured BER as a function of the reflector distance for the ℓ = +1 channel. (c) Measured BER as a function of the reflector distance for the ℓ = +3. (d) Measured BER as a function of the SNR for the ℓ = +1 channel. (e) Measured BER as a function of the SNR for the ℓ = +3. 52 Figures 4.9(b) and (c) show the BERs as functions of the reflector distance h for ℓ = +1 and ℓ = +3, respectively (the SNR for both channels is ~22 dB in the absence of a reflector). The BER of an OAM channel with ℓ = +1 significantly increases when the reflector is close to the link. On the basis of the results of the previous intra- and inter- channel crosstalk measurement, we believe that a high BER is mostly caused by the inter-channel crosstalk from an OAM channel with ℓ = +3. Figure 8(c) shows a similar BER variation, as observed in Fig. 4.7 (b), for the OAM channel with ℓ = +3. Given that the measured crosstalk from the OAM channel with ℓ = +1 to ℓ = +3 is low, we believe that the intra-channel crosstalk mainly contributes to the BER variation of the OAM channel with ℓ = +3. The results shown in Figs. 9(d) and 9(e) show the BER as functions of the SNR for channels with ℓ = +1 and ℓ = +3 at different reflector distances h. In the case of the channel with ℓ = +1, the BER error floor increases when the reflector is closer to the link, clearly indicating that the ℓ = +1 channel received more crosstalk from the ℓ = +3 channel when the multipath effects were stronger. In the case of the ℓ = +3 channel, however, we observe that the BER-SNR curves at different distances are quite similar. This observation indicates that the inter-channel crosstalk from the ℓ = +1 channel is relatively low and has limited influence on the performance of the ℓ = +3 channel. 4.5 Discussion The multipath OAM channels with higher values tend to have stronger intra- channel crosstalk. In the case of OAM beams with higher ℓ values, a low amount of power will be received from the direct path. Although more power is collected by the receiver aperture, the receiver SPP serves as a filter and helps reduce the power from the reflected path. OAM channels with higher ℓ values also cause an increase in the inter-channel crosstalk with the other OAM channels in the presence of multipath effects. The results of our investigation primarily focus on the fundamental effect of multipath in an OAM multiplexing scenario and thus consider a single specular 53 reflector. This scenario is not only practically relevant as being similar to a ground/wall reflection but also provides insights into the interaction between the direct and the reflected components. Further work may investigate the combined impact of multiple reflectors and diffused reflection. 4.6 Reference [1] A. F. Molisch, Wireless Communications 2nd edn (Wiley, 2011) [2] J. D. Persons, The Mobile Radio Propagation Channel 2nd edn (Wiley, 2010) [3] W. C. Jakes, Microwave Mobile Communications 2nd edition (Wiley-IEEE Press, 1994) [4] J. B. Andersen, T. S. Rappaport, and S. Yoshida. "Propagation measurements and models for wireless communications channels." Communications Magazine, IEEE 33 42-49 (1995) [5] M. F. Iskander, and Z. Yun. "Propagation prediction models for wireless communication systems." Microwave Theory and Techniques, IEEE Transactions on, 50, 662-673 (2002). [6] Y. Yan, G. Xie, M. Lavery. H. Huang, N. Ahmed, et al., “High-capacity millimetre- wave communications with orbital angular momentum multiplexing,” Nature Communications vol 5, pp. 4876 (2014) [7] S. H. Byun, G. A. Hajj, and L. E. Young, “Development and application of GPS signal multipath simulator,” Radio Science, vol 37, no.6, pp. 1098, 2002 [8] G. Xie, L. Li, Y. Ren, H. Huang, Y. Yan et al., “Performance Metrics and Design Parameters for a Free-space Communication Link Based on Multiplexing of Multiple Orbital-Angular-Momentum Beams,” accepted to IEEE Global Telecommunications Conference 2014 54 Chapter 5 Multicasting in Spatial Division- Multiplexing System based on Optical Orbital Angular Momentum 5.1 Introduction It has long been a desire of the optical communications community to enable efficient processing of an optical data signal while still in the optical domain.[1] One such processing function is multicasting (i.e., fanout), in which data on a single data channel can be duplicated onto multiple channels. These channels would be orthogonal, in the sense that each channel would represent, for example, a different user in a multi-user communication system.[2,3] Previously, time slots and wavelength channels have been multicasted in order to replicate the data such that they co-propagate and can be subsequently split for different potential user destinations.[4] Previously published results for SDM have shown data transmission in optical fiber and free space.[2] However, to our knowledge, there has been little reported on the network and signal processing function for SDM system, such as multicasting of data from one spatial mode onto multiple spatial modes. In this chapter, I will show the design and demonstration of spatial-mode multicasting of a single 100-Gbit/s (50- Gbaud quadrature-phase-shift-keyed (QPSK)) OAM mode onto multiple OAM modes. Multicasting function in an OAM multiplexing system aims to distribute the information from one OAM beam to multiple other OAM beams. The concept of OAM multicasting is shown in Fig.5.1. The charge number of incoming OAM channel is l 2 which carries signal Data2. The multicasting function would distribute the power of 55 the incoming OAM beam of charge l 2 to multiple OAM beams with different charge numbers, such as l 1 and l 3 , and therefore multicast Data2 from the input OAM channel onto multiple OAM channels. We design sliced phase patterns to achieve the OAM multicasting function. 5.2 Design and results Figure 5.1. Concept of multicasting function in a OAM multiplexing system. Figure 5.2. Approach of OAM multicasting using all-phase pattern with combination of two amplitude and phase sliced pattern. Left: input OAM state spectrum. Middle: Amplitude and phase patterns for multicasting. Right: OAM spectrum after multicasting. The approach is shown in Fig.5.2. We assume the incoming single OAM channel has charge number $. It has been shown that an angular amplitude mask of central angle 5 T with U-fold rotational symmetry can distribute energy from the input OAM beam of charge $ to multiple OAM beams, having charges of ..., −VU+$, ..., −U+$, $, $+U, ..., $+VU, ... (V is an integer). The power spectrum of OAM state can be expressed as [5]: |7 $+W$ | ) = ( 5 T 2L ) ) X/@Y ) 5 T 2 W$ , W$ = 0,±U,±2U… 0, [\ℎ]3 ^/X] 56 Figure 5.2a shows an amplitude mask when 5 T = 60° with 3-fold rotational symmetry. After passing through the mask, the light beam’s energy is distributed from the OAM channel of charge $ to three OAM channels of charges $−3, $, and $+3. However, there are two drawbacks to simply use an amplitude mask for multicasting. First, there is significant power loss because part of the beam energy is blocked. Second, the power distribution of the resulting OAM beams is not uniform, but has a sinc ) -like profile. In order to obtain an OAM multicasting function with minimal power loss and equalized channels power, we designed sliced phase-only patterns which is shown in the third row of Fig.5.2 (Fig.5.2c.); Notice that in Fig.5.2a, the mask's transmission part has a constant phase value d 5 = " # , which results in a sinc ) -like OAM charge spectrum, centered at input OAM charge $. In the second row (Fig.5.2b), the mask's transmission part is complementary to that of the first mask, and it has a spiral phase d 5 = −65, producing a sinc ) -like OAM charge spectrum centered at $−6. In Fig.2c, the sliced phase pattern can be viewed as the superposition of the transmission parts of the above two amplitude-phase masks. As a result, the output spectrum of the sliced phase pattern is a coherent addition of those two previous output spectra. The parameter " # is optimized such that most of the multicast OAM channels have equalized power except for two channels at the wings. 57 Figure 5.3. Experiment setup of OAM mode multicasting of 100 Gbit/s QPSK. SLM: Spatial Light Modulator. Col: Collimator. PC: polarization controller. The OAM multicasting block includes: OAM mode generation, multicasting input single mode to multiple OAM mode and OAM mode de- multiplexing. Figure 5.3 shows the experiment setup of OAM multicasting. An optical 100Gbit/s QPSK signal at wavelength of 1550 nm from a single mode fiber is sent into a fiber collimator. In the OAM mode multicasting block, the first spatial light modulator (SLM-1) loaded with a spiral phase pattern φ(θ)=15θ (0<θ<2π) converts the input Gaussian beam into a OAM beam of l=+15. A second SLM-2 loaded with a designed sliced phase pattern is used for OAM multicasting, i.e., distribute the energy from the input OAM beam onto multiple OAM beams. The third SLM-3 loaded with is used to de-multiplex one of the multicast OAM beam. The spiral phase pattern loaded on the SLM-3 is φ=-nθ, which converts OAM beam of l=n to the Gaussian beam of l=0. The converted Gaussian beam is then coupled into a single mode fiber and sent to coherent box for coherent detection of 100Gbit/s QPSK signals. 58 Figure 5.4. Data multicasting from one (OAM +15 ) to 5 OAM channels (OAM +6 , OAM +9 , OAM +12 , OAM +15 , OAM +18 ). (a), Left: intensity of input OAM +15 beam; middle, phase pattern used for multicasting; right: beam intensity after OAM multicasting. (b) The standard deviation over mean of the power of 5 multicast OAM channels as a function of " # . (c) Experimental results of OAM power spectrum before OAM multicasting. (d) Theoretical and experimental results of OAM power spectrum after 5-OAM multicasting. (e), BER curves of data channels after 5-OAM multicasting. Figure 5.4 shows the results of 5-OAM channel multicasting. As shown in Fig. 5.4a, a sliced phase pattern with θ e = 60°, β # = 70° which consists of two 3-fold rotational symmetry patterns is loaded on SLM2. The phase-only pattern for multicasting has the phase distribution as function of angle 5, d 5 = " # 3]h/[@ 1 −65 3]h/[@ 2 0≤5 <360° The input OAM n(o beam carries 100-Gbit/s QPSK signal. The designed multicast channels charge numbers are $ = 6,9,12,15,18, in which OAM nr ,OAM n() ,and OAM n(o beams have equalized power. Figure 5.4a also shows the intensity of input OAM n(o beam and the multicast OAM beams after reflecting 59 from the sliced phase pattern. The intensity of the beam changes from a ring into a triangle, indicating that the reflected beam becomes a superposition of multiple OAM beams. Notice that β 0 is the parameter that needs to be optimized to obtain the equalized power of multicasting channel. In order to evaluate the equalization of the channel power, we define u as the ratio of standard deviation (σ) and the mean value (µ) of the power of the multicast channels v ( ,v ) ,…,v o . u(" # )= w(v ( ,v ) ,v x ,v y ,v o ) z(v ( ,v ) ,v x ,v y ,v o ) Figure 5.5. Data multicasting from one to multiple OAM beams. a, Intensity input OAM channel ($ = 15 ), phase pattern, and intensity of beam after multicasting. b2, OAM charge spectrum before multicasting. b. Beam intensity after de-multiplexing by SLM-3 c, Theoretical and experimental results of OAM charge spectrum after multicasting using phase pattern from b1. b4, BER curve of seven OAM multicast channels with <3-dB power variation. e, The standard deviation over mean of the power of 7 multicast OAM channels as a function of " ( and " ) which are shown in Fig.5.5b1. As shown in Fig.5.5c, two sets of values (" ( = 322°," ) = 33°) and (" ( = 33°," ) = 322°) are found to obtain the minimum value u = 0.02. In Fig. 5.5, we show the results of (" ( = 322°," ) = 33° ). Fig. 5.5c shows the theoretical and 60 measured OAM power spectra after multicasting. In theory, 7 OAM channels should have <1-dB power variation. In the experiment, the measured power at the spectrum wings is found to be smaller than the theoretical predictions, and the power variation among the seven OAM channels is approximately 3 dB. The decline in measured power at the wings is also observed. We believe it is mainly due to the different free- space to fiber coupling efficiency of the demultiplexed Gaussian beams converted from OAM beams of different charges. Since the experiment setup is calibrated for OAM beam of charge number l=+15, the coupling efficiency from free space to fiber are expected to vary for different OAM charge number. The efficiency is lower especially when the OAM charge number is further away from l=+15. The crosstalk between the 7 multicasting OAM channels and other non-multicasting OAM channels is less than -20 dB. Fig. 5.5e shows the BER of seven multicasting OAM channels. 5.3 Reference [1] S. Watanabe, “Optical signal processing using nonlinear fibers,” J. Opt. Fiber. Commun. Rep. 3(1), 1–24 (2005) [2] A. Biberman, B. G. Lee, A. C. Turner-Foster, M. A. Foster,M. Lipson, A. L. Gaeta, and K. Bergman, Wavelength multicasting in silicon photonic nanowires, Opt. Express 18, 18047 (2010) [3] G. Contestabile, M. Presi, and E. Ciaramella, Multiple wavelength conversion for WDM multicasting by FWM in an SOA, IEEE Photon. Technol. Lett. 16, 1775 (2004). [4] J. P. Macker, J. E. Klinker, and M. S. Corson, “Reliable multicast data delivery for military networking,” Proc. of IEEE MILCOM’96 McLean, USA, 2, 399-403 (1996). 61 [5] B. Jack, M. J. Padgett and S. Franke-Arnold, Angular diffraction, New. J. Phys, 10, 103013, (2008) 62 Chapter 6 Fiber Coupler for Generating Orbital Angular Momentum Modes 6.1 Introduction An optical beam that carries orbital angular momentum (OAM) is characterized as having a helical phase wavefront of exp(±inφ) [1]. OAM beam with a topological charge number of n has 2nπ phase change. Each + or - charge is spatially orthogonal to all other charges. OAM has wide applications in atom trapping, optical tweezers and photon entanglement [2]. Recently, OAM has gained much interest for increasing the transmission capacity in optical communication systems given by the ability to carry independent data streams on orthogonal modes [3]. A key challenge has been the generation of OAM modes from the conventional Gaussian modes. This has typically been accomplished by using discrete spatial-light modulators (SLMs) [3], which are generally bulky and expensive. A laudable goal would be to use compact fiber technology for efficiently generating OAM modes in an integrated photonic component. Reports have shown the generation of a single OAM mode in a conventional multi-mode fiber using stress or acoustic waves [4, 5]. In this chapter, I show by simulation a new approach for generating OAM modes in a fiber coupler consisting of a central ring and four external cores. By designing the size of external cores and controlling the polarization state and phase of the four input lights, one can selectively generate OAM modes with odd charge numbers of n in the central ring. The purity of the OAM modes and generation efficiency can exceed 99% using <2mm-long fiber. This 63 fiber coupler could serve as an integrated OAM light source, and could also be used as transmitter and receiver in spatial-mode multiplexing and de-multiplexing system. 6.2 Design and results Figure 6.1. The structure of fiber coupler and the phase and polarization state of input lights. The red electric field components have a phase difference of 90 degrees relative to the blue ones. Figure 6.2. Blue line: Effective refractive index of the fundamental mode in the external cores as a function of the core radius. Red line: Effective refractive index of HE n,1 mode in the central ring. Figure 6.1 shows the structure of the proposed fiber coupler. The glass materials are: (i) background: Schott LLF1 with n L =1.53, (ii) ring: SF6 with n H1 =1.76, and (iii) external cores: SF4 with n H2 =1.71 (at 1550nm). The fabrication of this kind of high- contrast index fiber has been demonstrated [6]. The ring size is fixed with an inner radius of 5µm and an outer radius of 6.1 µm. The size of the external cores is designed to excite the OAM mode with different charge numbers of n. Figure 6.2 shows the effective refractive index (n eff ) of the fundamental mode as a function of the external core radius. On top of n eff of HE n,1 modes in the ring fiber is plotted. At the intersection of the blue curve and red lines, strong coupling between the light in external cores and ring fiber will occur. 64 Figure 6.3. (left) N eff of the symmetric and asymmetric (even and odd) modes composed of HE 3,1 in the ring and HE 1,1 in the cores. White arrows illustrate the electric field vector direction. (right) Group velocity difference of asymmetric and symmetric modes ∆β 1 = β asym -β sym = 1/v g,asym -1/v g,asym . The OAM modes in a single ring waveguide are the HE n,m and EH n,m modes with their electrical fields E z , E φ and E r as having the expression of E r, φ, z (r) exp(±inφ) [5, 7]. Our approach to excite the OAM mode is schematically illustrated in Fig. 1. Four beams are launched into the four external cores to excite the fundamental mode HE 1,1 . When the effective refractive indices (N eff ) of the HE 1,1 mode in the external cores and HE n,1 mode in the ring are matched, the lights in the external cores would couple into the ring to excite the HE n,1 modes. Fig. 6.2 shows N eff of the HE-polarized modes in the ring (red line) and HE 1,1 mode in the external cores as a function of the core radius (blue line) at λ=1550nm, which is calculated by Finite Element Analysis in COMSOL software package. By changing the core size to match the N eff of modes in the ring and cores, one can generate OAM modes with different charge numbers of n. The conditions for the input lights include: (i) The input lights are elliptically polarized. The top and bottom input lights have the same polarization with ellipticity |E x |/|E y |=ε (ii) The left and right inputs have the same polarization with ellipticity |E x |/|E y |=1/ε. (iii) The phase difference ∆Ф between x-polarized and y-polarized E fields is ±90 degrees. ∆Ф=90 0 gives the positive charge number of n>0 while ∆Ф=-90 0 gives the negative charge number of n<0. When the lights are coupled from the external cores into the ring, the input y- polarized E field mainly contributes to even mode HE e m,1 while the input x-polarized E field 65 mainly contributes to the odd modes HE o m,1 . Finally, an OAM mode HE OAM m,1 = HE e m,1 +iHE o m,1 is obtained in the ring. The purpose of using the ring structure and high- contrast index is to remove the N eff degeneracy of HE and EH modes [8]. For example, Fig. 6.3 (a) shows the N eff of symmetric and asymmetric modes composed of HE 3,1 mode in the ring and HE 1,1 mode in the four cores where the insets show the power distribution and E field of the even and odd modes. Fig. 6.3(b) shows the group velocity of the symmetric and asymmetric modes with the external core radius r=2µm. The group velocity mismatch between the symmetric and asymmetric modes ∆β 1 = β asym -β sym = 1/v g,asym -1/v g,asym is zero at λ=1550nm and does not exceed 0.1pm/mm at 1540~1560nm. This is small when the coupling length is about several millimeters. Figs.6. 1-3 illustrate the concept and physics of using an optical fiber coupler with multiple coherent input lights to generate OAM modes at the output, which is different with the physical principle of OAM generation methods in [4,5]. Figure 6.4. The intensity and phase of the azimuthal components of the generated OAM modes with odd charge numbers of +1, -3, +5, -7, and +9 in the central ring, respectively. We use full vector Beam Propagating Method to investigate the performance of the generated OAM modes. Figure 6.4 shows the intensity (Top) and phase (bottom) of the azimuthal component of the electric field at the input and the generated OAM modes with different charge numbers of n at the output. The phase increases clockwise for charge number n>0 while the phase decreases clockwise for n<0. The parameters used in the computation are: (i) the ring size with the inner radius of 5.0 µm, the outer 66 radius of 6.1 µm, and the radius of external cores for generating OAM modes with charge number of ±1, ±3, ±5, ±7 and ±9 are 2.2, 2.0, 1.6, 1.3, and 1.0 µm, respectively; (ii) the input polarization state of ε is 0, 0.3, 0.4, 0.6 and 0.8, respectively; and (iii) The offsets between the cores and ring is chosen to have the coupling length is around 1mm, which can be read out in Fig. 6.6 (b). The wavelength is λ=1550nm. Figure 6.5. Each column shows the OAM weight spectra of the generated OAM modes in the unit of dB with different charge number of n . The top is spectra of HE modes and the bottom is spectra of EH modes. The OAM weight determines the purity of the desired generated OAM mode. The purity of each OAM mode is >0.99 and the crosstalk among these modes is lower than -15dB. To estimate the purity of the generated OAM modes, we calculate the OAM power weight which is defined as ∫∫ = dxdy y x y x F C i i ) , ( ) , ( * ψ and ∑|C i | 2 =1 [9], where F(x,y) is the electric field of the generated OAM mode and ψ i (x,y) is the electrical field of the OAM eigenmode HE i,1 or EH i,1 in a ring. |C i | 2 is the OAM power weight with n=i, which is also known as the OAM spectra [10]. Fig. 6.5 shows the OAM weight spectra (in the unit of dB) of each OAM mode in Fig. 6.4. The top row is the HE-polarized OAM weight spectra and the bottom row is the EH-polarized OAM weight spectra. The desired OAM weight of the generated beam can be >0.99. The crosstalk among the different charges can be lower than -15 dB. This proves that the proposed approach could generate OAM modes with high purity. 67 We investigate the dependence of the OAM mode generation on the external core offset, input light polarization state, and wavelength. Fig. 6 shows the dependence of the coupling length on the offset. The offset is defined as the distance between the external core and the ring as shown in Fig.6. 1. Fig. 6.6 (a) shows the optical power (normalized to input power) in the external core as a function of propagation length for generating OAM mode with n=1. The extinction ratio is defined as ER=10×log 10 [(P 0 -P c )/P c ], where P 0 is the input power in one external core and P c is the power left in the core at the coupling length. Fig. 6.6(b) shows the coupling lengths of n=1, 3, 5, 7, 9 as the functions of the offset. The typical offset and coupling length are ~1.5µm and ~1mm, respectively. Figure 6.6. Dependence of OAM generation performance on the offset of external cores. (a) The light power in the external core as a function of the propagation distance for three offsets. (b) Coupling length as a function of the offset. The performance of OAM mode generation could be enhanced by controlling the polarization state of the input light. Fig. 7 shows the extinction ratio and OAM weight as a function of the polarization state with ellipticity ε=|E x |/|E y |. For each charge number of n, there is an optimized polarization state ε opt of the input light to achieve the maximum extinction ratio. As the charge number n increases from 1 to 9, ε opt also increases from 0 to 0.8. This can be explained as: E φ dominates for HE n,1 mode when n is small. As n increases, |E r |/|E φ | of mode HE n,1 increases. As ε increases, the input 68 lights provide more radial electric field components. When ε is approximately matched with the ratio |E r |/|E φ | of HE n,1 mode, one could obtain the maximum extinction ratio. The OAM weight is not influenced significantly by ε. It can be remained >0.9 in a relatively wide range of ε for each n. Figure 6.7. Dependence of OAM generation on the input polarization state. For some charge number with appropriate polarization, the weight could exceed 99% and the extinction ratio could exceed 20 dB (i.e., the generation efficiency is >0.99). Figure 6.8 shows the dependence of the OAM generation on wavelength by considering the waveguide dispersion. The 10-dB extinction ratio bandwidth for ten modes are 10~17 nm, and the 95% OAM weight bandwidth is 10~40 nm. The input light for each charge number n is set to be the optimized polarization according to results in Fig. 6.7. The performance of higher-order OAM modes are much sensitive to the wavelength variation since the higher-order OAM modes in the ring and the fundamental modes in the cores have much larger waveguide dispersion. As a result, the input light would excite other OAM modes in the vicinity of the desired OAM modes and therefore reduce the weight and extinction ratio of the desired OAM mode. 69 Figure 6.8. Dependence of OAM generation on the input wavelength. The higher-order OAM mode is much sensitive to the wavelength change because of the large waveguide dispersion. In conclusion, we show a new approach to generate OAM modes in a fiber-based coupler, which could be extended to spatial multiplexing and de-multiplexing systems. One may generate several OAM modes by placing more sets of external cores and inputs around the ring. Also this fiber coupler could work in a reciprocal way to de- multiplex OAM modes in the ring by coupling them back to the external cores. Further issues such as introducing the same source into four different external cores with high power efficiency and the group velocity matching of the designed coupler would be studied to achieve an Orbital-Angular-Momentum mode division transmitter. 6.3 Reference 1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman,Phys. Rev. A 45, 8185–8189 (1992). 2. S. Franke-Arnold, L. Allen, and M. Padgett, Laser & Photonics Reviews 2, 299– 313 (2008). 3. G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas'ko, S. Barnett, and S. Franke- Arnold, Opt. Express 12, 5448-5456 (2004). 70 4. D. McGloin, N. B. Simpson, and M. J. Padgett, Applied Optics, 37, 469-472 (1998). 5. P. Z. Dashti, F. Alhassen, and H. P. Lee, Phys. Rev. Lett. 96, 043604 (2006). 6. F. Poletti, X. Feng, G. M. Ponzo, M. N. Petrovich, W. H. Loh, and D. J. Richardson, Opt. Express 19 ,66-80 (2011). 7. Allan Snyder, Optical Waveguide Theory (1983) 8. S. Ramachandran, P. Kristensen, M. F. Yan, Opt. Letters, 34 , 2525-2527 (2009). 9. G. Molina-Terriza, J. P. Torres, and L. Torner, Phys. Rev. Lett. 88, 013601/1-4 (2002). 10. L. Torner, J. P. Torres, and S. Carrasco, Opt. Express 13, 873-881 (2005). 71 Chapter 7 A Novel Fiber Structure to Convert a Gaussian Beam to Higher-Order Optical Orbital Angular Momentum Modes 7.1 Introduction Light beams carrying optical orbital angular momentum (OAM) have a helical azimulthal wavefront phase of 2lπ, in which lℏ corresponds to the topological charge of the OAM states, representing that one photon carries an optical OAM with the value lℏ[1]. Recently, numerous research fields have become increasingly interested in beam carrying OAM, such as atom-light interaction, optical tweezers, photon entanglement, and optical communications [2,3]. A key challenge has been the generation of OAM beams. In free space, this has typically been accomplished by using a hologram [4] or phase plate [5]. Meanwhile, research interest in the generation and transmission of OAM beams and other vortex beams using optical fiber keeps increasing [6-14]. A primary goal could be the direct generation of OAM modes in an optical fiber from a regular fiber-mode input, rather than coupling the free space OAM modes into fiber with additional coupling setup and low coupling efficiency. Generating an OAM mode of | l |=1 in an optical fiber have been demonstrated or proposed by using stressed fiber [6], acoustic-optic interaction [7] and fiber grating [9]. A novel fiber structure [10] and integrated circuits [11] have been used to generate higher-order OAM modes in fiber from multiple coherent inputs. In this Letter, we propose and show, by using simulation, a new approach to converting a Gaussian beam into higher-order OAM modes in a fiber consisting of a square core and a ring refractive index profile. Since the square core breaks the circular symmetry of the waveguide, it allows the input of circularly polarized fundamental mode in the square 72 couple into the ring to generate higher-order OAM modes, performing a conversion between the photon’s spin-angular momentum (SAM) and OAM [15,16]. By designing the parameters of the fiber, one can selectively generate the odd order OAM modes (|l| ranging from 3 to 9, 8 modes total) in the ring from a circularly polarized Gaussian beam. We show that the purity of the generated OAM mode exceeds 96.4% and the extinction ratio reaches 30 dB. The fiber lengths for generation are less than 10mm. 7.2 Design and results (a) (b) Figure 7.1. (a) Cross section of the designed fiber coupler with a central square and a ring structure. (b) Refractive index of different regions of the designed fiber. Fig. 7.1(a) and (b) show the structure and refractive index of the designed fiber. The materials in the fiber consist of: (1) central square: Schott SF1 with n H1 =1.68, (2) ring: SF57 with n H1 =1.80, (3) material between the ring and the square: SF5 with n L1 =1.64, and (4) background: SF2 with n L2 =1.62 (at 1550nm). They were chosen because they have similar thermal and mechanical properties to have compatibility with each other in the fabrication [17]. 73 (a) (b) Figure 7.2 (a) The effective refractive index of the eigenmodes supported by the designed fiber. (b) Symmetric and asymmetric modes of HE 1,1 (core) and EH 3,1 (ring) in the designed fiber. The arrows show the direction of the electric field. OAM modes in a single ring fiber are characterized by HE l,n and EH l,n having electrical fields E z , E φ, and E r, with the expression of E r,φ,z (r)exp(±ilφ)exp(ikz) [10]. The input is a circularly polarized fundamental fiber mode (quasi-HE 1,1 ) in the square region. At some wavelengths the quasi-HE 1,1 mode in the square and the OAM mode (HE l,1 or EH l,1 ) in the ring have almost the same effective refractive index, and thereby the input circular polarized light in the central square will couple into the ring along the propagation and generate an OAM mode. This principle is similar to our previous work, in which the light beams in four external cores couple into a ring fiber and generate an OAM mode [10]. The fiber structure parameters for OAM mode generation of different orders are shown in Table 7.1. Fig. 7.2 shows the effective refractive index of the designed fiber’s eigenmodes composed of the fundamental mode quasi-HE 1,1 (square), EH 3,1 (ring), and HE 5,1 (ring). At two wavelengths λ c,3 =1517nm and λ c,5 =1573nm, the input circular polarized light in the square converts to l=3 EH polarized or l=5 HE polarized OAM modes in the ring along the propagation. Figure 7.2(b) shows the symmetric and asymmetric mode of HE 1,1 (core) and EH 3,1 (ring) when HE 1,1 is linearly polarized. When the input HE c 1,1 =HE x 1,1 ± i HE y 1,1 74 is circularly polarized, the mode in the ring will be EH OAM 3,1 =E r exp(±i3φ)+E φ exp(±i3φ). In addition, we design another fiber with a set of different structure parameters for the generation of the OAM modes of EH 7,1 and HE 9,1 at two other wavelengths (Table 1). EH 3,1 HE 5,1 EH 7,1 HE 9,1 Inner radius r 1 (µm) 4.0 4.0 4.5 4.5 Outer radius r 2 (µm) 4.5 4.5 5.3 5.3 Square width a (µm) 4.0 4.0 5.9 5.9 Coupling length L (mm) 4.0 3.6 7.1 5.6 Coupling wavelength (nm) 1517 1573 1522 1534 Table 7.1. Structural parameters for the OAM mode generation of different orders. Figure 7.3. The intensity and phase of the azimuthal electric field (HE polarized) and radial component (EH polarized) of the electric field of the generated OAM modes. 75 Figure 7.4. Optical power in the square core as the function of fiber length when generating the HE 5,1 OAM mode in the ring. We use the full-vector beam propagating method (BPM) [18] to study the process of generating OAM modes in the designed fiber. Figure 7.3 shows the input fundamental mode intensity and phase in the square core, and the intensity and phase of each generated OAM mode of | l | at their respective coupling lengths L (see Table 1). One can change the sign of the OAM charge l by changing the input’s polarization state (i.e., right-circular or left-circular polarized). The coupling length can be obtained from the effective refractive index difference Δn between the symmetric mode and asymmetric mode by using the relation L=λ c /(2Δn). For OAM mode HE 5,1 generation, we monitor the optical power in the square core at different fiber lengths as shown in Figure 7.4, during which three periods of power oscillation are observed. The most obvious one is due to the HE 1,1 (square) mode coupling with the HE 5,1 (ring) mode. By looking closely at the rapid and small fluctuations on the curve, two other faster power oscillations of different periods can be identified and attributed to the coupling of the square mode’s HE s 1,1 with the ring mode’s HE r 1,1 and EH r 1,1 . If the central core is circular, which means the whole fiber structure is strictly circularly symmetric, only the modes with the same value l can couple with each other between the core and the ring, and the coupling occurs at any wavelength. When the square core breaks the circular symmetry by some degrees, the coupling between the HE s 1,1 (square) mode 76 and the HE r 1,1 and EH r 1,1 (ring) mode still exists. The reason for the faster and smaller power oscillation is the very large effective refractive index Δn difference between the square mode and the ring modes, leading to a very short coupling length L=λ c /(2Δn) and a low extinction ratio. Figure 7.5. Each column shows the OAM charge power weight spectra of the generated OAM modes in units of dB of the different charge number l. The top is the spectra of the HE modes and the bottom is the spectra of the EH modes. To determine the mode purity of the generated OAM beam, we calculate the normalized power weight coefficients of the generated beam in the ring, which is defined as [10] ∫∫ = dxdy y x y x F C l l ) , ( ) , ( * ψ ∑|C l | 2 =1 (7.1) where F(x,y) is the electrical field of the generated beam in the ring, and ψ i (x,y) is the electrical field of the OAM eigenmodes in the ring. |C l | 2 is the normalized power weight of the OAM state l in the generated beam. Figure 7.5 shows the power weight coefficient |C l | 2 of the generated beam in Fig. 7.3, in units of dB. The results show that the mode purity can be above 96.4% (i.e., 17.5 dB higher than other charge numbers). The mode purity and extinction ratio of the OAM generation are evaluated under the wavelength variation as shown in Fig. 7.6. We define the extinction ratio as ER=10×log 10 [(P 0 -P c )/P c ], in which P 0 is the input power into the central square and P c 77 is the minimum power left in the central square at the coupling length L. The extinction ratio of l=3~9 are above 30 dB at their coupling wavelengths λ c,l, . Figure 7.6. The dependence of the (a) mode purity and (b) extinction ratio of the generated OAM modes on the wavelength. We further show the influence of the square core shape on the coupling between the core mode and the OAM mode in the ring. We assume the square has four fillets of radius r f as shown in the inset of Fig. 7.7 (a). As r f increases from 0 to a/2, the central core shape starts as a square and ends as a circle. Figure 7.7 (a) shows the influence of r f on the coupling wavelength λ c and the coupling length L in the generation of OAM mode HE 5,1 . The green dashed curve shows that as r f increases, the coupling wavelength λ c shifts toward the longer wavelength due to the decrease in the effective refractive index of the HE 1,1 in the central core. The blue curve shows as r f increases, the coupling length L increases, which indicates that the coupling effect has become weaker. When r f =a/2=2µm, there’s a sudden jump of coupling length L to infinity. It suggests that the more that the circular symmetry breaks, the stronger the coupling between the mode in the core and the OAM mode in the ring will be. 78 (a) (b) Figure 7.7. (a) Dependence of coupling length L and wavelength λ c on the fillet radius r f of the square core. (b) Dependence of the OAM mode purity (HE l=5) on the input offset d and tilt angle θ. We study the tolerance of the generated OAM mode purity to the unperfected input conditions (i.e., input offset d and input tilt angle θ). Figure 7.7(b) shows the results of l=5 and the HE polarized OAM mode. The input offset d is defined as the distance between the center of the input and the center of the square. The reason for the mode purity degradation is that the imperfect input conditions launch the higher-order mode in the central core that would couple with the ring and excite the other undesired ring modes. The fact that an OAM mode in the ring is generated from the fundamental mode in the square core can be understood by modes coupling effect between two waveguides [10]. We have shown the result of generating OAM modes HE l,n and EH l,n with n=1 and | l| up to 9. In principle, the generation of higher-order OAM mode with larger azimuthal number l and radial number n>1 should be possible if the fiber design enables the refractive index matching between the fundamental mode in the square core and those higher-order modes in the ring. Also, considering that the circular polarized states are associated with a photon’s spin angular momentum, this fiber structure actually performs a conversion between the spin angular momentum (SAM) and the orbital angular momentum (OAM). However, here the mechanism is not same as the spin-orbital conversion in isotropic medium [15,16]. In practice, this fiber 79 structure has a potential use as a compact device that is compatible with current step index fibers to efficiently generate higher-order OAM modes in a ring fiber. 7.3 Reference [1] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992). [2]S. Franke-Arnold, L. Allen, and M. Padgett, “Advances in optical angular momentum,” Laser & Photonics Reviews 2, 299–313 (2008). [3] J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur and A. E. Willner, “Terabit free-space data transmission employing orbital angular momentum multiplexing”, accepted to Nature Photonics, 2012 [4] J. Arlt, K, Dholakia, L. Allen, M. Padgett, “The production of multiringed Laguerre–Gaussian modes by computer-generated hologram”,J. Mod. Opt, 45, 1231, (1998) [5] K. Sueda, G. Miyaji, N. Miyanaga, and M. Nakatsuka, "Laguerre-Gaussian beam generated with a multilevel spiral phase plate for high intensity laser pulses," Opt. Express 12, 3548-3553 (2004) [6] D. McGloin, N. B. Simpson, and M. J. Padgett, Transfer of orbital angular momentum from a stressed fiber-optic waveguide to a light beam,” Applied Optics 37, 469-472 (1998). [7] P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of orbital angular momentum transfer between acoustic and optical vortices in optical fiber,” Phys. Rev. Lett. 96, 043604 (2006). 80 [8] S. Ramachandran, P. Kristensen, and M. F. Yan, “Generation and propagation of radially polarized beams in optical fibers,” Opt. Lett. 34, 2525-2527 (2009). [9] N. Bozinovic, P. Kristensen, and S. Ramachandran, Long-range fibre-transmission of photons with orbital angular momentum. paper CTuB1,CLEO 2011, Maryland. [10] Y. Yan, J. Wang, L. Zhang, J.Y. Yang, I. M. Fazal, N. Ahmed, B. Shamee, A. E. Willner, K. Birnbaum, and S. Dolinar, "Fiber coupler for generating orbital angular momentum modes," Opt. Lett. 36, 4269-4271 (2011) [11] C. R. Doerr, N. K. Fontaine, M. Hirano, T. Sasaki, L. L. Buhl, and P. J. Winzer, “Silicon photonic integrated circuit for coupling to a ring-core multimode fiber for space-division multiplexing,” ECOC 2011, paper Th.13.A.3. [12] Y. Yue, Y. Yan, N. Ahmed, J. Y Yang, L. Zhang, Y. Ren, H. Huang, K.M.Birnbaum, B.I. Erkmen, S. Dolinar, M. Tur, A. E. Willner , “Mode Properties and Propagation Effects of Optical Orbital Angular Momentum (OAM) Modes in a Ring Fiber”, IEEE Photonics Journal, 4, 535, (2012). [13] X. Ma, C. Liu, G. Chang, and A. Galvanauskas, Angular-momentum coupled optical waves in chirally-coupled-core fibers, Optics Express, 19, 26515, 2011 [14] Y. Yue, L. Zhang, Y. Yan, N. Ahmed, J-Y Yang, H. Huang, Y. Ren, S. Dolinar, M. Tur, and A.E. Willner, "Octave-spanning supercontinuum generation of vortices in an As 2 S 3 ring photonic crystal fiber," Opt. Lett 37, 1889, (2012) [15] Y. Zhao, J. S. Edgar, G. D. M. Jeffries, D. McGloin and D. T. Chiu, Spin-to- orbital angular momentum conversion in a strongly focused optical beam, Phys. Rev. Lett. 99 073901, (2007) [ 1 6] L. Marrucci, C. Manzo, and D. Paparo, Optical Spin-to-Orbital Angular Momentum Conversion in Inhomogeneous Anisotropic Media, PRL 96, 163905 (2006) 81 [17] F. Poletti, X. Feng, G. M. Ponzo, M. N. Petrovich, W. H. Loh, and D. J. Richardson, “All-solid highly nonlinear singlemode fibers with a tailored dispersion profile,” Opt. Express 19 ,66-80 (2011). [18] W. P. Huang, C. L. Xu, Simulation of three-dimensional optical waveguides by a full- vector beam propagation method , Quantum Electronics, IEEE Journal of, 29, 2639-2649, (1993). 82 Chapter 8 The Efficient Generation and Multiplexing of Optical Orbital Angular Momentum Modes in a Ring Fiber by Using Multiple Coherent Inputs 8.1 Introduction An optical beam having a helical wavefront phase of 2lπ is associated with an optical orbital angular momentum of lℏ of a photon [1]. Orbital angular momentum (OAM) modes of a different topological charge number l, which can be either a positive or negative integer, are spatially orthogonal to each other. Recently, OAM modes have gained much interest for increasing transmission capacity and spectral efficiency in optical communication systems because of their ability to carry independent data streams on orthogonal modes [2], which could be used in mode- division multiplexing (MDM) in free-space and fiber communication systems [3].The efficient generation and multiplexing of OAM modes has been a key challenge. Generation has typically been accomplished by using spatial-light modulators (SLMs) [4] or other free-space components, which are generally bulky and expensive. The multiplexing of multiple spatial modes is usually achieved by using beam splitters [2,3], which is inherently lossy, especially when the number of modes increase. One of the promising approaches to efficiently generate and multiplex multiple OAM modes without significant loss is to make use of the interference of multiple coherent Gaussian modes in the far field in free space [5]. More recently, following a similar 83 idea, integrated circuits for OAM mode generation in free space by using circular grating have been demonstrated [6,7]. However, the mode purity of the generated OAM state resulting from this approach may not be high enough due to the discontinuity of the input conditions, which may not be preferable to some applications, such as optical communications that require low crosstalk among the spatial channels. In this chapter, a new approach to generate OAM modes of high purity in a ring fiber from multiple coherent inputs from a Gaussian mode is shown by simulation. The purity of the generated OAM modes can exceed 99% within a 2cm long ring fiber. The total optical power loss of the generation and multiplexing of 7 orthogonal modes is 35%. We further show that in an OAM-based mode-division multiplexing system, there is a discrete Fourier transform (DFT) relationship between the data modulation of the input Gaussian mode channels and the data modulation of the output OAM mode channels. 8.2 Design and Results (a) (b) Figure 8.1. (a) The generation of OAM modes of charge number l in a ring fiber with N coherent Gaussian inputs. (b) The refractive index of the ring fiber, which is designed to support OAM l,1 modes with l=-3~3. 84 Figure 8.1(a) illustrates the proposed approach to generate and multiplex OAM modes by using multiple coherent inputs (which could be from a multi-core fiber [8] or a grating coupler [9]) followed by a fiber with a ring refractive index profile [10]. The linear-polarized modes in the fiber can be expressed as OAM l,p =R p (r)exp(ilφ), in which l is the OAM charge number, representing an azimuthal phase change of 2πl, and p is the radial index, indicating there are p intensity peaks in the radial direction. OAM l,p modes with different values of p or l are spatially orthogonal to one another. Figure 8.1(b) shows the refractive index of the ring fiber. The background refractive index is n 1 =1.46 and the refractive index difference is Δn=0.12%×n 1 . The N inputs to the ring fiber are fundamental Gaussian modes in multiple single-mode fibers of the diameter d s and of the same linearly polarized state. They are evenly distributed around a circle of diameter D=(d in +d out ). To generate N (-N/2<l<N/2) orthogonal modes of high purity, the parameters of the designed ring fiber are required to have two properties: (i) only OAM l,p modes with p=1 can be supported by this fiber, which means it is a “single-mode” fiber in the radial direction; and (ii) the number of the total azimuthal modes is not larger than N (i.e., -N/2≤ l≤ N/2). Under these two conditions, when the N coherent inputs of the same power and polarization have the phase relationship of ΔФ=Ф n -Ф n-1 =2πl/N, a linearly polarized OAM l,1 mode will be generated. Here we show, by using simulation, the generation 7 OAM modes of l ranging from -3 to +3 when N=8, d out =44µm, d in =32µm, n 1 =1.46, Δn=0.12%×n 1 =0.002, D=38µm, and d s =8µm. At 1550nm, the designed fiber only supports OAM l,1 from l=-3 to 3. The modes of l>3 are cut off. 85 (a) (b) 86 (c) Figure 8.2 (a). Left: Intensity of 8 Gaussian input modes. Right: The output intensity and phase patterns of the generated OAM l,1 modes with l=0~3 and LP 4,1 mode. (b) The normalized optical power of the generated OAM modes with l=0~3 and LP 4,1 mode in a 2 cm long ring fiber. (c) Comparison of the mode purity and crosstalk of the generated OAM modes in a ring fiber or a multi-mode fiber with a step refractive index. We use the full-vector beam propagation method (BPM) [11] to simulate the generation of the OAM mode of a different l in the ring fiber by controlling the phase difference of ΔФ=2πl/N. The fiber length in the simulation is 2cm. Figure 8.2(a) shows the intensity of 8 coherent inputs of a fundamental fiber mode at the input of the ring fiber (left), and the intensity and phase of the generated OAM l,1 when l=0~3, and the superposition of OAM +4,1 and OAM -4,1 having the expression R(r)cos(4×φ) when l=4 at the output of the ring fiber (right). Figure 8.2(b) shows the normalized total power in the ring fiber for generating OAM modes of a different l. The power for generating OAM l,1 modes of l=0~3 decreases at first, and is then sustained at a constant level. In contrast, the power of the cut-off mode OAM +4,1 +OAM -4,1 keeps falling along the propagation. We then calculate the mode purity of the generated beam, which is 87 determined by the overlap integration of the electrical field of the generated beam and the linearly polarized modes supported by the ring fiber. * ,, (, ) (, ) lp lp C F x y x y dxdy ψ = ∫∫ (1) where the F(x,y) is the normalized electric field of the generated beam. Ψ l,p (x,y) is the normalized electric field of the linearly polarized mode supported by the ring fiber, which has the expression of ψ l,p =R p (r)exp (ilφ). |C l,p | 2 is the power weight of each eigenmode in F(x,y) and ∑|C l,p | 2 =1, which could be used to determine mode purity [12]. As shown in Fig. 8.2(c), the mode purity of all the generated OAM l,1 modes is above 99%. To see how the conditions (i) and (ii) of the fiber design are crucial for the generation of the high-purity OAM modes, we replaced the ring fiber with a regular step-index multi-mode fiber and then performed the simulation and calculated the mode purity again. The diameter of the step-index fiber is the same as the outer ring diameter d out =44µm. The refractive index is Δn= 0.36% n. The step-index fiber totally supports 38 spatial modes of | l | up to 8 and | p | up to 4. As shown in Fig. 8.2(c), the purity of the OAM l,1 modes obtained by a step fiber is much lower (i.e., <60%) because the inputs also excite many other undesired modes supported by the step fiber. The lower mode purity leads to higher crosstalk among the spatial modes. Figure 8.2(c) also shows the comparison of mode crosstalk defined as 10×log(1-|C m,n | 2 ). When using the ring fiber, the crosstalk is <-30dB, while the step fiber crosstalk is >-5dB. The principle of generating high-purity OAM modes through this approach can be explained as follows: due to the periodic input condition Φ m -Φ m-1 =2πl/N (|l|<N/2), the input field is the superposition of the OAM modes with charge l, l±N, l±2N,···l±mN,···. Since the ring fiber is designed to support OAM modes with a charge of |l|<N/2 only, those higher-order modes l±N, ···l±mN··· are cut off in the ring fiber and their power dissipates along the propagation. Actually, the ring fiber acts as a spatial mode filter to maintain the desired OAM modes and to filter out the higher- order undesired modes, resulting in high purity of the generated OAM modes. Figure 88 8.3 shows an example of l=-3, which shows the power of the ring fiber’s OAM states at (a) z=0cm and (b) z=2cm. The ring fiber helps increase the mode purity ~40%. Figure 8.3. The power weight of the OAM state l at (a) the ring fiber input z=0cm and (b) ring fiber output z=2cm. We further show the tolerance of the generated OAM l,1 mode purity to the imperfect launching conditions. Figure 8.4(a) shows the generated OAM mode purity and crosstalk dependence on the offset (i.e., misalignment) between the centers of the ring fiber and the center position of the 8 inputs. Figure 8.5(b) shows the dependence on the phase error of the inputs. As illustrated in the inset, we assume that the phase of the N inputs has deviation α and –α alternately from the exact phase of Φ m =2πml/N that they are supposed to be. One can see that the purity and crosstalk of higher-order OAM modes are more sensitive to the imperfect launching conditions than the lower- order ones. 89 Figure 8.4 (a) The dependence of the generated OAM mode (l= 0~3) purity and crosstalk on the (a) offset between the centers of the ring fiber and the multiple inputs and (b) the phase error of the inputs. By controlling both the amplitude and phase of the multiple inputs, this approach can generate multiple OAM modes simultaneously without further power loss in the multiplexing stage. For example, if we try to generate and multiplex 7 modes of the same power, from Figure 8.2(b) we can calculate that the total loss is only ~35%, when compared with the multiplexing scheme using beam splitters [2,3]. Our scheme shows good scalability in terms of power loss (~[1-(1/2) N-1 ]×100%), especially when the mode number N increases. 90 Figure 8.5. The system diagram of an OAM mode-based MDM transmitter using multiple coherent inputs that generate and multiplex multiple OAM modes, in which each one carries independent data D l . D: the transmitted independent data channel from different OAM modes; M: modulator; d: data from each coherent input. Figure 8.5 shows a transmitter diagram in an OAM-based mode-division multiplexing system, which uses multiple coherent inputs to generate and multiplex N OAM modes in which each one carries an independent data channel D l (t). First, the light from a single source is split into N paths by a 1xN coupler. In order to generate mode OAM l,1 carrying data D l (t), the m th input is exp(iΦ m )·D l (t)=exp(i2πml/N)·D l (t). For multiplexing N spatial modes, the m th input should be /2 /2 1 ( ) exp( 2 / ) ( ) N ml lN dt i lmND t π =− + = ∑ (2) where the vector d l (t)= [d 0 , d 1 , … , d N/2 , d -1 , … d -(N/2-1) ] T is actually a discrete Fourier transform (DFT) of the vector D l (t)= [D 0 , D 1 , … , D N/2 , D -1 , … D -(N/2-1) ] T . The DFT can be performed by the digital signal processing (DSP) in the electrical domain and control modulators to generate N input optical signals. These N inputs carrying d l (t) 91 are mixed into the ring fiber and then evolve to multiple orthogonal OAM l , 1 mode carried data D l (t). Mathematically, the ring fiber performs an inverse DFT operation, which transforms d l (t) on the spatial basis of N separated Gaussian modes back to D l (t) on the spatial basis of OAM l,1 . 8.3 Reference [1] L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185–8189 (1992). [2] J. Wang, J. Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. Huang, Y. Ren, Y. Yue, S. Dolinar, M. Tur and A. E. Willner , accepted to Nature Photonics, 2012 [3] R. Ryf, S. Randel, A. H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, René-Jean Essiambre, P. J. Winzer, D. W. Peckham, A. H. McCurdy, and R. Lingle., J. Lightwave Technol. 30, 521–531 (2012). [4]J. Arlt, K, Dholakia, L. Allen, M. Padgett, J. Mod. Opt, 45, 1231, (1998) [5] B. Thidé, H. Then, J. Sjöholm, K. Palmer, J. Bergman, T. D. Carozzi, Ya. N. Istomin, N. H. Ibragimov, and R. Khamitova, Phys. Rev. Lett. 99, 087701 (2007) [6] C. R. Doerr, N. K. Fontaine, M. Hirano, T. Sasaki, L. L. Buhl, and P. J. Winzer, ECOC 2011, Th.13.A.3. [7] Tiehui Su, Ryan P. Scott, Stevan S. Djordjevic, Nicolas K. Fontaine, David J. Geisler, Xinran Cai, and S. J. B. Yoo , Optics Express, 20, 9396-9402, 2012. [8] B. Zhu, T. F. Taunay, M. F. Yan, J. M. Fini, M. Fishteyn, E. M. Monberg, and F. V.Dimarcello, Optics Express, 18 11117-11122 (2010) 92 [9] N. K. Fontaine, C. R. Doerr, M. A. Mestre, R. Ryf, P. Winzer, L. Buhl, Y. Sun, X. Jiang, and R. Lingle, Optical Fiber Communication Conference (OFC), PDP5B, 2012 [10] S. Ramachandran, P. Kristensen, and M. F. Yan, Opt. Lett. 34, 2525-2527 (2009) [11] W. P. Huang, C. L. Xu, Quantum Electronics, IEEE Journal of, 29, 2639-2649, (1993). [12] Y. Yan, J. Wang, L. Zhang, J.Y. Yang, I. M. Fazal, N. Ahmed, B. Shamee, A. E. Willner, K. Birnbaum, and S. Dolinar, Opt. Lett. 36, 4269-4271 (2011)

Abstract (if available)

Abstract One property of electromagnetic (EM) waves that has recently been explored is the ability to multiplex multiple beams carrying orbital angular momentum (OAM) such that each beam has a unique helical phase front. Such OAM-based multiplexing can potentially increase the system capacity and spectral efficiency of RF wireless and optical fiber communication links by transmitting multiple coaxial data streams. My dissertation will introduce the basic concept of OAM and the principle of using OAM for spatial multiplexing in RF/optical communications and my research in this area, including: high capacity mm-wave communications using OAM multiplexing, multipath effects of wireless OAM communications, multicasting network function in optical OAM communications, and different designs to generate and multiplexing OAM channels in optical fibers.

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Creator Yan, Yan (author)

Core Title High capacity optical and RF/mm-wave communications utilizing orbital angular momentum multiplexing

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 07/25/2016

Defense Date 04/04/2016

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

Tag mm-wave communications,multicasting,OAI-PMH Harvest,optical communications,optical fiber,orbital angular momentum,RF communications,spatial division multiplexing

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Advisor Willner, Alan E. (committee chair), Armani, Andrea M. (committee member), Molisch, Andreas F. (committee member)

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