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Orbital angular momentum based spatially multiplexed optical and millimeter-wave communications

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Orbital angular momentum based spatially multiplexed optical and millimeter-wave communications

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Content ORBITAL ANGULAR MOMENTUM BASED SPATIALLY MULTIPLEXED OPTICAL AND MILLIMETER–WAVE COMMUNICATIONS by Nisar Ahmed 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) May 2016 Copyright 2016 Nisar Ahmed To my parents and my wife, for their love and support ii Acknowledgments I would like to express my deepest gratitude for the guidance and support of my adviser and dissertation committee chair, Prof. Alan Willner. Without his phenomenal mentorship and persistent encouragement, the work presented in this dissertation would not have been possible. I have learned many personal lessons which will prove to be important in my professional career. I would also like to extend my deepest appreciation to Prof. Andreas Molisch forhisguidanceduringmydoctoralstudiesandbeingamemberonmydissertation and qualifying exam committees. My sincerest thanks to Prof. Stephan Haas for serving on my dissertation committee, and Prof. Alexander Sawchuck, Prof. Todd Brun, and Prof. Andrea Armani for being the members of my qualifying exam committee. I sincerely thank Prof. Moshe Tur from Tel Aviv University for taking personal interest and sharing his vast experience during my research. I am deeply indebted to Prof. Imran Hayee for his support and guidance throughout my grad school studies. I also acknowledge the support of Dr. Samuel Dolinar, Dr. Kevin M. Birnbaum, and Dr. Baris I. Erkmen from JPL for their guidance during my doctoral studies. Many thanks to my wonderful colleagues and researchers at the Optical Communications Lab (OC Lab) for their tremendous help in my research, iii Irfan Fazal, Hao Huang, Salman Khaleghi, Omer Faruk Yilmaz, Jeng-yuan Yang, Guodong Xie, Bishara Shamee, Morteza Ziyadi, Asher Maor, Mohammad Chitgarha, Yongxiong Ren, Yan Yan, Yang Yue, Zhe Zhao, Amirhossein Ariaei, Ahmed Almaiman, Long Li, Changjing Bao, Yinwen Cao, Peicheng Liao, Zhe Wang, Lin Zhang, Scott Noccio, Xiaoxia Wu, Martin Lavery, Jian Wang, and Antonella Bogoni. I also appreciate the help offered by the EE staff Diane Demetras, Anita Fung, Corine Wong, Gerrielyn Ramos, and Tim Boston. iv Contents Acknowledgments iii List of Figures vii Abstract x 1 Introduction 1 1.1 Generation, (De)Multiplexing, & Detection of OAM Beams . . . . . 2 1.2 OAM–based Optical Networking Functions . . . . . . . . . . . . . . 5 1.3 Optical & Millimeter–wave Bessel Beams . . . . . . . . . . . . . . . 6 1.4 Emulation of Free–space Distance . . . . . . . . . . . . . . . . . . . 7 1.5 Apodized Apertures for OAM Beams . . . . . . . . . . . . . . . . . 7 2 OAM–based Add–Drop Multiplexer 9 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Concept of the Add–Drop Multiplexer . . . . . . . . . . . . . . . . 10 2.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 12 3 OAM-based 2×2 Switch 15 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 Concept of the 2×2 OAM–switch . . . . . . . . . . . . . . . . . . . 16 3.3 Mode Up & Down Conversion . . . . . . . . . . . . . . . . . . . . . 17 3.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 Mode–division Multiplexing of Bessel–Gaussian Beams 23 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 Concept of Obstruction–tolerant Links . . . . . . . . . . . . . . . . 25 4.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 v 5 Millimeter–wave OAM–carrying Bessel Beams 37 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.2 Millimeter–wave Obstruction Tolerant Links . . . . . . . . . . . . . 38 5.3 Generation of Millimeter-wave Bessel Beams . . . . . . . . . . . . . 41 5.4 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 5.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 45 6 Emulation of Free–space Distance 50 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 6.2 Emulator Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 6.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 6.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 55 7 Apodized Apertures for OAM Beams 57 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 7.2 Concept of Apodized Apertures . . . . . . . . . . . . . . . . . . . . 58 7.3 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 7.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . 59 References 62 vi List of Figures 1.1 Transverse intensity and phase profiles of different OAM beams. . . 2 1.2 Concept of combining OAM multiplexing with WDM and PDM . . 2 1.3 OAM generation using Spiral Phase Plate . . . . . . . . . . . . . . 3 1.4 Block diagram of OAM generation using SLM . . . . . . . . . . . . 4 1.5 Concept of OAM–multiplexed data channels . . . . . . . . . . . . . 4 1.6 Detection of OAM–multiplexed data channels . . . . . . . . . . . . 5 2.1 Concept of the add–drop multiplexer for OAM beams . . . . . . . . 10 2.2 Experimental setup for the OAM add–drop multiplexer . . . . . . . 11 2.3 Phase masks and measured intensity profiles . . . . . . . . . . . . . 13 2.4 Measured BER for add–drop ` = +6 and -6 . . . . . . . . . . . . . 13 2.5 Measured BER for add–drop ` = +8 and -8 . . . . . . . . . . . . . 14 3.1 Concept of the 2×2 switch for OAM multiplexed channels . . . . . 16 3.2 Functional block diagram of the 2×2 OAM-based switch . . . . . . 17 3.3 Experiment setup for the 2×2 OAM–based switch . . . . . . . . . . 18 3.4 BER measurements for the output port A of the 2×2 switch . . . . 20 3.5 BER measurements for the output port B of the 2×2 switch . . . . 21 3.6 Interferograms of different OAM beams after the 2×2 switch . . . . 21 3.7 Measured crosstalk among different OAM beams after 2×2 switch . 22 vii 4.1 Concept of a link using multiplexed Bessel–Gaussian beams . . . . 25 4.2 Experimental setup for optical Bessel–Gaussian beams . . . . . . . 26 4.3 Measured intensity cross-sections of Bessel–Gaussian beams . . . . 27 4.4 Pictorial depiction of an obstructed BG beam . . . . . . . . . . . . 28 4.5 Intensity profiles of obstructed and unobstructed BG beam . . . . . 30 4.6 Received power as a function of normalized obstruction position . . 32 4.7 Crosstalk among different obstructed beams . . . . . . . . . . . . . 33 4.8 Measured BER for the multiplexed Bessel–Gaussian beams . . . . . 34 5.1 Diffraction effects on obstructed Gaussian and OAM beams . . . . . 39 5.2 Intensity and phase profiles of Bessel beams . . . . . . . . . . . . . 40 5.3 Simulated evolution of different obstructed Bessel beams . . . . . . 41 5.4 Design parameters of metamaterials–based conical lens . . . . . . . 42 5.5 Schematic structure of the metamaterials–based conical lens . . . . 43 5.6 Experiment setup for mm–wave obstruction–tolerant link . . . . . . 45 5.7 Received power vs. normalized obstruction position . . . . . . . . . 46 5.8 BER as a function of received SNR for OAM channel ` = +1 . . . . 47 5.9 BER as a function of received SNR for OAM channel ` = +3 . . . . 48 5.10 Measured intensity of obstructed and unobstructed Bessel beams . . 49 6.1 Concept of a Free-space emulator . . . . . . . . . . . . . . . . . . . 51 6.2 Numerical and analytic results for the Free–space emulator design . 53 6.3 Numerical simulation results for mode purity calculations . . . . . . 53 6.4 Experimental setup for the Free–space Emulator . . . . . . . . . . . 54 6.5 Measured intensity profile of LG beams . . . . . . . . . . . . . . . . 55 6.6 Results for different LG beams propagated over 5 and 25 m . . . . . 56 6.7 Experimentally retrieving phase of the LG beams . . . . . . . . . . 56 viii 7.1 Concept and experiment setup for the apodized apertures . . . . . . 59 7.2 Received power in the desired and neighboring modes . . . . . . . . 60 7.3 Received power in different modes . . . . . . . . . . . . . . . . . . . 60 7.4 Measured BER for different multiplexed OAM modes . . . . . . . . 61 ix Abstract Increasing capacity demands on optical networks have encouraged the researchers to explore new dimensions for multiplexing optical data channels. While wavelength–division multiplexing (WDM) and polarization–division multiplexing (PDM) have been extensively deployed in existing networks, one recently explored approach to significantly increase the capacity of communication systems is to employ spatial–division multiplexing (SDM) of multiple orthogonal modes using a single transmitter/receiver aperture pair, in which each mode carries an independent data stream. Orthogonality ensures that the modes can be efficiently multiplexed at the transmitter, spatially co-propagated, and demultiplexed at the receiver with minimal modal crosstalk. Of particular interest are the SDM techniques that multiplex data channels using optical beams that carry orbital angular momentum (OAM). In recent years, OAM has emerged as a potential approach to multiplex many spatially collocated optical data-carrying beams. Since OAM beams with different mode orders are orthogonal to each other, it is possible to efficiently multiplex and de– multiplex data channels carried by such beams. Recent optical and millimeter– wave(mm–wave)demonstrationshaveshownthatstructuredbeamscarryingOAM can serve as a modal basis set for enhancing the capacity and spectral efficiency of free–space communication systems. x The research work presented in this dissertation focuses on enabling OAM multi- plexed communication systems for high–speed and high–capacity optical and mm– wave communications, and includes analysis and experimental demonstrations of: (1) optical networking functions for OAM multiplexed data channels; (2) mode– division multiplexing of Bessel beams for obstruction–tolerant optical, and (3) millimeter–wave links; (4) emulation of free–space distance for OAM beams; and (5) apodized apertures to reduce crosstalk among neighboring modes. xi Chapter 1 Introduction In 1992, it was discovered that OAM can be carried by an electromagnetic (EM) beam if its phase front ‘twists’ in a helical fashion [1]. The wavefront of an OAM– carrying optical beam is described by its azimuthal phase dependence given bye i`θ , in which` is an integer describing the azimuthal mode order and θ is the azimuth angle. A characteristic property of the helical wavefront is that it twists along the direction of propagation creating a phase singularity on the propagation axis, which in turn manifests as a central intensity null. Such beams are also commonly known as ‘vortex’ beams [2]. The value of azimuthal mode order` describes the amount of OAM and defines the number of 2π phase shifts that occur in the azimuthal direction. Figure 1.1 shows few examples of transverse intensity and phase profiles of different OAM beams. As shown in this figure, the OAM beams with similar ` values but with different signs have same size of the intensity ring, however, wavefront have either right–handed or left–handed twisting arms. OAM beams with different mode orders (i.e. different ` values) form an orthogonal modal set such that each beam can carry independent data channel [3–6]. 1 -π π -π π -π π -π π -π π l = 0 l = +1 l = +2 l = -2 l = -1 Transverse Intensity Transverse Phase Figure 1.1: Transverse intensity and phase profiles of different OAM beams. Owing to its orthogonality, multiple OAM beams can therefore be used to spatially multiplex distinct data channels. Moreover, to achieve ‘multi–dimensional’ multiplexing, OAM multiplexing can be combined with other types of well known multiplexing schemes such as wavelength–division multiplexing (WDM) and polarization–division multiplexing (PDM) thereby allowing for higher system capacity and spectral efficiency [7]. The concept of combining different multiplexing techniques is depicted in Fig. 1.2. 90 o Y X X&Polarization Y&Polarization ... λ 1 λ 2 λ N λ 3 90 o Y X ... λ 1 λ 2 λ N λ 3 X&Polarization Y&Polarization Wavelength & Polarization Multiplexing Wavelength, Polarization & OAM Multiplexing Figure 1.2: Concept of combining OAM multiplexing with wavelength–division and polarization–division multiplexing techniques. 1.1 Generation, (De)Multiplexing, & Detection of OAM Beams In order to generate OAM beams, various methods are available which include both intra–cavity and mode transformation methods [8]. For the work presented 2 in this dissertation, OAM beams were generated by using the mode transformation methods as described below. Mode transformation methods convert the mode of a fundamental Gaussian beam into a desired mode using beam shaping methods. Two widely used mode transformation methods employ either spiral phase plate (SPP) or liquid–crystal– on–silicon (LCoS) spatial light modulators (SLM) [2,8]. Figure 1.3 shows the concept of using an SPP for OAM generation. The SPP is designed for a specific wavelength and it has azimuthally varying thickness such that its height can be given by [8]: Δh ` =`λθ/2π(n− 1) (1.1) in which, h is the height of the SPP, n is the refractive index of the material, and λ is the design wavelength. Such an SPP can be placed in the path of a plane wavefront to transform it into a helical wavefront. Similarly, same SPP can be used in reverse direction to transform an incoming helical wavefront with opposite sign of `, into a plane wavefront. While SPP are fixed optical elements (i.e. work only for a specific OAM mode andthewavelengthforwhichitisdesigned), SLMsareprogrammableLCoSdevices Antenna SPP ! = -3 OAM !"= -3 Spiral Phase Plate (SPP) Δh R Figure 1.3: OAM generation using Spiral Phase Plate. 3 -6 -4 -2 0 2 4 6 #10 -3 -6 -4 -2 0 2 4 6 #10 -3 -3 -2 -1 0 1 2 3 SLM with phase mask l = +3 Fourier Lens Input Gaussian Beam l = 0 Output OAM Beam l = +3 Intensity Phase Intensity Phase Figure 1.4: Block diagram of OAM generation using SLM. providing the ability to be reconfigured. Figure 1.4 depicts the concept of OAM generation using an SLM. An incoming Gaussian beam (` = 0) is passed through an SLM programmed with phase mask ` = +3. A Fourier lens placed after the SLM transforms the incoming Gaussian beam into an OAM beam with ` = +3 in its Fourier plane. Owing to the reciprocity of the process, similar setup can be used in reverse to transform a incoming OAM beam into a Gaussian beam. Inordertomultiplexdata–carryingOAMbeams, acascadedstructureofbeam- splitters can be used (see Fig. 1.5). A continuous–wave (CW) laser is modulated using desired modulation format and amplified using an optical amplifier. The SLM 1 PC 90 o$ PRBS Laser EDFA BPF SMF Ch. 1 Ch. 2 PC Col. BS λ/2 SLM 2 SLM N-1 SLM N Ch. N OAM Multiplexed Data Channels Modulator Figure 1.5: Concept of multiplexing OAM data channels using cascaded beamsplitters. PC: polarization controller, PRBS: pseudo random bit sequence, BPF: band–pass filter, Col: collimator, BS: beamsplitter 4 SMF BPF OSA$ 1% 90 o$ Hybrid$ LO EDFA EDFA Col. D S P BS OAM Multiplexed Data Channels SLM Figure 1.6: Detection of multiplexing OAM data channels. OSA: optical spectrum ana- lyzer, LO: local oscillator, ADC: analog–to–digital converter, DSP: digital signal pro- cessing amplified signal is first band–pass filtered and then split into multiple branches. In each branch, signals are de–correlated using a piece of single mode fiber (SMF) to introduce delay in each path. The signals from each branch are then coupled into free–space and reflected off the SLM programmed with phase mask of desired OAM mode. The generated OAM modes are then combined using cascaded beam- splitters such that the output of the last beamsplitter is a superposition of all the signals. At the receiver end, multiplexed OAM channels are demultiplexed one at a time using an SLM that is programmed with the inverse phase mask to select desired OAM mode. As described above, the purpose of the inverse phase mask is to convert OAM mode back into a fundamental Gaussian beam such that it can be coupled into an SMF. The signal coupled into the fiber is sent to the optical receiver for bit-error rate (BER) measurements (see Fig. 1.6). In the remainder of this chapter, a brief overview of the research work presented in this dissertation is provided. 1.2 OAM–based Optical Networking Functions Although high bit–rate, free-space, and fiber links have been demonstrated in a point–to–point setup using multiple spatially overlapping OAM beams, many 5 tools are required to implement reconfigurable optical networks in a multiuser environment. Thisissimilarinconcepttotheadvantagesthatwavelength-selective networking and switching historically provided on top of simple wavelength multiplexed high-capacity, point-to-point links [9–12]. Two experimental demonstrations of optical networking functions for OAM multiplexed channels are presented in this dissertation. In Chapter 2, an experimental demonstration of an add-drop multiplexer for OAM–multiplexed 100 Gb/s QPSK channels is presented [13]. Chapter 3 describes a 2×2 OAM based optical switch capable of switching OAM-multiplexed 50 Gbaud QPSK channels [14]. 1.3 Optical & Millimeter–wave Bessel Beams Free-spacetransmissionlinksgenerallyrelyonline-of-sight(LoS)operation[15–17]. Completely blocking the beam path may result in link outages. However, the link might still be able to function if the beam path is partially blocked. In the case where multiple optical or mm–wave OAM multiplexing is implemented, an interestingquestionarisesastotheeffectofanobstructionthatpartiallyblocksthe beams. For example, the phase-front of a wave becomes critical in differentiating the different data channels, and a partial obstruction may increase the crosstalk between different data channels. One potential approach to such an issue could be using “propagation- invariant” Bessel beams to tolerate partial obstructions [18]. Chapter 4 describes an experimental demonstration of a partially obstructed optical link that utilizes two multiplexed Bessel–Gaussian beams each carrying 50 Gbaud QPSK channels [19]. In chapter 5, we present self–healing of the multiplexed mm–wave OAM carrying Bessel beams and evaluate the link 6 performance in terms of power penalty and crosstalk among neighboring channels [20]. 1.4 Emulation of Free–space Distance Under free-space propagation over longer distances, OAM beams diverge and evolve to a larger beam spot. For some applications, e.g., communications, it is important to experimentally show different effects on beam evolution under free-space propagation. Unfortunately, this is difficult in a lab environment for anything more than a few meters, and yet many important types of experiments require extensive lab equipment. A laudable goal would be to have a free-space emulator for lab use that can correctly reproduce the intensity and phase charac- teristics of a propagating beam over longer distances. Chapter 6 describes a free-space emulator that can emulate free–space propagation of different distances for OAM beams [21]. We propagate OAM beams of different mode orders through the emulator and measure the spot size and curvature of the beams at the output. 1.5 Apodized Apertures for OAM Beams In an ideal system, receivers with sufficiently large apertures are needed such that the entire field is captured, thereby ensuring the recovery of the highest signal power for good signal-to-noise ratio, and the most phase change across the beam for good modal separation and reduced crosstalk. However, given beam divergence and limited aperture sizes, the entire beam may not be readily recoverable. There may be cases in which the aperture is not located at the beam center (i.e., not at the location of highest OAM phase change) in order to increase the total signal power being detected. In such a scenario, the limited aperture will distribute light 7 from the input OAM beams into neighboring OAM beams, producing potential crosstalk with other incoming data-carrying beams [22,23]. Just like with other types of apertures for which apodization might reduce negative effects, there exists a question as to the benefit of using an apodized aperture for OAM recovery. Chapter7explorestheuseofanapodizedapertureforreceivingadata-carrying OAM beam [24]. The performance of apodized aperture is compared with a hard aperture by varying aperture sizes and locations. 8 Chapter 2 OAM–based Add–Drop Multiplexer 2.1 Introduction Reconfigurable networking functions have been commonly used in traditional time- divisionmultiplexing(TDM)andWDMnetworks. Thusfar,communicationsusing OAMhavefocusedonestablishinghighcapacitystatic, point-to-pointtransmission linkssuchthatasinglelinkisemployedbetweenthetransmitterandreceiver [3–8]. However, similar to the historical advancements made in WDM systems [9–12], reconfigurable manipulation of spatially overlapping OAM data channels for a multi-user environment might extend the usefulness of OAM-based communication systems. This chapter describes experimental demonstrations of an OAM-based add-drop multiplexer [13]. 9 2.2 Concept of the Add–Drop Multiplexer Theconceptoftheadd–dropmultiplexerforOAMmultiplexedchannelsisshownin Fig. 2.1. Two multiplexed OAM beams with topological chargesl = +1 and -1 are sent to the spatial add-drop multiplexer. In mode down-conversion block, an SLM is programmed with the phase mask having topological charge m, where m has opposite sign to that of the mode to be dropped. Mode down-conversion process transforms one of the incoming OAM beams (a donut-like transverse intensity profile with helical wave front) into a Gaussian-like beam with l = 0 (a spot-like transverse intensity profile with planar wave front). The Gaussian-like non-OAM beam is then spatially filtered to separate it from the passthrough beam and is sent to the drop port. After reflecting off the SLM in the down-conversion process, the topological charge of the passthrough beam also changes and is given by (l+m). In the passthrough branch, another spatial filter is used to allow only the passthrough beams in that direction. The “add” channel on a Gaussian beam is then spatially Add–Drop Multiplexer Input Multiplexed OAM Beams Down Conversion Up Conversion Dropped Mode Added Mode SLM SLM Output Multiplexed OAM Beams Figure 2.1: Concept of the add-drop multiplexer for OAM beams. An SLM is used to down-convert the OAM beam carrying the data channel to be dropped. Passthrough channel is spatially filtered and combined with the add channel. 10 multiplexed with the passthrough OAM beam. These multiplexed beams are up– converted to their original mode orders by reflecting off an SLM programmed with the phase mask having topological charge opposite to that of the SLM in down– conversion block. 2.3 Experimental Setup The experimental setup is shown in Fig.2.2. A 100-Gbit/s RZ-QPSK signal at 1550 nm is split into two paths and is de-correlated by using a piece of fiber. Both signals are then launched into two collimators to shine two Gaussian beams (w o = 1.5 mm) onto SLM-1 and SLM-2, programmed with the phase masks to generate two OAM beams with topological charges of opposite signs. Both of the beams are multiplexed using a beam splitter and passed through a Fourier transforming lens. The two multiplexed modes are then reflected off SLM-3, which acts as a mode EDFA BPF EDFA OSA 1% tap Att. BS SLM-‐1 HWP Fiber SLM-‐2 HWP Col. SLM-‐3 HWP QPSK Tx. Col. Camera SLM-‐4 SLM-‐5 HWP HWP Col. SF-‐2 SF-‐1 BS EDFA I/Q Mod. BPF PC 2 x 50 Gbit/s MZM PC 50 GHz Clock QPSK OC 1x2 Fiber Col. f Coherent Detection f BS f f Col. QPSK Transmitter Polarization Diversity 90 0 -‐Hybrid ADC ADC ADC ADC Off-‐line DSP Coherent Detection LO Generation, Multiplexing, Add/Drop and Demultiplexing of OAM Modes SF-‐1 SF-‐2 Figure 2.2: Experimental setup for the OAM add–drop multiplexer. QPSK transmitter block shows the setup for generation of 100-Gbit/s RZ-QPSK. Free-space optics block illustrates the setup for generation, multiplexing, down-conversion, up-conversion and demultiplexing of OAM modes. Coherent detection block shows the setup for coherently detecting 100-Gbit/s RZ-QPSK signals. PC: polarization controller; BPF: band-pass filter; MZM: Mach-Zehnder Modulator; OC: optical coupler; Col.: collimator; HWP: half-wave plate; SLM: spatial light modulator; BS: beam splitter; SF: spatial filter; f: lens focal length; Att. Attenuator; LO: local oscillator. 11 down-converter. After reflection from SLM-3, the drop beam becomes a non-OAM, Gaussian-like beam in the Fourier plane of a lens. The topological charge of the pass-through beam also changes but remains an OAM beam. The beam is then split into two branches using a beam splitter. In the drop branch, spatial filter (SF- 1) allows only the down-converted beam, which is then coupled to the fiber and sent to the coherent detection setup. In the passthrough branch, the passthrough beam is also spatially filtered using spatial filter (SF-2) and multiplexed with the Gaussian beam carrying the data channel to be added using a beam-splitter. These multiplexed modes are then shone onto SLM-4, which acts as a mode up- converter. SLM-5 is used to spatially demultiplex beams at the receive end. The demultiplexed beam is then sent to the coherent detection setup. Please note that the topological charges of the input multiplexed beams are maintained at the output port of the spatial add-drop multiplexer. 2.4 Results and Discussion To demonstrate the functionality of the OAM–based add–drop multiplexer, we first multiplexed OAM beams with ` =+6 and -6 each carrying the 100 Gbit/s RZ–QPSK channel. The phase masks to generate such beams are shown in Fig- ure 2.3(a1)–(a2). The remaining panels of the top row in the same figure (i.e. Fig- ure 2.3(a3)–(a7)) show the transverse intensity profiles of the multiplexed beams, down–converted beams, only the passthrough beam, passthrough beam with the added beam, and the up–converted beams at different locations in the add-drop setup, respectively. We then repeat the experiment by sending multiplexed ` = +8 and -8 beams. The phase masks to generate these beams and the transverse intensity profiles at different locations in the setup are shown in the bottom panels of Figure 2.3 (i.e. Figure 2.3(b1)–(b7)). 12 Figure 2.3: Phase masks and measured Intensity profiles. Top row: phase masks with topological charges (a1) +6, (a2) -6; intensity of (a3) multiplexed +6 and -6, (a4) - 6 down-converted to 0 and +6 to +12, (a5) passthrough beam ` = +12, (a6) added Gaussianbeammultiplexedwithpassthroughbeam,(a7)up-convertedmultiplexedbeam ` = +6 and -6. Bottom row: Phase masks with topological charges (b1) +8, (b2) -8; intensity of (b3) multiplexed +8, -8, (b4) -8 down-converted to 0, +8 to +16, (b5) passthrough beam +16, (b6) added Gaussian beam multiplexed with passthrough beam, (b7) up-converted multiplexed +8, -8. In Figure 2.4(a), we show the measured BER performances for add, drop and passthrough beams when ` = +6 was dropped and then added, while ` = -6 was allowed to pass through. Similarly, Fig. 2.4(b) shows the measured BER when ` = -6 was dropped and then added while` = +6 was allowed to pass through. The BER curves corresponding to dropped and added l = +8 and dropped and added OAM-8 are shown in Figs. 2.5(a)-(b), respectively. Figure2.4: MeasuredBERfor(a)add–drop`=+6modeforadatalinkwithmultiplexed ` = +6 and -6 modes. (b) add–drop OAM -6 beam for a data link with multiplexed +6 and -6 OAM beams. 13 For the dropped and the added channels, an OSNR penalty < 1.6 dB is observed at BER of 2× 10 −3 . For the passthrough channel, the OSNR penalty is < 2.9 dB. Higher penalty for the passthrough channel is due to it experiencing both down- and up–conversion operations. (a) (b) Figure 2.5: Measured BER for (a) add–drop OAM +8 beam for a data link with multi- plexed +8 and -8 beams. (b) add–drop OAM -8 beam for a data link with multiplexed +8 and -8 beams. 14 Chapter 3 OAM-based 2×2 Switch 3.1 Introduction Various OAM–based networking functions have recently been demonstrated, including reconfigurable add–drop multiplexing [13,25], selectively switching a single OAM beam [26], and multicasting [27]. With the emerging research in using OAM–multiplexing to extend the capacity, it might be useful to extend the employability of an OAM–based communications link in a multi–user environment, in which different OAM data channels could be selectively manipulated. In this chapter, we describe reconfigurable 2×2 OAM–based switching of 50 Gbaud QPSK channels [14]. We use multiple reflective SLMs to spatially sepa- rate multiplexed OAM beams. After spatial separation, the beams are redirected, recombined, and sent toward the desired output port. The switch can be reconfig- ured to selectively redirect different combinations of the OAM beams to different output ports. 15 3.2 Concept of the 2×2 OAM–switch A 2×2 OAM-based switch is analogous to a 2×2 WDM switch. In WDM networks, a 2×2 switch either redirects one of the input wavelength channels to appear at the opposite output port (‘cross’ state) or allows a wavelength channel to simply pass through the switch without being redirected (‘bar’ state). As shown in Fig. 3.1, the concept of a 2×2 OAM-based switch is similar to a 2×2 WDM switch. Each input port of the switch receives two multiplexed OAM beams and depending on the switch state, OAM beams can be redirected to appear at a desired output port. The switch can be reconfigured to operate in either ‘cross’ or ‘bar’ state for each of the input OAM beams. As shown in Fig. 3.1, an exemplary switch configuration would be to switch l 2 with l 4 , while l 1 and l 3 simply pass through the switch. If switching is not desired, then the switch could be configured in the ‘bar’ state for all of the input OAM beams, in which case all of the input OAM beams simply pass through the switch without being redirected. Figure 3.2 depicts a functional block diagram of the 2×2 OAM-based switch. In each path, multiplexed OAM beams go through a mode down-conversion stage. Figure 3.1: The concept of the 2×2 OAM switch. One of the two OAM beams at any input port can be redirected toward one of the output ports. 16 3.3 Mode Up & Down Conversion Mode down-conversion refers to transforming one of the incoming OAM beams (a donut-like transverse intensity profile with helical wave front) into a Gaussian- like beam with ` = 0 (a spot-like transverse intensity profile with planar wave front). After passing through the down-conversion stage, the second collinear OAM beam in the multiplexed pair remains as an OAM beam (shown as a circle in Fig. 3.2), but with a different OAM mode order. Once spatially separated, the beams are passed through a programmable mode-dependent beam-steering element. This element spatially separates the two collinear beams by redirecting the inner Gaussian-like beam and outer OAM beam in different directions, such that the Gaussian-like beam from one path aligns with the OAM beam from the other path. The mode orders of the newly aligned beams are corrected at the up-conversion stage such that out going OAM beams have similar OAM values as their corresponding input beams. Up-conversion process is opposite of down- conversionprocessasittransformsanincomingGaussianbeamintoanOAMbeam with desired OAM value. After up-conversion stage, the beams are sent toward the corresponding output ports. Figure 3.2: Functional block diagram of the 2×2 OAM-based switch. Switching is performed with the help of mode down-conversion, programmable beam-steering, and mode up-conversion stages. 17 3.4 Experimental Setup The experiment setup is shown in Fig. 3.3. A 50 Gbaud NRZ-QPSK signal was split into four paths and de-correlated by using optical fibers of different lengths. Collimators were then used to generate four beams with Gaussian intensity profiles and a beam waist (w o ) of 1.1 mm. Two SLMs, SLM-1 and SLM-2 (512×512 pixels, 15 micron pixel pitch), were used to generate the OAM beams at the two input ports for the switch. The phase mask on each SLM was designed to have two different fork holograms [16] in two different regions, so that beam shined on each region transforms into the desired OAM beam. Half-wave plates (HWP) were used to align the incoming polarization to maximize the diffraction through SLMs. The two OAM beams were superposed with a 3 dB non-polarizing beam-splitter. An afocal setup with unity magnification (f = 200 mm) was used to keep the beams collimated over a longer distance. A pinholewas used inthe Fourier plane of the first lens in the afocal setup to suppress the zeroth order diffracted beam. This constitutes the setup for one of the two input ports of the switch. A similar setup generated the second input for the switch. SLM-3 was used inside the 2×2 switch to perform the down-conversion operation. For this purpose, we used an SLM with larger dimensions (600× 792 pixels, 20 micron pixel pitch). The hologram EDFA BPF Att EDFA Polarization Diversity 90 0 -‐Hybrid ADC ADC ADC ADC Off-‐line DSP LO OSA 1% tap From Col. # 5 Coherent Detection EDFA I/Q Mod. Laser BPF PC 100 Gbit/s Pattern Generator QPSK PC To Col. #1 To Col. #2 To Col. #3 To Col. #4 QPSK Transmitter Col. # 2 HWP HWP Col. # 1 M SLM 1 BS Col. # 4 HWP HWP Col. # 3 M SLM 2 BS M M M M M M SLM 3 SLM 4 SLM 6 SLM 5 f f f f HWP PH f f HWP Col. # 5 f f HWP HWP HWP f f PH M M M M HWP f f PH Multiplexed Input B Multiplexed Input A 2×2 OAM SWITCH Output A Output B Figure3.3: ExperimentSetup. SLMspatiallightmodulator; Col.: collimator; M:mirror; f: lens; BS: beam-splitter; PH: pinhole to select the 1st order diffracted beams; HWP: half-wave plate. 18 on SLM-3 was also divided into two spatial regions, so that each region could down-convert one of the multiplexed OAM beams coming from each input port. After down-conversion, the beams were passed through another afocal system (f = 200 mm) and were shone onto SLM-4. SLM-4 was used as a mode-dependent, variable beam-steering element and was programmed with a phase mask having two different blazed grating regions. The incoming beams from the two input ports were made to have different incidence angles at SLM-4. In conjunction with the steering angles of the blazed gratings, these incidence angles allowed redirection of the beams [26], such that the down-converted beam from one input port aligned and propagated collinearly with the OAM beam from the other input port. Mode up-conversion was performed by SLM-5. Like SLM-3 it was divided into two parts, each serving one output port. Each part was programmed with a phase mask having two regions to properly up-convert the incident ‘bar’ and ‘crossed’ beams. In the experiments reported below, SLM-5 was implemented with an available SLM whose pixel count was not enough to simultaneously handle both output ports but only one at a time. SLM-5 formed the last stage of the 2×2 switch. Power levels at the input and output ports of the switch were 19 dBm and 8 dBm (for both the ‘bar’ and ‘cross’ states and for all four modes within 0.5 dB), respectively, representing an insertion loss of 11 dB, whose main source was the SLMs’ diffraction efficiencies. As shown below, these power levels were sufficient to achieve error levels below the FEC limit. At the receive end, SLM-6 was used to select only one of the incoming OAM beams and transform it into a Gaussian- like beam with ` = 0. The selected beam was coupled into a single-mode fiber (SMF) by using a collimator. The received signal was then sent to the coherent detection setup for bit error rate (BER) measurements. Two multiplexed pairs of 19 OAM beams with ` = +4, -4, and +2, -6 were used for the input ports A and B, respectively. 3.5 Results and Discussion The capability of operating the switch in ‘cross’ and/or ‘bar’ state for each of the four OAM beams suggests five configurations as shown in Table 1. In order to implement output ports, we first aligned SLM-5 to up-convert the beams directed toward output port A. In the meantime, all data channels were transmitted simultaneously. The measured BER performances for the four switch configu- rations (C1 – C4) for output port A are shown in Figs. 3.4(a) and 3.4(b). An optical signal to noise ratio (OSNR) penalty < 2.5 dB was observed at a BER of 2× 10 −3 for the switched channels appearing at output port A. 6 8 10 12 14 16 18 5 4 3 2 Output Port A Back-to-Back C1(+4) C2(+4) C3(-4) C4(-4) -Log(BER) OSNR (dB) EFEC Threshold 6 8 10 12 14 16 18 5 4 3 2 Output Port A Back-to-Back C1 (-6) C2 (+2) C3 (-6) C4 (+2) -Log(BER) OSNR (dB) EFEC Threshold 6 8 10 12 14 16 18 5 4 3 2 Output Port A Back-to-Back C5 (+4) C5 (-4) -Log(BER) OSNR (dB) EFEC Threshold C1 C2 C3 C4 C5 (d) (a) (b) (c) Figure 3.4: Measured BER for beams appearing at output port A for different switch configurations: (a) channels from input port A; (b) channels from input port B; (c) channels from input port A while switch was configured in ‘bar’ state. 20 6 8 10 12 14 16 18 5 4 3 2 Output Port B Back-to-Back C1 (+2) C2 (-6) C3 (+2) C4 (-6) -Log(BER) OSNR (dB) EFEC Threshold 6 8 10 12 14 16 18 5 4 3 2 Output Port B Back-to-Back C1(-4) C2(-4) C3(+4) C4(+4) -Log(BER) OSNR (dB) EFEC Threshold 6 8 10 12 14 16 18 5 4 3 2 No Switching - Port B Back-to-Back C5 (+2) C5 (-6) -Log(BER) OSNR (dB) EFEC Threshold C2 C1 C3 C4 C5 (d) (a) (b) (c) Figure 3.5: Measured BER for beams appearing at output port B for different switch configurations: (a) channels from input port A; (b) channels from input port B; (c) channels from input port B while switch was configured in state. Figure 3.4(c) shows the measured BER performance for the channels appearing at output port A, while the switch was configured in the ‘bar’ state. SLM-5 and SLM-6 were then realigned to form output port B. Figures 3.5(a) and (b) show the BER for the four switching configurations for output port B. An OSNR penalty < 1 dB at a BER of 2× 10 −3 was observed for channels appearing at output port B. Measured BER performance for port B, while switch was operated in the ‘bar’ state, is shown in Fig. 3.5(c). To observe the transverse phase profiles of different OAM beams, OAM beams were interfered with a diverging Gaussian beam after up-conversion stage. The Interferograms shown in the Fig. 3.6 verified the correct up-conversion of OAM beams for different switch configurations. The crosstalk among different OAM beams was also measured after de– multiplexingthe desiredOAMbeam andmeasuring thereceivedpowerby coupling it into an SMF, while other undesired OAM beams were blocked. The difference Figure 3.6: Interferograms of different OAM beams after beam-steering and up- conversion stages. OAM mode order of each beam is shown in parenthesis. 21 between the received power in the desired OAM beam and the sum of the power leaked from each of the undesired OAM beams is shown as the crosstalk in Fig. 3.7. (+4) (-4) (+2) (-6) -35 -30 -25 -20 -15 Configuration C5 Crosstalk (dB) OAM Beams (l) Output Port A Output Port B (+4) (-6) (+2) (-4) -35 -30 -25 -20 -15 Crosstalk (dB) OAM Beams (l) Output Port A Output Port B Configuration C1 (+4) (+2) (-4) (-6) -35 -30 -25 -20 -15 Configuration C2 Crosstalk (dB) OAM Beams (l) Output Port A Output Port B (-4) (-6) (+4) (+2) -30 -25 -20 -15 Configuration C3 Crosstalk (dB) OAM Beams (l) Output Port A Output Port B (+2) (-4) (+4) (-6) -35 -30 -25 -20 -15 Configuration C4 Crosstalk (dB) OAM Beams (l) Output Port A Output Port B Figure 3.7: Measured crosstalk among different OAM beams after coupling into single mode fiber. 22 Chapter 4 Mode–division Multiplexing of Bessel–Gaussian Beams 4.1 Introduction Line-of-sight (LoS) wireless communication links are important for many applications, including backhaul, access, military and data center environments [15,28,29], and can exist through air or vacuum. A critical issue with all free– space LoS data links is the ability to overcome obstructions in the beam path [16]. Such a problem is exacerbated significantly in an OAM–multiplexed system, since obstructions will produce phase distortions in each beam’s phase front, thereby reducing the orthogonality and causing some amount of energy contained in a particular modal state to be coupled into other neighboring modes [22]. Such distortion and crosstalk fundamentally affects the OAM–multiplexed systems since themodedetermineseachuniquechannel, andphaseperturbationscandestroythe ability to recover each transmitted data channel uniquely. However, a key advance in multiple–channel, free–space links would be if the basic phase and amplitude 23 profiles of orthogonal beams remain intact even if there is an obstruction in their path. One potential approach could be to use Bessel-Gaussian (BG) beams for transmission over short ranges [18–20,30–36]. BG beams are propagation invariant over a length determined by the generation method and can extend to a few tens of meters [37–39]. BG beams have the unique property to reconstruct or ‘self- heal’ the transverse intensity and phase profiles after experiencing an obstruction. The self–healing property of the BG beams may have important applications in short–range, free–space communication links. In this chapter, we explore use of mode multiplexed BG beams in a free–space communication system that is fundamentally more tolerant to path obstructions [19]. We demonstrate an MDM system of multiple BG beams carrying OAM for obstruction–tolerant optical high-speed links. Two BG beams (` = +1 and +3), are multiplexed and transmitted over a distance of 1.4 m. Each beam carried 50–Gbaud quadrature–phase–shift–keying (QPSK) data. To emulate obstructed beam paths, opaque circular obstructions are used at various transverse positions. The ‘self-healing’ of BG beams is observed such that BG beams are able to reconstruct in the receiver planes. We recover data channels and assess system performance in terms of crosstalk from neighboring modes. As a comparison, the tolerance of non–Bessel OAM beams is investigated, under similarly obstructed path conditions. It is observed that under certain conditions, BG beams are more tolerant to obstructions than non–Bessel OAM beams. As an example, when obstructions are placed at the beam center, optical BG beams with ` = +1 and ` = +3 show >6 dB and >1.5 dB reduction in crosstalk as compared to optical OAM beams, respectively. When obstructions are placed off–centre, under similar 24 conditions, optical BG beams with ` = +1 and ` = +3 show >3 dB and >8 dB reduction in crosstalk in comparison with optical LG beams, respectively. 4.2 Concept of Obstruction–tolerant Links Figure 4.1 shows a conceptual block diagram of a free-space link that uses spatially multiplexed data–carrying beams. N distinct input OAM beams, each carrying a distinct data channel, are spatially multiplexed and transmitted through an axicon to be transformed into BG beams. Within the ‘Bessel–region’, beams are propagation invariant and, therefore, can sustain partial obstructions. At the end of the Bessel–region, an exit axicon having opposite cone angles is placed to remove conical phases. Finally, an OAM mode demultiplexer separates each OAM beam. The ability of BG beams to sustain the adverse effects of an object inadvertently blocking the beam path within the Bessel region might be beneficial in recovering data channels. Mode MUX Mode 1 Mode 2 Mode N Mode DMUX Axicon Bessel Region Axicon Mode 1 Mode 2 Mode N Obstruction Before Obstruction After Obstruction Self-healed Figure 4.1: Conceptual block diagram of a multimodal link using multiplexed BG beams. ‘Bessel-region’ is the distance over which BG beams are propagation invariant and retain their profile. Insets depict transverse intensity profiles of a BG beam before and after an opaque disk and in the receiver plane. 25 4.3 Experimental Setup Figure 4.2 shows the experimental setup to demonstrate an obstruction– tolerant optical transmission. Two fundamental Gaussian beams, each carrying uncorrelated 50–Gbaud QPSK channels, are shone on two different regions of SLM A to be transformed into Laguerre–Gaussian (LG) beams with` = +1 and` = +3. The two LG beams are then multiplexed using a beamsplitter and passed through a 3× magnifying setup such that in the plane of SLM–B, LG beams with ` = +1 and ` = +3 have spot sizes of 3.8 mm and 4.1 mm, respectively. Note that the spot size of an LG beam can be calculated by determining the second moment of intensity as [40]: Coherent Detection System QPSK Transmitter SLM A M M BS SLM C ATT 1 nm M f f Col. Col. Col. 3x Obstruction on X-Y Stage SLM B (b). After Obstruction (a). Before Obstruction (c). In the plane of SLM C Figure 4.2: Experimental setup. Phase mask on SLM–B transforms multiplexed LG beams into BG beams. Opaque circular obstruction placed on the x–y stage partial obstructed multiplexed BG beams. Insets (a), (b), and (c) show measured transverse intensity profiles of BG beam (` = 1) at different locations along the propagation direc- tion, before and after an opaque circular obstruction of radius 1.5 mm. Col.: Collima- tor, HWP: Half-wave plate, M: Mirror, SLM: Spatial light modulator, BS: Beamsplitter, ATT: Attenuator. 26 σ 2 r = 2 RR r 2 |E(r,φ,z)| 2 rdrdφ RR |E(r,φ,z)| 2 rdrdφ (4.1) in which,(r,φ) are the radial and azimuthal coordinates of a plane at a distance z. Since LG beams can be transformed into BG beams by placing an axicon in the beam path [37], SLM–B is used for this purpose. SLM–B is programmed with a phase mask whose transmittance function can be given by: T B (r) =exp j2πr ρ ! (4.2) in whichρ is an adjustable parameter controlling the cone angle of the holographic axicon [39,41–43]. After reflecting from SLM–B, the two LG beams transform into BG beams of similar mode orders (` = +1 and` = +3) and propagate to SLM–C. At a distance of 0.55 m from SLM–B, an opaque circular obstruction is placed on a linear translation stage such that it blocks the beam path at various transverse positions across the beam cross–section. Figures 4.3(a) and 4.3(b) show both experimentally measured and numerically calculated intensity cross–sections of the generated BG beams (` = +1, and ` = Figure 4.3: Measured intensity cross-sections of BG beams in the obstruction plane. BG beam with (a) ` = 1, and (b) ` = 3. Solid curves show numerically calculated cross–sections. 27 (b) Normalized Intensity 0 1 Obstruction Bessel Beam r r Obs r Beam (a) Figure4.4: PictorialdepictionofanobstructedBGbeam. (a)Transverseintensityprofile of a BG beam. Dashed circle signifies opaque circular obstruction placed at the beam center. In experiment, the obstruction is laterally moved to block the beam at various transverse positions. (b) Cross-section of BG beam intensity. Dashed line represents the cross–section of the opaque obstruction. Here r Obs is the radius of the obstruction, andr Beam is the spot size of the beam determined by calculating the second moment of intensity. +3) in the obstruction plane. The total distance between SLM–B and SLM–C is 1.4 m. To demultiplex one of the BG beams, SLM–C is programmed with a phase mask whose transmittance function is given by: T C (r) =exp (−j(`θ + 2πr/ρ)) (4.3) The phase mask on SLM–C effectively is a superposition of two phase masks: 1) The `θ term removes the helical phases associated with OAM beams; and 2) to remove conical phases associated with BG beams. The second term in the above transmission function represents a reversed axicon and is needed to remove conical phases associated with the BG beams such that the desired BG beam is converted back into Gaussian beam for coupling into single mode fiber (SMF). A set of two lenses is placed after SLM–C to couple the demultiplexed beam into an SMF. The received signal is then fed to a coherent receiver for detection and bit-error rate (BER) measurements. 28 Table 2. Obscuration Ratios for the Optical Link Beam Type Mode Order Spot Size Obscuration Ratio ( ζ ) BG l = 1 1 mm 1 (r Obs = 1 mm) 1.5 (r Obs = 1.5 mm) l = 3 2.1 mm 0.47 (r Obs = 1 mm) 0.71 (r Obs = 1.5 mm) LG l = 1 0.71 mm 1 (r Obs = 0.7 mm) 1.4 (r Obs = 1 mm) l = 3 1 mm 0.7 (r Obs = 0.7 mm) 1 (r Obs = 1 mm) As obstructions, the experiment uses opaque disks whose transmittance function is given by: T Obs (r) = 1−Circ r r Obs (4.4) in whichCirc(r/r Obs ) = 1 forr>r Obs , andCirc(r/r Obs ) = 0 forr<r Obs [57], and r Obs is the radius of the obstruction (see Figure 4.4). In the obstruction plane, the spot sizes of the BG beams` = +1 and` = +3 are 1 mm and 2.1 mm, respectively. Note that these values are calculated using same expression as that used for LG beams. We define obscuration ratio ξ as the ratio of obstruction radius to the radius of a BG beam (i.e. ξ = r Obs /r Beam ). We chose two different opaque disks having radiir Obs = 1 mm and 1.5 mm such that the obscuration ratios for` = +1 are 1 and 1.5, and for ` = +3 are 0.47 and 0.71, respectively (see Table 2). 4.4 Results While the link is operating, the obstructions are laterally traversed across the multiplexed beams. We observe that after encountering the obstructions, BG beams self–heal and reconstruct their transverse intensity profiles in the plane of SLM–C. As an example, Figure 4.5(a) shows a comparison between the transverse intensity profiles of the obstructed and unobstructed BG beam with ` = +3. In this figure, an obstruction of radius 1.5 mm (ξ = 0.71) is placed at the beam 29 center and transverse intensity profiles are taken at various locations along the propagation direction. A comparison of obstructed and unobstructed beams in the plane of SLM–C (right–most column) reveals reconstruction of the BG beam. Furthermore, Figure 4.5(b) shows the cross–sections of the normalized intensity profiles of both obstructed and unobstructed BG ` = +3 in the plane of SLM– C. In this figure, we observe that the outer rings of the obstructed beam have diminished due to the power loss incurred by the obstruction. Nonetheless, the inner rings of the two beams are in good agreement. To assess the link performance under obstructed path conditions, power in the desired mode and power leaked into the neighboring modes (after coupling into the (a) (b) 0.55 m 0.75 m 0.85 m 0.95 m 1.41 m Unobstructed Obstructed #10 -3 -3 -2 -1 0 1 2 3 0 0.5 1 #10 -3 -2 0 2 0 0.5 1 Obstructed Unobstructed Horizontal Cross-section (mm) 0 1 2 3 -1 -2 -3 0.0 0.5 1.0 Norm. Intensity D = 3 mm Obstruction Plane SLM-C Plane r Obs !=!1.5!mm! Figure 4.5: Transverse intensity profiles of obstructed and unobstructed BG beam ` = +3 after an obstruction of radius 1.5 mm (ξ = 0.71) at different locations along the propagation direction. (b) Normalized intensity cross–sections of obstructed and unobstructed beam in the plane of SLM–C. 30 SMF) are measured by using appropriate phase masks on SLM–C, one at a time. Forexample, tomeasurepowerfrom`=+3to`=+3(P 33 , hereP ij referstopower received in the j-th mode while i-th mode is transmitted) and power leaked into ` = +1 (P 31 ), we transmit only one beam with ` = +3 and place the obstruction in the beam center. As the obstruction moves laterally across the beam, the power is measured in ` = +3 (the desired mode). To measure the leakage power P 31 , SLM–C is loaded with the phase mask to select the undesired mode (` = +1). The same procedure is repeated for the case when ` = +1 is transmitted, and received power is measured in the desired mode ` = +1 (P 11 ) and in the undesired mode ` = +3 (P 13 ). Figures 4.6(a) and 4.6(b) show the measured power as a function of normal- ized obstruction position. In these figures, the position r of the obstructions is normalized by the radius of beam spot size, that is, r o = r Beam . In the same figure, we also show the results obtained by the numerical simulations. There is a slight mismatch between the numerical and experimental data in the absolute sense, however, the overall trend matches. The slight mismatch is due to the ideal condition considered in the model whereby the coupling to the SMF is neglected and the power received in each mode is calculated by the overlap integral between obstructed and unobstructed beams in the receiver plane. To compare the performance of obstructed BG beams with obstructed LG beams, a second experiment is performed by slightly modifying the existing exper- imental setup. For the second experiment, the magnification setup is removed and SLM–B is replaced with a simple mirror. Additionally, in the absence of conical phases, the a simpler phase mask is used on the SLM–C to convert LG beams back into Gaussian beams for coupling into SMF. The transfer function of the phase mask on the SLM–C can be given by T C (r) = exp(−j`θ). In this experiment, 31 P 33 P 31 P 33 P 31 Figure 4.6: Received power as a function of normalized obstruction position for BG ` = +3 obstructed by two obstructions of radii (a) 1 mm (ξ = 0.47), and (b) 1.5 mm (ξ = 0.71). Hollow triangles represent experimentally measured data and solid lines represent numerically calculated results. P ij refers to the power received in j-th mode while i-th mode is transmitted. the spot size of the LG beams with ` = +1 and ` = +3 in the obstruction plane are 0.71 mm and 1 mm, respectively. To have obstructed path conditions similar to the BG case, the obstruction of radii 0.7 mm and 1 mm are chosen such that obscuration ratios for ` = +1 are 1 and 1.4, and those for ` = +3 are 0.7 and 1, respectively (see Table 2). Figure 4.7 shows the measured crosstalk for the BG and LG cases. The crosstalk between the two modes is calculated by taking the ratio between the powers in the undesired and the desired mode. For the case in which BG beam with ` = +1 is obstructed by an obstruction of radius 1 mm (ξ = 1) placed at the beam center, we observe a crosstalk of -19 dB from` = +1 to` = +3 (Figure 4.7(a)). The crosstalk in this case is 6.73 dB lower than the corresponding case for the LG beam as shown in Figure 4.7(b). Higher crosstalk between LG modes can be explained by the inability of LG modes to reconstruct their beam profiles. As the obstruction size is increased to 1.5 mm (ξ = 1.5) for BG beam ` = +1, a 2.7 dB increase in the crosstalk is seen. For the corresponding LG case, although obscuration ratio is 32 smaller than the BG case (i.e. ξ = 1.4), system suffers from increased crosstalk. As the obstruction moves away from the beam center, crosstalk decreases steadily. In terms of maximum value of crosstalk, BG system has 11.8 dB lower crosstalk than the LG system. Figures 4.7(c) and 4.7(d) show the leakage power from beams with ` = +3 to ` = +1 when the two different obstructions are used to block BG and LG beams, respectively. Since the beams with ` = +3 have larger spot size, obstructions of fixed radii result in smaller obscuration ratios. When the obstructions are placed in the beam center, the BG system withξ = 0.71 has a 1.6 Figure 4.7: Crosstalk between different beams as a function of normalized obstruction position. CT ij refers to power leaked from i-th mode to the j-th mode. For comparison between BG and LG systems, obscuration ratios are also shown. The curves show crosstalk for: (a) BG ` = +1 to ` = +3, (b) LG ` = +1 to ` = +3, (c) BG ` = +3 to ` = +1, and (d) LG ` = +3 to ` = +1. 33 dB lower crosstalk than the similar LG system. Similarly, the maximum crosstalk for the BG is ~17 dB, while for the LG system, the maximum crosstalk is ~9 dB. Figures 4.8(a) and 4.8(b) show BER measurements for the unobstructed and the obstructed BG beams. First, the BER for both channels is measured without obstructingthebeampath. Then, theBERismeasuredagain, placingtheobstruc- tion in the beam center, a location where maximum power loss occurs. As shown in Figures 4.8(a) and 4.8(b), each channel can achieve a raw BER of 3.8× 10 −3 . At a BER of 3.8× 10 −3 , the optical signal–to–noise ratio (OSNR) penalties for the 1 mm and 1.5 mm obstructions are < 2.3 dB and < 1.7 dB, respectively. The large penalty for the channel on the` = +1 beam could be explained by noting that the ` = +1 beam has a smaller spot size and, therefore, encounters a relatively larger obstruction as compared with the` = +3 beam. Note that for the LG system, the significant amount of crosstalk introduced by the obstructed beam path prevents the link from operating, and a BER measurement is therefore not possible. (l = 1) (l = 3) (l = 1) (l = 3) (l = 1) (l = 3) (l = 1) (l = 3) Figure 4.8: Measured BER for the multiplexed BG beams obstructed by obstructions of radii: (a) r Obs = 1 mm, and (b) r Obs = 1.5 mm. 34 4.5 Discussion When considering the future use of BG beams in free-space communication links, it would be beneficial to address some additional issues, including the following. Achievable link length: In its simplest form, the length of the Bessel region for which there is self-healing is directly proportional to the spot size of the input beam and inversely related to the cone angle of the axicon [35], meaning that a larger input beam spot size and a smaller axicon angle results in a longer Bessel region. Although many experiments have demonstrated the possibility of Bessel regions from a few meters to up to 100 m [38,39], so-called ‘Bessel–like beams. also exist with distance–dependent cone angles, allowing for long non–diffracting regions [44]. Relative location of obstruction along the link: After experiencing an obstruction, the BG beam requires a certain distance to achieve self–healing such that the BG beam reconstructs its phase and amplitude profile. In the simple geometrical–ray optics limit, the minimum distance required by beams to self–heal after encountering an obstruction of diameter D can be given by D/(2tan(θ)), in which D is the obstruction diameter and θ is related to the apex angle of the axicon [35]. In general, an obstruction may exist at any location along the beam path within the Bessel region, but there will likely exist scenarios with obstructions near the receiver for which full self-healing has yet to occur and that will produce a system penalty. Obstructions of different shapes: In this work, we have assumed obstructions to be of circular symmetry, however, in reality obstruction of any shape could exist. Based on the ray-optics principle, the self–healing of BG beams would occur for obstruction of any shape as long as there are unobstructed conical rays that interfere in some plane at a distance after the obstruction [45]. The effects of 35 obstruction size and location across the beam cross–section has to be investigated and may produce system performance different than shown in this work. Alternate Methods: While BG modes rely on their self–healing property to sustain the obstructions, other methods may also be possible when the beams used are not propagation invariant. For such beams, one such method could be use of MIMO–DSP which estimates the received data based on the information contained in the neighboring modes. Effectiveness of such methods has been demonstrated in optical links suffering from atmospheric turbulence; however, its performance in an obstructed path scenario has yet to be investigated. Atmospheric Turbulence: Atmospheric turbulence tends to distort both the intensity and phase profile of the OAM carrying beams [46]. Various methods have been proposed along with proof-of-concept lab experiments [47,48]. Although we have not considered the effects of atmospheric turbulence on BG beams, however, schemes similar to shown in the above referenced work could be considered to mitigate turbulence induced distortion and scintillation effects. In addition, it is important to mention that other forms of propagation– invariant ‘Bessel–like’ beams also exist. For example, recently, it has been shown that LG beams with higher–order radial indices also possess self–healing properties [49]. Therefore, an investigation similar to the one reported in this paper could be performed with such other types of beams, evaluating relative performance gains. 36 Chapter 5 Millimeter–wave OAM–carrying Bessel Beams 5.1 Introduction Free-space communication links using multiplexing of multiple OAM beams have been demonstrated in both RF and optical domains [4,5,50,51]. In the case of single-beam systems, many “regular” Gaussian–beam based non-OAM free– space wireless transmission links rely on line-of-sight operation [16]. If there is an obstruction, the link will cease to operate properly when the mm-wave beam is completely blocked but may still be able to function if the beam is only partially blocked. For the case in which multiple mm-wave OAM beams are multiplexed, an interesting question arises regarding the effect of an obstruction that partially blocks the beams. For example, the phase-front of a wave becomes critical in distinguishing the different data channels, and a partial obstruction may degrade the phase orthogonality among the different OAM beams [3,52]. A laudable goal would be to demonstrate a type of transmission link for which partial obstructions 37 can be tolerated by an OAM based free-space mm-wave link. Specifically, it was shown in the optical domain that Bessel beams have the “self healing” feature such that the beam’s phase can be “reconstituted” after passing through a partial obstruction [20,32]. In this chapter, we describe an obstruction–tolerant 8 Gbit/s free–space mm– wave communications link [20]. Two 1 Gbaud 16–QAM channels are multiplexed using OAM carrying Bessel beams (` = +1 and` = +3) at 28 GHz and transmitted over a distance of 1.4 m. In order to emulate obstructed beam path, opaque circular obstructions of different radii carved out of mm–wave absorbers are used. After passing through the obstructions, Bessel beams reconstruct their field at the receiver end and are demultiplexed to recover individual channel. The tolerance of Bessel beams to different obstruction sizes is investigated by measuring the crosstalk. A power penalty < 10 dB and < 3 dB is observed for Bessel beams with ` = +l and +3, respectively. 5.2 Millimeter–wave Obstruction Tolerant Links In line-of-sight (LoS) links, an obstacle moving across the beam path causes loss of powerduetoblockageofthebeampath. Linkmayexperienceoutagesandretrans- mission of data might be required [16]. In links which utilize spatial multiplexing of multiple modes to carry distinct data channels, such as OAM multiplexing, the effects of the obstructed beam path are twofold. First, similar to single-mode links, part of the energy in the multiplexed modes scatters and does not reach the receiver. Secondly, since the information contained in the transverse field is crucial for discriminating different received modes, an obstruction inadvertently blocking the beam path will change the spatial profiles of the multiplexed beams due to diffraction effects. Thus, power leakage among neighboring modes occurs, which 38 in turn gives rise to inter-channel crosstalk. As an example, Fig. 5.1 shows simu- lated beam evolution of an obstructed fundamental Gaussian beam and an OAM (` = +1) beam propagated in free-space. In this figure, two cases of obstructed and unobstructed beams in the obstruction plane and in the observation plane (i.e. receiver plane) are shown. In case of the unobstructed beams, although the spot size increases due to beam divergence, the received Gaussian and OAM beams still retain their shape. For the case of obstructed beams, an opaque circular obstruc- tion having diameter equal to the half of the spot diameter of each beam is used to block the beam path. As can be seen in the observation plane, the two beams have undergone significant changes in their transverse profiles. One potential approach to resolve such an issue could be using Bessel beams which have their electric fields proportional toJ l (k r r)e jlθ , in whichJ l is the Bessel function of order `, k r is the radial wave-vector, and (r,θ) are the polar coordi- nates. Similar to OAM carrying beams (e.g. Laguerre-Gaussian beams), Bessel Before Obstruction After Obstruction Obsv. Plane un −Obstructed Obsv. Plane Obstructed Before Obstruction After Obstruction Obsv. Plane un −Obstructed Obsv. Plane Obstructed Obstruction Plane Observation Plane Obstructed* Unobstructed* Obstructed* Unobstructed* l*=*0* l*=*+1* Figure 5.1: Diffraction effects on the transverse intensity profiles of the obstructed Gaus- sianandOAM(`=+1)beamswhenanopaquecircularobstructionisplacedatthebeam center. In the observation plane, the difference in the transverse profiles of obstructed and unobstructed beams is clearly visible. All images have same dimensions and are calculated at a frequency of 28 GHz. 39 l = +3 l = 0 l = +1 Transverse Intensity Transverse Phase 0.5 1 1.5 2 2.5 3 3.5 4 x 10 5 0 2 4 6 8 10 12 14 x 10 4 1 2 3 4 5 6 7 x 10 4 0 0.5 1 1.5 2 2.5 3π 0 −3 −2 −1 0 1 2 3 -π π −3 −2 −1 0 1 2 3 -π π Figure 5.2: Transverse intensity and phase profiles of different Bessel beams. Ideal Bessel beams have infinite number of rings, each containing same amount of energy. beams having ` > 0 carry OAM and therefore can be used as carriers for spatial multiplexing of different data channels. The transverse intensity and phase pro- files of few Bessel beams are shown in Fig. 5.2. Ideal Bessel beams have infinite number of rings surrounding the inner-most spot (` = 0) or ring (` > 0) with each ring having same energy. An interesting feature of Bessel beams is their ‘non- diffracting’ nature. Ideal Bessel beams can propagate over long distances without diverging (i.e. are propagation invariant) and therefore retain their transverse field profiles at any point along the propagation path. Owing to this non-diffracting nature, Bessel beams are able to reconstruct their fields even after encountering an obstruction. To show the self-healing property of obstructed approximate Bessel beams and compareitwithGaussianandOAMbeams, inFig.5.3weshowsimulatedevolution of obstructed Bessel beams. In each case, the central spot (for` = 0) or the inner- most ring (for` > 0) are obstructed by opaque circular obstructions. A comparison 40 between the images of obstructed and unobstructed beams in the observation plane reveals the self-healing of the obstructed Bessel beams. 5.3 Generation of Millimeter-wave Bessel Beams The generation of Bessel beam in mm-wave regime has been reported by using con- ical lens [31]. For our experimental demonstrations, we have developed a method that uses metamaterials-based conical lens. Using standard printed circuit board (PCB) processing, we etch spatially varying slit arrays in a copper film on a one- layer PCB [53]. The design parameters for the metamaterials-based conical lens Obstruction Plane Observation Plane After Obstruction Obstructed* Before Obstruction Unobstructed* Obsv. Plane Obstructed Obstructed* Obsv. Plane un −Obstructed Unobstructed* l*=*0* Before Obstruction After Obstruction l*=*+1* Obsv. Plane un −Obstructed Obsv. Plane Obstructed Before Obstruction After Obstruction Obsv. Plane un −Obstructed Obsv. Plane Obstructed l*=*+3* Figure 5.3: Simulated evolution of different obstructed Bessel beams along propagation direction. In all cases, the inner-most rings of Bessel beams are obstructed by an opaque circular obstruction. A comparison of the obstructed and unobstructed beams in the observation plane reveals the reconstruction of the inner-most ring. All images have same dimensions and are calculated at a frequency of 28 GHz. 41 are shown in Fig. 5.4. The period of each unit is 3.5 mm and the size of each sub- wavelength rectangular slit is 3.06× 0.68 mm. The top layer of the designed PCB is covered with tin-lead-solder copper with thickness of 0.04 mm. The substrate is made of laminate with thickness of 1.5 mm and refractive index of 2.04 at 28 GHz. The subwavelength rectangular slit arrays can enhance the transmission of linearly polarized light (perpendicular to the slit direction, i.e., angle α relative to x direction). Each unit can be considered as a localized linear polarizer and hence introduces a so-called Pancharatnam–Berry phase to the output beams [53,54]. The angle α can be tuned to control the phase of the output beams. Assuming a linearly polarized beam with Jones vector E in = [1 0] T as the input beam, the output beam can be given by: E out = 0.25{e i2α [1 −i] T +e −i2α [1 i] T } + 0.5 [1 0] T (5.1) From equation 5.1, it is clear that the output beam consists of three polariza- tion states, namely right circular polarization [1 −i] T , left circular polarization X Figure 5.4: Design parameters of metamaterials–based conical lens for Bessel beam generation. The orientation angle is α = πx/Λ +πηρ, in which, Λ = 2.72 cm and η = 0.303 cm −1 . The size of the subwavelength rectangular slits is 3.06×0.68 mm 42 [1 i] T , and a linear polarization state [1 0] T . Also, the output components of right circular polarization and left circular polarization have angleα related phases e i2α ande −i2α , respectively, suggesting that we can use rectangular slit arrays with spatially varying angle as a phase mask. Figure 5.5 shows a schematic structure of the metamaterials-based conical lens. The conical lens can be regarded as a phase mask with phase shift decreasing linearly with the radial distance ρ. The transmission function of the conical lens can be written as [55]: T =exp −iβρ λ ! (5.2) in whichλ is the wavelength of the incident light. The parameterβ determines the dependence of phase shift on the radial distanceρ. In order to use the rectangular slit arrays as a conical lens, we set the angle α as a function of the radial distance ρ, i.e. α =βρ/2λ. Consequently, if E in = [1 0] T is used as the input beam, the output beam becomes: E out = 0.25{e iβρ λ [1 −i] T +e −iβρ λ [1 i] T } + 0.5 [1 0] T (5.3) + = Gra$ng' Conical'Lens' Conical'Lens'with'Gra$ng' Figure 5.5: Schematic structure of the metamaterials–based conical lens used to generate mm-wave Bessel beams at a frequency of 28 GHz. 43 Equation 5.3 shows that the proposed conical lens introduces a phase shift of exp (−iβρ/2λ) to the left circularly polarized components such that left circularly polarized output beam forms a Bessel beam. In order to separate the generated Bessel beam from other components of the output beam, a grating structure is introduced in the x direction (see Fig. 5.5). As a result, the angle α can be expressed as α = πx/Λ +πηρ, in which, Λ = 2.72 cm and η = 0.303 cm −1 . By using this arrangement, a normally incident 28 GHz beam is transformed into a Bessel beam and emerges at 23.6 degrees relative to the incident direction. 5.4 Experimental Setup The experiment setup is shown in Fig. 5.6. A 28 GHz clock source is used to gen- erate the carrier signal. Two arbitrary wavefrom generators (AWG 1 and AWG 2) are used to generate two 1-Gbaud 16-QAM signals. The two signals are then fed to two lensed horn antennas (Ant. 1 and Ant. 2). Both of the horn antennas have an exit aperture of 15 cm and are equipped with 15 cm diameter spherical lens. Two spiral phase plates (SPPs) fabricated using high-density polyethylene (HDPE) (refractive index n = 1.52 at 28 GHz), are used to transform the Gaus- sian beams generated by the lensed horn antennas into OAM beams with ` = +1 and ` = +3. A PCB based beamsplitter combines the two OAM beams such that both beams propagate collinearly. The beamsplitter is fabricated by spatially varying the reflectivity of the surface and is similar to polka-dot beamsplitter used in optical regime. In order to transform ` = +1 and ` = +3 OAM beams into Bessel beams, the metamaterials-based conical lens as described above is used. After passing through the conical lens, in the first diffraction order, multiplexed OAM beams transform into Bessel beams of similar mode orders and propagate toward the lensed-horn 44 28 GHz Source I/Q Mixer AWG 1 I/Q Mixer AWG 2 Real Time Scope Off-line Processing BS Metamaterials-based Conical Lens SPP (l = +3) Lensed Horn Ant. 2 Lensed Horn Ant. 1 Lensed Horn Ant. 3 2D Linear Stage SPP (l = +1) Amp. Amp. DMUX SPP (l = -1 or -3) Bessel Region Metamaterials*based- Axicon- Figure 5.6: Experiment setup. 28 GHz Gaussian beams generated by lensed horn anten- nas are converted into OAM beams (` = +1 and +3) by using spiral phase plates and multiplexed by using a beamsplitter. Metamaterials-based conical lens transforms mul- tiplexed OAM beams into Bessel beams of similar mode orders. Received beams are demultiplexed one at a time by using an SPP having opposite spiral phase. Amp: amplifier, AWG: Arbitrary waveform generator, Ant.: Gaussian lensed horn antenna, BS: beamsplitter, SPP: spiral phase plate. antenna (Ant. 3) placed 1.4 m away from the conical lens. A 2D linear translation stage holding the mm-wave absorbers of different radii is placed at 0.4 m away from the conical lens in the beam path. In order to demultiplex one of the received beams into the receiver antenna, an SPP having opposite mode order is placed before the receiver antenna. The received signals are recorded and digitized by a 80-Gsamples/s real-time scope which has an analogue bandwidth of 32 GHz. An offline post-processing is carried out to calculate the bit-error rates of both the channels. 5.5 Results and Discussion To characterize the power transfer matrix between the OAM channels when the obstruction is traversed across the beam, three opaque circular obstructions with radii 5 cm, 6 cm, and 8 cm are used. The obstructions are carved out of mm-wave 45 Figure 5.7: Received power vs. normalized obstruction position across the beam. Pij refers to the power received in jth mode while ith mode is being transmitted. absorbers. Note that in our experiment, the radius of the Bessel beam ` = +1 is 13.8 cm while the radius of the Bessel beam ` = +3 is 22.2 cm. We define the obscuration ratio as the ratio between obstruction radius r OBS and the radius of the Bessel beams r B . The received power is measured in the following way: we first transmit a 28 GHz CW signal over OAM ` = +1 (or` = +3) while the other channel OAM ` = +3 (or ` = +1) is turned ‘OFF’. Then we record the power coupled into the receiver antenna by an RF spectrum analyzer. Figure 5.7 shows the measured power of OAM channel ` = +3 as a function of the normalized positions of the obstruction. The position r of the obstruction is normalized by the radius of the incoming Bessel beam (r B ) (i.e. r 0 =r B ). For example, a position of ‘0’ refers to obstruction being aligned with the beam center. As shown in Figs. 5.7(a)-(c), the received power of transmitted OAM mode` = +3 increases steadily under different positions of the obstruction. As the obstruction movesawayfromthebeamcenter, thepowercoupledinto`=+1firstincreasesand then decreases (after reaching its maxima). The crosstalk between OAM channels ` = +1 and ` = +3 can thus be calculated by taking the difference between the received power of the unwanted mode and that of the desired mode. As the size of the obstruction increases, higher power loss as well as increased crosstalk is observed. We believe this occurs due to limitation in self-healing because of the 46 Figure 5.8: BER as a function of received SNR for OAM channel ` = +1 with and without obstruction. In all the cases, the obstructions are placed at the beam center. The radii of obstruction (r OBS ) are 5 cm, 6 cm, and 8 cm. fixed distance between conical lens and the receiver antenna. As obstruction size increases, more distance is needed for the obstructed Bessel beam to reconstruct itself [33]. To investigate the influence of beam obstruction on the power penalty of the OAMmultiplexedmm-wavesystem, Fig.5.8showsthebit-error-rate(BER)curves for channel on Bessel beam` = +1 with and without obstruction. In all the cases, the obstructions are placed at the beam center, i.e., the point where maximum power loss occurs for the desired channel. Each channel can achieve a raw BER of 3.8×10 −3 , a level that allows achieving a very low error rates by using appropriate forward error correction (FEC). We observe that the power penalty for OAM channel ` = +1 at FEC limit of 3.8× 10 −3 is 2.3 dB for the 5 cm obstruction. Figure 5.9 shows the measured BER for the channel on Bessel beam with ` = +3. For this case, power penalty of 1.7 dB is observed for the 5 cm obstruction. Also, 47 Figure 5.9: BER as a function of received SNR for OAM channel ` = +3 with and without obstruction. In all the cases, the obstructions are placed at the beam center. The radii of obstruction (r OBS ) are 5 cm, 6 cm, and 8 cm. the power penalties for both the channels increase with increasing obstruction sizes but remain < 10 dB and < 3 dB for Bessel beams with ` = +1 and ` = +3, respectively. Note that due to the smaller spot size of ` = +1 beam, channel on it experiences larger power penalty than the channel on ` = +3 for various obstruction diameter cases. Finally, Fig. 5.10 shows transverse profiles of obstructed and unobstructed Bessel beam ` = +1 at different distances. The obstruction radius in this case is 5 cm. The intensity images of unobstructed beams are also shown below each image of each of the obstructed beam for comparison. 48 Unobstructed l = + 1 Obstruction Plane Receiver Plane Obstructed l = + 1 Figure 5.10: Measured intensity images of obstructed and unobstructed Bessel beam ` = +1. Obstruction radius is 5 cm, and obstruction and receiver planes are 1 m apart. A comparison between the obstructed and unobstructed beams in the receiver plane shows reconstruction of the Bessel beam. 49 Chapter 6 Emulation of Free–space Distance 6.1 Introduction The amount of OAM carried by a beam can be identified by the number of 2π phase shifts, `, that occur across the wavefront, such that the phase is twisting in a helical fashion as it propagates [2]. The beam itself has an interesting structure, suchthat: (i)theintensityformsadoughnutringwithlittlepowerinthecenterand that grows with larger `, (ii) the phase changes in an azimuthal fashion according to the` number, and (iii) the beam itself diverges faster with a larger `. All these properties mean that the beam is complex and exhibits unique behavior. For applications such as communications, an important aspect is to experimentally evaluate the beam evolution under free-space propagation. The short distances (over few meters) available in a controlled lab environment make such analysis extremely difficult. Keeping in view these requirements, a laudable goal would be to have a bench–top setup that could emulate propagation of optical beams over a desired free-space distance by correctly reproducing the intensity and phase 50 M M BS BS Reference Beam Free-space Emulator M d f 1 f 2 SLM 2x HWP Col. λ 1550 nm Camera EDFA Free-space distance of length L Transmitted OAM Beam Received OAM Beam Free-space Emulator Input OAM Beam Output OAM Beam (a) (b) (c) M M BS BS Reference Beam Free-space Emulator M d f 1 f 2 SLM 2x HWP Col. λ 1550 nm Camera EDFA Free-space distance of length L Transmitted OAM Beam Received OAM Beam Free-space Emulator Input OAM Beam Output OAM Beam (a) (b) (c) Figure6.1: ConceptofaFree-spaceemulator. (a)OpticalbeampropagatedinFree-space of distance L; (b) A bench-top Free-space emulator to perform propagation experiments over long distances. characteristics of the optical beam. Figures 6.1(a)–(b) show the concept of free- space emulator, a system whose output field matches both in intensity and phase to a beam propagated over a specific distance in free-space. In this chapter, we describe a free-space emulator that can emulate free-space propagation of different distances for OAM beams [21]. To confirm our design, we propagate Laguerre Gaussian (LG) beams of different mode orders through the emulator and measure the spot size and curvature of the beams at the output. 6.2 Emulator Design We begin with LG beam (` = 0) having a beam waist w 0 . Due to the divergence, the beam waist at a distance z can be given by: w(z) =w 0 (1 + (z/z r ) 2 ) 0.5 (6.1) We define the ratio ofw(z) tow 0 as the magnification M of the free-space length, i.e.: M = w(z) w 0 (6.2) The divergence angle θ of such a beam can be calculated by using the relation: θ = 2 tan −1 [(w(z)−w 0 )/2z] (6.3) 51 Using these relations, we can calculate the focal length of an equivalent lens as: f eff =w 0 /θ (6.4) Next, we work out the focal lengths of two lenses (f 1 and f 2 ) required to match this system and the spacing d between the two lenses. We fix the focal length of the second lens f 2 and calculate the spacing d using the relation: d =f eff f 2 (M + 1)/ (M f eff +f 2 ) (6.5) The focal length of the first lens is then given by: f 1 =d/ (M + 1) (6.6) These calculations are verified by numerical simulations. The numerical simulations were carried out by propagating the optical beams using Huygens–Fresnel diffraction integral. Figure 6.2(a) compares the beam waist of a beam numerically propagated through the emulator with analytic results. As can be seen in this figure, beam spot sizes are in good agreement with the analytic results. We then calculate the radius of curvature (ROC) of the optical beam at the output of the emulator and compare it with analytic values of ROC. As shown in the Figure 6.2(b), for short distances the ROC values have large error. We believethatthisisduetoourdesignbeingbasedonsimplifiedgeometricalanalysis. To optimize the design parameters, a small adjustment is made in d in order to minimize the curvature error, by numerically propagating the beam (under thin-lens approximation) using Huygens–Fresnel diffraction integral. The resulting 52 Beam Waist Emulated Distance (m) 0 5 10 25 50 100 Beam Waist (mm) 0 10 20 30 40 Emulator Model Analytic Beam Waist (mm) Propagation Distance (m) l = 0 l = 1 l = 2 Emulated Distance (m) 0 5 10 25 50 100 Error (%) 0 20 40 60 80 100 120 140 160 180 200 Error (%) Emulated Distance (m) Before Optimization After Optimization Radius of Curvature Emulated Distance (m) (a) (b) Figure 6.2: (a) Numerical and analytic results for beam waist of the optical beam prop- agated through the emulator for various distances.(b) Percent error for the radius of curvature of optical beams at the output of the emulator with and without optimization of the emulator design parameters. ROC values after this optimization are also shown in the 6.2(b) which shows that minimization of the error. Figure 6.3 shows the numerical simulation results for the mode purity of different LG beams propagated through the emulator for different free-space distances using thedesignedsystemwithandwithoutoptimizationoftheemulatorparameters. As Emulated Distance (m) 0 5 10 25 50 100 Mode Purity (Simulation) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Mode Purity (Simulation) Propagation Distance (m) Before Optimization After Optimization l = 0 l = 1 l = 2 Figure 6.3: Numerical simulation results for mode purity calculations with and without optimizing the emulator parameters. The spacing between lenses is optimized to achieve a theoretical mode purity > 0.99. 53 can be seen in this figure, a mode purity > 0.99 is achievable through optimization of the parameters. 6.3 Experimental Setup In order to validate our design, an experiment was also performed for two distances (5 m and 25 m). For emulating a distance of 5 m, the parameters for f 1 , d (after optimization), andf 2 are 353 mm, 748 mm , and 400 mm, respectively. Figure 6.4 shows the experimental setup. A 1550 nm laser is split into two paths to form the two arms of a Mach-Zehnder Interferometer (MZI). In one arm, a phase-only reflective SLM is used to generate LG beams such that the beam waist of ` = 0 beam at the input of the free-space emulator is 2.2 mm. The path lengths of the two arms are matched such that in the absence of emulator, a plane wavefront is acquired at the output. We also compensate for system aberrations by digitally applying Zernike polynomials on the SLM. An InGaAs camera is used to record the intensity and interference profiles of the output beam. M M BS BS Reference Beam Free-space Emulator M d f 1 f 2 SLM 2x HWP Col. λ 1550 nm Camera EDFA Figure 6.4: Experimental setup for the Free–space emulator Col. Collimator; HWP: Half-wave plate; BS: Beamsplitter; SLM: spatial light modulator; M: Mirror; f: lens; L: distance between two lenses. 54 6.4 Results and Discussion In order to measure the beam waist of the output beam, the measured intensity is fitted to the intensity of the corresponding LG beam. Figures. 6.5(a)–(c) show the measured transverse intensity profiles for LG beams with ` = 0, +1, and +2, respectively, propagated over an emulated distance of 5 m. The second row in the same figure shows the cross–section of each propagated beam fitted to the theoretically predicted beams at same distance. Figure 6.6 shows the results for the measured beam waist along with the analytic results for different LG beams for distances of 5 and 25 m. For ROC measurements, the effective ROC of the output beam is estimated by first retrieving the phase of the output beam. For phase retrieval, we used the method of phase stepping interferometry. This method requires recording 16 different interferograms of the propagated beam by ‘stepping’ the SLM phase with a step of 2π/16. By post–processing the 16 interferograms, we are able to retrieve the overall wavefront of the propagated beam as shown in the right most panel X Slice (mm) -12 -6 0 6 12 Normalized Intensity 0 0.2 0.4 0.6 0.8 1 Measured Fit l = 0 (a) X Slice (mm) -12 -6 0 6 12 Normalized Intensity 0 0.2 0.4 0.6 0.8 1 Measured Fit l = 1 l = 1 X Slice (mm) -12 -6 0 6 12 Normalized Intensity 0 0.2 0.4 0.6 0.8 1 Measured Fit l = 2 (b) (c) l = 0 Figure 6.5: (a)–(c) Experimentally measured intensity profiles of LG beam with ` = 0, +1, and +2, respectively. Second row shows the intensity fit of measured intensity of different LG beams. 55 OAM Mode Order l 0 1 2 Beam Waist (mm) 2 4 6 8 10 Beam Waist (mm) OAM Mode Order l 25 m 5 m Analytic Experiment Figure 6.6: Theoretical and experimental results for beam waists of different LG beams propagated over 5 and 25 m. of Figure 6.7. After phase retrieval, the effective ROC of the propagated beam is calculated using following relation [56]: 1 R(z) = −λ 2πσ 2 (z) ∞ Z −∞ x ∂φ(x,z) ∂x |E(x,z)| 2 dx (6.7) in which, R(z) is the radius of curvature, λ is the wavelength, φ is the retrieved transverse phase, and E is the electric field. For the 5 m case, we measured an effective radius of curvature for Gaussian beam to be 23.57 m, a 0.028 % error. In order to emulate a distance of 25 m, we built a 3× reduction system to image the output plane of the free-space emulator and accounted for the demagnification in our calculations. For this arrangement, we measured an error of 0.037% for ` = 0 beam’s ROC. -5 0 5 -5 0 5 -3 -2 -1 0 1 2 3 Phase Stepped Interferograms Retrieved Phase Figure 6.7: Experimentally retrieving phase of the LG beams propagated through the emulator for a distance of 25 m using phase stepping interferometry that required record- ing 16 different interferograms. 56 Chapter 7 Apodized Apertures for OAM Beams 7.1 Introduction In free–space propagation, optical beams diverge due to diffraction effects. Depending upon the initial spot size of the optical beam at the transmitter and the propagation distance, the received beam may have a spot size larger than the receiving aperture thereby resulting in the truncation of the received beam. The truncation of an incoming OAM beam through an aperture smaller than the incoming beam results in power leakage to neighboring modes [22]. Just like with other types of apertures for which apodization might reduce negative effects, there exists a question as to the benefit of using an apodized aperture for OAM recovery. In this chapter, we experimentally demonstrate an apodized aperture for receivingadata–carryingOAMbeam. Inouranalysis,theperformanceofapodized aperture is compared with a hard aperture by varying aperture sizes and locations. 57 7.2 Concept of Apodized Apertures The concept of receiving an OAM beam through a limiting aperture is shown in Figures7.1(a)-(c). InanOAMmultiplexedlink,thepowerleakageintoneighboring modes could cause crosstalk among data channels, and depending upon the shape of the limiting aperture, the crosstalk could be reduced. In our analysis, we have chosen two limiting apertures: 1) in most optics, a common limiting aperture is a circular ‘hard aperture’, that represents the finite extent of the receiver; and 2) an apodized aperture to avoid hard edge effects. Hard aperture was represented by [57]: T H =Circ ρ ρ o ! (7.1) in which ρ is the radial coordinate and ρ o is the width of the aperture. For the apodized aperture,|Besinc| 2 was used as defined below: Besinc = 2J 1 (πρ) πρ (7.2) in which J 1 is the Bessel function of first kind. The two aperture functions were scaled to have same equivalent areas [57]. 7.3 Experimental Setup The experiment setup is shown in Fig. 7.1(d). SLM-A was used to generate OAM beams. The generated OAM beams were expanded by a 2× magnifying setup. The spiral phase mask on SLM-B was used to convert OAM beams having helical phase fronts back into beams with flat phase fronts for coupling into SMF. A single OAM mode` = +2 was first transmitted through the setup. In the plane of the SLM-B, 58 Figure 7.1: Concept of using apodized aperture and Experiment setup. (a) Field in front of the limiting aperture; (b) limiting aperture, two cases analyzed are ‘Hard’ aperture, and apodized aperture; (c) Field behind the limiting aperture; (d) Experiment setup. SLM: spatial light modulator; Col.: collimator; HWP: Half-wave plate; M: Mirror; BS: Beamsplitter. ` = +2 had its annulus (radial distance of intensity peak) of 2.4 mm. The received power in ` = +2 was determined by using a spiral phase mask of ` = -2. In order to realize the limiting aperture, spiral phase mask on SLM-B was multiplied with either hard or apodized aperture function as described above. The power leakage to the neighboring modes (` = 0, +1, +3 and +4) due the limiting apertures was measured by changing spiral phase mask to ` = -3 and -4, respectively. 7.4 Results and Discussion Our goal was to determine optimal position and width of the limiting apertures so that the power in ` = +2 is maximized while minimizing the power leaked to the neighboring modes. We first traversed the hard and the apodized apertures of fixed width (20% of the spot size) along the beam cross-section. Figs. 7.2(a)-(b) show the power in` = +2 and neighboring modes vs. aperture position normalized by the annulus position (i.e. the position where intensity peak occurred). As can be seen in Figs. 7.2(a)-(b), the power in ` = +2 maximized when the apertures were placed at the annulus of the OAM beam, however the leakage power in the neighboring modes was also higher. As the apertures move toward the beam 59 -1.0 -0.5 0.0 0.5 1.0 -45 -40 -35 -30 -25 -20 -15 Power (dBm) Aperture Position / Annulus Position Apodized (+2) Hard (+2) Apodized (+3) Hard (+3) -1.0 -0.5 0.0 0.5 1.0 -45 -40 -35 -30 -25 -20 -15 Power (dBm) Aperture Position / Annulus Position Apodized (+2) Hard (+2) Apodized (+4) Hard (+4) l = +2 l = +3 l = +2 l = +4 (a) (b) Norm.&Aprt.&Width&=&0.21& Norm.&Aprt.&Width&=&0.21& Figure 7.2: Received power in the desired and neighboring modes (a) ` = +2 and +3; (b)` = +2 and +4 for hard (squares), and apodized apertures (circles) of fixed size ( 21 % of spot size). center, the power in ` = +2 decreased slower than the power in the neighboring modes. The higher power in l = +2 and lower leakage to the neighboring modes was observed when an apodized aperture was used. This could be explained by larger support required by the apodization function, which allowed for more spatial information of the phase front to pass through. Similar trend was observed for ` = 0 and +1. Fig. 7.3(a) shows the power in modes ` = +2, +3 and +4 when the (a) (b) 0.4 0.6 0.8 1.0 -60 -50 -40 -30 -20 Power (dBm) Aperture Size / Annulus Size Apodized (+2) Hard (+2) Apodized(+3) Hard (+3) Apodized(+4) Hard (+4) Norm.&Aprt.&Pos.&=&&0.25& 0.4 0.6 0.8 1.0 -40 -30 -20 -10 0 Crosstalk (dB) Aperture Size / Annulus Size Hard (+3) Hard (+4) Apodized (+3) Apodized (+4) Norm.&Aprt.&Pos.&=&0.25& Figure 7.3: (a) Received power in modes ` = +2, +3 and +4, when hard and apodized apertures of different sizes are placed at fixed position; (b) Crosstalk in neighboring modes. 60 apertures of varying sizes were positioned at a fixed location. The power leaked from ` = +2 to +3 and +4 is shown as crosstalk in Fig. 7.3(b). In order to analyze the performance of apodized aperture for data carrying OAM beams, two OAM modes ` = +2 and +4 each carrying 50-Gbaud QPSK channels were multiplexed and transmitted through the two apertures with fixed size and position. Initially, unrestricted spiral phase masks (full aperture case) were used to represent the ideal case. The spiral phase masks were then multiplied with respective aperture functions of different widths and positions. The measured BER for the received modes are shown in Figures 7.4(a)-(d). The improvement due to apodized aperture varied depending on the aperture size and location. (a) (b) (c) (d) 10 12 14 16 18 20 22 5 4 3 2 Norm. Aprt. Width = 0.42 Norm. Aprt. Pos. = 0.25 Full Aprt. (+2) Full Aprt. (+4) Apodized (+4) Hard (+4) -Log(BER) OSNR (dB) EFEC Threshold 10 12 14 16 18 20 22 5 4 3 2 Norm. Aprt. Width = 0.42 Norm. Aprt. Pos. = 0.25 Full Aprt. (+2) Full Aprt. (+4) Apodized (+2) Hard (+2) -Log(BER) OSNR (dB) EFEC Threshold 10 12 14 16 18 20 22 5 4 3 2 Norm. Aprt. Width = 0.38 Norm. Aprt. Pos. = 0.17 Full Aprt. (+2) Full Aprt. (+4) Apodized (+2) Hard (+2) -Log(BER) OSNR (dB) EFEC Threshold 10 12 14 16 18 20 22 5 4 3 2 Norm. Aprt. Width = 0.38 Norm. Aprt. Pos. = 0.17 Full Aprt. (+2) Full Aprt. (+4) Apodized (+4) Hard (+4) -Log(BER) OSNR (dB) EFEC Threshold (a) (b) (c) (d) 10 12 14 16 18 20 22 5 4 3 2 Norm. Aprt. Width = 0.42 Norm. Aprt. Pos. = 0.25 Full Aprt. (+2) Full Aprt. (+4) Apodized (+4) Hard (+4) -Log(BER) OSNR (dB) EFEC Threshold 10 12 14 16 18 20 22 5 4 3 2 Norm. Aprt. Width = 0.42 Norm. Aprt. Pos. = 0.25 Full Aprt. (+2) Full Aprt. (+4) Apodized (+2) Hard (+2) -Log(BER) OSNR (dB) EFEC Threshold 10 12 14 16 18 20 22 5 4 3 2 Norm. Aprt. Width = 0.38 Norm. Aprt. Pos. = 0.17 Full Aprt. (+2) Full Aprt. (+4) Apodized (+2) Hard (+2) -Log(BER) OSNR (dB) EFEC Threshold 10 12 14 16 18 20 22 5 4 3 2 Norm. Aprt. Width = 0.38 Norm. Aprt. Pos. = 0.17 Full Aprt. (+2) Full Aprt. (+4) Apodized (+4) Hard (+4) -Log(BER) OSNR (dB) EFEC Threshold Figure 7.4: Measured BER for multiplexed modes ` = +2 and +4 when hard and apodized apertures of fixed size are placed at fixed locations. Full aperture refers to the case when spiral phase mask was unrestricted to represent ideal case. 61 References [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 (1992). [2] A. M. Yao, and M. J. Padgett, “Orbital angular momentum: origins, behaviour and applications”, Adv. Opt. Photon. 3, 161 (2011). [3] G. Gibson, J. Courtial, M. Padgett, M. Vasnetsov, V. Pas’ko, S. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum”, Opt. Express, vol. 12, pp. 5448-5456, (2004). [4] J. Wang, J.-Y. Yang, I. M. Fazal, N. Ahmed, Y. Yan, H. 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Abstract (if available)

Abstract Increasing capacity demands on optical networks have encouraged the researchers to explore new dimensions for multiplexing optical data channels. While wavelength-division multiplexing (WDM) and polarization-division multiplexing (PDM) have been extensively deployed in existing networks, one recently explored approach to significantly increase the capacity of communication systems is to employ spatial-division multiplexing (SDM) of multiple orthogonal modes using a single transmitter/receiver aperture pair, in which each mode carries an independent data stream. Orthogonality ensures that the modes can be efficiently multiplexed at the transmitter, spatially co-propagated, and demultiplexed at the receiver with minimal modal crosstalk. ❧ Of particular interest are the SDM techniques that multiplex data channels using optical beams that carry orbital angular momentum (OAM). In recent years, OAM has emerged as a potential approach to multiplex many spatially collocated optical data-carrying beams. Since OAM beams with different mode orders are orthogonal to each other, it is possible to efficiently multiplex and de-multiplex data channels carried by such beams. Recent optical and millimeter-wave (mm-wave) demonstrations have shown that structured beams carrying OAM can serve as a modal basis set for enhancing the capacity and spectral efficiency of free-space communication systems. ❧ The research work presented in this dissertation focuses on enabling OAM multiplexed communication systems for high-speed and high-capacity optical and mm-wave communications, and includes analysis and experimental demonstrations of: (1) optical networking functions for OAM multiplexed data channels

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Creator Ahmed, Nisar (author)

Core Title Orbital angular momentum based spatially multiplexed optical and millimeter-wave communications

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Defense Date 01/28/2016

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

Tag OAI-PMH Harvest,OAM,optical communications,orbital angular momentum,spatial multiplexing

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Advisor Willner, Alan (committee chair), Haas, Stephan (committee member), Molisch, Andreas (committee member)

Creator Email nisar.jamali@gmail.com,nisarahm@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-217509

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