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Using beams carrying orbital angular momentum for communications and remote sensing

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Using beams carrying orbital angular momentum for communications and remote sensing

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Content Using Beams Carrying Orbital Angular Momentum for Communications and Remote Sensing by Guodong Xie A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulllment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (Electrical Engineering) May 2018 Copyright 2018 Guodong Xie Dedication This dissertation is dedicated to my outrageously loving wife, Fangxiang Wang, my supportive parents and parents-in-law, Yongbin Xie, Jilan Sun, Huaijun Wang, Xiulan Zhao, my farsighted grandpa, Jingyu Xie, and my respectful advisor, Professor Alan E. Willner. ii Acknowledgements First of all, I would like to thank my advisor Professor Alan E. Willner, who is a super great and excellent mentor. I enjoyed every femtosecond I spent with him. What he taught me is not only the knowledge and methodology for research but also the way how to be a good person. In Chinese, we say \One who be your teacher one day, is the father for the whole lifetime," which is also the best word I want to describe him. I would also like to thank Professor Moshe Tur from Tel-Aviv University in Israel, another mentor of mine. I have beneted a lot and will benet more from the wisdom I learnt from him. I would like to thank Professor Andrea M. Armani and Professor Andreas F. Molisch for severing on my dissertation and qualication exam. I would like to thank Professor Alexander A. Sawchuk and Professor Wei Wu for severing on my qualication exam. I sincerly acknowledge my previous and current colleagues, particularly Dr. Yang Yue, Dr. Salman Khaleghi, Dr. Mohammad R. Chitgarha, Dr. Hao Huang, Dr. Nisar Ahmed, Dr. Yan Yan, Dr. Yongxiong Ren, Dr. Morteza Ziyadi, Mr. Changjing Bao, Mr. Amirhossein M. Ariaei, Mr. Ahmed Almaiman, Mr. Yinwen Cao, Mr. Zhe Zhao, Mr. Long Li, Mr. Peicheng Liao, Mr. Moshe Willner, Mr. Zhe Wang, Ms. Cong Liu, Mr. Ahmad Fallahpour, Mr. Runzhou Zhang, Mr. Kai Pang, Mr. Asher Willner, Mr. Haoqian Song, Mr. Kaihang Zou, Mr. Hao Song, and Dr. Jing Du. I want to extend my deepest appreciation to my collaborators, particularly Professor Miles J. Padgett, Professor Robert W. Boyd, Professor Mark Neifeld, Professor Jerey H. Shapiro, iii Professor Jian Wang, Dr. Samuel J. Dolinar, Dr. Giovanni Milione, Dr. Robert Bock, Dr. Solyman Ashra, Dr. Dmitry Starodubov, Dr. Shilpa Talwar, and Dr. Soji Sajuyigbe. I also appreciate the tremendous support by the sta of the Electrical Engineering Department, particularly, Ms. Diane Demetras, Ms. Corine Wong, Ms. Gerrielyn Ramos, Ms. Susan Wiedem, Mr. Tim Boston, and Ms. Anita Fung. Finally, I would like to thank my family. I would say every piece of my achievements also belongs to you. iv Table of Contents Dedication ii Acknowledgements iii List Of Tables vii List Of Figures viii Abstract xiv Chapter 1: Introduction 1 1.1 Generation and Detection of OAM Beams . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 OAM Beams for Channel Multiplexing over Communications Links . . . . . . . . . 4 1.3 Using OAM Spectrum for Remote Sensing . . . . . . . . . . . . . . . . . . . . . . . 6 Chapter 2: Turbulence Eects and Compensation for OAM Multiplexed Commu- nications Links 8 2.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Algorithm, Approach, and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 Chapter 3: Link Design Guideline for OAM Multiplexed Communications Sys- tems 17 3.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.3 Signal Power Loss Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Channel Crosstalk Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.5 Power Penalty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.6 System Performance in the Presence of Both Lateral Displacement and Receiver Angular Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.7 Experimental Validation of the Model . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.8 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Chapter 4: Tunable Steering and Multiplexing of OAM Beams 38 4.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.2 Experimental System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.3 Simulation and Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 41 4.4 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 v Chapter 5: Multiplexing Laguerre-Gaussian Beams with Dierent Radial Indices 52 5.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5.2 Approach, Design, and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.3 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 Chapter 6: Object Parameters Measurements Using Complex OAM Spectrum 63 6.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2 Concept and Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.3 Approach, Design, and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.4 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 6.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 Chapter 7: Conclusion 78 Reference List 79 vi List Of Tables 3.1 Parameters in the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 vii List Of Figures 1.1 The wavefronts, intensity proles, and phase proles of (a) a Gaussian beam, OAM beam ` = 0, (b) OAM beam ` = 1, and (c) OAM beam ` = 2. . . . . . . . . . . . . 2 1.2 (a) Generation of an OAM beam using a hologram, (b) Convert an OAM beam to a Gaussian beam using a conjucate phase hologram. . . . . . . . . . . . . . . . . . 3 2.1 Concept of SPGD algorithm. SPGD: stochastic-parallel-gradient-descent. . . . . . 9 2.2 SPGD algorithm for the (k + 1) th iteration. . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Experimental setup. Col.: collimator; HWP: half wave plate; SLM: spatial light modulator; M: mirror; BS: beam splitter; IR: infrared. . . . . . . . . . . . . . . . . 12 2.4 Far-eld intensity proles and phase interference patterns for various OAM beams before and after phase correction under a specic turbulence realization (r 0 = 1mm). The phase interference patterns are generated by interfering a Gaussian beam with the uncorrected/corrected OAM beams. . . . . . . . . . . . . . . . . . . 14 2.5 (a) Intensity correlation and (b) received power of OAM +3 with and without phase correction when only branch (1) is turned on. (c) Intensity correlation and received power as a function of the iteration number (i.e., the number of repetitions that the algorithm loop executed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.6 (a) The received power of various OAM modes before/after phase correction when only OAM+3 is transmitted. The iteration number for the SPGD algorithm under this turbulence retaliation is 83. (b, c) Measured BERs as functions of optical signal-to-noise ratio (OSNR) when the phase pattern derived from OAM+3 is used to correct the phases of three simultaneously transmitted OAM beams (OAM+1, OAM+3 and OAM+5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.1 (a)Concept of OAM multiplexed FSO communication link.(b) Simulation schematic of an OAM multiplexed data link. (c) Conversion from a Gaussian beam into an OAM+3 beam using an SPP+3 which causes helical phase shift from 0 to 6. Mod.: modulator, Tx: transmitter; Rx: receiver; SPP: spiral phase plate. . . . . . . . . . 20 viii 3.2 Alignment between the transmitter and receiver for (a) perfectly aligned system, (b) system with lateral displacement, (c) system with receiver angular error, and (d) system with transmitter pointing error. Tx: transmitter; Rx: receiver; z: trans- mission distance; d: lateral displacement; ': receiver angular error; : pointing error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.3 (a) Simulated spot size (diameter) of dierent orders of OAM beams as a function of transmitted beam size given a link distance of 100 m. (b) Minimum spot sizes of dierent orders of OAM beams at dierent transmission link distances. (c) Relative transmitted beam size to achieve the minimum spot size at the receiver. Note that for this gures analysis only, the size of the receiver aperture is not considered. . . 24 3.4 Simulated power loss as a function of receiver aperture size (diameter) when only OAM+3 is transmitted under perfect alignment for (a) z=100 m, (b) z=1 km, and (c) z=10 km. D t : transmitted beam size; z: transmission distance. . . . . . . . . . 25 3.5 (a) Simulated power distribution among dierent OAM modes as a function of lateral displacement over a 100-m link for which only the OAM+3 mode is trans- mitted, the transmitted beam size is D t =3 cm and the receiver aperture size D a is 4.5 cm. (b, c) XT-1 and XT-2, respectively, as a function of lateral displace- ment for dierent transmission distances with dierent transmitted beam sizes. The receiver size is 1.5 times the transmitted beam size. XT-1: relative crosstalk to the nearest-neighboring mode (OAM+4). XT-2: the relative crosstalk to the second-nearest-neighboring mode (OAM+5). . . . . . . . . . . . . . . . . . . . . . 26 3.6 (a) Simulated power distribution among dierent OAM modes as a function of receiver angular error over a 100-m link for which only the OAM+3 is transmitted, the transmitted beam sizeD t is 3 cm and the receiver aperture sizeD a is 4.5 cm. (b, c) XT-1 and XT-2, respectively, as a function of receiver angular error for dierent transmission distances and transmitted beam sizes. The receiver size is 1.5 times the transmitted beam size. XT-1: relative crosstalk to the nearest-neighboring mode (OAM+4). XT-2: the relative crosstalk to the second-nearest-neighboring mode (OAM+5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.7 (a) Simulated power distribution among dierent OAM modes as a function of transmitter pointing error over a 100-m link for which only OAM+3 is transmitted, the transmitted beam size D t is 3 cm and the receiver aperture size D a is 4.5 cm. (b, c) XT-1 and XT-2, respectively, as a function of transmitter pointing error for dierent transmission distances and transmitted beam sizes. The receiver size is 1.5 times the transmitted beam size. XT-1: relative crosstalk to the nearest- neighboring mode (OAM+4). XT-2: the relative crosstalk to the second-nearest- neighboring mode (OAM+5). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.8 Simulated system power penalty as a function of lateral displacement when dif- ferent sets of OAM beams are transmitted over a 100-m link. Mode spacing=1: OAM+1, +2, +3, and +4 transmitted. Mode spacing=2: OAM+1, +3, +5, and +7 transmitted. Mode spacing=3: OAM+1, +4, +7, and +10 transmitted. (a) The transmitted beam size D t =3 cm and the receiver aperture size D a =4.5 cm. (b) D t =10 cm and D a =15 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ix 3.9 Simulated system power penalty as a function of receiver angular error when dif- ferent sets of OAM beams are transmitted in a 100-m link. (a) The transmitted beam sizeD t =3 cm and the receiver aperture size D a =4.5 cm. (b)D t =10 cm and D a =15 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 3.10 Simulated system power penalty as a function of transmitter pointing error when dierent sets of OAM beams are transmitted in a 100-m link. (a) The transmitted beam sizeD t =3 cm and the receiver aperture size D a =4.5 cm. (b)D t =10 cm and D a =15 cm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.11 Simulated system power penalty as a function of (a) lateral displacement, (b) re- ceiver angular error, (c) transmitter pointing error when the mode spacing is 2 and transmission distance is 100 m. D t : Transmitted beam size. The receiver aperture size D a is three times the size of the transmitted beam size (D a =1.5D t ). . . . . . 34 3.12 (a) Comparison between experimental and simulated power loss of dierent OAM beams as a function of receiver aperture size. The transmitter and receiver are perfectly aligned. (b) Comparison between experimental and simulated power dis- tribution among dierent OAM modes as a function of receiver aperture size with a lateral displacement of 0.2 mm. Lines and symbols are simulation and experiment results, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.1 Principle of the technique for the simultaneous generation and steering of multiple OAM beams. (a) The concept of using an antenna array for the simultaneous generation and steering of multiple OAM beams. x: horizontal direction;y: vertical direction; z: direction perpendicular to the antenna array plane; Mod: modulator; Comb.: beam combiner; : time delay. (b) The relative phase change in the azimuthal direction for OAM generation and the relative phase for horizontal beam steering. (c) The design of an antenna array for the generation and steering of two OAM beams. ( 11 , 12 , 13 , 14 , 15 , 16 , 17 ,and 18 , are designed for the generation and steering of one OAM beam, while 21 , 22 , 23 , 24 , 25 , 26 , 27 ,and 28 are designed for the generation and steering of another OAM beam). . . . . . . . . . . . . . . . 40 4.2 Simulation and experiment results of the generation of one OAM beam. (a) The intensity prole of the OAM+1 beam generated from an array of 8 antenna elements at 1.2 m. (b) The interference pattern of the generated OAM+1 beam with a Gaussian beam. (c1-c5) The intensity proles of OAM+1 beams generated from 8 antennas at 0.4 m, 0.6 m, 0.8 m, 1.0 m and 1.2 m, respectively. . . . . . . . . . . . 42 4.3 Simulation and experimental results of the OAM beam generation with dierent array radii. The intensity proles of the generated OAM +1 beams with array radii of (a) 4.5 cm, (b) 5.0 cm and (c) 6.0 cm. (d) Comparison between the simulation and experimental results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 x 4.4 The design of time delay. (a) The delays of the 8 antennas elements for the gen- eration of the OAM+1 and OAM-1 beams. (b) The delays for the 8 antennas elements for the steering of the beams. The bottom two panels show the delays applied to the 8 antennas elements for the simultaneous generation and steering of the OAM+1 (c) and OAM-1 beams (d). T c : the period of the carrier wave. In practice, only positive delays were implemented so that the element with the most negative delay (A3) was considered a reference, having zero delay, and all other elements were assigned relative delays according to the gures above. . . . . . . . . 44 4.5 Experimental and simulation results of the simultaneous generation and steering of one and two OAM beams. (a1-a6) Intensity proles of the steered OAM+1 beam with a steering angle of 0 , 5 , 10.2 , 15.5 , 21.3 , 27.8 , and 35.3 , respectively. (b1-b4) Simulation results of the steering of OAM+1 beam when the antenna el- ements have dierent emitting angle (EA). (c) Comparison between the designed steering angle and the measured steering angle. (d1-d4) Intensity proles of the steered superposition of the OAM+1 beam and OAM-1 beam with steering angles of 0 , 10.2 , 21.3 , and 35.3 , respectively. (e1-e4) The distribution of the received power among several modes when only the OAM+1 beam is generated with steering angles of 0 , 10.2 , 21.3 , and 35.3 , respectively. The measurement is performed at 1.2 m. The plot represents the average of multiple measurements, while the error bars indicate standard deviations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 4.6 BER measurement. (a1) BER performance when only one OAM beam is generated and steered with various steering angles. (b-e) BER performance when one or two OAM beams are transmitted at dierent steering angles. xt: crosstalk; w/: with; w/o: without; b2b: back to back, where the transmitter is connected to the receiver using a cable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.7 Measurement design. (a) The measurement setup to scan the intensity prole of the generated OAM beams without steering. (b) The measurement setup to scan the interference pattern of the generated OAM beams without steering. (c) The measurement setup to scan the intensity proles of the steered OAM beams. (d) The measurement setup to test the mode purity of the steered OAM beams. . . . 48 4.8 Simulation results for dierent antenna arrangements. (a) The arrangement of the antennas; L: number of ring layers, the cases of L = 1; 2; 3; 4 are investigated; N s : number of antennas on each of the rings, the cases ofN s = 8; 16; 32 are investigated. (b1-b6) The intensity proles of the generated OAM+1 beam for dierent L and Ns values. Eciency: the power on the inner-ring of the generated beam over the power of the whole beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 5.1 (a) Intensity proles and phase fronts of Laguerre-Gaussian (LG) beams with dif- ferent azimuthal indices and radial indices. . . . . . . . . . . . . . . . . . . . . . . 53 5.2 (a) Experimental setup of a free space optical communication system multiplexing LG `=0;p=0 andLG `=0;p=1 . QPSK: quadratic-phase-shift-keying; SLM: spatial light modulator; M: mirror; Col: collimator; BS: beam splitter. (b, c) De-multiplexing principle of LG `=0;p=0 and LG `=0;p=1 channels, respectively. . . . . . . . . . . . . 55 xi 5.3 Principle of using a single mode ber (SMF) as a lter to separate the LG `=0;p=1 andLG `=0;p=0 beam: the simulated power loss of anLG `=0;p=1 beam when ltered by an SMF. (a) The transmission distance varies from 0 to 100 mm. (b) The transmission distance varies from 0 to 100 um. Subgures in (b): the intensity proles of the LG `=0;p=1 beam at various distances in the ber. . . . . . . . . . . 56 5.4 (a1 and a2) Intensity proles of the generated LG `=0;p=0 and LG `=0;p=1 beams, respectively. (b1 and b2) Phase patterns for the de-multiplexing of LG `=0;p=0 and LG `=0;p=1 beams, respectively. C r : circle radius. . . . . . . . . . . . . . . . . . . . 57 5.5 Denition of the match and mismatch between the r of the de-multiplexing pattern and the size of the LG `=0;p=1 beam. . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.6 Crosstalk between the LG `=0;p=0 and LG `=0;p=1 channels as a function of circle radius for the de-multiplexing pattern. Tx: p=0; Rx: p=1 refers to the power received by the receiver for the LG `=0;p=1 beam while LG `=0;p=0 beams is trans- mitted. (a-f) The intensity proles of the beams in front of the single mode ber. C r : Circle radius on the pattern for channel de-multiplexing. Note that in the cases of Tx: p=0; Rx: p=0 and Tx: p=1; Rx: p=0, an all-zero pattern on SLM-3 is used for the channel de-multiplexing and the variation of C r does not change the pattern. 59 5.7 Bit error rate (BER) as a function of an optical signal to noise ratio (OSNR) with various circle radii of the de-multiplexing pattern for LG `=0;p=0 and LG `=0;p=1 channels, respectively. B2B: When only one channel is turned on with the other channel turned o. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.8 Crosstalk between the LG `=1;p=0 and LG `=1;p=1 channels as a function of circle radius for the de-multiplexing pattern. Tx: p=0; Rx: p=1 refers to the power received by the receiver forLG `=1;p=1 beam whileLG `=1;p=0 beams is transmitted. C r : Circle radius on the pattern for channel de-multiplexing. . . . . . . . . . . . . 61 5.9 Crosstalk among LG modes with dierent ` and/or dierent p values with `=0 or 1 and p=0 or 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6.1 The concept of using the OAM spectrum to measure an objects parameters. The beams intensity prole and OAM spectrum (a) before object truncation, (b) right after object truncation, and (c) some distance after object truncation. . . . . . . . 66 6.2 (a) Experimental setup. The SLM-3, L-1, L-2, collimator and power monitor form the complex OAM spectrum analyser. Col: collimator, HWP: half-wave plate, M: mirror, SLM: spatial light modulator, L: lens. (b) The shape and position of the object. : opening angle of the object; : orientation of the object relative to the y-axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 xii 6.3 (a1-a8) Various orientations (states) for an object having an opening angle of 2/3. (b1-b8) The image of the light beam truncated by the objects in (a1-a8), respec- tively. (c) The OAM intensity spectrum measured for the light truncated by the objects in (a1-a8). (d) The OAM intensity spectrum measured for the light trun- cated by objects having various opening angles. (e) The relationship between the opening angles and the rst-dip position in the OAM intensity spectrum. (f1-f3) Objects having 1, 2 and 3 slots, each of which has an opening angle of 2/6. (g1-g3) The images of the light beam truncated by the objects in (f1-f3). (h) The OAM intensity spectrum measured for the light truncated by the objects in (f1-f3). (i) The OAM intensity spectrum measured for the light truncated by an object having an opening angle of 2/3 when the distance from the object to the OAM spectrum analyser is 0 cm, 35 cm, 52 cm and 104 cm. (j) The OAM intensity spectrum matrix measured when various OAM beams are used as the probe. The distance from the object to the OAM spectrum analyser is 0 in c, d, e and h, j. . . . . . . . 69 6.4 (a) The OAM phase spectrum measured for the light truncated by an object having an opening angle of 2/3 but various negative orientation angles. (b) The OAM phase spectrum measured for the light truncated by an object having an opening angle of 2/3 but various positive orientation angles. (c) The relationship between the orientation angle of the object and the slope of the OAM phase specturm. (d) The OAM phase spectrum measured for the light truncated by an object having various opening angles and an orientation angle of /8. (e, f) The OAM phase spectrum measured for the light truncated by an object having an opening angle of 2/3 and an orientation angle of /4. (e) No pre-estimate for . (f) Pre-estimate of = =8. In this Figure, the distance from the object to the OAM spectrum analyser is 0. The lines show the simulation results, and the symbols show the experimental measurements. In the measurement, the data is calculated by an Arctan function, and the measured phases are between -/2 and /2. Therefore, we add a or phase shift to some of the measurements for the convenience of phase slop calculation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 6.5 The OAM intensity spectrum measured for the light truncated by an object having an opening angle of 2m=5, where m = 1; 2; 3; 4. . . . . . . . . . . . . . . . . . . . 74 6.6 Principle used to measure the OAM phase spectrum. T 0 , T 45 , T 90 , and T 135 are calculated according to` 1 and` 2 and are loaded to SLM-3 sequentially to measure the phases between OAM ` 1 and ` 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 xiii Abstract An electromagnetic (EM) wave can carry orbital angular momentum (OAM) if the wave has a helical transverse phase structure of exp(j`), in which is the transverse azimuthal angle and ` is an unbounded integer (the OAM state number). OAM beams with dierent ` values are mutually orthogonal to each other, allowing them (i) to be multiplexed together along the same beam axis and de-multiplexed with low crosstalk to establish a channel-multiplexed line-of-sight communications link (e.g., spaced division multiplexing or SDM), (ii) to form an orthogonal mode basis such that other beams could be decomposed over this basis to provide potential advantages for a remote sensing system. OAM based SDM links have been demonstrated in both the optical domain and the radio fre- quency (RF) domain. Specically, a 100-Tbit/s communication link combining SDM, wavelength division multiplexing (WDM), and polarization division multiplexing (PDM) was demonstrated at the wavelength1550 nm in 2013. In addition, a 32-Gbit/s communication link combining the SDM and PDM at a carrier frequency of 28 GHz was demonstrated in 2014. However, these proof-of-concept demonstrations did not take the potential system design challenges into con- sideration. In this thesis, the challenges and potential solutions for the OAM based free-space SDM system are discussed, including: (i) the atmospheric turbulence eects and the turbulence compensation approaches, (ii) system-level design guidelines for long-distance OAM multiplexed links, (iii) beam steering for OAM multiplexing based communication systems, and (iv) the use of Laguerre-Gaussian beams with dierent radial indices for channel multiplexing. xiv Besides, OAM beams' unique intensity and phase structures could potentially provide a pow- erful tool for remote sensing other than communications. As an example, this thesis also explores the measurements of an object's opening angle and the opening's orientation using the information provided by OAM complex spectrum. xv Chapter 1 Introduction Although electromagnetic (EM) waves have been studied for well over a century, one property of EM waves, namely orbital angular momentum (OAM), was discovered in the 1990's [1]. An EM wave carrying OAM has a helical transverse phase structure of exp(j`), in which is the transverse azimuthal angle and` is an unbounded integer (the OAM state number) [1, 2]. Figure 1.1 shows the wavefronts, the intensity proles, and the phase proles of a Gaussian beam (OAM ` = 0), and two OAM beams (OAM ` = 1; 2). Important characteristics of each OAM beam include: (i) the intensity has a \doughnut" shape with little power in the center [2], (ii) the diameter of the beam grows with a larger OAM order [2], and (iii) OAM relates to the spatial phase prole rather than to the state of polarization of the beam, which is associated with spin angular momentum [3]. OAM beams with dierent` values are mutually orthogonal to each other, allowing them (i) to be multiplexed together along the same beam axis and de-multiplexed with low crosstalk to establish a channel-multiplexed line-of-sight communication link (e.g., spaced division multiplexing or SDM) [2, 4, 5, 6, 7], (ii) to form an orthogonal mode basis such that other beams could be decomposed over this basis to provide potential advantages for a remote sensing system [8, 9, 10, 11]. 1 -10 -8 -6 -4 -2 0 2 4 6 8 x 10 -4 -10 -8 -6 -4 -2 0 2 4 6 8 x 10 -4 [m] [m] -10 -8 -6 -4 -2 0 2 4 6 8 x 10 -4 -10 -8 -6 -4 -2 0 2 4 6 8 x 10 -4 [m] [m] -10 -8 -6 -4 -2 0 2 4 6 8 x 10 -4 -10 -8 -6 -4 -2 0 2 4 6 8 x 10 -4 -10 -8 -6 -4 -2 0 2 4 6 8 x 10 -4 -10 -8 -6 -4 -2 0 2 4 6 8 x 10 -4 != 0 != 1 ‘No OAM’ != 2 Intensity Phase (a) (b) (c) Figure 1.1: The wavefronts, intensity proles, and phase proles of (a) a Gaussian beam, OAM beam ` = 0, (b) OAM beam ` = 1, and (c) OAM beam ` = 2. 1.1 Generation and Detection of OAM Beams Many approaches for creating OAM beams have been proposed and demonstrated. One could obtain a single or multiple OAM beams directly from the output of a laser cavity [12, 13] or by converting a fundamental Gaussian beam into an OAM beam outside a cavity. The converter could be a spiral phase plate, diractive phase holograms [14, 15, 16], metamaterials [17, 18, 19, 20], cylindrical lens pairs [21], q-plates [22, 23], or ber structures [24, 25]. There are also dierent ways to detect an OAM beam, such as using a converter that creates a conjugate helical phase, or using a plasmonic detector [26]. Among all external-cavity methods, perhaps the most straightforward one is to pass a Gaussian beam through a coaxially placed spiral phase plate (SPP) [27, 28] as shown in Figure 1.2(a). A limitation of using an SPP is that each OAM state requires a dierent plate. An SPP is an optical element with a helical surface. To produce an OAM beam with a state of `, the thickness prole of the plate should be machined as `=2(n 1), withn being the refractive index of the medium. As an alternative, recongurable diractive optical elements, e.g., a pixelated spatial light modulator (SLM), or a digital micro-mirror device can be programmed to function as any refractive element of choice at a given wavelength. It should also be noted that the aforementioned methods produce OAM beams with only modulating the phase of the incoming beam. To generate a pure mode, one must jointly control both the phase and the intensity of the incoming beam. This could be achieved using a phase-only SLM with a more complex phase hologram [29]. 2 Besides the above mentioned approach, some novel material structures, such as meta-surfaces, can also be used for OAM generation [17, 18, 19, 20]. A compact meta-surface could be made into a phase plate by manipulating the spatial phase response caused by the structure. Another interesting liquid crystal-based device named q-plate is also used as a mode converter which converts a circularly polarized beam into an OAM beam [22, 23]. A q-plate is essentially a liquid crystal slab with a uniform birefringent phase retardation of and a spatially varying transverse optical axis pattern. Along the path circling once around the center of the plate, the optical axis of the distributed crystal elements may have a number of rotations dened by the value of q. Almost all of the mode conversion approaches can also be used to detect an OAM beam. For example, an OAM beam can be converted back to a Gaussian-like non-OAM beam if the helical phase front is removed [4], e.g., by passing the OAM beam through a conjugate SPP or phase hologram as shown in Figure 1.2(b). Incoming Gaussianbeam ConvertedintoOAM beam Incoming OAMbeam Re-convertedinto Gaussian Holographic phasefilter (a) (b) Figure 1.2: (a) Generation of an OAM beam using a hologram, (b) Convert an OAM beam to a Gaussian beam using a conjucate phase hologram. 3 1.2 OAM Beams for Channel Multiplexing over Communications Links Waves transmitted and received using a single aperture pair (that is, the data pass through one aperture at the transmitter and are received by another aperture at the receiver) have been used for free-space optical and wireless communications long time ago. Although there have been many advances, such as advanced modulation format and polarization division multiplexing multiplexing (PDM), wavelength or frequency division multiplexing (WDM or FDM), the basic system architecture has remained fairly unchanged. Thus, OAM mode multiplexing of multiple orthogonal data-carrying beams multiplexed at the transmitter and de-multiplexed at the receiver using a single transmitter/receiver aperture pair would represent a signicant architectural change [5, 7]. Indeed, channel multiplexing using OAM beams has recently seen exciting progress in both optical domain and radio frequency (RF) domain. Optical systems have been demonstrated with the following selected results: (i)2-Tbit/s free-space transmission over 1 meter using OAM and PDM [7], (ii) 100-Tbit/s free-space transmission over 1 meter using OAM multiplexing, PDM, and WDM [30], (iii)400-Gbits/ free-space transmission over 120 meters using OAM [31], (iv) 2-Tbit/s using OAM and WDM over1.1-km of specially fabricated vortex ber for which the OAM modes exhibit relatively little power coupling as they propagate [32], and (v) using OAM to send data between two sites over 143 kilometers [33]. RF systems have been demonstrated with the following selected results: (i) A high-capacity mm-wave communication link by transmitting eight multiplexed OAM beams (four OAM beams on each of the two orthogonal polarizations), achieving a capacity of 32 Gbit/s at a single carrier frequency of 28 GHz [34], and (ii) 4-Gbps uncompressed video transmission over a 60-GHz OAM wireless channel [35]. This thesis focuses on the performance metrics investigation and system robustness improve- ment for OAM multiplexed communication link, which includes the following topics: 4 1. Phase correction for a distorted OAM beam using a Zernike polynomials based stochastic- parallel-gradient-descent (SPGD) algorithm [36]: an SPGD algorithm based on Zernike poly- nomials is proposed to generate the phase correction pattern for a distorted OAM beam. The Zernike-polynomial coecients for the correction pattern are obtained by monitor- ing the intensity prole of the distorted OAM beam through an iteration-based feedback loop. The experimental results show that the proposed approach improves the quality of the turbulence-distorted OAM beam. Moreover, the phase correction patterns derived from a probe OAM beam through emulated turbulence is used to correct other OAM beams transmitted through the same turbulence. The experimental results show the patterns de- rived this way simultaneously correct multiple OAM beams propagating through the same turbulence and the crosstalk among these modes is reduced by more than 5 dB. 2. Metrics and design considerations for a free-space optical OAM multiplexed communication [37]: The design considerations for an OAM multiplexed free-space data link is studied, an- alyzing power loss, channel crosstalk and power penalty of the link in the case of limited-size receiver apertures and misalignment between the transmitter and receiver. The trade-os for dierent transmitted beam sizes, receiver aperture sizes and mode spacing of the transmitted OAM beams under given lateral displacements or receiver angular errors are investigated. The simulation and experimental results show that (i) a system with a larger transmitted beam size and a larger receiver aperture is more tolerant to the lateral displacement but less tolerant to the receiver angular error; (ii) a system with a larger mode spacing, which uses larger OAM charges, suers more system power loss but less channel crosstalk; thus, a system with a small mode spacing shows lower system power penalty when system power loss dominates (e.g., small lateral displacement or receiver angular error) while that with a larger mode spacing shows lower power penalty when channel crosstalk dominates (e.g., larger lateral displacement or receiver angular error). 5 3. Demonstration of tunable steering and multiplexing of two 28 GHz data carrying OAM beams using antenna array [38]: The simultaneous generation and steering of two OAM beams utilizing a custom-designed circular antenna array at 28 GHz is simulated and experimentally demonstrated. A steering angle of up to 35 degrees from the antenna array normal is achieved. The resutls show that (i) the antenna array radius is inversely propositional to the divergence of the generated OAM beam; (ii) the steering angle of the generated OAM beams is limited by the emitting angle of the antenna elements; and (iii) a larger steering angle may degrade the mode purity of the generated OAM beams as well as increase power penalty caused by the inter-symbol-interference of the data channel. Besides, the transmission of two 1-Gbaud quadratic phase shift keying (QPSK) signal over the two steerable multiplexed OAM beams with both multiplexed channels achieved bit error rates (BERs) 3:8 10 3 with steering angles of up to 35.3 degrees is demonstrated. 4. Experimental demonstration of a 200 Gbit/s free-space optical link by multiplexing Laguerre- Gaussian (LG) beams with dierent radial indices [39]: A 200-Gbit/s SDM system using two Laguerre-Gaussian beams with dierent radial indices (LG `=0;p=0 ,LG `=0;p=1 ) is demon- strated. A single mode ber (SMF) is used to spatially separate the two multiplexed beams at the receiver. With a proper design of the radial change of the de-multiplexing pat- tern, the channel crosstalk could be minimized and both channels could achieve a BER of 3:8 10 3 . Moreover, the multiplexing of four LG beams with dierent azimuthal indices and dierent radial indices (e.g., LG `=0;p=0 , LG `=0;p=1 , LG `=1;p=0 , LG `=1;p=1 beams) are also demonstrated with a <12 dB channel crosstalk. 1.3 Using OAM Spectrum for Remote Sensing Besides, OAM beams' unique intensity and phase structures could potentially provide a powerful tool for remote sensing other than communications. As an example, this thesis also explores the 6 measurements of an object's opening angle and the opening's orientation using the information provided by OAM complex spectrum [40]: A light beam can be decomposed into a set of spatial modes that are taken from an orthogonal basis. As a proof-of-concept example, we measure an objects opening angle using orbital angular momentum (OAM) based complex spectrum (mode decomposation). The results show that the dip (i.e., notch) positions of the OAM intensity spectrum are dependent on an objects opening angle but independent of the openings angular orientation, whereas the slope of the OAM phase spectrum is dependent of the openings orientation but independent on the opening angle. 7 Chapter 2 Turbulence Eects and Compensation for OAM Multiplexed Communications Links 2.1 Background and Motivation Orbital angular momentum (OAM) multiplexing has emerged as an optical communication tech- nique for the simultaneous transmission of spatially overlapping orthogonal modes that can be eciently multiplexed at the transmitter and demultiplexed at the receiver [2, 7]. Terabit/s data transmissions using OAM multiplexing over short free-space ranges have been demonstrated with little distortion imposed by the propagation [7, 30]. However, the beams in a practical free- space link may be distorted during propagation due to factors such as atmospheric turbulence [41, 42, 43, 44, 45, 46]. In such cases phase-front distortions can severely degrade demultiplexing performance, motivating attempts to apply phase-front correction prior to demultiplexing. A critical challenge for phase-front correction is characterizing the wave-front distortion in- curred on each OAM beam. OAM beams with `6= 0 have a helical phase structure with a phase and power singularity at beam center. Standard wave-front sensors, such as the Shack-Hartman, rely on intensity dierentials between parts of a received beam to calculate the distortion [47, 48], this power singularity may cause compensation systems using these traditional sensors to fail [49]. 8 Turbulence compensation for a free-space OAM multiplexed system has been demonstrated in a system that used a Shack-Hartmann sensor to measure the wave-front distortion on a Gaussian beam (` = 0) that co-propagated with, but was polarized orthogonally to, the OAM communica- tion beams [50]. It would be desirable to design a sensing method whose probe beam could have a phase singularity (e.g. an OAM beam with `6= 0 where the probe beam could have more overlap with the information-carrying OAM beams during propagation). Rather than measuring the phase of the incoming distorted OAM beam, we propose to use its intensity pattern, together with a Zernike-polynomials-based stochastic-parallel-gradient-descent (SPGD) algorithm [47, 48, 51, 52] to derive the phase correction pattern (in [33], intensity prole monitoring was employed to recognize the order of the distorted OAM beams but the distortion was not nally corrected) as shown in Figure 2.1. The resulting phase correction pattern can eciently remove a signicant part of the acquired distortion on the probe beam. In addition, this pattern can be used as the phase correction of other OAM beams that passed through the same distorting transmission medium. Experimental results with three OAM beams show that the crosstalk of each OAM channel can be reduced by >5 dB using our SPGD phase correction technique. Distorted OAM Beam Wave-front Corrector After Correction SPGD Algorithm SPGD Algorithm Based Correction Loop Intensity Profile Figure 2.1: Concept of SPGD algorithm. SPGD: stochastic-parallel-gradient-descent. 9 2.2 Algorithm, Approach, and Results A 2-dimensional phase pattern on the unit disk can be approximated, in polar coordinates, by a linear combination of orthogonal Zernike polynomials Z n (r;) as [53, 54, 55]: '(r;;a 1 ;a 2 ;:::;a N ) = N X n=1 a n Z n (r;); 0r 1; 0 2 (2.1) Here a n is the coecient of the n th Zernike polynomial, and N is the number of terms (up to 14 terms in our experiment) in the approximation. The critical step in deriving a correction pattern for a distorted OAM beam is thus to determine the coecients of Zernike polynomial in a fast and accurate way. The proposed SPGD algorithm for determining these coecients is as follows, Feedback signals: As shown in [56], the mode purity of an OAM beam increases monotonically with increasing quality of its intensity prole, which is dened as correlation coecientC k between the far-eld intensity prole of the OAM beamI(r;) and its ideal intensity distributionI id (r;), namely, C k = Z 1 0 Z I(r;)I id (r;)ddr (2.2) The higher the correlation coecient the closer is the measured OAM beam to its unperturbed shape. This suggests that the intensity prole can be used to derive the error signal in the feedback loop to update the correction phase pattern. Initialization: The algorithm starts with a blank correction pattern: ' 0 ='(r;; 0; 0;:::; 0). Correction-pattern iteration: Given the k th (k 0) iteration, the process for the (k + 1) th iteration of the algorithm is shown in Figure 2.2. First, the current correction pattern ' k (r;) = '(r;;a 1;k ;a 2;k ;:::;a N;k ) is used to partially restore the OAM beam's wave-front and the intensity prole of the partially-restored beamI 1;k (r;) is recorded. The correlation coecientC 1;k between 10 I 1;k (r;) and its ideal theoretical proleI id (r;) is then calculated. Next, another phase pattern, ' k (r;) + (r;), is tried, where (r;) = N X n=1 s n Z n (r;) (2.3) with s n ;n = 1; 2;:::N, being a random sequence of1 values and is a small number, typically 0.01 (the smaller the better at the expense of longer convergence time). A new measurement of I 2;k (r;) is then obtained and a new correlation coecient C 2;k is calculated. Correction-pattern update: The correction pattern for each iteration is updated using the function ' k+1 (r;) = ' k (r;) +(C 2;k C 1;k )(r;), where is an empirically determined constant which is 200 in our experiment. A smaller leads to a lower learning rate and a larger leads to over-correction, both of which decrease the convergence speed. This update for Zernike- polynomial coecients is thus: ' k+1 (r;) ='(r;;a 1;k+1 ;a 2;k+1 ;:::;a N;k+1 ) a n;k+1 =a n;k +s n (C 1;k C 2;k ); n = 1; 2;:::; N (2.4) Current Correction Pattern φ k Correction Pattern with perpetuation φ k +Δ whereΔ=Σs n δZ n (θ,r) Measured Intensity Profile I 1,k Measured Intensity Profile I 2,k Phase Correction Pattern φ k+1 =φ k +η Δ(C 1,k -C 2,k ) Principle of stochastic-parallel-gradient-descent (SPGD) algorithm Intensity Correlation C 2,k n=1 N Intensity Correlation C 1,k Figure 2.2: SPGD algorithm for the (k + 1) th iteration. All the Zernike-polynomial terms are updated simultaneously, with the principal update being on the dominate term, i.e. the term causing the largest distortion, while the remaining terms 11 experience random walks (in the + or - directions) that eventually cancel out. In the experiments reported below, less than 100 cycles are required to obtain practical convergence, after which additional iterations had negligible eect on thefa n g coecients. SLM-5 Coupler Col. Col. HWP M BS Col. SLM-2 M SLM-4 M IR camera SPGD algorithm Signal generation Signal Detection Delay 0 lens SLM-1 Branches (2) and (3) are turned on only during BER measurement (1) SLM-3 M Emulated turbulence IR Camera Power meter (2) (3) Figure 2.3: Experimental setup. Col.: collimator; HWP: half wave plate; SLM: spatial light modulator; M: mirror; BS: beam splitter; IR: infrared. Figure 2.3 shows our experimental setup for SPGD phase correction. We rst generate a 50-Gbaud QPSK signal at 1550 nm. The signal is then copied into three branches, after being decorrelated by propagation through bers of dierent lengths. Collimators at the end of each branch couple the light into free space in the form of three Gaussian beams. These beams are then converted into OAM beams of dierent orders via spatial light modulators (SLMs) with dierent phase patterns. One beam (branch (1)) is converted into OAM+3 for use as the turbulence probe, while the others (branches (2) and (3)) are respectively converted into OAM+1 and OAM+5. Note that branches (2) and (3) are turned on only when bit-error-rates (BERs) are being measured. After they are combined by the beam splitters, the three beams are transmitted through a turbulence emulator, which is a phase-screen plate that emulates Kolmogorov-spectrum turbulence with an r 0 = 1 mm Fried parameter at 1550 nm [54]. Here, we verify our approach under static turbulence. However, in a practical system, the approach should be fast enough to track the dynamic varying of atmosphere turbulence. At the receiver, the intensity prole of the probe beam is recorded by an infrared (IR) camera, providing the feedback signal to the SPGD 12 algorithm. The phase-correction pattern, generated by the SPGD algorithm is loaded onto SLM-4 for wave-front restoration of one or more OAM beams. After that restoration, the desired OAM beam is down-converted into a Gaussian-like beam via SLM-5 and then coupled into a single mode ber for coherent detection. To monitor the performance of the SPGD-based phase correction, we measured both the far- eld intensity prole of each received OAM beam and its interference pattern with a Gaussian beam (before and after phase correction). The Fried parameter of the emulated turbulence is held asr 0 = 1mm, and the emulator's phase screen does not rotated during these measurements. Because OAM beams have`-dependent beam diameters,D ` , at the emulator, they thus experience a given turbulence dierently. For OAM+1 to OAM+5, D ` =r 0 assumes the values 1.5, 1.8, 2.1, 2.4, and 2.7, respectively. We rst transmit OAM+3 through the emulated turbulence and use the SPGD algorithm to generate the phase-correction pattern. This correction pattern is then used to correct the wave-front distortions of a Gaussian beam, as well as those of OAM+1 to OAM+5 beams. Figure 2.4 shows the intensity proles and interference patterns we obtained for the dierent OAM beams. Figures 2.4(c, d) show that the emulated turbulence distorts both the intensity prole of the OAM beams and the interference patterns. After phase correction, however, both the intensity proles and the interference patterns indicate that the improved mode purity (see Figures 2.4 (e, f)). Figure 2.5(a) shows the intensity correlations of OAM+3 with and without phase correction for various turbulence realizations under static turbulence conditions (D 3 =r 0 = 2:1 and C 2 n = 3:6 10 15 m 2=3 over an eective path length of 1 km, where C 2 n is the atmospheric structure constant [54]. Dierent realizations are obtained by illuminating dierent regions of the turbulence emulator). The results show that the algorithm improves the beams intensity correlation to> 0:9. Another measure of improvement is the received power coupled into the single-mode ber. Figure 2.5(b) shows an increase in the received power on OAM+3 of 4-15 dB following phase correction. 13 Gaussian OAM+1 OAM+2 OAM+3 OAM+4 OAM+5 (b) Phase Profile without Turbulence Gaussian OAM+1 OAM+2 OAM+3 OAM+4 OAM+5 (c) Intensity Profile with Turbulence without Compensation Gaussian OAM+1 OAM+2 OAM+3 OAM+4 OAM+5 (d) Phase Profile with Turbulence without Compensation Gaussian OAM+1 OAM+2 OAM+3 OAM+4 OAM+5 (e) Intensity Profile with Turbulence with Compensation Gaussian OAM+1 OAM+2 OAM+3 OAM+4 OAM+5 (f) Phase Profile with Turbulence with Compensation Gaussian OAM+1 OAM+2 OAM+3 OAM+4 OAM+5 (a) Intensity Profile without Turbulence Figure 2.4: Far-eld intensity proles and phase interference patterns for various OAM beams before and after phase correction under a specic turbulence realization (r 0 = 1mm). The phase interference patterns are generated by interfering a Gaussian beam with the uncorrected/corrected OAM beams. The convergence speed of an iteration-based phase correction approach is of great importance. Figure 2.5(c) shows the intensity correlation and received power for the OAM+3 probe beam as a function of the algorithm's iteration number. It shows that 50-100 iterations suce. During a single iteration, the algorithm can only guarantee evaluation of one dominate Zernike-polynomial term. Within 50-100 iterations, all 14 terms could be generated. In addition, the random walks performed by the non-dominate terms cancel each other out over 50-100 iterations. To achieve the synchronization among the phase corrector (SLM, 40 Hz refresh frequency), the detector (Camera, 50 Hz) and the SPGD algorithm (performed on a regular computer), delays are added among adjacent stages of the algorithm so that a single iteration takes around 1 second. Further upgrade of the hardware and the synchronization mechanism will improve the iteration speed of this approach. The turbulence induced wave-front distortion may lead to power leakage from a specic OAM mode to the neighboring modes. As shown in Figure 2.6 (a), without phase correction, most of the power of OAM+3 spreads to its neighboring modes (here, OAM +1 to +5 are measured), and that power distribution approximately uniform distribution. With phase correction, the majority 14 Improvement Improvement 0 2 4 6 8 10 0.4 0.5 0.6 0.7 0.8 0.9 1.0 w/o Compen. w/ Compen. Intiensity correlation Realization 0 2 4 6 8 10 -40 -35 -30 -25 -20 -15 Rx power (dBm) Realization w/o phase correction w/ phase correction (a) (b) 0 50 100 150 200 250 300 0.5 0.6 0.7 0.8 0.9 1.0 Relative coefficient Iteration Number Relative coefficient -30 -28 -26 -24 -22 -20 -18 Received power Received power (dBm) D 3 /r 0 =2.1 C n 2 =3.6×10 -15 m -2/3 Effective path length: 1 km (c) Figure 2.5: (a) Intensity correlation and (b) received power of OAM +3 with and without phase correction when only branch (1) is turned on. (c) Intensity correlation and received power as a function of the iteration number (i.e., the number of repetitions that the algorithm loop executed). of the power is concentrated in OAM+3, with much less power being leaked to the neighbor- ing modes. This implies that our SPGD-algorithm-based phase correction approach reduce the turbulence-induced crosstalk between the OAM channels of a multiplexed communication link. In a communication system, it is desirable to use a single phase correction for all the channels. To test the capability of SPGD-based wave-front sensing to supply such a phase correction, we applied the correction pattern derived from OAM+3 to a multiplexed beam, comprising three channels (OAM+1, +3, and +5), propagating through the same turbulence. Figures 2.6 (b, c) show the BERs of the OAM+3 channel before and after phase correction for two turbulence real- izations. Without phase correction, the BER can barely reach the forward error correction limit of 3:8 10 3 , due to the large amount of crosstalk from the other two channels. With the phase correction, the BER could achieve the FEC limit. The measured crosstalk from the OAM+1 and +5 channels to the OAM+3 channel is reduced from -7.03 dB to -17 dB and from -10.5 dB to -15.7 dB, respectively. This indicates that our approach could potentially be used in an optical communications system for turbulence compensation. 15 10 15 20 25 10 -4 10 -3 10 -2 w/o Compen. w/ Compen. w/o Turbulence Bit Error Rate OSNR (dB) 10 15 20 25 10 -4 10 -3 10 -2 w/o Compen. w/ Compen. w/o Turbulence Bit Error Rate OSNR (dB) FEC Limit 0 1 2 3 4 5 -45 -40 -35 -30 -25 -20 -15 -10 w/o Turbulence w/o Compen. w/ Compen. Rx Power (dBm) OAM Charge (a) (b) (c) Figure 2.6: (a) The received power of various OAM modes before/after phase correction when only OAM+3 is transmitted. The iteration number for the SPGD algorithm under this turbulence retaliation is 83. (b, c) Measured BERs as functions of optical signal-to-noise ratio (OSNR) when the phase pattern derived from OAM+3 is used to correct the phases of three simultaneously transmitted OAM beams (OAM+1, OAM+3 and OAM+5). 2.3 Summary and Discussion In conclusion, we have proposed and demonstrated a Zernike-polynomials based SPGD algorithm to derive a phase correction pattern for a distorted OAM beam based on intensity measurement. Specically, when the correction pattern derived from one probe OAM beam is applied to other multiplexed OAM beams transmitted through the same medium, the crosstalk between them is substantially reduced. 16 Chapter 3 Link Design Guideline for OAM Multiplexed Communications Systems 3.1 Background and Motivation Free-space optical (FSO) communication links can potentially benet from the simultaneous trans- mission of multiple spatially orthogonal beams through a single aperture pair, such that each beam carries an independent data stream and the total capacity is multiplied by the number of beams [2, 7, 34, 57, 58, 59]. Orthogonality of the beams enables ecient multiplexing and demultiplexing at the transmitter and receiver, respectively. The use of orbital angular momentum (OAM) beams as an orthogonal modal basis set for multiplexing has received recent interest [7, 34, 59]. We note that there are other orthogonal modal basis sets, such as Hermite-Gaussian (HG) modes [60], that could be used for multiplexing data channels in free space. While it is not straightforward to say which of these approaches is necessarily \better", OAM modes do oer the potential advantage of being conveniently matched to many optical subsystems due to their circular symmetry. Previous experimental reports [7] have included the demonstration of Terabit/s FSO data transmission using OAM multiplexing with a link distance of1 m. On the other hand, recent experiments have shown the feasibility of OAM beam transmission over distances of several km [33]. Given 17 the unique properties of OAM beams coupled with the recent scientic interest and the as-yet un- determined practical usefulness of OAM transmission, this chapter is intended to help assess the potential viability and technical challenges of using multiple OAM modes for enhancing free-space communications over non-trivial distances. With OAM, each beam has a phase front that \twists" in a helical fashion, and the beam's OAM order determines the number of 2 phase shifts across the beam [1]. Such OAM beams have a ring-shape intensity distribution and phase front of where is the topological charge and is azimuthal angle. Important characteristics of each OAM beam include: (i) the intensity has a \doughnut" shape with little power in the center, and (ii) the diameter of the beam grows with a larger OAM order. Moreover, the amount of phase change per unit area is greatest in the center of the beam, and phase distribution is critical for ensuring modal purity and beam orthogonality. For a practical system, the above characteristics of the OAM beam present several important challenges when designing an FSO communication link, such as: (i) sucient signal power and phase change needs to be recovered [61], and (ii) inter-modal crosstalk should be minimized [37, 41, 62, 63]. An important goal that has not been adequately explored in depth is to nd the systems limitations, trade-os and design parameters for an OAM multiplexed FSO communication link. In this chapter, we explore performance metrics and design considerations for an FSO commu- nication link using OAM multiplexing. The design issues for the transmitted beam size, receiver aperture size, and mode spacing are given through the investigation of system power loss, channel crosstalk, and system power penalty. By analyzing power loss of the desired OAM channel due to beam divergence under a given limited-size aperture, a design consideration for the transmitted beam size is proposed. Through studying the eects of the misalignment between the transmitter and receiver (lateral displacement or receiver angular error) on OAM channel crosstalk and sys- tem power penalty, proper aperture sizes and mode spacing of the transmitted OAM beams could be selected to reduce system performance degradation. Our simulations and some experiments indicate that: (i) a system with a larger beam size and a larger receiver aperture shows a better 18 tolerance to the lateral displacement but is less tolerant to the receiver angular error; (ii) the selection of mode spacing of such a system could be based on a trade-o between signal power loss and crosstalk. For instance, a system with small mode spacing shows a lower system power penalty under a small lateral displacement or receiver angular error, while a larger mode spac- ing shows a lower power penalty when the lateral displacement or receiver angular error is large. Mode-multiplexed communication systems using other orthogonal modal sets will also likely suer from signal power loss and inter-modal crosstalk in a relatively similar fashion as OAM modes but with dierent parameters governing the link; therefore, the methods in this chapter can be modi- ed and adapted to potentially be used to determine the performance of other mode-multiplexed systems. 3.2 System Model Figure 3.1(a) shows a schematic of an FSO communication link using OAM multiplexing. The multiplexed OAM beams diverge when transmitted through free space. By careful choice of the transmitted beam size, OAM mode spacing and the receiver aperture size, the system power loss, channel crosstalk and system power penalty could be reduced. Our simulation model of an OAM multiplexed FSO communication link is depicted in Figure 3.1(b). Independent data streams are carried by dierent collimated Gaussian beams at the same wavelength, each of which is coupled from a single mode ber to free space by a collimator. Each collimator is followed by a spiral phase plate (SPP) with a unique order to convert the Gaussian beam into a data-carrying OAM beam (see Figure 3.1(c)). An SPP is dened by its thickness, which varies azimuthally according to h() =`=2(n 1) (3.1) 19 Its maximum thickness dierence is h =`(n 1). Here, is the azimuthal angle varying from 0 to 2,n is the refractive index of the plate material, and is the wavelength of the laser beam. Dierent orders of OAM beams are then multiplexed to form a concentric-ring-shape and coaxi- ally transmitted through free space. The multiplexed OAM beams are numerically propagated by using the Kirchho-Fresnel diraction integral [64] to the receiver aperture located at a certain propagation distance. To investigate the signal power and crosstalk eect on neighboring OAM channels, the power distribution among the dierent OAM modes is analyzed through the modal decomposition approach, which corresponds to the case where the received OAM beams are de- multiplexed without power loss and the power of a desired OAM channel is completely collected by the receiver, which is innitely large and perfectly aligned with the transmitter [37, 63]. Transmitter Receiver Tx aperture Rx aperture Beam size Mode spacing Link distance Aperture size Power loss Crosstalk Power penalty OAM i OAM j Collimator SPP 1 Tx aperture Free space transmission Rx aperture Mod. SPP 2 SPP n M U X OAM beam anaylysis . . . . . . Laser Mod. Mod. Splitter data1 data2 data n . . . Gaussian beam OAM beam Intensity profile Phase front Intensity profile Phase front (a) (b) (c) Figure 3.1: (a)Concept of OAM multiplexed FSO communication link.(b) Simulation schematic of an OAM multiplexed data link. (c) Conversion from a Gaussian beam into an OAM+3 beam using an SPP+3 which causes helical phase shift from 0 to 6. Mod.: modulator, Tx: transmitter; Rx: receiver; SPP: spiral phase plate. An experiment with a transmitted beam size of 2.2 mm over a 1-m link is carried out to partially validate our system model. In the experiment, spatial light modulators (SLMs), which cause spiral phase delay to the incoming beam by loading by a spiral phase hologram, are used to function as SPPs at the transmitter. At the receiver, the beams are demultiplexed by another SLM loaded with an inverse spiral phase pattern of the desired mode to be detected and the resulting angularly at phase front beam is then coupled into a single mode ber for power measurement. 20 Assuming perfect ber coupling, this process of OAM beam detection closely corresponds to the modal decomposition approach in our simulation model. For the convenience of analysis, the following assumptions are made: 1. The wavelength of the laser source is 1550 nm. It should be noted that the specic values in the analyzed results using other wavelengths might be dierent. However, our fundamental approach remains valid. 2. All channels have the same transmitted power. 3. The collimator output at the transmitter is assumed to be a fundamental Gaussian beam (i.e. OAM 0) and all the OAM beams are generated from Gaussian beams with the same beam waist. 4. The SPP at the transmitter is assumed suciently large" to encompass the whole beam. 5. The transmitter aperture is considered to be the same size as the receiver aperture which is reasonable in a bidirectional link. However, the transmitter aperture in our analysis is always larger than the transmitted beam size and we assume it has no eect on the transmitted beam. Both the transmitter beam size and the receiver aperture size are parameters in the analysis. 6. The insertion loss of the multiplexer is not considered, although it adds a constant inser- tion loss in a practical system. Besides, the insertion loss of the SPP, which is assumed independent of the OAM order, is also ignored. 7. For calculations of spot size (beam diameter), the second moment of the intensity of an OAM or Gaussian beams, which is generally related to the beam waist, is employed, as given by the following equation: D = 2 s 2 R 2 0 R 1 0 r 2 I(r;)rdrd R 2 0 R 1 0 I(r;)rdrd where is the beam intensity prole and are polar coordinates [65]. 21 8. For the analysis of OAM carrying beams, we have considered Gaussian beams transformed into OAM beams by passing through SPPs (i.e. SPP based OAM beams). Most of the OAM beams used in previously reported communication links are similar to the SPP based OAM beams [7, 66, 67]. Although the OAM beams generated by passing Gaussian beams through SPPs are not exactly Laguerre-Gauss (LG) beams, such beams have similar characteristics in a communications link [4, 68] 9. We only analyzed the case of a single-polarized system. Since there is no obvious crosstalk between dierent polarizations for the beam transmitted through free space, most of results could also be applied to a dual-polarization system without further modications [7, 30]. In an ideal case, transmitter and receiver would be perfectly aligned, (i.e., the center of the receiver would overlap with the center of the transmitted beam, and the receiver would be per- pendicular to the line connecting their centers, as shown in Figure 3.2(a)). However, in a practical system, due to jitter and vibration of the transmitter/receiver platform, the transmitter and re- ceiver may have lateral shift relative to each other (i.e., lateral displacement) or may have angular shift (i.e., receiver angular error), as depicted in Figures 3.2(b) and (c), respectively. The lat- eral displacement and receiver angular error might occur simultaneously. A specic example is a pointing error at the transmitter that leads to both lateral displacement and angular error at the receiver, as depicted in Figure 3.2(d). In general, a practical link might use a tracking system to mitigate the random time-varying misalignment between the transmitter and receiver due to system vibration or long-term drift. For example, there is a commercially available tracking system with lateral resolution below 0.1 mm and angular resolution below 1 rad. We analyze the performance of an FSO communication link employing OAM multiplexing for the above scenarios. The parameters discussed are listed in Table 3.1. 22 Tx Rx Beam propagation direction Perfect aligned system Displacement d Rx Angular error φ Tx Pointing error (displacement + Rx angular error) φ θ z d (a) (b) (c) (d) Figure 3.2: Alignment between the transmitter and receiver for (a) perfectly aligned system, (b) system with lateral displacement, (c) system with receiver angular error, and (d) system with transmitter pointing error. Tx: transmitter; Rx: receiver; z: transmission distance; d: lateral displacement; ': receiver angular error; : pointing error. Table 3.1: Parameters in the Model D t Transmitted beam size (diameter) D a Receiver aperture size (diameter) z Transmission distance of the link d Lateral displacement ' Receiver angular error Transmitter pointing error 3.3 Signal Power Loss Analysis It is generally preferred to collect as much signal power as possible at the receiver in a com- munications link to ensure ample signal-to-noise ratio (SNR). Since OAM beams diverge while propagating in free space and available optical elements usually have limited aperture size due to the components cost, it would be desirable to choose a proper transmitted beam size when designing an OAM multiplexed FSO communication link over a certain transmission distance. In this section, we introduce approaches to design a suitable transmitted beam size by presenting our analyses of the OAM beam divergence and power loss over dierent transmission distances due to limited-size apertures. Given a xed transmitted beam size, an OAM beam with a higher order has a larger spot size over a given distance. Figure 3.3(a) shows the divergence of dierent OAM beams when they have dierent transmitted beam sizes over a 100-m link. Take OAM+4 as an example: when the transmitted beam size is less than 3 cm, the spot size at the receiver increases 23 0 5 10 15 20 0 5 10 15 20 25 30 Spot size at the Rx (cm) Transmitted beam size (cm) Gaussian Beam OAM+1 OAM+4 OAM+7 OAM+10 OAM+10,+7,+4, +1 and Gaussian (a) (b) 0 2 4 6 8 10 0 20 40 60 80 100 Tx beam size for Minimization (cm) OAM order 100 m 1 km 10 km (c) 0 2 4 6 8 10 0 20 40 60 80 100 Minimum spot size at Rx (cm) OAM order 100 m 1 km 10 km Figure 3.3: (a) Simulated spot size (diameter) of dierent orders of OAM beams as a function of transmitted beam size given a link distance of 100 m. (b) Minimum spot sizes of dierent orders of OAM beams at dierent transmission link distances. (c) Relative transmitted beam size to achieve the minimum spot size at the receiver. Note that for this gures analysis only, the size of the receiver aperture is not considered. when increasing the transmitted beam size. This is because smaller beams diract faster. How- ever, when the transmitted beam size is larger than 3 cm, further increasing the transmitted beam leads to larger spot size at the receiver. This is because the geometrical characteristic of the beam dominates over its diraction characteristic. Such a trade-o needs to be considered to control the size of the received beam at a proper range when designing a link. For a specic transmission distance and OAM mode order, there exists a transmitted beam size to achieve a minimum spot size at the receiver. The minimum spot size at the receiver and corresponding transmitted beam sizes for dierent link distances as a function of OAM mode order are shown in Figures 3.3(b) and (c), respectively. These results indicate that: (i) OAM beams with higher orders will have a larger minimum spot size given the same transmission distance; and (ii) the minimum spot size at the receiver grows approximately linearly with the increase of the required transmitted beam size. For system design considerations, it is desirable to select parameters for the transmitted beam that ensures generally a minimum spot size at the receiver for all of the modes simultaneously. Here, we choose 3 cm and 10 cm as the transmitted beam size for the 100-m link, 10 cm and 30 cm for the 1-km link, and 30 cm for the 10-km link as examples for system performance analysis; although we did not include larger aperture sizes for the 10-km link simply due to the current 24 state of sizes of practical of optical elements, we emphasize that our analysis can be extended to larger apertures and produce improved performance for longer-distance systems. One of the eects caused by a limited-size receive aperture is signal power loss of the system, because the spot size of the diverged beam is too large to be fully captured. Figure 3.4 shows the power loss of OAM+3 with dierent transmission distances and transmitted beam sizes. The power loss is directly related to the SNR of the received signal. We choose the receiver size to be 1.5 times the transmitted beam size; typically, this receiver size is large enough to capture sucient power from the transmitted beam (e.g., in our case, less than a 10-dB power loss for OAM+3). If a free-space multi-mode OAM system has a common single receiver aperture for all modes, then higher-order OAM beams that diverge more during propagation have more power loss than lower-order OAM beams. Although we considered modes up to +10 for which the system penalties can be signicant given the current state of practical aperture sizes, our analysis can be extended to higher-order modes. z=100 m z=1 km z=10 km (a) (b) (c) 0 5 10 15 20 0 10 20 30 40 50 60 70 80 Power loss (dB) Rx aperture size (cm) Dt= 3 cm Dt= 10 cm Dt= 30 cm 0 20 40 60 80 100 0 10 20 30 40 50 60 70 80 Power loss (dB) Rx aperture size (cm) Dt=3 cm Dt=10 cm Dt=30 cm 0 10 20 30 40 0 10 20 30 40 50 60 70 80 Power loss (dB) Rx aperture size (cm) Dt= 3 cm Dt= 10 cm Dt= 30 cm Figure 3.4: Simulated power loss as a function of receiver aperture size (diameter) when only OAM+3 is transmitted under perfect alignment for (a) z=100 m, (b) z=1 km, and (c) z=10 km. D t : transmitted beam size; z: transmission distance. 25 3.4 Channel Crosstalk Analysis If the transmitter and receiver are perfectly aligned (i.e., the received beam phase prole and the receiver aperture are concentric), then the power of the transmitted OAM mode does not spread into neighboring modes; this is due to the fact that orthogonality among dierent modes within a limited-size receiver aperture is still ensured based on recovering the full OAM phase change of the helical phase distribution [69]. However, in a practical system, the presence of lateral displacement or receiver angular error between the transmitter and receiver causes a phase-prole mismatch between incoming OAM beams and the receiver. This mismatch tends to reduce the received modal orthogonality, thereby leading to power leakage and crosstalk from the desired mode into adjacent modes. 0 1 2 3 -60 -50 -40 -30 -20 -10 0 Normalized power (dB) Lateral displacement (mm) OAM+1 OAM+2 OAM+3 OAM+4 OAM+5 XT-1 XT-2 (b) (a) (c) 0 5 10 15 20 25 30 -40 -35 -30 -25 -20 -15 -10 -5 0 1 km, 10 cm 1 km, 30 cm 10 km, 30 cm XT-2 (dB) Lateral displacement (mm) 100m,3cm 100m,10cm 100m,30cm 0 5 10 15 20 25 30 -40 -35 -30 -25 -20 -15 -10 -5 0 XT-1 (dB) Lateral displacement (mm) z=100 m, Dt=3 cm z=100 m, Dt=10 cm z=100 m, Dt=30 cm z=1 km, Dt=10 cm z=1 km, Dt=30 cm z=10 km, Dt=30 cm Figure 3.5: (a) Simulated power distribution among dierent OAM modes as a function of lateral displacement over a 100-m link for which only the OAM+3 mode is transmitted, the transmitted beam size is D t =3 cm and the receiver aperture size D a is 4.5 cm. (b, c) XT-1 and XT-2, respectively, as a function of lateral displacement for dierent transmission distances with dierent transmitted beam sizes. The receiver size is 1.5 times the transmitted beam size. XT-1: relative crosstalk to the nearest-neighboring mode (OAM+4). XT-2: the relative crosstalk to the second- nearest-neighboring mode (OAM+5). First, we investigate the eect of lateral displacement on the channel crosstalk by xing the receiver aperture size. Figure 3.5(a) shows the power distribution among dierent OAM modes due to a lateral displacement between transmitter and receiver when only OAM+3 is transmitted. The transmitted beam size D t =3 cm and the receiver aperture size D a =4.5 cm. As the lateral 26 displacement increases, the power leaked to the other modes increases while the power on OAM+3 decreases. This is because larger displacement causes larger mismatch between the received OAM beams and receiver. The power leaked to OAM+2 and OAM+4 is greater than that of OAM+1 and OAM+5 due to their smaller mode spacing with respect to OAM+3. One of the most important concerns is the power leaked to the nearest- and second-nearest-neighboring modes. Here we dene relative crosstalk to the nearest-neighboring mode XT-1 as the ratio of the power leaked to the nearest-neighboring mode (in our simulation, we examine OAM+4) to the power on the desired mode (OAM+3). Furthermore, XT-2 is dened as the relative crosstalk to the second-nearest-neighboring mode (OAM+5 in our case). Figures 3.5(b, c) shows the relative crosstalk XT-1 and XT-2, respectively, for dierent link distances with various transmitted beam sizes. The results indicate that: (i) larger transmitted beam size and longer transmission distances result in smaller XT-1 and XT-2; and (ii) a system with larger mode spacing is more tolerant to lateral displacement. Here, we use OAM+3 as an example to analyze the power leakage to the neighboring modes. However, it is expected that other modes would have similar performance trends. Besides lateral displacement, angular errors might also occur at the receiver. In the presence of a receiver angular error of magnitude ', the incoming phase front hitting the receiver has an additional tilt-related term and its values on the edges of the beam form `'D=2, where ` is the topological charge and is azimuthal angle and D is the spot size at the receiver. Clearly, these phase deviations from pure helicity are bound to introduce power leakage. Figure 3.6(a) shows the power distribution among dierent OAM modes under dierent re- ceiver angular errors when only OAM+3 is transmitted with D t =3 cm and D a =4.5 cm. With a xed receiver aperture size, a larger receiver angular error causes a higher power leakage to the other modes. Figures 3.6(b, c) shows that the system with a larger transmitted beam size and a longer range has higher XT-1 and XT-2. 27 0 3 6 9 12 15 -60 -50 -40 -30 -20 -10 0 OAM+3 OAM+4 OAM+5 Normalized power (dB) Rx angular error (urad) OAM+1 OAM+2 XT-1 XT-2 (c) (b) (a) 0 5 10 15 -30 -25 -20 -15 -10 -5 0 XT-1 (dB) Rx angular error (urad) z=100 m, Dt=3 cm z=100 m, Dt=10 cm z=100 m, Dt=30 cm z=1 km, Dt=10 cm z=1 km, Dt=30 cm z=10 km, Dt=30 cm 0 5 10 15 -30 -25 -20 -15 -10 -5 0 100m,3cm 100m,10cm 100m,30cm 1km,10cm 1km,30cm 10km,30cm XT-2 (dB) Rx angular error (urad) Figure 3.6: (a) Simulated power distribution among dierent OAM modes as a function of receiver angular error over a 100-m link for which only the OAM+3 is transmitted, the transmitted beam sizeD t is 3 cm and the receiver aperture sizeD a is 4.5 cm. (b, c) XT-1 and XT-2, respectively, as a function of receiver angular error for dierent transmission distances and transmitted beam sizes. The receiver size is 1.5 times the transmitted beam size. XT-1: relative crosstalk to the nearest- neighboring mode (OAM+4). XT-2: the relative crosstalk to the second-nearest-neighboring mode (OAM+5). In a practical system, lateral displacement and receiver angular error might occur simultane- ously, and the amounts of lateral displacement and receiver angular error might be random. The transmitter pointing error is an important parameter that determines the performance of a free- space OAM link. For our analysis, we consider the transmitter pointing error to be a functional combination of both lateral displacement and receiver angular error. A transmitter pointing error could be considered as the combination of a lateral displacement ofd = tan()z and a receiver angular error ' =, where z is the link distance. Figure 3.7(a) shows the power distribution among dierent OAM modes under dierent trans- mitter pointing errors when only OAM+3 is transmitted with D t =3 cm and D a =4.5 cm. Given a xed transmitter pointing error or receiver angular error, the power leakage in Figure 3.7(a) is higher than that in Figure 7(a) because a transmitter pointing error includes a lateral displace- ment in addition to the receiver angular error. Figures 3.7(b) and (c) show the XT-1 and XT-2, respectively, for a system with dierent transmission distances and transmitted beam sizes; we note that the trends of the results are similar to those shown in Figures 3.6(b) and (c). This trend appears because the receiver angular error becomes the dominated factor that aects the 28 system given a specic transmitter pointing error and transmitted beam size. To reiterate the key 0 5 10 15 -30 -25 -20 -15 -10 -5 0 XT-2 (dB) Tx pointing error (urad) 100m,3cm 100m,10cm 100m,30cm 1km,10cm 1km,30cm 10km,30cm 0 3 6 9 12 15 -60 -50 -40 -30 -20 -10 0 OAM+1 OAM+2 OAM+3 OAM+4 OAM+5 Normalized power (dB) Tx pointing error (urad) XT-1 XT-2 (a) (b) (c) 0 5 10 15 -30 -25 -20 -15 -10 -5 0 XT-1 (dB) Tx pointing error (urad) z=100 m, Dt=3 cm z=100 m, Dt=10 cm z=100 m, Dt=30 cm z=1 km, Dt=10 cm z=1 km, Dt=30 cm z=10 km, Dt=30 cm Figure 3.7: (a) Simulated power distribution among dierent OAM modes as a function of trans- mitter pointing error over a 100-m link for which only OAM+3 is transmitted, the transmitted beam size D t is 3 cm and the receiver aperture size D a is 4.5 cm. (b, c) XT-1 and XT-2, re- spectively, as a function of transmitter pointing error for dierent transmission distances and transmitted beam sizes. The receiver size is 1.5 times the transmitted beam size. XT-1: rela- tive crosstalk to the nearest-neighboring mode (OAM+4). XT-2: the relative crosstalk to the second-nearest-neighboring mode (OAM+5). points, a larger beam size at the receiver will result in two opposing eects: (i) a smaller lateral- displacement-induced crosstalk because the dierential phase-change per unit area is smaller, and (ii) a larger tilt-phase-error-induced crosstalk because the phase error scales with a larger optical path delay. 3.5 Power Penalty Analysis Eects that are critical to determining proper system performance and design criteria include the power loss due to beam divergence and the channel crosstalk due to both lateral displacement and angular error. A consequence of the signal power loss and channel crosstalk analyzed in the previous sections is the increase in system power penalty, which is the signal to noise ratio (SNR) dierence need to achieve a certain bit error rate (BER) by an OAM channel and ideal channel. It is used as metric to evaluate the system performance degradation. Signal power loss resulting from limited-size receiver aperture and channel crosstalk due to lateral displacement or 29 receiver angular error degrades the signal-to-interference-plus-noise ratio (SINR) of each channel, thus aecting the BER or power penalty performance. We simulated a four-channel OAM FSO communication link each channel transmitting a 16-QAM signal. With the background noise assumed to follow the Gaussian model, the error probability of a 16-QAM signal is: P e;16QAM = 3Q r 4 5 E agv N 0 !" 1 3 4 Q r 4 5 E agv N 0 # (3.2) where E avg =N 0 is the average SNR per bit. E avg is the average signal power per bit and N 0 is the power density of a Gaussian white noise and Q(:) is the complementary error function [70]. Equation (3.2) allows the calculations of the minimum required transmitted power P rq for a single channel (no crosstalk) to achieve a certain BER, given a Gaussian background noise. In our simulation, we choose an FEC limit of 3:8 10 3 as the BER threshold. With the assumption that all channels have the same transmitted power and channel crosstalk interferes with the signal in a similar way as noise at our BER threshold [71]. Considering crosstalk eects in a mode-multiplexed system, the required transmitted power for channel m can be expressed as: P rq;m =P rq P rq N 0 1 (3.3) where is the normalized signal power of the desired mode, and is the total crosstalk from all the undesired modes. The power penalty is dened as: P penalty = 10log 10 P rq;m P rq dB (3.4) To explore the in uence of limited-size receiver aperture and lateral displacement on power penalty, four channels are simulated in a 100-m OAM multiplexed FSO communication link. The similar approach also applies to other link distances. Power penalties for all four channels 30 might be dierent due to dierent OAM orders having dierent spots sizes. To ensure that ev- ery channel works, the largest power penalty among all channels is dened as the system power penalty. We simulate various sets of OAM beams to analyze system power penalty with dierent 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 2 4 6 8 10 Power penalty (dB) Lateral displacement (mm) Mode space=1 Mode space=2 Mode space=3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0 2 4 6 8 10 Power penalty (dB) Lateral displacement (mm) Mode space=1 Mode space=2 Mode space=3 (a) (b) Figure 3.8: Simulated system power penalty as a function of lateral displacement when dierent sets of OAM beams are transmitted over a 100-m link. Mode spacing=1: OAM+1, +2, +3, and +4 transmitted. Mode spacing=2: OAM+1, +3, +5, and +7 transmitted. Mode spacing=3: OAM+1, +4, +7, and +10 transmitted. (a) The transmitted beam sizeD t =3 cm and the receiver aperture size D a =4.5 cm. (b) D t =10 cm and D a =15 cm. mode spacing. Figure 3.8(a) shows the case of 3 cm transmitted beam size. When the lateral displacement is larger than 0.75 mm, the system with mode spacing of two (OAM+1,+3,+5,+7 transmitted) shows a lower power penalty than with mode spacing of one (OAM+1,+2,+3,+4 transmitted). This is because the channel crosstalk between adjacent OAM modes is higher than between OAM modes of a spacing of two. When the lateral displacement is less than 0.75 mm, the system with mode spacing of one shows less power penalty than with mode spacing of two. This is because the system with mode spacing of two has larger power loss due to the larger beam size at the receiver. Figure 3.8(b) shows the case when D t =10 cm and D a =15 cm. A comparison to the results in Figure 3.8(a) shows that a larger transmitted beam size, which leads a larger beam size at the receiver, could help reduce the system power penalty caused by lateral displacement. Similarly, the in uence of the receiver angular error on the system power penalty is also explored. Figure 3.9(a) shows when dierent sets of four OAM beams are transmitted over a 100-m link 31 (a) (b) 0 3 6 9 12 0 2 4 6 8 10 Power penalty (dB) Rx angular error (urad) Mode space=1 Mode space=2 Mode space=3 0 3 6 9 12 0 2 4 6 8 10 Power penalty (dB) Rx angular error (urad) Mode space=1 Mode space=2 Mode space=3 Figure 3.9: Simulated system power penalty as a function of receiver angular error when dierent sets of OAM beams are transmitted in a 100-m link. (a) The transmitted beam size D t =3 cm and the receiver aperture size D a =4.5 cm. (b) D t =10 cm and D a =15 cm. withD t =3 cm andD a =4.5 cm. The mode spacing of two has better performance than mode spac- ing of one when the receiver angular error is larger than 6 rad. In Figure 3.9(b), where D t =10 cm and D a =15 cm, the power penalty is slightly larger than that in Figure 3.9(a). Figure 3.10 shows the system power penalty when there is transmitter pointing error. Figure 3.10(a) shows a higher power penalty than does Figure 3.9(a) because the transmitter pointing error contains extra lateral displacement besides the receiver angular error. In addition, Figure 3.10(b) shows a trend similar to Figure 3.9(b) because when the transmitted beam size and receiver aperture size are large, the power penalty is mostly caused by the receiver angular error rather than the lateral displacement. The power penalty analysis indicate some selections rules for mode spacing: (a) a larger transmitted beam size and receiver aperture could increase the system tolerance to lateral displacement but decrease its tolerance to receiver angular error; (b) systems with larger mode spacing have higher order OAM beams, which leads to a higher signal power loss due to beam divergence; however, such systems also suers less channel crosstalk. As a trade-o between signal power loss and crosstalk, a system with a small mode spacing shows a lower system power penalty under a small lateral displacement or receiver angular error, while a larger mode spacing shows lower power penalty when the lateral displacement or receiver angular error is large. 32 (a) (b) 0 3 6 9 12 0 2 4 6 8 10 Power penalty (dB) Tx pointing error (urad) Mode space=1 Mode space=2 Mode space=3 0 3 6 9 12 0 2 4 6 8 10 Power penalty (dB) Tx pointing error (urad) Mode space=1 Mode space=2 Mode space=3 Figure 3.10: Simulated system power penalty as a function of transmitter pointing error when dierent sets of OAM beams are transmitted in a 100-m link. (a) The transmitted beam size D t =3 cm and the receiver aperture size D a =4.5 cm. (b) D t =10 cm and D a =15 cm. 3.6 System Performance in the Presence of Both Lateral Displacement and Receiver Angular Error In a practical system, the lateral displacement and receiver angular error might occur simulta- neously. It is shown in the previous section that when the transmitted beam size and receiver aperture size are larger, the system exhibits greater tolerance to the lateral displacement but lower tolerance to the angular error. Given certain lateral displacements and receiver angular errors, how to select the transmitted beam size and receiver aperture size to reduce the total power penalty would be an interesting question. In the section, we x the mode spacing to two. Dierent transmitted beam sizes of 5, 6, 8, 10, and 15 cm with corresponding receiver aperture size of 7.5, 9, 12, 15, and 22.5 cm are considered. Figure 3.11 shows the system power penalty for dierent transmitted beam sizes considering lateral displacement or receiver angular error. When the lateral displacement is 3 mm, a system with the transmitted beam size of 10 cm suers6 dB less power penalty than that with a transmitted beam size of 6 cm (see Figure 3.11(a)). However, the former suers 3 dB more power penalty than the latter when the receiver angular error is 10 rad (see Figure 3.11(b)). There 33 exists a trade-o between the eects of the lateral displacement and receiver angular error. For the parameters design of a practical system, one might need to select a proper beam size to reduce the system performance degradation considering this trade-o. Figure 3.11(c) shows the power penalty as a function of transmitter pointing error. A systems withD t =6 cm shows the lowest power penalty than the others for dierent transmitting pointing error. In terms of power penalty, a larger beam size at the receiver is more sensitive to the receiver angular error and less sensitive to the lateral displacement. In addition, the eects of lateral displacement and receiver angular error are related to the transmitter pointing error and the link distance. Therefore, for a given transmitter pointing error and link distance, trade-os exist when considering dierent deleterious eects and choosing a beam size at the transmitter that minimizes the power penalty. 0 2 4 6 0 2 4 6 8 10 Dt=10 cm Dt=15 cm Power penalty (dB) Lateral displacement (mm) Dt=5 cm Dt=6 cm Dt=8 cm 0 5 10 15 0 2 4 6 8 10 Dt=5 cm Dt=6 cm Dt=8 cm Dt=10 cm Dt=15 cm Power penalty (dB) Rx angular error(urad) (a) (b) (c) 0 2 4 6 8 10 12 0 2 4 6 8 10 Power penalty (dB) Tx pointing error(urad) Dt=5 cm Dt=6 cm Dt=8 cm Dt=10 cm Dt=15 cm Figure 3.11: Simulated system power penalty as a function of (a) lateral displacement, (b) receiver angular error, (c) transmitter pointing error when the mode spacing is 2 and transmission distance is 100 m. D t : Transmitted beam size. The receiver aperture size D a is three times the size of the transmitted beam size (D a =1.5D t ). 3.7 Experimental Validation of the Model As a partial validation of our link model, an experiment without lateral displacement between the transmitter and receiver is rst introduced. Figure 3.12(a) shows that the experimental results of 34 power loss of dierent OAM modes due to limited-size receiver aperture are in good agreement with the simulation results. Another validation of the simulation model considering a lateral displacement is shown in Figure 3.12(b). Over the 1-m link and a transmitted beam size of 2.2 mm, OAM+3 is transmitted with a lateral displacement of 0.2 mm. Measured and simulated power distribution with dierent receiver apertures show similar trends. Lines: simulation Symbols: experiment 1 2 3 4 5 6 0 10 20 30 40 50 60 Signlal power loss (dB) Aperture size (mm) OAM+1 OAM+2 OAM+3 OAM+4 OAM+5 (a) (b) 1 2 3 4 5 6 7 8 9 -60 -50 -40 -30 -20 -10 0 Normalized power (dB) Aperture size (mm) OAM+1 OAM+2 OAM+3 OAM+4 OAM+5 Lines: simulation Symbols: experiment Figure 3.12: (a) Comparison between experimental and simulated power loss of dierent OAM beams as a function of receiver aperture size. The transmitter and receiver are perfectly aligned. (b) Comparison between experimental and simulated power distribution among dierent OAM modes as a function of receiver aperture size with a lateral displacement of 0.2 mm. Lines and symbols are simulation and experiment results, respectively. 3.8 Summary and Discussion In this chapter, we explored performance metrics and design parameters for OAM multiplexed FSO communication links. The link distance, transmitted beam size, transmitter and receiver aperture sizes and OAM mode spacing were studied holistically. By analyzing the system power loss, channel crosstalk and system power penalty, a proper transmitted beam size, receiver aperture size, and OAM mode spacing could be selected for the system to handle lateral displacement, receiver angular error or transmitter pointing error between the transmitter and receiver. The following points are worth mentioning: 35 1. We only consider the use of OAM beams with plus charges for data transmission. Our design approach could be similarly applied to the system in which OAM beams with both plus and minus charges are used for multiplexing [7, 34, 72]. 2. Atmospheric turbulence might result in beam distortions in an OAM multiplexed FSO link [42, 50, 54, 73]. Of all the eects caused by turbulence, beam wandering and arrival angle uctuation could be considered as the lateral displacement and receiver angular error dis- cussed in our design approach. However, the intensity and phase uctuation of the received beam caused by atmospheric turbulence might lead to severe signal fading at the receiver [54]. Such eects are not considered in our approach, but may need further exploration. 3. Digital signal processing algorithms, such as multiple-input-multiple-output (MIMO) equal- ization that can be included in our simulations. The eects of the lateral displacement and receiver angular error could be reduced if using such algorithms [74, 75]. 4. In real-world systems, lateral displacement and receiver angular error are generally time- varying random processes that are typically described as beam jitter and beam wandering [54]. Our approach may help provide the analysis of the upper and lower bounds of system performance given a specic dynamic range of beam jitter and beam wandering. 5. In a practical system, we might need to couple the beam into a receiving optics (e.g., single mode ber) for signal detection. The angular error caused by beam deviation at the focal plane of such receiving optics has already been taken into account by mode decomposition. 6. The link analysis and design criteria of this chapter are provided for systems operating in the optical region of 1550 nm. We note that our general approach can also be applied to systems operating at other frequencies (e.g., mm-wave [34, 76]). Systems operating in other spectral ranges may have changes in dimensions as well as potential changes in performance trends. 36 7. We focused on the link distances of 100 m, 1 km and 10 km. For a longer link distance, a larger transmitter and receiver aperture size might be needed. For example, an aperture size greater than 1 m might be need for a 100-km link. 8. Our approach considered the system which uses collimated beams. Employing a beam forming technique at the transmitter, which controls the beam size after a certain distance, might be helpful to reduce the performance degradation due to the misalignment. 9. Although one can consider the tight alignment and aperture-size tolerances of the free- space OAM-multiplexed system to be a technical challenge, one can also envision these added requirements as potentially providing an added benet of increasing the diculty of eavesdropping by any o-axis receivers. 37 Chapter 4 Tunable Steering and Multiplexing of OAM Beams 4.1 Background and Motivation There is continual interest to increase both the capacity and spectral eciency in free-space radio frequency (RF) line-of-sight communication systems [70, 71, 77]. One technique for such an increase is to multiplex multiple data-carrying beams so that they can be transmitted through a single transmitter/receiver aperture pair. If each beam occupies a unique spatial mode in an orthogonal basis set, then channel multiplexing and transmission could occur with little inherent crosstalk and mode coupling. Such mode-division-multiplexing (MDM) has been demonstrated in free-space line-of-sight communication links using multiple orbital angular momentum (OAM) modes in RF [34, 35, 59, 76, 78, 79, 80] and optical regiems [2, 7]. OAM beams could be generated in several ways in the RF domain, including passing a con- ventional Gaussian beam through a spiral phase plate (SPP) [81, 82] and directly generating one or multiple OAM beams from a circular array of antenna elements [83, 84, 85]. In line-of-sight communication systems, accurate alignment between the transmitter and re- ceiver is important to guarantee sucient signal power at the receiver. Such alignment is even more important for OAM multiplexing systems since misalignment between the transmitter and receiver may cause crosstalk among channels [37]. Therefore, it would be desirable to design a 38 transmitter that can not only generate OAM beams but also steer the beams for active pointing. Previous reports have experimentally shown the ability to steer an optical OAM beam by shaping an input single Gaussian beam with a fork hologram that combines a spiral phase and a tilted phase front (e.g., a grating) [4, 86]. However, few reports have shown the steering of RF OAM beams using a circular antenna array [87, 38]. In this chapter, we demonstrate the simultaneous generation and tunable steering of multiple OAM beams at a 28-GHz carrier frequency using a circular array of antenna elements. Since beam steering and OAM generation both rely on delays among the multiple beams, by tailoring the delays among dierent elements, two OAM beams are generated with a variable steering angle of up to 35 degrees. We investigated the eects of the beam steering on the quality of the generated OAM beams. Moreover, we use the steerable OAM beams to establish a 4 Gbit/s communication link, with each OAM beam carrying a 1-Gbaud quadratic-phase-shift-keying (QPSK) signal. 4.2 Experimental System Design Figure 4.1(a) shows the principle of the simultaneous generation and steering of multiple OAM beams using a single antenna array. An RF sine-wave generator of frequency ! c is shared by M modulators (the case of M=2 is shown in the gure), which are fed by M data streams, fA m (t);m = 1;:::;Mg , to produce M modulated signals with the same carrier frequency: fA m (t) exp(j! c t);m = 1;:::;Mg. Each of the M signals is then split into N (the number of antenna elements in the array, N=8 is shown in the gure) branches, where each of the MN branch is assigned a pre-determined time delay mn ;m = 1;:::;M;n = 1;:::N, Each delay mn is the sum of two terms: mn = mn;S + mn;O . The rst term is designed for the steering and the second term is designed for the generation of an OAM beam of order ` m . The steering part is given by the well-known equation [88]: where mn;S =R cos( mn ) sin( m )=c, where R the array radius, is the distance from the array centre to the circularly placed antenna elements; m is the 39 beam steering angle of the m th beam; mb = 2(n 1)=N is the azimuthal angle of n th antenna element m from a plane, m , formed by a line pointing at the steering direction and the the an- tenna array normal; andc is the propagation speed of the electromagnetic wave in free space. The generation of the OAM beam of order ` m requires azimuthal phase delays, given by ' m =` m m , which, at carrier frequency ! c , can be achieved by the time delays mn;O =' m =! c =` m mn =! c . Multiple OAM beams of ordersf` m ;m = 1;:::;Mg can be simultaneously generated and at the same time individually steered to anglesf m ;m = 1;:::;Mg if the n th element of the antenna array is fed with the superposition: s n (t) = P M m=1 A m (t) exp(j! c (t mn ));n = 1;:::;N. As Figure 4.1: Principle of the technique for the simultaneous generation and steering of multiple OAM beams. (a) The concept of using an antenna array for the simultaneous generation and steering of multiple OAM beams. x: horizontal direction; y: vertical direction; z: direction perpendicular to the antenna array plane; Mod: modulator; Comb.: beam combiner;: time delay. (b) The relative phase change in the azimuthal direction for OAM generation and the relative phase for horizontal beam steering. (c) The design of an antenna array for the generation and steering of two OAM beams. ( 11 , 12 , 13 , 14 , 15 , 16 , 17 ,and 18 , are designed for the generation and steering of one OAM beam, while 21 , 22 , 23 , 24 , 25 , 26 , 27 ,and 28 are designed for the generation and steering of another OAM beam). shown in Figures 4.1 (b) and (c), such a superposition is implemented using N eld combiners, each with M inputs and a single summing output, where: (i) The output of the n th combiner is connected to the n th element of the antenna array; (ii) The m th inputs of the N combiners are 40 fed by the Ndelayed signals associated with the m th beam. Figure 4.1(c) shows our proposed design for the case of M=2 beams and N=8 antenna elements. The required time delays are implemented by \trombone-like tunable time delay lines. Thus, for the generation and steering of beam #1, the time delaysf 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 g, are applied to the correspond- ingly designated 8 delays in Figure 4.1(c), while beam #2 generated and steered by applying f 21 ; 22 ; 23 ; 24 ; 25 ; 26 ; 27 ; 28 g to the other group of 8 delay lines, which then feed the other input ports of the 8 eld combiners. Previous reports have indicated that the order of OAM beams that could be generated from a ring antenna array with N elements obeys 2j`j + 1 < N [76, 85, 89]. As an example, with 8 antenna elements used in our design, the order of the generated OAM beams obeys 2j`j 3. 4.3 Simulation and Experimental Results We rst investigate the generation of a single unmodulated OAM+1 beam using the designed antenna array. Figure 4.2(a) shows the intensity prole of the generated OAM+1 beam at the distance of 1.2 m from the antenna array (see the Method section for the measurement and simulation approaches) with an array radius R of 4.5 cm. The 8 array elements are excited with phase delays of 0, /4, /2, 3/4, , 5/4, 3/2 and 7/4. Our results agree well with the previous studies in the RF domain [83, 85] and in the optical domain [90]. Besides, the beams phase distribution is characterized through its interference (see Figure 4.2(b)), with a regular Gaussian beam. The clean rotating arm is in good match with the simulation result. In addition to the near eld and far eld intensity proles of the generated OAM+1 beam [90], we also investigate the evolution of the beams intensity proles at dierent distances. Figures 4.2(c1-c5) depict the intensity proles of the generated OAM+1 beam at various ranges from 0.4 to 1.2 m. Our experimental results agree well with the simulation predictions. At a very short distance, the ring shape of the generated OAM beam is somewhat unclear, while after 41 (a) 8antennas for OAM+1 (b) Interferogram of (a) simulation experiment simulation experiment 48 cm 40 cm (c5) distance 1.2 m simulation experiment simulation experiment simulation experiment simulation experiment simulation experiment 48 cm 48 cm 48 cm 48 cm 48 cm (c4) distance 1.0 m (c3) distance 0.8 m (c2) distance 0.6 m (c1) distance 0.4 m Figure 4.2: Simulation and experiment results of the generation of one OAM beam. (a) The intensity prole of the OAM+1 beam generated from an array of 8 antenna elements at 1.2 m. (b) The interference pattern of the generated OAM+1 beam with a Gaussian beam. (c1-c5) The intensity proles of OAM+1 beams generated from 8 antennas at 0.4 m, 0.6 m, 0.8 m, 1.0 m and 1.2 m, respectively. some propagation, the intensity prole evolves to a better ring shape. Due to the experimental limitation, we only measure the beam evolution at a distance up to 1.2 m. The distance that the generated OAM beam could propagate is determined by the divergence of the beam and its diraction limit [91]. An OAM beam diverges faster than a regular Gaussian beam with the same beam waist, while the receiver for the communication system might have a limited aperture size. Therefore, taking into account the divergence of the generated OAM beams is imperative for the system design. As expected, we nd that the OAM beams divergence is inversely proportional to the array radius R. Figures 4.3 (a-c) show the intensity proles of the generated OAM+1 beam using an 8-element antenna array after a 1.2-m propagation, when R is 4.5 cm, 5 cm, and 6 cm, respectively. Figure 4.3 (d) shows the simulated and measured beam sizes as a function of array radius R. Since the waist of the generated beam is proportional to the array radius, larger R would result in a smaller far-eld pattern. Therefore, for a practical implementation, the array radius needs to be properly designed considering the divergence of the generated OAM beams and the link distance. 42 Figure 4.3: Simulation and experimental results of the OAM beam generation with dierent array radii. The intensity proles of the generated OAM +1 beams with array radii of (a) 4.5 cm, (b) 5.0 cm and (c) 6.0 cm. (d) Comparison between the simulation and experimental results. Next, we investigate the simultaneous generation and steering of one, as well as multiple OAM beams using the 8-element antenna array, initially with no data modulation. Figure 4.4(a) shows the time delays among the 8 antennas only for the generation of the OAM+1 and OAM-1 beams ( m = 0;m = 1; 2), where these delays simply increase or decrease monotonically with the azimuthal location of the element. Figure 4.4(b) shows the relative time delays for beam steering with various angles without imparting an OAM charge to the beams. Since the steering is designed to be in the horizontal plane, the two elements in each of the pairsfA1;A5g,fA2;A4g andfA6;A8g (Figure 4.1(c)) share the same delay, regardless of the steering angle. Figure 4.4(c) shows the combined delays, which would allow the simultaneous generation and steering of the OAM+1 beam. Figure 4.4(d) shows the delays required for the simultaneous generation and steering of the OAM-1 beam. Moreover, the OAM+1 and OAM-1 beams could be generated and steered simultaneously if the dierent branches of the two beams are combined using beam combiners in Figure 4.1(a). Before presenting steering results, it is important to note that while Figure 4.4 calls for relative delays in access of 4-5 periods of the carrier (T c = 2=! c ), our delay-generating RF trombones could only cover a range of 0 to T c . Therefore, we set each of the delays to a value derived from 43 Figure 4.4: The design of time delay. (a) The delays of the 8 antennas elements for the generation of the OAM+1 and OAM-1 beams. (b) The delays for the 8 antennas elements for the steering of the beams. The bottom two panels show the delays applied to the 8 antennas elements for the simultaneous generation and steering of the OAM+1 (c) and OAM-1 beams (d). T c : the period of the carrier wave. In practice, only positive delays were implemented so that the element with the most negative delay (A3) was considered a reference, having zero delay, and all other elements were assigned relative delays according to the gures above. its true delay value modulo T c (e.g., a delay of 5.5 T c was implemented by a delay of 0.5 T c ). Since an integer number of periods represent a multiple of 2, this transformation is harmless as long as there is no modulation on the carrier, as in shown Figure 4.5 below. Figures 4.5(a1-a7) show the intensity proles of the steered OAM +1 beam (see Method section for the measurement approach) at various steering angles. When increasing the steering angle, distortions are observed in the resulting beam pattern. The divergence angle of the beam emitted from each antenna element is28 and the total coverage area of the antenna array is limited by this angle. Therefore, larger steering angle may cause the quality degradation of the steered OAM beam due to the limited coverage area as simulated in Figure 3.4 (b1-b4). Figure 3.4(c) shows the comparison between the designed steering angle and measured steering angle, which indicates our steering angle is within the error of less than 1 . As described above, the OAM+1 and OAM-1 beams could be simultaneously generated and steered using the same antenna array by feeding each antenna element with the sum of the respective contributions of the two OAM beams (Figure 4.1(a)). Figures 4.5(d1-d4) show the superposition of the OAM+1 and OAM-1 beams with steering angles of 0 , 10.2 , 21.3 , and 44 Figure 4.5: Experimental and simulation results of the simultaneous generation and steering of one and two OAM beams. (a1-a6) Intensity proles of the steered OAM+1 beam with a steering angle of 0 , 5 , 10.2 , 15.5 , 21.3 , 27.8 , and 35.3 , respectively. (b1-b4) Simulation results of the steering of OAM+1 beam when the antenna elements have dierent emitting angle (EA). (c) Comparison between the designed steering angle and the measured steering angle. (d1-d4) Intensity proles of the steered superposition of the OAM+1 beam and OAM-1 beam with steering angles of 0 , 10.2 , 21.3 , and 35.3 , respectively. (e1-e4) The distribution of the received power among several modes when only the OAM+1 beam is generated with steering angles of 0 , 10.2 , 21.3 , and 35.3 , respectively. The measurement is performed at 1.2 m. The plot represents the average of multiple measurements, while the error bars indicate standard deviations. 45 35.3 , respectively, clearly indicating that the designed structure has achieved the simultaneous generation and steering of multiple OAM beams. We note that by properly changing the relative time delays, a steering of -35.3 should also be achievable. For a communication system, OAM mode purity, characterized by the distribution of the received power among OAM modes other than the transmitted one, is one of the most important factors that aects the channel crosstalk. Figures 4.5(e1-e4) show the mode purity of the generated OAM+1 beam when the steering angle varies from 0 to 35.3 . The power leaked to neighbouring modes increases as the steering angle varies from 0 to 35.3 . The larger the steering angle of an OAM beam in our setup, the more distortions (Figure 4.5(a), and consequently, the stronger the leakage to neighbouring modes. For the investigation of the bit error rate (BER) performance of the proposed scheme, two 1-GBaud QPSK signal streams were used to modulate the multiplexed and steered OAM+1 and OAM-1 beams. After free space propagation of1.2 m, the signal was de-multiplexed and recorded by an oscilloscope for BER measurement. As shown in Figure 4.6(a), as the steering angle increases, the BER goes up for a specic received power even when only one OAM channel is transmitted (no crosstalk). This trend is attributed to the deviation from true time delay steering, introduced by the modulo transformation between the true delay value and the implemented one. At maximum steering of 35.3 , the required relative true time delay of 5.5 T c between antenna elements A3 and A7 was implemented by a 0.5 T c . This implementation created an unwanted time dierence of 5 T c (almost 18 of the baud duration) between the modulation information coming from these two elements, giving rise to inter-symbol interference and higher BER. While deteriorating with increasing steering, even at 35.3 BER performance of 3:8 10 3 could be achieved. As shown in Figure 4.6(b-e), when the steering angle is small, the BER of multiplexing two OAM channels is very close to the case when only a single OAM channel is transmitted due to the low channel crosstalk. When the steering angle is 21.3 and 35.3 , penalties of3 dB and 46 5 dB are observed for the multiplexing of two channels compared to that with only one channel. However, the system could still achieve a BER below 3:8 10 3 even at a steering angle of 35.3 . Figure 4.6: BER measurement. (a1) BER performance when only one OAM beam is generated and steered with various steering angles. (b-e) BER performance when one or two OAM beams are transmitted at dierent steering angles. xt: crosstalk; w/: with; w/o: without; b2b: back to back, where the transmitter is connected to the receiver using a cable. 4.4 Summary and Discussion We have demonstrated the simultaneous generation and steering of multiple OAM beams utilising a custom-designed circular antenna array at 28 GHz. Two OAM beams were generated with a steering angle of up to 35 . The following points are also worth mentioning: (i) For optimum performance it is important not only to control the time delays but also to minimize variations among the powers emitted from the dierent antenna elements. (ii) Only horizontal steering was demonstrated. Indeed, the proposed approach can achieve beam steering in any direction, at least within the tested cone of 70 apex angle. (iii) The same concept and implementation should work 47 equally well at other RF frequencies. (iv) The implemented tunable delay in our experiment was limited to one period of the RF carrier. The inter-symbol interference caused by large steering angles can be reduced when true time delay [92] operation is achieved by using tunable delay lines what is capable of covering the full required delay range. (v) In general, Laguerre-Gaussian (LG) modes represent a complete 2 dimensional modal basis set and can be described by two indices (i.e., an azimuthal index` and a radial indexp), and OAM can exist for LG modes with dierent p values. If only p=0 is used for multiple beams of dierent values, then this can be considered a subset of the fuller 2 dimensional set of LG modes. However, since OAM can exist for p6= 0 as well, then the use of dierent p values and dierent values can produce a fuller set of modes and theoretically a higher system capacity over a given spatial area [39, 93]. 4.5 Appendix Figure 4.7: Measurement design. (a) The measurement setup to scan the intensity prole of the generated OAM beams without steering. (b) The measurement setup to scan the interference pattern of the generated OAM beams without steering. (c) The measurement setup to scan the intensity proles of the steered OAM beams. (d) The measurement setup to test the mode purity of the steered OAM beams. 48 The method for the intensity prole measurement the generated beams: As shown in Figure 4.7(a), we measure the intensity proles of the generated beams using a probe antenna with a radius of2 mm, which is placed on a two-dimensional (2D) linear translation stage. The RF eld collected by the probe antenna is then sent to an RF spectrum analyser for power measurement. The stage can scan in horizontal and vertical directions, thus measuring the 2D intensity prole of the received beam. The method for the measurement of the interference pattern of the generated beams with a Gaussian beam: To measure the transverse phase of an OAM beam, we use a Gaussian beam to interfere with the OAM beam, as shown in Figure 4.7(b). The two beams are combined using a beam splitter, which was fabricated using a printed-circuit board with a spatially varying re ective surface. It has 50% transmission eciency at 28 GHz when the beam has a 45 incident angle. The method for the intensity prole measurement of the generated beams with some steering angle: When the generated OAM beam is steered, we rst estimate the propagation direction of the generated beam. As shown in Figure 4.7(c), the 2D stage is then placed 1.2 m after the circular antenna array to scan the intensity prole of the steered OAM beam. The 2D stage scans in a plane perpendicular to the estimated propagation direction of the beam to get the intensity proles of the steered OAM beams. The method for the mode purity measurement: As shown in Figure 4.7(d), to measure the received power distribution on dierent modes when a specic mode is transmitted (mode purity), we rst use a spiral phase plate (SPP) with the OAM order of ` to convert the OAM +` component of the received OAM beam into a Gaussian-like beam. A horn antenna is then followed to collect the Gaussian-like beam for power measurement. In our experiment setup, the SPP is dened by its thickness which varies azimuthally according toh () ==2`= ( 1), acquiring a maximum thickness dierence of h =`=( 1) ( is the azimuthal angle, is the refractive index of the plate material and is the wavelength of the millimeter-wave). When a specic beam 49 is received, we sequentially apply SPPs with dierent orders, thereby measuring the magnitude of parasitic OAM components. The simulation method: We have also simulated the generation and steering of OAM beams using antenna arrays. In the simulation, we rst generate the electrical eld of a Gaussian beam at the distance of z: f(x;y;z) = w 0 w(z) exp x 2 +y 2 w 2 (z) exp j kz +k x 2 +y 2 2(z) (z) (4.1) where (x;y) is the plane perpendicular to the transmission direction of the Gaussian beam at the distance z;k = 2= is the wave number and is the wavelength; w 0 is the beam waist; w(z) = p w 2 0 + 2 z 2 = (w 2 0 2 );(z) =z +w 4 0 2 = 2 z ; (z) = arctan z= w 2 0 . We then calculate the electrical eld of the whole beam emitted from the antenna array as: F (x;y;z) = M X m=1 N X n=1 f(x +R cos( mn );y +R sin( mn );z) exp (j! c mn ) (4.2) where M is the number of generated and steered OAM beams; N is the number of the antenna elements; R is the distance from the antenna elements to the array center; mn = 2(n 1)=N is the azimuthal angle ofn th antenna element from a plane; and is the time delay for then th branch of the m th generated beam. Generation eciency and antenna number: One important factor for the generation of OAM beams using an antenna array is its generation eciency, in terms of the ratio of the power on the most inner ring of the beam over the power of the whole beam. Figure 4.8(a) shows a few designed antenna array structures, where L is the number of concentric layers housing the antenna elements, and N s is the number of elements, equally distributed on the circumference of each layer. By carefully designing the structure of the array, better generation eciency could be achieved: (i) as the number of elements per layer increases, more power is concentrated on 50 Figure 4.8: Simulation results for dierent antenna arrangements. (a) The arrangement of the antennas; L: number of ring layers, the cases of L = 1; 2; 3; 4 are investigated; N s : number of antennas on each of the rings, the cases of N s = 8; 16; 32 are investigated. (b1-b6) The intensity proles of the generated OAM+1 beam for dierent L and Ns values. Eciency: the power on the inner-ring of the generated beam over the power of the whole beam. the inner ring, as shown in Figures 4.8(b1-b3); and (ii) increasing the number of layers could also improve the generation eciency, as shown in Figures 4.8(b3-b6). When the array has four layers, each containing 32 elements, a generation eciency of 58:26% is observed in the simulation. 51 Chapter 5 Multiplexing Laguerre-Gaussian Beams with Dierent Radial Indices 5.1 Background and Motivation There is continuing interest in increasing the capacity of a free-space optical communication system using a single transmitter and receiver aperture pair. Space division multiplexing (SDM) is a potential approach to improve a systems capacity by transmitting several spatially overlapping data-carrying beams simultaneously [94]. As a subset of SDM, mode division multiplexing (MDM) can be used, in which multiple orthogonal data-carrying beams can be eciently (de)multiplexed and co-propagate with little inherent crosstalk. One reported example of such a MDM system is the use of orbital angular momentum (OAM) beams as the modal basis; the transmission of multiple beams each carrying dierent OAM values has been demonstrated primarily in free space [2, 4, 6, 7, 30, 57, 95]. In general, an MDM system benets from an orthogonal modal basis. One of the most well known types of orthogonal beams is the Laguerre-Gaussian (LG) beam. An LG beam can be characterized by two indices: the radial indexp, which is a non-negative integer, and the azimuthal index `, which is an integer; we note that p and` are associated with the number of radial nodes (ring/spot on beams intensity) and the specic OAM values, respectively [1, 96]. Figure 5.1(a) 52 shows the intensity proles and phases of LG beams with various p and ` values. Given an LG beam, the number of nodes in the intensity is p+1, and the phase rotation in the azimuthal direction is equal to 2`. Intensity Profile Phase ℓ=0,p=0 ℓ=1,p=0 ℓ=2,p=0 ℓ=0,p=1 ℓ=0,p=2 ℓ=1,p=2 π -π 1 0 Figure 5.1: (a) Intensity proles and phase fronts of Laguerre-Gaussian (LG) beams with dierent azimuthal indices and radial indices. Importantly, two LG beams with the same beam waist are orthogonal to each other if they have either a dierent p value or a dierent ` value, indicating that dierent LG beams could be employed for ecient channel multiplexing and de-multiplexing. The \same beam waist is important for LG beams with the same ` but dierentp values since their orthogonality relies on the phase/ intensity change in the radial direction. Previous studies have typically focused on the multiplexing of data-carrying LG and LG-like beams with dierent ` values but the same p value (p=0) [4, 7, 30, 57, 95]. Alternatively, there has been little reported to the multiplexing of data-carrying LG beams with p > 0, either with the same or dierent ` values [39]. The possibility of utilizing the orthogonality of LG beams with dierent p values along with dierent ` values could provide more communication modes as well as form a complete orthogonal basis. In this chapter, we demonstrate a 200-Gbit/s SDM system using two LG beams with dierent radial indices (LG `=0;p=0 and LG `=0;p=1 ). We use a single mode ber (SMF) to spatially separate the two multiplexed beams at the receiver. With a proper design of the radial change of the de-multiplexing pattern, the channel crosstalk could be minimized and both channels could achieve a BER of 3:8 10 3 . Moreover, the multiplexing of four LG beams with dierent azimuthal indices and dierent radial indices (e.g., LG `=0;p=0 , 53 LG `=0;p=1 , LG `=1;p=0 , and LG `=10;p=1 beams) is also demonstrated with a <-12 dB channel crosstalk. 5.2 Approach, Design, and Results Generally, the electrical eld of an LG beam can be represented by [65, 97]: u (r;;`;p;w 0 ) = p 2r j`j w 0 j`j+1 exp r 2 w 0 2 +j` LP j`j p 2r 2 w 0 2 (5.1) wherew 0 is the beam waist,LP j`j p is the generated Laguerre polynomials andr; is the cylindrical coordinate. The orthogonality of LG beams with a unique beam waist to form a two-dimensional complete and orthonormal basis could be represented by [65, 97, 98]: Z 1 0 Z 2 0 u (r;;` 1 ;p 1 ;w 0 )u (r;;` 2 ;p 2 ;w 0 )rddr = 8 > > < > > : 0; ` 1 6=` 2 orp 1 6=p 2 1; ` 1 =` 2 andp 1 =p 2 (5.2) Figure ??(a) shows our experimental setup. First, a 50 Gbaud quadratic phase shift keying (QPSK) signal at 1550 nm is generated and splitted into two branches. One of the branches is relatively delayed by a 10-meter SMF for signal de-correlation. The two branches (branch 1 and branch 2 ) are then coupled into free space by two collimators. Two spatial light modulators (SLMs) are used to tailor these two branches into two LG beams in the designed azimuthal orders, radial orders, and beam waists. A beam splitter (BS) is used to multiplex the two LG beams. The multiplexed beams are then propagated in free space for1 meter. At the receiver, an SLM-3, which is loaded with a specically designed de-multiplexing phase pattern, is used to down-convert the required channel back into a Gaussian (LG `=0;p=0 ) beam sequentially. The down-converted beams are then coupled into an SMF using an objective lens. Finally, the signal is sent for a coherent detection and bit error rate (BER) measurement. 54 Delay fiber Col. HWP BS 50/50 Coupler 50Gbaud QPSK Signal SLM-2 SLM-1 Objective lens M SLM-3 Fiber Co-axis transmitted beams Can be coupled into SMF Phase on SLM-3 0 Receiver for LGℓ=0,p=0 Receiver for LGℓ=0,p=1 (b) 0 π (c) (a) Coherent Detection Phase on SLM-3 Cannot be coupled into SMF Cannot be coupled into SMF Can be coupled into SMF Figure 5.2: (a) Experimental setup of a free space optical communication system multiplexing LG `=0;p=0 and LG `=0;p=1 . QPSK: quadratic-phase-shift-keying; SLM: spatial light modulator; M: mirror; Col: collimator; BS: beam splitter. (b, c) De-multiplexing principle of LG `=0;p=0 and LG `=0;p=1 channels, respectively. The principle of de-multiplexing theLG `=0;p=0 , andLG `=0;p=1 beams is illustrated in Figures 5.2(b) and (c), respectively. As shown in Figure 5.2(b), to recover the LG `=0;p=0 channel (branch 1 ), an all-zero phase pattern is loaded onto SLM-3, and the LG `=0;p=0 and LG `=0;p=1 modes remain in the original mode after being re ected by SLM-3. TheLG `=0;p=0 beam is then coupled into an SMF, while theLG `=0;p=1 beam cannot. As shown in Figure 5.2, to recover theLG `=0;p=1 channel (branch 2 ), a phase pattern, which has a zero-phase circle inside and phase outside, is loaded onto SLM-3. This pattern converts the incoming LG `=0;p=0 beam into an LG `=0;p=1 beam as well convert the incoming LG `=0;p=1 beam into an LG `=0;p=0 beam. Therefore, only information from branch 2 could be coupled into the SMF. 55 The experimental design of using an SMF to separate theLG `=0;p=0 beam from theLG `=0;p=1 beam is based on the fact that LG `=0;p=1 could not be coupled into the SMF. To verify this, we simulate the propagation loss of a 1550 nm LG `=0;p=1 beam in a regular SMF (SMF 28e, the ber used in our experiment). The refractive index of the core and cladding is 1.45 and 1.4447, respectively. The core and cladding radii is 4.1 um and 50 um, respectively. The radius of the LG `=0;p=1 beam is 4.2 um (which is the same as the size of the beam after being focused by objective lens in the experiment). As shown in Figire 5.3(a), a 100-milimeter propagation leads to a >70 dB power loss for theLG `=0;p=1 beam. Figure 3b shows the power loss for the rst 100-um propagation and its intensity proles at 0 um, 20 um, 40 um and 80 um. These results indicate that an SMF could block theLG `=0;p=1 beam, therefore spatially separate theLG `=0;p=0 and the LG `=0;p=1 beam. 0 20 40 60 80 100 -80 -60 -40 -20 0 Distance (mm) Power in Fiber (dB) Propagation direction LGℓ=0,p=1 Distance Core Cladding 0 20 40 60 80 100 -15 -12 -9 -6 -3 0 Distance (um) Power in Fiber (dB) 1 0 0 um 20 um 40 um 80 um (a) (b) Figure 5.3: Principle of using a single mode ber (SMF) as a lter to separate the LG `=0;p=1 and LG `=0;p=0 beam: the simulated power loss of an LG `=0;p=1 beam when ltered by an SMF. (a) The transmission distance varies from 0 to 100 mm. (b) The transmission distance varies from 0 to 100 um. Subgures in (b): the intensity proles of the LG `=0;p=1 beam at various distances in the ber. Figures 5.4(a1) and (a2) show the intensity proles of theLG `=0;p=0 beam and theLG `=0;p=1 beam in the experiment, respectively. Figure 5.4(b1) shows the phase pattern on SLM-3 for the de-multiplexing of the LG `=0;p=0 channel, which is an all-zero phase pattern. Figure 5.4(b2) shows the phase patterns on SLM-3 for the de-multiplexing of the LG `=0;p=1 channel. These 56 0 0 Radius: Cr (a2) (a1) (b2) (b1) π Figure 5.4: (a1 and a2) Intensity proles of the generated LG `=0;p=0 and LG `=0;p=1 beams, respectively. (b1 and b2) Phase patterns for the de-multiplexing of LG `=0;p=0 and LG `=0;p=1 beams, respectively. C r : circle radius. patterns contain a circle at the center with the zero phase and phase outside. The radius of the circle is dened as C r . The orthogonality among LG beams with the same ` value but dierent p values relies on their phase changes in the radial direction [39]. Therefore, if the C r of the de-multiplexing pattern does not match the size of the incoming LG beam (see Figure 5.5), the orthogonality of the beams may potentially be degraded. Figure 6 shows the measured power and crosstalk of the LG `=0;p=0 and LG `=0;p=1 channels as a function of C r . When the LG `=0;p=0 beam is received, the power changes very little as C r varies from 0.4 mm to 1.8 mm. This is because the receiver for the LG `=0;p=0 beam uses an all-zero phase pattern for the mode de-multiplexing, regardless of C r . In addition, when the LG `=0;p=0 beam is transmitted, the receiver for the LG `=0;p=0 mode receives high power, while when the LG `=0;p=1 mode is transmitted, the receiver for the LG `=0;p=0 mode receives very little power. This is consistent with our previous analysis that the LG `=0;p=0 mode could be coupled into an SMF, while the LG `=0;p=1 mode could not. When LG `=0;p=0 is transmitted, the power leaked to the receiver for the LG `=0;p=1 mode decreases as C r increases from 0.4 mm to 1.0 mm and increases as C r increases from 1 mm to 1.8 mm. At C r =1.0 mm, a valley of -18 dB is achieved. In contrast, when LG `=0;p=1 is transmitted and the received power by the receiver for the LG `=0;p=1 mode increases as C r increases from 0.4 mm to 1.0 mm and decreases asC r increases from 1 mm to 1.8 mm. AtC r =1.0 mm, a peak is achieved. In the experiment, the LG `=0;p=1 beam shone on SLM-3 has the inner node radius of1 mm, 57 and a pattern withC r =1.0 mm matches the size of the beam in the radial direction, thus leading to the lowest channel crosstalk. Any mismatch in the radial direction (e.g., a dierent C r value) may lead to the increase of the channel crosstalk. Intensity profile of LG ℓ=0,p=1 De-multiplex pattern (matched) De-Mux pattern (mismatched) De-Mux pattern (mismatched) x x <x >x Figure 5.5: Denition of the match and mismatch between the r of the de-multiplexing pattern and the size of the LG `=0;p=1 beam. As shown in the sub-gures of Figure 5.6, the intensity proles of the beam for points (a-e) are also measured on the plane in front of the SMF. For points (e) and (f), the intensity proles show clear LG `=0;p=0 and LG `=0;p=1 beams, respectively. After being coupled into the SMF, most of the power of (e) is received while very little power of (f) is received. Points (a) and (b) show the intensity proles for the receiver of the LG `=0;p=1 beam when LG `=0;p=0 is transmitted. As indicated by point (a), when a proper C r is chosen, the incoming LG `=0;p=0 is converted to an LG `=0;p=1 beam. Therefore, very little power is captured by the SMF. However, as indicated by point (b), whenC r =1.7 mm, which is not a proper value, the incomingLG `=0;p=0 still has strong power in the center, which induces crosstalk to the LG `=0;p=1 channel. Points (c) and (d) show the power on the receiver of theLG `=0;p=1 beam whenLG `=0;p=1 is transmitted. As indicated by point (c), when a proper C r is chosen, most power of the incoming LG `=0;p=1 beam is converted to anLG `=0;p=0 -like beam, which could be captured by the SMF. However, as indicated by point (d), when C r =1.7 mm, which is not a proper value, the incoming LG `=0;p=1 is converted to a beam with three nodes, and very little of the power could be captured by the SMF. Figure 5.7 shows the BER measurement of the two channels. The de-multiplexing pattern for the LG `=0;p=0 channel is an all-zeros pattern regardless of C r . A dierent C r shows a similar BER performance. Compared to the back-to-back case (where only the LG `=0;p=0 channel is 58 Figure 5.6: Crosstalk between theLG `=0;p=0 andLG `=0;p=1 channels as a function of circle radius for the de-multiplexing pattern. Tx: p=0; Rx: p=1 refers to the power received by the receiver for the LG `=0;p=1 beam while LG `=0;p=0 beams is transmitted. (a-f) The intensity proles of the beams in front of the single mode ber. C r : Circle radius on the pattern for channel de- multiplexing. Note that in the cases of Tx: p=0; Rx: p=0 and Tx: p=1; Rx: p=0, an all-zero pattern on SLM-3 is used for the channel de-multiplexing and the variation ofC r does not change the pattern. transmitted, and the LG `=0;p=1 channel is turned o), there is a4 dB power penalty. This might be caused by the crosstalk due to the imperfection of the ber coupling. For the LG `=0;p=1 channel, C r =1.0 mm shows a better BER performance than those of C r =0.8 mm and C r =1.2 mm, because the pattern ofC r =1.0 mm matches the size of the incomingLG `=0;p=1 beam best. The above-mentioned demonstrations only considered the LG beams with ` = 0. We also investigate the orthogonality of LG beams with the same higher azimuthal order (` > 0) but dierent radial orders (p values). Take the multiplexing of theLG `=1;p=0 beam and theLG `=1;p=1 beam as an example. To receive the LG `=1;p=0 channel, a spiral phase with ` =1 (see Figure 5.8(b)) is loaded on SLM-3 to convert the LG `=1;p=0 beam into an LG `=0;p=0 beam, while the LG `=1;p=1 beam is converted into LG `=0;p=1 . Similarly, to receive the LG `=1;p=1 channel, a special spiral phase with` =1 (see Figure 5.8(c)) is loaded on SLM-3 to convert the LG `=1;p=1 59 Figure 5.7: Bit error rate (BER) as a function of an optical signal to noise ratio (OSNR) with vari- ous circle radii of the de-multiplexing pattern forLG `=0;p=0 andLG `=0;p=1 channels, respectively. B2B: When only one channel is turned on with the other channel turned o. beam into an LG `=0;p=0 beam, while the LG `=1;p=0 beam is converted into LG `=0;p=1 . This pattern is divided by a circle with the radius of C r into two sections, in which the inside section and outside section has a phase shift. Figure 5.8(a) shows the power and crosstalk for a system multiplexing LG `=1;p=0 and LG `=1;p=1 . The receiver for LG `=1;p=0 is not sensitive to C r , while the receiver for LG `=1;p=1 is sensitive to C r because the de-multiplexing of the LG `=1;p=1 beam relies on the phase change in the radial direction but the de-multiplexing of the LG `=1;p=0 beam does not. We also test the crosstalk among four LG modes with dierent azimuthal orders and/or dif- ferent radial orders, which are the LG `=0;p=0 , LG `=0;p=1 , LG `=1;p=0 , and LG `=1;p=1 beams, as shown in Figure 5.9. When two LG modes have dierent ` and dierent p values, the crosstalk is usually-20 dB because the beams orthogonality could be guaranteed in both the azimuthal direction and the radial direction. When two beams have the same ` value or the same p value, the crosstalk is-12 dB. This type of crosstalk may be caused by (i) the phase vibration on the surface of the SLM or (ii) the imperfection of free space to ber coupling. Such a system may potentially enable a 400-Gbit/s data transmission. 60 Cr (mm) 0.3 0.5 0.7 0.9 1.1 1.3 Normalized Power (dB) -18 -14 -10 -6 Tx:p=0; Rx:p=1 Tx:p=0; Rx:p=0 Tx:p=1; Rx:p=0 Tx:p=1; Rx:p=1 ℓ=1 (a) (b) (c) Figure 5.8: Crosstalk between theLG `=1;p=0 andLG `=1;p=1 channels as a function of circle radius for the de-multiplexing pattern. Tx: p=0; Rx: p=1 refers to the power received by the receiver for LG `=1;p=1 beam while LG `=1;p=0 beams is transmitted. C r : Circle radius on the pattern for channel de-multiplexing. 5.3 Summary and Discussion To summarize, a 200 Gbit/s SDM system using LG beams with dierent radial indices (LG `=0;p=0 and LG `=0;p=1 ) has been demonstrated with both channels achieved the BER of 3:8 10 3 . Moreover, the multiplexing of four LG beams with dierent azimuthal indices and dierent radial indices (e.g., LG `=0;p=0 ,LG `=0;p=1 ,LG `=1;p=0 , andLG `=1;p=1 beams) is also demonstrated with a <-12 dB channel crosstalk. The following points are worth mentioning (i) Other approaches to separate the LG `=0;p=0 beam and the LG `=0;p=1 beam could also potentially be employed. However, the SMF approach mentioned in this chapter does oer the benets of connecting the system to a ber-based coherent direction system. (ii) The free space to ber coupling system aims to couple the whole LG mode into the SMF. Therefore, the spot size of the beam at the focus plane of the objective lens should be controlled to match the radius of the ber. (iii) Compared to the system using only LG beams 61 0 -5 -10 -15 -20 Power (dB) Transmitted Mode Received Mode LGℓ=1,p=0 LGℓ=0,p=0 LGℓ=1,p=1 LGℓ=0,p=1 LGℓ=0,p=1 LGℓ=1,p=1 LGℓ=0,p=0 LGℓ=1,p=0 Figure 5.9: Crosstalk among LG modes with dierent ` and/or dierent p values with `=0 or 1 and p=0 or 1. with p=0, the proposed approach oers a more readily available spatial mode as well as provides a complete free-space mode basis. However, the beams orthogonality requires the phase match in the radial direction, which also increases the systems complexity. (iv) The multiplexing of LG beams with dierentp values may be limited by the link distance, because the beam orthogonality requires the capture of the whole beam, where in the OAM multiplexing system with dierent ` but p=0, a partial receiver at the beam center will not induce extra crosstalk. (v) An SLM, which has a limited resolution, is used for LG beam de-multiplexing. We believe that increasing the resolution of the devices would improve the system performance. 62 Chapter 6 Object Parameters Measurements Using Complex OAM Spectrum 6.1 Background and Motivation When a light beam passes through or is re ected o a physical object or medium, its intensity and phase can be uniquely aected [54]. The information about this object or media could be resolved by investigating the beams intensity as captured by a camera or its phase as obtained by an interferometer [91]. Analogous to time signals, which can be decomposed into multiple orthogonal frequency functions, a light beam can also be decomposed into a set of spatial modes that are taken from an orthogonal basis [99]. Such a decomposition can potentially provide a powerful tool for spatial spectrum analysis, which may enable stable, accurate, and robust extraction of physical object information that may not be readily achievable using traditional approaches. An arbitrary beam (e.g., an objet-truncated Gaussian beam) can also be described by its complex spatial spectrum. Such a complex spectrum is formed by the beams complex decomposi- tion coecients over a mutually orthogonal modal basis set [99, 100], such as Laguerre-Gaussian (LG `;p , with p = 0; 1; 2;:::;, as the radial order, and ` = 0;1;2;::: as the azimuthal order) modes [1, 4, 100]. Generally, the spectrum of a beam comprising a single pure mode peaks only 63 at the value corresponding to the order of the mode, while the spectrum of an arbitrary beam could have complex non-zero values for many mode orders. A subset of LG modes could be the orbital angular momentum (OAM) modes with zero radial index, e.g., LG `;0 . An OAM mode has a phase front of which twists in a helical fashion as it propagates [4], where` is also referred to as the OAM order, and is the azimuthal angle. OAM modes are incomplete in the radial coordinate but they could form a complete orthogonal basis in the azimuthal coordinate. Therefore, when they are used as the orthogonal basis on which to characterize an object-truncated beam, the coecient of various orders of OAM modes may represent the various azimuthal properties of the object [40, 69, 101, 102]. The properties of OAM modes have included the following: (i) The phase-change rate of an OAM mode is proportional to its order, meaning that a larger-order OAM beam has a smaller phase-change spatial periodicity [40, 102]; (ii) The complex OAM spectrum of an arbitrary beam could form a Fourier pair with its spatial-intensity distribution in the azimuthal direction [40, 99, 102]; (iii) The intensity of an OAM mode is circularly symmetric, which means the intensity of a beams OAM spectrum is generally rotation insensitive [1]; and (iv) An OAM mode is relatively stable in a homogeneous medium, which indicates that the amount of OAM of a beam could be constant during free-space propagation regardless of the beam diraction [54, 91]. The use of structured beams has recently been investigated in imaging [9, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116], sensing [40, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126], and communications [7, 6, 9, 100] and other applications [127], including using OAM basis for single-frequency imaging and sensing in the radio frequency domain [115] and the quantum domain [111]. It may be desirable to design a complex OAM spectrum analyzer in the classical optical domain and explore its potential to provide information that could not be readily obtained using the traditional approach. In this chapter, we demonstrate the use of OAM-based complex spectral analysis in the clas- sical optical domain to measure object parameters. The OAM intensity spectrum (amplitude of 64 the complex decomposition coecients) and OAM phase spectrum (angle of the complex decom- position coecients) of an object-truncated Gaussian beam are measured by carefully designing a phase mask in the OAM spectral analyzer. Using the measured complex OAM signature, we could identify the objects relative shape information from the OAM intensity spectrum and its position information from the OAM phase spectrum, where it is dicult for the traditional approach to measure. Specically, we explore the potential of a complex OAM spectrum based system to detect the opening angle and orientation of a sector-shaped object, achieving a >15 dB signal to noise ratio. Our results show the followings: (i) The OAM intensity spectrum is dependent on the opening angle of the object but insensitive to its orientation; (ii) The OAM intensity spectrum is relatively insensitive to free-space beam propagation in the lab, where the measured OAM intensity spectrum is nearly identical when the distance from the object to the OAM spectrum analyzer varies between 0 and 104 cm; and (iii) The OAM phase spectrum is dependent on the orientation of the object but insensitive to its opening angle. 6.2 Concept and Principle Figure 6.1 shows the concept of using an OAM spectral analyzer to measure object parameters. To measure an objects parameters, including shape, thickness, and temperature, a light beam (Gaussian beam, OAM beam, or other beams) could be shone onto an object, and by investigat- ing this beam after the object-truncation, the parameter of interest could be retrieved. In this work, we use the measurement of the opening angle and the openings orientation of an object as an example. When a camera is used to capture the intensity of the beam truncated by the object, the following problems might arise: (i) Due to the diraction caused by object truncation and beam propagation, the image may become too blurry to quantify the opening angle of the object, as shown in the intensity display in Figure 1c, especially when the object is not focused on the camera; (ii) If the object is rotating at a relatively high speed, the camera may not be 65 able to capture a steady image; (iii) A camera is usually a multi-pixel device, which may cost more than a single-pixel device; and (iv) A camera usually could not capture the phase informa- tion of the object-truncated light. However, the complex OAM spectrum based approach could Object Power OAM order Intensity ! Power OAM order Intensity ! Power OAM order Intensity ! Correlated spectrum (b) (c) (a) Figure 6.1: The concept of using the OAM spectrum to measure an objects parameters. The beams intensity prole and OAM spectrum (a) before object truncation, (b) right after object truncation, and (c) some distance after object truncation. potentially determine the opening angle and openings orientation of an object without suering the above-mentioned issues. This approach is based on the following facts: (i) Object truncation of the probe beam could change its OAM spectrum, and the OAM spectrum before and after the object truncation dier suciently that the truncation could be identied; (ii) The complex OAM spectrum of a beam forms a Fourier pair with its spatial distribution in the azimuthal direction [40, 101, 69, 102], in which an opening on the object would lead to a Sinc function in its OAM intensity spectrum; (iii) The light, propagating onward from the truncated object, maintains the powers of its constituent OAM modes so that its OAM intensity spectrum is unaected by the radially symmetric diraction kernel [91]; and (iv) The phase-change rate of an OAM is pro- portional to its order; therefore, rotating an object may cause dierent phase delays to dierent components on the OAM phase spectrum. 66 6.3 Approach, Design, and Results Figure 6.2(a) shows the experimental setup. A spatial light modulator (SLM-1) is used to generate the desired probe beam with a certain OAM order. We generally use a Gaussian beam (` = 0) as the probe, and we also show the cases when various orders of OAM beams are used as the probes to measure object parameters. SLM-2 is used to emulate objects with various parameters (opening angles, orientations and numbers of opening slot). 6.2(b) shows the properties of these objects. We made the following assumptions: i) The object has a larger size than the probe beam; (ii) The object has a sector-shaped opening that is characterized by ; and (iii) The orientation of the opening is dened as the angle between the left edge of the slot opening and the y-axis, which is characterized by . The beam truncated by the object is then collected by a complex OAM spectrum analyzer, which is composed of SLM-3, a subsystem coupling light from free space to single mode ber (SMF), and a power monitor. To obtain a complex OAM spectrum, we measure the OAM intensity spectrum and the OAM phase spectrum separately. To measure the OAM intensity spectrum, SLM-3 is loaded with various spiral phase patterns, so that the decomposition of incoming light onto various OAM orders is measured sequentially and together forms an OAM intensity spectrum. To measure the OAM phase spectrum, we measure the relative phase between OAM ` (` =8;7;:::; +7; +8) and OAM 0 (i.e., Gaussian mode) sequentially. For each phase measurement, four dierent specially designed patterns are loaded on SLM-3 sequentially, and together the four intensities measured by the power monitor enable calculation of the phase. In this experiment, the distance between the object (SLM-2) and the OAM spectrum analyzer (SLM- 3) varies from 0 to 35, 52 and 104 cm. We note that for 0 distance, SLM-2 serves only as a mirror, and the object emulation pattern and OAM spectrum analyzer pattern are combined and loaded on SLM-3. 67 SLM-1: Beam shaper L-1 Laser L-2 SLM-2: Object emulator SLM-3: Special pattern HWP Col Power monitor M Free space Fiber Col Complex OAM spectrum analyzer M M θ ! x y (b) (a) Figure 6.2: (a) Experimental setup. The SLM-3, L-1, L-2, collimator and power monitor form the complex OAM spectrum analyser. Col: collimator, HWP: half-wave plate, M: mirror, SLM: spatial light modulator, L: lens. (b) The shape and position of the object. : opening angle of the object; : orientation of the object relative to the y-axis. We rst use the the complex OAM spectrum based approach to measure the opening angle of an object. As Figure 6.3(a) shows, the sample object has an opening angle of 2pi/3 and is placed in the propagation path of a Gaussian probe beam. For comparison, we also measure the image of the object-truncated light with a lens-less camera. As Figure 6.3(b) shows, it is dicult to determine the opening angle of the object due to blurring. However, when we measure the OAM intensity spectrum, we observe several dips in the OAM spectrum, as shown in the Figure 6.3(c). The power dips appear on OAM ordersj`j = 3N(N is a non-zero integer), indicating that the object has an opening angle of. When an object, azimuthally opened with an angle of 2=3, truncates an OAM probe with order ` 1 (here ` 1 = 0), the probes spectral OAM component of order` 2 vanishes when` 1 ` 2 = 2N= [69, 101]. This is due to the vanishing eect of the overlap integral between` 1 and` 2 over an azimuthal angle of, that is, R 1 0 R 0 u(r;;` 1 )u (r;;` 2 )rddr when (` 1 ` 2 ) = 2N, whereu(:) is the electrical eld. The power dierence between a dip and its neighbors has a dierence of >15 dB, indicating that our approach has a high signal-to-noise ratio (SNR). 68 OAM Order -8 -6 -4 -2 0 2 4 6 8 10 Normalized Power (dB) -50 -40 -30 -20 -10 0 state 1 state 2 state 3 state 4 state 5 state 6 state 7 state 8 4.2 mm 4.2 mm a1 a2 a3 a4 a5 a6 a7 a8 b1 b2 b3 b4 b5 b6 b7 b8 (c) f2 f1 f3 g1 g2 g3 OAM Order -8 -6 -4 -2 0 2 4 6 8 Normalized Power (dB) -50 -40 -30 -20 -10 0 1 open slot 2 open slots 3 open slots (h) First Dip Position (OAM Order) 2 4 6 8 10 12 Openning Angle (rad) 0 1 2 3 Simulation Measurement OAM Order -8 -6 -4 -2 0 2 4 6 8 Normalized Power (dB) -50 -40 -30 -20 -10 0 3=2 :/3 3=2 :/4 3=2 :/5 3=2 :/6 (d) (e) OAM Order -8 -6 -4 -2 0 2 4 6 8 Normalized Power (dB) -50 -40 -30 -20 -10 0 dis: 0 cm dis: 35 cm dis: 52 cm dis: 104 cm θ=2!/3 (i) TX OAM Order -8 -6 -4 -2 0 2 4 6 8 RX OAM Order -8 -6 -4 -2 0 2 4 6 8 0 10 20 30 -46 -36 -26 -16 (j) (dB) Figure 6.3: (a1-a8) Various orientations (states) for an object having an opening angle of 2/3. (b1-b8) The image of the light beam truncated by the objects in (a1-a8), respectively. (c) The OAM intensity spectrum measured for the light truncated by the objects in (a1-a8). (d) The OAM intensity spectrum measured for the light truncated by objects having various opening angles. (e) The relationship between the opening angles and the rst-dip position in the OAM intensity spectrum. (f1-f3) Objects having 1, 2 and 3 slots, each of which has an opening angle of 2/6. (g1-g3) The images of the light beam truncated by the objects in (f1-f3). (h) The OAM intensity spectrum measured for the light truncated by the objects in (f1-f3). (i) The OAM intensity spectrum measured for the light truncated by an object having an opening angle of 2/3 when the distance from the object to the OAM spectrum analyser is 0 cm, 35 cm, 52 cm and 104 cm. (j) The OAM intensity spectrum matrix measured when various OAM beams are used as the probe. The distance from the object to the OAM spectrum analyser is 0 in c, d, e and h, j. When we rotate the object in Figure 6.3(a) with various angles (0,/4, /2, 3/4, , 5/4, 3/2 and 7/4) along its center (corresponding to the states 1 to 8 in Figure 6.3(b)), the measured 69 OAM spectrum is almost unchanged, as Figure 6.3(c) shows. We believe this is due to the circular symmetry of the intensity of an OAM beam. Therefore, our approach could potentially be used to determine the opening angle of the object even when the object is rotating. Furthermore, we verify the relationship between objects opening angle and the OAM intensity spectrum of the beam truncated by the object. As Figure 6.3(d) shows, when the objects opening angle is modied to 2/4, dips are observed on the modes withj`j = 4N (e.g.,` =8;4; +4; +8). When the opening angle is 2/5, dips are observed on the modes with j`j = 5N (e.g., ` = 5; +4; +5). To explore the relationship between the objects opening angle and the rst dip position that appears, we simulate a more general case, as shown in Figure 6.3(e). The rst dip position is inversely proportional to the objects opening angle. The object might have more than one opening slot, as shown in Figure 6.3(f). Here, we explore three dierent objects with one, two and three slots, each of which has an opening angle of = 2=6. As Figure 6.3(g) shows, the images measured using a camera are still blurred. However, the OAM spectrum shows clear characteristics that help determine both the objects opening angle and its slot number. As Figure 6.3(h) shows, when there are two slots, peaks appear at every other mode, while when there are three slots, peaks appear every three modes. This peak periodicity indicates the number of opening slots on the object. No matter how many slots the object has, deep dips show up on OAM +6 and OAM-6, indicating that the size of the opening is 2/6. As Figure 6.3(i) shows, the OAM spectrum varies within a small range when the spectrum analyzer is placed at various distances (0 cm, 35 cm, 52 cm and 104 cm) from the object. This is because the kernel of the diraction (the eect of propagating free space) is proportional to the phase-changing direction of the OAM beam. However, we observe some power uctuation at various distances. We believe this is because our system ignores LG modes with non-zero radial orders (p6= 0). When the beam is truncated by an object and propagates in free space, higher- 70 radial-order LG modes may arise whose power could not be collected by our OAM spectrum analyzer since it contains a free space to SMF coupling subsystem. We discuss the case when a Gaussian beam is used as the pilot to measure the object parameter. An OAM beam could also be used as the pilot beam. As Figure 6.3(j) shows, when various orders of OAM beams are used sequentially as probes, the magnitudes of the measured OAM spectral components form a circulant matrix. Such a matrix could provide multiple copies of information for measuring object parameters, thus potentially ensuring higher accuracy by averaging the various measurements. Besides the OAM intensity spectrum, the resultant OAM modes may also have various phases. Here, we sequentially measured the relative phase between OAM` and OAM 0 for` =8;7;:::; +7; +8, to get an OAM phase spectrum. In general, to measure the phase dierence between OAM ` 1 and OAM ` 2 , we load phase masks T 0 , T 45 , T 90 , and T 135 on SLM-3, where: 8 > > > > > > > > > > < > > > > > > > > > > : exp(jT 0 ) = exp(j` 1 ) + exp(j` 2 ) exp(jT 45 ) = exp(j` 1 ) +j exp(j` 2 ) exp(jT 90 ) = exp(j` 1 ) exp(j` 2 ) exp(jT 135 ) = exp(j` 1 )j exp(j` 2 ) (6.1) We dene the intensities measured by the power monitor when loading each phase mask T 0 , T 45 , T 90 , andT 135 on SLM-3 as I 0 , I 45 , I 90 , andI 135 , respectively. Then, the relative phase dierence between the two OAM modes is given by [120, 126, 128] ' = arctan ((I 0 I 90 )= (I 45 I 135 )) (6.2) We note that reports have shown the complex OAM spectrum measurement using an interferom- eter [124, 125, 129], which might be sensitive to the setup vibration since its two branches usually 71 OAM Order -8 -6 -4 -2 0 2 4 6 8 10 Phase (rad) -2 -1 0 1 2 /=- :/8 /=- :/16 /=0 OAM Order -8 -6 -4 -2 0 2 4 6 8 10 Phase (rad) -2 -1 0 1 2 /= :/8 /= :/16 /=0 OAM Order -8 -6 -4 -2 0 2 4 6 8 10 Phase (rad) -2 -1 0 1 2 3=2 :/3 3=2 :/4 3=2 :/5 3=2 :/6 OAM Order -8 -6 -4 -2 0 2 4 6 8 10 Phase (rad) -2 -1 0 1 2 /= :/4 OAM Order -8 -6 -4 -2 0 2 4 6 8 10 Phase (rad) -2 -1 0 1 2 /= :/4 Measured Slop -2 -1 0 1 2 Openning Orientation (rad) -2 -1 0 1 2 Simulation Measurment !=2"/3 !=2"/3 (a) (b) (c) #="/8 No # pre-estimation With a pi/8 # Pre-estimation (e) (f) (d) Figure 6.4: (a) The OAM phase spectrum measured for the light truncated by an object having an opening angle of 2/3 but various negative orientation angles. (b) The OAM phase spectrum measured for the light truncated by an object having an opening angle of 2/3 but various positive orientation angles. (c) The relationship between the orientation angle of the object and the slope of the OAM phase specturm. (d) The OAM phase spectrum measured for the light truncated by an object having various opening angles and an orientation angle of /8. (e, f) The OAM phase spectrum measured for the light truncated by an object having an opening angle of 2/3 and an orientation angle of /4. (e) No pre-estimate for . (f) Pre-estimate of ==8. In this Figure, the distance from the object to the OAM spectrum analyser is 0. The lines show the simulation results, and the symbols show the experimental measurements. In the measurement, the data is calculated by an Arctan function, and the measured phases are between -/2 and/2. Therefore, we add a or phase shift to some of the measurements for the convenience of phase slop calculation. have dierent optical paths. However, the approach in this work is based on non-interference [120, 126, 128] and may provide a more stable phase measurement. We measure the OAM phase spectrum of the object-truncated beam when the object has an opening angle of 2/3 but various orientation angles . As Figures 6.4(a) and (b) show, the orientation angle is closely related to the slope of the OAM phase spectrum. When the orientation angle of the object is, the slope of the measured phase is approximately, where a dierent sign of indicates that the object is oriented in a dierent direction relative to the y-axis. We believe this is because dierent OAM modes have dierent phase-change rates. Rotating the object by a 72 certain angle may cause dierent phase delays for dierent OAM modes, and such phase delays are proportional to the OAM orders (see Figure 6.4(c)). We also x the orientation of the object to be/8. When we change the objects opening angle from 2/3 to 2/4, 2/5 and 2/6, the slope of the OAM spectrum is nearly constant, as shown in Figure 6.4(d), indicating that the OAM phase spectrum is insensitive to the objects opening angle. Our phase measurement is based on calculating an Arctan function, which means we could only resolve a phase between -/2 and /2. This might cause phase-measurement degradation when the object has a large orientation angle (). As Figure 6.4(e) shows, when = /4, the measured phase shows periodicity every three or four OAM modes, making it dicult to determine the slope of the phase. However, this could be compensated for by calculating a pre-estimate 0 . Using a pre-estimate, the phase masks loaded to SLM-3 are calculated by: 8 > > > > > > > > > > < > > > > > > > > > > : exp(jT 0 ) = exp(j` 1 ) + exp(j` 2 +j` 2 0 ) exp(jT 45 ) = exp(j` 1 ) +j exp(j` 2 +j` 2 0 ) exp(jT 90 ) = exp(j` 1 ) exp(j` 2 +j` 2 0 ) exp(jT 135 ) = exp(j` 1 )j exp(j` 2 +j` 2 0 ) (6.3) Figure 6.4(f) shows the OAM phase spectrum measurement for an object with an opening angle of 2/3 and an orientation angle of = =4, when we pre-estimate 0 = =8. Using this pre-estimate, the slope of the measured phase exhibits less uctuation. 6.4 Summary and Discussion There are several issues that are valuable to consider when using a beams complex spectrum for object parameter measurement. 73 The approach outlined in this paper can be extended to provide higher resolution than we have experimentally shown. As one example, the object could have an opening angle of 2m=N instead of 2=N, where m is an integer. Figure 6.5 shows the cases when N = 5, and m = 1; 2; 3; 4. We nd that the position of the dip is determined by N (see Characteristic B), while the power of the transmitted probe beam (the peak or the ratio of peak over its neighbors) re ects the relative information of m (see Characteristic A) because larger m allows more received power. OAM Order -8 -6 -4 -2 0 2 4 6 8 Normalized Power (dB) -50 -40 -30 -20 -10 0 3=2 :/5 3=4 :/5 3=6 :/5 3=8 :/5 Characteristic A Characteristic B Figure 6.5: The OAM intensity spectrum measured for the light truncated by an object having an opening angle of 2m=5, where m = 1; 2; 3; 4. As a second example, the resolution of the measured opening angle could be further improved by increasing the number of OAM modes that have been investigated. As a proof-of-concept, we investigate the power and phase of OAM beams with orders of from -8 to +8. Evidently more modes could be used to improve the resolution of the measurement as indicated by the simulation results in Figure 6.3(e). OAM modes are used as basis for measuring the object opening angle and object openings orientation. The concept demonstrated in this work is extendable, and modal basis sets other than OAM, LG modes or Hermite-Gaussian (HG) mode, could also be applied for object parameter measurement, depending on the objects and modes properties. 74 6.5 Appendix To measure the OAM intensity spectrum of an object-truncated beam, we load the various spiral phases (`) on SLM-3 sequentially. Each pattern down-converts a specic OAM component (OAM+`) to a Gaussian-like beam that could be collected by the single mode ber (SMF), while the other component is converted to non-zero OAM modes, which could not be collected by the SMF. When ` varies between -8 and 8, an OAM spectrum is obtained. T 0 T 45 T 90 T 135 S 1 =OAM ! 1 ! 1 =+3 ! 2 =+0 T 0 T 45 T 90 T 135 S 2 =OAM ! 2 0 2" Figure 6.6: Principle used to measure the OAM phase spectrum. T 0 , T 45 , T 90 , and T 135 are calculated according to ` 1 and ` 2 and are loaded to SLM-3 sequentially to measure the phases between OAM ` 1 and ` 2 . Figure 6.6 shows the principle for measuring the phase between OAM ` 1 and OAM ` 2 . In general, measuring the phase dierence between two OAM modes is analogous to measuring the relative phase dierence between the left-circularly polarized component and the right-circularly polarized component of a light beam in Stokes polarimetry [128]. First, we dene a light beam using left-circularly polarized and right-circularly polarized light as the basis for E =E l ^ l +E r ^ r, where E is the electrical eld of the beam and E l and E r denote the components of the beam that are polarized left- and right-circularly, respectively. We could have x- and y-polarized light as another basis for representing the light beam, E =E x ^ x +E y ^ y, 8 > > < > > : E x =E l +E r E y =E l E r (6.4) 75 where E x and E y denote the x- and y-polarized component of the beam, respectively. We could also have another basis, which is light that is polarized 45-degrees (E w ) and 135-degrees (E v ), which obeys E =E w ^ w +E v ^ v where: 8 > > < > > : E w =E l +jE r E v =E l jE r (6.5) We use I 0 , I 45 , I 90 , and I 135 to characterize the intensity of E x , E y , E w , and E v , respectively. The ,I 0 , I 45 , I 90 , and I 135 of a light could be measured when we have a linear polarizer rotated 0, 45, 90 and 135 degrees, respectively, with respect to its transmission axis. Then, the relative phase dierence between components of the light that are polarized left- and right-circularly is given by [120, 126, 128]: ' = arctan ((I 0 I 90 )= (I 45 I 135 )) (6.6) Analogously, when we replace the original basis (beams that are polarized left- and right-circularly) with two OAM modes (with the orders of ` 1 and ` 2 ), we are able to measure the relative phase dierence between the two OAM modes. We load phase masks T 0 , T 45 , T 90 , and T 135 , which are analogous to a linear polarizer rotated 0, 45, 90 and 135 degrees, respectively, on SLM-3, where I 0 , I 45 , I 90 , and I 135 obey: 8 > > > > > > > > > > < > > > > > > > > > > : exp(jT 0 ) = exp(j` 1 ) + exp(j` 2 ) exp(jT 45 ) = exp(j` 1 ) +j exp(j` 2 ) exp(jT 90 ) = exp(j` 1 ) exp(j` 2 ) exp(jT 135 ) = exp(j` 1 )j exp(j` 2 ) (6.7) Similarly, we dene the intensities measured by the power monitor when loading each phase mask T 0 , T 45 , T 90 , and T 135 , on SLM-3 as I 0 , I 45 , I 90 , and I 135 , respectively. Then, the relative phase dierence between the two OAM modes is given by the same Equation 6.7. Figure 6.6 76 shows the phase proles for OAM 0 and OAM+3 and the T 0 , T 45 , T 90 , and T 135 phase patterns used to measure their relative phases. 77 Chapter 7 Conclusion The thesis has made summary and conclusion on each chapter. 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Abstract (if available)

Abstract An electromagnetic (EM) wave can carry orbital angular momentum (OAM) if the wave has a helical transverse phase structure of exp(jlφ), in which φ is the transverse azimuthal angle and l is an unbounded integer (the OAM state number). OAM beams with different l values are mutually orthogonal to each other, allowing them (i) to be multiplexed together along the same beam axis and de-multiplexed with low crosstalk to establish a channel-multiplexed line-of-sight communications link (e.g., spaced division multiplexing or SDM), (ii) to form an orthogonal mode basis such that other beams could be decomposed over this basis to provide potential advantages for a remote sensing system. ❧ OAM based SDM links have been demonstrated in both the optical domain and the radio frequency (RF) domain. Specifically, a 100-Tbit/s communication link combining SDM, wavelength division multiplexing (WDM), and polarization division multiplexing (PDM) was demonstrated at the wavelength ∼1550 nm in 2013. In addition, a 32-Gbit/s communication link combining the SDM and PDM at a carrier frequency of 28 GHz was demonstrated in 2014. However, these proof-of-concept demonstrations did not take the potential system design challenges into consideration. In this thesis, the challenges and potential solutions for the OAM based free-space SDM system are discussed, including: (i) the atmospheric turbulence effects and the turbulence compensation approaches, (ii) system-level design guidelines for long-distance OAM multiplexed links, (iii) beam steering for OAM multiplexing based communication systems, and (iv) the use of Laguerre-Gaussian beams with different radial indices for channel multiplexing. ❧ Besides, OAM beams’ unique intensity and phase structures could potentially provide a powerful tool for remote sensing other than communications. As an example, this thesis also explores the measurements of an object’s opening angle and the opening’s orientation using the information provided by OAM complex spectrum.

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Creator Xie, Guodong (author)

Core Title Using beams carrying orbital angular momentum for communications and remote sensing

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Publication Date 01/31/2018

Defense Date 11/29/2017

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

Tag OAI-PMH Harvest,optical communications,orbital angular momentum,remote sensing,spaced division multiplexing

Language English

Contributor Electronically uploaded by the author (provenance)

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

Creator Email guodongx@usc.edu,littletiger1926@gmail.com

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c40-467378

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Legacy Identifier etd-XieGuodong-5986.pdf

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Rights Xie, Guodong

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

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

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