About the Editors xv
List of Contributors xvii
Preface xxi
Part I Fundamentals and Basics of Electromagnetic Vortices 1
1 Fundamentals of Orbital Angular Momentum Beams: Concepts, Antenna Analogies, and Applications 3
Anastasios Papathanasopoulos and Yahya Rahmat-Samii
1.1 Electromagnetic Fields Carry Orbital Angular Momentum 3
1.2 OAM Beams; Properties and Analogies with Conventional Beams 4
1.2.1 Laguerre-Gaussian Modes 5
1.3 Communicating Using OAM: Potentials and Challenges 10
1.3.1 OAM Communication Link Scenarios and Technical Barriers 11
1.3.2 OAM Emerging Applications and Perspectives 14
1.3.2.1 Free-space Communications 14
1.3.2.2 Optical Fiber Communications 17
1.4 OAM Generation Methods 20
1.5 Summary and Perspectives 22
Appendix 1.A OAM Far-field Calculation 23
References 26
2 OAM Radio - Physical Foundations and Applications of Electromagnetic Orbital Angular Momentum in Radio Science and Technology 33
Bo Thidé and Fabrizio Tamburini
2.1 Introduction 33
2.2 Physics 34
2.2.1 The Classical Electromagnetic Field 34
2.2.2 Electrodynamic Observables 36
2.2.2.1 Behavior at Very Long Distances 41
2.3 Implementation 45
2.3.1 Wireless Information Transfer with Linear Momentum 46
2.3.2 Wireless Information Transfer with Angular Momentum 48
2.3.2.1 Spin Angular Momentum vs. Orbital Angular Momentum 50
2.3.2.2 Angular Momentum Transducers 50
2.3.2.3 Electric Hertzian Dipoles 52
2.3.3 Astronomy Applications 58
Appendix A 61
2.A.1 Theory 61
2.A.1.1 Classical Majorana-Oppenheimer Formalism and Its Affinity to First Quantization Formalism 61
2.A.1.1.1 Riemann-Silberstein Electromagnetic Potentials and Fields 63
A.1.1.1 Purely Electric Sources 66
A.1.1.2 Useful Approximations 67
A.1.2.1 The Paraxial Approximation 68
A.1.2.2 The Far-Zone Approximation 70
2.A.2 Poincaré Invariants and Conserved Quantities of the EM Field 74
A.2.1 Energy 74
A.2.2 Linear Momentum 76
A.2.2.1 Gauge Invariance 78
A.2.2.2 First Quantization Formalism 79
A.2.3 Angular Momentum 80
A.2.3.1 Gauge Invariance 82
A.2.3.2 First Quantization Formalism 83
References 84
Part II Physical Wave Phenomena of Electromagnetic Vortices 97
3 Generation of Microwave Vortex Beams Using Metasurfaces 99
Jia Yuan Yin and Tie Jun Cui
3.1 Introduction 99
3.2 Metasurfaces for Vortex-beam Generation 100
3.2.1 Reflective Metasurfaces for Vortex-beam Generation 101
3.2.2 Transmission Metasurfaces for Vortex-beam Generation 108
3.2.3 Planar Metasurfaces for Vortex-beam Generation 110
3.2.4 Metasurfaces for Modified Vortex-beam Generation 112
3.2.5 One-dimensional Metasurface for Vortex-beam Generation 113
3.3 Conclusion 114
Acknowledgments 114
References 115
4 Application of Transformation Optics and 3D Printing Technology in Vortex Wave Generation 121
Jianjia Yi, Shah Nawaz Burokur, and Douglas H. Werner
4.1 Introduction 121
4.2 Theoretical Basis of Transformation Optics and 3D Printing 121
4.2.1 The Concept and Development of Transformation Optics 121
4.2.2 An Overview of 3D Printing Techniques 125
4.3 Several Applications of Transformation Optics in Vortex Waves 128
4.3.1 All-Dielectric Transformed Material for the Generation of OAM Beams 128
4.3.2 All-dielectric Metamaterial Medium for Collimating OAM Vortex Waves 137
4.3.3 A Transformation Optics-Based Lens for Horizontal Radiation of OAM Vortex Waves 147
4.4 Conclusions 153
References 154
5 Millimeter-Wave Transmit-Arrays for High-Capacity and Wideband Generation of Scalar and Vector Vortex Beams 157
Zhi Hao Jiang, Lei Kang, Wei Hong, and Douglas H. Werner
5.1 Introduction 157
5.2 Vector Vortex Beams and Hybrid-Order PSs 159
5.3 Millimeter-Wave Transmit-Array Unit Cell Designs 161
5.3.1 Ka-Band CP Unit Cell Design 161
5.3.2 Q-Band CP Unit Cell Design 165
5.3.3 K-Band Dual-CP Unit Cell Design 166
5.4 Millimeter-Wave Transmit-Arrays for Vortex Beam Multiplexing 171
5.4.1 Far-Field Pattern Calculation for Transmit-Arrays 171
5.4.2 Multiplexing of Scalar Vortex Beams 172
5.4.3 Multiplexing of Vector Vortex Beams with Symmetry Constraints 176
5.4.4 Multiplexing of Vector Vortex Beams with Broken Symmetry 182
5.5 Conclusion 183
Acknowledgment 183
References 184
6 Twisting Light with Metamaterials 189
Natalia M. Litchinitser
6.1 Introduction 189
6.2 OAM Beams on the Nanoscale 194
6.3 Active OAM Sources 201
6.4 OAM Light in Engineered Nonlinear Colloidal Systems 206
6.5 Conclusion 214
References 214
7 Generation of Optical Vortex Beams 223
Yuanjie Yang and Cheng-Wei Qiu
7.1 Introduction 223
7.2 Basic Theory of Optical Vortex 224
7.3 Generation of Optical Vortex 225
7.3.1 Generation of Vortex Beams using Optical Elements 225
7.3.1.1 Spiral Phase Plate 225
7.3.1.2 Fork-grating Hologram 226
7.3.1.3 Spiral Zone Plate Holograms 226
7.3.2 Generation of Vortex Beams Using Digital Devices 227
7.3.3 Generation of Vortex Beams Based on Mode Conversion 229
7.3.4 Generation of Vortex Beams Based on the Superposition of Waves 230
7.3.5 Generation of Vortex Beams Based on Metasurfaces 231
7.4 Generation of Novel Vortex Beams 233
7.4.1 Perfect Vortex Beam 233
7.4.2 Fractional Vortex Beams 235
7.4.3 Anomalous Vortex Beam 237
7.4.4 Vortex Beams with Varying OAM 239
7.5 Conclusion 241
References 241
8 Orbital Angular Momentum Generation, Detection, and Angular Momentum Conservation with Second Harmonic Generation 245
Menglin L. N. Chen, Xiaoyan Y. Z. Xiong, Wei E. I. Sha, and Li Jun Jiang
8.1 Orbital Angular Momentum Generation and Detection 245
8.1.1 OAM Generation 246
8.1.1.1 Complementary Metasurfaces 247
8.1.1.2 Quasi-Continuous Metasurfaces 247
8.1.1.3 Photonic Crystals 250
8.1.2 OAM Detection 252
8.1.2.1 Modified Dynamic Mode Decomposition 252
8.1.2.2 Holographic Metasurfaces 254
8.2 AM Conservation: Nonlinear Optics 256
8.2.1 BEM for Nonlinear Optics 256
8.2.2 Verification of the Algorithm 258
8.2.3 Mixing of Spin and OAM 259
8.2.4 General Angular Momenta Conservation Law 261
8.3 Conclusion 263
References 264
Part III Engineering Applications of Electromagnetic Vortices 269
9 Orbital Angular Momentum Based Structured Radio Beams and its Applications 271
Xianmin Zhang, Shilie Zheng, Wei E. I. Sha, Li Jun Jiang, Xiaowen Xiong, Zelin Zhu, Zhixia Wang, Yuqi Chen, Jiayu Zheng, Xinyue Wang, and Menglin L. N. Chen
9.1 Introduction 271
9.2 PS-OAM Based Structured Beams 272
9.2.1 Plane Spiral OAM 272
9.2.2 Structured Radio Beam 273
9.3 Antennas for Structured Beams 276
9.3.1 Antennas for PS-OAM Waves 276
9.3.2 SIW-based Compact Antenna 279
9.3.3 Partial Arc Transmitting Scheme 284
9.4 Potential Applications 286
9.4.1 Radar Detection 286
9.4.2 MIMO System 287
9.4.3 Spatial Field Digital Modulation 289
9.5 Conclusion 291
References 291
10 OAM Multiplexing Using Uniform Circular Array and Microwave Circuit for Short-range Communication 295
Kentaro Murata and Naoki Honma
10.1 Introduction 295
10.2 OAM Multiplexing System and its Mechanism 297
10.2.1 Coaxial UCA Configuration 297
10.2.2 Circulant Channel Matrix 298
10.2.3 DFT/IDFT Beamformers 299
10.3 OAM Multiplexing for Short-range Communications 300
10.3.1 Achievable Rate 300
10.3.2 Array Topology 301
10.3.3 Optimal Array Radius 304
10.3.4 Butler Matrix 309
10.3.5 Performance Evaluation 312
10.4 Conclusion and Key Challenges 317
References 318
11 OAM Communications in Multipath Environments 321
Xiaoming Chen and Wei Xue
11.1 Introduction 321
11.1.1 Fading in Wireless Propagation 321
11.1.1.1 Pass Loss 322
11.1.1.2 Large-Scale Fading 322
11.1.1.3 Small-Scale Fading 322
11.1.2 Diversity and Multiplexing 323
11.1.3 MIMO Systems 324
11.2 OAM Communication in Line-of-sight Environment 325
11.2.1 Conventional OAM Multiplexing 325
11.2.2 OAM Multiplexing with Spatial Equalization 329
11.3 OAM Multiplexing in Multipath Environment 337
11.3.1 Specular Reflection 337
11.3.1.1 Intra-channel Interference 338
11.3.1.2 Inter-channel Interference 341
11.3.2 Indoor Environment 343
11.3.2.1 Inter-Symbol Interference (ISI) 343
11.3.2.2 Antenna misalignment 346
11.3.3 Highly Reverberant Environments 349
11.4 Conclusion 354
References 354
12 High-capacity Free-space Optical Communications Using Multiplexing of Multiple OAM Beams 357
Alan E. Willner, Runzhou Zhang, Kai Pang, Haoqian Song, Cong Liu, Hao Song, Xinzhou Su, Huibin Zhou, Nanzhe Hu, Zhe Zhao, Guodong Xie, Yongxiong Ren, Hao Huang, and Moshe Tur
12.1 Introduction 357
12.2 Challenges for an OAM Multiplexing Free-space Optical Communication System 359
12.2.1 Beam divergence 360
12.2.2 Misalignment 361
12.2.3 Atmospheric Turbulence Effects 362
12.2.4 Obstruction 364
12.2.5 Summary 364
12.3 Free-space Optical OAM Links 364
12.3.1 High-capacity OAM Multiplexed Communication Link Under Laboratory Conditions 365
12.3.2 OAM-based FSO Link Beyond Laboratory Distances 368
12.3.3 Summary 371
12.4 Inter-channel Crosstalk Mitigation Methods in OAM-multiplexed FSO Communications 371
12.4.1 Adaptive Optics for Crosstalk Mitigation 371
12.4.1.1 AO Using a Wavefront Sensor (WFS) and a Gaussian Probe Beam 372
12.4.1.2 AO Using WFS and Gaussian Probe Beam in a Quantum Communication Link 374
12.4.1.3 AO Using a Camera for Beam Intensity Measurement 376
12.4.2 Spatial Modes Manipulation for Crosstalk Mitigation 378
12.4.2.1 Turbulence Precompensation by OAM Mode Combination 378
12.4.2.2 Simultaneous Orthogonalizing and Shaping of Multiple LG Beams 380
12.4.3 Digital Signal Processing for Crosstalk Mitigation 381
12.4.3.1 MIMO Equalization for Crosstalk Mitigation in Laboratory 382
12.4.3.2 Turbulence-Resilient Beam Mixing for Crosstalk Mitigation 383
12.4.4 Summary 384
12.5 OAM Multiplexing for Unmanned Aerial Vehicle (UAV) Platforms 385
12.5.1 OAM System Design and Demonstrations for UAV Platforms 386
12.5.2 Multiple-Input-Multiple-Output (MIMO) Mitigation for Atmospheric Turbulence in UAV Platforms 389
12.5.3 Summary 390
12.6 OAM Multiplexing in Underwater Environments 391
12.6.1 Underwater Effects for OAM Beam Propagation 392
12.6.2 OAM Multiplexing Demonstrations in Underwater Environments 392
12.6.3 Multiple-Input-Multiple-Output (MIMO) Mitigation for Inter-Channel Crosstalk in Underwater Environments 394
12.6.4 Summary 394
12.7 Summary of this Chapter 394
Acknowledgment 396
References 396
Part IV Multidisciplinary Explorations of Electromagnetic Vortices 401
13 Theory of Vector Beams for Chirality and Magnetism Detection of Subwavelength Particles 403
Mina Hanifeh and Filippo Capolino
13.1 Characterization of Azimuthally and Radially Polarized Beams 403
13.2 Circular Dichroism for a Particle of Subwavelength Size 407
13.2.1 Helicity of an Azimuthally Radially Polarized Vector Beam 409
13.3 Photoinduced Force Microscopy at Nanoscale 411
13.3.1 Magnetic Photoinduced Force Microscopy by Using an APB 412
13.3.2 Chirality Photoinduced Force Microscopy 415
13.4 Conclusion 418
References 418
14 Quantum Applications of Structured Photons 423
Alessio D'Errico and Ebrahim Karimi
14.1 Introduction 423
14.2 Photonic Degrees of Freedom 424
14.3 Single Photon Source: SPDC 426
14.4 Generation and Detection of Structured Photon Quantum States 430
14.4.1 Generation of Structured Photon States 430
14.4.2 Detection of Structured Photons 433
14.5 Quantum Key Distribution 434
14.5.1 BB84 Protocol 436
14.5.2 Alignment-free QKD 437
14.5.3 High-dimensional QKD 438
14.6 Quantum Simulation with Quantum Walks 442
14.6.1 Quantum Walks in the OAM Space 443
14.6.2 Shaping the Walker Space: Cyclic Walks and Walks on 2D Lattices 444
14.6.3 Applications: Wavepacket Dynamics and Detection of Topological Phases 446
14.7 Outlook 450
References 450
Index 457
