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

<p><b>ZHI HAO JIANG, PHD,</b> is Professor at the State Key Laboratory of Millimeter Waves and Associate Dean of the School of Information Science and Engineering, Southeast University. He is the co-editor of <i>Electromagnetics of Body-Area Networks: Antennas, Propagation, and RF Systems</i>. </p> <p><b>DOUGLAS H. WERNER, PhD,</b> is Director of the Computational Electromagnetics and Antennas Research Lab, as well as a faculty member of the Materials Research Institute at Penn State. He is also Editor for the <i>IEEE Press Series on Electromagnetic Wave Theory & Applications</i>.

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