The Angular Momentum of Light

The angular momentum of light 1......Page 1
Copyright 4......Page 4
Contents 5......Page 5
List of contributors\tpage 12......Page 12
Preface\t15......Page 15
1.1 Introduction 17......Page 17
1.2 Mechanical properties of optical fields\t18......Page 18
1.2.1 Energy of the electromagnetic field 19......Page 19
1.2.2 Linear momentum of the electromagnetic field 21......Page 21
1.2.4 Spin and orbital angular momentum 22......Page 22
1.3.1 Helmholtz equation 24......Page 24
1.3.2 Paraxial approximation 27......Page 27
1.4.1 Paraxial light beams 28......Page 28
1.4.2 Non-paraxial light beams 36......Page 36
1.5.1 Angular momentum flux 37......Page 37
1.5.2 Azimuthal phase structure 40......Page 40
1.6 Generating light beams with orbital angular momentum 41......Page 41
References 43......Page 43
2.1 Introduction 47......Page 47
2.2 Optical vortices 48......Page 48
2.2.1 Morphology of noncanonical vortices 49......Page 49
2.2.2 Multiple vortices 51......Page 51
2.3 Equations of evolution for the isolated vortices 52......Page 52
2.3.1 Dynamics of a single vortex in linear homogeneous media 53......Page 53
2.3.2 Inversion of the topological charge of a vortex 54......Page 54
2.3.3 Experimental demonstration of the vortex inversion 57......Page 57
2.3.4 Vortices and angular momentum 58......Page 58
2.3.5 Another interpretation of the charge inversion 59......Page 59
2.4 Interactions between vortices 60......Page 60
2.4.1 GRaded INdex media as a laboratory for vortex interactions 61......Page 61
2.4.2 Vortex interaction in GRIN media 62......Page 62
2.4.3 Interaction of two vortices 63......Page 63
References 65......Page 65
3.1 Introduction 67......Page 67
3.2.1 Hermite-Gauss and Laguerre-Gauss beams 68......Page 68
3.2.2 Relations between first-order Hermite-Gaussand Laguerre-Gauss beams 69......Page 69
3.3 A review of polarization modes 71......Page 71
3.4.1 Theoretical description of first-order vector beams 73......Page 73
3.4.2 Higher-order vector beams 77......Page 77
3.5.1 Experimental methods of production 80......Page 80
3.5.2 Detection of vector beams by polarization projection 82......Page 82
3.6 Vector beams in non-classical states of light 84......Page 84
References 85......Page 85
4.1 Introduction 87......Page 87
4.1.1 Vortex solitons 88......Page 88
4.1.3 Two-soliton spiraling 90......Page 90
4.1.4 Soliton clusters and necklaces 92......Page 92
4.2 Azimuthons 93......Page 93
4.2.1 Theoretical results 94......Page 94
4.2.2 Experimental results 97......Page 97
4.3.1 Stabilization of nonlocal solitons 99......Page 99
4.3.2 Nonlocal azimuthons 101......Page 101
4.3.3 Self-transforming nonlocal solitons 103......Page 103
4.4 Suppression of collapse 105......Page 105
4.5 Conclusions 107......Page 107
References 108......Page 108
5.1 Introduction 114......Page 114
5.2.1 Quantum and classical evolution 116......Page 116
5.2.2 Ladder operators generating basis sets of Gaussian solutions 118......Page 118
5.2.3 Fundamental solution 119......Page 119
5.2.5 Stationary states and eigenenergies 121......Page 121
5.2.6 Basis transformations 122......Page 122
5.3 Quantum harmonic oscillator in two dimensions 123......Page 123
5.3.1 Hermite-Gaussian eigenstates 124......Page 124
5.3.2 Laguerre-Gaussian eigenstates 125......Page 125
5.3.3 General Hermite-Laguerre states 126......Page 126
5.3.5 Angular momentum 128......Page 128
5.3.6 Oscillation between Fourier transforms 129......Page 129
5.4.1 Propagation and ray operators 130......Page 130
5.4.2 Ladder operators and basis sets of modes 132......Page 132
5.5.1 Equivalence of free modes and oscillator states 134......Page 134
5.5.2 Classical oscillations and rays of light 136......Page 136
5.5.3 Shape-invariant modes 137......Page 137
5.5.4 Astigmatic modes 138......Page 138
5.5.5 Fourier relations of general paraxial beams 139......Page 139
5.6.1 Equivalent lens guide 140......Page 140
5.6.2 Stability condition of an optical resonator 141......Page 141
5.6.3 Structure and frequencies of resonator modes 142......Page 142
5.6.4 Rays as displaced modes 143......Page 143
5.6.5 Geometric mode 146......Page 146
5.7 Summary and conclusions 148......Page 148
References 149......Page 149
6.1 Introduction 151......Page 151
6.2.1 Angle and orbital angular momentum states 154......Page 154
6.2.2 Commutator for angle and orbital angular momentum 156......Page 156
6.2.3 Physical states 157......Page 157
6.3.1 Intelligent states 160......Page 160
6.3.2 Constrained minimum uncertainty product states 165......Page 165
6.3.3 Large angular uncertainties 167......Page 167
6.4 Fractional orbital angular momentum 168......Page 168
6.4.1 Construction of fractional orbital angular momentum states 169......Page 169
6.4.2 Overlap of fractional orbital angular momentum states 170......Page 170
6.4.3 Orbital angular momentum distribution of fractional states 172......Page 172
References 175......Page 175
7.1 Introduction 178......Page 178
7.2 Doppler shift 179......Page 179
7.3 Dynamical rotational frequency shift 181......Page 181
7.4 Atom on a turntable 183......Page 183
7.5 Observation of the dynamical rotational frequency shift 186......Page 186
References 188......Page 188
8.1.2 Angular momenta and spin-orbit interactions of light 190......Page 190
8.1.3 History and current motivation 192......Page 192
8.2.1 Transversality, rotations, and Berry phase 194......Page 194
8.2.2 Coordinate, momentum, and angular momentum 200......Page 200
8.2.3 Poynting energy flows 204......Page 204
8.3.1 Free-space solutions 206......Page 206
8.3.2 Focusing by a high-NA lens 212......Page 212
8.3.3 Scattering by small particles 215......Page 215
8.3.4 Imaging and microscopy 217......Page 217
8.4.1 Hall effects from tilt of the beam 222......Page 222
8.4.2 Beam shifts upon reflection and refraction 225......Page 225
8.4.3 Geometrodynamics of light in a gradient-index medium 236......Page 236
8.5 Conclusion 246......Page 246
References 249......Page 249
9.1 Introduction 262......Page 262
9.2 Quantum fields 263......Page 263
9.3 Interactions with matter 267......Page 267
9.4 Parametric and поп-parametric processes 270......Page 270
9.5 Parity issues 272......Page 272
9.6 Chirality 274......Page 274
9.7 Conclusion 275......Page 275
References 276......Page 276
10.1 Introduction 280......Page 280
10.2 Electric and magnetic fields of Bessel beams 281......Page 281
10.3 Motion of charged particles in a Bessel beam 283......Page 283
10.4 Ponderomotive potential 287......Page 287
10.5.1 Trapping of particles along a helix 291......Page 291
10.5.2 Trapping of particles by crossed Bessel beams 294......Page 294
10.5.3 Trapping of particles by a standing Bessel wave 295......Page 295
10.6 Outlook 297......Page 297
References 299......Page 299
11.1 Introduction 300......Page 300
11.2 Overview 301......Page 301
11.3 OAM transfer 302......Page 302
11.4.1 Two-level atoms in twisted light 303......Page 303
11.4.2 Transient forces 305......Page 305
11.5.1 Steady state light-induced torque 307......Page 307
11.5.2 Dipole potential 308......Page 308
11.5.3 Doppler shift 309......Page 309
11.5.4 Atom dynamics 310......Page 310
11.6 Multiple beams 311......Page 311
11.6.1 Optical molasses in two and three dimensions 314......Page 314
11.7 Three-level atoms in LG beams 315......Page 315
11.8 Surface plasmonic optical vortices (SPOVs) 320......Page 320
11.9 Conclusions 326......Page 326
References 327......Page 327
12.2 Conservation of OAM 330......Page 330
12.3 Single photon OAM and quantum correlations 332......Page 332
12.5 Analogy with polarisation 334......Page 334
12.6 Tests of quantum mechanics in higher-dimensional OAM spaces 337......Page 337
12.7 Spiral bandwidth and dimensionality 340......Page 340
12.8 Conclusions 342......Page 342
References 343......Page 343
13.1 Introduction 346......Page 346
13.2 Spinning trapped particles 347......Page 347
13.3 Counting spiral fringes 350......Page 350
13.4 Diffraction effects from apertures 352......Page 352
13.5 Diffractive holographic filters 355......Page 355
13.6 More complex holograms 356......Page 356
13.7 The rotational Doppler effect 359......Page 359
13.8 A Dove prism interferometer 360......Page 360
13.9 Optical transformation 362......Page 362
13.10 Summary 364......Page 364
References 365......Page 365
14.1 Introduction 368......Page 368
14.2 Far-field projection of helico-conical beams 369......Page 369
14.3 Projecting multiple optical twisters and twister-chains 373......Page 373
14.4 Optical setup 374 ......Page 374
14.5 Experimental results 376......Page 376
14.6 Conclusions 378......Page 378
References 379......Page 379
0. Firstenberg, M. Shuker, R. Pugatch and N. Davidson......Page 381
15.1 Coherent diffusion 382......Page 382
15.2 Diffusion of orbital angular momentum 383......Page 383
15.3 Laguerre-Gaussian modes of diffusion 387......Page 387
15.4.1 Expansion 391......Page 391
15.4.2 Contraction 393......Page 393
15.5 Fractional and varying OAM 395......Page 395
15.7 Summary 397......Page 397
References 398......Page 398
16.1 Introduction 401......Page 401
16.2 Theory of OAM entanglement 402......Page 402
16.3 Issues in OAM entanglement 405......Page 405
16.3.2 Dimensionalities (K2d vs Kaz) and optical etendue N 406......Page 406
16.4.1 Experimental tools for OAM measurement 407......Page 407
16.4.2 Production of phase plates 408......Page 408
16.4.3 OAM analysis with mode-selective detectors 409......Page 409
16.4.4 OAM analysis with bucket detectors 413......Page 413
16.5 Concluding discussion and challenges 419......Page 419
References 420......Page 420
Index 423\t......Page 423