The angular momentum of light

1 Light beams carrying orbital angular momentum 1
J. B. Gotte and S. M. Barnett 1
1.1 Introduction 1
1.2 Mechanical properties of optical fields 2
1.2.1 Energy of the electromagnetic field 3
1.2.2 Linear momentum of the electromagnetic field 5
1.2.3 Angular momentum of the electromagnetic field 6
1.2.4 Spin and orbital angular momentum 6
1.3 Wave equations 8
1.3.1 Helmholtz equation 8
1.3.2 Paraxial approximation 11
1.4 Paraxial and поп-paraxial optics 12
1.4.1 Paraxial light beams 12
1.4.2 Non-paraxial light beams 20
1.5 Angular momentum of light beams 21
1.5.1 Angular momentum flux 21
1.5.2 Azimuthal phase structure 24
1.6 Generating light beams with orbital angular momentum 25
References 27

2 Vortex transformations and vortex dynamics in optical fields 31
G. Molina-Terriza 31
2.1 Introduction 31
2.2 Optical vortices 32
2.2.1 Morphology of noncanonical vortices 33
2.2.2 Multiple vortices 35
2.3 Equations of evolution for the isolated vortices 36
2.3.1 Dynamics of a single vortex in linear homogeneous media 37
2.3.2 Inversion of the topological charge of a vortex 38
2.3.3 Experimental demonstration of the vortex inversion 41
2.3.4 Vortices and angular momentum 42
2.3.5 Another interpretation of the charge inversion 43
2.4 Interactions between vortices 44
2.4.1 GRaded INdex media as a laboratory for vortex interactions 45
2.4.2 Vortex interaction in GRIN media 46
2.4.3 Interaction of two vortices 47
2.5 Conclusion and discussion 49
References 49

3 Vector beams in free space 51
E. J. Galvez 51
3.1 Introduction 51
3.2 Spatial modes 52
3.2.1 Hermite-Gauss and Laguerre-Gauss beams 52
3.2.2 Relations between first-order Hermite-Gauss and Laguerre-Gauss beams 53
3.3 A review of polarization modes 55
3.4 Vector beams 57
3.4.1 Theoretical description of first-order vector beams 57
3.4.2 Higher-order vector beams 61
3.5 Experimental approaches 64
3.5.1 Experimental methods of production 64
3.5.2 Detection of vector beams by polarization projection 66
3.6 Vector beams in non-classical states of light 68
3.7 Conclusions 69
References 69

4 Optical beams with orbital angular momentum in nonlinear media 71
A. S. Desyatnikov and Y. S. Kivshar 71
4.1 Introduction 71
4.1.1 Vortex solitons 72
4.1.2 Azimuthal modulational instability 74
4.1.3 Two-soliton spiraling 74
4.1.4 Soliton clusters and necklaces 76
4.2 Azimuthons 77
4.2.1 Theoretical results 78
4.2.2 Experimental results 81
4.3 Solitons in nonlocal media 83
4.3.1 Stabilization of nonlocal solitons 83
4.3.2 Nonlocal azimuthons 85
4.3.3 Self-transforming nonlocal solitons 87
4.4 Suppression of collapse 89
4.5 Conclusions 91
References 92

5 Ray optics, wave optics and quantum mechanics 98
G. Nienhuis 98
5.1 Introduction 98
5.2 Evolution with a quadratic Hamiltonian 100
5.2.1 Quantum and classical evolution 100
5.2.2 Ladder operators generating basis sets of Gaussian solutions 102
5.2.3 Fundamental solution 103
5.2.4 Basis sets of the solutions of the Schrodinger equation 105
5.2.5 Stationary states and eigenenergies 105
5.2.6 Basis transformations 106
5.3 Quantum harmonic oscillator in two dimensions 107
5.3.1 Hermite-Gaussian eigenstates 108
5.3.2 Laguerre-Gaussian eigenstates 109
5.3.3 General Hermite-Laguerre states 110
5.3.4 Basis sets of non-stationary solutions of the harmonic oscillator 112
5.3.5 Angular momentum 112
5.3.6 Oscillation between Fourier transforms 113
5.4 Paraxial wave optics as a Hamiltonian system 114
5.4.1 Propagation and ray operators 114
5.4.2 Ladder operators and basis sets of modes 116
5.5 Basis sets of paraxial beams and harmonic oscillators 118
5.5.1 Equivalence of free modes and oscillator states 118
5.5.2 Classical oscillations and rays of light 120
5.5.3 Shape-invariant modes 121
5.5.4 Astigmatic modes 122
5.5.5 Fourier relations of general paraxial beams 123
5.6 Astigmatic optical resonators 124
5.6.1 Equivalent lens guide 124
5.6.2 Stability condition of an optical resonator 125
5.6.3 Structure and frequencies of resonator modes 126
5.6.4 Rays as displaced modes 127
5.6.5 Geometric mode 130
5.7 Summary and conclusions 132
References 133

6 Quantum formulation of angle and orbital angular momentum 135
J. B. Gotte and S. M. Barnett 135
6.1 Introduction 135
6.2 Quantum theory of rotation angles 138
6.2.1 Angle and orbital angular momentum states 138
6.2.2 Commutator for angle and orbital angular momentum 140
6.2.3 Physical states 141
6.3 Intelligent and minimum uncertainty states 144
6.3.1 Intelligent states 144
6.3.2 Constrained minimum uncertainty product states 149
6.3.3 Large angular uncertainties 151
6.4 Fractional orbital angular momentum 152
6.4.1 Construction of fractional orbital angular momentum states 153
6.4.2 Overlap of fractional orbital angular momentum states 154
6.4.3 Orbital angular momentum distribution of fractional states 156
References 159

7 Dynamical rotational frequency shift 162
Bialynicki-Birula and Z. Bialynicka-Birula 162
7.1 Introduction 162
7.2 Doppler shift 163
7.3 Dynamical rotational frequency shift 165
7.4 Atom on a turntable 167
7.5 Observation of the dynamical rotational frequency shift 170
7.6 Conclusions 172
References 172

8 Spin-orbit interactions of light in isotropic media 174
K. Y. Bliokh, A. Aiello and M. A. Alonso 174
8.1 Introduction 174
8.1.1 Spin-orbit interaction in quantum physics 174
8.1.2 Angular momenta and spin-orbit interactions of light 174
8.1.3 History and current motivation 176
8.2 Geometric phases, angular momenta, and energy flows 178
8.2.1 Transversality, rotations, and Berry phase 178
8.2.2 Coordinate, momentum, and angular momentum 184
8.2.3 Poynting energy flows 188
8.3 Spin-orbit interactions in поп-paraxial fields 190
8.3.1 Free-space solutions 190
8.3.2 Focusing by a high-NA lens 196
8.3.3 Scattering by small particles 199
8.3.4 Imaging and microscopy 201
8.4 Spin-orbit interactions in locally paraxial fields 206
8.4.1 Hall effects from tilt of the beam 206
8.4.2 Beam shifts upon reflection and refraction 209
8.4.3 Geometrodynamics of light in a gradient-index medium 220
8.5 Conclusion 230
References 233

9 Quantum electrodynamics, angular momentum and chirality 246
D. L. Andrews and Mohamed Babiker 246
9.1 Introduction 246
9.2 Quantum fields 247
9.3 Interactions with matter 251
9.4 Parametric and поп-parametric processes 254
9.5 Parity issues 256
9.6 Chirality 258
9.7 Conclusion 259
References 260

10 Trapping of charged particles by Bessel beams 264
I. Bialynicki-Birula, Z. Bialynicka-Birula and N. Drozd 264
10.1 Introduction 264
10.2 Electric and magnetic fields of Bessel beams 265
10.3 Motion of charged particles in a Bessel beam 267
10.4 Ponderomotive potential 271
10.5 Trapping of particles by superpositions of Bessel beams 275
10.5.1 Trapping of particles along a helix 275
10.5.2 Trapping of particles by crossed Bessel beams 278
10.5.3 Trapping of particles by a standing Bessel wave 279
10.6 Outlook 281
References 283

11 Theory of atoms in twisted light 284
M. Babiker, D. L Andrews and V. E. Lembessis 284
11.1 Introduction 284
11.2 Overview 285
11.3 OAM transfer 286
11.4 Radiation forces-cooling and trapping 287
11.4.1 Two-level atoms in twisted light 287
11.4.2 Transient forces 289
11.5 Steady state forces 291
11.5.1 Steady state light-induced torque 291
11.5.2 Dipole potential 292
11.5.3 Doppler shift 293
11.5.4 Atom dynamics 294
11.6 Multiple beams 295
11.6.1 Optical molasses in two and three dimensions 298
11.7 Three-level atoms in LG beams 299
11.8 Surface plasmonic optical vortices (SPOVs) 304
11.9 Conclusions 310
References 311

12 An experimentalist’s introduction to orbital angular momentum for quantum optics 314
J. Romero, D. Giovannini, S. Franke-Arnold and M. J. Padgett 314
12.1 Introduction 314
12.2 Conservation of OAM 314
12.3 Single photon OAM and quantum correlations 316
12.4 Two-photon state 318
12.5 Analogy with polarisation 318
12.6 Tests of quantum mechanics in higher-dimensional OAM spaces 321
12.7 Spiral bandwidth and dimensionality 324
12.8 Conclusions 326
References 327

13 Measurement of light’s orbital angular momentum 330
M. P. J. Lavery, J. Courtial and M. J. Padgett 330
13.1 Introduction 330
13.2 Spinning trapped particles 331
13.3 Counting spiral fringes 334
13.4 Diffraction effects from apertures 336
13.5 Diffractive holographic filters 339
13.6 More complex holograms 340
13.7 The rotational Doppler effect 343
13.8 A Dove prism interferometer 344
13.9 Optical transformation 346
13.10 Summary 348
References 349

14 Efficient generation of optical twisters using helico-conical beams 352
V. R. Daria, D. Palima and J. Gliickstad 352
14.1 Introduction 352
14.2 Far-field projection of helico-conical beams 353
14.3 Projecting multiple optical twisters and twister-chains 357
14.4 Optical setup 358
14.5 Experimental results 360
14.6 Conclusions 362
References 363

15 Self-similar modes of coherent diffusion with orbital angular momentum 365
O. Firstenberg, M. Shuker, R. Pugatch and N. Davidson 365
15.1 Coherent diffusion 366
15.2 Diffusion of orbital angular momentum 367
15.3 Laguerre-Gaussian modes of diffusion 371
15.4 Experiments with vortex modes 375
15.4.1 Expansion 375
15.4.2 Contraction 377
15.5 Fractional and varying OAM 379
15.6 Non-diffusing modes with OAM 381
15.7 Summary 381
References 382
16 Quantum entanglement of orbital angular momentum 385
M. P. Van Exter, E. R. Eliel and J. P. Woerdman 385
16.1 Introduction 385
16.2 Theory of OAM entanglement 386
16.3 Issues in OAM entanglement 389
16.3.1 OAM conservation and Schmidt modes 390
16.3.2 Dimensionalities (K2d vs Kaz) and optical etendue N 390
16.3.3 Generated vs detectable entanglement 391
16.4 Experimental analysis of OAM states 391
16.4.1 Experimental tools for OAM measurement 391
16.4.2 Production of phase plates 392
16.4.3 OAM analysis with mode-selective detectors 393
16.4.4 OAM analysis with bucket detectors 397
16.5 Concluding discussion and challenges 403
References 404

Index 407