Coherent x-ray optics

Contents......Page 10
1 X-ray wave-fields in free space......Page 14
1.1 Vacuum wave equations for electromagnetic fields......Page 15
1.2 Spectral decomposition and the analytic signal......Page 18
1.3 Angular spectrum of plane waves......Page 19
1.4 Fresnel diffraction......Page 23
1.4.1 Operator formulation......Page 24
1.4.2 Convolution formulation......Page 25
1.5 Fraunhofer diffraction......Page 29
1.6.1 Kirchhoff diffraction integral......Page 31
1.6.2 Rayleigh–Sommerfeld diffraction integrals......Page 36
1.7.1 Random variables and random processes......Page 39
1.7.2 Intermediate states of coherence......Page 42
1.7.3 Spatial coherence......Page 43
1.7.4 Temporal coherence......Page 49
1.8 The mutual coherence function......Page 50
1.9 Propagation of two-point correlation functions......Page 59
1.9.1 Vacuum wave equations......Page 60
1.9.2 Operator formulation......Page 63
1.9.3 Green function formulation......Page 66
1.9.4 Van Cittert–Zernike theorem......Page 71
1.10 Higher-order correlation functions......Page 72
1.11 Summary......Page 73
2 X-ray interactions with matter......Page 77
2.1 Wave equations in the presence of scatterers......Page 78
2.2 The projection approximation......Page 84
2.3 Point scatterers and the outgoing Green function......Page 90
2.3.1 First method for obtaining Green function......Page 92
2.3.2 Second method for obtaining Green function......Page 93
2.4 Integral-equation formulation of scattering......Page 96
2.5 First Born approximation for kinematical scattering......Page 98
2.5.1 Fraunhofer and first Born approximations......Page 100
2.5.2 Angular spectrum and first Born approximation......Page 102
2.5.3 The Ewald sphere......Page 103
2.6 Born series and dynamical scattering......Page 110
2.7 Multislice approximation......Page 112
2.8 Eikonal approximation and geometrical optics......Page 114
2.9 Scattering, refractive index, and electron density......Page 121
2.10.1 Compton scattering......Page 128
2.10.2 Photoelectric absorption and fluorescence......Page 132
2.11.1 Scattered monochromatic fields......Page 135
2.11.2 Scattered polychromatic fields......Page 140
2.12 Summary......Page 143
3 X-ray sources, optical elements, and detectors......Page 149
3.1.1 Brightness and emittance......Page 150
3.1.2 Fixed-anode and rotating-anode sources......Page 151
3.1.3 Synchrotron sources......Page 152
3.1.4 Free-electron lasers......Page 158
3.1.5 Energy-recovering linear accelerators......Page 162
3.1.6 Soft X-ray lasers......Page 164
3.2.1 Diffraction gratings......Page 165
3.2.2 Fresnel zone plates......Page 173
3.2.3 Analyser crystals......Page 182
3.2.4 Crystal monochromators......Page 189
3.2.5 Crystal beam-splitters and interferometers......Page 191
3.2.6 Bragg–Fresnel crystal optics......Page 196
3.2.7 Free space......Page 198
3.3.1 X-ray reflection from surfaces......Page 199
3.3.2 Capillary optics......Page 204
3.3.3 Square-channel arrays......Page 205
3.3.4 X-ray mirrors......Page 206
3.4.1 Prisms......Page 208
3.4.2 Compound refractive lenses......Page 211
3.5 Virtual optical elements......Page 216
3.6.1 Critical detector parameters......Page 218
3.6.2 Types of X-ray detector......Page 221
3.6.3 Detectors and coherence......Page 225
3.7 Summary......Page 229
4 Coherent X-ray imaging......Page 241
4.1.1 Imaging using coherent fields......Page 243
4.1.2 Imaging using partially coherent fields......Page 250
4.1.3 Cascaded systems......Page 251
4.2 Self imaging......Page 253
4.2.1 Talbot effect for monochromatic fields......Page 255
4.2.2 Talbot effect for polychromatic fields......Page 259
4.2.3 Montgomery effect for monochromatic fields......Page 262
4.2.4 Montgomery effect for polychromatic fields......Page 266
4.3.1 In-line holography......Page 267
4.3.3 Fourier holography......Page 271
4.4 Phase contrast......Page 274
4.4.1 Zernike phase contrast......Page 276
4.4.2 Differential interference contrast......Page 281
4.4.3 Analyser-based phase contrast......Page 283
4.4.4 Propagation-based phase contrast......Page 291
4.4.5 Hybrid phase contrast......Page 297
4.5 Phase retrieval......Page 302
4.5.1 Gerchberg–Saxton algorithm and extensions......Page 304
4.5.2 The transport-of-intensity equation......Page 308
4.5.3 One-dimensional phase retrieval......Page 314
4.6 Interferometry......Page 323
4.6.1 Bonse–Hart interferometer......Page 324
4.6.2 Young interferometer......Page 328
4.6.3 Intensity interferometer......Page 331
4.6.4 Other means for coherence measurement......Page 334
4.7.1 General remarks on virtual optics......Page 335
4.7.2 Example of virtual optics......Page 337
4.8 Summary......Page 340
5 Singular X-ray optics......Page 354
5.2 Nodal lines......Page 355
5.3 Nodal lines are vortex cores......Page 359
5.4 Polynomial vortex solutions to d’Alembert equation......Page 360
5.5.1 Vortex nucleation and annihilation......Page 364
5.5.2 Stability with respect to perturbations......Page 366
5.5.3 Vortex interaction with a background field......Page 367
5.6.1 Interference of three coherent plane waves......Page 370
5.6.2 Synthetic holograms......Page 376
5.6.3 Spiral phase masks......Page 383
5.6.4 Spontaneous vortex formation......Page 386
5.7 Domain walls and other topological phase defects......Page 393
5.8 Caustics and the singularity hierarchy......Page 395
5.9 Summary......Page 400
A.1 Fourier transforms in one and two dimensions......Page 406
A.2 Convolution theorem......Page 407
A.4 Fourier derivative theorem......Page 408
A.5 Sifting property of Dirac delta......Page 409
B: Fresnel scaling theorem......Page 410
C: Reciprocity theorem for monochromatic scalar fields......Page 414
C......Page 418
E......Page 419
G......Page 420
L......Page 421
P......Page 422
S......Page 423
Z......Page 424