Preface to the first edition ................................... XI
Glossary ....................................................... XV
1 Introduction ................................................. 1
1.1 Book Plan ............................................... 2
1.2 Introduction to Mellin Transforms ....................... 7
1.3 Higher Transcendental Functions ........................ 15
2 Basic Equations for Wave Propagation in Turbulence .......... 25
2.1 Turbulence Spectra ..................................... 26
2.2 Rytov Approximation .................................... 37
2.3 Phase and Log-Amplitude Variances ...................... 41
2.4 Power Spectral Density ................................. 56
2.5 Beam Shape and Strehl Ratio ............................ 57
2.5.1 Structure function dependent on aperture
position ........................................ 60
2.5.2 Structure function independent of aperture
position ........................................ 62
3 Filter Functions ............................................ 69
3.1 Circular Aperture Modes ................................ 70
3.2 Piston and Tilt on an Annulus .......................... 75
3.3 Finite Apertures and Focal Anisoplanatism .............. 78
3.4 Adaptive-Optics Systems ................................ 83
3.5 Structure Function for a Distributed Beacon ............ 84
3.6 Developing New Variance Filter Functions ............... 87
4 Zero-Parameter Problems ..................................... 95
4.1 Turbulence Models and Moments .......................... 96
4.2 Tilt and Piston for Collimated and Focused Beams ....... 99
4.3 Gradient Tilt ......................................... 102
4.4 Difference Between Gradient and Zernike Tilt .......... 103
4.5 Zernike Mode Variance ................................. 104
4.6 Piston and Tilt of a Gaussian Beam .................... 105
4.7 Beam Movement at a Target ............................. 107
4.8 Angle-of-Arrival Jitter ............................... 108
4.9 Scintillation for Collimated and Focused Beams ........ 110
4.10 Phase Variance with Finite Servo Bandwidth ............ 113
4.11 Variances for Beams Corrected by Adaptive Optics ...... 114
5 Integral Evaluation with Mellin Transforms ................. 117
5.1 Integral Evaluation with One Parameter ................ 118
5.2 Asymptotic Solutions .................................. 126
5.2.1 Alternate method of integral evaluation ........ 131
5.3 Multiple Poles ........................................ 132
5.3.1 Expansion of integrand functions ............... 132
5.3.2 Example 1 ...................................... 134
5.3.3 Example 2 ...................................... 135
5.3.4 Example 3 ...................................... 135
5.3.5 Example 4 ...................................... 136
5.3.6 Example 5 ...................................... 137
5.3.7 Example 6 ...................................... 138
6 Examples with a Single Positive Parameter .................. 145
6.1 Zernike Modes and Tilt for the von Karman Spectrum .... 146
6.2 Tilt for the Greenwood Spectrum ....................... 150
6.3 Tilt with Finite Inner Scale .......................... 152
6.4 Piston- and Tilt-Removed Phase Variance on an
Annulus ............................................... 153
6.5 Effect of Diffraction on Tilt ......................... 160
6.6 Tilt Anisoplanatism ................................... 164
6.7 Power Spectral Density of Tilt ........................ 172
6.8 Scintillation with Finite Apertures and Sources ....... 176
6.9 Scintillation with Finite Inner Scale ................. 181
6.10 Scintillation Anisoplanatism .......................... 184
6.11 Focus Anisoplanatism .................................. 186
6.12 Zernike Anisoplanatism ................................ 188
6.13 Focal Anisoplanatism for Point Sources ................ 195
6.14 Focal Anisoplanatism for Distributed Sources .......... 200
6.15 Focal Anisoplanatism for Offset Sources ............... 204
7 Strehl Ratio ............................................... 209
7.1 Strehl Ratio for Propagation Through Turbulence ....... 209
7.2 Strehl Ratio with Beam Jitter ......................... 214
7.3 Strehl Ratio with Anisoplanatism ...................... 219
7.4 Strehl Ratio for Various Anisoplanatic Effects ........ 225
7.4.1 Displacement anisoplanatism .................... 225
7.4.2 Angular anisoplanatism ......................... 225
7.4.3 Time-delay anisoplanatism ...................... 226
7.4.4 Chromatic anisoplanatism ....................... 228
7.4.5 Combined displacement anisoplanatism ......... 232
7.5 Strehl Ratio Using Numerical Integration .............. 234
7.5.1 Strehl ratio for uncorrected turbulence with
tilt present and removed ....................... 235
7.5.2 Strehl ratio for a finite beacon and focused
corrected beam ................................. 239
7.5.3 Strehl ratio for a finite beacon and
collimated corrected beam ...................... 245
8 Mellin Transforms with a Complex Parameter ................. 251
8.1 Mellin-Barnes Integrals with Complex Parameters ....... 251
8.2 Asymptotic Results with a Complex Parameter ........... 253
8.3 The Mellin Transform of an Exponential Times
a Bessel Function ..................................... 257
9 Finite Beam Characteristics As Examples with a Single
Complex Parameter .......................................... 261
9.1 Phase and Log-Amplitude Variances of Beam Waves ....... 261
9.2 Power Spectral Density of Beam Waves .................. 267
9.3 Scintillation on Beam Waves ........................... 267
9.4 Heuristic Scintillation Formulas ...................... 274
10 Mellin Transforms in N Complex Planes ...................... 277
10.1 Convergence of Multi-Parameter Series ................. 279
10.2 Path Closure at Infinity .............................. 282
10.3 Integration in Multiple Complex Planes ................ 284
10.3.1 Evaluation procedure for N complex planes ...... 295
10.4 Asymptotic Solution in Two or More Complex Planes ..... 295
11 Integral Evaluation with N Parameters ...................... 305
11.1 An Integral with Two Bessel Functions and
a Sinusoid ............................................ 306
11.2 An Integral with Three Bessel Functions ............... 308
11.3 Example in Three and N Complex Planes ................. 311
11.4 Effect of Outer Scale on Tilt Anisoplanatism .......... 314
11.5 Tilt with Inner and Outer Scale ....................... 317
11.6 Power Spectrum of Tilt with Outer Scale ............... 321
11.7 Structure and Correlation Functions with Inner and
Outer Scales .......................................... 328
12 Beam Shape ................................................. 337
12.1 General Formula for Beam Shape ........................ 337
12.2 Beam Shape for Uncorrected Turbulence ................. 340
12.3 Beam Shape with Tilt Jitter ........................... 342
12.4 Beam Shape with Anisoplanatism ........................ 345
Appendix A: Additional Mellin Transforms ...................... 351
Appendix B: Transcendental Functions .......................... 359
Index ......................................................... 363
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