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INDEX

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A

adaptive optics 4

filter function 83

focal anisoplanatism 80 195

variance 114

Zernike anisoplanatism 188

anisoplanatism

angular 225

beam shape 345

centroid 103

chromatic 228

cone 195

displacement 225

focal

distributed source 200

offset source 204

point source 195

focus 186 195

scintillation 184

Strehl ratio 219

tilt 164

outer scale 314

time delay 226

Zernike 188

annulus

piston

filter function 77

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annulus (Cont.)

variance 153

tilt

filter function 76

variance 153

aperture averaging

filter function 78

Gaussian beam 80

scintillation 176

Appell function 310

asterisk notation 10

asymptotic series

N complex planes 295

complex parameter 253

Poincaré 130

positive parameter 126

asymptotic solutions 126 253 295

atmospheric density 229

B

beam movement 107

beam shape 57 337

anisoplanatism 345

general solution 337

tilt jitter 342

uncorrected turbulence 340

beam wave

filter function 264

phase 261

power spectral density 267

scintillation 261

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Bessel function 10

complex argument 252

integrals 69

modified 260

third kind 12

Born approximation 39

Buckingham’s Pi theorem 27

Bufton wind model 227

C

Cauchy residue theorem 119

N complex planes 284

characteristic sizes

aperture 178

coherence diameter 97

isoplanatic angle 97

source angle 181

tilt anisoplanatism 171

time 227

coherence diameter 97 211

spherical wave 210

convergence of power series 279

convolution

N complex planes 14

one complex plane 13

correlation

beam wave 265

correlation function

density 37

focus 187

inner and outer scale 328

refractive index 37

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covariance function

filter function 55

D

diffraction

tilt 160

diffraction parameter 44

digamma function 133

distributed source 78 176

E

Euler’s constant 133

F

filter function 69

adaptive optics 83

anisoplanatism 54

annulus 75

beam wave 264

covariance function 55

finite aperture 78

focal 78

general formula 51

piston 72

annulus 77

power spectral density 56

beam wave 267

servo 57

single wave 52

tilt 72

annulus 76

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filter function (Cont.)

gradient 75

Zernike modes 71

focal anisoplanatism 195

focus 186

Fourier transform

definition 29

Fox H-function 17 120 123 212

Fresnel number 161

Fried’s coherence diameter 97

G

gamma function

definition 8

duplication formula 9

Gauss-Legendre formula 9

recursion relation 9

residue 8

Stirling’s formula 10

Gauss hypergeometric function

definition 15

examples 22

Gaussian beam

tilt 106

Gaussian beam 41 105

aperture averaging

filter function 80

piston 105

Strehl Ratio 64

Gegenbauer polynomials 220 356 361

generalized hypergeometric function 15

Greenwood frequency 114

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Greenwood spectrum 33

guidestar 195

H

Heaviside function 10

Hesse’s determinant 301

Hill spectrum 33

Horn function 280

Hufnagel-Valley turbulence 96

Huygens-Fresnel approximation 58

hypergeometric function

examples 22

Gauss 15 22

generalized 15

I

incomplete gamma function 96

inner scale 26 33 181 317

correlation function 328

definition 33

scintillation 176

tilt 152 317

intermittency 38

isoplanatic angle 97 225

K

Karhunen-Loève polynomials 74

Kolmogorov spectrum 31

L

Legendre function 220

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log-amplitude 46

log-polar transform 8

M

Macdonald function 12

Maréchal approximation 64

extended 64

Maxwell’s equations 38

Meijer G-function 17 212

Mellin convolution theorem 13

N complex planes 14

Mellin transform

N complex planes 277

definition 7

properties 7

Mellin-Barnes integral 16

complex parameter 251

Miller-Zieske 98

mirages 25

moment

temporal 227

turbulence 96

velocity 114

N

Navier-Stokes equations 27

O

outer scale 33 317

correlation function 328

definition 31

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outer scale (Cont.)

spectrum 33

tilt

Greenwood spectrum 150

von Kármán spectrum 146

turbulence spectrum 31

P

paraxial assumption 42

phase 46

beam wave 261

covariance function 55

piston

annulus 153

Gaussian beam 105

infin ite beams 99

Pochhammer symbol 15

Poincaré series 130

polylogarithm 132

power spectral density

beam wave 267

filter function 56

tilt 172 321

propagation parameter 43

psi function 133

R

Radar effects 26

refractive index 25 229

resolution 211

Reynolds number 28

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Riccati equation 39

Rytov approximation 37

S

scintillation 110

adaptive optics 114

anisoplanatism 184

aperture averaging 176

beam wave 261

covariance function 55

finite apertures 176

finite sources 176

focused beam 112

infinite beam 112

inner scale 181

saturated 38 41

servo system 57

beam wave 267

variance 113

SLCSAT Day turbulence 98

SLCSAT Night turbulence 98

steepest descent 126

N complex planes 296

complex parameter 253

Stirling’s formula 10

Strehl Ratio

Gaussian beam 64

top-hat beam 62

Strehl ratio 57 209

anisoplanatism 219 225

angular 225

chromatic 228

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Strehl ratio (Cont.)

combined effects 232

displacement 225

time delay 226

beam jitter 214

outer scale 213

tilt removed 234

uncorrected turbulence 209

Structure function 84

outer scale 214

structure function 55

density 28

inner and outer scale 328

log-amplitude 55 62

phase 55 62 328

velocity 27

T

Taylor frozen turbulence 56

temporal moment 227

thermal blooming 38

tilt

angle-of-arrival 108

anisoplanatism 164

outer scale 314

annulus 153

collimated 101

diffraction 160

focused 101

Gaussian beam 106

gradient 102

Greenwood spectrum 150

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tilt (Cont.)

inner and outer scale 317

inner scale 152

power spectral density 172

outer scale 321

von Kármán spectrum 146

Zernike 99

top-hat beam 42

transform

Fourier 29

log-polar 8

Mellin 7

trigamma function 133

turbulence

Strehl ratio 209

turbulence models 96

Hufnagel-Valley 96

SLC Day 98

SLC Night 98

turbulence moments 96

full 96

partial 96

turbulence spectra

Greenwood 33

Hill 33

Kolmogorov 31

Tatarski 33

von Kármán 31

modified 33

turbulence strength 28

two-thirds power law 27

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V

velocity moment 114

von Kármán spectrum 31 146

W

wind model 227

Z

Zernike anisoplanatism 188

Zernike modes 70 104

focus 186

piston 72

tilt 72 99