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This section contains the preface, table of contents, and glossary.

Library of Congress Cataloging-in-Publication Data

Tyson, Robert K., 1948-

Field guide to adaptive optics / Robert K. Tyson, Benjamin W. Frazier. -- 2nd ed

p. cm. -- (Spie field guides series; v. FG24)

Includes bibliographical references and index.

ISBN 978-0-8194-9017-9

1. Optics, Adaptive. 2. Optical detectors. 3. Optical measurements. I. Frazier, Benjamin W. (Benjamin West) II. Title.

TA1522.T93 2012

621.36′9--dc23

2011050389

Published by

SPIE

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Phone: +1.360.676.3290

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Printed in the United States of America.

First printing

Introduction to the Series

Welcome to the SPIE Field Guides—a series of publications written directly for the practicing engineer or scientist. Many textbooks and professional reference books cover optical principles and techniques in depth. The aim of the SPIE Field Guides is to distill this information, providing readers with a handy desk or briefcase reference that provides basic, essential information about optical principles, techniques, or phenomena, including definitions and descriptions, key equations, illustrations, application examples, design considerations, and additional resources. A significant effort will be made to provide a consistent notation and style between volumes in the series.

Each SPIE Field Guide addresses a major field of optical science and technology. The concept of these Field Guides is a format-intensive presentation based on figures and equations supplemented by concise explanations. In most cases, this modular approach places a single topic on a page, and provides full coverage of that topic on that page. Highlights, insights and rules of thumb are displayed in sidebars to the main text. The appendices at the end of each Field Guide provide additional information such as related material outside the main scope of the volume, key mathematical relationships and alternative methods. While complete in their coverage, the concise presentation may not be appropriate for those new to the field.

The SPIE Field Guides are intended to be living documents. The modular page-based presentation format allows them to be easily updated and expanded. We are interested in your suggestions for new Field Guide topics as well as what material should be added to an individual volume to make these Field Guides more useful to you. Please contact us at fieldguides@SPIE.org.

John E. Greivenkamp, Series Editor

College of Optical Sciences

The University of Arizona

The Field Guide Series

Keep information at your fingertips with all of the titles in the Field Guide Series:

Adaptive Optics, Second Edition, Robert K. Tyson & Benjamin W. Frazier

Atmospheric Optics, Larry C. Andrews

Binoculars and Scopes, Paul R. Yoder, Jr. & Daniel Vukobratovich

Diffractive Optics, Yakov G. Soskind

Geometrical Optics, John E. Greivenkamp

Illumination, Angelo Arecchi, Tahar Messadi, & R. John Koshel

Image Processing, Khan M. Iftekharuddin & Abdul A. Awwal

Infrared Systems, Detectors, and FPAs, Second Edition, Arnold Daniels

Interferometric Optical Testing, Eric P. Goodwin & James C. Wyant

Laser Pulse Generation, Rüdiger Paschotta

Lasers, Rüdiger Paschotta

Microscopy, Tomasz Tkaczyk

Optical Fabrication, Ray Williamson

Optical Fiber Technology, Rüdiger Paschotta

Optical Lithography, Chris A. Mack

Optical Thin Films, Ronald R. Willey

Polarization, Edward Collett

Probability, Random Processes, and Random Data Analysis, Larry C. Andrews & Ronald L. Phillips

Radiometry, Barbara G. Grant

Special Functions for Engineers, Larry C. Andrews

Spectroscopy, David W. Ball

Visual and Ophthalmic Optics, Jim Schwiegerling

Wave Optics, Dan Smith

Field Guide to Adaptive Optics, Second Edition

There have been a number of books and thousands of papers published with descriptions and mathematical expressions regarding adaptive optics. The material in this Field Guide is a summary of the methods for determining the requirements of an adaptive optics system, the performance of the system, and requirements for the components of the system. This second edition has a greatly expanded presentation of adaptive optics control system design and operation. Discussions of control models are accompanied by various recommendations for implementing the algorithms in hardware.

This book is not just another book on adaptive optics. There are already many fine volumes. This volume is intended for students, researchers, and practicing engineers who want a “go to” book when the calculation was “needed yesterday” (by a customer who won’t be paying for it until the next fiscal year).

Many of the expressions are in the form of integrals. When that is the case, we show the results graphically for a variety of practical values. Some of the material in this volume duplicates similar expressions found in other volumes of the Field Guide series. We have attempted to remain consistent with symbols of the other volumes. In some cases, however, we chose different symbols because they are well known within the adaptive optics literature.

Descriptions of the operation of subsystems and components and specific engineering aspects remain in the citations of the Bibliography.

This Field Guide is dedicated to the late Horace Babcock, whose pioneering ideas created the field of adaptive optics.

Robert K. Tyson

University of North Carolina at Charlotte

Ben W. Frazier

AOA Xinetics,

Northrop Grumman Aerospace Systems

Table of Contents

Glossary x

Introduction 1

Conventional Adaptive Optics System 1

Image Spread with Atmospheric Turbulence 2

The Principle of Phase Conjugation 3

Point Spread Function for an Astronomical Telescope 4

Modeling the Effect of Atmospheric Turbulence 5

Fried’s Coherence Length 5

Isoplanatic Angle 6

Kolmogorov Model 7

Atmospheric Turbulence Models 8

Coherence Length for Various Wavelengths and Turbulence Models 9

Greenwood Frequency 10

Wind Models 11

Scintillation 12

Zernike Polynomials 13

Legendre Polynomials 15

Angle of Arrival (Tilt) Fluctuations (Image Motion) 17

Modulation Transfer Function 18

Beam Propagation 19

System Performance Estimation 19

Modal and Zonal Fitting Error 20

Wavefront Sensors 21

Partial Correction 21

Shack–Hartmann Wavefront Sensor and Error 22

Shack–Hartmann Lenslet Array Selection 24

Curvature Wavefront Sensor and Error 25

Pyramid Wavefront Sensor and Error 26

Deformable Mirrors 27

Photodiodes 27

Photodiode Noise 28

Lateral-Effect Position-Sensing Detectors 29

Quad Cells 30

Noise Equivalent Angle 32

The Strehl Ratio: Laser Beam Propagation to the Far Field with Wavefront Error 33

Strehl Ratio 34

Laser “Brightness” 35

Laser Beam Quality 36

Astronomical “Brightness” 37

Spot Size for a Gaussian Beam 38

Spot Size for a Uniform Circular Aperture 39

Temporal Error 40

Focal Anisoplanatism (the “Cone Effect”) 41

Laser Guide Stars 42

Subsystem Requirements: The Wavefront Sensor 44

Angular Isoplanatic Error 45

Subsystem Requirements: Tilt Mirror 46

Subsystem Requirements: How Many Actuators?

Zonal or Modal Control 47

Subsystem Requirements: Deformable Mirror 48

Deformable Mirror Actuator Configurations 49

Ferroelectric Actuators 50

Electrostatic Actuators 52

Voice Coil Actuators 53

Deformable Mirror Influence Function Models 54

Bimorph and MEMS Mirrors 55

Segmented Deformable Mirrors 56

Control and Reconstruction 57

Actuator and Wavefront Sensor Layouts 57

Correctability and Flattening of a Deformable Mirror 58

Adaptive Optics System Feedback Configuration 59

Deformable Mirror Dynamic Model 60

Controller Dynamic Model 61

Wavefront Sensor Dynamic Model 62

Latency 63

One-Dimensional Sampling 64

Two-Dimensional Sampling 66

Temporal Sampling Rate Selection 67

System Stability 68

General Control-System Parameters 69

Sensitivity Functions 70

Bandwidth Estimation from Controller Gains 72

Poke Matrix 73

Example System Geometry 74

Singular-Value Decomposition of the Poke Matrix 76

Actuator and Subaperture Observability 77

Identification of Actuator Locations 79

Poke Matrix Smoothing 80

Actuator Slaving: Active Actuator Identification 81

Actuator Slaving: Slave Logic 82

Adding Slaving into the Reconstructor 83

Tilt Removal 84

Piston and Waffle Removal 85

Reconstructor Generation: Least Squares 86

Reconstructor Generation: Regularization 87

T-Filter 89

Modal Suppression 90

Interactuator Shear Suppression 91

Nullspace Suppression 92

Weighting Matrices 93

Modal Feedback 94

Reconstructor Generation: Procedure 96

Reconstructor Comparison 97

Slope Discrepancy 98

Offloads and Woofer-Tweeter Systems 99

Open-Loop Wavefront Estimation 100

Kalman Filtering 101

Multivariable System Performance 102

Disturbance Injection 104

Wavefront Sensor Calibration 105

Centroiding and Thresholding 106

Misregistration 107

Subaperture Spillover 108

Equation Summary 109

Bibliography 117

Index 121

Glossary

a

Width of segment gap

A

Structure constant at the surface

ADC

Analog-to-digital converter

AInfl

Influence function amplitude

a0

Piezoelectric constant

b

Size of mirror segment

B

Laser brightness

BAstro

Astronomical brightness

B

Poke matrix

Bc

Calibrated poke matrix

Br

Reduced poke matrix

Bs

Smoothed poke matrix

c

Speed of light (=3×108 m/s)

ca

Interactuator coupling

cnm

Fourier–Legendre coefficient

Cn2

Atmospheric turbulence structure constant

CCD

Charge-coupled device

d

Size of subaperture (in object space)

d

Separation of the membrane and the addressing electrode

d0

Characteristic distance of a laser guide star

D

Aperture diameter

DAC

Digital-to-analog converter

d

Vector of wavefront disturbances

e

Vector of calculated wavefront errors

E

Pulse energy of laser

en

Read-noise in electrons per pixel

f

Focal length

fBW

Closed-loop bandwidth

fc

Crossover frequency

fG

Greenwood frequency

F

Focal length of the system

FRayleigh

Return flux for Rayleigh guide star

FSodium

Return flux for sodium guide star

g1

Loop gain

g2

Leak gain

G

Gain

G

Generalized system plant model

h

Planck’s constant (=6.626×10−34 J s)

h

Altitude

HT

Height of the tropopause

H–V

Hufnagel–Valley

I(r)

Intensity distribution

IAper

Intensity at the circular aperture

Id

Dark current

IGap

Diffracted energy from gaps

IJ

Johnson noise current

In

Noise current

Ip

Photocurrent

Is

Shot noise current

I0

On-axis intensity

I

Identity matrix

J1

Bessel function

k

Wavenumber

k

Sample time index

kB

Boltzmann constant (= 1.38 × 10−23 J/K)

K

Aperture shape parameter for beam propagation

Kg

Increase in error at the null

K

Controller model

l0

Inner scale of turbulence

L

Propagation distance

LT

Thickness of the tropopause

L0

Outer scale of turbulence

m

Azimuthal index for Zernike polynomials

m

Number of wavefront measurements in influence matrix

mv

Visual magnitude of a star

M

Magnification

M2

Beam quality

M

Modal feedback matrix

Mr

Reduced modal feedback matrix

MTF

Modulation transfer function

n

Radial index for Zernike polynomials

n

Vector of measurement noise

na

Number of actuators

nB

Number of detected background photo-electrons per subaperture

nm

Number of measurements

np

Number of detected photoelectrons per subaperture

ns

Number of slaved actuators

nv

Number of active actuators

nR

Rayleigh scattering density

NAct

Number of actuators

ND

Number of pixels in a subaperture

NEA

Noise equivalent angle

NEP

Noise equivalent power

Np

Photon count

NZern

Number of Zernike modes

O

Offload matrix

OPD

Optical path difference

p

Curvature sensor image plane offset

P

Optical power

Pn

Legendre polynomial of order n

P

Error propagation covariance matrix

Pp

Piston projection matrix

Pt

Tilt projection matrix

Pw

Waffle projection matrix

PSD

Position sensing detector

PSF

Point spread function

q

Electron charge (= 1.6 × 10−19 C)

Qd

Atmospheric disturbance covariance

Qn

Measurement noise covariance

r

Radial coordinate

ra

Actuator pad radius

rc

Interactuator spacing

rm

Mirror radius

rs

Radius of supporting ring

r0

Coherence length of the atmosphere

RMS

Root mean square

r

Vector coordinate in the wavefront

R

Radius of circle

R

Photodiode responsivity

R

Reconstructor

Rr

Reduced reconstructor

RSS

Root sum square

Zernike radial polynomial

s

Shear distance

s

Summing index in the Zernike radial polynomial

s

Laplace transform variable

sAct

Distance between actuators

S

Strehl ratio

S

Sensitivity function

Sd

Slope discrepancy matrix

Sl

Slave logic matrix

Sw/jit

Strehl ratio including effects of jitter

SLC

Strategic laser communications

SNR

Signal-to-noise ratio

SVD

Singular value decomposition

t

Thickness of the bimorph

td

Latency

T

Matrix transpose operator

T

Transmission of the optics

T

Complementary sensitivity function

TA

Transmission of the atmosphere

Tm

Tension of the membrane

Ts

Sampling time

u

Vector of actuator commands

vG

Wind velocity at low altitude

vT

Wind velocity at the tropopause

vw

Wind velocity as a function of altitude

v(z)Bufton

Wind velocity of the Bufton model

V

Applied voltage

w

Gaussian beam radius

w0

Gaussian beam waist

W(x, y)

Wavefront

W

Wind velocity aloft

W

Weighting matrix

WCE

Wavefront control experiment

WFS

Wavefront sensor

x, y

Cartesian coordinates

y

Corrected output wavefront

z

Coordinate along propagation path

z

Altitude (if propagation path is vertical)

z

Z-transform variable

αjit

Root-mean-square average jitter

(Δφ)2

Wavefront error variance (distance squared)

δ

Hartmann spot shift

Δλ

Spectral bandwidth

ε0

Permittivity

ε0

Phase

φ

Deformable mirror influence function (surface deflection)

φ

Wind direction relative to the telescope aperture

Φ(r)

Phase

η

Detector efficiency

κ

Fitting constant

λ

Wavelength

λLGS

Laser wavelength

ν

Spatial frequency

νG

Wind velocity at low altitude

θ

Angular size of reference source

θd

Half-angle beam divergence

θ0

Isoplanatic angle

ρ

Discrete-time equivalent latency

ρCol

Sodium column abundance

σfitting

Wavefront fitting error

σNa

Resonant backscatter cross section

σR

Rayleigh scattering cross section

σTemp

Temporal wavefront error

σTilt

Wavefront tilt

σWFS

Wavefront sensor measurement error

σ2

Wavefront error variance (radians squared)

σI2

Intensity variance

σn2

Measurement error variance

σχ2

Log-amplitude variance

σφ

Root-mean-square wavefront error over a subaperture

τ

Actuator rise time

ΩN

Nyquist frequency

ζ

Zenith angle

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