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Library of Congress Cataloging-in-Publication Data

Tyson, Robert K. 1948-.

Field guide to adaptive optics / Robert K. Tyson, Benjamin

West Frazier.

p. cm.--(SPIE field guides)

Includes bibliographical references and index.

ISBN 0-8194-5319-6

1. Optics, Adaptive. 2. Optical detectors. 3. Optical

Measurements. I. Frazier, Benjamin West. II. Title. III.

Series.

TA1522.T93 2004

621.36’9—dc22

2004005336

Published by

SPIE—The International Society for Optical Engineering

P.O. Box 10

Bellingham, Washington 98227-0010 USA

Phone:+1 360 676 3290

Fax: +1 360 647 1445

Email: spie@spie.org

Web: http://spie.org

The content of this book reflects the work and thought of the author. Every effort has been made to publish reliable and accurate information herein, but the publisher is not responsible for the validity of the information or for any outcomes resulting from reliance thereon.

Printed in the United States of America.

Introduction to the Series

Welcome to the SPIE Field Guides! This volume is one of the first in a new 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.

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

Optical Sciences Center

The University of Arizona

Field Guide to Geometrical Optics

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 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

Xinetics, Inc.

Table of Contents

Glossaryix

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

Astronomical “Brightness” 6

Isoplanatic Angle 7

Zernike Polynomials 8

Atmospheric Turbulence Models 10

Coherence Length for Various Wavelengths and Turbulence Models 11

Wind Models 12

Kolmogorov Model 13

Greenwood Frequency 14

Angle of Arrival Fluctuations (Image Motion) 15

Modulation Transfer Function 16

Beam Propagation 17

Laser “Brightness” 17

The Strehl Ratio—Laser Beam Propagation to the Far Field with Wavefront Error 18

Strehl Ratio 19

Spot Size for a Gaussian Beam 20

Spot Size for a Uniform Circular Aperture 21

System Performance Estimation 22

System Performance Estimation 22

Modal and Zonal Fitting Error 23

Partial Correction 24

Temporal Error 25

Focal Anisoplanatism (the “Cone Effect”) 26

Laser Guide Stars 27

Scintillation 29

Wavefront Sensors 30

Wavefront Sensor Requirements 30

Shack-Hartmann Wavefront Sensor and Error 31

Lenslet Array Selection 33

Shearing Interferometer Wavefront Sensor and Error 34

Curvature Wavefront Sensor and Error 36

Deformable Mirrors 37

Tilt Mirror Requirements 37

Deformable Mirror Requirements 38

Deformable Mirror Actuator Configurations 39

Actuator and Wavefront Sensor Layouts 40

Actuators: Requirements for Zonal or Modal Control 41

Deformable Mirror Influence Function Models 42

Bimorph and MEMS Mirrors 43

Segmented Deformable Mirrors 44

Control and Reconstruction 45

Adaptive Optics System Control Model 45

Reconstructor 46

Wavefront Control 47

Influence Matrix 48

Kalman Filtering and Wavefront Estimation 49

Computational Latency 50

Predictor 51

Effect of Sampling Rate on Achievable Bandwidth 52

Hartmann Sensing Software Implementation 53

Appendix 54

Equation Summary 54

Notes 60

Bibliography 62

Index 65

Glossary

Unprimed variables and symbols are in object space. Primed variables and symbols are in image space.

Frequently used variables and symbols:

a

Actuator reference command

a

Width of segment gap

A

Structure constant at the surface

AInfl

Influence function amplitude

a0

Piezoelectric constant

b

Size of mirror segment

b

Vector of bias commands

B

Deformable mirror influence matrix

B

Laser brightness

BAstro

Astronomical brightness

bi

Command for biasing the ith actuator

c

Speed of light (= 3 × 108 m/s)

ca

Interactuator coupling

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

dsub

Subaperture size

D

Aperture diameter

d

Vector of wavefront input disturbances

E

Pulse energy of laser

en

Read-noise in electrons per pixel

f

Focal length

f3dB

Closed-loop bandwidth

fc

Crossover frequency

fG

Greenwood frequency

F

Focal length of the system

FRayieigh

Return flux for Rayleigh guide star

FSodium

Return flux for sodium guide star

G

Gain

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

I0

On-axis intensity

IAper

Intensity at the circular aperture

IGap

Diffracted energy from gaps

J1

Bessel function

k

Wavenumber

k

Sample time index

K

Aperture shape parameter for beam propagation

K

Shearing interferometer fringe contrast

Kg

Increase in error at the null

I0

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

MTF

Modulation transfer function

n

Radial index for Zernike polynomials

n

Number of actuators in influence matrix

nB

Number of detected background photoelectrons per subaperture

np

Number of detected photoelectrons per subaperture

nR

Rayleigh scattering density

NAct

Number of actuators

ND

Number of pixels in a subaperture

Np

Photon count

NZern

Number of Zernike modes

OPD

Optical path difference

p

Curvature sensor image plane offset

P

Optical power

P

Error covariance matrix

PSF

Point spread function

r

Radial coordinate

r0

Coherence length of the atmosphere

ra

Actuator pad radius

rc

Interactuator spacing

rm

Mirror radius

rs

Radius of supporting ring

rms

Root-mean-square

r

Vector coordinate in the wavefront

R

Radius of circle

R

Reconstructor matrix

Zernike radial polynomial

s

Shear distance

s

Summing index in the Zernike radial polynomial

Sact

Distance between actuators

S

Strehl ratio

Sd

Covariance matrix

SW/jit

Strehl ratio including effects of jitter

SNR

Signal-to-noise ratio

SLC

Strategic laser communications

SVD

Singular value decomposition

t

Thickness of the bimorph

T

Transmission of the optics

TA

Transmission of the atmosphere

Tm

Tension of the membrane

u

Command vector

V

Applied voltage

v

Vector of measurement noise

vT

Wind velocity at the tropopause

vw

Wind velocity as a function of altitude

VG

Wind velocity at low altitude

v(z)Bufton

Wind velocity of the Bufton model

w

Gaussian beam radius

w0

Gaussian beam waist

W(x,y)

Wavefront

W

Wind velocity aloft

WCE

Wavefront control experiment

x,y

Cartesian coordinates

y

Vector of measured aberrations

z

Coordinate along propagation path

z

Altitude (if propagation path is vertical)

αjit

Root-mean-square average jitter

(Δϕ)2

Wavefront error variance (distance squared)

δ

Hartmann spot shift

ε0

Permittivity

φ

Phase

φ

Deformable mirror influence function (surface deflection)

ϕ

Wind direction relative to the telescope aperture

Φ

Vector of wavefront aberrations

Φ(r)

Phase

η

Detector efficiency

κ

Fitting constant

Δλ

Spectral bandwidth

λLGS

Laser wavelength

νG

Wind velocity at low altitude

θ0

Isoplanatic angle

θ

Angular size of reference source

ρCol

Sodium column abundance

σR

Rayleigh scattering cross section

σNa

Resonant backscatter cross section

σ2

Wavefront error variance (radians squared)

σrms

Root-mean-square wavefront error

σTemp

Temporal wavefront error

σx2

Log-amplitude variance

σI2

Intensity variance

σWFS

Wavefront sensor measurement error

σϕ

Root-mean-square wavefront error over a subaperture

σSI

Shearing interferometer wavefront error

σTilt

Wavefront tilt

υ

Spatial frequency

ζ

Zenith angle

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