Field Guide to
Adaptive Optics Robert K.Tyson University of North Carolina at Charlotte
Benjamin W. Frazier Xinetics, Inc.
SPIE Field Guides Volume FG03 John E. Greivenkamp, Series Editor
SPIE PRESS A Publication of SPIE—The International Society for Optical Engineering Bellingham, Washington USA
Library of Congress
Introduction to t h e Series
Data
Tyson, Robert K. 1948guide to adaptive optics / Robert K. Tyson, Benjamin West Frazier. p. cm.-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 2004005336
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.
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[email protected]. John E. Greivenkamp, Series Editor Optical Sciences Center The University of Arizona
Field Guide to Adaptive 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 C o n t e n t s Glossary
ix
Introduction Conventional Adaptive Optics System Image Spread with Atmospheric Turbulence The Principle of Phase Conjugation Point Spread Function for an Astronomical Telescope Modeling the Effect of Atmospheric Turbulence Fried's Coherence Length Astronomical "Brightness" Isoplanatic Angle Polynomials Atmospheric Turbulence Models Coherence Length for Various Wavelengths and Turbulence Models Wind Models Kolmogorov Model Greenwood Frequency Angle of Arrival Fluctuations (Image Motion) Modulation Transfer Function Beam Propagation Laser "Brightness" The Strehl Ratio—Laser Beam Propagation to the Far Field with Wavefront Error Strehl Ratio Spot Size for a Gaussian Beam Spot Size for a Uniform Circular Aperture System Performance E s t i m a t i o n System Performance Estimation Modal and Zonal Fitting Error Partial Correction Temporal Error Focal Anisoplanatism (the "Cone Effect") Laser Guide Stars Scintillation
vii
1 1 2 3 4 5 5 6 7 8 10 11 12 13 14 15 16 17 18 19 20 22 22 23 24 25 26 27 29
Glossary
Table of C o n t e n t s (cont'd) Wavefront Sensors Wavefront Sensor Requirements Shack-Hartmann Wavefront Sensor and Error Lenslet Array Selection Shearing Interferometer Wavefront Sensor and Error Curvature Wavefront Sensor and Error Deformable Mirrors Tilt Mirror Requirements Deformable Mirror Requirements Deformable Mirror Actuator Configurations Actuator and Wavefront Sensor Layouts Actuators: Requirements for Zonal or Modal Control Deformable Mirror Influence Function Models Bimorph and MEMS Mirrors Segmented Deformable Mirrors
30 30 31 33 34 36 37 38 39 40 41 42 43 44
Control and R e c o n s t r u c t i o n Adaptive Optics System Control Model Reconstructor Wavefront Control Influence Matrix Kalman Filtering and Wavefront Estimation Computational Latency Predictor Effect of Sampling Rate on Achievable Bandwidth Hartmann Sensing Software Implementation
45 45 46 47 48 49 50 51 52 53
Appendix Equation Summary Notes
54 54 60
Bibliography
62
Index
65
Actuator reference command Width of segment gap Structure constant at the surface Influence function amplitude Piezoelectric constant Size of mirror segment Vector of bias commands Deformable mirror influence matrix Laser brightness Astronomical brightness Command for biasing the Speed of light (= 3 x
actuator m/s)
Interactuator coupling Atmospheric turbulence structure constant Charge-coupled device Size of subaperture (in object space) Separation of the membrane and the addressing electrode Characteristic distance of a laser guide star Subaperture size Aperture diameter Vector of wavefront input disturbances Pulse energy of laser Read-noise in electrons per pixel Focal length Closed-loop bandwidth Crossover frequency Greenwood frequency Focal length of the system Return flux for Rayleigh guide star Return flux for sodium guide star
Glossary (cont'd)
Glossary (cont'd) Gain
Rayleigh scattering density
Planck's constant (= 6.626 x Altitude
J • s)
Height of the tropopause
Number of actuators Number of pixels in a subaperture
Hufnagel-Valley
Photon count Number of Zernike modes
Intensity distribution
Optical path difference
On-axis intensity
Curvature sensor image plane offset
Intensity at the circular aperture
Optical power
Diffracted energy from gaps Bessel function
Error covariance matrix
Sample time index
Point spread function Radial coordinate Coherence length of the atmosphere
Aperture shape parameter for beam propagation
Actuator pad radius
Shearing interferometer fringe contrast Increase in error at the null
Mirror radius Radius of supporting ring
Inner scale of turbulence
Root-mean-square
Propagation distance
Vector coordinate in the wavefront
Thickness of the tropopause Outer scale of turbulence
Radius of circle
Interactuator spacing
Azimuthal index for Zernike polynomials
Reconstructor matrix Zernike radial polynomial
Number of wavefront measurements in influence matrix Visual magnitude of a star Magnification
Shear distance Summing index in the Zernike radial polynomial Distance between actuators
Modulation transfer function
Strehl ratio Covariance matrix Strehl ratio including effects of jitter
Radial index for Zernike polynomials Number of actuators in influence matrix Number of detected background photoelectrons per subaperture Number of detected photoelectrons per subaperture X
Signal-to-noise ratio Strategic laser communications Singular value decomposition
Glossary
Glossary (cont'd)
Thickness of the bimorph
Phase
Transmission of the optics
Detector efficiency
Transmission of the atmosphere Tension of the membrane
Fitting constant
Command vector
Laser wavelength
Applied voltage
Wind velocity at low altitude
Vector of measurement noise
Isoplanatic angle
Wind velocity at the tropopause
Angular size of reference source
Wind velocity as a function of altitude Wind velocity at low altitude
Sodium column abundance
Wind velocity of the Bufton model Gaussian beam radius
Resonant backscatter cross section
Spectral bandwidth
Rayleigh scattering cross section Wavefront error variance (radians squared)
Gaussian beam waist Wavefront
Root-mean-square wavefront error
Wind velocity aloft
Log-amplitude variance
Wavefront control experiment Cartesian coordinates Vector of measured aberrations
Temporal wavefront error Intensity variance Wavefront sensor measurement error
Coordinate along propagation path
Root-mean-square wavefront error over a subaperture
Altitude (if propagation path is vertical)
Shearing interferometer wavefront error
Root-mean-square average jitter
Wavefront tilt
Wavefront error variance (distance squared)
Spatial frequency
Hartmann spot shift Permittivity Phase Deformable mirror influence function (surface deflection) Wind direction relative to the telescope aperture Vector of wavefront aberrations
Zenith angle
Introduction
1
Conventional Adaptive Optics System A conventional (linear) adaptive optics system, whether it is used for imaging or whether it is used for laser beam propagation, consists of three principal subsystems: a wavefront sensor to detect the optical disturbance, an active mirror or deformable mirror to correct for the optical disturbance, and a control computer to monitor and decode the sensor information for the active mirror.
Adaptive optics imaging system.
Adaptive optics laser projection system.
2
Adaptive Optics Image Spread w i t h Atmospheric Turbulence
Linear systems theory shows how an image is composed of an object convolved with the point spread function PSF of the imaging system. Atmospheric turbulence degrades the PSF and smears the image. The PSF is the image of a point source of light. The imaging process experiences diffraction, and the object is convolved with the PSF. The resultant image is a blurred version of the true object. Point Spread Function
Adding aberrations to the optical system results in a broadening of the PSF and increased blurring. Adaptive optics can compensate for the aberrations and reduce blurring.
Object
What is adaptive optics? Answer for the common spheric turbulence screws up the image. Adaptive optics unscrews it. Image
Introduction
3
The Principle of P h a s e Conjugation All systems of adaptive optics generally use the principle of phase conjugation. An optical beam is made up of both an amplitude A and a phase component and is described mathematically by the electric field Adaptive optics reverses the phase to provide compensation for the phase distortion. The reversal of the phase, being in the exponent of the electric field vector, means changing the sign of the term behind the imaginary number. This mathematical conjugation corresponds to phase conjugation of the optical field, just what is needed to compensate for a distorted phase.
While Horace Babcock is generally thought to be the "inventor" of adaptive optics with his paper "The possibility of compensating astronomical seeing," Astron. 65, 229, (1953)] his exact idea was never put into practice. It wasn't until the technological developments in electro-optics in the late 1960s and early 1970s that made a working adaptive optics system possible.
4
Adaptive Optics P o i n t Spread F u n c t i o n for an Astronomical Telescope
For an uncompensated astronomical telescope the point spread function is limited by the diffraction of the optics and the atmospheric turbulence. The PSF spot has a central core width and an angular width proportional to X/D, where D is the telescope pupil diameter. A halo surrounding the core has a width with an angular size of roughly where represents the strength of atmospheric turbulence.
Modeling
the
Effect
of
Atmospheric
Turbulence
5
Fried's C o h e r e n c e Length Fried's c o h e r e n c e l e n g t h is a widely used descriptor of the level of atmospheric turbulence at a particular site. For a fixed wavelength X, astronomical s e e i n g is given by the angle For a known structure constant profile [ (z), where z is the altitude] and a flat-Earth assumption, the coherence length is given by
where k = 2n/X, is the z e n i t h angle (0 deg is straight overhead), and the integral is over the path to the groundbased telescope from the source of light. Under turbulence, the resolution is limited by Fried's coherence length rather than the diameter of the telescope. Since ranges from under 5 cm with poor seeing to more than 20 cm with good seeing, even in the best conditions, a large diameter telescope without adaptive optics does not provide any better resolution than a telescope with a smaller diameter.
6
Adaptive Optics Astronomical "Brightness"
The term b r i g h t n e s s represents the brightness of an object in the heavens. As the object such as a star is observed, the amount of light (number of photons) collected by an aperture (such as the human eye) per second is astronomical brightness. The visual magnitude of a star is a logarithmic measure of the star's brightness in the visible spectrum. Smaller numbers represent brighter stars; negative numbers represent even brighter stars. One expression that accounts for atmospheric absorption relates visual magnitude to brightness:
Modeling
the
Effect
of
Atmospheric
Turbulence
7
Isoplanatic Angle Light traveling from a wavefront b e a c o n should traverse the same atmosphere as the light from the object of interest. When the angular difference between the paths results in a mean-square wavefront error of 1.0 rad 2 , the angular difference is called the isoplanatic angle. For a given structure constant profile where z is the altitude), and a flat-Earth assumption, the isoplanatic angle is given by
where is the z e n i t h angle, and the integral is over the path from the ground-based telescope to the source of light above the surface. The graph illustrates the isoplanatic angle versus wavelength for the H-V m o d e l and the Stragic Laser C o m m u n i c a t i o n SLC model of turbulence.
6 4 Visual magnitude
2
It takes about one millisecond for light to pass vertically through the Earth's atmosphere.
8
Adaptive Optics
Modeling
the
Effect
of
Atmospheric
Turbulence
9
Zernike P o l y n o m i a l s
Zernike P o l y n o m i a l s (cont'd)
Optical p h a s e can be represented by a 2D surface over the aperture. The deviation from flat (or some other reference surface) is the wavefront error sensed by the wavefront sensor. A very useful infinite-series representation of the wavefront is the Zernike polynomial series. Radial (index n) and azimuthal (index m) polynomials are preceded by Zernike coefficients and that completely describe the wavefront up to the order specified by the largest n or m. The series is written
A few terms are graphed to visualize their relationship to third-order optical aberrations.
where the azimuthal polynomials are sines and cosines of multiple angles and the radial polynomial is
The series is especially useful in adaptive optics because the polynomials are orthogonal over a circle of radius R, common to many optical system geometries. For R normalized to unity, the first few radial terms are given here.
= = TO = = = TO
0 1 2 3 4
re = 0 1
n = 1
re = 2
re = 3
r r
3
re = 4 6 r 4 - 6r 2 + 1 4r 4 - 3r 2
Coma Zernike polynomials associated with low-order modes.
Reflecting t e l e s c o p e s of the Cassegrain design have a central obscuration, which requires an extremely large number of Zernike coefficients—more than can be adequately described. A set of annular Zernike polynomials may be obtained from Gram-Schmidt orthogonalization, and this series is generally used for optical systems with central obscurations.
10
Adaptive Optics Atmospheric Turbulence Models
One of the most widely used models for the atmospheric turbulence structure constant as a function of altitude is the H-V model:
Modeling the Effect of Atmospheric Turbulence
Coherence Length for Various Wavelengths and Turbulence Models The c o h e r e n c e l e n g t h is shown for two profiles, the H-V 5/7 and the SLC-Night, for wavelengths from 0.4 urn to 10.6 and three z e n i t h angles.
where h is the altitude in kilometers, and is in units of parameters A and W are adjustable for local conditions. For the most common H-V 5/7 model (leading to = 5 cm and = 7 the structure constant at the surface A is 1.7 x 10 14 , and the wind velocity aloft W is 21. For conditions other than the 5/7 model, one can calculate A and W from
where the c o h e r e n c e l e n g t h is in centimeters and the isoplanatic angle is in microradians. Other models are layered, such as the SLC-Night model: Altitude (above ground) h< 18.5m 18.5
11
Turbulence models are site-specific because climate and geography play an important role in the profile.
12
Adaptive
Optics
Modeling
the
Effect
of
Atmospheric
Turbulence
13
Wind Models
Kolmogorov Model
For calculations involving the atmospheric w i n d profile, one can use a general Gaussian model developed by Greenwood.
Large masses of heated air transfer kinetic energy to smaller and smaller scales. The largest size is called the outer scale When the scale size becomes small enough the kinetic energy of the air is dissipated as heat. This small size is called the inner scale L.
where the wind velocity is in m/s. In the expression, is the wind velocity at low altitude, is the wind velocity at the tropopause, is the z e n i t h angle, is the height of the tropopause, is the thickness of the tropopause layer, and is the wind direction relative to the telescope azimuth. The wind velocity at the tropopause typically ranges from 23 m/s to 46 m/s. The Bufton w i n d model assumes a 0° zenith angle and wind direction and applies the values: = 5 m/s; = 30 m/s; = 9.4 km; 4.8 km.
Three realizations of the wind model: (1) Greenwood: m/s; = 35 m/s, = 12.0 km, = 5 km; = 45; (2) Greenwood: = 0 ; and (3) Bufton.
14
Adaptive
Optics
Modeling
the
Effect
of
Atmospheric
Turbulence
15
Greenwood Frequency
Angle of Arrival F l u c t u a t i o n s (Image Motion)
The G r e e n w o o d frequency is a measure of the rate at which atmospheric turbulence changes with time. For the flat-Earth assumption
Turbulence causes j i t t e r in a beam or an image. The root mean square average jitter is computed from the expression
where
where D is the aperture diameter. If there is any optical system that imparts magnification on the beam, the jitter at the image plane (or more generally, the compact portion of the beam) must be multiplied by the magnification.
is the z e n i t h angle, of altitude.
is the wind velocity as a
The temporal power spectrum exhibits behavior in regimes related to the modes of optical turbulence. For low frequencies, those less than the crosso v e r frequency the tilt-included phase spectrum follows a (-2/3) power law. When Wavelength (urn) tilt is removed the spectrum follows a ( + 4/3) power law. For frequencies above the crossover, the Kolmogorov spectrum is followed with the (-8/3) power law. The crossover frequency can be approximated from the turbulence-weighted wind velocity v and the aperture diameter D:
The Greenwood frequency determines how quickly the adaptive optics system must respond in order to compensate for turbulence. As a rule-of-thumb, the closed-loop w i d t h should be at least 10 times the Greenwood frequency.
16
Adaptive Optics
17
Beam Propagation
Modulation Transfer F u n c t i o n
Laser "Brightness"
The m o d u l a t i o n transfer function MTF is a measure of the spatial frequency r e s p o n s e of an imaging system. Assuming linear response, the MTF can be composed of separate MTFs of various components or features of the system that are multiplied to find the total system MTF. The MTF for a diffraction-limited circular aperture of diameter D is
A common description of a laser beam, especially a highpower beam, is to describe its brightness. This is a measure of on-axis power with the explicit propagation distance removed. Using the definitions of jitter effects and Strehl ratio on the following pages, the brightness B is
where the spatial frequency v is in units of The MTF of the uncompensated atmosphere has various forms. The short-exposure MTF does not include temporal beam wander (tilt), which smears an image over time. The long-exposure MTF includes the effect of beam tilt and results in an MTF that is driven by the c o h e r e n c e l e n g t h of the atmosphere:
where K is an aperture shape parameter (K = 1 for a circular aperture), T is the transmission of all optics in the system, and P is the optical power (watts). Brightness is expressed in watts per steradian. The brightness can be used to calculate the energy density on a target by dividing brightness by the square of the distance from the beam propagation aperture to the target.
Note the improvement in imaging quality (increase in MTF) with the larger diameter, whereas in both cases, the atmosphere dominates the image degradation. For historical reasons, the radiometric terminology used by the astronomical community differs from the terminology that has been introduced by radiometric standards organizations. Brightness (W/sr) in standard notation is intensity. Intensity (W/m2) in standard notation is irradiance. The notation common to the community is used throughout this Guide.
These transfer functions relate directly to the diffractionlimited PSF and the halo seen on page 4.
18
Adaptive Optics
The Strehl P r o p a g a t i o n to the Far Field w i t h Wavefront Error The on-axis intensity of a uniform circular beam after it propagates distance L is
where L> >D, the diameter of the aperture, and is the intensity at the circular aperture (W/m2). With aberrations represented by a wavefront error variance (A(())2 (units of optical path distance), the reduction of onaxis intensity, the Strehl ratio, is approximately
Beam Propagation Strehl Ratio When b e a m jitter is present, the optic axis is swept over a small cone and the average intensity in the center of the beam is reduced. When the jitter is assumed to be Gaussian, where is the single-axis beam jitter (in radians), the intensity is multiplied by the factor
to find the further reduction in on-axis intensity. Sometimes the combined effects of wavefront error and beam or image jitter are combined into the Strehl ratio. Rewriting the wavefront error variance in radians squared,
The on-axis intensity with aberrations is then
Astronomical Strehl ratios without adaptive optics are typically very small. With adaptive optics, the Strehl ratio can be improved by orders of magnitude. For a wellconditioned beam in w e a k turbulence, the Strehl ratio without adaptive optics can be 20%, but improvable to 90% or better with adaptive optics.
The definition of Strehl ratio in the classic text Principles of Optics does not include defocus or tilt (jitter) terms in the definition. However, for adaptive optics system calculations, the absorption of these effects into the overall Strehl ratio is convenient.
Because the wavefront variance can be considered a Gaussian variable uncorrelated between various sources, spatial and temporal effects can be efficiently combined using this general definition of Strehl ratio. The Strehl Ratio was originally called Definitionshelligkeit in 1902 by K. Strehl. It is German for Definition of Brightness.
Adaptive Optics
Beam Propagation
21
Spot Size for a Gaussian B e a m
Spot Size for a Uniform Circular Aperture
A laser beam spot does not have a well-defined edge. For a Gaussian beam, the spot size is defined as the diameter 2w where the intensity is the value or 0.135 times the intensity on the optical axis.
The Fraunhofer diffraction pattern of a laser beam emitted from a circular aperture with uniform (constant) amplitude and no phase aberrations is
As a Gaussian beam with wavelength X propagates a distance the spot size changes according to:
where I(r) is the intensity distribution, r is the radial coordinate in the Fraunhofer plane, D is the aperture diameter, and J, is a Bessel function. The Bessel function reaches zero for a number of values of r, the first of which is
This value, where the intensity goes to zero, clearly defines an "edge" of the spot. Thus the spot size is
In atmospheric turbulence, with a constant distance the beam spot size grows according to
turbulence beam dominates the diffractive beam spreading.
over a This spot, called the Airy disk, contains 84% of the energy of the beam.
spreading
22
Adaptive Optics S y s t e m Performance E s t i m a t i o n
One method of determining how well an adaptive optics system performs is to evaluate the residual errors of the system components. Because a deformable mirror cannot exactly match the shape of Kolmogorov turbulence, fitting error results. Similarly, because a control s y s t e m cannot respond instantaneously to the disturbance, a temporal error results. When the source of the wavefront measurement (the wavefront beacon) is positioned away from the object of interest or an outgoing laser target position, the wavefront sensor measures slightly different turbulence, termed isoplanatic error. Noise within the wavefront sensor makes exact measurement impossible and sensor n o i s e error results. Assuming that these errors are uncorrelated and are essentially Gaussian random variables, their variances can be added to determine a system error. With a 2 in units of square radians, the system error is
When these parameters are correlated, the analysis becomes more complex. For example, sensor noise is often a function of integration time of the wavefront sensor, which is an important contributor to the closed-loop b a n d w i d t h and the temporal error.
Hindsight is 20/20. If the Hubble S p a c e Telescope had had an adaptive optics system on board, it would not have been necessary to correct the primary mirror aberration with a NASA-termed "emergency-servicing" mission.
System Performance Estimation
23
Modal and Zonal Fitting Error When a typical deformable mirror cannot exactly the spatial structure of stochastic atmospheric turbulence, modes are used and applied to the deformable mirror, causing the residual wavefront error to be reduced. Where is the number of completely corrected Zernike modes, and D is the aperture diameter, the wavefront error is found from
If many actuators across a continuous faceplate are used, the residual wavefront error can be reduced. The distance between actuators, in the same space as the measurement of is Fitting c o n s t a n t K is related to the stiffness of the deformable mirror faceplate:
24
Adaptive Optics
System Performance Estimation
25
Partial Correction
Temporal Error
To evaluate fitting error, determine the residual wavefront variance after the adaptive optics system removes Zernike modes from the disturbed wavefront. When Kolmogorov turbulence is assumed, calculate the wavefront variance as a function of the number of modes removed and the ratio. A few terms are:
The control s y s t e m must be able to keep up with changing disturbances. For Kolmogorov turbulence, residual wavefront error is related to the closed-loop bandwidth and the Greenwood frequency
For a larger number of modes, where is the total number of Zernike modes completely removed:
Great improvements are seen by correction of only a few Zernike modes, 95% of the energy of the aberrations in a Kolmogorov wavefront is contained within the first 13 modes.
The Greenwood frequency ranges from tens to hundreds of hertz. The adaptive optics system should be capable of responding faster than the turbulence to provide adequate compensation and remain stable, with a factor of 10, as a general rule. This means that the closed-loop width should range from hundreds to thousands of Hertz.
Astronomers F o y and Labeyrie independently considered using laser backscatter from the atmosphere as an artificial wavefront beacon—a laser guide star. However, when funding agencies were presented with the proposal, it was realized that the concept was already a few years old. Highly classified research based on ideas by Julius Feinlieb, Richard Hudgin, David Fried and others had been going on within laboratories in the U.S. Air Force, Navy, and numerous defense contractors. With the collapse of the Soviet Union in 1991, much of the research was declassified and available to astronomers worldwide.
Adaptive Optics
System Performance Estimation Laser Guide Stars
Focal Anisoplanatism (the "Cone Effect") When an object in the sky is too dim to make wavefront sensor measurements and no bright object is within the isoplanatic angle of the object of interest, artificial laser guide stars can be produced. Rayleigh scattering in the lower atmosphere (16 to 20 km) can send sufficient light back toward the wavefront sensor and it can be placed near the object of interest. Object of interest 80 km
atmosphere
Laser guide star
60 km
40 km
Sensed 20 km
aperture
Ground
When a laser guide star is placed a finite distance from the telescope and wavefront sensor, a portion of the atmosphere remains unsensed. The cone of atmosphere that is sensed results in c o n e effect. It is also called focal anisoplanatism, because the source of the wavefront is at a different focus than the object of interest.
The wavefront m e a s u r e m e n t errors caused by this effect were investigated by Fried resulting in the expression
where the character-istic distance varies with the atmospheric Cf profile and the altitude of the guide star, When the H-V 5/7 model of turbulence is assumed, and the altitude is expressed in kilometers, then (expressed in meters) can be approximated by
27
The laser radar e q u a t i o n for calculating the return flux (photons per square meter) for a laser of given pulse energy E is:
where
h is the detector efficiency, is atmospheric transmission up to the guide star altitude is the cross-sectiondensity product, is the laser wavelength, h is Planck's constant, and c is the speed of light.
In 1957, V.P. Linnik published "On the possibility of reducing the influence of atmospheric seeing on the image quality of stars" in the Russian journal Optics and Spectroscopy. It was the first mention of artificial laser guide stars, predating the invention of the laser by three years.
Volcanic aerosols can greatly enhance the backscatter useful for Rayleigh laser guide stars. Unfortunately, we cannot yet accurately predict an eruption. Fortunately, we cannot yet cause an eruption.
28
Adaptive
Optics
System
Performance
Estimation
Scintillation
Laser Guide Stars (cont'd)
The total amount of a t o m i c sodium in the mesosphere useful for laser guide stars could fit inside a phone booth.
To avoid the errors associated with focal anisoplanatism, a laser guide star is placed at a higher altitude. Around 90 km altitude, a layer of atomic sodium can produce backscatter at its resonant wavelength of 589.1583 nm. D e t e c t e d sodium-line p h o t o n flux:
where is the resonant backscatter cross-section and is the column abundance. The product of the crosssection and the abundance is about 0.02.
Dr. William Happer was the first to suggest resonant backscatter sodium laser guide stars when he served on the JASON committee, a group of scientists and engineers who advise agencies of the Federal Government on matters of defense, intelligence, energy policy and other technical problems. The committee is so named because it meets during the months of JulyAugust-September-October-November.
29
While a beam propagates, the effect of phase aberrations manifests into amplitude or intensity variations. The scintillation is easily seen as the twinkling of stars in the night sky. The amplitude fluctuation, represented by the log-amplitude variance is a function of wavelength, propagation length and the strength of atmospheric turbulence
Weak turbulence: Moderate turbulence S t r o n g turbulence
The variance in intensity that results from this variance in log-amplitude is given by
where the variable A represents the aperture averaging factor. For weak turbulence,
where k = and D is the aperture diameter. The effect of adaptive optics on the intensity variance is approximated by
where modes.
is the number of fully compensated Zernike
30
Adaptive Optics
Wavefront Sensors
31
Wavefront S e n s o r R e q u i r e m e n t s
Shack-Hartmann Wavefront S e n s o r a n d Error
With a large number of wavefront sensor subapertures, irrespective of the type of sensor, the accuracy for measuring the wavefront will increase. Assuming turbulence, the wavefront error v a r i a n c e from the wavefront sensor is
The Shack-Hartmann wavefront sensor is a pupil plane measurement of local wavefront slopes (the first derivative of the wavefront) within a subaperture defined typically by a l e n s l e t array. The positions of the Hartmann spots on the detector(s) are proportional to wavefront tilt or slope. Shack-Hartmann sensor error is dependent upon signalto-noise ratio, size of subapertures, gaps between pixels in the detector focal plane, and the finite size of the reference source.
for subapertures of size
Variables: SNR = signal-to-noise ratio - increase in error at the null due to gaps in the detector focal plane D = size of subaperture (in object space) = coherence length of the atmosphere (in object space) 8 = angular size of the reference source = root-mean-square wavefront error in radians measured over a subaperture
Further, assuming that the value of the wavefront measurement will always be between the dynamic range requirement for the wavefront sensor is
For the conditions where = 0.5 is shown on the next page.
= 10 cm, the error
Adaptive Optics
Wavefront Sensors
Shack-Hartmann Wavefront S e n s o r a n d Error (cont'd) For a conventional CCD c a m e r a used for wavefront slope measurement, the signal-to-noise ratio is
Lenslet Array S e l e c t i o n The lenslet array diameter should be chosen so that each subaperture experiences only local tilt. The dynamic range requirements can then be used to determine the necessary focal length of the lenslet array. The total spot shift 5 for an input tilt angle 9 (in radians) is given by
where f is the focal length of the lenslet array. The overall input tilt across a subaperture is found by where is equal to the number of detected photoelectrons per subaperture (sum of all pixels); is equal to the number of pixels in a subaperture (e.g., = 4 for quadcell); is equal to the number of detected background photoelectrons per subaperture; is equal to the read-noise in electrons per pixel; G equals the gain (G = 1 for a nonintensified CCD).
where is the diameter of the lenslet array and OPD is the optical path difference in waves. In waves, the spot shift 8 for an input tilt is
which relates the spot shift to the /-number of the lenslet. The pixel size should be chosen to optimize the dynamic range. A higher number of pixels per subaperture increases both the dynamic range and linearity of the sensor, while also increasing read noise. The subaperture size in pixels should be chosen so that the maximum spot shift will not drift into a neighboring subaperture. The required is then:
for a given dynamic range (in tilt)
Most commercially available lenslet arrays are square rather than circular, so the effective spot size is given as:
34
Adaptive
Optics
S h e a r i n g Interferometer Wavefront S e n s o r and Error The s h e a r i n g interferometer wavefront sensor is a pupil plane measurement of local wavefront slopes within a region defined by the interference p a t t e r n of overShear Plate lapping copies of the wavefront. The intensity of the light in the interference Interference pattern is proPattern portional to the local wavefront tilt or slope. S h e a r i n g interferometer sensor error is caused by signal-to-noise ratio, size of subapertures, shear distance, and fringe contrast. For a shearing interferometer, using a four-bin phase detection configuration, the wavefront error variance (in radians squared) is given by the expression
where K is the modulation (or fringe contrast), SNR is the signal-to-noise ratio at the detector, is the subaperture size, and s is the shear distance. The modulation can be computed from knowledge of the wavefront W(x,y), spectral bandwidth AX, and shape and size of the source: where
would be used for a shear in the x-direction, with a similar expression for shear in the y-direction.
Wavefront
35
Sensors
S h e a r i n g Interferometer Wavefront S e n s o r a n d Error (cont'd) The modulation decrease due to the spectral bandwidth of the source is
For a circular source:
For a rectangular source:
The first continuously operational facility with adaptive optics that used a shearing interferometer was the Compensated Imaging System installed in 1982 at the Air Force Maui Optical Station in Hawaii. That particular system has been decommissioned. A shearing interferometer measures the derivative or the slope of the phase by optically comparing a beam with a laterally shifted (sheared) replica of itself. It computes the slope by finding the intensity of the interferogram:
where s is the length of the shear. If one takes the limit of this quantity as s we have exactly the definition of a derivative as found in all introductory calculus textbooks.
36
Adaptive
Optics
Curvature Wavefront Sensor and Error The curvature sensor is an image-plane measurement of local wavefront curvature [the second derivative of the wavefront deduced from two specific out-of-focus images. A point-by-point Image plane subtraction of the images is proportional to the wavefront curvature term minus the derivative of the wavefront at the edge Curvature Measurement
To provide an accurate measurement of wavefront curvature, the blur from the turbulence must be small compared to the area where the curvature Image mage plane #2 measurement is taken. plane With p being the offset from the focal plane of the system and f as the focal length of the system, the blur requirement leads to
Deformable Mirrors
37
Tilt Mirror R e q u i r e m e n t s The amount of wavefront tilt is dependent upon the diameter of the full aperture D. The variance of the wavefront tilt (in radians) is
A tilt-corrector mirror is usually in a compact part of the beam train. The tilt angle is magnified in that part of the beam by the ratio
Variance of a single curvature measurement:
where N is the photon count. Because the curvature sensor directly measures the Laplacian of the wavefront, mirrors are generally used for closed-loop compensation, as they possess Laplacian influence functions.
The angular tilt corrected is twice the angle through which the tilt mirror moves, thus the angular stroke required for the tilt corrector mirror is
38
Adaptive Optics
Deformable Mirrors
39
Deformable Mirror R e q u i r e m e n t s
Deformable Mirror Actuator Configurations
How far must a deformable mirror move? The answer depends upon the strength of the aberrations. For Kolmogorov turbulence, the tilt-corrected variance (in units per waves squared) is
Deformable mirror actuators are typically placed into a square or a hexagonal array. Certain configurations are adjusted to fit a circular beam and certain "canonical" numbers survive. Square array:
where D is the aperture diameter, assuming that most of the disturbance will be within . Also, because of the reflection off the mirror surface, one unit of motion of the mirror results in two units of wavefront correction. The total required stroke of the deformable mirror will be
4, 9, 16, 21 (5 x 5 - 4 on corners), 37 (7 x 7 - 12 on corners), 69 (9 x 9 - 12 on corners), 97 (11 x 11 -24 on corners), 241 (17 x 1 7 - 4 8 on corners), 341 (19 x 19 - 20 on corners), 577 (25 - 4 8 on corners), 941 (35 x 35 - 284 on corners). Hexagonal array:
A typical commercially available actuator can provide 3.5 of stroke, although larger actuators are available that can provide up to 30 of stroke. The electronic driver rather than the actuator usually limits the bandwidth— each actuator acts like a capacitor, storing charge as it is excited and discharging as it is released. The action of charging and discharging requires current—larger actuators require more current than smaller ones, and higher frequencies require more current than lower ones. The more current required, the more difficult it is to supply. A higher current can also have the effect of heating up the actuators, which can degrade the performance and reduce stroke and bandwidth. Early deformable mirrors built for high-energy laser weapons needed to be water-cooled to keep from being damaged by the beam. Today's multilayer coatings enhance reflectivity and reduce absorption and possible damage.
The hexagonal structure of actuators is the most efficient use of space. With the hexagonal configuration, the maximum number of actuators can be placed in a given area. Honeybees, which predate mammals by millions of years, did the research first.
40
Adaptive Optics
Deformable Mirrors
41
Actuator and Wavefront S e n s o r Layouts
Actuators: R e q u i r e m e n t s for Zonal or Modal Control
The arrangement of actuators and wavefront sensor subapertures (the registration) affects the control algorithm and the stability of the control system. In the figure, the numbered small circles represent actuator positions and the dotted large circles represent the subapertures with the orthogonal slope measurements represented by the arrows.
For the case of zonal correction, each actuator and its associated wavefront sensor N are related to the spatialfitting error. Inverting the expressions for fitting error for zonal correction
where is the total number of active actuators, the aperture diameter is D, the desired Strehl ratio is S, and K is related to the mirror influence function. A typical value for is about 0.3, so the relationship reduces to
geometry
Wavefront control experiment
Southwell geometry
Fried geometry
Many other geometries, including random registration of the wavefront sensor and the deformable mirror can be used with proper calibration and mechanical stability. The actuator spacing is also a requirement in an adaptive optics system. The Nyquist s a m p l i n g t h e o r e m states that spatial frequencies greater than half the sampling frequency cannot be observed. This means that the actuator spacing must be less than half the smallest required spatial frequency period in order to provide the required correction.
If each actuator is recognized as one degree of freedom for the control system, that degree of freedom can be translated into a spatial mode. By inverting the spatial fitting error for Zernike modal correction we find
Thus, the number of actuators is roughly equal to the number of d e g r e e s of freedom that is roughly equal to the number of corrected Zernike modes. The values depend on actuator influence functions, the desired Strehl ratio, the completeness of compensating a specific Zernike mode, and the strength of turbulence given by the ratio.
42
Adaptive Optics
Deformable Mirrors
43
Deformable Mirror Influence F u n c t i o n Models
Bimorph and MEMS Mirrors
Two expressions are widely used to describe the influence function of a continuous faceplate deformable mirror.
The deflection of a bimorph mirror at distance r from the center of an actuator pad is
The deflection normal to the mirror surface can have a cubic relationship: and
where (x,y) are Cartesian coordinates and is the amplitude of the influence function. The origin is at the actuator location. The other expression has a Gaussian form:
where r is the polar radial coordinate in the mirror plane, is interactuator spacing, and is the c o u p l i n g between actuators expressed as a number between 0 and 1. The coupling is the movement of the surface at an unpowered actuator expressed as a fraction of the motion of its nearest-neighbor actuator.
V is the applied voltage, t is the 2 piezoelectric layers thickness of the with opposite polarity bimorph, is the actuator pad radius, is the radius of the supporting ring, is the mirror radius, and is the piezoelectric constant (related to the stress tensor). The deflection of a systems MEMS m e m b r a n e mirror at a distance r from the center of an actuator pad is
and
V is the applied voltage, d is the separation of the membrane and the addressing electrode, and the permittivity, 8.85 x 10 12 F Because the membrane is so thin, very little power is required to deflect the membrane. While voltages can be a few hundred volts, the electrical current is very small.
44
Adaptive
Optics
Control and Reconstruction
45
S e g m e n t e d Deformable Mirrors
Adaptive Optics System Control Model
S e g m e n t e d deformable mirrors exhibit a loss of light in the gaps between segments. In addition, the regular geometric pattern of segments acts like a diffraction grating.
The basic block diagram of a conventional adaptive optics system is shown below, with the optical signals dashed and the electrical signals solid.
Because the optical beam itself is being controlled, there are no dynamics in the system being controlled, therefore no memory of previous aberrations. To accommodate this, an adaptive optics system generally incorporates an integrator after the commands. The integrator artificially induces memory in the system, so that a state space representation of the system can be written as
The diffracted e n e r g y from the effect of the gaps proportional to the width of the gap, a, and the size of the segment, b, according to the relation
where is the vector of wavefront aberrations, B is the deformable mirror influence matrix, u is the vector of actuator c o m m a n d s , d is the vector of wavefront input disturbances (turbulence), v is the vector of measurement noise, y is the vector of measured aberrations, and k is the sample time index. In a discrete time system, integration corresponds to summation, and the integrator for the command vector u can be implemented as where R is the reconstructor.
46
Adaptive
Optics
Reconstructor The goal of an adaptive optics system is to create a regulator and drive all the states (aberrations) to zero for the model
This amounts to determining the inverse of the influence matrix, so that the actuator c o m m a n d s are given by and the states are driven to zero With n actuators and m wavefront measurements, the influence matrix has dimensions x n. Since the influence matrix is generally not a square matrix, a perfect inverse cannot be found and the reconstructor is determined by least-squares techniques. The simplest solution to this least-squares problem is the MooreP e n r o s e p s e u d o inverse, which for an overd e t e r m i n e d system is given as Some numerical difficulties that may be associated with the pseudo inverse, in particular, singularity. The only term that should cause the influence matrix to be singular is piston; therefore, the p i s t o n c o m p o n e n t can be compensated for by adding a row of to the bottom of the influence matrix. This provides an influence matrix with dimensions (m + 1) X n and a reconstructor with dimensions n x (m + 1). To further alleviate numerical difficulties, the singular value d e c o m p o s i t i o n SVD also provides a least squares solution for the generalized inverse and is often used in total least-squares problems. The SVD of a matrix B yields the following: In practice, the SVD solution is more numerically reliable, but the Moore-Penrose solution is easier to calculate.
Control and Reconstruction
47
Wavefront Control Once the reconstructor is calculated, it is possible to implement a closed-loop control system. However, the reconstructor itself is associates the command value used in calculating the influence matrix to unity (1). If is the actuator c o m m a n d value used in calculating the influence matrix, is the ith row of the reconstructor, and b, is the command for biasing the ith actuator, the scale for the individual commands follows:
The quantity - a) can be rewritten as a diagonal matrix and can be used to generate a reconstructor that provides correctly scaled commands. If is the new reconstructor and b is the vector of bias commands, the actuator commands are written in a simple form including the integrator
This provides commands that are in the correct numerical basis. In an adaptive optics system, the piston c o m p o n e n t of the commands accumulates and must be removed to preserve the dynamic range of the deformable mirror. Piston is typically removed from the commands by performing an intermediate step and removing the average value of the commands at each iteration:
Since piston in the commands translates to piston in the deformable mirror surface, removing the piston component has no effect on the performance of the adaptive optics system other than to preserve the dynamic range. For digital control systems, the commands must also be quantized to the word length of the system.
48
Adaptive
Optics
Influence Matrix The deformable mirror influence matrix describes how the actuator affect the surface of the deformable mirror as measured by the wavefront sensor. The influence matrix may also be thought of as an operator that translates n actuator commands to m wavefront measurements and is an X n matrix. The influence matrix is usually determined at run time by sending each actuator a command and then measuring the results with the wavefront sensor. Each set of measurements forms a column in the influence matrix. Since the actuator geometry is such that no two actuators can have the same effect on the surface of the deformable mirror, the influence matrix must have full column rank. Typically, an adaptive optics system is over d e t e r m i n e d - t h a t is, there are more wavefront measurements than actuators, so that > n. Therefore, the condition of fullrow rank is required for the influence matrix to be invertible and is limited by the fact that different piston values in the wavefront can produce identical wavefront measurements. The deformable mirror's continuous surface is interpolated by the face sheet from the positions of the discrete actuators. This interpolation means that there must be some c o u p l i n g in the influence functions between actuators, typically 5-15%. However, since actuators on opposite sides of the deformable mirror cannot affect the surface in the same manner, the influence matrix is generally block diagonal. This property can be leveraged with sparse matrix techniques to provide computationally efficient methods for calculating the reconstructor.
Control and Reconstruction
49
Filtering a n d Wavefront E s t i m a t i o n The Kalman filter is an optimal estimator used to counter the effects of noise in the wavefront measurement, which provides the minimum mean-square-error estimate of the wavefront and is based on the system model of
Using knowledge of atmospheric turbulence statistics with covariance matrix as well the measurement noise statistics with the covariance matrix wavefront e s t i m a t e s can be generated. First, find the steady-state solution for the algebraic equation for the error covariance matrix P:
Using this steady-state solution, the Kalman gain may be calculated from Finally, the wavefront estimates may be generated from the following state equations for the current estimate, : and the predicted estimate, (J) :
The actuator c o m m a n d s may then be generated by
where is the scaled reconstructor, and b is the vector of actuator bias commands. The Kalman filter also has a time varying solution, which may be useful in some cases dealing with nonstationary atmospheric turbulence.
50
Adaptive
Optics
51
Control and Reconstruction
Computational L a t e n c y
Predictor
A problem often encountered in adaptive optics systems using high-speed cameras is the effect of computational latency. The nature of the camera read-out is to continually transfer pixel information. A common solution in image-processing applications is to use double buffering so that one frame may be processed while the next frame is being acquired. When the time required to calculate wavefront estimates and implement the commands approaches the frame-transfer time of the camera, the measurements no longer accurately reflect the commands being generated and a delay in the system occurs. This delay introduces high-frequency p h a s e errors in the system as shown in the simulation below. This simulation shows the results of computational latency using an input aberration of frequency 1/50 the sampling frequency of the system. Solutions for computational latency are to sample the desired closedloop b a n d w i d t h much higher and use a predictor.
A method of dealing with a one-cycle computational delay is a predictor. Since the deformable mirror influence matrix and the commands previously generated are known, the wavefront estimates can be predicted. The predictor should have the form Note that the predictor uses R and not R'. The reconstructor used in the predictor should be so the influence matrix and the predictor have the same numerical basis. The simulation below again shows the effects of computational latency, this time using a predictor. This predictor does not yield an optimal estimate, however, the filter does provide an optimal predicted estimate that is updated with each iteration. Therefore, if the Kalman filter is used, the predicted estimate should be used in place of the current estimate .
52
Adaptive
Optics
Control
and
Reconstruction
53
Effect of Sampling Rate on Achievable B a n d w i d t h
H a r t m a n n S e n s i n g Software I m p l e m e n t a t i o n
Understanding the effect of the s a m p l i n g rate is critical in analyzing the performance of an adaptive optics system. The figures below show the effects of a sinusoidal signal sampled at the Nyquist rate (two times the frequency) and at eight times the frequency of the signal. Light red represents the input sinusoidal signal, black represents the sampled signal, and red represents the output commands. Looking at these figures, it is clear that there is a significant delay or error introduced even when sampling at eight times, the result of which is to degrade the stability of a closed-loop system. With this in mind, it is common practice to set the sampling frequency to 15-20 times higher than the desired closed-loop bandwidth.
Implementation of the H a r t m a n n sensor can present a number of computational difficulties. The size of the subapertures should be large enough to prevent neighboring spots from wandering in and small enough that the number of pixels to compute is kept to a minimum.
A common method of dealing with this is to use a set of fixed subapertures in conjunction with a set of dynamic tracking windows. The tracking windows should be just slightly larger than the spots but smaller than the subapertures, so the centroiding algorithm runs quickly and accurately. These windows should also move with the spots as shown in the figure. The tracking windows must also be able to cope with situations where the spot is on the edge of the window or where the spot is "lost." The centroiding algorithm can be either a simple center of mass calculation, or a more complicated morphological imageprocessing routine based on the size and/or shape of the spots. To compensate for losing a spot, a tracking window should either grow in size until it again finds the spot or attempt to move intelligently in the direction of travel of the spot.
54
Adaptive
Equation Summary Coherence length:
Optics
55
Appendix
Equation Summary (cont'd) On-axis intensity of uniform circular beam:
On-axis intensity with aberrations: Isoplanic angle:
Laser brightness: Greenwood frequency:
Hufnagel-Valley model:
Intensity distribution for a uniform circular aperture:
Strehl ratio with reduction of on-axis intensity:
Strehl ratio with wavefront error and jitter: Bufton wind model:
Wavefront error variance in radians squared: Root-mean-square average jitter:
56
Adaptive
Equation Summary (cont'd) Amplitude fluctuation:
Optics
57
Appendix
Equation Summary (cont'd) Coherence length of the atmosphere with modulation transfer function:
Gaussian spot beam waist change: Modal wavefront error:
Gaussian growth spot with turbulence: Zonal fitting error:
Zernike series: Shack-Hartmann sensor error:
Azimuthal polynomials: Shearing interferometer wavefront error variance: Modulation transfer function for a diffractionlimited circular aperture:
Wavefront slope measurement using a CCD camera:
58
Adaptive
Optics
59
Appendix
Equation Summary (cont'd)
Equation Summary (cont'd) Variance of a single curvature measurement:
Detected sodium-line photon flux:
Number of actuators required:
Inverted spatial fitting error for zonal correction:
Temporal error:
Wavefront sensor:
error
variance
assuming
wavefront
Inverted spatial fitting error for Zernike modal correction:
Wavefront measurement error:
Angular stroke for tilt-corrector mirror:
Visual magnitude related to brightness:
Variance of wavefront tilt:
Laser r a d a r equation:
Total required stroke for deformable mirror:
62 Adaptive
Optics
Bibliography (cont'd)
Bibliography Bar-Shalom, X. R. Li, and T. Kirubarajan, Estimation with Applications to Tracking and Navigation, Theory Algorithms and Software, (John Wiley and Sons, New York, New York, 2001). Born, M. and E. Wolf, Principles of Optics, (Pergamon Press, Oxford, 1975).
5 th Ed.
J. L., "Comparison of vertical profile turbulence structure with stellar observations," Opt. 12, 1785 (1973). Churnside, J. H., "Aperture averaging of optical scintillations in the turbulent atmosphere," Appl. Opt. 30, 1982 (1991). Forbes, F. F., 1114, 146 (1989).
63
Appendix
PZT active mirror,"
Goodman, J. W., Introduction to Fourier Optics, 2 nd Ed. (McGraw-Hill, New York, 1996.) Goodman, J. W., Statistical Optics (Wiley, New York, 1985). Greenwood, D. P., "Bandwidth specification for adaptive optics systems," J. Opt. Soc. Am. 67, 390 (1977). Grosso, R. P., and M. Yellin, "The membrane mirror as an adaptive optical element," J. Opt. Soc. Am. 67, 399 (1977). Hardy, J. W., Adaptive Optics for Astronomical Telescopes, (Oxford Univ. Press, Oxford, 1998). Hayes M. H., Statistical Digital Signal Processing and Modeling (John Wiley and Sons, New York, New York, 1996).
Franklin, G. F., J. D. Powell, and M. L. Workman, Digital Control of Dynamic Systems, 2 nd Ed. Reading, Massachusetts, 1990).
Hudgin, R. H., "Wave-front compensation error due to finite corrector-element size," J. Opt. Soc. Am. 67, 393 (1977).
Fried, D. L., "Focus anisoplanatism in the limit of infinitely many artificial-guide-star referrence spots," J. Opt. Am. A 12, 939 (1995).
Johnson, B. and D. V. Murphy, Thermal Blooming Laboratory Experiment, Part I, Lincoln Laboratory MIT Project Report BCP-2 (November 1988).
Fried, D. L., "Anisoplanatism in adaptive optics," J. Opt. Soc. Am. 72, 52 (1982).
Miller, M. G. and P. L. Zieske, "Turbulence environment characterization," ADA072379, Rome Air Development Center (1979).
Gardner, C. S., B. M. Welsh, and L. A. Thompson "Design and performance analysis of adaptive optical telescopes using laser guide stars," Proc. IEEE 78, 1721-1743 (1990). Gonzalez, R. C. and R. E. Woods, Digital Image Processing, 2 nd Ed. (Prentice Hall, Upper Saddle River, New Jersey, 2002)
Noll, R. J., "Zernike polynomials and turbulence," J. Opt. Soc. Am. 66, 207 (1976).
atmospheric
Roddier, F., "Curvature sensing and compensation: a new concept in adaptive optics," Appl. Opt. 27, 1223 (1988).
64
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Optics
Bibliography (cont'd) Sinha N. K. and B. Dynamic Systems (Van York, New York, 1983).
Index
Modeling and Identification of Reinhold Company, New
Taranenko, V. G., G. P. Koshelev, and N. S. Romanyuk, "Local deformations of solid mirrors and their frequency dependence," Sou. J. Opt. Technol. 48, 650 (1981). Trefethen L. N. and D. III, Numerical Linear Algebra, (Society for Industrial and Applied Mathematics, Philadelphia, Pennsylvania, 1997). Tyson, R. K., Introduction to Adaptive Optics WA, 2000).
Press,
Tyson, R. K., Principles of Adaptive Optics, (Academic Press, Boston, 1998). Tyson, R. K., "Adaptive optics and ground-to-space laser communications," Opt. 35, 3640-3646 (1996). P. B., profiles for values of the coherence length and isoplanatic patch angle," W. J. Schafer Associates, Arlington, VA (1988).
cone effect, 26 control computer, 1 control system, 22, 25, 40, 41, 47 conventional (linear) adaptive optics system, 1, 45 coupling, 42 cross-over frequency, 14 curvature sensor, 36
aberrations, 2, 18, 21, 24, 29, 38, 45, 46 actuator, 23, 40-42, 46, 48 actuator commands, 4 5 49 Airy disk, 21 astronomical brightness, 6 astronomical seeing, 3, 5 atomic sodium, 28 atmospheric turbulence, 2, 4, 5, 10, 20, 29 atmospheric turbulence statistics, 45, 49 atmospheric wind profile, 12
defocus, 18 deformable mirror, 1, 23, 42 degrees of freedom, 41 derivative, 31, 35, 36 detected sodium-line photon flux, 28 double buffering, 50 diffracted energy, 44 diffraction, 4 dynamic range, 30, 33, 47
Babcock, 3 beam jitter, 19 Bessel function, 21 bimorph mirror,36, 43 blur, 36 brightness, 6, 17, 19 wind model, 12
25 fitting constant, 23 fitting error, 22, 24 flat-Earth assumption, 5, 7, 14 focal anisoplanatism, 26, 28 Foy, 25 Fraunhofer diffraction pattern, 21 Fried, 25, 26 Fried geometry, 40 Fried's coherence length, 5 fringe contrast, 34
Cassegrain, 9 CCD camera, 32 circular source, 35 closed-loop bandwidth, 22, 25, 50, 52, closed-loop control system, 47 closed-loop system stability, 52 coherence length, 10, 11 computational latency, 50,51
65
Index Index
Gaussian beam, 20 Gaussian model, 12 Gram-Schmidt orthogonalization, 9 Greenwood frequency, 14,25
Kolmogorov atmospheric turbulence, 14, 22, 25, 30, 38 Labeyrie, 25 laser guide star, 26-28 laser radar equation, 27 lenslet array, 33 Linnik, 27 log-amplitude variance, 29 long-exposure MTF, 16
halo, 4 Happer, 28 Hartmann sensor, 53 high-frequency phase error, 50 Hubble Space Telescope, 22 Hudgin, 25 Hudgin geometry, 40 (H-V) model, 7, 10 5/7, 10,11,26
magnification, 15 membrane mirror, 43 moderate turbulence, 29 modulation transfer function (MTF), 16 Moore-Penrose pseudo inverse, 46
image degradation, 16 influence function, 41, 42,48 influence matrix, 45-48, 51 integrator, 45, 47 intensity, 17 intensity variation, 29 interference pattern, 34 irradiance, 17 isoplanatic angle, 7, 10, 26 isoplanatic error, 22
Nyquist rate, 52 Nyquist sampling theorem, 40 on-axis intensity, 18, 19 optical phase, 8 over determined, 48 over-determined system, 48 phase conjugation, 3 piston component, 46, 47 point spread function, 2, 4 predictor, 50, 51
jitter, 15, 17 filter, 49, 51
Rayleigh laser guide star, 27 66
strong turbulence, 29 subapertures, 30, 31-33, 40,53 system error, 22 temporal error, 22, 25 temporal power spectrum, 14 third-order optical aberration, 9 tilt, 14, 16, 33, 38 tracking windows, 53 tropopause, 12
reconstructor, 45, 48, 51 rectangular source, 35 reflecting telescope, 9 residual error, 22, 23 sampling rate, 52 scintillation, 29 segmented deformable mirror, 44 sensor noise error, 22 Shack-Hartmann wavefront sensor, 31 shearing interferometer sensor error, 34 shearing interferometer sensor error, 34 short-exposure MTF, 16 single curvature measurement variance, 36 singular value decomposition (SVD), 46 Strategic Laser Communication model (SLC), 7 SLC-night, 10, 11 slope, 31, 34, 40 signal-to-noise ratio (SNR), 32 Southwell geometry, 40 spatial frequency response, 16 spatial-fitting error, 41 spatial mode, 41 spot size, 20, 21, 33 Strehl ratio, 17, 18, 19, 41
visual magnitude, 6 volcanic aerosol, 27 wavefront beacon, 7, 22 wavefront error, 7, 8, 18, 19, 23, 25 wavefront error variance, 19, 30, 34 wavefront estimate, 49 wavefront measurement error, 26 wavefront sensor, 1, 8, 22, 26, 27, 30-31, 3436,48 wavefront sensor subaperture, 30, 40 wavefront variance, 19, 24 wavefront tilt, 31, 34, 37 weak turbulence, 18, 29 zenith angle, 5, 7, 12, 14 Zernike mode, 24, 29 Zernike polynomial series, 8 67
Robert K. Tyson is an Associate Professor of Physics and Optical Science at The University of North Carolina at Charlotte. He is a Fellow of - The International Society for Optical Engineering. He has a B.S. in physics from Penn State University and M.S. and Ph.D. degrees in physics from West Virginia University. He was a senior systems engineer with United Technologies Optical Systems from 1978 to 1987 and he was a senior scientist with Schafer Corporation until 1999. He is the author Principles of Adaptive Optics (Academic Press, 1st edition 1991, 2nd edition 1998) and Introduction to Adaptive Optics (SPIE Press, 2000) and the editor volumes on adaptive optics. Professor Tyson's current research interests include atmospheric turbulence studies, classical diffraction, novel wavefront sensing, and amplitude and phase manipulation techniques to enhance propagation, laser communications, and imaging.
Benjamin West Frazier is an Associate Electrical Engineer with Xinetics, Inc. in Devens, MA. He received his MSEE and BSEE degrees from The University of North Carolina at Charlotte, where he focused on robust H-infinity control of adaptive optics systems. His current duties include the development of real-time closedloop adaptive optics systems and the implementation of an automated testing process for qualifying deformable mirrors.
