Page 1

Adaptive optics implementation with a Fourier

reconstructor

Oded Glazer, Erez N. Ribak, and Leonid Mirkin

Adaptive optics takes its servo feedback error cue from the wavefront sensor. The common Hartmann–

Shack spot grid, representing the wavefront slopes, is usually analyzed by finding the spot centroids. In

a novel application, we used the Fourier decomposition of the spot pattern to find deviations from grid

regularity. This decomposition was performed either in the Fourier domain or in the image domain, as

a demodulation of the grid of spots. We analyzed the system, built a control loop for it, and tested it

thoroughly. This allowed us to close the loop on wavefront errors caused by turbulence in the optical

system.© 2007 Optical Society of America

OCIS codes:

010.1080, 010.7350, 100.2650, 100.5070.

1.

Adaptive optics deals with sensing of a distorted

wavefront and its correction. There is a variety of

applications for adaptive optics, for instance in astro-

nomical observations where we want to discern as-

tronomical objects. The optical signal passes through

free space without distortions, but once it encounters

the earth’s atmosphere the beam is affected and ab-

errated. There is a need to compensate for the refrac-

tive index variation produced by thermal air streams

that make the atmosphere turbulent.1Adaptive op-

tics is also being used for military and ophthalmic

applications.2

Wavefront correction is achieved by applying a se-

ries of calculated reconstruction commands on the

phase modulators. The phase modulator is a device

thatcanmodifythewavefrontofanincomingdistorted

beam by adding a refractive or reflective delay, in our

case using a membrane deformable mirror. The force

on the membrane is affected by a number of parallel

actuating capacitors to which voltage is applied. The

data for the reconstruction is based on a Hartmann–

Shack wavefront sensor which is essentially a lenslet

array, sampling the wavefront slope. This correction

Introduction

process must be achieved within a few milliseconds in

order to improve the image quality before the aberra-

tions change significantly. In the future, with the ad-

vent of sensitive and low-noise detectors, with the

development of larger aperture mirror telescopes with

thousands of actuators, large deformable secondary

mirrors, and huge telescope structures we need to de-

velop faster real-time compensating algorithms to

meet these demands, and more efficient schemes for

phase compensation must be built. This work is in-

tended as a step in this direction.

Currently adaptive optics systems use centroid

method reconstructors to convert gradient measure-

ments to wavefront phase estimates, where local

slopes move beams from their nominal positions and

correction is made according to this movement.3Most

works have utilized a least-squares control algorithm

and predetection compensation to minimize the wave-

front error in an optical system.4–7The purpose of

this work was to test for what is believed to be the

first time the implementation of two alternative

approaches: Fourier demodulation8and direct demo-

dulation of the spot pattern9in order to find the

wavefront slope.10This is multiplied, as in the con-

ventionalleast-squaresapproach,byareconstruction

matrix to produce a vector of mirror commands.

In general, the reconstruction process involves two

consecutive stages: (1) calculation of the wavefront

slopes using the Hartmann spots, or the wavefront

curvaturefromtheintensitytransportmeasurement,

and (2) calculation of the wavefront itself or the mir-

ror positions, if the deformable mirror is of the piston

type. For other phase modulators, such as the bi-

morph mirror, stage (2) provides the mirror com-

The authors are with Technion, Israel Institute of Technology,

Haifa 32000, Israel. O. Glazer and L. Mirkin are with the Faculty of

Mechanical Engineering. E. Ribak (eribak@physics.technion.ac.il) is

with the Faculty of Physics.

Received 13 April 2006; revised 3 October 2006; accepted 3 Oc-

tober 2006; posted 5 October 2006 (Doc. ID 69931).

0003-6935/07/040001-07$15.00/0

© 2007 Optical Society of America

1 February 2007 ? Vol. 46, No. 4 ? APPLIED OPTICS1

Page 2

mands. Until now, Fourier analysis was only applied

in the second stage for integration of the slopes,11,12

whereas now we test its real-time employment in the

first stage as well. A full Fourier approach is also

possible for well-behaved wavefronts where both

stages are performed in the Fourier domain: The

Hartmann data are transformed, the wavefront gra-

dient isolated and integrated, and the phase is ob-

tained by an inverse transform. This approach was

applied for wavefront calculation,13but still has to be

tested for adaptive optics. Its obvious advantage is

the absence of the laborious multiplication by the

reconstructor matrix, prohibitively large for extended

systems. Since our mirror has a polar and not a Car-

tesian symmetry, we did not employ this option.

We start our description with the introduction and

investigation of the techniques we used for the con-

trol scheme and also define a discrete formalism for

the deformable mirror in the linear-discrete regime.

Next we turn to the optical setup and the other main

topic of this paper, detailing the sensing analysis

methods that we applied.

2.

Our adaptive optical system is basically a laser beam,

which gets distorted in the optical system, reflects off

a deformable mirror corrector, and is measured in

a wavefront sensor. Our goal here is to flatten the

wavefront according to a premeasured reference. The

wavefront is measured through an image processing

algorithm described below. The control of the wave-

front is thus actually performed in terms of its

2 ? 25 ? 50 slope components, which are compared

with the corresponding slopes of the reference wave-

front. The corrections of the wavefront are achieved

by the use of a deformable mirror, the shape of which

can be affected by an array of 37 electromechanical

actuators, which, in turn, are controlled through in-

put voltage V. This system can be presented by the

block diagram in Fig. 1.

Note that although the electromechanical actua-

tors are nonlinear, this nonlinearity is known: The

stroke is proportional to the square of the input volt-

age. Thus we can linearize the actuators’ responses

by inverting (precompensating) the actuator nonlin-

earity in the control law. Therefore we analyze the

system assuming that the actuation part is linear

(the resulting controller is always complemented by

the inverse of the actuation nonlinearity). Also, the

dynamics and settling time of the actuators are much

faster than the achievable controller bandwidth,

which is limited by the computational time required

to perform the image processing step. Thus we can

safely neglect all dynamical effects in the actuators

and absorb the first two blocks in Fig. 1 into the

Control Scheme

controller. Consequently, the controller design below

will address only the last two blocks, the wavefront

sensor and the signal processing unit. These blocks

will be united into a single, static system, H (the

Hartmann–Shack sensor), which is a 50 ? 37 recon-

structor matrix.

Flattening the wavefront is complicated by the

presence of disturbances and unmodeled effects and

nonlinearities, such as residual mirror hysteresis,

membrane response to sound and vibrations, and

thermal effects. To cope with these effects and com-

pensate them, a feedback control law shall be used.

The most important property of feedback is its ability

to cope with modeling errors, external disturbances,

and uncertainties.

In our system, the feedback control can be imple-

mented by assigning the input voltage to the actua-

tors on the basis of the mismatch between the actual

and desired wavefronts, w and r. An important point

here is that we cannot generate any wavefront as we

have only 37 actuators to control 50 variables (an

underactuated system). We therefore can track ref-

erences from the 37-dimensional image space of H

only. This can be interpreted as imposing the con-

straint r ? Hraon the reference signal r, where rais

arbitrary. To simplify the design of the feedback con-

troller, or more precisely, to decouple the control loop,

we premultiply the tracking error, r ? w, by the

pseudoinverse of H (Ref. 14) H#? ?H?H??1H?, before

processing it by the controller. Since H#?Hra? w?

? ra? H#w, this is equivalent to tracking raby a

filtered wave front H#w.

Thus the data from the Hartmann–Shack sensor

are multiplied by H#, compared with the chosen ref-

erence ra, and this difference is processed by a digital

controller C1?z? acting at the sampling rate of 30 Hz,

as dictated by the video rate (Fig. 2). Signal d there

reflects external disturbances and unmodeled effects.

Signal n reflects the measurements noise, which is

brought about by the measurement device, namely,

the noise and the background signal in the CCD sen-

sor. Our goal here is to design C1?z?, which stabilizes

the closed-loop system and makes the tracking error

e ? ra? H#w small despite the presence of d (which

is typically relatively slow) and n (which is typically

relatively fast).

A.

The system in Fig. 2 is not yet determined as we do

not know the relation between the control variable u

and the processed wavefront w, the wavefront slopes

or their Fourier low frequencies. Thus the first step of

Calibration (Identification of H)

Fig. 1.The plant (controlled process) and the sensor.

Fig. 2.Block diagram of the feedback control system.

2APPLIED OPTICS ? Vol. 46, No. 4 ? 1 February 2007

Page 3

the proposed scheme is to identify this relation, i.e.,

the operator H.

The operator H is constructed in the calibration

process. We identify it by giving the maximal voltage

to each separate actuator, one at a time, and the

wavefront of each of these pokes is measured by cal-

culating the Fourier demodulation of the spot pat-

tern. The maximal voltage for each poke is chosen to

reduce errors attributable to the presence of the mea-

surement noise. Each step of this procedure immedi-

ately yields the column of H corresponding to the

exciting actuator. Notice that we simply use Fourier

modes, namely the Fourier components of the gradi-

ent of the wavefront, in contrast with other processes

employing gradients of Zernike modes, and in other

cases atmospheric modes.

B.

It is readily verified that the closed-loop tracking er-

ror e ? ra? y satisfies

Controller Design

E?z??S?z??Ra?z??D?z???T?z?H#N?z?,

where S?z? ? ?I ? C1?z???1and T?z? ? ?1 ?

C1?z???1C1?z? ? I ? S?z? are the sensitivity and com-

plementary sensitivity transfer matrices, respectively.

In the choice of the controller C1?z? the following re-

quirements were taken into account:

Simplicity: Since the feedback control law is to

be implemented in real time, we cannot afford com-

plicated (i.e., high-order) controllers, the implemen-

tation of which would be numerically demanding.

Y

Set-point tracking: Since the reference vector rz

is assumed constant, we require a zero steady state

error, limt→?ez?t? ? 0, for all constant rzand dz. This

amounts to requiring that the static gain of the sen-

sitivity transfer matrix be zero, S?1? ? 0, or, equiva-

lently, that the static gain of the controller, C1?1?, be

infinite.

Y

Low sensitivity to measurement noise: The

measurement noise nzis typically fast, so to prevent

its amplification by the feedback loop we require that

the magnitude of the complementary sensitivity fre-

quency response, T?ej??, ? ? ???, ??, at high frequen-

cies be low (smaller than 1). This requirement can be

cast as |T??1?| ? ?, for some ? ? 1, i.e., as a con-

straint on |T?z?| at the highest frequency.

Y

The first two requirements suggest the following

digital decentralized proportional-integral (PI) form

of the controller:

C1?z??z?1kp?1?

kaz

z?1?I,

where kpis the proportional gain, and kais the accu-

mulator (the presence of the integrator term guaran-

tees that the static gain of this controller is infinite).

Note that C1?z? has a pole at the origin to reflect

inevitable computational delay.

Some lengthy, albeit straightforward, algebra yields

then that the third requirements, combined with the

standard closed-loop stability requirement, imposes

the following constraints on the controller parameters:

kp??1,

kpka?0,

kp?2?ka??

2?

1??,

1??, if ??1

kp?2?ka???

2?

2.

The gray areas in Fig. 3 show admissible pairs ?kp, ka?

for several values of ?.

Note that the stability conditions are actually re-

covered as ? → ?. As ? decreases, the admissible

areas shrink. The requirement of a small high-

frequency gain is especially restrictive in the first

quadrant, where the proportional gain cannot be in-

creased much. Since a low proportional gain gives

rise to slow transient response, this essentially im-

plies that negative controller coefficients are prefer-

able. Indeed, in the third quadrant we can still use a

high proportional gain, even if ? is small.

3.

To test the system and control, we built a simple

optical system (Fig. 4). The measurement was per-

formed by using a lenslet array, and compensation

was produced by a membrane mirror (OKO Technol-

ogies 37-channel electrostatic mirror). This was a

part from a package from Adaptive Optics Associates,

Inc. (AOA), which also included a Tk?Tcl language

program for Hartmann–Shack wavefront sensing

and for adaptive optics closed-loop control and diag-

nostic graphics. We first blocked the deformable mir-

ror arm and took a reference image for calibration of

the wavefront sensor itself. Then we blocked the ref-

erence arm and measured the aberrations in the

Experiment

Fig. 3.

?kp, ka? plane.

Stability and high-frequency performance areas in the

1 February 2007 ? Vol. 46, No. 4 ? APPLIED OPTICS3

Page 4

other arm. To calibrate the system, we poked each

mirror element in turn and measured the wavefront.

Alternatively, we poked a whole mode (a combination

of elements) to find its response. We inverted the

responses by using singular value decomposition, to

construct a matrix response to any given wavefront

through application of voltage to the relevant actua-

tors. After testing the system by correcting for static

errors, we moved on to dynamic correction: A hair

dryer was operated at lower voltage to create weak

turbulence across the optical path in front of the

deformable mirror. In all cases full correction was

achieved.

A.

The common method today for constructing the wave-

front phase distortion is by sampling its local slopes

using a lenslet array and applying a digital closed-

loop process. The slopes are measured at the focal

plane of the lenslet array as foci shifts. A centroiding

algorithm3finds the location of each focal spot, from

which the whole map of slopes is constructed, to serve

as inputs for the control loop.

To calculate the local shifts of the foci we also used

two new methods: First, we performed a Fourier de-

modulation of the Hartmann–Shack pattern.8This

method extracts the essential data of the wavefront

behavior, namely the shifts of focal spots, by consid-

ering them as modulation of the full regular spot

pattern. Its disadvantage is the requirement for a full

Fourier transform at all possible frequencies, even

irrelevant ones. Second, we conducted a direct de-

modulation of the foci frequency only, without revert-

ing to Fourier transforms. This helped us to reduce

the processing times.9

For the first method, we used a 2D fast Fourier

transform (FFT) to obtain the slope of the wavefront

phase.1,15That slope appears in the complex side

lobes of the transform, where these side lobes corre-

spond to the lenslet period. One can try to retrieve it

Fourier Reconstruction

from the real, imaginary, amplitude, phase, or her-

mitian parts of the transform.8,13While the hermi-

tian option is theoretically the best, it requires an

integer frequency of spots, which was not possible

for us, or Fourier plane interpolation, not available

through the given software. Instead, we chose exper-

imentally the most suitable part of the transform

by realizing that it should have the highest rms

response to the same set of input wavefront aberra-

tions. The 200-odd Hartmann–Shack spots were im-

agedwithaCCDcameraholding480 ? 640 pixels.To

avoid aliasing we removed the side columns and

added empty rows to create a 512 ? 512 pixel array.

Not surprisingly, the amplitude part of the transform

yielded the highest response to the aberrations. Un-

fortunately, the loss of phase information inhibited

their use, and indeed an adaptive optics control loop

became unstable. Instead, we chose the imaginary

part of the transform, which proved to be much more

reliable and stable.

To build the reconstruction matrix, we tried a num-

ber of schemes, and settled on a poke reconstructor:

The response of the wavefront to a constant voltage

on each actuator resulted in a different transform

pattern. We took the imaginary part of the two side

lobes corresponding to the x and y slopes in the trans-

form and stacked them one on top of the other for

display purposes (Fig. 5), or staggered them into a

linear array as a column input. Each poked actuator

gave such a column, and their combination provided

the response of the system. That response matrix has

the width of the number of actuators (37), and the

height of the combined number of pixels in the two

side lobes (50). Each such matrix element contained

the imaginary part of the Fourier component of the

wavefront slope. From it was subtracted the refer-

ence response matrix without any voltage applied to

the actuators (the calibration frame). The arrange-

ment of the data in the slope vectors is not important

but should be kept consistent between calibration

Fig. 5.

current measurement of the wavefront (less a reference array).

Onlytheimaginarypartoftwolobesisshown,correspondingtothe

x and the y slopes. On top is a square of 25 pixels of the 0, 1 lobe,

below it the square of the 1, 0 lobe.

(Color online) Selected part of the Fourier transform of the

Fig. 4.

from the bottom right and splits (in beam splitter BS2) into either

a reference channel or a deformable mirror channel. Light from

the unblocked arm returns into the wavefront sensor (WFS:

Hartmann–Shack lenslet array and camera). An image of the cor-

rected laser beam is split (in beam splitter BS1) into another

camera. Turbulence is added between the lenses in front of the

deformable mirror.

The optical system. A collimated, wide laser beam enters

4APPLIED OPTICS ? Vol. 46, No. 4 ? 1 February 2007

Page 5

and operation. Next we used a singular value decom-

position (SVD) algorithm to determine the pseudoin-

verse of the response matrix. Thresholding removed

the weaker eigenvalues of the inverse. Thus the pro-

cedure to construct a reconstructor matrix is the

same as for centroiding, except that the Fourier

modes of the slopes are used and not the slopes them-

selves.

In running the adaptive optics loop, each Hartmann–

Shack frame is Fourier transformed, and the same

vector is formed from its two orthogonal side lobes

(again, the imaginary values of 25 elements around

the main frequency in each direction). This vector is

nowmultipliedbythereconstructormatrixtoyieldthe

next mirror commands for the controller.

B.

Time is a critical factor in a fast-changing system,

and thus a minimum of CPU cycles is required to

close the adaptive optics loop. Even though the Fou-

rier transform is very efficient, we have just seen that

we may use a very small number of Fourier compo-

nents out of the many calculated. For example, we

were able to close the loop with 25 x-lobe and

25 y-lobe Fourier components, out of the 5122?

262, 144 in the full frame. It makes more sense to

transformonlythenecessaryFouriercomponents,and

thiscanbeachievedbyphasedetection:Sinceweknow

the frequency k0of the Hartmann spots, we multiply

by ejk0x, and integrate by smoothing the complex

result. Smoothing can be achieved by convolution

with a kernel the width of the Hartmann pitch,

which is too slow for a real-time process. Instead,

we shifted and added the complex array to half its

values, and then again at half the pitch.9This is

equivalent to finding only the k0harmonic in one

direction. In a temporal sense this is similar to super-

heterodyne or locked-in detection. Finally we ex-

tracted the phase of the smoothed array. This phase

is the x component of the slope. The process was

Smoothing Reconstruction

repeated for the y direction. The results are shown in

Fig. 6.

The analysis is performed in the image domain and

hence is very fast.9As the resulting number of points

(slope components) is equal to the number of cam-

era pixels inside the aperture, much larger than the

number of actuators, coarser smoothing is possible.

Following the smoothing and phase extraction, we

simply sampled regularly the aperture for the slope

(once for each direction) in 2 ? 25 points as an input

for the control loop. From this point on we proceeded

exactly as in the centroiding scheme: The phases from

these 50 samples made a single slope vector, used to

create the response matrix for each of the 37 actua-

tors. SVD created the reconstructor matrix. Then each

new Hartmann–Shack pattern was twice multiplied,

smoothed, and down sampled to obtain the same

slopes vector, to be used in the mirror control loop.

Fig. 7.

to their final values within five iterations for kp ? ?0.3 and

ka? ?0.3. Insert on right: cuts through the laser input images for

three iterations, when they reduce to diffraction size (bottom

curve). To better see the width, the camera was overexposed, and

the top of diffraction images is saturated.

(Color online) Screen shot of the first six modes converging

Fig. 6.

diameter). The Hartmann pattern is grabbed, multiplied by the exponential of its base frequency, smoothed, and its phase extracted.

The slope is sampled at 25 spots inside this pattern (left panel). After subtraction of the reference image (flat surface) we get the control

signal input (right panel).

(Color online) Hartmann pattern filtered by smoothing: y-slope response to a single actuator (0.3 ?m stroke, 15 mm mirror

1 February 2007 ? Vol. 46, No. 4 ? APPLIED OPTICS5

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1.

UsingtheTk?TclandCprogramsprovidedbyAOAwe

were able to close the control loop in 80 ms (30 Hz

frame rate) in the classical centroid approach. In the

next stage we tried the Fourier and smoothing phase

extraction methods. The inefficiency of the FFT

algorithm and the shear frame size made the pro-

cess much slower, closing the loop in approximately

600 ms. At this rate we were still able to correct

aberrations due to slow turbulence (Fig. 7). No at-

tempt was made to subsample or bin down the frame,

or locate a more efficient FFT or similar algorithm.11

The reason is that the AOA software package is no

longer supported, and the inclusion of new C subrou-

tines is not possible. As a result, the smoothing algo-

rithm control loop exceeded 2500 ms. Its application

in C or MATLAB was, on the other hand, much faster

then the previous methods.9Using a different com-

puter, it was applied successfully to a living human

eye.16

We also verified the control schemes for different

parameters. By changing the hair dryer speed we

induced weak and strong turbulence changes, which

translated into corresponding temporal and spatial

aberration scales. Indeed we found that our model led

to areas of divergence and uncontrolled oscillations

and areas of stable operation, exactly matching our

model. We tested many points in the ?kp, ka? plane,

and we show one example in Fig. 8. Future work, on

a more advanced system, will test our theory on the

Timing and Control

propagation of noise17and further validate the con-

trol model.

4.

We operated what we believe to be the first adaptive

optics control system utilizing Fourier phase extrac-

tion. In one realization we used a full Fourier trans-

form of the Hartmann spots, and extricated the

phases (namely the wavefront slopes) from the or-

thogonal side lobes as control inputs. We were able to

close the adaptive optics loop in acceptable time,

which can be further reduced by improved FFT algo-

rithms or hardware. Fourier modes might be better

descriptors of turbulence-created phase and of de-

formable mirrors with Cartesian or hexagonally tiled

actuators. For very large systems they should be the

natural choice. Add to that the advantage that strong

aberrations are better tolerated in Fourier analysis,

as there is no need to contain the Hartmann–Shack

spots within the designated lenslet areas. Thus the

lenslets can have longer focal lengths, and increase

the phase sensitivity. Yet another advantage lies in

taking the first side lobes: High-frequency noise is

filtered out, making the use of a hardware aperture12

redundant.

All these advantages exist also in the second real-

ization, where we demodulated the same Hartmann

grid in the image domain, and sampled it inside the

deformable mirror. These sample points were also

used to close the loop and correct the wavefront for

Summary

Fig. 8.

a screen shot of the first six modes and their convergence under mild turbulence for kp? ?0.25 and ka? ?0.4.

(Color online) Comparison between the simulation of the control system response (right panel) and the experiment (left panel),

6 APPLIED OPTICS ? Vol. 46, No. 4 ? 1 February 2007

Page 7

different turbulence conditions. Our control system

functioned as predicted and was able to follow and

immediately correct even strong cases of turbulence.

In a separate experiment16we were able to close the

loop on a living eye using the same method.

There is an advantage to sampling the phase of the

interpolated wavefront: Sample locations can be ju-

diciously chosen according to criteria other than the

given lenslet positions. For example, their density

and spread can be guided by the turbulence spectrum

and by the wind: Sampling can be denser in the up-

wind direction to improve prediction.18In ocular

adaptive optics, spherical aberration can be domi-

nant, which also calls for more sampling around the

pupil periphery. Other factors can be the aperture

shape or boundary and the positions of the actuators.

It might be advantageous to sample where the slopes

of the influence functions of the actuators are maxi-

mal and the sensitivity is better. Sampling can also

be optimized experimentally as in the SVD process.

We intend to further investigate these points since it

might be possible to minimize or even avoid the ma-

trix multiplication in the control loop altogether if the

right sampling choice is taken.

Parts of this work were supported by the Israeli

Ministry of Science, and in conjunction with the EU

Sharp-Eye network.

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