Vol. 73, No. 12/December 1983/J. Opt. Soc. Am. 1771

Optimal wave-front correction using slope measurements

Edward P. Wallner

Optical Systems Division, Itek Corporation, 10 Maguire Road, Lexington, Massachusetts 02173

- Received July 26, 1982; revised manuscript received July 17,1982

In adaptive optical systems that compensate for random wave-front disturbances, a wave front is measured andcorrections are made to bring it to the desired shape. For most systems of this type, the local wave-front slope isfirst measured, the wave front is next reconstructed from the slope, and a correction is then fitted to the recon-structed wave front. Here a more realistic model of the wave-front measurements is used than in the previous liter-ature, and wave-front estimation and correction are analyzed as a unified process rather than being treated as sepa-rate and independent processes. The optimum control law is derived for an arbitrary array of slope sensors andan arbitrary array of correctors. Application of this law is shown to produce improved results with noisy measure-ments. The residual error is shown to depend directly on the density of the slope measurements, but the sensitivityto the precise location of the measurements that was indicated in the earlier literature is not observed.

INTRODUCTION

The wave-front correcting system treated here consists of anumber of sensors, each of which measures the weighted av-erage of wave-front slope over some region; a number of cor-rector actuators, each of which adds a corrective function tothe wave-front phase; and a control law relating actuatordisplacements to sensor measurements. The correction willbe imperfect because of the finite size and number of sensors,measurement noise (including photon noise in the sensors),the finite number of degrees of freedom in the corrector(whether distributed by zones or by modes), and possibledeficiencies in the control law.

The approach taken to the control law in the literature hasbeen first to reconstruct an estimate of the wave front fromthe slope data and then to fit the actuator functions to theestimate.1 A number of papers have dealt with the problemof wave-front reconstruction from noisy data.2 -10 Thesepapers generally treated the wave front as an array of discretepoints and the measurements as phase differences betweenpairs of points, and only Ref. 10 included the effect of inte-grating wave-front slope over a finite area. The least-mean-squares estimate of the wave-front phase based on noisymeasurements was derived, but no use was made in the pro-cess of any knowledge of the statistics of the wave front beingmeasured. The latter information is useful even in the caseof small measurement errors since there are always wave-frontmodes to which the sensor array is blind.

The problem of fitting a corrector to a wave front has alsobeen analyzed by several authors, with particular applicationto wave fronts with the Kolmogorov spectrum typical of at-mospheric turbulence.11-'5

In practice, all these problems are interdependent and thederivation of the optimal control law requires a unifiedtreatment of the overall system. The present paper gives sucha treatment for an arbitrary array of wave-front sensors, anarbitrary array of corrector functions, and an arbitrary dis-tribution of uncorrected wave fronts. The control law thatminimizes the mean-square residual error and the error itselfare evaluated.

SYSTEM DESCRIPTION AND ASSUMPTIONS

The overall correction system may be represented as a numberof sensors that measure wave-front slope in the plane of anaperture, a corrector with a number of actuators acting in thesame plane, and a control law connecting sensors to actuators.The aperture of the optical system can be described by aweighting function proportional to the intensity of light at thewave front. It is convenient to normalize this function sothat

L: dxWA(x) = 1, (1)

where x is the two-dimensional vector position in the apertureplane, WA (x) is the aperture weighting function, and S O' dxindicates integration over the entire aperture plane.

The uncorrected wave-front phase or optical path differenceis described by the function (x). Because any uniform phaseover the aperture does not affect optical performance, we canequivalently deal with the phase with aperture average re-moved, i(x). For this function

(2)E dx WA(X 0(X = 0,

where

0(x) = (x) - s- dx' WA(X') {(X')

and 4(x) is the input wave-front phase.The output of each wave-front sensor is taken as the

weighted average of the wave-front slope over some region ofthe aperture, which approximates the output of real wave-front slope sensors for small wave-front deviations and a fixedwave-front intensity pattern. The noise on the measurementsis represented by a random additive function of position,which permits a convenient treatment of photon noise. Thesignal out of the nth sensor is then

sn = dx Ws8 (x)[Os (X) + v(x)], (3)

0030-3941/83/121771-06$01.00 © 1983 Optical Society of America

Edward P. Wallner

1772 J. Opt. Soc. Am./Vol. 73, No. 12/December 1983

where

Sn is the signal from the nth sensor,Wen (x) is the weighting function for the nth sensor,q5 (x) is the slope of wave front in the direction of sensitivity

of nth sensor, andv (x) is the noise distribution over the aperture.

The signal can be expressed in terms of the wave-front phaseby integrating Eq. (3) by parts:

Sn = dx[-W-n(x) 0x) + W, (x)v(x)I, (4)

where Wsn(x) is the derivative of Wsn(x) in the direction ofslope sensitivity of the nth sensor. In Eq. (4) use has beenmade of the fact that Wsn (x) goes to zero at infinity.

The noise is assumed to be unbiased and uncorrelated withthe wave-front phase:

= E E E TjnMj'n' SnSn ')i j' n n'

x f dx WA (x)rj(x)rj (X)

-2 L_ L Mjn r dxWA(x~rj(x)(SnO(X))i n -

+ 3' dxWA(x)(0 2(x)). (11)

These expressions may be simplified by defining additionalterms. The integral involving the product of two sensorfunctions is defined as the matrix Snn':

Snn'= (Snsns)

= f dx' 4' dx-[Wsn(X')WS.(X')(O(X')Xx))

+ Wsn(X) WVn'(X/")(V(x')V(x'))]. (12)

(v(x)) = (v(x)o(x')) = 0. (5)

The control law for the system generates a command to eachactuator of the wave-front corrector based on all the sensoroutputs. For a linear control law, the command to the jthactuator is

cj = F Mjnsnx (6)

where cj is the command to the jth actuator and Mjn is theweighting of nth sensor signal in jth actuator command.

Finally, the responses of the actuators are assumed tocombine linearly to form the total wave-front correction,

W(x = YE cjryjx, (7)

where 5(x) is the total wave-front correction and rj(x) is theresponse of the jth actuator to a unit command.

EVALUATION OF MEAN-SQUARE RESIDUALERROR

The residual wave-front error may be expressed by using Eq.(7) and expanded by using Eqs. (4) and (6):

e(x) = 0(X) - OWx

= Z rj(x) E MjnSn - 0 .(x). (8)i n

The mean-square residual error at x is formed by using Eq.(8) and assuming that the sensor noise and the wave-frontphase are uncorrelated:

(e2(x)) = Z E E F rj(x)rj'(x)MjnMj'n'(snsn')i j/ n n

-2 E rj(x)Mjn(sn. (x)) + (02(x)),i n

(9)

The integral involving the product of two actuator responsefunctions is defined as the matrix Rjj':

Sjj = ' dxWA(x)rj(x)rj'(x). (13)

The integral involving products of sensor and actuator re-sponse functions is defined as the matrix Aj,:

Ain = Jf dxWA (x)rj(x)(sn b(x))

= dx J' dx'WA(x)rj(x)Wsn(x')(0(x)0(x')).

(14)

The average mean-square uncorrected error is defined as(E 2):

(e)= 3' dxWA(x)(02 (x)). (15)

Substituting these definitions into Eq. (11) gives the simpli-fied form

(e2) = E E E MjnMj'n'Snn'Rjj' - 2 E i MjnAjn + (e2)-jj' n n' j n

(16)

MINIMIZATION OF RESIDUAL ERROR

Equation (16) gives the average mean-square residual errorfor arbitrary weights in the control matrix Min. To optimizesystem performance we wish to choose the weights that min-imize the error.

Differentiating Eq. (16) with respect to element Mjyn' andequating to zero yields

d( ') -2 E Z MjnSnn'Rjj' -2Aj'ndM1 'n' i n~

= 0. (17)

where (f) is the expected value of f.The overall measure of performance is the expected

mean-square error averaged over the aperture:

(E) = S dxWA(x)(C2(x)) (10)

By letting repeated indices indicate summation and makinguse of the symmetry of Rjj1 , Eq. (17) may be written moreconcisely as

Rj'jMjnSnn' = Aj1 ,,. (18)

Edward P. Wallner

Vol. 73, No. 12/December 1983/J. Opt. Soc. Am. 1773

The solution to this equation is the minimizing control matrixMn-

Min = RVAj'n'Sn .n- (19)

The R matrix will not be singular unless there are redundantactuators that allow the same correction to be made withdifferent combinations of actuator commands. In that casethe generalized inverse of the matrix may be used at the ex-pense of additional computation, or the redundant functionscan be eliminated with equivalent results.

Alternatively, a cost function depending on the actuatorcommands can be added to the quantity being minimized. Ifthe cost rises more than linearly with the actuator command,the correction will be distributed over the redundant modes,and R will not be singular. The S matrix will not be singular,even with redundant sensors, if there is independent noise oneach measurement, as will be the case in any real system.

The value of the minimum average mean-square residualerror, (eJ), is found by substituting Eq. (19) in Eq. (16):

(e2) = (E2) -Mj*Aj.

= (c2-R R]1Aj'n'STAj

This expression, with Eqs. (12)-(14), gives the optimal per-formance for an arbitrary system and arbitrary wave-frontstatistics.

The overall performance will depend on the magnitude ofthe sensor noise relative to the expected signal that is due tofluctuations of the input wave front. If the noises in all sen-sors, ad, are equal and are large compared with the signalvariance, the matrix S` approaches Ila', the correction termin Eq. (20) becomes small, and the residual error approachesthe uncorrected error.

For small measurement errors, the residual error will ap-proach a limit set by the inability of the finite arrays of sensorsand actuators to fit the wave front perfectly.

In the transition region where the residual error is smallcompared with the uncorrected error but large compared withthe fitting error, the performance will depend directly on themeasurement errors and the error propagation of the controlmatrix.

REPRESENTATION BY STRUCTUREFUNCTION

In the case of atmospheric turbulence, the statistics of theinput wave front are defined in terms of a structure func-tion,

D(x, x') = ([4(X) - i(X')]2). (21)

The correlation of phases with averages removed required inthe computation of S, A, and (ed) can be derived directly byusing Eq. (21). The average removed phase is

OW(x) = K(x) - f dx"WA(x")P(x")

= f dx"WA(x")W'(x) - (x")]. (22)

(20)

The correlation required is

(0(x)0(x')) = 5 dx" 5 dx"''WA(xI)WA (x')

X ([44x) - (x")][(x') - (x')])

= -l/2 5 dx" 5 dx "'WA (x" ) WA (x"')

X [D(x, X') - D(X, X"') - D(x", x')+ D(x", x"')]

= -1/2 D(x, x') + g(x) + g(x') - a, (2

where

g(x) = 1/2 5 dx"WA(x")D(x, x"), (

a = 1/2 J dx" dx"' WA (x") WA (x"' )D (x", X"')f-. -0

23)

24)

(25)

Substituting into Eq. (15),

(e2) = J dxWA(x)(02(x))

= a. (26)

In the evaluation of S and A, all terms on the right-hand sideof Eq. (23) except the first can be seen to integrate to zero,leaving

Snn'= r dx' | dx"[-l/2WSn(x')W'sn'(x")D(x', x")

+ Wsnm(X) WSn'(X)(V(X') VV(X "]) ) ], (27)

Ajn =-1/2 dx 5 dx'WA(x)rj(x) WMM(x)D(x, x').

(28)

Equations (26)-(28) may be used with Eqs. (19) and (20) toexpress the optimal control law and evaluate its performancein terms of the structure function of the wave front beingcorrected.

PHOTON NOISE

In many systems of interest, the limit on performance is setby the noise that is due to photon statistics in the wave-frontmeasurements. The photon noise for a single wave front canbe represented by making v (x) a spatially white-noise func-tion. The correlation of the noise on two wave-front mea-surements depends in detail on the configuration of thewave-front sensor used. If the measurements are made onseparate wave fronts, either in sequence or separated by abeam splitter, the noises will be uncorrelated, even thoughthey refer to the same area in the aperture.

Measurements of orthogonal slopes made simultaneouslyon the same wave front will also have uncorrelated noises. Toreflect these conditions the correlation of the noise functionsmay be written as

(v(x)v(x')) = knn'aUno(x-x'), (29)

Edward P. Wallner

1774 J. Opt. Soc. Am./Vol. 73, No. 12/December 1983

where

a2 is the photon noise density,knn' =n ln-' for n and n' measured simultaneously on the

same area of the same wave front butknn' is 0 otherwise,In is a unit vector in the direction of slope sensitivity of the

nth measurement, andI(x) is the Dirac delta function.

The noise term in Eq. 10 or 27 may then be written as

J dx' E_ dx"Wsn(x') Wsn(x")(V(x')V(X"))

= knn'Uan ldX'Wsn(x')Wsn'(x'). (30)

In most cases the basic measurements will be disjoint, andknn' will be the Kronecker delta if each measurement is treatedseparately. It may be convenient to combine a number of thebasic measurements before processing in order to reduce theorder of the matrices involved in the computation of thecontrol law. In this case the more general form of knn' wouldbe needed.

CLOSED-LOOP RECONSTRUCTION

Most wave-front correction systems that have been imple-mented up to the present time have used a feedback-controlloop in which the actuators are driven to null the wave-frontsensor signals generated by the input wave front. Such asystem is analyzed here for comparison with the optimumsystem derived above.

If a set of commands cj is sent to the actuators, the phasecorrection created will be given by Eq. (7). Letting repeatedindices indicate summation, we may write this as

0(x) = cjrj(x). (31)

If this correction is subtracted from the input wave front, thesignal from the nth sensor as given in Eq. (4) becomes

Sn' = Sn - J' dx[-WSn(x)0(x)]

= Sn - f dx[-WSn(x)cjrj(x)]- (32)

The integral on the right-hand side of Eq. (32) for a unit ac-tuator command defines the matrix Pnj, which relates the nthsensor output to the jth actuator input:

Pnj = ef dx[-Wsn(x)rj(x)].

Substituting in Eq. (32), we find that

Sn = Sn - Pnjcj.

(33)

(34)In the closed-loop operation considered here, the command

vector c; is chosen to drive the corrected signal vector sn tozero. This requires that

= PI.sci nisn, (35)

where PI1 is the generalized inverse of Pnj. The generalizedinverse minimizes the sum of the squares of the residual sig-

Edward P. Wallner

nals and sets to zero any actuator modes not detected by thesensors.

The expected mean-square error averaged over the apertureis found by substituting the control matrix Pjn for Mjn in Eq.(16):

(eC) =(E)-2P'njAjn + Pn+jPn+jJSnnRjJ.i (36)For a system with a given configuration of sensors and actu-ators, Eq. (36) allows the performance using simple closed-loop control to be compared with that obtained using optimalcontrol.

COMPUTATIONAL RESULTS

The theory derived here has been applied to a system con-sisting of a square aperture with uniform illumination and asquare array of identical actuators with Gaussian responsefunctions. The array is positioned with an actuator at eachcorner of the aperture, and the ratio of the side of the apertureto the actuator spacing, N, is varied in the computations. Theactuator response falls to Ile at the adjacent actuator.

The wave front being corrected is assumed to have theKolmogorov structure function typifying atmospheric tur-bulence:

D(Y, X') = 6.88391(x - ')/roI 5/3, (37)where D(x, x') is the phase-structure function, rad2 , and rois Fried's coherence length, m. The mean-square uncorrectedphase error over the full aperture for this structure functionis

(E0) = 1.3103(A/ro)5/3, (38)

where (,E2) is the uncorrected phase error, rad2 , and A is a sideof aperture m.

The wave-front sensor investigated averages the wave-frontslope with uniform weight over a square subaperture of di-mensions equal to the actuator spacing. Two different con-figurations of such sensors are considered.

In the first configuration, shown in Fig. 1, X and Y slopesare measured in displaced subapertures. This configurationis typically used in the shearing interferometer sensor and wastreated by Rimmer 2 and Hudgin.4 The mean-square slopemeasured over a square subaperture of side L for the Kol-mogorov structure function is

a2= 6.4051L-/3rf5/13, (39)

where asc is the mean-square slope signal, rad2 m-2 , and L =A/N is a subaperture dimension m. The noise on this mea-surement is assumed to be pure photon noise for which the

Fig. 1. Displaced subaperture wave-front sensor configuration.

Vol. 73, No. 12/December 1983/J. Opt. Soc. Am. 1775

mean-square slope error is inversely proportional to thenumber of photons used in the measurements.

In this configuration, a sensor subaperture located at anedge of the aperture parallel to the slope being measured willhave only half of its area illuminated and will therefore havetwice the mean-square slope error of a fully illuminatedsubaperture.

For a uniform illumination yielding p photons per squaremeter, the noise will be

2 K KanM pA=

where

ao is the mean-square slope noise, rad2 m-2,K is a constant, rad2 m-2 ,M is the number of photoelectrons used,p is the photoelectron density m-2 , andA, is the subaperture area m2

.

Fig. 2. Performancetion.

0.5

Figure 2 shows the residual error for this configuration as afunction of photoelectron density and actuator spacing. Theaperture size A and ro, where each is assumed to be 1 m, anda photon noise constant of 1204.6 were used. (This constantwould apply to a shearing interferometer with optimum shearimaging a disk of 5-Arad diameter in light of 0.55-Am wave-length.)

The three regions of operation are easily distinguished in-Fig. 2. For densities much less than 100, the measurementnoise is so large that the a priori estimate of zero phase is moreaccurate than the measurement. For high densities, the erroris determined by actuator and sensor spacing and is againindependent of irradiance. This fitting error varies as (AlNro)5 /3 when N is sufficiently large that edge effects may beneglected.

In the transition region, where the photon error is smallcompared with the uncorrected error but large compared withthe fitting error, the performance depends primarily onphotoelectron density and depends relatively weakly on ac-tuator spacing, in agreement with the results of Fried3 andHudgin.4

Figure 3 compares the performance of the optimal recon-structor with that of the closed-loop reconstructor for N = 4.At low photoelectron densities, the closed-loop reconstructorfaithfully follows the noise input, leading to large errors. Athigh densities, the performance is close to that of the opti-mum, but the actuators are not fitted to the wave-front quiteas well as possible. The optimal reconstructor thus improvesperformance at all illumination levels.

The second sensor configuration considered is shown inFig. 4. Here the x and y slopes are measured in a commonsubaperture that has an actuator at each corner. This con-figuration characterizes the Hartmann sensor and was ana-lyzed by Fried.3

The performance of the two configurations is compared inFig. 5. The top dashed curve is for a single subaperture withfour actuators, one at each corner. The fitting error in thiscase is slightly larger than for the displaced subapertureconfiguration with the same subaperture size, which uses twiceas many slope measurements to link the four actuators. Forlarger numbers of actuators for which edge effects are reduced,

of displaced subaperture sensor configura-

0.21-

0.1

0.05 r0.02 -

101 10o 103 10 lo, lo0

Photoimalrn Dcnsdty . ctuor

Fig. 3. Comparison of optimal and closed-loop reconstructors.

Fig. 4. Common subaperture wave-front sensor configuration.

N - A/L

l

32

100

l AA

).2- - - ---- -

t.1-

.05 * Displ..ed subaprb-r -onfigurati-n

.02 -- Connen subapert... configuration,

., I I I - - . I. A h_, . I C 1 #

0\ lo, 103 jo4

Photiiisonronfunalvaption

Fig. 5. Comparison of sensor configurations.

the performance of the two configurations is virtually indis-tinguishable. The difference in performance between theseconfigurations as given by Fried3 and Hudgin4 does not appearin these results.

10 ~Displaced senso

2 - confi g-rati on ]

.' = 1,20D4.6/(,2

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= A/L

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Edward P. Wallner

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1776 J. Opt. Soc. Am./Vol. 73, No. 12/December 1983

As was discussed by Flerrmann,8 when the wave-front re-construction algorithm based on the average of the phases atthe four nearest neighbors plus the average of the four con-necting phase differences is used with the common subap-erture configuration, two interleaved arrays of points areproduced with an arbitrary phase difference between them.Fundamentally, the approach given here resolves the ambi-guity by using the statistics of the wave fronts being estimatedto imply the unobserved mode.

These results also contradict those of Southwell, 9 who showsa mean-square error for the common subaperture configura-tion approximately twice that of the displaced subapertureconfiguration. In his model, slopes were measured at discretepoints with equal noise on all measurements, and the wave-front error was evaluated only over an array of discretepoints.

With more-realistic sensors that average the slopes overfinite subapertures, with a performance measure that evalu-ates the error over all the aperture, and with use of the sta-tistics of the wave-front phase in the estimation process, theextreme sensitivity of performance to the precise sensorconfiguration reported by Southwell is not observed.

The same conclusion was reached in a separate paper' 6 thatdealt with the same sensor configurations but allowed actuatorresponse functions to be chosen arbitrarily. Again, except foredge effects, performance was the same for the two configu-rations treated here as well as the configuration consideredby Southwell in which X and Y slopes are measured in com-mon subapertures centered at the actuator locations.

The results of the performance computations indicate thatappreciable improvement in performance can be obtained byusing the optimal reconstructor and that performance de-pends on sensor dimensions but not significantly on the po-sition of the X or Y sensor array with respect to the actuatorarray.

Edward P. Wallner

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4. R. H. Hudgin, "Wave-front reconstruction for compensatedimaging," J. Opt. Soc. Am. 67, 375-377 (1977).

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12. B. L. McGlamery and P. E. Silva (Visibility Laboratory, ScrippsInstitution of Oceanography, University of California, San Diego,California 92152), "A preliminary comparison of wave-front errormeasurement devices for use in compensated imaging systems"(unpublished report, March 1975).

13. D. L. Fried, "Required number of degrees-of-freedom for anadaptive optics system," Optical Sciences Consultants Rep.TR-191 (October 1975).

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15. J. Y. Wang and J. K. Markey, "Modal compensation of atmo-spheric turbulence phase distortion," J. Opt. Soc. Am. 68, 78-88(1978).

16. E. P. Wallner, "Comparison of wave-front sensor configurationsusing optimal reconstruction and correction," Proc. Soc. Photo-Opt. Instrum. Eng. 351, 42-53 (1982).