40. Nosach, V. Popovichev, V. Ragul'skii, and F. Faisullov, JETP Lett.16, 435 (1972).

"V. Wang and C. R. Giuliano, Opt. Lett. 2, 4 (1978).riR. W. Hollwarth, J. Opt. Soc. Am. 67, 7 (1977).7 A. Yariv and D. M. Pepper, Opt. Lett. 1, 16 (1978).8 D. M. Bloom and G. C. Bjorklund, Appl. Phys. Lett. 31, 592

(1977).9 S. M. Jensen and R. W. Hellwarth, Appl. Phys. Lett. 32, 166

(1978).10D. M. Bloom, P. E. Liao, and N. P. Economou, Opt. Lett. 2, 58

(1978).lip. F. Liao, D. M. Bloom, and N. P. Economou, Appl. Phys. Lett. 32,

813 (1978).

12 D. M. Pepper, D. Fekete, and A. Yariv, Appl. Phys. Lett. 33, 41(1978).

13R. L. Abrams and R. C. Lind, Opt. Lett. 2, 94 (1978).14p P. Iano and D. M. Bloom, Opt. Lett. 3, 4 (1978).15 E. E. Bergmann, I. J. Bigio, B. J. Feldman, and R. A. Fisher, Opt.

Lett. 3, 82 (1978).16While resonant enhancement can be used to increase the cubic

nonlinearity (four-wave mixing), it tends to be associated withabsorption processes that are absent in the three-wave case.

17C. V. Heer and P. F. McManamon, Opt. Commun. 23, 49 (1977).18N. C. Griffen and C. V. Heer, Appl. Phys. Lett. 33, 865 (1978).19F. Zernike and J. E. Midwinter, Applied Nonlinear Optics (Wiley,

New York, 1973), p. 42.

Modal wave-front estimation from phase derivativemeasurements

Ronald CubalchiniHughes Aircraft Company, Culver City, California 90230

(Received 6 November 1978)

Modal estimation of wave-front phase from phase derivatives is discussed. It is shown that it isdesirable to minimize the number of modes estimated and the number of measurements used tomaintain the quality of the estimates of low-order modes. It is also shown that mode cross couplingoccurs when one tries to estimate modes higher than astigmatism.

INTRODUCTION

Certain wave-front sensors (e.g., shearing interferometers,Hartmann sensors) measure local tilts at several locationsdistributed over the sensor entrance pupil. In applicationsit is often necessary to estimate the wave-front phase at thepupil using this measured data. Two methods by which thewave-front phase may be represented are: (i) zonal estima-tion, and (ii) modal estimation. In either case, a least squaresestimation algorithm can be used to construct the wave-frontphase.

Zonal estimation of the wave-front phase and the problemof propagating measurement variances to the zonal estimatesare discussed by Rimmerl and Fried.2 In this paper modalestimation of the wave-front phase and the propagation ofmeasurement variances to the modal estimates are dis-cussed.

The definition of terms and assumptions made concerningthe problem are presented in the following section. The leastsquares estimator is derived in Sec. II. The propagation ofmeasurement variances to the estimated modal coefficientsis derived in Sec. III. Several examples are presented in Sec.IV. The examples shown have been chosen to illustrate thecare that must be taken to preserve the quality of the wave-front phase estimates. Several observations and conclusionsare presented in Sec. V.

1. DEFINITIONS AND ASSUMPTIONS

The primary assumption concerning the wave-front phase0 (X, Y) at the sensor pupil is that it can be represented by an

infinite sum expansion of orthogonal functions. In the ex-amples presented, the set of orthogonal functions used areZernike polynomials. However, the least squares estimatorderived in the succeeding section is correct for any set of or-thogonal functions defined over the pupil. Thus, the wave-front phase is given by

(1)0 (X, Y) = E aKZK (X, Y).K

The wave-front phase is presented in rectangular coordi-nates, rather than polar coordinates, to accommodate theusual design of wave-front sensors that measure tilt in thesubapertures in the X and Y directions, rather than p and 0directions.

It is assumed that the wave-front sensor measures the X-directional derivative of the phase, P, or the Y-directionalderivative of the phase, Q, or both at a particular location ina subaperture. That is

PX(XY)i = ,o X Y)i + v' (2)

and/or

Q(XY)i = 0(XY), + pi, (3)

where (X,Y)i are the coordinates of the ith measurement andPi and Ai are the measurement noise associated with thesamples. No real sensor measures the directional derivativeat a point. In a real sensor the aperture is partitioned intosubapertures, and the measurement is actually an average of

972 J. Opt. Soc. Am., Vol. 69, No. 7, July 1979 � 1979 Optical Society of America 9720030-3941/79/070972-06$00.50

the directional derivative (tilt) over the subaperture associatedwith the measurement. Since the wave-front phase is analyticover the subaperture, a noise-free measurement would rep-resent the phase at some point in the subaperture (by themean-value theorem). However, the measurement is ascribedto be the tilt at the center of the subaperture. Thus, an in-strumental error will exist in the estimation of the wave front,since the measurements are not the tilts at the sampling lo-cations. No attempt is made to bound this error source;however, its magnitude is clearly a function of the spatialfrequency content of the wave-front phase and the numberof subapertures across the diameter of the pupil. To minimizethe instrumental error, one maximizes the number of subap-ertures. In this paper, this instrumental error is assumed tobe zero.

Some wavefront sensors (e.g., shearing interferometers) donot measure the wavefront phase directional derivative.Instead, they measure a finite phase difference. 3 In this paperit is assumed for these sensors that

and

60MXY) = K1[k(X + SY) - O(XY)]

y )= K2[ (XY+ S) - (S,Y)]

(4)

(5)

where K1 and K2 are proportionality constants and S is theshear.

11. LEAST SQUARES ESTIMATOR

Consider the measurement vector P, where

P = IP(X,Y)1,P(X,Y)2, . . - P(XY)K, Q(X,Y)1,Q(X,Y)2 . . Q(X,Y)LIT. (6)

The purpose of the estimator is to select a set of constantsrepresented by the vector A based upon the measurementvector P,

The estimate of the wave-front phase is given by

I$(X,Y) = E diZi(XY).

i=2(8)

The term d1 Z1 (X,Y) is a constant and can be omitted. Thecriterion for choosing A is to minimize error squared betweenthe estimates and the measurements. The only differencehere with respect to the normal least squares estimationproblem is that the measurements correspond to the X andY directional derivatives of the estimate. Thus the minimi-zation functional contains the derivatives of the estimate.The directional derivatives of the estimate are simply givenby

and

b1 I ? J Z1(X,Y)ox= E di o

axl~ I AZ (Xy)by i2 y

(9)

(10)

Thus the least squares minimization functional is given by

F E(P(X,Y)k - I d, 'Zi(XvY)k)2k=1 i=2 (X

(11)

Now F is minimized with respect to A. This results in thefollowing set of equations:

E (P(XY)k iZ v )k) ± E (QXi y Zi(XY)i)k=1 o X 1=1 a

E lej ( E bZj M Y)k bZi M Y)k

- EZj((XkY)Y Z Zi(XY)k I1=1 C by

where j = 2,3,. . , I. This expression can be writteicompactly as

DP = EA,

(12)

n more

A = 162,d3 . * *, dl}.

(13)

(7) where

/Z2(X,Y)l bZ2(X,Y)2 (Z2(X,Y)K bZ2(X,Y)l C)Z2(X, Y)L_)X ax X by by

D ) = Z(X, Y)l (Z3(X, Y)2 b)Z3(X, Y)K bZ3(X, Y)l Z(X )

_)X ... XbYb

(ZA XY)l (ZI (X, Y)2 _)ZI (X Y)K _)ZI (X Y) 1 -)ZI (X, Y)L\ x ax ox Cy bY /

and

E = DDT.

Hence the estimator vector can be found explicitly,

A = E- 1DP = (DDT)-'DP.

(15)

(16)

The least squares estimator of each weighting coefficient isa linear combination of the measurements. In turn, theweighting coefficients in the estimator are determined by the

(14)

directional derivatives of the basis functions evaluated at thesampling locations.

III. PROPAGATION OF VARIANCES

Linear System Theory can now easily be applied to the es-timator to provide the propagation of variance equation.4

The covariance matrix for the estimated coefficients is

CA = (DDT)-1DCpDT(DDT)-1, (17)

973 J. Opt. Soc. Am., Vol. 69, No. 7, July 1979

,I �Q(XY)j _ I _ Zi(XY)112+ F_ Y_ a ' by ) -

1=1 i=2

Ronald Cubalchini 973

where

(2622

CA °0 d3d2

2

2 2aa2d3 ...d2d \2~ 2d 3d3 ... * 6 2d

W13 .. WI 83d

E = o 1|

so that

(18)

and

/'2 O2 2/P(X, Y)iP(X, Y)1 P(X, Y)IP(X, Y)2 * -. -P(X, Y)IQ(X, Y)L

Cp= P(X,Y)2P(X,Y)1 OP(X,Y)2P(X,Y)2 ..... 7 P(XY)2Q(XY)L

2 2 a...l2)\Q(X, Y)LP(X, Y)l (Q(X, Y)LP(X, Y)2 * Q(X, Y)LQ(X, Y)L

(19)

Note that if the noises for all measurements are independentand have the same variance, the propagation of varianceequation becomes

CA = (DDT)Y 1 4U = E' (20)

where

= SP(XY)1lP(XY)l = X, Y)2(P(X9 Y)2=v

2

= .=°Q(X, Y)LQ(X, p)L-

The expression in Eq. (20) provides the basis for the examplesthat follow.

IV. EXAMPLES

In the examples that follow, the set of orthogonal functionsthat will be used is given in Table I. This is the standard setof Zernike polynomials except for the normalizing parameters.The choice of normalizing coefficients just scales the estima-tion vector. Hence standard normalization is not required.

The motivation behind the selection of these polynomialsfor use in the examples is two-fold: (i) they are widely usedin the field of optics; and (ii) the low-order modes correspondto simple shapes that can be generated with reasonable ac-curacy by deformable mirrors in adaptive optics systems.Thus, practical applications of the estimator described abovewill most likely involve use of the Zernike polynomials.

A. Two measurements, two modesIn this example, assume that the X-directional derivative

and the Y-directional derivative are sampled at the origin. Inthis case

2 2 20d2d2 = dTad3 = a7p. (21)

This is, of course, the expected result. Since the first twoZernike modes represent tilts, this example simply describesthe propagation of variances in a tracking or angle of arrivalsensor. The variance in the tilt estimate is the same as thevariance in the phase measurements in the aperture. In thecase of a tracker the subaperture is the full aperture. So localtilts (i.e., phase measurements) and global tilts (i.e., d2 anda3) are the same.

B. Four measurements. two modesIn this example, the X-directional derivatives will be

sampled at ('/2, 0), (-'/2, 0) and the Y-directional derivativeswill be sampled at (0, '/2), (0, -1/2). In this case

E st2 00 2'

so that

a2d2 = 23d3 = P/2 (22)

Again the result is as expected. An inspection of the esti-mation equation, Eq. (16), indicates that the estimate d2 (i.e.,the X-directional tilt over the aperture) depends only on theX-directional derivative measurements. Likewise, the esti-mate d3 (i.e., the Y-directional tilt over the aperture) dependsonly on the Y-directional measurements. Thus, the estimatesOf Q2 and 63 each depend on two measurements. The varianceof these two estimates is decreased by the number of mea-surements that affect each of them. In a real system, however,there is a pitfall to be avoided. If the total sensor aperturearea is held constant, as the number of measurements in-creases, the variance ap of each measurement increases as isshown in the appendix. In a sensor whose field-of-view isdiffraction limited

P = N2 TOTAL, (23)

where N is the number of subapertures into which the aper-ture is divided, and uTOTAL is the variance of the tilt mea-surement that would have been made if the area of the entireaperture were used for making the measurement.5

In a sensor whose field-of-view is not limited by diffraction

P = NTOTAL- (24)

TABLE 1. Zernike polynomials and directional derivatives.

Mode Polynomial Polynomial X Ynumber (cylindrical) (cartesian) derivative derivative

1 1.0 1.0 0.0 0 02 r cosO X 1.0 0.03 rsinO Y 0.0 1.04 2r 2 -1 2X2 + 2Y 2 -1 4X 4Y5 r 2cos2O X2 - y2 2X -2Y6 r 2 sin 20 2XY 2Y 2X7 (3r3 - 2r) cosO 3X(X 2 + y2) - 6X 3[3X2 + Y 2 - 2] 6XY8 (3r 3 - 2r) sinO 3Y(X2 + y2) - 6Y 6XY 3[X2 + 3y 2 - 2]9 r3 cos3O X(X 2 - 3Y2 ) 3(X 2 - y2) -6XY

10 r 3 sin3O Y(3X2 - y2 ) 6XY 3(X2 - y2 )

974 J. Opt. Soc. Am., Vol. 69, No. 7, July 1979 Ronald Cubalchini 974

In a sensor whose field-of-view is limited by diffraction, as thenumber of subapertures increases the variance increases as

da2d 2 = ai3d3 = 2N OTAL- (25)

In a sensor whose field-of-view is not limited by diffraction,the variance of the tilt measurement remains the same (i.e.independent of N)

ad262 = ad3d3 = 2 TOTAL- (26)

Of course, 0TOTAL for a diffraction-limited field-of-view sensoris smaller than aTOTAL for a non-diffraction-limited field-of-view sensor (everything else being equal) since the non-diffraction-limited sensor "sees" more background photonsdue to its larger field-of-view.

C. Four measurements, three modesIn this example, the sampling geometry is identical to that

in Example B; however, one more mode will be extracted fromthe measurements. In this case

2 0 0

E= 0 2 00 0 16

so that

d22 = °a3 d3 = t7P/2,

pled at (-%,2/1/2), (0,1/2), (2/3,1/2), (-.2/3,-1/2), (0,-1/2), (2/3,-1/2).The Y-directional derivatives will be sampled at (1/2,-2/3),(1/2,0), (1/2A2/3), (-1/2,-2/3), (-1/2,0), (-1/2,2/a). In this case

-6 0 0 0 00 6 0 0 0

E= 0 0- 0 01,9

1280 0 0 12 0

-00 0 0 12

so that

and

2q = 26 _= 2 62d2 3a3d3 P/

aq24, = 9aP/512 - 4/56.89,

a2q5a5 = 9a2/128 - a/14.22,

(32)

(33)

(34)

2C~~6= cTP/12

(35)

If the sampling locations are changed so that they lie equal-ly distributed on a circle of radius one half (i.e., X-directionalderivatives sampled at (-v/'/4,'/ 4 ), (0,/2), (X-/4,1/4 ), (-V3-/4- 1/4) (0,-1/2), (V3-/4,-114 ) and Y-directional derivatives

(27) sampled at (-1/2,0), (-1/4,V/-/4), (Q/4,V-3/4), (1/2,0), (Q/4 ,-V-3/4),(-1/4,-V/-/4), then the E matrix is modified to the fol-lowing

a2 4 = aP/16. (28)

The significance of this example is that the variance of the tiltmodes is not changed by the additional estimation of the focusmode.

D. Eight measurements, five modesIn this example, both X- and Y-directional derivatives will

be sampled at (1/2,1/2), (-1/2,1/2), ('/2, -1/2), and (-1/2,-1/2). Inthis case

~40 0 0 0-0 4 0 0 0

E= 0 0 32 0 00 0 0 8 00 0 0 0 8

so that

d2d2 2 d3 = /4,

a04d4 = 7p/32,

°2 = 2a5605 = d60 6 = C /8

(29)

(30)

(31)

-6 00 6

E = 0O 0

LO O

002400

00060

010

0 ,061

so that

2 ya62 62 = =P/6,

Ud4d4 = aP/24,

2 ? 2aa5 d5 =a 2 6 66 = a'P/6.

and

(36)

(37)

(38)

If the sampling locations are changed so that they lie equal-ly distributed on a circle of radius two thirds (i.e., X-direc-tional derivatives sampled at (-/v'/3,1/ 3 ), (0,2/3), (\/-/3,1/ 3 ),(-V/33,-113), (0,-2/), (V3/13,-'/3 ) and Y-directional deriva-tives sampled at (-%2/,0), (-1/3,V/3), (1/3 ,\3/3), (2/30),(1/3 ,-V/-13), (-1/3,--¶/3/3), then the E matrix is modified tothe following:

Again the variance of the tilt is as expected. Note that thevariance of the focus is behaving in the same way as the tilt.That is, for a diffraction-limited field-of-view sensor, as thenumber of subapertures is increased (but maintaining thesame total collecting area), the variance of the focus mea-surement increases. In the non-diffraction-limited field-of-view case, the variance is independent of the number ofsubapertures.

E. Twelve measurements, five modesIn this example, the X-directional derivatives will be sam-

-6 0 0 00 6 0 0

0 0 384 09

0 0 0 323

0 0 0 0

00

0

0

32

3-

so that

aU2d2 =` 33 3 =363 /6, (39)

975 J. Opt. Soc. Am., Vol. 69, No. 7, July 1979

-

Ronald Cubalchini 975

°d4d4 = (9oP/384) (ap/42.67),

a25d5 = dO6a6 = 3a6/32 -ay/10.67.

(40)

(41)

This set of three different sampling geometries serve to il-lustrate that the estimation of tilt is unaffected by the locationof the sampling points. However, for the higher-order terms,the estimation variance decreases as one moves the samplingpoints to the edges of the aperture. Also note that Eqs. (34)and (35) indicate that when the geometry of sampling pointsdoes not contain the same symmetry as the modes to be esti-mated, the propagation of variances will affect differently twomodes that occur in quadrature.

F. Twelve measurements, nine modesIn this example, the X-directional derivatives will be

sampled at (-2/3,1/2), (0,1/2), (2/3,1/2), (-2/ 3 ,-1/ 2 ), (0,-1/2), and(2h,- 1/2). The Y-directional derivatives will be sampled at(1/2/3), (1/2,0), (1/2,2/3), (-1/2,-2/3), (-1/2,0), and (-1/2,2/3). Inthis case

-0.3590000

0.0760

0.0290

00.426

000

00.096

00.077

00

0.018000000

000

0.07000000

6 0 0 0

0 6 0 0

5120 0-

9

0 0

0 0-620

462

0 -6-4

5 0

6

0 -6

0

128

62 50 --0 -0

4 6

0 0 - 0 - 54 6

0 0 0 0 0

0 - 0 09

0 0 12 0

0 0 0 6199

0 0 0 0

2650 0 0 -2

24

0 0 0 0

0

0

0

619

8

0

265

24

0

0265

24

0

1331

72

0

This implies

0000

0.0830

000

0.076000

00.030

00.015

0

00.096

00

00

0.0360

0.026

0.0290000

0.0150

0.0620

so that

ad2d2 = Oa/2.79,

ad3di3 = ap2/2.35,

a464 = ap/55.56,

d5d5 = Ur/14.29,

2q 2da6d6 = ao/12,

d7d7 = P/33.33,

ad8d8 = fp2/27.78,

dodg= =O/16.13,

adlodio = UP/13.70.

This is the first incidence of cross correlation between theestimated modes. This cross correlation between modal es-timates is mentioned by Noll6 and by Wang and Markey7 intheir work on modal compensation of turbulence. Here it isseen that the cross correlation between modes is not specifi-cally a turbulence related phenomenon. In this case, thenonorthogonality of the derivatives of the Zernike polyno-mials, which is independent of the statistics of the wave-frontphase, causes the cross correlation to occur.

The effect of the cross correlation is rather dramatic. Thiscan be seen by comparing the present example with exampleE using the first geometry of samplers. These two cases uti-lize the same measurements. The sole difference is thenumber of modes that are being estimated with the data.Note that the result of attempting to estimate the comatic

terms in the Zernike expansion degrades the variance of thetilt terms by more than a factor of 2.

The obvious question is whether the total wave-front esti-mate is better with or without the estimate of the comaticterms. Unfortunately, this, in general, cannot be answered.If one ascribes a particular noise spectrum to the wave-frontphase to be measured an answer can be given, however, as wasdone by Fried. 2

Another question is whether by proper placing of thesamplers the cross coupling can be removed. The answer isno, as long as Zernike polynomials are used. In order for thecross coupling to disappear, (Z 7 /6X and oZgl/X have to bezero at all the X-directional derivative sampling points andbZ8/b Y and oZ1o/6Y have to be zero at all Y-directional de-rivative sampling points. This condition occurs only at(d1/<V, +i/x/2) for both directional derivative samples.This provides a maximum of eight samples. If one attemptsto estimate nine modes with eight samples, the estimatormatrix E becomes singular. The cross coupling occurs be-cause the derivatives of the Zernike polynomials are not or-thogonal to one another. The crosscoupling could be elimi-nated if an orthogonal set of polynomials were used in-stead.

VI. CONCLUSIONS

The analysis presented herein shows that modal estimatesof wave-front phase by means of least squares estimation using

wave-front phase directional derivative measurements aresensitive to the number of measurements used and their ge-

976 J. Opt. Soc. Am., Vol. 69, No. 7, July 1979

0

0

265

24

0

1331

72 -

00.077

000

00.026

00.073-

E-1 =

Ronald Cubalchini 976

ometry. It has been shown based upon signal-to-noise con-siderations that increasing the number of measurements doesnot decrease the variance of the estimates. In fact, in manycases the variance of the estimates will increase. It also hasbeen shown that in order to minimize the variance of higher-order Zernike modes (at least Z4 through Z10), the samplesmust be taken as far from the center of the aperture as possi-ble. It also has been shown that when one attempts to esti-mate Zernike modes higher than astigmatism, it is done onlyat the expense of the estimation ability of the lower-ordermodes. This is true because of the nonorthogonality of thederivatives of the Zernike polynomials. In order to avoid thisproblem, one can instead choose a set of polynomials whosederivatives are orthogonal over the locations of the measure-ments. Thus, it is concluded that, in constructing a wave-front estimator that utilizes measurements of the wave-front-phase directional derivatives, one should minimize thenumber of measurements used to estimate a fixed number ofmodes.

And it is also concluded that, as one increases the numberof modes to be estimated, the variance of the individual modesincreases. Further analysis is required to determine the op-timum number of modes that should be estimated in a givensituation. However, the present analysis indicates that theoptimum number tends to be minimized by signal-to-noiseconsiderations.

ACKNOWLEDGMENT

The author is grateful to C. B. Hogge for suggesting theproblem. The many helpful discussions with H. W. Brewer,W. P. Brown, Jr., and G. Valley are also acknowledged. Thiswork was performed under AF Contract F29601-77-C-0087,Kirtland AFB

APPENDIX: VARIANCE OF PHASE DERIVATIVEMEASUREMENTS AS A FUNCTION OF THENUMBER OF SUBAPERTURES

In this appendix the dependence of the variance of phasederivatives on the number of subapertures is derived. Thisderivation is a radiometric analysis of a background limitedsensor.

It is assumed that the wave-front sensor has an aperturewith area A which is divided into N subapertures, each ofequal area A , where

As = A/N.

The signal photons collected by a subaperture during onemeasurement time is

Qs = (HA/hv)(A/N)TO??qtd,

where HA is the irradiance at the aperture, h is Planck's con-stant, v is the frequency of the photons, ro is the transmittanceof the optics, f7q is the quantum efficiency of the detector, tdis the measurement time.

The noise photons collected by a subaperture during onemeasurement time is

QN = [(O-dNb/hv)(A/N)7qtd] 1/2,

where (<d is the solid field-of-view and Nb is the backgroundradiance.The signal to noise ratio is thus

SNR= Qs= HATO (7qtd '1/2( AQN hvNb \dN

Let

/ = HATO ("lq td/hv Nb) 1/2

so

SNR = / (A/wdN)1 1 2.

Now it is desired to determine the variance of the phase di-rectional derivatives, when given the above signal-to-noiseratio. But the tilt variance depends on the signal-processingtechniques used to make the measurement. However, re-gardless of the signal-processing technique, the variance isinversely proportional to the square of the signal to noiseratio8 ; that is,

U = K/SNR2 = KWdN/I 2A,

where K is a constant. Thus, as long as the size of the field-of-view is not affected by diffraction, the phase variance isproportional to the number of subapertures. However, thesize of the field-of-view can be limited by diffraction. Theminimum size field-of-view associated with a subapertureis

Wd = K1NX 2/A,

where K1 is a function of the shape of the aperture. As anexample of K1, for circular apertures

K1 = (1.22) 2 7r.

Thus, when diffraction limits the size of the field-of-view,

Yp = KK 1 X2 N2 / 2 A2 .

The variance is proportional to the square of the number ofsubapertures.

'M. P. Rimmer, "Method for evaluating lateral shearing interfero-grams," Appl. Opt. 13, 623-629 (1974).

2D. L. Fried, "Least-square fitting a wave-front distortion estimateto an array of phase difference measurements," J. Opt. Soc. Am.67, 370-375 (1977).

3 J. C. Wyant, "Use of an ac heterodyne lateral shear interferometerwith real-time wavefront correction systems," Appl. Opt. 14,2622-2626 (1975).

4P. B. Liebelt, Art Introduction to Optimal Estimation (Addison-Wesley, Reading, Massachusetts, 1967).

5W. P. Brown, Jr. (private communication).6R. J. Noll, "Zernike polynomials and atmospheric turbulence," J.

Opt. Soc. Am. 66, 207-211 (1976).7 J. Y. Wang and J. K. Markey, "Modal compensation of atmospheric

turbulence phase -distortion," J. Opt. Soc. Am. 68, 78-87 (1978).8R. D. Hudson, Jr., Infrared System Engineering (Wiley, New York,1969).

977 J. Opt. Soc. Am., Vol. 69, No. 7, July 1979 Ronald Cubalchini 977