Centroid anisoplanatism

Vol. 2, No. 5/May 1985/J. Opt. Soc. Am. A 765

Centroid anisoplanatism

H. T. Yura and M. T. Tavis

Electronics Research Laboratory, The Aerospace Corporation, P.O. Box 92957, Los Angeles, California 90009

Received September 7, 1984; accepted January 15, 1985

A new and potentially serious optical-system beam degradation is discussed. The degradation, which we definehere as centroid anisoplanatism, deals with the errors and corresponding on-axis intensity reduction that are ob-tained when centroid or wave-front gradient tracking systems are employed to determine the overall atmospheric-turbulence-induced tilt in short-term imaging and/or laser transmitting systems. The error between overall tiltand centroid measurements becomes more important both at shorter wavelengths and for large-diameter optics.It is also exacerbated by point-ahead limitations and scintillation. Specifically, it is shown that a Strehl ratio ofless than 3 X 10-2 results for D/ro > 100, where D is the optics aperture diameter and ro is the turbulence-inducedlateral coherence length.

1. INTRODUCTION

For many laser systems operating in the atmosphere, atmo-spheric-turbulence effects are severe. Unless some sort ofadaptive-optics corrections are employed, these effects willgreatly reduce the viability of such systems. This is especiallythe case for short-wavelength and large receiving/transmittingaperture systems. The performance of any adaptive-opticsimaging or transmitting system can be expected to improvesignificantly if the atmospherically induced phase errors as-sociated with the desired propagation path are known. Inconventional phase-conjugate adaptive optics, a beacon signalthat originates at the object of the imaging system or the aimpoint of the laser transmitter system is used to provide thedesired information. When the instantaneous phase of thebeacon signal is measured by appropriate wave-front sensors,the required adaptive-optics correction is then given by thenegative of this phase (i.e., the complex conjugate of the field).In practice, the term instantaneous phase measurements re-fers to measurements that are taken over time periods of theorder of less than 1 msec.

As is well known, there are still many factors that can po-tentially degrade system performance. 1

-4 For example, in

some cases a convenient beacon can be provided by reflectedlight from the object of interest. Because of the finite velocityof light, the object of interest has moved to a new positionwhen beacon information is received. For an object of interestmoving fast enough, the propagation-path direction from thebeacon to the transmitter and the return-path direction forwhich the adaptive-optics correction is to be provided can besignificantly different. Then the wave-front distortionsmeasured by observing the beacon do not accurately reflectwhat adaptive-optics wave-front correction is needed tocompensate for the turbulence encountered on the target'strue propagation path. As a result, the adaptive-optics sys-tem performance can be significantly less than diffractionlimited, and we say that the system is suffering from (angular)anisoplanatic effects.2 '3

We deal with a new type of optical-system degradation(referred to here as centroid anisoplanatism) that arises whenfull-aperture centroid tracking (i.e., Hartmann-type sensing)is employed to obtain overall wave-front-tilt information. 5

Specifically, we define centroid anisoplanatism to mean the

optical-system errors associated with using a centroid mea-surement as an estimate of wave-front tilt projection. Assuch, this definition extends the usual classical-optics defi-nition of anisoplanatism. As will be apparent in what follows,wave-front centroid measurements are sometimes referredto in the literature as wave-front gradient measurements. Weinvestigate the impact of wave-front centroid measurementson system performance, because this is closest to what manywave-front sensors actually measure. The impact of subap-erture centroid measurements on the general wave-frontsensing problem was considered previously.6 7 Here, however,we consider for the first time the important special case ofusing full-aperture centroid measurements as an estimatorfor overall wave-front tilt and its corresponding impact onsystem performance. It will be shown in what follows that theuse of a full-aperture centroid tracker will result in a seriousreduction of mean on-axis irradiance for short-wavelength-large-diameter adaptive-optics systems.

We consider an adaptive-optics system in which the de-termination of the turbulence-induced optical wave-frontdistortion over an aperture of diameter D is accomplished bytwo separate subsystems. One measures the overall wave-front tilt over the entire aperture, whereas the other (sepa-rately) measures all the higher-order phase distortions (i.e.,the wave-front figure). It is assumed here that overall tiltinformation is obtained from a centroid-type sensor. Thedetails of the figure-sensor subsystem are of no concern here.To illustrate the effects of centroid anisoplanatism, we furtherassume that the figure sensor yields highly accurate infor-mation of the higher-order phase distortions across the ap-erture. Given these assumptions, we show below how centroidanisoplantic effects arise and under what conditions theyadversely affect system performance.

The physical basis for centroid anisoplanatism is relatedto the fact that the true overall instantaneous tilt angle is notcompletely correlated to the corresponding angular positionof the centroid position in the focal plane, as is obtained in awave-front gradient measurement. As a result, if one useswave-front gradient measurements to correct for overallwave-front tilt, a residual phase error results, which translates,for example, to a corresponding reduction in on-axis irra-diance. The effect is small for systems where Diro < 10,where ro is the turbulence-induced lateral coherence length

0740-3232/85/050765-09$02.00 © 1985 Optical Society of America


766 J. Opt. Soc. Am. A/Vol. 2, No. 5/May 1985

of the atmosphere. However, for Diro >> 10, the effect israther severe. For example, as is shown below, for a clearcircular aperture with Diro > 100, the mean reduction in on-axis irradiance relative to that of a diffraction-limited systemin the absence of turbulence (i.e., the Strehl ratio) is 3 X10-2.

In its most basic form, centroid anisoplanatism arises whenwe consider a fixed target and a beacon collocated on the opticaxis at some distance z from an optical system transmitting/receiving at wavelength X. We first neglect irradiance fluc-tuations (i.e., scintillation effects) and consider only purephase distortions. Scintillation and point-ahead limitations,which are due to the finite velocity of light, exacerbate theeffect of centroid anisoplanatism. We include point-aheadeffects later on in the analysis. However, scintillation effects,neglected here, will be treated in a forthcoming study.

In Section 2, we present the theoretical analysis requiredfor this study. Sections 3-5 present the significant resultsconcerning the performance of an adaptive-optics systemwhen it is degraded by centroid anisoplanatism and whenpoint-ahead limitations are included. The conclusions of thisstudy are summarized in Section 6.

2. GENERAL CONSIDERATIONS

Throughout the analysis, the normalized antenna gain (i.e.,the Strehl ratio) is the major quantity of interest. We areparticularly interested in the performance of an adaptive-optics laser transmitter, although the results are equally valid(through reciprocity) for an adaptive-optics imaging system.The normalized antenna gain Irel is defined as

rel ((O) )IDL ()

(1)

W(r) = for eD r D

2 2

otherwise

For example, for e = 0, we obtain from Eqs. (3) and (4)

K(x) = 2D2 [cos'1 x - x( -2)1/2],

where x = rD.The generalized structure function is given by

D(ri - r2) = (0(ri, t) - &P(r2, t)]2 ),

(4)

(5)

(6)

where 60(r, t) is the residual phase error at coordinate r andtime t. The residual phase error is simply the difference be-tween the phase that is due to turbulence and the phase cor-rection actually applied by the adaptive-optics system. Wehave (suppressing the explicit time dependence for simplicityin notation)

60(r) = P(r; p) - 0B (r; p'), (7)

where 0(r; p) is the turbulence-induced phase distortion thatresults from a point source located at the object aim point (p,z) of the laser transmitter, fk(r; p') is the estimate of thephase distortion obtained from a point beacon source locatedat (p', z), and z is the propagation distance. In practice, p andp' are not equal for many scenarios of interest. We firstconsider the case p = p', i.e., the beacon and the object aimpoint are collocated (and fixed). Below we generalize toscenarios where p p' and calculate the resulting reductionin on-axis irradiance.

Next, we express (r) and kB (r) as

qO(r) = kr Ot + 6k(r)

Here, IDL (0) refers to the on-axis irradiance for a diffrac-tion-limited system not degraded by turbulence, and (I(0))refers to the main on-axis irradiance that results from theadaptive-optics system in the presence of turbulence (anglebrackets denote an ensemble average). A value of Irel nearunity indicates performance near that of the diffraction-limited system, whereas an Irel << 1 indicates poor systemperformance.

In order to proceed, we consider a uniformly illuminatedtransmitting aperture of diameter D and obscuration ratio e.Furthermore, we assume far-field and/or focused propagationconditions, since this corresponds to most cases of practicalconcern. Under these conditions, it can be shown that4

Irel = d2 r exp[-'/ 2 D(r)]K(r), (2)

where r is a two-dimensional vector in the plane of thetransmitting aperture,

K(r) = f d2 R W(R + /2r)W(R - /2 r) (3)I fd 2R W(R) 12

W(r) is the aperture function associated with the aperture ofinterest, and D(r) is the generalized phase structure functionfor the adaptive-optics system under consideration.4 For auniformly illuminated aperture we have

(8)

and

,OB(r) = kr t + Q(r). (9)

Here, Ot and 6k are the actual overall turbulence-induced tiltangle and figure distortion over the aperture, respectively, andk is the wave number (=2ir/X). Similarly, Ot and 5Q are theestimates of these quantities, as obtained from measurementsof the beacon. In Eqs. (8) and (9) we omit an overall constant(i.e., piston) from the optical phase because it is unmeasurableand does not affect system performance. If one representsthe phase across an aperture in terms of orthonormal Zernikepolynomials Zp as 7

(10)P(r)= F apZp(r),p=l

then the first term on the right-hand side of Eq. (8) corre-sponds to p = 1, 2 in Eq. (10). Similarly, 60(r) is given byY-;=3 aZp (r).

We now assume that two separate sensing subsystems areused to obtain Ot and Q(r). Furthermore, we assume thatthe figure sensor is essentially perfect such that

Q(r) = 60(r). (11)

Although this can never be obtained in practice, we assumethat whatever residual figure error remains is so small thatnegligible impact on system performance is obtained. Sub-



stituting Eqs. (8), (9), and (11) into Eq. (6) yields that thegeneralized structure function is given by

k2r2 -2

(12)

where r = ri-r2 and statistical isotropy has been assumed.Expanding the square in Eq. (12) yields

k 2r2

D(r) = 2 ((6t2) + (2) - 2(O - Ot)), (13)2 I

It is shown in Appendix A for. W(r) given by Eq. (4) and theKolmogorov spectrum of atmospheric turbulence that

(0et2) = (Oto2 ) [(1 + E23/3 )(1 - E4 2

- E4F(17/6)r(29 6 ) 2F1('/6, - 11/6, 3, E2)]7(14/)

(22)

where I is the gamma function and 2 F1 is the hypergeometricfunction. The quantity

where ( t 2 ) and ( 2) are the variance of the true tilt an'

estimate of the tilt, respectively, and (O- Ot ) is the correlabetween them.

Substituting Eq. (13) into Eq. (2) and rearranging tEyields

Irel = f d2 rK(r)exp(-'/ 2 k2 0o¢2r 2),

where

o,02 = ( 2 ),y'

= (0t2) 1_ ((9t2)) 1/27Y = /2 1 + (921 to 2)) R,

and

[ thetion

0 2 (5/3 ( 1 )2(0to 2 ) = 14.35 r-I D- (23)

is the (two-axis) variance of wave-front tilt across a clear ap-erms erture of diameter D (Ref. 3) and ro the turbulence-induced

lateral coherence length. 8

(14) Next, we assume that the overall tilt of the beacon wavefront is obtained from a centroid tracker. That is, we assumethat our estimator of tilt is given by

(15) Ot = , (24)

(16) where O, is the instantaneous angular position of the centroidof the intensity distribution in the focal plane of the apertureof diameter D. This quantity is defined as9"10

R A O(t -Oe)((0t2))1/2(( 0t2))1/2

is the normalized correlation coefficient between the overalltilt angle and its estimator. The quantity ap can be regardedas an effective residual tilt phase error over the aperture dueto centroid anisoplanatism.

The integral appearing in Eq. (14) cannot be performedanalytically for general K(r). However, an accurate usefulapproximation is readily obtained for large 0,2. In this casethe main contribution to the integral is obtained near r = 0,and hence

27rK(0)Irel - (18)

From this asymptotic limit, a useful engineering expressioncan be constructed that is a good approximation to Irel for allvalues of ¢,2. Noting that Irel - 1 as a02 - 0, we obtain

Ire _ 1/1 + [k20f 2 /27rK(0)]. (19)

For example, for a circular aperture of diameter D and ob-scuration ratio E we have from Eqs. (3) and (4) that K(0) =4/7rD 2( -

2) and hence

(20)Irel _ 2 1/1 + (/D) 2

In the next section we calculate explicitly the statisticalquantities indicated in Eqs. (15)-(17) for various cases of in-terest.

3. TILT AND CENTROID STATISTICS

Now it can be shown that the wave-front tilt over the entireaperture can be expressed as7'8

ro(r)W(r)d2 r (21)/2 k f r2W(r)d2r

where 0 is the angular coordinate in the focal plane as mea-sured from the center of the aperture and I(0) is the instan-taneous focal-plane irradiance distribution.

Now it can be shown, when scintillation effects are ne-glected, that 9'10

(26)= f Vo(r) I W(r) 12d2rk I W(r) 1 2 d 2

r

Owing to the presence of the gradient operator in the nu-merator of Eq. (26), centroid measurements are sometimesreferred to as wave-front gradient measurements. Under thesame assumptions that led to Eq. (22) it is shown in AppendixA that

(9~'2) = (Oco02) [i+E 13(1 - 2 )2 [

_ 2E2F(17/6)r(11/6 ) 2F1(/6, -5/6,2, E2)

F'(8/3) II(27)

and

(0,02) = 13.39 I5/3 ( II 2I ~roI ' kD .

(28)

is the two-axis variance of the centroid angle in the focal planeof a telescope of clear-aperture diameter D.9 "10 ComparingEqs. (16) and (21) reveals that the variance of wave-front tiltis about 7% greater than the corresponding variance in thecentroid angle.

The normalized correlation coefficient between overallwave-front tilt and its estimator is given by ( = 0,)

(R * 0 0)((°t2) (,2))1/2

(29)

On substituting Eqs. (21) and (26) into Eq. (29) it is shown ipAppendix A that

(17)= f I(O)0d20

X I(O)d 2 0(25)


U,52 o.


1.

Re

0.

O.' 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Fig. 1. The tilt-centroid correlation coefficient as a function of theobscuration ratio e.

Table 1. R versus

0 R

0.0 0.99190.1 0.99120.2 0.98890.3 0.98500.4 0.97980.5 0.97380.6 0.96730.7 0.96070.8 0.95420.9 0.9474

RE = (11/6)1/ 2 A,/(A 2 A3 )'/2 ,

where

Al = r(1/ 3 )(1 + E 7 /3 ) - 2 Fl(-5 / 6 , l/6, 3, e2 )

6e2

- - 2F,(-1"/6, '16, 2, 2 ),

(30a)

and are plotted in Fig. 1 and listed in Table 1. Similar resultsfor R(E) were obtained previously by a different method.' 0

We note that the two-axis pointing error is given by Aof =[((Ot - 0c )2)]1/2 = [(0tE 2

)ye]1/2. This quantity is independentof wavelength and proportional to D-1 /6. For example, fore = 0, X/D = 10-7, and Diro = 100, we obtain that Aoc 260nrad.

4. STREHL RATIO:AND TARGET

COLLOCATED BEACON

Explicit expressions for Irel can be readily obtained by sub-stituting Eqs. (22), (23), (27), (28), and (30) into Eqs. (14)-(17).We find that

Irel = fo dxxM(x)expF- k 2D2 x2], (32)

where

C"P2 = (e 2),y, (33a)

1 2 1 + (O cE2)) - ((9cE 2) 1/2l ', (°ee)) ~.(9t,2)) RE, (33b)and Me(x) is given explicitly by Eq. (8) of Ref. 10 multipliedby 16/r(1 - E2 )2 . We have

M,(x) = 16 2)2 (cos x - x(1 -x2)1/2

- [E2 cos'1 (x/E) - x(E2 - x2)1/2]H(E - x)

- 7E2H( + e x

+ aCe2 - 0 + 4X2 sin2 a[cot(a - )-cot a]j

XH(x - 2 )H(12 x)), (34a)

where

(I + 4X2 - E2U = Cos-' (1+ - ,

a = cos'1 (1 - 4X2 - 2c~~~~~Ex = Io~(30b)

(34b)

(34c)

A 2 = /3) (1 + E23 3 ) -4

2 F,('/ 6 , - 1/6, 3, E2),

(30c)

and

A3 = (8/3)(1 + Ell/ 3) - 2E22F,(/ 6, - 5/6, 2, E2), (30d)

IRELwhere R is the normalized correlation coefficient for a clearcircular aperture and is given by

Ro =

| dxx- /3J,(x)J 2(x)

dxx - 4 /3J 22 (X) X dxx-8/3J2(x)12

8 X 23 1/2= 11 X 17 = 0.9919. (31)

Values for R(E) can be readily obtained from Eqs. (30) and (31)

1000D/rQ

Fig. 2. Strehl ratio as a function of D/ro for various values of theobscuration ratio .

. I I I . I . I - I - I .. ~I , I I

0-

9

A I I, 1 1 , 1I1 , 1 , I, I, I ,



Table 2. 1.8f(e) versus E

e 1.8f(E)

0.0 0.0150

0.1 0.01610.2 0.01950.3 0.02460.4 0.03030.5 0.03470.6 0.03640.7 0.03430.8- 0.02780.9 0.0166

and

(34d)W = . X <

whereO~X ly 2 = (t 2 )?yx y I

'Yx' = 1/2(1 + (0t2) )- (f:-)1 RxY,

(40)

(41)

and

=((0t2)X'Y)/2((0,2)y )1/2 (42)

are the x and y components of the normalized correlationcoefficients, respectively. Here we assume, as is the usualcase, that the atmosphere is statistically homogeneous andlocally isotropic, from which it follows that (t,c

2(p)) =

(Otc 2(p')) and (t,c 2)x = (t,c 2) = 1/2(0tc 2 ).

For spherically symmetric aperture functions W(r) weobtain from Eq. (39)

Irel = f dxxME(x)exp [ k2D

2( °t 2) (yX + yy)X 21

Equation (32) has been integrated numerically arplotted in Fig. 2 as a function of Diro for various values

The engineering approximation given by expression (2(

Irel 1/1 + 1.8(D/ro) 5/3f(e),

where

nd isof E.

X Io[kD ( 0t 2)( yx - yy)X2]14II

(43)

)) is where Io is the modified Bessel function of the first kind oforder zero.

(35) In Appendix B it is shown for a circular aperture of diameter

D and obscuration ratio e that

dxx - 3Fxs (x) [J,(x) - EJ1(Ex)][J2(x) -E2J 2(Ex)]Rx, (a) = R. 4 , ,

I f dxx-1/13[J1(x) - EJl(Ex)][J2(x) - E2J2(Ex)](44)

E dxx-1 4 /3 [J 2 (x) - 2 J2 (eX)]20o

- (36)

where R, is the correlation coefficient, given by Eqs. (30), for

the zero point-ahead angle,

(1 - E4)2 E dxx-14/3J22(X)

and ye iS given by Eq. (3b). Numerical values of f(e) are given

in Table 2 for various values of E.For the special case of e = 0 we find that

Irel!_ 1/1 + 0.0150(D/ro) 5 /3.

= y fpth [(2xaz) F J2 (2x )]C2(Z)dZ,

I = r C, 2 (Z)dz.path

(37)

For example, for D/ro > 100 both numerical integration ofEqs. (32) and (37) yield Irel < 0.03, which indicates that cen-

troid anisoplanatism is a serious effect here.EGI.

G5n

UJG7

G

Lt

5. POINT-AHEAD EFFECTS

In some applications, the beacon and the target of interest are

separated by an angular position a [the angle subtended atthe transmitter between the vector positions p and p' of Eq.(7)]. For definiteness we assume that a is parallel to the x

axis. Here, azimuthal symmetry is lost, and the generalized

structure function, assuming that the estimate of tilt is givenby a centroid measurement, is

D(x, y) = k 21x2 ( [t(p) - Oc(P')]x2)+ Y2 ([Ot(p) - c(p')]y2) (38)

The corresponding Strehl ratio is given by

Irel = Sf dxdyK(x, y)exp[-(1/2)k 2 (ao>x2x2 + afiY2y2)]I

(39)

(45)

(46)

ALTITUDE ABOVE SITE ImI

Fig. 3. The daytime and nighttime Navy/DARPA models for C' 2versus altitude.

f(E) = yE(l - E2)



D/ro

Fig. 4. Strehl ratio as a function of D/ro for various values of thepoint-ahead angle and .

Table 3. 1.84(e) versus eE a = 50 rad a = 100,4rad

0.0 0.0158 0.01800.1 0.0169 0.01900.2 0.0203 0.02220.3 0.0253 0.02710.4 0.0332 0.03490.5 0.0352 0.03640.6 0.0369 0.03780.7 0.0346 0.03530.8 0.0280 0.02850.9 0.0167 0.0169

C. (z) is the refractive-index structure constant profile alongthe propagation path of interest, and Jo is the zero-orderBessel function of the first kind. Numerical integration ofEqs. (44) and (45) is complicated by the oscillating nature andslow falloff of the Bessel function with large argument. Wegive an alternative expression for Eq. (44) in Appendix B. Wealso show a technique for expressing the product of threeBessel functions in terms of the hypergeometric function,which permits easier calculation of Eq. (44) directly.

In order to obtain numerical results, we employ the Navy/DARPA Cn2 modelll shown in Fig.3. Figure 4 describes plotsof Irel as a function of Dro for various values of the point-ahead angle and .

In this case, it is also possible to derive an engineering for-mula for Irel. We find that

effects change Irel only slightly when compared with the effectsof centroid anisoplanatism. Although tilt anisoplanatic ef-fects were treated previously, 3 it can be shown that, strictlyspeaking, they apply to optical tilt projection systems only.In particular, this treatment cannot be used for optical sys-tems that employ overall centroid tracking.

6. CONCLUSION

In this study, we have demonstrated explicitly that the use ofinstantaneous centroid-tracking information is not a validestimator for the corresponding overall wave-front tilt. Forshort-wavelength large-diameter optical systems the effectcan be particularly severe, representing the largest contri-bution to the reduction of on-axis irradiance. Point-aheadeffects have been included, and it has been shown that in thepresence of centroid anisoplantism they are of second orderin importance. Finally, we note that tilt anisoplanatism, in-cluding point-ahead limitations, was treated previously.3However, the analysis of Ref. 3 does not apply to systems thatemploy centroid/gradient trackers to obtain overall wave-fronttilt information. For such cases, the analysis presented in thisstudy must be used.

APPENDIX A

In this appendix, the statistical quantities that appear inSection 3 are evaluated.

Tilt VarianceFrom Eq. (21) it follows directly that

(Ot2) = At- 2 Sf d2rid2 r2r, r2 BO(r - r2)W(r1)W(r2),(Al)

where

Bo(r - r2 ) = ((r,),O(r2 )) (A2)

is the phase-correlation function and the constant At is givenby the denominator of Eq. (21). To proceed we use Fourierintegral techniques. Let

Bo(r) = d2q P(q)exp(iq r)(2ir)2

(A3)

and

rW(r) = - d2) Ht(q)exp(iq r),(27w)2

(A4)

where the phase spectrum and the tilt filter function are givenby

fE(ca) = [x(t)yy(a)J1/2(l - E2)

X

and yxy are given by Eq. (41). Numerical values of fE(a) aregiven in Table 3 for a = 50 and a = 100 rad for various valuesof e. It is easily seen from Fig. 4 and Table 3 that point-ahead

(47) Po,,(q) =. d2rBO(r)exp(-iq r) (A5)

and

Ht(q) = f d2rrW(r)exp(-iq r), (A6)

respectively.Substituting Eqs. (A4) and (AS) into Eq. (Al) and per-

forming the integration over r, r2 , and one of the q variablesyields

(0t 2) At -2 d 2 q )12 (

whereIrel - 1/1 + .8(Dro)5/ 3fE(a),

E dxx -14/3 [J2() - E2J 2(Ex)] 2

(1 - 4 )2 dxx -14/3J22(X)



Now for a circular aperture of diameter D and obscurationratio E, Ht is readily obtained from Eqs. (4) and (A6) as

Ht(q) = -i 2 [J2(qD/2) - E2 J2 (EqD/2)], (A8)2q

where

u = q/q

is a unit vector along q.The phase power spectrum that corresponds to the Kol-

mogorov spectrum is given, in the normalization used here,by

P4,(q) = 8.186k2 1q-1/3,

-tJXJ,(at)J,(at)dt

(a/2)x-lF(X)F( + v X + 1)

2F V -AU+ + lrtv + + + rt - + A + 1)R(,t+v+1)>R(X)>0, a>0.

Reduction of Eq. (A12) yields Eq. (22) of Section 3.

Centroid VarianceFrom Eq. (26) it follows directly that

(A15)

(A9) (0,2) = A,- 2 rr d2rjd2r2V .V2B0(rj - r2)IW(rD)W(r2 )1,(A16)

where

I= .fpai C 2 (z)dz (A10).fpath

is the integral of the structure constant C 2 along the propa-gation path-of interest.

By substituting Eqs. (A8) and (A9) into Eq. (A7), changingto dimensionless variables, and introducing the atmosphericlateral coherence length 8

ro = (0.423k 21)-3 /5 , (All)

Eq. (A7) becomes

(t 2) f dxx 4 3 [J2(x) E2J2(Ex)]2

dxx - 4 /3 J22 (x)

(A12)where (to 2 ) is given by Eq. (23).

The infinite integrals over the Bessel functions may be re-duced to hypergeometric functions through the use of thefollowing formula' 2 :

| t-XJ,(at)J(bt)dt =

(b/a)P(a/2)-lry 1+ v + 1)

2F(v + 1 )( - v + X + 1)

.(,+v-A+1 v-,u-A+l1 b2~~~'X 2Fi A + - , + V-A- +IIv + 1, 2

R(,u+v-X+l)>0, R(A)>-1, 0<b<a,

where the constant A, is given by the denominator in Eq.(26). Application of Fourier integral transform techniques

yields

(A17)(C2) = A.-2 r 2q) q2Pq(-q)jHc(q)j2(2.7) 2

where

H, (q) = 5 d2 rl W(r)j2 exp(-iq r), (A18)

which for a circular aperture bf diameter D and obscurationratio is

Hc(q) = - [J(qD2) - EJj(EqD/2)]-q

(A19)

Substituting Eqs. (A9) and (A19) into Eq. (A17), changingto dimensionless variables, and introducing the parameter royields

(6 co2 ) J J'o dxx- 8/ 3[J,(X) - eJl(Ex)]21

(O 2)

2 dxx-8/ 3J, 2 (x)

(A20)

where (0co2) is given by Eq. (28). Use of Eqs. (A13)-(A15)

yields Eq. (27) of Section 3.

Tilt-Centroid CorrelationFrom Eqs. (21) and (26) it follows directly that

(Ot 0,) = (AtA0 )-' f d2rjd 2r 2W(rI W(r2 )t2

X ri * V2[B(r, - r2)1.

A13) Using Fourier integral transform techniques yields

(A21)

(a/b),u(b/2)X-1rtZ (+ -A +

S t-XJ,(at)J(bt)dt = 2b/) ( + \ + 121'(g + 1)F(v - 2 + X + 1)

(,+V-X+1 ,-v-X+1 a2X 2F, , , + 1,

2 2 b2J

R(, + v-X + 1) > 0, R(X) >-1, O< a <b,(A14)

and

(Ot * ) = (AtA0)-' 5 d2 P,(-q)q- Ht(q)H0(q).(2ir)2

(A22)

Hence from Eqs. (29), (A7), (A17), and (A22) we obtain

R = S d 2 qP¢(-q)lq- Ht(q)H 0(q)l[5 d2 qP0(-q)q 2 jH0 (q)j2 5 d2qP,6(-q)jHt(q)j2]112

(A23)

Substituting Eqs. (A8), (A9), and (A19) into Eq. (A23) andchanging to dimensionless variables yields



dxx-11/3 [J,(x) - EJl(ex)][J2(x) - E2J2((X)]RE = Ro d

{Sdxx l/[J2(X) sJ2(EX)] Xf dxx 11Jlx-~le)2

(A24)

Again, through the use of Eqs. (A13)-(A15), Eq. (A24) maybe reduced to yield Eqs. (30) of Section 3.

APPENDIX B

When the object of interest and the beacon are spatially sep-arated the only statistical quantity that is different from thosederived in Appendix A is the tilt-centroid correlation func-tions ( .* a ) . This quantity is now given by

(Ot O = (AtA 0 )-1 fy d 2rld 2 r 2 W(r1)W(r 2 )12

X r V2 [B(r - r2; P - )], (B1)where p - p' is the projection of the vector coordinates of thepositions of the object and the beacon in a plane perpendicularto the line of sight.

To evaluate Eq. (B1) we use the Fourier integral techniquesemployed in Appendix A. The only change is to the powerspectrum of the phase correlation function. It can be shownhere that

Pb(q; a) = 8.186k 2 q-11/3 ,f dzC' 2 (z)exp(izq. a).

(B2)

Assuming that a is parallel to the x axis yields for = 0

(At tc)xy = constant f dzCn2(z)G(z).,y, (B3)

where

G(z)xwy = J dqq"-1/3 J1(Dq/2)J2 (Dq/2)

x 2 r do kCOs 2 O) exp(izqa coso)

= X 5 dqq"-1/ 3J1(Dq/2)J 2 (Dq/2)

X [JO(qaz) F J 2 (qaz)], (B4)

from which the results in Eqs. (44) and (45) follow directly.Evaluation of Eq. (44) of Section 5 can be difficult without

a technique of reducing the product of three Bessel functions.First note that

where

w = (a2+ b 2 - 2ab cos )1/2

Using Eq. (B7) and Eqs. (A13)-(A15), Eq. (44) may be reducedto finite integrals over the hypergeometric functions. Thesehypergeometric functions do not oscillate and vary slowly overthe integration range leading to relatively quick calculations.The resulting expressions are lengthy because of the differentbranches expressed in Eqs. (A13)-(A15). For that reason,they are not listed here.

An alternative expression for Eq. (44) may be obtained ina manner similar to techniques used in Ref. 10. The corre-lation corresponding to Eq. (44) is

Rx,y(a) =

R, f C 2(h)F.,,,(h)dh

I I p8/3 M,(p)dp

where I is given by Eq. (46) and

F.(a) = 2 pME(p) -------- [2F(l/ 6, 1/2, 1 z)f W ~~+ 0l1/3

- 1/2 2F,(1/6, /2, 3, z)] 6(P + )l3

X 2 F,(7/6, /2, 3, z)} dp,

F (at) = 1 (p + (p)'3

2F,(/ 6, /2, 3, z)dp,

haA=DD

P1

(p + )2

and Me(p) is given by Eqs. (34).

REFERENCES

(B8)

(B9)

(B10)

(B11)

Jo(x) + J2(x) = 2J,(x)/x,

Jo(x) - J2 (x) = 2J,(x) _2J2(X),x

(B5)

(B6)

so that Eq. (B4) reduces to a single product of three Besselfunctions for the y tilt and that the x tilt is given by the y tiltminus an integral over a different triple product of Besselfunctions. In both cases, the triple product contains a Besselfunction of an order that is repeated, although with differentargument. Watson' 3 shows that

[abl

J,(ax)J,(bx) H f J,(Wx)sin 2v odo

XV r(v + 1/2)r(1/2) O WV

(B7)

1. D. L. Fried, "Varieties of isoplanatism," Proc. Soc. Photo-Opt.Instrum. Eng. 75, 20 (1976).

2. D. L. Fried, "Anisoplanatism in adaptive optics," J. Opt. Soc. Am.72, 52 (1982).

3. G. C. Valley, "Isoplanatic degradation of tilt correction andshort-term imaging systems," Appl. Opt. 19, 574 (1980).

4. G. A. Tyler, "Turbulence-induced adaptive-optics performancedegradation: evaluation in the time domain," J. Opt. Soc. Am.A 1, 251 (1984).

5. J. W. Hardy, "Active optics: a new technology for the control oflight," IEEE Proc. 66, 651 (1978).

6. J. Herrmann, "Least squares wavefront errors of minimumnorm," J. Opt. Soc. Am. 70, 28-35 (1980).

7. G. A. Tyler and D. L. Fried, "Wavefront sensing a new approachto wavefront reconstruction," Rep. No. TR-514 (Optical ScienceCompany, Placentia, Calif., May 1983).

8. D. L. Fried, "Optical resolution through a randomly inhomo-geneous medium for very long and very short exposures," J. Opt.Soc. Am. 56.1372 (1966).


H. T. Yura and M. T. Tavis Vol. 2, No. 5/May 1985/J. Opt. Soc. Am. A 773

9. V. I. Tatarskii, The Effects of the Turbulent Atmosphere onWave Propagation (National Technical Information Services,Springfield, Va., 1971).

10. M. T. Tavis and H. T. Yura, "Strong turbulence effects on shortwavelength lasers," Rep. SD-TR-79-24 (Aerospace Corporation,Los Angeles, Calif., December 15, 1979) [note that in Eq. (8) theterm 02 should be M].

11. R. R. Jones, J. W. Rockway, L. B. Stotts, D. W. Hansen, and A.J. Julian, "Submarine laser communication evaluation algo-rithm," Tech. Rep. 673 (Naval Ocean Systems Center, San Diego,Calif., May 1981).

12. Y. L. Luke, Integrals of Bessel Functions (McGraw-Hill, NewYork, 1962), p. 324.

13. G. N. Watson, A Treatise of the Theory Bessel Functions(Cambridge U. Press, Cambridge, 1945).

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Centroid anisoplanatism