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Statistical Accuracy in Stellar SpeckleInterferometry at Low Light LevelsJ.G. Walker aa Royal Signals and Radar Establishment, Malvern,Worcestershire WR14 3PS, EnglandPublished online: 14 Nov 2010.

To cite this article: J.G. Walker (1981) Statistical Accuracy in Stellar Speckle Interferometryat Low Light Levels, Optica Acta: International Journal of Optics, 28:7, 885-905, DOI:10.1080/713820640

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OPTICA ACTA, 1981, VOL . 28, NO. 7, 885-905

Statistical accuracy in stellar speckle interferometryat low light levels

J . G . WALKERRoyal Signals and Radar Establishment, Malvern,Worcestershire WR14 3PS, England

(Received 26 August 1980)

Abstract . An expression for the statistical accuracy of a measurement of theangular diameter of a faint astronomical object, made using the speckle techniquecombined with a least squares fitting procedure, is derived . The expression isused to find optimum values for the exposure time, filter bandwidth and detectorcell size . The dependence of the accuracy on object and telescope size is shownand limiting magnitudes are given .

1 . IntroductionStellar speckle interferometry [1-4] is a technique for obtaining high angular

resolution information about centro-symmetrical astronomical objects despite thepresence of the turbulent atmosphere . The technique consists in estimating thepower spectrum, or, equivalently, the spatial autocorrelation function of the imageintensity; the ensemble is taken over a number of short exposure images (specklepatterns) . To date, the power spectrum data processing has been most widely used[5] . However, power spectrum data processing cannot, easily, be carried out in realtime. Real-time analysis has the important advantage that a very large number ofexposures can be processed without data storage problems . For faint objects a verylarge number of exposures must be processed as the signal-to-noise ratio of theinformation in each exposure is very low . A number or real-time speckle systemsbased on an actual or effective array of photon detectors, linked to a digital vectorautocorrelator, are currently in use [6, 7] and being developed [8] .

For a well resolved disc object, the autocorrelation function of the imageintensity, C(~), may take the forms shown in figures 1 (a) and (b), depending on theseeing conditions . In the case of very bad seeing an estimate of the object diametercould be deduced from, say, a measurement of the width of the initial decrease in theautocorrelation function, see figure 1 (a) . However, in the case of good seeing, it isdifficult to identify the initial decrease and hence obtain a reliable estimate of theobject diameter . Generally, the shape of the autocorrelation function will be affectedby the seeing conditions .

The effect of the seeing conditions may be calibrated out of the autocorrelationfunction in a number of ways . Labeyrie [9] and co-workers use an extrapolationprocedure . Worden and co-workers [10,11] have proposed and implementedanother method in which as well as estimating the autocorrelation function, C(~), across-correlation function, C'(~), between exposures separated in time is alsoestimated . The time separation must be long enough for the speckles in the cross-correlated exposures to be statistically uncorrelated .

0030 309 81 28()7 0585 S02 00 1 1981 '1%,, lo,

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C

0 0

C

2

3

4

(a)

Vd

00

2

4 Vd

(b)

Figure 1 . Form of the speckle spatial autocorrelation function for a well resolved disc objectin the case of (a) very bad and (b) very good seeing conditions .

The method of Welter and Worden [10] consists in estimating the differencebetween the autocorrelation and cross-correlation functions . It is shown inAppendix A that the difference between the autocorrelation and cross-correlationfunctions, c(~), where

C(c)=C(O - C'(0 ,

(1)

is proportional to the diffraction limited image of the autocorrelation of the objectintensity, which for a well resolved disc object has the form shown in figure 2 . Theform of c(~) is independent of the seeing conditions and so may be used to estimatereliably the angular diameter of the object .

In this paper the following case is considered : the object is a faint uniform disc,the image intensity is sampled by a two-dimensional array of adjacent square photondetectors, the data processing method suggested by Welter and Worden is used, then

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0

1 UdFigure 2. Form of the speckle spatial autocorrelation function for a well resolved disc object

after processing by the method of Welter and Worden [10] .

the object angular diameter is estimated by fitting a curve of known form to the datapoints using the method of least squares . The problem, with which this paper isconcerned, can be stated as follows : for a given object, telescope, observation period,size of photon detectors, exposure time, filter bandwidth and atmospheric con-ditions, what is the accuracy of the estimated value for the angular diameter of theobject?

The problem is similar to that of estimating the accuracy of spectral linewidthsmeasured by the technique of intensity fluctuation spectroscopy and the analysisgiven here is similar, in some respects, to an analysis of that problem due to Jakemanet al . [12] . The solution involves a first-order approximation and consequently theresult is only valid for percentage accuracies of about 10 per cent and better .

Previous treatments [13-16] of the speckle technique in the autocorrelation modehave included signal-to-noise ratio estimates. The signal-to-noise ratios, as definedin these treatments, are best described as point-wise signal-to-noise ratios, as theyare measures of the accuracy of the individual estimated points in the autocorrelationfunction. These point-wise signal-to-noise ratio expressions have been used topredict limiting magnitudes for the speckle technique [14,15] . However, as they donot take account of the fact that the accuracy with which a curve can be fitted to a setof points, depends not only on the statistical accuracy of each point but also on thenumber of data points on the curve, the present treatment allows a more quantitativedescription of limiting magnitude to be made .

2 . Theory2.1 . Sampling scheme

The autocorrelation function, C(~), and the cross-correlation function, C'(~), aredefined by

C(0= I(x)I(x+ . )dx),

(2)\ J ~

C'(o=C

I(x)I'(x+ )dx),

(3)

J ~

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where I(x) is the short exposure image intensity, I'(x) is an exposure recorded someshort time after I(x) and ( • > indicates an average over an ensemble of shortexposures . Although the analysis applies to a two-dimensional intensity distri-bution, for simplicity only one spatial dimension will be shown . The case, in whichthe image intensity is sampled by a P x P array of adjacent square photon detectorcells, each of area A„ is considered . The array is assumed large enough to contain thewhole seeing disc. Owing to the finite size of the detector cells, the sampled quantityis E(x) rather than I(x) . E(x) is the image intensity integrated over the square area A,centred at x . Consequently, using this sampling scheme, a true estimator for C(~)and C'(~) cannot be found. However, estimators for C EO and C'E(~) defined by

CEG) _ \J

E(x)E(x+) dx ),

(4)

C'E(,)=~J~~E(x)E'(x+ )dx ),

(5)

can be found . CE (O and C'E(~) are equal to C(~) and C'(~) convolved with theautocorrelation of -the function equal to one within the area of a detector cell and zerootherwise. As such, CEG) and CE(~) are somewhat smeared out versions of C(~) andC'(~) ; the amount of smearing being dependent on the size of the detector cells .Unnormalized estimators for CEG) and CE(~) are given by CM(~) and C11 (~) definedby

M P

CM(Sk) = ~,

n' ( 1 1i+v -6v,0)> (6)j=1 i=1

andM P

CM(Sk) =

, nini+V,

( 7 )j=1 i=1

where

bk =xi+v - Xi,

n is the number of photons detected during the jth exposure by the detector centredat x, M is the number of exposures used and h is a small non-zero integer chosen tosatisfy the conditions stated in Appendix A . It should be noted that in order to reducecomputation time it is usual to `clip' the photon counts . The normal method is thatcounts greater than unity are reduced to unity . In general this leads to distortion ofthe auto- and cross-correlations but at low light levels, <n j> << 1, the functions areonly significantly affected for Sk = 0. For this reason if clipping is being used the datapoint at Sk=0 is not used in the fitting procedure .

Using the sampling scheme outlined above, the data processing method ofWelter and Worden [10, 11] consists in estimating the function c E(~ k ) for a number ofk values,

CE(O = Cg(S) -C E(0 .

(8)

It is shown in Appendix B that c E (~) is equal to c(~) convolved with theautocorrelation of the function equal to unity within the area of a detector cell andzero otherwise . An unnormalized estimator for C E(~ k ) obtained in any one obser-vation is given by C k ,

Ck = CM(~k) - CM( k) .

(9)

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2 .2 . Statistics of the estimator ekIf an observation were repeated a number of times then the estimators CM(~k),

CM(~k) and hence ek measured in each case would differ owing to the statistical natureof the signal. In this section the mean and variance of e k averaged over an ensemble ofstatistically equivalent observations are examined . The statistics of CM( k) have beenexamined in a previous treatment [16] ; for the sake of brevity some of the resultsobtained in that treatment will be used here .

Let the mean value of ek be denoted by ck. Substituting equations (6) and (7) into(9) and averaging yields

M P

Ck- <Ck> = Y_ Y <nj(n~+~-S,o)>-<n;n!+,,

(10)j=1 i=1

Assuming n to be a stationary ergodic process and using the fact that n and ni+v arestatistically uncorrelated, ck can be written as

P

Ck = MY <ni(ni+v-&,O)>-<ni><ni+v> .

(11 )i=1

Using the relationship [17],

<EiE1>

<ni(nj - Vi .j)>

<Ei> <Ej>

<n i > <nj>where Ei is the integrated intensity over the area of the detector centred at x i duringan exposure, equation (11) can be expressed in the form

ck=M ~ <EiEi+~>-<Ei> <Ei+,,~<ni> <ni+,> .

i=1

<Ei><Ei+v>

Making the reasonable assumption that <E(x)>, <E(x)E(x+ ~)> and <n(x)> are slowlyvarying over the area of any one of the detector cells, c k can be approximated by theintegral

Ck=MArJ

<E(x)E(x + ~k) >- <E(x)> <E(x+Sk)><n(x)> <(n(x+~k)> dx, (14)

<E(x)> <E(x+~0>

where <n(x)> is the average number of photons detected per unit area at x in oneexposure. To a good approximation <n(x)> can be written as a gaussian function [18],

<n(x)>=A

exp(-x 2 /y 2 ),

(15)

where nph is the average number of detected photons per exposure, y is a measure ofthe radius of the seeing disc and A(=ny 2 ) is the effective area of the seeing disc .Using equation (15) the mean of ek can be written as

Ck=<CM(bk)>-<CM(Sk)>

MnPhAr

<E(x)E(x+' k)>-<E(x)> <E(x+4)>A Z f

<E(x)> <E(x+ bk)~

x exp_xZ +(xS)2 dx.

(16)1

Y 2

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The variance of ck will now be examined. Ck is the difference between the tworandom variables CGk) and The conditions stated in Appendix A aresufficient to ensure that Cy(~k ) and C'M(~k) are statistically uncorrelated, so using thevariance law [19] the variance of ek is given by

Var Ck =Var CM(~ k)+ Var C ''M(`,k) .

(17)

It was shown in a previous paper [18], by a simple but rather lengthy analysis, that inthe limit of low light levels, such that <n i > << 1, the variance of CM(~k ) is given by

VarCM(bk) = <CM(bk)>, ~k# 0 .

(18)

Using a similar analysis it can be shown that the variance of CM(ck ) is given by

Var CM(Sk)=<C'M(4)> .

(19)Substitution of equations (18) and (19) into (17) yields

Varck=<CM(bk)>+<C'M( 4 )>,

(20)Comparison of equations (16) and (20) yields

Varck=MnphAr ~~ <E(x)E(x+~k)>+<E(x)><E(x+~k)>A

<E(x)> (E(x + 4) >

x exp -Xz + (x+ ~k)zdx .

(21)Y 2

1In almost all cases of practical interest the inequality,

<E(x)E(x + f )> - <E(x)> <E(x + ~)> «1

(22)<E(x)> <E(x+ f )>

holds well and, to a good approximation, equation (21) reduces to

Var ck= 2M phA r

exp x

~~a

C Z+(x2~k)Z

dx .

(23)Iz

-A

Y

Performing the integral yields

Mn

zVarck= A"exp -k2

(24)2Y

2.3 . Least squares fitThe factor ek is an unnormalized estimator for CE(~k ) and so using equation (B 4),

Ck can be expressed asCk o-J(' k , d, 1, Ar ),

(25)

where J is equal to the autocorrelation of the geometrical image of the objectconvolved with the diffraction limited point spread function of the telescope, allconvolved with the autocorrelation of the function which defines the area of onedetector cell, d is the diameter of the geometrical image of the object, l is thediffraction limited resolution of the telescope and Ar is the area of a detector cell . Fora given telescope, central wavenumber and detector array J can be written as afunction of Sk and d only . It is convenient to normalize J(~ k , d) by setting J(0, d) =1and write

ck=BJ(~k, d) .

(26)

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An expression for B is found by writing

B - ck, bk -0 .

Using equation (16) this can be expressed in the formz

B=M;AA ' Q,

(28)

where Q defined by

(27)

zQ=A

Q(x) eXp ~- YZ/ dx,

(29)

is the weighted average of Q(x), which is the mean square contrast of E(x) definedby

Q(X) =<Ez(x)>-<E(x)>2 .

(30)(E(x))

The factor Q(x), and hence B, may depend not only on the object size, telescope sizeand detector cell size but also on the exposure time, filter bandwidth and theatmospheric conditions .

As the dependence of equation (26) on bk is not affected by the value of B, for thepurposes of fitting a curve to the experimental points B can be regarded as anarbitrary constant independent of the object size . Suppose that an observation isperformed and ee" k is computed at every point on a T x T square matrix . The principleof least squares, as applied to the present problem, is the minimizing of the quantityS defined by

TS= I

wk(Ck - BJ(~k, d)) 2 ,

(31)k=1 .4k*0

where d is the estimator for d, B is the estimator for B, and wk is the weightingassigned to the measurement of ee" k . The data point at Sk=0 is not included in thesummation as the value, of ee" k at Sk=0 is affected by any form of `clipping' asmentioned in § 2.1 . The minimization of S is achieved by making variations in d andB so when S is a minimum,

where

J(Sk, d) =d)k, d)

ad

and the limits of the summations are omitted for the sake of simplicity .Eliminating B from equations (32) and (33) yields

Y, wkJ2(Sk> d) - Y, WkJ'(~k, d)Ck -> wkJ(4, d)J'(Sk, d) . Y_ Wk J(Ck,d)Ck .k

k

k

k

as =ad

0= wk(2BZJ(4, d)J'(~k , d) - 2BJ'(4, d)ck),

as _aB 0=Y-wk(2BJz(4,d) - 2J(~k,d)ck),

k

(32)

(33)

(34)

(35)

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Making the first order approximations,

Ck - Ck - SCk,

d=d-Sd,

expanding, neglecting second-order terms and using equation (26) gives

where

J'(4,d)=OA( k, d)

cad

The variance of d can now be written as

Var d=<Sd2>

_Y2 wkJ(~k, d)J'(Sk, d)'Y_Y_Wkwk'J(4, d)J(4, d) <6464'>k

k k'

+[,,2 wkJ2(~k, d) •Y Y- wkwk'J' (Sk, d)J'(~k', d) <SCkSCk'>k

k k'

Y_ wkJ(4, d)J (Sk, d) • Y_ WkJ(4, d)SCk - Y_ wkJ2(Sk, d) •E WkJ (~k, d)SCk

Sd=kkkk,

( 37)

B{y-2 wkJ(Sk, d)J (4, d)-I,WkJ2(~k, d)' Y,WkJ 2(Sk, d) }k

k

)k

- 2 Y_ wkJ(Sk, d)Jl (Sk, d) ' Y, wkJZ(Sk, d) 'Y_L wkwk'A( k, d)J~(4', d) <SCkSCk'>k

k k'

B2 2WkJ(~k, d)J~(~k, d) -Y WkJ2(Sk, d) •Y_WkJ

r2(~ k , d)}2{ .ll~k

,

k

Sk

(36)

(38)

(39)

In the low light level limit, <nI> << 1, the main source of noise is photon noise . Asthere is no correlation between the photon noise in any two detector cells, the errorsin Ck are uncorrelated with the errors in Ck', So

<6C04 '> = < SCk >Sk .k' = Var CkSk,k, .

(40)

Substituting equation (40) into (39) and using the previously derived expressions forVarCk (equation (24)) and B (equation (28)) gives

A

~'

2Var d=

MnphArQ 2L

{ 2WJJ .W2J

kk Z exp

2T2 )

+Y 2 wkJ2 •Y wkJ' 2exp(`~Z12y

- 2E wkJJ' • wkJZ ~ wkif expC- 2v2

/)

{Y2 W kJJ'_Y_ wkJ2'E WkJ 2i2 , (41)

where the summations are over the range k = 1, T ; Sk # 0 and the parameters of J andJ', bk and d, are omitted for the sake of brevity .

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2.4 . Signal-to-noise ratioThe signal-to-noise ratio (SNR) for a measurement of an angular diameter, d, is

defined here as

SNR=

d 1/2 .

(42)(Var d)

If, as is usual in least squares fitting, a weighting inversely proportional to thevariance of the data point is used,

constantwk

Var ckthen the SNR can be expressed as

ml /2n hAr /2QSNR=

A1 2dF(d, T, A„ l, y),

where

F(d, T, A,,, 1, y) =2

2

2Y2 JJ' exp

(2Y2 /-Y J 2 exp (2Y2

/Y J 2 exp

C2Y2 /

(43)

(44)

(44 a)(((

2

x2 Sk

2

Sk

1/21J y2 J2 exp (~k2 /

J'2 exp(

Sk2

)-E2 JJ' exp

( -2 ) •Y J exp

(2/ll

2y

2y

2y

2y

and the summations are over the range k =1, T ; ~k :A 0 and where, to recapitulate,M is the number of exposures used,npi, is the average number of detected photons per exposure,Ar is the area of the individual photon detectors, which are assumed to be

square and adjacent in an array large enough to contain the whole seeingdisc,

A is the effective area of the seeing profile, which is assumed to be agaussian of width 2y at the 1/e points, A=ny2,is equal to the autocorrelation of the geometrical image of the objectconvolved with the diffraction limited point spread function of thetelescope (Airy pattern) all convolved with the autocorrelation of thefunction which defines the area of a detector cell J(0, d) = 1,

d is the diameter of the geometrical image of the object,T is the square root of the number of data points, assumed to lie in a square

matrix,is the diffraction limited resolution of the telescope, andis the weighted average of mean square contrast of the integratedintensity .

J(Sk, d)

lQ

3. Optimum exposure parameters and limiting magnitudesIn this section the dependence of the SNR expression (equation (44)) on the

exposure time, filter bandwidth and detector cell size is examined and the choice ofthe optimum values of these parameters is discussed. The optimum values are takento be those corresponding to the maximum SNR . The dependence of the SNR on theobject size, and the telescope size is also examined . Finally the limiting magnitudefor an observation of the angular diameter of a disc object is considered .

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The dependences of most of the factors in the SNR expression on the parametersmentioned above are simple power law dependences. However, the dependence of Qthe mean square contrast of the integrated intensity in the image is morecomplicated. It is convenient to consider the dependence of Q before the overalldependence of the SNR expression is considered .

3 .1 . Integrated speckle contrastA speckle pattern formed in polychromatic light generally exhibits a radial

structure, speckles towards the edge of the pattern being elongated in the radialdirection. This effect is due to the wavelength dependence of the Fourier transformrelation between the complex amplitude in the pupil and image planes of thetelescope . A result of this effect is that the speckle contrast, Q(x), decreases withdistance away from the centre of the seeing disc . The dependence of Q(x) on x may beeliminated by using either a narrow bandwidth filter or by imaging the patternthrough a lens, such as that designed by Wynne [20], which compensates for thewavelength dependence of the Fourier transform . As it is probable that compensat-ing lenses will be widely used in future speckle interferometry work, it will beassumed that a compensating lens is used so the x dependence of Q(x) can beneglected and Q replaced by Q,

Q=<E2>-<E>

(45)<E>

where E is the intensity integrated over the area of any one of the detector cells duringan exposure .

The short exposure image intensity recorded at time t with an exposure time of Atmay be expressed as

fx t+&

I(x) =

J

S(k)O(x' -x)i(x', t, k) dx'dtdk,

(46)o

r

where S(k) is the spectral distribution of the filter, O(x) is the geometrical image ofthe object and i(x, t, k) is intensity in the speckle pattern image due to a point object .The integrated intensity, E, sampled by one detector cell which without loss ofgenerality may be taken to be the one centred at the origin, can be written as

E = J A(x)I(x)dx,

,

(47)x

where A,(x) is equal to unity within the area of the detector cell centred at the originand zero otherwise . Substituting equation (46) into (47) and performing the xintegral gives

E= J

x

ft+ot f

S(k)G(x)i(x, t, k) dxdtdk, (48)o

t

where G(x) is the convolution of the geometrical image of the object with the detectorcell aperture,

G(Y)= fAp(x')O(x_x')dx' .

(49)x

It is worth noting that the measured intensity of a speckle pattern formed by a finiteobject and sampled by a finite aperture is the same as that either formed by an object

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of form equal to the convolution of the finite object and the finite detector andsampled by a point detector or formed by a point object and sampled by a detectorcell of form equal to the convolution of the finite object and finite detector.

A reasonable model for a stellar speckle pattern, imaged through the sort ofwavelength compensating lens described by Wynne, due to a point object, is

i(x, t, k)=i*(x, t, k) •L (x, k),

(50)

where i*(x, t, k) is a random gaussian speckle pattern extending to infinity aild havingstatistical properties which are stationary with respect to changes in x and t, as wouldbe formed under conditions of arbitrarily bad seeing and L(x, k) is the local averageof i(x, t, k) which represents the envelope of the seeing disc and is assumed to bestationary in time . Substituting equation (50) into (48) and (45) the mean squarecontrast of the integrated intensity can be written as

t+et t+et

m

1+Q=

S(kl)S(k2)G(xl)G(x2)f 0 o t

t

J -« -Mx <i*(xl, tl, ki)i*(x2, t2, kz)>L(xi)L(x2) dx l dx2dt l dt 2dk l dk2

) z .o

+ fi

<S(k)i*(x, t, k)>G(x)L(x) dxdtdk

(51)-X

Assuming that L(x, k) is slowly varying in x compared with G(x) and slowlyvarying in k compared with S(k) equation (51) may be expressed, to a goodapproximation, in the form

ft+et t+et

Q= I

J

f-

fXS(kl)S(k2)G(xl)G(x2)

0

o

t

t

x

x 52(x1, x2, t1, t2, kl, k2) dx ldx2dt l dt 2dk l dk2

f00

2=A; Ao

S(k) dkl) , (52)

0

where Ao is the area of the geometrical image of the object and S2 is the sixth-ordercorrelation function defined by

K(x1, x2, t1, t2, k1, k2) -<i*.(xi, t1,)'*(x2,t2, k2)>

-L

(53)

The first assumption is a good approximation if .the size of the object and the size of adetector cell are both much less than the size of the seeing disc, which is the usual casein speckle interferometry . The validity of the second assumption depends on thespectral emissivity of the object as well as the atmospheric conditions, but is usually afair approximation for filters of bandwidth up to about 200 nm .

The assumption is now made that the function Q may be factorized,

whereO(x1, x2, t1, t2, k1, k2)=) x(xl, x2)K2t(t1, t2)f2k(kl, k2),

=<i*(xl)i*(x2)> -1x(xl, X2)

<i*>2

fl i(t1, t2)=<1(tl)Z*(t2)> -1,<i*>2

(54)

(55)

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Qk(kl,k2)=<i*(kl)i*(k2)> -1 .

(57)<i*>2

In some recent experiments [21] the temporal correlation function S2 t (t 1i t 2 ) wasmeasured, under varying degrees of spectral and spatial averaging and for a range ofwavenumbers and positions within the seeing disc, and showed no significantvariation with these changes. These results, at least, indicate that the assumption ofwriting equation (54) is reasonable . Using the form of S2 given by equation (54) theintegral of equation (52) may be separated and the mean square contrast written inthe form

where QD is the contrast reduction due to the finite size of the object and the finite sizeof the detector cells, Qt is the contrast reduction due to the finite exposure time andQk is the contrast reduction due to the finite filter bandwidth . Using the fact thatS2 x (x 1 , x2 ) depends only on the difference x 1 -x 2 1, QD may be written in the form

1Q D = 2 2

S2x(x)CG(x)dx,

(59)Ar Ao f - x

where CG is the autocorrelation function of G(x) defined by

CG(X) =J

G(x')G(x'+x)dx' .

(60)

For a gaussian speckle pattern it is a well-known result that the spatial autocorre-lation function of the intensity, Qx(x), formed by a telescope with a hard edgedaperture is equal to the diffraction limited point spread function of the telescope . Fora circular pupil of diameter D, Q D may be written as

4J,2 2Dx/k)QD = A1A2

CG(x) (2Dx/k )2 dx .

(61)r o f-

Using the fact that fl,(t 1 , t2 ) depends only on the difference ltl - t 2 1, Qt may be writtenas

Qt=2 °t

A

(At - t)S2,t (t) dt .

(62)0

It was recently shown [21] that to a good approximation Qt(t) may be expressed as agaussian function,

Q = QDQtQk,

Qp) =exp (- t 2 /2 2 ),

(58)

(63)

where 2 is a measure of time-scale or typical `lifetime' of the speckles in the image .The quantity c is dependent on atmospheric conditions . Using this form for S2 t (t), Q tcan be written as

Q t =V7Tt erf(At/2)+ 222 (exp (-At2/22)-1),

(64)At

At

where erf is the error function . For simplicity the filter is assumed to have arectangular spectral transmittance,

S(k)=1, k a <k<kb ,

S(k) = 0, otherwise .(65)

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Using this form for S(k) and assuming that Qk(k l , k 2 ) depends only on the differenceIk 1 -k2 1, Qk may be written as

2

ekQk=2

(4k-k)S2k(k)dk,

(66)Ak o

where Ak = kb - ka is the bandwidth of the filter . Recent experiments [22, 23] haveshown that to a good approximation f2 k(k) may be expressed as a gaussian function,

f2k(k)=exp ( - a2k 2 ),

(67)where a is a measure of the degree of correlation between the speckle patterns atdifferent wavenumbers and is dependent on atmospheric conditions . Using thisform for 1Zk(k), Qk can be written as

z

zQk=aAkerf(aAk)+eXp (

_

0Ak2k )f1.

( 68)

3 .2 . Exposure timeThree of the factors in the SNR expression are dependent on the exposure time ;

the mean photon count per exposure, the number of exposures that may be made andthe contrast reduction. The mean photon count per exposure is simply proportionalto the exposure time .

n pe octlt .

(69)

The maximum number . of exposures that may be made in a given observation time isinversely proportional to the exposure time,

where Q, is given by equation (64) . The optimum exposure time, AtOPT is thatcorresponding to the maximum value of the SNR . Equation (71) has been computedand it is found that the maximum SNR occurs for

AtOPT =1 , 6T,

(72)at which point Q,=0 . 7 .

Values of r have been measured on a number of nights at the cassegrain focus ofthe 91 cm telescope of the Royal Greenwich Observatory [21-23] . It was found that iwas generally stable for periods of an hour or more, measured values were in therange 2 to 8 ms and the most frequently measured values were between 3 and 4 ms .With T=3 .5 ms the optimum exposure time, AT oPT is about 5 ms .

Of course, these quoted values are only representative of one telescope at one site .Nevertheless, as the principal mechanism responsible for the evolution of thespeckles is the evolution of the phase and intensity variations in the pupil plane[21, 23], it would seem unlikely that significantly different results would have beenobtained with a larger telescope at the same site . However, these results cannotnecessarily be considered as representative of other sites . Using the optimumexposure time, as given by equation (72), the SNR scales as the square root of the T,the typical lifetime of the speckles in the image . It is worth noting that previous

1Moc (70)

4t

Substituting equations (69) and (70) into (44) gives

SNR (At) ocAt i " 2 Q,, (71)

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analyses [14,15] of speckle interferometry have assumed that the exposure timeshould be sufficiently short to `freeze the atmosphere', so that the speckle contrast isnot significantly decreased by the finite exposure time . That is they have assumedAt«i. This analysis shows that freezing the atmosphere does not in fact lead to theoptimum condition at low light levels, which might be described as occurring whenthe atmosphere is `partially frozen' . As i is dependent on atmospheric conditions itshould ideally be measured prior to performing the speckle interferometry, so thatthe optimum exposure time can be used .

3 .3 . Filter bandwidthTwo factors in the SNR are dependent on the filter bandwidth ; the mean photon

count and the contrast reduction . For a rectangular spectral transmittance the meanphoton count is simply proportional to the bandwidth,

np , ccAk

(73)

Substituting equation (73) into (44) gives

SNR (Ak) ccAkQk ,

( 74)where Qk is given by equation (68) .

Equation (74) does not show a maximum as Ak is increased, rather it increasesalmost linearly, initially, and then tends asymptotically to a maximum value . Thisresult suggests that, at low light levels, speckle interferometry is best performed inwhite light. However, restricting the bandwidth may have some advantages . Thevalue of a has been measured [22, 23] to be of the order of 0 .4µm, using the 91 cmtelescope of the Royal Greenwich Observatory . For this value of a, equation (74)increases only slightly for Ak greater than 3 µm -1 and certain effects not so farconsidered may cause the SNR to be decreased if the bandwidth is increased beyondthis value . The performance of the compensating lens and of the prisms used tocorrect for atmospheric dispersion [5, 8] will decrease with increasing bandwidth .Also increasing the photon count increases the time required to compute theautocorrelation function of each exposure, if the computation time exceeds theexposure time then some delay between exposures is necessary with a resultantwastage of information and consequent drop in SNR . Thus, for a =0 .4 µm, abandwidth of Ak = 3 ym -1 might be taken as being optimal. For these values Q k has avalue of 0 . 8 . For a bandwidth chosen by the method described above, the SNR scalesinversely with a . As a is dependent on the atmospheric conditions it should ideally bemeasured prior to performing the speckle interferometry both to help in choosingthe filter to be used and to assess the limiting magnitude of the technique for thoseconditions. Apparatus for measuring a is described in a previous paper [22] .

3 .4 . Detector cell sizeThe detector cell size enters directly into the SNR expression as the square root

of the area of one cell. It also affects the contrast reduction QD and the final term, asthe size of the detector cells defines the spacing of the computed data points in theautocorrelation function . The final term is also dependent on the object size ; thediffraction limited point spread function, the number of computed data points andthe seeing conditions .

The number of data points which may be computed, T 2 , is usually limited by thenumber of stores available for storing the auto- and cross-correlation functions . The

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SNR expression has been computed as a function of detector cell size, for the case ofd=0.5 arc second, y=1 arc second, 1=0 .03 arc second and a number of T values. Toreduce computation time the slight broadening of J due to the finite size of l wasneglected anti an approximate form for the autocorrelation of a disc, which isaccurate to ± 3 per cent, was used,

J(~, d, 0, 0) =1-1 . 25 ~/d+ 0 .25~4/d4 , 0

d.

(75)

The results are shown in figure 3 . The value of A, for which the SNR is a maximumdepends on the value of T . The optimum detector cell size is given, roughly, by theequation

which corresponds to the case in which the autocorrelation function occupies thewhole of the area covered by the matrix of data points . In the usual case, in which theobject is significantly smaller than the seeing disc, the choice of the optimumdetector cell size is only weakly dependent on the seeing .

4dzAOPT- T2 ' (76)

0 .02 0 .04 0. 06 0.08 0 .1 0 .12

C , arc second

Figure 3. SNR (arbitrary units) as a function of detector cell size, ,lA,, for the case ofd=0 .5 arc second, y= larc second, 1=0 . 03 arc second and a number of T values .

3.5 . Object sizeThe object size enters the SNR directly as the signal . It also affects the contrast

reduction Q p and the last term . The SNR has been computed as a function of theobject diameter for the case of y =1 arc second, 1=0 .03 arc second, T=41 and theoptimum value of the detector cell size . To reduce computation time the slightbroadening of J due to the finite size of A, was neglected . The results are shown infigure 4 . For a well resolved object, d>5 1, the SNR is approximately inverselyproportional to the object diameter . This result differs from that of Barnett andParry [13], which states that the signal-to-noise ratio scales inversely with the cube ofthe object diameter . The difference arises because theirs is a point-wise signal-to-noise ratio and there is an error in their analysis which was discussed in a previouspaper [16] .

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SNR -

5-

00

0.2

0 .4

0. 6

0. 8

1

d, arc second

Figure 4 . SNR (arbitrary units) as a function of object diameter, d, for the case ofy=1 arc second, 1=0 .03 arc second, T=41 and the optimum value of the detector cell size .

3.6. Telescope sizeThe telescope size affects the mean photon count, the speckle contrast QD and the

final term in equation (44) . For a circular telescope aperture the mean photon countvaries as

nph Cr-1/h,

where l is the distance to the first zero in the Airy pattern . The SNR has beencomputed as a function of telescope resolving power, 1, for the case of a circularaperture, T= 1 arc second, d=0 .1 arc second, T=41 and the optimum value of A, . Toreduce computation time the slight broadening of J due to the finite size of A, wasneglected. The results are shown in figure 5 . Although the SNR increases with

SNR

5-

0 . 02

0 .04

0.06

0.08

0. 1

I , arc second

Figure 5 . SNR (arbitrary units) as a function of telescope resolving power, 1, for the case of acircular telescope aperture, y=1 arc second, d=0 . 1 arc second and the optimum valueof the detector cell size .

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increasing telescope size, the increase is not significant beyond the point at which theobject is well resolved by the telescope (d' 51) . The reason is that the gain in SNRdue to the increasing photon count is almost cancelled by the decrease due to thereduced speckle contrast . It is clear from this that those wishing to improve theaccuracy of a measurement of an angular diameter would gain more by carefulselection of exposure time, filter bandwidth and detector cell size on the sametelescope than by using the same experimental arrangement on a larger instrument .

3 .7 . Limiting magnitudesAs stated in the Introduction the way in which the SNR is defined in this

treatment allows a more quantitative definition of limiting magnitude to be made .Defining the limiting magnitude as that at which the SNR, as defined in thistreatment, falls below some level implies that the fractional accuracy of ameasurement made at the limiting magnitude is simply the inverse of the chosenSNR level. Previous treatments [14,15] have somewhat arbitrarily chosen somepoint-wise SNR to estimate a limiting magnitude ; this approach, however, gives noindication of the accuracy of a measurement made at this magnitude .

In calculating limiting magnitudes it will be assumed that the telescope has acircular aperture 4 m in diameter and that the central wavelength is 500 nm whichimplies a Rayleigh resolution of 0 .03 arc second . It will also be assumed that theoptimum values for the exposure time, filter bandwidth and detector cell size areused. Under these conditions the SNR may be expressed as

3aT ''2T 1~2SNR

2A112o dQDA T1 2 F(d T, AOPT, l),

(77)

where a is a constant, dependent on the star brightness, and the photon detection

m

20-

18-

16-

14-

12-4

10' 1°h

0.1°/°accuracy

0 .01%

Figure 6. Limiting magnitude, m, as a function of the desired fractional accuracy (which isthe inverse of the desired SNR) for the case of a 4 m telescope, a central wavelength of500 nm, an overall light transmittance through optics and filter of 50 per cent, a 10 percent quantum efficiency of the detectors, an observation period of 2000 s, a speckle timescale of r=3 . 5ms, a speckle spectral correlation coefficient of o=0 . 4 µm, two objectsizes and the optimum values of exposure time, filter bandwidth and detector cell size .

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efficiency of the telescope and optical system defined by

nph = oAtAk,

(78)

and To is the observation time . For a 4m telescope it is estimated [14] that a zeromagnitude object should give 10 s photons/s/A. Assuming an overall light transmit-tance through the optics and filter of 50 per cent and detectors with a 10 per centquantum efficiency implies that a can be written as

a=a, 2 . 5

(79)

where a 1 = 2 x 106 pm/ms and m is the magnitude of the object . Using this estimatefor a and equation (77) an expression for the limiting magnitude m can be written as

M=lo

3aiTo 2 i i z dAoeri12QDF(d T AOPT 1)

80g 2 .5

2A i ~ 26

SNR

( )

Equation (80) has been computed as a function of SNR for the case of T o = 2000 s,i = 3 . 5 ms, v =0.4 µm, y = 1 arc second (A =7ty2 ) and a number of object sizes . Theresults are shown in figure 6 .

4. DiscussionThe results given in the preceding section should be useful both to those wishing

to optimize and determine the limitations of an exciting speckle system and to thosetrying to decide between speckle and Michelson interferometry for accurate highresolution astronomical studies . The present paper deals only with measurements ofthe diameters of disc objects. A similar treatment which also deals with binaryseparation measurements has been given elsewhere [23] . An interesting possibilitywould be to extend the present treatment to give estimates of the accuracies to whichlimb darkening parameters could be measured by the speckle technique .

AcknowledgmentsThe major part of this work was carried out while the author was at Imperial

College, London . The author is grateful to Professor W . T . Welford and Drs . A. H .Greenaway and G . Parry for a number of useful discussions and to the ScienceResearch Council for financial support while at Imperial College .

Appendix AThe purpose of this Appendix is to show that the function c is proportional to the

diffraction limited image of the autocorrelation of the object intensity . Let us model ashort exposure image due to a point object as

z'= i*L,

(A 1)

where i* is a unit mean speckle pattern extending to infinity and having stationarystatistical properties as would be formed by the telescope under conditions ofarbitrarily bad seeing and L is the local average of i and represents the envelope of theshort exposure seeing disc . For a finite object the short exposure image, I, can bewritten as

I =i x®O,

(A 2)

where 0 is the geometrical image of the object and Q denotes convolution .

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The function c can be written as

c=<I ©I) -<.100,

where < > denotes an average over an ensemble of short exposures and I' is an imagerecorded some time after I . Substituting (A 1) and (A 2) into (A 3) and making use ofthe commutative property of convolution gives

c={<i*L®x i*L>-<i*L®x i*'L'>}®x {O(O} .

(A 4)

As i* and i*' are statistically stationary and independent of L and L', c can be writtenas

c={<i*i2> <LQ L>-<i*iz'> <LQ L'>}Q {OQO},

(A 5)

where the subscripts indicate spatial positions in the image plane . If the time delaybetween the recording of I and I' is sufficiently long that iK and ii' are completelyuncorrelated but short enough so that L and L' are fully correlated, then

(A 6)

<L( L'>=<L( L) .

(A7)

In practice the speckles within stellar images have a correlation time of the order of5 ms whereas the seeing envelope evolves on a time scale of the order of a second [21] .So using a time delay between I and I' of about 50 ms should satisfy equations (A 6)and (A 7) to a good approximation . Substituting equations (A 6) and (A 7) into (A 5)gives

c={(<i?i2>-<i*`>2)<L©L>}©{O®O} .

(A 3 )

(A 8)

It is a well-known result that for a hard edged aperture telescope the term<i*i2>-<i > 2 is proportional to the diffraction limited point spread function of thetelescope . The term <L(& L> is the average autocorrelation of the short exposureimage envelope. For . a large telescope this term is usually very slowly varyingcompared with the diffraction limited point spread function and can to a goodapproximation may be assumed to be a constant over the area of the diffractionlimited point spread function .

Making this assumption leads tocoo{<i~iz)-<i )2 }Q {OQO},

(A9)which is the desired result .

It is worth noting that this result differs from that of Welter and Worden [10] .They claim that c is "proportional to the autocorrelation of the image effectivelyuninfluenced by a turbulent atmosphere" . It may be seen that their result is equal tothe result of this analysis convolved with the diffraction limited point spreadfunction. Their result is clearly incorrect in the limit of a point object for which it iswell known that the autocorrelation of the speckle is proportional to the diffractionlimited point spread function not the autocorrelation of this, as is predicted by theiranalysis . This difference may arise because of an error in deriving equation (13) intheir paper concerning the limits of the integrations .

Appendix BThe purpose of this Appendix is to show that the function ce is equal to the

function c convolved with the autocorrelation of the function equal to unity withinthe area of one detector cell and zero otherwise, denoted by AP .

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The function CE can be written as

CE =<EOE>-<EOE'>, (B 1)

where < > denotes an average over an ensemble of short exposures and E' is theintegrated intensity in an image recorded some time after E. The integrated intensityE can be expressed as

E=IOAp .

( B 2)

Substituting equation (B 2) into (B 1) and making use of the commutative property ofconvolution gives

cE ={APOAP}Qx {<I ®I> (B 3)

Using equation (A 3), (B 3) can be written as

CE = {Ap OAp} (9c,

(B 4)

which is the desired result .

On etablit une expression pour la precision statistique d'une mesure de diametre angulaired'un objet astronomique faible, en utilisant la technique du speckle combinee a un processusd'ajustage par moindres carres . L'expression est utilisee pour les valeurs optimales du tempsd'exposition, de la largeur de bande du filtre et des dimensions du detecteur . La dependancede la precision sur les dimensions de l'objet et du telescope est montree et on donne lesmagnitudes limites .

Es wird ein Ausdruck fur die statistische Genauigkeit des mittels Speckle-Technik andeiner Anpassung nach dem Prinzip der kleinsten Quadrate gemessenen Winkeldurchmesserseines schwachen astronomischen Objekts hergeleitet . Dieser Ausdruck wird zur Bestimmungder optimalen Werte fur Belichtungszeit, Filterbandbreite and Empfanger-Zellengrollebenutzt. Es wird die Abhangigkeit der Genauigkeit von Objekt- and Teleskopgrol3e gezeigtand Grenzwerte werden angegeben .

References[1] LABEYRIE, A ., 1976, Progress in Optics XIV, edited by E . Wolf (Amsterdam : North-

Holland) .[2] DAINTY, J . C . (editor), 1975, Topics in Applied Physics, Vol. 9 (Berlin : Springer-Verlag) .[3] WORDEN, S . P ., 1977, Vistas Astr ., 20, 301 .[4] DAVIS, J ., and TANGO, W . J . (editors), 1979, High Angular Resolution Stellar

Interferometry, I .A.U. Colloquium No . 50 (International Astronomical Union) .[5] BEDDOES, D . R ., DAINTY, J . C ., MORGAN, B. L., and SCADDAN, R . J ., 1976, J . opt . Soc .

Am., 66, 1247 .[6] BLAZIT, A., BONNEAU, D ., KOECHLIN, L ., and LABEYRIE, A., 1977, Astrophys . J . Lett .,

214, L79 .[7] SCHMIDT, G . D ., ANGEL, J . R . P ., and HARMS, R ., 1977, Publ . astr . Soc . Pacific, 89,410 .[8] SCADDAN, R . J ., MORGAN, B . L., and DAINTY, J . C ., 1979, High Angular Resolution

Stellar Interferometry, I .A.U . Colloquium No. 50 (International AstronomicalUnion), p . 27-1 .

[9] LABEYRIE, A ., 1978, Ann . Rev. Astr . Astrophys ., 16, 77 .[10] WELTER, G . L., and WORDEN, S . P ., 1978, J. opt. Soc. Am ., 68, 1271 .[11] HUBBARD, G ., REED, M., STRITTMATTER, P ., HEGE, K., and WORDEN, S . P ., 1979, High

Angular Resolution Stellar Interferometrv, I .A .U . Colloquium No . 50 (InternationalAstronomical Union), p . 28-1 .

[12] JAKEMAN, E ., PIKE, E . R., and SWAIN, S ., 1971, J. Phys . A, 4, 517 .[13] BARNETT, M . E., and PARRY, G ., 1977, Optics Commun ., 21, 60 .[14] DAINTY, J . C ., 1978, Mon. Not . R . astr . Soc ., 183, 223 .

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[15] DAINTY, J . C ., and GREENAWAY, A . H., 1979, High Angular Resolution StellarInterferometry, I .A.U. Colloquium No. 50, International Astronomical Union), p . 23-1 .

[16] WALKER, J . G., 1979, Optics Commun ., 29, 273 .[17] JAKEMAN, E ., 1974, Photon Correlation and Light Beating Spectroscopy, edited by H . Z .

Cummins and E . R. Pike (London : Plenum Press), p . 45 .[18] DAINTY, J . C ., and SCADDAN, R . J ., 1975, Mon. Not. R . astr . Soc ., 170, 519 ..[19] ARLEY, N ., and BUCH, K . R ., 1950, Introduction to the Theory of Probability, and Statistics

(New York : Wiley), p . 75 .[20] WYNNE, C . G., 1979, Optics Commun ., 28, 21 .[21] PARRY, G., WALKER, J. G., and SCADDAN, R . J ., 1979, Optica Acta, 26, 563 .[22] WALKER, J . G., 1979, High Angular Resolution Stellar interferometry, I .A.U .

Colloquium No. 50, International Astronomical Union), p . 25-1 .[23] WALKER, J . G., 1980, Ph .D . Thesis, University of London .

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