Towards high resolution imaging by speckle interferometry

Volume 26, number 1

OPTICS COMMUNICATIONS

July 1978

TOWARDS HIGH RESOLUTION IMAGING BY SPECKLE INTERFEROMETRY R.H.T. BATES, M.O. MILNER, G.I. LUND * and A.D. SEAGAR Electrical Engineering Department, University of Canterbury, Christchurch, New Zealand Received 8 November 1977 Revised manuscript received 15 December 1977

It is shown how it should be possible, under arbitrary seeing conditions, to image star clusters, whose apparent angular extension is less than the seeing disc, by combining Labeyrie's original speckle interferometry and the extension of it due to Lynds, Worden and Harvey together with Patterson analysis as developed by X-ray crystallographers. An encouraging simulation in an optical laboratory is reported. We discuss a further experimental result which suggests that our idea could be used with an array of telescopes whose focal fields are combined coherently.

We present the results of an optical simulation of an imaging technique which is an amalgam of three methods being actively developed in several laboratories at the present time. An extended preamble is necessary to put our ideas into context. Before this decade it was thought that the turbulence in the earth's atmosphere made it virtually impossible to achieve diffraction-limited imaging with a large (i.e. diameter greater than 0.5 m) earth-bound telescope. In 1970, Labeyrie [1] showed how the intensity of the visibility of a celestial object could be obtained from a large number of statistically independent, short exposures of the object (each exposure is less than 20 ms, and successive exposures are separated by more than 300 ms - the light forming the image is prefiltered to a bandwidth of less than 200 A). Practical astronomical use was made [2] in 1972 of Labeyrie's technique (called speckle interferometry because of the mottled appearance of the short exposures, which have a character similar to that of laser speckle [3] ; so the short exposures are called

speckle images). Labeyrie [4] has since indicated how to resolve faint objects. It seems that the required information can be recovered from recorded speckle images quantised to two levels [5]. * Now at Universit~ de Nice, Parc Valrose, 06034 Nice, France.

22

Since the visibility of an object is the Fourier transform of its brightness distribution, Labeyrie's speckle interferometry usually permits only the autocorrelation of the brightness distribution to be recovered. So, the size of an object can be determined, as can the distribution of the sizes of features on an object, such as details on the surface of the Sun [6]. Only in special cases can a true image of an object be formed [7]. It is convenient here to introduce certain definitions. A resolvable object is one whose disc is larger than the Airy disc of the telescope. The seeing disc is a long (several seconds or more) exposure of the image of an unresolvable object, which is an object too small to be resolved by the telescope under perfect seeing conditions. It follows that a speckle image of an unresolvable object consists of a number of distorted Airy discs scattered over an area of the image plane of the size of the seeing disc. There is an extension of speckle interferometry, due to Knox and Thompson [8], which in principle allows a true image of any object to be formed. It seems to be very difficult to apply in practice and so far has only been used to infer details on the surface of the Sun [9]. In searching for an imaging technique easier to apply than Knox and Thompson's method, Lynds, Worden and Harvey [10] hit on the idea that each speckle in a short exposure of a resolvable object is a

Volume 26, number 1

OPTICS COMMUNICATIONS

July 1978

o

G Fig. 1. Speckle interferometry simulated on an optical bench: (a) simulated star cluster, (b) typical speckle image, (c) speckle mask, (d) result of optically correlating (using spatially incoherent light) the speckle image and the speckle mask (scale 6 : 1 compared with (a) above). randomly distorted image of that object. So, when the speckles are superimposed, the imperfections of the individual images tend to average out. Acting on this idea, they formed an image of Betelgeuse [ 10] which we attempted to improve [ 11 ]. To explain the essence of Lynds, Worden and Harveys's method we suppose for the moment that the speckle image shown in fig. lb is a short exposure of a single, resolvable star. So, each of the bright speckles in the image can be taken to be a distorted disc of the star. We then place tracing paper over the image and mark a small black dot on what seems to be the

brightest part of each bright speckle. A negative of this tracing paper (called the speckle mask) looks like fig. lc, each bright point of which lies at what seems to be the "centre" of one of the speckles in fig. lb. By correlating the speckle image with the speckle mask we superimpose all the distorted discs of the star. The autocorrelation of a celestial object is somewhat similar to the Patterson map of the electron density in a regular crystal. The Patterson map is actually the autocorrelation of the whole crystal, which means that it contains cross-correlations between the electron densities in adjacent unit cells as well as the autocorre23

Volume 26, number 1

OPTICS COMMUNICATIONS

lation of the electron density in a single unit cell. The Patterson map is three-dimensional, whereas the autocorrelation of a celestial object is two-dimensional. Even though the autocorrelation is considerably simpler, Baldwin and Warner [12] have found that some crystallographic approaches to the interpretation of Patterson maps can be applied with advantage in radio astronomical interferometry for situations where accurate estimates of the visibility phase cannot be obtained. We have realised that Baldwin and Warner's ideas can be combined with Labeyrie's and Lynds, Worden and Harvey's forms of speckle interferometry to provide a method of imaging star clusters under severe seeing conditions. We have performed an optical simulation of our proposed technique, using an experimental approach which we developed previously [13]. Fig. la shows a simulated star cluster, a typical speckle image of which (under seeing conditions such that no detail of the cluster would be apparent in a long exposure) is shown in fig. lb. The autocorrelation of the cluster can be obtained by Labeyrie's original speckle interferometry (for instance, refer to fig. 4f of [13]). What Baldwin and Warner [12] have recognised is that, if one star in the cluster is appreciably brighter than the others, the autocorrelation is dominated by the cross-correlations of this bright star with the others. This means that the autocorrelation a(x, y) can be approximated usefully by the expression

a(x,y) = ~(0, O) + c(x,y) + c(-x, -y) , where 6(0, 0) represents the bright central spot, and

c(x, y) is the cross-correlation of the bright star with the rest of the cluster. The bright star (or more accurately, its autocorrelation) forms 6(0, 0), together with the autocorrelations of the other stars. Without further measurements or extra a priori information or protracted iterative procedures (such as the X-ray crystallographers have developed), c(x, y) cannot be separated from c(-x, -y). Suppose for the moment that there are three stars in the cluster, the fainter ones being situated at (Xl, Yl) and (x2, Y2), taking the coordinate origin to coincide with the bright star. Then inspection of a(x, y) reveals that there are, in general, four equally likely pairs of positions for the fainter stars: (x 1, Yl) and ( - x 1, -Y2); (-Xl, Yl) and (x 2 , Y2); (-Xl, - Y l ) and ( - x 2, -Y2); as well as the actual positions. Baldwin and Warner's 24

July 1978

[ 12] fig. 2 illustrates this very clearly. Suppose that an image of the two brightest stars in the cluster can be formed. It is now convenient to interpret the autocorrelation of the cluster rather more precisely than before:

a(x, y) = A(0, O) + b(x,y) + b(-x, -y) where A(0, 0) represents the bright spectral spot and the cross-correlation of the two brightest stars, and b(x, y) is the cross-correlation of the two brightest stars with the rest of the cluster. Note that b(x, y) consists of (N - 2) replicas of the two brightest stars (where N is the number of stars in the cluster). The intensity and position of each of these replicas is proportional to the intensity and position of one of the fainter stars. This means that b(x,y) can be distinguished from b(-x, -y) by inspection, provided that the intensities of the two brightest stars are appreciably different. The replicas that make up b(-x, -y) are mirror images of the replicas in b(x, y). The most powerful aspect of this concept is that the image of the two brightest stars need only be accurate enough to distinguish between b(x, y) and b(-x, -y). Once this has been done, b(x, y) itself provides ( N - 2) accurate estimates of the relative intensities and positions of the two brightest stars. Our proposal depends upon one of the stars in the cluster being noticeably brighter than any of the others (as in fig. 1a). We interpret the brightest parts of each of the bright speckles in fig. lb as distorted images of the brightest star. Consequently, when the speckle mask of fig. lc is correlated with the speckle image, these distorted images are superimposed. It follows that the other stars in the cluster should be superimposed. Fig. 1d shows the result of using spatially incoherent light to correlate fig. lb with fig. lc. Fig. 1d is a rather poor reconstruction of the cluster. In fact, only the two brightest stars are reproduced, and not very faithfully. However, it is clear from fig. ld that the fainter of the two brightest stars is below the brighter one, rather than above it. This, together with the estimate of the separation of the two brightest stars available from fig. 1d, is sufficient, when inspecting a(x, y), to distinguish b(x, y) from

b(-x, -y). This optical simulation has encouraged us sufficiently to start experimenting with electro-optic devices interfaces to a digital computer. We find that we

Volume 26, number 1

OPTICS COMMUNICATIONS

can thereby process many speckle images quite rapidly and carry out various averaging procedures. We are performing an extended series of experiments which we hope to report at a later date. Labeyrie [14] has demonstrated fringe formation in the combined light gathered by a pair of widely separated telescopes. This has encouraged him to propose speckle interferometry arrays of, say, a dozen telescopes (each of about 1 m daimeter) covering an area of at least 1000 m 2. Such an arrangement would provide an order of magnitude improvement in resolution over what is presently available. However, the transfer function (or radiation pattern) of the sparsely filled aperture would have many, comparatively large, minor lobes which would tend to degrade images formed with the array. Nevertheless, it must be remembered that much of the effect of these lobes could be removed by averaging, since the positions of the minor lobes vary as the array's pointing direction is changed relative to the zenith. There is another effect which is likely to make it easier to process images formed with a speckle interferometry array. We came upon it by accident in our optical laboratory. Fig. 2 is a record of a typical illustration of the effect. Fig. 2a shows a speckle image formed with a simulated telescope having an undisturbed circular aperture, three times larger than the one which was used to obtain the speckle image shown in fig. lb. The simulated seeing conditions were the same for figs. lb and 2a. Fig. 2b shows how the speckle image appears when the single circular aperture is replaced by six much smaller apertures distributed at random within the area formerly occupied by the single, larger aperture. The individual speckles in fig. 2b are much more distinct, making it easier to place the dots on the speckle mask. An important question, which should be investigated, is whether the signal-to-noise limits [ 15,16] which apply to Labeyrie's speckle interferometry also apply to the method of imaging devised by Lynds, Worden and Harvey [10]. We suspect that they do not, because Labeyrie's speckle interferometry is second order interferometry, whereas the technique of Lynds, Worden and Harvey is effectively first order interferometry, where we have invoked our definition of the order of an interferometer [7].

July 1978

Fig. 2. Typical speckle images obtained with a telescope simulated on an optical bench: (a) single telescope with "large" circular aperture, (b) six telescopes with "small" circular apertures distributed randomly over an area equal to that of the "large" aperture; the light from six telescopes is combined coherently.

References [1] A. Labeyrie, Astron. Astrophys. 6 (1970) 85. [2] D.Y. Gezari, A. Labeyrie and R.V. Stachnik, Astrophys. J. 173 (1972) L1. [3] J.C. Dainty, AoE. Ennos, M. Francon, J.W. Goodman, T.S. McKechnie and G. Parry, Laser speckle and related phenomena (Springer-Verlag, Berlin-Heidelber-N.Y., 1975). [4] A. Labeyrie, Nouv. Rev. Optique 5 (1974) 141. 25

Volume 26, number 1

OPTICS COMMUNICATIONS

[5] R.H.T. Bates, Mon. Not. Roy. Astr. Soc., in press. [6] J.W. Harvey and J.B. Breckinridge, Astrophys. J. 182 (1973)137. [7] R.H.T. Bates and P.T. Gough, IEEE 1rans. Computers C-24 (1975) 449. [8] K.T. Knox and B.J. Thompson, Astrophys. J. 193 (1974) L45. [9] R.V. Stachnik, P. Nisenson, D.C. Ehn, R.H. Hudgin and V.E. Schirf, Nature 266 (1977) 149. [10] C.R. Lynds, S.P. Worden and J.W. Harvey, Astrophys. J. 207 (1976) 174.

26

July 1978

[ 11 ] M.J. McDonnell and T.H.R. Bates, Astrophys. J. 208 (1976) 443. [12] J.E. Baldwin,and P.J. Warner, Mon. Not. Roy. Astr. Soc. 175 (1976) 345. [13] P.T. Gough and R.H.T. Bates, Optica Acta 21 (1974) 243. [14] A. Labeyrie, Astrophys. J. 196 (1975) L71. [15] F. Roddier, Opt. Commun. 10 (1974) 103. [16] C. Aim~ and F. Roddier, Opt. Commun. 19 (1976) 57.