Terrestrial optical aperture synthesis technique (TOAST)

Volume 58, number 3

OPTICS COMMUNICATIONS

1 June 1986

T E R R E S T R I A L O P T I C A L A P E R T U R E S Y N T H E S I S T E C H N I Q U E (TOAST) A.H. G R E E N A W A Y Centre for Theoretical Studies, Royal Signals and Radar Establishment, St. Andrews Road, Malvern WR14 3PS, UK Received 22 November 1985

An aperture-synthetic, high-resolution imaging technique, for obtaining near diffraction-limited images from a single telescope is described. The scheme yields model-independent reconstructions of the object under study and is suitable for operation simultaneouslywith other observingprogrammessuch as spectroscopicstudies or speckle imaging. A specificscheme suitable for use on the 2m5 Isaac Newton Telescopeis presented, which would have a magnitude limit of - 11.

1. Introduction

2. Redundant spacings calibration (RSC)

The principles of aperture synthesis [1-3] are well known, as are many of the associated data processing techniques [ 3 - 5 ] . These standard approaches yield a parametric solution to the imaging problem, the parameters being determined using algorithms that amount to model building. Modifications facilitating model-independent image reconstructions have been proposed and analysed [6-9] and are here presented in a form that is suitable for use on a ground-based telescope. As applied to such a telescope, the method uses a mask to define several apertures (with dimensions of the order of the correlation length of the atmospherically distorted wavefront), a high frequency phase shifting device (working at ~1 kHz) and a highquality, spatially-sensitive, photon-counting detector. Only a small fraction of the flux collected by the telescope primary passes through the sub-apertures defined by the mask, most of the flux is therefore available for other observing purposes, such as speckle imaging [10-12] or spectroscopic studies. This advantage is tempered somewhat by the bright magnitude limit associated with the technique, which will be shown here to be approximately +I 1. However, the experiment proposed should be seen as a prototype for a very high-resolution spacecraft-borne instrument [9] and as an independent validation of the various speckle imaging techniques.

The principles of RSC as a means for avoiding the parametric solution associated with aperture synthesis using an array of non-redundantly spaced apertures, have been described elsewhere [9]. The method is capable of producing very high dynamic range images of the object under study [6]. The Fourier transform of the brightness distribution of a self-luminous object may be determined by observing the cross-correlation between the complex amplitudes at a series of spatially separated points [13]. In an optical system, a convenient way to achieve this would be to form an image of the object through a mask consisting of a set of small apertures [2], by means of which the sum of the interference of the radiation passing through every aperture pair may be measured. In any real experiment these data are corrupted by imperfections in the instrument, by atmospherically induced phase and amplitude errors and by measurement noise. The task facing the experimenter is to eliminate, or at least mitigate, the effects of these errors on the synthesised image. The first two errors mentioned above corrupt the measurements in a way that may be described by combinations of a small set of parameters, one for each of the apertures. A set of N apertures provides up to N ( N - 1)/2 measurements of the object Fourier transform (the number of ways in which an aperture pair may be selected). However, the number of un149

Volume 58, number 3

OPTICS COMMUNICATIONS

knowns in the system (object spectrum plus aperture errors) is N(N + 1)/2, slightly greater than the number of measurements. Since there are more unknowns than measurements, there exists an infinity (i.e. a continuum) of possible solutions in a multi-dimensional space, even in the absence of measurement noise. That is, the problem is ill-posed. Conventional aperture synthesis, as applied in radio astronomy, selects one of these solutions by imposing a priori conditions on the reconstruction and by building models of the object that satisfy these conditions and are consistent with the measured data. Such an approach may allow for measurement noise by stipulating that the model need only generate the measured data to within the known measurement tolerance. Unfortunately, a change in the a priori assumptions could result in a qualitatively different solution but one which was equally acceptable in a minimum norm sense. One way to identify such ambiguities is to make several reconstructions with variations in the a priori assumptions, but such a procedure is computationally expensive. RSC recognises the fact that if two aperture pairs have the same vector spacing they represent measurement of the same object parameter, thus reducing by one the number of unknowns in the problem. I f N aperture pairs correspond to repeated (i.e. redundant) spacings the number of unknowns and the number of potential measurements are equal, and thus one may expect to obtain a solution without recourse to a priori models. The solution thus obtained is unique [9]. If there are more redundant spacings than the minimum required to match the number of measurements and unknowns, the solution is unique in the minimum norm sense. If one attempts to apply RSC to a simple aperturemask imaging experiment, one finds that the idea cannot be applied directly because aperture pairs with redundant spacings provide only one measurement, in which information from the redundant aperture pairs is inextricably mixed together. To circumvent this difficulty proposals have been made to use specially designed interferometers that re-map the information and permit one to unscramble the interference between aperture pairs with the same spacing [9,14,15]. These proposals have advantages over the technique suggested here, since they permit one to make all the interference fringes parallel and thus use a dispersive 150

1 June 1986

device to make measurements simultaneously at all wavelengths. However, such interferometers are not easy to construct. By contrast, TOAST replaces the specially designed interferometers with a high frequency vibrating mirror. This mirror needs to be driven repeatably over an excursion 1/am and at frequencies ideally in excess of 1 kHz.

3. Redundant spacings calibration using TOAST

Consider an obscured objective like that of the Isaac Newton Telescope of the La Palma Observatory. Suppose that the objective of the telescope is reimaged at some suitable point in an optical system and obscured with a mask of the form shown in fig. 1. An image of the object formed through such a mask will consist of many superposed fringe patterns, one for each aperture pair in the mask. The mask of fig. 1 contains 27 pairs of apertures with identical vector spacings. Each repeated spacing occurs exactly twice

Fig. 1. A 15 aperture TOAST scheme suitable for the Isaac Newton Telescope. The inner circle (cross hatched) represents the area obscured by the telescope secondary mirror. The outermost circle indicates the edge of the 2m5 primary mirror. The intermediate circle (vcrtical shading) represents the area covered by the vibrating mirror referred to in the text. The 15 small circles represent apertures in a mask that otherwise obscures the telescope primary. This mask could be a pierced mirror and it is this that permits TOAST to be operated simultaneously with other observing programmes.

Volume 58, number 3

OPTICS COMMUNICATIONS

and each aperture participates in at least two of the 27 repeated spacings. Referring to apertures as inner or outer according to whether they lie inside or outside the shaded area of fig. 1, each repeated spacing involves one inner/outer pair and either an outer/ outer pair or an inner/inner pair. Now, assume that any instrumental and atmospherically induced errors are frozen over the time interval under consideration. Light passing through the aper-

1 June 1986

ture mask is reflected off a mirror which has been divided into two concentric regions (fig. 1). The inner region covers the central obscuration of the telescope objective and the three innermost holes in the mask. The outer region of this mirror covers the remaining 12 apertures defined by the mask. Consider what happens if the inner region is piezo-electricaUy driven to produce a phase shift, over the three central apertures, that varies periodically with respect to the outer 12 apertures. One may then identify two distinct sorts of fringe systems produced by interference between the light passing through the various aperture pairs. (i) Inner/inner and outer/outer aperture pairs produce fringes that are fixed in space. That is, the positions, orientations and visibilities of such fringes will be unaffected by the displacement of the inner region of the mirror in a direction perpendicular to the plane of the aperture mask. Thus spatial properties of this fringe pattern are constant in time (remember, the atmosphere and instrumental defects are presumed frozen), and the amplitudes and relative phases of the fringes may be deduced from the spatial variations in the intensity pattern falling on the detector photocathode. If one examines the output of the detector at a fixed point on the photocathode as a function of time, these fringes contribute only a constant background. (ii) Conversely, inner/outer aperture pairs produce fringes that are subject to a periodic displacement of their position on the detector photocathode. As such these fringes will, if the phase shift has an amplitude exceeding 21r, "wash out" in time and be reduced to a uniform background. However, if one were to look at Fig. 2. Spatial frequency coversge for the aperture scheme of fig. 1. (a) The spatial frequencies present in the three dimensional Fourier transform of the detector output. The points represent spatial frequencies that appear only in the d.c. image ( ~(~, 0)) in eq. (3)). The open circles represent spatial frequencies that only appear in time-varying image ( 9 07, ±co)in eq. (3)). (b) The intrinsic response function of the TOAST system described. The grating response associated with the aperture grid is clearly vis1~ole.The l'mal point spread achieved is the product of this with the Fourier transform of one of the apertures in the mask. The intensity scale is linear. (c) The point spread function achieved by TOAST assuming individual apertures of 20 cm diameter. Use of a second piezo stack would allow one t O add more apertures to the scheme and significantly reduCe the side-lobes,barely visible in the plot. On the right is the!impulse response of a perfect 2.5 m dish, for comparison. The intensity scale is again linear. 151

Volume 58, number 3

OPTICS COMMUNICATIONS

the detector output at a fixed position on the detector, one would be able to 'see' the fringe contrast and phase as a function of time. Further, for any given spatial fringe period, the temporal signal to which it gives rise will be 7r out of phase at points on the detector separated by exactly half the spatial period, and exactly in phase at points separated by the spatial fringe period. Thus one has one set of fringes that are only visible in the long-time average image and another set only visible as a function of both time and space. These fringe patterns may be disentangled by taking the three-dimensional (time and space) Fourier transform of the detector output. This task is not too demanding, since the computation may be restricted to 69 spatial frequencies in the long-time average image and 36 in the time-varying image. The 105 terms thus defined permit one to uniquely identify the interference effects associated with each aperture pair. The static and time dependent terms, the spatial frequency coverage and the point spread function that the method yields are illustrated in fig. 2.

4. An algebraic analysis of TOAST The intensity detected in a TOAST experiment may be written

l(x, t) = f a g

-

× A*(~') exp[ix(~ - ~')],

t) M*(~', t) (1)

where g is the co-ordinate in the telescope pupil of plane, 0(~) is the Fourier spectrum (complex degree of coherence) of the object under study, and A(~) represents the frozen turbulent atmosphere and any mirror or other optical imperfections, and M(g, t) = P(~) + Q(~) exp(i6ot),

(2)

1 June 1986

represents the TOAST aperture mask and phase shifter The three-dimensional Fourier transform of I(x, t) may be written

9(7,

=f

dt I(x, t) exp [i(x .r/+ tv)].

Substituting (1) and (2) in the above yields

9 (n,

=f

dt

O(r/)

+

+ Q(~) Q*(~ + r/)] A(~)A*(~ + rl) exp(itv)

+

dt O(rl) (Q(~)P*(~ + 7?) exp [it(v + co)]

+ P(~) Q*(~ + r/) exp [it(v - w)] } A (~) A*(~ + rT). (3) Within the assumptions that have been made, the first term on the right hand side of (3) is zero V v :/: 0. For v = 0 the first term is non-zero only for those values of ~7for which the sum of the autocorrelations P ® P and Q ® Q are non-zero. Conversely, the second term on the right hand side of equation (3) is zero V v 4: +~. For v = +~, it is non-zero at those values of for which the cross-correlation of P and Q are non-zero. As required, this proves that the TOAST encoding scheme permits one to uniquely identify the 105 interference terms discussed in the previous section. Naturally, the atmospheric function A(~) is not really frozen in time, but the analysis given here is an adequate approximation if the integration over time in eq. (3) is ~vindowed' to restrict it to an appropriate range, i.e. processed in 'frames' chosen according to the atmospheric relaxation time.

5. Accuracy and limiting magnitudes for TOAST

with P(~) = 1, = 0,

Q(~) = 1,

for the 12 apertures not affected by the phase shifter, elsewhere, for the 3 apertures that are subject to a periodic phase shift with angular frequency ¢O,

= 0, 152

elsewhere,

Measurement noise, and its effect on the method proposed, has thus far been ignored. Such noise cannot be described by a small set of parameters that may be identified with individual apertures and thus cannot be corrected by RSC (although the use of redundant spacings may permit one to ameliorate the effects of such error by least squares or other statistical techniques). The main source of measurement noise

Volume 58, number 3

OPTICS COMMUNICATIONS

will be due to the small number of photons detected during the short time interval in which it may be presumed that the atmosphere is frozen. The effect of photon statistics on the estimation of the phase of the Fourier transform of an intensity distribution has already been dealt with [16,17] and it has been shown that the standard deviation on the phase may be approximated by N o = 2AqzAXIO(~)I2108_0.4 m

1 June 1986 STELLAR MAfiNITUDE

16 I

iJ

i

l

i

10

I

12

o" ~15

~

1.C

z

~,/10

)1/2 radians

(4)

;~/SO ;VIO0 I0s

where N is the number of apertures in the system (here 15),A is the area of each aperture (m2), q is the optical efficiency (optics plus detector), z is the integration time (s), AX is the bandpass used (nm), O(~) is the Fourier transform of the object brightness distribution, normalized such that O(0) = 1, and m is the visual magnitude of the object under study. The maximum value for the integration time will be dictated by the atmospheric relaxation time (typically 10 -2 s) and the phase shifter acting on the three inner apertures will need to move through at least 2~t several times during this interval. It is this that dictates a frequency ~1 kHz and an excursion of 1 #m for the piezo-driven mirror suggested for the phase shifting device. The phase variance has been plotted in fig. 3, for point objects, as a function of the visual magnitude of the object under study and for the following parameters - subapertures area 0.04 m 2, optical efficiency 0.05, a ban@ass 100 nm and an atmospheric relaxation time 10 -2 s. If the variance on the phase at any point is one radian or less it should be possible to average successive reconstructions to improve the quality of the final image of the object. This point may, therefore, be taken to represent the limiting magnitude for the scheme which, from the figure, is m~+ll. From tables [18] there are, on average, 11 objects brighter than this per square degree of sky. Thus, although this is a very bright magnitude limit there are, none the less, many objects that could be studied with TOAST. Since RSC has been used there is no ambiguity in the phase deduced, even if the object overfills the field. In consequence, TOAST could also be used to image the solar surface.

I0 z'

I03

102

I#

DETEZTEO PHOTONSPERAPERTURE

Fig. 3. Effect of photon statistics on phase determination in a 15 aperture system. The statistical error in the phase is shown in radians on the left and as a fraction of the wavelength on the right. On the lower axis the mean number of detected photons passing through each aperture is shown. These three axes form a graph applicable to any 15 aperture scheme. On the upper axis the equivalent object magnitude, corresponding to the parameters discussed in the text, is shown.

6. Discussion TOAST can be used to obtain near diffractionlimited images of objects brighter than visual magnitude about +11, and is suitable for simultaneous application with other observing programmes on bright objects. In terms of the spatial resolution and data rate the technique will work with detectors currently available, but its potential for the highest signal-tonoise will only be realised when photon-counting detectors capable of rates up to 108 photons s-1 become available. The data reduction techniques required, although intrinsically amenable to hard-wired real-time solution [9], will probably be beyond the real-time capacity of current, general-purpose systems. As a result of the bright limiting magnitude it may be doubted whether the TOAST will enable much new astronomy to be done, but it will form a very necessary ground-based 'proof-of-concept' for more ambitious satellite-based proposals [9,14,15] and also offers a useful tool for studying the effects of the atmosphere on a propagating wave. The aperture arrangement proposed here is neither unique nor is it claimed to be optimal. Note that the 153

Volume 58, number 3

OPTICS COMMUNICATIONS

use of redundancy permits one to use more apertures within the telescope objective than would be possible with non-redundant schemes. This greater density of apertures permits one to synthesise a better point spread function. At the cost of the additional complication of adding a second piezo stack to drive another segment of the mirror at a different temporal frequency, more apertures could be added to the scheme and the shape of the impulse response function further improved.

Acknowledgements The author acknowledges extensive discussion with A.W.S. Williams. Discussion with his colleagues on the OASIS project [9], with J. Bregman and U. Schwarz has been facilitated by NATO Research Grant 723/84.

References [1] R.C. Jennison, Mon. Not. R. Astr. Soe. 118 (1958) 276. [2] W.T. Rhodes and J.W. Goodman, J. Opt. Soe. Am. 63 (1973) 647.

154

1 June 1986

[3] A.R. Thompson and L.R. D'Addario, Synthesis mapping, National Radio Astronomy Observatory, October 1982. [4] U.J. Sehwazz, Astron. Astrophys. 65 (1978) 345. [5] C. Van Sehoonoveld, Image formation from coherence functions in astronomy (Reidel, Dordreeht, 1979). [6] J.E. Noordam and A.G. de Bruyn, Nature 299 (1982) 597. [7] A~I. Grcenaway, Optics Comm. 42 (1982) 157. [8] N.R. Arnot, Optics Comm. 45 (1983) 380. [9] N.R. Arnot, P.D. Atherton, A.H. Greenaway and J.E. Noordarn, Traitement du Signal 2 (1985) 129. [10] K.T. Knox and B.J. Thompson, Astrophys. J. 193 (1974) L45. [11] G.P. Weigelt, Optics Comm. 21 (1977) 55. [12] J.G. Walker, Appl. Optics 21 (1982) 3132. [13] M. Born and E. Wolf, Principles of optics (Pergamon, London, 1959). [14] J~E. Noordam, P.D. Atherton and A.H. Grcenaway, in: Colloquium on Kflometric arrays in space, European Space Agency, 1984, pp. 63-69. [15 ] P.D. Atherton, A.H. Greenaway and J.E. Noordam, in: Colloquium on kilometric arrays in space, European Space Agency, 1984, pp. 141-144. [16] J.F. Walkup and J.W. Goodman, J. Opt. Soc. Am. 63 (1973) 399. [17] A.H. Greenaway, Optics Comm. 54 (1985) 75. [ 18 ] C.W. Allen, Astrophysical quantities (Athlone Press, London, 1973).