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Speckle Imaging Through the Atmosphere W.I. Beavers, D.E. Dudgeon, J.W. Beletic. and M. T. Lane Speckle Imaging through the Atmosphere The atmosphereis the limitingfactor inhigh-resolution ground-based optical telescope observations of objects in space. New speckle-imaging techniques allow astronomers to overcome atmospheric distortion and achieve the goal ofdiffraction-limited ground­ based telescope performance. Studies and experiments at Lincoln Laboratory utilize speckle imaging for observation of near-earth satellites. Thousands of separate exposures, each 2 to 5 ms in duration, are collected within a few seconds. A computationally intensive algorithm is then- used to reconstruct a single diffraction­ limited image from the collection of separate exposures. The image-reconstruction process effectively removes the dis.tortion imposed by the atmosphere. Photon noise, which limits the quality ofimage reconstruction, must be properly compensated bythe actual detector calibration. A textbook prediction of telescope resolving the seeingdisk, the wandering ofthe image, and power, or limiting detectable angularseparation the twinkling or scintillation are all conse­ (between double stars, for example), rarely de­ quences ofthe passage ofthe beam through the scribes actual telescope performance. Theoreti- atmosphere. "cally, the smallest resolvable angle efor a tele­ Until recent years, attempts to overcome the scopewith objective lens ormirrorofdiameterD, problems in seeing were directed primarily in visible light, is toward finding optimal telescope sites. A good telescope environment is uniform in tempera­ ture and free from turbulent air. Observatory domes are unheated, and the telescope mirror, where D is in meters and eis in seconds of arc. which is usually in an open-tube framework, If no atmosphere were present, ewould be the receives special ventilation. In spite ofextensive limit of resolution for any given telescope. effort and great care to increase resolution, However, turbulence in the atmosphere seeing disks in the range of 1 to 2 arc sec are the causes a point oflight in space, such as a star, general rule at most observatories, with rarer to appear through telescopes on earth as a periods of sub-are-sec seeing. puffed image, or seeing disk. The turbulence at any instantproduces variations in atmospheric The Earthbound View density, temperature, andindexofrefraction. As an example, the seeing disk undergoes excur­ A modelproposed byF. Roddier describes the sions or apparent rapid movement due to in­ seeingconditionsinatmosphericturbulence [1]. stantaneous tilts ofthe wavefronts reaching the Byconsidering time intervals ofless than 10ms, telescope. Furthermore, the twinkling, or scin­ the turbulenceinthe atmosphere can betreated tillation, of light, which is visible to the naked as a frozen pattern ofphase variations. Figure 1 eye, occurs in the signal of any detector at the shows an appropriate frozen pattern during a focal plane ofthe telescope. The twinkling repre­ brieftime interval. Theincomingwavefrontfrom sents changes in the instantaneous brightness a distant object above the atmosphere is sepa­ ofa starorpoint-source object atfrequencies up rated into constantphase cells ofdimension roin to a few hundred hertz. The angular spread of the planeofthe telescope objective. (Iftherewere The Lincoln Laboratory Journal. Volume 2. Number 2 (1989) 207 Beavers et aI. - Speckle Imaging through the Atmosphere quasi-monochromatic field in the focal plane Aperture leads to an illumination, or image intensity, of N 2 N 2 2 I 0/ 1 = I L o/i I = L I o/i 1 + L L 0/; 'Ij. (1) i = 1 i = 1 i '" j j The instantaneous image intensity results from the sum of two different terms in Eq. 1. The first tenn on the right-hand side ofthe equation is the incoherent superposition of the intensities of all ---,,;.---+----.........---~ IlfII 2 the individual subapertures. Since the subaper­ tures are assumed to be ro in size, the diffrac­ tion limit 8 for a subaperture is given by d _ 0.12 N Subapertures ed--- '0 This contribution to the image intensity, which deSCribes the seeing disk, contains no high­ N spatial-frequency information. Figure 2 shows 2 ~ 1lf11 + ~ ~ lfI; lfI.* the relationship between spatial frequencies in ;=1 ;*ii J the focal plane and distances in the objective Sum of planes. Uncorrelated The second tenn in Eq. 1, a sum of cross Seeing Random products. describes the interference between Disk Phasors the subapertures. In effect, the subapertures Applicable ---------------- Spatial Frequencies Low High form a multiaperture interferometer with ran­ dom phase differences between the separate Intensity ~ N N 1/2 elements. The second term contains high-spa­ Power ~ N 2 N tial-frequency information from all the various Normalized Power ~ 1 1/ N orientation and baseline combinations present in the subaperture pairs. Fig. 1-The division of the telescope aperture into N effec­ tive subapertures. The image results from two independent The combination of the two terms in Eg. 1 terms with different dependences on N. The first term produces an instantaneous image of the puffed describes the superposition of the intensities of al/ the seeing disk, with two primary features. The individual subaperlures, andthe second term describes the interference between the subapertures. image fills the seeing disk, described by the first tenn, within which there is a fine scale structure of bright spots or patches called spec/des, de­ scribed by the second tenn. The speckles con­ no atmosphere, the incoming wavefront would tain spatial frequency information up to the have constant phase over the entire objective diffraction limit ofthe telescope, but the speckle surface.) In subsequent time intervals, a new pattern changes rapidly, generally lasting for cell pattern emerges that is different in detail but time durations less than 10 to 20 ms. similar in statistical properties such as the size and number of the constant-phase cells or effec­ Modern Solutions to Atmospheric tive subapertures. Diffraction theory is used to Distortion calculate the resulting image in the telescope focal plane as the sum of the contributions from Attempts to overcome the seeing problem all the individual subapertures. With a total ofN created by atmospheric distortion have taken subapertures, each contributing a complex two approaches. In the first approach, mechani­ amplitude !P., the complex amplitude lfI of the cal compensation of a mirror in the light path I 208 The Lincoln Laboratory Journal. Volume 2. Number 2 (1989) Beavers et aI. - Speckle Imaging through the Atmosphere Telescope Aperture Plane u = Q ~ = Spatial Frequency A f Fig. 2-The relationship between spatial frequencies in the image plane, and separations in the telescope aperture plane. produces real-time corrections to the wavefront made the first major contributionwhen he noted distortions [21. In the second approach (de­ that the speckle structure in very-short-expo­ scribed in this article). a reconstructed image is sure astronomical images was similar to the obtained by calculating the Fourier transform of structure observed with laser-illuminated dif­ the combination of amplitude and phase map fusers [3J. He applied a formalism to the problem produced by the evaluation ofthe power spectra (similar to what follows below) and extracted and bispectra for thousands of noisy short high-spatial-frequency information not previ­ exposures. ously available to astronomers. The process he developed is known as speckle interferometry. Speckle Interferometry Let i (x) represent the observed two-dimen­ sional image i (x, y). and let 0 (x) represent the Astronomers in the past two decades devel­ corresponding object 0 (x. y). The combined tele­ oped several techniques to minimize the effects scope-atmosphere point-spread function t (x) of atmospheric distortion. In 1970 A. Labeyne describes the light distribution when a point of The Lincoln Laboratory Journal. Volume 2. Number 2 (1989) 209 Beavers et aI. - Speckle Imaging through the Atmosphere light is imaged by the telescope. In this isopla­ the object's power spectrum contains informa­ natic case (see the box. "Isoplanatism·'). tion only about amplitudes. and not about the object's phases. With simple objects such as i(x) = a(x) * L(x) (2) double stars. the power spectrum reveals only where * denotes the two-dimensional convolu­ the angular separation. orientation (with redun­ tion for the imaging process. In the Fourier dancy of 180°), and magnitude difference be­ transform. or spatial-frequency representation. tween the star pair. With more complicated of Eq. 2. the isoplanatic condition is written as objects. such as satellites, speckle interferom­ etry is not appropriate, and considerable effort J(u) = O(u) • T(u) (3) has gone into finding methods to obtain phase where u is the two-dimensional spatial-fre­ information for the optical imaging task [5]. quency variable corresponding to the spatial variable x. Phase-Recovery Techniques Labeyrie faced an experimental dilemma. A highly magnified image ofa point source such as The Knox-Thompson (KT) method. one of the a star could be obtained in a brief photographic first studies to provide impressive results on the exposure. However. the high magnification recovery of object phases. appeared in 1974 [61. spread the light on his detector so that little total The KT method utilizes a general second-order exposure resulted. Clearly. a single exposure moment. or cross spectrum. instead of a power that froze the turbulence was not sufficient. spectrum. By substituting expressions like Additional exposures could be made. but they <f(u)I*(u + ~u» for the power spectrum. Eq. 4 produced different samples of the turbulence. becomes and averaging many exposures directly would simply reproduce the bad seeing result. Instead. < I(u) l*(u + t.u) > = O(u) O*(u + t.u) Labeyrie concentrated his attention on the • < T(u) T'(u + t.u) > (5) power spectra of the set of frames. The time average of the power in Eq. 3 is where only spatial-frequency differences up to a value of half the seeing limit, or I~ul :::; 0.5 ral A. are considered. When telescope aberrations are negligible. the cross-spectral transfer function The left-hand side of Eq.
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