Fourier-transform spectral imaging: retrieval of source...

94 J. Opt. Soc. Am. A/Vol. 3, No. 1/January 1986

Fourier-transform spectral imaging:retrieval of source information from three-dimensional

spatial coherence

Kazuyoshi Itoh and Yoshihiro Ohtsuka

Department of Engineering Science, Hokkaido University, Sapporo 060, Japan

Received April 3, 1985; accepted July 16, 1985

A method is described for efficiently obtaining the comprehensive information of a polychromatic radiator. Undercertain conditions both the spatial and spectral details of the radiative object can be recovered simultaneously fromthe three-dimensional spatial coherence function in the diffraction region. The recovery of object information isbased on a Fourier-transform relationship derived from the basic formula [E. Wolf and W. H. Carter, J. Opt. Soc.Am. 68, 953-964 (1978)] describing the field correlation function in terms of the source correlation function. A newtype of interferometer is proposed for the efficient collection of the spatial coherence data. Experimental results ofthe spectral-image recovery are also presented.

INTRODUCTION

The spatial and spectral power densities of a light source aresometimes deduced from the second-order correlation func-tions of the optical radiation. Each of the density functionsand the corresponding correlation function are known toform a Fourier-transform pair. These relationships are re-ferred to as Wiener-Khintchine and Van Cittert-Zerniketheorems'-4 and provide, respectively, the bases for Fouierspectroscopy 5 ,6 and interferometric imaging.7-14

Unfortunately, however, general radiative objects cannotuniquely be identified by these separate densities. Forunique identification, the general power density G(x, y, v)that is dependent on both spatial position (x, y) and opticalfrequency v is desired. This is the motivation for the variousapproaches1 5-2 0 to spectral imaging or imaging spectroscopy.The simplest approach to spectral imaging might be to takemany photographs through a series of spectral filters. Forprecise measurements, more elaborate filters such as thetriple Fabry-Perot interferometers1 5 may be employed. Ac-cording to circumstances one may take the alternative ap-proaches-spectroscopy at each pixel16 or line by line.1 7 Inparticular applications, these simple approaches have beenproved to be useful,15-' 7 but in general they lack efficiency.This is because most of the incident flux is rejected by thespectral filter or the spectrometer slit. The multiplexingtechnique'8 employed in Hadamard-transform spectrom-eters may improve the efficiency. The Hadamard-trans-form imaging spectrometers' 8' 19 literally aim to measureG(x, y, v), yet they have an additional purpose of using asingle detector with the doubly multiplexing technique.Another interesting approach is speckle spectroscopy,2 0

which offers diffraction-limited prism spectra of stellar ob-jects.

In this paper, we present a method of spectral imagingbased on correlation measurements. It is shown that undercertain conditions G(x, y, v) can be related to the three-dimensional (3-D) spatial coherence in a simple Fourier-transform relationship. This relationship rests on the gen-

eral integral equation derived by Wolf and Carter.2 1 Theresult can be thought of as a unification of Wiener-Khint-chine and Van Cittert-Zernike theorems and may serve togive an insight into the spatial coherence properties2 2 -25 of apartially coherent, polychromatic wave field.

The new formulation of 3-D Fourier synthesis can be in-corporated with an optical correlator to realize an efficientspectral imager. A crude version of the spectral imager hasbeen constructed by modifying a wave-front folding inter-ferometer.7' 26 The experimental results of spectral imagerecovery using a two-color point object are presented.

Because of the similarity of situations, readers should bereferred to the series of work by Baltes and co-workers.2 7-32

They studied theoretically2 7 -30 and experimentally31' 32 theutility of the far-zone intensity and correlation function fordetecting a hidden structure such as a grating within a fluc-tuating partially coherent source. The space-time intensitycorrelation technique was also studied by Newman andDainty.3 3 In this paper, we are concerned not with suchstructured partially coherent sources but with polychromat-ic natural incoherent sources.

FORMULATIONConsider a plane polychromatic incoherent source (a) in theplane of z = 0 in a Cartesian coordinate system as shown inFig. 1(a). Let the general power-density function associatedwith this planar source be denoted by G(P, v), where P is atwo-dimensional (2-D) position vector (x, y) and v is theoptical frequency. We are interested in the relationshipbetween G(P, v) and the spatial coherence function in thediffraction region. Let the mutual coherence function attwo points in the diffraction region be denoted by r(Q1, Q2,T), where Q, and Q2 are 3-D position vectors [Q1 = (xi, yi, z,),Q2 = (x2, Y2, Z2 )] and T stands for the time delay. The cross-spectral density function of the primary source can be ex-pressed in terms of the general power density by G(P1, P2, ii)= G(P,, Pe)6(P, - P 2), where Pi = (xi, Y1), P 2 = (x2, Y2) and6( ) is the Dirac delta function.

The propagation equation for the mutual coherence func-

0740-3232/86/010094-07$02.00 © 1986 Optical Society of America

K. Itoh and Y. Ohtsuka

Vol. 3, No. 1/January 1986/J. Opt. Soc. Am. A 95

1, 2, and dP = dxdy. The density function [G(P, v)] com-pletely vanishes outside the source area.

To obtain a simpler expression we assume here that thefollowing equation holds:

(2)

where

k = k(Z - P)/|Z -PI,Z (0, 0, Z),

r = -R2 = Q- Q2.Fortunately, this approximation conforms to rather com-mon situations where the incident flux can be regarded as anincoherent superposition of plane waves. Such a require-ment can largely be met by the usual astronomical observa-tions. Let the observation points Q, and Q2 be locatedinside a sphere that has a radius a and is centered on (0, 0, Z)as shown in Fig. 1(a). The incident wave fronts are wellapproximated by an ensemble of plane waves if the followingcondition holds:

Z >> 7ra2/x,

L

Qi

(b)

Fig. 1. Geometry: (a) free-space propagation and (b) propagationthrough a positive lens. The mutual coherence function propagatesfrom polychromatic incoherent source a to an observation area,where the mutual coherence function at Qi and Q2 without the timedelay is evaluated.

tion of a polychromatic partially coherent field is wellknown.1-4 The equation can be used for obtaining the ap-proximate relationship. However, we adopt here the recentresult due to Wolf and Carter. 2' They have drawn a cleardistinction between the source and field descriptions andderived the exact relationship between the source and fieldcross-spectral densities.

The equation associated with a completely incoherentsource [Eq. (2.21) of Ref. 21] is relevant to the present situa-tion. Integration of the equation for the field cross-spectraldensity with respect to v gives the mutual coherence func-tion with r = 0,

r(Q1, Q2 , 0) = J J J (IR,11R 2 1)-'G(Pv)

X exphk(C}R1 -1R 2 1)jdvdP, (1)

where j = VJET, k is the wave number, Rm = Qm-OP form =

where X is the wavelength. This plane-wave approximationis somewhat related to the Fraunhofer approximation. Ifthe sphere is replaced by a circular aperture of the samediameter and the aperture is illuminated from behind by aplane monochromatic wave, expression (3) is exactly thecondition for observing the Fraunhofer diffraction patternon the source plane. Note that our approximation does notrequire the source size to be small.

Now that the incident light can be regarded as an ensem-ble of plane waves, the position vector (P) of an elementary

Z source determines the angle of arrival of the plane wavesemanating therefrom. Let o and 6 be defined as in Fig. 1(a).Then position vector P can be represented in terms of 1 andO as

P = (Z tan 0 cos O, Z tan40 sin 0).

It is reasonable to assume that a >> X. Then we have fromexpression (3) Z >> a, and it follows that

I R1j _ I R 2I1 _-Z/cos X

Substitution of expression (2) and Eqs. (4) and (5) into Eq.(1) gives

r"r) =- f Z G(Z tan 0 cos 6, Z tan k sin 0, v)Jo J=r/2 Jo

X exp(jk -r)tan 0 cos OdvdodO,

where r(r) = r(Q + r, Q, 0). Since Eq. (6) can be thought ofas an integral equation in the 3-D polar coordinate systemv-0k-0, we can obtain the alternative formula in Cartesiancoordinate system u-v-w. Let s denote a position vector (u,v, w). Then Eq. (6) can be rewritten as

rF(t) = JJJ G,(s)exp(-27rt - s)ds,

where t = r/c, c is the velocity of light, ds = dudvdw, and

r,(t) = r(ct),

G,(s) = G(Zu/w, Zv/w, Is l )/(-I s ) (8b)

y

0

P

X

so

a,(a)

Y

I0

(3)

(4)

(5)

(6)

(7)

(8a)


k(JR11 -JR21)


S V

U

W

Fig. 2. Spectral image of a monochromatic point object. Thespatial information is preserved in the angular position of image,while the spectral information is encoded in the distance betweenthe origin and the point of image.

This is the final result expressing the simple relationshipbetween the 3-D spatial coherence and the power density.The equation can readily be inverted to give an expressionfor GC(s), which contains all the information about the densi-ty function. Recovery of GC(s) based on this Fourier-trans-form relationship may be called the Fourier-transform spec-tral imaging.

Next we consider another common situation as depictedin Fig. 1(b). The planar source is located in the front focalplane of positive lens L. In this case, we also arrive at theidentical expression for Ft(t) as in Eq. (7).

We first rewrite Eq. (1) according to the Huygens-Fresnelprinciple as

r(Q1, Q2,0 ) = JJ K(Q1, P, v)

X K*(Q 2 , PI v)G(P, V)dvdP, (9)

where * denotes the complex conjugate and K(Qm, PV, v) isthe optical response at Qm to a monochromatic elementarypoint source of frequency v located at P for m = 1, 2. In theprevious situation, as shown in Fig. 1(a), the response func-tion was taken to be a spherical wave:

K(QmP, ) = JRm. exp(k I Rm) I). (10)

In the present situation, as is clear from Fig. 1(b), the opticalwave field behind the lens is exactly an incoherent superpo-siton of plane waves. Let the focal length be Z and 0 and Obe defined as in the figure. Then the amplitudes of theplane waves may be approximated by (-Z/cos 0)-k. Thenew response function is thus given by

K'(Qm, P, v) = -Z-Z expUj(k -R'm - kZ/cos k)]cos 0, (11)

where k is defined in the same way as in Eq. (2), R'm = Rm -PO' for m = 1, 2. Since position vector P can also be repre-sented by Eq. (4), we obtain the same equation as Eqs. (6)and (7). In this case the final result, identical with Eq. (7),

holds exactly, provided that the lens is perfect and placed atthe correct position.

Finally the interrelationship between the real object andthe reconstructed image should be clarified. Referring toFig. 1(a), consider a monochromatic point source of opticalfrequency v located at P = (x, y) in the z = 0 plane. Wemeasure the mutual coherence function around a point Z =(0, 0, Z). Since the observing site is fixed, location P can bedesignated by the two values of 0 and O. The reconstructedimage will in this case appear as in Fig. 2. The angularposition of the image (S) is designated again by 0 and 6, butthe distance from the origin to S represents optical frequen-cy v. Note that v is different from w. This is because thepath difference associated with an obliquely incident planewave is greater than the actual longitudinal path differenceby a factor of (cos 0)-l. Thus, in contrast to Fourier spec-troscopic data, the off-axis effect is automatically incorpo-rated in the theory.

Representation of images in the angular coordinate sys-tem is not new but customary in astronomy. Radio imagesof Earth-rotation synthesis telescopes are also mapped onthe celestial sphere. It is interesting that the synthesistelescopes generally provide the information about the 3-Dspatial coherence function of radio wave field.34

36 TheEarth's rotation necessarily introduces 3-D loci of antennaspacing vectors, which are determined by the array geometryand the object position. It is even possible to synthesizeradio spectral images using an appropriately designed arrayand the Earth's rotation.

CORRELATOR

Our prototype instrument proposed for the 3-D correlationmeasurement is a simple combination of Michelson andwave-front folding interferometers. The Michelson inter-ferometer is a well-known instrument that exhibits the greatefficiency of longitudinal correlation measurement when ap-plied to Fourier spectroscopy.5,6 The wave-front foldinginterferometer, which was to the authors' knowledge firstsuggested by Murty, 2 6 is also an efficient instrument forlateral correlation measurement and extensively used forinterferometric imaging applications 9"'1-' 4' 37 and also for cal-ibration of atmospheric seeing. 38 40

The new instrument, called the volume interferometer, isshown schematically in Fig. 3. The incident light beam issplit by beam splitter BS into two parts; one, which will becalled the first beam, is reflected by right-angle prism P, andthe other, called the second beam, by P2. We take a Carte-sian coordinate system as shown in the figure. Let theoptical wave field at the z = 0 plane created by only the firstbeam be denoted by V(x, y, 0)/2. If the two beams areequally split by BS, the optical wave field produced by thesecond beam is V(-x, -y, 0)/2. Superposition of the twobeams simply results in folding of the wave front. Then alongitudinal motion of P2 produces an axial path difference.If the displacement of P 2 is denoted by L, the wave field overthe z = 0 plane is the superposition of V(x, y, 0)/2 and V(-x,y, -2L)/2. The interferogram obtained over the z = 0 planemight be a function of three parameters x, y, and L. Forconsistent notation, the last parameter L may be replaced byz without confusion, and then the interferogram can be rep-resented by'



Z "k

Fig. 3. Schematic of the volume interferometer. The light beamssplit by BS (beam splitter) are reflected back by P1 and P2 (right-angle prisms) and combined again at BS. The superposed wavefield suffers from lateral shear of 1800 rotation. The motion of P2

further introduces longitudinal shear, and thus a 3-D shear dis-placement is achieved.

I(r) = Jr(o) + [F(2r) + r(-2r)]/21/2, (12)

where the spatial stationarity of the mutual coherence func-tion is assumed. The (1/2) factor outside the curly bracesaccounts for the fact that one half of the incident flux isreflected by the interferometer. Note that the interfero-gram is mapped over the volume of an imagined space whose

x and y coordinates correspond to those of the actual spacebut the z coordinate to the displacement of the prism (rever-sal of the x coordinate due to P, is neglected for simplicity).

Image recovery is straightforward. The interferogram isappropriately scaled such that

It(t) = I(ct) = {Fr(0) + [rF(2t) + rt(-2t)]/21/2. (13)

From Eq. (7), the Fourier transform of It(t) is

It(s) = IG06(s) + [G,(s/2)/8 + G,(-s/2)/8]/21/2, (14)

where

Go = JJJ G,(s)ds.

The first and the third terms in Eq. (14) are spurious images,similar to those encountered in holography. However, incontrast to holography, these spurious images do not overlapthe true image because of the positivity of optical frequency.Since j is known to be ir/2 < 0 < 7r, discrimination betweenGC(s/2) and Gc(-s/2) is apparent. Thus extended objectscan safely be located anywhere in the field of view without

any confusion. It should also be mentioned for the mini-mum redundant measurements that sampling of the volumeinterferogram [I(r)] can be single sided or, in other words,

can be restricted to the region of x > 0, y > 0, or z > 0. This

is due to the reality of G,(s). Further discussions on the

techniques for more efficient sampling might almost be par-allel to those of Fourier spectroscopy5'6 and therefore areomitted.

LIMITATIONS

To discuss the performance limits of the volume interferom-eter, we consider a specific instrument shown in Fig. 3. Weassume that the 2-D interference pattern produced over thez = 0 plane, which is limited by a rectangular aperture of L,X L1, is detected by a space-filling array of N, X N, rectangu-lar detectors of size 1l X 1, and that the maximum displace-ment L3 of prism P2 is partitioned into N3 intervals of length13. The photocurrent generated in each detector element isintegrated during the time when P2 moves in each displace-ment interval with a constant velocity. As a result, thevolume interferogram measurable over a range of 2L, X 2L,X 2L3 is smoothed by a detection cell of rectangular solid 21,

X 21, X 213.

Resolution LimitsFrom the Fourier-transform relationship [Eq. (14)] betweenthe inteferograms and images, several limitations arise ow-ing to the maximum measurable range and the finite cell sizeof the volume interferogram. Most of them are familiar, butsome are peculiar to the volume interferometer.

The most fundamental are the resolution limits, which aregiven in terms of u, v, and w coordinates; bu = =v = c/2L, andbw = c/2L3. If L, >> L 3, as is usually the case, the spectralresolution can be approximated by Pv _ 6w, and spatialangular resolution by b6 4 bu/v = X/2L,. These limitationsare, respectively, equivalent to those in Fourier spectroscopyand of conventional telescopes. The wavelength depen-dence of the angular resolution implies its shift-variantproperty in the image space.

Under the same conditions, the measurable spectral rangeis also limited such that AP < c/413, in the same way as in

Fourier spectroscopy. The symmetric character of the spec-tral image space imposes also the similar limitations on thefield of view; the angular field of view is restricted by AX <X/21l. The throughput, 5' 6 which is proportional to the col-

lectable radiation power and is defined by the product of thearea of the entrance pupil and the acceptable solid angle, istherefore approximately (XN,/4)2 .

Finally it should be mentioned that the finite volume ofthe detection cell results in the instrument profile in theimage space, i.e., the spectral images suffer from position-dependent attenuations. The profile is, however, knownbeforehand to be a 3-D sinc function from the Fourier-transform relationship and may readily be compensated.

Noise LimitsThe signal-to-noise ratios (SNR's) of spectral images areprimarily limited by the signal dynamic range. This limita-tion, usually set by the quantization process, has alreadybeen studied5'6 in Fourier spectroscopy. Here we brieflydiscuss its consequences in Fourier-transform spectral imag-ing.

We consider a monochromatic cosinusoidal volume inter-ferogram. The final image is to be recovered as a 3-D dis-

crete Fourier transform (DFT) of the interferogram, whichis digitized and assumed to contain a small amount of quan-

K. itoh and Y. Ohtsuka


tization noise. Let the variance of the quantization noiseand the number of detection or resolution cells be denotedby Ur2 and M(=Nl 2 N3 ), respectively. For simplicity, weassume that the noise in the spectral image is a zero-meancircular complex Gaussian noise.41' 42 The variances of itsreal and imaginary parts are equal and known to be one halfof the second moment of its modulus.41 In this case, thenoise fluctuations along the real axis are relevant becausethe SNR is assumed to be large. Since it can be shown fromParseval's relation for the DFT4 3 that the second moment ofthe modulus is Maq2, the noise variance in the image isMaq2/2. On the other hand, the image signal component ofthe point object is one half of the total input energy, which isindependent of M provided that the area of the entrancepupil and the total observation time are fixed. Thus theSNR for the spectral image is EO/(aqx/~iM), where Eo is thetotal input energy. It can also be shown that the same resultis obtained when multiple objects are present. This result isessentially the same as the former analyses5' 6'42 in one di-mension. Nevertheless, it should be kept in mind that thenumber of resolution cells for the Fourier-transform spectralimaging tends to be extremely large compared with those forits one-dimensional analogs. A very high dynamic range ofinput signal is required for moderate SNR and resolution.The requisite high dynamic range is the price for the signifi-cant gain in efficiency. The ultimate performance will beachieved by using a photon-noise-limited detection system.

We now discuss the limitations due to the statistical fluc-tuations in a number of detected photoevents of signal andbackground radiation.

We first assume that we observe a cosinusoidal volumeinterferogram created by a monochromatic point object, asbefore. However the interferogram only contains the pho-ton noise. Here we use the result due to Walkup and Good-man.4 2 The rms SNR for the monochromatic point object isgiven by4 2

pO = No/[2(Nb + No)]'12, (15)

where No and Nb are, respectively, the expected total num-bers of signal and background photons detected over theentire volume of the interferogram. If multiple objects arepresent, Eq. (15) is slightly modified. Let the rms SNR of apoint object placed at s = so among the other objects bedenoted by po'. It can be shown that

po' = NO/[2(Nb + Nt)]"12 , (16)

where Nt is the expected total number of signal photonsregistered over the interferogram but No denotes only thecontribution from the point object located at s = so.

Comparison of these SNR's suggests that if the back-ground noise is predominant (Nb >> Nt, No), multiple objectscan be analyzed simultaneously with negligible loss of SNR.The simultaneous observation leads to the increase of obser-vation time for each object, and thus we gain the multiplexor Fellgett's advantage.5' 6 On the other hand, if the photonnoise predominates (Nb << No, Nt), the multiplex advantageis neutralized by the increase of the photon noise (No < Nt).

To derive a measure of practical performance limit, weconsider a situation where many stars scattered in the fieldof view are simultaneously analyzed. A limitation will bededuced in terms of the minimum observation time required

for achieving a desired SNR. We neglect the backgroundnoise (Nb = 0) and assume that the stars are equally brightand nearly white. We also neglect the atmospheric effects,which will be discussed later. The expected total number ofphotons to be registered over the interferogram can be relat-ed to apparent visual magnitude m, of the star at a wave-length of 5560 A by44

Nt = 7ABN3T 103-04m , (17)

where n is the detector quantum efficiency, A is the area ofthe entrance aperture in square centimeters, B is the spec-tral passband in angstroms, Ns is the number of stars in thefield of view, and T is the total observation time in seconds.Let 1 = 0.1, A = 104 cm2 , B = 103 , Ns = 102, and m, = 10.Then we can expect Nt = 107 T. If we assume that a spectralresolution of 6B = 10 A is necessary, the ratio between Nt andNo is fixed to be Nt/No = N3 (B/bB) = 104. Finally we take aSNR of 10 as the desired ratio. Then from Eq. (16) we haveT = 2 X 103 sec. Although the present estimation is basedon the relatively low SNR and poor spectral resolution, it isremarkable that it takes only less than an hour to analyzeboth the spatial distribution and spectral details of hun-dreds of star of tenth magnitude. Note that the photonnoise is uniformly distributed in the image space. If theradiation is not white but is composed of strong emissionlines, the SNR's at the corresponding image points are sig-nificantly enhanced. The volume interferometer may evencompete with the Medusa spectrograph,'6 one of the mostefficient and possibly most elaborate spectral imagers.

Atmospheric LimitsIf the present technique is directly applied to the earth-bound large telescopes in the visible range, the advantage ofthe large aperture size may not fully be gained owing toatmospheric turbulence. The size of the primary mirrormight effectively be reduced to typically 10 cm in diameter.This loss of effective aperture size is similar to that of con-ventional astrophotography. The wave-front fluctuationscaused by atmospheric turbulence smear out the high-spa-tial frequency information in the photograph or in the inter-ferogram. Some modification is necessary to take advan-tage of the large aperture size.

Rapid scanning for the longitudinal path difference isessential to realize a sturdy system against atmospheric dis-turbances. If the signal integration time for each detectioncell of the interferogram is less than 10 msec, the smearingwill largely be avoided.4 5 Scanning time of less than 10 sec istherefore short enough to record a high-contrast interfero-gram of size N3 = 103. To recover the high-quality image,however, repeated rapid scanning followed by additionaldata processing will be necessary. This problem of estimat-ing a high-quality image from an ensemble of phase-dis-turbed Fourier spectra or their moduli is one of the topics ofcurrent research activities.10-14 ,45,46 Here we only point outthe possibilities of making use of the earthbound large tele-scopes by the repeated rapid-scanning technique,

EXPERIMENT

The principle has been verified by an experiment. Sincereconstruction of a 2-D spectral image is sufficient for thedemonstration, a 2-D version of the volume interferometer



U

w

w

Fig. 4. Reconstructed spectral image of a point object composed ofHe-Cd (4416-A) and He-Ne (6328-A) laser spectra. Two different

aspects are shown along with the spurious images. The two spectralcomponents lie on the same radial line, which indicates the angularposition of object.

was constructed. The interferometer will be explained byreferring to the original shown in Fig. 3. The first prism (Pl)

was replaced by a plane mirror, and only one prism (P2) was

involved in the 2-D version. A linear image sensor was

placed in the z = 0 plane along the x axis. Thus the cross

section of y = 0 rather than the entire volume interferogramis observed. From the 3-D Fourier-transform relationship,we know that the 2-D Fourier transform of this cross section

represents the projection of the image onto the v = 0 plane.Recovery of a 2-D spectral image is achieved in this way,provided that the object has the spatial structure along the xaxis.

A point object with two spectral components was pro-

duced by focusing a light beam composed of He-Cd (4416-A)

and He-Ne (6328-A) laser spectra. To obtain these spectralcomponents of equal magnitude in the final image, the de-

tector spectral response was compensated by adjusting thelaser power. The object was placed slightly off the opticalaxis of the interferometer in the x direction. A positive lens

was placed between the object and interferometer to fill thecondition for case (b). in Fig. 1. The translator for the

longitudinal scanning was controlled by a microcomputer,and the data acquisition from the image sensor was synchro-nized with the motion of the translator; the motion wasstepwise rather than continuous.

The recovered image is shown in Fig. 4 along with the

spurious images [Eq. (14)]. Two different aspects of the

same image are shown. As the object is a single point with

two spectral components, we expect two impulses on a single

radial line. To emphasize this feature, the image is elongat-ed along the u axis by a factor of 103. The two spectral

components are clearly resolved and lie on a single radial

line.

CONCLUDING REMARKS

It has been shown that under realistic conditions the spatialand spectral information of a distant radiative object canefficiently be recorded in the volume interferogram and laterbe retrieved by a simple Fourier transformation. The inter-ferogram literally comprises the real part of the 3-D spatialcoherence function and is measured by the new interferome-ter, a combination of wave-front folding and Michelson in-terferometers. The principle of this method has been con-fired by experiment.

The brief assessment of limitations shows the utility of theproposed technique but at the same time suggests the prob-lems to be solved for the particular applications. Provided

that the photon flux is sufficient, the technique can be ap-plied directly to the simultaneous spectroscopy of manydifferent objects. The multiplex advantage can be gained,and the measurement time will be saved. Ultimate perfor-mance will, however, be achieved by a photon-noise-limitedsystem. For astronomical applications associated withearthbound large telescopes, it is necessary that an optimumalgorithm be developed to retrieve the high-resolution spec-tral image from the atmospherically disturbed volume inter-ferograms. The development is promising because most ofthe techniques for stellar speckle interferometry 45 are in

principle applicable.Applications in the infrared (IR) regions depend on the

availability of suitable IR image sensors. Rapid progress inthis field will soon make it possible to construct IR~spectralimagers in diverse spectral regions. In connection with IRastronomical applications, it should be emphasized that thecapability of spatial discrimination can significantly reducethe effects of sky background; if the object is a collection of

unresolvable stars, the signal of the individual star is affect-ed only by the small fraction of sky background inside the

same resolution cell.Finally we propose that the present technique be used in

combination with a space telescope. The nonredundancyinherent in the wave-front folding type of interferometermakes it easy to correct for the fixed aberrations.14 A highefficiency may be expected even in photon-noise-limitedshort-wavelength regions because the high throughput canbe realized by a high-density detector array.

ACKNOWLEDGMENTS

Part of this work was presented at the ICO-13 Congress in

Sapporo, August 20-24, 1984. K. Itoh wishes to thank G. P.Weigelt and W. T. Welford for the valuable comments relat-ed to Refs. 20 and 40, respectively. The authors would alsolike to thank the reviewers for valuable comments.

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14. K. Itoh and Y. Ohtsuka, "Interferometric image reconstructionthrough the turbulent atmosphere," Appl. Opt. 20, 4239-4244(1981).

15. See, for example, R. E. Loughhead, R. J. Bray, and N. Brown,"Instrument profile of a triple Fabry-Perot interferometer foruse in solar spectroscopy," Appl. Opt. 17, 415-419 (1978).

16. See, for example, J. M. Hill, J. R. P. Angel, J. S. Scott, D.Lindley, and P. Hintzen, "Multiple object spectroscopy: theMedusa spectrograph," Astrophys. J. 242, L69-L72 (1980).

17. See, for example, J. B. Breckinridge, N. A. Page, R. R. Shannon,and J. M. Rodgers, "Reflecting Schmidt imaging spectrome-ters," Appl. Opt. 22, 1175-1180 (1983).

18. M. Harwit and N. J. A. Sloane, Hadamard Transform Optics(Academic, New York, 1979), Chaps. 3-5.

19. H. Sakai, "Doubly multiplex Fourier spectroscopy," J. Opt. Soc.Am. 73, 1947 (1983).

20. G. Weigelt, "Speckle interferometry, speckle holography,speckle spectroscopy, and reconstruction of high-resolution im-ages from space telescope," in Proceedings, ESO Conference onScientific Importance of High Angular Resolution at IR andOptical Wavelengths, M. H. Ulrich and K. Kjar, eds. (EuropeanSpace Organization, Garching, 1981), pp. 95-113.

21. E. Wolf and W. H. Carter, "Coherence and radiant intensity inscalar wave fields generated by fluctuating primary planar sour-ces," J. Opt. Soc. Am. 68, 953-964 (1978).

22. E. Wolf, and A. J. Devaney, "On a relationship between spectralproperties and spatial coherence properties of light," Opt. Lett.6, 168-170 (1981).

23. A. W. Lohmann and J. Ojeda-Castaneda, "Spatial periodicitiesin partially coherent fields," Opt. Acta 30, 475-479 (1983).

24. E. Wolf, A. J. Devaney, and F. Gori, "Relationship betweenspectral properties and spatial properties in one-dimensionalfree fields," Opt. Commun. 46, 4-8 (1983).

25. G. Indebetouw, "Spatially periodic wave fields: an experimen-

tal demonstration of the relationship between the lateral andthe longitudinal spatial frequencies," Opt. Commun. 49, 86-90(1984).

26. M. V. R. K. Murty, "Interference between wavefronts rotated orreversed with respect to each other and its relation to spatialcoherence," J. Opt. Soc. Am. 54, 1187-1190 (1964).

27. H. P. Baltes, H. A. Ferwerda, A. S. Glass, and B. Steinle, "Re-trieval of structural information from the far-zone intensity andcoherence of scattered radiation," Opt. Acta 28, 11-28 (1981).

28. H. P. Baltes, A. S. Glass, and K. M. Jauch, "Multiplexing ofcoherence by beam-splitters," Opt. Acta 28, 873-876 (1981).

29. H. P. Baltes and K. M. Jauch, "Multiplex version of the VanCittert-Zernike theorem," J. Opt. Soc. Am. 71, 1434-1439(1981).

30. A. S. Glass and H. P. Baltes, "The significance of far-zonecoherence for sources or scatters with hidden periodicity," Opt.Acta 29, 169-185 (1982).

31. K. M. Jauch and H. P. Baltes, "Coherence of radiation scatteredby grating covered by a diffuser," Opt. Acta 28, 1013-1015(1981).

32. K. M. Jauch and H. P. Baltes, "Reversing-wave-front interfer-ometry of radiation from a diffusely illuminated phase grating,"Opt. Lett. 7, 127-129 (1982).

33. D. Newman and J. C. Dainty, "Detection of gratings hidden bydiffusers using intensity interferometry," J. Opt. Soc. Am. A 1,403-411 (1984).

34. E. B. Fomalont, "Earth-rotation aperture synthesis," Proc.IEEE 61, 1211-1218 (1973).

35. R. N. Bracewell, "Computer image processing," Ann. Rev. As-tron. Astrophys. 17, 113-134 (1979).

36. P. J. Napier, A. R. Thompson, and R. D. Eckers, "The very largearray: design and performance of a modern synthesis radiotelescope," Proc. IEEE 71, 1295-1320 (1983).

37. 0. Bryngdahl and A. W. Lohmann, "Variable magnification inincoherent holography," Appl. Opt. 9, 231-232 (1970).

38. J. C. Dainty and R. J. Scaddan, "A coherence interferometer fordirect measurement of the atmospheric transfer function,"Mon. Not. R. Astron. Soc. 167, 69-73 (1974).

39. J. B. Breckinridge, "Measurement of the amplitude of phaseexcursions in the earth's atmosphere," J. Opt. Soc. Am. 66,143-144 (1976).

40. J. W. O'Byrne, School of Physics, University of Sydney, Sydney,Australia (personal communication).

41. J. W. Goodman, Statistical Optics (Wiley, New York, 1985),Chap. 2.

42. J. F. Walkup and J. W. Goodman, "Limitations of fringe param-eter estimation at low light levels," J. Opt. Soc. Am. 63, 399-407(1973).

43. A. V. Oppenheim and R. W. Schafer, Digital Signal Processing(Prentice-Hall, Englewood Cliffs, N.J., 1975), p. 125.

44. A. D. Code, "Stellar energy distribution," in Stellar Atmo-sphere, J. L. Greenstein, ed. (U. Chicago Press, Chicago, Ill.,1960). p. 85.

45. J. C. Dainty, "Stellar speckle interferometry," in Laser Speckleand Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 286-306.

46. K. Itoh and Y. Ohtsuka, "Phase estimation based on the maxi-mum likelihood criterion," Appl. Opt. 22, 3054-3057 (1983).


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