MIT 2 717 - Fourier-transform spectral imaging

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94 J. Opt. Soc. Am. A/Vol. 3, No. 1/January 1986Fourier-transform spectral imaging:retrieval of source information from three-dimensionalspatial coherenceKazuyoshi Itoh and Yoshihiro OhtsukaDepartment of Engineering Science, Hokkaido University, Sapporo 060, JapanReceived April 3, 1985; accepted July 16, 1985A 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.INTRODUCTIONThe 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'-4and provide, respectively, the bases for Fouierspectroscopy5,6and interferometric imaging.7-14Unfortunately, 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 variousapproaches15-2 0to 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 interferometers15may be employed. Ac-cording to circumstances one may take the alternative ap-proaches-spectroscopy at each pixel16or line by line.1 7Inparticular applications, these simple approaches have beenproved to be useful,15-'7but in general they lack efficiency.This is because most of the incident flux is rejected by thespectral filter or the spectrometer slit. The multiplexingtechnique'8employed in Hadamard-transform spectrom-eters may improve the efficiency. The Hadamard-trans-form imaging spectrometers'8'19literally 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,20which 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.21 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 properties22-25of 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.7'26The 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.27-32They studied theoretically27-30and experimentally31'32theutility 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.33In 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, -P2), where Pi = (xi, Y1), P2= (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 AmericaK. Itoh and Y. OhtsukaVol. 3, No. 1/January 1986/J. Opt. Soc. Am. A 951, 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)wherek = 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,LQi(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


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