1084 OPTICS LETTERS / Vol. 23, No. 14 / July 15, 1998

Display of spatial coherence

David Mendlovic and Gal Shabtay

Faculty of Engineering, Tel Aviv University, 69978 Tel Aviv, Israel

Adolf W. Lohmann*

Department of Physics of Complex Systems, Weizmann Institute of Science, 76100 Rehovot, Israel

Naim Konforti

Faculty of Engineering, Tel Aviv University, 69978 Tel Aviv, Israel

Received March 30, 1998

The mutual-intensity function plays a major role in characterizing quasi-monochromatic, partially coherentoptical signals. We demonstrate an optical system for displaying the mutual intensity of a one-dimensionalinput beam. The experimental system is based on the fact that the mutual intensity of a signal can beexpressed as the ensemble averaging of a cross-correlation operation between two related optical signals.The setup consists of a Sagnac interferometer followed by an optoelectronic joint transform correlator.Experimental results demonstrate the capabilities of the mutual-intensity analyzer. 1998 Optical Society ofAmerica

OCIS codes: 030.1640, 070.4550, 070.2580, 070.2590.

In a partially coherent system the signal travels asthe mutual intensity (sometimes called the coherencefunction).1 – 8 We address the problem of how to dis-play mutual intensity at the exit of a partially co-herent system. To that end we have designed andtested an interferometric system. We also designedand used an auxiliary system for generating mutual in-tensity, which we then displayed by our interferometricsystem. A general case of such correlation functionsinvolves the space-frequency domain, in which, foreach temporal frequency, the spatial correlation prop-erties of the field are considered by the cross-spectraldensity.3 It was shown6 that for most sources thecross-spectral density W sR1, R2, nd at two space pointscan be expressed as the ensemble average of realiza-tions of monochromatic f ields at temporal frequencyn:

W sR1, R2, nd ­ kV sR1, ndVpsR2, ndl , (1)

where R1 and R2 denote the radius vectors of twospace points and V is the associated analytical signal(complex amplitude) whose real part is the electri-cal f ield component. Equation (1) means that mostfeatures of partially coherent light fields can bedescribed through some features of monochromaticfields.

Because of the major importance of the quasi-monochromatic case and the fact that nowadaysexperiments are usually done with laser light, in thisLetter we concentrate on the quasi-monochromaticcase. Thus the temporal behavior of V is of the form

V sR, td ­ usR, tdexpsi2pntd , (2)

0146-9592/98/141084-03$15.00/0

where usR, td is slowly varying in time with respect tothe mean period of the f ield snd. Substituting this ex-pression into Eq. (1), dropping the temporal frequencynotation, and restricting ourselves to one-dimensionalsignals, we find that Eq. (1) can be rewritten in asimpler form, which is usually referred to as the mu-tual intensity:

Gsx1, x2d ­ W sx1, x2, nd ­ kusx1, tdupsx2, tdl . (3)

We shall not restrict ourselves to partial coherentfields, which are produced by an incoherent source(the Van Cittert–Zernike theorem4). Therefore, in ourdiscussion the mutual intensity is in general a complexfunction of two variables: x1 and x2. The goal ofthis Letter is to invent an optical setup that providesthe mutual-intensity distribution of one-dimensionalsignals as its output.

Several ways of observing mutual intensity arebased on the following equations:

kjusx1, td 1 usx2, tdj2l ­ G11 1 G22 1 G12 1 G21 , (4a)

Gij ­ kusxi, tdupsxj , tdl , i, j ­ 1, 2 . (4b)

Note that in this notation G means G12. One way ofobserving mutual intensity is by noticing that the realpart of G is related to the fringe pattern observed intwo-beam interferograms. Recently,9 the imaginarypart of G was also extracted by a special interfero-metric technique based on the Sagnac interferometer.Another way of observing mutual intensity is based ona combination of four measurements of the type basedon Eq. (4a) with various phase shifts (0±, 90±, 180±, and

1998 Optical Society of America

July 15, 1998 / Vol. 23, No. 14 / OPTICS LETTERS 1085

270±).10 The measurement results are

lmsx1, x2d ­

øÇusx1, td 1 usx2, tdexp

µi

mp

2

∂ Ç2¿,

m ­ 0, 1, 2, 3 , (5)

and the coherence function is extracted by the followingpostdetection calculation:

4Gsx1, x2d ­ 4G12sx1, x2d ­3X

m­0lmsx1, x2dexp

µ2i

mp

2

∂.

(6)

Eventually, three measurements with phase shifts of(0±, 120±, and 240±) should be enough to achieve thesame result. The main drawback of all previous meth-ods for measuring mutual intensity is that Gsx1, x2d ismeasured for each point pair sx1, x2d separately.

In the following we propose a method for observingthe mutual intensity of all point pairs (on a line)simultaneously. We make use of the fact that a typicaloptical system has two coordinates, x and y. We let xplay the role of x1 and y the role of x2. The methodis based on the fact that mutual intensity can beexpressed in terms of a correlation operation:

Gsx, 2yd ­ kus y, tddsxd p usx, tdds ydl , (7)

where p stands for a two-dimensional spatial correla-tion operation.

Given a complex amplitude signal usx, td, we nowwish to display its mutual-intensity function. Theproposed system for handling this task is divided intotwo main parts. The first part accepts us y, tddsxd asits input and provides f sx0, y 0, td as its output, wherethe sx0, y 0d coordinate system is connected to sx, yd via45± rotation:

f sx0, y 0, td ­ us y 0, tddsx0 1 x0d 1 usx0 2 x0, tdds y 0d .

(8)

Equation (8) describes two shifted and rotated ver-sions of the input complex amplitude, one along thex0 axis and the other along the y 0 axis. To generatef sx0, y 0, td, we used a Sagnac interferometer (Fig. 1).The main advantage of using the Sagnac interferome-ter is the fact that both rays pass through the same op-tical path, thus making it possible to have imaging onthe same plane without changing the coherence charac-teristics of the input light beam. Inside the interfero-meter we used a rotator (e.g., a Dove prism) to rotateone beam by 45± and the other beam by 245±. Becausea rotator changes the polarization, we used a polarizerat the output of the Sagnac interferometer to allow in-terference later. From a practical viewpoint, the po-larizer has another advantage, which corresponds tothe fact that the beam splitter is not perfect; i.e., itstransmission coefficient is not equal to its ref lection co-

efficient. One can overcome this drawback by chang-ing the polarization at the output of the Sagnac inter-ferometer. Misaligning one of the mirrors results inspatial separation between the two beams at the out-put of the Sagnac interferometer (a separation distancethat is equal to

p2 times the input beam length is suf-

ficient). The change in the optical axis that is causedby this misalignment is handled afterward by use of agrating.

After realizing f sx0, y 0, td, we approach the secondpart of the proposed analyzer. According to Eq. (7),one notices that to generate G one should perform acorrelation operation between the two terms in Eq. (8).A joint transform correlator11,12 (JTC) can do this.f sx0, y 0, td, which appears in Eq. (8), serves as input forthe JTC, which consists of two cycles. In the f irst cyclea Fourier transformation is performed. The modulussquare of the result is recorded (by a CCD) and usedas input for the second cycle, which also consists of aFourier transformation (that is done by a computer inthis case). At the output a correlation between the twoterms of Eq. (8) is obtained along the f irst diffractionorder. The time-averaging operation that is desiredfor obtaining G is achieved automatically during theintensity recording at the end of the f irst cycle. Sincethe Fourier transform is a linear operation, averagingat the end of the f irst cycle is equivalent to averagingthe output. Therefore, at the output plane, at the f irstdiffraction order one obtains the time average of thecorrelation between the two terms of Eq. (8), which cor-responds to Eq. (7).

It should be noted that the imaging lens in Fig. 1imparts a quadratic phase to the object f ield. Thisundesired phase factor could be eliminated by place-ment of a compensating lens in the image (or input)plane or by a change in the location of the CCD to theexact Fourier plane (which is displaced owing to thequadratic phase). Locating the grating midway be-tween the input plane of the JTC and the CCD ensuresthat the misaligned rays will overlap in the Fourierdomain (the CCD plane). This location is done by

Fig. 1. Proposed experimental setup.

1086 OPTICS LETTERS / Vol. 23, No. 14 / July 15, 1998

Fig. 2. Results obtained for the coherent illumina-tion: (a) joint power spectrum and (b) mutual intensity.

Fig. 3. Results obtained for the incoherent illumina-tion: (a) joint power spectrum and (b) mutual intensity.

adjustment of the grating such that its 21 and 11 or-ders overlap.

Figure 1 presents the entire optical part of themutual-intensity analyzer, i.e., the Sagnac interfero-meter, the Fourier-transforming lens (which belongsto the f irst cycle of the JTC), and the CCD. A com-puter performs the second cycle of the JTC (Fouriertransformation). The fact that a computer is usedfor the second Fourier transformation is advan-tageous since the mutual intensity is generally acomplex function. If the f inal display is obtainedthrough an optical processor, only the magnitudecan be displayed, whereas a computer can provideboth the real and the imaginary parts of the mutualintensity.

The proposed system (Fig. 1) was constructed andtested for two cases: spatially coherent and spa-tially incoherent quasi-monochromatic illumination.The coherent illumination is realized by a simple laserlight source; passing this source through a rotating dif-fuser simulates the incoherent illumination. In bothcases the intensity profile of the input is a line, i.e.,rectsxyDxd.

In this special case the mutual-intensity functionsfor coherent and incoherent illuminations are given by

Gcohsx, yd ­ rectsxyDxdrects yyDxd ,

Gincsx, yd ­ rectsxyDxddsx 2 yd . (9)

Figure 2 shows the results obtained for the coherentcase. Figure 2(a) is the joint power spectrum cap-tured by the CCD for the coherent case, and Fig. 2(b)shows the first diffraction order of the digitally calcu-lated Fourier transform of Fig. 2(a); i.e., Fig. 2(b) is ac-tually the measured mutual intensity (where only themagnitude is displayed). Note that this measured mu-tual intensity is very similar to the expected result Gcoh[Eqs. (9)] but is rotated by 45± owing to the change ofcoordinates from sx, yd to sx0, y 0d. Figure 3 shows thesame results for the incoherent illumination; again theobtained results match the expected ones.

In conclusion, a novel method for measuring anddisplaying mutual intensity has been presented. Theconcept is based on performing a correlation operation,which is natural in optics. The experimental setupconsists of a Sagnac interferometer and joint trans-form correlator architecture, whose second cycle isperformed by a computer. Mutual intensity is thendisplayed and measured in a one-shot fashion; i.e., noshifting of the object or use of a slit for each point pairis needed as in previous methods.

*Permanent address: Physikalisches Institut, Er-langen-Nurnberg University, Erwin Rommelstrasse 1,91056 Erlangen, Germany.

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