Imaging 2-D objects with a grating interferometer: twomethods

Jianhong Wu, Linsen Chen, and Yaguang Jiang

Grating interferometric imaging has some unique advantages. The theory developed has been for 1-D objectsonly. The 2-D imaging methods reported are either complex or have low SNR. Two new methods for 2-Dinterferometric imaging are presented in this paper. The first is that an object composed of discrete points isimaged with a grating interferometer composed of three or more 1-D grating interferometers of which thegrating line directions are different. The second is that a 2-D continuous distribution object is imaged with a1-D grating interferometer by sampling line byline. These two methods are simple and may be practical forreal time processing. Key words: Interferometric imaging, grating interferometer.

1. IntroductionThe properties of the grating interferometer have

been analyzed in detail.1-4 In particular, the Fouriertransform relation between the fringe pattern contrastand source structure for an incoherent source is ofgreat significance because it leads to interesting appli-cations, such as image formation through scattering orphase distorting media.5 6 Also the azimuthal positionand angular velocity of an object can be measuredsimultaneously. 7 Both real time Fourier transforma-tion and image formation by successive Fourier trans-formation have been reported. 8 9 Some of these appli-cations would also apply to interferometers in generalrather than just grating interferometers. However,the grating interferometer has some unique advan-tages, the most significant of which are that it can formperfect, full contrast fringes in completely incoherentlight and that it is achromatic, allowing most of theabove operations to be carried out in perfectly whitelight.

The theory of grating interferometric imaging thusdeveloped has been for 1-D objects only. It would bemuch more significant to do interferometric imagingwith 2-D objects, and two methods have been proposedfor this. The first is with an interferometer formedfrom circular gratings. The 2-D image is generatedeither by performing a correlation operation on a coni-cal surface within the fringe box (the volume in whichthe two interfering beams overlap and, thus, interfere)or else performing a 3-D, or volume, correlation withinthe fringe box.' 0 In either case, the calculation iscomplicated and the calculation volume is enormous.Thus, the method is at best inconvenient and probably

The authors are with Suzhou University, Laser Research Section,Suzhou, Jiangsu Province, China.

Received 14 October 1988.0003-6935/90/081225-05$02.00/0.© 1990 Optical Society of America.

impossible for real time imaging. In addition, themanufacture of high quality circular gratings is diffi-cult. A second method, passive synthetic apertureimaging using a grating interferometer, has also beendescribed."

The interferometric imaging systems have a prob-lem known as bias buildup, which arises because thefringe patterns from individual object points combineincoherently, with a reduction in fringe contrast pro-portional to the square root of the number of objectpoints. Since extension to 2-D typically means a largeincrease in the number of object points, the bias build-up is exacerbated and the SNR is greatly reduced.There is also a second problem; there is an ambiguity inthat object points at +0 and -0 produce identicalfringe patterns and thus reconstruct the same image.This ambiguity is akin to the twin-image problem ofin-line holography.

Two new methods for 2-D interferometric imagingare presented here. In the first method, which re-quires that the object consists of discrete points, aninterferometer composed of three or more 1-D gratinginterferometers is used, in which the grating line (orruling) directions are different. In the second method,a 2-D continuous distribution object is imaged with a1-D grating interferometer by sampling line by line.

II. One-Dimensional Grating InterferometerThe imaging properties of the 1-D grating interfer-

ometer have been described4 and are briefly reviewedhere. As shown in Fig. 1, a point source at infinity in adirection sends a plane wave to an interferometercomposed of two diffraction gratings with spatial fre-quency f, and 2ff. Grating G, produces various dif-fracted orders but only the +1 and -1 orders are used.The +1 order propagates to the second grating G2,where it is redirected by diffraction into the -1 orderof G2; similarly, the -1 order of G, is redirected into the+1 order of G2. These two beams are brought togeth-er, thus forming the overlap region that we have de-

10 March 1990 / Vol. 29, No. 8 / APPLIED OPTICS 1225

scribed as the fringe box. The fringes, to within aconstant, have the form

I = 1/2 + /2 cos4irfl(x-z sin). (1)

These fringes are achromatic and nonlocalized. Foran extended object, the total intensity within thefringe box is obtained by integrating Eq. (1) over 0:

I = 1/2 f S(O)dO + 1/2 f S(O) cos[4irf(x - z sinO)]dO, (2)

where S(0) is the source intensity distribution as afunction of 0. We make the approximation sinO = 0.To avoid the ambiguity that objects at +0 and -0produce the same projection on the z-axis, we canrecord fringes along a plane tilted with respect to the z-axis.

Figure 2 shows a grating interferometer fringe box.The fringes are recorded along the plane x' = 0. Theangle between x- and x'-axes is a. By using the rela-tion between the corrdinate systems xoz and x'oz', andalso Eq. (2), the fringe intensity recorded is found to be

I(z') = I, + 1/2 X S(0) cos[4-fl(sina - 0 cosa)z']dO,

e ~~~~~

G1

G2

Fig. 1. Grating interferometer.

x

Z

(3)

where, I = 1/2 S S(0)d0. From Eq. (3) we can see thatthe object resides on the carrier frequency 2fi sina. Ifwe inverse Fourier transform this fringe pattern in anoptical processor (Fig. 3), we obtain

51-1{I(z')1= I(z') exp( 27r xzi) dz'

2 2 (F) + 2S (2fTX -.Ftan)

(4)

where, F is the focal length of the Fourier-transformlens L. The first term of Eq. (4) is the undiffractedlight energy and each of the two remaining terms is animage of the object. We utilize the second term only.We choose a coordinate system centered at t; thus theimage is located at xf = t + 2f1XF sina. The field at theinverse-Fourier plane is given by

UQt) = Cs (t) (5)

where a = 2f1X cosaF and C is a constant. Eq. (5)expresses the image of the object S(0).

111. Extension To 2-D ObjectsThe grating imaging method can be extended to two

dimensions, although not with complete generality; aswill be seen, the two dimensional extensions depend onthe distribution of discrete points or continuous ob-jects.

A. Imaging Point Objects with 3-Grating Interferometer

Figure 4 shows a grating interferometer that canimage, subject to certain restrictions, objects com-posed of discrete points. The interferometer com-prises four gratings. Grating G1 is a superimpositionof three sets of lines (i.e., three sets of recorded fringe

Fig. 2. Fringe box of a grating interferometer.

Z r

L

A

V.---- F -

Fig. 3. Formation of an image from the interferogram.

y

axI

Cl /~

P3

Fig. 4. Grating interferometer composed of coaxial 1-D gratinginterferometer.

patterns) all of the same spatial frequency, but ofdifferent orientations. Two sets are parallel to the x-and y-axes, and the third makes an angle y with the x-axis. Gratings G2, G3, and G4 each have a single grat-ing line pattern, twice the spatial frequency of the

1226 APPLIED OPTICS / Vol. 29, No. 8 / 10 March 1990

f k

- Al

4

+ 1 S _ ' + tana2 (�fX Xco..F I

grating G1 lines and each has a line direction paralleone of the line patterns of G1. Since these gratingsin different planes, the fringe boxes P1, P2, andproduced by grating G interactions are separa,along the z-axis.

As with the 1-D imaging case, an object is placedthe focal plane of a collimator and is thus at infinityseen from the interferometer. A point on the objproduces at G1, the field

u= VS(x,OY) exp(121r A x) exp(12r Y)

where S is the source intensity distribution as a fu]tion of the angles Ox and Oy, the angles between the wEvector ko and the planes x = 0 and y = 0, respectiveAn analysis comparable to that for the 1-D gratiinterferometer yields the fringe patterns:

I = I + 1/2 f S(Ox,Oy) cos[4irfl(x - O.z)]dO.d6Y,

12 = I + 1/2 C S(OxOy) cos[4rfl(y - yz)JdOxdOY,

I3 = I + '/2 X S(O.4,y) cosJ47rf 1 [sinyx + cosyy

- z(sinyOx + cos-y)]JdOd6Y,

where I = 1/2 S(OX)O)dOxdOy, with each pattern fall-ing in the appropriate fringe box. Inspection of theseequations shows that they are not simple Fouriertransforms of the object distributions, as in the 1-Dcase. The three sets of patterns contain considerableinformation about the object; enough so that a recon-struction of S(OxOy) is sometimes possible.

Let the object be composed of N points distributedtwo-dimensionally:

S(xOy) = E S(OX - 0xk

0y - yk)- (10)

k

Also, suppose

0x1 x2*-- ON y1 y2 *-OyN, (11)

a condition that can be satisfied by properly choosingthe coordinate system. Combining Eq. (10) with Eqs.(7-9) yields:

I = Io + /2 2 Sk cos[4rfl(x - Oxkz)], (12)

12 = I + /2 2 Sk cos[4rf1(y - Oykz)], (13)

I3 = I + V/2 z Sk cosJ47rf1 [sinyx + cosyy

- (sinyOxk + cosyOyk)ZI}- (14)

As in the one 1-D case, these three fringe sets arerecorded on tilted planes. The recording planes in P1and P2 are parallel to the y- and x-axes, respectively,and make an angle a with the z-axis. The recordingplane in P3 is parallel to the line y = tanyx and is also atan angle a with the z-axis. Each record is treated as a1-D Fourier transform hologram and reconstructs withresolution in only one direction, namely, the directionnormal to the grating lines of the gratings G2-G4.Thus, the three records produce, respectively, thethree fields

UJQ) = C1 E Ske(- aOxk),k

U2(7) = C2 E Skb(7- a+yk) +k

U3(,nr) = C3 E Skb(-q - tany - aOyk + tan-yaOk).k

(15)

(16)

(17)

Each image set consists of N lines, with an orientationparallel to the lines of the grating that produced them.These three images are equivalent to the object pro-

sic- jected onto three axes, the x- and y-axes, and a thirdlve axis normal to y = tanyx. The problem is then to!nY. reconstruct the original object distribution from theseng projections.

Multiplication of ul, u2, and U3 produces:(7)

(8) u(Qn) = C E ( k d-baOXk)(7 - ayk)6 (6 - tany -aO k

+ tan-yaOk) + E S'S1 Q(t - aXk)5(7 -a~y)(

kol

- tan-y - aOy + tanya0, 1) + E SiSSk(t - aOx)

sj #k

X (n -a0y)(n- tan-ye -ak + tanyaOxk)] - (18)

In Eq. (18), the three sets of terms are each theproduct of three -functions. The first two -func-tions together represent a point in the t - q plane andthe final one is a straight line. The product of thethree 6-functions is nonzero only when the point fallson a line. If y is chosen so that

tany 51 0Yk yj ij,k = 1, 2, . . ., N andOxk -0,d i F o d , or i = o k ,

(19)

the first summation of Eq. (18) is nonzero only, yield-ing

UiQ'7) = CE S6(t - aOXk,77 - aOyk).k

(20)

This is the correct reconstruction of the original 2-Dobject.

If the object is imaged in only two directions, forexample, x and y, then al and U2 will be obtained, andthe product will be N2 points, consisting of the Ncorrect ones and N(N - 1) false reconstructions; that is

U'Q(,) = C Sk(t - aXk)a(7 - ayk)w ~~~~~~~~k

+ I SiSj6( - a,i)S(,7 - aOy1)1

i#1

(21)

If the condition of Eq. (11) is not satisfied there will bea partial overlap of the image points; consequently,relative intensities of the image points will not corre-spond to those of the object points. The purpose of

10 March 1990 / Vol. 29, No. 8 / APPLIED OPTICS 1227

imaging in the third direction is thus the elimination ofthe N(N - 1) false image points. The condition of Eq.(19) ensures that a three-directional imaging processwill eliminate the false responses. Also, the two condi-tions of Eq. (11) and Eq. (19) together are sufficient,but not necessary, conditions for giving a correct re-construction, both in the number of image points andtheir relative intensities.

The constraints imposed by the conditions of Eqs.(11) and (19) lead to increased difficulty as the numberof object points is increased. In rotating the coordi-nate system of Eq. (21), we see that the positions of thefalse images depend on the selection of the rotation,whereas the position of the true image points do not.Therefore, we want to choose the rotations so thatwhen the images are multiplied, the false terms willbecome zero. Also, more directions in the imagingprocess will help in eliminating the false terms. Wemay, for example, image in four directions instead ofthree, and combine the output in pairs, forming oneimage from two directions and a second from the othertwo directions. Multiplication of these two imageswill then give false image rejection better than thethree-directional imaging process. The following ex-planation helps in understanding the superior falseimage rejection capability of this method. In thethree-directional imaging method, the false images arerejected if the point images of the second term of Eq.(21) do not fall on the line images formed from thethird direction. In the four-directional imaging case,each of the two images is an array of points, and falseimages are rejected if the points of one image do not fallon the points of the other. Since the possibility of acoincidence between two points is much less than be-tween a line and a point, the probablity of a false imageis far less than in the former case. Therefore, theoperation eliminating the false images in the four-directional imaging method is performed more easilythan the three-directional imaging method.

B. Sampling 2-D Objects Line by LineAnother imaging method is to sample a 2-D object

line by line. The 2-D imaging process is thus changedinto a succession of 1-D imaging operations. Let theintensity of the object be S(O,,Oy). A 1-D angle sampleis placed in front of G1, the sampler is a set of narrowslats that allow light from only one direction to betransmitted. The sampler is aimed in a sequence ofdirections Oy = 01,02, .. ,ON. A series of 1-D images,SI(O.,01),S2(0x.,02),.. .,SN(OX,ON), are obtained with a 1-D grating interferometer. These can be synthesizedinto a 2-D image. A variant of this is to place a sam-pling slit at the back focal plane of the first lens of thetelescope system, which is laid in front of 1-D gratinginterferometer. Because angular direction in the ob-ject plane corresponds to position in the Fourier trans-form plane, the two methods are equivalent.

IV. Experimental ResultsExperimental verification of the previous theory was

obtained by the grating interferometer arrangement

Fig. 5. Optical arrangement of the experiments.

(a) (c)

Fig. 6. Imaging a circular array of eight points with a gratinginterferometer by rotating the object twice.

shown in Fig. 5. The light source was a laser withspatial coherence destroyed by the usual rotatingground glass method. Since the system is perfectlyachromatic, a white light source could also have beenused. The object is placed near the ground glass, andthe light is collimated by lens L. The gratings, withspatial frequencies of 150 and 300 lines/mm, were fab-ricated by recording an interference pattern on dichro-mated gelatin. In our experiments, the multidirec-tional grating interferometer of Fig. 4 was replaced bya simple 1-D grating interferometer, with the objectrotated to different positions.

The first object [Fig. 6(a)] is a circular array of eightpoints. Although the conditions of Eqs. (11) and (19)are not satisfied, the reconstruction is still correct.The object was rotated to three different positions,thus producing fringes corresponding to Eqs. (12-14).The tilt angle of the recording plate was 450.

The reconstructions were carried out with a stan-dard, laser-illuminated Fourier transform system.The incident plate was at non-normal incident to satis-fy the Bragg angle, since Bragg effects were non-negli-gible. The image of Fig. 6(b) is formed when the objectis in the orientation of Fig. 6(a). Similar images areobtained for the other rotational positions, 450 and90°. Multiplication of the three images produced theimage shown in Fig. 6(c), which is indeed a correctreconstruction.

Another example is shown in Fig. 7(a), in which atwelve-point circularly symmetric object is imaged bya 1-D grating interferometer at four rotational posi-tions: 0, 300, 90°, and 1200. Fig. 7(b) shows animage when the object is at the position shown in Fig.7(a). By multiplying the four images with four rota-tional positions, the image of Fig. 7(c) is obtained.

Finally, a line by line scan can be used in an arrange-ment similar to that shown in Fig. (5). A sampling slitis placed at the object. The sampling slit is movednormal to its length and successive exposures aremade. The 2-D object is thus converted into a succes-sion of 1-D objects, which can then be assembled toform the complete 2-D image. If the object is a contin-uous distribution instead of a collection of discrete

1228 APPLIED OPTICS / Vol. 29, No. 8 / 10 March 1990

(a) (b) (c)

Fig. 7. Imaging a 12-point circularly symmetric object with a grat-ing interferometer by rotating the object three times.

(a) (b)

(c) (d)Fig.8. Imaging a continuous distribution object with a 1-D grating

interferometer by sampling line by line.

points, the sampling plate can be broken into an arrayof discrete points. This will reduce the bias buildup.Fig. 8(a) shows a continuous object; Fig. 8(b) is thesampling plate, a 1-D linear array of nine samplingpoints. This plate scans the object, moving verticallyacross the object. Eight 1-D images are obtained, oneof which is shown in Fig. 8(c). The synthesis of theseimages is shown in Fig. 8(d).

V. DiscussionThe imaging method using the system composed of

three of more 1-D grating interferometers is suitablefor objects composed of discrete points. If the object iscontinuous, it can be sampled with an angle sampler ora system similar to a telescope in which the 2-D sam-pling points are placed at the focal plane of the firstlens, changing the object into discrete points. Giventhe constraints of Eqs. (11) and (19), imaging 2-Dcontinuous distribution objects is difficult.

The two methods of imaging 2-D objects in thispaper can be combined with the technique of succes-sive Fourier transformation by using two grating inter-ferometers in tandem. For the first method, the sec-ond grating interferometer can be linked to the firstgrating interferometer with mirrors. These Fouriertransforms are carried out in grating interferometers.The images in this method can be obtained in real time.If sampling line by line is carried out in real time with a1-D angle sampler or with the telescope system inwhich the sampling line is controlled electronically,these 1-D images can also be obtained in real time.

Therefore, real time imaging is more conveniently ob-tained with the systems described here than with theproposed circular grating interferometer and the pas-sive synthetic aperture imaging methods.

For the line by line sampling method, the object issampled in rectangular coordinates. The bias back-ground is reduced by this sampling method, since thenumber of simultaneous objects is reduced from N2 toN and the background is, therefore, much weaker thanwhen the imaging is done by passive synthetic apertureimaging, and the SNR is, therefore, better.

If the object is sampled with the telescope system,the two methods in this paper have another advantage.Since the object is at infinity and its spatial frequencyis low, the sampler at the focal plane of the first lenscan be considered as a low frequency filter. When theobject light passes through scattering media, the highfrequency components produced by scattering lightare filtered and the bias background formed by scat-tering light in the fringe box is reduced. Therefore,the SNR is increased. This method is similar to thatof Ref. (12).

The authors wish to thank E. N. Leith for his help inpreparing this manuscript. We also acknowledgeWeimin Shen for his assistance with the experiments.This work was supported by Natural Science Founda-tion of China.

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4. E. N. Leith and B. J. Chang, "Image Formation with an Achro-matic Interferometer," Opt. Commun. 23, 217-219 (1977).

5. B. J. Chang, J. S. Chang, and E. N. Leith, "Imaging ThroughScattering Media with an Achromatic Interferometer," Opt.Lett. 4, 118-120 (1979).

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10. E. N. Leith, G. Collins, I. Khoo and T. Wynn, "CorrelationImage Formation with an Axicon," J. Opt. Soc. Am. 70,141-145(1980).

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