Volume 72, number 1,2 OPTICS COMMUNICATIONS I July 1989

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1 Volume 72, number 1,2 OPTICS COMMUICATIOS I July 1989 LAU RIGS: I-REGISTER ICOHERET SUPERPOSITIO OF RADIAL SELF-IMAGES P. ADRI~S a, E. TEPICHI b and J. OJEDA-CASTAlqEDA b Departamento de Optica. Universidad de Valencia, Burjassot, Spain b Istituto acional deastrofisica, Optica y Electrbnica, A.P. 216, Puebla Pue.,Mbxico Received 21 ovember 1988 We describe an optical method for obtaining in-register, incoherent superposition of self-images, with radial symmetry. That is, the Lau effect is implemented, either at infinity or at finite distances, in the form of bright and dark rings of high visibility. This is applied for visualizing radially phase structures, with good signal-to-noise ratio. 1. Introduction Complex operations for optical signal processing can be implemented with high signal-to-noise ratio, if suitable codified incoherent sources are employed [1,2]. For the implementation of the complex operations, with incoherent light, it is important to obtain several mutually incoherent, amplitude processed pictures. Sometimes, it is necessary to obtain in-register superposition of several incoherent versions of the same picture. The Lau effect is a diffraction experiment, performed with incoherent light, in which the key requirement of in-register superposition is applied to the self-imaging phenomenon [3-15]. The Lau effect has received wide attention in the last years, and many of the interpretations are linked to different applications. Among others we mention those of interferometry [6-11] and spatial filtering [12-15]. In refs. [ 9 ] and [ 10 ], attempts were made to implement the Lau effect in 2-D. However, the setup of Barteit and Li [9], which is useful in interferomerry, only produces two mutually perpendicular, I- D Lau bands; consequently, there is no radial symmetry. Colautti et al. [10] obtained radial patterns with very low visibility. The purpose of this publication is to indicate that the Lau effect can be obtained with very high visibility, in 2-D, with radial symmetry. We apply this result for implementing a new interferometer, which renders visible, in a radial fashion, transparent structures. In section 2, we discuss the formation of Lau rings, both at infinity and at finite distances. In section 3, we report some experimental verifications of our theory. Finally, in section 4, we apply the above results for visualizing, in a radial fashion a simple transparent structure. 2. Basic theory A simple description, in heuristic terms, can be stated by using the concept of virtual Fraunhofer diffraction pattern [5] (see also [14] and [16]) as follows. If the object is illuminated with a spherical wavefront, coming from a point source So, then the Fraunhofer diffraction pattern, of the object, appears in a virtual fashion, at the plane of So. The virtual diffraction pattern can be thought of as illuminated by a spherical wavefront that converges towards the object; see fig. l a. ow, if we consider that the object is a screen whose transmittance is proportional to the Montgomery rings [ 5,17 ], then a Fraunhofer diffraction pattern (with radial symmetry and high visibility) appears at the plane of So. This virtual diffraction /89/$03.50 Elsevier Science Publishers B.V. ( orth-holland Physics Publishing Division ) 47

2 Volume 72, number 1,2 OPTICS COMMUICATIOS I July 1989 / a) /'~ - ~ ' "...,-" ~ ~ ~ OB ECT _ 4 ~ ' ~., ~ ' ~ ~ ' ~ k SE_F-'MAGES OF THE D,FFRACT,O / h... ~ '{\\\~/5/~,::,.',~ 1 VIRTUAL. FRAUHOFER I "''~.I. : DIFFRACTIO PATTER z ~ i!i~':,:,~:` > ~ / " 0 BJ ECT ~--" '""'+",,~-~,~'7-', -r Fig. 1. Source arrangement for obtaining Lau rings: (a) initial point source So, ( b ) point source S) located at a self-image plane generated by So. pattern exhibits a set of self-images; some of them are virtual, others are real. If we place another point source Sj, say with j= l, aligned with So, then we generate a second set of selfimages. The two sets of self-images are mutually incoherent if the two point sources So and S, are incoherent. Furthermore, the two sets of self-images can be located at the same planes, and have the same magnification, if the point source Sj is placed at one of the self-image planes created by So. Consequently, it is possible to bring in-register the two sets of selfimages. The above operation can be repeated using several point sources S,, say with j--2, 3... and in this way we can obtain the in-register, incoherent superposit ion of several mutually incoherent self-images of the Fraunhofer diffraction pattern of the Montgomery rings. The resultant irradiance being the in-register, incoherent superposition of self-images with high visibility and radial symmetry, will then exhibit high visibility and radial symmetry. This type of irradiance pattern is here called Lau rings Diffraction theory The above description can be stated using diffraction theory as follows. The amplitude transmittance of the object that contains Montgomery's rings is,v t(r2)= ~. 5(r2-np2). (1) n=o In eq. (1), r is the radial coordinate in the object plane, and p2 is the spacing, in r 2, of the Montgomery rings. ow, if the object in eq. (1) is illuminated with a point source So, aligned with the center of the object, at a distance Zo, then the complex amplitude just behind the object is u(r)=exp(iltr2/)tzo) t(r 2 ) = ~ exp(iltnp2/).zo) 5(r2-np2). (2) n=o At a distance z from the object, the Fresnel diffraction pattern of u(r) is 48

3 1/zj. Volume 72, number 1,2 OPTICS COMMUICATIOS 1 July 1989 v(r' ) =exp(ittr'2/2z) oc 2rt ~ Jo(2nrr'/2z) u(r) exp(ircr2/2z) rdr, 0 = exp ( ircr' 2 / 2z ) 2rip ~, x/ n Jo(2nx/~ pr' / 2z ) n=0 exp[ (innpz/;.)( l/zo+ l/z)], (3) where Jo denotes the Bessel function of the first kind, and zero order. We note that the phase factor, inside the sum, in eq. (3) becomes irrelevant if l/zo+l/z=2m2/p 2, m=0, +1, (4) The value m = 0 describes the formation of the virtual, Fraunhofer pattern at the source plane z = -Zo. For m :~ 0, eq. (4) specifies the positions of the selfimages of the same diffraction pattern associated with Montgomery's rings. The irradiance distribution at any of the self-image plane is, according with eq. (3), l(r'/~.z)=(21tp) 2 ~. ~ x//nq n~o q=0 X Jo(2rtx/n pr'/2z) Jo(2nv/qpr'/2z). (5) We use now another point source Sj, say with j= 1, to create another virtual, Fraunhofer diffraction pattern of the same Montgomery's rings; and consequently, another set of self-images of the Fraunhofer pattern. The point source is located at a distance zj from the object, as shown in fig. lb. The irradiance distribution at the detection plane, specified by the distance z from the object, is similar to that in eq. (5) if l/zj+l/z=2rn'2/p 2, m'=o, +1, (6) ote that both sets of self-images, one generated by So and the other by S, are mutually incoherent if the point sources So and Sj are mutually incoherent. Both sets of self-images coincide at the same detection plane if 1/z= 2m2/p 2-1/z o = 2m')./p2 _ Or, equivalently, if (7) l/zo+ l/ (-zj)=2(m-m')2/p 2. (8) We claim, by comparison between eqs. (4) and (8), that both sets of self-images coincide if the point source Sj is placed at one of the self-image planes of the pattern generated by So. That is, if 1/Zo-1/zj=2j2/p 2, j=+l, +2, +3,... (9) ow, since the scale of the Fraunhofer diffraction pattern depends only on z, see eq. (5), both sets of self-images have the same magnification at the same detection plane. We can then superpose two mutually incoherent patterns with the same irradiance profile and the same magnification. The above arguments can, of course, be repeated for any arbitrary number of point sources Sj, with j= 1, 2, 3,..., to set Lau rings. We remark that the technique here described is lensless, and that the Lau rings are set at finite distances. However, it is equally possible to set the detection plane at infinity. By considering that z--,oo in eq. (4), a self-image is obtained at infinity if zo =p2/22m. (10) In this case we must observe at the back focal plane of a positive lens, which is equivalent to set z=fin eq. (5); as is common in the classical Lau effect [ 3 ]. Combining eqs. (4) and (10) we note that self-image planes coincide with the focal planes associated with Montgomery's rings, as a zone plate. In other words, now the consonance condition, eq. (9), implies that the sources Sj coincide with the focal points of the zone plate. Therefore, all focal planes become Lau planes and, then, Lau rings can be observed at any foci, and in particular at infinity. We consider next some experimental verifications. 3. Experimental results In lines a, b, and c, of fig. 2, we show the irradiance distribution associated with three different point sources Sj, j=0, 1 and 2. Fig. 2d is the resultant irradiance superposition, if the three sources are mutually incoherent and in-register. In fig. 3 we display the irradiance profiles associated with fig. 2. ote that the radial patterns are practically equal between any of them, and the resultant superposition denoted as Lau rings at infinity. The experimental results in figs. 2 and 3 were ob- 49

4 ,? b C d Fig. 2. Irradiance distributions of rings created by the following point sources: (a) So, (b) S~, (c) $2. In (d) we show the incoherent superposition of (a), (b) and ( c ). Fig. 3. Irradiance distribution profiles of the patterns in fig. 2 50

5 I LASER2 M 2 MI -~ --- [ LASER I ] ~L2 A OBJECT LASER3 S l'-~z 0 ---,4 I f.~ Fig. 4. Experimental setup. The object used in our experiment has a spacing between rings, of p= 1 ram, and the distance Zo was set to ram. tained with the setup shown in fig. 4. Three different He-e lasers, 2=632.8 nm, were employed for obtaining three mutually incoherent point sources. The results were taken on the back focal plane of L3; as indicated in fig. 4. So, clear Lau rings at infinity were achieved. Our above experimental results, and the theory, were compared against the following result. The three mutually incoherent point sources were displaced from the positions indicated in eq. (9), that is, to out-of-register positions. The resultant superposition is shown in fig. 5. ote that the resultant superposition is dissimilar, even when it exhibits 2-D radially symmetry, as the pattern in fig. 2d. The way to achieve the closest similarity (between patterns) is by setting the point sources at the distances given in eq. (9). 4. Application: visualization of phase objects The above experimental results were applied to visualize phase structures in a radial fashion as follows. The Lau rings were imaged over a transparent structure, which consists of a bleached photograph of three lines, each with the same five letters over a uniform background. At the Fraunhofer plane, of the phase object, we place a screen that acts as spatial filter. Then, we form the image of the object with the spatially filtered spectrum (see fig. 6). Since at the Fraunhofer plane, the spatial filter is a photographic negative of Montgomery's rings, the method is called dark-field visualization. ~"~ /OBJECT~ "X~LAE FRAUHOFER IMAGE Fig. 5. Irradiance distribution created by the same point sources as in fig. 2; but now located at out-of-register positions. Fig. 6. Optical setup for visualizing radially 2-D phase structures, under noncoherent illumination. The Fraunhofer plane is conjugated to the object plane of fig

6 Fig. 7 shows our experimental results. The picture in fig. 7a displays the image irradiance that appears if the phase structure is uniform and the spatial filter is absent. Fig. 7b shows the image irradiance of fig. 7a for a uniform transparent object and with the spatial filter. Fig. 7c is the picture taken at the image plane of a bleached photograph, without using the spatial filter. Finally, fig. 7d shows the image irradiance for a bleached photograph when using the spatial filter. ote that now the phase structure is rendered visible, in a radial fashion, with good signal-to-noise ratio. 5. Conclusions We discussed the diffraction theory of the formation, with noncoherent light, of bright and dark rings with high visibility, here called Lau rings. We indicated that the Lau rings can be formed as the inregister incoherent superposition of the Fraunhofer diffraction pattern of the Montgomery rings, which exhibits self-images. This superposition can take place either at infinity or at finite distances. The theoretical treatment was verified by some experiments. We indicated, finally, how to apply these results for visualizing radially phase structures, in dark-field, with good signal-to-noise ratio. Acknowledgement J. Ojeda-Castafieda acknowledges the financial support of the Minsterio de Educaci6n y Ciencia (Programa de Cooperaci6n Cientifica con Iberoam- 6rica), Spain. The work was financially supported mainly by the Direcci6n General de Investigaci6n Cientifica y T6cnica (grant PB ), Ministerio de Educaci6n y Ciencia, Spain. Fig. 7. Image irradiances for: (a) a uniform phase object without spatial filter, (b) uniform phase object and spatial filter, (c) bleached photograph without spatial filter, (d) bleached photograph and spatial filter. 52

7 References [ 1 ] F.T.S. Yu, White light optical signal processing (J. Wiley, ew York, 1985) p [2] W.T. Rhodes and A.A. Sawchuk, in: Optical information processing, ed. S.H. Lee (Springer-Verlag, Berlin, 1981 ) p. 69. [ 3 ] J. Jahns and A.W. Lohmann, Optics Comm. 28 ( 1979 ) 263. [4] K. Patorski, Optica Acta 30 (1983) 745. [5]J. Jahns, A.W. Lohmann and J. Ojeda-Castafieda, Optica Acta 31 (1984) 313. [6] B.J. Chang, R. Alferness and E.. Leith, Appl. Optics 14 (1975) [7] H.O. Bartelt and J. Jahns, Optics Comm. 30 (1979) 268. [8] E.. Leith and G.J. Swanson, Appl. Optics 19 (1980) 638. [9] H.O. Bartelt and Y. Li, Optics Comm. 48 (1983) I. [ 10] C. Colautti, L.M. Zerbino, E.E. Sicre and M. Garavaglia, Appl. Optics 26 (1987) [ 11 ]. Bolognini, J. Ojeda-Castafieda and E.E. Sicre, Optica Acta 32 (1985) 409. [12] C. Colautti, E.E. Sicre and M. Garavaglia, Optics Comm. 57 (1986) 16. [ 13 ] J. Ojeda-Castai~eda and E.E. Sicre, Optics Comm. 59 (1986) 87. [ 14 ] P. Andr6s, J. Ojeda-Castafieda and J. Ibarra, Optics Comm. 60 (1986) 206. [ 15 ] J. Ojeda-Castafieda, P. Andr6s and J. Ibarra, Optics Comm. 67 (1988) 256. [16]A.W. Lohmann, ed., Optical Information Processing (Erlangen, 1978) p [17]W.D. Montgomery, J. Opt. Soc. Am. 57 (1967) 772; 58 (1968) [ 18] A.W. Lohmann, J. Ojeda-Castafleda and. Streibl. Optica Acta 30 (1983)