Formation of the Holographic Image of a Diffuse Object in Second-Harmonic Radiation Generated by a Nonlinear Medium

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1 Optics and Spectroscopy, Vol. 89, No. 3, 000, pp Translated from Optika i Spektroskopiya, Vol. 89, No. 3, 000, pp Original Russian Text Copyright 000 by Denisyuk, Andreoni, Bondani, Potenza. GOLOGRAPHY Formation of the Holographic Image of a Diffuse Object in Second-Harmonic Radiation Generated by a Nonlinear Medium Yu. N. Denisyuk*, A. Andreoni**, M. Bondani**, and M. A. S. Potenza* *Ioffe Physicotechnical Institute, Russian Academy of Sciences, St. Petersburg, Russia **Department of Physics, Chemistry, and Mathematics, Insubrian University, Como, 100 Italy; Milan University, Milan, 0133 Italy Received March 15, 000 Abstract Results of experiments on recording three-dimensional holographic images of extended diffuse objects using an SHG hologram generating the second harmonic are presented. In this case, the object image is formed by the second-harmonic radiation whose wavelength is smaller than the wavelength of object and reference waves recorded on a hologram by a factor of two. Elements of the theory of an SHG hologram are considered. A holographic image of a transparency object illuminated with diffuse light is obtained. It is shown that the resolving power of this image is close to the limit determined by diffraction effects. An experiment on defocusing the reconstructed image showed that it was localized in one spatial plane and, therefore, was threedimensional. 000 MAIK Nauka/Interperiodica. INTRODUCTION The study of various transformations of the spatial structure of light fields has been one of the main lines of investigation in optics from its onset. It is undeniable that holography, which represents a method based on recording the interference pattern formed by object and reference waves in a photosensitive material [1], is one of the substantial achievements in this area of study. Recently, a radically new method of transformation of the spatial structure of wave fields was proposed, which makes it possible to transform waves fields without any time delay [, 3]. This method, which is called correlation in the second harmonic, is based on mixing object and reference waves in a nonlinear crystal with a large second-order nonlinearity. The spatial structure of the resulting radiation at the double frequency represents cross-correlation of initial wave fronts. Note that, in this case, the so-called noncollinear SHG scheme is used in which two spatially separated waves interfering in a crystal produce conditions for SHG. This scheme has long been known in nonlinear optics [4]. It was used mainly to separate the second harmonic from radiation at the fundamental frequency. Later on, the nonlinear scheme was used to double the frequency of ultrashort pulses because it is able to eliminate the mismatch of group velocities of the waves travelling in a crystal. However, it was never used for transforming the spatial structure of wave fields. In essence, the scheme used in the method of correlation in the second harmonic is similar to the hologram recording scheme, and therefore it is reasonable to expect that one can also obtain three-dimensional holographic images by this method. Initially, this possibility was theoretically analyzed using the approaches typical of nonlinear optics, but this way met with serious problems because the methods of analysis adopted in nonlinear optics are insufficiently suitable for the description of transformations of wave fronts with a complex spatial configuration. The solution of the image formation problem was found using the approaches used in holography [5]. In particular, it was shown that each thin layer of a nonlinear crystal in this case could be treated as a hologram reconstructing the image in accordance with the wellknown laws of holography. Because of this, the structure formed in a nonlinear material by the cross-correlation of wave fields was named the hologram generating the second harmonic (the SHG hologram). The theory of an SHG hologram developed on the basis of laws of holography and lens optics enables us to predict the position, scale, and resolution of an image reconstructed by an SHG hologram. In particular, in spite of some predictions, this theory showed that the resolution of the reconstructed image in the horizontal and vertical directions was the same. The theory was verified in the experiments, which completely supported its validity. The only drawback of the experiment was that a point light source was used as an object. This drawback was caused by the fact that the energy of the Nd:YAG laser used by us was low, and, therefore, it was impossible to record an image of an object with a rather complex structure. In this paper, we present results of the experiments in which we over X/00/ $ MAIK Nauka/Interperiodica

2 FORMATION OF THE HOLOGRAPHIC IMAGE OF A DIFFUSE OBJECT 477 came these difficulties and obtained images of extended diffuse objects. ELEMENTS OF THE THEORY OF AN SHG HOLOGRAM An optical scheme for recording an SHG hologram is shown in Fig. 1. In substance, it is totally similar to the scheme of recording a conventional off-axis hologram [6]. In this case, the interference pattern formed by the wave E O of radiation scattered by an object O and the spherical reference wave E R emitted by a reference point source R is recorded in a nonlinear photomaterial H characterized by the second-order nonlinearity. In the subsequent analysis of the SHG hologram recording, we assume that a nonlinear photosensitive material has a negligibly small thickness. In the case of real recording in a material with a certain finite thickness, one can use this analysis to describe the processes taking place in one of the sections of a material (see section S in Fig. 1). The electric fields of the object wave (E O ) and the reference wave (E R ) in the plane S of the hologram H can be written in the form E O = A O ( r 1 ) exp{ i[ F O ( x 1, y 1, λ 1 ) + ωt] }, (1) E R = A R exp{ i[ k 1 L R ( x 1, y 1, λ 1 ) + ωt] }, () where A 0 is the amplitude of a complex object wave, which depends on the spatial coordinate r 1 ; F O (x 1, y 1, λ 1 ) is the phase distribution for the object wave in the plane S; L R (x 1, y 1, λ 1 ) is the eikonal of the reference wave E R, which describes the distribution of optical paths of rays travelling from the reference source R to the hologram surface S and determines the form of the reference wave; k 1 = π/λ is the wave number; and ω is the field frequency. The resulting value of the electric field E ω in the section S is given by the sum of the fields E O and E R : E ω = A O ( r 1 ) exp{ i[ F O ( x 1, y 1, λ 1 ) + ωt] } + A R exp{ i[ k 1 L R ( x 1, y 1, λ 1 ) + ωt] }. (3) The nonlinear material H illuminated by the wave E ω generates the secondary wave with frequency ω. According to the laws of nonlinear optics, the wave produced in a medium with the second-order nonlinear susceptibility χ () is described by the expression [7] E ω = me ω E ω, (4) where m is a certain constant, E ω is the initial electric field interacting with the medium, and E ω is the secondary wave field produced in the nonlinear medium. Substituting E ω from Eq. (3) into Eq. (4), we obtain an equation for the second-harmonic field, which consists of three components. As in the case of a conventional hologram, the object image should be described by the component representing the product of the object wave E O and the reference wave E R : E ω E ω = A O ( r 1 )A R exp{ i[ F O ( x 1, y 1, λ 1 ) + k 1 L R ( x 1, y 1, λ 1 ) + ωt ] }, (5) where is that component of the electric field reconstructed by the SHG hologram that is assumed to form the object image (Fig. 1). From Eq. (5), it follows that the oscillation frequency of the wave field reconstructed by the SHG holograms exceeds the frequency of the electric field recorded in it by a factor of two. In particular, this means that the wave vector k of the reconstructed wave exceeds the wave vector k 1 of radiation recorded in the hologram by a factor of two: k = k 1, λ 1 = λ. (6) (7) Comparing Eq. (5) with Eqs. (1) and (), one can easily see that the components of the phase distribution of the reconstructed wave coincide with the phase distribution of initial object and reference waves. The only difference in this case is that the oscillation frequency of the reconstructed wave is equal to twice the oscillation frequency of the waves being recorded. The question arises: What kind of image can be formed by this wave and is it possible to treat the field distribution recorded by the hologram as an image? O R E O E R S H E I ω Fig. 1. Schematic diagram of SHG recording. (O) object; (R) reference source; (S) arbitrary plane cross section in the body of an SHG hologram H; (E O, E R ) wave fronts of the I I object and reference waves; ( E ω, E ω, E ω ) beams with frequency ω generated by the hologram; the beam E ω represents the object wave reconstructed by the hologram. E ω E ω I

3 478 DENISYUK et al. Let us analyze the sense of the first phase component F O (x 1, y 1, λ 1 ) in Eq. (5), which coincides with the phase distribution of the object wave [Eq. (1)] but belongs to the wave field with the double frequency. As the first step of solving this problem, we decrease all dimensions in the SHG hologram recording scheme by a factor of two. As a result, the dimensions of the scaled object O S and the scaled hologram S S and the distance between them are decreased by a factor of two in comparison with the corresponding dimensions of the initial SHG recording scheme. The wavelength λ S of light illuminating the scaled object will also be smaller by a factor of two than the wavelength λ 1 of light recorded on the hologram. One can easily see that the scaled wavelength λ S will be equal to the wavelength λ generated by the SHG hologram. The scaled hologram recording scheme is defined in the coordinate system (x S, y S, z S ), which is also scaled by a factor of two with respect to the initial coordinate system (x 1, y 1, z 1 ): x 1 = x S, y 1 = y S, z 1 = z S. (8) The wavelength λ S of the scaled system is determined by the relation λ S = λ = 0.5λ 1, (9) where λ 1 and λ are the wavelengths of the wave field being recorded and of the reconstructed field. According to the physical sense of phases of the wave field, they are proportional to the ratio of distance to wavelength. Because the wavelength of light scattered by the object O, as well as the object dimensions and all distances in the optical scheme used for hologram recording, are decreased by one and the same factor (by a factor of two), the phase distribution F S (x S, y S, λ S ) of the wave scattered by the scaled object O S, measured in the scaled coordinate system (x S, y S ) on the surface of the scaled hologram S S, will coincide with the phase distribution F O (x 1, y 1, λ 1 ) of light scattered by the initial object O and determined in the initial coordinate system on the surface of the initial hologram S: F S ( x S, y S, λ S ) = F O ( x 1, y 1, λ 1 ). (10) One can reduce the phase distribution F S (x S, y S, λ S ) to the initial coordinate system (x 1, y 1 ) using a twofold magnification of all dimensions by a two-power telescope (f 1 + f ) with f = f 1. As for a telescopic optical system, it is known that this system, when magnifying an object found in the plane (x, y), does not change phases of the corresponding points of the object. In our case, an object reconstructed by the hologram S S is characterized by the phase distribution F S (x S, y S ). Taking into account expressions (8) and (9) and the fact that a telescopic system does not change phases of the corresponding points of an object being magnified, one can write F M S ( x 1, y 1, λ ) = F S ( x S, y S, λ S ), (11) where F S (x S, y S, λ S ) is the phase distribution in the wave M field of the scaled object, and F S (x 1, y 1, λ ) represents the result of twofold magnification of F S (x S, y S, λ S ) along the x and y directions. Substituting the value of F S (x S, y S, λ S ) from equation (11) into equation (10) gives F M S ( x 1, y 1, λ ) = F O ( x 1, y 1, λ 1 ). (1) From Eq. (1), it follows that the phase distribution of light scattered by the scaled object O S, which was further magnified by a telescopic system by a factor of two, coincides with the phase distribution of the wave field reconstructed by the SHG hologram [see equation (5)]. Starting from this fact, one can easily determine the configuration of the image reconstructed by the SHG hologram. Indeed, magnifying the phase distribution F S (x S, y S, λ S ) in the wave field of the radiation scattered by the scaled object by a factor of two along the x- and y-axes, the telescopic system simultaneously magnifies all components of this field. In particular, transverse dimensions of the scaled object O S will also increase by a factor of two and, therefore, will coincide with transverse dimensions of the initial object O. Because the longitudinal magnification of any optical system is equal to the square of the transverse magnification, all dimensions of the scaled object O S will be increased along the z-axis by a factor of four. As a result, the image of the object reconstructed by the SHG hologram coincides with the object in the transverse direction but is magnified by a factor of two in the longitudinal direction. Let us pass onto the analysis of the second component of phase distribution in Eq. (5) ϕ R = k 1 L R ( x 1, y 1, λ 1 ). (13) The character of transformations introduced by this component into the image reconstructed by the hologram is rather easy to analyze. Substituting the value of k 1 from equation (6) into Eq. (13), we obtain L ϕ R k R ( r 1 λ 1 ) = (14) From (14), it follows that the phase component ϕ R of the wave field reconstructed by the hologram is characterized by the eikonal L R (r 1 λ ), which is smaller than the eikonal of the initial reference wave E R [see Eq. ()] by a factor of two: L R L ( r 1 λ ) R ( r 1 λ 1 ) = (15)

4 FORMATION OF THE HOLOGRAPHIC IMAGE OF A DIFFUSE OBJECT 479 In particular, this means that, in the case where the wave front of the reference wave represents a plane wave travelling at a certain angle to the surface S of the hologram, the eikonal L R (r 1, λ ) corresponds to the plane whose points are two times closer to the surface S than the corresponding points of the reference wave. As a result, the plane surface describing the phase distribution corresponding to the eikonal L R (r 1, λ ) is tilted to the hologram surface by the angle that is smaller than the angle of the reference wave by a factor of two. After suation with the first phase component if O (x 1, y 1, λ 1 ) of equation (5), the phase distribution ik L R (r 1, λ ) causes the corresponding change of the direction of propagation of the object wave E O, which is described by the first phase component of Eq. (5). In the case of a spherical reference wave, its eikonal divided by two corresponds to the sphere whose radius exceeds the radius of the reference wave by a factor of two. This phase distribution, when sued with the phase component of equation (5), causes focusing of the reconstructed object wave similar to the one introduced by the corresponding lens. Both the change in the direction of propagation of the object wave and its additional focusing can be rather easily calculated using the laws of optics of lens systems. EXPERIMENTAL As mentioned above, the theory of an SHG hologram was verified in the experiment, with a point light source being used as an object [5]. The experiment supported theoretical data on the resolution of the reconstructed image and its position in space. However, it was evident that an experiment of this kind was insufficient to give a final verification of the ability of an SHG hologram to reconstruct the image of a real object consisting of a set of points scattering light with different phases and amplitudes. Moreover, it was clear that it is impossible to verify in an experiment with a point object the statement that the transverse magnification of the image reconstructed by an SHG hologram is equal to unity because an ideal point source has no size at all. All these drawbacks of the experiment were visible from the outset because our choice of an object was restricted by the laser energy that could be used for hologram recording. Indeed, the laser available at our disposal produced light pulses (λ 1 = 1.06 µm) 18 ns long with repetition rate of 10 Hz. The pulse energy was 0.8 J, but it was distributed between several modes. To separate out one axial mode, we used a special system of apertures. Upon filtering, the pulse energy was about 70 mj. This energy was quite sufficient to transform an initial plane wave with wavelength λ 1 to the second harmonic with wavelength λ using a BBO crystal, but it was insufficient for recording a hologram of a real object. At the first stage of the experiment, we tried to record an SHG hologram by introducing an object (a transparency with an image draw on it) in one of the initial plane waves, which intersected in a crystal. However, soon it became clear that the images that could be observed in this case represented shadows of the transparency, which were projected by the initial directed laser beam onto the crystal surface and traveled further with the second-harmonic wave generated by the crystal. To prove the fact of holographic image formation, it was necessary to illuminate the transparency with diffuse light so that each point of a photosensitive material be illuminated by all points of an object drawn on the transparency. However, the realization of this recording is associated with a very strong loss of light used for hologram recording. In particular, a ground glass coonly used as a diffuser scatters light in an angle of about 1 rad. The geometry of our recording scheme and the properties of a BBO crystal allowed us to record only that light that propagated within the limits of an angle of 0.05 rad. Therefore, only 0.5% of light scattered by an object would be involved in this case of hologram recording. To eliminate the loss of light caused by the mismatch of the angular aperture of the optical system and the angular aperture of light scattered by an object, we illuminated the transparency using special diffusers fabricated in Vavilov State Optical Institute. They were produced by etching ground glass with hydrofluoric acid. After this processing, the initial rough surface profile of ground glass became more smooth, and the angle of light scattering by a diffuser became considerably smaller. We measured the angular distribution of light scattered by such diffusers. In these experiments, diffusers were transilluminated by a 1.06-µm laser beam, and the distribution of scattered-light intensity was measured with a CCD camera whose sensitive element was found at a certain distance L from the diffuser surface. The experiments showed that most of the energy of scattered light was concentrated within the limits of a certain circle. Outside this circle, the intensity of scattered light was very low. Inside the circle, the intensity distribution had the form of a usual speckle pattern, and the speckle size was determined by the diameter of the light beam incident on a diffuser, the distance from the CCD camera to the diffuser, and the laser radiation wavelength. It is substantial that in the case where the diffuser was translated along its surface, the maxima of all speckles displaced too, and none of them remained in the initial position. The absence of iovable maxima means that these diffusers did not transmit the so-called zero order, and, therefore, the formation of shadow images in this case was completely excluded.

5 480 DENISYUK et al é T CCD L 5 f = +75 ÇÇé L 4 f = +100 The angle of light scattering by diffusers was determined by dividing the diameter of the circle of scattered light on the sensitive element of the CCD camera by the distance from this element to the diffuser surface. The experiment showed that two diffusers scattered light into an angle of 0.01 rad. For two other diffusers, the angle was 0.0 rad. Using these directional diffusers, we started experiments on SHG hologram recording using the offaxis recording scheme (Fig. 1). We used several types of transparencies as objects. The images produced by modern printers and the images produced on conventional color films were found to be completely unsuitable for our purpose because all their areas were totally transparent in IR light with wavelength of 1.06-µm. Finally, we found an 430 S L 3 f = 10 o F Z é' Z Nd:YAG 1 r L 1 D 1 D D 3 L α B S 90 T Lamp Fig.. Schematic optical diagram of the experimental setup. (Nd:YAG) Nd:YAG laser; ( o ) light beam used for illuminating an object O drawn on a transparency T; (S) diffuser; (BBO) crystal in which an SHG hologram is recorded; ( r ) light beam forming the reference wave; (D 1 D 4 ) system of apertures filtering the initial laser beam; (L 3, L 4 ) lenses of a telescopic system expanding the beam of the reference wave; (CCD) CCD camera recording the image reconstructed by the hologram; (O') position of the object recorded on the hologram T in the case where the object itself and not the image reconstructed by the hologram is recorded. 0 D 4 Z 3 old negative with an image of a matreshka, which was recorded on a usual black-and-white silver halide film. This transparency looked in IR light the same as in the visible light. The object (the matreshka s head) was 5 in size. It was drawn by lines 0. thick. The object was placed at a distance of 00 from the BBO crystal. The object and reference waves were formed from single-mode radiation of a Nd:YAG laser obtained by filtering its emission. Upon filtering, radiation was considerably lower in power than the initial multimode beam. The reference beam was 1 in diameter, and therefore the recorded SHG had the same diameter. It was evident that a hologram of this size was unable to provide sufficient angular resolution of the object recorded on it. However, the energy of pulses of the filtered laser beam was too small and an increase of the laser beam diameter would inevitably cause a decrease in the efficiency of conversion of primary radiation to the second harmonic. An experiment made using this recording scheme showed that an SHG hologram reconstructed the object image, but this image was strongly distorted and blurred because of the insufficiently high resolving power of the hologram. Indeed, the angular resolution of a hologram with a pupil 1 in diameter for light with wavelength λ 1 = 1.06 µm is rad. For the 00- distance from a hologram to an object, an optical system with this resolution is able resolve on an object a spot (the so-called scattering spot) 0. in diameter. At the same time, the picture on the transparency was drawn by lines 0. thick. As a result, the lines of the reconstructed image were strongly blurred and distorted by a speckle pattern, and the average speckle diameter in this case was also 0.. The next step in improving the quality of the image reconstructed by the SHG hologram was based on special properties of recording using the second-order nonlinearity. Indeed, a usual hologram records and stores in a photosensitive material the intensity distribution formed by the interference of two waves. If the interference pattern was recorded by radiation with different wavelengths, the patterns corresponding to these wavelengths overlap and partially delete the suary interference pattern. An SHG hologram based on recording in a medium with second-order nonlinearity is able to generate light in the case where the wavelengths of interfering fields differ from one another. Taking into account this property of an SHG hologram, we used a beam split from primary multimode laser radiation as a reference beam. This beam was 6 in diameter, and the pulse energy was 80 mj. The reference beam was formed from single-mode radiation, which was obtained by filtering primary multimode laser radiation. This beam was expanded in diameter up to 6 using a telescopic

6 FORMATION OF THE HOLOGRAPHIC IMAGE OF A DIFFUSE OBJECT (a) 3 (b) (c) (d) Fig. 3. Images of the object. (a) Image reconstructed by the SHG hologram exposed to a single laser pulse; (b) intensity distribution in the plane found at a distance of 43.5 from the plane in which the reconstructed image is localized; (c) result of averaging images recorded on the SHG hologram by 50 laser pulses in the case of a moving diffuser; (d) image recorded by the CCD camera for the transparency object placed at the site that was previously occupied by the images reconstructed by the SHG hologram. system. The energy of pulses travelling along the reference beam was 50 mj. Using expanded laser beams, it became possible to illuminate the whole area of the BBO crystal, and the recorded hologram and the crystal became the same size (5 5 ). In accordance with an increase of the hologram pupil, its angular resolving power increased by a factor of five and became equal to rad. Moreover, we moved the object closer to the BBO crystal, and the distance between them became equal to 100. All these means provided an increase in linear resolution of the image reconstructed by the hologram by a factor of ten (from 0. to 0.0 ). The schematic diagram of the final experimental optical setup is presented in Fig.. An initial beam 6 in diameter emitted by a Nd:YAG laser is split by a semitransparent mirror Z 1 into two rays o and r. The ray r, which is used as a reference ray, is filtered by a system of apertures D 1 D 4. After that, it is expanded by a factor of ten by a telescopic system consisting of lenses L 3 and L 4. Finally, the reference wave, which is about 6 in diameter and contains 50-mJ pulses, illuminates the surface of the BBO crystal where it meets the object wave. The light ray o, which is used for illuminating the object, is split from the initial laser ray by the mirror Z 1 and then passes through a neutral filter F, which is used to prevent damage of the BBO crystal. Upon reflection from the mirror Z, the ray o is directed onto a diffuser S. In this case, the diffuser represents a sandwich consisting of two diffusers. The angle of light scattering by each of these diffusers was 0.0 rad. The suary scattering angle for the diffuser S was 0.04 rad.

7 48 DENISYUK et al. An object represented a transparency with a picture O (the matreshka s face). The object was drawn by transparent lines 0. thick on the black background. The object was 4 in size, and its distance from the BBO crystal was 95. One can easily calculate that in this hologram recording scheme the angle of light scattering by the diffuser S (α S = 0.04 rad) was sufficient to provide the situation when each point of the BBO crystal is illuminated by all points of the object O. In our experiments, we used a BBO 1 crystal (a β- barium borate crystal with type I phase matching). The crystal of square cross section 5 5 was 0.3 thick. It was fabricated by AKADIMPEX, Budapest, Hungary. As shown earlier in [5], the phase mismatch of waves produced in different sections of the crystal was quite acceptable when the angle between the interfering beams was rad. In this experiment, it was also rad. Using the setup presented in Fig., we performed several experiments on the SHG hologram recording. An experiment on recording the object image by a single laser pulse showed that the reconstructed image had a rather high quality, although it was somewhat distorted by the speckle pattern, which is inevitable in the case of image recording by laser radiation (Fig. 3a). The image was found on the bisectrix B S of the angle between the object and reference beams. It was found at a distance of 35 from the lens L 5, which focused the image on the sensor of the CCD camera. The experimental value of this distance was somewhat different from a theoretical value of 30, which was calculated on the basis of the theoretical statement that the distance from the reconstructed image to the BBO crystal exceeds the distance from the recorded object to the crystal by a factor of two. Thus, the experiment supported the fact that the longitudinal magnification of the image reconstructed by the SHG hologram is equal to two [5]. In the next experiment, we shifted the sensitive element of the CCD camera by 17.5 far from the lens projecting the image reconstructed by the hologram onto the sensor. In this way we recorded the light intensity distribution in the plane found at a distance of 43 from the plane with the object image reconstructed by the hologram. One can see from Fig. 3b that the image in this shifted plane was strongly blurred. This experiment supported the statement that the image reconstructed by the SHG hologram is three-dimensional because is has a certain place in space and does not represent a shadow image without a certain localized position. We also performed an experiment on recording and averaging 50 images obtained for different positions of the moving diffuser S. One can see from Fig. 3c that the distortions introduced by the speckle pattern were totally absent in this case, which makes it possible to estimate the natural resolving power of the SHG hologram. According to our estimates, the resolution of the reconstructed image was close to the limit determined by diffraction effects. In the experiment, we also compared the image reconstructed by the SHG hologram with the object recorded on this hologram. In this case, all elements of the optical system had the same positions as in the case of hologram recording (Fig. ). The transparency T illuminated with emission of a filament lamp l was positioned at the site for which its image on the display of the CCD camera was the sharpest and occupied the position that was previously occupied by the image reconstructed by the hologram. The position found in this way for the transparency T was close to the position in which the image reconstructed by the SHG hologram was found. In particular, the transparency T was positioned on the bisectrix B S of the angle between the object and reference rays. The distance from the transparency to the BBO crystal was 10 instead of 190 predicted by the theory and 195 measured in the experiment. The transverse size of the image on the display of the CCD camera was almost equal to the transverse size of the image reconstructed by the SHG hologram (Fig. 3d). This experiment supported the theoretical prediction that the transverse magnification of the image reconstructed by the SHG hologram is equal to unity. On the whole, the results of the experiments illustrated in Figs. 3a 3d showed that the SHG hologram is able to form high-quality images of extended diffuse objects. CONCLUSIONS Thus, we experimentally proved for the first time that a hologram recorded using the second-order nonlinearity (an SHG hologram) is able to form threedimensional holographic images of extended diffuse objects. The experiment also supported the main results of the theory, which predicted the values of longitudinal and transverse magnifications of the reconstructed image, its resolution, and position in space. The ability of an SHG hologram to form high-quality images is of considerable interest from the viewpoint of producing switchable interlinks for computers and counication lines. In this case, an SHG hologram can be used for projecting matrices of luminous points onto output faces of fiber glass counication lines. Because of an extremely short response time of an SHG hologram, it is able to switch interlinks almost without time delay. ACKNOWLEDGMENTS We are grateful to A.A. Leshchev and M.V. Vasil ev for placing at our disposal directed diffusers. Yu.N. Denisyuk is grateful to the CARIPLO Founda-

8 FORMATION OF THE HOLOGRAPHIC IMAGE OF A DIFFUSE OBJECT 483 tion for financial support of the work, and A.M. Potenza is grateful to the ENEA Foundation. REFERENCES 1. D. Gabor, Proc. R. Soc. London, Ser. A 197, 454 (1949).. A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, J. Nonlinear Opt. Phys. Mater. 8, 55 (1999). 3. A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, Eur. Phys. J. D 8, 111 (000). 4. T. R. Zhang, Heung Ro Choo, and M. C. Downer, Appl. Opt. 9 (7), 397 (1990). 5. Y. N. Denisyuk, A. Andreoni, and M. A. C. Potenza, J. Opt. Mem. Neural Netw. 8, 13 (1999). 6. E. N. Leith and J. Upatnieks, J. Opt. Soc. Am. 54 (11), 195 (1964). 7. B. E. Saleh and M. C. Teich, in Fundamentals of Photonics, Wiley Series in Pure and Applied Optics, Ed. by J. W. Goodman (Wiley, New York, 1991), pp Translated by A. Kirkin