Change in Size of 3D Image using a Periodic Fourier Transform Hologram

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1 Change in Sie of 3D Image using a Periodic Fourier Transform Hologram Kunihiro Sato *, Satoshi Nakaaki, Masakau Morimoto Kensaku Fujii Dept. of Electrical Eng. and Computer Science, Graduate School of Engineering, Himeji Institute of Technology, Shosha 2167, Himeji, Hyogo, , Japan ABSTRACT A method of changing in sie of a three-dimensional (3D) image using a Fourier transform hologram (FTH) or a periodic FTH is described. Here, the periodic FTH is made for information reduction in hologram by superimposing a number of identical FTHs. The second hologram for reconstruction of magnified 3D images is reproduced by connecting a number of small holograms clipped from the original FTH or from the periodic FTH. Numerical calculations and experiments using the computer-generated hologram (CGH) have show that numbers of rays of light illuminated from the second hologram produce a magnified similar 3D image. Distortion of the image owing to the magnification can be avoided by reconstructing the image from a number of small holograms. Resolution, which depends on the magnifying power of the image and on the sie of small holograms, is estimated by the numerical calculation. Keywords: : Fourier transform hologram, magnification, distortion, computer-generated hologram, resolution I. INTRODUCTION There has been significant interest in real-time holographic display systems in resent years 1-5. In order for such display systems to become practical, it is necessary that 3D images of various sies of a same object are reconstructed from holograms of various sies. One way to change the sie of the 3D image is to change the distance between an illuminating point source and a hologram. When looking at the hologram, the observer sees magnified 3D images with distortion and blur caused by the magnification. Distortion and blur cannot be avoided if magnified image is reconstructed from the ordinary hologram. This is a disadvantageous point of the holographic display. In this paper, a method to change the sie of a 3D image without distortion is described. A large number of small holograms are clipped partially from an original Fourier transform hologram, and are connected reproduce a hologram for reconstruction of a magnified 3D image. Distortion and blur of magnified images can be avoided by reproducing the second hologram from a number of small holograms. The price to be paid is a loss of resolution. The relation between the magnifying power of images and loss of resolution is discussed in Section II. Resolution depends not only the magnifying power, but also on the sie of small holograms and on depth of objects. Experimental demonstration is describes in Section III. Experiments are carried out to demonstrate reconstruction of magnified 3D images using computer-generated holograms (CGHs). II. MAGNIFICATION OF 3D IMAGES If a holographic image is in the Fourier transform plane of the hologram, the position of the image is invariant with respect to the translation of the hologram in its plane 6. Because of this invariant relation, it is possible to translate some part of the hologram without apparent motion of the image. On the other hand, when each eye sees the appropriate image of a given pair of stereoscopic images, the observer perceives a 3D scene. In the similar sense, when an eye instantaneously sees some stereoscopic images more than two images of the same object, it is expected that the observer * Correspondence: * 398 Practical Holography XVII and Holographic Materials IX, Tung H. Jeong, Sylvia H. Stevenson, Editors, Proceedings of SPIE-IS&T Electronic Imaging, SPIE Vol (2003) 2003 SPIE-IS&T X/03/$15.00

2 O ( o x o, ) H 1 D m x δ 2 O 1 ( o mx o, m ) H 2 D x c ( r x r, ) 1 c 1 ( r mx r, m ) (a) (b) Figure 1. (a) Reconstruction of the point image from the original hologram H1 and (b) reconstruction of the magnified point image from the reproduced second hologram H2. A small hologram with width D near the position (mx, 0) in the hologram H2 is identical to one near the position (x, 0) in the hologram H1. can perceives depth of the 3D image through focusing adjustment. One of the physical bases of this One of the physical bases of this method is the invariant relation between the position of the holograms and the image, and another is the perception of depth of the image mentioned above. Reconstruction of the point image from the original hologram is illustrated in Fig. 1(a), where the point C (x r, r ) of illuminating point source is the same point of the reference point source. The magnified point image m times as large as the original sie is reconstructed at the point O 1 (mx 0, m 0 ) by illuminating the second hologram with the point source located at the point C 1 (mx r, m r ) as illustrated in Fig. 1(b). Here, the second hologram H2 is reproduced by connecting a large number of small holograms clipped partially from the original FTH. Formation of the second hologram is illustrated in Fig. 2(a). A small hologram with width D near the position (mx,0) in the second hologram H2 is identical to one near the position (x,0) in the original hologram H1. A number of small parts of the second hologram H2 reconstruct a number of magnified images of a point object on the plane at = 1 as illustrated in Fig. 1(b). The observer can perceive the reproduced image at the collect point O 1 (mx 0, m 0 ) if his eye instantaneously sees some stereoscopic images of the point object. A merit of this method is that the distortion of magnified images can be avoided by reproducing the second hologram from a number of small holograms. The price is, of course, a limited resolution of images. Each small hologram reconstructs a magnified image without distortion. Resolution δ 1 of the image reconstructed at =m 0 is estimated by δ, (1) 1 = ( m 0 D)λ which is inversely proportional to the width D of small holograms. On the contrary, spread δ 2 of the magnified point image produced at the point O 1 (mx 0, m 0 ) is proportional to D under the condition δ 2 >> δ 1. From Fig. 1(b), the spread δ 2 is easily estimated by m0 δ 2 = 1 D. (2) 1 Proc. of SPIE Vol

3 Here, the distance 1 in Eq. (2) is given by the formula = + 0 m r 1 r. (3) We curried out numerical calculation to examine reconstruction of magnified point image at the correct position and to evaluate spatial spread of the images. Magnified images are numerically calculated from interference fringes recorded on the hologram H2. Figure 3 shows results obtained from the numerical calculation, where parameters are chosen as r =63.3cm, 0 =83.3cm, and m=4.0. Half width δ of the spread point image has a minimum value for a value of D at which δ 2 δ 1 is satisfied. H 1 H 2 H H 2 3 D T D = NT (a) (b) Figure 2. (a) Connecting a large number of small holograms clipped from the original hologram H1 to form the second hologram H2 for a magnified image, and (b) superimposing a small hologram of H2 for formation of a periodic third hologram H3 with a period T.. We proposed a method of information reduction in hologram by superimposing a number of identical Fourier transform holograms 8. Number of identical holograms are superimposed to form the continuous periodic third hologram H3 with a period T as illustrated in Fig. 2(b), where identical hologram deviate each other with a pitch T horiontally and vertically. Small bands of spatial frequencies are integrated and recorded in a small area with width T of the periodic hologram H3. Spread δ 3 of the magnified point image reconstructed from the hologram H3 is estimated by m = D δ r which is derived under the condition δ 3 >> δ 1 in the similar manner of the case of δ 2. Results obtained from the numerical calculation are shown in Fig. 4, where half width δ is evaluated from magnified point images. The point images are calculated numerically from interference fringes on the hologram H3. (4) 400 Proc. of SPIE Vol. 5005

4 Figure 3. Half width of the magnified point image reconstructed from the second hologram H2 as a function of the width D of small holograms. δ 1 =(m 0 /D)λ is the resolving power, and δ2 is expressed by Eq. (2). The point object is located at 0 =83.3cm, the reference point source is at r =63.3cm, the illuminating point source is at c = 4 r, and then the magnification of the image becomes m = c / r =4. Figure 4. Half width of the magnified point image reconstructed from the third hologram H3 as a function of the width D of small holograms. δ1 =(m 0 /D)λ is the resolving power, and δ3 is expressed by Eq. (4). The original point object is at 0 =83.3cm, the reference point source is at r =63.3cm, the illuminating point source is at c = 4 r, and then the magnification of the image becomes m = 4. Proc. of SPIE Vol

5 III. EXPERIMENTAL DEMONSTRATION Experiments for a 2D object illustrated in Fig. 5(a) and for a 3D object illustrated in Fig. 5(b) have been carried out to prove the method described above. Setups for calculating the CGH are illustrated in Figs. 5(a) and 5(b. In the experiments, we drew CGH patterns by a high-resolution printer with 1440 dot per inch and made square CGHs with width 25mm by taking a photograph of the CGH patterns on a high contrast film with the resolution 320 lines/mm. A 5mW He-Ne laser is used as a coherent source of monochromatic red light. The beam is spatially filtered, expanded, and collimated by a microscope, a pinhole, and a collimating lens. 2cm 2.5cm 2cm 2.5cm = 150cm 100cm 50cm =51cm (b) =0cm (a) =0cm Figure 5. (a) 2D object and (b) 3D for calculating the CGH. The 2D object consists of point light sources and the 3D object is composed of a series of a point light sources. The 3D object shown in Fig. 5(b) is composed of a series of point light sources. Width, height, and depth of the objects are 2cm, 2cm and 100cm, respectively. Since the lateral magnification is different from the longitudinal magnification, distortion of the images is not avoidable when the magnified images are reconstructed from the original FTH. In order to show the distortion of the image, the magnified image is compared with the similar image m times as large as the original object in Figs. 7(b) 7(e). We recognie different two images in Fig. 7(b) or in Fig. 7(d). One is the similar image and another is the magnified image with distortion. Distortion of the magnified image from the original FTH is evident on comparing two images in Figs. 7(b) and 7(d). On the contrary, we see only one virtual image in Fig. 7(c) or in Fig. 7(e). The magnified images without distortion are reconstructed from the reproduced second hologram H2 or from the third hologram H3. We can see from this result that the observer perceive similar magnified 3D image without distortion when his eye instantaneously sees some stereoscopic images reconstructed from a number of small parts of the hologram H2 or H Proc. of SPIE Vol. 5005

6 (a) (d) (b) (e) (c) (f) Figure 6. Photographs of real images of the 2D object reconstructed from the second hologram for the illuminating point source located at (a) c = 2 r, at (b) c = r, and at (c) c = r /2. Photographs of real images of the 2D object reconstructed from the third hologram for the illuminating point source located at (d) c = 2 r, at (e) c = r, and at (f) c = r /2. The 2D object and the reference point source are at the equal distance from the hologram, 0 = r = 51cm. In this case, the magnification of the image is given by m = c / r. The 3D object shown in Fig. 5(b) is composed of a series of point light sources. Width, height, and depth of the objects are 2cm, 2cm and 100cm, respectively. Since the lateral magnification is different from the longitudinal magnification, distortion of the images is not avoidable when the magnified images are reconstructed from the original FTH. In order to show the distortion of the image, the magnified image is compared with the similar image m times as large as the original object in Figs. 7(b) 7(e). We recognie different two images in Fig. 7(b) or in Fig. 7(d). One is the similar image and another is the magnified image with distortion. Distortion of the magnified image from the original FTH is evident on comparing two images in Figs. 7(b) and 7(d). On the contrary, we see only one virtual image in Fig. 7(c) or in Fig. 7(e). The magnified images without distortion are reconstructed from the reproduced second hologram H2 or from the third hologram H3. We can see from this result that the observer perceive similar magnified 3D image without distortion when his eye instantaneously sees some stereoscopic images reconstructed from a number of small parts of the hologram H2 or H3. Proc. of SPIE Vol

7 (b) (c) (a) (d) (e) Figure 7. (a) Photographs of a reconstructed image of the 3D object from the original hologram for the illuminating point source located at c = r = 100 cm. (b) Comparison of a magnified image reconstructed from the original hologram for c = 2 r with a reconstructed similar image two times as large as the original object. (c) Comparison of a magnified similar image reconstructed from the original hologram for c = r /2 with a reconstructed similar image half times as large as the original object. (d) Comparison of a magnified image reconstructed from the second hologram for c = 2 r with a reconstructed similar image two times as large as the original object. (e) Comparison of a magnified image reconstructed from the second hologram for c = r /2 with a reconstructed similar image half times as large as the original object. 404 Proc. of SPIE Vol. 5005

8 IV. CONCLUSION A method of changing in sie of a tree-dimensional (3D) image using a Fourier transform hologram (FTH) has been presented. Connection of a number of small holograms, which are clipped from the original FTH, reproduces the second hologram for reconstruction of the magnified 3D image. Numerical calculations and experiments using the CGH have shown that numbers of rays of light illuminated from a number of small holograms produce a magnified similar 3D image. Distortion of the image is avoidable by reconstructing the image from a number of small holograms. This method is particularly useful for the visual display of the 3D holographic image of various sies. Since spatial frequencies of a hologram can be changed over the wide range by scaling down an original hologram reproduced by this method, the method described in this paper may also be applicable to transformation from a long-wavelength hologram, such as the ultrasound hologram, to the optical hologram for reconstructing a visible image of the object. REFERENCES 1. P. St. Hilaire, S. A. Benton, and M. Lucente, Synthetic aperture holography : a novel approach to three-dimensional displays, J. Opt. Soc. Am. A9, pp , M. Lucente, R. Pappu, C. J. Sparrell, and S. A. Benton, Progress in holographic video with the acousto-optical modulator display, Proc. SPIE, 2577, pp. 2-7, N. Hashimoto, K. Hoshino, and S. Morokawa, Improved real-time holography system with LCDs, Proc. SPIE, 1667, pp. 2-7, K. Sato, Characteristics of kinoform by LCD and its application to display the animated color 3D image, Proc. SPIE, 2176, pp , Maeno, N. Fukaya, O. Nishikawa, K. Sato, and T. Honda, Electro-holographic display using 15 Mega pixels LCD, Proc. SPIE, 2652, pp , L. H. Lin, A Method of Hologram Information Reduction by Spatial Frequency Sampling, Appl. Opt., 7, pp , P. C. Mehta and V. V. Rampal, Lasers and Holography, ch. 4. World Science, S. Nakaaki, K. Sato, M. Morimoto, K. Fujii, Information reduction in hologram by superimposing spatial frequency bands, Proc. SPIE, EI5005A-39, Proc. of SPIE Vol

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