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1 HODIC Circular Vol.23, No.4 pp.17 27(Nov. 2003) Programming Library for Wave-Optical Simulation and Digitally Synthetic Holograms Kyoji Matsushima Department of Electrical Engineering, Kansai University Yamate-cho , Suita, Osaka / matsu@kansai-u.ac.jp Abstract Recent evolution of digitally synthetic holograms as well as diffractive optical elements is surveyed with emphasis on its algorithm/software. The implement of these algorithms lead to necessity of a common programming library that capsules a two-dimensional array of complex values given as sampled complex amplitude of the wave field. The LightWave library designed to meet the requirement is intoroduced. 1 (Computer-Generated Hologram) (Digitally Synthetic (Diffracted Optical Hologram) Element, DOE) (2 ) DOE
2 Object Model Synthesis of Object Wave Coding Print or Display 2 [3] 2 µm 4 µm pixel 1 1 DOE 2.1 LightWave Lohmann D (DXF, VRML ) CG [1] CG 3 [2] [5, 6]
![= >? N H N N @ N 3 @ 2 E J5 K H? A @ @ G H N 0 C H= [4] 6.25 µm 6.](/docs-images/100/144643441/images/3-0.jpg "25 µm 4800 4800pixel 5 Hologram y Object r p z x 4 MIT Benton [7, 8] 2 3 [4, 9] ( ) M N T total = τ a MN (1) τ a [sec/pixel/point] CPU [6,10] [10] [11] CG [12,13] CPU [14] [15] [16] [15] 5 p n+1 = p")
3 = >? N H N N 2 E J5 K G H N 0 C H= [4] 6.25 µm 6.25 µm pixel 5 Hologram y Object r p z x 4 MIT Benton [7, 8] 2 3 [4, 9] ( ) M N T total = τ a MN (1) τ a [sec/pixel/point] CPU [6,10] [10] [11] CG [12,13] CPU [14] [15] [16] [15] 5 p n+1 = p n (2 r n+1 p n ), r n+1 = r n + s n p n, (2) s n+1 = s n + c 1. p 0 1/r 0, r 0 d x 02, (3) s 0 (x 0 + δx/2)δx, c 1 δx 2. Alpha 21164A(600MHz) τ a =47 [nsec/pixel/point] [15]
![] 2.1.](/docs-images/100/144643441/images/4-4.jpg "2 表面モデル光波回折法 点光源では どのような工夫をしようとも基本 的にその計算時間は式 (1)")
![[17 19] こ れは筆者らが開発した 傾いた平面間における光 波の回折伝搬の計算理論 [20,](/docs-images/100/144643441/images/4-6.jpg "21] を応用したもの (-1,-1,-1) (+1,-1,-1) (+2,-4,-1) である")
4 (a ) (b ) y y S y S x x y y x j z f ( x, y ) b x z = d g ( x, y ) z q q a z R g ( x, y ) q x q q R f ( x, y ) 図 6 傾いた平面間における光波の回折伝搬 Amplitude 図 8 表面モデル光波回折法 Phase (0, 0, 0) (0, 60, 0) (0, 60, 30 ) (0, 60, 60 ) 図 7 60 傾いた平面上で計算した正方形開口からの回 折光波 ただし搬送波成分は除去してある 計算精度的には不利になるが 基本的に加算のみ で計算可能であるため固定小数点演算 (整数演算) が利用でき Pentium II(450MHz) プロセッサで も τa = 0.6 [nsec/pixel/point] に達する [16] 表面モデル光波回折法 点光源では どのような工夫をしようとも基本 的にその計算時間は式 (1) に従うことは免れない そこで 筆者らは全く新しい物体光波合成法とし 図 9 表面モデル光波回折法で作成した傾いた平面の ホログラムの再生像 表面モデル光波回折法を提案している [17 19] こ れは筆者らが開発した 傾いた平面間における光 波の回折伝搬の計算理論 [20, 21] を応用したもの (-1,-1,-1) (+1,-1,-1) (+2,-4,-1) である この回折伝搬計算法は波面展開法を発展 させたものであり 図 6 に示したように 平面 S 上で定義される光波の複素振幅から平面 R 上で の複素振幅を求めることができ このとき S と R (a) (b) (c) は任意の傾きを持っていて良い これを用いると 例えば 図 7 のように 正方形開口からの回折光 図 10 表面モデル光波回折法で作成した立体物のホロ 波をその開口に対して非平行な面で計算すること グラムの再生像 ができる
5 D N O D N O 2 E JI K H? A NX OX = 5 5 > A? J N O 0 C H= > A? J > I J=? A 0 C H= X NX OX D 2 2 X D NX OX = > > 11 N O X NX OX 8 [22] [23]. 12 [25 27] 11(a) CG z- Lohmann [1] FFT buffer [2, 24] [28, 29] 11(b)
![13 (1) 14 (2) 15 [31]](/docs-images/100/144643441/images/6-0.jpg "2 [32] 2 O(x, R(x,")
6 13 (1) 14 (2) 15 [31] 2 [32] 2 O(x, R(x, I(x, = O(x, +R(x, 2 (4) t [30] 0 I(x, 4 FFT 12 O(x, t 0 I(x, (5) () t 0 R(x, I(x, O(x, (5) ( bipolar intensit [32]
![DOE DOE 2 DOE DOE Direct Binary Search (DBS) [35] (Iterative Fourier Transformation](/docs-images/100/144643441/images/7-1.jpg "Algorithm, IFTA) [36, 37] [38, 39] 2.")
7 出力 u in ( x, t( x, フーリエ変換 U ( x, DOE 位相分布の制約 像面での制約 U ( x, t ( x, = U ( x, uin ( x, 逆フーリエ変換 u sig ( x, 入力 (a) [33] (b) [31] 2 GA [34] 16(a) GA [33] (b) (a) DOE DOE 2 DOE DOE Direct Binary Search (DBS) [35] (Iterative Fourier Transformation Algorithm, IFTA) [36, 37] [38, 39] IFTA (DOE) HOE / DOE 17 IFTA ( ) DOE ( 1) FFT 18 IFTA DOE (a)
8 = 18 (a) (b) (b) TEM 00 > # 3 LightWave 1: #include <LightWave.h> 2: void main(void) 3: { 4: LightWave a(512, 512, 2e-6); 5: a.setrect(0.5e-3, 0.5e-3); 6: a.spwprop(10e-3); 7: a.saveasbmp("phase.bmp", LW_PHASE); 8: a.normalize(); 9: a.saveasbmp("amp.bmp", LW_AMPLITUDE); 10: } 20 2 LightWave LightWave (1) 2 (2) (3) (4) (5) (6) y LightWaveオブジェクト x 逆伝搬 伝搬 z LightWave 3.1 LightWave 19 LightWave LightWave LightWave 2 µm 5 0.5mm
9 BMP LightWave LightWave 130 I LightWave WaveFront GUI CLR LightWave C#.NET WaveFront WaveFront 22 LightWave LightWave Intel 32bit CPU Windows 3.3 LightWave LightWave 32bit CPU OS Visual C++ Dynamic Link Library 32bit (DLL) DLL 2GB Microsoft.NET LightWave Common Language Runtime (CLR) CLR C++.NET DOE C#.NET VisualBasic.NET 64bit CPU WaveFront GUI LightWave 1 LightWave GUI CPU MPI
10 I LightWave I/O 19 FFT LightWave Windows Linux/Unix 3.4 LightWave / LightWave LightWave [1] A. W. Lohmann: Three-dimensional properties of wave-fields, Optik, 51, pp (1978). [2],,,, 32, pp (2003). [3],,, 56, p (2002). [4], 3,, 56, p. 986 (2002). [5] J. P. Waters: Holographic image synthesis utilizing theoretical methods, Appl. Phys. Lett., 9, pp (1966). [6] A. D. Stein, Z. Wang and J. J. S. Leigh: Computer-generated holograms: A simplified ray-tracing approach, Computers in Physics, 6, pp (1992). [7] P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler and H. Yoshikawa: Electronic display system for computational holography, SPIE Proc. Practical Holography IV, 1212, pp (1990). [8] P. St.-Hilaire, S. A. Benton, M. Lucente, J. Underkoffler and H. Yoshikawa: Real-time holographic display: Improvements using a multichannel acousto-optic modulator and holographic optical elements, SPIE Proc. Practical Holography V, 1461, pp (1991). [9], 3, 3 HODIC ( 20, AIT96-39), pp (1996). [10] M. Lucente: Interactive computation of holograms using a look-up table, J. Electronic Imaging, 2, pp (1993). [11] J. L. Juárez-Pérez, A. Olivares-Pérez and R. Berriel-Valdos: Nonredundant calculation for creating digital Fresnel holograms, Appl. Opt., 36, pp (1997).
11 [12] A. Ritter, J. Böttger, O. Deussen, M. König and T. Strothotte: Hardware-based rendering of full-parallax synthetic holograms, Appl. Opt., 38, pp (1999). [13] C. Petz and M. Magnor: Fast hologram synthesis for 3D geometry models using graphics hardware, SPIE Proc. Practical Holography XVII and Holographic Materials IX, #5005, pp (2003). [14], HORN-4, , pp (2001). [15] K. Matsushima and M. Takai: Recurrence formulas for fast creation of synthetic threedimensional holograms, Appl. Opt., 39, pp (2000). [16] H. Yoshikawa, S. Iwase and T. Oneda: Fast computation of Fresnel holograms employing difference, SPIE Proc. Practical Holography XIV and Holographic Materials VI, #3956, pp (2000). [17], H. Schimmel, F. Wyrowski CGH, , pp (2001). [18] K. Matsushima, H. Schimmel and F. Wyrowski: New creation algorithm for digitally synthesized holograms in surface model by diffraction from tilted planes, SPIE Proc. Practical Holography XVI, #4659, p. 53 (2002). [19], CGH II, , pp (2002). [20], H. Schimmel, F. Wyrowski, Optics Japan 2001, pp (2001). [21] K. Matsushima, H. Schimmel and F. Wyrowski: Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves, J. Opt. Soc. Am., A20, pp (2003). [22] K. Matsushima and A. Kondoh: Wave optical algorithm for creating digitally synthetic holograms of three-dimensional surface objects, SPIE Proc. Practical Holography XVII and Holographic Materials IX, #5005, pp (2003). [23],,, pp (2002). [24], CG, 3 99, pp (1999). [25] J. Underkoffler: Occlusion processing and smooth surface shading for fully computed synthetic holography, SPIE Proc. Practical Holography XI, 3011, pp (1997). [26] T. Hamano and M. Kitamura: Computergenerated holograms for reconstructing multi-3-d images by space-division recording method, Proc. of SPIE, pp (2000). [27],, z,, 57, pp (2003). [28], CGH Babinet, , pp (2002). [29],, J85-D-II, pp (2002). [30], 3, , pp (2003). [31] S. Weissbach, F. Wyrowski and O. Bryngdahl: Digital phase hologram: Coding and quantization with an error diffusion concept, Opt. Commun., 72, pp (1989). [32],,,, 2003, pp. D (2003). [33],, CGH 2, Optics Japan 2003 (). [34] GA,, 54, pp (2000). [35] M. A. Seldwitz, J. P. Allebach and D. W. Sweedney: Synthesis of digital holograms by direct binary search, Appl. Opt., 26, pp (1987). [36] F. Wyrowski: Iterative quantization of digital amplitude holograms, Appl. Opt., 28, pp (1989). [37] F. Wyrowski and O. Bryndahl: Iterative Fourier-transform algorithm applied to computer holography, J. Opt. Soc. Am., A5, pp (1988). [38],, 109, pp (2001). [39],, Optics Japan 2002, pp (2002).
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