設計繞射光學元件之模擬退火法的研究

1. 設計繞射光學元件之模擬退火法的研究 林正峰及邱華楠 南台科技大學光電系2005.12.10

2. 繞射光學元件 Definition of Diffractive Optical Elements (DOE) 我們所定義的繞射元件是一般純相位的元件，其會因元件表面的起伏變化或內部的折射率的變化，而對於射入光的波前產生相位調變。 Several μm ~ several tens ofμm d =~ 1m m 1D or 2D 表面起伏型週期性相位光柵 DOE 成像系統示意圖 Diffraction pattern Period: M x N phase elements Laser beam (plane wave) Lens DOE Transmittance function g(x,y)=exp( i f (x,y)) Intensity | F{g(x,y)} |2 References: [1]. ”Introduction to Fourier Optics”, Joseph W. Goodman , McGRAW-HELL (1996)

3. 模擬退火法 模擬退火法的基本流程 start initialization T0 ,L0 T1 ,L1 Tn ,Ln End 1. Input diffraction pixel matrix (N 2) 2. Set design pattern 3. Set parameters : a , b , g0 , m . 4. Generate a transmittance function (g(x,y)) with random phase 5. Compute initial cost function (Cini ) 6. Generate initial temperature (T0) 7. Algorithm stop criteria. A sequence of Markov chain 使用模擬退火法的目的 The cost function is globally minimized. If the elapsed time is infinite, simulated annealing can obtain global minimum value. The real simulated annealing can only obtain near global minimum value . References: [2].”A quantitative analysis of the simulated annealing algorithm: A case study for the traveling salesman problem.”, E. H. L. Aarts, J. H. M. Korst, P. J. M. van Laarhoven, Journ. of Statistical Physics 50, p189-206. (1988) [3]. ”計算方法叢書－非數值並行算法（第一冊）模擬退火算法”, 康立山等著, 科學出版社. (1997)

4. T High T Low A Markov chain start Temperature, Cold= Cini 1.Randomly select a pixel and randomly change its phase to a different value 2.Compute new cost function Probability P(ΔC)=exp(-ΔC/T) DC > 0 Cold = C new DC < 0 N <= P(ΔC ) Accept the change N > P(ΔC ) No Unacceptable change The iteration to reach length of Markov chain The N is a uniformly distributed random number in [0,1] The cost function is reconstruction error Yes End References: [2].”A quantitative analysis of the simulated annealing algorithm: A case study for the traveling salesman problem.”, E. H. L. Aarts, J. H. M. Korst, P. J. M. van Laarhoven, Journ. of Statistical Physics 50, p189-206. (1988) [3]. ”計算方法叢書－非數值並行算法（第一冊）模擬退火算法”, 康立山等著, 科學出版社. (1997)

5. DC (+) ： DC >0時的DC平均值。 γ0：較差代價函數值的轉移接受率，一般訂定γ0為0.99 。 DC (+) T0= ln(g0 -1) N-1 N-1 N-1 N-1 Cost＝ΣΣ(|G(p,q)|2－b hlimSpq)2 ΣΣ Spq＝1 p=0 q=0 p=0 q=0 模擬退火法設計之要點 Important points in the algorithm 1. Definition of the Cost function G(p,q)：正規化的複數振幅 hlim：效率上限 ，b：校正效率上限係數 2. Initial TemperatureT0 3. Cooling Schedule ( Decrement rule ) T k + 1＝a T k ，k =0,1,2, …　，一般訂定α為0.9 4. Length of Markov chain ( Stop criteria for each Markov chain ) L=mN 2，一般訂定m為5 5. Algorithm Stop Criteria No better configuration is found in a single Markov chain. ( Near-global minimum value). References: [2].”A quantitative analysis of the simulated annealing algorithm: A case study for the traveling salesman problem.”, E. H. L. Aarts, J. H. M. Korst, P. J. M. van Laarhoven, Journ. of Statistical Physics 50, p189-206. (1988) [3]. ”計算方法叢書－非數值並行算法（第一冊）模擬退火算法”, 康立山等著, 科學出版社. (1997)

6. ηs min SNRmin＝10×log10 (dB) ηn max ηs max－ηs min Uniformity＝ ×100% ηs max＋ηs min ηeff ＝ Σ ηmn×100% (m,n) Ss 如何評估DOEs SNRnim Uniformity 理想光點強度分佈 Grating Efficiency Corresponding normalized intensity ηmn＝|G(m,n)|2 G(m,n)：正規化的複數振幅 實際光點強度分佈 References: [4]. ”Design of diffractive optical elements with optimization of signal-to-noise ratio and without a dummy area.”, Jeng-Feng Lin and Alexander A. Sawchuk , Applied Optics /Vol. 36 No.14 /10 May 1997.

7. 設計繞射光學元件之模擬退火法的特性分析 配置空間 代價函數值及b 值與繞射效果的關係 平衡時配置出現的機率分佈 準平衡出現的機率分佈

8. 2 Nc = Z N ，Nc = 2 256 ≒ 1.2×1077. 配置空間 Configuration of two-dimensional DOEs N2 Pixel matrix, Z phase level If N=16, Z=2 Total no. of configurations Configuration of one-dimensional DOEs 2D DOE (ar2h) N=16, Z=2 Nc = Z N ，Nc = 2 16 = 65536. 配置(configuration)=DOE穿透函數 Set 1D-DOE pattern (thre) Set 1D-DOE pattern (sa01)

9. 平衡時配置出現的機率分佈 Equilibrium distribution exp {[Cmin－C(i)／c]} qi(c)＝ c : 溫度, i：配置, R：所有配置所成的集合 Σ qi(c)＝ 1. Σj Rexp {[Cmin－C(i)／c]} T =3.2494 T 0.3200 T =0.0186 T =0.0065 T =0.0012 T =0.0008 此圖代價函數全域最小值的總數 References: [2]. ”Simulated Annealing: theory and Applications”, P. J. M. van Laarhoven and E. H. L. Aarts , D. Reidel Publishing Company. (1992) [5].”A quantitative analysis of the simulated annealing algorithm: A case study for the traveling salesman problem.”, E. H. L. Aarts, J. H. M. Korst, P. J. M. van Laarhoven, Journ. of Statistical Physics 50, p189-206. (1988)

10. m =1, 100 times distance = 0.172770 m =3, 100 times distance = 0.115155 m =5, 100 times distance = 0.096626 m =3, 10000 times distance = 0.0325 m =5, 10000 times distance = 0.0249 m =1, 10000 times distance = 0.0273 準平衡出現的機率分佈 Quasi-equilibrium Distance= || a (Lk,ck)－ qi(ck)||＜ε, Lk Length of Markov chain , ck temperature , e small positive value. T=0.0065 T=0.0012 T=0.0008 T=0.0065 T=0.0012 T=0.0008 References: [2]. ”Simulated Annealing: theory and Applications”, P. J. M. van Laarhoven and E. H. L. Aarts , D. Reidel Publishing Company. (1992) [5].”A quantitative analysis of the simulated annealing algorithm: A case study for the traveling salesman problem.”, E. H. L. Aarts, J. H. M. Korst, P. J. M. van Laarhoven, Journ. of Statistical Physics 50, p189-206. (1988)

11. 模擬的繞射圖案 Simulated Diffraction pattern Ar2h Cir2 Plus Sa01 Stut Farm (32x32) Mesh (32x32) 1D-DOE pattern (Thre) 1D-DOE pattern (Sa01) 2-phase-level pattern : 2D-DOE pattern : ar2h, cir2, plus, sa01, farm, mesh. 1D-DOE pattern : thre,sa01 8-phase-level pattern : stut.

12. 參考資料 [1]. Introduction to Fourier Optics, Joseph W. Goodman , McGRAW-HELL (1996) [2]. Simulated Annealing: theory and Applications, P. J. M. van Laarhoven and E. H. L. Aarts , D. Reidel Publishing Company. (1992) [3]. 計算方法叢書－非數值並行算法（第一冊）模擬退火算法, 康立山等著, 科學出版社. (1997) [4]. ”Design of diffractive optical elements with optimization of signal-to-noise ratio and without a dummy area.”, Jeng-Feng Lin and Alexander A. Sawchuk , Applied Optics /Vol. 36 No.14 /10 May 1997. [5]. ”A quantitative analysis of the simulated annealing algorithm: A case study for the traveling salesman problem.”, E. H. L. Aarts, J. H. M. Korst and P. J. M. van Laarhoven, Journ. of Statistical Physics 50, p189-206. (1988) [6]. ”Efficency limit of spatially quantized fourier-type periodic diffraction optical elements”, Jeng-Feng Lin, Optics and photonics Taiwan’99, Dec. 16-17, p1094-1096, (1999). [7]. ”Verification and comparison of two efficiency limits of spatially quantized fourier-type periodic diffraction optical elements.”, Optics and photonics Taiwan’99, p1158-1159, Jeng-Feng Lin, (2001). [8]. ”Iterative simulated quenching for designing irregular-spot-array generators.” , Jean-Numa Gillet and Yunlong Sheng , Applied Optics /Vol. 39 No.20 /10 July 2000. [9]. ”Simulated Quenching with Temperature Rescaling for Designing Diffractive Optical Elements” , Yunlong Sheng ,工研院光電所89年微光電元件技術研討會. [10]. ”Spectrum leveling by an iterative algorithm with a dummy area for synthesizing the kinoform”, Hiroshi Akahori, Applied Optical/ Vol.25, No.5/ p802-811, 1 March 1988 [11]. Optics, Eugene Hecht, ADDISON WESLEY , (1998). [12]. ”Introduction to Diffractive Optical Elements (DOEs)”, Jeng-Feng Lin , STUT , May 29 2001