设计绕射光学元件之模拟退火法的研究

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繞射光學元件繞射光學元件Definition of Diffractive Optical Elements (DOE)1D or 2D 表面起伏型週期性相位光柵Several m several tens ofm d = 1m m Laser beam(plane wave)Period:M x N phase elementsDOETransmittance functiong(x,y)=exp( i f (x,y)Intensity| Fg(x,y) |2 LensDiffraction patte

2、rnReferences:1. ”Introduction to Fourier Optics”, Joseph W. Goodman , McGRAW-HELL (1996)DOE 成像系統示意圖成像系統示意圖我們所定義的繞射元件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，而對於射入射入光的波前產生相位調變光的波前產生相位調變。startinitializationT0 ,L0T1 ,L1Tn ,LnEndA sequence of Markov chain1. Input diffraction pixel matrix (N 2)2. Set design

3、pattern 3. Set parameters : a , b , g0 , m .4. Generate a transmittance function (g(x,y) with random phase5. Compute initial cost function (Cini )6. Generate initial temperature (T0) 7. Algorithm stop criteria.模擬退火法模擬退火法模擬退火法的基本流程模擬退火法的基本流程 References:2.”A quantitative analysis of the simulated anne

4、aling 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)The cost function is globally minimized.If the elapsed time is infinite, si

5、mulated annealing can obtain global minimum value.The real simulated annealing can only obtain near global minimum value .使用模擬退火法的目的使用模擬退火法的目的 1.Randomly select a pixel and randomlychange its phase to a different value2.Compute new cost function Probability P(C)=exp(-C/T) Accept the changeCold = C n

6、ewUnacceptable changeEndA Markov chainN P(C ) The N is a uniformly distributed random number in 0,1The cost function is reconstruction errorYesNoThe iteration to reach length of Markov chainstartD DC 0 D DC 0時的DC平均值。0：較差代價函數值的轉移接受率，一般訂定0為0.99 。如何評估如何評估DOEsSNRnim Uniformity Grating Efficiency Corresp

7、onding normalized intensityReferences: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.SNRmin10log10 (dB)s minn maxUniformity 100%s maxs mins maxs mine

8、ff mn 100%(m,n) Ssmn|G(m,n)|2 G(m,n)：正規化的複數振幅 實際光點強度分佈實際光點強度分佈理想光點強度分佈理想光點強度分佈設計繞射光學元件之模擬退火法的特性分析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配置空間配置空間 Configuration of two-dimensional DOEsN2 Pixel matrix, Z phase levelIf N=16, Z=2Total no. of configurations Configuration of one-dime

9、nsional DOEs2D DOE (ar2h) N=16, Z=2Nc = Z N ， Nc = 2 256 1.21077.2Nc = Z N ， Nc = 2 16 = 65536.Set 1D-DOE pattern (thre)Set 1D-DOE pattern (sa01)配置(configuration)=DOE穿透函數Equilibrium distributionc : 溫度, i：配置, R：所有配置所成的集合 qi(c) 1.T =3.2494T =0.0065T 0.3200T =0.0012T =0.0186T =0.0008References:2. ”Simu

10、lated 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

11、. of Statistical Physics 50, p189-206. (1988)qi(c)exp CminC(i)cj R exp CminC(i)c平衡時配置出現的機率分佈平衡時配置出現的機率分佈 此圖代價函數全域最小值的總數Quasi-equilibrium Distance = | a (Lk,ck) qi(ck)|, Lk Length of Markov chain , ck temperature , e e small positive value. 準平衡出現的機率分佈準平衡出現的機率分佈 References:2. ”Simulated Annealing: the

12、ory 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 Phy

13、sics 50, p189-206. (1988)m =1, 100 times distance = 0.172770 m =3, 100 times distance = 0.115155 m =5, 100 times distance = 0.096626 m =1, 10000 times distance = 0.0273m =3, 10000 times distance = 0.0325 m =5, 10000 times distance = 0.0249 T=0.0065T=0.0012T=0.0008T=0.0065T=0.0012T=0.0008Ar2h Cir2 Pl

14、us Sa01 Stut Farm (32x32) Mesh (32x32) 1D-DOE pattern (Thre) 1D-DOE pattern (Sa01)Simulated Diffraction pattern2-phase-level pattern : 2D-DOE pattern : ar2h, cir2, plus, sa01, farm, mesh. 1D-DOE pattern : thre,sa018-phase-level pattern : stut.模擬的繞射圖案模擬的繞射圖案參考資料參考資料1. Introduction to Fourier Optics,

15、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 rati

16、o 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

17、Statistical Physics 50, p189-206. (1988)6. ”Efficency limit of spatially quantized fourier-type periodic diffraction optical elements”, Jeng-Feng Lin, Optics and photonics Taiwan99, Dec. 16-17, p1094-1096, (1999). 7. ”Verification and comparison of two efficiency limits of spatially quantized fourie

18、r-type periodic diffraction optical elements.”, Optics and photonics Taiwan99, 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