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
Definition of Diffractive Optical Elements (DOE)
Several μm ~
several tens ofμm
d =~ 1m m
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。
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))
References:
[1]. ”Introduction to Fourier Optics”, Joseph W. Goodman , McGRAW-HELL (1996)
Intensity
| F{g(x,y)} |2
模擬退火法
模擬退火法的基本流程
start
initialization
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.
T0 ,L0
T1 ,L1
Tn ,Ln
End
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)
A Markov chain
start
T High
Temperature, Cold= Cini
1.Randomly select a pixel and randomly
change its phase to a different value
2.Compute new cost function
T Low
Probability
P(ΔC)=exp(-ΔC/T)
DC > 0
N <= P(ΔC )
DC  Cnew  Cold
Cold = C new
DC < 0
Accept the change
N > P(ΔC )
Unacceptable change
The N is a uniformly distributed random
number in [0,1]
The cost function is reconstruction error
The iteration to reach
length of Markov chain
No
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)
模擬退火法設計之要點
Important points in the algorithm
1. Definition of the Cost function
N-1 N-1
Cost＝ΣΣ(|G(p,q)|2－b hlim Spq)2
p=0 q=0
N-1 N-1
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
ΣΣ Spq＝1
p=0 q=0
2. Initial Temperature T0
DC (+)
T0= ln(g -1)
0
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 。
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)
如何評估DOEs
SNRnim
ηs min
SNRmin＝10×log10
(dB)
ηn max
Uniformity
ηs max－ηs min
Uniformity＝η
×100%
s max＋ηs min
理想光點強度分佈
Grating Efficiency
ηeff ＝ Σ ηmn ×100%
(m,n) Ss
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.
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
Configuration of two-dimensional DOEs
N2 Pixel matrix, Z phase level
If N=16, Z=2
Total no. of configurations Nc = Z N
2
， Nc = 2 256 ≒ 1.2×1077.
Configuration of one-dimensional DOEs
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)
2D DOE (ar2h)
N=16, Z=2
平衡時配置出現的機率分佈
Equilibrium distribution
exp {[Cmin－C(i)／c]}
qi(c)＝
Σj R exp {[Cmin－C(i)／c]}
c : 溫度, i：配置, R：所有配置所成的集合
Σ qi(c) ＝ 1.
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)
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
m =1, 100 times distance = 0.172770
T=0.0065
m =1, 10000 times distance = 0.0273
T=0.0012
m =3, 100 times distance = 0.115155
T=0.0008
m =5, 100 times distance = 0.096626
T=0.0012
m =3, 10000 times distance = 0.0325
T=0.0008
m =5, 10000 times distance = 0.0249
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)
模擬的繞射圖案
Simulated Diffraction pattern
Ar2h
Farm (32x32)
Cir2
Plus
Mesh (32x32)
1D-DOE pattern (Thre)
2-phase-level pattern :
2D-DOE pattern : ar2h, cir2, plus, sa01, farm, mesh.
1D-DOE pattern : thre,sa01
8-phase-level pattern : stut.
Sa01
Stut
1D-DOE pattern (Sa01)
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, p189206. (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