-
分别用GS算法和模拟退火算法设计傅里叶型DOE,如图 1所示。已知输出的衍射像面Σ2上目标图像的振幅分布为Buv,(u, v)为输出面点的坐标,且DOE为相位型元件,即输入的物面Σ1上振幅分布Axy=1,(x, y)为输入面点的坐标,FT为傅里叶变换(Fourier transform),IFT为傅里叶逆变换(inverse Fourier transform),设计结果为Σ1面上的相位分布。
GS算法迭代过程如下:(1)生成随机相位分布φ0,与已知振幅Axy(Axy=1)组成初始物面分布,即f=Axyexp(jφ0);(2)将复振幅分布f代入(1)式进行傅里叶变换,得到衍射像分布F;(3)用目标图像振幅Buv代替F中的振幅,与F中的相位组成新的衍射像分布F′,即F′=Buvexp[jangle(F)],angle()代表计算相位;(4)将F′代入(2)式进行逆傅里叶变换,得到物面分布f′,用振幅1代替f′中振幅,与其相位组成新的物面分布f,即f=1exp[jangle(F')];(5)回到第(2)步,进行下一次循环,直到误差函数达到给定精度ε,或循环次数达到阈值停止迭代,即可得到所需的DOE相位分布φ=angle(f′) [20]:
$ \begin{array}{l} F\left( {u, v} \right) = \int {\int_{ - \infty }^{ + \infty } {f(x, y){\rm{exp}}\left[ { - {\rm{j2 \mathit{ π} }}\left( {ux} \right.} \right. + } } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {\mathit{uy}} \right)} \right]{\rm{d}}x{\rm{d}}y \end{array} $
(1) $ \begin{array}{l} f\prime \left( {x, y} \right) = \int {\int_{ - \infty }^{ + \infty } {F\prime (u, v){\rm{exp}}\left[ {{\rm{j2 \mathit{ π} }}\left( {ux} \right.} \right. + } } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {\left. {\mathit{uy}} \right)} \right]{\rm{d}}u{\rm{d}}v \end{array} $
(2) 模拟退火算法作为局部搜索算法的扩展,其核心思想是在每一次修改模型的过程中,在最优状态下的邻域内随机产生1个新的状态模型,然后以一定的概率选择邻域中评价函数更好的状态,这种接受新模型的方式使其成为一种全局最优算法。
模拟退火算法的迭代过程如下:(1)给定初始相位φ0,初始温度t0,最低温度tend;(2)对当前相位φ0做邻域扰动,生成新相位解:φ1=φ0+S(x, y),S(x, y)为邻域扰动因子;(3)初始相位φ0和新相位解φ1分别与给定振幅Axy(Axy=1)组合,计算衍射像分布F0和F1;(4)计算F0和F1与目标图像振幅Buv间的均方(mean square, MS)误差EMS, 0和EMS, 1;(5)判断两个误差的大小,若EMS, 1 < EMS, 0,用衍射像分布F1逆运算得到DOE分布f1,否则重复步骤(2)~步骤(5),直到达到当前温度下的最大循环;(6)降低温度进行下一次迭代,直到达到最低温度tend,输出优化后的相位,即所求相位[15]。
模拟中使用均方误差EMS判断输出面振幅与目标图像振幅间的误差,它常用来判断设计结果是否满足要求,EMS定义如下:
$ {E_{{\rm{MS}}}} = \frac{{\sum\limits_{u = 1}^U {\sum\limits_{v = 1}^V {{{\left[ {\left| {{F_{uv}}} \right| - {B_{uv}}} \right]}^2}} } }}{{\sum\limits_{u = 1}^U {\sum\limits_{v = 1}^V {{B_{uv}}^2} } }} $
(3) DOE衍射像期望区域的光强不均匀性IRMS(均方根root mean square, RMS)与振幅不均匀性ARMS是DOE设计的另一重要评价标准,它们的定义如下:
$ \left\{ \begin{array}{l} {I_{{\rm{RMS}}}} = \sqrt {\frac{{\sum\limits_{u = 1}^U {\sum\limits_{v = 1}^V {{{\left[ {\frac{{{I_{uv}}}}{{\bar I}} - 1} \right]}^2}} } }}{{UV - 1}}} {\rm{ }}\\ \bar I = \frac{{\sum\limits_{u = 1}^U {\sum\limits_{v = 1}^V {{I_{uv}}} } }}{{UV}} \end{array} \right. $
(4) $ \left\{ \begin{array}{l} {A_{{\rm{RMS}}}} = \sqrt {\frac{{\sum\limits_{u = 1}^U {\sum\limits_{v = 1}^V {{{\left[ {\frac{{{A_{uv}}}}{{\bar A}} - 1} \right]}^2}} } }}{{UV - 1}}} {\rm{ }}\\ \bar A = \frac{{\sum\limits_{u = 1}^U {\sum\limits_{v = 1}^V {{A_{uv}}} } }}{{UV}} \end{array} \right. $
(5) (3) 式~(5)式中,(u, v)为输出面Σ2上采样点的坐标; U, V为行与列的采样点数,计算中均取512;Iuv和Auv分别为Σ2上期望区域点(u, v)处的光强和振幅; I和A为期望区域的平均光强与平均振幅。
-
用标量衍射理论来阐明改进方法的原理。1678年, 惠更斯为了描述波的传播过程提出关于子波的设想; 1818年, 菲涅耳引入干涉概念补充了惠更斯原理,提出了计算衍射光场分布U(P)的方法,如图 2所示。其表达式为:
$ U\left( P \right)=C\iint{U\left( {{P}_{0}} \right)K\left( \theta \right)\text{exp}\left( \text{j}kr \right){{r}^{-1}}\text{d}S} $
(6) 式中,S为积分曲面,C为常数,U(P0)为衍射面上任一点P0处光场的复振幅,θ为PP0与过P0点波面的法线n之间的夹角,K(θ)表示子波源P0对P点光场的贡献,它与角度θ有关,k是波数,r是P与P0之间的距离。但此表达式仅基于子波源和相干叠加的假说,缺乏严格的波动理论基础支撑,C与K(θ)的具体形式也无法确定。
1882年, 基尔霍夫从亥姆霍兹方程出发,利用格林定理求解波动方程,导出了基尔霍夫衍射公式:
$ U(P)=\frac{1}{4\text{ }\!\!\pi\!\!\text{ }}\iint{\left[ G\cdot \frac{\partial U\left( {{P}_{0}} \right)}{\partial n}-U({{P}_{0}})\cdot \frac{\partial G}{\partial n} \right]\text{d}S} $
(7) 式中, 格林函数G=exp(jkr)/r,推导中用到两点假设:(1)在衍射孔径Σ上(如图 2所示)场分布U(P0)及其偏导数$\partial$U(P0)/$\partial$n与没有衍射屏时完全相同;(2)在位于衍射屏几何阴影区S1的那部分上,场分布U及其偏导数$\partial$U(P0)/$\partial$n恒为0。以此为基础推导出了著名的菲涅耳-基尔霍夫衍射公式:
$ \begin{align} & \ \ \ \ \ U\left( P \right)=\frac{1}{\text{j}\lambda }\iint{U\left( {{P}_{0}} \right)}\times \\ & \frac{\text{cos}(\vec{n}, \vec{r})-\text{cos}(\vec{n}, \vec{r}\prime )}{2}\frac{\text{exp}\left( \text{j}kr \right)}{r}dS~ \\ \end{align} $
(8) 式中, λ是光波波长,λ=2π/k。正是这两点假设备受质疑[21],因为对于波动方程,如果它的一个解在任意非无限小的面上,光场的复振幅和它的法向导数都为零,则这个解在整个空间都为零。直接与这个基本假设相矛盾。为此索末菲选择了与基尔霍夫不同的格林函数G,从而在边界条件中不必规定屏后光场及其法向导数均为零,克服了基尔霍夫边界条件的矛盾。
实际情况是,衍射屏会在一定程度上影响孔径附近的场分布,屏后的阴影区光场分布不会完全为零,光场会泄漏到屏后孔径之外[21]。比如常用的直边菲涅耳衍射,其衍射图样的强度呈减幅振荡分布,不存在光强为零的区域。所以在空间中任意处有光源存在,便不会有光强分布为0的区域,反之亦然。于是,在衍射光学元件设计中必须考虑到这一特性,继而作相应的改进:目标图像的背景不应该为零,必须增加一个振幅自由度。最简单的方法是将目标图像的背景振幅设置为不为零的均匀振幅分布。但注意到衍射是一个低通滤波的过程,通常会引起如直边衍射的振铃现象,所以增加的振幅分布取成减幅振荡与均匀振幅的混合会更合理。
振幅自由度对衍射光学元件成像的影响研究
Effect of amplitude freedom on imaging of diffractive optical elements
-
摘要: 为了在衍射光学元件设计中提高再现像光强分布均匀性, 改善成像质量, 利用光场传播过程中一定存在泄漏的现象, 为目标图像增加了混合型振幅自由度, 优化了设计算法, 并设计了傅里叶变换型衍射光学元件, 验证了方法的有效性。结果表明, 增加振幅自由度后设计的元件的再现像光强不均匀性从14%左右降到了1%以下, 均方误差从10%左右降到了0.1%以下。该方法对Gerchberg & Saxton(GS)算法和模拟退火算法均适用。Abstract: In order to improve the uniformity of intensity distribution of the reconstructed images and improve the imaging quality in the design of the diffractive optical elements, the design algorithm is optimized by adding a mixed amplitude degree of freedom to the target image due to the fact that there must be leakage in the light field propagation process, and the effectiveness of the method was verified by designing a Fourier transform type diffractive optical element. The simulation results show that, compared without amplitude degree of freedom, the reconstructed image's intensity uniformity of the elements designed by this method is reduced from about 14% to less than 1%, and the mean square error is reduced from about 10% to less than 0.1%, and the method is applicable to both Gerchberg & Saxton(GS) algorithm and simulated annealing algorithm.
-
Key words:
- diffraction /
- diffractive optical element /
- amplitude degree of freedom /
- uniformity
-
-
[1] JIN G F, YAN Y B, WU M X, et al. Binary optics.Beijing: National Defense Industry Press, 1998: 1-3(in Chinese). [2] GERCHBERG R W, SAXTON W O. A practical algorithm for the determination of phase from image and diffraction plane pictures[J]. Optik, 1972, 35(2):237-250. [3] GU B Y, DONG B Zh, YANG G Zh, et al. Iterative optimization approach for the design of diffractive phase elements simultaneously implementing several optical functions[J]. Journal of the Optical Society of America, 1994, A11(5):1632-1640. [4] SANG T, WANG R L. Optic diffraction transform pair and its application to design of diffractive phase elements[J]. Acta Photonica Sinica, 1997, 26(11):1020-1027. [5] CORMIER G, BOUDREAU R, THERIAULT S. Real-coded genetic algorithm for Bragg grating parameter synthesis[J]. Journal of the Optical Society of America, 2001, B18(12):1771-1776. [6] WYROWSKI F. Diffractive optical elements: Iterative calculation of quantized, blazed phase structures[J]. Journal of the Optical Society of America, 1990, A7(6):1738-1748. [7] SOIFER V, KOTLYAR V, DOSKOLOVICH L. Iterative methods for diffractive optical elements computation[M]. New York, USA: Taylor & Francis, 1997: 66-80. [8] HUANG L X, YAO X, CAI D M, et al. A high accuracy and fast iterative algorithm for phase retrieval. Chinese Journal of Lasers, 2010, 37(5):1218-1221(in Chinese). [9] PENG J M, DU Sh J, JIANG P Zh. Phase retrieval based on improved GS algorithm[J]. High Power Laser and Particle Beams, 2013, 25(2):315-318(in Chinese). doi: 10.3788/CJL20103705.1218 [9] WANG H Ch, YUE W R, SONG Q, et al. A hybrid Gerchberg Saxton-like algorithm for DOE and CGH calculation[J]. Optics & Lasers in Engineering, 2017, 89:109-115. [10] GU X, XU K Sh. Some improvements on the GS algorithm in phase retrieval problem[J]. Journal of Fudan University(Natural Science Edition), 2000, 39(2):205-211(in Chinese). [11] GU X, XU K Sh. Some improvements on the GS algorithm in phase retrieval problem[J]. Journal of Fudan University(Natural Science Edition), 2000, 39(2):205-211(in Chinese). [12] ZOU J Y, LU Y X, WANG L, et al. An improvements on GS algorithm for design of computer optical elements. Journal of Optoelectronics·Laser, 2007, 18(10):1180-1183(in Chinese). [13] CHANG Ch L. Study on the algorithm of computer generated hologram based on diffraction theory[D]. Nanjing: Southeast University, 2015: 65-74(in Chinese). [14] LIANG Ch Y, ZHANG W, RUI D W, et al. Speckle reduction method with phase range limited diffractive optical element[J]. Acta Photonica Sinica, 2017, 46(1):0105002(in Chinese). doi: 10.3788/gzxb20174601.0105002 [15] LI M, TANG J B, LIU J. Improved GS algorithm of BOE based on simulated annealing principle[J]. Ship Electronic Engineering, 2013, 33(10):52-55(in Chinese). [16] ZHANG W, LIANG Ch Y, LI J, et al. Design of optical elements for beam shaping and uniform illumination in laser digital projection display system[J]. Acta Optica Sinica, 2015, 35(8):0805001(in Chinese). doi: 10.3788/AOS201535.0805001 [17] QU W D, GU H R, TAN Q F. Design of refractive/diffractive hybrid optical elements for beam shaping with large diffraction pattern[J]. Chinese Optics Letters, 2016, 14(3):031404. doi: 10.3788/COL201614.031404 [18] YANG M X, KONG Zh, TAN Q F, et al. Precise design of diffractive optical elements for annular beam shaping. Acta Optica Sinica, 2019, 39(3):0305002(in Chinese). doi: 10.3788/AOS201939.0305002 [19] ZOU J Y, LU Y X, HUANG Z Q, et al, The design of diffraction optical elements for beam shaping with improved GS algorithm[J]. Infrared and Laser Engineering, 2006, 35(s2):48-52(in Chinese). [20] YAN Sh H. Design of diffractive micro-optics[M].Beijing: National Defense Industry Press, 2011: 66-71(in Chinese). [21] LÜ N G. Fourier optics[M]. Beijing: China Machine Press, 2016: 74-86(in Chinese).