-
双向反射分布函数散射光空间关系如图1所示,函数被定义为单位立体角ΔΩ内的散射功率Δ
$ {P}_{s} $ 比入射功率$ {P}_{i} $ ,定义式为:$${\rm BRDF}\cdot {\rm cos}{\theta }_{s}=\frac{\Delta {P}_{s}\left({\theta }_{s}\right)}{\Delta {\varOmega }_{s}{P}_{i}}$$ (1) 光学薄膜表面光散射,需要考虑干涉效应,其双向反射分布函数可表示为[20]:
$${\rm BRDF}\cdot {\rm cos}{\theta }_{s}=\sum\limits_{i=1}^{N}\sum\limits_{j=0}^{N}{F}_{i}{F}_{j}^{*}{\rm PSD}_{ij}({f}_{x},{f}_{y})$$ (2) 式中:功率谱密度(PSD)描述了薄膜界面粗糙度(i=j)以及多层薄膜不同界面之间的粗糙度相关性(i≠j);
$ {F}_{j} $ 是第j个界面的光学因子。如果假设多层膜中所有界面有相同的PSD,则BRDF完全取决于光学因子。光学因子
$ {F}_{j} $ 与光照条件、观测条件、膜层折射率、膜层厚度等因素有关。由于光学因子的表达式比较广泛,这里给出了一个简单的表达式:$$ {F}_{j}=\left({\varepsilon }_{j}-{\varepsilon }_{j-1}\right)E\left({{\textit{z}}}_{j},{\theta }_{0}\right)E\left({{\textit{z}}}_{j},{\theta }_{s}\right)$$ (3) 式中:
$ {\varepsilon }_{j} $ 和${\varepsilon }_{j-1}$ 是第j层和第j−1层涂层材料的介电常数;$ E{\left({\textit{z}}\right)}_{j} $ 是第j层界面的电场振幅;$ {\theta }_{0} $ 为入射角;$ {\theta }_{s} $ 为散射角。根据公式(3)可知,介电常数的差值和界面处的电场强度可以调控光学因子。因此,可以通过薄膜设计来实现减小光学元件存在的光散射问题[21]。光学薄膜的特征矩阵为:
$${M_j} = \left[ {\begin{array}{*{20}{c}} {{\rm cos}{W_j}}&{iZ_j^{ - 1}{\rm sin}{W_j}}\\ {i{Z_j}{\rm sin}{W_j}}&{{\rm cos}{W_j}} \end{array}} \right]$$ (4) 式中:
${W}_{j}=\dfrac{2\pi }{\lambda }{n}_{j}{d}_{j}{\rm cos}{\theta }_{j}$ ,是相位差;Z表示修正导纳,${Z}_{j}={n}_{j}{\rm cos}{\theta }_{j}$ (TE),${Z}_{j}={n}_{j}/{\rm cos}{\theta }_{j}$ (TM),$ {\theta }_{j} $ 是折射角;j表示第j层膜。L层膜的特征矩阵为:$$M = \prod\limits_{j = 1}^L {{M_j}} = \left[ {\begin{array}{*{20}{c}} {{m_{11}}}&{i{m_{12}}}\\ {i{m_{21}}}&{{m_{22}}} \end{array}} \right]$$ (5) 由此可得:
$${E}_{0}^++{E}_{0}^-=\left({m}_{11}+{im}_{12}{Z}_{s}\right){E}_{s}^+$$ (6) $${E}_{0}^+-{E}_{0}^-=\left(\frac{{im}_{21}}{{Z}_{0}}+\frac{{m}_{22}{Z}_{s}}{{Z}_{0}}\right){E}_{s}^+$$ (7) 式中:
$ {E}_{0}^{+} $ 和$ {E}_{0}^{-} $ 分别代表入射界面的入射光和反射光的电场强度;$ {E}_{s}^{+} $ 代表基底内入射光的电场强度;$ {Z}_{0} $ ,$ {Z}_{s} $ 分别是入射介质和基底的修正导纳,由公式(6)、(7)可得:
$${\left|{E}_{0}^+\right|}^{2}=0.25\left[{\left({m}_{11}+\frac{{m}_{22}{Z}_{s}}{{Z}_{0}}\right)}^{2}+{\left(\frac{{m}_{21}}{{Z}_{0}}+{m}_{12}{Z}_{s}\right)}^{2}\right]{\left|{E}_{s}^+\right|}^{2}$$ (8) $$ {\left|{E}_{0}^-\right|}^{2}=0.25\left[{\left({m}_{11}-\frac{{m}_{22}{Z}_{s}}{{Z}_{0}}\right)}^{2}+{\left(\frac{{m}_{21}}{{Z}_{s}}+{m}_{12}{Z}_{0}\right)}^{2}\right]{\left|{E}_{s}^+\right|}^{2}$$ (9) 第一层膜中距离薄膜入射界面下
${\rm{\Delta }}{{z}}$ 处的逆矩阵为:$$\left[ {\begin{array}{*{20}{c}} {m_{11}^{\rm{'}}}&{m_{12}^{\rm{'}}}\\ {m_{21}^{\rm{'}}}&{m_{22}^{\rm{'}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {{\rm cos}{\rm{\Delta }}U}&{ - iZ_1^{ - 1}{\rm sin}{\rm{\Delta }}U}\\ { - i{Z_1}{\rm sin}{\rm{\Delta }}U}&{{\rm cos}{\rm{\Delta }}U} \end{array}} \right]$$ (10) 式中:
${\rm{\Delta }}U=2\pi n{\rm{\Delta }}z{\rm cos}{\theta }_{1}/\lambda$ 。该点电场的平方为:
$$ {\left|E\right|}^{2}=\left[{\left({m}_{11}^{''}\right)}^{2}+{\left({{Z}_{s}m}_{11}^{''}\right)}^{2}\right]{\left|{E}_{s}^+\right|}^{2}$$ (11) 其中,
$ \left[ {\begin{array}{*{20}{c}} {m_{11}^{{\rm{''}}}}&{m_{12}^{\rm{'}}}\\ {m_{21}^{{\rm{''}}}}&{m_{22}^{{\rm{''}}}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} {m_{11}^{\rm{'}}}&{im_{12}^{\rm{'}}}\\ {im_{21}^{\rm{'}}}&{m_{22}^{\rm{'}}} \end{array}} \right] \times \left[ {\begin{array}{*{20}{c}} {{m_{11}}}&{i{m_{12}}}\\ {i{m_{21}}}&{{m_{22}}} \end{array}} \right]$ 。以此类推可求得膜内任一点处电场强度的平方。
由公式(8)、(11)可得电场强度平方相对值的分布为:
$${{N}}=\frac{{\left|E\right|}^{2}}{{\left|{E}_{0}^+\right|}^{2}}$$ (12) 总散射损耗(S)与双向反射分布函数之间的关系可表示为[22]:
$$ S =2\pi {\int }_{0}^{\frac{\pi }{2}}{\rm BRDF}\cdot {\rm cos}{\theta }_{s}\cdot {\rm sin}{\theta }_{s}\cdot {\rm d}{\theta }_{s}$$ (13) Bobbert-Vlieger(B-V)模型基于米氏散射理论,是用来描述光学表面球状粒子的散射[23],其适用条件为粒子半径与入射光波长相当,并且该模型也可以用来分析薄膜表面球状粒子的光散射[24]。
-
光学元件的散射损耗主要来源于光学元件表面粗糙度、划痕、污染物等引起的表面散射以及内部缺陷等引起的体散射。文中的研究重点是降低光学元件表面污染物引起的光散射,因此,在研究中暂不考虑光学元件和光学薄膜表面粗糙度和内部缺陷等引起的光散射,同时将表面粒子污染物等效为球状粒子。
当光在薄膜内部传播时,入射光和反射光会因为干涉效应在光学薄膜内部形成驻波场,而此驻波场分布可通过在光学元件表面设计光学薄膜来进行调控。因此,可通过在光学元件表面设计单层光学薄膜来实现对光学元件表面上方粒子污染物球心处的驻波场场强进行调控,当粒子球心处电场强度最小时,依据公式(2)和(3)可知,此时可获得最小的粒子散射损耗。其物理模型如图2所示。
如图2所示,光学表面上各种球体表示大小不一的粒子污染物。左侧为光学元件表面上存在粒子污染物时的情况,当入射光束照射在粒子污染物上时,由于表面粒子污染物的存在,会引起一定的光散射损耗。右侧为镀有单层光学薄膜的光学表面,其表面存在与左侧K9玻璃基底相同的表面粒子污染物。
为了有效调控污染物粒子球心处的电场强度,笔者选用高折射率TiO2和低折射率的SiO2分别作为膜层材料,以K9玻璃为基底,分别设计两种不同折射率的单层光学薄膜,通过调整和优化单层薄膜的厚度进而调控粒子污染物所在处的电场强度。为了保证粒子污染物所在处电场极小值对应的薄膜厚度的准确性,需要粒子的大小尽可能在入射光波长的二分之一之内,即小于316.4 nm(
$ {\rm{\lambda }} $ =632.8 nm),文中选择半径为100 nm的球状粒子污染物作为理论分析对象,该尺寸也是千级超净环境下粒子污染物半径的统计值。图3中,入射光波长
$ {\rm{\lambda }} $ =632.8 nm,在K9玻璃基底上设计单层光学薄膜,球状粒子污染物位于薄膜的上表面,且球状底部与薄膜表面的距离为0。图中不同曲线分别表示半径为158.2 nm和半径为100 nm粒子污染物球心处的电场强度分布。可以看出粒子球心处的电场强度会随着薄膜厚度的变化而变化。在得到粒子球心处电场强度分布后,可将粒子球心处的电场强度极小值对应的薄膜厚度作为降低粒子污染物球心处电场强度分布的最佳优化膜层厚度,在此基础上,进一步分析表面粒子污染物的散射特性,以及单层光学薄膜对表面粒子污染物散射的调控特性。
Single-layer film regulation characteristics of particle pollutant scattering from optical surfaces
-
摘要: 为了降低超精密低损耗光学元件表面粒子污染物的光散射损耗,文中提出通过在光学表面沉积单层薄膜来调控表面场强分布,从而降低散射损耗的方法。理论分析了K9玻璃超光滑光学表面不同厚度单层二氧化硅(SiO2)和单层二氧化钛(TiO2)薄膜表面上方半径为100 nm粒子污染物所在处的电场强度,理论分析结果发现,当SiO2薄膜厚度为137.4 nm,TiO2薄膜厚度为12.3 nm时,表面粒子污染物所在处的电场强度最小。在此基础上分别计算了光学元件表面沉积厚度为137.4 nm的单层SiO2薄膜以及厚度为12.3 nm的单层TiO2薄膜,表面粒子污染物的总散射损耗(S)和双向反射分布函数(BRDF),计算结果表明,在波长为632.8 nm的光垂直入射时,单层SiO2薄膜和单层TiO2薄膜可有效降低其表面粒子的BRDF,且可将K9玻璃表面的总散射分别降低12.40%和25.04%。实验验证了单层SiO2薄膜对于表面粒子污染物散射降低的有效性。
-
关键词:
- 光学薄膜 /
- 光散射 /
- 双向反射分布函数(BRDF) /
- 总散射损耗
Abstract: In order to reduce the light scattering loss of particle pollutants on the surface of ultra-precision low-loss optical elements, a method of adjusting the surface field intensity distribution by depositing a monolayer film on the optical surface was proposed. The electric field intensity of particle pollutants with a radius of 100 nm above the surface of monolayer SiO2 and monolayer TiO2 thin films of different thicknesses on the ultra-smooth optical surface of K9 glass was theoretically analyzed. The theoretical analysis results show that when the thickness of SiO2 thin film is 137.4 nm and the thickness of TiO2 thin film is 12.3 nm, the surface particle contaminant has the lowest electric field intensity. On this basis, the total scattering loss (S) and bidirectional reflection distribution function (BRDF) of particle pollutants on the surface of optical element, films for SiO2 with a thickness of 137.4 nm and TiO2 with a thickness of 12.3 nm were calculated respectively. The calculated results show that when the light with a wavelength of 632.8 nm is vertically incident, monolayer SiO2 film and monolayer TiO2 film can effectively reduce the BRDF of the surface particles, and the total scattering on the surface of K9 glass can be reduced by 12.40% and 25.04% respectively. The effectiveness of single-layer SiO2 thin film for reducing the scattering of surface particle pollutants is verified by experiments. -
-
[1] Tribble A C, Boyadjian B, Davis J, et al. Contamination control engineering design guide-lines for the aerospace community[C]//Proceedings of the SPIE, 1996, 2864: 4-15. [2] Sun Tengfei, Zhang Jun, Lv Haibing, et al. Influence of optical mirror surface contaminants on lasertransmission characteristics [J]. Infrared and Laser Engineering, 2014, 43(5): 1444-1448. (in Chinese) doi: 10.3969/j.issn.1007-2276.2014.05.017 [3] Meng Lingqiang, Wang Qing, Zhang Liping, et al. Laser-induced damage threshold test system based on light scattering and grey level of image [J]. Optical Instruments, 2016, 38(6): 534-538. (in Chinese) [4] 段利华. 光学薄膜激光损伤及散射检测研究[D]. 重庆大学, 1005. Duan Lihua. Study on laser damage and scattering detection of optical thin films[D]. Chongqing: Chongqing University, 2005. (in Chinese) [5] Zhao Xue, Zhou Yanping, Liu Haigang. Effect of optical system caused by space organism pollution [J]. Optical Technology, 2004(1): 113-115, 118. (in Chinese) [6] Ma Jing, Zhu Funan, Zhou Yanping, et al. Detection device of optical surface pollution for satellite and ground [J]. Optics and Precision Engineering, 2016, 24(8): 1878-1883. (in Chinese) doi: 10.3788/OPE.20162408.1878 [7] Xiao Jing, Zhang Bin. Influence of the optical components contamination on the signal to noise ratio in infrared optical systems [J]. Infrared and Laser Engineering, 2012, 41(4): 1010-1016. (in Chinese) doi: 10.3969/j.issn.1007-2276.2012.04.033 [8] Murotani H. Influence of the surface-roughness of the substrate on the light scatteringof optical thin films [J]. Journal of the Japan Society of Precision Engineering, 2014, 80(6): 519-523. doi: 10.2493/jjspe.80.519 [9] Young R P. Mirror scatter degradation by particulate contamination[C]//Proceedings of the SPIE, Optical System Contamination: Effects, Measurement, Control II, 1990, 1329: 246-254. [10] Facey T A, Nonnenmacher A L. Measurement of total hemispherical emissivity contaminated mirror surfaces[C]//Proceedings of the SPIE, 1989, 967: 308-313. [11] Nahm K, Spyak P R, Wolfe W L. Scattering from contaminated surfaces[C]//33rd Annual Techincal Symposium. International Society for Optics and Photonics, 1990: 294-305. [12] Nahm K. Light scattering by polystyrene spheres on a conducting plane[D]. Arizona: The University of Arizona, 1985. [13] Spyak P R, Wolfe W L. Scatter from particulate-contaminated mirors. part 2: Theory and experiment for dust and lambda= 0.6328 μm [J]. Optical Engineering, 1992, 31(8): 1757-1763. doi: 10.1117/12.58709 [14] Spyak P R. A cryogenic scatterometer and scatter from particulate contaminants on minors[D]. Arizona: The University of Arizona, 1990. [15] Huang Cong, You Xinghai, Zhang Bin. Influence of surface cleanliness of optical element on its surface scattering characteristics [J]. Infrared and Laser Engineering, 2019, 48(1): 0120002. (in Chinese) doi: 10.3788/IRLA201948.0120002 [16] Gao Pingping, Lu Min, Wang Zhile, et al. Differentiation of polarization scattering characteristics of surface nanoparticle defects [J]. Chinese Optics, 2020, 13(5): 975-987. (in Chinese) doi: 10.37188/CO.2020-0083 [17] Kim H W, Reif R. Ex situ wafer surface cleaning by HF dipping for low temperature silicon epitaxy [J]. Thin Solid Films, 1997, 305(1-2): 280-285. doi: 10.1016/S0040-6090(97)00122-3 [18] 张志国. 光学元件表面的洁净风刀冲扫技术研究[D]. 哈尔滨: 哈尔滨工业大学, 2015, 41-50. Zhang Zhiguo. Research on clean air knife rinse technology for optical component surface[D]. Harbin: Harbin Institute of Technology, 2015: 41-50. (in Chinese) [19] Wang Zemin, Zeng Xiaoyan, Huang Weiling. Status and prospect of laser cleaning procedure [J]. Laser Technology, 2000, 24(2): 68-73. (in Chinese) [20] Trost M, Herffurth T, Schröder S, et al. Scattering reduction through oblique multilayer deposition [J]. Appl Opt, 2014, 53(4): 197-204. doi: 10.1364/AO.53.00A197 [21] Zhang Jinlong, Wu Han, Jiao Hongfei, et al. Reducing light scattering in high-reflection coatings through destructive interference at fully correlated interfaces [J]. Optics Letters, 2017, 42(23): 5046. doi: 10.1364/OL.42.005046 [22] Yang Chen, Pan Yongqiang. Light scattering properties of three layers broadband anti-reflective films [J]. Optics and Optoelectronic Technology, 2018, 16(1): 11-15. (in Chinese) [23] Bobbert P A, Vlieger J. Light scattering by a sphere on a substrate [J]. Physica A: Statistical Mechanics and Its Applications, 1986, 137(1-2): 209-242. doi: 10.1016/0378-4371(86)90072-5 [24] Kim J H, Ehrman S H, Mulholland G W, et al. Polarized light scattering by dielectric and metallic spheres on oxidized silicon surfaces [J]. Applied Optics, 2004, 43(3): 585-591. doi: 10.1364/ao.43.000585 [25] Pan Y, Liu J, Lei G, et al. Reducing light scattering of single-layer TiO2 and single-layer SiO2 optical thin films [J]. Optik - International Journal for Light and Electron Optics, 2021, 231: 166380. doi: 10.1016/j.ijleo.2021.166380