极紫外光栅波导中TE_(0)模式有效折射率分析  

Investigation of EUV Grating Waveguide Effective Refraction Index for TE_(0)Mode

作　　者：毛亚星 曹清[1] 程长杰 朱吉瑞 陈文轩 肖晓琳 Mao Yaxing;Cao Qing;Cheng Changjie;Zhu Jirui;Chen Wenxuan;Xiao Xiaolin(Department of Physics,Shanghai University,Shanghai 200444,China;Zhejiang Hangzhou No.7 High School,Hangzhou 310024,Zhejiang,China;Precision Optical Manufacturing and Testing Center,Shanghai Institute of Optics and Fine Mechanics,Chinese Academy of Sciences,Shanghai 201800,China)

机构地区：[1]上海大学理学院物理系,上海200444 [2]浙江省杭州第七中学,浙江杭州310024 [3]中国科学院上海光学精密机械研究所精密光学制造与检测中心,上海201800

出　　处：《中国激光》2024年第12期457-464,共8页Chinese Journal of Lasers

摘　　要：将极紫外光刻掩模板的吸收层视为光栅波导,对光栅波导的最低阶横电模式(TE_(0)模式)进行理论分析。根据光栅波导电场分布的对称性,首先指出其电场分布应为双曲余弦函数cosh(·),并提供了有效折射率的零阶近似值n_(eff,0)。利用有限元软件对TE_(0)模式进行仿真,然后将模拟结果与解析表达式计算结果进行对比,验证理论猜测的准确性。在吸收材料为金和银的情况下,零阶近似值(n_(eff,0))的相对误差已分别小至0.75%和0.96%。在此基础上,进一步推导出了一个简单的适用于非亚波长光栅波导有效折射率(neff)的高阶迭代公式,并用其来减小零阶近似的相对误差。数值模拟结果表明,经过7次迭代后,相对误差已减小到0.11%;经过23次迭代后,相对误差小于10^(-5)。Objective The imaging phenomena in extreme ultraviolet(EUV)lithography must be elaborated from more than one perspective.Traditionally,previous studies on waveguide methods have considered the cladding electric field distribution of the absorber as an evanescent field,which is similar to a single parallel-plate waveguide.However,these studies ignore the periodicity of the absorbers.In this study,the absorber of the EUV lithography mask is regarded as a grating waveguide.Owing to the periodicity of the absorber,two adjacent periods must affect the field distribution.Therefore,the electric field of the absorber is a linear superposition of the adjacent periodic field distributions.We propose that the electric field distribution in the absorber in the lowest-order transverse electric(TE_(0))mode is a hyperbolic cosine function cosh(·).We provide the zero-order approximation value n_(eff,0)of the effective refractive index neff for the TE_(0)mode.To further decrease the relative error of n_(eff,0)according to the boundary conditions,we derive the eigenvalue equation for the grating waveguide.To obtain a good approximation,we derive an iterative formula of n_(eff,m)and use the iteration method to decrease relative error.Methods According to the line types in the grating waveguide,we assumed a fitting curve function for the electric field distribution of the TE_(0)mode and provided theαvalue range of the grating waveguide.According to waveguide theory,the field distribution in the core should be cos(·).Owing to the periodicity of the absorber,the electric field distribution in the cladding must not be in an exponential decay form,which will be cosh(·).Furthermore,we assumed a zero-order approximation value for the effective refractive index.To verify the accuracy of n_(eff,0)and the feasibility of cosh(·),we used the COMSOL Multiphysics software to simulate the TE_(0)mode in the grating waveguide.We selected Au and Ag as absorber materials for the simulation,and the findings indicated that there were small errors

关 键 词：物理光学 极紫外 光栅波导 有效折射率 零阶近似值 高阶数值迭代 

分 类 号：O436[机械工程—光学工程]