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

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 between the simulated and theoretical results.To increase the accuracy further,the eigenvalue equation of the grating waveguide was obtained according to the boundary conditions.We also derived an iterative formula for the mth-order effective refractive index(n_(eff,m)).As a special example,we selected Au as the cladding material to further verify the iterative formula.An iterative relation equation and iteration method were used to decrease the relative error.Results and Discussions The relative errors of n_(eff,0)for Au and Ag are 0.75%and 0.96%,respectively(Table 1).The accuracy of n_(eff,0)is extremely high,but there is still a slight error between the theoretical and simulated field distributions(Fig.3).To further increase the accuracy,we selected Au to verify the iterative formula.The relative error changes with the number of iterations(Fig.4);with an increase in m,the relative error decreases.When m=23,the relative error decreases to less than 10^(-5).In this case,the field distribution also shows very good agreement with the simulated result shown in Fig.5.It can be observed that with the iterative formula,n_(eff,23)can describe the TE_(0)mode of the grating waveguide accurately.Conclusions We consider the absorber of the EUV lithography mask as a grating waveguide and perform a rigorous simulation to describe the TE_(0)mode of the absorber.Owing to the periodicity of the grating waveguide,the electric field in the cladding of the absorber is a linear superposition of two adjacent periodic field distributions,which is the cosh(·)function.We propose a zero-order approximation value for the effective refractive index.The accuracy of the zero-order approximation value is verified by selecting Au and Ag absorbers for the simulation using COMSOL.The relative errors in n_(eff,0)for the two materials are 0.75%and 0.96%,respectively.The relative errors were already small initally,and we use an iterative method to further increase the accuracy of n_(eff,0).The eigenvalue equation for the grating waveguide is derived based on the boundary conditions.Subsequently,a simple iterative formula with a high accuracy is obtained.As a specific example,the Au absorber material is selected to verify the feasibility of the iterative formula.After seven iterations,the relative error of n_(eff,7)decreases to 0.11%.After 23 iterations,n_(eff,23)converges to the simulation value,and the relative error decreases to less than 10^(-5).The feasibility and accuracy of the zero-order approximation value and iterative formula are verified.