Investigation of EUV Grating Waveguide Effective Refraction Index for TE0 Mode
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(TE0)mode is a hyperbolic cosine function cosh(·).We provide the zero-order approximation value neff,a of the effective refractive index neff for the TE0 mode.To further decrease the relative error of neff,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 neff,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 TE0 mode and provided the α value 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 neff,0 and the feasibility of cosh(·),we used the COMSOL Multiphysics software to simulate the TE0mode 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(neff,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 neff,0 for Au and Ag are 0.75％and 0.96％,respectively(Table 1).The accuracy ofneff,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,neff,23 can describe the TE0 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 TE0 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 neff,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 neff,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 neff,7 decreases to 0.11％.After 23 iterations,neff,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.
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