Analysis of effective refractive index of TE 0 mode in extreme ultraviolet grating waveguide.
Abstract
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 n eff 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.
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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 n eff 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.
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