Adjoint-State Operator for Periodic Structure Scattering
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Solution Overview
Problem
Current methods for calculating electromagnetic scattering properties and structural parameters of periodic structures in lithographic processes are computationally intensive, requiring numerous linear system solutions and derivative computations, especially when using finite-difference approximations.
Innovation Solution
The method employs an adjoint-state operator and variable to numerically solve a linear system, reducing the number of computations by determining scattering coefficients and derivatives through optimized expressions, allowing for faster calculation of electromagnetic scattering properties and structural parameters.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If conventional numerical solvers are used to compute electromagnetic scattering properties for multiple incident-wave directions, then accurate scattering coefficients can be obtained, but the computational time and complexity increase significantly
Solution Approach 1:
The method pre-computes the adjoint-state variable vn that captures the essential scattering characteristics of the periodic structure. This preliminary computation enables rapid determination of scattering coefficients for multiple incident-wave directions without re-solving the full electromagnetic problem, thus resolving the contradiction between accuracy and computation time
Solution Approach 2:
The adjoint-state variable vn serves as an intermediary that bridges the complex electromagnetic scattering problem and the simplified coefficient determination. By introducing this intermediate representation, the method avoids direct computation for each incident direction while maintaining accuracy
2Manufacturing precision
If finite-difference approximations are used for derivative computations, then structural parameters can be determined, but the number of computations and computational burden increase
Solution Approach 1:
The method changes the computational approach by using adjoint-state variables and optimized expressions that reduce the number of required computations. Instead of computing derivatives through multiple finite-difference evaluations, the invention uses a transformed mathematical representation that requires fewer computational operations while maintaining precision
3Reliability
If multiple linear systems are solved for each angle of incidence, then complete scattering properties can be obtained, but productivity and processing speed decrease
Solution Approach 1:
The method merges the computation of scattering coefficients for multiple incident-wave directions into a single adjoint-state variable computation. By combining these calculations, the invention maintains complete scattering property information while significantly improving processing speed and productivity
Data Source
AI summary
Numerical calculation of electromagnetic scattering properties and structural parameters of periodic structures is disclosed. A reflection coefficient has a representation as a bilinear or sesquilinear form. Computations of reflection coefficients and their derivatives for a single outgoing direction can benefit from an adjoint-state variable. Because the linear operator is identical for all angles of incidence that contribute to the same outgoing wave direction, there exists a single adjoint-state variable that generates all reflection coefficients from all incident waves that contribute to the outgoing wave. This adjoint-state variable can be obtained by numerically solving a single linear system, whereas one otherwise would need to solve a number of linear systems equal to the number of angles of incidence.