BTTB Preconditioner for Lithographic Structure Reconstruction
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Solution Overview
Problem
Current methods for calculating electromagnetic scattering properties in lithographic processes are computationally burdensome, especially for real-time reconstruction of complex structures, due to the nonlinear nature of the inverse scattering problem and the high computational demand of existing numerical techniques.
Innovation Solution
A method involving the design of a preconditioner for a multi-level matrix system with a Toeplitz structure is introduced, which includes a Block-Toeplitz-Toeplitz-Block (BTTB) structure, to efficiently solve the linear system and reconstruct parameters of lithographic structures, using an inverse generating function approach and Kronecker product approximation.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If known numerical techniques are used to model electromagnetic scattering, then measurement precision is improved, but computing time increases excessively
Solution Approach 1:
The patent segments the scattering calculation problem into distinct computational components that can be processed separately and efficiently combined. The scattering problem is divided into fundamental solutions, basis function expansions, and matrix operations that can be optimized independently, reducing overall computational burden while maintaining accuracy.
Solution Approach 2:
The patent transforms the scattering calculation from direct numerical simulation to a parameter-based approach using basis function expansions. By changing the mathematical representation parameters and using analytical solutions where possible, the computation achieves both speed and precision without requiring excessive computational resources.
2Productivity
If real-time reconstruction is desired for complex structures, then productivity is improved, but device complexity increases
Solution Approach 1:
The patent performs preliminary computational work by pre-calculating and storing fundamental solutions and basis function expansions. This preliminary action creates a library of computational building blocks that can be rapidly assembled during real-time reconstruction, achieving high productivity without requiring the full computational complexity to be resolved during the actual measurement and reconstruction process.
Data Source
AI summary
Disclosed is a method for reconstructing a parameter of a lithographic process. The method comprises the step of designing a preconditioner suitable for an input system comprising the difference of a first matrix and a second matrix, the first matrix being arranged to have a multi-level structure of at least three levels whereby at least two of said levels comprise a Toeplitz structure. One such preconditioner is a block-diagonal matrix comprising a BTTB structure generated from a matrix-valued inverse generating function. A second such preconditioner is determined from an approximate decomposition of said first matrix into one or more Kronecker products.