TCC Kernel Extraction via Iterative Rescaling for Lithography
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Solution Overview
Problem
Current methods for computing transmission cross-coefficient (TCC) kernels in lithographic simulations are computationally expensive, leading to slow simulations and increased time in the mask making process due to their high complexity.
Innovation Solution
The method involves forming a sampling matrix from a random, pseudorandom, or structured matrix, iteratively multiplying the TCC matrix against it, and adaptively rescaling until convergence to form a low-rank basis, which is used to compute a reduced TCC matrix through eigenvalue or singular value decomposition, allowing for the extraction and expansion of TCC kernels for model calibration and mask optimization.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If standard methods are used to compute TCC kernels from the TCC matrix, then accurate lithographic simulation results are obtained, but computational complexity increases and simulation time is extended
Solution Approach 1:
The patent segments the computation process into two stages: first computing the TCC matrix using Hopkins theory, then extracting TCC kernels through eigenvalue decomposition. This segmentation allows for optimized computation of each stage, reducing overall computational complexity from O(N^3) to O(N^2) or O(N log N) while maintaining accurate simulation results for lithographic performance prediction
Solution Approach 2:
The patent computes only the necessary TCC kernels required for lithographic simulation rather than performing complete TCC matrix operations. By identifying and computing only the dominant eigenvalues and eigenvectors needed for accurate simulation, the method reduces computational burden while maintaining sufficient precision for mask making and optical proximity correction
2Reliability
If complete TCC matrix computation is performed, then comprehensive optical properties are captured, but computational resources and processing time are excessively consumed
Solution Approach 1:
The patent extracts only the essential TCC kernels from the complete TCC matrix through eigenvalue decomposition. By taking out and computing only the dominant kernels that capture the most significant optical properties, the method maintains reliable optical model accuracy for lithographic simulations while dramatically improving mask making efficiency by avoiding computation of redundant matrix elements
Solution Approach 2:
The patent changes the computational parameters from operating on the complete TCC matrix to operating on extracted TCC kernels represented by their eigenvalues and eigenvectors. This parameter transformation reduces the dimensional complexity from N×N matrix operations to smaller kernel computations, enhancing productivity in mask making processes while preserving the reliability needed for accurate optical modeling
Data Source
AI summary
A method and an apparatus for computing feature kernels for optical model simulation are provided. In the method, a feature matrix mathematically describing a plurality of properties of an optical imaging system is identified. A sampling matrix comprising at least one vector serving as input to form a low-rank basis for the feature matrix is generated. The sampling matrix is iteratively multiplied by the feature matrix and a multiplication result is adaptively rescaled according to numerical stability until a convergence condition is met. The iteration results are used to form a reduced feature matrix. Decomposition values of the reduced feature matrix are computed and a plurality of feature kernels are extracted from the computed decomposition values.