Electromagnetic Scattering Calculation Using Chebyshev Expansion
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Solution Overview
Problem
Current methods for calculating electromagnetic scattering properties of structures, particularly in lithographic processes, face challenges with computational burden and stability issues when dealing with complex structures, leading to long computation times and high memory requirements.
Innovation Solution
A method involving pseudo-spectral polynomial expansions and regularized linear systems is employed to solve volume integral equations for electromagnetic scattering, using Chebyshev and Legendre expansions to achieve efficient matrix-vector products and improve convergence rates, while stabilizing the numerical scheme through regularization techniques.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If known numerical procedures are used to model scattering, then scattering properties can be predicted, but computational burden becomes impractical for real-time reconstruction and complicated structures
Solution Approach 1:
The patent transforms the electromagnetic scattering problem by changing the mathematical parameters and formulation. It uses a volume integral equation approach with contrast source inversion, transforming the traditional Maxwell's equations into a form that can be solved more efficiently. The key parameter change is representing the scattered field as an integral over the volume of the object with a contrast function, which enables faster numerical solution while maintaining accuracy for complicated structures.
2Measurement precision
If known numerical procedures are used to model scattering, then scattering properties can be predicted, but memory requirements become excessive for real-time reconstruction
Solution Approach 1:
The patent segments the scattering problem into manageable components by using a volume integral formulation. The object is divided into volume elements with a contrast function χ(r), and the scattered field is computed as an integral over these segmented volume elements. This segmentation approach reduces memory requirements compared to traditional full-wave methods, as it only requires storage of the contrast function and intermediate integral results rather than complete field distributions throughout space.
3Measurement precision
If nonlinear solver is used for inverse scattering reconstruction, then structural parameters can be determined, but computational complexity increases significantly
Solution Approach 1:
The patent introduces an intermediate contrast function χ(r) as a mediator between the measured scattered field data and the unknown structural parameters. The contrast source inversion method uses this intermediate function to linearize the inverse scattering problem. By representing the scattered field as an integral involving the contrast function and using Fourier transform techniques, the complex nonlinear inverse problem is transformed into a more tractable form that can be solved efficiently with reduced computational complexity.
Data Source
AI summary
A method of calculating electromagnetic scattering properties of a structure, the structure including materials of differing properties and the structure being periodic in at least one lateral direction and extending in a vertical direction, comprises: numerically solving a volume integral equation for electromagnetic scattering for a plurality of modes in the at least one lateral direction, by performing, for each respective mode, integration (1350) of a pseudo-spectral polynomial (Chebyshev) expansion in the vertical direction multiplied by a ID Green's function using the same sample points in the orthogonal direction for all of the plurality of modes. The integration is performed by solving a regularized linear system of equations between first (1116) and second (1120) discrete transformation steps to compute (1118) values of a regularized Chebyshev expansion coefficient vector (γ). Electromagnetic scattering properties of the structure are calculated using the results of the numerical solution.