Mask Pattern Generation Using Shifted Pupil Functions
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Solution Overview
Problem
Existing methods for calculating mask patterns in photolithography require long calculation times due to the complexity of convolution integrals and double integrals involved in transmission cross coefficient calculations, which is inefficient for modern semiconductor manufacturing needs.
Innovation Solution
A method that shifts pupil functions based on point light sources on the pupil plane, generates shifted pupil functions, and calculates an optical image using a matrix product of transposed and complex-conjugated vectors, eliminating the need for convolution integrals and significantly reducing calculation time.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If transmission cross coefficient calculation methods are used to generate mask patterns, then manufacturing precision is improved, but calculation time increases significantly
Solution Approach 1:
The patent segments the continuous pupil function into discrete sampling points on the pupil plane. By representing the illumination spectrum as a set of discrete point light sources rather than a continuous distribution, the calculation is divided into manageable discrete elements that can be processed efficiently through matrix operations rather than continuous integrals.
Solution Approach 2:
The patent replaces the traditional mechanical/mathematical approach of performing convolution integrals and double integrals with an optical-inspired matrix multiplication approach. The vector-matrix multiplication inherently performs the convolution operation more efficiently, substituting a computationally intensive mathematical mechanism with a more efficient linear algebra operation.
2Measurement precision
If conventional optical image calculation methods are used, then accuracy is maintained, but computational complexity increases
Solution Approach 1:
The patent changes the parameter representation from continuous functions requiring convolution integrals to discrete vectors and matrices. By sampling the pupil function at discrete points and representing it as a vector, the calculation complexity is reduced while maintaining the essential optical imaging characteristics through the matrix multiplication operation.
Solution Approach 2:
The patent creates a simplified mathematical model that copies the essential behavior of the complex optical system. The vector-matrix multiplication approach replicates the convolution operation's effect on the aerial image without requiring the full computational machinery of traditional optical calculation methods.
Data Source
AI summary
In a method for generating, with a computer, a pattern of a mask, a pattern on an object plane of a projection optical system is set, shifted plural pupil functions are generated, a matrix containing the generated plural pupil functions is defined, an image of the pattern on the object plane is calculated by generating a vector obtained by transposing and complex-conjugating a vector containing, as components, values of the pupil functions at origin coordinates on a pupil plane from among components of the matrix, and performing convolution integral between the pattern on the object plane and a Fourier transform of a product of the vector and the matrix, an assist pattern for the pattern on the object plane is generated using the calculated image, and a pattern of the mask including the pattern on the object plane and the assist pattern is generated.