Lithographic Control Interface Using Periodic Base Functions
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Solution Overview
Problem
High-order polynomial-based control interfaces in lithographic apparatuses suffer from instability at the edges of the exposure field due to Runge's effect, limiting their ability to represent parameter data at high resolutions.
Innovation Solution
The use of periodic base functions with different frequencies and periods smaller than the exposure field dimensions, combined with lower-order polynomial base functions, to represent control parameter data, thereby avoiding oscillatory behavior and enhancing resolution control.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Manufacturing precision
If high-order polynomial base functions are used to represent control parameter data, then manufacturing precision is improved, but reliability deteriorates due to Runge's effect causing instability at field edges
Solution Approach 1:
The patent changes the mathematical basis from polynomial functions to periodic functions (such as Fourier series or wavelet transforms). This parameter change in the functional form eliminates Runge's effect while maintaining high-resolution control capability, as periodic functions naturally oscillate within bounded ranges rather than diverging at field edges
Solution Approach 2:
The patent employs a composite mathematical representation by combining multiple periodic base functions with different frequencies and periods. This composite approach类似于 using composite materials, where different periodic functions with specific frequency characteristics are combined to achieve both high resolution and stability across the exposure field
2Manufacturing precision
If the control interface is adapted to higher resolution control capabilities, then manufacturing precision is improved, but device complexity increases due to incorporation of increasingly higher order polynomial terms
Solution Approach 1:
The patent fundamentally changes the parameterization approach from polynomial order to periodic function frequency content. This allows achieving high resolution control through the spectral content of periodic functions rather than increasing polynomial order, thereby reducing control interface complexity while maintaining precision
Data Source
AI summary
A method for representing control parameter data for controlling a lithographic apparatus during a scanning exposure of an exposure field on a substrate, the method including: obtaining a set of periodic base functions, each base function out of the set of periodic base functions having a different frequency and a period smaller than a dimension associated with the exposure field across which the lithographic apparatus needs to be controlled; obtaining the control parameter data; and determining a representation of the control parameter data using the set of periodic base functions.