Orientation Zernike Polynomials for Polarization Description
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Solution Overview
Problem
Current microlithographic optical systems face challenges in completely describing the imaging properties, especially for high numerical aperture systems, due to polarization dependence, which is not adequately addressed by scalar Zernike polynomials.
Innovation Solution
The introduction of orientation Zernike polynomials allows for a complete and systematic description of polarized imaging in microlithography lenses by expanding diattenuation and retardance, enabling the assessment and optimization of optical systems through Jones pupil decomposition and singular value decomposition.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If scalar Zernike polynomials are used to describe optical aberrations, then the description is simple and intuitive, but polarization effects cannot be adequately addressed in high numerical aperture systems
Solution Approach 1:
The patent segments the complex Jones pupil description into two parts: scalar Zernike polynomials for intensity/phase information and orientation Zernike polynomials for polarization information (diattenuation and retardance). This segmentation allows each component to be described with appropriate mathematical tools, improving overall description accuracy while maintaining manageable complexity.
Solution Approach 2:
The patent extends the traditional scalar Zernike polynomial approach (2D spatial description) by adding an orientation dimension through orientation Zernike polynomials. This dimensional extension enables the description of polarization effects by incorporating angular dependence, allowing complete characterization of imaging properties including diattenuation and retardance orientation.
2Reliability
If orientation Zernike polynomials are introduced to describe polarization effects, then complete specification of imaging properties is achieved, but the mathematical complexity increases
Solution Approach 1:
The orientation Zernike polynomials serve multiple functions: they describe diattenuation orientation, describe retardance orientation, and can be combined with scalar Zernike polynomials to provide complete imaging property specification. This multi-functionality justifies the additional mathematical complexity by enabling comprehensive optical system characterization.
Solution Approach 2:
The patent uses orientation Zernike polynomials as an intermediary between the Jones pupil matrix description and the physical optical properties. Instead of directly analyzing the complex Jones matrix, the orientation Zernike polynomials provide an intermediate mathematical representation that simplifies the extraction and interpretation of polarization effects like diattenuation and retardance.
3Manufacturing precision
If complete polarization description is implemented, then imaging quality control is improved, but assessment and optimization become more difficult
Solution Approach 1:
The patent replaces complex physical measurements and assessments with a mathematical modeling approach using orientation Zernike polynomials. Instead of performing intricate polarization measurements and analyses, the system uses computational expansion and coefficient analysis to assess imaging properties, simplifying the detection and measurement process while improving precision.
Data Source
AI summary
The present disclosure relates to specification, optimization and matching of optical systems by use of orientation Zernike polynomials. In some embodiments, a method for assessing the suitability of an optical system of a microlithographic projection exposure apparatus is provided. The method can include determining a Jones pupil of the optical system, at least approximately describing the Jones pupil using an expansion into orientation Zernike polynomials, and assessing the suitability of the optical system on the basis of the expansion coefficient of at least one of the orientation Zernike polynomials in the expansion.