Orthonormal Polynomials for Noncircular Pupil Wavefront Analysis
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Solution Overview
Problem
Existing optical systems with noncircular pupils, such as elliptical or hexagonal pupils, face challenges in wavefront analysis due to the use of Zernike circle polynomials, which are not orthogonal over discrete data sets, leading to errors in aberration determination and wavefront fitting.
Innovation Solution
The development of methods and systems to determine a set of orthonormal polynomials over noncircular pupils using analytical and numerical approaches, including the selection of a complete set of polynomials, calculation of a conversion matrix, and application of techniques like Gram-Schmidt orthogonalization or matrix transformation to derive orthonormal polynomials suitable for wavefront analysis.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Measurement precision
If Zernike circle polynomials are used for wavefront analysis, then orthogonality is achieved over circular pupils, but accuracy deteriorates for noncircular pupils and discrete data sets
Solution Approach 1:
The patent changes the fundamental parameter of polynomial orthogonality from circular domain to noncircular domain. By developing orthonormal polynomials specifically tailored to noncircular pupil shapes (elliptical, rectangular, hexagonal, etc.), the system maintains measurement precision while adapting to diverse pupil geometries. This involves transforming the basis functions to match the specific pupil boundary conditions.
Solution Approach 2:
The patent segments the wavefront analysis process into two distinct parts: (1) determination of orthonormal polynomials specific to the pupil shape, and (2) application of these polynomials to the discrete aberration data. This segmentation allows the system to first establish the appropriate mathematical basis for the specific pupil geometry, then apply it to the measurement data, thereby resolving the contradiction between circular polynomial orthogonality and noncircular pupil adaptability.
2Loss of information
If a complete set of polynomials is used to express aberration function, then representation completeness is achieved, but orthogonality advantages are lost and coefficient independence deteriorates
Solution Approach 1:
The patent transforms the polynomial set from non-orthogonal complete basis to orthonormal basis through Gram-Schmidt orthogonalization or similar processes. This parameter change in the mathematical structure of the basis functions maintains the completeness of aberration representation while simultaneously establishing orthogonality relationships that ensure coefficient independence. The orthonormal polynomials are constructed to satisfy both completeness and orthogonality conditions.
Solution Approach 2:
The patent performs preliminary orthogonalization of the polynomial basis set before applying it to the wavefront data. By pre-processing the complete set of polynomials to establish orthogonality relationships, the system ensures that subsequent coefficient determination from discrete data points yields independent, reliable results. This preliminary action of orthogonalization prevents the loss of coefficient independence that would otherwise occur with non-orthogonal polynomials.
3Stability of the object's composition
If Zernike circle polynomials are used over discrete data points, then circular pupil orthogonality is maintained, but measurement accuracy deteriorates due to lack of orthogonality over discrete sets
Solution Approach 1:
The patent applies local quality by creating polynomials that are specifically optimized for the local characteristics of noncircular pupils and discrete data distributions. Rather than using universal circular Zernike polynomials, the system develops orthonormal polynomials adapted to the specific pupil geometry and data sampling pattern, thereby maintaining both orthogonality stability and measurement precision for the particular application.
Solution Approach 2:
The patent introduces dynamics by making the polynomial basis adaptable to different pupil shapes and data configurations. The orthonormal polynomials are determined dynamically based on the specific pupil geometry (circular, elliptical, rectangular, hexagonal, etc.) and the discrete data distribution, rather than being fixed to circular symmetry. This dynamic adaptation ensures orthogonality is maintained for the specific configuration being analyzed.
Data Source
AI summary
Systems, methods, and software for determining a set of analytical or numerical polynomials that is orthonormal over circular or noncircular pupils are described. Closed-form orthonormal polynomials for circular, annular, hexagonal, elliptical, rectangular, and square pupils are derived. Such techniques can be applied to ray tracing as in the optical design and wavefront fitting from measurement as in the optical testing. These approaches can also be applied to wavefront reconstruction in adaptive optics.