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Aspherical surface construction methods based on extended Jacobi polynomials.

Wangxuan Sun, Yunhan Huang, Zhen Zhang

Journal of the Optical Society of America. A, Optics, Image Science, and Vision

|June 10, 2026

Summary

This summary is machine-generated.

We developed a new aspherical surface representation using extended Jacobi polynomials. This method improves optical system accuracy and stability, especially for large-aperture systems.

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Area of Science:

Optical Engineering
Applied Mathematics
Computational Optics

Background:

Conventional polynomial representations struggle with high-order oscillations and edge-fitting in large-aperture annular optical systems.
Existing methods lack independent control over aperture range and fitting weights, impacting numerical stability.
Accurate surface representation is crucial for advanced optical system design.

Purpose of the Study:

To introduce an aspherical surface representation using extended Jacobi polynomials.
To overcome limitations of conventional methods in large-aperture annular systems.
To enhance numerical stability and edge-fitting accuracy.

Main Methods:

Utilized extended Jacobi polynomials with an extended orthogonal interval.
Introduced adjustable weighting parameters for independent control of aperture range and fitting weights.
Applied the method to a refractive-reflective panoramic imaging system.

Main Results:

Successfully mitigated high-order oscillations and improved edge-fitting capability for both large and small apertures.
Achieved enhanced numerical stability through independent control of fitting parameters.
Demonstrated effective suppression of oscillations and superior edge-fitting accuracy.

Conclusions:

The proposed extended Jacobi polynomial method offers a robust solution for aspherical surface representation in complex optical systems.
This approach significantly enhances performance metrics like modulation transfer function (MTF) in panoramic imaging.
The method provides a valuable tool for designing high-performance large-aperture optical systems.