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    We generalized orthonormal elliptical polynomials for optical wavefront analysis, enabling precise characterization of aberrations in systems with vignetted pupils. This advancement aids in understanding complex optical performance.

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    Area of Science:

    • Optics and Photonics
    • Mathematical Physics

    Background:

    • Orthonormal polynomials are crucial for describing optical wavefronts.
    • Existing formulations often lack generality for arbitrary aspect ratios and orientations.

    Purpose of the Study:

    • To generalize the analytical form of orthonormal elliptical polynomials.
    • To extend their applicability to arbitrary aspect ratios and orientations.
    • To provide expressions up to the 4th order.

    Main Methods:

    • Generalization of the analytical form of orthonormal elliptical polynomials.
    • Derivation of expressions for arbitrary aspect ratios and orientations.
    • Application of the generalized polynomials to wavefront expansion.

    Main Results:

    • The generalized analytical form of orthonormal elliptical polynomials is derived.
    • Expressions are provided up to the 4th order.
    • Wavefront expansion up to the 8th order is demonstrated for two specific cases.

    Conclusions:

    • The generalized polynomials offer a powerful tool for optical wavefront analysis.
    • The method is effective for systems with vignetted pupils and off-axis wavefronts.
    • This work enhances the capability to analyze complex optical aberrations.