Zernike like functions on spherical cap: principle and applications in optical surface fitting and graphics rendering

Area of Science:

• Optics and Photonics
• Computational Science

Background:

• Zernike circular polynomials (ZCP) are standard for optical surface analysis but struggle with highly curved spherical caps.
• Existing methods face limitations in aperture angle and surface curvature, hindering accurate characterization.

Purpose of the Study:

• To develop novel Zernike-like functions applicable to all spherical cap types, including highly curved surfaces.
• To provide a systematic method for deriving and calculating these new functions.

Main Methods:

• Utilized the Gram-Schmidt algorithm to derive analytical expressions for three new function sets.
• Developed hemispherical harmonics (HSH), Zernike spherical functions (ZSF), and longitudinal spherical functions (LSF).

Main Results:

• HSH, ZSF, and LSF functions were derived, offering complete and orthogonal performance.
• ZSF and LSF demonstrate aperture-invariant orthogonality, suitable for arbitrary spherical caps.
• The new functions address limitations of ZCP for wide aperture angles and high curvature.

Conclusions:

• The derived HSH, ZSF, and LSF functions provide robust tools for spherical cap surface characterization.
• These functions are valuable for diverse applications including optical design, aberration analysis, and scientific modeling.
• The new functions also enhance efficiency in graphics rendering for virtual reality and gaming.