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

Zernike circular polynomials (ZCP) are widely used in optical testing, fabrication and adaptive optics. However, their ability to characterize information on spherical cap is often limited by aperture angle, with highly curved surface being particularly challenging. In this study, we propose a simple and systematic process to derive Zernike like functions that are applicable to all types of spherical cap. The analytical expressions of three function sets are calculated using Gram-Schmidt algorithm. They are hemispherical harmonics (HSH), Zernike spherical function (ZSF) and longitudinal spherical function (LSF). HSH satisfies Laplacian equation and composes a subset of spherical harmonics (SH). ZSF and LSF can be applied to arbitrary spherical cap and their orthogonality is invariant to aperture. The achieved functions, with their complete and orthogonal performance, can serve as essential tools for surface characterization required for a wide range of applications like large-angle lenses description in illumination design, aberration analysis in high aperture systems, human cornea measurement fitting, geomagnetic field modelling, etc. Moreover, they are important for graphics rendering in virtual reality and games by solving the reflectance equation efficiently.