Orthogonal basis with a conicoid first mode for shape specification of optical surfaces

TL;DR

This paper introduces a rigorous theoretical framework for creating orthogonal basis functions to accurately represent optical surfaces, especially using a conicoid first mode, applicable to various shapes and initial conditions.

Contribution

It develops a general method for deriving orthogonal shape modes for optical surfaces, including aspheres and freeform shapes, using transformations of Legendre polynomials and spherical harmonics.

Findings

The proposed basis system outperforms standard polynomials in representing optical surfaces.

Numerical comparisons show improved accuracy with the new basis functions.

The method is versatile for different initial shapes and polynomial types.

Abstract

A rigorous and powerful theoretical framework is proposed to obtain systems of orthogonal functions (or shape modes) to represent optical surfaces. The method is general so it can be applied to different initial shapes and different polynomials. Here we present results for surfaces with circular apertures when the first basis function (mode) is a conicoid. The system for aspheres with rotational symmetry is obtained applying an appropriate change of variables to Legendre polynomials, whereas the system for general freeform case is obtained applying a similar procedure to spherical harmonics. Numerical comparisons with standard systems, such as Forbes and Zernike polynomials, are performed and discussed.