Computation of 2D Fourier transforms and diffraction integrals using Gaussian radial basis functions

TL;DR

This paper introduces a fast and accurate method for computing 2D Fourier and diffraction integrals using Gaussian radial basis functions, improving efficiency in optical system analysis.

Contribution

The authors develop a Gaussian RBF-based series expansion method for rapid and precise calculation of diffraction integrals, outperforming existing approaches.

Findings

Achieves rapid convergence and high accuracy in integral computation.

Demonstrates improved performance over Nijboer-Zernike theory.

Enables efficient evaluation for multiple defocus parameters.

Abstract

We implement an efficient method of computation of two dimensional Fourier-type integrals based on approximation of the integrand by Gaussian radial basis functions, which constitute a standard tool in approximation theory. As a result, we obtain a rapidly converging series expansion for the integrals, allowing for their accurate calculation. We apply this idea to the evaluation of diffraction integrals, used for the computation of the through-focus characteristics of an optical system. We implement this method and compare it performance in terms of complexity, accuracy and execution time with several alternative approaches, especially with the extended Nijboer-Zernike theory, which is also outlined in the text for the reader's convenience. The proposed method yields a reliable and fast scheme for simultaneous evaluation of such kind of integrals for several values of the defocus parameter, as required in the characterization of the through-focus optics.