Efficient high-order accurate Fresnel diffraction via areal quadrature and the nonuniform FFT

10/12/2020
by   Alex H. Barnett, et al.
0

We present a fast algorithm for computing the diffracted field from arbitrary binary (hard-edged) planar apertures and occulters in the scalar Fresnel approximation, for up to moderately high Fresnel numbers (≲ 10^3). It uses a high-order areal quadrature over the aperture, then exploits a single 2D nonuniform fast Fourier transform (NUFFT) to evaluate rapidly at target points (of order 10^7 such points per second, independent of aperture complexity). It thus combines the high accuracy of edge integral methods with the high speed of Fourier methods. Its cost is 𝒪(n^2 log n), where n is the linear resolution required in source and target planes, to be compared with 𝒪(n^3) for edge integral methods. In tests with several aperture shapes, this translates to between 2 and 5 orders of magnitude acceleration. In starshade modeling for exoplanet astronomy, we find that it is roughly 10^4 × faster than the state of the art in accurately computing the set of telescope pupil wavefronts. We provide a documented, tested MATLAB/Octave implementation. An appendix shows the mathematical equivalence of the boundary diffraction wave, angular integration, and line integral formulae, then analyzes a new non-singular reformulation that eliminates their common difficulties near the geometric shadow edge. This supplies a robust edge integral reference against which to validate the main proposal.

READ FULL TEXT

page 13

page 14

11/25/2021

Computing weakly singular and near-singular integrals in high-order boundary elements

We present algorithms for computing weakly singular and near-singular in...
04/16/2021

An implementation of an efficient direct Fourier transform of polygonal areas and volumes

Calculations of the Fourier transform of a constant quantity over an are...
03/10/2020

The Smooth Forcing Extension Method: A High-Order Technique for Solving Elliptic Equations on Complex Domains

High-order numerical methods for solving elliptic equations over arbitra...
07/08/2021

Fast accurate approximation of convolutions with weakly singular kernel and its applications

In this article, we present an O(N log N) rapidly convergent algorithm f...
08/22/2022

Rapid evaluation of Newtonian potentials on planar domains

The accurate and efficient evaluation of Newtonian potentials over gener...