Efficient high-order accurate Fresnel diffraction via areal quadrature and the nonuniform fast Fourier transform

Abstract

We present a fast algorithm for computing the diffracted field from arbitrary binary (sharp-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 and then exploits a single 2D nonuniform fast Fourier transform to evaluate rapidly at target points (on the order of 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 Oðn 2 log nÞ, where n is the linear resolution required in the source and target planes, to be compared with Oðn 3 Þ for edge integral methods. In tests with several aperture shapes, this translates to between two and five 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 formulas and then the analysis of a new nonsingular 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. © The Authors. Published by SPIE under a Creative Commons Attribution 4.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

A fast, high-order scheme for evaluating volume potentials on complex 2D geometries via area-to-line integral conversion and domain mappings

“…The integrals resulting from Lemma 1 have at times been referred to as dilation integrals (with dilation with respect to the star point), with e.g. reference [48] relying precisely (up to a fixed geometrydependent translation) on the relation (3.14) to generate accurate quadratures of smooth functions.…”

“…We also note a variety of volumetric-via-surface-integral quadratures that, like ours, use ideas from classical proofs of the Poincaré lemma (which, in brief, is applicable to domains which are star-shaped with respect to some interior point-but not necessarily to all points-and leads to representations of volume integrals as surface integrals with an iterated integral whose associated physical integration points lay on rays from the boundary to the star-point). Recent work of [48] results in integral representations similar in some respects to our own but only smooth integrands are considered there. Other works such as [49] have focused on addressing the Newton potential problem specifically, but a target-centered coordinate system with the star-point as the origin results in quadrature points that lay outside the domain for regions that are non-convex (i.e.…”

Lightweight starshade position sensing with convolutional neural networks and simulation-based inference

“…Simulated images are generated using the latest diffraction models 18 that have been experimentally validated at high contrast. 9 The obstructions from NGRST's secondary mirror and its support structure are superimposed on each image and simulated photon and detector noise are added.…”

cuFINUFFT: a load-balanced GPU library for general-purpose nonuniform FFTs

“…They thus have a wealth of applications in engineering and scientific computing, including image reconstruction from off-grid Fourier data (e.g. MRI gridding [1], optical coherence tomography [2], cryo electron microscopy [3]- [7], radioastronomy [8], coherent diffraction X-ray imaging [9]); wave diffraction [10]; partial differential equations [11], [12]; and long-range interactions in molecular [13] and particle dynamics [14]. For reviews, see [15]- [18].…”