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

Abstract

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.

Calculating non-scalar diffraction efficiently via merging Braunbek method and Bluestein method

“…The derivation ultimately led to the famous Kirchhoff diffraction equation that matched remarkably well with experiments of the time. Furthermore, the equation was similar to that of a Fourier transform (FT), which shed light on subsequent research searching for efficient calculations of diffraction [7,8].…”