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 (≲103). 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 107 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 ( n2 log n ) , where n is the linear resolution required in the source and target planes, to be compared with O ( n3 ) 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 104 × 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 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.
Highlights
All codes are written in MATLAB R2017a, apart from FINUFFT, which is a parallel C++ library with MEX interface
We start by setting up a simple areal quadrature for the interior of a simple smooth closed curve
nonuniform fast Fourier transform (NUFFT) t 1 (ε 1⁄4 10−6) 1.9 × 10−8 3.7 × 10−6 10.4 to z → ∞, the Fraunhofer limit is reached in a stable fashion [here the first factor in Eq (8) should be discarded, whereas the third factor tends to unity]
Summary
The numerical modeling of wave diffraction from thin two-dimensional (2D) screens and apertures in the Fresnel regime has many applications in optics[1] and acoustics (Ref. 2, Sec. 8.4), including instrument modeling,[3,4] lithography mask design,[5] Fourier optics,[6] coherent x-rays,[7] acoustic emission,[8] computer-generated binary holograms,[9] starshades,[10] and Fresnel zone plate imagers.[11,12] This usually involves a plane or spherical wavefront hitting a binary (“0–1”) mask of given shape, continuous opacity/phase variation is possible. The occulter shape and distance are optimized to give a deep shadow region, with a relative intensity on the order of 10−10 across the telescope pupil, throughout a given wavelength range, while minimizing the occulter’s physical size (for practical reasons) and the angular size at the telescope. This has led to shapes with “petals” that emulate a continuous radial apodization, with radii on the order of 10 m and distances on the order of 107 m.
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