Efficient high-order accurate Fresnel diffraction via areal quadrature and the nonuniform FFT. (arXiv:2010.05978v2 [astro-ph.IM] UPDATED)
<a href="http://arxiv.org/find/astro-ph/1/au:+Barnett_A/0/1/0/all/0/1">Alex H. Barnett</a>
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 ($lesssim 10^3$). 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 ${mathcal O}(n^2 log n)$, where $n$ is the
linear resolution required in source and target planes, to be compared with
${mathcal O}(n^3)$ 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 $10^4
times$ 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.
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 ($lesssim 10^3$). 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 ${mathcal O}(n^2 log n)$, where $n$ is the
linear resolution required in source and target planes, to be compared with
${mathcal O}(n^3)$ 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 $10^4
times$ 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.
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