论文标题
有效的高阶准确精确菲涅尔衍射通过面积正交和非均匀的FFT
Efficient high-order accurate Fresnel diffraction via areal quadrature and the nonuniform FFT
论文作者
论文摘要
我们提出了一种快速算法，用于计算来自标量菲涅尔近似值中的任意二进制（锋利的）平面孔和起义者，以计算衍射场，以最大为中度高的壁画数（$ \ \ lyssim 10^3 $）。它在孔径上使用高阶面积正交，然后利用单个2D非均匀快速傅立叶变换（NUFFT）在目标点快速评估（每秒$ 10^7 $ over forder of Aperture复杂性）。因此，它结合了边缘积分方法的高精度与傅立叶方法的高速。它的成本为$ {\ MATHCAL O}（n^2 \ log n）$，其中$ n $是源和目标平面中所需的线性分辨率，与$ {\ Mathcal o}（n^3）$相比，用于边缘积分方法。在具有几种光圈形状的测试中，这转化为2至5个数量级加速度。在针对系外星天文学的星际模型中，我们发现它比准确地计算望远镜学生波前的集合时比最新的$ 10^4 \ times $ $。我们提供经过记录的，测试的MATLAB/八度实现。 附录显示了边界衍射波，角积分和线积分公式的数学等效性，然后分析了一种新的非单明一向的重新印度，从而消除了它们在几何阴影边缘附近的共同困难。这提供了强大的边缘积分参考，以验证主要提案。
We present a fast algorithm for computing the diffracted field from arbitrary binary (sharp-edged) planar apertures and occulters in the scalar Fresnel approximation, for up to moderately high Fresnel numbers ($\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.
