Wang Zongzhao, Baladron-Zorita Olga, Hellmann Christian, Wyrowski Frank
Opt Express. 2020 Mar 30;28(7):10552-10571. doi: 10.1364/OE.388022.
The introduction of the fast Fourier transform (FFT) constituted a crucial step towards a faster and more efficient physio-optics modeling and design, since it is a faster version of the Discrete Fourier transform. However, the numerical effort of the operation explodes in the case of field components presenting strong wavefront phases-very typical occurrences in optics- due to the requirement of the FFT that the wrapped phase be well sampled. In this paper, we propose an approximated algorithm to compute the Fourier transform in such a situation. We show that the Fourier transform of fields with strong wavefront phases exhibits a behavior that can be described as a bijective mapping of the amplitude distribution, which is why we name this operation "homeomorphic Fourier transform." We use precisely this characteristic behavior in the mathematical approximation that simplifies the Fourier integral. We present the full theoretical derivation and several numerical applications to demonstrate its advantages in the computing process.
快速傅里叶变换（FFT）的引入是朝着更快、更高效的物理光学建模与设计迈出的关键一步，因为它是离散傅里叶变换的更快版本。然而，由于FFT要求对包裹相位进行良好采样，在光学中非常典型的场分量呈现强波前相位的情况下，该运算的数值计算量会激增。在本文中，我们提出了一种在这种情况下计算傅里叶变换的近似算法。我们表明，具有强波前相位的场的傅里叶变换呈现出一种行为，这种行为可描述为幅度分布的双射映射，这就是我们将此运算命名为“同胚傅里叶变换”的原因。我们在简化傅里叶积分的数学近似中精确地利用了这种特征行为。我们给出了完整的理论推导和几个数值应用，以证明其在计算过程中的优势。
