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.

中文翻译：

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用于光学仿真的亚纯傅里叶变换的理论和算法

快速傅立叶变换（FFT）的推出是朝着更快，更高效的物理光学建模和设计迈出的关键一步，因为它是离散傅立叶变换的更快版本。但是，由于FFT要求对包裹相位进行良好采样，因此在场分量呈现强波前相位（光学中非常典型的情况）的情况下，运算的数值工作会激增。在本文中，我们提出了一种近似算法来计算这种情况下的傅立叶变换。我们表明具有强波前相位的场的傅立叶变换表现出一种可以描述为幅度分布的双射映射的行为，这就是为什么我们将此操作称为“同胚傅立叶变换”的原因。我们在简化傅立叶积分的数学近似中精确地使用了这种特征行为。我们介绍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。