Zhao Wanli, Wei Chenlu, Yuan Caojin, Chang Chenliang, Ma Jun, Zhu Rihong
Opt Lett. 2020 Nov 1;45(21):5937-5940. doi: 10.1364/OL.405257.
Fast Fourier transform (FFT) is the most commonly used mathematical method in numerical calculation, and the FFT-based angular spectrum method (ASM) is also used widely in diffraction calculation. However, the frequency and spatial sampling rules in FFT limit the effective propagation distance and the observation window range of ASM. A novel method for calculating the angular spectrum based on the matrix product is proposed in this Letter. This method realizes the fast calculation of discrete Fourier transform (DFT) based on the matrix product, in which the sampling matrix is orthogonally decomposed into two vectors. Instead of FFT, angular spectrum diffraction calculation is carried out based on the matrix product, which is named the matrix product ASM. The method in this Letter uses a simple mathematical transformation to achieve maximum compression of the sampling interval in the frequency domain, which significantly increases the effective propagation distance of the angular spectrum. Additionally, the size of the observation window can be enlarged to obtain a wider calculation range by changing the spatial sampling of the output plane.
快速傅里叶变换（FFT）是数值计算中最常用的数学方法，基于FFT的角谱方法（ASM）在衍射计算中也被广泛应用。然而，FFT中的频率和空间采样规则限制了ASM的有效传播距离和观测窗口范围。本文提出了一种基于矩阵乘积计算角谱的新方法。该方法基于矩阵乘积实现了离散傅里叶变换（DFT）的快速计算，其中采样矩阵被正交分解为两个向量。基于矩阵乘积而非FFT进行角谱衍射计算，该方法被称为矩阵乘积ASM。本文方法通过简单的数学变换实现了频域采样间隔的最大压缩，显著增加了角谱的有效传播距离。此外，通过改变输出平面的空间采样，可以扩大观测窗口的大小以获得更宽的计算范围。
