利用傅里叶变换的可分割性计算夫琅禾费衍射图样

CALCULATION OF FRAUNHOFER DIFFRACTION BASED ON THE SEPARABILITY OF FOURIER TRANSFORM

本文揭示了由傅里叶变换模拟夫琅禾费衍射图像的过程具有可分割性与可加性,将衍射现象的物理过程与其傅里叶变换的数学表现相统一。通过对比基尔霍夫公式与二维傅里叶变换,验证了夫琅禾费衍射图像与衍射孔的二维傅里叶变换等价。类比夫琅禾费衍射是无数个子波的相干叠加,具有可分割性与可加性,得到由二维傅里叶变换模拟衍射图像这一过程也具有可分割性与可加性。这表明夫琅禾费衍射图像等价于其衍射孔任意分割后各部分傅里叶变换的叠加,且分割后各部分的衍射图像与其傅里叶变换一一对应。以正n边形为例,我们使用本方法推导出的衍射图像的Matlab仿真结果与现有文献中其他方法给出的结果一致。最后,我们给出了衍射现象与傅里叶变换等价的物理解释。

In this paper,we prove that the process of simulating Fraunhofer diffraction image by Fourier transform is separable and additive.The physical process of diffraction phenomenon is unified with the mathematical expression of Fourier transform.Firstly,it is verified that Fraunhofer diffraction image is equivalent to two-dimensional Fourier transform of pinhole diffraction by comparing Kirchhoff formula with two-dimensional Fourier transform.Secondly,compared with Fraunhofer diffraction,which is the coherent superposition of numerous wavelet and has the separability and additivity,it is concluded that the process of simulating diffraction image by two-dimensional Fourier transform is also separable and additive.It shows that Fraunhofer diffraction image is equivalent to the superposition of Fourier transform of each part of diffraction hole with arbitrary division.To validate the theory,taking the regular n-polygon as an example,the Matlab simulation results of diffraction image derived by this method are consistent with existing results given by other methods in the existing literatures.Finally,we give a physical explanation about equivalence of diffraction phenomenon and Fourier transform.