Abstract
Special affine Fourier transformation (SAFT) can be considered as an extension of the fractional Fourier transformation (FRT) and the most general linear mapping in phase space. A general formula for SAFT in frequency-domain is derived, which gives a direct relationship between the input and output spatial frequency spectra of a light field. It shows that the SAFT has similar and symmetric feature in both space- and frequency-domains. As its special cases, Collins formula in frequency-domain, the spatial frequency representations of the almost-FRT, almost-Fresnel and almost-Fourier transformations are explicitly obtained. These formulae may provide a tool for investigating the performance of a lossless optical system including small deformations in both domains in a unified way within the framework of linear theory.
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