Collins formula in frequency-domain and fractional Fourier transforms

Collins formula in frequency-domain has been derived. This formula gives the direct relationship between input and output spatial frequency spectra of a light field. Furthermore, by applying this formula the fractional Fourier transform (FRT) in frequency-domain can be investigated. It has been found that an arbitrary reciprocally symmetric ABCD optical system (RSOS) can implement the FRT in both space-domain and frequency-domain, with the same order. In other words, the FRT is a symmetric transform that performs the same operation in both space-domain and frequency-domain.