Numerical implementation of the Fresnel transform, and its application in linear optical systems

Abstract

The numerical calculation of the Fresnel Transform (FST) presents significant challenges due to the high sampling rate associated with the chirp function in the kernel. The development of a general algorithm is further complicated due the fact that the output extent of the FST is dependant on the propagation distance. In this paper we implement a novel technique for calculating the FST in which we use the Wigner Distribution Function (WDF) to choose an optimal sampling rate. This method is suitable for all propagation distances. Our method can also be used to describe the effect of a lens in an optical system. A lens in an optical system is modeled as a Chirp Modulation Transform (CMT), which acts on the spatial frequency distribution of a signal in a manner that is analogous to the effect an FST has on the spatial distribution of a signal. Combining our results for the FST and the CMT we numerically calculate the light distribution at the output of both a Cai-Wang and a Lohmann Type-I Optical Fractional Fourier Transform (OFRT) system. Analytic solutions for the OFRT of a square (1-D) and circular (1-D) aperture are presented. The numerical results are compared to the analytically derived solution.