Kernel Expansions of Generalized Fresnel Transform

Abstract

The structural analysis of generalized Fresnel transform space is presented. It is considered that essential properties of generalized Fresnel transform space are included in generalized Fresnel kernel. Systems of orthonormal functions and biorthonormal functions in generalized Fresnel transform space are derived from the generalized Fresnel kernel which is the generating function for those functions. The system of orthonormal functions, which are also called mode functions, is closely related to the modes of propagation in nonorthogonal optical systems. In the special case where the system is orthogonal, these functions can be expressed as the product of two Hermite polynomialss and exponential function. Moreover, it is shown that any generalized Fresnel transform may be expanded in a series by the system of these functions.