Discrete numerical implementation of the Fresnel transform in Cartesian and cylindrical coordinate systems

The Fresnel transform can be used to calculate the distribution of the complex field within the paraxial approximation. However not all the diffraction integrals can be solved analytically, and so numerical solutions (direct integration of the diffraction integral) are sometimes required to solve the diffraction problem. The distribution in the input plane can be expressed in either a Cartesian coordinate system (x, y) or in a cylindrical coordinate system (r, θ). Discretizing the input field in a Cartesian or in a cylindrical coordinate system produces an infinite set of replicas in the diffraction plane. However the properties of these replicas are fundamentally different from each other. In a Cartesian coordinate system replicas appear periodically along the x, y axis, however in the cylindrical coordinate system, replicas appear along concentric circles located about the optical axis. In this manuscript we examine the characteristics of these numerical integration approaches.