Derivation and application of a Green function propagator suitable for nonparaxial propagation over a two-dimensional domain.

The goal of optical simulation is to determine the performance characteristics of an optical system from a knowledge of its physical construction and how it affects light sent through it. To produce meaningful results efficiently, two simulation approaches are available for passing light through a system, geometrical raytracing and wave optics. Within the wave optics realm, there are many techniques for determining the optical fields within a system, both numerical and analytical. A few of the numerical techniques are finite-difference, finite-element, and FFT-based; analytical techniques include modal expansions, coupled wave theory, series expansions, and Green function propagators. A propagator is a function that gives the light fields at any specified location if they are known at a source location; this is possible because the light fields, electric and magnetic, satisfy a differential equation, in the case of time harmonic fields, the Helmholtz equation. The propagator is a transfer function for the fields and often takes the form of an integral, in which case, the integrand is a product of the transfer function with the source field distribution, and the integration is performed over the source field coordinates. The integrand transfer function, also known as a Green function or propagation kernel, is a solution of the Helmholtz equation. An approximation is often used in finding a solution to the Helmholtz equation, called the paraxial approximation, in which the second derivative in the propagation direction is dropped. If no approximation is made, and all second derivatives are kept, the solution is nonparaxial. In the present paper, a Green function for the propagator of the Helmholtz equation over two-dimensional domains is derived, differing in functional form from previous work on two-dimensional propagation. An angular spectrum integral is evaluated and the resulting Green function, the propagator kernel, is a nonparaxial analytic solution of the Helmholtz equation. The propagator could be applied directly to the electric and magnetic field components; instead, it is applied to the Hertz vector components. The Hertz vector is a potential function, similar to the vector potential, defined such that the electric and magnetic fields are found by taking derivatives of it. An advantage of the Hertz vector is that only it needs be propagated, versus two, electric and magnetic, vectors. In this paper, the derived propagator is applied to Hertz vector components defined by Legendre polynomial expansions, and derivatives are taken of the propagated Hertz vector components to calculate the associated electric and magnetic fields. The Green function propagator and all field quantities produced by its application are closed form analytic expressions.