When Parabolized Propagation Fails: A Matrix Square Root Propagator for EM Waves

摘要

A Helmholtz equation containing cross-polarization terms is factored to produce the operator form of a one-way, full vector, wave equation. A direct discretization of the solution to this operator equation produces a matrix evolution equation that preserves the full (forward arc) propagation spectrum of the original Helmholtz equation. Thus it naturally includes both the evanescent and high-angle waves that are always absent in the standard parabolization of the Helmholtz equation. The first implementation of this propagator in the scalar case produces an accurate algorithm, but memory and execution times required by calculating large matrix square roots and exponentials become unwieldly as grid sizes grow. A Newton iteration for the matrix square root successfully decreases execution time, but only marginally reduces memory usage. A second implementation, dependent upon calculating the Frechet derivative of the matrix square root in the first algorithm, separates the effects of refractive index variations from freespace propagation, but ultimately suffers from similar memory issues associated with calculating large matrix exponentials. Two-dimensional computational examples comparing differences and similarities with a standard parabolized propagator include wavepackets with high modulation wavenumber, and gaussian beams passing through media boundaries and random refractive index fields. Three-dimensional examples display vector modal development in a circular waveguide, explicitly displaying cross-polarization effects.