A Vector Helmholtz Electromagnetic Wave Propagator for Inhomogeneous Media

The vector electric-field Helmholtz equation, containing cross-polarization terms, is factored to produce both pseudo-differential and exponential operator forms of a three-dimensional, one-way, vector, wave equation for propagation through inhomogeneous media. From this operator factorization we develop a high-order approximate, vector Helmholtz propagator that correctly handles forward-arc, high-angle scattering and diffraction from inhomogeneities at all resolved length scales, and seamlessly includes evanescent waves. Our implementation of the exponential operator form of the one-way propagator is discussed extensively. A rational approximation/partial fraction decomposition of the exponential operator converts the propagator into a moderate number of large, sparse, linear solves whose results are summed together at each step to advance the electric field in space. We use a new AAA-Lawson rational interpolant for this approximation, rather than the more common Padé expansions that have appeared in the seismic and ocean acoustics literature previously. GMRES is used to solve these large systems. A direct-solve, free space propagation method proves to be an effective preconditioner for GMRES, but can also serve as a standalone propagator in homogeneous media. Scalar computational examples shown include plane wave diffraction by a circular aperture and Gaussian beam propagation through sine-product and homogeneous refractive index fields. The sine-product example compares its results to that of paraxial propagation through the same media, and demonstrates the substantial differences between these propagator paradigms when the scale of the inhomogeneities is of the order of the fundamental wavelength in the Helmholtz equation. We also examine the convergence of the homogeneous media beam results to fields generated by Clenshaw-Curtis evaluation of the first Rayleigh-Sommerfeld integral for the same initial conditions.