Babich-Like Ansatz for Three-Dimensional Point-Source Maxwell's Equations in an Inhomogeneous Medium at High Frequencies

We propose a novel Babich-like ansatz consisting of an infinite series of dyadic coefficients (three-by-three matrices) and spherical Hankel functions for solving point-source Maxwell’s equations in an inhomogeneous medium so as to produce the so-called dyadic Green’s function. Using properties of spherical Hankel functions, we derive governing equations for the unknown asymptotics of the ansatz including the traveltime function and dyadic coefficients. By proposing matching conditions at the point source, we rigorously derive asymptotic behaviors of these geometrical-optics ingredients near the source so that their initial data at the source point are well-defined. To verify the feasibility of the proposed ansatz, we truncate the ansatz to keep only the first two terms, and we further develop partial differential equation–based Eulerian approaches to compute the resulting asymptotic solutions. Since the system of governing equations for each dyadic coefficient is strongly coupled, we introduce auxiliary variables to transform these strongly coupled systems into decoupled scalar equations. Furthermore, we develop high-order Lax–Friedrichs weighted essentially nonoscillatory schemes for computing these auxiliary variables so that the Green’s function can be constructed. Numerical examples demonstrate that our new ansatz yields a uniform asymptotic solution in the region of space containing a point source but no other caustics.