Initial Potential-based Time Domain Surface Integral Equations for Dielectric Regions

Abstract

The A-Φ formulation has been proposed as a new paradigm for deriving computational electromagnetics methods with no low frequency breakdown, and that are more directly applicable to coupling into quantum physics problems of interest. This formulation utilizes equations developed in terms of the magnetic vector potential (A) and electric scalar potential (Φ), which are deemed more fundamental quantities for quantum applications than the electric and magnetic fields. Further, computational electromagnetics solvers developed from the A-Φformulation have been successful at overcoming many of the inherent multiscale limitations that exist for field-based solvers. This has been shown in recent work, where time domain integral equations (TDIEs) applicable to perfect electric conductor (PEC) objects were shown to be stable and accurate over broad frequency ranges. However, many quantum electromagnetics applications of interest are pursued at optical frequencies where approximating metals as PEC is no longer acceptable. Instead, these regions are more appropriately described with a Drude-Lorentz-Sommerfeld model; and so computational electromagnetics solvers applicable to multiscale geometries that can analyze these regions over very broad frequency ranges are needed. This current work begins to address this need by presenting for the first time in either frequency or time domain a set of A-Φ formulation surface integral equations applicable to simple, loss-free dielectric regions. To do this, we derive TDIEs based on recently developed integral representation of solutions to the wave equations for A and Φ. A rigorous functional framework that was used to analyze the stability of the PEC A-Φ formulation TDIEs is leveraged to determine appropriate combinations of integral equations and unknowns so that stable marching-on-in-time discretization approaches for the dielectric TDIEs can be developed here. The result is a set of A-Φ formulation TDIEs with appropriate discretization guidelines that can accurately and stably analyze dielectric regions at middle frequencies. Although not applicable at very low frequencies, these equations provide an essential step toward this goal by determining a baseline for combinations of equations and unknowns for dielectric regions within this formulation.