An Examination of the Behaviour of Lorentz-Gauge Vector and Scalar Potentials at Material Discontinuities

The problem of reflection and refraction of electromagnetic plane waves, obliquely incident upon a planar interface where the complex permittivity ε and permeability μ are discontinuous, is considered. Instead of the classical treatment using electric and magnetic fields E and H, the analysis is presented in terms of scalar electric potential V and vector magnetic potential A subject to the Lorentz-gauge condition. The results are used to verify a procedure [1], namely specifying A · n/V arbitrarily at material interfaces, proposed for rendering A and V unique in an all-frequency Galerkin finite-element analysis. It is confirmed that arbitrary parameters do arise in the Lorentz-potential solutions here. With A and V continuous everywhere, the wave solutions correspond to tangential E automatically being continuous across the planar interface, whereas continuity of tangential Η needs to be set independently, say from a weak Galerkin condition. It is shown that for TE incidence the procedure forces a degeneration into the Coulomb gauge whereas for TM cases the Lorentz gauge is effectively fixed. The precise nature of the procedure [1] has been clarified and the behaviour of Lorentz potentials at electric and magnetic walls explicitly considered. The tests of [1] applied to a dielectric-loaded TE,₁₀₁ cavities, [2,3], are shown to correspond to a pure TE case with predetermined Coulomb gauge.