Electrodynamic calculations with Hermite interpolation: Role of symmetry and degeneracies of fields in a cavity

We show that the present approaches for the solution of Maxwell’s equations in complex geometries have limitations that can be overcome using Hermite interpolation polynomials. Our approach of calculating the field yields better accuracy by several orders of magnitude than comparable applications of the edge-element based commercial software. We note that the vector finite element that is widely used yield pixellated solutions, and ill-defined vector solutions at nodes. Our solutions have smooth representation within and across the elements, and well defined directions for the fields at the nodes. We investigate fields in an empty cubic metallic cavity and explain the level degeneracy that is larger than what is to be expected from the geometrical octahedral symmetry. This behavior is identified as an example of “accidental degeneracy,” and is explained by displaying additional operators that form a larger covering group. We show that the inclusion of a smaller dielectric cube within the cubic cavity leads to the removal of this accidental degeneracy. The proposed method should be effective in obtaining results for scalar-vector coupled field problems such as in modeling quantum well cavity lasers and in plasmonics modeling, while allowing multi-scale physical calculations.