Cavity electrodynamics with Hermite interpolation: Role of symmetry and degeneracies

Abstract

We show that the present approaches for the solution of Maxwell’s equations in complex geometries have limitations that can be overcome through the use of C(1)-continuous Hermite interpolation polynomials. Our approach of calculating fields using the Hermite finite element method yields better accuracy by several orders of magnitude than comparable applications of the edge-based vector finite element method. We note that the vector finite element that is widely used yields pixelated solutions and ill-defined vector solutions at nodes. Our solutions have a smooth representation within and across the elements and well defined directions for the fields at the nodes. We reexamine the issue of removing spurious zero-frequency solutions. 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 Oh symmetry of the cube. This behavior is identified as an example of “accidental degeneracy” and is explained in detail. We show that the inclusion of a smaller dielectric cube of relative permittivity ϵ2 within the cubic cavity leads to the removal of this accidental degeneracy so that the eigenfields have the symmetry Oh. A further reduction of symmetry is obtained by allowing the dielectric function in the enclosed cube to have a linear dependence on z. 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.