Waveguides, Accidental Degeneracy in Cavities, and photon dispersion in Escher Photonic Crystals
I 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. The new approach (Hermite-FEM) of calculating fields, in the finite element method, yields better accuracy by several orders of magnitude than comparable applications of the edge-based vector finite element method (VFEM). Note that the VFEM that are widely used yield pixellated solutions, and ill-defined vector directions at nodes. The solutions in HFEM have smooth representation everywhere, and well-defined directions for the fields at the nodes. The methodology is tested with empty and dielectrically loaded waveguides, with photonic crystals, and with 3D EM cavities. “The Horsemen,” a famous drawing by M. C. Escher is treated as a 2D photonic crystal and the EM fields are mapped out for such a complex periodic structure that has the symmetry of the non-symmorphic group “pg”. This demonstrates the power and flexibility of the method.
I reexamine the issue of removing spurious zero-frequency solutions that occur with all discretization schemes. We investigate fields in an empty cubic metallic cavity and explain the level degeneracy that is far 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 using group theory. 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 geometrical symmetry of the cube, Oh.
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.
Undergraduate & graduate students are especially invited to attend – details of the theory will be made manageable.
Refreshments will be served in Olin Hall 118 at 3:30 P.M