Electromagnetic calculations for multiscale and multiphysics simulations: a new perspective

Abstract

Present day electromagnetic field calculations have basic limitations since they employ techniques based on edge-based discretization methods. While these vector finite element methods (VFEM) solve the issues of tangential continuity of fields and the removal of spurious solutions, the resulting fields do not have a unique directionality at nodes in the discretization mesh. This review presents three calculations of electromagnetic fields: (i) waveguides, (ii) cavity fields, and (iii) photonic crystals. We have developed Hermite interpolation polynomials and node-based finite element methods in the framework of variational principles. We show that the Hermite-finite element method (HFEM) better accuracy with lower computational cost, and provides field directional continuity across the discretized space. It also permits multiscale calculations with mixed physics. For example, the nanoscale modeling of quantum well semiconductor laser structures has to be done together with cavity electrodynamics at the micron scale in vertical-cavity surface emitting lasers and other optoelectronic systems. These can be treated with high accuracy with the new HFEM, which is also applicable to quantum mechanical simulations in meso-and nano-scale systems. We use group representation theory to derive the HFEM polynomial basis set in two dimensions. In three dimensions we derive these polynomials using usual methods. We show that degeneracies in the frequency spectrum in a cubic cavity can be denumerably large even though the symmetry of the cube, O h , supports only singlets, doublets, or triplets. The additional operators available for the problem explains the origin of this "accidental degeneracy." We discuss this remarkable degeneracy and its reduction in detail. We consider photonic crystals corresponding to a 2D checkerboard superlattice structure, and the Escher drawing of "The Horsemen" which satisfies the nonsymmorphic group pg. We show that HFEM is able to deliver high accuracy in such spatially complex examples with far less computational effort than Fourier expansion methods. Finite element analysis employs geometric discretization and hence transcends geometrical limitations. Techniques explained here can be immediately extended to realistic and geometrically complex structures. The new algorithms developed here hold the promise of successful modeling of multi-physics systems. This general method is applicable to a broad class of physical systems, e.g., to semiconducting lasers which require simultaneous modeling of transitions in quantum wells or dots together with EM cavity calculations, to modeling plasmonic structures in the presence of EM field emissions, and to on-chip propagation within monolithic integrated circuits.