Efficient High Accuracy Electromagnetic Field Calculations - Modeling fields in Waveguides and Photonic Crystals

The electromagnetic field equations for waveguides were solved using analytic variational methods by Schwingernearly 70 years ago. Since that time, interest has evolved in solving field equations computationally in ever more complex systems, such as dielectrically loaded waveguides, optical fibers, photonic crystals, and meta-aterials with negative refractive index. The use of vector finite elements for representing electromagnetic fields is standard procedure for modern simulations and is a staple of academic instruction. Properly constructed edge elements offer specific mathematical advantages such as tangential field component continuity, the suppression of spurious solutions, and the satisfaction of the Nedelec conditions to prevent pathological representations. 

I propose new alternative polynomial basis functions, derived using group theory, for use in electromagnetic field calculations that can be used with triangular elements on a simple nodal mesh. We employ scalar, fifth-order Hermite interpolation polynomials to solve Maxwell's equations in two dimensions in the finite element method. Each field component is represented by scalar Hermite elements while maintaining field and derivative continuity for treating boundary conditions and avoiding spurious modes. We have analyzed (i) homogeneous conducting waveguides in order to benchmark the results, (ii) inhomogeneously loaded waveguides to ensure that all modes are captured in the 
analysis, and (iii) photonic crystals to show the advantages of the new finite elements. 

The Hermite interpolation functions provide greater accuracy than vector finite elements of equal polynomial order, while bypassing the issue of spurious modes observed when using Lagrange polynomials. Comparisons are made to edge vector element calculations with comparable number of degrees of freedom. In addition to highly accurate eigenvalue computation, the calculated spatial fields representing the eigenmodes do not suffer degradation from mixed polynomial order. The impact is that few elements can be used to resolve the same level of 
accuracy. 

I show examples of photonic crystal calculations displaying the efficacy of the new method in capturing details when the periodic system has considerable spatial complexity in dielectric distributions.