Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals

Abstract

We analyze discontinuous Galerkin finite element discretizations of the Maxwell equations with periodic coefficients. These equations are used to model the behavior of light in photonic crystals, which are materials containing a spatially periodic variation of the refractive index commensurate with the wavelength of light. Depending on the geometry, material properties and lattice structure these materials exhibit a photonic band gap in which light of certain frequencies is completely prohibited inside the photonic crystal. By Bloch/Floquet theory, this problem is equivalent to a modified Maxwell eigenvalue problem with periodic boundary conditions, which is discretized with a mixed discontinuous Galerkin (DG) formulation using modified Nédélec basis functions. We also investigate an alternative primal DG interior penalty formulation and compare this method with the mixed DG formulation. To guarantee the non-pollution of the numerical spectrum, we prove a discrete compactness

Analysis of a mixed discontinuous Galerkin method for the time-harmonic Maxwell equations with minimal smoothness requirements

“…REMARK 3.1 The main difference between the mixed DG formulation (3.6)-(3.7) and those discussed in (Houston et al, 2004;Lu et al, 2017) is the use of the lifting operator R F . Two main benefits of using the lifting operator are:…”

Computational nanophotonics in 3D

“…Accurate eigenvalue computations, however, require that the DG discretization satisfies the divergence constraint in the time-harmonic Maxwell equations [129,132,[134][135][136][137][138]. For the Maxwell eigenvalue problem, the neglect of the divergence condition leads to a large number of zero eigenvalues, which belong to the null space of the curl-curl operator.…”

“…Therefore, iterative eigenvalue solvers have difficulty to find the physical eigenvalues with the smallest frequencies due to the large spurious null space. An important advantage of the mixed DG method is that it provides a spectrum of electric field solutions that is (nearly) free of zero eigenvalues, and for some discretizations free of spurious (unphysical) modes [129,132,[135][136][137][138].…”

“…In the present Thesis, we employ a mixed DG discretization with Brezzi-type numerical fluxes and element-wise curl-conforming elements [128,129,132,[135][136][137][138]140]. The mixed method is based on the work by Houston et al and Lu et al [132,135]. The advantage of using Brezzi-type numerical fluxes is that they allow a straightforward estimate of the penalty parameters that appear in the DG discretization [141].…”

“…In contrast, the stability of the interior penalty method depends on a careful selection of the penalty parameters dependent on the mesh resolution h, and suitable estimates have not yet been derived for the discretization described in Ref. [135]. We have implemented the mixed DG discretization using Brezzi-type numerical fluxes for the Maxwell equations with periodic boundary conditions in a fully parallelized code and tested it on 3D computations of photonic crystals [142].…”