Bloch Modes In Photonic Crystals Research Articles

Discontinuous Galerkin Approximations for Computing Electromagnetic Bloch Modes in Photonic Crystals

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 Nedelec 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 property for the corresponding DG space. The convergence rate of the numerical eigenvalues is twice the minimum of the order of the polynomial basis functions and the regularity of the solution of the Maxwell equations.We present both 2D and 3D numerical examples to verify the convergence rate of the mixed DG method and demonstrate its application to computing the band structure of photonic crystals.

Cathodoluminescence microscopy: Optical imaging and spectroscopy with deep-subwavelength resolution

Abstract

Preface to BIT 55:1

This special issue of BIT contains articles related to the Workshop on Computational Electromagnetics andAcoustics, held at theMathematisches Forschungsinstitut Oberwolfach, Germany, in January 2013. This workshop was organized by R. Hiptmair (ETH Zurich, Switzerland), R. H.W. Hoppe (Univ. Augsburg, Germany and Houston, Texas), P. Joly (INRIA, Paris, France), and U. Langer (Univ. Linz, Austria). It was attended by 52 invited researchers in the field and had its focus on recent developments of mathematical theory and algorithms. A total of 26 presentations were given at the workshop, their topics covering major current developments in the field of Computational Electromagnetics and Acoustics. Extended abstracts of all contributions have been collected in the Oberwolfach Report 03/2013 (http://www.ems-ph.org/journals/ show_issue.php?issn=1660-8933&vol=10&iss=1, doi:10.4171/OWR/2013/03). Now, several participants of theworkshop, all leaders in their fields, have heeded the organizers’ call for papers for this special issue of the journal BIT. Their manuscripts provide a glimpse of current progress in the mathematical and numerical analysis of problems arising in acoustics or electromagnetics. The contribution P. Monk and J. Chen is in the very topical area of time-domain boundary integral equation methods, and contains deep numerical analysis of a convolution quadrature based scheme for transient electromagnetics. The article by X. Claeys, E. Spindler and myself proposes and analyses a novel boundary integral formulation for scattering transmission problems, whose Galerkin boundary discretization will always yield well conditioned linear systems In their paper M. Dauge, R. A. Norton, and R. Scheichl study the regularity of Bloch modes in photonic crystals, which arise as solutions of eigenvalue problems. They obtain new results in elliptic regularity theory, which are important for designing good approximation spaces for numerical methods.

Lasing characteristics of a GaN photonic crystal nanocavity light source

Lasing characteristics from photonic crystal defects fabricated on bulk GaN are investigated. The device demonstrates multimode lasing with linewidths as narrow as 2–3Å, and an enhanced spontaneous emission factor β∼0.045. The emission spectra indicate that the laser emission is initiated horizontally in the defect nanocavity and then coupled to the vertical radiation, possibly via photonic crystal Bloch modes or by scattering.

Adiabatic matching stage for coupling of light to extended Bloch modes of photonic crystals

In this letter, we present a matching stage for reflection reduction based on the principle of gradual change to efficiently couple light to propagating modes of photonic crystals (PCs). Basic physical considerations in designing these matching stages are investigated and a systematic yet simple design procedure is suggested. We show that matching stages obtained using this method are wideband in frequency, have a wide acceptance angle, and are robust against fabrication imperfections. Therefore, they are the preferred choice in general-purpose matching stages to be used along with dispersion-based PC devices.

From scattering or impedance matrices to Bloch modes of photonic crystals.

The dispersion relation of Bloch waves is derived from the properties of a single grating layer. A straightforward way to impose the Bloch condition leads to the calculation of the eigenvalues of the transfer matrix through the single grating layer. Unfortunately, the transfer-matrix algorithm is known to be unstable as a result of the growing evanescent waves. This problem appears again in the calculation of the eigenvalues, making unusable the transfer matrix in numerous practical problems. We propose two different algorithms to circumvent this problem. The first one takes advantage of scattering matrices, while the second one takes advantage of impedance matrices. Numerical evidence of the efficiency of the algorithms is given. Dispersion diagrams of simple cubic and woodpile photonic crystals are obtained by using, respectively, the scattering and impedance matrices.