Wave mechanics in media pinned at Bravais lattice points

The propagation of waves through microstructured media with periodically arranged inclusions has applications in many areas of physics and engineering, stretching from photonic crystals through to seismic metamaterials. In the high-frequency regime, modelling such behaviour is complicated by multiple scattering of the resulting short waves between the inclusions. Our aim is to develop an asymptotic theory for modelling systems with arbitrarily-shaped inclusions located on general Bravais lattices. We then consider the limit of point-like inclusions, the advantage being that exact solutions can be obtained using Fourier methods, and go on to derive effective medium equations using asymptotic analysis. This approach allows us to explore the underlying reasons for dynamic anisotropy, localisation of waves, and other properties typical of such systems, and in particular their dependence upon geometry. Solutions of the effective medium equations are compared with the exact solutions, shedding further light on the underlying physics. We focus on examples that exhibit dynamic anisotropy as these demonstrate the capability of the asymptotic theory to pick up detailed qualitative and quantitative features

Field representation for optical defect resonances in multilayer microcavities using quasi-normal modes

Quasi-normal modes are used to characterize transmission resonances in 1D optical defect cavities and the related field approximations. We specialize to resonances inside the bandgap of the periodic multilayer mirrors that enclose the defect cavities. Using a template with the most relevant QNMs a variational principle permits to represent the field and the spectral transmission close to resonances

Outgoing wave conditions in photonic crystals and transmission properties at interfaces

We analyze the propagation of waves in unbounded photonic crystals, the waves are described by a Helmholtz equation with $x$-dependent coefficients. The scattering problem must be completed with a radiation condition at infinity, which was not available for $x$-dependent coefficients. We develop an outgoing wave condition with the help of a Bloch wave expansion. Our radiation condition admits a (weak) uniqueness result, formulated in terms of the Bloch measure of solutions. We use the new radiation condition to analyze the transmission problem where, at fixed frequency, a wave hits the interface between free space and a photonic crystal. We derive that the vertical wave number of the incident wave is a conserved quantity. Together with the frequency condition for the transmitted wave, this condition leads (for appropriate photonic crystals) to the effect of negative refraction at the interface

A Bloch wave numerical scheme for scattering problems in periodic wave-guides

We present a new numerical scheme to solve the Helmholtz equation in a wave-guide. We consider a medium that is bounded in the $x_2$-direction, unbounded in the $x_1$-direction and $\varepsilon$-periodic for large $|x_1|$, allowing different media on the left and on the right. We suggest a new numerical method that is based on a truncation of the domain and the use of Bloch wave ansatz functions in radiation boxes. We prove the existence and a stability estimate for the infinite dimensional version of the proposed problem. The scheme is tested on several interfaces of homogeneous and periodic media and it is used to investigate the effect of negative refraction at the interface of a photonic crystal with a positive effective refractive index.Comment: 25 pages, 10 figure

Asymptotic network models of subwavelength metamaterials formed by closely packed photonic and phononic crystals

We demonstrate that photonic and phononic crystals consisting of closely spaced inclusions constitute a versatile class of subwavelength metamaterials. Intuitively, the voids and narrow gaps that characterise the crystal form an interconnected network of Helmholtz-like resonators. We use this intuition to argue that these continuous photonic (phononic) crystals are in fact asymptotically equivalent, at low frequencies, to discrete capacitor-inductor (mass-spring) networks whose lumped parameters we derive explicitly. The crystals are tantamount to metamaterials as their entire acoustic branch, or branches when the discrete analogue is polyatomic, is squeezed into a subwavelength regime where the ratio of wavelength to period scales like the ratio of period to gap width raised to the power 1/4; at yet larger wavelengths we accordingly find a comparably large effective refractive index. The fully analytical dispersion relations predicted by the discrete models yield dispersion curves that agree with those from finite-element simulations of the continuous crystals. The insight gained from the network approach is used to show that, surprisingly, the continuum created by a closely packed hexagonal lattice of cylinders is represented by a discrete honeycomb lattice. The analogy is utilised to show that the hexagonal continuum lattice has a Dirac-point degeneracy that is lifted in a controlled manner by specifying the area of a symmetry-breaking defect

Lorenz-Mie theory for 2D scattering and resonance calculations

This PhD tutorial is concerned with a description of the two-dimensional generalized Lorenz-Mie theory (2D-GLMT), a well-established numerical method used to compute the interaction of light with arrays of cylindrical scatterers. This theory is based on the method of separation of variables and the application of an addition theorem for cylindrical functions. The purpose of this tutorial is to assemble the practical tools necessary to implement the 2D-GLMT method for the computation of scattering by passive scatterers or of resonances in optically active media. The first part contains a derivation of the vector and scalar Helmholtz equations for 2D geometries, starting from Maxwell's equations. Optically active media are included in 2D-GLMT using a recent stationary formulation of the Maxwell-Bloch equations called steady-state ab initio laser theory (SALT), which introduces new classes of solutions useful for resonance computations. Following these preliminaries, a detailed description of 2D-GLMT is presented. The emphasis is placed on the derivation of beam-shape coefficients for scattering computations, as well as the computation of resonant modes using a combination of 2D-GLMT and SALT. The final section contains several numerical examples illustrating the full potential of 2D-GLMT for scattering and resonance computations. These examples, drawn from the literature, include the design of integrated polarization filters and the computation of optical modes of photonic crystal cavities and random lasers.Comment: This is an author-created, un-copyedited version of an article published in Journal of Optics. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from i