The homogenisation of Maxwell's equations with applications to photonic crystals and localised waveforms on metafilms

An asymptotic theory is developed to generate equations that model the global behaviour of electromagnetic waves in periodic photonic structures when the wavelength is not necessarily long relative to the periodic cell dimensions; potentially highly-oscillatory short-scale detail is encapsulated through integrated quantities. The theory we develop is then applied to two topical examples, the first being the case of aligned dielectric cylinders, which has great importance in the modelling of photonic crystal fibres. We then consider the propagation of waves in a structured metafilm, here chosen to be a planar array of dielectric spheres. At certain frequencies strongly directional dynamic anisotropy is observed, and the asymptotic theory is shown to capture the effect, giving highly accurate qualitative and quantitative results as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour

Dynamic homogenisation of Maxwell’s equations with applications to photonic crystals and localised waveforms on gratings

A two-scale asymptotic theory is developed to generate continuum equations that model the macroscopic be- haviour of electromagnetic waves in periodic photonic structures when the wavelength is not necessarily long relative to the periodic cell dimensions; potentially highly-oscillatory short-scale detail is encapsulated through integrated quantities. The resulting equations include tensors that represent effective refractive indices near band edge frequencies along all principal axes directions, and these govern scalar functions providing long-scale mod- ulation of short-scale Bloch eigenstates, which can be used to predict the propagation of waves at frequencies outside of the long wavelength regime; these results are outside of the remit of typical homogenisation schemes. The theory we develop is applied to two topical examples, the first being the case of aligned dielectric cylin- ders, which has great importance in modelling photonic crystal fibres. Results of the asymptotic theory are veri- fied against numerical simulations by comparing photonic band diagrams and evanescent decay rates for guided modes. The second example is the propagation of electromagnetic waves localised within a planar array of di- electric spheres; at certain frequencies strongly directional propagation is observed, commonly described as dy- namic anisotropy. Computationally this is a challenging three-dimensional calculation, which we perform, and then demonstrate that the asymptotic theory captures the effect, giving highly accurate qualitative and quantitative comparisons as well as providing interpretation for the underlying change from elliptic to hyperbolic behaviour

Mesoscopic magnetism in dielectric photonic crystal meta materials: topology and inhomogeneous broadening

We consider meta materials made from a two-dimensional dielectric rod-type photonic crystal. The magnetic response is studied within the recently developed homogenization theory and we in particular study the effects of topology and inhomogeneous broadening. While topology itself mainly affects the Mie resonance frequency we find that the dispersion in the topological radius R of the dielectric rods may lead to significant inhomogeneous broadening and suppression of the negative-mu phenomena for dR/R0 >> epsilon''/epsilon', with epsilon=epsilon'+i*epsilon'' being the relative dielectric function of the rods.Comment: 13 pages including 1 table and 5 figure

Phase diagram for the transition from photonic crystals to dielectric metamaterials

Photonic crystals and metamaterials represent two seemingly different classes of artificial electromagnetic media but often they are composed of similar structural elements arranged in periodic lattices. The important question is how to distinguish these two types of periodic photonic structures when their parameters, such as dielectric permittivity and lattice spacing, vary continuously. Here, we discuss transitions between photonic crystals and all-dielectric metamaterials and introduce the concept of a phase diagram and an order parameter for such structured materials, based on the physics of Mie and Bragg resonances. We show that a periodic photonic structure transforms into a metamaterial when the Mie gap opens up below the lowest Bragg bandgap where the homogenization approach can be justified and the effective permeability becomes negative. Our theoretical approach is confirmed by detailed microwave experiments for a metacrystal composed of a square lattice of glass tubes filled with heated water. This analysis yields deep insight into the properties of periodic photonic structures, and it also provides a useful tool for designing different classes of electromagnetic materials in a broad range of parameters.Comment: 7 pages, 6 figure

Transformation Optics with Photonic Band Gap Media

We introduce a class of optical media based on adiabatically modulated, dielectric-only, and potentially extremely low-loss, photonic crystals. The media we describe represent a generalization of the eikonal limit of transformation optics (TO). The foundation of the concept is the possibility to fit frequency isosurfaces in the k-space of photonic crystals with elliptic surfaces, allowing them to mimic the dispersion relation of light in anisotropic effective media. Photonic crystal cloaks and other TO devices operating at visible wavelengths can be constructed from optically transparent substances like glasses, whose attenuation coefficient can be as small as 10 dB/km, suggesting the TO design methodology can be applied to the development of optical devices not limited by the losses inherent to metal-based, passive metamaterials.Comment: 4 pages, 4 figure

Sub-Wavelength Plasmonic Crystals: Dispersion Relations and Effective Properties

We obtain a convergent power series expansion for the first branch of the dispersion relation for subwavelength plasmonic crystals consisting of plasmonic rods with frequency-dependent dielectric permittivity embedded in a host medium with unit permittivity. The expansion parameter is $\eta=kd=2\pi d/\lambda$, where $k$ is the norm of a fixed wavevector, $d$ is the period of the crystal and $\lambda$ is the wavelength, and the plasma frequency scales inversely to $d$, making the dielectric permittivity in the rods large and negative. The expressions for the series coefficients (a.k.a., dynamic correctors) and the radius of convergence in $\eta$ are explicitly related to the solutions of higher-order cell problems and the geometry of the rods. Within the radius of convergence, we are able to compute the dispersion relation and the fields and define dynamic effective properties in a mathematically rigorous manner. Explicit error estimates show that a good approximation to the true dispersion relation is obtained using only a few terms of the expansion. The convergence proof requires the use of properties of the Catalan numbers to show that the series coefficients are exponentially bounded in the $H^1$ Sobolev norm