Photonic Band Gaps: Noncommuting Limits and the "Acoustic Band"

Unveiling Extreme Anisotropy in Elastic Structured Media

Periodic structures can be engineered to exhibit unique properties observed at symmetry points, such as zero group velocity, Dirac cones, and saddle points; identifying these and the nature of the associated modes from a direct reading of the dispersion surfaces is not straightforward, especially in three dimensions or at high frequencies when several dispersion surfaces fold back in the Brillouin zone. A recently proposed asymptotic high-frequency homogenization theory is applied to a challenging time-domain experiment with elastic waves in a pinned metallic plate. The prediction of a narrow high-frequency spectral region where the effective medium tensor dramatically switches from positive definite to indefinite is confirmed experimentally; a small frequency shift of the pulse carrier results in two distinct types of highly anisotropic modes. The underlying effective equation mirrors this behavior with a change in form from elliptic to hyperbolic exemplifying the high degree of wave control available and the importance of a simple and effective predictive model.

Classical homogenization to analyse the dispersion relations of spoof plasmons with geometrical and compositional effects

We show that the classical homogenization is able to describe the dispersion relation of spoof plasmons in structured thick interfaces with periodic unit cell being at the subwavelength scale. This is because the interface in the real problem is replaced by a slab of an homogeneous birefringent medium, with an effective mass density tensor and an effective bulk modulus. Thus, explicit dispersion relation can be derived, corresponding to guided waves in the homogenized problem. Contrary to previous effective medium theories or retrieval methods, the homogenization gives effective parameters depending only on the properties of the material and on the geometry of the microstructure. Although resonances in the unit cell cannot be accounted for within this low-frequency homogenization, it is able to account for resonances occurring because of the thickness of the interface and thus, to capture the behaviour of the spoof plasmons. Beyond the case of simple grooves in a hard material, we inspect the influence of tilting the grooves and the influence of the material properties.

Asymptotics for metamaterials and photonic crystals

Metamaterial and photonic crystal structures are central to modern optics and are typically created from multiple elementary repeating cells. We demonstrate how one replaces such structures asymptotically by a continuum, and therefore by a set of equations, that captures the behaviour of potentially high-frequency waves propagating through a periodic medium. The high-frequency homogenization that we use recovers the classical homogenization coefficients in the low-frequency long-wavelength limit. The theory is specifically developed in electromagnetics for two-dimensional square lattices where every cell contains an arbitrary hole with Neumann boundary conditions at its surface and implemented numerically for cylinders and split-ring resonators. Illustrative numerical examples include lensing via all-angle negative refraction, as well as omni-directive antenna, endoscope and cloaking effects. We also highlight the importance of choosing the correct Brillouin zone and the potential of missing interesting physical effects depending upon the path chosen.

Homogenization of Maxwell's equations in periodic composites: boundary effects and dispersion relations

We consider the problem of homogenizing the Maxwell equations for periodic composites. The analysis is based on Bloch-Floquet theory. We calculate explicitly the reflection coefficient for a half space and derive and implement a computationally efficient continued-fraction expansion for the effective permittivity. Our results are illustrated by numerical computations for the case of two-dimensional systems. The homogenization theory of this paper is designed to predict various physically measurable quantities rather than to simply approximate certain coefficients in a partial differential equation.

Bloch waves in periodic multi-layered acoustic waveguides

The band spectrum and associated Floquet–Bloch eigensolutions, arising in acoustic and electromagnetic waveguides, which have periodic structure along the guide while remaining of finite width, are found. Homogeneous Dirichlet or Neumann conditions along the guide walls, or an alternation of them, are taken. Importantly, in some cases, a total stop band at zero frequency is identified providing space for low-frequency localized modes; geometric defects in the structured waveguide also create these modes. Numerical and asymptotic techniques identify dispersion curves and trapped modes. Some cases demonstrate maxima and minima of the spectral edges within the Brillouin zone and also allow for ultraslow light or sound. Imaging applications using anomalous dispersion to generate subwavelength resolution are possible and are demonstrated.

Bloch–Floquet bending waves in perforated thin plates

This paper presents a mathematical model describing propagation of bending waves in a perforated thin plate. It is assumed that the holes are circular and form a doubly periodic square array. A spectral problem for the biharmonic operator is formulated in a unit cell containing a single defect, and its analytical solution is constructed using a multipole method. The overall system for the coefficients in the multipole expansion is then solved numerically. We generate dispersion diagrams for the two cases where the boundaries of holes are either clamped or free. We show that in the clamped case, there is a total low-frequency band gap in the limit of inclusions of zero radius, and give a simple formula describing the corresponding band diagram in this limit. We show that in the free-edge case, the band diagram of the vibrating plate is much closer to that of plane waves in a uniform plate than for the clamped case.

Diamagnetic response of metallic photonic crystals at infrared and visible frequencies

We show analytically and numerically that diamagnetic response (effective magnetic permeability mue<1) at infrared and visible frequencies can be achieved in photonic crystals composed of metallic nanowires or nanospheres when the wavelength is much larger than the lattice constant a (lambda approximately 2000a). When lambda approximately100a, the metallic photonic crystals will exhibit strong diamagnetic response (mue<0.8), leading to many interesting phenomena such as the unusual Brewster angle for s waves and incident-angle-and-polarization-independent reflection and transmission.

Refraction of water waves by periodic cylinder arrays

We show that in the long wavelength limit, water waves propagate through an array of bottom-mounted vertical cylinders as if the water has an effective depth and effective gravitational constant that depends on the filling ratio of the cylinders, leading to refraction phenomena that can be described by analytic formulas. The results are obtained with rigorous homogenization techniques, as well as the multiple scattering formalism that gives full dispersion relationships. This phenomenon provides a mechanism to control the flow of ocean wave energy, as exemplified by a water-wave focusing lens.

Homogenization of magnetodielectric photonic crystals

We calculate the low-frequency index of refraction of a medium which is homogeneous along axis z and possesses a periodic dependence of the permittivity epsilon(r) and permeability micro(r) in the x-y plane (2D magnetodielectric photonic crystal). Exact analytical formulas for the effective index of refraction for two eigenmodes with vector E or H polarized along axis z are obtained. We show that, unlike nonmagnetic photonic crystals where the E mode is ordinary and the H mode is extraordinary, now both modes exhibit extraordinary behavior. Because of this distinction, the magnetodielectric photonic crystals exhibit optical properties that do not exist for natural crystals. We also discuss the limiting case of perfectly conducting cylinders and clarify the so-called problem of noncommuting limits, omega-->0 and epsilon--> infinity.

Density of states functions for photonic crystals

We discuss density of states functions for photonic crystals, in the context of the two-dimensional problem for arrays of cylinders of arbitrary cross section. We introduce the mutual density of states (MDOS), and show that this function can be used to calculate both the local density of states (LDOS), which gives position information for emission of radiation from photonic crystals, and the spectral density of states (SDOS), which gives angular information. We establish the connection between MDOS, LDOS, SDOS and the conventional density of states, which depends only on frequency. We relate all four functions to the band structure and propagating states within the crystal, and give numerical examples of the relation between band structure and density of states functions.

Photonic band structure calculations using scattering matrices

We consider band structure calculations of two-dimensional photonic crystals treated as stacks of one-dimensional gratings. The gratings are characterized by their plane wave scattering matrices, the calculation of which is well established. These matrices are then used in combination with Bloch's theorem to determine the band structure of a photonic crystal from the solution of an eigenvalue problem. Computationally beneficial simplifications of the eigenproblem for symmetric lattices are derived, the structure of eigenvalue spectrum is classified, and, at long wavelengths, simple expressions for the positions of the band gaps are deduced. Closed form expressions for the reflection and transmission scattering matrices of finite stacks of gratings are established. A new, fundamental quantity, the reflection scattering matrix, in the limit in which the stack fills a half space, is derived and is used to deduce the effective dielectric constant of the crystal in the long wavelength limit.