The term photonic crystals appears because of the analogy between electron waves in crystals and the light waves in artificial periodic dielectric structures. During the recent years the investigation of one-, two-and three-dimensional periodic structures has attracted a widespread attention of the world optics community because of great potentiality of such structures in advanced applied optical fields. The interest in periodic structures has been stimulated by the fast development of semiconductor technology that now allows the fabrication of artificial structures, whose period is comparable with the wavelength of light in the visible and infrared ranges.

http://ieeexplore.ieee.org/iel5/6354/16969/00781867.pdf

Photonic crystals

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Variational Principles Of Continuum Mechanics

Dispersion of elastic waves in a strongly inhomogeneous three-layered plate

We develop a continuum model, valid at high frequencies, for wave propagation through elastic media that contain periodic, or nearly periodic, arrangements of traction free, or clamped, inclusions. The homogenisation methodology we create allows for wavelengths and periodic spacing to potentially be of similar scale and therefore is not limited to purely long-waves and low frequency. We treat in-plane elasticity, with coupled shear and compressional waves and therefore a full vector problem, demonstrating that a two-scale asymptotic approach using a macroscale and microscale results in effective scalar continuum equations posed entirely upon the macroscale; the vector nature of the problem being incorporated on the microscale. This rather surprising result is comprehensively verified by comparing the resultant asymptotics to full numerical simulations for the Bloch problem of perfectly periodic media. The dispersion diagrams for this Bloch problem are found both numerically and asymptotically. Periodic media exhibit dynamic anisotropy, e.g. strongly directional fields at specific frequencies, and both finite element computations and the asymptotic theory predict this. Periodic media in elasticity can be related to the emergent fields of metamaterials and photonic crystals in electromagnetics and relevant analogies are drawn. As an illustration we consider the highly anisotropic cases and show how their existence can be predicted naturally from the homogenisation theory.

https://spiral.imperial.ac.uk/bitstream/10044/1/18823/2/revised.pdf

Homogenisation for elastic photonic crystals and dynamic anisotropy

Anti-plane dynamic shear of a strongly inhomogeneous dynamic laminate with traction-free faces is analysed. Two types of contrast are considered, including those for composite structures with thick...

https://journals.sagepub.com/doi/pdf/10.1177/1081286518790804

Asymptotic analysis of an anti-plane dynamic problem for a three-layered strongly inhomogeneous laminate:

http://spiral.imperial.ac.uk/bitstream/10044/1/18822/2/clear.pdf

Long-wave asymptotic theories: The connection between functionally graded waveguides and periodic media

Reflection from a semi-infinite stack of layers using homogenization

High-frequency homogenization is applied herein to develop asymptotics for waves propagating along line defects in lattices; the approaches developed are anticipated to be of wide application to many other systems that exhibit surface waves created or directed by microstructure. With the aim being to create a long-scale continuum representation of the line defect that nonetheless accurately incorporates the microscale information, this development uses the microstructural information embedded within, potentially high-frequency, standing wave solutions. A two-scaled approach is utilized for a simple line defect and demonstrated versus exact solutions for quasi-periodic systems and versus numerical solutions for line defects that are themselves perturbed or altered. In particular, Rayleigh--Bloch states propagating along the line defect, and localized defect states, are identified both asymptotically and numerically. Additionally, numerical simulations of large-scale lattice systems illustrate the physics u...

Asymptotics for rayleigh-bloch waves along lattice line defects

The isothermal annular Poiseuille flow of a weakly compressible Newtonian liquid with constant shear and bulk viscosities is considered. A linear equation of state is assumed and a perturbation analysis in terms of the primary flow variables is performed up to the first order using the isothermal compressibility as the perturbation parameter. The effects of compressibility, the bulk viscosity, the radii ratio, the aspect ratio, and the Reynolds number on the velocity and pressure fields are studied.

L.M. Joseph

Papers

Long-wave asymptotic theories: The connection between functionally graded waveguides and periodic media

Reflection from a semi-infinite stack of layers using homogenization

Asymptotics for rayleigh-bloch waves along lattice line defects

Perturbation solution of the compressible annular Poiseuille flow of a viscous fluid