Abstract:
We derive exact strong-contrast expansions for the effective dielectric tensor $\epeff$ of electromagnetic waves propagating in a two-phase composite random medium with isotropic components explicitly in terms of certain integrals over the $n$-point correlation functions of the medium. Our focus is the long-wavelength regime, i.e., when the wavelength is much larger than the scale of inhomogeneities in the medium. Lower-order truncations of these expansions lead to approximations for the effective dielectric constant that depend upon whether the medium is below or above the percolation threshold. In particular, we apply two- and three-point approximations for $\epeff$ to a variety of different three-dimensional model microstructures, including dispersions of hard spheres, hard oriented spheroids and fully penetrable spheres as well as Debye random media, the random checkerboard, and power-law-correlated materials. We demonstrate the importance of employing $n$-point correlation functions of order higher than two for high dielectric-phase-contrast ratio. We show that disorder in the microstructure results in an imaginary component of the effective dielectric tensor that is directly related to the {\it coarseness} of the composite, i.e., local volume-fraction fluctuations for infinitely large windows. The source of this imaginary component is the attenuation of the coherent homogenized wave due to scattering. We also remark on whether there is such attenuation in the case of a two-phase medium with a quasiperiodic structure.

Abstract:
We study long range propagation of electromagnetic waves in random waveguides with rectangular cross-section and perfectly conducting boundaries. The waveguide is filled with an isotropic linear dielectric material, with randomly fluctuating electric permittivity. The fluctuations are weak, but they cause significant cumulative scattering over long distances of propagation of the waves. We decompose the wave field in propagating and evanescent transverse electric and magnetic modes with random amplitudes that encode the cumulative scattering effects. They satisfy a coupled system of stochastic differential equations driven by the random fluctuations of the electric permittivity. We analyze the solution of this system with the diffusion approximation theorem, under the assumption that the fluctuations decorrelate rapidly in the range direction. The result is a detailed characterization of the transport of energy in the waveguide, the loss of coherence of the modes and the depolarization of the waves due to cumulative scattering.

Abstract:
We analyze electromagnetic field propagation through a random medium which consists of hyperbolic metamaterial domains separated by regions of normal "elliptic" space. This situation may occur in a problem as common as 9 micrometer light propagation through a pile of sand, or as exotic as electromagnetic field behavior in the early universe immediately after the electro-weak phase transition. We demonstrate that spatial field distributions in random hyperbolic and random "elliptic" media look strikingly different. This effect may potentially be used to evaluate the magnitude of magnetic fields which existed in the early universe.

Abstract:
The objective of this paper is to establish the properties of the electromagnetic wave propagation in a diversity of situations in material media with magnetic monopoles and even in the situations of entities simultaneously containing electric and magnetic charges. This analysis requires the knowledge and solutions of the ``Maxwell" equations in material media compatible with the existence of magnetic monopoles and the extended concepts of linear responses (conductivity, split-charge susceptibility, kinetic susceptibility, permittivity and magnetic permeability) in the case of presence of electric and magnetic charges. This analysis can facilitate insights and suggestions for electrical and optical experiments according a better knowledge of the materials whose behaviour can be analyzed under the consideration of the existence of entities with equivalent properties of the magnetic monopoles.

Abstract:
Maxwell's equations are cast in the form of the Schr\"{o}dinger equation. The Lanczos propagation method is used in combination with the fast Fourier pseudospectral method to solve the initial value problem. As a result, a time-domain, unconditionally stable, and highly efficient numerical algorithm is obtained for the propagation and scattering of broad-band electromagnetic pulses in dispersive and absorbing media. As compared to conventional finite-difference time-domain methods, an important advantage of the proposed algorithm is a dynamical control of accuracy: Variable time steps or variable computational costs per time step with error control are possible. The method is illustrated with numerical simulations of extraordinary transmission and reflection in metal and dielectric gratings with rectangular and cylindrical geometry.

Abstract:
We show that a signal can propagate in a particular direction through a model random medium regardless of the precise state of the medium. As a prototype, we consider a point particle moving on a one-dimensional lattice whose sites are occupied by scatterers with the following properties: (i) the state of each site is defined by its spin (up or down); (ii) the particle arriving at a site is scattered forward (backward) if the spin is up (down); (iii) the state of the site is modified by the passage of the particle, i.e. the spin of the site where a scattering has taken place, flips ($\uparrow \Leftrightarrow \downarrow $). We consider one dimensional and triangular lattices, for which we give a microscopic description of the dynamics, prove the propagation of a particle through the scatterers, and compute analytically its statistical properties. In particular we prove that, in one dimension, the average propagation velocity is $ = 1/(3-2q)$, with $q$ the probability that a site has a spin $\uparrow$, and, in the triangular lattice, the average propagation velocity is independent of the scatterers distribution: $ = 1/8$. In both cases, the origin of the propagation is a blocking mechanism, restricting the motion of the particle in the direction opposite to the ultimate propagation direction, and there is a specific re-organization of the spins after the passage of the particle. A detailed mathematical analysis of this phenomenon is, to the best of our knowledge, presented here for the first time.

Abstract:
Explicit formulas for fundamental and generalized solutions of the Cauchy problem for Maxwell's system are obtained for the case when the dielectric permeability is a symmetric positive definite matrix, the magnetic permeability is a positive constant, the conductivity vanished. The visualization of electromagnetic wave propagation made using these formulas by MatLAB, C++.

Abstract:
Propagation of the extremely short electromagnetic pulse in non-linear dielectric media without the slowly varying envelope approximation is discussed. The models under consideration take into account both resonant and not-resonant excitations of non-linear medium, and polarisation states of electromagnetic wave. Steady state solutions of the relevant equations are presented for certain of these models.

Abstract:
Characteristics of wave propagation in some time-varying media have been studied. Using the separation of variables and hybrid method, accurate solutions of electromagnetic waves are obtained when and are monotone and periodic functions with respect to time. Furthermore, the classes of and on which wave equation has analytical solution are studied and listed. In time-varying media conditions, some time-varying parameters such as frequency shift, change of phase velocity, wave impedance are also discussed. The examples and rationale for the results are given.

Abstract:
In a communication scheme, there exist points at the transmitter and at the receiver where the wave is reduced to a finite set of functions of time which describe amplitudes and phases. For instance, the information is summarized in electrical cables which preceed or follow antennas. In many cases, a random propagation time is sufficient to explain changes induced by the medium. In this paper we study models based on stable probability laws which explain power spectra due to propagation of different kinds of waves in different media, for instance, acoustics in quiet or turbulent atmosphere, ultrasonics in liquids or tissues, or electromagnetic waves in free space or in cables. Physical examples show that a sub-class of probability laws appears in accordance with the causality property of linear filters.