One-dimensional Inhomogeneous Medium Research Articles

Propagation of electromagnetic waves in stationary anisotropic media

In this paper we have investigated the fundamental properties of the solutions of Maxwell’s equations describing free electromagnetic waves in stationary anisotropic media, with tensor characteristics along the Z axis. A matrix of coefficients, the structure of the matrix of Maxwell’s equations, dispersion equations and indicatrix curves for a transparent anisotropic one-dimensional inhomogeneous conductive medium have been obtained. Exact analytical solutions in case of homogeneous anisotropic media are constructed on the basis of the matrix structure.

Bremmer series for the multimodal sound propagation in inhomogeneous waveguides

The method of Bremmer series is presented and implemented for the multimodal sound propagation in a waveguide with varying cross-section. The solution is constructed iteratively by summing terms of increasing order, with each order representing the number of scattering events. This formulation generalizes to higher dimensions the series decomposition proposed by Bremmer in one-dimensional inhomogeneous media, and it also gives an insight on the complex wave scattering of the coupled guided modes. The accuracy and convergence of the solution are inspected and a comparison is made with the one-way coupled mode equation derived assuming no reflected waves. It is notably shown that the first-order Bremmer series, simply obtained by quadrature, is a relevant alternative to classical WKB or one-way approximations.

Analytic solution to field distribution in one-dimensional inhomogeneous media

A new general expression of a solution to Maxwell׳s equations derived recently has been applied to a 1-D inhomogeneous medium. It is shown that it solves any inhomogeneous refractive index profile. Its main advantage is that it does not require integration of either the differential wave equation or the refractive index profile, as it is the case with other methods. The solution is expressed in a closed form. The obtained numerical results compare favorably with both the slow-varying envelope method as well as with the exact numerical method.

Transient analysis of 1D inhomogeneous media by dynamic inhomogeneous finite element method

The dynamic inhomogeneous finite element method is studied for use in the transient analysis of onedimensional inhomogeneous media. The general formula of the inhomogeneous consistent mass matrix is established based on the shape function. In order to research the advantages of this method, it is compared with the general finite element method. A linear bar element is chosen for the discretization tests of material parameters with two fictitious distributions. And, a numerical example is solved to observe the differences in the results between these two methods. Some characteristics of the dynamic inhomogeneous finite element method that demonstrate its advantages are obtained through comparison with the general finite element method. It is found that the method can be used to solve elastic wave motion problems with a large element scale and a large number of iteration steps.

Reflection and dissipation of Alfvén waves in interstellar clouds

Context: Supersonic nonthermal motions in molecular clouds are often interpreted as long-lived magnetohydrodynamic (MHD) waves. The propagation and amplitude of these waves is affected by local physical characteristics, most importantly the gas density and the ionization fraction. Aims: We study the propagation, reflection and dissipation of Alfv\'en waves in molecular clouds deriving the behavior of observable quantities such as the amplitudes of velocity fluctuations and the rate of energy dissipation. Methods: We formulated the problem in terms of Els\"asser variables for transverse MHD waves propagating in a one-dimensional inhomogeneous medium, including the dissipation due to collisions between ions and neutrals and to a nonlinear turbulent cascade treated in a phenomenological way. We considered both steady-state and time-dependent situations and solved the equations of the problem numerically with an iterative method and a Lax-Wendroff scheme, respectively. Results: Alfv\'en waves incident on overdense regions with density profiles typical of cloud cores embedded in a diffuse gas suffer enhanced reflection in the regions of the steepest density gradient, and strong dissipation in the core's interior. These effects are especially significant when the wavelength is intermediate between the critical wavelength for propagation and the typical scale of the density gradient. For larger wave amplitudes and/or steeper input spectra, the effects of the perpendicular turbulent cascade result in a stronger energy dissipation in the regions immediately surrounding the dense core. Conclusions: The results may help to interpret the sharp decrease of line width observed in the environments of low-mass cloud cores in several molecular transitions.

Transformation and absorption of MHD oscillations in plane-stratified models of the Earth’s magnetosphere

The process of resonant transformation of fast magnetosonic (FMS) waves originating from the solar wind in Alfven and slow magnetosonic (SMS) oscillations in the Earth’s magnetosphere is investigated. The study of this process in a one-dimensional inhomogeneous medium model shows that the presence of a resonant surface for SMS oscillations in the plasma configuration significantly increases the absorption of the energy of FMS waves incident on the magnetosphere. The spatial distribution of the absorption rate of an energy flux of FMS waves is examined. In numerical calculations, the Kolmogorov spectrum was used for FMS waves, which is typical of the waves in the transition layer between a shock wave and the magnetopause. It is shown that the rate of energy absorption of FMS waves due to the resonant excitation of SMS oscillations is by several orders of magnitude greater than the absorption of their energy associated with resonant Alfven waves at the same magnetic shells.

A local limit theorem for a transient chaotic walk in a frozen environment

This paper studies particle propagation in a one-dimensional inhomogeneous medium where the laws of motion are generated by chaotic and deterministic local maps. Assuming that the particle’s initial location is random and uniformly distributed, this dynamical system can be reduced to a random walk in a one-dimensional inhomogeneous environment with a forbidden direction. Our main result is a local limit theorem which explains in detail why, in the long run, the random walk’s probability mass function does not converge to a Gaussian density, although the corresponding limiting distribution over a coarser diffusive space scale is Gaussian.

Inverse scattering problem for the wave equation in a one-dimensional inhomogeneous medium

A one-dimensional inverse scattering problem for the Helmholtz wave equation is considered. The problem consists in reconstruction of the refractive index of transparent inhomogeneous media from a specified complex reflection spectrum. A method previously proposed by the authors for reconstruction of the Bragg gratings in the coupled mode approximation is used. Numerical simulations of the inverse scattering problem for the classical Rayleigh layer and for an exponentially smooth transitional layer are performed. The solutions obtained demonstrate the high efficiency of the proposed approach. The accuracy of refractive index reconstruction turns out to be essentially dependent on the degree of layer smoothness. The proposed approach can be applied in problems of the development of coated and interference optics, in synthesis of the Bragg gratings and multilayer optical mirrors, and in radiophysics and acoustics.

Inverse radiation problem in one-dimensional slab by time-resolved reflected and transmitted signals

A time-domain inverse approach is proposed for estimating the distribution of absorbing and scattering coefficients in one-dimensional inhomogeneous media. The temporal reflected and transmitted signals are detected when an ultra-short pulse irradiates on the boundary of semi-transparent scattering media. Forward computation and inverse algorithm employ the least-squares finite element method and conjugate gradient method, respectively. As the prevalent diffusion approximation is not employed in our model, the present approach can be extended to more comprehensive application. The investigation about detected signals indicates that the reflected signals play a significant role in reconstructing optical properties; the signals in early sampling time are more important than those at long-time logarithm slope, and so, more attention should be paid to the early signals in the solution of inverse radiation problem. Three different inverse radiation problems are investigated to show the ability of the present approach to deal with the two-layer, three-layer and continuous inhomogeneous media. The effect of measured errors on the accuracy of reconstruction is investigated by adding artificial random errors. The results indicate that accurate reconstruction depends on not only precise numerical simulation but also quality of detected data.

Method of Propagating Waves for a One-Dimensional Inhomogeneous Medium

The aim of this work is to develop a method of propagating waves based on the idea of a wave as a changing state of a medium. This method allows us to represent a solution of the one-dimensional wave equation in an inhomogeneous medium as the sum of two constantly deformed waves, the “right wave” and the “left wave,” transported from point to point with coefficients depending on the points and the transport time. By the propagating-wave method we obtain explicit (as far as possible) formulas for solutions of the mixed problem with homogeneous and inhomogeneous boundary conditions and solutions of the Goursat problem. The derivation of these formulas is based on special convolution formulas for the transport coefficients that are similar to the addition identities for trigonometric functions.

Theory of spontaneous parametric down-conversion from photonic crystals

An effective-medium approach is used to derive a set of coupled-mode equations describing quadratic parametric interactions in a one-dimensional inhomogeneous medium of finite length. The solutions for both copropagating and counterpropagating interactions are used to study the quantum-mechanical process of spontaneous parametric down-conversion (SPDC). An example is studied in which the inhomogeneous medium is a photonic crystal. The effect that device geometry has on both the efficiency and bandwidth of SPDC is explored. With appropriate design, we find that it is possible not only to enhance the efficiency of SPDC, but also to generate broadband copropagating and narrowband counterpropagating down-converted light. A setup is proposed with the potential to act as a source of entangled photon pairs.

PT-invariant Helmholtz optics and its applications to slab waveguides

In optical Integrated circuits (OIC) and optical device design (passive and active), the index of refraction profile is a coordinate dependent parameter. Specifically in slab waveguides which is a basic element in optical engineering, the guided modes and its behavior with respect to waveguide parameters can be extracted from the exact solution of the Helmholtz equation. In the general case, the numerical approach is applied for determination of the electric and magnetic fields. In this paper a suitable algorithm, using the PT-symmetric quantum mechanical approach has been given for light transmission through one-dimensional inhomogeneous media (PT-symmetric index of refraction).

A traveling wave formula in a one-dimensional inhomogeneous medium

A traveling wave formula in a one-dimensional inhomogeneous medium

Direct and inverse problems of wave propagation through a one-dimensional inhomogeneous medium

This paper deals with a wave process induced by an acoustic impulse in a vertically inhomogeneous half-space. The solvability of the corresponding direct initial boundary value problem is proved. We also consider a class of one-dimensional inverse problems including, in particular, the classical inverse problem of the scattering theory, the transmission and reflection problems of seismology, etc. It is shown that all these inverse problems are equivalent in a certain sense, and therefore can be solved by identical methods. Theoretical results are illustrated by numerical examples.

Closed-form solutions for one-dimensional inhomogeneous anisotropic medium in a special case. I. Direct scattering problem

An analytical formulation is presented for the direct scattering problem of one-dimensional (1-D) inhomogeneous anisotropic medium in a special case. This method gives nonlinear differential equations for the reflection coefficients of the anisotropic medium. Applying the discontinuity condition of the dielectric parameters at the interface of the medium with free space, approximate closed-form solutions for the reflection coefficients are obtained. Numerical examples show the validity of the method.

Closed-form solutions for one-dimensional inhomogeneous anisotropic medium in a special case. II. Inverse scattering problem

For pt. I see ibid., vol.45, no.5, p.936, 1997. An analytical method is presented for the inverse scattering problem of a one-dimensional (1-D) inhomogeneous anisotropic medium in a special case. Using the closed-form formulations for the reflection coefficients derived in the first part of this paper, reconstruction formulas are obtained for the wave impedance profiles, permittivity profiles or permeability profiles of the anisotropic medium, all of which are given in closed form. In the meantime, a partial inverse scattering method for the electric parameters at the interface of the medium with free space is also investigated by using a Wedtzel-Kramers-Brillouin (WKB) approximation. Numerical examples show the validity of the methods.

Determination of electrophysical properties of a layered structure with a statistically rough surface via an inversion method.

The existing methods for nonevasive determination of the electrophysical properties of plane-layered media are applicable only to the structures with an even boundary. Meanwhile, the irregular boundaries are often encountered in thin-film research, plasma physics, remote sensing. Here, an approach to reconstruct the permittivity profile of a one-dimensional inhomogeneous medium with a slightly rough surface is proposed. What differentiates it from the existing approaches is that it makes use of the noncoherent scattering data due to random roughness. It is assumed that the unperturbed position of medium's boundary, as well as the correlation function of random roughness, are known a priori. The available data consist of the backscattering cross-section (BCS) measurements at a discrete set of frequencies for an incident linearly polarized time-harmonic plane wave. The profile reconstruction is performed by successive minimization of a squared error of BCS via the Newton-Kantorovich iterative technique. The modeled values of BCS are calculated via the small perturbation method. The numerical experiments show that the information concerning the underlying medium, which is contained in the noncoherent scattering data due to surface roughness, is quite sufficient to permit feasible reconstruction of the medium's electrophysical properties.

Novel applications of an approximate profile inversion for one-dimensional medium

In one-dimensional inverse scattering, the approximate formulas to reconstruct the permittivity profiles are very interesting because of their closed forms. Nevertheless, as the result of Born approximation, they are invalid under the strong-scattering conditions which restrict their applications. In this paper, a novel approximation for inverting the permittivity profile of a one-dimensional inhomogeneous medium is derived by using an equivalent microwave networking technique. Compared with the published formulas of Ladoucer and Jordan (1985), the novel approximation is not only more general in the form, but suitable for both weak-scattering and strong-scattering conditions. Reconstruction examples show that when the relative permittivity reaches 80, the novel formula is effective, which show the validity of this scheme. >

Inverse scattering method for one-dimensional inhomogeneous lossy medium by using a microwave networking technique

The formulation of reflection coefficients from an inhomogeneous lossy medium illuminated by TE and TM waves is approximately derived, in closed form, by using a microwave network method. From the formulation, a novel inverse scattering scheme to reconstruct simultaneously the permittivity and conductivity profiles, is proposed. This scheme is suitable for both continuous and discontinuous profiles, under both the weak scattering and strong scattering conditions. It has also been shown that when the conductivity of the medium equals zero, the reconstructed result of this scheme will reduce to the one formulated by Ladouceur and Jordan (1985). Numerical and closed-form reconstruction examples show the validity of the scheme.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Traffic Flow Problems in One-Dimensional Inhomogeneous Media

Traffic flow problems in one-dimensional inhomogeneous media are studied using cellular automaton models. The effects of (i) a single impurity site, at which a car takes longer time to pass through, in an otherwise uniform medium, (ii) the addition of a finite concentration of overtaking sites that can accomodate two cars at the same time, and (iii) the presence of both an impurity site and overtaking sites are studied. Analytic expressions for the average car velocity and flux in the long time limit as a function of car density are derived.