Planar Stratified Media Research Articles

Infrared modulation via coupling gratings lying on III–V multiple quantum wells

We develop a theory to study the diffraction and absorption properties of a lamellar grating etched on top of a plane stratified medium which is an inorganic multiple quantum well made of isotropic and anisotropic materials. It is shown that the grating characteristics can be optimized in such a way that even when illuminated under normal incidence at 10.6 μm wavelength, the multiple quantum well may absorb 40% of the incident energy. Changing the permittivity tensor of the components of the superlattice by acting on the intraband transitions via an electrical or optical action (Stark effect for example) opens a way to infrared modulation and detection.

Finite difference time-domain analysis of a gaussian wave propagating in planar stratified media

The finite difference time-domain method is applied to analyse a gaussian wave propagating in planar stratified dielectric media. To investigate the modulating effects of geometry to propagation signals, we have studied various configurations of dielectric layers. Numerical results show that the spatial distances between the dielectric layers can significantly affect the reflected and transmitted electromagnetic waves. There exist window frequencies at which an incident field almost completely penetrates a three-layered dielectric system.

On the treatment of the propagation of electromagnetic waves in an isotropic inhomogeneous dielectric medium as waves in a medium with effective anisotropy described by the permeability tensor

We show that plane E and H waves in a smoothly inhomogeneous isotropic plane-stratified medium (plasma) can be considered as normal waves in a uniaxial medium with isotropic permittivity and effective-permeability tensor.

Localization and mode conversion for elastic waves in randomly layered media II

This paper is Part II of a two part work in which we derive localization theory for elastic waves in plane-stratified media, a multimode problem complicated by the interconversion of shear and compressional waves, both in propagation and in backscatter. We consider the low frequency limit, i.e., when the randomness constitutes a microstructure. In Part I, we set up the general suite of problems and derived the probability density and moments for the fraction of reflected energy which remains in the same mode (shear or compressional) as the incident field. Our main mathematical tool was a limit theorem for stochastic differential equations with a small parameter. In this part we use the limit theorem of Part I and the Oseledec theorem, which establishes the existence of the localization length and other structural information, to compute: the localization length and another deterministic length, called the equilibration length, which gives the scale for equilibration of shear and compressional energy in propagation; and the probability density of the ratio of shear to compressional energy in transmission through a large slab. This last quantity is shown to be asymptotically independent of the incident field. We also extend the results to the small fluctuation, rather than the low frequency case.

Localization and mode conversion for elastic waves in randomly layered media I

This paper is Part I of a two-part work in which we derive localization theory for elastic waves in plane-stratified media, a multimode problem complicated by the interconversion of shear and compressional waves, both in propagation and in backscatter. We consider the low frequency limit, i.e., when the randomness constitutes a microstructure. In this part, we set up the general suite of problems and derive the probability density and moments for the fraction of reflected energy which remains in the same mode (shear or compressional) as the incident field. Our main mathematical tool is a limit theorem for stochastic differential equations with a small parameter. In Part II we will use the limit theorem of Part I and the Oseledec Theorem, which establishes the existence of the localization length and other structural information, to compute: the localization length and another deterministic length, called the equilibration length, which gives the scale for equilibration of shear and compressional energy in propagation; and the probability density of the ratio of shear to compressional energy in transmission through a large slab. This last quantity is shown to be asymptotically independent of the incident field. We also extend the results to the small fluctuation, rather than the low frequency case.

A boundary integral method for calculating displacement fields in three-dimensional viscoelastic piecewise-homogeneous media. II. Matrix representation solutions

A method of calculating stationary displacement fields in three-dimensional viscoelastic media, consisting of homogeneous regions with curvilinear separation boundaries, is developed on the basis of the boundary integral method. As a result of using the fundamental solutions corresponding to the sum of converging and diverging waves, the field prolongation through a homogeneous region of the medium reduces to simple multiplication of stress-strain matrices. This makes it possible to construct an effective calculation scheme of the displacement field at any point of the medium, similar to the matrix scheme used in seismology for planar stratified media.

The triple coalescence of coupling points for waves in stratified media

For waves propagating in a plane stratified medium coupling between two characteristic waves occurs near points where the eigenvalues for the two waves are equal, and these are called coupling points. More complicated effects occur where two coupling points coincide. This is called a coalescence of coupling points. It has been previously studied for the case where the same pair of eigenvalues is equal at each of the coalescing coupling points. This paper studies the case where the equal eigenvalues at the two coupling points are not the same pair, but have one eigenvalue in common so that at the coalescence three eigenvalues are equal. This is called a triple coalescence and it can occur for radio waves in the ionosphere. The basic theory is given and tested by some computed solutions. It brings to light some cases where the expected ½π phase advance of a reflected wave is replaced by a ½π phase retardation. A possible generalization of the theory to deal with more complicated coalescences is briefly discussed.

Lloyd's mirage: a variant of Lloyd's mirror

We consider an extension of the classical experiment known as `Lloyd's mirror' in which an interference pattern is formed by the superposition of light coming directly from a point source with that coming from the specularly reflected image of the same source in a mirror. In the present experiment the mirror is replaced by a planar stratified medium, whose refractive index monotonically decreases away from the surface. This produces a mirage-like reflection. The interference is then the superposition of light coming directly from the source with that from its mirage-like image. An approximate treatment of the positions of the resultant interference fringes is given and a simple experimental demonstration is described.

Can a Travelling Electromagnetic Wave be Present in a Layered Medium?

Abstract The problem of the presence of a travelling electromagnetic wave in an inhomogeneous medium is addressed. An argument is presented in favour of the option that the Poynting vector determines whether the wave is travelling or not. A strict harmonic solution to the Maxwell equations in a plane stratified medium is shown not to be a travelling wave. This solution can be decomposed into two travelling fields which separately do not satisfy the Maxwell equations and which are identified as the transmitted and return fields. The direction of the Poynting vector for the transmitted field satisfies the generalized Snell's law. The reflectance of a layer of such a medium is calculated and shown to be zero for some incidence angles. Phases of the two travelling fields are considered and shown to be different from those of geometric optics.

Supernova remnants in plane-stratified media: predictions for H -emitting regions

Numerical models for the time-dependent evolution of adiabatic supernova remnants in plane-stratified media are presented. These models approximate SNR evolution in the density distributions found near to the edges of dense molecular clouds or diffuse stellar wind bubbles. Use of a simple T −1/2 cooling law enables cooling surfaces within the remnants to be identified and their Hα emission estimated. Projection of this emission on to the plane of the sky at a variety of angles of inclination, together with a consideration of the kinematics, allows comparisons with observed remnants to be made. It is found that the phenomenon of fast Hα emission, as observed in the Cygnus Loop and N49 and N63 in the Large Magellanic Cloud, can be easily explained as a consequence of supernova remnant evolution in regions having high density contrasts

The reflection of light by planar stratified media: the groupoid of amplitudes and a phase 'Thomas precession'

The reflection coefficient of light on stratified planar structures can be obtained by postulating the use of a complex generalization of Einstein's addition theorem for parallel velocities. The algebraic properties of the 'composition law of amplitudes' show that the set of all complex amplitudes of the electromagnetic field in heterostructures forms a weakly associative-commutative groupoid. The first concrete application of this abstract concept was found only in 1988 in special relativity. This work exhibits another example in a quite different field of physics. It also puts into evidence that the 'phase rotation' of light in stratified planar structures is to be considered as a 'Thomas rotation'.

Plane electromagnetic wave propagating parallel to the gradient of the refractive index

A known exact solution to Maxwell’s equations in a plane-stratified medium is shown not to be a traveling wave. The solution can be decomposed into two traveling fields that do not satisfy Maxwell’s equations separately. These fields are identified as the transmitted and return fields. The transmitted field is compared with the geometric-optics approximation: The phases of the two fields do not coincide. The reflectance of a layer of such a medium is calculated and shown to be the same as that obtained by the approximation of a graded profile.

A generalized Gel'fand-Levitan-Marchenko integral equation

This study focuses on the Gel'fand-Levitan and Marchenko equations with the view of unifying them via a general integral equation formulation which is pertinent to the electromagnetic inverse scattering problem. The approach used to derive this integral equation is based on a time-domain formulation, as opposed to the traditional spectral-domain approach, which is formulated in the frequency domain. The integral equation allows the inversion of a planar stratified, lossless, and non-magnetic medium excited by a point dipole source that gives rise to either transverse-electric or transverse-magnetic polarized waves.

On the orthogonality of surface wave eigenfunctions in cylindrical coordinates

SUMMARY The orthogonality of the Rayleigh wave eigenfunctions in laterally homogeneous, plane-stratified media is guaranteed by the structure of the ordinary differential equations describing elastic wave motion in cylindrical coordinates. This coupled first-order system is identical to that which characterizes 2-D plane wave propagation in a Cartesian coordinate reference frame. The orthogonality relation in 2-D can also be derived from energy considerations; however, an analogous argument in cylindrical coordinates has not hitherto been made. We derive the orthogonality relations for Rayleigh waves in 3-D from energy considerations and demonstrate that the standard (2-D) expression is, in fact, the generalization of a slightly more specific form. In addition, the cylindrical coordinate formulation permits the derivation of a functional orthogonality relation between Love and Rayleigh waves. The normalization of Love and Rayleigh wave eigenfunctions in cylindrical coordinates is shown to be related to the energy transport of a given outgoing Fourier-Bessel component across any surface which wholly encompasses the z-axis.

Computation of the surface electron-energy-loss spectrum in specular geometry for an arbitrary plane-stratified medium

In recent years, high-resolution electron-energy-loss spectroscopy (EELS) has been successfully applied to the study of vibrational properties of surfaces and interfaces of materials having a layered structure. At the same time the theory of long-wavelength electromagnetic properties of surfaces and interfaces and their role in the inelastic scattering processes of electrons at the surface has also developed to a point where detailed quantitative comparisons between experimental and theoretical vibrational spectra can now be made routinely. The computer program described here allows to calculate the EELS spectrum of a large variety of target materials, going from a homogeneous thick substrate to periodic superlattices or any arbitrary stacking of layers.

An induction gradiometer for conductivity contrast detection: Theoretical considerations

We present a full wave theoretical model for an induction gradiometric sensor employed for detection of conductivity contrast. The sensor is composed of a wire antenna radiating from a rectangular waveguide, with receivers in the waveguide positioned symmetrically on either side of the transmitter. We first solve the canonical problem of an open‐ended rectangular waveguide in a planar stratified medium and then apply reciprocity and the Born approximation to deduce the response of the sensor in the presence of an arbitrary external medium. We compare our results against experimentally measured data and find excellent agreement. We present a discussion of these results together with an analysis of the numerical considerations involved in the computation.

Marching and spatial-spectral filtering of localized phase-space representations of wave fields in layered media

Spatial and spectral (directional) information of the wave data are of equal importance in the analysis and synthesis of wave phenomena. Their simultaneous collaboration in a phase-space format may be achieved by incorporating windowing techniques into (conventional) spectral integrals. The result is a local-spectral representation of the field that is concentrated inherently around physically meaningful regions in phase space. This representation is most naturally suited to perform simultaneous spatial and spectral processing of wave data and therefore may serve as a powerful tool in a variety of applications such as remote sensing, imaging, field reconstruction, and more. In this work a method to propagate the local spectra of wave fields through plane stratified media is developed and discussed. An adaptive marching algorithm that permits control of the localization process will be presented. The ability of the new scheme to perform simultaneous spatial and spectral data filtering of wave phenomena, and its resolving power in phase space, will be discussed and demonstrated by means of numerical examples.

Electromagnetic scattering and radiation by surfaces of arbitrary shape in layered media. I. Theory

An accurate and general procedure for the analysis of electromagnetic radiation and scattering by perfectly conducting objects of arbitrary shape embedded in a medium consisting of an arbitrary number of planar dielectric layers is developed. The key step in this procedure is a formulation of the so-called mixed-potential electric field integral equation (MPIE) that is amenable to an existing advanced solution technique developed for objects in free space and that employs the method of moments in conjunction with a triangular-patch model of the arbitrary surface. Hence, the goal is to immediately increase analysis capabilities in electromagnetics, yet remain compatible with the large existing base of knowledge concerning the solution of surface integral equations. Three alternative forms of the MPIE in plane-stratified media are developed, and their properties are discussed. One of the developed MPIEs is used to analyze scatterers and antennas of arbitrary shape that penetrate the interface between contiguous dielectric half-spaces. >

An interactive demonstration of electromagnetic wave propagation using time-domain finite differences

The finite difference time-domain (FDTD) method is one of the most widely used computational methods in electromagnetics. Using FDTF, Maxwell's equations are solved directly in the time domain via finite differences and time stepping. the basic approach is relatively easy to understand and is an alternative to the more usual frequency-domain approaches. In order to take advantage of this, an interactive personal computer program based on FDTD has been developed. The program directly solves Maxwell's equation via finite differences. The solution is for one dimension, corresponding to normal incidence propagation through a planar stratified medium. The program displays an electromagnetic pulse as it propagates through the medium. Since Maxwell's equations are solved directly, the reflected and transmitted pulse amplitudes demonstrate how the reflection and transmission coefficients determine reflected and transmitted wave amplitudes. Since lossy material layers can be included, frequency dispersion can be demonstrated. >

Rayleigh-Love wave coupling in an azimuthally anisotropic medium.

We present the generalized representation of the equations of motion for an elastic solid in a concise vector form with arbitrary orthogonal curvilinear coordinates and no assumption on symmetry of elastic moduli. For the particular case of the Cartesian coordinates, this representation leads to the generalized y-method for eigenvalues of surface wave dispersion in an anisotropic plane-stratified medium. The generalized y-method is the integration method to find eigenvalues of surface wave dispersion by iterating numerical depth-integration of first-order ordinary differential equations that are derived from the generalized representation of the equations of motion and Hookean law. Based on the generalized y-method, we present some numerical results for dispersion curves and azimuthal variations of surface wave velocities to fully display Rayleigh-Love wave coupling in Kawasaki's (1986) azimuthally anisotropic model for the upper mantle beneath the Pacific ocean. When Rayleigh-Love wave coupling takes place in a particular period range between a pair of nearby modes, surface waves display the following distinct singularities: (1) a difference of phase velocities of the pair of the modes is smaller than about 0.5 km/s, (2) a pair of dispersion curves of group velocities cross each other, (3) polarizations of particle motion directions are twisted along the pair of dispersion curves from Rayleigh- to Love-types and from Love- to Rayleigh-types. These sigularities are dependent on the relative depth- and azimuthal-distribution of the upper mantle anisotropy. When Rayleigh-Love wave coupling does not take place, the first-order perturbation theory works well.