Planar Layered Media Research Articles

Gaussian beams in inhomogeneous media: A review

The paper outlines the most important results of the paraxial complex geometrical optics (CGO) in respect to Gaussian beams diffraction in the smooth inhomogeneous media and discusses interrelations between CGO and other asymptotic methods, which reduce the problem of Gaussian beam diffraction to the solution of ordinary differential equations, namely: (i) Babich’s method, which deals with the abridged parabolic equation and describes diffraction of the Gaussian beams; (ii) complex form of the dynamic ray tracing method, which generalizes paraxial ray approximation on Gaussian beams and (iii) paraxial WKB approximation by Pereverzev, which gives the results, quite close to those of Babich’s method. For Gaussian beams all the methods under consideration lead to the similar ordinary differential equations, which are complex-valued nonlinear Riccati equation and related system of complex-valued linear equations of paraxial ray approximation. It is pointed out that Babich’s method provides diffraction substantiation both for the paraxial CGO and for complex-valued dynamic ray tracing method. It is emphasized also that the latter two methods are conceptually equivalent to each other, operate with the equivalent equations and in fact are twins, though they differ by names. The paper illustrates abilities of the paraxial CGO method by two available analytical solutions: Gaussian beam diffraction in the homogeneous and in the lens-like media, and by the numerical example: Gaussian beam reflection from a plane-layered medium.

Synthesis and Analysis of Reflective and Mixed Active Multifunctional Plane Layered Media

Synthesis and Analysis of Reflective and Mixed Active Multifunctional Plane Layered Media

An Improved Weak-Form BCGS-FFT Combined With DCIM for Analyzing Electromagnetic Scattering by 3-D Objects in Planarly Layered Media

Fast algorithms for electrically large objects buried in layered media are mainly hindered by two time-consuming processes. One is the table filling of Green's function, and the other is the solving of the impedance matrix equation. For the first, to accelerate the evaluation of the time-consuming Sommerfeld integral in the dyadic Green's function (DGF), the discrete complex image method (DCIM) is introduced to get a closed-form DGF. To further accelerate the calculation of DGF for the volume electric field integral equation (EFIE), DGF is split before applying DCIM. For the second, the iterative solver stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) is combined with DCIM for solving the matrix equations. Meanwhile, the closed-form DGF enables the spherical-mean Green's function, which eliminates the singularity of Green's function. Numerical results show that the weaker singularity results in a faster and steadier convergence rate for iterative solvers

Waveguide probe structures for a multilayer magnetodielectric: An inverse problem

Direct and inverse problems are considered for diffraction by an open end of a rectangular waveguide (RW) with a flange that adjoins a piecewise inhomogeneous planar layered lossy medium. Also considered are similar diffraction problems for a junction of an RW and a rectangular resonator filled with a multilayer medium and a junction of two RWs, one of which contains a multilayer plate. Such open and shielded waveguide probe structures (WPSs) are used for determination and nondestructive testing of parameters of multilayer samples. The direct problem is formulated on the basis of admittance and impedance algorithms with consideration for losses existing in the medium, flange, and screens. In this case, the approximation of the given aperture field allows obtainment of explicit solutions for open and shielded WPSs in the form of integrals. Solution of the inverse problem that lies in determining thicknesses, permittivities, and permeabilities of the layers from measured values of the magnitude of the reflection coefficient is obtained by minimizing the corresponding least-squares error and by constructing artificial neural networks. In order to increase the accuracy, it is proposed to use a two-port of errors and perform measurements for several positions of the sample with respect to the flange and different impedance conditions behind the sample.

A coaxial probe for the control of parameters of a multilayer dielectric: Direct and inverse problems

Diffraction of an electromagnetic wave by an open end of a coaxial line with an infinite flange adjoining a piecewise-uniform plane-layered lossy medium is considered. The direct problem is solved on the basis of admittance and impedance algorithms taking into account losses in the flange and the medium. For an infinite conductivity of the flange, a stationary functional for the input admittance is used to find an explicit solution in the form of an integral; numerical results are obtained. The inverse problem consisting in determination of the layer parameters (thickness, permittivity, and permeability) from known (experimental) values of the magnitude of the reflection coefficient at specified frequencies via minimization of the corresponding residual of the least-squares method is considered. It is proposed to increase the amount of experimental data through performance of measurements for several positions of the sample relative to the flange and for different impedance conditions behind the sample. In order to solve the inverse problem, a perceptron neural network is constructed. This approach reduces the solution time by several orders of magnitude.

A radio-wave method for detecting and identifying permeable inclusions in stratified inhomogeneous media

The problems associated with the efficiency of practical application of a numerical algorithm for detecting, locating, and identifying inhomogeneous dielectric inclusions in plane-layered media are considered. The results of computational experiments on the determination of both physical and geometric parameters of inhomogeneities and the effect of the measurement error on the quality of the inspection procedure are presented. This method can be used in the nondestructive testing of the quality of industrial dielectric articles.

A Combined Method for Calculating the Field of a Point Source in a Waveguide (Plane-Layered Media)

The two-dimensional problem of propagating the waves generated by a point source, in a depth plane-layered waveguide is investigated. The Dirichlet condition is given on the boundary of the waveguide and also two consistency conditions are defined on the interface between the fluid and the bottom. The velocity of acoustic waves is assumed to be dependent only on the depth of the waveguide. With the help of cumbersome transformations, the solution is represented as a sum of geometric-optical waves, normal waves, and a remainder. Sufficient conditions on the total amount of detached normal and geometric-optical waves are obtained. The remainder is expressed by a simple formula. Bivliography: 22 titles.

Mode conversion in plasmas with two-dimensional inhomogeneities

Most of the mode conversion theories considered so far assume only a plane-layered medium, i.e. a medium where the parameters depend on one spatial coordinate. We generalize the mode-conversion method of Cairns and Lashmore-Davies to plasmas with two-dimensional inhomogeneities. In the method presented here, the frequencies ω 1 and ω 2 of the uncoupled modes belonging to two different dispersion equations are considered as functions of the space variable r and the wave vector k and are coupled together via a small quantity η. We calculate the energy transmission and conversion coefficients analytically by solving two coupled wave amplitude equations in the electron cyclotron range of frequencies. The results are applicable to electron Bernstein wave heating of plasmas with two-dimensional inhomogeneity, e.g. spherical tokamaks.

Near-field microwave diagnostics for medical applications

We study both theoretically and experimentally the possibilities of detecting malignant tumor inside a human body using near-field microwave probing. Our theoretical analysis is based on the developed theory of near-field diagnostics of plane-layered media. We verified this theory experimentally under controlled conditions of water-medium probing. Model calculations of recorded tumor contrast as a function of the size and depth of the tumor are performed. Two-dimensional images of an object imitating the tumor are obtained for different depths of object immersion in water.

Characterization of Photonic Crystals Based on Opal–Semiconductor Composites by Bragg Reflection Spectroscopy

The Bragg reflection spectra of photonic crystals based on the opal-semiconductor (GaP, GaN) composites are studied. An approach is developed which provides a means for characterizing the opal-like structures. The approach is based on examination of the Bragg reflection spectral features and of the conditions for multiwave Bragg diffraction. The shape of the Bragg spectra is analyzed in the context of the approximation of a periodic planar-layered medium taking into account sintering and axial compression of the a-SiO2 spheres that constitute the three-dimensional opal lattice. The diameters and the coefficient of compression of the spheres are determined with regard to the spectroscopic structural invariant suggested in the study for defining the conditions for observation of multiwave Bragg diffraction at the system of the {111} planes.

On the source type of integral equations for scattering by an object in a layered medium

This paper is concerned with the source type of integral equation to represent the scattering by an inhomogeneous object in a homogeneous layer of a planar layered medium in the frequency domain. By decomposing the scattered field into a particular and a general constituent the structure of the integral operator of the integral equation is identified. The particular constituent represents the scattered field inside the layer, that embodies the contrasting object, due to the presence of virtual contrast sources inside the inhomogeneous object, while the general constituent represents the interaction with the other layers due to the presence of source distributions on each side of the layer that embodies the contrasting object. The part due to the particular constituent has a convolution structure in all spatial directions. The part due to the general constituent consists of two terms, one has again a convolution structure with respect to all spatial coordinates, while the other has a convolution structure with respect to the horizontal coordinates and a correlation structure in the vertical coordinates. These properties facilitate a fast and efficient computation of the integral operator with the help of fast Fourier transforms. For the full 3D electromagnetic case, where the inhomogeneous object exhibits both an electric and magnetic contrast, the source type of integral equations are derived and their structure is discussed.

Planarly layered diffraction tomography with accurate Green function

Diffraction tomography (DT) imaging techniques require knowledge of the background Green function. Due to its simplicity, it is standard practice to use a homogeneous medium Green function. We have developed a model of a planarly layered Green function that can be used in DT imaging of planarly layered media where the layer acoustical properties and dimensions are known. We present the theory and applications. [Work performed under the auspices of the U.S. Department of Energy by University of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.]

Postseismic viscoelastic‐gravitational half space computations: Problems and solutions

We consider the problem of surface deformation arising from a fault in a semi‐infinite, elastic‐gravitational, and/or viscoelastic‐gravitational, plane‐layered medium, subject to an externally imposed gravitational acceleration g. Rundle [1981, 1982] presented a calculation in which self‐gravitation, represented by terms proportional to G are neglected, and the externally imposed acceleration due to gravity, g, is considered constant in the medium. Because of the recent strong interest in computations of this type, we examine the assumptions involved in these computations. We show that these assumptions are not likely to have serious consequences in the relatively near‐field viscoelastic displacements, where the earth's curvature is neglected. We also show that the approximation described by Rundle [1981, 1982], which was technically not regular as z → ∞, can easily be regularized using a new approach without appreciable change in the resulting displacement field.

Dyadic Eigenfunctions and Natural Modes for Hybrid Waves in Planar Media

A new formulation for studying electromagnetic wave propagation in an open, planarly layered medium is presented based on eigenfunctions of the Hertzian potential dyadic Green's function operator. Due to the complicated coupling of scalar components of potential at material interfaces, elevation of the usual vector eigenfunction problem to dyadic level is found to lead to a convenient, compact representation of wave propagation phenomena. Although we study the source-free problem here, the three columns of the eigenfunction dyadics represent (up to an excitation-dependent amplitude) the vector fields excited by a given three-dimensional source. The general theory of dyadic eigenfunctions is presented, including orthogonality and the possibility of associated functions (root functions) at modal degeneracies, and an example of propagation in a grounded dielectric slab environment is provided.

Approximations of reflection travel times with high accuracy at all offsets

Available high-order travel time equations allow us to flatten seismic gathers in a larger range of offsets than the normal moveout hyperbola, but they require the estimation of at least one extra parameter and they are still insufficiently accurate in certain situations, e.g. when strong ray bending occurs. To approximate reflection travel times very accurately with as few parameters as possible, I have derived new time-offset series, following a simple model-based approach: using information on velocities and on the recording configuration available for a given survey, I build a set of reference velocity models and calculate the associated travel time-offset functions. These functions define the vectorial space of all realistic time-offset curves. A set of basis functions for this vectorial space can be obtained by singular value decomposition. The new travel time series basically consist of weighted sums of these basis functions. A simple test in a plane-layered medium with strong heterogeneity and large aperture shows that the new series may be more efficient than other types of high-order travel time equations: with less coefficients, they provide more accurate approximations of the exact travel time curve (also at large offsets), and more reliable estimates of parameters of the velocity model, such as the RMS velocity and the quartic coefficient a4 of the Taner and Koehler series. On a practical level, the estimation of coefficients is simplified because the bases of functions are othogonal. The proposed procedure can be applied to many types of problems, e.g. to handle non-hyperbolic travel times resulting from anisotropy, wave conversions, high heterogeneity, wide apertures, etc. It should be useful for optimizing velocity analysis and different processing techniques that use travel time approximations, e.g. moveout correction, multiple attenuation and time migration.

Evaluation of layered media Green's functions via rational function fitting

An efficient rational function fitting methodology, called VECTFIT, is utilized toward the closed-form evaluation of the Sommerfeld integrals associated with electromagnetic Green's functions in planar layered media. VECTFIT approximates the component of the spectrum of the Green's function that remains after the extraction of the primary source contribution and the quasistatic part with a rational function, thus enabling a robust and expedient closed-form evaluation of the Sommerfeld integral for electromagnetic potentials and associated field quantities.

Synthesis and Analysis of Interfacing-Filtering Devices on Distributed Elements with the Use of Their Resistance Matrix

An algorithm for synthesis of interfacing-filtering devices by the criterion of preset factors of the standing wave in the pass band and blackout band was developed. By means of circuit simulation, the frequency characteristics of an interfacing-filtering device in the form of a plane layered medium synthesized under the above criterion were studied.

A fast volume integral equation solver for electromagnetic scattering from large inhomogeneous objects in planarly layered media

A newly developed iterative method, the stabilized biconjugate gradient fast Fourier transform (BCGS-FFT) method is applied to simulate electromagnetic scattering from large inhomogeneous objects embedded in a planarly layered medium. In this fast solver, the weak-form formulation is applied to obtain a less singular discretization of the volume electric field integral equation. Several techniques are utilized to speed up the dyadic function evaluation. To accelerate the operation of the dyadic function on an induced current (i.e., the Green's operation), the function is split into convolutional and correlational components so that FFT can be applied. The CPU time and memory cost of this BCGS-FFT method is O(NlogN) and O(N), respectively, where N is the number of unknowns, significantly more efficient than the method of moments (MoM). As a result, this method is capable of solving large-scale electromagnetic scattering problems in a planarly layered background. A large-scale scattering problem in a layered medium with more than three million unknowns has been solved on a Sun Ultra 60 workstation with 1.2 GBytes memory.

Leaky-wave analysis of transient fields due to sources in planarly layered media

The transient field due to canonical sources in planarly layered media is obtained using a leaky-wave analysis. By proper choice of integration paths in both the complex frequency and complex wavenumber planes, transient fields are obtained exactly as a temporal inversion integral over a discrete sum of residues. The residues include both proper and improper surface-wave modes, analytically continued into the complex frequency plane. The method is applicable for all times of interest, although for certain source-receiver locations an early-time period is identified which encompasses the specular reflection from the nearest interface, and during which time the residue series requires special treatment. The presented analysis leads to a computationally simple and efficient method for obtaining transient fields due to sources in layered media. Results are shown for the transient potential due to line and point sources over a grounded dielectric slab, although the technique is applicable to multiple planar layers.

On the modeling of 3-D inclusions in conductive media using extended Born models in the diffusive regime

Diffusive fields in 3-D configurations, e.g., as those faced when searching for inclusions in conductive planar layered media (a metal slab, a half-space subsoil) at low frequencies, can be modeled via Green's integral formulations that require Green dyads. Such formulations make good use of extended Born approximations in order to express the secondary sources existing within the probed inclusion from the primary fields at same location. Here it is sketched how several such approximations perform in cases of interest by comparison with full solutions and benchmark measurements. The purpose of this contribution is the analysis of the electromagnetic behavior of 3-D bounded inclusions affecting a conductive, non-magnetic and isotropic planar-layered metal probed by a low- frequency current source operated nearby in air with an application to eddy current Non-destructive Evaluation. A comparison between measured data and data obtained from exact and approximated models will be discussed. Among all methods available to model eddy currents in a damaged conductive structure, an integral formulation, resulting from the application of the Green's theorem onto the equations governing the fields propagation, has been chosen. This formulation requires the expressions of the Green dyads for layered structures which are known in closed-form in the spectral domain. Two integral equations are solved: the coupling equation which links the total field in the inclusion to itself, the incident field and the contrast through an appropriate Green dyad, and the observation (or data) equation which links the scattered field on the receivers to the total field in the inclusion and the contrast through another Green dyad. The integral equations are discretized using a Method of Moments and the discrete counterpart of the coupling equation is solved with a Conjugate-Gradient-FFT-based method taking into account its convolutional structure. This is very demanding in terms of computational time and partially-linearized approximations pioneered in (1) and investigated also in (2,3) are introduced and their effectiveness is discussed (one should also refer to earlier works (4)).