Iterative determination of permittivity and conductivity profiles of a dielectric slab in the time domain

A numerical method is presented to compute one unknown constitutive parameter of an inhomogeoeous lossy dielectric slab from the reflected field in the time domain. The method is based upon a space-time discretization of the integral equation for the reflected field. In the inversion, especially those space-time points where the numerical computation of the electric-field strength in the slab is most accurate are taken into account. This is achieved by computing the unknown parameter iteratively. Alternately solving equations for an approximate direct-scattering problem and an approximate inverse-scattering problem yields successive approximations for the electric field in the slab and the unknown constitutive coefficient. Both problems lead to an infinite system of linear equations from which a finite subsystem is selected. General criteria for this selection are presented. Various profiles have been reconstructed numerically from the reflected field due to a sine-squared incident pulse.     

Iterative approach to the frequency-domain solution of the inverse-scattering problem for an inhomogeneous lossless dielectric slab

A previously developed time-domain iterative inverse-scattering method is generalized to the problem of reconstructing the susceptibility of an inhomogeneous lossless dielectric slab from frequency-domain reflection data. In subsequent iteration steps, approximations for the electric field inside the slab and the unknown susceptibility profile are obtained by alternately solving an approximate direct-scattering problem and an approximate inverse-scattering problem. Numerical results are presented and discussed.     

Linear and nonlinear EMP diffusion through a ferromagnetic conducting slab

The penetration of an EMP field, such as a plane-wave incident upon and a surge-arrestor current terminated at a shielding plate, through a ferromagnetic conducting slab made of iron or steel is investigated. The diffusion of the electromagnetic field in the highly conducting slab is complicated by the presence of the nonlinear saturation of the ferromagnetic permeabilitymu, due to the large amplitude of the incident EMP. Such a saturation, compared to the no-saturation constantmucase, makes the field diffuse faster in the slab and admits a stronger penetration field inside the shielding if the slab is thin and the pulse duration is long, as expected. On the contrary, if the slab is thick and the puslewidth is short, as the practical cases are, the saturation reduces the penetrated field but maintains its time shape. In this report, first we solve analytically the one-dimension plane-wave incidence problem for a slab with a constantmu. Then we use the results to partly predict and to interpret the numerical values obtained by using a finite difference code for the case of a nonlinearmu. These comparisons reveal excellent agreement. Second, we solve the constantmu, cylindrical-wave incidence problem by an approximate but extremely useful analysis, with its validity parameters clearly established. The results, shown to bear a simple relationship to those for the one-dimension problem, enable one to make use of the one-dimension results and predict easily the penetrated fields caused by a cylindrical incident current. Then for the nonlinear-mucase, we justify and extend that relationship by which the behaviors of the cylindrically diffused fields are obtained from those numerical results of the one-dimensional problem.     

Scattering characteristics of a rectangular groove in a reactivesurface

The diffraction of a plane electromagnetic wave by a rectangular groove in a reactive plane is studied rigorously through the Fourier transform technique. The corresponding boundary value problem is formulated into a modified Wiener-Hopf equation of the third kind whose approximate solution involves two sets of infinite number of unknown constants satisfying two infinite systems of linear algebraic equations. Numerical solutions of these systems are obtained for various values of the surface reactance and the sizes of the groove through which the effects of these parameters on the diffraction phenomenon are studied     

Fringe integral equation method for a truncated grounded dielectricslab

The problem of scattering by a semi-infinite grounded dielectric slab illuminated by an arbitrary incident TM_z polarized electric field is studied by solving a new set of “fringe” integral equations (F-IEs), whose functional unknowns are physically associated to the wave diffraction processes occurring at the truncation. The F-IEs are obtained by subtracting from the surface/surface integral equations pertinent to the truncated slab, an auxiliary set of equations obtained for the canonical problem of an infinite grounded slab illuminated by the same source. The F-IEs are solved by the method of moments by using a set of subdomain basis functions close to the truncation and semi-infinite domain basis functions far from it. These latter functions are properly shaped to reproduce the asymptotic behavior of the diffracted waves, which is obtained by physical inspection. The present solution is applied to the case of an electric line source located at the air-dielectric interface of the slab. Numerical results are compared with those calculated by a physical optics approach and by an alternative solution, in which the integral equation is constructed from the field continuity through an aperture orthogonal to the slab. The applications of the solution to an array of line currents are also presented and discussed     

Three-Dimensional Through-Wall Imaging Under Ambiguous Wall Parameters

A through-wall imaging problem for a 3-D geometry is considered. Scatterers are located beyond a wall represented by a dielectric slab whose features are unknown or known with some degree of uncertainty. A two-step imaging procedure is presented. First, the thickness and the dielectric permittivity of the wall are estimated by a simple procedure which takes into account that actual measurements concern the total scattered field (i.e., the field reflected by the wall plus the one scattered by the obscured scatterers). Then, the problem is cast as a linear inverse scattering problem and solved by means of a truncated-singular value decomposition algorithm. In particular, a 2-D sliced approach is employed to obtain the 3-D scene. Numerical examples are shown to assess the effectiveness of the reconstruction procedure.     

Diffraction of guided waves by the end face of a dielectric slab waveguide terminated with a finite conductive strip

The diffraction of guided waves by the end face of a dielectric slab waveguide short circuited with a finite conductive strip is analyzed. An integral equation technique is employed to formulate the corresponding boundary problem. The unknown term in this integral equations is the electric field E(x) on the terminal plane of the waveguide. The homogeneous term is determined from the incident guided wave. A method of moments technique is employed to compute approximately the electric field E(x) by using Laguerre functions as describing and testing functions. The reflection coefficients of the guided waves are computed by using the approximate expression of the E(x) field. Numerical results are given for several guide and conductor plate dimensions.     

Realization of Impedance Boundary

Impedance boundary is generally considered as an approximate model for material interfaces. Considering an electromagnetic field in a certain wave-guiding anisotropic material it is shown that a slab of such a material backed by a perfect electric conductor (PEC) plane can be exactly represented by impedance-boundary conditions. As a result of the analysis, a novel explicit relation is derived between the surface admittance dyadic of the impedance boundary and the material and geometric parameters of the anisotropic slab which can be used to realize a given admittance dyadic. The relation is verified with results known for the perfect electromagnetic (PEMC) boundary and bi-axial material, considered as special cases of the theory     

EM plane wave diffraction by a a planar junction of two thinmaterial half-planes-oblique incidence

A uniform geometrical theory of diffraction (UTD) solution is developed for analyzing the high-frequency diffraction of an obliquely incident plane wave by a two-part, thin, planar, transparent material slab. The solution is obtained by appropriately combining two solutions for the two related configurations involving perfectly conducting electric and magnetic ground plane bisections of the original slab. The analysis is based on the Wiener-Hopf technique, and each of the grounded material half-planes is assumed to be electrically thin so that it can be modeled by a generalized impedance boundary condition of 0(t), where t is the corresponding slab thickness. To solve the boundary value problem completely, an additional condition related to the field behavior at the junction of the two material half-planes is imposed. This junction condition is determined by matching an approximate quasi-static solution, which is developed in the proximity of the discontinuity, with the corresponding external Wiener-Hopf solution in the common region of overlap. The solution thus obtained automatically satisfies reciprocity     

A New Stacked Two-Dimensional Spectral Iterative Technique (SIT) for Analyzing Microwave Power Deposition in Biological Media

Conventional numerical methods for analyzing power deposition in biological media have been restricted to bodies which are relatively small electrically. A new, stacked-two-dimensional-spectral-iterative-technique (SIT), presented below, does not involve the generation and inversion of a matrix and is capable of analyzing larger bodies. It is based on modeling the body by a set of planar parallel slabs and utilizing the simple (convolution-type) relationship between a current distribution on any slab and the field due to this current. This invertible relationship is conveniently formulated in the transform domain in a strictly algebraic fashion. The interactions between the various slabs are also simple and algebraic in the spectral domain. The solution is generated in an iterative manner by applying these relationships sequentially over the slabs until convergence is achieved. Discussion on convergence and numerical examples are given.     

Surface waves on a uniaxial plasma slab--Their group velocity and power flow

The relation between group velocity and the velocity of energy transport for surface waves in plane-stratified, anisotropic, dispersive media, which was derived in [1], is verified by direct calculation for the case of surface waves on a uniaxial, cold plasma slab located in free space. A superimposed dc magnetic field of infinite strength and parallel to the interfaces generates the uniaxial anisotropy in the slab. Surface waves having an arbitrary direction of propagation with respect to the dc magnetic field are considered. A useful graphical presentation of the dispersion relation is given, from which the direction of propagation of "surface wave rays" is directly obtained.     

TE-wave scattering by a dielectric cylinder of arbitrary cross-section shape

The theory and equations are developed for the scattering pattern of a dielectric cylinder of infinite length and arbitrary cross-section shape. The harmonic incident wave is assumed to have its electric vector perpendicular to the axis of the cylinder, and the fields are assumed to have no variations along this axis. Although some investigators have approximated the field within the dielectric body by the incident field, a more accurate solution is obtained here by treating the field as an unknown function which is determined by solving a system of linear equations. Scattering patterns obtained by this method are presented for dielectric shells of circular and semicircular cross section, and for a thin plane dielectric slab of finite width. The results for the circular shell agree accurately with the exact classical solution. The effects of surface-wave excitation and mutual interaction among the various portions of the shell are included automatically in this solution.     

Optimization techniques and inverse problems: Reconstruction of conductivity profiles in the time domain

The iterative probing of an inhomogeneous lossy dielectric slab, whose conductivity is unknown, is discussed. The probing is done in the time domain from the measurements of the field on the interface when this slab is illuminated by a known field. An exact integral formulation is used. Minimization of a cost function, characteristic of the discrepancy between the measured field and the field which would be scattered by a known slab, is specified by the optimization theory. The notion of the adjoint state of the field is introduced. The influence of some parameters of this minimization is studied. Great importance is given to the sensitivity of the probing as function of the amplitude of errors in the data. Such an iterative probing appears fast, accurate, and efficient, even in the case of large errors.     

Asymptotic solutions for the scattered field of plane wave by a cylindrical obstacle buried in a dielectric half-space

Two-dimensional scattering of a plane wave by an embedded conducting strip is formulated rigorously using the concept of Kobayashi potential, in which potential or wave function is expressed in terms of infinite integrals including the Bessel function in the integrand. By imposition of the required boundary conditions at the interface of the dielectric half-space and on the strip, the problem is reduced to a dual integral equation (DIE). Using the discontinuous properties of Weber-Schafheitlin's integrals, DIE is transformed into a matrix equation with infinite unknowns whose elements are expressed by infinite integrals. Asymptotic solutions for the matrix elements are derived when the separation between the interface of the different media and the obstacle is large compared to the wavelength. Using these results, the expression for the scattered field is derived in a general form which can be applied to an arbitrary cylindrical obstacle. Some numerical results are given for conducting strip and circular cylinder to see the effect of inhomogeneity on the surrounding medium, size of the obstacle, and the angle of incidence on the scattered field.     

Variational analysis of the dielectric rib waveguide using theconcept of `transition function' and including edge singularities

The authors focus on the pure LSE/LSM cases. They derive a highly accurate transverse resonance diffraction variational solution of the problem, of order 1 (a scalar dispersion equation), by assuming at the transverse step discontinuity a single function trial field which incorporates the physical properties of the solution. This is, in fact, the surface wave mode of a slab waveguide of a height intermediate between that of the rib and that of the cladding slab, including dielectric edge singularities in the LSM case. The height of the intermediate guide is obtained by optimizing the overlapping integral with the slab mode in the rib and in the cladding. This criterion turns out to be equivalent to choosing an intermediate guide whose effective dielectric constant (EDC) is the geometric mean of those of the rib and cladding. Numerical results are in excellent agreement with those obtained by finite difference, even at cutoff, where the EDC fails and most methods tend to overestimate the value of β     

Analysis of the distorted Born approximation for subsurface reconstruction: truncation and uncertainties effects

The problem of determining the dielectric permittivity profile of buried objects starting from the knowledge of the scattered field is considered in the two-dimensional geometry when incomplete near-zone data are collected at a single frequency under a multiview/multistatic measurement configuration. In particular, attention is paid to the practical issues of the truncated observation domain and the presence of uncertainties on data. The problem is tackled with reference to the scalar polarization by linearization of the mathematical relationship between the unknown dielectric permittivity profile and the scattered field. A homogeneous, possibly lossy, half-space geometry for the subsurface modeling is adopted, thus leading to the so-called distorted Born approximation (DBA). A thorough investigation of the class of unknown functions that can be reliably retrieved is performed by dealing with singular value decomposition of the relevant linear operator. It results that even if sources and receivers are located at the interface, a very restricted set of profile variations can be reconstructed by a stable inversion algorithm. In particular, reduced vertical features of the buried objects with respect to the horizontal ones can be reconstructed under DBA. Moreover, the truncation of the observation domain further restricts this set, affecting mainly the vertical resolution. Numerical results confirming the validity of the analysis are also provided.     

Analysis of guided modes in multilayer/multiconductor structures bythe “boundary integral-resonant mode expansion method”

This paper describes a new method for the analysis of modes propagating in shielded waveguides consisting of many dielectric and perfectly conducting layers of different widths. The eigenvalue problem inherent to the mode determination is formulated subdividing the structure into layered elementary wave-guides EWGs and matching the fields generated in adjacent EWGs by unknown equivalent sources placed at the interfaces. The special representation of the EWG field, consisting of boundary integrals and mode expansions, leads to a linear matrix eigenvalue problem involving a limited number of variables. Thanks to this peculiarity the method permits one to determine many modes in short computing times. The method was implemented in a flexible and fully automatic computer code, whose reliability and efficiency has been confirmed by many tests     

Efficient low-frequency inversion of 3-D buried objects with large contrasts

An efficient inversion method is proposed using Cui et al.'s high-order extended Born approximations to reconstruct the conductivity object function of three-dimensional dielectric objects buried in a lossy Earth. High-order solutions of the object function are obtained, which have closed-form relations to the linear inverse-scattering solution. Because such relations can be evaluated quickly using the fast Fourier transform, the high-order solutions have similar simplicity as the linear inversion. When the contrasts of buried objects are large, the high-order solutions are much more accurate due to the approximate consideration of multiple-scattering effects within the objects. Hence, good-resolution images can be obtained for large-contrast objects using the new method by only solving a linear inverse problem. Numerical experiments have shown the validity and efficiency of the proposed method.     

Diffraction from an abruptly terminated dielectric slab waveguide in the presence of a cylindrical scatterer

The efficiency of using an adaptor lens in front of an abruptly terminated symmetric slab optical waveguide is examined analytically in this paper. The adaptor lens is assumed to be of cylindrical shape with a constant refractive index. The coupling of the modes propagating inside the slab guide into radiation waves in the presence of the lens is treated by integral equation methods. In the first step an integral equation is derived for the Green's functionG(r/r')of an abruptly terminated slab waveguide. The integral equation is solved approximately by an iterative procedure giving accurate results when the difference of the refractive indices between the slab waveguide and substrate-cover regions is small. The Green's function is then used to formulate another integral equation for the unknown field inside the adaptor lens. The latter integral equation for the cylindrical lens cross section area is solved by adopting a cylindrical partial wave expansion for the unknown interior field. After determining this field, the reflection-coupling coefficients for the guided modes propagating in the opposite direction of the incident wave, are computed by using the Green's functionG(r/r'). Radiation patterns are also derived in the far field region for an incident slab guided mode. Numerical results are computed and presented for several guide dimensions, lens radii, and refractive indices.     

Electromagnetic Prospection via Homogeneous and Inhomogeneous Plane Waves: The Case of an Embedded Slab

This paper deals with the linear approximation of the inverse scattering problem for a dielectric slab embedded into a homogeneous half-space. After a proper normalization of the scattered field, the problem at hand is recognized to be related to a Fourier transform inversion with limited data. This permits to resort to the well-known results available in signal processing in order to discuss about the class ofthe retrievable unknowns and the resolution limits achievable by inversion. In particular, the role of both homogeneous and inhomogeneous incident plane waves in carrying out information about the profile under test is investigated and the role played by the dielectric permittivity of the host medium in determining the information content of the scattered field is pointed out.