Reflection Kernel Research Articles

An Optimization Approach To the Design of Weakly Reflecting Slabs for Obliquely Incident Transient Electromagnetic Waves

An optimization approach is used in the design of weakly reflecting plane-stratified slabs where transient plane waves impinges obliquely. The slabs are dispersive with a spatially varying impedance and conductivity. In the optimization process, the direct problem is solved iteratively by a time-domain Green functions technique which deals with scattering kernels. Given the prescribed reflection and transmission kernels, an objective functional is minimized for a time larger than one roundtrip and for given incident directions. By introducing dual functions, the exact and explicit expressions for the gradients are derived for the TE and TM modes.

Time-domain wave splitting and propagation in dispersive media

Time-domain left- and right-moving waves are identified in a homogeneous dispersive medium through a diagonalization of Maxwell’s equations. This exact physical wave splitting is then used in the derivation of a differential equation for the propagator kernel, which can be easily solved numerically by a simple integration. Numerical results for two particular dispersion models (Lorentz and Debye models) are presented. It is also shown how the physical wave splitting can be used in the determination of the scattering of a plane-wave pulse from a dispersive half-space. Explicit expressions are given for the reflection kernels for the Lorentz and Debye models.

Transient waves in nonstationary media

This paper treats propagation of transient waves in nonstationary media, which has many applications in, for example, electromagnetics and acoustics. The underlying hyperbolic equation is a general, homogeneous, linear, first-order 2×2 system of equations. The coefficients in this system depend on one spatial coordinate and time. Furthermore, memory effects are modeled by integral kernels, which, in addition to the spatial dependence, are functions of two different time coordinates. These integrals generalize the convolution integrals, frequently used as a model for memory effects in the medium. Specifically, the scattering problem for this system of equations is addressed. This problem is solved by a generalization of the wave splitting concept, originally developed for wave propagation in media which are invariant under time translations, and by an imbedding or a Green’s functions technique. More explicitly, the imbedding equation for the reflection kernel and the Green’s functions (propagator kernels) equations are derived. Special attention is paid to the problem of nonstationary characteristics. A few numerical examples illustrate this problem.

Design of reflectionless media for transient electromagnetic waves

A method for modelling reflectionless conductive dispersive media is presented. The media are either half-spaces or finite dispersive slabs with a spatially varying impedance. The problem of finding reflectionless media for plane waves at normal incidence is formulated as an inverse problem where the constitutive relation is to be determined as a function of depth given a reflection kernel which is zero. The inverse problem is solved by a time-domain Green function technique. It is seen that non-reflecting half-spaces can be constructed in a number of different ways, whereas non-reflecting slabs can only be found if the back-wall is non-conducting.

Closed-form solution for the transient reflected pressure for a point source above an acoustic half-space with an exponentially stratified density

The reflection problem for a transient point source over an acoustic half-space with an exponentially stratified density is considered. A spatial Hankel transform is used to obtain a one-dimensional wave equation. The transient reflected pressure on the surface in the Hankel-transformed space is expressed by a convolution of a reflection kernel and the transformed incident pressure, and the exact expression for the reflection kernel is obtained by means of a Laplace transform. Closed-form solution of the transient reflected pressure in physical space is obtained through an inverse Hankel transform. Closed-form solution for a generalization of the exponential density profile is also given. Numerical examples are given to illustrate these closed-form solutions.

Inverse scattering for anisotropic mirror image symmetric media

An inverse medium scattering problem for a homogeneous slab of anisotropic dispersive medium is presented. The inverse problem is to recover the dyadic susceptibility kernel from the dyadic reflection kernels for obliquely incident transient electromagnetic plane waves. A time domain technique, based on wave splitting and invariant imbedding, is used. Mirror images enter into the formalism in a natural way and are therefore discussed. This leads to a simplified problem for a class of mirror image symmetric media. This is solved numerically and the stability of the algorithm is tested using noisy data.

Direct and inverse scattering from dispersive media

Direct and inverse scattering from dispersive media are studied in the time domain. The media are one-dimensional and homogeneous but can be impedance mismatched to the surrounding media. A time-domain method is introduced where the reflection and transmission kernels are obtained from Volterra equations of the second kind. Numerical examples are given based upon both synthetic and measured data for the inverse problem.

Transient electromagnetic wave propagation in anisotropic dispersive media

Transient electromagnetic wave propagation in a stratified, anisotropic, dispersive medium is considered. Specifically, the direct scattering problem is addressed. The dispersive, anisotropic medium is modeled by constitutive relations (a 3 × 3 matrix-valued susceptibility operator) containing time convolution integrals. In the general case, nine different susceptibility kernels characterize the medium. An incident plane wave impinges obliquely upon a finite slab consisting of a stratified anisotropic medium. The scattered fields are obtained as time convolutions of the incident field with the scattering kernels. The scattering (reflection and transmission) kernels are uniquely determined by the slab and are independent of the incident field. The scattering problem is solved by a wave-splitting technique. Two different methods for determining the scattering kernels are presented: an embedding and a Green’s function approach. Explicit analytic expressions of the wave front are given for a special class of media. Some numerical examples illustrate the analysis.

A Simplified Time-Domain Inverse Approach to Nonplanar Stratified Media

The scalar wave equation in a curvilinear coordinate system is considered first as a model for wave propagation in nonplanar stratified media. The phase velocity is assumed to vary in a direction normal to the stratification of the scattering object. A spatial integral transform is used to reduce the problem to a problem with one spatial dimension. A nonphysical wave splitting is introduced which leads to simplifications for the inverse scattering problem. The imbedding equation for the associated reflection kernel is obtained. Numerical results for the reconstruction of the phase velocity for a stratified circular cylinder and a stratified sphere are presented. Electromagnetic scattering from a stratified circular cylinder or sphere is then treated in vector form by an analogous approach. In an inverse problem the permittivity is reconstructed.

Non-reflecting dispersive media

The modeling of non-reflecting one-dimensional dispersive media is discussed. The media are temporal dispersive with a spatially varying impedance. It is shown that the effects from the variation of the impedance can be matched by the temporal dispersive effects so that the media do not reflect any field regardless of the shape of the incident transient field. The problem of finding reflectionless media is formulated as an inverse problem where the constitutive relation is to be determined as a function of depth given a reflection kernel which is zero. A time-domain Green functions technique is used to solve the inverse problem.

An integrodifferential equation for the two-dimensional reflection kernel

Faber, Seth, and Wing [l] applied the method of doubling to the determination of the two-dimensional invariant imbedding operators (reflection, transmission, right-turn and left-turn) for a variety of discrete-directional models, including one in which motion of the underlying particles is allowed only in the two opposing directions along each of two mutually orthogonal axes (the four-compass-point model). In this work the application of the addition/doubling formulas was initiated at some “fundamental square,” for which the invariant-imbeclcling functions were determined on the physical basis of considering interactions only to first order in collisions (i.e., the possibility of an incident particle engendering two or more collisions in the fundamental square was neglected). The present work is intended as a continuation of that of [l], in that our

Factorization of a dissipative wave equation and the Green functions technique for axially symmetric fields in a stratified slab

The time domain scattering problem for axially symmetric fields and a dissipative stratified slab is considered. Explicit representations are derived for the wave splitting operators that factorize the dissipative Hankel transformed wave equation. The Green functions approach associated with this splitting is developed. Partial differential equations (PDEs) for the Green functions are derived and used to derive the imbedding equations for the reflection and transmission kernels. Numerical results for the internal fields are given in the direct problem, and the phase velocity is reconstructed in the inverse problem.

Scattering of Transient Electromagnetic Waves in Reciprocal Bi-Isotropic Media

In this paper propagation of transient electromagnetic waves in a reciprocal bi-isotropic medium is presented. The constitutive relations are convolution integrals with two susceptibility kernels that model the medium. The propagation problem is solved by the introduction of a wave-splitting technique. This wave-splitting is used to solve the propagation problem using either an imbedding approach or a Green's function technique. In particular, the scattering problem of an electromagnetic wave that impinges normally on a slab of finite or infinite extent is solved. The slab is assumed to be inhomogeneous with respect to depth. The scattering problem consists of finding the reflected and the transmitted fields and the generic quantities are the reflection and the transmission kernels of the medium. Explicit expressions for the rotation and the attenuation of the wave front is presented for the inhomogeneous slab. In the special case of a homogeneous infinite slab it is proved that the reflection kernel satisfies a nonlinear Volterra equation of the second kind, very suitable for numerical calculations. It is also proved that no cross polarization contribution appears for the homogeneous slab. Several numerical computations illustrate the analysis.

Scattering of Transient Electromagnetic Waves in Reciprocal Bi-Isotropic Media

In this paper propagation of transient electromagnetic waves in a reciprocal biisotropic medium is presented. The constitutive relations are convolution integrals with two susceptibility kernels that model the medium. The propagation problem is solved by the introduction of a wave-splitting technique. This wave-splitting is used to solve the propagation problem using either an imbedding approach or a Green's function technique. In particular, the scattering problem of an electromagnetic wave that impinges normally on a slab of finite or infinite extent is solved. The slab is assumed to be inhomogeneous with respect to depth. The scattering problem consists of finding the reflected and the transmitted fields and the generic quantities are the reflection and the transmission kernels of the medium. Explicit expressions for the rotation and the attenuation of the wave front is presented for the inhomogeneous slab. In the special case of a homogeneous infinite slab it is proved that the reflection kernel satisfies a nonlinear Volterra equation of the second kind, very suitable for numerical calculations. It is also proved that no cross polarization contribution appears for the homogeneous slab. Several numerical computations illustrate the analysis.

Time Domain Wave Splitting Approach to Transmission Along a Nonuruform LCRG Line

The calculation of the transient voltage and current along a nonuniform LCRG transmission line of finite length is considered. The model equations form a system of first order coupled differential equations for the voltage and current, and it is solved by a combination of a wave splitting technique and an invariant imbedding approach as well as a Green's functions approach. Partial differential equations for the reflection kernel, transmission kernel and the Green's functions are derived. Results from two different types of splittings are compared. Numerical results for the direct and also for particular inverse problems are presented. Finally it is noted that the system of equations obtained in a special case corresponds to Maxwell's equations for plane wave propagation in a slab where the permittivity, conductivity, and permeability vary with the depth.

Abstract Riccati equations in an L1 space of finite measure and applications to transport theory

AbstractWe consider operator‐valued Riccati initial‐value problems of the form R′(t) + TR(t) + R(t)T = TA(t) + TB(t)R(t) + R(t)TC(t) + R(t)TD(t)R(t), R(0) = R0. Here A to D and R0 have values as non‐negative bounded linear operators in L1 (μ), where μ is a finite measure, and T is a closed non‐negative operator in L1 (μ) satisfying additional technical conditions. For such problems the notion of strongly mild solutions is defined, and local existence and uniqueness theorems for such solutions are established. The results of the analysis are applied to the reflection kernels with both isotropically scattering homogeneous and anisotropically scattering inhomogeneous medium.

Factorization of the dissipative wave equation and inverse scattering

A decomposition of the solution of the dissipative wave equation into incoming and outgoing components across a smooth surface in a homogeneous region is presented. (The proof of the decomposition is given only for the plane surface.) This is then applied to the factorization of the dissipative wave equation into incoming and outgoing components in a planar-stratified medium. The Ricatti integral–differential equation for the reflection operator that relates the two components is obtained. It is shown how the zeroth and second moments (with respect to the transverse variable in a planar-stratified medium) of the reflection kernel can be used in the inverse problem to recover the velocity and dissipation coefficient from knowledge of the scattered field.

A Banach space ODE framework for inverse scattering via invariant imbedding

Using invariant imbedding, Corones, Davison, and Krueger ( J. Math. Anal. Appl., 2 (1983) , 393–415; J. Acoust. Soc. Amer., 74 (1983) , 1535–1541; “Inverse Optics, Proc. SPIE 413” (A. J. Devaney, Ed.), pp. 102–106, SPIE, Bellingham, Wash., 1983 have approached the inverse scattering problem for an inhomogeneous medium by deriving an integro-differential equation whose initial data is the reflection kernel for the medium and whose solution yields the material properties of the medium. In this paper we pose this integro-differential equation as an ODE in the Banach space C[0, 1] to obtain local existence, uniqueness, continuous dependence, and differentiability of the solution with respect to the initial data. In addition, we show that the Frechet derivative of the mapping taking initial data to the material properties has the form −4 I + K, where I is the identity and K is a compact operator on the Banach space.

Direct and inverse scattering in the time domain via invariant imbedding equations

This paper examines the implementation of a new approach to direct and inverse scattering problems in the time domain, which is applied here to the case of a one-dimensional lossless medium. The method is based on an integro-differential equation which is satisfied by the reflection kernel and has been derived elsewhere using invariant imbedding techniques. An example of an exact solution to this equation is given. Numerical schemes for solving the direct and inverse problems are derived by discretizing this equation. The behavior of the resulting algorithms is then tested on several examples. The results suggest that the inversion method is stable in the presence of additive noise.

Inverse problems for nonabsorbing media with discontinuous material properties

One-dimensional electromagnetic and elastic inverse problems are formulated for media with discontinuous material properties. In addition, an impedence mismatch between source and medium is allowed. The measured data for either problem is shown to generate a reflection kernel which is used in the solution of the inverse problem. The solution algorithm itself is a time domain technique which is a special case of previously obtained results for absorbing media.