Staffan Ström

Matrix formulation of acoustic scattering from an arbitrary number of scatterers

The transition matrix formulation of acoustic scattering given previously by Waterman [J. Acoust. Soc. Am. 45, 1417 (1969)] is extended to the case of an arbitrary number of scatterers. The resulting total transition matrix is expressed in terms of the individual transition matrices and in terms of functions which describe a translation of the origin for the spherical (and cylindrical) wave solutions of Helmholtz equation. Explicit formulas are given for the case of two and three scatterers and the (finite) iteration scheme for the general case is described. Some numerical calculations concerning some aspects of the scattering from two spheres are also reported.

Scattering from buried inhomogeneities - a general three-dimensional formalism

In the present article we give a general three‐dimensional formalism for scattering in two half spaces, one of which contains a bounded inhomogeneity. Our formalism consists of an extension of the transition matrix method which has been given by Waterman, a method which applies equally well to acoustic, electromagnetic, and elastic scattering. The formalism is here developed in detail for the case when the source and inhomogeneity are situated in different half‐spaces. However, the same method works for other source positions as well, and the basic equations are given also for the case when the source and the inhomogeneity lie in the same half space. In the final expression for the total scattered field, that part (the so‐called anomalous scattered field) which depends on the presence of the inhomogeneity, can be separated, and the physical meaning of the various quantities which determine this anomalous scattered field can be identified. The inhomogeneity enters through its T matrix, and previous results...

The electromagnetic scattering problem in the time domain for a dissipative slab and a point source using invariant imbedding

A vertical magnetic dipole source above an inhomogeneous slab is considered where the permittivity e=e(z) and conductivity σ=σ(z) vary with the depth z. Using the axial symmetry of the problem, a Hankel transform is introduced. Besides physical scattering kernels, ‘‘nonphysical’’ scattering kernels are introduced and the direct and inverse transformed problem is solved by means of wave splitting in an invariant imbedding approach. The imbedding equations for the nonphysical scattering kernels are simpler than the corresponding equations for the physical kernels and they are solved using existing techniques for the imbedding equations for normally incident plane waves. The solution for the physical kernels is obtained by establishing relations between the two sets of kernels. These relations are given by Volterra equations of the second kind and thus are very easy to solve numerically. Some applications and extensions of the results derived in the paper are discussed.

Convergence of the T-matrix approach in scattering theory. II

The T‐matrix numerical scheme is widely used in practice. Convergence of this scheme was not proved. A proof of convergence is given in this paper.

Matrix formulation of acoustic scattering from multilayered scatterers

A T−matrix formulation of both acoustic and electromagnetic scattering has been given by Waterman for the case of one homogeneous scatterer, and this formulation has subsequently been extended to the case of several such scatterers by the present authors. In the present article we show that the matrix formulation is also well suited for the treatment of acoustic scattering from scatterers consisting of an arbitrary number of consecutively enclosing homogeneous layers with different propagation constants. We also show how the earlier results on the matrix formulation of acoustic scattering can be combined with these new results so as to apply to more general types of multilayered scatterers. Some numerical applications are given.Subject Classification: 20.30.

Simultaneous reconstruction of two parameters from the transient response of a nonuniform LCRG transmission line

A time-domain method to determine two of the physical parameters L, C, R, and G of a nonuniform transmission line, from the knowledge of the remaining two parameters and the transient response of the line, is presented. The method is based on the recently introduced compact Green functions approach. It is shown that any pair of the parameters can be reconstructed if L and C are considered in terms of the characteristic impedance Z and the wavefront speed c. The impedance is assumed to be continuous along the nonuniform line, but may be discontinuous at the end. It is shown that a discontinuity at the end both simplifies the reconstruction and makes the reconstruction less sensitive to noise. The parameters R and G, which both represent losses, are the most difficult to reconstruct. It is found that reconstructions involving c as well as reconstructions of R and G fail under certain conditions, at certain points along the line. The method is tested numerically for all combinations of parameters, and the fa...

Modern Theory of Gratings

The principal results of modern electromagnetic theory of gratings are reviewed briefly in this chapter. The model initial boundary value problems and boundary value problems are formulated and supplied with basic equations, domains of analysis, boundary and initial conditions, and the condition providing their unambiguous resolution are determined. The analytic relations between problems in time and frequency domains are found out. The problems connected with the consideration of gratings as open periodic resonators and waveguides are formulated and analyzed. The actual essential results of spectral theory of gratings and resonant scattering theory by periodic structures are presented in concise form. 1.1 The Formulation of Boundary Value and Initial Boundary Value Problems in the Theory of Diffraction Gratings 1.1.1 Fundamental Equations The initial boundary value problems and boundary value problems for the system of differential Maxwell equations form the corner stone of time domain and frequency domain electromagnetic theory. The solutions to these problems provide us with results, describing physical phenomena of spatio–temporal and spatial–frequency transformations of electromagnetic fields occurring in a large variety of structures: gratings, wave-guiding units, open resonators, radiating elements in antennas, etc. The adequacy and accuracy of the results depend in an essential way on the quality of the mathematical problems formulation and on the possibility of detailed analytic investigation of the solutions before the start of their numerical determination. In this chapter, we describe the problems of electromagnetic theory of gratings resulting from following system of equations: η0rot H = ε E ∂t + σ E + J, rot E = −η0μ H ∂t , (1.1) div ( μ H = 0 , div ε E = ρ, (1.2) 1 Y.K. Sirenko, S. Ström (eds.), Modern Theory of Gratings, Springer Series in Optical Sciences 153, DOI 10.1007/978-1-4419-1200-8_1, C

The electromagnetic inverse problem in the time domain for a dissipative slab and a point source using invariant imbedding: reconstruction of the permittivity and conductivity

Abstract He, S. and S. Strom, The electromagnetic inverse problem in the time domain for a dissipative slab and a point source using invariant imbedding: Reconstruction of the permittivity and conductivity, Journal of Computational and Applied Mathematics 42 (1992) 137-155. We consider the electromagnetic inverse problem for a point source above an inhomogeneous dissipative slab of permittivity ∈(z) and conductivity σ(z), where z is the depth. Two inversion algorithms based on the invariant imbedding equations derived in previous work are used to reconstruct both the permittivity and the conductivity. Both algorithms use two-sided reflection data and one of them also uses transmission data. Results are presented for clean and noisy data.

The null-field approach to electromagnetic resonance of composite objects

Abstract In the present article, the null-field approach to electromagnetic resonance properties of three-dimensional composite objects is reviewed. In this approach the resonance problem is solved by searching zeroes of the determinant of the total Q -matrix for the composite object. The derivation of the Q -matrix for three main classes of composite objects is given. It is noted that for each class, several alternative null-field approaches are usually available. The numerical implementation of the Q -matrix for homogeneous and composite objects is discussed, with special emphasis on consistency checks for the results. It is found that numerical convergence is obtained in a frequency interval that often contains ten or more resonance modes. The resonance frequencies and quality factors for some axially symmetric composite objects are compared with published numerical and experimental data whenever possible. In general, good agreement is found in these comparisons.

Time domain Green functions technique for a point source over a dissipative stratified half-space with a phase velocity mismatch at the surface

Abstract A point source over a dissipative stratified half-space with a discontinuity for both the phase velocity and the dissipation coefficient at the surface is considered. The axial symmetry of the problem is exploited in a spatial Hankel transform. The Green functions technique based on wave splitting is used to solve both the direct and inverse problems in the time domain. A Green tensor is introduced which directly maps the physical incident field outside the inhomogeneous half-space to the internal split fields. The PDEs for the Green functions are derived. In the direct problem, the internal fields are calculated as well as the reflected field. The phase velocity is reconstructed in the inverse problem. Simultaneous reconstruction of the phase velocity and dissipation coefficient using two different values of the Hankel transform parameter is analyzed. Two special cases, namely a point source over a homogeneous non-dissipative half-space and over a homogeneous dissipative half-space, are discussed and it is shown that the solution for the reflected fields is given by Volterra equations of the second kind.