Generalization of the optical theorem for a multipole based on integral transforms

Based on integral transforms for wave fields, we obtain a generalization of the optical theorem for the case in which a local inhomogeneity is excited by a multipole source of arbitrary order. This generalization permits determining the total scattered and absorbed energy analytically by computing the derivatives of the scattered field at a single point. This relation can be used to compute the absorption cross-section in problems related to plasmonic structures and also to test computer modules when multipole radiation is scattered by transparent bodies.     

Generalization of the optical theorem to a singular source in the presence of a half-space

The scattering of the field of an electric dipole by a local penetrable body in the presence of a transparent half-space is considered. A relation is obtained that is similar to the optical theorem in the case of an incident plane wave. With this generalization, the fluorescence enhancement factor or the efficiency of an optical antenna can be calculated without computing the absorbed energy, which considerably reduces the computational costs.     

A uniqueness theorem in inverse scattering of elastic waves

The scattering of plane elastic waves in an isotropic inhomogeneousmedium is considered. The existence and uniqueness of the directproblem is stated, and a reciprocity principle for the far fieldsof the scattered waves formulated. Finally, it is proved thatthe knowledge of the far-field patterns for a bounded sequenceof different frequencies and certain sets of incoming planewaves uniquely determines the density of the medium.     

Recent developments in the qualitative approach to inverse electromagnetic scattering theory

We consider the inverse scattering problem of determining both the shape and some of the physical properties of the scattering object from a knowledge of the (measured) electric and magnetic fields due to the scattering of an incident time-harmonic electromagnetic wave at fixed frequency. We shall discuss the linear sampling method for solving the inverse scattering problem which does not require any a priori knowledge of the geometry and the physical properties of the scatterer. Included in our discussion is the case of partially coated objects and inhomogeneous background. We give references for numerical examples for each problem discussed in this paper.     

A convergence theorem for the fast multipole method for 2 dimensional scattering problems

The Fast Multipole Method (FMM) designed by V. Rokhlin rapidly computes the field scattered from an obstacle. This computation consists of solving an integral equation on the boundary of the obstacle. The main result of this paper shows the convergence of the FMM for the two dimensional Helmholtz equation. Before giving the theorem, we give an overview of the main ideas of the FMM. This is done following the papers of V. Rokhlin. Nevertheless, the way we present the FMM is slightly different. The FMM is finally applied to an acoustic problem with an impedance boundary condition. The moment method is used to discretize this continuous problem.

     

The Atkinson-Wilcox expansion theorem for electromagnetic chiral waves

Consider the problem of scattering of a time-harmonic electromagnetic wave by a three-dimensional bounded and smooth obstacle. The infinite space outside the obstacle is filled by a homogeneous isotropic chiral medium. In the region exterior to a sphere that includes the scatterer, any solution of the generalized Helmholtz's equation that satisfies the Silver-Müller radiation condition has a uniformly and absolutely convergent expansion in inverse powers of the radial distance from the center of the sphere. The coefficients of the expansion can be determined from the leading coefficient, “the radiation pattern”, by a recurrence relation.     

Optical theorem for the scattering of waves in an elastic plate

The scattering of waves in an elastic plate is investigated. A generalization of the optical theorem is found for the case of a noncompact scatterer. The formulas obtained can be used for indirect control of the correctness of computing diffraction fields.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 104, pp. 20–23, 1981.The author is grateful to D. P. Kouzov for his constant attention to the work.     

On the scattering of spherical electromagnetic waves by a layered sphere

A spherical electromagnetic wave is scattered by a layered sphere.The exact expressions of the scattered and interior fields areobtained by solving the corresponding boundary-value problem,by means of a combination of Sommerfeld's and T-matrix methods.A recursive algorithm with respect to the number of layers isextracted for the computation of the fields in every layer.The far-field pattern and the scattering cross-sections aredetermined in terms of the physical and geometrical characteristicsof the scatterer. As the point-source tends to infinity, theknown results for plane wave incidence are recovered. Numericalresults are presented for several cases and various parametersof the layered spherical scatterer.