Transfer matrix of a spherical scatterer.

It was recently recognized that the internal structure of scatterers can significantly alter the picture of wave propagation and localization in random media. When the size of scatterers cannot be neglected, scattering is anisotropic and a ‘‘dwell time’’ for waves appears in addition to the time of flight between scattering. This substantially modifies such parameters describing wave propagation as the diffusion coefficient, transport mean free path, and transport velocity. The microscopic approach to the problem of the renormalization of the diffusion coefficient is based on the Bethe-Salpeter equation for the field-field correlation function. This equation and the solution obtained involve the transfer matrix for a single scatterer, tkk8 v , where k,k8 are outgoing and incident momenta, respectively, and v is the frequency. This matrix contains all information relating to a scatterer’s structure, its influence on the anisotropy of scattering, and on internal resonances. A great deal of this information is ignored in the commonly used on-shell approximation for the transfer matrix where both momenta k and k8 are taken on the ‘‘mass shell,’’ k5v. The off-shell transfer matrix, originally arising in the Bethe-Salpeter equation, is not restricted by this condition. This matrix resolves the structure of a scattered field at any distance away from a scatterer, while the on-shell matrix gives the far-zone asymptote of a field only. The on-shell approximation formally enters the theory of wave transport in random media via the commonly accepted d-function approximation for the imaginary part of the Green function, ImGk ,v}d(v 2cp k2). The d approximation confines all transfer matrices in the expression of the diffusion coefficient on the mass shell. However, the presence of derivatives of tkk8 v does not allow one to set v5k automatically. Even within the on-shell approximation, one has to first calculate the derivatives and then to allow k→v . Neglecting this fact can lead to unexpected results. For instance, numerical calculations of the influence of weak absorption within the on-shell approximation show that the diffusion coefficient becomes larger than its value in the medium without microstructure. This effect disappears if the off-shell matrix is used. The off-on-shell matrix for a scatterer with an infinite permittivity was first used in Ref. 1. Such a matrix gives a far-zone asymptote of a scattered field when the source is located at a finite distance from a scatterer. This destroys the symmetry between the incident and scattered momenta which exists due to the reciprocal principal for the Green function. Analysis of the renormalization of the diffusion coefficient employing the off-on-shell transfer matrix of a scatterer with a finite permittivity has shown a strong enhancement of the previously obtained corrections to D and qualitatively agrees with experimental data. Numerical agreement can be attained with the use of an exact off-shell matrix. In this paper we derive the transfer matrix of a dielectric sphere and analyze some of its general properties.