Scattering Matrix

Abstract

An incident wave, upon impinging on a scatterer, is linearly transformed into a scattered one and this linear transformation is mathematically expressed, for example, by the Franz integral representation as discussed in Section 1.5, Chapter 1. The scattering process may also be interpreted as a polarization transformation in which the incident polarization is transformed in general by the scatterer into a different one at each field point in space (cf. Aksenov [3.1], for example). It is a mathematical fact that every linear transformation can be represented in matrix and, therefore, so can the Franz integral representation for the scattered field. From the standpoint of polarization transformation, however, there is a compelling, natural reason to cast this incident-scattered field transformation relationship in matrix form. The scattering matrix (S-matrix) arises when the scattered field in the far zone is represented in matrix with reference to prescribed bases at the transmitter and receiver positions. Mathematically speaking then, the S-matrix is in essence nothing more than a tranformation matrix that maps the incident field to the scattered one in the far zone. It is this matrix operator form, however, that is not only mathematically most natural but best suited for extracting information concerning the polarization transformation properties in electromagnetic scattering. For example, in the given scattering geometry and the transmitting and the receiving antenna systems, the S-matrix depends only on the features of the scatterer and is independent of the incident polarization chosen. This property can be exploited to advantage in detection of scatterers by choosing characteristic incident polarizations for enhancing or suppressing received powers (cf. next chapter). This is in analogy with the situation in two-port electric network theory in which the scattering matrix of the network depends only on the system itself, and is independent of the input source and the terminating load. Throughout the chapter we assume that both the transmitter and receiver are in the far zone and that the incident wave is an arbitrarily polarized plane wave.