The Jones Matrix

If a scatterer is illuminated by an electromagnetic plane wave transmitted by an antenna, such that, at the scatterer the incidenct wave is given by

$$\vec{E}^{tr} =E_h^{tr}\vec{e}_h + E_v^{tr}\vec{e}_v$$ (3.1)

and that induces currents in the scatterer, which in turn reradiate a scattered wave. In the far zone of the scatterer, the scattered wave can be considered as a plane wave. The scattering process can be modeled as a linear transformation, described by a matrix S. The received field is then given by [Sinclair48], [Sinclair50], [Kennaugh51]

$$\vec{E}^{re} = \left[ S \right] \vec{E}^{tr} = \left[\begin{array... ...ray}\right] \left[\begin{array}{c} E_{h}^{tr} \\ E_{v}^{tr} \end{array}\right]$$ (3.2)

The [S]-matrix, sometimes referred to as Jones matrix [Jones41], is a complex $2 \times 2$ matrix, containing the information about the scatterer. The elements of [S] are denoted as the complex scattering amplitudes $S_{ij}=\vert S_{ij}\vert e^{i\phi_{ij}} , i,j \in \{h,v\}$. $S_{hh}$ and $S_{vv}$ are called co-polar and $S_{hv}$ and $S_{vh}$ cross-polar components. Due to the reciprocity theorem for monostatic systems and reciprocal media, the cross-polar components are equal $S_{hv} \stackrel{-}{=} S_{vh} = S_{x}$ [Ulaby90]. Therefore, the [S] reduces to

$$[S] = e^{i\phi_0} \left[\begin{array}{cc} \vert S_{hh}\vert & \ve... ...phi_x -\phi_0)} & \vert S_{vv}\vert e^{i(\phi_{vv} -\phi_0)} \end{array}\right]$$ (3.3)

Thus the [S]-matrix yields 5 independent parameters (3 amplitudes and 2 phases).