In optics, polarization mixing refers to changes in the relative strengths of the Stokes parameters caused by reflection or scattering—see vector radiative transfer—or by changes in the radial orientation of the detector.
Example: A sloping, specular surface
The definition of the four Stokes components are, in a fixed basis:
[ I Q U V ] = [ | E v | 2 + | E h | 2 | E v | 2 − | E h | 2 2 Re ⟨ E v E h ∗ ⟩ 2 Im ⟨ E v E h ∗ ⟩ ] , {\displaystyle \left[{\begin{array}{c}I\\Q\\U\\V\end{array}}\right]=\left[{\begin{array}{c}|E_{v}|^{2}+|E_{h}|^{2}\\|E_{v}|^{2}-|E_{h}|^{2}\\2\operatorname {Re} \left\langle E_{v}E_{h}^{*}\right\rangle \\2\operatorname {Im} \left\langle E_{v}E_{h}^{*}\right\rangle \end{array}}\right],}
where Ev and Eh are the electric field components in the vertical and horizontal directions respectively. The definitions of the coordinate bases are arbitrary and depend on the orientation of the instrument. In the case of the Fresnel equations, the bases are defined in terms of the surface, with the horizontal being parallel to the surface and the vertical in a plane perpendicular to the surface. When the bases are rotated by 45 degrees around the viewing axis, the definition of the third Stokes component becomes equivalent to that of the second, that is the difference in field intensity between the horizontal and vertical polarizations. Thus, if the instrument is rotated out of plane from the surface upon which it is looking, this will give rise to a signal. The geometry is illustrated in the above figure: θ {\displaystyle \theta } is the instrument viewing angle with respect to nadir, θ e f f {\displaystyle \theta _{\mathrm {eff} }} is the viewing angle with respect to the surface normal and α {\displaystyle \alpha } is the angle between the polarisation axes defined by the instrument and that defined by the Fresnel equations, i.e., the surface. Ideally, in a polarimetric radiometer, especially a satellite mounted one, the polarisation axes are aligned with the Earth's surface, therefore we define the instrument viewing direction using the following vector:
v ^ = ( sin θ , 0 , cos θ ) . {\displaystyle \mathbf {\hat {v}} =(\sin \theta ,~0,~\cos \theta ).}
We define the slope of the surface in terms of the normal vector, n ^ {\displaystyle \mathbf {\hat {n}} } , which can be calculated in a number of ways. Using angular slope and azimuth, it becomes:
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