If a linearly polarized classical monochromatic electromagnetic radiation undergoes a scattering, does the scattered electric field have the same polarization as the incident electric field? I am looking for an answer (or deduce the conclusion mathematically) from classical electromagnetic theory of scattering.
It depends on what the wave is scattered from. The simplest case to study (as an exercise to get intuition) is scattering of a monochromatic EM wave in vacuum from an infinite metal plane: in this case the boundary condition is that the component of the electric field along the surface should be zero. This may add a phase of $\pi$ to one or both components of the electric field, but no other transformations occur.
In other words:
- linear polarization remains linear polarization (taking account that it is polarized perpendicular to the direction of the wave propagation, which is generally not the same for the incident and the scattered waves)
- circular polarization might change its direction of rotation - I can't be sure without doing a calculation that I described above
- obviously, if the waves are not in vacuum, or if the scattering surface has complex geometry, or if it is not a metallic surface, the answer may become a lot more complicated.
This is a huge subject in radar and it has a vast literature but there are no simple answers besides that depolarization (i.e., the generation of the orthogonal polarized reflected/scattered wave) is dependent on frequency, incident angle, statistics of the reflecting surface, etc. I only mention here one application, namely meteorological radar, where the transmitted signal is linearly polarized but the receiver measures both polarizations of the reflected wave, this is especially important to detect/classify rain.