 Solution

Unpolarized incident light has equal, mutually incoherent $y$- and $z$-polarized intensities. Both components scatter into the same direction $\mathbf{v}$, so their coherency matrices add toThis matrix has rank one when the signal is nonzero. The light is therefore fully linearly polarized along $\mathbf{v}$, withA fixed highly anisotropic needle can act as a polarizing scatterer even for unpolarized input. The equality of total intensities with (c) does not mean the incident states were identical. These conclusions use the leading longitudinal-polarizability approximation; finite transverse response can change the polarization purity.