Rank-one needle polarizability (${\alpha }_{\parallel }\mathbf{u}{\mathbf{u}}^{\mathsf{T}}$)

A slender high-permittivity particle responds mainly along its axis $\mathbf{u}$, giving $\mathbf{\alpha }\simeq {\alpha }_{\parallel }\mathbf{u}{\mathbf{u}}^{\mathsf{T}}$. Far-field projection into an observation plane leaves one real polarization direction. Every nonzero output is therefore linearly polarized in this leading approximation, even for circular or unpolarized input.