Rank-one needle polarizability
= Rank-one needle polarizability
{title2=$\alpha_\parallel\mathbf u\mathbf u^{\mathsf T}$}
A slender high-permittivity particle responds mainly along its axis $\mathbf u$, giving $\boldsymbol\alpha\simeq\alpha_\parallel\mathbf u\mathbf u^{\mathsf T}$. 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.