Polarization purity and cross-channel intensity correlations

We consider the question of monitoring polarization purity, that is, measuring deviations from orthogonality <i>δ</i>_τ and <i>δ</i>_ϵ of an ostensibly orthogonal polarization basis with a reference channel of ellipticity <i>ϵ</i> and tilt <i>τ</i>. A simple result was recently derived for a phase-sensitive receiver observing unpolarized radiation [IEEE Trans. Geosci. Remote Sens.62, 2003610 (2024)10.1109/TGRS.2024.3380531]: with <i>ρ</i>⁽¹⁾ denoting the Pearson complex correlation coefficient between channel <i>complex fields</i>, it states that ∓<i>cos</i>⁡(2<i>ϵ</i>)<i>δ</i>_τ±<i>i</i><i>δ</i>_ϵ≈<i>ρ</i>⁽¹⁾ when <i>δ</i>_{<i>τ</i>,<i>ϵ</i>}≪1. However, phase-sensitive (in-phase and quadrature) data are seldom available at optical frequencies. To that end, here we generalize the result by deriving a new equation for the polarization "alignment" error: <i>cos</i>²(2<i>ϵ</i>)<i>δ</i><i>τ</i>2+<i>δ</i>_ϵ²≈<i>ρ</i>⁽²⁾, where <i>ρ</i>⁽²⁾ is the intensity cross-correlation coefficient. Only the measurement of the <i>(real) intensity</i> cross-correlation coefficient is needed when observing unpolarized light. For the special case of a linearly polarized basis, the tilt error is simply <i>δ</i>_τ≈<i>ρ</i>⁽²⁾, and for the circular basis case, with ellipticity deviation <i>δ</i>_ϵ from circular helicity <i>π</i>/4 (the reference channel of opposite helicity), <i>δ</i>_ϵ≈<i>ρ</i>⁽²⁾. These results provide simple means to gauge the quality of polarimeters and depolarizers.