Polarization purity and cross-channel intensity correlations

We consider the question of monitoring polarization purity, that is, measuring deviations from orthogonality δ_τ and δ_ϵ of an ostensibly orthogonal polarization basis with a reference channel of ellipticity ϵ and tilt τ. 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 ρ⁽¹⁾ denoting the Pearson complex correlation coefficient between channel complex fields, it states that ∓cos⁡(2ϵ)δ_τ±iδ_ϵ≈ρ⁽¹⁾ when δ_τ,ϵ≪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: cos²(2ϵ)δτ2+δ_ϵ²≈ρ⁽²⁾, where ρ⁽²⁾ is the intensity cross-correlation coefficient. Only the measurement of the (real) intensity cross-correlation coefficient is needed when observing unpolarized light. For the special case of a linearly polarized basis, the tilt error is simply δ_τ≈ρ⁽²⁾, and for the circular basis case, with ellipticity deviation δ_ϵ from circular helicity π/4 (the reference channel of opposite helicity), δ_ϵ≈ρ⁽²⁾. These results provide simple means to gauge the quality of polarimeters and depolarizers.