Relating the statistics of the angle of linear polarization to measurement uncertainty of the Stokes vector

This work shows an analytic solution to the central moments of the angle of linear polarization (AoLP) when the linear Stokes parameters are independent and Gaussian distributed with different means but equal variance. Such a result is useful for distinguishing AoLP features from noise in polarimetry. When the DoLP is high relative to the measurement uncertainty of the linear Stokes vector, AoLP statistics have been shown to be well approximated by a Gaussian distribution. When the DoLP is zero, AoLP values are uniformly distributed. In general, the probability density function (PDF) of AoLP does not have a closed-form solution and this is the first report, to our knowledge, on an exact analytic form for the central moments of the AoLP. This analytic form will be useful when the AoLP is of interest even when the DoLP is low and the corresponding PDF on the AoLP is in between the extreme cases of a Gaussian or a uniform distribution. We also show that a simple propagation of error (PE) analysis underestimates the AoLP variance at extremely low DoLP but is verified for cases of DoLP that are high relative to the Stokes measurement uncertainty. An example use of the AoLP variance in imaging polarimetry is presented.