Calculation of the Degree and Orientation of the Polarization

The use of Stokes parameters to describe polarization has mathematical advantages, but is not easy to represent in a graphical manner. For human interpretation therefore, polarization is usually described by the degree of polarization, $p$, (i.e. the ratio of polarized to total intensity), and the orientation of the plane of polarization, $\theta$. Note, $\theta$ is the angle between the plane of polarization and the reference direction of the polarimeter. To convert this to a position angle on the sky, the position angle of the reference direction must be known. The simplest way to derive these parameters from the Stokes parameters is as follows:

$$\begin{eqnarray*} I_{p} & = & \sqrt{ Q^{2} + U^{2} } \\ p & = & I_{p}/I \\ \\ \theta & = & 0.5.\arctan (U/Q) \end{eqnarray*}$$

where $I_{p}$ is the polarized intensity. However, the estimation of $p$ is complicated by the non-symmetric noise statistics produced by squaring and adding $Q$ and $U$. For low polarizations, the squaring of the noise will tend to shift the mean of the distribution of $p$ to higher values, thus resulting in an over-estimation of $p$. The use of the following expression for $I_{p}$ reduces the effect of this statistical bias:

$$\begin{eqnarray*} I_{p} & = & \sqrt{ Q^{2} + U^{2} - \sigma^{2}} \\ \end{eqnarray*}$$

where $\sigma^{2}$ is the variance on $Q$ or $U$ (which are assumed equal). A description of the statistical behaviour of polarization parameters is given by Serkowski (Advances in Astronomy and Astrophysics, ed. Z. Kopal, Academic Press, New York, London (1962), 1, 304).