3 The measurement of sky polarization

3.1 Observational method

The measurement with one polarimeter of the linear polarization of a wave coming from a direction $\boldsymbol{\hat{n}}$ on the sky, requires at least three measurements with different polarimeter orientations. Since the Stokes parameters Q and U are not invariant under rotations, we define them at each point $\boldsymbol{\hat{n}}$ with respect to a reference frame of tangential vectors ${(\hat{e}_\lambda,\hat{e}_\beta)}$. The output signal given by a polarimeter looking at point $\boldsymbol{\hat{n}}$ is:

$$\!\!\!\!\!\!\!\!\! M_{\rm polar}=\frac{1}{2} ( I+Q \cos 2\Psi+U \sin 2\Psi )$$ (2)

where $\Psi$ is the angle between the polarimeter and $\hat{e}_\lambda$. In the following, we choose the longitude-latitude reference frame as the fixed reference frame on the sky (see Fig. 6).

Figure 6: The reference frame used to define the Stokes parameters and angular position $\Psi$ of a polarimeter. $\Psi$ lies in the plane ${(\hat{e}_\lambda,\hat{e}_\beta)}$

3.2 Destriping method

The destriping method consists in using redundancies at the intersections between circle pairs to estimate, for each circle iand each polarimeter p, the offsets O_i^p on polarimeter measurements. For each circle intersection, we require that all three Stokes parameters in a fixed reference frame in that direction of the sky, as measured on each of the intersecting circles, be the same. A $\chi^2$minimization leads to a linear system whose solution gives the offsets. By subtracting these offsets, we can recover the Stokes parameters corrected for low-frequency noise.

3.3 Formalism

We consider a mission involving n circles. The set of all circles that intercept circle i is denoted by $\mathcal{I}(i)$ and contains $N_{\mathcal{I}(i)}$ circles. For any pair of circles i and j, we denote the two points where these two circles intersect (if any) by ${\{i,j,\delta\}}$. In this notation i is the circle currently scanned, j the intersecting circle in set $\mathcal{I}(i)$, and $\delta$indexes the two intersections ( $\delta = 1 (-1)$ indexes the first (second) point encountered from the northernmost point on the circle) so that the points ${\{j,i, -\delta\}}$ and ${\{i,j,\delta\}}$ on the sky are identical.

The Stokes parameters at point ${\{i,j,\delta\}}$, with respect to a fixed global reference system, are denoted by a 3-vector $\boldsymbol{S}_{i,j,\delta}$, with

$$\!\!\!\!\!\!\!\!\! \boldsymbol{S}_{i,j,\delta} = \boldsymbol{... ...\left( \boldsymbol{\hat{n}} \equiv \{i,j,\delta \} \right) .$$ (3)

At intersection ${\{i,j,\delta\}}$, the set of measurements by hpolarimeters travelling along the scanning circle i is a h-vector denoted by $\boldsymbol{M}_{i,j,\delta}$, and is related to the Stokes parameters at this point by (see Eq. 2):

$$\!\!\!\!\!\!\!\!\! \boldsymbol{M}_{i,j,\delta} = \boldsymbol{\mathcal{A}}_{i,j,\delta} \boldsymbol{S}_{i,j,\delta}$$ (4)

where $\boldsymbol{\mathcal{A}}_{i,j,\delta}$ is the $h \times 3$ matrix:

$$\!\!\!\!\!\!\!\!\! \boldsymbol{\mathcal{A}}_{i,j,\delta} = \... ...\sin 2\Psi_{ h}(i,j,\delta)\\ \end{array}\right). \nonumber$$

$\Psi_{ p}(i,j,\delta) \in [0,\pi]$ is the angle between the orientation of polarimeter p and the reference axis in the fixed global reference frame (see Fig. 6). The matrix $\boldsymbol{\mathcal{A}}_{i,j,\delta}$ can be factorised as

$$\!\!\!\!\!\!\!\!\! \boldsymbol{\mathcal{A}}_{i,j,\delta} = \boldsymbol{\mathcal{A}} \boldsymbol{R}_{i,j,\delta}.$$ (5)

The constant $h \times 3$ matrix $\boldsymbol{\mathcal{A}}$ characterizes the geometrical setup of the h polarimeters in the focal reference frame:

$$\!\!\!\!\!\!\!\!\!\boldsymbol{\mathcal{A}} =\frac{1}{2} \left... ...\ 1&\cos 2 \Delta_{h}&\sin 2 \Delta_h\\ \end{array}\right)$$ (6)

where $\Delta_{ p}$ is the angle between the orientations of polarimeters p and 1, so we have $\Psi_{ p}=\Psi_1+\Delta_{p}$ and $\Delta_1=0$. The rotation matrix $\boldsymbol{R}_{i,j,\delta}$ brings the focal plane to its position at intersection ${\{i,j,\delta\}}$ when scanning along circle i:

$$\!\!\!\!\!\!\!\!\!\boldsymbol{R}_{i,j,\delta}= \left( \begi... ...i_1(i,j,\delta)&\cos 2\Psi_1(i,j,\delta) \end{array}\right).$$ (7)