1 Introduction

This paper originates from preparatory studies for the Planck satellite mission. This Cosmic Microwave Background (CMB) mapping satellite is designed to be able to measure the polarisation of the CMB in several frequency channels with the sensitivity needed to extract the expected cosmological signal. Several authors (see for instance Rees 1968; Bond & Efstathiou 1987; Melchiorri & Vittorio 1996; Hu & White 1997; Seljak & Zaldarriaga 1998), have pointed out that measurements of the polarisation of the CMB will help to discriminate between cosmological models and to separate the foregrounds. In the theoretical analyses of the polarised power spectra, it is in general assumed (explicitly or implicitly) that the errors are uncorrelated between the three Stokes parameters I, Q and U in the reference frame used to build the polarised multipoles (Zaldarriaga & Seljak 1997; Ng & Liu 1997). However, the errors in the three Stokes parameters will in general be correlated, even if the noise of the three or more measuring polarimeters are not, unless the layout of the polarimeters is adequately chosen. In this paper we construct configurations of the relative orientations of the polarimeters, hereafter called "Optimised Configurations'' (OC), such that, if the noise in all polarised bolometers have the same variance and are not correlated, the measurement errors in the Stokes parameters I, Q and U are independent of the direction of the focal plane and decorrelated. Moreover, the volume of the error box is minimised. The properties of decorrelation and minimum error are maintained when one combines redundant measurements of the same point of the sky, even when the orientation of the focal plane is changed between successive measurements. Finally, when combining unpolarised and data from OC's, the resulting errors retain their optimised properties.

In general, the various polarimeters will not have the same levels of noise and will be slightly cross-correlated. Assuming that these imbalances and cross-correlations are small, we show that for OC's the resulting correlations between the errors on I, Q and U are also small and easily calculated to first order. This remains true when one combines several measurements of the same point of the sky, the correlations get averaged but do not cumulate.

Finally, we calculate the error matrix between E and B multipolar amplitudes and show that it is also simpler in OC's.

1.1 General considerations

In the reference frame where the Stokes parameters I, Q, and U are defined, the intensity detected by a polarimeter rotated by an angle $\alpha$ with respect to the x axis is:

$$\begin{eqnarray} I_\alpha=\frac{1}{2} (I + Q\, \cos 2\alpha + U\, \sin 2\alpha).\end{eqnarray}$$ (1)

Because polarimeters only measure intensities, angle $\alpha$ can be kept between 0 and $\pi$. To be able to separate the 3 Stokes parameters, at least 3 polarised detectors are needed (or 1 unpolarised and 2 polarised), with angular separations different from multiples of $\pi/2$. If one uses $n \ge 3$ polarimeters with orientations $\alpha_{ p},\ 1 \le p \le n$ for a given line of sight, the Stokes parameters will be estimated by minimising the $\chi^2$:

$$\begin{eqnarray} \chi^2 = \left(\bf{M}- \bf{A}\bf{S}\right)^T\, \bf{N}^{-1}\,\left(\bf{M}- \bf{A} \bf{S}\right) \end{eqnarray}$$ (2)

where $\bf{M}= \left( \begin{array} {c} m_1\\ \vdots \\ m_p\\ \vdots \\ m_n \end{array}\right)$is the vector of measurements, and $\bf{N}$ is their $n\times n$noise autocorrelation matrix. The $n\times 3$ matrix

$$\begin{eqnarray} {\bf A} =\frac{1}{2} \left(\begin{array} {ccc} 1&\cos 2 \alpha_... ... & \vdots\\ 1&\cos 2 \alpha_n&\sin 2 \alpha_n\\ \end{array}\right)\end{eqnarray}$$ (3)

relates the results of the n measurements to the vector of the Stokes parameters $\bf{S}= \left( \begin{matrix} I\\ Q\\ U \end{matrix}\right)$ in a given reference frame, for instance a reference frame fixed with respect to the focal instrument. If one looks in the same direction of the sky, but with the instrument rotated by an angle $\psi$ in the focal plane, the matrix A is simply transformed with a rotation matrix of angle $2\, \psi$:

$$\begin{eqnarray} \bf{A}\rightarrow \bf{A}\ \bf{R}(\psi), \mbox{ with } \bf{R}(\p... ...os 2\psi&\sin 2\psi\\ 0&-\sin 2\psi&\cos 2\psi \end{array}\right).\end{eqnarray}$$ (4)

As is well known, the resulting estimation for the Stokes parameters and their variance matrix $\bf{V}$ are: and

$$\begin{eqnarray} {\bf V} = \left({\bf A}^T\, {\bf N}^{-1}\, {\bf A}\right)^{-1}.\end{eqnarray}$$ (5)