2 Optimised configurations

2.1 The ideal case

If we assume that the measurements $m_p\ (1 \le p \le n)$ have identical and decorrelated errors
(${\bf N}_{ p q} = <\delta m_{ p}\ \delta m_{ q}\gt = \sigma_0^2\, \delta_{ p q}$), the $\chi^2$ is simply:

$$\chi^2 = \frac{1}{\sigma^2} \sum_{p=1}^n\left[m_{ p}- \frac{1}{2}(I + Q\,\cos 2\alpha_{ p} + U\,\sin 2\alpha_{p})\right]^2,$$ (6)

and the inverse of the covariance matrix of the Stokes parameters is given by:  

$${\bf V}^{-1}=\frac{1}{\sigma^2}\,{\bf X}, \qquad\qquad {\bf ... ... - \sum_{1}^n\,\cos 4\alpha_{ p}) \end{array}\right). \nonumber$$

It is shown in the appendix that, if the orientations of the polarimeters are evenly distributed on 180$^\circ$:

$$\begin{eqnarray} \alpha_p = \alpha_1 + (p-1)\,\frac{\pi}{n},\ p = 1...\ n,\mbox{ with } n \ge 3,\end{eqnarray}$$ (7)

the matrix $\bf{V}$ takes the very simple diagonal form:

$$\begin{eqnarray} {\bf V}_0 = \sigma^2 {{\bf X}_0}^{-1}, \mbox{ with } {{\bf X}_0... ... {ccc} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \\ \end{array}\right), \end{eqnarray}$$ (8)

independent of the orientation of the focal plane, while its determinant is minimised. In other words, at the same time, the errors on the Stokes parameters get decorrelated, their error matrix becomes independent of the orientation of the focal plane and the volume of the error ellipsoid takes its smallest possible value: $\frac{\pi}{3} \left(\frac{4\sigma}{\sqrt{n}}\right)^3$.

The "Optimised Configurations'' (OC) are the sets of polarimeters which satisfy condition (8), (see Fig. 1).

  

Figure 1: The relative orientations of polarimeters in "Optimised Configurations'' with 3, 4 and 5 detectors

They are hereafter referred to by the subscript as in Eq. 9. The smallest OC involves three polarimeters with relative angle $\pi/3$. With 4 polarimeters, the angular separation must be $\pi/4$, and so on. Note that a configuration with one unpolarised detector and 2 polarised detectors can never measure the Stokes parameters with uncorrelated errors, because this would require polarimeters oriented 90$^\circ$ apart from each other, which would not allow the breaking of the degeneracy between Q and U.

If one combines several OC's with several unpolarised detectors, all uncorrelated with each other, the resulting covariance matrix for the Stokes parameters remains diagonal and independent of the orientation of the various OC's. More precisely, when combining the measurements of n _T unpolarised detectors (temperature measurements), with n _P polarised detectors arranged in OC's, the covariance matrix of the Stokes parameters reads:  

$${\bf V}\, =\, \frac{4\,\sigma_{ P}^2}{{n}_{ P}} \left(\begin... ... 0 \\ 0 & ~~~2~~~ & 0 \\ 0 & 0 & ~~~2~~~ \\ \end{array}\right),$$ (9)

where we have introduced inverse average noise levels, $\sigma_{ T}$ and $\sigma_{ P}$, for the unpolarised and polarised detectors respectively:

$$\frac{1}{{\sigma_{ T}}^2} = \left\langle \frac{1}{\sigma_{\... ... \left\langle\frac{1}{\sigma_{\rm polarised}^2}\right\rangle .$$ (10)

Note that the levels of noise can be different from one OC to the other and from those of the unpolarised detectors.

2.2 A more realistic description of the measurements

In general one expects that there will be some slight imbalance and cross-correlation between the noise of the detectors. The noise matrix of the measurements will in general take the form:

$$\begin{eqnarray} {\bf N} = \sigma^2 ({\mathchoice {\rm 1\mskip-4mu l} {\rm 1\msk... ...\mskip-4.5mu l} {\rm 1\mskip-5mu l}}+ \hat{\beta}+ \hat{\gamma}) ,\end{eqnarray}$$ (11)

where the imbalance $\hat{\beta}$ and cross-correlation $\hat{\gamma}$ matrices

$$\begin{eqnarray} & \hat{\beta}= \left(\begin{array} {cccc} \beta_{1\,1}&0 &0 &\... ...}&0 &\ldots\\ \vdots &\vdots &\vdots &\ddots\\ \end{array}\right),\end{eqnarray}$$ (12)

are "small'', that is will be treated as first order perturbations in the following, and therefore

$${\bf N}^{-1} = \frac{1}{\sigma^2}\, ({\mathchoice {\rm 1\msk... ...skip-4.5mu l} {\rm 1\mskip-5mu l}}- \hat{\beta}- \hat{\gamma}).$$

To this order, the variance matrix of the Stokes parameters is easily obtained from Eq. (5):

$$\begin{eqnarray} {\bf V} = \sigma^2 \left[{\bf X}^{-1} + \hat{\mathcal{B}}+ \hat{\mathcal{G}}\right],\end{eqnarray}$$ (13)

where the matrix $\bf{X}$ is given by Eq. 7 and the first order corrections to $\bf{V}$, matrices $\hat{\mathcal{B}}$ and $\hat{\mathcal{G}}$, read:

$$\begin{eqnarray} \left(\begin{array} {l} \hat{\mathcal{B}}\\ \hat{\mathcal{G}}\e... ...t{\beta}\\ \hat{\gamma}\end{array}\right) \,{\bf A}\ {\bf X}^{-1}.\end{eqnarray}$$ (14)

In an OC, the matrix ${\bf X}^{-1}$ takes the simple diagonal form ${{\bf X}_0}^{-1}$ of Eq. (9), and the non diagonal parts, $\hat{\mathcal{B}}$ and $\hat{\mathcal{G}}$ remain of order 1 in $\hat{\beta}$ and $\hat{\gamma}$. For instance, if we consider an OC with 3 polarimeters, and polarimeter number 1 is oriented in the x direction,

$${\hat{\mathcal{B}}} = \frac{4}{3}\ \left(\begin{array} {ccc}... ...ta_{2\,2}}{\sqrt{3}} & - \beta_{1\,1} \\ \end{array}\right),\,$$

where $\beta_{2\,2} + \beta_{3\,3} = - \beta_{1\,1}$, and

$$\begin{eqnarray} \hat{\mathcal{G}}= \frac{4}{3} &\times \nonumber \\ & \left(\b... ...\gamma_{1\,3})}{\sqrt{3}} & - 2\,\gamma_{2\,3} \end{array}\right).\end{eqnarray}$$ (15)

Note that $\hat{\mathcal{B}}$ and $\hat{\mathcal{G}}$ transform under a rotation of the focal plane by a rotation $\bf{R}(\psi)$:

$$\begin{eqnarray} \left(\begin{array} {l} \hat{\mathcal{B}}\\ \hat{\mathcal{G}}\e... ...t{\mathcal{B}}\\ \hat{\mathcal{G}}\end{array}\right) \bf{R}(\psi).\end{eqnarray}$$ (16)

Because $\bf{V}_0$ is invariant, as long as $\hat{\beta}$ and $\hat{\gamma}$ are small the dependence of $\bf{V}$ on the orientation of the focal plane remains weak.