C Dual-beam Flat-field Corrections, and the E and F-Factors

This section describes the flat-fielding process for dual-beam data, and the E and F-factor corrections in mathematical terms.

Let the intensity in the $O$ and $E$ beams transmitted by the analyser be $I_o\left(\alpha \right)$ and $I_e\left(\alpha \right)$, where $\alpha$ is the effective analyser angle (i.e. twice the half-wave plate rotation angle). Malus’ law (see appendix B) gives:

where $I_p$ is the polarized intensity, $I_u$ is the unpolarized intensity, and $\theta$ is the angle between the plane of polarization and the reference direction.

The signals measured by the detector (before flat-fielding) are:

where $S_o$ and $S_e$ are the sensitivities of the detector to the $O$ and $E$ rays (these are independent of $\alpha$, but vary across the detector), and $E_{\alpha }$ is an exposure factor which takes into account any differences in exposure time, sky transparency, etc. Note, it is assumed that the $O$ and $E$ ray images are in a fixed position with respect to the detector in all exposures.

For target exposure $T_0$ (for which $\alpha$ is zero), the transmitted intensities are denoted as $I_o^t\left(0\right)$ and $I_e^t\left(0\right)$ and the measured signals as $M_o^t\left(0\right)$ and $M_e^t\left(0\right)$.

If the polarization of the flat-field surface is spatially constant, then the measured signals in the master flat-field will be proportional to the detector sensitivity functions $S_o$ and $S_e$. If the constants of proportionality for the $O$ and $E$ ray images are $K_o$ and $K_e$, then the measured signals in the master flat-field can be written as:

Target exposure $T_0$ is flat-fielded by dividing it by the master flat-field. Thus, the measured intensities after the flat-field correction ($M_o^c\left(0\right)$ and $M_e^c\left(0\right)$) are:

The target exposure $T_{45}$ is taken with an analyser angle of 90°  (accomplished by rotating the half-wave plate by 45° ), and the corresponding $O$ and $E$ ray intensities are $I_o^t\left(90\right)$ and $I_e^t\left(90\right)$, where:

In other words, exposure $T_{45}$ records the same intensities as exposure $T_0$, but swapped so that the $O$ ray becomes the $E$ ray, and vice versa. The measured target signals at this new analyser angle are:

These measured signals are flat-fielded to give the following corrected signals:

To simplify the notation, put $s_1=M_o^c\left(0\right)$, $s_2=M_e^c\left(0\right)$, $s_3=M_o^c\left(90\right)$ and $s_4=M_e^c\left(90\right)$. In other words, $s_1$ and $s_2$ are the flat-fielded $O$ and $E$ ray signals from exposure $T_0$, and $s_3$ and $s_4$ are the corresponding signals from exposure $T_{45}$. In order to calculate the polarization we need signals which are proportional to the incoming intensities, with a common constant of proportionality. In order to achieve this, we need to estimate the ratio of the exposure factors, $E_0$ and $E_{90}$, and the “F-factor”, $F$, where:

From the above expressions for the flat-fielded signals, it can be seen that:

We use this value of $F$ to correct the measured $E$ ray flat-fielded signals, $s_2$ and $s_4$ to get:

Summing the $O$ and corrected $E$ rays signals for exposure $T_0$ ($s_1$ and $s_2^{\prime }$) gives:

where $I_T$ is the total intensity (equal to the sum of the $O$ and $E$ ray intensities). Likewise, summing the $O$ and corrected $E$ rays signals for exposure $T_{45}$ ($s_3$ and $s_4^{\prime }$) gives:

From this, the ratio of the exposure factors $E_0$ and $E_{90}$ can be found by dividing these expression:

This ratio, together with the F-factor found earlier, allow the measured signals $s_1$ to $s_4$ to be corrected so that they all have a common calibration. An identical procedure can be applied to the other pair of target exposures ($T_{22.5}$ and $T_{67.5}$), leading to estimates of their exposure factors, and another estimate of the F factor.

Note, each pair of target exposures must be flat-fielded using the same master flat-field frame.