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 Io(α) and Ie(α), where α is the effective analyser angle (i.e. twice the half-wave plate rotation angle). Malus’ law (see appendix B) gives:

Io(α) = Ip. cos 2(αθ) + Iu 2 Ie(α) = Ip. sin 2(αθ) + Iu 2

where Ip is the polarized intensity, Iu is the unpolarized intensity, and θ is the angle between the plane of polarization and the reference direction.

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

Mo(α) = So.Io(α).Eα Me(α) = Se.Ie(α).Eα

where So and Se are the sensitivities of the detector to the O and E rays (these are independent of α, but vary across the detector), and Eα 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 T0 (for which α is zero), the transmitted intensities are denoted as Iot(0) and Iet(0) and the measured signals as Mot(0) and Met(0).

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 So and Se. If the constants of proportionality for the O and E ray images are Ko and Ke, then the measured signals in the master flat-field can be written as:

Mol = So.Ko Mel = Se.Ke

Target exposure T0 is flat-fielded by dividing it by the master flat-field. Thus, the measured intensities after the flat-field correction (Moc(0) and Mec(0)) are:

Moc(0) = Mot(0)/Mol = So.Iot(0).E0 So.Ko = Iot(0).E0 Ko Mec(0) = Met(0)/Mel = Se.Iet(0).E0 Se.Ke = Iet(0).E0 Ke

The target exposure T45 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 Iot(90) and Iet(90), where:

Iot(90) = Ip. cos 2(90 θ) + Iu 2 = Ip. sin 2(θ) + Iu 2 = Iet(0) Iet(90) = Ip. sin 2(90 θ) + Iu 2 = Ip. cos 2(θ) + Iu 2 = Iot(0)

In other words, exposure T45 records the same intensities as exposure T0, but swapped so that the O ray becomes the E ray, and vice versa. The measured target signals at this new analyser angle are:

Mot(90) = So.Iot(90).E90 = So.Iet(0).E90 Met(90) = Se.Iet(90).E90 = Se.Iot(0).E90

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

Moc(90) = Mot(90)/Mol(0) = So.Iet(0).E90 So.Ko = Iet(0).E90 Ko Mec(90) = Met(90)/Mel(0) = Se.Iot(0).E90 Se.Ke = Iot(0).E90 Ke

To simplify the notation, put s1 = Moc(0), s2 = Mec(0), s3 = Moc(90) and s4 = Mec(90). In other words, s1 and s2 are the flat-fielded O and E ray signals from exposure T0, and s3 and s4 are the corresponding signals from exposure T45. 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, E0 and E90, and the “F-factor”, F, where:

F = Ke/Ko

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

F = s1 s4 .s3 s2

We use this value of F to correct the measured E ray flat-fielded signals, s2 and s4 to get:

s2 = s2.F = Iet(0).E0 Ke . Ke Ko = Iet(0).E0 Ko s4 = s4.F = Iot(0).E90 Ke . Ke Ko = Iot(0).E90 Ko

Summing the O and corrected E rays signals for exposure T0 (s1 and s2) gives:

s1 + s2 = Iot(0).E0 Ko + Iet(0).E0 Ko = (Iot(0) + Iet(0)).E0 Ko = IT.E0 Ko

where IT 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 T45 (s3 and s4) gives:

s3 + s4 = Iet(0).E90 Ko + Iot(0).E90 Ko = (Iet(0) + Iot(0)).E90 Ko = IT.E90 Ko

From this, the ratio of the exposure factors E0 and E90 can be found by dividing these expression:

s3 + s4 s1 + s2 = IT.E90 Ko . Ko IT.E0 = E90 E0

This ratio, together with the F-factor found earlier, allow the measured signals s1 to s4 to be corrected so that they all have a common calibration. An identical procedure can be applied to the other pair of target exposures (T22.5 and T67.5), 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.