Reducing the four frames, with two object spectra and two sky spectra each, to a polarization spectrum is a fairly simple matter. However, be aware of the fact that you are generally trying to obtain information on a small fraction of a large signal. The difficult steps will be sky subtraction and correction for scattered light. Experiment with your data to find the procedure which best suits your requirements.
Since polarimetric analysis uses relative photometry of two spectra (the o and e-ray), it is essential to avoid unintentional loss of photons along the way. You will probably collapse the spectrum of the o and e-ray in the spatial direction; make sure you get all the counts in your extracted spectra, even if the star moved or the seeing changed during your observation. If the boundaries of your collapsing operation coincide with strong gradients in the spatial direction (collapse region too narrow, but possibly of use for maximising object/sky contrast), you may find some false polarization, since the o an e images will not be sampled identically by the CCD pixel structure and this mismatch may vary from one exposure to another. However, such false polarization will not be a fast function of wavelength and may not matter in your application; if you cannot live with it, you must widen the collapse region or correct your data for individual pixel gains first, as discussed in the section on Flat Fields.
The basic reduction steps leading to the Q and U spectrum (i.e. cosmic ray removal, extraction of object and sky spectra, sky subtraction, dividing spectra as outlined in Section 3.2 and 3.3) can all be done within FIGARO. However, there are software packages which make data reduction and assessment of the results much easier and faster as well. These software packages handle 1-dimensional spectra and are therefore not appropiate for pixel-by-pixel analysis of the data. Firstly there is the STARLINK software package Time Series Polarimetry (TSP, by J. Bailey). Secondly there is a set of FIGARO-based routines available written by J. Walsh. We have gained experience with the latter package and found it very easy to use, easy to install and adapt if necessary. Both packages are available on the La Palma data-reduction VAX. Before using these packages make sure you understand and agree with the way Q, U, P and are derived and handled.
Slightly modified versions of the above Walsh routines have now been
installed on the La Palma IRAF system and we consider this the most
convenient of the on-site alternatives. A description will be issued
as a La Palma Technical Note soon; in case of doubt, contact
Note : an easy mistake to make is to average the degree of
polarization P (e.g. co-adding polarization spectra or binning up
a polarization spectrum).
Remember that P is the modulus of a vector, and that adding moduli is
not equivalent to true vector addition.
Always use Stokes Q and U or and
to do arithmetic. Finally, re-compute P and .