We have seen in Sect. 2.1 that observations for at least two positions
of the half-wave plate are required to determine the three
unknown parameters namely, the total,
intensity (I), fraction of light in linearly polarized condition (p),
and position angle of the plane of polarization (
).
However, the situation is not so simple in reality because - (i) the responsivity of the system to the two orthogonal
polarization
components may not be the same, and (ii) the responsivity of the CCD
is a function of the position on its surface. Due to these
effects the signals which are actually measured in the two images
(
and 
) are given by
where 
and 
represent the
effects mentioned above.
In order to ensure that and do not change during
observations of an object, the positions of the two images are kept fixed on
the CCD. Further, as the analyzer is fixed with respect to the
detector, the orientation of the ordinary and extraordinary polarizations
also remain fixed with respect to it, independent of the polarization vector
of the incident beam. Therefore
the ratio of the factors and can be
estimated as
by making use of the fact that a rotation of the half-wave plate by 45
simply leads to an interchange of the signals in the ordinary and extraordinary
images. Now the actual ratio of the fluxes in the two images may be recovered
as
This ratio is substituted in Eq. (2) and a cosine curve is fitted to the four
values of 
obtained so as to make the best estimates of p and
.
Since CCDs exhibit both intrapixel as well as pixel to pixel sensitivity
variations, ideally the images should be stable to within a
small fraction of a pixel during exposures at different positions of the
half-wave plate. But since this is difficult to achieve, it is advisable to
dither the telescope pointing atleast by 
pixels along each axis
and to get an FWHM of greater than 3 pixels.
Since the grid placed at the telescope focal plane covers almost half the field, several exposures at slightly different telescope orientations might be required to cover the entire object field. Alternatively, in the case of stellar fields with slowly changing (in intensity and polarization) background, the grid may be removed during observations, provided the field is not too crowded.
A polarimetry package has been developed for data analysis within the IRAF environment using a mixture of standard IRAF tasks, custom-made CL scripts and FORTRAN routines. PSF fitting tasks of the DAOPHOT package are used to determine accurately the centroids of the stellar images. The intensity estimates are, however made using aperture photometry covering a diameter greater than 2 FWHM so as to integrate more than 90% of the signal.
In order to keep a check on the errors, it is useful to take multiple
exposures for each position of the half-wave plate.
To ensure that the normalized Stoke's parameters follow a normal
distribution as closely as possible, the signals measured at each position of
the half-wave plate, should
be averaged together before taking their ratios to give the normalized
Stoke's parameters. However, this method has its demerits too. Firstly, it
leaves the non-linear 
fitting technique with only one degree of
freedom, to estimate the two parameters p and from the four
values of . Secondly, it does not allow the estimation of errors
in the measurement of the Stoke's parameters, since there are not enough
degrees of freedom. Therefore, it appears optimum to use a fitting technique
which estimates p and , from all the values of obtained
from individual exposures.