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3. Sources of errors

In addition to the fundamental limit due to photon statistics, the measurement of polarization is affected by several factors some of which are discussed in the following subsections. However, the effects of flat-field errors and their remedy needs particular attention and is treated in detail in Sect. 5.

3.1. Atmospheric effects

In IMPOL, two orthogonal polarization components are measured simultaneously and the Stoke's parameter is obtained from the ratio of the fluxes in these components (Eq. 2). Since the atmosphere is not birefringent, this eliminates the effects due to atmospheric scintillation, or for that matter any effect which changes both the polarization components by the same factor - like variations in the effective exposure times of observations, presence of thin clouds etc.

3.2. Distortions in polarization

All the optical elements are anti-reflection coated to minimize the polarization efffects due to reflection at their surfaces, and care is taken to minimize the stray light, reflected from the walls etc., reaching the detector. In particular, all the aperture stops in the beam path are chosen to be non-metallic, including the grid in the focal plane, so as to avoid polarization of light scattered from these (Pospergelis 1965). The mounts for field lens, the half-wave plate and the Wollaston prism have been designed to minimize stress-birefringence due to differential thermal expansion.

The retardance introduced by the half-wave plate could deviate from 180tex2html_wrap1285\ either because of the finite angle of incidence or because of the chromatic effects. If the beam is incident at a small angle i, the maximum change in retardance is given by (derived from expression in Serkowski 1975)
where tex2html_wrap_inline1317 and tex2html_wrap_inline1319 are the refractive indices of the material for ordinary and extraordinary rays. For the aperture used, the maximum angle of incidence is about 5tex2html_wrap1285 so that tex2html_wrap_inline1323 rad. The chromatic effects give tex2html_wrap_inline1325 rad for the wideband and tex2html_wrap_inline1327 rad for the V-band. It can be shown (Serkowski 1974) that the depolarization tex2html_wrap_inline1331 due to an uncertainty of tex2html_wrap_inline1333 rad in the retardance is to the lowest order given by tex2html_wrap_inline1335, p being the fractional polarization. Therefore even for the wideband tex2html_wrap_inline1339. Further, circular polarization in the incident light is converted to a linear polarization of magnitude tex2html_wrap_inline1341, where V is the circularity parameter. But in typical observations this does not pose a serious problem because the circular polarization is usually less than the linear polarization.

If the position angle of the half-wave plate fast axis changes with wavelength, it renders the measurement of tex2html_wrap_inline1257 erroneous. The achromatic half-wave plate used in IMPOL does not produce any appreciable dispersion in its fast-axis position angle over the wavelength range of interest.

The uncertainty tex2html_wrap_inline1347 involved in the positioning of the half-wave plate leads to a maximum error in the measurement of linear polarization given by tex2html_wrap_inline1349. Thus, tex2html_wrap_inline1351 for tex2html_wrap_inline1353.

3.3. Photon noise

From Eq. (3) it can be shown that the variance due to photon noise in the measurement of each of the Stoke's parameters tex2html_wrap_inline1361 is given by
where tex2html_wrap_inline1363 and tex2html_wrap_inline1365 are the number of photoelectrons and k is the ratio of the flux from the background to that from the source. For small values of p, tex2html_wrap_inline1363 and tex2html_wrap_inline1365 are approximately equal and denoting tex2html_wrap_inline1375, the standard deviation of p can be written as
where NB = kN for observations with the grid and tex2html_wrap_inline1381 for observations without the grid (see Sect. 5). Similarly, we can also show that

It is worth noting here that p, as defined above, is a positive definite quantity and follows the Rice probability distribution given by

Here, p0 is the true value of fractional polarization being estimated by p and I0 is the modified Bessel function of order zero. Since p is a biased estimator, several schemes have been suggested (Simmons & Stewart 1984 and references therein) for debiasing, but none of them is fully satisfactory. For values of tex2html_wrap_inline1393 larger than about 4, almost all the debiasing schemes agree and reduce approximately to the relation
where tex2html_wrap_inline1395 is the "debiased" estimate of p0. Thus it is desirable to work with the normalized Stoke's parameters as far as possible and use the quantities p and tex2html_wrap_inline1257 only to present the final results. However, in general, the normalized Stoke's parameters themselves might not be normally distributed and might have bias (Clarke et al. 1983). Besides, for photon-noise dominated measurements, the positive kurtosis of the distribution will lead to erroneous estimates of the confidence levels unless more than a few thousand photoelectrons are collected. Thus, in order to arrive at an optimum procedure, it is essential to carefully study the nature of the dominant sources of noise in the measurement and their effects on the analysis (see Sect. 5).

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