3. Instrumental polarization at the prime focus

Once the angle of incidence i (in general) for a particular ray is known we can find the reflectivities and corresponding to the p and s component of the electric vector of the incident ray. These reflectivities are complex numbers and can be expressed as (Born & Wolf 1957):

where r is the angle of refraction, determined from the relation

is the complex refractive index of the telescope surface. In the present case, we shall assume , the complex refractive index of aluminium at wavelength = 5893 Angström (Sodium D line) (Born & Wolf 1957).

However, when the angle of incidence i is 0, we use the relation

The amplitudes of the reflected ray in the p and s-directions (denoted by and ) are related to the corresponding amplitudes for the incident ray ( and ) by the following relations:

Here it is to be noted that the p-directions of the vector for R and E are not same.

3.1. Case of an unpolarized star

For unpolarized incident light the electric vector of incident ray does not have any preference for a particular directions. Therefore we assume

so that the two components are equal in magnitude and the total intensity is 1. For any electromagnetic wave (with two orthogonal amplitude vectors), one generally defines a set of four Stokes parameters for the analysis of polarization (Shurcliff 1962). In the present case for and orthogonal components, the Stokes parameters in the p-s coordinate frame can be written as follows:

where and as the phase angles of and components. The fourth Stokes parameter v is related to the circular polarization which will not be considered in the present case.

Further, any set of three stokes parameters (I,Q,U) are related to the degree of linear polarization (P) and position angle of linear polarization (PA) by the following relations (Shurcliff 1962):

Now in order to find the net instrumental polarization combining all the rays, we should express all the Stokes parameters in a common coordinate frame. The choice obviously is the XY coordinate frame. This is because the polarizers and analyzers of polarimeter are normally placed in a plane perpendicular to the telescope axis (which in our case is the XY plane) and all the polarization measurements are done with respect to that plane.

Thus to transform the individual set of Stokes parameters into the XY frame, one should rotate the p-s frame by an angle . However, strictly speaking, this is not correct as the reflected rays are not parallel to the Z axis and accordingly the ps-plane is not parallel to the XY-plane. But in present day polarimeters the optical components (polarizers, analysers etc.) are supposed to take care of this oblique incidence of rays on them. Most of them have a correction facility (up to a few degrees of angle of incidence) and polarization measurements are made by the polarimeter as though all the rays are entering the polarimeter parallel to telescope axis. Therefore we can assume by rotating the ps-frame by an angle we can transform the Stokes parameter to an XY-frame. The new set of Stokes parameters under such a rotation can be expressed as (Chandrasekhar 1960)

In our case since each ray is unpolarized, we assume there is no systematic phase relation between its p and s components. Accordingly the time average value of can be assumed to be zero. The above i_XY, q_XY and u_XY values are functions of and field angle . These values are determined for each ray individually and the corresponding Stokes parameters are added to get the resultant Stokes parameter values ( and U) for the entire beam.

From this I, Q, U values we estimate the instrumental polarization and position angle by using relation (15a,b). An expression for instrumental polarization so produced at the prime focus can be written as:

where

The expressions for and are available in (10a,b).

3.2. Case of a polarized star

With the star on the axis of the telescope (i.e. ), we assume that the polarization vector (denoted by the corresponding electric vector "'') makes an angle with the X axis. For this ray the d.c. of the electric vector() will be . Now we assume the position of the star is rotated from the z axis and moved towards the x axis in the xz plane by an angle . In other words, if the field angle of the star is now , the new d.c. of the electric vector () will be now . Here, and are the two unknown parameters to be determined. Since the incident ray and the electric vector(), are orthogonal to each other we should have

Again for the electric vector () we should have

Combining Eqs. (18) and (19), we finally get

Thus knowing the position angle of polarization vector we can find its d.c. . With and , as the components of -vector in the directions (pl_pi, pm_pi, pn_pi) and , they can be expressed by the relations:

Now substituting these values of and in (13a,b) we can get the values of and , which can in turn be used to calculate the Stokes parameters from relation (14a,b,c). But in the present case we shall not assume the Stokes parameter to be zero as the light is already polarized. Finally we can calculate the polarization values observed for a polarized star, as a measure of depolarization effect. When the field angle () is zero, for any value of such observed polarization value (P) can be expressed as:

where is the phase difference between the reflectivities and .