### 3 Single-beam Polarimetry

This section gives a brief introduction to the general principles of single-beam polarimetry. It explains words and concepts used later, and defines the data model used by POLPACK.

Single waveband polarimetry is described here but the principles can be extended to imaging spectropolarimetry (see section 4).

#### 3.1 The Polarimeter

An exposure from a single-beam polarimeter consists of an image of the sky in a single state of polarization. Some form of analyser within the polarimeter removes all but the required state of polarization from the incoming light, which then goes on to be recorded on a suitable imaging device such as a CCD camera. A series of exposures is usually taken with a different state of polarization being recorded by each exposure. These exposures allow both the degree and orientation of the polarization to be estimated.

Several analyser arrangements are commonly used. POLPACK can handle data from the following forms:

(1)
A single analyser which passes only light polarized parallel to a specified axis. The analyser is rotated to measure light polarized in different directions.
(2)
Several fixed analysers, each of which passes only light polarized parallel to its axis. The required analyser is inserted into the light path to measure light polarized in different directions.
(3)
A single fixed analyser which passes light polarized parallel to its axis. A half-wave plate is placed in the light path in front of the analyser to rotate the plane of polarization of the incoming light before it reaches the analyser. Light polarized in different directions can thus be measured by rotating the half-wave plate.

A half-wave plate has a preferred axis. Light polarized parallel to this axis passes through the half-wave plate unchanged. Light polarized perpendicular to the axis is retarded by half a wavelength. The net effect of this is to rotate the plane of polarization of the light so that the axis of the half-wave plate bisects the angle between the planes of polarization in the incoming and outgoing light.

This form of polarimeter is shown diagrammatically in Figure 4.

There will be a reference direction within the focal plane. The orientation of the rotating analyser or half-wave plate is specified by giving the angle between the reference direction and the analyser (or half-wave plate). In POLPACK, the reference direction is specified by giving the anti-clockwise angle from the first image axis to the reference direction. This angle is usually referred to as $ANGROT$, and is specified in degrees.

The combination of a fixed analyser and a rotating half-wave plate can be thought of as equivalent to a single rotating analyser, in which the analyser rotates twice as fast as the half-wave plate. If the anti-clockwise angle from the reference direction to the half-wave plate is $WPLATE$, then the effective analyser position is given by:

$\begin{array}{rcll}\varphi & =& 2.WPLATE& \text{}\end{array}$

This is the angle from the reference direction to a hypothetical analyser which would give the same effect as the combination of the fixed analyser and half-wave plate (see Figure 5).

The recorded intensity for an analyser position (or effective analyser position) of $\varphi$ will be:

$\begin{array}{rcll}{I}_{rec}& =& \frac{t}{2}\left(I+ϵ.\left(Q.cos2\varphi +U.sin2\varphi \right)\right)& \text{(1)}\text{}\text{}\end{array}$

Here $I$, $Q$ and $U$ are the Stokes parameters describing the incoming partially plane polarized light. $I$ is the total intensity, i.e. the sum of the polarized and unpolarized intensities. If a fraction $p$ of the incoming light is totally plane polarized, then:

$\begin{array}{rcll}{I}_{p}& =& I.p& \text{}\\ I& =& {I}_{p}+{I}_{u}& \text{}\end{array}$

where ${I}_{p}$ is the polarized intensity, and ${I}_{u}$ is the unpolarized intensity. Q measures the intensity polarized parallel to the reference direction, and U measures the intensity polarized perpendicular to the reference direction. They are given by:

$\begin{array}{rcll}Q& =& {I}_{p}.cos2\theta & \text{}\\ U& =& {I}_{p}.sin2\theta & \text{}\end{array}$

Here, $\theta$ is the anti-clockwise angle from the reference direction to the direction of polarization of the incoming light.

The two other values, $t$ and $ϵ$ in the above expression for the record intensity, ${I}_{rec}$, are the analyser transmission and the analyser efficiency. The transmission measures the total throughput of the analyser, and the efficiency measures the ability of the analyser to select a single state of polarization. A perfect analyser would have a value of $1.0$ for both. A perfectly bad analyser such as a piece of high quality glass, would have a transmission of $2.0$ and an efficiency of zero. If you know the transmission and efficiency of your analysers, then POLPACK can take account of them when estimating the values of the Stokes parameters, $I$, $Q$ and $U$. If you do not known them, then don’t worry... just use the default values of $1.0$ for both. These will probably be acceptable since instrument makers usually go to some trouble to make their analysers as perfect as possible. If in doubt, you could try re-processing your data with different values for the analyser transmission and efficiency. This will give you some idea of how sensitive the final results are to the values used.

#### 3.2 The Observational Procedure

At least three exposures with different analyser or half-wave plate positions are required to estimate the degree and orientation of the polarization. However, if you can, you should take some extra analyser positions. Not only does this reduce the noise, but it also allows consistency checks to be performed, since the recorded intensity should be a sinusoidal function (see equation 1) of analyser or half-wave plate position. POLPACK provides facilities to reject data values which are more than a given distance from the best fitting sine curve (see appendix E).

#### 3.3 The Data Reduction

There are many similarities between the reduction of single-beam dual-beam data. The basic differences are that each target exposure contains only a single image of the sky, instead of the two images produced by a dual-beam polarimeter, and that rotation between images is allowed. The various steps involved are summarised below. For more details see section 2.3. Details of how POLPACK can be used to implement these ideas in practice are given in later sections of this document.