Introduction

i.e. the X and Y components of the electric field differ in phase by a factor which does not depend on time. It can be shown

As you have already seen in chapter 1, signals in
radio astronomy are not monochromatic waves, but are better described
as quasi-monochromatic plane waves^{15.3}. Further, the quantity that is typically
measured in radio astronomy is not related to the field (i.e. a voltage),
but rather a quantity that has units of voltage squared, i.e. related
to some correlation function of the field (see chapter 4).
For these reasons, it is usual to characterize the polarization properties
of the incoming radio signals using
quantities called *Stokes * parameters. Recall that for a
quasi monochromatic wave, the electric field **E** could be considered
to be the real part of a complex analytical signal
.
If the X and Y components of this complex analytical signal are
, and
, respectively, then
the four Stokes parameters are defined as:

Interestingly, equations (15.1.3) are formally identical to equations (15.1.2) apart from the following transformations viz. , , , where the superscript indicates linear polarized co-ordinates and circular polarized co-ordinates. Although these two co-ordinate systems are the ones most frequently used, the Stokes vector could in principle be written in any co-ordinate system based on two linearly independent (but not necessarily orthogonal) polarization states. In fact, as we shall see, such non orthogonal co-ordinate systems will arise naturally when trying to describe measurements made with non ideal radio telescopes.

The degree of polarization of the wave is defined as

(15.1.4) |

(15.1.5) |

It is also instructive to examine the Stokes parameters separately for
the special case of a monochromatic plane wave. We have (see
equations (15.1.1) and (15.1.2)):

i.e. for a linearly polarized wave () we have V = 0, and for a circularly polarized wave ( ) we have . So and measure linear polarization, and measures circular polarization. This intepretation continues to be true in the case of partially polarized waves.

- ... shown
^{15.1} - See for example, Born & Wolf `Principles of Optics', Sixth Edition, Section 1.4.2
- ... sense
^{15.2} - Note that there is an additional ambiguity here, i.e. are you looking along the direction of propagation of the wave, or against it? To keep things interesting neither convention is universally accepted, although in principle one should follow the convention adopted by the IAU (Transactions of the IAU Vol. 15B, (1973), 166.)
- ... waves
^{15.3} - Recall that as all astrophysically interesting sources are distant, the plane wave approximation is a good one
- ... value
^{15.4} - Strictly speaking this is the ensemble average. However, as always, we will assume that the signals are ergodic, i.e. the ensemble average can be replaced with the time average.
- ... system
^{15.5} - These polaraization co-ordinate systems are of course in some abstract polarization space and not real space