next up previous contents
Next: Polarization in Radio Astronomy Up: Polarimetry Previous: Polarimetry   Contents


Consider the simplest kind of electromagnetic wave, i.e. a plane monochromatic wave of frequency $\nu$ propagating along the +Z axis of a cartesian co-ordinate system. Since electro-magnetic waves are transverse, the electric field E must lie in the X-Y plane. Further since the wave is mono-chromatic one can write
{\bf E}(t) = E_x \cos(2 \pi \nu t) {\bf e}_x +
E_y \cos(2 \pi \nu t + \delta ) {\bf e}_y,
\end{displaymath} (15.1.1)

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 shown15.1 that the implication of this is that over the course of one period of oscillation, the tip of the electric field vector in general traces out an ellipse. There are two special cases of interest. The first is when $\delta = 0$. In this case the tip of the electric field vector traces out a line segment, and the wave is said to be linearly polarized. The other special case is when $E_x = E_y$ and $\delta = \pm \pi/2$. In this case the electric field vector traces out a circle in the X-Y plane, and depending on the sense15.2 in which this circle is traversed the wave is called either left circular polarized or right circular polarized.

As you have already seen in chapter 1, signals in radio astronomy are not monochromatic waves, but are better described as quasi-monochromatic plane waves15.3. 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 $\mathcal{E}(t)$. If the X and Y components of this complex analytical signal are $\mathcal{E}_x(t)$, and $\mathcal{E}_y(t)$, respectively, then the four Stokes parameters are defined as:

I &=& <\mathcal{E}_x \mathcal{E}^*_x> + <...
<\mathcal{E}^*_x \mathcal{E}_y> &=& (U - iV)/2.
\end{array}\end{displaymath} (15.1.2)

where the angle brackets indicate taking the average value15.4. The Stokes parameters as defined in equation (15.1.2) clearly depend on the orientation of the co-ordinate system. In radio astronomy it is conventional (see chapter 10) to take the +X axis to point north and the +Y axis to point east. It is important to realize that the Stokes parameters are descriptors of the intrinsic polarization state of the electro-magnetic wave, i.e. the Stokes vector $(I~Q~U~V)^T$ is a true vector. The equations (15.1.2) simply give its components in a particular co-ordinate system, the linear polarization co-ordinate system15.5. One would instead work in a circularly polarized reference frame, i.e. where the electric field is decomposed into two circularly polarized components, $\mathcal{E}_r(t)$, and $\mathcal{E}_l(t)$. The relation between these components and the Stokes parameters are:
I &=& <\mathcal{E}_r \mathcal{E}^*_r> + <...
<\mathcal{E}^*_r \mathcal{E}_l > &=& (Q-iU)/2.
\end{array}\end{displaymath} (15.1.3)

Interestingly, equations (15.1.3) are formally identical to equations (15.1.2) apart from the following transformations viz. $Q^{+} \rightarrow V^{\odot}$, $U^{+} \rightarrow Q^{\odot}$, $V^{+} \rightarrow U^{\odot}$, where the superscript $+$ indicates linear polarized co-ordinates and $\odot$ 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

P = {\sqrt{ Q^2 + U^2 +V^2} \over I}.
\end{displaymath} (15.1.4)

From equation (15.1.2) we have
I^2 - Q^2 -U^2 -V^2 =
- \left<\mathcal{E}_x \mathcal{E}_y\right>^2 \right)
\end{displaymath} (15.1.5)

and hence from the Schwarz inequality it follows that $0 \le P \le 1$ and that $P =1$ iff $\mathcal{E}_x = c \mathcal{E}_y$, where $c$ is some complex constant. For a mono-chromatic plane wave (equation (15.1.1)) therefore, $P =1$ or equivalently $I^2 = Q^2 +U^2 +V^2 $, i.e. there are only three independent Stokes parameters. For a general quasi mono-chromatic wave, $P < 1$, and the wave is said to be partially polarized.

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^2_x + E^2_y\\
Q &=& E^2_x - E^2_...
...x E_y \cos(\delta)\\
V &=& 2 E_x E_y \sin(\delta),

i.e. for a linearly polarized wave ($\delta = 0$) we have V = 0, and for a circularly polarized wave ( $E_x=E_y, \delta = \pm\pi/2$) we have $Q=U=0$. So $Q$ and $U$ measure linear polarization, and $V$ measures circular polarization. This intepretation continues to be true in the case of partially polarized waves.


... shown15.1
See for example, Born & Wolf `Principles of Optics', Sixth Edition, Section 1.4.2
... sense15.2
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.)
... waves15.3
Recall that as all astrophysically interesting sources are distant, the plane wave approximation is a good one
... value15.4
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.
... system15.5
These polaraization co-ordinate systems are of course in some abstract polarization space and not real space

next up previous contents
Next: Polarization in Radio Astronomy Up: Polarimetry Previous: Polarimetry   Contents