Generalized Faraday rotation

The polarization of radiation changes as it propagates through any birefringent medium. Birefringence implies that there are two natural wave modes which may be described by their polarizations, which are necessarily orthogonal to each other, and by , the difference in their wavenumbers. In a cold plasma the natural wave modes may be assumed circularly polarized for present purposes. The propagation effect is then Faraday rotation, which causes the plane of any linear polarization to rotate and which does not alter the degree of circular polarization. In a medium whose natural modes are linearly or elliptically polarized, the counterpart of Faraday rotation, which we refer to as ``generalized Faraday rotation'', can lead to a partial conversion of linear into circular polarization. Such conversion is the basis for the alternative mechanism for the production of circular polarization discussed in this paper.

Arbitrarily polarized radiation may be separated into an unpolarized component and a completely polarized component. In general the polarized component is elliptical, as illustrated in Figure 1. The directions and in Figure 1 define the major and minor axes of the polarization ellipse. An arbitrary elliptical polarization can be represented by a point, P, on the Poincaré sphere, as illustrated in Figure 2. The north and south pole correspond to opposite circular polarizations, and points on the equator separated by correspond to orthogonal linear polarizations, as indicated in Figure 2a. The cartesian components of the point P are related to the Stokes parameters, I, Q, U, V, through , , , which also define the parameters and , as illustrated in Figure 2b. Faraday rotation corresponds to changing at constant . Partial conversion of linear into circular polarization is possible through any process that causes to change.

  
Figure: (a) The Poincaré sphere, with the circular polarizations indicated at the poles and the linear polarizations at the equator; (b) the parameters , , q, u, v for an arbitrary point P on the sphere.

The natural wave modes of the medium are orthogonally polarized and hence they correspond to points at the opposite ends of a diagonal through the center of the Poincaré sphere. This diagonal defines an axis that is characteristic of the natural modes of the medium. Generalized Faraday rotation causes the point P to rotate at constant latitude relative to the axis defined by the natural modes of the medium. Faraday rotation, as usually defined, corresponds to circularly polarized natural modes, and then the axis about which this rotation occurs is the vertical axis. This causes the angle in Figure 1 to rotate at a rate per unit distance, s, along the ray path. Generalized Faraday rotation corresponds to modes with elliptical or linear polarizations, and then (provided the point P is not at one or other end of the axis defined by the two modes) the parameter also changes periodically along the ray path, implying a cyclic partial conversion of linear into circular polarization.

The transfer equation for the Stokes parameters due to generalized Faraday rotation is of the form (e.g., Melrose & McPhedran 1991, p. 188)
 

 
where s denotes distance along the ray path, and where T is the axial ratio of the polarization ellipse of one of the modes. (Interchange of the modes corresponds to , .)

The case where the natural modes are linearly polarized ( or T=0), as in a uniaxial crystal, is familiar in another context: a quarter-wave plate. For linearly polarized modes the axis defined by the two modes is in the equatorial plane of the Poincaré sphere. If the polarization point is initially on the equator at a longitude to this axis, then generalized Faraday rotation causes P to rotate about a great circle that passes through both the north and south poles. A quarter-wave plate uses this effect, with the thickness of the plate adjusted such that the rotation is through just of this great circle, so that the initial linear polarization (at to the planes of polarization of the two natural modes) is converted into circular polarization.

  
Figure 3: The representative point for radiation in an anisotropic medium rotates about the diagonal joining the points for the two natural modes.

An example of a more general case is illustrated in Figure 3 where the natural modes are highly elliptical () and a sample polarization point moves around the solid path, which is a circle at constant latitude relative to the axis shown by the solid arrow directed radially from the center of the sphere. It is apparent from this example that if the wave modes are elliptical or linear, then radiation that is initially linearly polarized develops a circularly polarized component as the polarization changes in a periodic manner along the ray path (e.g., Pacholczyk & Swihart 1970). The wave modes of a cold plasma are significantly elliptically polarized for a very small range of angles () about propagation perpendicular to the direction of the magnetic field. However, a more likely cause of significant elliptical polarization of the natural modes is the relativistic electrons themselves.

© Copyright Astronomical Society of Australia 1997