# Synchrotron Polarization

## Cyclotron Polarization

Recall for cyclotron radiation, the emitted power pattern as a function of angle looks like a torus centered around the acceleration vector (right), that is if the center hole’s size is infinitely small. This behavior is due to the ${\displaystyle \sin(\theta )^{2}}$ term in the Larmor Formula. The polarization of Larmor (or cyclotron) radiation is linear and directed along the “kink” in the electric field created by the acceleration of the charge. For Larmor radiation, the polarization is therefore parallel to the acceleration vector (i.e. along the red line in the figure to the right). One way to remember this is that polarization of Larmor radiation is linear and follows a “donut cut” along the emitted torus-shaped power profile (red line), as opposed to a bagel cut (blue line).

Now imagine one non-relativistic electron spiralling around a magnetic field. If we were to look at this electron along the magnetic field direction we would see the electron carve out a circular path around the magnetic field lines (i.e. face-on). Now try to picture the behavior of the donut-shaped power pattern as the electron spirals around the magnetic field. If the electron’s orbit radius is small enough, it’s almost as if we took the donut and spun it on its side, like a nickel on a table top. If the red-line in the figure above represents the linear polarization of Larmor radiation, then we can see that Cyclotron radiation viewed face-on is circularly polarized. Remember that observing face-on refers to being not face-on with respect to the donut, but being face-on with respect to the circular path the electron traces out in its trajectory (which are perpendicular to each other!). This is similar to how a spinning nickel viewed from above has an observed edge that rotates over time.

Now image we view the same non-relativistic electron spiralling around a magnetic field but view it from the side, such that it traces out a trajectory that resembles a dipole (edge-on). You can picture the Larmor radiation pattern and see that the observed polarization is always the same throughout one cycle of the electron: it has linear polarization that is constant over time. Another way to state this is that the polarization of Larmor radiation is linear along the direction of acceleration, and because the projected acceleration vector always lies along a single line for cyclotron emission viewed edge-on, the polarization is also linear and constant over time.

If we don’t view the electron directly edge-on we get elliptical polarization that depends on the inclination angle of our observation.

## Synchrotron Polarization

Synchrotron emission due to highly relativistic electrons focuses the donut-shaped emission pattern into highly collimated beams. While we typically work with diagrams that show the beaming to occur in the plane of the orbit, in practice highly relativistic electrons also have large velocities along the magnetic field lines, which means their beamed radiation is viewed neither directly face-on nor edge-on (see Figure 6.5 in Rybicki and Lightman). This means that synchrotron polarization, for a single electron, is generally elliptically polarized. The handed-ness of the polarization, however, depends on whether or not us as observers lie directly above or below the center-line of the focused beam: the point of "maximal radiation" (again Figure 6.5). Because synchrotron polarization is highly collimated, for any reasonable distribution of particles (that also have a distribution of pitch angles) we can assume that we will view synchrotron radiation above for the center-line for some electrons and below it for other electrons, meaning we will be seeing both left-handed and right-handed polarized emission at the same time. These work to cancel themselves out to form mostly linearly polarized emission. For an ensemble of particles following a power-law distribution in energy to the ${\displaystyle -p}$ degree, the intensity of radiation that is linearly polarized can be shown to be about 70 percent of the total intensity: ${\displaystyle \Pi =(p+1)/(p+7/3)}$. Knowing that in practice ${\displaystyle p}$ ranges from 2 to 3, we get back that the degree of polarization is roughly 70 percent (Rybicki and Lightman pg 181). To derive this relationship yourself, see problem 6.5 in Rybicki and Lightman.