Polarization information

When radiation=synchrotron_spectrum SOFT will also store information about the polarization of the detected radiation. Using the polimage and polspectrometer sycouts, it is possible to generate output files containing the polarization information in image or spectrum format. In this section usage and interpretation of the data will be briefly be discussed.

What information does SOFT store?

SOFT stores the four Stokes parameters, \(S\), \(Q\), \(U\) and \(V\), averaged over the relevant parameters (depending on which model is being used). The emitted synchrotron power per unit frequency, per unit solid angle, can be written in terms of the two quantities \(A_\parallel\) and \(A_\perp\) as

\[\frac{\mathrm{d}^2 P}{\mathrm{d}\omega\mathrm{d}\Omega} \propto \left| -\boldsymbol{\epsilon}_\parallel A_\parallel + \boldsymbol{\epsilon}_\perp A_\perp \right|^2.\]

where \(\boldsymbol{\epsilon}_\parallel\) is a vector corresponding to polarization in the gyration plane, and \(\boldsymbol{\epsilon}_\perp\) to polarization in the plane orthogonal to that. It can be shown that the Stokes parameters can be expressed using \(A_\parallel\) and \(A_\perp\) through

\[\begin{split}I &\propto A_\parallel^2 + A_\perp^2,\\ Q &\propto \left( A_\perp^2 - A_\parallel^2 \right)\cos 2\beta,\\ U &\propto \left( A_\perp^2 - A_\parallel^2 \right)\sin2\beta,\\ V &\propto 2A_\parallel A_\perp \cos 2\beta.\end{split}\]

The angle \(\beta\) is the angle between the plane of parallel polarization and the plane in which the horizontal polarization is measured. The first Stokes parameter, \(I\), is just the intensity of the radiation as obtained also from the SOFT image sycout.

The fourth Stokes parameter \(V\) is often quoted as identically zero in the literature, a result stemming from that the object \(A_\parallel A_\perp\) is odd in the angle \(\psi\) between the guiding-center’s emission cone and a line-of-sight. When averaged over all emission angles, the contribution to \(V\) therefore cancels identically. In the angular and spectral distribution implemented in SOFT however, we do not neglect the finite emission width, and therefore obtain a finite contribution to the \(V\) parameter, since it is possible for only part of the emission cone to overlap the detector (corresponding to “cut-offs” in the integration over emission angle).

For a derivation of the full \(\frac{\mathrm{d}^2 P}{\mathrm{d}\omega\mathrm{d}\Omega}\) , see for example Jackson’s “Electrodynamics”, Landau-Lifshitz “The Classical Theory of Fields” or Mathias Hoppe’s Master’s thesis (link).

File format

The polimage sycout of SOFT outputs a variable-based file (such as SDT, HDF5 or Matlab) containing the following variables:

Variable	Description
detectorPosition	Vector specifying the position of the detector
detectorDirection	Central viewing direction of the detector
detectorVisang	Vision angle of the detector
StokesI	Stokes \(I\) parameter
StokesQ	Stokes \(Q\) parameter
StokesU	Stokes \(U\) parameter
StokesV	Stokes \(V\) parameter
wall	Wall data used for the simulation