  Polarization measurements from relativistic outflows are a valuable tool to
probe the geometry of the emission region and the microphysics of the particle
distribution. Indeed, the polarization level depends on: (i) the local magnetic
field orientation, (ii) the geometry of the emitting region with respect to the
line of sight, and (iii) the electron pitch-angle distribution. Here we
consider optically thin synchrotron emission and we extend the theory of
circular polarization from a point source to an extended radially expanding
relativistic jet. We present numerical estimates for both linear and circular
polarization in such systems. We consider different configurations of the
magnetic field, spherical and jetted outflows, isotropic and anisotropic
pitch-angle distributions, and outline the difficulty in obtaining the reported
high level of circular polarization observed in the afterglow of GRB 121024A.
We conclude that the origin of the observed polarization cannot be intrinsic to
an optically thin synchrotron process, even when the electron pitch-angle
distribution is extremely anisotropic.
