is shown in Fig. \[FigStokesQ\], where $\bar{I}_a$ is the phase-averaged ALP intensity.

In the absence of ALPs, astrophysical modeling of thermal and hard X-rays from magnetars predicts mainly X-mode polarization, for which the electric field is perpendicular to the plane containing the external magnetic field and the direction of propagation. For the strong magnetic fields of magnetars, the polarization in the X-mode is expected to be especially pronounced due to the vacuum birefringence effect. The polarization radius, which is the location where the polarization vector stops tracking the magnetic field, is large for strong magnetic fields and the overall polarization is enhanced.

The unique observational signature of ALP-photon conversion is the change in the predicted polarization pattern. ALPs add to the astrophysical picture described above by producing O-mode photons, for which the electric field is parallel to the plane containing the external magnetic field and the direction of propagation. We compute the radius of conversion, where the probability of conversion becomes significant, and find that it is typically of the same order or larger than the polarization radius, implying an overall O-mode superposed on the X-mode coming purely from astrophysics. These results are displayed in terms of the Stokes parameter 