through an astrophysical sector consisting of the production region (either inside or closely by the magnetar), the plasma region and the region where the dipolar approximation is not valid. The amplitudes at the end of this sector are then used as initial amplitudes for the particles evolving through a vacuum sector consisting of the dipolar magnetic field of the magnetar with negligible plasma effects. It is now clear that our work focuses on the vacuum sector.

On the one hand, the ALP-to-photon conversion probability $P_{a\to\gamma}$ for pure ALP initial state in the limit of dipolar magnetic field depends on six ALP and magnetar parameters corresponding to the ALP energy $\omega$, the ALP mass $m_a$, the ALP-photon coupling constant $g$, the magnetar’s radius $r_0$, the (dimensionless) magnetar magnetic field at the surface $b_0=B_0/B_c$ and the angle between the direction of propagation and the magnetic field $\theta$. On the other hand, the normalized surface-subtracted Stokes parameter $R(\chi_0)\equiv R(\chi_0,\infty)$ depends also on the mixed initial state through $\chi_0$.

Fig. \[FigR\] shows the normalized surface-subtracted Stokes parameter for two points in the ALP and magnetar parameter space in function of $\chi_0$. A comparison with $P_{a\to\gamma}\cos(2\chi_0)$ is also shown, confirming the validity of (they are 