@Umeda:1997da; @Paul:2018msp; @Maruyama:2017xzl] while the latter originates from the magnetar itself and interactions with the plasma in the magnetar’s atmosphere (*e.g.* parallel-perpendicular mode conversion in inhomogeneous magnetar’s atmosphere) [@Lai:2006af].

Hence, it is expected that at the magnetar’s surface, where the ALP and parallel photon fields start oscillating into one another as they travel outwards, their amplitudes can differ substantially depending on the actual values of the ALP parameters. Moreover, their phase difference is effectively random.

Thus, for given initial amplitudes corresponding to specific ALP parameters, *i.e.* for a specific $A$ and $\chi(1)=\chi_0$, it is natural to average over the initial phase difference $\Delta\phi(1)=\Delta\phi_0$ to determine the intensities away from the magnetar. This implies that, at a distance $xr_0$ from the magnetar’s surface, the averaged intensities $\bar{I}(x)$ for initial $\chi(1)=\chi_0$ are \[EqnIb\] Due to the singularities appearing for pure initial states and the equality of the intensities for the two different choices of phase difference, there is no averaging for pure initial states and only one initial phase difference (say $\Delta\phi_0=0$) is necessary to determine the averaged intensities at position $xr_0$.

Since the intensities transform as in under the transformation and under the transformation , the averaged intensities also behave as 