the normalized surface-subtracted Stokes parameter $R(\chi_0,x)$.

Normalized Surface-Subtracted Stokes Parameter
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Since we do not rely on any specific models for the ALP and photon production mechanisms, the values of $A$, $\chi_0$ and $A_\perp$ which determine the initial ALP and photon amplitudes at the magnetar’s surface are not fixed. However, from it is clear that the differences of the Stokes parameters at infinity and at the magnetar’s surface does not depend on $A_\perp$. Moreover, since $\bar{I}_\parallel(\chi_0,x)\propto A^2$, their dependence on $A$ is simple. It is thus convenient in the following to study the normalized surface-subtracted Stokes parameter \[EqnR\] Alternatively, one can set $A=1$ in to compute the normalized surface-subtracted Stokes parameter . Apart from the implicit dependence on the ALP and magnetar parameters, the normalized surface-subtracted Stokes parameter is thus a function of the boundary condition $\chi_0$ only instead of the boundary conditions $A$, $\chi_0$ and $A_\perp$.

The transformation properties of the normalized surface-subtracted Stokes parameter (and its non-normalized counterpart) are easily obtained. They are \[EqnTR\] from the invariance of under the transformations $\chi(x)\to\chi(x)\pm\pi$, and respectively. Therefore implies that the normalized surface-subtracted Stokes parameter is a periodic function of $\chi_0$ with period $\pi$ which is even in $\chi_0$ with respect 