to $\pi/2$ and odd in $\chi_0$ with respect to $\pi/4$. As such, it can be expanded in Fourier modes as \[EqnRF\] with the modes given by \[EqnRFM\] The latter can be computed directly from since the integral is $\chi_0$-independent. Indeed, averaging over $\Delta\phi_0$ eliminates all dependence on the boundary conditions (this statement is highly non-trivial, for a proof the reader is referred to Appendix \[SApp\]). Then the Fourier modes are simply \[EqnRFMSoln\] which are $\chi_0$-independent and thus the normalized surface-subtracted Stokes parameter is \[EqnRFSoln\] It can be written in a more familiar form by setting $\chi_0=0$. Indeed, since is $\chi_0$-independent, we can set $\chi_0=0$ and replace the average over $\Delta\phi_0$ by $\Delta\phi_0=0$. Analogously, we can set $\chi_0=0$ directly in and compare both sides. Both approaches lead to our final result \[EqnRSoln\] where is the ALP-to-photon conversion probability at a distance $xr_0$ for pure ALP initial state. Therefore, the normalized surface-subtracted Stokes parameter factorizes into two parts, the ALP-to-photon conversion probability for pure ALP initial state $P_{a\to\gamma}(x)$ that is a function of the ALP and magnetar parameters, and $\cos(2\chi_0)$ which encodes the dependence on the mixed initial state. In addition, from it is clear that $-1\leq R(\chi_0,x)\leq1$ with $R(\pi/4,x)=0$ as proven 