pieces: the ALP-to-photon conversion probability and a factor that encodes the composition of the initial state. Specfically, normalizing the surface-subtracted Stokes parameter and defining the quantity where $A$ is related to the initial amplitudes of the ALP and parallel photon states, we find that \[EqnRfact\] This is our main result. It has several implications. Firstly, the spatial dependence of the Stokes parameters, as well as their dependence on the properties of the magnetar and the ALP parameters like the mass and coupling constant, are all encoded in the ALP-to-photon conversion probability $P_{a\to\gamma}$. Secondly, given the conversion probability, one can obtain the value of $R$ simply by scaling with the appropriate initial condition $\cos(2\chi_0)$. Most importantly, the factorization of $R$ in can be utilized to provide an analytic expression for the Stokes parameter $Q$ in , which is the parameter we prefer when discussing the observational aspects of our work.

We utilize the methods of our previous paper [@Fortin:2018ehg] to display the dependence of $R$ on the photon energy as well as ALP parameters. As a benchmark point, we take the CAST-allowed values for the ALP mass and coupling, $m_a=10^{-8}\,\text{keV}$ and $g/e=5\times10^{-17}\,\text{keV}^{-1}$, respectively. The dependence of $Q$ on $\bar{I}_\parallel/\bar{I}_\perp$ and $\bar{I}_a/\bar{I}_\perp$ 