before.

Using , the surface-subtracted Stokes parameters are \[EqnStokesDiffSoln\] which implies that the Stokes parameters and the averaged intensities are \[EqnStokesSoln\] and \[EqnIbSoln\] respectively.

The behavior on the mixture of the initial state has been completely uncovered in , , and . Indeed, the dependences on $A$, $\chi_0$ and $A_\perp$ are exhibited explicitly. Hence it is easy to determine the behavior of the averaged intensities or the Stokes parameters under changes of the mixed initial state. For example, the Stokes parameter $Q(\chi_0,x)$ at a distance $xr_0$ is always smaller (larger) than or equal to the Stokes parameter $Q(\chi_0,1)$ at the magnetar’s surface if $\chi_0\in[0,\pi/4]$ ($\chi_0\in[\pi/4,\pi/2]$), *i.e.* if the initial state is a mixture dominated by ALP (parallel photon).

Moreover, the ALP-to-photon conversion probability for pure ALP initial state is the only necessary quantity to compute in order to determine all the remaining quantities of interest. This important observation simplifies greatly the problem since $P_{a\to\gamma}(x)$ is independent of $\chi_0$ and no averaging on $\Delta\phi_0$ is necessary.

The remarkable factorization of the dependence on the initial state in the averaged intensities is a consequence of the averaging over the initial phase difference. If the average had not been performed, there would have 