\[EqnTpiIb\] and \[EqnTpi2Ib\] Moreover, from the invariance under the transformation $\chi(x)\to\chi(x)\pm\pi$, the averaged intensities are periodic functions of $\chi_0$ with period $\pi$. Hence from the averaged intensities are even functions of $\chi_0$ with respect to $\pi/2$ while from the averaged intensities for some initial amplitudes are related to the averaged intensities for the inverted initial amplitudes. As a corollary of , it is straightforward to see that $\bar{I}_a(\pi/4,x)=\bar{I}_\parallel(\pi/4,x)=A^2/2$. Therefore, for a half-and-half initial state, the averaged intensities are exactly $1/2$ of the sum of the ALP and parallel photon initial intensities. Thus, it is only necessary to compute the averaged intensities for initial $\chi(1)\in[0,\pi/4)$ to determine the averaged intensities for all possible initial states. In fact, the next section demonstrates that we can do better.

Obviously, averaging over the perpendicular photon initial phase is of no consequence for the averaged perpendicular photon intensity, hence $\bar{I}_\perp(x)=A_\perp^2$ and the averaged perpendicular photon intensity remains constant.

Photon Polarizations {#SPP}
====================

This section introduces the Stokes parameters describing the polarization state of the photon signal. The effect of ALP-photon coupling is discussed qualitatively, with a more quantitative analysis for a particular magnetar presented in the next section.

Stokes Parameters
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At a given position 