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This fixes severe errors in the implementation of the Zeeman effect from my 2014 paper. It also fixes severe errors from the paper itself - I will contact JQSRT to ask how to submit an errata.

All the problems are listed:

  1. The paper writes equation (20) as $w(z) = F_a + iF_b$. This is false, $w(z) = F_a + 2iF_b$. This should be clear from equation (19), where that extra division by two is present. This factor propagates to/from equation (5) where a scaling factor is missing. The code implemented equation (5) and (20) as written, resulting in a factor 2 too high of a magneto-optical effect.
  2. The matrices (11)-(14) are wrong. This can be understood from the paper itself. $\pm45$ polarization is rotated clockwise 45 degrees from vertical polarization and horizontal polarization, respectively. However, I apply matrix (A.7) to perform this rotation. I create the matrices (11)-(14) via $\mathrm{K}=\mathrm{L}\mathrm{K}_m\mathrm{L}^{-1}$ but as written in the paper should have applied $\mathrm{K}=\mathrm{L}^{-1}\mathrm{K}_m\mathrm{L}$, since the inverse of a rotation matrix is a rotation in the inverse direction. This causes what matrix (15) describes as $A_U$ and $B_U$ to flip sign.
  3. The matrices (11)-(14) are probably wrong yet again. The sign of the $B_V$ component should be the opposite that of the $A_V$ component. But they have the same sign in the paper.
  4. The code has decayed. I do not know when it happened, but at some point I made enough changes to the code that the $\sigma_-$ lines had become $\sigma_+$ and vice verse.
  5. The code has decayed even more. I have been rotating the linear polarization counter-clockwise. So ARTS have been representing 135/45 polarization rather than $\pm45$ in the Zeeman module. Well, except for the previous error in rotation which means it is not clear exactly what polarization mode has been used - just not the right one.

Overall, all of these errors are subtle and you would have problems spotting them in $I \pm Q$, though you will be hard pressed to make $I \pm Q$ work as well as $I \pm U$ and $I \pm V$.

This should be merged ASAP, but it should be known that we might flip the sign on the $V$ polarization if we find that it disagrees with scattering calculations. The current implementation is right-handed.

@olemke olemke changed the title Fix Zeeman effect clockwise angles and a spurious ratio [ARTS-2.6] Fix Zeeman effect clockwise angles and a spurious ratio Oct 7, 2025
@olemke olemke merged commit b4d10af into atmtools:v2.6.x Oct 7, 2025
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