# Introduction to Spectropolarimetric Measurements of Magnetic Fields

Most spectral lines split into several components if they are formed in a magnetic field. This was discovered by the Dutch physicist P. Zeeman, and after him, this splitting is called Zeeman-effect.

For the measurements of the magnetic field, those lines are best suited that split into one unshifted and two symmetrically shifted components – so-called Zeeman triplets. Such a case is shown in Figure 1.

The splitting follows the equation: Δλ [pm] = (geff ⁄ 854.7) (λ [nm] ⁄ 500)2 H [G], where λ is the wavelength, H the strength of the magnetic field measured in Gauss, and geff is a splitting factor (Landé factor) depending on the specific spectral line. Condition for a proper measurement is that the magnetic field is strong enough for a clear separation of the components. The splitting increases with wavelength, therefore spectral lines in the near infrared are often preferred.

An advantage for the observer is the fact that the light in the split components is differently polarized (). It depends on the direction of the magnetic field how the components are polarized. If the line-of-sight is parallel to the magnetic field, the light in the split components of a Zeeman triplet is circularly polarized, and the sense of rotation is opposite. At the same time, the central component disappears.

If the line-of-sight is perpendicular to the magnetic field,  all components are linearly polarized. Now, the central component is polarized in a plane perpendicular to that of the split components.

In most cases, one will observe the magnetic field under a certain angle, and one will measure a mixture of  circularly and linearly polarized light.

When the ratio of circularly and linearly polarized light is determined, and  the orientation of the plane of linear polarization is known, one can determine the direction of the magnetic field vector. Only an ambiguity remains, because one cannot distinguish a plane from another one rotated by 180°.

The polarization is described by the Stokes-parameters I, Q, U, and V named after the British physicist G. Stokes. I is the unpolarized part of the light, Q and U describe two different states of linear polarization, and V is the circular polarization (see polarized light). The Stokes parameters are demonstrated in Figure 2.

Unfortunately, the amplitudes of the Stokes parameters depend not only on the magnetic field, but also on temperature and other thermodynamical parameters. Therefore, the problem to determine the magnetic vector field cannot be solved analytically, but one has to proceed iteratively with so-called inversion codes.