Voigt effect

The Voigt effect is a magneto-optical phenomenon which rotates and elliptizes linearly polarised light sent into an optically active medium.

[1] The effect is named after the German scientist Woldemar Voigt who discovered it in vapors.

is the difference of refraction indices depending on the Voigt parameter

As usual, from this tensor, magneto-optical phenomena are described mainly by the off-diagonal elements.

the electric field and a homogenously in-plane magnetized sample

is the rotation of polarization due to the coupling of the light with the magnetization.

is experimentally a small quantity of the order of mrad.

We emphasized with the fact that it is because the light propagation vector is perpendicular to the magnetization plane that it is possible to see the Voigt effect.

Following the notation of Hubert,[2] the generalized dielectric cubic tensor

two cubic constants describing magneto-optical effect depending on

When the magnetization is perpendicular to the propagation wavevector, on the contrary to the Kerr effect,

A way to simplify the problem consists to use the electric field displacement vector

The inverse dielectric tensor can seem complicated to handle, but here the calculation was made for the general case.

Eigenvalues and eigenvectors are found by solving the propagation equation on

After a straightforward calculation of the system's determinant, one has to make a development on 2nd order in

Knowing the eigenvectors and eigenvalues inside the material, one have to calculate

the reflected electromagnetic vector usually detected in experiments.

Inside the medium, the electromagnetic field is decomposed on the derived eigenvectors

In practice, this configuration is not used in general for ferromagnetic samples since the absorption length is weak in this kind of material.

However, the use of transmission geometry is more common for paramagnetic liquid or cristal where the light can travel easily inside the material.

The calculation for a paramagnetic material is exactly the same with respect to a ferromagnetic one, except that the magnetization is replaced by a field

For convenience, the field will be added at the end of calculation in the magneto-optical parameters.

In the approximation of no absorption, one obtains for the Voigt rotation in transmission geometry:

As an illustration of the application of the Voigt effect, we give an example in the magnetic semiconductor (Ga,Mn)As where a large Voigt effect was observed.

A typical hysteresis cycle containing the Voigt effect is shown in figure 1.

This cycle was obtained by sending a linearly polarized light along the [110] direction with an incident angle of approximately 3° (more details can be found in [4]), and measuring the rotation due to magneto-optical effects of the reflected light beam.

This cycle was obtained with a light incidence very close to normal, and it also exhibits a small odd part; a correct treatment has to be carried out in order to extract the symmetric part of the hysteresis corresponding to the Voigt effect, and the asymmetric part corresponding to the longitudinal Kerr effect.

In the case of the hysteresis presented here, the field was applied along the [1-10] direction.

The switching mechanism is as follows: The simulation of this scenario is given in the figure 2, with

As one can see, the simulated hysteresis is qualitatively the same with respect to the experimental one.

Schematic of the polar Kerr effect, longitudinal Kerr effect and the Voigt effect
Framework and coordinate system for the derivation of Voigt effect. , and are referring to the incident, reflected and transmitted electromagnetic field.
Fig 1 : a) Experimental hysteresis cycle on a planar (Ga,Mn)As sample b) Voigt hysteresis cycle obtained by extracting the symmetric part of (a). c) Longitudinal Kerr obtained by extracting the asymmetric part of (a)
Fig 2 : a) Switching mechanism of an in-plane (Ga,Mn)As sample for a magnetic field applied along the [1-10] axis at 12 K. b) Voigt signal simulated from the mechanism showed in a)