“Voigt effect”的意思、由来-开放百科全书

The Voigt effect is a magneto-optical phenomenon which rotates and elliptizes linearly polarised light sent into an optically active medium.^[1] Unlike many other magneto-optical effects such as the Kerr or Faraday effect which are linearly proportional to the magnetization (or to the applied magnetic field for a non magnetized material), the Voigt effect is proportional to the square of the magnetization (or square of the magnetic field) and can be seen experimentally at normal incidence. There are several denominations for this effect in the literature: the Cotton–Mouton effect (in reference to French scientists Aimé Cotton and Henri Mouton), the Voigt effect (in reference to the German scientist Woldemar Voigt), and magnetic-linear birefringence. This last denomination is closer in the physical sense, where the Voigt effect is a magnetic birefringence of the material with an index of refraction parallel (

) and perpendicular

) to the magnetization vector or to the applied magnetic field.

For an electromagnetic incident wave linearly polarized

and an in-plane polarized sample

, the expression of the rotation in reflection geometry is

is :

where

is the difference of refraction indices depending of the Voigt parameter

(same as for the Kerr effect),

the material refraction indices and

the parameter responsible of the Voigt effect and so proportional to the

in the case of a paramagnetic material.

Theory

As with the other magneto-optical effects, the theory is developed in a standard way with the use of an effective dielectric tensor from which one calculates systems eigenvalues and eigenvectors. As usual, from this tensor, magneto-optical phenomena are described mainly by the off-diagonal elements.

Here, one considers an incident polarisation propagating in the z direction:

the electric field and a homogenously in-plane magnetized sample

where

is counted from the [100] crystallographic direction. The aim is to calculate

where

is the rotation of polarization due to the coupling of the light with the magnetization. Let us notice that

is experimentally a small quantity of the order of mrad.

is the reduced magnetization vector defined by

the magnetization at saturation. 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.

Dielectric tensor

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

take the following form :

where

is the material dielectric constant,

the Voigt parameter,

and

two cubic constants describing magneto-optical effect depending of

is the reduce

. Calculation is made in the spherical approximation with

. At the present moment,{{When|date=July 2016}} there is no evidence that this approximation is not valid, as the observation of Voigt effect is rare because extremely small with respect to the Kerr effect.

Eigenvalues and eigenvectors

To calculate the eigenvalues and eigenvectors, we consider the propagation equation derived from the Maxwell equations, with the convention

. :

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

may have all his three components equals to zero making calculations rather more complicated and making Fresnels equations no longer valid. A way to simplify the problem consists to use the electric field displacement vector

. Since

and

we have

. The inconvenient is to deal with the inverse dielectric tensor which may be complicated to handle. Here calculation is made in the general case which is complicated mathematically to handle, however one can follow easily the demonstration by considering

Eigenvalues and eigenvectors are found by solving the propagation equation on

which gives the following system of equation :

where

represents the inverse

element of the dielectric tensor

, and

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

and first order of

. This led to the two eigenvalues corresponding the two refraction indices :

Reflection geometry

Continuity relation

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

the reflected electromagnetic vector usually detected in experiments. We use the continuity equations for

and

where

is the induction defined from Maxwell's equations by

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

. The system of equation to solve is :

Calculation of rotation angle

The rotation angle

and the ellipticity angle

are defined from the ratio

with the two following formulae:

where

and

represent the real and imaginary part of

. Using the two previously calculated components, one obtains:

where

is the difference of refraction indices. Consequently, one obtains something proportional to

and which depends on the incident linear polarisation. For proper

no Voigt rotation can be observed.

is proportional to the square of the magnetization since

and

Transmission geometry

The calculation of the rotation of the Voigt effect in transmission is in principle equivalent to the one of the Faraday effect. 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 replace by a field

(

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

Consider the transmitted electromagnetic waves

propagating in a medium of length L. From equation (5), one obtains for

and

where

and

are the eigenvectors previously calculated, and

is the difference for the two refraction indices. The rotation is then calculated from the ratio

, with devellopement in first order in

and second order in

. This gives :

Again we obtain something proportional to

and

, the light propagation length. Let us notice that

is proportional to

in the same way with respect to the geometry in reflexion for the magnetization. In order to extract the Voigt rotation, we consider

and

real. Then we have to calculate the real part of (14). The resulting expression is then inserted in (8). In the approximation of no absorption, one obtains for the Voigt rotation in transmission geometry:

Illustration of Voigt effect in GaMnAs

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.^[3] At low temperatures (in general for

) for a material with an in-plane magnetization, (Ga,Mn)As exhibits a biaxial anisotropy with the magnetization aligned along (or close to) <100> directions.

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. In contrast to the common longitudinal/polar Kerr effect, the hysteresis cycle is even with respect to the magnetization, which is a signature of the Voigt effect. 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 follow :

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

are approximately twice of

Theory

Dielectric tensor

Eigenvalues and eigenvectors

Reflection geometry

Continuity relation

Calculation of rotation angle

Transmission geometry

Illustration of Voigt effect in GaMnAs

See also

References

Further reading