Magneto-optic Effect - Gyrotropic Permittivity

Gyrotropic Permittivity

In particular, in a magneto-optic material the presence of a magnetic field (either externally applied or because the material itself is ferromagnetic) can cause a change in the permittivity tensor ε of the material. The ε becomes anisotropic, a 3×3 matrix, with complex off-diagonal components, depending of course on the frequency ω of incident light. If the absorption losses can be neglected, ε is a Hermitian matrix. The resulting principal axes become complex as well, corresponding to elliptically-polarized light where left- and right-rotating polarizations can travel at different speeds (analogous to birefringence).

More specifically, for the case where absorption losses can be neglected, the most general form of Hermitian ε is:

\varepsilon = \begin{pmatrix} \varepsilon_{xx}' & \varepsilon_{xy}' + i g_z & \varepsilon_{xz}' - i g_y \\ \varepsilon_{xy}' - i g_z & \varepsilon_{yy}' & \varepsilon_{yz}' + i g_x \\ \varepsilon_{xz}' + i g_y & \varepsilon_{yz}' - i g_x & \varepsilon_{zz}' \\ \end{pmatrix}

or equivalently the relationship between the displacement field D and the electric field E is:

where is a real symmetric matrix and is a real pseudovector called the gyration vector, whose magnitude is generally small compared to the eigenvalues of . The direction of g is called the axis of gyration of the material. To first order, g is proportional to the applied magnetic field:

where is the magneto-optical susceptibility (a scalar in isotropic media, but more generally a tensor). If this susceptibility itself depends upon the electric field, one can obtain a nonlinear optical effect of magneto-optical parametric generation (somewhat analogous to a Pockels effect whose strength is controlled by the applied magnetic field).

The simplest case to analyze is the one in which g is a principal axis (eigenvector) of, and the other two eigenvalues of are identical. Then, if we let g lie in the z direction for simplicity, the ε tensor simplifies to the form:

\varepsilon = \begin{pmatrix} \varepsilon_1 & + i g_z & 0 \\ - i g_z & \varepsilon_1 & 0 \\ 0 & 0 & \varepsilon_2 \\ \end{pmatrix}

Most commonly, one considers light propagating in the z direction (parallel to g). In this case the solutions are elliptically polarized electromagnetic waves with phase velocities (where μ is the magnetic permeability). This difference in phase velocities leads to the Faraday effect.

For light propagating purely perpendicular to the axis of gyration, the properties are known as the Cotton-Mouton effect and used for a Circulator.

Read more about this topic:  Magneto-optic Effect