Magneto-optics

When a magnetic field H is applied to a magnetic medium (crystal), a change in the magnetization M within

the medium will occur as described by the constitution relation of the Maxwell equations M ¼ w

·H where w

is the magnetic susceptibility tensor of the medium. The change in magnetization can in turn induce a

perturbation in the complex optical permittivity tensor e

. This phenomenon is called the magneto-optic

effect. Mathematically, the magneto-optic effect can be described by expanding the permittivity tensor as a

series in increasing powers of the magnetization (Torfeh et al., 1977) as follows:

where

Here, j is the imaginary number. M

, M

, and M

are the magnetization components along the principal

crystal axes X, Y, and Z, respectively. e

is the permittivity of free space. e

is the relative permittivity of the

medium in the paramagnetic state (i.e., M ¼ 0), f

is the first-order magneto-optic scalar factor, f

is the

second-order magneto-optic tensor factor, d

is the Kronecker delta, and e

is the antisymmetric alternate

index of the third order. Here we have used Einstein notation of repeated indices and have assumed that the

medium is quasi-transparent so that e

is a Hermitian tensor. Moreover, we have also invoked the Onsager

relation in thermo-dynamical statistics, i.e., e

(M) ¼ e

(M). The consequences of Hermiticity and Onsager

relation are that the real part of the permittivity tensor is an even function of M whereas the imaginary part is

an odd function of M. For a cubic crystal, such as yttrium-iron-garnet (YIG), the tensor f

reduces to only

three independent terms. In terms of Voigt notation, they are f

, f

, and f

. In a principal coordinate system,

the tensor can be expressed as

where Df ¼ f

f

2f

:

In the principal crystal axes [100] coordinate system, the magneto-optic permittivity reduces to the

following forms:

where * denotes complex conjugate operation. The elements are given by

paramagnetic state

Faraday rotation

Cotton-Mouton effect

In order to keep the discussion simple, analytic complexities due to optical absorption of the magnetic

medium have been ignored. Such absorption can give rise to magnetic circular dichroism (MCD) and

magnetic linear dichroism (MLD). Interested readers can refer to Hellwege (1978) and Arecchi and Schulz-

DuBois (1972) for more in-depth discussions on MCD and MLD.