ABSTRACT
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.