Magnetization and the dielectric
permittivity tensor
We saw in part 9 of this tutorial that the propagation
equation for the electric field of an electromagnetic wave in vacuum and in a
dielectric material differ by a single quantity - the dielectric permittivity
tensor. In part 10, we obtained a very simple form of this equation by
evaluating it in the Fourier domain. Now, we need the expression of the
dielectric tensor. So, what is this thing anyway?
The difference between Maxwell's equations in vacuum and in
the dielectric medium is the induced polarization, which was described in part 8
of the tutorial. The induced polarization, or density of electric dipoles, gives
rise to an electric field that opposes the one from the light wave. If the
polarization is very large, then we can imagine a situation where it could
completely nullify the propagation of an electromagnetic wave in the medium. We
see then that the permittivity acts as an "attenuator" of the propagating light,
i.e. it can "permit" it to go through and specifies how much of it is
"permitted" to go through.
Furthermore, since it is the only quantity that accounts for
the interaction of light and matter, we expect it to contain any information
regarding the presence and direction of magnetization in the material. This is
easy to understand intuitively - a magnetic field will influence an electric
dipole via the Lorenz force:
(Eq.11.1)
where q is the charge, E the electric field,
B is the magnetic induction and v is the instantaneous velocity.
Consequently, it affects the induced polarization and thereby it must appear in
the dielectric permittivity tensor.
Then, for an arbitrary direction of the magnetization
can be written as:
(Eq.11.2)
The diagonal components of this tensor are nonmagnetic and
the off-diagonal ones contain the magnetization contribution. We will now
consider the case of the Magneto-Optical Kerr Effect (MOKE) since it is more
general than the treatment of the Faraday effect.
Fig. 10 shows the
schematic orientation of the experiment.
Fig. 10 Schematic
illustration of the Kerr effect in the three different configurations.
There are three different Kerr configurations with respect to the optical
(y-z) plane: transverse with M along (1,0,0), longitudinal with M
along (0,1,0) and polar with M along (0,0,1), see Fig. 10. In the
longitudinal case, it can be said that the magnetization along the y direction
results from electrical current loops in the x-z plane (see Fig. 11).
Fig.
11 In the longitudinal Kerr configuration, the magnetization along y results
from current loops in the xz-plane.
Since the medium is
isotropic, mirror reflection operations should not affect it. If we now
perform a mirror reflection operation in the x-y plane, we can see that this
will result in reversing the direction of the current loop producing the
magnetization, i.e. the magnetization direction will be reversed. This
operation reduces the symmetry and therefore all components of the dielectric
tensor associated with it (namely ) should be zero. The same argument applies for the mirror
reflection in the y-z plane, i.e. . Only the mirror operation in the x-z plane conserves the
direction of M and therefore
(see Fig 12).
Fig.
12 Mirror reflections in the xy (left on the figure) and yz (right on the
figure) result in a reversal of the direction of magnetization.
By following a similar reasoning the tensor components for
the polar and transverse configuration can be found.
The wave equation
in Fourier domain (Eq. 10.13 ) becomes then:
(Eq.11.3)
We can simplify further this identity by expressing n
in terms of the angle of refraction, see Fig. 13.
(Eq.11.4)
Fig.
13 Reflection (incidence) and transmission angles.
Eq. 11.3 can now be rewritten as:
(Eq.11.5)
And this is fairly easy to solve for the three different
geometries of the Kerr effect.
