In this part of the tutorial we shall establish mathematical
expressions for the different polarizations of incident light.
First of all, remember that in part 6, we introduced the
complex polarization as a quantity with a real part equal to the optical
rotation and with an imaginary part equal to the ellipticity, see Eq.6.10. Under
the effect of an applied magnetic field to the sample, the real and imaginary
part of the complex polarization will change. This will therefore result in a
different complex polarization. Hence, overall we can say that the complex
polarization has rotated by a certain complex angle. It makes sense to call this
angle - the complex Kerr angle . So, to summarize:
(Eq.14.1)
Let us consider a coordinate system in the
plane of polarization, since this is where the action takes place, see Fig. 14.
Note that in the left panel is exactly the same figure as Fig. 13; I have only
added the z' axis, which is the z axis in our plane of polarization. Thus, in
the right panel you can see the effect of the Kerr effect on the complex
polarization.
Fig.
14 The coordinate system in the polarization plane.
It is immediately apparent that:
(Eq.14.2)
(Eq.14.3)
Plugging those expressions into Eq. 14.2, we obtain:
(Eq.14.4)
Note that this formula is valid for any
incident light polarization. In Fig. 14, we have represented P incoming
polarization but we can easily have used S as well. In that case, z' would have
been along Ei and we would have introduced a x' axis instead
of x.
In order to proceed further with Eq. 14.4,
we now need some way to express the values of the reflected field amplitudes for
left and right circularly polarized light. And the Fresnel formula can do just
that.
(Eq.14.5)
As we have seen in the very beginning of this tutorial (part
1), the linearly polarized light can be represented as a combination of left and
right circularly polarized light waves. Furthermore, those are equal. Therefore,
eliminating Ei, Eq 14.4 becomes:
(Eq.14.6)
Eq. 12.6 gives us the value of the refractive index for the
longitudinal MOKE. This can be approximated by:
(Eq.14.7)
And, we shall introduce the quantities A0
and A1 in order to simplify the notations:
(Eq.14.8)
With those conventions, Eq.14.6 reads:
(Eq.14.9)
We can now introduce the susceptibilities:
(Eq.14.10)
And we obtain the formula for the magneto-optical Kerr effect
in the longitudinal configuration, on S-polarized light:
(Eq.14.11)
14.2 S-polarized light, polar MOKE geometry
In the polar case, we simply need to give different values to A1 according to Eqs.13.1. Then, very similarly, we can simply
proceed from Eq. 14.9:
(Eq.14.12)
And we can express the magneto-optical Kerr effect in the
longitudinal configuration, on S-polarized light as:
For the P-polarized light, we simply have to start with the
appropriate Fresnel formula and the rest of the steps are exactly the same as
those presented above. I will therefore not go into all the mathematical
details. The final result is:
(Eq.14.14)
14.4 P-polarized light, polar MOKE geometry
Following the same reasoning, the formula for the polar case is:
