Jones Matrix Formalism for Faraday Effect and
Different Arrangements of MOKE
Ravi Prakash Singh
1811122
15/03/2021
Abstract
1
Objective
• To theoretically find the Verdet constant of a magnetic sample due to
Faraday effect.
• To theoretically analyse different configuration of Magneto-optic Kerr
Effect(MOKE) and to find is there exist any component of Kerr rotation or Kerr ellipticity.
2
Rough
E=nH depends on material
Knowing E is enough
3
Jones matrix
The polarisation of light can be described by two components namely s and
p components of polarisations . The s and p component of the polarisation
corresponds to the vertical and horizontal component with respect to the
plane of incidence. When light passes through any optical instrument the
output light can again be described by the s and p components . Using
this fact a square matrix which operates on a column vector giving another
column vector where the column vector hold the information about the light
and the operator signifies how the s and p component changes when light
passes through the instrument associated with the matrix .And the intensities of the output light can be calculated using square of the matrix as
I ∝ E 2 . This is known as the Jones matrix formalism .
1
The two magneto-optic effects we will see here and use Jone matric on them,
one is Faraday effect and other is magneto-optic Kerr effect(MOKE).
Also how can the jones matrix formalism can be used to theoretically calculate the effect of these effects on light(EMW) .
4
Optical Elements and its Jones Matrix
Jones matrix formalism is a theoretical tool which is used to describe the
effect of an optical instrument when light passes through it using a 2X2
square matrix acting on a 2X1 column matrix . We can multiply matrix
of the instrument used in the order they are kept in the experiment. the
resultant column vector is the detected light¿ the output matrix is squared
to obtain the intensity of the output light. The output intensity is used to
calculate the kerr rotation and ellipticity.
A laser with certain wavelength of light is emitted on the polarizer which is
then passing through magnetic sample, Photo-elastic modulator, and then
going through a analyzer it is collected with photo detector and the data can
be analysed with computer. A electromagnet is used to give magnetic field to
the sample. A schematic representation of longitudinal MOKE arrangement
is shown in the fig.
The incident linearly polarised light can be represented as
E=
4.1
Ep
Es
i
(1)
Polarizer
Polarizer is a optical filter which allows light waves of specific polarization
and blocks the others. In Jones matric formalism it is expressed as
cos2 (α)
sin(α) cos(α)
P =
(2)
sin(α) cos(α)
sin2 (α)
where α is the angle of polarization.
4.2
Magnetic sample
A magnetic sample is used where the polarized light is getting reflected.
Usually a ferromagnetic sample is used. The Jones martix for sample is
rp eiδp
rps eiδps
S=
(3)
−rps eiδps rs eiδs
The diagonal terms are magnetization independent and called as fresnel
reflection coefficient and the off-diagonal terms are responsible for MOKE.
2
The coefficients rij are the ratio of the incident j polarized electric field and
reflected i polarized electric field. The complex can be written as
p
ΘpK = θK
+ iεpK =
−rps eiδps
−rps i(δps −δp )
e
=
iδ
p
rp
rp e
(4)
rps eiδps
rps i(δps −δs )
=
e
iδ
s
rs
rs e
(5)
s
ΘsK = θK
+ iεsK =
So,the cos components are Kerr rotation and the sin components are Kerr
ellipticity.
5
Faraday Effect
Faraday Effect is the rotation of the polarization plane of linearly polarized
light when the light propagates through a medium along the direction of an
applied magnetic field. Every material shows Faraday rotation. The extent
of the rotation can be shown with the following equation:
∆ϕ = V BL
(6)
where ∆ϕ is the rotation angle in degrees
B is the magnetic field in Tesla
L is the sample length in meters
V is a material-dependent constant known as the Verdet constant.
The Faraday effect is caused by left and right circularly polarized waves
propagating at slightly different speeds, a property known as circular birefringence. Since a linear polarization can be decomposed into the superposition of two equal-amplitude circularly polarized components of opposite
handedness and different phase, the effect of a relative phase shift, induced
by the Faraday effect, is to rotate the orientation of a wave’s linear polarization. In this experiment our goal is to find the Verdet constant.
When a plane polarised light falls on an magnetic material the angle of
polarization get rotated.
Let ϕ be the angle of rotation of plane polarised light by the magnetic
sample. So, ϕ = V B0 Lcos(ωt).
And let the polarizer is at an random angle θ such that 0o < θ < 90o .
I is the resultant intensity after passing through the analyzer and I0 is the
initial intensity of the plane polarized light.
3
From Malu’s law we have
I = I0 cos2 (ϕ + θ)
= I0 (cos ϕ cos θ − sin ϕ sin θ)2
1
2
2
2
2
= I0 cos ϕ cos θ + sin ϕ sin θ − sin 2ϕ sin 2θ
2
2
2
= I0 cos θ − ϕ sin 2θ + O ϕ
(∵ ϕ ≪ 1)
2
= I0 cos θ − V B0 L cos(ωt) sin 2θ
⇒
(7)
V B0 L sin 2θ
IAC
=
IDC
cos2 θ
= 2V B0 L tan θ
So, the Verdet constant(V) can be written as following:
V =
cot θ IAC
2B0 L IDC
(8)
Now the verdet constant can be written in terms of θ, IAC and IDC .
Now for different cases of polariser angles:
if θ =√30 degree.
AC
V = 2B03L IIDC
if θ =√45 degree.
AC
V = 2B03L IIDC
if θ = 90 degree.
IAC
1
V = 2√3B
L IDC
0
4