.
We will continue with this Magneto-optic Effect.
Ah ah with the. Faraday rotation, now this
ah transmission mode look at the microscopic
description picture, Zeeman splitting, then
other configurations like longitudinal configuration,
circular magnetic birefringence, Faraday Effect
we will continue with that. Then the transverse
configuration, linear magnetic ah birefringence,
Voight defect, reflection mode. This MOKE,
polar, longitudinal and equatorial mode, absorption
mode, measurement of rotation angles .
So, as I mentioned that the parallel configuration
is known as the Faraday Effect which we will
again continue for ah the perpendicular configuration
Voight effect will we will take up next .
So, this magneto optic effect phenomenologically
this is related to the normal Zeeman effect.
Magnetic field causes the splitting of the
energy levels ah which shifts the original
resonant frequency nu of an absorption line
to slightly higher and lower frequencies nu
1 and nu 2.
Therefore, a double dispersion effect is generated
LCP light is dispersed at a at a frequency
nu 1 whereas, the right circularly polarized
light will be dispersed at a frequency nu
2. So, effectively the 2 frequencies of the
light will be different .
Therefore, there will be a birefringence.
See that when it is unperturbed then you have
this ah wave function ah .
And ah when it is when it is perturbed then
this wave function in presence of this there
is there has been a shift, there has been
a small ah dipole which will be the superposition
of this S like and P like orbitals. So, this
is expansion of the unperturbed orbitals .
Now, this P orbitals px and py they will correspond
to this px which is p plus which is equal
to px plus ip y and p minus these are the
2 different sense of rotation you can see
that px this is minus of ip y, this is plus
of ip y.
So, these 2 and they will correspond to the
transition which are this ah right circularly
and left circularly polarized light will be
the interaction.
So, now the spin orbit coupling will split
this ah L equal to 1 into effectively ah 4
lines ah 4 energy levels which are given by
this. These are already are known and there
will be transitions from this to this which
corresponds to this right circularly, from
here to here this corresponds to this right
circularly.
Whereas, this transition will correspond to
this ah left circularly and similarly this
transition will also correspond to the left
circularly. So, without magnetization these
are the ah the um energy level and then with
exchange splitting you have each of them split
into 2 energy levels with upward and downward
transition and then this is the exchange plus
spin orbit coupling, it results into ah this
4 transitions .
Now, this causes the LCP an RCP componency,
different refractive indices thus a retardation
of one component relative to the other that
gives rise to a net phase difference between
the 2 lights. And when this these lights are
emerging these components combine to give
an elliptical polarization whose major axis
is now rotated relative to the original polarization
direction.
So, the angle is called the Faraday rotation
angle .
You can see this you have an input polarized
light and the magnetic field is applied along
the length of the medium. And you can see
that the plane of polarization slightly rotates
and when emerging from the crystal this has
undergone a neat rotation of this much that
is represented by Faraday theta f which is
due to the . And this depends on the interaction
length of the magnetic field with the optical
light that is l and the magnetic field.
So, this we have seen. Now, Faraday rotation
is directly proportional to the magnetization
M ah and it is also directly proportional
to the length of the ah light path within
the material. So, this we have represented
ah in terms of the external magnetic field
b and now it is in terms of the magnetization
of the material. This proportionality constant
as I have mentioned is called the Verdet constant,
it can be represented by the related parameters.
Because of the linear dependence of theta
on the internal magnetization or in most cases
on the external magnetic field Faraday Effect
is is called an odd effect because, it linearly
the defect the the rotation is is linearly
proportional to the um magnetization M or
the magnetic field b. So, it is linearly proportional.
The faraday effect is referred to as the circular
magnetic birefringence or magnetic Dichroism
CMB .
Faraday rotation this we have seen that this
Verdet constant it depends on the the magnet
medium properties, ambient temperature and
also the wavelength of of the light that is
used to interact with the magnetic field within
the medium .
Now, we will continue this magnetization the
transmission mode magnetization in the sample
could be due to the presence of external magnetic
field or it could be because of the internal
magnetization as I have mentioned. This magnetization
ah could be parallel and perpendicular, now
we will consider the perpendicular configuration
which is the Voight effect.
And ah when the magnetic field is perpendicular
to the optical path the incident optical beam
is linearly polarized then a different type
of interaction with the material occurs.
This effect of perpendicular configuration
is known as the this is the this is the definition
of this Voight effect .
Instead of now, this mechanism for this we
will look at in a slightly different way that
instead of looking at the decomposition of
a linear or a plane polarized light into the
left circularly polarized light and right
circularly polarized components we will see
that the electric field of the light is now
either parallel or perpendicular to the magnetic
field.
The parallel component is no way affected
by the magnetic field as you know that this
electric field which is parallel to the magnetic
field ah will not affect this. Whereas, the
cross magnetic and electric field that is
the 2 perpendicular components they will oppositely
interact with.
So, this effect was discovered later in 1899
the unequal absorption or phase shift of the
linear polarizations which are parallel and
perpendicular to the magnetization direction.
So, the light beam or the polarization of
the light beam is parallel and this is perpendicular,
it may be it may be somewhere in between this.
So, we will decompose this polarization of
the linear polarized light into parallel and
perpendicular components. They the relevant
ah permittivity of the medium in presence
of the magnetic field we will not ah talk
about this. This is the direction of the magnetization
and the this effect is quadratic in the magnetization
that is the M square.
And this is also called the cotton mouton
effect or linear magnetic Dichroism. So, in
that case this ah intensity the difference
of the intensity due to the perpendicular
polarization and parallel polarization will
be the effective intensity available when
the light beam is um exiting from the medium
of interaction.
So, this interaction results again in a difference
of the refractive indices that is plus minus
delta n.
Thus a a a birefringence of the 2 perpendicular
components will be observed. This will lead
to a rotation ah of the plane of polarization
that is theta V ah this ah which will be equal
to ah your K 0 twice pi upon lambda delta
n and ah is proportional to the length of
the medium ah in which the optical beam interacts
with the magnetic field.
This effect results in a linear magnetic birefringence
and is independent of the orientation of the
field which is axial. It is proportional to
the square of the magnetic field M square
ah if it is we call the if we look at the
magnetization of the material and if we ah
consider the ah field, external magnetic field
then it is proportional to H square. And that
is why this effect is called the even effect
.
So, in the reflection mode now we have reflected
light from the the surface of the magnetized
material in all different configurations of
the direction of magnetization with respect
to the optical path. This effect is known
as the the Kerr effect magneto optic Kerr
effect.
You see in Kerr effect measured in the reflected
light and it was discovered in the year 1876
an incident beam is incident and is reflected
ah in presence of the magnetic field ah which
is ah ah intersecting with this material.
And the result will be the change in the polarization
properties. Whereas, in the Faraday Effect
which we have seen that a an optical beam
is passing through the medium and the medium
properties will be affected the optical properties
of the medium will be influenced by the external
magnetic field.
So, this magneto optic Kerr effect are called
MOKE this ah you have an incident at as.
Suppose, you have an incident s polarized
light we have learnt what is s polarized light
and p polarized light in the beginning of
this ah hs. s polarized light ah this is the
input polarization and the perpendicular component
is the p polarized light. So, when they are
incident on the surface of a material or or
of a medium ah which is in in a magnetic field
then the reflected light will also contain
p polarized light. Because, there was no p
polarized light initially it was purely s
polarized light and now that on reflection
we can see a small component of the p polarization
has appeared because of the reflection.
And there has been an effective rotation of
the plane of polarization. You can see that
there has been an effective rotation of the
plane polarized light. On reflection at magnetized
sample p polarized wave also appears as a
result of this . 3 different configurations
of the light with respect to the magnetization
in the sample, this tells you that when the
light is incident there could be 3 possible
configurations with respect to the magnetic
field, magnetic field could be along this,
could be along this or could be along this.
So, you have 3 possible orientation of. The
magnetic field these are these are the ah
orientations let us suppose we call this this
is x, y and z axis. So, along z axis if the
magnetic field exists then and the incident
light is being incident on this surface at
this point and getting reflected. So, this
configuration is the is the polar ah configuration.
Whereas, if the magnetic field is is along
this direction that is along y direction and
the light beam is incident and reflected this
configuration is called the longitudinal configuration.
It is it is along the along the plane of the
incidence of the light beam. The other configuration
is the transverse MOKE configuration, that
is in which the magnetic field is now perpendicular
to the plane of incidence and you have the
optical beam which is incident here and reflected
this. So, these 3 effects are in general different.
So, these are the corresponding permittivity
relations which are used to analyze the properties
of the reflected light ah in presence of the
ah in presence of magnetization of the material
in 3 different orientations.
So, analysis of the reflection mode is ah
actually quite involved because of the oblique
nature of incidence and one expects a a mixture
of all possible polarizations in each direction.
A simplified analysis which is based on the
electromagnetic boundary conditions at the
reflecting surface for polar polar configurations
at normal incidence give some insight into
the effect. For example, if you consider the
Fresnel coefficient for this case for the
case of polar incidence you see this one this
polar incidence you have incident light and
you have reflected light. It can have s polarized
light and p polarized light both the components
could be present in that case because of reflection
the Fresnel coefficients for normal incidence
ah the reflected amplitude will contain this
refractive index term.
And this gives the hints. Dependence of r
on n the reflectivity , 
this this reflection coefficient it depends
on n that gives you the hints that the difference
in the value of n for the LCP and RCP because,
LCP and RCP they will be again composed of
the ah polarizations and as a result there
will be a ah difference in the amplitude and
phase shift between the 2 lights.
If the incident light is now linearly polarized
the reflected light will be elliptically polarized
because of the difference the difference in
the amplitude. And also the phase ah since
the 2 circular components that compose the
incident beam will no longer be of equal amplitude
because of the reflection. So, because of
the reflection coefficient the 2 components
will have different amplitudes on reflection
from the surface and as a result their amplitudes
are different and also they are ah in phase
not equal in phase and this will lead to that
is ah as it has happened in the case of Faraday
Effect. So, there will be a change in the
polarization .
The main axis of the ellipse now rotates by
an angle ah theta K, this K is for Kerr rotation.
So, which is relative to the original direction
of our polarization angle theta K is is ah
very small of the order of minutes .
In the equatorial configuration the components
of the of the reflected lights as calculated
show that ah the only component parallel to
the field is affected by magnetization. And
and the the reflected light ah will the amplitude
of the reflected light is is linearly proportional
to the the strength of magnetization that
is the parallel component of the reflected
light the amplitude would be proportional
to the the magnetization value.
So, this is unlike this quadratic Voight effect
and which has a thus configuration is similar,
but the effect is different .
Because, of the uniaxiality in the reflection
mode for this equatorial configuration when
magnetization direction is reversed that is
anti parallel then effect this directly the
intensity of the reflected light. And this
change can be detected without using an analyzer
which means that considerably increase in
the efficiency.
This intensity measurement will will be will
be ah visible and that tells that the the
rotation is quite large a typical value of
theta K in the visible range is about 10 minutes
is about 10 minutes.
Now, this measurement of rotation angles ah
theta F, theta V and theta K. So, rotation
they can be measured by different methods
static method involves using Babinet compensator
polarizer and analyzer setup ah as a function
of the change in the magnetic field.
It could be by a dynamic method ah in which
the magnetization is periodically switched
on and off and ah one can use a lock in amplifier
that records the intensity variation as a
as a function of the change in the magnetic
field.
So, this absorption mode ah in this case the
this effect is the difference in the absorption
coefficient of the polarizations that is polarized
light that is LCP and RCP components of the
light. This ah the difference of the polarization
ah they slightly they are absorbed by different
amount by sample of ah magnetized ah which
is in the beam direction.
Then linear magnetic Dichroism that also exists
which is similar to the Voight effect and
ah magnetic field is transverse to the light
path in this in this particular ah configuration.
CMD and LMD that is this circular magnetic
Dichroism, linear magnetic Dichroism they
can give some insight on the they are used
to can be used as ah in experiments to over
have an insight of the band structures of
crystals.
And the magnetically induced transition levels
to study this property they are used this
circular magnetic Dichroism and linear magnetic
Dichroism properties.
Magnetic field used problem, now this ah we
will just look at the ah few methods of magneto
optic measurements using cross polarizer,
vibrating polarizer, ah measurement of ellipticity.
So, this ah this crossed nicol prism technique
here you have the source of light ah then
you have a pair of nicol prisms, one is the
polarizer, the other one is the analyzer.
And you have a magnetic field which is perpendicular
to the direction of the beam path and ah you
measure the intensity of the light because
this because of this crossed nicol prism um.
You know for this kind of amplitude modulation
setup we have a pair of ah polarizer polarizers,
one is called polarizer the other one is analyzer.
And if there is a rotation of the of the polarization
axis of the optical beam then the intensity
varies which is detected by a detector placed
in the path of the exiting light. And the
intensity variation as a as a function of
the rotation of the plane of polarization
of the optical beam is shown here.
So, this is how this is the rotation angle
and this is the intensity. So, it goes through
a maxima and then it becomes a minima and
this will be periodically as you keep on increasing
the magnetic field. So, this is again the
ah depends on the length of interaction of
the magnetic field with the um ah with the
optical beam .
Vibrating polarizer technique here also you
have a pair of polarizer and analyzer. But,
this time this magnetic field is along the
along the direction of the light propagation.
You can see this, you have a crystal ah magneto
optic crystal and the light is traveling through
it and ah you measure the detected intensity
um as a function of the magnetic field ah
by a detector at the light output.
The rotating analyzer technique in this case
you have the polarizer which is fixed and
the analyzer is as a as a function of magnetic
field because, this is again the um you know
parallel configuration Faraday configuration
in which the magnetic field is along the ah
direction of the light propagation and as
a function of the strength of the magnetic
field the rotation occurs and ah because of
the rotation there will be a fall in the light
intensity.
But, if you rotate the analyzer then you can
measure the same intensity at different rotation
angle of the analyzer. So, and you note down
the ah angle of rotation of the analyzer.
So, from there we can make an estimate of
the ah estimate of the magnetic field the
rotation as a function of the magnetic field
.
Then this Faraday rotation this is also a
very good and interesting ah um laboratory
experiment. You have a polarizer um and the
source of light is passing through the polarizer.
So, therefore, the exiting light is a plane
polarized light. You have a ah magneto optic
material and or else it could be a current
passing through ah current passing through
a a solenoid which is given by I equal to
I 0 plus delta I sin pt. And as a result if
the the because of the variation of the current
ah the plane of polarization variation of
the current the magnetic field also varies.
And the which will result in the change in
the periodic change in the plane of polarization
which is detected by a detector. Ah this magnetic
field and ah this ah detector current they
are fed into an Op-amp which is again connected
to the system.
So, in this way which is a which is called
a Faraday modulator in which as a function
of the input current the magnetic field changes.
Magnetic field in turn changes varies the
plane of polarization and ah because you have
an analyzer. So, this is placed between a
polarizer and analyzer therefore, the intensity
also changes.
This change in the intensity will change the
detector current which will ah which will
be um added through an Op-amp and then in
the differential mode and you get this Faraday
rotation measurement .
Now, this ellipticity measurement in this
setup you have the electric field E 0 which
of this ah um of this elliptically polarized
which can be decomposed into E 0 ah sign eta
and E 0 cosine eta.
This one and you have another component. So,
this goes through a quarter wave plate the
resulting light can again be decomposed. So,
this is the direction of the electric field
E prime which can be written as this because,
it passes through a quarter wave plate we
add a phase E to the power of minus I pi by
2 and effectively this becomes ah cosine eta
I plus sin eta i.
So, this is the direction of the rotation
of this plane polarized light. So, in this
way we can measure the ellipticity of the
um elliptically polarized light which is the
due to the ah as a function of the magnetic
field .
So, we discussed this ah transmission mode
ah the microscopic description ah um from
the Zeeman splitting longitudinal and circular
magnetic birefringence, Faraday rotation,
Voight effect in the 3 different configurations
of the ah reflection mode polar longitudinal
and equatorial mode, absorption mode measurement
of the rotation angles because of the magnetic
field.
Thank you very much .
