Electro-optic effect. Earlier we have discussed
the electromagnetic waves propagation an isotropic
and anisotropic medium. Now, we will see when
the medium is influenced by an external electric
field, what happens to the birefringence,
whether it is isotropic or anisotropic, there
may be a change in the optical properties.
So, we have organized this discussion, first
we will look at the understanding point of
electro-optic effect, then the index ellipsoid
its distortion under electric field corresponding
induced birefringence, then we will categorize
the two kinds of electro-optic effects the
Pockels effect and Kerr effect. We will look
at a certain class of crystals centrosymmetric
crystal ah which will remain unaffected as
regards this Pockels effect, but will exhibit
Kerr effect then we will look at the electro-optic
tensor.
So, as I have mentioned that earlier we studied
the propagation of light in isotropic and
anisotropic media, in anisotropic media there
may be a state of polarization change with
the propagation depending on the orientation
of the principal axes with respect to the
direction of propagation. But in presence
of an external electric field the refractive
index properties of the medium may undergo
change. There may be some changes in the the
refractive index properties of the medium.
And as a result it may induce birefringence
in otherwise isotropic medium and it may alter
the birefringence properties of the medium
which is already a birefringence medium.
That is to say that if it is an isotropic
medium it may become anisotropic, with certain
changes in the refractive index properties
and if it is already an anisotropic medium
there may be changes in the refractive indices.
So, that the new medium under the influence
of external field is again anisotropic, but
the properties are different. These effect
that is the effect caused by an external electric
field on a medium with respect to a propagating
electromagnetic waves in terms of the changes
in the permittivity that is the refractive
index properties of the medium is called electro-optic
effect. .
Light modulator, electro-optic effect causes
changes in the refractive index of a medium
with regard to an externally applied electric
field and if the medium is placed with right
configuration or orientation with a suitable
ah positioning of the medium with respect
to the propagating electromagnetic waves.
The birefringence that will be induced in
the medium can be electrically controlled
by the external by change in the magnitude
of the external electric field.
In the case of anisotropic medium this electrically
controlled anisotropic may altered the state
of polarization of the electromagnetic wave
that is propagating through the medium. A
half wave plate placed between two crossed
polarisers can yield an amplitude modulation,
that is if we have an analyser and the polariser
and in between there is a half wave plate
then if the property of the half wave plate
is modulated the amplitude of the light which
is exiting from the analyser will be modulated.
A retardation of the wave plate can be controlled
electrically, the wave plate that will be
placed which in this case will be an electro-optic
crystal, electro-optic medium that can be
controlled electrically and that will result
in optical modulation switching off light.
We will see this things in details.
Now, let us consider a medium a crystal which
is electro-optic crystal that is if I apply
a voltage that is an electric field across
the crystal and the light is propagating in
these direction with a polariser and an analyser
who are mutually at cross position. So, they
make an angle of 90 degree with each other
then we can see that this system can be used
as a as an amplitude modulator. Refractive
index of the electro-optic medium is a function
of the applied electric field.
And this effect when this strange refractive
index profile will cause a change in the modulation
of the light which are polarized linearly
polarized at one end and you look at the light
coming out of the analyser which is which
used passed pass axis used at 90 degree with
this. So, this configuration is very useful
and very widely used in amplitude modulation.
Now, let us try to understand ah simply how
this electro-optic effect comes into the picture.
Let us suppose you have a medium these are
the fixed nuclear and you have outer electrons,
this is the crystal lattice. And if you view
this crystal from two orthogonal directions
so lattice from this side or from this side
it appears identical when there is no electric
field. So, there is no effect of the electric
field on to the crystal as a result when you
look at the crystal from any direction it
does not appear to be different, so it looks
same when viewed different orthogonal directions.
On the contrary in presence of an electric
field there will be distortion in the electron
clouds that is simply looking at the electron
orbits the there has been a modification there
has been a change in the shape of this. So,
if you view from the two orthogonal directions
that is from this side and from this side
they will appear different. If you are applied
the electric field in this direction the modification
the distortion of the of the electron cloud
will will appear like this. So, it looks different
when viewed from the different orthogonal
directions. So, this electric field causes
distortion of the electron clouds which indices
an anisotropic response to the optical field.
So, this is a very simple way of understanding
the basic mechanism of electro-optic effect
you have a field, you have no field, there
is no change you have a field there is a change
in the in the electron cloud and as a result
you look at the same crystal from different
directions will appear to be different.
Now, we have already learnt the index ellipsoid
associated with the with the medium, with
an optical medium and if it is an isotropic
medium, and that to in absence of an electric
field the index ellipsoid is a sphere - n
x, n y and n z all of them are equal. So,
it represents a sphere that is in all directions
the electromagnetic waves will see the same
refractive index so which is represented by
this equation. But in presence of an electric
field E the same medium may become anisotropic
there may be distortion in the index ellipsoid
as shown here and consequently there may be
a change in all the n x, n y and n z values
representing the refractive indices along
x, y and z direction. As it is obvious from
this picture in that case the index ellipsoid
will be represented by this equation the which
is the general ellipsoid equation, ok.
This was the case when the medium was isotropic
and it may become anisotropic under the influence
of an external electric field, but if you
already have an anisotropic medium in absence
of the electric field it may be that all the
it may be a biaxial system that is n x not
equal to n y, not equal to n z, then it will
represent the ellipsoid equation by this,
equation n x, n y n z all are different but
this is in the principal axes system.
Now, if I apply an external electric field
along these direction there may be a distortion
of the ellipsoid. And there may be a corresponding
change in the n x, n y and n z the new values
are n x prime, n y prime and n z prime all
of them may be un equal and the general equation
of the ellipsoid it will be represented by
this in this case. From here we can again
find out the principal refractive indices
by making some transformation of this equation
we will learn and we will see in details with
examples.
The distortion of the ellipsoid when a steady
electric field is applied to the medium there
is a change in 1 by n square term, that is
the change will be represented by delta 1
by n square that occurs in all 6 terms of
the ellipsoid. Look at this we have all 6
terms, everywhere there may be a change in
the property of 1 by n x, n y n z square.
For example, if I have an isotropic medium
which is represented by this because all the
refractive indices principal refractive indices
are n 0, n 0, n 0 same they may respond to
the electric field by modifying the ellipsoid.
A very simple example to understand that there
will be an additional incremental change in
the refractive index value in this way in
this form, in this from. So, it is associated
with all the 6 terms in the index ellipsoid.
For an anisotropic medium in principal axes
system this equation will be like this n x,
n y n z maybe in general all of them are different;
n x, n y if there equal then will call uniaxial
if all of you have seen that and if all of
them are different that is a biaxial system.
So, this anisotropic medium may respond to
the electric field by modifying the ellipsoid
in this form you have incremental change in
the which is because of the presence of the
electric field delta of 1 by n square x y,
z, y z, z x. So, these are the terms which
will be which are the new coming into the
index ellipsoid. So, be it an isotropic or
anisotropic the electric field modifies the
ellipsoid by altering the refractive indices
associated with a direction.
So, this is the summary that whether it is
an isotropic system or an anisotropic system
the presence of the electric field may alter
the refractive index properties. So, isotropic
may become an anisotropic, anisotropic may
become again anisotropic, but the properties
are now different from the one which was in
absence of the electric field.
So, the new principal refractive indices and
hence the birefringence that can be evaluated
from this equation because under the influence
of the electric field we have the ellipsoid
equation, and from there we know how to find
out the principal axes system and that the
corresponding principal refractive induces.
So, now once having this equation in hand
we can look for the new principal axes system
by suitably rotating the coordinate axes.
And this can be performed by diagonalisation
of the ellipsoid matrix. So, this is again
associated with a 3 by 3 matrix which we have
seen and you have also seen an example how
this matrix can be diagonalized as a tutorial
in the anisotropic cases. Otherwise by inspecting
and providing by inspecting this equation
looking at this equation and providing suitable
Euler angle rotation of the axes we can again
find out the new refractive induces of the
medium and hence the corresponding birefringence.
So, by doing this we can find out the new
refractive induces in the principal axes system.
Now, the effect of the external electric field
can be categorized in two ways we will experience
two situations if the changes in the refractive
induces ah proportional to the external field
simply linearly proportional that is delta
n is proportional to E then such effect is
known as Pockels effect after the name of
the discoverer, Pockels effect. This is also
known as linear electro-optic effect because
the change is proportional to the electric
field, but in some situations the refractive
induces are proportional to the square of
the electric field that is delta n is now,
proportional to E square. The effect is known
as Kerr effect after the name of the discoverer.
This is also known as the quadratic electro-optic
effect. So, under this electro-optic effect
we have two groups of effects study that is
Pockels effect and effect Kerr effect. So,
we will first discuss the Pockels effect that
is the linear electro-optic effect.
So, under this Pockels effect or Kerr effect
certain geometrical configurations of the
medium are available. So, that if one applied
an electric field then this electric field
may act this in a different way on two orthogonal
linearly polarized waves passing through the
medium. So, let us suppose that there are
two orthogonal linearly polarized waves which
are passing through the medium and an electric
field is acting on it, in that case the two
polarized wave to linearly polarized waves
will be affected in different waves and as
a result ah there will be an electric field
dependent retardation because there will be
an induced birefringence because of this and
which will cause a delay between the two linearly
polarized waves.
If there is a delay then by controlling the
electric field we can actually modulate the
amplitude of the light according to the to
the magnitude of the applied electric field.
So, thus electro-optic effect provides a means
to encode information and data in the signal,
the light modulation can occur at very high
frequencies that is the beauty of this electro-optic
effect it can be in the in the range of megahertz
to gigahertz. Hence the modulators are of
great use infact product commercially available
and use ah electro-optic devices in the optical
network and optical communication system.
Many more applications in the devices like
directional coupler, optical switches are
also commercially available in the integrated
optics. And this electro-optic system is used
ah in signal processing phase amplitude and
frequency modulation of light we will study
some very interesting example cases in the
following sections.
Additional there are many more applications
like lens with variable refractive index with
a controllable focal length, prism with variable
beam bending optical scanning devices, controllable
phase shifter, optical phase modulator birefringent
crystal with variable refractive indices,
then intensity modulators optical switches
and many more applications .
So, in general the electro-optic medium is
a function of the medium is a function of
the refractive index is a function of the
electric field and one can apply a Taylor
of expansion to look at how the electric field
is causing the changes in the refractive index
property.
So, if you apply a Taylor expansion and written
only the first 3 terms then you can see that
changed delta n can be written as n E minus
n o which is equal to a 1 ah written a 1 for
this and a 2 for this del 2 n del E square
at E equal to 0. So, we can represent this
change in the refractive index by writing
this E and E square dependent on (Refer Time:
21:18).
So now, let us now defined another quantity
that is impermeability which is one upon n
square and ah evidently this is a tensor and
this so the of the n impermeability change
in the impermeability is delta n is equal
to 1 by n square which can be written as if
you take this derivative of this 1 by n square
this gives you this, and there will be two
terms. As a result this because delta n value
I will pick up from here to pick up from here
and put it here then if I substitute for this
quantity twice n cube a 1 and this as r equal
to this and s equal to this then you can write
delta n is equal to this, and delta eta that
is changed in the impermeability equal to
r into E and s into E square.
So, you see that the impermeability change
is a function of is a combination superposition
of the field dependent and the square of the
field dependent terms.
You can look at the same equation in a in
a direct way that if I put this equations
in this form then I take 1 by n square which
can be represented in this form, and if I
directly substitute this values for as writing
as r and s equal to this and this then also
we end up with this equation. So, looking
at the same expression from two different
waves basically they have the same origin.
So, Pockels effect in many materials this
s term is negligible that is discard effect
is negligible ah ah, but this r term is quite
prominent. So, in that case this r is the
Pockels coefficient typical values of r is
in the range of this 10 power minus 12 to
10 power minus 10 or 1 to 100 picometre volt.
Example if you take one that for an electric
field of the order of 10 power 6 volt per
meter then change in the refractive index
is only 10 power minus 6 to 10 power minus
4 which is very small but the phase change
associated with this may be very large and
appreciable. We will see that example in future.
So, there are any important electro-optic
materials crystals KDP potassium dihydrogen
phosphate, then ADP ammonium dihydrogen phosphate
lithium niobate lithium tantalite. These are
very useful and commercially used electro-optic
material crystals.
Then the second effect is the Kerr effect
if the material is centrosymmetric then even
and even function n of E is an even function
then r equal to 0 in that case this pockel
effect is absent, but Kerr effect is present
and delta n will be represented by this equation.
The typical values of s will be of the order
of 10 power minus 18 to 10 power minus 14
and electric field of the order of 10 power
of minus 6 will create a change in the refractive
index which is a order of 10 power minus 6
to minus 2 which is appreciable.
So, some polar liquids examples nitrotoluene,
nitrobenzene they exhibit very large Kerr
constants. Centrosymmetric there is crystal
there is one class of crystal which is very
interesting it has some inversion symmetry
and all the r ij coefficients associated with
the electro-optic tensors are identically
0. And for inversion symmetry the direction
x y z and the reverse direction that is minus
x minus y minus z are indistinguishable. So,
there is no change with respect to the crystal.
So, as a result and applied field E and minus
E must induce the same electro-optic response.
To look at this if I write r ij and then I
reverse the field minus ij then I will get
minus r ij E j put together this will be equal
to 0 that means, r ij this. So, for centrosymmetric
crystal all the coefficients r ij will be
equal 0 and there is no Pockels effect. So,
if there is an s missing here, so there is
no Pockels effect. And centrosymmetric crystal
even though it does not so any Pockels effect,
but it may exhibit Kerr effect we will see
example at the lateritic.
So, general index ellipsoid is ah for an anisotropic
material is this and in other coordinate system
we can represent this like this in the compact
form it comes out like this. We may write
this general index ellipsoid in this form,
ok.
Now, with this electro when a steady electric
field E 1, E 2, E 3 are applied a change in
this that is this delta n occurs for all 6
components and then we can write this impermeability
as the impermeability with no field. And then
sum of the additional terms and we can write
this equation in this form because we have
represented this as the Pockels coefficient
and this quantity as the Kerr coefficient.
So, you can write this change in the impermeability
in presence of electric field by this equation.
So, this minus this will give you both the
effects of by represented by in these two
terms.
So, these are the compact equation and you
can see that this will give you twenty 7 coefficients
Pockels coefficients and this will give you
81 coefficients E k and E k. So, n is symmetric
n ij equal eta ij equal to eta ji, s and r
are invariant to permutations of i and j.
As a result r ij k equal to r jik and similarly
s ijkl equal to s jikl. And in the same way
s is also in variant under permutations of
k and l. So, you can write in this form k
and l are reverse.
Therefore because there invariant two permutations
of i and j and k and l. So, you can shrink
this this indices and actually they will represent
the same coefficient. So, the number of coefficients
reduces to 18 and 36 for the Kerr coefficient
or pockel it is 18.
So, just to understand how it shrinks to 1
because 1 1 will be represented by 1, 2 2
will be represented by 2 and so on. 2 3 will
be the 2 3 and 3 2 there again same, so you
write this equal to 4 and similarly 3 1, 5
and 2 1, 6 and this is have we have seen this
in the index ellipsoid matrix the same ah
notation and similarly replacing k l with
single index n we can write this k and l will
be represented by this. So, s k l m n, so
there can be an again shrink to this. So,
r 112 will be represented by r 12, s 11 12
will be represented by 1, 6 1 and 2 gives
you 6 you can see that; 1 and 2 gives 6 and
1 1 gives you 1. So, it becomes s 16 and similarly
12 31 will become s 65. So, effectively we
have 18 and 36 coefficients.
So, with without field you have this equation
in the principal axes system, and with the
field you have this equation in the principal
axes system. And the changes are in the impermeability
is represented by this equation. So, you have
delta n square which is this is for the pockel
coefficient. Only if you pick up the pockel
term then you have this pockel coefficient
which is represented by this.
An example Pockels coefficients ah you have
this delta n 1 delta n square 1, 2, 3 etcetera,
and you have this tensor which looks like
this and this is the electric field. So, this
compact equation ah represents this matrix
equation, and you have 18 terms in this Pockels
coefficient tensor.
So, in this discussion will continue will
continue with this tensor for various examples
like isotropic medium, gallium arsenide, anisotropic
medium ah like KDP, etcetera and then will
continue our discussion.
Thank you.
