So, we have seen that the electromagnetic
waves in free space the electric and magnetic
field vectors how they are related, how they
are oriented, what is the relation amongst
them .
And now, we will see the polarisation property
very interesting property of the electromagnetic
waves and that will first take up in the case
of electromagnetic waves in free space.
So, let us consider a situation that the the
wave is propagating along z direction. So,
you can write that the k vector as z unit
vector into k, where k is the magnitude of
the propagation vector and you also assume
that the electric field E is along x. So,
in the earlier occasion also we assume the
same convention that the propagation is along
z direction and the electric field is oriented
along x . So, having known that we can write
the electric field in this way E equal to
E 0 sin omega t x unit vector because, the
electric field vectors are along the x direction.
Now, by writing this equation we have considered
the imaginary part of the complex vector solution
that is E equal to E 0 e to the power of i
omega t minus kz. So, in the rest part of
the discussion we will consider only in the
sin cosine form. If I have considered E equal
to x unit vector E 0 sin omega t minus kz,
then the B will be oriented along the y direction
we have seen as well. So, B equal to y unit
vector B 0 sin omega t minus kz and you can
see from these two equations that E and B
are in phase. This situation, the orientation
when the electron electric is restricted only
in one particular coordinate direction and
magnetic field is also restricted in one particular
coordinate direction; such a situation of
the electromagnetic waves called that linearly
polarised wave.
So, this is how we represent an x polarized
wave. So, the situation where the electric
field vectors are along the x direction we
call such a wave as a x polarised wave, x
polarised linearly polarised wave .
Then we consider a situation where because,
our intention is to study the other variation
the other properties polarisation properties.
So, we consider another situation where the
wave is y polarized, in that case how do I
write this wave equation? E should be equal
to y unit vector E 0 sin omega t minus kz
and B will be equal to minus k minus x unit
vector B 0 sin omega t minus kz. Look at this
figure you see y along y the electric field
is oriented and along minus x you have the
electrical field oriented. So that means,
this y polarised light means the electric
fields will be along the y direction and magnetic
fields will be along minus x direction.
So, by that convention if we consider this
situation then we can take up this particular
situation that we add an additional phase
of pi by 2 to this wave because, we started
off with this wave which has a 0 at at z equal
to 0. So, these a sin wave, but if I add a
phase of pi by 2 to the power each of the
waves, the wave will advance by a phase of
pi by 2 and the other orientation of the electric
and magnetic fields will remain same.
So, in that case how do I write this equation?
For electric field everything remains same
I just add a phase of pi by 2 and for the
magnetic field as well as add a phase of pi
by 2. As a result you can write this equation
in this form y unit vector E 0 cosine of omega
t kz. And for the for the magnetic field we
can express as minus of x unit vector B 0,
indeed these as become a cosine function earlier
it used to be a sin function.
Now, recall that Maxwell's equations are linear
and a linear superposition of their solutions
the x polarised wave and y polarised waves
will be considered. And that is also valid
when I take this x polarised light which I
have considered first, that is given by the
set of these two equations. And the y polarised
light which I have considered by which are
given by the set of these two two equations.
This is y polarised light, but there has been
an advancement of phase by pi by 2 from the
starting point. So, at z equal to 0 we will
consider what happens to the space, but before
that you we will have to consider the superposition
of these two.
So, the linear superposition of this x and
y polarised waves will give you the total
electric field E equal to this x polarised
part and x component of the x polarised part
and y component of the polarize part. Similarly,
for the magnetic field we can write these
equations. So, basically these are taken from
these two, I just add take the superposition
linear superposition of E and this E. So,
this quantity plus this quantity put together
and likewise for the magnetic field this quantity
and this quantity will put together to get
this equation in this form right.
Now, we will consider what happens when we
see the wave at z equal to 0. So, at z equal
to 0 I put this z equal to 0 so, this become
0, this becomes 0 and all together all the
in all 4 places it becomes 0. So, it becomes
only a time dependent osculating electric
field and indeed it is true at a given point
for a travelling wave electric field vector
suggest osculating at the same point and the
osculation is purely simple harmonic type.
So, E x can be represented as E 0 sin omega
t and E y will be represented as E 0 cosine
omega t. And similarly, for the magnetic field
you can write this equation.
Now, look at this situation that if you take
if you look for the amplitude of the total
wave you can write because, if you take the
square and add them you get this equation
E square, the resultant amplitude square will
be equal to E 0 E 0 square which is constant.
And this, equation basically gives you the
it reminds you the equation of a circle E
x square plus E y square equal to sum constant
square. And the same thing will have seen
in the case of rotating magnetic field and
many other occasions E x square plus E y square
equal to E z square will be represented by
a circle, where the radius of the circle is
the amplitude of the resultant wave.
For the case of magnetic field so, it will
be a circle of different magnitude B 0 because,
B 0 and E 0 they are related by a factor of
c which is the velocity of the electromagnetic
wave. So, they will describe a circle and
the superposition will give you that the tip
of the electric field and the magnetic field
vectors rotate on the circumference of a circle.
This particular situation, this particular
orientation of the electric field or the magnetic
field in a travelling electromagnetic wave
is known as the circularly polarised wave.
And this is of tremendous importance for various
parts of optics and also it is very interesting
to analyze different situations for circularly
polarised light, linearly polarised light,
their production, their analysis and these
are all very interesting findings.
So, we we have seen that the tip of the electric
field vector is describing a circle, but there
is an aspect that we have to consider whether
it is rotating in the clockwise sense or in
the anticlockwise sense. So, that should be
decided by the orientation of the electric
fields and the magnetic fields, their relative
orientation of the electric and magnetic we
will consider a particular situation. So,
the electric filed vector the tip of the electric
field vector, when you consider different
positions along the z axis they will be advancing
in the form of an helix.
So, E and B vectors traces an advancing helix
as it is shown here and this situation is
the truth three-dimensional picture for a
circularly polarised wave. Even though it
describes a circle at one plane at one position
which is perpendicular one plane which is
perpendicular to the z axis, but the tip is
advancing because the wave itself is advancing.
So, the tip will be described will be described
in the trace of and the helix.
Next we consider this situation that, how
we analyze whether it is the tip is rotating
tip of the result and electronic ah result
and electric field is rotating in the clockwise
sense or anticlockwise sense. In order to
do that we considered that at z equal to 0,
what happens to this electric field and the
magnetic field. Let us consider the electric
field E x equal to E 0 sin omega t and E y
equal to E 0 cosine omega t.
Now, at time t equal to 0 that is at the initial
time and the position is also at z equal to
0 for all the all the discussion, we have
considered that the position is fixed at z
equal to 0. That is why we would write the
equation E x equal to E 0 sin omega t, E y
equal to E 0 cosine omega t . So, at z equal
to 0 you get that E x equal to 0, but E y
equal to E 0 because, cosine omega t will
become 1 and you get that because E y equal
to the maximum amplitude E 0. So, this is
your your so, the orientation of the resultant
electric field is along this
Then next we consider at time t equal to pi
by 4 times 1 upon omega. So, at this time
if I substitute this value of t in this equation,
then this will give you E x equal to 1 by
root 2 E 0 because, pi by 4 into 1 by omega
will give you only pi by 4. So, sin pi by
4 we give you 1 upon root 2. So, E x will
have the value 1 by root 2 E 0 and E y will
have the value 1 by root 2 E 0; when you substitute
this t value here because, cosine and sin
at 45 degree will give you the same value.
That means, if you take the resultant of the
electric field component and the magnetic
field component that is E x square plus E
y square which will be equal to E 0 square
their position will now, be along this direction.
Next, if you consider twice this time at an
interval of pi by 4 into 1 by omega t then
E x because now, this quantity will become
1 whereas, this quantity will become 0. So,
the result is that we will get E x equal to
E 0, but E y equal to 0; that means, E x the
entire amplitude is along the x axis that
is along these direction. So, it tells if
you proceed further with the same analysis,
with the values for intervals of pi by 4 1
by omega t, you can see that the electric
field will next time orient along this direction
and next time it will come along this direction
and so on.
So that means, the tip of the vector is now
rotating in this sense . So, this is called
a right circularly polarised light. So, we
could very nicely explain that how the tip
of the electric field is is rotating on the
circumference of a circle and representing
a right circularly polarised light.
So, looking at the source this is how we we
distinguish the right-circularly polarised
light and the left-circularly polarised light.
So, looking at the source E of the wave coming
towards you is seen rotating anticlockwise
the wave is circularly polarised light. And
looking against the direction of propagation
that is looking at the source, if E is seen
rotating in the clockwise sense the wave will
be left-circularly polarised light. So, in
the earlier case in the earlier case what
we have seen the the electric field vects
vector is rotating in the in the clockwise
sense. But this is ah wave from the source
looking against the source, looking towards
the source this will be right-circularly polarised
light.
Let us next consider ah situation to look
for left-circularly polarised light. For that
if you considered the combination of x and
y polarised waves in this form, that is E
equal to E 0 cosine omega t which is the x
component and y component it will be minus
E 0 sin omega t minus kz. For B similarly,
we will have a cosine for the y component
and the sin for the x component. Then this
equation, if you do the same analysis that
is at time at z at at position z equal to
0; we write down this equation for this which
will be E 0 cosine omega t and this will be
E 0 sin omega t.
For the magnetic field this will be B 0 cosine
omega t and B 0 sine omega t, but this is
minus and this is plus. So, we can take the
superposition and we can find the resultant
of this wave at different positions that is
at t equal to 0, t equal to pi by 4 into 1
upon omega and 2 pi by 4 into 1 upon omega
and so on and so forth. Then we can see that
the tip of the of the electric field is rotating
in the anticlockwise sense, in the counter
clockwise sense and such a wave will be referred
to as a left-circularly polarised light.
When the amplitude of E x and E y are not
the same because so, for throughout the discussion
we have considered that the electric field
amplitude and magnetic field amplitudes; electric
field ampli component the the magnitude of
the electric field for the the x component
and that for the y component they were same.
Then it was a it was a circularly polarised
light because, the contributions from E x
and E y where same, but let us suppose in
a situation where the E x and E y they are
different then instead of a circle now, the
tip of the electric field will describe an
ellipse.
The ellipticity will depend on the relative
magnitude of the E x and E y component and
in the same way the tip of the electric field
will advance as along the direction of propagation.
So, the ellipse will down represent the circle
in the case of elliptically polarised and
such a situation is called elliptically polarised
wave. So, we have seen in this discussion
that starting from the basic electromagnetic
wave equation for free space or vacuum how
we write the solutions plane wave solutions.
And from the plane wave solutions we use the
Maxwell's equation: the first equation, the
second equation. From there we arrive at that
the dot product of k and E, k and B they are
are 0. So, meaning that k and B will be perpendicular
to each other, E and k there also perpendicular
to each other. But, it is not sure till then
that whether E and B are also perpendicular
to each other. And for that, a more rigorous
analysis we use that the Maxwell's third and
fourth equation the curl equation; where we
could constitute the the cross product of
the k and E. And also, the cross product of
k and B to show that all the 3 components,
all the 3 vectors that is the electric field,
the magnetic field and the propagation vector
three of them are oriented mutually perpendicular
to each other.
And they, we also found the relative magnitude
of the electric and magnetic field, they are
they are connected by a factor c which is
the speed of light in the case of free space.
Then we analyzed how the how the electric
field is oriented and how we achieve the linearly
polarised light, circularly polarised light.
And we have also seen, if the amplitude of
the x and y component of the electric fields
are different then we end up with and elliptically
polarised wave .
Thank you.
