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We have seen that ray theory has its limitations.
It cannot accurately predict the propagation
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characteristics of an optical fiber, particularly
when the light confinement dimensions are
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of the order of or comparable to the wavelength
of light. In that case we will have to use
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wave theory. In wave theory light is treated
as an electromagnetic wave. And therefore,
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we need to understand how these electromagnetic
waves propagate in an optical fiber. Before
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doing that we would like to first understand
how an electromagnetic wave propagates you
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know free space or in a dielectric medium
which is of infinite extent for example, this
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room. So, first we would understand how light
propagates in this room or in an infinitely
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extended dielectric medium.
So, we will do this in this lecture. Since
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we want to understand propagation of light
in infinitely extended dielectric medium,
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and light is an electromagnetic wave. So,
we should first consider the Maxwell’s equations
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in infinite dielectric medium.
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So, the simplest case if we take, then it
is a homogenous linear isotropic charge free
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and current free dielectric medium. One example
of such a medium is glass. What is homogenous
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medium? Homogenous medium means that the refractive
index of the medium is the same at every point.
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It does not depend upon x, y and z. Linear
medium means that if I propagate light of
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frequency omega then it propagates as frequency
omega itself, it does not generate new frequencies.
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Or if I propagate light in a medium then the
refractive index of the medium remains independent
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of the intensity of light. So, this is a linear
medium. Isotropic means that if I have two
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independent orthogonal polarizations for example,
this and this. Then these polarizations see
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the same refractive index of the medium. And
since it is a dielectric medium, so there
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are no free charges and no free currents.
So, let me write down the Maxwell’s equations,
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which is the starting point for us. So, the
first Maxwell’s equation is del dot D is
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equal to 0, which is nothing but the differential
form of Gauss’s Law. The second equation
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is del dot B is equal to 0, which simply tells
that magnetic monopoles do not exist. Third
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is del cross E is equal to minus del B over
del t, which is nothing but Faraday's law
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in differential form. And the last one is
del cross H is equal to del D over del t,
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which is nothing but Ampere's law. Apart from
these four Maxwell’s equations we also have
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constitutive relations which are D is equal
to epsilon E. And B is equal to mu H if I
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consider a dielectric medium, such as glass
then it is a non-magnetic medium and in a
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non-magnetic medium mu is approximately equal
to mu naught.
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So, at some places I will use this approximation
also. I will use the convention throughout
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my lectures that curly letters wherever I
use curly letters it means that they are the
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functions of x, y, z and t, while the straight
letters do not have any time dependence they
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are the functions of only the special coordinates.
So, what I want to do? I want to find out
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how electromagnetic wave propagates in a medium.
And electromagnetic wave has associated electric
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field and magnetic field. So, what I want
to do is basically I want to find out how
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the electric and magnetic fields associated
with the light waves, vary with special coordinates
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and with time. For that I need to form a differential
equation in E and a differential equation
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in H, which can tell me how E and H vary with
x, y, z and t.
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In order to form the differential equation,
let me again look at the four Maxwell’s
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equations and then let me do some mathematical
manipulations. So, what I do? I take the curl
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of equation 3. So, I get del cross del cross
E is equal to minus del del cross B over del
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t.
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Now I use the vector identities which gives
me del of del this is equal to del cross del
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cross is equal to gradient of divergence of
E minus del square E. And if I look at this
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the del cross B and I look at equation 4,
if I multiply here by mu and here also then
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it becomes del cross b. So, del cross B is
mu times del D over del t, So I put it here.
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Now since this is a homogenous medium and
in a homogenous medium what I have del dot
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D is equal to 0 and D is equal to epsilon
E. So, in a homogenous medium since epsilon
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is not a function of x, y and z, then that
epsilon will come out of this del divergence
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operator. Which means that in a homogenous
medium del dot D is equal to 0 will translate
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to del dot E is equal to 0.
So, this term goes to 0. And again I put D
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is equal to epsilon E then this equation simply
becomes del square E is equal to mu epsilon
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del 2 E over del t square. So, here you can
see that this del square is nothing but del
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square over del x square plus del over del
y square plus del square over del z square.
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So, this simply tells me that how E varies
with x y and z and with time. So, I need to
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solve this differential equation. And get
the functional form of E. So, in order to
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solve this equation, first what I would do
I will consider a very simple case, simple
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case of one dimension. Which means that I
assume that E varies only with z and t, if
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I assume this then this equation will now
become del 2 E over del z square is equal
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to mu epsilon del 2 E over del t square.
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And since E is the vector Ex x cap plus Ey
y cap plus Ez z cap, which means that this
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is note single equation, but it constitutes
3 equations - one in Ex, one in Ey and one
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in Ez.
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And so, so I have equation 3 equations in
scalar components. So, if I draw this vector
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sign and write the scalar equation then it
would be del 2 E over del z square is equal
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to mu epsilon del 2 E over del t square where
E is nothing but it can be either Ex or Ey
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or Ez. So now, I need to solve this equation.
How do I solve it? Well since E is a function
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of z and t and I also see that mu epsilon
is independent of z and t, then I can use
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the method of separation of variables. What
is the method of separation of variables?
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Well, I can represent this E of z t is equal
to E of z and t of t. So, I separate them
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out. If I put this E into this equation, then
I get an equation t times del 2 E over del
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z square is equal to mu epsilon Ez del 2 t
over del t square.
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I divide this with this E times t and then
I transform this equation into this form.
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What I have done essentially? I have separated
out the variables. On the left hand side I
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have terms which contain only z, which are
the function of z only and on the right hand
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side I have terms which contain t which are
the function of t only.
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Now, to solve this what I do? The most natural
thing is that I quit each of them to some
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constant. And since it is a second order differential
equation, then that constant I will take in
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the form of his of a square. So, that I avoid
square roots.
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So, I quit it to some constant k square. Let
us now first solve the t part with this please
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see that k, k is simply a constant, a purely
a mathematical constant. What does it signify?
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Physically we will see later on. So, if I
now take this t part, then it is mu epsilon
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del 2 t over del t square is equal to k square
t, and let me write this mu epsilon as 1 over
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v square, mu epsilon is equal to 1 over v
square. Why I am doing this? Because this
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will then come here and I am taking in the
form of a square, so that I have everywhere
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square itself.
So, this will become now this and v square
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will come this side. So, what I do? I have
now v square k square I represented by another
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constant omega square. So, in this way the
t equation becomes del 2 t over del t square
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is equal to omega square times t. So, please
again look that this k, this v and this omega
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there. I write now do not have any physical
and interpretation for them, but later on
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I will see what is the physical interpretation.
What is the solution of this equation now?
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I can immediately see that the solution of
this equation is of the form e to the power
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plus minus omega t. Which means that s t goes
to plus or minus infinity the solution blows
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up which means that this is not a physically
viable solution. It is a solution it is a
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mathematically correct solution, but I do
not have any use for such kind of solution
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because it is not physically viable. What
kind of solution I am looking for? Well, I
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am looking for a solution which represents
a wave. And a wave we will have an oscillatory
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solution, which means sin omega t or cosine
omega t or a combination of these.
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So, I can immediately see that if I get a
solution which is of the form of e to the
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power plus minus i omega t then it is an oscillatory
solution. And for that I need to have minus
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omega square here, which means that I should
have minus k square here. So, the most natural
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choice that occurred to me to take this as
plus k square, does not give me any physically
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a viable solution. So, I take this as minus
k square. So, I take it as minus k square
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and now if I do the same thing and get the
solution then it is of course, e to the power
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plus minus i omega t, which is an oscillatory
solution.
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Now let us look at z part. So, z part is 1
over E del 2 E over del z square is equal
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to minus k square or del 2 E over del z square
is equal to minus k square Ez. Then the solution
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of this equation is simply of the form e to
the power plus minus i kz.
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So, if I now combine these z part and t parts,
then the complete solution is Ez t is equal
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to E 0 e to the power i plus minus kz plus
minus omega t. This is the most general solution,
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where E 0 is the constant. It does not depend
upon x, y, z or t. So, this is the most general
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solution of this equation. And this represents
a wave. Now I can choose the sign shear appropriately
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So that it can represent a wave in a particular
fashion. For example, if I choose the signs
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like this I represent it like this E 0 e to
the power i omega t minus kz then this is
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the wave propagating in positive z direction.
How it is propagating in positive z direction?
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I will see. What else I now see here is if
I take a particular position z. If I fix z,
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let us say z is equal to 0. And I look at
the solution in time then I see that the solution
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is oscillating in time. And it is oscillating
in time with frequency omega after a certain
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time which is 2 pi over omega, it comes back
to the same position.
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So, omega is nothing but the frequency or
angular frequency of oscillation. So now,
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I interpret this omega which I had represented
here, as the frequency. Now let us take a
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snapshot of this, is snapshot of this means
that I freeze frame in time. Let us I fix,
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let us take that time is t is equal to 0,
then the solution is e to the power minus
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i kz. Now if I look at this I plot it. So,
it would be an oscillatory wave like this.
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In z a sinusoidal function in z. And I see
that it repeats itself after a distance 2
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pi over k. Then k is nothing but the wave
vector. So, here the k which I had taken just
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this constant purely mathematical constant
now I see that it is nothing but the wave
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vector. And the omega which came out as v
times k is nothing but the frequency. What
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is v I still need to see? Let us examine the
nature of this wave what kind of wave it is.
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If I look at this phase of this wave then
I find out what are the surfaces of constant
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phase, then what I find that if I put omega
t minus kz is equal to constant so that I
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get the surface of constant phase.
Then at a particular time t I get the surface
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z is equal to constant. z is equal to constant
is an equation of a plane, which means that
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the surface of constant phase is a plane.
And so, this kind of wave is known as plane
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wave. Let me find out at what velocity this
surface of constant phase is moving. So, for
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that what I do I take the derivative of this,
I differentiate this. When I differentiate
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this then I get omega Dt minus kz is equal
to 0, which means Dz over Dt is equal to omega
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over k and omega over k is nothing but v,
which means that v which I had from here is
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nothing but the velocity of the wave, the
phase velocity. So, this is a plane wave moving
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with phase velocity omega over and that omega
over k is nothing but v; the constant which
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we represent it here.
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Now, let me generalized this for 3 D case.
So, the 3 D wave equation is this. And so,
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the solution would be E x y z t is equal to
E0, E0 is again a constant times e to the
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power i omega t. So, the t solution remains
the same. And since now I have xyz all of
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them. So, the solution would you minus kx
x minus ky y minus kz z. What is k? k is nothing
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but kx x cap plus ky y cap plus kz z cap.
And if I represent the position vector in
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vector form then it is x x cap plus y y cap
plus z z cap. So, this is nothing but this
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is nothing but k dot r. So, I represent it
as E 0 e to the power i omega t minus k dot
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r. And from here you can immediately see that
k square is kx square plus ky square plus
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kz square.
Now, for a given k I can have infinite sets
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of kx ky kz. So, infinite sets of kx ky kz
can give me the same value of k. What is the
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meaning of those infinite sets? If you look
at this is k dot r, in what direction this
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is moving? It is moving in one particular
direction, which is represented by the values
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of kx ky and kz. kx ky and kz are nothing
but the projections of vector k on x y and
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z axis. So, the values of kx ky and kz will
give me at what angle this wave is moving.
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And with the propagation constant k the magnitude
of propagation constant is always this. So,
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for a given magnitude of propagation constant
k, I can have various angles possible. So,
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this tells me that if a light wave is allowed
to go in an infinitely extended dielectric
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medium, then it can go in any direction. There
is no restriction on it, it can go in this
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direction, this direction, this direction,
this direction, this direction, this direction
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any direction is possible for the same value
of k.
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I can do the same analysis for the magnetic
field. And I will get the solution as H x
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y z t is equal to H 0 e to the power i omega
t minus kx x minus ky y minus kz z or H 0
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e to the power i omega t minus k dot r. So,
for a light wave I have got now the associated
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electric field which is represented by this,
and associated magnetic field which is represented
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by this. These are of course, by scalar components.
The same solutions are for E x y z and Hx
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Hy Hz. Well, so let me now consider a wave
which is going in certain direction and let
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me choose does that direction as z direction.
If I choose the direction is z direction,
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then mathematically I can write the electric
field associated with this as E 0 e to the
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power i omega t minus kz. Of course, this
constitutes 3 equations, one is in Ex another
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is in Ey and yet another one is in Ez.
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Similarly, the magnetic field can be written
like this, where H is Hx x Hy y and Hz z cap.
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Now let me do one thing with this, let me
take the divergence of this and since this
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is a homogenous medium, then del dot E would
be equal to 0. If I put del dot E is equal
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to 0 it means that del Ex over del x plus
del Ey over del y plus del Ez over del z is
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equal to 0. If I pick up Ex components from
here, then it would be E 0 x e to the power
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i omega t minus kz. E 0 x is the constant
and here you do not see any term of x which
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means del Ex over del x is equal to 0. Similarly,
del Ey over del y is equal to 0, which means
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that if these 2 are 0 then this implies that
del Ez over del z should be equal to 0, one
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thing. Second I do del dot B is equal to 0
I use this Maxwell’s equation del dot B
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is equal to 0. So, similarly it will also
give me the del Hz over del z is equal to
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0.
So, these solutions of the differential equations
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the wave equations which I got earlier. Now
if I put these solutions into these Maxwell’s
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equations then they give me del Ez over del
z is equal to 0, and del Hz over del z is
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equal to 0. What does it mean? If I look at
Ez, from here then Ez would be E 0 z E 2 the
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power i omega t minus kz. If I look at Hz
from here then it would be H 0 z e to the
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power i omega t minus kz. Now for this to
be 0 there are 2 possibilities. One is Ez
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is constant and another is that the amplitude
of Ez itself is 0. Let me first look at the
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first possibility that Ez is constant. If
Ez is constant which means this whole thing
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is constant, E 0 z itself is a constant. So,
which means that this term has to be a constant,
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if this term is a constant then there is no
wave. Because then even the x component and
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y components they will become constant they
will not vary with z and t.
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So, there is no wave solution, which means
that I cannot put e to the power i omega t
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minus kz is constant. So, in order to satisfy
this equation the only possibilities E 0 z
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0 which means that Ez is equal to 0 that is
there is no longitudinal component because
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z is the direction of propagation itself.
So, Ez is nothing but the longitudinal component.
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So, there is no longitudinal component. Similarly,
this equation gives me that H 0 H should be
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equal to 0, or in this way Hz is equal to
0. So, there is no longitudinal component
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of magnetic field also which means that the
solutions of the wave equation that I got
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are the waves which do not have any longitudinal
components, which means that they are the
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transverse waves.
Now if the wave is propagating in z direction,
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then there is no Ez there is no Hz. Which
means that E cannot vibrate along z and H
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cannot vibrate along z. So, they can only
vibrate in x and y directions. That is they
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00:30:52,821 --> 00:30:58,809
can vibrate in a plane perpendicular to the
direction of propagation if z is the direction
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of propagation which is the direction of this
pointer then and then this is the transverse
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plane.
So, E and H can only vibrate in this plane.
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00:31:13,049 --> 00:31:25,020
So, which means that your, if this is the
plane, then E can vibrate along this or along
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00:31:25,020 --> 00:31:32,590
this or along this or along this. And similarly
H can vibrator along this along this along
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00:31:32,590 --> 00:31:44,230
this, what does this direction of vibration
represent? This direction of vibration represents
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nothing but the polarization. The direction
of vibration of electric field vector represents
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00:31:51,070 --> 00:31:58,530
the direction of polarization. So, if I again
look at this wave which is propagating in
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positive z direction, then the non-vanishing
components of E and H would be Ex, Ey and
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00:32:10,160 --> 00:32:15,140
Hx, Hy like this.
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And now if I consider a particular case, where
I say let my electric field of light vibrate
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along x. And so, the y component is 0 because
x and y are orthogonal to each other. So,
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so if it is vibrating along x then there is
no y component.
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00:32:41,880 --> 00:32:50,740
So, electric field is vibrating along this,
and the wave is propagating like this. Then
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such a wave is known as linearly polarized
wave polarized in x direction. So, this is
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x polarized wave. You can understand it in
a very simple way that if you tie a string,
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one end of the string to tie on that side
of the wall. And one end you keep with you
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and shake it, let us say this is x direction,
if you shake it like this then a wave propagates
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like this, in the string. So, wave is going
in z direction while a point on the string
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00:33:32,910 --> 00:33:40,480
or particle on the string will oscillate in
x direction. If you take any point on the
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string the direction of oscillation of that
point will always be in x direction. So, this
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00:33:45,960 --> 00:33:55,040
is x polarized wave. If you vibrate it in
y-direction then it goes like this. So, particle
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00:33:55,040 --> 00:33:58,960
oscillates in y direction then it is y polarized
wave ok.
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00:33:58,960 --> 00:34:09,590
So, if I consider this then what is the corresponding
magnetic field? To get the corresponding magnetic
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00:34:09,590 --> 00:34:16,829
field I use the Maxwell’s equation which
relates the electric field to magnetic field.
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00:34:16,829 --> 00:34:30,190
Then if I take the x component here Hx then
del Hx over del t will be 0, if I use this.
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00:34:30,190 --> 00:34:43,069
And del Hy over del t times mu would be equal
to minus del Ex over del z. This gives me
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00:34:43,069 --> 00:34:53,220
that H 0 x is equal to 0 which means that
if Ex is non 0 then Hx would be 0 first thing.
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So, if the wave is polarized along x that
is electric field vector is vibrating along
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00:34:59,880 --> 00:35:08,869
x, then there is no component of magnetic
field vibration in x. Magnetic field vibrates
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00:35:08,869 --> 00:35:17,999
along y with what amplitude it comes from
here. So, if I put this Hy from here and Ex
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00:35:17,999 --> 00:35:26,890
from here and simplify this then the corresponding
magnetic field amplitude is k over omega mu
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00:35:26,890 --> 00:35:37,049
times E 0 x, or omega epsilon over k E 0 x.
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00:35:37,049 --> 00:35:48,459
So, in summary what are the electric and magnetic
fields associated with a light beam? Well
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00:35:48,459 --> 00:35:54,350
if I take the example of linearly polarized
wave which is polarized in x direction, and
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00:35:54,350 --> 00:36:04,150
propagating in z-direction. Then the electric
field associated with this is x cap E 0 e
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00:36:04,150 --> 00:36:09,220
to the power i omega t minus kz it is the
same equation as in the previous slide I have
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00:36:09,220 --> 00:36:19,630
just generalized it instead of putting E 0
x, I have now put E 0 and x cap.
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Then the corresponding magnetic field would
be in y direction. So, it would be y cap H
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00:36:26,329 --> 00:36:34,250
0 e to the power i omega t minus kz. And the
amplitude of H would be related to the amplitude
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00:36:34,250 --> 00:36:41,559
of E by this relation. I can also write it
down in terms of B this would be simplify
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00:36:41,559 --> 00:36:50,690
y cap B 0 e to the power i omega t minus kz
where B 0 would be nothing but k over omega
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00:36:50,690 --> 00:36:57,730
E 0 because if you take this mu on this site,
and combine this mu with H then it will become
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00:36:57,730 --> 00:37:03,349
B.
What is k over omega? k over omega is nothing
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00:37:03,349 --> 00:37:12,779
but 1 over v. So, B 0 is nothing but E 0 over
v. v is the velocity of electromagnetic wave.
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00:37:12,779 --> 00:37:20,200
If I consider the free space than this v is
nothing but c. The velocity of light in free
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00:37:20,200 --> 00:37:26,170
space, which is 3 into 10 to the power 8,
which is very large. The amplitude B 0 would
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00:37:26,170 --> 00:37:35,369
be much smaller than the amplitude E 0 which
means that the amplitude of magnetic field
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00:37:35,369 --> 00:37:41,400
is very, very small as compared to the electric
field associated with light. And that is why
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00:37:41,400 --> 00:37:49,960
it is the electric field that affects the
retina of our eye. And so, the direction of
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00:37:49,960 --> 00:37:57,220
polarization is associated with the direction
of electric field vibration.
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00:37:57,220 --> 00:38:07,609
In the next lecture we would see more carefully
about the polarization and we would also see
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how much power is associated with this kind
of wave.
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00:38:11,220 --> 00:38:12,160
Thank you.