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YEN-JIE LEE: All right, so let's get started.
So today, very happy to have you here.
We are going to talk about continuing
our discussion on the two-dimensional and
three-dimensional, waves.
So as I mentioned before, there are two interesting situation
which we can actually increase the dimensional.
So for example, I can have all the objects oscillating
in just one direction, but I changed the way
that I placed all those objects in the space.
For example, I can have particles
which are arranged in two or three-dimensional arrays.
And we were talking about how to understand this kind of system.
And all those objects are oscillating
in just one direction-- for example,
up and down in one direction.
That's actually one measure we can increase the dimension.
There's another interesting example
which we will talk about today is
to change the direction of the oscillation,
even rotate the natural electromagnetic waves,
for example.
And how can we achieve that and how
do we understand this phenomena, that's actually going to be
covered by the lecture today.
So before we move forward, we will
have a short review on what we have
learned from the last lecture.
And this is actually what we discussed.
If I have two materials, the left-hand side
and the right-hand side are two different kinds of materials,
or very thin membranes.
And if I have an incident wave coming
into the boundary of these two materials, which is just
x equal to 0, as we discussed last time,
basically the boundary condition require
that in order to make sure that the membrane doesn't break,
that means the k vector, the projection
in the y direction of the k vectors has to be the same.
That means the projection of the wave in the y direction, which
is along this line, the wavelength of all
the incidents refracted and transmitted wave,
the wavelengths should be all the same.
Otherwise, I can change y and make the membrane break,
or break the boundary condition.
So that's actually the first thing
which we learned from the math we were doing last time.
And also, the k value are not arbitrary, right?
So we were already assuming that all three waves, all the three
plane waves are oscillating at the same frequency.
If they are oscillating at different frequencies,
it doesn't work because I can change time, then
make the membranes break.
So therefore, all the three plane waves
will be oscillating at the same frequency-- omega.
And according to this formula, so we have defined
and which is actually the refraction index,
which is equal to some constant c divided
by the speed of the propagation, the phase velocity
of the left-hand side material.
And we also can define n prime, which is actually
equal to c, the same constant, divided by v prime, which
is the speed of propagation of the right-hand side material.
And according to the dispersion relation,
we can calculate what will be the length of the k
if omega is given.
The omega for the left-hand side and right-hand,
refracted, transmitted, and incident waves
should be the same.
Therefore, I can immediately write down
what would be the length of the k vector.
And that means the length of the k vector
would be equal to n times omega over c.
So c over n is actually just v. So basically it's
just the non-dispersive median dispersion relation.
And also you can go ahead and write down
what would be the length of the k prime.
And that is actually determined by n prime omega over c.
So based on these two interesting formations,
what we can conclude is that that means k prime and kr
cannot be arbitrary.
They have to be aligned to form a specific pattern.
The pattern is that the projection
in the right direction should be the same.
And also at the same time, the length of the k vector
is determined by the refraction index.
So once we fix all that and put them all together,
we can conclude that based on the projection in the y
direction, we will conclude that theta
will be equal to theta r, where theta is actually that incident
angle and theta r is the refraction angle, which
is describing the direction of the refractive wave.
And also the second very interesting information
we learned is that based on the boundary condition,
we can conclude n sine theta will be equal to n prime sine
theta prime.
So that's actually Snell's law, which some of you
actually learned from the high school days
or from the earlier lectures in physics.
And these two interesting results
form the basis of geometrical optics or laws.
That's two of the most important laws we
learn from geometrical optics.
So that actually gives you some examples
and gives you some more feeling about what
we are talking about.
So what we are talking about is that if I have an incident
wave coming into this boundary, and that incident angle is
theta 1, what I would expect based on what we have just
derived is that theta r, the refracted wave
direction, the refractive wave angle,
the refraction angle would be equal to the incident angle,
according to what we have just derived.
I was just rotating this by 90 degrees.
And also we can calculate what will be theta 2 according
to Snell's law--
n1 sine theta 1 equal to n2 sine theta 2.
And if I continue and propagate the incident wave,
here I assume that n2 is larger.
If you have a larger refraction index,
that means the speed of propagation is smaller.
Larger n value give you a smaller speed of propagation.
Therefore, you can see that the wavefront, which is actually
the position of the peak, actually
got delayed compared to the original projection.
And you can see that the red line is actually really
what you would see in the second median.
And then we can also continue to propagate
and you will see that, interesting,
that means the plane wave would change direction because
of the matching boundary condition,
the membranes doesn't break.
You can see that the peak position
match from the median number 1 and median number 2.
The peak position, which is the position of the line, match.
And also, due to the slower speed of propagation,
the plane wave actually change direction.
So that's actually how we can understand
the mathematical result which we derived last time
by this interesting example.
So that's considered a situation which
is maybe a little bit interesting to you.
So what will happen if I now shoot light
from inside some material which delay the light slightly--
for example, n1 is equal to 1.5.
I shoot something through the material,
and then the second material I have has higher speed of light,
which is actually having the speed of light
equal to light in a vacuum case.
So in this example, I have n1 equal to n2.
Since n1 sine theta 1 is equal to n2 sine theta 2,
so now I can calculate what will be the resulting theta
2 according to this equation.
So the resulting theta 2 will be sine theta 2 will
be equal to n1 over n2 sine theta 1.
In this case, n1 is greater than n2, therefore,
this term is greater than 1.
In this case, actually, this ratio is equal to 1.5.
So this is actually pretty interesting.
That means this factor is actually greater than 1.
I have sine theta 1, which is multiplied in this vector.
So that means if I increase theta 1, if I increase
theta 1, at some point I will not
be able to get theta 2 because theta 2 will
be arcsine 1.5 sine 1.
And at some point, 1.5 sine theta 1 will be greater than 1,
then I don't have a theta 2, which
can satisfy this equation.
So what will happen?
Maybe the whole system explodes, I don't know.
So we are going to do a simple experiment
to see what is going to happen.
So before we do this kind of dangerous experiment,
we are going to turn this light off and be prepared.
Hope everybody will survive.
So here I have a laser, which is actually shooting a laser beam
through this tank of water.
So this is water, with n value roughly 1.33.
And outside of water is air, so the speed of the light
is roughly equal to c.
So n value is actually roughly 1.
So that's actually exactly the situation we are looking for.
And let's turn on this experiment, careful.
Wow, look at this.
You can see that here there was no light coming out.
Why is that?
Because of the mathematics--
the mathematics say that, OK, sorry, guys,
theta 2 doesn't work.
Therefore, there will be no light really coming out.
You can see my hand, you can see the light here.
When I put my hand here, it's not burning my hand.
But that's OK.
All right, and you can see that nothing really comes out.
And all the energy are bounced back into the water.
The same thing also happened here.
You can see that the light is actually
bouncing back and forth and moving into the left-hand side
direction until it passed through here, maybe into there,
because it's actually still bouncing around
inside this tank.
So the good news is that no explosion, like what we
have here.
The interesting thing we find is that all the energy
will be bounced back if you have large enough incident
angle in this situation.
As you can see from here that all the light's energy are
bounced back into water.
So that's actually very, very interesting.
And that means we can probably make
use of this property to send light through some large n
material.
So that's actually how optical fiber works.
So basically you can actually shoot a light
into the optical fiber.
And the light is going to be bouncing back and forth
between the boundaries inside the fiber.
And then you can actually send those information through light
by this kind of mechanism.
So that's how optical fiber actually works.
And we have a setup, which I have a light here,
shooting light into the optical fiber.
And I have paper here which tried
to block part of the light.
So from here, you can see the text which I put on the paper,
because some of the light is blocked
by this paper, the text on the paper.
And this light goes through this optical fiber
and continues and propagates and got captured by the camera.
So you can see, can you read out loud
what is actually written there?
Can you see it?
The--
AUDIENCE: Last question on the
YEN-JIE LEE: The exam is--
oh, no, no, no, no, transmission interrupted.
Oh my god.
All right, so I'm sorry it didn't work,
but it worked in the beginning, right?
So you can see that this is a wonderful way
to send optical signal.
And we actually do that.
We send signal from US to Asia through all those crazy lines
under the sea.
So that's really cool.
And finally before we change the topic,
I'm sure that you will enjoy doing the practice in your PSET
number 8.
We are going to learn that this is actually highly related
to a beautiful phenomena we see in daily life--
the rainbow is actually really related to Snell's law,
and that you are going to solve this problem in the PSET.
All right, so any questions so far related
to Snell's law and refraction index, et cetera?
All right, so if not, we'll go ahead and talk
about the second example.
The second example is that instead
introducing more objects in this array,
we could change the direction of the oscillation
as a function of time and see what is going to happen.
For example, the direction of the electric field,
I can make it rotate or change as a function of time.
And that's actually called circularly polarized wave.
And that means we are going to talk about polarization today.
Before we start the real lecture,
what we are going to do is that I
will quote words from Feynman.
"It doesn't matter how beautiful your theory is,
it doesn't matter how smart you are,
but if it doesn't agree with experiment, is wrong."
So that's actually a very important lesson.
And I have been telling you that electromagnetic wave is
predicted to be oscillating in the transverse direction,
as you can see from that little figure.
So if the directional of propagation
is to the right-hand side, so that means electric field
and the magnetic field can only be oscillating
in a plane which is perpendicular to the direction
of propagation.
That's actually what we have learned
from Maxwell's equation.
But all those things are theory, right?
Do you believe those theory?
I'm not sure.
Maybe light is actually oscillating
in the longitudinal direction, right?
If you are a physicist, you should ask this question.
Maybe you'll find something which
is inconsistent with the theory, then you'll feel really happy.
So the question we are trying to answer
is how do we know the electromagnetic waves are
transverse waves.
And we are going to work through a few examples
to convince ourselves, maybe that's the case,
maybe we have some hint.
Of course, we cannot prove 100% but very likely this is
probably the case.
So let's go ahead and start the discussion today.
So instead of adding one more dimension
by arranging materials, we can actually
discuss how the pointing direction of the electric field
depends as a function of time.
So in order to do that, I need to remind you
how to write down electric field for the electromagnetic wave.
So in a previous lecture, if I organize myself
so that the electromagnetic field is
going in the z direction-- the electromagnetic field
is going in the z direction.
All right, if I choose that, then basically electric field
as a function of z and time will be
equal to the real part of some vector psi 0 times
exponential i kz minus omega t.
If I assume that this electric field,
the electromagnetic field is propagating to where--
the z direction, right?
So by now, you should get used to this already.
This is going to the positive z direction.
Where I intentionally write psi 0
there is actually a vector which contains two components-- psi 1
in the x direction and psi 2 in the y direction.
And you can see from here that you can immediately
recognize that this can be returned
as a superposition of two waves.
So one is actually electromagnetic wave
with electric field in the x direction.
And the other wave, the other electromagnetic wave,
is actually oscillating with electric field in the y
direction and with amplitude psi 2.
So that is one you can get immediately because
of the superposition principle.
So it's a superposition of two electromagnetic waves.
And of course, in this case, I take
the real part of this vector.
So in general, psi 1 can be a1 exponential i phi 1.
And in general psi 2 can be a2 exponential i phi 2.
So in this notation basically, we write everything
in terms of vector.
And sometimes we write those things in terms of matrix,
and that sometimes serve a better purpose for calculation.
So we can also rewrite this thing in matrix form.
So I have a matrix E which have two components, E x and E y.
And this is actually equal to a real part of some Z matrix,
which also contain two components, times a scalar,
which is exponent i kz minus omega t.
So as can see, I am now just setting up
the language we want to speak so that we communicate.
So in this case, the z matrix is written as psi 1 and psi 2,
which are the two components.
One is in the x direction, the other one
is in the y direction.
And we are going to use this language
and see what we learn from it.
So you can see that we have been discussing
electromagnetic wave, which is propagating
toward the positive z direction.
And I have two component which I can have--
x direction electric field and y direction electric field.
So let me try to go through some example
and see what we can actually learn from this.
So the first example I would like to talk about
is that if I have two waves, E1.
The first wave is actually E0 cosine kz
minus omega t in the x direction.
And then the second wave is E2 equal to E0 cosine
kz minus omega t, in the y direction.
Be careful-- the direction of the electric field
is different for the first and the second plane wave.
And if you actually notice from this expression,
these two waves are in phase.
So in-phase means that they reach maxima at the same time.
So in this case, if I add two waves with no phase difference,
what is going to happen?
So I am going to have a E vector, which is E1 plus E2.
And if I plug the locus of this kind of electric field
in the two-dimensional space, x and y, so now I am plotting,
so I am fixing my z position for example at 0.
And I would like to see how the locus of the E
vary as a function of time.
In this case, since E1 and E2 have the same amplitude, which
is E0, and also they have no phase difference,
that means they reach maxima simultaneously.
So that means originally, for example, in the beginning
the electric field is 0.
And this electric field projection to the xy plane
will increase until some maximum.
At the maxima, this will be E0, the x and y position
will be E0.
Then it goes back to minus E0, minus E0,
in the x and y direction position.
And it goes back and forth.
So in this case, this is actually still not
very different from what we had discussed
before because the electric field is still
oscillating up and down.
But the difference is that it's not
oscillating at the x-axis or y-axis,
but in the axis which is actually 45 degree with respect
to the x or y-axis.
And now we can do a exercise to write it down
in the matrix notation.
Now I have this E matrix will be equal to the real part of E0,
E0.
And both E0's are real, exponential i kz minus omega t.
And basically I can conclude that z will be equal to E0, E0,
and this will be E0, 1, 1.
In this kind of situation, the electric field
is varying as a function of time.
The position to xy plane is a line.
When I happens, we call it linearly polarized.
So this is actually just a name.
But indeed the locus of the electric field on the xy plane
is a linear line, so that we call it linearly polarized.
And of course you can say, OK, this is just one example.
I can have many, many examples which you also create
a line when you plot the locus.
So for example, I can have z equal to E0, 1, 0.
Can somebody tell me what does that notation mean?
AUDIENCE: That just means one wave electromagnetic wave
in one direction.
YEN-JIE LEE: That's right.
So that means you only have electric field
in the x direction.
So this is the x direction, this is y direction.
And this is also linearly polarized.
Of course you can have very similarly E0, 0, 1.
And in this case, you only have electric field
in the y direction.
Just want to tell you my language
I'm introducing here so that we communicate.
And of course, you can have z equal to E0 cosine theta
sine theta.
What does that mean?
This means that when I plot the locus on the xy plane,
I'm going to have a straight line still,
but now the angle between this line and the x-axis
is going to be theta.
So all those examples, you can see that in the first case
the oscillation is in the x direction.
And in the second case the oscillation
is in the y direction.
And then in the third case it can be in an arbitrary line--
theta angle away from the x-axis.
So all those things are linearly polarized.
So so far, there is nothing to surprise you.
So that's what we have been talking about.
And you can say that, Professor Lee,
you were just not doing a very good job.
You didn't rotate the axis right so that all the electric field
is in the x direction.
So in principle for the discussion of the linearly
polarized wave, you can do a good job
by rotating your axis so that you only
have one component in the x direction.
So I say, OK, yes, I agree, but this is useful discussion.
All right, so now maybe you're bored.
They have no phase difference, right?
So how about we introduce some phase difference and see what
will happen.
So now if I consider the second situation--
I have E1 is equal to E0 cosine kz minus omega
t in the x direction.
Now I consider E2, this would be E0 sine kz minus omega t
in the y direction.
I hope you can see it.
And then you can see that now they reach the x component
and y component electric field.
They reach maxima at different times because of the phase
difference.
How big is the phase difference?
Can somebody actually remind me?
AUDIENCE: Pi over 2.
YEN-JIE LEE: Pi over 2, very good.
All right, so this I can write it
down as E0 cosine kz minus omega t minus pi
over 2, as you already figured out.
And now I can also again write it in the language I like,
the matrix format.
Before I achieve that, I can write it as a real part of E0
in the x direction plus E0 exponential minus i pi over 2,
because this minus pi over 2 sign,
the factor here for the y direction.
And all those things are multiplied by exponential i kz
minus omega t.
Both of them are actually oscillating at the same omega
angular frequency.
And you can see that now I can collect this phase difference
back into a complex factor here, exponential minus i pi over 2.
And exponential minus i pi over 2 is actually just minus i.
So basically you can figure out that it's just minus i.
So then I can write down my expression for the E matrix.
This one look like real part E0, 1,
minus i, exponential i kz minus omega t.
And in this case, z vector will be 1 minus i
in the language which I introduced today.
Everybody is following?
Am I going too fast?
Very good.
Thank you for the feedback.
So now I'm going to do the same thing
to plot the locus of the electric field
as a function of time on the xy plane.
So now you can see that it's pretty interesting.
Since I have a cosine and sine, I assume that t is equal to 0,
z is equal to 0, then I will only
have electric field in the x direction.
So this is actually when t is equal to 0.
All right, and if I increase time,
I fix the z to be equal to 0.
If I increase the time, this locus
is going to do an interesting thing.
It's going to be rotating until it finish one period.
Why is that happening?
Because what I'm plotting is the locus of the electric field.
And you can see that if I set z equal to 0, when
z is equal to 0, this will give you
E0 cosine omega t in the x direction
and E0 sine omega t in the y direction.
So E0 cosine omega t, E0 sine omega t-- wow,
this reminds you about the previous discussion.
If you have a cosine and a sine, they
work together in different directions, that's
going to get you a circle.
So that's why in this locus you see a circle.
And it's rotating in a counter-clockwise direction.
And the speed of the rotation is related to omega t.
And now if I increase the time, then this point
is going to be going in a counter-clockwise direction.
And you may be super surprised because what I have been doing
is to add two linearly polarized waves together.
The only thing which I always say,
the magic I have been doing, is to introduce a phase
difference.
And you can see that instead of going up and down,
up and down as a function of time, now
it's actually doing rotation.
So what is going to happen is that as a function of the time
the direction of the electric field
is going to rotate as a function of time.
So we call this situation circularly polarized.
By the way, because of the initial two components
I put in, both of them have amplitude E0.
Therefore, it's circularly polarized.
I can also try to do something different.
For example, I can change the x direction.
So the third situation is that I can
change the first x direction amplitude from E0 to E0 over 2.
Then what I'm going to get is something like this.
The only thing which I change with respect to 2
is that I changed the amplitude of the electric field
in the x direction by a factor of 2.
Can somebody actually tell me what
is going to happen in this situation?
What will happen to the locus?
AUDIENCE: Gets squished.
YEN-JIE LEE: Yeah, it gets squished in,
so in the x direction.
Yeah, that's right, very good.
So what you are going to get is that instead of a circle
you get something like this.
So this is the x direction and this is the y direction.
And this will give you a maxima x equal to E0 over 2.
And of course the original amplitude in the y direction
didn't change, and that gives you a maximum value of E0.
And this becomes an elliptically polarized wave.
So this kind of situation is called elliptically polarized.
How do we actually visualize this situation?
So that means, as in the beginning,
when the electric field is aligned with the x-axis,
it's pointing in the x-axis.
It has a shorter length.
And when it rotate, rotate as a function of time to the y-axis,
the amplitude becomes bigger.
And it gets smaller again, and increase again,
it becomes bigger.
So you can see the amplitude is actually
changing as a function of time.
When you have this kind of situation,
they have different amplitudes, although they have a fixed
phase difference, so pi over 2.
Any questions so far?
OK, so if you noticed that there's actually
another way to produce elliptically polarized wave.
What we could do is that instead of changing the phase
by a factor of pi over 2, we can change the phase difference.
The phase difference can be delta
phi, which is the phase difference,
can be a different value, arbitrary value, not equal
to pi over 2 or 3 pi over 2.
If the delta phi, which is the phase difference between x
and the y direction, EM waves--
electric field.
If the phase difference is not pi over 2 or not 2 pi over 2,
then you can also create an elliptically polarized wave.
So that starts from the original figure,
which we actually discussed--
situation number 1.
Situation number 1, I have delta phi,
the phase difference equal to 0.
When that happened, you have a linearly polarized wave.
Basically what you are going to get
is a line in this two-dimensional space.
If I increase, so now I slightly delayed.
If I slightly delay the electric field in the y direction.
So if I slightly delay electric field it in the y direction
so that delta phi now is greater than 1, what is going to happen
is that it's going to look like this.
This means that they will not reach maxima really
simultaneously.
There will be because of the phase difference.
And you'll see that originally when
there were no phase difference, you
would be oscillating back and forth in this blue line.
When you increase the delta phi slightly,
then you are going to get also a elliptical shape,
but now is tilted with some degree, which
is 45 degree in this case.
So therefore you can see that now I can also
create this is also elliptically polarized wave,
I can also create elliptically polarized wave
by adding two components, which they have some slight phase
difference.
So that is actually how I can create a something which
is called circularly polarized or elliptically polarized.
Originally before you come into this class,
it may look really completely bizarre
that, oh, I can have electric field going up
and down as a function of time.
But how could I rotate this thing, right?
Looks really strange.
How can I see this from the Maxwell's equation?
But now you get the idea.
Basically, that's because I can now overlap two components.
Both components individually are linearly polarized.
But if I introduce a phase difference,
then the superposition of these two components
become something which is actually
rotating as a function of time.
And that's pretty interesting.
So let's visualize what we have learned so far.
The first one is linearly polarized.
Actually, it doesn't surprise you--
that's the example which we have been
using in the previous lectures.
And this is the situation of circularly polarized wave.
So let's focus on the figure at z equal to 0.
So you can see that in this case the direction
of the electric field is actually
rotating as a function of time.
And of course as we discussed, I can actually
add two electric fields with different amplitudes
or introduce slightly different phase,
then you will see that not only that the direction is changing,
but also the amplitude is changing.
And in this case we call it elliptically polarized wave.
Any questions so far?
Yes?
AUDIENCE: So the magnitude always is constant with time
now?
YEN-JIE LEE: Yes, so the magnitude,
or say the intensity, is proportional to E squared,
right?
So it's actually a constant.
So now let me add more excitement here.
So now suppose I have a perfect conductor, all right.
So what I am going to do is that I'm
going to introduce some more excitement by shooting
this linearly polarized wave through some material.
If I have a perfect conductor where all those strips,
I have only strips of perfect conductor instead of a plate--
we were talking about plate before.
And the lesson we learned is that the plate
is going to refract the electromagnetic wave
because that electrons on the plate is so busy,
it's trying to make sure that the electric field
in the surface of the plate is equal to 0
because that all those electrons are really moving around.
So that's making sure the electric field is canceled.
And therefore, it's going to refract
the electromagnetic wave.
So how about I restrict the direction
of the movement of the electron so that it can only move
in the horizontal direction?
What is going to happen is that in this case,
the electric field is actually oscillating up and down.
And the electrons see this field and they will say,
no, no, no, this is not why we work.
This is not we are going to vote for.
And I'm going to rearrange ourselves to compensate that,
all right.
And they were asking, can I move.
Oh yes, I can move in the horizontal direction.
So they jump up and down, then they can actually
cancel this electric field.
Therefore, what is going to happen
when we have this perfect conductor is
that the electromagnetic field is going to be bounced back
like what we had before when we talked about a metal plate,
all right.
So that's actually pretty nice.
Now suppose I have another perfect conductor.
If I have another perfect conductor,
this time the perfect conductor is arranged such
that all those charges can only move inside or outside
of the board instead of going up and down.
Now I have the incident wave, which is actually polarized
in the horizontal direction.
So you can see that this time the electrons
are really nervous about this.
And oh no, I have to do something,
but they cannot move up and down.
Therefore, what is going to happen
is that there will be no cancellation of electric field
at this boundary.
And this polarized wave can pass through
this so-called polarizer or perfect conductor
without getting stopped or get refracted.
And as you can see that in these two examples,
I have so called the easy axis.
So you can see that the easy axis, as the name
should tell you, so easy axis is the axis which is easy.
All right, so what does that mean?
That means if I have electric field aligned
with the easy axis, it will pass through.
And the easy axis is perpendicular to the direction
of all those streets I have in the perfect conductor.
Let's look at the first example.
It tells you that the electric field is not
aligned with the easy axis.
That means it's not easy, so it got
refracted-- oh, life is hard.
And the second electric field is smarter.
It got aligned with the easy axis.
So life is easy, it passed through.
That's how I remember this so-called easy axis.
Any questions?
So we have been talking about all those crazy names
of polarized light.
And I hope that you also have learned about unpolarized light
before.
Most of the lights are unpolarized.
For example, light emitted from those light bulbs
are not polarized because every emission of a photon
can be aligned in different direction.
Therefore, as I was talking about,
there are an infinite number of photons hitting my face
and bouncing back to your eye.
All those things are not aligned to the same direction
as what we have been discussing before.
In the case of linearly polarized wave,
all the electric fields are pointing to the same direction.
And apparently those lights hitting my face is not aligned
and it's actually pointing to random directions.
So that brings me to the fifth situation, which
I would like to talk about.
The fifth situation is that I could have unpolarized light.
So what do I mean by unpolarized light?
That means electromagnetic waves which
were produced independently by a large number
of uncorrelated source.
You may ask, OK, wait, wait, wait--
unpolarized means that we have a lot of different light source
pointing to different angles.
Shouldn't it just look like this?
So you have pointing this direction, all the directions,
all the possible directions, pointing
to all different possible directions, like this.
Shouldn't that diagram actually tell you
something about unpolarized light?
Life is hard.
If everything is aligned and also
arriving at the same time like this, what is going to happen?
They cancel, right?
Because all those things are vectors.
If they reach maxima at the same time,
this means that they are emitting always
at the same time, but in random direction.
And emitting at the same wavelength,
then they are going to cancel each other.
So apparently that's not what I mean by unpolarized light.
What I mean is that they can be pointing
to different directions, but that emission
time and the wavelength of all those emitted electromagnetic
waves can be slightly different, otherwise they
will just cancel each other completely.
Any questions about what I just said?
So very good.
So now what I am going to do is to show you some examples
related to polarizer.
Before that, I will take a five minute break before we
come back to the demonstration.
So we will come back at 36.
OK, so welcome back.
So we would continue the discussion of the polarization
and the polarized wave and the polarizers.
So before that, I already introduced the polarizer
by the perfect conductor arranged in the x direction
or y direction, many of them are strips,
aligned in the x direction and the y direction.
And usually we use a simple diagram
like this, a circle and an arrow,
to indicate the easy axis to tell you
about the property of the polarizer.
So in this case, if I have a circle
and I have an arrow, which is pointing toward the x-axis,
in this case my coordinate system is here,
horizontal is x, vertical is y, that
means my easy axis is in the x direction.
And I can actually write down what
would be the matrix presentation of p0, which would present
the effect of this polarizer.
The p0 will be a matrix in this form, 1, 0, 0, 0.
If I have this P0 acting on the z,
for example, I can have the p0 acting on the z, then
basically what would happen after the light passed
through this polarizer will become z multiplied by p0.
So basically all this p0 matrix does
is to extract the position of the field in the x-axis.
But I'm just writing it down in the matrix format.
And on the other hand, for example, I
can have this easy axis, which is 90 degrees, which
affects the x-axis, which is in the y direction,
then I write this down in the form of p pi over 2.
This pi over 2 is the angle between the easy axis
and the x direction.
And in this case, you are going to get 0, 0, 0,
1 because all this matrix set notation
does is to extract the component which is actually
in the y direction.
And you can of course multiply--
when you have an incident electromagnetic wave
which you can extract the polarization z matrix, you just
multiply p pi over 2 and z.
Basically you can actually get the resulting polarization
of the electromagnetic field after passing
through this polarizer.
What is going to happen is that only the y component
will survive.
In the case of easy axis aligned in the x direction,
only the x component will survive.
So now let me go through a short example here.
So suppose initially I have an electromagnetic field
polarized, it's linearly polarized in the y direction.
And I have that electromagnetic field
passing through a polarizer, which is actually theta
degree away from the y-axis.
So what is going to happen is that all the components which
is parallel to the axis survive.
All the projected components in the axis perpendicular
to the easy axis didn't make it because the electrons are going
to be oscillating up and down like crazy
to compensate and then refract that component.
So what is going to happen is that after passing
through the easy axis, the direction of the polarization
will be altered such that it's actually
in line with the easy axis of the polarizer.
So you can see that, after passing
through this polarizer, the direction of the polarization,
this is still linearly polarized,
the direction is changed by theta degree.
And also the amplitude also changed
because only the component parallel to the easy axis
survived.
Therefore, the magnitude of the E vector
becomes E0 cosine of cosine theta.
And therefore, the amplitude of the light got reduced.
It becomes I0 cosine theta squared
because the intensity of the light
is proportional to E squared.
So after all this, this is time to take a look
at some demonstration here.
So here I have a setup.
I have emission of linearly polarized microwave
from the left-hand side, which is the source.
And I have a receiver which is connected to this scope.
You can see the result on the scope.
You can see that indeed they are energy
passing from this source.
And it got accepted and then recorded by the scope.
Now I have all of those--
OK, all are not so perfect, but those
are metals which have many, many strips
or many, many little rods here.
And my initial linearly polarized wave
is actually pointing in the vertical direction.
So if I have my polarizer arranged in this direction,
can you predict what will happen to the readout on the scope?
Will I see signal or not?
How many of you think we will see signal
if I arrange that the electromagnetic wave is
polarized in the up and down direction?
OK, one, two, three--
only four people think so.
How many of you think nothing will be observed by the scope?
Most of you actually think so.
So let's really do this experiment.
Look at what is happening.
Can you see it?
Do I see a signal?
No.
All the signals are canceled, right?
Why?
Because all the hard work of all those electrons in the metal,
they are like crazy, oh my god, this is a disaster.
I am going to oscillate up and down-- cancel, cancel, cancel.
Then it got cancelled.
On the other hand, I can save all those electrons.
So now I'm going to rotate this thing by 90 degrees
and see what is going to happen.
Can you see it?
Nothing happened, right?
Because those electrons-- oh my god, crisis coming,
but now there's nothing I can do because I cannot move up
and down.
Therefore, there will be no refraction.
So they have just to accept the fact
that this electromagnetic field went through.
All right, and that actually does
something really interesting.
Look at this-- so now I can actually
make electromagnetic field completely destroyed,
because originally the electromagnetic field is
linearly polarized, up and down in this direction.
So if I have a plate which is 45 degrees
and I put in another one here--
it doesn't really work very well.
The signal probably is too small.
How about we do this demo in a different way.
So right now here I have my computer here.
Can you see my computer?
You can see the screen, right?
The screen essentially is made of LED.
Those LED screens emit polarized light.
And those are the polarizer, which
is the equivalent version of those metals
but arranged in a really fine grain--
so you cannot really see all those strips.
And they look pretty transparent.
But the idea is pretty similar.
And the polarizing axis is in the horizontal direction.
The easy axis is in horizontal direction.
So let me put this here.
Can you still see the screen?
You cannot, right.
But if I rotate this by 90 degrees, you can see it.
That is because all those light emitted
from the screen it linearly polarized
up and down direction.
So now if I have another identical polarizer which I
insert between these two, so now you cannot see anything
because all the light is pointing up and down,
or the electric field is pointing up and down.
And now if I insert another one, which is 45 degrees,
can you see something?
You can see it-- why?
That is because if I insert this additional polarizer, like what
I was discussing, it's going to take
the position to the direction of the easy axis.
And if I have another one, which is the easy axis pointing
toward the x direction, then again since this vector is
tilted already by some theta degree,
then you can see some residual component
which can pass through the second polarizer.
On the other hand, if I remove this polarizer in between,
then what is going to happen is that all the components
is pointing to the y direction.
Then you cannot see any component which pass through
this polarizer.
So that's actually pretty interesting.
And I can rotate this, and you can see
that the magnitude is changing.
Only when I have 45 degrees, I see a maximum intensity.
Any questions?
Very good.
So now what I'm going to do is to discuss with you
an interesting question, which was posted by Einstein.
Einstein said that, as I said, "God doesn't play dice
with the world."
So that's actually what he believed.
So we can do an interesting experiment, which
I have single photon source.
So what do I mean by single photon source?
I can emit one photon at a time--
just one, and then the second one, and the third one--
and have then pass through some imaginary polarizer.
So let's suppose I have unpolarized
light with intensity at 0.
So basically, after it passed through this polarizer
with a axis pointing to the x direction,
in this case the x direction is pointing
to this direction, what you are going to get
is that you are going to filter from all
those unpolarized direction, you are
going to filter only the light which is actually parallel
to the easy axis.
So therefore, after the unpolarized light source
passes through this polarizer, you
are going to have all the electric field
pointing to the x direction.
Now everybody can accept.
And of course if I have a second one, which
is in the y direction, since there will
be no component pointing to the y direction,
you get zero electric field.
That's very nice.
So Einstein was really happy that, oh,
that means unpolarized light maybe is just 50%
of the polarized light in the x direction
and 50% of the polarized light in the y direction.
Because each time I emit one photon, the first half of them
got stopped by the x direction easy axis polarizer.
And then the second half got stopped
by the second polarizer.
So that probably makes sense.
But how about-- as we did in the demonstration--
how about we rotate the second polarizer by 45 degrees?
What will be the intensity?
But now, hey, you cannot split a single photon because a photon
is a photon.
How can I split, right?
Because if I describe my unpolarized light
as 50% linearly polarized in x, the other half is linearly
polarized in y, then I am in trouble
because I don't know how to calculate
what is this intensity.
So basically maybe all of them pass through or all of them
doesn't pass through.
Then we can do this experiment.
And this is the result--
what you are going to get is that the intensity
is going to be I0 over 4.
That is actually not 0 or I over 2.
So that means really the nature of plate
dies because a single photon is equivalent to something which
can be described by a wave.
So that gives us some possible connection
to quantum mechanics, because that
means a single photon is not like a single object, which
is actually passing through all those polarizers.
But they actually act also like waves.
So that is something which we'll follow up with later lectures.
When we talk about interference, et cetera,
we are going to also discuss related issues
about the connection to quantum mechanics.
So today, we actually have learned
about polarization, linearly polarized, circularly
polarized, and elliptically polarized
electromagnetic waves, and also unpolarized light source.
And we have learned about how to produce
polarized electromagnetic wave with polarizer.
Basically have the unpolarized light
source pass through a polarizer, you'll have a polarized light.
We didn't cover quarter wave plate yet.
We are going to cover that next time.
And also, next time we will talk about how we actually can
generate electromagnetic field.
We have been talking about electromagnetic field
for a long time, but how are they actually
generated is an issue which we have actually touched.
And that is actually going to happen after the midterm
next week.
Thank you very much.
And if you have any questions, please let me know.
AUDIENCE: [INAUDIBLE]