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We have discussed earlier the
boundary conditions for ideal

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conductors, and today we are
going to aim at dielectrics.

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In conductors,
we couldn't have electric

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fields and we couldn't have a
changing magnetic field.

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The situation is very different
in dielectrics.

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The dielectric is an ideal
insulator.

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It has zero conductivity.
And there are electric fields.

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And changing magnetic fields
are certainly allowed inside the

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dielectrics.
One adjustment that we have to

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make, which we discussed
actually earlier,

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is that wherever we have
epsilon zero in Maxwell's

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equations, we have to replace it
by kappa of E epsilon zero,

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the dielectric constant.
And wherever we have mu zero,

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we have to replace this by
kappa of m times nu zero.

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That is the magnetic
permeability.

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And that has certain
consequences for the speed of

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propagation of electromagnetic
radiation.

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I want to stress,
though, that kappa of m,

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except for paramagnetic
materials which are rare,

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kappa of m is always very close
to 1.00000, except for

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paramagnetic materials.
We know that the speed of light

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of electromagnetic radiation in
vacuum was this.

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And so now that is going to
change.

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We have to take this into
account.

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Now we get v,
which is now the speed of

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light, speed of electromagnetic
radiation, which is now c

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divided by the square root of
kappa E times kappa m.

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And that is also c divided by
n, as in Nancy.

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And n is called the index of
refraction of the dielectric.

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The dielectric constant is a
strong functional frequency.

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I have mentioned that earlier.
That is the whole basic idea

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behind dispersion.
If I stick for now to light,

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it is about 5 times 10 to the
14 hertz, and I give you an

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example which you have seen
before for water.

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Kappa of E for light,
that means for that high

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frequency, is approximately
1.77.

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And so the index of refraction
is about 1.33.

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But it is different for the
different colors.

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A little bit different,
but it is different.

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And if you take glass,
there are many different kinds

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of glass with many different
indices of refraction,

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we will typically take today a
number of about 1.5.

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 03:32 
That stands for incident.

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What is going to be reflected I
am going to call r.

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That stands for reflected.
And what penetrates into the

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medium two, I will give the
letter t which stands for

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transmitted into.
We also call it refracted.

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But now you get reflection and
refracted, that is confusing,

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so I give it the letter t.
And so there is here a k

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vector.
And this then is k of the

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incident radiation.
I call this angle theta one.

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And then there will be a
reflection at this surface.

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And so I am going to put in
here k reflected.

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And this angle I will call
theta three.

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I will show you shortly that
theta one is the same as theta

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three.
And then here some of that

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radiation goes into the second
medium.

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And I have here k transmitted.
The magnitude of these two k

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vectors is, of course,
the same because lambda is 2 pi

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divided by k.
But the magnitude of the k

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vector here is different because
the two have different indices

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of refraction.
06:00

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And then, if you prefer,
I can write it down in complex

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notation, omega t minus k of i
dot r.

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And this is the vector.
If you would like to write it

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down as the cosine,
that is fine,

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too, of course.
And so this is the incident

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wave.
I can now change the i to an r

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here and to an r here.
And I can change this i to an r

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here.
This r has nothing to do with

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that r.
That is the position vector.

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This means reflected.
I can put a t here and a t here

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and a t here,
and I would have three

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equations then for the incident
one, for the reflective one and

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for the transmitted one.
And now comes a key point that

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at z equals zero,
that means really at the

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boundary these three waves have
to be inphase with each other,

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except for a possible 180
degree phase change.

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And this should remind you of
something that we have dealt

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with many times before.
You have a string which is

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connected to another string.
And at the junction they are

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inphase.
The incident one and the

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transmitted one are inphase with
each other.

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And the incident one and the
reflected one either inphase,

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mountain comes back as a
mountain, or out of phase 180

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degrees.
And the same situation you have

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here.
And that means that this dot

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product for these three cases,
ki dot r, kr dot r and kt dot r

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at z equals zero must be the
same, except for the 180

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degrees.
But the 180 degrees we can

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always deal with in terms of a
minus sign.

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If you massage this a little
bit further, which is really not

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much more than high school
algebra but has worked out in

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Bekefi and Barrett.
If you massage this a little

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further, you will be able to
demonstrate it is the

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consequence of the fact that
these three dot products are the

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same at z equals zero.
On page 472 this is worked out

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in great detail.
You will find then that theta

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one is theta three and that n1
times the sine of theta one

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becomes n2 times the sine of
theta two.

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We call theta one the angle of
incidence.

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We call theta three the angle
of reflections.

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The angle of incidence is
always the angle of reflection.

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This law you may remember from
your high school days,

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if you had any physics,
is Snell's law.

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And that is a law that does not
follow for Maxwell's equations.

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Many of you think that Snell's
law is the consequence of

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Maxwell's equations.
That is not true.

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This law also holds for water.
It holds for sound.

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It is simply the consequence of
the fact that there is a

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different velocity in the medium
here than there is in the medium

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there.
Maxwell's equations will come

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up later and have nothing to do
with Snell's law.

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Willebrord Snellius was a
mathematician.

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He was Dutch and he discovered
this relationship empirically,

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which is quite an amazing
thing, by the way.

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In 1621, he discovered it for
light at the time that it was

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not known that the velocity of
light in water was actually

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different from the velocity of
light, the speed of light in

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air.
Thank you very much.

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This is theta,
too.

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Thank you.
Much obliged.

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Now I am going to apply Snell's
law for two different

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situations.
I will first go from air to

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water.
Here we have air.

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And we will just assume that n
is very close to 1.00.

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And then here we have water.
This is n1.

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One is always where you are two
is where you are going.

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We have water.
And so n2 is 1.33.

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And so here we have light
coming in, and this angle is

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then theta one.
And here we have light being

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reflected.
And we know that the angle is

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also theta one.
And then here we have light.

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This is the incident one,
the reflected one and the

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transmitted one.
And this angle then is theta

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two.
And what you see here and what

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I will demonstrate is not the
most exciting thing.

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If you increase theta one then
you will see that theta two also

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increases.
That follows immediately from

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Snell's law.
You make this angle larger,

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and so this opens up and this
angle also opens up.

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What is interesting is that you
cannot make theta one any larger

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than 90 degrees.
And when you do make theta one

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90 degrees, theta two has a
maximum.

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And that maximum follows
immediately, of course,

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from Snell's law.
And, in this case,

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the angle theta two,
which is the maximum value that

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you can have,
is then 48.7 degrees.

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That is the consequence of the
indices of refraction the way I

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have chosen them.
And I will show you that.

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What you can also do,
which makes it way more

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interesting, is you can also
send light from water to air.

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Now you must be careful.
If you are in water that is now

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your n1.
If you go to air that is now

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your n2.
I will be repetitive,

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but I do that for a good
reason.

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N1 is 1.33.
And now this is air.

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And so now n2 is 1.00.
Here is now light coming in.

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This now, by my definition,
is theta one.

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That is the angle of incidence.
And so this now,

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by my definition,
is the angle of reflection,

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which is also theta one.
And then here in this medium in

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air, you will have light coming
out at any angle which we now

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call theta two.
And you can apply Snell's law

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to find these angles.
And it is now immediately

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obvious that if you make theta
one larger than 48.7 degrees

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that there is no longer any
solution for Snell's law,

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because the sine of theta two
cannot be larger than one.

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And so now there is a crisis
for nature.

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And what nature now does is
reflects all the light off this

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surface.
Nothing comes out anymore in

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the air.
And we call it total internal

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reflection.
And that happens at a critical

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angle.
Theta critical which in this

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case is, of course,
the 48.7 degrees for this

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transition.
And you will find that value

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the sine of the critical angle
n2 over n1.

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And it is only meaningful if n1
is larger than n2.

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There is no critical angle in
this situation.

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There is only a critical angle
that means total internal

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reflection if n1 is larger than
n2.

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We say, as physicists,
you have to go from an optical

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dense medium to an optically
less dense medium.

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That is when you can have total
reflection here at the surface.

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And so I would like to
demonstrate that.

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We are going to do that here.
We are going to make it a

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little dark for you so that you
see it better.

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I have here a laser light
coming out.

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Shall we turn that off?
Yeah.

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Can you turn that off?
It would be nice.

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Thank you very much.
You have here light coming from

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above.
And it strikes the water.

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And you see that some of that
light is reflected.

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And one of my major goals
today, which we have not reached

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yet, is to actually calculate
the light intensity of that

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reflected light and see how
little is reflected.

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But for now it is enough that
you notice that the angle of

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incident is the angle of
reflection.

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And notice how much goes into
that second medium of water.

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That is that angle theta two
which is clearly small than the

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angle theta one.
What I can do now is increase

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the angle.
I can increase the angle theta

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00:16:02,208 --> 00:16:05,741
one, provided that I get my
hands out of the way.

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There we go.
And this is not so exciting.

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You still see reflected light
coming here.

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00:16:11,703 --> 00:16:15,088
And you see that the angle
theta two increases.

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00:16:15,088 --> 00:16:16,560
Is this exciting?
No.

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But this is a consequence of
Snell's law, the breaking of

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light in water.
Now I do something that is way

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00:16:23,994 --> 00:16:27,307
more interesting.
I now have light coming from

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below.
We manipulated the light using

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a prism.
It comes in now from below.

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00:16:32,967 --> 00:16:36,630
It doesn't come in from above,
but it comes in from the water

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here.
And that is the situation I had

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on the blackboard there on the
right side.

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00:16:41,330 --> 00:16:45,053
Now you see a reflection in the
water and you see some of that

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radiation coming out into the
air.

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The intensity of the light in
the air is actually quite

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strong.
Notice that.

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00:16:51,524 --> 00:16:55,308
But if now I make the angle of
incidents larger than 49 degrees

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then this, of course,
will approach 90 degrees and

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will then suddenly disappear and
100% will be reflected.

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And that is what is coming up
now.

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00:17:05,434 --> 00:17:08,606
I am going to make the angle
larger, larger.

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Notice that the light intensity
of the beam in air is already

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decreasing.
And now I reach that critical

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angle of 49 degrees.
And now we have 100% internal

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reflection.
All the light that came out of

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00:17:22,622 --> 00:17:26,901
here, 100% is now reflected at
that surface because we have

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exceeded the critical angle.
And this, as you can imagine,

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00:17:32,621 --> 00:17:35,695
plays a major and interesting
role.

234
00:17:35,695 --> 00:17:39,673
In fiber optics,
this allows you to transport

235
00:17:39,673 --> 00:17:44,646
images over thousands of miles
without any loss of light

236
00:17:44,646 --> 00:17:47,268
intensity.
But, in a nutshell,

237
00:17:47,268 --> 00:17:51,789
it comes down to this.
If I had here some kind of a

238
00:17:51,789 --> 00:17:56,942
glass fiber or plastic fiber,
index of refraction say 1.5,

239
00:17:56,942 --> 00:18:03,000
which would have a critical
angle of about 42 degrees.

240
00:18:03,000 --> 00:18:09,955
And if we send here light in
then it is very easy to arrange

241
00:18:09,955 --> 00:18:16,674
it that the angle there is
larger than the critical angle.

242
00:18:16,674 --> 00:18:23,158
And, if that is the case,
then it is for 100% reflected.

243
00:18:23,158 --> 00:18:29,878
But if it hits here it is,
again, larger than the critical

244
00:18:29,878 --> 00:18:34,239
angle.
It is again for 100% reflected.

245
00:18:34,239 --> 00:18:39,780
And you can send it five times
around the earth,

246
00:18:39,780 --> 00:18:43,434
and it will come out
unaffected.

247
00:18:43,434 --> 00:18:49,329
This is a very powerful way.
And there are a lot of

248
00:18:49,329 --> 00:18:56,284
companies who try to make money
out of this who use this for

249
00:18:56,284 --> 00:19:01,000
image transfer.
18:32

250
00:19:01,000 --> 00:19:04,750
And the light that comes out
here, you see it on the wall

251
00:19:04,750 --> 00:19:06,893
there, I can put a knock in
here.

252
00:19:06,893 --> 00:19:10,778
I can do anything with it I
want and it will not change the

253
00:19:10,778 --> 00:19:13,189
intensity.
I can bend it over in this

254
00:19:13,189 --> 00:19:14,662
direction.
There it is.

255
00:19:14,662 --> 00:19:17,140
I can do anything with this I
want to.

256
00:19:17,140 --> 00:19:20,891
Look how tightly this is wound.
I will see it here at the

257
00:19:20,891 --> 00:19:23,101
ceiling.
That is the idea of fiber

258
00:19:23,101 --> 00:19:25,445
optics.
And to show you that you can

259
00:19:25,445 --> 00:19:30,000
actually also send a message,
that means an image.

260
00:19:30,000 --> 00:19:33,061
And that is,
of course, where the big

261
00:19:33,061 --> 00:19:36,123
companies are becoming
interested in.

262
00:19:36,123 --> 00:19:38,759
I want to demonstrate that,
too.

263
00:19:38,759 --> 00:19:41,480
I have a special message for
you.

264
00:19:41,480 --> 00:19:45,818
We have here this special
message which I wrote this

265
00:19:45,818 --> 00:19:49,134
morning.
And then we have here the fiber

266
00:19:49,134 --> 00:19:52,366
optics, a few thousand of these
fibers.

267
00:19:52,366 --> 00:19:57,214
And you can put a knock in it.
The message will go through

268
00:19:57,214 --> 00:20:00,754
here.
The fibers are arranged in the

269
00:20:00,754 --> 00:20:03,269
following way.
This is about two centimeters

270
00:20:03,269 --> 00:20:06,135
by six millimeters,
and here are all these fibers.

271
00:20:06,135 --> 00:20:08,767
And they come out at the other
end, of course,

272
00:20:08,767 --> 00:20:11,400
in the same sequence.
You don't scramble them.

273
00:20:11,400 --> 00:20:13,973
If you scramble them then you
cannot read it,

274
00:20:13,973 --> 00:20:17,483
which the military people use,
by the way, in the old days to

275
00:20:17,483 --> 00:20:19,530
transmit messages that were
decoded.

276
00:20:19,530 --> 00:20:21,811
But, in any case,
we will not decode it.

277
00:20:21,811 --> 00:20:24,209
We have the same image that
comes in here,

278
00:20:24,209 --> 00:20:27,718
the fibers come out in the same
way there, and here we have a

279
00:20:27,718 --> 00:20:32,083
television camera.
And then you can see very

280
00:20:32,083 --> 00:20:35,916
shortly there the message that I
wrote for you.

281
00:20:35,916 --> 00:20:39,250
Let me see whether we can get
that there.

282
00:20:39,250 --> 00:20:42,166
There you see the individual
fibers.

283
00:20:42,166 --> 00:20:46,166
For each one of the little
honeycombs is a fiber.

284
00:20:46,166 --> 00:20:51,000
And here comes the message.
I cannot read it because I have

285
00:20:51,000 --> 00:20:54,666
to concentrate,
but maybe you can read it and

286
00:20:54,666 --> 00:20:58,083
let me know what it reads.
Can I hear you?

287
00:20:58,083 --> 00:21:01,481
Oh, boy.
You can read it?

288
00:21:01,481 --> 00:21:06,296
Well, we knew that anyhow.
You see, it is not so bad,

289
00:21:06,296 --> 00:21:10,277
the quality.
And this is only a few thousand

290
00:21:10,277 --> 00:21:13,518
fibers.
This is actually a very nice

291
00:21:13,518 --> 00:21:18,333
way of showing you that you can
transport that image.

292
00:21:18,333 --> 00:21:22,870
And, as I do this,
I hope you realize that nothing

293
00:21:22,870 --> 00:21:25,833
changes.
You see no change there.

294
00:21:25,833 --> 00:21:30,000
And I make a crazy knot in
here.

295
00:21:30,000 --> 00:21:34,488
It will all go through and
there is no loss of light

296
00:21:34,488 --> 00:21:38,096
intensity.
Now comes the key question that

297
00:21:38,096 --> 00:21:43,023
is really at the heart of this
lecture, and that is light

298
00:21:43,023 --> 00:21:46,544
intensities.
I now am interested in light

299
00:21:46,544 --> 00:21:49,799
intensities.
We have an incident beam,

300
00:21:49,799 --> 00:21:54,640
we have a reflected beam and we
have a transmitted beam.

301
00:21:54,640 --> 00:21:58,688
And now, in order to get the
light intensities,

302
00:21:58,688 --> 00:22:03,000
I now must use Maxwell's
equations.

303
00:22:03,000 --> 00:22:14,400
And the first thing we have to
do is we have to derive the

304
00:22:14,400 --> 00:22:24,799
boundary conditions at the
surface of the dielectric.

305
00:22:24,799 --> 00:22:35,200
We did that before for
conductors, but now we have to

306
00:22:35,200 --> 00:22:45,200
do that for dielectrics.
Here we have the boundary.

307
00:22:45,200 --> 00:22:53,599
This is medium one,
index of refraction n1,

308
00:22:53,599 --> 00:22:58,400
kappa of E1,
kappa of m1.

309
00:22:58,400 --> 00:23:03,680
22:32
This is E1.

310
00:23:03,680 --> 00:23:12,697
That means in that medium one.
That is the total E vector of

311
00:23:12,697 --> 00:23:21,254
the electromagnetic wave coming
in, highly time variable,

312
00:23:21,254 --> 00:23:27,825
frequency omega,
but that speaks for itself.

313
00:23:27,825 --> 00:23:35,924
And I am going to decompose
this now, as I did before,

314
00:23:35,924 --> 00:23:43,107
into two components.
One that is parallel to the

315
00:23:43,107 --> 00:23:48,455
surface.
We call that the tangential

316
00:23:48,455 --> 00:23:53,957
component.
I am going to call this E1

317
00:23:53,957 --> 00:23:59,000
tangential.
23:30

318
00:23:59,000 --> 00:24:01,695
This n has nothing to do with
that n.

319
00:24:01,695 --> 00:24:05,214
That means normal.
I still feel bad about my two

320
00:24:05,214 --> 00:24:08,434
c's last time,
but there is not always a way

321
00:24:08,434 --> 00:24:11,579
that you can bypass using the
same symbols.

322
00:24:11,579 --> 00:24:14,574
I could have done that last
time, though.

323
00:24:14,574 --> 00:24:18,318
And now, in medium two,
when the waves get through,

324
00:24:18,318 --> 00:24:21,538
there is an electric vector
which is now E2.

325
00:24:21,538 --> 00:24:25,956
And I am going to decompose
that now also into one component

326
00:24:25,956 --> 00:24:30,000
tangential.
This is now E2 tangential.

327
00:24:30,000 --> 00:24:34,778
And this is now E2 normal.
Now, we have four Maxwell's

328
00:24:34,778 --> 00:24:37,754
equations, two dips and two
curls.

329
00:24:37,754 --> 00:24:42,532
The dips mean pill boxes.
The curls mean closed loops.

330
00:24:42,532 --> 00:24:45,327
Last time I did two out of
four.

331
00:24:45,327 --> 00:24:50,467
Today, I am a little greedier
and will do one out of four.

332
00:24:50,467 --> 00:24:55,336
And you do the other three.
None of them are difficult.

333
00:24:55,336 --> 00:25:00,836
It is just a straight-forward
8.02 application of the dips and

334
00:25:00,836 --> 00:25:05,111
the curls.
I will do only one.

335
00:25:05,111 --> 00:25:09,333
I will do the curl of E is
minus dB/dt.

336
00:25:09,333 --> 00:25:14,000
It is my favorite one.
It is Faraday's law.

337
00:25:14,000 --> 00:25:18,888
It runs our economy.
Whenever I see a chance,

338
00:25:18,888 --> 00:25:24,000
I use Faraday's law.
That means the closed loop

339
00:25:24,000 --> 00:25:29,777
integral of E dot dL,
dL is a little vector along the

340
00:25:29,777 --> 00:25:36,000
path that I take in the
direction of my path.

341
00:25:36,000 --> 00:25:40,947
It is a dot product.
It has to be a closed loop.

342
00:25:40,947 --> 00:25:46,105
That now equals minus d phi.
Let me put a B there.

343
00:25:46,105 --> 00:25:51,789
It is a magnetic flux change.
This is the magnetic flux

344
00:25:51,789 --> 00:25:58,105
change through a surface that is
open and that is attached to

345
00:25:58,105 --> 00:26:02,380
that loop.
You can choose any loop you

346
00:26:02,380 --> 00:26:05,112
want to.
You can even choose any

347
00:26:05,112 --> 00:26:09,520
surface, provided it is
everywhere attached to that

348
00:26:09,520 --> 00:26:12,517
loop.
And then Faraday's law holds.

349
00:26:12,517 --> 00:26:16,220
An amazing law,
truly incredible and always

350
00:26:16,220 --> 00:26:19,217
holds.
Here is going to be my path,

351
00:26:19,217 --> 00:26:24,066
no different from what I did
earlier when we derived the

352
00:26:24,066 --> 00:26:27,063
boundary conditions for
conductors.

353
00:26:27,063 --> 00:26:32,000
And I am going to march around
like this.

354
00:26:32,000 --> 00:26:36,094
And let this be here dz.
I will avoid the letter L

355
00:26:36,094 --> 00:26:38,852
because we already have a dL
here.

356
00:26:38,852 --> 00:26:42,779
I call it just z.
And let this from here to here

357
00:26:42,779 --> 00:26:46,707
be just capital L.
And so now we are going to do

358
00:26:46,707 --> 00:26:50,300
the closed loop integral.
Follow me closely.

359
00:26:50,300 --> 00:26:55,147
I start here and I go here.
Little dL is in this direction.

360
00:26:55,147 --> 00:27:00,161
This one here in medium one has
no effect because it is a dot

361
00:27:00,161 --> 00:27:03,639
product.
But this one has an effect.

362
00:27:03,639 --> 00:27:06,493
dL and this component are in
the same direction,

363
00:27:06,493 --> 00:27:10,016
but when I go down here it is
exactly the same value with a

364
00:27:10,016 --> 00:27:12,809
minus sign because dL is in the
down direction.

365
00:27:12,809 --> 00:27:16,453
This contribution here and here
will exactly kill each other.

366
00:27:16,453 --> 00:27:18,518
There is no need writing them
down.

367
00:27:18,518 --> 00:27:20,643
And the same is true for medium
two.

368
00:27:20,643 --> 00:27:23,923
When I go from here to here,
I get one component of the

369
00:27:23,923 --> 00:27:27,141
normal vector with the dL.
And here I get another one.

370
00:27:27,141 --> 00:27:30,360
The two will cancel out each
other because it is a dot

371
00:27:30,360 --> 00:27:36,425
product.
All I have to worry about is

372
00:27:36,425 --> 00:27:44,010
the path from here to here and
from here to here.

373
00:27:44,010 --> 00:27:53,335
And the path from here to here,
this component has no effect

374
00:27:53,335 --> 00:28:02,185
because it is at 90 degrees,
so it is only the horizontal

375
00:28:02,185 --> 00:28:11,351
component that I deal with.
And so I get that E1 tangential

376
00:28:11,351 --> 00:28:16,566
times L that is in this
direction.

377
00:28:16,566 --> 00:28:21,940
This is my positive direction
of E.

378
00:28:21,940 --> 00:28:30,000
But now I go in this direction.
28:01

379
00:28:30,000 --> 00:28:32,524
But now I am going to make dz
zero.

380
00:28:32,524 --> 00:28:36,757
That is always the trick we do
when we go to the boundary.

381
00:28:36,757 --> 00:28:41,064
We must make this zero because
we want to know exactly what

382
00:28:41,064 --> 00:28:44,034
changes at the boundary.
We make dz zero.

383
00:28:44,034 --> 00:28:47,301
But, once you make dz zero,
the surface here,

384
00:28:47,301 --> 00:28:51,237
that is the surface through
which I want to measure my

385
00:28:51,237 --> 00:28:53,910
changing magnetic field goes to
zero.

386
00:28:53,910 --> 00:28:57,920
So this part goes to zero.
And so now you see that this

387
00:28:57,920 --> 00:29:03,130
equals zero.
And now we have derived our

388
00:29:03,130 --> 00:29:08,579
first boundary condition,
and the first boundary

389
00:29:08,579 --> 00:29:14,144
condition them for a dielectric
is the following,

390
00:29:14,144 --> 00:29:18,666
that E1 tangential equals E2
tangential.

391
00:29:18,666 --> 00:29:23,304
It is not too different from a
conductor.

392
00:29:23,304 --> 00:29:29,565
The only difference is that E
tangential was zero for a

393
00:29:29,565 --> 00:29:36,488
conductor.
And so it was zero on both

394
00:29:36,488 --> 00:29:42,011
sides.
This one and this one look

395
00:29:42,011 --> 00:29:52,714
alike, provided you keep in mind
that for a conductor it had to

396
00:29:52,714 --> 00:29:59,101
be zero.
Now there are three Maxwell's

397
00:29:59,101 --> 00:30:05,660
equations left.
They are in your court.

398
00:30:05,660 --> 00:30:11,011
And I will give you the
results.

399
00:30:11,011 --> 00:30:19,815
But you can derive them.
It is really not difficult,

400
00:30:19,815 --> 00:30:26,375
no more difficult than what I
just did.

401
00:30:26,375 --> 00:30:31,545
29:58
However, there is a huge

402
00:30:31,545 --> 00:30:34,248
difference.
This rho S was time variable as

403
00:30:34,248 --> 00:30:36,309
the electromagnetic wave came
in.

404
00:30:36,309 --> 00:30:40,236
It changed like crazy with the
frequency of the incoming wave.

405
00:30:40,236 --> 00:30:43,776
This one cannot change because
it is an ideal insulator.

406
00:30:43,776 --> 00:30:47,768
There is no conductivity so no
charges can move on the surface.

407
00:30:47,768 --> 00:30:50,343
In other words,
this rho S is the surface

408
00:30:50,343 --> 00:30:53,948
charge density that somehow
ended up on that surface that

409
00:30:53,948 --> 00:30:57,618
you may have put there yourself.
And it will never change.

410
00:30:57,618 --> 00:31:02,065
It can also be zero.
There is a big difference.

411
00:31:02,065 --> 00:31:05,799
But it is, of course,
physically the same rho S,

412
00:31:05,799 --> 00:31:08,341
namely Coulombs per square
meter.

413
00:31:08,341 --> 00:31:11,042
But the function is very
different.

414
00:31:11,042 --> 00:31:15,411
It is very easy to do a
transition whereby this is zero.

415
00:31:15,411 --> 00:31:20,098
But if you put a lot of charge
at that surface then this is,

416
00:31:20,098 --> 00:31:24,546
of course, a large number.
And so that is the number one.

417
00:31:24,546 --> 00:31:27,009
And then we have B1n equals
B2n.

418
00:31:27,009 --> 00:31:31,696
And then we have the last one
which is B1 tangential divided

419
00:31:31,696 --> 00:31:42,644
by kappa m1.
It is the same as B2 tangential

420
00:31:42,644 --> 00:31:54,634
divided by kappa m2.
Well, let me use a more

421
00:31:54,634 --> 00:32:08,298
consistent notation.
This is the two and this is m

422
00:32:08,298 --> 00:32:18,615
and this is the one and this is
the m.

423
00:32:18,615 --> 00:32:27,817
That is the notation I have
there.

424
00:32:27,817 --> 00:32:34,727
32:00
You have them on your handout.

425
00:32:34,727 --> 00:32:39,818
They are also on the Web.
And I will derive two the four.

426
00:32:39,818 --> 00:32:43,363
And I will leave you with the
other two.

427
00:32:43,363 --> 00:32:46,454
For that, I have to make some
room.

428
00:32:46,454 --> 00:32:50,636
And I think this we really
don't need any more.

429
00:32:50,636 --> 00:32:53,454
Let's do it on the center
board.

430
00:32:53,454 --> 00:32:58,636
This is not done as thoroughly
and as well as I would have

431
00:32:58,636 --> 00:33:03,000
liked to see in Bekefi and
Barrett.

432
00:33:03,000 --> 00:33:07,494
They do not derive in all its
glory all four Fresnel

433
00:33:07,494 --> 00:33:11,989
equations, yet they are
absolutely fundamental to an

434
00:33:11,989 --> 00:33:17,189
understanding of what happens
with the light when it strikes

435
00:33:17,189 --> 00:33:22,125
from one dielectric to another.
Try to follow me closely.

436
00:33:22,125 --> 00:33:26,708
I will try to go slowly,
but if it confuses you I can

437
00:33:26,708 --> 00:33:31,297
understand that.
Here we have again the

438
00:33:31,297 --> 00:33:35,161
transition from one medium to
another.

439
00:33:35,161 --> 00:33:41,322
We have here medium one with
index of refraction n1 and here

440
00:33:41,322 --> 00:33:46,857
we have medium number two with
index of refraction n2.

441
00:33:46,857 --> 00:33:51,139
We have incident radiation
coming in here,

442
00:33:51,139 --> 00:33:56,256
and this angle is theta one.
And we have reflected

443
00:33:56,256 --> 00:34:02,000
radiation, and this angle is
also theta one.

444
00:34:02,000 --> 00:34:07,819
And then we have radiation that
goes into this medium,

445
00:34:07,819 --> 00:34:11,882
and this angle is theta one.
Incident.

446
00:34:11,882 --> 00:34:14,298
Reflected.
Transmitted.

447
00:34:14,298 --> 00:34:19,568
The radiation that comes in,
the E vector must be

448
00:34:19,568 --> 00:34:24,509
perpendicular to the direction
of propagation,

449
00:34:24,509 --> 00:34:30,000
which is this.
It must be perpendicular.

450
00:34:30,000 --> 00:34:34,361
It must be somewhere in this
plane perpendicular to the

451
00:34:34,361 --> 00:34:37,672
blackboard and perpendicular to
this line.

452
00:34:37,672 --> 00:34:41,306
That is nonnegotiable.
Therefore, I can always

453
00:34:41,306 --> 00:34:46,071
decompose that if I want to.
And you will see shortly why we

454
00:34:46,071 --> 00:34:48,817
do that.
I can always decompose any

455
00:34:48,817 --> 00:34:53,420
electromagnetic wave that comes
into this direction in two

456
00:34:53,420 --> 00:34:56,085
components.
One component which is

457
00:34:56,085 --> 00:35:01,173
perpendicular to the blackboard,
which we will give the symbol E

458
00:35:01,173 --> 00:35:05,914
of i perpendicular.
And one component in the

459
00:35:05,914 --> 00:35:09,584
blackboard, which we will call E
of i parallel.

460
00:35:09,584 --> 00:35:14,609
And we run out of symbols so we
try to do all kinds of different

461
00:35:14,609 --> 00:35:18,119
ways to give names.
This is a component of an

462
00:35:18,119 --> 00:35:22,746
incoming electromagnetic wave.
It has an electric vector in

463
00:35:22,746 --> 00:35:26,096
some direction.
It could even be circularly

464
00:35:26,096 --> 00:35:29,766
polarized light.
I can always decompose that in

465
00:35:29,766 --> 00:35:35,159
two components.
And I can always decompose it

466
00:35:35,159 --> 00:35:41,369
one that oscillates like this
and one that oscillates like

467
00:35:41,369 --> 00:35:45,073
this.
I will now march through this

468
00:35:45,073 --> 00:35:51,501
problem and I will now show you
that I can predict what this

469
00:35:51,501 --> 00:35:58,474
component will come out here and
how this component will come out

470
00:35:58,474 --> 00:36:02,058
here.
And then I will let you do the

471
00:36:02,058 --> 00:36:05,694
job on this component,
how that comes out here and how

472
00:36:05,694 --> 00:36:09,467
this component comes out here.
And then the job is done,

473
00:36:09,467 --> 00:36:13,515
because you can always then
combine this component with this

474
00:36:13,515 --> 00:36:16,054
component.
And that will tell you then

475
00:36:16,054 --> 00:36:18,523
what the transmitted wave looks
like.

476
00:36:18,523 --> 00:36:22,434
And you can do the same here.
And so that is my next task.

477
00:36:22,434 --> 00:36:25,933
I am going to do it for the
perpendicular component.

478
00:36:25,933 --> 00:36:29,020
Once we have done it,
for these two components

479
00:36:29,020 --> 00:36:33,000
independently,
we are safe for all cases.

480
00:36:33,000 --> 00:36:37,129
Whether elliptically polarized
light comes in,

481
00:36:37,129 --> 00:36:42,177
circularly polarized light or
linearly polarized lights,

482
00:36:42,177 --> 00:36:45,297
we can then tackle any
possibility.

483
00:36:45,297 --> 00:36:49,610
Follow me closely.
Here is an enlargement of the

484
00:36:49,610 --> 00:36:55,117
incident electromagnetic wave.
This is the direction it comes

485
00:36:55,117 --> 00:36:57,962
in.
I will only do the component

486
00:36:57,962 --> 00:37:04,317
perpendicular to the blackboard.
I am only looking at this

487
00:37:04,317 --> 00:37:09,695
component E of i perpendicular.
That component must have an

488
00:37:09,695 --> 00:37:14,145
associated B vector.
It is married to a B vector.

489
00:37:14,145 --> 00:37:19,152
The two go hand in hand.
One does not exist without the

490
00:37:19,152 --> 00:37:22,026
other.
And the E cross B of that

491
00:37:22,026 --> 00:37:25,271
component must go in this
direction.

492
00:37:25,271 --> 00:37:30,000
And the B vector there must lie
like so.

493
00:37:30,000 --> 00:37:34,440
There is no other possibility.
This must be the B vector of

494
00:37:34,440 --> 00:37:37,885
the incident B.
But it just so happens that it

495
00:37:37,885 --> 00:37:40,564
lies in the plane of the
blackboard.

496
00:37:40,564 --> 00:37:43,703
Therefore, we give it the
symbol parallel.

497
00:37:43,703 --> 00:37:47,454
Because, if the E vector is
oscillating like this,

498
00:37:47,454 --> 00:37:52,047
the only way that the wave can
propagate in this direction is

499
00:37:52,047 --> 00:37:54,803
if the B vector oscillates like
this.

500
00:37:54,803 --> 00:37:58,937
It is the B parallel component
that is married to the E

501
00:37:58,937 --> 00:38:05,040
perpendicular component.
This angle is theta one.

502
00:38:05,040 --> 00:38:09,601
I am now going to decompose
this one.

503
00:38:09,601 --> 00:38:15,556
I will do it in green.
B of i parallel times the

504
00:38:15,556 --> 00:38:21,638
cosine theta one,
and the vertical component is B

505
00:38:21,638 --> 00:38:26,832
of i parallel times the sine of
theta one.

506
00:38:26,832 --> 00:38:32,246
This is the one.
When it reaches z equals zero,

507
00:38:32,246 --> 00:38:35,765
that is this one.
That is why I am doing it that

508
00:38:35,765 --> 00:38:38,311
way.
I want to get a component that

509
00:38:38,311 --> 00:38:42,729
hits the surface that I can
handle with Maxwell's equations.

510
00:38:42,729 --> 00:38:46,173
And, when this component hits
the surface here,

511
00:38:46,173 --> 00:38:48,944
that is also the tangential
component.

512
00:38:48,944 --> 00:38:52,388
That is this one.
You see now why I am doing it

513
00:38:52,388 --> 00:38:55,159
this way.
I need components that I can

514
00:38:55,159 --> 00:38:57,705
handle with the boundary
condition.

515
00:38:57,705 --> 00:39:03,420
That is the incoming stuff.
Now we will do the reflected

516
00:39:03,420 --> 00:39:06,273
stuff.
Again, thing angle is theta

517
00:39:06,273 --> 00:39:09,213
one.
And I am only dealing with the

518
00:39:09,213 --> 00:39:13,536
perpendicular component.
Here is that perpendicular

519
00:39:13,536 --> 00:39:16,043
component.
That is E reflected

520
00:39:16,043 --> 00:39:19,933
perpendicular.
Now I want to know the B vector

521
00:39:19,933 --> 00:39:23,824
with which it is married.
This is the B vector

522
00:39:23,824 --> 00:39:29,097
perpendicular in the blackboard
that is associated with that E

523
00:39:29,097 --> 00:39:32,974
vector.
It is being reflected but it is

524
00:39:32,974 --> 00:39:34,884
parallel.
It is in the plane of

525
00:39:34,884 --> 00:39:37,113
incidence.
The plane of incidence is

526
00:39:37,113 --> 00:39:41,061
always defined as the incident
light and the normal through the

527
00:39:41,061 --> 00:39:43,481
surface.
The blackboard is what we call

528
00:39:43,481 --> 00:39:46,856
the plane of incident.
Now I can decompose that into a

529
00:39:46,856 --> 00:39:50,676
horizontal component and I can
decompose that into a vertical

530
00:39:50,676 --> 00:39:52,969
component.
I am not interested in the

531
00:39:52,969 --> 00:39:58,000
vertical component because I am
only going to use this equation.

532
00:39:58,000 --> 00:40:03,093
I will only do the component in
this direction.

533
00:40:03,093 --> 00:40:09,627
And so this one here is then B
reflected perpendicular times

534
00:40:09,627 --> 00:40:15,496
the cosine of theta one.
Now we have to go into medium

535
00:40:15,496 --> 00:40:18,818
two.
There is one more to come.

536
00:40:18,818 --> 00:40:23,137
There it is.
And this angle now is theta

537
00:40:23,137 --> 00:40:26,791
two.
And the light goes like this,

538
00:40:26,791 --> 00:40:33,059
electromagnetic radiation.
I am only dealing with E

539
00:40:33,059 --> 00:40:36,662
perpendicular,
which is what you are to do.

540
00:40:36,662 --> 00:40:41,381
This is the E perpendicular
component of the transmitted

541
00:40:41,381 --> 00:40:43,612
wave.
This is E transmitted

542
00:40:43,612 --> 00:40:47,215
perpendicular.
It is married to a B vector,

543
00:40:47,215 --> 00:40:50,905
and that B vector must be in
this direction.

544
00:40:50,905 --> 00:40:56,224
To make sure that E cross B is
in the direction of propagation,

545
00:40:56,224 --> 00:41:03,336
this is B transmitted parallel.
And the horizontal component is

546
00:41:03,336 --> 00:41:09,470
then B transmitted parallel
times the cosine of theta two.

547
00:41:09,470 --> 00:41:13,452
All of this we need.
But we are close.

548
00:41:13,452 --> 00:41:17,434
I will give you a minute to
digest it.

549
00:41:17,434 --> 00:41:24,000
I will erase this so that we
stay on the center board.

550
00:41:24,000 --> 00:41:29,000



551
00:41:29,000 --> 00:41:32,219
Now I go to the boundary
conditions.

552
00:41:32,219 --> 00:41:37,462
Before I go to the boundary
conditions, there is one thing

553
00:41:37,462 --> 00:41:42,890
that I want you to remember.
That is there is a relationship

554
00:41:42,890 --> 00:41:47,213
between B and E.
They are married to each other.

555
00:41:47,213 --> 00:41:50,617
And the ratio of their
amplitude is c.

556
00:41:50,617 --> 00:41:53,560
Well, not quite.
It used to be c,

557
00:41:53,560 --> 00:42:00,000
but now it is v because we are
now working in dielectrics.

558
00:42:00,000 --> 00:42:04,224
Since B is married to E,
we can always write down,

559
00:42:04,224 --> 00:42:08,793
and we are going to need that,
that the magnitude of B

560
00:42:08,793 --> 00:42:13,362
parallel must always be the same
as the magnitude of E

561
00:42:13,362 --> 00:42:18,275
perpendicular divided by v,
which is also the magnitude of

562
00:42:18,275 --> 00:42:21,379
E perpendicular divided by c
times n.

563
00:42:21,379 --> 00:42:24,051
That is the index of
refraction.

564
00:42:24,051 --> 00:42:30,000
And we are going to need this.
This is completely kosher.

565
00:42:30,000 --> 00:42:34,663
This is what we knew from
Maxwell's equations way before

566
00:42:34,663 --> 00:42:37,377
we dealt with this.
We are close.

567
00:42:37,377 --> 00:42:42,040
At the boundary condition,
I now must make the E vector,

568
00:42:42,040 --> 00:42:46,789
the tangential component on the
left side the same as the

569
00:42:46,789 --> 00:42:50,350
tangential component,
not on the left side,

570
00:42:50,350 --> 00:42:54,166
on the top side the same as on
the medium two.

571
00:42:54,166 --> 00:42:59,000
Medium one and medium two must
have the same.

572
00:42:59,000 --> 00:43:04,445
But medium one has two.
Medium one has this component

573
00:43:04,445 --> 00:43:09,787
when it reaches here,
and it has this component when

574
00:43:09,787 --> 00:43:14,395
it comes back.
I always must add the incident

575
00:43:14,395 --> 00:43:18,165
and the reflected E field,
of course.

576
00:43:18,165 --> 00:43:23,192
We get E incidence
perpendicular plus E reflected

577
00:43:23,192 --> 00:43:27,277
perpendicular,
must now be E transmitted

578
00:43:27,277 --> 00:43:31,542
perpendicular.
Of course, transmitted

579
00:43:31,542 --> 00:43:34,556
perpendicular component means in
this plane.

580
00:43:34,556 --> 00:43:38,341
That is that tangential thing.
All these components are

581
00:43:38,341 --> 00:43:41,214
tangential.
You see now my problem with t,

582
00:43:41,214 --> 00:43:42,967
a, n and t.
It is awkward.

583
00:43:42,967 --> 00:43:45,700
But this is the perpendicular
component.

584
00:43:45,700 --> 00:43:48,995
That is a given.
That is the boundary condition.

585
00:43:48,995 --> 00:43:51,448
Now I go to this boundary
condition.

586
00:43:51,448 --> 00:43:55,093
And, just for simplicity,
I will leave the kappas out

587
00:43:55,093 --> 00:44:00,000
because the kappas are one,
as far as I am concerned.

588
00:44:00,000 --> 00:44:02,752
If you want to put them in,
be my guest.

589
00:44:02,752 --> 00:44:06,635
What are those components?
Well, I already eluded you to

590
00:44:06,635 --> 00:44:08,400
that.
This is one of them,

591
00:44:08,400 --> 00:44:10,729
a medium one.
This is another one,

592
00:44:10,729 --> 00:44:13,764
a medium one.
And this is one in medium two.

593
00:44:13,764 --> 00:44:17,788
And the two in medium one
together must be the same as the

594
00:44:17,788 --> 00:44:21,176
one in medium two to meet the
boundary condition.

595
00:44:21,176 --> 00:44:23,505
I am going to write this one
down.

596
00:44:23,505 --> 00:44:27,811
And so I am going to get that B
of i parallel times the cosine

597
00:44:27,811 --> 00:44:32,869
of theta one.
That is in this direction.

598
00:44:32,869 --> 00:44:38,719
Then I have to subtract this
one, because it is in the

599
00:44:38,719 --> 00:44:44,127
opposite direction,
minus B of r parallel time the

600
00:44:44,127 --> 00:44:48,541
cosine of theta one.
That is on one side.

601
00:44:48,541 --> 00:44:55,163
Plus and minus must now be the
same as B transmitted parallel

602
00:44:55,163 --> 00:45:03,000
times the cosine of theta two.
And now I am going to use this.

603
00:45:03,000 --> 00:45:12,070
I want to convert all my b's
into E perpendicular.

604
00:45:12,070 --> 00:45:22,065
And, once I have done this,
I am spitting distance away

605
00:45:22,065 --> 00:45:31,506
from knowing what the ratios of
those E vectors are.

606
00:45:31,506 --> 00:45:40,761
And that is my goal.
If you know the ratio of the E

607
00:45:40,761 --> 00:45:50,202
vectors over the reflected one
and the incident one,

608
00:45:50,202 --> 00:46:00,012
you know the ratio of
intensities, because that is the

609
00:46:00,012 --> 00:46:10,378
pointing vector and you get
squares and things like that.

610
00:46:10,378 --> 00:46:20,744
I am going to replace now the
parallel component by the E

611
00:46:20,744 --> 00:46:30,000
perpendicular divided by v.
45:30

612
00:46:30,000 --> 00:46:33,671
And look at this equation.
If you know n1,

613
00:46:33,671 --> 00:46:37,611
n2 and theta one,
Snell's law gives you theta

614
00:46:37,611 --> 00:46:40,746
two.
Now you have two equations with

615
00:46:40,746 --> 00:46:43,432
three unknowns.
This intensity,

616
00:46:43,432 --> 00:46:47,731
this electric field,
this electric field and that

617
00:46:47,731 --> 00:46:51,850
electric field.
What you can do now is find the

618
00:46:51,850 --> 00:46:57,492
ratios of these electric fields,
and that is the only thing that

619
00:46:57,492 --> 00:47:01,259
is meaningful.
Because, clearly,

620
00:47:01,259 --> 00:47:05,555
you can make the intensity of
the incident one as strong as

621
00:47:05,555 --> 00:47:08,740
you want to.
By combining this equation with

622
00:47:08,740 --> 00:47:11,777
this equation,
it is clearly a high school

623
00:47:11,777 --> 00:47:15,777
exercise to now find what is E
transmitted divided by E

624
00:47:15,777 --> 00:47:18,814
incident, of course,
for the perpendicular

625
00:47:18,814 --> 00:47:21,703
component?
I admit that I have only done

626
00:47:21,703 --> 00:47:25,481
the perpendicular component.
You are going to do the

627
00:47:25,481 --> 00:47:30,074
parallel component.
And I can also find now what

628
00:47:30,074 --> 00:47:34,592
the E vector of the reflected on
is divided by the E vector of

629
00:47:34,592 --> 00:47:38,074
the incident one,
but only for the perpendicular

630
00:47:38,074 --> 00:47:40,296
component.
And that is my goal.

631
00:47:40,296 --> 00:47:44,888
Once I have ratios of E vector,
I can calculate light intensity

632
00:47:44,888 --> 00:47:47,111
ratio.
And that is when we were

633
00:47:47,111 --> 00:47:51,555
looking at the demonstration
there, when I wanted to show you

634
00:47:51,555 --> 00:47:55,703
Snell's law, there is no way
that we had to calculate why

635
00:47:55,703 --> 00:47:59,703
there was so little light being
reflected in that first

636
00:47:59,703 --> 00:48:05,090
demonstration.
And now you take your hands out

637
00:48:05,090 --> 00:48:09,376
and look at page one.
It is also on the Web.

638
00:48:09,376 --> 00:48:13,064
There you see,
in all glorious detail,

639
00:48:13,064 --> 00:48:18,745
the four Fresnel equations,
two of which I was in spitting

640
00:48:18,745 --> 00:48:22,434
distance.
To convert this to these two

641
00:48:22,434 --> 00:48:27,118
ratios is no more than another
three minute job,

642
00:48:27,118 --> 00:48:33,000
but it is so trivial that I
don't want to do it.

643
00:48:33,000 --> 00:48:37,880
At the top you see Snell's law.
That means if you tell me what

644
00:48:37,880 --> 00:48:41,639
n1 is and what n2 is,
and you give me theta one,

645
00:48:41,639 --> 00:48:45,239
I will immediately tell you
what theta two is.

646
00:48:45,239 --> 00:48:48,039
That is nonnegotiable.
That is easy.

647
00:48:48,039 --> 00:48:50,519
Once you know n1,
theta one, n2,

648
00:48:50,519 --> 00:48:55,320
theta two, these four equations
will tell you what the ratios

649
00:48:55,320 --> 00:49:01,000
are of the incident electric
vector over the reflected one.

650
00:49:01,000 --> 00:49:03,950
But there is a parallel
component and there is a

651
00:49:03,950 --> 00:49:07,089
perpendicular component.
And so the only two that I

652
00:49:07,089 --> 00:49:11,044
derived, or I came very close to
deriving, that was this one and

653
00:49:11,044 --> 00:49:14,058
that was this one.
I was within spitting distance

654
00:49:14,058 --> 00:49:16,820
of these two.
Had I made it one step further,

655
00:49:16,820 --> 00:49:19,582
you would have seen this
equation coming out.

656
00:49:19,582 --> 00:49:23,035
The fact that I put a zero here
means that I made it the

657
00:49:23,035 --> 00:49:26,237
magnitude of the E vector.
And I give it a shorthand

658
00:49:26,237 --> 00:49:31,676
notation, a little parallel.
Little r is the ratio of r over

659
00:49:31,676 --> 00:49:34,785
i and the t is the ratio of t
over i.

660
00:49:34,785 --> 00:49:39,103
This is shorthand notation.
I give you two columns.

661
00:49:39,103 --> 00:49:43,075
They are the same,
except that one is sometimes

662
00:49:43,075 --> 00:49:47,047
easier to use than the other.
This is extremely

663
00:49:47,047 --> 00:49:51,192
consumer-friendly,
though you may not think that.

664
00:49:51,192 --> 00:49:54,041
It is extremely
consumer-friendly.

665
00:49:54,041 --> 00:50:00,000
You cannot go wrong if you
simply apply these equations.

666
00:50:00,000 --> 00:50:04,181
No one in the world could just
in five minutes derive these

667
00:50:04,181 --> 00:50:06,344
equations.
It is a lot of work.

668
00:50:06,344 --> 00:50:10,453
I spent 20 minutes on one,
and I only came within spitting

669
00:50:10,453 --> 00:50:12,832
distance.
I didn't even finish it.

670
00:50:12,832 --> 00:50:16,365
It is a lot of time.
As long as you know what they

671
00:50:16,365 --> 00:50:19,537
are, you just use them.
I gave you theta one,

672
00:50:19,537 --> 00:50:21,555
I gave you n1,
I give you n2,

673
00:50:21,555 --> 00:50:25,160
you calculate theta two,
and out come these ratios.

674
00:50:25,160 --> 00:50:29,197
There are things in here which
are amazing which are very

675
00:50:29,197 --> 00:50:33,259
opaque.
They are not very transparent.

676
00:50:33,259 --> 00:50:37,402
And I am going to work with you
on that after the break.

677
00:50:37,402 --> 00:50:40,189
I just want you to know you
have them.

678
00:50:40,189 --> 00:50:43,654
They are worth gold.
They are not in Bekefi and

679
00:50:43,654 --> 00:50:46,592
Barrett.
Bekefi and Barrett only mention

680
00:50:46,592 --> 00:50:49,906
one and a half or so for a very
special case.

681
00:50:49,906 --> 00:50:53,898
This is the complete set.
And so I would have thought,

682
00:50:53,898 --> 00:50:57,966
but let me check my notes,
that this is an ideal moment

683
00:50:57,966 --> 00:51:02,148
for a break.
And then, after the break,

684
00:51:02,148 --> 00:51:06,942
we will try to digest some
remarkable consequences of these

685
00:51:06,942 --> 00:51:09,917
equations.
And so, if you can help me

686
00:51:09,917 --> 00:51:12,975
handing out the mini-quiz number
nine.

687
00:51:12,975 --> 00:51:15,785
The idea was,
but I could be wrong,

688
00:51:15,785 --> 00:51:18,925
that you can answer this in 30
seconds.

689
00:51:18,925 --> 00:51:22,727
That was my goal.
And, if it takes you any more

690
00:51:22,727 --> 00:51:26,363
than 30 seconds,
you might as well not do it,

691
00:51:26,363 --> 00:51:30,000
believe me.
You will see why.

692
00:51:30,000 --> 00:51:32,581
If you can help me handing it
out.

693
00:51:32,581 --> 00:51:37,273
Wait until the whistle blows.
Can you help me handing it out?

694
00:51:37,273 --> 00:51:40,245
And you?
I hope you all have a handout.

695
00:51:40,245 --> 00:51:44,000
There are handouts at all three
entrances.

696
00:51:44,000 --> 00:51:49,000



697
00:51:49,000 --> 00:51:52,857
As I promised,
I now want to give you a little

698
00:51:52,857 --> 00:51:56,457
bit of insight,
but I will only stretch the

699
00:51:56,457 --> 00:52:01,000
surface of the consequences of
Fresnel equations.

700
00:52:01,000 --> 00:52:10,450
And let's first do a case
whereby theta one is theta there

701
00:52:10,450 --> 00:52:19,072
and is therefore zero.
Snell's law tells you if theta

702
00:52:19,072 --> 00:52:25,538
one is zero then theta two is
also zero.

703
00:52:25,538 --> 00:52:33,829
We give that a name.
We call that normal incidence.

704
00:52:33,829 --> 00:52:41,621
It then so happens,
which is by no means obvious

705
00:52:41,621 --> 00:52:50,077
when you look at these
equations, that the first two

706
00:52:50,077 --> 00:52:56,875
equations get you exactly the
same result.

707
00:52:56,875 --> 00:53:05,000
And so you will find then --
52:30

708
00:53:05,000 --> 00:53:11,142
And now I take a special case
which is on your page number

709
00:53:11,142 --> 00:53:14,913
two.
That is the transition from air

710
00:53:14,913 --> 00:53:19,008
to glass.
That is the upper panel going

711
00:53:19,008 --> 00:53:23,857
from air to glass.
Where you are is always n1.

712
00:53:23,857 --> 00:53:30,000
Never forget that.
N1 is then very close to 1.00.

713
00:53:30,000 --> 00:53:35,698
And n2 is very close to 1.5,
if we take 1.5 for glass.

714
00:53:35,698 --> 00:53:41,612
We will find then that for this
case, this is minus 0.2.

715
00:53:41,612 --> 00:53:45,053
What is the meaning of the
minus?

716
00:53:45,053 --> 00:53:50,000
Come on.
We have seen that with strings.

717
00:53:50,000 --> 00:53:55,000



718
00:53:55,000 --> 00:53:58,407
It means that there is 180
phase flip.

719
00:53:58,407 --> 00:54:02,828
That is all it means.
It means that the E vector,

720
00:54:02,828 --> 00:54:07,065
if it is in a positive
direction as it arrives,

721
00:54:07,065 --> 00:54:09,460
it reflects,
it flips over.

722
00:54:09,460 --> 00:54:14,250
That minus sign is what we see
very often when we hit

723
00:54:14,250 --> 00:54:17,381
boundaries where there is a
change.

724
00:54:17,381 --> 00:54:21,618
This is a 180 phase flip.
And this is plus 0.8.

725
00:54:21,618 --> 00:54:27,236
There is no phase flip in going
from medium one to medium two?

726
00:54:27,236 --> 00:54:31,677
Of course not.
That was the same with the

727
00:54:31,677 --> 00:54:34,000
string.
If you have a string which is

728
00:54:34,000 --> 00:54:37,354
connected to another string,
if the junction goes up,

729
00:54:37,354 --> 00:54:41,161
then the wave that goes into
the second medium also goes up.

730
00:54:41,161 --> 00:54:44,000
There is no way that it could
down, remember?

731
00:54:44,000 --> 00:54:46,903
It is the reflected one that
can change phase,

732
00:54:46,903 --> 00:54:50,451
but not the transmitted one.
That is not too surprising,

733
00:54:50,451 --> 00:54:55,121
by the way.
If now you look at that upper

734
00:54:55,121 --> 00:54:59,750
panel, which is a plot of
Fresnel equations,

735
00:54:59,750 --> 00:55:05,348
four Fresnel equations have
been plotted here for the

736
00:55:05,348 --> 00:55:10,838
transition from air to glass
horizontally theta one,

737
00:55:10,838 --> 00:55:16,866
and vertically is the ratio of
the E vector in those four

738
00:55:16,866 --> 00:55:19,558
cases.
Reflected parallel,

739
00:55:19,558 --> 00:55:24,725
reflected perpendicular,
transmitted parallel and

740
00:55:24,725 --> 00:55:30,000
transmitted perpendicular.
And look.

741
00:55:30,000 --> 00:55:33,352
Theta one is zero.
What do you see there?

742
00:55:33,352 --> 00:55:37,124
You see minus 0.2,
which I just calculated for

743
00:55:37,124 --> 00:55:40,895
you, and plus 0.8 at an angle of
zero degrees.

744
00:55:40,895 --> 00:55:45,338
Now, clearly we can also
calculate the light intensity

745
00:55:45,338 --> 00:55:47,098
now.
That was my goal.

746
00:55:47,098 --> 00:55:51,959
I don't give a damn whether
that light changes phase by 180

747
00:55:51,959 --> 00:55:55,060
degrees or not.
That is a later issue.

748
00:55:55,060 --> 00:55:59,000
I want to know light
intensities.

749
00:55:59,000 --> 00:56:03,407
And light intensities means I
deal with pointing vectors,

750
00:56:03,407 --> 00:56:05,925
E cross B.
Well, B, in magnitude,

751
00:56:05,925 --> 00:56:09,939
is always E divided by v.
Whenever you compare light

752
00:56:09,939 --> 00:56:14,740
intensities, all you have to do
is take E squared and then you

753
00:56:14,740 --> 00:56:18,361
divide it by v.
But since the reflected one and

754
00:56:18,361 --> 00:56:23,240
the incident one are both in the
same medium, I don't even have

755
00:56:23,240 --> 00:56:27,333
to worry about v because v is
the same in both cases,

756
00:56:27,333 --> 00:56:33,000
which is not true when I go
from medium one to medium two.

757
00:56:33,000 --> 00:56:37,216
Therefore, I can now calculate
for you the intensity,

758
00:56:37,216 --> 00:56:41,432
i stands for intensity,
of the reflected wave divided

759
00:56:41,432 --> 00:56:44,432
by the intensity of the incident
wave.

760
00:56:44,432 --> 00:56:49,216
And I claim that it is the E
vector reflected divided by the

761
00:56:49,216 --> 00:56:53,675
E vector incident squared.
That is all it is because the

762
00:56:53,675 --> 00:56:58,621
velocity in that medium is the
same for reflected and incident

763
00:56:58,621 --> 00:57:01,956
wave.
This takes everything into

764
00:57:01,956 --> 00:57:04,815
account.
Mu zero is taken into account.

765
00:57:04,815 --> 00:57:08,953
B is taken into account.
The fact that we have a square,

766
00:57:08,953 --> 00:57:13,016
that is the result of the B.
Remember, it is E cross B?

767
00:57:13,016 --> 00:57:16,327
And so this is 0.04.
That is a famous number.

768
00:57:16,327 --> 00:57:18,885
Every astronomer knows that
number.

769
00:57:18,885 --> 00:57:22,949
That means when light strikes a
glass surface at normal

770
00:57:22,949 --> 00:57:26,786
incidence 4% comes back.
There is nothing you can do

771
00:57:26,786 --> 00:57:31,000
about that.
4% of that radiation is lost.

772
00:57:31,000 --> 00:57:36,257
If you want to get it through a
lens there is 4% that comes

773
00:57:36,257 --> 00:57:39,520
back.
Therefore, in order to conserve

774
00:57:39,520 --> 00:57:43,599
energy, i of t divided by i of i
must be 0.96.

775
00:57:43,599 --> 00:57:46,862
Well, you do that.
The answer is yes,

776
00:57:46,862 --> 00:57:50,578
it is 0.96.
But I want you to calculate it

777
00:57:50,578 --> 00:57:55,836
because now you must take into
account that the velocity in

778
00:57:55,836 --> 00:58:02,000
medium two is different from the
velocity in medium one.

779
00:58:02,000 --> 00:58:05,567
If you overlook that,
you will never find 0.96.

780
00:58:05,567 --> 00:58:10,299
If you take this number and you
square it, you don't get 0.96,

781
00:58:10,299 --> 00:58:15,030
but you cannot do that because
in medium two the B vector is E

782
00:58:15,030 --> 00:58:18,520
divided by its own v.
And that own velocity is

783
00:58:18,520 --> 00:58:21,700
different from the velocity in
medium one.

784
00:58:21,700 --> 00:58:24,570
I have set you off on the right
track.

785
00:58:24,570 --> 00:58:30,000
This is still correct but you
cannot just take this ratio.

786
00:58:30,000 --> 00:58:36,894
There is something so
incredibly bizarre about Fresnel

787
00:58:36,894 --> 00:58:41,837
equations.
Look at equation number one,

788
00:58:41,837 --> 00:58:49,642
and look at the right column.
Suppose you make theta one plus

789
00:58:49,642 --> 00:58:56,406
theta two 90 degrees.
I will show you that it is very

790
00:58:56,406 --> 00:59:02,000
easy.
Then that thing goes to zero.

791
00:59:02,000 --> 00:59:06,055
Because the tangent of 90
degrees is infinitely high.

792
00:59:06,055 --> 00:59:09,487
R parallel, the way I have
called that there,

793
00:59:09,487 --> 00:59:11,749
goes to zero.
That is amazing.

794
00:59:11,749 --> 00:59:15,181
That means there must be one
angle theta one,

795
00:59:15,181 --> 00:59:18,924
if I hit it just right,
that none of the parallel

796
00:59:18,924 --> 00:59:22,434
component is reflected.
The only thing that is

797
00:59:22,434 --> 00:59:25,944
reflected is then the
perpendicular component.

798
00:59:25,944 --> 00:59:30,000
Now, it may not be reflected
for 100%.

799
00:59:30,000 --> 00:59:36,373
But some of it is reflected.
Therefore, the reflected light

800
00:59:36,373 --> 00:59:41,758
is now 100% polarized.
There is only one angle for

801
00:59:41,758 --> 00:59:47,032
which that is happens,
and that angle has a name.

802
00:59:47,032 --> 00:59:50,439
We call that the Brewster
angle.

803
00:59:50,439 --> 00:59:56,153
And the tangent of theta
Brewster, that is that theta

804
00:59:56,153 --> 1:00:01,684
one, is n2 divided by n1.
I want to remind you,

805
1:00:01,684 --> 1:00:06,336
one is always where you are and
two is where you are going.

806
1:00:06,336 --> 1:00:11,310
And I can easily show you that
this is an immediate consequence

807
1:00:11,310 --> 1:00:13,796
of that.
I will prove it to you.

808
1:00:13,796 --> 1:00:17,326
For one thing,
if theta one plus theta two is

809
1:00:17,326 --> 1:00:22,058
90 degrees then the cosine of
theta one is the sine of theta

810
1:00:22,058 --> 1:00:23,983
two.
That is high school.

811
1:00:23,983 --> 1:00:30,000
But n1 times sine theta one is
n2 times the sine of theta two.

812
1:00:30,000 --> 1:00:34,360
That is my countryman Snell.
And, if you combine these two,

813
1:00:34,360 --> 1:00:36,766
in 30 seconds you can prove
this.

814
1:00:36,766 --> 1:00:41,353
And so this is the angle that
then will automatically give you

815
1:00:41,353 --> 1:00:44,586
that theta one plus theta two is
90 degrees.

816
1:00:44,586 --> 1:00:48,421
And that, of course,
is so wonderful to demonstrate.

817
1:00:48,421 --> 1:00:51,578
And I can demonstrate that in
various ways.

818
1:00:51,578 --> 1:00:56,090
And look at that upper panel
that you have on your handout on

819
1:00:56,090 --> 1:01:00,000
page two.
Look at that Brewster angle.

820
1:01:00,000 --> 1:01:06,068
If you go from air to glass,
so now we go from air to glass,

821
1:01:06,068 --> 1:01:10,902
and we know what n2 is,
n2 is 1.5 and n1 is one,

822
1:01:10,902 --> 1:01:16,251
then you will find that theta
Brewster, which is this

823
1:01:16,251 --> 1:01:22,114
equation, is 56.3 degrees.
Of course, that is only correct

824
1:01:22,114 --> 1:01:26,845
if n2 is exactly 1.5,
which is almost never is.

825
1:01:26,845 --> 1:01:32,708
And, if you use Snell's law,
then you will find that theta

826
1:01:32,708 --> 1:01:38,274
two is 33.7 degrees.
Of course the sum must be 90,

827
1:01:38,274 --> 1:01:41,411
and it is 90 if I did not make
a mistake.

828
1:01:41,411 --> 1:01:45,647
And then you get the crazy
situation that r parallel is

829
1:01:45,647 --> 1:01:48,235
zero.
And so now you can calculate

830
1:01:48,235 --> 1:01:53,019
what is now our perpendicular
for which you have to go to your

831
1:01:53,019 --> 1:01:56,862
Fresnel equations.
There is no way that anyone can

832
1:01:56,862 --> 1:02:02,458
immediately see what that is.
Now you have to go to this

833
1:02:02,458 --> 1:02:08,037
equation and substitute in here
the values for theta one and

834
1:02:08,037 --> 1:02:13,616
theta two and for n1 and n2.
And you will find then that our

835
1:02:13,616 --> 1:02:17,776
perpendicular,
and I will just make sure that

836
1:02:17,776 --> 1:02:20,707
I have it right,
is minus 0.385.

837
1:02:20,707 --> 1:02:25,246
That is what it is.
If we want to know intensity,

838
1:02:25,246 --> 1:02:31,108
all we have to do is square it
because we are still in the same

839
1:02:31,108 --> 1:02:36,724
medium.
The reflected medium and the

840
1:02:36,724 --> 1:02:42,178
incident medium is one in the
same medium.

841
1:02:42,178 --> 1:02:50,293
So 0.385 squared is about 0.15.
What this means is that if you

842
1:02:50,293 --> 1:02:57,077
have an E component
perpendicular that comes in then

843
1:02:57,077 --> 1:03:03,806
15% of that will be reflected.
And zero percent will be

844
1:03:03,806 --> 1:03:06,440
reflected of the parallel
component.

845
1:03:06,440 --> 1:03:10,806
That means if unpolarized light
comes in, unpolarized light

846
1:03:10,806 --> 1:03:14,946
means that half of it is
perpendicular and half of it is

847
1:03:14,946 --> 1:03:19,537
parallel, 7.5% of the incoming
unpolarized light will come out

848
1:03:19,537 --> 1:03:21,946
100% polarized in this
direction.

849
1:03:21,946 --> 1:03:24,053
Imagine.
You now have a wave.

850
1:03:24,053 --> 1:03:26,462
You start with unpolarized
light.

851
1:03:26,462 --> 1:03:30,000
You shine it onto a glass
panel.

852
1:03:30,000 --> 1:03:34,973
And, if the angle of incidence
is close to 56 degrees,

853
1:03:34,973 --> 1:03:38,351
7.5% reflects of the total
intensity.

854
1:03:38,351 --> 1:03:44,076
And it is 100% polarized in the
direction perpendicular to the

855
1:03:44,076 --> 1:03:48,768
plane of incidence.
And that is what you see there.

856
1:03:48,768 --> 1:03:53,272
On this plot you see the
Brewster angle of 56.3%.

857
1:03:53,272 --> 1:03:57,120
You notice that our parallel
goes to zero?

858
1:03:57,120 --> 1:04:02,000
Therefore, everything that
comes back --

859
1:04:02,000 --> 1:04:05,041
Of course, some of it
penetrates into medium,

860
1:04:05,041 --> 1:04:07,185
too.
I didn't calculate how much

861
1:04:07,185 --> 1:04:09,812
that is.
But here it is the minus 0.385

862
1:04:09,812 --> 1:04:12,923
that I just calculated that is
this component.

863
1:04:12,923 --> 1:04:16,172
That is refracted.
And you can imagine that this

864
1:04:16,172 --> 1:04:19,007
is, of course,
great stuff to demonstrate.

865
1:04:19,007 --> 1:04:22,809
Before I demonstrate that,
I want you to appreciate that

866
1:04:22,809 --> 1:04:27,165
if you go from glass to air that
there is also a Brewster angle.

867
1:04:27,165 --> 1:04:32,586
There is a different one.
Of course, because then n1 and

868
1:04:32,586 --> 1:04:36,034
n2 flip roles.
And so, if you look at the

869
1:04:36,034 --> 1:04:41,034
bottom panel of your page two,
you will see that there is a

870
1:04:41,034 --> 1:04:45,862
Brewster angle at 33.7 degrees.
That means if you are now

871
1:04:45,862 --> 1:04:51,034
inside glass and you bounce the
light off the medium with air

872
1:04:51,034 --> 1:04:55,086
that then, at an incident angle
of 33.7 degrees,

873
1:04:55,086 --> 1:05:00,000
then you can create also 100%
polarized light.

874
1:05:00,000 --> 1:05:03,209
Notice also that in this case,
but only in this case,

875
1:05:03,209 --> 1:05:05,740
there is such a thing as a
critical angle.

876
1:05:05,740 --> 1:05:09,382
Remember, earlier I mentioned
you only have a critical angle

877
1:05:09,382 --> 1:05:13,271
of total internal reflection if
you go from optical dense medium

878
1:05:13,271 --> 1:05:16,481
to a less dense medium.
That is the case where you go

879
1:05:16,481 --> 1:05:19,320
from glass to air.
There was no such thing when

880
1:05:19,320 --> 1:05:22,530
you go from air to glass.
And that is what you see in

881
1:05:22,530 --> 1:05:25,123
this panel.
You see the moment you hit this

882
1:05:25,123 --> 1:05:30,000
critical angle that 100% of the
radiation is totally reflected.

883
1:05:30,000 --> 1:05:33,709
You see, this is 1.0 and the r
parallel is 1.0.

884
1:05:33,709 --> 1:05:37,016
Nothing penetrates into the
second medium.

885
1:05:37,016 --> 1:05:40,806
The Fresnel equations,
I want you to appreciate,

886
1:05:40,806 --> 1:05:44,354
are unbelievably powerful.
They are extremely

887
1:05:44,354 --> 1:05:48,548
consumer-friendly because you
can now calculate light

888
1:05:48,548 --> 1:05:53,306
intensities of reflected and
transmitted light for any angle

889
1:05:53,306 --> 1:05:57,661
of incidence theta one.
All you have to know is what n1

890
1:05:57,661 --> 1:06:02,228
and what n2 is.
Whatever the incoming light is,

891
1:06:02,228 --> 1:06:06,326
if it is elliptically polarized
you decompose it into an E

892
1:06:06,326 --> 1:06:09,274
perpendicular and an E parallel
component.

893
1:06:09,274 --> 1:06:12,294
And then you turn the crank
four equations.

894
1:06:12,294 --> 1:06:16,248
And then you get the reflected
component and you get the

895
1:06:16,248 --> 1:06:19,483
transmitted component.
You can do that for any

896
1:06:19,483 --> 1:06:23,150
incoming light whether it is
circularly polarized or

897
1:06:23,150 --> 1:06:26,313
elliptically polarized or
linearly polarized.

898
1:06:26,313 --> 1:06:30,411
And you can now also evaluate
to what degree the reflected

899
1:06:30,411 --> 1:06:35,250
light is polarized.
So far we always said 100%

900
1:06:35,250 --> 1:06:39,750
polarized or it is unpolarized.
Life is not that way.

901
1:06:39,750 --> 1:06:44,336
There is also something,
of course, of being partially

902
1:06:44,336 --> 1:06:47,798
polarized.
And I am going to give you now

903
1:06:47,798 --> 1:06:52,557
one possible way of defining the
degree of polarization,

904
1:06:52,557 --> 1:06:56,365
the degree of linear,
I stress, polarization.

905
1:06:56,365 --> 1:07:02,012
And I will define that for you.
I give it the capital letter v.

906
1:07:02,012 --> 1:07:04,024
I don't think that is
universal.

907
1:07:04,024 --> 1:07:07,853
I just gave it the letter E.
It is a degree of polarization.

908
1:07:07,853 --> 1:07:11,033
And I define that as the
intensity of the parallel

909
1:07:11,033 --> 1:07:14,862
component minus the intensity of
the perpendicular component

910
1:07:14,862 --> 1:07:18,627
divided by the intensity of the
parallel component plus the

911
1:07:18,627 --> 1:07:21,223
intensity of the perpendicular
component.

912
1:07:21,223 --> 1:07:24,923
And I take the absolute value.
This number can be anywhere

913
1:07:24,923 --> 1:07:27,995
from zero to one.
If it is zero,

914
1:07:27,995 --> 1:07:31,126
it is totally unpolarized.
And, if it is one,

915
1:07:31,126 --> 1:07:33,544
then it is 100% linearly
polarized.

916
1:07:33,544 --> 1:07:36,888
You can have a situation that
the answer is 0.5,

917
1:07:36,888 --> 1:07:40,516
that it is partially polarized.
It is not completely

918
1:07:40,516 --> 1:07:43,006
unpolarized.
That means one of these

919
1:07:43,006 --> 1:07:46,776
components then dominates.
And then you have partially

920
1:07:46,776 --> 1:07:50,119
polarized light.
And when you have your linearly

921
1:07:50,119 --> 1:07:54,530
polarizer, you can actually tell
that, as you rotate it around,

922
1:07:54,530 --> 1:07:57,091
there is a change in light
intensity.

923
1:07:57,091 --> 1:08:02,000
Only if this is one is it 100%
linearly polarized.

924
1:08:02,000 --> 1:08:05,482
A small test.
This test may come too early

925
1:08:05,482 --> 1:08:09,558
for you, but just see how you
react to this test.

926
1:08:09,558 --> 1:08:13,890
You know that there are
sunglasses which are made of

927
1:08:13,890 --> 1:08:17,967
polarized polarizers.
And so here is one of those

928
1:08:17,967 --> 1:08:22,893
sunglasses and here is another
one and here is another one.

929
1:08:22,893 --> 1:08:27,479
And you are the manufacturer.
You have to decide on the

930
1:08:27,479 --> 1:08:33,000
direction of polarization of the
linearly polarizers.

931
1:08:33,000 --> 1:08:39,697
Would you do it this way or
that way or this way or any

932
1:08:39,697 --> 1:08:42,922
other way?
And, if so, why?

933
1:08:42,922 --> 1:08:48,503
Now, think about it.
What would be the goal of

934
1:08:48,503 --> 1:08:52,968
having sunglasses that are
polarized?

935
1:08:52,968 --> 1:08:58,798
What would be the goal?
Why would you want that?

936
1:08:58,798 --> 1:09:05,000
Just because it looks sexy and
it is in?

937
1:09:05,000 --> 1:09:07,672
No.
There really is a reason for

938
1:09:07,672 --> 1:09:10,000
that.
What is the reason?

939
1:09:10,000 --> 1:09:20,000



940
1:09:20,000 --> 1:09:25,064
That is the point.
The goal of polarizers is that

941
1:09:25,064 --> 1:09:30,550
light that has been reflected,
for instance of water,

942
1:09:30,550 --> 1:09:34,981
when you drive your car and
there is water,

943
1:09:34,981 --> 1:09:40,889
or ice, that the reflected
light is linearly polarized to

944
1:09:40,889 --> 1:09:44,964
some degree.
Now, if the light comes in at

945
1:09:44,964 --> 1:09:48,079
the Brewster angle,
it would be 100% polarized.

946
1:09:48,079 --> 1:09:51,804
That would be very special.
But suppose the sun is there

947
1:09:51,804 --> 1:09:56,003
and I am driving my car here and
the sunlight hits water and it

948
1:09:56,003 --> 1:10:00,000
comes in this direction then it
is highly polarized.

949
1:10:00,000 --> 1:10:04,268
At the very minimum partially
polarized in the perpendicular

950
1:10:04,268 --> 1:10:07,886
direction, perpendicular to the
plane of incidence.

951
1:10:07,886 --> 1:10:12,010
This is the plane of incidence.
It is polarized like this,

952
1:10:12,010 --> 1:10:15,555
to a large degree.
So how do I want my sunglasses?

953
1:10:15,555 --> 1:10:17,943
Like this.
Because now I kill this

954
1:10:17,943 --> 1:10:20,909
component.
And that is exactly the way you

955
1:10:20,909 --> 1:10:23,586
make them.
This is the correct answer.

956
1:10:23,586 --> 1:10:28,000
And that, of course,
calls for a demonstration.

957
1:10:28,000 --> 1:10:33,047
I have here a light beam and
here I have a pane of glass.

958
1:10:33,047 --> 1:10:36,383
And the light reflects off that
glass.

959
1:10:36,383 --> 1:10:40,890
And we have aimed it roughly at
the Brewster angle.

960
1:10:40,890 --> 1:10:45,667
And you cannot do that,
of course, exactly the beam is

961
1:10:45,667 --> 1:10:50,174
not so well defined.
And so when I look here at the

962
1:10:50,174 --> 1:10:54,591
light that reflects off that
plane, it is painful.

963
1:10:54,591 --> 1:11:00,000
I am not even joking.
It is enormously annoying.

964
1:11:00,000 --> 1:11:05,784
There is a tremendous amount of
light that reflects from it.

965
1:11:05,784 --> 1:11:10,588
Who has seen the Matrix?
You have seen the Matrix?

966
1:11:10,588 --> 1:11:13,921
Remember me?
[LAUGHTER] Now I look,

967
1:11:13,921 --> 1:11:18,627
and I have no problem.
I can just see there is an

968
1:11:18,627 --> 1:11:23,235
astronaut walking on the moon.
Now, you may say,

969
1:11:23,235 --> 1:11:27,450
sure, you can say that,
but what do we know?

970
1:11:27,450 --> 1:11:32,115
I accept that.
The light that comes back is

971
1:11:32,115 --> 1:11:36,574
polarized in this direction.
This is the plane of incidence.

972
1:11:36,574 --> 1:11:39,974
It is almost 100% polarized in
this direction.

973
1:11:39,974 --> 1:11:42,770
I am holding this in front of
my face.

974
1:11:42,770 --> 1:11:46,624
This is the direction of
polarization of this sheet.

975
1:11:46,624 --> 1:11:48,589
Look at me.
Now look at me.

976
1:11:48,589 --> 1:11:50,176
Big difference,
right?

977
1:11:50,176 --> 1:11:53,425
What I am doing now,
this is a super pair of

978
1:11:53,425 --> 1:11:57,355
sunglasses, very expensive and
specially made for me.

979
1:11:57,355 --> 1:12:01,933
This one is super.
It cuts out this component.

980
1:12:01,933 --> 1:12:05,727
And, when I wear them like
this, the sunglasses is not

981
1:12:05,727 --> 1:12:08,735
doing anything.
I mean, you can look at the

982
1:12:08,735 --> 1:12:11,527
wall there, you see the same
phenomenon.

983
1:12:11,527 --> 1:12:14,463
That is what sunglasses are
doing for you.

984
1:12:14,463 --> 1:12:18,615
And this is an amazing thing
that you cannot even read what

985
1:12:18,615 --> 1:12:22,696
is under that glass plate.
You cannot even see what it is,

986
1:12:22,696 --> 1:12:26,062
but the moment you do this you
have no problems.

987
1:12:26,062 --> 1:12:30,000
That is the power of the
Brewster angle.

988
1:12:30,000 --> 1:12:34,387
And then the linear polarizers
take care of that.

989
1:12:34,387 --> 1:12:38,225
I would like you take out of
your envelope,

990
1:12:38,225 --> 1:12:42,338
if you have it with you,
one linear polarizer.

991
1:12:42,338 --> 1:12:47,456
And all I want you to do is
relax for one minute and look

992
1:12:47,456 --> 1:12:51,844
around in this room.
And we have objects that are

993
1:12:51,844 --> 1:12:56,962
made of metal and we have
objects that are made of glass.

994
1:12:56,962 --> 1:13:01,806
If unpolarized light strikes
metal, it will not become

995
1:13:01,806 --> 1:13:06,082
polarized.
If unpolarized light strikes a

996
1:13:06,082 --> 1:13:09,552
dielectric, it can become
polarized to some degree.

997
1:13:09,552 --> 1:13:13,092
In an extreme case for 100%.
If you take your linear

998
1:13:13,092 --> 1:13:16,493
polarizer and you rotate it in
front of your eyes,

999
1:13:16,493 --> 1:13:20,588
I want you to see that when you
look at dielectrics like the

1000
1:13:20,588 --> 1:13:24,683
glass or varnished surfaces or
leather that you may actually

1001
1:13:24,683 --> 1:13:28,223
see it change in light
intensity, which you will not

1002
1:13:28,223 --> 1:13:32,818
see when you look at metal.
If you do see it at metal,

1003
1:13:32,818 --> 1:13:36,734
it means that the light that
struck the metal was already

1004
1:13:36,734 --> 1:13:39,531
partially polarized.
And then, of course,

1005
1:13:39,531 --> 1:13:42,818
it remains polarized.
Metal, an ideal conductor,

1006
1:13:42,818 --> 1:13:45,685
doesn't change the degree of
polarization.

1007
1:13:45,685 --> 1:13:49,251
It keeps it as it was.
But it is the dielectric that

1008
1:13:49,251 --> 1:13:51,000
can change it.


1009
1:13:51,000 --> 1:13:57,000


1010
1:13:57,000 --> 1:13:58,728
You have almost the Brewster
angle?

1011
1:13:58,728 --> 1:14:01,641
But it is only for you.
Not for other people.

1012
1:14:01,641 --> 1:14:04,377
It depends on where you are
located.

1013
1:14:04,377 --> 1:14:09,145
You are saying if you look into
the direction of the projector

1014
1:14:09,145 --> 1:14:11,568
there.
It doesn't do much for me

1015
1:14:11,568 --> 1:14:15,164
because the angle of view is
different from me.

1016
1:14:15,164 --> 1:14:19,072
You can see the effect.
I can see it now very well.

1017
1:14:19,072 --> 1:14:22,121
Absolutely.
One of those lamps goes out.

1018
1:14:22,121 --> 1:14:26,264
Boy, it is getting dark.
That is what you had in mind.

1019
1:14:26,264 --> 1:14:32,206
Thank you very much.
But look at some objects here.

1020
1:14:32,206 --> 1:14:34,986
Look at this,
for instance.

1021
1:14:34,986 --> 1:14:40,972
I bet if you rotate this that
certain lights change their

1022
1:14:40,972 --> 1:14:45,248
intensity.
And it also looks at varnished

1023
1:14:45,248 --> 1:14:49,524
surfaces.
Anything that is dielectric has

1024
1:14:49,524 --> 1:14:55,403
potentially this possibility.
And now comes the ultimate

1025
1:14:55,403 --> 1:15:02,371
demonstration of Brewster angle.
What I am going to do now is

1026
1:15:02,371 --> 1:15:07,538
create, out of 100% unpolarized
light, 7.5% linearly polarized

1027
1:15:07,538 --> 1:15:09,909
light.
The famous 7.5% that I

1028
1:15:09,909 --> 1:15:14,398
mentioned to you earlier.
And the way I am going to do

1029
1:15:14,398 --> 1:15:18,210
that is as follows.
We have here a light beam.

1030
1:15:18,210 --> 1:15:22,275
Actually, it goes in this
direction but big deal.

1031
1:15:22,275 --> 1:15:27,273
And this is unpolarized light.
And I am going to put in here

1032
1:15:27,273 --> 1:15:31,885
one and only one pane of glass.
Here it is.

1033
1:15:31,885 --> 1:15:35,187
One pane of glass.
And I bounce that light

1034
1:15:35,187 --> 1:15:39,778
unpolarized off this surface,
and I am going to project it

1035
1:15:39,778 --> 1:15:41,791
here.
And, when I do that,

1036
1:15:41,791 --> 1:15:44,369
then the light will go like
this.

1037
1:15:44,369 --> 1:15:49,281
This angle is near 56 degrees.
Then the light is polarized for

1038
1:15:49,281 --> 1:15:53,308
100% in the direction
perpendicular to the plane of

1039
1:15:53,308 --> 1:15:56,127
incidence.
That means, in this case,

1040
1:15:56,127 --> 1:16:00,718
since this is the plane of
incidence, that is the incident

1041
1:16:00,718 --> 1:16:04,664
light and the normal,
it will be polarized in this

1042
1:16:04,664 --> 1:16:08,744
direction.
And, as I rotate that pane of

1043
1:16:08,744 --> 1:16:11,934
glass, the reflected beam will
move on the board here.

1044
1:16:11,934 --> 1:16:14,280
And, when I approach the
Brewster angle,

1045
1:16:14,280 --> 1:16:17,409
I will show you that I have it
100% polarized in this

1046
1:16:17,409 --> 1:16:19,636
direction.
When I am over the Brewster

1047
1:16:19,636 --> 1:16:23,246
angle it is partially polarized.
When I am under the Brewster

1048
1:16:23,246 --> 1:16:26,916
angle it is partially polarized,
but I will come very close to

1049
1:16:26,916 --> 1:16:30,143
that.
Once I have done that you are

1050
1:16:30,143 --> 1:16:32,877
going to get even more for your
money.

1051
1:16:32,877 --> 1:16:36,277
I will show you something even
more remarkable.

1052
1:16:36,277 --> 1:16:38,715
Most of the light will go
through.

1053
1:16:38,715 --> 1:16:41,967
For one thing,
the parallel component doesn't

1054
1:16:41,967 --> 1:16:45,367
come out at all,
so that means it goes through.

1055
1:16:45,367 --> 1:16:48,323
And, of course,
most of the perpendicular

1056
1:16:48,323 --> 1:16:52,831
component will still go through
because only 7.5% of the total

1057
1:16:52,831 --> 1:16:56,452
radiation comes out here.
Most of it goes through.

1058
1:16:56,452 --> 1:17:02,008
Well, I put another plane here.
And then I put another one and

1059
1:17:02,008 --> 1:17:04,888
another one and another one and
another one.

1060
1:17:04,888 --> 1:17:07,633
And Markos and I don't even
know how many,

1061
1:17:07,633 --> 1:17:09,843
but I think there are 200 in
here.

1062
1:17:09,843 --> 1:17:13,191
And every time that the light
strikes that surface,

1063
1:17:13,191 --> 1:17:15,937
at the Brewster angle,
whatever comes out,

1064
1:17:15,937 --> 1:17:18,950
if anything comes out,
it is polarized in that

1065
1:17:18,950 --> 1:17:21,562
direction.
Every time it is polarized in

1066
1:17:21,562 --> 1:17:24,575
this direction.
What do you think then happens

1067
1:17:24,575 --> 1:17:27,723
with the light that ultimately
makes it through?

1068
1:17:27,723 --> 1:17:31,792
100% polarized.
I have sucked out every

1069
1:17:31,792 --> 1:17:36,077
perpendicular component.
And so what remains then is the

1070
1:17:36,077 --> 1:17:39,818
parallel component.
Imagine what I have done now.

1071
1:17:39,818 --> 1:17:42,545
I start with 100% unpolarized
light.

1072
1:17:42,545 --> 1:17:45,896
With one reflection,
I can get 7.5% linearly

1073
1:17:45,896 --> 1:17:48,935
polarized.
With hundreds of reflections,

1074
1:17:48,935 --> 1:17:53,064
without being too specific,
I can 100% polarized light

1075
1:17:53,064 --> 1:17:57,974
here, which you are going to see
there, and I get 100% polarized

1076
1:17:57,974 --> 1:18:01,685
light here.
The light that comes out here

1077
1:18:01,685 --> 1:18:04,939
is polarized in this direction.
That is the perpendicular

1078
1:18:04,939 --> 1:18:07,206
component.
The other component is in the

1079
1:18:07,206 --> 1:18:10,054
plane of incidence.
Remember, one is perpendicular

1080
1:18:10,054 --> 1:18:13,192
the other is incidence.
In the plane of incidence means

1081
1:18:13,192 --> 1:18:16,679
this is the normal to the glass,
this is the incident beam so

1082
1:18:16,679 --> 1:18:20,282
this is the plane of incidence.
It is in the plane of incidence

1083
1:18:20,282 --> 1:18:23,537
so it is polarized like this.
When I use those 300 or 400

1084
1:18:23,537 --> 1:18:26,384
glass plates in a row,
the light that you see here

1085
1:18:26,384 --> 1:18:31,362
will be polarized like this.
And I can show you that because

1086
1:18:31,362 --> 1:18:35,173
I have a big polarizer.
If we can lower the screen.

1087
1:18:35,173 --> 1:18:39,060
Markos, would you do that?
Then I will turn this on.

1088
1:18:39,060 --> 1:18:41,956
There it is.
We will make it very dark.

1089
1:18:41,956 --> 1:18:45,614
Not completely dark.
We are going to set it at TV

1090
1:18:45,614 --> 1:18:47,062
level.
First of all,

1091
1:18:47,062 --> 1:18:50,110
this only goes through one pane
of glass.

1092
1:18:50,110 --> 1:18:54,378
You cannot use your polarizers
because whenever the light

1093
1:18:54,378 --> 1:18:58,722
reflects off a screen it loses
its degree of polarization,

1094
1:18:58,722 --> 1:19:04,262
at least to a high degree.
This is the light that goes the

1095
1:19:04,262 --> 1:19:06,597
plane.
And 7.5%, if I am near the

1096
1:19:06,597 --> 1:19:08,786
Brewster angle,
comes out here.

1097
1:19:08,786 --> 1:19:11,997
93% go through and about 7.5%
comes out here,

1098
1:19:11,997 --> 1:19:14,551
if I am close to the Brewster
angle.

1099
1:19:14,551 --> 1:19:16,886
And I may actually be close
here.

1100
1:19:16,886 --> 1:19:19,732
Actually, let me purposely not
be close.

1101
1:19:19,732 --> 1:19:23,381
Now I am not very close.
Here you see it is already

1102
1:19:23,381 --> 1:19:27,029
polarized to some degree,
but you see clearly light

1103
1:19:27,029 --> 1:19:30,822
behind it.
Now I am going to go through

1104
1:19:30,822 --> 1:19:33,230
the Brewster angle.
And now I kill it.

1105
1:19:33,230 --> 1:19:35,899
I am very close to the Brewster
angle now.

1106
1:19:35,899 --> 1:19:38,177
This light is almost 100%
polarized.

1107
1:19:38,177 --> 1:19:41,497
This is the direction.
The direction of polarization

1108
1:19:41,497 --> 1:19:44,100
is like this.
That is the direction of my

1109
1:19:44,100 --> 1:19:45,988
polarizer.
And here I kill it.

1110
1:19:45,988 --> 1:19:49,047
And that, of course,
is hardly polarized at all.

1111
1:19:49,047 --> 1:19:52,887
I can show it by rotating it.
It is hardly polarized at all.

1112
1:19:52,887 --> 1:19:55,946
It has lost some of its
perpendicular component,

1113
1:19:55,946 --> 1:19:59,852
but not so much that you can
really see the difference when I

1114
1:19:59,852 --> 1:20:03,650
rotate this.
Now I am going to make the

1115
1:20:03,650 --> 1:20:07,523
light go through my hundreds of
panes, and that is going to be

1116
1:20:07,523 --> 1:20:10,507
even more interesting.
I think that is this one.

1117
1:20:10,507 --> 1:20:13,873
Now I do the same thing,
except I do not have one pane

1118
1:20:13,873 --> 1:20:16,349
of glass but I have many.
A few hundred,

1119
1:20:16,349 --> 1:20:18,380
Markos.
We don't want to open it.

1120
1:20:18,380 --> 1:20:22,063
You see, when you open it you
break it, so we have no idea.

1121
1:20:22,063 --> 1:20:25,746
But the idea is we agree that
it will be close the Brewster

1122
1:20:25,746 --> 1:20:28,285
angle here.
This now is polarized in this

1123
1:20:28,285 --> 1:20:32,148
direction.
That means if it is polarized

1124
1:20:32,148 --> 1:20:35,481
in this direction,
I should be able to kill it

1125
1:20:35,481 --> 1:20:36,962
like this.
And I can.

1126
1:20:36,962 --> 1:20:41,111
This is when I let it through
and this is when I kill it.

1127
1:20:41,111 --> 1:20:45,259
And I claim that this is
polarized now in this direction,

1128
1:20:45,259 --> 1:20:48,666
100% polarized.
That means I can let it through

1129
1:20:48,666 --> 1:20:51,481
like this, but I can kill it
like this.

1130
1:20:51,481 --> 1:20:54,296
There you go.
Now I have made 50% of my

1131
1:20:54,296 --> 1:20:57,777
incident light go out there,
50% comes out here,

1132
1:20:57,777 --> 1:21:02,000
roughly, 100% polarized,
100% polarized.

1133
1:21:02,000 --> 1:21:11,394
This is an absolutely
remarkable thing.

1134
1:21:11,394 --> 1:21:25,733
We now have a method to convert
unpolarized light into 100%

1135
1:21:25,733 --> 1:21:37,352
polarized radiation.
A component like this and a

1136
1:21:37,352 --> 1:21:49,466
component like this.
And one of the amazing things

1137
1:21:49,466 --> 1:22:02,074
of physics is that it makes the
impossible possible.

1138
1:22:02,074 --> 1:22:16,661
And that is the power and the
beauty of physics and is why I

1139
1:22:16,661 --> 1:22:25,313
love physics,
to make the impossible

1140
1:22:25,313 --> 1:22:37,283
possible.
It is like the impossible

1141
1:22:37,283 --> 1:22:45,828
dream.
It is also to make difficult

1142
1:22:45,828 --> 1:22:56,384
things easy.
Physics makes difficult things

1143
1:22:56,384 --> 1:22:57,641
easy.
See you Thursday.