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GEORGE BARBASTATHIS:
So does anybody
have questions from
the last lecture?
After some time to wake up,
anybody still have questions?
So I will start, and if
you remember of a question
that you had, please interrupt.
I think if you push
the button the button,
I will hear a sound over here.
So we will know that
there's a question.
So I'd like to
pick up the thread
from where I left last time.
We covered fairly quickly
because we ran out of time.
But we covered that
the law of reflection
and the law of refraction.
So I'd like to go back
to the low of reflection,
and remind you that it is a
very simple, very simple result.
The minimum path requirement
for the light rays
force those reflections
to be symmetric.
So this has a
strange consequence
that we're unfamiliar with
from when we look in the mirror
daily.
And that is the fact that
our left and right locations
in our body.
They flip when we
look at the mirror.
So we'd like to make
that a little bit more
quantitative by looking
at this diagram.
So to understand this,
suppose that I am--
I have an object
that is oriented.
You can think of it as
perhaps two pencils that
are sort of following
the ray paths
and then they get
reflected from the mirror.
So, for example, if you
look at the central ray,
it will be reflected
symmetrically.
So, again, what you see here is
the front view of the mirror.
And then I will draw a few
ray a paths in perspective.
So this is the ray paths
that start from the object.
I don't know if you
can see me over there,
but I'm trying to show what will
happen with an actual pencil.
So if this was a pencil that
is coming towards the mirror,
and this is the
surface of the mirror.
The pencil will go like this.
And then law of reflection
says that it will also
come out like this again.
This follows very easily if
you simply trace the rays.
In the next step in
my animation here, it
paints the sort of positions
of the pencils and their tops
as they get reflected
from the mirror.
So what happened here
is the following.
As you can see, the
pencil did not actually
change orientation.
This is a strange
thing about mirrors.
But what did change
is the following.
Imagine that you
are walking together
with this pair of pencils.
As you are walking
this way, you will
see that the one that is
horizontally oriented.
The top of it is
pointing to your right.
But if you walk all
the way to the mirror.
Go to the center of the
ray, and start going
backwards the other way around.
Then all of a sudden,
the top of the pencil
appeared on your left.
This is a simple consequence
of the law of reflection.
The way we interpret it
in everyday life when
we see through a mirror
is the following.
Of course, in everyday
life, we don't
talk about left handed
and right handed triads.
But the way we interpret
it is because normally when
we look at a good
quality mirror,
we do not know that
there is a mirror there.
We actually see a continuation
of that are being reflected
behind, back behind the mirror.
So what you're seeing then
if we look behind the mirror
is actually this triad, with the
pencils oriented as shown here.
And if we interpret it as
being seen from behind,
then, of course, left has
become right, and vice versa.
So another way to
think about it is
that when we look
through a mirror,
it is as if we're looking
from the image from the back.
So a better way
to think about it
is if you're
looking at something
that is written on my t-shirt.
If you look at it
from the mirror,
it would appear as
if I were hollow,
and you would see the back
of the writing on my t-shirt.
And you all know that if
you look at the ambulance
sign in ambulance
trucks, ambulance
whatever you call them, cars.
Because it is meant to be
visible through the mirror
of a driver, they actually
write the sign backwards.
So when you look at it, it is as
if you saw a transparency of it
from the back.
So this is a simple consequence
of the law of reflection.
This is a very
simple silly question
that sometimes is asked.
And it actually takes quite
a long explanation to answer.
And that question
goes like this.
If you look at the
mirror, yes, we
know that left and right
flip, but up and down do not
flip, right?
You don't see yourself
upside down in a flat mirror.
You see yourself flipped
from left to right.
What is the reason?
The reason is shown here.
The reason is because
of the flipping
of the relationship between the
left and right in the object
side, as you interpret
the projection
of the rays that are coming
from the opposite side
of the mirror.
So this requires a
little bit of thinking,
so I'll let you think about it.
Unless you have a question
now, please ask it.
If not, you can think
about it and come back
with more questions
on Wednesday.
Is there any immediate
question about this?
That's kind of a subtle
and elegant point.
I should have brought
a mirror with me.
Piper, do you have
a mirror over there?
AUDIENCE: No, I don't
have a mirror here.
GEORGE BARBASTATHIS:
No mirrors, OK.
But everybody has access
to mirrors, right?
It's the one optical element
that we can find very easily.
So I'll let you practice with
your mirror in your bathroom,
and then come back
and ask me questions.
OK.
AUDIENCE: I'm lying down.
You're horizontal.
GEORGE BARBASTATHIS:
That's right.
Our producer like this.
The other thing I
wanted to talk about
is expand a little bit
on the law of reflection.
Oh, I'm sorry, the
law of refraction
that we derived last time.
So the law of refraction
is the last equation
shown on this line.
It says that this quantity,
the index of refraction
multiplied by the sign of
the angle of incidence, where
the angle of incidence
is defined with respect
to the normal to the surface.
It is preserved.
So as you go through multiple
surfaces, the law of refraction
says that this quantity
must be preserved.
So there is a common
problem that I
have posted just a
couple of hours ago
that asks you to make an analogy
between the law of refraction,
and the problem of
a lifeguard who has
to save a person in the water.
So recall the reason the
law of refraction happens
is because the light must
minimize its path between two
points, P and P prime.
So a very similar
problem is if you
have a trajectory that you're
trying to design in a way
that you minimize the
time that you will
spend going on this trajectory.
So here, you have a swimmer
who is sitting on the beach.
I'm sorry, not a swimmer.
You have a lifeguard
sitting on the beach.
And the lifeguard sees a
person drowning in water
farther behind.
The lifeguard can run on the
beach at some velocity, v sub
r.
They can also swim in water
at some velocity v sub s.
Most people swim
slower than they run,
so we can assume here that
the running speed is faster
than the swimming speed.
And the question is
how should the swimmer
plan his path so that he can
reach the drowning person as
fast as possible?
Again, you can think that,
for example, the straight path
is not the best
because he's spending
too much time in water.
I mean, water, he's slower.
So he may want to spend
a little bit extra time
in the fast middle,
and a little bit
less time in the slow medium.
But, again, he cannot overdo it.
If he goes a really crazy
path, then, again, he
will end up with a longer time.
So light is trying to
do a similar thing.
It is time to optimize
the obstacle path
length, or
equivalently, the time
that it takes for the light ray
to reach from a starting point
to an ending point.
And, again, remember this is
almost an exact analogy here.
The speed of light
is faster in air,
and slower in a
dielectric medium.
So if you had the
air and glass here,
that would be a very
similar situation.
I'm going to skip the next line.
And I'm going to go to
this one, to number 34.
So what I'm trying to say here
is to point out two cases.
They're not really different.
They're just two cases
of the same situation.
In one, you are going from
a medium of lower index
to a medium of hire index.
And in this case,
obviously, because
of the law of refraction,
the angle of refraction
will increase as you
go from left to right,
from low index to high index.
The opposite will happen
if you go from high index
to low index.
OK, so this has
two consequences,
which you can think of as
you reach the extremes.
If you come in at the
maximum possible angle
here, 90 degrees,
then you can imagine
that you will not enter
at 90, but you will enter
at a slightly smaller angle.
So basically, if you
are coupling in light
from air to glass, or in
general from a medium of lower
index to a medium
of higher index,
you have a limited
column of approach
that you can couple light into.
And that is given by
this equation over here.
When the exterior angle
theta reaches 90 degrees,
then this is the maximum
angle, theta prime,
that you can access inside
the high index medium.
The opposite is perhaps
slightly more interesting,
is what happens when you
reach or exceed theta
prime equals to 90
degrees over here.
So if you look again at
the law of refraction,
you can realize that it
is possible to arrange
for a combination
of n and theta,
such that the product is bigger
than the index of refraction
at the medium side.
In order, now, to satisfy
the law of refraction,
you would have to require
that the sine of an angle
is bigger than 1.
So since we are limited to
deal with real angles here,
not complex, this cannot happen.
What really happens there is
that the light will actually
be reflected.
If you satisfy this
condition, this product
becomes bigger than
the index of refraction
outside in the medium.
Then when you satisfy
this condition,
then light will be reflected
inside the high index medium.
And that is known as total
internal reflection, or TIR.
So let's look at TIR in
slightly more detail over here.
So I have a glass
medium in there,
and imagine that I
have a wavefront that
is arriving from glass
towards the air interface.
So there is a combination
of index and angle,
where the product of the
index inside the glass
times the sine of the
angle equals exactly one.
What happens then is
the law of refraction
does not break down yet.
But what will happen is the
light will be refracted,
and it will propagate exactly
parallel to the interface.
This is known as a surface wave.
If you increase the angle
now, then the product
will become bigger than one.
The law of refraction
cannot be satisfied anymore.
So what will happen then.
Oh, and I should have said that
the angle where this happens
is called the critical angle.
Because it is the
angle just below which
I still have refraction.
If exceed this angle,
then I get this phenomena
of total internal reflection,
where all of the light
is reflected inside the glass,
inside the high index medium.
So it is almost as
if the interface here
abruptly changes,
and instead of being
mostly transmissible over here,
it becomes mostly reflective.
So it starts acting
like a mirror.
There's one difference
that makes it slightly
different than a mirror.
And the difference
is that if you
were to calculate the electric
field on the opposite side
of the interface, that
is, inside the medium
where light does not propagate.
You will discover that
there is some leakage.
The electric field has non-zero
values in the low index
medium over here.
Even though the electric
field that you find
is not propagating, it is
what is called evanescent.
It is in exponential decay,
but there is no wavefront.
There is no wavefront
of light propagating
in the vertical
direction like this.
This is called an
evanescent wave,
and we will revisit
it later when
we deal with electromagnetics.
Because right now,
the way I defined
it is not perhaps very
rigorous, or very quantitative.
But I wanted to give
you a sort of a heads
up that something slightly
more than geometrical optics
prediction happens here.
But as far as
geometrical optics goes,
that we will be dealing
for the next few lectures,
there is a reflection.
This says what I just
mentioned, that we
will talk more about
these evanescent waves
a little bit later.
One more thing that I want to
say about evanescent waves.
One way you can sort of
realize the existence
of evanescent waves
is with a sort
a related phenomenon called
frustrated total internal
reflection, also known as FTIR,
because that's quite a mouthful
to pronounce.
So FTIR happens if you
have this situation,
where you're beyond
that critical angle,
and therefore, your
total internal reflecting
into the medium.
But you bring near
the interface,
you're bringing
another piece of glass,
another piece of
high index medium
in a way that an appreciable
amount of the evanescent wave
is allowed to enter inside
the high index medium.
If that happens, as we've
said, the TIR is frustrated.
What it really means is that
the TIR stops happening now.
What would happen
is a small amount
of light will still be
reflected, of course.
You cannot avoid that.
But the significant
portion of the light
will couple out into
the next material,
and it will actually
be transmitted.
So this is a very
interesting phenomenon,
because if you think
about it, the light is
forbidden to enter this region.
Snell's law says that light
cannot cross into air,
yet because of the proximity,
the light can actually couple
out.
And again, we will see a much
more rigorous explanation
and quantitative description
of this phenomenon
later, when we're doing
it through magnetics.
Some of you who may have
taken electronics or quantum
mechanics, there's
a similar effect
called tunneling in potential
barriers in quantum mechanics.
The equations are very similar
that describe this phenomenon.
In both cases, you
have a wave that
is crossing a forbidden
region in order
to pass into an
allowable region again.
This may be actually
be a good find
for Piper to solve a demo of
the total internal reflection.
And Piper will actually solve
it in the context of a prism.
Piper, maybe you
can start setting up
while I give a brief
description of prisms.
You all are familiar, I suppose.
They're pieces of glass that
are cut into various triangular
and other polygonal shapes.
And typically, prisms are--
they're arranged either so
that the light passes through,
as shown in the top diagram.
Or if you bring the
light from the bottom,
and you manage to exceed
the critical angle
in the interface, then you
can also total internally
reflect the light, and create
a situation like this that
is known as a retro-reflector.
Now, a rule of thumb that is
useful to know for glass, which
has index of refraction 1.5.
The critical angle
is about 42 degrees.
So if you are incident
at 45, as shown
in this case of what is called
isosceles triangle, then--
actually, I'm sorry.
This is equilateral, isn't it?
If you're incident at 45
in an equilateral triangle,
then you will satisfy
the TIR condition,
and you get this
retro-reflector.
And there's more
complicated prisms
that allow you-- for example,
this is the pentaprism.
After two bounces, the light
will exit at 90 degrees angle.
So Piper, maybe you
can do the demo now?
PROFESSOR: Sure, sure.
So just before
showing the demo, I'm
going to pass around
this prism that
has in these two
surfaces, two images.
So then what you see is
that in this window here,
you tilt it like this.
You're going to see when
you get a hold of it.
You actually see
two different images
due to total
internal reflection.
So at a given
angle, you basically
get the light reflecting
from one of the surfaces.
And another angle,
you get the other one.
So it's actually very
interesting to see.
OK, so we're seeing here
the top view of the demo.
We put together a wide light
source and a laser source.
And before showing you
what actually happens,
let me just introduce
some of the components
that we are going to be seeing
in several demos from now on.
First of all, we have the white
light source, it's a lamp.
And then this component
here, it's a regular lens
that you're familiar with.
So the job of these
lens is to collimate--
that's the term that we use--
to convert this light close
to a plane wave, similar
to the light that
is coming from the sun.
So the sun, again, would be
like a point source of light
far away.
So then these
lenses are basically
transforming in this
light into parallel rays
from the geometrical
optics point of view.
So we use these lens.
This is an iris, similar
to the iris that controls
the aperture in your eye.
And this is just used to
split the light into two--
I'm sorry, to reduce the
diameter of the beam.
Then this component,
here this one.
It's what we call
the beam splitter,
or more specifically, the
non-polarized beam splitter.
And it's a component
that allows us to split
the light into two paths.
So here, this is one
path, and another path.
So once we split the light in
two paths, and in this case,
it has equal ratios.
So it's 50% to one
side, 50% to the other.
Then let's follow
one of these parts.
This part illuminates
one of these prisms here.
And again, there's going to
be some refraction following
the law of refraction
that we saw.
But in addition
to that, as we'll
see in the next
couple of slides also.
There is a phenomenon
called dispersion
that you're familiar
with when you
see rainbows in a rainy day.
And basically, that
has to do with the fact
that different wavelengths see
a different index of refraction.
So if you go back
to the Snell's law,
they will basically
bend in a different way.
So therefore, you
see a rainbow effect.
So I don't know if you can see
the side view in the camera
please?
Yeah, so that's a picture of it.
OK, but here in the
classroom too, please.
I don't know if you can see it.
We're going to try to
show it also in this.
But I have here two components.
This is a prism
that is doing these,
what we call the normal
dispersion, which basically has
that the biggest angle that
bends is the blue light,
or the shorter wavelength.
And then we have
another component here
that we haven't
talked about yet,
but we'll see it in
the next few slides.
So this is just an introductory
introduction to this element.
It's called
transmission grating.
And this component is
used in several systems.
They're a different principle.
It's not refraction anymore.
It's using the diffraction
property of the light.
And using that diffraction
property can also
create this rainbow that you
could see here in the back,
and hopefully, you
can see the picture.
What we are going to see
here is that actually
the opposite trend happens.
The red angles bend more
than the blue angles,
and that's called
anomalous dispersion.
So we have normal dispersion
in the prism case,
like in the one shown
here in transparency,
and anomalous dispersion
showed in the grating.
So I'm going to tilt
this a little bit.
OK, it's fine.
I'm going to try to see.
So I don't know
if you can see it.
Can you see the rainbow
here from the back?
If you're in the classroom?
GEORGE BARBASTATHIS:
Yeah, we can see it.
PROFESSOR: All right, excellent.
So this is just the
rainbow from the grating.
After class, you can just come
here and play with the demo.
And you're going to
see the two cases.
You can actually
trace the rays and see
which color is bending
more, and distinguish
between anomalous and
normal dispersion.
So the last thing that
I want to show here
is the total internal
reflection principle.
In this case, this piece of
acrylic here that we have.
You can see that it's forming--
it's basically having a
laser light coupling into one
of the sides, so similar to
that exit sign over there.
And what we have is
that the acrylic--
the piece of acrylic
acts like a wave guide.
So it conducts a light inside.
And the reason the
light doesn't escape
is because it's
basically suffering
total internal reflection at the
interface between the acrylic,
which has a higher index
than air, so stays inside.
Now, in order to
couple the light out,
I put some tape here, as you
can see forming the letters MIT.
And that basically
frustrates the light,
allows it to break
the incidence angle.
So a ray now instead of getting
into a very flat surface
at an angle that is larger
than the critical angle,
it basically reaches
a surface that
has a diffuser type of angle,
so maybe it can escape out.
So then, you can see
all this light diffusing
out forming like either this
image, or the image of the exit
sign.
So that one, you can see
it also in Singapore?
The frustrated?
GEORGE BARBASTATHIS: Yeah.
PROFESSOR: OK.
GEORGE BARBASTATHIS: It
looks frustrated to us.
PROFESSOR: From the
side view, I guess.
So I don't know if you want
to add anything, George.
GEORGE BARBASTATHIS: So
what you see on the slide
that I'm projecting it
now is an application
of the same principle
that Piper just showed.
It has an application
in a bunch of commonly
used conventional devices,
namely, fingerprint sensors.
Where instead of the tape
that Piper put over there,
usually what you do is
they place their finger
touching the side of the prism.
And then because our buddy,
this may be surprising to you.
It was surprising to me
when I first heard it.
Our body is composed mostly
of water, about 75% is water.
So therefore, the
refractive index of our skin
is close to 1.3, which
is, of course, higher
than the refractive
index of air.
So what happens here is
over here, for example,
you have a glass and air.
So therefore, light will be
totally internally reflected.
But at the ridges of the
finger, of the fingerprint,
you have glass and water.
That is 1.5 to 1.33 or so.
So therefore, the ridges appear
dark because they frustrate
the total internal reflection.
The light couples
into your finger.
And therefore, a surprisingly
sharp image of the finger
appears in the camera.
Actually, what you see
here, it does a disservice.
The projector and the
pixelation of my computer
does a disservice to the
quality of the fingerprint image
that you get.
So nowadays, most
fingerprint sensors
are based on this principle.
In fact, we have improved it.
Instead of using
the prism in places
like laptops that have
fingerprint security,
it is the same principle,
but you slide the finger.
But still, the ridge
of the fingerprints
as you slide the
finger over the sensor
is captured by the principle
of total internal reflection.
Any questions about that?
About TIR and FTIR?
OK, one other use of TIR
is in another very useful--
another extremely
useful property of light
is that you can actually capture
it, almost like in a wire.
And you can guide the light
over a very long distance.
Now, why this is
very important is
because we know from
experience, and we'll also
learn later as the
Huygens principle,
that light does not
like to be confined.
Generally, light, once you
generate light in a source,
the light would like to expand.
It would like to open up and
propagate in an expansive way.
For example, the sun,
the stars, and so on.
They propagate isotropically,
all around them.
And you know the same from
the light bulbs and so on.
The light expands.
There's an exception, of
course, called lasers.
Lasers can be quite collimated.
But even lasers,
they tend to expand.
If you leave a laser
beam by itself,
and you propagate it
for a long distance,
eventually, it will expand.
It will become quite big.
So the way to undo this
property of light--
if you want to transmit
light over a long distance
without expansion-- is
to use a wave guide.
So wave guides typically,
they use this phenomenon
of total internal reflection.
In the simplest case, you
have a slab of high index--
dielectric middle sandwiched
between two other pieces
of lower index medium.
And what happens there provided
that the light is incident
at the sharp enough angle that
is beyond the critical angle
between the two media.
Then the light will sort
of bounce back and forth
between the two interfaces.
And this way, you can
actually transmit it
over a very long distance.
So, of course, if
it is not true.
If the light arrives at
a shallower angle, then,
of course, it will
actually couple out,
and it will not
be guided anymore.
The way you establish whether
the light will be guided or not
is by using these properties
called the and numerical
aperture of the wave guide.
So this is a term,
numerical aperture,
that we'll hear again and
again in this class, at least
in three different contexts.
But they all mean
the same thing.
Actually, they mean
an angle of acceptance
of an optical system.
So in this context
here of a wave guide,
compare the two rays.
One is sort of the
solid ray, and the other
is the dotted ray.
The solid ray comes
in from air, then
is refracted at the
vertical interface.
And because this
angle is fairly small,
by the time it gets into
the middle, the angle
it makes through the
perpendicular surface
of the interface between
the slab and the cladding.
It actually satisfies the
TIR condition over here.
You can see a little bit
if you familiarize yourself
with the way Snell's law works.
You can see that as you
increase this angle over here,
this angle over here
actually decreases.
So the dotted ray
actually can arrive
at below the critical angle.
So therefore, the
daughter ray is not
guided where the
solid ray is guided.
So the numerical aperture is
the maximum angle, theta naught,
that you can tolerate
before you stop
satisfying that TIR condition.
And therefore, the
numerical aperture
is the maximum angle that you
can couple into the wave guide.
If you tried to bring light
at a higher angle than that,
it will actually not be guided.
It will escape into the
cladding, and it will get lost.
It will disappear.
So with a little bit of algebra,
which I haven't done here.
I will let you do
it by yourselves.
In years past, I used to
give this as a homework,
but I didn't do it this time.
But anyway, with a little
bit of algebra and playing
with Snell's law,
you can find out
that the numerical
aperture in this case
is given by this
quantity over here.
The square root
of the difference
of squares between the
two indices of refraction.
And as I mentioned earlier,
physically what it means.
This quantity is the
angle of acceptance
of the wave guide for the light
that you want to couple in.
Typically, wave
guides in practice,
they have a very
small difference
between the index of the core,
where the light is guided,
and the index of the cladding.
This difference is typically in
the order of 10 to the minus 3,
or 10 to the minus 4.
So therefore, the
numerical aperture is what?
It is small or large for
a typical wave guide?
If the index difference
is very small,
is the numerical appearance
or a small or large?
Small, right?
We're going to set
up a competition here
between Singapore and Cambridge.
So you guys, when you have an
answer, please push the button.
On either side, please
push the button.
So the numerical aperture
is actually very small,
as our colleagues
here correctly said.
And what is on the
slide is the opposite.
If you have a high
index contrast,
then you get a high
numerical aperture.
There is one more type of
wave guide, which is actually
the same principle,
but a slightly
different implementation.
It's called a gradient
index wave guide.
And the way to understand it.
Imagine that I stack a bunch
of different slabs of glass
with index that varies
from a small value.
This I denoted as the
sort of light gray.
Then to darker gray, denoting
higher index of refraction,
and then back to light gray.
I mean, back to low
index of refraction.
So if you imagine the ray
coming in from the top here.
It will refract into the
guide, and then as it goes in,
it will keep getting refracted.
Now, we can adjust
the numbers here,
so that one of these
interfaces, the light
will exceed the critical angle.
And therefore, it will be
totally internally reflected
at this interface.
And if that is
true, the same thing
will happen also at
the top interface.
As I mentioned before, this
quantity, n sine theta,
is preserved.
So therefore, if this
quantity, n sine theta,
was such that the TIR condition
was satisfied over here,
then the same will
happen over here.
Because everything else
is symmetric, correct?
The reflections are symmetric,
and the quantity, n sine theta,
is preserved.
So therefore, if you put a light
in this kind of arrangement,
it would actually follow
a periodic trajectory.
The light will sort of
be periodically reflected
from these interfaces, and will
follow a periodic trajectory
down the stack of slabs.
Therefore, this is also
a kind of wave guide.
It is commonly referred to
as green, where green is not
for the facial expression,
but stands for gradient index.
And there's a special case of
a gradient index where they
don't normally do it this way.
The way they do it is with
a continuous variation.
And they manufacture
it with diffusion.
It's very interesting.
They take a piece of glass,
and they diffuse ions.
Because the ions change
the index of refraction
of the glass, then
they can sort of
get a continuous profile of
gradually variable index.
And in the special case where
this profile is quadratic,
it turns out that the
trajectory of the light paths
is kind of like a helix.
It become sinusoidal.
And the light sort of
bounces in sinusoidal fashion
between the two interfaces,
this one and this one.
We will do this in more
detail four lectures later.
If you look at your
syllabus, there's
something called
Hamiltonian optics.
We will actually
see in action how
this sinusoidal periodic
trajectory comes about.
Yes.
AUDIENCE: So what
is the [INAUDIBLE]??
GEORGE BARBASTATHIS: Yeah.
That's a very good question.
PROFESSOR: Can you
repeat the question?
Because they didn't
press the button here.
Can you repeat the
question, please?
GEORGE BARBASTATHIS: I'm sorry.
Could you repeat
with the button?
Yeah.
AUDIENCE: Yeah.
What is the advantage of
using this type of wave guide
compared to step
index wave guide?
GEORGE BARBASTATHIS:
So the advantage,
which I cannot describe yet
because we have not done wave
optics.
But in wave guides, there's a
phenomenon called dispersion.
It is very similar to the
dispersion from a prism
that Piper showed before.
But in telecommunications,
when you transmit signal down
a wave guide, it has the
effect of basically lowering
the speed, the effective
speed at which you
can transmit information.
So it turns out that
the step index wave
guide has a higher dispersion
than the gradient index wave
guide.
So you get a much higher speed
in fibers of gradient index.
So that's one reason
why people use it.
This will take a while.
And, of course, more
practical wave guides that
are shaped like a wire.
Literally, they are
known as optical fibers.
Again, you have two
types of fibers.
You have the step index fiber,
where you have a higher index
core, and the light is kind
of bouncing back and forth
between the core
and the cladding.
And there's also
gradient index fibers,
where the light is following
a helical trajectory.
The phenomenon of
wave guiding, again, I
have to defer to
electromagnetics.
It is much easier to describe
with electromagnetics
than it is with wave optics.
But for now, we can get a sort
of a preliminary description
with the means that we
have available to us.
As a sort of a
curiosity, it turns out
that these gradient
index wave guides.
They appear in the nature
in certain animals.
I mean, insects, actually.
Their eyes, they're composed
of several wave guides,
each one of which is actually
a gradient index wave guide.
And the way the animal eye
works is it captures-- remember,
the wave guide has a
limited numerical aperture.
So each one of these guides,
each one of these small eyes,
they're called ommatidia,
these little wave guides.
So each one of those
captures a very narrow angle
of light sort of within
the field of view
of the eye of the insect.
And then sends a signal down to
the optic nerve of the insect.
So basically, the insect
with this kind of eye, it
forms a very bloody
picture of the background,
because it integrates a
relatively large range
of angles.
But still, the range of
angles is small enough
that it allows it to
quote unquote see.
Now, insects, of
course, they don't
see the way we see, at least
from our everyday experience.
And the reason, of course,
is that the insects
have a very limited brain.
A typical insect might
have about 10,000 neurons
in his brain.
Show of hands, does
anybody know how many
neurons we have in our brain?
10 to the 11.
So we have about eight orders
of magnitude more neurons.
In case you are
wondering, whether you
are smart or
educated, it does not
have to do with a number
of neurons that you have,
but it has to do with the
connections between neurons.
Neurons are
connected with wires.
It turns out an
educated person has
approximately 10
times more connections
than an uneducated person.
The same number of neurons,
but more connections.
And so it's a sad fact of
life that every day, adults--
that is, after age six or
so, even at childhood--
would begin to lose neurons.
In fact, each one
of us every day
will lose about 100,000 neurons.
Nothing to worry about,
because we have 10 to the 11.
So even with all this
loss, we can still
survive until a fairly old age.
But anyway, it is true.
So [INAUDIBLE].
Anyway, the insect,
on the other hand,
has about 10,000
neurons altogether.
So it has to make do with
these 10,000 neurons.
It has to move.
It has to feed.
It has to mate and
all of these things.
So the way they handle it is
they get very simple vision,
and they navigate according
to differences in lighting.
So the typical example is an
insect flying toward a tree.
Nature has evolved the insect
to avoid this situation,
because if it flies into the
tree it will crash and die.
But if it flies towards a
tree, the insect sees a dark
background-- the
trunk of the tree--
surrounded by light,
which is sort of leaking
on the sides of the tree.
As it flies by, it
sees an edge of light
that is very rapidly
expanding, because it
is approaching the tree.
So the insect is wired.
It as actually automatic.
The insect doesn't think, oh
my god, I'm going to crash.
Let me turn.
It's automatic.
As soon as the
neurons of the insect
register a difference
in lighting
between successive
ommatidia over here,
they turn on the motor--
the flies or whatever, the
legs and so on of the insect,
the navigation of the insect.
And the insect turns
and avoids the obstacle.
So this is what I
say about insect.
More precisely, I'm referring
to the fruit fly, which
has been very broad,
very extensively studied
in this context.
But anyway, this is a very,
very interesting story
of how the insects
use what we now
consider as a
rather sophisticated
optical instrument,
a green wave guide
in order to generate a
very simple type of vision
based navigation.
The next thing I
was going to say,
Piper already mentioned it.
The index of refraction
of most dielectric media
turns out to be a strong
fraction of the wavelength.
So I stole from
a book, actually,
from the Soto website.
Soto is a glass manufacturer.
They make glasses
that are used very
commonly in optical instrument
lenses and such, and prisms
and so on.
So they have this
picture on online
of the index of
refraction as a function
of wavelength for a relatively
large range of wavelengths
going along the wave
from ultraviolet
into the deep
infrared over here.
So you can see that the
index varies quite a bit.
Also, with the plot
here, the absorption
coefficient of the material.
And you can see that they are
kind of correlated in the sense
that when the index does
something interesting,
the absorption also seems
to do something interesting.
It is not coincidentally.
It turns out to have
a very interesting
theoretical foundation.
I will go into it.
Perhaps I will go into
it later in the class.
But anyway, the point I've
been trying to make here
is that you can see
that the index can
have quite a bit of variation.
So because of that, if
you send broadband light
that contains multiple
colors into an element,
such as a prism.
Then you can observe these
phenomena that Piper showed.
Different wavelengths,
different colors,
they experience different index,
and therefore, the Snell's law
applies differently to them.
That is why you have this--
it's called analysis
of white light.
It becomes a rainbow.
For most materials, it is true.
Longer wavelengths actually have
a lower index of refraction.
Now let's see, does
this make sense?
If you look at this
picture over here.
Does it make sense what I said?
Which wavelength apparently has
the lower index, blue or red?
It better be right, or I--
either I made the wrong slide.
But anyway, I have taught
this class for several years.
So you would think that if
I had made the wrong slide,
I would have fixed
it by now, right?
So the slide is correct.
The way to figure
it out is you have
to imagine a normal to
the surface over here.
So which wavelength appears to
have the stronger refraction?
Blue or red?
Blue, right?
So the blue wavelength
suffers a strong refraction.
That is, the blue
wavelength has what index?
Higher or lower?
Higher.
And indeed, the
blue wavelength is
softer than the red wavelength.
So this is consistent with
the curve that you see here.
The blue wavelength is
probably somewhere around here.
The red wavelength is
somewhere around here.
It is not a dramatic variation
in the visible range.
And that is typical
for most glasses.
In the visible range, they have
a relatively slow variation
of the index of refraction.
But nevertheless, it is there.
And you saw evidence
of it in the experiment
that Piper just
did with the prism.
And the last thing
that I want to say.
I don't want to
belabor this point.
People use various quantities
to characterize dispersion.
And typically, they
characterize them with respect
to the various emission lights--
emission lines,
from atomic spectra.
So they use this as
reference [INAUDIBLE]..
The reason, I suppose.
The reason is that
back when people
developed these measures,
lasers were not available.
So the best way to define
wavelength standards
was with emission lines.
So they use typically
the hydrogen C line and F
line, and the sodium D line.
And then they define
these quantities,
the dispersive power and
the dispersive index,
which are defined according
to the index at these three
different wavelengths.
So this is very useful for
people who do optical design.
And it gives you sort of an idea
of how dispersive is a glass.
These quantities are actually
inverse relative to each other.
And this an example.
For ground glass,
typically, you want
the V number, the
dispersive power, to be low,
if you want a
dispersion free element.
OK, any questions?
OK, so I'm not going to go
over the second lecture.
We will postpone
it for Wednesday.
Basically, we have slid back by
about an hour, but that's OK.
We'll catch up later.
But what I'll do is I would
like to get you started thinking
about next Wednesday's lecture.
So next Wednesday,
we'll basically
see a bunch of applications
of Fermat's principle.
Namely, the principle
that says that light
chooses its trajectory trying
to minimize the optical path
length.
So we saw already
two applications,
one in the law of
reflection, and the other
in the law of refraction.
So the next applications
will be in focusing.
So the question we'll
ask the next Wednesday
is how can we design a surface,
or reflect a surface such
that if light is arriving
from infinity in parallel rays
like this, this
surface upon reflection
focuses all the rays.
So they pass from the
same common focal point F.
So you can look it
up into the notes,
and then I will go over
it again on Wednesday,
how we can use the Fermat's
principle in order to design.
You can actually design.
We can come up with an
analytic expression that
has to be a parabola for
the sacrifice that gives
the perfect focus onto a point.
So the homework has been posted.
The first three
problems you can do
without a need for any of this.
Actually, I think the
first four problems.
You don't need any of this.
The problems are not due until
actually the next Wednesday.
Not this Wednesday,
but Wednesday the 18th,
nine days from today.
So you're in good shape with
regards to the homework.
