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GEORGE BARBASTATHIS: So today,
we'll start with a demo,

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00:00:25,550 --> 00:00:31,430
so Pepe will be on stage.

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00:00:31,430 --> 00:00:35,357
And then we'll continue with
the lectures on the gratings.

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00:00:35,357 --> 00:00:36,940
Pepe, whenever you're
ready, you can--

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00:00:36,940 --> 00:00:39,140
JOSE DOMINGUEZ-CABALLERO: OK.

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Can we-- so today,
the goal today

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is to show you a couple of
very interesting experiments.

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00:00:46,520 --> 00:00:48,230
And again, the idea
of this is for you

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00:00:48,230 --> 00:00:53,600
to see all these derivations
that we've been doing actually

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00:00:53,600 --> 00:00:55,730
applied in a real experiment
and see the results

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00:00:55,730 --> 00:00:56,690
that you would expect.

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00:00:56,690 --> 00:00:59,450
And for instance, these
last [? pieces, ?]

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00:00:59,450 --> 00:01:02,960
you were supposed to compute the
interference between a couple--

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00:01:02,960 --> 00:01:05,570
two plane waves propagating
at a different angle,

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00:01:05,570 --> 00:01:08,730
so you were supposed
to also calculate

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00:01:08,730 --> 00:01:09,950
analytical solution of that.

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00:01:09,950 --> 00:01:13,140
Now, we're going to see the
real evidence of that solution.

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00:01:13,140 --> 00:01:15,980
So for that, we actually built,
in this case, a little more

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00:01:15,980 --> 00:01:19,490
elaborate setup than the ones
that we've been showing before.

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00:01:19,490 --> 00:01:22,500
This actually a
Mach-Zehnder interferometer

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00:01:22,500 --> 00:01:26,210
that Professor Sheppard
described last time,

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and I will explain
it very briefly.

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00:01:28,640 --> 00:01:31,770
Can we zoom out, please?

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00:01:31,770 --> 00:01:34,270
Controls?

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00:01:34,270 --> 00:01:36,620
I guess maybe
that's the maximum.

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00:01:36,620 --> 00:01:38,440
All right, so we start
here with a laser.

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00:01:41,553 --> 00:01:42,970
So the laser comes
from this side,

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00:01:42,970 --> 00:01:46,970
and up to this section
here is where we just

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00:01:46,970 --> 00:01:49,960
collimate the
light, so basically,

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00:01:49,960 --> 00:01:52,780
what I'm going to focus right
now is on this other section.

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00:01:52,780 --> 00:01:56,020
So after this part here,
we have a nice plane wave.

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00:01:56,020 --> 00:01:57,765
This component
here is a polarizer

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00:01:57,765 --> 00:02:00,340
that, right now, just
take it as an element that

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00:02:00,340 --> 00:02:03,490
allows me to control
the intensity such

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that the CCD doesn't
receive too much light.

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00:02:07,090 --> 00:02:12,287
Now, these two pieces of glass
here are the beam splitters.

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00:02:12,287 --> 00:02:14,620
So we've seen that these beam
splitters-- basically what

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00:02:14,620 --> 00:02:17,980
they do is that they will
split a section of the beam

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00:02:17,980 --> 00:02:20,740
to one side and a section of
the beam to the other side.

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00:02:20,740 --> 00:02:24,370
In this case, it's
50/50 percent.

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00:02:24,370 --> 00:02:26,170
Now, then, we have two mirrors.

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So as you can see,
there's one path here,

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and the other path here.

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00:02:31,180 --> 00:02:34,240
The two paths that
now are going to be

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combined by this second
beam splitter here.

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Sometimes, this is
called a beam combiner.

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And then they co-propagate here,
pass through this iris here,

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00:02:44,470 --> 00:02:46,720
and then in this case,
they're reflecting-- again,

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00:02:46,720 --> 00:02:50,620
these mirror are
going to the CCD.

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So essentially, what we have
here is these two plane waves

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that first are splitted and
then are combined back again

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together.

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00:02:58,640 --> 00:03:01,660
And if the two plane waves
are exactly in line--

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we see that we're not
supposed to see fringes

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because they are
exactly in line,

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but we learned also that if we
introduce a little angle here,

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now the planar wavefronts of
each plane wave will interfere,

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and then we're going to see
some sinusoidal pattern.

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So this is exactly the same
as in the [? pieces. ?] One

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of the problems that had what
would be the interference

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pattern in an xy
plane in this case.

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In an xy plane,
[INAUDIBLE] two plane waves

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propagating with a slightly
different k vectors.

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So now, let's look at
the interference pattern.

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Now, this is here in
my computer screen.

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So this is a video, real
time, of this interferometer.

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So as you can see, you
get very nice fringes

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of very bright and
very dark, so we

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know that the
peaks are the proof

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of constructive interference,
the null destructive

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interference.

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And you also see
some vibration here,

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because we said that
these type of instruments

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are very, very, very sensitive
to environmental conditions,

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and I'm actually going
to tap the table.

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Even if I just blow
some air, but I

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want to get some moisture
in there, in the optics,

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but you can see
variations of the fringes.

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So then these type
of interferometers

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are used for
different measurements

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to calculate, for
instance, phases or measure

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different profiles of
reflective optics, et cetera.

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So there are many
applications of

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these Mach-Zehnder
interferometers,

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and the reason being
the sensitivity.

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Now, we also talked
about another thing--

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the period of this sinusoidal
pattern that we see

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is proportional to the
angle of the plane waves.

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So in this case, I'm going to
move one of the mirrors-- say,

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this mirror here--

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to change one of the beam paths.

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The other one is
going to remain fixed.

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So let me tilt it a little bit
and see what happens first.

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So we see that the period of
the plane wave is changing.

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In this case, I'm moving
it in in the y direction.

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00:05:05,020 --> 00:05:07,530
So now the period is larger.

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00:05:07,530 --> 00:05:10,980
Does anyone want to tell me why
do you think it's happening?

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00:05:10,980 --> 00:05:12,870
Are the plane waves
more close to be

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parallel, or further away?

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00:05:30,020 --> 00:05:31,548
Yep?

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STUDENT: Appears inversely
proportional to theta,

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00:05:33,590 --> 00:05:36,440
so your period's smaller data.

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00:05:36,440 --> 00:05:38,710
JOSE DOMINGUEZ-CABALLERO:
Yes, exactly.

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00:05:38,710 --> 00:05:41,080
So essentially,
here, by doing this,

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I'm making the plane waves
to be as in line as I can.

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And of course, since I'm
increasing the period,

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also the oscillations
will start increasing,

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and now, we get these
artifacts due to the frame

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rate of the camera that are
going to be very noticeable,

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I suppose, in Singapore.

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That's why I started
with a small fringe.

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I can also change the
tilting of the fringe

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to do nice things like
this by basically changing

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the orientation of the mirrors.

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All right, so now
what I'm going to do

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is I'm going to block
one of these beam paths.

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So we have a nice
plane wave here,

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and now, we are going
to repeat the aperture,

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this lead experiment
that I showed last time.

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But today, we're going to
do it in different L regime.

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So basically, we're going
to try to mimic physically

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the experiment that
Professor Barbastathis showed

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00:06:35,090 --> 00:06:38,270
in the video that he showed
in, I think, last lecture.

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All right, so we
have this slate here.

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Let me get it down a little bit.

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And I need to control
the intensity,

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so I'm going to move
this closer to the CCD.

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This is as close as I can get.

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So as you can see, there
is a plane wave hitting--

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let me just reduce the exposure.

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There is a plane wave
hitting this slate,

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and you can see the
diffraction here happening

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in the discontinuity
when basically it

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hits the metal piece.

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And I can tune the size
here, and of course,

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if I make this larger,
the diffraction effects

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appear less noticeable.

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But for a smaller slate--

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let's do that-- it's
actually pretty nice.

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We see the nice
diffraction, but I'm

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going to increase
intensity, so you can see

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the fringes that happen there.

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Now, let's see what
happens if I have it--

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I fixed position, so
the size of this slate,

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and now I'm going to move it
along the optical axis back.

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So I'm propagating back.

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Let me just reduce the intensity
here, so we are somewhere here.

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So now, I'm going
to go back, and you

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00:08:22,700 --> 00:08:25,310
can see how the diffraction
pattern changes.

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And this is
essentially the video.

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GEORGE BARBASTATHIS:
Can you hear me?

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JOSE DOMINGUEZ-CABALLERO: Yes.

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GEORGE BARBASTATHIS:
Can you go back

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where the fringe in
the center was dark?

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Yeah, something like that.

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This is the equivalent of
the Poisson blinking spot

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that we mentioned.

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Go a little bit
closer to the camera.

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JOSE DOMINGUEZ-CABALLERO:
Go a little bit what?

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00:08:44,700 --> 00:08:46,060
I'm sorry.

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00:08:46,060 --> 00:08:48,352
GEORGE BARBASTATHIS: A little
bit closer to the camera.

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Yeah.

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Try to have the center dark.

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JOSE DOMINGUEZ-CABALLERO: Oh,
I'm going to show the Poisson

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00:08:56,910 --> 00:08:58,702
spot with a circular
aperture a bit later--

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GEORGE BARBASTATHIS:
But I want to make

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00:09:00,327 --> 00:09:01,420
a point about this case.

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00:09:01,420 --> 00:09:04,190
In this case, you can see,
again, the Poisson spot,

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00:09:04,190 --> 00:09:08,010
but it's actually an entire
dark line in the center.

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00:09:08,010 --> 00:09:10,830
That's because the
scattering of the two edges

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is actually out of phase,
so that the interference

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00:09:13,740 --> 00:09:16,872
destructively at the center,
and you see an entire dark band.

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00:09:16,872 --> 00:09:19,330
And then of course, in this
case, with a circular aperture,

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00:09:19,330 --> 00:09:21,732
you see the Poisson
spot as a spot.

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00:09:21,732 --> 00:09:23,190
In this case, it
is a vertical bar.

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00:09:23,190 --> 00:09:23,580
JOSE DOMINGUEZ-CABALLERO:
There you go.

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GEORGE BARBASTATHIS:
Yeah, that's perfect.

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00:09:24,612 --> 00:09:25,710
Perfect, yeah.

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00:09:25,710 --> 00:09:28,043
JOSE DOMINGUEZ-CABALLERO: So
here, you see the blinking.

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So this is dark, and
this would be bright.

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This would be the
blinking line, I suppose.

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00:09:38,430 --> 00:09:44,670
Now, let me change it to
the other video that we saw,

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00:09:44,670 --> 00:09:49,681
that was the circular aperture.

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So this is a circular
aperture of a fixed size.

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In this case, you can see,
when it's close to the camera,

193
00:10:00,220 --> 00:10:03,340
again, as close as
I can get it to,

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00:10:03,340 --> 00:10:05,975
you see refraction patterns.

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00:10:05,975 --> 00:10:07,600
So now, this is very
nice, because what

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00:10:07,600 --> 00:10:09,017
we're supposed to
see here, again,

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is going to be that famous
Poisson spot or blinking spot.

198
00:10:13,382 --> 00:10:15,570
And before we see the
results that we, again,

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00:10:15,570 --> 00:10:20,340
saw in the movie, there's a nice
historical story about these.

200
00:10:20,340 --> 00:10:27,270
In [INAUDIBLE],, Fresnel spent
a very nice paper to a contest

201
00:10:27,270 --> 00:10:28,812
about the refraction of light.

202
00:10:28,812 --> 00:10:30,270
So at the time,
remember that there

203
00:10:30,270 --> 00:10:34,680
was all this controversy of
is light a wave or a particle.

204
00:10:34,680 --> 00:10:38,610
So in the committee
judging these papers,

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00:10:38,610 --> 00:10:40,560
Poisson, a very
respected scientist,

206
00:10:40,560 --> 00:10:42,360
was actually in the committee.

207
00:10:42,360 --> 00:10:44,310
And he saw Fresnel's
paper, and he was

208
00:10:44,310 --> 00:10:45,630
like, no, this has to be wrong.

209
00:10:45,630 --> 00:10:48,400
And he wanted to disprove it,
so he actually went and did

210
00:10:48,400 --> 00:10:51,300
some calculations
and then proved

211
00:10:51,300 --> 00:10:56,370
that, if Fresnel was
right, what this means

212
00:10:56,370 --> 00:11:00,450
is that if you have, say, an
opening like this, somewhere

213
00:11:00,450 --> 00:11:03,510
behind that opening, after
shining it with a plane wave,

214
00:11:03,510 --> 00:11:06,080
you're supposed to
see a dark spot.

215
00:11:06,080 --> 00:11:07,980
And that dark spot is
very counterintuitive

216
00:11:07,980 --> 00:11:09,775
because, basically,
you're supposed to see,

217
00:11:09,775 --> 00:11:12,150
from the geometrical optics,
all the light going straight

218
00:11:12,150 --> 00:11:12,650
through.

219
00:11:12,650 --> 00:11:17,850
So how could there be a very
dark spot in the center?

220
00:11:17,850 --> 00:11:20,220
So then basically, he
said, no, this is the proof

221
00:11:20,220 --> 00:11:23,670
that Fresnel's theory is wrong.

222
00:11:23,670 --> 00:11:27,540
But then another member in the
committee, Dominique Arago,

223
00:11:27,540 --> 00:11:30,300
he actually went ahead
and did the experiment,

224
00:11:30,300 --> 00:11:33,450
and he saw this Poisson
spot experimentally.

225
00:11:33,450 --> 00:11:35,070
And then thanks to
that, of course,

226
00:11:35,070 --> 00:11:36,780
Fresnel won the contest.

227
00:11:36,780 --> 00:11:39,480
And now this, I guess, as
opposed to-- like as occurs

228
00:11:39,480 --> 00:11:40,380
to Poisson--

229
00:11:40,380 --> 00:11:43,100
it's called a Poisson spot.

230
00:11:43,100 --> 00:11:46,440
All right, let's see the
Poisson spot in action.

231
00:11:46,440 --> 00:11:51,400
So I'm going to go-- here,
you see it in bright, dark.

232
00:11:51,400 --> 00:11:53,790
I'm going to go
even further away.

233
00:11:56,430 --> 00:11:59,950
So I'm moving it back, and
you see the blinking spot.

234
00:11:59,950 --> 00:12:00,925
Trying to do it slowly.

235
00:12:04,100 --> 00:12:04,970
Going to go back.

236
00:12:04,970 --> 00:12:06,304
Can you see it in Singapore?

237
00:12:11,512 --> 00:12:12,470
I guess that was a yes.

238
00:12:17,160 --> 00:12:18,742
Cool.

239
00:12:18,742 --> 00:12:20,950
GEORGE BARBASTATHIS: They're
very beautiful actually.

240
00:12:20,950 --> 00:12:22,992
JOSE DOMINGUEZ-CABALLERO:
Yes, they're very nice.

241
00:12:25,620 --> 00:12:29,100
So can we switch to the
front camera, please?

242
00:12:35,370 --> 00:12:37,600
Control?

243
00:12:37,600 --> 00:12:39,490
Can we switch to
the front camera?

244
00:12:39,490 --> 00:12:42,220
So now, the next thing
that I'm going to show--

245
00:12:42,220 --> 00:12:43,790
it's a diffraction grating.

246
00:12:43,790 --> 00:12:46,300
So diffraction gratings--
it's a very interesting

247
00:12:46,300 --> 00:12:47,952
optical component.

248
00:12:47,952 --> 00:12:48,910
The front view, please.

249
00:12:48,910 --> 00:12:50,993
I'm going to show something
with a piece of paper.

250
00:12:53,777 --> 00:12:55,790
Or I guess if not, I'm
going to show it here.

251
00:12:55,790 --> 00:12:56,788
STUDENT: [INAUDIBLE]

252
00:12:56,788 --> 00:12:58,288
JOSE DOMINGUEZ-CABALLERO:
I'm sorry?

253
00:12:58,288 --> 00:13:00,560
STUDENT: [INAUDIBLE]
explained [INAUDIBLE]..

254
00:13:00,560 --> 00:13:01,768
JOSE DOMINGUEZ-CABALLERO: OK.

255
00:13:03,063 --> 00:13:04,730
So the next thing
that I'm going to show

256
00:13:04,730 --> 00:13:07,820
is an optical component,
that is very used

257
00:13:07,820 --> 00:13:11,000
in practicing different
optical setups, that

258
00:13:11,000 --> 00:13:12,740
is called a grating.

259
00:13:12,740 --> 00:13:15,410
So grating-- you can think of
it as a material, in this case,

260
00:13:15,410 --> 00:13:16,700
that--

261
00:13:16,700 --> 00:13:19,130
the one that I'm going to show
is called a phase grating.

262
00:13:19,130 --> 00:13:22,520
Basically, the index or
refraction within the material

263
00:13:22,520 --> 00:13:26,470
varies in a given way-- in
this case, sinusoidally.

264
00:13:26,470 --> 00:13:29,500
So it's a periodic variation
of the index of refraction,

265
00:13:29,500 --> 00:13:33,790
and then when the light
hits this element, looks

266
00:13:33,790 --> 00:13:37,220
like a flat piece of
glass, but in reality, it

267
00:13:37,220 --> 00:13:39,070
has this index
varying, it basically

268
00:13:39,070 --> 00:13:41,500
decomposes in multiple angles.

269
00:13:41,500 --> 00:13:44,060
And what I'm going to
show here, in this case,

270
00:13:44,060 --> 00:13:48,430
is a normal laser
pointer, but it actually

271
00:13:48,430 --> 00:13:54,130
has two set of gratings
in this little end.

272
00:13:54,130 --> 00:13:55,900
So what we are going
to see, again, now,

273
00:13:55,900 --> 00:13:58,478
is we're going to see some
beautiful array of patterns.

274
00:13:58,478 --> 00:13:59,020
You see them?

275
00:13:59,020 --> 00:14:01,650
And actually, I'm going
to project them here.

276
00:14:01,650 --> 00:14:03,470
This, I'm not sure
if the Singapore

277
00:14:03,470 --> 00:14:07,070
side is going to see it or not.

278
00:14:07,070 --> 00:14:09,075
So now, I have to
cross linear gratings.

279
00:14:11,780 --> 00:14:13,700
I actually can put
them in things,

280
00:14:13,700 --> 00:14:16,800
so I can see very nice spots.

281
00:14:16,800 --> 00:14:23,890
I can open them, and you can do
all sorts of fun things here.

282
00:14:23,890 --> 00:14:25,650
So the optical
component actually

283
00:14:25,650 --> 00:14:28,000
looks something like this.

284
00:14:28,000 --> 00:14:30,360
This is the second time
that I showed a grating.

285
00:14:30,360 --> 00:14:32,130
The first time was in
the very first demo,

286
00:14:32,130 --> 00:14:34,800
when I used the grating
to disperse light,

287
00:14:34,800 --> 00:14:35,970
similar to a prism.

288
00:14:35,970 --> 00:14:39,900
So if you remember, I
had a prism dispersing

289
00:14:39,900 --> 00:14:41,580
in what is called a
normal dispersion,

290
00:14:41,580 --> 00:14:42,840
and a grating with
what is called

291
00:14:42,840 --> 00:14:43,882
the anomalous dispersion.

292
00:14:43,882 --> 00:14:45,930
So this also, if you
shine it with white light,

293
00:14:45,930 --> 00:14:48,210
will form a very,
very nice rainbow.

294
00:14:48,210 --> 00:14:52,675
So in this case, it's a bit hard
to show this guy with a camera,

295
00:14:52,675 --> 00:14:55,050
but I'm going to try, if we
can put-- the overhead camear

296
00:14:55,050 --> 00:14:56,710
I think is on.

297
00:14:56,710 --> 00:14:59,790
So what I'm going to do is
going to increase the intensity.

298
00:15:03,350 --> 00:15:08,470
And use the aperture,
and I'm going

299
00:15:08,470 --> 00:15:12,880
to try to show it in
a piece of paper here.

300
00:15:12,880 --> 00:15:14,760
So you see here,
the three spots,

301
00:15:14,760 --> 00:15:17,660
in this case, of
this element being,

302
00:15:17,660 --> 00:15:19,530
in this case, more orders.

303
00:15:19,530 --> 00:15:23,820
So each of these spot we
call a diffraction order.

304
00:15:23,820 --> 00:15:26,430
The one that goes straight
through would be the 0 order.

305
00:15:26,430 --> 00:15:29,640
Then we go to plus/minus 1,
plus/minus 2, so on and so

306
00:15:29,640 --> 00:15:32,160
forth, and I guess
after these, we're

307
00:15:32,160 --> 00:15:34,320
going to do the
actual derivation,

308
00:15:34,320 --> 00:15:36,750
the mathematical
derivation of this.

309
00:15:36,750 --> 00:15:40,157
You want to add something else,
George, for the grating side?

310
00:15:40,157 --> 00:15:41,740
GEORGE BARBASTATHIS:
I think this only

311
00:15:41,740 --> 00:15:46,990
had the plus/minus 1,
plus/minus 3, and a very weak 0?

312
00:15:46,990 --> 00:15:48,675
JOSE DOMINGUEZ-CABALLERO:
Let's see.

313
00:15:48,675 --> 00:15:50,800
GEORGE BARBASTATHIS: Because
I only saw four spots,

314
00:15:50,800 --> 00:15:51,720
so it must have been--

315
00:15:51,720 --> 00:15:54,050
JOSE DOMINGUEZ-CABALLERO:
There's more, I think.

316
00:15:54,050 --> 00:15:55,852
GEORGE BARBASTATHIS: Oh, OK.

317
00:15:55,852 --> 00:15:56,560
Not very visible.

318
00:15:56,560 --> 00:15:57,518
Oh, yeah, that's right.

319
00:15:57,518 --> 00:15:58,320
There's more.

320
00:15:58,320 --> 00:15:59,700
JOSE DOMINGUEZ-CABALLERO: Yeah.

321
00:15:59,700 --> 00:16:01,325
GEORGE BARBASTATHIS:
0 is also present,

322
00:16:01,325 --> 00:16:03,835
but you can see that the
plus/minus 2, plus/minus 4

323
00:16:03,835 --> 00:16:04,560
are missing.

324
00:16:04,560 --> 00:16:06,360
We'll talk about
this in class later.

325
00:16:06,360 --> 00:16:09,810
The point to see here is that
you see the pattern is not

326
00:16:09,810 --> 00:16:10,560
quite regular.

327
00:16:10,560 --> 00:16:12,900
It looks like there
is one missing

328
00:16:12,900 --> 00:16:16,020
order between the last--

329
00:16:18,640 --> 00:16:22,540
if you label the central
one 0, then it goes 1,

330
00:16:22,540 --> 00:16:25,010
and then it keeps 2,
and then it goes to 3.

331
00:16:25,010 --> 00:16:28,038
So that's the point
I wanted to make.

332
00:16:28,038 --> 00:16:29,330
JOSE DOMINGUEZ-CABALLERO: Good.

333
00:16:33,620 --> 00:16:35,450
So I think we switch
back to you, George.

334
00:16:38,498 --> 00:16:40,040
GEORGE BARBASTATHIS:
Thank you, Pepe.

335
00:16:40,040 --> 00:16:41,748
JOSE DOMINGUEZ-CABALLERO:
You're welcome.

336
00:16:42,057 --> 00:16:43,890
GEORGE BARBASTATHIS: I
feel jealous actually

337
00:16:43,890 --> 00:16:47,280
because this is much better
to see it in real life

338
00:16:47,280 --> 00:16:48,780
than it is to see it in math.

339
00:16:48,780 --> 00:16:53,250
But hopefully,
this will motivate

340
00:16:53,250 --> 00:16:57,760
you to tolerate
the math, because

341
00:16:57,760 --> 00:17:00,240
if the physical phenomena
are so pretty, maybe

342
00:17:00,240 --> 00:17:05,660
it is worthwhile to go to the
effort of understanding them,

343
00:17:05,660 --> 00:17:06,240
I suppose.

344
00:17:20,910 --> 00:17:23,050
So today, I will talk--

345
00:17:23,050 --> 00:17:26,460
I will basically pick
up where Pepe left,

346
00:17:26,460 --> 00:17:30,000
and I will talk about
diffraction gratings.

347
00:17:46,750 --> 00:17:49,430
Oh, before I forget,
some of you asked if you

348
00:17:49,430 --> 00:17:52,410
could have the movies from--

349
00:17:52,410 --> 00:17:56,730
the animations that they
played in class the other day.

350
00:17:56,730 --> 00:17:59,720
So if you go to the website,
I have created a new section

351
00:17:59,720 --> 00:18:01,310
called Animations.

352
00:18:01,310 --> 00:18:03,620
And I've put the
movies as AVI files,

353
00:18:03,620 --> 00:18:06,290
so you can download
them from there.

354
00:18:06,290 --> 00:18:09,770
And also I have
revised the notes.

355
00:18:09,770 --> 00:18:14,180
Last time, if you recall, we
caught a couple of factors of 2

356
00:18:14,180 --> 00:18:16,170
that were missing, and
so on and so forth.

357
00:18:16,170 --> 00:18:18,300
So I fixed those.

358
00:18:18,300 --> 00:18:20,790
If you find any more errors
in the notes, please let me

359
00:18:20,790 --> 00:18:22,290
know so that I can
fix them as well.

360
00:18:29,090 --> 00:18:31,100
So today, we'll talk about--

361
00:18:31,100 --> 00:18:33,890
so basically, I will
describe, mathematically,

362
00:18:33,890 --> 00:18:37,820
what Pepe showed in real life.

363
00:18:37,820 --> 00:18:41,340
So a grating basically,
as Pepe mentioned,

364
00:18:41,340 --> 00:18:45,260
is a periodic,
thin transparency.

365
00:18:45,260 --> 00:18:47,390
So we've talked about thin
transparencies before.

366
00:18:47,390 --> 00:18:50,360
We mentioned that they
modulate in general.

367
00:18:50,360 --> 00:18:52,840
Thin transparencies
modulate the amplitude

368
00:18:52,840 --> 00:18:56,840
and the phase of the
optical field, so

369
00:18:56,840 --> 00:18:59,690
in that particular case,
when the thin transparency's

370
00:18:59,690 --> 00:19:03,950
a periodic function of space,
then we'll call it a grating.

371
00:19:03,950 --> 00:19:08,930
And I guess for clarity
of presentation,

372
00:19:08,930 --> 00:19:13,970
it is customary to classify
gratings as amplitude or phase,

373
00:19:13,970 --> 00:19:17,120
depending on whether
the grating acts

374
00:19:17,120 --> 00:19:21,740
upon the magnitude of
the electromagnetic field

375
00:19:21,740 --> 00:19:24,650
or the phase of the
electromagnetic field.

376
00:19:24,650 --> 00:19:26,750
Now, this can cause a
little bit of confusion

377
00:19:26,750 --> 00:19:30,080
because we have
agreed to use the term

378
00:19:30,080 --> 00:19:34,610
amplitude to denote the complex
amplitude of the optical field.

379
00:19:34,610 --> 00:19:38,270
Confusingly enough, when people
talk about amplitude gratings,

380
00:19:38,270 --> 00:19:43,730
they should be calling
them magnitude gratings,

381
00:19:43,730 --> 00:19:47,060
because as you can see
in the case on the left,

382
00:19:47,060 --> 00:19:51,260
the transmission function
of an amplitude grating

383
00:19:51,260 --> 00:19:52,893
is a positive real number.

384
00:19:52,893 --> 00:19:54,310
So therefore, this
kind of grating

385
00:19:54,310 --> 00:19:59,990
acts directly onto the
magnitude of the optical field.

386
00:19:59,990 --> 00:20:03,530
Nevertheless, for
historical reasons,

387
00:20:03,530 --> 00:20:05,030
people call these are gratings--

388
00:20:05,030 --> 00:20:08,690
I suppose they call
them magnitude--

389
00:20:08,690 --> 00:20:10,880
I'm sorry, they call
them amplitude gratings.

390
00:20:10,880 --> 00:20:13,700
So hopefully, this will
not create confusion.

391
00:20:13,700 --> 00:20:16,250
But anyway, the simplest
possible modulation

392
00:20:16,250 --> 00:20:18,890
is sinusoidal, and
you can see here

393
00:20:18,890 --> 00:20:22,190
sort of juxtaposed in
a sinusoidal amplitude

394
00:20:22,190 --> 00:20:25,760
grating and the
sinusoidal phase grating.

395
00:20:25,760 --> 00:20:28,940
So starting with the
amplitude grating,

396
00:20:28,940 --> 00:20:34,290
the complex transmission
function of the grating

397
00:20:34,290 --> 00:20:35,860
is actually not complex at all.

398
00:20:35,860 --> 00:20:39,210
It is a real positive
number, and what I've done

399
00:20:39,210 --> 00:20:42,460
is, on top of it, I've
plot its magnitude,

400
00:20:42,460 --> 00:20:44,760
which is, of course,
itself since this

401
00:20:44,760 --> 00:20:48,570
is a real positive
function, and its phase.

402
00:20:48,570 --> 00:20:51,900
And of course, the phase
is 0 because the phase

403
00:20:51,900 --> 00:20:53,640
of a complex number,
which happens

404
00:20:53,640 --> 00:20:58,260
to be real and positive, is 0.

405
00:20:58,260 --> 00:21:02,490
The quantities to mention here
are the period of the grating.

406
00:21:02,490 --> 00:21:07,470
We use the upper case
Greek symbol lambda, which

407
00:21:07,470 --> 00:21:12,150
looks like a hat, I guess.

408
00:21:12,150 --> 00:21:14,710
That symbol is called lambda.

409
00:21:14,710 --> 00:21:17,420
So this is the period of the
grating, so in this case,

410
00:21:17,420 --> 00:21:21,180
you can see the grating
seems to fulfill

411
00:21:21,180 --> 00:21:24,360
one undulation
between approximately

412
00:21:24,360 --> 00:21:27,660
minus 5 units and plus 5 units.

413
00:21:27,660 --> 00:21:31,460
Therefore, the period
here is 10 units.

414
00:21:31,460 --> 00:21:34,980
And actually, the unit I've
picked on the horizontal axis

415
00:21:34,980 --> 00:21:38,640
is the wavelength of the
light, so this grating

416
00:21:38,640 --> 00:21:41,700
has a period of 10 wavelengths.

417
00:21:41,700 --> 00:21:44,760
So if the light were a visible--

418
00:21:44,760 --> 00:21:48,390
green light has a wavelength
of approximately half

419
00:21:48,390 --> 00:21:53,760
a micrometer, so in real
units for the green light,

420
00:21:53,760 --> 00:21:58,530
the period here would have been
approximately 5 micrometers

421
00:21:58,530 --> 00:22:02,210
or 10 wavelengths.

422
00:22:02,210 --> 00:22:05,450
Then m-- we saw
the symbol m also

423
00:22:05,450 --> 00:22:07,670
when we talk about interference.

424
00:22:07,670 --> 00:22:14,520
We called m the contrast
between the brightest

425
00:22:14,520 --> 00:22:18,510
and the darkest part of a fringe
in the case of an interference

426
00:22:18,510 --> 00:22:19,440
pattern.

427
00:22:19,440 --> 00:22:21,420
In this case, we still
call it contrast,

428
00:22:21,420 --> 00:22:26,580
but it is between the
most transmissive part

429
00:22:26,580 --> 00:22:30,330
of the grating and the
darkest part of the grating.

430
00:22:30,330 --> 00:22:33,930
So this case, you can see that
it goes between approximately

431
00:22:33,930 --> 00:22:36,990
0.9 and 0.1.

432
00:22:36,990 --> 00:22:39,180
The maximum
transmissivity is 0.9,

433
00:22:39,180 --> 00:22:41,850
and the minimum
transmissivity is 0.1.

434
00:22:41,850 --> 00:22:45,570
So therefore, of the value of
m for the grating shown here

435
00:22:45,570 --> 00:22:48,420
is approximately 0.8.

436
00:22:48,420 --> 00:22:52,620
If the grating was swinging
between maximum 100%

437
00:22:52,620 --> 00:22:57,730
transmission, totally
bright and totally dark,

438
00:22:57,730 --> 00:22:59,100
that would have been 0.

439
00:22:59,100 --> 00:23:00,810
Then the contrast
would've been 1.

440
00:23:00,810 --> 00:23:05,380
So this is the function
of this m parameter here.

441
00:23:05,380 --> 00:23:08,670
And finally, we allow a phase
shift, which basically says,

442
00:23:08,670 --> 00:23:10,290
if the maximum--

443
00:23:10,290 --> 00:23:13,560
since the grating is written
as a cosine function here,

444
00:23:13,560 --> 00:23:18,030
phi tells me if the maximum
transmission, the maximum value

445
00:23:18,030 --> 00:23:21,930
of the cosine, is
centered with x equals 0

446
00:23:21,930 --> 00:23:23,380
on the horizontal axis--

447
00:23:23,380 --> 00:23:25,950
so you can see that, in this
case, it is indeed center.

448
00:23:25,950 --> 00:23:29,310
So therefore, the
phase phi would equal 0

449
00:23:29,310 --> 00:23:33,440
in the grating
shown in this case.

450
00:23:33,440 --> 00:23:35,970
So that's what I had
to say about that.

451
00:23:35,970 --> 00:23:38,830
Under the grating, I should
go to the phase grating.

452
00:23:38,830 --> 00:23:40,660
It is actually kind
of the opposite.

453
00:23:40,660 --> 00:23:42,490
So the phase
grating is expressed

454
00:23:42,490 --> 00:23:46,030
as a complex exponential
with a purely--

455
00:23:46,030 --> 00:23:52,790
with a pure phase without
any amplitude modulation.

456
00:23:52,790 --> 00:23:55,090
So it's magnitude then is 1.

457
00:23:55,090 --> 00:23:59,090
So you see that the
magnitude is a flat function,

458
00:23:59,090 --> 00:24:04,330
so this means that this grating
transmits all of the light.

459
00:24:04,330 --> 00:24:08,480
What it does do, though,
is it modulates the phase,

460
00:24:08,480 --> 00:24:10,450
and you can see that
it modulates the phase

461
00:24:10,450 --> 00:24:15,322
in a sinusoidal fashion.

462
00:24:15,322 --> 00:24:17,530
So this is what you see now
in the bottom plot, which

463
00:24:17,530 --> 00:24:20,260
is that the phase, the angle--

464
00:24:20,260 --> 00:24:22,390
by the way, the
phase, sometimes we

465
00:24:22,390 --> 00:24:28,850
denote the phase as this angular
symbol of a complex number.

466
00:24:28,850 --> 00:24:32,680
So the phase is simply the
argument of the exponent here,

467
00:24:32,680 --> 00:24:35,630
and is, itself, sinusoidal.

468
00:24:35,630 --> 00:24:39,320
And it also has a
period and a contrast.

469
00:24:39,320 --> 00:24:41,570
The contrast in this case
is called the phase contrast

470
00:24:41,570 --> 00:24:43,762
because we're talking
about the phase,

471
00:24:43,762 --> 00:24:45,220
and you can see
that, in this case,

472
00:24:45,220 --> 00:24:49,270
it is swinging also
approximately between plus 0.4

473
00:24:49,270 --> 00:24:50,590
and minus 0.4.

474
00:24:50,590 --> 00:24:54,280
So the contrast is
also 0.8 in this case,

475
00:24:54,280 --> 00:24:58,830
but it could be anything,
really, between minus pi and pi

476
00:24:58,830 --> 00:25:04,010
to get a complete
phase contrast.

477
00:25:04,010 --> 00:25:05,740
The other two
parameters are the same.

478
00:25:05,740 --> 00:25:08,740
The period is also
about 10 wavelengths,

479
00:25:08,740 --> 00:25:12,370
and the phase shift is 0.

480
00:25:12,370 --> 00:25:14,230
We have to be a little
bit careful here not

481
00:25:14,230 --> 00:25:19,250
to confuse the phase of the
grating with the phase shift.

482
00:25:19,250 --> 00:25:21,580
So again, the phase
shift is telling us

483
00:25:21,580 --> 00:25:27,040
whether that transmission
is aligned or no

484
00:25:27,040 --> 00:25:28,900
with the origin of
the coordinates.

485
00:25:28,900 --> 00:25:31,150
So this case,
since I have a sine

486
00:25:31,150 --> 00:25:32,650
in the phase of the
grating, you see

487
00:25:32,650 --> 00:25:36,190
that it assumes the
value 0 at x equals 0,

488
00:25:36,190 --> 00:25:38,200
so again, there's
no phase shift.

489
00:25:38,200 --> 00:25:42,100
So this phase shift is not to
be confused with the phase delay

490
00:25:42,100 --> 00:25:45,910
that the grating imposes
upon the complex amplitude

491
00:25:45,910 --> 00:25:47,480
of the electromagnetic field.

492
00:25:47,480 --> 00:25:48,920
So that's another
important point,

493
00:25:48,920 --> 00:25:52,080
and I would like
to point it out.

494
00:25:52,080 --> 00:25:54,980
So how do these
things behave now?

495
00:25:54,980 --> 00:25:59,060
So what I will do is I
will play two movies, which

496
00:25:59,060 --> 00:26:01,610
I guess are not as
convincing as the movies

497
00:26:01,610 --> 00:26:04,750
that Pepe showed because
Pepe is doing an experiment.

498
00:26:04,750 --> 00:26:08,240
In this case, these are
simply Matlab calculations.

499
00:26:08,240 --> 00:26:13,050
But basically, the two movies
will show you two great things.

500
00:26:13,050 --> 00:26:15,680
One is an amplitude grating,
and one is a phase grating.

501
00:26:15,680 --> 00:26:21,310
And it will show you how the
field propagates after grating.

502
00:26:21,310 --> 00:26:23,990
And these movies are also
posted to the website

503
00:26:23,990 --> 00:26:26,790
so you can play them
yourselves again later.

504
00:26:26,790 --> 00:26:32,660
Now, I should say, the grating
grooves are oriented horizontal

505
00:26:32,660 --> 00:26:37,510
in this case, and you cannot
see them on the camera

506
00:26:37,510 --> 00:26:40,693
or on the video because the
pixel size is not big enough

507
00:26:40,693 --> 00:26:42,360
to allow you to see
them, unfortunately.

508
00:26:42,360 --> 00:26:45,410
This is an artifact
of the calculation.

509
00:26:45,410 --> 00:26:50,640
So what you see here
is some variation

510
00:26:50,640 --> 00:26:55,880
of the grating that is actually
not the grating itself.

511
00:26:55,880 --> 00:26:58,850
As I said it, is
called a moire pattern,

512
00:26:58,850 --> 00:27:01,150
if you are familiar
with the term.

513
00:27:01,150 --> 00:27:03,500
But anyway, I don't want to
go into moire theory here,

514
00:27:03,500 --> 00:27:05,900
but the reason you
can see this pattern

515
00:27:05,900 --> 00:27:10,160
is because the pixel size
of the camera in the display

516
00:27:10,160 --> 00:27:13,460
is not small enough to show
you the period of the grating.

517
00:27:13,460 --> 00:27:18,200
So the grating is actually
expressed by this function.

518
00:27:18,200 --> 00:27:22,610
X is the same in the
equation as in the axis,

519
00:27:22,610 --> 00:27:24,680
and the period of the
grating is 2 wavelengths.

520
00:27:24,680 --> 00:27:29,250
So this grating has a
very, very small period.

521
00:27:29,250 --> 00:27:31,530
The contrast I picked
to be 1 in this case,

522
00:27:31,530 --> 00:27:33,880
and there's no phase shift.

523
00:27:33,880 --> 00:27:37,030
So I will play the movie now,
and you will see what happens.

524
00:27:37,030 --> 00:27:40,350
So the field will
propagate along the z-axis,

525
00:27:40,350 --> 00:27:44,970
and you can see that the light
that originally illuminated

526
00:27:44,970 --> 00:27:48,660
the grating actually now
splits into three parts.

527
00:27:48,660 --> 00:27:52,290
These are, also Pepe mentioned,
called diffraction orders.

528
00:27:52,290 --> 00:27:54,780
One of them propagates
straight through.

529
00:27:54,780 --> 00:27:57,630
That is called the
0 diffraction order,

530
00:27:57,630 --> 00:28:03,120
and then you get a plus 1 above
and a minus 1 diffraction order

531
00:28:03,120 --> 00:28:04,140
below.

532
00:28:04,140 --> 00:28:09,000
So this is the typical behavior
of a sinusoidal amplitude

533
00:28:09,000 --> 00:28:10,350
grating.

534
00:28:10,350 --> 00:28:12,720
Now, if I play the same
thing for the first grating,

535
00:28:12,720 --> 00:28:14,170
there's two things to notice.

536
00:28:14,170 --> 00:28:16,480
First of all, there's
no moire at all here.

537
00:28:16,480 --> 00:28:18,330
That's because the
phase grating does not

538
00:28:18,330 --> 00:28:22,060
modulate the magnitude of
the electric field at all.

539
00:28:22,060 --> 00:28:24,600
So in other words,
at the entrance

540
00:28:24,600 --> 00:28:27,210
toward the grating
itself, the grating

541
00:28:27,210 --> 00:28:29,700
looks just like an
empty transparency.

542
00:28:29,700 --> 00:28:32,490
It has not changed the
magnitude of the field.

543
00:28:32,490 --> 00:28:34,890
What it has done, it
has changed the phase,

544
00:28:34,890 --> 00:28:36,930
and because the
phase of the field

545
00:28:36,930 --> 00:28:40,380
got modulated inside
this slit over here,

546
00:28:40,380 --> 00:28:42,780
what you will see when I
play the grating is, again,

547
00:28:42,780 --> 00:28:44,280
that it splits.

548
00:28:44,280 --> 00:28:45,940
But now, it splits
into more than one.

549
00:28:45,940 --> 00:28:49,620
It splits into, in this case,
you can see very clearly

550
00:28:49,620 --> 00:28:52,400
five diffraction orders.

551
00:28:52,400 --> 00:28:54,420
You can see very
clearly, there is

552
00:28:54,420 --> 00:28:59,850
one in the center, then
another one and another one.

553
00:28:59,850 --> 00:29:03,330
So these are the plus 1, plus
2, and then at the bottom,

554
00:29:03,330 --> 00:29:08,740
you have minus 1, minus
2 diffraction orders.

555
00:29:08,740 --> 00:29:13,810
So it is, in some ways,
similar to the amplitude case,

556
00:29:13,810 --> 00:29:18,090
but in this case, we'll get
more diffraction orders.

557
00:29:18,090 --> 00:29:21,240
And so I guess what I
would like to do now

558
00:29:21,240 --> 00:29:24,720
is I will try to explain
what is the difference,

559
00:29:24,720 --> 00:29:28,740
and why the phase grating gives
us more diffraction order.

560
00:29:28,740 --> 00:29:32,010
I should say, by the way,
that the sinusoidal--

561
00:29:32,010 --> 00:29:34,900
I want to emphasize the
sinusoidal amplitude

562
00:29:34,900 --> 00:29:37,610
grating gives us only three--

563
00:29:37,610 --> 00:29:40,005
the 0 order, the plus
1, and the minus 1.

564
00:29:40,005 --> 00:29:42,870
In general, other
types of gratings,

565
00:29:42,870 --> 00:29:46,290
for example binary or
phase gratings and so on,

566
00:29:46,290 --> 00:29:48,560
they give us more than
three diffraction orders.

567
00:29:52,980 --> 00:29:56,588
So first of all, let's look
at the physical picture.

568
00:29:56,588 --> 00:29:58,380
So I'll start with the
sinusoidal amplitude

569
00:29:58,380 --> 00:30:01,287
because this is the
simplest possible grating,

570
00:30:01,287 --> 00:30:03,870
and let's look at the physical
picture of what's going on here

571
00:30:03,870 --> 00:30:06,810
and why we get this
diffraction order.

572
00:30:06,810 --> 00:30:09,200
So if I have a plane
wave impinging-- oh,

573
00:30:09,200 --> 00:30:14,020
so again, this symbol, being
Greek, I have an advantage,

574
00:30:14,020 --> 00:30:14,520
I guess.

575
00:30:14,520 --> 00:30:16,020
I can recognize it.

576
00:30:16,020 --> 00:30:18,060
It is the uppercase
lambda, so we'll

577
00:30:18,060 --> 00:30:20,320
use the symbol lowercase--

578
00:30:20,320 --> 00:30:23,410
control, may I have the paper?

579
00:30:23,410 --> 00:30:25,180
Actually, I already
have the paper, do I?

580
00:30:25,180 --> 00:30:26,310
Yeah.

581
00:30:26,310 --> 00:30:31,040
So this symbol that we
use for the wavelength

582
00:30:31,040 --> 00:30:35,325
is the lowercase lambda.

583
00:30:38,520 --> 00:30:42,030
This symbol that we use
generally for a special period

584
00:30:42,030 --> 00:30:42,870
is the uppercase.

585
00:30:47,850 --> 00:30:49,350
It may sound a
little bit ridiculous

586
00:30:49,350 --> 00:30:53,870
that I emphasize it so much, but
believe me, in previous years,

587
00:30:53,870 --> 00:30:57,960
all the generations of students
have been endlessly confused

588
00:30:57,960 --> 00:30:59,185
by the lambda and the lambda.

589
00:30:59,185 --> 00:31:00,560
So I would like
to emphasize it--

590
00:31:00,560 --> 00:31:02,190
these are two different symbols.

591
00:31:02,190 --> 00:31:04,500
We use the lower case
for the wavelength

592
00:31:04,500 --> 00:31:08,370
and the upper case for
the spatial period.

593
00:31:08,370 --> 00:31:09,150
So that's it then.

594
00:31:09,150 --> 00:31:11,790
That's the period, and
the inverse of that--

595
00:31:11,790 --> 00:31:14,700
the inverse of a period
generally is a frequency.

596
00:31:14,700 --> 00:31:18,140
So since now we have a spatial
period, the inverse [? u0, ?]

597
00:31:18,140 --> 00:31:21,450
we'll call this
special frequency.

598
00:31:21,450 --> 00:31:23,700
And let's illuminate
this grating now

599
00:31:23,700 --> 00:31:26,040
with an incident plane wave.

600
00:31:26,040 --> 00:31:29,190
So throughout this lecture, I
will be illuminating gratings

601
00:31:29,190 --> 00:31:31,950
with plane waves on axis.

602
00:31:31,950 --> 00:31:34,740
In the homework that
I've posted today,

603
00:31:34,740 --> 00:31:37,290
and actually just became
visible a few minutes ago,

604
00:31:37,290 --> 00:31:39,120
in the homework,
you will see what

605
00:31:39,120 --> 00:31:42,510
happens if you illuminate the
grating with a tilted plane

606
00:31:42,510 --> 00:31:45,010
wave or with a spherical wave.

607
00:31:45,010 --> 00:31:46,170
This is also possible.

608
00:31:46,170 --> 00:31:49,380
You can illuminate a
grate in any way you like.

609
00:31:49,380 --> 00:31:51,520
Now, we'll start with
the simplest case,

610
00:31:51,520 --> 00:31:54,540
which is on axis plane wave.

611
00:31:54,540 --> 00:31:56,610
So we don't know yet
what's going to happen,

612
00:31:56,610 --> 00:31:57,650
but we know one thing.

613
00:31:57,650 --> 00:32:00,240
We know Huygen's principal,
and Huygen's says

614
00:32:00,240 --> 00:32:04,380
that, at each point where
the grating is transmissive,

615
00:32:04,380 --> 00:32:06,350
we'll get point sources.

616
00:32:06,350 --> 00:32:09,000
So of course, I do not
only get only these two,

617
00:32:09,000 --> 00:32:12,610
I get the point
source here, which

618
00:32:12,610 --> 00:32:15,820
is very strong because this is
the location where the grating

619
00:32:15,820 --> 00:32:17,560
has maximum transmission.

620
00:32:17,560 --> 00:32:20,930
Next to it, I will
get another point--

621
00:32:20,930 --> 00:32:24,820
another Huygen's point source
that is slightly attenuated.

622
00:32:24,820 --> 00:32:27,190
Then next to it, another
one that is slightly more

623
00:32:27,190 --> 00:32:29,620
attenuated, another
one, another one.

624
00:32:29,620 --> 00:32:32,170
There comes a point here where
the grating is totally black,

625
00:32:32,170 --> 00:32:36,070
that it is totally absorbing.

626
00:32:36,070 --> 00:32:39,730
Therefore, there's is a missing
Huygen's source over here,

627
00:32:39,730 --> 00:32:41,920
and then as I move on
along the optical axis,

628
00:32:41,920 --> 00:32:45,250
I will get more Huygen's point
sources that are progressively

629
00:32:45,250 --> 00:32:46,540
less attenuated.

630
00:32:46,540 --> 00:32:48,600
And then finally, I
get over here where

631
00:32:48,600 --> 00:32:51,550
I have another strong,
as strong as possible,

632
00:32:51,550 --> 00:32:53,590
Huygen's point source.

633
00:32:53,590 --> 00:32:55,800
So we derived an integral.

634
00:32:55,800 --> 00:33:01,300
Professor Sheppard last week
[INAUDIBLE] of an integral that

635
00:33:01,300 --> 00:33:05,350
allows us to basically
sum the contributions

636
00:33:05,350 --> 00:33:08,470
from all these
Huygen's point sources

637
00:33:08,470 --> 00:33:11,530
and derive different
L diffracted field

638
00:33:11,530 --> 00:33:13,910
after the grating.

639
00:33:13,910 --> 00:33:17,660
So of course, this integral,
we call the Fresnel integral,

640
00:33:17,660 --> 00:33:20,540
and here is another linguistic
lesson-- this time in French,

641
00:33:20,540 --> 00:33:22,230
not in Greek.

642
00:33:22,230 --> 00:33:27,310
This is pronounced "Freh-nell."

643
00:33:27,310 --> 00:33:29,080
The S is silent.

644
00:33:29,080 --> 00:33:32,350
So if you go to a conference,
and you say "Fres-nell,"

645
00:33:32,350 --> 00:33:35,560
people will conclude that
you don't know optics.

646
00:33:35,560 --> 00:33:37,030
So part of knowing
optics is part

647
00:33:37,030 --> 00:33:41,670
of knowing the French and the
Greek language, I suppose.

648
00:33:41,670 --> 00:33:44,080
I'm joking, of course,
but it's still actually,

649
00:33:44,080 --> 00:33:47,940
professionals are like
children, who are vicious.

650
00:33:47,940 --> 00:33:50,230
If we see someone make a
silly mistake like this,

651
00:33:50,230 --> 00:33:52,117
say "Fres-nell"
instead of "Freh-nell,"

652
00:33:52,117 --> 00:33:52,950
we make fun of them.

653
00:33:52,950 --> 00:33:55,590
And you don't want
to be made fun

654
00:33:55,590 --> 00:33:57,820
of in a professional setting.

655
00:33:57,820 --> 00:34:00,420
So always remember, this
is pronounced "Freh-nell,"

656
00:34:00,420 --> 00:34:02,400
not "Fres-nell."

657
00:34:02,400 --> 00:34:04,590
So the Fresnel
integral, therefore,

658
00:34:04,590 --> 00:34:07,830
allows us to compute the
field, the diffracted field,

659
00:34:07,830 --> 00:34:09,300
after the grating.

660
00:34:09,300 --> 00:34:12,989
But we will not do it here
because the Fresnel integral,

661
00:34:12,989 --> 00:34:14,670
in this case, is awful.

662
00:34:14,670 --> 00:34:16,920
It is very difficult to compute.

663
00:34:16,920 --> 00:34:18,870
However, I will show
you that, by using

664
00:34:18,870 --> 00:34:22,290
very simple physical
arguments, we can nevertheless

665
00:34:22,290 --> 00:34:25,020
derive the Fresnel
diffracted field anyway

666
00:34:25,020 --> 00:34:31,214
without making use of this
nasty Fresnel integral.

667
00:34:33,790 --> 00:34:37,860
So let's go back to our
Huygen's picture here.

668
00:34:37,860 --> 00:34:41,830
I would just consider this two
point sources of Huygen's which

669
00:34:41,830 --> 00:34:43,112
are the strongest.

670
00:34:43,112 --> 00:34:44,820
And this is, of course,
an approximation,

671
00:34:44,820 --> 00:34:46,780
a simplification, but
it will still give us

672
00:34:46,780 --> 00:34:48,489
the correct physical picture.

673
00:34:48,489 --> 00:34:50,230
That's why I'm doing it.

674
00:34:50,230 --> 00:34:52,630
So basically, over
here, you can see

675
00:34:52,630 --> 00:34:55,570
that these are point sources,
so they basically emit light

676
00:34:55,570 --> 00:34:57,970
in all possible directions.

677
00:34:57,970 --> 00:35:02,650
So I will pick one
arbitrary direction,

678
00:35:02,650 --> 00:35:05,320
and I will call it theta.

679
00:35:05,320 --> 00:35:08,560
So here is a ray, if you
wish, a ray emanating

680
00:35:08,560 --> 00:35:12,760
from this Huygen's point
source at angle theta,

681
00:35:12,760 --> 00:35:16,210
and here's another ray emanating
from the next Huygen's point

682
00:35:16,210 --> 00:35:18,970
source at the same angle theta.

683
00:35:18,970 --> 00:35:22,980
Now, these rays-- are they
the same or different?

684
00:35:22,980 --> 00:35:25,950
Geometrical optics
says they're the same,

685
00:35:25,950 --> 00:35:27,160
but now, we know better.

686
00:35:27,160 --> 00:35:30,900
We know that these
are not really bullets

687
00:35:30,900 --> 00:35:32,650
that propagate like particles.

688
00:35:32,650 --> 00:35:34,490
These are actually waves.

689
00:35:34,490 --> 00:35:37,410
And if you compare
the wave emitted here

690
00:35:37,410 --> 00:35:40,860
and the wave emitted
there, if you draw a line

691
00:35:40,860 --> 00:35:46,420
perpendicular to these waves,
this would be the wavefront.

692
00:35:46,420 --> 00:35:49,950
So you can see, easily now,
that if you compare the wave

693
00:35:49,950 --> 00:35:54,210
on the lower ray with a
wave with the upper ray,

694
00:35:54,210 --> 00:35:58,950
you can see that the wave in the
lower ray is actually delayed.

695
00:35:58,950 --> 00:36:00,900
It has traveled the
longer distance.

696
00:36:00,900 --> 00:36:04,920
Therefore, it is delayed by
a certain amount with respect

697
00:36:04,920 --> 00:36:06,950
to the ray that is below.

698
00:36:06,950 --> 00:36:09,300
And the way, of course,
to compute this delay

699
00:36:09,300 --> 00:36:14,070
is to draw an orthogonal
triangle here.

700
00:36:14,070 --> 00:36:20,400
This side of the triangle
is actually the wavefront.

701
00:36:20,400 --> 00:36:24,900
This side of the triangle
is the phase delay,

702
00:36:24,900 --> 00:36:28,420
and the hypotenuse of the
triangle is the period--

703
00:36:28,420 --> 00:36:30,270
is the special
period of the grating

704
00:36:30,270 --> 00:36:32,550
that we denoted as lambda.

705
00:36:32,550 --> 00:36:34,500
So by using the
Pythagoras theorem,

706
00:36:34,500 --> 00:36:39,100
then, we can actually
compute the phase delay,

707
00:36:39,100 --> 00:36:42,220
and we can compute--

708
00:36:42,220 --> 00:36:44,210
there's, of course, a typo here.

709
00:36:44,210 --> 00:36:47,000
Let me fix it right away, and
I will post a fix in the notes.

710
00:36:54,970 --> 00:37:00,080
Anyway, I think it's better
if I don't fix it now,

711
00:37:00,080 --> 00:37:04,980
but the correct equation over
there should say the optical

712
00:37:04,980 --> 00:37:09,150
path length difference--
that's what OPLD means--

713
00:37:09,150 --> 00:37:14,930
equals lambda, the grating
period, times sine theta.

714
00:37:14,930 --> 00:37:15,680
Sorry about that.

715
00:37:15,680 --> 00:37:17,810
These notes came from
another old [? convention ?]

716
00:37:17,810 --> 00:37:19,630
where the period
was denoted as d.

717
00:37:26,350 --> 00:37:28,180
So with this little
fix, that OPLD

718
00:37:28,180 --> 00:37:32,230
equals the period of the
grating times sine theta,

719
00:37:32,230 --> 00:37:36,640
we can see now that, if
this quantity of OPLD

720
00:37:36,640 --> 00:37:44,130
equals a wavelength, then the
two rays are actually in phase.

721
00:37:44,130 --> 00:37:46,060
And not only one
wavelength-- it might

722
00:37:46,060 --> 00:37:48,720
equal two wavelengths,
three wavelengths,

723
00:37:48,720 --> 00:37:51,070
any integral number
of wavelengths.

724
00:37:51,070 --> 00:37:52,770
Then these two
waves are in phase.

725
00:37:52,770 --> 00:37:56,130
Therefore, they
interfere constructively.

726
00:37:56,130 --> 00:37:57,520
And this is the key, now--

727
00:37:57,520 --> 00:38:02,260
the constructive interference
of waves in these specific,

728
00:38:02,260 --> 00:38:06,940
now, directions is what we
call diffraction orders,

729
00:38:06,940 --> 00:38:10,250
that Pepe very beautifully
showed you earlier.

730
00:38:10,250 --> 00:38:15,880
And you also saw them in
my simulation in Matlab.

731
00:38:15,880 --> 00:38:20,500
And so q here, the
quantity q, is an integer.

732
00:38:20,500 --> 00:38:22,620
It is meant to
denote an integer,

733
00:38:22,620 --> 00:38:26,660
and we can derive this now.

734
00:38:26,660 --> 00:38:33,200
So we can derive the sine
of the angle of the q,

735
00:38:33,200 --> 00:38:37,010
over the qth, I guess,
diffraction order where

736
00:38:37,010 --> 00:38:40,800
qth might be first,
second, third, and so on.

737
00:38:40,800 --> 00:38:43,040
So q runs 1, 2, 3, and so on.

738
00:38:43,040 --> 00:38:45,410
And of course,
because typically, we

739
00:38:45,410 --> 00:38:47,630
operate in the
paraxial approximation,

740
00:38:47,630 --> 00:38:50,600
we can also drop the
sign here and say simply

741
00:38:50,600 --> 00:38:53,530
that the plus/minus
1 diffraction

742
00:38:53,530 --> 00:38:55,760
order is propagating
at an angle that

743
00:38:55,760 --> 00:38:57,890
is given by this
ratio over there,

744
00:38:57,890 --> 00:39:02,420
the ratio of the
small Greek lambda

745
00:39:02,420 --> 00:39:06,560
to the big Greek lambda, that
is the ratio of the wavelength

746
00:39:06,560 --> 00:39:08,292
over the period.

747
00:39:08,292 --> 00:39:10,000
And you can see
immediately, for example,

748
00:39:10,000 --> 00:39:15,250
that if I make the
grating period smaller

749
00:39:15,250 --> 00:39:17,530
than the wavelength,
I will get in trouble

750
00:39:17,530 --> 00:39:20,770
because this sine over there
will attempt to become bigger

751
00:39:20,770 --> 00:39:21,940
than 1.

752
00:39:21,940 --> 00:39:23,810
Anybody knows what
happens in this case

753
00:39:23,810 --> 00:39:26,560
if I make the grating
period smaller than lambda,

754
00:39:26,560 --> 00:39:28,127
than the wavelength?

755
00:39:31,110 --> 00:39:34,450
Mr. [INAUDIBLE]?

756
00:39:34,450 --> 00:39:36,970
Yeah, actually, I can do--

757
00:39:36,970 --> 00:39:39,520
I can make a period
smaller than a wavelength.

758
00:39:39,520 --> 00:39:41,160
What will happen
then is I will get

759
00:39:41,160 --> 00:39:43,330
what is called an
evanescent wave,

760
00:39:43,330 --> 00:39:45,530
so I will not get a
diffraction order anymore.

761
00:39:45,530 --> 00:39:47,860
I'll get an evanescent
wave from the grating,

762
00:39:47,860 --> 00:39:50,540
and this is a term
that we've seen before.

763
00:39:50,540 --> 00:39:52,580
We haven't defined
yet, mathematically,

764
00:39:52,580 --> 00:39:56,350
but I want to alert you that,
basically, an evanescent

765
00:39:56,350 --> 00:39:59,500
wave is an exponential
decay of a diffracted field.

766
00:39:59,500 --> 00:40:04,120
So you do not quite get a
diffracted order anymore.

767
00:40:04,120 --> 00:40:08,350
For the purposes of the class,
in the next few lectures

768
00:40:08,350 --> 00:40:12,400
at least, definitely for the
next three or four weeks,

769
00:40:12,400 --> 00:40:17,260
we will assume that the
grating period is much larger

770
00:40:17,260 --> 00:40:18,370
than the wavelength.

771
00:40:18,370 --> 00:40:21,130
So therefore, this quantity
is always less than 1,

772
00:40:21,130 --> 00:40:24,490
and the sine is always
properly less than 1.

773
00:40:24,490 --> 00:40:28,900
And we can solve it, so we can
get the real and not evanescent

774
00:40:28,900 --> 00:40:29,780
diffraction order.

775
00:40:33,270 --> 00:40:34,543
I think my computer--

776
00:40:39,750 --> 00:40:41,705
so as I said
before, the plus 1--

777
00:40:41,705 --> 00:40:42,830
this is actually arbitrary.

778
00:40:42,830 --> 00:40:46,240
I can call plus 1--

779
00:40:46,240 --> 00:40:49,630
by convention, the plus
1 is the one that goes--

780
00:40:49,630 --> 00:40:54,995
if we denote the light--

781
00:40:54,995 --> 00:40:56,860
as in geometrical
optics, the light

782
00:40:56,860 --> 00:40:59,860
always goes from left to right.

783
00:40:59,860 --> 00:41:05,240
And the angle is
positive if it is acute

784
00:41:05,240 --> 00:41:08,280
measured counterclockwise
from the optical axis.

785
00:41:08,280 --> 00:41:11,420
So that is why this is the
plus first diffraction order.

786
00:41:11,420 --> 00:41:14,240
It has a positive propagation
angle, and then the one

787
00:41:14,240 --> 00:41:15,870
that has negative
propagation angle,

788
00:41:15,870 --> 00:41:20,910
we'll call it minus
first diffraction order.

789
00:41:20,910 --> 00:41:23,920
The 0th diffraction
order is the one

790
00:41:23,920 --> 00:41:29,710
that has theta equals 0
that is propagating on axis.

791
00:41:29,710 --> 00:41:32,470
That is also known as a DC term.

792
00:41:32,470 --> 00:41:35,120
Now, for those of you who
are electrical engineers,

793
00:41:35,120 --> 00:41:38,660
the term DC already alerts
you to some kind of a Fourier

794
00:41:38,660 --> 00:41:42,650
transform or some kind of
a frequency representation.

795
00:41:42,650 --> 00:41:45,940
We will see later that, indeed,
light propagating on axes

796
00:41:45,940 --> 00:41:50,440
can be thought of as the
equivalent of a DC component

797
00:41:50,440 --> 00:41:55,020
in Fourier series decomposition.

798
00:41:55,020 --> 00:42:00,930
For now, let's take it
as a term, as a sort of--

799
00:42:00,930 --> 00:42:04,270
what do you call
this-- as a jargon,

800
00:42:04,270 --> 00:42:06,830
and we'll see later why
we call this the DC term.

801
00:42:11,350 --> 00:42:13,750
This is a mathematical
derivation

802
00:42:13,750 --> 00:42:16,420
that we can do in order
to actually become

803
00:42:16,420 --> 00:42:20,020
a little bit more quantitative
and a little bit more correct.

804
00:42:20,020 --> 00:42:22,310
So the mathematical
definition is very simple.

805
00:42:22,310 --> 00:42:24,850
We start with the
expression for the grating.

806
00:42:28,020 --> 00:42:30,630
Then we decompose the
cosine into the two

807
00:42:30,630 --> 00:42:34,690
complex exponentials that
we can [? write ?] it.

808
00:42:34,690 --> 00:42:41,410
And then what we do is we
simply make a substitution

809
00:42:41,410 --> 00:42:44,380
according to the equation
that I saw before already.

810
00:42:44,380 --> 00:42:52,410
We say that 1 over lambda
equals sine theta upon lambda.

811
00:42:52,410 --> 00:42:54,550
This may seem arbitrary,
but I can always

812
00:42:54,550 --> 00:42:57,940
do it because lambda--

813
00:42:57,940 --> 00:43:01,010
I'm sorry [INAUDIBLE].

814
00:43:01,010 --> 00:43:02,180
Uh-oh, we're out of focus.

815
00:43:05,910 --> 00:43:07,470
Anyway, you cannot
see the question,

816
00:43:07,470 --> 00:43:10,290
but you can certainly see
this substitution here.

817
00:43:10,290 --> 00:43:12,310
And I cannot-- thank you.

818
00:43:12,310 --> 00:43:16,110
So we can always
do the substitution

819
00:43:16,110 --> 00:43:20,440
by defining sine theta to be
the ratio of the wavelength

820
00:43:20,440 --> 00:43:24,240
over the period,
and we can also make

821
00:43:24,240 --> 00:43:27,150
a substitution of [? u0, ?]
the special frequency as I

822
00:43:27,150 --> 00:43:28,740
defined it before.

823
00:43:28,740 --> 00:43:29,790
The point is not that.

824
00:43:29,790 --> 00:43:32,730
The point is that each
one of these terms

825
00:43:32,730 --> 00:43:35,910
actually is recognizable
as a plane wave.

826
00:43:35,910 --> 00:43:40,735
And now, we'll really
need my transparency here.

827
00:43:40,735 --> 00:43:45,810
So if you recall, when
we derived plane waves--

828
00:43:45,810 --> 00:43:47,330
we did this some time ago.

829
00:43:47,330 --> 00:43:49,500
Actually, I think
Professor Sheppard did it

830
00:43:49,500 --> 00:43:51,450
in the context of
electromagnetics.

831
00:43:51,450 --> 00:43:54,570
The way we write a
plane wave is like this.

832
00:44:31,163 --> 00:44:33,580
These are what you will see
if you go back into your notes

833
00:44:33,580 --> 00:44:36,620
by about two or three lectures.

834
00:44:36,620 --> 00:44:38,380
If you have a plane
wave propagating

835
00:44:38,380 --> 00:44:41,080
in the direction
of this arrow here,

836
00:44:41,080 --> 00:44:43,790
then the expression
that I put above

837
00:44:43,790 --> 00:44:48,460
is actually the phasor for
the electromagnetic field

838
00:44:48,460 --> 00:44:50,470
describing that plane wave.

839
00:44:50,470 --> 00:44:56,750
And if, in addition, we say that
the grating is at z equals 0--

840
00:44:56,750 --> 00:44:58,790
which I can always do.

841
00:44:58,790 --> 00:45:02,140
I can define my axis so that
the grating is located at

842
00:45:02,140 --> 00:45:03,230
z equals 0--

843
00:45:03,230 --> 00:45:05,540
then this term will
actually disappear.

844
00:45:05,540 --> 00:45:08,300
And you can recognize
now this expression

845
00:45:08,300 --> 00:45:12,890
that is left for the plane
wave is the same that you have

846
00:45:12,890 --> 00:45:17,120
already here with
the complex exponent,

847
00:45:17,120 --> 00:45:20,030
provided you make the
substitution that we

848
00:45:20,030 --> 00:45:23,400
had before, provided you
make the substitution--

849
00:45:23,400 --> 00:45:23,900
this one.

850
00:45:26,960 --> 00:45:29,360
With a substitution,
the grating itself

851
00:45:29,360 --> 00:45:32,370
becomes like the expression
for not just one, but actually

852
00:45:32,370 --> 00:45:33,800
three plane waves.

853
00:45:33,800 --> 00:45:36,020
And what are the
three plane waves?

854
00:45:36,020 --> 00:45:43,470
Well, here is one, which is the
plus first diffraction order.

855
00:45:43,470 --> 00:45:47,610
There's a plane wave
propagating at angle theta.

856
00:45:47,610 --> 00:45:50,850
Then the next one, the
constant is actually

857
00:45:50,850 --> 00:45:52,950
the DC term that
propagates on axis,

858
00:45:52,950 --> 00:45:58,500
because if you substitute
theta equals 0 here,

859
00:45:58,500 --> 00:46:01,560
then this entire thing
disappears, so you just

860
00:46:01,560 --> 00:46:02,610
have the constant.

861
00:46:02,610 --> 00:46:06,070
That is a plane wave
propagating on axis.

862
00:46:06,070 --> 00:46:09,143
And finally, this
term is propagating

863
00:46:09,143 --> 00:46:10,560
at the negative
angle, and this is

864
00:46:10,560 --> 00:46:14,970
what you call the minus
1 diffraction order.

865
00:46:14,970 --> 00:46:17,760
So this may seem a little bit
like mathematical trickery,

866
00:46:17,760 --> 00:46:20,400
but has a very nice
physical interpretation,

867
00:46:20,400 --> 00:46:24,600
if only you recall our
convention for the phasor

868
00:46:24,600 --> 00:46:28,820
expression of plane waves.

869
00:46:28,820 --> 00:46:32,980
The rest is relatively
straightforward.

870
00:46:32,980 --> 00:46:36,810
A sinusoidal phase
grating, as Pepe mentioned,

871
00:46:36,810 --> 00:46:39,270
is basically a
grating which does not

872
00:46:39,270 --> 00:46:42,250
modulate the amplitude, but
the phase of the optical field.

873
00:46:42,250 --> 00:46:43,930
And you can do it in two ways.

874
00:46:43,930 --> 00:46:45,420
One is what I saw here.

875
00:46:45,420 --> 00:46:48,520
That is known as a
surface relief grating,

876
00:46:48,520 --> 00:46:49,740
which has been processed.

877
00:46:49,740 --> 00:46:51,510
Typically, this is
done by chemical--

878
00:46:51,510 --> 00:46:55,530
by optical exposure and then
chemical development, that

879
00:46:55,530 --> 00:46:57,860
causes the surface
of, typically,

880
00:46:57,860 --> 00:47:04,200
a polymer or a properly prepared
colloidal, colloidal glass.

881
00:47:04,200 --> 00:47:08,700
It causes the surface to pick
up a sinusoidal modulation

882
00:47:08,700 --> 00:47:09,610
like this.

883
00:47:09,610 --> 00:47:14,160
So because the material
has a refractive index n,

884
00:47:14,160 --> 00:47:16,200
you can see that
light, depending

885
00:47:16,200 --> 00:47:19,860
if you take a ray, depending
on where it hits this material,

886
00:47:19,860 --> 00:47:23,920
it might suffer a
different phase delay.

887
00:47:23,920 --> 00:47:26,010
Another way that
Pepe mentioned is you

888
00:47:26,010 --> 00:47:28,650
might actually have a
flat piece of material

889
00:47:28,650 --> 00:47:31,230
and modulate the index
of refraction inside.

890
00:47:31,230 --> 00:47:33,390
Again, that would
have a similar effect.

891
00:47:33,390 --> 00:47:36,510
The complex transmission
function would be of this form.

892
00:47:36,510 --> 00:47:40,500
It would be an
exponential whose value

893
00:47:40,500 --> 00:47:45,240
depends upon the optical
thickness of the material

894
00:47:45,240 --> 00:47:46,680
that the light went through.

895
00:47:46,680 --> 00:47:52,150
You can see the period here,
and you can see the phase delay.

896
00:47:52,150 --> 00:47:54,750
So the phase delay,
of course, depends

897
00:47:54,750 --> 00:47:56,880
on the difference between
the index of refraction

898
00:47:56,880 --> 00:47:59,115
inside the material and
the index of refraction

899
00:47:59,115 --> 00:48:02,350
in the surrounding
middle that is air.

900
00:48:02,350 --> 00:48:05,270
So that's the
expression over there,

901
00:48:05,270 --> 00:48:07,360
and this is, of course,
the phase contrast

902
00:48:07,360 --> 00:48:10,410
that we mentioned before,
when we defined the grating.

903
00:48:10,410 --> 00:48:13,060
And in order, now,
to figure out what

904
00:48:13,060 --> 00:48:15,310
happens after this
grating, I will

905
00:48:15,310 --> 00:48:22,540
have to resort to mathematical,
again, trickery, I guess.

906
00:48:22,540 --> 00:48:24,880
If you go to books
of optical tables,

907
00:48:24,880 --> 00:48:28,840
you can find very easily
that this expression can also

908
00:48:28,840 --> 00:48:32,830
be written as a
sum, as a sum that

909
00:48:32,830 --> 00:48:37,300
involves some kind of
nasty Bessel functions.

910
00:48:37,300 --> 00:48:38,930
We don't even have
to worry about this,

911
00:48:38,930 --> 00:48:41,470
but what we will go
ahead and add in a second

912
00:48:41,470 --> 00:48:43,870
is that each one of
these Bessel functions

913
00:48:43,870 --> 00:48:49,000
corresponds to diffraction
order for this grating.

914
00:48:49,000 --> 00:48:52,600
And the way you can see this
is if you rewrite the grating

915
00:48:52,600 --> 00:48:54,200
like this.

916
00:48:54,200 --> 00:48:57,670
So this is now simply using
a mathematical formula,

917
00:48:57,670 --> 00:49:02,260
but now, if you actually do
the same trick I did here,

918
00:49:02,260 --> 00:49:06,550
if you recognize that each one
of these complex exponentials

919
00:49:06,550 --> 00:49:09,730
basically corresponds
to the plane wave,

920
00:49:09,730 --> 00:49:13,570
then you can see that
what I've written there,

921
00:49:13,570 --> 00:49:16,030
it came out of a book of
mathematical formulas,

922
00:49:16,030 --> 00:49:18,450
but physically, it
contains an amplitude

923
00:49:18,450 --> 00:49:21,760
that is given by the value
of a Bessel function.

924
00:49:21,760 --> 00:49:24,910
But then more importantly,
it contains a sequence

925
00:49:24,910 --> 00:49:27,900
of plane waves that are
known as diffraction orders,

926
00:49:27,900 --> 00:49:32,730
and the propagation angle of
the plane wave indexed by q

927
00:49:32,730 --> 00:49:35,470
is given by this explanation
here, the same expression

928
00:49:35,470 --> 00:49:36,740
that they had before.

929
00:49:36,740 --> 00:49:39,700
So these are then the
diffraction orders

930
00:49:39,700 --> 00:49:42,060
for the sinusoidal
phase grating.

931
00:49:45,270 --> 00:49:48,180
Even more generally,
if you don't

932
00:49:48,180 --> 00:49:50,670
have a sinusoidal
grating, but still

933
00:49:50,670 --> 00:49:57,800
some kind of a periodic
grating with arbitrary shape,

934
00:49:57,800 --> 00:50:00,800
and also possibly you
might even have absorption,

935
00:50:00,800 --> 00:50:06,410
so this grating truly
implements a complex amplitude

936
00:50:06,410 --> 00:50:09,230
transmission, which is
nevertheless periodic.

937
00:50:09,230 --> 00:50:12,350
You may recall from
your basic math

938
00:50:12,350 --> 00:50:15,050
that you have an arbitrary
complex periodic function,

939
00:50:15,050 --> 00:50:17,720
you can write it as
a Fourier series.

940
00:50:17,720 --> 00:50:19,460
And in this case,
the Fourier series

941
00:50:19,460 --> 00:50:21,440
also has the same
physical meaning

942
00:50:21,440 --> 00:50:25,220
as the one I did before,
the Fourier series

943
00:50:25,220 --> 00:50:28,450
expansion coefficients.

944
00:50:28,450 --> 00:50:31,900
And actually, the amplitudes
and the harmonics-- these

945
00:50:31,900 --> 00:50:33,760
are called harmonics,
if you recall.

946
00:50:33,760 --> 00:50:36,430
The harmonics of the
Fourier series themselves

947
00:50:36,430 --> 00:50:39,610
now correspond to plane
waves that were previously

948
00:50:39,610 --> 00:50:41,230
called diffraction orders.

949
00:50:41,230 --> 00:50:43,300
So then you can see
that each Fourier series

950
00:50:43,300 --> 00:50:47,140
coefficient actually corresponds
to a separate diffraction

951
00:50:47,140 --> 00:50:47,640
order.

952
00:50:50,890 --> 00:50:55,640
And in addition, recall, again,
Professor Sheppard last week.

953
00:50:55,640 --> 00:50:58,870
He mentioned that if you
have an optical field,

954
00:50:58,870 --> 00:51:02,770
and you take the magnitude
squared of its phasor,

955
00:51:02,770 --> 00:51:06,400
then what you get is actually
the amount of energy that

956
00:51:06,400 --> 00:51:10,970
propagates that is
carried by this field.

957
00:51:10,970 --> 00:51:14,290
So therefore, the magnitude
squared of the Fourier series

958
00:51:14,290 --> 00:51:17,530
coefficient actually
corresponds to the energy

959
00:51:17,530 --> 00:51:20,920
that goes into each
separate diffraction order.

960
00:51:20,920 --> 00:51:22,960
So in this case,
so they're called

961
00:51:22,960 --> 00:51:24,002
diffraction efficiencies.

962
00:51:24,002 --> 00:51:25,918
And I should have mentioned
earlier, actually,

963
00:51:25,918 --> 00:51:26,665
but I forgot--

964
00:51:42,590 --> 00:51:47,770
so in the case of the
sinusoidal amplitude grating,

965
00:51:47,770 --> 00:51:49,490
the diffraction
efficiencies are given

966
00:51:49,490 --> 00:51:51,980
by the squares of
the coefficients that

967
00:51:51,980 --> 00:51:54,410
go in front of each plane wave.

968
00:51:54,410 --> 00:51:59,050
So you can see that
approximately-- how much--

969
00:51:59,050 --> 00:52:04,390
25% of the energy is
going into the 0 order.

970
00:52:04,390 --> 00:52:06,650
It is propagating on axis.

971
00:52:06,650 --> 00:52:09,170
And let's assume that
the contrast is 1.

972
00:52:09,170 --> 00:52:13,420
In that case, approximately
12.5% of the energy

973
00:52:13,420 --> 00:52:16,750
is going into each
diffraction order.

974
00:52:16,750 --> 00:52:19,180
That is 1/8.

975
00:52:19,180 --> 00:52:21,690
If I sum them up, how
much do they transmit?

976
00:52:28,110 --> 00:52:33,120
25% plus 12.5% plus
[INAUDIBLE] 12.5%--

977
00:52:33,120 --> 00:52:34,480
50%, right?

978
00:52:34,480 --> 00:52:35,730
What happens to the other 50%?

979
00:52:43,856 --> 00:52:45,442
[INAUDIBLE]?

980
00:52:45,442 --> 00:52:47,150
STUDENT: I think there
possibly maybe had

981
00:52:47,150 --> 00:52:49,850
a plus/minus 3 order
or something, I mean,

982
00:52:49,850 --> 00:52:50,845
other orders?

983
00:52:50,845 --> 00:52:53,220
GEORGE BARBASTATHIS: That is
plausible, but in this case,

984
00:52:53,220 --> 00:52:55,800
we derived only
three orders-- the 0,

985
00:52:55,800 --> 00:52:58,833
the plus 1, and the minus 1,
so there's no higher order.

986
00:52:58,833 --> 00:53:00,500
What happened to the
rest of the energy?

987
00:53:03,770 --> 00:53:06,273
STUDENT: Is it absorbed
depending on the grating?

988
00:53:06,273 --> 00:53:07,690
GEORGE BARBASTATHIS:
That's right.

989
00:53:07,690 --> 00:53:11,510
The rest of them got absorbed
or actually reflected

990
00:53:11,510 --> 00:53:15,690
backwards by the opaque
parts of the grating.

991
00:53:15,690 --> 00:53:17,510
So remember, this is
an amplitude grating.

992
00:53:17,510 --> 00:53:19,280
It blocks part of the light.

993
00:53:19,280 --> 00:53:22,297
So in this case, it's
actually blocked about 50%

994
00:53:22,297 --> 00:53:22,880
of the energy.

995
00:53:35,790 --> 00:53:36,970
So as the last example--

996
00:53:36,970 --> 00:53:41,650
which I will not derive here,
but I will let you go over

997
00:53:41,650 --> 00:53:43,650
it yourselves at home--

998
00:53:43,650 --> 00:53:45,420
I did the case of a binary now.

999
00:53:45,420 --> 00:53:49,740
So this is a phase grating very
similar to the surface relief

1000
00:53:49,740 --> 00:53:52,000
that I showed before,
but in this case,

1001
00:53:52,000 --> 00:53:54,090
the phase is a binary function.

1002
00:53:54,090 --> 00:53:55,270
So why is it binary?

1003
00:53:55,270 --> 00:53:58,035
Because this a grating
which is either--

1004
00:54:00,935 --> 00:54:05,140
the light goes either through
a tall part or a short part.

1005
00:54:05,140 --> 00:54:07,800
The tall part, of course,
suffers more phase delay

1006
00:54:07,800 --> 00:54:10,890
because the light has to go
through glass for a little bit

1007
00:54:10,890 --> 00:54:13,080
longer than the short part.

1008
00:54:13,080 --> 00:54:16,680
Therefore, if I plot the phase
of the complex transmission,

1009
00:54:16,680 --> 00:54:18,320
expect to see
something like this.

1010
00:54:18,320 --> 00:54:21,540
The part of the grating that are
taller, they give me a longer

1011
00:54:21,540 --> 00:54:24,930
phase delay, and of course, the
phase delay can now be tried

1012
00:54:24,930 --> 00:54:29,020
normalized so that it is
0 at some reference value.

1013
00:54:29,020 --> 00:54:35,340
And then what is the phase
delay at the long part

1014
00:54:35,340 --> 00:54:39,635
actually depends on the
height of this groove.

1015
00:54:39,635 --> 00:54:41,010
So even the height
of this groove

1016
00:54:41,010 --> 00:54:53,860
is such that s equals
2 pi n minus 1 over--

1017
00:55:01,380 --> 00:55:03,150
let me get this straight.

1018
00:55:03,150 --> 00:55:12,430
So delta phi equals 2
pi upon lambda times

1019
00:55:12,430 --> 00:55:31,980
n times s minus 2 pi over lambda
times s2, where this is s1.

1020
00:55:31,980 --> 00:55:34,740
This is s2.

1021
00:55:34,740 --> 00:55:42,480
So if I say that s1 equals s2
plus s, then what I will get

1022
00:55:42,480 --> 00:55:48,600
is 2 pi over lambda
times n minus 1 times s.

1023
00:55:48,600 --> 00:55:50,250
So you can see from here that--

1024
00:56:04,260 --> 00:56:08,730
so if s is given by lambda
divided by twice the index

1025
00:56:08,730 --> 00:56:12,210
difference between the glass
and the surrounding medium,

1026
00:56:12,210 --> 00:56:18,870
then the phase delay implemented
by this grating is actually pi.

1027
00:56:18,870 --> 00:56:20,430
So if you have a
function like this,

1028
00:56:20,430 --> 00:56:25,570
then, phase delay pi means
that the complex amplitude

1029
00:56:25,570 --> 00:56:30,100
transmission is minus
1, and phase delay 0

1030
00:56:30,100 --> 00:56:31,670
means that it is plus 1.

1031
00:56:31,670 --> 00:56:34,510
So there's is another way
to write this function.

1032
00:56:34,510 --> 00:56:39,305
It is 1 at half of the period,
and minus 1 the other half

1033
00:56:39,305 --> 00:56:40,360
of the period.

1034
00:56:40,360 --> 00:56:42,620
So you can actually--

1035
00:56:42,620 --> 00:56:44,530
it's a little bit of a
[? grind. ?] Next time

1036
00:56:44,530 --> 00:56:46,600
I will show you a
faster way to derive it

1037
00:56:46,600 --> 00:56:52,000
that you can derive the
Fourier series for this kind

1038
00:56:52,000 --> 00:56:55,685
of a binary phase grating,
and you'll find actually

1039
00:56:55,685 --> 00:56:56,810
something very interesting.

1040
00:56:56,810 --> 00:56:59,650
I don't know if you noticed
the animation when I showed it.

1041
00:56:59,650 --> 00:57:01,090
Let me play it once again.

1042
00:57:01,090 --> 00:57:07,870
You'll find that even orders
of this grating are actually 0,

1043
00:57:07,870 --> 00:57:10,360
and next time, I will
actually say a little bit more

1044
00:57:10,360 --> 00:57:12,910
about this.

1045
00:57:12,910 --> 00:57:17,230
Anyway, you find the expression
for the diffracted orders

1046
00:57:17,230 --> 00:57:21,850
equals to something that looks
like sine of 5 times the order.

1047
00:57:21,850 --> 00:57:27,070
So of course, if the order is
integral, if the order is even,

1048
00:57:27,070 --> 00:57:29,750
then there's no diffraction.

1049
00:57:29,750 --> 00:57:31,360
It turns out that
this grating has

1050
00:57:31,360 --> 00:57:35,080
a relatively strong diffraction
efficiency of about 40%

1051
00:57:35,080 --> 00:57:37,090
into the first order.

1052
00:57:37,090 --> 00:57:38,950
STUDENT: I've got a question.

1053
00:57:38,950 --> 00:57:40,540
GEORGE BARBASTATHIS: Yes?

1054
00:57:40,540 --> 00:57:42,100
STUDENT: How does
the polarization

1055
00:57:42,100 --> 00:57:44,800
of the incident plane wave
change the diffraction

1056
00:57:44,800 --> 00:57:46,260
efficiencies?

1057
00:57:46,260 --> 00:57:48,260
GEORGE BARBASTATHIS: So
these guys were actually

1058
00:57:48,260 --> 00:57:50,660
in the scalar optics
approximation,

1059
00:57:50,660 --> 00:57:55,190
so we neglect the
effect of polarization.

1060
00:57:55,190 --> 00:57:57,770
To include it is actually
rather complicated business

1061
00:57:57,770 --> 00:58:02,150
that we may get to near
the end of this class,

1062
00:58:02,150 --> 00:58:07,580
but it requires something called
coupled wave theory, which

1063
00:58:07,580 --> 00:58:11,000
is relatively advanced.

1064
00:58:11,000 --> 00:58:14,360
A rule is that if that
period is relatively long,

1065
00:58:14,360 --> 00:58:16,820
a rule of thumb
about 10 wavelengths

1066
00:58:16,820 --> 00:58:20,380
or longer, the effect of
the polarization is minor.

1067
00:58:20,380 --> 00:58:23,900
You have a grating whose
period is 2 wavelengths,

1068
00:58:23,900 --> 00:58:25,713
3 wavelengths,
something like that,

1069
00:58:25,713 --> 00:58:27,380
then the effect is
actually significant,

1070
00:58:27,380 --> 00:58:29,420
and it cannot be neglected.

1071
00:58:29,420 --> 00:58:31,280
But in this class,
we will actually--

1072
00:58:31,280 --> 00:58:32,780
at least at the
beginning, we will

1073
00:58:32,780 --> 00:58:35,630
pretend that the
period is large enough

1074
00:58:35,630 --> 00:58:38,035
that we can neglect
polarization effects.

1075
00:58:38,035 --> 00:58:39,050
Very good question.

1076
00:58:39,050 --> 00:58:41,128
Practice actually
makes a big difference.

1077
00:58:44,800 --> 00:58:45,880
Other questions?

1078
00:58:52,750 --> 00:58:55,850
STUDENT: Just now, we show
one grating, which modifies

1079
00:58:55,850 --> 00:58:57,550
the amplitude of the wave.

1080
00:58:57,550 --> 00:59:00,513
So what's a physical
picture of such a grating?

1081
00:59:00,513 --> 00:59:02,680
GEORGE BARBASTATHIS: So for
example, what you can do

1082
00:59:02,680 --> 00:59:05,030
is you can Just take
a piece of glass,

1083
00:59:05,030 --> 00:59:09,040
and evaporate the aluminum or
some other metal into the areas

1084
00:59:09,040 --> 00:59:10,830
that you want to
block the light.

1085
00:59:10,830 --> 00:59:17,170
So that's one way
to make the grating,

1086
00:59:17,170 --> 00:59:23,530
So we take a piece
of glass, then

1087
00:59:23,530 --> 00:59:29,560
you can evaporate
aluminum uniformly here.

1088
00:59:29,560 --> 00:59:36,490
Then on top of it, you can put
photoresist and patterning.

1089
00:59:36,490 --> 00:59:37,620
OK.

1090
00:59:37,620 --> 00:59:42,070
Then you can etch.

1091
00:59:42,070 --> 00:59:45,550
So why when you etch, you will
actually remove these parts.

1092
00:59:45,550 --> 00:59:59,590
What you will end up
is something like this.

1093
00:59:59,590 --> 01:00:02,067
Then you also remove the
rest of the photoresist.

1094
01:00:10,346 --> 01:00:12,810
So that would be, now,
a binary amplitude

1095
01:00:12,810 --> 01:00:15,510
grating because you
have aluminum blocking

1096
01:00:15,510 --> 01:00:18,100
the light here and
then not in here,

1097
01:00:18,100 --> 01:00:21,190
so the light goes through.

1098
01:00:21,190 --> 01:00:24,180
Yeah, as I said, this is called
a binary amplitude grating,

1099
01:00:24,180 --> 01:00:26,880
and you will deal with
it in the homework.

1100
01:00:26,880 --> 01:00:29,730
To make a grayscale, like
the one that I showed

1101
01:00:29,730 --> 01:00:33,030
in my calculations here,
it's actually a little bit

1102
01:00:33,030 --> 01:00:37,500
difficult. To make this kind
of thing is rather hard.

1103
01:00:37,500 --> 01:00:42,270
It can be done with sort of
proper evaporation tools,

1104
01:00:42,270 --> 01:00:42,930
but very hard.

1105
01:00:42,930 --> 01:00:47,310
So this is a rather--

1106
01:00:47,310 --> 01:00:49,710
it's a simple construction
that they need in order

1107
01:00:49,710 --> 01:00:51,450
to describe the
physics, but it's not

1108
01:00:51,450 --> 01:00:52,950
very easy to implement
the practice.

1109
01:00:52,950 --> 01:00:54,950
STUDENT: And I guess,
just to tie it to the demo

1110
01:00:54,950 --> 01:00:56,940
for the phase counterpart,
you can do this

1111
01:00:56,940 --> 01:01:00,000
by-- you saw the
interference pattern produced

1112
01:01:00,000 --> 01:01:03,730
by the Mach-Zehnder, right, that
there were sinusoidal fringes.

1113
01:01:03,730 --> 01:01:06,470
So instead of the CCD, you
could imagine putting there,

1114
01:01:06,470 --> 01:01:08,310
for the sensitive
polymer, for example,

1115
01:01:08,310 --> 01:01:10,650
like photoresist,
that will expose

1116
01:01:10,650 --> 01:01:14,280
that material with these
bright and dark fringes.

1117
01:01:14,280 --> 01:01:16,560
That basically will cause
that refractive index

1118
01:01:16,560 --> 01:01:18,840
in a sinusoidal
pattern, and that

1119
01:01:18,840 --> 01:01:22,620
will produce the phase
version of this grating that

1120
01:01:22,620 --> 01:01:25,278
were shown before also.

1121
01:01:25,278 --> 01:01:26,820
GEORGE BARBASTATHIS:
As it turns out,

1122
01:01:26,820 --> 01:01:28,930
if you do what Pepe
just described-- let

1123
01:01:28,930 --> 01:01:30,720
me see if I get to it here.

1124
01:01:30,720 --> 01:01:33,762
If you do what Pepe just
described, you get two effects.

1125
01:01:33,762 --> 01:01:35,220
One is that the
index of refraction

1126
01:01:35,220 --> 01:01:40,290
changes in the exposed areas,
but also, because there is--

1127
01:01:40,290 --> 01:01:42,240
you can understand
this is a fluid effect.

1128
01:01:42,240 --> 01:01:44,670
The index of refraction
changing means

1129
01:01:44,670 --> 01:01:48,030
that the material becomes,
actually, slightly denser.

1130
01:01:48,030 --> 01:01:51,690
So this will result in
a surface undulation

1131
01:01:51,690 --> 01:01:55,890
in order to conserve its--

1132
01:01:55,890 --> 01:01:59,520
to conserve matter, the surface
will actually have to undulate.

1133
01:01:59,520 --> 01:02:02,590
So you end up with a grating
that is non-surface relief.

1134
01:02:02,590 --> 01:02:05,880
So these exposure processes
are relatively complicated.

1135
01:02:05,880 --> 01:02:12,750
So if we have time, I may go
into it later during the class.

1136
01:02:12,750 --> 01:02:17,800
But another way to do this, to
do the binary grating, which

1137
01:02:17,800 --> 01:02:20,860
is done very often
in practice, is using

1138
01:02:20,860 --> 01:02:23,266
the electron-beam lithography.

1139
01:02:26,530 --> 01:02:36,810
Where is my-- so
this kind of grating

1140
01:02:36,810 --> 01:02:39,400
is also done with
a similar process,

1141
01:02:39,400 --> 01:02:41,510
but using electron-beam
lithography.

1142
01:02:41,510 --> 01:02:43,180
So what you do is basically--

1143
01:02:46,450 --> 01:02:52,070
one way to do it is
you start with glass,

1144
01:02:52,070 --> 01:02:56,290
then you coat it with a
material that is called HSQ.

1145
01:02:56,290 --> 01:02:59,900
And actually, you can pattern
it using an electron beam.

1146
01:02:59,900 --> 01:03:07,100
You put an electron beam
resist on top, which is PMMA.

1147
01:03:07,100 --> 01:03:09,830
Then, using the electron beam,
you remove parts of the PMMA.

1148
01:03:09,830 --> 01:03:15,870
And then you etch
away these parts.

1149
01:03:15,870 --> 01:03:20,490
Then you end up with a
pattern that looks like this.

1150
01:03:29,200 --> 01:03:30,400
So HSQ is actually clear.

1151
01:03:30,400 --> 01:03:32,020
It's very similar to glass.

1152
01:03:32,020 --> 01:03:36,860
So this would be a physical
realization of the binary phase

1153
01:03:36,860 --> 01:03:37,960
grating.

1154
01:03:37,960 --> 01:03:40,690
The light is all
transparent, but the light

1155
01:03:40,690 --> 01:03:43,240
propagates a longer
distance inside the material

1156
01:03:43,240 --> 01:03:46,470
here, whereas here, it
propagates in the air.

1157
01:03:46,470 --> 01:03:48,430
So that's why we'll
get this effect.

1158
01:03:51,430 --> 01:03:54,055
STUDENT: [INAUDIBLE]

1159
01:03:54,055 --> 01:03:55,680
GEORGE BARBASTATHIS:
With a laser beam,

1160
01:03:55,680 --> 01:03:58,050
you can actually get
very small gratings.

1161
01:03:58,050 --> 01:04:02,370
If it is completely
periodic, then people

1162
01:04:02,370 --> 01:04:07,460
can make something like 100
nanometers or less easily.

1163
01:04:07,460 --> 01:04:10,120
So what I mean by 100 nanometers
is that this period over here

1164
01:04:10,120 --> 01:04:14,350
might be 100 nanometers,
even less, actually.

1165
01:04:14,350 --> 01:04:16,080
So of course, for
visible light, that

1166
01:04:16,080 --> 01:04:18,240
would be a
subwavelength grating.

1167
01:04:18,240 --> 01:04:19,710
It wouldn't do much.

1168
01:04:19,710 --> 01:04:24,150
But it could be useful, for
example, for ultraviolet light,

1169
01:04:24,150 --> 01:04:27,540
or for other applications where
you don't use it as a grating.

1170
01:04:27,540 --> 01:04:30,060
If it is nonperiodic,
very often--

1171
01:04:30,060 --> 01:04:31,590
we will see later in the class--

1172
01:04:31,590 --> 01:04:34,080
sometimes, we want to make
patterns that look like this.

1173
01:04:40,000 --> 01:04:44,800
OK, so this is, again, a phase
grating, but it is nonperiodic.

1174
01:04:44,800 --> 01:04:46,940
You can see that
the period changes.

1175
01:04:46,940 --> 01:04:49,125
So if the period changes
in the quadratic fashion,

1176
01:04:49,125 --> 01:04:50,500
this is a very
important element.

1177
01:04:50,500 --> 01:04:55,180
It's called a "Fresnel" Here
is, again, the French term--

1178
01:04:55,180 --> 01:04:58,700
"Fresnel zone plate."

1179
01:04:58,700 --> 01:05:00,280
This I will describe
in class later.

1180
01:05:00,280 --> 01:05:04,240
It's a very, very important
optical component.

1181
01:05:04,240 --> 01:05:07,600
But it turns out because
of this kind of pattern

1182
01:05:07,600 --> 01:05:10,580
it is nonperiodic, it is
not so easy to make using

1183
01:05:10,580 --> 01:05:12,250
electron-beam lithography.

1184
01:05:12,250 --> 01:05:14,950
So in that case, you
have to go to great pains

1185
01:05:14,950 --> 01:05:16,670
to make small features.

1186
01:05:16,670 --> 01:05:19,900
So in this case, it's
a little bit strange.

1187
01:05:19,900 --> 01:05:22,510
But because of the
nonperiodicity,

1188
01:05:22,510 --> 01:05:24,520
the feature sizes
here are limited

1189
01:05:24,520 --> 01:05:29,290
to bigger values-- so for
example, 200 to 300 nanometers.

1190
01:05:29,290 --> 01:05:33,010
This has to do with the
way electron beams scatter

1191
01:05:33,010 --> 01:05:35,080
from the photoresist.

1192
01:05:35,080 --> 01:05:37,600
Actually, some of
you are taking,

1193
01:05:37,600 --> 01:05:41,710
simultaneously, Professor
[? Bargrand's ?] class.

1194
01:05:41,710 --> 01:05:44,800
He goes into great detail
into this particular--

1195
01:05:44,800 --> 01:05:46,660
into why this is the case.

1196
01:05:50,620 --> 01:05:53,080
For visible light, it
is actually quite easy

1197
01:05:53,080 --> 01:05:56,380
to make reasonably sized
gratings with lambda

1198
01:05:56,380 --> 01:05:59,920
bigger than, say, five
or 10 wavelengths.

1199
01:05:59,920 --> 01:06:02,350
It is actually easy to
make these kind of gratings

1200
01:06:02,350 --> 01:06:05,740
either with optical exposure,
like Pepe described,

1201
01:06:05,740 --> 01:06:08,850
or with electron-beam
lithography.

1202
01:06:08,850 --> 01:06:12,520
There's also other
techniques, for example,

1203
01:06:12,520 --> 01:06:13,995
interference lithography.

1204
01:06:13,995 --> 01:06:15,370
There's a whole
set of techniques

1205
01:06:15,370 --> 01:06:17,078
that people use to
make this grating.

1206
01:06:17,078 --> 01:06:17,620
I don't know.

1207
01:06:17,620 --> 01:06:22,990
Pepe, do you know, the one that
you showed, how was it made?

1208
01:06:22,990 --> 01:06:25,210
STUDENT: Not really.

1209
01:06:25,210 --> 01:06:25,810
Not really.

1210
01:06:25,810 --> 01:06:27,250
GEORGE BARBASTATHIS: [INAUDIBLE]

1211
01:06:27,250 --> 01:06:28,833
STUDENT: It was the
one that you used.

1212
01:06:28,833 --> 01:06:31,115
Do you know?

1213
01:06:31,115 --> 01:06:32,240
I don't know, to be honest.

1214
01:06:32,240 --> 01:06:33,430
No.

1215
01:06:33,430 --> 01:06:36,688
I think it's a binary grating
and probably was done--

1216
01:06:36,688 --> 01:06:38,230
I don't think it
was done with e-beam

1217
01:06:38,230 --> 01:06:39,380
because it's pretty large.

1218
01:06:39,380 --> 01:06:40,380
GEORGE BARBASTATHIS: No.

1219
01:06:40,380 --> 01:06:40,540
No.

1220
01:06:40,540 --> 01:06:40,670
No.

1221
01:06:40,670 --> 01:06:41,430
STUDENT: It has to do with--

1222
01:06:41,430 --> 01:06:43,870
GEORGE BARBASTATHIS: [INAUDIBLE]
suspect that way also

1223
01:06:43,870 --> 01:06:47,190
of making these
gratings by molding.

1224
01:06:47,190 --> 01:06:48,190
That's another way.

1225
01:06:48,190 --> 01:06:51,580
So you basically create a
very expensive negative,

1226
01:06:51,580 --> 01:06:52,720
like a mold.

1227
01:06:52,720 --> 01:06:53,860
And then you stamp--

1228
01:06:53,860 --> 01:06:54,610
not quite glass.

1229
01:06:54,610 --> 01:06:57,700
It has to be some kind
of a pliable material.

1230
01:06:57,700 --> 01:06:58,450
That's one way.

1231
01:06:58,450 --> 01:07:01,990
And then they can
also make them by--

1232
01:07:01,990 --> 01:07:04,960
basically, by scratching
by very accurate machines.

1233
01:07:04,960 --> 01:07:08,260
So there are many, many
different ways to make--

1234
01:07:08,260 --> 01:07:11,452
STUDENT: The dip-pen lithography
or some kind of [INAUDIBLE]

1235
01:07:11,452 --> 01:07:13,410
GEORGE BARBASTATHIS: It's
reactive ion etching.

1236
01:07:13,410 --> 01:07:15,010
STUDENT: Dip-pen lithography?

1237
01:07:15,010 --> 01:07:16,385
GEORGE BARBASTATHIS:
Oh, Dip-pen.

1238
01:07:16,385 --> 01:07:17,648
STUDENT: Yeah.

1239
01:07:17,648 --> 01:07:18,690
GEORGE BARBASTATHIS: Yes.

1240
01:07:18,690 --> 01:07:21,345
Except dip-pen lithography
can allow you to do only--

1241
01:07:24,680 --> 01:07:26,160
where is it?

1242
01:07:26,160 --> 01:07:28,880
I lost my amplitude grating.

1243
01:07:28,880 --> 01:07:32,330
It can only make amplitude
gratings, not phase gratings.

1244
01:07:32,330 --> 01:07:34,070
Depends.

1245
01:07:34,070 --> 01:07:34,930
It's ink, right?

1246
01:07:34,930 --> 01:07:36,180
So good deposit ink.

1247
01:07:36,180 --> 01:07:38,390
And this way, you
can actually absorb.

1248
01:07:38,390 --> 01:07:40,460
You can get absorption bands.

1249
01:07:40,460 --> 01:07:43,700
But I don't know if you
can make phase gratings

1250
01:07:43,700 --> 01:07:44,970
with dip-pen lithography.

1251
01:07:44,970 --> 01:07:45,910
Have to look into it.

1252
01:07:50,310 --> 01:07:51,660
They're very popular nowadays.

1253
01:07:51,660 --> 01:07:55,320
Again, professor [? Bagram ?]
goes into this in more detail.

1254
01:07:55,320 --> 01:07:59,992
But a very popular
technique is-- it's called--

1255
01:07:59,992 --> 01:08:00,700
help me out here.

1256
01:08:00,700 --> 01:08:04,120
How do you call the
stamping technique?

1257
01:08:04,120 --> 01:08:05,745
STUDENT: Oh, we
call it nanoimprint.

1258
01:08:05,745 --> 01:08:07,370
GEORGE BARBASTATHIS:
Nanoimprint, yeah.

1259
01:08:07,370 --> 01:08:09,040
Nanoimprint lithography, yeah.

1260
01:08:09,040 --> 01:08:12,340
That's a very popular technique
to make these kind of elements

1261
01:08:12,340 --> 01:08:16,210
with very small
periods, actually.

1262
01:08:16,210 --> 01:08:18,790
People make [? 15 ?]
nanometers or less,

1263
01:08:18,790 --> 01:08:20,791
actually, with nanoimprint.

1264
01:08:25,604 --> 01:08:26,479
STUDENT: Sorry, Prof.

1265
01:08:26,479 --> 01:08:28,879
Do we have time for
one more question?

1266
01:08:28,879 --> 01:08:30,254
GEORGE BARBASTATHIS:
I have time,

1267
01:08:30,254 --> 01:08:31,609
but some people have classes.

1268
01:08:31,609 --> 01:08:33,080
But yeah, go ahead.

1269
01:08:33,080 --> 01:08:34,330
STUDENT: OK, I'm just curious.

1270
01:08:34,330 --> 01:08:36,880
Let's say, just now, you
mentioned the sine theta cannot

1271
01:08:36,880 --> 01:08:38,200
be greater than 1.

1272
01:08:38,200 --> 01:08:40,779
So what happens if we
shine the visible light

1273
01:08:40,779 --> 01:08:44,077
on to a grating with a period
[INAUDIBLE] 100 nanometer?

1274
01:08:44,077 --> 01:08:45,160
GEORGE BARBASTATHIS: Yeah.

1275
01:08:45,160 --> 01:08:46,430
So I actually said it before.

1276
01:08:46,430 --> 01:08:47,680
Let me repeat it again.

1277
01:08:47,680 --> 01:08:54,300
So he's asking, what will happen
if you have a grating here,

1278
01:08:54,300 --> 01:08:59,590
like this, for example,
where the period lambda now

1279
01:08:59,590 --> 01:09:02,662
is less than the wavelength?

1280
01:09:02,662 --> 01:09:03,330
OK.

1281
01:09:03,330 --> 01:09:09,649
So, if you recall, a
planewave is of this form--

1282
01:09:09,649 --> 01:09:17,050
e to the i kxx
plus kyy plus kzz--

1283
01:09:17,050 --> 01:09:20,710
and this, we saw when
we did electromagnetics.

1284
01:09:20,710 --> 01:09:25,189
And we also saw that because
of the electromagnetic wave

1285
01:09:25,189 --> 01:09:29,180
equation, the wave
vector components here--

1286
01:09:29,180 --> 01:09:30,819
they have to satisfy
this equation.

1287
01:09:38,319 --> 01:09:46,740
OK, so now, according to the
equation that we had before,

1288
01:09:46,740 --> 01:09:50,880
if this is the x direction--
and let's ignore y for now.

1289
01:09:50,880 --> 01:09:55,000
So basically, I can
neglect this term for now.

1290
01:09:55,000 --> 01:10:03,170
And what we have is that the
sine factor will equal lambda

1291
01:10:03,170 --> 01:10:05,710
upon the period.

1292
01:10:05,710 --> 01:10:07,220
And it's bigger than 1.

1293
01:10:07,220 --> 01:10:11,810
OK, mathematically, if
I have a complex angle,

1294
01:10:11,810 --> 01:10:13,430
the sine can be bigger than 1.

1295
01:10:13,430 --> 01:10:15,560
But what does this
physically mean?

1296
01:10:15,560 --> 01:10:18,740
Physically, if you plug
into this equation here,

1297
01:10:18,740 --> 01:10:27,440
you will get that the kz
squared equals 2 pi upon lambda

1298
01:10:27,440 --> 01:10:37,430
squared minus 2 pi upon lambda
squared based on this equation.

1299
01:10:37,430 --> 01:10:47,270
When I said that kx equals
2 pi upon lambda sine theta

1300
01:10:47,270 --> 01:10:52,680
equals to pi upon lambda times
lambda over lambda equals 2 pi

1301
01:10:52,680 --> 01:10:53,690
upon lambda--

1302
01:10:53,690 --> 01:10:55,480
OK.

1303
01:10:55,480 --> 01:11:01,490
So if I do that now, I will
get that the kz squared

1304
01:11:01,490 --> 01:11:02,465
equals 2 pi.

1305
01:11:11,640 --> 01:11:14,850
And the quantity inside the
square root is less than 0,

1306
01:11:14,850 --> 01:11:20,670
so we'll get that the kz
now will equal plus minus.

1307
01:11:20,670 --> 01:11:22,560
It will become imaginary.

1308
01:11:22,560 --> 01:11:25,980
[INAUDIBLE] plus minus i
the square root of something

1309
01:11:25,980 --> 01:11:26,882
positive.

1310
01:11:31,975 --> 01:11:33,100
So what does this mean now?

1311
01:11:33,100 --> 01:11:46,870
I have a wave whose amplitude
will be like e to the i kxx.

1312
01:11:46,870 --> 01:11:48,820
Nothing changes here.

1313
01:11:48,820 --> 01:11:56,220
But then it will be plus minus
i times something positive.

1314
01:11:56,220 --> 01:12:04,432
Let's call this "alpha"
for gravity here.

1315
01:12:04,432 --> 01:12:05,890
OK, this is what
the wave will look

1316
01:12:05,890 --> 01:12:09,010
like after the
subwavelength grating.

1317
01:12:09,010 --> 01:12:17,090
So now, clearly here,
if you multiply the two

1318
01:12:17,090 --> 01:12:20,100
imaginary units, we
will get a real number.

1319
01:12:20,100 --> 01:12:24,180
So we get either exponential
decay or exponential explosion.

1320
01:12:24,180 --> 01:12:27,380
Physically, you cannot expect
the wave to grow exponentially.

1321
01:12:27,380 --> 01:12:30,470
That would mean someone
is supplying power,

1322
01:12:30,470 --> 01:12:32,190
and there's nothing
like that here.

1323
01:12:32,190 --> 01:12:33,590
So therefore, you
get a wave that

1324
01:12:33,590 --> 01:12:38,440
will look like e to
the minus alpha z--

1325
01:12:38,440 --> 01:12:40,240
there's a z here
in [INAUDIBLE]----

1326
01:12:40,240 --> 01:12:44,180
times e to the minus
e to the plus ikxx.

1327
01:12:44,180 --> 01:12:45,495
So what is this now?

1328
01:12:45,495 --> 01:12:48,710
The e to the minus alpha
z's an exponential decay

1329
01:12:48,710 --> 01:12:50,510
away from the grating.

1330
01:12:50,510 --> 01:12:53,480
The e do the ikx is what
we call a "surface wave."

1331
01:12:53,480 --> 01:12:55,470
It propagates like this.

1332
01:12:55,470 --> 01:12:57,580
So actually, what you
get after this grating

1333
01:12:57,580 --> 01:13:02,150
is you get a wave that
propagates parallel

1334
01:13:02,150 --> 01:13:03,770
to the grating.

1335
01:13:03,770 --> 01:13:06,170
I don't know if the camera
can show me on this one--

1336
01:13:06,170 --> 01:13:06,730
probably not.

1337
01:13:06,730 --> 01:13:09,470
But anyway, as you get the
wave that propagates parallel

1338
01:13:09,470 --> 01:13:12,710
to the grating, it is
called the "surface wave,"

1339
01:13:12,710 --> 01:13:14,240
but it doesn't live very long.

1340
01:13:14,240 --> 01:13:18,060
It decays exponentially
away from the grating.

1341
01:13:18,060 --> 01:13:20,130
That is actually called
an "evanescent wave."

1342
01:13:30,070 --> 01:13:31,710
And it is not really
part of the class.

1343
01:13:36,070 --> 01:13:38,880
This is a very simplified
description of what happens.

1344
01:13:38,880 --> 01:13:40,590
To properly do it,
I would have to take

1345
01:13:40,590 --> 01:13:42,620
into account polarization.

1346
01:13:42,620 --> 01:13:44,490
I would basically have
to be more careful

1347
01:13:44,490 --> 01:13:46,410
solving Maxwell's equations.

1348
01:13:46,410 --> 01:13:49,770
And because this class
is simple, basic optics,

1349
01:13:49,770 --> 01:13:52,060
I generally stay away from
those kinds of gray things.

1350
01:13:52,060 --> 01:13:54,030
That's why I don't
describe them in detail.

1351
01:13:54,030 --> 01:13:58,283
But since you asked, I gave you
a very simplified description.

1352
01:14:03,490 --> 01:14:06,012
We may talk about this
a bit near the end,

1353
01:14:06,012 --> 01:14:08,220
depending on how much time
we have left near the end.

1354
01:14:08,220 --> 01:14:12,175
But if we do, I will talk
about this in more detail.

1355
01:14:12,175 --> 01:14:13,550
It's also related
to the question

1356
01:14:13,550 --> 01:14:16,820
that someone has asked
before about polarization.

1357
01:14:16,820 --> 01:14:19,520
If you have a
subwavelength grating,

1358
01:14:19,520 --> 01:14:23,090
then you definitely have to
take into account the incident

1359
01:14:23,090 --> 01:14:23,870
polarization.

1360
01:14:30,268 --> 01:14:31,310
STUDENT: Yeah, actually--

1361
01:14:31,310 --> 01:14:33,768
GEORGE BARBASTATHIS: I see our
audience is [INAUDIBLE] now.

1362
01:14:33,768 --> 01:14:35,650
So maybe it's time for
us to get our martinis

1363
01:14:35,650 --> 01:14:37,690
and the rest of the others
to get their coffee.

1364
01:14:41,580 --> 01:14:43,130
OK?