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GEORGE BARBASTATHIS:
So a little bit
00:00:22.310 --> 00:00:26.450
of housekeeping
before we continue.
00:00:26.450 --> 00:00:28.100
First of all, you
may have noticed
00:00:28.100 --> 00:00:32.840
that in the reading
assignments, I have started
00:00:32.840 --> 00:00:35.930
Boston from Goodman's book.
00:00:35.930 --> 00:00:39.080
So there is some pros
and cons about this.
00:00:39.080 --> 00:00:42.920
Goodman is very good if you
are an engineer, especially
00:00:42.920 --> 00:00:45.140
electrical or
mechanical engineer.
00:00:45.140 --> 00:00:48.500
Because then you are very used
to thinking about systems,
00:00:48.500 --> 00:00:52.320
block diagrams,
transforms, and so on.
00:00:52.320 --> 00:00:55.070
So it is very nice
living this way.
00:00:55.070 --> 00:00:57.440
But it's a little
bit mathematical.
00:00:57.440 --> 00:01:01.930
Hecht is more on
the physics side.
00:01:01.930 --> 00:01:05.930
So I feel Hecht is written for
junior or sophomore physics
00:01:05.930 --> 00:01:06.430
students.
00:01:09.150 --> 00:01:12.270
And, of course, they're
very nicely complimentary.
00:01:12.270 --> 00:01:15.390
The only real downside is that
they use different notation.
00:01:15.390 --> 00:01:19.770
So if you tried Stanley
from both Hecht and Goodman,
00:01:19.770 --> 00:01:21.480
you have to be a
little bit careful
00:01:21.480 --> 00:01:23.850
to keep their notation--
00:01:23.850 --> 00:01:26.220
I mean, the notation
is not consistent,
00:01:26.220 --> 00:01:29.640
but you have to keep yourself
from getting confused
00:01:29.640 --> 00:01:32.010
by the inconsistent notation.
00:01:32.010 --> 00:01:34.350
Nevertheless, the diagrams
are, of course, consistent
00:01:34.350 --> 00:01:36.930
because they both
calculate the same.
00:01:36.930 --> 00:01:39.020
Fresnel, the fraction
pattern is the same
00:01:39.020 --> 00:01:40.742
from [INAUDIBLE]
pattern and so on.
00:01:40.742 --> 00:01:43.200
But you have to be a little
bit mindful of the coordinates.
00:01:43.200 --> 00:01:45.940
For example, they may be
using different symbols.
00:01:45.940 --> 00:01:47.790
Anyway, I highly
recommend that you're
00:01:47.790 --> 00:01:49.190
starting from both books.
00:01:49.190 --> 00:01:52.410
Hecht also has a much more
intuitive explanations,
00:01:52.410 --> 00:01:54.990
and many more
figures, and so on.
00:01:54.990 --> 00:01:59.070
But Goodman is more
rigorous, and also better
00:01:59.070 --> 00:02:01.900
suited to an engineer's
way of thinking.
00:02:01.900 --> 00:02:04.640
So that's why I
use both textbooks.
00:02:04.640 --> 00:02:09.430
By now, it's actually closer
to Goodman from this point on.
00:02:09.430 --> 00:02:12.090
So that's an additional benefit.
00:02:12.090 --> 00:02:15.630
Anyway, the reading assignments
are from both books.
00:02:15.630 --> 00:02:17.730
If you decide to
follow just one book.
00:02:17.730 --> 00:02:22.010
For example, either Hecht by
itself or Goodman by itself.
00:02:22.010 --> 00:02:26.168
You don't miss significantly,
you can follow either book.
00:02:26.168 --> 00:02:28.710
But I think until you get the
complete picture, if you follow
00:02:28.710 --> 00:02:35.410
both books and in the way they
serve to reinforce each other.
00:02:35.410 --> 00:02:38.390
Anyway, so that's the
study about the textbooks.
00:02:38.390 --> 00:02:40.730
A little bit more housekeeping.
00:02:40.730 --> 00:02:42.760
I have posted the
slightly revised
00:02:42.760 --> 00:02:45.790
version of Monday's notes.
00:02:45.790 --> 00:02:48.040
One minor correction,
if you look at this.
00:02:48.040 --> 00:02:51.490
This is the very last
slide from Monday.
00:02:51.490 --> 00:02:54.730
There was an error, at
least one that they found.
00:02:54.730 --> 00:02:59.200
In the expression for the
Fourier coefficient c sub q,
00:02:59.200 --> 00:03:03.040
it was the function
sinc of 2 over 2.
00:03:03.040 --> 00:03:03.880
This is correct.
00:03:03.880 --> 00:03:08.380
On Monday, there was an
extra pi inside the argument
00:03:08.380 --> 00:03:09.540
of the same function.
00:03:09.540 --> 00:03:12.130
That pi should not have been
there, so I've removed it.
00:03:12.130 --> 00:03:16.170
We will see today later the
derivation of this expression,
00:03:16.170 --> 00:03:19.200
actually, a similar expression.
00:03:19.200 --> 00:03:23.180
So hopefully, that
will clarify matters.
00:03:23.180 --> 00:03:27.640
The thing I did
compare to Monday.
00:03:27.640 --> 00:03:32.470
Piper reminded me on Monday,
when he discussed the grading,
00:03:32.470 --> 00:03:34.930
about the dispersion.
00:03:34.930 --> 00:03:39.640
So if you look at the--
00:03:39.640 --> 00:03:42.160
go back a little bit
to the expressions,
00:03:42.160 --> 00:03:46.550
or the diffraction
angle from a grating.
00:04:10.440 --> 00:04:12.140
OK, that's a good one.
00:04:12.140 --> 00:04:16.620
OK, so last week, we
focused on a discussion
00:04:16.620 --> 00:04:18.029
about the effect of the period.
00:04:18.029 --> 00:04:22.060
So we said that if you
make the period smaller,
00:04:22.060 --> 00:04:25.680
the diffraction order
is spread out more,
00:04:25.680 --> 00:04:27.630
so the diffraction
angle is inversely
00:04:27.630 --> 00:04:29.560
proportional to the period.
00:04:29.560 --> 00:04:32.920
Actually, the sine of
the diffraction angle.
00:04:32.920 --> 00:04:35.490
We didn't say anything
about wavelength.
00:04:35.490 --> 00:04:37.860
So, of course, the wavelength
appears in the numerator
00:04:37.860 --> 00:04:41.730
there, which means that if you
have light of multiple colors,
00:04:41.730 --> 00:04:46.350
then longer wavelengths will
focus at the longer angle.
00:04:46.350 --> 00:04:48.270
So this is what you
see on the last slide
00:04:48.270 --> 00:04:52.410
that they posted today
in the revised notes.
00:04:52.410 --> 00:04:58.440
So I have a grating here which
is illuminated by white light.
00:04:58.440 --> 00:05:00.960
So, of course, white
light is composed
00:05:00.960 --> 00:05:03.120
of a broad spectrum
of colors ranging
00:05:03.120 --> 00:05:06.600
from somewhere in the
infinite to somewhere
00:05:06.600 --> 00:05:09.200
in the ultraviolet.
00:05:09.200 --> 00:05:11.160
Anyway, let's take
the discussion.
00:05:11.160 --> 00:05:13.680
Let's keep it to the
visible wavelength.
00:05:13.680 --> 00:05:19.185
So, of course, the red
wavelength has a longer color.
00:05:19.185 --> 00:05:22.500
I'm sorry, the red color
has a longer wavelength.
00:05:22.500 --> 00:05:25.900
So therefore, the red color
is diffracted at the larger
00:05:25.900 --> 00:05:28.200
angle than the blue.
00:05:28.200 --> 00:05:32.010
So this phenomenon is called
dispersion, very similar
00:05:32.010 --> 00:05:35.080
to the grating dispersion.
00:05:35.080 --> 00:05:37.690
But it is referred to
as anomalous, because it
00:05:37.690 --> 00:05:40.290
is the opposite of the grating.
00:05:40.290 --> 00:05:42.460
I'm sorry, it is the
opposite of a prism.
00:05:42.460 --> 00:05:43.420
What am I saying?
00:05:43.420 --> 00:05:44.800
Let me start over.
00:05:44.800 --> 00:05:47.950
So in the case of
a prism, as well as
00:05:47.950 --> 00:05:50.710
in the case of a
grating, the phenomenon
00:05:50.710 --> 00:05:53.350
of analysis of
white light, where
00:05:53.350 --> 00:05:55.960
white light after propagating
through the element gets
00:05:55.960 --> 00:05:58.890
decomposed into its
color component.
00:05:58.890 --> 00:06:00.850
And each one color
component propagates
00:06:00.850 --> 00:06:02.290
at the different angle.
00:06:02.290 --> 00:06:05.280
This phenomenon is
referred to as dispersion.
00:06:05.280 --> 00:06:07.420
If the two are different
in the sense that
00:06:07.420 --> 00:06:10.540
in the case of a
grating, the blue light
00:06:10.540 --> 00:06:15.190
is refracted at the smaller
angle than the red light.
00:06:15.190 --> 00:06:17.230
That is called the
normal dispersion.
00:06:17.230 --> 00:06:20.610
In the case of a prism, it
is not diffraction anymore.
00:06:20.610 --> 00:06:21.790
It is refraction.
00:06:21.790 --> 00:06:23.470
And we saw why it happens.
00:06:23.470 --> 00:06:25.300
This has to do
with the dependence
00:06:25.300 --> 00:06:30.000
of the index of refraction
of glass on wavelength.
00:06:30.000 --> 00:06:33.340
And the dependence is
such that the blue light
00:06:33.340 --> 00:06:38.570
is refracted at the larger
angle, in the case of a prism.
00:06:38.570 --> 00:06:41.570
That is called normal
dispersion, as opposed
00:06:41.570 --> 00:06:42.590
to the anomalous.
00:06:42.590 --> 00:06:46.460
Now, there is nothing anomalous
about the grating, I suppose.
00:06:46.460 --> 00:06:48.410
Historically, this had
to do with the fact
00:06:48.410 --> 00:06:53.660
that people observed this
phenomenon with prisms first,
00:06:53.660 --> 00:06:55.863
so they call it
normal dispersion.
00:06:55.863 --> 00:06:57.280
Then they noticed
that the grating
00:06:57.280 --> 00:07:00.560
does the opposite thing, so they
said, well, this is abnormal.
00:07:00.560 --> 00:07:02.340
They called it anomalous.
00:07:02.340 --> 00:07:03.950
Anomalous in Greek
means abnormal.
00:07:07.130 --> 00:07:11.470
Anyway, so this is what I wanted
to add to Monday's lecture.
00:07:11.470 --> 00:07:13.190
Are there any questions
about gratings?
00:07:45.470 --> 00:07:47.635
Actually, between today
and next Wednesday,
00:07:47.635 --> 00:07:51.740
we will cover the basics
of diffraction theory.
00:07:51.740 --> 00:07:54.230
And the rest of
the class will be
00:07:54.230 --> 00:07:56.510
basically applications
on what we
00:07:56.510 --> 00:07:59.850
cover in these three lectures.
00:07:59.850 --> 00:08:01.680
So needless to say,
these three lectures
00:08:01.680 --> 00:08:04.441
are very, very important.
00:08:04.441 --> 00:08:11.770
So let us start with a little
observation on our Fresnel
00:08:11.770 --> 00:08:13.360
propagation formula.
00:08:13.360 --> 00:08:15.970
So to remind you very
briefly, today, I
00:08:15.970 --> 00:08:21.130
will be using the
whiteboard a lot.
00:08:21.130 --> 00:08:24.340
But the equations
that I write are all
00:08:24.340 --> 00:08:29.680
of them, either in the
notes or in the textbooks.
00:08:29.680 --> 00:08:33.340
So feel free to copy them if
you like, but you don't have to.
00:08:33.340 --> 00:08:35.275
It may be better.
00:08:35.275 --> 00:08:37.900
It's up to you whether you want
to copy the derivations or not.
00:08:37.900 --> 00:08:40.792
But there will be a lot of them.
00:08:40.792 --> 00:08:42.250
Probably by the
end of the lecture,
00:08:42.250 --> 00:08:45.770
your wrist may be
a little bit tired.
00:08:45.770 --> 00:08:47.300
OK.
00:08:47.300 --> 00:08:55.410
So to remind you, we did
this about a week ago.
00:08:55.410 --> 00:09:02.200
We said that if you have a
complex field at the plane xy,
00:09:02.200 --> 00:09:04.940
and then you propagate
by a distance, z.
00:09:04.940 --> 00:09:08.060
Then at the output
plane x prime, y prime,
00:09:08.060 --> 00:09:11.410
the field is given by
this convolution integral,
00:09:11.410 --> 00:09:14.350
which I will rewrite here.
00:09:14.350 --> 00:09:22.750
So the field at the output
after propagating by distance z
00:09:22.750 --> 00:09:25.540
equals some constants--
00:09:30.185 --> 00:09:31.560
and we'll spend
some time talking
00:09:31.560 --> 00:09:35.280
about these constants--
times what we'll
00:09:35.280 --> 00:09:38.470
call the convolution integral.
00:09:38.470 --> 00:09:41.690
So the convolution integral
is really written something
00:09:41.690 --> 00:09:42.950
like this, e to the i pi.
00:09:55.930 --> 00:10:02.200
OK, so this is the integral
within the Fresnel and scalar
00:10:02.200 --> 00:10:03.710
approximations.
00:10:03.710 --> 00:10:06.110
Then what we can do is we can--
00:10:06.110 --> 00:10:09.920
let's take this
exponent, and expand it.
00:10:13.110 --> 00:10:16.660
So I will only do
it for the x case.
00:10:27.047 --> 00:10:28.130
That's not rocket science.
00:10:28.130 --> 00:10:31.580
They've just expanded
the binomial.
00:10:31.580 --> 00:10:34.640
Now, let me remind you.
00:10:34.640 --> 00:10:40.930
So the x prime coordinate
is at the output plane.
00:10:40.930 --> 00:10:43.900
And also, you can see that
it is not participating
00:10:43.900 --> 00:10:44.770
in the integration.
00:10:44.770 --> 00:10:47.350
The integration is
with respect to x.
00:10:47.350 --> 00:10:49.120
So the x prime
part will actually
00:10:49.120 --> 00:10:52.520
be thrown out of the integral.
00:10:52.520 --> 00:10:56.530
And then I have
this remaining part.
00:10:56.530 --> 00:11:00.010
So the next thing
I'm going to do
00:11:00.010 --> 00:11:05.590
is I'm going to assume that
the input field is finite.
00:11:05.590 --> 00:11:13.060
So finite means that the field,
if you look at the input field
00:11:13.060 --> 00:11:13.990
here.
00:11:13.990 --> 00:11:18.520
The field is non-zero in
a relatively small region
00:11:18.520 --> 00:11:20.200
near the optical axis.
00:11:20.200 --> 00:11:23.710
But then away from the optical
axis, the field becomes zero.
00:11:23.710 --> 00:11:26.510
That is a reasonable
assumption, because most objects
00:11:26.510 --> 00:11:30.867
that we're going to create
in real life are finite.
00:11:30.867 --> 00:11:32.200
We've said this before, I think.
00:11:32.200 --> 00:11:34.180
We deal with things
like plane waves
00:11:34.180 --> 00:11:37.140
and spherical waves in this
class, which are infinite.
00:11:37.140 --> 00:11:39.700
But these are idealizations
that we use in order
00:11:39.700 --> 00:11:42.220
to make the math simpler.
00:11:42.220 --> 00:11:46.960
In this case, let's use actually
a real life assumption, which
00:11:46.960 --> 00:11:50.740
is that the object is finite.
00:11:50.740 --> 00:11:54.430
And since the
object is finite, x
00:11:54.430 --> 00:11:59.000
will be confined to
relatively small values.
00:11:59.000 --> 00:12:03.050
So what I will do now is I
will look at this expression, x
00:12:03.050 --> 00:12:05.710
prime over lambda z.
00:12:05.710 --> 00:12:09.070
This is this part
of the exponent.
00:12:09.070 --> 00:12:13.930
And what I will do now is I will
allow z to become very large.
00:12:18.020 --> 00:12:18.990
So what does this mean?
00:12:18.990 --> 00:12:20.840
It means that I
start propagating
00:12:20.840 --> 00:12:24.950
further and further and further
away from the transparency.
00:12:24.950 --> 00:12:29.880
Of course, x is limited by
the size of the input field.
00:12:29.880 --> 00:12:35.210
So x square is also
limited, but z grows
00:12:35.210 --> 00:12:37.470
as I propagate further away.
00:12:37.470 --> 00:12:45.820
So there will come a point
where x square maximum
00:12:45.820 --> 00:12:49.210
will become less than lambda z.
00:12:49.210 --> 00:12:53.470
That is, the maximum value
of this fraction over here
00:12:53.470 --> 00:12:55.000
will become less than one.
00:12:55.000 --> 00:12:57.310
And in fact, if I keep
propagating further
00:12:57.310 --> 00:12:59.500
and further, this
term will actually
00:12:59.500 --> 00:13:02.850
grow less and less and less.
00:13:02.850 --> 00:13:06.770
OK, if that is the case,
then the coefficient--
00:13:06.770 --> 00:13:11.080
then this term over here
will become negligible,
00:13:11.080 --> 00:13:13.730
which means that I can drop it.
00:13:13.730 --> 00:13:18.230
OK, so this is what is written
in your transparency over here.
00:13:18.230 --> 00:13:23.120
Basically, I actually did
this for both x and y.
00:13:23.120 --> 00:13:25.820
I pulled out the
x prime, y prime.
00:13:25.820 --> 00:13:28.200
And I'm left with this
expression over here.
00:13:28.200 --> 00:13:32.190
So let me also do it
on my whiteboard here.
00:13:32.190 --> 00:13:34.640
So what I'm saying
is that the g out.
00:13:42.730 --> 00:13:46.420
In the whiteboard, I will
not write the y-coordinates,
00:13:46.420 --> 00:13:48.310
because it is too much.
00:13:48.310 --> 00:13:53.300
Well, maybe I write
them to avoid confusion.
00:13:53.300 --> 00:13:55.510
OK, so this will
now take this form.
00:14:07.820 --> 00:14:09.440
This got pulled
out of the integral
00:14:09.440 --> 00:14:12.360
because they don't participate
in the integration.
00:14:12.360 --> 00:14:15.610
And then what I have
inside looks like this.
00:14:34.470 --> 00:14:36.560
And what I was saying
is that if I let z
00:14:36.560 --> 00:14:39.620
become long enough,
then it can basically
00:14:39.620 --> 00:14:41.840
neglect those two terms.
00:14:41.840 --> 00:14:44.570
Now, why do I neglect
only these two terms,
00:14:44.570 --> 00:14:51.280
and not the products of y over
y times y prime and x times
00:14:51.280 --> 00:14:52.720
x prime.
00:14:52.720 --> 00:14:54.460
Why should they keep
those two terms?
00:14:54.460 --> 00:14:56.950
And, of course, I forgot
something in both cases.
00:14:56.950 --> 00:15:03.350
In here, I should have
g sub in of x comma y,
00:15:03.350 --> 00:15:04.840
and the same in here.
00:15:04.840 --> 00:15:06.230
OK, you see what I did?
00:15:06.230 --> 00:15:10.970
Over here, I need to put the
input field, and the same here.
00:15:18.880 --> 00:15:23.370
So why am I keeping this term,
and I am not dropping it?
00:15:30.780 --> 00:15:32.280
If I drop it, it's
very easy, right?
00:15:32.280 --> 00:15:35.040
If I drop it altogether,
then what I will get
00:15:35.040 --> 00:15:39.360
is the integral of g sub in.
00:15:39.360 --> 00:15:41.260
It's like the
average of the input.
00:15:41.260 --> 00:15:43.665
Why am I not doing that?
00:15:43.665 --> 00:15:45.290
AUDIENCE: If you
remove the cross term,
00:15:45.290 --> 00:15:48.860
then we actually neglect the
complete spatial variation
00:15:48.860 --> 00:15:52.200
of the input field,
and it becomes
00:15:52.200 --> 00:15:55.630
like a pointed [INAUDIBLE].
00:15:55.630 --> 00:15:56.630
GEORGE BARBASTATHIS: OK.
00:15:56.630 --> 00:15:59.510
So I would actually
disagree with that.
00:15:59.510 --> 00:16:03.740
If I neglect it, I will get
that g sub out of x comma
00:16:03.740 --> 00:16:11.030
y prime equals g sub
in of x comma y dx, dy.
00:16:11.030 --> 00:16:12.650
So this is a constant, right?
00:16:12.650 --> 00:16:16.170
It is like the
average of g sub in.
00:16:16.170 --> 00:16:18.460
So I haven't really
neglected it.
00:16:18.460 --> 00:16:24.392
I have said that it gets
averaged out at the output.
00:16:24.392 --> 00:16:25.600
And by the way, that's wrong.
00:16:25.600 --> 00:16:26.420
That's not true, right?
00:16:26.420 --> 00:16:27.378
This is a wrong result.
00:16:27.378 --> 00:16:30.470
My question is
why is that wrong?
00:16:30.470 --> 00:16:33.640
Why do I have to keep
the entire formula,
00:16:33.640 --> 00:16:36.512
z out of x prime, y prime, z.
00:16:36.512 --> 00:16:37.720
Let me write it properly now.
00:17:15.510 --> 00:17:19.579
So the question is why can
I neglect the quadratics,
00:17:19.579 --> 00:17:22.810
but I must keep the cross terms?
00:17:27.520 --> 00:17:29.695
Actually a very simple answer.
00:17:29.695 --> 00:17:31.070
All you have to
do is look at it.
00:17:37.922 --> 00:17:40.130
AUDIENCE: Does it mean you
are restricting the output
00:17:40.130 --> 00:17:42.553
field to a limited area?
00:17:42.553 --> 00:17:43.970
GEORGE BARBASTATHIS:
That's right.
00:17:43.970 --> 00:17:45.345
If you look at
these cross terms,
00:17:45.345 --> 00:17:48.760
they actually depend on the
output coordinate, x prime.
00:17:48.760 --> 00:17:52.000
I haven't made any
assumptions about x prime.
00:17:52.000 --> 00:17:54.700
x prime can be as
large as I want.
00:17:57.310 --> 00:18:00.820
So this term can actually
dominate the quadratic term.
00:18:00.820 --> 00:18:02.870
That's the point.
00:18:02.870 --> 00:18:08.710
This term, I can control by
making the input transparency
00:18:08.710 --> 00:18:15.140
relatively small, and
making z arbitrarily large.
00:18:15.140 --> 00:18:18.850
However, this term
has the x prime in it.
00:18:18.850 --> 00:18:22.100
And x prime can be
actually very large itself.
00:18:22.100 --> 00:18:23.680
So I can never
neglect this term.
00:18:26.190 --> 00:18:30.720
That's the point
of this argument.
00:18:30.720 --> 00:18:31.720
Everybody clear on that?
00:18:39.680 --> 00:18:43.300
So if I indeed neglect this
term, then what I will get
00:18:43.300 --> 00:18:47.900
is that g sub out--
00:18:47.900 --> 00:18:50.550
I think this marker is dead, so
I'll move on to the next one.
00:18:56.020 --> 00:18:58.380
It has some terms
which I will neglect,
00:18:58.380 --> 00:19:02.400
but it is proportional to a
quantity that looks like this.
00:19:18.970 --> 00:19:20.870
Which I can also rewrite as.
00:19:49.625 --> 00:19:50.500
I didn't do anything.
00:19:50.500 --> 00:19:53.260
All I did was I--
00:19:53.260 --> 00:19:57.288
these parameters here, if
you look at this coefficient.
00:20:00.960 --> 00:20:04.482
It does not depend on the
variable of integration.
00:20:04.482 --> 00:20:05.940
Therefore, I can
call it something.
00:20:05.940 --> 00:20:07.710
I call it u.
00:20:07.710 --> 00:20:10.810
And this integral now.
00:20:10.810 --> 00:20:13.470
It is probably familiar to you.
00:20:13.470 --> 00:20:16.740
If it were in one dimension, it
would be immediately familiar.
00:20:16.740 --> 00:20:18.630
It is a Fourier transform.
00:20:18.630 --> 00:20:21.330
But it is in two dimensions,
so it appears a little bit more
00:20:21.330 --> 00:20:23.430
complicated with two variables.
00:20:23.430 --> 00:20:26.787
But actually, well, it is
a convention to write--
00:20:26.787 --> 00:20:27.620
what happened here--
00:20:34.270 --> 00:20:37.150
to write Fourier
transforms as uppercase.
00:20:37.150 --> 00:20:42.540
So this is a g sub in of u
comma v. The Fourier transform
00:20:42.540 --> 00:20:44.100
computed at frequencies.
00:20:52.240 --> 00:20:55.660
Something magical happened
if I let the field propagate
00:20:55.660 --> 00:21:01.240
at a relatively long distance.
00:21:01.240 --> 00:21:05.780
They field that I get
at that output plane
00:21:05.780 --> 00:21:09.040
actually equals
approximately the
00:21:09.040 --> 00:21:13.350
Fourier transform of the input
field, which is interesting.
00:21:13.350 --> 00:21:16.670
And we'll see a lot
of ramifications
00:21:16.670 --> 00:21:20.340
of that in the next few hours.
00:21:20.340 --> 00:21:24.440
OK, let's see this
now in a calculation.
00:21:24.440 --> 00:21:28.010
Actually, Piper solved it
in practice, in real life.
00:21:28.010 --> 00:21:29.780
Last time at the
demo, you actually
00:21:29.780 --> 00:21:32.360
saw this kind of
thing happening.
00:21:32.360 --> 00:21:34.120
What I will show
now is some movies
00:21:34.120 --> 00:21:35.760
which are also posted
on the website,
00:21:35.760 --> 00:21:37.363
so you can go back
and produce them.
00:21:37.363 --> 00:21:38.780
So in the movie,
you will actually
00:21:38.780 --> 00:21:42.320
see the Fourier transform
slowly developing.
00:21:42.320 --> 00:21:44.380
In this case, it is a
rectangular aperture.
00:21:44.380 --> 00:21:47.910
So the Fourier transform
is very easy to compute.
00:21:47.910 --> 00:21:50.980
Of course, in the movie,
there's no Fourier transform.
00:21:50.980 --> 00:21:55.560
In the movie, all I did is
I convolved with a Fresnel
00:21:55.560 --> 00:21:59.820
propagation coordinate for
progressively longer distances.
00:21:59.820 --> 00:22:02.460
And then I put all those frames
together, and I made the movie.
00:22:02.460 --> 00:22:05.640
But you will see that
as z grows larger,
00:22:05.640 --> 00:22:08.790
the aperture develops
a small oscillation.
00:22:08.790 --> 00:22:14.580
But then eventually, it
develops this pattern that well,
00:22:14.580 --> 00:22:16.792
it is called the sinc function.
00:22:16.792 --> 00:22:18.750
And we'll see in a moment
that this is actually
00:22:18.750 --> 00:22:21.660
the special Fourier
transform of that function.
00:22:21.660 --> 00:22:24.955
Let me play that once again.
00:22:24.955 --> 00:22:26.330
If you notice
carefully, you will
00:22:26.330 --> 00:22:30.460
see that you start with a
nice, clear, sharp aperture.
00:22:30.460 --> 00:22:31.990
As we will propagate,
first, we'll
00:22:31.990 --> 00:22:35.420
see some diffraction ringing.
00:22:35.420 --> 00:22:39.180
Professor Sheppard described
in detail last time.
00:22:39.180 --> 00:22:42.860
But then this ringing slowly
gives away to this cross
00:22:42.860 --> 00:22:44.060
like looking pattern.
00:22:52.410 --> 00:22:53.390
So you can see it here.
00:22:53.390 --> 00:22:55.500
It has a very distinct pattern.
00:22:55.500 --> 00:22:59.030
It has a central lobe,
and then side lobes
00:22:59.030 --> 00:23:02.760
expanding along the
x and y dimension.
00:23:02.760 --> 00:23:08.430
OK, so what is this now,
and how did it come about?
00:23:08.430 --> 00:23:16.180
This function, the aperture
that I started with.
00:23:16.180 --> 00:23:18.840
I can describe it
mathematically as g sub
00:23:18.840 --> 00:23:26.940
in of x comma y equals 1 if--
00:23:26.940 --> 00:23:28.830
basically, let me go back.
00:23:31.970 --> 00:23:36.070
So this function
equals 1 if x and y
00:23:36.070 --> 00:23:37.930
are within this rectangle.
00:23:37.930 --> 00:23:42.190
Mathematically, we can
describe this as x less
00:23:42.190 --> 00:23:47.170
than the size of the
aperture, and y also
00:23:47.170 --> 00:23:51.160
less than the size
of the aperture.
00:23:51.160 --> 00:23:53.920
Generally, it may be a
rectangle, not a square.
00:23:53.920 --> 00:23:55.780
So use the different variables.
00:23:55.780 --> 00:23:57.250
And it is zero otherwise.
00:24:00.610 --> 00:24:03.640
OK, so it is convenient
to define a function.
00:24:17.470 --> 00:24:19.740
And actually, I goofed.
00:24:19.740 --> 00:24:22.530
If the size of the
aperture is x over naught,
00:24:22.530 --> 00:24:28.280
then the value of the function
is 1 for x less than x0.
00:24:28.280 --> 00:24:31.280
So I will explain
this in a second.
00:24:35.720 --> 00:24:39.080
OK, so if I defined the
function rect this way.
00:24:52.830 --> 00:24:54.030
So that's the function.
00:24:54.030 --> 00:24:55.655
Sometimes, it's also
known as a boxcar.
00:25:00.270 --> 00:25:04.560
And so this would be rect of x.
00:25:04.560 --> 00:25:12.320
And if I went to plot
rect of x over x0,
00:25:12.320 --> 00:25:18.730
then if I substitute
x over x0 0, 0 here.
00:25:18.730 --> 00:25:24.260
Then it is 1 if x over
x0 is less than 1/2.
00:25:24.260 --> 00:25:29.950
So basically, this extends from
minus x0 over 2 to x0 over 2.
00:25:29.950 --> 00:25:31.930
And it equals the
value 1 over there.
00:25:35.170 --> 00:25:42.890
And therefore, the total size of
the boxcar is x0 as advertised.
00:25:42.890 --> 00:25:48.480
OK, and, of course, if I define
the rect function this way,
00:25:48.480 --> 00:25:51.110
then my original
function, g sub in.
00:25:53.770 --> 00:25:57.775
I can simply write it as
was done here, as a product.
00:26:11.470 --> 00:26:14.770
So now, what is the Fourier
transform of that product?
00:26:14.770 --> 00:26:17.340
Let me write it down to the
Fourier transform definition.
00:26:45.670 --> 00:26:48.328
OK, that's the Fourier
transform definition.
00:26:48.328 --> 00:26:50.370
The first thing that I
notice in this case, which
00:26:50.370 --> 00:26:53.130
is very convenient, is that
the integral is separable.
00:26:53.130 --> 00:26:56.260
I can write it, really, as
a product of two integrals.
00:26:56.260 --> 00:26:58.590
One of them is in
the x dimension.
00:27:07.890 --> 00:27:11.116
And the other looks very
similar in the y dimension.
00:27:11.116 --> 00:27:13.250
This is not always the case.
00:27:13.250 --> 00:27:16.695
Many, or actually, most 2D
functions are not like that.
00:27:16.695 --> 00:27:17.945
But in this case, we're lucky.
00:27:25.268 --> 00:27:27.060
Means you don't have
to do the whole thing.
00:27:27.060 --> 00:27:30.092
We can just do the integral
for one coordinate,
00:27:30.092 --> 00:27:32.300
and then we immediately have
the answer for the other
00:27:32.300 --> 00:27:35.590
coordinate as well.
00:27:35.590 --> 00:27:37.840
OK, so let's write it then.
00:27:42.500 --> 00:27:44.625
So I will just write the
one dimensional integral.
00:28:02.630 --> 00:28:03.130
OK.
00:28:05.990 --> 00:28:11.380
So now, let's do one
more simplification.
00:28:11.380 --> 00:28:17.640
I will assume that
x0 is unit, is unity.
00:28:20.360 --> 00:28:22.520
We'll come back and
rectify this one.
00:28:22.520 --> 00:28:24.890
But for now, I will
just assume it this way
00:28:24.890 --> 00:28:28.200
to make my life a
little bit easier.
00:28:28.200 --> 00:28:31.550
So if that is the
case, if x0 is one.
00:28:31.550 --> 00:28:35.580
Then [INAUDIBLE] what is the
function rect of x over one.
00:28:35.580 --> 00:28:37.680
It is this one over here.
00:28:37.680 --> 00:28:41.750
So it is 0 outside,
and it is only 1
00:28:41.750 --> 00:28:46.250
between x equals
minus 1/2 and 1/2.
00:28:46.250 --> 00:28:49.045
So in that case,
then, the integral.
00:29:02.060 --> 00:29:04.860
That's the integral.
00:29:04.860 --> 00:29:06.520
Now, there's a simple
one to calculate.
00:29:20.500 --> 00:29:22.360
I don't know how I
picked up a naught here.
00:29:22.360 --> 00:29:24.120
This would not be no naught.
00:29:24.120 --> 00:29:24.760
Drop this one.
00:29:24.760 --> 00:29:26.948
Just e to the i to pi u, right?
00:29:26.948 --> 00:29:27.490
Nothing here.
00:29:30.410 --> 00:29:40.980
So I have u times 1/2 minus
e to the i 2 pi, u minus 1/2.
00:29:43.560 --> 00:29:50.650
And these two minuses cancel.
00:29:50.650 --> 00:29:53.590
And if I flip this
around, it is actually
00:29:53.590 --> 00:29:55.450
the definition of a sine.
00:29:55.450 --> 00:30:04.820
So this equals 1 over
minus i 2 pi u0, times
00:30:04.820 --> 00:30:09.260
minus 2i sine of what was here.
00:30:09.260 --> 00:30:11.690
The twos also cancel.
00:30:11.690 --> 00:30:16.530
And I get pi u.
00:30:16.530 --> 00:30:20.430
So finally, after dropping
the remaining constants.
00:30:20.430 --> 00:30:21.390
I still got a u0.
00:30:21.390 --> 00:30:22.560
There's no u0.
00:30:22.560 --> 00:30:25.200
For some reason, my brain
puts a naught there.
00:30:25.200 --> 00:30:26.250
There shouldn't be.
00:30:26.250 --> 00:30:28.580
So after finally canceling
whatever is left,
00:30:28.580 --> 00:30:36.030
I get sine of pi u over pi u.
00:30:36.030 --> 00:30:39.430
OK, so that function, by
definition, is called a sinc.
00:30:58.490 --> 00:31:02.353
And I'll jump ahead in
the naught a little bit.
00:31:02.353 --> 00:31:04.020
You can also look it
up in the textbook.
00:31:07.940 --> 00:31:10.830
The sinc function
looks like this.
00:31:10.830 --> 00:31:12.020
It is in page--
00:31:12.020 --> 00:31:16.550
I forget which page,
between page 12 and page 14
00:31:16.550 --> 00:31:18.212
of the Goodman textbook.
00:31:18.212 --> 00:31:19.295
This is the sinc function.
00:31:21.800 --> 00:31:24.620
One argument equals 0.
00:31:24.620 --> 00:31:28.490
2 and it has a peak and
then it kind of oscillates,
00:31:28.490 --> 00:31:30.320
but the amplitude
of its oscillation
00:31:30.320 --> 00:31:35.330
drops inversely proportional
to the argument.
00:31:35.330 --> 00:31:37.700
So the oscillation
comes from the sine.
00:31:37.700 --> 00:31:43.258
The inversely proportional comes
from the u in the denominator.
00:31:43.258 --> 00:31:45.050
So this may be a little
bit boring for you.
00:31:45.050 --> 00:31:47.133
For those of you who have
taken signal processing,
00:31:47.133 --> 00:31:50.643
you're probably ready
to go to sleep now.
00:31:50.643 --> 00:31:52.560
For reasons of completeness,
we have to do it,
00:31:52.560 --> 00:31:55.020
to go through it.
00:31:55.020 --> 00:31:58.840
Then I will not compute
too many Fourier integrals.
00:31:58.840 --> 00:32:01.760
But in any case, if you are to
compute one Fourier transform,
00:32:01.760 --> 00:32:04.470
that's the one to compute.
00:32:04.470 --> 00:32:05.370
So that's it, then.
00:32:05.370 --> 00:32:08.400
This is also the definition
of the original rect function
00:32:08.400 --> 00:32:09.960
that we had.
00:32:09.960 --> 00:32:14.400
And its Fourier transform
is the sinc function.
00:32:14.400 --> 00:32:17.790
Now, we're not done yet, because
I made one more simplifying
00:32:17.790 --> 00:32:19.170
assumption.
00:32:19.170 --> 00:32:21.360
I said that x0 equals 1.
00:32:21.360 --> 00:32:24.150
So what do we do about this x0?
00:32:24.150 --> 00:32:29.840
Well, does anybody know
what I can do about this x0?
00:32:42.978 --> 00:32:44.645
OK what I'll do is a
change of variable.
00:32:48.000 --> 00:32:49.750
And I will do it in
the general case.
00:32:53.170 --> 00:32:57.970
Let's say that
they have a G of u
00:32:57.970 --> 00:33:02.200
equals the Fourier transform of
some general function, g of x.
00:33:10.450 --> 00:33:16.780
So this is then the Fourier
transform of g of x.
00:33:16.780 --> 00:33:24.100
What is the Fourier transform
of a scale version of g of x?
00:33:24.100 --> 00:33:25.780
Well, that would be
something like this.
00:33:25.780 --> 00:33:31.285
It would be from minus
infinity to infinity, g of ax,
00:33:31.285 --> 00:33:36.930
e to the minus i two pi ux dx.
00:33:36.930 --> 00:33:40.350
And to get rid of
this ugly thing here,
00:33:40.350 --> 00:33:42.870
I can make a change
of coordinates.
00:33:42.870 --> 00:33:46.950
For example, let's
say that c equals ax.
00:33:46.950 --> 00:33:51.480
Then this means that
dx e equals a dx.
00:33:51.480 --> 00:33:54.890
And I can write the integral.
00:33:54.890 --> 00:33:57.330
So they actually
become dx e over a.
00:33:57.330 --> 00:34:01.620
I pick up 1 over a out here.
00:34:01.620 --> 00:34:02.970
Nothing happens to infinity.
00:34:02.970 --> 00:34:04.470
It remains infinity.
00:34:04.470 --> 00:34:08.025
This would be g of c, e
to the minus i two pi.
00:34:10.920 --> 00:34:17.690
Now, x is also c upon a, big C.
00:34:17.690 --> 00:34:18.620
I haven't cheated.
00:34:18.620 --> 00:34:21.889
This is the transformed--
the integral.
00:34:21.889 --> 00:34:24.730
So the 1 over a basically
keeps me honest here.
00:34:24.730 --> 00:34:29.480
Makes sure that the area of
the differential is preserved.
00:34:29.480 --> 00:34:31.520
It's also known as a Jacobian.
00:34:31.520 --> 00:34:33.750
But anyway, that's what it is.
00:34:33.750 --> 00:34:39.199
And I can do one more
little manipulation here.
00:34:39.199 --> 00:34:40.889
I can rewrite it like this.
00:34:55.800 --> 00:34:59.120
And we can recognize
now that this integral--
00:34:59.120 --> 00:35:01.120
let me see if I can fit
them both on the screen.
00:35:04.550 --> 00:35:09.250
OK, so recognize
that this integral
00:35:09.250 --> 00:35:13.710
is the same as this
integral, except
00:35:13.710 --> 00:35:17.580
with a different variable,
with a different argument.
00:35:17.580 --> 00:35:24.290
So therefore, what I derived
here is one over a G of,
00:35:24.290 --> 00:35:26.590
u over a.
00:35:30.280 --> 00:35:33.240
OK, so this is a property
of Fourier transforms
00:35:33.240 --> 00:35:34.650
known as the scaling theorem.
00:35:39.480 --> 00:35:42.180
Or sometimes, people call
it the similarity theorem.
00:35:48.640 --> 00:35:54.760
And let's see how we can apply
it to the question at hand.
00:35:54.760 --> 00:36:02.190
We derived that the
rectangle function.
00:36:02.190 --> 00:36:05.280
If you Fourier transform it,
you get the sinc function.
00:36:07.950 --> 00:36:11.640
OK, what I really wanted to
get is the Fourier transform
00:36:11.640 --> 00:36:17.620
of a rectangle,
which has a size, x0.
00:36:17.620 --> 00:36:20.290
Now let me write down
the scaling theorem.
00:36:20.290 --> 00:36:25.840
It says that g of ax
Fourier transforms to 1
00:36:25.840 --> 00:36:29.290
over a G of u over a.
00:36:29.290 --> 00:36:33.112
So in this case, a is
identical to 1 over x0.
00:36:33.112 --> 00:36:34.570
So therefore, the
Fourier transform
00:36:34.570 --> 00:36:39.470
will be x0, sinc of x0 times u.
00:36:39.470 --> 00:36:44.050
And this is intuitively
satisfying because the units
00:36:44.050 --> 00:36:45.910
here inside the sink.
00:36:45.910 --> 00:36:48.550
The units are naught.
00:36:48.550 --> 00:36:54.100
x0 has dimensions
of space, meters.
00:36:54.100 --> 00:36:58.690
u is a frequency, so it has in
dimensions of inverse meters.
00:36:58.690 --> 00:37:01.060
So therefore, what I
have inside the argument
00:37:01.060 --> 00:37:03.940
has no dimensions at all,
which is, of course, of the way
00:37:03.940 --> 00:37:06.050
it should be.
00:37:06.050 --> 00:37:08.860
OK, so this is them.
00:37:22.080 --> 00:37:25.890
So [INAUDIBLE]
for one last time.
00:37:25.890 --> 00:37:29.850
This is how we obtain this
function with the central lobe.
00:37:29.850 --> 00:37:34.470
But the side lobe is actually
not quite the sinc function
00:37:34.470 --> 00:37:37.230
itself, because I'm
blocking the intensity here.
00:37:37.230 --> 00:37:40.830
It is actually sinc squared.
00:37:40.830 --> 00:37:43.830
But anyway, this is
where this came from.
00:37:43.830 --> 00:37:46.590
So, of course, if you
multiply the two dimensions,
00:37:46.590 --> 00:37:50.100
you get a sinc in
the x dimension,
00:37:50.100 --> 00:37:52.230
and sinc in the y dimension,
and then, of course,
00:37:52.230 --> 00:37:53.940
you get the product.
00:37:53.940 --> 00:37:55.830
And the Fourier
transform theorem
00:37:55.830 --> 00:38:00.630
says that the final field
will actually be the Fourier
00:38:00.630 --> 00:38:06.100
transform, but with
the coordinate,
00:38:06.100 --> 00:38:08.450
the special frequency
coordinate replaced
00:38:08.450 --> 00:38:12.455
by x prime over lambda z.
00:38:12.455 --> 00:38:14.170
This is where this came from.
00:38:14.170 --> 00:38:17.180
I substituted u with
x prime over lambda z.
00:38:19.690 --> 00:38:22.930
So the bottom line is that
this is perhaps easier
00:38:22.930 --> 00:38:26.440
if you look at it heads up.
00:38:26.440 --> 00:38:28.540
So here is a
rectangular function.
00:38:28.540 --> 00:38:32.440
I only saw the x dimension
here with a size of x0.
00:38:32.440 --> 00:38:36.460
Then you can see that
the Fourier transform.
00:38:36.460 --> 00:38:38.830
Actually this square, the
intensity of the Fourier
00:38:38.830 --> 00:38:39.580
transform.
00:38:39.580 --> 00:38:41.560
It has this characteristic
sinc pattern
00:38:41.560 --> 00:38:44.410
with a central lobe
and then side lobes.
00:38:44.410 --> 00:38:47.650
And the size of the
central lobe is inversely
00:38:47.650 --> 00:38:50.380
proportional to the
size of the rectangle.
00:38:50.380 --> 00:38:52.750
So if I make this
rectangle smaller,
00:38:52.750 --> 00:38:55.030
this size will become bigger.
00:38:58.580 --> 00:39:03.260
So this is then our first
Fraunhofer diffraction pattern.
00:39:03.260 --> 00:39:04.940
The Fraunhofer
diffraction pattern
00:39:04.940 --> 00:39:08.360
of a rectangular function.
00:39:08.360 --> 00:39:11.010
Of course, there is
many different apertures
00:39:11.010 --> 00:39:13.602
that are of interest
in this business.
00:39:13.602 --> 00:39:15.810
Oh, and this, by the way,
is called the sinc pattern,
00:39:15.810 --> 00:39:18.090
as I already mentioned.
00:39:18.090 --> 00:39:20.460
So there's many different
patterns of interest.
00:39:20.460 --> 00:39:23.100
For example, very
often in optics,
00:39:23.100 --> 00:39:25.520
we use circular apertures.
00:39:25.520 --> 00:39:30.450
Lenses, irises, in cameras,
most optical systems
00:39:30.450 --> 00:39:33.210
have a circular aperture.
00:39:33.210 --> 00:39:37.150
In this case, we talked
about the blinking
00:39:37.150 --> 00:39:38.795
or the Poisson spot here.
00:39:38.795 --> 00:39:40.420
But that's not what
I'm interested now.
00:39:40.420 --> 00:39:44.000
I'm interested in the far
field diffraction pattern.
00:39:44.000 --> 00:39:48.430
And in this case, you also
get a pattern with a--
00:39:48.430 --> 00:39:53.090
kind of looks like a sinc, but
a sinc with circular symmetry.
00:39:53.090 --> 00:39:55.780
It is not exactly a sinc.
00:39:55.780 --> 00:39:58.840
It is given by a rather
nasty formula here.
00:39:58.840 --> 00:40:00.640
It is the ratio.
00:40:00.640 --> 00:40:03.270
First of all, it is all
done in polar coordinates.
00:40:03.270 --> 00:40:05.170
So you see that you
get the square root
00:40:05.170 --> 00:40:09.760
of the sum of the Cartesian
coordinates squared.
00:40:09.760 --> 00:40:11.950
But the function itself
is given by the ratio
00:40:11.950 --> 00:40:14.950
of a Bessel function
of the first kind
00:40:14.950 --> 00:40:19.810
and order 1 divided
by its argument.
00:40:19.810 --> 00:40:23.100
I will not go into to the
detail of the derivation here.
00:40:23.100 --> 00:40:25.570
Goodman describes
it in great detail.
00:40:25.570 --> 00:40:29.820
So if you're interested, you can
go and check it out over there.
00:40:29.820 --> 00:40:32.890
I do want to emphasize
a couple of things.
00:40:32.890 --> 00:40:36.240
First of all, that is
this sometimes by analogy
00:40:36.240 --> 00:40:38.050
to the sinc.
00:40:38.050 --> 00:40:41.970
This pattern is
referred to as a jinc.
00:40:41.970 --> 00:40:45.090
So the J comes, of
course, from the Bessel J.
00:40:45.090 --> 00:40:48.300
So we call it a jinc function.
00:40:48.300 --> 00:40:52.480
And more commonly it is
referred to as the Airy pattern.
00:40:52.480 --> 00:40:56.730
Airy not because it sucks
air or something like that.
00:40:56.730 --> 00:40:59.220
Actually, it is named after
someone, some Englishman,
00:40:59.220 --> 00:41:01.560
whose name was Airy.
00:41:01.560 --> 00:41:03.840
So Airy pattern.
00:41:03.840 --> 00:41:06.480
And if you compare it
with the previous one.
00:41:08.990 --> 00:41:09.850
The previous one.
00:41:14.190 --> 00:41:17.280
I'm sorry, you have endure
this animation again.
00:41:17.280 --> 00:41:19.550
OK, so the previous one.
00:41:19.550 --> 00:41:24.680
The null actually occurred
that lambda l divided
00:41:24.680 --> 00:41:27.800
by the size of the aperture.
00:41:27.800 --> 00:41:35.720
In the case of the
jinc, there's a factor
00:41:35.720 --> 00:41:39.830
of 1.22 that [INAUDIBLE]
the calculation.
00:41:39.830 --> 00:41:43.640
So the null basically occurs
at the very similar looking
00:41:43.640 --> 00:41:44.750
variable.
00:41:44.750 --> 00:41:50.690
If you make the diameter
shorter, the size of the jinc
00:41:50.690 --> 00:41:51.780
will grow.
00:41:51.780 --> 00:41:56.120
But the null, the zero of the
jinc, occurs at this function,
00:41:56.120 --> 00:41:59.690
at this value, 1.22,
which, of course, comes
00:41:59.690 --> 00:42:02.840
from the zero of
the Bessel function.
00:42:02.840 --> 00:42:04.900
So there's no intuition here.
00:42:04.900 --> 00:42:10.820
It's just where this function
happens to reach value zero.
00:42:10.820 --> 00:42:11.320
OK.
00:42:14.550 --> 00:42:16.380
Let me skip this
slide, and perhaps you
00:42:16.380 --> 00:42:18.950
can go over it and
talk about it later.
00:42:18.950 --> 00:42:22.080
It basically elaborates a
little bit on the issue of--
00:42:22.080 --> 00:42:25.800
I said before that in order
to observe the Fraunhofer
00:42:25.800 --> 00:42:29.150
diffraction pattern, have
to let z become long enough.
00:42:29.150 --> 00:42:31.980
Have to propagate
the field far enough.
00:42:31.980 --> 00:42:35.610
So this slide answers the
question, well, how far is far?
00:42:35.610 --> 00:42:36.910
Let me skip it four now.
00:42:36.910 --> 00:42:40.920
And if we have time later,
I will come back to it.
00:42:40.920 --> 00:42:42.660
But what I would like
to get started now
00:42:42.660 --> 00:42:46.620
is a few comments on
Fourier transform the cells,
00:42:46.620 --> 00:42:49.640
and how they apply to
different apertures.
00:42:49.640 --> 00:42:53.160
So calling the Fourier transform
is a topic in applied math,
00:42:53.160 --> 00:42:54.360
really.
00:42:54.360 --> 00:42:58.590
I don't want to
convert this to 18085,
00:42:58.590 --> 00:43:01.710
or whatever it is at MIT that
you'll learn those things.
00:43:01.710 --> 00:43:05.440
But I will remind you of
some of the basic properties.
00:43:05.440 --> 00:43:09.340
So one is the definition
of the Fourier transform.
00:43:09.340 --> 00:43:11.020
I already wrote it down.
00:43:11.020 --> 00:43:13.330
Many of you are more
familiar with the time domain
00:43:13.330 --> 00:43:16.960
definition, where the
Fourier variable is actually
00:43:16.960 --> 00:43:20.440
a frequency measured in hertz.
00:43:20.440 --> 00:43:23.500
Of course, because here,
we're talking about signals
00:43:23.500 --> 00:43:25.180
in the space domain.
00:43:25.180 --> 00:43:28.690
The frequency variable
is the spatial frequency,
00:43:28.690 --> 00:43:31.510
so the units are
actually inverse meters.
00:43:31.510 --> 00:43:33.530
Hertz is inverse second.
00:43:33.530 --> 00:43:36.610
The units here are
inverse meters.
00:43:36.610 --> 00:43:38.920
And, of course,
because we're dealing
00:43:38.920 --> 00:43:41.778
with two dimensional
special variables,
00:43:41.778 --> 00:43:43.945
it is a two dimensional
Fourier transform because it
00:43:43.945 --> 00:43:45.070
is a double integral.
00:43:45.070 --> 00:43:46.727
But other than that,
it's very similar.
00:43:46.727 --> 00:43:48.310
The other thing I
wanted to remind you
00:43:48.310 --> 00:43:51.580
is that there is an inverse
Fourier transform which
00:43:51.580 --> 00:43:55.780
looks very similar, except
for a minus sign, so
00:43:55.780 --> 00:43:57.205
into the exponent here.
00:43:57.205 --> 00:43:59.080
And, of course, the
inverse Fourier transform
00:43:59.080 --> 00:44:02.020
takes you back to the
original function.
00:44:02.020 --> 00:44:02.950
So it's like a dance.
00:44:02.950 --> 00:44:04.780
You start with a
initial function.
00:44:04.780 --> 00:44:06.670
You compute the
Fourier transform,
00:44:06.670 --> 00:44:13.420
then you plug it into the into
the inverse Fourier transform,
00:44:13.420 --> 00:44:15.580
and you get back
what you started.
00:44:15.580 --> 00:44:19.960
That is sometimes referred to
as the Fourier integral instead
00:44:19.960 --> 00:44:21.675
of an inverse Fourier transform.
00:44:24.730 --> 00:44:28.210
So what is this really,
this Fourier transform?
00:44:28.210 --> 00:44:31.920
If you look at its surreal
part, and if you have a real
00:44:31.920 --> 00:44:33.250
function here.
00:44:33.250 --> 00:44:35.230
Basically, what the
Fourier transform does
00:44:35.230 --> 00:44:38.480
is it multiplies this function.
00:44:38.480 --> 00:44:41.980
It is denoted as
red here, g of x.
00:44:41.980 --> 00:44:44.550
It multiplies with a sinusoid.
00:44:44.550 --> 00:44:47.960
The real part of this complex
exponential is a cosine.
00:44:47.960 --> 00:44:50.020
So you multiply the
function with this cosine,
00:44:50.020 --> 00:44:52.270
and then you integrate.
00:44:52.270 --> 00:44:55.300
OK, so why do you do
something like that?
00:44:55.300 --> 00:44:58.720
Actually, does anybody
know why Fourier came up
00:44:58.720 --> 00:45:01.420
with this kind of transform?
00:45:04.190 --> 00:45:06.860
What was the context
that Fourier--
00:45:06.860 --> 00:45:08.270
what was Fourier?
00:45:08.270 --> 00:45:10.190
Fourier was a French
mathematician,
00:45:10.190 --> 00:45:12.200
or a French applied
physicist, I guess.
00:45:12.200 --> 00:45:14.490
And he was trying to solve
a particular problem.
00:45:14.490 --> 00:45:16.190
Does anybody know
what's the problem
00:45:16.190 --> 00:45:18.620
he was trying to solve when
he came up with this business?
00:45:23.610 --> 00:45:26.120
OK, it was a problem
of heat transfer.
00:45:26.120 --> 00:45:29.300
Fourier was trying to
solve the problem of what
00:45:29.300 --> 00:45:34.130
is that temperature distribution
between two hot plates, one
00:45:34.130 --> 00:45:37.680
of them at temperature t1,
that at temperature of t2.
00:45:37.680 --> 00:45:40.180
And actually, the answer is not
given by a Fourier integral.
00:45:40.180 --> 00:45:42.660
It is given by a Fourier series.
00:45:42.660 --> 00:45:46.130
And if you make the plates go.
00:45:46.130 --> 00:45:48.620
If you increase the
distance between the plates,
00:45:48.620 --> 00:45:50.810
the Fourier series
becomes an integral.
00:45:50.810 --> 00:45:55.010
So this entire
mathematical arsenal here,
00:45:55.010 --> 00:45:58.010
it actually came from the
field of heat transfer,
00:45:58.010 --> 00:45:59.640
interestingly enough.
00:45:59.640 --> 00:46:01.810
Anyway, that is of no
concern to us here.
00:46:01.810 --> 00:46:04.490
The Fourier transform,
as many of you know--
00:46:04.490 --> 00:46:07.580
especially those who
do acoustics or signal
00:46:07.580 --> 00:46:10.010
processing-- it has
tremendous applications
00:46:10.010 --> 00:46:11.320
in signal processing nowadays.
00:46:11.320 --> 00:46:14.140
And, of course, it is still
used in heat transfer.
00:46:14.140 --> 00:46:18.740
But in our context here, it
is more signal processing
00:46:18.740 --> 00:46:20.630
that we will use it.
00:46:20.630 --> 00:46:23.090
OK, so why do we
multiply by a sinusoid?
00:46:23.090 --> 00:46:25.020
Well, the reason
is the following.
00:46:25.020 --> 00:46:31.760
Suppose that G, our transformed
function, is itself a sinusoid.
00:46:31.760 --> 00:46:34.900
OK, so here is G with a
particular frequency, u0.
00:46:39.670 --> 00:46:42.640
So G is the red sinusoid.
00:46:42.640 --> 00:46:48.280
The Fourier transform
kernel is another sinusoid.
00:46:48.280 --> 00:46:52.360
And in general, they
have different frequency,
00:46:52.360 --> 00:46:53.843
like shown here.
00:46:53.843 --> 00:46:55.510
So what does the value
of this integral?
00:46:55.510 --> 00:46:56.010
Do you know?
00:47:01.270 --> 00:47:03.100
If the two frequencies
are different.
00:47:03.100 --> 00:47:06.700
If you multiply two
sinusoids and integrate them
00:47:06.700 --> 00:47:09.370
over a very long distance,
actually infinite.
00:47:12.160 --> 00:47:13.967
Actually, by convention
in this class.
00:47:13.967 --> 00:47:15.550
I don't know if I
mentioned it before.
00:47:15.550 --> 00:47:19.710
The convention, if I don't
put bounds to an integral,
00:47:19.710 --> 00:47:23.030
that mean it goes from minus
infinity to plus infinity.
00:47:23.030 --> 00:47:26.470
So this is an infinite
integral of two sinusoids
00:47:26.470 --> 00:47:30.250
with different
frequency multiplied.
00:47:30.250 --> 00:47:31.240
What is the answer?
00:47:31.240 --> 00:47:32.850
AUDIENCE: Zero?
00:47:32.850 --> 00:47:34.200
GEORGE BARBASTATHIS: Zero, yeah.
00:47:34.200 --> 00:47:39.030
Because the various oscillations
that will cancel eventually.
00:47:39.030 --> 00:47:41.990
So you'll get nothing.
00:47:41.990 --> 00:47:47.240
OK, however, there's
a singular case when
00:47:47.240 --> 00:47:50.900
the frequencies are the same.
00:47:50.900 --> 00:47:54.370
And what is the value of
this integral in this case?
00:47:54.370 --> 00:47:55.360
Well, infinite, right?
00:47:55.360 --> 00:47:59.300
Because if you multiply
them, this will be positive.
00:47:59.300 --> 00:48:01.620
This will also be positive,
because you are multiplying
00:48:01.620 --> 00:48:03.530
two negative quantities.
00:48:03.530 --> 00:48:06.970
So you actually get infinity,
which is not very good.
00:48:06.970 --> 00:48:08.560
But in mathematics,
we have a way
00:48:08.560 --> 00:48:11.450
of dealing with this kind
of abrupt infinities.
00:48:11.450 --> 00:48:13.730
We call them delta functions.
00:48:13.730 --> 00:48:17.350
And, of course, I'm severely
abusing the mathematics here.
00:48:17.350 --> 00:48:19.840
The way the delta
function comes up.
00:48:19.840 --> 00:48:21.130
Does anybody know?
00:48:21.130 --> 00:48:22.025
It comes as a limit.
00:48:22.025 --> 00:48:23.650
The way you get a
delta function is you
00:48:23.650 --> 00:48:28.000
actually bound this integral,
so that you get a finite value.
00:48:28.000 --> 00:48:30.670
And then you let the
bound go to infinity,
00:48:30.670 --> 00:48:33.280
and the limit is
a delta function.
00:48:33.280 --> 00:48:36.910
Anyway, without going into
these mathematical intricacies,
00:48:36.910 --> 00:48:41.710
we can represent this
situation here as--
00:48:41.710 --> 00:48:45.170
OK, forget for forget the second
delta function for a moment.
00:48:45.170 --> 00:48:49.120
But this situation where
the value of the integral
00:48:49.120 --> 00:48:53.220
is zero for all
frequencies except one.
00:48:53.220 --> 00:48:56.010
Because the integral
assumes a huge value.
00:48:56.010 --> 00:48:59.390
Then we write it as
a delta function.
00:48:59.390 --> 00:49:01.460
And the why we get
two delta functions.
00:49:01.460 --> 00:49:03.530
Well, we'll get
two delta functions
00:49:03.530 --> 00:49:08.050
because the way this works
is if you take the Fourier
00:49:08.050 --> 00:49:16.310
transform of an
exponential, this
00:49:16.310 --> 00:49:19.760
is a single delta function.
00:49:19.760 --> 00:49:21.060
Now, if you taking the cosine.
00:49:34.050 --> 00:49:37.190
Of course, the cosine is a sum
of two complex exponentials.
00:49:48.930 --> 00:49:50.580
And now we know how
to deal with this.
00:49:50.580 --> 00:49:53.520
It's one of those that's given
by an expression like this one.
00:49:53.520 --> 00:49:56.130
So you actually get two
symmetric delta functions.
00:50:03.710 --> 00:50:06.210
OK, so what is
the one half here?
00:50:06.210 --> 00:50:11.390
Well, the one half is
actually the energy contained
00:50:11.390 --> 00:50:12.440
in this delta function.
00:50:16.730 --> 00:50:17.925
So that's normal.
00:50:17.925 --> 00:50:19.550
The thing is that is
a little bit weird
00:50:19.550 --> 00:50:21.920
about this is that
this sort of situation
00:50:21.920 --> 00:50:25.640
implies that there's such a
thing as negative frequency.
00:50:25.640 --> 00:50:28.130
Of course, there's no
negative frequencies.
00:50:28.130 --> 00:50:30.290
The frequencies can
only be positive.
00:50:30.290 --> 00:50:32.360
The reason that we need
a negative frequency
00:50:32.360 --> 00:50:35.300
is actually for
mathematical rigor,
00:50:35.300 --> 00:50:38.100
because we insisted
on using phasors.
00:50:38.100 --> 00:50:40.610
You remember a
long time ago, when
00:50:40.610 --> 00:50:42.080
we started talking about waves.
00:50:42.080 --> 00:50:45.230
We said that waves are
real, so they are actually
00:50:45.230 --> 00:50:47.030
cosine functions.
00:50:47.030 --> 00:50:49.400
But for mathematical
convenience,
00:50:49.400 --> 00:50:52.970
in order to avoid complicated
trigonometric calculations,
00:50:52.970 --> 00:50:55.640
it would represent
this cosine function
00:50:55.640 --> 00:50:57.620
as a complex exponential.
00:50:57.620 --> 00:51:02.070
Well, if you really had the
simple cosine transform.
00:51:02.070 --> 00:51:07.430
So you use the cosine into
the kernel for the integral.
00:51:07.430 --> 00:51:10.160
That is known as a
Fourier cosine transform,
00:51:10.160 --> 00:51:12.530
and then it contains only
positive frequencies.
00:51:12.530 --> 00:51:13.940
But it's nice to calculate.
00:51:13.940 --> 00:51:15.650
Gives you very ugly formulas.
00:51:15.650 --> 00:51:18.020
So that's why we'll use
the complex exponential.
00:51:18.020 --> 00:51:20.780
It is simpler formulas,
but the price we pay
00:51:20.780 --> 00:51:22.880
is this weird
negative frequency.
00:51:22.880 --> 00:51:24.590
So there's nothing
to worry about.
00:51:24.590 --> 00:51:27.200
It is not wrong
physics in any way.
00:51:27.200 --> 00:51:29.930
It is simply a matter of
mathematical convenience
00:51:29.930 --> 00:51:32.926
that leads to these
negative frequencies.
00:51:37.770 --> 00:51:43.350
And, of course, I will not go
through all these derivations
00:51:43.350 --> 00:51:44.160
over here.
00:51:44.160 --> 00:51:46.860
But several functions,
their Fourier transforms
00:51:46.860 --> 00:51:49.600
can be computed in
[INAUDIBLE] form.
00:51:49.600 --> 00:51:53.010
In fact, all of these functions,
you can go ahead if you like,
00:51:53.010 --> 00:51:55.260
and do the Fourier
transform by yourselves.
00:51:55.260 --> 00:51:59.280
It is relatively simple
mathematical exercise.
00:51:59.280 --> 00:52:01.260
So we will use some
of these very often.
00:52:01.260 --> 00:52:03.930
Mostly, we'll use the
rectangular function.
00:52:03.930 --> 00:52:05.600
I already talked about this one.
00:52:05.600 --> 00:52:06.970
We'll use the circular function.
00:52:06.970 --> 00:52:08.960
I talked a bit very briefly.
00:52:08.960 --> 00:52:11.940
Then there is the
triangular function, which
00:52:11.940 --> 00:52:14.320
has a shot of a grayscale.
00:52:14.320 --> 00:52:17.370
It starts from zero, then
progressively it goes to one,
00:52:17.370 --> 00:52:19.500
and then drops
back down to zero.
00:52:19.500 --> 00:52:24.550
In linear fashion, and the com.
00:52:24.550 --> 00:52:26.650
The composite sequence
of delta functions, that
00:52:26.650 --> 00:52:29.530
is very useful in sampling.
00:52:29.530 --> 00:52:32.410
I don't use it very much
in this class, actually.
00:52:32.410 --> 00:52:35.350
I sort of bypass the
issue of sampling.
00:52:35.350 --> 00:52:38.200
But I'm sure all of you are
familiar with Nyquist sampling
00:52:38.200 --> 00:52:40.690
rates, Nyquist frequencies,
and so on and so forth.
00:52:40.690 --> 00:52:43.280
So these all can be explained
by the com function.
00:52:43.280 --> 00:52:45.785
And Goodman has a
section in the book.
00:52:45.785 --> 00:52:46.660
I forget where it is.
00:52:46.660 --> 00:52:48.660
It's a section two
point something.
00:52:53.400 --> 00:52:56.858
Yeah, section 2.4, two
dimensional sampling
00:52:56.858 --> 00:52:57.900
theory that goes over it.
00:52:57.900 --> 00:53:00.350
I will not go over
it in the class.
00:53:00.350 --> 00:53:05.550
But it may be good idea
for you to review it.
00:53:05.550 --> 00:53:08.040
OK.
00:53:08.040 --> 00:53:12.560
So as I said, there's
several functions here
00:53:12.560 --> 00:53:16.230
who's Fourier transforms
can be computed.
00:53:16.230 --> 00:53:19.560
I will not go through all of
these, but it is good for you
00:53:19.560 --> 00:53:21.990
to know where this kind
of thing is in the book,
00:53:21.990 --> 00:53:24.060
so when necessary,
you can refer to them,
00:53:24.060 --> 00:53:27.520
and you can get the answers
for values [INAUDIBLE]..
00:53:27.520 --> 00:53:30.450
So for example, here is
the rectangle function
00:53:30.450 --> 00:53:32.670
that we computed before.
00:53:32.670 --> 00:53:35.340
And, of course, it
gives the sinc response.
00:53:35.340 --> 00:53:41.490
Another one worth remembering is
Gaussian, a Gaussian function.
00:53:41.490 --> 00:53:45.190
Actually, also Fourier
transforms to Gaussian,
00:53:45.190 --> 00:53:46.560
which is interesting.
00:53:46.560 --> 00:53:52.230
And another useful one that
we will deal with later
00:53:52.230 --> 00:53:53.430
is this one.
00:53:53.430 --> 00:53:56.560
You should look at
them all before last.
00:53:56.560 --> 00:54:00.860
It also looks like a
Gaussian, but with a J here.
00:54:00.860 --> 00:54:01.840
So this we recognize.
00:54:01.840 --> 00:54:03.310
Physically, what
is this function?
00:54:03.310 --> 00:54:06.100
It is a complex
quadratic exponential.
00:54:06.100 --> 00:54:07.870
Physically, what did we call it?
00:54:18.513 --> 00:54:20.305
If I write it in a
slightly different form,
00:54:20.305 --> 00:54:21.680
you will recognize
it right away.
00:54:27.438 --> 00:54:27.980
What is this?
00:54:42.860 --> 00:54:44.497
AUDIENCE: Spherical
wave along z?
00:54:44.497 --> 00:54:46.330
GEORGE BARBASTATHIS:
It is a spherical wave,
00:54:46.330 --> 00:54:48.280
propagating a distance z.
00:54:48.280 --> 00:54:49.880
So what you see over
there is actually
00:54:49.880 --> 00:54:52.810
a spherical wave with a
slightly weird definition,
00:54:52.810 --> 00:54:56.280
a squared equals
1 over lambda z.
00:54:56.280 --> 00:54:58.900
So this expression here
in the row before last
00:54:58.900 --> 00:55:00.460
is a spherical wave.
00:55:00.460 --> 00:55:05.137
So a Fourier transform
is also a spherical wave.
00:55:05.137 --> 00:55:06.970
And we will use this
Fourier transform quite
00:55:06.970 --> 00:55:10.375
a bit in the next two lectures.
00:55:10.375 --> 00:55:12.500
It might be good if you
start studying, by the way,
00:55:12.500 --> 00:55:15.102
if you don't know what this is,
it means you haven't studied.
00:55:15.102 --> 00:55:16.810
And I don't know how
you did the homework
00:55:16.810 --> 00:55:20.745
without studying, possibly by
copying from the last year.
00:55:20.745 --> 00:55:22.120
But I strongly
recommend that you
00:55:22.120 --> 00:55:26.020
don't do that, because
you're presumably here
00:55:26.020 --> 00:55:27.028
in order to learn.
00:55:27.028 --> 00:55:28.570
And you don't learn
unless you study.
00:55:32.290 --> 00:55:37.222
So it is about time, not
because of the quiz, but anyway.
00:55:37.222 --> 00:55:39.180
The quiz is also coming
up, so it is about time
00:55:39.180 --> 00:55:41.380
to start studying it.
00:55:41.380 --> 00:55:43.470
So this is like a
friendly advice,
00:55:43.470 --> 00:55:46.110
I guess, from an older guy.
00:55:46.110 --> 00:55:48.100
Study.
00:55:48.100 --> 00:55:48.600
OK.
00:55:59.500 --> 00:56:03.910
[INAUDIBLE] that the
Fourier transform has.
00:56:03.910 --> 00:56:07.870
Once we have this basic Fourier
transforms that are shown here,
00:56:07.870 --> 00:56:10.690
then we can compute even
more Fourier transforms
00:56:10.690 --> 00:56:17.530
by using the various properties
of the Fourier transform.
00:56:17.530 --> 00:56:19.330
So one of those we
wanted to derive.
00:56:19.330 --> 00:56:21.100
This is the scaling theorem.
00:56:21.100 --> 00:56:23.680
I did this at the
beginning of the class.
00:56:23.680 --> 00:56:28.510
And it tells you that if
you scale the argument that
00:56:28.510 --> 00:56:30.670
goes inside the
Fourier transform,
00:56:30.670 --> 00:56:33.920
then the Fourier transform
itself scales the opposite way.
00:56:33.920 --> 00:56:37.270
So for example, in the case
of the Fraunhofer diffraction,
00:56:37.270 --> 00:56:39.990
it says that if you make
an aperture smaller,
00:56:39.990 --> 00:56:43.610
its Fraunhofer diffraction
pattern becomes larger.
00:56:43.610 --> 00:56:46.990
So this is the scaling theorem,
physical and mathematical.
00:56:46.990 --> 00:56:49.450
Physical, it tells you that
the Fraunhofer diffraction
00:56:49.450 --> 00:56:50.620
becomes bigger.
00:56:50.620 --> 00:56:54.040
Mathematically, it comes
from this scaling property
00:56:54.040 --> 00:56:55.870
of the Fourier transforms.
00:56:55.870 --> 00:56:58.990
Another important one is
the scaling theorem, which
00:56:58.990 --> 00:57:02.530
will prove a little bit later.
00:57:02.530 --> 00:57:06.400
But it's also very
important one.
00:57:06.400 --> 00:57:09.140
Actually, all of these
properties are very important.
00:57:09.140 --> 00:57:12.340
Number four is actually
energy conservation.
00:57:12.340 --> 00:57:15.280
It relates the modulus--
00:57:15.280 --> 00:57:18.810
the integral of the
modulus of a function.
00:57:18.810 --> 00:57:20.270
We recognize this as energy.
00:57:20.270 --> 00:57:23.350
If you look at number
four, magnitude g squared
00:57:23.350 --> 00:57:25.940
is actually intensity.
00:57:25.940 --> 00:57:30.460
And if you integrate intensity
over the entire plane,
00:57:30.460 --> 00:57:35.147
then you'll get, of course,
energy flux, you get power.
00:57:35.147 --> 00:57:36.730
And power has to be
conserved, so this
00:57:36.730 --> 00:57:40.950
is what this theorem says, very
important, Parseval's theorem.
00:57:40.950 --> 00:57:45.290
And the convolution theorem
is also very important.
00:57:45.290 --> 00:57:47.710
We'll see an application
a little bit later today,
00:57:47.710 --> 00:57:50.290
or maybe Monday if we
run out of time today.
00:57:50.290 --> 00:57:53.304
But anyway, all of these
are very important.
00:57:59.730 --> 00:58:03.520
OK, so I will show
you some Fourier
00:58:03.520 --> 00:58:08.963
transforms to sort of give
you some of the properties.
00:58:13.860 --> 00:58:16.910
Are we still on?
00:58:16.910 --> 00:58:17.608
AUDIENCE: Yes.
00:58:17.608 --> 00:58:18.900
GEORGE BARBASTATHIS: Thank you.
00:58:27.080 --> 00:58:32.210
So this is a sinusoid.
00:58:32.210 --> 00:58:36.175
Of course, this is not
a physical transparency.
00:58:36.175 --> 00:58:38.300
Well, I can make a physical
transparency like this,
00:58:38.300 --> 00:58:40.550
but this assumes
negative values, which
00:58:40.550 --> 00:58:43.520
means that to make a physical
transparency like this one,
00:58:43.520 --> 00:58:46.490
you would have to have
a phase delay, as well
00:58:46.490 --> 00:58:53.870
as a grayscale variation.
00:58:53.870 --> 00:59:00.090
What I'm trying to say is
that if you have a cosine,
00:59:00.090 --> 00:59:01.070
two pi ux.
00:59:04.350 --> 00:59:10.110
Its magnitude goes like this.
00:59:16.620 --> 00:59:17.325
And its phase.
00:59:24.400 --> 00:59:25.150
What is the phase?
00:59:33.267 --> 00:59:34.600
What is the phase of the cosine?
00:59:52.500 --> 00:59:54.410
What is the phase of a
positive real number?
00:59:58.910 --> 01:00:01.040
What is the phase button?
01:00:01.040 --> 01:00:03.700
Someone said zero here.
01:00:03.700 --> 01:00:05.360
And that's correct.
01:00:05.360 --> 01:00:07.240
What is the phase of a
negative real number?
01:00:14.240 --> 01:00:17.800
Someone here suggesting zero.
01:00:17.800 --> 01:00:18.850
Negative real number.
01:00:21.710 --> 01:00:23.543
AUDIENCE: 180 degrees?
01:00:23.543 --> 01:00:25.210
GEORGE BARBASTATHIS:
Fine, that's right.
01:00:25.210 --> 01:00:29.830
So therefore, the
phase of the cosine
01:00:29.830 --> 01:00:35.340
is zero, where the
cosine is positive,
01:00:35.340 --> 01:00:39.346
and jumps to pi, where
the cosine is negative.
01:00:55.102 --> 01:00:57.560
OK, so that would be a very
difficult transparency to make,
01:00:57.560 --> 01:00:57.770
right?
01:00:57.770 --> 01:01:00.250
Because you would have to
have the grayscale variation
01:01:00.250 --> 01:01:04.270
like this to impose
the variation
01:01:04.270 --> 01:01:05.362
the amplitude modulation.
01:01:05.362 --> 01:01:07.570
And then you would have to
impose some variable phase
01:01:07.570 --> 01:01:14.360
delay also, in order to
impose the phase delay.
01:01:14.360 --> 01:01:16.590
So that is difficult to do.
01:01:16.590 --> 01:01:19.310
But anyway, mathematically,
we can write anything we like.
01:01:19.310 --> 01:01:22.580
So this is the cosine, and its
Fourier transform, of course,
01:01:22.580 --> 01:01:25.580
consists of two delta functions.
01:01:25.580 --> 01:01:29.210
So this is what these bright
dots indicate, delta functions
01:01:29.210 --> 01:01:35.130
whose spacing equals the inverse
of the period of the cosine.
01:01:35.130 --> 01:01:39.700
And, of course, if you
squeeze the cosine,
01:01:39.700 --> 01:01:43.470
since the spacing
equals the period,
01:01:43.470 --> 01:01:47.510
then the two delta functions
will go further away.
01:01:47.510 --> 01:01:49.700
Another way to describe
the same is, of course,
01:01:49.700 --> 01:01:51.230
by the scaling theorem.
01:01:51.230 --> 01:01:53.500
If you squeeze, it's
equivalent to scaling
01:01:53.500 --> 01:01:59.020
by a quantity larger than one.
01:01:59.020 --> 01:02:01.420
And therefore, the
spacing will also
01:02:01.420 --> 01:02:05.290
scale by a quantity
larger than one.
01:02:05.290 --> 01:02:07.420
What is a more
physical transparency
01:02:07.420 --> 01:02:08.950
that we saw in the
previous lecture?
01:02:15.600 --> 01:02:17.160
I said that this
is difficult to do,
01:02:17.160 --> 01:02:20.220
because you have to impose
both amplitude and phase
01:02:20.220 --> 01:02:24.047
variation on the transparency.
01:02:24.047 --> 01:02:25.380
AUDIENCE: A binary transparency?
01:02:25.380 --> 01:02:27.213
GEORGE BARBASTATHIS: A
binary, that's right.
01:02:27.213 --> 01:02:28.230
That's right.
01:02:28.230 --> 01:02:33.640
If you add a transparency whose
magnitude looks like this.
01:02:41.200 --> 01:02:43.010
Goes between zero and one.
01:02:43.010 --> 01:02:43.850
That is fine, right?
01:02:43.850 --> 01:02:47.730
I can do it very simply by
taking a piece of glass.
01:02:47.730 --> 01:02:51.050
And I can deposit some
metal, for example, aluminum,
01:02:51.050 --> 01:02:56.070
or chromium, or more something
like that in these regions.
01:02:56.070 --> 01:02:58.240
Oops, so you can not
see what they were.
01:02:58.240 --> 01:02:58.740
Yeah.
01:03:10.922 --> 01:03:11.630
Sorry about that.
01:03:11.630 --> 01:03:15.210
I pushed a button here that
I should not have pushed.
01:03:15.210 --> 01:03:18.110
So in these regions where I have
the [INAUDIBLE] to the metal,
01:03:18.110 --> 01:03:20.630
the transmissivity goes to zero.
01:03:20.630 --> 01:03:24.170
Another transparency that we
saw, and it is also physical,
01:03:24.170 --> 01:03:24.920
is this one.
01:03:27.643 --> 01:03:30.060
That was actually the first
example that we did on Monday.
01:03:39.360 --> 01:03:41.616
OK, how do I express
this transparency?
01:03:44.740 --> 01:03:45.355
Is it cosine?
01:04:21.570 --> 01:04:25.133
AUDIENCE: Is it 1/2
plus 1/2 cosine?
01:04:25.133 --> 01:04:26.216
GEORGE BARBASTATHIS: Yeah.
01:04:31.440 --> 01:04:33.630
Because it swings between
zero and one, right?
01:04:33.630 --> 01:04:34.720
So this, this will do it.
01:04:39.880 --> 01:04:41.980
Each Fourier
transform of this one.
01:04:41.980 --> 01:04:43.930
How would it be different
than the Fourier
01:04:43.930 --> 01:04:47.970
transform of the cosine that
I have on my slide here?
01:04:54.560 --> 01:04:57.740
What is the Fourier
transform of this one?
01:04:57.740 --> 01:05:00.650
AUDIENCE: So you
have a DC component.
01:05:00.650 --> 01:05:05.330
You have a DC component,
and then yeah.
01:05:05.330 --> 01:05:08.210
The magnitude of that
frequency is half of it.
01:05:19.650 --> 01:05:28.800
GEORGE BARBASTATHIS: So
in this representation,
01:05:28.800 --> 01:05:33.015
I would still have the tool
delta functions at spacing.
01:05:42.550 --> 01:05:44.260
But also in addition,
I would have
01:05:44.260 --> 01:05:47.030
an extra spot in the center.
01:05:47.030 --> 01:05:48.820
And this part would be brighter.
01:05:48.820 --> 01:05:53.320
So the power that goes into
this part correspondingly
01:05:53.320 --> 01:06:01.610
would be 1/2, 1/4, 1/4 squared.
01:06:10.060 --> 01:06:12.490
So the spot that
goes into the center
01:06:12.490 --> 01:06:16.140
is what you very correctly
refer to as the DC thermal.
01:06:18.810 --> 01:06:20.860
And now, of course, we
know why we call it DC.
01:06:20.860 --> 01:06:22.790
I think I mentioned
it also last time.
01:06:22.790 --> 01:06:25.730
It's because it corresponds
to zero frequency.
01:06:25.730 --> 01:06:29.870
So in electrical signals,
the zero frequency
01:06:29.870 --> 01:06:35.930
is known as the direct
current, or DC, DC component.
01:06:35.930 --> 01:06:38.290
OK, now without
cheating, that is
01:06:38.290 --> 01:06:43.960
without looking at the
next page of the notes.
01:06:43.960 --> 01:06:46.210
I would like to ask you, and
see if someone can guess.
01:06:46.210 --> 01:06:51.070
If I rotate this
grating by some angle
01:06:51.070 --> 01:06:53.100
what will happen to
the Fourier transform?
01:07:04.120 --> 01:07:04.620
Yeah?
01:07:08.090 --> 01:07:09.950
AUDIENCE: If you rotate
it by 90 degrees,
01:07:09.950 --> 01:07:13.573
I'd expect the frequencies to
rotate by 90 degrees, as well.
01:07:13.573 --> 01:07:14.990
GEORGE BARBASTATHIS:
That's right.
01:07:14.990 --> 01:07:16.730
So if you rotate
by 90 degrees, you
01:07:16.730 --> 01:07:20.930
expect the two spots to appear
along the V-axis rather than
01:07:20.930 --> 01:07:23.040
the U-axis.
01:07:23.040 --> 01:07:26.787
If you rotate
somewhere in between,
01:07:26.787 --> 01:07:28.120
where would this [INAUDIBLE] go?
01:07:28.120 --> 01:07:31.110
They will also rotate
in what fashion?
01:07:34.050 --> 01:07:35.730
OK, so the observation
to make here
01:07:35.730 --> 01:07:39.640
is that the two
spots if you draw
01:07:39.640 --> 01:07:46.030
a line that connects the
two Fourier delta functions.
01:07:46.030 --> 01:07:48.600
These lines would be
perpendicular to the fringes
01:07:48.600 --> 01:07:50.450
of the grating.
01:07:50.450 --> 01:07:53.710
And this will remain true as
you rotate the grating then,
01:07:53.710 --> 01:07:56.110
because actually, the
Fourier transform does not
01:07:56.110 --> 01:07:58.810
know what the coordinates are.
01:07:58.810 --> 01:08:01.900
So the Fourier transform knows
that you have a variation
01:08:01.900 --> 01:08:03.650
along this direction.
01:08:03.650 --> 01:08:05.320
And that gives rise
to the two delta
01:08:05.320 --> 01:08:09.160
function in this direction.
01:08:09.160 --> 01:08:12.700
In the vertical direction,
there's no variation.
01:08:12.700 --> 01:08:14.170
So therefore, the
Fourier transform
01:08:14.170 --> 01:08:16.939
is confined to the
zero frequency.
01:08:16.939 --> 01:08:20.350
So if you rotate the
grating, then these spots
01:08:20.350 --> 01:08:24.189
will rotate so that the
line connecting them
01:08:24.189 --> 01:08:28.120
remains perpendicular
to the fringes.
01:08:28.120 --> 01:08:34.970
This may not show quite right
because the projector actually
01:08:34.970 --> 01:08:37.640
squeezes my slide.
01:08:37.640 --> 01:08:40.495
So it may not show quite right.
01:08:40.495 --> 01:08:42.870
But if you think about it,
you should convince yourselves
01:08:42.870 --> 01:08:46.279
that the two spots
should be located
01:08:46.279 --> 01:08:52.270
along a line perpendicular to
the grooves of this gradient.
01:08:52.270 --> 01:08:54.189
And, of course, if you
squeeze the grooves
01:08:54.189 --> 01:08:56.770
in this rotated grating,
then the two spots
01:08:56.770 --> 01:09:00.399
will also move away,
again, along the same line
01:09:00.399 --> 01:09:01.750
perpendicular to the grooves.
01:09:05.338 --> 01:09:06.380
Any questions about that?
01:09:13.370 --> 01:09:15.500
The other property of the
Fourier transform farm
01:09:15.500 --> 01:09:17.990
which is listed in
the table of formulas
01:09:17.990 --> 01:09:21.710
that they showed
earlier is linearity.
01:09:21.710 --> 01:09:25.340
And linearity says that if
you have a function that
01:09:25.340 --> 01:09:30.779
is the linear superposition
of two functions whose Fourier
01:09:30.779 --> 01:09:33.859
transform you know, then
the Fourier transform
01:09:33.859 --> 01:09:38.660
of this function is the linear
superposition of the two
01:09:38.660 --> 01:09:39.800
Fourier transforms.
01:09:39.800 --> 01:09:41.300
So for example,
here is a function
01:09:41.300 --> 01:09:46.149
consisting of two gratings of
period lambda one and lambda
01:09:46.149 --> 01:09:46.649
two.
01:09:49.850 --> 01:09:51.500
Which one is the
Fourier transform?
01:09:51.500 --> 01:09:54.140
That's the Fourier transform
of the long period, right?
01:09:54.140 --> 01:09:56.120
Because the two spots
are close together.
01:09:56.120 --> 01:09:59.610
If you take the Fourier
transform of the short period,
01:09:59.610 --> 01:10:02.710
the Fourier transforms
are further apart.
01:10:02.710 --> 01:10:04.960
If you do the superposition now.
01:10:04.960 --> 01:10:07.930
What you get, well, it is a bit.
01:10:07.930 --> 01:10:10.430
If you superimpose
two frequencies,
01:10:10.430 --> 01:10:11.950
you get the bit pattern.
01:10:11.950 --> 01:10:13.420
Here it is.
01:10:13.420 --> 01:10:15.200
Looks kind of messy.
01:10:15.200 --> 01:10:17.280
The Fourier transform
is relatively cleaner.
01:10:17.280 --> 01:10:20.440
It is the two dots that
you get from this one,
01:10:20.440 --> 01:10:22.870
plus the two dots that you
got from the other one.
01:10:22.870 --> 01:10:25.330
So therefore, you
get four dots total.
01:10:25.330 --> 01:10:28.630
That's what the
superposition theorem says.
01:10:28.630 --> 01:10:31.360
Of course, you can generalize.
01:10:31.360 --> 01:10:33.570
I don't know if you can
see in both of them.
01:10:33.570 --> 01:10:38.520
On the top right here,
there's a bunch of dots.
01:10:38.520 --> 01:10:41.000
These dots actually,
each one of those.
01:10:41.000 --> 01:10:42.630
They're symmetrical
along the axis,
01:10:42.630 --> 01:10:45.840
so therefore, they
correspond to sinusoids.
01:10:45.840 --> 01:10:48.720
And the superposition
of sinusoids
01:10:48.720 --> 01:10:49.800
looks very messy here.
01:10:49.800 --> 01:10:52.425
It is still periodic, but messy.
01:10:52.425 --> 01:10:54.550
Of course, if you look at
it in the Fourier domain,
01:10:54.550 --> 01:10:57.000
each one of those
is it represented
01:10:57.000 --> 01:11:02.850
by its own individual
pair of delta functions.
01:11:02.850 --> 01:11:04.620
But, of course, this
is discrete now.
01:11:04.620 --> 01:11:07.830
What is even more
interesting is that if you
01:11:07.830 --> 01:11:11.280
were to connect all
these delta functions
01:11:11.280 --> 01:11:14.250
and get the continuous
Fourier transform, then
01:11:14.250 --> 01:11:16.890
your original pattern over
here, this page domain
01:11:16.890 --> 01:11:18.210
would become nonperiodic.
01:11:18.210 --> 01:11:19.740
So you can see very clearly.
01:11:19.740 --> 01:11:21.270
Here, I have discrete--
01:11:21.270 --> 01:11:25.160
a discrete Fourier
transform that corresponds
01:11:25.160 --> 01:11:26.540
to periodic pattern.
01:11:29.113 --> 01:11:31.030
AUDIENCE: Could you draw
the Fourier transform
01:11:31.030 --> 01:11:32.397
in the overhead projector?
01:11:32.397 --> 01:11:33.480
Because we can not see it.
01:11:33.480 --> 01:11:35.350
It looks dark,
totally dark here.
01:11:35.350 --> 01:11:36.400
GEORGE BARBASTATHIS: OK.
01:11:36.400 --> 01:11:41.520
I cannot quite draw it, but
I can sort of cartoon it.
01:11:41.520 --> 01:11:44.090
So the cartoon would
be dots like this.
01:12:01.797 --> 01:12:02.630
Something like this.
01:12:05.610 --> 01:12:12.760
So each pair of dots
corresponds to a cosine.
01:12:12.760 --> 01:12:14.840
And then what you
see on the top left
01:12:14.840 --> 01:12:17.555
is actually the superposition
of all of these cosines.
01:12:21.290 --> 01:12:21.790
OK.
01:12:24.840 --> 01:12:26.250
And, of course,
you can have sort
01:12:26.250 --> 01:12:32.380
of more general transparencies.
01:12:32.380 --> 01:12:34.720
You guys are too young
to remember this,
01:12:34.720 --> 01:12:40.610
but about in 2006, I believe.
01:12:40.610 --> 01:12:45.740
The Boston baseball
team beat the Yankees
01:12:45.740 --> 01:12:49.900
After 85 or 86 years.
01:12:49.900 --> 01:12:52.190
They finally managed
to beat them again.
01:12:52.190 --> 01:12:56.000
And the night of the game,
this is the Prudential Tower.
01:12:56.000 --> 01:12:58.800
For those of you who live
in Singapore, this is the--
01:12:58.800 --> 01:13:01.430
I think it's the tallest
building in Boston.
01:13:01.430 --> 01:13:05.600
And so that night, they lit
up their lights in the offices
01:13:05.600 --> 01:13:08.190
in a way that if you
looked at the pattern,
01:13:08.190 --> 01:13:11.970
you could see the sign,
Go Sox, the Boston famous
01:13:11.970 --> 01:13:13.970
called Red Sox.
01:13:13.970 --> 01:13:17.360
And, of course, the Fox
25 is the TV channel
01:13:17.360 --> 01:13:20.460
that sponsored the match.
01:13:20.460 --> 01:13:22.580
So I took a picture
with my camera.
01:13:22.580 --> 01:13:25.860
I can see this tower from
where I used to live in Boston.
01:13:25.860 --> 01:13:26.870
So this is a picture.
01:13:26.870 --> 01:13:30.420
And if you represent
it as a transparency,
01:13:30.420 --> 01:13:33.140
so that is the bright
spots correspond
01:13:33.140 --> 01:13:34.820
to transmission of light.
01:13:34.820 --> 01:13:37.790
The dark spots correspond
to blocking the light.
01:13:37.790 --> 01:13:40.000
Then you can think
it's Fourier transform.
01:13:40.000 --> 01:13:42.680
And you can see sort of
a more general pattern
01:13:42.680 --> 01:13:44.388
that looks like this.
01:13:44.388 --> 01:13:46.430
What is interesting is
that if you look carefully
01:13:46.430 --> 01:13:47.300
at this pattern.
01:13:47.300 --> 01:13:49.790
And I don't know if you
can see it in Boston.
01:13:49.790 --> 01:13:53.270
But the pattern here
looks kind of diffuse.
01:13:53.270 --> 01:13:56.300
But there's some distinct
spots, actually quite
01:13:56.300 --> 01:13:59.420
a few of these spots.
01:13:59.420 --> 01:14:01.810
Can anybody guess where
these spots came from?
01:14:09.510 --> 01:14:11.610
AUDIENCE: Some of the
features in the image,
01:14:11.610 --> 01:14:18.840
I guess, have straight lines
that kind of act like a box
01:14:18.840 --> 01:14:20.790
function, but not completely.
01:14:23.720 --> 01:14:24.720
Sorry, other way around.
01:14:24.720 --> 01:14:31.170
You're seeing basically
periodic structure in the image
01:14:31.170 --> 01:14:34.680
on the left gets reflected as
spots in the Fourier domain
01:14:34.680 --> 01:14:36.390
on the right.
01:14:36.390 --> 01:14:38.760
GEORGE BARBASTATHIS:
That's right.
01:14:38.760 --> 01:14:42.820
The building has irregular
spacing between the windows.
01:14:42.820 --> 01:14:45.480
So you see a very clear
periodic pattern here.
01:14:45.480 --> 01:14:51.090
It is modulated by the
Go Sox illumination,
01:14:51.090 --> 01:14:53.430
but nevertheless,
even the dark windows
01:14:53.430 --> 01:14:56.010
are visible in the
picture, right?
01:14:56.010 --> 01:14:56.642
Dark windows.
01:14:56.642 --> 01:14:58.600
Some of the windows, they
turned on the lights,
01:14:58.600 --> 01:14:59.560
some they didn't.
01:14:59.560 --> 01:15:02.130
But still, you can
see the windows,
01:15:02.130 --> 01:15:05.100
even if the lights were off.
01:15:05.100 --> 01:15:07.158
So this gives rise to
a periodic pattern.
01:15:07.158 --> 01:15:08.700
And, of course, the
Fourier transform
01:15:08.700 --> 01:15:11.910
of a periodic pattern
as I said before is
01:15:11.910 --> 01:15:15.720
a sequence of dots corresponding
to the Fourier series
01:15:15.720 --> 01:15:16.910
coefficients.
01:15:16.910 --> 01:15:22.110
So that's why you see this very
nice distinct dots over here.
01:15:22.110 --> 01:15:26.220
It is the windows
in the high rise.
01:15:26.220 --> 01:15:27.910
There's also more periodicity.
01:15:27.910 --> 01:15:31.870
This is a roof that
also is periodic.
01:15:31.870 --> 01:15:33.250
You can see a grating here.
01:15:39.630 --> 01:15:41.030
can you still hear me?
01:15:41.030 --> 01:15:43.510
I keep dropping
their microphone.
01:15:43.510 --> 01:15:44.010
Thanks.
01:15:48.910 --> 01:15:51.700
The grating here
should be visible as--
01:15:51.700 --> 01:15:53.710
it must be one of
these pairs of dots
01:15:53.710 --> 01:15:56.320
that do not correlate
with the building.
01:15:56.320 --> 01:16:00.790
That is the pattern
on the roof over here.
01:16:00.790 --> 01:16:02.380
This is a roof of
another building.
01:16:07.920 --> 01:16:12.560
Now, let's look at
the various theorems.
01:16:12.560 --> 01:16:14.360
I've already said
this before, so that's
01:16:14.360 --> 01:16:15.830
the similarity theorem.
01:16:15.830 --> 01:16:18.450
If you compare the Fourier
transform of two rectangles,
01:16:18.450 --> 01:16:20.400
one small, one big.
01:16:20.400 --> 01:16:23.210
The Fourier transform will
have the opposite behavior.
01:16:23.210 --> 01:16:25.040
The small rectangle
will give rise
01:16:25.040 --> 01:16:29.920
to a large Fourier transform.
01:16:29.920 --> 01:16:31.930
The other one that
I wanted to describe
01:16:31.930 --> 01:16:36.450
is this one, which
is the shift theorem.
01:16:36.450 --> 01:16:42.100
So the shift theorem, we
briefly glanced over it
01:16:42.100 --> 01:16:44.200
in the earlier slide.
01:16:44.200 --> 01:16:47.090
So let me remind you what
this earlier slide said.
01:16:50.310 --> 01:16:53.820
So the shift theorem
goes like this.
01:16:53.820 --> 01:16:56.400
I will do [INAUDIBLE]
in one dimension only.
01:16:56.400 --> 01:17:04.720
Then let's say that g of x has
a Fourier transform, G of u.
01:17:04.720 --> 01:17:14.380
The question is now, if I shift
g of x by some amount, x0,
01:17:14.380 --> 01:17:17.940
what is the Fourier transform?
01:17:17.940 --> 01:17:20.860
OK, so we do the same thing.
01:17:20.860 --> 01:17:24.060
We know that since this is true.
01:17:24.060 --> 01:17:28.815
Since this is true, we
know that G of u equals--
01:17:36.790 --> 01:17:40.790
then that is the definition
of the Fourier transform.
01:17:40.790 --> 01:17:44.600
Now, the question
is what is this one?
01:17:44.600 --> 01:17:46.570
So this one, the
Fourier transform
01:17:46.570 --> 01:17:50.110
of the shifted function will be
given by something like this.
01:17:57.205 --> 01:17:58.580
This, of course,
the same Fourier
01:17:58.580 --> 01:18:01.880
transform, but now I plugged
in the shifted function.
01:18:01.880 --> 01:18:05.600
And in order to
bring it to order,
01:18:05.600 --> 01:18:07.460
again, I will do a
coordinate transform.
01:18:11.720 --> 01:18:14.490
And this is very
easy because, again,
01:18:14.490 --> 01:18:17.280
the bounds of the integral
are minus infinity, infinity.
01:18:17.280 --> 01:18:20.780
They don't change upon
the transformation.
01:18:20.780 --> 01:18:24.120
The integral doesn't
change either.
01:18:24.120 --> 01:18:27.240
I mean, the differential
doesn't change either.
01:18:27.240 --> 01:18:28.760
The only thing that
changes is here.
01:18:28.760 --> 01:18:30.075
So you'll get the integral.
01:18:37.630 --> 01:18:41.640
So x equals c plus x0, right?
01:19:20.170 --> 01:19:24.470
Because this you recognize
is the same as this.
01:19:24.470 --> 01:19:26.090
OK, so this is
the shift theorem.
01:19:33.600 --> 01:19:35.430
So now, why is it
related to this one?
01:19:35.430 --> 01:19:40.900
Well, this one, if I
do a cross section.
01:19:55.010 --> 01:19:56.070
It will look like this.
01:19:56.070 --> 01:19:58.280
What I did is I
drew a cross section
01:19:58.280 --> 01:20:00.450
along the vertical axis.
01:20:00.450 --> 01:20:03.440
So let's call the
vertical axis x.
01:20:03.440 --> 01:20:05.510
So this is one.
01:20:05.510 --> 01:20:06.330
What is this?
01:20:06.330 --> 01:20:07.460
Well, this is a rectangle.
01:20:10.750 --> 01:20:11.840
OK, we know this one.
01:20:11.840 --> 01:20:14.240
And we computed this
Fourier transform.
01:20:14.240 --> 01:20:17.930
If you look at this one,
it is also rectangle.
01:20:22.090 --> 01:20:25.870
The size is the same.
01:20:25.870 --> 01:20:30.570
If this is x0, this is also x0.
01:20:35.410 --> 01:20:37.950
But it is displaced.
01:20:37.950 --> 01:20:39.710
Let's use a symbol
for this displacement.
01:20:39.710 --> 01:20:40.700
Let's call it a.
01:20:45.690 --> 01:20:47.070
Actually, this would be minus a.
01:20:50.718 --> 01:20:51.930
And this one.
01:21:05.560 --> 01:21:06.060
OK.
01:21:08.800 --> 01:21:12.270
So let's see if we can
apply the shift theorem.
01:21:17.530 --> 01:21:19.930
Actually, we have to
apply two theorems here,
01:21:19.930 --> 01:21:24.108
the shift theorem and
the scaling theorem.
01:21:24.108 --> 01:21:25.400
Which one should I apply first?
01:21:38.710 --> 01:21:40.150
OK, let me start.
01:21:40.150 --> 01:21:41.422
Let's do one thing at a time.
01:21:41.422 --> 01:21:42.880
So let me write
this function down.
01:21:42.880 --> 01:21:46.340
So g of x equals.
01:22:07.358 --> 01:22:09.400
Each one of those corresponds
to the three rects.
01:22:17.860 --> 01:22:20.580
OK.
01:22:20.580 --> 01:22:22.568
Now, I want to take
the Fourier transform.
01:22:28.920 --> 01:22:32.050
So one, I've already done.
01:22:32.050 --> 01:22:33.620
It's this one.
01:22:33.620 --> 01:22:34.890
I guess I use the red pen.
01:22:39.690 --> 01:22:41.910
This one, we concluded earlier.
01:22:41.910 --> 01:22:47.260
It is x0, sinc of ux0.
01:22:49.968 --> 01:22:51.010
What about the other one?
01:23:11.630 --> 01:23:13.700
First of all, linearity
says that I can just
01:23:13.700 --> 01:23:15.360
add them, right?
01:23:15.360 --> 01:23:16.070
So that's easy.
01:23:19.090 --> 01:23:20.780
Yes?
01:23:20.780 --> 01:23:24.640
AUDIENCE: They're going to
be the x0, sinc of u of x0.
01:23:24.640 --> 01:23:33.393
But shifted by e to the i 2 pi,
a, and all that other stuff.
01:23:33.393 --> 01:23:34.810
GEORGE BARBASTATHIS:
That's right.
01:23:34.810 --> 01:23:46.460
So I will get this one for
this term times the shift
01:23:46.460 --> 01:23:50.140
according to the shift theorem.
01:23:56.120 --> 01:24:07.360
And similarly,
with a minus sign.
01:24:07.360 --> 01:24:12.870
Because here, the shift is
in the negative direction.
01:24:12.870 --> 01:24:13.370
I'm sorry.
01:24:13.370 --> 01:24:16.360
The minus sign belongs here,
and the plus belongs here.
01:24:20.580 --> 01:24:23.190
OK, so get a common
term in all of this.
01:24:50.480 --> 01:24:51.390
OK, one is a plus.
01:24:51.390 --> 01:24:53.760
That is a minus.
01:24:53.760 --> 01:24:56.590
OK, so does this explain
now what you see here?
01:25:12.030 --> 01:25:13.683
AUDIENCE: It's a 1 plus cosine.
01:25:13.683 --> 01:25:15.100
GEORGE BARBASTATHIS:
That's right.
01:25:15.100 --> 01:25:15.730
This is.
01:25:30.930 --> 01:25:33.952
And indeed, in this
calculated pattern.
01:25:33.952 --> 01:25:35.160
Actually, the way I did this.
01:25:35.160 --> 01:25:39.680
I used the fft2
function in Matlab.
01:25:39.680 --> 01:25:41.550
And you can see
that fft2 correctly
01:25:41.550 --> 01:25:43.800
produced as a
sinusoidal modulation
01:25:43.800 --> 01:25:48.230
here, which is imposed by
the shift theorem, really.
01:25:48.230 --> 01:25:49.930
So that's very interesting.
01:25:49.930 --> 01:25:56.420
If you translate the
original function,
01:25:56.420 --> 01:25:59.410
you get this
sinusoidal modulation
01:25:59.410 --> 01:26:00.760
in the Fourier transform.
01:26:00.760 --> 01:26:02.990
And now because we
have a superposition--
01:26:02.990 --> 01:26:04.710
an interference, really--
01:26:04.710 --> 01:26:10.090
of sinusoidal modulations,
that's why we'll get the--
01:26:10.090 --> 01:26:14.680
well, these fringes in the
Fourier transform pattern.
01:26:17.190 --> 01:26:21.540
And, of course, if you
rotate this pattern.
01:26:21.540 --> 01:26:24.150
Then the fringes also
rotate by the same token
01:26:24.150 --> 01:26:25.380
we said before, right?
01:26:25.380 --> 01:26:29.160
Because now in this case, the
displacement is both x and y.
01:26:29.160 --> 01:26:32.090
So you will get the
complex exponential
01:26:32.090 --> 01:26:33.410
in the rotated case.
01:26:46.620 --> 01:26:50.100
When you go to
the Fourier space,
01:26:50.100 --> 01:26:52.050
due to the shift
theorem, you will
01:26:52.050 --> 01:26:55.930
get the complex exponential
of the form, e to the minus i
01:26:55.930 --> 01:27:02.420
two pi ax plus by.
01:27:02.420 --> 01:27:04.970
Where, for example, this is a.
01:27:08.435 --> 01:27:09.680
And this is b.
01:27:21.930 --> 01:27:23.430
So when you do
this, a preposition
01:27:23.430 --> 01:27:25.470
of these complex
exponentials, you
01:27:25.470 --> 01:27:30.190
will get rotated fringes in
the Fourier transform pattern.
01:27:36.310 --> 01:27:36.810
OK.
01:27:46.090 --> 01:27:47.232
Any questions?
01:28:00.270 --> 01:28:02.340
The last thing before
we quit for tonight,
01:28:02.340 --> 01:28:06.430
or for this morning, is
the evolution theorem.
01:28:06.430 --> 01:28:08.970
And that's a really,
really important one
01:28:08.970 --> 01:28:11.490
that you probably remember.
01:28:11.490 --> 01:28:13.750
I don't know, maybe you
remember it with horror.
01:28:13.750 --> 01:28:15.540
Or maybe you remember
it with fondness.
01:28:15.540 --> 01:28:18.000
But anyway, whatever
the case may be.
01:28:18.000 --> 01:28:21.550
You may remember from
your systems classes.
01:28:21.550 --> 01:28:26.100
So the convolution
theorem says that if you
01:28:26.100 --> 01:28:41.720
have a system whose
input is g sub in of x,
01:28:41.720 --> 01:28:46.800
and the output is g
sub out of x prime.
01:28:46.800 --> 01:28:49.020
A linear system is
actually-- the output
01:28:49.020 --> 01:28:50.918
is expressed as a convolution.
01:29:04.610 --> 01:29:07.270
And you may be more familiar
with seeing these convolutions
01:29:07.270 --> 01:29:10.690
in the time domain,
but it doesn't matter.
01:29:10.690 --> 01:29:12.820
In the case of
optics, we're dealing
01:29:12.820 --> 01:29:14.900
with space domain signals.
01:29:14.900 --> 01:29:17.470
So we simply swap t with x.
01:29:17.470 --> 01:29:19.005
But it's actually the same idea.
01:29:19.005 --> 01:29:20.380
So one of the
[INAUDIBLE] example
01:29:20.380 --> 01:29:25.140
of this convolution in the
case of Fresnel propagation.
01:29:25.140 --> 01:29:27.210
If you remember,
Fresnel propagation
01:29:27.210 --> 01:29:32.320
was g out of x prime,
y prime, proportional.
01:29:32.320 --> 01:29:34.420
It had some additional
terms in front.
01:29:34.420 --> 01:29:37.650
But the integral of that we
got goes something like this.
01:29:37.650 --> 01:29:40.420
G sub in of x comma y.
01:29:53.370 --> 01:29:57.520
So it is earlier convolution,
where this function, h of xy.
01:30:05.500 --> 01:30:06.310
What is this again?
01:30:15.370 --> 01:30:16.950
What is this physically?
01:30:16.950 --> 01:30:18.308
AUDIENCE: It's a spherical wave.
01:30:18.308 --> 01:30:19.600
GEORGE BARBASTATHIS: Thank you.
01:30:32.920 --> 01:30:34.420
So the convolution
here emphasize
01:30:34.420 --> 01:30:39.640
that if you take Fourier
transforms of everybody.
01:30:39.640 --> 01:30:43.710
So you Fourier
transform this one.
01:30:43.710 --> 01:30:46.660
You call it G sub in of u.
01:30:46.660 --> 01:30:48.230
You Fourier transform this one.
01:30:48.230 --> 01:30:52.350
You call it G sub out of u.
01:30:52.350 --> 01:30:53.730
You Fourier transform this one.
01:31:05.130 --> 01:31:06.040
OK.
01:31:06.040 --> 01:31:09.910
Then the convolution theorem
says that this equals
01:31:09.910 --> 01:31:15.660
G sub in of u times H of u.
01:31:15.660 --> 01:31:17.250
OK, that's the
convolution theorem.
01:31:17.250 --> 01:31:20.850
So it says that in
the space domain,
01:31:20.850 --> 01:31:23.280
you have a convolution
relationship.
01:31:23.280 --> 01:31:27.347
Then in the Fourier domain, you
simply get a multiplication.
01:31:27.347 --> 01:31:29.430
And actually, that also
goes the other way around.
01:31:29.430 --> 01:31:32.430
If you have a multiplication
in the space domain,
01:31:32.430 --> 01:31:35.820
you have a convolution
in the frequency domain.
01:31:35.820 --> 01:31:37.770
We'll get to use that
a little bit later.
01:31:40.420 --> 01:31:42.010
Does anybody want
me to prove this?
01:31:42.010 --> 01:31:43.677
Do you believe it,
or should I prove it?
01:31:50.880 --> 01:31:53.933
Well, let me prove it.
01:31:53.933 --> 01:31:55.600
Since we're in the
mood of a must today.
01:32:01.650 --> 01:32:03.855
So let me write the
convolution integral.
01:32:03.855 --> 01:32:05.730
Actually, before I do
that, let me write down
01:32:05.730 --> 01:32:07.158
the Fourier transforms.
01:32:27.576 --> 01:32:28.790
OK, similarly.
01:32:50.330 --> 01:32:52.480
OK, so these are
really all the same.
01:33:00.760 --> 01:33:02.225
Now, let me write the output.
01:33:08.950 --> 01:33:12.330
OK, and by the same token, these
are the Fourier transforms.
01:33:12.330 --> 01:33:14.910
I can also write the
Fourier integrals
01:33:14.910 --> 01:33:16.050
in the inverse fashion.
01:33:16.050 --> 01:33:18.240
So for example, I
can neither g sub
01:33:18.240 --> 01:33:28.330
in of x equals integral G sub in
of u, e to the plus i two pi ux
01:33:28.330 --> 01:33:29.280
du.
01:33:29.280 --> 01:33:33.090
If you recall, we call this
the inverse Fourier transform,
01:33:33.090 --> 01:33:35.670
or the Fourier integral.
01:33:35.670 --> 01:33:38.040
And by the same
token, I have h of x
01:33:38.040 --> 01:33:42.210
equals a similar looking
integral for H of u.
01:33:42.210 --> 01:33:48.180
And g sub out, again,
similar looking integral
01:33:48.180 --> 01:33:50.680
for G sub out of u.
01:33:53.230 --> 01:33:54.540
OK, these are just definitions.
01:33:54.540 --> 01:33:57.930
So far, I haven't
really done anything.
01:33:57.930 --> 01:34:00.230
Now, let me write out
the convolution integral.
01:34:16.217 --> 01:34:18.050
What I will do now is
a little bit horrible,
01:34:18.050 --> 01:34:21.110
but you will see the
logic of it in a second.
01:34:21.110 --> 01:34:24.567
I will substitute the
Fourier transform.
01:34:24.567 --> 01:34:25.400
Actually, I'm sorry.
01:34:25.400 --> 01:34:27.560
I will substitute
the Fourier integral
01:34:27.560 --> 01:34:30.038
inside this relationship.
01:34:30.038 --> 01:34:31.330
So how many integrals do I get?
01:34:31.330 --> 01:34:32.180
I get three, right?
01:34:32.180 --> 01:34:34.730
I get one that I had, and
then each one of those
01:34:34.730 --> 01:34:36.670
will be written as an integral.
01:34:36.670 --> 01:34:38.308
So here are the three integrals.
01:34:41.730 --> 01:34:44.170
That's the original one.
01:34:44.170 --> 01:34:48.140
Then for g sub in of x, I
substitute its own Fourier
01:34:48.140 --> 01:34:48.640
integral.
01:35:02.930 --> 01:35:04.240
And the same for h.
01:35:10.357 --> 01:35:11.690
Have to be a little bit careful.
01:35:11.690 --> 01:35:15.320
h is computed in this
shifted coordinate,
01:35:15.320 --> 01:35:19.010
so it is x prime minus x du.
01:35:22.580 --> 01:35:23.660
OK.
01:35:23.660 --> 01:35:26.630
Now, what I'll do is assuming
that these functions are
01:35:26.630 --> 01:35:28.570
well-behaved and
so on and so forth,
01:35:28.570 --> 01:35:32.090
I will actually interchange
the order of integration.
01:35:36.780 --> 01:35:39.120
Let's see if I can do it in
a way that it all fits here.
01:35:59.500 --> 01:36:01.690
OK, let me be a little
bit more careful here.
01:36:01.690 --> 01:36:04.990
[INAUDIBLE] variable, u is in
the same in the two integrals.
01:36:04.990 --> 01:36:08.830
So to avoid confusion, I
will actually label them.
01:36:08.830 --> 01:36:12.060
I will call this
u1, and this u2.
01:36:15.250 --> 01:36:21.100
OK, so now, I have the
du1, du2 integrals.
01:36:21.100 --> 01:36:25.580
What's inside g sub in of u1?
01:36:25.580 --> 01:36:28.410
H of u2.
01:36:28.410 --> 01:36:33.550
And all of this is
multiplied by a x integral.
01:36:47.600 --> 01:36:48.470
So what do I have?
01:36:48.470 --> 01:36:51.590
So for x, I have
u1 from this term,
01:36:51.590 --> 01:36:53.990
and minus u2 from this term.
01:36:58.380 --> 01:36:59.670
And what's left?
01:36:59.670 --> 01:37:01.110
This thing left over, right?
01:37:01.110 --> 01:37:06.850
So let me not forget it, e
to the i two pi u2 x prime.
01:37:06.850 --> 01:37:09.450
x prime, of course,
is not plain.
01:37:09.450 --> 01:37:12.850
So I'll just leave it there.
01:37:12.850 --> 01:37:14.710
It is not plain in the
integration, that is.
01:37:14.710 --> 01:37:15.210
OK.
01:37:19.400 --> 01:37:21.730
So now, what is this?
01:37:21.730 --> 01:37:24.650
I put one too many dx's.
01:37:24.650 --> 01:37:25.410
So what is this?
01:37:42.110 --> 01:37:46.250
It is the Fourier transform
of an exponential.
01:37:46.250 --> 01:37:49.230
Remember, these
integrals without bounds,
01:37:49.230 --> 01:37:52.520
they really go from minus
infinity to infinity, right?
01:37:52.520 --> 01:37:55.360
So if I integrate an
exponential from minus infinity
01:37:55.360 --> 01:37:56.950
to infinity, what do I get?
01:37:56.950 --> 01:37:58.970
We said it earlier this morning.
01:38:19.540 --> 01:38:22.810
Your tuition is ticking
away one second at a time.
01:39:00.560 --> 01:39:02.490
Well, it's 9:25,
according to my clock.
01:39:02.490 --> 01:39:04.681
So I guess we stop here.
01:39:04.681 --> 01:39:06.870
And I'll let you ponder
this on your own.
01:39:10.310 --> 01:39:12.780
See you on Monday.