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COLIN SHEPPARD: OK, so for
the ones that just arrived,

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I'm going to be giving
the lecture today.

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00:00:25,680 --> 00:00:27,090
But George is here.

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So if you've got
questions, you can ask me.

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00:00:30,660 --> 00:00:32,720
I don't mind being
interrupted at any time.

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00:00:32,720 --> 00:00:36,917
But also, if you want
George to answer something,

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00:00:36,917 --> 00:00:38,000
he's sitting here as well.

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00:00:38,000 --> 00:00:41,450
So he can also contribute.

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00:00:41,450 --> 00:00:47,090
So last time, George was looking
at the Fourier transforming

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property of a lens.

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00:00:49,330 --> 00:00:55,070
And he looks at the--
derived the 1D case, where

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you've just got a function of, x
rather than a function of x, y.

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You've got this
transparency a distance z

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in front of the lens.

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00:01:06,100 --> 00:01:09,990
And then you've got a
screen placed a distance

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00:01:09,990 --> 00:01:12,650
f behind the lens.

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00:01:12,650 --> 00:01:16,150
And so the question is, what
do you get on that screen?

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00:01:16,150 --> 00:01:20,590
And I'm not going to go
through all the algebra of that

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00:01:20,590 --> 00:01:22,780
again because George
did it last time.

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00:01:22,780 --> 00:01:23,950
This was the calculus.

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00:01:23,950 --> 00:01:25,390
This was the answer.

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What you find is that what
you get in this plane here

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00:01:31,660 --> 00:01:35,050
is actually the
Fourier transform

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00:01:35,050 --> 00:01:39,330
of the thin transparency.

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00:01:39,330 --> 00:01:43,060
So g of x is this
transparency here.

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00:01:43,060 --> 00:01:46,320
You get the Fourier
transformer that.

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00:01:46,320 --> 00:01:51,970
And multiplied by
a parabolic phase

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00:01:51,970 --> 00:01:56,710
factor, which depends on the
position of this screen--

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00:01:56,710 --> 00:01:59,980
of the transparency.

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00:01:59,980 --> 00:02:02,200
What is the value of this z?

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00:02:02,200 --> 00:02:08,620
You'll notice that you get this
factor here, 1 minus z over f.

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00:02:08,620 --> 00:02:12,430
So if z equals f,
this is going to be 0.

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00:02:12,430 --> 00:02:15,110
And this phase factor
is going to disappear.

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00:02:15,110 --> 00:02:18,170
So for the special
case where z equals

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00:02:18,170 --> 00:02:23,390
f, all you see here is just
the Fourier transform of this.

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00:02:23,390 --> 00:02:26,580
Very simple-- very
simple answer.

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00:02:26,580 --> 00:02:30,550
And people actually have known
this for quite a long time.

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00:02:30,550 --> 00:02:32,900
And at one time, it did
look really attractive

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00:02:32,900 --> 00:02:35,600
that you could
actually imagine using

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00:02:35,600 --> 00:02:39,620
this as a real way of
doing Fourier transforms.

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00:02:39,620 --> 00:02:42,910
Especially as we're
now going to do,

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00:02:42,910 --> 00:02:48,230
you can immediately generalize
that to the 2D case.

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00:02:48,230 --> 00:02:52,040
So you can do a 2D
Fourier transform.

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00:02:52,040 --> 00:02:56,810
And when you think that the
light propagates through this

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at the speed of light, you
can do a 2D Fourier transform

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00:02:59,750 --> 00:03:01,610
at the speed of light.

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00:03:01,610 --> 00:03:03,530
And when people
first realized this,

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00:03:03,530 --> 00:03:05,765
which was back in the 1950s--

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00:03:05,765 --> 00:03:11,540
'50s or early '60s, computers
were very slow then.

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00:03:11,540 --> 00:03:16,020
So doing a 2D Fourier
transform was not trivial.

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00:03:16,020 --> 00:03:19,760
So a lot of effort was put into
trying to design instruments

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to do that.

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00:03:20,270 --> 00:03:20,820
Yes.

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00:03:20,820 --> 00:03:22,190
GEORGE BARBASTATHIS: Do you
know who was the first one

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to realize this?

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Was it Vander Lugt?

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COLIN SHEPPARD: The
question is, who was

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the first to recognize that?

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00:03:29,960 --> 00:03:36,845
Yeah Vander Lugt was
at Michigan, wasn't he?

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00:03:36,845 --> 00:03:40,580
And interestingly,
the holography guys

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00:03:40,580 --> 00:03:42,733
were also in the same lab.

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00:03:42,733 --> 00:03:43,650
What's the guy's name.

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00:03:43,650 --> 00:03:45,790
The one-- Emmett Leith.

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00:03:45,790 --> 00:03:50,240
And interestingly, Emmett
Leith and Vander Lugt

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both published virtually the
same paper in the same year.

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00:03:54,920 --> 00:03:58,295
It's a very, very interesting
historical thing there.

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They're both from the
same lab, and they both

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00:04:00,170 --> 00:04:04,100
invented almost the same
idea as far as I can see.

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And neither of them
referred to the other guy.

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Very peculiar.

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Anyway, I don't know who really
came up with this idea first.

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00:04:13,030 --> 00:04:21,440
But actually, I suspect that
actually, even Ernst Abbe

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realized this.

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Ernst Abbe was the scientist who
designed the Zeiss microscopes

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120, 130 years ago.

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And he came up with a
theory for image formation

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in a microscope,
which is actually

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00:04:39,940 --> 00:04:41,690
what we're going to
be getting onto pretty

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well, the next thing--

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00:04:43,250 --> 00:04:46,940
the imaging with a 4F F system.

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00:04:46,940 --> 00:04:50,130
He understood that
130 years ago.

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00:04:50,130 --> 00:04:52,940
And I don't think he's often
given as much credit is

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00:04:52,940 --> 00:04:54,180
he should get.

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00:04:54,180 --> 00:04:56,600
Anyway, you can
use this as a way

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00:04:56,600 --> 00:05:00,130
of doing a Fourier transform,
and even a 2D Fourier

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00:05:00,130 --> 00:05:00,860
transform.

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00:05:00,860 --> 00:05:02,780
So you see, going
from this stage

95
00:05:02,780 --> 00:05:06,000
to this stage is
trivial, really.

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00:05:06,000 --> 00:05:09,500
All we've done here
is assuming that it's

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00:05:09,500 --> 00:05:13,400
a separable function
of x and y so you

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00:05:13,400 --> 00:05:16,700
get exactly the same
sort of integral for y

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00:05:16,700 --> 00:05:18,860
as you get for x.

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00:05:18,860 --> 00:05:21,920
So it now becomes
a double integral.

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00:05:21,920 --> 00:05:25,430
This is the two-dimensional
Fourier transform object

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00:05:25,430 --> 00:05:28,200
of g of x, y.

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00:05:28,200 --> 00:05:34,460
And you can see, you get
again the phase factor.

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00:05:34,460 --> 00:05:37,100
Here, because it's
separable, you've

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00:05:37,100 --> 00:05:40,310
got this e to the
i pi x squared,

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00:05:40,310 --> 00:05:43,170
and you got an e to
the i pi y squared.

107
00:05:43,170 --> 00:05:46,700
And of course, x squared
plus y squared very nicely

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00:05:46,700 --> 00:05:48,390
equals r squared.

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00:05:48,390 --> 00:05:54,920
So this becomes a
spherical type wave.

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00:05:54,920 --> 00:06:00,760
So we've still got this 1 minus
z over f So if z equals f,

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00:06:00,760 --> 00:06:04,750
then again, this
phase factor vanishes,

112
00:06:04,750 --> 00:06:09,410
and we end up with just
the pure Fourier transform.

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00:06:09,410 --> 00:06:11,100
Yeah, as I say,
so a lot of people

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00:06:11,100 --> 00:06:14,100
thought that this
was quite a neat way

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00:06:14,100 --> 00:06:16,200
of doing Fourier transforms.

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00:06:16,200 --> 00:06:19,590
There was a lot of work
on optical computers

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00:06:19,590 --> 00:06:23,280
all the way through
the '60s and '70s.

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00:06:23,280 --> 00:06:26,700
But the trouble is that, as
fast as they could develop

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00:06:26,700 --> 00:06:30,810
these systems, the
computer people developed

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00:06:30,810 --> 00:06:32,800
their computers even faster.

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00:06:32,800 --> 00:06:36,040
So they were always one
step behind, I think.

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00:06:36,040 --> 00:06:37,740
So I think in the
end, people realized

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00:06:37,740 --> 00:06:41,880
it was a waste of time,
and that it was never

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00:06:41,880 --> 00:06:49,850
really going to become
a practical way of doing

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00:06:49,850 --> 00:06:51,300
real calculations.

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00:06:51,300 --> 00:06:52,970
But in the early days, it was.

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00:06:52,970 --> 00:06:55,850
And Emmett Leith we
were just mentioning,

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00:06:55,850 --> 00:06:59,420
he first developed these
systems for actually

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00:06:59,420 --> 00:07:07,730
analyzing the images they
took from a synthetic aperture

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00:07:07,730 --> 00:07:14,360
They'd fly an aeroplane over
North Korea, taking pictures.

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00:07:14,360 --> 00:07:19,370
And then the idea was that
the plane takes many pictures

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00:07:19,370 --> 00:07:23,960
as it's traveling along, and
they're coherent with respect

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00:07:23,960 --> 00:07:24,600
to each other.

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00:07:24,600 --> 00:07:29,880
So you can actually combine
them digitally afterwards

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00:07:29,880 --> 00:07:33,380
and get an image, which
has got amazing resolution.

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00:07:33,380 --> 00:07:36,770
And you can see the guy in North
Korea reading his newspaper

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00:07:36,770 --> 00:07:39,260
or whatever.

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00:07:39,260 --> 00:07:42,860
So they really did that
sort of thing way back then.

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00:07:42,860 --> 00:07:46,700
Right, so that's the
equation here then.

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00:07:46,700 --> 00:07:50,120
And yeah, so this is
just going through how

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00:07:50,120 --> 00:07:52,340
that thing was derived.

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00:07:52,340 --> 00:08:00,300
So we've got the screen
here, the g of x.

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00:08:00,300 --> 00:08:03,990
By the time we get
here, it's convolved

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00:08:03,990 --> 00:08:07,930
with some parabolic
phase function.

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00:08:07,930 --> 00:08:09,630
And then after it
goes through here,

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00:08:09,630 --> 00:08:13,110
it's multiplied by another
parabolic phase function.

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00:08:13,110 --> 00:08:15,830
And then as it
propagates here, it's

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00:08:15,830 --> 00:08:19,330
convolved with another
parabolic phase function.

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00:08:19,330 --> 00:08:24,050
And if we have that magic
condition, that this is f

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00:08:24,050 --> 00:08:28,080
and this is f, most of these
parabolic phases cancel out

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00:08:28,080 --> 00:08:31,300
and we're left with this
very simple final answer.

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00:08:31,300 --> 00:08:33,090
So that's basically
what's happening.

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00:08:33,090 --> 00:08:37,260
And George went through the
various stages of doing this.

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00:08:37,260 --> 00:08:41,429
Of course, you end up with a
couple of integrals of products

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00:08:41,429 --> 00:08:43,100
of three phase functions here.

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00:08:43,100 --> 00:08:49,350
So the algebra process
was quite complicated.

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00:08:49,350 --> 00:08:53,270
Well, not really complicated,
but just lengthy.

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00:08:53,270 --> 00:08:56,720
But everything in
the end cancels.

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00:08:56,720 --> 00:09:00,240
If you haven't got z
equals 2f, though, then you

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00:09:00,240 --> 00:09:05,400
can see that you have
got this phase factor.

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00:09:05,400 --> 00:09:08,960
So you don't actually get
exactly the Fourier transform

162
00:09:08,960 --> 00:09:10,780
in this plane in that case.

163
00:09:10,780 --> 00:09:13,290
You get the Fourier
transform multiplied

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00:09:13,290 --> 00:09:15,630
by this phase factor.

165
00:09:15,630 --> 00:09:18,890
And that can be
important in some cases.

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00:09:18,890 --> 00:09:22,750
That's something that you
need to take into account.

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00:09:22,750 --> 00:09:26,490
There's another
special case, actually.

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00:09:26,490 --> 00:09:29,620
That's the case when z equals 0.

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00:09:29,620 --> 00:09:33,500
The case when z equals
0 is if this screen

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00:09:33,500 --> 00:09:35,030
coincides with the lens--

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00:09:35,030 --> 00:09:37,656
immediately close
up to the lens.

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00:09:37,656 --> 00:09:41,060
And I think Goodman does
that one as well, doesn't he.

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00:09:41,060 --> 00:09:45,130
And so if z equals 0, this goes.

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00:09:45,130 --> 00:09:47,330
And so we're still left
with this phase factor,

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00:09:47,330 --> 00:09:52,797
e to the i pi x double
dash squared over lambda f.

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00:09:52,797 --> 00:09:54,380
GEORGE BARBASTATHIS:
And also, Goodman

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00:09:54,380 --> 00:09:56,570
does the case of
the transparency

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00:09:56,570 --> 00:09:59,247
after the lens, which I
did not do in the notes.

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00:09:59,247 --> 00:10:00,330
COLIN SHEPPARD: Right, OK.

180
00:10:00,330 --> 00:10:04,320
Yeah, so Goodman does this
case of the transparency placed

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00:10:04,320 --> 00:10:04,820
in here.

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00:10:04,820 --> 00:10:06,695
GEORGE BARBASTATHIS:
That this is z negative.

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00:10:06,695 --> 00:10:11,015
COLIN SHEPPARD: Yeah, but
it's very similar answers.

184
00:10:14,250 --> 00:10:22,510
OK, so this is just
writing down finally, then.

185
00:10:22,510 --> 00:10:29,650
We're defining then the Fourier
transform of little g is big G.

186
00:10:29,650 --> 00:10:33,220
So the final answer is that
you've got this parabolic phase

187
00:10:33,220 --> 00:10:37,210
function times the
Fourier transform of g.

188
00:10:47,880 --> 00:10:48,380
OK.

189
00:10:48,380 --> 00:10:55,030
So this is pointing
out that we've now

190
00:10:55,030 --> 00:10:58,960
looked at two different ways
you can do a Fourier transform.

191
00:10:58,960 --> 00:11:04,030
One way is to just propagate
the light a very large distance.

192
00:11:04,030 --> 00:11:07,060
You remember, if you have
[? frame ?] of a diffraction,

193
00:11:07,060 --> 00:11:13,600
if your screen is a very long
way away from the diffracting

194
00:11:13,600 --> 00:11:16,960
screen, object,
whatever it was called--

195
00:11:16,960 --> 00:11:20,410
transparency then again,
you get this Fourier

196
00:11:20,410 --> 00:11:22,120
transform relationship.

197
00:11:22,120 --> 00:11:25,660
But of course, here he's
put a proportional side.

198
00:11:25,660 --> 00:11:28,150
There's some things
out the front there.

199
00:11:28,150 --> 00:11:31,930
And in particular, there's going
to be, again, a parabolic phase

200
00:11:31,930 --> 00:11:33,070
term, isn't there?

201
00:11:33,070 --> 00:11:37,180
Because in this case here,
what you'd expect actually

202
00:11:37,180 --> 00:11:38,950
is that this
Fourier transform is

203
00:11:38,950 --> 00:11:43,840
going to be actually on the
surface of some curved surface,

204
00:11:43,840 --> 00:11:46,870
not of a plane surface.

205
00:11:46,870 --> 00:11:50,800
So if the screen
is plane, then it's

206
00:11:50,800 --> 00:11:55,960
going to be multiplied by
some parabolic phase term.

207
00:11:55,960 --> 00:11:59,200
And now we've just looked
at this other case where

208
00:11:59,200 --> 00:12:03,280
we don't have to have a
very large lab in order

209
00:12:03,280 --> 00:12:05,560
to do this experiment.

210
00:12:05,560 --> 00:12:08,020
We can actually make the
experiment quite small

211
00:12:08,020 --> 00:12:12,100
so that it fits on top of the
table just by using a lens.

212
00:12:12,100 --> 00:12:16,420
And really what you can see,
that that is actually doing,

213
00:12:16,420 --> 00:12:19,630
according to your
geometrical optics ideas now,

214
00:12:19,630 --> 00:12:22,390
you can see what this is doing.

215
00:12:22,390 --> 00:12:27,680
This is producing an
image of this at infinity.

216
00:12:27,680 --> 00:12:30,800
So you can see that these two
things are actually equivalent.

217
00:12:30,800 --> 00:12:33,550
Well, what it's
doing is bringing

218
00:12:33,550 --> 00:12:38,680
what would have been at
infinity to some finite plane.

219
00:12:38,680 --> 00:12:42,190
So that's all it's doing there.

220
00:12:42,190 --> 00:12:45,070
But again, we've got
this [? string ?]

221
00:12:45,070 --> 00:12:49,580
here, this proportional
sign, which

222
00:12:49,580 --> 00:12:53,340
means there's some other
constant factors out there,

223
00:12:53,340 --> 00:12:58,940
including in general this
parabolic phase factor, which

224
00:12:58,940 --> 00:13:06,150
for this particular case, when
you got f and f, it disappears.

225
00:13:06,150 --> 00:13:09,390
The parabolic phase
function disappears

226
00:13:09,390 --> 00:13:13,830
when you've got this very, very
special case with the two f's

227
00:13:13,830 --> 00:13:14,330
there.

228
00:13:19,390 --> 00:13:21,720
OK, now this one.

229
00:13:21,720 --> 00:13:23,640
What this is really
trying to show

230
00:13:23,640 --> 00:13:25,335
is that, really,
there's two ways

231
00:13:25,335 --> 00:13:26,460
of looking at this problem.

232
00:13:30,270 --> 00:13:32,460
These two diagrams
here are really

233
00:13:32,460 --> 00:13:37,950
trying to show the
same case, but trying

234
00:13:37,950 --> 00:13:42,360
to get you thinking about
a different way of viewing

235
00:13:42,360 --> 00:13:43,450
the problem.

236
00:13:43,450 --> 00:13:46,500
So in this case
here, you can think

237
00:13:46,500 --> 00:13:55,180
of any point on the
screen on the transparency

238
00:13:55,180 --> 00:13:58,440
is going to act
like a point source.

239
00:13:58,440 --> 00:14:02,430
So it's going to produce a
spherical expanding wave.

240
00:14:02,430 --> 00:14:06,270
And then this is a distance
f, so that spherical wave

241
00:14:06,270 --> 00:14:07,950
is going to be
converted into a plane

242
00:14:07,950 --> 00:14:12,240
wave The angle of propagation
of this plane wave

243
00:14:12,240 --> 00:14:15,960
depends on the distance at
this point from the origin,

244
00:14:15,960 --> 00:14:17,880
of course.

245
00:14:17,880 --> 00:14:22,440
And so you can think, then,
of what you see on the screen

246
00:14:22,440 --> 00:14:27,360
as being made up of
what's called an angular

247
00:14:27,360 --> 00:14:31,500
spectrum of plane waves, where
the strength of each of those

248
00:14:31,500 --> 00:14:35,940
and get a spectrum components
comes about from the value

249
00:14:35,940 --> 00:14:39,410
of at that point.

250
00:14:39,410 --> 00:14:42,680
So that's one way of
thinking about it.

251
00:14:42,680 --> 00:14:44,640
And this way [INAUDIBLE]
thinking about it

252
00:14:44,640 --> 00:14:45,600
the other way.

253
00:14:45,600 --> 00:14:48,550
This way, you're
thinking of the object.

254
00:14:48,550 --> 00:14:50,070
It's going to diffract.

255
00:14:50,070 --> 00:14:55,230
If you imagine your transparency
as being like a grating,

256
00:14:55,230 --> 00:14:57,985
it produces a diffraction.

257
00:14:57,985 --> 00:15:01,230
It's producing
light grating orders

258
00:15:01,230 --> 00:15:04,830
For example, if this
was a very simple case

259
00:15:04,830 --> 00:15:08,460
of a sinusoidal
grating, you'd expect it

260
00:15:08,460 --> 00:15:11,100
to produce some grating orders.

261
00:15:11,100 --> 00:15:15,840
And these grating orders
correspond to plane waves

262
00:15:15,840 --> 00:15:18,000
traveling in particular angles.

263
00:15:18,000 --> 00:15:21,240
So this shows the
angular spectrum

264
00:15:21,240 --> 00:15:29,040
of the diffraction of the light
given by the transparency.

265
00:15:29,040 --> 00:15:35,040
And then the lens collects
these different plane

266
00:15:35,040 --> 00:15:36,750
white components.

267
00:15:36,750 --> 00:15:39,300
All of them are
going to be converted

268
00:15:39,300 --> 00:15:42,750
into a spherical
converging wave that

269
00:15:42,750 --> 00:15:45,420
converges a distance f away.

270
00:15:45,420 --> 00:15:49,440
It doesn't say that, but
this must be [? there. ?]

271
00:15:49,440 --> 00:15:55,360
And the plane wave's
moving at different angles

272
00:15:55,360 --> 00:16:00,240
are going to converge to a
point which, of course, is

273
00:16:00,240 --> 00:16:05,970
proportional to the sine of
the angle of the propagation.

274
00:16:05,970 --> 00:16:09,390
So here, at any
point on the screen,

275
00:16:09,390 --> 00:16:13,410
you're going to sum up
again these angular spectra

276
00:16:13,410 --> 00:16:16,740
components, each of which is
going to give a spherical wave.

277
00:16:16,740 --> 00:16:20,670
So you can see, looking
at that diagram again,

278
00:16:20,670 --> 00:16:23,280
they're like inverses
of each other.

279
00:16:23,280 --> 00:16:26,760
I think this is another
example where, very often, you

280
00:16:26,760 --> 00:16:30,865
can think of a
reciprocal type behavior.

281
00:16:33,890 --> 00:16:38,620
You could just flip
this over, and it would

282
00:16:38,620 --> 00:16:42,180
look the same the other way.

283
00:16:42,180 --> 00:16:44,160
OK, so this way
here, then, we're

284
00:16:44,160 --> 00:16:53,070
thinking of this as just
like the Huygens sense

285
00:16:53,070 --> 00:16:54,750
that we spoke about
when we were first

286
00:16:54,750 --> 00:17:05,150
doing diffraction [? there. ?]

287
00:17:05,150 --> 00:17:08,520
OK, so this is giving
some nice pictures.

288
00:17:08,520 --> 00:17:12,150
I'm going to steal this
presentation afterwards,

289
00:17:12,150 --> 00:17:15,150
because there's so many
good displays in this thing,

290
00:17:15,150 --> 00:17:18,329
that I'm going to use it in
my own lectures in the future.

291
00:17:18,329 --> 00:17:22,069
So this has given
me some good ideas.

292
00:17:22,069 --> 00:17:27,140
So we start off, then,
this is a rectangle.

293
00:17:27,140 --> 00:17:31,530
Oh, I'm going to say
something in a minute, though.

294
00:17:31,530 --> 00:17:32,910
So this is a distance f.

295
00:17:32,910 --> 00:17:35,080
This is a distance f.

296
00:17:35,080 --> 00:17:37,170
So this length, then,
as we just described,

297
00:17:37,170 --> 00:17:39,750
is going to produce a
Fourier transform of that.

298
00:17:39,750 --> 00:17:45,420
So the Fourier transform
of rect is a is a sinc.

299
00:17:45,420 --> 00:17:51,148
So you'd expect to get this a
sinc in x times a sinc in y.

300
00:17:51,148 --> 00:17:53,190
Now, the thing I was going
to say though, George,

301
00:17:53,190 --> 00:17:55,710
this looks to me that
this is longer than that.

302
00:17:55,710 --> 00:17:58,170
This is longer this
way than that way.

303
00:17:58,170 --> 00:18:01,576
So this should be longer that
way than this way, I think.

304
00:18:01,576 --> 00:18:03,993
GEORGE BARBASTATHIS: That's
the aspect ratio of the screen

305
00:18:03,993 --> 00:18:04,493
actually.

306
00:18:04,493 --> 00:18:06,682
COLIN SHEPPARD: Yeah, the
aspect ratio of the screen

307
00:18:06,682 --> 00:18:07,890
ought to be 90 degrees round.

308
00:18:07,890 --> 00:18:09,030
GEORGE BARBASTATHIS: There's
not a [? way of the ?]

309
00:18:09,030 --> 00:18:10,155
Fourier transform, I guess.

310
00:18:10,155 --> 00:18:10,800
[LAUGHS]

311
00:18:10,800 --> 00:18:15,630
COLIN SHEPPARD: Anyway, so
this conversation does bring it

312
00:18:15,630 --> 00:18:18,120
home, I think, and
a very important

313
00:18:18,120 --> 00:18:21,990
point that we've mentioned
before in the course.

314
00:18:21,990 --> 00:18:26,340
That if you make a
function smaller,

315
00:18:26,340 --> 00:18:29,490
it's Fourier transform is wider.

316
00:18:29,490 --> 00:18:32,080
If you make the function
wider, this Fourier transfer

317
00:18:32,080 --> 00:18:34,830
is smaller.

318
00:18:34,830 --> 00:18:38,070
And you'll notice, then, these
products of these two sincs

319
00:18:38,070 --> 00:18:42,090
gives a bright
spot in the center,

320
00:18:42,090 --> 00:18:46,170
a series of bright
spots along the axes.

321
00:18:46,170 --> 00:18:49,740
There are also some bright
spots in this region here.

322
00:18:49,740 --> 00:18:52,710
But they're so weak that
you can't actually see them.

323
00:18:52,710 --> 00:18:56,300
Really, you can only see
the ones along the axes.

324
00:18:56,300 --> 00:18:59,310
All right, so that's
the first example.

325
00:18:59,310 --> 00:19:04,770
Then this one's showing how,
if we make the transparency

326
00:19:04,770 --> 00:19:10,840
smaller, the rectangle smaller,
so this pattern gets bigger,

327
00:19:10,840 --> 00:19:13,210
now actually we can
just about manage

328
00:19:13,210 --> 00:19:17,230
to maybe see some ones
other than along the axis.

329
00:19:17,230 --> 00:19:21,400
And then this one is
an interesting one.

330
00:19:21,400 --> 00:19:27,620
So now we've got not just one
aperture, but three apertures.

331
00:19:27,620 --> 00:19:33,580
And so how can we work out
what the Fourier transform

332
00:19:33,580 --> 00:19:36,140
of three apertures would be?

333
00:19:36,140 --> 00:19:37,800
Anyone got some ideas?

334
00:19:37,800 --> 00:19:41,890
I'm sure [INAUDIBLE]
will have an idea.

335
00:19:41,890 --> 00:19:42,940
Anyone?

336
00:19:42,940 --> 00:19:45,800
How would anyone think of
doing this sort of thing?

337
00:19:53,530 --> 00:19:56,663
Have you come across a thing
called the convolution theorem?

338
00:19:56,663 --> 00:19:58,330
Probably you should
have done somewhere,

339
00:19:58,330 --> 00:19:59,280
I would have thought.

340
00:19:59,280 --> 00:20:01,030
So there's the thing
in Fourier transforms

341
00:20:01,030 --> 00:20:03,250
called convolution theorem.

342
00:20:03,250 --> 00:20:09,580
So this is actually, like,
three spikes convolved

343
00:20:09,580 --> 00:20:12,010
with a square, which
really all it means

344
00:20:12,010 --> 00:20:16,270
is we place a square on
each of the three spikes.

345
00:20:16,270 --> 00:20:22,210
And the convolution theorem
says that the Fourier transform

346
00:20:22,210 --> 00:20:26,570
of a convolution is the product
of the Fourier transforms.

347
00:20:26,570 --> 00:20:30,890
So what you'd expect is that the
Fourier transform that you're

348
00:20:30,890 --> 00:20:33,560
going to see here is
the Fourier transform

349
00:20:33,560 --> 00:20:40,170
of the rectangle, which is this,
multiplied by something else,

350
00:20:40,170 --> 00:20:43,890
which is the Fourier
transform of the three spikes.

351
00:20:43,890 --> 00:20:49,070
And the three spikes is
like a delta function

352
00:20:49,070 --> 00:20:51,810
and a plus or minus
delta function.

353
00:20:51,810 --> 00:20:55,170
The plus or minus delta
function together,

354
00:20:55,170 --> 00:20:58,140
e to the i sampling plus
e to the minus i sampling

355
00:20:58,140 --> 00:20:59,730
gives cause.

356
00:20:59,730 --> 00:21:03,300
And the one in the center
just gives a constant.

357
00:21:03,300 --> 00:21:07,500
So the Fourier transform of
these three delta functions

358
00:21:07,500 --> 00:21:12,830
is 1 plus a cosine, or
something like that.

359
00:21:12,830 --> 00:21:15,660
And so that's what we
expect to see, then.

360
00:21:15,660 --> 00:21:19,800
This thing, this pattern
that we saw before,

361
00:21:19,800 --> 00:21:24,690
is going to be multiplied
by a cosine function which

362
00:21:24,690 --> 00:21:28,000
comes about because of
the separation of this.

363
00:21:28,000 --> 00:21:31,650
And of course, because
this distance is actually

364
00:21:31,650 --> 00:21:35,290
much bigger than the
width of the rectangle,

365
00:21:35,290 --> 00:21:41,220
the spacing of the cosine is
much finer than the Fourier

366
00:21:41,220 --> 00:21:43,120
transform of the rectangle.

367
00:21:43,120 --> 00:21:44,610
And so that's why
these fringes--

368
00:21:44,610 --> 00:21:49,440
you see these fringes,
very fine fringes,

369
00:21:49,440 --> 00:21:55,260
they become a result of the
fact that these components here

370
00:21:55,260 --> 00:21:57,690
are quite well-separated.

371
00:21:57,690 --> 00:22:00,810
So that's another
very nice example.

372
00:22:00,810 --> 00:22:08,790
And so all these theorems that
you learn in Fourier transforms

373
00:22:08,790 --> 00:22:12,030
can all be directly
taken over into optics.

374
00:22:12,030 --> 00:22:15,240
And sometimes, they
can save a lot of work,

375
00:22:15,240 --> 00:22:18,450
because you can just use the
standard forms for things

376
00:22:18,450 --> 00:22:21,680
that you can look up
in a book of tables.

377
00:22:21,680 --> 00:22:26,710
And it saves you a lot of
algebra, or even worse,

378
00:22:26,710 --> 00:22:28,870
arithmetic.

379
00:22:28,870 --> 00:22:30,920
Right, so this is
another example.

380
00:22:30,920 --> 00:22:34,000
So now we're back to the
bigger rectangle, or something

381
00:22:34,000 --> 00:22:35,815
close to the bigger rectangle.

382
00:22:35,815 --> 00:22:41,180
The Fourier transform of the
rectangle, we saw before.

383
00:22:41,180 --> 00:22:45,490
But now this one, you see this
is multiplied by some fringes.

384
00:22:45,490 --> 00:22:48,670
So this is like the opposite
of this one, really.

385
00:22:48,670 --> 00:22:52,150
So this is a convolution.

386
00:22:52,150 --> 00:22:53,980
This is a product.

387
00:22:53,980 --> 00:22:57,470
So we use the same
convolution theorem again.

388
00:22:57,470 --> 00:23:00,910
The fringes, of
course, correspond

389
00:23:00,910 --> 00:23:02,870
just like we said before.

390
00:23:02,870 --> 00:23:05,110
The Fourier transform of
the three delta functions

391
00:23:05,110 --> 00:23:06,670
was the fringes.

392
00:23:06,670 --> 00:23:09,100
So the Fourier transform
with the fringes

393
00:23:09,100 --> 00:23:12,050
is like three delta functions.

394
00:23:12,050 --> 00:23:14,350
So what we'd expect,
this is a product

395
00:23:14,350 --> 00:23:16,790
of the fringes
and the rectangle.

396
00:23:16,790 --> 00:23:21,070
So we expect here to see the
convolution of the Fourier

397
00:23:21,070 --> 00:23:23,320
transform of the
fringes and the Fourier

398
00:23:23,320 --> 00:23:25,390
transform of the rectangle.

399
00:23:25,390 --> 00:23:26,630
And that's what we see.

400
00:23:26,630 --> 00:23:33,440
So we see three of these
diffraction patterns situated

401
00:23:33,440 --> 00:23:36,650
on these three spots
corresponding to the Fourier

402
00:23:36,650 --> 00:23:38,870
transform of that grating.

403
00:23:38,870 --> 00:23:42,170
So a very nice
display, that one.

404
00:23:42,170 --> 00:23:42,850
I like that one.

405
00:23:48,260 --> 00:23:53,450
OK, so this is now
introducing the idea

406
00:23:53,450 --> 00:23:56,910
of what's called the 4F system.

407
00:23:56,910 --> 00:24:01,950
So we're going to do this not
just once, but do it twice.

408
00:24:01,950 --> 00:24:04,310
So this just shows
what we've had so far.

409
00:24:04,310 --> 00:24:08,270
If we've got some
particular transparency,

410
00:24:08,270 --> 00:24:12,560
we get some particular
diffraction pattern

411
00:24:12,560 --> 00:24:15,300
in the back focal plane.

412
00:24:15,300 --> 00:24:19,660
And then in the 4F system,
we do the same again.

413
00:24:19,660 --> 00:24:23,110
We now have another
lens which is

414
00:24:23,110 --> 00:24:25,630
going to do exactly the same.

415
00:24:25,630 --> 00:24:28,330
It's going to look
at this pattern

416
00:24:28,330 --> 00:24:31,060
and produce it's
Fourier transform.

417
00:24:31,060 --> 00:24:36,550
So remember, if this is f
and this is f and this is f

418
00:24:36,550 --> 00:24:39,070
and this is f, we won't
have to worry anything

419
00:24:39,070 --> 00:24:41,890
about these parabolic phases.

420
00:24:41,890 --> 00:24:45,430
It's just straight Fourier
transforms with nothing else.

421
00:24:45,430 --> 00:24:47,860
And so you can see what you get.

422
00:24:47,860 --> 00:24:49,060
It's very simple.

423
00:24:49,060 --> 00:24:51,440
You start off with this object.

424
00:24:51,440 --> 00:24:56,410
So here is Fourier transform
And if you take the Fourier

425
00:24:56,410 --> 00:24:59,100
transform of the
Fourier transform,

426
00:24:59,100 --> 00:25:01,560
what do you get if you do a
Fourier transform of a Fourier

427
00:25:01,560 --> 00:25:02,060
transform?

428
00:25:02,060 --> 00:25:02,970
AUDIENCE: [INAUDIBLE]

429
00:25:02,970 --> 00:25:04,818
COLIN SHEPPARD: Yeah,
it's not quite, is it?

430
00:25:04,818 --> 00:25:05,360
You remember?

431
00:25:05,360 --> 00:25:06,530
You invert it.

432
00:25:06,530 --> 00:25:10,160
That's right, Yeah,
so someone got that.

433
00:25:10,160 --> 00:25:14,160
If you remember, if you do a
Fourier transform of a Fourier

434
00:25:14,160 --> 00:25:17,960
transform, you get the
original function inverted.

435
00:25:17,960 --> 00:25:20,270
You have to take the Fourier
transform four times,

436
00:25:20,270 --> 00:25:22,880
and then you get back
to where you start.

437
00:25:22,880 --> 00:25:24,980
And so that's what
we can see anyway.

438
00:25:24,980 --> 00:25:27,380
So this is the inverted.

439
00:25:27,380 --> 00:25:29,650
Well, of course, inverted,
that's just what we expect,

440
00:25:29,650 --> 00:25:30,150
isn't it?

441
00:25:30,150 --> 00:25:33,620
Because we know that when
we have an imaging system,

442
00:25:33,620 --> 00:25:35,730
we get an inverted image.

443
00:25:35,730 --> 00:25:39,060
So everything fits together.

444
00:25:39,060 --> 00:25:41,840
It gives exactly
the same results

445
00:25:41,840 --> 00:25:45,320
that we expect from
our geometrical optics.

446
00:25:45,320 --> 00:25:48,050
And so in this case,
of course, this

447
00:25:48,050 --> 00:25:49,920
is a rather symmetrical object.

448
00:25:49,920 --> 00:25:53,120
So the fact that this
has been inverted,

449
00:25:53,120 --> 00:25:54,950
I guess you can't
really even see.

450
00:26:03,560 --> 00:26:06,870
OK, so this is
saying, then, that you

451
00:26:06,870 --> 00:26:11,400
can make a 4F system
which produces,

452
00:26:11,400 --> 00:26:15,930
then, an inverted version
of the original object.

453
00:26:15,930 --> 00:26:18,180
You might wonder what
the point of that is.

454
00:26:18,180 --> 00:26:21,660
I mean, I guess
in a way, there's

455
00:26:21,660 --> 00:26:24,600
no point in making an
image which is much

456
00:26:24,600 --> 00:26:27,150
the same as the object, really.

457
00:26:27,150 --> 00:26:28,960
But one reason why
you might want--

458
00:26:28,960 --> 00:26:30,310
well, there's two reasons.

459
00:26:30,310 --> 00:26:35,220
One is that it's actually
produced this image

460
00:26:35,220 --> 00:26:36,780
in a different plane.

461
00:26:36,780 --> 00:26:41,790
So it's actually transported
this distribution of light

462
00:26:41,790 --> 00:26:44,580
here from this
plane to this plane.

463
00:26:44,580 --> 00:26:47,070
So that's one type
of application

464
00:26:47,070 --> 00:26:50,490
where you might want to
do something like this.

465
00:26:50,490 --> 00:26:54,330
And the second thing, of course,
is that although this is always

466
00:26:54,330 --> 00:26:58,710
called a 4F system,
one we obviously

467
00:26:58,710 --> 00:27:02,790
have to have this and this
[INAUDIBLE] both equal for this

468
00:27:02,790 --> 00:27:04,090
to work.

469
00:27:04,090 --> 00:27:08,190
We have to add this and this
f to be equal for it to work.

470
00:27:08,190 --> 00:27:12,110
But this f doesn't have
to be the same as that f.

471
00:27:12,110 --> 00:27:14,530
So you can actually
have a system

472
00:27:14,530 --> 00:27:17,980
like this, where you actually
produce a magnified image

473
00:27:17,980 --> 00:27:19,965
or a demagnified image.

474
00:27:19,965 --> 00:27:22,355
And now, that, of course, is
getting a lot more useful.

475
00:27:25,440 --> 00:27:29,340
So that's how you
can produce an image

476
00:27:29,340 --> 00:27:33,310
of an object in another plane.

477
00:27:33,310 --> 00:27:38,620
Now, the next thing
to note is, of course,

478
00:27:38,620 --> 00:27:42,200
so far we've assumed
that all this light gets

479
00:27:42,200 --> 00:27:46,710
through this optical
system and gets here.

480
00:27:46,710 --> 00:27:51,530
So if that was the case,
you get a perfect image.

481
00:27:51,530 --> 00:27:54,260
If all the information
from the objects

482
00:27:54,260 --> 00:27:58,140
was getting to the image, then
you wouldn't lose anything.

483
00:27:58,140 --> 00:28:02,360
But in practice, of course,
that's never going to be true.

484
00:28:02,360 --> 00:28:05,480
And, well, one way it's not
going to be true, of course,

485
00:28:05,480 --> 00:28:07,520
is because in practice,
these lenses are never

486
00:28:07,520 --> 00:28:11,360
going to be infinitely large.

487
00:28:11,360 --> 00:28:15,050
But also, we could,
if we wanted to,

488
00:28:15,050 --> 00:28:19,070
actually put some sort
of mask in this plane

489
00:28:19,070 --> 00:28:22,940
here to actually stop
some of that light getting

490
00:28:22,940 --> 00:28:25,940
through to the image plane.

491
00:28:25,940 --> 00:28:30,800
And so now we're going to look
at what happens if we do that.

492
00:28:30,800 --> 00:28:34,330
This is much the
same overall effect

493
00:28:34,330 --> 00:28:37,600
as George spoke about in
previous lectures, actually.

494
00:28:37,600 --> 00:28:40,960
Not imaging case, but he
spoke about-- you remember

495
00:28:40,960 --> 00:28:44,080
the results with the
building with the windows

496
00:28:44,080 --> 00:28:52,360
that you could actually, just
by doing digital processing

497
00:28:52,360 --> 00:28:54,400
on the Fourier
transforms, you can

498
00:28:54,400 --> 00:28:57,010
manipulate the
strength of the grating

499
00:28:57,010 --> 00:28:59,990
components in the final image.

500
00:28:59,990 --> 00:29:02,260
So the same is true
here, except now we're

501
00:29:02,260 --> 00:29:06,910
doing it optically
rather than digitally.

502
00:29:06,910 --> 00:29:10,940
So what you'd expect, then, is
if we put something in here--

503
00:29:10,940 --> 00:29:14,080
so here, this is
showing a passband.

504
00:29:14,080 --> 00:29:16,455
So let's say you
put in here some--

505
00:29:16,455 --> 00:29:20,110
a [? paint ?] screen
with a small hole in.

506
00:29:20,110 --> 00:29:25,240
The hole is this passband, and
the screen is the block band.

507
00:29:25,240 --> 00:29:28,690
So it only let's through
the region in the center.

508
00:29:28,690 --> 00:29:32,620
According to how big
that passband is,

509
00:29:32,620 --> 00:29:36,940
you'll get different
answers in the image plane.

510
00:29:40,460 --> 00:29:43,280
Yeah, of course, it
says the transparency

511
00:29:43,280 --> 00:29:45,440
may be a grayscale.

512
00:29:45,440 --> 00:29:49,610
It doesn't have to be a binary
type thing, black and white.

513
00:29:49,610 --> 00:29:51,620
Or it could even
be a phase mask.

514
00:29:51,620 --> 00:29:55,580
It could be that this
is changing the phase

515
00:29:55,580 --> 00:30:00,350
by, let's say, pi by 2 relative
to this or pi relative to this,

516
00:30:00,350 --> 00:30:01,160
or whatever.

517
00:30:01,160 --> 00:30:03,890
And you get different
answers according

518
00:30:03,890 --> 00:30:09,050
to what you choose, obviously,
for the geometry of this

519
00:30:09,050 --> 00:30:10,235
what we call spatial filter.

520
00:30:17,910 --> 00:30:22,250
OK, so yeah, this is
going back to this idea

521
00:30:22,250 --> 00:30:26,190
of these two ways of thinking
about the same problem.

522
00:30:26,190 --> 00:30:30,980
So this is the first way,
where we think of our objects

523
00:30:30,980 --> 00:30:34,640
as being made up of lots
of point scatterers Each

524
00:30:34,640 --> 00:30:40,730
of these scatterers gives
rise to, like, Huygens'

525
00:30:40,730 --> 00:30:43,730
spherical wavelets.

526
00:30:43,730 --> 00:30:46,610
Each of these is
collected by lens.

527
00:30:46,610 --> 00:30:50,030
The lens convert
that to a plane wave.

528
00:30:50,030 --> 00:30:53,960
The plane wave reaches
the second lens.

529
00:30:53,960 --> 00:30:56,030
And because it's
a plane wave, it's

530
00:30:56,030 --> 00:30:59,120
converted back to a
spherical wave that converges

531
00:30:59,120 --> 00:31:02,220
at distance f away from that.

532
00:31:02,220 --> 00:31:05,000
And now we've actually
introduced the fact

533
00:31:05,000 --> 00:31:07,220
that these f's
can be different--

534
00:31:07,220 --> 00:31:08,445
so f1, f2.

535
00:31:12,200 --> 00:31:13,910
This has to be equal
to this, and this

536
00:31:13,910 --> 00:31:15,980
has to be equal to this.

537
00:31:15,980 --> 00:31:20,120
Otherwise, the phases
won't cancel out properly.

538
00:31:20,120 --> 00:31:24,230
And so you can see
that what you'd expect,

539
00:31:24,230 --> 00:31:27,740
then, is that because
in this diagram here,

540
00:31:27,740 --> 00:31:33,020
F1 is bigger than F2, this
image is actually smaller

541
00:31:33,020 --> 00:31:35,540
than the original object.

542
00:31:35,540 --> 00:31:38,180
And note now, you
can also see directly

543
00:31:38,180 --> 00:31:42,180
the geometry showing
the inversion,

544
00:31:42,180 --> 00:31:47,180
the point above the axis has
come to an image below the axis

545
00:31:47,180 --> 00:31:48,220
here.

546
00:31:48,220 --> 00:31:51,560
So that's the one way
of thinking about it.

547
00:31:51,560 --> 00:31:54,290
And then the other way
is to think in terms

548
00:31:54,290 --> 00:31:56,300
of spatial frequencies.

549
00:31:56,300 --> 00:32:00,230
We think of this object as
being like a grating, which

550
00:32:00,230 --> 00:32:02,270
is going to produce
grating orders which

551
00:32:02,270 --> 00:32:04,550
travel in different directions.

552
00:32:04,550 --> 00:32:09,660
Each of these grating
orders is like a plane wave.

553
00:32:09,660 --> 00:32:11,570
And when it reaches
the lens, it's

554
00:32:11,570 --> 00:32:16,820
going to produce a converging
spherical wave, which

555
00:32:16,820 --> 00:32:22,530
comes to a focus at
distance f behind the lens.

556
00:32:22,530 --> 00:32:24,550
And then after that, of
course, it doesn't stop.

557
00:32:24,550 --> 00:32:25,830
But it carries on.

558
00:32:25,830 --> 00:32:29,970
It diverges again,
reaches the second lens.

559
00:32:29,970 --> 00:32:32,100
And then the second
lens, because this

560
00:32:32,100 --> 00:32:35,730
is a spherical wave coming
from a distance f away,

561
00:32:35,730 --> 00:32:38,450
is converted into
a plane wave again.

562
00:32:38,450 --> 00:32:44,790
So again, you've got this nice
symmetry between what goes in

563
00:32:44,790 --> 00:32:46,950
and what comes out.

564
00:32:46,950 --> 00:32:50,930
So there are these
really different ways

565
00:32:50,930 --> 00:32:53,510
of thinking about how
this is all going on.

566
00:32:57,620 --> 00:32:58,420
End of show.

567
00:32:58,420 --> 00:33:01,143
AUDIENCE: [INAUDIBLE]

568
00:33:01,143 --> 00:33:02,268
COLIN SHEPPARD: Right then.

569
00:33:02,268 --> 00:33:03,492
AUDIENCE: [INAUDIBLE]

570
00:33:03,492 --> 00:33:05,030
COLIN SHEPPARD: Right, OK.

571
00:33:05,030 --> 00:33:08,700
So all that was just a warm-up.

572
00:33:08,700 --> 00:33:11,190
That was last week's lecture--
last Monday's lecture

573
00:33:11,190 --> 00:33:12,000
we've done so far.

574
00:33:39,770 --> 00:33:41,030
Right.

575
00:33:41,030 --> 00:33:42,290
Oh, yes.

576
00:33:42,290 --> 00:33:44,540
Lots of interesting
things coming up later.

577
00:33:49,590 --> 00:33:52,780
OK, so we're going to
carry on straight away

578
00:33:52,780 --> 00:33:56,350
with more of this 4F system
and spatial filtering.

579
00:33:56,350 --> 00:33:59,750
So we're just getting
started on this.

580
00:33:59,750 --> 00:34:01,960
And so there's lots of
very interesting things

581
00:34:01,960 --> 00:34:03,760
you can do with these.

582
00:34:03,760 --> 00:34:06,490
So this is back to the same
thing again [INAUDIBLE]

583
00:34:06,490 --> 00:34:09,730
showing again, but a
bit more detail now,

584
00:34:09,730 --> 00:34:13,030
because we're actually
going to really look

585
00:34:13,030 --> 00:34:16,600
at more detail of where
these orders are formed

586
00:34:16,600 --> 00:34:18,739
and what you can do with them.

587
00:34:18,739 --> 00:34:24,090
So we have a grating object.

588
00:34:24,090 --> 00:34:27,540
Let's imagine this is just
a 1 plus a cosine, which

589
00:34:27,540 --> 00:34:30,270
means then it
produces a zero order,

590
00:34:30,270 --> 00:34:33,120
and then a plus
and minus 1 order.

591
00:34:33,120 --> 00:34:35,080
So three plane waves.

592
00:34:35,080 --> 00:34:37,320
And as we said, each
of those plane waves

593
00:34:37,320 --> 00:34:43,050
is going to be converted to
a converging spherical wave.

594
00:34:43,050 --> 00:34:46,199
And so in this back
focal plane, you'll

595
00:34:46,199 --> 00:34:51,310
see three spots corresponding
to the three orders.

596
00:34:51,310 --> 00:34:55,760
And then carrying
on, these plane waves

597
00:34:55,760 --> 00:34:58,610
then converted back
into a spherical wave.

598
00:34:58,610 --> 00:35:02,120
And you get an image which
is back to the grating again.

599
00:35:02,120 --> 00:35:06,410
So at the moment,
the optical system

600
00:35:06,410 --> 00:35:08,990
has transmitted all
the information.

601
00:35:08,990 --> 00:35:12,740
And so what we really
expect is that the image

602
00:35:12,740 --> 00:35:16,190
is exactly the same as the
object, apart from the fact

603
00:35:16,190 --> 00:35:21,020
it's inverted and it might be
magnified, or the opposite.

604
00:35:21,020 --> 00:35:24,870
But the resolution you'd
expect would be the same.

605
00:35:24,870 --> 00:35:29,520
All the detail in here,
you'd expect to see in there.

606
00:35:29,520 --> 00:35:32,870
So the next thing
to look at is what

607
00:35:32,870 --> 00:35:35,600
happens if we now
place something

608
00:35:35,600 --> 00:35:39,260
in this plane that changes
the relative strength

609
00:35:39,260 --> 00:35:41,060
of these different orders.

610
00:35:41,060 --> 00:35:43,640
And there are lots of
things that we could try.

611
00:35:43,640 --> 00:35:47,030
The first one, the
simplest one to look at,

612
00:35:47,030 --> 00:35:51,140
is what happens if we just
put in something, a mask

613
00:35:51,140 --> 00:35:58,160
here that lets through the
zero order and not the others.

614
00:35:58,160 --> 00:36:03,560
The plus 1 and the minus 1 order
are obstructed by this screen.

615
00:36:03,560 --> 00:36:07,490
So all we let through
is this single order,

616
00:36:07,490 --> 00:36:11,520
which is going to be
converted to a plane wave.

617
00:36:11,520 --> 00:36:15,990
And so on your screen here, all
you'll see is the plane wave.

618
00:36:15,990 --> 00:36:18,510
So the image that you see--

619
00:36:18,510 --> 00:36:19,680
well, I call it an image.

620
00:36:19,680 --> 00:36:21,810
But it's not really
much of an image.

621
00:36:21,810 --> 00:36:24,420
It's completely featureless.

622
00:36:24,420 --> 00:36:29,340
It's just like a gray spot--

623
00:36:29,340 --> 00:36:30,900
a gray region.

624
00:36:30,900 --> 00:36:32,500
There's no detail there.

625
00:36:32,500 --> 00:36:37,230
So we've lost that structure.

626
00:36:37,230 --> 00:36:41,280
Actually, because we
put this mask in here,

627
00:36:41,280 --> 00:36:46,740
the structure is beyond
the resolution limit

628
00:36:46,740 --> 00:36:48,640
of the optical system.

629
00:36:48,640 --> 00:36:49,770
And so we can't see it.

630
00:36:54,230 --> 00:36:58,840
OK, so this is what we might
call a low-pass filter.

631
00:36:58,840 --> 00:37:02,610
So if you think of these
as being like these grating

632
00:37:02,610 --> 00:37:06,420
orders basically represent
different spatial frequencies.

633
00:37:06,420 --> 00:37:11,370
So just in analogy with what
you have in electricity,

634
00:37:11,370 --> 00:37:14,220
where you've got
frequencies and time,

635
00:37:14,220 --> 00:37:20,290
in space, we've got distance and
we've got spatial frequencies.

636
00:37:20,290 --> 00:37:25,120
So this is why we could talk
about low-pass filtering.

637
00:37:25,120 --> 00:37:27,530
We're filtering the
spatial frequencies.

638
00:37:27,530 --> 00:37:31,440
And just like in
time, you remember

639
00:37:31,440 --> 00:37:34,800
when you've got
your hi-fi system,

640
00:37:34,800 --> 00:37:38,070
if you turn down
the treble, that's

641
00:37:38,070 --> 00:37:41,160
acting like a low-pass filter.

642
00:37:41,160 --> 00:37:44,880
It's cutting out the
high-frequency sound

643
00:37:44,880 --> 00:37:47,850
and letting through the
low-frequency sound only,

644
00:37:47,850 --> 00:37:48,690
isn't it?

645
00:37:48,690 --> 00:37:51,340
So that's what we call
a low-pass filter.

646
00:37:51,340 --> 00:37:53,680
And this is exactly the same.

647
00:37:53,680 --> 00:37:55,140
We've put in the mask.

648
00:37:55,140 --> 00:37:59,280
It's a bit different in a
way from the electronic case,

649
00:37:59,280 --> 00:38:01,560
because in optics, we've
got a lot more freedom

650
00:38:01,560 --> 00:38:03,240
as to what we can do.

651
00:38:03,240 --> 00:38:05,550
We can actually put in
something here which

652
00:38:05,550 --> 00:38:08,220
has got real sharp edges to it.

653
00:38:08,220 --> 00:38:12,170
In electronics, if you're
trying to make a filter,

654
00:38:12,170 --> 00:38:15,240
there's always a
limit to how much

655
00:38:15,240 --> 00:38:18,730
you can make the roll
off of the filter.

656
00:38:18,730 --> 00:38:20,940
So you can't get something
that suddenly cuts off

657
00:38:20,940 --> 00:38:21,980
like that very easily.

658
00:38:26,710 --> 00:38:31,030
OK, so this is going
through the maths of this.

659
00:38:31,030 --> 00:38:36,610
Our original transparency,
then, is a 1 plus cosine.

660
00:38:36,610 --> 00:38:40,600
And the 1 plus cosine,
we Fourier transform

661
00:38:40,600 --> 00:38:44,750
that to find what we
call the object spectrum.

662
00:38:44,750 --> 00:38:50,080
And so in this case, finding
the Fourier transform of cosine

663
00:38:50,080 --> 00:38:52,690
is very straightforward,
because we just

664
00:38:52,690 --> 00:38:56,860
expand cosine as e to
the i something plus e

665
00:38:56,860 --> 00:38:59,730
on the minus i something over 2.

666
00:38:59,730 --> 00:39:04,360
And each of those e
to the i type terms

667
00:39:04,360 --> 00:39:09,670
will correspond to
a single frequency.

668
00:39:09,670 --> 00:39:13,660
So we end up now with three
terms at different frequencies.

669
00:39:13,660 --> 00:39:18,670
The one Fourier transforms to
a delta function of u, which

670
00:39:18,670 --> 00:39:22,180
is a spike where u equals 0.

671
00:39:22,180 --> 00:39:25,030
And the two components
of the cosine

672
00:39:25,030 --> 00:39:32,350
correspond to another 2 delta
functions, each situated--

673
00:39:32,350 --> 00:39:36,210
this one's a u
equals some value u0.

674
00:39:36,210 --> 00:39:40,150
And this one is some
value u equals minus u0.

675
00:39:40,150 --> 00:39:44,320
So you've got these three spikes
that we've been talking about

676
00:39:44,320 --> 00:39:46,750
for the last few slides now.

677
00:39:46,750 --> 00:39:51,790
So this is the object spectrum.

678
00:39:51,790 --> 00:39:55,340
It's the angular spectrum
that we've got here.

679
00:39:55,340 --> 00:39:59,590
So actually, you can see that
u here basically represents

680
00:39:59,590 --> 00:40:04,263
the angle that these plane
waves are traveling in.

681
00:40:04,263 --> 00:40:05,680
And there's three
different angles

682
00:40:05,680 --> 00:40:09,630
that the light can travel in.

683
00:40:09,630 --> 00:40:16,660
And so yeah, that
will propagate, then.

684
00:40:16,660 --> 00:40:19,360
And as we've said
before, you can

685
00:40:19,360 --> 00:40:24,320
see that these rays
are moving at an angle.

686
00:40:24,320 --> 00:40:29,980
And so as you go
along a distance f,

687
00:40:29,980 --> 00:40:32,770
you get a shift
in the x position

688
00:40:32,770 --> 00:40:37,230
which is given by, of
course, the slope of this.

689
00:40:37,230 --> 00:40:41,170
The more the slope, the
more it deflects by the time

690
00:40:41,170 --> 00:40:43,150
you travel a distance f.

691
00:40:43,150 --> 00:40:47,150
So this then gives the
shifts of these two things.

692
00:40:47,150 --> 00:40:51,700
So that's before
the screen, then--

693
00:40:51,700 --> 00:40:53,350
sorry, before the mask.

694
00:41:01,820 --> 00:41:04,324
[SIDE CONVERSATION]

695
00:41:39,030 --> 00:41:41,110
Yeah, so this is showing, then--

696
00:41:41,110 --> 00:41:46,250
so these raise, then,
as they come down here.

697
00:41:46,250 --> 00:41:48,190
We put in this
pupil mask which is

698
00:41:48,190 --> 00:41:49,990
going to get rid of these two.

699
00:41:49,990 --> 00:41:51,520
And it just leaves
the other one.

700
00:41:51,520 --> 00:41:55,660
So we just left them
with our single beam,

701
00:41:55,660 --> 00:41:59,350
which as we said before gives
an image which is not really

702
00:41:59,350 --> 00:41:59,850
an image.

703
00:42:02,470 --> 00:42:05,500
Right, yeah, so
after the pupil mask,

704
00:42:05,500 --> 00:42:06,990
we multiplied by the mask.

705
00:42:06,990 --> 00:42:08,080
Gets rid of those two.

706
00:42:08,080 --> 00:42:10,090
Just leaves the other one.

707
00:42:10,090 --> 00:42:14,920
And finally, then, the
final image you'll see

708
00:42:14,920 --> 00:42:17,860
is going to be the Fourier
transformer of that.

709
00:42:17,860 --> 00:42:22,400
And the Fourier transform of the
delta function is a constant.

710
00:42:22,400 --> 00:42:26,530
And so you see something
which has got no detail in it.

711
00:42:31,800 --> 00:42:35,250
OK, so this gives
another example, then.

712
00:42:37,980 --> 00:42:41,970
Here, we're looking at some
binary amplitude grating.

713
00:42:41,970 --> 00:42:44,820
So this is a
particular example--

714
00:42:44,820 --> 00:42:46,060
another example.

715
00:42:46,060 --> 00:42:48,810
So last time, we looked
at the cosine type thing.

716
00:42:48,810 --> 00:42:51,010
So this one is now
a binary grating,

717
00:42:51,010 --> 00:42:53,250
which means that the
object is something

718
00:42:53,250 --> 00:42:56,080
which is either black or white.

719
00:42:56,080 --> 00:42:57,790
And we shine a light onto this.

720
00:42:57,790 --> 00:43:01,420
So it's like a
square wave grating.

721
00:43:01,420 --> 00:43:06,010
And so unlike the previous
case where cosine, of course,

722
00:43:06,010 --> 00:43:10,690
just produced those
three grating components,

723
00:43:10,690 --> 00:43:14,380
in this case, there's going
to be a whole load of grating

724
00:43:14,380 --> 00:43:15,040
components.

725
00:43:15,040 --> 00:43:17,380
They're going to go on forever.

726
00:43:17,380 --> 00:43:19,600
But they get weaker and weaker.

727
00:43:19,600 --> 00:43:23,050
And so you can see here,
it shows a few of them.

728
00:43:23,050 --> 00:43:25,180
The first, the
second, so there'll

729
00:43:25,180 --> 00:43:27,230
be the third, the
fourth, and so on.

730
00:43:27,230 --> 00:43:29,490
And then minus 1,
and so on and so on.

731
00:43:29,490 --> 00:43:33,340
And each of these is moving
at a different angle.

732
00:43:33,340 --> 00:43:37,960
So when the plane wave
gets to the pupil mask--

733
00:43:37,960 --> 00:43:44,040
sorry, when the plane wave
is focused to the pupil mask,

734
00:43:44,040 --> 00:43:45,880
it's going to reach there.

735
00:43:45,880 --> 00:43:49,360
At a point, the distance
from the axis, of course,

736
00:43:49,360 --> 00:43:55,000
is going to depend on the angle
of this plane wave component,

737
00:43:55,000 --> 00:43:56,890
the grating component.

738
00:43:56,890 --> 00:43:59,440
So if you look
here, you're going

739
00:43:59,440 --> 00:44:02,710
to see all these different
grating orders all

740
00:44:02,710 --> 00:44:05,420
arrayed in this position here.

741
00:44:05,420 --> 00:44:12,310
And so by putting some
sort of mask here,

742
00:44:12,310 --> 00:44:17,650
we can monitor and change the
relative strength or phrase

743
00:44:17,650 --> 00:44:21,100
or whatever we like of
those different components.

744
00:44:21,100 --> 00:44:23,410
And then they'll
be added together,

745
00:44:23,410 --> 00:44:28,040
produce some nice effect that
we might want to look at.

746
00:44:28,040 --> 00:44:31,880
So let's see what we're
going to get in this case.

747
00:44:31,880 --> 00:44:34,660
So in this particular
case in the picture,

748
00:44:34,660 --> 00:44:37,570
the size of this
aperture is enough to let

749
00:44:37,570 --> 00:44:41,950
through the central three
components, the zero order

750
00:44:41,950 --> 00:44:44,860
and the plus and minus 1 order.

751
00:44:44,860 --> 00:44:48,430
But it stopped all
the other ones.

752
00:44:48,430 --> 00:44:53,650
So what you will see here is
the Fourier transform of this.

753
00:44:53,650 --> 00:44:55,260
[INAUDIBLE] before.

754
00:44:55,260 --> 00:44:58,510
It's these three spikes again.

755
00:44:58,510 --> 00:45:03,580
The Fourier transform of
the three spikes is just a 1

756
00:45:03,580 --> 00:45:04,960
plus a cosine.

757
00:45:04,960 --> 00:45:07,270
The relative strength of
these might change according

758
00:45:07,270 --> 00:45:11,160
to the properties of these--

759
00:45:11,160 --> 00:45:13,860
of the transmission of the mask.

760
00:45:13,860 --> 00:45:20,740
But anyway, what you see finally
is an object which is like--

761
00:45:20,740 --> 00:45:28,210
you see the 1 plus cosine
type variation in amplitude.

762
00:45:28,210 --> 00:45:32,530
And so you notice that
this thing here started

763
00:45:32,530 --> 00:45:36,740
with these very square edges.

764
00:45:36,740 --> 00:45:39,860
We no longer see the
very square edges.

765
00:45:39,860 --> 00:45:46,490
So this is an example of
how we've lost resolution.

766
00:45:46,490 --> 00:45:50,660
The optical system has got
a certain finite resolution.

767
00:45:50,660 --> 00:45:53,810
There's detail in
this original object

768
00:45:53,810 --> 00:45:56,720
that we can't actually
see in the final image.

769
00:45:56,720 --> 00:45:58,655
And this is because
not all the light

770
00:45:58,655 --> 00:45:59,780
has got through the system.

771
00:46:07,150 --> 00:46:12,010
OK, so we're going to look
at a particular case now.

772
00:46:12,010 --> 00:46:14,830
This is the bit I was saying
I wasn't very keen on.

773
00:46:14,830 --> 00:46:16,090
No, no, it's all right.

774
00:46:16,090 --> 00:46:18,520
But anyway, there's
a load of maths

775
00:46:18,520 --> 00:46:22,360
here, which I might
actually gloss over a bit,

776
00:46:22,360 --> 00:46:24,950
and let you look at
it yourself later.

777
00:46:24,950 --> 00:46:26,690
But basically,
then, we're looking

778
00:46:26,690 --> 00:46:29,890
at a particular example.

779
00:46:29,890 --> 00:46:34,855
It's a binary grating with
perfect contrast, m equals 1--

780
00:46:34,855 --> 00:46:36,700
i.e., this goes
from black to white.

781
00:46:36,700 --> 00:46:37,990
That's what that means.

782
00:46:37,990 --> 00:46:42,040
It's got a period of 10 microns.

783
00:46:42,040 --> 00:46:44,650
The spacing, the
period, is the distance

784
00:46:44,650 --> 00:46:48,550
between the repeat
on this grating.

785
00:46:48,550 --> 00:46:52,810
And it's got a duty
cycle of a third, which

786
00:46:52,810 --> 00:46:59,220
means that the white bit is
a third of the black bit.

787
00:46:59,220 --> 00:47:00,370
Is that what that means?

788
00:47:00,370 --> 00:47:00,850
AUDIENCE: [INAUDIBLE]

789
00:47:00,850 --> 00:47:01,920
COLIN SHEPPARD: A
third of the whole.

790
00:47:01,920 --> 00:47:02,795
AUDIENCE: [INAUDIBLE]

791
00:47:02,795 --> 00:47:05,570
COLIN SHEPPARD: Yeah,
a third of the period.

792
00:47:05,570 --> 00:47:11,550
OK, and then we're also told
that the 4F system consists

793
00:47:11,550 --> 00:47:16,720
of two identical lenses of focal
length f equals 20 centimeters.

794
00:47:16,720 --> 00:47:20,040
So both f1 and f2
are 20 centimeters.

795
00:47:20,040 --> 00:47:23,760
And we put a pupil
mask of diameter

796
00:47:23,760 --> 00:47:29,150
3 centimeters here
at the Fourier plane.

797
00:47:29,150 --> 00:47:33,467
What is the intensity observed
at the output image plane?

798
00:47:33,467 --> 00:47:34,425
So that's the question.

799
00:47:42,780 --> 00:47:46,890
OK, the secrets to solve this
kind of problem, what you've

800
00:47:46,890 --> 00:47:49,500
got to do is to
work out what you're

801
00:47:49,500 --> 00:47:52,230
going to get in this
plane, which is basically

802
00:47:52,230 --> 00:47:54,780
the Fourier transform of this.

803
00:47:54,780 --> 00:48:00,330
And then you're going to
have to multiply by the mask

804
00:48:00,330 --> 00:48:02,490
and see which of those
orders get through.

805
00:48:02,490 --> 00:48:07,920
So the relative scaling of this
diffraction pattern in the mask

806
00:48:07,920 --> 00:48:10,380
is obviously all
important for that.

807
00:48:10,380 --> 00:48:12,870
If you don't get that
right, then you'll

808
00:48:12,870 --> 00:48:17,870
get the wrong answer,
obviously, because you

809
00:48:17,870 --> 00:48:19,870
might get orders getting
through that shouldn't,

810
00:48:19,870 --> 00:48:22,110
or the other way around.

811
00:48:22,110 --> 00:48:28,440
And so once we've done
that, we know now what

812
00:48:28,440 --> 00:48:34,720
the grating orders, the Fourier
components in the final image,

813
00:48:34,720 --> 00:48:35,840
are going to be.

814
00:48:35,840 --> 00:48:38,400
And we then do another
Fourier transfer.

815
00:48:38,400 --> 00:48:39,960
And we've got the
image amplitude.

816
00:48:47,420 --> 00:48:48,640
There we are.

817
00:48:48,640 --> 00:48:53,878
Again, we go [? do ?] scaling
to get it the right size.

818
00:48:53,878 --> 00:48:55,420
Normally, of course,
at the very end,

819
00:48:55,420 --> 00:48:57,920
we would be interested in
intensity, not amplitude,

820
00:48:57,920 --> 00:48:58,420
as well.

821
00:49:06,390 --> 00:49:09,720
OK, so this is our
example low-pass

822
00:49:09,720 --> 00:49:13,230
filtering, a binary
amplitude grating.

823
00:49:13,230 --> 00:49:17,040
So this is this grating
as we've chosen.

824
00:49:17,040 --> 00:49:22,980
And so you can see this, that
the dimension's in microns.

825
00:49:22,980 --> 00:49:27,820
So this is quite a fine grating.

826
00:49:27,820 --> 00:49:30,930
So you can see that it's
not symmetrical, by the way.

827
00:49:30,930 --> 00:49:35,890
The duty cycle is
not a half, right?

828
00:49:35,890 --> 00:49:39,730
So the width of
the bright regions

829
00:49:39,730 --> 00:49:42,940
is smaller than the width
of the dark regions.

830
00:49:42,940 --> 00:49:44,380
And that's important
when it comes

831
00:49:44,380 --> 00:49:46,990
to looking at the
Fourier transform,

832
00:49:46,990 --> 00:49:49,120
and therefore the
spectrum of that.

833
00:49:49,120 --> 00:49:53,430
And then the pupil
mask that we're

834
00:49:53,430 --> 00:49:56,020
going to look at looks
something like that.

835
00:49:56,020 --> 00:49:59,860
We let through the
frequencies near the center.

836
00:49:59,860 --> 00:50:02,770
And we keep out--

837
00:50:02,770 --> 00:50:06,090
reject the frequencies which
are further from the center.

838
00:50:12,370 --> 00:50:17,820
OK, so for the Fourier
transformer of that,

839
00:50:17,820 --> 00:50:19,590
this is a repetitive function.

840
00:50:19,590 --> 00:50:23,670
So the Fourier transform just
becomes a Fourier series.

841
00:50:23,670 --> 00:50:28,930
And this is an expression for
the Fourier series of this.

842
00:50:28,930 --> 00:50:34,490
It basically consists
of two parts.

843
00:50:34,490 --> 00:50:39,070
It's got a sort of
envelope, which is a sinc.

844
00:50:39,070 --> 00:50:44,720
Basically, this is going to
consist of an infinite number

845
00:50:44,720 --> 00:50:48,910
of discrete frequencies.

846
00:50:48,910 --> 00:50:52,420
It's going to consist of
a constant term, which

847
00:50:52,420 --> 00:50:53,110
is the zero.

848
00:50:53,110 --> 00:50:58,860
It is going to consist of
a first harmonic and then

849
00:50:58,860 --> 00:51:02,590
a second harmonic and a
third harmonic, and so on.

850
00:51:02,590 --> 00:51:04,840
If the duty cycle
has been chosen

851
00:51:04,840 --> 00:51:09,820
to be a half, i.e. it was a
square wave, then by symmetry,

852
00:51:09,820 --> 00:51:11,740
it turns out that
the second harmonic

853
00:51:11,740 --> 00:51:15,160
and the fourth harmonic
and all those all vanish.

854
00:51:15,160 --> 00:51:18,520
So you'd only end up with the
zero, the first, the third,

855
00:51:18,520 --> 00:51:21,140
the fifth, and so on.

856
00:51:21,140 --> 00:51:25,540
But in general, there will
be all the orders there.

857
00:51:25,540 --> 00:51:29,230
And the relative
strength of those orders

858
00:51:29,230 --> 00:51:35,470
here, you can see the
size depends on the value

859
00:51:35,470 --> 00:51:37,360
of this sinc function.

860
00:51:37,360 --> 00:51:41,630
Actually, that's a very easy way
of seeing about the even orders

861
00:51:41,630 --> 00:51:42,940
canceling.

862
00:51:42,940 --> 00:51:48,640
Because what happens is, if
the duty factor is a half,

863
00:51:48,640 --> 00:51:52,960
it turns out that
the even orders when

864
00:51:52,960 --> 00:51:57,550
q equals even number
coincide with the zeros

865
00:51:57,550 --> 00:51:58,750
of the sinc function.

866
00:51:58,750 --> 00:52:01,720
And that's why they vanish.

867
00:52:01,720 --> 00:52:09,240
So where this comes
from is that--

868
00:52:09,240 --> 00:52:13,260
so we're saying that
this function is a--

869
00:52:16,350 --> 00:52:20,060
maybe it's easier if I go on
to the next slide, I think,

870
00:52:20,060 --> 00:52:21,610
which is the Fourier transform.

871
00:52:21,610 --> 00:52:26,570
Yeah, so the Fourier
transform, the strength

872
00:52:26,570 --> 00:52:30,510
of these components is
what we've got to find out.

873
00:52:30,510 --> 00:52:34,700
And so this function
Fourier transforms

874
00:52:34,700 --> 00:52:38,720
to something which is a
series of delta functions

875
00:52:38,720 --> 00:52:42,860
corresponding to the different
orders, different values, of q.

876
00:52:42,860 --> 00:52:46,670
And the strength of the
different components

877
00:52:46,670 --> 00:52:48,380
is given by the
value of the sync.

878
00:52:50,910 --> 00:52:56,850
So you can see again, if there
are some values of alpha q

879
00:52:56,850 --> 00:53:02,862
which can be 0,
and therefore would

880
00:53:02,862 --> 00:53:04,820
mean that those components
[? will ?] be there.

881
00:53:11,850 --> 00:53:20,890
OK, so when it gets to the pupil
mask, all that's happening--

882
00:53:20,890 --> 00:53:24,490
maybe we ought to
go back to the--

883
00:53:24,490 --> 00:53:29,300
we'll go back to this diagram.

884
00:53:29,300 --> 00:53:34,250
So you can see that
what we need to know

885
00:53:34,250 --> 00:53:38,270
is, this is coming to a
focus at this point here.

886
00:53:38,270 --> 00:53:41,900
But we need to know what
that distance here is.

887
00:53:41,900 --> 00:53:47,690
But we need to know that
relative scaling of where

888
00:53:47,690 --> 00:53:49,100
that diffraction order appears.

889
00:54:06,725 --> 00:54:09,224
Oh, that was well
up here [INAUDIBLE]..

890
00:54:15,300 --> 00:54:19,503
OK, so this is this lambda
f that comes in here,

891
00:54:19,503 --> 00:54:21,170
because it's going
through a distance f.

892
00:54:21,170 --> 00:54:24,655
So that gives the scaling
of this distribution

893
00:54:24,655 --> 00:54:26,810
that you get there.

894
00:54:26,810 --> 00:54:34,190
And so that's where u now is
replaced by this x double dash

895
00:54:34,190 --> 00:54:35,600
over lambda f--

896
00:54:35,600 --> 00:54:40,570
the scaling of that size of
that diffraction pattern.

897
00:54:43,210 --> 00:54:46,800
And then this is put in
the answer, [INAUDIBLE],,

898
00:54:46,800 --> 00:54:47,955
is that you get this--

899
00:54:51,010 --> 00:54:56,660
oh, yeah, we're told the
alpha is the juicy fat cycle.

900
00:54:56,660 --> 00:54:58,730
So that's the third.

901
00:54:58,730 --> 00:55:01,270
So alpha is a third in here.

902
00:55:05,310 --> 00:55:09,795
And where do we get the 1
centimeter from, George?

903
00:55:09,795 --> 00:55:12,410
AUDIENCE: [INAUDIBLE]

904
00:55:21,300 --> 00:55:23,630
COLIN SHEPPARD: Yeah,
I think I saw something

905
00:55:23,630 --> 00:55:26,080
like this on the crib sheet.

906
00:55:26,080 --> 00:55:27,520
Which he will hide, by the way.

907
00:55:31,290 --> 00:55:37,680
So we've got-- all
right here, we've

908
00:55:37,680 --> 00:55:48,770
got lambda equals 0.5 microns,
f equals 20 centimeters,

909
00:55:48,770 --> 00:55:54,640
lambda period is
equal to 10 microns.

910
00:55:54,640 --> 00:56:00,550
So that gives that lambda
f over [INAUDIBLE] lambda

911
00:56:00,550 --> 00:56:07,645
is 0.5 times 20 over 10.

912
00:56:12,630 --> 00:56:15,422
No, that's not--
this is microns.

913
00:56:15,422 --> 00:56:16,380
AUDIENCE: That's right.

914
00:56:16,380 --> 00:56:19,640
Lambda-- both lambdas
are in microns.

915
00:56:19,640 --> 00:56:21,290
COLIN SHEPPARD: Oh yeah.

916
00:56:21,290 --> 00:56:22,670
These are in microns.

917
00:56:22,670 --> 00:56:23,600
These are in microns.

918
00:56:23,600 --> 00:56:25,230
These are in centimeters.

919
00:56:25,230 --> 00:56:27,140
So this is in centimeters.

920
00:56:27,140 --> 00:56:30,700
And this is equal
to 1 centimeter.

921
00:56:30,700 --> 00:56:35,620
Right, OK, so that's
what's going on there.

922
00:56:35,620 --> 00:56:37,367
So that's the 1 centimeter.

923
00:56:41,940 --> 00:56:46,970
OK, so that's just plotting,
then, these different orders.

924
00:56:46,970 --> 00:56:53,560
And so you can see here the
zero order plus 1 plus 2 plus 3,

925
00:56:53,560 --> 00:56:54,980
so on.

926
00:56:54,980 --> 00:57:00,950
But the screen blocks
out all the ones,

927
00:57:00,950 --> 00:57:02,420
except for these central ones.

928
00:57:05,510 --> 00:57:10,370
And so this thing, these
relative strengths,

929
00:57:10,370 --> 00:57:12,290
must come from--
well, there's a factor

930
00:57:12,290 --> 00:57:15,200
of third at the
front is this one.

931
00:57:15,200 --> 00:57:18,280
And then this is
multiplied by the value

932
00:57:18,280 --> 00:57:24,210
of sinc of a third, which must
give the rest of this thing

933
00:57:24,210 --> 00:57:26,094
here.

934
00:57:26,094 --> 00:57:27,770
GEORGE BARBASTATHIS:
And so I have

935
00:57:27,770 --> 00:57:32,030
posted in the website a
supplement to this lecture.

936
00:57:32,030 --> 00:57:33,890
So in the supplement,
I have worked

937
00:57:33,890 --> 00:57:36,110
out all these
calculations-- how you

938
00:57:36,110 --> 00:57:40,470
compute the sinc of 1 over 1/3,
sinc of 2 over 1/3, and so on.

939
00:57:40,470 --> 00:57:43,420
So you can see some more
details in the website.

940
00:57:43,420 --> 00:57:44,660
COLIN SHEPPARD: Yeah, OK.

941
00:57:44,660 --> 00:57:46,430
You remember a
sinc, you've defined

942
00:57:46,430 --> 00:57:49,057
the sine of pi x over pi x.

943
00:57:49,057 --> 00:57:49,640
Remember that?

944
00:57:54,420 --> 00:57:57,310
So that's the way that
people will usually

945
00:57:57,310 --> 00:58:05,310
define a sinc of x equals
sine of pi x over pi x.

946
00:58:05,310 --> 00:58:11,580
All right, so it's
defined in that way,

947
00:58:11,580 --> 00:58:15,697
because then lots of nasty
pi's and things cancel out

948
00:58:15,697 --> 00:58:16,530
and everything else.

949
00:58:27,090 --> 00:58:31,000
All right, so we've
got to multiply that

950
00:58:31,000 --> 00:58:36,010
by the size of the mask.

951
00:58:36,010 --> 00:58:37,780
So as it shows
here, the mask going

952
00:58:37,780 --> 00:58:43,240
from minus 1.5 centimeters to
plus 1.5 centimeters, which

953
00:58:43,240 --> 00:58:45,100
means that this grating
component, which

954
00:58:45,100 --> 00:58:48,340
is at 2 centimeters,
doesn't get through.

955
00:58:48,340 --> 00:58:54,760
So that's what gets
through the aperture.

956
00:58:54,760 --> 00:58:59,170
And then we Fourier
transform that again.

957
00:58:59,170 --> 00:59:03,480
And so the delta
function transforms

958
00:59:03,480 --> 00:59:07,200
to an exponential-- a
shifted delta function,

959
00:59:07,200 --> 00:59:09,780
and corresponds to exponential.

960
00:59:15,440 --> 00:59:16,905
And therefore, our output--

961
00:59:19,850 --> 00:59:21,830
this f was the
same in this case.

962
00:59:21,830 --> 00:59:24,600
The two f's of the two
[INAUDIBLE] lens were the same.

963
00:59:24,600 --> 00:59:29,020
So we get 1 over
lambda f again there.

964
00:59:29,020 --> 00:59:33,650
AUDIENCE: What will happen if
we make the width of this mask

965
00:59:33,650 --> 00:59:35,540
minus 2 to plus 2?

966
00:59:35,540 --> 00:59:39,806
Will there be any diffraction
due to this second order?

967
00:59:39,806 --> 00:59:43,660
COLIN SHEPPARD: Well, if it was
exactly minus 2 to the plus 2,

968
00:59:43,660 --> 00:59:48,290
then I guess if it
really was exactly there,

969
00:59:48,290 --> 00:59:52,160
you'd expect half of it
to get through this order.

970
00:59:52,160 --> 00:59:55,740
I mean normally, we would
assume it either lets it through

971
00:59:55,740 --> 00:59:58,200
or doesn't let it through.

972
00:59:58,200 --> 01:00:04,160
So if you made the size of the
aperture 2.5, then of course

973
01:00:04,160 --> 01:00:07,440
this next order would
get through, wouldn't it?

974
01:00:07,440 --> 01:00:09,190
AUDIENCE: But if it
is exact [INAUDIBLE],,

975
01:00:09,190 --> 01:00:10,800
that means do we
have to consider

976
01:00:10,800 --> 01:00:13,638
the diffraction of [INAUDIBLE]?

977
01:00:13,638 --> 01:00:18,470
COLIN SHEPPARD: No, I
think that effectively, I

978
01:00:18,470 --> 01:00:21,983
think you have to say that
half of it got through.

979
01:00:21,983 --> 01:00:23,900
GEORGE BARBASTATHIS:
Actually, that's correct.

980
01:00:23,900 --> 01:00:26,575
Because in actuality,
the grating is finite.

981
01:00:26,575 --> 01:00:27,950
So these are not
delta functions.

982
01:00:27,950 --> 01:00:29,520
They're sinc patterns.

983
01:00:29,520 --> 01:00:32,810
So if you place the
grating at exactly 2,

984
01:00:32,810 --> 01:00:35,720
then you allow 1/2 of the
sinc to go through, right?

985
01:00:35,720 --> 01:00:37,370
So therefore, 1/2
is [INAUDIBLE]..

986
01:00:37,370 --> 01:00:39,745
COLIN SHEPPARD: Yeah, that's
a very good way of thinking.

987
01:00:39,745 --> 01:00:41,900
Here, we started off
by saying that this

988
01:00:41,900 --> 01:00:46,700
is a continuous
repetitive grating.

989
01:00:46,700 --> 01:00:49,640
And so we can think of it in
terms of a Fourier series.

990
01:00:49,640 --> 01:00:53,340
But of course, it's not
actually infinitely big.

991
01:00:53,340 --> 01:00:57,560
I guess it's not going to be
more than a few light years

992
01:00:57,560 --> 01:00:58,800
across.

993
01:00:58,800 --> 01:01:05,990
So this is multiplied by
another rect function, really.

994
01:01:05,990 --> 01:01:11,300
So the Fourier transform of this
whole thing multiplied by rect

995
01:01:11,300 --> 01:01:13,550
is something
convolved with a sinc.

996
01:01:13,550 --> 01:01:17,450
So each of these things is
not really a delta function.

997
01:01:17,450 --> 01:01:20,038
But it's really a
very narrow sinc,

998
01:01:20,038 --> 01:01:21,746
which depends on the
size of our grating.

999
01:01:21,746 --> 01:01:22,246
.

1000
01:01:22,246 --> 01:01:24,590
So the point George
is making is,

1001
01:01:24,590 --> 01:01:27,920
if you think of it as
a sinc, then it really

1002
01:01:27,920 --> 01:01:29,810
would be letting
through half of it.

1003
01:01:32,930 --> 01:01:35,420
I think mathematically, it'd
have to be like that as well.

1004
01:01:35,420 --> 01:01:37,010
You get used to
these when you do--

1005
01:01:37,010 --> 01:01:42,080
you must remember from solving
partial differential equations,

1006
01:01:42,080 --> 01:01:43,580
when you do the
boundary conditions,

1007
01:01:43,580 --> 01:01:46,750
you always have to do
that sort of thing.

1008
01:01:46,750 --> 01:01:48,480
It seems a bit of a cheat, but--

1009
01:01:53,980 --> 01:01:56,160
OK, so this is putting
the numbers in, then.

1010
01:01:56,160 --> 01:02:00,180
So this is the three components.

1011
01:02:00,180 --> 01:02:03,860
These are these root 3
over 2 pi's we had there.

1012
01:02:03,860 --> 01:02:08,070
And then finally, these
two combined together

1013
01:02:08,070 --> 01:02:10,190
to make a cosine.

1014
01:02:10,190 --> 01:02:13,860
And our final
amplitude of our image

1015
01:02:13,860 --> 01:02:17,370
is going to be a
constant plus of cosine.

1016
01:02:17,370 --> 01:02:19,290
You notice that it's not--

1017
01:02:19,290 --> 01:02:21,420
the amplitude of
this is not the same

1018
01:02:21,420 --> 01:02:24,870
as the amplitude of that, of
course, because of the fact

1019
01:02:24,870 --> 01:02:28,920
that it was a this
rectangular grating.

1020
01:02:31,900 --> 01:02:36,550
Yeah, and finally, though,
as I just mentioned briefly

1021
01:02:36,550 --> 01:02:38,230
a little while ago,
what you actually

1022
01:02:38,230 --> 01:02:39,910
see is not the amplitude.

1023
01:02:39,910 --> 01:02:43,030
What you see is the
intensity, which

1024
01:02:43,030 --> 01:02:46,160
is the modulus square
of the amplitude.

1025
01:02:46,160 --> 01:02:50,410
So you remember, we
said that we normally

1026
01:02:50,410 --> 01:02:53,170
are going to call the intensity
just the modulus square

1027
01:02:53,170 --> 01:02:54,380
of the amplitude.

1028
01:02:54,380 --> 01:02:57,400
There perhaps should be
some halves and funny things

1029
01:02:57,400 --> 01:02:59,920
out there if we were
really doing it according

1030
01:02:59,920 --> 01:03:03,520
to the Poynting vector.

1031
01:03:03,520 --> 01:03:05,770
But normally, we don't
bother with that.

1032
01:03:05,770 --> 01:03:08,150
We just say it's
the modulus square.

1033
01:03:08,150 --> 01:03:11,170
So the modulus
square of this, we're

1034
01:03:11,170 --> 01:03:17,050
going to get this squared and
twice the product of these two,

1035
01:03:17,050 --> 01:03:18,580
and then this one squared.

1036
01:03:18,580 --> 01:03:22,190
So we get this term
in Cos squared.

1037
01:03:22,190 --> 01:03:26,570
And then you have to remember
your formulae that you learn

1038
01:03:26,570 --> 01:03:33,060
in maths, Cos 2 theta equals 1
minus 2 cosine squared theta,

1039
01:03:33,060 --> 01:03:35,390
which allows us the right
Cos squared theta is

1040
01:03:35,390 --> 01:03:38,120
1 plus Cos theta over 2.

1041
01:03:38,120 --> 01:03:45,370
So this Cos squared, we can
now replace by a constant plus

1042
01:03:45,370 --> 01:03:48,320
a cosine of twice the angle.

1043
01:03:48,320 --> 01:03:53,090
And so this is the
constant bit, and this is

1044
01:03:53,090 --> 01:03:56,390
the cosine of twice the angle.

1045
01:03:56,390 --> 01:04:01,210
All right, so that gives a
another harmonic in the image.

1046
01:04:01,210 --> 01:04:03,770
So that's something that
people are a bit confused

1047
01:04:03,770 --> 01:04:05,690
by sometimes, first of all.

1048
01:04:05,690 --> 01:04:08,000
Because we've gone
through a lot of talk

1049
01:04:08,000 --> 01:04:11,420
about how it's only the first
time on it that's gone through.

1050
01:04:11,420 --> 01:04:14,990
But actually now, when
we look at the intensity,

1051
01:04:14,990 --> 01:04:17,600
we've now got a second
harmonic that's appearing

1052
01:04:17,600 --> 01:04:20,280
in the final intensity.

1053
01:04:20,280 --> 01:04:23,130
So note, the second
harmonic term intensity

1054
01:04:23,130 --> 01:04:26,800
due to the magnitude
square operation.

1055
01:04:26,800 --> 01:04:30,640
This term explains the ringing
in coherent low-pass filtering

1056
01:04:30,640 --> 01:04:32,230
systems.

1057
01:04:32,230 --> 01:04:45,660
So the ringing is the wiggles
you get on the sharp edge.

1058
01:04:45,660 --> 01:04:48,800
Right, and that's a sketch
of what it looks like.

1059
01:04:48,800 --> 01:04:52,990
So it doesn't look quite
like a cosine grating

1060
01:04:52,990 --> 01:04:57,100
anymore, because
when you square it,

1061
01:04:57,100 --> 01:05:02,170
the low values get depressed
more than the high values.

1062
01:05:02,170 --> 01:05:06,030
So it makes it look
more peaky, like that.

1063
01:05:14,570 --> 01:05:16,560
OK, so perhaps
we'll start again.

1064
01:05:16,560 --> 01:05:19,080
I was just waiting for one
of our people to come back.

1065
01:05:19,080 --> 01:05:20,340
Ah, well, there we are.

1066
01:05:25,330 --> 01:05:27,560
Just in time.

1067
01:05:27,560 --> 01:05:32,960
OK, so yeah, before we carry on,
then, there was a question just

1068
01:05:32,960 --> 01:05:34,140
before the break.

1069
01:05:34,140 --> 01:05:35,570
So we're going to
have a question.

1070
01:05:35,570 --> 01:05:38,540
AUDIENCE: OK, actually,
I have two questions.

1071
01:05:38,540 --> 01:05:41,060
The first question regarding
the physical meaning

1072
01:05:41,060 --> 01:05:45,610
of the low-pass filter, just
now in the previous slide.

1073
01:05:45,610 --> 01:05:47,660
COLIN SHEPPARD: Yeah,
OK, so the question

1074
01:05:47,660 --> 01:05:50,690
is about the low-pass filter.

1075
01:05:50,690 --> 01:05:53,640
So you remember in--

1076
01:05:53,640 --> 01:05:57,380
perhaps I'll draw
something from electronics.

1077
01:05:57,380 --> 01:06:03,140
So you remember from
your hi-fi amplifier.

1078
01:06:03,140 --> 01:06:05,890
So this is frequency.

1079
01:06:05,890 --> 01:06:11,540
So these are high frequencies
and these are low frequencies.

1080
01:06:11,540 --> 01:06:15,260
So this corresponds to
[? ah! ?] And this corresponds

1081
01:06:15,260 --> 01:06:17,260
to [? uh. ?]

1082
01:06:17,260 --> 01:06:22,850
And if we multiply this by some
function that looks like that,

1083
01:06:22,850 --> 01:06:24,970
this is what we call
a low-pass filter,

1084
01:06:24,970 --> 01:06:27,670
because which we let
through the low frequencies

1085
01:06:27,670 --> 01:06:31,870
and we stop the
high frequencies.

1086
01:06:31,870 --> 01:06:33,530
So the same is true here.

1087
01:06:33,530 --> 01:06:40,030
So when we say low-pass,
we mean that we

1088
01:06:40,030 --> 01:06:44,540
let through the spots which
are close to the axis,

1089
01:06:44,540 --> 01:06:47,642
and stop the ones that
are far from the axis.

1090
01:06:47,642 --> 01:06:48,850
So that's the first question.

1091
01:06:48,850 --> 01:06:50,248
Then there's another one.

1092
01:06:50,248 --> 01:06:51,790
AUDIENCE: The second
one is regarding

1093
01:06:51,790 --> 01:06:56,080
the Abbe's rule because
Abbe's rule tell us if we can

1094
01:06:56,080 --> 01:07:00,240
[INAUDIBLE] two orders of the
wave, we can form an image.

1095
01:07:00,240 --> 01:07:05,020
But just [INAUDIBLE] even
with three orders of the wave,

1096
01:07:05,020 --> 01:07:08,530
we still get very [INAUDIBLE]
compared to that original one.

1097
01:07:08,530 --> 01:07:10,705
So I wanted to check
out what is the reason.

1098
01:07:10,705 --> 01:07:13,210
COLIN SHEPPARD:
Right, OK, so this

1099
01:07:13,210 --> 01:07:18,020
is going back to Abbe's
rule, which I don't know--

1100
01:07:18,020 --> 01:07:20,530
I've never really heard
what he said, actually.

1101
01:07:20,530 --> 01:07:23,650
But have you got that
in a book or something?

1102
01:07:23,650 --> 01:07:26,850
It's probably in Chinese, is it?

1103
01:07:26,850 --> 01:07:31,470
But anyway, the Chinese
Abbe came up with this rule

1104
01:07:31,470 --> 01:07:33,170
that you must have at least--

1105
01:07:33,170 --> 01:07:35,130
well, we gave an
example just now,

1106
01:07:35,130 --> 01:07:37,836
didn't we, where you
only had one beam.

1107
01:07:37,836 --> 01:07:42,653
One order got through, and
we didn't get an image,

1108
01:07:42,653 --> 01:07:44,070
because you're
obviously not going

1109
01:07:44,070 --> 01:07:47,310
to get any interference if
you've only got one beam.

1110
01:07:47,310 --> 01:07:53,730
So according to Abbe, you've got
to have two beams interfering

1111
01:07:53,730 --> 01:07:59,000
in order to get some variation
in amplitude in the image.

1112
01:07:59,000 --> 01:08:02,730
And so he says at
least two, I guess.

1113
01:08:02,730 --> 01:08:10,800
So if you think of these,
you could let throw the--

1114
01:08:10,800 --> 01:08:14,850
last time we had the 0 and
the plus 1 and the minus 1,

1115
01:08:14,850 --> 01:08:16,979
which is three beams.

1116
01:08:16,979 --> 01:08:19,500
And the point that was
made in the question

1117
01:08:19,500 --> 01:08:22,020
was that, although we
got an image, it was

1118
01:08:22,020 --> 01:08:24,069
wasn't a very good image.

1119
01:08:24,069 --> 01:08:28,979
So I think that's
probably right.

1120
01:08:28,979 --> 01:08:35,850
I guess Abbe's rule is
all to do with resolution

1121
01:08:35,850 --> 01:08:37,229
of an optical system--

1122
01:08:37,229 --> 01:08:39,158
the smallest detail you can see.

1123
01:08:39,158 --> 01:08:40,950
But it doesn't necessarily
mean that you're

1124
01:08:40,950 --> 01:08:43,080
going to get a good image.

1125
01:08:43,080 --> 01:08:45,060
And changing the
relative strengths,

1126
01:08:45,060 --> 01:08:48,420
we're going to see this in
lots of examples now, actually.

1127
01:08:48,420 --> 01:08:50,580
I guess the whole of
the rest of this lecture

1128
01:08:50,580 --> 01:08:52,770
will be about these--

1129
01:08:52,770 --> 01:08:57,990
more examples of stopping
beams and so on, and changing

1130
01:08:57,990 --> 01:09:00,390
beams and seeing what
it does to the pattern.

1131
01:09:00,390 --> 01:09:02,729
And so very often,
what you'll find,

1132
01:09:02,729 --> 01:09:06,779
then, is you get something
which is maybe not a good image

1133
01:09:06,779 --> 01:09:09,279
of the original object.

1134
01:09:09,279 --> 01:09:15,120
But it's an image, which is
much better than nothing.

1135
01:09:15,120 --> 01:09:21,300
OK, so does that
answer the question?

1136
01:09:21,300 --> 01:09:25,350
OK, so we now looking
at a bandpass filter.

1137
01:09:25,350 --> 01:09:27,720
So going back to what
I was saying here,

1138
01:09:27,720 --> 01:09:31,370
the bandpass filter,
of course, is

1139
01:09:31,370 --> 01:09:34,069
something that looks like this.

1140
01:09:34,069 --> 01:09:38,300
A bandpass filter stops the
low frequencies and also

1141
01:09:38,300 --> 01:09:41,010
the high frequencies.

1142
01:09:41,010 --> 01:09:45,920
And so you can see here, the
optical equivalent of this,

1143
01:09:45,920 --> 01:09:48,620
it's now stopping
the zero order,

1144
01:09:48,620 --> 01:09:52,260
but allowing the first
orders to get through.

1145
01:09:52,260 --> 01:09:56,680
And then the later ones
would also not get through.

1146
01:09:56,680 --> 01:09:59,240
Oh yeah, that was
another point that George

1147
01:09:59,240 --> 01:10:01,130
mentioned to me during
the break that I

1148
01:10:01,130 --> 01:10:03,980
hadn't appreciated fully.

1149
01:10:03,980 --> 01:10:06,770
And that is that this
particular example, which

1150
01:10:06,770 --> 01:10:12,260
had this duty cycle of
a third, actually, it

1151
01:10:12,260 --> 01:10:15,680
turns out that the third
order of this one vanishes.

1152
01:10:15,680 --> 01:10:19,010
So I guess the third, the
sixth, the ninth, all these.

1153
01:10:19,010 --> 01:10:20,900
And this is just like we said.

1154
01:10:20,900 --> 01:10:25,620
That for a square wave grating,
all the even orders cancel.

1155
01:10:25,620 --> 01:10:29,540
So it's all to do with the
zeros of that sinc function.

1156
01:10:29,540 --> 01:10:36,210
We add sinc of alpha q, where
alpha is the duty cycle.

1157
01:10:36,210 --> 01:10:41,600
So for any duty cycle, there
will be particular values

1158
01:10:41,600 --> 01:10:48,830
of the particular integers for
the order, which would vanish

1159
01:10:48,830 --> 01:10:51,860
in the Fourier transform.

1160
01:10:51,860 --> 01:10:56,720
Right, so here, we're
letting through the plus 1

1161
01:10:56,720 --> 01:10:57,710
and the minus 1.

1162
01:11:03,177 --> 01:11:04,510
Did something [INAUDIBLE] there?

1163
01:11:04,510 --> 01:11:07,814
AUDIENCE: Yeah, [INAUDIBLE].

1164
01:11:07,814 --> 01:11:09,230
COLIN SHEPPARD: Ah.

1165
01:11:09,230 --> 01:11:09,900
Right, yeah.

1166
01:11:09,900 --> 01:11:15,360
OK, so the amplitude is
going to be a 1 plus cosine.

1167
01:11:15,360 --> 01:11:17,940
And intensity, we're going to
get a second harmonic image,

1168
01:11:17,940 --> 01:11:20,235
just like we had with
the previous case.

1169
01:11:22,840 --> 01:11:27,590
0 and then now we're going to
go through this in more detail.

1170
01:11:27,590 --> 01:11:29,770
So it's the same
grating as before.

1171
01:11:29,770 --> 01:11:33,600
So we haven't got to work
out all that stuff again.

1172
01:11:33,600 --> 01:11:37,240
You remember, 1 centimeter
was where the first order was.

1173
01:11:37,240 --> 01:11:43,870
So we've got this series
of these different grating

1174
01:11:43,870 --> 01:11:45,460
components.

1175
01:11:45,460 --> 01:11:50,230
And these ones are 0,
as I just described.

1176
01:11:50,230 --> 01:11:52,690
The next one wouldn't be 0.

1177
01:11:52,690 --> 01:11:55,450
And we're putting this
masking, which is just

1178
01:11:55,450 --> 01:11:56,890
going to let throw these two.

1179
01:12:03,970 --> 01:12:08,200
And so mathematically,
we can write mask

1180
01:12:08,200 --> 01:12:11,520
as being the sum
of two rectangles--

1181
01:12:11,520 --> 01:12:13,500
rectangle, another rectangle.

1182
01:12:19,580 --> 01:12:22,660
So then it lets through
those two components

1183
01:12:22,660 --> 01:12:24,800
and stops the others.

1184
01:12:24,800 --> 01:12:29,350
So we've got these
two delta functions.

1185
01:12:29,350 --> 01:12:33,180
And then we just got to do
a Fourier transform again.

1186
01:12:33,180 --> 01:12:40,800
Each of the delta functions
produces a plane wave--

1187
01:12:40,800 --> 01:12:42,420
[INAUDIBLE] complex exponential.

1188
01:12:47,080 --> 01:12:50,560
We're then going to
add those together.

1189
01:12:50,560 --> 01:12:54,660
And we've got to change
the [? interdistance, ?] of

1190
01:12:54,660 --> 01:12:59,880
course, and then combine
these two complex exponentials

1191
01:12:59,880 --> 01:13:01,620
to make a cosine.

1192
01:13:01,620 --> 01:13:04,320
And then finally, we've
got to square it again

1193
01:13:04,320 --> 01:13:07,610
in order to get the
intensity [? out. ?]

1194
01:13:07,610 --> 01:13:12,410
So notice what we get now.

1195
01:13:12,410 --> 01:13:15,506
You see we get
the constant term.

1196
01:13:15,506 --> 01:13:18,040
You see, here, it's just cosine.

1197
01:13:18,040 --> 01:13:20,530
There's no constant term anymore
because the constant term

1198
01:13:20,530 --> 01:13:21,460
is gone.

1199
01:13:21,460 --> 01:13:26,270
So because when we square
it, we get Cos squared.

1200
01:13:26,270 --> 01:13:29,440
The Cos squared, we go
into double angles again.

1201
01:13:29,440 --> 01:13:32,560
We get a constant plus a cosine.

1202
01:13:32,560 --> 01:13:36,970
And so you see that actually,
we get an image where

1203
01:13:36,970 --> 01:13:40,390
the first harmonic
doesn't appear at all,

1204
01:13:40,390 --> 01:13:42,170
but the second harmonic does.

1205
01:13:42,170 --> 01:13:47,080
So this comes back to
your question in a way.

1206
01:13:47,080 --> 01:13:50,050
You end up with getting
an image which is actually

1207
01:13:50,050 --> 01:13:51,700
really not like the object.

1208
01:13:51,700 --> 01:13:54,730
Because you see that this
has got twice the frequency

1209
01:13:54,730 --> 01:14:01,830
of this, although Abbe says
that you can get an image,

1210
01:14:01,830 --> 01:14:04,730
he doesn't say that you
can get a correct image.

1211
01:14:04,730 --> 01:14:08,130
And in this case, you could
be very confused by this.

1212
01:14:08,130 --> 01:14:11,490
You could look in your
microscope at something

1213
01:14:11,490 --> 01:14:13,800
and you could see this.

1214
01:14:13,800 --> 01:14:18,630
And you'd be completely
mistaken about the object that

1215
01:14:18,630 --> 01:14:22,140
was actually producing this,
because it's actually twice as

1216
01:14:22,140 --> 01:14:24,610
big as what you expect.

1217
01:14:24,610 --> 01:14:25,110
So--

1218
01:14:25,110 --> 01:14:26,360
AUDIENCE: I've got a question.

1219
01:14:26,360 --> 01:14:27,740
COLIN SHEPPARD: Oh, yeah.

1220
01:14:27,740 --> 01:14:29,580
You frightened me.

1221
01:14:29,580 --> 01:14:31,710
AUDIENCE: If you grab
just the zero order

1222
01:14:31,710 --> 01:14:33,990
and say the first order,
but not the minus 1,

1223
01:14:33,990 --> 01:14:36,000
you get the same period
as the original grating?

1224
01:14:39,655 --> 01:14:41,490
COLIN SHEPPARD:
Yeah, I guess so.

1225
01:14:41,490 --> 01:14:42,980
Would that be right, George?

1226
01:14:42,980 --> 01:14:47,850
If you take this one and
this one, so you've got--

1227
01:14:47,850 --> 01:14:51,230
and then you square it.

1228
01:14:51,230 --> 01:14:53,070
You're going to get
1 plus an e to the i.

1229
01:14:53,070 --> 01:14:58,920
You're going to-- yeah,
he's working it out.

1230
01:14:58,920 --> 01:14:59,945
He's working it out.

1231
01:14:59,945 --> 01:15:01,687
We'll have a race, shall we?

1232
01:15:04,610 --> 01:15:08,318
You're going to have a 1
plus e to the [INAUDIBLE] i

1233
01:15:08,318 --> 01:15:09,110
something or other.

1234
01:15:12,060 --> 01:15:16,290
And then this module is squared.

1235
01:15:16,290 --> 01:15:21,200
And then you can take out an
e to the i something over 2.

1236
01:15:21,200 --> 01:15:22,400
You've got the answer?

1237
01:15:22,400 --> 01:15:22,900
Yeah.

1238
01:15:28,750 --> 01:15:30,790
Something like that.

1239
01:15:30,790 --> 01:15:31,740
So this is cosine.

1240
01:15:31,740 --> 01:15:34,050
So you're going to end
up with cosine squared.

1241
01:15:36,560 --> 01:15:38,200
It's the same as this, isn't it?

1242
01:15:38,200 --> 01:15:38,700
[INAUDIBLE]?

1243
01:15:38,700 --> 01:15:40,830
It would just be the phase term.

1244
01:15:40,830 --> 01:15:43,936
No, it can't be, because
it's half as far apart.

1245
01:15:43,936 --> 01:15:45,280
AUDIENCE: [INAUDIBLE]

1246
01:15:45,280 --> 01:15:48,300
COLIN SHEPPARD: Yeah, you
got the first harmonic,

1247
01:15:48,300 --> 01:15:51,260
because it's the
square of half of it.

1248
01:15:51,260 --> 01:15:52,010
Yeah.

1249
01:15:52,010 --> 01:15:54,660
OK, so it's double half of it.

1250
01:15:54,660 --> 01:15:57,300
So you do get the
first harmonic, yeah.

1251
01:15:57,300 --> 01:15:59,040
You're quite correct.

1252
01:15:59,040 --> 01:16:01,620
But it might work
for this object.

1253
01:16:01,620 --> 01:16:06,090
But I guess there'd be other
objects, which would also

1254
01:16:06,090 --> 01:16:08,660
produce funny results.

1255
01:16:08,660 --> 01:16:10,960
And of course, as we know--

1256
01:16:10,960 --> 01:16:13,080
well, some of us know.

1257
01:16:13,080 --> 01:16:14,040
Most of you won't know.

1258
01:16:14,040 --> 01:16:15,930
But for example,
[INAUDIBLE] will

1259
01:16:15,930 --> 01:16:22,260
know that if you have a
mask, which is actually

1260
01:16:22,260 --> 01:16:24,660
metric like that, it's
going to have some quite

1261
01:16:24,660 --> 01:16:26,170
interesting properties.

1262
01:16:26,170 --> 01:16:26,922
So I don't know
whether you're going

1263
01:16:26,922 --> 01:16:28,589
to go through these
at some later point.

1264
01:16:28,589 --> 01:16:30,492
AUDIENCE: [INAUDIBLE]

1265
01:16:30,492 --> 01:16:32,160
COLIN SHEPPARD:
Maybe in the exam.

1266
01:16:32,160 --> 01:16:36,930
OK, so everyone [INAUDIBLE]
up on asymmetric gratings.

1267
01:16:36,930 --> 01:16:38,705
AUDIENCE: [INAUDIBLE]

1268
01:16:38,705 --> 01:16:42,150
COLIN SHEPPARD: Yes,
single-sideband modulation.

1269
01:16:42,150 --> 01:16:43,115
Yeah.

1270
01:16:43,115 --> 01:16:44,490
And what's the
other word for it?

1271
01:16:44,490 --> 01:16:52,010
What's this, when they
measure the flow of the--

1272
01:16:52,010 --> 01:16:53,110
in aerodynamics?

1273
01:16:53,110 --> 01:16:55,030
Schlieren.

1274
01:16:55,030 --> 01:16:57,990
It's basically Schlieren.

1275
01:16:57,990 --> 01:17:04,660
OK, and now we're looking
at a tilted illumination.

1276
01:17:04,660 --> 01:17:08,110
So same sort of system
as the last one,

1277
01:17:08,110 --> 01:17:12,190
but we're now not illuminating
it where the plane wave along

1278
01:17:12,190 --> 01:17:13,780
the axis.

1279
01:17:13,780 --> 01:17:16,220
We're going to tilt this.

1280
01:17:16,220 --> 01:17:23,230
And so what we find if
we've got a grating,

1281
01:17:23,230 --> 01:17:24,700
and we illuminate
it with a plane

1282
01:17:24,700 --> 01:17:30,220
wave, what do we find happens
if we tilt the plane wave?

1283
01:17:30,220 --> 01:17:31,810
Have you come across this--

1284
01:17:31,810 --> 01:17:33,930
you did this
earlier on, did you?

1285
01:17:33,930 --> 01:17:35,920
I don't think I was
there for this one.

1286
01:17:35,920 --> 01:17:38,320
But actually, all that
happens is it just

1287
01:17:38,320 --> 01:17:40,840
tilts the diffraction pattern.

1288
01:17:40,840 --> 01:17:42,400
Very simple.

1289
01:17:42,400 --> 01:17:46,270
So you can see
now, the zero order

1290
01:17:46,270 --> 01:17:49,220
is still in this direction.

1291
01:17:49,220 --> 01:17:53,860
And all of the other orders
just move [INAUDIBLE]..

1292
01:17:53,860 --> 01:17:56,530
They all rotate
through the same angle.

1293
01:17:56,530 --> 01:17:59,560
This is actually only true
if you can think of this

1294
01:17:59,560 --> 01:18:01,930
as being a thin grating.

1295
01:18:01,930 --> 01:18:07,870
And if you go into theory of
volume holograms and things

1296
01:18:07,870 --> 01:18:09,760
like that, then you'll
find that things

1297
01:18:09,760 --> 01:18:11,350
become much more complicated.

1298
01:18:11,350 --> 01:18:14,620
The relative strengths of the
gratings can change and so on.

1299
01:18:14,620 --> 01:18:21,220
But this is a thin grating, so
that you can use this property

1300
01:18:21,220 --> 01:18:24,250
that you just multiply it
by the transmit transmission

1301
01:18:24,250 --> 01:18:28,280
of the grating, then
that will be the case.

1302
01:18:28,280 --> 01:18:30,610
So you can see what
that's going to do.

1303
01:18:30,610 --> 01:18:34,870
All the gratings move up,
which means that they all

1304
01:18:34,870 --> 01:18:38,020
get moved up there as well.

1305
01:18:38,020 --> 01:18:42,340
So you see the zero
order grating, which

1306
01:18:42,340 --> 01:18:48,250
was appearing down at the
origin, is now tilted upwards

1307
01:18:48,250 --> 01:18:50,360
and appears at this point here.

1308
01:18:50,360 --> 01:18:55,090
So with this bandpass
filter that we've chosen,

1309
01:18:55,090 --> 01:18:57,370
we now get the
zero order through.

1310
01:18:57,370 --> 01:19:03,010
So we're getting the
0 to through this one.

1311
01:19:03,010 --> 01:19:06,040
And we're getting the minus
2 order through this one.

1312
01:19:12,760 --> 01:19:19,480
Right, so we illuminate the
grating with an off-axis plane

1313
01:19:19,480 --> 01:19:26,410
wave at an angle of 2.865
degrees, which is obviously

1314
01:19:26,410 --> 01:19:32,210
very carefully designed, so that
this grating this grating order

1315
01:19:32,210 --> 01:19:36,460
moves through from here to here.

1316
01:19:36,460 --> 01:19:37,135
So what happens?

1317
01:19:42,050 --> 01:19:48,120
OK, so it's just describing
what I just said.

1318
01:19:48,120 --> 01:19:50,810
Yeah, this is just describing
why that happens, then,

1319
01:19:50,810 --> 01:19:53,548
in terms of multiplying the--

1320
01:19:57,710 --> 01:20:00,200
this tilted wave is
just going to have

1321
01:20:00,200 --> 01:20:08,070
a complex exponential
phase variation like this.

1322
01:20:08,070 --> 01:20:10,830
So you multiply
that by the grating.

1323
01:20:10,830 --> 01:20:13,680
And then it's just
[INAUDIBLE] this result

1324
01:20:13,680 --> 01:20:15,110
of rotating the thing.

1325
01:20:20,990 --> 01:20:26,910
OK, so there are
the grating orders.

1326
01:20:26,910 --> 01:20:31,720
The zero order one is this one.

1327
01:20:31,720 --> 01:20:37,090
And this is the plus and minus
1 and the plus and minus 2.

1328
01:20:37,090 --> 01:20:40,900
And so we let through those two.

1329
01:20:40,900 --> 01:20:47,350
So we're now letting
through the zero

1330
01:20:47,350 --> 01:20:50,870
order and the minus 2 order.

1331
01:20:50,870 --> 01:20:53,130
So it's a bit like the
one in the question.

1332
01:20:53,130 --> 01:20:59,800
But now it's the minus 2
order, not the minus 1 order.

1333
01:20:59,800 --> 01:21:02,740
So this is going to be another
example where it doesn't give

1334
01:21:02,740 --> 01:21:13,670
a very good image of it from
that result. OK, so yeah, let's

1335
01:21:13,670 --> 01:21:15,156
just [INAUDIBLE].

1336
01:21:17,934 --> 01:21:18,950
Here we are.

1337
01:21:18,950 --> 01:21:20,610
So it's 1 centimeters.

1338
01:21:24,480 --> 01:21:27,000
That magic angle was
designed so that they're all

1339
01:21:27,000 --> 01:21:28,410
shifted by 1 centimeter.

1340
01:21:32,320 --> 01:21:34,000
So after passing
through the mask,

1341
01:21:34,000 --> 01:21:36,970
we just get those
two components.

1342
01:21:36,970 --> 01:21:39,490
And so these are
the two components.

1343
01:21:44,041 --> 01:21:45,770
Ah, but [INAUDIBLE]
no, it isn't.

1344
01:21:45,770 --> 01:21:47,562
Yeah, that's right,
because there's still--

1345
01:21:47,562 --> 01:21:50,590
it's a bit more complicated
than where I was--

1346
01:21:50,590 --> 01:21:52,710
it's tricky, this.

1347
01:21:52,710 --> 01:21:57,560
So not only of course that
we let through the zero

1348
01:21:57,560 --> 01:22:00,650
order and the minus 2 order.

1349
01:22:00,650 --> 01:22:05,060
But those orders have got
this complex phase shift.

1350
01:22:05,060 --> 01:22:05,690
Is that right?

1351
01:22:05,690 --> 01:22:09,106
AUDIENCE: [INAUDIBLE]

1352
01:22:17,900 --> 01:22:20,270
COLIN SHEPPARD: Yeah,
exactly, exactly.

1353
01:22:20,270 --> 01:22:22,610
Because of the phase,
because of the--

1354
01:22:22,610 --> 01:22:23,480
yeah.

1355
01:22:23,480 --> 01:22:25,460
OK, so that's what I'm saying.

1356
01:22:25,460 --> 01:22:30,010
Although this is a zero
order and a minus 2 order,

1357
01:22:30,010 --> 01:22:32,010
they're actually shifted.

1358
01:22:32,010 --> 01:22:34,910
And so when you do
the Fourier transform,

1359
01:22:34,910 --> 01:22:38,180
they're just as
though they were there

1360
01:22:38,180 --> 01:22:40,920
plus 1 and the minus 1 order.

1361
01:22:40,920 --> 01:22:42,380
And that's neat.

1362
01:22:42,380 --> 01:22:43,130
Very interesting.

1363
01:22:43,130 --> 01:22:44,010
AUDIENCE: But with
different amplitudes.

1364
01:22:44,010 --> 01:22:46,135
COLIN SHEPPARD: But with
different amplitudes given

1365
01:22:46,135 --> 01:22:47,690
by these two strengths here.

1366
01:22:51,950 --> 01:23:00,690
And then-- yeah, so then we
have to put the values of u

1367
01:23:00,690 --> 01:23:02,580
in terms of the x.

1368
01:23:02,580 --> 01:23:04,860
And then we combine
the two exponentials.

1369
01:23:04,860 --> 01:23:06,060
Well, can't do that anymore.

1370
01:23:06,060 --> 01:23:06,640
Can we?

1371
01:23:06,640 --> 01:23:08,290
Oh yes, we can.

1372
01:23:08,290 --> 01:23:12,250
AUDIENCE: Well, you can,
but [INAUDIBLE] sine and the

1373
01:23:12,250 --> 01:23:14,230
cosine.

1374
01:23:14,230 --> 01:23:16,230
COLIN SHEPPARD: Yeah, exactly.

1375
01:23:16,230 --> 01:23:18,470
So have you actually
done that, then?

1376
01:23:18,470 --> 01:23:20,770
AUDIENCE: [INAUDIBLE]

1377
01:23:20,770 --> 01:23:23,330
COLIN SHEPPARD: Right,
so you've got to do

1378
01:23:23,330 --> 01:23:25,553
the modulus square of this.

1379
01:23:25,553 --> 01:23:27,845
So you're going to have this
squared plus this squared.

1380
01:23:27,845 --> 01:23:28,655
AUDIENCE: Plus [INAUDIBLE].

1381
01:23:28,655 --> 01:23:30,530
COLIN SHEPPARD: Oh yeah,
OK, so [INAUDIBLE]..

1382
01:23:30,530 --> 01:23:31,640
Yeah, of course you are.

1383
01:23:31,640 --> 01:23:36,370
So then twice the
products of these two.

1384
01:23:36,370 --> 01:23:38,402
And-- OK.

1385
01:23:38,402 --> 01:23:41,090
Is that right?

1386
01:23:41,090 --> 01:23:41,590
No, sorry.

1387
01:23:41,590 --> 01:23:42,585
That's not right, is it?

1388
01:23:42,585 --> 01:23:44,585
AUDIENCE: The magnitude
of the first [INAUDIBLE]

1389
01:23:44,585 --> 01:23:47,558
the magnitude of the
second plus [INAUDIBLE]..

1390
01:23:47,558 --> 01:23:49,060
COLIN SHEPPARD:
Yeah, that's right.

1391
01:23:49,060 --> 01:23:50,700
AUDIENCE: [INAUDIBLE]

1392
01:23:50,700 --> 01:23:51,790
COLIN SHEPPARD: Yeah.

1393
01:23:51,790 --> 01:23:54,188
AUDIENCE: [INAUDIBLE]

1394
01:23:57,900 --> 01:24:00,120
COLIN SHEPPARD: OK, and
this is what it looks like.

1395
01:24:00,120 --> 01:24:00,620
Ta-da.

1396
01:24:03,910 --> 01:24:07,090
OK, so the field is
the first harmonic.

1397
01:24:07,090 --> 01:24:09,130
The intensity is
the second harmonic

1398
01:24:09,130 --> 01:24:10,210
because of the squaring.

1399
01:24:10,210 --> 01:24:12,790
And the contrast is 0.7.

1400
01:24:12,790 --> 01:24:16,930
Right, so again we end up with
a bad image as it's shifted,

1401
01:24:16,930 --> 01:24:19,270
because it's double frequency.

1402
01:24:19,270 --> 01:24:22,060
And also, the contrast
is reduced in this case,

1403
01:24:22,060 --> 01:24:26,410
because the strength of the
two beams was not the same.

1404
01:24:26,410 --> 01:24:28,780
This is calculating
here the contrast.

1405
01:24:28,780 --> 01:24:33,250
This is the normal definition
of contrast fringes--

1406
01:24:33,250 --> 01:24:36,160
the difference over the sum.

1407
01:24:36,160 --> 01:24:38,050
And it comes to 0.7062.

1408
01:24:38,050 --> 01:24:43,630
In the previous case, of
course, the contrast was 1.

1409
01:24:47,640 --> 01:24:51,670
Right, another example--
a new pupil mask

1410
01:24:51,670 --> 01:24:56,200
consisting of two holes,
each of diameter 1 centimeter

1411
01:24:56,200 --> 01:24:59,980
and centered further away
from the axis plus or minus

1412
01:24:59,980 --> 01:25:04,070
2 centimeters from the
optical axis respectively.

1413
01:25:04,070 --> 01:25:08,350
What is the intensity observed
at the output image plane?

1414
01:25:08,350 --> 01:25:11,320
Right, so at least it's
told us the 2 centimeters.

1415
01:25:11,320 --> 01:25:12,880
We know where that is.

1416
01:25:12,880 --> 01:25:15,100
We don't have to worry
about working out

1417
01:25:15,100 --> 01:25:20,380
what 0.3479 degrees is.

1418
01:25:20,380 --> 01:25:24,880
We know that the first
order was at 1 centimeter.

1419
01:25:24,880 --> 01:25:30,310
So the second order is
where the aperture is.

1420
01:25:30,310 --> 01:25:32,520
So the second order is
going to get through,

1421
01:25:32,520 --> 01:25:34,940
the plus 2 and the minus 2.

1422
01:25:40,590 --> 01:25:44,940
And did not appear,
then, up there?

1423
01:25:44,940 --> 01:25:48,540
Right, so the second harmonic
is going to get through.

1424
01:25:48,540 --> 01:25:50,443
And when you do
the square of that,

1425
01:25:50,443 --> 01:25:52,110
you're going to get
the fourth harmonic.

1426
01:25:55,090 --> 01:25:58,610
So this is going through
the mass of that, then.

1427
01:25:58,610 --> 01:26:02,090
So we've got just
these two components.

1428
01:26:04,850 --> 01:26:06,730
So these have now
got 2 centimeters

1429
01:26:06,730 --> 01:26:09,460
in, rather than 1 centimeter.

1430
01:26:09,460 --> 01:26:14,050
So that means that you
get a 2 multiplied in here

1431
01:26:14,050 --> 01:26:16,160
that wasn't there before.

1432
01:26:16,160 --> 01:26:21,300
So that's where the
second harmonic comes in.

1433
01:26:21,300 --> 01:26:23,820
And then when you do the
square, again, you're

1434
01:26:23,820 --> 01:26:26,700
only going to get the
cosine and no constants.

1435
01:26:26,700 --> 01:26:31,500
And then you do Cos squared and
change that into double angles,

1436
01:26:31,500 --> 01:26:34,530
or fourth angles, actually.

1437
01:26:34,530 --> 01:26:37,350
And now we're going to
see what it looks like.

1438
01:26:37,350 --> 01:26:42,490
And now, it's even worse
an image of what we wanted.

1439
01:26:42,490 --> 01:26:45,920
So I guess this is maybe
telling us what we shouldn't do,

1440
01:26:45,920 --> 01:26:47,220
rather than what we should do.

1441
01:26:47,220 --> 01:26:48,610
AUDIENCE: Got another question.

1442
01:26:48,610 --> 01:26:51,010
Is this is still
assuming f1 equals f2?

1443
01:26:51,010 --> 01:26:54,722
And what happens if
f2 is smaller than f1?

1444
01:26:54,722 --> 01:26:56,680
COLIN SHEPPARD: All it
would do, if you changed

1445
01:26:56,680 --> 01:27:02,140
the relative values of an
f1 an f2, all it would do

1446
01:27:02,140 --> 01:27:06,790
is change the size
of the final image.

1447
01:27:06,790 --> 01:27:09,760
If you think about this system--

1448
01:27:09,760 --> 01:27:13,280
let's go back to one where
it shows that-- there we are.

1449
01:27:13,280 --> 01:27:25,060
You can see that the relative
distances from the axis only

1450
01:27:25,060 --> 01:27:28,240
depend on this part
of the optical system.

1451
01:27:28,240 --> 01:27:32,410
And then after the
pupil mask, we've

1452
01:27:32,410 --> 01:27:34,570
already decided,
then, what orders

1453
01:27:34,570 --> 01:27:36,820
get into the final image.

1454
01:27:36,820 --> 01:27:39,370
And then the final part,
all that's going to do

1455
01:27:39,370 --> 01:27:43,150
is just change the relative
size of the final image you see.

1456
01:27:57,080 --> 01:27:59,510
Right, now we're going to do
another bit more complicated

1457
01:27:59,510 --> 01:28:00,100
one--

1458
01:28:00,100 --> 01:28:02,750
a phase pupil mask.

1459
01:28:02,750 --> 01:28:07,070
And so what this
is shown as is, you

1460
01:28:07,070 --> 01:28:11,300
can see it's got an opaque
part of the screen here.

1461
01:28:11,300 --> 01:28:14,690
And then in the
central part, it's

1462
01:28:14,690 --> 01:28:21,560
transparent, but it's got
some sort of structure there--

1463
01:28:21,560 --> 01:28:25,160
some diffractive
dielectric element,

1464
01:28:25,160 --> 01:28:31,160
which changes the relative
phase of this order compared

1465
01:28:31,160 --> 01:28:34,180
with the other orders.

1466
01:28:34,180 --> 01:28:39,060
So the question is, what's
that going to do to your image?

1467
01:28:39,060 --> 01:28:41,430
So the rest of the system
is completely the same.

1468
01:28:44,860 --> 01:28:48,517
So this is what we are
considering putting in there.

1469
01:28:48,517 --> 01:28:50,350
It doesn't matter what
the thickness of this

1470
01:28:50,350 --> 01:28:53,590
is, because all
that's going to do

1471
01:28:53,590 --> 01:28:57,040
is change the relative
phase of all of these.

1472
01:28:57,040 --> 01:28:59,638
So it won't affect how
they interfere with each.

1473
01:28:59,638 --> 01:29:01,930
It will just change the phase
of the final image, which

1474
01:29:01,930 --> 01:29:03,580
you don't see anyway.

1475
01:29:03,580 --> 01:29:08,080
But what is important
is this bit here.

1476
01:29:08,080 --> 01:29:14,530
The thickness of this
s equals 0.25 microns,

1477
01:29:14,530 --> 01:29:20,680
and the wavelength was
half of 0.5 microns.

1478
01:29:20,680 --> 01:29:23,470
So that's lambda by 2.

1479
01:29:23,470 --> 01:29:29,210
So that means it changes
the phase by 180 degrees.

1480
01:29:29,210 --> 01:29:32,110
AUDIENCE: [INAUDIBLE]

1481
01:29:32,110 --> 01:29:33,661
COLIN SHEPPARD: Ah, n minus 1.

1482
01:29:33,661 --> 01:29:35,870
AUDIENCE: [INAUDIBLE]
the next slide.

1483
01:29:35,870 --> 01:29:37,465
COLIN SHEPPARD: Yeah, n minus 1.

1484
01:29:37,465 --> 01:29:39,478
So it's not going to do that.

1485
01:29:39,478 --> 01:29:40,020
Take it back.

1486
01:29:40,020 --> 01:29:42,054
AUDIENCE: [INAUDIBLE]

1487
01:29:42,054 --> 01:29:44,600
COLIN SHEPPARD:
It's [INAUDIBLE]..

1488
01:29:44,600 --> 01:29:51,770
OK, this is known as a phase
pupil mask or a pupil phase

1489
01:29:51,770 --> 01:29:54,285
mask, or a mask pupil phase.

1490
01:29:56,840 --> 01:29:59,140
AUDIENCE: [INAUDIBLE]

1491
01:29:59,140 --> 01:30:01,060
COLIN SHEPPARD:
Yeah, that's right.

1492
01:30:01,060 --> 01:30:02,740
So that must be in
the next lecture,

1493
01:30:02,740 --> 01:30:04,640
because it's not in this one.

1494
01:30:04,640 --> 01:30:06,640
So let's get down to
the real application.

1495
01:30:06,640 --> 01:30:11,560
OK, so George said that I
would have just failed the exam

1496
01:30:11,560 --> 01:30:15,250
because I forgot the minus 1.

1497
01:30:15,250 --> 01:30:19,000
Of course, this light
goes through the glass.

1498
01:30:19,000 --> 01:30:22,660
But this light goes through
the glass and then the air.

1499
01:30:22,660 --> 01:30:26,110
So to work out the
relative phases of these,

1500
01:30:26,110 --> 01:30:31,000
you have to look at the relative
change in the refractive index

1501
01:30:31,000 --> 01:30:33,010
times the optical thickness.

1502
01:30:33,010 --> 01:30:41,620
So this is 0.25 microns,
and this is at 0.25 microns

1503
01:30:41,620 --> 01:30:43,420
divided by the wavelength.

1504
01:30:43,420 --> 01:30:48,295
And it's going to be
multiplied by 1.5 minus 1.

1505
01:30:48,295 --> 01:30:50,860
So this is assuming it's glass.

1506
01:30:50,860 --> 01:30:57,390
So this means that the
phase change is pi over 2.

1507
01:30:57,390 --> 01:31:00,400
So not pi, but pi over 2.

1508
01:31:05,700 --> 01:31:08,438
OK, so exactly the same grating.

1509
01:31:08,438 --> 01:31:09,480
These are all the orders.

1510
01:31:12,480 --> 01:31:18,210
And so you see, first of all,
we get rid of these outer ones.

1511
01:31:18,210 --> 01:31:20,790
We let through just these three.

1512
01:31:20,790 --> 01:31:25,410
But of those, we're
also going to change

1513
01:31:25,410 --> 01:31:30,360
the relative phase of this one
relative to the other, too.

1514
01:31:30,360 --> 01:31:33,900
That doesn't matter, of
course, in your sum whether you

1515
01:31:33,900 --> 01:31:35,970
change the phase of
this or you change

1516
01:31:35,970 --> 01:31:37,230
the phase of the other two.

1517
01:31:37,230 --> 01:31:40,770
Certainly, the relative phase,
which is the important thing.

1518
01:31:50,740 --> 01:31:53,070
OK, so these are the
three conditions, then.

1519
01:31:53,070 --> 01:31:56,410
For the ones that are not
don't get through at all,

1520
01:31:56,410 --> 01:31:58,420
the strength of those is zero.

1521
01:31:58,420 --> 01:32:05,150
Then here, we're taking these
ones to be of strength one,

1522
01:32:05,150 --> 01:32:11,630
and this one in the center to
be strength of e to the i phi.

1523
01:32:11,630 --> 01:32:16,790
And we know that
phi is pi over 2.

1524
01:32:16,790 --> 01:32:21,520
So this is e to the i pi
over 2, which is just i.

1525
01:32:21,520 --> 01:32:23,840
Is that right?

1526
01:32:23,840 --> 01:32:27,310
Why have we got this i
minus 1 then, George?

1527
01:32:27,310 --> 01:32:32,380
GEORGE BARBASTATHIS:
[INAUDIBLE] So one way

1528
01:32:32,380 --> 01:32:36,970
to write this pupil function
is as a rect that is as big

1529
01:32:36,970 --> 01:32:37,765
as the opening.

1530
01:32:37,765 --> 01:32:39,790
COLIN SHEPPARD: Yeah, OK.

1531
01:32:39,790 --> 01:32:41,800
Sorry, yeah.

1532
01:32:41,800 --> 01:32:51,280
So you can think of this as
being the whole thing minus--

1533
01:32:51,280 --> 01:32:54,900
well, this on top of
the whole rectangle.

1534
01:32:54,900 --> 01:32:58,660
Otherwise, you'd have to do this
little rectangle and this bit

1535
01:32:58,660 --> 01:33:02,120
a rectangle and this bit of
rectangle as three things.

1536
01:33:02,120 --> 01:33:05,050
So you could do it either
of those ways, I guess.

1537
01:33:05,050 --> 01:33:07,810
But the way that George
has done it here,

1538
01:33:07,810 --> 01:33:10,060
it's the rectangle, of
the whole [? lot, ?]

1539
01:33:10,060 --> 01:33:14,410
it's all of this
minus the central one,

1540
01:33:14,410 --> 01:33:17,140
plus the central one
with the phase change.

1541
01:33:17,140 --> 01:33:19,450
So that's the minus
the one that's

1542
01:33:19,450 --> 01:33:24,740
not there plus the one that is
there where the phase changed.

1543
01:33:24,740 --> 01:33:27,820
And yeah, so now--

1544
01:33:27,820 --> 01:33:32,060
well of course, we still get
all these delta functions.

1545
01:33:32,060 --> 01:33:35,600
But now we've got
some i's coming in.

1546
01:33:35,600 --> 01:33:36,100
Sorry?

1547
01:33:36,100 --> 01:33:36,975
AUDIENCE: Just one i.

1548
01:33:36,975 --> 01:33:39,850
COLIN SHEPPARD:
Just one i, yeah.

1549
01:33:39,850 --> 01:33:47,120
And then these two combine
to produce cosine still.

1550
01:33:47,120 --> 01:33:50,160
I guess if you change the phase
of one relative to the other,

1551
01:33:50,160 --> 01:33:51,900
it'd be more complicated.

1552
01:33:51,900 --> 01:33:55,080
But these two produce a cosine.

1553
01:33:55,080 --> 01:33:58,110
So we've got an i plus a cosine.

1554
01:33:58,110 --> 01:34:00,980
And then we do the
modulus square of that.

1555
01:34:00,980 --> 01:34:03,210
And the modulus square
of that, of course,

1556
01:34:03,210 --> 01:34:06,590
is you're not going to get a
cross product term are you,

1557
01:34:06,590 --> 01:34:10,860
because these two are
in phase quadrature.

1558
01:34:10,860 --> 01:34:14,527
So when you do the modulus
square, you're only going--

1559
01:34:14,527 --> 01:34:15,110
is that right?

1560
01:34:15,110 --> 01:34:15,610
Yeah.

1561
01:34:15,610 --> 01:34:17,800
You're going to get this
squared plus this squared.

1562
01:34:17,800 --> 01:34:22,590
And this squared, you can
write as these two terms

1563
01:34:22,590 --> 01:34:26,090
when you do the double angle.

1564
01:34:26,090 --> 01:34:27,670
And I guess the next
one, we're going

1565
01:34:27,670 --> 01:34:29,390
to see what it looks like.

1566
01:34:29,390 --> 01:34:32,660
Ta-da, this is
what it looks like.

1567
01:34:32,660 --> 01:34:36,650
First harmonic in field,
second harmonic in intensity

1568
01:34:36,650 --> 01:34:38,780
because of the squaring.

1569
01:34:38,780 --> 01:34:42,950
And again, the contrast is
now much reduced, actually,

1570
01:34:42,950 --> 01:34:44,930
in this case--

1571
01:34:44,930 --> 01:34:48,160
down to 0.25.

1572
01:34:48,160 --> 01:34:51,455
So I guess this hasn't turned
out to be a very [INAUDIBLE]

1573
01:34:51,455 --> 01:34:52,540
strategy.

1574
01:34:52,540 --> 01:34:55,030
This lecture seems to
be mainly showing you

1575
01:34:55,030 --> 01:34:57,500
what you shouldn't do when
you make your microscope.

1576
01:34:57,500 --> 01:34:59,263
AUDIENCE: [INAUDIBLE]

1577
01:34:59,263 --> 01:35:00,706
COLIN SHEPPARD: [LAUGHS]

1578
01:35:00,706 --> 01:35:03,592
AUDIENCE: [INAUDIBLE]

1579
01:35:03,592 --> 01:35:09,590
COLIN SHEPPARD: OK,
so now we're going

1580
01:35:09,590 --> 01:35:11,840
to go on to think
of this in terms

1581
01:35:11,840 --> 01:35:14,660
of what we call a
point spread function

1582
01:35:14,660 --> 01:35:16,490
of the low-pass filter.

1583
01:35:16,490 --> 01:35:19,400
So do you remember, going
back to our analogies

1584
01:35:19,400 --> 01:35:24,740
with electronics, we can
think of-- any electronics you

1585
01:35:24,740 --> 01:35:30,140
can either think of
your hi-fi amplifier

1586
01:35:30,140 --> 01:35:32,610
as being like a
black box, don't you?

1587
01:35:32,610 --> 01:35:37,090
And you put in something
and you get out something.

1588
01:35:37,090 --> 01:35:39,690
And you can either
think of what happens

1589
01:35:39,690 --> 01:35:45,040
was inside there as being like
an h of t, an impulse response.

1590
01:35:45,040 --> 01:35:50,770
Or you can think of it as
being a frequency response.

1591
01:35:50,770 --> 01:35:54,670
So there are two
different ways of thinking

1592
01:35:54,670 --> 01:35:56,020
about this black box.

1593
01:35:56,020 --> 01:35:58,240
The same is true in this case.

1594
01:35:58,240 --> 01:36:00,040
So up until now,
we've been thinking

1595
01:36:00,040 --> 01:36:04,420
of this in terms of the
frequency response, what

1596
01:36:04,420 --> 01:36:06,990
frequencies get into the image.

1597
01:36:06,990 --> 01:36:09,370
And now we're going to think
of what that means in terms

1598
01:36:09,370 --> 01:36:11,055
of the point spread function.

1599
01:36:11,055 --> 01:36:13,180
We've come across the point
spread function before,

1600
01:36:13,180 --> 01:36:14,890
of course.

1601
01:36:14,890 --> 01:36:20,820
So the point spread
function is basically

1602
01:36:20,820 --> 01:36:24,660
the image of a point object.

1603
01:36:24,660 --> 01:36:27,780
And our point objects, we
make by having a screen

1604
01:36:27,780 --> 01:36:30,300
with a very small hole in.

1605
01:36:30,300 --> 01:36:32,445
So this, then, is
going to expand.

1606
01:36:36,220 --> 01:36:38,200
It reached the lens.

1607
01:36:38,200 --> 01:36:40,310
It's going to be
collimated because we

1608
01:36:40,310 --> 01:36:43,180
were seeing this on the axis.

1609
01:36:43,180 --> 01:36:50,210
This collimated beam is going
to arrive at our pupil mask.

1610
01:36:50,210 --> 01:36:51,890
Part of this collimated
beam is going

1611
01:36:51,890 --> 01:36:53,500
to go through the aperture.

1612
01:36:53,500 --> 01:36:55,700
It's going to get to a lens.

1613
01:36:55,700 --> 01:36:58,420
It's then going
to be focused down

1614
01:36:58,420 --> 01:37:06,540
and produce a converging wave
to form a spot on the screen.

1615
01:37:06,540 --> 01:37:09,110
So this is what we think of as--

1616
01:37:09,110 --> 01:37:12,230
the final spot is going to be
what we call the point spread

1617
01:37:12,230 --> 01:37:14,947
function h of x, y.

1618
01:37:18,930 --> 01:37:24,610
OK, so how do we work out
this point spread function?

1619
01:37:24,610 --> 01:37:28,440
So what we do is, we can
again work through this system

1620
01:37:28,440 --> 01:37:30,450
in terms of Fourier transforms.

1621
01:37:30,450 --> 01:37:32,590
This is a delta function.

1622
01:37:32,590 --> 01:37:35,160
So the Fourier transform
of a delta function

1623
01:37:35,160 --> 01:37:37,020
is just a constant.

1624
01:37:37,020 --> 01:37:42,600
We multiply the constant
by the pupil mask.

1625
01:37:42,600 --> 01:37:47,080
And then we've got
another system here,

1626
01:37:47,080 --> 01:37:50,070
which does another
Fourier transform.

1627
01:37:50,070 --> 01:37:54,930
So what we're going to get
in the final plane, the image

1628
01:37:54,930 --> 01:37:57,390
of a point object, is
going to be the Fourier

1629
01:37:57,390 --> 01:37:59,235
transform of the pupil mask.

1630
01:38:02,070 --> 01:38:04,710
And yeah, so it
makes the point here

1631
01:38:04,710 --> 01:38:08,520
that we have to be careful, of
course, how we scale things.

1632
01:38:08,520 --> 01:38:10,980
And of course, the
scaling is going

1633
01:38:10,980 --> 01:38:14,898
to depend on the relative
sizes of these f1 and f2

1634
01:38:14,898 --> 01:38:15,690
as well, of course.

1635
01:38:21,150 --> 01:38:25,620
So, example-- if we got
the low pass filter,

1636
01:38:25,620 --> 01:38:27,630
so 0 is in the center.

1637
01:38:27,630 --> 01:38:29,940
It lets through the
low frequencies,

1638
01:38:29,940 --> 01:38:33,450
stops the outer frequencies.

1639
01:38:33,450 --> 01:38:37,740
So this is the pupil mask.

1640
01:38:37,740 --> 01:38:43,490
What does the point spread
function of that give?

1641
01:38:43,490 --> 01:38:51,530
And so we're assuming that
the input transparency

1642
01:38:51,530 --> 01:38:54,540
is delta x on the axis.

1643
01:38:54,540 --> 01:38:57,460
So this is illuminated on-axis.

1644
01:38:57,460 --> 01:39:02,920
The field after the pupil
mask is just the same

1645
01:39:02,920 --> 01:39:05,670
as the pupil mask.

1646
01:39:05,670 --> 01:39:08,480
And then we Fourier
transform that,

1647
01:39:08,480 --> 01:39:11,240
and it's going to give a sinc.

1648
01:39:11,240 --> 01:39:14,800
That the Fourier
transform of rect is sinc.

1649
01:39:14,800 --> 01:39:18,650
And here, we've got the
relative sizes of this

1650
01:39:18,650 --> 01:39:21,590
as well, so that we
know how big it is.

1651
01:39:24,710 --> 01:39:29,000
And so that the amplitude
point spread function, then,

1652
01:39:29,000 --> 01:39:36,000
is this Fourier transform
rescaled with this x dashed

1653
01:39:36,000 --> 01:39:36,960
over lambda f.

1654
01:39:36,960 --> 01:39:41,280
We replace the u by the
x [INAUDIBLE] lambda f.

1655
01:39:41,280 --> 01:39:44,010
And this finally,
then, is our answer.

1656
01:40:02,980 --> 01:40:05,720
Which three times
is it saying there?

1657
01:40:05,720 --> 01:40:07,437
This three, or is that three?

1658
01:40:07,437 --> 01:40:08,520
GEORGE BARBASTATHIS: Yeah.

1659
01:40:08,520 --> 01:40:12,300
COLIN SHEPPARD: Yeah,
OK, because the energy

1660
01:40:12,300 --> 01:40:14,310
has to be conserved
by this thing.

1661
01:40:14,310 --> 01:40:16,780
So you remember,
there's this theorem

1662
01:40:16,780 --> 01:40:18,840
in Fourier transforms
again that says

1663
01:40:18,840 --> 01:40:22,560
the area under the
function squared

1664
01:40:22,560 --> 01:40:26,870
is equal to the area under
the Fourier transform squared.

1665
01:40:26,870 --> 01:40:30,470
So it's conservation of energy.

1666
01:40:30,470 --> 01:40:36,670
And so this is the amplitude
point spread function.

1667
01:40:36,670 --> 01:40:40,450
It's sinc, so it goes negative.

1668
01:40:40,450 --> 01:40:43,300
Sinc is sine pi x over pi x.

1669
01:40:43,300 --> 01:40:44,900
So it goes negative.

1670
01:40:44,900 --> 01:40:46,400
So that's why it's plotted here.

1671
01:40:46,400 --> 01:40:49,140
This is 0.

1672
01:40:49,140 --> 01:40:58,760
It wiggles up and down
above the 0 [? light. ?]

1673
01:40:58,760 --> 01:41:04,100
And so of course, if you looked
at the image of a point object,

1674
01:41:04,100 --> 01:41:07,400
you'd see the modulus
square of this.

1675
01:41:07,400 --> 01:41:12,550
Just like when we looked at
our examples of the grating,

1676
01:41:12,550 --> 01:41:15,860
we finally look at the
actual intensity, rather than

1677
01:41:15,860 --> 01:41:16,910
the amplitude.

1678
01:41:16,910 --> 01:41:19,590
The intensity, of course,
is always positive.

1679
01:41:19,590 --> 01:41:22,010
So you'd actually be looking
at the square of that.

1680
01:41:22,010 --> 01:41:26,380
But if you're trying to think
of this thing as being--

1681
01:41:26,380 --> 01:41:28,670
what we're going to carry
on to say, of course,

1682
01:41:28,670 --> 01:41:33,080
is in Fourier space,
or in Fourier domain,

1683
01:41:33,080 --> 01:41:35,830
the spatial frequencies
of the object

1684
01:41:35,830 --> 01:41:39,340
are multiplied by the mask.

1685
01:41:39,340 --> 01:41:47,570
In terms of real space, you're
going to get a convolution.

1686
01:41:47,570 --> 01:41:49,680
The product becomes
a convolution.

1687
01:41:49,680 --> 01:41:54,230
So we're going to convolve our
objects with an amplitude point

1688
01:41:54,230 --> 01:41:56,083
spread function.

1689
01:41:56,083 --> 01:41:58,250
But we have to do it with
the amplitude point spread

1690
01:41:58,250 --> 01:42:01,100
function, find the final
the total amplitude

1691
01:42:01,100 --> 01:42:04,460
in the image, and then
the modulus square

1692
01:42:04,460 --> 01:42:08,210
to find the final the intensity.

1693
01:42:08,210 --> 01:42:11,090
OK, and this is an example
of the point spread

1694
01:42:11,090 --> 01:42:14,960
function of this phase
filter we looked at.

1695
01:42:14,960 --> 01:42:19,610
The pupil mask, this is
the same as we had before.

1696
01:42:19,610 --> 01:42:21,020
So this is its modulus.

1697
01:42:21,020 --> 01:42:23,280
This is the phase.

1698
01:42:23,280 --> 01:42:27,980
And if you calculate the
point spread function of this,

1699
01:42:27,980 --> 01:42:28,745
it comes to this.

1700
01:42:28,745 --> 01:42:30,080
This is the modulus.

1701
01:42:30,080 --> 01:42:32,090
This is the phase.

1702
01:42:32,090 --> 01:42:35,810
So you can see that that is why
we got a pretty horrible image.

1703
01:42:35,810 --> 01:42:39,530
It's because it's got a very
nasty point spread function.

1704
01:42:39,530 --> 01:42:41,870
[INAUDIBLE] point spread
function, for a start,

1705
01:42:41,870 --> 01:42:44,460
it's got these very
large side lobes,

1706
01:42:44,460 --> 01:42:46,145
which normally we don't like.

1707
01:42:46,145 --> 01:42:49,070
We normally like to have
our point spread function

1708
01:42:49,070 --> 01:42:54,110
to be nicely narrow and smooth,
and without the big side lobes.

1709
01:42:54,110 --> 01:42:55,610
And then more than
that, it's also

1710
01:42:55,610 --> 01:42:59,015
got these horrible phase
jumps, which I think

1711
01:42:59,015 --> 01:43:02,780
are very likely to produce
a pretty awful image, which

1712
01:43:02,780 --> 01:43:04,890
is probably what we saw.

1713
01:43:04,890 --> 01:43:08,010
So this goes through the
sum of actually doing that.

1714
01:43:08,010 --> 01:43:11,100
So this is the same as
we had before, then--

1715
01:43:11,100 --> 01:43:13,370
the big rectangle minus
the little rectangle

1716
01:43:13,370 --> 01:43:15,140
with the phase.

1717
01:43:15,140 --> 01:43:20,070
And then we've got to do the
Fourier transform of that.

1718
01:43:20,070 --> 01:43:22,850
And so the way we've
done it here, then,

1719
01:43:22,850 --> 01:43:25,940
we just got the sum
of two rectangles.

1720
01:43:25,940 --> 01:43:28,490
Each of them becomes a sinc.

1721
01:43:28,490 --> 01:43:32,960
But you've got this i here
you have to be careful of.

1722
01:43:32,960 --> 01:43:38,510
And then finally, we work
out the modulus square

1723
01:43:38,510 --> 01:43:41,570
of the part of the
point spread function

1724
01:43:41,570 --> 01:43:44,780
and the phase of the point
spread function right so

1725
01:43:44,780 --> 01:43:48,590
this is the modulus square
of the point spread function.

1726
01:43:48,590 --> 01:43:51,170
So these side lobes are
actually quite huge--

1727
01:43:54,220 --> 01:43:55,120
quite big.

1728
01:44:03,530 --> 01:44:06,470
OK, yeah, that's just looking at
the difference between the two,

1729
01:44:06,470 --> 01:44:07,640
then.

1730
01:44:07,640 --> 01:44:12,690
So if we have our
low-pass filter--

1731
01:44:12,690 --> 01:44:15,650
so this is what I was saying
about these being huge.

1732
01:44:15,650 --> 01:44:20,380
You see the low-pass
filter, these side lobes

1733
01:44:20,380 --> 01:44:22,490
are pretty small by comparison.

1734
01:44:22,490 --> 01:44:28,160
I guess you could use some sort
of shaded aperture, or maybe

1735
01:44:28,160 --> 01:44:31,670
an aperture with some structure
that would allow you to make

1736
01:44:31,670 --> 01:44:35,870
those side lobes even smaller.

1737
01:44:35,870 --> 01:44:40,300
This is this process that's
sometimes called apodization.

1738
01:44:40,300 --> 01:44:44,410
Apodization means
to cut off the feet.

1739
01:44:44,410 --> 01:44:47,150
I think if George
was giving the talk,

1740
01:44:47,150 --> 01:44:49,100
he'd say something
about the Greek

1741
01:44:49,100 --> 01:44:52,840
for feet being something
to do with pod.

1742
01:44:52,840 --> 01:44:53,710
Right, there we are.

1743
01:44:58,690 --> 01:45:02,350
OK, we are over time, really.

1744
01:45:02,350 --> 01:45:03,510
[INAUDIBLE] stop now?

1745
01:45:03,510 --> 01:45:05,120
AUDIENCE: [INAUDIBLE]

1746
01:45:05,120 --> 01:45:07,120
COLIN SHEPPARD: Yeah,
maybe that's a good point.

1747
01:45:07,120 --> 01:45:08,340
I think that's good.

1748
01:45:08,340 --> 01:45:11,562
You've got the concept of
the point spread function.

1749
01:45:11,562 --> 01:45:13,770
Now you've got a week to
think about the point spread

1750
01:45:13,770 --> 01:45:14,400
function.

1751
01:45:14,400 --> 01:45:18,460
And then next time,
it will be very clear.

1752
01:45:18,460 --> 01:45:22,500
OK, so any questions
from either side?