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GEORGE BARBASTATHIS:
OK, good morning.
9
00:00:24,770 --> 00:00:28,250
So I'd like to pick up from
where we left off last time.
10
00:00:28,250 --> 00:00:31,850
So we were discussing
the various properties
11
00:00:31,850 --> 00:00:33,600
of the Fourier
transform and what
12
00:00:33,600 --> 00:00:37,850
they mean in terms of the
Fraunhofer diffraction
13
00:00:37,850 --> 00:00:39,032
patterns.
14
00:00:39,032 --> 00:00:41,240
So what we're doing last
time when we ran out of time
15
00:00:41,240 --> 00:00:45,300
was we were proving the
convolution theorem.
16
00:00:45,300 --> 00:00:47,090
So this is the proof here.
17
00:00:47,090 --> 00:00:49,610
And just to remind
you very briefly,
18
00:00:49,610 --> 00:00:53,240
we start with an expression
that looks like a convolution.
19
00:00:53,240 --> 00:00:55,610
We have an output
function equals an input
20
00:00:55,610 --> 00:00:58,880
function times a kernel.
21
00:00:58,880 --> 00:01:04,010
Then what we did is we wrote
each one of those, the input
22
00:01:04,010 --> 00:01:07,490
as well as the kernel, we wrote
those as Fourier transforms.
23
00:01:07,490 --> 00:01:10,490
Here they are respectively.
24
00:01:10,490 --> 00:01:16,460
Then we rearranged the integrals
and the order of integration.
25
00:01:16,460 --> 00:01:26,800
And we noticed that this
expression here is also
26
00:01:26,800 --> 00:01:28,360
known as a delta function.
27
00:01:32,050 --> 00:01:35,590
The integral of an exponential,
of a complex exponential
28
00:01:35,590 --> 00:01:38,395
from minus infinity to
infinity is a delta function.
29
00:01:40,930 --> 00:01:43,295
Oh, too bright.
30
00:01:43,295 --> 00:01:44,920
OK, I think that's
a little bit better.
31
00:01:50,950 --> 00:01:51,940
I can rewrite this.
32
00:01:51,940 --> 00:01:53,970
Let me do it one step at a time.
33
00:01:53,970 --> 00:01:57,850
So I can rewrite now
this big integral.
34
00:02:02,560 --> 00:02:05,860
Unfortunately, the black
marker has run out of steam,
35
00:02:05,860 --> 00:02:06,970
so I'll switch color.
36
00:02:32,420 --> 00:02:34,550
So since we have a double
integral with respect
37
00:02:34,550 --> 00:02:39,590
to the two frequency variables
and a delta function inside,
38
00:02:39,590 --> 00:02:43,280
the delta function will knock
out one of the integrals.
39
00:02:43,280 --> 00:02:45,385
And we simply replace the--
40
00:02:52,050 --> 00:02:54,470
well, let's pick--
we can pick whichever
41
00:02:54,470 --> 00:02:57,500
variable we want to
integrate, so let's pick u1.
42
00:02:57,500 --> 00:03:05,730
So we'll simply get du1 times
G sub in of u1 H of u1 one
43
00:03:05,730 --> 00:03:10,100
e to the i 2 pi u1 x prime.
44
00:03:10,100 --> 00:03:12,170
And now we realize and
remember what this is.
45
00:03:12,170 --> 00:03:15,230
If you go back to the
beginning of the derivation,
46
00:03:15,230 --> 00:03:18,060
this is the outcome
of the convolution.
47
00:03:18,060 --> 00:03:20,120
So this is actually-- let
me write it out again.
48
00:03:20,120 --> 00:03:24,290
This is G sub out of x prime.
49
00:03:27,950 --> 00:03:30,230
What we've got now is
a Fourier transform.
50
00:03:30,230 --> 00:03:33,638
Because, you see, here is
the Fourier transform kernel.
51
00:03:33,638 --> 00:03:35,180
Actually-- I'm
sorry-- what we've got
52
00:03:35,180 --> 00:03:36,730
is a Fourier integral.
53
00:03:36,730 --> 00:03:40,400
We've got to write the
outcome of the convolution
54
00:03:40,400 --> 00:03:42,170
as a Fourier
integral, where this
55
00:03:42,170 --> 00:03:46,180
is the Fourier integral
kernel or the inverse Fourier
56
00:03:46,180 --> 00:03:47,480
transform kernel.
57
00:03:47,480 --> 00:03:49,680
The two terms mean
the same thing.
58
00:03:49,680 --> 00:03:52,580
And this is the actual
inverse Fourier transform.
59
00:03:52,580 --> 00:03:56,090
So, therefore, this result
is equivalent to G sub
60
00:03:56,090 --> 00:04:04,070
out of u equals G sub
in of u times H of u.
61
00:04:04,070 --> 00:04:06,860
So this is, then, the proof of
the convolution theorem, which
62
00:04:06,860 --> 00:04:13,700
says that if two
functions are related
63
00:04:13,700 --> 00:04:16,970
as a convolution
with a kernel, then
64
00:04:16,970 --> 00:04:20,390
the equivalent relationship
in the Fourier domain
65
00:04:20,390 --> 00:04:23,570
is actually a product of the
Fourier transform of the input
66
00:04:23,570 --> 00:04:26,990
function times the Fourier
transform of the kernel.
67
00:04:26,990 --> 00:04:31,340
So this mathematical-- this
analytical result is what
68
00:04:31,340 --> 00:04:34,440
you've seen in the simulations
that I have on this screen
69
00:04:34,440 --> 00:04:37,820
here, where the two--
70
00:04:37,820 --> 00:04:39,320
I will do this in one dimension.
71
00:04:39,320 --> 00:04:41,240
The simulations
are obviously 2D,
72
00:04:41,240 --> 00:04:45,390
but it's a little bit easier
to do everything in 1D here
73
00:04:45,390 --> 00:04:48,360
so we don't spend too
much time writing.
74
00:04:48,360 --> 00:04:53,300
So the two functions that
we have here are actually--
75
00:04:53,300 --> 00:04:56,130
one of them is a sinusoid.
76
00:04:56,130 --> 00:04:58,025
So if I have a sinusoid--
77
00:05:00,930 --> 00:05:04,980
I keep using the blue
marker, [INAUDIBLE]..
78
00:05:04,980 --> 00:05:06,275
So the sinusoid is--
79
00:05:10,980 --> 00:05:15,020
let me just write it
with full contrast
80
00:05:15,020 --> 00:05:16,960
to save some writing here.
81
00:05:20,490 --> 00:05:22,470
This is a sinusoid.
82
00:05:22,470 --> 00:05:25,110
Then it's Fourier
transform will actually--
83
00:05:25,110 --> 00:05:27,480
as we have said
several times, it
84
00:05:27,480 --> 00:05:29,430
will consist of three
delta functions.
85
00:05:52,750 --> 00:05:56,130
Then we have another function
which looks like a rectangle.
86
00:05:56,130 --> 00:05:58,720
That's on the right-hand
side, top right.
87
00:05:58,720 --> 00:06:00,470
And we've already seen this one.
88
00:06:00,470 --> 00:06:03,880
So the rectangle is a
rectangular function.
89
00:06:03,880 --> 00:06:10,750
If it has a width, let's say,
a, then the Fourier transform
90
00:06:10,750 --> 00:06:11,865
will look like this.
91
00:06:20,992 --> 00:06:22,450
And we call this
the sinc function.
92
00:06:27,860 --> 00:06:33,230
So the height of this
function is a by virtue
93
00:06:33,230 --> 00:06:35,270
of the scaling theorem.
94
00:06:35,270 --> 00:06:41,690
And then the nulls, they go
like 1/a, 2/a, and so on.
95
00:06:41,690 --> 00:06:43,970
And then, symmetrically,
on the negative side,
96
00:06:43,970 --> 00:06:48,750
minus 1/a, minus 2/a,
and so on and so forth.
97
00:06:48,750 --> 00:06:51,380
So these are two Fourier
transform relationships.
98
00:06:51,380 --> 00:06:54,800
Now, what we see
on the bottom side
99
00:06:54,800 --> 00:06:59,450
is basically a grating
which is truncated.
100
00:06:59,450 --> 00:07:02,320
So I take my infinite grating.
101
00:07:09,040 --> 00:07:11,480
It continues on and on.
102
00:07:11,480 --> 00:07:12,590
And then I truncate it.
103
00:07:17,800 --> 00:07:19,260
That is, I multiply--
104
00:07:32,924 --> 00:07:38,790
I multiply it by a rectangular
function which sets the side--
105
00:07:38,790 --> 00:07:43,510
which sets the aperture
of the truncation.
106
00:07:43,510 --> 00:07:47,370
So the way I've written it,
the width of this boxcar
107
00:07:47,370 --> 00:07:54,190
would be a, and the period
of the grating itself
108
00:07:54,190 --> 00:07:55,890
would be uppercase lambda.
109
00:07:55,890 --> 00:07:57,640
So, of course, we can
go ahead and compute
110
00:07:57,640 --> 00:08:00,317
this Fourier transform
analytically,
111
00:08:00,317 --> 00:08:01,650
but that would be quite painful.
112
00:08:01,650 --> 00:08:05,420
We'd have to do a lot of
changes of variable and so on.
113
00:08:05,420 --> 00:08:09,250
However, by virtue of
the convolution theorem,
114
00:08:09,250 --> 00:08:11,620
because this is a product,
this Fourier transform
115
00:08:11,620 --> 00:08:13,520
will be a convolution.
116
00:08:13,520 --> 00:08:15,700
So, basically, what we
have to do in order--
117
00:08:15,700 --> 00:08:23,480
when we multiply these two
results in this domain,
118
00:08:23,480 --> 00:08:27,650
in this domain we
have to convolve them.
119
00:08:27,650 --> 00:08:31,130
So what is now-- so
the assumption here
120
00:08:31,130 --> 00:08:34,340
is that the size of the
boxcar is actually bigger
121
00:08:34,340 --> 00:08:35,720
by the size of the period.
122
00:08:35,720 --> 00:08:37,250
It does not have
to be much bigger,
123
00:08:37,250 --> 00:08:40,309
but it's nice if it
is bigger for what I'm
124
00:08:40,309 --> 00:08:42,000
about to draw to be correct.
125
00:08:42,000 --> 00:08:46,220
So what we get if we can
convolve this pattern
126
00:08:46,220 --> 00:08:50,210
with the three delta
functions, each delta
127
00:08:50,210 --> 00:08:53,610
will produce a replica
of this function centered
128
00:08:53,610 --> 00:08:55,610
at the location of
the delta function.
129
00:08:55,610 --> 00:08:57,590
So, therefore, we'll
get three replicas
130
00:08:57,590 --> 00:09:01,550
of this sinc pattern centered
at the corresponding locations.
131
00:09:01,550 --> 00:09:03,650
So let me draw this a
little bit carefully here.
132
00:09:10,050 --> 00:09:11,900
So I'll try to be as
accurate as I can.
133
00:09:17,830 --> 00:09:18,630
OK.
134
00:09:18,630 --> 00:09:21,450
So here are the two--
135
00:09:21,450 --> 00:09:22,960
well, there's three
delta functions.
136
00:09:22,960 --> 00:09:24,150
One is at the origin.
137
00:09:24,150 --> 00:09:27,860
One is at minus 1 up on
lambda, and the other
138
00:09:27,860 --> 00:09:30,000
is at plus 1 up on lambda.
139
00:09:30,000 --> 00:09:32,400
And then, on each
one of those, I'm
140
00:09:32,400 --> 00:09:35,380
going to center one
of the sinc functions.
141
00:09:35,380 --> 00:09:42,410
So here is one sinc
function, and here
142
00:09:42,410 --> 00:09:50,160
is another sinc function,
and another one.
143
00:09:55,500 --> 00:09:57,360
So I tried to do it carefully.
144
00:09:57,360 --> 00:09:58,385
Why is this taller?
145
00:09:58,385 --> 00:09:59,760
Because, if you
recall, the delta
146
00:09:59,760 --> 00:10:02,130
function that was at
the origin is actually
147
00:10:02,130 --> 00:10:05,590
twice the size of the
other two delta functions.
148
00:10:05,590 --> 00:10:10,170
So the size of this one
would be a/2 actually,
149
00:10:10,170 --> 00:10:11,610
because, if you
recall, we're also
150
00:10:11,610 --> 00:10:15,810
picking up one factor of
a from the sinc itself.
151
00:10:15,810 --> 00:10:20,490
Then the size of this would
be a/4, and the size of this
152
00:10:20,490 --> 00:10:23,070
would also be a/4.
153
00:10:23,070 --> 00:10:26,580
When I say the size,
I mean the height.
154
00:10:26,580 --> 00:10:28,350
And then where are
the nulls located?
155
00:10:28,350 --> 00:10:29,830
Well, these nulls are simple.
156
00:10:29,830 --> 00:10:34,370
They're still at 1/a, minus 1/a.
157
00:10:34,370 --> 00:10:35,340
Where is this null?
158
00:10:35,340 --> 00:10:39,885
For example, this null would
be at 1 over lambda plus 1/a.
159
00:10:39,885 --> 00:10:42,990
And then I have another
null over here at 1
160
00:10:42,990 --> 00:10:46,900
over lambda minus 1/a,
and so on and so forth.
161
00:10:46,900 --> 00:10:50,670
And then I have more nulls
at 1 over lambda minus 2/a,
162
00:10:50,670 --> 00:10:53,850
minus 3/a, and so
on and so forth.
163
00:10:53,850 --> 00:10:55,230
And now my assumption
that I have
164
00:10:55,230 --> 00:10:57,710
several periods of the
grating with the boxcar,
165
00:10:57,710 --> 00:11:00,150
you can see it is quite
convenient because I can draw
166
00:11:00,150 --> 00:11:03,420
these sinc functions and
I can kind of ignore what
167
00:11:03,420 --> 00:11:05,550
happens in between over here.
168
00:11:05,550 --> 00:11:07,260
As you can imagine,
something complicated
169
00:11:07,260 --> 00:11:10,650
will happen over here because
the sincs are actually
170
00:11:10,650 --> 00:11:14,190
adding coherently, so
something funky will happen.
171
00:11:14,190 --> 00:11:16,170
But if they're
pretty far apart, you
172
00:11:16,170 --> 00:11:18,240
can see that the envelope
of the sinc function
173
00:11:18,240 --> 00:11:19,620
actually decays quite fast.
174
00:11:19,620 --> 00:11:22,640
So I can more or less ignore
what is happening over here,
175
00:11:22,640 --> 00:11:25,460
and I can simply draw
the sinc function.
176
00:11:25,460 --> 00:11:28,260
Of course, this is because
I'm doing it by hand.
177
00:11:28,260 --> 00:11:31,920
If I use a computational tool,
such as MATLAB or Mathematical,
178
00:11:31,920 --> 00:11:34,780
it will do it for me.
179
00:11:34,780 --> 00:11:36,730
So this is the
one-dimensional calculation,
180
00:11:36,730 --> 00:11:38,660
that is easier to do by hand.
181
00:11:38,660 --> 00:11:41,420
As a bonus, I think you
guys have already computed--
182
00:11:41,420 --> 00:11:43,150
or, if you haven't
already, you will
183
00:11:43,150 --> 00:11:46,300
compute some time
between Tuesday midnight
184
00:11:46,300 --> 00:11:48,280
and Wednesday at 8:00 AM.
185
00:11:48,280 --> 00:11:53,380
Some time you will compute
the same convolution
186
00:11:53,380 --> 00:11:55,640
for the two-dimensional case.
187
00:11:55,640 --> 00:11:57,350
And you can see the result here.
188
00:11:57,350 --> 00:11:59,330
So, in effect, this
saves you from--
189
00:12:02,290 --> 00:12:04,880
well, at least you know if your
result is correct or wrong.
190
00:12:04,880 --> 00:12:07,090
If you derive something
that looks like this,
191
00:12:07,090 --> 00:12:09,140
then you know you've
done correctly.
192
00:12:09,140 --> 00:12:10,630
So you see what's happened.
193
00:12:10,630 --> 00:12:12,840
If you had the infinite
grating, the infinite grating
194
00:12:12,840 --> 00:12:15,850
gives you three
delta functions whose
195
00:12:15,850 --> 00:12:19,690
axis is kind of perpendicular
to the fringes of the grating,
196
00:12:19,690 --> 00:12:22,270
and the one that the
origin is stronger.
197
00:12:22,270 --> 00:12:23,950
Of course, the
rectangular function
198
00:12:23,950 --> 00:12:25,768
produces a sinc pattern.
199
00:12:25,768 --> 00:12:27,310
And then when you
convolve them, when
200
00:12:27,310 --> 00:12:35,600
you have a tilted grating now
and multiply it by a rect,
201
00:12:35,600 --> 00:12:39,290
then you see that you get
three sinc patterns oriented
202
00:12:39,290 --> 00:12:42,230
along the same axis, which is
perpendicular to the fringes
203
00:12:42,230 --> 00:12:43,310
of the grating.
204
00:12:43,310 --> 00:12:45,290
But the sinc pattern
itself is actually
205
00:12:45,290 --> 00:12:48,740
perpendicular to the
rectangular aperture.
206
00:12:48,740 --> 00:12:51,110
So this is what the
convolution theorem says.
207
00:12:51,110 --> 00:12:53,670
And, of course, you
can turn it around.
208
00:12:53,670 --> 00:12:57,050
You can actually turn
the grating around
209
00:12:57,050 --> 00:12:58,730
and rotate the aperture.
210
00:12:58,730 --> 00:13:02,990
In this case now, again, you
will see the sincs oriented
211
00:13:02,990 --> 00:13:06,710
parallel to the fringes again,
but now the rectangular pattern
212
00:13:06,710 --> 00:13:09,860
itself is rotated, and therefore
the sinc patterns themselves
213
00:13:09,860 --> 00:13:12,730
are rotated.
214
00:13:12,730 --> 00:13:15,985
So the reason we use these
properties is because--
215
00:13:15,985 --> 00:13:17,360
again, as you can
imagine, if you
216
00:13:17,360 --> 00:13:19,700
were to write this as
an explicit integral,
217
00:13:19,700 --> 00:13:21,410
it can be done and
you would-- of course
218
00:13:21,410 --> 00:13:23,300
you would still get
the correct result,
219
00:13:23,300 --> 00:13:25,220
but it would be quite
painful to compute.
220
00:13:30,447 --> 00:13:32,030
So now that we have
all these results,
221
00:13:32,030 --> 00:13:33,790
we can actually apply them.
222
00:13:33,790 --> 00:13:35,670
So far, these were
mathematically results.
223
00:13:35,670 --> 00:13:38,700
I could pretend
I'm teaching 18085,
224
00:13:38,700 --> 00:13:41,660
or whatever it is that
you learn those things.
225
00:13:41,660 --> 00:13:45,420
But this is also-- of course,
they actually become Fraunhofer
226
00:13:45,420 --> 00:13:49,200
diffraction patterns if you
simply perform a scale--
227
00:13:49,200 --> 00:13:51,420
a coordinate change.
228
00:13:51,420 --> 00:13:54,820
So the Fourier transforms,
we compute them with respect
229
00:13:54,820 --> 00:13:59,730
to the frequency variables,
u and v we call them.
230
00:13:59,730 --> 00:14:02,640
Goodman calls them
F sub x, F sub y,
231
00:14:02,640 --> 00:14:05,010
but it is actually
the same variable.
232
00:14:05,010 --> 00:14:08,520
Well, if you substitute
a special variable
233
00:14:08,520 --> 00:14:11,093
and you multiply by the
scaling factor lambda z,
234
00:14:11,093 --> 00:14:13,260
that we saw the derivation
in the previous lecture--
235
00:14:13,260 --> 00:14:14,550
I'm not going to do it again--
236
00:14:14,550 --> 00:14:17,330
then you actually get the
Fraunhofer diffraction pattern.
237
00:14:17,330 --> 00:14:19,200
So it's a simple
scaling argument.
238
00:14:19,200 --> 00:14:21,630
And you can tell that--
239
00:14:21,630 --> 00:14:23,540
well, at least it is
plausibly correct.
240
00:14:23,540 --> 00:14:27,080
Here, the units are
correct, because u and v
241
00:14:27,080 --> 00:14:28,500
are frequency variables.
242
00:14:28,500 --> 00:14:31,300
So, therefore, their
units are inverse meters.
243
00:14:31,300 --> 00:14:33,260
And then multiply
it by meter squared.
244
00:14:33,260 --> 00:14:35,490
The wavelength is meter,
the distance is meter.
245
00:14:35,490 --> 00:14:37,380
So, therefore, I get meters.
246
00:14:37,380 --> 00:14:40,380
So the units are correct.
247
00:14:40,380 --> 00:14:42,640
And this is one
example that we did.
248
00:14:42,640 --> 00:14:45,920
Then here's another
one where I shrink
249
00:14:45,920 --> 00:14:47,240
the rectangular aperture.
250
00:14:47,240 --> 00:14:50,130
And, of course, the Fraunhofer
diffraction pattern will grow.
251
00:14:50,130 --> 00:14:54,060
We call this a similarity
or scaling theorem.
252
00:14:54,060 --> 00:14:57,090
Then I can, for example,
use the shift theorem
253
00:14:57,090 --> 00:15:00,110
to calculate the
Fraunhofer diffraction
254
00:15:00,110 --> 00:15:03,150
pattern from three rectangles.
255
00:15:03,150 --> 00:15:04,710
As we discussed
the last time, this
256
00:15:04,710 --> 00:15:07,960
will give rise to a sinusoidal
modulation in the Fraunhofer
257
00:15:07,960 --> 00:15:08,850
domain.
258
00:15:08,850 --> 00:15:11,460
And you can also do
the convolution theorem
259
00:15:11,460 --> 00:15:13,362
in this case with a
truncated aperture,
260
00:15:13,362 --> 00:15:14,320
and so on and so forth.
261
00:15:20,900 --> 00:15:26,360
So this basically concludes the
discussion of the Fraunhofer
262
00:15:26,360 --> 00:15:29,100
diffraction pattern.
263
00:15:29,100 --> 00:15:34,640
But since we've been
discussing Fourier transforms,
264
00:15:34,640 --> 00:15:37,880
there's another very
basic property of Fourier
265
00:15:37,880 --> 00:15:41,600
transforms that I would
like to introduce here,
266
00:15:41,600 --> 00:15:44,660
and then we will see it in
full glory for the next two
267
00:15:44,660 --> 00:15:49,010
lectures, and that
is spatial filtering.
268
00:15:49,010 --> 00:15:51,380
So spatial filtering is
basically the following.
269
00:15:51,380 --> 00:15:54,070
It says if you go
to this Fraunhofer
270
00:15:54,070 --> 00:15:57,440
domain, or, in general,
in the transform domain--
271
00:15:57,440 --> 00:15:59,120
which, we will see
a little bit later,
272
00:15:59,120 --> 00:16:01,940
that we don't need to have
to go very far actually.
273
00:16:01,940 --> 00:16:04,970
By using a lens, we can produce
a Fraunhofer diffraction
274
00:16:04,970 --> 00:16:07,270
pattern at the back
focal plane of the lens.
275
00:16:07,270 --> 00:16:08,870
That's very convenient.
276
00:16:08,870 --> 00:16:11,720
But if I go here and I
do some modification,
277
00:16:11,720 --> 00:16:14,480
and then take another Fourier
transform, then, of course,
278
00:16:14,480 --> 00:16:16,580
the signal I reconstruct
is not identical
279
00:16:16,580 --> 00:16:18,260
to my original
signal, but it will
280
00:16:18,260 --> 00:16:21,290
have changed because I've
modified the frequency
281
00:16:21,290 --> 00:16:22,560
spectrum.
282
00:16:22,560 --> 00:16:24,960
So this is called
spatial filtering.
283
00:16:24,960 --> 00:16:27,720
So here's an example
that I have constructed.
284
00:16:27,720 --> 00:16:31,140
So in this case, I've
contacted the signal
285
00:16:31,140 --> 00:16:34,690
in the space domain that
looks like three sinusoids.
286
00:16:34,690 --> 00:16:38,150
Now, you cannot tell very
clearly from this pattern that
287
00:16:38,150 --> 00:16:41,140
you have three sinusoids, but if
you take the Fourier transform,
288
00:16:41,140 --> 00:16:44,090
then you see three
spots here, three dots.
289
00:16:44,090 --> 00:16:45,740
Actually, you see six.
290
00:16:45,740 --> 00:16:48,410
But you recall that each
sinusoid corresponds
291
00:16:48,410 --> 00:16:49,850
to two dots.
292
00:16:49,850 --> 00:16:52,130
So the conjugate dots here,
this one and this one,
293
00:16:52,130 --> 00:16:53,330
they are one sinusoid.
294
00:16:53,330 --> 00:16:55,040
This and this one,
another sinusoid.
295
00:16:55,040 --> 00:16:57,860
This and this one are
yet another sinusoid.
296
00:16:57,860 --> 00:17:01,010
So spatial filtering, a
very simple occurrence
297
00:17:01,010 --> 00:17:04,550
of spatial filtering is what
happens if, for example, you
298
00:17:04,550 --> 00:17:09,369
go in with some black
marker or some opaque screen
299
00:17:09,369 --> 00:17:12,859
in the case of optics and
you remove one of these dots.
300
00:17:12,859 --> 00:17:15,770
If you do that in the transform
domain, then you will see it.
301
00:17:15,770 --> 00:17:18,300
Now, watch as I
transition the slide.
302
00:17:18,300 --> 00:17:21,139
You will see that the
spatial pattern also changes.
303
00:17:24,079 --> 00:17:24,829
OK.
304
00:17:24,829 --> 00:17:27,920
So now it becomes
kind of horizontal,
305
00:17:27,920 --> 00:17:30,950
and it is horizontal because
the two dominant-- the two
306
00:17:30,950 --> 00:17:35,340
dominant sinusoids are actually
along the horizontal axis.
307
00:17:35,340 --> 00:17:36,810
So, therefore,
your grooves are--
308
00:17:36,810 --> 00:17:37,310
I'm sorry.
309
00:17:37,310 --> 00:17:40,330
Your grooves are vertical
because the two dots here
310
00:17:40,330 --> 00:17:42,980
are on the horizontal axis.
311
00:17:42,980 --> 00:17:44,840
But there's also
a weaker sinusoid
312
00:17:44,840 --> 00:17:47,930
that gives rise to
these weak diagonal
313
00:17:47,930 --> 00:17:49,730
fringes that you see over here.
314
00:17:49,730 --> 00:17:51,230
But, basically, you
can see that one
315
00:17:51,230 --> 00:17:53,820
of the three spatial
frequencies has vanished here.
316
00:17:53,820 --> 00:17:57,740
So this is the simplest
case of spatial filtering.
317
00:17:57,740 --> 00:17:59,640
And, of course, you
can generalize it.
318
00:17:59,640 --> 00:18:02,710
Here's again the Red Sox--
319
00:18:02,710 --> 00:18:07,430
or I should say the GO SOX
pattern on the Boston high rise
320
00:18:07,430 --> 00:18:09,110
that I showed last time.
321
00:18:09,110 --> 00:18:16,070
And this is the spatial
frequency representation
322
00:18:16,070 --> 00:18:18,320
or the Fourier transform
of this pattern.
323
00:18:18,320 --> 00:18:20,010
And then we can apply
various filters.
324
00:18:20,010 --> 00:18:21,980
For example, if I
go with a filter
325
00:18:21,980 --> 00:18:24,560
and I block all the
high frequencies,
326
00:18:24,560 --> 00:18:27,260
then you can see that my
pattern appears blurred.
327
00:18:27,260 --> 00:18:28,910
In fact, it is
more than blurred.
328
00:18:28,910 --> 00:18:33,050
The windows have kind of
disappeared of the building.
329
00:18:33,050 --> 00:18:35,730
And that is because the
windows, if I go back,
330
00:18:35,730 --> 00:18:36,970
you will see that the--
331
00:18:36,970 --> 00:18:37,730
hi, Colin.
332
00:18:37,730 --> 00:18:38,530
You're back.
333
00:18:38,530 --> 00:18:39,410
COLIN: Yes.
334
00:18:39,410 --> 00:18:40,160
GEORGE BARBASTATHIS: Welcome.
335
00:18:40,160 --> 00:18:41,077
COLIN: Sorry I'm late.
336
00:18:41,077 --> 00:18:42,932
GEORGE BARBASTATHIS:
Oh, no problem.
337
00:18:42,932 --> 00:18:44,390
I thought you were
still in Poland.
338
00:18:44,390 --> 00:18:45,480
COLIN: No, I got back.
339
00:18:45,480 --> 00:18:46,480
GEORGE BARBASTATHIS: Oh.
340
00:18:46,480 --> 00:18:48,920
Oh, OK, welcome back.
341
00:18:48,920 --> 00:18:51,530
And so the windows,
if you recall,
342
00:18:51,530 --> 00:18:55,740
the windows are kind of
periodic in this high rise here.
343
00:18:55,740 --> 00:18:59,210
So they correspond to these
dots in the frequency domain,
344
00:18:59,210 --> 00:19:01,160
kind of like delta functions.
345
00:19:01,160 --> 00:19:04,670
And because, in this case,
I have blocked the dots,
346
00:19:04,670 --> 00:19:08,898
you see that the windows
disappear from the high rise.
347
00:19:08,898 --> 00:19:11,190
And, of course, you can do
other funky kind of filters.
348
00:19:11,190 --> 00:19:13,290
This is called a
band pass filter.
349
00:19:13,290 --> 00:19:15,200
And, in this case,
the windows reappear
350
00:19:15,200 --> 00:19:19,280
because now I center this
doughnut, this annulus.
351
00:19:19,280 --> 00:19:22,940
I centered it so that some of
these dots in the frequency
352
00:19:22,940 --> 00:19:24,210
domain, they survive.
353
00:19:24,210 --> 00:19:26,480
And you can see that,
of course, the--
354
00:19:26,480 --> 00:19:28,715
well, it is not
fully reconstructed,
355
00:19:28,715 --> 00:19:30,090
the original
building, of course,
356
00:19:30,090 --> 00:19:32,480
because there's still
spatial frequencies missing.
357
00:19:32,480 --> 00:19:35,810
But you can see that the pattern
of windows of the high rise
358
00:19:35,810 --> 00:19:37,610
has kind of reappeared.
359
00:19:37,610 --> 00:19:40,340
Now, what happened
to the sign GO SOX?
360
00:19:40,340 --> 00:19:43,100
It vanished, and it vanished
because, in this case,
361
00:19:43,100 --> 00:19:45,440
I have blocked the
lower frequencies.
362
00:19:45,440 --> 00:19:50,970
The GO SOX sign, it has
survived the low pass filter
363
00:19:50,970 --> 00:19:55,440
because this is a relatively
slow-varying signal, right?
364
00:19:55,440 --> 00:20:00,930
So its frequency content, you
expect it to be centered--
365
00:20:00,930 --> 00:20:03,960
I'm sorry-- to be concentrated
near the center of the Fourier
366
00:20:03,960 --> 00:20:05,580
domain where the
frequencies are low.
367
00:20:05,580 --> 00:20:08,550
When we do the band pass
filter, the GO SOX vanishes.
368
00:20:11,480 --> 00:20:14,030
And that is because I blocked
the low frequencies where
369
00:20:14,030 --> 00:20:16,610
this signal was represented.
370
00:20:16,610 --> 00:20:20,750
So you can see that you can do
quite interesting manipulations
371
00:20:20,750 --> 00:20:24,620
on images using this concept
of spatial frequency.
372
00:20:24,620 --> 00:20:29,480
And, actually, the GO SOX
signal has not quite vanished.
373
00:20:29,480 --> 00:20:33,470
If you look carefully, there is
a little bit of evidence of it
374
00:20:33,470 --> 00:20:35,070
here, but it's
quite hard to see.
375
00:20:35,070 --> 00:20:37,070
And, of course, there's
a little bit of evidence
376
00:20:37,070 --> 00:20:41,063
because there is a little bit
of the frequency content leaking
377
00:20:41,063 --> 00:20:42,480
into the intermediate
frequencies.
378
00:20:42,480 --> 00:20:44,330
So, therefore, some
of it has survived,
379
00:20:44,330 --> 00:20:46,280
but mostly it is gone.
380
00:20:46,280 --> 00:20:48,080
So that is the--
381
00:20:48,080 --> 00:20:52,130
so it is not a perfect filter,
but it works quite well.
382
00:20:52,130 --> 00:20:54,650
Of course, the other thing
that vanished is the average.
383
00:20:54,650 --> 00:20:56,640
You can see that
the sky, that used
384
00:20:56,640 --> 00:20:59,750
to be kind of an average
gray, it is gone also.
385
00:20:59,750 --> 00:21:02,120
Because the average--
of course, the average
386
00:21:02,120 --> 00:21:05,610
is presented at the zero spatial
frequency, and I blocked it.
387
00:21:05,610 --> 00:21:08,180
So, therefore, the
average is gone.
388
00:21:08,180 --> 00:21:13,080
So this is called
a spatial filter.
389
00:21:15,860 --> 00:21:17,240
OK.
390
00:21:17,240 --> 00:21:20,630
So we're still one
lecture behind.
391
00:21:20,630 --> 00:21:25,040
So this is what I was supposed
to have done last Wednesday.
392
00:21:25,040 --> 00:21:29,800
And if you look ahead, there is
some discussion of the transfer
393
00:21:29,800 --> 00:21:31,750
function of a
Fresnel propagation,
394
00:21:31,750 --> 00:21:34,570
and then something
called the Talbot effect.
395
00:21:34,570 --> 00:21:36,370
So I will not do this right now.
396
00:21:36,370 --> 00:21:39,115
I will postpone it, if
I may, for next week.
397
00:21:39,115 --> 00:21:40,660
What I would like
to do is I would
398
00:21:40,660 --> 00:21:45,740
like to switch to the lecture
that I posted today online,
399
00:21:45,740 --> 00:21:48,550
and that is Lecture 10A.
400
00:22:00,995 --> 00:22:02,120
The reason I'm doing that--
401
00:22:02,120 --> 00:22:04,280
I will go back and talk
about the Talbot effect.
402
00:22:04,280 --> 00:22:05,150
Don't worry.
403
00:22:05,150 --> 00:22:06,980
But the reason
I'm doing that now
404
00:22:06,980 --> 00:22:09,260
is because I would
like to press on
405
00:22:09,260 --> 00:22:11,390
with the concept of
spatial frequencies
406
00:22:11,390 --> 00:22:13,100
and spatial filtering.
407
00:22:13,100 --> 00:22:15,770
Because it is quite an important
one, and I think the sooner
408
00:22:15,770 --> 00:22:17,340
you learn it, the better.
409
00:22:17,340 --> 00:22:20,870
Talbot effect, well
you can learn later.
410
00:22:20,870 --> 00:22:24,140
But this business of spatial
filtering, in my experience,
411
00:22:24,140 --> 00:22:26,150
it takes quite a bit
of time to digest,
412
00:22:26,150 --> 00:22:29,202
so I would like to do it
sooner rather than later.
413
00:22:29,202 --> 00:22:30,410
So I already alluded to that.
414
00:22:30,410 --> 00:22:34,090
I said that this Fraunhofer
diffraction pattern
415
00:22:34,090 --> 00:22:36,320
is a Fourier
transform, but we don't
416
00:22:36,320 --> 00:22:39,320
have to go to infinity to watch
the Fraunhofer diffraction
417
00:22:39,320 --> 00:22:42,740
pattern if we want to generate
a Fourier transform optically.
418
00:22:42,740 --> 00:22:44,887
We can also do it
by using a lens.
419
00:22:44,887 --> 00:22:46,220
So this is what I will do today.
420
00:22:46,220 --> 00:22:48,410
For the rest of the
lecture, I will show you
421
00:22:48,410 --> 00:22:53,330
how a lens can produce a
spatial Fourier transform,
422
00:22:53,330 --> 00:22:56,792
and what can we do with it.
423
00:22:56,792 --> 00:22:59,900
So, very briefly, to remind
you, from geometrical optics,
424
00:22:59,900 --> 00:23:00,770
this is--
425
00:23:00,770 --> 00:23:04,450
we did this some time ago,
maybe about a month ago.
426
00:23:04,450 --> 00:23:07,040
So, to remind you,
a lens is a device
427
00:23:07,040 --> 00:23:13,820
that looks like a, well, at
least one curved glass surface,
428
00:23:13,820 --> 00:23:17,150
typically more than one.
429
00:23:17,150 --> 00:23:19,880
And it is a device
that we can use
430
00:23:19,880 --> 00:23:22,430
to focus or collimate light.
431
00:23:22,430 --> 00:23:25,100
So, for example, if you
illuminate a lens with a plane
432
00:23:25,100 --> 00:23:28,400
wave, then the lens will
focus that plane wave
433
00:23:28,400 --> 00:23:31,240
at one focal distance
to the right.
434
00:23:31,240 --> 00:23:34,340
On the other hand, if
you place a point source
435
00:23:34,340 --> 00:23:36,620
at one focal
distance to the left,
436
00:23:36,620 --> 00:23:38,810
the lens will collimate
it, will produce
437
00:23:38,810 --> 00:23:41,180
a plane wave,
which we also refer
438
00:23:41,180 --> 00:23:42,742
to as an image at infinity.
439
00:23:42,742 --> 00:23:44,450
And, of course, what
I am discussing here
440
00:23:44,450 --> 00:23:48,305
is for the case of a
positive spherical lens.
441
00:23:48,305 --> 00:23:49,680
There's other
lenses that we saw,
442
00:23:49,680 --> 00:23:53,270
negative lenses that would do
something slightly different.
443
00:23:53,270 --> 00:23:55,460
But I don't want to do a
full review of lenses here,
444
00:23:55,460 --> 00:23:58,280
just to remind you what is
relevant to our discussion
445
00:23:58,280 --> 00:23:58,943
here.
446
00:23:58,943 --> 00:24:00,860
And, of course, the other
thing that lenses do
447
00:24:00,860 --> 00:24:04,010
is they can produce images
as finite conjugates.
448
00:24:04,010 --> 00:24:08,750
If you place an object
at some distance s sub o,
449
00:24:08,750 --> 00:24:11,120
then the lens will form
an image at a distance
450
00:24:11,120 --> 00:24:16,340
s sub i, which is related
to s sub o by the lens law.
451
00:24:16,340 --> 00:24:19,170
So we did the stuff to that
when we did geometrical optics,
452
00:24:19,170 --> 00:24:26,070
so I don't want to produce
recurrent nightmares to you
453
00:24:26,070 --> 00:24:27,800
by repeating it here.
454
00:24:27,800 --> 00:24:30,080
So what I will do now is
I will describe the lens
455
00:24:30,080 --> 00:24:33,030
in the context of wave optics.
456
00:24:33,030 --> 00:24:34,960
So, of course, in the
context of wave optics,
457
00:24:34,960 --> 00:24:39,060
we have to describe the lens
as some kind of a transparency,
458
00:24:39,060 --> 00:24:41,380
as some kind of a
phase function that is
459
00:24:41,380 --> 00:24:43,970
applied to the optical field.
460
00:24:43,970 --> 00:24:46,818
So I don't want to go into
the details of this one.
461
00:24:46,818 --> 00:24:47,860
Is there a question, or--
462
00:24:51,720 --> 00:24:53,430
someone has a
microphone on, so I can
463
00:24:53,430 --> 00:24:54,867
hear you shifting on your seat.
464
00:24:54,867 --> 00:24:56,284
Anyway, it doesn't
matter, though.
465
00:24:59,297 --> 00:25:01,130
If you have a question,
please interrupt me.
466
00:25:01,130 --> 00:25:02,570
Of course, this applies always.
467
00:25:08,977 --> 00:25:09,810
So what is this now?
468
00:25:09,810 --> 00:25:15,310
So we will do a very
crude approximation here.
469
00:25:15,310 --> 00:25:18,310
We will actually neglect
the thickness of the lens.
470
00:25:18,310 --> 00:25:20,760
We did this again when we
did geometrical optics.
471
00:25:20,760 --> 00:25:23,730
And that is, of course, because
it is not, strictly speaking,
472
00:25:23,730 --> 00:25:27,420
correct, but the results
that we get are pretty good,
473
00:25:27,420 --> 00:25:29,800
and it makes our
mathematics pretty simple.
474
00:25:29,800 --> 00:25:34,800
So the combination of the
two is a good justification
475
00:25:34,800 --> 00:25:35,850
to make an approximation.
476
00:25:35,850 --> 00:25:39,330
If you have one of
the two reasons,
477
00:25:39,330 --> 00:25:41,160
it is not good to
make an approximation.
478
00:25:41,160 --> 00:25:43,810
For example, if it
makes your math simple
479
00:25:43,810 --> 00:25:47,820
but the answer is wrong, then
you don't do the approximation.
480
00:25:47,820 --> 00:25:52,477
If you get the correct answer
but the math is not simplified,
481
00:25:52,477 --> 00:25:54,060
again we don't make
the approximation.
482
00:25:54,060 --> 00:25:56,940
You might as well go with
the accurate calculation.
483
00:25:56,940 --> 00:25:58,590
But, in this case,
we get two bonuses.
484
00:25:58,590 --> 00:26:01,950
And if we have both bonuses,
then we do the approximation.
485
00:26:04,670 --> 00:26:10,310
So what is happening here,
if you take a field--
486
00:26:10,310 --> 00:26:13,730
imagine like Huygens
wavelets impinging
487
00:26:13,730 --> 00:26:15,650
on the lens from the left.
488
00:26:15,650 --> 00:26:19,720
The wavelet that
impinges in the center
489
00:26:19,720 --> 00:26:23,300
will actually see the
thickest part of the lens,
490
00:26:23,300 --> 00:26:27,450
so it will sustain the
longer phase delay because it
491
00:26:27,450 --> 00:26:29,040
propagates through glass.
492
00:26:29,040 --> 00:26:32,670
If you take a Huygens wavelet
that actually impinges away
493
00:26:32,670 --> 00:26:36,670
from the axis, it will see a
thinner portion of the lens.
494
00:26:36,670 --> 00:26:38,640
Therefore, it will
have less phase delay
495
00:26:38,640 --> 00:26:42,360
because it propagates a
shorter distance in glass.
496
00:26:42,360 --> 00:26:45,390
It will still propagate
some distance in air
497
00:26:45,390 --> 00:26:48,610
on the left and the
right of the lens.
498
00:26:48,610 --> 00:26:53,750
So if you compute that
now, the difference is,
499
00:26:53,750 --> 00:26:58,940
of course, it is given by
the spherical calculation.
500
00:26:58,940 --> 00:27:00,540
I don't want to go through this.
501
00:27:00,540 --> 00:27:02,590
You can go back and
do it yourselves.
502
00:27:02,590 --> 00:27:04,970
It's a very simple
geometrical calculation,
503
00:27:04,970 --> 00:27:08,880
with the addition of the
paraxial approximation.
504
00:27:08,880 --> 00:27:11,420
So even if you glance at
this here, you see that--
505
00:27:11,420 --> 00:27:13,790
actually, this I
copied from Goodman,
506
00:27:13,790 --> 00:27:16,533
so the equations are
verbatim from Goodman's book.
507
00:27:16,533 --> 00:27:17,450
It's a scan, actually.
508
00:27:19,800 --> 00:27:20,300
I'm sorry.
509
00:27:20,300 --> 00:27:21,140
It is not a scan.
510
00:27:21,140 --> 00:27:23,060
It is a scan from my own
notes from last year.
511
00:27:23,060 --> 00:27:25,960
But, anyway, it is verbatim
copied from Goodman.
512
00:27:25,960 --> 00:27:28,730
And you can see that I replaced
the square root with a Taylor
513
00:27:28,730 --> 00:27:29,570
expansion.
514
00:27:29,570 --> 00:27:31,573
So it is a paraxial
approximation.
515
00:27:31,573 --> 00:27:32,990
And the result
that you get, which
516
00:27:32,990 --> 00:27:35,630
is what I really
wanted to do, is
517
00:27:35,630 --> 00:27:37,250
something that looks like this.
518
00:27:58,510 --> 00:28:00,210
OK.
519
00:28:00,210 --> 00:28:01,130
This is what we get.
520
00:28:01,130 --> 00:28:05,910
So we'll get the complex
amplitude transmittance
521
00:28:05,910 --> 00:28:09,360
of the lens if you
express it in wave optics.
522
00:28:09,360 --> 00:28:11,350
It is actually a
quadratic phase delay.
523
00:28:11,350 --> 00:28:11,850
That's if.
524
00:28:15,466 --> 00:28:18,600
And in this quadratic
phase delay,
525
00:28:18,600 --> 00:28:20,610
a magical distance happens.
526
00:28:20,610 --> 00:28:24,770
A magical distance
appears, which we recognize
527
00:28:24,770 --> 00:28:26,080
to be the focal length.
528
00:28:26,080 --> 00:28:27,930
This, if we recall from
geometrical optics,
529
00:28:27,930 --> 00:28:32,840
we used to call this
the lensmaker's formula.
530
00:28:36,640 --> 00:28:38,540
So, basically, we
recover the expression
531
00:28:38,540 --> 00:28:43,805
for the focal length of the
lens, but now we have a wave--
532
00:28:46,723 --> 00:28:48,140
I don't want to
say wave function.
533
00:28:48,140 --> 00:28:49,920
That sounds like
quantum mechanics.
534
00:28:49,920 --> 00:28:54,500
But now we have actually--
we have a complex amplitude
535
00:28:54,500 --> 00:28:55,610
transmittance.
536
00:28:55,610 --> 00:28:59,130
That's what we have
associated with that one.
537
00:28:59,130 --> 00:29:04,280
So now let's see why this is the
same result as we had before.
538
00:29:07,500 --> 00:29:10,410
So the trick here is that
we replace the lens--
539
00:29:10,410 --> 00:29:17,710
when we have a
situation like this one,
540
00:29:17,710 --> 00:29:20,640
we replace the lens
with its amplitude--
541
00:29:20,640 --> 00:29:22,662
complex amplitude transmittance.
542
00:29:34,150 --> 00:29:35,438
So forget about the curvature.
543
00:29:35,438 --> 00:29:37,230
Forget about the glass
and everything else.
544
00:29:37,230 --> 00:29:40,480
We just replace it with
a thin transparency.
545
00:29:40,480 --> 00:29:42,520
And then we illuminate
it with something.
546
00:29:45,350 --> 00:29:48,440
Let's start by choosing this
something to be a plane wave.
547
00:29:52,660 --> 00:29:55,500
So here's the wave vector
of this plane wave.
548
00:29:55,500 --> 00:30:00,640
And because it is propagating at
an angle, we write it as g sub
549
00:30:00,640 --> 00:30:01,140
in.
550
00:30:04,382 --> 00:30:09,470
Actually, we write
it g sub t sub minus,
551
00:30:09,470 --> 00:30:11,630
because it is the
field immediately
552
00:30:11,630 --> 00:30:21,760
to the left of the
transparency, of x comma y.
553
00:30:21,760 --> 00:30:22,820
It is a plane wave.
554
00:30:22,820 --> 00:30:25,130
Let's call this angle theta--
555
00:30:25,130 --> 00:30:26,120
what did they call it?
556
00:30:26,120 --> 00:30:26,620
Theta 0.
557
00:30:30,700 --> 00:30:34,160
So since it is a plane wave,
the proper expression for it
558
00:30:34,160 --> 00:30:43,165
is e to i 2 pi sine theta 0 up
on lambda x plus cosine theta
559
00:30:43,165 --> 00:30:45,430
0 up on lambda z.
560
00:30:48,790 --> 00:30:50,360
And I'm going to
do two things here.
561
00:30:50,360 --> 00:30:54,540
First of all, I'm going to place
the transparency at z equal 0.
562
00:30:54,540 --> 00:30:58,550
So that knocks out this
factor, because equal 0.
563
00:30:58,550 --> 00:31:00,900
And then I'm going to make
the paraxial approximation,
564
00:31:00,900 --> 00:31:02,940
so that knocks out the sine.
565
00:31:02,940 --> 00:31:05,760
So, basically,
the field incident
566
00:31:05,760 --> 00:31:09,090
upon the transparency,
upon the lens, that this,
567
00:31:09,090 --> 00:31:16,370
is simply e to the i 2 pi
theta 0 x up on lambda.
568
00:31:19,720 --> 00:31:22,850
So what I would
like to do now is
569
00:31:22,850 --> 00:31:29,870
to compute the field after the
transparency, g sub t plus.
570
00:31:29,870 --> 00:31:39,040
So the rule that we described
when we did thin transparencies
571
00:31:39,040 --> 00:31:40,580
is that we multiply.
572
00:31:40,580 --> 00:31:41,080
I'm sorry.
573
00:31:41,080 --> 00:31:44,360
I'm using a slightly different
notation on the whiteboard
574
00:31:44,360 --> 00:31:45,410
than in the notes.
575
00:31:45,410 --> 00:31:47,505
You don't have a
minus and a plus,
576
00:31:47,505 --> 00:31:51,330
but it's basically
the same thing.
577
00:31:51,330 --> 00:32:06,170
So g sub t plus multiplied
by the transparency itself.
578
00:32:11,770 --> 00:32:16,240
So the plus stands for after,
the minus stands for before,
579
00:32:16,240 --> 00:32:19,055
and the nothing stands for
the transparency itself.
580
00:32:19,055 --> 00:32:23,755
Oh, and another thing that I
did in the notes is I defined--
581
00:32:30,900 --> 00:32:37,220
I defined this quantity u0
equals theta 0 up on lambda.
582
00:32:37,220 --> 00:32:40,340
And this now we recognize
as a spatial frequency
583
00:32:40,340 --> 00:32:43,010
because it has units
of inverse meters.
584
00:32:46,530 --> 00:32:48,000
So what do we get now?
585
00:32:48,000 --> 00:32:50,970
What do we do is a
little bit of algebra,
586
00:32:50,970 --> 00:32:55,470
but it actually results
in an physically
587
00:32:55,470 --> 00:32:58,320
intuitive, physically
meaningful result.
588
00:32:58,320 --> 00:33:01,080
So that justifies the
algebra, I suppose.
589
00:33:01,080 --> 00:33:02,110
So let me write it out.
590
00:33:02,110 --> 00:33:05,370
Let me write out this
product over here.
591
00:33:20,370 --> 00:33:22,800
OK, that's it.
592
00:33:22,800 --> 00:33:25,400
So now I have to
do something that--
593
00:33:25,400 --> 00:33:27,155
let me leave it
here so you can see.
594
00:33:27,155 --> 00:33:29,660
We have to do something that
you may remember from horror,
595
00:33:29,660 --> 00:33:32,300
from your high school
or elementary school.
596
00:33:32,300 --> 00:33:34,202
I don't know where you
learned these things.
597
00:33:34,202 --> 00:33:35,660
It's called to
complete the square.
598
00:33:38,778 --> 00:33:40,570
What is the square that
I want to complete?
599
00:33:40,570 --> 00:33:42,640
If you look at
the exponents, you
600
00:33:42,640 --> 00:33:44,910
have an expression
that looks like this.
601
00:33:44,910 --> 00:33:49,400
I'll knock out the minus sign.
602
00:33:49,400 --> 00:33:58,022
x square over lambda
f minus 2 u0 x.
603
00:33:58,022 --> 00:33:58,730
Can you see that?
604
00:33:58,730 --> 00:34:00,880
I've neglected the y's
and everything else,
605
00:34:00,880 --> 00:34:02,800
and the pi's, and so on.
606
00:34:02,800 --> 00:34:06,530
If I can do that, then I
can take care of the rest.
607
00:34:06,530 --> 00:34:09,020
So how can I complete the sign?
608
00:34:09,020 --> 00:34:11,449
Well, I tend to get
confused with this,
609
00:34:11,449 --> 00:34:13,690
so let me knock out the
1 over lambda f also.
610
00:34:23,889 --> 00:34:24,800
Now it looks better.
611
00:34:24,800 --> 00:34:25,467
Now I can do it.
612
00:34:35,510 --> 00:34:37,000
Basically, to
complete the sign, I
613
00:34:37,000 --> 00:34:39,889
have to add and subtract the
square of this business here.
614
00:34:52,260 --> 00:34:54,340
And now I can write it as--
615
00:35:08,810 --> 00:35:10,790
OK, now I can go
back and substitute
616
00:35:10,790 --> 00:35:12,140
into my original expression.
617
00:35:12,140 --> 00:35:15,860
I'm done with
manipulating the exponent.
618
00:35:15,860 --> 00:35:18,980
And my original
expression was this one.
619
00:35:18,980 --> 00:35:25,235
So I can rewrite out now, g sub
t plus of x comma y equals--
620
00:35:28,466 --> 00:35:32,400
first of all, I have
this constant term.
621
00:35:32,400 --> 00:35:33,250
That is constant.
622
00:35:33,250 --> 00:35:37,520
It means it does not depend
on x, the spatial variable.
623
00:35:37,520 --> 00:35:38,750
So I'll just take it out.
624
00:35:43,110 --> 00:35:44,570
I should not forget my pi's.
625
00:35:44,570 --> 00:35:45,950
So there's a pi over here.
626
00:35:56,310 --> 00:36:01,810
So-- no, I don't like
the way this came out.
627
00:36:01,810 --> 00:36:04,378
I was expecting to divide,
but I had multiplied.
628
00:36:09,856 --> 00:36:12,100
OK, that looks better.
629
00:36:12,100 --> 00:36:15,280
So I'm doing OK here,
because what are the units?
630
00:36:15,280 --> 00:36:15,860
No units.
631
00:36:15,860 --> 00:36:18,130
I have a spatial
frequency squared times
632
00:36:18,130 --> 00:36:21,520
distance squared, so no units.
633
00:36:21,520 --> 00:36:23,358
And what I have
left is something
634
00:36:23,358 --> 00:36:24,400
that looks like this now.
635
00:36:39,804 --> 00:36:40,304
OK.
636
00:36:42,820 --> 00:36:45,240
So the first part, I don't
have to worry too much about.
637
00:36:45,240 --> 00:36:47,750
This is just a constant
factor, as I said.
638
00:36:47,750 --> 00:36:52,460
But this one, now I recognize
that's a spherical wave.
639
00:36:52,460 --> 00:36:57,050
It is a spherical wave because
it contains quadratic phases
640
00:36:57,050 --> 00:36:58,700
in the exponent.
641
00:36:58,700 --> 00:37:01,690
It is a converging spherical
wave because of the minus sign
642
00:37:01,690 --> 00:37:02,190
here.
643
00:37:07,790 --> 00:37:12,410
And it's not quite its
origin but its sink.
644
00:37:12,410 --> 00:37:15,880
The location where the
spherical wave becomes a point
645
00:37:15,880 --> 00:37:21,420
is actually shifted by
this factor over here.
646
00:37:24,890 --> 00:37:27,330
This is displacement.
647
00:37:27,330 --> 00:37:27,830
OK.
648
00:37:31,198 --> 00:37:32,740
So, basically, this
is what I've got.
649
00:37:32,740 --> 00:37:35,550
I've got a spherical
wave which converges.
650
00:37:35,550 --> 00:37:37,610
Oh, and where does it converge?
651
00:37:37,610 --> 00:37:40,900
Well, the distance that a
spherical wave converges
652
00:37:40,900 --> 00:37:44,630
is what multiplies the
wavelength in the denominator.
653
00:37:44,630 --> 00:37:48,652
So this is where it converges.
654
00:37:51,960 --> 00:37:54,050
So this is basically
what you see here.
655
00:37:54,050 --> 00:37:56,270
The spherical wave
after the lens
656
00:37:56,270 --> 00:37:59,800
converges to a
distance u0 lambda f.
657
00:37:59,800 --> 00:38:04,890
If you substitute the definition
for u0, it is theta 0 times
658
00:38:04,890 --> 00:38:07,250
f away from the axis.
659
00:38:07,250 --> 00:38:10,250
And the distance between
the lens and the focus
660
00:38:10,250 --> 00:38:11,730
is one focal distance.
661
00:38:11,730 --> 00:38:12,860
So this is not news.
662
00:38:12,860 --> 00:38:14,590
We knew this from
geometrical optics.
663
00:38:14,590 --> 00:38:19,238
We just rederived it using
the thin transparency.
664
00:38:19,238 --> 00:38:20,780
So this, I guess,
gives us conviction
665
00:38:20,780 --> 00:38:24,410
that our approach is correct,
because we rederived something
666
00:38:24,410 --> 00:38:28,550
from geometrical optics.
667
00:38:28,550 --> 00:38:30,800
I will not do the next one.
668
00:38:30,800 --> 00:38:33,020
The next one, I'll let
you do by yourselves.
669
00:38:33,020 --> 00:38:37,340
You can repeat a similar
procedure of completing squares
670
00:38:37,340 --> 00:38:39,850
in order to see what
happens to a diverging
671
00:38:39,850 --> 00:38:42,860
spherical wave placed
at one focal distance
672
00:38:42,860 --> 00:38:44,240
to the left of the lens.
673
00:38:44,240 --> 00:38:46,220
And you can convince
yourselves easily
674
00:38:46,220 --> 00:38:49,040
that this becomes a
plane wave propagating
675
00:38:49,040 --> 00:38:53,270
at an angle equal to the
ratio of the displacement
676
00:38:53,270 --> 00:38:54,628
over the focal length.
677
00:38:54,628 --> 00:38:57,170
So this is again something that
we saw in geometrical optics.
678
00:38:57,170 --> 00:38:59,880
It is not new.
679
00:38:59,880 --> 00:39:00,380
OK.
680
00:39:06,440 --> 00:39:08,600
The real result that I
want to derive here--
681
00:39:08,600 --> 00:39:11,300
and I will try to do it
carefully in the time that we
682
00:39:11,300 --> 00:39:12,800
have left--
683
00:39:12,800 --> 00:39:16,230
is the Fourier
transform property which
684
00:39:16,230 --> 00:39:17,483
I will do for a special case.
685
00:39:17,483 --> 00:39:19,400
Actually, I will not do
it for a special case.
686
00:39:19,400 --> 00:39:20,190
I'm going to--
687
00:39:20,190 --> 00:39:21,340
I take it back.
688
00:39:21,340 --> 00:39:25,480
I will do it for the
general case of a lens--
689
00:39:25,480 --> 00:39:28,010
I'm sorry-- of a
thin transparency
690
00:39:28,010 --> 00:39:32,830
placed at a distance z
to the left of the lens.
691
00:39:32,830 --> 00:39:34,230
Goodman does three cases.
692
00:39:34,230 --> 00:39:37,090
First, he does the
case z equals 0.
693
00:39:37,090 --> 00:39:40,860
Then he does the case z equals
f, and then another case.
694
00:39:40,860 --> 00:39:41,710
It doesn't matter.
695
00:39:41,710 --> 00:39:44,085
We'll just do it for the
general case, and we're covered.
696
00:39:49,180 --> 00:39:52,000
So what-- first
of all, let me do
697
00:39:52,000 --> 00:39:54,790
the derivation in one variable.
698
00:39:54,790 --> 00:39:56,110
So don't write too much.
699
00:39:56,110 --> 00:39:58,310
So we'll basically skip y.
700
00:39:58,310 --> 00:40:03,570
All of the derivations
will be just with x.
701
00:40:06,260 --> 00:40:08,240
So I have a thin
transparency g of x.
702
00:40:13,600 --> 00:40:15,550
And then what I will
do is I will propagate
703
00:40:15,550 --> 00:40:24,910
it distance z to the lens.
704
00:40:31,390 --> 00:40:37,420
Now, on the lens, my
coordinate is x prime.
705
00:40:37,420 --> 00:40:39,925
And since I'm
propagating a field--
706
00:40:42,850 --> 00:40:45,070
also, I forgot to
say-- that's quite
707
00:40:45,070 --> 00:40:47,170
important-- the
implicit assumption here
708
00:40:47,170 --> 00:40:51,340
is that the illumination is
an on-axis plane wave coming
709
00:40:51,340 --> 00:40:52,540
from the left.
710
00:40:52,540 --> 00:40:54,730
So that, if you recall,
we said a couple
711
00:40:54,730 --> 00:40:56,500
of times, that is simply--
712
00:40:56,500 --> 00:41:01,480
its complex amplitude is 1,
because I can choose that--
713
00:41:01,480 --> 00:41:02,680
a constant phase.
714
00:41:02,680 --> 00:41:04,690
And there's no x
variations with this one.
715
00:41:08,640 --> 00:41:12,209
So the Fresnel propagation
kernel, if I go from z--
716
00:41:16,041 --> 00:41:21,640
So g is-- let me maintain
my notation consistent here.
717
00:41:21,640 --> 00:41:31,220
So to the left of the
lens, L minus of x prime--
718
00:41:31,220 --> 00:41:34,340
so that is the field to
the left of the lens--
719
00:41:34,340 --> 00:41:38,730
is going to be given by a
Fresnel diffraction integral.
720
00:41:38,730 --> 00:41:45,195
And, actually, in my derivation,
I skipped the constant.
721
00:41:59,940 --> 00:42:02,390
And what is the
constant that I skipped?
722
00:42:02,390 --> 00:42:03,300
It is this one.
723
00:42:11,240 --> 00:42:13,160
And this constant
should be there,
724
00:42:13,160 --> 00:42:15,382
but it is not doing
anything significant for us
725
00:42:15,382 --> 00:42:17,090
in this case, so that's
why I skipped it.
726
00:42:17,090 --> 00:42:19,160
To save writing, basically.
727
00:42:19,160 --> 00:42:22,490
So from now on, we will
basically neglect this.
728
00:42:22,490 --> 00:42:25,410
Even though it is there,
we will simply neglect.
729
00:42:28,230 --> 00:42:36,470
Now, the field after the
lens equals the field
730
00:42:36,470 --> 00:42:40,570
before the lens times
the lens itself.
731
00:42:40,570 --> 00:42:42,480
And the lens itself is
something of the form
732
00:42:42,480 --> 00:42:47,200
e to the minus i pi x
squared up on lambda f.
733
00:42:50,350 --> 00:43:01,410
And, finally, I have this field,
and I have to propagate it.
734
00:43:07,280 --> 00:43:08,100
How long?
735
00:43:08,100 --> 00:43:10,160
Now I have to do
this part, which
736
00:43:10,160 --> 00:43:13,730
means I have to propagate
by distance f until I
737
00:43:13,730 --> 00:43:15,250
reach the back focal plane.
738
00:43:22,170 --> 00:43:27,010
And that is another
Fresnel integral.
739
00:43:27,010 --> 00:43:30,930
I will call it g sub f, I guess.
740
00:43:36,250 --> 00:43:39,120
Again, there is a factor
here which I will neglect.
741
00:43:56,530 --> 00:43:59,010
OK.
742
00:43:59,010 --> 00:44:00,645
So now let's put
everything together.
743
00:44:06,150 --> 00:44:12,590
I have two Fresnel
convolution integrals--
744
00:44:12,590 --> 00:44:15,140
one with respect to
the input coordinates,
745
00:44:15,140 --> 00:44:20,310
one with respect to
the lens coordinates.
746
00:44:20,310 --> 00:44:24,300
And what is left inside, I will
simply substitute all the rest.
747
00:44:24,300 --> 00:44:25,425
I have the input itself.
748
00:44:28,260 --> 00:44:30,990
Then I have the
propagation kernel
749
00:44:30,990 --> 00:44:33,120
from the input to the lens.
750
00:44:38,410 --> 00:44:39,420
Then I have the lens.
751
00:44:44,100 --> 00:44:46,140
And then I have the
propagation kernel
752
00:44:46,140 --> 00:44:48,100
from the lens to the
back focal plane.
753
00:44:56,884 --> 00:44:59,330
OK.
754
00:44:59,330 --> 00:45:00,990
That's what it is.
755
00:45:00,990 --> 00:45:02,660
This looks a little
scary, but part
756
00:45:02,660 --> 00:45:05,420
of the purpose of this
class is to teach you
757
00:45:05,420 --> 00:45:09,100
how to not be scared by
this kind of integrals.
758
00:45:09,100 --> 00:45:10,940
So the way you know
this, you don't
759
00:45:10,940 --> 00:45:12,910
get scared by this
sort of integral
760
00:45:12,910 --> 00:45:14,810
is you manipulate
the exponents here.
761
00:45:14,810 --> 00:45:16,340
And you try to--
the first thing you
762
00:45:16,340 --> 00:45:19,130
do when you reach an
integral of this kind
763
00:45:19,130 --> 00:45:22,220
is you try to collect terms.
764
00:45:22,220 --> 00:45:23,660
So I'll write the
exponents here.
765
00:45:23,660 --> 00:45:29,520
I will expand the exponents
and collect terms then.
766
00:45:29,520 --> 00:45:35,140
So if I expand the exponents,
I will get x prime squared
767
00:45:35,140 --> 00:45:41,650
plus x squared minus 2x
x prime over lambda z.
768
00:45:41,650 --> 00:45:44,320
This came from this
term over here.
769
00:45:44,320 --> 00:45:51,300
Then I have minus x
square over lambda f.
770
00:45:51,300 --> 00:45:52,335
And then I have plus--
771
00:46:01,030 --> 00:46:03,650
and I think I
missed a prime here.
772
00:46:03,650 --> 00:46:04,900
That should have been x prime.
773
00:46:07,810 --> 00:46:09,070
Yes, correct.
774
00:46:09,070 --> 00:46:11,180
The lens should also
be with an x prime.
775
00:46:11,180 --> 00:46:11,680
Thank you.
776
00:46:24,938 --> 00:46:26,730
It is very fortunate
that you corrected me,
777
00:46:26,730 --> 00:46:29,630
because if you hadn't I
would be kind of stuck here.
778
00:46:29,630 --> 00:46:30,130
OK.
779
00:46:32,902 --> 00:46:34,750
So the thing that
you notice first
780
00:46:34,750 --> 00:46:39,090
is that some of these
exponents get knocked out.
781
00:46:42,180 --> 00:46:46,100
This one kills this one.
782
00:46:46,100 --> 00:46:47,560
That's very pleasant.
783
00:46:53,140 --> 00:46:54,550
Now what do I have to do?
784
00:47:07,175 --> 00:47:09,750
I still need to make an
integration with respect
785
00:47:09,750 --> 00:47:10,490
to x prime.
786
00:47:15,250 --> 00:47:19,050
So x prime appears
here and here.
787
00:47:19,050 --> 00:47:21,810
And I have to make an
integration with respect to x.
788
00:47:24,740 --> 00:47:27,570
Well, here is x.
789
00:47:27,570 --> 00:47:28,880
Here is x.
790
00:47:32,060 --> 00:47:32,560
OK.
791
00:47:35,270 --> 00:47:37,945
So what-- any ideas?
792
00:47:37,945 --> 00:47:40,070
Anybody want to speculate
on what I should do here?
793
00:47:48,180 --> 00:47:50,157
Let me write the integral.
794
00:47:50,157 --> 00:47:51,990
That's a little bit
confusing the way it is.
795
00:47:51,990 --> 00:47:55,260
Let me rewrite it so you can see
what the integral looks like.
796
00:49:05,834 --> 00:49:07,050
Let me do it carefully.
797
00:49:07,050 --> 00:49:12,282
So I have e to the minus i
2 pi x prime up on lambda.
798
00:49:12,282 --> 00:49:13,740
That's common in
the two exponents.
799
00:49:13,740 --> 00:49:20,400
Inside, I have x up on z
plus x double prime up on f.
800
00:49:31,228 --> 00:49:32,270
So what should I do next?
801
00:50:03,220 --> 00:50:08,500
Is there any glaring sort of
integral that popped up here?
802
00:50:39,850 --> 00:50:42,190
What is-- here, we
have an integral
803
00:50:42,190 --> 00:50:45,790
of another exponential, right?
804
00:50:45,790 --> 00:50:58,490
And they-- let me
rewrite it like this.
805
00:51:40,960 --> 00:51:43,460
OK.
806
00:51:43,460 --> 00:51:45,470
So the glaring integral
that I was referring
807
00:51:45,470 --> 00:51:48,930
to before is this one.
808
00:51:48,930 --> 00:51:52,440
That's a Fourier transform.
809
00:51:52,440 --> 00:51:54,316
Whose Fourier transform?
810
00:51:54,316 --> 00:51:56,580
The Fourier
transform of whomever
811
00:51:56,580 --> 00:51:58,290
appears in this
location over here.
812
00:52:07,690 --> 00:52:09,940
And where is the Fourier
transform computed?
813
00:52:09,940 --> 00:52:15,580
Well, it is computed in this
spatial frequency, right?
814
00:52:15,580 --> 00:52:19,630
It is computed in whatever
spatial frequency multiplies
815
00:52:19,630 --> 00:52:23,880
the dummy variable
in the exponent.
816
00:52:23,880 --> 00:52:25,380
Now, what is this
Fourier transform?
817
00:52:25,380 --> 00:52:28,170
We don't know, but we have
our notes, or we have Goodman,
818
00:52:28,170 --> 00:52:30,270
or we have the
tables of formulas.
819
00:52:30,270 --> 00:52:35,910
So switching to Lecture 9B.
820
00:52:52,960 --> 00:52:56,690
This is our Fourier
transform pairs.
821
00:52:56,690 --> 00:53:01,130
I recognize this
integral, recognize
822
00:53:01,130 --> 00:53:02,620
this Fourier transform.
823
00:53:02,620 --> 00:53:06,380
It is the second
row from the bottom.
824
00:53:06,380 --> 00:53:13,910
If you look at this expression
and the expression over here,
825
00:53:13,910 --> 00:53:16,270
it is actually the
same Fourier transform.
826
00:53:16,270 --> 00:53:18,370
It is the Fourier
trans-- what you
827
00:53:18,370 --> 00:53:20,380
see here is the
Fourier transform
828
00:53:20,380 --> 00:53:24,220
of the quadratic phase
in the exponential.
829
00:53:24,220 --> 00:53:26,770
So we have the answer.
830
00:53:26,770 --> 00:53:28,740
The answer is right here.
831
00:53:28,740 --> 00:53:29,980
Again, I will neglect--
832
00:53:29,980 --> 00:53:31,810
actually, this constant
is quite important,
833
00:53:31,810 --> 00:53:33,800
but I will be neglect
it nevertheless.
834
00:53:36,580 --> 00:53:42,850
So, basically, the way to get
a one-to-one correspondence is
835
00:53:42,850 --> 00:53:45,990
to simply substitute
what would--
836
00:53:45,990 --> 00:53:49,840
what in the table is denoted
as a square is actually
837
00:53:49,840 --> 00:53:55,630
identical to 1 over
lambda z in our case.
838
00:53:55,630 --> 00:54:00,050
So I can write out
now the answer.
839
00:54:00,050 --> 00:54:10,850
This thing equals-- first of
all, before I do any further,
840
00:54:10,850 --> 00:54:13,930
we recognize that this does
not play in the integration.
841
00:54:13,930 --> 00:54:15,710
This has the output
plane coordinate,
842
00:54:15,710 --> 00:54:17,120
so I will simply
take it outside.
843
00:54:35,310 --> 00:54:36,620
And then I will write out the--
844
00:54:36,620 --> 00:54:39,260
in one shot, I will
write out the outcome
845
00:54:39,260 --> 00:54:42,390
of this Fourier transform.
846
00:54:42,390 --> 00:54:50,060
So that a square that I have
in the original function,
847
00:54:50,060 --> 00:54:52,840
it will go inverse
in the other one.
848
00:54:52,840 --> 00:54:56,560
So we'll get, then,
e to the what?
849
00:54:56,560 --> 00:54:58,210
I will get an extra minus sign.
850
00:54:58,210 --> 00:55:03,140
If I have plus j here,
I have minus j here.
851
00:55:03,140 --> 00:55:05,946
So this will then
become e to the minus--
852
00:55:05,946 --> 00:55:11,470
we'll have all the pi's and
so on-- minus i pi lambda z.
853
00:55:11,470 --> 00:55:15,520
And then I will get the square
of the spatial frequency.
854
00:55:15,520 --> 00:55:21,400
So it will be 1 over lambda
square x up on z plus x double
855
00:55:21,400 --> 00:55:25,370
prime up on f squared.
856
00:55:25,370 --> 00:55:27,496
OK, that's it.
857
00:55:27,496 --> 00:55:30,770
So now I can manipulate
it a little bit further.
858
00:55:46,150 --> 00:55:49,240
And now let me write
out all these exponents
859
00:55:49,240 --> 00:55:51,910
that come out of this square.
860
00:55:51,910 --> 00:55:53,590
So I will get--
861
00:55:53,590 --> 00:55:57,640
the first one will be x
square up on z square.
862
00:55:57,640 --> 00:55:59,320
So one z will be killed.
863
00:55:59,320 --> 00:56:01,540
One lambda has
already been killed.
864
00:56:01,540 --> 00:56:05,530
So I will get e to
the minus i pi x
865
00:56:05,530 --> 00:56:08,610
square over lambda z, right?
866
00:56:11,130 --> 00:56:12,390
Then I will get this term.
867
00:56:12,390 --> 00:56:16,960
That will be e to
the minus i pi--
868
00:56:16,960 --> 00:56:23,100
this is tricky-- z x double
prime square over lambda
869
00:56:23,100 --> 00:56:25,110
f square.
870
00:56:25,110 --> 00:56:27,130
This came out of the
square of this one.
871
00:56:27,130 --> 00:56:29,040
And I will also
get the cross term.
872
00:56:29,040 --> 00:56:33,562
So that will be
e to the minus i.
873
00:56:33,562 --> 00:56:35,170
And then we'll do it carefully.
874
00:56:35,170 --> 00:56:37,790
I will get 2 pi.
875
00:56:37,790 --> 00:56:40,070
And what is left here--
876
00:56:40,070 --> 00:56:41,670
one z will cancel.
877
00:56:41,670 --> 00:56:49,130
I will get x x double
prime up on lambda f.
878
00:56:49,130 --> 00:56:52,560
And now, happily, we see that
this additional quadratic that
879
00:56:52,560 --> 00:56:56,070
was very annoying over
here, this one, it got
880
00:56:56,070 --> 00:56:58,950
killed by this one.
881
00:56:58,950 --> 00:57:02,460
This one is not playing
in the integration either,
882
00:57:02,460 --> 00:57:04,490
so I can actually
take it out of here.
883
00:57:38,260 --> 00:57:40,770
And this is the result
that I was after.
884
00:57:40,770 --> 00:57:43,620
You see that I actually got
another Fourier transform.
885
00:57:43,620 --> 00:57:46,590
This is well recognizable as
a Fourier transform kernel.
886
00:57:50,590 --> 00:57:54,090
So what I have in
this part over here,
887
00:57:54,090 --> 00:57:56,250
it is actually the
Fourier transform
888
00:57:56,250 --> 00:58:02,615
of the transparency calculated
at these coordinates, that is
889
00:58:02,615 --> 00:58:05,250
the argument of the integral.
890
00:58:09,160 --> 00:58:10,930
Now, something
funny happened here,
891
00:58:10,930 --> 00:58:14,820
and this doesn't look
quite right to me.
892
00:58:14,820 --> 00:58:16,570
That should be f, right?
893
00:58:16,570 --> 00:58:19,110
OK.
894
00:58:19,110 --> 00:58:20,420
I don't know how this became z.
895
00:58:24,300 --> 00:58:25,050
Oh, yes, yes, yes.
896
00:58:25,050 --> 00:58:25,883
OK, I know now, yes.
897
00:58:25,883 --> 00:58:29,250
That should be f, yes.
898
00:58:29,250 --> 00:58:31,200
Somewhere in my
notes I converted
899
00:58:31,200 --> 00:58:34,560
this to f, but, thankfully,
not the physics.
900
00:58:34,560 --> 00:58:35,802
So this should not be z.
901
00:58:35,802 --> 00:58:36,510
This should be f.
902
00:58:43,340 --> 00:58:46,510
OK, so now it looks right.
903
00:58:46,510 --> 00:58:48,060
OK.