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GEORGE BARBASTATHIS:
So a little bit

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00:00:22,310 --> 00:00:26,450
of housekeeping
before we continue.

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00:00:26,450 --> 00:00:28,100
First of all, you
may have noticed

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00:00:28,100 --> 00:00:32,840
that in the reading
assignments, I have started

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00:00:32,840 --> 00:00:35,930
Boston from Goodman's book.

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00:00:35,930 --> 00:00:39,080
So there is some pros
and cons about this.

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00:00:39,080 --> 00:00:42,920
Goodman is very good if you
are an engineer, especially

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00:00:42,920 --> 00:00:45,140
electrical or
mechanical engineer.

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00:00:45,140 --> 00:00:48,500
Because then you are very used
to thinking about systems,

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00:00:48,500 --> 00:00:52,320
block diagrams,
transforms, and so on.

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00:00:52,320 --> 00:00:55,070
So it is very nice
living this way.

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00:00:55,070 --> 00:00:57,440
But it's a little
bit mathematical.

20
00:00:57,440 --> 00:01:01,930
Hecht is more on
the physics side.

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00:01:01,930 --> 00:01:05,930
So I feel Hecht is written for
junior or sophomore physics

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00:01:05,930 --> 00:01:06,430
students.

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00:01:09,150 --> 00:01:12,270
And, of course, they're
very nicely complimentary.

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00:01:12,270 --> 00:01:15,390
The only real downside is that
they use different notation.

25
00:01:15,390 --> 00:01:19,770
So if you tried Stanley
from both Hecht and Goodman,

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00:01:19,770 --> 00:01:21,480
you have to be a
little bit careful

27
00:01:21,480 --> 00:01:23,850
to keep their notation--

28
00:01:23,850 --> 00:01:26,220
I mean, the notation
is not consistent,

29
00:01:26,220 --> 00:01:29,640
but you have to keep yourself
from getting confused

30
00:01:29,640 --> 00:01:32,010
by the inconsistent notation.

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00:01:32,010 --> 00:01:34,350
Nevertheless, the diagrams
are, of course, consistent

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00:01:34,350 --> 00:01:36,930
because they both
calculate the same.

33
00:01:36,930 --> 00:01:39,020
Fresnel, the fraction
pattern is the same

34
00:01:39,020 --> 00:01:40,742
from [INAUDIBLE]
pattern and so on.

35
00:01:40,742 --> 00:01:43,200
But you have to be a little
bit mindful of the coordinates.

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00:01:43,200 --> 00:01:45,940
For example, they may be
using different symbols.

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00:01:45,940 --> 00:01:47,790
Anyway, I highly
recommend that you're

38
00:01:47,790 --> 00:01:49,190
starting from both books.

39
00:01:49,190 --> 00:01:52,410
Hecht also has a much more
intuitive explanations,

40
00:01:52,410 --> 00:01:54,990
and many more
figures, and so on.

41
00:01:54,990 --> 00:01:59,070
But Goodman is more
rigorous, and also better

42
00:01:59,070 --> 00:02:01,900
suited to an engineer's
way of thinking.

43
00:02:01,900 --> 00:02:04,640
So that's why I
use both textbooks.

44
00:02:04,640 --> 00:02:09,430
By now, it's actually closer
to Goodman from this point on.

45
00:02:09,430 --> 00:02:12,090
So that's an additional benefit.

46
00:02:12,090 --> 00:02:15,630
Anyway, the reading assignments
are from both books.

47
00:02:15,630 --> 00:02:17,730
If you decide to
follow just one book.

48
00:02:17,730 --> 00:02:22,010
For example, either Hecht by
itself or Goodman by itself.

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00:02:22,010 --> 00:02:26,168
You don't miss significantly,
you can follow either book.

50
00:02:26,168 --> 00:02:28,710
But I think until you get the
complete picture, if you follow

51
00:02:28,710 --> 00:02:35,410
both books and in the way they
serve to reinforce each other.

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00:02:35,410 --> 00:02:38,390
Anyway, so that's the
study about the textbooks.

53
00:02:38,390 --> 00:02:40,730
A little bit more housekeeping.

54
00:02:40,730 --> 00:02:42,760
I have posted the
slightly revised

55
00:02:42,760 --> 00:02:45,790
version of Monday's notes.

56
00:02:45,790 --> 00:02:48,040
One minor correction,
if you look at this.

57
00:02:48,040 --> 00:02:51,490
This is the very last
slide from Monday.

58
00:02:51,490 --> 00:02:54,730
There was an error, at
least one that they found.

59
00:02:54,730 --> 00:02:59,200
In the expression for the
Fourier coefficient c sub q,

60
00:02:59,200 --> 00:03:03,040
it was the function
sinc of 2 over 2.

61
00:03:03,040 --> 00:03:03,880
This is correct.

62
00:03:03,880 --> 00:03:08,380
On Monday, there was an
extra pi inside the argument

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00:03:08,380 --> 00:03:09,540
of the same function.

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00:03:09,540 --> 00:03:12,130
That pi should not have been
there, so I've removed it.

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00:03:12,130 --> 00:03:16,170
We will see today later the
derivation of this expression,

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00:03:16,170 --> 00:03:19,200
actually, a similar expression.

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00:03:19,200 --> 00:03:23,180
So hopefully, that
will clarify matters.

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00:03:23,180 --> 00:03:27,640
The thing I did
compare to Monday.

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00:03:27,640 --> 00:03:32,470
Piper reminded me on Monday,
when he discussed the grading,

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00:03:32,470 --> 00:03:34,930
about the dispersion.

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00:03:34,930 --> 00:03:39,640
So if you look at the--

72
00:03:39,640 --> 00:03:42,160
go back a little bit
to the expressions,

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00:03:42,160 --> 00:03:46,550
or the diffraction
angle from a grating.

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00:04:10,440 --> 00:04:12,140
OK, that's a good one.

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00:04:12,140 --> 00:04:16,620
OK, so last week, we
focused on a discussion

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00:04:16,620 --> 00:04:18,029
about the effect of the period.

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00:04:18,029 --> 00:04:22,060
So we said that if you
make the period smaller,

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00:04:22,060 --> 00:04:25,680
the diffraction order
is spread out more,

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00:04:25,680 --> 00:04:27,630
so the diffraction
angle is inversely

80
00:04:27,630 --> 00:04:29,560
proportional to the period.

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00:04:29,560 --> 00:04:32,920
Actually, the sine of
the diffraction angle.

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00:04:32,920 --> 00:04:35,490
We didn't say anything
about wavelength.

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00:04:35,490 --> 00:04:37,860
So, of course, the wavelength
appears in the numerator

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00:04:37,860 --> 00:04:41,730
there, which means that if you
have light of multiple colors,

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00:04:41,730 --> 00:04:46,350
then longer wavelengths will
focus at the longer angle.

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00:04:46,350 --> 00:04:48,270
So this is what you
see on the last slide

87
00:04:48,270 --> 00:04:52,410
that they posted today
in the revised notes.

88
00:04:52,410 --> 00:04:58,440
So I have a grating here which
is illuminated by white light.

89
00:04:58,440 --> 00:05:00,960
So, of course, white
light is composed

90
00:05:00,960 --> 00:05:03,120
of a broad spectrum
of colors ranging

91
00:05:03,120 --> 00:05:06,600
from somewhere in the
infinite to somewhere

92
00:05:06,600 --> 00:05:09,200
in the ultraviolet.

93
00:05:09,200 --> 00:05:11,160
Anyway, let's take
the discussion.

94
00:05:11,160 --> 00:05:13,680
Let's keep it to the
visible wavelength.

95
00:05:13,680 --> 00:05:19,185
So, of course, the red
wavelength has a longer color.

96
00:05:19,185 --> 00:05:22,500
I'm sorry, the red color
has a longer wavelength.

97
00:05:22,500 --> 00:05:25,900
So therefore, the red color
is diffracted at the larger

98
00:05:25,900 --> 00:05:28,200
angle than the blue.

99
00:05:28,200 --> 00:05:32,010
So this phenomenon is called
dispersion, very similar

100
00:05:32,010 --> 00:05:35,080
to the grating dispersion.

101
00:05:35,080 --> 00:05:37,690
But it is referred to
as anomalous, because it

102
00:05:37,690 --> 00:05:40,290
is the opposite of the grating.

103
00:05:40,290 --> 00:05:42,460
I'm sorry, it is the
opposite of a prism.

104
00:05:42,460 --> 00:05:43,420
What am I saying?

105
00:05:43,420 --> 00:05:44,800
Let me start over.

106
00:05:44,800 --> 00:05:47,950
So in the case of
a prism, as well as

107
00:05:47,950 --> 00:05:50,710
in the case of a
grating, the phenomenon

108
00:05:50,710 --> 00:05:53,350
of analysis of
white light, where

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00:05:53,350 --> 00:05:55,960
white light after propagating
through the element gets

110
00:05:55,960 --> 00:05:58,890
decomposed into its
color component.

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00:05:58,890 --> 00:06:00,850
And each one color
component propagates

112
00:06:00,850 --> 00:06:02,290
at the different angle.

113
00:06:02,290 --> 00:06:05,280
This phenomenon is
referred to as dispersion.

114
00:06:05,280 --> 00:06:07,420
If the two are different
in the sense that

115
00:06:07,420 --> 00:06:10,540
in the case of a
grating, the blue light

116
00:06:10,540 --> 00:06:15,190
is refracted at the smaller
angle than the red light.

117
00:06:15,190 --> 00:06:17,230
That is called the
normal dispersion.

118
00:06:17,230 --> 00:06:20,610
In the case of a prism, it
is not diffraction anymore.

119
00:06:20,610 --> 00:06:21,790
It is refraction.

120
00:06:21,790 --> 00:06:23,470
And we saw why it happens.

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00:06:23,470 --> 00:06:25,300
This has to do
with the dependence

122
00:06:25,300 --> 00:06:30,000
of the index of refraction
of glass on wavelength.

123
00:06:30,000 --> 00:06:33,340
And the dependence is
such that the blue light

124
00:06:33,340 --> 00:06:38,570
is refracted at the larger
angle, in the case of a prism.

125
00:06:38,570 --> 00:06:41,570
That is called normal
dispersion, as opposed

126
00:06:41,570 --> 00:06:42,590
to the anomalous.

127
00:06:42,590 --> 00:06:46,460
Now, there is nothing anomalous
about the grating, I suppose.

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00:06:46,460 --> 00:06:48,410
Historically, this had
to do with the fact

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00:06:48,410 --> 00:06:53,660
that people observed this
phenomenon with prisms first,

130
00:06:53,660 --> 00:06:55,863
so they call it
normal dispersion.

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00:06:55,863 --> 00:06:57,280
Then they noticed
that the grating

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00:06:57,280 --> 00:07:00,560
does the opposite thing, so they
said, well, this is abnormal.

133
00:07:00,560 --> 00:07:02,340
They called it anomalous.

134
00:07:02,340 --> 00:07:03,950
Anomalous in Greek
means abnormal.

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00:07:07,130 --> 00:07:11,470
Anyway, so this is what I wanted
to add to Monday's lecture.

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00:07:11,470 --> 00:07:13,190
Are there any questions
about gratings?

137
00:07:45,470 --> 00:07:47,635
Actually, between today
and next Wednesday,

138
00:07:47,635 --> 00:07:51,740
we will cover the basics
of diffraction theory.

139
00:07:51,740 --> 00:07:54,230
And the rest of
the class will be

140
00:07:54,230 --> 00:07:56,510
basically applications
on what we

141
00:07:56,510 --> 00:07:59,850
cover in these three lectures.

142
00:07:59,850 --> 00:08:01,680
So needless to say,
these three lectures

143
00:08:01,680 --> 00:08:04,441
are very, very important.

144
00:08:04,441 --> 00:08:11,770
So let us start with a little
observation on our Fresnel

145
00:08:11,770 --> 00:08:13,360
propagation formula.

146
00:08:13,360 --> 00:08:15,970
So to remind you very
briefly, today, I

147
00:08:15,970 --> 00:08:21,130
will be using the
whiteboard a lot.

148
00:08:21,130 --> 00:08:24,340
But the equations
that I write are all

149
00:08:24,340 --> 00:08:29,680
of them, either in the
notes or in the textbooks.

150
00:08:29,680 --> 00:08:33,340
So feel free to copy them if
you like, but you don't have to.

151
00:08:33,340 --> 00:08:35,275
It may be better.

152
00:08:35,275 --> 00:08:37,900
It's up to you whether you want
to copy the derivations or not.

153
00:08:37,900 --> 00:08:40,792
But there will be a lot of them.

154
00:08:40,792 --> 00:08:42,250
Probably by the
end of the lecture,

155
00:08:42,250 --> 00:08:45,770
your wrist may be
a little bit tired.

156
00:08:45,770 --> 00:08:47,300
OK.

157
00:08:47,300 --> 00:08:55,410
So to remind you, we did
this about a week ago.

158
00:08:55,410 --> 00:09:02,200
We said that if you have a
complex field at the plane xy,

159
00:09:02,200 --> 00:09:04,940
and then you propagate
by a distance, z.

160
00:09:04,940 --> 00:09:08,060
Then at the output
plane x prime, y prime,

161
00:09:08,060 --> 00:09:11,410
the field is given by
this convolution integral,

162
00:09:11,410 --> 00:09:14,350
which I will rewrite here.

163
00:09:14,350 --> 00:09:22,750
So the field at the output
after propagating by distance z

164
00:09:22,750 --> 00:09:25,540
equals some constants--

165
00:09:30,185 --> 00:09:31,560
and we'll spend
some time talking

166
00:09:31,560 --> 00:09:35,280
about these constants--
times what we'll

167
00:09:35,280 --> 00:09:38,470
call the convolution integral.

168
00:09:38,470 --> 00:09:41,690
So the convolution integral
is really written something

169
00:09:41,690 --> 00:09:42,950
like this, e to the i pi.

170
00:09:55,930 --> 00:10:02,200
OK, so this is the integral
within the Fresnel and scalar

171
00:10:02,200 --> 00:10:03,710
approximations.

172
00:10:03,710 --> 00:10:06,110
Then what we can do is we can--

173
00:10:06,110 --> 00:10:09,920
let's take this
exponent, and expand it.

174
00:10:13,110 --> 00:10:16,660
So I will only do
it for the x case.

175
00:10:27,047 --> 00:10:28,130
That's not rocket science.

176
00:10:28,130 --> 00:10:31,580
They've just expanded
the binomial.

177
00:10:31,580 --> 00:10:34,640
Now, let me remind you.

178
00:10:34,640 --> 00:10:40,930
So the x prime coordinate
is at the output plane.

179
00:10:40,930 --> 00:10:43,900
And also, you can see that
it is not participating

180
00:10:43,900 --> 00:10:44,770
in the integration.

181
00:10:44,770 --> 00:10:47,350
The integration is
with respect to x.

182
00:10:47,350 --> 00:10:49,120
So the x prime
part will actually

183
00:10:49,120 --> 00:10:52,520
be thrown out of the integral.

184
00:10:52,520 --> 00:10:56,530
And then I have
this remaining part.

185
00:10:56,530 --> 00:11:00,010
So the next thing
I'm going to do

186
00:11:00,010 --> 00:11:05,590
is I'm going to assume that
the input field is finite.

187
00:11:05,590 --> 00:11:13,060
So finite means that the field,
if you look at the input field

188
00:11:13,060 --> 00:11:13,990
here.

189
00:11:13,990 --> 00:11:18,520
The field is non-zero in
a relatively small region

190
00:11:18,520 --> 00:11:20,200
near the optical axis.

191
00:11:20,200 --> 00:11:23,710
But then away from the optical
axis, the field becomes zero.

192
00:11:23,710 --> 00:11:26,510
That is a reasonable
assumption, because most objects

193
00:11:26,510 --> 00:11:30,867
that we're going to create
in real life are finite.

194
00:11:30,867 --> 00:11:32,200
We've said this before, I think.

195
00:11:32,200 --> 00:11:34,180
We deal with things
like plane waves

196
00:11:34,180 --> 00:11:37,140
and spherical waves in this
class, which are infinite.

197
00:11:37,140 --> 00:11:39,700
But these are idealizations
that we use in order

198
00:11:39,700 --> 00:11:42,220
to make the math simpler.

199
00:11:42,220 --> 00:11:46,960
In this case, let's use actually
a real life assumption, which

200
00:11:46,960 --> 00:11:50,740
is that the object is finite.

201
00:11:50,740 --> 00:11:54,430
And since the
object is finite, x

202
00:11:54,430 --> 00:11:59,000
will be confined to
relatively small values.

203
00:11:59,000 --> 00:12:03,050
So what I will do now is I
will look at this expression, x

204
00:12:03,050 --> 00:12:05,710
prime over lambda z.

205
00:12:05,710 --> 00:12:09,070
This is this part
of the exponent.

206
00:12:09,070 --> 00:12:13,930
And what I will do now is I will
allow z to become very large.

207
00:12:18,020 --> 00:12:18,990
So what does this mean?

208
00:12:18,990 --> 00:12:20,840
It means that I
start propagating

209
00:12:20,840 --> 00:12:24,950
further and further and further
away from the transparency.

210
00:12:24,950 --> 00:12:29,880
Of course, x is limited by
the size of the input field.

211
00:12:29,880 --> 00:12:35,210
So x square is also
limited, but z grows

212
00:12:35,210 --> 00:12:37,470
as I propagate further away.

213
00:12:37,470 --> 00:12:45,820
So there will come a point
where x square maximum

214
00:12:45,820 --> 00:12:49,210
will become less than lambda z.

215
00:12:49,210 --> 00:12:53,470
That is, the maximum value
of this fraction over here

216
00:12:53,470 --> 00:12:55,000
will become less than one.

217
00:12:55,000 --> 00:12:57,310
And in fact, if I keep
propagating further

218
00:12:57,310 --> 00:12:59,500
and further, this
term will actually

219
00:12:59,500 --> 00:13:02,850
grow less and less and less.

220
00:13:02,850 --> 00:13:06,770
OK, if that is the case,
then the coefficient--

221
00:13:06,770 --> 00:13:11,080
then this term over here
will become negligible,

222
00:13:11,080 --> 00:13:13,730
which means that I can drop it.

223
00:13:13,730 --> 00:13:18,230
OK, so this is what is written
in your transparency over here.

224
00:13:18,230 --> 00:13:23,120
Basically, I actually did
this for both x and y.

225
00:13:23,120 --> 00:13:25,820
I pulled out the
x prime, y prime.

226
00:13:25,820 --> 00:13:28,200
And I'm left with this
expression over here.

227
00:13:28,200 --> 00:13:32,190
So let me also do it
on my whiteboard here.

228
00:13:32,190 --> 00:13:34,640
So what I'm saying
is that the g out.

229
00:13:42,730 --> 00:13:46,420
In the whiteboard, I will
not write the y-coordinates,

230
00:13:46,420 --> 00:13:48,310
because it is too much.

231
00:13:48,310 --> 00:13:53,300
Well, maybe I write
them to avoid confusion.

232
00:13:53,300 --> 00:13:55,510
OK, so this will
now take this form.

233
00:14:07,820 --> 00:14:09,440
This got pulled
out of the integral

234
00:14:09,440 --> 00:14:12,360
because they don't participate
in the integration.

235
00:14:12,360 --> 00:14:15,610
And then what I have
inside looks like this.

236
00:14:34,470 --> 00:14:36,560
And what I was saying
is that if I let z

237
00:14:36,560 --> 00:14:39,620
become long enough,
then it can basically

238
00:14:39,620 --> 00:14:41,840
neglect those two terms.

239
00:14:41,840 --> 00:14:44,570
Now, why do I neglect
only these two terms,

240
00:14:44,570 --> 00:14:51,280
and not the products of y over
y times y prime and x times

241
00:14:51,280 --> 00:14:52,720
x prime.

242
00:14:52,720 --> 00:14:54,460
Why should they keep
those two terms?

243
00:14:54,460 --> 00:14:56,950
And, of course, I forgot
something in both cases.

244
00:14:56,950 --> 00:15:03,350
In here, I should have
g sub in of x comma y,

245
00:15:03,350 --> 00:15:04,840
and the same in here.

246
00:15:04,840 --> 00:15:06,230
OK, you see what I did?

247
00:15:06,230 --> 00:15:10,970
Over here, I need to put the
input field, and the same here.

248
00:15:18,880 --> 00:15:23,370
So why am I keeping this term,
and I am not dropping it?

249
00:15:30,780 --> 00:15:32,280
If I drop it, it's
very easy, right?

250
00:15:32,280 --> 00:15:35,040
If I drop it altogether,
then what I will get

251
00:15:35,040 --> 00:15:39,360
is the integral of g sub in.

252
00:15:39,360 --> 00:15:41,260
It's like the
average of the input.

253
00:15:41,260 --> 00:15:43,665
Why am I not doing that?

254
00:15:43,665 --> 00:15:45,290
AUDIENCE: If you
remove the cross term,

255
00:15:45,290 --> 00:15:48,860
then we actually neglect the
complete spatial variation

256
00:15:48,860 --> 00:15:52,200
of the input field,
and it becomes

257
00:15:52,200 --> 00:15:55,630
like a pointed [INAUDIBLE].

258
00:15:55,630 --> 00:15:56,630
GEORGE BARBASTATHIS: OK.

259
00:15:56,630 --> 00:15:59,510
So I would actually
disagree with that.

260
00:15:59,510 --> 00:16:03,740
If I neglect it, I will get
that g sub out of x comma

261
00:16:03,740 --> 00:16:11,030
y prime equals g sub
in of x comma y dx, dy.

262
00:16:11,030 --> 00:16:12,650
So this is a constant, right?

263
00:16:12,650 --> 00:16:16,170
It is like the
average of g sub in.

264
00:16:16,170 --> 00:16:18,460
So I haven't really
neglected it.

265
00:16:18,460 --> 00:16:24,392
I have said that it gets
averaged out at the output.

266
00:16:24,392 --> 00:16:25,600
And by the way, that's wrong.

267
00:16:25,600 --> 00:16:26,420
That's not true, right?

268
00:16:26,420 --> 00:16:27,378
This is a wrong result.

269
00:16:27,378 --> 00:16:30,470
My question is
why is that wrong?

270
00:16:30,470 --> 00:16:33,640
Why do I have to keep
the entire formula,

271
00:16:33,640 --> 00:16:36,512
z out of x prime, y prime, z.

272
00:16:36,512 --> 00:16:37,720
Let me write it properly now.

273
00:17:15,510 --> 00:17:19,579
So the question is why can
I neglect the quadratics,

274
00:17:19,579 --> 00:17:22,810
but I must keep the cross terms?

275
00:17:27,520 --> 00:17:29,695
Actually a very simple answer.

276
00:17:29,695 --> 00:17:31,070
All you have to
do is look at it.

277
00:17:37,922 --> 00:17:40,130
AUDIENCE: Does it mean you
are restricting the output

278
00:17:40,130 --> 00:17:42,553
field to a limited area?

279
00:17:42,553 --> 00:17:43,970
GEORGE BARBASTATHIS:
That's right.

280
00:17:43,970 --> 00:17:45,345
If you look at
these cross terms,

281
00:17:45,345 --> 00:17:48,760
they actually depend on the
output coordinate, x prime.

282
00:17:48,760 --> 00:17:52,000
I haven't made any
assumptions about x prime.

283
00:17:52,000 --> 00:17:54,700
x prime can be as
large as I want.

284
00:17:57,310 --> 00:18:00,820
So this term can actually
dominate the quadratic term.

285
00:18:00,820 --> 00:18:02,870
That's the point.

286
00:18:02,870 --> 00:18:08,710
This term, I can control by
making the input transparency

287
00:18:08,710 --> 00:18:15,140
relatively small, and
making z arbitrarily large.

288
00:18:15,140 --> 00:18:18,850
However, this term
has the x prime in it.

289
00:18:18,850 --> 00:18:22,100
And x prime can be
actually very large itself.

290
00:18:22,100 --> 00:18:23,680
So I can never
neglect this term.

291
00:18:26,190 --> 00:18:30,720
That's the point
of this argument.

292
00:18:30,720 --> 00:18:31,720
Everybody clear on that?

293
00:18:39,680 --> 00:18:43,300
So if I indeed neglect this
term, then what I will get

294
00:18:43,300 --> 00:18:47,900
is that g sub out--

295
00:18:47,900 --> 00:18:50,550
I think this marker is dead, so
I'll move on to the next one.

296
00:18:56,020 --> 00:18:58,380
It has some terms
which I will neglect,

297
00:18:58,380 --> 00:19:02,400
but it is proportional to a
quantity that looks like this.

298
00:19:18,970 --> 00:19:20,870
Which I can also rewrite as.

299
00:19:49,625 --> 00:19:50,500
I didn't do anything.

300
00:19:50,500 --> 00:19:53,260
All I did was I--

301
00:19:53,260 --> 00:19:57,288
these parameters here, if
you look at this coefficient.

302
00:20:00,960 --> 00:20:04,482
It does not depend on the
variable of integration.

303
00:20:04,482 --> 00:20:05,940
Therefore, I can
call it something.

304
00:20:05,940 --> 00:20:07,710
I call it u.

305
00:20:07,710 --> 00:20:10,810
And this integral now.

306
00:20:10,810 --> 00:20:13,470
It is probably familiar to you.

307
00:20:13,470 --> 00:20:16,740
If it were in one dimension, it
would be immediately familiar.

308
00:20:16,740 --> 00:20:18,630
It is a Fourier transform.

309
00:20:18,630 --> 00:20:21,330
But it is in two dimensions,
so it appears a little bit more

310
00:20:21,330 --> 00:20:23,430
complicated with two variables.

311
00:20:23,430 --> 00:20:26,787
But actually, well, it is
a convention to write--

312
00:20:26,787 --> 00:20:27,620
what happened here--

313
00:20:34,270 --> 00:20:37,150
to write Fourier
transforms as uppercase.

314
00:20:37,150 --> 00:20:42,540
So this is a g sub in of u
comma v. The Fourier transform

315
00:20:42,540 --> 00:20:44,100
computed at frequencies.

316
00:20:52,240 --> 00:20:55,660
Something magical happened
if I let the field propagate

317
00:20:55,660 --> 00:21:01,240
at a relatively long distance.

318
00:21:01,240 --> 00:21:05,780
They field that I get
at that output plane

319
00:21:05,780 --> 00:21:09,040
actually equals
approximately the

320
00:21:09,040 --> 00:21:13,350
Fourier transform of the input
field, which is interesting.

321
00:21:13,350 --> 00:21:16,670
And we'll see a lot
of ramifications

322
00:21:16,670 --> 00:21:20,340
of that in the next few hours.

323
00:21:20,340 --> 00:21:24,440
OK, let's see this
now in a calculation.

324
00:21:24,440 --> 00:21:28,010
Actually, Piper solved it
in practice, in real life.

325
00:21:28,010 --> 00:21:29,780
Last time at the
demo, you actually

326
00:21:29,780 --> 00:21:32,360
saw this kind of
thing happening.

327
00:21:32,360 --> 00:21:34,120
What I will show
now is some movies

328
00:21:34,120 --> 00:21:35,760
which are also posted
on the website,

329
00:21:35,760 --> 00:21:37,363
so you can go back
and produce them.

330
00:21:37,363 --> 00:21:38,780
So in the movie,
you will actually

331
00:21:38,780 --> 00:21:42,320
see the Fourier transform
slowly developing.

332
00:21:42,320 --> 00:21:44,380
In this case, it is a
rectangular aperture.

333
00:21:44,380 --> 00:21:47,910
So the Fourier transform
is very easy to compute.

334
00:21:47,910 --> 00:21:50,980
Of course, in the movie,
there's no Fourier transform.

335
00:21:50,980 --> 00:21:55,560
In the movie, all I did is
I convolved with a Fresnel

336
00:21:55,560 --> 00:21:59,820
propagation coordinate for
progressively longer distances.

337
00:21:59,820 --> 00:22:02,460
And then I put all those frames
together, and I made the movie.

338
00:22:02,460 --> 00:22:05,640
But you will see that
as z grows larger,

339
00:22:05,640 --> 00:22:08,790
the aperture develops
a small oscillation.

340
00:22:08,790 --> 00:22:14,580
But then eventually, it
develops this pattern that well,

341
00:22:14,580 --> 00:22:16,792
it is called the sinc function.

342
00:22:16,792 --> 00:22:18,750
And we'll see in a moment
that this is actually

343
00:22:18,750 --> 00:22:21,660
the special Fourier
transform of that function.

344
00:22:21,660 --> 00:22:24,955
Let me play that once again.

345
00:22:24,955 --> 00:22:26,330
If you notice
carefully, you will

346
00:22:26,330 --> 00:22:30,460
see that you start with a
nice, clear, sharp aperture.

347
00:22:30,460 --> 00:22:31,990
As we will propagate,
first, we'll

348
00:22:31,990 --> 00:22:35,420
see some diffraction ringing.

349
00:22:35,420 --> 00:22:39,180
Professor Sheppard described
in detail last time.

350
00:22:39,180 --> 00:22:42,860
But then this ringing slowly
gives away to this cross

351
00:22:42,860 --> 00:22:44,060
like looking pattern.

352
00:22:52,410 --> 00:22:53,390
So you can see it here.

353
00:22:53,390 --> 00:22:55,500
It has a very distinct pattern.

354
00:22:55,500 --> 00:22:59,030
It has a central lobe,
and then side lobes

355
00:22:59,030 --> 00:23:02,760
expanding along the
x and y dimension.

356
00:23:02,760 --> 00:23:08,430
OK, so what is this now,
and how did it come about?

357
00:23:08,430 --> 00:23:16,180
This function, the aperture
that I started with.

358
00:23:16,180 --> 00:23:18,840
I can describe it
mathematically as g sub

359
00:23:18,840 --> 00:23:26,940
in of x comma y equals 1 if--

360
00:23:26,940 --> 00:23:28,830
basically, let me go back.

361
00:23:31,970 --> 00:23:36,070
So this function
equals 1 if x and y

362
00:23:36,070 --> 00:23:37,930
are within this rectangle.

363
00:23:37,930 --> 00:23:42,190
Mathematically, we can
describe this as x less

364
00:23:42,190 --> 00:23:47,170
than the size of the
aperture, and y also

365
00:23:47,170 --> 00:23:51,160
less than the size
of the aperture.

366
00:23:51,160 --> 00:23:53,920
Generally, it may be a
rectangle, not a square.

367
00:23:53,920 --> 00:23:55,780
So use the different variables.

368
00:23:55,780 --> 00:23:57,250
And it is zero otherwise.

369
00:24:00,610 --> 00:24:03,640
OK, so it is convenient
to define a function.

370
00:24:17,470 --> 00:24:19,740
And actually, I goofed.

371
00:24:19,740 --> 00:24:22,530
If the size of the
aperture is x over naught,

372
00:24:22,530 --> 00:24:28,280
then the value of the function
is 1 for x less than x0.

373
00:24:28,280 --> 00:24:31,280
So I will explain
this in a second.

374
00:24:35,720 --> 00:24:39,080
OK, so if I defined the
function rect this way.

375
00:24:52,830 --> 00:24:54,030
So that's the function.

376
00:24:54,030 --> 00:24:55,655
Sometimes, it's also
known as a boxcar.

377
00:25:00,270 --> 00:25:04,560
And so this would be rect of x.

378
00:25:04,560 --> 00:25:12,320
And if I went to plot
rect of x over x0,

379
00:25:12,320 --> 00:25:18,730
then if I substitute
x over x0 0, 0 here.

380
00:25:18,730 --> 00:25:24,260
Then it is 1 if x over
x0 is less than 1/2.

381
00:25:24,260 --> 00:25:29,950
So basically, this extends from
minus x0 over 2 to x0 over 2.

382
00:25:29,950 --> 00:25:31,930
And it equals the
value 1 over there.

383
00:25:35,170 --> 00:25:42,890
And therefore, the total size of
the boxcar is x0 as advertised.

384
00:25:42,890 --> 00:25:48,480
OK, and, of course, if I define
the rect function this way,

385
00:25:48,480 --> 00:25:51,110
then my original
function, g sub in.

386
00:25:53,770 --> 00:25:57,775
I can simply write it as
was done here, as a product.

387
00:26:11,470 --> 00:26:14,770
So now, what is the Fourier
transform of that product?

388
00:26:14,770 --> 00:26:17,340
Let me write it down to the
Fourier transform definition.

389
00:26:45,670 --> 00:26:48,328
OK, that's the Fourier
transform definition.

390
00:26:48,328 --> 00:26:50,370
The first thing that I
notice in this case, which

391
00:26:50,370 --> 00:26:53,130
is very convenient, is that
the integral is separable.

392
00:26:53,130 --> 00:26:56,260
I can write it, really, as
a product of two integrals.

393
00:26:56,260 --> 00:26:58,590
One of them is in
the x dimension.

394
00:27:07,890 --> 00:27:11,116
And the other looks very
similar in the y dimension.

395
00:27:11,116 --> 00:27:13,250
This is not always the case.

396
00:27:13,250 --> 00:27:16,695
Many, or actually, most 2D
functions are not like that.

397
00:27:16,695 --> 00:27:17,945
But in this case, we're lucky.

398
00:27:25,268 --> 00:27:27,060
Means you don't have
to do the whole thing.

399
00:27:27,060 --> 00:27:30,092
We can just do the integral
for one coordinate,

400
00:27:30,092 --> 00:27:32,300
and then we immediately have
the answer for the other

401
00:27:32,300 --> 00:27:35,590
coordinate as well.

402
00:27:35,590 --> 00:27:37,840
OK, so let's write it then.

403
00:27:42,500 --> 00:27:44,625
So I will just write the
one dimensional integral.

404
00:28:02,630 --> 00:28:03,130
OK.

405
00:28:05,990 --> 00:28:11,380
So now, let's do one
more simplification.

406
00:28:11,380 --> 00:28:17,640
I will assume that
x0 is unit, is unity.

407
00:28:20,360 --> 00:28:22,520
We'll come back and
rectify this one.

408
00:28:22,520 --> 00:28:24,890
But for now, I will
just assume it this way

409
00:28:24,890 --> 00:28:28,200
to make my life a
little bit easier.

410
00:28:28,200 --> 00:28:31,550
So if that is the
case, if x0 is one.

411
00:28:31,550 --> 00:28:35,580
Then [INAUDIBLE] what is the
function rect of x over one.

412
00:28:35,580 --> 00:28:37,680
It is this one over here.

413
00:28:37,680 --> 00:28:41,750
So it is 0 outside,
and it is only 1

414
00:28:41,750 --> 00:28:46,250
between x equals
minus 1/2 and 1/2.

415
00:28:46,250 --> 00:28:49,045
So in that case,
then, the integral.

416
00:29:02,060 --> 00:29:04,860
That's the integral.

417
00:29:04,860 --> 00:29:06,520
Now, there's a simple
one to calculate.

418
00:29:20,500 --> 00:29:22,360
I don't know how I
picked up a naught here.

419
00:29:22,360 --> 00:29:24,120
This would not be no naught.

420
00:29:24,120 --> 00:29:24,760
Drop this one.

421
00:29:24,760 --> 00:29:26,948
Just e to the i to pi u, right?

422
00:29:26,948 --> 00:29:27,490
Nothing here.

423
00:29:30,410 --> 00:29:40,980
So I have u times 1/2 minus
e to the i 2 pi, u minus 1/2.

424
00:29:43,560 --> 00:29:50,650
And these two minuses cancel.

425
00:29:50,650 --> 00:29:53,590
And if I flip this
around, it is actually

426
00:29:53,590 --> 00:29:55,450
the definition of a sine.

427
00:29:55,450 --> 00:30:04,820
So this equals 1 over
minus i 2 pi u0, times

428
00:30:04,820 --> 00:30:09,260
minus 2i sine of what was here.

429
00:30:09,260 --> 00:30:11,690
The twos also cancel.

430
00:30:11,690 --> 00:30:16,530
And I get pi u.

431
00:30:16,530 --> 00:30:20,430
So finally, after dropping
the remaining constants.

432
00:30:20,430 --> 00:30:21,390
I still got a u0.

433
00:30:21,390 --> 00:30:22,560
There's no u0.

434
00:30:22,560 --> 00:30:25,200
For some reason, my brain
puts a naught there.

435
00:30:25,200 --> 00:30:26,250
There shouldn't be.

436
00:30:26,250 --> 00:30:28,580
So after finally canceling
whatever is left,

437
00:30:28,580 --> 00:30:36,030
I get sine of pi u over pi u.

438
00:30:36,030 --> 00:30:39,430
OK, so that function, by
definition, is called a sinc.

439
00:30:58,490 --> 00:31:02,353
And I'll jump ahead in
the naught a little bit.

440
00:31:02,353 --> 00:31:04,020
You can also look it
up in the textbook.

441
00:31:07,940 --> 00:31:10,830
The sinc function
looks like this.

442
00:31:10,830 --> 00:31:12,020
It is in page--

443
00:31:12,020 --> 00:31:16,550
I forget which page,
between page 12 and page 14

444
00:31:16,550 --> 00:31:18,212
of the Goodman textbook.

445
00:31:18,212 --> 00:31:19,295
This is the sinc function.

446
00:31:21,800 --> 00:31:24,620
One argument equals 0.

447
00:31:24,620 --> 00:31:28,490
2 and it has a peak and
then it kind of oscillates,

448
00:31:28,490 --> 00:31:30,320
but the amplitude
of its oscillation

449
00:31:30,320 --> 00:31:35,330
drops inversely proportional
to the argument.

450
00:31:35,330 --> 00:31:37,700
So the oscillation
comes from the sine.

451
00:31:37,700 --> 00:31:43,258
The inversely proportional comes
from the u in the denominator.

452
00:31:43,258 --> 00:31:45,050
So this may be a little
bit boring for you.

453
00:31:45,050 --> 00:31:47,133
For those of you who have
taken signal processing,

454
00:31:47,133 --> 00:31:50,643
you're probably ready
to go to sleep now.

455
00:31:50,643 --> 00:31:52,560
For reasons of completeness,
we have to do it,

456
00:31:52,560 --> 00:31:55,020
to go through it.

457
00:31:55,020 --> 00:31:58,840
Then I will not compute
too many Fourier integrals.

458
00:31:58,840 --> 00:32:01,760
But in any case, if you are to
compute one Fourier transform,

459
00:32:01,760 --> 00:32:04,470
that's the one to compute.

460
00:32:04,470 --> 00:32:05,370
So that's it, then.

461
00:32:05,370 --> 00:32:08,400
This is also the definition
of the original rect function

462
00:32:08,400 --> 00:32:09,960
that we had.

463
00:32:09,960 --> 00:32:14,400
And its Fourier transform
is the sinc function.

464
00:32:14,400 --> 00:32:17,790
Now, we're not done yet, because
I made one more simplifying

465
00:32:17,790 --> 00:32:19,170
assumption.

466
00:32:19,170 --> 00:32:21,360
I said that x0 equals 1.

467
00:32:21,360 --> 00:32:24,150
So what do we do about this x0?

468
00:32:24,150 --> 00:32:29,840
Well, does anybody know
what I can do about this x0?

469
00:32:42,978 --> 00:32:44,645
OK what I'll do is a
change of variable.

470
00:32:48,000 --> 00:32:49,750
And I will do it in
the general case.

471
00:32:53,170 --> 00:32:57,970
Let's say that
they have a G of u

472
00:32:57,970 --> 00:33:02,200
equals the Fourier transform of
some general function, g of x.

473
00:33:10,450 --> 00:33:16,780
So this is then the Fourier
transform of g of x.

474
00:33:16,780 --> 00:33:24,100
What is the Fourier transform
of a scale version of g of x?

475
00:33:24,100 --> 00:33:25,780
Well, that would be
something like this.

476
00:33:25,780 --> 00:33:31,285
It would be from minus
infinity to infinity, g of ax,

477
00:33:31,285 --> 00:33:36,930
e to the minus i two pi ux dx.

478
00:33:36,930 --> 00:33:40,350
And to get rid of
this ugly thing here,

479
00:33:40,350 --> 00:33:42,870
I can make a change
of coordinates.

480
00:33:42,870 --> 00:33:46,950
For example, let's
say that c equals ax.

481
00:33:46,950 --> 00:33:51,480
Then this means that
dx e equals a dx.

482
00:33:51,480 --> 00:33:54,890
And I can write the integral.

483
00:33:54,890 --> 00:33:57,330
So they actually
become dx e over a.

484
00:33:57,330 --> 00:34:01,620
I pick up 1 over a out here.

485
00:34:01,620 --> 00:34:02,970
Nothing happens to infinity.

486
00:34:02,970 --> 00:34:04,470
It remains infinity.

487
00:34:04,470 --> 00:34:08,025
This would be g of c, e
to the minus i two pi.

488
00:34:10,920 --> 00:34:17,690
Now, x is also c upon a, big C.

489
00:34:17,690 --> 00:34:18,620
I haven't cheated.

490
00:34:18,620 --> 00:34:21,889
This is the transformed--
the integral.

491
00:34:21,889 --> 00:34:24,730
So the 1 over a basically
keeps me honest here.

492
00:34:24,730 --> 00:34:29,480
Makes sure that the area of
the differential is preserved.

493
00:34:29,480 --> 00:34:31,520
It's also known as a Jacobian.

494
00:34:31,520 --> 00:34:33,750
But anyway, that's what it is.

495
00:34:33,750 --> 00:34:39,199
And I can do one more
little manipulation here.

496
00:34:39,199 --> 00:34:40,889
I can rewrite it like this.

497
00:34:55,800 --> 00:34:59,120
And we can recognize
now that this integral--

498
00:34:59,120 --> 00:35:01,120
let me see if I can fit
them both on the screen.

499
00:35:04,550 --> 00:35:09,250
OK, so recognize
that this integral

500
00:35:09,250 --> 00:35:13,710
is the same as this
integral, except

501
00:35:13,710 --> 00:35:17,580
with a different variable,
with a different argument.

502
00:35:17,580 --> 00:35:24,290
So therefore, what I derived
here is one over a G of,

503
00:35:24,290 --> 00:35:26,590
u over a.

504
00:35:30,280 --> 00:35:33,240
OK, so this is a property
of Fourier transforms

505
00:35:33,240 --> 00:35:34,650
known as the scaling theorem.

506
00:35:39,480 --> 00:35:42,180
Or sometimes, people call
it the similarity theorem.

507
00:35:48,640 --> 00:35:54,760
And let's see how we can apply
it to the question at hand.

508
00:35:54,760 --> 00:36:02,190
We derived that the
rectangle function.

509
00:36:02,190 --> 00:36:05,280
If you Fourier transform it,
you get the sinc function.

510
00:36:07,950 --> 00:36:11,640
OK, what I really wanted to
get is the Fourier transform

511
00:36:11,640 --> 00:36:17,620
of a rectangle,
which has a size, x0.

512
00:36:17,620 --> 00:36:20,290
Now let me write down
the scaling theorem.

513
00:36:20,290 --> 00:36:25,840
It says that g of ax
Fourier transforms to 1

514
00:36:25,840 --> 00:36:29,290
over a G of u over a.

515
00:36:29,290 --> 00:36:33,112
So in this case, a is
identical to 1 over x0.

516
00:36:33,112 --> 00:36:34,570
So therefore, the
Fourier transform

517
00:36:34,570 --> 00:36:39,470
will be x0, sinc of x0 times u.

518
00:36:39,470 --> 00:36:44,050
And this is intuitively
satisfying because the units

519
00:36:44,050 --> 00:36:45,910
here inside the sink.

520
00:36:45,910 --> 00:36:48,550
The units are naught.

521
00:36:48,550 --> 00:36:54,100
x0 has dimensions
of space, meters.

522
00:36:54,100 --> 00:36:58,690
u is a frequency, so it has in
dimensions of inverse meters.

523
00:36:58,690 --> 00:37:01,060
So therefore, what I
have inside the argument

524
00:37:01,060 --> 00:37:03,940
has no dimensions at all,
which is, of course, of the way

525
00:37:03,940 --> 00:37:06,050
it should be.

526
00:37:06,050 --> 00:37:08,860
OK, so this is them.

527
00:37:22,080 --> 00:37:25,890
So [INAUDIBLE]
for one last time.

528
00:37:25,890 --> 00:37:29,850
This is how we obtain this
function with the central lobe.

529
00:37:29,850 --> 00:37:34,470
But the side lobe is actually
not quite the sinc function

530
00:37:34,470 --> 00:37:37,230
itself, because I'm
blocking the intensity here.

531
00:37:37,230 --> 00:37:40,830
It is actually sinc squared.

532
00:37:40,830 --> 00:37:43,830
But anyway, this is
where this came from.

533
00:37:43,830 --> 00:37:46,590
So, of course, if you
multiply the two dimensions,

534
00:37:46,590 --> 00:37:50,100
you get a sinc in
the x dimension,

535
00:37:50,100 --> 00:37:52,230
and sinc in the y dimension,
and then, of course,

536
00:37:52,230 --> 00:37:53,940
you get the product.

537
00:37:53,940 --> 00:37:55,830
And the Fourier
transform theorem

538
00:37:55,830 --> 00:38:00,630
says that the final field
will actually be the Fourier

539
00:38:00,630 --> 00:38:06,100
transform, but with
the coordinate,

540
00:38:06,100 --> 00:38:08,450
the special frequency
coordinate replaced

541
00:38:08,450 --> 00:38:12,455
by x prime over lambda z.

542
00:38:12,455 --> 00:38:14,170
This is where this came from.

543
00:38:14,170 --> 00:38:17,180
I substituted u with
x prime over lambda z.

544
00:38:19,690 --> 00:38:22,930
So the bottom line is that
this is perhaps easier

545
00:38:22,930 --> 00:38:26,440
if you look at it heads up.

546
00:38:26,440 --> 00:38:28,540
So here is a
rectangular function.

547
00:38:28,540 --> 00:38:32,440
I only saw the x dimension
here with a size of x0.

548
00:38:32,440 --> 00:38:36,460
Then you can see that
the Fourier transform.

549
00:38:36,460 --> 00:38:38,830
Actually this square, the
intensity of the Fourier

550
00:38:38,830 --> 00:38:39,580
transform.

551
00:38:39,580 --> 00:38:41,560
It has this characteristic
sinc pattern

552
00:38:41,560 --> 00:38:44,410
with a central lobe
and then side lobes.

553
00:38:44,410 --> 00:38:47,650
And the size of the
central lobe is inversely

554
00:38:47,650 --> 00:38:50,380
proportional to the
size of the rectangle.

555
00:38:50,380 --> 00:38:52,750
So if I make this
rectangle smaller,

556
00:38:52,750 --> 00:38:55,030
this size will become bigger.

557
00:38:58,580 --> 00:39:03,260
So this is then our first
Fraunhofer diffraction pattern.

558
00:39:03,260 --> 00:39:04,940
The Fraunhofer
diffraction pattern

559
00:39:04,940 --> 00:39:08,360
of a rectangular function.

560
00:39:08,360 --> 00:39:11,010
Of course, there is
many different apertures

561
00:39:11,010 --> 00:39:13,602
that are of interest
in this business.

562
00:39:13,602 --> 00:39:15,810
Oh, and this, by the way,
is called the sinc pattern,

563
00:39:15,810 --> 00:39:18,090
as I already mentioned.

564
00:39:18,090 --> 00:39:20,460
So there's many different
patterns of interest.

565
00:39:20,460 --> 00:39:23,100
For example, very
often in optics,

566
00:39:23,100 --> 00:39:25,520
we use circular apertures.

567
00:39:25,520 --> 00:39:30,450
Lenses, irises, in cameras,
most optical systems

568
00:39:30,450 --> 00:39:33,210
have a circular aperture.

569
00:39:33,210 --> 00:39:37,150
In this case, we talked
about the blinking

570
00:39:37,150 --> 00:39:38,795
or the Poisson spot here.

571
00:39:38,795 --> 00:39:40,420
But that's not what
I'm interested now.

572
00:39:40,420 --> 00:39:44,000
I'm interested in the far
field diffraction pattern.

573
00:39:44,000 --> 00:39:48,430
And in this case, you also
get a pattern with a--

574
00:39:48,430 --> 00:39:53,090
kind of looks like a sinc, but
a sinc with circular symmetry.

575
00:39:53,090 --> 00:39:55,780
It is not exactly a sinc.

576
00:39:55,780 --> 00:39:58,840
It is given by a rather
nasty formula here.

577
00:39:58,840 --> 00:40:00,640
It is the ratio.

578
00:40:00,640 --> 00:40:03,270
First of all, it is all
done in polar coordinates.

579
00:40:03,270 --> 00:40:05,170
So you see that you
get the square root

580
00:40:05,170 --> 00:40:09,760
of the sum of the Cartesian
coordinates squared.

581
00:40:09,760 --> 00:40:11,950
But the function itself
is given by the ratio

582
00:40:11,950 --> 00:40:14,950
of a Bessel function
of the first kind

583
00:40:14,950 --> 00:40:19,810
and order 1 divided
by its argument.

584
00:40:19,810 --> 00:40:23,100
I will not go into to the
detail of the derivation here.

585
00:40:23,100 --> 00:40:25,570
Goodman describes
it in great detail.

586
00:40:25,570 --> 00:40:29,820
So if you're interested, you can
go and check it out over there.

587
00:40:29,820 --> 00:40:32,890
I do want to emphasize
a couple of things.

588
00:40:32,890 --> 00:40:36,240
First of all, that is
this sometimes by analogy

589
00:40:36,240 --> 00:40:38,050
to the sinc.

590
00:40:38,050 --> 00:40:41,970
This pattern is
referred to as a jinc.

591
00:40:41,970 --> 00:40:45,090
So the J comes, of
course, from the Bessel J.

592
00:40:45,090 --> 00:40:48,300
So we call it a jinc function.

593
00:40:48,300 --> 00:40:52,480
And more commonly it is
referred to as the Airy pattern.

594
00:40:52,480 --> 00:40:56,730
Airy not because it sucks
air or something like that.

595
00:40:56,730 --> 00:40:59,220
Actually, it is named after
someone, some Englishman,

596
00:40:59,220 --> 00:41:01,560
whose name was Airy.

597
00:41:01,560 --> 00:41:03,840
So Airy pattern.

598
00:41:03,840 --> 00:41:06,480
And if you compare it
with the previous one.

599
00:41:08,990 --> 00:41:09,850
The previous one.

600
00:41:14,190 --> 00:41:17,280
I'm sorry, you have endure
this animation again.

601
00:41:17,280 --> 00:41:19,550
OK, so the previous one.

602
00:41:19,550 --> 00:41:24,680
The null actually occurred
that lambda l divided

603
00:41:24,680 --> 00:41:27,800
by the size of the aperture.

604
00:41:27,800 --> 00:41:35,720
In the case of the
jinc, there's a factor

605
00:41:35,720 --> 00:41:39,830
of 1.22 that [INAUDIBLE]
the calculation.

606
00:41:39,830 --> 00:41:43,640
So the null basically occurs
at the very similar looking

607
00:41:43,640 --> 00:41:44,750
variable.

608
00:41:44,750 --> 00:41:50,690
If you make the diameter
shorter, the size of the jinc

609
00:41:50,690 --> 00:41:51,780
will grow.

610
00:41:51,780 --> 00:41:56,120
But the null, the zero of the
jinc, occurs at this function,

611
00:41:56,120 --> 00:41:59,690
at this value, 1.22,
which, of course, comes

612
00:41:59,690 --> 00:42:02,840
from the zero of
the Bessel function.

613
00:42:02,840 --> 00:42:04,900
So there's no intuition here.

614
00:42:04,900 --> 00:42:10,820
It's just where this function
happens to reach value zero.

615
00:42:10,820 --> 00:42:11,320
OK.

616
00:42:14,550 --> 00:42:16,380
Let me skip this
slide, and perhaps you

617
00:42:16,380 --> 00:42:18,950
can go over it and
talk about it later.

618
00:42:18,950 --> 00:42:22,080
It basically elaborates a
little bit on the issue of--

619
00:42:22,080 --> 00:42:25,800
I said before that in order
to observe the Fraunhofer

620
00:42:25,800 --> 00:42:29,150
diffraction pattern, have
to let z become long enough.

621
00:42:29,150 --> 00:42:31,980
Have to propagate
the field far enough.

622
00:42:31,980 --> 00:42:35,610
So this slide answers the
question, well, how far is far?

623
00:42:35,610 --> 00:42:36,910
Let me skip it four now.

624
00:42:36,910 --> 00:42:40,920
And if we have time later,
I will come back to it.

625
00:42:40,920 --> 00:42:42,660
But what I would like
to get started now

626
00:42:42,660 --> 00:42:46,620
is a few comments on
Fourier transform the cells,

627
00:42:46,620 --> 00:42:49,640
and how they apply to
different apertures.

628
00:42:49,640 --> 00:42:53,160
So calling the Fourier transform
is a topic in applied math,

629
00:42:53,160 --> 00:42:54,360
really.

630
00:42:54,360 --> 00:42:58,590
I don't want to
convert this to 18085,

631
00:42:58,590 --> 00:43:01,710
or whatever it is at MIT that
you'll learn those things.

632
00:43:01,710 --> 00:43:05,440
But I will remind you of
some of the basic properties.

633
00:43:05,440 --> 00:43:09,340
So one is the definition
of the Fourier transform.

634
00:43:09,340 --> 00:43:11,020
I already wrote it down.

635
00:43:11,020 --> 00:43:13,330
Many of you are more
familiar with the time domain

636
00:43:13,330 --> 00:43:16,960
definition, where the
Fourier variable is actually

637
00:43:16,960 --> 00:43:20,440
a frequency measured in hertz.

638
00:43:20,440 --> 00:43:23,500
Of course, because here,
we're talking about signals

639
00:43:23,500 --> 00:43:25,180
in the space domain.

640
00:43:25,180 --> 00:43:28,690
The frequency variable
is the spatial frequency,

641
00:43:28,690 --> 00:43:31,510
so the units are
actually inverse meters.

642
00:43:31,510 --> 00:43:33,530
Hertz is inverse second.

643
00:43:33,530 --> 00:43:36,610
The units here are
inverse meters.

644
00:43:36,610 --> 00:43:38,920
And, of course,
because we're dealing

645
00:43:38,920 --> 00:43:41,778
with two dimensional
special variables,

646
00:43:41,778 --> 00:43:43,945
it is a two dimensional
Fourier transform because it

647
00:43:43,945 --> 00:43:45,070
is a double integral.

648
00:43:45,070 --> 00:43:46,727
But other than that,
it's very similar.

649
00:43:46,727 --> 00:43:48,310
The other thing I
wanted to remind you

650
00:43:48,310 --> 00:43:51,580
is that there is an inverse
Fourier transform which

651
00:43:51,580 --> 00:43:55,780
looks very similar, except
for a minus sign, so

652
00:43:55,780 --> 00:43:57,205
into the exponent here.

653
00:43:57,205 --> 00:43:59,080
And, of course, the
inverse Fourier transform

654
00:43:59,080 --> 00:44:02,020
takes you back to the
original function.

655
00:44:02,020 --> 00:44:02,950
So it's like a dance.

656
00:44:02,950 --> 00:44:04,780
You start with a
initial function.

657
00:44:04,780 --> 00:44:06,670
You compute the
Fourier transform,

658
00:44:06,670 --> 00:44:13,420
then you plug it into the into
the inverse Fourier transform,

659
00:44:13,420 --> 00:44:15,580
and you get back
what you started.

660
00:44:15,580 --> 00:44:19,960
That is sometimes referred to
as the Fourier integral instead

661
00:44:19,960 --> 00:44:21,675
of an inverse Fourier transform.

662
00:44:24,730 --> 00:44:28,210
So what is this really,
this Fourier transform?

663
00:44:28,210 --> 00:44:31,920
If you look at its surreal
part, and if you have a real

664
00:44:31,920 --> 00:44:33,250
function here.

665
00:44:33,250 --> 00:44:35,230
Basically, what the
Fourier transform does

666
00:44:35,230 --> 00:44:38,480
is it multiplies this function.

667
00:44:38,480 --> 00:44:41,980
It is denoted as
red here, g of x.

668
00:44:41,980 --> 00:44:44,550
It multiplies with a sinusoid.

669
00:44:44,550 --> 00:44:47,960
The real part of this complex
exponential is a cosine.

670
00:44:47,960 --> 00:44:50,020
So you multiply the
function with this cosine,

671
00:44:50,020 --> 00:44:52,270
and then you integrate.

672
00:44:52,270 --> 00:44:55,300
OK, so why do you do
something like that?

673
00:44:55,300 --> 00:44:58,720
Actually, does anybody
know why Fourier came up

674
00:44:58,720 --> 00:45:01,420
with this kind of transform?

675
00:45:04,190 --> 00:45:06,860
What was the context
that Fourier--

676
00:45:06,860 --> 00:45:08,270
what was Fourier?

677
00:45:08,270 --> 00:45:10,190
Fourier was a French
mathematician,

678
00:45:10,190 --> 00:45:12,200
or a French applied
physicist, I guess.

679
00:45:12,200 --> 00:45:14,490
And he was trying to solve
a particular problem.

680
00:45:14,490 --> 00:45:16,190
Does anybody know
what's the problem

681
00:45:16,190 --> 00:45:18,620
he was trying to solve when
he came up with this business?

682
00:45:23,610 --> 00:45:26,120
OK, it was a problem
of heat transfer.

683
00:45:26,120 --> 00:45:29,300
Fourier was trying to
solve the problem of what

684
00:45:29,300 --> 00:45:34,130
is that temperature distribution
between two hot plates, one

685
00:45:34,130 --> 00:45:37,680
of them at temperature t1,
that at temperature of t2.

686
00:45:37,680 --> 00:45:40,180
And actually, the answer is not
given by a Fourier integral.

687
00:45:40,180 --> 00:45:42,660
It is given by a Fourier series.

688
00:45:42,660 --> 00:45:46,130
And if you make the plates go.

689
00:45:46,130 --> 00:45:48,620
If you increase the
distance between the plates,

690
00:45:48,620 --> 00:45:50,810
the Fourier series
becomes an integral.

691
00:45:50,810 --> 00:45:55,010
So this entire
mathematical arsenal here,

692
00:45:55,010 --> 00:45:58,010
it actually came from the
field of heat transfer,

693
00:45:58,010 --> 00:45:59,640
interestingly enough.

694
00:45:59,640 --> 00:46:01,810
Anyway, that is of no
concern to us here.

695
00:46:01,810 --> 00:46:04,490
The Fourier transform,
as many of you know--

696
00:46:04,490 --> 00:46:07,580
especially those who
do acoustics or signal

697
00:46:07,580 --> 00:46:10,010
processing-- it has
tremendous applications

698
00:46:10,010 --> 00:46:11,320
in signal processing nowadays.

699
00:46:11,320 --> 00:46:14,140
And, of course, it is still
used in heat transfer.

700
00:46:14,140 --> 00:46:18,740
But in our context here, it
is more signal processing

701
00:46:18,740 --> 00:46:20,630
that we will use it.

702
00:46:20,630 --> 00:46:23,090
OK, so why do we
multiply by a sinusoid?

703
00:46:23,090 --> 00:46:25,020
Well, the reason
is the following.

704
00:46:25,020 --> 00:46:31,760
Suppose that G, our transformed
function, is itself a sinusoid.

705
00:46:31,760 --> 00:46:34,900
OK, so here is G with a
particular frequency, u0.

706
00:46:39,670 --> 00:46:42,640
So G is the red sinusoid.

707
00:46:42,640 --> 00:46:48,280
The Fourier transform
kernel is another sinusoid.

708
00:46:48,280 --> 00:46:52,360
And in general, they
have different frequency,

709
00:46:52,360 --> 00:46:53,843
like shown here.

710
00:46:53,843 --> 00:46:55,510
So what does the value
of this integral?

711
00:46:55,510 --> 00:46:56,010
Do you know?

712
00:47:01,270 --> 00:47:03,100
If the two frequencies
are different.

713
00:47:03,100 --> 00:47:06,700
If you multiply two
sinusoids and integrate them

714
00:47:06,700 --> 00:47:09,370
over a very long distance,
actually infinite.

715
00:47:12,160 --> 00:47:13,967
Actually, by convention
in this class.

716
00:47:13,967 --> 00:47:15,550
I don't know if I
mentioned it before.

717
00:47:15,550 --> 00:47:19,710
The convention, if I don't
put bounds to an integral,

718
00:47:19,710 --> 00:47:23,030
that mean it goes from minus
infinity to plus infinity.

719
00:47:23,030 --> 00:47:26,470
So this is an infinite
integral of two sinusoids

720
00:47:26,470 --> 00:47:30,250
with different
frequency multiplied.

721
00:47:30,250 --> 00:47:31,240
What is the answer?

722
00:47:31,240 --> 00:47:32,850
AUDIENCE: Zero?

723
00:47:32,850 --> 00:47:34,200
GEORGE BARBASTATHIS: Zero, yeah.

724
00:47:34,200 --> 00:47:39,030
Because the various oscillations
that will cancel eventually.

725
00:47:39,030 --> 00:47:41,990
So you'll get nothing.

726
00:47:41,990 --> 00:47:47,240
OK, however, there's
a singular case when

727
00:47:47,240 --> 00:47:50,900
the frequencies are the same.

728
00:47:50,900 --> 00:47:54,370
And what is the value of
this integral in this case?

729
00:47:54,370 --> 00:47:55,360
Well, infinite, right?

730
00:47:55,360 --> 00:47:59,300
Because if you multiply
them, this will be positive.

731
00:47:59,300 --> 00:48:01,620
This will also be positive,
because you are multiplying

732
00:48:01,620 --> 00:48:03,530
two negative quantities.

733
00:48:03,530 --> 00:48:06,970
So you actually get infinity,
which is not very good.

734
00:48:06,970 --> 00:48:08,560
But in mathematics,
we have a way

735
00:48:08,560 --> 00:48:11,450
of dealing with this kind
of abrupt infinities.

736
00:48:11,450 --> 00:48:13,730
We call them delta functions.

737
00:48:13,730 --> 00:48:17,350
And, of course, I'm severely
abusing the mathematics here.

738
00:48:17,350 --> 00:48:19,840
The way the delta
function comes up.

739
00:48:19,840 --> 00:48:21,130
Does anybody know?

740
00:48:21,130 --> 00:48:22,025
It comes as a limit.

741
00:48:22,025 --> 00:48:23,650
The way you get a
delta function is you

742
00:48:23,650 --> 00:48:28,000
actually bound this integral,
so that you get a finite value.

743
00:48:28,000 --> 00:48:30,670
And then you let the
bound go to infinity,

744
00:48:30,670 --> 00:48:33,280
and the limit is
a delta function.

745
00:48:33,280 --> 00:48:36,910
Anyway, without going into
these mathematical intricacies,

746
00:48:36,910 --> 00:48:41,710
we can represent this
situation here as--

747
00:48:41,710 --> 00:48:45,170
OK, forget for forget the second
delta function for a moment.

748
00:48:45,170 --> 00:48:49,120
But this situation where
the value of the integral

749
00:48:49,120 --> 00:48:53,220
is zero for all
frequencies except one.

750
00:48:53,220 --> 00:48:56,010
Because the integral
assumes a huge value.

751
00:48:56,010 --> 00:48:59,390
Then we write it as
a delta function.

752
00:48:59,390 --> 00:49:01,460
And the why we get
two delta functions.

753
00:49:01,460 --> 00:49:03,530
Well, we'll get
two delta functions

754
00:49:03,530 --> 00:49:08,050
because the way this works
is if you take the Fourier

755
00:49:08,050 --> 00:49:16,310
transform of an
exponential, this

756
00:49:16,310 --> 00:49:19,760
is a single delta function.

757
00:49:19,760 --> 00:49:21,060
Now, if you taking the cosine.

758
00:49:34,050 --> 00:49:37,190
Of course, the cosine is a sum
of two complex exponentials.

759
00:49:48,930 --> 00:49:50,580
And now we know how
to deal with this.

760
00:49:50,580 --> 00:49:53,520
It's one of those that's given
by an expression like this one.

761
00:49:53,520 --> 00:49:56,130
So you actually get two
symmetric delta functions.

762
00:50:03,710 --> 00:50:06,210
OK, so what is
the one half here?

763
00:50:06,210 --> 00:50:11,390
Well, the one half is
actually the energy contained

764
00:50:11,390 --> 00:50:12,440
in this delta function.

765
00:50:16,730 --> 00:50:17,925
So that's normal.

766
00:50:17,925 --> 00:50:19,550
The thing is that is
a little bit weird

767
00:50:19,550 --> 00:50:21,920
about this is that
this sort of situation

768
00:50:21,920 --> 00:50:25,640
implies that there's such a
thing as negative frequency.

769
00:50:25,640 --> 00:50:28,130
Of course, there's no
negative frequencies.

770
00:50:28,130 --> 00:50:30,290
The frequencies can
only be positive.

771
00:50:30,290 --> 00:50:32,360
The reason that we need
a negative frequency

772
00:50:32,360 --> 00:50:35,300
is actually for
mathematical rigor,

773
00:50:35,300 --> 00:50:38,100
because we insisted
on using phasors.

774
00:50:38,100 --> 00:50:40,610
You remember a
long time ago, when

775
00:50:40,610 --> 00:50:42,080
we started talking about waves.

776
00:50:42,080 --> 00:50:45,230
We said that waves are
real, so they are actually

777
00:50:45,230 --> 00:50:47,030
cosine functions.

778
00:50:47,030 --> 00:50:49,400
But for mathematical
convenience,

779
00:50:49,400 --> 00:50:52,970
in order to avoid complicated
trigonometric calculations,

780
00:50:52,970 --> 00:50:55,640
it would represent
this cosine function

781
00:50:55,640 --> 00:50:57,620
as a complex exponential.

782
00:50:57,620 --> 00:51:02,070
Well, if you really had the
simple cosine transform.

783
00:51:02,070 --> 00:51:07,430
So you use the cosine into
the kernel for the integral.

784
00:51:07,430 --> 00:51:10,160
That is known as a
Fourier cosine transform,

785
00:51:10,160 --> 00:51:12,530
and then it contains only
positive frequencies.

786
00:51:12,530 --> 00:51:13,940
But it's nice to calculate.

787
00:51:13,940 --> 00:51:15,650
Gives you very ugly formulas.

788
00:51:15,650 --> 00:51:18,020
So that's why we'll use
the complex exponential.

789
00:51:18,020 --> 00:51:20,780
It is simpler formulas,
but the price we pay

790
00:51:20,780 --> 00:51:22,880
is this weird
negative frequency.

791
00:51:22,880 --> 00:51:24,590
So there's nothing
to worry about.

792
00:51:24,590 --> 00:51:27,200
It is not wrong
physics in any way.

793
00:51:27,200 --> 00:51:29,930
It is simply a matter of
mathematical convenience

794
00:51:29,930 --> 00:51:32,926
that leads to these
negative frequencies.

795
00:51:37,770 --> 00:51:43,350
And, of course, I will not go
through all these derivations

796
00:51:43,350 --> 00:51:44,160
over here.

797
00:51:44,160 --> 00:51:46,860
But several functions,
their Fourier transforms

798
00:51:46,860 --> 00:51:49,600
can be computed in
[INAUDIBLE] form.

799
00:51:49,600 --> 00:51:53,010
In fact, all of these functions,
you can go ahead if you like,

800
00:51:53,010 --> 00:51:55,260
and do the Fourier
transform by yourselves.

801
00:51:55,260 --> 00:51:59,280
It is relatively simple
mathematical exercise.

802
00:51:59,280 --> 00:52:01,260
So we will use some
of these very often.

803
00:52:01,260 --> 00:52:03,930
Mostly, we'll use the
rectangular function.

804
00:52:03,930 --> 00:52:05,600
I already talked about this one.

805
00:52:05,600 --> 00:52:06,970
We'll use the circular function.

806
00:52:06,970 --> 00:52:08,960
I talked a bit very briefly.

807
00:52:08,960 --> 00:52:11,940
Then there is the
triangular function, which

808
00:52:11,940 --> 00:52:14,320
has a shot of a grayscale.

809
00:52:14,320 --> 00:52:17,370
It starts from zero, then
progressively it goes to one,

810
00:52:17,370 --> 00:52:19,500
and then drops
back down to zero.

811
00:52:19,500 --> 00:52:24,550
In linear fashion, and the com.

812
00:52:24,550 --> 00:52:26,650
The composite sequence
of delta functions, that

813
00:52:26,650 --> 00:52:29,530
is very useful in sampling.

814
00:52:29,530 --> 00:52:32,410
I don't use it very much
in this class, actually.

815
00:52:32,410 --> 00:52:35,350
I sort of bypass the
issue of sampling.

816
00:52:35,350 --> 00:52:38,200
But I'm sure all of you are
familiar with Nyquist sampling

817
00:52:38,200 --> 00:52:40,690
rates, Nyquist frequencies,
and so on and so forth.

818
00:52:40,690 --> 00:52:43,280
So these all can be explained
by the com function.

819
00:52:43,280 --> 00:52:45,785
And Goodman has a
section in the book.

820
00:52:45,785 --> 00:52:46,660
I forget where it is.

821
00:52:46,660 --> 00:52:48,660
It's a section two
point something.

822
00:52:53,400 --> 00:52:56,858
Yeah, section 2.4, two
dimensional sampling

823
00:52:56,858 --> 00:52:57,900
theory that goes over it.

824
00:52:57,900 --> 00:53:00,350
I will not go over
it in the class.

825
00:53:00,350 --> 00:53:05,550
But it may be good idea
for you to review it.

826
00:53:05,550 --> 00:53:08,040
OK.

827
00:53:08,040 --> 00:53:12,560
So as I said, there's
several functions here

828
00:53:12,560 --> 00:53:16,230
who's Fourier transforms
can be computed.

829
00:53:16,230 --> 00:53:19,560
I will not go through all of
these, but it is good for you

830
00:53:19,560 --> 00:53:21,990
to know where this kind
of thing is in the book,

831
00:53:21,990 --> 00:53:24,060
so when necessary,
you can refer to them,

832
00:53:24,060 --> 00:53:27,520
and you can get the answers
for values [INAUDIBLE]..

833
00:53:27,520 --> 00:53:30,450
So for example, here is
the rectangle function

834
00:53:30,450 --> 00:53:32,670
that we computed before.

835
00:53:32,670 --> 00:53:35,340
And, of course, it
gives the sinc response.

836
00:53:35,340 --> 00:53:41,490
Another one worth remembering is
Gaussian, a Gaussian function.

837
00:53:41,490 --> 00:53:45,190
Actually, also Fourier
transforms to Gaussian,

838
00:53:45,190 --> 00:53:46,560
which is interesting.

839
00:53:46,560 --> 00:53:52,230
And another useful one that
we will deal with later

840
00:53:52,230 --> 00:53:53,430
is this one.

841
00:53:53,430 --> 00:53:56,560
You should look at
them all before last.

842
00:53:56,560 --> 00:54:00,860
It also looks like a
Gaussian, but with a J here.

843
00:54:00,860 --> 00:54:01,840
So this we recognize.

844
00:54:01,840 --> 00:54:03,310
Physically, what
is this function?

845
00:54:03,310 --> 00:54:06,100
It is a complex
quadratic exponential.

846
00:54:06,100 --> 00:54:07,870
Physically, what did we call it?

847
00:54:18,513 --> 00:54:20,305
If I write it in a
slightly different form,

848
00:54:20,305 --> 00:54:21,680
you will recognize
it right away.

849
00:54:27,438 --> 00:54:27,980
What is this?

850
00:54:42,860 --> 00:54:44,497
AUDIENCE: Spherical
wave along z?

851
00:54:44,497 --> 00:54:46,330
GEORGE BARBASTATHIS:
It is a spherical wave,

852
00:54:46,330 --> 00:54:48,280
propagating a distance z.

853
00:54:48,280 --> 00:54:49,880
So what you see over
there is actually

854
00:54:49,880 --> 00:54:52,810
a spherical wave with a
slightly weird definition,

855
00:54:52,810 --> 00:54:56,280
a squared equals
1 over lambda z.

856
00:54:56,280 --> 00:54:58,900
So this expression here
in the row before last

857
00:54:58,900 --> 00:55:00,460
is a spherical wave.

858
00:55:00,460 --> 00:55:05,137
So a Fourier transform
is also a spherical wave.

859
00:55:05,137 --> 00:55:06,970
And we will use this
Fourier transform quite

860
00:55:06,970 --> 00:55:10,375
a bit in the next two lectures.

861
00:55:10,375 --> 00:55:12,500
It might be good if you
start studying, by the way,

862
00:55:12,500 --> 00:55:15,102
if you don't know what this is,
it means you haven't studied.

863
00:55:15,102 --> 00:55:16,810
And I don't know how
you did the homework

864
00:55:16,810 --> 00:55:20,745
without studying, possibly by
copying from the last year.

865
00:55:20,745 --> 00:55:22,120
But I strongly
recommend that you

866
00:55:22,120 --> 00:55:26,020
don't do that, because
you're presumably here

867
00:55:26,020 --> 00:55:27,028
in order to learn.

868
00:55:27,028 --> 00:55:28,570
And you don't learn
unless you study.

869
00:55:32,290 --> 00:55:37,222
So it is about time, not
because of the quiz, but anyway.

870
00:55:37,222 --> 00:55:39,180
The quiz is also coming
up, so it is about time

871
00:55:39,180 --> 00:55:41,380
to start studying it.

872
00:55:41,380 --> 00:55:43,470
So this is like a
friendly advice,

873
00:55:43,470 --> 00:55:46,110
I guess, from an older guy.

874
00:55:46,110 --> 00:55:48,100
Study.

875
00:55:48,100 --> 00:55:48,600
OK.

876
00:55:59,500 --> 00:56:03,910
[INAUDIBLE] that the
Fourier transform has.

877
00:56:03,910 --> 00:56:07,870
Once we have this basic Fourier
transforms that are shown here,

878
00:56:07,870 --> 00:56:10,690
then we can compute even
more Fourier transforms

879
00:56:10,690 --> 00:56:17,530
by using the various properties
of the Fourier transform.

880
00:56:17,530 --> 00:56:19,330
So one of those we
wanted to derive.

881
00:56:19,330 --> 00:56:21,100
This is the scaling theorem.

882
00:56:21,100 --> 00:56:23,680
I did this at the
beginning of the class.

883
00:56:23,680 --> 00:56:28,510
And it tells you that if
you scale the argument that

884
00:56:28,510 --> 00:56:30,670
goes inside the
Fourier transform,

885
00:56:30,670 --> 00:56:33,920
then the Fourier transform
itself scales the opposite way.

886
00:56:33,920 --> 00:56:37,270
So for example, in the case
of the Fraunhofer diffraction,

887
00:56:37,270 --> 00:56:39,990
it says that if you make
an aperture smaller,

888
00:56:39,990 --> 00:56:43,610
its Fraunhofer diffraction
pattern becomes larger.

889
00:56:43,610 --> 00:56:46,990
So this is the scaling theorem,
physical and mathematical.

890
00:56:46,990 --> 00:56:49,450
Physical, it tells you that
the Fraunhofer diffraction

891
00:56:49,450 --> 00:56:50,620
becomes bigger.

892
00:56:50,620 --> 00:56:54,040
Mathematically, it comes
from this scaling property

893
00:56:54,040 --> 00:56:55,870
of the Fourier transforms.

894
00:56:55,870 --> 00:56:58,990
Another important one is
the scaling theorem, which

895
00:56:58,990 --> 00:57:02,530
will prove a little bit later.

896
00:57:02,530 --> 00:57:06,400
But it's also very
important one.

897
00:57:06,400 --> 00:57:09,140
Actually, all of these
properties are very important.

898
00:57:09,140 --> 00:57:12,340
Number four is actually
energy conservation.

899
00:57:12,340 --> 00:57:15,280
It relates the modulus--

900
00:57:15,280 --> 00:57:18,810
the integral of the
modulus of a function.

901
00:57:18,810 --> 00:57:20,270
We recognize this as energy.

902
00:57:20,270 --> 00:57:23,350
If you look at number
four, magnitude g squared

903
00:57:23,350 --> 00:57:25,940
is actually intensity.

904
00:57:25,940 --> 00:57:30,460
And if you integrate intensity
over the entire plane,

905
00:57:30,460 --> 00:57:35,147
then you'll get, of course,
energy flux, you get power.

906
00:57:35,147 --> 00:57:36,730
And power has to be
conserved, so this

907
00:57:36,730 --> 00:57:40,950
is what this theorem says, very
important, Parseval's theorem.

908
00:57:40,950 --> 00:57:45,290
And the convolution theorem
is also very important.

909
00:57:45,290 --> 00:57:47,710
We'll see an application
a little bit later today,

910
00:57:47,710 --> 00:57:50,290
or maybe Monday if we
run out of time today.

911
00:57:50,290 --> 00:57:53,304
But anyway, all of these
are very important.

912
00:57:59,730 --> 00:58:03,520
OK, so I will show
you some Fourier

913
00:58:03,520 --> 00:58:08,963
transforms to sort of give
you some of the properties.

914
00:58:13,860 --> 00:58:16,910
Are we still on?

915
00:58:16,910 --> 00:58:17,608
AUDIENCE: Yes.

916
00:58:17,608 --> 00:58:18,900
GEORGE BARBASTATHIS: Thank you.

917
00:58:27,080 --> 00:58:32,210
So this is a sinusoid.

918
00:58:32,210 --> 00:58:36,175
Of course, this is not
a physical transparency.

919
00:58:36,175 --> 00:58:38,300
Well, I can make a physical
transparency like this,

920
00:58:38,300 --> 00:58:40,550
but this assumes
negative values, which

921
00:58:40,550 --> 00:58:43,520
means that to make a physical
transparency like this one,

922
00:58:43,520 --> 00:58:46,490
you would have to have
a phase delay, as well

923
00:58:46,490 --> 00:58:53,870
as a grayscale variation.

924
00:58:53,870 --> 00:59:00,090
What I'm trying to say is
that if you have a cosine,

925
00:59:00,090 --> 00:59:01,070
two pi ux.

926
00:59:04,350 --> 00:59:10,110
Its magnitude goes like this.

927
00:59:16,620 --> 00:59:17,325
And its phase.

928
00:59:24,400 --> 00:59:25,150
What is the phase?

929
00:59:33,267 --> 00:59:34,600
What is the phase of the cosine?

930
00:59:52,500 --> 00:59:54,410
What is the phase of a
positive real number?

931
00:59:58,910 --> 01:00:01,040
What is the phase button?

932
01:00:01,040 --> 01:00:03,700
Someone said zero here.

933
01:00:03,700 --> 01:00:05,360
And that's correct.

934
01:00:05,360 --> 01:00:07,240
What is the phase of a
negative real number?

935
01:00:14,240 --> 01:00:17,800
Someone here suggesting zero.

936
01:00:17,800 --> 01:00:18,850
Negative real number.

937
01:00:21,710 --> 01:00:23,543
AUDIENCE: 180 degrees?

938
01:00:23,543 --> 01:00:25,210
GEORGE BARBASTATHIS:
Fine, that's right.

939
01:00:25,210 --> 01:00:29,830
So therefore, the
phase of the cosine

940
01:00:29,830 --> 01:00:35,340
is zero, where the
cosine is positive,

941
01:00:35,340 --> 01:00:39,346
and jumps to pi, where
the cosine is negative.

942
01:00:55,102 --> 01:00:57,560
OK, so that would be a very
difficult transparency to make,

943
01:00:57,560 --> 01:00:57,770
right?

944
01:00:57,770 --> 01:01:00,250
Because you would have to
have the grayscale variation

945
01:01:00,250 --> 01:01:04,270
like this to impose
the variation

946
01:01:04,270 --> 01:01:05,362
the amplitude modulation.

947
01:01:05,362 --> 01:01:07,570
And then you would have to
impose some variable phase

948
01:01:07,570 --> 01:01:14,360
delay also, in order to
impose the phase delay.

949
01:01:14,360 --> 01:01:16,590
So that is difficult to do.

950
01:01:16,590 --> 01:01:19,310
But anyway, mathematically,
we can write anything we like.

951
01:01:19,310 --> 01:01:22,580
So this is the cosine, and its
Fourier transform, of course,

952
01:01:22,580 --> 01:01:25,580
consists of two delta functions.

953
01:01:25,580 --> 01:01:29,210
So this is what these bright
dots indicate, delta functions

954
01:01:29,210 --> 01:01:35,130
whose spacing equals the inverse
of the period of the cosine.

955
01:01:35,130 --> 01:01:39,700
And, of course, if you
squeeze the cosine,

956
01:01:39,700 --> 01:01:43,470
since the spacing
equals the period,

957
01:01:43,470 --> 01:01:47,510
then the two delta functions
will go further away.

958
01:01:47,510 --> 01:01:49,700
Another way to describe
the same is, of course,

959
01:01:49,700 --> 01:01:51,230
by the scaling theorem.

960
01:01:51,230 --> 01:01:53,500
If you squeeze, it's
equivalent to scaling

961
01:01:53,500 --> 01:01:59,020
by a quantity larger than one.

962
01:01:59,020 --> 01:02:01,420
And therefore, the
spacing will also

963
01:02:01,420 --> 01:02:05,290
scale by a quantity
larger than one.

964
01:02:05,290 --> 01:02:07,420
What is a more
physical transparency

965
01:02:07,420 --> 01:02:08,950
that we saw in the
previous lecture?

966
01:02:15,600 --> 01:02:17,160
I said that this
is difficult to do,

967
01:02:17,160 --> 01:02:20,220
because you have to impose
both amplitude and phase

968
01:02:20,220 --> 01:02:24,047
variation on the transparency.

969
01:02:24,047 --> 01:02:25,380
AUDIENCE: A binary transparency?

970
01:02:25,380 --> 01:02:27,213
GEORGE BARBASTATHIS: A
binary, that's right.

971
01:02:27,213 --> 01:02:28,230
That's right.

972
01:02:28,230 --> 01:02:33,640
If you add a transparency whose
magnitude looks like this.

973
01:02:41,200 --> 01:02:43,010
Goes between zero and one.

974
01:02:43,010 --> 01:02:43,850
That is fine, right?

975
01:02:43,850 --> 01:02:47,730
I can do it very simply by
taking a piece of glass.

976
01:02:47,730 --> 01:02:51,050
And I can deposit some
metal, for example, aluminum,

977
01:02:51,050 --> 01:02:56,070
or chromium, or more something
like that in these regions.

978
01:02:56,070 --> 01:02:58,240
Oops, so you can not
see what they were.

979
01:02:58,240 --> 01:02:58,740
Yeah.

980
01:03:10,922 --> 01:03:11,630
Sorry about that.

981
01:03:11,630 --> 01:03:15,210
I pushed a button here that
I should not have pushed.

982
01:03:15,210 --> 01:03:18,110
So in these regions where I have
the [INAUDIBLE] to the metal,

983
01:03:18,110 --> 01:03:20,630
the transmissivity goes to zero.

984
01:03:20,630 --> 01:03:24,170
Another transparency that we
saw, and it is also physical,

985
01:03:24,170 --> 01:03:24,920
is this one.

986
01:03:27,643 --> 01:03:30,060
That was actually the first
example that we did on Monday.

987
01:03:39,360 --> 01:03:41,616
OK, how do I express
this transparency?

988
01:03:44,740 --> 01:03:45,355
Is it cosine?

989
01:04:21,570 --> 01:04:25,133
AUDIENCE: Is it 1/2
plus 1/2 cosine?

990
01:04:25,133 --> 01:04:26,216
GEORGE BARBASTATHIS: Yeah.

991
01:04:31,440 --> 01:04:33,630
Because it swings between
zero and one, right?

992
01:04:33,630 --> 01:04:34,720
So this, this will do it.

993
01:04:39,880 --> 01:04:41,980
Each Fourier
transform of this one.

994
01:04:41,980 --> 01:04:43,930
How would it be different
than the Fourier

995
01:04:43,930 --> 01:04:47,970
transform of the cosine that
I have on my slide here?

996
01:04:54,560 --> 01:04:57,740
What is the Fourier
transform of this one?

997
01:04:57,740 --> 01:05:00,650
AUDIENCE: So you
have a DC component.

998
01:05:00,650 --> 01:05:05,330
You have a DC component,
and then yeah.

999
01:05:05,330 --> 01:05:08,210
The magnitude of that
frequency is half of it.

1000
01:05:19,650 --> 01:05:28,800
GEORGE BARBASTATHIS: So
in this representation,

1001
01:05:28,800 --> 01:05:33,015
I would still have the tool
delta functions at spacing.

1002
01:05:42,550 --> 01:05:44,260
But also in addition,
I would have

1003
01:05:44,260 --> 01:05:47,030
an extra spot in the center.

1004
01:05:47,030 --> 01:05:48,820
And this part would be brighter.

1005
01:05:48,820 --> 01:05:53,320
So the power that goes into
this part correspondingly

1006
01:05:53,320 --> 01:06:01,610
would be 1/2, 1/4, 1/4 squared.

1007
01:06:10,060 --> 01:06:12,490
So the spot that
goes into the center

1008
01:06:12,490 --> 01:06:16,140
is what you very correctly
refer to as the DC thermal.

1009
01:06:18,810 --> 01:06:20,860
And now, of course, we
know why we call it DC.

1010
01:06:20,860 --> 01:06:22,790
I think I mentioned
it also last time.

1011
01:06:22,790 --> 01:06:25,730
It's because it corresponds
to zero frequency.

1012
01:06:25,730 --> 01:06:29,870
So in electrical signals,
the zero frequency

1013
01:06:29,870 --> 01:06:35,930
is known as the direct
current, or DC, DC component.

1014
01:06:35,930 --> 01:06:38,290
OK, now without
cheating, that is

1015
01:06:38,290 --> 01:06:43,960
without looking at the
next page of the notes.

1016
01:06:43,960 --> 01:06:46,210
I would like to ask you, and
see if someone can guess.

1017
01:06:46,210 --> 01:06:51,070
If I rotate this
grating by some angle

1018
01:06:51,070 --> 01:06:53,100
what will happen to
the Fourier transform?

1019
01:07:04,120 --> 01:07:04,620
Yeah?

1020
01:07:08,090 --> 01:07:09,950
AUDIENCE: If you rotate
it by 90 degrees,

1021
01:07:09,950 --> 01:07:13,573
I'd expect the frequencies to
rotate by 90 degrees, as well.

1022
01:07:13,573 --> 01:07:14,990
GEORGE BARBASTATHIS:
That's right.

1023
01:07:14,990 --> 01:07:16,730
So if you rotate
by 90 degrees, you

1024
01:07:16,730 --> 01:07:20,930
expect the two spots to appear
along the V-axis rather than

1025
01:07:20,930 --> 01:07:23,040
the U-axis.

1026
01:07:23,040 --> 01:07:26,787
If you rotate
somewhere in between,

1027
01:07:26,787 --> 01:07:28,120
where would this [INAUDIBLE] go?

1028
01:07:28,120 --> 01:07:31,110
They will also rotate
in what fashion?

1029
01:07:34,050 --> 01:07:35,730
OK, so the observation
to make here

1030
01:07:35,730 --> 01:07:39,640
is that the two
spots if you draw

1031
01:07:39,640 --> 01:07:46,030
a line that connects the
two Fourier delta functions.

1032
01:07:46,030 --> 01:07:48,600
These lines would be
perpendicular to the fringes

1033
01:07:48,600 --> 01:07:50,450
of the grating.

1034
01:07:50,450 --> 01:07:53,710
And this will remain true as
you rotate the grating then,

1035
01:07:53,710 --> 01:07:56,110
because actually, the
Fourier transform does not

1036
01:07:56,110 --> 01:07:58,810
know what the coordinates are.

1037
01:07:58,810 --> 01:08:01,900
So the Fourier transform knows
that you have a variation

1038
01:08:01,900 --> 01:08:03,650
along this direction.

1039
01:08:03,650 --> 01:08:05,320
And that gives rise
to the two delta

1040
01:08:05,320 --> 01:08:09,160
function in this direction.

1041
01:08:09,160 --> 01:08:12,700
In the vertical direction,
there's no variation.

1042
01:08:12,700 --> 01:08:14,170
So therefore, the
Fourier transform

1043
01:08:14,170 --> 01:08:16,939
is confined to the
zero frequency.

1044
01:08:16,939 --> 01:08:20,350
So if you rotate the
grating, then these spots

1045
01:08:20,350 --> 01:08:24,189
will rotate so that the
line connecting them

1046
01:08:24,189 --> 01:08:28,120
remains perpendicular
to the fringes.

1047
01:08:28,120 --> 01:08:34,970
This may not show quite right
because the projector actually

1048
01:08:34,970 --> 01:08:37,640
squeezes my slide.

1049
01:08:37,640 --> 01:08:40,495
So it may not show quite right.

1050
01:08:40,495 --> 01:08:42,870
But if you think about it,
you should convince yourselves

1051
01:08:42,870 --> 01:08:46,279
that the two spots
should be located

1052
01:08:46,279 --> 01:08:52,270
along a line perpendicular to
the grooves of this gradient.

1053
01:08:52,270 --> 01:08:54,189
And, of course, if you
squeeze the grooves

1054
01:08:54,189 --> 01:08:56,770
in this rotated grating,
then the two spots

1055
01:08:56,770 --> 01:09:00,399
will also move away,
again, along the same line

1056
01:09:00,399 --> 01:09:01,750
perpendicular to the grooves.

1057
01:09:05,338 --> 01:09:06,380
Any questions about that?

1058
01:09:13,370 --> 01:09:15,500
The other property of the
Fourier transform farm

1059
01:09:15,500 --> 01:09:17,990
which is listed in
the table of formulas

1060
01:09:17,990 --> 01:09:21,710
that they showed
earlier is linearity.

1061
01:09:21,710 --> 01:09:25,340
And linearity says that if
you have a function that

1062
01:09:25,340 --> 01:09:30,779
is the linear superposition
of two functions whose Fourier

1063
01:09:30,779 --> 01:09:33,859
transform you know, then
the Fourier transform

1064
01:09:33,859 --> 01:09:38,660
of this function is the linear
superposition of the two

1065
01:09:38,660 --> 01:09:39,800
Fourier transforms.

1066
01:09:39,800 --> 01:09:41,300
So for example,
here is a function

1067
01:09:41,300 --> 01:09:46,149
consisting of two gratings of
period lambda one and lambda

1068
01:09:46,149 --> 01:09:46,649
two.

1069
01:09:49,850 --> 01:09:51,500
Which one is the
Fourier transform?

1070
01:09:51,500 --> 01:09:54,140
That's the Fourier transform
of the long period, right?

1071
01:09:54,140 --> 01:09:56,120
Because the two spots
are close together.

1072
01:09:56,120 --> 01:09:59,610
If you take the Fourier
transform of the short period,

1073
01:09:59,610 --> 01:10:02,710
the Fourier transforms
are further apart.

1074
01:10:02,710 --> 01:10:04,960
If you do the superposition now.

1075
01:10:04,960 --> 01:10:07,930
What you get, well, it is a bit.

1076
01:10:07,930 --> 01:10:10,430
If you superimpose
two frequencies,

1077
01:10:10,430 --> 01:10:11,950
you get the bit pattern.

1078
01:10:11,950 --> 01:10:13,420
Here it is.

1079
01:10:13,420 --> 01:10:15,200
Looks kind of messy.

1080
01:10:15,200 --> 01:10:17,280
The Fourier transform
is relatively cleaner.

1081
01:10:17,280 --> 01:10:20,440
It is the two dots that
you get from this one,

1082
01:10:20,440 --> 01:10:22,870
plus the two dots that you
got from the other one.

1083
01:10:22,870 --> 01:10:25,330
So therefore, you
get four dots total.

1084
01:10:25,330 --> 01:10:28,630
That's what the
superposition theorem says.

1085
01:10:28,630 --> 01:10:31,360
Of course, you can generalize.

1086
01:10:31,360 --> 01:10:33,570
I don't know if you can
see in both of them.

1087
01:10:33,570 --> 01:10:38,520
On the top right here,
there's a bunch of dots.

1088
01:10:38,520 --> 01:10:41,000
These dots actually,
each one of those.

1089
01:10:41,000 --> 01:10:42,630
They're symmetrical
along the axis,

1090
01:10:42,630 --> 01:10:45,840
so therefore, they
correspond to sinusoids.

1091
01:10:45,840 --> 01:10:48,720
And the superposition
of sinusoids

1092
01:10:48,720 --> 01:10:49,800
looks very messy here.

1093
01:10:49,800 --> 01:10:52,425
It is still periodic, but messy.

1094
01:10:52,425 --> 01:10:54,550
Of course, if you look at
it in the Fourier domain,

1095
01:10:54,550 --> 01:10:57,000
each one of those
is it represented

1096
01:10:57,000 --> 01:11:02,850
by its own individual
pair of delta functions.

1097
01:11:02,850 --> 01:11:04,620
But, of course, this
is discrete now.

1098
01:11:04,620 --> 01:11:07,830
What is even more
interesting is that if you

1099
01:11:07,830 --> 01:11:11,280
were to connect all
these delta functions

1100
01:11:11,280 --> 01:11:14,250
and get the continuous
Fourier transform, then

1101
01:11:14,250 --> 01:11:16,890
your original pattern over
here, this page domain

1102
01:11:16,890 --> 01:11:18,210
would become nonperiodic.

1103
01:11:18,210 --> 01:11:19,740
So you can see very clearly.

1104
01:11:19,740 --> 01:11:21,270
Here, I have discrete--

1105
01:11:21,270 --> 01:11:25,160
a discrete Fourier
transform that corresponds

1106
01:11:25,160 --> 01:11:26,540
to periodic pattern.

1107
01:11:29,113 --> 01:11:31,030
AUDIENCE: Could you draw
the Fourier transform

1108
01:11:31,030 --> 01:11:32,397
in the overhead projector?

1109
01:11:32,397 --> 01:11:33,480
Because we can not see it.

1110
01:11:33,480 --> 01:11:35,350
It looks dark,
totally dark here.

1111
01:11:35,350 --> 01:11:36,400
GEORGE BARBASTATHIS: OK.

1112
01:11:36,400 --> 01:11:41,520
I cannot quite draw it, but
I can sort of cartoon it.

1113
01:11:41,520 --> 01:11:44,090
So the cartoon would
be dots like this.

1114
01:12:01,797 --> 01:12:02,630
Something like this.

1115
01:12:05,610 --> 01:12:12,760
So each pair of dots
corresponds to a cosine.

1116
01:12:12,760 --> 01:12:14,840
And then what you
see on the top left

1117
01:12:14,840 --> 01:12:17,555
is actually the superposition
of all of these cosines.

1118
01:12:21,290 --> 01:12:21,790
OK.

1119
01:12:24,840 --> 01:12:26,250
And, of course,
you can have sort

1120
01:12:26,250 --> 01:12:32,380
of more general transparencies.

1121
01:12:32,380 --> 01:12:34,720
You guys are too young
to remember this,

1122
01:12:34,720 --> 01:12:40,610
but about in 2006, I believe.

1123
01:12:40,610 --> 01:12:45,740
The Boston baseball
team beat the Yankees

1124
01:12:45,740 --> 01:12:49,900
After 85 or 86 years.

1125
01:12:49,900 --> 01:12:52,190
They finally managed
to beat them again.

1126
01:12:52,190 --> 01:12:56,000
And the night of the game,
this is the Prudential Tower.

1127
01:12:56,000 --> 01:12:58,800
For those of you who live
in Singapore, this is the--

1128
01:12:58,800 --> 01:13:01,430
I think it's the tallest
building in Boston.

1129
01:13:01,430 --> 01:13:05,600
And so that night, they lit
up their lights in the offices

1130
01:13:05,600 --> 01:13:08,190
in a way that if you
looked at the pattern,

1131
01:13:08,190 --> 01:13:11,970
you could see the sign,
Go Sox, the Boston famous

1132
01:13:11,970 --> 01:13:13,970
called Red Sox.

1133
01:13:13,970 --> 01:13:17,360
And, of course, the Fox
25 is the TV channel

1134
01:13:17,360 --> 01:13:20,460
that sponsored the match.

1135
01:13:20,460 --> 01:13:22,580
So I took a picture
with my camera.

1136
01:13:22,580 --> 01:13:25,860
I can see this tower from
where I used to live in Boston.

1137
01:13:25,860 --> 01:13:26,870
So this is a picture.

1138
01:13:26,870 --> 01:13:30,420
And if you represent
it as a transparency,

1139
01:13:30,420 --> 01:13:33,140
so that is the bright
spots correspond

1140
01:13:33,140 --> 01:13:34,820
to transmission of light.

1141
01:13:34,820 --> 01:13:37,790
The dark spots correspond
to blocking the light.

1142
01:13:37,790 --> 01:13:40,000
Then you can think
it's Fourier transform.

1143
01:13:40,000 --> 01:13:42,680
And you can see sort of
a more general pattern

1144
01:13:42,680 --> 01:13:44,388
that looks like this.

1145
01:13:44,388 --> 01:13:46,430
What is interesting is
that if you look carefully

1146
01:13:46,430 --> 01:13:47,300
at this pattern.

1147
01:13:47,300 --> 01:13:49,790
And I don't know if you
can see it in Boston.

1148
01:13:49,790 --> 01:13:53,270
But the pattern here
looks kind of diffuse.

1149
01:13:53,270 --> 01:13:56,300
But there's some distinct
spots, actually quite

1150
01:13:56,300 --> 01:13:59,420
a few of these spots.

1151
01:13:59,420 --> 01:14:01,810
Can anybody guess where
these spots came from?

1152
01:14:09,510 --> 01:14:11,610
AUDIENCE: Some of the
features in the image,

1153
01:14:11,610 --> 01:14:18,840
I guess, have straight lines
that kind of act like a box

1154
01:14:18,840 --> 01:14:20,790
function, but not completely.

1155
01:14:23,720 --> 01:14:24,720
Sorry, other way around.

1156
01:14:24,720 --> 01:14:31,170
You're seeing basically
periodic structure in the image

1157
01:14:31,170 --> 01:14:34,680
on the left gets reflected as
spots in the Fourier domain

1158
01:14:34,680 --> 01:14:36,390
on the right.

1159
01:14:36,390 --> 01:14:38,760
GEORGE BARBASTATHIS:
That's right.

1160
01:14:38,760 --> 01:14:42,820
The building has irregular
spacing between the windows.

1161
01:14:42,820 --> 01:14:45,480
So you see a very clear
periodic pattern here.

1162
01:14:45,480 --> 01:14:51,090
It is modulated by the
Go Sox illumination,

1163
01:14:51,090 --> 01:14:53,430
but nevertheless,
even the dark windows

1164
01:14:53,430 --> 01:14:56,010
are visible in the
picture, right?

1165
01:14:56,010 --> 01:14:56,642
Dark windows.

1166
01:14:56,642 --> 01:14:58,600
Some of the windows, they
turned on the lights,

1167
01:14:58,600 --> 01:14:59,560
some they didn't.

1168
01:14:59,560 --> 01:15:02,130
But still, you can
see the windows,

1169
01:15:02,130 --> 01:15:05,100
even if the lights were off.

1170
01:15:05,100 --> 01:15:07,158
So this gives rise to
a periodic pattern.

1171
01:15:07,158 --> 01:15:08,700
And, of course, the
Fourier transform

1172
01:15:08,700 --> 01:15:11,910
of a periodic pattern
as I said before is

1173
01:15:11,910 --> 01:15:15,720
a sequence of dots corresponding
to the Fourier series

1174
01:15:15,720 --> 01:15:16,910
coefficients.

1175
01:15:16,910 --> 01:15:22,110
So that's why you see this very
nice distinct dots over here.

1176
01:15:22,110 --> 01:15:26,220
It is the windows
in the high rise.

1177
01:15:26,220 --> 01:15:27,910
There's also more periodicity.

1178
01:15:27,910 --> 01:15:31,870
This is a roof that
also is periodic.

1179
01:15:31,870 --> 01:15:33,250
You can see a grating here.

1180
01:15:39,630 --> 01:15:41,030
can you still hear me?

1181
01:15:41,030 --> 01:15:43,510
I keep dropping
their microphone.

1182
01:15:43,510 --> 01:15:44,010
Thanks.

1183
01:15:48,910 --> 01:15:51,700
The grating here
should be visible as--

1184
01:15:51,700 --> 01:15:53,710
it must be one of
these pairs of dots

1185
01:15:53,710 --> 01:15:56,320
that do not correlate
with the building.

1186
01:15:56,320 --> 01:16:00,790
That is the pattern
on the roof over here.

1187
01:16:00,790 --> 01:16:02,380
This is a roof of
another building.

1188
01:16:07,920 --> 01:16:12,560
Now, let's look at
the various theorems.

1189
01:16:12,560 --> 01:16:14,360
I've already said
this before, so that's

1190
01:16:14,360 --> 01:16:15,830
the similarity theorem.

1191
01:16:15,830 --> 01:16:18,450
If you compare the Fourier
transform of two rectangles,

1192
01:16:18,450 --> 01:16:20,400
one small, one big.

1193
01:16:20,400 --> 01:16:23,210
The Fourier transform will
have the opposite behavior.

1194
01:16:23,210 --> 01:16:25,040
The small rectangle
will give rise

1195
01:16:25,040 --> 01:16:29,920
to a large Fourier transform.

1196
01:16:29,920 --> 01:16:31,930
The other one that
I wanted to describe

1197
01:16:31,930 --> 01:16:36,450
is this one, which
is the shift theorem.

1198
01:16:36,450 --> 01:16:42,100
So the shift theorem, we
briefly glanced over it

1199
01:16:42,100 --> 01:16:44,200
in the earlier slide.

1200
01:16:44,200 --> 01:16:47,090
So let me remind you what
this earlier slide said.

1201
01:16:50,310 --> 01:16:53,820
So the shift theorem
goes like this.

1202
01:16:53,820 --> 01:16:56,400
I will do [INAUDIBLE]
in one dimension only.

1203
01:16:56,400 --> 01:17:04,720
Then let's say that g of x has
a Fourier transform, G of u.

1204
01:17:04,720 --> 01:17:14,380
The question is now, if I shift
g of x by some amount, x0,

1205
01:17:14,380 --> 01:17:17,940
what is the Fourier transform?

1206
01:17:17,940 --> 01:17:20,860
OK, so we do the same thing.

1207
01:17:20,860 --> 01:17:24,060
We know that since this is true.

1208
01:17:24,060 --> 01:17:28,815
Since this is true, we
know that G of u equals--

1209
01:17:36,790 --> 01:17:40,790
then that is the definition
of the Fourier transform.

1210
01:17:40,790 --> 01:17:44,600
Now, the question
is what is this one?

1211
01:17:44,600 --> 01:17:46,570
So this one, the
Fourier transform

1212
01:17:46,570 --> 01:17:50,110
of the shifted function will be
given by something like this.

1213
01:17:57,205 --> 01:17:58,580
This, of course,
the same Fourier

1214
01:17:58,580 --> 01:18:01,880
transform, but now I plugged
in the shifted function.

1215
01:18:01,880 --> 01:18:05,600
And in order to
bring it to order,

1216
01:18:05,600 --> 01:18:07,460
again, I will do a
coordinate transform.

1217
01:18:11,720 --> 01:18:14,490
And this is very
easy because, again,

1218
01:18:14,490 --> 01:18:17,280
the bounds of the integral
are minus infinity, infinity.

1219
01:18:17,280 --> 01:18:20,780
They don't change upon
the transformation.

1220
01:18:20,780 --> 01:18:24,120
The integral doesn't
change either.

1221
01:18:24,120 --> 01:18:27,240
I mean, the differential
doesn't change either.

1222
01:18:27,240 --> 01:18:28,760
The only thing that
changes is here.

1223
01:18:28,760 --> 01:18:30,075
So you'll get the integral.

1224
01:18:37,630 --> 01:18:41,640
So x equals c plus x0, right?

1225
01:19:20,170 --> 01:19:24,470
Because this you recognize
is the same as this.

1226
01:19:24,470 --> 01:19:26,090
OK, so this is
the shift theorem.

1227
01:19:33,600 --> 01:19:35,430
So now, why is it
related to this one?

1228
01:19:35,430 --> 01:19:40,900
Well, this one, if I
do a cross section.

1229
01:19:55,010 --> 01:19:56,070
It will look like this.

1230
01:19:56,070 --> 01:19:58,280
What I did is I
drew a cross section

1231
01:19:58,280 --> 01:20:00,450
along the vertical axis.

1232
01:20:00,450 --> 01:20:03,440
So let's call the
vertical axis x.

1233
01:20:03,440 --> 01:20:05,510
So this is one.

1234
01:20:05,510 --> 01:20:06,330
What is this?

1235
01:20:06,330 --> 01:20:07,460
Well, this is a rectangle.

1236
01:20:10,750 --> 01:20:11,840
OK, we know this one.

1237
01:20:11,840 --> 01:20:14,240
And we computed this
Fourier transform.

1238
01:20:14,240 --> 01:20:17,930
If you look at this one,
it is also rectangle.

1239
01:20:22,090 --> 01:20:25,870
The size is the same.

1240
01:20:25,870 --> 01:20:30,570
If this is x0, this is also x0.

1241
01:20:35,410 --> 01:20:37,950
But it is displaced.

1242
01:20:37,950 --> 01:20:39,710
Let's use a symbol
for this displacement.

1243
01:20:39,710 --> 01:20:40,700
Let's call it a.

1244
01:20:45,690 --> 01:20:47,070
Actually, this would be minus a.

1245
01:20:50,718 --> 01:20:51,930
And this one.

1246
01:21:05,560 --> 01:21:06,060
OK.

1247
01:21:08,800 --> 01:21:12,270
So let's see if we can
apply the shift theorem.

1248
01:21:17,530 --> 01:21:19,930
Actually, we have to
apply two theorems here,

1249
01:21:19,930 --> 01:21:24,108
the shift theorem and
the scaling theorem.

1250
01:21:24,108 --> 01:21:25,400
Which one should I apply first?

1251
01:21:38,710 --> 01:21:40,150
OK, let me start.

1252
01:21:40,150 --> 01:21:41,422
Let's do one thing at a time.

1253
01:21:41,422 --> 01:21:42,880
So let me write
this function down.

1254
01:21:42,880 --> 01:21:46,340
So g of x equals.

1255
01:22:07,358 --> 01:22:09,400
Each one of those corresponds
to the three rects.

1256
01:22:17,860 --> 01:22:20,580
OK.

1257
01:22:20,580 --> 01:22:22,568
Now, I want to take
the Fourier transform.

1258
01:22:28,920 --> 01:22:32,050
So one, I've already done.

1259
01:22:32,050 --> 01:22:33,620
It's this one.

1260
01:22:33,620 --> 01:22:34,890
I guess I use the red pen.

1261
01:22:39,690 --> 01:22:41,910
This one, we concluded earlier.

1262
01:22:41,910 --> 01:22:47,260
It is x0, sinc of ux0.

1263
01:22:49,968 --> 01:22:51,010
What about the other one?

1264
01:23:11,630 --> 01:23:13,700
First of all, linearity
says that I can just

1265
01:23:13,700 --> 01:23:15,360
add them, right?

1266
01:23:15,360 --> 01:23:16,070
So that's easy.

1267
01:23:19,090 --> 01:23:20,780
Yes?

1268
01:23:20,780 --> 01:23:24,640
AUDIENCE: They're going to
be the x0, sinc of u of x0.

1269
01:23:24,640 --> 01:23:33,393
But shifted by e to the i 2 pi,
a, and all that other stuff.

1270
01:23:33,393 --> 01:23:34,810
GEORGE BARBASTATHIS:
That's right.

1271
01:23:34,810 --> 01:23:46,460
So I will get this one for
this term times the shift

1272
01:23:46,460 --> 01:23:50,140
according to the shift theorem.

1273
01:23:56,120 --> 01:24:07,360
And similarly,
with a minus sign.

1274
01:24:07,360 --> 01:24:12,870
Because here, the shift is
in the negative direction.

1275
01:24:12,870 --> 01:24:13,370
I'm sorry.

1276
01:24:13,370 --> 01:24:16,360
The minus sign belongs here,
and the plus belongs here.

1277
01:24:20,580 --> 01:24:23,190
OK, so get a common
term in all of this.

1278
01:24:50,480 --> 01:24:51,390
OK, one is a plus.

1279
01:24:51,390 --> 01:24:53,760
That is a minus.

1280
01:24:53,760 --> 01:24:56,590
OK, so does this explain
now what you see here?

1281
01:25:12,030 --> 01:25:13,683
AUDIENCE: It's a 1 plus cosine.

1282
01:25:13,683 --> 01:25:15,100
GEORGE BARBASTATHIS:
That's right.

1283
01:25:15,100 --> 01:25:15,730
This is.

1284
01:25:30,930 --> 01:25:33,952
And indeed, in this
calculated pattern.

1285
01:25:33,952 --> 01:25:35,160
Actually, the way I did this.

1286
01:25:35,160 --> 01:25:39,680
I used the fft2
function in Matlab.

1287
01:25:39,680 --> 01:25:41,550
And you can see
that fft2 correctly

1288
01:25:41,550 --> 01:25:43,800
produced as a
sinusoidal modulation

1289
01:25:43,800 --> 01:25:48,230
here, which is imposed by
the shift theorem, really.

1290
01:25:48,230 --> 01:25:49,930
So that's very interesting.

1291
01:25:49,930 --> 01:25:56,420
If you translate the
original function,

1292
01:25:56,420 --> 01:25:59,410
you get this
sinusoidal modulation

1293
01:25:59,410 --> 01:26:00,760
in the Fourier transform.

1294
01:26:00,760 --> 01:26:02,990
And now because we
have a superposition--

1295
01:26:02,990 --> 01:26:04,710
an interference, really--

1296
01:26:04,710 --> 01:26:10,090
of sinusoidal modulations,
that's why we'll get the--

1297
01:26:10,090 --> 01:26:14,680
well, these fringes in the
Fourier transform pattern.

1298
01:26:17,190 --> 01:26:21,540
And, of course, if you
rotate this pattern.

1299
01:26:21,540 --> 01:26:24,150
Then the fringes also
rotate by the same token

1300
01:26:24,150 --> 01:26:25,380
we said before, right?

1301
01:26:25,380 --> 01:26:29,160
Because now in this case, the
displacement is both x and y.

1302
01:26:29,160 --> 01:26:32,090
So you will get the
complex exponential

1303
01:26:32,090 --> 01:26:33,410
in the rotated case.

1304
01:26:46,620 --> 01:26:50,100
When you go to
the Fourier space,

1305
01:26:50,100 --> 01:26:52,050
due to the shift
theorem, you will

1306
01:26:52,050 --> 01:26:55,930
get the complex exponential
of the form, e to the minus i

1307
01:26:55,930 --> 01:27:02,420
two pi ax plus by.

1308
01:27:02,420 --> 01:27:04,970
Where, for example, this is a.

1309
01:27:08,435 --> 01:27:09,680
And this is b.

1310
01:27:21,930 --> 01:27:23,430
So when you do
this, a preposition

1311
01:27:23,430 --> 01:27:25,470
of these complex
exponentials, you

1312
01:27:25,470 --> 01:27:30,190
will get rotated fringes in
the Fourier transform pattern.

1313
01:27:36,310 --> 01:27:36,810
OK.

1314
01:27:46,090 --> 01:27:47,232
Any questions?

1315
01:28:00,270 --> 01:28:02,340
The last thing before
we quit for tonight,

1316
01:28:02,340 --> 01:28:06,430
or for this morning, is
the evolution theorem.

1317
01:28:06,430 --> 01:28:08,970
And that's a really,
really important one

1318
01:28:08,970 --> 01:28:11,490
that you probably remember.

1319
01:28:11,490 --> 01:28:13,750
I don't know, maybe you
remember it with horror.

1320
01:28:13,750 --> 01:28:15,540
Or maybe you remember
it with fondness.

1321
01:28:15,540 --> 01:28:18,000
But anyway, whatever
the case may be.

1322
01:28:18,000 --> 01:28:21,550
You may remember from
your systems classes.

1323
01:28:21,550 --> 01:28:26,100
So the convolution
theorem says that if you

1324
01:28:26,100 --> 01:28:41,720
have a system whose
input is g sub in of x,

1325
01:28:41,720 --> 01:28:46,800
and the output is g
sub out of x prime.

1326
01:28:46,800 --> 01:28:49,020
A linear system is
actually-- the output

1327
01:28:49,020 --> 01:28:50,918
is expressed as a convolution.

1328
01:29:04,610 --> 01:29:07,270
And you may be more familiar
with seeing these convolutions

1329
01:29:07,270 --> 01:29:10,690
in the time domain,
but it doesn't matter.

1330
01:29:10,690 --> 01:29:12,820
In the case of
optics, we're dealing

1331
01:29:12,820 --> 01:29:14,900
with space domain signals.

1332
01:29:14,900 --> 01:29:17,470
So we simply swap t with x.

1333
01:29:17,470 --> 01:29:19,005
But it's actually the same idea.

1334
01:29:19,005 --> 01:29:20,380
So one of the
[INAUDIBLE] example

1335
01:29:20,380 --> 01:29:25,140
of this convolution in the
case of Fresnel propagation.

1336
01:29:25,140 --> 01:29:27,210
If you remember,
Fresnel propagation

1337
01:29:27,210 --> 01:29:32,320
was g out of x prime,
y prime, proportional.

1338
01:29:32,320 --> 01:29:34,420
It had some additional
terms in front.

1339
01:29:34,420 --> 01:29:37,650
But the integral of that we
got goes something like this.

1340
01:29:37,650 --> 01:29:40,420
G sub in of x comma y.

1341
01:29:53,370 --> 01:29:57,520
So it is earlier convolution,
where this function, h of xy.

1342
01:30:05,500 --> 01:30:06,310
What is this again?

1343
01:30:15,370 --> 01:30:16,950
What is this physically?

1344
01:30:16,950 --> 01:30:18,308
AUDIENCE: It's a spherical wave.

1345
01:30:18,308 --> 01:30:19,600
GEORGE BARBASTATHIS: Thank you.

1346
01:30:32,920 --> 01:30:34,420
So the convolution
here emphasize

1347
01:30:34,420 --> 01:30:39,640
that if you take Fourier
transforms of everybody.

1348
01:30:39,640 --> 01:30:43,710
So you Fourier
transform this one.

1349
01:30:43,710 --> 01:30:46,660
You call it G sub in of u.

1350
01:30:46,660 --> 01:30:48,230
You Fourier transform this one.

1351
01:30:48,230 --> 01:30:52,350
You call it G sub out of u.

1352
01:30:52,350 --> 01:30:53,730
You Fourier transform this one.

1353
01:31:05,130 --> 01:31:06,040
OK.

1354
01:31:06,040 --> 01:31:09,910
Then the convolution theorem
says that this equals

1355
01:31:09,910 --> 01:31:15,660
G sub in of u times H of u.

1356
01:31:15,660 --> 01:31:17,250
OK, that's the
convolution theorem.

1357
01:31:17,250 --> 01:31:20,850
So it says that in
the space domain,

1358
01:31:20,850 --> 01:31:23,280
you have a convolution
relationship.

1359
01:31:23,280 --> 01:31:27,347
Then in the Fourier domain, you
simply get a multiplication.

1360
01:31:27,347 --> 01:31:29,430
And actually, that also
goes the other way around.

1361
01:31:29,430 --> 01:31:32,430
If you have a multiplication
in the space domain,

1362
01:31:32,430 --> 01:31:35,820
you have a convolution
in the frequency domain.

1363
01:31:35,820 --> 01:31:37,770
We'll get to use that
a little bit later.

1364
01:31:40,420 --> 01:31:42,010
Does anybody want
me to prove this?

1365
01:31:42,010 --> 01:31:43,677
Do you believe it,
or should I prove it?

1366
01:31:50,880 --> 01:31:53,933
Well, let me prove it.

1367
01:31:53,933 --> 01:31:55,600
Since we're in the
mood of a must today.

1368
01:32:01,650 --> 01:32:03,855
So let me write the
convolution integral.

1369
01:32:03,855 --> 01:32:05,730
Actually, before I do
that, let me write down

1370
01:32:05,730 --> 01:32:07,158
the Fourier transforms.

1371
01:32:27,576 --> 01:32:28,790
OK, similarly.

1372
01:32:50,330 --> 01:32:52,480
OK, so these are
really all the same.

1373
01:33:00,760 --> 01:33:02,225
Now, let me write the output.

1374
01:33:08,950 --> 01:33:12,330
OK, and by the same token, these
are the Fourier transforms.

1375
01:33:12,330 --> 01:33:14,910
I can also write the
Fourier integrals

1376
01:33:14,910 --> 01:33:16,050
in the inverse fashion.

1377
01:33:16,050 --> 01:33:18,240
So for example, I
can neither g sub

1378
01:33:18,240 --> 01:33:28,330
in of x equals integral G sub in
of u, e to the plus i two pi ux

1379
01:33:28,330 --> 01:33:29,280
du.

1380
01:33:29,280 --> 01:33:33,090
If you recall, we call this
the inverse Fourier transform,

1381
01:33:33,090 --> 01:33:35,670
or the Fourier integral.

1382
01:33:35,670 --> 01:33:38,040
And by the same
token, I have h of x

1383
01:33:38,040 --> 01:33:42,210
equals a similar looking
integral for H of u.

1384
01:33:42,210 --> 01:33:48,180
And g sub out, again,
similar looking integral

1385
01:33:48,180 --> 01:33:50,680
for G sub out of u.

1386
01:33:53,230 --> 01:33:54,540
OK, these are just definitions.

1387
01:33:54,540 --> 01:33:57,930
So far, I haven't
really done anything.

1388
01:33:57,930 --> 01:34:00,230
Now, let me write out
the convolution integral.

1389
01:34:16,217 --> 01:34:18,050
What I will do now is
a little bit horrible,

1390
01:34:18,050 --> 01:34:21,110
but you will see the
logic of it in a second.

1391
01:34:21,110 --> 01:34:24,567
I will substitute the
Fourier transform.

1392
01:34:24,567 --> 01:34:25,400
Actually, I'm sorry.

1393
01:34:25,400 --> 01:34:27,560
I will substitute
the Fourier integral

1394
01:34:27,560 --> 01:34:30,038
inside this relationship.

1395
01:34:30,038 --> 01:34:31,330
So how many integrals do I get?

1396
01:34:31,330 --> 01:34:32,180
I get three, right?

1397
01:34:32,180 --> 01:34:34,730
I get one that I had, and
then each one of those

1398
01:34:34,730 --> 01:34:36,670
will be written as an integral.

1399
01:34:36,670 --> 01:34:38,308
So here are the three integrals.

1400
01:34:41,730 --> 01:34:44,170
That's the original one.

1401
01:34:44,170 --> 01:34:48,140
Then for g sub in of x, I
substitute its own Fourier

1402
01:34:48,140 --> 01:34:48,640
integral.

1403
01:35:02,930 --> 01:35:04,240
And the same for h.

1404
01:35:10,357 --> 01:35:11,690
Have to be a little bit careful.

1405
01:35:11,690 --> 01:35:15,320
h is computed in this
shifted coordinate,

1406
01:35:15,320 --> 01:35:19,010
so it is x prime minus x du.

1407
01:35:22,580 --> 01:35:23,660
OK.

1408
01:35:23,660 --> 01:35:26,630
Now, what I'll do is assuming
that these functions are

1409
01:35:26,630 --> 01:35:28,570
well-behaved and
so on and so forth,

1410
01:35:28,570 --> 01:35:32,090
I will actually interchange
the order of integration.

1411
01:35:36,780 --> 01:35:39,120
Let's see if I can do it in
a way that it all fits here.

1412
01:35:59,500 --> 01:36:01,690
OK, let me be a little
bit more careful here.

1413
01:36:01,690 --> 01:36:04,990
[INAUDIBLE] variable, u is in
the same in the two integrals.

1414
01:36:04,990 --> 01:36:08,830
So to avoid confusion, I
will actually label them.

1415
01:36:08,830 --> 01:36:12,060
I will call this
u1, and this u2.

1416
01:36:15,250 --> 01:36:21,100
OK, so now, I have the
du1, du2 integrals.

1417
01:36:21,100 --> 01:36:25,580
What's inside g sub in of u1?

1418
01:36:25,580 --> 01:36:28,410
H of u2.

1419
01:36:28,410 --> 01:36:33,550
And all of this is
multiplied by a x integral.

1420
01:36:47,600 --> 01:36:48,470
So what do I have?

1421
01:36:48,470 --> 01:36:51,590
So for x, I have
u1 from this term,

1422
01:36:51,590 --> 01:36:53,990
and minus u2 from this term.

1423
01:36:58,380 --> 01:36:59,670
And what's left?

1424
01:36:59,670 --> 01:37:01,110
This thing left over, right?

1425
01:37:01,110 --> 01:37:06,850
So let me not forget it, e
to the i two pi u2 x prime.

1426
01:37:06,850 --> 01:37:09,450
x prime, of course,
is not plain.

1427
01:37:09,450 --> 01:37:12,850
So I'll just leave it there.

1428
01:37:12,850 --> 01:37:14,710
It is not plain in the
integration, that is.

1429
01:37:14,710 --> 01:37:15,210
OK.

1430
01:37:19,400 --> 01:37:21,730
So now, what is this?

1431
01:37:21,730 --> 01:37:24,650
I put one too many dx's.

1432
01:37:24,650 --> 01:37:25,410
So what is this?

1433
01:37:42,110 --> 01:37:46,250
It is the Fourier transform
of an exponential.

1434
01:37:46,250 --> 01:37:49,230
Remember, these
integrals without bounds,

1435
01:37:49,230 --> 01:37:52,520
they really go from minus
infinity to infinity, right?

1436
01:37:52,520 --> 01:37:55,360
So if I integrate an
exponential from minus infinity

1437
01:37:55,360 --> 01:37:56,950
to infinity, what do I get?

1438
01:37:56,950 --> 01:37:58,970
We said it earlier this morning.

1439
01:38:19,540 --> 01:38:22,810
Your tuition is ticking
away one second at a time.

1440
01:39:00,560 --> 01:39:02,490
Well, it's 9:25,
according to my clock.

1441
01:39:02,490 --> 01:39:04,681
So I guess we stop here.

1442
01:39:04,681 --> 01:39:06,870
And I'll let you ponder
this on your own.

1443
01:39:10,310 --> 01:39:12,780
See you on Monday.