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PROFESSOR: Welcome
back, everybody to 8.03.

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00:00:27,980 --> 00:00:30,080
Very happy to see you again.

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00:00:30,080 --> 00:00:32,610
So as you can see
from the slides,

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00:00:32,610 --> 00:00:34,960
we will continue the
discussion from last time.

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00:00:34,960 --> 00:00:38,770
We were talking about
interference phenomena,

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00:00:38,770 --> 00:00:44,170
which involve two or
multiple point light source.

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00:00:44,170 --> 00:00:46,990
And they actually
interact with each other

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00:00:46,990 --> 00:00:49,780
and produce interesting
phenomenon, which

16
00:00:49,780 --> 00:00:53,820
we see with laser,
with water ripples,

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00:00:53,820 --> 00:01:00,400
and also we discussed how to
design a phased radar together.

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00:01:00,400 --> 00:01:04,280
And one thing which we learned
is that if you, for example,

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00:01:04,280 --> 00:01:07,420
have two slit interference, OK.

20
00:01:07,420 --> 00:01:12,280
And if you look at the intensity
of the resulting interference

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00:01:12,280 --> 00:01:16,390
pattern as a function
of angle, you

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00:01:16,390 --> 00:01:21,350
will see that there are
peaks, periodic peaks

23
00:01:21,350 --> 00:01:23,020
as a function of angle.

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00:01:23,020 --> 00:01:26,650
And we also know
how to calculate

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00:01:26,650 --> 00:01:29,775
where would be the
principal maxima, what

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00:01:29,775 --> 00:01:31,930
would be the minima,
which will have

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00:01:31,930 --> 00:01:35,871
destructive interference between
the two point light source.

28
00:01:35,871 --> 00:01:36,370
OK?

29
00:01:36,370 --> 00:01:40,810
So as usual, we go from
one electromagnetic wave

30
00:01:40,810 --> 00:01:45,160
to two electromagnetic waves
and two unelectromagnetic waves.

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00:01:45,160 --> 00:01:49,360
And today we are going
to do infinite number

32
00:01:49,360 --> 00:01:51,010
of electromagnetic
waves and they

33
00:01:51,010 --> 00:01:54,210
are going to interact
with each other

34
00:01:54,210 --> 00:01:57,880
or superpose an infinite
number of electromagnetic waves

35
00:01:57,880 --> 00:01:59,080
all together.

36
00:01:59,080 --> 00:02:04,360
And that brings us to that
discussion of diffraction.

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00:02:04,360 --> 00:02:05,260
OK?

38
00:02:05,260 --> 00:02:08,550
So what are we going
to talk about today is,

39
00:02:08,550 --> 00:02:12,790
for example, a point light
source, a laser pointer.

40
00:02:12,790 --> 00:02:17,410
And what would the image of
a laser pointer look like?

41
00:02:17,410 --> 00:02:23,420
When these lasers pass through
a single slit or just the laser

42
00:02:23,420 --> 00:02:23,920
itself.

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00:02:23,920 --> 00:02:27,510
The laser beam itself, what
will happen to this laser beam?

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00:02:27,510 --> 00:02:32,320
And also we will make some
comments on the Star Trek,

45
00:02:32,320 --> 00:02:33,006
for example.

46
00:02:33,006 --> 00:02:33,505
Right?

47
00:02:33,505 --> 00:02:38,250
They have this super weapon
which they shoot enemy

48
00:02:38,250 --> 00:02:39,840
with this laser beam.

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00:02:39,840 --> 00:02:44,710
And we'll see how practical that
is by the end of this course.

50
00:02:44,710 --> 00:02:48,380
And the third thing is that
it's related to resolution.

51
00:02:48,380 --> 00:02:51,630
So we are going to
design a phone, screen

52
00:02:51,630 --> 00:02:56,230
of your mobile phone together to
see what is actually practical,

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00:02:56,230 --> 00:02:59,470
what is actually not practical.

54
00:02:59,470 --> 00:03:03,370
If Yen-Jie is opening a new
company to develop jPhone,

55
00:03:03,370 --> 00:03:07,030
what should be the requirement
for the screen, for example?

56
00:03:07,030 --> 00:03:08,290
Which, I'm not going to do it.

57
00:03:11,140 --> 00:03:16,030
So this is actually what we
are going to discuss today.

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00:03:16,030 --> 00:03:20,470
So we are interested
in a situation where

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00:03:20,470 --> 00:03:24,700
you have plane waves, and those
plane waves are approaching

60
00:03:24,700 --> 00:03:29,140
from the left-hand side of the
screen toward a single slit.

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00:03:29,140 --> 00:03:33,010
So basically, the
setup is like this.

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00:03:33,010 --> 00:03:38,290
So you have those
wavefront basically

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00:03:38,290 --> 00:03:41,560
is traveling to the right-hand
side, the plane waves.

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00:03:41,560 --> 00:03:48,910
And on the wall, there's a
slit or hole, which is actually

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00:03:48,910 --> 00:03:51,730
a opening, and the
waves can actually

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00:03:51,730 --> 00:03:53,980
penetrate through this hole.

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00:03:53,980 --> 00:04:02,140
The width of this hole is
denoted by, or essentially

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00:04:02,140 --> 00:04:05,080
given to you, which
is actually D.

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00:04:05,080 --> 00:04:09,305
And we were wondering what
is going to happen to--

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00:04:12,210 --> 00:04:16,720
what are we going to observe on
the screen, which is actually

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00:04:16,720 --> 00:04:19,089
pretty far away from the wall.

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00:04:19,089 --> 00:04:29,950
And this screen is
actually used to observe

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00:04:29,950 --> 00:04:32,560
the pattern of the
interference pattern

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00:04:32,560 --> 00:04:39,110
of the electromagnetic
wave passing this hole.

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00:04:39,110 --> 00:04:45,600
So, as we discussed last time,
due to Huygens' principle,

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00:04:45,600 --> 00:04:52,730
every point is actually
like a point-like source

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00:04:52,730 --> 00:04:55,190
of spherical waves.

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00:04:55,190 --> 00:04:57,560
So, as you can see
now, we actually

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00:04:57,560 --> 00:05:00,620
consider the size of our slit.

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00:05:00,620 --> 00:05:07,740
Therefore, there must be a lot
of point-like source inside,

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00:05:07,740 --> 00:05:09,470
in this slit.

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00:05:09,470 --> 00:05:12,950
When this wavefront actually
pass through this wall,

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00:05:12,950 --> 00:05:17,240
there should be infinite
number of point-like source.

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00:05:17,240 --> 00:05:20,680
And all of them, due
to Huygens' principle,

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00:05:20,680 --> 00:05:25,520
is going to be like point-like
source of spherical waves.

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00:05:25,520 --> 00:05:30,280
And they are all emitting from
all those possible location,

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00:05:30,280 --> 00:05:32,810
and that they are
overlapping each other

88
00:05:32,810 --> 00:05:36,450
and they have constructive
or destructive interference

89
00:05:36,450 --> 00:05:37,650
with each other.

90
00:05:37,650 --> 00:05:39,510
So that is actually
what is happening

91
00:05:39,510 --> 00:05:43,970
with this single-slit
experiment.

92
00:05:43,970 --> 00:05:47,670
And we call that diffraction.

93
00:05:47,670 --> 00:05:50,930
So you may be wondering, why
do I call it diffraction?

94
00:05:50,930 --> 00:05:53,280
Why not interference?

95
00:05:53,280 --> 00:05:57,110
Because it's basically the
same phenomenon, right?

96
00:05:57,110 --> 00:05:59,810
I think it's just a
matter of wording.

97
00:05:59,810 --> 00:06:02,300
Feynman actually
commented on this,

98
00:06:02,300 --> 00:06:07,340
and he said that, nobody
was able to define

99
00:06:07,340 --> 00:06:10,960
the difference between
diffraction and interference

100
00:06:10,960 --> 00:06:13,040
in a satisfactory way.

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00:06:13,040 --> 00:06:14,060
Which is actually true.

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00:06:14,060 --> 00:06:15,780
So it's just a
matter of wording.

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00:06:15,780 --> 00:06:20,780
So we are looking at exactly the
same phenomena when we actually

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00:06:20,780 --> 00:06:23,960
discuss this experiment.

105
00:06:23,960 --> 00:06:26,570
So what I am going
to do today is now

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00:06:26,570 --> 00:06:29,840
to introduce to you the
way we can deal with this.

107
00:06:29,840 --> 00:06:34,010
I'm sure you have seen this
experiment before, maybe

108
00:06:34,010 --> 00:06:37,130
in 8.02 or in high school days.

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00:06:37,130 --> 00:06:39,350
On the other hand, what
we are going to do today

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00:06:39,350 --> 00:06:43,160
is to really make use
of the mathematics which

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00:06:43,160 --> 00:06:48,440
we have learned from 18.03
or from the previous lectures

112
00:06:48,440 --> 00:06:50,300
to attack this problem.

113
00:06:50,300 --> 00:06:51,950
So what is actually
the mathematics

114
00:06:51,950 --> 00:06:54,560
I am going to use today?

115
00:06:54,560 --> 00:06:57,680
So the mathematics
which I would like

116
00:06:57,680 --> 00:07:01,040
to use to attack this
problem is to use

117
00:07:01,040 --> 00:07:05,240
a two-dimensional
Fourier transform.

118
00:07:05,240 --> 00:07:08,690
I think by now you should
not be afraid of Fourier

119
00:07:08,690 --> 00:07:09,904
transform any more.

120
00:07:09,904 --> 00:07:11,070
It should be pretty natural.

121
00:07:11,070 --> 00:07:14,540
It's just integration,
and you evaluate, and then

122
00:07:14,540 --> 00:07:17,940
you are going to get
the corresponding number

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00:07:17,940 --> 00:07:19,160
whatsoever.

124
00:07:19,160 --> 00:07:22,790
But the cool thing is
that 18.03 give them

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00:07:22,790 --> 00:07:25,340
physical meaning
of those numbers,

126
00:07:25,340 --> 00:07:27,950
and I'm going to
talk about that.

127
00:07:27,950 --> 00:07:32,020
So what is actually the Fourier
transform I'm going to use?

128
00:07:32,020 --> 00:07:35,570
So I am going to
evaluate C, which

129
00:07:35,570 --> 00:07:41,060
is a function of kx and ky.

130
00:07:41,060 --> 00:07:43,520
And what is actually
this C function?

131
00:07:43,520 --> 00:07:48,650
This C function is equal to
1 over 4 pi squared, which

132
00:07:48,650 --> 00:07:50,180
I really don't care too much.

133
00:07:50,180 --> 00:07:52,400
It's just a constant.

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00:07:52,400 --> 00:07:55,730
And I do the integration
from minus infinity

135
00:07:55,730 --> 00:08:01,910
to infinity for dx, and I do a
integration from minus infinity

136
00:08:01,910 --> 00:08:08,270
to infinity dy, a
small letter scale, dy.

137
00:08:08,270 --> 00:08:11,210
And I have a f function,
which I will introduce you

138
00:08:11,210 --> 00:08:15,420
what the f function mean, what
does the f function actually

139
00:08:15,420 --> 00:08:16,820
represent.

140
00:08:16,820 --> 00:08:26,420
And exponential minus i, k is
the vector which is actually

141
00:08:26,420 --> 00:08:29,330
telling you the direction
of the propagation

142
00:08:29,330 --> 00:08:35,400
of the spherical wave,
times r, which is actually

143
00:08:35,400 --> 00:08:39,440
a function of x and y.

144
00:08:39,440 --> 00:08:41,630
And this is actually
the kind of integration

145
00:08:41,630 --> 00:08:45,530
which we will employ in order
to attack the problem we

146
00:08:45,530 --> 00:08:50,570
are interested in this lecture.

147
00:08:50,570 --> 00:08:53,740
So, what does this
integration mean?

148
00:08:53,740 --> 00:09:03,160
So we have basically some kind
of this two-dimensional Fourier

149
00:09:03,160 --> 00:09:04,290
transform.

150
00:09:04,290 --> 00:09:08,200
The f function is
actually telling you

151
00:09:08,200 --> 00:09:11,710
the shape of the source.

152
00:09:11,710 --> 00:09:14,170
So basically, this is
actually telling you

153
00:09:14,170 --> 00:09:19,840
about the shape of the source.

154
00:09:24,920 --> 00:09:27,620
As we discussed
before, the shape

155
00:09:27,620 --> 00:09:34,700
of the source, every
point on this shape

156
00:09:34,700 --> 00:09:41,360
is a source of spherical
wave, by Huygens' principle.

157
00:09:41,360 --> 00:09:44,740
So that is actually
telling you where should I

158
00:09:44,740 --> 00:09:48,410
do the integration.

159
00:09:48,410 --> 00:09:53,690
This one, exponential i
k dot r, what is that?

160
00:09:53,690 --> 00:10:00,000
This is actually telling you
about the spherical wave.

161
00:10:00,000 --> 00:10:05,290
So remember, we were doing
two-slit interference before,

162
00:10:05,290 --> 00:10:08,270
and we have actually two
exponential function, if you

163
00:10:08,270 --> 00:10:11,030
remember from last lecture.

164
00:10:11,030 --> 00:10:13,190
So now, this is
actually put there

165
00:10:13,190 --> 00:10:15,350
because each source
you are going

166
00:10:15,350 --> 00:10:18,400
to get exponential
function, which is actually

167
00:10:18,400 --> 00:10:23,060
presenting the propagation
of the electromagnetic wave.

168
00:10:23,060 --> 00:10:25,420
You can say that,
oh, wait, wait, wait.

169
00:10:25,420 --> 00:10:28,340
The omega t disappeared, right?

170
00:10:28,340 --> 00:10:30,770
There's no omega t here, right?

171
00:10:30,770 --> 00:10:33,980
But I don't really care
because everybody is actually

172
00:10:33,980 --> 00:10:38,000
oscillating at the same
frequency, the same phase.

173
00:10:38,000 --> 00:10:40,670
Therefore, I factorize out.

174
00:10:40,670 --> 00:10:43,700
After I have done
all the calculation,

175
00:10:43,700 --> 00:10:47,520
I can multiply the whole thing
by cosine omega t, and probably

176
00:10:47,520 --> 00:10:48,500
some phi.

177
00:10:48,500 --> 00:10:53,998
Then that is actually modulating
and oscillating up and down

178
00:10:53,998 --> 00:10:59,140
as the plane wave, as
you approach the wall.

179
00:10:59,140 --> 00:11:03,950
So, therefore, I actually
already factorize it out.

180
00:11:03,950 --> 00:11:08,510
So this is actually telling
you about the electric field.

181
00:11:08,510 --> 00:11:11,120
And what is actually here?

182
00:11:11,120 --> 00:11:16,530
This is actually the
unit area you are

183
00:11:16,530 --> 00:11:20,110
performing this integration.

184
00:11:20,110 --> 00:11:22,470
And you can actually
do integration

185
00:11:22,470 --> 00:11:25,500
over the full universe.

186
00:11:25,500 --> 00:11:27,930
So you have a plane
which actually extend

187
00:11:27,930 --> 00:11:29,820
to the whole full universe.

188
00:11:29,820 --> 00:11:32,190
But what is actually
really contributing

189
00:11:32,190 --> 00:11:35,310
is defined by this f
function, which is actually

190
00:11:35,310 --> 00:11:37,470
the shape of the source.

191
00:11:37,470 --> 00:11:41,160
And some normalization
factor, which I don't really

192
00:11:41,160 --> 00:11:44,430
care too much.

193
00:11:44,430 --> 00:11:49,760
So this looks really fancy,
but is actually not that fancy.

194
00:11:49,760 --> 00:11:53,910
And what product you are
getting here is our C function,

195
00:11:53,910 --> 00:11:58,120
is that C is actually a
function of kx and ky.

196
00:11:58,120 --> 00:12:00,210
What is kx and ky?

197
00:12:00,210 --> 00:12:05,110
It's actually telling you
the direction of propagation.

198
00:12:05,110 --> 00:12:07,590
The k vector is actually
telling you the direction

199
00:12:07,590 --> 00:12:08,970
of the propagation.

200
00:12:08,970 --> 00:12:18,660
If you evaluate C with a
specific given kx and ky,

201
00:12:18,660 --> 00:12:27,600
basically you are evaluating
the total electric field

202
00:12:27,600 --> 00:12:32,640
going some direction, which is
actually defined by kx and ky.

203
00:12:32,640 --> 00:12:34,320
So the big picture
is the following.

204
00:12:34,320 --> 00:12:37,115
So basically you
have some source.

205
00:12:37,115 --> 00:12:41,610
It can look like
this in the xy plane.

206
00:12:41,610 --> 00:12:44,000
This is x and y plane.

207
00:12:49,990 --> 00:12:54,310
All those things,
all those points,

208
00:12:54,310 --> 00:12:59,610
all those little
areas inside this hole

209
00:12:59,610 --> 00:13:04,210
is spherical wave source.

210
00:13:04,210 --> 00:13:11,230
And the f function actually
define the shape of this hole.

211
00:13:11,230 --> 00:13:16,090
And this integration is
actually integrating over

212
00:13:16,090 --> 00:13:19,020
all those little areas.

213
00:13:19,020 --> 00:13:21,030
And then calculate
the contribution

214
00:13:21,030 --> 00:13:26,720
from each small area,
sum them together.

215
00:13:26,720 --> 00:13:30,780
Then, finally, you are getting
something which is actually

216
00:13:30,780 --> 00:13:32,760
a function of kx and ky.

217
00:13:32,760 --> 00:13:34,700
What is kx and ky?

218
00:13:34,700 --> 00:13:39,480
It's actually giving you
the direction of propagation

219
00:13:39,480 --> 00:13:46,680
from this point-like
source to observer P.

220
00:13:46,680 --> 00:13:50,280
And this C function is
actually proportional

221
00:13:50,280 --> 00:13:54,450
to the total electric field.

222
00:13:54,450 --> 00:13:57,330
So you can see that,
hah, we have learned

223
00:13:57,330 --> 00:14:01,440
this Fourier transform
from the math department,

224
00:14:01,440 --> 00:14:04,860
and we give life
to this function.

225
00:14:04,860 --> 00:14:08,250
Now actually we understand
what we are doing now.

226
00:14:08,250 --> 00:14:12,430
We are actually really
summing all over the,

227
00:14:12,430 --> 00:14:17,910
summing over the available
point-like source.

228
00:14:17,910 --> 00:14:21,000
And add all the contribution
of the electromagnetic wave

229
00:14:21,000 --> 00:14:21,850
together.

230
00:14:21,850 --> 00:14:24,980
Then what we are
getting, the C function

231
00:14:24,980 --> 00:14:34,034
is actually proportional to
the total electric field.

232
00:14:36,790 --> 00:14:40,060
So that is actually
the big picture.

233
00:14:40,060 --> 00:14:43,640
Any questions so far?

234
00:14:43,640 --> 00:14:46,400
I hope you can actually
understand what we are doing.

235
00:14:46,400 --> 00:14:52,160
So now what I am going to do
is to really use this formula

236
00:14:52,160 --> 00:14:56,120
and attack the problem
which we are actually

237
00:14:56,120 --> 00:15:00,050
trying to understand,
the single-slit problem.

238
00:15:00,050 --> 00:15:05,900
So suppose I have a single
slit which looks like this.

239
00:15:05,900 --> 00:15:12,950
I'm zooming in this thing
maybe 100 times, 1,000 times.

240
00:15:12,950 --> 00:15:15,725
And this is actually a wall.

241
00:15:19,880 --> 00:15:21,170
It's very, very long.

242
00:15:23,910 --> 00:15:28,894
I would like to define
first my coordinate system.

243
00:15:28,894 --> 00:15:32,260
The x direction, as I
actually drew from there,

244
00:15:32,260 --> 00:15:34,540
is actually pointing upward.

245
00:15:34,540 --> 00:15:41,560
The y direction is actually
parallel to the wall.

246
00:15:41,560 --> 00:15:44,560
And the z direction
is actually going

247
00:15:44,560 --> 00:15:47,080
to where the screen
which I am trying

248
00:15:47,080 --> 00:15:52,540
to display the outcome
of this experiment.

249
00:15:52,540 --> 00:15:56,285
And the distance
between these two walls

250
00:15:56,285 --> 00:16:01,380
is actually D, which is
actually given there.

251
00:16:01,380 --> 00:16:04,830
So I would like to
actually understand

252
00:16:04,830 --> 00:16:09,010
what is going to happen when
the plane waves pass through

253
00:16:09,010 --> 00:16:12,420
with this single slit.

254
00:16:12,420 --> 00:16:17,520
Therefore, before I calculate
the C function proportional

255
00:16:17,520 --> 00:16:21,960
to the total electric
field, what I really need

256
00:16:21,960 --> 00:16:29,010
is a functional form, f, which
describe this single slit.

257
00:16:29,010 --> 00:16:33,830
And just to make sure that
everybody is on the same page,

258
00:16:33,830 --> 00:16:40,680
this wall is actually infinitely
long, from minus infinity in y

259
00:16:40,680 --> 00:16:43,570
to positive infinity y.

260
00:16:43,570 --> 00:16:45,870
So it's actually
a super long wall.

261
00:16:45,870 --> 00:16:53,010
And these two edge is
actually-- the distance

262
00:16:53,010 --> 00:16:57,180
between the edges is actually D.

263
00:16:57,180 --> 00:17:00,570
So what would be the f
function which describes

264
00:17:00,570 --> 00:17:05,780
the shape of the light source?

265
00:17:05,780 --> 00:17:09,030
f function is a
function of x and y.

266
00:17:09,030 --> 00:17:13,710
And I define a f
function, and I give you

267
00:17:13,710 --> 00:17:17,160
this function to describe
the experimental setup.

268
00:17:17,160 --> 00:17:21,420
So the f function can be
either 1, which actually shows

269
00:17:21,420 --> 00:17:24,329
that there are
point-like source there,

270
00:17:24,329 --> 00:17:29,150
or 0 when I am talking
about things on the wall,

271
00:17:29,150 --> 00:17:32,060
because there's no
point-like source there.

272
00:17:32,060 --> 00:17:38,780
Because the wall is
actually blocking the light.

273
00:17:38,780 --> 00:17:41,130
So it can be either 1 or 0.

274
00:17:41,130 --> 00:17:43,200
When is that equal to 1?

275
00:17:43,200 --> 00:17:47,460
When minus D over 2.

276
00:17:47,460 --> 00:17:51,660
If I define-- this is
actually x equal to 0.

277
00:17:51,660 --> 00:17:56,310
The middle of the slit
is actually x equal to 0.

278
00:17:56,310 --> 00:18:00,780
Then it is actually equal
to 1 when minus D over 2

279
00:18:00,780 --> 00:18:08,200
smaller than or equal to x,
smaller or equal to D over 2.

280
00:18:08,200 --> 00:18:13,710
So that will give you a slit
with width of capital D.

281
00:18:13,710 --> 00:18:17,760
On the other hand, if
the absolute value of x

282
00:18:17,760 --> 00:18:22,740
is greater than D
over 2, then I get 0.

283
00:18:22,740 --> 00:18:24,940
So now you can see
that is actually

284
00:18:24,940 --> 00:18:26,880
the meaning of f function.

285
00:18:26,880 --> 00:18:29,130
f function is
actually giving you

286
00:18:29,130 --> 00:18:34,530
a map of the point-like source.

287
00:18:34,530 --> 00:18:38,850
And what I am going to do now
is to really do the integration

288
00:18:38,850 --> 00:18:45,180
to sum over all the spherical
electromagnetic waves coming

289
00:18:45,180 --> 00:18:46,950
from all those
point-like source,

290
00:18:46,950 --> 00:18:50,670
and to calculate the
total electric field.

291
00:18:50,670 --> 00:18:55,140
So now I can go ahead and
calculate C function, which

292
00:18:55,140 --> 00:18:59,220
is actually a function
of kx and ky, related

293
00:18:59,220 --> 00:19:03,390
to the direction of
propagation, or, say,

294
00:19:03,390 --> 00:19:09,030
the relative position
of the observer

295
00:19:09,030 --> 00:19:12,630
and the overall
point-like source.

296
00:19:12,630 --> 00:19:16,540
And this is actually equal
to 1 over 4 pi squared,

297
00:19:16,540 --> 00:19:18,690
according to my formula.

298
00:19:18,690 --> 00:19:23,040
And now I'm going to do an
integration from minus infinity

299
00:19:23,040 --> 00:19:24,510
to infinity.

300
00:19:24,510 --> 00:19:28,490
But I found that there's
a shortcut I can take.

301
00:19:28,490 --> 00:19:35,160
f x y is only nonzero between
minus D over 2 and plus

302
00:19:35,160 --> 00:19:38,010
D over 2 in the x direction.

303
00:19:38,010 --> 00:19:41,260
Therefore, this integration
becomes integration

304
00:19:41,260 --> 00:19:48,094
from minus D over 2 to
positive D over 2 dx,

305
00:19:48,094 --> 00:19:53,830
exponential minus i kx times x.

306
00:19:53,830 --> 00:20:00,990
I'm taking part of the k vector
dot r out of this formula.

307
00:20:00,990 --> 00:20:06,290
The relevant part related
to x direction integration

308
00:20:06,290 --> 00:20:11,430
is exponential i kx times x.

309
00:20:11,430 --> 00:20:16,320
And now I can actually do the
integration in the y direction.

310
00:20:16,320 --> 00:20:19,230
So you can see that,
in the y direction,

311
00:20:19,230 --> 00:20:24,230
this slit is infinity
long, covering

312
00:20:24,230 --> 00:20:27,977
the from the left-hand
side edge of the universe

313
00:20:27,977 --> 00:20:29,810
to the right-hand side
edge of the universe.

314
00:20:29,810 --> 00:20:31,650
Really long.

315
00:20:31,650 --> 00:20:33,060
Super long.

316
00:20:33,060 --> 00:20:41,110
Minus infinity to infinity
in the y direction.

317
00:20:41,110 --> 00:20:45,600
The relevant part of
the exponential minus k

318
00:20:45,600 --> 00:20:51,530
dot r is exponential
minus i ky times y.

319
00:20:56,600 --> 00:21:00,470
So, before I do
this integration,

320
00:21:00,470 --> 00:21:04,820
I would like to remind you
one thing which is actually

321
00:21:04,820 --> 00:21:08,180
we have learned from the
past, from the help of math

322
00:21:08,180 --> 00:21:10,030
department.

323
00:21:10,030 --> 00:21:15,860
So we know delta function
x minus a is actually equal

324
00:21:15,860 --> 00:21:22,950
to 1 over 2 pi, integration
from minus infinity to infinity,

325
00:21:22,950 --> 00:21:30,950
exponential i p x minus a dp.

326
00:21:30,950 --> 00:21:35,890
So we know about this formula.

327
00:21:35,890 --> 00:21:43,180
So that means I can easily
evaluate this function.

328
00:21:43,180 --> 00:21:47,940
So this function, I'm actually
doing the integration over y.

329
00:21:47,940 --> 00:21:50,460
Therefore, what I'm
going to get is,

330
00:21:50,460 --> 00:21:54,060
I take 1 over 2 pi out of this.

331
00:21:54,060 --> 00:21:56,610
I take 2 pi out of this.

332
00:21:56,610 --> 00:21:59,040
Then, basically, I
can actually arrive

333
00:21:59,040 --> 00:22:02,710
expression, which is
actually delta function

334
00:22:02,710 --> 00:22:05,520
is a function of ky.

335
00:22:05,520 --> 00:22:10,960
After you do this integration
using this formula here.

336
00:22:10,960 --> 00:22:16,710
So p here is actually y in
my integration I'm doing.

337
00:22:16,710 --> 00:22:18,880
And what I'm going
to get is actually

338
00:22:18,880 --> 00:22:23,600
ky equal to 0, minus
0, and I simplify

339
00:22:23,600 --> 00:22:27,660
that to be delta function of ky.

340
00:22:27,660 --> 00:22:29,320
So basically, you
are going to get

341
00:22:29,320 --> 00:22:33,150
the ky contribution is going
to give you a delta function.

342
00:22:35,880 --> 00:22:41,490
So how about the
integration which

343
00:22:41,490 --> 00:22:44,590
is the other part
of the integration?

344
00:22:44,590 --> 00:22:46,780
The other part of
the integration

345
00:22:46,780 --> 00:22:52,660
is related to x
direction, is here.

346
00:22:52,660 --> 00:22:55,090
So, basically, what
is actually left over?

347
00:22:55,090 --> 00:22:58,750
I took already 1
over 2 pi from here.

348
00:22:58,750 --> 00:23:05,800
Therefore, I have 1 over 2
pi, and I do the integration.

349
00:23:05,800 --> 00:23:07,500
It's just a
exponential function.

350
00:23:07,500 --> 00:23:09,580
I'm not super worried.

351
00:23:09,580 --> 00:23:17,910
Basically, I get
1 over minus i kx.

352
00:23:17,910 --> 00:23:23,470
Exponential minus i kx x.

353
00:23:23,470 --> 00:23:29,870
And evaluated at D over
2, x equal to D over 2,

354
00:23:29,870 --> 00:23:33,740
and x equal to minus D over 2.

355
00:23:37,460 --> 00:23:40,210
I hope this part is
straightforward enough.

356
00:23:40,210 --> 00:23:41,670
Any questions so far?

357
00:23:41,670 --> 00:23:44,710
Everybody's following?

358
00:23:44,710 --> 00:23:45,530
All right.

359
00:23:45,530 --> 00:23:46,490
Very good.

360
00:23:46,490 --> 00:23:48,620
So I will continue the red part.

361
00:23:48,620 --> 00:23:51,010
So I will just look
at the red part

362
00:23:51,010 --> 00:23:53,280
and then continue on this board.

363
00:23:53,280 --> 00:23:56,260
I'm using the red pen, right.

364
00:23:56,260 --> 00:23:58,420
So, basically, what
I am going to get

365
00:23:58,420 --> 00:24:10,790
is, basically you have 1
over 2 pi, 1 over minus i kx,

366
00:24:10,790 --> 00:24:23,680
exponential minus i kx D over
2, minus exponential i kx D

367
00:24:23,680 --> 00:24:26,270
over 2.

368
00:24:26,270 --> 00:24:30,770
So the red part of the
left function become this.

369
00:24:30,770 --> 00:24:33,620
And you can actually
easily realize

370
00:24:33,620 --> 00:24:36,740
that this is
actually proportional

371
00:24:36,740 --> 00:24:39,110
to a sine function, right?

372
00:24:39,110 --> 00:24:49,750
So basically I'm going to get
1 over 2 pi minus 2i sine kx

373
00:24:49,750 --> 00:24:58,110
D over 2, divided by i kx.

374
00:25:02,010 --> 00:25:04,210
This is actually
coming from there.

375
00:25:04,210 --> 00:25:06,640
And this is actually
coming from-- this minus

376
00:25:06,640 --> 00:25:12,000
2i sine function is coming
from the exponential function

377
00:25:12,000 --> 00:25:15,500
I can cancel this
2, and basically I

378
00:25:15,500 --> 00:25:23,670
get 1 over pi kx sine
kx D divided by 2.

379
00:25:27,360 --> 00:25:31,260
So if I put everything
together, so basically

380
00:25:31,260 --> 00:25:45,270
what you are getting is delta
function of ky, 1 over pi kx

381
00:25:45,270 --> 00:25:49,040
sine kx D over 2.

382
00:25:53,300 --> 00:25:54,990
Am I going too fast?

383
00:25:54,990 --> 00:25:56,627
Everybody is following?

384
00:25:59,340 --> 00:26:02,230
So I hope this mathematics
is straightforward enough.

385
00:26:02,230 --> 00:26:05,970
And don't forget
what we are doing.

386
00:26:05,970 --> 00:26:08,530
So what we are doing
is the following.

387
00:26:08,530 --> 00:26:11,900
So we have this two-dimensional
Fourier transform.

388
00:26:11,900 --> 00:26:16,840
And the goal is to sum
over all the waves coming

389
00:26:16,840 --> 00:26:20,340
from a shape defined
by f function.

390
00:26:20,340 --> 00:26:22,980
And I'm going to
evaluate the C function,

391
00:26:22,980 --> 00:26:25,410
and the C function
is proportional

392
00:26:25,410 --> 00:26:29,650
to the total electric field.

393
00:26:29,650 --> 00:26:33,244
C is a function of kx and ky.

394
00:26:33,244 --> 00:26:36,850
kx and ky give you
the information

395
00:26:36,850 --> 00:26:41,530
about the direction,
relative position

396
00:26:41,530 --> 00:26:48,660
of the source and the observer
P. And from this exercise,

397
00:26:48,660 --> 00:26:50,590
what we actually
learn from here is

398
00:26:50,590 --> 00:26:55,810
that the C function
is a function of y,

399
00:26:55,810 --> 00:27:02,410
but essentially only nonzero
when ky is equal to what?

400
00:27:05,880 --> 00:27:06,380
AUDIENCE: 0.

401
00:27:06,380 --> 00:27:07,990
PROFESSOR: 0, right?

402
00:27:07,990 --> 00:27:10,890
Does that surprise you?

403
00:27:10,890 --> 00:27:12,950
No, probably not.

404
00:27:12,950 --> 00:27:14,300
Why is that?

405
00:27:14,300 --> 00:27:16,470
Why should we expect that?

406
00:27:16,470 --> 00:27:23,810
Because in the y direction
this slit is infinitely long.

407
00:27:23,810 --> 00:27:30,410
So if you have contribution
of many, many spherical wave,

408
00:27:30,410 --> 00:27:34,760
and this slit is
infinity long, the sum

409
00:27:34,760 --> 00:27:37,040
of all those spherical
wave is going

410
00:27:37,040 --> 00:27:40,940
to be still like a wavefront.

411
00:27:40,940 --> 00:27:43,160
You can do this in your head.

412
00:27:43,160 --> 00:27:50,690
So that means the direction,
if I choose a direction which

413
00:27:50,690 --> 00:27:56,320
is actually pointing to
somewhere which is actually

414
00:27:56,320 --> 00:28:00,020
with a ky not equal to 0--

415
00:28:00,020 --> 00:28:03,140
so that means I have
a specific direction--

416
00:28:03,140 --> 00:28:06,980
what I'm going to get is
that the electric field,

417
00:28:06,980 --> 00:28:10,410
the total electric field,
will be equal to 0.

418
00:28:14,030 --> 00:28:18,320
And, of course, you
can actually also

419
00:28:18,320 --> 00:28:23,070
talk about what will
happen in the x direction.

420
00:28:23,070 --> 00:28:26,750
So that is actually the
dependence of the C function

421
00:28:26,750 --> 00:28:29,700
to the kx.

422
00:28:29,700 --> 00:28:32,690
And we found
interesting dependence.

423
00:28:32,690 --> 00:28:37,620
It's actually sine kx
D over 2 divided by kx.

424
00:28:40,760 --> 00:28:45,190
So what I'm going to do is to
make our life slightly easier

425
00:28:45,190 --> 00:28:53,020
by defining something which is
actually easier to understand.

426
00:28:53,020 --> 00:28:55,270
But before that, I
would like to say

427
00:28:55,270 --> 00:29:00,310
that the electric field,
as I mentioned before,

428
00:29:00,310 --> 00:29:04,860
is going to be proportional
to the C function.

429
00:29:04,860 --> 00:29:08,980
And now I would like to
drop the y direction,

430
00:29:08,980 --> 00:29:11,270
because it's just
a delta function.

431
00:29:11,270 --> 00:29:14,150
Therefore, I can actually
drop it in the discussion.

432
00:29:14,150 --> 00:29:17,410
Then I will say that
this electric field

433
00:29:17,410 --> 00:29:25,150
is going to be proportional to
the sine kx D over 2, divided

434
00:29:25,150 --> 00:29:26,550
by kx.

435
00:29:29,900 --> 00:29:32,230
Since we have the
electric field,

436
00:29:32,230 --> 00:29:34,550
the magnitude of
the electric field,

437
00:29:34,550 --> 00:29:38,600
then I can actually calculate
what will be the intensity.

438
00:29:38,600 --> 00:29:40,880
Intensity is actually
what we care.

439
00:29:40,880 --> 00:29:44,420
It's going to be
proportional to E square,

440
00:29:44,420 --> 00:29:49,040
and that is actually
proportional to C square.

441
00:29:49,040 --> 00:29:51,310
And what is actually that value?

442
00:29:51,310 --> 00:29:57,920
That is going to be proportional
to sine square kx D divided

443
00:29:57,920 --> 00:30:04,250
by 2, divided by kx squared.

444
00:30:04,250 --> 00:30:05,250
Any questions so far?

445
00:30:09,820 --> 00:30:16,610
So remember what is
actually we are discussing.

446
00:30:16,610 --> 00:30:20,960
So we are discussing
about a single slit,

447
00:30:20,960 --> 00:30:28,610
and we were wondering what
will happen to observer point P

448
00:30:28,610 --> 00:30:34,050
when they actually do get,
when these observer do get

449
00:30:34,050 --> 00:30:39,200
the interference pattern of
all the point-like source

450
00:30:39,200 --> 00:30:41,010
between these two walls.

451
00:30:43,680 --> 00:30:47,250
We can actually make it
much more understandable

452
00:30:47,250 --> 00:30:50,250
by using angle,
which is actually

453
00:30:50,250 --> 00:30:54,060
theta, which is
the measure of AP,

454
00:30:54,060 --> 00:30:58,170
which is the direction of the--

455
00:30:58,170 --> 00:31:01,980
which is a vector connecting
the slit to the observer--

456
00:31:01,980 --> 00:31:04,960
to the horizontal direction.

457
00:31:04,960 --> 00:31:09,390
And I can define the
displacement with respect

458
00:31:09,390 --> 00:31:13,490
to the center to be x.

459
00:31:13,490 --> 00:31:18,840
And I can actually
also express AP

460
00:31:18,840 --> 00:31:21,060
by a vector which is r vector.

461
00:31:24,410 --> 00:31:27,190
Basically, after
this definition,

462
00:31:27,190 --> 00:31:33,800
we can actually calculate
or express sine theta.

463
00:31:33,800 --> 00:31:40,970
Since the distance between
the screen and the wall

464
00:31:40,970 --> 00:31:47,060
is very, very large, therefore
the theta angle is very small.

465
00:31:47,060 --> 00:31:52,250
Therefore, I can safely assume
that sine theta is actually

466
00:31:52,250 --> 00:31:53,720
x divided by r.

467
00:31:56,450 --> 00:31:59,050
And also, at the same
time, this is actually

468
00:31:59,050 --> 00:32:02,940
equal to kx divided by k.

469
00:32:05,480 --> 00:32:08,660
Because the k vector
is actually telling you

470
00:32:08,660 --> 00:32:12,260
the direction of propagation.

471
00:32:12,260 --> 00:32:14,790
So, therefore, I can
actually rewrite this.

472
00:32:14,790 --> 00:32:18,780
This will become kx.

473
00:32:18,780 --> 00:32:23,270
The magnitude of k
vector is actually

474
00:32:23,270 --> 00:32:25,760
basically 2 pi over lambda.

475
00:32:25,760 --> 00:32:28,330
So, therefore, you can
actually calculate that,

476
00:32:28,330 --> 00:32:35,010
and you will get kx times
lambda divided by 2 pi.

477
00:32:35,010 --> 00:32:38,850
Therefore, the goal
is to rewrite kx

478
00:32:38,850 --> 00:32:42,660
in a form which we
understand, which is theta.

479
00:32:42,660 --> 00:32:44,960
So now we have achieved that.

480
00:32:44,960 --> 00:32:46,593
What is actually kx?

481
00:32:46,593 --> 00:32:54,070
kx is actually equal to 2 pi
sine theta divided by lambda.

482
00:32:59,700 --> 00:33:04,940
And this means that
my intensity, which

483
00:33:04,940 --> 00:33:13,401
I appended there, will be
proportional to sine square pi

484
00:33:13,401 --> 00:33:21,995
D divided by lambda sine
theta, divided by 2 pi sine

485
00:33:21,995 --> 00:33:26,300
theta divided by
lambda, squared.

486
00:33:26,300 --> 00:33:31,440
So basically what I'm doing
is to replace kx and then

487
00:33:31,440 --> 00:33:35,660
write it in terms of theta.

488
00:33:35,660 --> 00:33:44,090
If I define beta to be
equal to pi D sine theta

489
00:33:44,090 --> 00:33:49,610
over lambda, if I define
this, basically you

490
00:33:49,610 --> 00:33:54,290
are getting sine
square beta, this

491
00:33:54,290 --> 00:33:57,670
will be proportional to
sine squared beta divided

492
00:33:57,670 --> 00:34:01,070
by beta squared.

493
00:34:01,070 --> 00:34:09,830
And this beta is actually
proportional to theta and D.

494
00:34:09,830 --> 00:34:12,000
Any questions so far?

495
00:34:12,000 --> 00:34:17,239
I'm just doing a replace, I'm
just replacing the variables

496
00:34:17,239 --> 00:34:20,060
so that it's actually
in terms of theta

497
00:34:20,060 --> 00:34:23,594
and in terms of some variable
which actually simplify

498
00:34:23,594 --> 00:34:25,870
the expression dramatically.

499
00:34:28,570 --> 00:34:31,130
So, that's very good.

500
00:34:31,130 --> 00:34:37,090
So we have actually
evaluated the intensity,

501
00:34:37,090 --> 00:34:44,719
the resulting intensity which
will show up on the screen.

502
00:34:44,719 --> 00:34:46,310
And then we found
that essentially

503
00:34:46,310 --> 00:34:51,690
proportional to sine square pi
D divided by lambda sine theta,

504
00:34:51,690 --> 00:34:53,659
divided by something squared.

505
00:34:53,659 --> 00:34:56,560
And then I called
this constant, sorry,

506
00:34:56,560 --> 00:35:00,110
I called this expression,
I defined this expression

507
00:35:00,110 --> 00:35:01,310
to be beta.

508
00:35:01,310 --> 00:35:04,340
Then the functional form
become much simpler.

509
00:35:04,340 --> 00:35:07,460
It's become sine square
beta divided by beta square.

510
00:35:10,790 --> 00:35:17,330
So what I am going to do now
is to visualize this result.

511
00:35:17,330 --> 00:35:24,050
So what I'm trying to do
now is to plot the intensity

512
00:35:24,050 --> 00:35:30,290
I as a function of sine
theta, for example, using

513
00:35:30,290 --> 00:35:32,610
this expression.

514
00:35:32,610 --> 00:35:35,960
So what I'm going to get is
something which is actually

515
00:35:35,960 --> 00:35:39,830
going to be decreasing.

516
00:35:39,830 --> 00:35:41,720
Something is going
to be decreasing

517
00:35:41,720 --> 00:35:43,640
as a function of beta.

518
00:35:43,640 --> 00:35:45,740
So that's the dashed line.

519
00:35:45,740 --> 00:35:49,460
This dashed line is
actually proportional to 1

520
00:35:49,460 --> 00:35:50,990
over beta squared.

521
00:35:53,760 --> 00:36:00,810
And sine theta very
small, you actually

522
00:36:00,810 --> 00:36:03,630
reach a maximum value of I0.

523
00:36:06,630 --> 00:36:14,460
When you move away
from theta equal to 0,

524
00:36:14,460 --> 00:36:20,460
you actually will hit a
minimum when the sine theta

525
00:36:20,460 --> 00:36:27,790
is equal to lambda over D.
Because if sine theta is

526
00:36:27,790 --> 00:36:33,200
equal to lambda over D, then
this expression become what?

527
00:36:33,200 --> 00:36:35,860
Become what value when
sine theta is actually

528
00:36:35,860 --> 00:36:36,550
lambda over D?

529
00:36:39,590 --> 00:36:40,540
Pi.

530
00:36:40,540 --> 00:36:43,130
Sine pi is 0, right?

531
00:36:43,130 --> 00:36:49,160
Therefore, you have a
destructive interference.

532
00:36:49,160 --> 00:36:50,830
This point is
really interesting.

533
00:36:50,830 --> 00:36:52,070
Why?

534
00:36:52,070 --> 00:36:57,430
Because that means all
the point-like source,

535
00:36:57,430 --> 00:37:02,120
all of them between
these two walls,

536
00:37:02,120 --> 00:37:09,480
are working together so nicely
such that the total field is

537
00:37:09,480 --> 00:37:12,270
completely cancelled.

538
00:37:12,270 --> 00:37:14,290
Isn't that remarkable?

539
00:37:14,290 --> 00:37:19,070
That's really, really
crazy when this happens.

540
00:37:19,070 --> 00:37:22,970
Takes a lot of work, infinite
number of source, to do that.

541
00:37:22,970 --> 00:37:26,960
Then, if you actually increase
further the sine theta,

542
00:37:26,960 --> 00:37:32,090
move away from the
center of the screen,

543
00:37:32,090 --> 00:37:35,930
basically you see that this
will increase again and reach

544
00:37:35,930 --> 00:37:41,960
a smaller maxima,
and again reach 0

545
00:37:41,960 --> 00:37:46,340
when this is actually
equal to 2 lambda over D.

546
00:37:46,340 --> 00:37:48,530
And this pattern continues.

547
00:37:48,530 --> 00:37:53,740
And, of course,
because of the symmetry

548
00:37:53,740 --> 00:37:56,780
we observe in this
expression, everything

549
00:37:56,780 --> 00:38:00,950
is actually proportional
to sine squared something.

550
00:38:00,950 --> 00:38:04,720
Therefore, this distribution
is actually symmetric.

551
00:38:04,720 --> 00:38:09,430
So you have minus lambda over
D, minus 2 lambda over D,

552
00:38:09,430 --> 00:38:12,915
et cetera, et cetera.

553
00:38:12,915 --> 00:38:14,400
Any questions so far?

554
00:38:20,340 --> 00:38:26,620
So what you can see here is
something really interesting.

555
00:38:26,620 --> 00:38:30,880
Sine theta, if you
multiply that by r,

556
00:38:30,880 --> 00:38:36,620
is telling you
the position which

557
00:38:36,620 --> 00:38:39,470
you will see on the screen.

558
00:38:39,470 --> 00:38:41,360
So this is actually--

559
00:38:41,360 --> 00:38:45,660
if you are interested in some
place, point of interest P,

560
00:38:45,660 --> 00:38:50,570
and this actually just
r times sine theta.

561
00:38:50,570 --> 00:38:52,310
And this is actually the slit.

562
00:38:55,420 --> 00:38:58,580
And I will move this
thing closer here.

563
00:38:58,580 --> 00:39:04,360
And the size of this
slit is called D.

564
00:39:04,360 --> 00:39:07,990
So one thing which is actually
very interesting in this result

565
00:39:07,990 --> 00:39:18,080
is that, if we look at the
width of the central principal

566
00:39:18,080 --> 00:39:19,780
maxima.

567
00:39:19,780 --> 00:39:21,770
The width is
actually the measure

568
00:39:21,770 --> 00:39:24,610
between the center
and the first minima,

569
00:39:24,610 --> 00:39:29,430
where you have complete
destructive interference.

570
00:39:29,430 --> 00:39:34,600
What you actually see here is
that this is actually something

571
00:39:34,600 --> 00:39:35,860
very interesting is happening.

572
00:39:35,860 --> 00:39:39,260
When you increase
D, if you increase

573
00:39:39,260 --> 00:39:42,400
D, what is going to
happen to the position

574
00:39:42,400 --> 00:39:46,590
of the first principal
minima, of our first minima?

575
00:39:46,590 --> 00:39:49,800
It's going to what?

576
00:39:49,800 --> 00:39:53,360
Going to become smaller.

577
00:39:53,360 --> 00:39:54,370
Right?

578
00:39:54,370 --> 00:40:01,180
So suppose I have a gap here and
I'm shooting a gun like crazy,

579
00:40:01,180 --> 00:40:03,250
boo-boo-boo-boo boo-boo-boo-boo.

580
00:40:03,250 --> 00:40:10,090
And I produce huge amount
of bullet, which I don't

581
00:40:10,090 --> 00:40:13,960
recommend to do that, for sure.

582
00:40:13,960 --> 00:40:16,140
What I'm going to do,
what I'm going to get

583
00:40:16,140 --> 00:40:21,230
is a distribution like this,
which are the bullets passing

584
00:40:21,230 --> 00:40:23,630
through this wall.

585
00:40:23,630 --> 00:40:27,230
If I increase the
size of the wall,

586
00:40:27,230 --> 00:40:31,150
the distribution I'm
getting is becoming what?

587
00:40:31,150 --> 00:40:32,490
Wider.

588
00:40:32,490 --> 00:40:33,290
Right?

589
00:40:33,290 --> 00:40:36,200
But the result here is
actually surprising.

590
00:40:36,200 --> 00:40:38,030
Why?

591
00:40:38,030 --> 00:40:43,020
When you increase
the width, when

592
00:40:43,020 --> 00:40:48,360
you increase the width of the D,
this function becomes smaller.

593
00:40:48,360 --> 00:40:53,240
That means the central
maxima will become narrower,

594
00:40:53,240 --> 00:40:56,680
as you can see from
this demonstration.

595
00:40:56,680 --> 00:41:01,170
So the left-hand side is an
experimental setup which you

596
00:41:01,170 --> 00:41:05,010
have a very, very narrow slit.

597
00:41:05,010 --> 00:41:09,120
And basically you get a
very wide distribution

598
00:41:09,120 --> 00:41:15,600
in the intensity as a function
of position on the screen.

599
00:41:15,600 --> 00:41:19,800
Right-hand side is
another situation where

600
00:41:19,800 --> 00:41:23,640
you have wider distribution.

601
00:41:23,640 --> 00:41:26,220
I'm sorry, wider
slit, and you are

602
00:41:26,220 --> 00:41:29,950
going to get a narrower
central maxima.

603
00:41:29,950 --> 00:41:34,950
Which is actually different
from the other experiment

604
00:41:34,950 --> 00:41:39,000
which we were actually doing.

605
00:41:39,000 --> 00:41:42,900
So that's the first thing
which we learn from here.

606
00:41:42,900 --> 00:41:46,650
And, also, the
distance, the distance

607
00:41:46,650 --> 00:41:52,920
between the maxima
and the minima

608
00:41:52,920 --> 00:41:56,430
is proportional to wavelength.

609
00:41:56,430 --> 00:42:01,230
So that means I can
measure wavelength

610
00:42:01,230 --> 00:42:05,340
by using the position
of the minima.

611
00:42:05,340 --> 00:42:10,020
And we are going to do that
to measure the wavelength

612
00:42:10,020 --> 00:42:13,210
of the laser beam.

613
00:42:13,210 --> 00:42:15,330
And, finally, the last
thing which we learn

614
00:42:15,330 --> 00:42:23,280
is that, in the central region,
you have a maxima of I0,

615
00:42:23,280 --> 00:42:27,570
and this intensity is
going to be going down,

616
00:42:27,570 --> 00:42:31,950
proportional to 1 over beta
squared, where beta is actually

617
00:42:31,950 --> 00:42:32,670
defined here.

618
00:42:32,670 --> 00:42:35,340
It's proportional
to D sine theta

619
00:42:35,340 --> 00:42:41,500
and inversely proportional
to wavelength.

620
00:42:41,500 --> 00:42:48,190
So now what I'm going
to do is experiment

621
00:42:48,190 --> 00:42:51,040
which I would like
to measure what would

622
00:42:51,040 --> 00:42:55,510
be the wavelength of my laser.

623
00:42:55,510 --> 00:42:58,040
So I have a laser here.

624
00:42:58,040 --> 00:42:58,540
Oh.

625
00:42:58,540 --> 00:43:01,720
OK, I don't want
to hurt anybody.

626
00:43:01,720 --> 00:43:04,300
So I have a laser here.

627
00:43:04,300 --> 00:43:08,770
And I have a slit, which you
cannot see, unfortunately.

628
00:43:08,770 --> 00:43:11,780
And I can read off the
width of the slit for you.

629
00:43:11,780 --> 00:43:14,020
The width of the
slit is carefully

630
00:43:14,020 --> 00:43:21,620
designed to be really
small, is 0.16 millimeter.

631
00:43:21,620 --> 00:43:23,900
This is my width.

632
00:43:23,900 --> 00:43:30,800
The D is actually equal
to 0.16 millimeter.

633
00:43:30,800 --> 00:43:36,440
And on the screen, you can see
that there's a pattern formed

634
00:43:36,440 --> 00:43:40,220
here, which you probably
cannot see very, very clearly,

635
00:43:40,220 --> 00:43:46,450
so I will try to lower the
intensity of the other source.

636
00:43:46,450 --> 00:43:49,450
So you can see,
then, see that there

637
00:43:49,450 --> 00:43:53,840
is an interference pattern
or diffraction pattern which

638
00:43:53,840 --> 00:43:54,920
is actually showing here.

639
00:43:57,680 --> 00:44:05,660
So what I really need in order
to calculate the wavelength

640
00:44:05,660 --> 00:44:08,450
is the sine theta angle.

641
00:44:08,450 --> 00:44:10,370
Which I will really need
the sine theta angle.

642
00:44:10,370 --> 00:44:11,786
Then I can actually
calculate what

643
00:44:11,786 --> 00:44:15,110
will be the wavelength
of this laser.

644
00:44:15,110 --> 00:44:19,580
So that means I will need
help from a volunteer.

645
00:44:19,580 --> 00:44:22,370
Who volunteer to
help me to measure

646
00:44:22,370 --> 00:44:26,090
the distance between this
slit and the large screen?

647
00:44:26,090 --> 00:44:27,590
Can somebody volunteer?

648
00:44:27,590 --> 00:44:28,210
Yes, please.

649
00:44:32,320 --> 00:44:36,220
So we are going to
measure the distance.

650
00:44:36,220 --> 00:44:36,970
Can you hold this?

651
00:44:36,970 --> 00:44:39,720
And can you actually put it?

652
00:44:39,720 --> 00:44:43,330
OK, try to pull this thing,
and we will try our best

653
00:44:43,330 --> 00:44:47,330
to make it straight.

654
00:44:47,330 --> 00:44:50,130
Thank you very much.

655
00:44:50,130 --> 00:44:52,345
We don't want to destroy
the experiment as well.

656
00:44:56,360 --> 00:44:57,360
This is not working?

657
00:45:01,050 --> 00:45:02,790
Let me do this in the other way.

658
00:45:02,790 --> 00:45:06,310
So how about-- trial
and error, right?

659
00:45:08,910 --> 00:45:10,240
How about this.

660
00:45:10,240 --> 00:45:14,690
You hold that thing, and I'm
going to actually measure

661
00:45:14,690 --> 00:45:16,036
the distance from here.

662
00:45:18,980 --> 00:45:24,520
And I need to really make it
really carefully, measure this

663
00:45:24,520 --> 00:45:25,810
very carefully.

664
00:45:25,810 --> 00:45:29,890
And I don't want to
destroy anything,

665
00:45:29,890 --> 00:45:30,980
which is very possible.

666
00:45:33,910 --> 00:45:37,670
So what I'm getting?

667
00:45:37,670 --> 00:45:42,230
I get 7.5 meter.

668
00:45:42,230 --> 00:45:47,720
So that's actually the distance
between the screen, the screen

669
00:45:47,720 --> 00:45:49,260
and the source.

670
00:45:49,260 --> 00:45:50,600
Hold that for a second.

671
00:45:50,600 --> 00:45:55,820
I am going to measure the width
of, the distance between two

672
00:45:55,820 --> 00:45:56,750
minima.

673
00:45:56,750 --> 00:46:04,170
The distance between two
minima is 7 centimeter.

674
00:46:04,170 --> 00:46:06,490
Thank you very much.

675
00:46:06,490 --> 00:46:08,620
Thank you for your help.

676
00:46:08,620 --> 00:46:12,170
So we have now
everything we need

677
00:46:12,170 --> 00:46:14,070
to calculate the wavelength.

678
00:46:17,276 --> 00:46:19,200
I'm going to clean
this up first.

679
00:46:24,380 --> 00:46:27,560
We have what?

680
00:46:27,560 --> 00:46:29,290
We have the distance now.

681
00:46:29,290 --> 00:46:34,970
The distance between the source
and the screen is 7.6 meter.

682
00:46:38,190 --> 00:46:41,860
So now I would like to calculate
what will be the lambda.

683
00:46:44,700 --> 00:46:47,600
And also I know the distance--

684
00:46:47,600 --> 00:46:55,830
the distance between these
two minima is 7 centimeter.

685
00:46:55,830 --> 00:47:01,303
So that means this
will be 3.5 centimeter.

686
00:47:04,270 --> 00:47:10,770
So lambda divided by D is
actually equal to sine theta.

687
00:47:10,770 --> 00:47:17,040
Which is actually small
d, which is the distance

688
00:47:17,040 --> 00:47:18,225
between the minima.

689
00:47:18,225 --> 00:47:20,550
The small d is here.

690
00:47:20,550 --> 00:47:24,860
The small d is the
distance between the minima

691
00:47:24,860 --> 00:47:26,580
and the center.

692
00:47:26,580 --> 00:47:31,190
Divided by r, which is the
distance between the source

693
00:47:31,190 --> 00:47:33,620
and the screen.

694
00:47:33,620 --> 00:47:36,300
Therefore, I can
have lambda will

695
00:47:36,300 --> 00:47:42,800
be equal to capital D
times small d divided by r.

696
00:47:42,800 --> 00:47:44,450
So what is actually the answer?

697
00:47:44,450 --> 00:47:48,140
So basically I have
capital D, which

698
00:47:48,140 --> 00:47:53,420
is actually 0.16 millimeter.

699
00:47:53,420 --> 00:47:57,080
So that is actually shown
there but you cannot see it.

700
00:47:59,720 --> 00:48:05,750
So I will use a different
board for this calculation.

701
00:48:05,750 --> 00:48:09,710
So, basically, we
will actually get

702
00:48:09,710 --> 00:48:17,825
lambda is equal to capital D
times small d divided by r.

703
00:48:17,825 --> 00:48:24,120
Capital D is 0.16 times
10 to the minus 3 meter.

704
00:48:27,710 --> 00:48:29,780
And what is actually
the small d?

705
00:48:29,780 --> 00:48:34,834
The small d is actually
3.5 centimeters.

706
00:48:40,620 --> 00:48:48,030
And, finally, I have 7.6 meter,
which is actually the small r.

707
00:48:48,030 --> 00:48:51,220
Divided by 7.6 meter.

708
00:48:56,140 --> 00:48:58,180
Can somebody actually
calculate this for me?

709
00:49:00,930 --> 00:49:02,460
Anybody have a smartphone?

710
00:49:07,320 --> 00:49:10,480
This means that I haven't
done this experiment myself,

711
00:49:10,480 --> 00:49:12,830
and we will see what
is going to happen.

712
00:49:12,830 --> 00:49:13,700
I hope it will work.

713
00:49:17,390 --> 00:49:20,337
What is actually the value?

714
00:49:20,337 --> 00:49:24,330
AUDIENCE: 7.368 times
10 to the negative 7.

715
00:49:24,330 --> 00:49:29,330
PROFESSOR: 7.368 times
10 to the minus 7.

716
00:49:29,330 --> 00:49:34,280
This is actually equal
to 7.37 times 10--

717
00:49:34,280 --> 00:49:34,790
oh, wait.

718
00:49:34,790 --> 00:49:40,050
This is actually 737 nanometer.

719
00:49:40,050 --> 00:49:43,880
Actually, the
wavelength of the red

720
00:49:43,880 --> 00:49:48,180
is actually between 620 and 750.

721
00:49:48,180 --> 00:49:51,791
And actually we are actually
getting the correct value.

722
00:49:51,791 --> 00:49:52,290
You see?

723
00:49:52,290 --> 00:49:55,770
So, actually, now you can
actually tell your friends

724
00:49:55,770 --> 00:49:59,400
that, although the
wavelength is so small,

725
00:49:59,400 --> 00:50:04,490
but I can't measure it
with such a square feet

726
00:50:04,490 --> 00:50:07,700
experimental setup.

727
00:50:07,700 --> 00:50:10,220
So that's a
successful experiment.

728
00:50:10,220 --> 00:50:12,550
So that is actually
telling you that it's

729
00:50:12,550 --> 00:50:16,400
a proof that this
formula, which we actually

730
00:50:16,400 --> 00:50:19,110
do all the crazy
work of this Fourier

731
00:50:19,110 --> 00:50:22,355
transform in
two-dimensional integration,

732
00:50:22,355 --> 00:50:24,470
it should really work.

733
00:50:24,470 --> 00:50:29,750
And the result is actually
not really far from what

734
00:50:29,750 --> 00:50:33,770
you can get from Wikipedia.

735
00:50:33,770 --> 00:50:37,190
So, at this point, I would like
to take a five-minute break

736
00:50:37,190 --> 00:50:38,750
to take some questions.

737
00:50:38,750 --> 00:50:43,790
And then we are going
to come back in, at 31,

738
00:50:43,790 --> 00:50:47,340
and we are going to discuss
another very interesting issue,

739
00:50:47,340 --> 00:50:48,471
resolution.

740
00:50:53,770 --> 00:50:55,540
So welcome back.

741
00:50:55,540 --> 00:50:59,680
So there are a few
questions about--

742
00:51:02,590 --> 00:51:08,420
there were a few questions
about the pattern here,

743
00:51:08,420 --> 00:51:09,790
which is interesting.

744
00:51:09,790 --> 00:51:12,970
So you can see that what we
actually concluded from here

745
00:51:12,970 --> 00:51:18,700
is that the width of the
central principal maxima

746
00:51:18,700 --> 00:51:23,950
is actually two times of the
width of the secondary maxima.

747
00:51:23,950 --> 00:51:27,330
So you can see that the width
here between these two points

748
00:51:27,330 --> 00:51:31,840
is actually lambda over D. But
the width between these two

749
00:51:31,840 --> 00:51:34,070
points, which actually
give you the width

750
00:51:34,070 --> 00:51:38,690
of the central principal
maxima, is actually 2 times

751
00:51:38,690 --> 00:51:40,900
of lambda over D.

752
00:51:40,900 --> 00:51:44,190
And now this actually can be
seen from the experiment there.

753
00:51:44,190 --> 00:51:47,090
Maybe not easy for the moment.

754
00:51:47,090 --> 00:51:50,340
But this is actually the width,
and the smaller structure

755
00:51:50,340 --> 00:51:58,090
is actually having a width
half of the central peak.

756
00:51:58,090 --> 00:52:00,950
So that is actually something
which is interesting,

757
00:52:00,950 --> 00:52:04,550
and I would like to share
that with everybody.

758
00:52:04,550 --> 00:52:07,870
So now we actually come back
to the original question

759
00:52:07,870 --> 00:52:11,800
we were actually
discussing last time.

760
00:52:11,800 --> 00:52:13,960
So one interesting
thing we observed

761
00:52:13,960 --> 00:52:17,770
in this two-slit
interference experiment

762
00:52:17,770 --> 00:52:22,330
is that you not only see all
those little structures, which

763
00:52:22,330 --> 00:52:24,850
is actually kind of
periodic structure,

764
00:52:24,850 --> 00:52:28,300
and that they are coming from
the two-slit interference.

765
00:52:28,300 --> 00:52:31,000
And you also see
this larger structure

766
00:52:31,000 --> 00:52:34,750
which is showing up there, which
is actually going up and down,

767
00:52:34,750 --> 00:52:38,530
and also it produce minima
at some specific point.

768
00:52:38,530 --> 00:52:42,890
Now we understand what
is actually happening.

769
00:52:42,890 --> 00:52:49,960
Suppose I have two-slit
interference experiment, where

770
00:52:49,960 --> 00:52:57,200
I have the width of the slits to
be capital D, to be very small.

771
00:52:57,200 --> 00:52:59,185
D is very, very small.

772
00:53:01,890 --> 00:53:04,500
And the distance
between the slit

773
00:53:04,500 --> 00:53:06,460
is actually called small d.

774
00:53:06,460 --> 00:53:09,870
Which is kind of
weird, but you have

775
00:53:09,870 --> 00:53:13,260
to accept that because
it's on my note.

776
00:53:13,260 --> 00:53:15,830
And you can see
that, interestingly,

777
00:53:15,830 --> 00:53:21,240
if this is the situation, then
you have this periodic pattern

778
00:53:21,240 --> 00:53:25,630
and you will see no
decrease in amplitude

779
00:53:25,630 --> 00:53:30,150
as a function of distance with
respect to the central point

780
00:53:30,150 --> 00:53:33,390
of the screen.

781
00:53:33,390 --> 00:53:35,490
So that's actually very nice.

782
00:53:35,490 --> 00:53:42,570
However, if you consider
a realistic situation,

783
00:53:42,570 --> 00:53:50,360
where the size, or say the width
of the slit is not negligible,

784
00:53:50,360 --> 00:53:52,040
is sizable.

785
00:53:52,040 --> 00:53:54,710
And what is going
to happen is that--

786
00:53:54,710 --> 00:54:00,170
OK, let's forget about the
second one for a moment.

787
00:54:00,170 --> 00:54:05,060
We already learned that
the output intensity

788
00:54:05,060 --> 00:54:10,250
of a single slit is already
varying as a function of angle.

789
00:54:10,250 --> 00:54:12,300
So I have this pattern.

790
00:54:12,300 --> 00:54:20,570
Therefore, if you have these
two realistic slit interacting

791
00:54:20,570 --> 00:54:23,570
with each other, have
interference pattern,

792
00:54:23,570 --> 00:54:29,120
what you are going to
expect is that you are going

793
00:54:29,120 --> 00:54:35,810
to have the two-slit
interference pattern modulated

794
00:54:35,810 --> 00:54:39,670
by diffraction pattern.

795
00:54:39,670 --> 00:54:42,850
Because, originally,
coming from a single slit,

796
00:54:42,850 --> 00:54:47,740
you already have a
varying intensity

797
00:54:47,740 --> 00:54:54,500
as a function of sine theta,
as we already discussed there.

798
00:54:54,500 --> 00:55:00,130
So, if we put all those
information together,

799
00:55:00,130 --> 00:55:03,940
we are going to get I.
The intensity is going

800
00:55:03,940 --> 00:55:13,390
to be equal to I0, which is
some maxima, sine beta divided

801
00:55:13,390 --> 00:55:22,790
by beta, square of that,
sine N delta divided by 2,

802
00:55:22,790 --> 00:55:29,610
divided by sine delta
divided by 2, squared.

803
00:55:29,610 --> 00:55:32,130
So basically what
I'm talking about

804
00:55:32,130 --> 00:55:36,210
is that, if you have
N-slit experiment,

805
00:55:36,210 --> 00:55:40,950
each slit have the same width.

806
00:55:40,950 --> 00:55:42,540
And what you are
going to get is--

807
00:55:42,540 --> 00:55:47,720
this is actually the N-slit
interference pattern.

808
00:55:56,510 --> 00:56:05,288
And that is actually modulated
by diffraction pattern.

809
00:56:09,680 --> 00:56:13,760
Where beta, just a
reminder, in this summary

810
00:56:13,760 --> 00:56:19,850
is pi capital D divided
by lambda sine theta.

811
00:56:19,850 --> 00:56:22,700
And the delta, which is
the optical path length

812
00:56:22,700 --> 00:56:28,880
difference we defined before,
is k times d sine theta,

813
00:56:28,880 --> 00:56:35,160
and that is actually equal
to 2 pi times d sine theta

814
00:56:35,160 --> 00:56:38,390
divided by lambda.

815
00:56:38,390 --> 00:56:42,290
So that is actually
why, when we perform

816
00:56:42,290 --> 00:56:50,510
the experiment of a double-slit
experiment in the last lecture,

817
00:56:50,510 --> 00:56:55,520
we get complicated
interference pattern like this,

818
00:56:55,520 --> 00:56:58,400
and it has a very
complicated structure.

819
00:56:58,400 --> 00:57:02,340
And now we actually understand
why the structure is like this.

820
00:57:02,340 --> 00:57:06,710
The small structure in
this case is actually

821
00:57:06,710 --> 00:57:10,970
coming from interference,
two-slit interference.

822
00:57:10,970 --> 00:57:15,270
And the additional structure,
larger-scale structure,

823
00:57:15,270 --> 00:57:18,300
is actually coming
from diffraction,

824
00:57:18,300 --> 00:57:22,350
is coming from the varying
intensity of a single slit

825
00:57:22,350 --> 00:57:26,180
as a function of sine theta.

826
00:57:26,180 --> 00:57:27,180
Any questions so far?

827
00:57:30,920 --> 00:57:33,500
We are making a lot of progress.

828
00:57:33,500 --> 00:57:36,710
So what I would
like to move on is

829
00:57:36,710 --> 00:57:39,390
to discuss with you
something really interesting.

830
00:57:39,390 --> 00:57:48,590
So we discussed and learned how
to explain why we have actually

831
00:57:48,590 --> 00:57:52,130
colorful soap bubble.

832
00:57:52,130 --> 00:57:55,100
So I have something
totally unrelated.

833
00:57:55,100 --> 00:58:00,160
So we have a soap bubble
also in the space, which

834
00:58:00,160 --> 00:58:03,500
is the Soap Bubble nebula.

835
00:58:03,500 --> 00:58:05,720
Which is really interesting,
and you can actually

836
00:58:05,720 --> 00:58:08,580
Google it and see what is
actually happening there.

837
00:58:08,580 --> 00:58:11,290
But, actually, that's
actually not my point.

838
00:58:11,290 --> 00:58:16,340
Then my point is that you
really need very good resolution

839
00:58:16,340 --> 00:58:21,050
telescope so that you can
actually observe those really

840
00:58:21,050 --> 00:58:26,030
beautiful objects which are
already there and cannot be

841
00:58:26,030 --> 00:58:28,250
made by human.

842
00:58:28,250 --> 00:58:29,450
Made by somebody else.

843
00:58:32,130 --> 00:58:36,840
So this is actually
what I'm getting into.

844
00:58:36,840 --> 00:58:41,220
So the resolution is
really something important.

845
00:58:41,220 --> 00:58:45,000
So when you take a
look at this picture,

846
00:58:45,000 --> 00:58:47,670
the resolution is not very good.

847
00:58:47,670 --> 00:58:52,470
So as you can see, now the peak
position of two nearby peak

848
00:58:52,470 --> 00:58:55,060
is actually connecting
to each other.

849
00:58:55,060 --> 00:58:58,370
Then what do we see
from this picture?

850
00:58:58,370 --> 00:59:00,320
You see maybe a lion?

851
00:59:00,320 --> 00:59:01,190
I don't know.

852
00:59:01,190 --> 00:59:04,010
Maybe, maybe not.

853
00:59:04,010 --> 00:59:10,000
But if you improve the
resolution, what do you see?

854
00:59:10,000 --> 00:59:12,800
It's actually zebra.

855
00:59:12,800 --> 00:59:14,750
So this is actually
the kind of thing

856
00:59:14,750 --> 00:59:18,590
which we would like
to discuss with you.

857
00:59:18,590 --> 00:59:22,790
We are actually touching
this important phenomenon,

858
00:59:22,790 --> 00:59:25,475
which is actually
needed for observing

859
00:59:25,475 --> 00:59:31,850
an interesting phenomena which
is actually happening really

860
00:59:31,850 --> 00:59:35,600
far away from the Earth.

861
00:59:35,600 --> 00:59:39,480
What is actually the resolution?

862
00:59:39,480 --> 00:59:41,740
And we are going to
talk about that as well.

863
00:59:41,740 --> 00:59:46,120
And I would like to show you
another interesting example.

864
00:59:46,120 --> 00:59:52,200
So this is a comparison
between not so serious picture

865
00:59:52,200 --> 00:59:55,440
and the picture from
Hubble telescope.

866
00:59:55,440 --> 01:00:01,360
So I was using some telescope
with D equal to 40 centimeter.

867
01:00:01,360 --> 01:00:02,850
And that's actually
the best thing

868
01:00:02,850 --> 01:00:09,720
which I can achieve, shooting
the same planetary nebula M57.

869
01:00:09,720 --> 01:00:13,650
That object is actually
2,500 light year away.

870
01:00:13,650 --> 01:00:17,490
And you can see that I cannot
get really a lot of detail from

871
01:00:17,490 --> 01:00:19,350
this image.

872
01:00:19,350 --> 01:00:23,310
And now, if you
compare that to D equal

873
01:00:23,310 --> 01:00:27,810
to 240 centimeter
Hubble telescope,

874
01:00:27,810 --> 01:00:29,400
and also, at the
same time, this thing

875
01:00:29,400 --> 01:00:33,324
is actually above
the atmosphere.

876
01:00:33,324 --> 01:00:34,990
So that's actually
very, very important.

877
01:00:34,990 --> 01:00:39,430
And you can see that you do get
a much, much better resolution,

878
01:00:39,430 --> 01:00:43,090
and you can actually see all
the fine detail, very, very

879
01:00:43,090 --> 01:00:46,690
fine detail of this image.

880
01:00:46,690 --> 01:00:50,630
And we are in the
position to understand

881
01:00:50,630 --> 01:00:59,075
the resolution and the
limit which we can have

882
01:00:59,075 --> 01:01:01,540
due to diffraction, actually.

883
01:01:01,540 --> 01:01:12,220
So if I consider now a
pinhole with diameter equal

884
01:01:12,220 --> 01:01:18,170
to D. So right now what
we are actually doing

885
01:01:18,170 --> 01:01:24,250
is not a single slit any
more, but a hole with radius

886
01:01:24,250 --> 01:01:27,220
D over 2.

887
01:01:27,220 --> 01:01:34,150
And we can do the same,
exactly the same calculation

888
01:01:34,150 --> 01:01:36,220
using this formula.

889
01:01:36,220 --> 01:01:39,800
But I'm not going to do
that for the sake of time.

890
01:01:39,800 --> 01:01:44,680
So we can do exactly that
same C function calculation.

891
01:01:44,680 --> 01:01:50,920
And what we are going to get
is I as a function of theta

892
01:01:50,920 --> 01:02:01,120
is equal to I0 J1 beta
divided by beta, squared.

893
01:02:01,120 --> 01:02:07,870
Where J1 is the Bessel
function of the first kind.

894
01:02:07,870 --> 01:02:11,200
Sounds really scary,
but it's actually not.

895
01:02:11,200 --> 01:02:20,440
So what I really need is the
zeros of the Bessel function,

896
01:02:20,440 --> 01:02:24,160
so that I can actually
extract the interference

897
01:02:24,160 --> 01:02:29,170
pattern and the width
of the central maximum.

898
01:02:29,170 --> 01:02:31,880
So now, since we are
having a pinhole,

899
01:02:31,880 --> 01:02:33,940
basically all of
those things are,

900
01:02:33,940 --> 01:02:37,720
all those patterns are
actually two-dimensional.

901
01:02:37,720 --> 01:02:46,660
And I was wondering what will be
the needed beta value so that I

902
01:02:46,660 --> 01:02:50,490
can actually reach the minima.

903
01:02:50,490 --> 01:02:52,610
Why is that important?

904
01:02:52,610 --> 01:02:55,800
That is actually
telling you the limit

905
01:02:55,800 --> 01:02:58,810
of the optical resolution.

906
01:02:58,810 --> 01:03:02,780
If I have two peaks
which are actually

907
01:03:02,780 --> 01:03:06,960
placed too close to each
other, like what we actually

908
01:03:06,960 --> 01:03:09,000
see in the previous
slide, then we

909
01:03:09,000 --> 01:03:15,510
can actually not separate very
well these two light source.

910
01:03:15,510 --> 01:03:19,590
On the other hand, if the
distance between these two peak

911
01:03:19,590 --> 01:03:23,430
is larger than the
first minima, then I

912
01:03:23,430 --> 01:03:25,221
can actually be very safe.

913
01:03:25,221 --> 01:03:26,220
I can actually separate.

914
01:03:26,220 --> 01:03:28,770
I can say that, ha, this
is really two peaks.

915
01:03:28,770 --> 01:03:31,080
Two stars, two light source.

916
01:03:31,080 --> 01:03:33,200
I can tell.

917
01:03:33,200 --> 01:03:36,060
So that is actually why
this is actually important.

918
01:03:36,060 --> 01:03:39,750
And where is
actually the minima?

919
01:03:39,750 --> 01:03:42,990
And I can already
solve that for you.

920
01:03:42,990 --> 01:03:47,130
And that is actually when
x is equal to roughly,

921
01:03:47,130 --> 01:03:53,790
the numerical value
is roughly 3.83.

922
01:03:53,790 --> 01:03:55,290
So that's actually
not important.

923
01:03:55,290 --> 01:03:57,180
Those numbers are not important.

924
01:03:57,180 --> 01:04:01,210
The important result is
really the conclusion.

925
01:04:01,210 --> 01:04:06,810
So beta is equal to 3.83,
and that is actually

926
01:04:06,810 --> 01:04:13,470
equal to pi D sine
theta divided by lambda.

927
01:04:13,470 --> 01:04:16,280
So that is actually our
original definition.

928
01:04:16,280 --> 01:04:21,390
And I can solve what will be the
sine theta, which is actually

929
01:04:21,390 --> 01:04:25,560
telling you the
position of the minima.

930
01:04:25,560 --> 01:04:33,580
So sine theta will be equal to
actually 1.22 lambda divided

931
01:04:33,580 --> 01:04:36,290
by D.

932
01:04:36,290 --> 01:04:38,040
So what does that mean?

933
01:04:38,040 --> 01:04:44,160
That means the position where
you have the first minima

934
01:04:44,160 --> 01:04:47,610
is actually happening
when you have sine theta--

935
01:04:52,480 --> 01:04:54,250
this is the theta--

936
01:04:54,250 --> 01:04:58,980
when you have sine theta equal
to 1.22 times lambda divided

937
01:04:58,980 --> 01:05:04,280
by D.

938
01:05:04,280 --> 01:05:07,130
So that is actually very nice.

939
01:05:07,130 --> 01:05:09,830
Doing exactly the same
exercise, and we now

940
01:05:09,830 --> 01:05:14,680
understand where my minima is.

941
01:05:14,680 --> 01:05:16,790
Then that is
actually telling you

942
01:05:16,790 --> 01:05:19,994
something about the resolution.

943
01:05:26,910 --> 01:05:29,480
So what I'm going
to try to get into

944
01:05:29,480 --> 01:05:34,910
is that, now, let's design a
phone together, a mobile phone

945
01:05:34,910 --> 01:05:36,470
together.

946
01:05:36,470 --> 01:05:41,820
So what is actually the
width of the human pupil?

947
01:05:44,780 --> 01:05:49,410
The width is actually
roughly 2 to 4 millimeter--

948
01:05:49,410 --> 01:05:54,480
when narrow, when you see a lot
of light all over the place--

949
01:05:54,480 --> 01:05:58,000
or 3 to 8 millimeter.

950
01:05:58,000 --> 01:06:03,860
So that is actually the
typical length when wide.

951
01:06:03,860 --> 01:06:06,910
So that is actually the
width of the pinhole.

952
01:06:09,590 --> 01:06:14,960
So, typically, the visible
light, as we calculated,

953
01:06:14,960 --> 01:06:18,610
is something like 500 nanometer.

954
01:06:21,820 --> 01:06:25,940
And the width of
the human pupil, we

955
01:06:25,940 --> 01:06:31,430
can actually take a
number of 5 millimeter.

956
01:06:31,430 --> 01:06:37,700
And now we can actually try to
give input to the phone design.

957
01:06:37,700 --> 01:06:42,650
So what will be the resolution
if we take these two

958
01:06:42,650 --> 01:06:44,510
parameter together?

959
01:06:44,510 --> 01:06:47,750
So, basically, the
resolution of your eye, we

960
01:06:47,750 --> 01:06:49,910
can now calculate that.

961
01:06:49,910 --> 01:06:51,520
So what is that?

962
01:06:51,520 --> 01:07:01,540
That is actually 1.22 times
500 nanometer divided by--

963
01:07:01,540 --> 01:07:03,890
OK, my function is D--

964
01:07:03,890 --> 01:07:06,980
so divided by D is
equal to 5 millimeter.

965
01:07:06,980 --> 01:07:10,700
5 millimeter.

966
01:07:10,700 --> 01:07:13,070
Basically, what you
are going to get

967
01:07:13,070 --> 01:07:17,900
is 1.22 times 10 to the minus 4.

968
01:07:17,900 --> 01:07:20,080
This is actually the resolution.

969
01:07:20,080 --> 01:07:23,670
Sine theta, roughly
equal to theta,

970
01:07:23,670 --> 01:07:28,430
is actually equal to 1.22
times 10 to the minus 4.

971
01:07:33,020 --> 01:07:40,240
I have a iPhone 6 or
7, whatever you have.

972
01:07:40,240 --> 01:07:44,850
Basically is 401 ppi.

973
01:07:44,850 --> 01:07:48,280
401 ppi is actually
what is that?

974
01:07:48,280 --> 01:07:51,250
Pixel per inch.

975
01:07:51,250 --> 01:07:53,200
So what is actually the delta x?

976
01:07:53,200 --> 01:07:57,680
So if I have a phone, OK,
it has a camera there.

977
01:07:57,680 --> 01:08:00,640
That is my phone.

978
01:08:00,640 --> 01:08:03,010
And this is my eye.

979
01:08:03,010 --> 01:08:05,740
Looks like an eye.

980
01:08:05,740 --> 01:08:09,670
The distance is 20 centimeter.

981
01:08:09,670 --> 01:08:11,500
I do this, which is unusual.

982
01:08:17,200 --> 01:08:18,850
We have 400 ppi.

983
01:08:18,850 --> 01:08:24,189
So what is actually the delta x,
the delta x between the pixels?

984
01:08:24,189 --> 01:08:32,109
The delta x is equal to 2.54
centimeter divided by 401,

985
01:08:32,109 --> 01:08:36,220
and that will give you
something like 6.3 times 10

986
01:08:36,220 --> 01:08:38,098
to the minus 3 centimeter.

987
01:08:40,970 --> 01:08:45,930
If I am trying to be
healthy and I do this, then

988
01:08:45,930 --> 01:08:47,649
what is actually
the delta theta?

989
01:08:47,649 --> 01:08:52,540
The delta theta is delta x
divided by 20 centimeter,

990
01:08:52,540 --> 01:08:55,660
and that is 3 times
10 to be minus 4.

991
01:08:59,420 --> 01:09:04,939
If you compare this
value to the limit

992
01:09:04,939 --> 01:09:11,090
I calculated here, you can see
that, what is the conclusion?

993
01:09:11,090 --> 01:09:16,010
Can I resolve the
pixels on the phone?

994
01:09:16,010 --> 01:09:19,640
The answer is yes.

995
01:09:19,640 --> 01:09:22,950
So that means this phone
is not good enough.

996
01:09:22,950 --> 01:09:24,080
They have to do more work.

997
01:09:26,600 --> 01:09:29,689
And now I'm going
to design a jPhone.

998
01:09:34,330 --> 01:09:38,630
Maybe at some point
I got really crazy

999
01:09:38,630 --> 01:09:41,779
and I decided to
open a company, which

1000
01:09:41,779 --> 01:09:44,450
is Yen-Jie's phone company.

1001
01:09:44,450 --> 01:09:50,990
And, of course, I will say this
is jPhone because it's Yen-Jie.

1002
01:09:50,990 --> 01:10:01,600
And I'm going to put
40,000 ppi in this phone.

1003
01:10:01,600 --> 01:10:03,424
Will you buy it?

1004
01:10:03,424 --> 01:10:03,965
AUDIENCE: No.

1005
01:10:03,965 --> 01:10:04,590
AUDIENCE: Sure.

1006
01:10:04,590 --> 01:10:05,950
AUDIENCE: How much?

1007
01:10:05,950 --> 01:10:07,257
PROFESSOR: $1.

1008
01:10:07,257 --> 01:10:07,840
You'll buy it?

1009
01:10:07,840 --> 01:10:10,440
We'll see.

1010
01:10:10,440 --> 01:10:14,760
Maybe you will buy it
because you are my student.

1011
01:10:14,760 --> 01:10:16,420
But it's not worth it.

1012
01:10:16,420 --> 01:10:18,080
Why is that?

1013
01:10:18,080 --> 01:10:26,520
Because you cannot resolve this
kind of fine or small distance

1014
01:10:26,520 --> 01:10:27,780
between pixels.

1015
01:10:27,780 --> 01:10:29,850
So it's actually useless.

1016
01:10:29,850 --> 01:10:31,100
So what is actually the limit?

1017
01:10:31,100 --> 01:10:34,590
You can also probably
give that to your friends.

1018
01:10:34,590 --> 01:10:40,720
2,000 pixel per inch
is roughly the limit.

1019
01:10:40,720 --> 01:10:46,140
Beyond that, maybe the next
generation of our students

1020
01:10:46,140 --> 01:10:49,790
will be using this like this.

1021
01:10:49,790 --> 01:10:52,580
Then it works, and
it is worth it.

1022
01:10:52,580 --> 01:10:56,230
You can actually
read this distance.

1023
01:10:56,230 --> 01:11:00,980
It doesn't work for old people
like me, but for young people

1024
01:11:00,980 --> 01:11:03,260
it works.

1025
01:11:03,260 --> 01:11:04,340
So very good.

1026
01:11:04,340 --> 01:11:06,800
So that's another thing
which you have learned.

1027
01:11:06,800 --> 01:11:10,670
So, finally, as I
promised you, we

1028
01:11:10,670 --> 01:11:15,650
are going to go back to
this business of designing

1029
01:11:15,650 --> 01:11:18,860
the Enterprise for Star Trek.

1030
01:11:18,860 --> 01:11:22,770
So what does Enterprise
do to their friends?

1031
01:11:22,770 --> 01:11:26,120
They shoot laser beam.

1032
01:11:26,120 --> 01:11:29,780
And they try to attack
the other ships.

1033
01:11:29,780 --> 01:11:33,836
And what I'm going to do
now is to calculate for you

1034
01:11:33,836 --> 01:11:35,294
what is going to happen.

1035
01:11:35,294 --> 01:11:38,700
OK, now I have this
laser beam here.

1036
01:11:38,700 --> 01:11:41,690
And, in principle,
before you take 8.03,

1037
01:11:41,690 --> 01:11:47,010
you are going to say,
aha, I can shoot the moon.

1038
01:11:47,010 --> 01:11:50,360
And this light is going
to be really narrow

1039
01:11:50,360 --> 01:11:53,800
and it's going to hit the moon,
a very small area on the moon.

1040
01:11:56,930 --> 01:11:59,750
Do you believe that now?

1041
01:11:59,750 --> 01:12:02,780
I hope the answer is not.

1042
01:12:02,780 --> 01:12:05,660
How crazy is this idea?

1043
01:12:05,660 --> 01:12:07,950
What is the size of the spot?

1044
01:12:07,950 --> 01:12:09,560
Can you guess?

1045
01:12:09,560 --> 01:12:12,980
Is that 1 millimeter?

1046
01:12:12,980 --> 01:12:15,500
10 meter?

1047
01:12:15,500 --> 01:12:18,880
Or 200 kilometer?

1048
01:12:18,880 --> 01:12:23,860
How many of you think
by now is 1 millimeter?

1049
01:12:23,860 --> 01:12:24,980
Nobody?

1050
01:12:24,980 --> 01:12:26,830
Fortunately.

1051
01:12:26,830 --> 01:12:30,330
How about 10 meter?

1052
01:12:30,330 --> 01:12:34,660
One, two, three.

1053
01:12:34,660 --> 01:12:35,320
OK.

1054
01:12:35,320 --> 01:12:36,490
Three of you.

1055
01:12:36,490 --> 01:12:39,305
How about 200 kilometer?

1056
01:12:39,305 --> 01:12:41,100
You believe that?

1057
01:12:41,100 --> 01:12:42,980
Really?

1058
01:12:42,980 --> 01:12:49,680
The answer is really
200 kilometer.

1059
01:12:49,680 --> 01:12:53,130
It's the size of Missouri state.

1060
01:12:56,590 --> 01:13:00,310
So now you can see that this
is not practical at all,

1061
01:13:00,310 --> 01:13:04,120
and you have to really do what?

1062
01:13:04,120 --> 01:13:07,216
Increase or decrease the radius?

1063
01:13:07,216 --> 01:13:08,070
AUDIENCE: Increase.

1064
01:13:08,070 --> 01:13:09,960
PROFESSOR: Increase.

1065
01:13:09,960 --> 01:13:13,490
By the end of this lecture
everybody get this idea.

1066
01:13:13,490 --> 01:13:18,820
Thank you very much
for the attention.

1067
01:13:18,820 --> 01:13:21,800
And I hope you
enjoyed this lecture.