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COLIN SHEPPARD: Right.

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00:00:21,480 --> 00:00:24,850
So you can see it's me again.

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00:00:24,850 --> 00:00:27,360
So George can have
a holiday again.

11
00:00:27,360 --> 00:00:29,490
But he's not really
having a real holiday

12
00:00:29,490 --> 00:00:31,970
because he's
actually sitting in.

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00:00:31,970 --> 00:00:35,135
And I guess one
reason why we decided

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00:00:35,135 --> 00:00:37,950
that he had to do that
was because I have

15
00:00:37,950 --> 00:00:40,770
to use his computer
because he's written

16
00:00:40,770 --> 00:00:45,360
these talks in this
strange application that

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00:00:45,360 --> 00:00:46,485
doesn't run on my computer.

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00:00:49,290 --> 00:00:50,940
So now we going to--

19
00:00:50,940 --> 00:00:56,010
today, we're just about getting
into interesting stuff now.

20
00:00:56,010 --> 00:00:59,950
Seems rather a long time since
the beginning of this course

21
00:00:59,950 --> 00:01:04,319
before we get into the
real wave optics, which

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00:01:04,319 --> 00:01:08,940
I'm sure everyone would agree
is the most interesting part.

23
00:01:08,940 --> 00:01:16,430
So this lecture is
about interferometry.

24
00:01:16,430 --> 00:01:18,102
And then the next lecture--

25
00:01:18,102 --> 00:01:20,310
I don't know how much I'm
going to get through today,

26
00:01:20,310 --> 00:01:24,600
but the next lecture's
about diffraction.

27
00:01:24,600 --> 00:01:29,580
So these are two
techniques, two phenomena

28
00:01:29,580 --> 00:01:33,840
that are obviously very
closely related to each other.

29
00:01:33,840 --> 00:01:37,950
So in this talk really two
types of interferometer--

30
00:01:37,950 --> 00:01:40,980
the Michelson interferometer
and the Mach-Zehnder

31
00:01:40,980 --> 00:01:44,330
interferometer.

32
00:01:44,330 --> 00:01:49,990
So it took me a few minutes
to understand this picture.

33
00:01:49,990 --> 00:01:53,040
But anyway, it
looks a bit messy.

34
00:01:53,040 --> 00:01:55,030
But in the end, I--

35
00:01:55,030 --> 00:01:57,740
George drew this,
obviously, otherwise

36
00:01:57,740 --> 00:01:59,680
I wouldn't be saying that.

37
00:01:59,680 --> 00:02:01,910
But what this is
showing, then, is

38
00:02:01,910 --> 00:02:08,039
trying to show how a wave varies
both in distance and in time.

39
00:02:08,039 --> 00:02:11,220
So the wave's propagating
along this z-direction.

40
00:02:11,220 --> 00:02:13,410
So if you look
along this z-axis,

41
00:02:13,410 --> 00:02:17,480
you will see this
amplitude variation

42
00:02:17,480 --> 00:02:19,290
given by the black line.

43
00:02:19,290 --> 00:02:22,460
So a cosine-type variation.

44
00:02:22,460 --> 00:02:28,220
And then at any point, any
value of z, you can see,

45
00:02:28,220 --> 00:02:32,330
plotted in red, how the
amplitude will change

46
00:02:32,330 --> 00:02:36,260
with time along this axis here.

47
00:02:36,260 --> 00:02:42,170
So you can see that you
get a sinusoidal variation

48
00:02:42,170 --> 00:02:45,190
both in time and in space.

49
00:02:45,190 --> 00:02:52,870
And a reminder here, then,
how you can express this wave.

50
00:02:52,870 --> 00:02:57,200
This is a
sinusoidal-shaped wave,

51
00:02:57,200 --> 00:02:59,890
which is moving in the
positive z-direction.

52
00:02:59,890 --> 00:03:03,700
And you can either express
it in this form, which

53
00:03:03,700 --> 00:03:07,180
is what you would actually
measure if you could really

54
00:03:07,180 --> 00:03:10,480
measure it-- except that, as we
said before, you can't really

55
00:03:10,480 --> 00:03:13,690
measure it because it's
moving too quickly.

56
00:03:13,690 --> 00:03:19,150
But normally we do
the manipulations

57
00:03:19,150 --> 00:03:23,560
on the expressions using this
complex notation because it

58
00:03:23,560 --> 00:03:26,050
makes it easier to do the sums.

59
00:03:28,540 --> 00:03:29,040
Yeah.

60
00:03:29,040 --> 00:03:31,660
So this is showing, then--

61
00:03:31,660 --> 00:03:34,940
this is a particular
value of distance,

62
00:03:34,940 --> 00:03:38,140
then, and showing
how it changes here

63
00:03:38,140 --> 00:03:43,720
the value of the phase of
this cosine wave at the time t

64
00:03:43,720 --> 00:03:46,780
equals 0 according
to the value of z.

65
00:03:46,780 --> 00:03:48,300
I guess what that's
trying to show.

66
00:03:52,900 --> 00:03:55,780
And then let's think
now what happens

67
00:03:55,780 --> 00:04:00,630
if we've got some plane
wave propagating along

68
00:04:00,630 --> 00:04:03,160
at an angle with the axis.

69
00:04:03,160 --> 00:04:06,370
So here's this
wave coming along,

70
00:04:06,370 --> 00:04:12,490
which it's going along a line--

71
00:04:12,490 --> 00:04:15,970
the direction of the
propagation of the wave

72
00:04:15,970 --> 00:04:18,190
is along this
direction, which is

73
00:04:18,190 --> 00:04:21,490
an angle theta with the axis.

74
00:04:21,490 --> 00:04:28,720
And so you can see that when
this wave reaches this plane z

75
00:04:28,720 --> 00:04:33,160
equals fixed here-- so this is
some plane where we might put

76
00:04:33,160 --> 00:04:37,030
a screen or detector
or something--

77
00:04:37,030 --> 00:04:41,290
you can see that the different
parts of this phase front

78
00:04:41,290 --> 00:04:47,510
are going to strike this
plane at different times.

79
00:04:47,510 --> 00:04:52,570
Or equivalent, you can say that
if you look at this plane here,

80
00:04:52,570 --> 00:04:54,910
this wave will experience--

81
00:04:54,910 --> 00:04:58,180
exhibit-- different
phase at different points

82
00:04:58,180 --> 00:05:00,200
along this line.

83
00:05:00,200 --> 00:05:04,390
And so this is what we're
looking at here then.

84
00:05:04,390 --> 00:05:09,460
As it says here, the path delay
increases linearly with x.

85
00:05:09,460 --> 00:05:12,040
So there'll be
somewhere where this

86
00:05:12,040 --> 00:05:17,140
arrives at a phase of 0, which
I guess is this point here.

87
00:05:17,140 --> 00:05:19,870
And then on one
side, the phase is

88
00:05:19,870 --> 00:05:22,120
going to be positive
and negative, according

89
00:05:22,120 --> 00:05:23,980
to which way you go.

90
00:05:23,980 --> 00:05:28,040
So mathematically, this
is how we express this.

91
00:05:28,040 --> 00:05:34,900
So this is our wave, the
same as the previous slide,

92
00:05:34,900 --> 00:05:39,080
but now the wave is moving at
this angle theta to the axis.

93
00:05:39,080 --> 00:05:40,180
So we get this term.

94
00:05:40,180 --> 00:05:42,100
This is our-- you
remember, we expressed this

95
00:05:42,100 --> 00:05:50,380
before as k.r for a wave moving
in general in a direction r.

96
00:05:50,380 --> 00:05:58,620
And so this-- so we
could write this then.

97
00:05:58,620 --> 00:06:04,200
This is what is in terms of
our phasor representation.

98
00:06:04,200 --> 00:06:10,000
So as before, we write this
as an e to the i type thing.

99
00:06:10,000 --> 00:06:11,710
And then we also--

100
00:06:11,710 --> 00:06:17,020
this e to the minus i omega
t, we just missed that out

101
00:06:17,020 --> 00:06:19,680
for clarity, just
because we don't

102
00:06:19,680 --> 00:06:22,070
want to have to write
that every time.

103
00:06:22,070 --> 00:06:25,210
So there's an e to
the minus i omega t

104
00:06:25,210 --> 00:06:28,210
understood that we don't write.

105
00:06:28,210 --> 00:06:31,690
And k, of course,
is 2 pi over lambda.

106
00:06:31,690 --> 00:06:38,870
So k is related inversely to
the wavelength of the light.

107
00:06:38,870 --> 00:06:43,600
And so this thing here,
this complex number here,

108
00:06:43,600 --> 00:06:47,400
you could write this
as e to the i phi,

109
00:06:47,400 --> 00:06:52,890
like this, where phi is the
phase of the wave, which

110
00:06:52,890 --> 00:06:56,520
is a function of position.

111
00:06:56,520 --> 00:06:59,520
And it can be written like this.

112
00:06:59,520 --> 00:07:03,690
So if we look at a
particular value of z,

113
00:07:03,690 --> 00:07:05,205
then this z is
going to be fixed.

114
00:07:07,800 --> 00:07:12,210
So this part, e to the i 2
pi over lambda z cos theta,

115
00:07:12,210 --> 00:07:14,100
is going to be fixed.

116
00:07:14,100 --> 00:07:16,230
And the phase
variation is only going

117
00:07:16,230 --> 00:07:18,660
to be because of
this other part here.

118
00:07:25,560 --> 00:07:29,750
So this is another example.

119
00:07:29,750 --> 00:07:37,700
And so this shows spherical
waves or rather distorted

120
00:07:37,700 --> 00:07:41,630
spherical waves they become
because of the advances

121
00:07:41,630 --> 00:07:45,290
in modern technology that
love to distort everything

122
00:07:45,290 --> 00:07:48,260
you draw and make
it so that it's

123
00:07:48,260 --> 00:07:50,290
not the right shape anymore.

124
00:07:50,290 --> 00:07:53,510
But anyway, this is
a point source which

125
00:07:53,510 --> 00:07:56,460
is giving out spherical waves.

126
00:07:56,460 --> 00:08:00,830
And you can see that here
we put our screen again

127
00:08:00,830 --> 00:08:04,130
or whatever it is
that z equals z0.

128
00:08:04,130 --> 00:08:07,430
So you can see that these
waves, as they travel out here,

129
00:08:07,430 --> 00:08:10,610
they're traveling a
different distance

130
00:08:10,610 --> 00:08:13,170
before they get to the screen.

131
00:08:13,170 --> 00:08:16,920
So you can see that's
going to do two things.

132
00:08:16,920 --> 00:08:22,040
One is, of course, the amplitude
is going to decay as 1 over r.

133
00:08:22,040 --> 00:08:25,280
So the amplitude of the wave
when it reaches out here

134
00:08:25,280 --> 00:08:30,920
is going to be smaller
than the amplitude here.

135
00:08:30,920 --> 00:08:34,370
But much more important
is the phase effect

136
00:08:34,370 --> 00:08:38,539
because you can see
the phase actually

137
00:08:38,539 --> 00:08:40,039
might change tremendously.

138
00:08:40,039 --> 00:08:43,400
Because the
wavelength is so small

139
00:08:43,400 --> 00:08:45,740
compared with
these distances, it

140
00:08:45,740 --> 00:08:49,040
means that you only have to
go quite a small distance

141
00:08:49,040 --> 00:08:52,790
before you get a big
change in the phase.

142
00:08:52,790 --> 00:08:56,690
So that's what we want
to be able to quantify.

143
00:08:56,690 --> 00:09:00,200
And so here we're riding
our spherical wave,

144
00:09:00,200 --> 00:09:02,250
just like we did before.

145
00:09:02,250 --> 00:09:06,480
So George has kept this the
same as he's said before.

146
00:09:06,480 --> 00:09:09,020
You'll notice that it's
got this funny pi over 2

147
00:09:09,020 --> 00:09:14,350
in there, which he
introduced that before.

148
00:09:14,350 --> 00:09:17,630
It has the effect of making
this like a sine, of course.

149
00:09:17,630 --> 00:09:22,350
Something minus pi
by 2 is like sine.

150
00:09:22,350 --> 00:09:26,240
This is actually like
a sinc type function.

151
00:09:26,240 --> 00:09:30,260
And that's the reason why
that pi over 2 is there.

152
00:09:30,260 --> 00:09:33,350
It's so that it
doesn't blow up when

153
00:09:33,350 --> 00:09:36,380
you look at what
happens at x equals

154
00:09:36,380 --> 00:09:39,980
y equals z equals t equals 0.

155
00:09:39,980 --> 00:09:41,780
If you didn't have
the pi over 2 there,

156
00:09:41,780 --> 00:09:43,400
you'd have horrible infinities.

157
00:09:43,400 --> 00:09:48,180
So you avoid those by
putting this pi over 2 there.

158
00:09:48,180 --> 00:09:49,290
And yeah.

159
00:09:49,290 --> 00:09:50,900
So this is the radius.

160
00:09:50,900 --> 00:09:53,160
r is just the square
root of x squared

161
00:09:53,160 --> 00:09:57,090
plus y squared plus z squared.

162
00:09:57,090 --> 00:10:00,030
So this is kr minus omega t.

163
00:10:00,030 --> 00:10:03,650
And this is our r at the bottom.

164
00:10:03,650 --> 00:10:07,680
And now we make
an approximation.

165
00:10:07,680 --> 00:10:10,080
The approximation
we're going to make

166
00:10:10,080 --> 00:10:13,260
is that we're not actually
really looking at this geometry

167
00:10:13,260 --> 00:10:16,980
as shown in the picture, but
we're looking at the geometry

168
00:10:16,980 --> 00:10:20,430
when, actually, we're only
looking in a small region

169
00:10:20,430 --> 00:10:22,470
near to the axis.

170
00:10:22,470 --> 00:10:26,370
What that really, in
physical practice,

171
00:10:26,370 --> 00:10:33,150
that means is that this
distance z0 becomes very large.

172
00:10:33,150 --> 00:10:36,000
We're a long way
away from the source.

173
00:10:36,000 --> 00:10:43,230
And so we can think of this
value z being a very large.

174
00:10:43,230 --> 00:10:43,730
Sorry.

175
00:10:43,730 --> 00:10:47,050
You understand my
funny English language,

176
00:10:47,050 --> 00:10:49,040
don't you, when I
talk about "zeds"?

177
00:10:49,040 --> 00:10:50,840
What you call "zees."

178
00:10:50,840 --> 00:10:53,330
I just suddenly realized that.

179
00:10:53,330 --> 00:10:55,260
Anyway.

180
00:10:55,260 --> 00:10:59,480
So we've already had this before
in earlier lectures as well.

181
00:10:59,480 --> 00:11:05,660
This is a square root that
we can expand by the binomial

182
00:11:05,660 --> 00:11:10,820
and assume that z is large
much larger than x and y.

183
00:11:10,820 --> 00:11:14,990
So you take out a factor
z, and the square root

184
00:11:14,990 --> 00:11:20,300
becomes 1 plus a half-something.

185
00:11:20,300 --> 00:11:24,330
And so eventually you
get this sort of thing.

186
00:11:24,330 --> 00:11:29,480
So that's what we do
inside this phase term.

187
00:11:29,480 --> 00:11:33,260
But the thing at
the bottom, we say

188
00:11:33,260 --> 00:11:35,330
we don't have to be
so careful with that

189
00:11:35,330 --> 00:11:39,540
because it's not so
sensitive to this amplitude

190
00:11:39,540 --> 00:11:43,900
because all it does
is just changes

191
00:11:43,900 --> 00:11:48,890
the magnitude of the
wave by a small amount.

192
00:11:48,890 --> 00:11:51,500
Because we've said
x and y is small.

193
00:11:51,500 --> 00:11:56,090
But up here, you have to be
more careful because we're

194
00:11:56,090 --> 00:11:59,540
talking about phases.

195
00:11:59,540 --> 00:12:02,690
So anyway, at the bottom, we
can just neglect the x and y.

196
00:12:02,690 --> 00:12:05,310
So that's where the z
has come from there.

197
00:12:05,310 --> 00:12:08,720
And when we do this
binomial thing on the top,

198
00:12:08,720 --> 00:12:11,090
we get this
expression here then.

199
00:12:11,090 --> 00:12:16,520
So the dependence on
x and y, you can see,

200
00:12:16,520 --> 00:12:19,400
just becomes a
quadratic type term--

201
00:12:19,400 --> 00:12:22,530
x squared plus y
squared over 2z.

202
00:12:22,530 --> 00:12:26,210
And so this is what this
picture is drawing here.

203
00:12:26,210 --> 00:12:31,460
The path delay increases
quadratically with x.

204
00:12:31,460 --> 00:12:33,950
So as you go further,
this point here

205
00:12:33,950 --> 00:12:37,040
corresponds to the
center of the screen.

206
00:12:37,040 --> 00:12:43,050
And then as you go
further away, this

207
00:12:43,050 --> 00:12:47,850
increases with some parabolic
variation like this.

208
00:12:50,440 --> 00:12:54,790
So that's doing it in
terms of the cosines.

209
00:12:54,790 --> 00:13:00,190
If we go in terms of the
complex exponentials,

210
00:13:00,190 --> 00:13:02,860
much the same sort of approach.

211
00:13:02,860 --> 00:13:07,060
You can just expand,
again, the square root

212
00:13:07,060 --> 00:13:08,830
inside the exponential now.

213
00:13:08,830 --> 00:13:10,990
Exactly the same, really.

214
00:13:10,990 --> 00:13:14,410
And you end up getting
this expression here.

215
00:13:14,410 --> 00:13:17,140
And so this is an
example now, you

216
00:13:17,140 --> 00:13:21,850
can see, of the power of
this complex exponential form

217
00:13:21,850 --> 00:13:32,390
though because this complex
exponential of here two, terms

218
00:13:32,390 --> 00:13:35,270
or it might be more
terms in general,

219
00:13:35,270 --> 00:13:38,510
you can just expand
them, can't you?

220
00:13:38,510 --> 00:13:41,210
You can just say that
multiplied together.

221
00:13:41,210 --> 00:13:43,710
Whereas if it was
cosine, of course,

222
00:13:43,710 --> 00:13:48,140
you'd have to remember all
these things-- cos of a plus

223
00:13:48,140 --> 00:13:51,320
b plus c.

224
00:13:51,320 --> 00:13:53,580
Horrible.

225
00:13:53,580 --> 00:13:56,910
I don't know why I'm
making funny noises.

226
00:13:56,910 --> 00:13:59,550
Maybe I shouldn't keep
getting too near the speakers.

227
00:13:59,550 --> 00:14:02,260
Is that what it is?

228
00:14:02,260 --> 00:14:02,760
Yeah.

229
00:14:02,760 --> 00:14:05,010
So with the complex
exponentials,

230
00:14:05,010 --> 00:14:07,290
we can break it up very simply.

231
00:14:07,290 --> 00:14:11,730
And this inside the braces,
then, is our phi before,

232
00:14:11,730 --> 00:14:13,240
or i phi.

233
00:14:13,240 --> 00:14:16,950
So we can say this is e0 e
to the i phi, where the phase

234
00:14:16,950 --> 00:14:24,540
phi has just got this parabolic
term, plus some constant term,

235
00:14:24,540 --> 00:14:26,740
which corresponds to
this, which corresponds,

236
00:14:26,740 --> 00:14:28,705
of course to this distance--

237
00:14:28,705 --> 00:14:31,080
the phase that you get through
traveling in that distance

238
00:14:31,080 --> 00:14:33,390
there.

239
00:14:33,390 --> 00:14:35,830
So that's how you
get a phase delay.

240
00:14:35,830 --> 00:14:37,435
AUDIENCE: I've got a question.

241
00:14:37,435 --> 00:14:38,310
COLIN SHEPPARD: Yeah.

242
00:14:38,310 --> 00:14:42,148
AUDIENCE: What happened to
the pi over 2 in the phasor?

243
00:14:42,148 --> 00:14:44,690
COLIN SHEPPARD: What happened
to the pi over 2 in the phasor?

244
00:14:48,530 --> 00:14:50,420
So we've got good--

245
00:14:57,340 --> 00:14:59,120
it's there, isn't it?

246
00:14:59,120 --> 00:15:00,910
And then it's disappeared.

247
00:15:00,910 --> 00:15:03,390
[INTERPOSING VOICES]

248
00:15:03,390 --> 00:15:04,990
COLIN SHEPPARD:
Ah, the i outside.

249
00:15:04,990 --> 00:15:05,620
There it is.

250
00:15:05,620 --> 00:15:07,540
It's that one.

251
00:15:07,540 --> 00:15:08,110
Yeah.

252
00:15:08,110 --> 00:15:08,610
OK.

253
00:15:08,610 --> 00:15:11,560
So it's an i outside.

254
00:15:11,560 --> 00:15:16,092
And yeah.

255
00:15:16,092 --> 00:15:18,300
That's an interesting-- I
think George said something

256
00:15:18,300 --> 00:15:20,310
about that before
because effectively it

257
00:15:20,310 --> 00:15:25,830
makes this wave 90
degrees out of phase.

258
00:15:25,830 --> 00:15:31,680
And I like to think
of that as being

259
00:15:31,680 --> 00:15:36,110
related to some sort of
resonance-type phenomenon.

260
00:15:36,110 --> 00:15:40,060
If you excite something
that resonates,

261
00:15:40,060 --> 00:15:44,520
then it's going to resonate 90
degrees out of phase, isn't it?

262
00:15:44,520 --> 00:15:51,630
So if you've got a point source
and you excite it with a wave,

263
00:15:51,630 --> 00:15:56,730
then the scattered wave is going
to be 90 degrees out of phase

264
00:15:56,730 --> 00:16:00,780
with the driving force, just
because it's like a resonance.

265
00:16:00,780 --> 00:16:04,860
And I guess that's also
true in the Huygens' thing

266
00:16:04,860 --> 00:16:08,200
that we come on in
the next lecture.

267
00:16:08,200 --> 00:16:08,700
But yeah.

268
00:16:08,700 --> 00:16:09,030
OK.

269
00:16:09,030 --> 00:16:10,238
Thanks for pointing that out.

270
00:16:10,238 --> 00:16:14,150
I'd forgotten that.

271
00:16:14,150 --> 00:16:16,520
OK.

272
00:16:16,520 --> 00:16:18,630
Other than that
it's fine, isn't it?

273
00:16:18,630 --> 00:16:19,130
I hope.

274
00:16:22,320 --> 00:16:24,810
So we're now going on
to discuss something

275
00:16:24,810 --> 00:16:27,010
about the significance there.

276
00:16:27,010 --> 00:16:27,510
Yeah.

277
00:16:27,510 --> 00:16:31,080
So the point is, I
mentioned that there

278
00:16:31,080 --> 00:16:36,000
is this modulus term,
intensity-type term.

279
00:16:36,000 --> 00:16:38,790
But this is a very weak change.

280
00:16:38,790 --> 00:16:42,810
So you can imagine, if you're
trying to do an experiment,

281
00:16:42,810 --> 00:16:46,740
to actually measure the
properties of that wave field

282
00:16:46,740 --> 00:16:50,970
by looking at the intensity
would be quite tough because we

283
00:16:50,970 --> 00:16:54,030
said that these two
terms are very much

284
00:16:54,030 --> 00:16:55,930
smaller than that one.

285
00:16:55,930 --> 00:16:59,820
So looking at the
phase of the signal

286
00:16:59,820 --> 00:17:05,829
is a much more sensitive way
to look at that wave front.

287
00:17:05,829 --> 00:17:10,000
And as it says, the
phase can tell us

288
00:17:10,000 --> 00:17:12,849
all sorts of things
about the wave,

289
00:17:12,849 --> 00:17:17,349
like where the wave has come
from, how far it's traveled,

290
00:17:17,349 --> 00:17:22,510
the materials it's gone
through, and generally

291
00:17:22,510 --> 00:17:26,890
the optical path,
which you remember

292
00:17:26,890 --> 00:17:30,130
we had optical path
before, integral ndz.

293
00:17:30,130 --> 00:17:32,590
So that will change the phase.

294
00:17:32,590 --> 00:17:35,410
And so if we can
measure the phase,

295
00:17:35,410 --> 00:17:39,870
we can find out information
about those things.

296
00:17:39,870 --> 00:17:41,660
The problem, though,
as we've said before,

297
00:17:41,660 --> 00:17:44,630
is that the phase is not
something you can actually see

298
00:17:44,630 --> 00:17:47,690
directly, because it's so fast.

299
00:17:47,690 --> 00:17:49,520
So 10 to the minus--

300
00:17:49,520 --> 00:17:52,370
10 to the 15 hertz is typical.

301
00:17:52,370 --> 00:17:53,180
Very, very fast.

302
00:17:53,180 --> 00:17:56,510
Much faster than you can
measure with any electronics.

303
00:17:56,510 --> 00:17:58,350
So how do we do this?

304
00:17:58,350 --> 00:18:03,260
We do it by some form
of interferometry,

305
00:18:03,260 --> 00:18:08,150
which is basically mixing
it with a reference beam.

306
00:18:08,150 --> 00:18:13,670
And then you look at the
intensity of the interference

307
00:18:13,670 --> 00:18:15,200
between the two waves.

308
00:18:15,200 --> 00:18:18,410
So this is how we do it.

309
00:18:18,410 --> 00:18:20,810
So I said that
we're going to look

310
00:18:20,810 --> 00:18:23,000
at two types of interferometer.

311
00:18:23,000 --> 00:18:25,770
The first one is the
Michelson interferometer.

312
00:18:25,770 --> 00:18:28,610
So this is how you make a
Michelson interferometer.

313
00:18:28,610 --> 00:18:32,660
You get light from some
source and split it

314
00:18:32,660 --> 00:18:36,740
into two parts with
a beam splitter.

315
00:18:36,740 --> 00:18:42,110
And one of those beams is
what we call the signal arm.

316
00:18:42,110 --> 00:18:44,250
This is what we're going
to try and measure.

317
00:18:44,250 --> 00:18:49,310
So here, this shows this light
going through some object,

318
00:18:49,310 --> 00:18:53,660
and then being
reflected by a mirror,

319
00:18:53,660 --> 00:18:56,780
and then coming
back to a detector.

320
00:18:56,780 --> 00:19:00,710
And then the other arm
is the reference arm.

321
00:19:00,710 --> 00:19:03,860
So this is going to
produce a reference signal.

322
00:19:03,860 --> 00:19:06,710
So effectively, we're
going to be comparing

323
00:19:06,710 --> 00:19:09,260
the phases in those two arms.

324
00:19:09,260 --> 00:19:11,580
So the reference
beam comes down also.

325
00:19:11,580 --> 00:19:13,880
So these add together.

326
00:19:13,880 --> 00:19:17,150
So we can add them as phasors.

327
00:19:17,150 --> 00:19:18,890
This is for the reference beam.

328
00:19:18,890 --> 00:19:22,340
Ar e to the i phi r.

329
00:19:22,340 --> 00:19:26,960
For the signal beam we've
got As e to the i phi s.

330
00:19:26,960 --> 00:19:31,490
We add the amplitude because
we're assuming it's coherent.

331
00:19:31,490 --> 00:19:34,790
Maybe I ought to stress
that a bit, of course.

332
00:19:34,790 --> 00:19:39,020
So this is assuming that
this is a coherent--

333
00:19:39,020 --> 00:19:42,740
these two beams a coherent
with respect to each other.

334
00:19:42,740 --> 00:19:50,120
So if you had light here from
a laser, that would be true.

335
00:19:50,120 --> 00:19:53,450
And we'll probably
come onto cases where

336
00:19:53,450 --> 00:19:55,850
it might not be true later on.

337
00:19:55,850 --> 00:19:58,010
Once we've got the
amplitude, then we just

338
00:19:58,010 --> 00:20:02,070
have to find the modulus
square to find the intensity.

339
00:20:02,070 --> 00:20:03,980
And if you remember,
there's actually

340
00:20:03,980 --> 00:20:05,840
a proportional sign here.

341
00:20:05,840 --> 00:20:09,520
There's some funny
constants with epsilon

342
00:20:09,520 --> 00:20:14,260
a half and things, which we
just don't usually worry about.

343
00:20:14,260 --> 00:20:21,550
So these phases tell
us the optical path

344
00:20:21,550 --> 00:20:24,580
that the light has gone
through in these two arms

345
00:20:24,580 --> 00:20:27,260
in order to get to the detector.

346
00:20:27,260 --> 00:20:29,710
And so what we're
measuring is the difference

347
00:20:29,710 --> 00:20:33,580
in the optical path
in those two arms.

348
00:20:33,580 --> 00:20:36,740
And so this shows
how that can work.

349
00:20:36,740 --> 00:20:42,130
So this is an example where
these two waves arrive in phase

350
00:20:42,130 --> 00:20:43,180
with each other.

351
00:20:43,180 --> 00:20:46,780
So then they'll
add constructively

352
00:20:46,780 --> 00:20:49,730
and you'll get a big
signal like that.

353
00:20:49,730 --> 00:20:53,020
On the other hand, you could
imagine that they might not

354
00:20:53,020 --> 00:20:53,860
arrive like that.

355
00:20:53,860 --> 00:20:59,200
They might arrive in anti-phase
like this, in which case

356
00:20:59,200 --> 00:21:01,930
you can see the two waves
now are going to cancel out.

357
00:21:01,930 --> 00:21:05,200
And you're not going to
see now any interference.

358
00:21:05,200 --> 00:21:08,390
In practice, of course, it could
be anything in between here.

359
00:21:08,390 --> 00:21:08,890
All right.

360
00:21:08,890 --> 00:21:11,080
So as you change--

361
00:21:11,080 --> 00:21:15,460
you can change the relative
phase of this light

362
00:21:15,460 --> 00:21:20,800
by changing the position
of this mirror or the size

363
00:21:20,800 --> 00:21:23,650
or refractive index
of this object

364
00:21:23,650 --> 00:21:26,410
that's in there, or the
wavelength of the light,

365
00:21:26,410 --> 00:21:27,397
or whatever.

366
00:21:27,397 --> 00:21:29,230
It's going to depend
on all of these things.

367
00:21:33,580 --> 00:21:37,330
So example, measuring distance.

368
00:21:37,330 --> 00:21:39,490
So we've said that
effectively what we're doing

369
00:21:39,490 --> 00:21:42,610
is measuring the optical path.

370
00:21:42,610 --> 00:21:45,490
So optical path depends on
a few things, doesn't it?

371
00:21:45,490 --> 00:21:48,580
Optical path depends
on the distance,

372
00:21:48,580 --> 00:21:51,340
it depends on the
wavelength, and it depends

373
00:21:51,340 --> 00:21:53,420
on the refractive index.

374
00:21:53,420 --> 00:21:56,080
So these are all things
that you can measure

375
00:21:56,080 --> 00:21:58,550
using this sort of system.

376
00:21:58,550 --> 00:22:00,650
So here it's talking
as an example.

377
00:22:00,650 --> 00:22:03,230
There's no object
in this one anymore.

378
00:22:03,230 --> 00:22:05,790
So this is just free space.

379
00:22:05,790 --> 00:22:10,780
And so in this case,
then, what you can measure

380
00:22:10,780 --> 00:22:16,060
is the difference in the optical
path in this arm and this arm.

381
00:22:16,060 --> 00:22:17,200
And the optical path--

382
00:22:17,200 --> 00:22:19,740
well, the refractive index
is now the same in each,

383
00:22:19,740 --> 00:22:21,730
the wavelength's
the same in each,

384
00:22:21,730 --> 00:22:25,840
so you could measure
the distance.

385
00:22:25,840 --> 00:22:31,000
So this is an interferometer
for measuring distance.

386
00:22:31,000 --> 00:22:34,360
So where could you use this?

387
00:22:34,360 --> 00:22:38,500
Well, imagine, this might not
actually be a flat mirror.

388
00:22:38,500 --> 00:22:42,220
It might actually have some
structure there, in which case

389
00:22:42,220 --> 00:22:45,160
you could actually measure
what that structure is,

390
00:22:45,160 --> 00:22:47,830
or if it was a mirror,
you could actually

391
00:22:47,830 --> 00:22:51,820
use this to accurately measure
the distance of this mirror.

392
00:22:51,820 --> 00:22:58,060
How far away it is from the
instrument, so to speak.

393
00:22:58,060 --> 00:23:02,690
So the interesting thing--
another interesting thing--

394
00:23:02,690 --> 00:23:04,870
this is the Michelson
interferometer.

395
00:23:04,870 --> 00:23:08,440
And the Michelson
interferometer,

396
00:23:08,440 --> 00:23:12,040
as the name says, was
invented by Michelson.

397
00:23:12,040 --> 00:23:14,290
Well, I think he
probably invented it.

398
00:23:14,290 --> 00:23:18,010
He certainly has his
name attached to it.

399
00:23:18,010 --> 00:23:22,090
But he mainly used it
for measuring wavelength.

400
00:23:22,090 --> 00:23:26,000
So we'll come back to that after
I've gone through the matter

401
00:23:26,000 --> 00:23:26,830
bit, actually.

402
00:23:26,830 --> 00:23:28,720
I think it's probably best.

403
00:23:28,720 --> 00:23:29,500
Anyway.

404
00:23:29,500 --> 00:23:33,070
So what we've said is we add
together these two phases,

405
00:23:33,070 --> 00:23:37,300
find the modulus square, and
then we've got to expand that.

406
00:23:37,300 --> 00:23:40,370
So this is a square of
a sum of two things.

407
00:23:40,370 --> 00:23:43,790
So we're going to get the square
of this, the square of this,

408
00:23:43,790 --> 00:23:50,210
and then we're going to
get some cross terms.

409
00:23:50,210 --> 00:23:51,940
And this is what
it comes out to be.

410
00:23:51,940 --> 00:23:55,615
The two cross terms, one
is the complex conjugate

411
00:23:55,615 --> 00:24:00,060
of the other, which means that
when you add them together,

412
00:24:00,060 --> 00:24:02,880
the final result is real.

413
00:24:02,880 --> 00:24:04,830
That has to be true,
of course, because this

414
00:24:04,830 --> 00:24:10,630
is an expression for intensity,
and intensity has to be real.

415
00:24:10,630 --> 00:24:14,110
And so we've written
it in this nice form,

416
00:24:14,110 --> 00:24:21,610
Ar squared plus As squared
plus 2 Ar As cos of something.

417
00:24:21,610 --> 00:24:24,730
And this thing
here, the something,

418
00:24:24,730 --> 00:24:29,590
is the difference between
the phases of the two beams.

419
00:24:29,590 --> 00:24:32,990
So this next line
puts it into a sort

420
00:24:32,990 --> 00:24:36,250
of general term, general form.

421
00:24:36,250 --> 00:24:41,670
It's of the forms a
constant plus a cosine.

422
00:24:41,670 --> 00:24:47,600
And then these quantities,
i0 m and delta phi,

423
00:24:47,600 --> 00:24:51,320
you can also write in terms
of these other parameters

424
00:24:51,320 --> 00:24:53,880
we had earlier.

425
00:24:53,880 --> 00:24:59,640
So as it says, when
both paths are clear,

426
00:24:59,640 --> 00:25:02,790
then these phases
adjust directly

427
00:25:02,790 --> 00:25:08,400
proportional to the
distance ls and lr.

428
00:25:08,400 --> 00:25:13,800
And so the phase difference
is just k times the difference

429
00:25:13,800 --> 00:25:16,020
in the two paths.

430
00:25:16,020 --> 00:25:20,910
So therefore, if you
measure the signal,

431
00:25:20,910 --> 00:25:23,860
if you know what
the wavelength is,

432
00:25:23,860 --> 00:25:25,860
you can measure what the
difference in those two

433
00:25:25,860 --> 00:25:28,230
distances is.

434
00:25:28,230 --> 00:25:31,380
There's a bit of
a proviso there,

435
00:25:31,380 --> 00:25:36,060
which is quite a big proviso,
that you can only do this,

436
00:25:36,060 --> 00:25:38,970
of course, modulo 2 pi.

437
00:25:38,970 --> 00:25:44,040
So because this is a cosine,
it's going to keep repeating.

438
00:25:44,040 --> 00:25:48,030
And you won't know which
of the cycles you're on.

439
00:25:48,030 --> 00:25:52,410
So it means that interferometry
is usually very sensitive.

440
00:25:52,410 --> 00:25:59,040
You can measure very, very small
distances, easily much smaller

441
00:25:59,040 --> 00:26:00,720
than the wavelength.

442
00:26:00,720 --> 00:26:04,350
And people sometimes talk
about measuring hundredth

443
00:26:04,350 --> 00:26:08,400
of an Angstrom and
amazing small quantities

444
00:26:08,400 --> 00:26:10,480
using interferometry.

445
00:26:10,480 --> 00:26:15,160
The LIGO project, I don't
know what they're claiming.

446
00:26:15,160 --> 00:26:18,780
10 to the minus 15 meters
or something they're

447
00:26:18,780 --> 00:26:24,345
trying to measure to detect
these gravitational waves.

448
00:26:26,890 --> 00:26:27,540
Anyway.

449
00:26:27,540 --> 00:26:29,680
Yeah.

450
00:26:29,680 --> 00:26:33,200
So you only measure
it modulo 2 pi.

451
00:26:33,200 --> 00:26:36,410
So that means that if you're
trying to use this method

452
00:26:36,410 --> 00:26:39,470
to measure-- for
example, I'm looking

453
00:26:39,470 --> 00:26:42,680
at the door at the back
of the lecture theater.

454
00:26:42,680 --> 00:26:44,180
If I'm trying to
use this to measure

455
00:26:44,180 --> 00:26:48,860
the distance of that door,
I can measure the distance

456
00:26:48,860 --> 00:26:51,920
to an accuracy,
maybe, of something

457
00:26:51,920 --> 00:26:53,960
smaller than a micron.

458
00:26:53,960 --> 00:26:58,840
But I wouldn't really know
whether it had added to that 10

459
00:26:58,840 --> 00:27:01,250
meters or whatever.

460
00:27:01,250 --> 00:27:05,330
So that actually has been what--

461
00:27:05,330 --> 00:27:07,280
a lot of people have
put effort into trying

462
00:27:07,280 --> 00:27:09,770
to develop interferometric
methods that you

463
00:27:09,770 --> 00:27:15,560
can get that extra information
so that you can actually

464
00:27:15,560 --> 00:27:18,880
do what we call unwrapping,
because this is like--

465
00:27:18,880 --> 00:27:21,930
they sometimes call
this phase wrapping.

466
00:27:21,930 --> 00:27:24,680
The phase wraps
because of the cosine.

467
00:27:24,680 --> 00:27:28,140
And so you don't know which
of these lobes you're on.

468
00:27:28,140 --> 00:27:28,710
Yeah.

469
00:27:28,710 --> 00:27:32,140
Well, going back to Michelson.

470
00:27:32,140 --> 00:27:35,730
So Michelson did this
the other way around.

471
00:27:35,730 --> 00:27:40,400
He knows this distance
and uses this expression

472
00:27:40,400 --> 00:27:45,910
to find the wavelength or
the k value or whatever.

473
00:27:45,910 --> 00:27:48,770
And that's, of course, what
you do in an interferometer,

474
00:27:48,770 --> 00:27:49,970
isn't it?

475
00:27:49,970 --> 00:27:53,510
If you do a Fourier
transform interferometer,

476
00:27:53,510 --> 00:27:58,970
you're using that as
a way of measuring

477
00:27:58,970 --> 00:28:01,880
the wavelength of the light.

478
00:28:01,880 --> 00:28:07,370
In the case of Michelson,
he looked at various things.

479
00:28:07,370 --> 00:28:11,060
He didn't look at lasers because
there weren't any lasers then.

480
00:28:11,060 --> 00:28:14,600
But he looked at light
from discharge tubes

481
00:28:14,600 --> 00:28:17,970
and various different
types of lamp and so on.

482
00:28:17,970 --> 00:28:21,140
And he looked at
how the signal would

483
00:28:21,140 --> 00:28:24,610
vary as you move this mirror.

484
00:28:24,610 --> 00:28:27,870
And it turns out that if
you do move the mirror

485
00:28:27,870 --> 00:28:30,630
and you know accurately how
you're moving the mirror,

486
00:28:30,630 --> 00:28:33,150
you can actually
work out from that

487
00:28:33,150 --> 00:28:37,920
what the spectral
distribution of the source is.

488
00:28:37,920 --> 00:28:46,120
That also suggests a way
around this wrapping problem.

489
00:28:46,120 --> 00:28:49,390
The way around the wrap or a
way around the wrapping problem

490
00:28:49,390 --> 00:28:51,700
is to use not just one
wavelength but many

491
00:28:51,700 --> 00:28:53,150
wavelengths.

492
00:28:53,150 --> 00:28:56,380
So even two wavelengths
is a big help, actually.

493
00:29:04,730 --> 00:29:05,230
Yeah.

494
00:29:05,230 --> 00:29:07,330
It's quite interesting
that Michelson--

495
00:29:07,330 --> 00:29:09,670
I think I've spoken about
this with some of my students

496
00:29:09,670 --> 00:29:12,970
before, that Michelson, he wrote
quite a lot about measuring

497
00:29:12,970 --> 00:29:15,700
wavelengths using
that interferometer.

498
00:29:15,700 --> 00:29:20,200
And then many, many
years later, people

499
00:29:20,200 --> 00:29:24,370
came up with this
bright idea of using

500
00:29:24,370 --> 00:29:26,980
what they call the low
coherence interferometer

501
00:29:26,980 --> 00:29:29,380
or the white light
interferometer

502
00:29:29,380 --> 00:29:31,030
to measure distances.

503
00:29:31,030 --> 00:29:36,130
But as far as I know,
Michelson never suggested this.

504
00:29:36,130 --> 00:29:38,780
I've never found it in
any of these papers.

505
00:29:38,780 --> 00:29:39,280
Anyway.

506
00:29:39,280 --> 00:29:41,800
So this is another application.

507
00:29:41,800 --> 00:29:45,370
Here we're going to
look at a real object

508
00:29:45,370 --> 00:29:48,080
that we put in there and try
and find out something about it.

509
00:29:48,080 --> 00:29:53,910
So we put in this lump
of material, length

510
00:29:53,910 --> 00:29:57,010
l, a certain refractive index.

511
00:29:57,010 --> 00:30:02,220
And you can see now then that
the phase of this light when

512
00:30:02,220 --> 00:30:09,480
it gets there is going to be
changed by the light traveling

513
00:30:09,480 --> 00:30:11,490
through that material.

514
00:30:11,490 --> 00:30:18,760
And have we taken
into account, George,

515
00:30:18,760 --> 00:30:23,710
that it's gone
through twice here,

516
00:30:23,710 --> 00:30:25,020
or even on the previous one?

517
00:30:29,580 --> 00:30:31,560
There ought to be a 2 in there.

518
00:30:31,560 --> 00:30:34,520
There ought to be a 2
in here, oughtn't there?

519
00:30:34,520 --> 00:30:35,020
All right.

520
00:30:35,020 --> 00:30:39,760
So the phase difference
is actually twice this

521
00:30:39,760 --> 00:30:44,650
because it has this amount of
phase difference on the way

522
00:30:44,650 --> 00:30:47,880
in and on the way out.

523
00:30:47,880 --> 00:30:50,200
So I hadn't noticed
that before either.

524
00:30:50,200 --> 00:30:53,500
But it applies to
this next one as well.

525
00:30:53,500 --> 00:30:58,940
So this one, the light goes
through this object here.

526
00:30:58,940 --> 00:31:02,030
And so there's this
extra phase delay

527
00:31:02,030 --> 00:31:05,790
that comes because of this
object, which is this bit here.

528
00:31:05,790 --> 00:31:07,610
So again, there ought
to be a factor of 2

529
00:31:07,610 --> 00:31:09,890
for all of these terms.

530
00:31:09,890 --> 00:31:13,010
And so there we are, though.

531
00:31:13,010 --> 00:31:16,970
We've got this
extra optical path

532
00:31:16,970 --> 00:31:22,880
and so now we can measure the
optical path of that object.

533
00:31:22,880 --> 00:31:26,160
Again, modulo 2 pi.

534
00:31:26,160 --> 00:31:31,110
So if this is many
wavelengths, you

535
00:31:31,110 --> 00:31:34,920
might have trouble
to know exactly how

536
00:31:34,920 --> 00:31:38,230
long it is, of course, because
of the wrapping of the phase

537
00:31:38,230 --> 00:31:38,730
again.

538
00:31:42,540 --> 00:31:49,530
So this modulo 2 pi we
can actually measure--

539
00:31:49,530 --> 00:31:51,800
this is the optical
path difference.

540
00:31:51,800 --> 00:31:54,330
n minus 1 times L.

541
00:31:54,330 --> 00:31:57,330
The minus, of course,
is because the other--

542
00:31:57,330 --> 00:31:59,310
the reference beam
has gone through air

543
00:31:59,310 --> 00:32:01,470
when this has gone
through this material.

544
00:32:01,470 --> 00:32:07,020
So what we measure is n minus 1
times L. If you knew what n is,

545
00:32:07,020 --> 00:32:09,570
we can know what L is.

546
00:32:09,570 --> 00:32:13,140
If we knew what L is,
we could know what n is.

547
00:32:13,140 --> 00:32:17,010
But as it says there, if
you don't know either,

548
00:32:17,010 --> 00:32:18,740
then you can't find either.

549
00:32:18,740 --> 00:32:23,100
All you know is
what n minus 1 L is.

550
00:32:23,100 --> 00:32:26,550
Anyone got any ideas
about how you might--

551
00:32:26,550 --> 00:32:29,700
what might you do
to get both of those

552
00:32:29,700 --> 00:32:33,430
if you wanted to
know both of those?

553
00:32:33,430 --> 00:32:36,170
Any ideas how you could do that?

554
00:32:36,170 --> 00:32:39,947
AUDIENCE: [INAUDIBLE]

555
00:32:39,947 --> 00:32:41,530
COLIN SHEPPARD: Based
on what, though?

556
00:32:41,530 --> 00:32:44,440
You'd have to have
two measurements.

557
00:32:44,440 --> 00:32:46,963
GEORGE BARBASTATHIS:
Press the button, please.

558
00:32:46,963 --> 00:32:52,390
AUDIENCE: [INAUDIBLE] Yeah.

559
00:32:52,390 --> 00:32:54,702
Someone says it's to use
two wavelengths to set up

560
00:32:54,702 --> 00:32:55,660
simultaneous equations.

561
00:32:55,660 --> 00:32:56,350
COLIN SHEPPARD: Exactly.

562
00:32:56,350 --> 00:32:56,880
Good one.

563
00:32:56,880 --> 00:32:57,380
Yeah.

564
00:32:57,380 --> 00:32:58,613
Who said it?

565
00:32:58,613 --> 00:32:59,780
Someone in the back said it.

566
00:32:59,780 --> 00:33:01,200
AUDIENCE: I heard it somewhere.

567
00:33:01,200 --> 00:33:02,030
COLIN SHEPPARD:
You're not getting

568
00:33:02,030 --> 00:33:03,072
the full credit for that.

569
00:33:03,072 --> 00:33:03,697
AUDIENCE: Yeah.

570
00:33:03,697 --> 00:33:04,240
Yeah.

571
00:33:04,240 --> 00:33:06,340
COLIN SHEPPARD:
Anyway, someone got it.

572
00:33:06,340 --> 00:33:07,090
So yeah.

573
00:33:07,090 --> 00:33:11,680
So if you measured this for
two different wavelengths,

574
00:33:11,680 --> 00:33:14,500
then you could have a pair
of simultaneous equations,

575
00:33:14,500 --> 00:33:18,070
and you could solve
for both of those.

576
00:33:18,070 --> 00:33:22,480
That, of course, assuming
that there is no dispersion.

577
00:33:22,480 --> 00:33:26,550
If n changes with
wavelength, then you

578
00:33:26,550 --> 00:33:28,960
are still back to square
one, unless you do

579
00:33:28,960 --> 00:33:31,670
some more wavelengths I guess.

580
00:33:31,670 --> 00:33:32,490
But anyway, yeah.

581
00:33:32,490 --> 00:33:34,900
So it just shows
you, though, you

582
00:33:34,900 --> 00:33:39,530
can do a lot of these
experiments very easily.

583
00:33:39,530 --> 00:33:41,260
So now let's sort
of-- these are used--

584
00:33:41,260 --> 00:33:44,120
this Michelson
interferometer is used

585
00:33:44,120 --> 00:33:49,060
a lot in metrology,
industrial sort of metrology.

586
00:33:49,060 --> 00:33:54,310
And all countries have their--

587
00:33:54,310 --> 00:33:56,500
in the US, it used to
be called the National

588
00:33:56,500 --> 00:33:58,030
Bureau of Standards, didn't it?

589
00:33:58,030 --> 00:34:00,595
But I don't know that
that exists anymore.

590
00:34:00,595 --> 00:34:01,720
AUDIENCE: It's called NIST.

591
00:34:01,720 --> 00:34:02,320
COLIN SHEPPARD: NIST.

592
00:34:02,320 --> 00:34:02,820
NIST.

593
00:34:02,820 --> 00:34:04,070
That's right.

594
00:34:04,070 --> 00:34:04,570
It changed.

595
00:34:04,570 --> 00:34:06,350
I don't know why it
changed its name.

596
00:34:06,350 --> 00:34:08,920
But anyway, so all
countries have got this.

597
00:34:08,920 --> 00:34:14,409
They have this
standard measures.

598
00:34:14,409 --> 00:34:18,100
They have procedures for
calibrating distances

599
00:34:18,100 --> 00:34:19,760
and so on.

600
00:34:19,760 --> 00:34:25,900
And the idea is that if a
company like General Motors

601
00:34:25,900 --> 00:34:33,940
is making parts for a
car, that those parts will

602
00:34:33,940 --> 00:34:39,280
fit as a spare for another car
that's made by another factory.

603
00:34:43,790 --> 00:34:47,440
So say you make
some shaft that's

604
00:34:47,440 --> 00:34:51,699
supposed to be 1
inch in diameter,

605
00:34:51,699 --> 00:34:56,440
you want it to be
accurately 1 inch diameter.

606
00:34:56,440 --> 00:34:59,230
Otherwise, the
bit from one place

607
00:34:59,230 --> 00:35:01,630
wouldn't fit the
bit from the other.

608
00:35:01,630 --> 00:35:04,290
There is actually a
true story about that.

609
00:35:04,290 --> 00:35:06,060
A thing called the Vickers inch.

610
00:35:06,060 --> 00:35:09,550
Vickers is an engineering
company in the UK.

611
00:35:09,550 --> 00:35:13,990
And this Vickers
inch was a big thing

612
00:35:13,990 --> 00:35:16,840
that came up during
the Second World War,

613
00:35:16,840 --> 00:35:19,780
where lots of
factories all over UK

614
00:35:19,780 --> 00:35:23,140
were converted to
make munitions.

615
00:35:23,140 --> 00:35:27,250
And somewhere down
the line, people

616
00:35:27,250 --> 00:35:30,430
realized that Vickers had a
different inch from everyone

617
00:35:30,430 --> 00:35:32,080
else.

618
00:35:32,080 --> 00:35:34,690
So when they made
a shell and tried

619
00:35:34,690 --> 00:35:39,300
putting it in the barrel of
the gun, it didn't quite fit.

620
00:35:39,300 --> 00:35:42,130
So eventually that
was sorted out.

621
00:35:42,130 --> 00:35:44,750
They worked out what
the problem was.

622
00:35:44,750 --> 00:35:48,280
And I guess it's that sort
of problem that makes it

623
00:35:48,280 --> 00:35:50,720
so that the companies now--

624
00:35:50,720 --> 00:35:54,650
obviously the countries keep
these standards and so on.

625
00:35:54,650 --> 00:35:58,210
So the sort of
thing you might do

626
00:35:58,210 --> 00:36:00,540
is measure the height of this.

627
00:36:00,540 --> 00:36:02,860
Let's say you put in--

628
00:36:02,860 --> 00:36:05,338
they have these things
called gauge blocks.

629
00:36:05,338 --> 00:36:06,880
A lot of you are
mechanical engineers

630
00:36:06,880 --> 00:36:08,900
so you must know
about gauge blocks.

631
00:36:08,900 --> 00:36:12,760
So you can buy these
standard lumps of metal that

632
00:36:12,760 --> 00:36:14,870
have got a certain thickness.

633
00:36:14,870 --> 00:36:18,850
And so you put this
lump of metal in here

634
00:36:18,850 --> 00:36:21,886
and measure the thickness of
it using an interferometer.

635
00:36:24,780 --> 00:36:27,310
So these standards,
of course, they

636
00:36:27,310 --> 00:36:29,260
have these what they
call primary standards,

637
00:36:29,260 --> 00:36:32,320
and then they're handed down.

638
00:36:32,320 --> 00:36:36,250
I'm not quite sure how
the length works anymore,

639
00:36:36,250 --> 00:36:39,070
but it's all defined in terms
of the wavelength of something,

640
00:36:39,070 --> 00:36:41,570
isn't it?

641
00:36:41,570 --> 00:36:44,350
And then all of the
countries around the world,

642
00:36:44,350 --> 00:36:47,890
they all have to
compare their meter

643
00:36:47,890 --> 00:36:51,300
to make sure that they've
all got the same meter.

644
00:36:51,300 --> 00:36:53,050
And then within each country--

645
00:36:53,050 --> 00:36:58,710
so in the US, NIST would then
be measuring these gauge blocks

646
00:36:58,710 --> 00:37:02,430
that would be sent to
them from other companies

647
00:37:02,430 --> 00:37:05,430
to make sure that they've
got the right size.

648
00:37:05,430 --> 00:37:07,140
And then in the
companies, then they

649
00:37:07,140 --> 00:37:13,320
would compare those gauge
blocks against other gauge

650
00:37:13,320 --> 00:37:15,720
blocks and other things.

651
00:37:15,720 --> 00:37:20,540
So this is what you use-- one
of the things you use this for.

652
00:37:20,540 --> 00:37:23,280
So now, fringe visibility.

653
00:37:23,280 --> 00:37:23,780
Yeah.

654
00:37:23,780 --> 00:37:26,638
So we've said that
you get these fringes.

655
00:37:26,638 --> 00:37:28,930
Perhaps I ought to go back
and explain that a bit more.

656
00:37:32,210 --> 00:37:34,670
Back to this one.

657
00:37:34,670 --> 00:37:37,990
So you can see here
we've got a constant term

658
00:37:37,990 --> 00:37:42,370
plus a sinusoidal-type term
or cosinusoidal-type term.

659
00:37:42,370 --> 00:37:49,180
And you can see that this
term m is called the contrast

660
00:37:49,180 --> 00:37:51,160
or fringe visibility.

661
00:37:51,160 --> 00:37:55,720
So if m is 1, then you can see
that they are going to be--

662
00:37:55,720 --> 00:37:59,020
for particular
values of delta phi,

663
00:37:59,020 --> 00:38:02,780
this intensity is
going to go to 0.

664
00:38:02,780 --> 00:38:04,740
So that's when you get--

665
00:38:04,740 --> 00:38:08,500
you've got very high
fringe visibility.

666
00:38:08,500 --> 00:38:10,600
And you can see
that this visibility

667
00:38:10,600 --> 00:38:18,610
depends on this, Ar 2 Ar As
over ir squared plus As squared.

668
00:38:18,610 --> 00:38:20,890
And you can show
quite easily that that

669
00:38:20,890 --> 00:38:24,600
is going to reach a maximum.

670
00:38:24,600 --> 00:38:28,690
It's going to be equal
to 1 when Ar equals As.

671
00:38:28,690 --> 00:38:32,500
So if you ensure
that Ar equals As,

672
00:38:32,500 --> 00:38:36,820
then you're going to get
your visibility of 1.

673
00:38:36,820 --> 00:38:39,190
I guess it can't be bigger
than 1 because that would

674
00:38:39,190 --> 00:38:43,070
mean that the intensity
would go negative,

675
00:38:43,070 --> 00:38:45,290
which is not possible.

676
00:38:45,290 --> 00:38:48,070
So this plots what it is--

677
00:38:48,070 --> 00:38:53,510
what you would see as
a function of delta phi

678
00:38:53,510 --> 00:38:57,110
and for a particular value of n.

679
00:38:57,110 --> 00:39:01,660
So there's some background,
sometimes called the bias.

680
00:39:01,660 --> 00:39:05,480
And then you've got some
wiggles on top of that.

681
00:39:05,480 --> 00:39:09,460
And so this plot is
plotted against delta phi.

682
00:39:09,460 --> 00:39:12,260
Delta phi could change by
a number of different ways,

683
00:39:12,260 --> 00:39:13,060
of course.

684
00:39:13,060 --> 00:39:16,450
One way it could change is if
you actually move the mirror.

685
00:39:16,450 --> 00:39:19,090
Another way it could change
is if you tilted the mirror.

686
00:39:19,090 --> 00:39:21,250
If you tilted the
mirror, then there'd

687
00:39:21,250 --> 00:39:24,070
be different delta phis
at different transverse

688
00:39:24,070 --> 00:39:26,050
positions on the mirror.

689
00:39:26,050 --> 00:39:32,390
And if we take Ar equal As, then
you've got perfect contrast,

690
00:39:32,390 --> 00:39:34,190
m equals 1.

691
00:39:34,190 --> 00:39:36,640
You see it goes
right down to zero.

692
00:39:36,640 --> 00:39:40,400
And that's the high
visibility of the fringes

693
00:39:40,400 --> 00:39:41,480
that you can get.

694
00:39:41,480 --> 00:39:47,030
If, on the other hand,
you find that if you take,

695
00:39:47,030 --> 00:39:49,610
let's say, the
reference beam to be

696
00:39:49,610 --> 00:39:52,940
much stronger than
the signal beam,

697
00:39:52,940 --> 00:39:55,610
then you can see that m
is going to tend to zero.

698
00:39:55,610 --> 00:40:00,020
The contrast is going
to become very weak.

699
00:40:00,020 --> 00:40:05,090
Actually, sometimes
people do do that.

700
00:40:05,090 --> 00:40:09,050
Contrast is not necessarily
the most important thing

701
00:40:09,050 --> 00:40:16,770
because if you notice what
we had before, you see that--

702
00:40:23,485 --> 00:40:25,367
let's go back to
the previous one.

703
00:40:28,880 --> 00:40:29,380
Yeah.

704
00:40:29,380 --> 00:40:31,120
Let's go back to this one here.

705
00:40:31,120 --> 00:40:34,580
You can see that you've
got these three terms.

706
00:40:34,580 --> 00:40:36,220
So what we're saying
is if you make

707
00:40:36,220 --> 00:40:41,970
Ar much bigger than As, then
you make the contrast is going

708
00:40:41,970 --> 00:40:44,170
to get smaller and
smaller because obviously

709
00:40:44,170 --> 00:40:47,500
this term is going to become
much bigger than this one.

710
00:40:47,500 --> 00:40:52,150
But nevertheless, it does
actually make this term bigger.

711
00:40:52,150 --> 00:40:55,210
So if actually what
you're trying to do

712
00:40:55,210 --> 00:40:59,470
is measure this signal in
some background of noise,

713
00:40:59,470 --> 00:41:04,450
it's sometimes a good thing
to make Ar bigger than As

714
00:41:04,450 --> 00:41:07,960
because as you increase Ar,
it increases the strength

715
00:41:07,960 --> 00:41:12,370
of this interference term.

716
00:41:12,370 --> 00:41:14,400
And that's the
principle that's used

717
00:41:14,400 --> 00:41:18,290
in heterodyne interferometry.

718
00:41:18,290 --> 00:41:24,110
So not always true that you try
and make the contrast as big

719
00:41:24,110 --> 00:41:25,160
as possible.

720
00:41:25,160 --> 00:41:26,407
Yeah.

721
00:41:26,407 --> 00:41:28,490
GEORGE BARBASTATHIS: There
is one more factor of 2

722
00:41:28,490 --> 00:41:30,020
missing here.

723
00:41:30,020 --> 00:41:33,410
On the left-hand side bottom,
the swing of the fringes

724
00:41:33,410 --> 00:41:37,920
should be 2m times
i0 times-- yeah.

725
00:41:37,920 --> 00:41:41,240
There's a factor of 2 missing.

726
00:41:41,240 --> 00:41:42,260
That's quite unusual.

727
00:41:42,260 --> 00:41:43,620
Three slides in a row.

728
00:41:43,620 --> 00:41:46,060
They're all missing
a factor of 2.

729
00:41:46,060 --> 00:41:47,810
COLIN SHEPPARD: And
they don't cancel out.

730
00:41:47,810 --> 00:41:47,960
GEORGE BARBASTATHIS: Yeah.

731
00:41:47,960 --> 00:41:48,460
I guess.

732
00:41:53,098 --> 00:41:53,890
COLIN SHEPPARD: OK.

733
00:41:53,890 --> 00:41:56,920
So that's the Michelson
interferometer.

734
00:41:56,920 --> 00:42:00,280
So the Michelson interferometer
is basically, you can see,

735
00:42:00,280 --> 00:42:02,890
a sort of reflection
interferometer.

736
00:42:02,890 --> 00:42:06,008
And we're now going to
go into the Mach-Zehnder

737
00:42:06,008 --> 00:42:08,050
interferometer, which is
basically a transmission

738
00:42:08,050 --> 00:42:09,230
interferometer.

739
00:42:09,230 --> 00:42:12,620
So the principle
is very similar.

740
00:42:12,620 --> 00:42:15,160
You split the light
into two parts.

741
00:42:15,160 --> 00:42:18,200
The two paths-- two light beams
go through different path,

742
00:42:18,200 --> 00:42:24,640
and then you combine
them onto a detector.

743
00:42:24,640 --> 00:42:27,680
So this works in
transmission mode.

744
00:42:27,680 --> 00:42:30,920
Here you're looking
through an object.

745
00:42:30,920 --> 00:42:33,080
So now you're not going
to get that factor of 2,

746
00:42:33,080 --> 00:42:37,400
of course, because the light
is only going through it once.

747
00:42:37,400 --> 00:42:39,890
So the light goes through here.

748
00:42:39,890 --> 00:42:43,550
You notice that the way
this has been drawn,

749
00:42:43,550 --> 00:42:47,750
it's been drawn so that this
light is going like this,

750
00:42:47,750 --> 00:42:50,040
this light is going like this.

751
00:42:50,040 --> 00:42:56,710
So you can see this path and
this path are roughly equal.

752
00:42:56,710 --> 00:43:01,720
You might equally-- well, you
could have thought of making

753
00:43:01,720 --> 00:43:05,540
this by doing it more like--

754
00:43:05,540 --> 00:43:06,280
couldn't you?

755
00:43:06,280 --> 00:43:14,500
You could have
done it like that.

756
00:43:17,840 --> 00:43:23,240
So where now you can see, if
we'd have made it like that,

757
00:43:23,240 --> 00:43:28,930
this path is obviously
longer than that path.

758
00:43:28,930 --> 00:43:35,250
So I guess nowadays if
you did that experiment

759
00:43:35,250 --> 00:43:39,240
with a laser which has
got a very long coherence

760
00:43:39,240 --> 00:43:42,390
length, something like a
frequency double YAG laser,

761
00:43:42,390 --> 00:43:44,040
this would work fine.

762
00:43:44,040 --> 00:43:48,780
If you tried doing this with
an arc lamp or something

763
00:43:48,780 --> 00:43:54,060
like that it wouldn't work
because this distance is

764
00:43:54,060 --> 00:43:58,410
much longer than the coherence
length of the source.

765
00:43:58,410 --> 00:44:02,400
So you're all used to the
idea that actually you

766
00:44:02,400 --> 00:44:04,380
don't see interference
fringes all the time.

767
00:44:04,380 --> 00:44:07,890
In the normal,
regular, everyday life

768
00:44:07,890 --> 00:44:10,620
you don't see interference
fringes very much.

769
00:44:10,620 --> 00:44:13,620
And that's because
in everyday life,

770
00:44:13,620 --> 00:44:17,640
normally the light is
incoherent rather than coherent.

771
00:44:17,640 --> 00:44:23,700
So if you make this system here
or the Michelson interferometer

772
00:44:23,700 --> 00:44:28,620
we showed before and you make
it so that these paths are

773
00:44:28,620 --> 00:44:32,840
accurately the same
length, then you

774
00:44:32,840 --> 00:44:37,085
could get interference
even with light from a--

775
00:44:37,085 --> 00:44:40,475
you can get interference even
from light from a tungsten lamp

776
00:44:40,475 --> 00:44:41,920
bulb.

777
00:44:41,920 --> 00:44:46,060
And they produce really
nice, beautiful fringes

778
00:44:46,060 --> 00:44:47,470
with different colors.

779
00:44:47,470 --> 00:44:52,810
Maybe my students maybe
have seen these in the lab.

780
00:44:52,810 --> 00:44:54,480
But anyway.

781
00:44:54,480 --> 00:44:57,330
So here you can use this
to measure the optical path

782
00:44:57,330 --> 00:44:59,890
of some object.

783
00:44:59,890 --> 00:45:03,050
And yeah.

784
00:45:03,050 --> 00:45:08,960
So this is now, again,
looking at this idea

785
00:45:08,960 --> 00:45:10,720
of tilting something.

786
00:45:10,720 --> 00:45:14,450
I mentioned about tilting
with regard to the Michelson

787
00:45:14,450 --> 00:45:15,260
interferometer.

788
00:45:15,260 --> 00:45:17,600
I said, if you tilt
one of those mirrors,

789
00:45:17,600 --> 00:45:23,990
you would actually get the
optical path varying linearly

790
00:45:23,990 --> 00:45:26,430
with transverse distance.

791
00:45:26,430 --> 00:45:28,700
So if you actually
looked at an image

792
00:45:28,700 --> 00:45:31,010
of that interference,
what you would see

793
00:45:31,010 --> 00:45:34,580
would be a series of fringes.

794
00:45:34,580 --> 00:45:37,700
And the spacing
between the fringes

795
00:45:37,700 --> 00:45:41,490
would increase as you
increase the angle.

796
00:45:41,490 --> 00:45:43,440
So this is the same here.

797
00:45:43,440 --> 00:45:48,110
We're getting the offset
between these two paths--

798
00:45:48,110 --> 00:45:55,220
these two beams-- by rotating
this mirror slightly.

799
00:45:55,220 --> 00:45:58,730
And then as you can see,
these two beams arrive

800
00:45:58,730 --> 00:46:01,470
on a CCD camera, it shows here.

801
00:46:01,470 --> 00:46:03,740
So you will actually--
in this case,

802
00:46:03,740 --> 00:46:07,910
you would get some
variation in intensity

803
00:46:07,910 --> 00:46:09,980
on this detector, which
you could actually

804
00:46:09,980 --> 00:46:11,960
measure with the detector.

805
00:46:11,960 --> 00:46:13,520
So it would look
something like this.

806
00:46:13,520 --> 00:46:17,150
This would now be,
then, a function of x

807
00:46:17,150 --> 00:46:20,870
rather than just being a
function of the optical path

808
00:46:20,870 --> 00:46:22,540
difference.

809
00:46:22,540 --> 00:46:27,590
So this bit here, I guess, is
explaining, is deriving that.

810
00:46:27,590 --> 00:46:30,540
So we're adding
two wave together.

811
00:46:30,540 --> 00:46:33,050
One is arriving normally.

812
00:46:33,050 --> 00:46:36,720
The other one is tilted
through an angle theta.

813
00:46:36,720 --> 00:46:44,150
And so this one here is going
to have a linear phase variation

814
00:46:44,150 --> 00:46:48,575
as a function of x, which will
then appear inside this cosine.

815
00:46:51,980 --> 00:46:55,350
So finally what
we get here then,

816
00:46:55,350 --> 00:46:59,870
the value of this bit
here, the z cos theta,

817
00:46:59,870 --> 00:47:03,000
is just this constant here.

818
00:47:03,000 --> 00:47:07,850
So all that does is it shifts
these fringes this way.

819
00:47:07,850 --> 00:47:11,880
But normally, if you were
doing this experiment,

820
00:47:11,880 --> 00:47:15,050
one thing you could
do then very simply

821
00:47:15,050 --> 00:47:19,890
is to measure what this spatial
wavelength of these fringes is.

822
00:47:19,890 --> 00:47:23,850
And from that you
could, for example,

823
00:47:23,850 --> 00:47:25,250
if you knew all
the other things,

824
00:47:25,250 --> 00:47:27,680
you could measure what
this rotation theta was,

825
00:47:27,680 --> 00:47:28,820
couldn't you?

826
00:47:28,820 --> 00:47:31,700
If you knew what the
rotation theta is,

827
00:47:31,700 --> 00:47:33,980
then I don't know what
you could measure.

828
00:47:33,980 --> 00:47:37,250
But you could measure
something, probably.

829
00:47:37,250 --> 00:47:40,850
So the Mach-Zehnder,
I mentioned that this

830
00:47:40,850 --> 00:47:43,640
is like a transmission
interferometer.

831
00:47:43,640 --> 00:47:50,180
The Michelson is like a
reflection interferometer.

832
00:47:50,180 --> 00:47:52,400
Both of these sorts
of interferometers

833
00:47:52,400 --> 00:47:55,430
are used in
microscopy, actually.

834
00:47:55,430 --> 00:47:58,880
And so you can make an
interference microscope

835
00:47:58,880 --> 00:48:01,330
based on this technique.

836
00:48:01,330 --> 00:48:04,220
And you could put
in here, let's say,

837
00:48:04,220 --> 00:48:08,540
a biological slide with a bit
of tissue or something on it.

838
00:48:08,540 --> 00:48:11,480
And you could use this to
measure the obstacle path

839
00:48:11,480 --> 00:48:14,210
in going through that object.

840
00:48:14,210 --> 00:48:17,320
It's not trivial to do that.

841
00:48:17,320 --> 00:48:20,570
You'd normally, if you want
to do it in a microscope,

842
00:48:20,570 --> 00:48:22,610
you'd have to place some
microscope objectives

843
00:48:22,610 --> 00:48:24,650
and things in here.

844
00:48:24,650 --> 00:48:27,110
And usually it's found
that if you do that, that

845
00:48:27,110 --> 00:48:28,640
messes everything up.

846
00:48:28,640 --> 00:48:31,580
So you normally have to put
the microscope objectives

847
00:48:31,580 --> 00:48:33,500
in this one as well.

848
00:48:33,500 --> 00:48:37,160
So you end up really having
to have two microscopes,

849
00:48:37,160 --> 00:48:39,180
and the whole thing
becomes very expensive.

850
00:48:39,180 --> 00:48:43,670
So that sort of method has
really gone out of fashion,

851
00:48:43,670 --> 00:48:46,550
although many years
ago Zeiss and people

852
00:48:46,550 --> 00:48:49,230
used to make
instruments like that.

853
00:48:49,230 --> 00:48:52,453
The same goes for a Michelson
type interferometer.

854
00:48:52,453 --> 00:48:53,870
You can measure
that for looking--

855
00:48:53,870 --> 00:48:58,940
use that in a microscope
for looking at surfaces--

856
00:48:58,940 --> 00:49:00,550
surface height
changes, and so on.

857
00:49:06,170 --> 00:49:07,700
OK.

858
00:49:07,700 --> 00:49:11,480
So this shows, again, how
this comes about, then.

859
00:49:11,480 --> 00:49:15,080
You've got this one wave, which
is coming along this angle

860
00:49:15,080 --> 00:49:16,420
here.

861
00:49:16,420 --> 00:49:18,780
And so this
represents the phases.

862
00:49:18,780 --> 00:49:21,400
This is a movie.

863
00:49:21,400 --> 00:49:25,920
Do I just click it again and
it'll do something, will it?

864
00:49:25,920 --> 00:49:27,010
Yes.

865
00:49:27,010 --> 00:49:27,960
There we are.

866
00:49:27,960 --> 00:49:28,460
OK.

867
00:49:28,460 --> 00:49:31,890
So it didn't go very far.

868
00:49:31,890 --> 00:49:33,180
Anyway.

869
00:49:33,180 --> 00:49:35,560
Probably it was
quick enough to see.

870
00:49:35,560 --> 00:49:38,820
It was a wave that's
moving this way.

871
00:49:38,820 --> 00:49:47,380
But the important thing, if
I go back and do it again--

872
00:49:47,380 --> 00:49:48,050
OK.

873
00:49:48,050 --> 00:49:50,350
So you've all got
to watch this time.

874
00:49:50,350 --> 00:49:53,350
And there's two things
I want you to watch.

875
00:49:53,350 --> 00:49:56,860
One is the wave moving this way.

876
00:49:56,860 --> 00:50:01,720
But more importantly, look
what happens on this line

877
00:50:01,720 --> 00:50:04,990
here where you've got
the camera placed.

878
00:50:04,990 --> 00:50:07,653
GEORGE BARBASTATHIS: [INAUDIBLE]

879
00:50:07,653 --> 00:50:08,570
COLIN SHEPPARD: Right.

880
00:50:08,570 --> 00:50:08,770
Yeah.

881
00:50:08,770 --> 00:50:09,270
Yeah.

882
00:50:09,270 --> 00:50:10,060
Yeah.

883
00:50:10,060 --> 00:50:12,910
Otherwise, I think people
have to be very quick.

884
00:50:31,180 --> 00:50:31,720
Yes.

885
00:50:31,720 --> 00:50:33,570
Look at that.

886
00:50:33,570 --> 00:50:34,080
OK.

887
00:50:34,080 --> 00:50:34,750
So good.

888
00:50:38,100 --> 00:50:42,320
So this wave is moving out here.

889
00:50:42,320 --> 00:50:45,980
But you can see, if you look
at where the detector is,

890
00:50:45,980 --> 00:50:48,310
it's as though the fringes--

891
00:50:48,310 --> 00:50:52,960
the phase is actually
moving continuously

892
00:50:52,960 --> 00:50:54,940
in one direction like that.

893
00:50:54,940 --> 00:51:07,490
So right.

894
00:51:14,645 --> 00:51:16,020
GEORGE BARBASTATHIS:
So the point

895
00:51:16,020 --> 00:51:20,190
is that the reference has
a constant phase because it

896
00:51:20,190 --> 00:51:25,090
is normal on the vertical axis.

897
00:51:25,090 --> 00:51:30,810
So as the incident wave
is passing through,

898
00:51:30,810 --> 00:51:38,220
the peaks are the
locations where

899
00:51:38,220 --> 00:51:41,020
the intensity accumulates.

900
00:51:41,020 --> 00:51:46,210
So you end up getting a
peak, whereas in the nulls,

901
00:51:46,210 --> 00:51:49,140
it means that as the
wave is passing through,

902
00:51:49,140 --> 00:51:50,790
the phases are not lining up.

903
00:51:50,790 --> 00:51:53,070
So you end up getting a null.

904
00:51:53,070 --> 00:51:53,950
That's how you--

905
00:51:53,950 --> 00:51:54,550
COLIN SHEPPARD: Yeah.

906
00:51:54,550 --> 00:51:55,060
Yeah.

907
00:51:55,060 --> 00:51:56,440
OK.

908
00:51:56,440 --> 00:51:56,940
Yeah.

909
00:51:56,940 --> 00:52:01,340
So I guess the reference
beam, as you say,

910
00:52:01,340 --> 00:52:02,650
is coming along this way.

911
00:52:02,650 --> 00:52:05,000
But of course, that's
got fringes as well.

912
00:52:05,000 --> 00:52:05,500
So--

913
00:52:05,500 --> 00:52:05,990
GEORGE BARBASTATHIS:
That's right.

914
00:52:05,990 --> 00:52:08,200
COLIN SHEPPARD:
--the peaks of that,

915
00:52:08,200 --> 00:52:10,420
when they match up with
the peaks of the other,

916
00:52:10,420 --> 00:52:14,640
would give the maxima in the--

917
00:52:14,640 --> 00:52:17,320
GEORGE BARBASTATHIS: There's
locations where the peaks

918
00:52:17,320 --> 00:52:19,840
always arrive at the same time.

919
00:52:19,840 --> 00:52:21,230
But a little bit
below, the peaks

920
00:52:21,230 --> 00:52:22,940
will be actually out of phase.

921
00:52:22,940 --> 00:52:24,190
So they end up getting a null.

922
00:52:28,940 --> 00:52:31,357
COLIN SHEPPARD: End of show.

923
00:52:31,357 --> 00:52:32,440
So that's the end of that.

924
00:52:32,440 --> 00:52:34,420
That sounds like a
good place for a break.

925
00:52:37,010 --> 00:52:39,443
So let's make a break.

926
00:52:39,443 --> 00:52:41,360
Unless anyone's got some
questions, of course.

927
00:52:41,360 --> 00:52:42,110
Yeah, of course.

928
00:52:42,110 --> 00:52:44,547
[INAUDIBLE] always
got questions.

929
00:52:44,547 --> 00:52:46,130
AUDIENCE: In Michelson
interferometer,

930
00:52:46,130 --> 00:52:52,430
the object would affect the
intensity according to whatever

931
00:52:52,430 --> 00:52:54,290
phase delay it introduces.

932
00:52:54,290 --> 00:52:56,960
But in the Mach-Zehnder
interferometer,

933
00:52:56,960 --> 00:53:00,500
it will affect the
spacing of the fringes.

934
00:53:00,500 --> 00:53:02,420
Is that correct?

935
00:53:02,420 --> 00:53:04,310
It will change the spacing--

936
00:53:04,310 --> 00:53:10,687
we added this constant
slope in the signal beam

937
00:53:10,687 --> 00:53:11,520
in the Mach-Zehnder.

938
00:53:11,520 --> 00:53:13,250
COLIN SHEPPARD: I
think this tilting--

939
00:53:13,250 --> 00:53:15,410
the tilting the one beam
relative to another you

940
00:53:15,410 --> 00:53:18,662
can doing either sort
of interferometer.

941
00:53:18,662 --> 00:53:20,120
That's not a
fundamental difference

942
00:53:20,120 --> 00:53:22,120
between the two types.

943
00:53:22,120 --> 00:53:24,650
AUDIENCE: Oh.

944
00:53:24,650 --> 00:53:29,900
So I was thinking, when you have
a tilted plane, tilted mirror,

945
00:53:29,900 --> 00:53:33,320
then you would measure the
spacing between fringes

946
00:53:33,320 --> 00:53:38,360
to find out the object's
optical path length.

947
00:53:38,360 --> 00:53:41,420
But if you didn't
have that, then you

948
00:53:41,420 --> 00:53:46,640
would just measure the
change in intensity

949
00:53:46,640 --> 00:53:50,433
to get the wrapped phase
and then apply unwrapping.

950
00:53:50,433 --> 00:53:51,100
Is that correct?

951
00:53:54,163 --> 00:53:56,580
COLIN SHEPPARD: Well, it depends
what you're trying to do.

952
00:53:56,580 --> 00:53:58,530
AUDIENCE: To find out the phase.

953
00:53:58,530 --> 00:53:58,770
COLIN SHEPPARD: Yeah.

954
00:53:58,770 --> 00:53:59,270
Yeah.

955
00:53:59,270 --> 00:54:01,790
Yeah.

956
00:54:01,790 --> 00:54:03,500
You're thinking of it varying.

957
00:54:03,500 --> 00:54:05,353
It's a spatially-varying
quantity

958
00:54:05,353 --> 00:54:07,020
that you're trying
to measure, isn't it?

959
00:54:07,020 --> 00:54:07,300
AUDIENCE: Yeah.

960
00:54:07,300 --> 00:54:07,800
Yeah.

961
00:54:07,800 --> 00:54:10,240
COLIN SHEPPARD: So I mean,
very often what you do,

962
00:54:10,240 --> 00:54:12,510
of course, is you
actually-- very

963
00:54:12,510 --> 00:54:15,510
often you want the fringes
because they give you something

964
00:54:15,510 --> 00:54:17,270
to actually look at.

965
00:54:17,270 --> 00:54:22,800
So normally, if you don't
do the tilting thing,

966
00:54:22,800 --> 00:54:27,210
what you'd see if the two
paths were equal would be,

967
00:54:27,210 --> 00:54:30,210
let's say, completely
featureless, bright,

968
00:54:30,210 --> 00:54:31,590
wouldn't it?

969
00:54:31,590 --> 00:54:32,500
AUDIENCE: Yeah.

970
00:54:32,500 --> 00:54:36,090
COLIN SHEPPARD: And so let's
say you put some object in there

971
00:54:36,090 --> 00:54:38,490
which has got some
phase change, you

972
00:54:38,490 --> 00:54:42,600
would just see a small change
from that bright, which

973
00:54:42,600 --> 00:54:43,793
might not be very visible.

974
00:54:43,793 --> 00:54:44,460
AUDIENCE: I see.

975
00:54:44,460 --> 00:54:45,030
Yeah.

976
00:54:45,030 --> 00:54:51,900
COLIN SHEPPARD: But for example,
then, if you tilted the system,

977
00:54:51,900 --> 00:54:53,800
then you would get some fringes.

978
00:54:53,800 --> 00:54:55,890
And the sort of thing you
might see, of course--

979
00:54:55,890 --> 00:55:00,410
let's say you put in a
disk of material like this

980
00:55:00,410 --> 00:55:03,240
and you're trying to measure
the thickness of this disk

981
00:55:03,240 --> 00:55:04,920
of material--

982
00:55:04,920 --> 00:55:08,970
so what you would find is if
it was nicely uniformly thick,

983
00:55:08,970 --> 00:55:12,210
you'd find that you've got
these regular sort of fringes

984
00:55:12,210 --> 00:55:17,480
across here, whereas
if it wasn't, then

985
00:55:17,480 --> 00:55:19,410
you might find that
you get fringes

986
00:55:19,410 --> 00:55:24,310
that have got some shape that
looks something like this.

987
00:55:24,310 --> 00:55:27,770
So by looking at the
geometry of those fringes,

988
00:55:27,770 --> 00:55:32,930
you can quite easily measure
the changes and the deformations

989
00:55:32,930 --> 00:55:33,940
in the shape of it.

990
00:55:37,820 --> 00:55:41,930
And as you change the tilt,
it would change the separation

991
00:55:41,930 --> 00:55:44,710
between these fringes.

992
00:55:44,710 --> 00:55:47,810
And so if you make
the tilt very small,

993
00:55:47,810 --> 00:55:51,500
then you'd just effectively be
looking at just a single fringe

994
00:55:51,500 --> 00:55:53,720
or no fringes, really.

995
00:55:53,720 --> 00:55:55,530
So then it would be
actually very difficult

996
00:55:55,530 --> 00:56:00,110
to see the changes
in the thickness

997
00:56:00,110 --> 00:56:05,241
unless you actually made
very sensitive measurements.

998
00:56:10,940 --> 00:56:11,440
OK.

999
00:56:11,440 --> 00:56:12,650
So everyone back again.

1000
00:56:18,180 --> 00:56:18,680
Yeah.

1001
00:56:18,680 --> 00:56:22,160
I thought of something
else that perhaps I

1002
00:56:22,160 --> 00:56:26,600
could mention following
on from this diagram.

1003
00:56:26,600 --> 00:56:34,100
So in the lectures, the
system, as you remember,

1004
00:56:34,100 --> 00:56:38,800
was arranged to do
this sort of thing.

1005
00:56:38,800 --> 00:56:43,370
So the idea here is
that the two paths now--

1006
00:56:43,370 --> 00:56:48,090
this path and this path--
are roughly equal in size.

1007
00:56:48,090 --> 00:56:50,070
In the old days,
when people were

1008
00:56:50,070 --> 00:56:56,220
trying to do this with
non-coherent sources,

1009
00:56:56,220 --> 00:56:58,590
then you had to
be really careful.

1010
00:56:58,590 --> 00:57:02,520
And for example, this is a beam
splitter, which is actually,

1011
00:57:02,520 --> 00:57:07,410
of course, a lump of glass
maybe with some coating on it

1012
00:57:07,410 --> 00:57:08,910
on one side.

1013
00:57:08,910 --> 00:57:12,030
But you can see this beam
has gone through this glass

1014
00:57:12,030 --> 00:57:15,740
but this beam hasn't
gone through that glass.

1015
00:57:15,740 --> 00:57:18,230
And so these are all
things that you also

1016
00:57:18,230 --> 00:57:19,590
have to take into account.

1017
00:57:19,590 --> 00:57:25,730
And so in order to correct for
the optical effects of, say,

1018
00:57:25,730 --> 00:57:28,910
for instance, that
slab of glass there,

1019
00:57:28,910 --> 00:57:32,870
you might actually end up
putting a slab of glass

1020
00:57:32,870 --> 00:57:35,310
in here--

1021
00:57:35,310 --> 00:57:41,050
some sort of compensating
piece of glass--

1022
00:57:41,050 --> 00:57:45,360
so that this would produce
the same dispersion

1023
00:57:45,360 --> 00:57:47,560
in the two arms.

1024
00:57:47,560 --> 00:57:51,600
So there was a lot of
effort put into trying

1025
00:57:51,600 --> 00:57:53,220
to design these
systems so that they

1026
00:57:53,220 --> 00:57:56,320
would work with the
broadband sources and so on.

1027
00:57:56,320 --> 00:58:00,000
Nowadays, I guess, most of the
time we're doing it with lasers

1028
00:58:00,000 --> 00:58:02,890
and we don't have to worry
about any of these things,

1029
00:58:02,890 --> 00:58:05,010
but if you look in any
of the standard books

1030
00:58:05,010 --> 00:58:07,950
on interferometry,
there are a number

1031
00:58:07,950 --> 00:58:09,960
of textbooks on
interferometry that

1032
00:58:09,960 --> 00:58:11,640
will give you all
the information

1033
00:58:11,640 --> 00:58:14,060
if you want to go more into it.

1034
00:58:14,060 --> 00:58:20,490
So now we're going to change
step slightly from interference

1035
00:58:20,490 --> 00:58:22,680
to diffraction.

1036
00:58:22,680 --> 00:58:25,290
And so what's the difference
between interference

1037
00:58:25,290 --> 00:58:26,670
and diffraction?

1038
00:58:26,670 --> 00:58:30,030
The answer is really not a
lot, except that normally

1039
00:58:30,030 --> 00:58:34,980
interference we're talking
about two beams interfering,

1040
00:58:34,980 --> 00:58:39,450
whereas in diffraction we're
normally talking about lots

1041
00:58:39,450 --> 00:58:40,980
of beams interfering.

1042
00:58:40,980 --> 00:58:45,930
So you can think of the two
as really being effectively

1043
00:58:45,930 --> 00:58:49,620
the same phenomenon, actually.

1044
00:58:49,620 --> 00:58:55,310
And so we're going to look
at Huygens' principle.

1045
00:58:55,310 --> 00:58:58,460
We're going to go on to
the Young's interferometer.

1046
00:58:58,460 --> 00:59:01,220
So this really does
show, doesn't it,

1047
00:59:01,220 --> 00:59:04,940
how interferometry is
related to diffraction

1048
00:59:04,940 --> 00:59:07,670
because this lecture is
really about diffraction.

1049
00:59:07,670 --> 00:59:11,210
And then we're going to go
to a more complicated case

1050
00:59:11,210 --> 00:59:13,670
of Fresnel diffraction.

1051
00:59:16,600 --> 00:59:21,370
So first of all
Huygens' principle.

1052
00:59:21,370 --> 00:59:25,580
So I think at the beginning
of the lecture course,

1053
00:59:25,580 --> 00:59:29,840
George said a bit about
the history of how

1054
00:59:29,840 --> 00:59:34,130
Huygens' came up with this
idea quite a long time ago,

1055
00:59:34,130 --> 00:59:34,630
wasn't it?

1056
00:59:34,630 --> 00:59:36,580
It was even before
Newton or around--

1057
00:59:36,580 --> 00:59:39,880
I think slightly before but
around the same time as Newton.

1058
00:59:39,880 --> 00:59:43,900
So when people
still hadn't really

1059
00:59:43,900 --> 00:59:47,920
decided whether light
was a wave or a particle.

1060
00:59:47,920 --> 00:59:49,860
And of course, we
still haven't decided.

1061
00:59:49,860 --> 00:59:53,260
So times don't change, really.

1062
00:59:53,260 --> 00:59:56,680
But anyway so probably
now, looking back,

1063
00:59:56,680 --> 01:00:01,130
we could say that Newton was
the first quantum opticist.

1064
01:00:01,130 --> 01:00:03,010
But anyway.

1065
01:00:03,010 --> 01:00:06,220
So this was this
was Huygens' idea.

1066
01:00:06,220 --> 01:00:10,570
He was saying that if you
think of a light beam--

1067
01:00:10,570 --> 01:00:14,620
a wave propagating in space,
then at some moment in time

1068
01:00:14,620 --> 01:00:18,050
this is a wavefront.

1069
01:00:18,050 --> 01:00:21,890
The wavefront is a
surface of constant phase.

1070
01:00:21,890 --> 01:00:29,900
And he came up with this model
for how light propagates where

1071
01:00:29,900 --> 01:00:35,420
he assumed that all the
points on this wavefront

1072
01:00:35,420 --> 01:00:38,030
can be thought of as
being a source of what

1073
01:00:38,030 --> 01:00:40,080
he called secondary wavelets.

1074
01:00:40,080 --> 01:00:44,690
So each of these points
radiates like a spherical wave.

1075
01:00:44,690 --> 01:00:48,650
And then if you sum those
all up and calculate

1076
01:00:48,650 --> 01:00:52,950
what the effect is at some
distance in front here,

1077
01:00:52,950 --> 01:00:58,110
then that would then predict
what that wavefront--

1078
01:00:58,110 --> 01:01:02,750
what the wave had done
at a slightly later time.

1079
01:01:02,750 --> 01:01:04,940
So that's the principle.

1080
01:01:04,940 --> 01:01:09,440
And so quite a
simple idea, really.

1081
01:01:09,440 --> 01:01:16,100
And you can come up with a very
simple mathematical description

1082
01:01:16,100 --> 01:01:17,240
of it.

1083
01:01:17,240 --> 01:01:19,940
Unfortunately, it doesn't
work marvelously well.

1084
01:01:19,940 --> 01:01:23,420
And people put a lot
of effort into trying

1085
01:01:23,420 --> 01:01:25,610
to improve the theory.

1086
01:01:25,610 --> 01:01:30,560
And they came up with
some rather arbitrary--

1087
01:01:30,560 --> 01:01:32,480
they seem rather arbitrary--

1088
01:01:32,480 --> 01:01:35,420
factors that they had
to include in order

1089
01:01:35,420 --> 01:01:36,960
to get it to work properly.

1090
01:01:36,960 --> 01:01:40,250
So I think nowadays we probably
look back over all this

1091
01:01:40,250 --> 01:01:42,500
and think it was just history.

1092
01:01:42,500 --> 01:01:46,310
And there are better ways of
really tackling the problem.

1093
01:01:46,310 --> 01:01:50,240
But anyway, at the very
simple level though,

1094
01:01:50,240 --> 01:01:55,350
it's still a nice simple
theory for how you do that.

1095
01:01:55,350 --> 01:01:58,070
So if you want to calculate
what the field is at this point

1096
01:01:58,070 --> 01:02:00,200
here, say, you look
at all these waves

1097
01:02:00,200 --> 01:02:02,840
that come from all these
points on the wavefront,

1098
01:02:02,840 --> 01:02:05,980
add them all together
as phases, of course,

1099
01:02:05,980 --> 01:02:08,110
and that will give you
the field at that point.

1100
01:02:11,640 --> 01:02:14,270
And there it's showing
how you do that.

1101
01:02:14,270 --> 01:02:17,420
And then you can do it
again and again and again

1102
01:02:17,420 --> 01:02:19,040
and show how the
wave propagates.

1103
01:02:21,790 --> 01:02:25,930
So example, a hole
in an opaque screen.

1104
01:02:25,930 --> 01:02:28,900
So I guess this is pretty
well the simplest example

1105
01:02:28,900 --> 01:02:31,660
you could possibly think of.

1106
01:02:31,660 --> 01:02:33,790
Well, no, I suppose the
opaque screen with no hole

1107
01:02:33,790 --> 01:02:35,050
in would be even simpler.

1108
01:02:35,050 --> 01:02:37,390
Then you wouldn't see anything.

1109
01:02:37,390 --> 01:02:41,350
But what we do is we just have
a very small hole in this.

1110
01:02:41,350 --> 01:02:45,130
And so this is going to
act like a source that's

1111
01:02:45,130 --> 01:02:49,810
going to radiate like
a spherical wave.

1112
01:02:49,810 --> 01:02:54,190
And what we'd see on this
screen that we place here,

1113
01:02:54,190 --> 01:02:58,030
you'd see that is
virtually what we've

1114
01:02:58,030 --> 01:03:00,700
been looking at before anyway.

1115
01:03:00,700 --> 01:03:02,180
We've done that before.

1116
01:03:02,180 --> 01:03:03,100
So there we are.

1117
01:03:03,100 --> 01:03:05,990
There's the spherical wave.

1118
01:03:05,990 --> 01:03:10,653
And then we can-- are we going
to get some more on that?

1119
01:03:10,653 --> 01:03:12,070
Did something
happen then, George?

1120
01:03:12,070 --> 01:03:13,306
GEORGE BARBASTATHIS: Yes.

1121
01:03:13,306 --> 01:03:16,167
[INAUDIBLE]

1122
01:03:16,167 --> 01:03:17,500
COLIN SHEPPARD: Ah! e to the ir.

1123
01:03:17,500 --> 01:03:17,870
Right.

1124
01:03:17,870 --> 01:03:18,370
Sorry.

1125
01:03:18,370 --> 01:03:20,500
I couldn't see
anything happening.

1126
01:03:20,500 --> 01:03:24,200
So this is the plane
wave coming in.

1127
01:03:24,200 --> 01:03:29,600
And now something's come up--

1128
01:03:29,600 --> 01:03:31,023
the hole is a delta function.

1129
01:03:31,023 --> 01:03:31,940
Did that just come up?

1130
01:03:31,940 --> 01:03:32,982
GEORGE BARBASTATHIS: Yep.

1131
01:03:35,755 --> 01:03:37,880
COLIN SHEPPARD: And now
something else has come up.

1132
01:03:37,880 --> 01:03:39,910
This is the spherical wave now.

1133
01:03:39,910 --> 01:03:41,450
And this is the
same expression we

1134
01:03:41,450 --> 01:03:44,820
had for the spherical wave in
the previous lecture, then.

1135
01:03:44,820 --> 01:03:50,300
So you remember this
parabolic type approximation

1136
01:03:50,300 --> 01:03:54,360
we made for a spherical wave.

1137
01:03:54,360 --> 01:03:58,550
This is called the
Fresnel approximation,

1138
01:03:58,550 --> 01:04:02,030
named after the guy Fresnel
who I guess must have come up

1139
01:04:02,030 --> 01:04:04,280
with this in the first place.

1140
01:04:04,280 --> 01:04:06,860
And you remember it's true--

1141
01:04:06,860 --> 01:04:12,260
it's going to be approximately
true if the x and y are small

1142
01:04:12,260 --> 01:04:15,030
compared with the distance l.

1143
01:04:15,030 --> 01:04:17,960
So you need that in order
to apply the binomial.

1144
01:04:20,730 --> 01:04:21,230
Right.

1145
01:04:21,230 --> 01:04:22,580
So there it is.

1146
01:04:22,580 --> 01:04:26,540
I'll just mention again
this i that someone from MIT

1147
01:04:26,540 --> 01:04:28,090
asked about earlier.

1148
01:04:28,090 --> 01:04:34,190
So that i then at the bottom is
saying that this spherical wave

1149
01:04:34,190 --> 01:04:38,870
is actually 90 degrees
out of phase with the wave

1150
01:04:38,870 --> 01:04:41,870
that arrives at this point.

1151
01:04:41,870 --> 01:04:43,320
So that's quite interesting.

1152
01:04:43,320 --> 01:04:47,350
So you can think of it
being like a resonance

1153
01:04:47,350 --> 01:04:48,500
sort of effect, I think.

1154
01:04:50,965 --> 01:04:53,090
GEORGE BARBASTATHIS: I
guess this is something that

1155
01:04:53,090 --> 01:04:55,970
Huygens' principle cannot
really explain, right?

1156
01:04:55,970 --> 01:04:58,430
This is why you have to
do more than Huygens'.

1157
01:04:58,430 --> 01:05:00,470
But we take it for
granted in the class,

1158
01:05:00,470 --> 01:05:02,285
so we don't get into it.

1159
01:05:02,285 --> 01:05:03,160
COLIN SHEPPARD: Yeah.

1160
01:05:03,160 --> 01:05:03,590
OK.

1161
01:05:03,590 --> 01:05:03,890
Yeah.

1162
01:05:03,890 --> 01:05:04,390
Yeah.

1163
01:05:04,390 --> 01:05:07,610
So I think George is really
saying that Huygens' didn't

1164
01:05:07,610 --> 01:05:11,420
actually have that i there.

1165
01:05:11,420 --> 01:05:13,335
So we're cleverer than he is.

1166
01:05:13,335 --> 01:05:14,960
GEORGE BARBASTATHIS:
We don't know why.

1167
01:05:18,920 --> 01:05:20,990
COLIN SHEPPARD: So
there we are then.

1168
01:05:20,990 --> 01:05:25,110
And oh.

1169
01:05:25,110 --> 01:05:26,540
Now we've got two holes.

1170
01:05:26,540 --> 01:05:28,210
I knew something had happened.

1171
01:05:28,210 --> 01:05:31,226
And the spherical wave is now
hidden behind here partly.

1172
01:05:31,226 --> 01:05:32,726
GEORGE BARBASTATHIS:
More will come.

1173
01:05:32,726 --> 01:05:34,820
COLIN SHEPPARD: More will come.

1174
01:05:34,820 --> 01:05:36,560
So now we've got two holes.

1175
01:05:36,560 --> 01:05:39,290
Each of these is
a spherical wave.

1176
01:05:39,290 --> 01:05:41,543
And of course, when
we get two of them,

1177
01:05:41,543 --> 01:05:43,460
there's going to be two
spherical waves, which

1178
01:05:43,460 --> 01:05:45,140
I think is going to-- oh, yes.

1179
01:05:45,140 --> 01:05:46,115
Another spherical wave.

1180
01:05:46,115 --> 01:05:47,870
Ah, and another equation.

1181
01:05:47,870 --> 01:05:48,740
There we are.

1182
01:05:48,740 --> 01:05:52,550
So these are the
equations of what

1183
01:05:52,550 --> 01:05:56,330
you see on the screen for each
of these two spherical waves.

1184
01:05:56,330 --> 01:05:57,980
And if you look at
them, you'll notice

1185
01:05:57,980 --> 01:06:04,827
that these are distance x0 and
minus x0 relative to the axis.

1186
01:06:04,827 --> 01:06:07,160
And you can see the only
difference in these expressions

1187
01:06:07,160 --> 01:06:11,080
is this sign, plus
or minus there.

1188
01:06:11,080 --> 01:06:13,795
So we now want to
add those together.

1189
01:06:16,600 --> 01:06:23,068
And so here we're looking
at some observation point.

1190
01:06:23,068 --> 01:06:24,735
You've got to add
those things together.

1191
01:06:27,500 --> 01:06:28,000
Yeah.

1192
01:06:28,000 --> 01:06:31,900
So of course, as
it explains here,

1193
01:06:31,900 --> 01:06:35,040
if this distance-- we
look at this distance

1194
01:06:35,040 --> 01:06:39,850
and this distance, if the
path difference between those

1195
01:06:39,850 --> 01:06:42,460
is an exact number
of wavelengths,

1196
01:06:42,460 --> 01:06:44,140
then they're going
to add up in phase.

1197
01:06:44,140 --> 01:06:46,990
If they're exact number
of half wavelengths,

1198
01:06:46,990 --> 01:06:50,500
then you're going to get
destructive interference.

1199
01:06:50,500 --> 01:06:53,890
And you're going to
get a black region.

1200
01:06:58,610 --> 01:06:59,360
So there they are.

1201
01:06:59,360 --> 01:07:02,210
There's the distances d1 and d2.

1202
01:07:02,210 --> 01:07:05,290
This is the
expressions for those.

1203
01:07:05,290 --> 01:07:09,060
And again, we can use
this Fresnel approximation

1204
01:07:09,060 --> 01:07:11,130
to expand the square root.

1205
01:07:11,130 --> 01:07:15,150
And then d2 minus d1 is
going to be this minus this.

1206
01:07:15,150 --> 01:07:17,360
And you see the
only difference--

1207
01:07:17,360 --> 01:07:19,140
there's a sine there, George.

1208
01:07:22,190 --> 01:07:27,640
And so yeah.

1209
01:07:27,640 --> 01:07:30,830
So there's going to be
4x0 x dashed over 2l.

1210
01:07:30,830 --> 01:07:35,170
So one of the twos
cancels and you get this.

1211
01:07:35,170 --> 01:07:41,360
So the optical path length,
optical path difference,

1212
01:07:41,360 --> 01:07:42,220
is this.

1213
01:07:42,220 --> 01:07:47,080
2x0 x dashed over l.

1214
01:07:47,080 --> 01:07:50,830
So x0 is this distance here.

1215
01:07:50,830 --> 01:07:53,600
x dashed is this distance here.

1216
01:07:53,600 --> 01:07:57,020
So both of those are
small quantities.

1217
01:07:57,020 --> 01:08:01,700
And l, of course,
is a big quantity.

1218
01:08:01,700 --> 01:08:06,700
So this is actually, then, in
terms of distance, of course.

1219
01:08:06,700 --> 01:08:09,730
You know two distances
up the top and a distance

1220
01:08:09,730 --> 01:08:11,120
at the bottom.

1221
01:08:11,120 --> 01:08:15,130
This has got the right
dimensions of length

1222
01:08:15,130 --> 01:08:16,080
for optical path.

1223
01:08:19,930 --> 01:08:20,430
Right.

1224
01:08:20,430 --> 01:08:25,050
So as we said, that should give
constructive and destructive

1225
01:08:25,050 --> 01:08:26,229
interference.

1226
01:08:26,229 --> 01:08:30,660
What you're going to see
is just fringes like this.

1227
01:08:30,660 --> 01:08:34,890
So there's going to be
positions where these two add up

1228
01:08:34,890 --> 01:08:40,784
in phase and positions where you
get destructive interference.

1229
01:08:44,300 --> 01:08:51,240
So if the optical path
difference is equal to lambda,

1230
01:08:51,240 --> 01:08:54,689
then if you put lambda
equal to this thing,

1231
01:08:54,689 --> 01:09:01,140
then you get that x dashed
is lambda l over 2x0.

1232
01:09:01,140 --> 01:09:03,918
So this then tells you--

1233
01:09:03,918 --> 01:09:05,460
if you made this n
lambda, of course,

1234
01:09:05,460 --> 01:09:07,439
you'd have an n in here.

1235
01:09:07,439 --> 01:09:10,649
But this distance here to--

1236
01:09:10,649 --> 01:09:17,430
this distance to there is given
by this lambda l over 2x0.

1237
01:09:17,430 --> 01:09:26,819
So you can see the smaller x0
is, the bigger x dashed is.

1238
01:09:26,819 --> 01:09:30,590
The smaller you make this,
the bigger you make this.

1239
01:09:42,337 --> 01:09:42,920
What happened?

1240
01:09:42,920 --> 01:09:46,160
GEORGE BARBASTATHIS: [INAUDIBLE]

1241
01:09:46,160 --> 01:09:47,029
COLIN SHEPPARD: Ah.

1242
01:09:47,029 --> 01:09:49,319
That's facing there.

1243
01:09:49,319 --> 01:09:56,490
So this the spacing
of the fringes, then.

1244
01:09:56,490 --> 01:09:59,700
It's this lambda l over 2x0.

1245
01:10:03,990 --> 01:10:04,770
OK.

1246
01:10:04,770 --> 01:10:08,560
And so this is going
through the maths of that.

1247
01:10:08,560 --> 01:10:11,820
So these are the two waves
we've added together.

1248
01:10:11,820 --> 01:10:14,790
The only difference
between them is this sign.

1249
01:10:14,790 --> 01:10:18,060
So we can take out the
other things as factors.

1250
01:10:18,060 --> 01:10:22,500
And so these are the two terms
with the sign difference.

1251
01:10:22,500 --> 01:10:29,020
So minus i something plus e to
the i something gives cosine.

1252
01:10:29,020 --> 01:10:32,420
And so that is our final answer.

1253
01:10:32,420 --> 01:10:36,670
What you see, the intensity
is the modulus square of this.

1254
01:10:36,670 --> 01:10:40,120
These are all phase terms so
we don't worry about them.

1255
01:10:40,120 --> 01:10:42,730
And I guess in this
expression here

1256
01:10:42,730 --> 01:10:45,130
we don't worry about any of
these other things either,

1257
01:10:45,130 --> 01:10:46,270
these constants.

1258
01:10:46,270 --> 01:10:51,860
The only thing that's
important is this term here.

1259
01:10:51,860 --> 01:10:55,480
And we seem to have kept the 4.

1260
01:10:55,480 --> 01:10:57,480
2 squared.

1261
01:10:57,480 --> 01:10:59,790
I guess-- yeah.

1262
01:10:59,790 --> 01:11:01,730
Why have we kept that?

1263
01:11:01,730 --> 01:11:02,230
Anyway.

1264
01:11:02,230 --> 01:11:03,820
4 cos squared something.

1265
01:11:03,820 --> 01:11:06,340
And then we say that
cos squared of something

1266
01:11:06,340 --> 01:11:11,090
is equal to 1 plus cos 2 theta.

1267
01:11:11,090 --> 01:11:15,400
And so therefore our
final result, then, this

1268
01:11:15,400 --> 01:11:17,680
is then our expression.

1269
01:11:17,680 --> 01:11:19,750
Very similar to what we
had before, of course,

1270
01:11:19,750 --> 01:11:22,660
for the interferometry.

1271
01:11:22,660 --> 01:11:28,360
A background with a cosine
fringes on top of it.

1272
01:11:28,360 --> 01:11:30,130
So here it is.

1273
01:11:30,130 --> 01:11:33,100
It's very similar to
what we had before.

1274
01:11:33,100 --> 01:11:36,250
The m here is 1.

1275
01:11:36,250 --> 01:11:39,670
And that of course is
just like with the case

1276
01:11:39,670 --> 01:11:43,030
of the interferometer
that we described before.

1277
01:11:43,030 --> 01:11:45,040
You remember we
got m equals 1 when

1278
01:11:45,040 --> 01:11:47,950
the strength of the object
beam and the reference beam

1279
01:11:47,950 --> 01:11:49,480
were equal.

1280
01:11:49,480 --> 01:11:53,300
So in this case they're
equal, aren't they,

1281
01:11:53,300 --> 01:11:58,450
because the two slits
in the screen are equal.

1282
01:11:58,450 --> 01:12:01,510
If you made those two slightly
different widths or something,

1283
01:12:01,510 --> 01:12:04,030
then that would not be true.

1284
01:12:04,030 --> 01:12:06,580
Then you might end up with
something which didn't

1285
01:12:06,580 --> 01:12:10,240
have this visibility of 1.

1286
01:12:14,340 --> 01:12:16,890
So that's the first example.

1287
01:12:16,890 --> 01:12:18,780
So that one really
was showing how

1288
01:12:18,780 --> 01:12:21,720
interferometry and diffraction
are really the same.

1289
01:12:21,720 --> 01:12:26,490
Now we're going to go on to
some more about diffraction

1290
01:12:26,490 --> 01:12:30,770
and introduce the idea
of a transparency.

1291
01:12:30,770 --> 01:12:34,110
So all we're doing here
is we're introducing

1292
01:12:34,110 --> 01:12:37,350
the concept of a
transparency that

1293
01:12:37,350 --> 01:12:44,310
can change the magnitude
and phase of a wave

1294
01:12:44,310 --> 01:12:46,560
by transmitting through it.

1295
01:12:46,560 --> 01:12:53,410
And so this shows
here our transparency.

1296
01:12:53,410 --> 01:12:58,650
And you can see here
it shows that the--

1297
01:12:58,650 --> 01:13:00,885
I don't know what this
color, whether it means--

1298
01:13:00,885 --> 01:13:04,260
GEORGE BARBASTATHIS: [INAUDIBLE]

1299
01:13:04,260 --> 01:13:09,160
COLIN SHEPPARD: So this means
the absorption of the screen.

1300
01:13:09,160 --> 01:13:11,400
But it might, I guess,
also change the phrase.

1301
01:13:11,400 --> 01:13:13,980
It might have some
refractive index as well.

1302
01:13:13,980 --> 01:13:14,990
Yeah.

1303
01:13:14,990 --> 01:13:18,400
And it changes in
thickness as well.

1304
01:13:18,400 --> 01:13:21,240
So you have to
work out, just like

1305
01:13:21,240 --> 01:13:23,730
before, the optical
path as the light

1306
01:13:23,730 --> 01:13:25,350
goes through this structure.

1307
01:13:25,350 --> 01:13:27,540
And it might change
both the amplitude

1308
01:13:27,540 --> 01:13:30,470
and the phase of the light.

1309
01:13:30,470 --> 01:13:37,350
So we can think of
this thin transparency

1310
01:13:37,350 --> 01:13:43,470
as just having a multiplying
effect if it's thin enough.

1311
01:13:43,470 --> 01:13:49,620
So you can use your
Huygens' type principle

1312
01:13:49,620 --> 01:13:52,830
to say that when the light
has gone through there,

1313
01:13:52,830 --> 01:13:58,290
it produces some
wavefront on the far side.

1314
01:13:58,290 --> 01:14:01,590
So the assumptions-- the
thickness I've just said.

1315
01:14:01,590 --> 01:14:05,460
Also, the features
on the transparency--

1316
01:14:05,460 --> 01:14:06,760
what's that mean?

1317
01:14:06,760 --> 01:14:07,620
What's that mean?

1318
01:14:07,620 --> 01:14:11,607
The features on the transparency
are smaller than lambda.

1319
01:14:11,607 --> 01:14:12,940
GEORGE BARBASTATHIS: [INAUDIBLE]

1320
01:14:12,940 --> 01:14:14,023
COLIN SHEPPARD: Ah, right.

1321
01:14:14,023 --> 01:14:14,600
OK.

1322
01:14:14,600 --> 01:14:15,683
That's why I was confused.

1323
01:14:19,300 --> 01:14:23,390
So this transparency can
change the attenuation.

1324
01:14:23,390 --> 01:14:26,720
So the black bits are
going to absorb the light.

1325
01:14:26,720 --> 01:14:29,600
But it could also
change the phase.

1326
01:14:29,600 --> 01:14:31,400
So phase delay.

1327
01:14:31,400 --> 01:14:34,160
And both of these
are going to be--

1328
01:14:34,160 --> 01:14:37,610
both of these effects are going
to be dependent on the material

1329
01:14:37,610 --> 01:14:41,930
properties of this transparency,
of course, and also

1330
01:14:41,930 --> 01:14:43,310
its thickness.

1331
01:14:43,310 --> 01:14:48,540
And as it points out
here, in some cases,

1332
01:14:48,540 --> 01:14:51,620
you might be able to think
of these quantities--

1333
01:14:51,620 --> 01:14:54,590
the attenuation and
the phase delay--

1334
01:14:54,590 --> 01:14:58,010
as being binary quantities.

1335
01:14:58,010 --> 01:15:01,380
They might only have
two different values.

1336
01:15:01,380 --> 01:15:04,040
For example, you might
have black and white,

1337
01:15:04,040 --> 01:15:08,930
or they might be gray scale-type
continuously varying-type

1338
01:15:08,930 --> 01:15:11,550
quantities.

1339
01:15:11,550 --> 01:15:15,740
And then the phase also
you can think of that--

1340
01:15:15,740 --> 01:15:19,820
maybe it's continuously
varying phase change,

1341
01:15:19,820 --> 01:15:22,880
or it might have
just two values,

1342
01:15:22,880 --> 01:15:28,910
binary values, or it might even
have multiple values maybe,

1343
01:15:28,910 --> 01:15:29,870
say eight values.

1344
01:15:29,870 --> 01:15:35,570
You can code-- people often
use transparencies like this

1345
01:15:35,570 --> 01:15:42,740
to code binary numbers
up to some basis of eight

1346
01:15:42,740 --> 01:15:45,970
or something like that.

1347
01:15:45,970 --> 01:15:51,970
So if these assumptions
are true, then

1348
01:15:51,970 --> 01:15:57,310
according to Huygens' principle
the propagation through this,

1349
01:15:57,310 --> 01:16:02,530
all it's going to do
basically is change the--

1350
01:16:02,530 --> 01:16:04,670
this is going to change to this.

1351
01:16:04,670 --> 01:16:06,430
This is going to change to this.

1352
01:16:06,430 --> 01:16:09,670
And you're not going
to get this affecting

1353
01:16:09,670 --> 01:16:14,410
this very much if this
assumption of this thickness

1354
01:16:14,410 --> 01:16:20,340
being very thin is valid,
in which case all it does--

1355
01:16:20,340 --> 01:16:22,180
all this transparency does--

1356
01:16:22,180 --> 01:16:28,000
is multiplies the field
going in by the transmission

1357
01:16:28,000 --> 01:16:33,250
of the grating to give
the amplitude of the wave

1358
01:16:33,250 --> 01:16:34,360
as it comes out.

1359
01:16:34,360 --> 01:16:39,010
So g minus is the wave-- is the
amplitude on the way in. g plus

1360
01:16:39,010 --> 01:16:43,000
is the amplitude of the
wave on the way out.

1361
01:16:43,000 --> 01:16:46,240
And this transmittances,
we call this

1362
01:16:46,240 --> 01:16:49,210
the amplitude transmittance
of the grating.

1363
01:16:49,210 --> 01:16:52,310
And you see it's a
function of position.

1364
01:16:52,310 --> 01:16:58,300
And as we just described, it can
be broken up into an absorption

1365
01:16:58,300 --> 01:17:01,830
term and a phase term.

1366
01:17:01,830 --> 01:17:05,640
And as we just said, those
two might either be--

1367
01:17:05,640 --> 01:17:08,340
they could be binary or they
could be continuously variable.

1368
01:17:15,920 --> 01:17:16,420
Yes.

1369
01:17:16,420 --> 01:17:19,180
So this is saying how--

1370
01:17:19,180 --> 01:17:20,650
you can think of
it as being very

1371
01:17:20,650 --> 01:17:24,890
similar to the Young's
interferometer, the two slit

1372
01:17:24,890 --> 01:17:26,770
expert experiment.

1373
01:17:26,770 --> 01:17:37,220
But now you can see
here we're thinking

1374
01:17:37,220 --> 01:17:39,470
of it as being many beams.

1375
01:17:39,470 --> 01:17:42,230
So you're breaking
this up into not just

1376
01:17:42,230 --> 01:17:45,670
two beams but many beams.

1377
01:17:45,670 --> 01:17:48,440
And each of these is
going to propagate.

1378
01:17:48,440 --> 01:17:52,890
But I'm not quite sure what
this is trying to show.

1379
01:17:56,190 --> 01:17:56,690
Yeah.

1380
01:17:56,690 --> 01:17:57,190
OK.

1381
01:17:57,190 --> 01:17:59,640
So it's pretty well
what I said before.

1382
01:17:59,640 --> 01:18:04,400
So if this thing is thin
enough, then this point

1383
01:18:04,400 --> 01:18:06,680
here is only going to be
affected by this one and not

1384
01:18:06,680 --> 01:18:08,660
by the others.

1385
01:18:08,660 --> 01:18:13,010
So then you will get this
simple multiplicative

1386
01:18:13,010 --> 01:18:14,925
sort of relationship.

1387
01:18:18,110 --> 01:18:24,080
So you can think of diffraction
then as being like interference

1388
01:18:24,080 --> 01:18:25,430
with many, many beams.

1389
01:18:36,010 --> 01:18:36,510
Yeah.

1390
01:18:36,510 --> 01:18:38,090
So this is saying,
just like we said

1391
01:18:38,090 --> 01:18:44,090
before, we are describing
this interference

1392
01:18:44,090 --> 01:18:45,740
of these many, many beams now.

1393
01:18:45,740 --> 01:18:52,610
When we did the Young's
experiment, each of the slits

1394
01:18:52,610 --> 01:18:54,560
was described by
a delta function

1395
01:18:54,560 --> 01:18:57,050
at a particular position.

1396
01:18:57,050 --> 01:18:59,510
And now we're saying
that in general, we

1397
01:18:59,510 --> 01:19:04,970
can think of this as not being
just two slits but an integral

1398
01:19:04,970 --> 01:19:09,680
over many, many apertures--

1399
01:19:09,680 --> 01:19:15,890
very small apertures--
placed at these points x1 y1.

1400
01:19:15,890 --> 01:19:18,860
Sorry, placed at the
points xy, I guess.

1401
01:19:18,860 --> 01:19:23,450
And then we're integrating
over all those points in order

1402
01:19:23,450 --> 01:19:28,880
to get the total effect of
all those spherical waves.

1403
01:19:28,880 --> 01:19:31,250
This is now we're
saying that g--

1404
01:19:33,830 --> 01:19:39,140
so this is g minus after the--

1405
01:19:39,140 --> 01:19:42,980
it's just saying that g--

1406
01:19:42,980 --> 01:19:46,807
GEORGE BARBASTATHIS: [INAUDIBLE]

1407
01:19:46,807 --> 01:19:48,890
COLIN SHEPPARD: This is
just the shifting theorem.

1408
01:19:48,890 --> 01:19:51,040
And now we're going to
state that in there, are we?

1409
01:19:51,040 --> 01:20:01,175
GEORGE BARBASTATHIS: [INAUDIBLE]

1410
01:20:01,175 --> 01:20:02,050
COLIN SHEPPARD: Yeah.

1411
01:20:02,050 --> 01:20:05,185
GEORGE BARBASTATHIS: [INAUDIBLE]

1412
01:20:05,185 --> 01:20:06,060
COLIN SHEPPARD: Yeah.

1413
01:20:06,060 --> 01:20:06,430
Yeah.

1414
01:20:06,430 --> 01:20:06,980
OK.

1415
01:20:06,980 --> 01:20:07,290
Yeah.

1416
01:20:07,290 --> 01:20:07,780
Yeah.

1417
01:20:07,780 --> 01:20:08,280
Yeah.

1418
01:20:20,390 --> 01:20:23,180
So this is what goes in then.

1419
01:20:23,180 --> 01:20:29,690
So this is our single point
on the wavefront going in.

1420
01:20:29,690 --> 01:20:33,710
And this gives rise to
a wave that comes out,

1421
01:20:33,710 --> 01:20:37,100
a spherical wave that comes
out, which is of this form.

1422
01:20:37,100 --> 01:20:39,950
It's the same as this
multiplied by the transmission

1423
01:20:39,950 --> 01:20:41,450
of the source.

1424
01:20:41,450 --> 01:20:46,312
And then this then radiates
like a spherical wave.

1425
01:20:46,312 --> 01:20:48,770
And then you're going to get
some distribution of amplitude

1426
01:20:48,770 --> 01:20:52,040
on this observation plane,
which is going to be exactly

1427
01:20:52,040 --> 01:20:54,120
the same as we had before.

1428
01:20:54,120 --> 01:20:55,360
And we've got this--

1429
01:20:55,360 --> 01:20:57,950
again, we've got
this parabolic phase

1430
01:20:57,950 --> 01:21:03,600
function that comes from
the Fresnel approximation.

1431
01:21:03,600 --> 01:21:08,920
So it assumes that x dashed y
dashed are much smaller than z.

1432
01:21:13,030 --> 01:21:16,320
So that's for a single point.

1433
01:21:16,320 --> 01:21:19,610
And now we do-- now we
integrate over all of these.

1434
01:21:19,610 --> 01:21:22,790
And the final result
for the amplitude

1435
01:21:22,790 --> 01:21:26,590
we'll see on the screen is
given by this expression then.

1436
01:21:26,590 --> 01:21:29,090
So this is the
incident wave, this

1437
01:21:29,090 --> 01:21:31,760
is the transmission
of the screen,

1438
01:21:31,760 --> 01:21:35,210
and this is the
bit that describes

1439
01:21:35,210 --> 01:21:38,420
the propagation of
the spherical waves

1440
01:21:38,420 --> 01:21:41,600
according to the
Fresnel approximation.

1441
01:21:45,940 --> 01:21:46,760
So yeah.

1442
01:21:46,760 --> 01:21:51,170
So this result is known as the
Fresnel diffraction integral,

1443
01:21:51,170 --> 01:21:53,090
or simply the Fresnel integral.

1444
01:21:53,090 --> 01:22:00,850
And so I mentioned earlier that
this approximation, replacing

1445
01:22:00,850 --> 01:22:02,690
the square root by
the squared terms,

1446
01:22:02,690 --> 01:22:05,000
is called the Fresnel
approximation.

1447
01:22:05,000 --> 01:22:06,260
So all of these--

1448
01:22:06,260 --> 01:22:08,300
Fresnel gets the name
of all of these things.

1449
01:22:15,490 --> 01:22:15,990
Yeah.

1450
01:22:15,990 --> 01:22:20,170
This is just pointing out that
this expression we've just

1451
01:22:20,170 --> 01:22:24,010
written down, the form
of this is exactly

1452
01:22:24,010 --> 01:22:27,620
the form of a convolution.

1453
01:22:27,620 --> 01:22:33,400
And so that's a very
important result.

1454
01:22:33,400 --> 01:22:36,800
A convolution, of course, means
it's space invariant, doesn't

1455
01:22:36,800 --> 01:22:37,300
it?

1456
01:22:37,300 --> 01:22:41,350
It means that you
can write this--

1457
01:22:41,350 --> 01:22:45,100
it's what goes in convolved
with something which is

1458
01:22:45,100 --> 01:22:48,140
independent of the coordinates.

1459
01:22:48,140 --> 01:22:51,530
And so if you think of
this as a convolution,

1460
01:22:51,530 --> 01:22:56,080
the thing you have to
convolve with is this thing.

1461
01:22:56,080 --> 01:23:01,900
So you can think of
this as being like--

1462
01:23:01,900 --> 01:23:03,640
well, it's called
the point spread

1463
01:23:03,640 --> 01:23:06,260
function or the amplitude
point spread function.

1464
01:23:06,260 --> 01:23:12,310
So if you put in some amplitude
and convolve it with this,

1465
01:23:12,310 --> 01:23:13,720
then you'll get something out.

1466
01:23:27,100 --> 01:23:29,560
So this is explaining
what I just said.

1467
01:23:29,560 --> 01:23:33,610
So this is the sort
of behavior that you

1468
01:23:33,610 --> 01:23:38,410
think of when you're looking
at a mechanical or electrical

1469
01:23:38,410 --> 01:23:38,980
systems.

1470
01:23:38,980 --> 01:23:42,910
You can think of a system
of being like a black box,

1471
01:23:42,910 --> 01:23:45,490
or in this case, a blue box.

1472
01:23:45,490 --> 01:23:49,420
And something goes in
and something comes out.

1473
01:23:49,420 --> 01:23:53,650
And we don't have to worry
too much about what's in here.

1474
01:23:53,650 --> 01:23:56,920
But you can express what goes
in, of course, either in terms

1475
01:23:56,920 --> 01:24:01,540
of some sort of impulse
response function,

1476
01:24:01,540 --> 01:24:04,150
or as we'll come to
in later lectures,

1477
01:24:04,150 --> 01:24:08,390
as some sort of transfer
function-type behavior.

1478
01:24:08,390 --> 01:24:11,650
But at the moment, we're
doing in the spatial domain.

1479
01:24:11,650 --> 01:24:17,120
So if we convolve this with
this point spread function,

1480
01:24:17,120 --> 01:24:19,630
then we'll get this
output, like this.

1481
01:24:26,900 --> 01:24:30,970
Now we go on to do
some examples of this.

1482
01:24:30,970 --> 01:24:34,450
And this is probably the
most important example--

1483
01:24:34,450 --> 01:24:36,250
comes up all the time--

1484
01:24:36,250 --> 01:24:41,660
because lots of optical
components are circular.

1485
01:24:41,660 --> 01:24:47,240
So lenses and so on, very often
they've got this circular form.

1486
01:24:47,240 --> 01:24:51,980
And so this comes up
time and time again.

1487
01:24:51,980 --> 01:24:56,890
And so all we've got to
do now then is we put in--

1488
01:24:56,890 --> 01:25:02,570
we say that our amplitude
after the light's gone

1489
01:25:02,570 --> 01:25:09,200
through this circular aperture
is 1 inside the circle

1490
01:25:09,200 --> 01:25:10,845
and 0 outside the circle.

1491
01:25:13,540 --> 01:25:16,960
And so that Fresnel
diffraction integral

1492
01:25:16,960 --> 01:25:20,140
that we have to work out
for this circular aperture

1493
01:25:20,140 --> 01:25:25,150
is just 1 inside and 0 outside.

1494
01:25:25,150 --> 01:25:28,360
So it's just-- you
just have to do

1495
01:25:28,360 --> 01:25:30,430
the integral over the inside.

1496
01:25:30,430 --> 01:25:34,390
So this is just over
the values of x and y

1497
01:25:34,390 --> 01:25:36,820
that are inside the
bright part here.

1498
01:25:39,330 --> 01:25:43,560
Now, so unfortunately, that
integral is a bit hard to do.

1499
01:25:43,560 --> 01:25:47,670
So we're not going to do it
at the moment in general.

1500
01:25:47,670 --> 01:25:52,710
But we will do a very simple
example, a special case

1501
01:25:52,710 --> 01:25:55,510
of what happens along the axis.

1502
01:25:55,510 --> 01:26:03,410
So if we put x and y
both zero, now we get--

1503
01:26:03,410 --> 01:26:07,670
we put x dashed equals
0, y dashed equals 0.

1504
01:26:07,670 --> 01:26:10,560
The integral comes down to this.

1505
01:26:10,560 --> 01:26:14,450
And now that is an
integral we can do.

1506
01:26:14,450 --> 01:26:16,625
And it's an integral that
probably a lot of you

1507
01:26:16,625 --> 01:26:18,410
have come across before.

1508
01:26:18,410 --> 01:26:22,740
The way to solve that is to
go into polar coordinates.

1509
01:26:22,740 --> 01:26:27,390
And so if you change
into polar coordinates,

1510
01:26:27,390 --> 01:26:29,730
you've got a theta and a phi--

1511
01:26:29,730 --> 01:26:33,030
a theta and a rho part.

1512
01:26:33,030 --> 01:26:36,230
The theta part is just an
integral of a constant,

1513
01:26:36,230 --> 01:26:38,640
so that just becomes 2 pi.

1514
01:26:38,640 --> 01:26:42,760
And the integral of this
other thing you can now do.

1515
01:26:42,760 --> 01:26:46,310
Integral of rho e to
the i rho squared over

1516
01:26:46,310 --> 01:26:49,400
something you can do
because the rho outside

1517
01:26:49,400 --> 01:26:54,550
is the differential
of the bit inside.

1518
01:26:54,550 --> 01:26:58,270
So to actually do
that properly, you

1519
01:26:58,270 --> 01:27:01,310
can make a substitution then.

1520
01:27:01,310 --> 01:27:04,910
Make a substitution that
xi is equal to this thing

1521
01:27:04,910 --> 01:27:06,740
inside here.

1522
01:27:06,740 --> 01:27:09,980
And after you've made
that substitution,

1523
01:27:09,980 --> 01:27:14,690
you now just get integral
of e to the i xi.

1524
01:27:14,690 --> 01:27:20,690
So that's something
you can just do.

1525
01:27:20,690 --> 01:27:22,520
And you get an answer.

1526
01:27:34,630 --> 01:27:35,130
Yeah.

1527
01:27:35,130 --> 01:27:36,660
Notice that you get--

1528
01:27:36,660 --> 01:27:41,490
so this final result
here, this term

1529
01:27:41,490 --> 01:27:45,510
here is just the phase
term as a function of z.

1530
01:27:45,510 --> 01:27:49,320
So that's just what you would
get if it was just a wave

1531
01:27:49,320 --> 01:27:51,570
moving in free space.

1532
01:27:51,570 --> 01:27:54,210
And then these other
two terms, notice,

1533
01:27:54,210 --> 01:27:55,920
they're exactly the same form.

1534
01:27:55,920 --> 01:28:00,300
This is an e to the i
something, and this is a sine

1535
01:28:00,300 --> 01:28:01,410
of exactly the same thing.

1536
01:28:08,120 --> 01:28:10,920
So that's what we get.

1537
01:28:10,920 --> 01:28:13,470
As I say, if you
go back to this,

1538
01:28:13,470 --> 01:28:15,360
if you wanted to do
this integral here,

1539
01:28:15,360 --> 01:28:17,670
it's a bit more complicated.

1540
01:28:17,670 --> 01:28:20,870
And so we're not going
to do that at the moment.

1541
01:28:20,870 --> 01:28:25,488
But I guess eventually we
have to do something about it.

1542
01:28:25,488 --> 01:28:28,103
GEORGE BARBASTATHIS: [INAUDIBLE]

1543
01:28:28,103 --> 01:28:29,020
COLIN SHEPPARD: Right.

1544
01:28:29,020 --> 01:28:33,520
So if we look at this
amplitude along the axis,

1545
01:28:33,520 --> 01:28:37,750
then, you can see it's got this
sinusoidal sort of behavior.

1546
01:28:37,750 --> 01:28:40,840
If you looked at the
intensity, this would just go,

1547
01:28:40,840 --> 01:28:42,850
and you just get sine
squared of something.

1548
01:28:42,850 --> 01:28:47,350
Sine squared as a function
of a constant over z.

1549
01:28:47,350 --> 01:28:50,620
So as it points out
here, then, it's

1550
01:28:50,620 --> 01:28:54,820
going to have peaks
and nulls, maxima

1551
01:28:54,820 --> 01:28:59,650
and minima, corresponding to
these particular values of z.

1552
01:28:59,650 --> 01:29:04,210
So you'll get a
sinusoidal variation

1553
01:29:04,210 --> 01:29:08,140
in amplitude along the axis.

1554
01:29:08,140 --> 01:29:10,465
The other thing
to note is that--

1555
01:29:13,640 --> 01:29:14,660
where's it gone?

1556
01:29:19,980 --> 01:29:21,450
Yeah.

1557
01:29:21,450 --> 01:29:26,790
Notice that this z and
this z canceled out.

1558
01:29:26,790 --> 01:29:32,700
This z here appeared when
we made this substitution.

1559
01:29:32,700 --> 01:29:36,960
And so these z's finally cancel.

1560
01:29:36,960 --> 01:29:41,940
And so the final result
has got no z in it.

1561
01:29:41,940 --> 01:29:45,570
Sorry, no z in the
modulus part term.

1562
01:29:45,570 --> 01:29:48,450
So you've got this
series of fringes

1563
01:29:48,450 --> 01:29:54,060
which are of constant
amplitude however

1564
01:29:54,060 --> 01:29:56,340
far away from this
aperture you go,

1565
01:29:56,340 --> 01:29:58,860
which might seem
rather surprising.

1566
01:29:58,860 --> 01:30:02,280
You might think that as you
got closer to the aperture,

1567
01:30:02,280 --> 01:30:06,540
it would get
brighter, might you?

1568
01:30:06,540 --> 01:30:07,800
But it doesn't, it seems.

1569
01:30:10,688 --> 01:30:13,230
I'm just trying to think quickly
why that should be the case.

1570
01:30:17,010 --> 01:30:18,610
But I'm sure it is true.

1571
01:30:22,400 --> 01:30:24,950
GEORGE BARBASTATHIS: Notice
also the pi over 2 phase shift

1572
01:30:24,950 --> 01:30:28,440
has disappeared.

1573
01:30:28,440 --> 01:30:30,415
It got eaten up by the sine.

1574
01:30:30,415 --> 01:30:31,290
COLIN SHEPPARD: Yeah.

1575
01:30:31,290 --> 01:30:31,800
That's true.

1576
01:30:31,800 --> 01:30:32,130
Sorry.

1577
01:30:32,130 --> 01:30:33,040
I didn't mention that.

1578
01:30:33,040 --> 01:30:33,460
Yeah.

1579
01:30:33,460 --> 01:30:33,960
OK.

1580
01:30:33,960 --> 01:30:37,710
So when you put in the limits
of the integration, then

1581
01:30:37,710 --> 01:30:39,340
you've got this minus this.

1582
01:30:39,340 --> 01:30:41,550
And this we replaced by sine.

1583
01:30:41,550 --> 01:30:45,580
But of course we need to get
an extra i to get the sine.

1584
01:30:45,580 --> 01:30:46,080
Yeah.

1585
01:30:46,080 --> 01:30:49,260
So the pi over 2 phase
shift is also gone.

1586
01:30:49,260 --> 01:30:52,620
The other thing that perhaps
is worth mentioning also

1587
01:30:52,620 --> 01:30:56,760
is that when we got down
to this integral of course,

1588
01:30:56,760 --> 01:30:58,740
this is just the
Fourier transform of 1,

1589
01:30:58,740 --> 01:31:00,860
isn't it, within this region.

1590
01:31:00,860 --> 01:31:05,710
It's the Fourier transform
of a rect function.

1591
01:31:05,710 --> 01:31:10,780
And even if we looked at
more complicated example,

1592
01:31:10,780 --> 01:31:13,290
if you add this circular
symmetry, whatever

1593
01:31:13,290 --> 01:31:19,500
this amplitude transmittance
of this aperture had been,

1594
01:31:19,500 --> 01:31:22,620
all we'd have to do is to
do the Fourier transform

1595
01:31:22,620 --> 01:31:26,400
of that amplitude
transmittances of that aperture

1596
01:31:26,400 --> 01:31:30,810
as a function of
this coordinate xi.

1597
01:31:30,810 --> 01:31:34,320
So that's a sort of trick
you can use to calculate

1598
01:31:34,320 --> 01:31:36,600
these things easily sometimes.

1599
01:31:36,600 --> 01:31:39,630
I think we're here
for the movies now.

1600
01:31:39,630 --> 01:31:44,130
So MATLAB has calculated
these things for us.

1601
01:31:44,130 --> 01:31:46,740
And when I press the
button, it will go.

1602
01:31:46,740 --> 01:31:49,230
But I'm not going to press
it until you're ready, just

1603
01:31:49,230 --> 01:31:51,010
in case it's very quick.

1604
01:31:51,010 --> 01:31:54,150
So this is a big aperture.

1605
01:31:54,150 --> 01:31:56,700
So what you'll see
is how the wave

1606
01:31:56,700 --> 01:31:59,640
propagates from that aperture.

1607
01:31:59,640 --> 01:32:01,720
This is actually
quite a big aperture.

1608
01:32:01,720 --> 01:32:05,230
So you don't expect to see a
lot of diffraction in this case.

1609
01:32:05,230 --> 01:32:08,220
When we come to this one,
this is a small aperture.

1610
01:32:08,220 --> 01:32:10,920
And as we said before, the
smaller it is to start,

1611
01:32:10,920 --> 01:32:13,110
the more it's going to spread.

1612
01:32:13,110 --> 01:32:16,440
So you're going to see a
lot more effect on that one.

1613
01:32:16,440 --> 01:32:18,900
And so you have to
look quite closely

1614
01:32:18,900 --> 01:32:23,630
to see what happens
as it propagates.

1615
01:32:23,630 --> 01:32:26,900
So this is a radius
of 40 lambda.

1616
01:32:26,900 --> 01:32:29,120
Now you can see some sort
of fringes appearing,

1617
01:32:29,120 --> 01:32:35,230
and it becomes like a bright
halo around the edges.

1618
01:32:35,230 --> 01:32:36,760
And if you look
very closely, you'll

1619
01:32:36,760 --> 01:32:39,700
see quite a lot of
fine structure on this.

1620
01:32:42,780 --> 01:32:47,870
And then the small one,
this is 10 wavelengths.

1621
01:32:47,870 --> 01:32:50,530
So this one changes
a lot quicker.

1622
01:32:50,530 --> 01:32:53,598
You see it's doing
quite drastic things.

1623
01:32:53,598 --> 01:32:55,390
By the time it gets to
the end, you've even

1624
01:32:55,390 --> 01:32:56,820
got a black spot in the center.

1625
01:33:03,190 --> 01:33:05,730
The fact that, of course--

1626
01:33:08,600 --> 01:33:11,650
we said before that the
smaller the aperture,

1627
01:33:11,650 --> 01:33:14,770
the more it spreads, but
also the quicker it spreads.

1628
01:33:14,770 --> 01:33:16,780
So you can see that
actually this one

1629
01:33:16,780 --> 01:33:19,000
has got a lot further
in the development

1630
01:33:19,000 --> 01:33:20,920
of the pattern than this one.

1631
01:33:20,920 --> 01:33:24,820
This one still looks roughly
similar to what it started as.

1632
01:33:24,820 --> 01:33:27,024
This one is looking
very, very different.

1633
01:33:35,217 --> 01:33:36,550
It calls this the blinking spot.

1634
01:33:36,550 --> 01:33:38,550
I've never heard of it
called the blinking spot.

1635
01:33:38,550 --> 01:33:39,270
But anyway.

1636
01:33:42,120 --> 01:33:43,550
GEORGE BARBASTATHIS: [INAUDIBLE]

1637
01:33:43,550 --> 01:33:43,940
COLIN SHEPPARD: Well.

1638
01:33:43,940 --> 01:33:44,420
Yeah.

1639
01:33:44,420 --> 01:33:45,212
I'm not even sure--

1640
01:33:45,212 --> 01:33:48,110
I always think of
the Poisson spot

1641
01:33:48,110 --> 01:33:51,770
as being what you get
from an opaque object,

1642
01:33:51,770 --> 01:33:54,980
rather than for--

1643
01:33:54,980 --> 01:33:57,512
but maybe that's the Arago spot.

1644
01:33:57,512 --> 01:33:58,970
GEORGE BARBASTATHIS:
But Binet says

1645
01:33:58,970 --> 01:34:00,785
the two are the same, right?

1646
01:34:00,785 --> 01:34:01,660
COLIN SHEPPARD: Yeah.

1647
01:34:01,660 --> 01:34:02,200
Yeah.

1648
01:34:02,200 --> 01:34:03,770
Yeah.

1649
01:34:03,770 --> 01:34:04,270
OK.

1650
01:34:04,270 --> 01:34:07,140
So now the rectangular aperture.

1651
01:34:07,140 --> 01:34:11,980
The rectangular aperture--
actually it turns out you

1652
01:34:11,980 --> 01:34:13,810
can't get a simple
expression for what

1653
01:34:13,810 --> 01:34:17,500
happens along the axis
of the rectangular case,

1654
01:34:17,500 --> 01:34:19,660
unlike the circular case.

1655
01:34:19,660 --> 01:34:22,380
So that's probably why
George hasn't done it.

1656
01:34:22,380 --> 01:34:26,980
And so you can get expressions
in terms of Fresnel integrals

1657
01:34:26,980 --> 01:34:29,440
for everywhere, in
fact, for this case.

1658
01:34:29,440 --> 01:34:31,090
So the square case, then.

1659
01:34:31,090 --> 01:34:32,530
Let's see what that does.

1660
01:34:32,530 --> 01:34:33,970
The big square.

1661
01:34:33,970 --> 01:34:38,790
Again, you can see-- you can
see some sort of fringes.

1662
01:34:38,790 --> 01:34:41,380
And by the time it's
got to the end here,

1663
01:34:41,380 --> 01:34:45,460
you can see bright lines around
the edges, and also bright

1664
01:34:45,460 --> 01:34:48,480
spots in the four corners.

1665
01:34:48,480 --> 01:34:54,130
And then if we look
at the small square,

1666
01:34:54,130 --> 01:34:56,470
again, it's changing
more quickly.

1667
01:34:56,470 --> 01:34:59,710
And you'll notice that
we've developed by then--

1668
01:34:59,710 --> 01:35:03,010
we've got quite strong sort
of streaking along these two

1669
01:35:03,010 --> 01:35:04,900
axes here.

1670
01:35:04,900 --> 01:35:10,140
And in fact, you see
this final pattern there

1671
01:35:10,140 --> 01:35:18,250
that is the diffraction
pattern of the square aperture.

1672
01:35:18,250 --> 01:35:19,880
What does it say?

1673
01:35:19,880 --> 01:35:25,107
Well, we haven't done Fraunhofer
diffraction yet, have we?

1674
01:35:25,107 --> 01:35:26,440
GEORGE BARBASTATHIS: [INAUDIBLE]

1675
01:35:26,440 --> 01:35:28,600
COLIN SHEPPARD: This is
introduction to it. yeah.

1676
01:35:28,600 --> 01:35:32,860
This is an example of
Fraunhofer diffraction then.

1677
01:35:32,860 --> 01:35:34,810
If you're a long way
away, you get what's

1678
01:35:34,810 --> 01:35:36,460
called Fraunhofer diffraction.

1679
01:35:36,460 --> 01:35:38,740
This is what it looks like.

1680
01:35:38,740 --> 01:35:41,920
And as we're going
to show later,

1681
01:35:41,920 --> 01:35:45,190
this is very simply
related to the Fourier

1682
01:35:45,190 --> 01:35:48,490
transform of the
structure, the object

1683
01:35:48,490 --> 01:35:51,260
that we're going to look at.

1684
01:35:51,260 --> 01:35:51,810
OK.

1685
01:35:51,810 --> 01:35:52,310
Yeah.

1686
01:35:52,310 --> 01:35:57,440
So this case here, eventually
this one would look like this.

1687
01:35:57,440 --> 01:36:00,760
But we haven't gone
anywhere near far enough

1688
01:36:00,760 --> 01:36:02,650
for it to look like that.

1689
01:36:02,650 --> 01:36:04,720
But eventually it
would get like that.

1690
01:36:04,720 --> 01:36:08,710
The distance have to go depends
on how big the aperture is.

1691
01:36:13,510 --> 01:36:15,360
End of show.