1 00:00:02,490 --> 00:00:04,830 The following content is provided under a Creative 2 00:00:04,830 --> 00:00:06,250 Commons license. 3 00:00:06,250 --> 00:00:08,460 Your support will help MIT OpenCourseWare 4 00:00:08,460 --> 00:00:12,550 continue to offer high-quality educational resources for free. 5 00:00:12,550 --> 00:00:15,090 To make a donation or to view additional materials 6 00:00:15,090 --> 00:00:19,020 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:19,020 --> 00:00:20,280 at ocw.mit.edu. 8 00:00:23,784 --> 00:00:24,450 YEN-JIE LEE: OK. 9 00:00:24,450 --> 00:00:29,010 So welcome back, everybody, to 8.03. 10 00:00:29,010 --> 00:00:31,200 So before we start the lecture today, 11 00:00:31,200 --> 00:00:34,710 we will give you, as usual, a short review on what we 12 00:00:34,710 --> 00:00:39,600 have learned, and also, an introduction about what we 13 00:00:39,600 --> 00:00:42,490 are going to learn today. 14 00:00:42,490 --> 00:00:45,410 So last lecture, we were discussing 15 00:00:45,410 --> 00:00:50,760 an interesting phenomenon, which is seeing film interference 16 00:00:50,760 --> 00:00:51,660 pattern. 17 00:00:51,660 --> 00:00:54,120 As you can see from this slide, we 18 00:00:54,120 --> 00:00:59,690 were wondering why the soap bubbles are colorful. 19 00:00:59,690 --> 00:01:02,160 And in the end of the class, we actually 20 00:01:02,160 --> 00:01:07,080 recognized that the reason why the soap bubbles are colorful 21 00:01:07,080 --> 00:01:10,130 is because of the interference phenomenon 22 00:01:10,130 --> 00:01:16,470 between the refracted light on the bubble. 23 00:01:16,470 --> 00:01:20,210 One puzzle path is that the light goes into the-- 24 00:01:20,210 --> 00:01:25,330 goes refracted directly from the surface of the soap film. 25 00:01:25,330 --> 00:01:29,250 The other possible optical path is 26 00:01:29,250 --> 00:01:34,290 to get refracted by the inner surface of the film. 27 00:01:34,290 --> 00:01:37,410 Therefore, the interference between these two paths 28 00:01:37,410 --> 00:01:44,390 actually created a colorful pattern on the bubble. 29 00:01:44,390 --> 00:01:51,790 So we also learned about how thick is the soap film. 30 00:01:51,790 --> 00:01:54,670 And I think just a quick reminder, actually, 31 00:01:54,670 --> 00:02:01,300 we concluded that in order to see a colorful pattern, 32 00:02:01,300 --> 00:02:05,020 the thickness of the wall or, say, the film, 33 00:02:05,020 --> 00:02:09,269 should be something like in the order of 100 nanometer. 34 00:02:09,269 --> 00:02:10,810 So that's actually pretty remarkable, 35 00:02:10,810 --> 00:02:14,970 because that's already in the order of the size of a virus. 36 00:02:14,970 --> 00:02:15,540 OK. 37 00:02:15,540 --> 00:02:19,120 OK that's actually pretty cool. 38 00:02:19,120 --> 00:02:22,210 So what are we going to do today? 39 00:02:22,210 --> 00:02:24,940 What we are going to do today is to continue the discussion 40 00:02:24,940 --> 00:02:28,300 that all kinds of different phenomenons, 41 00:02:28,300 --> 00:02:31,810 which can be explained by interference. 42 00:02:31,810 --> 00:02:36,580 We will learn interference phenomenon with a double slit 43 00:02:36,580 --> 00:02:42,820 experiment and using, for example, laser or water, 44 00:02:42,820 --> 00:02:46,040 and which I have a water tank here, 45 00:02:46,040 --> 00:02:48,740 which I will show you the interference pattern. 46 00:02:48,740 --> 00:02:51,820 And also, the second thing we are going to learn today 47 00:02:51,820 --> 00:02:58,650 is how does a phased radar actually works. 48 00:02:58,650 --> 00:02:59,520 OK? 49 00:02:59,520 --> 00:03:03,470 So by the end of the lecture today, you 50 00:03:03,470 --> 00:03:07,200 should be able to learn why we should construct 51 00:03:07,200 --> 00:03:13,420 the radar in the way and how to actually 52 00:03:13,420 --> 00:03:17,140 focus on the electromagnetic wave 53 00:03:17,140 --> 00:03:19,390 to work one specific direction. 54 00:03:19,390 --> 00:03:21,760 So that's essentially what we are going to learn today. 55 00:03:21,760 --> 00:03:24,130 The third goal is that we are going 56 00:03:24,130 --> 00:03:29,590 to make a connection to quantum mechanics from the lecture 57 00:03:29,590 --> 00:03:31,250 today. 58 00:03:31,250 --> 00:03:31,750 All right. 59 00:03:31,750 --> 00:03:34,570 So let's immediately get started. 60 00:03:34,570 --> 00:03:39,520 So before we start the discussion of a double slit 61 00:03:39,520 --> 00:03:42,580 experiment, I would like to remind everybody 62 00:03:42,580 --> 00:03:46,510 about Huygens' Principle, which you may already learned it 63 00:03:46,510 --> 00:03:49,970 from 8.02 or in the high school days. 64 00:03:49,970 --> 00:03:52,490 So what essentially is this principle? 65 00:03:52,490 --> 00:03:55,630 So this principle is saying that if I 66 00:03:55,630 --> 00:04:02,980 take a look at all the points in the wavefront, basically, 67 00:04:02,980 --> 00:04:08,080 you can treat all those points on the wavefront a point 68 00:04:08,080 --> 00:04:09,040 source. 69 00:04:09,040 --> 00:04:11,770 And this point source, essentially, a point source 70 00:04:11,770 --> 00:04:14,830 of a spherical wave. 71 00:04:14,830 --> 00:04:19,390 And it's immediate from all the points on the wavefront. 72 00:04:19,390 --> 00:04:23,130 So you can see from this slide, basically, 73 00:04:23,130 --> 00:04:28,240 if we choose to focus on the yellow point on the wavefront, 74 00:04:28,240 --> 00:04:30,670 you can see that from each yellow point, 75 00:04:30,670 --> 00:04:37,900 you can actually treat that as a spherical wave point source. 76 00:04:37,900 --> 00:04:41,500 And then what you actually need to do in order 77 00:04:41,500 --> 00:04:45,610 to calculate what would be the total electric field, 78 00:04:45,610 --> 00:04:49,240 for example, is to add up all those contribution 79 00:04:49,240 --> 00:04:50,590 from each point. 80 00:04:50,590 --> 00:04:53,610 And then you will be able to actually explain 81 00:04:53,610 --> 00:04:59,530 the interference pattern, which we see in the experiment. 82 00:04:59,530 --> 00:05:05,200 You may wonder where is this Huygens' Principle coming from? 83 00:05:05,200 --> 00:05:08,130 And although we are not going to derive that directly 84 00:05:08,130 --> 00:05:11,380 in the lecture today, but I can actually safely 85 00:05:11,380 --> 00:05:14,530 tell you that essentially, it can be derived 86 00:05:14,530 --> 00:05:16,290 from Maxwell's equation. 87 00:05:16,290 --> 00:05:16,870 OK? 88 00:05:16,870 --> 00:05:19,300 I will link some document, which actually 89 00:05:19,300 --> 00:05:23,060 shows the proof of the principle on the website 90 00:05:23,060 --> 00:05:25,870 and for your reference. 91 00:05:25,870 --> 00:05:29,450 The other thing which you may or you may not know 92 00:05:29,450 --> 00:05:38,050 is that we are really lucky so that we can use this Huygens' 93 00:05:38,050 --> 00:05:40,890 Principle in our universe. 94 00:05:40,890 --> 00:05:42,220 Why is that? 95 00:05:42,220 --> 00:05:45,280 Because if you look at the mathematical proof 96 00:05:45,280 --> 00:05:51,310 of this principle, it is because the number of dimension, number 97 00:05:51,310 --> 00:05:54,880 of spatial dimension is odd, which 98 00:05:54,880 --> 00:05:56,830 is three in our universe-- 99 00:05:56,830 --> 00:05:59,610 Or in my universe also is yours, OK, 100 00:05:59,610 --> 00:06:02,300 [LAUGHS] happened to be yours, as well-- 101 00:06:02,300 --> 00:06:06,280 such that the Huygens' principle actually works. 102 00:06:06,280 --> 00:06:11,680 On the other hand, if the number of dimension is even, 103 00:06:11,680 --> 00:06:13,720 there's no Huygens' principle, actually. 104 00:06:13,720 --> 00:06:15,700 So that's pretty interesting in that we 105 00:06:15,700 --> 00:06:19,780 are really lucky that it actually works in our universe. 106 00:06:19,780 --> 00:06:24,580 But I will not go into detail in 8.03. 107 00:06:24,580 --> 00:06:28,180 So let's get started with a concrete example, which 108 00:06:28,180 --> 00:06:31,630 we would like to further investigate to understand 109 00:06:31,630 --> 00:06:34,120 the interference phenomena. 110 00:06:34,120 --> 00:06:36,370 And those will prepare ourselves to the understanding 111 00:06:36,370 --> 00:06:40,000 of the design of the radar, for example. 112 00:06:40,000 --> 00:06:40,630 All right. 113 00:06:40,630 --> 00:06:44,440 So suppose I have experimental set 114 00:06:44,440 --> 00:06:50,350 up here which contain a wall where on the wall, 115 00:06:50,350 --> 00:06:55,450 there are two slit, A and a B. The upper one is A. The lower 116 00:06:55,450 --> 00:06:58,120 one is B, as designed here. 117 00:06:58,120 --> 00:07:04,570 And from the left inside there's an insert plane wave 118 00:07:04,570 --> 00:07:08,460 with a wavelength lambda, which is showing here. 119 00:07:08,460 --> 00:07:11,780 And this plane wave, plane electromagnetic wave, 120 00:07:11,780 --> 00:07:15,760 or can be water wave, et cetera, essentially 121 00:07:15,760 --> 00:07:20,020 approaching the wall with these two slits there. 122 00:07:20,020 --> 00:07:24,040 And we were wondering what would be the resulting 123 00:07:24,040 --> 00:07:26,750 pattern on the screen. 124 00:07:26,750 --> 00:07:30,250 This screen is actually pretty far away 125 00:07:30,250 --> 00:07:34,720 from the experimental setup, the wall on the left-hand side. 126 00:07:34,720 --> 00:07:37,030 How far is that? 127 00:07:37,030 --> 00:07:41,700 The distance between the screen, which shows the resulting 128 00:07:41,700 --> 00:07:46,710 interference pattern, and that the wall is actually defined. 129 00:07:46,710 --> 00:07:48,260 It's actually given here. 130 00:07:48,260 --> 00:07:51,720 It's actually called L, capital L. 131 00:07:51,720 --> 00:07:56,660 And in this experimental setup L essentially pretty, pretty 132 00:07:56,660 --> 00:08:01,960 large and is much, much larger than the d, where d, small d, 133 00:08:01,960 --> 00:08:05,180 is the distance between the two slits. 134 00:08:05,180 --> 00:08:06,700 OK? 135 00:08:06,700 --> 00:08:10,810 So our job now is to understand what will be-- 136 00:08:10,810 --> 00:08:13,660 and to predict what is going to be the interference 137 00:08:13,660 --> 00:08:19,390 pattern coming from the electromagnetic wave which 138 00:08:19,390 --> 00:08:23,170 pass through point A and point B, 139 00:08:23,170 --> 00:08:26,030 and what is going to happen over, say, 140 00:08:26,030 --> 00:08:29,790 what would be the result which we will observe on the screen. 141 00:08:29,790 --> 00:08:30,600 OK? 142 00:08:30,600 --> 00:08:32,100 So the first thing which we can do 143 00:08:32,100 --> 00:08:36,190 is that we can now assign observer, 144 00:08:36,190 --> 00:08:39,809 which is called P, one of the point of interest 145 00:08:39,809 --> 00:08:42,840 on the screen, which is located here. 146 00:08:42,840 --> 00:08:45,160 And then we can link or, say, the 147 00:08:45,160 --> 00:08:49,530 connect the point A, which is the location of the first slit 148 00:08:49,530 --> 00:08:53,310 and the location of the second slit, which 149 00:08:53,310 --> 00:08:58,260 is called B. We can link those points together by a line. 150 00:08:58,260 --> 00:09:03,550 And that is actually denoted by AP and the BP, these two lines. 151 00:09:03,550 --> 00:09:08,080 Since we are talking about L, which is essentially very, very 152 00:09:08,080 --> 00:09:12,270 large, assuming that the distance, the length 153 00:09:12,270 --> 00:09:15,250 scale of the distance between the wall and the screen 154 00:09:15,250 --> 00:09:17,650 is much, much larger than the length 155 00:09:17,650 --> 00:09:21,490 scale of the distance between the two slit, which is d. 156 00:09:21,490 --> 00:09:28,495 Therefore, I can safely assume that AP and the BP 157 00:09:28,495 --> 00:09:31,260 are almost parallel to each other. 158 00:09:31,260 --> 00:09:32,960 Right? 159 00:09:32,960 --> 00:09:38,620 And I can also try to express the location of the P point 160 00:09:38,620 --> 00:09:44,140 by using the angle between BP and the horizontal direction. 161 00:09:44,140 --> 00:09:45,160 OK? 162 00:09:45,160 --> 00:09:47,070 And the horizontal direction is actually 163 00:09:47,070 --> 00:09:49,330 showing there's a dash line here. 164 00:09:49,330 --> 00:09:52,300 And the angle between BP and the horizontal direction 165 00:09:52,300 --> 00:09:54,990 it's called theta here. 166 00:09:54,990 --> 00:09:55,660 OK. 167 00:09:55,660 --> 00:10:01,380 So since AP and the BP are almost parallel to each other, 168 00:10:01,380 --> 00:10:05,850 I can now calculate what would be the optical path length 169 00:10:05,850 --> 00:10:09,600 difference between AP and the BP. 170 00:10:09,600 --> 00:10:10,440 Right? 171 00:10:10,440 --> 00:10:13,590 So in order to actually calculate 172 00:10:13,590 --> 00:10:19,170 the phase difference between the electromagnetic wave coming 173 00:10:19,170 --> 00:10:25,320 from slit A compared to slit B, I need to calculate-- 174 00:10:25,320 --> 00:10:33,865 again, like what we did last time-- optical path length 175 00:10:33,865 --> 00:10:34,811 difference. 176 00:10:37,500 --> 00:10:38,000 OK? 177 00:10:38,000 --> 00:10:44,150 In this case, I can call the distance between A and P, rA. 178 00:10:44,150 --> 00:10:47,930 And then I can also call the distance between B 179 00:10:47,930 --> 00:10:50,570 and the P, rB. 180 00:10:50,570 --> 00:10:53,270 Then the optical path length difference 181 00:10:53,270 --> 00:10:57,405 is called rB minus rA. 182 00:10:57,405 --> 00:10:58,970 And then we can actually calculate 183 00:10:58,970 --> 00:11:04,400 that because we have already given you the angle between BP 184 00:11:04,400 --> 00:11:06,380 and the horizontal direction. 185 00:11:06,380 --> 00:11:08,900 And basically, we can safely conclude 186 00:11:08,900 --> 00:11:14,780 that the path length difference is actually this line here. 187 00:11:14,780 --> 00:11:17,900 Therefore, I can actually calculate and get 188 00:11:17,900 --> 00:11:19,780 the optical path length difference, 189 00:11:19,780 --> 00:11:25,850 the difference between rB and the rA to be d sine theta. 190 00:11:25,850 --> 00:11:27,310 OK? 191 00:11:27,310 --> 00:11:30,650 Once we have that, it's actually pretty straightforward 192 00:11:30,650 --> 00:11:33,666 to calculate what would be the phase difference. 193 00:11:37,560 --> 00:11:44,940 The phase difference between the field coming from slit A, which 194 00:11:44,940 --> 00:11:48,640 I will call it EA here, and the field 195 00:11:48,640 --> 00:11:55,790 coming from the slit B, which I will call it EB here. 196 00:11:55,790 --> 00:12:00,750 The phase difference, as you define a lot of time 197 00:12:00,750 --> 00:12:06,390 to be delta, delta can be calculated by the optical path 198 00:12:06,390 --> 00:12:10,460 length difference, d sin theta, divided 199 00:12:10,460 --> 00:12:16,170 by lambda, which essentially telling you how many period 200 00:12:16,170 --> 00:12:23,130 have passed when the light have to actually overcome this-- 201 00:12:23,130 --> 00:12:27,396 or say have to pass through this optical path length difference. 202 00:12:27,396 --> 00:12:28,770 And, of course, these things need 203 00:12:28,770 --> 00:12:32,930 to be modified by 2 pi in order to translate 204 00:12:32,930 --> 00:12:35,970 from a number of period to a phase difference. 205 00:12:35,970 --> 00:12:41,220 Therefore, you get the phase difference between AP and BP 206 00:12:41,220 --> 00:12:44,280 to be delta equal to d sine theta 207 00:12:44,280 --> 00:12:48,520 divided by lambda times 2 pi. 208 00:12:48,520 --> 00:12:49,020 OK. 209 00:12:49,020 --> 00:12:52,020 So you can see that all those calculations 210 00:12:52,020 --> 00:12:53,290 are pretty straightforward. 211 00:12:53,290 --> 00:12:57,885 Maybe you have already seen that before in an earlier class. 212 00:12:57,885 --> 00:13:01,110 But what I want to say is that it is actually 213 00:13:01,110 --> 00:13:04,560 because of Huygens' Principle, such 214 00:13:04,560 --> 00:13:09,000 that you can't expect something which will show up at point P, 215 00:13:09,000 --> 00:13:11,790 right? 216 00:13:11,790 --> 00:13:13,920 If you don't have Huygens' Principle 217 00:13:13,920 --> 00:13:15,690 what is going to happen? 218 00:13:15,690 --> 00:13:19,380 What is going to happen is that the light 219 00:13:19,380 --> 00:13:23,100 passing through this slit will just go straight. 220 00:13:23,100 --> 00:13:25,510 And they will never overlap each other. 221 00:13:25,510 --> 00:13:26,010 OK? 222 00:13:26,010 --> 00:13:33,920 So that's actually why, because of the Huygens' principle, 223 00:13:33,920 --> 00:13:38,820 all the points on the wavefront are treated as a point 224 00:13:38,820 --> 00:13:41,570 source of a spherical wave. 225 00:13:41,570 --> 00:13:42,090 OK? 226 00:13:42,090 --> 00:13:46,140 So that is essentially why you can expect that something will 227 00:13:46,140 --> 00:13:51,720 hit the P point, which is because, in this case, 228 00:13:51,720 --> 00:13:54,810 we have two points, two point source. 229 00:13:54,810 --> 00:14:00,420 And they are emitting spherical waves 230 00:14:00,420 --> 00:14:02,390 coming from these two points. 231 00:14:02,390 --> 00:14:02,970 OK? 232 00:14:02,970 --> 00:14:06,600 So it is really because of Huygens' Principle, which 233 00:14:06,600 --> 00:14:13,380 applies here, such that we can actually observe the phenomenon 234 00:14:13,380 --> 00:14:15,510 at the P. And now, we have managed 235 00:14:15,510 --> 00:14:17,550 to calculate the phase difference, 236 00:14:17,550 --> 00:14:20,670 which is delta, presented here. 237 00:14:20,670 --> 00:14:26,620 So what the next question is, what would be the intensity? 238 00:14:26,620 --> 00:14:32,200 Since we have already calculated the phase difference delta, 239 00:14:32,200 --> 00:14:35,720 what would be the intensity observed at P? 240 00:14:35,720 --> 00:14:40,370 So for that, we have already prepared ourselves 241 00:14:40,370 --> 00:14:42,580 from the last few lectures. 242 00:14:42,580 --> 00:14:44,320 So now, we can actually calculate 243 00:14:44,320 --> 00:14:47,610 what would be the total E. The total E will 244 00:14:47,610 --> 00:14:54,130 be equal to EA plus EB. 245 00:14:54,130 --> 00:14:57,350 And here, I'm going to use complex notation just 246 00:14:57,350 --> 00:14:59,010 for simplicity. 247 00:14:59,010 --> 00:15:01,940 And basically, you can rewrite EA 248 00:15:01,940 --> 00:15:14,650 and the EB as E0 exponential i omega t minus k times rA 249 00:15:14,650 --> 00:15:25,160 plus E0 exponential i omega t minus k rB. 250 00:15:25,160 --> 00:15:29,120 The first term is actually telling you the contribution 251 00:15:29,120 --> 00:15:34,610 from the first slit, slit A. And the second term is actually 252 00:15:34,610 --> 00:15:39,832 telling you the contribution coming from slit B. 253 00:15:39,832 --> 00:15:43,910 In this set up, I'm telling you that I have the plane 254 00:15:43,910 --> 00:15:48,080 wave coming from the left hand side of the experiment 255 00:15:48,080 --> 00:15:50,330 and actually hitting the wall. 256 00:15:50,330 --> 00:15:52,420 And you can see that from the drawing. 257 00:15:52,420 --> 00:15:57,525 Actually, the wavefront, essentially, 258 00:15:57,525 --> 00:16:02,450 actually telling you that the direction of the electric field 259 00:16:02,450 --> 00:16:08,960 is actually in the Z direction in my coordinate system shown 260 00:16:08,960 --> 00:16:09,980 on the board. 261 00:16:09,980 --> 00:16:14,180 So basically, the Z direction is actually pointing to you guys. 262 00:16:14,180 --> 00:16:18,020 And that means the electric field is actually 263 00:16:18,020 --> 00:16:20,120 oscillating in this direction. 264 00:16:20,120 --> 00:16:20,720 OK? 265 00:16:20,720 --> 00:16:24,140 So therefore, I have to be careful of those vectors. 266 00:16:24,140 --> 00:16:26,990 So therefore, I need to give it other direction. 267 00:16:26,990 --> 00:16:31,360 And in this case, it's actually the Z direction. 268 00:16:31,360 --> 00:16:34,280 And also, you can see that the amplitude is actually 269 00:16:34,280 --> 00:16:41,620 denoted by E0 because I always assuming that both slit have 270 00:16:41,620 --> 00:16:44,120 the same finite width. 271 00:16:44,120 --> 00:16:47,060 For the moment, ignore the width of the slit. 272 00:16:47,060 --> 00:16:49,740 And also, they are coming from the same plane wave. 273 00:16:49,740 --> 00:16:54,980 Therefore, the amplitude is all denoted by E0. 274 00:16:54,980 --> 00:16:55,820 OK? 275 00:16:55,820 --> 00:16:58,060 So now, I have the expression here. 276 00:16:58,060 --> 00:17:01,610 And I can now go ahead and simplify this expression 277 00:17:01,610 --> 00:17:04,020 and rewrite that in this form. 278 00:17:04,020 --> 00:17:06,190 So I can now extract the E0. 279 00:17:06,190 --> 00:17:09,349 And also, I extract the common factors here, 280 00:17:09,349 --> 00:17:15,950 which essentially the exponential i omega t 281 00:17:15,950 --> 00:17:19,235 and also, minus k rA. 282 00:17:19,235 --> 00:17:22,310 I can actually factorize some part 283 00:17:22,310 --> 00:17:25,220 of the exponential function out. 284 00:17:25,220 --> 00:17:27,869 So the choice I made is that I actually 285 00:17:27,869 --> 00:17:33,320 could factorize out exponential i omega t minus k times r. 286 00:17:33,320 --> 00:17:34,670 Basically, I take these out. 287 00:17:34,670 --> 00:17:41,900 And I get this term showing here, omega t minus k rA. 288 00:17:41,900 --> 00:17:43,860 I take this out. 289 00:17:43,860 --> 00:17:46,760 Then basically, what you are doing to get inside 290 00:17:46,760 --> 00:17:52,770 will be 1 plus exponential minus i delta, actually. 291 00:17:55,656 --> 00:17:57,100 times z. 292 00:17:57,100 --> 00:17:57,920 OK? 293 00:17:57,920 --> 00:17:59,240 Why is that delta? 294 00:17:59,240 --> 00:18:04,010 Because once you factorize out or take out exponential i omega 295 00:18:04,010 --> 00:18:07,310 t minus k rA, basically, you are left 296 00:18:07,310 --> 00:18:13,200 with something proportional to exponential i minus k 297 00:18:13,200 --> 00:18:15,600 rB minus rA, right? 298 00:18:15,600 --> 00:18:20,310 And that is actually the optical path length difference here. 299 00:18:20,310 --> 00:18:25,650 And also, of course, you can always rewrite lambda 300 00:18:25,650 --> 00:18:27,560 over 2 pi, right? 301 00:18:27,560 --> 00:18:34,800 Basically, you write this to be k times d times theta. 302 00:18:34,800 --> 00:18:35,300 Right? 303 00:18:35,300 --> 00:18:39,740 So therefore, you can actually immediately identify 304 00:18:39,740 --> 00:18:43,830 the second term is to essentially exponential 305 00:18:43,830 --> 00:18:45,670 minus i delta. 306 00:18:45,670 --> 00:18:47,270 OK? 307 00:18:47,270 --> 00:18:49,330 Any questions here? 308 00:18:49,330 --> 00:18:50,280 OK. 309 00:18:50,280 --> 00:18:54,810 Because d sine theta essentially is just rB minus rA, 310 00:18:54,810 --> 00:18:58,530 therefore, I safely replace that by delta here. 311 00:18:58,530 --> 00:18:59,361 OK? 312 00:18:59,361 --> 00:18:59,860 All right. 313 00:18:59,860 --> 00:19:02,430 So since everybody's on the same page, 314 00:19:02,430 --> 00:19:07,440 I can now, again, factorize out not only the omega t 315 00:19:07,440 --> 00:19:11,320 minus kA term, but I can actually 316 00:19:11,320 --> 00:19:14,960 do a trick to factorize out, also, exponential 317 00:19:14,960 --> 00:19:18,200 minus i delta divided by 2 out. 318 00:19:18,200 --> 00:19:19,830 And basically, what I'm going to get 319 00:19:19,830 --> 00:19:26,480 is exponential i delta over 2 plus exponential minus i delta 320 00:19:26,480 --> 00:19:29,910 over 2. 321 00:19:29,910 --> 00:19:33,360 This reason why I'm doing this is because, huh, now, 322 00:19:33,360 --> 00:19:36,180 I have this term identified. 323 00:19:36,180 --> 00:19:40,210 And this is actually just 2 times cosine delta 324 00:19:40,210 --> 00:19:41,210 divided by 2. 325 00:19:41,210 --> 00:19:43,860 All right? 326 00:19:43,860 --> 00:19:45,240 OK? 327 00:19:45,240 --> 00:19:50,320 So now, I'm really pretty close to the intensity. 328 00:19:50,320 --> 00:19:52,590 So what would be the intensity coming out 329 00:19:52,590 --> 00:19:55,200 of this electric field? 330 00:19:55,200 --> 00:20:00,090 That is actually going to be average intensity, 331 00:20:00,090 --> 00:20:03,450 as we discussed last time in the lecture. 332 00:20:03,450 --> 00:20:08,220 The average intensity is proportional to square 333 00:20:08,220 --> 00:20:09,740 of E vector. 334 00:20:09,740 --> 00:20:10,650 Right? 335 00:20:10,650 --> 00:20:12,810 In the complex notation, how do we 336 00:20:12,810 --> 00:20:17,940 evaluate the absolute value of E vector square? 337 00:20:17,940 --> 00:20:20,260 In the complex notation, basically, you 338 00:20:20,260 --> 00:20:28,530 get basically, E times E star, where E is actually 339 00:20:28,530 --> 00:20:33,790 the amplitude, which is the size of the E vector, the magnitude 340 00:20:33,790 --> 00:20:35,560 of the E vector. 341 00:20:35,560 --> 00:20:37,800 Then, basically, you will see that this 342 00:20:37,800 --> 00:20:45,035 will be proportional to cosine square delta divided by 2. 343 00:20:45,035 --> 00:20:46,510 Right? 344 00:20:46,510 --> 00:20:51,010 Because you can see that if I calculate EE star, 345 00:20:51,010 --> 00:20:56,380 then all the terms with related to exponential i something 346 00:20:56,380 --> 00:20:57,730 actually got cancelled. 347 00:20:57,730 --> 00:20:58,480 Right? 348 00:20:58,480 --> 00:21:01,470 So therefore, you can see the "aha" very, very quickly. 349 00:21:01,470 --> 00:21:04,180 We can show that the intensity will 350 00:21:04,180 --> 00:21:08,590 be proportional to cosine square delta divided by 2, 351 00:21:08,590 --> 00:21:12,280 where delta is the phase difference 352 00:21:12,280 --> 00:21:15,160 between the first path and the second path. 353 00:21:15,160 --> 00:21:16,090 OK? 354 00:21:16,090 --> 00:21:17,080 Any questions so far? 355 00:21:20,890 --> 00:21:22,580 OK. 356 00:21:22,580 --> 00:21:30,050 So we can see that the intensity essentially changing really 357 00:21:30,050 --> 00:21:34,266 rapidly as a function of delta. 358 00:21:34,266 --> 00:21:35,040 Right? 359 00:21:35,040 --> 00:21:41,210 So when I have a situation where delta is equal to 0-- 360 00:21:41,210 --> 00:21:47,420 let's actually stop here a bit and enjoy what we have 361 00:21:47,420 --> 00:21:48,800 as you learn from here. 362 00:21:48,800 --> 00:21:49,310 All right? 363 00:21:49,310 --> 00:21:53,310 So if you have delta equal to 0, what does that mean? 364 00:21:53,310 --> 00:21:56,630 That means there's no phase difference 365 00:21:56,630 --> 00:21:58,810 between the first and second electric field. 366 00:21:58,810 --> 00:22:02,580 Therefore, when you add them together-- 367 00:22:02,580 --> 00:22:06,050 just a reminder about the notation we were using before. 368 00:22:06,050 --> 00:22:10,130 So if you draw the vector in a complex frame, what 369 00:22:10,130 --> 00:22:13,790 you are doing is that you are actually adding EA 370 00:22:13,790 --> 00:22:20,180 and the EB together in the most efficient way, right? 371 00:22:20,180 --> 00:22:23,270 Because the delta is equal to 0, the phase differences 372 00:22:23,270 --> 00:22:24,470 is equal to 0. 373 00:22:24,470 --> 00:22:27,610 Therefore, you are actually adding them in a straight line. 374 00:22:27,610 --> 00:22:28,430 OK? 375 00:22:28,430 --> 00:22:32,400 So that actually will give you the maxima intensity. 376 00:22:32,400 --> 00:22:37,111 Because when delta is equal to 0, cosine 0 is 1. 377 00:22:37,111 --> 00:22:37,610 Right? 378 00:22:37,610 --> 00:22:42,210 Therefore, you are reaching the maxima in the intensity. 379 00:22:42,210 --> 00:22:45,940 So now, I can always increase my delta 380 00:22:45,940 --> 00:22:49,940 until a number which is actually pi. 381 00:22:49,940 --> 00:22:52,610 What is going to happen is that if I still 382 00:22:52,610 --> 00:22:56,870 use the notation which I was using for the complex frame, 383 00:22:56,870 --> 00:22:58,650 what it does this is that, huh. 384 00:22:58,650 --> 00:23:04,670 Now, I am actually completely cancel the electric field, 385 00:23:04,670 --> 00:23:08,670 because the phase difference now is pi, right? 386 00:23:08,670 --> 00:23:11,480 So therefore, in the complex frame, 387 00:23:11,480 --> 00:23:15,710 you are adding the two vectors in way such 388 00:23:15,710 --> 00:23:18,140 that they completely cancel each other. 389 00:23:18,140 --> 00:23:20,300 The magnitude of the two vectors are 390 00:23:20,300 --> 00:23:24,020 the same, as shown here, which is actually E0, right? 391 00:23:24,020 --> 00:23:27,440 Therefore, what you are going to get, as you expect, 392 00:23:27,440 --> 00:23:32,030 is going to be 0, because they completely cancel. 393 00:23:32,030 --> 00:23:32,880 OK? 394 00:23:32,880 --> 00:23:36,770 You can also see that from this formula we did right here. 395 00:23:36,770 --> 00:23:41,000 When delta is equal to pi, then essentially, cosine pi over 2. 396 00:23:41,000 --> 00:23:44,210 Then you get intensity equal to 0. 397 00:23:44,210 --> 00:23:44,990 OK? 398 00:23:44,990 --> 00:23:47,330 Everybody accept this? 399 00:23:47,330 --> 00:23:48,790 All right. 400 00:23:48,790 --> 00:23:53,380 Now, I can still continue and increase the delta, 401 00:23:53,380 --> 00:23:56,750 for example, until delta is equal to 2 pi. 402 00:23:56,750 --> 00:23:58,370 Then you are getting this again. 403 00:23:58,370 --> 00:24:04,210 Basically, you have EA and the EB, again, line up each other. 404 00:24:04,210 --> 00:24:08,530 And the difference is that this EB actually 405 00:24:08,530 --> 00:24:14,890 rotated maybe 360 degree. 406 00:24:14,890 --> 00:24:20,540 And basically, you will see that, again, the intensity 407 00:24:20,540 --> 00:24:22,200 become the maxima again. 408 00:24:22,200 --> 00:24:23,350 OK? 409 00:24:23,350 --> 00:24:26,110 So that is actually how we can actually 410 00:24:26,110 --> 00:24:30,190 understand this result. And, of course, you 411 00:24:30,190 --> 00:24:34,450 can also go ahead and plot or simulate 412 00:24:34,450 --> 00:24:42,670 this result in the computer and really draw the amplitude, 413 00:24:42,670 --> 00:24:47,560 really draw the intensity as a function of angle here, 414 00:24:47,560 --> 00:24:49,630 or, say, the delta here. 415 00:24:49,630 --> 00:24:52,520 As you can see from here, that the intensity 416 00:24:52,520 --> 00:24:58,450 is actually reaching the maximum in the center. 417 00:24:58,450 --> 00:24:59,470 Why is that? 418 00:24:59,470 --> 00:25:04,500 In the center, if I have observer here 419 00:25:04,500 --> 00:25:07,270 in the center, what is going to happen 420 00:25:07,270 --> 00:25:10,210 is that the path length, optical path 421 00:25:10,210 --> 00:25:14,860 length between AP prong and the BP prong 422 00:25:14,860 --> 00:25:18,520 is going to be the same by symmetry, 423 00:25:18,520 --> 00:25:21,010 because it's actually in the optical center. 424 00:25:21,010 --> 00:25:26,900 Therefore, you will expect that delta is actually equal to 0. 425 00:25:26,900 --> 00:25:27,400 OK? 426 00:25:27,400 --> 00:25:31,170 So that's essentially why you see the maxima there. 427 00:25:31,170 --> 00:25:34,210 And if you start to move away from there, 428 00:25:34,210 --> 00:25:37,730 you will see that the delta start to increase. 429 00:25:37,730 --> 00:25:41,170 And at some point, you'll reach a minima, which 430 00:25:41,170 --> 00:25:44,630 you can see that on the plot. 431 00:25:44,630 --> 00:25:47,670 And that is actually because now, 432 00:25:47,670 --> 00:25:50,320 due to the increasing optical path length 433 00:25:50,320 --> 00:25:52,310 difference and the phase difference, 434 00:25:52,310 --> 00:25:54,550 the two electric field is starting 435 00:25:54,550 --> 00:25:57,640 to cancel each other, which actually produce 436 00:25:57,640 --> 00:26:00,220 the black pattern there. 437 00:26:00,220 --> 00:26:06,790 And finally, after it pass delta equal to pi, 438 00:26:06,790 --> 00:26:10,210 then these two electric fields start to work together again. 439 00:26:10,210 --> 00:26:10,910 All right? 440 00:26:10,910 --> 00:26:12,610 They're collaborating again. 441 00:26:12,610 --> 00:26:14,050 And you can see that again. 442 00:26:14,050 --> 00:26:18,010 You would get another maxima afterward. 443 00:26:18,010 --> 00:26:18,850 OK? 444 00:26:18,850 --> 00:26:23,260 And here, you can see that is actually my calculation. 445 00:26:23,260 --> 00:26:28,000 And, of course, I can do a demonstration to you 446 00:26:28,000 --> 00:26:29,560 to really show that this is actually 447 00:26:29,560 --> 00:26:35,760 what we are going to see based on the demonstration we 448 00:26:35,760 --> 00:26:37,060 are going to show here. 449 00:26:37,060 --> 00:26:39,260 So now, I am going to turn the light off. 450 00:26:44,630 --> 00:26:49,790 And here, I have a device which actually contain a water tank. 451 00:26:49,790 --> 00:26:53,210 And I need to actually turn this thing up. 452 00:26:59,310 --> 00:27:04,310 On the water tank I have two vibrator, which is actually 453 00:27:04,310 --> 00:27:07,090 acting as a point source. 454 00:27:07,090 --> 00:27:10,540 So basically, those vibrator vibrating up and down 455 00:27:10,540 --> 00:27:15,640 to create waves in this tank. 456 00:27:15,640 --> 00:27:16,240 OK? 457 00:27:16,240 --> 00:27:18,910 So basically, you can see that, huh, really, 458 00:27:18,910 --> 00:27:21,350 you have two point-like source. 459 00:27:21,350 --> 00:27:26,350 And you can see spherical waves is actually really generated 460 00:27:26,350 --> 00:27:30,220 and is really propagating away from the point source. 461 00:27:30,220 --> 00:27:31,540 OK? 462 00:27:31,540 --> 00:27:34,930 And what I can do now, you can see 463 00:27:34,930 --> 00:27:38,380 that this picture is really dynamic, because we 464 00:27:38,380 --> 00:27:40,510 can see that wavefront essentially moving 465 00:27:40,510 --> 00:27:41,900 as a function of time. 466 00:27:41,900 --> 00:27:45,700 So what I'm going to do is to really change 467 00:27:45,700 --> 00:27:48,730 the frequency of the light, which is actually 468 00:27:48,730 --> 00:27:52,280 shining on this water, so that you can actually 469 00:27:52,280 --> 00:27:55,150 see the fixed pattern here. 470 00:27:55,150 --> 00:28:01,520 And now, I am going to change the light frequency. 471 00:28:01,520 --> 00:28:05,870 You can see now I only shine the water tank 472 00:28:05,870 --> 00:28:10,460 at the specific time which match the speed of the propagation 473 00:28:10,460 --> 00:28:11,840 of the water wave. 474 00:28:11,840 --> 00:28:13,790 And you can see, aha, I've actually managed 475 00:28:13,790 --> 00:28:16,370 to freeze the wavefront. 476 00:28:16,370 --> 00:28:16,951 We see? 477 00:28:16,951 --> 00:28:17,450 OK. 478 00:28:17,450 --> 00:28:23,610 So you can see, now, really, you can see coming from the source, 479 00:28:23,610 --> 00:28:28,210 they are circular wavefront, which 480 00:28:28,210 --> 00:28:32,840 actually mimicking the result from Huygens' Principle. 481 00:28:32,840 --> 00:28:35,810 And you can see that they are complicated 482 00:28:35,810 --> 00:28:39,240 interference pattern forming. 483 00:28:39,240 --> 00:28:41,600 You can see that at some point they 484 00:28:41,600 --> 00:28:43,520 have constructive interference. 485 00:28:43,520 --> 00:28:46,100 If you focus on the central part, 486 00:28:46,100 --> 00:28:50,840 you can see that the maxima is actually reach there. 487 00:28:50,840 --> 00:28:53,720 On the other hand, if you move away, a little bit 488 00:28:53,720 --> 00:28:57,770 away from the center, you can see that really, the intensity 489 00:28:57,770 --> 00:28:58,550 drop. 490 00:28:58,550 --> 00:29:03,885 And at some point, you will also see that, OK, again, I 491 00:29:03,885 --> 00:29:06,190 am changing the procedure in such 492 00:29:06,190 --> 00:29:10,250 that the phase difference between the contribution 493 00:29:10,250 --> 00:29:14,320 of our source A and the B essentially equal to 2 pi. 494 00:29:14,320 --> 00:29:21,340 In that case, you will be able to see that another maxima is 495 00:29:21,340 --> 00:29:22,660 actually created again. 496 00:29:28,600 --> 00:29:30,660 So now, we can actually also show you 497 00:29:30,660 --> 00:29:37,890 that a lot, in fact, based on this glorious pattern, 498 00:29:37,890 --> 00:29:40,630 let's actually take a look at the projector here. 499 00:29:40,630 --> 00:29:47,280 So if I look at on the individual slide, which 500 00:29:47,280 --> 00:29:51,160 I have here, you can see that those are actually 501 00:29:51,160 --> 00:29:55,450 a point-light source and is creating a circular pattern. 502 00:29:55,450 --> 00:29:59,130 And now, I can actually overlap with two patterns together. 503 00:29:59,130 --> 00:30:09,270 And you can see that when I have the center of the two circles 504 00:30:09,270 --> 00:30:12,850 pretty close to each other, you can see that really, you 505 00:30:12,850 --> 00:30:14,610 have very small d. 506 00:30:14,610 --> 00:30:17,520 In this case, you have very small distance 507 00:30:17,520 --> 00:30:20,610 between source number one and number two. 508 00:30:20,610 --> 00:30:25,350 Then basically, based on our expression, 509 00:30:25,350 --> 00:30:30,390 so you can see that delta is equal to d sine theta divided 510 00:30:30,390 --> 00:30:32,820 by lambda times two pi, right? 511 00:30:32,820 --> 00:30:36,120 And you can actually calculate sine theta 512 00:30:36,120 --> 00:30:42,051 will be equal to delta divided by k times t. 513 00:30:42,051 --> 00:30:42,550 OK? 514 00:30:53,060 --> 00:30:56,890 When delta is equal to pi, that is going to give you 515 00:30:56,890 --> 00:31:01,030 a minima where, essentially, also showing here, 516 00:31:01,030 --> 00:31:04,510 the minima is shown as the black pattern here. 517 00:31:04,510 --> 00:31:05,240 OK? 518 00:31:05,240 --> 00:31:07,660 You can see from on here. 519 00:31:07,660 --> 00:31:10,270 So what this says, your formula is showing you 520 00:31:10,270 --> 00:31:15,280 that when I have d, which is very small, 521 00:31:15,280 --> 00:31:16,840 what is going to happen is that I'm 522 00:31:16,840 --> 00:31:22,870 going to get sine theta to be very large when d is actually 523 00:31:22,870 --> 00:31:23,770 very small. 524 00:31:23,770 --> 00:31:25,370 And that can be shown here. 525 00:31:25,370 --> 00:31:28,930 When I have d, which is the distance between the center 526 00:31:28,930 --> 00:31:32,510 of these two point source, very small, 527 00:31:32,510 --> 00:31:38,140 you can see that the place you get the minima 528 00:31:38,140 --> 00:31:44,710 is really far away from the center, which is actually here. 529 00:31:44,710 --> 00:31:45,550 OK? 530 00:31:45,550 --> 00:31:48,910 Now, what I'm going to do is to increase the distance 531 00:31:48,910 --> 00:31:50,200 between these two source. 532 00:31:50,200 --> 00:31:53,410 According to our position, what is going to happen 533 00:31:53,410 --> 00:32:00,220 is that the central maxima will decrease. 534 00:32:00,220 --> 00:32:04,810 The position where you get a minima will be moving closer 535 00:32:04,810 --> 00:32:07,870 to the center, according to that formula, 536 00:32:07,870 --> 00:32:10,210 because it's proportional to 1 over d. 537 00:32:10,210 --> 00:32:12,400 And we can do this really carefully 538 00:32:12,400 --> 00:32:14,900 to see if I can succeed. 539 00:32:14,900 --> 00:32:19,900 And you can see that really, when I am moving 540 00:32:19,900 --> 00:32:22,960 these two slides away from each other, 541 00:32:22,960 --> 00:32:25,590 you can see that the pattern is changing, right? 542 00:32:25,590 --> 00:32:34,390 And the center maxima, or, say, this Gaussian-like curve there 543 00:32:34,390 --> 00:32:37,020 becoming narrower and narrower. 544 00:32:37,020 --> 00:32:37,600 OK? 545 00:32:37,600 --> 00:32:40,058 So that essentially what we can actually observe form here. 546 00:32:40,058 --> 00:32:45,822 And our calculation really works very well here. 547 00:32:48,720 --> 00:32:49,500 Very good. 548 00:32:49,500 --> 00:32:53,480 So do we have any questions regarding 549 00:32:53,480 --> 00:32:55,030 the demonstration we have here? 550 00:32:59,360 --> 00:33:00,300 OK. 551 00:33:00,300 --> 00:33:04,560 So all those things seems to be pretty straightforward to you. 552 00:33:04,560 --> 00:33:08,700 And what we are actually now is seeing a position 553 00:33:08,700 --> 00:33:13,030 where we can actually discuss how we actually 554 00:33:13,030 --> 00:33:22,570 can understand the radar, which is how actually radar works. 555 00:33:22,570 --> 00:33:25,090 So here is actually how radar works. 556 00:33:25,090 --> 00:33:28,860 Suppose you have some unknown object, 557 00:33:28,860 --> 00:33:31,350 which is like an airplane, OK? 558 00:33:31,350 --> 00:33:35,220 And you would like to know where is this object. 559 00:33:35,220 --> 00:33:41,550 What you do, actually, is to shoot whatever radio waves 560 00:33:41,550 --> 00:33:43,840 toward some direction and see if there 561 00:33:43,840 --> 00:33:45,300 are something coming back. 562 00:33:45,300 --> 00:33:45,960 Right? 563 00:33:45,960 --> 00:33:49,290 Then you know there's something on the sky 564 00:33:49,290 --> 00:33:51,770 because you can detect the refracted wave. 565 00:33:51,770 --> 00:33:52,760 Right? 566 00:33:52,760 --> 00:33:56,070 So we shoot this airplane. 567 00:33:56,070 --> 00:33:59,050 And then something is going to come back. 568 00:33:59,050 --> 00:34:01,630 And now, we can say OK. 569 00:34:01,630 --> 00:34:03,870 In that direction I have something coming back. 570 00:34:03,870 --> 00:34:06,370 That means there's something there. 571 00:34:06,370 --> 00:34:09,090 And I can also measure the time it takes 572 00:34:09,090 --> 00:34:10,590 for the wave to come back. 573 00:34:10,590 --> 00:34:13,211 Then I know where it's actually that object. 574 00:34:13,211 --> 00:34:13,710 Right? 575 00:34:13,710 --> 00:34:18,750 So that's actually a pretty straightforward thing to do. 576 00:34:18,750 --> 00:34:23,170 However, there's one difficulty. 577 00:34:23,170 --> 00:34:27,790 So this is actually the radiation pattern 578 00:34:27,790 --> 00:34:32,920 of oscillating dipole which we actually learned before. 579 00:34:32,920 --> 00:34:36,280 So the problem is that, OK, what we really 580 00:34:36,280 --> 00:34:40,600 need is electromagnetic wave, which is actually very, very 581 00:34:40,600 --> 00:34:44,920 narrow in angle and pointing to some specific direction. 582 00:34:44,920 --> 00:34:46,449 And then I would like to see if I 583 00:34:46,449 --> 00:34:49,920 can get some refractive wave coming from that direction. 584 00:34:49,920 --> 00:34:50,710 OK? 585 00:34:50,710 --> 00:34:53,770 The problem is that, look! 586 00:34:53,770 --> 00:34:57,040 if I oscillate some charge up and down, 587 00:34:57,040 --> 00:35:00,590 the radiation I'm getting is really, really broad. 588 00:35:00,590 --> 00:35:01,090 Right? 589 00:35:01,090 --> 00:35:04,090 So it's going toward all kinds of different direction. 590 00:35:04,090 --> 00:35:07,240 So if you use this to detect things, 591 00:35:07,240 --> 00:35:10,810 you are always going to get something coming back, 592 00:35:10,810 --> 00:35:14,080 because it's actually shooting the electromagnetic wave 593 00:35:14,080 --> 00:35:16,450 into random direction. 594 00:35:16,450 --> 00:35:20,530 And you are not sure any more where 595 00:35:20,530 --> 00:35:23,570 is actually this object you are trying to detect. 596 00:35:23,570 --> 00:35:24,070 OK? 597 00:35:24,070 --> 00:35:27,890 So that's actually apparently a problem. 598 00:35:27,890 --> 00:35:33,520 And what we can actually do is to make use of the interference 599 00:35:33,520 --> 00:35:35,440 phenomenon, which we can actually 600 00:35:35,440 --> 00:35:39,400 learn from here to actually try to make sure 601 00:35:39,400 --> 00:35:41,470 that the electromagnetic wave is actually 602 00:35:41,470 --> 00:35:46,130 pointing to some specific direction we want. 603 00:35:46,130 --> 00:35:51,100 So let's actually go ahead consider a three slit 604 00:35:51,100 --> 00:35:52,099 experiment. 605 00:35:56,230 --> 00:35:59,130 I have this setup changed. 606 00:35:59,130 --> 00:36:00,900 Originally, I have two slits. 607 00:36:00,900 --> 00:36:05,440 And now, I drew it in three holes on the wall. 608 00:36:05,440 --> 00:36:11,140 And, again, I have the distance between the slits to be d. 609 00:36:11,140 --> 00:36:16,060 And I call this slit number 1, 2, and 3. 610 00:36:16,060 --> 00:36:19,150 And we were wondering what would be the interference 611 00:36:19,150 --> 00:36:23,380 pattern on the screen, which is actually 612 00:36:23,380 --> 00:36:28,540 far away from the wall, as a distance of L. 613 00:36:28,540 --> 00:36:32,020 And I'm interested in their intensity 614 00:36:32,020 --> 00:36:34,130 at the point P on this screen. 615 00:36:34,130 --> 00:36:35,500 OK? 616 00:36:35,500 --> 00:36:39,310 So what I am going to do is to basically repeat 617 00:36:39,310 --> 00:36:43,080 what we have done in the previous example. 618 00:36:43,080 --> 00:36:46,540 I'm trying to connect 1 to the P, 2 P, 619 00:36:46,540 --> 00:36:50,150 and the 3 P, basically, connect the slit 620 00:36:50,150 --> 00:36:54,400 to the point of interest on the screen. 621 00:36:54,400 --> 00:36:58,390 And I can actually also-- 622 00:36:58,390 --> 00:37:03,520 you know this angle, this 1 P to the horizontal direction, 623 00:37:03,520 --> 00:37:09,280 this angle is called theta in my notation. 624 00:37:09,280 --> 00:37:11,850 Then clearly, I can go ahead and calculate 625 00:37:11,850 --> 00:37:15,280 what will be the optical path length 626 00:37:15,280 --> 00:37:17,710 difference between of the light coming 627 00:37:17,710 --> 00:37:21,650 from slit number 1, slit number 2 and the slit number 3. 628 00:37:21,650 --> 00:37:22,205 OK? 629 00:37:22,205 --> 00:37:27,020 And in this case, what I'm interested 630 00:37:27,020 --> 00:37:33,530 is delta 1, 2 and delta 1, 3. 631 00:37:33,530 --> 00:37:34,850 Right? 632 00:37:34,850 --> 00:37:39,770 Since the screen is really far away from the wall, 633 00:37:39,770 --> 00:37:42,860 therefore, I can actually savor the assurance 634 00:37:42,860 --> 00:37:49,640 that these two angle is actually theta because the three lines, 635 00:37:49,640 --> 00:37:52,210 due to the large distance, this L 636 00:37:52,210 --> 00:37:54,030 is actually really, really large. 637 00:37:54,030 --> 00:37:56,870 Therefore, they are actually almost parallel to each other. 638 00:37:56,870 --> 00:37:57,800 OK? 639 00:37:57,800 --> 00:38:02,270 So what is going to happen is that delta 1, 2, which 640 00:38:02,270 --> 00:38:05,170 is the phase difference between light 641 00:38:05,170 --> 00:38:08,590 from the first slit and second slit, 642 00:38:08,590 --> 00:38:17,330 is actually going to be equal to delta 2, 3. 643 00:38:17,330 --> 00:38:19,100 It's going to be equal to the phase 644 00:38:19,100 --> 00:38:22,250 difference between the second slit, the light 645 00:38:22,250 --> 00:38:24,450 from second slit and third slit. 646 00:38:24,450 --> 00:38:26,330 And what is actually that number? 647 00:38:26,330 --> 00:38:31,275 This number is going to be equal to d sine theta 648 00:38:31,275 --> 00:38:34,432 divided by lambda times 2 pi. 649 00:38:34,432 --> 00:38:38,150 It's exactly the same as what we actually 650 00:38:38,150 --> 00:38:39,840 get from the first example. 651 00:38:39,840 --> 00:38:40,340 OK? 652 00:38:43,310 --> 00:38:45,790 Therefore, what is going to happen 653 00:38:45,790 --> 00:38:51,190 is that no matter what theta I choose, 654 00:38:51,190 --> 00:38:54,700 the phase difference between nearby slit 655 00:38:54,700 --> 00:38:58,000 is actually a constant, which is actually this one. 656 00:38:58,000 --> 00:39:01,710 And I will call this phase difference to be delta. 657 00:39:04,650 --> 00:39:08,670 I would like to ask you a question now. 658 00:39:08,670 --> 00:39:13,530 The question is, how do we choose the delta 659 00:39:13,530 --> 00:39:22,030 here such that I have completely destructive interference? 660 00:39:22,030 --> 00:39:29,802 Now, I have three vectors, vector E1, vector E2, 661 00:39:29,802 --> 00:39:31,295 and the vector E3. 662 00:39:37,600 --> 00:39:41,500 The phase difference between E1, E2, and E3, 663 00:39:41,500 --> 00:39:47,330 the nearby phase difference is actually delta. 664 00:39:47,330 --> 00:39:51,190 So the question is, how do I actually completely cancel 665 00:39:51,190 --> 00:39:54,070 the electric field so that I have completely 666 00:39:54,070 --> 00:39:55,600 destructive interference? 667 00:39:55,600 --> 00:39:59,520 Can somebody help me here? 668 00:39:59,520 --> 00:40:04,890 The hint is that you can actually use this vector sum 669 00:40:04,890 --> 00:40:08,244 idea in the complex frame. 670 00:40:08,244 --> 00:40:09,240 STUDENT: [INAUDIBLE] 671 00:40:09,240 --> 00:40:10,610 PROFESSOR: Yes, very good. 672 00:40:10,610 --> 00:40:14,460 To form a triangle in the complex frame, right? 673 00:40:14,460 --> 00:40:21,150 So what we can do is now choose the phase difference 674 00:40:21,150 --> 00:40:29,630 delta to be such that E1, E2, and E3 actually 675 00:40:29,630 --> 00:40:32,410 form a triangle. 676 00:40:32,410 --> 00:40:34,930 You see what I mean? 677 00:40:34,930 --> 00:40:37,570 Therefore, you can actually already get 678 00:40:37,570 --> 00:40:41,650 what would be the required delta value. 679 00:40:41,650 --> 00:40:49,110 The required delta value is going to be 2 pi divided by 3. 680 00:40:49,110 --> 00:40:50,520 Right? 681 00:40:50,520 --> 00:40:52,350 OK? 682 00:40:52,350 --> 00:40:53,030 So very good. 683 00:40:53,030 --> 00:40:55,320 So now, we are not afraid anymore. 684 00:40:55,320 --> 00:40:57,720 So how about four slit experiment? 685 00:40:57,720 --> 00:41:01,890 I just add another slit, d essentially the distance 686 00:41:01,890 --> 00:41:04,670 between the fourth slit and the third slit. 687 00:41:04,670 --> 00:41:13,430 What will be the delta required to have 688 00:41:13,430 --> 00:41:15,114 destructive interference? 689 00:41:19,090 --> 00:41:21,386 Anybody can help me? 690 00:41:21,386 --> 00:41:23,240 STUDENT: [INAUDIBLE] 691 00:41:23,240 --> 00:41:24,820 YEN-JIE LEE: Very good. 692 00:41:24,820 --> 00:41:31,150 So if you have four slit, based on this intuition, 693 00:41:31,150 --> 00:41:37,460 which we developed from the complex notation vector 694 00:41:37,460 --> 00:41:42,010 sum, what is going to happen is that if you have four slit, 695 00:41:42,010 --> 00:41:49,251 the delta will be equal to 2 pi divided by 4. 696 00:41:49,251 --> 00:41:49,750 OK? 697 00:41:49,750 --> 00:41:52,450 So what does this tell us? 698 00:41:52,450 --> 00:41:55,950 So remember, the sine theta, sine 699 00:41:55,950 --> 00:41:58,360 theta is telling you the location where 700 00:41:58,360 --> 00:41:59,710 you get the minima. 701 00:41:59,710 --> 00:42:00,640 OK? 702 00:42:00,640 --> 00:42:04,630 So this is actually the power profile, or, say, 703 00:42:04,630 --> 00:42:07,101 the intensity profile. 704 00:42:07,101 --> 00:42:07,600 OK? 705 00:42:07,600 --> 00:42:10,330 And this is actually equal to 0. 706 00:42:10,330 --> 00:42:11,940 And this is actually delta. 707 00:42:11,940 --> 00:42:13,030 OK? 708 00:42:13,030 --> 00:42:19,410 The place which you get zero intensity 709 00:42:19,410 --> 00:42:25,790 is actually becoming closer and closer to zero. 710 00:42:25,790 --> 00:42:26,290 Right? 711 00:42:26,290 --> 00:42:29,290 Because sine theta, which is the angle 712 00:42:29,290 --> 00:42:33,850 between horizontal direction and this observer P, 713 00:42:33,850 --> 00:42:36,340 is proportional to delta. 714 00:42:36,340 --> 00:42:40,510 When you have destructive interference at angle 715 00:42:40,510 --> 00:42:45,400 which is smaller, smaller, and smaller, that means what? 716 00:42:45,400 --> 00:42:51,310 That means the central Gaussian-like structure 717 00:42:51,310 --> 00:42:55,315 is going to be becoming narrower and narrower. 718 00:42:58,730 --> 00:43:00,125 Does that make sense? 719 00:43:02,980 --> 00:43:03,650 Very good. 720 00:43:03,650 --> 00:43:07,270 So at least we found something interesting now. 721 00:43:07,270 --> 00:43:13,330 That means, ha, one idea to get very narrow 722 00:43:13,330 --> 00:43:16,870 electromagnetic wave pointing to some direction 723 00:43:16,870 --> 00:43:22,180 is to have a huge number of point light source 724 00:43:22,180 --> 00:43:26,830 and slit experiment such that I can actually 725 00:43:26,830 --> 00:43:32,200 construct something which is actually very narrow in angle. 726 00:43:32,200 --> 00:43:36,070 And I can use that to shoot the object which 727 00:43:36,070 --> 00:43:38,282 I would like to detect. 728 00:43:38,282 --> 00:43:40,690 You see what I mean? 729 00:43:40,690 --> 00:43:43,800 Does that make sense? 730 00:43:43,800 --> 00:43:46,551 OK? 731 00:43:46,551 --> 00:43:47,050 All right. 732 00:43:47,050 --> 00:43:47,920 So that's very good. 733 00:43:47,920 --> 00:43:53,901 So now, let's actually consider an N slit interference pattern 734 00:43:53,901 --> 00:43:54,400 OK? 735 00:43:54,400 --> 00:43:58,720 So suppose, now, I have not only 1, 2, 3, and then many 736 00:43:58,720 --> 00:44:01,610 more until N slit. 737 00:44:01,610 --> 00:44:03,010 All right? 738 00:44:03,010 --> 00:44:07,600 I can now go ahead and calculate the E total, which 739 00:44:07,600 --> 00:44:15,130 is the total electric field coming from all of the slit 740 00:44:15,130 --> 00:44:17,260 we have. 741 00:44:17,260 --> 00:44:24,430 Basically, this will be equal to E0 exponential i omega t 742 00:44:24,430 --> 00:44:36,110 minus kR where I define r1 is roughly capital R. OK? 743 00:44:36,110 --> 00:44:37,920 That's essentially the contribution 744 00:44:37,920 --> 00:44:41,980 from slit number 1 OK? 745 00:44:41,980 --> 00:44:44,170 And this contribution from slit number 1 746 00:44:44,170 --> 00:44:51,790 is going to be looking like exponential i omega t minus kR 747 00:44:51,790 --> 00:44:54,670 minus delta, right, because there is a phase 748 00:44:54,670 --> 00:44:57,640 difference between the light coming 749 00:44:57,640 --> 00:45:02,440 from first slit and the second slit, which is actually delta. 750 00:45:02,440 --> 00:45:03,640 All right? 751 00:45:03,640 --> 00:45:05,150 So what would be the third term? 752 00:45:05,150 --> 00:45:07,652 So these actually coming from slit number 2. 753 00:45:07,652 --> 00:45:08,860 What would be the third term? 754 00:45:08,860 --> 00:45:15,310 Exponential i omega t minus kR minus what? 755 00:45:15,310 --> 00:45:16,130 STUDENT: 2 delta. 756 00:45:16,130 --> 00:45:18,370 YEN-JIE LEE: 2 delta, yeah, because you 757 00:45:18,370 --> 00:45:21,730 can see that coming from here, seems 758 00:45:21,730 --> 00:45:25,190 the distance between theta as constant, which is d. 759 00:45:25,190 --> 00:45:30,510 Therefore, the phase difference between nearby slits 760 00:45:30,510 --> 00:45:32,440 is actually a constant. 761 00:45:32,440 --> 00:45:36,910 Therefore, I accumulating the phase difference now. 762 00:45:36,910 --> 00:45:39,682 I get 2 delta here. 763 00:45:39,682 --> 00:45:41,140 And this is actually a contribution 764 00:45:41,140 --> 00:45:42,040 from the first slit. 765 00:45:42,040 --> 00:45:48,140 And the et cetera, et cetera, until the Nth slit, 766 00:45:48,140 --> 00:45:50,350 which is actually going to be exponential 767 00:45:50,350 --> 00:45:58,810 i omega t minus kR minus N minus 1 delta. 768 00:45:58,810 --> 00:46:01,830 And summing all those things together, and all of them 769 00:46:01,830 --> 00:46:05,700 are in the Z direction. 770 00:46:05,700 --> 00:46:08,030 OK? 771 00:46:08,030 --> 00:46:10,520 So I'm now going to calculate this dimension. 772 00:46:10,520 --> 00:46:17,280 So basically, you are getting E0 exponential i omega t minus kR. 773 00:46:17,280 --> 00:46:21,820 I can actually factorize these factor out. 774 00:46:21,820 --> 00:46:28,910 And what am I going to get is 1 plus exponential minus i delta 775 00:46:28,910 --> 00:46:32,220 plus exponential minus exponential minus i 776 00:46:32,220 --> 00:46:35,430 2 delta plus blah, blah, blah. 777 00:46:35,430 --> 00:46:39,160 And basically, you will get exponential minus i 778 00:46:39,160 --> 00:46:42,730 minus 1 delta in the first term. 779 00:46:42,730 --> 00:46:48,050 And all those things are pointing to the Z direction. 780 00:46:48,050 --> 00:46:50,830 And this, I know how to actually calculate. 781 00:46:50,830 --> 00:46:52,040 Right? 782 00:46:52,040 --> 00:46:56,120 Just a reminder, basically, if you calculate summation 783 00:46:56,120 --> 00:47:01,130 N equal to 0 to N minus 1 r to the Nth. 784 00:47:01,130 --> 00:47:03,530 And these will give you 1 minus r 785 00:47:03,530 --> 00:47:08,510 to the N divided by phi minus r. 786 00:47:08,510 --> 00:47:09,440 OK? 787 00:47:09,440 --> 00:47:13,010 So basically, I can now go ahead and calculate this. 788 00:47:13,010 --> 00:47:22,640 And this will basically give you 1 minus-- 789 00:47:22,640 --> 00:47:26,220 OK, so the small r here has been replaced by exponential 790 00:47:26,220 --> 00:47:28,550 minus i delta, right? 791 00:47:28,550 --> 00:47:30,770 So therefore, what I'm going to get 792 00:47:30,770 --> 00:47:40,070 is 1 minus exponential minus i delta N for the upper part. 793 00:47:40,070 --> 00:47:44,320 And then I have 1 minus exponential 794 00:47:44,320 --> 00:47:47,530 minus i delta in the lower part. 795 00:47:47,530 --> 00:47:48,950 OK? 796 00:47:48,950 --> 00:47:53,710 So that actually make use of this formula, which are here. 797 00:47:53,710 --> 00:47:57,321 And, again, it should be simplify these series. 798 00:47:57,321 --> 00:47:57,820 All right? 799 00:48:00,330 --> 00:48:02,550 As usual, what I'm going to do is 800 00:48:02,550 --> 00:48:08,050 to use the trick similar to what I have done there 801 00:48:08,050 --> 00:48:11,040 to actually get cosine function out 802 00:48:11,040 --> 00:48:13,500 of the exponential functions. 803 00:48:13,500 --> 00:48:14,250 All right? 804 00:48:14,250 --> 00:48:17,460 So what I'm going to do is to factorize out 805 00:48:17,460 --> 00:48:23,470 exponential minus i delta N over 2 for the upper part. 806 00:48:23,470 --> 00:48:26,870 So basically, I get exponential minus i delta 807 00:48:26,870 --> 00:48:34,320 N divided by 2, exponential i delta N divided by 2 808 00:48:34,320 --> 00:48:39,683 minus exponential minus i delta N divided by 2. 809 00:48:39,683 --> 00:48:40,182 OK? 810 00:48:43,360 --> 00:48:46,980 This is actually divided by exponential 811 00:48:46,980 --> 00:48:52,620 minus i delta over 2 exponential i delta 812 00:48:52,620 --> 00:48:58,908 over 2 minus exponential minus i delta over 2. 813 00:48:58,908 --> 00:49:00,270 All right? 814 00:49:00,270 --> 00:49:02,460 The reason I'm doing this is because I 815 00:49:02,460 --> 00:49:06,600 would like to actually make this a cosine function. 816 00:49:06,600 --> 00:49:08,550 OK? 817 00:49:08,550 --> 00:49:10,020 Any questions so far? 818 00:49:14,830 --> 00:49:16,150 OK. 819 00:49:16,150 --> 00:49:18,340 So if no question, then basically, 820 00:49:18,340 --> 00:49:22,010 this expression can be, again, rewritten 821 00:49:22,010 --> 00:49:28,030 as exponential minus i delta N minus 1 divided 822 00:49:28,030 --> 00:49:33,340 by 2, because I have this denominator nominator 823 00:49:33,340 --> 00:49:38,020 exponential i delta N over 2 and that exponential minus i delta 824 00:49:38,020 --> 00:49:39,321 divided by 2. 825 00:49:39,321 --> 00:49:39,820 OK? 826 00:49:39,820 --> 00:49:42,460 Therefore, I can combine them all together 827 00:49:42,460 --> 00:49:45,950 and then get this expression here. 828 00:49:45,950 --> 00:49:49,180 And this is actually exponential minus exponential. 829 00:49:49,180 --> 00:49:52,640 Therefore, I am going to get sine out of it. 830 00:49:52,640 --> 00:49:56,520 And basically, I get sine and delta 831 00:49:56,520 --> 00:50:02,265 divided by 2 divided by sine delta over 2. 832 00:50:06,510 --> 00:50:07,030 OK. 833 00:50:07,030 --> 00:50:09,670 So now, I can actually go ahead and calculate 834 00:50:09,670 --> 00:50:13,270 what will be the resulting intensity. 835 00:50:13,270 --> 00:50:14,200 Right? 836 00:50:14,200 --> 00:50:19,290 The resulting intensity is going to be 837 00:50:19,290 --> 00:50:24,710 proportional to the square of the electric field. 838 00:50:24,710 --> 00:50:25,210 Right? 839 00:50:25,210 --> 00:50:30,510 So basically, the intensity will be proportional to E square. 840 00:50:30,510 --> 00:50:36,166 And that is actually equal to E times E star. 841 00:50:36,166 --> 00:50:39,300 E and the E is a complex conjugate. 842 00:50:39,300 --> 00:50:41,070 And basically, you will see that this 843 00:50:41,070 --> 00:50:49,810 will be proportional to sine and delta divided by 2 divided 844 00:50:49,810 --> 00:50:56,160 by sine delta over 2 square. 845 00:50:56,160 --> 00:51:01,950 Therefore, the intensity will be equal to i 846 00:51:01,950 --> 00:51:08,880 0 times sine and delta divided by 2 847 00:51:08,880 --> 00:51:12,860 divided by sine delta over 2. 848 00:51:12,860 --> 00:51:16,256 And then square that. 849 00:51:16,256 --> 00:51:17,684 Any questions? 850 00:51:21,020 --> 00:51:26,090 So after all this work, we have arrived at expression which 851 00:51:26,090 --> 00:51:28,592 is very hard to understand. 852 00:51:28,592 --> 00:51:29,092 Right? 853 00:51:29,092 --> 00:51:33,080 [LAUGHS] So what I'm going to do to help you 854 00:51:33,080 --> 00:51:38,830 is to really plot the result as a function of delta 855 00:51:38,830 --> 00:51:40,440 on the screen. 856 00:51:40,440 --> 00:51:44,050 You can see there are four plots here. 857 00:51:44,050 --> 00:51:46,380 The first one is N equal to 3. 858 00:51:46,380 --> 00:51:48,860 The upper left one is N equal to 3. 859 00:51:48,860 --> 00:51:52,270 So you can see that the pattern looks like this. 860 00:51:52,270 --> 00:51:56,140 So at delta equal to 0, surprise nobody, 861 00:51:56,140 --> 00:51:57,700 you are going to get maxima. 862 00:51:57,700 --> 00:51:58,200 Right? 863 00:51:58,200 --> 00:52:02,850 Because delta is equal to 0, you are adding N vectors 864 00:52:02,850 --> 00:52:04,700 the most efficient way. 865 00:52:04,700 --> 00:52:07,020 Therefore, you are going to get the maxima, which 866 00:52:07,020 --> 00:52:09,440 is i equal to i 0. 867 00:52:09,440 --> 00:52:10,980 OK? 868 00:52:10,980 --> 00:52:16,290 And if you move away from the center, delta equal to 0, 869 00:52:16,290 --> 00:52:20,970 and you see that is a small bump in between. 870 00:52:20,970 --> 00:52:22,800 Then you can continue and continue. 871 00:52:22,800 --> 00:52:26,820 And you see that there's another big peak again. 872 00:52:26,820 --> 00:52:27,540 You see? 873 00:52:27,540 --> 00:52:29,010 So that's essentially the structure 874 00:52:29,010 --> 00:52:33,660 if you plot this result, i equal to something proportional 875 00:52:33,660 --> 00:52:37,890 to sine square this expression there. 876 00:52:37,890 --> 00:52:43,850 And that's essentially what you will get when N is equal to 3. 877 00:52:43,850 --> 00:52:45,510 OK? 878 00:52:45,510 --> 00:52:49,380 And this is essentially how I remember this pattern. 879 00:52:49,380 --> 00:52:50,760 OK? 880 00:52:50,760 --> 00:52:55,710 So when N is equal to 3, you have a family 881 00:52:55,710 --> 00:52:58,935 of two adult and one child. 882 00:52:58,935 --> 00:52:59,920 [LAUGHTER] 883 00:52:59,920 --> 00:53:00,420 Right? 884 00:53:00,420 --> 00:53:03,170 So basically, you have two big peak. 885 00:53:03,170 --> 00:53:06,920 And between them, there's a small peak. 886 00:53:06,920 --> 00:53:07,420 OK? 887 00:53:07,420 --> 00:53:09,850 That's actually how I remember this pattern. 888 00:53:09,850 --> 00:53:12,880 And I think it's pretty nice, right? 889 00:53:12,880 --> 00:53:15,555 So you can have N equal to 4. 890 00:53:15,555 --> 00:53:17,560 It's a bigger family. 891 00:53:17,560 --> 00:53:19,570 You have two adults. 892 00:53:19,570 --> 00:53:22,680 The adults are slimmer, OK? 893 00:53:22,680 --> 00:53:23,626 All right? 894 00:53:23,626 --> 00:53:25,810 [LAUGHTER] 895 00:53:25,810 --> 00:53:28,780 Because they have a lot of work to do. 896 00:53:28,780 --> 00:53:30,600 Then they have two child. 897 00:53:30,600 --> 00:53:32,286 All right? 898 00:53:32,286 --> 00:53:37,002 N equal to 5, how many children do we have? 899 00:53:37,002 --> 00:53:37,960 STUDENT: We have three. 900 00:53:37,960 --> 00:53:39,260 YEN-JIE LEE: Three. 901 00:53:39,260 --> 00:53:43,910 Therefore, the adults are really frustrated. 902 00:53:43,910 --> 00:53:49,560 So they are even slimmer in a happy way, making it positive. 903 00:53:49,560 --> 00:53:52,020 And N equal to 6, woo. 904 00:53:52,020 --> 00:53:55,830 Oh my god, I have four children in the family. 905 00:53:55,830 --> 00:53:56,670 All right? 906 00:53:56,670 --> 00:54:00,610 So there are two things which we learned from here. 907 00:54:00,610 --> 00:54:06,540 The first one is that the number of big peak, which 908 00:54:06,540 --> 00:54:13,770 I would call it principal maxima, the number 909 00:54:13,770 --> 00:54:16,580 of principal maxima is actually pretty 910 00:54:16,580 --> 00:54:22,116 similar as a function of delta. 911 00:54:22,116 --> 00:54:26,630 But the number of secondary maxima 912 00:54:26,630 --> 00:54:30,710 increase as a function of N value. 913 00:54:30,710 --> 00:54:35,150 N value is actually telling you how many slits 914 00:54:35,150 --> 00:54:38,020 you have in the experiment. 915 00:54:38,020 --> 00:54:44,060 And also, you can see that the delta is actually becoming-- 916 00:54:44,060 --> 00:54:48,130 the first minima, the delta value 917 00:54:48,130 --> 00:54:51,580 is actually decreasing as a function of N value. 918 00:54:51,580 --> 00:54:52,580 Right? 919 00:54:52,580 --> 00:54:55,660 So the parents are getting slimmer. 920 00:54:55,660 --> 00:54:56,390 All right? 921 00:54:56,390 --> 00:54:59,120 So therefore, you can see that if I 922 00:54:59,120 --> 00:55:04,130 would like to have a radar which is actually pointing 923 00:55:04,130 --> 00:55:06,920 to a very specific direction, what 924 00:55:06,920 --> 00:55:12,870 essentially the choice of N value which we will need? 925 00:55:12,870 --> 00:55:14,870 Infinity or a very large number. 926 00:55:14,870 --> 00:55:15,420 OK? 927 00:55:15,420 --> 00:55:18,300 For sure in your life, we cannot do infinity. 928 00:55:18,300 --> 00:55:22,410 But now, we have found a way to actually design 929 00:55:22,410 --> 00:55:28,890 our radar since sine theta is actually proportional to delta. 930 00:55:28,890 --> 00:55:30,990 Therefore, what we actually really need 931 00:55:30,990 --> 00:55:36,390 to do is to really maximize the number of slits 932 00:55:36,390 --> 00:55:39,650 we have so that actually we can create a radar which 933 00:55:39,650 --> 00:55:45,390 would really point toward the direction of the enemy, which 934 00:55:45,390 --> 00:55:48,700 is shown there, invading the earth. 935 00:55:48,700 --> 00:55:49,200 OK. 936 00:55:49,200 --> 00:55:50,670 [LAUGHTER] 937 00:55:50,670 --> 00:55:52,630 And we can actually detect it. 938 00:55:52,630 --> 00:55:53,130 OK. 939 00:55:53,130 --> 00:55:56,880 So we will take a five minute break before we actually 940 00:55:56,880 --> 00:56:01,530 go to the last part of the course, which is the connection 941 00:56:01,530 --> 00:56:04,360 to quantum mechanics. 942 00:56:04,360 --> 00:56:05,880 So we come back at 35. 943 00:56:05,880 --> 00:56:09,880 [SIDE CONVERSATIONS] 944 00:56:12,380 --> 00:56:13,880 [SIDE CONVERSATIONS] 945 00:56:13,880 --> 00:56:17,080 YEN-JIE LEE: OK so welcome come back from the break. 946 00:56:17,080 --> 00:56:21,370 So before we move to the connection 947 00:56:21,370 --> 00:56:24,250 to quantum mechanics, I would like 948 00:56:24,250 --> 00:56:27,085 to talk some more about what we have learned 949 00:56:27,085 --> 00:56:29,110 from the design of the radar. 950 00:56:29,110 --> 00:56:30,040 OK? 951 00:56:30,040 --> 00:56:33,940 So this essentially what we actually get. 952 00:56:33,940 --> 00:56:40,410 The position of the minima that required the phase difference 953 00:56:40,410 --> 00:56:45,100 delta is actually equal to 2 pi divided by N value, 954 00:56:45,100 --> 00:56:48,820 because it was this delta value. 955 00:56:48,820 --> 00:56:53,080 The N vectors is going to cancel each other. 956 00:56:53,080 --> 00:56:56,600 And you are going to form something like a circle 957 00:56:56,600 --> 00:57:02,920 if you choose delta equal to 2 pi divided by capital N. OK? 958 00:57:02,920 --> 00:57:05,090 And don't forget why this is actually delta. 959 00:57:05,090 --> 00:57:12,600 The delta is actually d sine theta divided by lambda. 960 00:57:12,600 --> 00:57:13,650 Right? 961 00:57:13,650 --> 00:57:14,470 OK? 962 00:57:14,470 --> 00:57:17,900 And times 2 pi. 963 00:57:17,900 --> 00:57:19,221 OK? 964 00:57:19,221 --> 00:57:19,720 Right? 965 00:57:19,720 --> 00:57:24,100 So therefore, you can see that the sine theta is actually 966 00:57:24,100 --> 00:57:28,580 proportional to lambda divided by N times d. 967 00:57:28,580 --> 00:57:31,030 OK? 968 00:57:31,030 --> 00:57:34,960 And in this case, you can see that if you increase 969 00:57:34,960 --> 00:57:38,900 N value, the resolution or the width 970 00:57:38,900 --> 00:57:43,270 of the central principal maxima is 971 00:57:43,270 --> 00:57:46,180 going to be decreasing as a function, though, 972 00:57:46,180 --> 00:57:47,860 N value you're putting. 973 00:57:47,860 --> 00:57:53,530 So in short, how do I actually design a high-resolution radar? 974 00:57:53,530 --> 00:57:59,720 What I really need is to have lambda to be small. 975 00:57:59,720 --> 00:58:00,310 OK? 976 00:58:00,310 --> 00:58:04,930 So that means I need to use high-frequency electromagnetic 977 00:58:04,930 --> 00:58:05,920 wave. 978 00:58:05,920 --> 00:58:09,310 I can maximize the N value. 979 00:58:09,310 --> 00:58:11,960 I can actually make d very large. 980 00:58:11,960 --> 00:58:15,980 That means I'm going to have a very large radar design. 981 00:58:15,980 --> 00:58:16,840 Right? 982 00:58:16,840 --> 00:58:20,480 Then I can have a very good resolution. 983 00:58:20,480 --> 00:58:21,110 OK. 984 00:58:21,110 --> 00:58:23,070 So we are almost done with radar. 985 00:58:23,070 --> 00:58:25,900 But there's a problem. 986 00:58:25,900 --> 00:58:30,680 The problem is that if you look at this, if this is actually 987 00:58:30,680 --> 00:58:34,760 the position of the principal minima, 988 00:58:34,760 --> 00:58:36,950 you can see that is always pointing 989 00:58:36,950 --> 00:58:43,520 to the center of the radar where the delta is equal to 0. 990 00:58:43,520 --> 00:58:44,240 OK? 991 00:58:44,240 --> 00:58:49,360 And then that means I can only scan in one direction. 992 00:58:49,360 --> 00:58:54,970 There is a reason why those radar are called phased radar. 993 00:58:54,970 --> 00:58:57,160 That is because now I can actually 994 00:58:57,160 --> 00:59:02,020 change the relative phase of all those point source emitted 995 00:59:02,020 --> 00:59:05,680 from the radar so that I can shift 996 00:59:05,680 --> 00:59:09,711 the direction of the central principal maxima. 997 00:59:09,711 --> 00:59:10,210 OK? 998 00:59:10,210 --> 00:59:13,630 So what is actually done here is like this. 999 00:59:13,630 --> 00:59:17,500 So basically, I can have introduced 1000 00:59:17,500 --> 00:59:21,760 before emitting the electromagnetic wave, 1001 00:59:21,760 --> 00:59:26,550 I can introduce a zero additional phase difference. 1002 00:59:26,550 --> 00:59:29,830 And for the second one, I introduce additional phase 1003 00:59:29,830 --> 00:59:31,690 difference of phi. 1004 00:59:31,690 --> 00:59:32,360 OK? 1005 00:59:32,360 --> 00:59:35,290 And for the third one, I introduce 1006 00:59:35,290 --> 00:59:40,630 additional phase difference between the third slit-- 1007 00:59:40,630 --> 00:59:44,430 or say the third emitter and the first emitter by 2 delta. 1008 00:59:44,430 --> 00:59:48,730 And for N's emitter, I introduce a phase difference 1009 00:59:48,730 --> 00:59:51,720 of N minus 1 phi. 1010 00:59:51,720 --> 00:59:53,560 OK? 1011 00:59:53,560 --> 01:00:01,040 If I add this phase difference into the setup, what 1012 01:00:01,040 --> 01:00:03,440 I'm going to get is like this. 1013 01:00:03,440 --> 01:00:11,280 So basically, delta will become 2 pi divided by lambda d sine 1014 01:00:11,280 --> 01:00:15,892 theta minus phi angle. 1015 01:00:15,892 --> 01:00:17,720 All right? 1016 01:00:17,720 --> 01:00:23,180 And this phi is actually the artificial eddy phase 1017 01:00:23,180 --> 01:00:25,490 difference between those source. 1018 01:00:25,490 --> 01:00:26,330 OK? 1019 01:00:26,330 --> 01:00:29,270 And that means I will require-- 1020 01:00:29,270 --> 01:00:40,030 and this will be equal to 2 pi divided by N value, such 1021 01:00:40,030 --> 01:00:42,790 as you have completely destructive interference. 1022 01:00:42,790 --> 01:00:43,930 OK? 1023 01:00:43,930 --> 01:00:47,620 I can now make this phi to be time-dependent. 1024 01:00:47,620 --> 01:00:49,390 For example, it's increasing as a function 1025 01:00:49,390 --> 01:00:53,530 of time, phi times t, right? 1026 01:00:53,530 --> 01:01:00,180 Then what is going to happen is that as a function of time, 1027 01:01:00,180 --> 01:01:03,400 I'm going to change the sine theta value 1028 01:01:03,400 --> 01:01:07,690 so that I can get a complete cancellation, 2 pi over N. 1029 01:01:07,690 --> 01:01:08,620 Right? 1030 01:01:08,620 --> 01:01:12,220 So effectively, I'm changing the angle 1031 01:01:12,220 --> 01:01:19,840 of the central principal maxima by introducing 1032 01:01:19,840 --> 01:01:23,860 additional artificial phase difference between all 1033 01:01:23,860 --> 01:01:25,590 those point source. 1034 01:01:25,590 --> 01:01:26,290 OK? 1035 01:01:26,290 --> 01:01:28,630 And this is actually the way we can actually 1036 01:01:28,630 --> 01:01:34,180 rotate the place we are scanning up and down 1037 01:01:34,180 --> 01:01:39,320 and get a very nice result to detect the enemy. 1038 01:01:39,320 --> 01:01:40,620 OK? 1039 01:01:40,620 --> 01:01:41,513 Any questions? 1040 01:01:44,411 --> 01:01:45,821 No? 1041 01:01:45,821 --> 01:01:46,320 OK. 1042 01:01:46,320 --> 01:01:50,230 So now, I'm going to move on and discuss 1043 01:01:50,230 --> 01:01:54,280 a very interesting experiment. 1044 01:01:54,280 --> 01:01:59,250 So this is very exciting experiment content, 1045 01:01:59,250 --> 01:02:02,640 billiard balls and the two slits. 1046 01:02:02,640 --> 01:02:03,996 OK? 1047 01:02:03,996 --> 01:02:05,370 And we will wonder, then, what is 1048 01:02:05,370 --> 01:02:08,040 going to happen when those balls especially 1049 01:02:08,040 --> 01:02:09,300 pass through the slit. 1050 01:02:09,300 --> 01:02:12,220 Can anybody actually tell me what she is going to happen? 1051 01:02:12,220 --> 01:02:15,060 And what will be the statistics, or say, 1052 01:02:15,060 --> 01:02:19,740 the count, which I am going to go on to get in the receiver 1053 01:02:19,740 --> 01:02:20,710 later? 1054 01:02:20,710 --> 01:02:23,930 Anybody can actually tell me? 1055 01:02:23,930 --> 01:02:27,950 If I actually shoot a lot of balls through this slit-- 1056 01:02:27,950 --> 01:02:30,410 don't be shy, right? 1057 01:02:30,410 --> 01:02:31,730 It's easy. 1058 01:02:31,730 --> 01:02:34,210 No? 1059 01:02:34,210 --> 01:02:35,664 Nobody wants-- 1060 01:02:35,664 --> 01:02:37,452 STUDENT: They make [INAUDIBLE] 1061 01:02:37,452 --> 01:02:39,180 YEN-JIE LEE: Yeah, that's right. 1062 01:02:39,180 --> 01:02:40,630 Right? 1063 01:02:40,630 --> 01:02:42,628 Doesn't surprise nobody, right? 1064 01:02:42,628 --> 01:02:46,500 [LAUGHS] Yeah, too afraid of answering questions. 1065 01:02:46,500 --> 01:02:51,980 OK, you can see that they make two path, right? 1066 01:02:51,980 --> 01:02:52,820 No? 1067 01:02:52,820 --> 01:02:53,450 Right? 1068 01:02:53,450 --> 01:02:54,050 OK. 1069 01:02:54,050 --> 01:02:55,190 Very good. 1070 01:02:55,190 --> 01:02:59,630 So now, this is the exciting part. 1071 01:02:59,630 --> 01:03:06,110 Now, instead of shooting billiard balls, what I'm going 1072 01:03:06,110 --> 01:03:09,500 to do is to shoot electrons. 1073 01:03:09,500 --> 01:03:12,680 So I can actually prepare an electron source 1074 01:03:12,680 --> 01:03:16,160 and heat it up, such that it start to emit electrons. 1075 01:03:16,160 --> 01:03:20,510 And I have two slits and have them pass through this slits. 1076 01:03:20,510 --> 01:03:22,670 And I have a screen, which actually 1077 01:03:22,670 --> 01:03:25,670 have an electron detector to count 1078 01:03:25,670 --> 01:03:30,630 the number of electron which I am going to get on the screen. 1079 01:03:30,630 --> 01:03:34,200 The reason why I call it single electron source 1080 01:03:34,200 --> 01:03:41,040 is because each time I control my experiment such that it only 1081 01:03:41,040 --> 01:03:45,524 emit one electron every time. 1082 01:03:45,524 --> 01:03:46,940 OK? 1083 01:03:46,940 --> 01:03:52,750 The question I'm trying to ask is, will I 1084 01:03:52,750 --> 01:03:55,330 see some pattern, which is actually 1085 01:03:55,330 --> 01:04:04,460 light like billiard balls, and they form two piles in a pack? 1086 01:04:04,460 --> 01:04:07,420 That's actually option number one. 1087 01:04:07,420 --> 01:04:12,410 Or I'm going to see really something crazy? 1088 01:04:12,410 --> 01:04:16,480 It's the electron is going to be interfere-- 1089 01:04:16,480 --> 01:04:21,150 it's going through the interference with itself. 1090 01:04:21,150 --> 01:04:23,950 And that essentially option number two. 1091 01:04:23,950 --> 01:04:24,450 OK? 1092 01:04:24,450 --> 01:04:27,900 The lure of 8.03 is that everybody had to choose one. 1093 01:04:27,900 --> 01:04:28,830 OK? 1094 01:04:28,830 --> 01:04:36,640 So how many of you think what is going to happen is number one? 1095 01:04:36,640 --> 01:04:38,410 Come on. 1096 01:04:38,410 --> 01:04:41,680 I have only one electron each time. 1097 01:04:41,680 --> 01:04:43,445 Nobody think so? 1098 01:04:43,445 --> 01:04:43,945 Wow. 1099 01:04:47,620 --> 01:04:49,283 Maybe all of you are wrong. 1100 01:04:49,283 --> 01:04:55,110 [LAUGHS] How about the second option? 1101 01:04:55,110 --> 01:04:55,867 STUDENT: [LAUGHS] 1102 01:04:55,867 --> 01:04:57,450 YEN-JIE LEE: Hey, some of you actually 1103 01:04:57,450 --> 01:04:58,580 didn't raise your hand. 1104 01:04:58,580 --> 01:04:59,080 Come on. 1105 01:04:59,080 --> 01:04:59,580 Come on. 1106 01:04:59,580 --> 01:05:00,130 [LAUGHTER] 1107 01:05:00,130 --> 01:05:02,260 OK, everybody. 1108 01:05:02,260 --> 01:05:03,630 Wow. 1109 01:05:03,630 --> 01:05:06,080 What is actually happening to you brain? 1110 01:05:06,080 --> 01:05:08,310 [LAUGHTER] 1111 01:05:08,310 --> 01:05:10,410 My brain is not functional like this. 1112 01:05:10,410 --> 01:05:11,140 OK. 1113 01:05:11,140 --> 01:05:16,200 So I really hope that I can bring the experiment to here. 1114 01:05:16,200 --> 01:05:17,630 But unfortunately, that's actually 1115 01:05:17,630 --> 01:05:19,910 going to be difficult. OK? 1116 01:05:19,910 --> 01:05:21,440 So what I'm going to do is that I'm 1117 01:05:21,440 --> 01:05:25,500 going to show you the experimental result, 1118 01:05:25,500 --> 01:05:26,870 this video. 1119 01:05:26,870 --> 01:05:30,680 And we are going to see what is going to happen. 1120 01:05:30,680 --> 01:05:33,660 You see that there are dots popping out. 1121 01:05:33,660 --> 01:05:34,830 What are those? 1122 01:05:34,830 --> 01:05:40,340 Those are the detected electron one-by-one on screen. 1123 01:05:40,340 --> 01:05:41,090 OK? 1124 01:05:41,090 --> 01:05:45,630 So basically, you can see that the number of dots 1125 01:05:45,630 --> 01:05:47,810 are increasing as a function of time. 1126 01:05:47,810 --> 01:05:52,010 And I actually-- I mean, speeding up things a bit 1127 01:05:52,010 --> 01:05:54,480 so that actually you can see the pattern quicker. 1128 01:05:54,480 --> 01:05:54,980 OK. 1129 01:05:54,980 --> 01:05:57,021 So you can see that there are more and more dots. 1130 01:05:57,021 --> 01:05:59,000 And each time, you can see that I only get 1131 01:05:59,000 --> 01:06:04,373 one electron per image here. 1132 01:06:04,373 --> 01:06:05,260 Right? 1133 01:06:05,260 --> 01:06:07,010 So you can see now there are more and more 1134 01:06:07,010 --> 01:06:09,880 and more and more, and accumulating more data, 1135 01:06:09,880 --> 01:06:12,800 like what we actually done in The Large Hadron Collider. 1136 01:06:12,800 --> 01:06:15,590 We wait there, collect more data. 1137 01:06:15,590 --> 01:06:18,410 And we are speeding things up. 1138 01:06:18,410 --> 01:06:21,320 And you can see that, wow, something's 1139 01:06:21,320 --> 01:06:24,250 actually developing. 1140 01:06:24,250 --> 01:06:24,960 What is that? 1141 01:06:28,410 --> 01:06:30,120 Can you see it? 1142 01:06:30,120 --> 01:06:34,270 Now, you are speeding up like 1,000 times faster. 1143 01:06:34,270 --> 01:06:37,026 You can see what pattern? 1144 01:06:37,026 --> 01:06:38,874 STUDENT: Interference pattern. 1145 01:06:38,874 --> 01:06:40,290 YEN-JIE LEE: Interference pattern. 1146 01:06:40,290 --> 01:06:43,510 What is going on? 1147 01:06:43,510 --> 01:06:44,660 You are not surprised? 1148 01:06:44,660 --> 01:06:45,620 STUDENT: No. 1149 01:06:45,620 --> 01:06:46,600 YEN-JIE LEE: Oh my god. 1150 01:06:46,600 --> 01:06:47,320 What is going on? 1151 01:06:47,320 --> 01:06:50,250 [LAUGHTER] 1152 01:06:50,250 --> 01:06:53,490 I'm so surprised. 1153 01:06:53,490 --> 01:06:54,750 Look at this. 1154 01:06:54,750 --> 01:07:00,300 So I have emission of one electron each time. 1155 01:07:00,300 --> 01:07:04,815 And that is actually the four snapshot which I took-- 1156 01:07:04,815 --> 01:07:07,480 which actually this experiment, Hitachi Group 1157 01:07:07,480 --> 01:07:08,940 actually did this experiment. 1158 01:07:08,940 --> 01:07:13,880 You can actually click on this link to the more detail. 1159 01:07:13,880 --> 01:07:17,600 And they took four snapshots of the experiment. 1160 01:07:17,600 --> 01:07:20,240 And you can see that in the beginning, 1161 01:07:20,240 --> 01:07:23,570 you can see clearly each time you only get 1162 01:07:23,570 --> 01:07:27,230 one electron out of the source. 1163 01:07:27,230 --> 01:07:28,760 OK? 1164 01:07:28,760 --> 01:07:31,140 But as a function of time, you're 1165 01:07:31,140 --> 01:07:33,420 accumulating more and more. 1166 01:07:33,420 --> 01:07:35,580 And you see that clearly, there's 1167 01:07:35,580 --> 01:07:43,410 a pattern forming, which, is actually consistent with what 1168 01:07:43,410 --> 01:07:47,540 we see in this calculation. 1169 01:07:47,540 --> 01:07:48,040 OK? 1170 01:07:48,040 --> 01:07:52,470 So I think that's actually truly amazing. 1171 01:07:52,470 --> 01:07:55,050 And what does that mean? 1172 01:07:55,050 --> 01:08:02,011 That means the electron is playing with itself. 1173 01:08:02,011 --> 01:08:04,115 It's interfering with itself. 1174 01:08:06,620 --> 01:08:07,220 Right? 1175 01:08:07,220 --> 01:08:08,960 That's really strange. 1176 01:08:08,960 --> 01:08:09,980 What is going to happen? 1177 01:08:09,980 --> 01:08:11,130 What is going on? 1178 01:08:11,130 --> 01:08:17,630 So one single electron pass through both slit, which is 1179 01:08:17,630 --> 01:08:19,050 actually the option you choose. 1180 01:08:19,050 --> 01:08:21,220 Surprise me. 1181 01:08:21,220 --> 01:08:25,630 And then they interfere like waves. 1182 01:08:25,630 --> 01:08:31,020 And they produce the pattern which we see on the screen. 1183 01:08:31,020 --> 01:08:35,850 That is actually really crazy to me. 1184 01:08:35,850 --> 01:08:40,359 What is actually even more crazy is this situation. 1185 01:08:40,359 --> 01:08:47,160 So now, if I make measurement in front of the slit, OK, so now, 1186 01:08:47,160 --> 01:08:50,970 I puts on a little device. 1187 01:08:50,970 --> 01:08:54,750 When the electron pass through one of the slit, 1188 01:08:54,750 --> 01:08:57,701 I say, send me a signal. 1189 01:08:57,701 --> 01:08:58,200 OK? 1190 01:08:58,200 --> 01:09:01,950 So now, I can clearly know that which 1191 01:09:01,950 --> 01:09:05,939 slit the electron is actually going through 1192 01:09:05,939 --> 01:09:07,200 in the experiment. 1193 01:09:07,200 --> 01:09:08,220 OK? 1194 01:09:08,220 --> 01:09:13,170 And the crazy thing is that if I do that, then it 1195 01:09:13,170 --> 01:09:16,359 becomes two piles. 1196 01:09:16,359 --> 01:09:17,054 OK? 1197 01:09:17,054 --> 01:09:19,220 Of course, maybe there are some diffraction pattern. 1198 01:09:19,220 --> 01:09:22,370 But it really changes the pattern 1199 01:09:22,370 --> 01:09:25,279 of the experimental result. And that is actually 1200 01:09:25,279 --> 01:09:27,319 really very strange. 1201 01:09:27,319 --> 01:09:30,500 And we are going to talk about that briefly 1202 01:09:30,500 --> 01:09:33,960 in the next lecture. 1203 01:09:33,960 --> 01:09:37,365 So before the end, I'm going to show you 1204 01:09:37,365 --> 01:09:39,920 an additional demonstration which 1205 01:09:39,920 --> 01:09:43,189 motivate the discussion what we are going 1206 01:09:43,189 --> 01:09:47,109 to have in the next lecture. 1207 01:09:47,109 --> 01:09:55,920 So now, I can actually turn off the light again and also hide 1208 01:09:55,920 --> 01:09:57,950 the image. 1209 01:09:57,950 --> 01:09:58,460 OK. 1210 01:09:58,460 --> 01:10:00,978 I hope I can find the pattern. 1211 01:10:00,978 --> 01:10:04,620 [LAUGHS] All right. 1212 01:10:04,620 --> 01:10:08,290 So here, I have two laser. 1213 01:10:08,290 --> 01:10:11,940 So I'm going to turn up the first laser. 1214 01:10:11,940 --> 01:10:17,970 And this laser is going to pass through a two slit-- 1215 01:10:17,970 --> 01:10:24,480 a two really nearby slit and form an interference pattern. 1216 01:10:24,480 --> 01:10:26,490 As you can see on the wall-- 1217 01:10:26,490 --> 01:10:29,850 I hope you can see, I don't know if you can see clearly-- 1218 01:10:29,850 --> 01:10:35,760 that you can see there are many, many dots, nearby dots, 1219 01:10:35,760 --> 01:10:38,670 which actually shows you the position 1220 01:10:38,670 --> 01:10:43,270 of the principal maximas, right, because are actually 1221 01:10:43,270 --> 01:10:45,120 two slit experiment. 1222 01:10:45,120 --> 01:10:49,880 Therefore, how many children do we have in the family? 1223 01:10:49,880 --> 01:10:50,890 Zero, right? 1224 01:10:50,890 --> 01:10:52,720 Because they are-- 1225 01:10:52,720 --> 01:10:54,580 OK, they just got married, maybe. 1226 01:10:54,580 --> 01:10:56,460 [LAUGHS] All right. 1227 01:10:56,460 --> 01:10:59,470 So therefore, you will see only adults. 1228 01:10:59,470 --> 01:11:04,210 And that is actually the principal maximas. 1229 01:11:04,210 --> 01:11:07,420 You can see many, many nearby dots. 1230 01:11:07,420 --> 01:11:10,000 They are almost equally bright. 1231 01:11:10,000 --> 01:11:10,900 OK? 1232 01:11:10,900 --> 01:11:15,050 But there's something happening to this pattern as well. 1233 01:11:15,050 --> 01:11:17,620 And you can see that-- wait, wait, wait a second. 1234 01:11:17,620 --> 01:11:21,760 In the calculation we get the principle maxima 1235 01:11:21,760 --> 01:11:24,040 to have the same height, right? 1236 01:11:24,040 --> 01:11:28,250 That means you are going to get exactly those same intensity 1237 01:11:28,250 --> 01:11:30,790 for all the maximas. 1238 01:11:30,790 --> 01:11:33,440 But you don't see that here. 1239 01:11:33,440 --> 01:11:38,940 You can see that if you move away from the center too much, 1240 01:11:38,940 --> 01:11:42,210 the intensity is decreasing. 1241 01:11:42,210 --> 01:11:43,470 You see at the edge? 1242 01:11:43,470 --> 01:11:49,440 It actually even goes to zero. 1243 01:11:49,440 --> 01:11:50,750 Right? 1244 01:11:50,750 --> 01:11:52,350 What is actually happening? 1245 01:11:52,350 --> 01:11:57,490 Something clearly is actually missing in our calculation. 1246 01:11:57,490 --> 01:12:01,780 And that missing part is actually 1247 01:12:01,780 --> 01:12:06,270 diffraction, which we will talk about that in the next lecture. 1248 01:12:06,270 --> 01:12:14,800 So if you compare this pattern to the second demo, 1249 01:12:14,800 --> 01:12:17,410 you can see in the right hand side 1250 01:12:17,410 --> 01:12:20,560 setup, which I have here, which I should give you 1251 01:12:20,560 --> 01:12:22,540 a projection on the wall, which is actually 1252 01:12:22,540 --> 01:12:30,190 lower part of the demo, you can see that this laser actually 1253 01:12:30,190 --> 01:12:32,800 pass through a single slit. 1254 01:12:32,800 --> 01:12:36,340 But this slit is actually pretty wide. 1255 01:12:36,340 --> 01:12:37,120 OK? 1256 01:12:37,120 --> 01:12:42,400 And you can see that indeed, you see the laser coming out, 1257 01:12:42,400 --> 01:12:46,730 but essentially, not a single spot. 1258 01:12:46,730 --> 01:12:49,160 And it has some kind of pattern, which 1259 01:12:49,160 --> 01:12:52,130 is actually popping out there. 1260 01:12:52,130 --> 01:12:56,000 And this is also related to interference 1261 01:12:56,000 --> 01:12:58,740 between infinite number of source. 1262 01:12:58,740 --> 01:12:59,240 OK? 1263 01:12:59,240 --> 01:13:04,700 And you can see that the pattern seems to really pretty 1264 01:13:04,700 --> 01:13:11,060 similar to the pattern we see in the upper demo, 1265 01:13:11,060 --> 01:13:15,380 except that upper demo have individual similar structure, 1266 01:13:15,380 --> 01:13:19,820 which is the principal maxima from the two slit interference. 1267 01:13:19,820 --> 01:13:25,400 And we are going to solve the mystery in the lecture 1268 01:13:25,400 --> 01:13:26,830 next time. 1269 01:13:26,830 --> 01:13:27,330 OK. 1270 01:13:27,330 --> 01:13:29,170 So thank you very much. 1271 01:13:29,170 --> 01:13:33,300 And if you have any questions related to the lecture today, 1272 01:13:33,300 --> 01:13:35,206 I will be here to answer your questions. 1273 01:13:43,850 --> 01:13:46,920 So this is a demo which we would like 1274 01:13:46,920 --> 01:13:52,080 to show you, Single Slit and the Double Slit Interference 1275 01:13:52,080 --> 01:13:52,991 Pattern. 1276 01:13:52,991 --> 01:13:53,490 OK? 1277 01:13:53,490 --> 01:13:56,730 So the first scene is the setup. 1278 01:13:56,730 --> 01:14:00,210 So we have a laser beam, which is actually 1279 01:14:00,210 --> 01:14:10,730 passing through this either single slit or double slit 1280 01:14:10,730 --> 01:14:11,790 experiment. 1281 01:14:11,790 --> 01:14:17,030 And then the laser beam will be going through this 1282 01:14:17,030 --> 01:14:20,550 and interfere and show interesting pattern 1283 01:14:20,550 --> 01:14:22,410 on the screen. 1284 01:14:22,410 --> 01:14:25,050 And there are two setup. 1285 01:14:25,050 --> 01:14:30,130 The left-hand side one is two slit interference experiment. 1286 01:14:30,130 --> 01:14:36,940 And right-hand side is a single slit diffraction experiment. 1287 01:14:36,940 --> 01:14:40,860 So you can see left-hand side one, I already turned it on. 1288 01:14:40,860 --> 01:14:43,290 Laser beam passed through two slits. 1289 01:14:43,290 --> 01:14:47,990 And they form complicated pattern on the screen. 1290 01:14:47,990 --> 01:14:51,400 And you can see there are two kinds of structure here. 1291 01:14:51,400 --> 01:14:54,000 The first one is the very fine structure, 1292 01:14:54,000 --> 01:14:58,070 which you can see that it's like some row of dots 1293 01:14:58,070 --> 01:15:00,120 in the center of the pattern. 1294 01:15:00,120 --> 01:15:05,740 And there are larger scale pattern as well, 1295 01:15:05,740 --> 01:15:10,100 which you can see that the overall intensity of all 1296 01:15:10,100 --> 01:15:13,160 those little dots are also variating 1297 01:15:13,160 --> 01:15:17,880 as a function of distance with respect to the center. 1298 01:15:17,880 --> 01:15:21,190 So during the lecture, we were wondering what actually 1299 01:15:21,190 --> 01:15:23,500 cause this kind of pattern. 1300 01:15:23,500 --> 01:15:25,230 And the answer is that this is actually 1301 01:15:25,230 --> 01:15:30,390 coming from the effect of single slit interference. 1302 01:15:30,390 --> 01:15:33,400 The reason why we have this pattern 1303 01:15:33,400 --> 01:15:37,000 is because the two slit is actually not 1304 01:15:37,000 --> 01:15:41,060 infinitely narrow in my setup. 1305 01:15:41,060 --> 01:15:45,360 Therefore, within a single slit, there 1306 01:15:45,360 --> 01:15:49,800 is already a interference pattern coming out of it. 1307 01:15:49,800 --> 01:15:55,710 Therefore, the compound effect, results in a very complicated 1308 01:15:55,710 --> 01:15:57,570 structure we see on the screen. 1309 01:15:57,570 --> 01:16:00,600 So to demonstrate this effect, now, I'm 1310 01:16:00,600 --> 01:16:05,160 going to turn on the right-hand side setup. 1311 01:16:05,160 --> 01:16:06,870 In the right-hand side setup, I am 1312 01:16:06,870 --> 01:16:10,140 going to have the laser beam, which 1313 01:16:10,140 --> 01:16:15,840 you see emitting from here, pass through a single slit. 1314 01:16:15,840 --> 01:16:19,500 I actually set it up so that they 1315 01:16:19,500 --> 01:16:24,300 have the same width between the single slit experiment 1316 01:16:24,300 --> 01:16:26,730 and double slit experiment. 1317 01:16:26,730 --> 01:16:35,960 And then you can see after I turn it on, 1318 01:16:35,960 --> 01:16:41,450 you can see that now, we have two sets of pattern. 1319 01:16:41,450 --> 01:16:46,580 The lower set is actually coming from a single slit interference 1320 01:16:46,580 --> 01:16:47,540 experiment. 1321 01:16:47,540 --> 01:16:52,760 And you can see very nicely that first of all, it 1322 01:16:52,760 --> 01:16:56,690 has a similar pattern, like what we see in the double slit 1323 01:16:56,690 --> 01:16:58,080 experiment. 1324 01:16:58,080 --> 01:17:02,460 Secondly, you can see that basically, we carefully tune 1325 01:17:02,460 --> 01:17:05,060 these two experiments so that the distance 1326 01:17:05,060 --> 01:17:09,750 between the slit and the screen is roughly the same. 1327 01:17:09,750 --> 01:17:12,920 Finally, we also set it up, as I've mentioned before, 1328 01:17:12,920 --> 01:17:16,120 such that the width of the individual slit 1329 01:17:16,120 --> 01:17:19,490 in the double and the single slit experiment are the same. 1330 01:17:19,490 --> 01:17:22,850 And you can see that with single slit experiment, 1331 01:17:22,850 --> 01:17:27,140 we also see a very similar pattern 1332 01:17:27,140 --> 01:17:29,900 that you have a central maxima. 1333 01:17:29,900 --> 01:17:40,850 You have a high-intensity light going toward the center 1334 01:17:40,850 --> 01:17:42,590 of the pattern. 1335 01:17:42,590 --> 01:17:49,320 And the intensity actually decrease dramatically really 1336 01:17:49,320 --> 01:17:52,250 quickly as a function of distance. 1337 01:17:52,250 --> 01:17:55,460 And also, you can see that the pattern actually 1338 01:17:55,460 --> 01:17:59,780 matches with what you see in the double slit experiment 1339 01:17:59,780 --> 01:18:00,680 very well. 1340 01:18:00,680 --> 01:18:03,710 And that is actually pretty remarkable. 1341 01:18:03,710 --> 01:18:08,240 And from these two experiment, we 1342 01:18:08,240 --> 01:18:12,510 understand why we have also a complicated structure 1343 01:18:12,510 --> 01:18:15,620 in the double slit experiment, not 1344 01:18:15,620 --> 01:18:18,700 just like many, many little maximas, 1345 01:18:18,700 --> 01:18:20,510 many, many little dots. 1346 01:18:20,510 --> 01:18:23,620 But also, you have this overall modulation 1347 01:18:23,620 --> 01:18:25,370 in the light intensity. 1348 01:18:25,370 --> 01:18:29,060 And that is actually mainly coming from the single slit 1349 01:18:29,060 --> 01:18:30,910 diffraction pattern.