1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high-quality educational resources for free. 5 00:00:10,120 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,650 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,650 --> 00:00:17,536 at ocw.mit.edu. 8 00:00:21,697 --> 00:00:24,030 GEORGE BARBASTATHIS: So you guys walked in just in time. 9 00:00:28,780 --> 00:00:36,907 So today's topic is slightly different, but before we start, 10 00:00:36,907 --> 00:00:38,490 you asked a question last time, right? 11 00:00:38,490 --> 00:00:39,350 AUDIENCE: Yes. 12 00:00:39,350 --> 00:00:42,220 GEORGE BARBASTATHIS: And what's your name again? 13 00:00:42,220 --> 00:00:43,947 AUDIENCE: [INAUDIBLE] 14 00:00:43,947 --> 00:00:45,280 GEORGE BARBASTATHIS: [INAUDIBLE] 15 00:00:45,280 --> 00:00:46,416 AUDIENCE: [INAUDIBLE] 16 00:00:46,416 --> 00:00:48,200 GEORGE BARBASTATHIS: [? Kalpesh, ?] 17 00:00:48,200 --> 00:00:51,270 OK, again, so [? Kalpesh, ?] my friend here, asked 18 00:00:51,270 --> 00:00:52,440 me a question. 19 00:00:52,440 --> 00:00:56,520 And I think you asked it in the lecture before the quiz. 20 00:00:56,520 --> 00:00:57,870 So I answered it. 21 00:00:57,870 --> 00:00:59,310 I guess you were not convinced. 22 00:00:59,310 --> 00:01:03,500 So he asked me again just before the quiz started. 23 00:01:03,500 --> 00:01:08,265 I told him, well, wait until actually we, the class, 24 00:01:08,265 --> 00:01:11,265 have convened, so we can all discuss it together. 25 00:01:11,265 --> 00:01:13,140 So I was thinking how to answer his question. 26 00:01:13,140 --> 00:01:15,970 And then I decided it was a really interesting question. 27 00:01:15,970 --> 00:01:19,470 So I made some additional slides to discuss the topic 28 00:01:19,470 --> 00:01:20,853 that he asked. 29 00:01:20,853 --> 00:01:22,520 And these slides are not on the website, 30 00:01:22,520 --> 00:01:24,330 so don't look for them. 31 00:01:24,330 --> 00:01:27,120 You cannot find them, even though I will probably post 32 00:01:27,120 --> 00:01:27,690 them anyway. 33 00:01:27,690 --> 00:01:29,933 But for now, you just have to watch 34 00:01:29,933 --> 00:01:31,350 and participate in the discussion, 35 00:01:31,350 --> 00:01:33,350 and see what happens. 36 00:01:33,350 --> 00:01:35,400 So [? Kalpesh ?] asked the following, 37 00:01:35,400 --> 00:01:39,330 when I started, actually, the class, I 38 00:01:39,330 --> 00:01:45,030 claimed that optical rays have to satisfy Fermat's principle. 39 00:01:45,030 --> 00:01:49,590 So therefore, they must always go where their path is minimal. 40 00:01:49,590 --> 00:01:54,670 So in the case of a GRIN, I use the notation here 41 00:01:54,670 --> 00:01:59,320 where the gray area in the slide corresponds 42 00:01:59,320 --> 00:02:01,590 to high index of refraction. 43 00:02:01,590 --> 00:02:04,650 The sort of whitest area is low in index. 44 00:02:04,650 --> 00:02:06,490 So this is like a typical GRIN. 45 00:02:06,490 --> 00:02:09,520 You can imagine this profile may be quadratic or something 46 00:02:09,520 --> 00:02:11,380 like that. 47 00:02:11,380 --> 00:02:15,460 And what we said in the class, and you all 48 00:02:15,460 --> 00:02:17,770 believed me except for [? Kalpesh, ?] I guess. 49 00:02:17,770 --> 00:02:21,040 What I said in the class is that if you have a ray that enters 50 00:02:21,040 --> 00:02:25,740 this medium sort of off axis, like shown here, 51 00:02:25,740 --> 00:02:27,730 the ray will actually bend inwards. 52 00:02:27,730 --> 00:02:33,240 So all of the rays will come to a focus somewhere around there. 53 00:02:33,240 --> 00:02:38,200 But [? Kalpesh ?] asked, why doesn't it do this? 54 00:02:38,200 --> 00:02:40,488 I mean, the rays here seems to actually-- 55 00:02:40,488 --> 00:02:42,280 it seems to do the wrong thing, doesn't it? 56 00:02:42,280 --> 00:02:46,020 It seems to go towards the high-index area, 57 00:02:46,020 --> 00:02:48,150 whereas the minimum path principle 58 00:02:48,150 --> 00:02:51,440 would suggest that the ray should go outwards like this, 59 00:02:51,440 --> 00:02:53,040 towards the low-index area. 60 00:02:53,040 --> 00:02:54,860 So the question is, what will the ray do? 61 00:02:54,860 --> 00:02:57,270 Will it do this or that? 62 00:02:57,270 --> 00:02:58,680 We know from experience-- 63 00:02:58,680 --> 00:03:02,760 I mean, we know from experience, from experiment, I should say-- 64 00:03:02,760 --> 00:03:04,320 that gain optics do focus. 65 00:03:04,320 --> 00:03:07,470 So we know that physics actually behaves like this. 66 00:03:07,470 --> 00:03:10,200 But the question is, how is that consistent with our description 67 00:03:10,200 --> 00:03:11,260 of the physical system? 68 00:03:11,260 --> 00:03:12,177 What is going on here? 69 00:03:14,980 --> 00:03:16,480 So that is [? Kalpesh's ?] question. 70 00:03:16,480 --> 00:03:21,790 And what I said when he asked it in class, more or less-- 71 00:03:21,790 --> 00:03:23,740 I cannot remember exactly what I said-- 72 00:03:23,740 --> 00:03:27,400 but I said to properly contemplate the answer, 73 00:03:27,400 --> 00:03:30,372 we must consider not just this ray. 74 00:03:30,372 --> 00:03:32,080 If we just have this ray, we don't really 75 00:03:32,080 --> 00:03:34,000 know whether the ray will go. 76 00:03:34,000 --> 00:03:38,860 But we also have another ray, the one that is on axis. 77 00:03:38,860 --> 00:03:40,510 Now these two rays, they actually 78 00:03:40,510 --> 00:03:42,700 originated at the same point. 79 00:03:42,700 --> 00:03:44,020 Where did they originate? 80 00:03:44,020 --> 00:03:47,870 At infinity, somewhere far out to the left over here. 81 00:03:47,870 --> 00:03:52,930 And what I said is that basically these rays, 82 00:03:52,930 --> 00:03:57,760 because this ray goes through a longer optical path, 83 00:03:57,760 --> 00:04:04,060 because it is entered in a high-index medium, 84 00:04:04,060 --> 00:04:07,840 this has too much, its optical path. 85 00:04:07,840 --> 00:04:11,320 So this ray actually enters a lower index. 86 00:04:11,320 --> 00:04:14,380 So therefore, it starts with a sort of predicament 87 00:04:14,380 --> 00:04:17,410 that points to a smaller optical path. 88 00:04:17,410 --> 00:04:18,970 So in order to catch up, in order 89 00:04:18,970 --> 00:04:23,380 to match the longer optical path of this ray, 90 00:04:23,380 --> 00:04:28,270 this ray, the sort of off-axis ray, 91 00:04:28,270 --> 00:04:30,880 must follow along at a geometrical distance 92 00:04:30,880 --> 00:04:35,770 so that the longer geometrical distance times the lower index 93 00:04:35,770 --> 00:04:38,426 kind of will balance, will compensate. 94 00:04:40,990 --> 00:04:43,720 So this is where we left it. 95 00:04:43,720 --> 00:04:51,510 But then [? Kalpesh ?] came back to me at the quiz, 96 00:04:51,510 --> 00:04:56,410 and he told me, I don't believe it. 97 00:04:56,410 --> 00:04:58,512 You want to rephrase your question? 98 00:04:58,512 --> 00:04:59,680 AUDIENCE: No, [INAUDIBLE]. 99 00:04:59,680 --> 00:05:02,180 GEORGE BARBASTATHIS: OK, so I presented your objection well. 100 00:05:04,790 --> 00:05:09,170 So first of all, does anybody else from here or from Boston 101 00:05:09,170 --> 00:05:11,610 have some additional insight about this? 102 00:05:11,610 --> 00:05:13,220 I can go back to the original. 103 00:05:13,220 --> 00:05:15,230 We'll call it the [? "Kalpesh ?] Question." 104 00:05:15,230 --> 00:05:16,390 Yes. 105 00:05:16,390 --> 00:05:28,170 AUDIENCE: [INAUDIBLE] 106 00:05:28,170 --> 00:05:30,728 AUDIENCE: We didn't hear the question. 107 00:05:30,728 --> 00:05:32,520 GEORGE BARBASTATHIS: Is your button pushed? 108 00:05:32,520 --> 00:05:33,020 Yes. 109 00:05:33,020 --> 00:05:35,400 You have to make sure that the red light is on. 110 00:05:35,400 --> 00:05:36,330 AUDIENCE: OK. 111 00:05:36,330 --> 00:05:39,940 Yeah, in the standard case of an interface, 112 00:05:39,940 --> 00:05:43,450 we have a ray propagating from left to right 113 00:05:43,450 --> 00:05:46,810 and we have an interface, which is top to bottom. 114 00:05:46,810 --> 00:05:48,970 But in this case, the interface could 115 00:05:48,970 --> 00:05:52,390 be considered to be left to right, couldn't we? 116 00:05:52,390 --> 00:05:56,990 Because the refractive index change is from top to bottom. 117 00:05:56,990 --> 00:05:59,020 So the interface-- 118 00:05:59,020 --> 00:06:02,170 I'm just thinking that if we consider the interface to be 119 00:06:02,170 --> 00:06:04,060 from left to right, then the normal 120 00:06:04,060 --> 00:06:05,680 will be from top to bottom. 121 00:06:05,680 --> 00:06:09,567 And still, the Snell's law will lead us to the correct answer, 122 00:06:09,567 --> 00:06:10,400 correct [INAUDIBLE]. 123 00:06:10,400 --> 00:06:11,442 GEORGE BARBASTATHIS: Yes. 124 00:06:11,442 --> 00:06:14,270 You can actually-- by the time you 125 00:06:14,270 --> 00:06:17,510 reach here whether the ray has already been, 126 00:06:17,510 --> 00:06:18,690 you can make that argument. 127 00:06:18,690 --> 00:06:20,320 Because now the ray is really-- 128 00:06:20,320 --> 00:06:21,445 [INTERPOSING VOICES] 129 00:06:21,445 --> 00:06:22,070 AUDIENCE: Yeah. 130 00:06:22,070 --> 00:06:22,640 GEORGE BARBASTATHIS: But the question 131 00:06:22,640 --> 00:06:25,580 is, why should the ray start bending inwards to begin with? 132 00:06:25,580 --> 00:06:27,410 Why should it not bend this way? 133 00:06:27,410 --> 00:06:28,220 AUDIENCE: Yeah. 134 00:06:28,220 --> 00:06:29,520 GEORGE BARBASTATHIS: And if it starts bending this way, 135 00:06:29,520 --> 00:06:32,160 then Snell's law also said that it should go like this. 136 00:06:32,160 --> 00:06:35,270 So there's something amiss here. 137 00:06:35,270 --> 00:06:36,030 AUDIENCE: Yeah. 138 00:06:36,030 --> 00:06:37,030 GEORGE BARBASTATHIS: OK. 139 00:06:41,183 --> 00:06:42,850 So there's no doubt-- as I said, there's 140 00:06:42,850 --> 00:06:45,580 no doubt that the correct physics is this one. 141 00:06:45,580 --> 00:06:48,520 The question is, is our model correct? 142 00:06:48,520 --> 00:06:51,760 Is our claim, all this stuff we developed about optical paths 143 00:06:51,760 --> 00:06:54,640 and so on, does it really agree with the physics? 144 00:06:58,430 --> 00:07:03,530 So I don't hear any-- any other comments or objections 145 00:07:03,530 --> 00:07:04,060 about this? 146 00:07:04,060 --> 00:07:13,280 Or-- so let's go with a little bit more care 147 00:07:13,280 --> 00:07:16,190 through this argument. 148 00:07:16,190 --> 00:07:17,820 What is happening. 149 00:07:17,820 --> 00:07:20,210 Let's take out the medium, so there's nothing now, 150 00:07:20,210 --> 00:07:21,840 just free space. 151 00:07:21,840 --> 00:07:24,770 So the rays started at infinity. 152 00:07:24,770 --> 00:07:27,380 They're heading towards infinity. 153 00:07:27,380 --> 00:07:31,710 Everybody's happy with a straight path, because clearly, 154 00:07:31,710 --> 00:07:34,370 the straight line in uniform space 155 00:07:34,370 --> 00:07:37,700 is the shortest between two points. 156 00:07:37,700 --> 00:07:40,130 And clearly, these two rays-- 157 00:07:40,130 --> 00:07:43,830 they never really meet, but we say they meet at infinity 158 00:07:43,830 --> 00:07:47,510 at the left and they also meet again at infinity to the right. 159 00:07:47,510 --> 00:07:52,430 And they will have traversed the same optical path by the same-- 160 00:07:52,430 --> 00:07:54,470 by the time they meet again at infinity. 161 00:07:54,470 --> 00:07:58,640 Now, you might ask, well, what does it really mean? 162 00:07:58,640 --> 00:08:00,680 I mean, after all, the path itself is infinite. 163 00:08:00,680 --> 00:08:03,890 What does it mean they travel equal infinite paths? 164 00:08:03,890 --> 00:08:06,260 Well, I can get, actually, around this problem 165 00:08:06,260 --> 00:08:10,550 pretty easily because I can actually put some boundaries 166 00:08:10,550 --> 00:08:11,870 to the left and to the right. 167 00:08:11,870 --> 00:08:15,170 And I can keep these boundaries at a finite distance. 168 00:08:15,170 --> 00:08:18,210 Then you can definitely measure the optical path. 169 00:08:18,210 --> 00:08:20,240 Then you can start moving the boundaries 170 00:08:20,240 --> 00:08:23,990 farther away to the left and farther away to the right. 171 00:08:23,990 --> 00:08:27,500 The optical paths remain the same as I move the boundaries. 172 00:08:27,500 --> 00:08:30,570 And I can continue this process until the boundaries 173 00:08:30,570 --> 00:08:32,360 are really, really very far. 174 00:08:32,360 --> 00:08:35,900 This is mathematically really what we mean by "infinity." 175 00:08:35,900 --> 00:08:37,789 So I can get around this difficulty. 176 00:08:37,789 --> 00:08:41,510 I can certainly talk about equal paths, 177 00:08:41,510 --> 00:08:45,590 even if the paths are actually infinite. 178 00:08:45,590 --> 00:08:47,810 This is sort of a technicality, but nevertheless 179 00:08:47,810 --> 00:08:53,870 an important one so we know that we're on solid ground. 180 00:08:53,870 --> 00:08:56,075 So I hope I have convinced you that Fermat 181 00:08:56,075 --> 00:08:59,420 is happy in this case of free space, 182 00:08:59,420 --> 00:09:02,450 because the rays indeed propagate 183 00:09:02,450 --> 00:09:05,000 through the minimum path, and indeed, they're 184 00:09:05,000 --> 00:09:06,740 all equal in terms of path. 185 00:09:12,710 --> 00:09:15,980 Now, this is really not anything new. 186 00:09:15,980 --> 00:09:18,120 This is what I said before. 187 00:09:18,120 --> 00:09:23,420 And basically, this says that the longer geometrical 188 00:09:23,420 --> 00:09:28,550 path actually gets to be multiplied by a lower index, 189 00:09:28,550 --> 00:09:33,980 so therefore, by the time the rays meet again here, 190 00:09:33,980 --> 00:09:36,710 they still have equal optical paths. 191 00:09:36,710 --> 00:09:39,980 Because the path to the left is equal-- they both came 192 00:09:39,980 --> 00:09:43,040 from infinity-- then the two trajectories here, 193 00:09:43,040 --> 00:09:45,710 if you sort integrate the optical path all along, 194 00:09:45,710 --> 00:09:47,060 you would get equal answer. 195 00:09:52,597 --> 00:09:53,180 Are you happy? 196 00:09:57,270 --> 00:09:59,714 Maybe push the button in. 197 00:09:59,714 --> 00:10:06,740 AUDIENCE: [INAUDIBLE] Instead of a wave coming 198 00:10:06,740 --> 00:10:09,590 towards a whole cross-section of the GRIN lens, 199 00:10:09,590 --> 00:10:13,940 suppose if you have a small spot coming only towards the top. 200 00:10:13,940 --> 00:10:17,950 So will it go towards up in that case? 201 00:10:17,950 --> 00:10:19,450 GEORGE BARBASTATHIS: So the question 202 00:10:19,450 --> 00:10:21,370 is, suppose I take out this ray. 203 00:10:21,370 --> 00:10:25,800 Will the ray-- will this thing also focus? 204 00:10:25,800 --> 00:10:26,550 What do you think? 205 00:10:34,390 --> 00:10:38,210 AUDIENCE: I think we have only a single ray towards the top, 206 00:10:38,210 --> 00:10:42,200 which is coming towards the lower refractive index, 207 00:10:42,200 --> 00:10:44,980 then it should go towards the lower refractive index 208 00:10:44,980 --> 00:10:50,680 because it doesn't need to match with the same phase 209 00:10:50,680 --> 00:10:53,960 as the center wave. 210 00:10:53,960 --> 00:10:57,700 GEORGE BARBASTATHIS: OK, let's not quite 211 00:10:57,700 --> 00:11:00,040 argue about a single ray, because there's 212 00:11:00,040 --> 00:11:02,170 no such thing as a single ray. 213 00:11:02,170 --> 00:11:07,630 If you illuminate a very, real infinitesimal optical spot 214 00:11:07,630 --> 00:11:11,260 over here, as we will learn in wave optics, what you will get 215 00:11:11,260 --> 00:11:17,020 is not a sort of straight line of light, 216 00:11:17,020 --> 00:11:18,860 but you will get a very strong diffraction. 217 00:11:18,860 --> 00:11:21,610 So what will happen if you illuminate a single spot here 218 00:11:21,610 --> 00:11:24,310 is the light will spread violently into the GRIN. 219 00:11:24,310 --> 00:11:27,400 And then what will happen is a kind of a complicated issue. 220 00:11:27,400 --> 00:11:34,870 So the way, essentially by claiming the fraction, 221 00:11:34,870 --> 00:11:36,700 I got around the problem that you're 222 00:11:36,700 --> 00:11:40,870 posing, because as long as I have a finite pencil of rays 223 00:11:40,870 --> 00:11:43,780 coming in here, then the same thing will happen, 224 00:11:43,780 --> 00:11:47,620 because I can always compare two rays that enter 225 00:11:47,620 --> 00:11:49,090 at slightly different indices. 226 00:11:49,090 --> 00:11:50,770 Then the same argument follows. 227 00:11:50,770 --> 00:11:55,130 So this pencil will also kind of focus in the same-- 228 00:11:55,130 --> 00:12:01,510 so basically, I can remove parts of that ray diagram 229 00:12:01,510 --> 00:12:03,970 and the same thing will still happen according 230 00:12:03,970 --> 00:12:05,130 to geometrical optics. 231 00:12:05,130 --> 00:12:07,180 Diffraction says differently, but according 232 00:12:07,180 --> 00:12:10,930 to geometrical optics, I can remove portions of these rays 233 00:12:10,930 --> 00:12:15,362 and still have focusing. 234 00:12:15,362 --> 00:12:17,620 There's another objection that I haven't had yet. 235 00:12:21,200 --> 00:12:23,890 I'm not finished. 236 00:12:23,890 --> 00:12:26,000 I am not satisfied that I answered 237 00:12:26,000 --> 00:12:27,572 [? Kalpesh's ?] question yet. 238 00:12:27,572 --> 00:12:29,530 And I see that [? Kalpesh ?] is also skeptical, 239 00:12:29,530 --> 00:12:31,570 and I hope someone-- 240 00:12:31,570 --> 00:12:33,850 I have another slide to bring up the objection, 241 00:12:33,850 --> 00:12:38,190 but I'm wondering if someone will bring it up, 242 00:12:38,190 --> 00:12:39,320 or maybe another objection. 243 00:12:39,320 --> 00:12:41,940 At least there's one objection that I thought of, 244 00:12:41,940 --> 00:12:43,140 but there may be others. 245 00:12:47,460 --> 00:12:49,000 AUDIENCE: Question. 246 00:12:49,000 --> 00:12:50,390 GEORGE BARBASTATHIS: Yeah. 247 00:12:50,390 --> 00:12:54,260 AUDIENCE: Why is it that the ray on top bends 248 00:12:54,260 --> 00:12:57,260 to slow down to meet the ray that's 249 00:12:57,260 --> 00:13:00,050 traveling through the denser medium versus both of them 250 00:13:00,050 --> 00:13:03,188 traveling to the less dense optical medium, 251 00:13:03,188 --> 00:13:04,230 because that could also-- 252 00:13:04,230 --> 00:13:04,820 [INTERPOSING VOICES] 253 00:13:04,820 --> 00:13:06,570 GEORGE BARBASTATHIS: That's the objection. 254 00:13:06,570 --> 00:13:08,420 That's exactly the objection. 255 00:13:08,420 --> 00:13:13,160 So the objection is who says that the central ray should 256 00:13:13,160 --> 00:13:15,530 go straight? 257 00:13:15,530 --> 00:13:20,150 I sort of silently took it for granted 258 00:13:20,150 --> 00:13:23,050 that the central ray here goes straight, 259 00:13:23,050 --> 00:13:25,010 and therefore, this other ray has 260 00:13:25,010 --> 00:13:27,620 to race in order to meet it. 261 00:13:27,620 --> 00:13:29,970 Why don't the two rays go like this? 262 00:13:29,970 --> 00:13:31,730 Is this your question? 263 00:13:31,730 --> 00:13:33,170 I mean, yes. 264 00:13:33,170 --> 00:13:36,493 Why don't the two rays-- why don't they both do like this? 265 00:13:36,493 --> 00:13:38,160 They seem to be doing the correct thing. 266 00:13:38,160 --> 00:13:42,735 They seem to be both going toward the lower index. 267 00:13:46,350 --> 00:13:50,430 We know again-- we can argue that this would happen, 268 00:13:50,430 --> 00:13:53,440 but we know from physics that this does not happen. 269 00:13:53,440 --> 00:13:58,790 So what is the physical-- 270 00:13:58,790 --> 00:14:03,400 AUDIENCE: That can't possibly happen by symmetry. 271 00:14:03,400 --> 00:14:06,160 GEORGE BARBASTATHIS: OK, thank you. 272 00:14:06,160 --> 00:14:08,500 Accurately, it is not a question of symmetry 273 00:14:08,500 --> 00:14:11,800 because indeed, in this case, I assume 274 00:14:11,800 --> 00:14:15,760 that the refractive index is symmetric. 275 00:14:15,760 --> 00:14:19,870 However, now let me see if I can do this without revealing 276 00:14:19,870 --> 00:14:20,830 my next slide. 277 00:14:27,910 --> 00:14:31,750 However, I can make an asymmetric index profile. 278 00:14:31,750 --> 00:14:34,390 So this is a simulation of the GRIN 279 00:14:34,390 --> 00:14:36,840 with a symmetric quadratic index. 280 00:14:36,840 --> 00:14:39,850 Here, I made the profile asymmetric. 281 00:14:39,850 --> 00:14:43,540 So basically, this is the equation here. 282 00:14:43,540 --> 00:14:48,430 I should simply draw it, except I don't have a marker. 283 00:14:51,020 --> 00:14:54,410 Anyway, basically here, the index of refraction 284 00:14:54,410 --> 00:14:57,290 is a parabola that has a different slope 285 00:14:57,290 --> 00:15:00,590 above and different slope below. 286 00:15:00,590 --> 00:15:04,190 So you see that even though this is asymmetric, stubbornly, 287 00:15:04,190 --> 00:15:07,400 the central ray still goes straight. 288 00:15:07,400 --> 00:15:10,130 Therefore, it is not symmetry which makes 289 00:15:10,130 --> 00:15:12,011 the previous case impossible. 290 00:15:18,610 --> 00:15:22,150 AUDIENCE: But, well, I said "symmetry," but if you go 291 00:15:22,150 --> 00:15:27,310 to your next one, that ray that's going straight 292 00:15:27,310 --> 00:15:30,630 along the center-- 293 00:15:30,630 --> 00:15:31,130 is it gone? 294 00:15:31,130 --> 00:15:34,948 Yeah, that ray goes straight, doesn't it? 295 00:15:34,948 --> 00:15:35,990 GEORGE BARBASTATHIS: Yep. 296 00:15:35,990 --> 00:15:37,640 AUDIENCE: And that's because there's 297 00:15:37,640 --> 00:15:43,900 no transverse gradient in the refractive index where 298 00:15:43,900 --> 00:15:46,210 it's propagating, I guess. 299 00:15:46,210 --> 00:15:47,628 GEORGE BARBASTATHIS: OK, yes. 300 00:15:47,628 --> 00:15:48,170 That is true. 301 00:15:48,170 --> 00:15:50,360 There's no movement to push it. 302 00:15:50,360 --> 00:15:52,580 Yeah, and that, I agree with. 303 00:15:52,580 --> 00:15:55,880 However, that really still does not answer his question, 304 00:15:55,880 --> 00:16:01,165 because the reason momentum works, 305 00:16:01,165 --> 00:16:03,040 the reason we apply Snell's Law, for example, 306 00:16:03,040 --> 00:16:07,030 is because we claim path minimization. 307 00:16:07,030 --> 00:16:09,670 So I suppose you could argue this way. 308 00:16:09,670 --> 00:16:11,110 I have another argument, actually, 309 00:16:11,110 --> 00:16:15,073 why this ray must go straight. 310 00:16:15,073 --> 00:16:16,990 Actually, I don't know if it must go straight, 311 00:16:16,990 --> 00:16:23,290 but then another argument why basically this is impossible, 312 00:16:23,290 --> 00:16:25,065 why this is physically incorrect. 313 00:16:32,830 --> 00:16:37,260 AUDIENCE: If you do that, isn't it true that the bottom ray's 314 00:16:37,260 --> 00:16:40,860 path is still longer than the top ray's path because 315 00:16:40,860 --> 00:16:45,750 of the index of refraction is low, and then it's just low 316 00:16:45,750 --> 00:16:47,290 all the way, and-- 317 00:16:47,290 --> 00:16:48,542 yeah. 318 00:16:48,542 --> 00:16:49,750 GEORGE BARBASTATHIS: Exactly. 319 00:16:49,750 --> 00:16:57,190 If you do that, there is no way ever 320 00:16:57,190 --> 00:17:00,550 that this ray will catch up. 321 00:17:00,550 --> 00:17:03,910 Because this ray now, not only is it 322 00:17:03,910 --> 00:17:07,060 going through a higher index of refraction, 323 00:17:07,060 --> 00:17:11,619 it is also following a longer geometrical trajectory. 324 00:17:11,619 --> 00:17:15,970 So therefore, if these two rays ever meet again-- 325 00:17:15,970 --> 00:17:17,555 they might meet at a finite distance. 326 00:17:17,555 --> 00:17:19,180 The way I drew them here, at some point 327 00:17:19,180 --> 00:17:22,690 they will meet again, or they might meet at infinity. 328 00:17:22,690 --> 00:17:25,599 They might somehow become parallel and meet again 329 00:17:25,599 --> 00:17:26,530 at infinity. 330 00:17:26,530 --> 00:17:29,410 But no matter what, this ray over here 331 00:17:29,410 --> 00:17:32,650 has traveled the longer optical path. 332 00:17:32,650 --> 00:17:35,680 And now that is a clear violation of Fermat. 333 00:17:35,680 --> 00:17:38,740 So we basically arrived at an impossibility. 334 00:17:38,740 --> 00:17:42,190 So this is physically, then, incorrect. 335 00:17:42,190 --> 00:17:50,790 That is the reason why the rays have to bend inwards. 336 00:17:50,790 --> 00:17:55,080 And so the proper interpretation of Fermat's principle-- 337 00:17:55,080 --> 00:17:57,270 and actually, the book has an interesting discussion 338 00:17:57,270 --> 00:17:59,300 about this-- 339 00:17:59,300 --> 00:18:04,260 we can claim, yes, that Fermat minimizes the optical path, 340 00:18:04,260 --> 00:18:08,520 but perhaps, at least linguistically a more precise 341 00:18:08,520 --> 00:18:13,110 definition is that Fermat requires 342 00:18:13,110 --> 00:18:17,520 that the path of the rays is stationary. 343 00:18:17,520 --> 00:18:20,190 Now, "stationary" in mathematics-- 344 00:18:20,190 --> 00:18:24,480 what it means is that if you perturb it in some way, 345 00:18:24,480 --> 00:18:27,030 you will get a bigger answer-- 346 00:18:27,030 --> 00:18:28,800 if you perturb it in anyway. 347 00:18:28,800 --> 00:18:31,485 So remember-- I wish I had the marker. 348 00:18:34,120 --> 00:18:35,823 I think I can just use a pen, though. 349 00:18:39,960 --> 00:18:42,570 Let's try with a pen, and you guys tell me in Boston 350 00:18:42,570 --> 00:18:43,410 if you can see it. 351 00:18:43,410 --> 00:18:47,670 Remember with Fermat, we did this analogy. 352 00:18:47,670 --> 00:18:51,090 We said-- you can see it, right-- 353 00:18:51,090 --> 00:18:56,440 we said-- I'm sorry, I should have 354 00:18:56,440 --> 00:18:59,220 said remember with Snell's Law. 355 00:18:59,220 --> 00:19:04,470 Snell's Law says that the ray must do something 356 00:19:04,470 --> 00:19:10,670 like this in order to connect this point to this point. 357 00:19:10,670 --> 00:19:14,480 So mathematically, what we derive 358 00:19:14,480 --> 00:19:17,300 is that any other possibility-- 359 00:19:17,300 --> 00:19:27,640 this, this, this, this, and also, why not this? 360 00:19:27,640 --> 00:19:31,510 These are all different possible ray trajectories. 361 00:19:31,510 --> 00:19:32,930 There's a mathematical tool. 362 00:19:32,930 --> 00:19:34,590 It's called "calculus of variation," 363 00:19:34,590 --> 00:19:36,550 which does a miraculous thing. 364 00:19:36,550 --> 00:19:40,570 It somehow compares all of these possible trajectories, 365 00:19:40,570 --> 00:19:42,920 and then it arrives at the minimum trajectory. 366 00:19:42,920 --> 00:19:44,830 And it tells us, no, no. 367 00:19:44,830 --> 00:19:47,010 No matter what-- these are infinite possibilities. 368 00:19:47,010 --> 00:19:48,190 That you can imagine. 369 00:19:48,190 --> 00:19:50,530 But this tool, calculus of variations, 370 00:19:50,530 --> 00:19:54,490 actually tells us eventually that all of these 371 00:19:54,490 --> 00:19:57,220 give us a longer total optical path. 372 00:19:57,220 --> 00:20:01,120 And the only true one, if you wish, the one that minimizes, 373 00:20:01,120 --> 00:20:04,600 the stationary one, is the one that I drew kind of bolder 374 00:20:04,600 --> 00:20:06,910 here. 375 00:20:06,910 --> 00:20:09,160 So this is the proper way to interpret Fermat. 376 00:20:09,160 --> 00:20:14,560 It is not that a single ray will actually 377 00:20:14,560 --> 00:20:17,410 look for the lower index of refraction. 378 00:20:17,410 --> 00:20:18,940 It is actually that when you have 379 00:20:18,940 --> 00:20:22,270 a bundle of rays that share a common beginning 380 00:20:22,270 --> 00:20:26,140 and a common end, then the trajectory 381 00:20:26,140 --> 00:20:30,700 that these rays will follow will be such that they're all equal, 382 00:20:30,700 --> 00:20:33,850 and that this trajectory now is stationary, in the sense 383 00:20:33,850 --> 00:20:36,620 that if you were to perturb it in any way-- 384 00:20:36,620 --> 00:20:39,070 if you were to move some of these rays up, or down, 385 00:20:39,070 --> 00:20:41,020 or bend them, or whatever-- you would end up 386 00:20:41,020 --> 00:20:45,220 getting a longer trajectory. 387 00:20:45,220 --> 00:20:47,860 That's the proper interpretation of Fermat. 388 00:20:47,860 --> 00:20:52,120 So this is actually a very interesting topic. 389 00:20:52,120 --> 00:20:55,100 And I confess that when I was a student, 390 00:20:55,100 --> 00:20:56,290 I was also bothered by this. 391 00:20:56,290 --> 00:20:59,190 I mean, the rays seems to be doing the wrong thing, 392 00:20:59,190 --> 00:20:59,960 after all. 393 00:20:59,960 --> 00:21:03,070 It seems to be going really the way it shouldn't go. 394 00:21:03,070 --> 00:21:03,700 Why? 395 00:21:03,700 --> 00:21:07,000 The reason is as we just described-- because the path 396 00:21:07,000 --> 00:21:10,170 has to be stationary. 397 00:21:10,170 --> 00:21:12,170 The bottom line out of this, which will actually 398 00:21:12,170 --> 00:21:15,730 be useful in our ensuing discussion 399 00:21:15,730 --> 00:21:20,080 of Hamiltonian optics, is that the index of refraction 400 00:21:20,080 --> 00:21:23,713 is acting a little bit like gravity. 401 00:21:23,713 --> 00:21:25,130 Actually, it's acting like gravity 402 00:21:25,130 --> 00:21:26,480 more than you would think. 403 00:21:26,480 --> 00:21:32,150 The index of refraction seems to be attracting the rays, which 404 00:21:32,150 --> 00:21:34,850 is, again, a little bit counterintuitive if you 405 00:21:34,850 --> 00:21:37,210 interpret Fermat as a minimum path. 406 00:21:37,210 --> 00:21:48,590 But after we went through this explanation, it makes sense. 407 00:21:48,590 --> 00:21:54,790 The index of refraction pretty much collects rays near it. 408 00:21:54,790 --> 00:21:55,892 Yes, there's a question. 409 00:21:55,892 --> 00:21:56,475 AUDIENCE: Yes. 410 00:21:56,475 --> 00:22:01,640 George, I think if you consider the interfaces left to right, 411 00:22:01,640 --> 00:22:03,540 the geometric works out. 412 00:22:03,540 --> 00:22:11,090 And so if we assume any ray coming like this, 413 00:22:11,090 --> 00:22:15,470 going from left bottom to top right, then it will-- 414 00:22:15,470 --> 00:22:19,580 it is going from a higher index to lower index, which 415 00:22:19,580 --> 00:22:22,670 means it will tend to increase its angle with respect 416 00:22:22,670 --> 00:22:23,240 to normal. 417 00:22:23,240 --> 00:22:25,400 So it will keep bending like this. 418 00:22:25,400 --> 00:22:27,990 And at some point, there will be total internal reflection, 419 00:22:27,990 --> 00:22:29,545 and it will bend towards the-- 420 00:22:29,545 --> 00:22:30,110 GEORGE BARBASTATHIS: That's right, yeah. 421 00:22:30,110 --> 00:22:31,430 Doesn't happen here, for example. 422 00:22:31,430 --> 00:22:32,040 If this ray continues-- 423 00:22:32,040 --> 00:22:33,020 AUDIENCE: And then again, it will-- 424 00:22:33,020 --> 00:22:33,650 GEORGE BARBASTATHIS: It will kind of bend. 425 00:22:33,650 --> 00:22:33,860 AUDIENCE: Yeah. 426 00:22:33,860 --> 00:22:35,270 GEORGE BARBASTATHIS: Yeah. 427 00:22:35,270 --> 00:22:36,380 AUDIENCE: So maybe-- 428 00:22:36,380 --> 00:22:38,833 I mean, that's just another way of looking at the problem, 429 00:22:38,833 --> 00:22:40,250 that if you consider the interface 430 00:22:40,250 --> 00:22:45,420 to be normal to the change in the refractive index, then-- 431 00:22:45,420 --> 00:22:45,957 yeah. 432 00:22:45,957 --> 00:22:46,790 [INTERPOSING VOICES] 433 00:22:46,790 --> 00:22:48,582 GEORGE BARBASTATHIS: Actually, that's true. 434 00:22:48,582 --> 00:22:51,080 Snell's Law also justifies this attraction. 435 00:22:51,080 --> 00:22:55,190 I mean, the point that is really hard to swallow 436 00:22:55,190 --> 00:22:59,270 is why this ray starts bending inwards at this interface. 437 00:22:59,270 --> 00:23:01,760 After it has started, then it has to fall, 438 00:23:01,760 --> 00:23:04,850 because Snell says that it should fall. 439 00:23:04,850 --> 00:23:07,210 So that is consistent. 440 00:23:07,210 --> 00:23:08,560 So really, this acts-- 441 00:23:08,560 --> 00:23:11,440 you can think of a mechanical analogy, which 442 00:23:11,440 --> 00:23:14,230 would be like imagine that you take a tube 443 00:23:14,230 --> 00:23:15,360 and you cut it in half. 444 00:23:15,360 --> 00:23:18,700 So now you have these hollow sphere. 445 00:23:18,700 --> 00:23:22,540 And then you launch a billiard ball into that surface. 446 00:23:22,540 --> 00:23:25,480 What will the billiard ball do? 447 00:23:25,480 --> 00:23:26,680 It will spin. 448 00:23:26,680 --> 00:23:30,070 It will go down, and then up again, and then down again, 449 00:23:30,070 --> 00:23:31,870 and spin, and spin, and spin. 450 00:23:31,870 --> 00:23:35,710 So basically, the surface in this sort 451 00:23:35,710 --> 00:23:39,760 of mechanical analogy, the equivalent of the surface shape 452 00:23:39,760 --> 00:23:41,605 is the index of refraction here. 453 00:23:41,605 --> 00:23:44,230 And of course, the billiard ball is subject to its own gravity. 454 00:23:44,230 --> 00:23:47,080 And the surface-- how do you call it, 455 00:23:47,080 --> 00:23:49,840 the [INAUDIBLE] contact from the surface, 456 00:23:49,840 --> 00:23:50,990 and so on, and so forth. 457 00:23:50,990 --> 00:23:57,550 But the bottom line is that this is a mechanical analog. 458 00:23:57,550 --> 00:23:58,930 I'm sorry, this is-- 459 00:23:58,930 --> 00:24:01,690 whatever, the mechanical analogy for this 460 00:24:01,690 --> 00:24:03,885 is gravity, attraction. 461 00:24:07,861 --> 00:24:12,153 AUDIENCE: [INAUDIBLE] 462 00:24:12,153 --> 00:24:13,570 GEORGE BARBASTATHIS: Yes, provided 463 00:24:13,570 --> 00:24:16,852 you define the gravitational potential appropriately 464 00:24:16,852 --> 00:24:18,310 in terms of the initial refraction. 465 00:24:18,310 --> 00:24:20,395 And this will actually come up near the end of the Hamiltonian 466 00:24:20,395 --> 00:24:21,280 lecture. 467 00:24:21,280 --> 00:24:23,750 So I will show you exactly the formula. 468 00:24:23,750 --> 00:24:26,350 Did everybody hear his question? 469 00:24:26,350 --> 00:24:27,345 AUDIENCE: No. 470 00:24:27,345 --> 00:24:29,220 GEORGE BARBASTATHIS: OK, could you repeat it? 471 00:24:29,220 --> 00:24:31,510 AUDIENCE: Yeah, I was asking, is the analogy 472 00:24:31,510 --> 00:24:35,000 between the gravitational force and the reflective [INAUDIBLE] 473 00:24:35,000 --> 00:24:35,500 complete? 474 00:24:37,567 --> 00:24:39,650 GEORGE BARBASTATHIS: Yeah, and you have an answer. 475 00:24:39,650 --> 00:24:44,920 OK, now that I have a marker, let me also answer-- 476 00:24:44,920 --> 00:24:50,070 so in this case, what I did is on the left-hand side, 477 00:24:50,070 --> 00:24:54,370 we have the familiar quadratic index profile. 478 00:24:56,890 --> 00:25:02,130 So this is n0 minus the some coefficient times x squared. 479 00:25:02,130 --> 00:25:06,520 On the right-hand side, what I did is 480 00:25:06,520 --> 00:25:09,460 I chose an index that [INAUDIBLE] like this. 481 00:25:13,440 --> 00:25:15,480 I'm exaggerating, but basically, it 482 00:25:15,480 --> 00:25:17,550 is still a parabola on the left and the right, 483 00:25:17,550 --> 00:25:20,280 but has a different curvature on the left and the right. 484 00:25:20,280 --> 00:25:22,847 And the point here is that the central ray still 485 00:25:22,847 --> 00:25:23,430 goes straight. 486 00:25:23,430 --> 00:25:25,320 Of course, this is a terrible lens. 487 00:25:25,320 --> 00:25:26,900 It does something really horrible 488 00:25:26,900 --> 00:25:28,472 back here, does not focus properly, 489 00:25:28,472 --> 00:25:29,430 and so on and so forth. 490 00:25:29,430 --> 00:25:35,670 But the only reason I did this is to show that the ray here 491 00:25:35,670 --> 00:25:38,370 goes straight not because the index of refraction 492 00:25:38,370 --> 00:25:43,300 is symmetrical, even though Colin's point is well taken. 493 00:25:43,300 --> 00:25:45,550 I mean, the reason it goes straight here-- 494 00:25:45,550 --> 00:25:48,070 again, you can think of it in terms of Snell's Law. 495 00:25:48,070 --> 00:25:50,350 It does not want to go this way, because it 496 00:25:50,350 --> 00:25:58,880 would bend to a longer optical path then the ray next to it. 497 00:25:58,880 --> 00:26:02,320 It does not want to go down, because again, it 498 00:26:02,320 --> 00:26:05,320 would end up with a longer optical path with respect 499 00:26:05,320 --> 00:26:06,820 to the ray below it. 500 00:26:06,820 --> 00:26:09,810 So therefore, it has to go straight. 501 00:26:09,810 --> 00:26:12,360 Can you imagine a situation-- 502 00:26:12,360 --> 00:26:13,380 is it always straight? 503 00:26:13,380 --> 00:26:16,620 Can I actually bend this ray? 504 00:26:16,620 --> 00:26:19,140 What would I have to do to make the central ray bend? 505 00:26:23,680 --> 00:26:25,180 Actually, Colin answered it already. 506 00:26:30,730 --> 00:26:32,440 Yeah, they cannot see you. 507 00:26:32,440 --> 00:26:34,010 Can you-- 508 00:26:34,010 --> 00:26:36,287 AUDIENCE: Curve the whole profile. 509 00:26:36,287 --> 00:26:38,620 GEORGE BARBASTATHIS: So you have to put a gradient along 510 00:26:38,620 --> 00:26:40,480 z in the index of refraction. 511 00:26:40,480 --> 00:26:44,650 Then if the ray sees a tilted interface, which basically 512 00:26:44,650 --> 00:26:48,100 means a gradient of the index of refraction here, it will bend. 513 00:26:48,100 --> 00:26:52,380 Because again, Snell tells it to bend. 514 00:26:52,380 --> 00:26:53,660 But since there is no-- 515 00:26:53,660 --> 00:26:57,350 since here, the index has no z dependency, 516 00:26:57,350 --> 00:26:58,700 then the ray must go straight. 517 00:27:36,330 --> 00:27:41,430 So first of all, thanks to [? Kalpesh ?] for asking 518 00:27:41,430 --> 00:27:42,930 this question, because it turned out 519 00:27:42,930 --> 00:27:46,650 to be a really, really helpful one. 520 00:27:51,480 --> 00:27:56,750 And in addition to that, today we have kind of two more things 521 00:27:56,750 --> 00:27:57,650 to do. 522 00:27:57,650 --> 00:28:02,120 So I need to really finish the Hamiltonian optics discussion, 523 00:28:02,120 --> 00:28:05,360 and then we'll start talking about wave optics. 524 00:28:05,360 --> 00:28:08,540 So we'll see how far we go. 525 00:28:12,090 --> 00:28:14,640 And I have to make some announcements near the end. 526 00:28:14,640 --> 00:28:18,160 So I will try to save about five minutes near the end 527 00:28:18,160 --> 00:28:20,750 for my announcements. 528 00:28:20,750 --> 00:28:24,830 So last time, we were discussing gradient index optics. 529 00:28:27,890 --> 00:28:37,860 And just to briefly remind you, this 530 00:28:37,860 --> 00:28:40,630 is actually what prompted the [? Kalpesh ?] question. 531 00:28:40,630 --> 00:28:44,100 We went through this derivation where we had the gradient index 532 00:28:44,100 --> 00:28:44,670 element. 533 00:28:44,670 --> 00:28:50,970 I will stop saying gradient index it's difficult for me 534 00:28:50,970 --> 00:28:52,180 to pronounce-- 535 00:28:52,180 --> 00:28:53,180 a GRIN light. 536 00:28:53,180 --> 00:28:58,050 So we had the GRIN element, and we did this simple derivation 537 00:28:58,050 --> 00:29:00,000 which basically said that this GRIN 538 00:29:00,000 --> 00:29:03,100 element is acting like a lens. 539 00:29:03,100 --> 00:29:05,850 And again, with our newfound wisdom, 540 00:29:05,850 --> 00:29:08,250 it is acting like a lens by virtue 541 00:29:08,250 --> 00:29:11,795 of the attraction of the rays by the origin 542 00:29:11,795 --> 00:29:16,430 of high index near the center of this element. 543 00:29:16,430 --> 00:29:25,280 And we also went through this other case of axial GRIN. 544 00:29:25,280 --> 00:29:29,300 And we also said that this is-- 545 00:29:29,300 --> 00:29:31,340 now that's interesting. 546 00:29:31,340 --> 00:29:35,300 If you only have z gradient, then it is still not a lens. 547 00:29:35,300 --> 00:29:37,050 I'll let you think about this by yourself. 548 00:29:37,050 --> 00:29:39,140 In order to make it a lens, you have 549 00:29:39,140 --> 00:29:42,848 to have an x gradient, the index of refraction. 550 00:29:42,848 --> 00:29:44,390 So this case is somewhere in between, 551 00:29:44,390 --> 00:29:48,890 where you have a z gradient-- so the index is higher 552 00:29:48,890 --> 00:29:52,680 in the front and lower in the back-- 553 00:29:52,680 --> 00:29:54,720 but in order to get optical power, that 554 00:29:54,720 --> 00:29:58,530 is to get focusing power, in this case you actually grind. 555 00:29:58,530 --> 00:30:00,990 In addition, you grind it to a sphere. 556 00:30:00,990 --> 00:30:04,800 And we said that the reason you do this axial GRIN here 557 00:30:04,800 --> 00:30:06,750 is because you're going to correct 558 00:30:06,750 --> 00:30:08,400 for spherical aberration. 559 00:30:08,400 --> 00:30:14,400 So if you compare the paths now of this ray to this ray, 560 00:30:14,400 --> 00:30:18,210 in this case, this ray goes through a longer optical path, 561 00:30:18,210 --> 00:30:21,310 because the index is uniformly high here. 562 00:30:21,310 --> 00:30:23,250 Therefore, it bends faster. 563 00:30:23,250 --> 00:30:25,950 In this case, by lowering the index of refraction 564 00:30:25,950 --> 00:30:27,910 that this ray goes, you actually force 565 00:30:27,910 --> 00:30:31,170 it to go to a longer geometrical trajectory. 566 00:30:31,170 --> 00:30:34,380 Therefore, you're pushing it further to the back. 567 00:30:34,380 --> 00:30:38,130 This is another way to interpret the way this element corrects. 568 00:30:38,130 --> 00:30:40,080 You can interpret it with Snell's law. 569 00:30:40,080 --> 00:30:41,920 This is what we did last time. 570 00:30:41,920 --> 00:30:44,400 But now, with our newfound wisdom 571 00:30:44,400 --> 00:30:48,210 on optical paths and Fermat, we can also explain it this way. 572 00:30:48,210 --> 00:30:50,730 I lowered the optical path by virtue 573 00:30:50,730 --> 00:30:52,530 of lowering the index here. 574 00:30:52,530 --> 00:30:55,110 Therefore, the ray has to go outwards in order 575 00:30:55,110 --> 00:31:00,800 to increase its travel distance, and still 576 00:31:00,800 --> 00:31:04,910 maintain the same path as the central ray. 577 00:31:04,910 --> 00:31:10,550 So the net effect is that you're correcting the bad quality 578 00:31:10,550 --> 00:31:15,410 focus here that was caused by the spherical aberration. 579 00:31:15,410 --> 00:31:17,480 And I believe where we left it last time is 580 00:31:17,480 --> 00:31:20,920 we asked this question, what if I 581 00:31:20,920 --> 00:31:24,600 have an arbitrary index of refraction 582 00:31:24,600 --> 00:31:26,410 as a function of position? 583 00:31:26,410 --> 00:31:31,300 What if this is not quadratic, this is not axial, 584 00:31:31,300 --> 00:31:35,820 this is some general function of r, the Cartesian coordinate? 585 00:31:35,820 --> 00:31:36,540 What then? 586 00:31:36,540 --> 00:31:38,660 What is the-- 587 00:31:44,655 --> 00:31:45,780 I lost my train of thought. 588 00:31:45,780 --> 00:31:47,190 What is the path of the-- 589 00:31:47,190 --> 00:31:49,160 what is the trajectory of the rays 590 00:31:49,160 --> 00:31:53,270 that go through this variable index medium. 591 00:31:53,270 --> 00:31:55,940 And I don't remember how far I went here, 592 00:31:55,940 --> 00:31:58,580 but we basically took a turn here, 593 00:31:58,580 --> 00:32:03,950 and we started talking about mechanics probably 594 00:32:03,950 --> 00:32:06,170 to some of your surprise. 595 00:32:06,170 --> 00:32:07,670 Can you remind me, how far did I go? 596 00:32:07,670 --> 00:32:09,420 Did I actually finish this example or not? 597 00:32:09,420 --> 00:32:10,963 AUDIENCE: Yeah, I think so. 598 00:32:10,963 --> 00:32:11,880 Yeah, we covered this. 599 00:32:13,817 --> 00:32:16,400 GEORGE BARBASTATHIS: So I don't feel like going into it again, 600 00:32:16,400 --> 00:32:22,260 anyway, because this is sort of good, old physics 601 00:32:22,260 --> 00:32:25,170 101 kind of thing. 602 00:32:25,170 --> 00:32:31,490 What might be a little bit unfamiliar is this development 603 00:32:31,490 --> 00:32:35,510 here, which some of you may have seen, some of you 604 00:32:35,510 --> 00:32:37,420 may not have seen. 605 00:32:37,420 --> 00:32:41,380 So this is really not doing anything spectacular. 606 00:32:41,380 --> 00:32:44,680 This is a simple mathematical manipulation. 607 00:32:44,680 --> 00:32:48,840 So we start with the equation of motion 608 00:32:48,840 --> 00:32:50,680 for the mechanical oscillator. 609 00:32:50,680 --> 00:32:53,340 So the equation of motion for the mechanical-- 610 00:32:53,340 --> 00:32:58,890 I'm sorry not the equation of motion, 611 00:32:58,890 --> 00:33:01,660 but the energy for the mechanical oscillator. 612 00:33:01,660 --> 00:33:03,870 So the energy consists, of course, of two terms. 613 00:33:03,870 --> 00:33:08,310 One of them is, if you wish, the elastic energy, 614 00:33:08,310 --> 00:33:13,890 the potential that is due to the restoring force of the spring. 615 00:33:13,890 --> 00:33:15,450 And the other is the kinetic energy. 616 00:33:18,030 --> 00:33:24,230 And v is the velocity, so v is basically x dot, 617 00:33:24,230 --> 00:33:26,925 the first time derivative of the position. 618 00:33:29,700 --> 00:33:31,810 So this is one thing that we know. 619 00:33:31,810 --> 00:33:33,270 This is the energy of the system. 620 00:33:33,270 --> 00:33:36,420 And since in our model, there is no dissipation, 621 00:33:36,420 --> 00:33:39,630 the energy must be conserved. 622 00:33:39,630 --> 00:33:43,770 I can also write an equation of motion for this system. 623 00:33:43,770 --> 00:33:48,000 And since it is a simple spring force kind of system, 624 00:33:48,000 --> 00:33:51,810 I expect my equation of motion to be second order, 625 00:33:51,810 --> 00:33:53,700 so there would be something of this sort-- 626 00:33:53,700 --> 00:33:59,880 m x double dot plus kx equals the force, the excitation. 627 00:34:02,390 --> 00:34:05,710 So let's say that this system is free. 628 00:34:05,710 --> 00:34:07,100 There is no force. 629 00:34:07,100 --> 00:34:12,620 So if it is moving, it is moving because of a prior excitation, 630 00:34:12,620 --> 00:34:14,659 which if I start it [INAUDIBLE] it 631 00:34:14,659 --> 00:34:17,230 will go on forever, because there is no dissipation. 632 00:34:17,230 --> 00:34:18,913 There is no place for the energy to go. 633 00:34:18,913 --> 00:34:21,080 If you get this thing moving, it will go on forever. 634 00:34:21,080 --> 00:34:22,730 But now, there's no force. 635 00:34:22,730 --> 00:34:25,310 So basically, I can write down the equation of motion 636 00:34:25,310 --> 00:34:29,068 like this, as a homogeneous equation. 637 00:34:29,068 --> 00:34:30,860 So there's is a trick that those of you who 638 00:34:30,860 --> 00:34:34,630 use Matlab to solve differential equations are very familiar. 639 00:34:34,630 --> 00:34:36,350 And the trick is the following-- 640 00:34:36,350 --> 00:34:39,830 you break this higher order differential equation. 641 00:34:39,830 --> 00:34:44,449 You break it down into as many as you need first order 642 00:34:44,449 --> 00:34:46,010 differential equations. 643 00:34:46,010 --> 00:34:48,679 Since this is a second order equation, 644 00:34:48,679 --> 00:34:51,602 you only need two first order equations. 645 00:34:51,602 --> 00:34:53,060 And the way you do it is basically, 646 00:34:53,060 --> 00:34:55,550 you write down an equation that looks kind of stupid. 647 00:34:55,550 --> 00:35:03,890 The equation says that x dot equals v. This is one equation. 648 00:35:03,890 --> 00:35:07,100 And then the other equation comes from this one-- 649 00:35:07,100 --> 00:35:11,270 since x dot equals v, it means that x double dot equals v dot. 650 00:35:11,270 --> 00:35:13,708 This is the acceleration. 651 00:35:13,708 --> 00:35:15,500 So the second equation now comes from this, 652 00:35:15,500 --> 00:35:25,230 that is, v dot equals minus whatever k over m times x. 653 00:35:25,230 --> 00:35:34,500 Now, I can define the vector x v. And what 654 00:35:34,500 --> 00:35:37,400 this set of equations here has done, 655 00:35:37,400 --> 00:35:41,330 it has actually connected the first derivative of this vector 656 00:35:41,330 --> 00:35:44,040 with the vector itself via matrix. 657 00:35:44,040 --> 00:35:46,730 And here is the vector in its first derivative. 658 00:35:46,730 --> 00:35:50,610 Here is the vector itself. 659 00:35:50,610 --> 00:35:54,230 And what I have to put here in order to make this look correct 660 00:35:54,230 --> 00:35:56,180 is I have to put 1 here, and then 661 00:35:56,180 --> 00:36:02,130 minus k over m here, and then 0s, and then I'm done. 662 00:36:02,130 --> 00:36:04,970 So again, if you ever used Matlab 663 00:36:04,970 --> 00:36:07,790 to solve an ordinary differential equation, 664 00:36:07,790 --> 00:36:08,990 you're familiar with this. 665 00:36:08,990 --> 00:36:12,800 Matlab actually forces you to rewrite your equation this way. 666 00:36:12,800 --> 00:36:14,720 But it turns out that this kind of method 667 00:36:14,720 --> 00:36:18,740 actually preceded Matlab by about a century, 668 00:36:18,740 --> 00:36:23,270 because this is a standard way of writing the equation in-- 669 00:36:23,270 --> 00:36:25,820 actually, I believe it was Poincare who actually first 670 00:36:25,820 --> 00:36:26,620 came up with this. 671 00:36:26,620 --> 00:36:33,540 It's the standard way to derive the dynamics of a system. 672 00:36:33,540 --> 00:36:37,280 And this is not-- 673 00:36:37,280 --> 00:36:39,290 I don't want to spend too much time with this. 674 00:36:39,290 --> 00:36:42,980 But what this slide basically points out to you is 675 00:36:42,980 --> 00:36:48,120 the following-- that if you take this expression here, 676 00:36:48,120 --> 00:36:51,648 which I will call the energy, but now we'll call it-- 677 00:36:51,648 --> 00:36:53,690 of course, you cannot see it because it's hidden. 678 00:36:53,690 --> 00:36:55,150 Now you can see it. 679 00:36:55,150 --> 00:36:58,940 So this expression is to call the energy 680 00:36:58,940 --> 00:37:01,220 but to that I will call it the Hamiltonian 681 00:37:01,220 --> 00:37:03,680 and it will become apparent in a second why, 682 00:37:03,680 --> 00:37:07,190 but if you take this expression, this is now 683 00:37:07,190 --> 00:37:10,040 a function of the two dynamical variables, the same two 684 00:37:10,040 --> 00:37:13,850 variables that appeared in this expression over here. 685 00:37:13,850 --> 00:37:18,827 They actually get into this equation, x comma v. 686 00:37:18,827 --> 00:37:20,660 It's a bit of a funny way to think about it, 687 00:37:20,660 --> 00:37:26,110 but certainly you see x here and you see v here. 688 00:37:26,110 --> 00:37:28,970 And actually, I don't like v very much, 689 00:37:28,970 --> 00:37:32,180 because if I keep using v, I will not 690 00:37:32,180 --> 00:37:34,410 come up with the correct set of equations. 691 00:37:34,410 --> 00:37:37,140 I will replace v with the momentum. 692 00:37:37,140 --> 00:37:39,470 So that's a minor cheating. 693 00:37:39,470 --> 00:37:44,770 Instead of v, I will replace it with p. 694 00:37:44,770 --> 00:37:47,400 So p, the momentum, is simply m times v. 695 00:37:47,400 --> 00:37:49,270 So I haven't cheated in any major way. 696 00:37:49,270 --> 00:37:53,750 I just did the simple multiplication. 697 00:37:53,750 --> 00:37:58,840 So if you replace p here, then the equation 698 00:37:58,840 --> 00:38:00,280 will look like this. 699 00:38:00,280 --> 00:38:02,140 Nothing changes here. 700 00:38:02,140 --> 00:38:08,500 This part will look like 1/2 p squared over m. 701 00:38:08,500 --> 00:38:10,610 Oh, sorry. 702 00:38:10,610 --> 00:38:11,110 OK. 703 00:38:13,650 --> 00:38:15,660 So nothing major here. 704 00:38:15,660 --> 00:38:18,030 The magic now-- it's not really magic. 705 00:38:18,030 --> 00:38:20,700 There's a good reason for it, but it 706 00:38:20,700 --> 00:38:24,420 looks like magic is that if I do the following, if I take 707 00:38:24,420 --> 00:38:28,560 the two derivatives of this expression with respect 708 00:38:28,560 --> 00:38:30,180 to x and p-- 709 00:38:30,180 --> 00:38:37,940 so if I take dh dx and if I take dh dp. 710 00:38:37,940 --> 00:38:40,147 Can you still see? 711 00:38:40,147 --> 00:38:40,730 What do I get? 712 00:38:40,730 --> 00:38:44,540 If I get dh dx, what is that? 713 00:38:44,540 --> 00:38:45,110 It is kx. 714 00:38:47,900 --> 00:38:54,550 And if I take dh dp, I will get p over m. 715 00:39:00,140 --> 00:39:07,060 So if I look carefully here, this is p over m. 716 00:39:09,710 --> 00:39:12,090 That's the velocity. 717 00:39:12,090 --> 00:39:20,500 And this is minus kp, isn't it? 718 00:39:20,500 --> 00:39:29,570 Wait a minute-- of course, I know why. 719 00:39:29,570 --> 00:39:30,560 I have v dot here. 720 00:39:30,560 --> 00:39:34,100 Instead of v dot, let me write it in terms of p dot. 721 00:39:34,100 --> 00:39:37,040 If I do in terms of p dot, I multiply 722 00:39:37,040 --> 00:39:43,190 by m both sides of the equation, I will get minus kx. 723 00:39:43,190 --> 00:39:45,260 I don't have to do anything else. 724 00:39:45,260 --> 00:39:46,250 So what do I get? 725 00:39:46,250 --> 00:39:50,060 I get that dh dp, which equals p over m, 726 00:39:50,060 --> 00:39:52,550 is this term in this equation. 727 00:39:52,550 --> 00:39:56,870 And if I get dh dx, which is kx, it's 728 00:39:56,870 --> 00:39:59,280 actually this term in this equation. 729 00:39:59,280 --> 00:40:01,700 So basically, by taking the partial derivatives 730 00:40:01,700 --> 00:40:04,580 of this expression, the Hamiltonian, 731 00:40:04,580 --> 00:40:07,370 with respect to my two dynamical variables' 732 00:40:07,370 --> 00:40:11,840 position and momentum, I came up with the equation of motion 733 00:40:11,840 --> 00:40:15,350 decomposed in the dynamical variables, Matlab 734 00:40:15,350 --> 00:40:18,050 style, or actually Poincare's probably 735 00:40:18,050 --> 00:40:19,910 shifting in his grave-- 736 00:40:19,910 --> 00:40:23,270 Poincare style, in the way [INAUDIBLE] dynamical systems. 737 00:40:27,000 --> 00:40:29,520 So this is really what I say in the top of the slide. 738 00:40:29,520 --> 00:40:32,495 Of course, I say it in a slightly more formal way. 739 00:40:32,495 --> 00:40:33,870 And then the rest of what I do is 740 00:40:33,870 --> 00:40:37,070 I simply solve this equation, but I will not do it here. 741 00:40:37,070 --> 00:40:39,103 This is a simple harmonic oscillator. 742 00:40:39,103 --> 00:40:40,770 The mechanical system, as I said before, 743 00:40:40,770 --> 00:40:44,940 is a car that will move forever, eternally 744 00:40:44,940 --> 00:40:46,770 in a sinusoidal fashion. 745 00:40:46,770 --> 00:40:49,350 And it's kind of interesting, I guess, 746 00:40:49,350 --> 00:40:57,330 that the energy alternates between 100% kinetic, which 747 00:40:57,330 --> 00:41:00,810 happens when the cart, the particle 748 00:41:00,810 --> 00:41:02,910 moves through the rest position, because then it 749 00:41:02,910 --> 00:41:05,800 is the maximum velocity and has no potential. 750 00:41:05,800 --> 00:41:07,950 It is the rest position of the spring. 751 00:41:07,950 --> 00:41:11,010 And it converts to completely elastic when 752 00:41:11,010 --> 00:41:14,220 it goes to the edge of the range of motion, 753 00:41:14,220 --> 00:41:16,260 because over there, it has to reverse. 754 00:41:16,260 --> 00:41:20,040 Therefore, it will go velocity zero momentarily, 755 00:41:20,040 --> 00:41:22,590 and the kinetic energy has vanished at this point. 756 00:41:22,590 --> 00:41:24,480 It has become all potential. 757 00:41:24,480 --> 00:41:27,713 This is probably boring, because this stuff is very well known. 758 00:41:27,713 --> 00:41:29,130 What I really want to do is I want 759 00:41:29,130 --> 00:41:32,250 to establish a connection between this very nice, very 760 00:41:32,250 --> 00:41:38,820 well-known mechanical stuff with optics. 761 00:41:38,820 --> 00:41:41,830 So basically, what I will do now is I will derive-- 762 00:41:41,830 --> 00:41:44,610 I will not really derive, but I will show you 763 00:41:44,610 --> 00:41:47,880 a set of Hamiltonian equations that actually 764 00:41:47,880 --> 00:41:50,720 describe ray trajectories. 765 00:41:50,720 --> 00:41:53,950 And what is really spectacular is 766 00:41:53,950 --> 00:41:55,580 that, as you pointed out earlier, 767 00:41:55,580 --> 00:42:01,000 they will turn out to have an exact mechanical analogy if you 768 00:42:01,000 --> 00:42:03,440 use a certain equation for the index of refraction. 769 00:42:03,440 --> 00:42:05,880 So we'll get to that. 770 00:42:05,880 --> 00:42:09,650 So there's two ways to derive the Hamiltonian equations. 771 00:42:09,650 --> 00:42:13,430 Does anybody know, in the field of mechanics? 772 00:42:13,430 --> 00:42:16,990 Does anybody know how we get the Hamiltonian equations, what 773 00:42:16,990 --> 00:42:19,480 is the process? 774 00:42:25,180 --> 00:42:27,950 Well, I sort of just described it over here, 775 00:42:27,950 --> 00:42:30,100 but I guess what I'm trying to get 776 00:42:30,100 --> 00:42:32,800 to is, how did this come about? 777 00:42:32,800 --> 00:42:35,410 Of course, it came about from my physical intuition, 778 00:42:35,410 --> 00:42:38,470 because I know that to have a kinetic-- 779 00:42:38,470 --> 00:42:41,710 someone told me 1,000 years ago when I was a student in school 780 00:42:41,710 --> 00:42:44,030 that this is the case. 781 00:42:44,030 --> 00:42:44,530 But-- 782 00:42:51,810 --> 00:42:54,510 AUDIENCE: It's the invariant property of the system. 783 00:42:54,510 --> 00:42:57,173 It's the total energy we put in at the start. 784 00:42:57,173 --> 00:42:58,590 GEORGE BARBASTATHIS: Yes, so it is 785 00:42:58,590 --> 00:43:02,640 the invariance of the system, and also, mechanical systems, 786 00:43:02,640 --> 00:43:05,770 they also have a Fermat principle of their own. 787 00:43:05,770 --> 00:43:08,580 If you recall, mechanical systems 788 00:43:08,580 --> 00:43:14,070 also try to minimize their potential. 789 00:43:14,070 --> 00:43:16,950 The trajectory that they follow tries 790 00:43:16,950 --> 00:43:19,590 to minimize their potential. 791 00:43:19,590 --> 00:43:24,090 So a good example of that is actually, 792 00:43:24,090 --> 00:43:26,220 forgetting that this is optics, imagine 793 00:43:26,220 --> 00:43:31,290 that what you see here in this sort of fancy-looking diagram, 794 00:43:31,290 --> 00:43:36,370 this is not the index of refraction, but this is a star. 795 00:43:36,370 --> 00:43:40,160 Imagine that the center of this [INAUDIBLE] massive object, 796 00:43:40,160 --> 00:43:43,030 a massive gravitational attraction, 797 00:43:43,030 --> 00:43:45,730 like actually the sun. 798 00:43:45,730 --> 00:43:50,140 And imagine that it's not a ray anymore, 799 00:43:50,140 --> 00:43:57,572 but imagine that that trajectory is the path of the satellite. 800 00:43:57,572 --> 00:43:58,780 This is convincing, isn't it? 801 00:43:58,780 --> 00:44:06,010 If you allow the grayscale to denote gravity now, 802 00:44:06,010 --> 00:44:08,470 not index of refraction but potential 803 00:44:08,470 --> 00:44:14,980 due to the gravitational field exerted by the planet, 804 00:44:14,980 --> 00:44:20,260 then if I launch a spacecraft, or a satellite, 805 00:44:20,260 --> 00:44:24,790 or whatever in the vicinity of this gravitational field, 806 00:44:24,790 --> 00:44:32,950 as you know if you are a sort of amateur astronomer or aspiring 807 00:44:32,950 --> 00:44:36,460 astronaut-- if you are, then you know 808 00:44:36,460 --> 00:44:40,210 that depending on how you launch your spacecraft, sometimes, 809 00:44:40,210 --> 00:44:44,770 it will actually sort of circle once and then escape. 810 00:44:44,770 --> 00:44:47,030 That's very bad, because it means that say, 811 00:44:47,030 --> 00:44:50,950 if you're an astronaut, you're going to space forever. 812 00:44:50,950 --> 00:44:53,200 But also you know that if you launch it properly, then 813 00:44:53,200 --> 00:44:55,420 you can actually get into orbit. 814 00:44:55,420 --> 00:44:57,140 And orbit means that the dynamics here 815 00:44:57,140 --> 00:45:01,420 is such that you will actually start circling this object. 816 00:45:01,420 --> 00:45:04,098 In fact, we never launch spacecraft to stars. 817 00:45:04,098 --> 00:45:06,390 Typically, we launch them to planets nearby, but then-- 818 00:45:09,290 --> 00:45:12,970 So it turns out that the trajectories 819 00:45:12,970 --> 00:45:16,410 that these spacecraft follow, actually 820 00:45:16,410 --> 00:45:20,563 they also minimize their "optical path," where there's 821 00:45:20,563 --> 00:45:21,730 no index of refraction then. 822 00:45:21,730 --> 00:45:23,050 It is just gravity. 823 00:45:23,050 --> 00:45:26,645 But they actually do follow the minimum path around the star. 824 00:45:26,645 --> 00:45:29,020 And in fact, this has been known for more than a century. 825 00:45:29,020 --> 00:45:32,290 I mean, Copernicus even hinted at that in his work, 826 00:45:32,290 --> 00:45:40,450 when he computed the orbits of the planets 827 00:45:40,450 --> 00:45:42,250 in the solar system. 828 00:45:42,250 --> 00:45:44,620 So these are very well known stuff. 829 00:45:44,620 --> 00:45:47,982 So what I will do now, then, is I will actually drive you, 830 00:45:47,982 --> 00:45:49,690 without really proving anything, but that 831 00:45:49,690 --> 00:45:52,360 will drive you to a set of Hamiltonian equations 832 00:45:52,360 --> 00:45:56,970 that describe the same analogy, but for optics now. 833 00:45:56,970 --> 00:45:59,990 So in order to derive our Hamiltonian equations, 834 00:45:59,990 --> 00:46:02,412 we actually need two postulates. 835 00:46:02,412 --> 00:46:03,870 Now what does it mean, "postulate?" 836 00:46:03,870 --> 00:46:05,180 Do you know that word? 837 00:46:05,180 --> 00:46:07,434 Yes, [INAUDIBLE]. 838 00:46:07,434 --> 00:46:12,330 AUDIENCE: [INAUDIBLE] I have a doubt regarding 839 00:46:12,330 --> 00:46:13,920 definition of Hamiltonian. 840 00:46:13,920 --> 00:46:18,270 Why can't it be some other invariant quantity 841 00:46:18,270 --> 00:46:21,300 like angular momentum or linear momentum, 842 00:46:21,300 --> 00:46:23,853 and energy in this case? 843 00:46:23,853 --> 00:46:25,770 GEORGE BARBASTATHIS: So you're actually there. 844 00:46:25,770 --> 00:46:30,270 You're forcing me to say what I didn't say before. 845 00:46:30,270 --> 00:46:33,300 How did this come about? 846 00:46:33,300 --> 00:46:35,970 This comes about-- and I have to apologize now, 847 00:46:35,970 --> 00:46:40,080 because I will throw a lot of jargon at you. 848 00:46:40,080 --> 00:46:42,900 This comes about from something called the Lagrangian 849 00:46:42,900 --> 00:46:45,030 principle. 850 00:46:45,030 --> 00:46:46,830 So the Lagrangian is very-- actually, 851 00:46:46,830 --> 00:46:51,833 if you write the Lagrangian, it looks very similar to this one. 852 00:46:51,833 --> 00:46:52,750 Looks similar to this. 853 00:46:55,490 --> 00:46:57,490 The question you're asking the mechanical system 854 00:46:57,490 --> 00:47:00,487 is, what is the trajectory that the system will follow? 855 00:47:00,487 --> 00:47:02,320 So we have a particle or a set of particles, 856 00:47:02,320 --> 00:47:05,170 and you ask what is the trajectory. 857 00:47:05,170 --> 00:47:08,080 There Lagrangian actually looks very similar to this, 858 00:47:08,080 --> 00:47:09,130 but with a minus sign. 859 00:47:15,900 --> 00:47:19,850 And I never quite remember if you put the minus 860 00:47:19,850 --> 00:47:24,960 sign here or here, but anyway, one of those has a minus sign. 861 00:47:24,960 --> 00:47:28,510 And so what you do then is you put this Lagrangian 862 00:47:28,510 --> 00:47:29,360 into this integral. 863 00:47:29,360 --> 00:47:35,030 And you say that the trajectory that the particle will 864 00:47:35,030 --> 00:47:37,640 follow actually minimizes this integral. 865 00:47:37,640 --> 00:47:42,710 Now, this has a very good physical justification, 866 00:47:42,710 --> 00:47:44,660 because this is actually Newton's Law. 867 00:47:44,660 --> 00:47:46,670 The way you get this Lagrangian is you 868 00:47:46,670 --> 00:47:49,250 start with good, old Newton, which says that-- 869 00:47:57,265 --> 00:47:58,015 Then you multiply. 870 00:48:07,440 --> 00:48:08,190 I haven't cheated. 871 00:48:08,190 --> 00:48:09,620 I've multiplied it with a quantity 872 00:48:09,620 --> 00:48:11,250 that is hopefully non-zero. 873 00:48:11,250 --> 00:48:13,300 And then you can write it down as 1 and-- 874 00:48:30,977 --> 00:48:32,810 actually not quite 0 any more, but constant. 875 00:48:38,672 --> 00:48:40,380 Now confuse my mind with science somehow. 876 00:48:48,580 --> 00:48:51,052 Well, anyway. 877 00:48:51,052 --> 00:49:02,213 AUDIENCE: [INAUDIBLE] 878 00:49:02,213 --> 00:49:03,880 GEORGE BARBASTATHIS: That's a good plan, 879 00:49:03,880 --> 00:49:08,035 because this is a restoring force. 880 00:49:11,501 --> 00:49:14,180 But anyway, how do we actually derive the Hamiltonian? 881 00:49:14,180 --> 00:49:16,030 So I don't know what has happened, 882 00:49:16,030 --> 00:49:18,790 but if you think about it, this is now the Hamiltonian, 883 00:49:18,790 --> 00:49:20,490 isn't it? 884 00:49:20,490 --> 00:49:23,807 So by a simple trick on Newton's Law, it derive-- 885 00:49:23,807 --> 00:49:25,390 that's not quite what I was expecting, 886 00:49:25,390 --> 00:49:26,608 but it happened anyway. 887 00:49:26,608 --> 00:49:28,150 The answer to the question is simpler 888 00:49:28,150 --> 00:49:30,190 than I was expecting it to be. 889 00:49:30,190 --> 00:49:34,320 So there is the Hamiltonian. 890 00:49:34,320 --> 00:49:37,125 now why that and why not the momentum? 891 00:49:37,125 --> 00:49:39,375 That really has to do with the Lagrangian, but let's-- 892 00:49:46,780 --> 00:49:48,460 It's clear what I did here, right? 893 00:49:48,460 --> 00:49:52,130 This is just a simple manipulation that I learned 894 00:49:52,130 --> 00:49:54,408 from Professor [? Jerry ?] [? Wickham ?] at Caltech, 895 00:49:54,408 --> 00:49:54,908 actually. 896 00:50:10,910 --> 00:50:12,530 I think I skipped a slide. 897 00:50:12,530 --> 00:50:14,980 Here we go. 898 00:50:14,980 --> 00:50:17,920 So we were at this word, "postulate." 899 00:50:17,920 --> 00:50:19,840 What does "postulate" mean? 900 00:50:27,645 --> 00:50:29,270 AUDIENCE: We accept something without-- 901 00:50:31,840 --> 00:50:34,083 we say something and we accept something. 902 00:50:34,083 --> 00:50:35,750 GEORGE BARBASTATHIS: That's right, yeah. 903 00:50:35,750 --> 00:50:39,030 So the postulate is a good way to develop a scientific theory 904 00:50:39,030 --> 00:50:40,760 in the following sense-- 905 00:50:40,760 --> 00:50:43,740 first of all, postulates cannot be very complicated. 906 00:50:43,740 --> 00:50:48,510 So it is something simple that we accept with two caveats-- 907 00:50:48,510 --> 00:50:51,590 one is that we accept it because it 908 00:50:51,590 --> 00:50:53,660 has justified the observation. 909 00:50:53,660 --> 00:50:56,270 So postulate had better be something that has never 910 00:50:56,270 --> 00:50:58,410 been violated to our knowledge. 911 00:50:58,410 --> 00:51:01,160 For example, the sun rises every day. 912 00:51:01,160 --> 00:51:03,740 As far as humanity has recorded history, 913 00:51:03,740 --> 00:51:05,930 this has happened every day, except, of course, 914 00:51:05,930 --> 00:51:09,470 in the north above the arctic, where some days the sun 915 00:51:09,470 --> 00:51:10,130 does not rise. 916 00:51:12,820 --> 00:51:14,300 So that's one. 917 00:51:14,300 --> 00:51:16,070 And the second is that the theory 918 00:51:16,070 --> 00:51:19,640 that follows from the postulate has to be self consistent. 919 00:51:19,640 --> 00:51:22,370 So it has to be mathematically consistent. 920 00:51:22,370 --> 00:51:25,760 So typically the postulates are things that I don't quite feel 921 00:51:25,760 --> 00:51:29,630 like proving, but they're good assumptions, 922 00:51:29,630 --> 00:51:33,880 like assumptions that cannot be very easily challenged. 923 00:51:33,880 --> 00:51:35,480 So the assumptions here, one of them 924 00:51:35,480 --> 00:51:39,590 is that rays are continuous. 925 00:51:39,590 --> 00:51:41,750 That really cannot be very easily challenged, 926 00:51:41,750 --> 00:51:48,230 because if a ray were to jump, that would be really strange. 927 00:51:48,230 --> 00:51:50,680 And I can quote a number of physical reasons 928 00:51:50,680 --> 00:51:52,570 why this would be really strange. 929 00:51:52,570 --> 00:51:56,410 For one is that relativity-- 930 00:51:56,410 --> 00:52:01,360 it means that somehow, a photon, if we believe that photons 931 00:52:01,360 --> 00:52:03,190 follow the ray trajectories, it means 932 00:52:03,190 --> 00:52:09,880 that somehow light within zero time, actually 933 00:52:09,880 --> 00:52:11,680 traverses a finite distance. 934 00:52:11,680 --> 00:52:15,400 That is impossible, so that is one good reason. 935 00:52:18,110 --> 00:52:20,720 So if you accept, then, that the rays must be continuous, 936 00:52:20,720 --> 00:52:22,650 actually, it is only continuity. 937 00:52:22,650 --> 00:52:24,830 I am not saying that the rays are differentiable. 938 00:52:24,830 --> 00:52:27,120 So the rays can bend like this. 939 00:52:27,120 --> 00:52:31,280 So here, actually, obviously, it is non-differentiable, 940 00:52:31,280 --> 00:52:32,900 but it's still continuous. 941 00:52:32,900 --> 00:52:35,510 So the postulate actually talks about continuity, 942 00:52:35,510 --> 00:52:37,870 not differentiability. 943 00:52:37,870 --> 00:52:39,970 Piece-wise differentiable, yes, it 944 00:52:39,970 --> 00:52:44,770 is, because it is certainly piece-wise differentiable here, 945 00:52:44,770 --> 00:52:46,330 piece-wise differentiable there. 946 00:52:46,330 --> 00:52:48,370 This is actually a technicality. 947 00:52:48,370 --> 00:52:51,240 Functions that are not piece-wise differentiable 948 00:52:51,240 --> 00:52:55,240 are really nasty beasts, like Weierstrass functions and stuff 949 00:52:55,240 --> 00:52:58,300 like that, which normally we don't deal with in engineering. 950 00:52:58,300 --> 00:53:00,960 So basically, the second condition here 951 00:53:00,960 --> 00:53:02,980 is a mathematical technicality. 952 00:53:02,980 --> 00:53:04,400 So we don't have to dwell on that. 953 00:53:07,620 --> 00:53:11,430 Now here's something that probably as engineers, we've 954 00:53:11,430 --> 00:53:15,300 got turned off by when we learned it in calculus. 955 00:53:15,300 --> 00:53:18,660 But it's actually a useful thing that is called the "mean value 956 00:53:18,660 --> 00:53:20,190 theorem." 957 00:53:20,190 --> 00:53:22,150 And the mean value theorem says the following-- 958 00:53:22,150 --> 00:53:26,920 says that if you are given a continuous piece-wise 959 00:53:26,920 --> 00:53:30,080 differentiable function-- 960 00:53:30,080 --> 00:53:30,910 so here it is-- 961 00:53:36,150 --> 00:53:37,340 so this is x. 962 00:53:37,340 --> 00:53:38,840 This is f of-- 963 00:53:38,840 --> 00:53:42,230 so this curve here is f of x. 964 00:53:42,230 --> 00:53:53,710 This is some function of x1, x2, f of x1, f of x2. 965 00:53:53,710 --> 00:53:56,590 The mean value theorem says that according 966 00:53:56,590 --> 00:54:01,570 to the conditions I just stated, there is a point somewhere 967 00:54:01,570 --> 00:54:05,020 in this interval satisfying the following property, 968 00:54:05,020 --> 00:54:17,240 that f of x2 minus x1 over x2 minus x1 equals f prime of x. 969 00:54:17,240 --> 00:54:23,000 So basically, this ratio, I can pick a point xi 970 00:54:23,000 --> 00:54:26,690 somewhere in this interval such that the value 971 00:54:26,690 --> 00:54:29,240 of the derivative of a function at this point 972 00:54:29,240 --> 00:54:30,770 equals this ratio. 973 00:54:30,770 --> 00:54:32,570 And of course, you can realize immediately 974 00:54:32,570 --> 00:54:36,025 that if you make the interval very, very tiny, then 975 00:54:36,025 --> 00:54:38,150 of course, you get the definition of the derivative 976 00:54:38,150 --> 00:54:38,930 itself. 977 00:54:38,930 --> 00:54:42,440 But the mean value theorem is a little bit more general. 978 00:54:42,440 --> 00:54:45,980 And it is the mean value theorem that I use here in order 979 00:54:45,980 --> 00:54:47,540 to write this equation. 980 00:54:47,540 --> 00:54:49,970 This is basically the mean value theorem, 981 00:54:49,970 --> 00:54:54,980 where the quantity of interest that I applied 982 00:54:54,980 --> 00:54:57,593 is the ray trajectory. 983 00:54:57,593 --> 00:54:58,760 Let me back up for a second. 984 00:54:58,760 --> 00:55:00,510 I should have started with something else. 985 00:55:00,510 --> 00:55:02,400 What am I really plotting here? 986 00:55:02,400 --> 00:55:05,420 So the red curve is the ray trajectory. 987 00:55:05,420 --> 00:55:09,620 So I describe the ray trajectory as a vector, Q. 988 00:55:09,620 --> 00:55:11,690 So the vector is a position vector. 989 00:55:11,690 --> 00:55:14,990 This is the origin of the coordinates 990 00:55:14,990 --> 00:55:16,830 and Q is the position vector. 991 00:55:16,830 --> 00:55:19,610 And then I parameterize it with a variable s. 992 00:55:19,610 --> 00:55:21,080 So basically, s is an index. 993 00:55:21,080 --> 00:55:23,900 As I increase, as I step through s, 994 00:55:23,900 --> 00:55:28,310 I basically walk along this trajectory. 995 00:55:28,310 --> 00:55:30,830 So you can think of s as time, as time 996 00:55:30,830 --> 00:55:33,960 goes forward, the light particles, if you wish, 997 00:55:33,960 --> 00:55:35,810 are moving along this curve. 998 00:55:38,410 --> 00:55:42,650 And at each point on the curve, since it is continuous 999 00:55:42,650 --> 00:55:46,010 and piece-wise differentiable, at each point I can define 1000 00:55:46,010 --> 00:55:47,410 a tangent-- 1001 00:55:47,410 --> 00:55:49,480 except, of course, the point where 1002 00:55:49,480 --> 00:55:52,570 it is not differentiable, but let's skip those for a moment. 1003 00:55:52,570 --> 00:55:54,070 In the point that is differentiable, 1004 00:55:54,070 --> 00:55:55,570 we can define a tangent. 1005 00:55:55,570 --> 00:55:58,630 And this tangent is the vector P. 1006 00:55:58,630 --> 00:56:05,230 So this vector P is actually the momentum of the ray. 1007 00:56:05,230 --> 00:56:10,900 So since the tangent is associated with a derivative-- 1008 00:56:10,900 --> 00:56:12,940 in fact, in high-dimensional space, 1009 00:56:12,940 --> 00:56:16,090 it is even more obvious because the tangent really points 1010 00:56:16,090 --> 00:56:17,070 towards the-- 1011 00:56:19,920 --> 00:56:20,670 it is the tangent. 1012 00:56:23,500 --> 00:56:25,570 Then this really is an expression 1013 00:56:25,570 --> 00:56:29,230 of the mean value theorem for the continuous and piece-wise 1014 00:56:29,230 --> 00:56:30,580 differentiable ray. 1015 00:56:30,580 --> 00:56:34,930 And of course, I can write it down as a differential equation 1016 00:56:34,930 --> 00:56:37,080 by taking the limit. 1017 00:56:37,080 --> 00:56:42,270 And there is one element that is sort of missing here. 1018 00:56:42,270 --> 00:56:43,620 That's the normalization. 1019 00:56:43,620 --> 00:56:48,148 Why did I pick to normalize by the value of the tangent? 1020 00:56:48,148 --> 00:56:49,440 Let's leave that alone for now. 1021 00:56:49,440 --> 00:56:52,110 Let's assume that it is correct, and see 1022 00:56:52,110 --> 00:56:55,590 if it will lead us to an acceptable physical result. 1023 00:56:55,590 --> 00:56:59,250 So what I'm trying to say is that the denominator here 1024 00:56:59,250 --> 00:57:00,000 is arbitrary. 1025 00:57:00,000 --> 00:57:01,110 I don't really need it. 1026 00:57:01,110 --> 00:57:02,480 I can throw it out. 1027 00:57:02,480 --> 00:57:05,880 It is a normalization parameter, and I put it there 1028 00:57:05,880 --> 00:57:09,570 because with hindsight, it would lead me 1029 00:57:09,570 --> 00:57:14,370 to a physically acceptable result. 1030 00:57:14,370 --> 00:57:17,130 The second postulate is a little bit easier 1031 00:57:17,130 --> 00:57:20,070 to swallow, thanks to [? Kalpesh's ?] question, 1032 00:57:20,070 --> 00:57:23,550 because the second postulate says that the ray, 1033 00:57:23,550 --> 00:57:27,520 very similar to a spacecraft approaching a planet, 1034 00:57:27,520 --> 00:57:31,650 a ray actually is attracted by a high index of refraction. 1035 00:57:31,650 --> 00:57:34,860 And the way we quantify this assumption is we 1036 00:57:34,860 --> 00:57:39,060 say that if the ray enters a region where 1037 00:57:39,060 --> 00:57:41,400 the index of refraction is variable, 1038 00:57:41,400 --> 00:57:43,770 that this has a gradient. 1039 00:57:43,770 --> 00:57:46,980 So that's the gradient, in case you have forgotten-- 1040 00:57:46,980 --> 00:57:47,880 I hope not. 1041 00:57:47,880 --> 00:57:50,490 So if there's a gradient in index of refraction, 1042 00:57:50,490 --> 00:57:52,590 the ray will receive a kick. 1043 00:57:52,590 --> 00:57:55,140 And the kick of the ray-- 1044 00:57:55,140 --> 00:57:59,220 what the kick really means is that its momentum will change. 1045 00:57:59,220 --> 00:58:01,590 Therefore, the trajectory will bend 1046 00:58:01,590 --> 00:58:05,250 and the kick that is a change in momentum 1047 00:58:05,250 --> 00:58:07,560 will be proportional to the gradient. 1048 00:58:07,560 --> 00:58:08,970 That is the postulate. 1049 00:58:08,970 --> 00:58:12,800 And I will not do it here, but if you-- 1050 00:58:12,800 --> 00:58:17,800 it is a little bit of a long derivation, I have to warn you. 1051 00:58:17,800 --> 00:58:19,240 I went through it and I decided-- 1052 00:58:19,240 --> 00:58:20,740 actually, I went through it and then 1053 00:58:20,740 --> 00:58:22,540 I decided not to put it in the notes, 1054 00:58:22,540 --> 00:58:24,670 because it is a little bit involved. 1055 00:58:24,670 --> 00:58:29,560 But this actually turns out to be equivalent to Snell's Law. 1056 00:58:29,560 --> 00:58:33,450 So if you want, I can give you a copy of my notes 1057 00:58:33,450 --> 00:58:36,700 where I derived it, but if you don't 1058 00:58:36,700 --> 00:58:40,000 feel like going through two pages of calculus 1059 00:58:40,000 --> 00:58:44,080 of variations, then please take my word for it. 1060 00:58:44,080 --> 00:58:47,770 This is actually Snell's Law, but written in a funny way. 1061 00:58:51,430 --> 00:58:56,650 So that's, then, the sort of variational version. 1062 00:58:56,650 --> 00:58:58,870 And we can write it down as a differential equation, 1063 00:58:58,870 --> 00:58:59,650 and we get this. 1064 00:59:03,720 --> 00:59:06,810 So now, we're kind of going backwards. 1065 00:59:06,810 --> 00:59:08,940 Now, we have two differential equations. 1066 00:59:08,940 --> 00:59:09,650 Yes. 1067 00:59:09,650 --> 00:59:11,770 AUDIENCE: Why the position variables 1068 00:59:11,770 --> 00:59:16,700 are different on both sides, s and s dash? 1069 00:59:16,700 --> 00:59:18,450 GEORGE BARBASTATHIS: You mean the s prime? 1070 00:59:18,450 --> 00:59:19,350 AUDIENCE: Yeah, s prime. 1071 00:59:19,350 --> 00:59:20,850 GEORGE BARBASTATHIS: Oh, the s prime 1072 00:59:20,850 --> 00:59:24,252 is actually the xi from the mean value theorem. 1073 00:59:24,252 --> 00:59:24,960 AUDIENCE: Oh, OK. 1074 00:59:24,960 --> 00:59:26,543 GEORGE BARBASTATHIS: When you write it 1075 00:59:26,543 --> 00:59:29,702 in the variational form, s prime can you be anywhere in here. 1076 00:59:29,702 --> 00:59:30,660 AUDIENCE: Right, right. 1077 00:59:30,660 --> 00:59:32,050 GEORGE BARBASTATHIS: I don't know exactly where it is. 1078 00:59:32,050 --> 00:59:33,780 And very similar to here, actually again here 1079 00:59:33,780 --> 00:59:34,980 I invoke the mean value theorem. 1080 00:59:34,980 --> 00:59:35,920 I didn't say that then. 1081 00:59:35,920 --> 00:59:36,410 AUDIENCE: Yeah. 1082 00:59:36,410 --> 00:59:36,780 Yeah. 1083 00:59:36,780 --> 00:59:37,170 GEORGE BARBASTATHIS: But of course, 1084 00:59:37,170 --> 00:59:38,587 when you do the differential form, 1085 00:59:38,587 --> 00:59:40,650 it vanishes, because the interval collapses. 1086 00:59:48,040 --> 00:59:50,560 So now what I have is actually two differential equations. 1087 00:59:50,560 --> 00:59:52,960 One of them contains a trajectory, 1088 00:59:52,960 --> 00:59:55,000 the other contains a momentum. 1089 00:59:55,000 --> 00:59:59,740 So this is basically the two-dimensional analog 1090 00:59:59,740 --> 01:00:03,920 to the dynamical system that I wrote here. 1091 01:00:03,920 --> 01:00:06,340 I have an equation for the position. 1092 01:00:06,340 --> 01:00:07,413 This is it. 1093 01:00:07,413 --> 01:00:09,580 And I have a differential equation for the momentum. 1094 01:00:09,580 --> 01:00:10,390 This is it. 1095 01:00:10,390 --> 01:00:12,620 This is the equation for the position. 1096 01:00:12,620 --> 01:00:14,380 This is the equation for the momentum. 1097 01:00:14,380 --> 01:00:17,740 I used q for the position here, because that's a convention 1098 01:00:17,740 --> 01:00:19,330 in geometrical optics. 1099 01:00:19,330 --> 01:00:23,670 People use q, not x, but q is the position vector. 1100 01:00:26,900 --> 01:00:27,730 Well, that's fine. 1101 01:00:27,730 --> 01:00:32,310 The question is, is there anything 1102 01:00:32,310 --> 01:00:33,450 that is conserved here? 1103 01:00:33,450 --> 01:00:39,900 Did I derive any sort of Hamiltonian? 1104 01:00:39,900 --> 01:00:43,140 It turns out-- and again, this requires a bit of hindsight, 1105 01:00:43,140 --> 01:00:51,410 but I can pick the Hamiltonian to be of this form, 1106 01:00:51,410 --> 01:00:55,820 to be equal to the magnitude of the momentum 1107 01:00:55,820 --> 01:00:59,280 minus the index of refraction. 1108 01:00:59,280 --> 01:01:03,130 Let's write down the two equations again. 1109 01:01:03,130 --> 01:01:05,340 So this is the equations. 1110 01:01:05,340 --> 01:01:08,100 And the Hamiltonian that I picked 1111 01:01:08,100 --> 01:01:13,880 is H equals p minus n of q. 1112 01:01:13,880 --> 01:01:18,030 You can see now that if I do the gradient-- 1113 01:01:18,030 --> 01:01:21,450 now in the high-dimensional case, instead of 1114 01:01:21,450 --> 01:01:23,000 the derivatives of the Hamiltonian, 1115 01:01:23,000 --> 01:01:24,900 I have to take the gradient. 1116 01:01:24,900 --> 01:01:29,760 So if I do the gradient of the Hamiltonian with respect to-- 1117 01:01:29,760 --> 01:01:40,958 let's do the first one-- with respect to q, 1118 01:01:40,958 --> 01:01:42,500 this is really what I'm going to get. 1119 01:01:47,500 --> 01:01:49,875 And if I do the Hamiltonian with respect to p-- 1120 01:01:55,550 --> 01:01:56,405 what is p? 1121 01:02:05,620 --> 01:02:09,160 So for example, if you do dH dp x, 1122 01:02:09,160 --> 01:02:13,830 you will get p x over the square root. 1123 01:02:13,830 --> 01:02:23,930 And then you can follow dH dp y and so on. 1124 01:02:23,930 --> 01:02:30,305 So therefore, the gradient, which is defined as a vector-- 1125 01:03:07,580 --> 01:03:10,400 So you can basically see from this simple derivation 1126 01:03:10,400 --> 01:03:17,180 here that by this choice of a Hamiltonian, the Hamiltonian 1127 01:03:17,180 --> 01:03:20,960 that I wrote here, I can actually derive consistently 1128 01:03:20,960 --> 01:03:24,260 the set of Hamiltonian equations. 1129 01:03:24,260 --> 01:03:27,200 So I basically find that the dq ds 1130 01:03:27,200 --> 01:03:30,800 equals the derivative or the gradient of the Hamiltonian 1131 01:03:30,800 --> 01:03:33,920 with respect to the conjugate variable, the momentum, 1132 01:03:33,920 --> 01:03:36,800 and the other way around for dp ds. 1133 01:03:40,330 --> 01:03:43,720 So this equation, then, is the Hamiltonian. 1134 01:03:43,720 --> 01:03:45,670 It is not the only one, by the way. 1135 01:03:45,670 --> 01:03:47,140 Hamiltonians, they're not unique. 1136 01:03:47,140 --> 01:03:50,830 Clearly for example, I can add an arbitrary constant here. 1137 01:03:50,830 --> 01:03:53,400 I can get another Hamiltonian. 1138 01:03:53,400 --> 01:03:55,400 And we know that for energy also. 1139 01:03:55,400 --> 01:03:56,440 Energy is not unique. 1140 01:03:56,440 --> 01:04:00,172 I can add constants to it, and it is still remains energy. 1141 01:04:00,172 --> 01:04:01,630 So you need the reference basically 1142 01:04:01,630 --> 01:04:04,370 to decide which way to go. 1143 01:04:04,370 --> 01:04:06,836 But anyway, this is one possible Hamiltonian. 1144 01:04:10,320 --> 01:04:13,100 How do we explain it physically? 1145 01:04:13,100 --> 01:04:16,440 Well, the momentum is a vector. 1146 01:04:16,440 --> 01:04:20,790 And what the Hamiltonian says, that the difference 1147 01:04:20,790 --> 01:04:24,900 between the magnitude of the vector 1148 01:04:24,900 --> 01:04:27,570 at the given position of the index of refraction, 1149 01:04:27,570 --> 01:04:29,670 the difference has to be constant. 1150 01:04:29,670 --> 01:04:32,130 I can pick this constant to be 0. 1151 01:04:32,130 --> 01:04:34,560 Since it is arbitrary, let me pick it to be 0. 1152 01:04:34,560 --> 01:04:37,350 If that is the case, what I'm really saying 1153 01:04:37,350 --> 01:04:41,640 is that the momentum, the vector of the momentum 1154 01:04:41,640 --> 01:04:44,890 is constrained to lie on a sphere. 1155 01:04:44,890 --> 01:04:48,770 And that actually is known as Descartes' sphere, 1156 01:04:48,770 --> 01:04:50,580 but it is not really-- 1157 01:04:50,580 --> 01:04:53,820 Descartes was not the first one to make this observation. 1158 01:04:53,820 --> 01:04:56,770 It was an Arab scientist whose name is, I believe, 1159 01:04:56,770 --> 01:05:00,770 al-Haytham, who actually made this observation 1160 01:05:00,770 --> 01:05:02,150 for the first time. 1161 01:05:02,150 --> 01:05:05,150 And actually, al-Haytham also derived Snell's Law 1162 01:05:05,150 --> 01:05:07,950 for the first time. 1163 01:05:07,950 --> 01:05:11,430 So basically what the Hamiltonian that we derived 1164 01:05:11,430 --> 01:05:15,120 says is that the momentum of the array 1165 01:05:15,120 --> 01:05:17,270 is constrained to lie on a sphere. 1166 01:05:23,040 --> 01:05:28,820 Now I did not go through the slides when we did Snell's law, 1167 01:05:28,820 --> 01:05:31,970 but it is in your notes. 1168 01:05:31,970 --> 01:05:34,550 And you may have noticed it if you were diligent when 1169 01:05:34,550 --> 01:05:36,860 you were studying the notes. 1170 01:05:36,860 --> 01:05:38,620 Because I actually showed-- 1171 01:05:38,620 --> 01:05:42,420 this slide is another way to derive Descartes' Law. 1172 01:05:42,420 --> 01:05:46,340 So what is happening here is I have the familiar situation 1173 01:05:46,340 --> 01:05:47,620 of Snell's Law. 1174 01:05:47,620 --> 01:05:53,620 I have a medium of index n to the left 1175 01:05:53,620 --> 01:05:56,810 and the medium of index n prime to the right. 1176 01:05:56,810 --> 01:06:02,720 And in each medium, I associate a sphere with a radius 1177 01:06:02,720 --> 01:06:05,090 equal to what Descartes predicts. 1178 01:06:05,090 --> 01:06:09,410 So the sphere has radius n on the left and another sphere 1179 01:06:09,410 --> 01:06:11,870 with radius n prime on the right. 1180 01:06:11,870 --> 01:06:13,720 In this case, n prime is bigger than n, 1181 01:06:13,720 --> 01:06:17,470 so the sphere is bigger on the right-hand side. 1182 01:06:17,470 --> 01:06:22,290 So Snell's Law here is derived by applied momentum 1183 01:06:22,290 --> 01:06:25,680 observation, which actually, Professor Sheppard mentioned 1184 01:06:25,680 --> 01:06:27,590 in the beginning when we were discussing 1185 01:06:27,590 --> 01:06:29,340 [? Kalpesh's ?] question. 1186 01:06:29,340 --> 01:06:33,610 And momentum observation says the following-- 1187 01:06:33,610 --> 01:06:45,450 that in the vertical direction, up and down, clearly, 1188 01:06:45,450 --> 01:06:48,100 this geometry is invariant. 1189 01:06:48,100 --> 01:06:49,350 There's no change. 1190 01:06:49,350 --> 01:06:52,990 So therefore, the momentum must be conserved. 1191 01:06:52,990 --> 01:06:55,610 The momentum on the left-hand side, 1192 01:06:55,610 --> 01:06:59,570 which we denote as p vertical, and the momentum 1193 01:06:59,570 --> 01:07:03,290 on the right-hand side, p prime vertical, must be equal. 1194 01:07:03,290 --> 01:07:10,350 And since the momenta controls Descartes' calculation 1195 01:07:10,350 --> 01:07:14,080 equal to this, then basically, these vertical components 1196 01:07:14,080 --> 01:07:17,700 are obtained by projecting the radius of the sphere 1197 01:07:17,700 --> 01:07:19,583 onto the vertical axis. 1198 01:07:19,583 --> 01:07:21,000 And of course, if you do this now, 1199 01:07:21,000 --> 01:07:26,310 you immediately get Snell's Law, because this angle is theta. 1200 01:07:26,310 --> 01:07:30,510 So therefore, this length equals n sine theta 1201 01:07:30,510 --> 01:07:33,960 on the left-hand side, and n prime sine theta 1202 01:07:33,960 --> 01:07:36,030 prime on the right-hand side. 1203 01:07:36,030 --> 01:07:40,240 So that is yet another way to obtain Snell's Law. 1204 01:07:40,240 --> 01:07:41,950 And of course, you might wonder what 1205 01:07:41,950 --> 01:07:45,130 happens to the momentum on the [? back, ?] 1206 01:07:45,130 --> 01:07:47,200 on the longitudinal dimension. 1207 01:07:47,200 --> 01:07:47,920 That's OK. 1208 01:07:47,920 --> 01:07:49,300 You can have a change in momentum 1209 01:07:49,300 --> 01:07:53,560 there because there is a change in media. 1210 01:07:53,560 --> 01:07:56,290 So the change in media actually corresponds 1211 01:07:56,290 --> 01:07:58,030 to a change in momentum. 1212 01:07:58,030 --> 01:08:00,520 And momentum is conserved, with actually 1213 01:08:00,520 --> 01:08:04,170 means that the medium here, if you actually 1214 01:08:04,170 --> 01:08:06,800 use the proper electrodynamics, there is actually 1215 01:08:06,800 --> 01:08:09,050 a force applied in this medium here 1216 01:08:09,050 --> 01:08:11,050 due to the change in momentum of the light. 1217 01:08:11,050 --> 01:08:13,370 But this is beyond the scope of the class, 1218 01:08:13,370 --> 01:08:17,760 so we'll not discuss it in any greater detail. 1219 01:08:17,760 --> 01:08:21,319 So this kind of picture here, with the Descartes sphere, 1220 01:08:21,319 --> 01:08:25,020 it also applies to a GRIN medium. 1221 01:08:25,020 --> 01:08:28,430 So here is a medium whose index of refraction is variable. 1222 01:08:28,430 --> 01:08:30,630 So you go from low to high. 1223 01:08:30,630 --> 01:08:34,790 Again, the convention is gray means higher index. 1224 01:08:34,790 --> 01:08:35,960 And you can break it down. 1225 01:08:35,960 --> 01:08:38,390 Like we did, again, in the beginning of the class, 1226 01:08:38,390 --> 01:08:42,710 we can break it down into slices of progressively higher index. 1227 01:08:42,710 --> 01:08:46,510 And you can apply Snell's Law, using 1228 01:08:46,510 --> 01:08:52,279 either Snell itself or the succession of Descartes sphere. 1229 01:08:52,279 --> 01:08:55,020 You can apply it as the ray goes down. 1230 01:08:55,020 --> 01:08:58,160 And then either because it has too much momentum 1231 01:08:58,160 --> 01:09:01,970 or because it has to satisfy Snell's Law, 1232 01:09:01,970 --> 01:09:04,939 these two statements are exactly equivalent, 1233 01:09:04,939 --> 01:09:07,529 the ray must bend its path. 1234 01:09:07,529 --> 01:09:13,979 So this is yet another way to justify why the ray is bending. 1235 01:09:13,979 --> 01:09:16,529 It is bending because it must conserve 1236 01:09:16,529 --> 01:09:18,420 the value of this Hamiltonian. 1237 01:09:18,420 --> 01:09:21,550 It must lie on the Descartes sphere. 1238 01:09:21,550 --> 01:09:23,040 But of course, in the GRIN medium, 1239 01:09:23,040 --> 01:09:24,750 we have to be a little bit careful 1240 01:09:24,750 --> 01:09:27,569 because this sphere has a variable radius, 1241 01:09:27,569 --> 01:09:29,580 as you can see in this diagram over here. 1242 01:09:34,520 --> 01:09:38,510 Now let's come to the other question that was asked. 1243 01:09:38,510 --> 01:09:42,407 Is there an analogy between mechanics and optics? 1244 01:09:42,407 --> 01:09:44,240 And because we're running a little bit late, 1245 01:09:44,240 --> 01:09:46,729 I don't want to spend too long with this analogy. 1246 01:09:46,729 --> 01:09:47,990 It's a very interesting one. 1247 01:09:47,990 --> 01:09:50,250 I'll let you study it. 1248 01:09:50,250 --> 01:09:55,690 And if you'd like, we can also discuss it later. 1249 01:09:55,690 --> 01:09:57,800 But basically, the analogy comes from this-- 1250 01:09:57,800 --> 01:10:02,290 we go back and rewrite the Hamiltonian equation 1251 01:10:02,290 --> 01:10:05,200 of the mechanical system as a sort 1252 01:10:05,200 --> 01:10:08,230 of unassailable kinetic energy term. 1253 01:10:08,230 --> 01:10:09,725 The first term over here is clearly 1254 01:10:09,725 --> 01:10:11,350 the kinetic energy, the term that looks 1255 01:10:11,350 --> 01:10:13,510 like momentum squared over 2m. 1256 01:10:13,510 --> 01:10:18,130 Then the potential-- we wrote the potential before. 1257 01:10:18,130 --> 01:10:19,900 In the case of the elastic spring, 1258 01:10:19,900 --> 01:10:28,060 the elastic potential was v of q equals 1/2 k q [INAUDIBLE].. 1259 01:10:28,060 --> 01:10:32,020 That's one special case of a restoring force, 1260 01:10:32,020 --> 01:10:33,700 of an elastic potential. 1261 01:10:33,700 --> 01:10:35,330 But it could be more general. 1262 01:10:35,330 --> 01:10:40,020 So we write it as v of q. 1263 01:10:40,020 --> 01:10:42,790 And then the rest of the definitions 1264 01:10:42,790 --> 01:10:45,680 are basically straightforward. 1265 01:10:45,680 --> 01:10:49,170 The momentum equals m times the time derivative 1266 01:10:49,170 --> 01:10:52,020 of the trajectory. 1267 01:10:52,020 --> 01:10:53,430 Anyway, like I said, I don't want 1268 01:10:53,430 --> 01:10:56,070 to go through this in great detail. 1269 01:10:56,070 --> 01:11:00,810 But basically, if as a starting force 1270 01:11:00,810 --> 01:11:05,280 we use the index of refraction squared divided 1271 01:11:05,280 --> 01:11:08,610 by 2m, where m is like a fictitious mass 1272 01:11:08,610 --> 01:11:10,380 that we have to assign to the photon, 1273 01:11:10,380 --> 01:11:13,290 then we would basically get the one-to-one analogy. 1274 01:11:13,290 --> 01:11:15,360 This is quite interesting, because it is really 1275 01:11:15,360 --> 01:11:18,810 acting like a spring or like gravity, 1276 01:11:18,810 --> 01:11:20,760 but gravity is not exactly the same, 1277 01:11:20,760 --> 01:11:22,500 because gravity is like 1 over r. 1278 01:11:22,500 --> 01:11:24,000 So you have to make an approximation 1279 01:11:24,000 --> 01:11:24,890 and blah, blah, blah. 1280 01:11:24,890 --> 01:11:29,370 But if you think of it as a spring, 1281 01:11:29,370 --> 01:11:32,080 then you get basically a one-to-one analogy. 1282 01:11:32,080 --> 01:11:33,426 So it's kind of interesting. 1283 01:11:37,320 --> 01:11:39,030 And the last thing I want to say, 1284 01:11:39,030 --> 01:11:42,900 because it has a practical significance, 1285 01:11:42,900 --> 01:11:49,950 is that once we write down the Hamiltonian equations, 1286 01:11:49,950 --> 01:11:51,120 then it is very convenient. 1287 01:11:51,120 --> 01:11:54,370 In addition to all this nice physics, and intuition, 1288 01:11:54,370 --> 01:11:56,070 and so on, it is very convenient, 1289 01:11:56,070 --> 01:12:00,270 because we can just plug them in Matlab and solve them. 1290 01:12:00,270 --> 01:12:03,420 So if someone is giving us a graded index medium-- 1291 01:12:03,420 --> 01:12:10,260 now whether it is quadratic, or cubic, or whatever, 1292 01:12:10,260 --> 01:12:14,780 27th polynomial, or anything for that matter, 1293 01:12:14,780 --> 01:12:20,130 we can just solve it using this set of equations. 1294 01:12:20,130 --> 01:12:24,600 And here, what I did is I did an example of a quadratic index 1295 01:12:24,600 --> 01:12:26,973 medium on-axis, off-axis. 1296 01:12:26,973 --> 01:12:28,890 You can see that on-axis actually pretty good. 1297 01:12:28,890 --> 01:12:31,020 It is not completely free of spherical, 1298 01:12:31,020 --> 01:12:32,850 but pretty close to free. 1299 01:12:32,850 --> 01:12:35,700 You can see it does come to a very nice geometrical focus. 1300 01:12:35,700 --> 01:12:39,032 Of course, off-axis, it has a terrible aberration 1301 01:12:39,032 --> 01:12:39,990 that we've called what? 1302 01:12:39,990 --> 01:12:42,840 How do we call this particular aberration that we see here? 1303 01:12:42,840 --> 01:12:43,500 AUDIENCE: Coma. 1304 01:12:43,500 --> 01:12:44,583 GEORGE BARBASTATHIS: Coma. 1305 01:12:51,374 --> 01:12:51,874 Yes. 1306 01:12:56,260 --> 01:12:59,050 AUDIENCE: Can you use the index of refraction 1307 01:12:59,050 --> 01:13:01,690 to correct for coma? 1308 01:13:01,690 --> 01:13:05,470 GEORGE BARBASTATHIS: Yes, of course. 1309 01:13:05,470 --> 01:13:08,890 In fact, believe it or not, this is a research project 1310 01:13:08,890 --> 01:13:12,520 in my group, which says that suppose 1311 01:13:12,520 --> 01:13:15,820 you wanted to define an arbitrary index 1312 01:13:15,820 --> 01:13:18,640 distribution here in order to correct 1313 01:13:18,640 --> 01:13:20,817 for a given set of aberrations. 1314 01:13:20,817 --> 01:13:22,650 So someone is giving you the specifications. 1315 01:13:22,650 --> 01:13:24,640 They say, I want to eliminate the coma 1316 01:13:24,640 --> 01:13:26,080 for this angle of incidence. 1317 01:13:26,080 --> 01:13:29,060 I want to eliminate other aberrations, for example 1318 01:13:29,060 --> 01:13:32,320 astigmatism and so on, then in principle, you 1319 01:13:32,320 --> 01:13:36,190 can come up with some index of refraction, 1320 01:13:36,190 --> 01:13:39,670 n of r, which if properly chosen, 1321 01:13:39,670 --> 01:13:44,090 will eliminate this specified aberration. 1322 01:13:44,090 --> 01:13:45,965 Now of course, that's pretty easy, actually. 1323 01:13:45,965 --> 01:13:47,090 That doesn't need research. 1324 01:13:47,090 --> 01:13:50,050 It's kind of like a computational problem. 1325 01:13:50,050 --> 01:13:54,400 But how do you implement an arbitrary index of refraction? 1326 01:13:54,400 --> 01:13:56,200 That's difficult. So we're actually 1327 01:13:56,200 --> 01:13:58,240 working on that in my group. 1328 01:13:58,240 --> 01:14:00,290 But yeah, in principle, you can. 1329 01:14:00,290 --> 01:14:02,862 In practice, with commercial GRINs, 1330 01:14:02,862 --> 01:14:04,070 it a bit difficult, actually. 1331 01:14:04,070 --> 01:14:07,360 You would probably have to use something more than a GRIN, 1332 01:14:07,360 --> 01:14:09,150 as far as I know. 1333 01:14:09,150 --> 01:14:11,540 Yeah. 1334 01:14:11,540 --> 01:14:15,200 AUDIENCE: On the previous slide, you said n has to be-- 1335 01:14:15,200 --> 01:14:15,700 [INAUDIBLE] 1336 01:14:15,700 --> 01:14:17,492 AUDIENCE: Can you press the button, please? 1337 01:14:19,260 --> 01:14:22,590 Can you press the button? 1338 01:14:22,590 --> 01:14:28,600 AUDIENCE: He is pressing, but it's broken. 1339 01:14:32,023 --> 01:14:36,710 [INAUDIBLE] 1340 01:14:36,710 --> 01:14:38,270 AUDIENCE: On the previous slide, you 1341 01:14:38,270 --> 01:14:41,604 said something about physical-- 1342 01:14:41,604 --> 01:14:45,580 n has to be greater than 0. 1343 01:14:45,580 --> 01:14:48,420 GEORGE BARBASTATHIS: You caught me. 1344 01:14:48,420 --> 01:14:48,920 Yes. 1345 01:15:00,090 --> 01:15:03,305 Yes, I said that n has to be bigger than 1, 1346 01:15:03,305 --> 01:15:04,610 AUDIENCE: [INAUDIBLE] 1347 01:15:04,610 --> 01:15:06,110 GEORGE BARBASTATHIS: Better than 1. 1348 01:15:06,110 --> 01:15:10,880 The question is, can n be less than 1 and less than 0? 1349 01:15:13,560 --> 01:15:17,320 From the point of view of geometrical optics, 1350 01:15:17,320 --> 01:15:22,570 n is the speed of light in a bulk medium. 1351 01:15:22,570 --> 01:15:29,110 And in ordinary bulk media that we use in geometrical optics, 1352 01:15:29,110 --> 01:15:31,210 light is always slower than in vacuum, 1353 01:15:31,210 --> 01:15:33,700 so n is always bigger than 1. 1354 01:15:33,700 --> 01:15:37,480 In a few lectures, when we do proper electromagnetics-- also, 1355 01:15:37,480 --> 01:15:39,470 let me say that so far, n has been 1356 01:15:39,470 --> 01:15:41,980 a phenomenological quantity. 1357 01:15:41,980 --> 01:15:43,870 That's another funny word like "postulate." 1358 01:15:43,870 --> 01:15:47,530 What does it mean, "phenomenological." 1359 01:15:47,530 --> 01:15:49,390 It means a quantity that I have not 1360 01:15:49,390 --> 01:15:51,330 justified its physical origin. 1361 01:15:51,330 --> 01:15:52,950 I just pulled it out of a hat. 1362 01:15:52,950 --> 01:15:58,180 And I said, hey, you know what, the speed of light 1363 01:15:58,180 --> 01:16:00,580 in the medium is less than the speed in a vacuum. 1364 01:16:00,580 --> 01:16:02,080 So therefore, it is bigger than 1. 1365 01:16:02,080 --> 01:16:04,690 Now, when we do the proper physical origins of n, 1366 01:16:04,690 --> 01:16:09,360 then we will see that indeed, n can be less than 1. 1367 01:16:09,360 --> 01:16:11,970 It can even be negative, and that's another hot topic 1368 01:16:11,970 --> 01:16:15,310 in research optics nowadays. 1369 01:16:15,310 --> 01:16:16,390 So it is not necessary. 1370 01:16:16,390 --> 01:16:17,440 That's true. 1371 01:16:17,440 --> 01:16:20,350 But if you take it into the mechanical analogy, 1372 01:16:20,350 --> 01:16:22,540 interestingly enough, if you allow 1373 01:16:22,540 --> 01:16:28,330 n to go into these sort of strange modern sort of regions, 1374 01:16:28,330 --> 01:16:33,240 where it is less than 1 and even negative, then 1375 01:16:33,240 --> 01:16:34,990 in the mechanical analogy you would end up 1376 01:16:34,990 --> 01:16:38,008 with a negative energy, which is kind of interesting. 1377 01:16:38,008 --> 01:16:39,050 It doesn't mean anything. 1378 01:16:39,050 --> 01:16:40,050 This is just an analogy. 1379 01:16:40,050 --> 01:16:44,200 It doesn't invalidate anything, but it is kind of interesting. 1380 01:16:44,200 --> 01:16:50,810 And let me say upfront, n becoming smaller than 1 1381 01:16:50,810 --> 01:16:53,570 does not mean that I violate relativity. 1382 01:16:53,570 --> 01:16:56,900 It does not mean that something moves faster 1383 01:16:56,900 --> 01:16:59,210 than the speed of light. 1384 01:16:59,210 --> 01:17:02,610 And as we will see, just today, actually, n 1385 01:17:02,610 --> 01:17:04,710 is what we call the "phase velocity." 1386 01:17:04,710 --> 01:17:07,160 And the phase velocity of wave can be much greater 1387 01:17:07,160 --> 01:17:09,010 than the speed of light, because the phase 1388 01:17:09,010 --> 01:17:10,880 carries no information. 1389 01:17:10,880 --> 01:17:14,210 That's sort of the classical way to justify it. 1390 01:17:14,210 --> 01:17:17,990 So I can allow, actually, n to be less than 1, 1391 01:17:17,990 --> 01:17:19,740 and am perfectly fine. 1392 01:17:19,740 --> 01:17:24,060 So it is not necessary, but I should put a qualification 1393 01:17:24,060 --> 01:17:24,560 here. 1394 01:17:27,080 --> 01:17:30,700 Not physically allowable, but how you say-- 1395 01:17:30,700 --> 01:17:34,960 every day intuition allowable, perhaps. 1396 01:17:34,960 --> 01:17:37,360 But of course, physicists can do much better 1397 01:17:37,360 --> 01:17:38,903 than everyday intuition. 1398 01:17:45,050 --> 01:17:48,360 So let's resume for another 20 minutes or so, 1399 01:17:48,360 --> 01:17:50,490 and we'll continue on Monday. 1400 01:17:54,320 --> 01:17:56,390 So what I would like to do today is start 1401 01:17:56,390 --> 01:18:00,440 the description of wave optics. 1402 01:18:00,440 --> 01:18:06,780 So basically, we followed geometrical optics 1403 01:18:06,780 --> 01:18:09,870 and it was good enough to explain 1404 01:18:09,870 --> 01:18:13,920 a number of interesting things for us, imaging systems, 1405 01:18:13,920 --> 01:18:15,860 gradient indices, and so on. 1406 01:18:15,860 --> 01:18:18,640 But geometrical optics also has some serious failures. 1407 01:18:18,640 --> 01:18:24,030 The most important of those is that when 1408 01:18:24,030 --> 01:18:27,980 light interacts with very small features, 1409 01:18:27,980 --> 01:18:29,170 geometrical optics fails. 1410 01:18:29,170 --> 01:18:31,980 So for example, I used it when I answered 1411 01:18:31,980 --> 01:18:33,450 [? Kalpesh's ?] question. 1412 01:18:33,450 --> 01:18:37,370 If you limit light to a single ray-- that 1413 01:18:37,370 --> 01:18:40,920 is to an infinitesimal, small spot-- than geometrical optics 1414 01:18:40,920 --> 01:18:42,250 pattern fails. 1415 01:18:42,250 --> 01:18:45,485 So the picture that it gives us are completely wrong. 1416 01:18:45,485 --> 01:18:46,860 So in order to deal with that, we 1417 01:18:46,860 --> 01:18:50,650 have to deal with light as a wave phenomenon. 1418 01:18:50,650 --> 01:18:52,920 And there's an interesting history there. 1419 01:18:52,920 --> 01:18:56,270 Newton was a big proponent of geometrical or particle 1420 01:18:56,270 --> 01:18:58,450 description of light. 1421 01:18:58,450 --> 01:19:00,510 Some other contemporaries of his, 1422 01:19:00,510 --> 01:19:05,660 like Huygens, Fresnel, and sort of the Continental school, 1423 01:19:05,660 --> 01:19:10,470 they were proponents of the wave theory. 1424 01:19:10,470 --> 01:19:13,650 And for many years, the question remained 1425 01:19:13,650 --> 01:19:17,270 unsettled until Maxwell-- 1426 01:19:17,270 --> 01:19:21,920 in the end, I think 1899, or 1896, or something like that, 1427 01:19:21,920 --> 01:19:26,150 James Clerk Maxwell, actually a Professor of Fluid Mechanics, 1428 01:19:26,150 --> 01:19:29,390 came up with a set of equations that proved very convincingly 1429 01:19:29,390 --> 01:19:32,900 that the light is a wave. 1430 01:19:32,900 --> 01:19:35,600 And then 10 years later, quantum mechanics 1431 01:19:35,600 --> 01:19:37,880 proved that light is also a particle. 1432 01:19:37,880 --> 01:19:38,870 So both were correct. 1433 01:19:38,870 --> 01:19:43,340 Both Newton and the Continental school were correct. 1434 01:19:43,340 --> 01:19:47,030 So anyway, so you can think now that we're basically now moving 1435 01:19:47,030 --> 01:19:49,550 from England to the continental Europe, 1436 01:19:49,550 --> 01:19:52,850 because we're abandoning the geometrical optics theory, 1437 01:19:52,850 --> 01:19:55,860 and we're moving to wave optics. 1438 01:19:55,860 --> 01:20:05,000 Now, waves, of course, are a very pervasive phenomenon 1439 01:20:05,000 --> 01:20:07,550 that we see around us. 1440 01:20:07,550 --> 01:20:12,540 And they manifest themselves in many different forms. 1441 01:20:12,540 --> 01:20:19,730 So we're familiar with waves in seawater or rivers, and so on. 1442 01:20:19,730 --> 01:20:21,190 We're familiar with sound waves. 1443 01:20:21,190 --> 01:20:23,510 It's actually Singapore's founder, 1444 01:20:23,510 --> 01:20:25,760 Lee Kuan Yew giving a speech. 1445 01:20:25,760 --> 01:20:28,350 So we can communicate by sound waves. 1446 01:20:28,350 --> 01:20:30,890 That's another form of a material wave. 1447 01:20:30,890 --> 01:20:33,630 So like water waves, sound waves are material waves. 1448 01:20:33,630 --> 01:20:38,990 You set the air particles in motion when you speak. 1449 01:20:38,990 --> 01:20:43,430 We're also familiar with electromagnetic or radio waves, 1450 01:20:43,430 --> 01:20:44,930 not because we can see them. 1451 01:20:44,930 --> 01:20:47,960 These are actually not something that you can touch. 1452 01:20:47,960 --> 01:20:52,580 But all of us, we use cell phones, old-fashioned radios, 1453 01:20:52,580 --> 01:20:53,610 and so on. 1454 01:20:53,610 --> 01:20:54,210 We use them. 1455 01:20:54,210 --> 01:20:56,260 This is actually an old-fashioned antenna. 1456 01:20:56,260 --> 01:20:59,280 It's not that old-fashioned if you go to places like Europe, 1457 01:20:59,280 --> 01:21:00,710 like Greece, for example. 1458 01:21:00,710 --> 01:21:03,980 Many homes still have those antennas in their roof. 1459 01:21:03,980 --> 01:21:07,460 So this is the way people used to receive television signals 1460 01:21:07,460 --> 01:21:09,400 before cable TV. 1461 01:21:09,400 --> 01:21:12,665 And finally, there's other waves that they're sort of not quite 1462 01:21:12,665 --> 01:21:13,790 obvious that they're waves. 1463 01:21:13,790 --> 01:21:17,490 This is a very interesting geological phenomenon 1464 01:21:17,490 --> 01:21:18,450 in the Philippines. 1465 01:21:18,450 --> 01:21:20,270 It's called the "Chocolate Hills." 1466 01:21:20,270 --> 01:21:25,400 There's a number, it's actually hundreds of very small hills, 1467 01:21:25,400 --> 01:21:27,510 not very tall. 1468 01:21:27,510 --> 01:21:29,700 You might say, what kind of a wave is that? 1469 01:21:29,700 --> 01:21:31,050 The hills are not moving. 1470 01:21:31,050 --> 01:21:33,150 Actually, they're not moving in our time scale, 1471 01:21:33,150 --> 01:21:38,250 but in the geological terms, they actually move. 1472 01:21:38,250 --> 01:21:39,780 And sometimes, this motion-- 1473 01:21:39,780 --> 01:21:43,260 as you know, the motion of the earth-- 1474 01:21:43,260 --> 01:21:45,990 it can actually happen in timescales 1475 01:21:45,990 --> 01:21:49,022 that are perceived by humans, and then it's 1476 01:21:49,022 --> 01:21:49,980 called an "earthquake." 1477 01:21:49,980 --> 01:21:51,900 It's a very nasty phenomenon. 1478 01:21:51,900 --> 01:21:56,310 So the point is here that waves manifest themselves 1479 01:21:56,310 --> 01:21:58,170 in many different forms. 1480 01:21:58,170 --> 01:22:02,040 Of course, I omitted another important manifestation, 1481 01:22:02,040 --> 01:22:03,510 which is quantum mechanics. 1482 01:22:03,510 --> 01:22:06,480 Quantum mechanics says that everything is actually wave, 1483 01:22:06,480 --> 01:22:08,700 including yourselves and me. 1484 01:22:08,700 --> 01:22:11,830 And every particle is also described as a wave. 1485 01:22:11,830 --> 01:22:14,920 So electrons can be thought of as waves, and so on, and so 1486 01:22:14,920 --> 01:22:15,420 forth. 1487 01:22:15,420 --> 01:22:23,330 Anyway, the waves are in fact so pervasive that there's 1488 01:22:23,330 --> 01:22:25,250 no proper definition. 1489 01:22:25,250 --> 01:22:30,750 And to paraphrase something that an artist said once about art, 1490 01:22:30,750 --> 01:22:34,430 he said that art cannot be defined, 1491 01:22:34,430 --> 01:22:37,610 but you know it when you see it. 1492 01:22:37,610 --> 01:22:39,870 That's actually a very good way to put it, 1493 01:22:39,870 --> 01:22:41,970 at least as far as art is concerned. 1494 01:22:41,970 --> 01:22:44,430 But it's also true about waves. 1495 01:22:44,430 --> 01:22:47,310 Because there is a great variation 1496 01:22:47,310 --> 01:22:53,790 in the way waves occur, there's no proper definition. 1497 01:22:53,790 --> 01:22:57,060 What I attempted here is the simplest possible, 1498 01:22:57,060 --> 01:23:01,080 all-inclusive definition as a traveling disturbance. 1499 01:23:01,080 --> 01:23:03,630 As far as I can tell, it is certainly 1500 01:23:03,630 --> 01:23:05,850 inclusive of all waves. 1501 01:23:05,850 --> 01:23:07,470 Of course, it includes other things 1502 01:23:07,470 --> 01:23:10,320 that may not be waves, like a car moving down the road 1503 01:23:10,320 --> 01:23:12,000 is also a traveling disturbance. 1504 01:23:12,000 --> 01:23:13,680 It is clearly not a wave. 1505 01:23:13,680 --> 01:23:20,250 But anyway, another way to perhaps describe waves 1506 01:23:20,250 --> 01:23:22,330 is if you-- 1507 01:23:22,330 --> 01:23:25,170 there's some evidences of wave behavior. 1508 01:23:25,170 --> 01:23:28,198 And a very prominent one is what you call "interference." 1509 01:23:28,198 --> 01:23:29,990 So what you're seeing here is, we will see, 1510 01:23:29,990 --> 01:23:32,160 these are called "spherical waves." 1511 01:23:32,160 --> 01:23:34,650 And we see two point sources that 1512 01:23:34,650 --> 01:23:36,480 meet in these spherical waves. 1513 01:23:36,480 --> 01:23:38,640 And as they propagate out, you see 1514 01:23:38,640 --> 01:23:42,810 this very characteristic alternation 1515 01:23:42,810 --> 01:23:45,600 between bright and dark spots. 1516 01:23:45,600 --> 01:23:48,090 I have not even really said what this thing is. 1517 01:23:48,090 --> 01:23:52,180 Is it a sound wave, a water wave, an electromagnetic wave? 1518 01:23:52,180 --> 01:23:54,240 In fact, it could be all of the above. 1519 01:23:54,240 --> 01:23:57,990 In all of these cases, if I represent 1520 01:23:57,990 --> 01:24:01,320 as bright the high value of the disturbance 1521 01:24:01,320 --> 01:24:04,530 and as black the low value of the disturbance, 1522 01:24:04,530 --> 01:24:08,100 then these sort of not quite periodic, 1523 01:24:08,100 --> 01:24:12,860 but periodic-like looking behavior 1524 01:24:12,860 --> 01:24:14,280 is called "interference." 1525 01:24:14,280 --> 01:24:17,970 And when you observe this sort of interference, 1526 01:24:17,970 --> 01:24:20,790 then it is a pretty good indication 1527 01:24:20,790 --> 01:24:23,380 that you are actually observing a wave phenomenon. 1528 01:24:23,380 --> 01:24:26,820 In fact, the reason Fresnel in the Continental school 1529 01:24:26,820 --> 01:24:29,730 insisted that light is a wave is because they actually 1530 01:24:29,730 --> 01:24:32,760 observed light interference in experiments. 1531 01:24:32,760 --> 01:24:35,400 They observed the light doing something 1532 01:24:35,400 --> 01:24:36,850 like that in the experiment. 1533 01:24:36,850 --> 01:24:38,990 So they said, well, wait, it must be a wave, then. 1534 01:24:41,730 --> 01:24:44,550 Another phenomenon which is-- 1535 01:24:44,550 --> 01:24:47,040 actually, it is related to interference. 1536 01:24:47,040 --> 01:24:49,500 When we cover it, it has quite a different name. 1537 01:24:49,500 --> 01:24:51,030 It's called "diffraction." 1538 01:24:51,030 --> 01:24:53,640 But when we cover it, we will see that clearly, it 1539 01:24:53,640 --> 01:24:55,200 is not a new phenomenon. 1540 01:24:55,200 --> 01:24:58,560 It is interference in this guise. 1541 01:24:58,560 --> 01:25:03,840 And diffraction basically says that the geometrical definition 1542 01:25:03,840 --> 01:25:07,290 of shadows is incorrect. 1543 01:25:07,290 --> 01:25:11,820 So what you see here is a wave that is impinging on a block. 1544 01:25:11,820 --> 01:25:13,890 So it is blocked above and below. 1545 01:25:13,890 --> 01:25:17,160 And a small portion of it is allowed to pass through. 1546 01:25:17,160 --> 01:25:18,900 Geometrical optics would have you 1547 01:25:18,900 --> 01:25:21,500 believe that you would basically get 1548 01:25:21,500 --> 01:25:25,200 a continuation, unperturbed continuation 1549 01:25:25,200 --> 01:25:29,312 of the wave in the geometrical shadow of this slit. 1550 01:25:29,312 --> 01:25:30,770 You can think of it as a slit there 1551 01:25:30,770 --> 01:25:32,650 that the wave goes through. 1552 01:25:32,650 --> 01:25:34,620 But diffraction says that's not the case. 1553 01:25:34,620 --> 01:25:38,370 As you can see, there's a rather more complicated pattern 1554 01:25:38,370 --> 01:25:41,820 that emerges here. 1555 01:25:41,820 --> 01:25:45,710 And that is another evidence of a wave phenomenon 1556 01:25:45,710 --> 01:25:51,730 that we will describe quantitatively in great detail 1557 01:25:51,730 --> 01:25:52,230 later. 1558 01:25:55,470 --> 01:25:59,920 Let's take a look a little bit in some more waves. 1559 01:25:59,920 --> 01:26:02,770 So for example, the first one-- 1560 01:26:02,770 --> 01:26:05,770 it meets my definition as a traveling disturbance, 1561 01:26:05,770 --> 01:26:09,190 but not a very interesting one, because it is just a shape 1562 01:26:09,190 --> 01:26:10,640 moving. 1563 01:26:10,640 --> 01:26:13,750 The one that is on the right is perhaps more interesting, 1564 01:26:13,750 --> 01:26:15,860 because now you see sort of-- 1565 01:26:15,860 --> 01:26:17,890 it is more believable that it is a wave. 1566 01:26:17,890 --> 01:26:22,810 It has some oscillation in addition to the traveling 1567 01:26:22,810 --> 01:26:23,650 disturbance. 1568 01:26:23,650 --> 01:26:26,530 By the way, oscillatory behavior is also 1569 01:26:26,530 --> 01:26:29,500 characteristic of waves. 1570 01:26:29,500 --> 01:26:31,300 But it is not necessarily periodic. 1571 01:26:31,300 --> 01:26:33,237 In fact, you can have a periodic wave, 1572 01:26:33,237 --> 01:26:34,570 but that's not very interesting. 1573 01:26:34,570 --> 01:26:36,850 It is the non-periodic waves that 1574 01:26:36,850 --> 01:26:38,140 are the most interesting ones. 1575 01:26:42,510 --> 01:26:47,628 Now, if you let me go back and play this movie once again, 1576 01:26:47,628 --> 01:26:49,920 that's not very interesting, but what you will see here 1577 01:26:49,920 --> 01:26:51,337 is you will see that it's actually 1578 01:26:51,337 --> 01:26:55,110 the same envelope that you saw moving on this diagram. 1579 01:26:55,110 --> 01:26:57,810 You will see it is the same envelope moving over here, 1580 01:26:57,810 --> 01:27:00,780 but inside, you have some wiggles that are also moving. 1581 01:27:00,780 --> 01:27:03,610 And if you observe it very carefully, 1582 01:27:03,610 --> 01:27:05,680 you'll see that the envelope is moving 1583 01:27:05,680 --> 01:27:09,150 with a different velocity than the wiggles. 1584 01:27:09,150 --> 01:27:10,650 The wiggles have a name, by the way. 1585 01:27:10,650 --> 01:27:11,608 They're called fringes. 1586 01:27:11,608 --> 01:27:12,520 Can you see? 1587 01:27:12,520 --> 01:27:14,880 So the wiggles are actually moving faster 1588 01:27:14,880 --> 01:27:16,440 than the envelope. 1589 01:27:16,440 --> 01:27:18,180 So this brings us to the definition 1590 01:27:18,180 --> 01:27:23,050 of what I used once Professor Sheppard's question about n 1591 01:27:23,050 --> 01:27:23,970 earlier. 1592 01:27:23,970 --> 01:27:26,560 I used the term "phase velocity." 1593 01:27:26,560 --> 01:27:29,710 So what you saw in the previous movie, 1594 01:27:29,710 --> 01:27:32,250 in the one that was in the right-hand side, 1595 01:27:32,250 --> 01:27:35,160 was an envelope that was moving. 1596 01:27:35,160 --> 01:27:36,220 So here is the blue one. 1597 01:27:36,220 --> 01:27:37,410 That's the envelope. 1598 01:27:37,410 --> 01:27:40,600 And inside, there was a sinusoid. 1599 01:27:40,600 --> 01:27:43,880 The sinusoid is multiplied by the envelope. 1600 01:27:43,880 --> 01:27:47,620 A wave that is composed of a sinusoid multiplied 1601 01:27:47,620 --> 01:27:50,960 by an envelope is said to have a carrier. 1602 01:27:50,960 --> 01:27:55,720 So the carrier is a sinusoid itself. 1603 01:27:55,720 --> 01:28:04,610 The carrier can move at its own velocity, which, surprisingly, 1604 01:28:04,610 --> 01:28:08,300 is usually different than the velocity 1605 01:28:08,300 --> 01:28:10,170 that the envelope is moving. 1606 01:28:10,170 --> 01:28:13,730 Why I say surprisingly, well, some simple sort of [INAUDIBLE] 1607 01:28:13,730 --> 01:28:16,380 might have said, they better move at the same velocity, 1608 01:28:16,380 --> 01:28:19,130 shouldn't they, when in fact, the truth is the opposite. 1609 01:28:19,130 --> 01:28:20,890 More often than not-- 1610 01:28:20,890 --> 01:28:23,080 in fact, you have to try very hard to make them move 1611 01:28:23,080 --> 01:28:25,080 with the same velocity, but more often than not, 1612 01:28:25,080 --> 01:28:26,710 they move at different velocities. 1613 01:28:26,710 --> 01:28:28,730 So the velocity at which the envelope is moving 1614 01:28:28,730 --> 01:28:30,770 is called the "group velocity." 1615 01:28:30,770 --> 01:28:32,840 The velocity at which the carrier is moving 1616 01:28:32,840 --> 01:28:34,550 is called the "phase velocity." 1617 01:28:34,550 --> 01:28:38,000 And right now, we'll just leave those like this. 1618 01:28:38,000 --> 01:28:41,030 Later, we will actually derive them properly. 1619 01:28:41,030 --> 01:28:43,450 And we will see how they come about, 1620 01:28:43,450 --> 01:28:45,950 and how they are defined. 1621 01:28:45,950 --> 01:28:49,310 But I can say for now, without any proof, 1622 01:28:49,310 --> 01:28:58,210 that what we wrote in geometrical optics, c over n, 1623 01:28:58,210 --> 01:29:00,490 that is the phase velocity of the light. 1624 01:29:09,040 --> 01:29:11,590 If you were to put an envelope on the light somehow, 1625 01:29:11,590 --> 01:29:16,660 which is done in special lasers called "mode-locked lasers," 1626 01:29:16,660 --> 01:29:19,940 you can have a light wave that looks like this. 1627 01:29:19,940 --> 01:29:22,460 It has an envelope as it propagates. 1628 01:29:22,460 --> 01:29:26,890 In those, the group velocity is generally 1629 01:29:26,890 --> 01:29:29,710 different than the phase velocity of the light. 1630 01:29:34,970 --> 01:29:37,640 Of course I said that earlier that the other waves are not 1631 01:29:37,640 --> 01:29:38,600 very interesting. 1632 01:29:38,600 --> 01:29:43,370 Nevertheless, for educational and analysis purposes, 1633 01:29:43,370 --> 01:29:44,510 they're simple. 1634 01:29:44,510 --> 01:29:46,970 So therefore, we start with periodic waves 1635 01:29:46,970 --> 01:29:51,130 when we analyze waves, and periodic is also 1636 01:29:51,130 --> 01:29:53,810 known as "harmonic waves." 1637 01:29:53,810 --> 01:29:55,970 And of course, in a harmonic wave, 1638 01:29:55,970 --> 01:29:58,173 there's no envelope to speak of. 1639 01:29:58,173 --> 01:29:59,840 And therefore, there's no group velocity 1640 01:29:59,840 --> 01:30:02,860 to speak of because the envelope is flat. 1641 01:30:02,860 --> 01:30:06,210 There's no profile going anywhere. 1642 01:30:06,210 --> 01:30:08,990 The only thing you have is a sinusoidal disturbance 1643 01:30:08,990 --> 01:30:12,770 moving with the phase velocity. 1644 01:30:12,770 --> 01:30:15,980 Here you only have the carrier, so the only velocity 1645 01:30:15,980 --> 01:30:18,860 to speak of is the phase velocity. 1646 01:30:18,860 --> 01:30:21,040 That's a characteristic of the harmonic wave. 1647 01:30:21,040 --> 01:30:23,550 And I will start putting some notation here. 1648 01:30:23,550 --> 01:30:28,620 So the time, you can see that time is run in index here. 1649 01:30:28,620 --> 01:30:31,340 And the horizontal axis, I will call it z. 1650 01:30:31,340 --> 01:30:37,020 So we will stick to z as the actual wave 1651 01:30:37,020 --> 01:30:38,310 propagation for a while. 1652 01:30:41,280 --> 01:30:43,360 So now let's do some math. 1653 01:30:43,360 --> 01:30:49,090 Since this is a harmonic wave, I can expect 1654 01:30:49,090 --> 01:30:52,280 to describe it as a sinusoid. 1655 01:30:52,280 --> 01:30:54,890 And the sinusoid, in order to characterize it, 1656 01:30:54,890 --> 01:30:57,530 I need to set a number of quantities, 1657 01:30:57,530 --> 01:30:59,840 which have certain buzzwords. 1658 01:30:59,840 --> 01:31:02,450 So we'd better start defining our buzzwords right now, 1659 01:31:02,450 --> 01:31:05,000 so we don't get confused when we refer to them 1660 01:31:05,000 --> 01:31:06,600 later in the class. 1661 01:31:06,600 --> 01:31:09,440 So the constant that goes in front of the sinusoid 1662 01:31:09,440 --> 01:31:12,780 is called the "amplitude." 1663 01:31:12,780 --> 01:31:15,410 The reason I use e as a symbol for the constant 1664 01:31:15,410 --> 01:31:19,800 is because electromagnetic waves are usually electric fields. 1665 01:31:19,800 --> 01:31:23,200 We will see that later, after we do Maxwell's equations. 1666 01:31:23,200 --> 01:31:25,070 For now, this could be any symbol, actually. 1667 01:31:25,070 --> 01:31:28,370 In fact, very quickly I will refer to a different symbol. 1668 01:31:28,370 --> 01:31:31,610 But in any case, this quantity that appears here 1669 01:31:31,610 --> 01:31:33,350 is called the amplitude. 1670 01:31:33,350 --> 01:31:37,550 Then the other quantity that we need to describe a sinusoid 1671 01:31:37,550 --> 01:31:38,420 is the frequency. 1672 01:31:38,420 --> 01:31:41,810 But now the frequency, we have to be a little bit careful 1673 01:31:41,810 --> 01:31:44,150 because there's two frequencies going on. 1674 01:31:44,150 --> 01:31:46,430 You can see very clearly-- 1675 01:31:46,430 --> 01:31:49,900 you can think of this as a snapshot of the wave. 1676 01:31:49,900 --> 01:31:52,110 If I take this picture and I freeze it, 1677 01:31:52,110 --> 01:31:54,710 I will get something that looks like this. 1678 01:31:54,710 --> 01:31:59,320 So in this snapshot, the axis is space coordinate. 1679 01:31:59,320 --> 01:32:00,880 It is z. 1680 01:32:00,880 --> 01:32:03,490 So the period that you observe here, 1681 01:32:03,490 --> 01:32:05,410 the period that you observe in the spatial 1682 01:32:05,410 --> 01:32:09,050 axis, it is called the "wavelength." 1683 01:32:09,050 --> 01:32:09,920 We saw that before. 1684 01:32:09,920 --> 01:32:13,220 We did the definition of the wavelength in, I believe, 1685 01:32:13,220 --> 01:32:16,370 lecture number one. 1686 01:32:16,370 --> 01:32:17,990 Now suppose I do like this-- suppose 1687 01:32:17,990 --> 01:32:21,410 I fix my position somewhere on this axis, 1688 01:32:21,410 --> 01:32:26,060 and I observe the value of the wave as it is passing by me. 1689 01:32:26,060 --> 01:32:29,750 So now I'm fixed at the given z and observe the Wave. 1690 01:32:29,750 --> 01:32:31,850 What do I observe if I stand over here? 1691 01:32:35,230 --> 01:32:36,960 I will have another sinusoid, don't I? 1692 01:32:36,960 --> 01:32:40,940 Because if I am sitting here, the wave that I see 1693 01:32:40,940 --> 01:32:41,820 will be oscillating. 1694 01:32:41,820 --> 01:32:44,700 It will go in the sinusoidal fashion. 1695 01:32:44,700 --> 01:32:46,920 It will oscillate from a positive value 1696 01:32:46,920 --> 01:32:49,300 to a negative value. 1697 01:32:49,300 --> 01:32:55,050 So that actually associates with a temporal frequency 1698 01:32:55,050 --> 01:32:59,290 of the wave, for which we simply use the term "frequency." 1699 01:32:59,290 --> 01:33:03,970 So we'll say frequency, we mean the temporal frequency. 1700 01:33:03,970 --> 01:33:07,290 And some people use f. 1701 01:33:07,290 --> 01:33:11,550 I prefer to use the symbol nu for that quantity. 1702 01:33:11,550 --> 01:33:17,130 Finally, I have not defined my time axis yet. 1703 01:33:17,130 --> 01:33:20,820 So when I'm looking at the oscillation, if I'm fixed 1704 01:33:20,820 --> 01:33:25,000 and I look at the oscillation of the wave, at time 0, 1705 01:33:25,000 --> 01:33:29,150 the wave might be absolute positive. 1706 01:33:29,150 --> 01:33:31,790 In that case, it means the wave is a cosine. 1707 01:33:31,790 --> 01:33:33,730 Or it may be 0. 1708 01:33:33,730 --> 01:33:36,490 In that case, it means it is a sine. 1709 01:33:36,490 --> 01:33:40,210 And how do I correct for this discrepancy? 1710 01:33:40,210 --> 01:33:43,080 By adding a phase delay, by adding, for example 1711 01:33:43,080 --> 01:33:44,590 in this case, pi over 2. 1712 01:33:44,590 --> 01:33:46,370 So this additional phase that I need 1713 01:33:46,370 --> 01:33:49,930 to add in order to fix the origin of time 1714 01:33:49,930 --> 01:33:51,570 is called the "phase delay." 1715 01:33:51,570 --> 01:33:57,960 And for a while, I will use the symbol phi for the phase delay. 1716 01:33:57,960 --> 01:34:01,080 So we defined basically all of the quantities that we need, 1717 01:34:01,080 --> 01:34:06,930 except nu is not anywhere to be seen in this equation. 1718 01:34:06,930 --> 01:34:10,830 What I've written instead is this expression z minus ct. 1719 01:34:10,830 --> 01:34:12,876 And what is the meaning of this expression? 1720 01:34:23,580 --> 01:34:24,658 AUDIENCE: Traveling wave. 1721 01:34:24,658 --> 01:34:26,450 GEORGE BARBASTATHIS: It's a traveling wave, 1722 01:34:26,450 --> 01:34:29,640 or another way to put it is a traveling coordinate system. 1723 01:34:29,640 --> 01:34:44,570 If I write-- we have no doubt what this is. 1724 01:34:44,570 --> 01:34:46,340 That's a sinusoid. 1725 01:34:46,340 --> 01:34:50,180 Now, that's a sinusoid in an axis of coordinate xi. 1726 01:34:55,150 --> 01:34:59,830 If I now say xi equals z minus c t, 1727 01:34:59,830 --> 01:35:03,920 where t is time, what I have done is I've taken this axis 1728 01:35:03,920 --> 01:35:07,120 and I've started moving it with respect to its rest 1729 01:35:07,120 --> 01:35:09,310 position that equals 0. 1730 01:35:09,310 --> 01:35:12,940 So this is what I've done in the expression for the wave. 1731 01:35:12,940 --> 01:35:16,540 I've put this traveling coordinate system 1732 01:35:16,540 --> 01:35:18,340 into the sinusoid. 1733 01:35:18,340 --> 01:35:22,310 And I don't really have the frequency in here, 1734 01:35:22,310 --> 01:35:25,030 but you can see that time, the time variable, 1735 01:35:25,030 --> 01:35:27,040 is in the expression for the sinusoid. 1736 01:35:27,040 --> 01:35:29,340 So I can get the frequency very easily. 1737 01:35:29,340 --> 01:35:31,690 And the way I get it is by observing 1738 01:35:31,690 --> 01:35:35,710 that since the wave is periodic in the time domain, 1739 01:35:35,710 --> 01:35:38,320 it will repeat one period later. 1740 01:35:38,320 --> 01:35:41,480 So the period is the inverse of the frequency. 1741 01:35:41,480 --> 01:35:46,560 So if I look at this expression over here, 1742 01:35:46,560 --> 01:35:47,720 let me rewrite the wave. 1743 01:36:03,419 --> 01:36:05,010 I can actually write it-- 1744 01:36:09,860 --> 01:36:15,550 I haven't done anything, just expanded the parameters. 1745 01:36:15,550 --> 01:36:24,840 So when this quantity becomes 2 pi or any integral multiple 1746 01:36:24,840 --> 01:36:28,020 thereof, the wave will repeat, won't it? 1747 01:36:28,020 --> 01:36:32,070 So that will happen when the time 1748 01:36:32,070 --> 01:36:34,680 equals the period of the wave. 1749 01:36:34,680 --> 01:36:38,440 Or another way to put it is that 2 pi over lambda times 1750 01:36:38,440 --> 01:36:40,980 c over nu, where nu is the frequency, 1751 01:36:40,980 --> 01:36:43,110 the inverse of the period, here comes the pi. 1752 01:36:43,110 --> 01:36:45,840 Of course, the pi goes away, and we derive something 1753 01:36:45,840 --> 01:36:49,680 that we've seen before, c equals lambda nu. 1754 01:36:49,680 --> 01:36:54,990 And that is what we used to call the "dispersion relation." 1755 01:36:54,990 --> 01:36:58,560 So this is how we connect the phase 1756 01:36:58,560 --> 01:37:04,874 velocity of the wave with the wavelength and the frequency. 1757 01:37:08,590 --> 01:37:15,210 And the phase delay is actually a very important property 1758 01:37:15,210 --> 01:37:17,820 of waves, but it's not a real big deal. 1759 01:37:17,820 --> 01:37:19,330 This is what I was saying before. 1760 01:37:19,330 --> 01:37:19,860 AUDIENCE: George. 1761 01:37:19,860 --> 01:37:20,902 GEORGE BARBASTATHIS: Yes. 1762 01:37:20,902 --> 01:37:23,610 AUDIENCE: Why is it called "dispersion relation?" 1763 01:37:23,610 --> 01:37:25,770 GEORGE BARBASTATHIS: Yes, there's no dispersion 1764 01:37:25,770 --> 01:37:28,540 to be seen here, isn't there? 1765 01:37:28,540 --> 01:37:34,760 However, if I put, for example an index of refraction, then 1766 01:37:34,760 --> 01:37:36,760 the index of refraction is a function of lambda, 1767 01:37:36,760 --> 01:37:38,680 so then you'll get dispersion. 1768 01:37:38,680 --> 01:37:40,780 And of course, another case is this 1769 01:37:40,780 --> 01:37:43,540 is the dispersion relation in free space. 1770 01:37:43,540 --> 01:37:46,640 If you put boundary conditions to the wave-- for example, 1771 01:37:46,640 --> 01:37:48,460 if you confine it in a wave guide-- 1772 01:37:48,460 --> 01:37:51,070 the dispersion relation changes. 1773 01:37:51,070 --> 01:37:52,780 I will not do this in this class, 1774 01:37:52,780 --> 01:37:55,220 but that's another way where the wave becomes dispersive. 1775 01:37:58,613 --> 01:38:00,530 So that is the explanation of the phase delay. 1776 01:38:00,530 --> 01:38:02,730 I don't want to spend too much time. 1777 01:38:02,730 --> 01:38:05,560 If it equals 0, you see this. 1778 01:38:05,560 --> 01:38:09,000 This means the wave is a cosine, the phase delay is 0. 1779 01:38:09,000 --> 01:38:12,630 And then, depending on what you see at t equals 0, 1780 01:38:12,630 --> 01:38:14,730 you may have different phase delays. 1781 01:38:14,730 --> 01:38:19,367 Now, the reason the phase delay is an important quantity 1782 01:38:19,367 --> 01:38:21,450 is because I want you to look at it very carefully 1783 01:38:21,450 --> 01:38:24,860 and tell me what you see. 1784 01:38:24,860 --> 01:38:30,233 So anybody want to volunteer what has happened here? 1785 01:38:30,233 --> 01:38:31,650 I'm going to play this once again. 1786 01:38:41,097 --> 01:38:43,430 How are the two different, on the left and on the right? 1787 01:38:53,610 --> 01:38:57,910 On the left, again, look at it very carefully. 1788 01:38:57,910 --> 01:39:02,860 On the left, the two spherical waves, 1789 01:39:02,860 --> 01:39:05,950 they're always bright together or dark together. 1790 01:39:05,950 --> 01:39:08,750 So they oscillate, as we say, in phase. 1791 01:39:08,750 --> 01:39:12,020 On the right, the two waves are out of phase. 1792 01:39:12,020 --> 01:39:13,690 So if you look at it carefully, you 1793 01:39:13,690 --> 01:39:17,290 will see that when one is bright, the other is dark, 1794 01:39:17,290 --> 01:39:18,280 and vice versa. 1795 01:39:18,280 --> 01:39:20,095 I'll play it again so you can see it. 1796 01:39:26,740 --> 01:39:30,670 And because of this, again, I'm going to play it once again. 1797 01:39:30,670 --> 01:39:34,460 And now don't look at the centers of the waves anymore, 1798 01:39:34,460 --> 01:39:37,210 but look at the general patterns. 1799 01:39:37,210 --> 01:39:39,340 You will see that the general patterns are actually 1800 01:39:39,340 --> 01:39:40,660 different. 1801 01:39:40,660 --> 01:39:43,510 They have the same general shape, 1802 01:39:43,510 --> 01:39:49,000 but whether the pattern is dark or bright at any given instant 1803 01:39:49,000 --> 01:39:52,450 actually depends on how the two waves started. 1804 01:39:52,450 --> 01:39:56,470 So when we have-- this is quantum superposition, 1805 01:39:56,470 --> 01:40:01,010 because I have waves originating from two sources 1806 01:40:01,010 --> 01:40:02,680 simultaneously. 1807 01:40:02,680 --> 01:40:05,200 When we have a superposition, the structure 1808 01:40:05,200 --> 01:40:07,660 of the overall wave that we observe 1809 01:40:07,660 --> 01:40:13,420 depends very strongly on the relative phase between the two, 1810 01:40:13,420 --> 01:40:14,710 the two sources in this case. 1811 01:40:17,420 --> 01:40:19,280 So with that, I think I will stop, 1812 01:40:19,280 --> 01:40:22,220 and we will continue on Monday. 1813 01:40:22,220 --> 01:40:24,330 And I will not post any new notes for Monday, 1814 01:40:24,330 --> 01:40:27,250 because what I think we're far enough behind that we 1815 01:40:27,250 --> 01:40:29,860 should catch up on Monday. 1816 01:40:29,860 --> 01:40:34,410 What I would like to say is that it is about time to start 1817 01:40:34,410 --> 01:40:38,320 the project for the graduate class, for 2.710. 1818 01:40:38,320 --> 01:40:43,330 So soon, but not for-- 1819 01:40:43,330 --> 01:40:47,670 basically by Thursday morning Boston time and hopefully 1820 01:40:47,670 --> 01:40:51,250 earlier, I will post in the website 1821 01:40:51,250 --> 01:40:53,980 a list of possible projects and some instructions 1822 01:40:53,980 --> 01:40:55,180 on what to do. 1823 01:40:55,180 --> 01:40:58,720 And then basically, what I would like you guys to do 1824 01:40:58,720 --> 01:41:02,530 is spend the next week selecting your project, 1825 01:41:02,530 --> 01:41:05,500 and also contacting your mentors. 1826 01:41:05,500 --> 01:41:09,700 Each project will be mentored by one of my graduate students, 1827 01:41:09,700 --> 01:41:11,200 and some by me. 1828 01:41:11,200 --> 01:41:12,970 So possibly, I don't know if Colin, you 1829 01:41:12,970 --> 01:41:14,512 might be interested in mentoring one, 1830 01:41:14,512 --> 01:41:17,070 but anyway, we'll figure it out. 1831 01:41:17,070 --> 01:41:18,770 Maybe Sam from Singapore, maybe not. 1832 01:41:18,770 --> 01:41:25,520 But anyway, we will assign mentors for the projects, 1833 01:41:25,520 --> 01:41:26,985 and then we'll get them started. 1834 01:41:26,985 --> 01:41:30,700 The presentation is in May, May 6 or May 7. 1835 01:41:30,700 --> 01:41:33,570 But anyway, the sooner we start, the better.