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 --> 00:00:12,660 To make a donation or to view additional materials 6 00:00:12,660 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,640 at ocw.mit.edu. 8 00:00:21,040 --> 00:00:22,040 GEORGE BARBASTATHIS: OK. 9 00:00:22,040 --> 00:00:27,480 So I don't know if everybody's awake, but I will start. 10 00:00:30,120 --> 00:00:41,000 So today, the topic is actually Hamiltonian optics, 11 00:00:41,000 --> 00:00:45,720 but we had some leftover from last time, leakage. 12 00:00:45,720 --> 00:00:47,570 So I'd like to go through that very briefly. 13 00:00:47,570 --> 00:00:52,510 And then I will handover to Se Baek and Pepe, 14 00:00:52,510 --> 00:00:56,540 who will give a demonstration of aberrations 15 00:00:56,540 --> 00:00:59,240 in sort of experimental demonstration, 16 00:00:59,240 --> 00:01:04,090 and also how optical design software works. 17 00:01:04,090 --> 00:01:05,840 And then we'll get into Hamiltonian optics 18 00:01:05,840 --> 00:01:08,880 the second hour, and we'll see how far we go. 19 00:01:08,880 --> 00:01:09,380 OK. 20 00:01:09,380 --> 00:01:12,500 So last time we talked about how many aberrations? 21 00:01:12,500 --> 00:01:17,293 We talked about spherical, coma, that's about it. 22 00:01:17,293 --> 00:01:19,210 So we talked, actually, about two aberrations. 23 00:01:19,210 --> 00:01:21,700 And we said that there is five of them-- 24 00:01:21,700 --> 00:01:26,230 the five so-called Seidel aberrations or side order 25 00:01:26,230 --> 00:01:27,460 aberrations. 26 00:01:27,460 --> 00:01:31,570 And today, I'll cover the other three. 27 00:01:31,570 --> 00:01:36,600 So the first one in line is astigmatism, 28 00:01:36,600 --> 00:01:38,780 which is actually-- 29 00:01:38,780 --> 00:01:42,240 for me, at least, it is the most difficult to visualize. 30 00:01:44,870 --> 00:01:49,290 The best schematic I have seen actually is in your textbook. 31 00:01:49,290 --> 00:01:51,300 It's the middle picture over here. 32 00:01:51,300 --> 00:01:55,040 But before we get to that and we see what happens 33 00:01:55,040 --> 00:02:00,810 is, let's define to make sure that we know the terminology. 34 00:02:00,810 --> 00:02:07,500 So astigmatism happens-- look at the top left diagram. 35 00:02:07,500 --> 00:02:10,759 Astigmatism happens with an object point 36 00:02:10,759 --> 00:02:14,130 that is off center. 37 00:02:14,130 --> 00:02:17,700 And since it is off center, imagine that there's 38 00:02:17,700 --> 00:02:19,440 a fundamental asymmetry. 39 00:02:19,440 --> 00:02:24,990 If you take the chief ray that goes through the center 40 00:02:24,990 --> 00:02:25,890 of the lens-- 41 00:02:25,890 --> 00:02:28,860 and again, here the assumption is that the aperture 42 00:02:28,860 --> 00:02:30,720 is at the lens itself. 43 00:02:30,720 --> 00:02:35,010 So the chief ray goes to the center of the lens. 44 00:02:35,010 --> 00:02:39,150 So now you can take two ray-- you 45 00:02:39,150 --> 00:02:42,550 can think of two marginal rays. 46 00:02:42,550 --> 00:02:45,060 One is the marginal ray that goes 47 00:02:45,060 --> 00:02:49,110 through the meridional plane. 48 00:02:49,110 --> 00:02:54,610 So the meridional plane is the one shown like this, 49 00:02:54,610 --> 00:02:56,540 and the other is the marginal ray 50 00:02:56,540 --> 00:02:59,580 that goes through the sagittal plane, perpendicular 51 00:02:59,580 --> 00:03:02,820 to the meridional plane. 52 00:03:02,820 --> 00:03:08,310 So astigmatism occurs because the two-- 53 00:03:08,310 --> 00:03:11,890 now take cross-sections of the rays in the two planes-- 54 00:03:11,890 --> 00:03:14,540 the sagittal and the meridional plane. 55 00:03:14,540 --> 00:03:17,670 That two ray bundles, they do not necessarily 56 00:03:17,670 --> 00:03:18,870 focus at the same place. 57 00:03:21,510 --> 00:03:25,810 In the sagittal plane, actually, if you 58 00:03:25,810 --> 00:03:27,650 tried to visualize it in the sagittal plane, 59 00:03:27,650 --> 00:03:30,940 there is no difference than normal incidence. 60 00:03:30,940 --> 00:03:31,860 So you basically get-- 61 00:03:35,540 --> 00:03:38,380 so whereas in the meridional plane, because you 62 00:03:38,380 --> 00:03:40,848 are off-axis, you have this property 63 00:03:40,848 --> 00:03:42,390 that we discussed before, that you'll 64 00:03:42,390 --> 00:03:47,410 get different magnification at different points in the field. 65 00:03:47,410 --> 00:03:50,400 So as a result, these two might actually 66 00:03:50,400 --> 00:03:52,660 yield different focal distances. 67 00:03:52,660 --> 00:03:55,680 So this is depicted here, actually, where you're seeing-- 68 00:03:55,680 --> 00:03:57,320 it's actually a pretty good diagram. 69 00:03:57,320 --> 00:04:02,020 You can see the sagittal plane and the meridional plane. 70 00:04:02,020 --> 00:04:06,630 And you can see that the sagittal plane actually 71 00:04:06,630 --> 00:04:08,670 focuses out here. 72 00:04:08,670 --> 00:04:11,410 It is F sub S for sagittal. 73 00:04:11,410 --> 00:04:13,860 Whereas the meridional plane focuses here. 74 00:04:13,860 --> 00:04:17,459 It is called F sub T for tangential. 75 00:04:17,459 --> 00:04:21,060 So if an optical system is subject to this problem, 76 00:04:21,060 --> 00:04:22,689 we just call astigmatic. 77 00:04:27,090 --> 00:04:29,820 The typical way that people characterize astigmatism 78 00:04:29,820 --> 00:04:31,530 is with a spoke wheel. 79 00:04:31,530 --> 00:04:35,125 And most of you must have taken eye exams. 80 00:04:35,125 --> 00:04:36,750 They give you a similar test, actually, 81 00:04:36,750 --> 00:04:38,890 to test if your eyes are astigmatic 82 00:04:38,890 --> 00:04:40,290 and you need correction. 83 00:04:40,290 --> 00:04:43,020 They show you something like this, a spoke wheel. 84 00:04:43,020 --> 00:04:48,810 And they ask you, basically, if the spokes are simultaneously 85 00:04:48,810 --> 00:04:57,040 in focus with the circle. 86 00:04:57,040 --> 00:05:00,330 The outline circle. 87 00:05:00,330 --> 00:05:04,440 So if the system is astigmatic, then 88 00:05:04,440 --> 00:05:09,910 as you move a piece of paper along the optical axis, 89 00:05:09,910 --> 00:05:12,300 you will see one of them come to a focus first, 90 00:05:12,300 --> 00:05:14,960 and the sagittal come to the focus later. 91 00:05:14,960 --> 00:05:15,460 OK. 92 00:05:22,770 --> 00:05:23,880 So that's astigmatism. 93 00:05:30,430 --> 00:05:32,290 I hear some voices, but I think they're 94 00:05:32,290 --> 00:05:34,820 from the next classroom, not from Boston. 95 00:05:34,820 --> 00:05:38,120 Did anybody say anything there? 96 00:05:38,120 --> 00:05:38,850 I guess not. 97 00:05:38,850 --> 00:05:39,350 OK. 98 00:05:42,630 --> 00:05:48,040 A related aberration, which has to do with the fact that-- 99 00:05:48,040 --> 00:05:49,450 well, let me say first. 100 00:05:49,450 --> 00:05:52,840 So a related aberration is called caricature of field. 101 00:05:52,840 --> 00:05:56,230 This one is actually easy to understand. 102 00:05:56,230 --> 00:05:57,160 It says the following. 103 00:05:57,160 --> 00:06:01,660 It says that if you derive the imaging condition for a point 104 00:06:01,660 --> 00:06:06,690 on-axis, then imagine you go off-axis, 105 00:06:06,690 --> 00:06:10,680 it says that the image may not necessarily 106 00:06:10,680 --> 00:06:18,650 focus on the same plane as the image on-axis. 107 00:06:18,650 --> 00:06:22,100 In fact, what this aberration is about 108 00:06:22,100 --> 00:06:28,780 is that the image actually comes to focus on a curved surface. 109 00:06:28,780 --> 00:06:31,730 Center of the optical axis. 110 00:06:31,730 --> 00:06:33,350 Now, why do we care about the plane? 111 00:06:33,350 --> 00:06:35,450 Why do we want this to be a plane here? 112 00:06:35,450 --> 00:06:37,640 Well, in old fashion cameras, one 113 00:06:37,640 --> 00:06:41,470 used to put film at the image plane of the camera. 114 00:06:41,470 --> 00:06:45,050 So we have to be sure that the entire image is focused 115 00:06:45,050 --> 00:06:46,670 simultaneously on the film. 116 00:06:46,670 --> 00:06:48,050 Nowadays, we don't use film. 117 00:06:48,050 --> 00:06:51,200 We use a digital chip. 118 00:06:51,200 --> 00:06:53,810 But in both cases, they're flat. 119 00:06:53,810 --> 00:06:55,580 So curvature of field is a problem then 120 00:06:55,580 --> 00:07:04,770 because it says that actually the off-axis image points will 121 00:07:04,770 --> 00:07:06,530 be out of focus because they will focus 122 00:07:06,530 --> 00:07:08,387 in front of the final plane. 123 00:07:08,387 --> 00:07:10,470 So these are the kind of things that if you could, 124 00:07:10,470 --> 00:07:14,688 for example, make a curve chip surface, 125 00:07:14,688 --> 00:07:15,980 you could solve it immediately. 126 00:07:15,980 --> 00:07:19,610 All we have to do is match the surface of the chip 127 00:07:19,610 --> 00:07:23,300 to the surface of the curvature of the field. 128 00:07:23,300 --> 00:07:25,087 But in fact, actually, people are trying 129 00:07:25,087 --> 00:07:26,170 to do something like that. 130 00:07:26,170 --> 00:07:28,570 There's a lot of work on flexible electronics. 131 00:07:28,570 --> 00:07:31,340 And that's one of the motivations. 132 00:07:31,340 --> 00:07:35,766 But it is not yet widely available. 133 00:07:41,960 --> 00:07:45,050 Nevertheless, there is one condition 134 00:07:45,050 --> 00:07:50,420 where the surfaces that the field comes to focus actually 135 00:07:50,420 --> 00:07:51,640 becomes flat. 136 00:07:51,640 --> 00:07:53,720 This was derived by a mathematician. 137 00:07:53,720 --> 00:07:54,470 I don't know when. 138 00:07:54,470 --> 00:07:58,220 I think it was in the early 1900s. 139 00:07:58,220 --> 00:07:59,660 His name is Petzval. 140 00:07:59,660 --> 00:08:03,680 So this condition, it has a more complicated expression. 141 00:08:03,680 --> 00:08:07,100 But for two lenses, it works out to something like this. 142 00:08:07,100 --> 00:08:12,560 If the sum of these quantities where 143 00:08:12,560 --> 00:08:15,790 n1 is the index of refraction of the lens. 144 00:08:15,790 --> 00:08:17,245 And f1 is the focal length. 145 00:08:17,245 --> 00:08:18,620 So it's a little bit unusual now. 146 00:08:18,620 --> 00:08:21,950 We multiply the focal length by an index of refraction. 147 00:08:21,950 --> 00:08:23,160 We've never done that before. 148 00:08:23,160 --> 00:08:25,520 But this is what this guy derived. 149 00:08:25,520 --> 00:08:28,130 If they two sum up to zero, these quantities 150 00:08:28,130 --> 00:08:35,360 for a pair of lenses, then the field of curvature vanishes. 151 00:08:35,360 --> 00:08:44,430 And what is even more sort of remarkable about this relation 152 00:08:44,430 --> 00:08:46,980 is that it does not involve the space in between the lenses. 153 00:08:46,980 --> 00:08:52,430 No matter where they are, if they satisfy this condition, 154 00:08:52,430 --> 00:08:56,530 the field of curvature will be eliminated. 155 00:08:56,530 --> 00:09:01,390 And so for example, if they are composed 156 00:09:01,390 --> 00:09:05,020 of the same index of refraction, then the index of refraction 157 00:09:05,020 --> 00:09:07,430 also drops out of this equation. 158 00:09:07,430 --> 00:09:12,130 And then you can treat them as a composite, as a thick lens, 159 00:09:12,130 --> 00:09:14,650 basically, which this is a relationship that we already 160 00:09:14,650 --> 00:09:18,340 derived for a composite element. 161 00:09:18,340 --> 00:09:22,270 And you can make sure that one of them 162 00:09:22,270 --> 00:09:27,370 is the negative of the other so that you can make this quantity 163 00:09:27,370 --> 00:09:28,180 vanish. 164 00:09:28,180 --> 00:09:30,737 That is, f1 is equal to minus f2. 165 00:09:30,737 --> 00:09:32,320 And then you can find the focal length 166 00:09:32,320 --> 00:09:34,020 from this equation over here. 167 00:09:34,020 --> 00:09:37,360 So it's kind of cool because it says that you can actually 168 00:09:37,360 --> 00:09:38,465 eliminate-- 169 00:09:38,465 --> 00:09:39,840 if it is really important to you, 170 00:09:39,840 --> 00:09:45,500 you can eliminate this problem by using the Petzval condition. 171 00:09:45,500 --> 00:09:47,560 And again, there's a more complicated condition 172 00:09:47,560 --> 00:09:50,380 that applies to multilens systems that 173 00:09:50,380 --> 00:09:51,910 have more than two lenses. 174 00:09:51,910 --> 00:09:54,710 But I didn't bother to draw it here. 175 00:09:54,710 --> 00:09:56,720 And what is even more remarkable is 176 00:09:56,720 --> 00:10:00,950 that the Petzval surface and the field curvature 177 00:10:00,950 --> 00:10:04,820 is also related to astigmatism. 178 00:10:04,820 --> 00:10:10,280 So what you see in this plot here, also from the textbook, 179 00:10:10,280 --> 00:10:15,460 is the focal surfaces-- that is, the surfaces in which different 180 00:10:15,460 --> 00:10:17,870 points from a flat-- 181 00:10:17,870 --> 00:10:19,430 if you take a flat object and you 182 00:10:19,430 --> 00:10:22,160 take different points along this object, 183 00:10:22,160 --> 00:10:24,350 these three surfaces are correspondingly 184 00:10:24,350 --> 00:10:27,860 the tangential focus for the astigmatic system, 185 00:10:27,860 --> 00:10:37,450 the sagittal focus, and finally the Petzval surface. 186 00:10:37,450 --> 00:10:38,920 Yeah. 187 00:10:38,920 --> 00:10:40,680 Let me go for [INAUDIBLE]. 188 00:10:40,680 --> 00:10:44,090 So what we discussed here is actually the Petzval condition. 189 00:10:44,090 --> 00:10:47,840 I haven't said yet what is the Petzval surface. 190 00:10:47,840 --> 00:10:50,810 So the Petzval surface is the following. 191 00:10:50,810 --> 00:10:54,770 It turns out that as you change the stopping 192 00:10:54,770 --> 00:10:57,115 and optical system-- that is, as you move 193 00:10:57,115 --> 00:11:00,890 the stop in the axial direction, you 194 00:11:00,890 --> 00:11:03,230 can achieve the condition where the tangential 195 00:11:03,230 --> 00:11:06,350 and the sagittal astigmatic focal 196 00:11:06,350 --> 00:11:10,100 surfaces, they collapse into one surface. 197 00:11:10,100 --> 00:11:13,430 This surface is called the Petzval surface, 198 00:11:13,430 --> 00:11:15,380 not to be confused with the Petzval condition, 199 00:11:15,380 --> 00:11:18,120 even though, of course, they are related. 200 00:11:18,120 --> 00:11:19,730 So what you see here is basically, 201 00:11:19,730 --> 00:11:21,980 as you move the stop in an optical system-- 202 00:11:21,980 --> 00:11:25,325 this is borrowed from another textbook, not from-- 203 00:11:25,325 --> 00:11:26,700 I should have recorded, actually. 204 00:11:26,700 --> 00:11:30,890 This is from Jenkins and White, Fundamentals of Optics. 205 00:11:30,890 --> 00:11:35,510 So as you move the stop, you can find the situation 206 00:11:35,510 --> 00:11:39,050 where the two surfaces coincide, and that is the Petzval surface 207 00:11:39,050 --> 00:11:39,770 of the system. 208 00:11:39,770 --> 00:11:43,070 And of course, this one does not satisfy the Petzval condition 209 00:11:43,070 --> 00:11:45,980 because the Petzval surface is curved. 210 00:11:45,980 --> 00:11:50,150 If I could somehow satisfy both the Petzval condition 211 00:11:50,150 --> 00:11:54,560 and collapse that to surfaces-- tangential and sagittal-- 212 00:11:54,560 --> 00:11:59,210 onto the Petzval, then I would have a flat system 213 00:11:59,210 --> 00:12:03,380 with a flat field, which would also be free of astigmatism. 214 00:12:07,920 --> 00:12:13,540 So obviously we cannot get into too much detail about this 215 00:12:13,540 --> 00:12:16,360 without becoming quantitative. 216 00:12:16,360 --> 00:12:19,910 And this is quite difficult to deal with algebra, 217 00:12:19,910 --> 00:12:22,090 so I'm just being descriptive. 218 00:12:22,090 --> 00:12:23,900 And you will see some of this in action 219 00:12:23,900 --> 00:12:30,820 when Se Baek does the demo of the software in a little bit. 220 00:12:30,820 --> 00:12:34,367 And the light distortion that I will discuss 221 00:12:34,367 --> 00:12:36,200 has to do with something slightly different. 222 00:12:36,200 --> 00:12:40,240 So the field of curvature occurs and astigmatism 223 00:12:40,240 --> 00:12:45,370 occurs because if you take object points off-axis, 224 00:12:45,370 --> 00:12:47,990 they fail to focus at the same plane. 225 00:12:47,990 --> 00:12:52,480 They focus at different planes and on different surfaces, 226 00:12:52,480 --> 00:12:55,010 the sagittal and tangential. 227 00:12:55,010 --> 00:12:57,010 The distortion occurs because even 228 00:12:57,010 --> 00:13:00,550 if you manage to focus at one plane, 229 00:13:00,550 --> 00:13:02,200 the magnification is different. 230 00:13:02,200 --> 00:13:05,240 The magnification-- and this you can see very easily, actually, 231 00:13:05,240 --> 00:13:08,710 just looking at this diagram over here. 232 00:13:08,710 --> 00:13:12,510 If you take the on-axis point, the imaging and condition 233 00:13:12,510 --> 00:13:15,520 of the magnification are given by these distances. 234 00:13:15,520 --> 00:13:17,490 For example, the lateral magnification 235 00:13:17,490 --> 00:13:22,040 is the ratio of this distance to this distance. 236 00:13:22,040 --> 00:13:23,320 But what about this point now? 237 00:13:23,320 --> 00:13:26,900 For this point, the lateral magnification 238 00:13:26,900 --> 00:13:31,670 will be the ratio of this distance to this distance. 239 00:13:31,670 --> 00:13:33,890 And for those of you in Boston, I'm 240 00:13:33,890 --> 00:13:38,290 pointing out to the tilted chief ray. 241 00:13:38,290 --> 00:13:41,270 So I have to measure the distance 242 00:13:41,270 --> 00:13:43,332 from the lens on the chief ray. 243 00:13:43,332 --> 00:13:45,290 And you can see very easily that this length is 244 00:13:45,290 --> 00:13:47,690 different than the length of the chief ray 245 00:13:47,690 --> 00:13:49,730 for the on-axis point. 246 00:13:49,730 --> 00:13:52,970 So because of that, the magnification on-axis 247 00:13:52,970 --> 00:13:55,520 is different than the magnification off-axis. 248 00:13:55,520 --> 00:13:59,240 Therefore, if I attempt to image a checkerboard, 249 00:13:59,240 --> 00:14:02,720 like this, if the magnification is different, 250 00:14:02,720 --> 00:14:10,410 I will see this sort of-- 251 00:14:10,410 --> 00:14:12,540 I don't want to use the word "distorted." 252 00:14:12,540 --> 00:14:15,530 You will see this sort of altered-- 253 00:14:15,530 --> 00:14:18,120 let's say altered-- image. 254 00:14:18,120 --> 00:14:20,250 And this is the definition of distortion. 255 00:14:20,250 --> 00:14:22,440 So in optics we have to be a little careful. 256 00:14:22,440 --> 00:14:25,020 Distortion, colloquially, means everything. 257 00:14:25,020 --> 00:14:27,990 All of the stuff that I described before. 258 00:14:27,990 --> 00:14:30,660 In everyday English, you can call them distortions. 259 00:14:30,660 --> 00:14:32,760 Well, in optics, in professional optics, 260 00:14:32,760 --> 00:14:37,080 when we mention distortion, we automatically mean this one. 261 00:14:37,080 --> 00:14:39,420 So we have to be careful not to use the word 262 00:14:39,420 --> 00:14:41,060 "distortion" in the colloquial sense 263 00:14:41,060 --> 00:14:46,390 because it is reserved to mean exactly that. 264 00:14:46,390 --> 00:14:50,170 And I guess it has two forms. 265 00:14:50,170 --> 00:14:53,350 It can be of this type, where the magnification is 266 00:14:53,350 --> 00:14:55,300 higher off-axis. 267 00:14:55,300 --> 00:14:57,310 That is called pincushion. 268 00:14:57,310 --> 00:15:00,370 Or lower, that is called barrel. 269 00:15:00,370 --> 00:15:05,170 And they believe, in this case, the top one will give rise 270 00:15:05,170 --> 00:15:10,420 to barrel because in this case, because the stop is here, 271 00:15:10,420 --> 00:15:19,720 the aperture stop is here, the chief ray on the object side 272 00:15:19,720 --> 00:15:22,470 becomes longer off-axis. 273 00:15:22,470 --> 00:15:24,820 Now, if you recall, the magnification 274 00:15:24,820 --> 00:15:30,580 is the lateral magnification equals 275 00:15:30,580 --> 00:15:36,880 S on the image side over S on the object side with a minus 276 00:15:36,880 --> 00:15:37,380 sign. 277 00:15:37,380 --> 00:15:40,240 But the minus is irrelevant here. 278 00:15:40,240 --> 00:15:42,560 So since I made-- 279 00:15:42,560 --> 00:15:46,320 so in this case, S on the image side 280 00:15:46,320 --> 00:15:49,120 is approximately the same as on-axis. 281 00:15:49,120 --> 00:15:52,120 But S on the object side became longer 282 00:15:52,120 --> 00:15:54,650 by putting the stop here. 283 00:15:54,650 --> 00:15:57,640 So therefore, I get a lower magnification off-axis. 284 00:15:57,640 --> 00:16:00,730 Therefore, I get this type of distortion. 285 00:16:00,730 --> 00:16:03,100 If I put the stop on the other side, 286 00:16:03,100 --> 00:16:05,650 then it is the opposite case. 287 00:16:05,650 --> 00:16:09,460 Now I make this length longer. 288 00:16:09,460 --> 00:16:14,860 Therefore, I get pincushion distortion. 289 00:16:14,860 --> 00:16:17,770 And of course, the way to eliminate distortion altogether 290 00:16:17,770 --> 00:16:21,250 is to make the system like this. 291 00:16:21,250 --> 00:16:23,650 This is actually a telecentric system. 292 00:16:23,650 --> 00:16:26,620 And to make sure that the stop is exactly 293 00:16:26,620 --> 00:16:28,810 at the center of the system. 294 00:16:28,810 --> 00:16:31,420 And also-- yes. 295 00:16:31,420 --> 00:16:38,050 And so if you do that, then basically 296 00:16:38,050 --> 00:16:41,880 a little bit of thought will convince you that in this case 297 00:16:41,880 --> 00:16:45,780 the variation between this length is actually very small. 298 00:16:45,780 --> 00:16:50,010 So therefore, there is some distortion, 299 00:16:50,010 --> 00:16:51,480 but it is high order. 300 00:16:51,480 --> 00:16:54,550 The third order distortion is completely eliminated here. 301 00:16:54,550 --> 00:16:57,180 So you basically get a distortion-free image. 302 00:17:00,330 --> 00:17:00,830 OK. 303 00:17:03,460 --> 00:17:11,720 So before I turn over to Se Baek and Pepe, 304 00:17:11,720 --> 00:17:14,245 last time I mentioned caustics. 305 00:17:18,876 --> 00:17:21,290 So I would like to show what caustics are. 306 00:17:21,290 --> 00:17:24,349 So this is actually from the last page of today's notes, 307 00:17:24,349 --> 00:17:26,900 which I haven't done yet. 308 00:17:26,900 --> 00:17:30,890 And the details here are not very important. 309 00:17:30,890 --> 00:17:32,510 By the end of the lecture today, you 310 00:17:32,510 --> 00:17:34,460 will know what has happened here. 311 00:17:34,460 --> 00:17:38,720 This is a special type of lens called a gradient index lens. 312 00:17:38,720 --> 00:17:42,200 And I simulated it with off-axis incidence. 313 00:17:42,200 --> 00:17:44,820 So therefore, in this case, we have coma. 314 00:17:44,820 --> 00:17:48,860 So if you look at the ray, these are ray traces of this lens. 315 00:17:48,860 --> 00:17:53,540 For now, please accepted the fact that the rays are curved. 316 00:17:53,540 --> 00:17:55,360 But what I wanted to show you here, 317 00:17:55,360 --> 00:17:59,490 what is the caustic that I mentioned in the last lecture. 318 00:17:59,490 --> 00:18:03,660 You can see that in the bottom side of this diagram 319 00:18:03,660 --> 00:18:06,740 that ray paths, they kind of collapse onto each other. 320 00:18:06,740 --> 00:18:09,050 So you see that the rays are relatively uniformly 321 00:18:09,050 --> 00:18:11,870 distributed up to this point approximately. 322 00:18:11,870 --> 00:18:15,740 Over here, they begin to accumulate near one side. 323 00:18:15,740 --> 00:18:17,540 So what you would expect in this case, 324 00:18:17,540 --> 00:18:18,920 if you put a piece of paper here, 325 00:18:18,920 --> 00:18:23,360 is you would expect very strong light intensity 326 00:18:23,360 --> 00:18:26,880 near the bottom of the beam. 327 00:18:26,880 --> 00:18:29,390 So that is what we call a caustic. 328 00:18:29,390 --> 00:18:34,730 And I think in the examples that the guys will show you 329 00:18:34,730 --> 00:18:37,590 in the demo, you will also see some caustics form 330 00:18:37,590 --> 00:18:38,330 in real life. 331 00:18:38,330 --> 00:18:40,580 But this is what I was trying to describe last time. 332 00:18:40,580 --> 00:18:41,680 OK. 333 00:18:41,680 --> 00:18:43,490 So Pepe and Se Baek, I don't know 334 00:18:43,490 --> 00:18:44,740 if you guys are ready but-- 335 00:18:44,740 --> 00:18:46,760 PROFESSOR: Yeah. 336 00:18:46,760 --> 00:18:47,760 GEORGE BARBASTATHIS: OK. 337 00:18:47,760 --> 00:18:49,490 PROFESSOR: Can you help me with that. 338 00:18:49,490 --> 00:18:50,907 GEORGE BARBASTATHIS: Take it away. 339 00:19:15,145 --> 00:19:17,770 PROFESSOR: So this is a similar demo to the one that we showed. 340 00:19:17,770 --> 00:19:19,645 Now we're going to focus on a different thing 341 00:19:19,645 --> 00:19:21,730 of the aberrations that we've been learning. 342 00:19:21,730 --> 00:19:27,460 So so far we've seen two type of aberrations in the system 343 00:19:27,460 --> 00:19:29,890 that we can identify very easily. 344 00:19:29,890 --> 00:19:32,170 But first, let me describe the components. 345 00:19:32,170 --> 00:19:34,500 So can you go back here. 346 00:19:34,500 --> 00:19:37,490 So for this one here we start with a laser, as usual. 347 00:19:37,490 --> 00:19:40,510 And now we know that we have a collimating system so used 348 00:19:40,510 --> 00:19:43,450 to condition the light and make a nice plane wave at the output 349 00:19:43,450 --> 00:19:44,740 here. 350 00:19:44,740 --> 00:19:47,980 And then this plane wave illuminates these lens, 351 00:19:47,980 --> 00:19:50,990 which is a cylindrical lens, shown here. 352 00:19:50,990 --> 00:19:54,640 And then we are just looking at the rays going on. 353 00:19:54,640 --> 00:19:55,930 And this is for a green laser. 354 00:19:55,930 --> 00:19:58,660 It's a wavelength of 532 nanometers. 355 00:19:58,660 --> 00:20:00,940 So now let's think what's happening, actually. 356 00:20:00,940 --> 00:20:04,170 So we can reduce exposure. 357 00:20:04,170 --> 00:20:05,560 OK. 358 00:20:05,560 --> 00:20:15,190 So we can see at this point how the rays start converging. 359 00:20:15,190 --> 00:20:19,900 But it's a bit hard to see here, but they do not 360 00:20:19,900 --> 00:20:22,540 focus in a tight point. 361 00:20:22,540 --> 00:20:24,610 They actually focus at different points. 362 00:20:24,610 --> 00:20:26,250 And as we'll see in the software, 363 00:20:26,250 --> 00:20:28,870 this is basically the experimental version 364 00:20:28,870 --> 00:20:30,970 of a spherical aberration that we can see. 365 00:20:30,970 --> 00:20:34,510 And in this case, I'm illuminating the lens 366 00:20:34,510 --> 00:20:36,550 with an on-axis plane wave. 367 00:20:36,550 --> 00:20:38,290 So therefore, we can see its condition. 368 00:20:38,290 --> 00:20:42,910 Now let's try to look at the case of coma 369 00:20:42,910 --> 00:20:45,800 as seen from the top. 370 00:20:45,800 --> 00:20:48,775 So what I'm doing is that I'm rotating the lens. 371 00:20:52,850 --> 00:20:58,210 And then we can see how the rays start 372 00:20:58,210 --> 00:21:00,500 forming an asymmetric pattern. 373 00:21:00,500 --> 00:21:02,335 And let me try to get this closer together. 374 00:21:10,750 --> 00:21:13,950 So now for this off-axis plane wave, 375 00:21:13,950 --> 00:21:18,590 we see that that's some of the rays go straight through 376 00:21:18,590 --> 00:21:22,110 and some of the rays form these caustics here that we just 377 00:21:22,110 --> 00:21:22,830 mentioned-- 378 00:21:22,830 --> 00:21:25,650 George mentioned in the slides. 379 00:21:25,650 --> 00:21:30,660 If we were to put a page perpendicular to this one 380 00:21:30,660 --> 00:21:34,097 that I have here, which is hard because we need actually 381 00:21:34,097 --> 00:21:36,180 a camera in order to show you that with very, very 382 00:21:36,180 --> 00:21:39,210 tiny pixels because the actual point produces 383 00:21:39,210 --> 00:21:43,000 or the focus produced by this lens would be very small. 384 00:21:43,000 --> 00:21:45,900 But if we do that, we would see the comma shape 385 00:21:45,900 --> 00:21:46,920 that we described. 386 00:21:46,920 --> 00:21:51,600 And we're actually going to see that in the simulation software 387 00:21:51,600 --> 00:21:53,697 that we're going to show later. 388 00:21:53,697 --> 00:21:54,780 GEORGE BARBASTATHIS: Pepe? 389 00:21:54,780 --> 00:21:55,530 PROFESSOR: Yep? 390 00:21:55,530 --> 00:21:57,280 GEORGE BARBASTATHIS: Is it possible to put 391 00:21:57,280 --> 00:21:59,010 the other business card-- can you 392 00:21:59,010 --> 00:22:02,270 put the back of the business card there? 393 00:22:02,270 --> 00:22:04,397 PROFESSOR: You're saying perpendicular? 394 00:22:04,397 --> 00:22:05,480 GEORGE BARBASTATHIS: Yeah. 395 00:22:05,480 --> 00:22:07,520 PROFESSOR: You don't see anything, George. 396 00:22:07,520 --> 00:22:09,050 This is a cylindrical lens. 397 00:22:09,050 --> 00:22:10,337 This is cylindrical. 398 00:22:10,337 --> 00:22:11,420 GEORGE BARBASTATHIS: Yeah. 399 00:22:11,420 --> 00:22:12,410 PROFESSOR: Yeah. 400 00:22:12,410 --> 00:22:15,720 And you see what I'm saying? 401 00:22:15,720 --> 00:22:18,350 These are cylindrical lens. 402 00:22:18,350 --> 00:22:23,390 And even with a CCD, it's very, very small. 403 00:22:23,390 --> 00:22:26,090 You would need something, in order to measure it accurately, 404 00:22:26,090 --> 00:22:28,880 something that is called a Foucault knife edge test, 405 00:22:28,880 --> 00:22:32,657 for example, where you actually can measure what is called 406 00:22:32,657 --> 00:22:33,740 the point spread function. 407 00:22:33,740 --> 00:22:36,450 And we're going to learn that in the next part of the course. 408 00:22:36,450 --> 00:22:36,950 All right. 409 00:22:36,950 --> 00:22:41,240 So let me show you another type of aberration. 410 00:22:41,240 --> 00:22:44,240 Now imaging our transparency. 411 00:22:44,240 --> 00:22:46,700 In this case, I just have a white light source 412 00:22:46,700 --> 00:22:49,320 that is illuminating this object, which is just 413 00:22:49,320 --> 00:22:53,180 a transparency with some GRIN. 414 00:22:53,180 --> 00:22:56,360 In this case, it's going to be just the GRIN that George was 415 00:22:56,360 --> 00:22:59,000 showing that is regular GRIN. 416 00:22:59,000 --> 00:23:01,920 And I have a single lens here. 417 00:23:01,920 --> 00:23:07,800 It's a spherical lens, biconvex, with a short focal length. 418 00:23:07,800 --> 00:23:11,100 So therefore, the curvature is pretty high, 419 00:23:11,100 --> 00:23:12,970 and we expect to have a lot of aberrations. 420 00:23:12,970 --> 00:23:14,720 But the one that we're going to see here-- 421 00:23:14,720 --> 00:23:16,520 we can decrease exposure-- we're going 422 00:23:16,520 --> 00:23:18,220 to see the evidence of distortion, 423 00:23:18,220 --> 00:23:20,540 the aberration of distortion. 424 00:23:20,540 --> 00:23:23,240 And as you can see here-- 425 00:23:23,240 --> 00:23:25,610 let's just focus in. 426 00:23:25,610 --> 00:23:26,840 You see how the lines-- 427 00:23:26,840 --> 00:23:31,250 similar to the slide that George just showed, 428 00:23:31,250 --> 00:23:34,070 the lines are actually curved by this lens. 429 00:23:34,070 --> 00:23:36,920 Although these are supposed to be straight lines, 430 00:23:36,920 --> 00:23:37,790 they are curved. 431 00:23:37,790 --> 00:23:39,560 Let me just magnify a little bit this. 432 00:23:43,011 --> 00:23:44,976 There you go. 433 00:23:44,976 --> 00:23:45,476 Increase. 434 00:23:53,390 --> 00:23:56,420 Now if we were to replace these with another lens-- 435 00:23:56,420 --> 00:23:59,240 so for example, this one that has a longer focal length-- 436 00:23:59,240 --> 00:24:01,820 the radius of curvature is larger. 437 00:24:01,820 --> 00:24:04,080 So it's basically not as curved as this one. 438 00:24:07,280 --> 00:24:10,310 And we're trying to do the same. 439 00:24:10,310 --> 00:24:13,280 Now what I'm going to do is I'm going to use this lens to image 440 00:24:13,280 --> 00:24:15,440 my transparency. 441 00:24:29,000 --> 00:24:33,530 Now we can see how the lines look much more straight. 442 00:24:33,530 --> 00:24:36,810 That distortion aberration got reduced. 443 00:24:36,810 --> 00:24:39,200 So what type of aberration did you guys see? 444 00:24:39,200 --> 00:24:42,380 It was pincushions or barrel aberration, 445 00:24:42,380 --> 00:24:45,590 the one that you saw before? 446 00:24:45,590 --> 00:24:49,430 I'm going to put it again while you guys think. 447 00:25:09,100 --> 00:25:09,850 Which one is this? 448 00:25:16,300 --> 00:25:16,970 I can hear. 449 00:25:16,970 --> 00:25:18,220 Just press the button, please. 450 00:25:20,960 --> 00:25:21,980 AUDIENCE: Pincushion? 451 00:25:21,980 --> 00:25:22,930 PROFESSOR: Yeah. 452 00:25:22,930 --> 00:25:23,430 Yeah. 453 00:25:23,430 --> 00:25:25,800 We see the evidence of the magnification of the lenses. 454 00:25:25,800 --> 00:25:28,560 It's maybe hard for you being able to see. 455 00:25:28,560 --> 00:25:33,820 But here, the lines basically get curved to outside. 456 00:25:33,820 --> 00:25:36,450 So in the break, if you guys want to come on and just 457 00:25:36,450 --> 00:25:39,450 play with the demo and change the focal length 458 00:25:39,450 --> 00:25:42,570 and see where the images get formed, and also this one here, 459 00:25:42,570 --> 00:25:43,920 you're welcome to do so. 460 00:25:43,920 --> 00:25:46,260 Now we're going to switch to our numerical software 461 00:25:46,260 --> 00:25:49,350 to show how we actually deal with these in practice 462 00:25:49,350 --> 00:25:54,030 to optimize an imaging system. 463 00:26:07,425 --> 00:26:08,050 SE BAEK OH: OK. 464 00:26:11,930 --> 00:26:14,990 So people, before we move on to next topic, which 465 00:26:14,990 --> 00:26:17,720 is a GRIN medium and Hamiltonian optics, 466 00:26:17,720 --> 00:26:20,570 we want to show how we actually build or design 467 00:26:20,570 --> 00:26:25,010 the actual imaging optical system in practice. 468 00:26:25,010 --> 00:26:28,860 Of course, so far we learn a lot of stuff. 469 00:26:28,860 --> 00:26:32,690 The Snell's law, and reflection and refraction, 470 00:26:32,690 --> 00:26:35,270 and the lens maker's law, and paraxial optics, 471 00:26:35,270 --> 00:26:36,350 and blah, blah, blah. 472 00:26:36,350 --> 00:26:41,220 But in real life, we usually use a computer software. 473 00:26:41,220 --> 00:26:44,960 So just to give you an example, since most of you 474 00:26:44,960 --> 00:26:48,050 are mechanical engineer, including myself. 475 00:26:48,050 --> 00:26:50,070 So let's think about this. 476 00:26:50,070 --> 00:26:53,290 If you want to build some toy or some mechanical model 477 00:26:53,290 --> 00:26:59,930 for your bike or car or toy, what do you do 478 00:26:59,930 --> 00:27:02,920 is you start with sketching by hand. 479 00:27:02,920 --> 00:27:04,850 So make the conceptual model. 480 00:27:04,850 --> 00:27:07,850 And then you go to computer. 481 00:27:07,850 --> 00:27:08,810 Use the CAD program. 482 00:27:08,810 --> 00:27:11,660 So your build the 2D or 3D model. 483 00:27:11,660 --> 00:27:14,390 And if you have multiple element, 484 00:27:14,390 --> 00:27:16,760 you can even assembly, in SolidWorks. 485 00:27:16,760 --> 00:27:19,970 And you can export the 3D model. 486 00:27:19,970 --> 00:27:24,330 And if your system is complicated enough, 487 00:27:24,330 --> 00:27:27,170 you can also use finite element model 488 00:27:27,170 --> 00:27:32,330 to analyze the stress or strain or the temperature variations, 489 00:27:32,330 --> 00:27:33,320 blah, blah, blah. 490 00:27:33,320 --> 00:27:34,310 Same idea. 491 00:27:34,310 --> 00:27:40,160 You start with the conceptual model by drawing a few rays. 492 00:27:40,160 --> 00:27:46,100 And then you apply lens law and the 1 over object distance 493 00:27:46,100 --> 00:27:49,010 plus 1 over image distance should be 1 over f. 494 00:27:49,010 --> 00:27:53,150 And sometime you only use the [INAUDIBLE].. 495 00:27:53,150 --> 00:27:55,010 But after that you go to computer 496 00:27:55,010 --> 00:27:58,520 and use probably one of these commercialized software, which 497 00:27:58,520 --> 00:28:02,600 is Code V. So it's not V. It's a Roman number. 498 00:28:02,600 --> 00:28:06,551 So Code V or ZEMAX and Oslo. 499 00:28:06,551 --> 00:28:11,243 So the Code V and ZEMAX, they are industrial standards. 500 00:28:11,243 --> 00:28:12,660 And they are pretty much the same. 501 00:28:12,660 --> 00:28:14,710 So if you can handle one of these, 502 00:28:14,710 --> 00:28:17,230 then it's very easy to do-- 503 00:28:17,230 --> 00:28:19,660 easy to use the other one. 504 00:28:19,660 --> 00:28:23,680 So essentially what it does is it draws some rays for you 505 00:28:23,680 --> 00:28:28,740 but very fast, and it's automated fashion. 506 00:28:28,740 --> 00:28:32,470 Because if you have CAD program, you can use actually 507 00:28:32,470 --> 00:28:33,850 do this ray tracing. 508 00:28:33,850 --> 00:28:35,750 So you just draw a ray. 509 00:28:35,750 --> 00:28:38,110 And whenever you meet the interface, 510 00:28:38,110 --> 00:28:41,260 you complete the incident angle to the surface normal 511 00:28:41,260 --> 00:28:44,290 and apply the Snell's law and compute refraction angle 512 00:28:44,290 --> 00:28:46,150 and do the same thing. 513 00:28:46,150 --> 00:28:49,570 But it's very complicated, especially 514 00:28:49,570 --> 00:28:51,710 you have the 3D model. 515 00:28:51,710 --> 00:28:55,100 So this program does that for you. 516 00:28:55,100 --> 00:28:59,080 And it actually has very different method 517 00:28:59,080 --> 00:29:00,470 to do the ray tracing. 518 00:29:00,470 --> 00:29:03,030 So first of all it's just-- 519 00:29:03,030 --> 00:29:05,440 actually, we learn the ray transfer matrices. 520 00:29:05,440 --> 00:29:11,420 So it does ray tracing based on the paraxial region. 521 00:29:11,420 --> 00:29:16,450 But probably one of the reasons that we use the software. 522 00:29:16,450 --> 00:29:18,060 It does the real ray tracing. 523 00:29:18,060 --> 00:29:23,670 Just as I described, it just trace every ray 524 00:29:23,670 --> 00:29:26,570 based on Snell's law. 525 00:29:26,570 --> 00:29:32,570 And you can use it-- 526 00:29:32,570 --> 00:29:36,720 and actually, it also has many built-in features 527 00:29:36,720 --> 00:29:40,170 to analyze performance of optical system, 528 00:29:40,170 --> 00:29:41,710 like aberration. 529 00:29:41,710 --> 00:29:45,540 So we learn about the five geometric aberrations. 530 00:29:45,540 --> 00:29:46,860 You can analyze it. 531 00:29:46,860 --> 00:29:48,840 And also, point spread function or spot 532 00:29:48,840 --> 00:29:53,460 diagram, which means if you have ideal imaging system and lets 533 00:29:53,460 --> 00:29:57,740 say we have point object, the images 534 00:29:57,740 --> 00:30:00,840 should be also infinitely small point, assuming 535 00:30:00,840 --> 00:30:02,190 that they are conjugated. 536 00:30:02,190 --> 00:30:06,510 But in real life, it never be like this 537 00:30:06,510 --> 00:30:14,460 because you always have the finite size of focused image. 538 00:30:14,460 --> 00:30:16,680 So point spread function or spot diagram 539 00:30:16,680 --> 00:30:21,780 tells you how big is your focus at the image plane. 540 00:30:21,780 --> 00:30:25,570 And it also analyze MTF or OTF, which is frequency analysis. 541 00:30:25,570 --> 00:30:29,820 We cover later of the course. 542 00:30:29,820 --> 00:30:34,260 And probably-- actually, another reason 543 00:30:34,260 --> 00:30:36,030 that we have to use this software 544 00:30:36,030 --> 00:30:40,140 is to optimize the optical system. 545 00:30:40,140 --> 00:30:43,110 I prepared some examples, so we will see. 546 00:30:54,970 --> 00:30:56,850 So this is the-- 547 00:30:56,850 --> 00:30:59,250 so we are using the ZEMAX. 548 00:30:59,250 --> 00:31:02,060 Just the one of the industry's standard. 549 00:31:02,060 --> 00:31:06,880 And if you open the ZEMAX in this, 550 00:31:06,880 --> 00:31:12,070 we have this table, which basically are lens data editor. 551 00:31:12,070 --> 00:31:14,740 So filling in this table means you 552 00:31:14,740 --> 00:31:16,010 specify your optical system. 553 00:31:16,010 --> 00:31:18,210 I'll describe how to do it. 554 00:31:18,210 --> 00:31:19,570 But once you have this-- 555 00:31:19,570 --> 00:31:21,580 once you fill in this table, then you 556 00:31:21,580 --> 00:31:25,990 can analyze very different stuff. 557 00:31:25,990 --> 00:31:30,890 So for example, it shows the actual layout of a system, 558 00:31:30,890 --> 00:31:32,050 like this one. 559 00:31:32,050 --> 00:31:33,740 This 2D layout. 560 00:31:33,740 --> 00:31:39,200 And it even made the 3D model for you, like this one. 561 00:31:39,200 --> 00:31:42,620 So it can check the actual shape of the system. 562 00:31:51,800 --> 00:31:56,280 So this is called [INAUDIBLE]. 563 00:31:56,280 --> 00:32:01,290 So actually, the horizontal axis is actually your pupil plane, 564 00:32:01,290 --> 00:32:05,500 and vertical axis is deviation of the ray at the image plane. 565 00:32:05,500 --> 00:32:06,720 So if you have-- 566 00:32:06,720 --> 00:32:08,670 so if your system is aberration free, 567 00:32:08,670 --> 00:32:11,460 then these lines should be just horizontal here 568 00:32:11,460 --> 00:32:14,280 because all the rays meet at the point. 569 00:32:14,280 --> 00:32:18,180 But this is practical system so you have the deviation, which 570 00:32:18,180 --> 00:32:21,140 means you have aberration. 571 00:32:21,140 --> 00:32:28,780 And these guys are spot diagram. 572 00:32:28,780 --> 00:32:32,190 So if you have point object in your object space, 573 00:32:32,190 --> 00:32:36,150 then you'll probably get this kind of focus 574 00:32:36,150 --> 00:32:37,740 at your image plane. 575 00:32:42,300 --> 00:32:45,510 And this is same thing but for different wavelengths 576 00:32:45,510 --> 00:32:49,320 because the glass is dispersive, which 577 00:32:49,320 --> 00:32:53,482 means the refractive index is also 578 00:32:53,482 --> 00:32:54,690 a function of the wavelength. 579 00:32:54,690 --> 00:32:59,280 So if you change the wavelength, then the shape of these spots 580 00:32:59,280 --> 00:32:59,880 also change. 581 00:32:59,880 --> 00:33:04,680 So it analyzed this dispersive property. 582 00:33:10,970 --> 00:33:12,340 OK. 583 00:33:12,340 --> 00:33:17,090 So let me explain how to fit into those tables. 584 00:33:17,090 --> 00:33:20,210 So let's say we have this two lens system. 585 00:33:20,210 --> 00:33:25,310 So what you do is you just decompose your system 586 00:33:25,310 --> 00:33:26,930 to multiple surface. 587 00:33:26,930 --> 00:33:29,400 So for example, here I have the object. 588 00:33:29,400 --> 00:33:32,300 So this is by default surface zero. 589 00:33:32,300 --> 00:33:35,150 And the first surface should be the front surface 590 00:33:35,150 --> 00:33:37,430 of the first lens here. 591 00:33:37,430 --> 00:33:43,610 And surface two is the back surface of the first lens. 592 00:33:43,610 --> 00:33:46,580 And the surface three is the front surface 593 00:33:46,580 --> 00:33:49,650 of the second lens, and so on and so forth. 594 00:33:49,650 --> 00:33:53,720 So in the table we have the different-- the multiple roles. 595 00:33:53,720 --> 00:33:57,380 So each role represent the one surface. 596 00:33:57,380 --> 00:34:00,500 And the radius means the radius curvature of the surface. 597 00:34:00,500 --> 00:34:04,640 So for example, here, since this object is flat, 598 00:34:04,640 --> 00:34:07,390 the radius curvature is infinity. 599 00:34:07,390 --> 00:34:10,449 And thickness is the distance from the surface 600 00:34:10,449 --> 00:34:11,690 to the next surface. 601 00:34:11,690 --> 00:34:15,310 So in this case, d0 is the distance from here 602 00:34:15,310 --> 00:34:16,780 to the first lens. 603 00:34:16,780 --> 00:34:20,380 And in some case, if you deal with the infinite conjugate, 604 00:34:20,380 --> 00:34:24,400 like telescope, then this d0 could be infinite 605 00:34:24,400 --> 00:34:27,760 because your object is at infinity. 606 00:34:27,760 --> 00:34:30,879 And at the surface one, I have the lens which 607 00:34:30,879 --> 00:34:33,400 has the radius curvature R1. 608 00:34:33,400 --> 00:34:34,600 So I have R1. 609 00:34:34,600 --> 00:34:39,310 And thickness d1 represent the thickness of the actual length 610 00:34:39,310 --> 00:34:42,580 from here to here. 611 00:34:42,580 --> 00:34:45,699 And after the surface one, I have the glass. 612 00:34:45,699 --> 00:34:51,060 So I just can specify the type of glass, which is BK7 here. 613 00:34:51,060 --> 00:34:54,520 This is the most common optical glass. 614 00:34:54,520 --> 00:34:56,300 And you just keep repeating this one. 615 00:34:56,300 --> 00:35:00,650 So once you have this and you have the-- 616 00:35:00,650 --> 00:35:04,300 you just fill in the table in the ZEMAX. 617 00:35:04,300 --> 00:35:06,730 And there are also other parameters you have to specify. 618 00:35:06,730 --> 00:35:09,580 So as I said, the wavelength. 619 00:35:09,580 --> 00:35:14,770 And you also need to choose the field parameter, which 620 00:35:14,770 --> 00:35:16,880 basically this means object information. 621 00:35:16,880 --> 00:35:19,390 So if your object is in finite distance, 622 00:35:19,390 --> 00:35:25,240 then field means object height or size of your object, 623 00:35:25,240 --> 00:35:30,400 or if it is infinite, you can set the incident angle 624 00:35:30,400 --> 00:35:32,921 to the system, just like the telescope. 625 00:35:35,690 --> 00:35:41,020 So first example I prepare is actually 626 00:35:41,020 --> 00:35:43,760 the example the lecture three. 627 00:35:43,760 --> 00:35:46,340 So we had the two lens. 628 00:35:46,340 --> 00:35:50,930 Both of them have the focal length of 10 here and here. 629 00:35:50,930 --> 00:35:52,970 And if you went through the derivation, 630 00:35:52,970 --> 00:35:56,960 we had the several different ways to derive. 631 00:35:56,960 --> 00:35:59,128 But the image distance-- actually, it's 632 00:35:59,128 --> 00:35:59,920 not image distance. 633 00:35:59,920 --> 00:36:02,750 The imaging is located at 30 millimeter 634 00:36:02,750 --> 00:36:05,900 behind the second length, and the lateral magnification 635 00:36:05,900 --> 00:36:08,660 was negative 4. 636 00:36:08,660 --> 00:36:12,890 And also, if you did the wavelength of matrices, 637 00:36:12,890 --> 00:36:17,090 then effective focal length was 20 over 3, which is about 7. 638 00:36:17,090 --> 00:36:19,530 And the front focal length and back focal length 639 00:36:19,530 --> 00:36:21,710 was 10 over 3. 640 00:36:21,710 --> 00:36:23,450 So about 3, right? 641 00:36:23,450 --> 00:36:30,615 So in ZEMAX, I already filled in the table. 642 00:36:41,720 --> 00:36:45,930 So this is the how the system looks like. 643 00:36:45,930 --> 00:36:49,020 I have the object here, and I have the image here. 644 00:36:51,870 --> 00:36:55,800 And if you analyze the system, then it 645 00:36:55,800 --> 00:36:59,050 gives you all the specifications here. 646 00:36:59,050 --> 00:37:03,260 So here, the effective focal length is 6-point-- 647 00:37:03,260 --> 00:37:05,690 I don't know you can see, but the effective focal length 648 00:37:05,690 --> 00:37:07,600 is 6.92. 649 00:37:07,600 --> 00:37:08,490 So about 7. 650 00:37:08,490 --> 00:37:11,120 So we have a good agreement here. 651 00:37:11,120 --> 00:37:13,640 And the back focal length is about 3.05. 652 00:37:13,640 --> 00:37:16,490 So about 3. 653 00:37:23,290 --> 00:37:28,270 And the paraxial magnification is negative 4.8. 654 00:37:28,270 --> 00:37:34,720 So we have a slightly bigger magnification in real system, 655 00:37:34,720 --> 00:37:36,020 and so on and so forth. 656 00:37:36,020 --> 00:37:37,970 And also, I put the aperture stuff here. 657 00:37:37,970 --> 00:37:42,120 So you can compute the diameter-- 658 00:37:42,120 --> 00:37:44,890 the size and position of the entrance pupil and exit pupil 659 00:37:44,890 --> 00:37:45,962 as well. 660 00:37:45,962 --> 00:37:47,170 GEORGE BARBASTATHIS: Se Baek? 661 00:37:47,170 --> 00:37:47,837 SE BAEK OH: Yes? 662 00:37:47,837 --> 00:37:50,295 GEORGE BARBASTATHIS: I think the reason for the discrepancy 663 00:37:50,295 --> 00:37:53,193 is that the lenses have finite thickness in your model, right? 664 00:37:53,193 --> 00:37:53,860 SE BAEK OH: Yes. 665 00:37:53,860 --> 00:37:58,090 So in real model, as you can see in the layout, 666 00:37:58,090 --> 00:37:59,660 this is not in length. 667 00:37:59,660 --> 00:38:04,860 And I just put the brief numbers-- 668 00:38:04,860 --> 00:38:06,060 the radius of the lens. 669 00:38:06,060 --> 00:38:08,170 So I just put the 10.5. 670 00:38:08,170 --> 00:38:11,990 And the thickness of the lens in 1 millimeter in this case. 671 00:38:11,990 --> 00:38:13,240 So it's not the real length. 672 00:38:13,240 --> 00:38:15,600 So it gives you slightly different numbers. 673 00:38:22,650 --> 00:38:25,500 And the next example is actually the Hubble Space Telescope. 674 00:38:25,500 --> 00:38:28,140 So Professor Barbastathis already 675 00:38:28,140 --> 00:38:32,970 mentioned the last lecture about this telescope. 676 00:38:32,970 --> 00:38:40,200 So actually, NASA realized after they launched this telescope 677 00:38:40,200 --> 00:38:43,830 and find out the image is not good as it's supposed to be. 678 00:38:43,830 --> 00:38:47,460 And they form a committee to investigate what's the problem. 679 00:38:47,460 --> 00:38:52,450 And they eventually figured out, and they fix it. 680 00:38:52,450 --> 00:38:55,300 But let's just start with the structure here. 681 00:38:55,300 --> 00:39:00,810 So this is a typical classic type telescope. 682 00:39:00,810 --> 00:39:02,760 We have two mirrors. 683 00:39:02,760 --> 00:39:04,800 So one is primary mirror, which is 684 00:39:04,800 --> 00:39:09,600 a concave hyperboloid mirror, and we 685 00:39:09,600 --> 00:39:14,070 have the secondary mirror, which is convex hyperboloidal mirror. 686 00:39:14,070 --> 00:39:18,660 And the size of the mirror, the diameter is about 2.4 meters. 687 00:39:18,660 --> 00:39:20,040 The primary mirror. 688 00:39:20,040 --> 00:39:25,860 And the second one is about, I guess, 60 centimeters or so. 689 00:39:25,860 --> 00:39:30,300 And the distance between these two mirror was 4.94 meters. 690 00:39:30,300 --> 00:39:33,240 So it's a huge telescope. 691 00:39:33,240 --> 00:39:37,240 And after extensive investigation, 692 00:39:37,240 --> 00:39:43,080 the problem was, actually, the shape of the primary mirror 693 00:39:43,080 --> 00:39:45,750 is not same as the intended shape. 694 00:39:45,750 --> 00:39:47,680 So the overall shape-- 695 00:39:47,680 --> 00:39:50,190 I mean, the curvature of the mirror 696 00:39:50,190 --> 00:39:52,410 was slightly different than the design. 697 00:39:52,410 --> 00:39:55,410 And it turns out that even though-- 698 00:39:55,410 --> 00:39:59,650 actually, the diameter of this primary lens is 2.4 meter. 699 00:39:59,650 --> 00:40:04,550 But at the edge, the difference between the design 700 00:40:04,550 --> 00:40:07,910 and the actual shape was only 4 micrometer. 701 00:40:07,910 --> 00:40:13,110 So it's very small, the physical difference. 702 00:40:13,110 --> 00:40:14,940 So just give you an idea. 703 00:40:14,940 --> 00:40:17,640 So the black hair, like the Asians, 704 00:40:17,640 --> 00:40:20,990 like myself, the diameter of the thick hair 705 00:40:20,990 --> 00:40:22,860 is about 100 micrometer. 706 00:40:22,860 --> 00:40:25,560 So 4 micrometer is very, very small 707 00:40:25,560 --> 00:40:28,490 distance compared to the diameter. 708 00:40:28,490 --> 00:40:32,340 But in general, it introduced a huge speckle aberration, 709 00:40:32,340 --> 00:40:35,520 which means ideally you have-- 710 00:40:35,520 --> 00:40:38,220 all these incoming rays should meet at one point, 711 00:40:38,220 --> 00:40:41,900 but it has a different extended focal point. 712 00:40:41,900 --> 00:40:47,250 So the effect was too catastrophic. 713 00:40:47,250 --> 00:40:51,980 So the image of the galaxy it looked like this. 714 00:40:51,980 --> 00:40:54,900 But after they fix, it looked like this. 715 00:40:54,900 --> 00:41:03,510 So does anyone know how NASA fixed this problem? 716 00:41:03,510 --> 00:41:05,100 The telescope was already in orbit, 717 00:41:05,100 --> 00:41:07,020 so you cannot take it back to the ground. 718 00:41:07,020 --> 00:41:13,310 And to fabricate another big mirror also need huge resource. 719 00:41:13,310 --> 00:41:16,740 So how can you fix? 720 00:41:22,450 --> 00:41:22,950 Yeah. 721 00:41:22,950 --> 00:41:24,600 So actually what they did is-- 722 00:41:28,100 --> 00:41:29,420 this is huge telescope. 723 00:41:29,420 --> 00:41:31,130 And they had a focal point. 724 00:41:31,130 --> 00:41:33,860 And they have different cameras, because you 725 00:41:33,860 --> 00:41:38,030 deal with different spectrum, like visible or the IR or UV. 726 00:41:38,030 --> 00:41:40,460 And what they did, actually-- and all these cameras 727 00:41:40,460 --> 00:41:42,600 show the same focal plane. 728 00:41:42,600 --> 00:41:45,710 So what they did, they actually sacrificed another camera 729 00:41:45,710 --> 00:41:48,410 and put another mirror system here 730 00:41:48,410 --> 00:41:51,800 to introduce this speckle aberration 731 00:41:51,800 --> 00:41:54,380 but in opposite fashion. 732 00:41:54,380 --> 00:41:58,040 So they kind of canceled the speckle aberration. 733 00:41:58,040 --> 00:42:10,780 So I prepared an example in ZEMAX with the real parameter. 734 00:42:10,780 --> 00:42:16,000 So this is the intended design of the Hubble Space Telescope. 735 00:42:16,000 --> 00:42:18,850 And as you can see, the surface number two 736 00:42:18,850 --> 00:42:20,440 and surface number three, they are 737 00:42:20,440 --> 00:42:22,570 primary and secondary mirror. 738 00:42:22,570 --> 00:42:24,370 And this is hyperboloidal mirror. 739 00:42:24,370 --> 00:42:35,140 So you have-- we have here. 740 00:42:35,140 --> 00:42:38,480 So conic coefficient. 741 00:42:38,480 --> 00:42:42,580 And this is the simulation result if you have-- 742 00:42:42,580 --> 00:42:44,440 simulation result of image if you 743 00:42:44,440 --> 00:42:50,050 have the letter F. But due to the manufacturing 744 00:42:50,050 --> 00:42:54,020 process, this conic coefficient was not this one. 745 00:42:54,020 --> 00:43:01,360 So actually it was negative 1.02324. 746 00:43:01,360 --> 00:43:05,180 So overall shape it doesn't change that much. 747 00:43:05,180 --> 00:43:10,780 But the image analysis looks very blurred. 748 00:43:10,780 --> 00:43:16,430 So that's why we had the very blurred image of galaxy. 749 00:43:16,430 --> 00:43:20,105 And that's why you need to deal with all these aberration 750 00:43:20,105 --> 00:43:23,410 with this computer software because you can analyze 751 00:43:23,410 --> 00:43:25,020 easily and compensate easily. 752 00:43:28,750 --> 00:43:29,250 OK. 753 00:43:32,630 --> 00:43:35,610 So last example is optimization. 754 00:43:35,610 --> 00:43:39,320 So for example, let's think about this one. 755 00:43:39,320 --> 00:43:42,860 I want to design some optical system which 756 00:43:42,860 --> 00:43:45,540 has three lenses here. 757 00:43:45,540 --> 00:43:51,590 So these two positive lens is made by crown glass, 758 00:43:51,590 --> 00:43:54,830 and this negative lens is actually flint glass. 759 00:43:54,830 --> 00:43:56,570 And if you want to design this one, 760 00:43:56,570 --> 00:44:01,250 then you have many parameters you can tune. 761 00:44:01,250 --> 00:44:04,100 So the radius curvature of each length-- 762 00:44:04,100 --> 00:44:08,030 so R1/R2, R3/R4, and R5/R6. 763 00:44:08,030 --> 00:44:12,590 And also the thickness of the lens, and also the distance 764 00:44:12,590 --> 00:44:13,760 between the lenses. 765 00:44:13,760 --> 00:44:16,650 So we have another six parameters here. 766 00:44:16,650 --> 00:44:19,850 So you have total 12 parameters you can choose. 767 00:44:19,850 --> 00:44:26,270 And it's hard to find the best parameter because you need 768 00:44:26,270 --> 00:44:30,430 some specification to satisfy. 769 00:44:30,430 --> 00:44:34,190 So what you do is you can just run the optimization routine 770 00:44:34,190 --> 00:44:35,370 here. 771 00:44:35,370 --> 00:44:40,490 So first process-- first procedure is select variable. 772 00:44:40,490 --> 00:44:43,610 As I just described, you can tune the radius 773 00:44:43,610 --> 00:44:46,190 of curvature of the lens and the thickness 774 00:44:46,190 --> 00:44:48,470 and the distance between lenses. 775 00:44:48,470 --> 00:44:51,670 So you want to choose the variable you can tune. 776 00:44:51,670 --> 00:44:54,230 And this column, you define the merit function, 777 00:44:54,230 --> 00:44:57,230 or sometimes it is called object function or cross function. 778 00:44:57,230 --> 00:45:02,150 Basically, you just select what you want to minimize. 779 00:45:02,150 --> 00:45:04,400 For example, you want to minimize 780 00:45:04,400 --> 00:45:07,650 some speckle aberration for given object point, 781 00:45:07,650 --> 00:45:12,580 or maybe you can minimize a spot diagram, 782 00:45:12,580 --> 00:45:17,470 like size or focus for different field points. 783 00:45:17,470 --> 00:45:21,590 And also, you can add some constraint, like your total-- 784 00:45:21,590 --> 00:45:25,160 physical system should be like this. 785 00:45:25,160 --> 00:45:28,580 It cannot longer than certain distance. 786 00:45:28,580 --> 00:45:31,760 Or the thickness of the lens can be constrained due 787 00:45:31,760 --> 00:45:33,180 to the manufacturing process. 788 00:45:33,180 --> 00:45:36,440 So you can add some constraints. 789 00:45:36,440 --> 00:45:40,340 And then what you do is compute the value of merit function 790 00:45:40,340 --> 00:45:43,430 with the different combination of the parameter. 791 00:45:43,430 --> 00:45:46,940 And this is the automated-- this is iterative process 792 00:45:46,940 --> 00:45:49,082 so ZEMAX can do it for you. 793 00:46:03,860 --> 00:46:05,630 So this is the layout of the system. 794 00:46:05,630 --> 00:46:08,870 I start with just three each value-- 795 00:46:08,870 --> 00:46:12,150 I just choose the random value of this-- 796 00:46:12,150 --> 00:46:13,990 I mean, there's 12 values here. 797 00:46:13,990 --> 00:46:16,400 So we have the three lenses here. 798 00:46:16,400 --> 00:46:18,710 And parameter with the [INAUDIBLE],, 799 00:46:18,710 --> 00:46:20,310 they are variables. 800 00:46:20,310 --> 00:46:22,040 So we have the 12 variables here. 801 00:46:26,040 --> 00:46:28,910 And this is the result of the aberration 802 00:46:28,910 --> 00:46:31,940 at the initial state. 803 00:46:31,940 --> 00:46:36,490 And the spot diagram through the focus. 804 00:46:36,490 --> 00:46:39,660 So as the bottom left figure, I have four different incident 805 00:46:39,660 --> 00:46:40,160 angles. 806 00:46:40,160 --> 00:46:41,500 So 1, 2, 3, 4. 807 00:46:41,500 --> 00:46:43,440 The different colors here. 808 00:46:43,440 --> 00:46:47,400 So these four rows in spot diagram 809 00:46:47,400 --> 00:46:53,550 means they're corresponding to this different incident angle. 810 00:46:53,550 --> 00:46:58,550 And these five column means I had five different defocus. 811 00:46:58,550 --> 00:47:02,210 So center one is focused, and they are slightly 812 00:47:02,210 --> 00:47:03,530 defocused in front or back. 813 00:47:07,680 --> 00:47:12,040 I also have the spot diagram with different wavelength. 814 00:47:12,040 --> 00:47:15,350 So this one is about 500 nanometer. 815 00:47:15,350 --> 00:47:18,500 And it's about 480 nanometer. 816 00:47:18,500 --> 00:47:21,170 I guess this one is 550 or so. 817 00:47:21,170 --> 00:47:23,810 50 or so. 818 00:47:23,810 --> 00:47:31,870 And this is merit function. 819 00:47:31,870 --> 00:47:34,340 I already defined the merit function. 820 00:47:34,340 --> 00:47:38,280 Actually, I used the default merit function in this case. 821 00:47:38,280 --> 00:47:40,125 So there are many, many different parameter 822 00:47:40,125 --> 00:47:43,146 you can tune, but the essence is you 823 00:47:43,146 --> 00:47:45,790 want to minimize some aberration at the image plane. 824 00:47:54,240 --> 00:48:01,590 So I'm going to run this optimization with the auto 825 00:48:01,590 --> 00:48:03,220 update. 826 00:48:03,220 --> 00:48:10,370 So you're going to see the actual shape of this analysis 827 00:48:10,370 --> 00:48:14,230 for every iteration. 828 00:48:14,230 --> 00:48:21,650 So we start with this situation, and I run 50 cycles here. 829 00:48:21,650 --> 00:48:24,750 So every iteration is update the system. 830 00:48:24,750 --> 00:48:28,350 And it actually improve the performance of system. 831 00:48:34,550 --> 00:48:35,400 So yeah. 832 00:48:35,400 --> 00:48:39,490 So after 50 iterations, this focus is much nicer. 833 00:48:39,490 --> 00:48:40,500 Focus is much nicer. 834 00:48:40,500 --> 00:48:43,630 And it doesn't change that much. 835 00:48:43,630 --> 00:48:45,880 But actually, the scales are different. 836 00:48:45,880 --> 00:48:49,210 So actually it improved a lot. 837 00:48:49,210 --> 00:48:51,260 So after this is optimized-- so this 838 00:48:51,260 --> 00:48:53,450 is what usually optical engineers 839 00:48:53,450 --> 00:48:58,520 do in industry when you want to design some optical system. 840 00:48:58,520 --> 00:49:02,870 But is this the best design? 841 00:49:02,870 --> 00:49:06,810 I mean, can you say this is this the best design? 842 00:49:06,810 --> 00:49:11,260 Probably not because it's just one of the minimum. 843 00:49:11,260 --> 00:49:13,440 It can be just global minimum. 844 00:49:13,440 --> 00:49:20,820 So it's kind of sensitive to initial value or constraint 845 00:49:20,820 --> 00:49:21,780 or whatever. 846 00:49:21,780 --> 00:49:25,230 And also, there are many, many issues-- 847 00:49:25,230 --> 00:49:27,420 I mean practical issues-- we have to consider. 848 00:49:27,420 --> 00:49:39,680 For example, in this case, let's see this third lens. 849 00:49:39,680 --> 00:49:40,980 The last lens. 850 00:49:40,980 --> 00:49:47,790 The radius of curvature of this lens is 60.8 on this one. 851 00:49:47,790 --> 00:49:53,640 And the second radius is negative 68.3. 852 00:49:53,640 --> 00:49:55,860 So they are very similar, which means 853 00:49:55,860 --> 00:49:58,020 it looks like convex lens. 854 00:49:58,020 --> 00:50:01,490 But they are very deep. 855 00:50:01,490 --> 00:50:04,370 They are slightly different, but they are not same. 856 00:50:04,370 --> 00:50:06,230 So if you're on an assembly system, 857 00:50:06,230 --> 00:50:09,770 then you need to figure out which surfaces from surfaces 858 00:50:09,770 --> 00:50:12,050 shown is the back surface. 859 00:50:12,050 --> 00:50:14,840 But it's very similar, so it's hard to figure out. 860 00:50:14,840 --> 00:50:16,460 So sometimes you need to consider 861 00:50:16,460 --> 00:50:19,200 many, many practical issues because in this case, 862 00:50:19,200 --> 00:50:23,060 it's not good design because they are so similar. 863 00:50:23,060 --> 00:50:25,940 You better have the plane of convex 864 00:50:25,940 --> 00:50:27,980 to figure out which surface is front 865 00:50:27,980 --> 00:50:30,440 and which surface back surface. 866 00:50:30,440 --> 00:50:33,170 So ZEMAX does that for you. 867 00:50:33,170 --> 00:50:34,520 I mean, the optimization. 868 00:50:34,520 --> 00:50:37,370 But you need to interpret the result, 869 00:50:37,370 --> 00:50:42,170 and you need to take into account many practical issues. 870 00:50:42,170 --> 00:50:47,080 So basically, which means it means nothing. 871 00:50:47,080 --> 00:50:50,730 So that's why we learn many, many different things. 872 00:50:50,730 --> 00:50:53,390 I mean, the geometrical optics, you need to interpret it. 873 00:50:53,390 --> 00:50:55,625 And you need the experience and insight. 874 00:50:58,460 --> 00:50:58,960 Yeah. 875 00:50:58,960 --> 00:51:01,090 That's what I prepared today. 876 00:51:01,090 --> 00:51:03,860 PROFESSOR: The last example that we're going to show 877 00:51:03,860 --> 00:51:07,500 is on a gradient refractive index lens. 878 00:51:07,500 --> 00:51:10,490 So these are a very interesting type of lenses. 879 00:51:10,490 --> 00:51:12,340 I have this microphone here. 880 00:51:12,340 --> 00:51:17,262 Very interesting type of lenses because the index of refraction 881 00:51:17,262 --> 00:51:19,470 as opposed to normal lenses that you're familiar with 882 00:51:19,470 --> 00:51:21,595 and we've been showing over and over, which is just 883 00:51:21,595 --> 00:51:23,780 a piece of glass with a fixed index of refraction, 884 00:51:23,780 --> 00:51:26,150 in this case, as we've been mentioning, 885 00:51:26,150 --> 00:51:29,570 the index of refraction changes as a function, in this case, 886 00:51:29,570 --> 00:51:30,450 of radius. 887 00:51:30,450 --> 00:51:33,990 So the lens itself, it looks like this rod here. 888 00:51:33,990 --> 00:51:35,190 So this is a 3D model. 889 00:51:35,190 --> 00:51:40,820 Just like a cylinder of glass. 890 00:51:40,820 --> 00:51:44,230 But its properties, or its optical properties, 891 00:51:44,230 --> 00:51:46,560 especially a refractive index, as you can see, changes. 892 00:51:46,560 --> 00:51:50,018 And in this case, commercially available green lenses, 893 00:51:50,018 --> 00:51:51,560 the way they're manufactured, they're 894 00:51:51,560 --> 00:51:54,590 constrained to be with a parabolic index of refraction 895 00:51:54,590 --> 00:51:55,230 variation. 896 00:51:55,230 --> 00:51:58,520 So at the center-- so this is basically distance from the-- 897 00:51:58,520 --> 00:52:00,530 radial distance is at the center of the lens. 898 00:52:00,530 --> 00:52:01,760 Optical axis. 899 00:52:01,760 --> 00:52:03,500 And this as we propagate to one-- 900 00:52:03,500 --> 00:52:06,710 as we move to one of the extremes of the lens. 901 00:52:06,710 --> 00:52:09,620 And we see that we have the highest index of refraction 902 00:52:09,620 --> 00:52:10,580 at the center. 903 00:52:10,580 --> 00:52:14,160 And then the case parabolically to the sides. 904 00:52:14,160 --> 00:52:17,150 And if you can notice here the variation from top to bottom 905 00:52:17,150 --> 00:52:18,460 is maybe hard to read. 906 00:52:18,460 --> 00:52:19,950 But it's very, very small. 907 00:52:19,950 --> 00:52:24,262 So it goes from something like 1.6 to 1.59 908 00:52:24,262 --> 00:52:25,220 or something like that. 909 00:52:25,220 --> 00:52:29,450 So it's very, very small, the index variation. 910 00:52:29,450 --> 00:52:32,630 However, that is enough to guide the light 911 00:52:32,630 --> 00:52:34,670 and cause the rays to bend. 912 00:52:34,670 --> 00:52:36,777 So one thing that we're going to see here 913 00:52:36,777 --> 00:52:39,110 is that Snell's law is basically applied in a little bit 914 00:52:39,110 --> 00:52:40,400 different way. 915 00:52:40,400 --> 00:52:43,220 Before, we're interested in curved surfaces. 916 00:52:43,220 --> 00:52:46,160 And a ray hitting a given curved surface 917 00:52:46,160 --> 00:52:48,050 will bend accordingly to Snell's law 918 00:52:48,050 --> 00:52:49,520 and the index of refraction. 919 00:52:49,520 --> 00:52:54,310 In this case, you see that the faces of the lens to both sides 920 00:52:54,310 --> 00:52:55,550 are flat. 921 00:52:55,550 --> 00:52:58,493 So if the index of these was basically uniform, 922 00:52:58,493 --> 00:53:00,910 the plane wave would just go straight through like a piece 923 00:53:00,910 --> 00:53:02,772 of glass, like in a window. 924 00:53:02,772 --> 00:53:05,230 But the fact that this has a parabolic index of refraction, 925 00:53:05,230 --> 00:53:08,560 then it makes these rays to smoothly 926 00:53:08,560 --> 00:53:12,100 be guided and bent towards a focus. 927 00:53:12,100 --> 00:53:13,930 In this case, on-axis or off-axis. 928 00:53:13,930 --> 00:53:17,740 This is a 10 degree off-axis plane wave incidence. 929 00:53:17,740 --> 00:53:20,230 So what is the nice thing about these things? 930 00:53:20,230 --> 00:53:22,180 It's that instead of only using, you 931 00:53:22,180 --> 00:53:26,440 can think of the refractive power of these two surfaces. 932 00:53:26,440 --> 00:53:29,380 This type of lens uses the entire volume 933 00:53:29,380 --> 00:53:32,770 of the lens to guide the rays. 934 00:53:32,770 --> 00:53:35,740 So therefore, if I put a screen here and 935 00:53:35,740 --> 00:53:38,973 see the intersections of all these rays in that screen-- 936 00:53:38,973 --> 00:53:41,140 this is the spot diagram that we've been seeing also 937 00:53:41,140 --> 00:53:42,010 in the other ones-- 938 00:53:42,010 --> 00:53:43,690 we see that for the on-axis point, 939 00:53:43,690 --> 00:53:47,513 we have a tight focus, which typically is smaller. 940 00:53:47,513 --> 00:53:49,930 In this case, the focal length, the effective focal length 941 00:53:49,930 --> 00:53:52,630 of this system is very small. 942 00:53:52,630 --> 00:53:55,330 So if you try to do this with a normal refractive 943 00:53:55,330 --> 00:53:59,200 lens with curved surfaces, you would 944 00:53:59,200 --> 00:54:04,010 need to have a really, really curved lens. 945 00:54:04,010 --> 00:54:07,210 And as we know from the demos and from the discussions, 946 00:54:07,210 --> 00:54:09,310 those type of lenses will be really aberrated. 947 00:54:09,310 --> 00:54:11,890 However, these, even though it has a shorter focal length, 948 00:54:11,890 --> 00:54:14,710 it still focuses into a very tight focus 949 00:54:14,710 --> 00:54:17,440 because it uses, as I said, the entire volume. 950 00:54:17,440 --> 00:54:20,470 Now here, this is the example of coma. 951 00:54:20,470 --> 00:54:22,570 So if I actually show-- 952 00:54:22,570 --> 00:54:27,880 I can even focus a little bit into how the rays are coming. 953 00:54:27,880 --> 00:54:33,370 And we see the similar caustics that you saw in the demo 954 00:54:33,370 --> 00:54:35,960 and in the slides before. 955 00:54:35,960 --> 00:54:38,170 But if we put a screen here, this 956 00:54:38,170 --> 00:54:42,460 is the actual shape of the rays intersecting that plane. 957 00:54:42,460 --> 00:54:44,880 So these two are the circles of confusion. 958 00:54:44,880 --> 00:54:48,690 And just to give you a preview, this 959 00:54:48,690 --> 00:54:50,460 is basically for the on-axis point. 960 00:54:50,460 --> 00:54:52,440 So this is what geometrical optics predict. 961 00:54:52,440 --> 00:54:54,060 The bottom one. 962 00:54:54,060 --> 00:54:56,160 So the geometrical optics predict just how 963 00:54:56,160 --> 00:54:58,370 rays intersect a given plane. 964 00:54:58,370 --> 00:55:00,510 It doesn't take into account diffraction 965 00:55:00,510 --> 00:55:04,627 that we're going to learn in the next part of the course. 966 00:55:04,627 --> 00:55:06,960 However, diffraction is very important in optical system 967 00:55:06,960 --> 00:55:08,500 because it regulates resolution. 968 00:55:08,500 --> 00:55:11,070 And many of you have been asking about numerical aperture 969 00:55:11,070 --> 00:55:12,960 and how that depends on resolution 970 00:55:12,960 --> 00:55:16,350 and what happens changing the aperture stop and so on and so 971 00:55:16,350 --> 00:55:17,440 forth. 972 00:55:17,440 --> 00:55:21,013 Well, this is also very related to this point spread function 973 00:55:21,013 --> 00:55:21,930 that you can see here. 974 00:55:21,930 --> 00:55:24,900 And as you can see, it's a bright spot, 975 00:55:24,900 --> 00:55:27,730 and then there's concentric rings 976 00:55:27,730 --> 00:55:31,900 which is the way you can see the diffraction 977 00:55:31,900 --> 00:55:33,298 effect of that point. 978 00:55:33,298 --> 00:55:34,590 So this is geometrical version. 979 00:55:34,590 --> 00:55:37,660 This is the diffraction version or the more realistic version 980 00:55:37,660 --> 00:55:39,710 of how light would diffract. 981 00:55:39,710 --> 00:55:43,480 And in order to compute this, we'll 982 00:55:43,480 --> 00:55:45,432 basically need the math and the tools 983 00:55:45,432 --> 00:55:47,890 that we're going to learn in the second part of the course. 984 00:55:47,890 --> 00:55:51,940 But this is just giving you the type of science and analysis 985 00:55:51,940 --> 00:55:55,490 that we could do in this software. 986 00:55:55,490 --> 00:55:57,830 So any questions before-- 987 00:55:57,830 --> 00:55:59,757 because we're going to break, right, George? 988 00:55:59,757 --> 00:56:00,840 GEORGE BARBASTATHIS: Yeah. 989 00:56:03,880 --> 00:56:05,630 PROFESSOR: Any questions before the break? 990 00:56:11,610 --> 00:56:13,735 GEORGE BARBASTATHIS: I think it's time to continue. 991 00:56:21,260 --> 00:56:27,418 Before I move on, are there any questions about-- 992 00:56:27,418 --> 00:56:27,960 I don't know. 993 00:56:27,960 --> 00:56:28,585 About anything. 994 00:56:28,585 --> 00:56:33,300 About the homeworks, about the lectures, about imaging, 995 00:56:33,300 --> 00:56:34,320 about focusing? 996 00:56:38,280 --> 00:56:39,530 A bit of a milestone, I guess. 997 00:56:39,530 --> 00:56:41,160 I don't believe in quizzes. 998 00:56:41,160 --> 00:56:43,220 Actually, I hate giving exams. 999 00:56:43,220 --> 00:56:45,470 But I have to produce grades. 1000 00:56:45,470 --> 00:56:47,780 The register asks for grades. 1001 00:56:47,780 --> 00:56:53,390 So the accepted, I guess, established way is with exams. 1002 00:56:53,390 --> 00:56:55,130 But anyway. 1003 00:56:55,130 --> 00:56:57,900 And I also hate reviews because of the same reason 1004 00:56:57,900 --> 00:57:00,290 because a review suggests somehow 1005 00:57:00,290 --> 00:57:01,760 that the quiz is important. 1006 00:57:01,760 --> 00:57:03,260 Of course, the quiz is not important 1007 00:57:03,260 --> 00:57:06,410 except for the register. 1008 00:57:06,410 --> 00:57:08,420 The only significance, perhaps, exams 1009 00:57:08,420 --> 00:57:12,048 have is that they really force you to study. 1010 00:57:12,048 --> 00:57:14,340 I mean, if you didn't have an exam, then you would say, 1011 00:57:14,340 --> 00:57:15,770 well, maybe I'll study later. 1012 00:57:15,770 --> 00:57:17,930 Maybe I will study in the summer. 1013 00:57:17,930 --> 00:57:21,840 When I go out water skiing, I can study optics 1014 00:57:21,840 --> 00:57:24,620 while I water ski. 1015 00:57:24,620 --> 00:57:28,830 But anyway. 1016 00:57:28,830 --> 00:57:31,380 So I try to-- 1017 00:57:31,380 --> 00:57:34,290 in my classes, I try to de-emphasize exams 1018 00:57:34,290 --> 00:57:35,250 as much as possible. 1019 00:57:35,250 --> 00:57:37,598 And the assumption is that if throughout the course 1020 00:57:37,598 --> 00:57:39,390 of the class you have learned the material, 1021 00:57:39,390 --> 00:57:41,910 then you will do well in the exam. 1022 00:57:41,910 --> 00:57:43,470 Whereas to do the opposite, to try 1023 00:57:43,470 --> 00:57:48,480 to satisfy the needs of the exam is very limited because then, 1024 00:57:48,480 --> 00:57:49,780 what you get out of it? 1025 00:57:49,780 --> 00:57:53,290 You're wasting your time. 1026 00:57:53,290 --> 00:57:55,900 So that's why I try to not mention exams. 1027 00:57:55,900 --> 00:57:58,660 As far as I can help it, I try not to mention them. 1028 00:57:58,660 --> 00:57:59,620 But I know-- 1029 00:57:59,620 --> 00:58:01,100 I've been a student also. 1030 00:58:01,100 --> 00:58:04,930 At the time I would also be very worried about exams. 1031 00:58:04,930 --> 00:58:05,430 Anyway. 1032 00:58:05,430 --> 00:58:06,290 So this is the time. 1033 00:58:06,290 --> 00:58:10,330 Is there any sort of-- 1034 00:58:10,330 --> 00:58:11,120 whatever. 1035 00:58:11,120 --> 00:58:12,280 Review questions or-- 1036 00:58:31,090 --> 00:58:32,910 OK. 1037 00:58:32,910 --> 00:58:38,760 I'm going to count to from 3 down. 1038 00:58:38,760 --> 00:58:46,605 3, 2, how they do in auctions, 1. 1039 00:58:49,230 --> 00:58:51,340 You can still interrupt. 1040 00:58:51,340 --> 00:58:54,960 Also, we have the class forum. 1041 00:58:54,960 --> 00:58:58,300 I mean the online chat room or whatever it's called. 1042 00:58:58,300 --> 00:59:00,350 The forum the call it. 1043 00:59:00,350 --> 00:59:03,850 So of course, you can post questions there. 1044 00:59:03,850 --> 00:59:05,440 I will be diligently checking. 1045 00:59:08,270 --> 00:59:11,332 Also, I sent an announcement by email. 1046 00:59:11,332 --> 00:59:12,540 You may have not seen it yet. 1047 00:59:12,540 --> 00:59:16,020 But we've posted some solved problems. 1048 00:59:16,020 --> 00:59:19,390 Actually, the problems have been posted since Monday. 1049 00:59:19,390 --> 00:59:21,858 So the problems are solved. 1050 00:59:21,858 --> 00:59:23,400 So you don't really have to-- so this 1051 00:59:23,400 --> 00:59:25,170 is the homework for next week. 1052 00:59:25,170 --> 00:59:27,940 You don't have to actually turn in the homework on the quiz 1053 00:59:27,940 --> 00:59:28,440 day. 1054 00:59:28,440 --> 00:59:29,970 That would have been cruel. 1055 00:59:29,970 --> 00:59:32,980 But we actually gave you some solved problems. 1056 00:59:32,980 --> 00:59:34,620 So you can go over them. 1057 00:59:34,620 --> 00:59:37,860 And of course the recommendation is 1058 00:59:37,860 --> 00:59:41,060 to really pretend that you don't have the solution. 1059 00:59:41,060 --> 00:59:43,020 Try to solve it yourself. 1060 00:59:43,020 --> 00:59:44,580 And then check it-- 1061 00:59:44,580 --> 00:59:47,490 check your solution against ours. 1062 00:59:47,490 --> 00:59:50,610 And of course, you don't have to turn in your solution. 1063 00:59:50,610 --> 00:59:52,620 And I'll stop talking about the exam 1064 00:59:52,620 --> 00:59:56,810 now, unless someone has a question now or later. 1065 00:59:56,810 --> 00:59:58,550 So Pepe already gave an introduction 1066 00:59:58,550 --> 01:00:04,820 to the topic of gradient index or GRIN optics. 1067 01:00:04,820 --> 01:00:07,280 And he mentioned that the motivation is basically 1068 01:00:07,280 --> 01:00:11,660 that if you can focus the light by using 1069 01:00:11,660 --> 01:00:15,950 a non-uniform material, a material whose 1070 01:00:15,950 --> 01:00:20,570 index of refraction is variable, perhaps you can get away 1071 01:00:20,570 --> 01:00:23,600 by also using a material whose surface is flat. 1072 01:00:23,600 --> 01:00:27,080 So you don't have to use this awkward spherical shape that we 1073 01:00:27,080 --> 01:00:28,400 know from everyday optics. 1074 01:00:28,400 --> 01:00:30,230 Maybe you can have something that 1075 01:00:30,230 --> 01:00:34,880 looks like a slab or a plate. 1076 01:00:34,880 --> 01:00:38,100 And nevertheless, this flat-looking plate 1077 01:00:38,100 --> 01:00:40,850 might have the ability to focus light. 1078 01:00:40,850 --> 01:00:43,170 So the way you can do-- first of all, 1079 01:00:43,170 --> 01:00:44,430 can you do this kind of thing? 1080 01:00:44,430 --> 01:00:47,910 Can you make an element with variable index or the fraction. 1081 01:00:47,910 --> 01:00:50,490 So there's two-- well, there's many ways. 1082 01:00:50,490 --> 01:00:53,840 But in industry, in manufacturing practices, 1083 01:00:53,840 --> 01:00:55,940 there's two ways that are being used. 1084 01:00:55,940 --> 01:00:58,690 And I will describe them. 1085 01:00:58,690 --> 01:01:02,150 One is called ion exchange. 1086 01:01:02,150 --> 01:01:05,440 And what they do is they take a glass rod, 1087 01:01:05,440 --> 01:01:08,560 which is doped with a certain kind of ions-- 1088 01:01:08,560 --> 01:01:12,280 typically, they use sodium in the glass rod. 1089 01:01:12,280 --> 01:01:15,820 Then they have the glass meld at the very high temperature, 1090 01:01:15,820 --> 01:01:18,100 of course, with a different kind of ion. 1091 01:01:18,100 --> 01:01:19,810 Typically lithium. 1092 01:01:19,810 --> 01:01:26,450 And what happens is they slowly dip the glass rod 1093 01:01:26,450 --> 01:01:28,240 into the glass meld. 1094 01:01:28,240 --> 01:01:30,950 And over a period of several hours, what's happens 1095 01:01:30,950 --> 01:01:35,690 is the lithium ions, they kick out some of the sodium ions 1096 01:01:35,690 --> 01:01:36,710 from the glass rod. 1097 01:01:36,710 --> 01:01:38,390 They basically penetrate the rod, 1098 01:01:38,390 --> 01:01:42,357 and they kick out some of the sodium ions. 1099 01:01:42,357 --> 01:01:43,940 But of course, the index of refraction 1100 01:01:43,940 --> 01:01:48,320 depends on what kind of ions you have inside of the glass. 1101 01:01:48,320 --> 01:01:52,070 And also, because the ions, they propagate 1102 01:01:52,070 --> 01:01:54,230 at a finite speed, what will happen 1103 01:01:54,230 --> 01:01:56,990 here is in the exterior of the glass rod, 1104 01:01:56,990 --> 01:01:58,490 you get a lot of ion exchange. 1105 01:01:58,490 --> 01:02:00,470 So the index changes a lot. 1106 01:02:00,470 --> 01:02:03,230 But near the center, if you get your timing right, 1107 01:02:03,230 --> 01:02:06,050 near the center you don't get any exchange at all. 1108 01:02:06,050 --> 01:02:07,730 So you basically have-- 1109 01:02:07,730 --> 01:02:10,730 in the center you only have sodium ions. 1110 01:02:10,730 --> 01:02:13,430 Near the exterior to give both sodium and lithium. 1111 01:02:13,430 --> 01:02:16,190 So you can get a situation where the index of refraction 1112 01:02:16,190 --> 01:02:21,750 is different in the center than it is at the edge. 1113 01:02:21,750 --> 01:02:24,310 And then what they do, of course, 1114 01:02:24,310 --> 01:02:25,860 is after they finish this process, 1115 01:02:25,860 --> 01:02:31,990 they chop this glass rod into slabs. 1116 01:02:31,990 --> 01:02:35,600 And then each one of those slabs is to become a GRIN lens. 1117 01:02:35,600 --> 01:02:37,800 And as I said before, they do it typically 1118 01:02:37,800 --> 01:02:41,250 so that the index is higher in the center 1119 01:02:41,250 --> 01:02:44,910 and lower at the edges. 1120 01:02:44,910 --> 01:02:47,170 So this kind of thing I will do in detail later. 1121 01:02:47,170 --> 01:02:51,120 But if you look at it in cross-section, 1122 01:02:51,120 --> 01:02:54,540 it focuses like so if you illuminate it 1123 01:02:54,540 --> 01:03:00,090 with parallel bundle of rays. 1124 01:03:00,090 --> 01:03:01,950 You can think of the index of refraction 1125 01:03:01,950 --> 01:03:04,740 as an attraction of sorts for rays. 1126 01:03:04,740 --> 01:03:08,070 So the ray that enters at the lower index, it gets 1127 01:03:08,070 --> 01:03:09,750 attracted by the high index. 1128 01:03:09,750 --> 01:03:11,040 It bends. 1129 01:03:11,040 --> 01:03:13,620 And as a result, by the time the rays come out, 1130 01:03:13,620 --> 01:03:16,340 they actually come out as a focusing bundle. 1131 01:03:16,340 --> 01:03:21,230 So you can use this element to focus light. 1132 01:03:21,230 --> 01:03:25,020 So just like a lens, but it has a flat surface. 1133 01:03:25,020 --> 01:03:30,690 And because the ionic change is actually diffusion-driven-- 1134 01:03:30,690 --> 01:03:33,150 I don't want to go into the mathematics of diffusion. 1135 01:03:33,150 --> 01:03:40,050 But it turns out that if you get the specific form of the index 1136 01:03:40,050 --> 01:03:41,880 of refraction that you get in this case, 1137 01:03:41,880 --> 01:03:44,860 it's quadratic or parabolic. 1138 01:03:44,860 --> 01:03:47,320 So it is maximum in the center. 1139 01:03:47,320 --> 01:03:50,850 And then it drops as the square-- 1140 01:03:50,850 --> 01:03:59,730 as the radius square, where r is the radius in the slab times 1141 01:03:59,730 --> 01:04:02,010 some coefficient alpha. 1142 01:04:02,010 --> 01:04:05,760 And I will show in a second that this coefficient alpha 1143 01:04:05,760 --> 01:04:09,360 and the thickness of the lens, the product of the two 1144 01:04:09,360 --> 01:04:11,770 actually specifies the focal length. 1145 01:04:11,770 --> 01:04:13,350 So this looks like a funny formula. 1146 01:04:13,350 --> 01:04:15,930 But if you think about it, alpha must 1147 01:04:15,930 --> 01:04:19,560 have dimensions that are inverse distance 1148 01:04:19,560 --> 01:04:24,090 square so that the product alpha times r squared 1149 01:04:24,090 --> 01:04:25,740 is non-dimensional. 1150 01:04:25,740 --> 01:04:26,920 And d is distance. 1151 01:04:26,920 --> 01:04:29,490 So therefore, the units here are distance. 1152 01:04:29,490 --> 01:04:32,490 So indeed, the quantity over there is the distance. 1153 01:04:32,490 --> 01:04:35,920 It is the focal length. 1154 01:04:35,920 --> 01:04:37,890 So let's see now why this is the case. 1155 01:04:41,650 --> 01:04:44,500 So I guess we've done it several times. 1156 01:04:44,500 --> 01:04:46,240 So you must be tired of it by now. 1157 01:04:46,240 --> 01:04:49,470 I just lost my-- 1158 01:04:49,470 --> 01:04:50,970 oh, maybe I'll use the-- oh, thanks. 1159 01:04:54,540 --> 01:04:56,222 I also must not forget this. 1160 01:04:59,460 --> 01:05:00,500 OK. 1161 01:05:00,500 --> 01:05:04,220 So in typical fashion, we will use Fermat's principle 1162 01:05:04,220 --> 01:05:07,410 in order to calculate what is happening here. 1163 01:05:07,410 --> 01:05:11,960 So Fermat says that if you-- 1164 01:05:11,960 --> 01:05:12,965 here is my element. 1165 01:05:16,130 --> 01:05:21,360 And I have two rays that both started at infinity. 1166 01:05:21,360 --> 01:05:29,360 So the rays share their origin, the starting point. 1167 01:05:29,360 --> 01:05:33,680 And then my desire here, my intended outcome, 1168 01:05:33,680 --> 01:05:36,510 is that these two rays meet because I want 1169 01:05:36,510 --> 01:05:39,590 the element to focus the light. 1170 01:05:39,590 --> 01:05:46,340 So Fermat says that the optical path length here on-axis 1171 01:05:46,340 --> 01:05:49,500 and the optical path length out here off-axis, 1172 01:05:49,500 --> 01:05:50,780 they must be equal. 1173 01:05:50,780 --> 01:05:53,040 Because the rays started at the same point, 1174 01:05:53,040 --> 01:05:56,270 they end up at the same point, the optical path lengths 1175 01:05:56,270 --> 01:05:57,752 must be equal. 1176 01:05:57,752 --> 01:05:59,210 So what is the optical path length? 1177 01:05:59,210 --> 01:06:03,570 So this derivation here is very highly handwaving and very, 1178 01:06:03,570 --> 01:06:05,150 very inaccurate. 1179 01:06:05,150 --> 01:06:06,530 But bear with me. 1180 01:06:06,530 --> 01:06:09,080 In the next half hour, I will tell you 1181 01:06:09,080 --> 01:06:12,200 how this calculation can be done exactly. 1182 01:06:12,200 --> 01:06:14,240 But for now, this is kind of an interesting way 1183 01:06:14,240 --> 01:06:16,540 to think about GRIN, so we'll do it anyway. 1184 01:06:16,540 --> 01:06:17,870 It's also in the book. 1185 01:06:17,870 --> 01:06:20,720 So you don't really have to copy my derivation here. 1186 01:06:20,720 --> 01:06:22,530 You already have it in two places. 1187 01:06:22,530 --> 01:06:25,180 One is in the book, and the other is the notes. 1188 01:06:25,180 --> 01:06:28,550 So maybe you can just look at it and don't 1189 01:06:28,550 --> 01:06:30,770 bother to write it down. 1190 01:06:30,770 --> 01:06:32,060 So let's compare the two rays. 1191 01:06:32,060 --> 01:06:36,980 Let's compare the on-axis ray, which basically goes straight. 1192 01:06:36,980 --> 01:06:39,790 This is the optical axis. 1193 01:06:39,790 --> 01:06:45,250 And the off-axis ray, which as Pepe already mentioned, 1194 01:06:45,250 --> 01:06:47,650 the off-axis ray will actually do something like this. 1195 01:06:47,650 --> 01:06:48,690 It will bend. 1196 01:06:48,690 --> 01:06:50,940 And then, of course, after it comes out in free space, 1197 01:06:50,940 --> 01:06:52,330 it will go straight. 1198 01:06:52,330 --> 01:06:53,620 We know that for sure. 1199 01:06:53,620 --> 01:06:56,590 But inside the gradient index medium, the ray will bend. 1200 01:06:56,590 --> 01:07:05,390 It will follow a continuous kind of trajectory. 1201 01:07:05,390 --> 01:07:07,740 This is also counterintuitive because in our experience 1202 01:07:07,740 --> 01:07:09,550 light rays go straight. 1203 01:07:09,550 --> 01:07:10,960 But such is life. 1204 01:07:10,960 --> 01:07:13,720 Physics sometimes have unexpected outcomes. 1205 01:07:13,720 --> 01:07:16,450 If you put a ray in a gradient index medium, 1206 01:07:16,450 --> 01:07:20,900 indeed it will bend onto a curved trajectory. 1207 01:07:20,900 --> 01:07:22,850 So let's put some notation here. 1208 01:07:22,850 --> 01:07:24,850 So we'll call this r. 1209 01:07:24,850 --> 01:07:28,320 This is the radial direction. 1210 01:07:28,320 --> 01:07:30,235 We'll call this d. 1211 01:07:30,235 --> 01:07:33,190 This is the thickness of the GRIN. 1212 01:07:33,190 --> 01:07:34,420 We'll call this f. 1213 01:07:34,420 --> 01:07:35,920 This is the focal length. 1214 01:07:39,090 --> 01:07:43,050 And for the sake of this derivation, 1215 01:07:43,050 --> 01:07:45,570 let me actually redraw it. 1216 01:07:45,570 --> 01:07:46,800 I will redraw it. 1217 01:07:46,800 --> 01:07:49,080 Forgive me. 1218 01:07:49,080 --> 01:07:53,208 I just want to emphasize this caricature. 1219 01:07:59,910 --> 01:08:03,460 So we'll actually neglect this distance. 1220 01:08:03,460 --> 01:08:06,480 This space over there, I will pretend it's not there. 1221 01:08:06,480 --> 01:08:10,580 And this is consistent with a paraxial approximation. 1222 01:08:10,580 --> 01:08:15,930 Here is r, here is d again, and here is f. 1223 01:08:15,930 --> 01:08:17,160 And that's the optical axis. 1224 01:08:22,760 --> 01:08:23,260 OK. 1225 01:08:29,130 --> 01:08:31,510 So well, let's do, first of all, the optical path 1226 01:08:31,510 --> 01:08:35,800 length for the on-axis ray. 1227 01:08:35,800 --> 01:08:39,479 The optical path length equals the index 1228 01:08:39,479 --> 01:08:43,810 times the distance integrated over the path of the ray. 1229 01:08:43,810 --> 01:08:49,340 So this would equal the index at radius equal 0. 1230 01:08:49,340 --> 01:08:52,430 That is the index over here at the center 1231 01:08:52,430 --> 01:08:55,410 of the lens times the thickness of the lens 1232 01:08:55,410 --> 01:08:58,040 plus the focal length. 1233 01:08:58,040 --> 01:09:01,399 So of course, outside the lens, the index we assume to be 1. 1234 01:09:01,399 --> 01:09:05,270 You can have an imaginary lens, but let's not do that now. 1235 01:09:05,270 --> 01:09:06,790 So this is where it comes from. 1236 01:09:06,790 --> 01:09:07,630 AUDIENCE: Excuse me. 1237 01:09:07,630 --> 01:09:08,630 GEORGE BARBASTATHIS: Yeah? 1238 01:09:08,630 --> 01:09:10,380 AUDIENCE: According to Fermat's principle, 1239 01:09:10,380 --> 01:09:13,340 it says that the ray travels in such a way 1240 01:09:13,340 --> 01:09:17,300 that it tries to minimize the path. 1241 01:09:17,300 --> 01:09:22,359 So why does the ray goes from lower index to higher index? 1242 01:09:22,359 --> 01:09:25,680 Means if it goes into higher index material, 1243 01:09:25,680 --> 01:09:29,020 actually it is increasing its optical path length? 1244 01:09:29,020 --> 01:09:30,020 GEORGE BARBASTATHIS: Ah. 1245 01:09:30,020 --> 01:09:32,270 But it will bend. 1246 01:09:32,270 --> 01:09:33,950 That's what Snell says. 1247 01:09:33,950 --> 01:09:38,210 It will bend so that the total path length actually decreases. 1248 01:09:43,423 --> 01:09:45,840 Did I answer your question or did you have something else? 1249 01:09:45,840 --> 01:09:48,240 AUDIENCE: I think-- well, I don't know, 1250 01:09:48,240 --> 01:09:52,120 but maybe the answer to your question is that although-- 1251 01:09:52,120 --> 01:09:55,630 I think what he says is that it will actually 1252 01:09:55,630 --> 01:10:00,190 travel through a longer optical path in the medium 1253 01:10:00,190 --> 01:10:02,650 because it goes into the bigger index. 1254 01:10:02,650 --> 01:10:03,650 GEORGE BARBASTATHIS: Ah. 1255 01:10:03,650 --> 01:10:04,150 OK. 1256 01:10:04,150 --> 01:10:04,780 I see. 1257 01:10:04,780 --> 01:10:08,050 AUDIENCE: But I think that's counteracted 1258 01:10:08,050 --> 01:10:12,280 by the extra that would be outside, isn't it, in the air, 1259 01:10:12,280 --> 01:10:12,780 I think. 1260 01:10:12,780 --> 01:10:13,210 GEORGE BARBASTATHIS: Yeah. 1261 01:10:13,210 --> 01:10:15,277 And you will see that in the derivation. 1262 01:10:15,277 --> 01:10:17,360 Because it will have to go through this hypotenuse 1263 01:10:17,360 --> 01:10:19,750 over here, it will actually-- yeah. 1264 01:10:19,750 --> 01:10:23,110 So the overall path is actually minimized. 1265 01:10:23,110 --> 01:10:25,487 And this is the only way it can be minimized. 1266 01:10:28,340 --> 01:10:28,840 OK. 1267 01:10:28,840 --> 01:10:32,620 So lo and behold, then, let's do what you just said. 1268 01:10:32,620 --> 01:10:39,230 So what about the ray that arrives 1269 01:10:39,230 --> 01:10:41,240 at the higher elevation? 1270 01:10:41,240 --> 01:10:43,470 So here I'm going to cheat big time. 1271 01:10:43,470 --> 01:10:45,980 I'm going to pretend that all of this-- 1272 01:10:45,980 --> 01:10:48,530 the ray bends, of course. 1273 01:10:48,530 --> 01:10:52,490 But I'm going to pretend that throughout all this travel 1274 01:10:52,490 --> 01:10:56,900 here, the ray sees the same index of refraction. 1275 01:10:56,900 --> 01:11:00,450 This is a gross approximation, but I will do it. 1276 01:11:00,450 --> 01:11:02,120 So therefore, this is-- 1277 01:11:02,120 --> 01:11:05,780 so n0, another name for it is-- 1278 01:11:05,780 --> 01:11:07,190 it is also equal to n max. 1279 01:11:10,190 --> 01:11:14,448 The optical part for n1 will be equal n max. 1280 01:11:14,448 --> 01:11:18,690 1 minus alpha r square over 2, which 1281 01:11:18,690 --> 01:11:22,440 is the index of refraction at the elevated position. 1282 01:11:22,440 --> 01:11:24,870 And then, as you pointed out, it will 1283 01:11:24,870 --> 01:11:28,050 go through this increased distance, which is 1284 01:11:28,050 --> 01:11:32,760 the hypotenuse of the triangle. 1285 01:11:32,760 --> 01:11:37,450 So it is really r squared plus f squared. 1286 01:11:37,450 --> 01:11:42,840 So again, I pretended that this r and this r are the same. 1287 01:11:42,840 --> 01:11:45,970 I want to emphasize that this is an approximation. 1288 01:11:45,970 --> 01:11:47,680 I'm cheating. 1289 01:11:47,680 --> 01:11:50,340 OK. 1290 01:11:50,340 --> 01:11:52,288 The next is actually easier from now on. 1291 01:11:52,288 --> 01:11:54,580 All I'm going to do-- and the reason I'm going to do it 1292 01:11:54,580 --> 01:11:56,070 is to remind you. 1293 01:11:56,070 --> 01:11:58,650 I'm going to do the paraxial approximation, which basically 1294 01:11:58,650 --> 01:12:01,360 says that r is much less than f. 1295 01:12:01,360 --> 01:12:06,720 So that means that square root can be written as-- 1296 01:12:09,480 --> 01:12:12,000 I pull the f outside. 1297 01:12:12,000 --> 01:12:17,040 1 plus r square over f square approximately equal f1 1298 01:12:17,040 --> 01:12:20,730 plus r square over 2f square. 1299 01:12:20,730 --> 01:12:31,670 And this is actually equal to f plus r square over 2f. 1300 01:12:36,100 --> 01:12:42,430 So now let's write down Fermat's principle. 1301 01:12:42,430 --> 01:12:48,140 Fermat says that OPL 0 must be equal to OPL 1. 1302 01:12:48,140 --> 01:12:56,650 So OPL 0 was n max times d plus f. 1303 01:12:56,650 --> 01:13:00,040 And OPL 1 was-- 1304 01:13:00,040 --> 01:13:02,890 again, if you look at it from this equation, 1305 01:13:02,890 --> 01:13:07,540 OPL 1 was this expression times d, of course. 1306 01:13:07,540 --> 01:13:09,670 I forgot to put the d here. 1307 01:13:09,670 --> 01:13:12,190 Times this distance. 1308 01:13:12,190 --> 01:13:21,010 So it is equal to n max 1 minus alpha r square over 2 times 1309 01:13:21,010 --> 01:13:25,510 d plus f plus-- 1310 01:13:25,510 --> 01:13:31,060 what was it-- r square over 2f. 1311 01:13:31,060 --> 01:13:32,670 Now a number of things will cancel. 1312 01:13:32,670 --> 01:13:36,020 This will eat this one. 1313 01:13:36,020 --> 01:13:39,750 This will eat this one. 1314 01:13:39,750 --> 01:13:47,490 And all that is left is that n max times alpha r 1315 01:13:47,490 --> 01:13:56,130 square over 2d with a minus sign plus r square over 2f equals 0. 1316 01:13:56,130 --> 01:13:58,920 And now we can see that the quadratic dependence 1317 01:13:58,920 --> 01:14:01,080 for the index of refraction that we draw 1318 01:14:01,080 --> 01:14:04,290 is convenient because the same quadratic came out 1319 01:14:04,290 --> 01:14:07,570 of the paraxial approximation of the square root. 1320 01:14:07,570 --> 01:14:10,830 So I can just cancel the two quadratics. 1321 01:14:10,830 --> 01:14:13,470 And the twos also cancel. 1322 01:14:13,470 --> 01:14:16,950 And I get that f equals the inverse 1323 01:14:16,950 --> 01:14:23,690 of all of that, 1 over n max times alpha times d. 1324 01:14:23,690 --> 01:14:26,600 This is how we find the focal length of the GRIN. 1325 01:14:26,600 --> 01:14:29,420 So this appears a little bit mysterious what happened here. 1326 01:14:29,420 --> 01:14:31,430 How did I pull this out of the hat? 1327 01:14:31,430 --> 01:14:32,640 Well, what it really means-- 1328 01:14:32,640 --> 01:14:35,180 the fact that this has a solution 1329 01:14:35,180 --> 01:14:39,260 means that this element actually succeeds in focusing. 1330 01:14:39,260 --> 01:14:40,650 What I mean is the following. 1331 01:14:40,650 --> 01:14:42,800 Suppose that what came out of this equation 1332 01:14:42,800 --> 01:14:45,830 was not the nice quadratic that canceled, but suppose 1333 01:14:45,830 --> 01:14:46,950 I got something like this. 1334 01:14:46,950 --> 01:14:51,680 Suppose I got minus n max alpha r cube, for example, 1335 01:14:51,680 --> 01:14:58,820 over 2 times d plus r square over 2f equals 0. 1336 01:14:58,820 --> 01:15:01,130 Well, if this is what I had gotten, I'm out of luck. 1337 01:15:01,130 --> 01:15:03,170 This is not a proper lens. 1338 01:15:03,170 --> 01:15:04,940 So what I'm trying to say is suppose 1339 01:15:04,940 --> 01:15:09,020 that I made the GRIN with a cubic dependence 1340 01:15:09,020 --> 01:15:11,410 of the index of refraction, another quadratic. 1341 01:15:11,410 --> 01:15:14,030 Then this is the question I would have arrived instead. 1342 01:15:14,030 --> 01:15:16,580 Now I cannot cancel r anymore. 1343 01:15:16,580 --> 01:15:21,140 So it means that each ray goes to a different focal point. 1344 01:15:21,140 --> 01:15:23,390 So therefore, this element does not focus. 1345 01:15:23,390 --> 01:15:27,560 So this would not be a good lens. 1346 01:15:27,560 --> 01:15:29,210 Actually, the cubic dependence has 1347 01:15:29,210 --> 01:15:31,250 some other interesting properties, 1348 01:15:31,250 --> 01:15:37,760 but these are perhaps for a more advanced class. 1349 01:15:37,760 --> 01:15:39,380 Nevertheless, the quadratic dependence 1350 01:15:39,380 --> 01:15:41,690 is very convenient because as you can see, 1351 01:15:41,690 --> 01:15:43,010 it drops out of here. 1352 01:15:43,010 --> 01:15:45,500 It means that I have a unique focal length 1353 01:15:45,500 --> 01:15:47,960 at which all the rays focus, at least 1354 01:15:47,960 --> 01:15:49,610 in the paraxial approximation. 1355 01:15:52,230 --> 01:15:56,100 So that is the story of the quadratic GRIN, 1356 01:15:56,100 --> 01:16:01,460 how it comes about, and how we derive its focal length. 1357 01:16:01,460 --> 01:16:05,000 And I should also say that even though my derivation here 1358 01:16:05,000 --> 01:16:10,500 was cheating, basically the way I 1359 01:16:10,500 --> 01:16:14,840 cheated is I assumed that d, the thickness of the lens, 1360 01:16:14,840 --> 01:16:16,250 is also very small. 1361 01:16:16,250 --> 01:16:19,880 So it is very similar to the thin lens approximation. 1362 01:16:19,880 --> 01:16:23,090 Nevertheless, this equation is actually true, even 1363 01:16:23,090 --> 01:16:24,590 for thicker lenses. 1364 01:16:24,590 --> 01:16:27,280 So that's kind of interesting. 1365 01:16:27,280 --> 01:16:29,170 OK. 1366 01:16:29,170 --> 01:16:32,380 The second way people make GRIN optics is actually 1367 01:16:32,380 --> 01:16:35,730 a combination of melding and grinding. 1368 01:16:35,730 --> 01:16:40,240 And the way they do that results in a different profile 1369 01:16:40,240 --> 01:16:46,450 that now the index of refraction varies along the longitudinal-- 1370 01:16:46,450 --> 01:16:49,050 the axial dimension. 1371 01:16:49,050 --> 01:16:51,460 So what does this mean? 1372 01:16:51,460 --> 01:16:54,790 What they do first is they start with a stack 1373 01:16:54,790 --> 01:16:58,630 of glasses of different indices over refraction. 1374 01:16:58,630 --> 01:17:04,210 So here and in the rest of this lecture, the sort of darkness 1375 01:17:04,210 --> 01:17:06,020 indicates index of refraction. 1376 01:17:06,020 --> 01:17:09,850 So dark gray means high index. 1377 01:17:09,850 --> 01:17:12,520 Light gray means low index. 1378 01:17:12,520 --> 01:17:14,470 So they stack them. 1379 01:17:14,470 --> 01:17:17,050 Then they meld them, which basically 1380 01:17:17,050 --> 01:17:21,810 means that they mold and meld. 1381 01:17:25,480 --> 01:17:29,400 So after they meld them, basically 1382 01:17:29,400 --> 01:17:31,300 the index of refraction, the variation 1383 01:17:31,300 --> 01:17:37,000 becomes kind of continuous because as they meld, 1384 01:17:37,000 --> 01:17:39,680 the different indices will kind of diffuse into each. 1385 01:17:39,680 --> 01:17:42,880 So you will get a quasi-continuous variational 1386 01:17:42,880 --> 01:17:43,930 index. 1387 01:17:43,930 --> 01:17:47,290 And the final thing they do is they actually polish. 1388 01:17:47,290 --> 01:17:50,290 This stack that they've got with a continuous index variation, 1389 01:17:50,290 --> 01:17:51,913 they polish it. 1390 01:17:51,913 --> 01:17:53,830 So now this thing actually has two properties. 1391 01:17:53,830 --> 01:17:55,950 First of all, it is a lens in a traditional sense 1392 01:17:55,950 --> 01:18:02,070 because it has a spherical surface, so it focuses 1393 01:18:02,070 --> 01:18:04,080 by virtue of refraction. 1394 01:18:04,080 --> 01:18:07,390 But in addition, it has an axial index profile. 1395 01:18:07,390 --> 01:18:10,150 So here the motivation is actually 1396 01:18:10,150 --> 01:18:14,550 a little bit different than what Pepe and I described earlier. 1397 01:18:14,550 --> 01:18:18,550 So clearly I did not end up with a slab-looking element. 1398 01:18:18,550 --> 01:18:20,640 I still have a curvature. 1399 01:18:20,640 --> 01:18:22,620 But the motivation is different. 1400 01:18:22,620 --> 01:18:25,500 It's a little bit similar to-- 1401 01:18:25,500 --> 01:18:27,240 in the previous lecture I think we 1402 01:18:27,240 --> 01:18:30,780 mentioned something called the Smith correction or the Smith 1403 01:18:30,780 --> 01:18:32,220 telescope. 1404 01:18:32,220 --> 01:18:33,900 So it's a very similar motivation. 1405 01:18:33,900 --> 01:18:40,200 If you take a uniform index sphere, 1406 01:18:40,200 --> 01:18:42,870 we saw this a couple of times that in general, this 1407 01:18:42,870 --> 01:18:49,230 produces an aberrated image, even if it is oriented 1408 01:18:49,230 --> 01:18:49,840 properly. 1409 01:18:49,840 --> 01:18:53,340 This is supposed to be a plane or convex lens oriented 1410 01:18:53,340 --> 01:18:54,750 in the correct way. 1411 01:18:54,750 --> 01:18:58,140 That is the plane wave goes with the spherical surface. 1412 01:18:58,140 --> 01:19:00,150 But still, this does not completely 1413 01:19:00,150 --> 01:19:01,160 cancel the aberration. 1414 01:19:01,160 --> 01:19:04,390 There is still some residue. 1415 01:19:04,390 --> 01:19:09,320 So what do you do then, if you modify the index of refraction 1416 01:19:09,320 --> 01:19:15,400 in the interior of the surface, what happens is this ray-- 1417 01:19:15,400 --> 01:19:17,650 let's take the paraxial ray over here. 1418 01:19:17,650 --> 01:19:22,240 It hits at a relatively high index, whereas this ray, 1419 01:19:22,240 --> 01:19:25,300 it hits at a relatively lower index. 1420 01:19:25,300 --> 01:19:27,220 What this means, if you want to look 1421 01:19:27,220 --> 01:19:29,770 at it from the geometrical optics point of view-- 1422 01:19:29,770 --> 01:19:32,740 there's also a wave optics point of view that is interesting, 1423 01:19:32,740 --> 01:19:34,855 but from the geometrical optics point of view, 1424 01:19:34,855 --> 01:19:37,420 it means that the Snell's law of refraction 1425 01:19:37,420 --> 01:19:42,610 here will be milder for this ray than for this ray. 1426 01:19:42,610 --> 01:19:46,570 Therefore, this ray, who's used to focus kind of violently 1427 01:19:46,570 --> 01:19:49,300 and produce a spherical aberration here, 1428 01:19:49,300 --> 01:19:52,570 now it will be pulled outwards. 1429 01:19:52,570 --> 01:19:55,510 And therefore, it will be forced to go 1430 01:19:55,510 --> 01:20:00,130 through the proper focal point over here as the paraxial lens. 1431 01:20:00,130 --> 01:20:04,730 So this is a clever way to fix spherical aberration. 1432 01:20:04,730 --> 01:20:06,520 And I don't know who invented this first. 1433 01:20:06,520 --> 01:20:09,280 I believe-- and maybe, Colin, you again correct with this. 1434 01:20:09,280 --> 01:20:13,270 I believe in was Duncan Moore at the Rochester 1435 01:20:13,270 --> 01:20:17,520 who came up with this particular-- 1436 01:20:17,520 --> 01:20:19,450 at least I've seen a paper of his 1437 01:20:19,450 --> 01:20:21,910 which is as old as I am, 1971, that 1438 01:20:21,910 --> 01:20:24,765 discusses this optimization. 1439 01:20:30,250 --> 01:20:35,210 So in order to solve the basic principles of GRINs, 1440 01:20:35,210 --> 01:20:38,125 I would like to spend a little bit of time-- 1441 01:20:38,125 --> 01:20:39,500 I don't think we'll finish today. 1442 01:20:39,500 --> 01:20:44,530 We'll probably have to leak into next Wednesday after the quiz. 1443 01:20:44,530 --> 01:20:50,350 But I'll get us started on, how do we properly compute things 1444 01:20:50,350 --> 01:20:51,220 like index? 1445 01:20:51,220 --> 01:20:57,190 I mean, like focal length for a gradient index structure. 1446 01:20:57,190 --> 01:21:01,180 So we'd like to ask the question more generally. 1447 01:21:01,180 --> 01:21:06,250 If I have a medium whose index of refraction 1448 01:21:06,250 --> 01:21:08,590 varies as a function of position. 1449 01:21:08,590 --> 01:21:12,120 So here, this cloud is meant to indicate a variable index. 1450 01:21:12,120 --> 01:21:15,120 So you have high index, low index, in different positions. 1451 01:21:15,120 --> 01:21:18,330 So this is like a generalized GRIN if you wish. 1452 01:21:18,330 --> 01:21:19,370 We saw two types. 1453 01:21:19,370 --> 01:21:23,070 You saw quadratic and you saw axial. 1454 01:21:23,070 --> 01:21:24,270 But it could be really-- 1455 01:21:24,270 --> 01:21:26,490 in principle, I should be able to fabricate something 1456 01:21:26,490 --> 01:21:26,990 like this. 1457 01:21:26,990 --> 01:21:30,540 Not that it is readily doable, but let's 1458 01:21:30,540 --> 01:21:33,710 consider it for the sake of value. 1459 01:21:33,710 --> 01:21:35,340 So you've seen this diagram before. 1460 01:21:35,340 --> 01:21:37,560 In fact, I copied this from an earlier slide. 1461 01:21:37,560 --> 01:21:39,500 One of the previous-- 1462 01:21:39,500 --> 01:21:42,980 I think it was in the first lecture or second lecture. 1463 01:21:42,980 --> 01:21:46,490 So we don't know really what path 1464 01:21:46,490 --> 01:21:49,970 a ray will follow through these variable index medium, 1465 01:21:49,970 --> 01:21:51,050 but we know one thing. 1466 01:21:51,050 --> 01:21:55,160 We know that if you integrate this quantity, if you integrate 1467 01:21:55,160 --> 01:22:00,530 the index of refraction times the length, 1468 01:22:00,530 --> 01:22:03,680 the elemental length along the path of the ray, 1469 01:22:03,680 --> 01:22:07,710 we know that Fermat requires that these quantities will 1470 01:22:07,710 --> 01:22:08,790 be minimized. 1471 01:22:14,440 --> 01:22:17,940 So how do we deal with that? 1472 01:22:17,940 --> 01:22:20,240 I mean, what can we do about this? 1473 01:22:20,240 --> 01:22:26,080 Now, some of you who have taken advanced mechanics classes, 1474 01:22:26,080 --> 01:22:29,010 this sounds a little bit familiar. 1475 01:22:29,010 --> 01:22:32,130 What is the name that comes to mind when you 1476 01:22:32,130 --> 01:22:33,790 see this kind of expression? 1477 01:22:33,790 --> 01:22:35,310 So let me repeat again. 1478 01:22:35,310 --> 01:22:37,740 I have a situation where something-- 1479 01:22:37,740 --> 01:22:39,360 in this case, a light ray, but it 1480 01:22:39,360 --> 01:22:41,730 could be something else, a particle 1481 01:22:41,730 --> 01:22:45,243 or some mechanical analogy-- 1482 01:22:45,243 --> 01:22:47,160 can follow a number of different trajectories. 1483 01:22:47,160 --> 01:22:53,540 It can go this way or it can go this way or this way and so on. 1484 01:22:53,540 --> 01:22:55,840 To give you another sort of trivial example, 1485 01:22:55,840 --> 01:23:00,040 if I drop my marker here, it goes down straight. 1486 01:23:00,040 --> 01:23:03,540 So you may have wondered, well, why does it go down straight? 1487 01:23:03,540 --> 01:23:05,320 Why does it not do something like this 1488 01:23:05,320 --> 01:23:07,080 or like this or whatever? 1489 01:23:07,080 --> 01:23:09,630 Why does it not go up, for that matter? 1490 01:23:09,630 --> 01:23:11,550 Well, it goes straight because-- 1491 01:23:11,550 --> 01:23:15,110 well, because gravity pulls it down right. 1492 01:23:15,110 --> 01:23:17,280 Well, we're very happy with that. 1493 01:23:17,280 --> 01:23:21,300 We know from experience that things fall straight. 1494 01:23:21,300 --> 01:23:23,310 But we also know that gravity does not always 1495 01:23:23,310 --> 01:23:25,260 produce straight trajectories. 1496 01:23:25,260 --> 01:23:27,930 For example, the Earth. 1497 01:23:27,930 --> 01:23:31,320 Thankfully, the Earth follows a circular, 1498 01:23:31,320 --> 01:23:34,657 or actually elliptical, trajectory around the sun. 1499 01:23:34,657 --> 01:23:36,990 And that's very fortunate because if the Earth had a way 1500 01:23:36,990 --> 01:23:38,970 to escape from the sun, then we would all be-- 1501 01:23:38,970 --> 01:23:41,940 we would not even exist. 1502 01:23:41,940 --> 01:23:47,510 So we know that gravity can produce straight trajectories. 1503 01:23:47,510 --> 01:23:52,520 It can produce elliptical trajectories, 1504 01:23:52,520 --> 01:23:56,530 hyperbolical trajectories, and so on and so forth. 1505 01:23:56,530 --> 01:24:00,250 So there's more to it than the straight falling 1506 01:24:00,250 --> 01:24:08,022 of the apple in Newton's case or of the marker in my case. 1507 01:24:08,022 --> 01:24:10,230 So anyway, so the reason I'm giving all this preamble 1508 01:24:10,230 --> 01:24:13,740 is because this may be familiar for you 1509 01:24:13,740 --> 01:24:16,470 from the field of mechanics. 1510 01:24:16,470 --> 01:24:18,960 And the buzzword that comes to mind 1511 01:24:18,960 --> 01:24:22,590 when you have this kind of integral with something inside 1512 01:24:22,590 --> 01:24:26,620 and you demand that the integral itself is maximized 1513 01:24:26,620 --> 01:24:28,440 is Lagrangian. 1514 01:24:28,440 --> 01:24:32,910 It is the Lagrangian principle that you may have seen. 1515 01:24:32,910 --> 01:24:37,655 I don't think those of you took 2003 saw it. 1516 01:24:37,655 --> 01:24:39,030 Did you or did you not, actually? 1517 01:24:39,030 --> 01:24:41,390 I don't know. 1518 01:24:41,390 --> 01:24:43,225 Is anybody in the class who took 2003? 1519 01:24:43,225 --> 01:24:46,000 You should because 2004 is a prerequisite. 1520 01:24:46,000 --> 01:24:47,500 And as a principle, if you are here, 1521 01:24:47,500 --> 01:24:57,120 it means you have taken both 2003 and 2004. 1522 01:24:57,120 --> 01:25:02,523 We have lost our undergraduates, or they lost their voice, 1523 01:25:02,523 --> 01:25:03,440 or they're too sleepy. 1524 01:25:03,440 --> 01:25:05,615 Anyway. 1525 01:25:05,615 --> 01:25:07,240 Or there's another possibility, that we 1526 01:25:07,240 --> 01:25:13,550 don't remember if we were taught Lagrangians in 2003. 1527 01:25:13,550 --> 01:25:15,610 Anyway. 1528 01:25:15,610 --> 01:25:16,860 But I will not do it this way. 1529 01:25:16,860 --> 01:25:20,760 So it turns out I can solve this problem using a Lagrangian 1530 01:25:20,760 --> 01:25:21,820 formulation. 1531 01:25:21,820 --> 01:25:23,070 But I will not do it this way. 1532 01:25:23,070 --> 01:25:25,420 Did someone want to speak? 1533 01:25:25,420 --> 01:25:26,420 I saw the camera moving. 1534 01:25:26,420 --> 01:25:29,785 Was someone about to speak? 1535 01:25:29,785 --> 01:25:31,410 AUDIENCE: I was just going to say, yes, 1536 01:25:31,410 --> 01:25:32,680 we did have Lagrangian. 1537 01:25:32,680 --> 01:25:34,060 GEORGE BARBASTATHIS: Ah, you did. 1538 01:25:34,060 --> 01:25:34,560 OK. 1539 01:25:34,560 --> 01:25:36,221 Thank you. 1540 01:25:36,221 --> 01:25:37,155 Thank you. 1541 01:25:42,120 --> 01:25:44,440 Well, even if you did, I guess I will not 1542 01:25:44,440 --> 01:25:47,230 do it with the Lagrangians because, well, the math is 1543 01:25:47,230 --> 01:25:54,080 too complicated and would kind of cause a detour here. 1544 01:25:54,080 --> 01:25:56,250 Nevertheless, I will use a mechanical analogy. 1545 01:25:56,250 --> 01:25:58,850 So for a moment, we'll take a break from optics. 1546 01:25:58,850 --> 01:26:02,660 And I guess to justify why 2004 is 1547 01:26:02,660 --> 01:26:06,800 a prerequisite for this class, I will show this. 1548 01:26:06,800 --> 01:26:09,350 And I will spend, I guess, the next 15 or so minutes 1549 01:26:09,350 --> 01:26:12,410 to establish a connection between this mechanical system 1550 01:26:12,410 --> 01:26:14,940 and light rays. 1551 01:26:14,940 --> 01:26:16,905 For now, let's forget about light rays. 1552 01:26:16,905 --> 01:26:19,860 I've given you-- it's actually a particle. 1553 01:26:19,860 --> 01:26:22,570 So it looks like a car but it's really small. 1554 01:26:22,570 --> 01:26:25,590 So I don't have to worry about moments of inertia 1555 01:26:25,590 --> 01:26:28,480 and all of that stuff. 1556 01:26:28,480 --> 01:26:31,720 It is also frictionless, so I don't have any dissipation 1557 01:26:31,720 --> 01:26:33,440 at all in the system. 1558 01:26:33,440 --> 01:26:36,400 This particle is attached to a rigid-- 1559 01:26:36,400 --> 01:26:43,660 to an immobile, rigid wall with a spring. 1560 01:26:43,660 --> 01:26:45,520 And basically what I can do is I can 1561 01:26:45,520 --> 01:26:49,870 inject some energy into the particles by pulling it. 1562 01:26:49,870 --> 01:26:53,620 So if I do that, then we know that this system 1563 01:26:53,620 --> 01:26:56,860 has two forms of energy. 1564 01:26:56,860 --> 01:26:59,530 When I actually pull it, what I do is I 1565 01:26:59,530 --> 01:27:02,410 store energy in the spring. 1566 01:27:02,410 --> 01:27:05,530 The spring is not happy having its energy stored. 1567 01:27:05,530 --> 01:27:07,450 It will want to give it away. 1568 01:27:07,450 --> 01:27:09,900 What does it mean to give away the energy? 1569 01:27:09,900 --> 01:27:11,470 It means to convert it to kinetic. 1570 01:27:11,470 --> 01:27:15,250 It means the spring will start pulling the particle back. 1571 01:27:15,250 --> 01:27:18,340 As the particle is pulled back, it will accelerate. 1572 01:27:18,340 --> 01:27:20,480 It will reach a maximum velocity-- 1573 01:27:20,480 --> 01:27:23,320 we know that from classical mechanics-- 1574 01:27:23,320 --> 01:27:28,210 approximately where the spring has zero displacement. 1575 01:27:28,210 --> 01:27:29,770 And at this point, all of the energy 1576 01:27:29,770 --> 01:27:32,170 has been converted to kinetic. 1577 01:27:32,170 --> 01:27:34,868 And of course it will not stop there because it is moving, 1578 01:27:34,868 --> 01:27:36,160 and there's nothing to stop it. 1579 01:27:36,160 --> 01:27:37,630 It will keep moving. 1580 01:27:37,630 --> 01:27:39,190 But if it moves, now it is beginning 1581 01:27:39,190 --> 01:27:41,740 to push, to squeeze this spring. 1582 01:27:41,740 --> 01:27:44,830 So now the particle is giving back its energy 1583 01:27:44,830 --> 01:27:48,900 into the spring until all of the energy has been given. 1584 01:27:48,900 --> 01:27:50,760 Now the spring is squeezed. 1585 01:27:50,760 --> 01:27:52,020 Doesn't like to be squeezed. 1586 01:27:52,020 --> 01:27:56,100 It starts giving energy back to the particle. 1587 01:27:56,100 --> 01:27:57,810 And of course, on and on they go. 1588 01:27:57,810 --> 01:28:00,810 This will execute and oscillate in motion 1589 01:28:00,810 --> 01:28:07,210 forever because I neglected all dissipation. 1590 01:28:07,210 --> 01:28:09,490 And of course, what is happening in this case is 1591 01:28:09,490 --> 01:28:13,860 that at any given moment, the particle will have a mix-- 1592 01:28:13,860 --> 01:28:18,640 the system will have a mix of kinetic energy and energy 1593 01:28:18,640 --> 01:28:21,550 stored into the spring, which is referred 1594 01:28:21,550 --> 01:28:23,960 to as potential energy. 1595 01:28:23,960 --> 01:28:27,270 So of course, the kinetic energy is given by 1/2 times 1596 01:28:27,270 --> 01:28:29,520 the mass times the velocity squared. 1597 01:28:29,520 --> 01:28:31,920 If you remember the definition of momentum, 1598 01:28:31,920 --> 01:28:35,050 momentum equals mass times velocity. 1599 01:28:35,050 --> 01:28:36,960 So I can also write it like this. 1600 01:28:36,960 --> 01:28:40,470 Momentum squared over 2 times the mass. 1601 01:28:40,470 --> 01:28:41,910 And the potential energy-- 1602 01:28:41,910 --> 01:28:45,240 actually, the potential energy can have very different forms 1603 01:28:45,240 --> 01:28:49,500 depending on what is exactly the energy storage element. 1604 01:28:49,500 --> 01:28:50,960 In the case of a spring-- 1605 01:28:50,960 --> 01:28:54,230 actually, in the case of a hook spring, a linear spring, 1606 01:28:54,230 --> 01:28:58,110 it is simply equal to the square of the displacement 1607 01:28:58,110 --> 01:29:03,400 of the spring with respect to its rest position. 1608 01:29:03,400 --> 01:29:05,800 So anyway, no matter what these things are, 1609 01:29:05,800 --> 01:29:08,620 the sum must be constant because, again, 1610 01:29:08,620 --> 01:29:11,380 energy cannot be gained or lost by its system. 1611 01:29:11,380 --> 01:29:13,960 So the sum of the potential and kinetic energy 1612 01:29:13,960 --> 01:29:17,340 is always conserved. 1613 01:29:17,340 --> 01:29:19,530 So that is one thing, and it's very good. 1614 01:29:22,250 --> 01:29:26,900 Now, let me play a game. 1615 01:29:30,190 --> 01:29:38,595 So first of all, I will write the energy. 1616 01:29:38,595 --> 01:29:40,470 I will basically repeat what is on the slide. 1617 01:29:40,470 --> 01:29:44,100 I will repeat it on the whiteboard here. 1618 01:29:44,100 --> 01:29:45,658 So let me write the energy. 1619 01:29:45,658 --> 01:29:46,950 I will write it in a funny way. 1620 01:29:46,950 --> 01:29:52,430 I will write it as a function of the momentum p and the position 1621 01:29:52,430 --> 01:29:53,160 q. 1622 01:29:53,160 --> 01:29:55,460 And now, in the previous slide, these 1623 01:29:55,460 --> 01:30:00,920 were actually scalar quantities because this thing moves only 1624 01:30:00,920 --> 01:30:03,180 along a straight line. 1625 01:30:03,180 --> 01:30:07,130 But in general, of course, I can have positions and momenta 1626 01:30:07,130 --> 01:30:09,110 that are vectors. 1627 01:30:09,110 --> 01:30:10,500 So I switched to vector notation. 1628 01:30:10,500 --> 01:30:13,880 I hope that doesn't confuse you horribly. 1629 01:30:13,880 --> 01:30:14,900 So let me write it down. 1630 01:30:14,900 --> 01:30:20,510 So in the vector case, this will equal the momentum vector 1631 01:30:20,510 --> 01:30:27,020 divided by 2m plus 1/2 m-- 1632 01:30:27,020 --> 01:30:27,700 wait a minute. 1633 01:30:27,700 --> 01:30:36,250 1/2 k, the spring constant, times the position vector 1634 01:30:36,250 --> 01:30:38,400 square. 1635 01:30:38,400 --> 01:30:40,890 Now what I'm going to do, for fun, 1636 01:30:40,890 --> 01:30:45,000 is I'm going to take derivatives of this quantity with respect 1637 01:30:45,000 --> 01:30:48,270 to the position and the momentum. 1638 01:30:48,270 --> 01:30:50,340 Of course, since these became vectors, 1639 01:30:50,340 --> 01:30:52,080 now these are not exactly derivatives, 1640 01:30:52,080 --> 01:30:54,000 they are gradients. 1641 01:30:54,000 --> 01:31:00,570 So basically what I will do is I will compute the gradient of H 1642 01:31:00,570 --> 01:31:01,830 with respect to p. 1643 01:31:08,010 --> 01:31:09,240 This is a matter of notation. 1644 01:31:09,240 --> 01:31:15,090 I will denote it as dH dp. 1645 01:31:15,090 --> 01:31:16,440 So what is dH dp? 1646 01:31:16,440 --> 01:31:24,600 Well, what I have to do is I have to compute-- 1647 01:31:24,600 --> 01:31:28,620 clearly this term does not play because it only depends on q. 1648 01:31:28,620 --> 01:31:31,540 But this term contains the momentum. 1649 01:31:31,540 --> 01:31:33,970 So if I do it, it will actually-- you can do it. 1650 01:31:33,970 --> 01:31:38,770 It is actually p times its-- 1651 01:31:43,020 --> 01:31:44,900 p over m. 1652 01:31:44,900 --> 01:31:45,400 That's it. 1653 01:31:45,400 --> 01:31:45,942 Nothing else. 1654 01:31:48,370 --> 01:31:54,038 And then I'm going to do the same with respect to q. 1655 01:31:56,970 --> 01:32:01,220 And I'm going to call it dH dq. 1656 01:32:01,220 --> 01:32:04,130 Again, it is a gradient with respect to q. 1657 01:32:04,130 --> 01:32:06,050 And for the same reasons as the other one, 1658 01:32:06,050 --> 01:32:08,350 it will turn out to be k times 2. 1659 01:32:11,720 --> 01:32:12,220 OK. 1660 01:32:12,220 --> 01:32:14,630 Then again for fun-- 1661 01:32:14,630 --> 01:32:16,980 I'm not justifying what I'm doing. 1662 01:32:16,980 --> 01:32:20,440 It's just coming sort of some-- 1663 01:32:20,440 --> 01:32:24,220 coming out of some sort of epiphany. 1664 01:32:24,220 --> 01:32:28,210 I will write down two differential equations. 1665 01:32:28,210 --> 01:32:30,540 One of them, I will write this one. 1666 01:32:30,540 --> 01:32:32,940 It is already on your screen there. 1667 01:32:32,940 --> 01:32:36,870 dq dt. 1668 01:32:36,870 --> 01:32:37,830 What is that? 1669 01:32:37,830 --> 01:32:40,730 That's the time derivative of the displacement. 1670 01:32:40,730 --> 01:32:43,140 Therefore, it is the velocity. 1671 01:32:43,140 --> 01:32:51,870 So we'll take it to be equal to the gradient of the quantity 1672 01:32:51,870 --> 01:32:58,010 H of the energy, that we called it H for some strange reason, 1673 01:32:58,010 --> 01:33:00,020 with respect to the opposite variable. 1674 01:33:00,020 --> 01:33:01,655 This was the derivative of q. 1675 01:33:01,655 --> 01:33:04,280 I will take its gradient with respect to p. 1676 01:33:07,620 --> 01:33:08,870 And let's see what I get. 1677 01:33:08,870 --> 01:33:14,920 This way I get p over m. 1678 01:33:14,920 --> 01:33:18,300 That's correct. 1679 01:33:18,300 --> 01:33:22,540 Actually, this equation is not particularly insightful. 1680 01:33:22,540 --> 01:33:25,770 This is the definition of momentum, isn't it? 1681 01:33:25,770 --> 01:33:32,910 If you recall-- this means recall, like in cartoons. 1682 01:33:32,910 --> 01:33:36,200 The little cloud. 1683 01:33:36,200 --> 01:33:43,410 The momentum is defined as m times dq dt. 1684 01:33:43,410 --> 01:33:45,750 That's the definition of momentum. 1685 01:33:45,750 --> 01:33:48,050 So by doing this trick, by taking 1686 01:33:48,050 --> 01:33:53,660 the gradient of the energy with respect to the momentum 1687 01:33:53,660 --> 01:34:00,920 and setting it equal to the velocity, I got a true result. 1688 01:34:00,920 --> 01:34:03,420 I got a result that happens to be true. 1689 01:34:03,420 --> 01:34:06,860 Now let's see if I can pull the opposite trick. 1690 01:34:06,860 --> 01:34:10,070 If I can take the derivative with respect 1691 01:34:10,070 --> 01:34:14,970 to time of the momentum, that's like an acceleration quantity 1692 01:34:14,970 --> 01:34:17,790 because momentum is velocity. 1693 01:34:17,790 --> 01:34:20,170 So its derivative is acceleration. 1694 01:34:20,170 --> 01:34:22,610 And let's see if I can get away by putting 1695 01:34:22,610 --> 01:34:27,550 it equal to the derivative of this quantity H 1696 01:34:27,550 --> 01:34:29,770 with respect to the other variable. 1697 01:34:29,770 --> 01:34:32,170 In this case, I cannot quite get away with it. 1698 01:34:32,170 --> 01:34:35,620 I have to put a minus sign. 1699 01:34:35,620 --> 01:34:37,460 And what do I get if I put a minus sign? 1700 01:34:37,460 --> 01:34:37,960 kq. 1701 01:34:41,853 --> 01:34:42,520 What's this now? 1702 01:34:48,510 --> 01:34:50,730 kq? 1703 01:34:50,730 --> 01:34:54,020 It is k times displacement. 1704 01:34:54,020 --> 01:34:59,020 So what are the units of Hooke's constant time displacement? 1705 01:34:59,020 --> 01:35:00,970 Force. 1706 01:35:00,970 --> 01:35:02,220 What is on the left-hand side? 1707 01:35:06,280 --> 01:35:06,780 OK. 1708 01:35:06,780 --> 01:35:07,800 I said acceleration. 1709 01:35:07,800 --> 01:35:09,840 Actually, I should have corrected myself, 1710 01:35:09,840 --> 01:35:13,120 and I should have said, proportional to acceleration. 1711 01:35:13,120 --> 01:35:15,220 This is the derivative of the momentum. 1712 01:35:15,220 --> 01:35:17,900 So therefore, it is force. 1713 01:35:17,900 --> 01:35:19,980 So what is this then, the equation that I got? 1714 01:35:23,820 --> 01:35:26,960 It is the force balance on the-- 1715 01:35:26,960 --> 01:35:30,000 it is Newton's law, actually, on the particle. 1716 01:35:30,000 --> 01:35:34,170 It says that the force pulling the particle, 1717 01:35:34,170 --> 01:35:38,300 it is the restoring force due to the spring. 1718 01:35:38,300 --> 01:35:41,630 And what are the two equations combined? 1719 01:35:41,630 --> 01:35:43,650 Actually, the two equations combined, 1720 01:35:43,650 --> 01:35:47,020 they are what we call the equations of motion. 1721 01:35:47,020 --> 01:35:49,440 The equations of motion, by definition, 1722 01:35:49,440 --> 01:35:52,650 are first order differential equations 1723 01:35:52,650 --> 01:35:55,620 that link the dynamical variables that 1724 01:35:55,620 --> 01:36:00,250 is the position, the displacement, and the velocity. 1725 01:36:00,250 --> 01:36:01,270 So this is what I did. 1726 01:36:01,270 --> 01:36:03,960 I got a set of differential equations 1727 01:36:03,960 --> 01:36:08,430 that link the displacement to the momentum-- that 1728 01:36:08,430 --> 01:36:09,540 is the velocity. 1729 01:36:09,540 --> 01:36:14,370 And again, the derivative of the displacement back to the-- 1730 01:36:14,370 --> 01:36:16,120 I'm sorry, the derivative of the velocity, 1731 01:36:16,120 --> 01:36:19,200 the derivative of the momentum, back to the displacement 1732 01:36:19,200 --> 01:36:21,120 itself. 1733 01:36:21,120 --> 01:36:28,460 So if you look at it, if you omit this middle part, 1734 01:36:28,460 --> 01:36:31,210 if you look at it like this, these are simply-- 1735 01:36:31,210 --> 01:36:34,600 as I said, these are the equations of motion. 1736 01:36:34,600 --> 01:36:39,630 If you look at this part over here, this is of course 1737 01:36:39,630 --> 01:36:42,950 a more general result. That's why I bothered to do it. 1738 01:36:42,950 --> 01:36:44,700 And this is where the connection to optics 1739 01:36:44,700 --> 01:36:46,980 will come from next week. 1740 01:36:46,980 --> 01:36:50,550 This set of equations is called Hamiltonian. 1741 01:36:54,940 --> 01:36:58,190 And I will put in brackets "canonical." 1742 01:36:58,190 --> 01:36:59,790 I don't particularly like this term. 1743 01:36:59,790 --> 01:37:01,970 But anyway, some people use it. 1744 01:37:01,970 --> 01:37:05,539 Hamiltonian canonical equations. 1745 01:37:10,030 --> 01:37:12,410 So the point, then, which I will say just this, 1746 01:37:12,410 --> 01:37:14,760 and then I will stop and continue next time, 1747 01:37:14,760 --> 01:37:18,200 the point is that this is now a much more general property. 1748 01:37:18,200 --> 01:37:22,400 Even though I showed that it is true for the specific case 1749 01:37:22,400 --> 01:37:26,450 of a mass spring system without damper-- 1750 01:37:26,450 --> 01:37:27,800 I showed that to be true. 1751 01:37:27,800 --> 01:37:30,260 But this is a more general property 1752 01:37:30,260 --> 01:37:36,630 for any dynamical system that has a conserved quantity. 1753 01:37:36,630 --> 01:37:38,460 So in the case of the mechanical system, 1754 01:37:38,460 --> 01:37:40,900 the conserved quantity is energy. 1755 01:37:40,900 --> 01:37:42,670 If this conserved quantity can be 1756 01:37:42,670 --> 01:37:46,540 written as a function of the dynamical variables, 1757 01:37:46,540 --> 01:37:52,010 in this case, of the mechanical system position and momentum-- 1758 01:37:52,010 --> 01:37:52,510 OK. 1759 01:37:52,510 --> 01:37:54,880 So I started a long sentence. 1760 01:37:54,880 --> 01:37:57,460 Let me close my sentence. 1761 01:37:57,460 --> 01:38:02,170 So if a dynamical system has a conserved quantity, 1762 01:38:02,170 --> 01:38:07,880 then I can write down a set of Hamiltonian equations 1763 01:38:07,880 --> 01:38:09,770 for this dynamical system in terms 1764 01:38:09,770 --> 01:38:13,280 of the gradients of this conserved quantity. 1765 01:38:13,280 --> 01:38:16,490 And that will be the equations of motion. 1766 01:38:16,490 --> 01:38:17,520 OK. 1767 01:38:17,520 --> 01:38:20,270 So what we will do then next time 1768 01:38:20,270 --> 01:38:23,120 is we will actually show, in almost one 1769 01:38:23,120 --> 01:38:25,340 to one analogy, that-- 1770 01:38:25,340 --> 01:38:27,470 oh, and by the way, this conserved quantity 1771 01:38:27,470 --> 01:38:30,890 that we called it energy in the case of the mechanical system, 1772 01:38:30,890 --> 01:38:33,410 in general it's called the Hamiltonian. 1773 01:38:33,410 --> 01:38:38,540 So H itself is the Hamiltonian. 1774 01:38:38,540 --> 01:38:40,460 So what we'll do next is we will actually 1775 01:38:40,460 --> 01:38:47,060 show we can derive a Hamiltonian for rays for an optical system. 1776 01:38:47,060 --> 01:38:50,600 And of course, when we write equations of motion for rays, 1777 01:38:50,600 --> 01:38:53,550 we'll get the ray trajectories. 1778 01:38:53,550 --> 01:38:55,483 Therefore, we will be able to ray tracing. 1779 01:38:55,483 --> 01:38:56,900 And the reason this is interesting 1780 01:38:56,900 --> 01:39:00,410 because the potential actually will turn out 1781 01:39:00,410 --> 01:39:02,670 to be the index of refraction. 1782 01:39:02,670 --> 01:39:05,630 And in the same way, the potential here, 1783 01:39:05,630 --> 01:39:08,540 if you go back to my equation-- 1784 01:39:08,540 --> 01:39:12,930 the potential turned out to be a function of position. 1785 01:39:12,930 --> 01:39:16,860 In the gradient index optics that we mentioned just now, 1786 01:39:16,860 --> 01:39:19,820 the index of refraction is a function of position. 1787 01:39:19,820 --> 01:39:23,820 So we will see that in the case of the GRIN element, 1788 01:39:23,820 --> 01:39:27,000 the index of refraction actually has the role of a potential. 1789 01:39:27,000 --> 01:39:29,050 And it is this potential that draws the-- 1790 01:39:29,050 --> 01:39:30,750 very similar to gravity. 1791 01:39:30,750 --> 01:39:32,850 The potential that draws the sun-- 1792 01:39:32,850 --> 01:39:36,630 I mean, that draws the Earth to rotate around the sun and not 1793 01:39:36,630 --> 01:39:38,350 the other way around. 1794 01:39:38,350 --> 01:39:40,560 The same way the potential draws the Earth 1795 01:39:40,560 --> 01:39:43,710 to rotate around the sun, it is a similar, 1796 01:39:43,710 --> 01:39:46,170 or I should say analogous, kind of potential 1797 01:39:46,170 --> 01:39:49,980 that draws rays inwards in a GRIN element 1798 01:39:49,980 --> 01:39:53,610 and causes them to focus. 1799 01:39:53,610 --> 01:39:57,820 So we'll pick up on that next time. 1800 01:39:57,820 --> 01:40:00,970 And obviously, don't worry about this for the quiz. 1801 01:40:00,970 --> 01:40:03,570 So I'm not crazy. 1802 01:40:03,570 --> 01:40:05,040 So yeah. 1803 01:40:05,040 --> 01:40:08,350 So we'll properly conclude this topic next time. 1804 01:40:08,350 --> 01:40:10,500 It will be next week's homework.