1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,690 To make a donation, or to view additional materials 6 00:00:12,690 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,550 at ocw.mit.edu. 8 00:00:21,930 --> 00:00:27,070 GEORGE BARBASTATHIS: OK, so today I would like to cover-- 9 00:00:27,070 --> 00:00:29,070 to start, well, I will do a lot of things today. 10 00:00:29,070 --> 00:00:33,860 But to begin with, I would like to solve two applications 11 00:00:33,860 --> 00:00:35,960 of Fermat's principle. 12 00:00:35,960 --> 00:00:39,800 And the simplest possible thing that we try to do in optics, 13 00:00:39,800 --> 00:00:42,030 and that is to focus. 14 00:00:42,030 --> 00:00:44,670 So the one focus will come up again and again. 15 00:00:44,670 --> 00:00:46,440 It basically refers to the idea-- 16 00:00:46,440 --> 00:00:48,570 hi, Colin. 17 00:00:48,570 --> 00:00:54,150 It refers to the idea that we have rays, a bundle of rays 18 00:00:54,150 --> 00:00:56,610 that are propagating. 19 00:00:56,610 --> 00:00:59,840 And focusing means that we try to force these rays to meet 20 00:00:59,840 --> 00:01:01,342 at one point. 21 00:01:01,342 --> 00:01:03,300 So this, of course, is very useful for imaging, 22 00:01:03,300 --> 00:01:06,570 but even simpler, if you want to collect 23 00:01:06,570 --> 00:01:09,570 all of the light energy, for example, 24 00:01:09,570 --> 00:01:12,450 for a solar concentrator, or a satellite dish, 25 00:01:12,450 --> 00:01:14,998 or something like that. 26 00:01:14,998 --> 00:01:16,790 Then basically, this is what you try to do. 27 00:01:16,790 --> 00:01:19,900 You try to get the rays that are coming from the source, 28 00:01:19,900 --> 00:01:21,880 and focus them into a point. 29 00:01:21,880 --> 00:01:24,970 And this is valid, of course, in the entire regime 30 00:01:24,970 --> 00:01:27,390 of electromagnetic waves-- 31 00:01:27,390 --> 00:01:33,040 whether microwaves, or visible light, or even higher-- 32 00:01:33,040 --> 00:01:38,020 provided that the element that we use to focus the rays 33 00:01:38,020 --> 00:01:40,210 is much bigger than the wavelength. 34 00:01:40,210 --> 00:01:42,850 And here, I'm leaving it deliberately vague 35 00:01:42,850 --> 00:01:44,200 how much bigger. 36 00:01:44,200 --> 00:01:50,900 But think about it that is maybe a few thousand of wavelengths, 37 00:01:50,900 --> 00:01:52,690 or few hundred of wavelengths. 38 00:01:52,690 --> 00:01:55,610 In the microwave case, that's not exactly true. 39 00:01:55,610 --> 00:01:58,270 The RF wavelength maybe a couple of centimeters. 40 00:01:58,270 --> 00:02:01,330 The typical satellite dish is about the meter, 41 00:02:01,330 --> 00:02:03,790 so we're talking about 50 to 100 wavelength. 42 00:02:03,790 --> 00:02:07,480 But anyway, still the approximations are valid. 43 00:02:07,480 --> 00:02:11,140 So the simplest way to do this kind of operation 44 00:02:11,140 --> 00:02:15,700 to focus the light into a point so it can be detected 45 00:02:15,700 --> 00:02:22,210 is with a reflector, a mirror whose surface is curved. 46 00:02:22,210 --> 00:02:24,940 And you know from experience if you look at the spoon 47 00:02:24,940 --> 00:02:28,240 that, well, funny things happen if you look at the spoon. 48 00:02:28,240 --> 00:02:33,850 We will talk about that perhaps a little bit later. 49 00:02:33,850 --> 00:02:36,085 But you know that the spoon forms images, 50 00:02:36,085 --> 00:02:41,380 so this is basically a simplification or idealization 51 00:02:41,380 --> 00:02:43,010 of a spoon. 52 00:02:43,010 --> 00:02:45,880 So let's say that we have this surface. 53 00:02:45,880 --> 00:02:50,170 And for now, even though the label says parabolic reflector, 54 00:02:50,170 --> 00:02:53,180 I will pretend that I don't know the shape of the surface. 55 00:02:53,180 --> 00:02:56,740 So s of x is an unknown to be determined. 56 00:02:56,740 --> 00:03:00,530 And the goal, of course, is to have 57 00:03:00,530 --> 00:03:07,030 rays that are coming as parallel rays onto the parabol line, 58 00:03:07,030 --> 00:03:08,710 onto this unknown shape. 59 00:03:08,710 --> 00:03:11,320 And our goal is to focus these rays 60 00:03:11,320 --> 00:03:16,190 into a point on the axis of the surface. 61 00:03:16,190 --> 00:03:17,750 And what you see. 62 00:03:17,750 --> 00:03:24,047 You see here that I put the symbol of infinity. 63 00:03:24,047 --> 00:03:25,880 I will mention that also a little bit later. 64 00:03:25,880 --> 00:03:31,220 If we have rays that are parallel, you can think-- 65 00:03:31,220 --> 00:03:34,820 according to Euclidean geometry, parallel rays never meet. 66 00:03:34,820 --> 00:03:37,820 Or another way to say it is that parallel rays, if they ever 67 00:03:37,820 --> 00:03:40,760 meet, that would be at infinity, at a very, very long distance 68 00:03:40,760 --> 00:03:41,490 away. 69 00:03:41,490 --> 00:03:43,640 So when the object of the imaging system 70 00:03:43,640 --> 00:03:48,320 is very far away, such as the sun, or a star, 71 00:03:48,320 --> 00:03:51,140 or a really remote source. 72 00:03:51,140 --> 00:03:53,540 Then we can say that the ray is coming 73 00:03:53,540 --> 00:03:55,250 from that object that are parallel, 74 00:03:55,250 --> 00:03:58,020 or that are coming from infinity. 75 00:03:58,020 --> 00:04:00,070 So the objective here is, again, to determine 76 00:04:00,070 --> 00:04:02,940 the shape of the surface, s of x, such 77 00:04:02,940 --> 00:04:07,003 that the rays all focus at a point, f. 78 00:04:07,003 --> 00:04:08,670 So there's a number of ways of doing it. 79 00:04:08,670 --> 00:04:12,490 One is for example, I can apply the law of reflection anywhere 80 00:04:12,490 --> 00:04:13,120 in the surface. 81 00:04:13,120 --> 00:04:14,920 I have the array appearing here. 82 00:04:14,920 --> 00:04:17,500 I can compute the normal to the ray-- 83 00:04:17,500 --> 00:04:19,089 I'm sorry, the normal to the surface 84 00:04:19,089 --> 00:04:20,980 at that point of existence of the ray. 85 00:04:20,980 --> 00:04:22,600 And I can apply the law of reflection. 86 00:04:22,600 --> 00:04:24,383 I can solve the problem this way. 87 00:04:24,383 --> 00:04:26,800 But that is actually a very complicated way of solving it. 88 00:04:29,340 --> 00:04:30,780 Much work. 89 00:04:30,780 --> 00:04:32,500 So being, I guess, lazy. 90 00:04:35,010 --> 00:04:36,630 The best way to solve this problem 91 00:04:36,630 --> 00:04:39,000 is to invoke the Fermat's principle. 92 00:04:39,000 --> 00:04:41,790 And Fermat's principle says, again, to remind you, 93 00:04:41,790 --> 00:04:47,490 says that rays that depart from a common point 94 00:04:47,490 --> 00:04:50,130 and arrive at another point. 95 00:04:50,130 --> 00:04:54,550 They have to follow the minimum optical path. 96 00:04:54,550 --> 00:04:57,360 A corollary to that is that if you 97 00:04:57,360 --> 00:04:59,940 have two rays that start at the same point 98 00:04:59,940 --> 00:05:02,130 and meet again at the same point, 99 00:05:02,130 --> 00:05:04,800 then these rays must follow the same path. 100 00:05:04,800 --> 00:05:08,130 Because if one of those rays follows a shorter path, 101 00:05:08,130 --> 00:05:10,800 it means that the other ray that followed the longer path 102 00:05:10,800 --> 00:05:12,760 is violating Fermat's principle. 103 00:05:12,760 --> 00:05:14,590 And that cannot happen. 104 00:05:14,590 --> 00:05:17,280 So the way we're thinking about this here 105 00:05:17,280 --> 00:05:19,560 is to take, for example, the center 106 00:05:19,560 --> 00:05:25,170 ray that is going directly on axis, that started at infinity. 107 00:05:25,170 --> 00:05:26,340 We go to the reflector. 108 00:05:26,340 --> 00:05:28,440 It will be reflected exactly backwards, 109 00:05:28,440 --> 00:05:31,070 and will go through the focal point, f. 110 00:05:31,070 --> 00:05:33,840 If we take another ray that is away 111 00:05:33,840 --> 00:05:38,350 from the central axis of the unknown surface. 112 00:05:38,350 --> 00:05:40,800 Then this ray also started at infinity. 113 00:05:40,800 --> 00:05:44,340 It gets reflected, and then, again, meets at that point, f. 114 00:05:44,340 --> 00:05:47,990 So therefore, the pattern that these two rays followed 115 00:05:47,990 --> 00:05:51,020 must be the same. 116 00:05:51,020 --> 00:05:53,130 So how do we compute this path? 117 00:05:53,130 --> 00:05:56,290 Well, if you look at the central ray. 118 00:06:01,960 --> 00:06:04,300 The central ray is coming from infinity. 119 00:06:04,300 --> 00:06:07,650 I will use the plane passing through the focal point 120 00:06:07,650 --> 00:06:09,340 as my reference. 121 00:06:09,340 --> 00:06:13,660 And I will calculate the path back and forth 122 00:06:13,660 --> 00:06:16,180 from the focal point to the reflector and back. 123 00:06:16,180 --> 00:06:18,310 So I basically calculate this path. 124 00:06:18,310 --> 00:06:21,610 This path obviously equals 2 times f. 125 00:06:21,610 --> 00:06:26,680 By f, I denote the focal distance of the reflector. 126 00:06:26,680 --> 00:06:30,730 That is the distance at which the rays come to a focus. 127 00:06:30,730 --> 00:06:32,500 And the other path I will compute 128 00:06:32,500 --> 00:06:38,140 is I will ignore the path of the ray up to the reference plane. 129 00:06:38,140 --> 00:06:42,600 And then I will compute this path over here. 130 00:06:42,600 --> 00:06:44,030 So how do you compute this path? 131 00:06:44,030 --> 00:06:49,390 Well, the shape of the paraboloid is s of x. 132 00:06:49,390 --> 00:06:54,560 So basically, s of x is the elevation of the unknown shape 133 00:06:54,560 --> 00:06:57,920 with respect to the other reference plane 134 00:06:57,920 --> 00:07:01,700 that passes through the bottom of the unknown shape. 135 00:07:01,700 --> 00:07:05,990 So therefore, this distance here is f minus s, 136 00:07:05,990 --> 00:07:10,960 because it equals the focal distance minus the elevation 137 00:07:10,960 --> 00:07:13,580 of the reflector. 138 00:07:13,580 --> 00:07:19,220 And the other-- this distance over here. 139 00:07:19,220 --> 00:07:20,720 OK, so let me write it down. 140 00:07:20,720 --> 00:07:23,510 So it is f minus s. 141 00:07:23,510 --> 00:07:28,790 This is the first segment of the ray part. 142 00:07:28,790 --> 00:07:32,600 And the other is the hypotenuse of this orthogonal triangle, 143 00:07:32,600 --> 00:07:40,400 where one side of the triangle equals x. 144 00:07:40,400 --> 00:07:43,930 The other side of the triangle equals f plus s. 145 00:07:43,930 --> 00:07:46,820 And all I have to do is I have to apply Pythagoras' theorem. 146 00:07:50,450 --> 00:07:56,690 OK, so this is the path of the ray that arrived away 147 00:07:56,690 --> 00:07:58,340 from the optical axis. 148 00:07:58,340 --> 00:08:00,410 And, of course, this must equal the path 149 00:08:00,410 --> 00:08:05,940 of the ray that is exactly on the axis that equals to 2f. 150 00:08:05,940 --> 00:08:09,940 So now, we can play with his formula a little bit. 151 00:08:09,940 --> 00:08:11,860 I skip the derivation in the notes. 152 00:08:11,860 --> 00:08:14,790 Actually, I didn't quite skip the derivation of the notes. 153 00:08:14,790 --> 00:08:20,740 So instead of deriving it here, I will just animate it. 154 00:08:20,740 --> 00:08:23,350 And if we do the algebra here, we 155 00:08:23,350 --> 00:08:26,890 can basically eliminate one of the focal distances, 156 00:08:26,890 --> 00:08:28,780 bring that to the other side. 157 00:08:28,780 --> 00:08:31,780 And then do a little bit more algebraic manipulation. 158 00:08:31,780 --> 00:08:35,679 And after not too much work, we have this expression over here 159 00:08:35,679 --> 00:08:38,500 for the elevation of the unknown shape. 160 00:08:38,500 --> 00:08:41,232 And clearly, what you see over that is a parabola. 161 00:08:41,232 --> 00:08:42,940 Of course, it turned out to be a parabola 162 00:08:42,940 --> 00:08:46,450 because I did the calculation in the cross section-- 163 00:08:46,450 --> 00:08:49,600 in one cross section of the optical element. 164 00:08:49,600 --> 00:08:51,400 In reality, all of these elements 165 00:08:51,400 --> 00:08:56,700 are rotationally symmetric surfaces of revolution. 166 00:08:56,700 --> 00:08:59,560 So the reason I call it a paraboloid 167 00:08:59,560 --> 00:09:02,620 and not a parabola is because I take this shape, 168 00:09:02,620 --> 00:09:05,050 and then I spin it around the optical axis 169 00:09:05,050 --> 00:09:07,120 in order to get the familiar ball 170 00:09:07,120 --> 00:09:09,070 shape of the paraboloid this. 171 00:09:09,070 --> 00:09:13,570 But to do a little less math here, 172 00:09:13,570 --> 00:09:16,330 and to keep my life simple, I will only do these calculations 173 00:09:16,330 --> 00:09:18,380 in one plane. 174 00:09:18,380 --> 00:09:21,440 So turns out to be a simple parabola. 175 00:09:21,440 --> 00:09:25,040 So then this is your final answer. 176 00:09:25,040 --> 00:09:28,490 This is the shape of a reflector that 177 00:09:28,490 --> 00:09:31,630 will take a set of parallel rays, 178 00:09:31,630 --> 00:09:35,660 and focus them into a focal point in front 179 00:09:35,660 --> 00:09:38,670 of the reflector. 180 00:09:38,670 --> 00:09:39,720 So these are very nice. 181 00:09:39,720 --> 00:09:43,560 And we actually got our first imaging system. 182 00:09:43,560 --> 00:09:46,110 We managed to compute its focal length 183 00:09:46,110 --> 00:09:49,820 as a function of the shape of the element. 184 00:09:49,820 --> 00:09:52,070 There's a few things that I would like to remark here. 185 00:09:52,070 --> 00:09:55,420 One is, for example, that this works very nicely when 186 00:09:55,420 --> 00:10:00,430 the rays are indeed incident parallel to the normal 187 00:10:00,430 --> 00:10:02,720 at the center of the parabola here. 188 00:10:02,720 --> 00:10:06,530 If you imagine that I take this set of rays and I tilt it. 189 00:10:06,530 --> 00:10:09,160 Then a little bit of thought will convince yourself 190 00:10:09,160 --> 00:10:11,620 that it doesn't work so nicely anymore. 191 00:10:11,620 --> 00:10:15,030 If you were to then come in parallel rays, 192 00:10:15,030 --> 00:10:19,150 then the rays will kind of focus, 193 00:10:19,150 --> 00:10:24,020 but they would fail to meet all at once at the focal point, f. 194 00:10:24,020 --> 00:10:26,020 So this is a perfect focusing element 195 00:10:26,020 --> 00:10:29,200 for on axis incident rays. 196 00:10:29,200 --> 00:10:30,880 But it would be not so perfect anymore 197 00:10:30,880 --> 00:10:34,690 if the rays are arriving off axis. 198 00:10:34,690 --> 00:10:36,320 That's one observation to keep in mind. 199 00:10:36,320 --> 00:10:39,990 This will come up again later. 200 00:10:39,990 --> 00:10:41,740 One thing I would like to say is that this 201 00:10:41,740 --> 00:10:43,930 is a very nice element, indeed. 202 00:10:43,930 --> 00:10:45,910 But in some cases, it is not practical 203 00:10:45,910 --> 00:10:49,510 because as you can see, the focus is actually 204 00:10:49,510 --> 00:10:50,600 in front of the element. 205 00:10:50,600 --> 00:10:52,630 So if you were to detect the light, 206 00:10:52,630 --> 00:10:56,230 you would have to put, for example, a photo diode 207 00:10:56,230 --> 00:10:59,500 or some transducer at this location, which 208 00:10:59,500 --> 00:11:03,263 means you are actually blocking the path of the light. 209 00:11:03,263 --> 00:11:04,680 AUDIENCE: I have a quick question. 210 00:11:10,435 --> 00:11:12,560 GEORGE BARBASTATHIS: I'm sorry, I can not hear you. 211 00:11:12,560 --> 00:11:14,960 AUDIENCE: I can understand how the rays have 212 00:11:14,960 --> 00:11:16,910 a minimal optical path. 213 00:11:16,910 --> 00:11:18,410 But I don't understand why they have 214 00:11:18,410 --> 00:11:20,390 to have the same optical path. 215 00:11:20,390 --> 00:11:23,990 For example, you let it equal to 2f, which 216 00:11:23,990 --> 00:11:26,150 was at the center distance. 217 00:11:26,150 --> 00:11:28,910 So isn't it possible that the minimum optical path does not 218 00:11:28,910 --> 00:11:29,810 equal 2f? 219 00:11:29,810 --> 00:11:31,240 Or why is that the case? 220 00:11:33,292 --> 00:11:34,750 GEORGE BARBASTATHIS: Take two rays. 221 00:11:34,750 --> 00:11:38,650 In our case, one ray [INAUDIBLE] from infinity, and they both 222 00:11:38,650 --> 00:11:43,280 meet at F. So the rays can take different optical paths. 223 00:11:43,280 --> 00:11:45,650 One can go like this, the other can go like this. 224 00:11:45,650 --> 00:11:47,930 And doing our rays in general. 225 00:11:47,930 --> 00:11:53,050 So imagine that this has path l1, and this has path l2. 226 00:11:53,050 --> 00:11:54,980 And imagine, for the sake of argument, 227 00:11:54,980 --> 00:11:58,450 that l1 is less than l2. 228 00:11:58,450 --> 00:12:06,700 Fermat's principle says that if the ray, number one, 229 00:12:06,700 --> 00:12:10,240 goes through the first point from the origin 230 00:12:10,240 --> 00:12:11,890 to the destination. 231 00:12:11,890 --> 00:12:13,570 It follows a certain path. 232 00:12:13,570 --> 00:12:15,700 This path has to be minimal. 233 00:12:15,700 --> 00:12:17,780 But now, look at ray number two. 234 00:12:17,780 --> 00:12:22,090 This ray also goes from the same origin to the same destination, 235 00:12:22,090 --> 00:12:25,000 but follows a longer optical path. 236 00:12:25,000 --> 00:12:29,240 Therefore, the second ray has to violate Fermat's principle. 237 00:12:29,240 --> 00:12:32,657 So this is what we call mathematics abduction to-- 238 00:12:32,657 --> 00:12:33,490 what do you call it? 239 00:12:33,490 --> 00:12:35,020 To absurdum. 240 00:12:35,020 --> 00:12:36,880 That's right, abduction to absurdum. 241 00:12:36,880 --> 00:12:39,580 In other words, by assuming that the rays have 242 00:12:39,580 --> 00:12:43,060 unequal optical paths, we arrived 243 00:12:43,060 --> 00:12:46,170 at a conclusion that is inconsistent with Fermat's 244 00:12:46,170 --> 00:12:46,990 principle. 245 00:12:46,990 --> 00:12:49,990 So therefore, this has to be discarded. 246 00:12:49,990 --> 00:12:53,110 The only possibility is that l1 equals l2. 247 00:12:53,110 --> 00:12:56,080 So because the two rays start at the same origin 248 00:12:56,080 --> 00:12:58,000 and arrive at the same destination, 249 00:12:58,000 --> 00:12:59,850 they must have the same optical path. 250 00:13:02,680 --> 00:13:03,673 Did that make sense? 251 00:13:06,982 --> 00:13:09,190 AUDIENCE: For example, in the homework problem, where 252 00:13:09,190 --> 00:13:15,193 you have the swimmer and the lifeguard, well, I guess-- 253 00:13:15,193 --> 00:13:16,610 I'll think about it a little more. 254 00:13:16,610 --> 00:13:16,820 Yeah, makes sense. 255 00:13:16,820 --> 00:13:17,320 Thanks. 256 00:13:24,060 --> 00:13:25,060 GEORGE BARBASTATHIS: OK. 257 00:13:27,730 --> 00:13:31,840 So for the cases where the paraboloid is not convenient. 258 00:13:31,840 --> 00:13:33,580 Actually, I should say for the cases 259 00:13:33,580 --> 00:13:36,850 where the reflector is not convenient, 260 00:13:36,850 --> 00:13:39,520 because of the problem of blocking the ray 261 00:13:39,520 --> 00:13:42,760 path at the focal point. 262 00:13:42,760 --> 00:13:44,920 It is perhaps more convenient to attempt 263 00:13:44,920 --> 00:13:48,700 to use a refractive optical element that is a dielectric. 264 00:13:48,700 --> 00:13:53,510 And try to focus the rays using now, again, a curved surface. 265 00:13:53,510 --> 00:13:57,900 But in this case, it is a curved surface refractive index n. 266 00:13:57,900 --> 00:13:59,440 And for simplicity, I will assume 267 00:13:59,440 --> 00:14:02,380 that the rays are, again, are coming from free space, 268 00:14:02,380 --> 00:14:04,790 from vacuum, or from air. 269 00:14:04,790 --> 00:14:06,640 So the index of refraction is actually 270 00:14:06,640 --> 00:14:09,090 one on the left hand side over here. 271 00:14:09,090 --> 00:14:11,910 On the right hand side, it is n. 272 00:14:11,910 --> 00:14:15,450 So in this case, again, I have the same problem. 273 00:14:15,450 --> 00:14:20,390 I have an unknown shape, s of x, that I should follow, 274 00:14:20,390 --> 00:14:25,050 that I must somehow shape my refractor. 275 00:14:25,050 --> 00:14:27,090 And one could assume that this is anything. 276 00:14:27,090 --> 00:14:28,480 It could be, again, a parabola. 277 00:14:28,480 --> 00:14:30,872 It could be a circle, I mean, a sphere. 278 00:14:30,872 --> 00:14:32,580 It could be a number of different things. 279 00:14:32,580 --> 00:14:36,010 The question, what is it really. 280 00:14:36,010 --> 00:14:38,190 So the question is actually the same as the question 281 00:14:38,190 --> 00:14:40,700 that I asked before. 282 00:14:40,700 --> 00:14:44,900 And to answer it, we will actually follow once again 283 00:14:44,900 --> 00:14:46,480 the same argument. 284 00:14:46,480 --> 00:14:50,630 We will use Fermat to argue that since the two rays started 285 00:14:50,630 --> 00:14:52,860 at the same point, infinity. 286 00:14:52,860 --> 00:14:56,510 And they meet at the same point, F, at the same target, 287 00:14:56,510 --> 00:14:59,970 F. They must follow the same optical path. 288 00:14:59,970 --> 00:15:04,130 So in this case, I will apply Fermat with the reference plane 289 00:15:04,130 --> 00:15:10,850 that is tangential at the apex of the known refractive-- 290 00:15:10,850 --> 00:15:14,600 I'm sorry, of the unknown refractor shape. 291 00:15:14,600 --> 00:15:18,080 And I will compute the path of the own axis 292 00:15:18,080 --> 00:15:20,550 ray, which is very easy. 293 00:15:20,550 --> 00:15:23,610 So this path, you have to be a little bit careful now 294 00:15:23,610 --> 00:15:26,300 when we define the optical path. 295 00:15:26,300 --> 00:15:30,710 If you remember, the optical path in general. 296 00:15:30,710 --> 00:15:33,830 Optical path length equals the integral 297 00:15:33,830 --> 00:15:43,390 along the ray trajectory of the index of refraction times dl. 298 00:15:43,390 --> 00:15:45,790 So what this means now in our case 299 00:15:45,790 --> 00:15:49,510 is that the optical path length for this segment 300 00:15:49,510 --> 00:15:53,680 of the rays between the apex and the focal point 301 00:15:53,680 --> 00:16:01,600 equals f, the distance, times the index of refraction, 302 00:16:01,600 --> 00:16:05,310 in order to convert it to proper optical path. 303 00:16:05,310 --> 00:16:06,157 OK, so that's easy. 304 00:16:06,157 --> 00:16:07,740 Now, we have to compute the other one. 305 00:16:11,953 --> 00:16:14,120 So here's, again, I will exaggerate it a little bit. 306 00:16:14,120 --> 00:16:22,160 Here is, again, my unknown shape, s of x. 307 00:16:22,160 --> 00:16:29,810 And here is the ray that is arriving at some distance, x, 308 00:16:29,810 --> 00:16:32,270 from the apex. 309 00:16:32,270 --> 00:16:36,260 And this ray, according to my postulate here, 310 00:16:36,260 --> 00:16:41,990 will meet the axis at distance small f. 311 00:16:41,990 --> 00:16:46,862 To reach the point, uppercase f, that is the focal point. 312 00:16:46,862 --> 00:16:49,070 So again, I have to do a little bit of geometry here. 313 00:16:49,070 --> 00:16:51,980 This distance is s of x. 314 00:16:51,980 --> 00:16:55,610 Because this distance, again, is the elevation of the refractor 315 00:16:55,610 --> 00:16:58,920 shape relative to my reference plane over here. 316 00:17:04,170 --> 00:17:09,750 So this equals s plus this distance, which again, is 317 00:17:09,750 --> 00:17:11,984 the hypotenuse of a triangle. 318 00:17:11,984 --> 00:17:13,859 But I have to be, again, a little bit careful 319 00:17:13,859 --> 00:17:17,930 because this segment over here now, 320 00:17:17,930 --> 00:17:20,270 again, is inside the refractor. 321 00:17:20,270 --> 00:17:21,890 So before I do anything else, again, I 322 00:17:21,890 --> 00:17:24,390 have to multiply by the index of refraction. 323 00:17:24,390 --> 00:17:26,930 So it has to be n, the index of refraction, 324 00:17:26,930 --> 00:17:31,910 times the hypotenuse length, which 325 00:17:31,910 --> 00:17:34,353 equals the square root of the two sides, 326 00:17:34,353 --> 00:17:35,395 of the two sides squared. 327 00:17:42,743 --> 00:17:43,410 What am I doing? 328 00:17:43,410 --> 00:17:44,780 I'm sorry. 329 00:17:44,780 --> 00:17:48,840 x squared plus f minus s squared. 330 00:17:48,840 --> 00:17:50,304 That's the correct expression. 331 00:17:57,890 --> 00:17:59,840 OK, and [INAUDIBLE] in fact I could see it, 332 00:17:59,840 --> 00:18:01,423 because my computer has a preview. 333 00:18:01,423 --> 00:18:02,840 So I can see the equation, but I'm 334 00:18:02,840 --> 00:18:06,290 trying not to see it when I'm deriving things here. 335 00:18:06,290 --> 00:18:08,170 OK, so that is the equation, then. 336 00:18:08,170 --> 00:18:09,670 So again, according to the argument. 337 00:18:09,670 --> 00:18:11,795 Actually, let me repeat it once again, because this 338 00:18:11,795 --> 00:18:13,740 is a-- it's a very basic point. 339 00:18:13,740 --> 00:18:15,020 We have two rays here. 340 00:18:15,020 --> 00:18:17,220 One is going on axis. 341 00:18:17,220 --> 00:18:20,650 There is no refraction here, because it is incident exactly 342 00:18:20,650 --> 00:18:23,250 normally on the surface. 343 00:18:23,250 --> 00:18:25,390 And then it propagates at distance, f. 344 00:18:25,390 --> 00:18:29,720 So the optical path length of this ray equals n times f. 345 00:18:29,720 --> 00:18:32,560 Then I have another ray that propagated a little bit further 346 00:18:32,560 --> 00:18:36,840 in there because it arrives at the elevated portion 347 00:18:36,840 --> 00:18:39,220 of the refractor. 348 00:18:39,220 --> 00:18:42,410 And then it bends in some way that we don't know yet. 349 00:18:42,410 --> 00:18:44,680 But what we do know is that we want, 350 00:18:44,680 --> 00:18:47,310 we demand, we require, so to speak, 351 00:18:47,310 --> 00:18:50,050 this ray to go through the same point, F. 352 00:18:50,050 --> 00:18:53,150 So the fact that the ray goes through the same point, F. 353 00:18:53,150 --> 00:18:58,090 It means that it must follow the same path as the one axis ray. 354 00:18:58,090 --> 00:19:02,320 If this ray, for example, had followed a shorter path. 355 00:19:02,320 --> 00:19:04,620 Then there's no reason for this ray to go this way. 356 00:19:04,620 --> 00:19:06,670 It should have gone this way as well, 357 00:19:06,670 --> 00:19:09,280 because if it hadn't, it would be 358 00:19:09,280 --> 00:19:11,860 violating Fermat's principle. 359 00:19:11,860 --> 00:19:13,130 That is the argument. 360 00:19:13,130 --> 00:19:14,830 So therefore, there are two paths. 361 00:19:14,830 --> 00:19:17,530 This path, and this path. 362 00:19:17,530 --> 00:19:19,240 They must be exactly equal. 363 00:19:19,240 --> 00:19:22,930 So this is what this equation says over that. 364 00:19:22,930 --> 00:19:25,910 OK, this equation now. 365 00:19:25,910 --> 00:19:30,504 Does anybody know what this equation represents? 366 00:19:33,203 --> 00:19:35,120 If you have looked at the notes ahead of time, 367 00:19:35,120 --> 00:19:36,443 you know the answer. 368 00:19:36,443 --> 00:19:38,360 If you ever looked at the notes, this equation 369 00:19:38,360 --> 00:19:40,040 doesn't mean much. 370 00:19:40,040 --> 00:19:42,020 However, it can be manipulated. 371 00:19:42,020 --> 00:19:44,670 And I did not do all the algebra in the notes. 372 00:19:44,670 --> 00:19:47,790 I skipped a few steps. 373 00:19:47,790 --> 00:19:50,090 So the first thing you do is you basically try 374 00:19:50,090 --> 00:19:52,010 to get rid of the square root. 375 00:19:52,010 --> 00:19:55,140 So you bring this s on that side. 376 00:19:55,140 --> 00:19:56,780 You take squares of both sides. 377 00:19:56,780 --> 00:19:59,300 You do a little bit of algebra. 378 00:19:59,300 --> 00:20:01,530 You bring it to this form. 379 00:20:01,530 --> 00:20:04,170 Then this-- it still doesn't look very nice. 380 00:20:04,170 --> 00:20:06,870 So what you do is you try to write it to complete 381 00:20:06,870 --> 00:20:09,070 the sum of squares here. 382 00:20:09,070 --> 00:20:12,570 So that's a little bit messy. 383 00:20:12,570 --> 00:20:15,210 I will let you do it by yourselves, if you're 384 00:20:15,210 --> 00:20:16,217 interested and curious. 385 00:20:16,217 --> 00:20:18,300 If you get stuck, and you don't know how to do it, 386 00:20:18,300 --> 00:20:21,600 I will post the complete derivation on the website. 387 00:20:21,600 --> 00:20:24,030 But anyway, after a little bit more algebra, 388 00:20:24,030 --> 00:20:26,470 you get a result that looks like this. 389 00:20:26,470 --> 00:20:30,480 It looks like a square of the surface shape, s, 390 00:20:30,480 --> 00:20:33,570 minus the displacement, times another square 391 00:20:33,570 --> 00:20:37,210 of the coordinate x equals a constant. 392 00:20:37,210 --> 00:20:40,230 And this type of equation in general, 393 00:20:40,230 --> 00:20:44,860 an equation of the form s square over constant, 394 00:20:44,860 --> 00:20:50,610 plus x square over constant equals another constant, which 395 00:20:50,610 --> 00:20:52,290 might as well be one. 396 00:20:52,290 --> 00:20:55,650 This equation represents an ellipse. 397 00:20:55,650 --> 00:20:57,800 It is still not quite like this. 398 00:20:57,800 --> 00:21:04,440 It is of the form s minus some constant square over constant 399 00:21:04,440 --> 00:21:06,930 plus x square over constant equals 1. 400 00:21:06,930 --> 00:21:11,850 That is still in ellipse, but a displaced ellipse, right? 401 00:21:11,850 --> 00:21:15,300 What this really means is that the ellipse 402 00:21:15,300 --> 00:21:19,120 is centered at this location. 403 00:21:19,120 --> 00:21:21,860 OK, this comes from the displacement term 404 00:21:21,860 --> 00:21:23,560 in the square. 405 00:21:23,560 --> 00:21:26,380 And the eccentricity of the ellipse-- that is, 406 00:21:26,380 --> 00:21:30,130 the size of the minor axis relative to the major axis-- 407 00:21:30,130 --> 00:21:33,710 is given by the inverse of this quantity over here. 408 00:21:33,710 --> 00:21:36,430 So it is n minus 1 over n plus 1f. 409 00:21:36,430 --> 00:21:39,310 What is interesting to note is that the focal point 410 00:21:39,310 --> 00:21:43,160 is to the right hand side of the center of the ellipse. 411 00:21:43,160 --> 00:21:48,040 And in fact, this happens to be the focal point of the ellipse. 412 00:21:48,040 --> 00:21:50,600 If you look at the definition of an ellipse 413 00:21:50,600 --> 00:21:54,160 in the form of pure geometry, ellipses 414 00:21:54,160 --> 00:21:57,610 are characterized by two focal points. 415 00:21:57,610 --> 00:22:01,660 And it turns out in this case, the focus of the rays 416 00:22:01,660 --> 00:22:03,438 is one of the focal points of the ellipse. 417 00:22:03,438 --> 00:22:04,980 And that, of course, is symmetrically 418 00:22:04,980 --> 00:22:06,630 located on the other side. 419 00:22:06,630 --> 00:22:12,420 OK, so this is a bit of a mathematical trickery. 420 00:22:12,420 --> 00:22:14,640 But what I want to emphasize, which is actually 421 00:22:14,640 --> 00:22:17,760 kind of important to know, is that the focal point 422 00:22:17,760 --> 00:22:20,910 is actually past the center of the ellipse. 423 00:22:28,210 --> 00:22:29,340 That's about it, really. 424 00:22:36,860 --> 00:22:41,810 The result of this is sort of an ideal elliptical shape. 425 00:22:41,810 --> 00:22:43,700 That's something worth remembering. 426 00:22:43,700 --> 00:22:51,860 That if you have a ideal set of parallel rays coming 427 00:22:51,860 --> 00:22:53,780 in-- or as we will call it a little bit later, 428 00:22:53,780 --> 00:22:54,930 a plane wave-- 429 00:22:54,930 --> 00:22:59,450 and we wish to focus it at the final distance inside 430 00:22:59,450 --> 00:23:00,710 of the electric material. 431 00:23:00,710 --> 00:23:02,750 The best possible shape to achieve 432 00:23:02,750 --> 00:23:05,333 that is an elliptical shape. 433 00:23:05,333 --> 00:23:06,500 Now, again, there's caveats. 434 00:23:06,500 --> 00:23:09,410 For example, if I tilt this incoming ray, 435 00:23:09,410 --> 00:23:11,940 then the focus will not be perfect out here. 436 00:23:11,940 --> 00:23:15,230 So this is the same situation as in the parabol line. 437 00:23:15,230 --> 00:23:18,500 It only works so nice and so perfectly 438 00:23:18,500 --> 00:23:21,620 if the incidence is exactly normal. 439 00:23:24,130 --> 00:23:28,290 Out of curiosity, can anybody imagine-- 440 00:23:28,290 --> 00:23:31,020 so if you compare this with a paraboloid, 441 00:23:31,020 --> 00:23:32,580 one obvious advantage is, of course, 442 00:23:32,580 --> 00:23:35,770 that I'm not blocking the light path anymore. 443 00:23:35,770 --> 00:23:40,223 What is a possible disadvantage of this kind of approach? 444 00:23:40,223 --> 00:23:41,390 There are several, actually. 445 00:23:41,390 --> 00:23:42,410 Tell me at least one. 446 00:23:42,410 --> 00:23:43,730 Yeah. 447 00:23:43,730 --> 00:23:46,646 Push the button. 448 00:23:46,646 --> 00:23:49,202 AUDIENCE: The detector is inside the vector. 449 00:23:49,202 --> 00:23:51,410 GEORGE BARBASTATHIS: That's one obvious disadvantage. 450 00:23:51,410 --> 00:23:53,420 If I wanted to really use this, I 451 00:23:53,420 --> 00:23:55,020 would have to put the detector here. 452 00:23:55,020 --> 00:23:57,250 And, of course, it doesn't have to be a full ellipse. 453 00:23:57,250 --> 00:23:59,930 I can always clip the ellipse over here. 454 00:23:59,930 --> 00:24:02,720 The rays have no way of knowing that this happened to them, 455 00:24:02,720 --> 00:24:05,250 so I can always stick the detector over here. 456 00:24:05,250 --> 00:24:07,580 So maybe I can overcome that problem. 457 00:24:07,580 --> 00:24:09,510 But it is a problem, yeah. 458 00:24:09,510 --> 00:24:13,340 What is the second problem that kind of is sticking out? 459 00:24:19,143 --> 00:24:20,560 AUDIENCE: In the case of if you're 460 00:24:20,560 --> 00:24:22,268 trying to build a telescope, or something 461 00:24:22,268 --> 00:24:24,070 where you have to worry about the structure 462 00:24:24,070 --> 00:24:26,000 of your optical setup. 463 00:24:26,000 --> 00:24:28,000 You could think that maybe a solid element would 464 00:24:28,000 --> 00:24:30,310 be much more bulky or hard to abrogate 465 00:24:30,310 --> 00:24:34,730 than a reflecting element, like paraboloid? 466 00:24:34,730 --> 00:24:36,870 GEORGE BARBASTATHIS: That's correct. 467 00:24:36,870 --> 00:24:39,480 If you tried to make a big element, 468 00:24:39,480 --> 00:24:42,630 a big lens, a big focusing concentrator, 469 00:24:42,630 --> 00:24:45,542 like a solid concentrate, or a satellite dish. 470 00:24:45,542 --> 00:24:47,250 That's a very inconvenient shape to have. 471 00:24:47,250 --> 00:24:50,520 It is heavy-- absolutely, I agree. 472 00:24:50,520 --> 00:24:52,319 Tell me a third disadvantage. 473 00:25:00,470 --> 00:25:06,190 The paraboloid can go on a very long distance. 474 00:25:06,190 --> 00:25:10,870 This one is a finite shape because it's an ellipse. 475 00:25:10,870 --> 00:25:14,410 As it curves at some point, it will reach a derivative equal 476 00:25:14,410 --> 00:25:15,220 to zero. 477 00:25:15,220 --> 00:25:16,460 It will stop there. 478 00:25:16,460 --> 00:25:19,540 So basically, this element captures only a finite set 479 00:25:19,540 --> 00:25:22,880 of rays up to the minor axis of the ellipse. 480 00:25:22,880 --> 00:25:25,570 It is missing the rest. 481 00:25:25,570 --> 00:25:29,770 Tell me a fourth and final disadvantage of this setup 482 00:25:29,770 --> 00:25:31,350 that at least I could think of. 483 00:25:36,510 --> 00:25:39,190 I'll give you a hint. 484 00:25:39,190 --> 00:25:43,255 Last time, Piper did a very nice demonstration with a prism. 485 00:25:46,447 --> 00:25:48,530 Do you remember something that his demonstration-- 486 00:25:52,270 --> 00:25:55,060 something about this demonstration [INAUDIBLE].. 487 00:25:55,060 --> 00:25:57,970 AUDIENCE: There can be total internal reflection. 488 00:25:57,970 --> 00:26:00,168 Like on this 489 00:26:00,168 --> 00:26:01,585 GEORGE BARBASTATHIS: [INAUDIBLE].. 490 00:26:01,585 --> 00:26:04,815 AUDIENCE: [INAUDIBLE] 491 00:26:04,815 --> 00:26:07,190 GEORGE BARBASTATHIS: Is there a possibility for PIR here? 492 00:26:11,475 --> 00:26:12,850 AUDIENCE: The index of refraction 493 00:26:12,850 --> 00:26:15,144 isn't necessarily the same for all wavelengths? 494 00:26:16,687 --> 00:26:18,770 GEORGE BARBASTATHIS: So first of all, let's answer 495 00:26:18,770 --> 00:26:21,090 the previous question. 496 00:26:21,090 --> 00:26:25,640 Here, you're right to think about the possibility of PIR. 497 00:26:25,640 --> 00:26:28,440 But if you look at it in a little bit more detail, 498 00:26:28,440 --> 00:26:31,830 you will see that PIR is not a problem in this case. 499 00:26:31,830 --> 00:26:34,860 At least for the rays going up to here, 500 00:26:34,860 --> 00:26:37,040 there's no problem of PIR. 501 00:26:37,040 --> 00:26:40,520 Because the rays come from a low index 502 00:26:40,520 --> 00:26:42,860 medium to a high index medium. 503 00:26:42,860 --> 00:26:48,020 And therefore, they will always refract inside the [INAUDIBLE].. 504 00:26:48,020 --> 00:26:50,980 What is interesting to happen to the ray that 505 00:26:50,980 --> 00:26:55,100 is incident exactly tangential to the ellipse over here. 506 00:26:55,100 --> 00:26:57,290 It will still be refracted, if you think about it. 507 00:26:57,290 --> 00:26:58,370 It's Snell's law. 508 00:26:58,370 --> 00:27:01,100 And it will give you sort of the maximum acceptance 509 00:27:01,100 --> 00:27:05,840 cone of the interface that we discussed last week, 510 00:27:05,840 --> 00:27:10,130 I mean, on Monday in relation to Snell's law. 511 00:27:10,130 --> 00:27:13,000 So then out of the two demos that Piper did last week, 512 00:27:13,000 --> 00:27:15,110 the PIR is not the problem. 513 00:27:15,110 --> 00:27:17,330 It is the other one, as you correctly pointed out. 514 00:27:17,330 --> 00:27:18,890 So you want to repeat? 515 00:27:18,890 --> 00:27:19,840 Who has spoken before? 516 00:27:24,100 --> 00:27:26,025 Someone mentioned the index of refraction. 517 00:27:26,025 --> 00:27:27,400 AUDIENCE: The index of refraction 518 00:27:27,400 --> 00:27:30,350 isn't the same for all wavelengths. 519 00:27:30,350 --> 00:27:32,618 GEORGE BARBASTATHIS: That's right. 520 00:27:32,618 --> 00:27:33,160 That's right. 521 00:27:33,160 --> 00:27:35,770 So since the index of refraction is not 522 00:27:35,770 --> 00:27:37,960 the same for all wavelengths, what this means 523 00:27:37,960 --> 00:27:41,590 is that the focal distance now becomes a function 524 00:27:41,590 --> 00:27:43,420 of the index of refraction. 525 00:27:43,420 --> 00:27:45,340 And I may tune it. 526 00:27:45,340 --> 00:27:47,630 Basically, what I do is I manufacture this shape. 527 00:27:47,630 --> 00:27:50,560 So I can manufacture it for a certain index 528 00:27:50,560 --> 00:27:51,990 for a standard wavelength. 529 00:27:51,990 --> 00:27:53,990 If the index changes, this relation 530 00:27:53,990 --> 00:27:57,400 will not be satisfied anymore, and therefore 531 00:27:57,400 --> 00:28:01,690 the refractive element will become imperfect. 532 00:28:01,690 --> 00:28:05,590 How do we call this phenomenon of dependence of index 533 00:28:05,590 --> 00:28:10,100 of refraction on wavelength? 534 00:28:10,100 --> 00:28:12,290 Dispersion. 535 00:28:12,290 --> 00:28:13,250 Dispersion. 536 00:28:13,250 --> 00:28:18,060 And in the context that we're discussing here. 537 00:28:18,060 --> 00:28:21,010 OK, it is dispersion. 538 00:28:21,010 --> 00:28:23,410 In the context of focusing, when we 539 00:28:23,410 --> 00:28:28,070 get imperfect focusing because of dispersion, 540 00:28:28,070 --> 00:28:30,050 It has also another special name. 541 00:28:30,050 --> 00:28:35,246 It is called chromatic aberration. 542 00:28:38,650 --> 00:28:42,480 The term aberration we'll see again in the next hour. 543 00:28:42,480 --> 00:28:45,940 It generally refers to failures of optical systems 544 00:28:45,940 --> 00:28:48,640 to focus light as they're supposed to. 545 00:28:48,640 --> 00:28:51,380 So this is the first such failure that we encounter. 546 00:28:51,380 --> 00:28:53,660 The term chromatic, I have an advantage over you. 547 00:28:53,660 --> 00:28:54,440 I'm Greek. 548 00:28:54,440 --> 00:28:57,640 And this turns out to be a Greek word. 549 00:28:57,640 --> 00:29:02,780 Chroma in Greek means color. 550 00:29:02,780 --> 00:29:04,630 And this is the Greek spelling. 551 00:29:04,630 --> 00:29:08,090 If you want to spell it out in Latin so you can pronounce it, 552 00:29:08,090 --> 00:29:09,400 it is chroma. 553 00:29:09,400 --> 00:29:12,550 And that's where the word chromatic is derived from. 554 00:29:12,550 --> 00:29:15,010 It literally means separation due to color. 555 00:29:18,370 --> 00:29:21,680 Any questions on any of this so far? 556 00:29:33,570 --> 00:29:34,820 I'm sorry, there's a question. 557 00:29:34,820 --> 00:29:42,020 AUDIENCE: If someone thought of a lenslike element 558 00:29:42,020 --> 00:29:43,880 in which the focusing takes place 559 00:29:43,880 --> 00:29:47,130 on the other side of the element. 560 00:29:47,130 --> 00:29:49,495 Will it have a [INAUDIBLE] power surface? 561 00:29:51,880 --> 00:29:53,880 GEORGE BARBASTATHIS: That's coming up, actually, 562 00:29:53,880 --> 00:29:55,420 in a few slides. 563 00:29:55,420 --> 00:29:59,530 So yeah, I mean, the answer to your question is generally, 564 00:29:59,530 --> 00:30:04,330 if you want to focus a point from the left 565 00:30:04,330 --> 00:30:07,030 to a point on the right, at the opposite side of element. 566 00:30:07,030 --> 00:30:10,180 You need two hyperbolic surface. 567 00:30:10,180 --> 00:30:12,880 I'll cover that in a second. 568 00:30:12,880 --> 00:30:15,940 Of course, hyperbolas are difficult to manufacture. 569 00:30:15,940 --> 00:30:19,690 People typically make spherical surfaces. 570 00:30:19,690 --> 00:30:22,550 You can also have a sphere, nowadays, by injection molding. 571 00:30:22,550 --> 00:30:26,050 I will go into that also later. 572 00:30:26,050 --> 00:30:30,220 But even that is only perfect for on axis points. 573 00:30:30,220 --> 00:30:33,100 If you move points off axis, then it fails again. 574 00:30:33,100 --> 00:30:37,910 So it becomes the whole problem of optical design. 575 00:30:37,910 --> 00:30:42,270 OK, so I will switch now to today's, actually, 576 00:30:42,270 --> 00:30:43,050 set of lectures. 577 00:30:52,110 --> 00:30:54,830 And you realized that we were kind of behind 578 00:30:54,830 --> 00:30:56,090 by almost one lecture. 579 00:30:56,090 --> 00:30:58,340 I think we're beginning to catch up now. 580 00:30:58,340 --> 00:31:01,400 But what I would like to do today for the rest of the two 581 00:31:01,400 --> 00:31:05,360 hours that you have is to go relatively slowly 582 00:31:05,360 --> 00:31:11,100 over a number of different topics, which 583 00:31:11,100 --> 00:31:12,780 are also related to focusing. 584 00:31:12,780 --> 00:31:15,610 But as we go along, we will generalize 585 00:31:15,610 --> 00:31:18,795 focusing to more and more complicated situations. 586 00:31:18,795 --> 00:31:20,670 So I wanted to solve two cases of-- actually, 587 00:31:20,670 --> 00:31:24,060 one case of focusing using two different types of elements, 588 00:31:24,060 --> 00:31:27,850 a reflector and a refractor. 589 00:31:27,850 --> 00:31:30,700 In order to make this stuff a little bit more specific, 590 00:31:30,700 --> 00:31:33,340 let me remind you of some definitions 591 00:31:33,340 --> 00:31:37,420 that we discussed a little bit at the first lecture. 592 00:31:37,420 --> 00:31:41,200 But I would like to remind you again with a bit more detail. 593 00:31:41,200 --> 00:31:45,780 So what is the concept of a spherical wave. 594 00:31:45,780 --> 00:31:49,460 So spherical wave is what you get 595 00:31:49,460 --> 00:31:53,150 if you have light or originating at the point source. 596 00:31:53,150 --> 00:31:56,085 It is also referred to as a point object. 597 00:31:56,085 --> 00:31:57,710 So what happens then is you get the fan 598 00:31:57,710 --> 00:31:59,960 of rays that is divergent. 599 00:31:59,960 --> 00:32:04,220 And if you plug the normals to these rays, these normals, 600 00:32:04,220 --> 00:32:07,200 they look like spheres. 601 00:32:07,200 --> 00:32:10,590 So, of course, the wavefronts expand as the spherical wave 602 00:32:10,590 --> 00:32:13,110 propagates. 603 00:32:13,110 --> 00:32:17,820 And because these wavefronts are spherical surfaces, 604 00:32:17,820 --> 00:32:21,690 that's why we call it a spherical wave. 605 00:32:21,690 --> 00:32:23,870 Again, very important to remember by definition. 606 00:32:23,870 --> 00:32:27,260 The wavefront is normal to the rays. 607 00:32:27,260 --> 00:32:31,550 So what I have done here in principle is I took each ray. 608 00:32:31,550 --> 00:32:33,830 I computed the normal to the ray. 609 00:32:33,830 --> 00:32:38,060 I connected all the normals, and I get this sphere over here. 610 00:32:38,060 --> 00:32:40,520 I don't know if I did it very well in my graphic over here, 611 00:32:40,520 --> 00:32:43,820 but that's what this is supposed to denote. 612 00:32:43,820 --> 00:32:48,970 OK, you can flip this situation and create 613 00:32:48,970 --> 00:32:52,040 a convergent spherical wave, where basically, you 614 00:32:52,040 --> 00:32:57,150 have rays that are propagating towards the point. 615 00:32:57,150 --> 00:32:59,490 If that is the case, it is not a point source anymore. 616 00:32:59,490 --> 00:33:00,930 You call it a point image. 617 00:33:03,510 --> 00:33:06,870 In most optical systems-- at least in optical systems that 618 00:33:06,870 --> 00:33:08,850 we will consider here-- 619 00:33:08,850 --> 00:33:10,860 this is really the goal of an optical system. 620 00:33:10,860 --> 00:33:14,460 You try to take rays of light that 621 00:33:14,460 --> 00:33:18,120 are given to you by some source that you do not control, 622 00:33:18,120 --> 00:33:21,530 such as an object, or a star far out of reach. 623 00:33:21,530 --> 00:33:24,280 And what you try to do is you try to arrange your optics, 624 00:33:24,280 --> 00:33:27,130 select your lenses, space them appropriately, 625 00:33:27,130 --> 00:33:28,480 and so on and so forth. 626 00:33:28,480 --> 00:33:32,280 So that you can converge these incoming rays 627 00:33:32,280 --> 00:33:34,810 into point images. 628 00:33:34,810 --> 00:33:42,130 So if you wish this is our given, or our customer. 629 00:33:42,130 --> 00:33:44,000 And then this is what the customer wants. 630 00:33:44,000 --> 00:33:47,860 The customer wants to form point images. 631 00:33:47,860 --> 00:33:51,380 So a good deal of working the field of optics 632 00:33:51,380 --> 00:33:56,390 has to do with this problem, how you can image-- 633 00:33:56,390 --> 00:33:58,820 however you can create point images. 634 00:33:58,820 --> 00:34:00,530 I'll take this out, because it does not 635 00:34:00,530 --> 00:34:01,880 relate to this discussion. 636 00:34:01,880 --> 00:34:03,850 OK. 637 00:34:03,850 --> 00:34:05,860 So again, the wavefronts are spherical 638 00:34:05,860 --> 00:34:10,427 here with a common center at the point image. 639 00:34:10,427 --> 00:34:12,219 But in this case, of course, the wavefronts 640 00:34:12,219 --> 00:34:14,710 are contracting as you go towards the point image, 641 00:34:14,710 --> 00:34:17,350 because the rays are converging. 642 00:34:17,350 --> 00:34:21,820 And, of course, [AUDIO OUT] wavefronts ideally collapse 643 00:34:21,820 --> 00:34:23,010 into a point. 644 00:34:23,010 --> 00:34:24,820 This doesn't sound very physical. 645 00:34:24,820 --> 00:34:31,030 And a little bit later, when we do the theory of diffraction. 646 00:34:31,030 --> 00:34:34,790 We will see in a bit more detail what happens at that point. 647 00:34:34,790 --> 00:34:38,530 It is not really that the wavefront ideally collapses. 648 00:34:38,530 --> 00:34:40,659 Something else happens that we will see. 649 00:34:40,659 --> 00:34:43,330 Within the geometrical optics approximation, 650 00:34:43,330 --> 00:34:45,850 let's accept it for now that this is the case. 651 00:34:45,850 --> 00:34:48,550 That indeed, the rays can converge to a single point, 652 00:34:48,550 --> 00:34:52,980 and then the wavefront over there collapses. 653 00:34:52,980 --> 00:34:58,750 OK, this is true for pretty much every case of imaging. 654 00:34:58,750 --> 00:35:01,770 However, in a number of situations, 655 00:35:01,770 --> 00:35:04,940 it is very convenient to take one extreme case 656 00:35:04,940 --> 00:35:10,920 where the point source, or the point object is very far away. 657 00:35:10,920 --> 00:35:13,530 I mentioned that already about half an hour ago. 658 00:35:13,530 --> 00:35:17,100 And I called it the point source of infinity, 659 00:35:17,100 --> 00:35:18,810 or a point object of infinity. 660 00:35:18,810 --> 00:35:21,570 A very good approximation for that is a star. 661 00:35:21,570 --> 00:35:25,288 If we look at stars in the night sky, they look like points. 662 00:35:25,288 --> 00:35:26,580 Of course, a star is humongous. 663 00:35:26,580 --> 00:35:29,610 It is probably thousands of times bigger than the earth. 664 00:35:29,610 --> 00:35:33,480 But because it is at a distance of thousands or millions 665 00:35:33,480 --> 00:35:35,640 of light years away, it appears to us 666 00:35:35,640 --> 00:35:38,340 to be like a point, a point source. 667 00:35:38,340 --> 00:35:40,640 So again, there's another important point to emphasize. 668 00:35:40,640 --> 00:35:43,260 When we talk about the point object at infinity, 669 00:35:43,260 --> 00:35:46,580 we don't necessarily imply that it is mathematically 670 00:35:46,580 --> 00:35:48,560 a point, that is, it has zero dimension. 671 00:35:48,560 --> 00:35:50,690 What we really mean is it is so far 672 00:35:50,690 --> 00:35:54,980 away that the angle that I subtend towards that 673 00:35:54,980 --> 00:35:57,380 far away point is minimal. 674 00:35:57,380 --> 00:35:59,870 If that is true, if that assumption is true, 675 00:35:59,870 --> 00:36:01,610 then I can call it a point. 676 00:36:01,610 --> 00:36:04,620 I mean, I can call it a point source at infinity. 677 00:36:04,620 --> 00:36:08,100 And what happens if you really went near that point 678 00:36:08,100 --> 00:36:11,570 far at infinity, you would still see divergent rays. 679 00:36:11,570 --> 00:36:14,660 But because these rays propagate at a very long distance, 680 00:36:14,660 --> 00:36:18,440 you only get to see the set of rays near the axis 681 00:36:18,440 --> 00:36:20,128 that departed from that point. 682 00:36:20,128 --> 00:36:21,920 And because you're [INAUDIBLE],, these rays 683 00:36:21,920 --> 00:36:26,750 are really tightly collapsed near the central ray over here. 684 00:36:26,750 --> 00:36:29,760 They all look parallel. 685 00:36:29,760 --> 00:36:32,700 So a set of rays that I propagate and they're parallel. 686 00:36:32,700 --> 00:36:36,070 Like so on here, we call it a plane wave. 687 00:36:36,070 --> 00:36:38,520 If you go back to your notes from the first lecture, 688 00:36:38,520 --> 00:36:44,030 we called it the plane wave, or the planar wavefront. 689 00:36:44,030 --> 00:36:46,280 But we can also call it-- 690 00:36:46,280 --> 00:36:48,200 parallel [INAUDIBLE] rays, we can also 691 00:36:48,200 --> 00:36:52,215 call it an object at infinity. 692 00:36:57,280 --> 00:36:59,230 Of course, because the rays are parallel. 693 00:36:59,230 --> 00:37:03,130 If I draw the normals to them, they will form planes. 694 00:37:03,130 --> 00:37:05,670 They form ideal planar surfaces. 695 00:37:05,670 --> 00:37:07,830 And that's why this is called the planar 696 00:37:07,830 --> 00:37:11,290 wavefront, or a plane wave. 697 00:37:11,290 --> 00:37:14,377 For our purposes, the two terms are identical. 698 00:37:14,377 --> 00:37:16,210 And, of course, this goes the other way too. 699 00:37:16,210 --> 00:37:20,650 If I had a planar wave also propagating like this. 700 00:37:20,650 --> 00:37:26,980 I can also claim that I create a point image at infinity. 701 00:37:26,980 --> 00:37:29,170 Because if I take parallel lines and I 702 00:37:29,170 --> 00:37:32,140 propagate them for an infinitely long distance. 703 00:37:32,140 --> 00:37:34,520 OK, what I'm about to say is not mathematically correct, 704 00:37:34,520 --> 00:37:36,850 but in optics, we say it all the time. 705 00:37:36,850 --> 00:37:39,460 It is that these parallel rays will meet at infinity. 706 00:37:39,460 --> 00:37:42,040 Of course, parallel rays never meet. 707 00:37:42,040 --> 00:37:44,810 But you can simply to justify it in your mind, 708 00:37:44,810 --> 00:37:46,640 flip to this picture, and take these rays, 709 00:37:46,640 --> 00:37:49,300 propagate them backwards. 710 00:37:49,300 --> 00:37:51,003 At a very, very far away distance, 711 00:37:51,003 --> 00:37:53,170 you cannot tell the difference, whether they started 712 00:37:53,170 --> 00:37:57,640 as parallel rays, or they started as a tightly convergent 713 00:37:57,640 --> 00:37:59,930 small ray bundle. 714 00:37:59,930 --> 00:38:05,560 OK, so this [AUDIO OUT] because very often, 715 00:38:05,560 --> 00:38:09,520 it simplifies calculations and gives 716 00:38:09,520 --> 00:38:11,590 certain kinds of intuition. 717 00:38:11,590 --> 00:38:15,145 And again, we define it in the geometric optic sense. 718 00:38:18,230 --> 00:38:21,300 When we do diffraction about a month from now, 719 00:38:21,300 --> 00:38:24,020 we will define it more rigorously in a way 720 00:38:24,020 --> 00:38:27,260 that is probably much more convincing than I do now. 721 00:38:27,260 --> 00:38:29,480 The reason I'm apologizing for this definition 722 00:38:29,480 --> 00:38:32,870 is that there's no such thing as mathematical infinity 723 00:38:32,870 --> 00:38:33,600 in real life. 724 00:38:33,600 --> 00:38:36,380 Even a star, it is a very long distance away, 725 00:38:36,380 --> 00:38:38,970 but it is not an infinite distance away. 726 00:38:38,970 --> 00:38:40,360 So what do I mean by infinity? 727 00:38:40,360 --> 00:38:44,340 For example, [INAUDIBLE],, who is sitting here on the front row. 728 00:38:44,340 --> 00:38:46,760 Is he at infinity with respect to me, or no? 729 00:38:46,760 --> 00:38:50,620 Well, what about you guys who are sitting in the cam bridge? 730 00:38:50,620 --> 00:38:52,610 Well, OK, the surface of the earth is curved, 731 00:38:52,610 --> 00:38:54,633 so that creates an obvious problem. 732 00:38:54,633 --> 00:38:56,300 But suppose I could draw a straight line 733 00:38:56,300 --> 00:38:59,120 through the center of the earth from Singapore to Boston. 734 00:38:59,120 --> 00:39:01,950 Would you guys be at infinity with respect to me? 735 00:39:01,950 --> 00:39:05,760 OK, it actually depends on a lot of different things. 736 00:39:05,760 --> 00:39:09,980 It depends on the relative sizes that I have and you have. 737 00:39:09,980 --> 00:39:12,933 It depends on the wavelength that the light is propagating. 738 00:39:12,933 --> 00:39:14,600 And actually, that's about it, according 739 00:39:14,600 --> 00:39:15,680 to diffraction theory. 740 00:39:15,680 --> 00:39:19,030 So we will see a rigorous definition of what is infinity 741 00:39:19,030 --> 00:39:20,450 when we do diffraction. 742 00:39:20,450 --> 00:39:23,930 For now, let's take it on faith that if I 743 00:39:23,930 --> 00:39:26,630 have a set of rays that appear [AUDIO OUT],, 744 00:39:26,630 --> 00:39:32,680 we'll call it image of infinity. 745 00:39:32,680 --> 00:39:35,210 So sometimes, we use the term collimated. 746 00:39:38,642 --> 00:39:39,600 That's a better marker. 747 00:39:42,320 --> 00:39:45,550 Collimated rays. 748 00:39:45,550 --> 00:39:49,280 So collimated is from the word colinear, right? 749 00:39:49,280 --> 00:39:52,500 Creating parallel to each other. 750 00:39:52,500 --> 00:39:54,850 OK. 751 00:39:54,850 --> 00:39:59,610 So I included this slide, but as a reminder, and that 752 00:39:59,610 --> 00:40:03,060 as a reference, because for the next-- 753 00:40:03,060 --> 00:40:06,300 I don't know, for the next semester really, from now on, 754 00:40:06,300 --> 00:40:07,850 we'll be using these terms a lot. 755 00:40:07,850 --> 00:40:11,610 Spherical waves, plane waves, spherical wavefronts, 756 00:40:11,610 --> 00:40:13,960 and so on and so forth. 757 00:40:13,960 --> 00:40:16,920 The concept of infinity applies to 758 00:40:16,920 --> 00:40:20,810 both convergent and divergent spherical waves at infinity. 759 00:40:25,860 --> 00:40:27,350 All right. 760 00:40:27,350 --> 00:40:32,780 So the two examples that we saw at the beginning. 761 00:40:32,780 --> 00:40:36,730 These are basically examples of imaging sources at infinity. 762 00:40:36,730 --> 00:40:38,620 Because we had plane waves coming 763 00:40:38,620 --> 00:40:43,210 into the paraboloidal reflector and ellipsoidal 764 00:40:43,210 --> 00:40:45,890 refractor respectively. 765 00:40:45,890 --> 00:40:47,410 And then these two elements managed 766 00:40:47,410 --> 00:40:52,660 to convert these plane waves, these parallel 767 00:40:52,660 --> 00:40:59,690 ray bundles into perfect converging spherical waves. 768 00:40:59,690 --> 00:41:03,290 Therefore, they created perfect point images 769 00:41:03,290 --> 00:41:05,810 at the corresponding focal points. 770 00:41:05,810 --> 00:41:07,370 And then wavefronts, I also put down 771 00:41:07,370 --> 00:41:14,050 the two equations, the equations of the two elements. 772 00:41:14,050 --> 00:41:21,920 Now, let's flip the coin a little bit, and ask 773 00:41:21,920 --> 00:41:26,550 how can I create a point image at infinity? 774 00:41:26,550 --> 00:41:29,910 Now, I start with a point source that is within my range. 775 00:41:29,910 --> 00:41:31,770 Here is my point source. 776 00:41:31,770 --> 00:41:33,380 Here is my point source. 777 00:41:33,380 --> 00:41:36,770 And I'm trying to create an image 778 00:41:36,770 --> 00:41:38,415 of that source of infinity. 779 00:41:38,415 --> 00:41:40,790 Well, another way to say it is that what I'm trying to do 780 00:41:40,790 --> 00:41:46,150 is I'm trying to collimate the light. 781 00:41:52,050 --> 00:41:53,520 So in the case of the paraboloid, 782 00:41:53,520 --> 00:41:55,780 it's actually very simple. 783 00:41:55,780 --> 00:41:59,580 All I have to do is reverse the paths of the rays. 784 00:41:59,580 --> 00:42:02,190 This may also sound arbitrary, but in electromagnetics, 785 00:42:02,190 --> 00:42:05,940 there's a very basic principle that says that you can do that. 786 00:42:05,940 --> 00:42:07,990 I will not go into that in detail, 787 00:42:07,990 --> 00:42:11,370 but I can assure you that from the rigorous electromagnetic 788 00:42:11,370 --> 00:42:13,560 point of view, this is permissible. 789 00:42:13,560 --> 00:42:17,190 You can reverse the ray paths, and you still 790 00:42:17,190 --> 00:42:19,310 get the value of the electromagnetic field. 791 00:42:19,310 --> 00:42:23,010 Therefore, a paraboloidal reflector works both ways. 792 00:42:23,010 --> 00:42:25,620 You can use it either as a receiver, 793 00:42:25,620 --> 00:42:27,870 so you can focus a source at infinity, 794 00:42:27,870 --> 00:42:29,813 and you can detect it here. 795 00:42:29,813 --> 00:42:30,730 Or it's a transmitter. 796 00:42:30,730 --> 00:42:33,870 You can have a point source at this finite distance, 797 00:42:33,870 --> 00:42:38,723 and you can broadcast a number of cars, but you can beam it. 798 00:42:38,723 --> 00:42:39,390 That's the word. 799 00:42:39,390 --> 00:42:43,320 You can beam it to a very narrow-- 800 00:42:43,320 --> 00:42:46,810 to a target that is at a very long distance away. 801 00:42:49,690 --> 00:42:58,030 In the case of a refractor, the answer is not as simple. 802 00:42:58,030 --> 00:43:05,390 Because in the process of reversing the situation, 803 00:43:05,390 --> 00:43:09,950 now, the source is in air. 804 00:43:09,950 --> 00:43:14,360 And the collimated light is in glass, is in the dielectric. 805 00:43:14,360 --> 00:43:16,700 So therefore, the ellipse doesn't necessarily 806 00:43:16,700 --> 00:43:19,180 provide the same answer anymore. 807 00:43:19,180 --> 00:43:21,380 So basically, you have to solve the problem again. 808 00:43:21,380 --> 00:43:23,360 And indeed, we find that the ellipse is not 809 00:43:23,360 --> 00:43:25,260 the answer in this case. 810 00:43:25,260 --> 00:43:26,690 The answer is a hyperbola. 811 00:43:26,690 --> 00:43:29,010 A hyperboloidal surface. 812 00:43:29,010 --> 00:43:30,790 Given by this equation over here. 813 00:43:30,790 --> 00:43:32,800 If you compare it with the case of the ellipse, 814 00:43:32,800 --> 00:43:34,813 the difference is a minus sign. 815 00:43:34,813 --> 00:43:36,480 So again, I will not do this derivation. 816 00:43:36,480 --> 00:43:38,400 I will let you do it by yourselves 817 00:43:38,400 --> 00:43:39,740 if you have the inclination. 818 00:43:39,740 --> 00:43:44,570 Or if we have time at the end today, we might do it. 819 00:43:44,570 --> 00:43:45,870 It's a very similar derivation. 820 00:43:45,870 --> 00:43:47,760 We apply Fermat's principle. 821 00:43:47,760 --> 00:43:50,430 We basically demand that the path 822 00:43:50,430 --> 00:43:54,660 from here to some reference point equals the path from here 823 00:43:54,660 --> 00:43:58,800 to the same reference plane off axis. 824 00:43:58,800 --> 00:44:00,390 When you apply this principle, you 825 00:44:00,390 --> 00:44:06,640 end up with this equation after some algebraic manipulation. 826 00:44:06,640 --> 00:44:08,802 So the word confusion. 827 00:44:08,802 --> 00:44:11,890 I actually put together a table that 828 00:44:11,890 --> 00:44:15,580 contains all the possible cases of first 829 00:44:15,580 --> 00:44:22,140 of all, reflective versus refractive focusing elements. 830 00:44:22,140 --> 00:44:26,550 And point sources, or point images at infinity. 831 00:44:26,550 --> 00:44:29,550 And there's actually six possible cases. 832 00:44:29,550 --> 00:44:32,700 For the kinds of a reflector, we saw that it doesn't really 833 00:44:32,700 --> 00:44:33,450 make a difference. 834 00:44:33,450 --> 00:44:36,390 In both cases, a paraboloid is the answer. 835 00:44:36,390 --> 00:44:40,320 Because I computed already the focusing 836 00:44:40,320 --> 00:44:42,840 of an object at infinity, which is 837 00:44:42,840 --> 00:44:46,020 focused at the focal point of the paraboloid. 838 00:44:46,020 --> 00:44:48,610 And if I reverse the rays, the same paraboloid 839 00:44:48,610 --> 00:44:52,880 will take a point source at F, and image that point source 840 00:44:52,880 --> 00:44:54,150 at infinity. 841 00:44:54,150 --> 00:44:56,090 So that is very simple. 842 00:44:56,090 --> 00:44:59,740 In the case of a dielectric, I can do two things. 843 00:44:59,740 --> 00:45:01,820 First of all, I start with the ellipsoid. 844 00:45:01,820 --> 00:45:05,400 When we derive the case of the ellipsoid, 845 00:45:05,400 --> 00:45:08,250 focusing a plane wave coming from infinity. 846 00:45:08,250 --> 00:45:12,720 And we saw that it focuses at the focal point of the ellipse, 847 00:45:12,720 --> 00:45:14,400 inside the ellipse. 848 00:45:14,400 --> 00:45:16,050 If I reverse the array paths, then I 849 00:45:16,050 --> 00:45:18,570 have this situation where the light originates 850 00:45:18,570 --> 00:45:20,030 at the center of the ellipse. 851 00:45:20,030 --> 00:45:22,470 And then it gets collimated as it comes out. 852 00:45:22,470 --> 00:45:24,450 Now, the reason the ellipse is still the answer 853 00:45:24,450 --> 00:45:28,830 here is because the light starts at the dielectric. 854 00:45:28,830 --> 00:45:30,420 So really, now, I'm correct. 855 00:45:30,420 --> 00:45:32,400 I can still revise the optical path, 856 00:45:32,400 --> 00:45:34,510 and I get the correct answer. 857 00:45:34,510 --> 00:45:39,950 So this has led to cases of objects at infinity, 858 00:45:39,950 --> 00:45:42,090 and image at infinity. 859 00:45:42,090 --> 00:45:48,470 But have to be careful that the object is in air. 860 00:45:48,470 --> 00:45:51,380 In this case, the object of infinity is in air. 861 00:45:51,380 --> 00:45:53,810 The focus is inside the dielectric. 862 00:45:53,810 --> 00:45:55,820 In this case, the source of the object 863 00:45:55,820 --> 00:45:59,030 is inside the dielectric, and the image at infinity 864 00:45:59,030 --> 00:45:59,870 is in air. 865 00:46:02,710 --> 00:46:04,540 The other case is the one that I just 866 00:46:04,540 --> 00:46:07,000 mentioned, and did not derive, but I 867 00:46:07,000 --> 00:46:08,840 let you derive by yourselves. 868 00:46:08,840 --> 00:46:10,810 And that is the case of a source that 869 00:46:10,810 --> 00:46:12,770 is now at a finite distance. 870 00:46:12,770 --> 00:46:17,530 And I wish the image to be at infinity inside the dielectric. 871 00:46:17,530 --> 00:46:22,250 So this case, this is hyperboloidal surface. 872 00:46:22,250 --> 00:46:23,753 And, of course, you might ask, well, 873 00:46:23,753 --> 00:46:25,170 what the heck does it really mean? 874 00:46:25,170 --> 00:46:29,428 Can I really make an infinitely large refractor? 875 00:46:29,428 --> 00:46:31,470 Well, that says you don't have to, because if any 876 00:46:31,470 --> 00:46:33,940 of the refractor is cut here. 877 00:46:33,940 --> 00:46:36,770 Since the rays are coming out normal, 878 00:46:36,770 --> 00:46:38,860 they do not refract at all at this interface. 879 00:46:38,860 --> 00:46:42,700 So really, you can chop the hyperbola, 880 00:46:42,700 --> 00:46:47,890 and then you can still have your point, your image at infinity. 881 00:46:47,890 --> 00:46:49,090 So that's fine. 882 00:46:49,090 --> 00:46:51,642 And, of course, you can also reverse the ray paths here. 883 00:46:51,642 --> 00:46:53,350 And you can create the opposite situation 884 00:46:53,350 --> 00:46:55,720 where whenever you have an object at infinity, 885 00:46:55,720 --> 00:46:57,820 but inside the dielectric. 886 00:46:57,820 --> 00:47:01,632 And you have a point image at a finite distance in air. 887 00:47:01,632 --> 00:47:03,340 And that could also be hyperbolic, right? 888 00:47:03,340 --> 00:47:08,200 Because this, I derive simply by flipping the ray paths 889 00:47:08,200 --> 00:47:09,320 from one case to the next. 890 00:47:16,510 --> 00:47:19,960 OK, any questions about this? 891 00:47:19,960 --> 00:47:21,540 AUDIENCE: I've got a question. 892 00:47:21,540 --> 00:47:26,620 If your point sources within a green lens, 893 00:47:26,620 --> 00:47:31,032 what is the corresponding index of refraction function? 894 00:47:31,032 --> 00:47:32,740 GEORGE BARBASTATHIS: Within a green lens? 895 00:47:32,740 --> 00:47:33,365 AUDIENCE: Yeah. 896 00:47:36,123 --> 00:47:38,040 GEORGE BARBASTATHIS: I will cover green lenses 897 00:47:38,040 --> 00:47:39,530 in about two weeks. 898 00:47:42,460 --> 00:47:46,170 We will do Hamiltonian optics, so yeah. 899 00:47:46,170 --> 00:47:49,020 AUDIENCE: I also have a question. 900 00:47:49,020 --> 00:47:54,000 So if you said with the hyperboloidal refractor, 901 00:47:54,000 --> 00:47:55,560 if you cut off the back end, the rays 902 00:47:55,560 --> 00:47:57,390 will continue to come out parallel. 903 00:47:57,390 --> 00:47:59,100 Does that work in the opposite direction? 904 00:47:59,100 --> 00:48:01,683 Like if you have a lens that's flat on one end and a hyperbola 905 00:48:01,683 --> 00:48:04,680 on the other, can you focus it to a random point? 906 00:48:04,680 --> 00:48:05,263 Or not random. 907 00:48:05,263 --> 00:48:06,305 GEORGE BARBASTATHIS: Yes. 908 00:48:06,305 --> 00:48:08,080 This case, for example, that would work. 909 00:48:08,080 --> 00:48:09,300 AUDIENCE: Yeah, I can't see. 910 00:48:09,300 --> 00:48:11,550 GEORGE BARBASTATHIS: If you have rays coming in, yeah. 911 00:48:11,550 --> 00:48:13,078 AUDIENCE: [INAUDIBLE] right, yeah. 912 00:48:13,078 --> 00:48:14,870 GEORGE BARBASTATHIS: Of course, the problem 913 00:48:14,870 --> 00:48:16,680 if you do it this way is that you have to cut off 914 00:48:16,680 --> 00:48:17,820 the hyperbol at some point. 915 00:48:17,820 --> 00:48:23,442 So you cannot extend your incoming bundle infinitely. 916 00:48:23,442 --> 00:48:24,150 AUDIENCE: Thanks. 917 00:48:25,578 --> 00:48:27,120 GEORGE BARBASTATHIS: Of course, these 918 00:48:27,120 --> 00:48:30,840 are all very highly idealized situations, I should emphasize. 919 00:48:35,990 --> 00:48:38,301 Any other questions? 920 00:48:46,010 --> 00:48:50,700 OK, so now that you have seen these, 921 00:48:50,700 --> 00:48:55,440 I would like to say a few things about wavefronts, 922 00:48:55,440 --> 00:48:57,360 and what happens to the wavefronts 923 00:48:57,360 --> 00:48:59,177 as they go through of these-- 924 00:48:59,177 --> 00:49:01,260 AUDIENCE: George, I think that's another question. 925 00:49:01,260 --> 00:49:02,000 GEORGE BARBASTATHIS: Oh yeah, I'm sorry. 926 00:49:02,000 --> 00:49:02,700 Yes. 927 00:49:02,700 --> 00:49:04,950 AUDIENCE: So would the ellipsoidal reflector 928 00:49:04,950 --> 00:49:08,430 also be more subject to spherical aberrations 929 00:49:08,430 --> 00:49:11,457 since the focal point is actually within the optic 930 00:49:11,457 --> 00:49:12,540 instead of in front of it? 931 00:49:14,670 --> 00:49:16,170 GEORGE BARBASTATHIS: Say that again? 932 00:49:16,170 --> 00:49:18,180 If it is in front? 933 00:49:18,180 --> 00:49:21,355 AUDIENCE: Yeah, it's comparing the-- 934 00:49:21,355 --> 00:49:22,980 GEORGE BARBASTATHIS: OK, I have not yet 935 00:49:22,980 --> 00:49:24,790 defined what is spherical aberration. 936 00:49:24,790 --> 00:49:27,600 Assuming that we know what spherical aberration is. 937 00:49:27,600 --> 00:49:30,165 All of these are free of spherical aberration. 938 00:49:32,790 --> 00:49:35,640 By construction, they focus light perfectly. 939 00:49:35,640 --> 00:49:39,190 However, a spherical aberration will come in, for example, 940 00:49:39,190 --> 00:49:41,055 if you-- 941 00:49:41,055 --> 00:49:41,930 let's take this case. 942 00:49:46,690 --> 00:49:50,340 Yeah, if you move the source off axis, 943 00:49:50,340 --> 00:49:53,530 then the plane wave of coming out will be aberrated. 944 00:49:53,530 --> 00:49:54,390 The same here. 945 00:49:54,390 --> 00:49:56,670 If you tilt this plane wave. 946 00:49:56,670 --> 00:50:00,840 Again, the point image will be aberrated. 947 00:50:00,840 --> 00:50:03,710 Actually, it would be aberrated in more than one way. 948 00:50:03,710 --> 00:50:08,220 It will contain comma as well as spherical 949 00:50:08,220 --> 00:50:12,870 So these are free of aberrations to the degree 950 00:50:12,870 --> 00:50:17,817 that they are also operated as design. 951 00:50:17,817 --> 00:50:19,650 But we'll say a little bit about aberrations 952 00:50:19,650 --> 00:50:20,780 in the coming slides. 953 00:50:20,780 --> 00:50:22,810 So maybe you can postpone that. 954 00:50:27,260 --> 00:50:28,410 Any other questions? 955 00:50:36,740 --> 00:50:38,720 So speaking of aberrations, here they come. 956 00:50:42,950 --> 00:50:45,910 First of all, let me say a few things about the wavefronts. 957 00:50:45,910 --> 00:50:50,960 So by definition, I will just look at this case, 958 00:50:50,960 --> 00:50:53,720 and then you can construct similar diagrams 959 00:50:53,720 --> 00:50:57,298 for all the six cases that were in the previous slide. 960 00:50:57,298 --> 00:50:58,340 I didn't want to do that. 961 00:50:58,340 --> 00:51:00,590 It actually takes a very long time to do such a slide, 962 00:51:00,590 --> 00:51:03,210 so I didn't want to do all of them. 963 00:51:03,210 --> 00:51:06,180 But this is pretty representative. 964 00:51:06,180 --> 00:51:08,840 So it conveys the idea, I think. 965 00:51:08,840 --> 00:51:11,290 So by definition, we have a plane wave coming in. 966 00:51:11,290 --> 00:51:14,960 So you have plane wavefronts arriving at the interface. 967 00:51:14,960 --> 00:51:18,230 And also by definition, because by construction 968 00:51:18,230 --> 00:51:23,200 we required all the rays, all the refractive rays. 969 00:51:23,200 --> 00:51:26,720 They are required to meet at the focal point over here. 970 00:51:26,720 --> 00:51:29,570 It means that what we have inside 971 00:51:29,570 --> 00:51:33,890 the refractor is converging spherical waves. 972 00:51:33,890 --> 00:51:38,130 And therefore, if you look at the wavefronts collapsing in 973 00:51:38,130 --> 00:51:42,150 towards the focal point, they are perfect spheres. 974 00:51:42,150 --> 00:51:46,180 This is all equal section, so these are perfect spheres 975 00:51:46,180 --> 00:51:49,533 collapsing towards uppercase F. It's 976 00:51:49,533 --> 00:51:51,450 kind of interesting to think what happens here 977 00:51:51,450 --> 00:51:53,170 at the interface. 978 00:51:53,170 --> 00:51:55,900 So [INAUDIBLE] the interface, you have two things happening. 979 00:51:55,900 --> 00:52:00,010 You have the ray arriving at the left, then sort of being 980 00:52:00,010 --> 00:52:02,178 refracted into the sphere. 981 00:52:02,178 --> 00:52:03,595 But the wavelength has to remain-- 982 00:52:08,870 --> 00:52:11,120 well, it has to remain continuous. 983 00:52:11,120 --> 00:52:13,190 So basically, what happens is the wavefronts, 984 00:52:13,190 --> 00:52:15,950 they curve in a continuous fashion, as shown here. 985 00:52:20,440 --> 00:52:25,320 You can think of it basically as the wave approaches 986 00:52:25,320 --> 00:52:28,490 from the left toward the refractor, 987 00:52:28,490 --> 00:52:30,740 as it starts entering the refractor, of course, 988 00:52:30,740 --> 00:52:32,840 it enters first at the apex. 989 00:52:32,840 --> 00:52:36,110 And then the wavefront starts bulging. 990 00:52:36,110 --> 00:52:38,730 It sort of creates this bulge. 991 00:52:38,730 --> 00:52:41,270 And then as it goes more and more in, 992 00:52:41,270 --> 00:52:49,130 the bulge progresses to create a segment of a spherical surface. 993 00:52:49,130 --> 00:52:53,860 Now, you can justify that with a number of different ways. 994 00:52:53,860 --> 00:52:57,190 One is, of course, Fermat's principle that we just applied. 995 00:52:57,190 --> 00:52:59,370 The other is if you think that light actually 996 00:52:59,370 --> 00:53:05,100 propagates slower in this medium than it propagates in air, 997 00:53:05,100 --> 00:53:07,770 then you can see why the wavefront starts bulging here. 998 00:53:07,770 --> 00:53:12,300 Because as soon as the light enters the medium, it gets 999 00:53:12,300 --> 00:53:13,230 slowed down. 1000 00:53:13,230 --> 00:53:14,730 So now that portion of the wavefront 1001 00:53:14,730 --> 00:53:17,370 that is still outside in air is actually 1002 00:53:17,370 --> 00:53:20,370 faster than the portion that is here. 1003 00:53:20,370 --> 00:53:22,650 Even if you compare, let's take this case. 1004 00:53:22,650 --> 00:53:25,210 If you compare the portion of the wavefront that is here 1005 00:53:25,210 --> 00:53:28,280 at the apex to the portion that is here, 1006 00:53:28,280 --> 00:53:29,880 to the portion that is here. 1007 00:53:29,880 --> 00:53:32,250 You can see that this one has traveled the longer 1008 00:53:32,250 --> 00:53:34,020 distance in the medium. 1009 00:53:34,020 --> 00:53:36,330 So that's why it is further behind. 1010 00:53:36,330 --> 00:53:38,760 This one has traveled slightly less, 1011 00:53:38,760 --> 00:53:41,250 so therefore, it is a little bit further ahead. 1012 00:53:41,250 --> 00:53:45,360 And this one is still in air, so it is even more ahead. 1013 00:53:45,360 --> 00:53:47,010 So this is another way to explain 1014 00:53:47,010 --> 00:53:49,680 why the wavefronts are bulging, and they 1015 00:53:49,680 --> 00:53:52,380 become spherical as the wave enters the medium. 1016 00:53:56,200 --> 00:54:02,600 Now, of course, they become exactly spherical. 1017 00:54:02,600 --> 00:54:07,480 It's very special to shape that we chose. 1018 00:54:07,480 --> 00:54:10,600 Because we demanded Fermat to hold, 1019 00:54:10,600 --> 00:54:14,500 the wavefronts that result from refraction 1020 00:54:14,500 --> 00:54:16,310 of this elliptical interface. 1021 00:54:16,310 --> 00:54:18,010 They actually become spherical. 1022 00:54:18,010 --> 00:54:20,260 But you can easily convince yourself 1023 00:54:20,260 --> 00:54:22,410 that if I chose a different surface. 1024 00:54:22,410 --> 00:54:25,490 Say, if I chose a sphere or a hyperbola 1025 00:54:25,490 --> 00:54:30,650 in this case, or some crazy polynomial, quartic, sixth 1026 00:54:30,650 --> 00:54:34,460 order, I don't know, some generalized surface. 1027 00:54:34,460 --> 00:54:36,430 Then the wavefronts that we get here. 1028 00:54:36,430 --> 00:54:39,640 They might still bulge, because the light would enter a slower 1029 00:54:39,640 --> 00:54:42,845 middle, but they wouldn't be spherical. 1030 00:54:42,845 --> 00:54:44,220 Now, the fact that the wavefronts 1031 00:54:44,220 --> 00:54:45,400 would not be spherical. 1032 00:54:45,400 --> 00:54:50,430 What it really implies is that the focus is not perfect 1033 00:54:50,430 --> 00:54:53,160 anymore. 1034 00:54:53,160 --> 00:54:54,940 If these surfaces of the wavefronts, 1035 00:54:54,940 --> 00:54:57,420 they deviate from spheres. 1036 00:54:57,420 --> 00:55:00,930 Then the rays would not really all be at [? F, ?] 1037 00:55:00,930 --> 00:55:04,240 but they would cross at various places. 1038 00:55:04,240 --> 00:55:08,220 And that is what you call an aberrated image, 1039 00:55:08,220 --> 00:55:09,920 or an aberrated wavefront. 1040 00:55:09,920 --> 00:55:20,550 So the two are related, because if, for example, the wavefront 1041 00:55:20,550 --> 00:55:24,300 deviates from a sphere, then you call it an aberrated wavefront. 1042 00:55:24,300 --> 00:55:28,380 If the image deviates from the ideal point image, where 1043 00:55:28,380 --> 00:55:30,990 all the rays meet to a sort of blurry image 1044 00:55:30,990 --> 00:55:34,680 where the rays fail to meet, then that's an aberrated image. 1045 00:55:34,680 --> 00:55:36,840 So I will show some examples of that 1046 00:55:36,840 --> 00:55:41,290 later on, today, and also later during the class. 1047 00:55:41,290 --> 00:55:46,470 So there's a special case when the-- 1048 00:55:46,470 --> 00:55:48,480 well, let me save that special case for later. 1049 00:55:51,030 --> 00:55:54,730 I will talk about this in a little bit. 1050 00:55:54,730 --> 00:55:58,410 What I would like to say next is why is it so significant to-- 1051 00:55:58,410 --> 00:56:01,740 why do we try so hard to create these point images? 1052 00:56:08,100 --> 00:56:11,660 We're surrounded by objects that are-- 1053 00:56:11,660 --> 00:56:14,870 actually, most of the objects that we see around us 1054 00:56:14,870 --> 00:56:17,400 are reflective. 1055 00:56:17,400 --> 00:56:18,650 They are opaque. 1056 00:56:18,650 --> 00:56:23,760 So light that is entered in from a light source, 1057 00:56:23,760 --> 00:56:26,640 for example, a light bulb, or the sunlight when 1058 00:56:26,640 --> 00:56:27,810 we're outdoors. 1059 00:56:27,810 --> 00:56:31,270 What happens to the light as it hits the objects that 1060 00:56:31,270 --> 00:56:32,700 are surrounding us. 1061 00:56:32,700 --> 00:56:34,350 It scatters. 1062 00:56:34,350 --> 00:56:37,320 So scattering really means that the light 1063 00:56:37,320 --> 00:56:39,660 that is arriving from a source. 1064 00:56:39,660 --> 00:56:43,800 Each point in the object, it creates a highly divergent 1065 00:56:43,800 --> 00:56:45,060 spherical wave. 1066 00:56:45,060 --> 00:56:46,920 So basically, each point in the object 1067 00:56:46,920 --> 00:56:50,160 becomes a secondary point source. 1068 00:56:50,160 --> 00:56:51,600 This is part of a big-- 1069 00:56:51,600 --> 00:56:54,540 of another principle in light propagation 1070 00:56:54,540 --> 00:56:56,640 called the Huygen's principle. 1071 00:56:56,640 --> 00:56:59,050 But we're not quite ready to see that principle yet. 1072 00:56:59,050 --> 00:57:02,610 It needs a little bit more careful definition. 1073 00:57:02,610 --> 00:57:06,000 So for now, one way to justify in your mind 1074 00:57:06,000 --> 00:57:10,040 why this might happen is that the objects surrounding us. 1075 00:57:10,040 --> 00:57:12,210 They are composed of atoms. 1076 00:57:12,210 --> 00:57:17,190 And each atom, when it gets hit by the light beam. 1077 00:57:17,190 --> 00:57:18,690 It will be set in motion. 1078 00:57:18,690 --> 00:57:20,280 And you can imagine the situation 1079 00:57:20,280 --> 00:57:24,120 where the motion of the atom will irradiate a new light 1080 00:57:24,120 --> 00:57:24,900 beam. 1081 00:57:24,900 --> 00:57:26,580 So these light beams that are coming out 1082 00:57:26,580 --> 00:57:33,380 of there of the atoms composing the surfaces surrounding us. 1083 00:57:33,380 --> 00:57:36,020 This is what we perceive as-- 1084 00:57:36,020 --> 00:57:40,047 I should say, what we receive as scattered light. 1085 00:57:40,047 --> 00:57:41,880 And it sounds a little bit like hocus pocus, 1086 00:57:41,880 --> 00:57:43,713 but it's actually not very far from reality. 1087 00:57:43,713 --> 00:57:45,870 You can justify the argument I just 1088 00:57:45,870 --> 00:57:50,300 made with some very basic principles. 1089 00:57:50,300 --> 00:57:51,883 So it is more or less correct. 1090 00:57:51,883 --> 00:57:53,550 The bottom line is that the illumination 1091 00:57:53,550 --> 00:57:56,880 becomes these divergent spherical wavefronts. 1092 00:57:56,880 --> 00:57:58,980 Now, if you are sitting here, and you're 1093 00:57:58,980 --> 00:58:02,130 observing all the rays coming from the object. 1094 00:58:02,130 --> 00:58:05,610 You can imagine that it's a mess. 1095 00:58:05,610 --> 00:58:08,310 For example, here, you have rays coming from this point, 1096 00:58:08,310 --> 00:58:10,440 and from this point, and from this point, 1097 00:58:10,440 --> 00:58:13,230 from everywhere, everywhere inside the object. 1098 00:58:13,230 --> 00:58:15,240 You have rays arriving here. 1099 00:58:15,240 --> 00:58:17,490 So it is very difficult to understand what's going on. 1100 00:58:17,490 --> 00:58:21,420 All of these rays, they create kind of a very blurred image, 1101 00:58:21,420 --> 00:58:25,450 very similar to what happens to those of us who have glasses. 1102 00:58:25,450 --> 00:58:28,500 When we take off our glasses, everything appears blurry. 1103 00:58:28,500 --> 00:58:31,380 Well, it is not as bad as it would have been in this case. 1104 00:58:31,380 --> 00:58:36,390 But the reason things are blurry in our eyes is because when we 1105 00:58:36,390 --> 00:58:40,330 take our glasses off, we cannot form that effect-- 1106 00:58:40,330 --> 00:58:42,480 well, it's never perfect, but we can not 1107 00:58:42,480 --> 00:58:48,842 form good enough quality images in our eye. 1108 00:58:48,842 --> 00:58:51,300 So what we need in order to design these kind of situations 1109 00:58:51,300 --> 00:58:54,720 is an optical system, or a sequence 1110 00:58:54,720 --> 00:58:57,210 of optical elements that will pick up 1111 00:58:57,210 --> 00:59:00,450 these divergent spherical wavefronts, 1112 00:59:00,450 --> 00:59:03,870 and will convert them into converging 1113 00:59:03,870 --> 00:59:05,730 spherical wavefronts. 1114 00:59:05,730 --> 00:59:08,130 And if we can somehow create an optical system 1115 00:59:08,130 --> 00:59:11,670 that will take each and every one of these wavefronts. 1116 00:59:11,670 --> 00:59:17,040 Now, we're talking, of course, about an infinitely large 1117 00:59:17,040 --> 00:59:22,920 number of point scatterers that are generated here. 1118 00:59:22,920 --> 00:59:25,530 But if you can somehow take each and every one of them, 1119 00:59:25,530 --> 00:59:31,290 and I can convert it into a converging spherical wave, then 1120 00:59:31,290 --> 00:59:34,840 when I put all these converging spherical waves together, 1121 00:59:34,840 --> 00:59:36,480 I will actually get an image, which 1122 00:59:36,480 --> 00:59:39,600 will be as close to perfect as I could possibly 1123 00:59:39,600 --> 00:59:41,820 want within the limits of approximation 1124 00:59:41,820 --> 00:59:45,240 of geometrical optics. 1125 00:59:45,240 --> 00:59:47,280 Another way to express it, which is really 1126 00:59:47,280 --> 00:59:50,580 like a fancy mathematical term, but it is quite appropriate 1127 00:59:50,580 --> 00:59:55,130 here is the imaging system creates a map. 1128 00:59:55,130 --> 01:00:00,820 Creates a map from point sources in the object space 1129 01:00:00,820 --> 01:00:05,090 to point images in the image space. 1130 01:00:05,090 --> 01:00:07,970 And this map is, of course, very desirable. 1131 01:00:07,970 --> 01:00:12,860 Because if it fails, if I tried to detect the image over here 1132 01:00:12,860 --> 01:00:17,270 that is before the rays have come to focus. 1133 01:00:17,270 --> 01:00:20,840 Then the map becomes ill-defined, 1134 01:00:20,840 --> 01:00:24,820 because each point over here is receiving arrays 1135 01:00:24,820 --> 01:00:26,970 from multiple points in the object. 1136 01:00:26,970 --> 01:00:28,760 This is exactly the same situation 1137 01:00:28,760 --> 01:00:32,270 I was referring to before, when we take off our glasses. 1138 01:00:32,270 --> 01:00:35,360 Those of us unfortunate enough to require glasses. 1139 01:00:35,360 --> 01:00:39,350 For the rest of you who do not need glasses in everyday life, 1140 01:00:39,350 --> 01:00:41,630 you can, of course, create that same situation 1141 01:00:41,630 --> 01:00:45,350 by putting on a pair of glasses that you can borrow 1142 01:00:45,350 --> 01:00:46,940 from one of your colleagues. 1143 01:00:46,940 --> 01:00:48,500 So in both cases, you can add what 1144 01:00:48,500 --> 01:00:50,360 we call the focus, which basically 1145 01:00:50,360 --> 01:00:54,680 means that the image is not just imperfect, but it is blurry. 1146 01:00:54,680 --> 01:00:58,290 It is very far from perfect. 1147 01:00:58,290 --> 01:01:01,650 However, from what you must have gathered already 1148 01:01:01,650 --> 01:01:05,100 from the previous discussion, this is a very difficult task. 1149 01:01:05,100 --> 01:01:07,170 In fact, it is an impossible task. 1150 01:01:07,170 --> 01:01:12,660 Because you saw that even in the case of a simple, 1151 01:01:12,660 --> 01:01:17,440 ellipsoidal, or hyperboloidal focusing element. 1152 01:01:17,440 --> 01:01:22,830 You could see that the one to one mapping works only on axis. 1153 01:01:22,830 --> 01:01:27,320 If you tried to focus simultaneously points off axis, 1154 01:01:27,320 --> 01:01:31,220 well, then, of course, it stops to work. 1155 01:01:31,220 --> 01:01:34,390 It is very easy to convince yourself in geometrical terms. 1156 01:01:34,390 --> 01:01:36,170 Even then, there is some special cases 1157 01:01:36,170 --> 01:01:39,110 where you can have an entire surface that is in focus. 1158 01:01:39,110 --> 01:01:41,300 But we will talk about this later. 1159 01:01:43,910 --> 01:01:46,520 So this is actually-- 1160 01:01:46,520 --> 01:01:50,060 you could think of it as a very unfortunate event. 1161 01:01:50,060 --> 01:01:51,920 I think it as a fortunate event, because it 1162 01:01:51,920 --> 01:01:54,720 keeps people like me, and Professor Sheppard, 1163 01:01:54,720 --> 01:01:57,680 and a lot of colleagues, it keeps us employed. 1164 01:01:57,680 --> 01:02:00,080 Because this is the whole reason people 1165 01:02:00,080 --> 01:02:03,140 need optical systems is they need people like us. 1166 01:02:03,140 --> 01:02:06,890 So it is a very fortunate fact. 1167 01:02:06,890 --> 01:02:11,930 But what I'm trying to say here is that this job of perfect 1168 01:02:11,930 --> 01:02:14,630 imaging cannot be done exactly. 1169 01:02:14,630 --> 01:02:17,210 So therefore, the science, or if you wish, 1170 01:02:17,210 --> 01:02:19,890 the engineering of designing optical systems 1171 01:02:19,890 --> 01:02:23,150 is to arrive at the best compromise within the cost, 1172 01:02:23,150 --> 01:02:26,630 constraints, the elements that you have available, 1173 01:02:26,630 --> 01:02:30,607 the physics, of course, this is definitely inviolable. 1174 01:02:30,607 --> 01:02:31,940 You can violate a budget, right? 1175 01:02:31,940 --> 01:02:33,830 You can run a budget into the red, 1176 01:02:33,830 --> 01:02:37,450 as Wall Street banks discovered in the last six months. 1177 01:02:37,450 --> 01:02:39,200 But you can certainly not violate physics. 1178 01:02:39,200 --> 01:02:41,540 Even bankers cannot violate physics. 1179 01:02:41,540 --> 01:02:44,600 So within all these constraints imposed by physics, economics, 1180 01:02:44,600 --> 01:02:47,270 and so on so forth, how can you make an imaging system that 1181 01:02:47,270 --> 01:02:51,150 does the best job possible? 1182 01:02:51,150 --> 01:02:56,100 OK, so let's go back to this question 1183 01:02:56,100 --> 01:03:00,610 of how can we create good enough quality images so that we 1184 01:03:00,610 --> 01:03:02,920 can design imaging systems? 1185 01:03:02,920 --> 01:03:05,630 So let's look at a slightly different case, now. 1186 01:03:05,630 --> 01:03:10,520 So far, we saw pairs of images and sources, where 1187 01:03:10,520 --> 01:03:12,020 one of the tools at infinity. 1188 01:03:12,020 --> 01:03:14,930 So objects at infinity, finite images. 1189 01:03:17,560 --> 01:03:23,210 I mean, finite distant sources and images at infinity. 1190 01:03:23,210 --> 01:03:25,460 We start to deal with this case, where we have a point 1191 01:03:25,460 --> 01:03:27,750 object and a point image. 1192 01:03:27,750 --> 01:03:30,350 So both are now at finite distances. 1193 01:03:30,350 --> 01:03:34,773 Based on what we saw so far, and assuming that you're not 1194 01:03:34,773 --> 01:03:35,690 looking at your notes. 1195 01:03:35,690 --> 01:03:37,190 The answer is already in your notes, 1196 01:03:37,190 --> 01:03:39,560 if you reach that page in the notes. 1197 01:03:39,560 --> 01:03:42,720 But if you haven't seen that, then your time to guess. 1198 01:03:42,720 --> 01:03:45,780 What would be the ideal imager for this case that 1199 01:03:45,780 --> 01:03:51,780 would take a point object and focus it perfectly 1200 01:03:51,780 --> 01:03:55,140 into a point image on axis? 1201 01:03:55,140 --> 01:03:56,960 Did we not answer already based on-- 1202 01:04:04,990 --> 01:04:09,340 AUDIENCE: The concatenation of two hyperbolic reflectors. 1203 01:04:09,340 --> 01:04:11,020 GEORGE BARBASTATHIS: That's right. 1204 01:04:11,020 --> 01:04:13,587 So that is not necessarily one way of doing it, 1205 01:04:13,587 --> 01:04:15,170 but it's certainly one way of doing it 1206 01:04:15,170 --> 01:04:16,900 based on what we learn. 1207 01:04:16,900 --> 01:04:20,200 If I put two refractive surfaces like this, 1208 01:04:20,200 --> 01:04:23,350 I can arrange the first one to be a hyperbola. 1209 01:04:23,350 --> 01:04:27,340 What it will do is it will collimate. 1210 01:04:27,340 --> 01:04:32,190 Yeah, it will collimate the incident divergent wavefront. 1211 01:04:32,190 --> 01:04:34,760 And then if I can arrange for the second one 1212 01:04:34,760 --> 01:04:40,000 to be also hyperbola, then that will focus. 1213 01:04:40,000 --> 01:04:43,330 The plane wave that was propagated here, 1214 01:04:43,330 --> 01:04:46,880 it will focus into a perfect point image. 1215 01:04:46,880 --> 01:04:49,120 So this is now, again, a perfect imager. 1216 01:04:49,120 --> 01:04:51,730 But now, it works for finite distances. 1217 01:04:51,730 --> 01:04:54,640 One focal distance in front of the hyperbola 1218 01:04:54,640 --> 01:04:58,610 to one focal distance after the hyperbola. 1219 01:04:58,610 --> 01:05:03,860 And, well, just for references, these 1220 01:05:03,860 --> 01:05:06,050 are the two questions of the hyperbola. 1221 01:05:06,050 --> 01:05:08,090 I was a little bit careful here to define 1222 01:05:08,090 --> 01:05:10,775 the axis, and the displacement, and so on, 1223 01:05:10,775 --> 01:05:11,900 so that the question works. 1224 01:05:11,900 --> 01:05:13,240 I don't want to belabor that. 1225 01:05:13,240 --> 01:05:16,930 I let you go back and convince yourself that it is correct, 1226 01:05:16,930 --> 01:05:19,460 or convince yourself that is not correct, in which case, 1227 01:05:19,460 --> 01:05:22,140 please let me know so that I can correct it. 1228 01:05:22,140 --> 01:05:23,680 But I think I got it right. 1229 01:05:23,680 --> 01:05:27,370 Anyway, so these are the two equations of the hyperboloids. 1230 01:05:27,370 --> 01:05:30,020 And this is what in optics is very often 1231 01:05:30,020 --> 01:05:32,880 referred to as an asphere. 1232 01:05:32,880 --> 01:05:36,930 Now, this can be slightly confusing. 1233 01:05:36,930 --> 01:05:39,490 So I'm going to say it a few times here, 1234 01:05:39,490 --> 01:05:41,780 and then I'm going to ask you to go back home, read it 1235 01:05:41,780 --> 01:05:44,260 a few times, then print it out. 1236 01:05:44,260 --> 01:05:46,760 Post it in front of your bathroom mirror 1237 01:05:46,760 --> 01:05:49,320 on top of your bed and so on, whatever. 1238 01:05:49,320 --> 01:05:52,740 So you can see it a few times, and memorize it. 1239 01:05:52,740 --> 01:05:57,230 OK, so this type of element whose surface 1240 01:05:57,230 --> 01:06:02,450 is optimized to give a desired focus and behavior 1241 01:06:02,450 --> 01:06:04,220 is called in asphere. 1242 01:06:04,220 --> 01:06:06,200 Again, I have a benefit being Greek. 1243 01:06:06,200 --> 01:06:10,220 This A in front of the sphere means not a sphere. 1244 01:06:15,710 --> 01:06:17,190 Give an example from everyday life. 1245 01:06:17,190 --> 01:06:18,580 I can only think technical terms, 1246 01:06:18,580 --> 01:06:21,530 and isotropic, aplanatic. 1247 01:06:21,530 --> 01:06:23,840 Give me an example of from everyday life. 1248 01:06:23,840 --> 01:06:24,655 AUDIENCE: Atypical? 1249 01:06:24,655 --> 01:06:26,780 GEORGE BARBASTATHIS: Atypical, there you are, yeah. 1250 01:06:26,780 --> 01:06:27,280 Thank you. 1251 01:06:29,990 --> 01:06:32,800 OK, so it is atypical as in asphere. 1252 01:06:32,800 --> 01:06:36,250 So another term that is very commonly used 1253 01:06:36,250 --> 01:06:37,750 is in aspheric lens. 1254 01:06:43,180 --> 01:06:47,800 Again, I want to emphasize that this kind of a sphere. 1255 01:06:47,800 --> 01:06:50,542 It works perfectly on axis. 1256 01:06:50,542 --> 01:06:52,000 But as you can imagine, if you were 1257 01:06:52,000 --> 01:06:56,230 to move this point object off axis, it would go off. 1258 01:06:56,230 --> 01:07:00,440 It could create an image, but it would be highly aberrated. 1259 01:07:00,440 --> 01:07:04,865 It would be subject to, well, definitely spherical and comma 1260 01:07:04,865 --> 01:07:05,365 possibly. 1261 01:07:08,438 --> 01:07:09,480 It's a complicated story. 1262 01:07:14,743 --> 01:07:16,910 You can imagine that this doesn't happen at the way. 1263 01:07:16,910 --> 01:07:20,220 There's probably a range over here for point sources at that 1264 01:07:20,220 --> 01:07:24,110 near enough the axis that this refractor would work actually 1265 01:07:24,110 --> 01:07:25,280 quite well. 1266 01:07:25,280 --> 01:07:27,230 So the question is can I make such an element? 1267 01:07:31,300 --> 01:07:33,060 To answer this question, I should tell you 1268 01:07:33,060 --> 01:07:36,690 a little bit about how optical elements are usually made. 1269 01:07:36,690 --> 01:07:39,480 So what people at least used to do 1270 01:07:39,480 --> 01:07:42,630 in the old days of expensive optical elements, 1271 01:07:42,630 --> 01:07:44,130 is they would take a piece of glass. 1272 01:07:44,130 --> 01:07:47,160 They would mount it on a rotary stage. 1273 01:07:47,160 --> 01:07:51,150 Then they would bring a diamond tip near the glass. 1274 01:07:51,150 --> 01:07:53,700 And they would start rotating the glass. 1275 01:07:53,700 --> 01:07:58,785 And they would rotate basically not at all at the center, 1276 01:07:58,785 --> 01:08:01,860 and they would let the diamond touch the glass longer 1277 01:08:01,860 --> 01:08:06,345 and longer as they went away from the tip of the glass. 1278 01:08:06,345 --> 01:08:07,720 The result of that is, of course, 1279 01:08:07,720 --> 01:08:09,630 that you create a spherical surface 1280 01:08:09,630 --> 01:08:13,050 if you move the diamond at a fixed velocity. 1281 01:08:13,050 --> 01:08:15,750 So a fixed velocity was very easy. 1282 01:08:15,750 --> 01:08:17,430 Because all you needed is a model that 1283 01:08:17,430 --> 01:08:20,430 moves a diamond tip linearly. 1284 01:08:20,430 --> 01:08:22,229 That's pretty easy to do. 1285 01:08:22,229 --> 01:08:25,770 To make a hyperbola, you have to plant the path of the diamond 1286 01:08:25,770 --> 01:08:31,970 tip in a way that results in this nonuniform curvature. 1287 01:08:31,970 --> 01:08:35,189 So this used to be thought as a very difficult task. 1288 01:08:35,189 --> 01:08:35,689 Come on in. 1289 01:08:35,689 --> 01:08:36,410 Someone's outside. 1290 01:08:36,410 --> 01:08:36,910 Come on in. 1291 01:08:36,910 --> 01:08:39,404 Don't be shy. 1292 01:08:39,404 --> 01:08:40,279 She changed her mind. 1293 01:08:45,710 --> 01:08:50,359 But in the last 20 years or so, the field 1294 01:08:50,359 --> 01:08:53,038 of mechatronics that many of you are familiar with then 1295 01:08:53,038 --> 01:08:53,830 had advanced a lot. 1296 01:08:53,830 --> 01:08:58,310 So people who are able to design diamond tools that 1297 01:08:58,310 --> 01:09:01,939 could very carefully plan the path of the diamond tip, 1298 01:09:01,939 --> 01:09:05,580 and actually create aspheric elements, such as this one. 1299 01:09:05,580 --> 01:09:07,140 But it still remains very expensive. 1300 01:09:07,140 --> 01:09:10,408 So it could be an order of magnitude difference in price. 1301 01:09:10,408 --> 01:09:12,200 If you need to compare a spherical element, 1302 01:09:12,200 --> 01:09:14,040 compare it to an asphere. 1303 01:09:14,040 --> 01:09:14,540 Another 1304 01:09:14,540 --> 01:09:18,140 Major invention that helped this business is injection molding. 1305 01:09:18,140 --> 01:09:20,330 Instead of actually making the optical element 1306 01:09:20,330 --> 01:09:22,160 itself, what people do nowadays is 1307 01:09:22,160 --> 01:09:24,859 they can make a very expensive negative. 1308 01:09:24,859 --> 01:09:28,069 And then they inject plastic into the negative. 1309 01:09:28,069 --> 01:09:30,439 They solidify the plastic, and then they take it out. 1310 01:09:30,439 --> 01:09:33,319 And they can get any arbitrary surface they want. 1311 01:09:33,319 --> 01:09:36,529 That's an OK way to make a spheric optical element. 1312 01:09:36,529 --> 01:09:39,605 Nevertheless, spherical elements remain very popular. 1313 01:09:39,605 --> 01:09:42,470 And there's a number of reasons for these, [INAUDIBLE] spheres. 1314 01:09:42,470 --> 01:09:45,500 They can work very well on axis, but they 1315 01:09:45,500 --> 01:09:47,870 can have other problems that are perhaps 1316 01:09:47,870 --> 01:09:51,500 much more severe when they go off axis, which is not a-- 1317 01:09:51,500 --> 01:09:53,479 it is not a shoe in, so to speak, 1318 01:09:53,479 --> 01:09:57,630 to use aspheric optical elements. 1319 01:09:57,630 --> 01:10:02,140 So for that reason, we'll spend quite a bit of time, 1320 01:10:02,140 --> 01:10:06,190 and also, I guess, because of tradition 1321 01:10:06,190 --> 01:10:09,310 to analyze what happens to light as it goes 1322 01:10:09,310 --> 01:10:12,947 through spherical surfaces. 1323 01:10:12,947 --> 01:10:14,280 So spherical surface, of course. 1324 01:10:14,280 --> 01:10:17,470 If you replace the hyperbolas here with a sphere. 1325 01:10:17,470 --> 01:10:20,800 Let's say I take a sphere. 1326 01:10:20,800 --> 01:10:23,050 I need to say this very clearly, because the terms can 1327 01:10:23,050 --> 01:10:24,370 become confusing. 1328 01:10:24,370 --> 01:10:28,960 If I replace the asphere with a perfect sphere, 1329 01:10:28,960 --> 01:10:30,292 that is, with a ball. 1330 01:10:30,292 --> 01:10:32,500 Then, of course, as you can imagine, what will happen 1331 01:10:32,500 --> 01:10:35,290 is the wavefront, even inside the sphere 1332 01:10:35,290 --> 01:10:37,100 will not be planar anymore. 1333 01:10:37,100 --> 01:10:39,370 You can convince yourself if you think about it 1334 01:10:39,370 --> 01:10:42,940 that because this sphere actually bends faster 1335 01:10:42,940 --> 01:10:46,690 than the hyperbola, the arrays that are away from the axes. 1336 01:10:46,690 --> 01:10:47,980 They will bend more. 1337 01:10:47,980 --> 01:10:52,323 So you get a slightly focusing in effect inside the sphere. 1338 01:10:52,323 --> 01:10:53,990 Because, of course, not a perfect focus. 1339 01:10:53,990 --> 01:10:56,312 It is itself very high aberrated. 1340 01:10:56,312 --> 01:10:57,770 And by the time they come out, they 1341 01:10:57,770 --> 01:10:59,150 become aberrated even more. 1342 01:10:59,150 --> 01:11:03,530 So you get a very poor quality focus in general. 1343 01:11:03,530 --> 01:11:06,010 There is a special case where this might not happen, 1344 01:11:06,010 --> 01:11:07,790 but I will skip that for now. 1345 01:11:07,790 --> 01:11:10,020 So in general, you get an aberrated image. 1346 01:11:10,020 --> 01:11:12,620 So this is what they meant before by aberration. 1347 01:11:12,620 --> 01:11:15,230 You see that the rays, they form a blur here. 1348 01:11:15,230 --> 01:11:17,750 They fail to meet at the perfect focus, 1349 01:11:17,750 --> 01:11:20,360 as it would have been required by the perfect optical system. 1350 01:11:23,470 --> 01:11:27,460 And this particular type of aberration 1351 01:11:27,460 --> 01:11:32,020 that occurs when you use a spherical surface 1352 01:11:32,020 --> 01:11:35,650 or axis in this system is called spherical aberration. 1353 01:11:35,650 --> 01:11:37,660 Someone asked earlier about spherical. 1354 01:11:37,660 --> 01:11:38,530 This is spherical. 1355 01:11:38,530 --> 01:11:40,728 AUDIENCE: I've got a question. 1356 01:11:40,728 --> 01:11:41,770 GEORGE BARBASTATHIS: Yes. 1357 01:11:41,770 --> 01:11:42,985 AUDIENCE: How do you pick z2? 1358 01:11:45,800 --> 01:11:48,760 Like where do you choose to pick z2? 1359 01:11:48,760 --> 01:11:51,730 Is it at the focus of the closest rays, 1360 01:11:51,730 --> 01:11:54,070 or the furthest rays? 1361 01:11:54,070 --> 01:11:56,153 GEORGE BARBASTATHIS: This came from the hyperbola. 1362 01:12:14,130 --> 01:12:16,350 z1, z2, they come from the hyperbola. 1363 01:12:16,350 --> 01:12:20,960 Depending on the curvature over here, define z1 and z2. 1364 01:12:24,520 --> 01:12:27,000 And then I selected the sphere to match 1365 01:12:27,000 --> 01:12:30,200 the curvature of the hyperbola near to the axis. 1366 01:12:30,200 --> 01:12:33,090 So if you look at the animation again. 1367 01:12:33,090 --> 01:12:36,920 Near the axis, you will see that they actually perfectly match. 1368 01:12:36,920 --> 01:12:39,900 But then, of course, the sphere carves faster 1369 01:12:39,900 --> 01:12:43,165 than the hyperbola. 1370 01:12:43,165 --> 01:12:44,540 AUDIENCE: I think the question is 1371 01:12:44,540 --> 01:12:48,170 in the spherically aberrated case where you can not 1372 01:12:48,170 --> 01:12:53,698 find all the points meeting, how can you define the z2? 1373 01:12:53,698 --> 01:12:55,990 GEORGE BARBASTATHIS: OK, that's a really good question. 1374 01:12:55,990 --> 01:12:58,670 So the way you defined z2 is you look 1375 01:12:58,670 --> 01:13:02,570 at the two rays that are really, really close to the axis. 1376 01:13:02,570 --> 01:13:04,380 For those two rays that are propagating 1377 01:13:04,380 --> 01:13:08,457 at the very small angle, then the same z2 applies. 1378 01:13:08,457 --> 01:13:10,040 Then, of course, the rest of the rays. 1379 01:13:10,040 --> 01:13:15,280 They go off very far from that point. 1380 01:13:15,280 --> 01:13:18,180 But anyways, z2 here is exactly the same as it was before. 1381 01:13:22,290 --> 01:13:23,540 Did that answer your question? 1382 01:13:27,627 --> 01:13:28,210 AUDIENCE: Yes. 1383 01:13:28,210 --> 01:13:33,970 Long 1384 01:13:33,970 --> 01:13:35,950 GEORGE BARBASTATHIS: Now, if you go off axis, 1385 01:13:35,950 --> 01:13:38,405 then other types of aberrations kick in. 1386 01:13:38,405 --> 01:13:39,780 So we're not mentioning them now. 1387 01:13:39,780 --> 01:13:41,980 We'll come back to this topic later, 1388 01:13:41,980 --> 01:13:45,420 and we find them in more detail. 1389 01:13:45,420 --> 01:13:48,590 But for now, keep this in mind, that if you 1390 01:13:48,590 --> 01:13:54,110 use a sphere on axis. 1391 01:13:54,110 --> 01:13:55,390 Let me repeat. 1392 01:13:55,390 --> 01:14:01,000 If you use a sphere to focus a point source on axis 1393 01:14:01,000 --> 01:14:04,460 into a point image, also an axis at the finite distance, 1394 01:14:04,460 --> 01:14:06,370 then you get spherical aberration in general. 1395 01:14:13,560 --> 01:14:19,670 Well, that is true, but if you put it over here. 1396 01:14:19,670 --> 01:14:21,990 Just move the axis, yes. 1397 01:14:21,990 --> 01:14:23,730 Well, yeah. 1398 01:14:23,730 --> 01:14:26,310 OK. 1399 01:14:26,310 --> 01:14:28,260 Yeah, thank you. 1400 01:14:28,260 --> 01:14:30,880 Let me define what I mean by the axis here. 1401 01:14:30,880 --> 01:14:35,550 So typically, film and camera is the elements 1402 01:14:35,550 --> 01:14:40,330 that we use in order to capture images flat. 1403 01:14:40,330 --> 01:14:46,800 So if I put here either my camera lens. 1404 01:14:46,800 --> 01:14:49,480 I should say the tip of the camera, the camera 1405 01:14:49,480 --> 01:14:54,120 dial, or a film. 1406 01:14:54,120 --> 01:14:57,210 That defines an axis that is perpendicular to the film. 1407 01:14:57,210 --> 01:14:59,700 So Colin is correct, that if I take a point here, 1408 01:14:59,700 --> 01:15:03,240 and then move it sort of radially through the sphere. 1409 01:15:03,240 --> 01:15:06,660 It will form a perfect image over here. 1410 01:15:06,660 --> 01:15:10,420 The fact that I try to detect the image not at this point, 1411 01:15:10,420 --> 01:15:13,740 but on the plane, means that I will be subject to aberrations. 1412 01:15:13,740 --> 01:15:17,930 This is the proper definition of aberration. 1413 01:15:17,930 --> 01:15:20,310 And, of course, even the on axis point 1414 01:15:20,310 --> 01:15:25,170 will have spherical aberration, as I mentioned earlier. 1415 01:15:28,535 --> 01:15:29,660 Did I answer that question? 1416 01:15:35,090 --> 01:15:38,610 OK, so let's see how we deal with the refraction 1417 01:15:38,610 --> 01:15:39,300 from a sphere. 1418 01:15:41,700 --> 01:15:43,200 So [INAUDIBLE],, as you can imagine, 1419 01:15:43,200 --> 01:15:45,990 is a quite complicated problem because here 1420 01:15:45,990 --> 01:15:51,150 is a ray that is arriving from some point source on axis. 1421 01:15:51,150 --> 01:15:52,770 It hits the sphere. 1422 01:15:52,770 --> 01:15:55,530 And then my job is to find which direction this ray 1423 01:15:55,530 --> 01:15:58,670 will be refracted inside the sphere. 1424 01:15:58,670 --> 01:16:01,010 So now, again, I can use one of the tools 1425 01:16:01,010 --> 01:16:02,300 that we have in our arsenal. 1426 01:16:02,300 --> 01:16:06,230 I can use Fermat's principle, or I can use Snell's law. 1427 01:16:06,230 --> 01:16:08,155 The two are, of course, identical. 1428 01:16:08,155 --> 01:16:09,780 The fact of the matter is in this case, 1429 01:16:09,780 --> 01:16:13,160 it would be quite complicated to do. 1430 01:16:13,160 --> 01:16:15,200 So we'll do simplification. 1431 01:16:15,200 --> 01:16:17,840 First of all, let's define some notation here. 1432 01:16:17,840 --> 01:16:21,340 I will denote as x the distance of the ray 1433 01:16:21,340 --> 01:16:23,550 intersects the sphere. 1434 01:16:23,550 --> 01:16:29,690 I will define as alpha left the angle 1435 01:16:29,690 --> 01:16:31,730 that the ray makes with the optical axis 1436 01:16:31,730 --> 01:16:34,860 to the left of the interface. 1437 01:16:34,860 --> 01:16:36,910 This is the interface. 1438 01:16:36,910 --> 01:16:38,900 And then alpha right is the angle 1439 01:16:38,900 --> 01:16:41,820 that the ray makes with respect to the horizontal, 1440 01:16:41,820 --> 01:16:47,260 the optical axis the right hand side of the interface. 1441 01:16:47,260 --> 01:16:48,730 OK, that's my notation. 1442 01:16:48,730 --> 01:16:51,710 And now for generality, I do not assume 1443 01:16:51,710 --> 01:16:53,440 that this is air anymore, I just assume 1444 01:16:53,440 --> 01:16:55,550 that I have two dielectrics. 1445 01:16:55,550 --> 01:16:58,490 One has index n sub left, and the other 1446 01:16:58,490 --> 01:17:00,850 has index n sub right. 1447 01:17:00,850 --> 01:17:03,350 It's a little bit cumbersome to carry left and right around, 1448 01:17:03,350 --> 01:17:06,650 but it is convenient because we will leave that for a mnemonic 1449 01:17:06,650 --> 01:17:08,947 later on. 1450 01:17:08,947 --> 01:17:10,280 So how do we solve this problem? 1451 01:17:10,280 --> 01:17:13,680 Well, the way I elected to solve it is a little bit easier, 1452 01:17:13,680 --> 01:17:15,460 it turns out. 1453 01:17:15,460 --> 01:17:17,590 So for the first time, today I will not 1454 01:17:17,590 --> 01:17:18,940 use Fermat's principle. 1455 01:17:18,940 --> 01:17:20,680 I will use Snell's law. 1456 01:17:20,680 --> 01:17:22,147 And, of course, Snell's law follows 1457 01:17:22,147 --> 01:17:23,980 from Fermat's principle, so I haven't really 1458 01:17:23,980 --> 01:17:25,060 changed anything. 1459 01:17:25,060 --> 01:17:29,020 It's just more convenient to do the math this way. 1460 01:17:29,020 --> 01:17:32,320 So to apply Snell's law, I have to figure out 1461 01:17:32,320 --> 01:17:36,070 which way is the perpendicular to the interface 1462 01:17:36,070 --> 01:17:40,840 at the cross section of the ray with a spherical surface. 1463 01:17:40,840 --> 01:17:44,170 And, of course, if I have a sphere conveniently, 1464 01:17:44,170 --> 01:17:48,510 the normal to the sphere goes through the center. 1465 01:17:48,510 --> 01:17:50,730 So here is the normal to the sphere. 1466 01:17:50,730 --> 01:17:54,480 It goes to the center, and some additional rotation. 1467 01:17:54,480 --> 01:17:58,650 I will denote as R, the radius of the sphere. 1468 01:17:58,650 --> 01:18:04,710 And phi, I will denote the angle that this particular normal, 1469 01:18:04,710 --> 01:18:07,920 this particular radius makes with, again, 1470 01:18:07,920 --> 01:18:09,840 with a horizontal axis. 1471 01:18:13,220 --> 01:18:16,520 I think this is all the notation that I needed. 1472 01:18:16,520 --> 01:18:18,150 Actually, I'm not done yet. 1473 01:18:18,150 --> 01:18:20,740 So I also to apply Snell's law. 1474 01:18:20,740 --> 01:18:23,660 So to apply Snell's law, I have to look at the angles that 1475 01:18:23,660 --> 01:18:25,370 enter Snell's law. 1476 01:18:25,370 --> 01:18:30,710 So if you recall, apply Snell relative to the normal 1477 01:18:30,710 --> 01:18:32,150 to the interface. 1478 01:18:32,150 --> 01:18:34,370 So here's the normal, here is the ray. 1479 01:18:34,370 --> 01:18:36,950 Therefore, this angle is theta that 1480 01:18:36,950 --> 01:18:40,820 will go on the left-hand side of Snell's law. 1481 01:18:40,820 --> 01:18:44,360 And the corresponding is theta sub right 1482 01:18:44,360 --> 01:18:47,990 that will go to the right-hand side of Snell's law. 1483 01:18:47,990 --> 01:18:52,880 So speaking of which, here is Snell's law-- 1484 01:18:52,880 --> 01:18:56,690 n left times the sine of the angle on the left-hand side 1485 01:18:56,690 --> 01:18:59,870 will equal n to the right times the sine of the angle 1486 01:18:59,870 --> 01:19:02,390 on the right-hand side. 1487 01:19:02,390 --> 01:19:05,890 And now I have to do a little a bit of geometry. 1488 01:19:05,890 --> 01:19:08,650 So these angles, they are not known to me yet. 1489 01:19:08,650 --> 01:19:12,160 But I can observe that alpha left 1490 01:19:12,160 --> 01:19:15,160 is the same as the angle from the horizontal 1491 01:19:15,160 --> 01:19:17,270 to the ray over here. 1492 01:19:17,270 --> 01:19:21,880 And the angle from the horizontal to the radius 1493 01:19:21,880 --> 01:19:23,780 equals 5. 1494 01:19:23,780 --> 01:19:27,730 So this means that I have this property over here-- 1495 01:19:27,730 --> 01:19:30,830 theta left equals alpha left plus 5 1496 01:19:30,830 --> 01:19:34,370 in the corresponding equation for the right-hand side. 1497 01:19:34,370 --> 01:19:41,970 And I can also substitute these two relationships 1498 01:19:41,970 --> 01:19:43,525 into Snell's law. 1499 01:19:43,525 --> 01:19:45,630 I can apply a little bit of trigonometry 1500 01:19:45,630 --> 01:19:48,540 in order to break the sign of the sum 1501 01:19:48,540 --> 01:19:50,910 into this rather complicated-looking expression 1502 01:19:50,910 --> 01:19:51,917 over here. 1503 01:19:51,917 --> 01:19:53,250 So I haven't made much progress. 1504 01:19:53,250 --> 01:19:55,650 So all I managed to do is write a complicated set 1505 01:19:55,650 --> 01:19:59,000 of trigonometric relationships. 1506 01:19:59,000 --> 01:20:01,820 In fact, if I were to do a numerical solution 1507 01:20:01,820 --> 01:20:05,690 to this problem, I would just stop here and feed it 1508 01:20:05,690 --> 01:20:07,640 into a computer and solve. 1509 01:20:07,640 --> 01:20:09,890 But I don't want to do that quite yet, because there's 1510 01:20:09,890 --> 01:20:11,840 quite a bit of intuition to be derived 1511 01:20:11,840 --> 01:20:17,315 before we go blindly into a computer and plug the numerics. 1512 01:20:17,315 --> 01:20:19,190 So of course, the thing that we do in physics 1513 01:20:19,190 --> 01:20:22,460 when we try to derive intuition from a very complicated formula 1514 01:20:22,460 --> 01:20:25,330 like this one is we make approximations. 1515 01:20:25,330 --> 01:20:27,500 And the reasonable approximation to make here 1516 01:20:27,500 --> 01:20:32,320 is that the rays that are arriving to the element 1517 01:20:32,320 --> 01:20:36,200 are limited in a relatively small angular range. 1518 01:20:36,200 --> 01:20:38,480 This turns out to be a very practical assumption, 1519 01:20:38,480 --> 01:20:42,110 because it is a very common situation in optics to have 1520 01:20:42,110 --> 01:20:49,050 this small angle approximation. 1521 01:20:49,050 --> 01:20:51,830 And it turns out that even when the small angle approximation 1522 01:20:51,830 --> 01:20:53,870 is not exactly right, you can still 1523 01:20:53,870 --> 01:20:57,080 do a first pass of the design of an optical system 1524 01:20:57,080 --> 01:20:58,310 with this approximation. 1525 01:20:58,310 --> 01:21:00,020 And then you can improve it by doing 1526 01:21:00,020 --> 01:21:02,562 sort of more complicated numerical analysis. 1527 01:21:02,562 --> 01:21:04,520 So it's a very good place to start the analysis 1528 01:21:04,520 --> 01:21:06,210 of an optical system. 1529 01:21:06,210 --> 01:21:09,320 So the key assumption here is that alpha sub left, this one, 1530 01:21:09,320 --> 01:21:10,850 is very small. 1531 01:21:10,850 --> 01:21:12,310 If that is true-- 1532 01:21:12,310 --> 01:21:14,130 and also, I need one more assumption, 1533 01:21:14,130 --> 01:21:16,850 which is that this distance over here, 1534 01:21:16,850 --> 01:21:20,990 the distance that a ray meets the spherical interface, 1535 01:21:20,990 --> 01:21:24,080 is much smaller than the radius of the sphere. 1536 01:21:24,080 --> 01:21:27,350 In other words, the sphere is not very small, 1537 01:21:27,350 --> 01:21:30,210 but it is a relatively small curvature. 1538 01:21:30,210 --> 01:21:30,710 All right? 1539 01:21:30,710 --> 01:21:34,640 It has a relatively large radius of curvature 1540 01:21:34,640 --> 01:21:38,170 or, equivalently, a small curvature. 1541 01:21:38,170 --> 01:21:39,740 So the first answers are correct. 1542 01:21:39,740 --> 01:21:42,915 It is also reasonable to assume that alpha right, so 1543 01:21:42,915 --> 01:21:44,290 the right-hand side of interface, 1544 01:21:44,290 --> 01:21:46,380 is also very small angle. 1545 01:21:46,380 --> 01:21:49,660 And if I make the set of assumptions, 1546 01:21:49,660 --> 01:21:53,660 the entire set is called the paraxial approximation. 1547 01:21:53,660 --> 01:21:57,710 So "paraxial" really means that everything is confined 1548 01:21:57,710 --> 01:22:02,480 to being near the optical axis. 1549 01:22:02,480 --> 01:22:03,480 Very well. 1550 01:22:03,480 --> 01:22:06,270 So if we do this approximation now, a lot of this 1551 01:22:06,270 --> 01:22:09,740 gets simplified because of this Taylor series. 1552 01:22:09,740 --> 01:22:13,970 The sine of the angle of a small angles equals the angle itself. 1553 01:22:13,970 --> 01:22:16,990 The cosine of a small angle equals 1. 1554 01:22:16,990 --> 01:22:21,998 And also, the sine of this angle 5 is equal x/r. 1555 01:22:21,998 --> 01:22:23,290 I don't have to touch that one. 1556 01:22:23,290 --> 01:22:25,330 This one is actually exact. 1557 01:22:25,330 --> 01:22:29,320 But I can also neglect the cosine of pi, because-- 1558 01:22:29,320 --> 01:22:31,090 well, because, again, we are making 1559 01:22:31,090 --> 01:22:33,310 the paraxial approximation. 1560 01:22:33,310 --> 01:22:35,750 So if I do that, and I do a little bit of rearrangement 1561 01:22:35,750 --> 01:22:38,240 here, I arrive at this explanation 1562 01:22:38,240 --> 01:22:41,540 that says that the index of the right, 1563 01:22:41,540 --> 01:22:45,350 the index to the right time the angle to the right 1564 01:22:45,350 --> 01:22:48,740 equals the index to the left times the angle 1565 01:22:48,740 --> 01:22:53,460 to the left times this expression over here. 1566 01:22:53,460 --> 01:22:54,960 You might wonder a number of things. 1567 01:22:54,960 --> 01:22:57,590 For example, why did I leave the index here? 1568 01:22:57,590 --> 01:22:58,910 I'm looking for alpha right. 1569 01:22:58,910 --> 01:23:02,570 So why don't I just divide along and get done with it? 1570 01:23:02,570 --> 01:23:03,350 There is a reason. 1571 01:23:03,350 --> 01:23:05,570 And it will become apparent in the moment. 1572 01:23:05,570 --> 01:23:07,100 But for now, I'm almost done, right? 1573 01:23:07,100 --> 01:23:09,340 I have managed to do what I wanted. 1574 01:23:09,340 --> 01:23:12,110 I have a pretty closed-form expression 1575 01:23:12,110 --> 01:23:16,980 for the angle of refraction inside the sphere. 1576 01:23:16,980 --> 01:23:20,760 Now I'm going to do something really strange. 1577 01:23:20,760 --> 01:23:22,690 I will derive a similar expression 1578 01:23:22,690 --> 01:23:25,493 for light propagating in uniform space. 1579 01:23:25,493 --> 01:23:27,160 Now, this sounds totally useless, right? 1580 01:23:27,160 --> 01:23:28,570 But let me do it anyway. 1581 01:23:28,570 --> 01:23:30,490 And bear with me for a while. 1582 01:23:30,490 --> 01:23:34,670 I have a ray which is propagating in free space. 1583 01:23:34,670 --> 01:23:38,810 For reasons of notation, I will define two refractive indices 1584 01:23:38,810 --> 01:23:40,070 that are equal. 1585 01:23:40,070 --> 01:23:42,050 One is n left, one is n right. 1586 01:23:42,050 --> 01:23:45,770 But they're both equal to the same index of refraction n. 1587 01:23:45,770 --> 01:23:50,110 And what I'm interested to do is I'm interested in relating 1588 01:23:50,110 --> 01:23:55,430 the angle of the ray on the left-hand side with respect 1589 01:23:55,430 --> 01:23:58,880 to this reference plane over here and the elevation 1590 01:23:58,880 --> 01:24:02,450 of the ray with respect to the same plane with the same 1591 01:24:02,450 --> 01:24:05,770 quantities at the second reference plane-- 1592 01:24:05,770 --> 01:24:09,114 the elevation and the angle of the right-hand side. 1593 01:24:09,114 --> 01:24:11,090 [? There ?] are other trivial problem, 1594 01:24:11,090 --> 01:24:15,500 because, for example, Fermat says that in free space, 1595 01:24:15,500 --> 01:24:18,380 the ray must propagate at a straight line. 1596 01:24:18,380 --> 01:24:20,960 So therefore, I can immediately derive that alpha left 1597 01:24:20,960 --> 01:24:22,770 equals alpha right. 1598 01:24:22,770 --> 01:24:24,110 Not very surprising, right? 1599 01:24:24,110 --> 01:24:26,210 I'm stating the obvious here, and I'm 1600 01:24:26,210 --> 01:24:27,450 making a big deal out of it. 1601 01:24:27,450 --> 01:24:30,040 But nevertheless, let me write it like this. 1602 01:24:30,040 --> 01:24:31,790 Since the indices are the same, I can also 1603 01:24:31,790 --> 01:24:34,500 multiply by those indices. 1604 01:24:34,500 --> 01:24:38,620 And what about the elevations? 1605 01:24:38,620 --> 01:24:41,180 Well, the elevations satisfy a simple equation 1606 01:24:41,180 --> 01:24:46,820 relating the tangent of the angle of propagation multiplied 1607 01:24:46,820 --> 01:24:49,520 by the propagation distance d. 1608 01:24:49,520 --> 01:24:52,250 So the elevation to the right equals the elevation 1609 01:24:52,250 --> 01:24:55,520 to the left times this-- 1610 01:24:55,520 --> 01:24:57,540 time this quantity. 1611 01:24:57,540 --> 01:25:00,547 And then in the spirit of the paraxial approximation again, 1612 01:25:00,547 --> 01:25:02,630 I'm going to drop the tangent, because the tangent 1613 01:25:02,630 --> 01:25:05,330 of a small angle equals the angle itself. 1614 01:25:05,330 --> 01:25:10,340 And I arrive at a set of equations, 1615 01:25:10,340 --> 01:25:14,240 x right equals x left plus d times alpha left. 1616 01:25:14,240 --> 01:25:16,670 And this equation that I got before. 1617 01:25:16,670 --> 01:25:18,290 So now let me put the two together. 1618 01:25:18,290 --> 01:25:21,360 Let me put the free space propagation that I just 1619 01:25:21,360 --> 01:25:25,430 derived, and relationship between the ray elevation 1620 01:25:25,430 --> 01:25:29,060 and the ray angle to the left and to the right 1621 01:25:29,060 --> 01:25:34,510 of a uniform space of distance d, 1622 01:25:34,510 --> 01:25:37,810 and the equation that I just derived earlier 1623 01:25:37,810 --> 01:25:44,610 for the refraction of a ray from a spherical surface, 1624 01:25:44,610 --> 01:25:50,480 spherical refractive surface, of radius r. 1625 01:25:50,480 --> 01:25:51,250 OK. 1626 01:25:51,250 --> 01:25:54,160 So these are the two sets of equations that we got. 1627 01:25:54,160 --> 01:25:58,780 This is the set of equations that I got immediately before. 1628 01:25:58,780 --> 01:26:02,230 And this is the set of equations that I saw that I got earlier 1629 01:26:02,230 --> 01:26:06,630 when I solved the refraction problem 1630 01:26:06,630 --> 01:26:08,488 at the spherical interface. 1631 01:26:08,488 --> 01:26:10,530 And again, I want to emphasize that both of these 1632 01:26:10,530 --> 01:26:12,750 are correct in the paraxial approximation. 1633 01:26:12,750 --> 01:26:14,850 These are not exact expressions. 1634 01:26:14,850 --> 01:26:16,650 The equal sign is a bit of an ambiguous 1635 01:26:16,650 --> 01:26:18,420 here, because I've sort of assumed 1636 01:26:18,420 --> 01:26:20,458 that the paraxial approximation is correct. 1637 01:26:20,458 --> 01:26:22,250 So we're going on with this assumption now. 1638 01:26:22,250 --> 01:26:25,650 That's why I'm putting equal signs. 1639 01:26:25,650 --> 01:26:26,440 OK. 1640 01:26:26,440 --> 01:26:28,180 And I added one more equation that I 1641 01:26:28,180 --> 01:26:31,840 didn't write before that says that x to the left 1642 01:26:31,840 --> 01:26:33,017 equals x to the right. 1643 01:26:33,017 --> 01:26:34,850 In this case-- and this is a trivial thing-- 1644 01:26:34,850 --> 01:26:37,610 it says that the ray is continuous. 1645 01:26:37,610 --> 01:26:41,200 It arrives at the interface at the center point 1646 01:26:41,200 --> 01:26:43,165 and leaves from the same point. 1647 01:26:43,165 --> 01:26:44,290 And that's kind of obvious. 1648 01:26:44,290 --> 01:26:45,850 The ray should not jump. 1649 01:26:45,850 --> 01:26:47,980 So what is remarkable about these equations 1650 01:26:47,980 --> 01:26:51,670 is that they're both fully there, 1651 01:26:51,670 --> 01:26:53,770 and the relations-- and the quantities 1652 01:26:53,770 --> 01:26:56,050 that they're linear with respect to 1653 01:26:56,050 --> 01:26:59,020 are these two elements, the elevation 1654 01:26:59,020 --> 01:27:02,580 and the angle of the ray. 1655 01:27:02,580 --> 01:27:04,970 So you can think of them as kind of transport equations. 1656 01:27:04,970 --> 01:27:10,900 They allow you to transport a ray from two 1657 01:27:10,900 --> 01:27:13,100 very basic optical elements. 1658 01:27:13,100 --> 01:27:15,340 One is free space propagation. 1659 01:27:15,340 --> 01:27:18,610 It allows you to transport a ray through uniform space. 1660 01:27:18,610 --> 01:27:21,370 And the other is a refractive interface. 1661 01:27:21,370 --> 01:27:25,090 Again, it allows you to transport the ray 1662 01:27:25,090 --> 01:27:28,870 from the left to the right of a refractive interface. 1663 01:27:28,870 --> 01:27:33,400 And because they're linear, we can also write them 1664 01:27:33,400 --> 01:27:35,810 in matrix form as follows. 1665 01:27:35,810 --> 01:27:37,090 This is not new. 1666 01:27:37,090 --> 01:27:39,700 I just took these equations and rewrote them. 1667 01:27:39,700 --> 01:27:43,780 And I rewrote them in a slightly funny way. 1668 01:27:43,780 --> 01:27:49,240 I kept the index of refraction times the angle. 1669 01:27:49,240 --> 01:27:52,720 Can anybody-- now that we see it in this form-- 1670 01:27:52,720 --> 01:27:54,220 I will go to the matrix in a moment. 1671 01:27:54,220 --> 01:27:55,762 But now that you see it in this form, 1672 01:27:55,762 --> 01:27:58,600 can anybody guess why did I go through all this trouble 1673 01:27:58,600 --> 01:28:01,870 to keep the index of refraction in this expression? 1674 01:28:01,870 --> 01:28:04,360 Why don't I just write the angle by itself? 1675 01:28:14,860 --> 01:28:16,990 Is there a case, a situation that we've 1676 01:28:16,990 --> 01:28:21,420 seen, where this quantity may be conserved, for example? 1677 01:28:35,090 --> 01:28:37,860 AUDIENCE: You have the dispersion thing again. 1678 01:28:37,860 --> 01:28:41,460 Or where it depends-- where you have different n's, I suppose, 1679 01:28:41,460 --> 01:28:45,750 and therefore different angles depending on your wavelengths. 1680 01:28:45,750 --> 01:28:47,500 GEORGE BARBASTATHIS: Yeah. 1681 01:28:47,500 --> 01:28:49,580 Even simpler, actually. 1682 01:28:49,580 --> 01:28:53,060 If you are think of Snell's law-- 1683 01:28:53,060 --> 01:28:55,380 yeah, you're on the right track there. 1684 01:28:55,380 --> 01:29:17,030 If you think of Snell's law, Snell's law 1685 01:29:17,030 --> 01:29:23,940 says that n left sine alpha left equals 1686 01:29:23,940 --> 01:29:30,240 n right sine alpha right. 1687 01:29:30,240 --> 01:29:32,450 And of course, in the paraxial approximation 1688 01:29:32,450 --> 01:29:34,470 I drop the sinusoids. 1689 01:29:34,470 --> 01:29:37,770 So therefore, I arrive to the conclusion 1690 01:29:37,770 --> 01:29:41,310 that this quantity at the first element of this vector 1691 01:29:41,310 --> 01:29:44,780 is actually conserved. 1692 01:29:44,780 --> 01:29:48,300 And [INAUDIBLE] of course, you can go over it 1693 01:29:48,300 --> 01:29:49,880 yourselves and convince yourselves 1694 01:29:49,880 --> 01:29:51,510 that the equation is correct. 1695 01:29:51,510 --> 01:29:53,510 This is basically just taking the same equation. 1696 01:29:56,100 --> 01:30:04,660 And these matrices, we'll use them in the next lecture 1697 01:30:04,660 --> 01:30:05,950 in order to do ray tracing. 1698 01:30:05,950 --> 01:30:08,620 So basically, an optical element can always 1699 01:30:08,620 --> 01:30:12,610 be decomposed in a sequence of refractive interfaces 1700 01:30:12,610 --> 01:30:16,420 and free space or uniform space propagation. 1701 01:30:16,420 --> 01:30:18,820 So the reason the matrices are convenient-- 1702 01:30:18,820 --> 01:30:20,740 sort of jumping ahead a little bit-- 1703 01:30:20,740 --> 01:30:25,090 is because if you have a cascade of optical elements, 1704 01:30:25,090 --> 01:30:28,660 then you can ray trace simply by multiplying these matrices. 1705 01:30:28,660 --> 01:30:32,290 I will do that in detail in the next lecture next Monday. 1706 01:30:32,290 --> 01:30:34,990 But this is why we went through all this trouble 1707 01:30:34,990 --> 01:30:37,270 to derive the paraxial approximation 1708 01:30:37,270 --> 01:30:39,660 and then write it out in this matrix form, 1709 01:30:39,660 --> 01:30:44,460 so that we can use matrix properties to do ray tracing. 1710 01:30:44,460 --> 01:30:46,290 Then there's one more thing before we quit. 1711 01:30:49,890 --> 01:30:54,600 Do you see Snell's law in any of these laws over here? 1712 01:30:54,600 --> 01:31:02,380 You can see Snell's law on the handwritten slide over here. 1713 01:31:02,380 --> 01:31:04,150 But how about-- what about here? 1714 01:31:04,150 --> 01:31:06,061 Can you see Snell's law somewhere? 1715 01:31:18,590 --> 01:31:20,750 Another way to ask my question-- 1716 01:31:20,750 --> 01:31:22,950 these are questions that I wrote here. 1717 01:31:22,950 --> 01:31:26,270 Can they capture Snell's law in some way? 1718 01:31:26,270 --> 01:31:27,110 Do they include it? 1719 01:31:27,110 --> 01:31:27,610 How? 1720 01:31:39,610 --> 01:31:42,000 AUDIENCE: The elevation doesn't change if we apply it 1721 01:31:42,000 --> 01:31:44,890 at a point the interfaces. 1722 01:31:44,890 --> 01:31:45,890 GEORGE BARBASTATHIS: OK. 1723 01:31:45,890 --> 01:31:46,690 That is correct. 1724 01:31:46,690 --> 01:31:47,350 Yeah, that's correct. 1725 01:31:47,350 --> 01:31:47,590 OK. 1726 01:31:47,590 --> 01:31:48,590 Maybe I should restate-- 1727 01:31:48,590 --> 01:31:51,040 I asked the question in a confusing way. 1728 01:31:51,040 --> 01:31:55,490 The Snell's law at the flat interface-- 1729 01:31:55,490 --> 01:31:58,298 the same that I rewrote over here-- 1730 01:31:58,298 --> 01:31:59,965 can I get it somehow from this equation? 1731 01:32:08,110 --> 01:32:11,930 What is the radius of curvature of this surface? 1732 01:32:11,930 --> 01:32:13,780 Infinity, right? 1733 01:32:13,780 --> 01:32:19,600 So if I plug in infinity, if I plugged r equals infinity, 1734 01:32:19,600 --> 01:32:21,880 in this case of course, physically what 1735 01:32:21,880 --> 01:32:25,030 happens is the interface becomes flat. 1736 01:32:25,030 --> 01:32:28,390 Is the equation over here, this term goes away. 1737 01:32:28,390 --> 01:32:29,100 And I get what? 1738 01:32:29,100 --> 01:32:44,430 I get-- n left alpha left, n right alpha right, 1739 01:32:44,430 --> 01:32:46,430 x left, x right. 1740 01:32:46,430 --> 01:32:49,970 So you write x left equals x right, 1741 01:32:49,970 --> 01:32:54,220 but also n left alpha left equals n right alpha right, 1742 01:32:54,220 --> 01:32:58,520 which is actually Snell's law. 1743 01:32:58,520 --> 01:33:03,530 So this-- again, I'm restating something that I said before-- 1744 01:33:03,530 --> 01:33:06,160 this is the reason why we put this product 1745 01:33:06,160 --> 01:33:13,830 in the top element of the vector when we write this expression. 1746 01:33:13,830 --> 01:33:14,800 OK. 1747 01:33:14,800 --> 01:33:15,940 So we've run out of time. 1748 01:33:15,940 --> 01:33:18,630 So any questions? 1749 01:33:18,630 --> 01:33:19,245 Yeah. 1750 01:33:19,245 --> 01:33:22,640 AUDIENCE: I think [INAUDIBLE] 1751 01:33:26,747 --> 01:33:28,330 GEORGE BARBASTATHIS: Oh, you're right. 1752 01:33:28,330 --> 01:33:28,830 I'm sorry. 1753 01:33:28,830 --> 01:33:30,650 This would be 1, 0. 1754 01:33:30,650 --> 01:33:31,590 Yeah. 1755 01:33:31,590 --> 01:33:32,720 Yeah. 1756 01:33:32,720 --> 01:33:34,040 I'll fix the notes. 1757 01:33:34,040 --> 01:33:35,070 There's a typo. 1758 01:33:35,070 --> 01:33:40,013 The first matrix should be 1, 0, and then the rest is correct. 1759 01:33:40,013 --> 01:33:40,680 Yeah, thank you. 1760 01:33:43,970 --> 01:33:45,050 Did everybody hear that? 1761 01:33:45,050 --> 01:33:47,630 There's a typo in the notes. 1762 01:33:47,630 --> 01:33:50,680 The elements of the first row of this matrix need to be swapped. 1763 01:33:54,580 --> 01:33:55,080 OK. 1764 01:33:55,080 --> 01:33:59,090 So I'll see you all next Monday.