1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,650 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,650 --> 00:00:17,530 at ocw.mit.edu. 8 00:00:22,230 --> 00:00:24,860 GEORGE BARBASTATHIS: So while we're waiting for the phone-- 9 00:00:24,860 --> 00:00:29,090 I mean, for the microphone, a little bit of catching up 10 00:00:29,090 --> 00:00:30,940 from the previous lecture-- 11 00:00:34,930 --> 00:00:38,210 on the slide over there is a telescope. 12 00:00:38,210 --> 00:00:40,590 So I don't want to spend too much time on it. 13 00:00:40,590 --> 00:00:45,200 You wanted to see the telescope at least in one homework 14 00:00:45,200 --> 00:00:48,420 problem of the previous set, so you're familiar with it. 15 00:00:48,420 --> 00:00:52,850 Then, once you alluded to it earlier. 16 00:00:52,850 --> 00:00:57,230 So it is kind of opposite than the microscope in the sense 17 00:00:57,230 --> 00:01:02,480 that, OK, both instruments are similar because they magnify, 18 00:01:02,480 --> 00:01:04,010 but they magnify what? 19 00:01:04,010 --> 00:01:07,040 In the case of a microscope, we see very small detail. 20 00:01:07,040 --> 00:01:07,540 Thank you. 21 00:01:10,128 --> 00:01:12,420 We've got a new battery now, so we're back in business. 22 00:01:18,110 --> 00:01:24,700 So in the microscope, the object appears small 23 00:01:24,700 --> 00:01:28,090 because it is physically very small, 24 00:01:28,090 --> 00:01:30,160 and it is available nearby. 25 00:01:30,160 --> 00:01:32,380 That is a definition of a microscope. 26 00:01:32,380 --> 00:01:34,950 In the case of a telescope, the object 27 00:01:34,950 --> 00:01:37,990 appears small, not because it is physically small, 28 00:01:37,990 --> 00:01:41,090 but it is-- but because it is very far away. 29 00:01:41,090 --> 00:01:43,250 So for example, a star-- a star is humongous, 30 00:01:43,250 --> 00:01:47,830 and it may be several thousand times bigger than the Earth, 31 00:01:47,830 --> 00:01:51,310 but it appears very small because it is also maybe 32 00:01:51,310 --> 00:01:54,100 thousands of light years away. 33 00:01:54,100 --> 00:01:56,230 So this is the situation where we typically-- 34 00:01:56,230 --> 00:01:59,180 where we use a telescope. 35 00:01:59,180 --> 00:02:03,300 So you see there the function of the telescope here. 36 00:02:03,300 --> 00:02:06,020 So if this is the case of a telescope 37 00:02:06,020 --> 00:02:09,600 with an object with a finite location, so in this case, 38 00:02:09,600 --> 00:02:11,540 the object is physically big. 39 00:02:11,540 --> 00:02:14,580 It appears small because it is far away. 40 00:02:14,580 --> 00:02:16,640 So the telescope, in this case, it 41 00:02:16,640 --> 00:02:18,990 is actually very similar to the microscope, 42 00:02:18,990 --> 00:02:20,240 to the way it works. 43 00:02:20,240 --> 00:02:25,400 It creates an image of this object in the following sense, 44 00:02:25,400 --> 00:02:30,770 that each point in the object becomes a parallel ray bundle 45 00:02:30,770 --> 00:02:32,970 at the output of the telescope. 46 00:02:32,970 --> 00:02:34,460 So therefore, the observer-- again, 47 00:02:34,460 --> 00:02:37,370 recall that the final, final image 48 00:02:37,370 --> 00:02:42,690 is produced by the lens of the observer's eye on the retina. 49 00:02:42,690 --> 00:02:47,090 So the eye of the observer picks up this parallel ray bundle 50 00:02:47,090 --> 00:02:49,140 and focuses it on the retina. 51 00:02:49,140 --> 00:02:51,320 Now, if you extend the rays backwards 52 00:02:51,320 --> 00:02:54,770 through the final eyepiece of the telescope, 53 00:02:54,770 --> 00:02:56,630 you can see that the telescope actually 54 00:02:56,630 --> 00:03:01,220 is creating a virtual image that is magnified with its back 55 00:03:01,220 --> 00:03:02,820 to the original object. 56 00:03:02,820 --> 00:03:06,830 And that's why-- that's how the telescope magnifies 57 00:03:06,830 --> 00:03:08,840 this remote object. 58 00:03:08,840 --> 00:03:11,690 It is more common to use a telescope 59 00:03:11,690 --> 00:03:14,990 as shown in the second case, where the object is actually 60 00:03:14,990 --> 00:03:16,470 at infinity. 61 00:03:16,470 --> 00:03:20,600 And that is the classical case of a telescope, where now, 62 00:03:20,600 --> 00:03:23,810 instead of having spherical waves arriving 63 00:03:23,810 --> 00:03:27,260 from points in the object, now I have plane waves. 64 00:03:27,260 --> 00:03:33,230 So to a good approximation, to an excellent approximation, 65 00:03:33,230 --> 00:03:35,200 star is located at infinity. 66 00:03:35,200 --> 00:03:38,870 But not necessarily-- even terrestrial objects 67 00:03:38,870 --> 00:03:41,840 that are located, say, more than 100 meters, 68 00:03:41,840 --> 00:03:44,150 so a kilometer away, they satisfy 69 00:03:44,150 --> 00:03:45,950 this approximation quite well. 70 00:03:45,950 --> 00:03:49,700 So they produce these almost parallel ray bundles. 71 00:03:49,700 --> 00:03:53,600 So the telescope, in this case, it is afocal in the sense 72 00:03:53,600 --> 00:03:56,540 that the output of the scalar telescope, again you 73 00:03:56,540 --> 00:03:58,647 produce a parallel ray bundle. 74 00:03:58,647 --> 00:04:00,230 And the reason, of course, is that you 75 00:04:00,230 --> 00:04:05,660 want the observer to form an image with unaccommodated eye. 76 00:04:05,660 --> 00:04:10,580 That means that still the image must be at infinity 77 00:04:10,580 --> 00:04:11,930 so that the eye-- 78 00:04:11,930 --> 00:04:15,800 without any strain, the eye can focus it to the retina. 79 00:04:15,800 --> 00:04:20,450 And since that is the case, it means that the total-- 80 00:04:20,450 --> 00:04:22,220 now, let's be a little bit careful 81 00:04:22,220 --> 00:04:27,350 with the terminology-- the total power of the telescope is zero. 82 00:04:27,350 --> 00:04:30,530 That is, the telescope has no optical power. 83 00:04:30,530 --> 00:04:32,870 Its focal length is infinity. 84 00:04:32,870 --> 00:04:35,660 However, it does have magnifying power. 85 00:04:35,660 --> 00:04:38,840 What a telescope does in this case, the angle 86 00:04:38,840 --> 00:04:43,010 between a chief ray and the axis, 87 00:04:43,010 --> 00:04:45,740 the telescope magnifies this angle. 88 00:04:45,740 --> 00:04:49,010 And this is now by definition the magnification 89 00:04:49,010 --> 00:04:50,630 of the telescope, and you can see 90 00:04:50,630 --> 00:04:53,270 that by virtue of magnifying this angle, 91 00:04:53,270 --> 00:04:57,590 the telescope also magnifies the final real image that 92 00:04:57,590 --> 00:05:00,470 will be produced on the retina. 93 00:05:00,470 --> 00:05:01,753 And in order to be afocal-- 94 00:05:01,753 --> 00:05:03,170 this was the topic of the homework 95 00:05:03,170 --> 00:05:05,657 that I gave you guys a few-- 96 00:05:05,657 --> 00:05:07,490 I don't know-- I think it was two weeks ago. 97 00:05:07,490 --> 00:05:10,910 In order to be afocal, the distance between the two lenses 98 00:05:10,910 --> 00:05:13,100 must equal the sum of the focal length. 99 00:05:13,100 --> 00:05:15,510 It is very easy to convince yourselves in this case 100 00:05:15,510 --> 00:05:22,430 that an instrument such as this one, then in the imaging, 101 00:05:22,430 --> 00:05:28,820 in the matrix of the instrument, the matrix will look like this. 102 00:05:28,820 --> 00:05:33,460 And it will have a 0 in the 1, 2 element. 103 00:05:33,460 --> 00:05:37,630 In the first row, second column, it will have a 0. 104 00:05:37,630 --> 00:05:40,330 If that is the case-- whoops-- 105 00:05:40,330 --> 00:05:43,030 if that is the case, then it is called afocal. 106 00:05:43,030 --> 00:05:49,330 So by satisfying this condition, d 107 00:05:49,330 --> 00:05:54,220 equals f objective plus f eyepiece, we 108 00:05:54,220 --> 00:05:57,690 ensure that this is the case. 109 00:05:57,690 --> 00:06:01,860 So there's a number of different types of telescopes. 110 00:06:01,860 --> 00:06:03,900 I just picked one here. 111 00:06:03,900 --> 00:06:05,520 The textbook lists all of them. 112 00:06:05,520 --> 00:06:08,730 There is the Keplerian, Galilean, astronomical 113 00:06:08,730 --> 00:06:09,570 telescope. 114 00:06:09,570 --> 00:06:14,910 They all have different combinations of lenses. 115 00:06:14,910 --> 00:06:18,060 I picked one here that is composed exclusively 116 00:06:18,060 --> 00:06:21,850 of mirrors, and it is very common actually 117 00:06:21,850 --> 00:06:25,050 in observation-- in observatories. 118 00:06:25,050 --> 00:06:26,940 It is called the Cassegrain. 119 00:06:26,940 --> 00:06:28,980 It is also the same type that, if you 120 00:06:28,980 --> 00:06:31,380 go to the Discovery Store, for example, 121 00:06:31,380 --> 00:06:34,470 and buy a telescope for your nephew or your niece, 122 00:06:34,470 --> 00:06:35,370 this is the type. 123 00:06:35,370 --> 00:06:37,030 You look at it, and, you know, the tube 124 00:06:37,030 --> 00:06:41,680 is oriented horizontally, and you look at it from the top. 125 00:06:41,680 --> 00:06:43,860 And this is the schematic here. 126 00:06:43,860 --> 00:06:48,570 The object, or the sky, is on this end, 127 00:06:48,570 --> 00:06:50,220 and you place your eye this way. 128 00:06:50,220 --> 00:06:53,910 So the telescope itself consists of a large first mirror 129 00:06:53,910 --> 00:06:59,270 that functions as the primary, and then there's 130 00:06:59,270 --> 00:07:00,130 a second mirror. 131 00:07:00,130 --> 00:07:01,260 So if you look at the-- 132 00:07:01,260 --> 00:07:03,960 from this side, if you look into the telescope, 133 00:07:03,960 --> 00:07:08,960 you will see a small block in the front face 134 00:07:08,960 --> 00:07:10,990 of the instrument. 135 00:07:10,990 --> 00:07:12,780 So that is the secondary mirror. 136 00:07:12,780 --> 00:07:15,390 So the combination of primary and secondary-- of course, 137 00:07:15,390 --> 00:07:18,060 you use two of them to get even higher magnification, 138 00:07:18,060 --> 00:07:20,820 and the combination is the objective. 139 00:07:20,820 --> 00:07:22,320 And then, there's an eyepiece, which 140 00:07:22,320 --> 00:07:23,820 is where you stick your eye, and you 141 00:07:23,820 --> 00:07:27,710 see the-- you see the image of the sky. 142 00:07:27,710 --> 00:07:31,400 So this is composed purely of mirrors. 143 00:07:31,400 --> 00:07:34,310 So last time, we're going to call this kind of instrument 144 00:07:34,310 --> 00:07:35,960 catoptric. 145 00:07:35,960 --> 00:07:38,308 Then, there's another type of telescope. 146 00:07:38,308 --> 00:07:39,600 Actually, it's not a telescope. 147 00:07:39,600 --> 00:07:43,370 It's a kind of a method for correcting optical systems 148 00:07:43,370 --> 00:07:45,830 called Schmidt-- 149 00:07:45,830 --> 00:07:49,445 Schmidt telescope or Schmidt optical system. 150 00:07:49,445 --> 00:07:51,230 So let me say why you use this. 151 00:07:51,230 --> 00:07:56,840 So you recall that the ideal mirror surface to give perfect 152 00:07:56,840 --> 00:08:00,710 focusing of an object at infinity is actually parabolic. 153 00:08:00,710 --> 00:08:03,380 Now, suppose that we cannot make a parabola, 154 00:08:03,380 --> 00:08:06,328 so we have to make a sphere instead of a parabola. 155 00:08:06,328 --> 00:08:08,120 Well, in that case, a little bit of thought 156 00:08:08,120 --> 00:08:12,260 will convince yourself that the sphere, because it 157 00:08:12,260 --> 00:08:17,000 is more highly curved than the parabola near the edges, 158 00:08:17,000 --> 00:08:19,280 away from the paraxial region, it 159 00:08:19,280 --> 00:08:23,650 will actually bend the rays that are far from the axis. 160 00:08:23,650 --> 00:08:27,050 They will bend more than they should. 161 00:08:27,050 --> 00:08:29,990 So the result is the failure to reach perfect focus. 162 00:08:29,990 --> 00:08:31,530 This is what we have already seen, 163 00:08:31,530 --> 00:08:34,909 and we'll call it spherical aberration. 164 00:08:34,909 --> 00:08:37,010 So Schmidt basically came up with an idea 165 00:08:37,010 --> 00:08:40,730 to correct this spherical aberration 166 00:08:40,730 --> 00:08:42,460 by sticking a slightly-- 167 00:08:47,680 --> 00:08:52,180 a glass surface at the front pupil here. 168 00:08:52,180 --> 00:08:59,310 He put a glass surface which is slightly shaped. 169 00:08:59,310 --> 00:09:02,352 It has toroidal shape, and the purpose-- 170 00:09:02,352 --> 00:09:04,185 I don't know if you can see it in the image. 171 00:09:04,185 --> 00:09:07,280 It was a-- perhaps I should have exaggerated more. 172 00:09:07,280 --> 00:09:08,570 It is too subtle. 173 00:09:08,570 --> 00:09:11,420 But what it does is it takes this ray 174 00:09:11,420 --> 00:09:16,400 and sends it slightly higher, so this kind of functions 175 00:09:16,400 --> 00:09:17,810 like a negative lens, actually. 176 00:09:17,810 --> 00:09:21,280 You can think of it this way, with a very small power. 177 00:09:21,280 --> 00:09:24,470 And then, because this hits the surface 178 00:09:24,470 --> 00:09:27,380 at a different point with a different angle, 179 00:09:27,380 --> 00:09:31,100 then, basically, by engineering the shape of the surface here, 180 00:09:31,100 --> 00:09:34,280 you can make sure that most-- that all of the rays 181 00:09:34,280 --> 00:09:38,790 will actually meet ideally at the focal point. 182 00:09:38,790 --> 00:09:41,090 So this is a very simple way to correct for this. 183 00:09:41,090 --> 00:09:42,965 I don't know if it is simple, but, anyway, it 184 00:09:42,965 --> 00:09:46,880 is a way to correct for spherical aberration. 185 00:09:46,880 --> 00:09:49,550 So this is called catadioptric because it 186 00:09:49,550 --> 00:09:54,980 contains a reflector, as well as a refractive element. 187 00:09:54,980 --> 00:10:00,250 So according to our terminology, that's what we call it. 188 00:10:00,250 --> 00:10:02,280 So that's all I had to say about telescopes. 189 00:10:02,280 --> 00:10:05,810 Are there any questions or-- 190 00:10:05,810 --> 00:10:07,820 either about telescopes or microscopes 191 00:10:07,820 --> 00:10:10,210 or any of the optical systems that we saw? 192 00:10:29,500 --> 00:10:35,650 So next, I'm going to shift gears a little bit, 193 00:10:35,650 --> 00:10:38,200 and I'm going to expand on the very last topic 194 00:10:38,200 --> 00:10:42,540 that we saw, the topic of aberration. 195 00:10:42,540 --> 00:10:45,240 So we saw quite extensively-- we mentioned several times 196 00:10:45,240 --> 00:10:47,880 this spherical aberration. 197 00:10:47,880 --> 00:10:52,150 It turns out it is not the only one, 198 00:10:52,150 --> 00:10:55,950 and we'll see a number of different types of aberrations 199 00:10:55,950 --> 00:10:57,730 today. 200 00:10:57,730 --> 00:11:00,150 So to do a little bit of organization 201 00:11:00,150 --> 00:11:03,940 and classification, we start by classifying them as two types. 202 00:11:03,940 --> 00:11:07,770 One is chromatic, and this we have also seen in the past, 203 00:11:07,770 --> 00:11:10,230 in the context of a prism primarily. 204 00:11:10,230 --> 00:11:12,990 And it is due to the fact that the index of refraction 205 00:11:12,990 --> 00:11:14,460 has a variation with wavelength. 206 00:11:14,460 --> 00:11:19,020 So if you apply the lens maker's formula, or any of those, 207 00:11:19,020 --> 00:11:22,200 in order to design an element with a given focal length 208 00:11:22,200 --> 00:11:24,900 at the given wavelength, you have no guarantee 209 00:11:24,900 --> 00:11:28,200 that the focal length will be the same 210 00:11:28,200 --> 00:11:30,275 at the different wavelength. 211 00:11:30,275 --> 00:11:33,660 That fact is called chromatic aberration. 212 00:11:33,660 --> 00:11:36,030 Again, this comes from a Greek word. 213 00:11:36,030 --> 00:11:39,750 I had to do this, and I sound like this movie, My Big Fat 214 00:11:39,750 --> 00:11:43,410 Greek Wedding, but since I'm Greek, I guess it is natural. 215 00:11:43,410 --> 00:11:50,050 So chroma-- in Latin, this would spell the chroma, 216 00:11:50,050 --> 00:11:51,190 with the accent here. 217 00:11:51,190 --> 00:11:56,360 It means color in Greek. 218 00:11:56,360 --> 00:11:59,000 So that's where the term chromatic comes from. 219 00:12:02,720 --> 00:12:05,270 The second time-- or for the second type of aberration 220 00:12:05,270 --> 00:12:07,190 is called geometrical. 221 00:12:07,190 --> 00:12:10,400 And there's a long definition here, but I will go back to it, 222 00:12:10,400 --> 00:12:14,030 and perhaps this long definition will make more sense. 223 00:12:14,030 --> 00:12:15,920 Let me start with chromatic because this 224 00:12:15,920 --> 00:12:17,900 is easy to deal with-- 225 00:12:17,900 --> 00:12:22,460 OK, not easy to deal with, but it is easy to understand. 226 00:12:22,460 --> 00:12:25,430 So as I said, we have seen this type of plot 227 00:12:25,430 --> 00:12:28,790 before, where the horizontal axis is the wavelength. 228 00:12:28,790 --> 00:12:31,670 The vertical axis is the index of refraction. 229 00:12:31,670 --> 00:12:34,160 And forget this dashed line for a moment. 230 00:12:34,160 --> 00:12:36,710 Just look at the solid lines. 231 00:12:36,710 --> 00:12:39,170 So these are the index of refraction, 232 00:12:39,170 --> 00:12:42,410 this function of wavelength, for different glasses. 233 00:12:42,410 --> 00:12:45,778 You have seen this before in the context of a prism. 234 00:12:45,778 --> 00:12:47,070 This is, of course, a disaster. 235 00:12:47,070 --> 00:12:51,850 It means that if you have a lens like this one, which 236 00:12:51,850 --> 00:12:54,290 you designed for a certain focal length, 237 00:12:54,290 --> 00:12:57,220 say, somewhere in the middle of the spectrum, 238 00:12:57,220 --> 00:13:00,880 say, near yellow, then this means that-- 239 00:13:00,880 --> 00:13:07,120 well, as you can see, the index in the red wavelength for sort 240 00:13:07,120 --> 00:13:10,760 of normal materials-- this is actually called normal 241 00:13:10,760 --> 00:13:12,000 dispersion-- 242 00:13:12,000 --> 00:13:16,630 the index is smaller at the red wavelengths. 243 00:13:16,630 --> 00:13:20,410 So if you remember the lens maker's formula, 244 00:13:20,410 --> 00:13:23,140 smaller index means that the focal length is what? 245 00:13:29,950 --> 00:13:35,590 OK, the lens maker's formula says that 1 over f 246 00:13:35,590 --> 00:13:41,440 equals n minus 1, the difference in radii. 247 00:13:41,440 --> 00:13:47,590 So if n goes down, it means that 1 over f goes down. 248 00:13:47,590 --> 00:13:49,670 It means that the f goes up. 249 00:13:49,670 --> 00:13:52,460 So it means that the focal length is longer. 250 00:13:52,460 --> 00:13:56,300 And this is also what you see in this picture over here. 251 00:13:56,300 --> 00:14:01,930 You'll see that if you have a white light coming 252 00:14:01,930 --> 00:14:05,560 from infinity, you'll see that the red rays focus 253 00:14:05,560 --> 00:14:09,280 at the longer distance than the blue rays. 254 00:14:13,060 --> 00:14:20,650 So that is the evidence of chromatic aberration, 255 00:14:20,650 --> 00:14:24,370 and it turns out there's a way of correcting it, 256 00:14:24,370 --> 00:14:28,630 which is to stick two lenses, one of them with a-- 257 00:14:28,630 --> 00:14:32,630 that is a positive lens and the other a negative lens. 258 00:14:32,630 --> 00:14:36,880 And first of all, they have to-- 259 00:14:36,880 --> 00:14:40,330 the two lenses have to have a slightly different indices. 260 00:14:40,330 --> 00:14:42,550 Otherwise, it wouldn't even make sense, right? 261 00:14:42,550 --> 00:14:44,950 Then this would be just one element. 262 00:14:44,950 --> 00:14:47,700 So they have different indices, different dispersion 263 00:14:47,700 --> 00:14:51,400 probably, so one can pick, for example, between these glasses. 264 00:14:51,400 --> 00:14:53,630 You could pick one lens out of this one, 265 00:14:53,630 --> 00:14:55,240 another out of this one. 266 00:14:55,240 --> 00:14:57,880 This is how one designs this kind of element. 267 00:14:57,880 --> 00:15:00,610 And then, the way to understand it is because one of them 268 00:15:00,610 --> 00:15:02,240 tries to focus the rays. 269 00:15:02,240 --> 00:15:04,360 The other, because it is negative, 270 00:15:04,360 --> 00:15:06,760 it will try to actually defocus them. 271 00:15:06,760 --> 00:15:10,570 You can imagine how one can compensate 272 00:15:10,570 --> 00:15:12,550 the action of the positive lens with the action 273 00:15:12,550 --> 00:15:14,290 of the negative lens and eventually 274 00:15:14,290 --> 00:15:17,290 cancel, or at least approximately cancel, 275 00:15:17,290 --> 00:15:19,900 a chromatic aberration over a relatively 276 00:15:19,900 --> 00:15:21,430 long range of wavelengths. 277 00:15:21,430 --> 00:15:23,600 I used to give you this problem as a homework. 278 00:15:23,600 --> 00:15:26,230 I don't do it anymore. 279 00:15:26,230 --> 00:15:29,650 But, anyway, it is a fairly straightforward thing to do. 280 00:15:29,650 --> 00:15:31,990 And if you go over the textbook, there's 281 00:15:31,990 --> 00:15:35,170 an entire section where he tells you in detail how 282 00:15:35,170 --> 00:15:36,997 this kind of thing can be done. 283 00:15:36,997 --> 00:15:38,330 Conceptually, it is very simple. 284 00:15:38,330 --> 00:15:40,670 All you do is you balance the focal length. 285 00:15:40,670 --> 00:15:43,510 So you basically make sure that the composite has 286 00:15:43,510 --> 00:15:47,430 a focal length that is the nominal focal length, 287 00:15:47,430 --> 00:15:49,620 and then you balance the chromatic 288 00:15:49,620 --> 00:15:54,540 by use of the two elements of the composite. 289 00:15:54,540 --> 00:15:56,190 The geometrical aberrations, they're 290 00:15:56,190 --> 00:16:00,720 more interesting because, conceptually, they're actually 291 00:16:00,720 --> 00:16:06,950 much richer than chromatic, and also, algebraically, they're 292 00:16:06,950 --> 00:16:08,400 much more difficult to handle. 293 00:16:08,400 --> 00:16:11,790 So people have come up with all kinds of clever tricks 294 00:16:11,790 --> 00:16:16,150 in order to understand and express geometric aberrations. 295 00:16:16,150 --> 00:16:20,140 So again, let me remind you that at the exit 296 00:16:20,140 --> 00:16:24,740 of an optical instrument, the ideal wavefront should 297 00:16:24,740 --> 00:16:25,930 be spherical. 298 00:16:25,930 --> 00:16:28,810 So this is the dashed line over here. 299 00:16:28,810 --> 00:16:32,980 If you draw normals to this spherical wavefront, of course, 300 00:16:32,980 --> 00:16:36,430 there will be the radii of the corresponding sphere, 301 00:16:36,430 --> 00:16:38,920 and they will meet at the center of the sphere. 302 00:16:38,920 --> 00:16:41,650 So if I actually produce a spherical wavefront, 303 00:16:41,650 --> 00:16:44,470 then I can expect a perfect geometrical focus 304 00:16:44,470 --> 00:16:47,045 at the center of that spherical wavefront. 305 00:16:47,045 --> 00:16:49,420 So the problem, of course, these are optical instruments, 306 00:16:49,420 --> 00:16:53,360 as we saw repeatedly in a number of cases in the past, 307 00:16:53,360 --> 00:16:56,640 including the spherical reflectors, 308 00:16:56,640 --> 00:16:59,450 spherical refractors, and so on and so forth, 309 00:16:59,450 --> 00:17:02,860 they don't produce spherical wavefronts at the action. 310 00:17:02,860 --> 00:17:05,140 They produce wavefronts that, in general, may 311 00:17:05,140 --> 00:17:08,599 be some complicated surface. 312 00:17:08,599 --> 00:17:10,609 So in this case, you see the eye is-- 313 00:17:10,609 --> 00:17:12,040 of course, are exaggerated-- 314 00:17:12,040 --> 00:17:14,859 I made it terribly aberrated, and this 315 00:17:14,859 --> 00:17:17,082 is the solid line over here. 316 00:17:17,082 --> 00:17:25,030 So the difference between the actual aberrated wavefront 317 00:17:25,030 --> 00:17:30,180 and the ideal wavefront, which is, of course, spherical, 318 00:17:30,180 --> 00:17:33,640 this difference is by definition the aberration. 319 00:17:33,640 --> 00:17:37,040 So I tried to be fairly loyal here. 320 00:17:37,040 --> 00:17:39,240 So I-- again, this is a cartoon, but I 321 00:17:39,240 --> 00:17:42,480 took the difference between the solid line and the dashed line, 322 00:17:42,480 --> 00:17:44,610 and this is what you would get. 323 00:17:44,610 --> 00:17:47,460 Typically, you measure it in optical path length, 324 00:17:47,460 --> 00:17:50,740 and it doesn't have to be very dramatic. 325 00:17:50,740 --> 00:17:55,030 The vertical axis here can be as small as a few wavelengths, 326 00:17:55,030 --> 00:17:57,520 and that is often sufficient to destroy 327 00:17:57,520 --> 00:18:01,130 the quality of the image formation in an optical system. 328 00:18:01,130 --> 00:18:07,620 In fact, the book has the story of the Hubble telescope, which 329 00:18:07,620 --> 00:18:09,540 is kind of famous, or I should say 330 00:18:09,540 --> 00:18:15,390 infamous study of a disaster, and heroic, in optical design. 331 00:18:15,390 --> 00:18:19,650 The telescope was sent to space with a serious error 332 00:18:19,650 --> 00:18:20,550 in the design-- 333 00:18:20,550 --> 00:18:23,220 actually, in the assembly, that caused 334 00:18:23,220 --> 00:18:26,280 a huge spherical aberration. 335 00:18:26,280 --> 00:18:29,680 And then, they had to send another spacecraft to fix it, 336 00:18:29,680 --> 00:18:30,180 and so on. 337 00:18:30,180 --> 00:18:33,700 Anyway, they actually succeeded, which is impressive. 338 00:18:33,700 --> 00:18:37,480 But, anyway, so what the book says is that the error was-- 339 00:18:37,480 --> 00:18:38,860 actually turned out to be-- 340 00:18:38,860 --> 00:18:40,780 I think it was half a wavelength or so. 341 00:18:40,780 --> 00:18:43,570 Whatever that had was sufficient to destroy 342 00:18:43,570 --> 00:18:47,102 the quality of the images that were beamed from the telescope. 343 00:18:47,102 --> 00:18:49,310 So one has to be very, very careful in this business. 344 00:18:52,312 --> 00:18:53,770 So as you can imagine, if you tried 345 00:18:53,770 --> 00:18:56,980 to express this kind of complicated surface 346 00:18:56,980 --> 00:18:59,110 or its difference from a sphere, you 347 00:18:59,110 --> 00:19:03,740 will probably get a very nasty trigonometric expression, 348 00:19:03,740 --> 00:19:05,720 so for a number of reasons. 349 00:19:05,720 --> 00:19:09,040 One of them has to do with the ability 350 00:19:09,040 --> 00:19:10,840 to compute analytically, which is 351 00:19:10,840 --> 00:19:14,770 kind of an old-fashioned problem because now we have computers. 352 00:19:14,770 --> 00:19:17,230 So that's one reason, but the second reason, 353 00:19:17,230 --> 00:19:19,990 because of the desire to get intuition 354 00:19:19,990 --> 00:19:22,660 about how these things work. 355 00:19:22,660 --> 00:19:25,660 People normally express aberrations 356 00:19:25,660 --> 00:19:28,610 in the form of a Taylor series. 357 00:19:28,610 --> 00:19:31,200 And, of course, you are dealing now 358 00:19:31,200 --> 00:19:37,450 in-- this is a surface, not just a curve, as I show here. 359 00:19:37,450 --> 00:19:41,440 The Taylor series is high dimensional. 360 00:19:41,440 --> 00:19:47,180 You actually have many terms for each order in the Taylor series 361 00:19:47,180 --> 00:19:48,810 expansion. 362 00:19:48,810 --> 00:19:53,870 So these terms are actually grouped into orders, 363 00:19:53,870 --> 00:19:55,040 and the aberrations that-- 364 00:19:55,040 --> 00:19:58,490 and the terms that we consider as a first step 365 00:19:58,490 --> 00:20:01,970 in an optical system are the third-order aberrations, 366 00:20:01,970 --> 00:20:05,270 that are also known as Seidel aberrations. 367 00:20:05,270 --> 00:20:09,140 And, of course, the second-order aberrations 368 00:20:09,140 --> 00:20:13,760 disappear because we assume rotationally symmetric optics. 369 00:20:13,760 --> 00:20:17,360 That kills all the second-order terms in the Taylor series 370 00:20:17,360 --> 00:20:18,860 by definition. 371 00:20:18,860 --> 00:20:22,330 And the first-order aberration is actually only one, 372 00:20:22,330 --> 00:20:23,790 and that is the focus. 373 00:20:23,790 --> 00:20:26,340 So the first-order aberration is actually paraxial. 374 00:20:26,340 --> 00:20:28,370 It is something we've already considered. 375 00:20:28,370 --> 00:20:31,400 And then, it is the third order that we 376 00:20:31,400 --> 00:20:34,017 deal with in optical design. 377 00:20:34,017 --> 00:20:36,350 So there's all-- after you go through all the symmetries 378 00:20:36,350 --> 00:20:39,350 and all that, it turns out that the further the aberrations, 379 00:20:39,350 --> 00:20:40,640 it's five of them. 380 00:20:40,640 --> 00:20:44,600 One we've already encountered, it is spherical, 381 00:20:44,600 --> 00:20:47,510 and the other four are coma, astigmatism, curvature 382 00:20:47,510 --> 00:20:49,960 of field, and distortion. 383 00:20:49,960 --> 00:20:53,840 And what you see here is the aberration wavefronts, 384 00:20:53,840 --> 00:20:56,230 so this probably doesn't mean very much. 385 00:20:56,230 --> 00:20:58,120 What we'll do next is we'll actually 386 00:20:58,120 --> 00:21:00,490 see a ray diagram of all of these aberrations 387 00:21:00,490 --> 00:21:03,500 and what they look like. 388 00:21:03,500 --> 00:21:07,070 So the spherical, these diagrams are straight out of the book. 389 00:21:07,070 --> 00:21:09,440 We have seen actually several times, 390 00:21:09,440 --> 00:21:13,880 and the way spherical aberration looks like, it basically 391 00:21:13,880 --> 00:21:17,570 happens when the rays that are away from the axis, 392 00:21:17,570 --> 00:21:21,970 they receive more curvature than rays that are near the axis, 393 00:21:21,970 --> 00:21:23,260 or more or less. 394 00:21:23,260 --> 00:21:24,910 So in both of these cases, you'll 395 00:21:24,910 --> 00:21:29,530 get the situation where the rays from away from the axis, 396 00:21:29,530 --> 00:21:35,040 they focus before or after the paraxial focus. 397 00:21:35,040 --> 00:21:38,300 And it is interesting, the way-- but 398 00:21:38,300 --> 00:21:40,880 if you look at this kind of system, 399 00:21:40,880 --> 00:21:45,670 you wonder what exactly is the focal spot? 400 00:21:45,670 --> 00:21:47,440 And that's not an easy question to answer, 401 00:21:47,440 --> 00:21:52,270 so if you look at the marginal ray over here, 402 00:21:52,270 --> 00:21:54,490 the size of the beam that you create 403 00:21:54,490 --> 00:21:56,800 is, of course, defined by the marginal ray. 404 00:21:56,800 --> 00:22:01,210 So you see that the marginal ray kind of focuses here, then 405 00:22:01,210 --> 00:22:02,890 starts to focus less. 406 00:22:02,890 --> 00:22:05,890 Then it reaches a minimum size, and then 407 00:22:05,890 --> 00:22:09,820 starts to expand again because now this aberrated ray takes 408 00:22:09,820 --> 00:22:11,000 over. 409 00:22:11,000 --> 00:22:13,900 So there's a minimum width in this optical beam, 410 00:22:13,900 --> 00:22:16,790 in this aberrated optical beam to produce, 411 00:22:16,790 --> 00:22:20,700 and that is called the least-- the circle of least confusion. 412 00:22:20,700 --> 00:22:23,590 It is kind of a compromise to see whether this system is 413 00:22:23,590 --> 00:22:26,900 approximately in focus. 414 00:22:26,900 --> 00:22:30,140 Another thing that I want to emphasize from this diagram 415 00:22:30,140 --> 00:22:37,010 is that, as you can see, near the edge here of this bundle, 416 00:22:37,010 --> 00:22:40,460 there is a congruence of rays that approach. 417 00:22:40,460 --> 00:22:43,100 So if you-- this-- you can only see a few rays here, 418 00:22:43,100 --> 00:22:46,405 but if you imagine that you fill up the space with rays, 419 00:22:46,405 --> 00:22:47,780 then you would see that you would 420 00:22:47,780 --> 00:22:51,740 get a very tight intersection of successive rays that 421 00:22:51,740 --> 00:22:54,420 trace a curve here. 422 00:22:54,420 --> 00:22:57,780 This curve is called a caustic, and, again, 423 00:22:57,780 --> 00:23:00,520 that's another Greek word. 424 00:23:12,870 --> 00:23:16,505 [GREEK] means something that burns. 425 00:23:29,490 --> 00:23:34,020 And the reason it is called caustic or the reason it 426 00:23:34,020 --> 00:23:35,150 alludes-- 427 00:23:35,150 --> 00:23:38,740 the word alludes to burning is because this congruence 428 00:23:38,740 --> 00:23:41,610 of rays, if you were to put a piece of paper 429 00:23:41,610 --> 00:23:44,700 there, if you wanted to put an element, 430 00:23:44,700 --> 00:23:48,120 it would create and increase the intensity of light 431 00:23:48,120 --> 00:23:52,410 near the edge of this bundle. 432 00:23:52,410 --> 00:23:54,510 And therefore, it would cause, presumably, 433 00:23:54,510 --> 00:23:57,540 elevated temperature since you have more light energy, 434 00:23:57,540 --> 00:24:01,200 and that might actually result in burning. 435 00:24:01,200 --> 00:24:03,390 Caustics are very interesting things. 436 00:24:03,390 --> 00:24:06,240 I will say a little bit more perhaps next time 437 00:24:06,240 --> 00:24:11,670 about caustics, but they occur often when 438 00:24:11,670 --> 00:24:15,000 you deviate from a perfect spherical wavefront. 439 00:24:15,000 --> 00:24:18,310 When you pick different wavefronts 440 00:24:18,310 --> 00:24:19,920 other than spherical, these caustics 441 00:24:19,920 --> 00:24:21,640 have a tendency to occur. 442 00:24:21,640 --> 00:24:23,640 And there's a very beautiful mathematical theory 443 00:24:23,640 --> 00:24:24,992 that describes that. 444 00:24:24,992 --> 00:24:26,700 It is a little bit advanced, so we're not 445 00:24:26,700 --> 00:24:30,300 going to-- it is kind of beyond the scope of this class. 446 00:24:30,300 --> 00:24:35,100 But I thought since also the book has mentioned it, 447 00:24:35,100 --> 00:24:37,140 I thought it would be interesting to point out 448 00:24:37,140 --> 00:24:38,940 the existence of this caustic. 449 00:24:41,550 --> 00:24:46,170 When it comes to spherical aberration, 450 00:24:46,170 --> 00:24:47,910 there's only several things one can 451 00:24:47,910 --> 00:24:51,930 do in order to control the spherical aberration. 452 00:24:51,930 --> 00:24:54,690 One is the actual shape of the lens. 453 00:24:54,690 --> 00:24:58,797 If you'll go back to the lens maker formula, 454 00:24:58,797 --> 00:24:59,630 this one over here-- 455 00:24:59,630 --> 00:25:00,960 I've already written it-- 456 00:25:00,960 --> 00:25:05,820 you can see very easily that you can get the same focal length 457 00:25:05,820 --> 00:25:09,660 with a number of different combinations of radii 458 00:25:09,660 --> 00:25:14,470 for the front and the back surface of the lens. 459 00:25:14,470 --> 00:25:19,930 So from the paraxial point of view, 460 00:25:19,930 --> 00:25:22,670 it makes no difference whether you pick-- 461 00:25:22,670 --> 00:25:27,700 for example, whether you pick a plano-convex, 462 00:25:27,700 --> 00:25:30,520 or whether you pick a biconvex lens, 463 00:25:30,520 --> 00:25:35,230 or whether you pick a meniscus, and so on, from the paraxial 464 00:25:35,230 --> 00:25:37,570 point of view, they're all the same because-- 465 00:25:37,570 --> 00:25:40,180 as long as you have the same focal length. 466 00:25:40,180 --> 00:25:42,310 However, because this theory now-- we've 467 00:25:42,310 --> 00:25:44,440 gone beyond the paraxial approximation-- 468 00:25:44,440 --> 00:25:47,180 this theory is non-paraxial. 469 00:25:47,180 --> 00:25:51,810 Actually, the shape factor of the lens makes a difference. 470 00:25:51,810 --> 00:25:53,500 And then, forget the definitions here. 471 00:25:53,500 --> 00:25:57,670 This plot is the most interesting one, 472 00:25:57,670 --> 00:25:59,560 where you can see different lens shapes. 473 00:25:59,560 --> 00:26:04,300 This is a meniscus, kind of that the-- a convex meniscus. 474 00:26:04,300 --> 00:26:06,130 Then, this is plano-convex. 475 00:26:06,130 --> 00:26:09,310 This is biconvex, again plano-convex, 476 00:26:09,310 --> 00:26:11,530 but flipped the other way around, 477 00:26:11,530 --> 00:26:15,160 and then a concave meniscus. 478 00:26:15,160 --> 00:26:17,950 And this plot was computed so that all 479 00:26:17,950 --> 00:26:19,660 of these different shapes, they have 480 00:26:19,660 --> 00:26:23,950 the same paraxial focal length, but on the vertical axis, 481 00:26:23,950 --> 00:26:27,070 you kind of see the amount of spherical aberration 482 00:26:27,070 --> 00:26:28,240 that you receive. 483 00:26:28,240 --> 00:26:31,900 And you can see that it is very different, of course, 484 00:26:31,900 --> 00:26:35,110 and it actually has a minimum. 485 00:26:35,110 --> 00:26:37,560 It is not quite zero, but it is a minimum, 486 00:26:37,560 --> 00:26:46,040 and the minimum occurs typically near the plano-convex shape 487 00:26:46,040 --> 00:26:48,800 for an object of infinity. 488 00:26:48,800 --> 00:26:49,950 So why is that? 489 00:26:49,950 --> 00:26:51,440 Well, the reason is-- 490 00:26:51,440 --> 00:26:54,800 this is kind of nicely illustrated by another diagram 491 00:26:54,800 --> 00:26:57,335 that I stole from your textbook. 492 00:27:00,656 --> 00:27:03,810 If you place the plano-convex lens such 493 00:27:03,810 --> 00:27:07,140 that its flat surface faces the plane wave coming 494 00:27:07,140 --> 00:27:09,810 from infinity, then, basically, refraction 495 00:27:09,810 --> 00:27:13,530 occurs only at this spherical interface. 496 00:27:13,530 --> 00:27:14,440 So you have no hope. 497 00:27:14,440 --> 00:27:16,590 You will get very strong spherical aberration 498 00:27:16,590 --> 00:27:20,070 because, as we learned earlier, this spherical surface 499 00:27:20,070 --> 00:27:24,430 is very far from ideal as a focusing element. 500 00:27:24,430 --> 00:27:26,820 So this case is very highly aberrated. 501 00:27:26,820 --> 00:27:28,320 Now, you can do a very simple thing. 502 00:27:28,320 --> 00:27:30,900 You can flip the lens so that this spherical wave 503 00:27:30,900 --> 00:27:31,890 meets this spherical-- 504 00:27:31,890 --> 00:27:33,570 I'm sorry-- so that the plane wave 505 00:27:33,570 --> 00:27:35,640 meets this spherical surface. 506 00:27:35,640 --> 00:27:37,080 If you do that, then you actually 507 00:27:37,080 --> 00:27:39,990 get refraction to occur in two steps. 508 00:27:39,990 --> 00:27:43,680 And because the first step produces 509 00:27:43,680 --> 00:27:47,655 an aberrated spherical wave, but the second step also 510 00:27:47,655 --> 00:27:51,870 incurs some spherical aberration because of the nonlinearity 511 00:27:51,870 --> 00:27:55,830 in the Snell's law, it turns out that, in this case, 512 00:27:55,830 --> 00:28:00,970 the second surface, not exactly, but closely 513 00:28:00,970 --> 00:28:03,420 compensates the spherical aberration. 514 00:28:03,420 --> 00:28:08,230 And you get a much better focusing quality over here. 515 00:28:08,230 --> 00:28:11,430 So this is very good, practical advice that we give to the sort 516 00:28:11,430 --> 00:28:15,290 of first-year graduate students when they go to the laser lab, 517 00:28:15,290 --> 00:28:17,580 is that when you have-- 518 00:28:17,580 --> 00:28:22,460 when you try to focus a plane wave, you always orient-- 519 00:28:22,460 --> 00:28:26,250 but for A, you use a plano-convex lens, not 520 00:28:26,250 --> 00:28:29,710 a biconvex, and B, that you place it in this way, 521 00:28:29,710 --> 00:28:35,310 so that the spherical surface of the lens 522 00:28:35,310 --> 00:28:37,368 meets the planar wavefront. 523 00:28:37,368 --> 00:28:39,660 And, of course, it goes-- it goes the other way around. 524 00:28:39,660 --> 00:28:43,657 If you have a point source, and you want to collimate it, then 525 00:28:43,657 --> 00:28:45,240 again, you have to make sure basically 526 00:28:45,240 --> 00:28:47,540 you can reverse the rays over here. 527 00:28:47,540 --> 00:28:49,890 And you have to orient the lens so 528 00:28:49,890 --> 00:28:55,770 that the planar part of the lens faces the point source. 529 00:28:55,770 --> 00:28:59,250 So this is an example of doing a very simple trick in order 530 00:28:59,250 --> 00:29:06,090 to not quite eliminate, but reduce 531 00:29:06,090 --> 00:29:09,300 the spherical aberration. 532 00:29:09,300 --> 00:29:11,380 It turns out that-- yes? 533 00:29:11,380 --> 00:29:13,470 Push the button, please. 534 00:29:13,470 --> 00:29:15,480 AUDIENCE: [INAUDIBLE] 535 00:29:15,480 --> 00:29:16,980 AUDIENCE: He didn't push the button. 536 00:29:16,980 --> 00:29:18,022 GEORGE BARBASTATHIS: Yes. 537 00:29:19,578 --> 00:29:21,120 AUDIENCE: In that graph, how did they 538 00:29:21,120 --> 00:29:24,600 actually measure the spherical aberration in different lens 539 00:29:24,600 --> 00:29:25,377 shapes? 540 00:29:25,377 --> 00:29:26,460 GEORGE BARBASTATHIS: Yeah. 541 00:29:26,460 --> 00:29:28,090 So the way to-- thank you. 542 00:29:28,090 --> 00:29:29,310 I should have mentioned that. 543 00:29:29,310 --> 00:29:31,880 So you can characterize a spherical aberration 544 00:29:31,880 --> 00:29:35,250 based on one of these parameters here. 545 00:29:35,250 --> 00:29:39,570 One is-- so it's called the longitudinal spherical 546 00:29:39,570 --> 00:29:41,670 aberration, and it is the distance 547 00:29:41,670 --> 00:29:45,120 on axis between the paraxial focus 548 00:29:45,120 --> 00:29:49,960 and the focal point of the last-- of the marginal ray. 549 00:29:49,960 --> 00:29:50,820 Right? 550 00:29:50,820 --> 00:29:53,100 Another way to do it is to basically measure 551 00:29:53,100 --> 00:29:56,052 the size of the least confusion. 552 00:29:56,052 --> 00:29:58,260 So any of these-- if you think about it, all of these 553 00:29:58,260 --> 00:30:00,030 are proportional to each other, so you 554 00:30:00,030 --> 00:30:02,940 can put any of these quantities on the vertical axis here. 555 00:30:02,940 --> 00:30:04,500 I'm not sure what he did here. 556 00:30:04,500 --> 00:30:06,810 I think it says axial here, so it was 557 00:30:06,810 --> 00:30:10,560 the longitudinal distance here. 558 00:30:10,560 --> 00:30:11,270 And I thank you. 559 00:30:11,270 --> 00:30:12,478 I should have mentioned that. 560 00:30:12,478 --> 00:30:16,520 This way of characterization is common for all 561 00:30:16,520 --> 00:30:21,135 of the aberrations that we'll see in the next two slides. 562 00:30:21,135 --> 00:30:21,635 Yeah? 563 00:30:21,635 --> 00:30:22,510 AUDIENCE: [INAUDIBLE] 564 00:30:22,510 --> 00:30:24,540 GEORGE BARBASTATHIS: Button. 565 00:30:24,540 --> 00:30:26,330 AUDIENCE: We knew which is the lengths. 566 00:30:26,330 --> 00:30:29,030 Do they specify paraxial or focal length, 567 00:30:29,030 --> 00:30:33,417 or circle of least confusion as a f? 568 00:30:33,417 --> 00:30:34,500 GEORGE BARBASTATHIS: Yeah. 569 00:30:34,500 --> 00:30:35,708 That is a real good question. 570 00:30:38,600 --> 00:30:44,200 If it is a sort of high-end lens, 571 00:30:44,200 --> 00:30:45,590 it should be well corrected. 572 00:30:45,590 --> 00:30:53,160 So then, what it will specify is actually the least confusion 573 00:30:53,160 --> 00:30:57,670 focal end, but it is pretty close to paraxial. 574 00:30:57,670 --> 00:31:00,640 If it is a poor lens, then most likely what they specify 575 00:31:00,640 --> 00:31:04,330 is the paraxial focus, and then it is kind of up to you 576 00:31:04,330 --> 00:31:05,210 to position it. 577 00:31:09,610 --> 00:31:10,490 Any other questions? 578 00:31:18,660 --> 00:31:19,160 The-- 579 00:31:19,160 --> 00:31:19,770 AUDIENCE: In the previous-- 580 00:31:19,770 --> 00:31:20,230 GEORGE BARBASTATHIS: Yes. 581 00:31:20,230 --> 00:31:21,440 AUDIENCE: Hi, George. 582 00:31:21,440 --> 00:31:23,810 In the previous slides, what is the curve 583 00:31:23,810 --> 00:31:29,833 in the third and previous at-- 584 00:31:29,833 --> 00:31:32,250 hold on. 585 00:31:32,250 --> 00:31:34,680 Yeah, what is the curve in the third graph? 586 00:31:37,537 --> 00:31:39,620 GEORGE BARBASTATHIS: The curve in the third graph? 587 00:31:39,620 --> 00:31:40,245 AUDIENCE: Yeah. 588 00:31:40,245 --> 00:31:43,550 So the vertical axis is H, and horizontal axis was-- 589 00:31:43,550 --> 00:31:44,800 GEORGE BARBASTATHIS: This one? 590 00:31:44,800 --> 00:31:46,430 AUDIENCE: Yeah. 591 00:31:46,430 --> 00:31:48,430 GEORGE BARBASTATHIS: I don't remember, actually. 592 00:31:48,430 --> 00:31:51,080 I think it is the-- 593 00:31:51,080 --> 00:31:54,010 I suppose it is the amount of spherical aberration 594 00:31:54,010 --> 00:31:55,780 as a function of position in the field. 595 00:31:55,780 --> 00:31:58,388 I'm not really sure. 596 00:31:58,388 --> 00:31:59,430 Maybe you can check that. 597 00:31:59,430 --> 00:32:00,750 Actually, I have the book. 598 00:32:00,750 --> 00:32:02,237 You can check the book, huh? 599 00:32:02,237 --> 00:32:03,820 This came-- is directly from the book. 600 00:32:07,270 --> 00:32:10,730 I think it is the amount of spherical aberration. 601 00:32:10,730 --> 00:32:12,253 It makes sense because-- here. 602 00:32:20,650 --> 00:32:21,888 Any other questions? 603 00:32:29,290 --> 00:32:34,090 So it turns out that what we said before, 604 00:32:34,090 --> 00:32:37,960 and, actually, Colin made that point when I was discussing 605 00:32:37,960 --> 00:32:41,750 ellipsoid, that the fractals and so on and so forth, 606 00:32:41,750 --> 00:32:48,400 and what we said before, that a spherical surface produces 607 00:32:48,400 --> 00:32:53,170 spherical aberration, it is not always true. 608 00:32:53,170 --> 00:32:58,030 There is one condition where a spherical surface does not 609 00:32:58,030 --> 00:33:02,320 produce spherical aberration. 610 00:33:02,320 --> 00:33:04,060 And this condition is given here. 611 00:33:04,060 --> 00:33:06,670 That's another diagram I got-- 612 00:33:06,670 --> 00:33:09,290 that I got from the book. 613 00:33:09,290 --> 00:33:13,070 And you can see here, basically, the condition 614 00:33:13,070 --> 00:33:17,240 says that if the radius of this field is R, 615 00:33:17,240 --> 00:33:20,600 the index of refraction of this field is n2. 616 00:33:20,600 --> 00:33:24,590 The index of refraction outside this sphere is n1, 617 00:33:24,590 --> 00:33:31,980 and you place a point source at the distance R n1 over n2. 618 00:33:31,980 --> 00:33:34,762 Then, of course, the point source is inside this sphere. 619 00:33:34,762 --> 00:33:36,720 If you think about it, this is a negative lens. 620 00:33:36,720 --> 00:33:39,120 It will produce a virtual image. 621 00:33:39,120 --> 00:33:42,690 This virtual image is located over here, 622 00:33:42,690 --> 00:33:45,990 and it is free of spherical aberration. 623 00:33:45,990 --> 00:33:47,010 I will not prove this. 624 00:33:47,010 --> 00:33:49,440 This is actually in the book. 625 00:33:49,440 --> 00:33:53,850 It is given as a problem in the textbook, 626 00:33:53,850 --> 00:33:56,760 and the problems in the textbook are solved 627 00:33:56,760 --> 00:33:58,380 at the end of the textbook. 628 00:33:58,380 --> 00:34:01,580 That's why I never assign problems from the textbook. 629 00:34:01,580 --> 00:34:03,330 And anyway, you can look at it afterwards. 630 00:34:03,330 --> 00:34:06,000 It is a very simple geometrical proof of this one. 631 00:34:06,000 --> 00:34:08,880 So we'll skip it here. 632 00:34:08,880 --> 00:34:12,810 But the point of this is that there is actually 633 00:34:12,810 --> 00:34:17,449 a single combination of image-- 634 00:34:17,449 --> 00:34:20,389 I'm sorry-- of object image pairs, 635 00:34:20,389 --> 00:34:22,670 for which the spherical aberration is canceled 636 00:34:22,670 --> 00:34:25,960 for this spherical surface. 637 00:34:25,960 --> 00:34:29,770 And again, I want to emphasize that if you place the object 638 00:34:29,770 --> 00:34:34,510 point elsewhere, it will still form 639 00:34:34,510 --> 00:34:36,280 an image in the paraxial sense. 640 00:34:36,280 --> 00:34:40,389 This lens will remain a lens, but it will not 641 00:34:40,389 --> 00:34:42,429 be free of spherical aberration. 642 00:34:42,429 --> 00:34:49,576 It will only be free if this condition here is satisfied. 643 00:34:49,576 --> 00:34:51,409 And actually, this distance that I mentioned 644 00:34:51,409 --> 00:34:52,826 is from the center of this sphere. 645 00:34:55,480 --> 00:35:00,110 All right, so the reason for that-- 646 00:35:00,110 --> 00:35:03,790 so this now has a number of implications. 647 00:35:03,790 --> 00:35:08,080 For example, this is one reason why meniscus lenses are 648 00:35:08,080 --> 00:35:10,930 very popular in photography, or at least in very 649 00:35:10,930 --> 00:35:15,700 old-fashioned cameras that were very expensive. 650 00:35:15,700 --> 00:35:19,330 Now, we're talking about cameras in the '20s and '30s, you know, 651 00:35:19,330 --> 00:35:22,630 the type of camera that the cameraman would 652 00:35:22,630 --> 00:35:26,110 go under a black hood and then going, and then click. 653 00:35:26,110 --> 00:35:30,130 And, you know, the family would be sitting with the-- 654 00:35:30,130 --> 00:35:33,295 whatever, with the children, all that stuff. 655 00:35:33,295 --> 00:35:36,280 And so these cameras, they would typically use a single meniscus 656 00:35:36,280 --> 00:35:41,500 lens, and the reason was exactly that the meniscus, at least 657 00:35:41,500 --> 00:35:45,050 if you designed it for the typical distance-- 658 00:35:45,050 --> 00:35:49,390 remember, again, this was a camera for a photography shop. 659 00:35:49,390 --> 00:35:52,750 So it has-- you can kind of predict 660 00:35:52,750 --> 00:35:56,350 the location of the object because the camera's already 661 00:35:56,350 --> 00:35:59,210 always sitting in the same distance from the-- wherever 662 00:35:59,210 --> 00:36:02,020 you place the family to take the family picture. 663 00:36:02,020 --> 00:36:04,450 And then, in that case, you can engineer the meniscus 664 00:36:04,450 --> 00:36:06,340 to minimize this spherical aberration 665 00:36:06,340 --> 00:36:07,763 for this particular distance. 666 00:36:07,763 --> 00:36:09,180 It turns out the meniscus also has 667 00:36:09,180 --> 00:36:10,597 some other interesting properties, 668 00:36:10,597 --> 00:36:13,010 but I will not go into that. 669 00:36:13,010 --> 00:36:15,510 So of course, in that case, you cannot really put the family 670 00:36:15,510 --> 00:36:19,800 in glass, so you're going to have exactly this situation 671 00:36:19,800 --> 00:36:20,792 over here. 672 00:36:20,792 --> 00:36:22,500 But, nevertheless, if you think about it, 673 00:36:22,500 --> 00:36:26,340 you don't need-- you know, you can satisfy this condition, 674 00:36:26,340 --> 00:36:29,850 even if you have two spherical surfaces, provided 675 00:36:29,850 --> 00:36:32,780 that, as you go from one surface to the next, 676 00:36:32,780 --> 00:36:34,860 you have to satisfy the same condition. 677 00:36:34,860 --> 00:36:39,625 It is the same condition that you have to satisfy twice. 678 00:36:39,625 --> 00:36:41,250 And I forgot to mention, this condition 679 00:36:41,250 --> 00:36:44,137 is called the aplanatic condition. 680 00:36:44,137 --> 00:36:46,470 Actually, that, I don't know why it is called aplanatic, 681 00:36:46,470 --> 00:36:48,660 but it sounds like a-- 682 00:36:48,660 --> 00:36:49,810 you know, Colin, why-- 683 00:36:49,810 --> 00:36:51,450 where that comes from? 684 00:36:51,450 --> 00:36:53,130 Aplanatic means non-planar. 685 00:36:53,130 --> 00:36:54,720 It makes no sense at all, but, anyway, 686 00:36:54,720 --> 00:36:55,762 that's what they call it. 687 00:36:58,976 --> 00:37:04,470 And it is also used in oil immersion microscopy. 688 00:37:04,470 --> 00:37:06,630 In this case, we mentioned that. 689 00:37:06,630 --> 00:37:09,540 Actually, it was also a topic of one of the homeworks. 690 00:37:09,540 --> 00:37:12,750 Oil immersion means that you put the liquid, the droplet, 691 00:37:12,750 --> 00:37:17,100 between the objective and the sample in the microscope. 692 00:37:17,100 --> 00:37:19,470 And in this case, again, you can kind of 693 00:37:19,470 --> 00:37:23,603 make sure that when you apply the aplanatic condition-- 694 00:37:23,603 --> 00:37:25,020 because, again, in the microscope, 695 00:37:25,020 --> 00:37:29,700 you have very good control over the distance between the exit 696 00:37:29,700 --> 00:37:31,430 pupil and the object. 697 00:37:31,430 --> 00:37:35,700 And you can make sure that the aplanatic condition is 698 00:37:35,700 --> 00:37:38,980 satisfied for this particular distance, 699 00:37:38,980 --> 00:37:42,690 and you can use the index of refraction of the oil 700 00:37:42,690 --> 00:37:47,260 in order to satisfy the aplanatic condition. 701 00:37:47,260 --> 00:37:49,200 So now, that's the case where you exactly 702 00:37:49,200 --> 00:37:52,645 cancel the spherical aberration, so it is very interesting. 703 00:37:52,645 --> 00:37:54,270 It is different than the previous case. 704 00:37:54,270 --> 00:37:56,940 Here, we exactly cancel it by using 705 00:37:56,940 --> 00:38:00,163 this particular condition. 706 00:38:03,870 --> 00:38:09,730 The next aberration I wanted to discuss is coma. 707 00:38:09,730 --> 00:38:12,940 So coma is different than spherical in the sense 708 00:38:12,940 --> 00:38:18,346 that spherical occurs even if you have on-axis incidence. 709 00:38:18,346 --> 00:38:25,840 Coma actually occurs for off-axis incident rays. 710 00:38:25,840 --> 00:38:30,190 So this is the typical image, or the typical sort of depiction 711 00:38:30,190 --> 00:38:31,990 of coma that you see in books. 712 00:38:31,990 --> 00:38:35,410 Again, you have a plane wave coming from infinity, 713 00:38:35,410 --> 00:38:39,190 but it is incident at an angle of focus. 714 00:38:39,190 --> 00:38:41,860 So it is likely quite possible that you 715 00:38:41,860 --> 00:38:44,883 might have spherical aberration as well in this case. 716 00:38:44,883 --> 00:38:46,300 But in this particular case, we're 717 00:38:46,300 --> 00:38:48,460 only concerned for the aberration 718 00:38:48,460 --> 00:38:52,720 due to the fact that you are off axis, and that is called coma. 719 00:38:52,720 --> 00:38:56,950 So if you work out sort of the non-paraxial imaging 720 00:38:56,950 --> 00:39:01,750 for this case, you will discover that at the focal plane, 721 00:39:01,750 --> 00:39:04,690 sort of up to here, what you see is 722 00:39:04,690 --> 00:39:11,230 a cross-section of the system in the meridional plane. 723 00:39:11,230 --> 00:39:16,570 And this is what you would see as a front view on the imaging 724 00:39:16,570 --> 00:39:19,600 plane, and it turns out that you can also see it here. 725 00:39:19,600 --> 00:39:22,090 This is perhaps a better picture. 726 00:39:22,090 --> 00:39:24,460 It looks like a teardrop, and this is 727 00:39:24,460 --> 00:39:26,810 where the term coma comes from. 728 00:39:26,810 --> 00:39:28,330 It is not a condition that you enter 729 00:39:28,330 --> 00:39:30,790 after a very serious accident, but it 730 00:39:30,790 --> 00:39:35,530 is rather from the punctuation symbol, a comma. 731 00:39:35,530 --> 00:39:38,290 In old-fashioned books, kind of the comma looks like this. 732 00:39:38,290 --> 00:39:42,410 So ah, I don't know how good my drawing is, 733 00:39:42,410 --> 00:39:45,580 but this teardrop shape is kind of 734 00:39:45,580 --> 00:39:50,930 reminiscent of the punctuation, a comma. 735 00:39:50,930 --> 00:39:52,550 So this is what it looks like. 736 00:39:52,550 --> 00:39:57,310 And now, the geometry of this is a little bit complicated. 737 00:39:57,310 --> 00:39:59,260 You can basically calculate the shape 738 00:39:59,260 --> 00:40:02,110 of the coma in the meridional plane 739 00:40:02,110 --> 00:40:06,160 and the sagittal plane, perpendicular to that. 740 00:40:06,160 --> 00:40:09,400 So the meridional plane is this one, 741 00:40:09,400 --> 00:40:11,950 the one that you see over here that 742 00:40:11,950 --> 00:40:14,950 is-- if you take the plane of incidence, 743 00:40:14,950 --> 00:40:18,700 it is the plane that cuts through the center of the lens. 744 00:40:18,700 --> 00:40:20,710 And then if you rotate by 90 degrees, 745 00:40:20,710 --> 00:40:22,697 that would be the sagittal plane. 746 00:40:22,697 --> 00:40:24,280 So according, because we are off axis, 747 00:40:24,280 --> 00:40:25,697 the two are not symmetric anymore. 748 00:40:25,697 --> 00:40:28,330 On axis, they are symmetric, but off axis, they're not, 749 00:40:28,330 --> 00:40:31,390 so that's what gives rise to the strongly asymmetric nature 750 00:40:31,390 --> 00:40:32,980 of the coma. 751 00:40:32,980 --> 00:40:35,680 And it turns out, if you do some simple geometry, 752 00:40:35,680 --> 00:40:40,480 you find out that the length of this comatic shape 753 00:40:40,480 --> 00:40:43,900 along the tangential, the meridional plane, it 754 00:40:43,900 --> 00:40:46,390 turns out to be three times the length 755 00:40:46,390 --> 00:40:49,060 along the sagittal plane. 756 00:40:49,060 --> 00:40:50,350 So this follows from geometry. 757 00:40:50,350 --> 00:40:53,560 To be honest, I never quite went through the derivation 758 00:40:53,560 --> 00:40:57,820 of this one, but, anyway, someone did it. 759 00:40:57,820 --> 00:41:00,210 And these derivations, by the way, 760 00:41:00,210 --> 00:41:02,200 they have a way of becoming very complicated, 761 00:41:02,200 --> 00:41:05,500 even when they give very simple results, 762 00:41:05,500 --> 00:41:06,760 like this case over here. 763 00:41:10,210 --> 00:41:12,015 Whether we'd like to-- there's a few things 764 00:41:12,015 --> 00:41:13,930 I would like to say about coma. 765 00:41:13,930 --> 00:41:16,210 One is, of course, that it also depends 766 00:41:16,210 --> 00:41:19,600 on the shape of the lens. 767 00:41:19,600 --> 00:41:25,010 So here you see what this spherical aberration and coma 768 00:41:25,010 --> 00:41:28,480 plotted for lenses of the same focal length, 769 00:41:28,480 --> 00:41:31,300 but different shape factor. 770 00:41:33,890 --> 00:41:35,870 And you can see that, fortuitously, there's 771 00:41:35,870 --> 00:41:39,200 a region over here, where, above the spherical-- actually 772 00:41:39,200 --> 00:41:41,630 the coma is completely eliminated, 773 00:41:41,630 --> 00:41:46,040 and this spherical is close to zero. 774 00:41:46,040 --> 00:41:48,020 So this means that this is a good design point, 775 00:41:48,020 --> 00:41:51,050 if you want to simultaneously minimize coma 776 00:41:51,050 --> 00:41:52,400 and this spherical aberration. 777 00:41:55,860 --> 00:41:57,360 Now, you might wonder-- 778 00:41:57,360 --> 00:41:59,490 let me skip for a moment the discussion 779 00:41:59,490 --> 00:42:00,790 of the bottom figure over here. 780 00:42:00,790 --> 00:42:01,800 I'll come back to it. 781 00:42:01,800 --> 00:42:07,230 But you may wonder how it can be that coma is completely 782 00:42:07,230 --> 00:42:08,330 eliminated. 783 00:42:11,530 --> 00:42:13,990 It turns out that-- 784 00:42:13,990 --> 00:42:18,880 and, again, this is something I will give without proof. 785 00:42:18,880 --> 00:42:22,780 It turns out that there is a condition that, indeed, you 786 00:42:22,780 --> 00:42:25,630 can cancel coma. 787 00:42:25,630 --> 00:42:28,200 So this is called the sine condition. 788 00:42:28,200 --> 00:42:30,290 Sine, now, this is not a Greek word. 789 00:42:30,290 --> 00:42:32,650 It is actually a Latin word, [LAUGHS] 790 00:42:32,650 --> 00:42:36,880 and it is the same sine as the sine we have in [INAUDIBLE].. 791 00:42:36,880 --> 00:42:39,650 And this requires a little bit of explanation, 792 00:42:39,650 --> 00:42:42,940 so we'll do that, and then we'll quit for today. 793 00:42:42,940 --> 00:42:47,215 So the sine condition follows from another result 794 00:42:47,215 --> 00:42:50,080 for the optical sine theorem. 795 00:42:50,080 --> 00:42:52,930 Interestingly, for those of you who are mechanical engineers 796 00:42:52,930 --> 00:42:55,150 and know thermodynamics, this theorem 797 00:42:55,150 --> 00:42:58,767 was proven by a kind of a hero of thermodynamics, Clausius. 798 00:42:58,767 --> 00:43:00,850 There was-- I think there's some kind of something 799 00:43:00,850 --> 00:43:03,480 called Clausius Inequality, or something like that. 800 00:43:03,480 --> 00:43:06,640 It's a very big, important result in thermodynamics. 801 00:43:06,640 --> 00:43:09,280 So he proved it first, and then two other people 802 00:43:09,280 --> 00:43:13,360 associated with optics more closely, Abbe and Helmholtz, 803 00:43:13,360 --> 00:43:17,420 they derived it independently 10 years later. 804 00:43:17,420 --> 00:43:19,780 So what is it this theorem say? 805 00:43:19,780 --> 00:43:25,580 Well, here's again the geometry of an optical system. 806 00:43:25,580 --> 00:43:31,670 We use the notation alpha as before to denote the angles, 807 00:43:31,670 --> 00:43:32,600 and y-- 808 00:43:32,600 --> 00:43:35,690 for some reason, the book uses y, not x, 809 00:43:35,690 --> 00:43:38,930 to denote the elevation of rays. 810 00:43:38,930 --> 00:43:41,410 So this is not an exact result. It is not quite axial, 811 00:43:41,410 --> 00:43:44,830 but it says that if you satisfy the imaging condition, 812 00:43:44,830 --> 00:43:53,020 then the index of refraction, the ray elevation, and the sine 813 00:43:53,020 --> 00:43:57,130 of the ray angle in the object space 814 00:43:57,130 --> 00:44:00,820 are the same as the product of this-- actually, 815 00:44:00,820 --> 00:44:02,800 I should've said the product of the three 816 00:44:02,800 --> 00:44:07,770 is the same as the product in the image space. 817 00:44:07,770 --> 00:44:11,970 So you can see these quantities over here, all of them, 818 00:44:11,970 --> 00:44:13,520 a sub i-- 819 00:44:13,520 --> 00:44:17,270 I'm sorry-- this is a sub i, a sub o, and so on and so forth. 820 00:44:20,140 --> 00:44:21,690 Why is this significant now? 821 00:44:21,690 --> 00:44:27,010 If you take the ratio of y at the image plane-- 822 00:44:27,010 --> 00:44:31,420 so this is y sub i over y sub o-- 823 00:44:31,420 --> 00:44:34,230 that ratio is the magnification. 824 00:44:34,230 --> 00:44:35,710 So you see a problem here. 825 00:44:35,710 --> 00:44:38,130 The problem is that if you calculate 826 00:44:38,130 --> 00:44:42,537 the ratio from this quantity, the ratio 827 00:44:42,537 --> 00:44:43,620 would turn out to be what? 828 00:44:43,620 --> 00:44:52,260 I could do-- turn out to be something like n i-- 829 00:45:00,290 --> 00:45:07,490 sine alpha i over n object sine alpha object. 830 00:45:07,490 --> 00:45:09,590 And because of the presence of the sines 831 00:45:09,590 --> 00:45:14,570 here, you can very easily deduce that the magnification is not 832 00:45:14,570 --> 00:45:16,160 the same. 833 00:45:16,160 --> 00:45:19,180 The magnification that you had depends 834 00:45:19,180 --> 00:45:21,820 on the size of the object, which, of course, 835 00:45:21,820 --> 00:45:24,200 very annoying. 836 00:45:24,200 --> 00:45:26,380 We did not see that in the paraxial approximation, 837 00:45:26,380 --> 00:45:28,310 but, of course, this is non-paraxial. 838 00:45:28,310 --> 00:45:31,040 That's why it happens, but it is very annoying. 839 00:45:31,040 --> 00:45:34,270 It is-- and it is the reason why coma happens. 840 00:45:34,270 --> 00:45:38,160 You can also see it over here. 841 00:45:38,160 --> 00:45:43,660 You can see a very-- you can see sort of the different rays. 842 00:45:43,660 --> 00:45:48,970 They attempt to reach-- to focus at different locations 843 00:45:48,970 --> 00:45:51,295 in the image plane. 844 00:45:51,295 --> 00:45:52,170 So that's the reason. 845 00:45:52,170 --> 00:45:57,100 So the sine theorem says, boy, if I could actually make-- 846 00:45:57,100 --> 00:46:02,050 I'm sorry-- if I could actually make this ratio, 847 00:46:02,050 --> 00:46:06,070 sine alpha in the object over sine alpha in the image, 848 00:46:06,070 --> 00:46:09,130 if I could make equal to the corresponding paraxial 849 00:46:09,130 --> 00:46:11,570 quantities, then I would be happy 850 00:46:11,570 --> 00:46:13,420 because then the magnification would 851 00:46:13,420 --> 00:46:19,000 be the same, independent of object elevation. 852 00:46:19,000 --> 00:46:21,870 And therefore, I would be free of coma. 853 00:46:21,870 --> 00:46:23,640 So that is the sine theorem. 854 00:46:33,984 --> 00:46:38,480 So basically, the reason coma can be eliminated here, 855 00:46:38,480 --> 00:46:42,970 it is because, for this particular lens configuration, 856 00:46:42,970 --> 00:46:45,550 the sine theorem is actually satisfying. 857 00:46:45,550 --> 00:46:50,500 So you basically get a complete elimination of the coma. 858 00:46:58,290 --> 00:47:02,170 So unless there's any questions, I think we should quit here, 859 00:47:02,170 --> 00:47:03,730 and we'll continue with the remaining 860 00:47:03,730 --> 00:47:06,930 aberrations on Wednesday. 861 00:47:06,930 --> 00:47:07,680 AUDIENCE: George-- 862 00:47:07,680 --> 00:47:08,010 GEORGE BARBASTATHIS: Yeah. 863 00:47:08,010 --> 00:47:09,960 AUDIENCE: --can I make a comment about the-- 864 00:47:09,960 --> 00:47:12,840 I've been thinking about this aplanatic word 865 00:47:12,840 --> 00:47:13,980 that you mentioned. 866 00:47:13,980 --> 00:47:18,298 If you go back a couple of slides to the-- 867 00:47:18,298 --> 00:47:19,590 GEORGE BARBASTATHIS: Maybe far? 868 00:47:19,590 --> 00:47:20,215 AUDIENCE: Yeah. 869 00:47:20,215 --> 00:47:23,410 That-- yeah, that one will do. 870 00:47:23,410 --> 00:47:26,300 I think that-- now I'm thinking where that word comes from. 871 00:47:26,300 --> 00:47:28,630 I've never really thought about it before, but the-- 872 00:47:28,630 --> 00:47:31,150 that, of course, the aplanatic condition 873 00:47:31,150 --> 00:47:34,090 will be satisfied not just at one point, 874 00:47:34,090 --> 00:47:37,720 but on a complete sphere, with center 875 00:47:37,720 --> 00:47:40,480 at the center of the spherical surface. 876 00:47:40,480 --> 00:47:45,130 And the perfect image will be formed on another sphere, also 877 00:47:45,130 --> 00:47:47,810 with its center at the center of the spherical surface. 878 00:47:47,810 --> 00:47:52,090 So I think maybe that's why it says not plane. 879 00:47:52,090 --> 00:47:56,770 It's because the image on a spherical surface 880 00:47:56,770 --> 00:47:59,633 is going to be formed perfectly on another spherical surface. 881 00:47:59,633 --> 00:48:02,050 GEORGE BARBASTATHIS: Yeah, because with a virtual surface, 882 00:48:02,050 --> 00:48:03,700 as we'll see a little bit later. 883 00:48:03,700 --> 00:48:04,200 Yeah. 884 00:48:04,200 --> 00:48:05,853 Yeah. 885 00:48:05,853 --> 00:48:06,770 Yep, that makes sense. 886 00:48:06,770 --> 00:48:08,320 AUDIENCE: Hmm.