1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high-quality educational resources for free. 5 00:00:10,120 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,650 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,650 --> 00:00:17,536 at ocw.mit.edu. 8 00:00:21,670 --> 00:00:27,300 GEORGE BARBASTATHIS: There's some comments from the grader 9 00:00:27,300 --> 00:00:28,960 who looked at your homeworks. 10 00:00:28,960 --> 00:00:31,880 So a couple of points to remember 11 00:00:31,880 --> 00:00:37,670 that they seem to be sort of pervasive in several homeworks. 12 00:00:37,670 --> 00:00:41,360 One is that when we do Snell's law, 13 00:00:41,360 --> 00:00:45,710 we have to apply the rule-- the rule goes 14 00:00:45,710 --> 00:00:52,070 like n sine theta equals n prime sine theta prime. 15 00:00:52,070 --> 00:00:56,030 So remember that theta prime is measured with respect 16 00:00:56,030 --> 00:00:58,410 to the normal to the interface. 17 00:00:58,410 --> 00:01:00,320 So if this line is the interface, 18 00:01:00,320 --> 00:01:03,170 I have, for example, air and the glass here. 19 00:01:03,170 --> 00:01:04,470 AUDIENCE: [INAUDIBLE] 20 00:01:04,470 --> 00:01:05,470 GEORGE BARBASTATHIS: Oh. 21 00:01:09,885 --> 00:01:11,010 AUDIENCE: Yes, that's good. 22 00:01:11,010 --> 00:01:12,620 GEORGE BARBASTATHIS: I think my hand is confusing it. 23 00:01:12,620 --> 00:01:14,203 Maybe I should turn off the autofocus. 24 00:01:18,680 --> 00:01:19,690 OK. 25 00:01:19,690 --> 00:01:22,090 So if this is the interface, air and glass-- yes, 26 00:01:22,090 --> 00:01:23,800 I think my hand was confusing the-- 27 00:01:26,970 --> 00:01:32,650 and the ray is incident, for example, like this, 28 00:01:32,650 --> 00:01:35,510 then we apply Snell's law with respect to the normal. 29 00:01:35,510 --> 00:01:39,370 So this is the angle theta, not this one out here, 30 00:01:39,370 --> 00:01:40,680 not the other way around. 31 00:01:40,680 --> 00:01:43,360 And for this one, of course, the same goes to the other side 32 00:01:43,360 --> 00:01:44,350 of the-- whoops-- 33 00:01:44,350 --> 00:01:45,683 the other side of the interface. 34 00:01:45,683 --> 00:01:48,530 The ray would go in kind of like this. 35 00:01:48,530 --> 00:01:50,170 That would be theta prime. 36 00:01:50,170 --> 00:01:52,990 So that was one comment from the homeworks. 37 00:01:56,105 --> 00:01:59,510 The other comment is about TIR. 38 00:01:59,510 --> 00:02:03,600 TIR actually happens for all angle of incidence 39 00:02:03,600 --> 00:02:06,240 that are larger than the critical angle. 40 00:02:06,240 --> 00:02:07,740 There seemed to be some confusion 41 00:02:07,740 --> 00:02:12,660 that somehow TIR happens only at the critical angle. 42 00:02:12,660 --> 00:02:13,990 That is not true. 43 00:02:13,990 --> 00:02:17,250 In fact, strictly speaking, at exactly the critical angle, 44 00:02:17,250 --> 00:02:18,900 TIR does not happen. 45 00:02:18,900 --> 00:02:21,810 What does happen is this surface wave 46 00:02:21,810 --> 00:02:24,690 that I mentioned in the notes. 47 00:02:24,690 --> 00:02:29,870 But for all angles that are greater 48 00:02:29,870 --> 00:02:34,090 than the critical angle, this is when we have the TIR 49 00:02:34,090 --> 00:02:37,650 condition for all those angles, from critical 50 00:02:37,650 --> 00:02:41,240 all the way to whatever it is, 90 degrees, I suppose. 51 00:02:44,460 --> 00:02:48,490 The third comment-- and, actually, this is the last one. 52 00:02:48,490 --> 00:02:53,680 The third comment is about the speed of light. 53 00:02:53,680 --> 00:02:56,010 Some of you-- and, actually, this 54 00:02:56,010 --> 00:03:00,250 is also in relation to the streamer problem. 55 00:03:00,250 --> 00:03:03,390 There was some confusion where the speed of light 56 00:03:03,390 --> 00:03:06,240 is proportional to the index of refraction 57 00:03:06,240 --> 00:03:08,040 or inversely proportional. 58 00:03:08,040 --> 00:03:10,830 It is, in fact, inversely proportional. 59 00:03:10,830 --> 00:03:16,310 The speed of light goes like c over n, 60 00:03:16,310 --> 00:03:18,650 where n is the index of refraction. 61 00:03:18,650 --> 00:03:20,840 And I don't know if there's a mnemonic for this, 62 00:03:20,840 --> 00:03:25,790 but one way to remember it is that c, 63 00:03:25,790 --> 00:03:28,550 the speed of light in vacuum, is the highest 64 00:03:28,550 --> 00:03:30,750 speed that anything can have. 65 00:03:30,750 --> 00:03:33,420 So, therefore, when light enters a material, 66 00:03:33,420 --> 00:03:36,890 it can only go slower. 67 00:03:36,890 --> 00:03:38,780 So, then, the index of refraction 68 00:03:38,780 --> 00:03:42,230 tells you the amount by which the speed of light 69 00:03:42,230 --> 00:03:46,970 becomes slower in a given dielectric. 70 00:03:46,970 --> 00:03:51,860 So these are some things that are 71 00:03:51,860 --> 00:03:54,270 useful to keep in mind as we move forward. 72 00:04:01,540 --> 00:04:03,510 These two dots don't mean anything. 73 00:04:03,510 --> 00:04:05,320 Just the ink from the markers that 74 00:04:05,320 --> 00:04:10,280 penetrated the piece of paper, so please ignore it. 75 00:04:10,280 --> 00:04:17,060 So, last time, the topic was composite optical elements. 76 00:04:17,060 --> 00:04:22,910 So we started with an example of two lenses, two thin lenses 77 00:04:22,910 --> 00:04:26,810 that were in tandem, and we did the imaging condition, 78 00:04:26,810 --> 00:04:30,750 magnification, and all those beautiful things with it. 79 00:04:30,750 --> 00:04:33,530 And then we also looked at the thin lens-- 80 00:04:33,530 --> 00:04:36,650 I'm sorry-- at the thick lens, where the thickness is 81 00:04:36,650 --> 00:04:38,060 defined-- 82 00:04:38,060 --> 00:04:40,220 as you recall, it is defined relative 83 00:04:40,220 --> 00:04:42,200 to the paraxial approximation. 84 00:04:42,200 --> 00:04:46,310 If the distance between the two curved surfaces 85 00:04:46,310 --> 00:04:50,720 exceeds the cosine theta, or whatever 86 00:04:50,720 --> 00:04:54,080 it is that the paraxial approximation limits 87 00:04:54,080 --> 00:04:59,600 for the negligible distance, then we call the lens thick. 88 00:04:59,600 --> 00:05:02,915 And we saw that there's some differences. 89 00:05:02,915 --> 00:05:04,790 For example, the focal length is a little bit 90 00:05:04,790 --> 00:05:07,250 different in the formula than it this 91 00:05:07,250 --> 00:05:09,680 for the case of the thin lens. 92 00:05:09,680 --> 00:05:11,750 But the primary reason why we did it 93 00:05:11,750 --> 00:05:14,060 is because I wanted to define this concept 94 00:05:14,060 --> 00:05:16,430 of principal planes. 95 00:05:16,430 --> 00:05:18,480 So the principal planes-- 96 00:05:18,480 --> 00:05:21,680 I'll move slowly so the camera can follow me. 97 00:05:21,680 --> 00:05:26,330 So the principle planes are basically the locations 98 00:05:26,330 --> 00:05:29,360 where the rays bend at once. 99 00:05:29,360 --> 00:05:32,700 So in a thick optical element, whether it is a thick lens 100 00:05:32,700 --> 00:05:36,830 or it is a composite of several lenses in tandem, 101 00:05:36,830 --> 00:05:39,770 as you can imagine, the rays will bend several times. 102 00:05:39,770 --> 00:05:41,930 They may bend up, downs, straight, 103 00:05:41,930 --> 00:05:44,190 and so on and so forth. 104 00:05:44,190 --> 00:05:48,740 But the idea is that if you take a parallel ray from infinity, 105 00:05:48,740 --> 00:05:51,410 after however many times it bends-- in this case, 106 00:05:51,410 --> 00:05:54,260 it bends twice, once here and once here-- 107 00:05:54,260 --> 00:05:58,490 eventually it comes out at the final propagation angle. 108 00:05:58,490 --> 00:06:00,290 So the principal plane is the plane 109 00:06:00,290 --> 00:06:03,200 that you obtain if you extend the incoming ray, 110 00:06:03,200 --> 00:06:08,000 you extend the outgoing ray, and you find where they meet. 111 00:06:08,000 --> 00:06:09,480 That is the principal plane. 112 00:06:09,480 --> 00:06:12,110 In fact, in general, it is a principal surface. 113 00:06:12,110 --> 00:06:15,440 If you really do the geometry right, it is not a plane. 114 00:06:15,440 --> 00:06:17,870 It turns out to be a curved surface, 115 00:06:17,870 --> 00:06:21,000 but within a paraxial approximation, it is a plane. 116 00:06:21,000 --> 00:06:24,680 So that's why we define it with sometimes as a principal plane 117 00:06:24,680 --> 00:06:27,610 or as a principal surface. 118 00:06:27,610 --> 00:06:29,760 I kind of try to stick to the term principal 119 00:06:29,760 --> 00:06:32,610 plane in the notes. 120 00:06:32,610 --> 00:06:35,070 And then, relative to the principal plane, 121 00:06:35,070 --> 00:06:38,760 as we did last time, you can define the various quantities. 122 00:06:38,760 --> 00:06:40,620 The effective focal length is simply 123 00:06:40,620 --> 00:06:45,220 the distance from that principal plane to the focal point. 124 00:06:45,220 --> 00:06:48,480 So, basically, as far as the rays that 125 00:06:48,480 --> 00:06:51,240 are coming from the left are concerned, 126 00:06:51,240 --> 00:06:57,630 the principal plane is acting like a thin lens, because the-- 127 00:06:57,630 --> 00:07:00,570 is acting like a thin lens with a focal length 128 00:07:00,570 --> 00:07:04,410 equal to the effective focal length of the composite, 129 00:07:04,410 --> 00:07:07,230 because this is the ray coming from infinity. 130 00:07:07,230 --> 00:07:09,780 And when it comes out, it is as if I 131 00:07:09,780 --> 00:07:14,040 had put a thin lens in this particular location. 132 00:07:14,040 --> 00:07:17,650 The opposite is true for images at infinity. 133 00:07:17,650 --> 00:07:20,160 So if you take a ray from this point here, 134 00:07:20,160 --> 00:07:22,740 it actually goes to the first principal plane, 135 00:07:22,740 --> 00:07:25,330 and then takes off to infinity. 136 00:07:25,330 --> 00:07:26,970 Now, of course, again, this is what 137 00:07:26,970 --> 00:07:32,010 I get when I take the incoming ray and the outgoing ray, 138 00:07:32,010 --> 00:07:33,630 and then find where they meet. 139 00:07:33,630 --> 00:07:37,560 In general, this ray also suffers multiple refractions. 140 00:07:37,560 --> 00:07:42,180 But if I take the first one and the very last one, 141 00:07:42,180 --> 00:07:45,060 then I can have a thin lens at the first principal plane. 142 00:07:45,060 --> 00:07:49,290 So this is the idea, and we can see the reason 143 00:07:49,290 --> 00:07:51,780 why they are convenient when we do 144 00:07:51,780 --> 00:07:56,520 the imaging condition for the composite optical element. 145 00:07:56,520 --> 00:07:57,710 So we'll not do this again. 146 00:07:57,710 --> 00:08:01,960 Se Baek went through it in great detail last time. 147 00:08:01,960 --> 00:08:04,800 But you can see that, basically, if we 148 00:08:04,800 --> 00:08:07,020 know the principal planes, then we 149 00:08:07,020 --> 00:08:09,570 don't have to worry about the rest of the physical system 150 00:08:09,570 --> 00:08:11,790 any more, because we can discover things 151 00:08:11,790 --> 00:08:13,890 like imaging condition, magnification, 152 00:08:13,890 --> 00:08:17,520 and so on and so forth simply by applying the rules 153 00:08:17,520 --> 00:08:20,220 that we learned in the case of the thin lens. 154 00:08:20,220 --> 00:08:23,580 For example, in this case, in order to find the image, 155 00:08:23,580 --> 00:08:25,560 I take a ray that is horizontal. 156 00:08:25,560 --> 00:08:28,230 Therefore, it would have come from infinity. 157 00:08:28,230 --> 00:08:31,500 I drag it all the way to the second principal plane. 158 00:08:31,500 --> 00:08:35,580 Then I know it has to go through the focus. 159 00:08:35,580 --> 00:08:37,419 And then I extend it further. 160 00:08:37,419 --> 00:08:40,890 When I take another one that goes through the first focus, 161 00:08:40,890 --> 00:08:43,710 then I know that at the first principal plane, 162 00:08:43,710 --> 00:08:48,240 this ray will have to become horizontal, and again 163 00:08:48,240 --> 00:08:50,080 take off towards infinity. 164 00:08:50,080 --> 00:08:51,960 So, therefore, this ray goes like this. 165 00:08:51,960 --> 00:08:53,460 And then where they meet is actually 166 00:08:53,460 --> 00:08:54,967 the location of the image. 167 00:08:54,967 --> 00:08:55,800 So this is the idea. 168 00:08:55,800 --> 00:09:00,120 And then all the rest of the formulas about the imaging 169 00:09:00,120 --> 00:09:04,230 distances, the magnifications, lateral, angular, so on 170 00:09:04,230 --> 00:09:08,370 and so forth, they all fall out from the similar triangles 171 00:09:08,370 --> 00:09:10,590 in this diagram, just in the same way 172 00:09:10,590 --> 00:09:13,480 we did them for the thin lens, except now we 173 00:09:13,480 --> 00:09:17,850 have these sort of additional space between the two 174 00:09:17,850 --> 00:09:19,060 principal planes. 175 00:09:19,060 --> 00:09:21,790 So we have to be careful when we deal with a ray that is 176 00:09:21,790 --> 00:09:23,830 coming horizontal at the input. 177 00:09:23,830 --> 00:09:26,910 It has to bend at the second principal plane. 178 00:09:26,910 --> 00:09:28,620 And when we think the ray that goes 179 00:09:28,620 --> 00:09:31,830 through the first focal point and comes out horizontal, 180 00:09:31,830 --> 00:09:34,830 that one has to go through the first principal plane. 181 00:09:34,830 --> 00:09:41,070 And the reason I'm emphasizing this is because-- 182 00:09:41,070 --> 00:09:43,560 this is the generic case, where the first principal is 183 00:09:43,560 --> 00:09:46,180 to the left, the second is to the right. 184 00:09:46,180 --> 00:09:48,120 But there is optical systems where 185 00:09:48,120 --> 00:09:49,620 they may actually be swapped. 186 00:09:49,620 --> 00:09:51,450 You may have the second principal plane 187 00:09:51,450 --> 00:09:54,270 to the left of the first principal plane. 188 00:09:54,270 --> 00:09:56,240 There's no rule that forbids that. 189 00:09:56,240 --> 00:09:57,690 It can happen. 190 00:09:57,690 --> 00:09:58,860 So that's fine. 191 00:09:58,860 --> 00:10:00,420 We still have to apply the same rule, 192 00:10:00,420 --> 00:10:03,240 but we have to be careful where we bend the ray. 193 00:10:03,240 --> 00:10:05,830 So one of the homeworks-- you will see the homeworks that we 194 00:10:05,830 --> 00:10:09,240 posted today and are due, actually, next week-- 195 00:10:09,240 --> 00:10:11,910 you will see that this happens in one of the optical systems 196 00:10:11,910 --> 00:10:13,140 that we give you there. 197 00:10:13,140 --> 00:10:16,110 The second principal plane is actually to the left. 198 00:10:16,110 --> 00:10:21,880 So I'll let you work that out by yourselves, 199 00:10:21,880 --> 00:10:24,210 but keep that in mind. 200 00:10:24,210 --> 00:10:27,420 Always identify which principal plane 201 00:10:27,420 --> 00:10:30,400 you're using for any given imaging condition. 202 00:10:37,310 --> 00:10:39,930 Any questions about all of this? 203 00:10:44,540 --> 00:10:46,210 Can someone say at least good morning, 204 00:10:46,210 --> 00:10:49,824 so I know if our sound has been restored? 205 00:10:49,824 --> 00:10:51,947 AUDIENCE: Hello, hello, can you hear us? 206 00:10:51,947 --> 00:10:53,530 GEORGE BARBASTATHIS: OK, either nobody 207 00:10:53,530 --> 00:10:55,610 wants to say good morning or there's no sound. 208 00:10:55,610 --> 00:10:56,152 AUDIENCE: No. 209 00:10:56,152 --> 00:10:58,010 No sound, I guess. 210 00:10:58,010 --> 00:10:59,240 GEORGE BARBASTATHIS: OK. 211 00:10:59,240 --> 00:11:01,922 I'll keep going, and, hopefully, some time at some point 212 00:11:01,922 --> 00:11:02,630 we'll have sound. 213 00:11:02,630 --> 00:11:04,220 Or at some time you'll wake up enough 214 00:11:04,220 --> 00:11:06,650 to be willing to say good morning. 215 00:11:09,720 --> 00:11:11,850 So, today, we'll deal with a topic 216 00:11:11,850 --> 00:11:15,480 called stops and apertures. 217 00:11:15,480 --> 00:11:17,850 And, basically, we'll spend the rest 218 00:11:17,850 --> 00:11:21,900 of the hour learning the meaning of all of these terms. 219 00:11:21,900 --> 00:11:23,810 So there's quite a few of them-- 220 00:11:23,810 --> 00:11:27,075 aperture, pupil, entrance, exit pupil. 221 00:11:27,075 --> 00:11:31,300 And then we field stop, window, entrance and exit, and so on. 222 00:11:31,300 --> 00:11:33,690 So we'll spend some time learning what these things are 223 00:11:33,690 --> 00:11:37,470 and why they're useful in an optical system. 224 00:11:37,470 --> 00:11:40,580 So starting with the first one, the aperture stop, 225 00:11:40,580 --> 00:11:42,800 the aperture stop has to do with the fact 226 00:11:42,800 --> 00:11:47,780 that any optical instrument, in practice, has a finite size. 227 00:11:47,780 --> 00:11:51,180 Therefore-- OK, let me back up. 228 00:11:51,180 --> 00:11:53,570 So the instrument has a finite size. 229 00:11:53,570 --> 00:11:56,775 But a point source, as we learned, it actually-- 230 00:11:56,775 --> 00:11:58,760 at least an ideal source-- 231 00:11:58,760 --> 00:12:05,570 it emits kind of isotropically, all the way 4 pi steradians 232 00:12:05,570 --> 00:12:08,730 around the location of the point source. 233 00:12:08,730 --> 00:12:12,410 So what this means, then, is that the optical instrument 234 00:12:12,410 --> 00:12:17,410 can only capture a fraction of the light that this-- 235 00:12:17,410 --> 00:12:19,730 that any given point sources is emitting. 236 00:12:19,730 --> 00:12:21,350 So the aperture actually helps us 237 00:12:21,350 --> 00:12:24,920 quantify how much is the light that an optical instrument can 238 00:12:24,920 --> 00:12:28,920 admit given the location of a point source. 239 00:12:28,920 --> 00:12:32,420 So it is defined, actually, with respect to a point source that 240 00:12:32,420 --> 00:12:35,685 is located at the optical axis, that is, 241 00:12:35,685 --> 00:12:39,320 at the center where all of the optics 242 00:12:39,320 --> 00:12:42,800 are rotationally symmetric about. 243 00:12:42,800 --> 00:12:45,600 And there's two concepts here. 244 00:12:45,600 --> 00:12:47,660 One is the physical stop. 245 00:12:47,660 --> 00:12:49,820 So the physical stop is-- you can think of it 246 00:12:49,820 --> 00:12:55,130 as the rim of the optics, the physical limit beyond which 247 00:12:55,130 --> 00:12:57,560 the rays that are coming from the source, 248 00:12:57,560 --> 00:13:00,380 they miss the optical elements in the system. 249 00:13:00,380 --> 00:13:03,710 Those of you who have cameras, you're also 250 00:13:03,710 --> 00:13:07,580 familiar with physical stops that, 251 00:13:07,580 --> 00:13:11,220 in relatively complicated optical systems-- 252 00:13:11,220 --> 00:13:13,670 for example the lens that goes with the camera-- 253 00:13:13,670 --> 00:13:16,700 inside there's a little hole that 254 00:13:16,700 --> 00:13:19,760 is very often adjustable depending on the light level 255 00:13:19,760 --> 00:13:20,510 and so on. 256 00:13:20,510 --> 00:13:23,270 This hole, nowadays automatically, 257 00:13:23,270 --> 00:13:26,150 in the old days with photographers 258 00:13:26,150 --> 00:13:28,280 having manual control, you would actually 259 00:13:28,280 --> 00:13:32,200 open and close that opening to admit more or less light. 260 00:13:32,200 --> 00:13:34,720 This is called the aperture stop. 261 00:13:34,720 --> 00:13:37,490 And we also are familiar with something else, 262 00:13:37,490 --> 00:13:39,470 the pupil of our eyes. 263 00:13:39,470 --> 00:13:42,033 If you look at someone's eyes, there's a little hole. 264 00:13:42,033 --> 00:13:43,700 And if you look in a bright environment, 265 00:13:43,700 --> 00:13:45,800 this hole is relatively small. 266 00:13:45,800 --> 00:13:47,510 If you move to a dark environment, 267 00:13:47,510 --> 00:13:49,700 our eyes have a mechanism that kind of 268 00:13:49,700 --> 00:13:51,770 measures the ambient light and opens up 269 00:13:51,770 --> 00:13:53,540 or closes down the pupil. 270 00:13:53,540 --> 00:14:00,360 So this is the aperture stop for the optical system of our eye. 271 00:14:00,360 --> 00:14:02,910 So the aperture stop, then, is the physical element 272 00:14:02,910 --> 00:14:06,540 that limits the acceptance of the light rays. 273 00:14:06,540 --> 00:14:08,120 So, in this case, the aperture stop, 274 00:14:08,120 --> 00:14:12,000 you can think of this sort of black, thick line here, 275 00:14:12,000 --> 00:14:14,010 you can think of it as a physical, for example 276 00:14:14,010 --> 00:14:17,730 a metal plate with a hole in the center. 277 00:14:17,730 --> 00:14:20,880 So that hole would be the actual aperture stop 278 00:14:20,880 --> 00:14:23,430 in this particular instrument. 279 00:14:23,430 --> 00:14:28,590 The numerical aperture is a mathematical quantity. 280 00:14:28,590 --> 00:14:34,430 It is the angle subtended from the point source 281 00:14:34,430 --> 00:14:38,970 to the rim of the physical aperture stop. 282 00:14:38,970 --> 00:14:42,120 And I was a little bit sloppy when I called the angle. 283 00:14:42,120 --> 00:14:45,070 It is not the actual definition. 284 00:14:45,070 --> 00:14:48,210 It's the sine of that angle multiplied 285 00:14:48,210 --> 00:14:50,010 by the index of refraction. 286 00:14:50,010 --> 00:14:53,250 Now, we've already seen Snell's law several times. 287 00:14:53,250 --> 00:14:56,340 You can guess immediately the reason people define 288 00:14:56,340 --> 00:15:00,450 the numerical aperture this way is so that this quantity is 289 00:15:00,450 --> 00:15:06,870 conserved when you go through flat refractive interfaces. 290 00:15:06,870 --> 00:15:10,750 So it's very convenient because, well, as we'll see later, 291 00:15:10,750 --> 00:15:14,430 the numerical aperture is a very important quantity 292 00:15:14,430 --> 00:15:16,750 in characterizing an optical system. 293 00:15:16,750 --> 00:15:19,500 So, therefore, by sticking the index of refraction 294 00:15:19,500 --> 00:15:21,750 in this definition here, we'll make sure 295 00:15:21,750 --> 00:15:24,150 that the numerical aperture is preserved 296 00:15:24,150 --> 00:15:29,980 as rays go through various elements in the system, 297 00:15:29,980 --> 00:15:33,850 at least elements without power, without optical power, that is, 298 00:15:33,850 --> 00:15:37,278 as they go through flat interfaces. 299 00:15:37,278 --> 00:15:38,320 So that's the definition. 300 00:15:38,320 --> 00:15:40,480 Now, of course, in many optical systems, 301 00:15:40,480 --> 00:15:44,800 the index where the object lives is air. 302 00:15:44,800 --> 00:15:46,690 So, therefore, n is 1. 303 00:15:46,690 --> 00:15:48,970 And also, in this class at least, 304 00:15:48,970 --> 00:15:51,310 we deal with the paraxial approximation, 305 00:15:51,310 --> 00:15:53,620 where the sine of the angle is approximately 306 00:15:53,620 --> 00:15:54,940 equal to the angle itself. 307 00:15:54,940 --> 00:15:57,340 So, therefore, indeed, the numerical aperture 308 00:15:57,340 --> 00:16:00,890 is approximately equal to the angle itself. 309 00:16:00,890 --> 00:16:03,280 However, as you have seen from the homework, 310 00:16:03,280 --> 00:16:05,890 if you already started it, in many optical systems 311 00:16:05,890 --> 00:16:13,090 the object lives in a space of refractive index 312 00:16:13,090 --> 00:16:14,260 different than 1. 313 00:16:14,260 --> 00:16:16,330 For example, in immersion microscopy, 314 00:16:16,330 --> 00:16:18,280 we put a droplet of oil. 315 00:16:21,090 --> 00:16:23,680 We drop a droplet of oil on top of the sample. 316 00:16:23,680 --> 00:16:26,080 Then we stick the objective so that there 317 00:16:26,080 --> 00:16:29,140 is oil in this space here between the sample 318 00:16:29,140 --> 00:16:29,980 and the objective. 319 00:16:29,980 --> 00:16:31,510 Well, in that case, we really should 320 00:16:31,510 --> 00:16:34,510 include the index of refraction in the definition 321 00:16:34,510 --> 00:16:37,450 of the optic-- of the numerical aperture. 322 00:16:37,450 --> 00:16:40,360 So, of course, you cannot speak yet, 323 00:16:40,360 --> 00:16:44,200 but I do want to ask a question myself now. 324 00:16:44,200 --> 00:16:46,842 And if you can answer-- 325 00:16:46,842 --> 00:16:48,550 well, those of you who can answer easily. 326 00:16:48,550 --> 00:16:50,467 Those of you in Boston, if you want to answer, 327 00:16:50,467 --> 00:16:53,620 just please wave and we'll figure out how to do it. 328 00:16:53,620 --> 00:16:55,320 But the question I wanted to ask, 329 00:16:55,320 --> 00:16:57,720 what do you think the value of the numerical aperture is? 330 00:16:57,720 --> 00:17:01,750 Is it smaller than 1, bigger than 1? 331 00:17:01,750 --> 00:17:03,960 What is it? 332 00:17:03,960 --> 00:17:06,300 Do this if you think that the numerical aperture is 333 00:17:06,300 --> 00:17:07,560 bigger than 1. 334 00:17:07,560 --> 00:17:10,319 Do this if you think that the numerical aperture is always 335 00:17:10,319 --> 00:17:12,338 less than 1. 336 00:17:12,338 --> 00:17:13,130 What's the verdict? 337 00:17:16,079 --> 00:17:19,010 Always less than 1. 338 00:17:19,010 --> 00:17:21,770 Anybody who thinks the opposite, that the numerical aperture can 339 00:17:21,770 --> 00:17:22,550 be bigger than 1? 340 00:17:26,861 --> 00:17:27,869 It's really strange. 341 00:17:27,869 --> 00:17:30,658 I feel like we're transported to a different century now 342 00:17:30,658 --> 00:17:31,574 [INAUDIBLE]. 343 00:17:31,574 --> 00:17:33,750 Anyway. 344 00:17:33,750 --> 00:17:38,780 So, actually, in principle-- and, in fact, people do it-- 345 00:17:38,780 --> 00:17:41,420 I can construct an optical instrument 346 00:17:41,420 --> 00:17:44,300 with a relatively large acceptance angle. 347 00:17:44,300 --> 00:17:49,520 For example, theta may be as high as 75 degrees, 80 degrees. 348 00:17:49,520 --> 00:17:53,810 It is not uncommon in high-end microscope objective lenses 349 00:17:53,810 --> 00:17:57,130 to have a very high acceptance angle. 350 00:17:57,130 --> 00:17:59,690 Of course, this would violate the paraxial approximation, 351 00:17:59,690 --> 00:18:01,400 but that is no problem. 352 00:18:01,400 --> 00:18:04,310 The fact that the instrument is not paraxial 353 00:18:04,310 --> 00:18:06,470 actually does not make it an invalid instrument, 354 00:18:06,470 --> 00:18:07,910 so it's fine. 355 00:18:07,910 --> 00:18:11,000 All it means is that our simple theory here does not apply, 356 00:18:11,000 --> 00:18:13,520 and we have to be more sophisticated in the analysis, 357 00:18:13,520 --> 00:18:15,440 but we can have a large acceptance angle. 358 00:18:15,440 --> 00:18:18,710 For example-- so, in that case, the sine of this angle 359 00:18:18,710 --> 00:18:22,910 may be close to 1, maybe 0.9 or 0.95. 360 00:18:22,910 --> 00:18:26,190 And, also, in addition, remember that we can use immersion. 361 00:18:26,190 --> 00:18:31,190 So we can put the object in a liquid of index of refraction-- 362 00:18:31,190 --> 00:18:34,190 for example, if it is water, that would be 1.3. 363 00:18:34,190 --> 00:18:36,260 If it is one of the immersion oils 364 00:18:36,260 --> 00:18:38,870 that are used in microscopy, what is the index, typically? 365 00:18:38,870 --> 00:18:40,270 1.4, 1 point-- 366 00:18:40,270 --> 00:18:41,300 AUDIENCE: [INAUDIBLE] 367 00:18:41,300 --> 00:18:42,300 GEORGE BARBASTATHIS: Oh. 368 00:18:42,300 --> 00:18:45,475 Just the same as glass? 369 00:18:45,475 --> 00:18:46,385 AUDIENCE: [INAUDIBLE] 370 00:18:46,385 --> 00:18:47,750 GEORGE BARBASTATHIS: OK. 371 00:18:47,750 --> 00:18:53,730 So even higher than for immersion instruments, 1.54-- 372 00:18:53,730 --> 00:18:54,230 0.14. 373 00:18:54,230 --> 00:18:54,730 I'm sorry. 374 00:18:54,730 --> 00:18:57,170 1.514. 375 00:18:57,170 --> 00:19:02,767 And so, in that case, if I do 1.514 times 0.9, of course 376 00:19:02,767 --> 00:19:04,100 it will be bigger than 1, right? 377 00:19:11,855 --> 00:19:13,980 I don't know what it is, but it would be definitely 378 00:19:13,980 --> 00:19:15,180 bigger than 1. 379 00:19:15,180 --> 00:19:17,820 So, actually, I can have a numerical aperture 380 00:19:17,820 --> 00:19:19,350 that is bigger than 1. 381 00:19:19,350 --> 00:19:23,840 That is kind of a useful point to remember. 382 00:19:23,840 --> 00:19:25,990 Related to the-- so the numerical aperture 383 00:19:25,990 --> 00:19:28,970 is used very commonly in microscopy 384 00:19:28,970 --> 00:19:30,950 when we characterize objective lenses. 385 00:19:30,950 --> 00:19:35,240 Typically, we characterize them by their numerical aperture. 386 00:19:35,240 --> 00:19:38,750 In photography, especially, people 387 00:19:38,750 --> 00:19:42,930 prefer to use an alternate quantity known as the f number. 388 00:19:42,930 --> 00:19:44,840 So the f number is kind of the inverse 389 00:19:44,840 --> 00:19:47,990 of the numerical aperture times a factor of 1/2, 390 00:19:47,990 --> 00:19:51,200 which always confuses the hell out of me. 391 00:19:51,200 --> 00:19:56,630 But, anyway, the f number, it is written like this. 392 00:19:56,630 --> 00:20:01,020 And people replace the hash symbol with the actual number. 393 00:20:01,020 --> 00:20:04,970 For example, if you look at the lens of a camera, 394 00:20:04,970 --> 00:20:06,650 it may something like f/8. 395 00:20:09,230 --> 00:20:13,680 This means that the f number of this camera lens is 8. 396 00:20:13,680 --> 00:20:16,730 So the rule of thumb is where does your money go? 397 00:20:16,730 --> 00:20:20,600 When you pay money for a very good quality microscope 398 00:20:20,600 --> 00:20:24,510 objective, one of those that cost $10,000 or so, 399 00:20:24,510 --> 00:20:25,910 typically you pay-- 400 00:20:25,910 --> 00:20:28,190 in a microscope objective, you pay 401 00:20:28,190 --> 00:20:31,010 for a large numerical aperture. 402 00:20:31,010 --> 00:20:33,500 When you walk into a store and you 403 00:20:33,500 --> 00:20:38,190 buy a high-end, expensive camera lens, one of those Nikon 404 00:20:38,190 --> 00:20:41,390 or a Zeiss, really expensive multi-element lenses 405 00:20:41,390 --> 00:20:45,020 that look about as big as my computer 406 00:20:45,020 --> 00:20:49,580 here, what you pay there is for a really low f number. 407 00:20:49,580 --> 00:20:53,060 So, generally, large numerical aperture is good. 408 00:20:53,060 --> 00:20:57,740 Or, equivalently, low f number is good, 409 00:20:57,740 --> 00:20:59,390 provided you can afford it, of course. 410 00:21:03,750 --> 00:21:07,050 So, at the moment, the way we defined it so far, 411 00:21:07,050 --> 00:21:10,440 the numerical aperture, it seems to be related only 412 00:21:10,440 --> 00:21:13,380 to the energy acceptance of the optical system, that 413 00:21:13,380 --> 00:21:14,970 is, how much-- 414 00:21:14,970 --> 00:21:19,410 how many watts of optical power the optical system admits. 415 00:21:19,410 --> 00:21:21,330 So this is of course important, but you 416 00:21:21,330 --> 00:21:23,310 might wonder why am I making such a big deal, 417 00:21:23,310 --> 00:21:26,460 and why people pay so much money for high NA. 418 00:21:26,460 --> 00:21:30,600 Well, again, the energy acceptance, as you can imagine, 419 00:21:30,600 --> 00:21:32,280 is incredibly important because it 420 00:21:32,280 --> 00:21:36,360 will determine how many useful photons from the object you 421 00:21:36,360 --> 00:21:39,890 admit into your system for imaging purposes. 422 00:21:39,890 --> 00:21:43,830 But, as we will learn later, the numerical aperture 423 00:21:43,830 --> 00:21:46,820 also defines the resolution of an optical system. 424 00:21:46,820 --> 00:21:49,780 And, right now, you probably know from everyday life 425 00:21:49,780 --> 00:21:51,060 what resolution means. 426 00:21:51,060 --> 00:21:53,300 People talk about the resolution in your camera, 427 00:21:53,300 --> 00:21:56,280 in your television, and so on and so forth. 428 00:21:56,280 --> 00:21:58,650 We will do a precise definition of resolution 429 00:21:58,650 --> 00:22:01,320 later, in about a month from now. 430 00:22:01,320 --> 00:22:03,510 We'll define it very precisely, and we'll 431 00:22:03,510 --> 00:22:08,040 see why and how it is related to this numerical aperture 432 00:22:08,040 --> 00:22:09,000 quantity. 433 00:22:09,000 --> 00:22:13,900 But, again, the rule is that the largest the numerical aperture, 434 00:22:13,900 --> 00:22:17,940 or, equivalently, the smaller the f number, 435 00:22:17,940 --> 00:22:22,290 the better the resolution of an optical system. 436 00:22:22,290 --> 00:22:25,370 So we will see this come up again and again. 437 00:22:28,350 --> 00:22:34,860 So that's the basic definition that we use. 438 00:22:34,860 --> 00:22:37,350 Now, related to this concept of aperture stop 439 00:22:37,350 --> 00:22:39,690 and numeric aperture, there is two sort 440 00:22:39,690 --> 00:22:42,360 of derivative definitions, and these 441 00:22:42,360 --> 00:22:46,410 are called the pupils, the entrance and exit pupils. 442 00:22:46,410 --> 00:22:48,540 So in order to define those, I kind of 443 00:22:48,540 --> 00:22:50,350 need a multi-element optical system, 444 00:22:50,350 --> 00:22:52,800 so I sort of concocted one over there. 445 00:22:52,800 --> 00:22:53,550 I just made it up. 446 00:22:53,550 --> 00:22:55,310 This is nothing in particular. 447 00:22:55,310 --> 00:22:59,010 I just stuck some lenses in tandem, 448 00:22:59,010 --> 00:23:02,430 And i put the aperture stop somewhere in the middle. 449 00:23:02,430 --> 00:23:07,710 So think of this again as a physical element, a hole 450 00:23:07,710 --> 00:23:10,360 that I put in there. 451 00:23:10,360 --> 00:23:13,560 So the reason we need these additional definitions 452 00:23:13,560 --> 00:23:18,450 of the pupils is that if the aperture stop is varied 453 00:23:18,450 --> 00:23:21,690 inside an optical instrument, as it happens, for example, 454 00:23:21,690 --> 00:23:23,100 in your camera lens-- if you look 455 00:23:23,100 --> 00:23:25,993 at your photographic camera that you have at home, 456 00:23:25,993 --> 00:23:27,660 you will see that the aperture is varied 457 00:23:27,660 --> 00:23:30,750 inside a relatively large lens barrel 458 00:23:30,750 --> 00:23:34,170 if you have a sophisticated camera. 459 00:23:34,170 --> 00:23:36,510 The aperture is a physical element. 460 00:23:36,510 --> 00:23:37,860 It is a hole, right? 461 00:23:37,860 --> 00:23:43,250 So if you look at your lens from the one side 462 00:23:43,250 --> 00:23:45,890 that looks towards the object, or from the opposite side that 463 00:23:45,890 --> 00:23:50,500 looks towards the CCD, you will see the aperture, 464 00:23:50,500 --> 00:23:53,730 but you will not see the aperture itself, 465 00:23:53,730 --> 00:23:57,050 you will see the image of that aperture stop 466 00:23:57,050 --> 00:24:00,830 through the elements that precede or succeed it. 467 00:24:00,830 --> 00:24:06,710 So because of that, people have defined this concept 468 00:24:06,710 --> 00:24:09,890 of the entrance and exit pupil. 469 00:24:09,890 --> 00:24:11,690 So the entrance pupil is what you 470 00:24:11,690 --> 00:24:14,090 get if you take this aperture stop 471 00:24:14,090 --> 00:24:17,210 and now you treat it as an object in an imaging system, 472 00:24:17,210 --> 00:24:22,010 and you image it through the preceding optical elements. 473 00:24:22,010 --> 00:24:25,850 Now, here, I've committed a mortal violation. 474 00:24:25,850 --> 00:24:29,450 I imaged from right to left. 475 00:24:29,450 --> 00:24:33,380 We said last time that every time we'll 476 00:24:33,380 --> 00:24:35,600 write down an optical system, the light 477 00:24:35,600 --> 00:24:40,478 is assumed to go onwards from left to right. 478 00:24:40,478 --> 00:24:41,770 So, here, this is an exception. 479 00:24:41,770 --> 00:24:43,660 When we deal with aperture stops, 480 00:24:43,660 --> 00:24:46,870 we actually image from right to left, 481 00:24:46,870 --> 00:24:50,710 because we really need to look at this as the object, 482 00:24:50,710 --> 00:24:55,000 and image it through the optical elements that precede it. 483 00:24:55,000 --> 00:24:57,705 So if we do that, then we get the entrance pupil. 484 00:24:57,705 --> 00:25:01,940 So, in this case, it would look something like this. 485 00:25:01,940 --> 00:25:05,980 So this is now not a physical component any more. 486 00:25:05,980 --> 00:25:08,980 It is the image of the physical component that 487 00:25:08,980 --> 00:25:11,560 was inside the system, and we imaged 488 00:25:11,560 --> 00:25:13,930 through the elements that preceded it 489 00:25:13,930 --> 00:25:16,360 in that optical system. 490 00:25:16,360 --> 00:25:19,630 Similarly, we can define the exit pupil 491 00:25:19,630 --> 00:25:21,340 if we image the physical aperture 492 00:25:21,340 --> 00:25:24,730 stop through the elements that succeed it. 493 00:25:24,730 --> 00:25:27,038 So the two definitions are kind of conjugate. 494 00:25:29,910 --> 00:25:37,310 Now, with respect to this entrance, 495 00:25:37,310 --> 00:25:41,820 exit pupils and aperture stops-- this is a mouthful-- 496 00:25:41,820 --> 00:25:45,530 you can define two significant rays that 497 00:25:45,530 --> 00:25:48,200 come up in the design of optical systems all the time, 498 00:25:48,200 --> 00:25:50,030 and we'll see why. 499 00:25:50,030 --> 00:25:54,510 The first one is called the chief ray, 500 00:25:54,510 --> 00:25:57,430 and the chief ray is defined as the ray that 501 00:25:57,430 --> 00:26:00,130 goes through the center of the aperture stop, 502 00:26:00,130 --> 00:26:01,480 the physical aperture stop. 503 00:26:05,850 --> 00:26:06,970 So that's one. 504 00:26:06,970 --> 00:26:10,200 The other is called the marginal ray, 505 00:26:10,200 --> 00:26:13,770 and it is the ray that goes through the rim 506 00:26:13,770 --> 00:26:15,840 of the physical aperture stop, that is, 507 00:26:15,840 --> 00:26:19,950 the ray that just clears the aperture. 508 00:26:19,950 --> 00:26:22,650 If you take the next ray, that would 509 00:26:22,650 --> 00:26:27,520 propagate at a slightly higher angle with respect to this one. 510 00:26:27,520 --> 00:26:31,560 This ray would actually hit the physical block 511 00:26:31,560 --> 00:26:34,830 that is located at the aperture plane and would miss. 512 00:26:34,830 --> 00:26:40,830 So this last ray just before the aperture stop 513 00:26:40,830 --> 00:26:42,750 blocks the rest of the light, this 514 00:26:42,750 --> 00:26:45,443 is called the marginal ray. 515 00:26:45,443 --> 00:26:47,360 So you can already see the importance of this, 516 00:26:47,360 --> 00:26:51,660 because the marginal ray kind of defines again 517 00:26:51,660 --> 00:26:54,920 the angle of acceptance of the optical system. 518 00:26:54,920 --> 00:26:56,630 Except, this time, notice that I've 519 00:26:56,630 --> 00:26:59,620 done this for an off-axis object point. 520 00:26:59,620 --> 00:27:02,060 Recall that a numerical aperture, the way I 521 00:27:02,060 --> 00:27:04,910 defined it before, it was still the angle of acceptance 522 00:27:04,910 --> 00:27:08,590 but for an on-axis object point. 523 00:27:08,590 --> 00:27:11,150 Here, the marginal and the chief rays, 524 00:27:11,150 --> 00:27:15,410 I can define them for any point I like, any point I 525 00:27:15,410 --> 00:27:17,930 like that is in the object space. 526 00:27:23,090 --> 00:27:24,830 And this step, let me sort of-- 527 00:27:24,830 --> 00:27:28,530 OK, I already decided that these rays help us 528 00:27:28,530 --> 00:27:30,650 define the angle of acceptance. 529 00:27:30,650 --> 00:27:33,200 Now, before I show the next slide, 530 00:27:33,200 --> 00:27:42,320 and since these two rays, these two important rays, 531 00:27:42,320 --> 00:27:46,340 where do you think they might go with respect to the entrance 532 00:27:46,340 --> 00:27:48,140 and exit pupils? 533 00:27:48,140 --> 00:27:52,310 By definition, they go through the center and edge 534 00:27:52,310 --> 00:27:54,170 of the aperture stop, respectively, 535 00:27:54,170 --> 00:27:57,020 the chief and marginal ray. 536 00:27:57,020 --> 00:28:00,510 So, therefore, with respect to the entrance and exit pupils, 537 00:28:00,510 --> 00:28:02,390 where do you think these rays would pass? 538 00:28:22,495 --> 00:28:23,870 So the answer is that they should 539 00:28:23,870 --> 00:28:26,030 pass through the center and an edge, 540 00:28:26,030 --> 00:28:29,360 respectively, as well, because the entrance and exit 541 00:28:29,360 --> 00:28:32,570 pupils are images of the aperture stop. 542 00:28:32,570 --> 00:28:34,220 So, for example, if the chief ray 543 00:28:34,220 --> 00:28:37,100 goes through the center of the aperture stop, 544 00:28:37,100 --> 00:28:40,760 then it should also go through the center of the entrance 545 00:28:40,760 --> 00:28:43,520 and exit pupils because these are 546 00:28:43,520 --> 00:28:47,270 images of the aperture stop through the preceding 547 00:28:47,270 --> 00:28:48,990 and succeeding elements. 548 00:28:48,990 --> 00:28:51,430 And, similarly, the marginal ray hits 549 00:28:51,430 --> 00:28:54,230 the rim of the aperture stop. 550 00:28:54,230 --> 00:28:58,520 Therefore, it should also hit the rim of the entrance 551 00:28:58,520 --> 00:29:01,227 and exit pupils, respectively. 552 00:29:01,227 --> 00:29:02,810 Now, for the marginal ray, there could 553 00:29:02,810 --> 00:29:04,380 be magnifications involved. 554 00:29:04,380 --> 00:29:07,610 So, for example, the distance from the optical axis 555 00:29:07,610 --> 00:29:12,680 to the rim, it may be different in the aperture stop plane 556 00:29:12,680 --> 00:29:16,340 than it is, for example, in the entrance pupil plane. 557 00:29:16,340 --> 00:29:18,200 And that is because, in this case, 558 00:29:18,200 --> 00:29:21,710 the optical system has magnified the aperture stop. 559 00:29:21,710 --> 00:29:23,330 So the entrance pupil in this case 560 00:29:23,330 --> 00:29:26,360 is bigger than the physical stop itself. 561 00:29:26,360 --> 00:29:28,340 So, therefore, the marginal ray, it 562 00:29:28,340 --> 00:29:31,080 hits the optical axis at a different location. 563 00:29:31,080 --> 00:29:33,080 And, also, as you can see, there's an inversion. 564 00:29:33,080 --> 00:29:35,777 It hits at the bottom at the aperture stop, 565 00:29:35,777 --> 00:29:37,860 and then it hits at the top of the entrance pupil. 566 00:29:37,860 --> 00:29:39,500 This is fine, but what I-- 567 00:29:39,500 --> 00:29:42,260 because of the way optical systems behave. 568 00:29:42,260 --> 00:29:44,820 They can produce inversions and magnifications, 569 00:29:44,820 --> 00:29:46,520 and as we have seen. 570 00:29:46,520 --> 00:29:48,410 But what I want to emphasize is that they 571 00:29:48,410 --> 00:29:51,560 go through the rim because of that property 572 00:29:51,560 --> 00:29:56,750 that the aperture is imaged at the entrance and exit pupils, 573 00:29:56,750 --> 00:29:59,990 respectively. 574 00:29:59,990 --> 00:30:01,910 And, of course, the two rays will also 575 00:30:01,910 --> 00:30:08,360 meet again at the corresponding image point at the image plane. 576 00:30:08,360 --> 00:30:14,240 Because the two rays emanated from the object, if the imaging 577 00:30:14,240 --> 00:30:17,120 system is good, that is, if it does form an image properly 578 00:30:17,120 --> 00:30:19,790 and we don't have some kind of a weird defocus 579 00:30:19,790 --> 00:30:21,920 or some other kind of inconvenience, 580 00:30:21,920 --> 00:30:25,430 then the two rays should better meet again at the image point, 581 00:30:25,430 --> 00:30:31,820 because the entire spherical ray bundle that emanated here, 582 00:30:31,820 --> 00:30:33,710 that diverged here, it would also 583 00:30:33,710 --> 00:30:36,027 converge by the time it arrives here. 584 00:30:36,027 --> 00:30:37,610 So, therefore, these two rays are just 585 00:30:37,610 --> 00:30:39,890 members of this spherical ray bundle, 586 00:30:39,890 --> 00:30:42,110 so they will meet again at the object point. 587 00:30:46,930 --> 00:30:55,275 So any questions? 588 00:30:55,275 --> 00:30:56,400 I know this is frustrating. 589 00:30:56,400 --> 00:30:59,020 Even if you have a question, you cannot speak your mind. 590 00:30:59,020 --> 00:31:00,570 But any questions? 591 00:31:00,570 --> 00:31:03,982 Can you just wave at least so I know to save time later 592 00:31:03,982 --> 00:31:04,940 to answer the question? 593 00:31:18,240 --> 00:31:22,801 AUDIENCE: So how do you define the distance between the-- 594 00:31:22,801 --> 00:31:23,843 I guess he can't hear me. 595 00:31:23,843 --> 00:31:24,810 Ah, right. 596 00:31:24,810 --> 00:31:25,560 GEORGE BARBASTATHIS: Let me move on 597 00:31:25,560 --> 00:31:27,090 with a slightly different concept, 598 00:31:27,090 --> 00:31:30,280 and then we'll come back to this topic again. 599 00:31:30,280 --> 00:31:32,340 So the next topic is a different kind 600 00:31:32,340 --> 00:31:36,260 of stop called the field stop. 601 00:31:36,260 --> 00:31:39,140 So remember the definition of chief ray. 602 00:31:39,140 --> 00:31:41,270 We said that the chief ray is a ray that 603 00:31:41,270 --> 00:31:45,332 goes through the center of the pupil. 604 00:31:45,332 --> 00:31:49,460 I mean the entrance pupil, the aperture stop, and so on. 605 00:31:49,460 --> 00:31:52,710 So here is a chief ray from a tall object, 606 00:31:52,710 --> 00:31:55,070 and here is a chief ray from a short object. 607 00:31:59,250 --> 00:32:00,780 | of the chief rays, they will go 608 00:32:00,780 --> 00:32:04,583 through the center of the aperture stop by definition. 609 00:32:04,583 --> 00:32:06,750 And, of course, they will also go through the center 610 00:32:06,750 --> 00:32:07,840 of the entrance pupil. 611 00:32:07,840 --> 00:32:10,518 The entrance pupil used to be here, if you remember. 612 00:32:10,518 --> 00:32:12,060 I'm not drawing it any more because I 613 00:32:12,060 --> 00:32:13,560 don't want to clutter this diagram, 614 00:32:13,560 --> 00:32:15,070 but it used to be here. 615 00:32:15,070 --> 00:32:17,460 So the two rays, they meet at this point 616 00:32:17,460 --> 00:32:20,690 because they're the images of the aperture. 617 00:32:20,690 --> 00:32:23,150 However, this ray will go off axis 618 00:32:23,150 --> 00:32:25,370 at other points in the optical system, 619 00:32:25,370 --> 00:32:28,430 and there will be some point in the optical system 620 00:32:28,430 --> 00:32:31,600 where one of these rays may get cut. 621 00:32:31,600 --> 00:32:34,800 This element in the optical system 622 00:32:34,800 --> 00:32:40,250 that limits the chief rays, that element is called the field 623 00:32:40,250 --> 00:32:42,020 stop. 624 00:32:42,020 --> 00:32:46,160 And the physical significance of the field stop 625 00:32:46,160 --> 00:32:49,990 is actually very important and even more intuitive 626 00:32:49,990 --> 00:32:51,560 than the aperture stop. 627 00:32:51,560 --> 00:32:55,090 The field stop is what defines the field of view. 628 00:32:55,090 --> 00:32:57,710 And now the field of view has a formal definition. 629 00:32:57,710 --> 00:33:00,530 It is this angle here that I drew. 630 00:33:00,530 --> 00:33:03,590 But even in colloquial language, when we say the field of view, 631 00:33:03,590 --> 00:33:05,330 we know exactly what we mean. 632 00:33:05,330 --> 00:33:07,880 In an image, the field of view is basically 633 00:33:07,880 --> 00:33:10,490 the size that can be visible. 634 00:33:10,490 --> 00:33:14,180 And outside the field of view, the image is not formed. 635 00:33:14,180 --> 00:33:17,600 So an example is if I look at you-- well, actually, 636 00:33:17,600 --> 00:33:20,800 this is great because I look at you through a camera, 637 00:33:20,800 --> 00:33:23,270 and the way the camera is pointed right now, 638 00:33:23,270 --> 00:33:29,150 I can only see the central row of desks out in Cambridge but I 639 00:33:29,150 --> 00:33:31,403 cannot see those of you who are seated on the left 640 00:33:31,403 --> 00:33:32,070 or on the right. 641 00:33:32,070 --> 00:33:34,160 So you might be picking your nose, for example, 642 00:33:34,160 --> 00:33:36,200 and I would not be able to see it. 643 00:33:36,200 --> 00:33:39,250 And that's because you are outside my field of view. 644 00:33:39,250 --> 00:33:43,640 And thank you for the control, for turning the camera. 645 00:33:43,640 --> 00:33:47,570 Now what is happening, my field of view is shifting. 646 00:33:47,570 --> 00:33:52,390 So the field stop in this case is actually 647 00:33:52,390 --> 00:33:55,985 the dye, the electronic dye that captures 648 00:33:55,985 --> 00:33:58,890 the image electronically on your side, 649 00:33:58,890 --> 00:34:01,490 then transmits it to us here in Singapore. 650 00:34:01,490 --> 00:34:05,760 This is the physical field stop that limits how many of you 651 00:34:05,760 --> 00:34:08,030 I can see at any given instance. 652 00:34:08,030 --> 00:34:11,770 So the field stop is second to the numerical aperture 653 00:34:11,770 --> 00:34:13,880 and the aperture stop, is another very important 654 00:34:13,880 --> 00:34:14,690 quantity. 655 00:34:14,690 --> 00:34:19,670 It tells us how big is the size of an object that we can see. 656 00:34:19,670 --> 00:34:22,250 So if you look, for example, at a telescope-- 657 00:34:22,250 --> 00:34:24,770 if you look at the night sky through a telescope, 658 00:34:24,770 --> 00:34:29,739 the field of view is basically how many stars can fit, 659 00:34:29,739 --> 00:34:34,420 what portion of the sky you can fit within your image 660 00:34:34,420 --> 00:34:36,790 when you look through the telescope. 661 00:34:36,790 --> 00:34:40,150 When you look through a microscope, similarly, the size 662 00:34:40,150 --> 00:34:43,150 of the specimen that you can observe in the microscope, 663 00:34:43,150 --> 00:34:45,010 that would be the field of view. 664 00:34:45,010 --> 00:34:48,460 Now, intuitively, we would like to define the field of view 665 00:34:48,460 --> 00:34:51,909 in physical size, in meters, or millimeters, or microns, 666 00:34:51,909 --> 00:34:53,800 or whatever the case might be. 667 00:34:53,800 --> 00:34:56,330 For reasons that will become apparent a little bit later, 668 00:34:56,330 --> 00:34:58,240 we prefer to define the field of view 669 00:34:58,240 --> 00:35:02,560 as an angle actually, the angle subtended towards the object. 670 00:35:02,560 --> 00:35:03,502 And you can guess-- 671 00:35:03,502 --> 00:35:04,960 I don't even have to make you wait. 672 00:35:04,960 --> 00:35:08,620 You can guess why we define angle instead of physical size. 673 00:35:08,620 --> 00:35:11,720 That is because, very often, optical systems 674 00:35:11,720 --> 00:35:14,260 have variable magnification. 675 00:35:14,260 --> 00:35:17,740 So with the same optical system might move a lens, 676 00:35:17,740 --> 00:35:21,550 and then the object would move, but the field of view 677 00:35:21,550 --> 00:35:23,200 might not change. 678 00:35:23,200 --> 00:35:26,620 So if I define it as a size and then move it here, 679 00:35:26,620 --> 00:35:30,700 then the size would be bigger, whereas the angle generally 680 00:35:30,700 --> 00:35:31,640 is preserved. 681 00:35:31,640 --> 00:35:33,460 So that's why we define the field of view 682 00:35:33,460 --> 00:35:38,140 as an angle in radians or degrees instead 683 00:35:38,140 --> 00:35:40,170 of physical distances. 684 00:35:40,170 --> 00:35:41,420 AUDIENCE: Hello, hello, hello. 685 00:35:41,420 --> 00:35:44,120 GEORGE BARBASTATHIS: So that is the meaning of the field stop. 686 00:35:44,120 --> 00:35:47,780 And, similarly, through the entrance and exit pupils 687 00:35:47,780 --> 00:35:50,120 that we saw for the aperture stop, 688 00:35:50,120 --> 00:35:53,990 we can define equivalent quantities for the field stop, 689 00:35:53,990 --> 00:35:56,540 but we'll call the windows. 690 00:35:56,540 --> 00:36:00,770 So if we image the field stop through the preceding elements, 691 00:36:00,770 --> 00:36:03,620 we get what is called the entrance window. 692 00:36:03,620 --> 00:36:06,950 If we image it through the succeeding elements, 693 00:36:06,950 --> 00:36:09,770 we'll get what is called the exit window. 694 00:36:09,770 --> 00:36:13,820 And, generally, in an optical system, these windows, 695 00:36:13,820 --> 00:36:19,110 they should coincide with your object and image planes 696 00:36:19,110 --> 00:36:20,420 respectively. 697 00:36:20,420 --> 00:36:25,790 So the entrance window should coincide with the object plane. 698 00:36:25,790 --> 00:36:28,792 The exit window should coincide with the image plane. 699 00:36:28,792 --> 00:36:30,500 If that's not the case, there's something 700 00:36:30,500 --> 00:36:31,520 wrong with the optical system. 701 00:36:31,520 --> 00:36:32,060 It can happen. 702 00:36:32,060 --> 00:36:33,740 It is not impossible, but there's something 703 00:36:33,740 --> 00:36:34,990 wrong with the optical system. 704 00:36:38,370 --> 00:36:42,240 So now that we did that, let's put all these definitions 705 00:36:42,240 --> 00:36:44,640 together. 706 00:36:44,640 --> 00:36:47,250 There's a lot of confusion in the diagram, 707 00:36:47,250 --> 00:36:48,840 but it shows everything. 708 00:36:48,840 --> 00:36:50,880 It shows the chief and marginal rays. 709 00:36:50,880 --> 00:36:55,080 It shows the two physical stops, the aperture stop 710 00:36:55,080 --> 00:36:56,670 and the field stop. 711 00:36:56,670 --> 00:36:59,340 And then it shows their images with respect 712 00:36:59,340 --> 00:37:01,800 to the preceding and succeeding elements, 713 00:37:01,800 --> 00:37:06,660 that are called the entrance and exit pupil and windows, 714 00:37:06,660 --> 00:37:07,890 respectively. 715 00:37:07,890 --> 00:37:11,880 I realize this is a bunch of terms that I threw at once. 716 00:37:14,520 --> 00:37:17,100 So treat this as an information session, 717 00:37:17,100 --> 00:37:20,040 then go back and make sure you read them again from the notes 718 00:37:20,040 --> 00:37:23,190 and from the book so you make sure they sort of solidify 719 00:37:23,190 --> 00:37:25,380 in your mind. 720 00:37:25,380 --> 00:37:26,870 But this diagram, actually, I spent 721 00:37:26,870 --> 00:37:29,180 a lot of time making this diagram because I actually 722 00:37:29,180 --> 00:37:30,950 drew it by hand. 723 00:37:30,950 --> 00:37:33,080 Instead of using an optical software, 724 00:37:33,080 --> 00:37:35,330 I did it by hand, so it took me a lot of time 725 00:37:35,330 --> 00:37:36,200 to make it accurate. 726 00:37:36,200 --> 00:37:38,330 But, to the best that I can-- 727 00:37:38,330 --> 00:37:41,215 I tried, it's pretty accurate, except for this lens here. 728 00:37:41,215 --> 00:37:42,590 This lens doesn't look very good. 729 00:37:42,590 --> 00:37:43,840 It's a negative lens. 730 00:37:43,840 --> 00:37:47,450 It should have sort of diverged this ray bundle even more. 731 00:37:47,450 --> 00:37:48,972 If I had done this properly, this 732 00:37:48,972 --> 00:37:50,930 would then become monstrous, so I didn't do it. 733 00:37:50,930 --> 00:37:52,220 But, anyway. 734 00:37:52,220 --> 00:37:55,400 But you can see how it works. 735 00:37:55,400 --> 00:37:56,380 Here's the chief ray. 736 00:37:56,380 --> 00:37:58,160 Let's trace it through the system. 737 00:37:58,160 --> 00:37:59,440 So here's the chief ray. 738 00:37:59,440 --> 00:38:02,620 It goes through the center of the entrance pupil. 739 00:38:02,620 --> 00:38:07,360 Then it goes through the edge of the field stops. 740 00:38:07,360 --> 00:38:11,560 That means that I chose a chief ray that propagates at an angle 741 00:38:11,560 --> 00:38:13,030 equal to what? 742 00:38:13,030 --> 00:38:15,130 Equal to the field of view, right? 743 00:38:15,130 --> 00:38:18,320 Because this is the last chief ray admitted through a system. 744 00:38:18,320 --> 00:38:23,110 If I put a chief ray at a bigger angle, that is a taller object, 745 00:38:23,110 --> 00:38:27,400 it would miss the field stop. 746 00:38:27,400 --> 00:38:29,260 Then the same chief ray goes on. 747 00:38:29,260 --> 00:38:32,350 It goes through a center of the aperture stop 748 00:38:32,350 --> 00:38:36,220 again, then again through the center of the exit pupil, 749 00:38:36,220 --> 00:38:40,000 and finally hits the edge of the exit window, which 750 00:38:40,000 --> 00:38:44,055 is also the tip of the image. 751 00:38:44,055 --> 00:38:44,930 That's the chief ray. 752 00:38:44,930 --> 00:38:48,710 Now let's go through the marginal ray. 753 00:38:48,710 --> 00:38:53,060 The marginal ray will hit the rim of the entrance pupil, 754 00:38:53,060 --> 00:38:56,180 then will hit the rim of the field stop again. 755 00:38:56,180 --> 00:38:57,910 Why? 756 00:38:57,910 --> 00:39:00,500 I'll give you 30 seconds to think about it. 757 00:39:00,500 --> 00:39:03,150 Why does the marginal ray hit the rim of the field stop 758 00:39:03,150 --> 00:39:03,650 again? 759 00:39:19,170 --> 00:39:20,560 My computer has actually a timer, 760 00:39:20,560 --> 00:39:23,326 so I can count the 30 seconds very precisely. 761 00:39:27,610 --> 00:39:28,570 AUDIENCE: [INAUDIBLE] 762 00:39:28,570 --> 00:39:29,500 GEORGE BARBASTATHIS: Oh, there is audio, 763 00:39:29,500 --> 00:39:31,010 so you can even speak your mind now. 764 00:39:33,968 --> 00:39:35,010 Thanks, by the way, for-- 765 00:39:35,010 --> 00:39:36,467 thanks, control, for fixing it. 766 00:39:40,290 --> 00:39:40,800 Anybody? 767 00:39:40,800 --> 00:39:44,580 Why does the marginal ray also go through the rim, 768 00:39:44,580 --> 00:39:45,740 the edge of the field stop? 769 00:39:54,110 --> 00:39:57,200 AUDIENCE: Because the entrance window 770 00:39:57,200 --> 00:39:58,850 is an image of the field stop? 771 00:39:58,850 --> 00:40:00,450 Or it's the other way around. 772 00:40:00,450 --> 00:40:01,700 So the marginal ray is-- 773 00:40:01,700 --> 00:40:03,367 I mean, it's the edge of window, so it's 774 00:40:03,367 --> 00:40:05,360 going to be on the edge of the field stop also. 775 00:40:05,360 --> 00:40:07,110 GEORGE BARBASTATHIS: That's exactly right. 776 00:40:07,110 --> 00:40:09,400 These two planes are images of each other. 777 00:40:09,400 --> 00:40:12,980 So since the marginal ray meets the chief ray here, 778 00:40:12,980 --> 00:40:16,910 then it should also meet the chief ray here again. 779 00:40:16,910 --> 00:40:18,950 Another way to put it is that this is actually 780 00:40:18,950 --> 00:40:24,400 an intermediate image plane for this particular optical system. 781 00:40:24,400 --> 00:40:26,400 So then the-- in other words, you 782 00:40:26,400 --> 00:40:29,070 would see an image of the object itself here 783 00:40:29,070 --> 00:40:31,890 if you were to stick your eye. 784 00:40:31,890 --> 00:40:33,540 Then the imaginary ray will go again. 785 00:40:33,540 --> 00:40:36,780 Now it goes again through the edge of the aperture stop 786 00:40:36,780 --> 00:40:37,980 again. 787 00:40:37,980 --> 00:40:39,870 And that is, of course, because it 788 00:40:39,870 --> 00:40:42,150 went through the edge of the pupil, 789 00:40:42,150 --> 00:40:44,410 and since the pupil and the aperture stop 790 00:40:44,410 --> 00:40:47,010 are images of each other, then it has to go through the edge 791 00:40:47,010 --> 00:40:48,440 again. 792 00:40:48,440 --> 00:40:49,690 Now it will become repetitive. 793 00:40:49,690 --> 00:40:54,320 It will hit again the rim of the exit pupil for the same reason. 794 00:40:54,320 --> 00:40:57,820 And then, finally, it would hit the edge of the exit window 795 00:40:57,820 --> 00:41:01,840 because the exit window is an image of the field stop. 796 00:41:01,840 --> 00:41:05,980 So these multiple imaging requirements, 797 00:41:05,980 --> 00:41:08,680 they sort of make this all makes sense, right? 798 00:41:08,680 --> 00:41:12,210 But this is how they work. 799 00:41:12,210 --> 00:41:15,100 The marginal ray, as the name suggests, 800 00:41:15,100 --> 00:41:18,530 hits the rims of everything-- the aperture stops, the pupils, 801 00:41:18,530 --> 00:41:21,210 the windows, and all that. 802 00:41:21,210 --> 00:41:23,680 The chief ray, it goes through the center 803 00:41:23,680 --> 00:41:28,150 of the pupils, entrance and exit, and the aperture stop, 804 00:41:28,150 --> 00:41:32,020 and hits the edges, the rims of the field stop 805 00:41:32,020 --> 00:41:34,170 and the associated entrance and exit windows. 806 00:41:36,690 --> 00:41:40,650 And, finally, now that we have everything together, here 807 00:41:40,650 --> 00:41:44,210 is again the definition of the field of view. 808 00:41:44,210 --> 00:41:49,050 It is angle between the last two chief rays admitted 809 00:41:49,050 --> 00:41:51,440 by the system, last two meaning that it 810 00:41:51,440 --> 00:41:56,080 is the last two that the field stop would admit. 811 00:41:56,080 --> 00:42:00,500 And the numerical aperture is the angle 812 00:42:00,500 --> 00:42:08,180 between the two marginal rays that depart from the center, 813 00:42:08,180 --> 00:42:10,820 from the center of the window, that is, 814 00:42:10,820 --> 00:42:13,530 from the optical axis itself. 815 00:42:13,530 --> 00:42:20,170 We do not define the numerical aperture from off-axis points. 816 00:42:20,170 --> 00:42:26,306 The reason is that, in poorly designed optical systems-- 817 00:42:26,306 --> 00:42:29,440 first of all, let me say, in an ideal optical system, 818 00:42:29,440 --> 00:42:34,540 the numerical aperture should be equal no matter where you are. 819 00:42:34,540 --> 00:42:36,670 Within the field of view, the numerical aperture 820 00:42:36,670 --> 00:42:38,580 should be the same. 821 00:42:38,580 --> 00:42:40,700 In practice, that is very difficult to do, 822 00:42:40,700 --> 00:42:46,080 and if you do a particularly poor job as a designer, 823 00:42:46,080 --> 00:42:51,750 the numerical aperture can decrease drastically 824 00:42:51,750 --> 00:42:53,400 when you go off axis here. 825 00:42:53,400 --> 00:42:55,320 In fact, in the next example that I will show, 826 00:42:55,320 --> 00:42:56,670 this will happen. 827 00:42:56,670 --> 00:42:59,760 The numerical aperture becomes drastically smaller 828 00:42:59,760 --> 00:43:02,215 as you go off axis. 829 00:43:02,215 --> 00:43:04,440 That is called vignetting. 830 00:43:04,440 --> 00:43:07,020 It's one of those French words that have crept in optics 831 00:43:07,020 --> 00:43:08,610 and is difficult to pronounce. 832 00:43:08,610 --> 00:43:10,150 You will see it written in a second. 833 00:43:10,150 --> 00:43:14,640 I will just write it down here, vignetting. 834 00:43:14,640 --> 00:43:17,300 It is very annoying, because if you have vignetting 835 00:43:17,300 --> 00:43:21,150 in an optical system, then the edges of the image, 836 00:43:21,150 --> 00:43:22,350 they appear darker. 837 00:43:22,350 --> 00:43:25,500 Remember, the numerical aperture is the amount of light 838 00:43:25,500 --> 00:43:27,600 that the optical system admits. 839 00:43:27,600 --> 00:43:31,020 Well, if I admit more light in the center, 840 00:43:31,020 --> 00:43:34,450 I have a larger numerical aperture in the center. 841 00:43:34,450 --> 00:43:38,100 A numerical aperture decreases as I go off axis, 842 00:43:38,100 --> 00:43:40,950 it means that the edges of the image will be darker. 843 00:43:40,950 --> 00:43:42,540 It will be dimmer. 844 00:43:42,540 --> 00:43:44,670 And that is, of course, very undesirable. 845 00:43:44,670 --> 00:43:48,070 And, generally, when we design microscopes, telescopes, 846 00:43:48,070 --> 00:43:52,580 and such, we try to make sure that this will not happen. 847 00:43:52,580 --> 00:43:55,710 But, anyway, because this could happen, 848 00:43:55,710 --> 00:43:58,290 that's why the numerical aperture is defined 849 00:43:58,290 --> 00:44:01,200 at the center, on axis, so that it 850 00:44:01,200 --> 00:44:03,120 has an unambiguous definition. 851 00:44:06,945 --> 00:44:08,570 So having said all that, let's look now 852 00:44:08,570 --> 00:44:10,960 at an example that will actually reveal 853 00:44:10,960 --> 00:44:14,360 this vignetting phenomenon, in addition to other things. 854 00:44:14,360 --> 00:44:16,670 So this example is relatively simple. 855 00:44:16,670 --> 00:44:20,470 It is just a single thin lens. 856 00:44:20,470 --> 00:44:22,710 And then, in front of the thin lens, 857 00:44:22,710 --> 00:44:27,970 we've put an aperture stop. 858 00:44:27,970 --> 00:44:29,470 You can think of it-- actually, this 859 00:44:29,470 --> 00:44:33,700 is a very simplified model of the eye, of the human eye. 860 00:44:33,700 --> 00:44:35,710 We have a lens in front of it. 861 00:44:35,710 --> 00:44:38,110 In the human eye, we call it the pupil, right? 862 00:44:38,110 --> 00:44:40,270 Of course, in the human eye, in actuality 863 00:44:40,270 --> 00:44:42,130 you also have an additional lens in front 864 00:44:42,130 --> 00:44:43,900 of the pupil called the cornea. 865 00:44:43,900 --> 00:44:48,400 But, anyway, that's why this is perhaps not an ideal model. 866 00:44:48,400 --> 00:44:51,700 But, anyway, that's the example that I chose to do. 867 00:44:51,700 --> 00:44:54,340 And, in addition, it has a field stop which 868 00:44:54,340 --> 00:44:56,680 is located at the image plane. 869 00:44:56,680 --> 00:45:00,430 Now, generally, if you see a stop at the image plane, 870 00:45:00,430 --> 00:45:02,640 this is a good candidate to be the field stop, right. 871 00:45:02,640 --> 00:45:09,183 Because, clearly, if I image anything outside this opening-- 872 00:45:09,183 --> 00:45:10,850 this is what aperture means, by the way. 873 00:45:10,850 --> 00:45:12,860 Literally, aperture means opening. 874 00:45:12,860 --> 00:45:16,130 So if I image anything outside this opening, 875 00:45:16,130 --> 00:45:19,460 then it will hit the block, and therefore it will not be image. 876 00:45:19,460 --> 00:45:22,290 So, therefore, this is very clearly the field stop. 877 00:45:22,290 --> 00:45:25,460 Physically, the field stop is, for example, 878 00:45:25,460 --> 00:45:27,470 as I mentioned earlier, it is the size 879 00:45:27,470 --> 00:45:30,970 of the chip that registers the digital image, right? 880 00:45:30,970 --> 00:45:35,150 It's a very simple instance of a field stop. 881 00:45:35,150 --> 00:45:38,390 In old-fashioned cameras that use film, 882 00:45:38,390 --> 00:45:42,290 then the field stop would be, in that case, the film itself. 883 00:45:42,290 --> 00:45:44,900 Wherever the film is chemically active, 884 00:45:44,900 --> 00:45:46,760 this is where you record the image. 885 00:45:46,760 --> 00:45:49,305 Outside, it's just lost. 886 00:45:49,305 --> 00:45:51,180 Whatever is imaged outside that area is lost. 887 00:45:53,780 --> 00:45:56,150 So, in this case, it is pretty clear who is the aperture 888 00:45:56,150 --> 00:45:58,630 and who is the field stop. 889 00:45:58,630 --> 00:46:02,170 So you can see very clearly that if you take an object 890 00:46:02,170 --> 00:46:08,770 point on the axis, then the physical aperture here 891 00:46:08,770 --> 00:46:11,170 will block the angle of acceptance 892 00:46:11,170 --> 00:46:13,540 from this particular point. 893 00:46:13,540 --> 00:46:15,940 And, therefore, this is the aperture stop. 894 00:46:15,940 --> 00:46:19,280 And, also, because there is no elements to the left, 895 00:46:19,280 --> 00:46:20,920 this is also the entrance pupil. 896 00:46:20,920 --> 00:46:24,460 If there is no element to image it through, the aperture stop 897 00:46:24,460 --> 00:46:28,720 will also be, simultaneously, the entrance pupil. 898 00:46:28,720 --> 00:46:31,360 What this means is that if you look at the imaging instrument, 899 00:46:31,360 --> 00:46:34,540 you will actually see the pupil itself before you ever 900 00:46:34,540 --> 00:46:35,440 see anything else. 901 00:46:35,440 --> 00:46:37,860 So that's why the two things are actually 902 00:46:37,860 --> 00:46:40,270 coincident in this case. 903 00:46:40,270 --> 00:46:42,210 What about the exit pupil? 904 00:46:42,210 --> 00:46:45,180 Well, the exit pupil is what I will see on this side 905 00:46:45,180 --> 00:46:52,810 if I image this element through the succeeding elements. 906 00:46:52,810 --> 00:46:55,570 So the succeeding element is this lens here. 907 00:46:55,570 --> 00:46:57,960 So what will I see if I image this through the lens? 908 00:46:57,960 --> 00:47:02,597 Well, this element now-- forget about the actual object 909 00:47:02,597 --> 00:47:03,180 of the system. 910 00:47:03,180 --> 00:47:06,690 When we question-- when we ask what are the entrance and exit 911 00:47:06,690 --> 00:47:09,390 pupils, this is now our object. 912 00:47:09,390 --> 00:47:11,730 We're trying to see where this would be imaged 913 00:47:11,730 --> 00:47:14,340 through the optical system. 914 00:47:14,340 --> 00:47:17,160 And we notice is that this is between the focal point 915 00:47:17,160 --> 00:47:18,092 and the lens. 916 00:47:18,092 --> 00:47:19,050 Here's the focal point. 917 00:47:19,050 --> 00:47:20,010 Here's the lens. 918 00:47:20,010 --> 00:47:21,060 This is in between. 919 00:47:21,060 --> 00:47:25,240 Therefore, this will form a virtual image. 920 00:47:25,240 --> 00:47:28,500 So the exit pupil would be somewhere out here. 921 00:47:28,500 --> 00:47:31,710 It would be actually to the left of the lens. 922 00:47:31,710 --> 00:47:32,710 That is not surprising. 923 00:47:32,710 --> 00:47:35,460 All it means is that if I go here and then 924 00:47:35,460 --> 00:47:40,260 look at the optical instrument this way, what I will see, 925 00:47:40,260 --> 00:47:43,830 I will see a virtual image of the pupil, 926 00:47:43,830 --> 00:47:45,480 very similar to the virtual images 927 00:47:45,480 --> 00:47:48,870 that you saw last time in the demo in the class 928 00:47:48,870 --> 00:47:54,030 when Pepe brought the magnifier objective. 929 00:47:54,030 --> 00:47:57,790 So that's the exit pupil over here. 930 00:47:57,790 --> 00:47:59,940 And what about the marginal and chief rays? 931 00:47:59,940 --> 00:48:04,420 Well, here is the-- 932 00:48:04,420 --> 00:48:05,500 here they are. 933 00:48:05,500 --> 00:48:06,610 Here's the marginal ray. 934 00:48:06,610 --> 00:48:08,020 Here's the chief ray. 935 00:48:08,020 --> 00:48:09,457 The chief ray, by definition, has 936 00:48:09,457 --> 00:48:11,540 to go out through the center of the aperture stop. 937 00:48:11,540 --> 00:48:13,570 So that is indeed. 938 00:48:13,570 --> 00:48:17,970 The marginal ray hits the rim. 939 00:48:17,970 --> 00:48:21,320 And what about the field of view? 940 00:48:21,320 --> 00:48:24,430 Well, the field of view will be defined by the pupil-- 941 00:48:24,430 --> 00:48:25,030 I'm sorry. 942 00:48:25,030 --> 00:48:26,940 It would be defined by the field stop. 943 00:48:26,940 --> 00:48:30,730 Now, notice that the field stop is also the exit window 944 00:48:30,730 --> 00:48:33,400 because there is no optical elements to its right. 945 00:48:33,400 --> 00:48:35,740 So, therefore, that is the exit window. 946 00:48:35,740 --> 00:48:39,100 The entrance window is at the object plane, 947 00:48:39,100 --> 00:48:40,750 since this is the object plane and this 948 00:48:40,750 --> 00:48:44,320 is the image plane that satisfy an imaging condition. 949 00:48:44,320 --> 00:48:47,620 That is, if you image the field stop through the preceding 950 00:48:47,620 --> 00:48:50,330 optical elements, you'll arrive here. 951 00:48:50,330 --> 00:48:52,420 So that's the entrance window. 952 00:48:52,420 --> 00:48:54,670 And, of course, the field of view 953 00:48:54,670 --> 00:49:00,360 is defined by the two chief rays that just clear the field stop, 954 00:49:00,360 --> 00:49:01,750 so here they are. 955 00:49:01,750 --> 00:49:04,150 And the angle between the two is the field of view. 956 00:49:04,150 --> 00:49:07,030 By the way, there is one ambiguity in the literature. 957 00:49:07,030 --> 00:49:10,690 Some people define the field of view as the full angle. 958 00:49:10,690 --> 00:49:13,420 Some people define it as the half angle. 959 00:49:13,420 --> 00:49:16,290 It's a matter of taste, how you want to define it. 960 00:49:16,290 --> 00:49:19,270 But just make sure you tell people what you really mean, 961 00:49:19,270 --> 00:49:22,580 if you really mean the full thing or not. 962 00:49:22,580 --> 00:49:27,850 And here they're all together, all these quantities. 963 00:49:27,850 --> 00:49:32,220 And-- yes? 964 00:49:36,150 --> 00:49:39,620 Oh, let me ask-- yes, never mind. 965 00:49:39,620 --> 00:49:41,880 And, in this case, I actually worked out the math. 966 00:49:41,880 --> 00:49:43,130 I will not go through it here. 967 00:49:43,130 --> 00:49:46,010 I will let you do it by yourselves. 968 00:49:46,010 --> 00:49:47,760 If I put some symbols, for example 969 00:49:47,760 --> 00:49:54,630 if I call the size of the aperture stop, I call it a. 970 00:49:54,630 --> 00:49:58,230 The size of the field stop, I call it w prime. 971 00:49:58,230 --> 00:50:02,490 Then I have distance z between the aperture stop and the lens, 972 00:50:02,490 --> 00:50:07,950 distance s prime between the lens and the field stop. 973 00:50:07,950 --> 00:50:13,140 And then this calculation shows you where-- 974 00:50:13,140 --> 00:50:17,160 how you find the other elements of the system. 975 00:50:17,160 --> 00:50:20,400 For example, when it comes to the entrance pupil, 976 00:50:20,400 --> 00:50:21,450 what should you to do? 977 00:50:21,450 --> 00:50:23,250 Well, for the entrance pupil, we have 978 00:50:23,250 --> 00:50:24,780 to consider this optical system. 979 00:50:27,590 --> 00:50:31,730 If that's our aperture stop, then I have distance z. 980 00:50:31,730 --> 00:50:33,560 Then I have the lens. 981 00:50:33,560 --> 00:50:36,780 And then-- well, I'm sorry. 982 00:50:36,780 --> 00:50:38,240 The entrance pupil is the-- 983 00:50:38,240 --> 00:50:39,875 I should have said the exit pupil. 984 00:50:39,875 --> 00:50:41,000 So where is the exit pupil? 985 00:50:41,000 --> 00:50:43,550 Well I have to image this aperture stop 986 00:50:43,550 --> 00:50:44,610 through the system. 987 00:50:44,610 --> 00:50:45,980 So how do I image it? 988 00:50:45,980 --> 00:50:50,720 Well, let's say that the exit pupil is at some distance z 989 00:50:50,720 --> 00:50:53,430 prime to the right of the lens. 990 00:50:53,430 --> 00:50:56,150 Well, how do we find z prime? 991 00:50:56,150 --> 00:51:01,340 We apply the imaging condition, 1 over z plus 1 over z prime 992 00:51:01,340 --> 00:51:03,030 equals 1 over f. 993 00:51:03,030 --> 00:51:04,490 We can solve that. 994 00:51:04,490 --> 00:51:08,810 1 over z prime equals 1 over f minus 1 over z. 995 00:51:08,810 --> 00:51:14,200 This is also known as z minus f over f times z. 996 00:51:14,200 --> 00:51:19,160 Or z prime equals f times z over z minus f. 997 00:51:19,160 --> 00:51:24,200 Now, I said that the aperture stop 998 00:51:24,200 --> 00:51:28,460 is between the focal point and the lens. 999 00:51:28,460 --> 00:51:35,980 That means that z is less than f in this case. 1000 00:51:35,980 --> 00:51:39,520 Therefore, this quantity-- since z is less than f, 1001 00:51:39,520 --> 00:51:41,470 this quantity turned out to be negative. 1002 00:51:41,470 --> 00:51:44,860 If it is negative, it means that my assumption 1003 00:51:44,860 --> 00:51:50,810 before that the exit pupil is to the right was incorrect. 1004 00:51:50,810 --> 00:51:52,400 My assumption would have been correct 1005 00:51:52,400 --> 00:51:55,790 if z prime had turned out to be a positive quantity. 1006 00:51:55,790 --> 00:51:58,160 Since this is turning out to be negative here, 1007 00:51:58,160 --> 00:52:01,980 it means that the actual exit pupil is to the right. 1008 00:52:01,980 --> 00:52:02,480 I'm sorry. 1009 00:52:02,480 --> 00:52:05,690 It is to the left, opposite from my assumption. 1010 00:52:05,690 --> 00:52:10,245 So, therefore, the exit pupil is over here, as I already said, 1011 00:52:10,245 --> 00:52:13,490 or as I already said before. 1012 00:52:13,490 --> 00:52:16,480 And, basically, this result, these 1013 00:52:16,480 --> 00:52:18,740 are all the quantities here. 1014 00:52:18,740 --> 00:52:22,510 You can see the marginal ray from the field edge, 1015 00:52:22,510 --> 00:52:25,032 the marginal ray from on axis, the chief ray, 1016 00:52:25,032 --> 00:52:25,990 and so on and so forth. 1017 00:52:25,990 --> 00:52:28,660 And the last comment here, I did not really 1018 00:52:28,660 --> 00:52:30,370 prove it mathematically. 1019 00:52:30,370 --> 00:52:31,630 You can do it if you like. 1020 00:52:31,630 --> 00:52:34,330 It's a bit of a more involved calculation. 1021 00:52:34,330 --> 00:52:36,310 But even visually here, you can see 1022 00:52:36,310 --> 00:52:40,660 that the angle between marginal rays on axis 1023 00:52:40,660 --> 00:52:43,900 is, even in my handwritten diagram here, 1024 00:52:43,900 --> 00:52:47,740 it is visibly larger than the angle between the two 1025 00:52:47,740 --> 00:52:49,818 marginal rays off axis. 1026 00:52:49,818 --> 00:52:51,610 This means that this particular system here 1027 00:52:51,610 --> 00:52:54,790 is subject to vignetting that I mentioned earlier. 1028 00:52:54,790 --> 00:52:56,980 That's the term here, vignetting, 1029 00:52:56,980 --> 00:53:01,600 which means it is not a very well-designed optical system. 1030 00:53:01,600 --> 00:53:03,650 In sophisticated optical systems, 1031 00:53:03,650 --> 00:53:07,150 there's rules that we follow in order to avoid vignetting. 1032 00:53:07,150 --> 00:53:11,170 When I describe the microscope next week-- 1033 00:53:11,170 --> 00:53:12,430 I mean, not next week. 1034 00:53:12,430 --> 00:53:14,860 Next Wednesday, in 2 days from today, 1035 00:53:14,860 --> 00:53:18,220 I will show you how microscopes are designed 1036 00:53:18,220 --> 00:53:22,030 in order to avoid vignetting. 1037 00:53:22,030 --> 00:53:24,700 And I see that I've already gone 2 minutes over time. 1038 00:53:24,700 --> 00:53:26,140 So let me conclude, unless there's 1039 00:53:26,140 --> 00:53:30,780 any questions that someone will ask, now that we have sound.