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 --> 00:00:17,550 at ocw.mit.edu. 8 00:00:21,060 --> 00:00:22,560 GEORGE BARBASTATHIS: So does anybody 9 00:00:22,560 --> 00:00:26,040 have questions from the last lecture? 10 00:00:37,870 --> 00:00:41,807 After some time to wake up, anybody still have questions? 11 00:00:45,690 --> 00:00:48,120 So I will start, and if you remember of a question 12 00:00:48,120 --> 00:00:49,500 that you had, please interrupt. 13 00:00:49,500 --> 00:00:51,250 I think if you push the button the button, 14 00:00:51,250 --> 00:00:54,160 I will hear a sound over here. 15 00:00:54,160 --> 00:00:57,680 So we will know that there's a question. 16 00:00:57,680 --> 00:00:59,140 So I'd like to pick up the thread 17 00:00:59,140 --> 00:01:01,970 from where I left last time. 18 00:01:01,970 --> 00:01:04,819 We covered fairly quickly because we ran out of time. 19 00:01:04,819 --> 00:01:07,180 But we covered that the law of reflection 20 00:01:07,180 --> 00:01:09,580 and the law of refraction. 21 00:01:09,580 --> 00:01:13,330 So I'd like to go back to the low of reflection, 22 00:01:13,330 --> 00:01:18,600 and remind you that it is a very simple, very simple result. 23 00:01:18,600 --> 00:01:22,040 The minimum path requirement for the light rays 24 00:01:22,040 --> 00:01:24,490 force those reflections to be symmetric. 25 00:01:24,490 --> 00:01:26,080 So this has a strange consequence 26 00:01:26,080 --> 00:01:31,300 that we're unfamiliar with from when we look in the mirror 27 00:01:31,300 --> 00:01:32,590 daily. 28 00:01:32,590 --> 00:01:38,680 And that is the fact that our left and right locations 29 00:01:38,680 --> 00:01:39,710 in our body. 30 00:01:39,710 --> 00:01:41,538 They flip when we look at the mirror. 31 00:01:41,538 --> 00:01:43,330 So we'd like to make that a little bit more 32 00:01:43,330 --> 00:01:46,190 quantitative by looking at this diagram. 33 00:01:46,190 --> 00:01:51,080 So to understand this, suppose that I am-- 34 00:01:51,080 --> 00:01:54,760 I have an object that is oriented. 35 00:01:54,760 --> 00:01:58,870 You can think of it as perhaps two pencils that 36 00:01:58,870 --> 00:02:04,300 are sort of following the ray paths 37 00:02:04,300 --> 00:02:06,190 and then they get reflected from the mirror. 38 00:02:06,190 --> 00:02:08,860 So, for example, if you look at the central ray, 39 00:02:08,860 --> 00:02:10,990 it will be reflected symmetrically. 40 00:02:10,990 --> 00:02:14,070 So, again, what you see here is the front view of the mirror. 41 00:02:14,070 --> 00:02:17,770 And then I will draw a few ray a paths in perspective. 42 00:02:17,770 --> 00:02:25,360 So this is the ray paths that start from the object. 43 00:02:25,360 --> 00:02:27,110 I don't know if you can see me over there, 44 00:02:27,110 --> 00:02:31,190 but I'm trying to show what will happen with an actual pencil. 45 00:02:31,190 --> 00:02:34,410 So if this was a pencil that is coming towards the mirror, 46 00:02:34,410 --> 00:02:36,110 and this is the surface of the mirror. 47 00:02:36,110 --> 00:02:38,420 The pencil will go like this. 48 00:02:38,420 --> 00:02:40,760 And then law of reflection says that it will also 49 00:02:40,760 --> 00:02:44,120 come out like this again. 50 00:02:44,120 --> 00:02:48,730 This follows very easily if you simply trace the rays. 51 00:02:48,730 --> 00:02:52,190 In the next step in my animation here, it 52 00:02:52,190 --> 00:02:59,960 paints the sort of positions of the pencils and their tops 53 00:02:59,960 --> 00:03:03,860 as they get reflected from the mirror. 54 00:03:03,860 --> 00:03:05,570 So what happened here is the following. 55 00:03:05,570 --> 00:03:07,550 As you can see, the pencil did not actually 56 00:03:07,550 --> 00:03:08,670 change orientation. 57 00:03:08,670 --> 00:03:10,520 This is a strange thing about mirrors. 58 00:03:10,520 --> 00:03:12,810 But what did change is the following. 59 00:03:12,810 --> 00:03:14,720 Imagine that you are walking together 60 00:03:14,720 --> 00:03:17,770 with this pair of pencils. 61 00:03:17,770 --> 00:03:19,550 As you are walking this way, you will 62 00:03:19,550 --> 00:03:26,060 see that the one that is horizontally oriented. 63 00:03:26,060 --> 00:03:28,730 The top of it is pointing to your right. 64 00:03:28,730 --> 00:03:31,400 But if you walk all the way to the mirror. 65 00:03:31,400 --> 00:03:33,860 Go to the center of the ray, and start going 66 00:03:33,860 --> 00:03:35,630 backwards the other way around. 67 00:03:35,630 --> 00:03:37,650 Then all of a sudden, the top of the pencil 68 00:03:37,650 --> 00:03:40,160 appeared on your left. 69 00:03:40,160 --> 00:03:45,380 This is a simple consequence of the law of reflection. 70 00:03:45,380 --> 00:03:48,380 The way we interpret it in everyday life when 71 00:03:48,380 --> 00:03:50,528 we see through a mirror is the following. 72 00:03:50,528 --> 00:03:52,070 Of course, in everyday life, we don't 73 00:03:52,070 --> 00:03:55,250 talk about left handed and right handed triads. 74 00:03:55,250 --> 00:03:58,670 But the way we interpret it is because normally when 75 00:03:58,670 --> 00:04:00,557 we look at a good quality mirror, 76 00:04:00,557 --> 00:04:02,390 we do not know that there is a mirror there. 77 00:04:02,390 --> 00:04:06,480 We actually see a continuation of that are being reflected 78 00:04:06,480 --> 00:04:09,830 behind, back behind the mirror. 79 00:04:09,830 --> 00:04:13,100 So what you're seeing then if we look behind the mirror 80 00:04:13,100 --> 00:04:19,300 is actually this triad, with the pencils oriented as shown here. 81 00:04:19,300 --> 00:04:23,510 And if we interpret it as being seen from behind, 82 00:04:23,510 --> 00:04:27,470 then, of course, left has become right, and vice versa. 83 00:04:27,470 --> 00:04:28,942 So another way to think about it is 84 00:04:28,942 --> 00:04:30,400 that when we look through a mirror, 85 00:04:30,400 --> 00:04:35,080 it is as if we're looking from the image from the back. 86 00:04:35,080 --> 00:04:36,580 So a better way to think about it 87 00:04:36,580 --> 00:04:39,700 is if you're looking at something 88 00:04:39,700 --> 00:04:42,070 that is written on my t-shirt. 89 00:04:42,070 --> 00:04:43,600 If you look at it from the mirror, 90 00:04:43,600 --> 00:04:45,850 it would appear as if I were hollow, 91 00:04:45,850 --> 00:04:48,580 and you would see the back of the writing on my t-shirt. 92 00:04:48,580 --> 00:04:52,060 And you all know that if you look at the ambulance 93 00:04:52,060 --> 00:04:55,090 sign in ambulance trucks, ambulance 94 00:04:55,090 --> 00:04:57,250 whatever you call them, cars. 95 00:04:57,250 --> 00:05:00,040 Because it is meant to be visible through the mirror 96 00:05:00,040 --> 00:05:03,880 of a driver, they actually write the sign backwards. 97 00:05:03,880 --> 00:05:07,540 So when you look at it, it is as if you saw a transparency of it 98 00:05:07,540 --> 00:05:09,700 from the back. 99 00:05:09,700 --> 00:05:13,440 So this is a simple consequence of the law of reflection. 100 00:05:13,440 --> 00:05:15,430 This is a very simple silly question 101 00:05:15,430 --> 00:05:17,630 that sometimes is asked. 102 00:05:17,630 --> 00:05:22,200 And it actually takes quite a long explanation to answer. 103 00:05:22,200 --> 00:05:24,410 And that question goes like this. 104 00:05:24,410 --> 00:05:26,450 If you look at the mirror, yes, we 105 00:05:26,450 --> 00:05:30,280 know that left and right flip, but up and down do not 106 00:05:30,280 --> 00:05:30,790 flip, right? 107 00:05:30,790 --> 00:05:33,790 You don't see yourself upside down in a flat mirror. 108 00:05:33,790 --> 00:05:36,388 You see yourself flipped from left to right. 109 00:05:36,388 --> 00:05:37,180 What is the reason? 110 00:05:37,180 --> 00:05:38,590 The reason is shown here. 111 00:05:38,590 --> 00:05:46,190 The reason is because of the flipping 112 00:05:46,190 --> 00:05:51,140 of the relationship between the left and right in the object 113 00:05:51,140 --> 00:05:53,480 side, as you interpret the projection 114 00:05:53,480 --> 00:05:56,090 of the rays that are coming from the opposite side 115 00:05:56,090 --> 00:05:56,980 of the mirror. 116 00:05:56,980 --> 00:05:58,730 So this requires a little bit of thinking, 117 00:05:58,730 --> 00:06:01,940 so I'll let you think about it. 118 00:06:01,940 --> 00:06:04,967 Unless you have a question now, please ask it. 119 00:06:04,967 --> 00:06:06,800 If not, you can think about it and come back 120 00:06:06,800 --> 00:06:08,810 with more questions on Wednesday. 121 00:06:08,810 --> 00:06:11,020 Is there any immediate question about this? 122 00:06:11,020 --> 00:06:13,040 That's kind of a subtle and elegant point. 123 00:06:17,285 --> 00:06:18,910 I should have brought a mirror with me. 124 00:06:18,910 --> 00:06:20,535 Piper, do you have a mirror over there? 125 00:06:25,320 --> 00:06:27,260 AUDIENCE: No, I don't have a mirror here. 126 00:06:27,260 --> 00:06:28,520 GEORGE BARBASTATHIS: No mirrors, OK. 127 00:06:28,520 --> 00:06:30,320 But everybody has access to mirrors, right? 128 00:06:30,320 --> 00:06:34,841 It's the one optical element that we can find very easily. 129 00:06:34,841 --> 00:06:38,020 So I'll let you practice with your mirror in your bathroom, 130 00:06:38,020 --> 00:06:40,475 and then come back and ask me questions. 131 00:06:43,270 --> 00:06:44,620 OK. 132 00:06:44,620 --> 00:06:47,610 AUDIENCE: I'm lying down. 133 00:06:47,610 --> 00:06:48,640 You're horizontal. 134 00:06:51,780 --> 00:06:54,430 GEORGE BARBASTATHIS: That's right. 135 00:06:54,430 --> 00:06:55,640 Our producer like this. 136 00:07:01,830 --> 00:07:03,680 The other thing I wanted to talk about 137 00:07:03,680 --> 00:07:06,980 is expand a little bit on the law of reflection. 138 00:07:06,980 --> 00:07:09,200 Oh, I'm sorry, the law of refraction 139 00:07:09,200 --> 00:07:11,210 that we derived last time. 140 00:07:18,900 --> 00:07:21,870 So the law of refraction is the last equation 141 00:07:21,870 --> 00:07:23,320 shown on this line. 142 00:07:23,320 --> 00:07:26,170 It says that this quantity, the index of refraction 143 00:07:26,170 --> 00:07:32,020 multiplied by the sign of the angle of incidence, where 144 00:07:32,020 --> 00:07:34,750 the angle of incidence is defined with respect 145 00:07:34,750 --> 00:07:40,000 to the normal to the surface. 146 00:07:40,000 --> 00:07:41,210 It is preserved. 147 00:07:41,210 --> 00:07:44,200 So as you go through multiple surfaces, the law of refraction 148 00:07:44,200 --> 00:07:47,630 says that this quantity must be preserved. 149 00:07:47,630 --> 00:07:49,900 So there is a common problem that I 150 00:07:49,900 --> 00:07:52,300 have posted just a couple of hours ago 151 00:07:52,300 --> 00:07:57,730 that asks you to make an analogy between the law of refraction, 152 00:07:57,730 --> 00:08:00,550 and the problem of a lifeguard who has 153 00:08:00,550 --> 00:08:02,860 to save a person in the water. 154 00:08:02,860 --> 00:08:05,350 So recall the reason the law of refraction happens 155 00:08:05,350 --> 00:08:10,480 is because the light must minimize its path between two 156 00:08:10,480 --> 00:08:13,660 points, P and P prime. 157 00:08:13,660 --> 00:08:15,520 So a very similar problem is if you 158 00:08:15,520 --> 00:08:19,630 have a trajectory that you're trying to design in a way 159 00:08:19,630 --> 00:08:21,910 that you minimize the time that you will 160 00:08:21,910 --> 00:08:24,130 spend going on this trajectory. 161 00:08:24,130 --> 00:08:27,730 So here, you have a swimmer who is sitting on the beach. 162 00:08:30,890 --> 00:08:32,320 I'm sorry, not a swimmer. 163 00:08:32,320 --> 00:08:34,600 You have a lifeguard sitting on the beach. 164 00:08:34,600 --> 00:08:37,549 And the lifeguard sees a person drowning in water 165 00:08:37,549 --> 00:08:39,799 farther behind. 166 00:08:39,799 --> 00:08:44,084 The lifeguard can run on the beach at some velocity, v sub 167 00:08:44,084 --> 00:08:44,960 r. 168 00:08:44,960 --> 00:08:48,710 They can also swim in water at some velocity v sub s. 169 00:08:48,710 --> 00:08:51,100 Most people swim slower than they run, 170 00:08:51,100 --> 00:08:56,960 so we can assume here that the running speed is faster 171 00:08:56,960 --> 00:08:58,110 than the swimming speed. 172 00:08:58,110 --> 00:08:59,860 And the question is how should the swimmer 173 00:08:59,860 --> 00:09:04,820 plan his path so that he can reach the drowning person as 174 00:09:04,820 --> 00:09:05,690 fast as possible? 175 00:09:08,650 --> 00:09:11,420 Again, you can think that, for example, the straight path 176 00:09:11,420 --> 00:09:15,270 is not the best because he's spending 177 00:09:15,270 --> 00:09:17,130 too much time in water. 178 00:09:17,130 --> 00:09:18,780 I mean, water, he's slower. 179 00:09:18,780 --> 00:09:21,660 So he may want to spend a little bit extra time 180 00:09:21,660 --> 00:09:23,850 in the fast middle, and a little bit 181 00:09:23,850 --> 00:09:25,320 less time in the slow medium. 182 00:09:25,320 --> 00:09:26,790 But, again, he cannot overdo it. 183 00:09:26,790 --> 00:09:29,160 If he goes a really crazy path, then, again, he 184 00:09:29,160 --> 00:09:30,840 will end up with a longer time. 185 00:09:30,840 --> 00:09:33,180 So light is trying to do a similar thing. 186 00:09:33,180 --> 00:09:35,910 It is time to optimize the obstacle path 187 00:09:35,910 --> 00:09:38,370 length, or equivalently, the time 188 00:09:38,370 --> 00:09:41,850 that it takes for the light ray to reach from a starting point 189 00:09:41,850 --> 00:09:42,990 to an ending point. 190 00:09:42,990 --> 00:09:46,020 And, again, remember this is almost an exact analogy here. 191 00:09:46,020 --> 00:09:49,500 The speed of light is faster in air, 192 00:09:49,500 --> 00:09:51,550 and slower in a dielectric medium. 193 00:09:51,550 --> 00:09:53,910 So if you had the air and glass here, 194 00:09:53,910 --> 00:09:56,160 that would be a very similar situation. 195 00:09:59,120 --> 00:10:00,780 I'm going to skip the next line. 196 00:10:03,530 --> 00:10:06,135 And I'm going to go to this one, to number 34. 197 00:10:10,170 --> 00:10:13,950 So what I'm trying to say here is to point out two cases. 198 00:10:13,950 --> 00:10:15,990 They're not really different. 199 00:10:15,990 --> 00:10:19,260 They're just two cases of the same situation. 200 00:10:19,260 --> 00:10:23,160 In one, you are going from a medium of lower index 201 00:10:23,160 --> 00:10:25,560 to a medium of hire index. 202 00:10:25,560 --> 00:10:27,960 And in this case, obviously, because 203 00:10:27,960 --> 00:10:30,930 of the law of refraction, the angle of refraction 204 00:10:30,930 --> 00:10:33,630 will increase as you go from left to right, 205 00:10:33,630 --> 00:10:35,940 from low index to high index. 206 00:10:35,940 --> 00:10:38,550 The opposite will happen if you go from high index 207 00:10:38,550 --> 00:10:40,650 to low index. 208 00:10:40,650 --> 00:10:43,380 OK, so this has two consequences, 209 00:10:43,380 --> 00:10:46,240 which you can think of as you reach the extremes. 210 00:10:46,240 --> 00:10:48,510 If you come in at the maximum possible angle 211 00:10:48,510 --> 00:10:50,940 here, 90 degrees, then you can imagine 212 00:10:50,940 --> 00:10:53,310 that you will not enter at 90, but you will enter 213 00:10:53,310 --> 00:10:55,110 at a slightly smaller angle. 214 00:10:55,110 --> 00:10:58,860 So basically, if you are coupling in light 215 00:10:58,860 --> 00:11:03,780 from air to glass, or in general from a medium of lower 216 00:11:03,780 --> 00:11:05,430 index to a medium of higher index, 217 00:11:05,430 --> 00:11:08,070 you have a limited column of approach 218 00:11:08,070 --> 00:11:11,180 that you can couple light into. 219 00:11:11,180 --> 00:11:13,320 And that is given by this equation over here. 220 00:11:13,320 --> 00:11:16,920 When the exterior angle theta reaches 90 degrees, 221 00:11:16,920 --> 00:11:19,650 then this is the maximum angle, theta prime, 222 00:11:19,650 --> 00:11:24,510 that you can access inside the high index medium. 223 00:11:24,510 --> 00:11:27,080 The opposite is perhaps slightly more interesting, 224 00:11:27,080 --> 00:11:31,950 is what happens when you reach or exceed theta 225 00:11:31,950 --> 00:11:34,863 prime equals to 90 degrees over here. 226 00:11:34,863 --> 00:11:36,780 So if you look again at the law of refraction, 227 00:11:36,780 --> 00:11:40,260 you can realize that it is possible to arrange 228 00:11:40,260 --> 00:11:43,160 for a combination of n and theta, 229 00:11:43,160 --> 00:11:47,370 such that the product is bigger than the index of refraction 230 00:11:47,370 --> 00:11:48,840 at the medium side. 231 00:11:48,840 --> 00:11:52,140 In order, now, to satisfy the law of refraction, 232 00:11:52,140 --> 00:11:54,870 you would have to require that the sine of an angle 233 00:11:54,870 --> 00:11:56,130 is bigger than 1. 234 00:11:56,130 --> 00:11:58,920 So since we are limited to deal with real angles here, 235 00:11:58,920 --> 00:12:01,350 not complex, this cannot happen. 236 00:12:01,350 --> 00:12:04,080 What really happens there is that the light will actually 237 00:12:04,080 --> 00:12:05,130 be reflected. 238 00:12:05,130 --> 00:12:08,880 If you satisfy this condition, this product 239 00:12:08,880 --> 00:12:11,460 becomes bigger than the index of refraction 240 00:12:11,460 --> 00:12:17,120 outside in the medium. 241 00:12:17,120 --> 00:12:18,740 Then when you satisfy this condition, 242 00:12:18,740 --> 00:12:23,760 then light will be reflected inside the high index medium. 243 00:12:23,760 --> 00:12:27,980 And that is known as total internal reflection, or TIR. 244 00:12:27,980 --> 00:12:32,100 So let's look at TIR in slightly more detail over here. 245 00:12:32,100 --> 00:12:33,880 So I have a glass medium in there, 246 00:12:33,880 --> 00:12:35,870 and imagine that I have a wavefront that 247 00:12:35,870 --> 00:12:41,060 is arriving from glass towards the air interface. 248 00:12:41,060 --> 00:12:44,920 So there is a combination of index and angle, 249 00:12:44,920 --> 00:12:47,800 where the product of the index inside the glass 250 00:12:47,800 --> 00:12:50,860 times the sine of the angle equals exactly one. 251 00:12:50,860 --> 00:12:54,160 What happens then is the law of refraction 252 00:12:54,160 --> 00:12:55,870 does not break down yet. 253 00:12:55,870 --> 00:12:58,840 But what will happen is the light will be refracted, 254 00:12:58,840 --> 00:13:01,870 and it will propagate exactly parallel to the interface. 255 00:13:01,870 --> 00:13:04,460 This is known as a surface wave. 256 00:13:04,460 --> 00:13:08,180 If you increase the angle now, then the product 257 00:13:08,180 --> 00:13:09,530 will become bigger than one. 258 00:13:09,530 --> 00:13:12,260 The law of refraction cannot be satisfied anymore. 259 00:13:12,260 --> 00:13:13,450 So what will happen then. 260 00:13:13,450 --> 00:13:16,040 Oh, and I should have said that the angle where this happens 261 00:13:16,040 --> 00:13:18,080 is called the critical angle. 262 00:13:18,080 --> 00:13:21,500 Because it is the angle just below which 263 00:13:21,500 --> 00:13:24,080 I still have refraction. 264 00:13:24,080 --> 00:13:26,780 If exceed this angle, then I get this phenomena 265 00:13:26,780 --> 00:13:33,140 of total internal reflection, where all of the light 266 00:13:33,140 --> 00:13:36,950 is reflected inside the glass, inside the high index medium. 267 00:13:36,950 --> 00:13:39,740 So it is almost as if the interface here 268 00:13:39,740 --> 00:13:42,680 abruptly changes, and instead of being 269 00:13:42,680 --> 00:13:46,040 mostly transmissible over here, it becomes mostly reflective. 270 00:13:46,040 --> 00:13:49,390 So it starts acting like a mirror. 271 00:13:49,390 --> 00:13:51,760 There's one difference that makes it slightly 272 00:13:51,760 --> 00:13:53,710 different than a mirror. 273 00:13:53,710 --> 00:13:56,770 And the difference is that if you 274 00:13:56,770 --> 00:14:01,380 were to calculate the electric field on the opposite side 275 00:14:01,380 --> 00:14:03,510 of the interface, that is, inside the medium 276 00:14:03,510 --> 00:14:05,940 where light does not propagate. 277 00:14:05,940 --> 00:14:08,650 You will discover that there is some leakage. 278 00:14:08,650 --> 00:14:13,710 The electric field has non-zero values in the low index 279 00:14:13,710 --> 00:14:15,450 medium over here. 280 00:14:15,450 --> 00:14:18,170 Even though the electric field that you find 281 00:14:18,170 --> 00:14:20,940 is not propagating, it is what is called evanescent. 282 00:14:20,940 --> 00:14:25,350 It is in exponential decay, but there is no wavefront. 283 00:14:25,350 --> 00:14:28,370 There is no wavefront of light propagating 284 00:14:28,370 --> 00:14:31,133 in the vertical direction like this. 285 00:14:31,133 --> 00:14:32,550 This is called an evanescent wave, 286 00:14:32,550 --> 00:14:35,130 and we will revisit it later when 287 00:14:35,130 --> 00:14:36,870 we deal with electromagnetics. 288 00:14:36,870 --> 00:14:39,120 Because right now, the way I defined 289 00:14:39,120 --> 00:14:44,780 it is not perhaps very rigorous, or very quantitative. 290 00:14:44,780 --> 00:14:47,590 But I wanted to give you a sort of a heads 291 00:14:47,590 --> 00:14:51,970 up that something slightly more than geometrical optics 292 00:14:51,970 --> 00:14:54,220 prediction happens here. 293 00:14:54,220 --> 00:14:56,140 But as far as geometrical optics goes, 294 00:14:56,140 --> 00:14:58,820 that we will be dealing for the next few lectures, 295 00:14:58,820 --> 00:15:00,242 there is a reflection. 296 00:15:05,380 --> 00:15:08,320 This says what I just mentioned, that we 297 00:15:08,320 --> 00:15:10,740 will talk more about these evanescent waves 298 00:15:10,740 --> 00:15:12,140 a little bit later. 299 00:15:12,140 --> 00:15:15,030 One more thing that I want to say about evanescent waves. 300 00:15:17,895 --> 00:15:20,700 One way you can sort of realize the existence 301 00:15:20,700 --> 00:15:24,280 of evanescent waves is with a sort 302 00:15:24,280 --> 00:15:27,210 a related phenomenon called frustrated total internal 303 00:15:27,210 --> 00:15:31,140 reflection, also known as FTIR, because that's quite a mouthful 304 00:15:31,140 --> 00:15:32,370 to pronounce. 305 00:15:32,370 --> 00:15:35,070 So FTIR happens if you have this situation, 306 00:15:35,070 --> 00:15:37,280 where you're beyond that critical angle, 307 00:15:37,280 --> 00:15:39,690 and therefore, your total internal reflecting 308 00:15:39,690 --> 00:15:41,160 into the medium. 309 00:15:41,160 --> 00:15:43,770 But you bring near the interface, 310 00:15:43,770 --> 00:15:45,630 you're bringing another piece of glass, 311 00:15:45,630 --> 00:15:47,790 another piece of high index medium 312 00:15:47,790 --> 00:15:53,010 in a way that an appreciable amount of the evanescent wave 313 00:15:53,010 --> 00:15:57,780 is allowed to enter inside the high index medium. 314 00:15:57,780 --> 00:16:02,190 If that happens, as we've said, the TIR is frustrated. 315 00:16:02,190 --> 00:16:05,730 What it really means is that the TIR stops happening now. 316 00:16:05,730 --> 00:16:07,507 What would happen is a small amount 317 00:16:07,507 --> 00:16:09,340 of light will still be reflected, of course. 318 00:16:09,340 --> 00:16:10,920 You cannot avoid that. 319 00:16:10,920 --> 00:16:12,660 But the significant portion of the light 320 00:16:12,660 --> 00:16:17,640 will couple out into the next material, 321 00:16:17,640 --> 00:16:20,662 and it will actually be transmitted. 322 00:16:20,662 --> 00:16:22,370 So this is a very interesting phenomenon, 323 00:16:22,370 --> 00:16:24,162 because if you think about it, the light is 324 00:16:24,162 --> 00:16:27,160 forbidden to enter this region. 325 00:16:27,160 --> 00:16:30,620 Snell's law says that light cannot cross into air, 326 00:16:30,620 --> 00:16:33,550 yet because of the proximity, the light can actually couple 327 00:16:33,550 --> 00:16:34,170 out. 328 00:16:34,170 --> 00:16:37,460 And again, we will see a much more rigorous explanation 329 00:16:37,460 --> 00:16:40,340 and quantitative description of this phenomenon 330 00:16:40,340 --> 00:16:42,440 later, when we're doing it through magnetics. 331 00:16:42,440 --> 00:16:46,010 Some of you who may have taken electronics or quantum 332 00:16:46,010 --> 00:16:49,190 mechanics, there's a similar effect 333 00:16:49,190 --> 00:16:56,300 called tunneling in potential barriers in quantum mechanics. 334 00:16:56,300 --> 00:16:59,030 The equations are very similar that describe this phenomenon. 335 00:16:59,030 --> 00:17:01,530 In both cases, you have a wave that 336 00:17:01,530 --> 00:17:03,650 is crossing a forbidden region in order 337 00:17:03,650 --> 00:17:06,710 to pass into an allowable region again. 338 00:17:09,842 --> 00:17:11,300 This may be actually be a good find 339 00:17:11,300 --> 00:17:19,819 for Piper to solve a demo of the total internal reflection. 340 00:17:19,819 --> 00:17:24,277 And Piper will actually solve it in the context of a prism. 341 00:17:24,277 --> 00:17:25,819 Piper, maybe you can start setting up 342 00:17:25,819 --> 00:17:28,010 while I give a brief description of prisms. 343 00:17:32,420 --> 00:17:34,840 You all are familiar, I suppose. 344 00:17:34,840 --> 00:17:38,220 They're pieces of glass that are cut into various triangular 345 00:17:38,220 --> 00:17:40,940 and other polygonal shapes. 346 00:17:40,940 --> 00:17:43,770 And typically, prisms are-- 347 00:17:43,770 --> 00:17:46,610 they're arranged either so that the light passes through, 348 00:17:46,610 --> 00:17:48,380 as shown in the top diagram. 349 00:17:48,380 --> 00:17:51,440 Or if you bring the light from the bottom, 350 00:17:51,440 --> 00:17:54,460 and you manage to exceed the critical angle 351 00:17:54,460 --> 00:17:58,140 in the interface, then you can also total internally 352 00:17:58,140 --> 00:18:00,920 reflect the light, and create a situation like this that 353 00:18:00,920 --> 00:18:03,470 is known as a retro-reflector. 354 00:18:03,470 --> 00:18:07,400 Now, a rule of thumb that is useful to know for glass, which 355 00:18:07,400 --> 00:18:10,240 has index of refraction 1.5. 356 00:18:10,240 --> 00:18:14,130 The critical angle is about 42 degrees. 357 00:18:14,130 --> 00:18:16,430 So if you are incident at 45, as shown 358 00:18:16,430 --> 00:18:21,717 in this case of what is called isosceles triangle, then-- 359 00:18:21,717 --> 00:18:22,550 actually, I'm sorry. 360 00:18:22,550 --> 00:18:24,600 This is equilateral, isn't it? 361 00:18:24,600 --> 00:18:28,190 If you're incident at 45 in an equilateral triangle, 362 00:18:28,190 --> 00:18:30,800 then you will satisfy the TIR condition, 363 00:18:30,800 --> 00:18:33,580 and you get this retro-reflector. 364 00:18:33,580 --> 00:18:35,120 And there's more complicated prisms 365 00:18:35,120 --> 00:18:38,600 that allow you-- for example, this is the pentaprism. 366 00:18:38,600 --> 00:18:43,900 After two bounces, the light will exit at 90 degrees angle. 367 00:18:43,900 --> 00:18:47,750 So Piper, maybe you can do the demo now? 368 00:18:47,750 --> 00:18:48,820 PROFESSOR: Sure, sure. 369 00:18:48,820 --> 00:18:50,110 So just before showing the demo, I'm 370 00:18:50,110 --> 00:18:51,820 going to pass around this prism that 371 00:18:51,820 --> 00:18:55,310 has in these two surfaces, two images. 372 00:18:55,310 --> 00:18:57,520 So then what you see is that in this window here, 373 00:18:57,520 --> 00:18:59,760 you tilt it like this. 374 00:18:59,760 --> 00:19:02,290 You're going to see when you get a hold of it. 375 00:19:02,290 --> 00:19:04,380 You actually see two different images 376 00:19:04,380 --> 00:19:06,230 due to total internal reflection. 377 00:19:06,230 --> 00:19:08,110 So at a given angle, you basically 378 00:19:08,110 --> 00:19:10,212 get the light reflecting from one of the surfaces. 379 00:19:10,212 --> 00:19:11,920 And another angle, you get the other one. 380 00:19:11,920 --> 00:19:13,628 So it's actually very interesting to see. 381 00:19:18,410 --> 00:19:21,340 OK, so we're seeing here the top view of the demo. 382 00:19:21,340 --> 00:19:25,150 We put together a wide light source and a laser source. 383 00:19:25,150 --> 00:19:27,157 And before showing you what actually happens, 384 00:19:27,157 --> 00:19:28,990 let me just introduce some of the components 385 00:19:28,990 --> 00:19:33,160 that we are going to be seeing in several demos from now on. 386 00:19:33,160 --> 00:19:36,700 First of all, we have the white light source, it's a lamp. 387 00:19:36,700 --> 00:19:40,060 And then this component here, it's a regular lens 388 00:19:40,060 --> 00:19:41,360 that you're familiar with. 389 00:19:41,360 --> 00:19:43,990 So the job of these lens is to collimate-- 390 00:19:43,990 --> 00:19:47,080 that's the term that we use-- to convert this light close 391 00:19:47,080 --> 00:19:49,180 to a plane wave, similar to the light that 392 00:19:49,180 --> 00:19:50,350 is coming from the sun. 393 00:19:50,350 --> 00:19:53,200 So the sun, again, would be like a point source of light 394 00:19:53,200 --> 00:19:54,460 far away. 395 00:19:54,460 --> 00:19:56,020 So then these lenses are basically 396 00:19:56,020 --> 00:19:58,570 transforming in this light into parallel rays 397 00:19:58,570 --> 00:20:01,850 from the geometrical optics point of view. 398 00:20:01,850 --> 00:20:02,940 So we use these lens. 399 00:20:02,940 --> 00:20:05,380 This is an iris, similar to the iris that controls 400 00:20:05,380 --> 00:20:08,170 the aperture in your eye. 401 00:20:08,170 --> 00:20:11,170 And this is just used to split the light into two-- 402 00:20:11,170 --> 00:20:13,840 I'm sorry, to reduce the diameter of the beam. 403 00:20:13,840 --> 00:20:17,680 Then this component, here this one. 404 00:20:17,680 --> 00:20:20,110 It's what we call the beam splitter, 405 00:20:20,110 --> 00:20:23,080 or more specifically, the non-polarized beam splitter. 406 00:20:23,080 --> 00:20:25,000 And it's a component that allows us to split 407 00:20:25,000 --> 00:20:26,710 the light into two paths. 408 00:20:26,710 --> 00:20:31,050 So here, this is one path, and another path. 409 00:20:33,980 --> 00:20:38,270 So once we split the light in two paths, and in this case, 410 00:20:38,270 --> 00:20:39,450 it has equal ratios. 411 00:20:39,450 --> 00:20:42,530 So it's 50% to one side, 50% to the other. 412 00:20:42,530 --> 00:20:46,850 Then let's follow one of these parts. 413 00:20:46,850 --> 00:20:49,640 This part illuminates one of these prisms here. 414 00:20:49,640 --> 00:20:52,940 And again, there's going to be some refraction following 415 00:20:52,940 --> 00:20:54,650 the law of refraction that we saw. 416 00:20:54,650 --> 00:20:56,360 But in addition to that, as we'll 417 00:20:56,360 --> 00:20:58,780 see in the next couple of slides also. 418 00:20:58,780 --> 00:21:00,883 There is a phenomenon called dispersion 419 00:21:00,883 --> 00:21:02,300 that you're familiar with when you 420 00:21:02,300 --> 00:21:04,940 see rainbows in a rainy day. 421 00:21:04,940 --> 00:21:07,550 And basically, that has to do with the fact 422 00:21:07,550 --> 00:21:11,640 that different wavelengths see a different index of refraction. 423 00:21:11,640 --> 00:21:13,730 So if you go back to the Snell's law, 424 00:21:13,730 --> 00:21:16,700 they will basically bend in a different way. 425 00:21:16,700 --> 00:21:19,430 So therefore, you see a rainbow effect. 426 00:21:19,430 --> 00:21:22,250 So I don't know if you can see the side view in the camera 427 00:21:22,250 --> 00:21:25,090 please? 428 00:21:25,090 --> 00:21:29,130 Yeah, so that's a picture of it. 429 00:21:29,130 --> 00:21:31,410 OK, but here in the classroom too, please. 430 00:21:37,678 --> 00:21:38,970 I don't know if you can see it. 431 00:21:38,970 --> 00:21:41,010 We're going to try to show it also in this. 432 00:21:41,010 --> 00:21:43,660 But I have here two components. 433 00:21:43,660 --> 00:21:45,300 This is a prism that is doing these, 434 00:21:45,300 --> 00:21:49,840 what we call the normal dispersion, which basically has 435 00:21:49,840 --> 00:21:53,760 that the biggest angle that bends is the blue light, 436 00:21:53,760 --> 00:21:55,870 or the shorter wavelength. 437 00:21:55,870 --> 00:21:57,600 And then we have another component here 438 00:21:57,600 --> 00:21:59,693 that we haven't talked about yet, 439 00:21:59,693 --> 00:22:01,360 but we'll see it in the next few slides. 440 00:22:01,360 --> 00:22:04,440 So this is just an introductory introduction to this element. 441 00:22:04,440 --> 00:22:08,070 It's called transmission grating. 442 00:22:08,070 --> 00:22:10,560 And this component is used in several systems. 443 00:22:13,120 --> 00:22:14,370 They're a different principle. 444 00:22:14,370 --> 00:22:16,200 It's not refraction anymore. 445 00:22:16,200 --> 00:22:19,500 It's using the diffraction property of the light. 446 00:22:19,500 --> 00:22:21,510 And using that diffraction property can also 447 00:22:21,510 --> 00:22:24,900 create this rainbow that you could see here in the back, 448 00:22:24,900 --> 00:22:26,570 and hopefully, you can see the picture. 449 00:22:30,343 --> 00:22:32,260 What we are going to see here is that actually 450 00:22:32,260 --> 00:22:34,030 the opposite trend happens. 451 00:22:34,030 --> 00:22:37,870 The red angles bend more than the blue angles, 452 00:22:37,870 --> 00:22:40,190 and that's called anomalous dispersion. 453 00:22:40,190 --> 00:22:42,310 So we have normal dispersion in the prism case, 454 00:22:42,310 --> 00:22:44,470 like in the one shown here in transparency, 455 00:22:44,470 --> 00:22:49,060 and anomalous dispersion showed in the grating. 456 00:22:49,060 --> 00:22:50,920 So I'm going to tilt this a little bit. 457 00:22:55,010 --> 00:22:55,910 OK, it's fine. 458 00:22:55,910 --> 00:22:57,110 I'm going to try to see. 459 00:22:57,110 --> 00:22:58,770 So I don't know if you can see it. 460 00:22:58,770 --> 00:23:00,920 Can you see the rainbow here from the back? 461 00:23:00,920 --> 00:23:02,550 If you're in the classroom? 462 00:23:02,550 --> 00:23:03,560 GEORGE BARBASTATHIS: Yeah, we can see it. 463 00:23:03,560 --> 00:23:04,910 PROFESSOR: All right, excellent. 464 00:23:04,910 --> 00:23:07,582 So this is just the rainbow from the grating. 465 00:23:07,582 --> 00:23:10,040 After class, you can just come here and play with the demo. 466 00:23:10,040 --> 00:23:12,950 And you're going to see the two cases. 467 00:23:12,950 --> 00:23:15,080 You can actually trace the rays and see 468 00:23:15,080 --> 00:23:18,860 which color is bending more, and distinguish 469 00:23:18,860 --> 00:23:21,540 between anomalous and normal dispersion. 470 00:23:21,540 --> 00:23:23,390 So the last thing that I want to show here 471 00:23:23,390 --> 00:23:26,660 is the total internal reflection principle. 472 00:23:26,660 --> 00:23:30,920 In this case, this piece of acrylic here that we have. 473 00:23:30,920 --> 00:23:32,300 You can see that it's forming-- 474 00:23:32,300 --> 00:23:35,060 it's basically having a laser light coupling into one 475 00:23:35,060 --> 00:23:38,450 of the sides, so similar to that exit sign over there. 476 00:23:38,450 --> 00:23:40,730 And what we have is that the acrylic-- 477 00:23:40,730 --> 00:23:43,070 the piece of acrylic acts like a wave guide. 478 00:23:43,070 --> 00:23:46,190 So it conducts a light inside. 479 00:23:46,190 --> 00:23:48,470 And the reason the light doesn't escape 480 00:23:48,470 --> 00:23:50,810 is because it's basically suffering 481 00:23:50,810 --> 00:23:53,840 total internal reflection at the interface between the acrylic, 482 00:23:53,840 --> 00:23:57,470 which has a higher index than air, so stays inside. 483 00:23:57,470 --> 00:24:00,320 Now, in order to couple the light out, 484 00:24:00,320 --> 00:24:05,110 I put some tape here, as you can see forming the letters MIT. 485 00:24:05,110 --> 00:24:08,480 And that basically frustrates the light, 486 00:24:08,480 --> 00:24:10,360 allows it to break the incidence angle. 487 00:24:10,360 --> 00:24:15,140 So a ray now instead of getting into a very flat surface 488 00:24:15,140 --> 00:24:17,860 at an angle that is larger than the critical angle, 489 00:24:17,860 --> 00:24:19,370 it basically reaches a surface that 490 00:24:19,370 --> 00:24:22,940 has a diffuser type of angle, so maybe it can escape out. 491 00:24:22,940 --> 00:24:25,700 So then, you can see all this light diffusing 492 00:24:25,700 --> 00:24:30,560 out forming like either this image, or the image of the exit 493 00:24:30,560 --> 00:24:31,940 sign. 494 00:24:31,940 --> 00:24:34,470 So that one, you can see it also in Singapore? 495 00:24:34,470 --> 00:24:36,317 The frustrated? 496 00:24:36,317 --> 00:24:37,400 GEORGE BARBASTATHIS: Yeah. 497 00:24:37,400 --> 00:24:38,960 PROFESSOR: OK. 498 00:24:38,960 --> 00:24:42,310 GEORGE BARBASTATHIS: It looks frustrated to us. 499 00:24:42,310 --> 00:24:45,250 PROFESSOR: From the side view, I guess. 500 00:24:45,250 --> 00:24:47,938 So I don't know if you want to add anything, George. 501 00:24:47,938 --> 00:24:49,980 GEORGE BARBASTATHIS: So what you see on the slide 502 00:24:49,980 --> 00:24:52,590 that I'm projecting it now is an application 503 00:24:52,590 --> 00:24:55,260 of the same principle that Piper just showed. 504 00:24:55,260 --> 00:24:58,470 It has an application in a bunch of commonly 505 00:24:58,470 --> 00:25:02,690 used conventional devices, namely, fingerprint sensors. 506 00:25:02,690 --> 00:25:05,860 Where instead of the tape that Piper put over there, 507 00:25:05,860 --> 00:25:08,490 usually what you do is they place their finger 508 00:25:08,490 --> 00:25:11,280 touching the side of the prism. 509 00:25:11,280 --> 00:25:16,410 And then because our buddy, this may be surprising to you. 510 00:25:16,410 --> 00:25:18,610 It was surprising to me when I first heard it. 511 00:25:18,610 --> 00:25:22,920 Our body is composed mostly of water, about 75% is water. 512 00:25:22,920 --> 00:25:25,440 So therefore, the refractive index of our skin 513 00:25:25,440 --> 00:25:29,800 is close to 1.3, which is, of course, higher 514 00:25:29,800 --> 00:25:31,950 than the refractive index of air. 515 00:25:31,950 --> 00:25:35,280 So what happens here is over here, for example, 516 00:25:35,280 --> 00:25:38,280 you have a glass and air. 517 00:25:38,280 --> 00:25:42,930 So therefore, light will be totally internally reflected. 518 00:25:42,930 --> 00:25:46,710 But at the ridges of the finger, of the fingerprint, 519 00:25:46,710 --> 00:25:49,200 you have glass and water. 520 00:25:49,200 --> 00:25:53,010 That is 1.5 to 1.33 or so. 521 00:25:53,010 --> 00:25:56,440 So therefore, the ridges appear dark because they frustrate 522 00:25:56,440 --> 00:25:58,210 the total internal reflection. 523 00:25:58,210 --> 00:26:00,300 The light couples into your finger. 524 00:26:00,300 --> 00:26:04,110 And therefore, a surprisingly sharp image of the finger 525 00:26:04,110 --> 00:26:05,460 appears in the camera. 526 00:26:05,460 --> 00:26:07,550 Actually, what you see here, it does a disservice. 527 00:26:07,550 --> 00:26:11,280 The projector and the pixelation of my computer 528 00:26:11,280 --> 00:26:13,980 does a disservice to the quality of the fingerprint image 529 00:26:13,980 --> 00:26:14,910 that you get. 530 00:26:14,910 --> 00:26:18,150 So nowadays, most fingerprint sensors 531 00:26:18,150 --> 00:26:19,732 are based on this principle. 532 00:26:19,732 --> 00:26:20,940 In fact, we have improved it. 533 00:26:20,940 --> 00:26:23,250 Instead of using the prism in places 534 00:26:23,250 --> 00:26:27,050 like laptops that have fingerprint security, 535 00:26:27,050 --> 00:26:29,490 it is the same principle, but you slide the finger. 536 00:26:29,490 --> 00:26:32,130 But still, the ridge of the fingerprints 537 00:26:32,130 --> 00:26:34,860 as you slide the finger over the sensor 538 00:26:34,860 --> 00:26:39,525 is captured by the principle of total internal reflection. 539 00:26:44,270 --> 00:26:45,350 Any questions about that? 540 00:26:45,350 --> 00:26:46,900 About TIR and FTIR? 541 00:27:00,080 --> 00:27:07,050 OK, one other use of TIR is in another very useful-- 542 00:27:07,050 --> 00:27:10,080 another extremely useful property of light 543 00:27:10,080 --> 00:27:15,160 is that you can actually capture it, almost like in a wire. 544 00:27:15,160 --> 00:27:17,933 And you can guide the light over a very long distance. 545 00:27:17,933 --> 00:27:19,350 Now, why this is very important is 546 00:27:19,350 --> 00:27:21,780 because we know from experience, and we'll also 547 00:27:21,780 --> 00:27:24,570 learn later as the Huygens principle, 548 00:27:24,570 --> 00:27:28,110 that light does not like to be confined. 549 00:27:28,110 --> 00:27:32,130 Generally, light, once you generate light in a source, 550 00:27:32,130 --> 00:27:33,720 the light would like to expand. 551 00:27:33,720 --> 00:27:37,590 It would like to open up and propagate in an expansive way. 552 00:27:37,590 --> 00:27:40,140 For example, the sun, the stars, and so on. 553 00:27:40,140 --> 00:27:43,170 They propagate isotropically, all around them. 554 00:27:43,170 --> 00:27:45,640 And you know the same from the light bulbs and so on. 555 00:27:45,640 --> 00:27:47,010 The light expands. 556 00:27:47,010 --> 00:27:49,400 There's an exception, of course, called lasers. 557 00:27:49,400 --> 00:27:51,150 Lasers can be quite collimated. 558 00:27:51,150 --> 00:27:53,000 But even lasers, they tend to expand. 559 00:27:53,000 --> 00:27:54,600 If you leave a laser beam by itself, 560 00:27:54,600 --> 00:27:56,550 and you propagate it for a long distance, 561 00:27:56,550 --> 00:27:57,860 eventually, it will expand. 562 00:27:57,860 --> 00:28:00,420 It will become quite big. 563 00:28:00,420 --> 00:28:03,980 So the way to undo this property of light-- 564 00:28:03,980 --> 00:28:06,170 if you want to transmit light over a long distance 565 00:28:06,170 --> 00:28:09,300 without expansion-- is to use a wave guide. 566 00:28:09,300 --> 00:28:12,440 So wave guides typically, they use this phenomenon 567 00:28:12,440 --> 00:28:14,030 of total internal reflection. 568 00:28:17,330 --> 00:28:22,610 In the simplest case, you have a slab of high index-- 569 00:28:22,610 --> 00:28:26,510 dielectric middle sandwiched between two other pieces 570 00:28:26,510 --> 00:28:28,970 of lower index medium. 571 00:28:28,970 --> 00:28:33,110 And what happens there provided that the light is incident 572 00:28:33,110 --> 00:28:37,040 at the sharp enough angle that is beyond the critical angle 573 00:28:37,040 --> 00:28:38,480 between the two media. 574 00:28:38,480 --> 00:28:41,330 Then the light will sort of bounce back and forth 575 00:28:41,330 --> 00:28:43,340 between the two interfaces. 576 00:28:43,340 --> 00:28:45,590 And this way, you can actually transmit it 577 00:28:45,590 --> 00:28:47,330 over a very long distance. 578 00:28:47,330 --> 00:28:48,790 So, of course, if it is not true. 579 00:28:48,790 --> 00:28:51,950 If the light arrives at a shallower angle, then, 580 00:28:51,950 --> 00:28:54,245 of course, it will actually couple out, 581 00:28:54,245 --> 00:28:57,380 and it will not be guided anymore. 582 00:28:57,380 --> 00:29:01,990 The way you establish whether the light will be guided or not 583 00:29:01,990 --> 00:29:04,970 is by using these properties called the and numerical 584 00:29:04,970 --> 00:29:07,820 aperture of the wave guide. 585 00:29:07,820 --> 00:29:11,000 So this is a term, numerical aperture, 586 00:29:11,000 --> 00:29:14,150 that we'll hear again and again in this class, at least 587 00:29:14,150 --> 00:29:16,280 in three different contexts. 588 00:29:16,280 --> 00:29:18,380 But they all mean the same thing. 589 00:29:18,380 --> 00:29:20,960 Actually, they mean an angle of acceptance 590 00:29:20,960 --> 00:29:24,610 of an optical system. 591 00:29:24,610 --> 00:29:27,600 So in this context here of a wave guide, 592 00:29:27,600 --> 00:29:28,710 compare the two rays. 593 00:29:28,710 --> 00:29:31,160 One is sort of the solid ray, and the other 594 00:29:31,160 --> 00:29:32,760 is the dotted ray. 595 00:29:32,760 --> 00:29:35,940 The solid ray comes in from air, then 596 00:29:35,940 --> 00:29:39,140 is refracted at the vertical interface. 597 00:29:39,140 --> 00:29:42,530 And because this angle is fairly small, 598 00:29:42,530 --> 00:29:44,990 by the time it gets into the middle, the angle 599 00:29:44,990 --> 00:29:47,300 it makes through the perpendicular surface 600 00:29:47,300 --> 00:29:50,270 of the interface between the slab and the cladding. 601 00:29:50,270 --> 00:29:53,660 It actually satisfies the TIR condition over here. 602 00:29:53,660 --> 00:29:56,330 You can see a little bit if you familiarize yourself 603 00:29:56,330 --> 00:29:58,650 with the way Snell's law works. 604 00:29:58,650 --> 00:30:02,060 You can see that as you increase this angle over here, 605 00:30:02,060 --> 00:30:04,970 this angle over here actually decreases. 606 00:30:04,970 --> 00:30:09,320 So the dotted ray actually can arrive 607 00:30:09,320 --> 00:30:11,480 at below the critical angle. 608 00:30:11,480 --> 00:30:13,070 So therefore, the daughter ray is not 609 00:30:13,070 --> 00:30:15,590 guided where the solid ray is guided. 610 00:30:15,590 --> 00:30:19,880 So the numerical aperture is the maximum angle, theta naught, 611 00:30:19,880 --> 00:30:22,850 that you can tolerate before you stop 612 00:30:22,850 --> 00:30:24,938 satisfying that TIR condition. 613 00:30:24,938 --> 00:30:26,480 And therefore, the numerical aperture 614 00:30:26,480 --> 00:30:30,050 is the maximum angle that you can couple into the wave guide. 615 00:30:30,050 --> 00:30:32,930 If you tried to bring light at a higher angle than that, 616 00:30:32,930 --> 00:30:34,250 it will actually not be guided. 617 00:30:34,250 --> 00:30:37,400 It will escape into the cladding, and it will get lost. 618 00:30:37,400 --> 00:30:40,920 It will disappear. 619 00:30:40,920 --> 00:30:43,670 So with a little bit of algebra, which I haven't done here. 620 00:30:43,670 --> 00:30:45,890 I will let you do it by yourselves. 621 00:30:45,890 --> 00:30:48,200 In years past, I used to give this as a homework, 622 00:30:48,200 --> 00:30:49,433 but I didn't do it this time. 623 00:30:49,433 --> 00:30:51,600 But anyway, with a little bit of algebra and playing 624 00:30:51,600 --> 00:30:53,390 with Snell's law, you can find out 625 00:30:53,390 --> 00:30:55,220 that the numerical aperture in this case 626 00:30:55,220 --> 00:30:57,170 is given by this quantity over here. 627 00:30:57,170 --> 00:30:59,030 The square root of the difference 628 00:30:59,030 --> 00:31:02,030 of squares between the two indices of refraction. 629 00:31:02,030 --> 00:31:05,750 And as I mentioned earlier, physically what it means. 630 00:31:05,750 --> 00:31:07,940 This quantity is the angle of acceptance 631 00:31:07,940 --> 00:31:11,310 of the wave guide for the light that you want to couple in. 632 00:31:11,310 --> 00:31:13,190 Typically, wave guides in practice, 633 00:31:13,190 --> 00:31:15,350 they have a very small difference 634 00:31:15,350 --> 00:31:21,010 between the index of the core, where the light is guided, 635 00:31:21,010 --> 00:31:22,560 and the index of the cladding. 636 00:31:22,560 --> 00:31:26,030 This difference is typically in the order of 10 to the minus 3, 637 00:31:26,030 --> 00:31:27,620 or 10 to the minus 4. 638 00:31:27,620 --> 00:31:29,630 So therefore, the numerical aperture is what? 639 00:31:29,630 --> 00:31:32,660 It is small or large for a typical wave guide? 640 00:31:36,560 --> 00:31:38,490 If the index difference is very small, 641 00:31:38,490 --> 00:31:40,490 is the numerical appearance or a small or large? 642 00:31:48,930 --> 00:31:50,543 Small, right? 643 00:31:50,543 --> 00:31:52,210 We're going to set up a competition here 644 00:31:52,210 --> 00:31:53,590 between Singapore and Cambridge. 645 00:31:53,590 --> 00:31:57,940 So you guys, when you have an answer, please push the button. 646 00:31:57,940 --> 00:32:00,520 On either side, please push the button. 647 00:32:00,520 --> 00:32:03,100 So the numerical aperture is actually very small, 648 00:32:03,100 --> 00:32:05,980 as our colleagues here correctly said. 649 00:32:10,342 --> 00:32:12,050 And what is on the slide is the opposite. 650 00:32:12,050 --> 00:32:13,467 If you have a high index contrast, 651 00:32:13,467 --> 00:32:17,650 then you get a high numerical aperture. 652 00:32:17,650 --> 00:32:20,720 There is one more type of wave guide, which is actually 653 00:32:20,720 --> 00:32:22,370 the same principle, but a slightly 654 00:32:22,370 --> 00:32:24,200 different implementation. 655 00:32:24,200 --> 00:32:27,102 It's called a gradient index wave guide. 656 00:32:27,102 --> 00:32:28,310 And the way to understand it. 657 00:32:28,310 --> 00:32:33,920 Imagine that I stack a bunch of different slabs of glass 658 00:32:33,920 --> 00:32:36,940 with index that varies from a small value. 659 00:32:36,940 --> 00:32:40,460 This I denoted as the sort of light gray. 660 00:32:40,460 --> 00:32:43,640 Then to darker gray, denoting higher index of refraction, 661 00:32:43,640 --> 00:32:45,320 and then back to light gray. 662 00:32:45,320 --> 00:32:48,360 I mean, back to low index of refraction. 663 00:32:48,360 --> 00:32:51,500 So if you imagine the ray coming in from the top here. 664 00:32:51,500 --> 00:32:55,880 It will refract into the guide, and then as it goes in, 665 00:32:55,880 --> 00:32:57,690 it will keep getting refracted. 666 00:32:57,690 --> 00:32:59,960 Now, we can adjust the numbers here, 667 00:32:59,960 --> 00:33:03,170 so that one of these interfaces, the light 668 00:33:03,170 --> 00:33:05,410 will exceed the critical angle. 669 00:33:05,410 --> 00:33:07,820 And therefore, it will be totally internally reflected 670 00:33:07,820 --> 00:33:09,280 at this interface. 671 00:33:09,280 --> 00:33:12,020 And if that is true, the same thing 672 00:33:12,020 --> 00:33:15,380 will happen also at the top interface. 673 00:33:15,380 --> 00:33:18,690 As I mentioned before, this quantity, n sine theta, 674 00:33:18,690 --> 00:33:19,880 is preserved. 675 00:33:19,880 --> 00:33:22,940 So therefore, if this quantity, n sine theta, 676 00:33:22,940 --> 00:33:27,320 was such that the TIR condition was satisfied over here, 677 00:33:27,320 --> 00:33:29,273 then the same will happen over here. 678 00:33:29,273 --> 00:33:31,190 Because everything else is symmetric, correct? 679 00:33:31,190 --> 00:33:36,020 The reflections are symmetric, and the quantity, n sine theta, 680 00:33:36,020 --> 00:33:38,880 is preserved. 681 00:33:38,880 --> 00:33:44,110 So therefore, if you put a light in this kind of arrangement, 682 00:33:44,110 --> 00:33:46,650 it would actually follow a periodic trajectory. 683 00:33:46,650 --> 00:33:51,480 The light will sort of be periodically reflected 684 00:33:51,480 --> 00:33:55,560 from these interfaces, and will follow a periodic trajectory 685 00:33:55,560 --> 00:33:58,560 down the stack of slabs. 686 00:33:58,560 --> 00:34:01,680 Therefore, this is also a kind of wave guide. 687 00:34:01,680 --> 00:34:05,040 It is commonly referred to as green, where green is not 688 00:34:05,040 --> 00:34:09,110 for the facial expression, but stands for gradient index. 689 00:34:09,110 --> 00:34:17,989 And there's a special case of a gradient index where they 690 00:34:17,989 --> 00:34:19,429 don't normally do it this way. 691 00:34:19,429 --> 00:34:22,230 The way they do it is with a continuous variation. 692 00:34:22,230 --> 00:34:23,929 And they manufacture it with diffusion. 693 00:34:23,929 --> 00:34:24,846 It's very interesting. 694 00:34:24,846 --> 00:34:29,120 They take a piece of glass, and they diffuse ions. 695 00:34:29,120 --> 00:34:31,340 Because the ions change the index of refraction 696 00:34:31,340 --> 00:34:33,440 of the glass, then they can sort of 697 00:34:33,440 --> 00:34:39,830 get a continuous profile of gradually variable index. 698 00:34:39,830 --> 00:34:42,800 And in the special case where this profile is quadratic, 699 00:34:42,800 --> 00:34:46,520 it turns out that the trajectory of the light paths 700 00:34:46,520 --> 00:34:47,570 is kind of like a helix. 701 00:34:47,570 --> 00:34:49,730 It become sinusoidal. 702 00:34:49,730 --> 00:34:53,810 And the light sort of bounces in sinusoidal fashion 703 00:34:53,810 --> 00:35:00,540 between the two interfaces, this one and this one. 704 00:35:00,540 --> 00:35:04,378 We will do this in more detail four lectures later. 705 00:35:04,378 --> 00:35:05,920 If you look at your syllabus, there's 706 00:35:05,920 --> 00:35:07,880 something called Hamiltonian optics. 707 00:35:07,880 --> 00:35:10,060 We will actually see in action how 708 00:35:10,060 --> 00:35:15,900 this sinusoidal periodic trajectory comes about. 709 00:35:15,900 --> 00:35:16,610 Yes. 710 00:35:16,610 --> 00:35:18,294 AUDIENCE: So what is the [INAUDIBLE]?? 711 00:35:20,917 --> 00:35:22,000 GEORGE BARBASTATHIS: Yeah. 712 00:35:22,000 --> 00:35:23,315 That's a very good question. 713 00:35:23,315 --> 00:35:24,940 PROFESSOR: Can you repeat the question? 714 00:35:24,940 --> 00:35:26,890 Because they didn't press the button here. 715 00:35:26,890 --> 00:35:28,390 Can you repeat the question, please? 716 00:35:28,390 --> 00:35:28,970 GEORGE BARBASTATHIS: I'm sorry. 717 00:35:28,970 --> 00:35:30,020 Could you repeat with the button? 718 00:35:30,020 --> 00:35:30,740 Yeah. 719 00:35:30,740 --> 00:35:31,390 AUDIENCE: Yeah. 720 00:35:31,390 --> 00:35:34,130 What is the advantage of using this type of wave guide 721 00:35:34,130 --> 00:35:37,367 compared to step index wave guide? 722 00:35:37,367 --> 00:35:38,950 GEORGE BARBASTATHIS: So the advantage, 723 00:35:38,950 --> 00:35:41,830 which I cannot describe yet because we have not done wave 724 00:35:41,830 --> 00:35:42,760 optics. 725 00:35:42,760 --> 00:35:47,630 But in wave guides, there's a phenomenon called dispersion. 726 00:35:47,630 --> 00:35:51,880 It is very similar to the dispersion from a prism 727 00:35:51,880 --> 00:35:53,770 that Piper showed before. 728 00:35:53,770 --> 00:35:57,260 But in telecommunications, when you transmit signal down 729 00:35:57,260 --> 00:36:01,270 a wave guide, it has the effect of basically lowering 730 00:36:01,270 --> 00:36:03,940 the speed, the effective speed at which you 731 00:36:03,940 --> 00:36:06,170 can transmit information. 732 00:36:06,170 --> 00:36:08,440 So it turns out that the step index wave 733 00:36:08,440 --> 00:36:12,410 guide has a higher dispersion than the gradient index wave 734 00:36:12,410 --> 00:36:13,060 guide. 735 00:36:13,060 --> 00:36:17,645 So you get a much higher speed in fibers of gradient index. 736 00:36:17,645 --> 00:36:19,270 So that's one reason why people use it. 737 00:36:23,242 --> 00:36:24,200 This will take a while. 738 00:36:38,820 --> 00:36:41,210 And, of course, more practical wave guides that 739 00:36:41,210 --> 00:36:42,460 are shaped like a wire. 740 00:36:42,460 --> 00:36:44,293 Literally, they are known as optical fibers. 741 00:36:46,590 --> 00:36:48,090 Again, you have two types of fibers. 742 00:36:48,090 --> 00:36:51,740 You have the step index fiber, where you have a higher index 743 00:36:51,740 --> 00:36:55,500 core, and the light is kind of bouncing back and forth 744 00:36:55,500 --> 00:36:57,870 between the core and the cladding. 745 00:36:57,870 --> 00:37:00,780 And there's also gradient index fibers, 746 00:37:00,780 --> 00:37:04,200 where the light is following a helical trajectory. 747 00:37:04,200 --> 00:37:07,470 The phenomenon of wave guiding, again, I 748 00:37:07,470 --> 00:37:09,120 have to defer to electromagnetics. 749 00:37:09,120 --> 00:37:12,210 It is much easier to describe with electromagnetics 750 00:37:12,210 --> 00:37:14,850 than it is with wave optics. 751 00:37:14,850 --> 00:37:19,440 But for now, we can get a sort of a preliminary description 752 00:37:19,440 --> 00:37:21,870 with the means that we have available to us. 753 00:37:25,670 --> 00:37:28,370 As a sort of a curiosity, it turns out 754 00:37:28,370 --> 00:37:31,780 that these gradient index wave guides. 755 00:37:31,780 --> 00:37:35,960 They appear in the nature in certain animals. 756 00:37:35,960 --> 00:37:38,030 I mean, insects, actually. 757 00:37:38,030 --> 00:37:41,750 Their eyes, they're composed of several wave guides, 758 00:37:41,750 --> 00:37:45,300 each one of which is actually a gradient index wave guide. 759 00:37:45,300 --> 00:37:48,500 And the way the animal eye works is it captures-- remember, 760 00:37:48,500 --> 00:37:51,590 the wave guide has a limited numerical aperture. 761 00:37:51,590 --> 00:37:55,740 So each one of these guides, each one of these small eyes, 762 00:37:55,740 --> 00:37:59,670 they're called ommatidia, these little wave guides. 763 00:37:59,670 --> 00:38:02,780 So each one of those captures a very narrow angle 764 00:38:02,780 --> 00:38:07,250 of light sort of within the field of view 765 00:38:07,250 --> 00:38:08,600 of the eye of the insect. 766 00:38:08,600 --> 00:38:13,280 And then sends a signal down to the optic nerve of the insect. 767 00:38:13,280 --> 00:38:15,440 So basically, the insect with this kind of eye, it 768 00:38:15,440 --> 00:38:17,720 forms a very bloody picture of the background, 769 00:38:17,720 --> 00:38:22,640 because it integrates a relatively large range 770 00:38:22,640 --> 00:38:23,390 of angles. 771 00:38:23,390 --> 00:38:25,970 But still, the range of angles is small enough 772 00:38:25,970 --> 00:38:30,332 that it allows it to quote unquote see. 773 00:38:30,332 --> 00:38:31,790 Now, insects, of course, they don't 774 00:38:31,790 --> 00:38:38,270 see the way we see, at least from our everyday experience. 775 00:38:38,270 --> 00:38:40,580 And the reason, of course, is that the insects 776 00:38:40,580 --> 00:38:43,670 have a very limited brain. 777 00:38:43,670 --> 00:38:46,550 A typical insect might have about 10,000 neurons 778 00:38:46,550 --> 00:38:48,300 in his brain. 779 00:38:48,300 --> 00:38:50,030 Show of hands, does anybody know how many 780 00:38:50,030 --> 00:38:51,238 neurons we have in our brain? 781 00:38:57,780 --> 00:38:59,770 10 to the 11. 782 00:38:59,770 --> 00:39:03,650 So we have about eight orders of magnitude more neurons. 783 00:39:03,650 --> 00:39:05,630 In case you are wondering, whether you 784 00:39:05,630 --> 00:39:08,420 are smart or educated, it does not 785 00:39:08,420 --> 00:39:10,880 have to do with a number of neurons that you have, 786 00:39:10,880 --> 00:39:13,565 but it has to do with the connections between neurons. 787 00:39:13,565 --> 00:39:15,320 Neurons are connected with wires. 788 00:39:15,320 --> 00:39:18,080 It turns out an educated person has 789 00:39:18,080 --> 00:39:20,480 approximately 10 times more connections 790 00:39:20,480 --> 00:39:22,490 than an uneducated person. 791 00:39:22,490 --> 00:39:26,060 The same number of neurons, but more connections. 792 00:39:26,060 --> 00:39:30,230 And so it's a sad fact of life that every day, adults-- 793 00:39:30,230 --> 00:39:36,060 that is, after age six or so, even at childhood-- 794 00:39:36,060 --> 00:39:37,280 would begin to lose neurons. 795 00:39:37,280 --> 00:39:38,870 In fact, each one of us every day 796 00:39:38,870 --> 00:39:41,102 will lose about 100,000 neurons. 797 00:39:41,102 --> 00:39:43,310 Nothing to worry about, because we have 10 to the 11. 798 00:39:43,310 --> 00:39:45,020 So even with all this loss, we can still 799 00:39:45,020 --> 00:39:48,590 survive until a fairly old age. 800 00:39:48,590 --> 00:39:50,090 But anyway, it is true. 801 00:39:50,090 --> 00:39:53,390 So [INAUDIBLE]. 802 00:39:53,390 --> 00:39:55,630 Anyway, the insect, on the other hand, 803 00:39:55,630 --> 00:39:57,860 has about 10,000 neurons altogether. 804 00:39:57,860 --> 00:40:00,025 So it has to make do with these 10,000 neurons. 805 00:40:00,025 --> 00:40:00,650 It has to move. 806 00:40:00,650 --> 00:40:01,490 It has to feed. 807 00:40:01,490 --> 00:40:04,400 It has to mate and all of these things. 808 00:40:04,400 --> 00:40:07,610 So the way they handle it is they get very simple vision, 809 00:40:07,610 --> 00:40:10,910 and they navigate according to differences in lighting. 810 00:40:10,910 --> 00:40:15,410 So the typical example is an insect flying toward a tree. 811 00:40:15,410 --> 00:40:18,320 Nature has evolved the insect to avoid this situation, 812 00:40:18,320 --> 00:40:21,470 because if it flies into the tree it will crash and die. 813 00:40:21,470 --> 00:40:26,040 But if it flies towards a tree, the insect sees a dark 814 00:40:26,040 --> 00:40:28,250 background-- the trunk of the tree-- 815 00:40:28,250 --> 00:40:30,740 surrounded by light, which is sort of leaking 816 00:40:30,740 --> 00:40:32,240 on the sides of the tree. 817 00:40:32,240 --> 00:40:35,780 As it flies by, it sees an edge of light 818 00:40:35,780 --> 00:40:38,720 that is very rapidly expanding, because it 819 00:40:38,720 --> 00:40:41,370 is approaching the tree. 820 00:40:41,370 --> 00:40:42,400 So the insect is wired. 821 00:40:42,400 --> 00:40:43,260 It as actually automatic. 822 00:40:43,260 --> 00:40:45,630 The insect doesn't think, oh my god, I'm going to crash. 823 00:40:45,630 --> 00:40:46,500 Let me turn. 824 00:40:46,500 --> 00:40:47,600 It's automatic. 825 00:40:47,600 --> 00:40:49,850 As soon as the neurons of the insect 826 00:40:49,850 --> 00:40:51,680 register a difference in lighting 827 00:40:51,680 --> 00:40:54,770 between successive ommatidia over here, 828 00:40:54,770 --> 00:40:56,800 they turn on the motor-- 829 00:40:56,800 --> 00:40:59,592 the flies or whatever, the legs and so on of the insect, 830 00:40:59,592 --> 00:41:00,800 the navigation of the insect. 831 00:41:00,800 --> 00:41:05,690 And the insect turns and avoids the obstacle. 832 00:41:05,690 --> 00:41:07,680 So this is what I say about insect. 833 00:41:07,680 --> 00:41:10,600 More precisely, I'm referring to the fruit fly, which 834 00:41:10,600 --> 00:41:14,180 has been very broad, very extensively studied 835 00:41:14,180 --> 00:41:15,660 in this context. 836 00:41:15,660 --> 00:41:18,770 But anyway, this is a very, very interesting story 837 00:41:18,770 --> 00:41:22,040 of how the insects use what we now 838 00:41:22,040 --> 00:41:23,600 consider as a rather sophisticated 839 00:41:23,600 --> 00:41:25,850 optical instrument, a green wave guide 840 00:41:25,850 --> 00:41:29,090 in order to generate a very simple type of vision 841 00:41:29,090 --> 00:41:30,500 based navigation. 842 00:41:35,033 --> 00:41:36,450 The next thing I was going to say, 843 00:41:36,450 --> 00:41:38,570 Piper already mentioned it. 844 00:41:38,570 --> 00:41:43,820 The index of refraction of most dielectric media 845 00:41:43,820 --> 00:41:47,360 turns out to be a strong fraction of the wavelength. 846 00:41:47,360 --> 00:41:49,520 So I stole from a book, actually, 847 00:41:49,520 --> 00:41:52,410 from the Soto website. 848 00:41:52,410 --> 00:41:54,530 Soto is a glass manufacturer. 849 00:41:54,530 --> 00:41:56,150 They make glasses that are used very 850 00:41:56,150 --> 00:42:00,210 commonly in optical instrument lenses and such, and prisms 851 00:42:00,210 --> 00:42:02,392 and so on. 852 00:42:02,392 --> 00:42:03,850 So they have this picture on online 853 00:42:03,850 --> 00:42:06,670 of the index of refraction as a function 854 00:42:06,670 --> 00:42:09,400 of wavelength for a relatively large range of wavelengths 855 00:42:09,400 --> 00:42:14,470 going along the wave from ultraviolet 856 00:42:14,470 --> 00:42:17,090 into the deep infrared over here. 857 00:42:17,090 --> 00:42:20,000 So you can see that the index varies quite a bit. 858 00:42:20,000 --> 00:42:22,750 Also, with the plot here, the absorption 859 00:42:22,750 --> 00:42:24,490 coefficient of the material. 860 00:42:24,490 --> 00:42:28,660 And you can see that they are kind of correlated in the sense 861 00:42:28,660 --> 00:42:32,430 that when the index does something interesting, 862 00:42:32,430 --> 00:42:36,518 the absorption also seems to do something interesting. 863 00:42:36,518 --> 00:42:37,560 It is not coincidentally. 864 00:42:37,560 --> 00:42:39,370 It turns out to have a very interesting 865 00:42:39,370 --> 00:42:40,990 theoretical foundation. 866 00:42:40,990 --> 00:42:42,030 I will go into it. 867 00:42:42,030 --> 00:42:44,140 Perhaps I will go into it later in the class. 868 00:42:47,600 --> 00:42:49,900 But anyway, the point I've been trying to make here 869 00:42:49,900 --> 00:42:51,483 is that you can see that the index can 870 00:42:51,483 --> 00:42:53,180 have quite a bit of variation. 871 00:42:53,180 --> 00:42:57,550 So because of that, if you send broadband light 872 00:42:57,550 --> 00:43:01,970 that contains multiple colors into an element, 873 00:43:01,970 --> 00:43:03,070 such as a prism. 874 00:43:03,070 --> 00:43:06,850 Then you can observe these phenomena that Piper showed. 875 00:43:06,850 --> 00:43:10,000 Different wavelengths, different colors, 876 00:43:10,000 --> 00:43:13,810 they experience different index, and therefore, the Snell's law 877 00:43:13,810 --> 00:43:15,650 applies differently to them. 878 00:43:15,650 --> 00:43:19,060 That is why you have this-- 879 00:43:19,060 --> 00:43:21,130 it's called analysis of white light. 880 00:43:21,130 --> 00:43:22,110 It becomes a rainbow. 881 00:43:25,480 --> 00:43:27,050 For most materials, it is true. 882 00:43:29,890 --> 00:43:34,570 Longer wavelengths actually have a lower index of refraction. 883 00:43:34,570 --> 00:43:37,060 Now let's see, does this make sense? 884 00:43:37,060 --> 00:43:39,160 If you look at this picture over here. 885 00:43:39,160 --> 00:43:41,150 Does it make sense what I said? 886 00:43:41,150 --> 00:43:45,870 Which wavelength apparently has the lower index, blue or red? 887 00:44:00,840 --> 00:44:02,760 It better be right, or I-- 888 00:44:02,760 --> 00:44:04,010 either I made the wrong slide. 889 00:44:04,010 --> 00:44:06,460 But anyway, I have taught this class for several years. 890 00:44:06,460 --> 00:44:08,190 So you would think that if I had made the wrong slide, 891 00:44:08,190 --> 00:44:09,810 I would have fixed it by now, right? 892 00:44:09,810 --> 00:44:11,100 So the slide is correct. 893 00:44:11,100 --> 00:44:12,600 The way to figure it out is you have 894 00:44:12,600 --> 00:44:17,110 to imagine a normal to the surface over here. 895 00:44:17,110 --> 00:44:24,350 So which wavelength appears to have the stronger refraction? 896 00:44:24,350 --> 00:44:26,820 Blue or red? 897 00:44:26,820 --> 00:44:28,950 Blue, right? 898 00:44:28,950 --> 00:44:32,470 So the blue wavelength suffers a strong refraction. 899 00:44:32,470 --> 00:44:35,860 That is, the blue wavelength has what index? 900 00:44:35,860 --> 00:44:39,360 Higher or lower? 901 00:44:39,360 --> 00:44:41,420 Higher. 902 00:44:41,420 --> 00:44:44,600 And indeed, the blue wavelength is 903 00:44:44,600 --> 00:44:46,230 softer than the red wavelength. 904 00:44:46,230 --> 00:44:49,010 So this is consistent with the curve that you see here. 905 00:44:49,010 --> 00:44:51,500 The blue wavelength is probably somewhere around here. 906 00:44:51,500 --> 00:44:53,930 The red wavelength is somewhere around here. 907 00:44:53,930 --> 00:44:57,600 It is not a dramatic variation in the visible range. 908 00:44:57,600 --> 00:44:59,620 And that is typical for most glasses. 909 00:44:59,620 --> 00:45:03,455 In the visible range, they have a relatively slow variation 910 00:45:03,455 --> 00:45:04,580 of the index of refraction. 911 00:45:04,580 --> 00:45:06,080 But nevertheless, it is there. 912 00:45:06,080 --> 00:45:08,270 And you saw evidence of it in the experiment 913 00:45:08,270 --> 00:45:12,027 that Piper just did with the prism. 914 00:45:12,027 --> 00:45:13,610 And the last thing that I want to say. 915 00:45:13,610 --> 00:45:15,980 I don't want to belabor this point. 916 00:45:15,980 --> 00:45:22,430 People use various quantities to characterize dispersion. 917 00:45:22,430 --> 00:45:26,810 And typically, they characterize them with respect 918 00:45:26,810 --> 00:45:29,390 to the various emission lights-- 919 00:45:29,390 --> 00:45:34,280 emission lines, from atomic spectra. 920 00:45:34,280 --> 00:45:36,400 So they use this as reference [INAUDIBLE].. 921 00:45:36,400 --> 00:45:37,880 The reason, I suppose. 922 00:45:37,880 --> 00:45:40,070 The reason is that back when people 923 00:45:40,070 --> 00:45:43,610 developed these measures, lasers were not available. 924 00:45:43,610 --> 00:45:47,180 So the best way to define wavelength standards 925 00:45:47,180 --> 00:45:50,930 was with emission lines. 926 00:45:50,930 --> 00:45:53,960 So they use typically the hydrogen C line and F 927 00:45:53,960 --> 00:45:56,690 line, and the sodium D line. 928 00:45:56,690 --> 00:45:58,550 And then they define these quantities, 929 00:45:58,550 --> 00:46:03,080 the dispersive power and the dispersive index, 930 00:46:03,080 --> 00:46:06,320 which are defined according to the index at these three 931 00:46:06,320 --> 00:46:09,230 different wavelengths. 932 00:46:09,230 --> 00:46:13,250 So this is very useful for people who do optical design. 933 00:46:13,250 --> 00:46:19,520 And it gives you sort of an idea of how dispersive is a glass. 934 00:46:19,520 --> 00:46:25,070 These quantities are actually inverse relative to each other. 935 00:46:25,070 --> 00:46:26,720 And this an example. 936 00:46:26,720 --> 00:46:31,320 For ground glass, typically, you want 937 00:46:31,320 --> 00:46:34,580 the V number, the dispersive power, to be low, 938 00:46:34,580 --> 00:46:38,830 if you want a dispersion free element. 939 00:46:41,780 --> 00:46:44,730 OK, any questions? 940 00:47:05,590 --> 00:47:09,640 OK, so I'm not going to go over the second lecture. 941 00:47:09,640 --> 00:47:12,780 We will postpone it for Wednesday. 942 00:47:12,780 --> 00:47:15,720 Basically, we have slid back by about an hour, but that's OK. 943 00:47:15,720 --> 00:47:18,720 We'll catch up later. 944 00:47:18,720 --> 00:47:21,840 But what I'll do is I would like to get you started thinking 945 00:47:21,840 --> 00:47:25,820 about next Wednesday's lecture. 946 00:47:25,820 --> 00:47:27,650 So next Wednesday, we'll basically 947 00:47:27,650 --> 00:47:32,300 see a bunch of applications of Fermat's principle. 948 00:47:32,300 --> 00:47:35,840 Namely, the principle that says that light 949 00:47:35,840 --> 00:47:39,320 chooses its trajectory trying to minimize the optical path 950 00:47:39,320 --> 00:47:40,640 length. 951 00:47:40,640 --> 00:47:42,260 So we saw already two applications, 952 00:47:42,260 --> 00:47:44,480 one in the law of reflection, and the other 953 00:47:44,480 --> 00:47:47,170 in the law of refraction. 954 00:47:47,170 --> 00:47:49,350 So the next applications will be in focusing. 955 00:47:49,350 --> 00:47:51,800 So the question we'll ask the next Wednesday 956 00:47:51,800 --> 00:47:57,920 is how can we design a surface, or reflect a surface such 957 00:47:57,920 --> 00:48:02,060 that if light is arriving from infinity in parallel rays 958 00:48:02,060 --> 00:48:05,320 like this, this surface upon reflection 959 00:48:05,320 --> 00:48:06,880 focuses all the rays. 960 00:48:06,880 --> 00:48:10,288 So they pass from the same common focal point F. 961 00:48:10,288 --> 00:48:11,830 So you can look it up into the notes, 962 00:48:11,830 --> 00:48:14,860 and then I will go over it again on Wednesday, 963 00:48:14,860 --> 00:48:18,750 how we can use the Fermat's principle in order to design. 964 00:48:18,750 --> 00:48:19,750 You can actually design. 965 00:48:19,750 --> 00:48:22,070 We can come up with an analytic expression that 966 00:48:22,070 --> 00:48:24,760 has to be a parabola for the sacrifice that gives 967 00:48:24,760 --> 00:48:26,890 the perfect focus onto a point. 968 00:48:29,490 --> 00:48:31,610 So the homework has been posted. 969 00:48:31,610 --> 00:48:33,870 The first three problems you can do 970 00:48:33,870 --> 00:48:37,380 without a need for any of this. 971 00:48:37,380 --> 00:48:39,130 Actually, I think the first four problems. 972 00:48:39,130 --> 00:48:41,860 You don't need any of this. 973 00:48:41,860 --> 00:48:45,610 The problems are not due until actually the next Wednesday. 974 00:48:45,610 --> 00:48:48,910 Not this Wednesday, but Wednesday the 18th, 975 00:48:48,910 --> 00:48:50,290 nine days from today. 976 00:48:50,290 --> 00:48:54,660 So you're in good shape with regards to the homework.