WEBVTT 00:00:00.090 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.820 Commons license. 00:00:03.820 --> 00:00:06.030 Your support will help MIT OpenCourseWare 00:00:06.030 --> 00:00:10.120 continue to offer high quality educational resources for free. 00:00:10.120 --> 00:00:12.690 To make a donation, or to view additional materials 00:00:12.690 --> 00:00:16.650 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.650 --> 00:00:17.550 at ocw.mit.edu. 00:00:21.060 --> 00:00:22.560 GEORGE BARBASTATHIS: So does anybody 00:00:22.560 --> 00:00:26.040 have questions from the last lecture? 00:00:37.870 --> 00:00:41.807 After some time to wake up, anybody still have questions? 00:00:45.690 --> 00:00:48.120 So I will start, and if you remember of a question 00:00:48.120 --> 00:00:49.500 that you had, please interrupt. 00:00:49.500 --> 00:00:51.250 I think if you push the button the button, 00:00:51.250 --> 00:00:54.160 I will hear a sound over here. 00:00:54.160 --> 00:00:57.680 So we will know that there's a question. 00:00:57.680 --> 00:00:59.140 So I'd like to pick up the thread 00:00:59.140 --> 00:01:01.970 from where I left last time. 00:01:01.970 --> 00:01:04.819 We covered fairly quickly because we ran out of time. 00:01:04.819 --> 00:01:07.180 But we covered that the law of reflection 00:01:07.180 --> 00:01:09.580 and the law of refraction. 00:01:09.580 --> 00:01:13.330 So I'd like to go back to the low of reflection, 00:01:13.330 --> 00:01:18.600 and remind you that it is a very simple, very simple result. 00:01:18.600 --> 00:01:22.040 The minimum path requirement for the light rays 00:01:22.040 --> 00:01:24.490 force those reflections to be symmetric. 00:01:24.490 --> 00:01:26.080 So this has a strange consequence 00:01:26.080 --> 00:01:31.300 that we're unfamiliar with from when we look in the mirror 00:01:31.300 --> 00:01:32.590 daily. 00:01:32.590 --> 00:01:38.680 And that is the fact that our left and right locations 00:01:38.680 --> 00:01:39.710 in our body. 00:01:39.710 --> 00:01:41.538 They flip when we look at the mirror. 00:01:41.538 --> 00:01:43.330 So we'd like to make that a little bit more 00:01:43.330 --> 00:01:46.190 quantitative by looking at this diagram. 00:01:46.190 --> 00:01:51.080 So to understand this, suppose that I am-- 00:01:51.080 --> 00:01:54.760 I have an object that is oriented. 00:01:54.760 --> 00:01:58.870 You can think of it as perhaps two pencils that 00:01:58.870 --> 00:02:04.300 are sort of following the ray paths 00:02:04.300 --> 00:02:06.190 and then they get reflected from the mirror. 00:02:06.190 --> 00:02:08.860 So, for example, if you look at the central ray, 00:02:08.860 --> 00:02:10.990 it will be reflected symmetrically. 00:02:10.990 --> 00:02:14.070 So, again, what you see here is the front view of the mirror. 00:02:14.070 --> 00:02:17.770 And then I will draw a few ray a paths in perspective. 00:02:17.770 --> 00:02:25.360 So this is the ray paths that start from the object. 00:02:25.360 --> 00:02:27.110 I don't know if you can see me over there, 00:02:27.110 --> 00:02:31.190 but I'm trying to show what will happen with an actual pencil. 00:02:31.190 --> 00:02:34.410 So if this was a pencil that is coming towards the mirror, 00:02:34.410 --> 00:02:36.110 and this is the surface of the mirror. 00:02:36.110 --> 00:02:38.420 The pencil will go like this. 00:02:38.420 --> 00:02:40.760 And then law of reflection says that it will also 00:02:40.760 --> 00:02:44.120 come out like this again. 00:02:44.120 --> 00:02:48.730 This follows very easily if you simply trace the rays. 00:02:48.730 --> 00:02:52.190 In the next step in my animation here, it 00:02:52.190 --> 00:02:59.960 paints the sort of positions of the pencils and their tops 00:02:59.960 --> 00:03:03.860 as they get reflected from the mirror. 00:03:03.860 --> 00:03:05.570 So what happened here is the following. 00:03:05.570 --> 00:03:07.550 As you can see, the pencil did not actually 00:03:07.550 --> 00:03:08.670 change orientation. 00:03:08.670 --> 00:03:10.520 This is a strange thing about mirrors. 00:03:10.520 --> 00:03:12.810 But what did change is the following. 00:03:12.810 --> 00:03:14.720 Imagine that you are walking together 00:03:14.720 --> 00:03:17.770 with this pair of pencils. 00:03:17.770 --> 00:03:19.550 As you are walking this way, you will 00:03:19.550 --> 00:03:26.060 see that the one that is horizontally oriented. 00:03:26.060 --> 00:03:28.730 The top of it is pointing to your right. 00:03:28.730 --> 00:03:31.400 But if you walk all the way to the mirror. 00:03:31.400 --> 00:03:33.860 Go to the center of the ray, and start going 00:03:33.860 --> 00:03:35.630 backwards the other way around. 00:03:35.630 --> 00:03:37.650 Then all of a sudden, the top of the pencil 00:03:37.650 --> 00:03:40.160 appeared on your left. 00:03:40.160 --> 00:03:45.380 This is a simple consequence of the law of reflection. 00:03:45.380 --> 00:03:48.380 The way we interpret it in everyday life when 00:03:48.380 --> 00:03:50.528 we see through a mirror is the following. 00:03:50.528 --> 00:03:52.070 Of course, in everyday life, we don't 00:03:52.070 --> 00:03:55.250 talk about left handed and right handed triads. 00:03:55.250 --> 00:03:58.670 But the way we interpret it is because normally when 00:03:58.670 --> 00:04:00.557 we look at a good quality mirror, 00:04:00.557 --> 00:04:02.390 we do not know that there is a mirror there. 00:04:02.390 --> 00:04:06.480 We actually see a continuation of that are being reflected 00:04:06.480 --> 00:04:09.830 behind, back behind the mirror. 00:04:09.830 --> 00:04:13.100 So what you're seeing then if we look behind the mirror 00:04:13.100 --> 00:04:19.300 is actually this triad, with the pencils oriented as shown here. 00:04:19.300 --> 00:04:23.510 And if we interpret it as being seen from behind, 00:04:23.510 --> 00:04:27.470 then, of course, left has become right, and vice versa. 00:04:27.470 --> 00:04:28.942 So another way to think about it is 00:04:28.942 --> 00:04:30.400 that when we look through a mirror, 00:04:30.400 --> 00:04:35.080 it is as if we're looking from the image from the back. 00:04:35.080 --> 00:04:36.580 So a better way to think about it 00:04:36.580 --> 00:04:39.700 is if you're looking at something 00:04:39.700 --> 00:04:42.070 that is written on my t-shirt. 00:04:42.070 --> 00:04:43.600 If you look at it from the mirror, 00:04:43.600 --> 00:04:45.850 it would appear as if I were hollow, 00:04:45.850 --> 00:04:48.580 and you would see the back of the writing on my t-shirt. 00:04:48.580 --> 00:04:52.060 And you all know that if you look at the ambulance 00:04:52.060 --> 00:04:55.090 sign in ambulance trucks, ambulance 00:04:55.090 --> 00:04:57.250 whatever you call them, cars. 00:04:57.250 --> 00:05:00.040 Because it is meant to be visible through the mirror 00:05:00.040 --> 00:05:03.880 of a driver, they actually write the sign backwards. 00:05:03.880 --> 00:05:07.540 So when you look at it, it is as if you saw a transparency of it 00:05:07.540 --> 00:05:09.700 from the back. 00:05:09.700 --> 00:05:13.440 So this is a simple consequence of the law of reflection. 00:05:13.440 --> 00:05:15.430 This is a very simple silly question 00:05:15.430 --> 00:05:17.630 that sometimes is asked. 00:05:17.630 --> 00:05:22.200 And it actually takes quite a long explanation to answer. 00:05:22.200 --> 00:05:24.410 And that question goes like this. 00:05:24.410 --> 00:05:26.450 If you look at the mirror, yes, we 00:05:26.450 --> 00:05:30.280 know that left and right flip, but up and down do not 00:05:30.280 --> 00:05:30.790 flip, right? 00:05:30.790 --> 00:05:33.790 You don't see yourself upside down in a flat mirror. 00:05:33.790 --> 00:05:36.388 You see yourself flipped from left to right. 00:05:36.388 --> 00:05:37.180 What is the reason? 00:05:37.180 --> 00:05:38.590 The reason is shown here. 00:05:38.590 --> 00:05:46.190 The reason is because of the flipping 00:05:46.190 --> 00:05:51.140 of the relationship between the left and right in the object 00:05:51.140 --> 00:05:53.480 side, as you interpret the projection 00:05:53.480 --> 00:05:56.090 of the rays that are coming from the opposite side 00:05:56.090 --> 00:05:56.980 of the mirror. 00:05:56.980 --> 00:05:58.730 So this requires a little bit of thinking, 00:05:58.730 --> 00:06:01.940 so I'll let you think about it. 00:06:01.940 --> 00:06:04.967 Unless you have a question now, please ask it. 00:06:04.967 --> 00:06:06.800 If not, you can think about it and come back 00:06:06.800 --> 00:06:08.810 with more questions on Wednesday. 00:06:08.810 --> 00:06:11.020 Is there any immediate question about this? 00:06:11.020 --> 00:06:13.040 That's kind of a subtle and elegant point. 00:06:17.285 --> 00:06:18.910 I should have brought a mirror with me. 00:06:18.910 --> 00:06:20.535 Piper, do you have a mirror over there? 00:06:25.320 --> 00:06:27.260 AUDIENCE: No, I don't have a mirror here. 00:06:27.260 --> 00:06:28.520 GEORGE BARBASTATHIS: No mirrors, OK. 00:06:28.520 --> 00:06:30.320 But everybody has access to mirrors, right? 00:06:30.320 --> 00:06:34.841 It's the one optical element that we can find very easily. 00:06:34.841 --> 00:06:38.020 So I'll let you practice with your mirror in your bathroom, 00:06:38.020 --> 00:06:40.475 and then come back and ask me questions. 00:06:43.270 --> 00:06:44.620 OK. 00:06:44.620 --> 00:06:47.610 AUDIENCE: I'm lying down. 00:06:47.610 --> 00:06:48.640 You're horizontal. 00:06:51.780 --> 00:06:54.430 GEORGE BARBASTATHIS: That's right. 00:06:54.430 --> 00:06:55.640 Our producer like this. 00:07:01.830 --> 00:07:03.680 The other thing I wanted to talk about 00:07:03.680 --> 00:07:06.980 is expand a little bit on the law of reflection. 00:07:06.980 --> 00:07:09.200 Oh, I'm sorry, the law of refraction 00:07:09.200 --> 00:07:11.210 that we derived last time. 00:07:18.900 --> 00:07:21.870 So the law of refraction is the last equation 00:07:21.870 --> 00:07:23.320 shown on this line. 00:07:23.320 --> 00:07:26.170 It says that this quantity, the index of refraction 00:07:26.170 --> 00:07:32.020 multiplied by the sign of the angle of incidence, where 00:07:32.020 --> 00:07:34.750 the angle of incidence is defined with respect 00:07:34.750 --> 00:07:40.000 to the normal to the surface. 00:07:40.000 --> 00:07:41.210 It is preserved. 00:07:41.210 --> 00:07:44.200 So as you go through multiple surfaces, the law of refraction 00:07:44.200 --> 00:07:47.630 says that this quantity must be preserved. 00:07:47.630 --> 00:07:49.900 So there is a common problem that I 00:07:49.900 --> 00:07:52.300 have posted just a couple of hours ago 00:07:52.300 --> 00:07:57.730 that asks you to make an analogy between the law of refraction, 00:07:57.730 --> 00:08:00.550 and the problem of a lifeguard who has 00:08:00.550 --> 00:08:02.860 to save a person in the water. 00:08:02.860 --> 00:08:05.350 So recall the reason the law of refraction happens 00:08:05.350 --> 00:08:10.480 is because the light must minimize its path between two 00:08:10.480 --> 00:08:13.660 points, P and P prime. 00:08:13.660 --> 00:08:15.520 So a very similar problem is if you 00:08:15.520 --> 00:08:19.630 have a trajectory that you're trying to design in a way 00:08:19.630 --> 00:08:21.910 that you minimize the time that you will 00:08:21.910 --> 00:08:24.130 spend going on this trajectory. 00:08:24.130 --> 00:08:27.730 So here, you have a swimmer who is sitting on the beach. 00:08:30.890 --> 00:08:32.320 I'm sorry, not a swimmer. 00:08:32.320 --> 00:08:34.600 You have a lifeguard sitting on the beach. 00:08:34.600 --> 00:08:37.549 And the lifeguard sees a person drowning in water 00:08:37.549 --> 00:08:39.799 farther behind. 00:08:39.799 --> 00:08:44.084 The lifeguard can run on the beach at some velocity, v sub 00:08:44.084 --> 00:08:44.960 r. 00:08:44.960 --> 00:08:48.710 They can also swim in water at some velocity v sub s. 00:08:48.710 --> 00:08:51.100 Most people swim slower than they run, 00:08:51.100 --> 00:08:56.960 so we can assume here that the running speed is faster 00:08:56.960 --> 00:08:58.110 than the swimming speed. 00:08:58.110 --> 00:08:59.860 And the question is how should the swimmer 00:08:59.860 --> 00:09:04.820 plan his path so that he can reach the drowning person as 00:09:04.820 --> 00:09:05.690 fast as possible? 00:09:08.650 --> 00:09:11.420 Again, you can think that, for example, the straight path 00:09:11.420 --> 00:09:15.270 is not the best because he's spending 00:09:15.270 --> 00:09:17.130 too much time in water. 00:09:17.130 --> 00:09:18.780 I mean, water, he's slower. 00:09:18.780 --> 00:09:21.660 So he may want to spend a little bit extra time 00:09:21.660 --> 00:09:23.850 in the fast middle, and a little bit 00:09:23.850 --> 00:09:25.320 less time in the slow medium. 00:09:25.320 --> 00:09:26.790 But, again, he cannot overdo it. 00:09:26.790 --> 00:09:29.160 If he goes a really crazy path, then, again, he 00:09:29.160 --> 00:09:30.840 will end up with a longer time. 00:09:30.840 --> 00:09:33.180 So light is trying to do a similar thing. 00:09:33.180 --> 00:09:35.910 It is time to optimize the obstacle path 00:09:35.910 --> 00:09:38.370 length, or equivalently, the time 00:09:38.370 --> 00:09:41.850 that it takes for the light ray to reach from a starting point 00:09:41.850 --> 00:09:42.990 to an ending point. 00:09:42.990 --> 00:09:46.020 And, again, remember this is almost an exact analogy here. 00:09:46.020 --> 00:09:49.500 The speed of light is faster in air, 00:09:49.500 --> 00:09:51.550 and slower in a dielectric medium. 00:09:51.550 --> 00:09:53.910 So if you had the air and glass here, 00:09:53.910 --> 00:09:56.160 that would be a very similar situation. 00:09:59.120 --> 00:10:00.780 I'm going to skip the next line. 00:10:03.530 --> 00:10:06.135 And I'm going to go to this one, to number 34. 00:10:10.170 --> 00:10:13.950 So what I'm trying to say here is to point out two cases. 00:10:13.950 --> 00:10:15.990 They're not really different. 00:10:15.990 --> 00:10:19.260 They're just two cases of the same situation. 00:10:19.260 --> 00:10:23.160 In one, you are going from a medium of lower index 00:10:23.160 --> 00:10:25.560 to a medium of hire index. 00:10:25.560 --> 00:10:27.960 And in this case, obviously, because 00:10:27.960 --> 00:10:30.930 of the law of refraction, the angle of refraction 00:10:30.930 --> 00:10:33.630 will increase as you go from left to right, 00:10:33.630 --> 00:10:35.940 from low index to high index. 00:10:35.940 --> 00:10:38.550 The opposite will happen if you go from high index 00:10:38.550 --> 00:10:40.650 to low index. 00:10:40.650 --> 00:10:43.380 OK, so this has two consequences, 00:10:43.380 --> 00:10:46.240 which you can think of as you reach the extremes. 00:10:46.240 --> 00:10:48.510 If you come in at the maximum possible angle 00:10:48.510 --> 00:10:50.940 here, 90 degrees, then you can imagine 00:10:50.940 --> 00:10:53.310 that you will not enter at 90, but you will enter 00:10:53.310 --> 00:10:55.110 at a slightly smaller angle. 00:10:55.110 --> 00:10:58.860 So basically, if you are coupling in light 00:10:58.860 --> 00:11:03.780 from air to glass, or in general from a medium of lower 00:11:03.780 --> 00:11:05.430 index to a medium of higher index, 00:11:05.430 --> 00:11:08.070 you have a limited column of approach 00:11:08.070 --> 00:11:11.180 that you can couple light into. 00:11:11.180 --> 00:11:13.320 And that is given by this equation over here. 00:11:13.320 --> 00:11:16.920 When the exterior angle theta reaches 90 degrees, 00:11:16.920 --> 00:11:19.650 then this is the maximum angle, theta prime, 00:11:19.650 --> 00:11:24.510 that you can access inside the high index medium. 00:11:24.510 --> 00:11:27.080 The opposite is perhaps slightly more interesting, 00:11:27.080 --> 00:11:31.950 is what happens when you reach or exceed theta 00:11:31.950 --> 00:11:34.863 prime equals to 90 degrees over here. 00:11:34.863 --> 00:11:36.780 So if you look again at the law of refraction, 00:11:36.780 --> 00:11:40.260 you can realize that it is possible to arrange 00:11:40.260 --> 00:11:43.160 for a combination of n and theta, 00:11:43.160 --> 00:11:47.370 such that the product is bigger than the index of refraction 00:11:47.370 --> 00:11:48.840 at the medium side. 00:11:48.840 --> 00:11:52.140 In order, now, to satisfy the law of refraction, 00:11:52.140 --> 00:11:54.870 you would have to require that the sine of an angle 00:11:54.870 --> 00:11:56.130 is bigger than 1. 00:11:56.130 --> 00:11:58.920 So since we are limited to deal with real angles here, 00:11:58.920 --> 00:12:01.350 not complex, this cannot happen. 00:12:01.350 --> 00:12:04.080 What really happens there is that the light will actually 00:12:04.080 --> 00:12:05.130 be reflected. 00:12:05.130 --> 00:12:08.880 If you satisfy this condition, this product 00:12:08.880 --> 00:12:11.460 becomes bigger than the index of refraction 00:12:11.460 --> 00:12:17.120 outside in the medium. 00:12:17.120 --> 00:12:18.740 Then when you satisfy this condition, 00:12:18.740 --> 00:12:23.760 then light will be reflected inside the high index medium. 00:12:23.760 --> 00:12:27.980 And that is known as total internal reflection, or TIR. 00:12:27.980 --> 00:12:32.100 So let's look at TIR in slightly more detail over here. 00:12:32.100 --> 00:12:33.880 So I have a glass medium in there, 00:12:33.880 --> 00:12:35.870 and imagine that I have a wavefront that 00:12:35.870 --> 00:12:41.060 is arriving from glass towards the air interface. 00:12:41.060 --> 00:12:44.920 So there is a combination of index and angle, 00:12:44.920 --> 00:12:47.800 where the product of the index inside the glass 00:12:47.800 --> 00:12:50.860 times the sine of the angle equals exactly one. 00:12:50.860 --> 00:12:54.160 What happens then is the law of refraction 00:12:54.160 --> 00:12:55.870 does not break down yet. 00:12:55.870 --> 00:12:58.840 But what will happen is the light will be refracted, 00:12:58.840 --> 00:13:01.870 and it will propagate exactly parallel to the interface. 00:13:01.870 --> 00:13:04.460 This is known as a surface wave. 00:13:04.460 --> 00:13:08.180 If you increase the angle now, then the product 00:13:08.180 --> 00:13:09.530 will become bigger than one. 00:13:09.530 --> 00:13:12.260 The law of refraction cannot be satisfied anymore. 00:13:12.260 --> 00:13:13.450 So what will happen then. 00:13:13.450 --> 00:13:16.040 Oh, and I should have said that the angle where this happens 00:13:16.040 --> 00:13:18.080 is called the critical angle. 00:13:18.080 --> 00:13:21.500 Because it is the angle just below which 00:13:21.500 --> 00:13:24.080 I still have refraction. 00:13:24.080 --> 00:13:26.780 If exceed this angle, then I get this phenomena 00:13:26.780 --> 00:13:33.140 of total internal reflection, where all of the light 00:13:33.140 --> 00:13:36.950 is reflected inside the glass, inside the high index medium. 00:13:36.950 --> 00:13:39.740 So it is almost as if the interface here 00:13:39.740 --> 00:13:42.680 abruptly changes, and instead of being 00:13:42.680 --> 00:13:46.040 mostly transmissible over here, it becomes mostly reflective. 00:13:46.040 --> 00:13:49.390 So it starts acting like a mirror. 00:13:49.390 --> 00:13:51.760 There's one difference that makes it slightly 00:13:51.760 --> 00:13:53.710 different than a mirror. 00:13:53.710 --> 00:13:56.770 And the difference is that if you 00:13:56.770 --> 00:14:01.380 were to calculate the electric field on the opposite side 00:14:01.380 --> 00:14:03.510 of the interface, that is, inside the medium 00:14:03.510 --> 00:14:05.940 where light does not propagate. 00:14:05.940 --> 00:14:08.650 You will discover that there is some leakage. 00:14:08.650 --> 00:14:13.710 The electric field has non-zero values in the low index 00:14:13.710 --> 00:14:15.450 medium over here. 00:14:15.450 --> 00:14:18.170 Even though the electric field that you find 00:14:18.170 --> 00:14:20.940 is not propagating, it is what is called evanescent. 00:14:20.940 --> 00:14:25.350 It is in exponential decay, but there is no wavefront. 00:14:25.350 --> 00:14:28.370 There is no wavefront of light propagating 00:14:28.370 --> 00:14:31.133 in the vertical direction like this. 00:14:31.133 --> 00:14:32.550 This is called an evanescent wave, 00:14:32.550 --> 00:14:35.130 and we will revisit it later when 00:14:35.130 --> 00:14:36.870 we deal with electromagnetics. 00:14:36.870 --> 00:14:39.120 Because right now, the way I defined 00:14:39.120 --> 00:14:44.780 it is not perhaps very rigorous, or very quantitative. 00:14:44.780 --> 00:14:47.590 But I wanted to give you a sort of a heads 00:14:47.590 --> 00:14:51.970 up that something slightly more than geometrical optics 00:14:51.970 --> 00:14:54.220 prediction happens here. 00:14:54.220 --> 00:14:56.140 But as far as geometrical optics goes, 00:14:56.140 --> 00:14:58.820 that we will be dealing for the next few lectures, 00:14:58.820 --> 00:15:00.242 there is a reflection. 00:15:05.380 --> 00:15:08.320 This says what I just mentioned, that we 00:15:08.320 --> 00:15:10.740 will talk more about these evanescent waves 00:15:10.740 --> 00:15:12.140 a little bit later. 00:15:12.140 --> 00:15:15.030 One more thing that I want to say about evanescent waves. 00:15:17.895 --> 00:15:20.700 One way you can sort of realize the existence 00:15:20.700 --> 00:15:24.280 of evanescent waves is with a sort 00:15:24.280 --> 00:15:27.210 a related phenomenon called frustrated total internal 00:15:27.210 --> 00:15:31.140 reflection, also known as FTIR, because that's quite a mouthful 00:15:31.140 --> 00:15:32.370 to pronounce. 00:15:32.370 --> 00:15:35.070 So FTIR happens if you have this situation, 00:15:35.070 --> 00:15:37.280 where you're beyond that critical angle, 00:15:37.280 --> 00:15:39.690 and therefore, your total internal reflecting 00:15:39.690 --> 00:15:41.160 into the medium. 00:15:41.160 --> 00:15:43.770 But you bring near the interface, 00:15:43.770 --> 00:15:45.630 you're bringing another piece of glass, 00:15:45.630 --> 00:15:47.790 another piece of high index medium 00:15:47.790 --> 00:15:53.010 in a way that an appreciable amount of the evanescent wave 00:15:53.010 --> 00:15:57.780 is allowed to enter inside the high index medium. 00:15:57.780 --> 00:16:02.190 If that happens, as we've said, the TIR is frustrated. 00:16:02.190 --> 00:16:05.730 What it really means is that the TIR stops happening now. 00:16:05.730 --> 00:16:07.507 What would happen is a small amount 00:16:07.507 --> 00:16:09.340 of light will still be reflected, of course. 00:16:09.340 --> 00:16:10.920 You cannot avoid that. 00:16:10.920 --> 00:16:12.660 But the significant portion of the light 00:16:12.660 --> 00:16:17.640 will couple out into the next material, 00:16:17.640 --> 00:16:20.662 and it will actually be transmitted. 00:16:20.662 --> 00:16:22.370 So this is a very interesting phenomenon, 00:16:22.370 --> 00:16:24.162 because if you think about it, the light is 00:16:24.162 --> 00:16:27.160 forbidden to enter this region. 00:16:27.160 --> 00:16:30.620 Snell's law says that light cannot cross into air, 00:16:30.620 --> 00:16:33.550 yet because of the proximity, the light can actually couple 00:16:33.550 --> 00:16:34.170 out. 00:16:34.170 --> 00:16:37.460 And again, we will see a much more rigorous explanation 00:16:37.460 --> 00:16:40.340 and quantitative description of this phenomenon 00:16:40.340 --> 00:16:42.440 later, when we're doing it through magnetics. 00:16:42.440 --> 00:16:46.010 Some of you who may have taken electronics or quantum 00:16:46.010 --> 00:16:49.190 mechanics, there's a similar effect 00:16:49.190 --> 00:16:56.300 called tunneling in potential barriers in quantum mechanics. 00:16:56.300 --> 00:16:59.030 The equations are very similar that describe this phenomenon. 00:16:59.030 --> 00:17:01.530 In both cases, you have a wave that 00:17:01.530 --> 00:17:03.650 is crossing a forbidden region in order 00:17:03.650 --> 00:17:06.710 to pass into an allowable region again. 00:17:09.842 --> 00:17:11.300 This may be actually be a good find 00:17:11.300 --> 00:17:19.819 for Piper to solve a demo of the total internal reflection. 00:17:19.819 --> 00:17:24.277 And Piper will actually solve it in the context of a prism. 00:17:24.277 --> 00:17:25.819 Piper, maybe you can start setting up 00:17:25.819 --> 00:17:28.010 while I give a brief description of prisms. 00:17:32.420 --> 00:17:34.840 You all are familiar, I suppose. 00:17:34.840 --> 00:17:38.220 They're pieces of glass that are cut into various triangular 00:17:38.220 --> 00:17:40.940 and other polygonal shapes. 00:17:40.940 --> 00:17:43.770 And typically, prisms are-- 00:17:43.770 --> 00:17:46.610 they're arranged either so that the light passes through, 00:17:46.610 --> 00:17:48.380 as shown in the top diagram. 00:17:48.380 --> 00:17:51.440 Or if you bring the light from the bottom, 00:17:51.440 --> 00:17:54.460 and you manage to exceed the critical angle 00:17:54.460 --> 00:17:58.140 in the interface, then you can also total internally 00:17:58.140 --> 00:18:00.920 reflect the light, and create a situation like this that 00:18:00.920 --> 00:18:03.470 is known as a retro-reflector. 00:18:03.470 --> 00:18:07.400 Now, a rule of thumb that is useful to know for glass, which 00:18:07.400 --> 00:18:10.240 has index of refraction 1.5. 00:18:10.240 --> 00:18:14.130 The critical angle is about 42 degrees. 00:18:14.130 --> 00:18:16.430 So if you are incident at 45, as shown 00:18:16.430 --> 00:18:21.717 in this case of what is called isosceles triangle, then-- 00:18:21.717 --> 00:18:22.550 actually, I'm sorry. 00:18:22.550 --> 00:18:24.600 This is equilateral, isn't it? 00:18:24.600 --> 00:18:28.190 If you're incident at 45 in an equilateral triangle, 00:18:28.190 --> 00:18:30.800 then you will satisfy the TIR condition, 00:18:30.800 --> 00:18:33.580 and you get this retro-reflector. 00:18:33.580 --> 00:18:35.120 And there's more complicated prisms 00:18:35.120 --> 00:18:38.600 that allow you-- for example, this is the pentaprism. 00:18:38.600 --> 00:18:43.900 After two bounces, the light will exit at 90 degrees angle. 00:18:43.900 --> 00:18:47.750 So Piper, maybe you can do the demo now? 00:18:47.750 --> 00:18:48.820 PROFESSOR: Sure, sure. 00:18:48.820 --> 00:18:50.110 So just before showing the demo, I'm 00:18:50.110 --> 00:18:51.820 going to pass around this prism that 00:18:51.820 --> 00:18:55.310 has in these two surfaces, two images. 00:18:55.310 --> 00:18:57.520 So then what you see is that in this window here, 00:18:57.520 --> 00:18:59.760 you tilt it like this. 00:18:59.760 --> 00:19:02.290 You're going to see when you get a hold of it. 00:19:02.290 --> 00:19:04.380 You actually see two different images 00:19:04.380 --> 00:19:06.230 due to total internal reflection. 00:19:06.230 --> 00:19:08.110 So at a given angle, you basically 00:19:08.110 --> 00:19:10.212 get the light reflecting from one of the surfaces. 00:19:10.212 --> 00:19:11.920 And another angle, you get the other one. 00:19:11.920 --> 00:19:13.628 So it's actually very interesting to see. 00:19:18.410 --> 00:19:21.340 OK, so we're seeing here the top view of the demo. 00:19:21.340 --> 00:19:25.150 We put together a wide light source and a laser source. 00:19:25.150 --> 00:19:27.157 And before showing you what actually happens, 00:19:27.157 --> 00:19:28.990 let me just introduce some of the components 00:19:28.990 --> 00:19:33.160 that we are going to be seeing in several demos from now on. 00:19:33.160 --> 00:19:36.700 First of all, we have the white light source, it's a lamp. 00:19:36.700 --> 00:19:40.060 And then this component here, it's a regular lens 00:19:40.060 --> 00:19:41.360 that you're familiar with. 00:19:41.360 --> 00:19:43.990 So the job of these lens is to collimate-- 00:19:43.990 --> 00:19:47.080 that's the term that we use-- to convert this light close 00:19:47.080 --> 00:19:49.180 to a plane wave, similar to the light that 00:19:49.180 --> 00:19:50.350 is coming from the sun. 00:19:50.350 --> 00:19:53.200 So the sun, again, would be like a point source of light 00:19:53.200 --> 00:19:54.460 far away. 00:19:54.460 --> 00:19:56.020 So then these lenses are basically 00:19:56.020 --> 00:19:58.570 transforming in this light into parallel rays 00:19:58.570 --> 00:20:01.850 from the geometrical optics point of view. 00:20:01.850 --> 00:20:02.940 So we use these lens. 00:20:02.940 --> 00:20:05.380 This is an iris, similar to the iris that controls 00:20:05.380 --> 00:20:08.170 the aperture in your eye. 00:20:08.170 --> 00:20:11.170 And this is just used to split the light into two-- 00:20:11.170 --> 00:20:13.840 I'm sorry, to reduce the diameter of the beam. 00:20:13.840 --> 00:20:17.680 Then this component, here this one. 00:20:17.680 --> 00:20:20.110 It's what we call the beam splitter, 00:20:20.110 --> 00:20:23.080 or more specifically, the non-polarized beam splitter. 00:20:23.080 --> 00:20:25.000 And it's a component that allows us to split 00:20:25.000 --> 00:20:26.710 the light into two paths. 00:20:26.710 --> 00:20:31.050 So here, this is one path, and another path. 00:20:33.980 --> 00:20:38.270 So once we split the light in two paths, and in this case, 00:20:38.270 --> 00:20:39.450 it has equal ratios. 00:20:39.450 --> 00:20:42.530 So it's 50% to one side, 50% to the other. 00:20:42.530 --> 00:20:46.850 Then let's follow one of these parts. 00:20:46.850 --> 00:20:49.640 This part illuminates one of these prisms here. 00:20:49.640 --> 00:20:52.940 And again, there's going to be some refraction following 00:20:52.940 --> 00:20:54.650 the law of refraction that we saw. 00:20:54.650 --> 00:20:56.360 But in addition to that, as we'll 00:20:56.360 --> 00:20:58.780 see in the next couple of slides also. 00:20:58.780 --> 00:21:00.883 There is a phenomenon called dispersion 00:21:00.883 --> 00:21:02.300 that you're familiar with when you 00:21:02.300 --> 00:21:04.940 see rainbows in a rainy day. 00:21:04.940 --> 00:21:07.550 And basically, that has to do with the fact 00:21:07.550 --> 00:21:11.640 that different wavelengths see a different index of refraction. 00:21:11.640 --> 00:21:13.730 So if you go back to the Snell's law, 00:21:13.730 --> 00:21:16.700 they will basically bend in a different way. 00:21:16.700 --> 00:21:19.430 So therefore, you see a rainbow effect. 00:21:19.430 --> 00:21:22.250 So I don't know if you can see the side view in the camera 00:21:22.250 --> 00:21:25.090 please? 00:21:25.090 --> 00:21:29.130 Yeah, so that's a picture of it. 00:21:29.130 --> 00:21:31.410 OK, but here in the classroom too, please. 00:21:37.678 --> 00:21:38.970 I don't know if you can see it. 00:21:38.970 --> 00:21:41.010 We're going to try to show it also in this. 00:21:41.010 --> 00:21:43.660 But I have here two components. 00:21:43.660 --> 00:21:45.300 This is a prism that is doing these, 00:21:45.300 --> 00:21:49.840 what we call the normal dispersion, which basically has 00:21:49.840 --> 00:21:53.760 that the biggest angle that bends is the blue light, 00:21:53.760 --> 00:21:55.870 or the shorter wavelength. 00:21:55.870 --> 00:21:57.600 And then we have another component here 00:21:57.600 --> 00:21:59.693 that we haven't talked about yet, 00:21:59.693 --> 00:22:01.360 but we'll see it in the next few slides. 00:22:01.360 --> 00:22:04.440 So this is just an introductory introduction to this element. 00:22:04.440 --> 00:22:08.070 It's called transmission grating. 00:22:08.070 --> 00:22:10.560 And this component is used in several systems. 00:22:13.120 --> 00:22:14.370 They're a different principle. 00:22:14.370 --> 00:22:16.200 It's not refraction anymore. 00:22:16.200 --> 00:22:19.500 It's using the diffraction property of the light. 00:22:19.500 --> 00:22:21.510 And using that diffraction property can also 00:22:21.510 --> 00:22:24.900 create this rainbow that you could see here in the back, 00:22:24.900 --> 00:22:26.570 and hopefully, you can see the picture. 00:22:30.343 --> 00:22:32.260 What we are going to see here is that actually 00:22:32.260 --> 00:22:34.030 the opposite trend happens. 00:22:34.030 --> 00:22:37.870 The red angles bend more than the blue angles, 00:22:37.870 --> 00:22:40.190 and that's called anomalous dispersion. 00:22:40.190 --> 00:22:42.310 So we have normal dispersion in the prism case, 00:22:42.310 --> 00:22:44.470 like in the one shown here in transparency, 00:22:44.470 --> 00:22:49.060 and anomalous dispersion showed in the grating. 00:22:49.060 --> 00:22:50.920 So I'm going to tilt this a little bit. 00:22:55.010 --> 00:22:55.910 OK, it's fine. 00:22:55.910 --> 00:22:57.110 I'm going to try to see. 00:22:57.110 --> 00:22:58.770 So I don't know if you can see it. 00:22:58.770 --> 00:23:00.920 Can you see the rainbow here from the back? 00:23:00.920 --> 00:23:02.550 If you're in the classroom? 00:23:02.550 --> 00:23:03.560 GEORGE BARBASTATHIS: Yeah, we can see it. 00:23:03.560 --> 00:23:04.910 PROFESSOR: All right, excellent. 00:23:04.910 --> 00:23:07.582 So this is just the rainbow from the grating. 00:23:07.582 --> 00:23:10.040 After class, you can just come here and play with the demo. 00:23:10.040 --> 00:23:12.950 And you're going to see the two cases. 00:23:12.950 --> 00:23:15.080 You can actually trace the rays and see 00:23:15.080 --> 00:23:18.860 which color is bending more, and distinguish 00:23:18.860 --> 00:23:21.540 between anomalous and normal dispersion. 00:23:21.540 --> 00:23:23.390 So the last thing that I want to show here 00:23:23.390 --> 00:23:26.660 is the total internal reflection principle. 00:23:26.660 --> 00:23:30.920 In this case, this piece of acrylic here that we have. 00:23:30.920 --> 00:23:32.300 You can see that it's forming-- 00:23:32.300 --> 00:23:35.060 it's basically having a laser light coupling into one 00:23:35.060 --> 00:23:38.450 of the sides, so similar to that exit sign over there. 00:23:38.450 --> 00:23:40.730 And what we have is that the acrylic-- 00:23:40.730 --> 00:23:43.070 the piece of acrylic acts like a wave guide. 00:23:43.070 --> 00:23:46.190 So it conducts a light inside. 00:23:46.190 --> 00:23:48.470 And the reason the light doesn't escape 00:23:48.470 --> 00:23:50.810 is because it's basically suffering 00:23:50.810 --> 00:23:53.840 total internal reflection at the interface between the acrylic, 00:23:53.840 --> 00:23:57.470 which has a higher index than air, so stays inside. 00:23:57.470 --> 00:24:00.320 Now, in order to couple the light out, 00:24:00.320 --> 00:24:05.110 I put some tape here, as you can see forming the letters MIT. 00:24:05.110 --> 00:24:08.480 And that basically frustrates the light, 00:24:08.480 --> 00:24:10.360 allows it to break the incidence angle. 00:24:10.360 --> 00:24:15.140 So a ray now instead of getting into a very flat surface 00:24:15.140 --> 00:24:17.860 at an angle that is larger than the critical angle, 00:24:17.860 --> 00:24:19.370 it basically reaches a surface that 00:24:19.370 --> 00:24:22.940 has a diffuser type of angle, so maybe it can escape out. 00:24:22.940 --> 00:24:25.700 So then, you can see all this light diffusing 00:24:25.700 --> 00:24:30.560 out forming like either this image, or the image of the exit 00:24:30.560 --> 00:24:31.940 sign. 00:24:31.940 --> 00:24:34.470 So that one, you can see it also in Singapore? 00:24:34.470 --> 00:24:36.317 The frustrated? 00:24:36.317 --> 00:24:37.400 GEORGE BARBASTATHIS: Yeah. 00:24:37.400 --> 00:24:38.960 PROFESSOR: OK. 00:24:38.960 --> 00:24:42.310 GEORGE BARBASTATHIS: It looks frustrated to us. 00:24:42.310 --> 00:24:45.250 PROFESSOR: From the side view, I guess. 00:24:45.250 --> 00:24:47.938 So I don't know if you want to add anything, George. 00:24:47.938 --> 00:24:49.980 GEORGE BARBASTATHIS: So what you see on the slide 00:24:49.980 --> 00:24:52.590 that I'm projecting it now is an application 00:24:52.590 --> 00:24:55.260 of the same principle that Piper just showed. 00:24:55.260 --> 00:24:58.470 It has an application in a bunch of commonly 00:24:58.470 --> 00:25:02.690 used conventional devices, namely, fingerprint sensors. 00:25:02.690 --> 00:25:05.860 Where instead of the tape that Piper put over there, 00:25:05.860 --> 00:25:08.490 usually what you do is they place their finger 00:25:08.490 --> 00:25:11.280 touching the side of the prism. 00:25:11.280 --> 00:25:16.410 And then because our buddy, this may be surprising to you. 00:25:16.410 --> 00:25:18.610 It was surprising to me when I first heard it. 00:25:18.610 --> 00:25:22.920 Our body is composed mostly of water, about 75% is water. 00:25:22.920 --> 00:25:25.440 So therefore, the refractive index of our skin 00:25:25.440 --> 00:25:29.800 is close to 1.3, which is, of course, higher 00:25:29.800 --> 00:25:31.950 than the refractive index of air. 00:25:31.950 --> 00:25:35.280 So what happens here is over here, for example, 00:25:35.280 --> 00:25:38.280 you have a glass and air. 00:25:38.280 --> 00:25:42.930 So therefore, light will be totally internally reflected. 00:25:42.930 --> 00:25:46.710 But at the ridges of the finger, of the fingerprint, 00:25:46.710 --> 00:25:49.200 you have glass and water. 00:25:49.200 --> 00:25:53.010 That is 1.5 to 1.33 or so. 00:25:53.010 --> 00:25:56.440 So therefore, the ridges appear dark because they frustrate 00:25:56.440 --> 00:25:58.210 the total internal reflection. 00:25:58.210 --> 00:26:00.300 The light couples into your finger. 00:26:00.300 --> 00:26:04.110 And therefore, a surprisingly sharp image of the finger 00:26:04.110 --> 00:26:05.460 appears in the camera. 00:26:05.460 --> 00:26:07.550 Actually, what you see here, it does a disservice. 00:26:07.550 --> 00:26:11.280 The projector and the pixelation of my computer 00:26:11.280 --> 00:26:13.980 does a disservice to the quality of the fingerprint image 00:26:13.980 --> 00:26:14.910 that you get. 00:26:14.910 --> 00:26:18.150 So nowadays, most fingerprint sensors 00:26:18.150 --> 00:26:19.732 are based on this principle. 00:26:19.732 --> 00:26:20.940 In fact, we have improved it. 00:26:20.940 --> 00:26:23.250 Instead of using the prism in places 00:26:23.250 --> 00:26:27.050 like laptops that have fingerprint security, 00:26:27.050 --> 00:26:29.490 it is the same principle, but you slide the finger. 00:26:29.490 --> 00:26:32.130 But still, the ridge of the fingerprints 00:26:32.130 --> 00:26:34.860 as you slide the finger over the sensor 00:26:34.860 --> 00:26:39.525 is captured by the principle of total internal reflection. 00:26:44.270 --> 00:26:45.350 Any questions about that? 00:26:45.350 --> 00:26:46.900 About TIR and FTIR? 00:27:00.080 --> 00:27:07.050 OK, one other use of TIR is in another very useful-- 00:27:07.050 --> 00:27:10.080 another extremely useful property of light 00:27:10.080 --> 00:27:15.160 is that you can actually capture it, almost like in a wire. 00:27:15.160 --> 00:27:17.933 And you can guide the light over a very long distance. 00:27:17.933 --> 00:27:19.350 Now, why this is very important is 00:27:19.350 --> 00:27:21.780 because we know from experience, and we'll also 00:27:21.780 --> 00:27:24.570 learn later as the Huygens principle, 00:27:24.570 --> 00:27:28.110 that light does not like to be confined. 00:27:28.110 --> 00:27:32.130 Generally, light, once you generate light in a source, 00:27:32.130 --> 00:27:33.720 the light would like to expand. 00:27:33.720 --> 00:27:37.590 It would like to open up and propagate in an expansive way. 00:27:37.590 --> 00:27:40.140 For example, the sun, the stars, and so on. 00:27:40.140 --> 00:27:43.170 They propagate isotropically, all around them. 00:27:43.170 --> 00:27:45.640 And you know the same from the light bulbs and so on. 00:27:45.640 --> 00:27:47.010 The light expands. 00:27:47.010 --> 00:27:49.400 There's an exception, of course, called lasers. 00:27:49.400 --> 00:27:51.150 Lasers can be quite collimated. 00:27:51.150 --> 00:27:53.000 But even lasers, they tend to expand. 00:27:53.000 --> 00:27:54.600 If you leave a laser beam by itself, 00:27:54.600 --> 00:27:56.550 and you propagate it for a long distance, 00:27:56.550 --> 00:27:57.860 eventually, it will expand. 00:27:57.860 --> 00:28:00.420 It will become quite big. 00:28:00.420 --> 00:28:03.980 So the way to undo this property of light-- 00:28:03.980 --> 00:28:06.170 if you want to transmit light over a long distance 00:28:06.170 --> 00:28:09.300 without expansion-- is to use a wave guide. 00:28:09.300 --> 00:28:12.440 So wave guides typically, they use this phenomenon 00:28:12.440 --> 00:28:14.030 of total internal reflection. 00:28:17.330 --> 00:28:22.610 In the simplest case, you have a slab of high index-- 00:28:22.610 --> 00:28:26.510 dielectric middle sandwiched between two other pieces 00:28:26.510 --> 00:28:28.970 of lower index medium. 00:28:28.970 --> 00:28:33.110 And what happens there provided that the light is incident 00:28:33.110 --> 00:28:37.040 at the sharp enough angle that is beyond the critical angle 00:28:37.040 --> 00:28:38.480 between the two media. 00:28:38.480 --> 00:28:41.330 Then the light will sort of bounce back and forth 00:28:41.330 --> 00:28:43.340 between the two interfaces. 00:28:43.340 --> 00:28:45.590 And this way, you can actually transmit it 00:28:45.590 --> 00:28:47.330 over a very long distance. 00:28:47.330 --> 00:28:48.790 So, of course, if it is not true. 00:28:48.790 --> 00:28:51.950 If the light arrives at a shallower angle, then, 00:28:51.950 --> 00:28:54.245 of course, it will actually couple out, 00:28:54.245 --> 00:28:57.380 and it will not be guided anymore. 00:28:57.380 --> 00:29:01.990 The way you establish whether the light will be guided or not 00:29:01.990 --> 00:29:04.970 is by using these properties called the and numerical 00:29:04.970 --> 00:29:07.820 aperture of the wave guide. 00:29:07.820 --> 00:29:11.000 So this is a term, numerical aperture, 00:29:11.000 --> 00:29:14.150 that we'll hear again and again in this class, at least 00:29:14.150 --> 00:29:16.280 in three different contexts. 00:29:16.280 --> 00:29:18.380 But they all mean the same thing. 00:29:18.380 --> 00:29:20.960 Actually, they mean an angle of acceptance 00:29:20.960 --> 00:29:24.610 of an optical system. 00:29:24.610 --> 00:29:27.600 So in this context here of a wave guide, 00:29:27.600 --> 00:29:28.710 compare the two rays. 00:29:28.710 --> 00:29:31.160 One is sort of the solid ray, and the other 00:29:31.160 --> 00:29:32.760 is the dotted ray. 00:29:32.760 --> 00:29:35.940 The solid ray comes in from air, then 00:29:35.940 --> 00:29:39.140 is refracted at the vertical interface. 00:29:39.140 --> 00:29:42.530 And because this angle is fairly small, 00:29:42.530 --> 00:29:44.990 by the time it gets into the middle, the angle 00:29:44.990 --> 00:29:47.300 it makes through the perpendicular surface 00:29:47.300 --> 00:29:50.270 of the interface between the slab and the cladding. 00:29:50.270 --> 00:29:53.660 It actually satisfies the TIR condition over here. 00:29:53.660 --> 00:29:56.330 You can see a little bit if you familiarize yourself 00:29:56.330 --> 00:29:58.650 with the way Snell's law works. 00:29:58.650 --> 00:30:02.060 You can see that as you increase this angle over here, 00:30:02.060 --> 00:30:04.970 this angle over here actually decreases. 00:30:04.970 --> 00:30:09.320 So the dotted ray actually can arrive 00:30:09.320 --> 00:30:11.480 at below the critical angle. 00:30:11.480 --> 00:30:13.070 So therefore, the daughter ray is not 00:30:13.070 --> 00:30:15.590 guided where the solid ray is guided. 00:30:15.590 --> 00:30:19.880 So the numerical aperture is the maximum angle, theta naught, 00:30:19.880 --> 00:30:22.850 that you can tolerate before you stop 00:30:22.850 --> 00:30:24.938 satisfying that TIR condition. 00:30:24.938 --> 00:30:26.480 And therefore, the numerical aperture 00:30:26.480 --> 00:30:30.050 is the maximum angle that you can couple into the wave guide. 00:30:30.050 --> 00:30:32.930 If you tried to bring light at a higher angle than that, 00:30:32.930 --> 00:30:34.250 it will actually not be guided. 00:30:34.250 --> 00:30:37.400 It will escape into the cladding, and it will get lost. 00:30:37.400 --> 00:30:40.920 It will disappear. 00:30:40.920 --> 00:30:43.670 So with a little bit of algebra, which I haven't done here. 00:30:43.670 --> 00:30:45.890 I will let you do it by yourselves. 00:30:45.890 --> 00:30:48.200 In years past, I used to give this as a homework, 00:30:48.200 --> 00:30:49.433 but I didn't do it this time. 00:30:49.433 --> 00:30:51.600 But anyway, with a little bit of algebra and playing 00:30:51.600 --> 00:30:53.390 with Snell's law, you can find out 00:30:53.390 --> 00:30:55.220 that the numerical aperture in this case 00:30:55.220 --> 00:30:57.170 is given by this quantity over here. 00:30:57.170 --> 00:30:59.030 The square root of the difference 00:30:59.030 --> 00:31:02.030 of squares between the two indices of refraction. 00:31:02.030 --> 00:31:05.750 And as I mentioned earlier, physically what it means. 00:31:05.750 --> 00:31:07.940 This quantity is the angle of acceptance 00:31:07.940 --> 00:31:11.310 of the wave guide for the light that you want to couple in. 00:31:11.310 --> 00:31:13.190 Typically, wave guides in practice, 00:31:13.190 --> 00:31:15.350 they have a very small difference 00:31:15.350 --> 00:31:21.010 between the index of the core, where the light is guided, 00:31:21.010 --> 00:31:22.560 and the index of the cladding. 00:31:22.560 --> 00:31:26.030 This difference is typically in the order of 10 to the minus 3, 00:31:26.030 --> 00:31:27.620 or 10 to the minus 4. 00:31:27.620 --> 00:31:29.630 So therefore, the numerical aperture is what? 00:31:29.630 --> 00:31:32.660 It is small or large for a typical wave guide? 00:31:36.560 --> 00:31:38.490 If the index difference is very small, 00:31:38.490 --> 00:31:40.490 is the numerical appearance or a small or large? 00:31:48.930 --> 00:31:50.543 Small, right? 00:31:50.543 --> 00:31:52.210 We're going to set up a competition here 00:31:52.210 --> 00:31:53.590 between Singapore and Cambridge. 00:31:53.590 --> 00:31:57.940 So you guys, when you have an answer, please push the button. 00:31:57.940 --> 00:32:00.520 On either side, please push the button. 00:32:00.520 --> 00:32:03.100 So the numerical aperture is actually very small, 00:32:03.100 --> 00:32:05.980 as our colleagues here correctly said. 00:32:10.342 --> 00:32:12.050 And what is on the slide is the opposite. 00:32:12.050 --> 00:32:13.467 If you have a high index contrast, 00:32:13.467 --> 00:32:17.650 then you get a high numerical aperture. 00:32:17.650 --> 00:32:20.720 There is one more type of wave guide, which is actually 00:32:20.720 --> 00:32:22.370 the same principle, but a slightly 00:32:22.370 --> 00:32:24.200 different implementation. 00:32:24.200 --> 00:32:27.102 It's called a gradient index wave guide. 00:32:27.102 --> 00:32:28.310 And the way to understand it. 00:32:28.310 --> 00:32:33.920 Imagine that I stack a bunch of different slabs of glass 00:32:33.920 --> 00:32:36.940 with index that varies from a small value. 00:32:36.940 --> 00:32:40.460 This I denoted as the sort of light gray. 00:32:40.460 --> 00:32:43.640 Then to darker gray, denoting higher index of refraction, 00:32:43.640 --> 00:32:45.320 and then back to light gray. 00:32:45.320 --> 00:32:48.360 I mean, back to low index of refraction. 00:32:48.360 --> 00:32:51.500 So if you imagine the ray coming in from the top here. 00:32:51.500 --> 00:32:55.880 It will refract into the guide, and then as it goes in, 00:32:55.880 --> 00:32:57.690 it will keep getting refracted. 00:32:57.690 --> 00:32:59.960 Now, we can adjust the numbers here, 00:32:59.960 --> 00:33:03.170 so that one of these interfaces, the light 00:33:03.170 --> 00:33:05.410 will exceed the critical angle. 00:33:05.410 --> 00:33:07.820 And therefore, it will be totally internally reflected 00:33:07.820 --> 00:33:09.280 at this interface. 00:33:09.280 --> 00:33:12.020 And if that is true, the same thing 00:33:12.020 --> 00:33:15.380 will happen also at the top interface. 00:33:15.380 --> 00:33:18.690 As I mentioned before, this quantity, n sine theta, 00:33:18.690 --> 00:33:19.880 is preserved. 00:33:19.880 --> 00:33:22.940 So therefore, if this quantity, n sine theta, 00:33:22.940 --> 00:33:27.320 was such that the TIR condition was satisfied over here, 00:33:27.320 --> 00:33:29.273 then the same will happen over here. 00:33:29.273 --> 00:33:31.190 Because everything else is symmetric, correct? 00:33:31.190 --> 00:33:36.020 The reflections are symmetric, and the quantity, n sine theta, 00:33:36.020 --> 00:33:38.880 is preserved. 00:33:38.880 --> 00:33:44.110 So therefore, if you put a light in this kind of arrangement, 00:33:44.110 --> 00:33:46.650 it would actually follow a periodic trajectory. 00:33:46.650 --> 00:33:51.480 The light will sort of be periodically reflected 00:33:51.480 --> 00:33:55.560 from these interfaces, and will follow a periodic trajectory 00:33:55.560 --> 00:33:58.560 down the stack of slabs. 00:33:58.560 --> 00:34:01.680 Therefore, this is also a kind of wave guide. 00:34:01.680 --> 00:34:05.040 It is commonly referred to as green, where green is not 00:34:05.040 --> 00:34:09.110 for the facial expression, but stands for gradient index. 00:34:09.110 --> 00:34:17.989 And there's a special case of a gradient index where they 00:34:17.989 --> 00:34:19.429 don't normally do it this way. 00:34:19.429 --> 00:34:22.230 The way they do it is with a continuous variation. 00:34:22.230 --> 00:34:23.929 And they manufacture it with diffusion. 00:34:23.929 --> 00:34:24.846 It's very interesting. 00:34:24.846 --> 00:34:29.120 They take a piece of glass, and they diffuse ions. 00:34:29.120 --> 00:34:31.340 Because the ions change the index of refraction 00:34:31.340 --> 00:34:33.440 of the glass, then they can sort of 00:34:33.440 --> 00:34:39.830 get a continuous profile of gradually variable index. 00:34:39.830 --> 00:34:42.800 And in the special case where this profile is quadratic, 00:34:42.800 --> 00:34:46.520 it turns out that the trajectory of the light paths 00:34:46.520 --> 00:34:47.570 is kind of like a helix. 00:34:47.570 --> 00:34:49.730 It become sinusoidal. 00:34:49.730 --> 00:34:53.810 And the light sort of bounces in sinusoidal fashion 00:34:53.810 --> 00:35:00.540 between the two interfaces, this one and this one. 00:35:00.540 --> 00:35:04.378 We will do this in more detail four lectures later. 00:35:04.378 --> 00:35:05.920 If you look at your syllabus, there's 00:35:05.920 --> 00:35:07.880 something called Hamiltonian optics. 00:35:07.880 --> 00:35:10.060 We will actually see in action how 00:35:10.060 --> 00:35:15.900 this sinusoidal periodic trajectory comes about. 00:35:15.900 --> 00:35:16.610 Yes. 00:35:16.610 --> 00:35:18.294 AUDIENCE: So what is the [INAUDIBLE]?? 00:35:20.917 --> 00:35:22.000 GEORGE BARBASTATHIS: Yeah. 00:35:22.000 --> 00:35:23.315 That's a very good question. 00:35:23.315 --> 00:35:24.940 PROFESSOR: Can you repeat the question? 00:35:24.940 --> 00:35:26.890 Because they didn't press the button here. 00:35:26.890 --> 00:35:28.390 Can you repeat the question, please? 00:35:28.390 --> 00:35:28.970 GEORGE BARBASTATHIS: I'm sorry. 00:35:28.970 --> 00:35:30.020 Could you repeat with the button? 00:35:30.020 --> 00:35:30.740 Yeah. 00:35:30.740 --> 00:35:31.390 AUDIENCE: Yeah. 00:35:31.390 --> 00:35:34.130 What is the advantage of using this type of wave guide 00:35:34.130 --> 00:35:37.367 compared to step index wave guide? 00:35:37.367 --> 00:35:38.950 GEORGE BARBASTATHIS: So the advantage, 00:35:38.950 --> 00:35:41.830 which I cannot describe yet because we have not done wave 00:35:41.830 --> 00:35:42.760 optics. 00:35:42.760 --> 00:35:47.630 But in wave guides, there's a phenomenon called dispersion. 00:35:47.630 --> 00:35:51.880 It is very similar to the dispersion from a prism 00:35:51.880 --> 00:35:53.770 that Piper showed before. 00:35:53.770 --> 00:35:57.260 But in telecommunications, when you transmit signal down 00:35:57.260 --> 00:36:01.270 a wave guide, it has the effect of basically lowering 00:36:01.270 --> 00:36:03.940 the speed, the effective speed at which you 00:36:03.940 --> 00:36:06.170 can transmit information. 00:36:06.170 --> 00:36:08.440 So it turns out that the step index wave 00:36:08.440 --> 00:36:12.410 guide has a higher dispersion than the gradient index wave 00:36:12.410 --> 00:36:13.060 guide. 00:36:13.060 --> 00:36:17.645 So you get a much higher speed in fibers of gradient index. 00:36:17.645 --> 00:36:19.270 So that's one reason why people use it. 00:36:23.242 --> 00:36:24.200 This will take a while. 00:36:38.820 --> 00:36:41.210 And, of course, more practical wave guides that 00:36:41.210 --> 00:36:42.460 are shaped like a wire. 00:36:42.460 --> 00:36:44.293 Literally, they are known as optical fibers. 00:36:46.590 --> 00:36:48.090 Again, you have two types of fibers. 00:36:48.090 --> 00:36:51.740 You have the step index fiber, where you have a higher index 00:36:51.740 --> 00:36:55.500 core, and the light is kind of bouncing back and forth 00:36:55.500 --> 00:36:57.870 between the core and the cladding. 00:36:57.870 --> 00:37:00.780 And there's also gradient index fibers, 00:37:00.780 --> 00:37:04.200 where the light is following a helical trajectory. 00:37:04.200 --> 00:37:07.470 The phenomenon of wave guiding, again, I 00:37:07.470 --> 00:37:09.120 have to defer to electromagnetics. 00:37:09.120 --> 00:37:12.210 It is much easier to describe with electromagnetics 00:37:12.210 --> 00:37:14.850 than it is with wave optics. 00:37:14.850 --> 00:37:19.440 But for now, we can get a sort of a preliminary description 00:37:19.440 --> 00:37:21.870 with the means that we have available to us. 00:37:25.670 --> 00:37:28.370 As a sort of a curiosity, it turns out 00:37:28.370 --> 00:37:31.780 that these gradient index wave guides. 00:37:31.780 --> 00:37:35.960 They appear in the nature in certain animals. 00:37:35.960 --> 00:37:38.030 I mean, insects, actually. 00:37:38.030 --> 00:37:41.750 Their eyes, they're composed of several wave guides, 00:37:41.750 --> 00:37:45.300 each one of which is actually a gradient index wave guide. 00:37:45.300 --> 00:37:48.500 And the way the animal eye works is it captures-- remember, 00:37:48.500 --> 00:37:51.590 the wave guide has a limited numerical aperture. 00:37:51.590 --> 00:37:55.740 So each one of these guides, each one of these small eyes, 00:37:55.740 --> 00:37:59.670 they're called ommatidia, these little wave guides. 00:37:59.670 --> 00:38:02.780 So each one of those captures a very narrow angle 00:38:02.780 --> 00:38:07.250 of light sort of within the field of view 00:38:07.250 --> 00:38:08.600 of the eye of the insect. 00:38:08.600 --> 00:38:13.280 And then sends a signal down to the optic nerve of the insect. 00:38:13.280 --> 00:38:15.440 So basically, the insect with this kind of eye, it 00:38:15.440 --> 00:38:17.720 forms a very bloody picture of the background, 00:38:17.720 --> 00:38:22.640 because it integrates a relatively large range 00:38:22.640 --> 00:38:23.390 of angles. 00:38:23.390 --> 00:38:25.970 But still, the range of angles is small enough 00:38:25.970 --> 00:38:30.332 that it allows it to quote unquote see. 00:38:30.332 --> 00:38:31.790 Now, insects, of course, they don't 00:38:31.790 --> 00:38:38.270 see the way we see, at least from our everyday experience. 00:38:38.270 --> 00:38:40.580 And the reason, of course, is that the insects 00:38:40.580 --> 00:38:43.670 have a very limited brain. 00:38:43.670 --> 00:38:46.550 A typical insect might have about 10,000 neurons 00:38:46.550 --> 00:38:48.300 in his brain. 00:38:48.300 --> 00:38:50.030 Show of hands, does anybody know how many 00:38:50.030 --> 00:38:51.238 neurons we have in our brain? 00:38:57.780 --> 00:38:59.770 10 to the 11. 00:38:59.770 --> 00:39:03.650 So we have about eight orders of magnitude more neurons. 00:39:03.650 --> 00:39:05.630 In case you are wondering, whether you 00:39:05.630 --> 00:39:08.420 are smart or educated, it does not 00:39:08.420 --> 00:39:10.880 have to do with a number of neurons that you have, 00:39:10.880 --> 00:39:13.565 but it has to do with the connections between neurons. 00:39:13.565 --> 00:39:15.320 Neurons are connected with wires. 00:39:15.320 --> 00:39:18.080 It turns out an educated person has 00:39:18.080 --> 00:39:20.480 approximately 10 times more connections 00:39:20.480 --> 00:39:22.490 than an uneducated person. 00:39:22.490 --> 00:39:26.060 The same number of neurons, but more connections. 00:39:26.060 --> 00:39:30.230 And so it's a sad fact of life that every day, adults-- 00:39:30.230 --> 00:39:36.060 that is, after age six or so, even at childhood-- 00:39:36.060 --> 00:39:37.280 would begin to lose neurons. 00:39:37.280 --> 00:39:38.870 In fact, each one of us every day 00:39:38.870 --> 00:39:41.102 will lose about 100,000 neurons. 00:39:41.102 --> 00:39:43.310 Nothing to worry about, because we have 10 to the 11. 00:39:43.310 --> 00:39:45.020 So even with all this loss, we can still 00:39:45.020 --> 00:39:48.590 survive until a fairly old age. 00:39:48.590 --> 00:39:50.090 But anyway, it is true. 00:39:50.090 --> 00:39:53.390 So [INAUDIBLE]. 00:39:53.390 --> 00:39:55.630 Anyway, the insect, on the other hand, 00:39:55.630 --> 00:39:57.860 has about 10,000 neurons altogether. 00:39:57.860 --> 00:40:00.025 So it has to make do with these 10,000 neurons. 00:40:00.025 --> 00:40:00.650 It has to move. 00:40:00.650 --> 00:40:01.490 It has to feed. 00:40:01.490 --> 00:40:04.400 It has to mate and all of these things. 00:40:04.400 --> 00:40:07.610 So the way they handle it is they get very simple vision, 00:40:07.610 --> 00:40:10.910 and they navigate according to differences in lighting. 00:40:10.910 --> 00:40:15.410 So the typical example is an insect flying toward a tree. 00:40:15.410 --> 00:40:18.320 Nature has evolved the insect to avoid this situation, 00:40:18.320 --> 00:40:21.470 because if it flies into the tree it will crash and die. 00:40:21.470 --> 00:40:26.040 But if it flies towards a tree, the insect sees a dark 00:40:26.040 --> 00:40:28.250 background-- the trunk of the tree-- 00:40:28.250 --> 00:40:30.740 surrounded by light, which is sort of leaking 00:40:30.740 --> 00:40:32.240 on the sides of the tree. 00:40:32.240 --> 00:40:35.780 As it flies by, it sees an edge of light 00:40:35.780 --> 00:40:38.720 that is very rapidly expanding, because it 00:40:38.720 --> 00:40:41.370 is approaching the tree. 00:40:41.370 --> 00:40:42.400 So the insect is wired. 00:40:42.400 --> 00:40:43.260 It as actually automatic. 00:40:43.260 --> 00:40:45.630 The insect doesn't think, oh my god, I'm going to crash. 00:40:45.630 --> 00:40:46.500 Let me turn. 00:40:46.500 --> 00:40:47.600 It's automatic. 00:40:47.600 --> 00:40:49.850 As soon as the neurons of the insect 00:40:49.850 --> 00:40:51.680 register a difference in lighting 00:40:51.680 --> 00:40:54.770 between successive ommatidia over here, 00:40:54.770 --> 00:40:56.800 they turn on the motor-- 00:40:56.800 --> 00:40:59.592 the flies or whatever, the legs and so on of the insect, 00:40:59.592 --> 00:41:00.800 the navigation of the insect. 00:41:00.800 --> 00:41:05.690 And the insect turns and avoids the obstacle. 00:41:05.690 --> 00:41:07.680 So this is what I say about insect. 00:41:07.680 --> 00:41:10.600 More precisely, I'm referring to the fruit fly, which 00:41:10.600 --> 00:41:14.180 has been very broad, very extensively studied 00:41:14.180 --> 00:41:15.660 in this context. 00:41:15.660 --> 00:41:18.770 But anyway, this is a very, very interesting story 00:41:18.770 --> 00:41:22.040 of how the insects use what we now 00:41:22.040 --> 00:41:23.600 consider as a rather sophisticated 00:41:23.600 --> 00:41:25.850 optical instrument, a green wave guide 00:41:25.850 --> 00:41:29.090 in order to generate a very simple type of vision 00:41:29.090 --> 00:41:30.500 based navigation. 00:41:35.033 --> 00:41:36.450 The next thing I was going to say, 00:41:36.450 --> 00:41:38.570 Piper already mentioned it. 00:41:38.570 --> 00:41:43.820 The index of refraction of most dielectric media 00:41:43.820 --> 00:41:47.360 turns out to be a strong fraction of the wavelength. 00:41:47.360 --> 00:41:49.520 So I stole from a book, actually, 00:41:49.520 --> 00:41:52.410 from the Soto website. 00:41:52.410 --> 00:41:54.530 Soto is a glass manufacturer. 00:41:54.530 --> 00:41:56.150 They make glasses that are used very 00:41:56.150 --> 00:42:00.210 commonly in optical instrument lenses and such, and prisms 00:42:00.210 --> 00:42:02.392 and so on. 00:42:02.392 --> 00:42:03.850 So they have this picture on online 00:42:03.850 --> 00:42:06.670 of the index of refraction as a function 00:42:06.670 --> 00:42:09.400 of wavelength for a relatively large range of wavelengths 00:42:09.400 --> 00:42:14.470 going along the wave from ultraviolet 00:42:14.470 --> 00:42:17.090 into the deep infrared over here. 00:42:17.090 --> 00:42:20.000 So you can see that the index varies quite a bit. 00:42:20.000 --> 00:42:22.750 Also, with the plot here, the absorption 00:42:22.750 --> 00:42:24.490 coefficient of the material. 00:42:24.490 --> 00:42:28.660 And you can see that they are kind of correlated in the sense 00:42:28.660 --> 00:42:32.430 that when the index does something interesting, 00:42:32.430 --> 00:42:36.518 the absorption also seems to do something interesting. 00:42:36.518 --> 00:42:37.560 It is not coincidentally. 00:42:37.560 --> 00:42:39.370 It turns out to have a very interesting 00:42:39.370 --> 00:42:40.990 theoretical foundation. 00:42:40.990 --> 00:42:42.030 I will go into it. 00:42:42.030 --> 00:42:44.140 Perhaps I will go into it later in the class. 00:42:47.600 --> 00:42:49.900 But anyway, the point I've been trying to make here 00:42:49.900 --> 00:42:51.483 is that you can see that the index can 00:42:51.483 --> 00:42:53.180 have quite a bit of variation. 00:42:53.180 --> 00:42:57.550 So because of that, if you send broadband light 00:42:57.550 --> 00:43:01.970 that contains multiple colors into an element, 00:43:01.970 --> 00:43:03.070 such as a prism. 00:43:03.070 --> 00:43:06.850 Then you can observe these phenomena that Piper showed. 00:43:06.850 --> 00:43:10.000 Different wavelengths, different colors, 00:43:10.000 --> 00:43:13.810 they experience different index, and therefore, the Snell's law 00:43:13.810 --> 00:43:15.650 applies differently to them. 00:43:15.650 --> 00:43:19.060 That is why you have this-- 00:43:19.060 --> 00:43:21.130 it's called analysis of white light. 00:43:21.130 --> 00:43:22.110 It becomes a rainbow. 00:43:25.480 --> 00:43:27.050 For most materials, it is true. 00:43:29.890 --> 00:43:34.570 Longer wavelengths actually have a lower index of refraction. 00:43:34.570 --> 00:43:37.060 Now let's see, does this make sense? 00:43:37.060 --> 00:43:39.160 If you look at this picture over here. 00:43:39.160 --> 00:43:41.150 Does it make sense what I said? 00:43:41.150 --> 00:43:45.870 Which wavelength apparently has the lower index, blue or red? 00:44:00.840 --> 00:44:02.760 It better be right, or I-- 00:44:02.760 --> 00:44:04.010 either I made the wrong slide. 00:44:04.010 --> 00:44:06.460 But anyway, I have taught this class for several years. 00:44:06.460 --> 00:44:08.190 So you would think that if I had made the wrong slide, 00:44:08.190 --> 00:44:09.810 I would have fixed it by now, right? 00:44:09.810 --> 00:44:11.100 So the slide is correct. 00:44:11.100 --> 00:44:12.600 The way to figure it out is you have 00:44:12.600 --> 00:44:17.110 to imagine a normal to the surface over here. 00:44:17.110 --> 00:44:24.350 So which wavelength appears to have the stronger refraction? 00:44:24.350 --> 00:44:26.820 Blue or red? 00:44:26.820 --> 00:44:28.950 Blue, right? 00:44:28.950 --> 00:44:32.470 So the blue wavelength suffers a strong refraction. 00:44:32.470 --> 00:44:35.860 That is, the blue wavelength has what index? 00:44:35.860 --> 00:44:39.360 Higher or lower? 00:44:39.360 --> 00:44:41.420 Higher. 00:44:41.420 --> 00:44:44.600 And indeed, the blue wavelength is 00:44:44.600 --> 00:44:46.230 softer than the red wavelength. 00:44:46.230 --> 00:44:49.010 So this is consistent with the curve that you see here. 00:44:49.010 --> 00:44:51.500 The blue wavelength is probably somewhere around here. 00:44:51.500 --> 00:44:53.930 The red wavelength is somewhere around here. 00:44:53.930 --> 00:44:57.600 It is not a dramatic variation in the visible range. 00:44:57.600 --> 00:44:59.620 And that is typical for most glasses. 00:44:59.620 --> 00:45:03.455 In the visible range, they have a relatively slow variation 00:45:03.455 --> 00:45:04.580 of the index of refraction. 00:45:04.580 --> 00:45:06.080 But nevertheless, it is there. 00:45:06.080 --> 00:45:08.270 And you saw evidence of it in the experiment 00:45:08.270 --> 00:45:12.027 that Piper just did with the prism. 00:45:12.027 --> 00:45:13.610 And the last thing that I want to say. 00:45:13.610 --> 00:45:15.980 I don't want to belabor this point. 00:45:15.980 --> 00:45:22.430 People use various quantities to characterize dispersion. 00:45:22.430 --> 00:45:26.810 And typically, they characterize them with respect 00:45:26.810 --> 00:45:29.390 to the various emission lights-- 00:45:29.390 --> 00:45:34.280 emission lines, from atomic spectra. 00:45:34.280 --> 00:45:36.400 So they use this as reference [INAUDIBLE].. 00:45:36.400 --> 00:45:37.880 The reason, I suppose. 00:45:37.880 --> 00:45:40.070 The reason is that back when people 00:45:40.070 --> 00:45:43.610 developed these measures, lasers were not available. 00:45:43.610 --> 00:45:47.180 So the best way to define wavelength standards 00:45:47.180 --> 00:45:50.930 was with emission lines. 00:45:50.930 --> 00:45:53.960 So they use typically the hydrogen C line and F 00:45:53.960 --> 00:45:56.690 line, and the sodium D line. 00:45:56.690 --> 00:45:58.550 And then they define these quantities, 00:45:58.550 --> 00:46:03.080 the dispersive power and the dispersive index, 00:46:03.080 --> 00:46:06.320 which are defined according to the index at these three 00:46:06.320 --> 00:46:09.230 different wavelengths. 00:46:09.230 --> 00:46:13.250 So this is very useful for people who do optical design. 00:46:13.250 --> 00:46:19.520 And it gives you sort of an idea of how dispersive is a glass. 00:46:19.520 --> 00:46:25.070 These quantities are actually inverse relative to each other. 00:46:25.070 --> 00:46:26.720 And this an example. 00:46:26.720 --> 00:46:31.320 For ground glass, typically, you want 00:46:31.320 --> 00:46:34.580 the V number, the dispersive power, to be low, 00:46:34.580 --> 00:46:38.830 if you want a dispersion free element. 00:46:41.780 --> 00:46:44.730 OK, any questions? 00:47:05.590 --> 00:47:09.640 OK, so I'm not going to go over the second lecture. 00:47:09.640 --> 00:47:12.780 We will postpone it for Wednesday. 00:47:12.780 --> 00:47:15.720 Basically, we have slid back by about an hour, but that's OK. 00:47:15.720 --> 00:47:18.720 We'll catch up later. 00:47:18.720 --> 00:47:21.840 But what I'll do is I would like to get you started thinking 00:47:21.840 --> 00:47:25.820 about next Wednesday's lecture. 00:47:25.820 --> 00:47:27.650 So next Wednesday, we'll basically 00:47:27.650 --> 00:47:32.300 see a bunch of applications of Fermat's principle. 00:47:32.300 --> 00:47:35.840 Namely, the principle that says that light 00:47:35.840 --> 00:47:39.320 chooses its trajectory trying to minimize the optical path 00:47:39.320 --> 00:47:40.640 length. 00:47:40.640 --> 00:47:42.260 So we saw already two applications, 00:47:42.260 --> 00:47:44.480 one in the law of reflection, and the other 00:47:44.480 --> 00:47:47.170 in the law of refraction. 00:47:47.170 --> 00:47:49.350 So the next applications will be in focusing. 00:47:49.350 --> 00:47:51.800 So the question we'll ask the next Wednesday 00:47:51.800 --> 00:47:57.920 is how can we design a surface, or reflect a surface such 00:47:57.920 --> 00:48:02.060 that if light is arriving from infinity in parallel rays 00:48:02.060 --> 00:48:05.320 like this, this surface upon reflection 00:48:05.320 --> 00:48:06.880 focuses all the rays. 00:48:06.880 --> 00:48:10.288 So they pass from the same common focal point F. 00:48:10.288 --> 00:48:11.830 So you can look it up into the notes, 00:48:11.830 --> 00:48:14.860 and then I will go over it again on Wednesday, 00:48:14.860 --> 00:48:18.750 how we can use the Fermat's principle in order to design. 00:48:18.750 --> 00:48:19.750 You can actually design. 00:48:19.750 --> 00:48:22.070 We can come up with an analytic expression that 00:48:22.070 --> 00:48:24.760 has to be a parabola for the sacrifice that gives 00:48:24.760 --> 00:48:26.890 the perfect focus onto a point. 00:48:29.490 --> 00:48:31.610 So the homework has been posted. 00:48:31.610 --> 00:48:33.870 The first three problems you can do 00:48:33.870 --> 00:48:37.380 without a need for any of this. 00:48:37.380 --> 00:48:39.130 Actually, I think the first four problems. 00:48:39.130 --> 00:48:41.860 You don't need any of this. 00:48:41.860 --> 00:48:45.610 The problems are not due until actually the next Wednesday. 00:48:45.610 --> 00:48:48.910 Not this Wednesday, but Wednesday the 18th, 00:48:48.910 --> 00:48:50.290 nine days from today. 00:48:50.290 --> 00:48:54.660 So you're in good shape with regards to the homework.