WEBVTT 00:00:02.550 --> 00:00:04.920 The following content is provided under a Creative 00:00:04.920 --> 00:00:06.310 Commons license. 00:00:06.310 --> 00:00:08.520 Your support will help MIT OpenCourseWare 00:00:08.520 --> 00:00:12.610 continue to offer high quality educational resources for free. 00:00:12.610 --> 00:00:15.150 To make a donation or to view additional materials 00:00:15.150 --> 00:00:19.110 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:19.110 --> 00:00:20.330 at ocw.mit.edu. 00:00:23.620 --> 00:00:27.580 YEN-JIE LEE: So I hope you can hear me. 00:00:27.580 --> 00:00:29.690 Welcome back, everybody. 00:00:29.690 --> 00:00:31.750 I hope you have the full energy back 00:00:31.750 --> 00:00:35.380 from Thanksgiving vacation. 00:00:35.380 --> 00:00:39.070 Everybody, welcome back. 00:00:39.070 --> 00:00:45.470 So before we start, we'll talk about where we are now. 00:00:45.470 --> 00:00:50.180 So this is actually the goal, which we said, for 8.03, 00:00:50.180 --> 00:00:51.580 at the beginning. 00:00:51.580 --> 00:00:54.520 So what we have been discussing is 00:00:54.520 --> 00:00:58.190 to learn about how to translate physical situations 00:00:58.190 --> 00:01:03.430 into mathematics, simple harmonic oscillator, coupled 00:01:03.430 --> 00:01:05.530 oscillators, et cetera. 00:01:05.530 --> 00:01:11.360 And we tried to put together infinite number of oscillators. 00:01:11.360 --> 00:01:17.380 And we found waves from this interesting exercise. 00:01:17.380 --> 00:01:21.280 And of course, we learned about Fourier decompositions of waves 00:01:21.280 --> 00:01:25.860 and also learned about how to put together physical systems. 00:01:25.860 --> 00:01:29.920 In order to do that, you need to define boundary conditions. 00:01:29.920 --> 00:01:32.350 And those conditions need to be satisfied, 00:01:32.350 --> 00:01:35.900 so that you can describe multiple physical systems 00:01:35.900 --> 00:01:37.550 all together. 00:01:37.550 --> 00:01:39.640 And the third part of the course, we 00:01:39.640 --> 00:01:44.320 have been focusing on many, many applications, 00:01:44.320 --> 00:01:46.840 for instance, for the phenomenon related 00:01:46.840 --> 00:01:50.320 to electromagnetic waves and also 00:01:50.320 --> 00:01:53.830 many practical applications in optics. 00:01:53.830 --> 00:01:59.020 And we are pretty close to the discussion between wave 00:01:59.020 --> 00:02:03.190 and vibrations and the future course, which 00:02:03.190 --> 00:02:06.900 is 8.04, the connections to quantum mechanics. 00:02:06.900 --> 00:02:10.520 And if we have time, we will talk about gravitational waves 00:02:10.520 --> 00:02:13.660 if we manage to do that. 00:02:13.660 --> 00:02:16.030 It depends on how fast we progress. 00:02:18.900 --> 00:02:21.380 So let's start the lecture today. 00:02:21.380 --> 00:02:25.250 So first, we will give you a short review. 00:02:25.250 --> 00:02:27.250 Before Thanksgiving, we were talking 00:02:27.250 --> 00:02:30.100 about polarizer filters. 00:02:30.100 --> 00:02:34.900 And we have been researching how to make a very good photo so 00:02:34.900 --> 00:02:37.070 that you can post it on Facebook, right? 00:02:37.070 --> 00:02:38.800 So that's essentially what we learned. 00:02:38.800 --> 00:02:44.230 So if you want to take a picture of the sky, which is deep blue, 00:02:44.230 --> 00:02:46.390 then you need to use a polarizer. 00:02:46.390 --> 00:02:47.620 And the reason? 00:02:47.620 --> 00:02:50.740 We also understand that it's because, if you 00:02:50.740 --> 00:02:56.380 look at the sky, which is actually, roughly a 45 to 90 00:02:56.380 --> 00:02:59.290 degree wave from the direction of the sun. 00:02:59.290 --> 00:03:03.870 Basically, what you get is that all those light fronts 00:03:03.870 --> 00:03:06.730 scattering, between the sunlight and the molecules 00:03:06.730 --> 00:03:12.170 or little dust in the sky, polarize the light. 00:03:12.170 --> 00:03:15.850 Therefore, you can actual filter them using the polarizer. 00:03:15.850 --> 00:03:18.670 Of course, you have to tune your polarizer carefully, 00:03:18.670 --> 00:03:21.140 so that you can actually minimize 00:03:21.140 --> 00:03:23.470 the light from the sky, so that you get 00:03:23.470 --> 00:03:26.710 a sharper image in your photo. 00:03:26.710 --> 00:03:31.630 And also, we discussed about, with a polarizer, polarization 00:03:31.630 --> 00:03:36.100 filter, we can actually filter out also the reflected light, 00:03:36.100 --> 00:03:42.100 for example, from the water or from the window of a car. 00:03:42.100 --> 00:03:46.090 And that is because something which is closely related 00:03:46.090 --> 00:03:51.250 to the boundary condition, which we learned about from Maxwell's 00:03:51.250 --> 00:03:52.960 equation in matter. 00:03:52.960 --> 00:03:54.950 And this is actually the four equations, 00:03:54.950 --> 00:03:56.670 which we discussed last time. 00:03:56.670 --> 00:04:01.370 And in that issue, we were using that to explain the incident 00:04:01.370 --> 00:04:07.840 light, from air to something which is denser, 00:04:07.840 --> 00:04:09.880 for example lighter gas. 00:04:09.880 --> 00:04:14.780 And then we found that, if we start with unpolarized light-- 00:04:14.780 --> 00:04:17.620 this incident wave unpolarized light-- 00:04:17.620 --> 00:04:20.589 what we found is that the transmitted wave, which 00:04:20.589 --> 00:04:25.070 is actually in the bottom of this diagram, 00:04:25.070 --> 00:04:30.250 is actually still pretty close to unpolarized 00:04:30.250 --> 00:04:34.900 light but slightly polarized because of the transmission 00:04:34.900 --> 00:04:36.880 and the boundary condition. 00:04:36.880 --> 00:04:40.900 And the reflected light, something very interesting 00:04:40.900 --> 00:04:43.060 happens. 00:04:43.060 --> 00:04:46.690 Only the component, which is actually 00:04:46.690 --> 00:04:51.430 polarizing the direction, such that the electric field is 00:04:51.430 --> 00:04:56.470 oscillating in a direction perpendicular 00:04:56.470 --> 00:05:01.210 to the surface of this light, will survive. 00:05:01.210 --> 00:05:07.010 And that actually gives you polarized light, 00:05:07.010 --> 00:05:09.260 which is actually reflected from the surface. 00:05:09.260 --> 00:05:14.220 And this interesting phenomenon reaches the maxima, 00:05:14.220 --> 00:05:17.860 where you get the fully polarized light, when 00:05:17.860 --> 00:05:23.020 you actually set the incident angle of the unpolarized light 00:05:23.020 --> 00:05:25.460 at so-called Brewster's angle. 00:05:25.460 --> 00:05:28.540 And this Brewster's angle is happening 00:05:28.540 --> 00:05:32.520 when the reflected light and this transmitted light 00:05:32.520 --> 00:05:37.030 direction actually are orthogonal to each other. 00:05:37.030 --> 00:05:39.940 And that actually gives you the maxima effect 00:05:39.940 --> 00:05:42.040 we are looking for. 00:05:42.040 --> 00:05:43.930 So that's actually what we have learned 00:05:43.930 --> 00:05:46.840 from electromagnetic wave in matter 00:05:46.840 --> 00:05:49.460 and, also, matching the boundary conditions 00:05:49.460 --> 00:05:57.670 between the electromagnetic waves inside the material 00:05:57.670 --> 00:06:01.300 and in the air, such that we actually learn about all 00:06:01.300 --> 00:06:03.250 those interesting phenomena. 00:06:03.250 --> 00:06:05.770 And basically, we have learned how 00:06:05.770 --> 00:06:08.830 to describe electromagnetic waves, how 00:06:08.830 --> 00:06:13.210 to add electromagnetic waves together, how they propagate 00:06:13.210 --> 00:06:16.120 from one position to the other position, 00:06:16.120 --> 00:06:19.090 and how the boundary condition works 00:06:19.090 --> 00:06:23.020 and your equation of motion, et cetera, and something 00:06:23.020 --> 00:06:26.230 related to dielectric material. 00:06:26.230 --> 00:06:28.990 And today what we're going to do is 00:06:28.990 --> 00:06:32.080 to put all the things we have learned together 00:06:32.080 --> 00:06:34.620 and see if we can actually explain 00:06:34.620 --> 00:06:38.090 a very interesting phenomenon. 00:06:38.090 --> 00:06:44.220 So before I start, I will show you 00:06:44.220 --> 00:06:48.580 a demonstration, which I'm not sure if I will be successful. 00:06:48.580 --> 00:06:50.250 It's very difficult, actually. 00:06:50.250 --> 00:06:52.890 And of course, during the break, you're 00:06:52.890 --> 00:06:55.970 welcome to come over and play with all those demos. 00:06:55.970 --> 00:06:59.710 And here, I have two sticks. 00:06:59.710 --> 00:07:04.690 And I am going to create a soap bubble from this soap 00:07:04.690 --> 00:07:05.670 and water. 00:07:05.670 --> 00:07:10.070 And let's see if I will make it or not. 00:07:10.070 --> 00:07:14.220 So basically, I put this into the soap water. 00:07:14.220 --> 00:07:19.450 And I will try to open it to see if I can create a bubble. 00:07:19.450 --> 00:07:20.470 Yeah. 00:07:20.470 --> 00:07:22.290 You can see. 00:07:22.290 --> 00:07:25.720 You can see that there's a colorful soap bubble created. 00:07:28.870 --> 00:07:30.770 You can see that it is not always easy. 00:07:35.260 --> 00:07:37.320 Oh, it's getting very messy now. 00:07:37.320 --> 00:07:40.110 I'm trying to destroy the classroom. 00:07:40.110 --> 00:07:44.880 But it's OK, because we are MIT. 00:07:44.880 --> 00:07:47.170 You can that it's really beautiful. 00:07:47.170 --> 00:07:49.385 It's colorful. 00:07:49.385 --> 00:07:53.630 It live for a while, then it breaks. 00:07:53.630 --> 00:07:57.060 And of course, during the break, you are welcome to do this. 00:07:57.060 --> 00:08:02.860 And it's actually non-trivial to create this size of bubble. 00:08:02.860 --> 00:08:04.380 So the success rate is like 50%. 00:08:07.080 --> 00:08:13.810 So as you can see from this demonstration, 00:08:13.810 --> 00:08:17.740 we see something really beautiful. 00:08:17.740 --> 00:08:19.735 This bubble is colorful. 00:08:23.420 --> 00:08:27.190 And I didn't actually shine this soap bubble 00:08:27.190 --> 00:08:31.570 by all kinds of different preset colors. 00:08:31.570 --> 00:08:34.659 So it appears automatically and just shiny 00:08:34.659 --> 00:08:36.740 with all kinds of different-- whatever, 00:08:36.740 --> 00:08:41.260 wavelengths I get from the lights that are in here. 00:08:41.260 --> 00:08:43.659 Pretty bright light, there, on my face. 00:08:43.659 --> 00:08:47.980 And you can see that it becomes colorful. 00:08:47.980 --> 00:08:50.780 And we are going to understand what is going on 00:08:50.780 --> 00:08:56.500 and where this color is coming from. 00:08:56.500 --> 00:09:00.280 And the good news is that, based on the knowledge we 00:09:00.280 --> 00:09:03.940 have learned, we are in a very good position 00:09:03.940 --> 00:09:06.770 to understand this phenomenon. 00:09:06.770 --> 00:09:11.870 So before we start to explain this phenomena, 00:09:11.870 --> 00:09:14.650 I would like to talk about a phenomenon, 00:09:14.650 --> 00:09:17.080 so-called interference. 00:09:17.080 --> 00:09:21.160 So suppose we have two electromagnetic waves. 00:09:21.160 --> 00:09:22.810 We can actually add them together 00:09:22.810 --> 00:09:25.160 because of superposition principle. 00:09:25.160 --> 00:09:27.160 So what we can do is that, suppose 00:09:27.160 --> 00:09:30.300 I have two electric fields. 00:09:30.300 --> 00:09:40.400 E1 is actually defined as A1 cosine omega t minus kz 00:09:40.400 --> 00:09:44.360 plus phi 1 in the x direction. 00:09:44.360 --> 00:09:47.340 So by now, you should know that this is actually 00:09:47.340 --> 00:09:52.710 the electric field propagating at angular frequency omega, 00:09:52.710 --> 00:10:00.580 with a wave number k, going toward positive z direction. 00:10:00.580 --> 00:10:04.180 And the electric field is perpendicular to the direction 00:10:04.180 --> 00:10:08.500 of propagation in the x direction. 00:10:08.500 --> 00:10:14.410 And also, this electric field have a phase of phi 1. 00:10:14.410 --> 00:10:18.310 So that's actually what we know already by now. 00:10:18.310 --> 00:10:20.710 What does this mean, this expression mean? 00:10:20.710 --> 00:10:25.540 And it's actually the harmonic oscillating electric field. 00:10:25.540 --> 00:10:28.960 And of course, since I am talking about interference, 00:10:28.960 --> 00:10:34.990 basically, I can add this, the first electric field 00:10:34.990 --> 00:10:37.660 and the second electric field together 00:10:37.660 --> 00:10:39.020 and see what will happen. 00:10:39.020 --> 00:10:42.300 So now, if you define the second electric field 00:10:42.300 --> 00:10:46.615 to be A2, which is the amplitude, cosine omega 00:10:46.615 --> 00:10:52.630 t minus kz, basically, they have the same wavelengths and also 00:10:52.630 --> 00:10:54.810 the angular frequency-- 00:10:54.810 --> 00:10:57.130 plus phi 2. 00:10:57.130 --> 00:10:59.710 But they have different phase. 00:10:59.710 --> 00:11:02.650 And of course, in this setup, I asked 00:11:02.650 --> 00:11:05.140 them to be pointing to the same direction, which 00:11:05.140 --> 00:11:07.240 is the x direction. 00:11:07.240 --> 00:11:08.920 So we were wondering, what is going 00:11:08.920 --> 00:11:15.510 to happen if I consider the superposition of these two 00:11:15.510 --> 00:11:18.600 electric fields. 00:11:18.600 --> 00:11:22.900 And the total electric field, which is called the E vector, 00:11:22.900 --> 00:11:26.110 is actually E1 plus E2. 00:11:29.380 --> 00:11:31.810 So before that, I would like to remind you 00:11:31.810 --> 00:11:37.970 about pointing vector and also the so-called intensity. 00:11:37.970 --> 00:11:40.510 So pointing factor is actually defined 00:11:40.510 --> 00:11:43.300 as this S vector, pointing vector, 00:11:43.300 --> 00:11:51.370 is equal to 1 over mu 0 E cross B. So this is actually 00:11:51.370 --> 00:11:56.290 the directional flux of energy per unit area. 00:11:56.290 --> 00:12:00.520 So that should be the pointing vector which 00:12:00.520 --> 00:12:03.720 we have been using for a while. 00:12:03.720 --> 00:12:09.630 And also, another reminder is that, given the electric field, 00:12:09.630 --> 00:12:13.780 which is actually a harmonic progressing wave, 00:12:13.780 --> 00:12:16.750 the corresponding B field would be 00:12:16.750 --> 00:12:23.800 equal to 1/v. v is actually the speed of the light 00:12:23.800 --> 00:12:26.800 in some specific material. 00:12:26.800 --> 00:12:30.180 And k hat is actually the direction 00:12:30.180 --> 00:12:35.940 of propagation across E. This will give you 00:12:35.940 --> 00:12:38.260 the magnitude and also the direction 00:12:38.260 --> 00:12:42.100 of the corresponding magnetic field 00:12:42.100 --> 00:12:46.720 for the electromagnetic waves. 00:12:46.720 --> 00:12:49.750 So now I'm interested in what would 00:12:49.750 --> 00:12:59.770 be the resulting intensity, I, if I try to superimpose this, 00:12:59.770 --> 00:13:05.140 to try to put together these two electric fields. 00:13:05.140 --> 00:13:08.200 As you can see, these two electric field 00:13:08.200 --> 00:13:09.370 have different phase. 00:13:09.370 --> 00:13:11.140 The first one has phase phi 1. 00:13:11.140 --> 00:13:14.270 The second one, has phase phi 2. 00:13:14.270 --> 00:13:18.920 So this means that they may reach maxima or minima 00:13:18.920 --> 00:13:23.000 at different position in space. 00:13:23.000 --> 00:13:25.970 In this case, it's in the z direction. 00:13:25.970 --> 00:13:33.710 And what I'm actually defining here is two plane waves. 00:13:33.710 --> 00:13:37.020 And so it really depends on what would 00:13:37.020 --> 00:13:41.230 be the relative phase for the first and second 00:13:41.230 --> 00:13:43.490 electromagnetic wave. 00:13:43.490 --> 00:13:48.920 In order to quantify how much they cancel each other 00:13:48.920 --> 00:13:53.000 or how much they enhance each other, what I'm going to do 00:13:53.000 --> 00:13:58.670 is to evaluate the intensity of the resulting 00:13:58.670 --> 00:14:00.470 electromagnetic waves. 00:14:00.470 --> 00:14:02.210 And what is actually intensity? 00:14:02.210 --> 00:14:05.310 Intensity is actually the amplitude 00:14:05.310 --> 00:14:07.490 of the pointing vector. 00:14:07.490 --> 00:14:14.170 So I write this as the length of the S vector, pointing vector. 00:14:14.170 --> 00:14:19.010 And I can, of course, calculate what will be the value of this 00:14:19.010 --> 00:14:21.440 or, say, the length of the pointing vector. 00:14:21.440 --> 00:14:24.530 This will be equal to 1 over mu 0, 00:14:24.530 --> 00:14:26.630 based on these equations here-- 00:14:26.630 --> 00:14:27.770 just a reminder. 00:14:27.770 --> 00:14:32.160 And I would like to know what would be the length of the E 00:14:32.160 --> 00:14:35.030 cross B field. 00:14:35.030 --> 00:14:38.600 That will give you the length of the pointing vector. 00:14:38.600 --> 00:14:47.900 And basically, what you're going to get is 1 over mu 0 times-- 00:14:47.900 --> 00:14:52.790 since the B field is actually highly related 00:14:52.790 --> 00:14:59.780 to the electric field, and it takes a hit of 1/v, right, 00:14:59.780 --> 00:15:01.670 in terms of the size of the amplitude. 00:15:01.670 --> 00:15:08.430 So basically, you're going to get 1/v, E squared. 00:15:12.280 --> 00:15:16.420 And this cross product actually is OK, 00:15:16.420 --> 00:15:21.550 because it become E squared, because B field and the E field 00:15:21.550 --> 00:15:24.930 are always orthogonal to each other. 00:15:24.930 --> 00:15:28.140 And I can now rewrite this v. v is actually 00:15:28.140 --> 00:15:32.830 the velocity of the speed of light in matter. 00:15:32.830 --> 00:15:35.120 So basically, what I can rewrite is 00:15:35.120 --> 00:15:42.790 that will become 1/v will become c/n, 00:15:42.790 --> 00:15:49.060 which is actually the refractive index of a specific material. 00:15:49.060 --> 00:15:52.560 And still I have E squared here. 00:15:52.560 --> 00:15:56.320 Finally, I can rewrite this formula, 00:15:56.320 --> 00:16:05.320 since c is equal to 1 over square root mu 0 epsilon 0. 00:16:05.320 --> 00:16:09.030 Therefore, I can rewrite this expression 00:16:09.030 --> 00:16:11.210 in terms of epsilon 0. 00:16:11.210 --> 00:16:16.190 And what I'm going to get is c times n time epsilon 0, 00:16:16.190 --> 00:16:19.120 There E squared. 00:16:19.120 --> 00:16:25.570 So basically, what I do is I multiply both numerator 00:16:25.570 --> 00:16:33.490 and the denominator by c, and also 00:16:33.490 --> 00:16:35.560 I actually use this expression. 00:16:35.560 --> 00:16:37.990 Then I can actually cancel the mu 0 00:16:37.990 --> 00:16:44.380 and then write everything in terms of c, n, and epsilon 0. 00:16:44.380 --> 00:16:47.750 So until here, there was basically no magic. 00:16:47.750 --> 00:16:52.290 Basically, it's just rewriting the length 00:16:52.290 --> 00:16:58.570 of the pointing vector in terms of n and also 00:16:58.570 --> 00:17:01.640 the electric field. 00:17:01.640 --> 00:17:06.829 So now, what I am going to do is to plug in this expression 00:17:06.829 --> 00:17:11.270 into that formula and see what we are going to get. 00:17:11.270 --> 00:17:16.163 So let me evaluate what would be the E squared, 00:17:16.163 --> 00:17:17.329 the length of the E squared. 00:17:17.329 --> 00:17:22.099 So basically, the definition of the E is shown here. 00:17:22.099 --> 00:17:25.760 It's the superposition of E1 and E2. 00:17:25.760 --> 00:17:28.910 Therefore, I can now quickly write down 00:17:28.910 --> 00:17:31.430 what would be E squared here. 00:17:35.450 --> 00:17:42.140 Basically, you are going to get A1 squared, cosine squared, 00:17:42.140 --> 00:17:47.660 omega t minus kz plus phi 1. 00:17:47.660 --> 00:17:53.300 Basically, that is coming from the E1 times E1. 00:17:53.300 --> 00:17:55.540 The second term, which I'm going to get, 00:17:55.540 --> 00:17:59.040 is actually E2 times E2. 00:17:59.040 --> 00:18:03.650 E2 times E3, you are going to get A2 squared, 00:18:03.650 --> 00:18:12.360 cosine squared, omega t minus kz plus phi 2. 00:18:12.360 --> 00:18:15.890 Basically based on this equation, and I square it, 00:18:15.890 --> 00:18:19.890 and then I get the second part here. 00:18:19.890 --> 00:18:22.340 And finally, what I am going to get 00:18:22.340 --> 00:18:25.360 is the third term, which is actually E1 times E2. 00:18:30.920 --> 00:18:36.995 Basically, you are going to get 2 times A1 A2 cosine omega 00:18:36.995 --> 00:18:47.201 t minus kz plus phi 1 cosine omega t minus kz 00:18:47.201 --> 00:18:53.150 plus phi 2, which is actually the cross term of this E 00:18:53.150 --> 00:18:55.420 vector squared. 00:18:55.420 --> 00:18:56.360 Any questions so far? 00:18:59.710 --> 00:19:02.830 I hope this is pretty straightforward to you. 00:19:02.830 --> 00:19:06.610 And of course, I can now rewrite this product 00:19:06.610 --> 00:19:09.740 with 2 cosine, cosine times cosine, right? 00:19:09.740 --> 00:19:11.920 Basically, I can rewrite this using 00:19:11.920 --> 00:19:14.960 the formula, which we have related 00:19:14.960 --> 00:19:16.730 to a cosine times a cosine. 00:19:16.730 --> 00:19:27.620 Basically, I can rewrite this as 1/2 cosine 2 omega t minus 2kz 00:19:27.620 --> 00:19:37.510 plus phi 1 plus phi 2 plus 1/2 cosine phi 1 minus phi 2. 00:19:37.510 --> 00:19:39.850 So basically, the first term is actually 00:19:39.850 --> 00:19:42.400 collecting the content of the two cosine 00:19:42.400 --> 00:19:44.560 and their length together. 00:19:44.560 --> 00:19:47.920 The second term is actually calculating 00:19:47.920 --> 00:19:52.240 the difference between the content of the cosine function. 00:19:52.240 --> 00:19:55.770 And what I'm going to get is actually phi 1 minus phi 2. 00:19:58.990 --> 00:20:03.910 Now, based on this definition, intensity, 00:20:03.910 --> 00:20:07.800 I is actually equal to the magnitude of that 00:20:07.800 --> 00:20:09.010 pointing vector. 00:20:09.010 --> 00:20:11.290 So remember, our goal is to evaluate 00:20:11.290 --> 00:20:15.860 what will be the resulting average intensity. 00:20:15.860 --> 00:20:20.740 So now I can calculate what would be the average intensity 00:20:20.740 --> 00:20:23.260 over one period. 00:20:23.260 --> 00:20:29.310 So this will be equal to 1/T, integration over 0 00:20:29.310 --> 00:20:38.830 to T, one period, and the instantaneous intensity, I, dt. 00:20:41.572 --> 00:20:44.170 And what I am going to get is that-- 00:20:44.170 --> 00:20:46.090 so we have three terms. 00:20:46.090 --> 00:20:49.480 The first term is here, which is actually A1 squared, 00:20:49.480 --> 00:20:51.200 cosine squared. 00:20:51.200 --> 00:20:54.400 It's actually related to omega t. 00:20:54.400 --> 00:20:56.530 And the second term is here. 00:20:56.530 --> 00:21:01.570 It's also proportional to cosine squared omega t. 00:21:01.570 --> 00:21:03.850 And finally, we have two terms here, 00:21:03.850 --> 00:21:09.780 which is actually proportional to cosine 2 omega t. 00:21:09.780 --> 00:21:16.210 And finally, the last term is actually independent of time. 00:21:16.210 --> 00:21:22.620 So what I'm going to do is to evaluate the individual terms. 00:21:22.620 --> 00:21:24.880 For the first term, basically, A1 00:21:24.880 --> 00:21:27.610 squared, cosine squared, omega t. 00:21:27.610 --> 00:21:32.290 So by now, it should be pretty straightforward for you 00:21:32.290 --> 00:21:36.560 if I integrate cosine squared over one period of time, 00:21:36.560 --> 00:21:40.210 basically, what you are going to get is 1/2. 00:21:40.210 --> 00:21:45.220 So is actually done several times in the p set. 00:21:45.220 --> 00:21:48.160 So basically, what you're going to get is-- 00:21:48.160 --> 00:21:51.550 I am going to collect all of those constants from here 00:21:51.550 --> 00:21:52.300 and copy here. 00:21:52.300 --> 00:21:57.500 So basically, you have c times n times epsilon 0, 00:21:57.500 --> 00:22:05.110 which is actually coming from the definition of intensity. 00:22:05.110 --> 00:22:10.630 And then for the first term, what I am going to get 00:22:10.630 --> 00:22:14.250 is A1 squared divided by 2. 00:22:14.250 --> 00:22:17.220 This 1/2 is actually just an integral 00:22:17.220 --> 00:22:19.630 related to cosine squared. 00:22:19.630 --> 00:22:23.960 Similarly, you are going to get the same result, a very similar 00:22:23.960 --> 00:22:26.350 result, for the second term. 00:22:26.350 --> 00:22:31.610 The second term is going to give you A2 squared divided by 2. 00:22:35.440 --> 00:22:38.850 Finally, you can have the third term. 00:22:38.850 --> 00:22:43.040 The third turn is going to give you what value? 00:22:43.040 --> 00:22:45.494 Can somebody help me? 00:22:45.494 --> 00:22:46.370 AUDIENCE: 0. 00:22:46.370 --> 00:22:47.500 YEN-JIE LEE: Yes, 0, right? 00:22:47.500 --> 00:22:50.450 Because this is actually cosine 2 omega t, right? 00:22:50.450 --> 00:22:52.900 So if you integrate over one period, 00:22:52.900 --> 00:22:54.540 you are going to get 0 plus 0. 00:22:54.540 --> 00:22:56.940 Each period will give you 0, right? 00:22:56.940 --> 00:22:59.410 So 0 plus 0 is 0, so therefore you get 0. 00:22:59.410 --> 00:23:01.030 Very good. 00:23:01.030 --> 00:23:07.042 How about the last term, anybody can help me? 00:23:07.042 --> 00:23:08.797 AUDIENCE: It should remain as it is. 00:23:08.797 --> 00:23:09.880 YEN-JIE LEE: That's right. 00:23:09.880 --> 00:23:11.680 Because it's a constant. 00:23:11.680 --> 00:23:16.040 So the average of a constant is a constant, which is actually 00:23:16.040 --> 00:23:27.080 giving you 1/2 times 2 times A1 times A2 cosine 00:23:27.080 --> 00:23:29.470 phi 1 minus phi 2. 00:23:29.470 --> 00:23:32.680 Of course, I can cancel this 1/2, which 00:23:32.680 --> 00:23:35.200 is actually coming from here, and the 2, 00:23:35.200 --> 00:23:36.742 which is coming from here. 00:23:36.742 --> 00:23:43.200 And basically, you are getting A1 A2 cosine phi 1 minus phi 2. 00:23:43.200 --> 00:23:47.250 And I need to close this bracket. 00:23:47.250 --> 00:23:49.790 Any questions so far? 00:23:49.790 --> 00:23:54.100 So what we have been doing is that I evaluated 00:23:54.100 --> 00:23:57.530 the total electric field. 00:23:57.530 --> 00:23:59.740 I basically calculated the superposition 00:23:59.740 --> 00:24:01.030 of the two fields. 00:24:01.030 --> 00:24:05.620 And then I am interested in what would be the average intensity 00:24:05.620 --> 00:24:07.630 coming from this field. 00:24:07.630 --> 00:24:11.000 And I write down E squared explicitly. 00:24:11.000 --> 00:24:15.490 There are four terms and only three of them actually survive. 00:24:15.490 --> 00:24:21.550 And basically, the expression I'm getting is like this. 00:24:21.550 --> 00:24:24.280 Basically, you have some constant multiplied 00:24:24.280 --> 00:24:27.790 by A1 squared over 2 plus A2 squared over 2 00:24:27.790 --> 00:24:32.570 plus A1 A2 cosine phi 1 minus phi 2. 00:24:32.570 --> 00:24:37.960 You can see that the intensity depends on phi 1 and phi 2, 00:24:37.960 --> 00:24:40.100 right? 00:24:40.100 --> 00:24:44.320 So this actually would change the resulting intensity. 00:24:44.320 --> 00:24:48.670 So in order to get some idea about what does that mean 00:24:48.670 --> 00:24:53.800 and also how does the average intensity change 00:24:53.800 --> 00:24:58.060 as a function of phi 1 minus phi2, what I'm going to do 00:24:58.060 --> 00:25:03.220 is to define phi 1 minus phi 2 to be delta, which 00:25:03.220 --> 00:25:06.100 I will call phase difference. 00:25:06.100 --> 00:25:11.890 Then I would like to plot the averaging intensity, I, 00:25:11.890 --> 00:25:17.280 as a function of delta and see what's going to happen. 00:25:17.280 --> 00:25:19.420 So this is actually the result. So if I 00:25:19.420 --> 00:25:22.840 have the x-axis to be delta, which 00:25:22.840 --> 00:25:29.780 is actually phi 1 minus phi 2, and the y-axis is intensity. 00:25:29.780 --> 00:25:33.190 Of course, I would like to take out the constant, which 00:25:33.190 --> 00:25:36.520 is c times n times epsilon 0. 00:25:36.520 --> 00:25:41.740 So I am plotting the y-axis' average intensity divided 00:25:41.740 --> 00:25:46.150 by c times n times epsilon 0. 00:25:46.150 --> 00:25:48.210 What I'm going to get is something 00:25:48.210 --> 00:25:52.690 which is actually oscillating up and down, like this. 00:26:00.070 --> 00:26:06.895 The maxima value happens when delta is equal to 0. 00:26:09.670 --> 00:26:13.840 When delta is equal to 0, what is going to happen? 00:26:13.840 --> 00:26:18.715 This means that cosine delta is equal to what? 00:26:18.715 --> 00:26:23.020 , Therefore what you are going to get is A1 squared over 2 00:26:23.020 --> 00:26:27.000 plus A2 squared over 2 plus A1 A2. 00:26:37.770 --> 00:26:41.620 This is actually when there delta is equal to 0. 00:26:41.620 --> 00:26:44.860 And the intensity, as you reach maxima, 00:26:44.860 --> 00:26:48.640 and the maxima values is actually 1/2 00:26:48.640 --> 00:26:55.870 A1 plus A2 squared, based on these calculations. 00:26:55.870 --> 00:26:58.860 On the other hand, you can expect 00:26:58.860 --> 00:27:02.790 that that intensity will reach a minima when 00:27:02.790 --> 00:27:06.220 delta is equal to which value? 00:27:06.220 --> 00:27:09.349 Anybody can help me. 00:27:09.349 --> 00:27:09.890 AUDIENCE: Pi. 00:27:09.890 --> 00:27:11.380 YEN-JIE LEE: Pi, yes. 00:27:11.380 --> 00:27:14.020 When delta is pi, what is going to happen? 00:27:14.020 --> 00:27:17.320 Cosine pi is minus 1. 00:27:17.320 --> 00:27:21.280 So therefore, what you are getting is 1/2 A1 00:27:21.280 --> 00:27:26.740 squared plus A2 squared minus 2 A1 A2. 00:27:26.740 --> 00:27:35.300 And that will give you 1/2 A1 minus A2 squared. 00:27:35.300 --> 00:27:40.960 You can see that, when the filter is equal to 0 00:27:40.960 --> 00:27:46.660 or when the filter is equal to 2 pi, for example, 00:27:46.660 --> 00:27:51.440 if you increase the phase difference large enough, 00:27:51.440 --> 00:27:56.890 or the filter is actually 4 pi, all of those number 00:27:56.890 --> 00:28:02.530 will keep you maxima constructive interference. 00:28:02.530 --> 00:28:03.490 So what does that mean? 00:28:03.490 --> 00:28:08.650 That means you are adding these two electric fields 00:28:08.650 --> 00:28:12.550 in the most efficient way. 00:28:12.550 --> 00:28:18.910 On the other hand, when the value of the filter 00:28:18.910 --> 00:28:26.800 is equal to pi or equal to 3 pi or equal to 5 pi, et cetera, 00:28:26.800 --> 00:28:31.390 the intensity, the average intensity reaches a minima. 00:28:31.390 --> 00:28:34.930 That means, instead of adding them, 00:28:34.930 --> 00:28:37.240 you are actually canceling them. 00:28:37.240 --> 00:28:40.330 You are canceling the electric field 00:28:40.330 --> 00:28:46.390 of the first and the second electromagnetic wave. 00:28:46.390 --> 00:28:51.730 And now I give you a maxima intensity, which is 1/2 A1 00:28:51.730 --> 00:28:53.860 minus A2 squared. 00:28:53.860 --> 00:28:57.610 Just a reminder, this A1 and A2 is actually 00:28:57.610 --> 00:29:02.410 the amplitude of the first and second electric field. 00:29:02.410 --> 00:29:07.170 So what will happen if I set A1 equal to A2? 00:29:10.110 --> 00:29:14.130 If I set A1 equal to A2, that means the minima 00:29:14.130 --> 00:29:15.570 would be equal to what? 00:29:15.570 --> 00:29:16.350 AUDIENCE: 0. 00:29:16.350 --> 00:29:18.330 YEN-JIE LEE: 0, yeah, very good. 00:29:18.330 --> 00:29:20.956 How about the maxima? 00:29:20.956 --> 00:29:21.830 AUDIENCE: [INAUDIBLE] 00:29:21.830 --> 00:29:25.240 YEN-JIE LEE: It will be A1 plus A2, right? 00:29:25.240 --> 00:29:30.130 So basically, you are going to get four times larger value 00:29:30.130 --> 00:29:35.330 compared to the intensity before you add them together. 00:29:35.330 --> 00:29:41.030 So individual intensity is I. And after adding them together, 00:29:41.030 --> 00:29:43.120 with delta equal to 0, you are going 00:29:43.120 --> 00:29:46.660 to get four times larger intensity 00:29:46.660 --> 00:29:50.980 if the amplitude of the first and second electric field 00:29:50.980 --> 00:29:53.650 is the same. 00:29:53.650 --> 00:29:55.500 So very good. 00:29:55.500 --> 00:29:58.310 So that's actually the result of the calculation. 00:29:58.310 --> 00:30:03.370 And you can see that the amount of intensity we can get out 00:30:03.370 --> 00:30:07.770 of this highly depends on the filter, which is actually 00:30:07.770 --> 00:30:09.790 the phase difference between the first 00:30:09.790 --> 00:30:12.650 and the second electric field. 00:30:12.650 --> 00:30:17.290 Can we actually get some more feeling about this addition? 00:30:17.290 --> 00:30:20.980 So what I am going to do is to, again, 00:30:20.980 --> 00:30:27.460 write everything down in terms of imaginary number 00:30:27.460 --> 00:30:30.160 or, say, a complex number. 00:30:30.160 --> 00:30:38.220 So if I rewrite the electric field, E1, 00:30:38.220 --> 00:30:45.570 as a real part of A1 exponential i phi 1, 00:30:45.570 --> 00:30:51.400 exponential i omega t minus kz. 00:30:51.400 --> 00:30:54.340 In the x direction. 00:30:54.340 --> 00:30:58.240 And I can also rewrite the expression 00:30:58.240 --> 00:31:00.730 for the second electric field to be 00:31:00.730 --> 00:31:07.225 the real part of A2 exponential i phi 2, exponential omega 00:31:07.225 --> 00:31:13.180 t minus kz, again, in the x direction 00:31:13.180 --> 00:31:18.430 So if I add these two fields together, what I am doing 00:31:18.430 --> 00:31:24.520 is like in the complex plane I have an imaginary number 00:31:24.520 --> 00:31:27.310 contribution in the y direction. 00:31:27.310 --> 00:31:32.580 And the real part is actually in the x direction. 00:31:32.580 --> 00:31:36.890 Suppose omega t minus kz is 0. 00:31:36.890 --> 00:31:41.260 At some instant of time omega t minus kz is equal to 0. 00:31:41.260 --> 00:31:46.090 So what I'm doing is I have the first vector, which is actually 00:31:46.090 --> 00:31:52.810 presenting the contribution of the first electric field. 00:31:52.810 --> 00:31:58.460 And this electric field is going to be pointing to a direction 00:31:58.460 --> 00:32:05.200 phi 1 away from the real axis with amplitude equal to A1. 00:32:05.200 --> 00:32:08.860 This is actually what we learned from the first lecture. 00:32:08.860 --> 00:32:13.560 And if I add the second electric field, what I'm going to get 00:32:13.560 --> 00:32:21.820 is another vector, which is actually A2 in length. 00:32:21.820 --> 00:32:29.320 And the angle is phi 2, here. 00:32:29.320 --> 00:32:35.560 So the resulting amplitude is actually 00:32:35.560 --> 00:32:42.570 when I take the real part of the first and second expression, 00:32:42.570 --> 00:32:44.950 adding them together. 00:32:44.950 --> 00:32:51.220 Basically, I am taking a projection to the real axis. 00:32:51.220 --> 00:32:56.800 And that is actually the resulting amplitude 00:32:56.800 --> 00:33:02.050 of the electric field, which is actually the superposition 00:33:02.050 --> 00:33:04.330 of the first and second field. 00:33:04.330 --> 00:33:16.150 So you can see that, when phi 1 is 00:33:16.150 --> 00:33:21.710 equal to phi 2, what is going to happen is the following. 00:33:21.710 --> 00:33:24.640 So basically, what you're actually going to get 00:33:24.640 --> 00:33:29.260 is that you are increasing the length 00:33:29.260 --> 00:33:35.710 of the routing vector, which is the addition of the two 00:33:35.710 --> 00:33:37.580 vectors. 00:33:37.580 --> 00:33:42.310 You are actually getting a maxima out of this addition. 00:33:42.310 --> 00:33:45.790 Because phi 1 is equal to phi 2, therefore, these two vectors 00:33:45.790 --> 00:33:49.060 form a straight line, therefore, you 00:33:49.060 --> 00:33:53.950 can actually add and get maximum amount 00:33:53.950 --> 00:33:59.080 of the amplitude out of this. 00:33:59.080 --> 00:34:05.300 On the other hand, when phi 1 minus phi 2 00:34:05.300 --> 00:34:14.949 is pi, which I define as delta, when this happens, 00:34:14.949 --> 00:34:19.510 what we are doing is like addition of two vectors, 00:34:19.510 --> 00:34:23.409 in a complex plane, but they are pointing 00:34:23.409 --> 00:34:25.870 to the opposite direction. 00:34:25.870 --> 00:34:27.790 So the first one will be like this. 00:34:27.790 --> 00:34:31.630 And the second one will be looking like that. 00:34:31.630 --> 00:34:35.260 And they are actually trying to cancel each other. 00:34:35.260 --> 00:34:39.300 So that's actually how you can understand what is happening 00:34:39.300 --> 00:34:41.310 with different delta values. 00:34:41.310 --> 00:34:44.330 In this case, delta is equal to 0. 00:34:44.330 --> 00:34:46.960 The phase difference is equal to 0. 00:34:46.960 --> 00:34:51.159 And in the second case, phase difference is equal to pi. 00:34:51.159 --> 00:34:55.790 And then what happened in between is like this. 00:34:55.790 --> 00:34:58.730 You are adding them sort of together but not 00:34:58.730 --> 00:35:02.830 in the most efficient way or the most destructive way. 00:35:02.830 --> 00:35:05.290 And you are actually evaluating what 00:35:05.290 --> 00:35:08.830 will be the resulting amplitude by looking 00:35:08.830 --> 00:35:13.120 at the vector sum of the first and second field. 00:35:13.120 --> 00:35:16.180 So I hope that this will give you a some more intuition 00:35:16.180 --> 00:35:19.450 about what we have been doing. 00:35:19.450 --> 00:35:20.380 Any questions so far? 00:35:24.860 --> 00:35:29.150 So now, we are actually in a very good position 00:35:29.150 --> 00:35:32.960 once we understand this superposition 00:35:32.960 --> 00:35:36.410 of the two electric fields and the interference. 00:35:36.410 --> 00:35:40.160 Basically, the size of the resulting intensity 00:35:40.160 --> 00:35:43.070 will be highly dependent on the phase difference 00:35:43.070 --> 00:35:45.170 between the two fields. 00:35:45.170 --> 00:35:47.630 Then we are in a very good position 00:35:47.630 --> 00:35:53.270 to discuss the phenomenon which we just see in the demo. 00:35:53.270 --> 00:35:58.730 So before I actually perform the calculation 00:35:58.730 --> 00:36:00.590 and give you the explanation, I would like 00:36:00.590 --> 00:36:03.560 to take a vote, as of usual. 00:36:03.560 --> 00:36:08.390 So the question we are asking is, in addition 00:36:08.390 --> 00:36:11.630 to what we see in the demo-- 00:36:11.630 --> 00:36:16.310 we see a colorful bubble-- 00:36:16.310 --> 00:36:22.550 how thick is the soap film such that you can see color 00:36:22.550 --> 00:36:25.070 from the reflected light? 00:36:25.070 --> 00:36:28.850 The first option is maybe it's like 1 millimeter, which 00:36:28.850 --> 00:36:30.290 is possible. 00:36:30.290 --> 00:36:35.090 And that is about the size of the head of a pin. 00:36:35.090 --> 00:36:40.580 Or it can be 100 micron, so that's actually about the size, 00:36:40.580 --> 00:36:45.020 the thickness is about the size of the human hair. 00:36:45.020 --> 00:36:50.930 Or 100 nanometer, which is the size of the virus. 00:36:50.930 --> 00:36:54.410 How many of you think the thickness 00:36:54.410 --> 00:36:56.990 is roughly 1 millimeter? 00:36:56.990 --> 00:36:59.700 Raise your hand. 00:36:59.700 --> 00:37:01.447 Nobody thinks so. 00:37:01.447 --> 00:37:01.946 Really? 00:37:08.520 --> 00:37:11.460 Actually nobody think that's the case. 00:37:11.460 --> 00:37:14.090 How about 100 micron? 00:37:14.090 --> 00:37:15.680 How many of you think so? 00:37:28.680 --> 00:37:31.250 How about 100 nanometer? 00:37:31.250 --> 00:37:32.998 Me How many of you? 00:37:44.400 --> 00:37:47.140 So that is actually the vote. 00:37:47.140 --> 00:37:49.570 And we are going to know the result very soon. 00:37:49.570 --> 00:37:51.190 And how about the rest? 00:37:53.950 --> 00:37:54.770 Cool. 00:37:54.770 --> 00:37:58.900 So now we are going to solve the puzzle. 00:37:58.900 --> 00:38:03.670 So just a quick reminder about what 00:38:03.670 --> 00:38:06.620 we have learned from the last lecture. 00:38:06.620 --> 00:38:11.090 So there is a reason why we have the lecture first 00:38:11.090 --> 00:38:15.410 on the reflection of an electromagnetic wave before we 00:38:15.410 --> 00:38:19.560 discuss the color of the bubble. 00:38:19.560 --> 00:38:26.200 So from the last lecture, suppose 00:38:26.200 --> 00:38:33.080 I have two materials, which form an interface between material 00:38:33.080 --> 00:38:36.640 number 1, with refractive index n1, 00:38:36.640 --> 00:38:41.760 and the second material has a refractive index n2. 00:38:41.760 --> 00:38:45.020 If I have an incident wave, incident 00:38:45.020 --> 00:38:50.670 electromagnetic plane wave, and the incident angle 00:38:50.670 --> 00:38:57.960 is actually, in this case, 0, that means this incident plane 00:38:57.960 --> 00:39:02.680 wave is actually propagating in a direction which is actually 00:39:02.680 --> 00:39:05.800 hitting the surface directly. 00:39:05.800 --> 00:39:08.950 So if the initial amplitude is A, 00:39:08.950 --> 00:39:11.140 what we have learned from last time 00:39:11.140 --> 00:39:13.780 is that there will be a reflective wave, which 00:39:13.780 --> 00:39:17.620 is actually R times A. R is actually 00:39:17.620 --> 00:39:21.970 reflective coefficient. 00:39:21.970 --> 00:39:25.850 And finally, you have also the transmitted wave, 00:39:25.850 --> 00:39:34.510 which I call T times A, where is the transmission coefficient. 00:39:34.510 --> 00:39:38.380 From the exercise, which we actually already 00:39:38.380 --> 00:39:47.270 done last time, R is equal to n1 minus n2 divided by n1 plus n2. 00:39:47.270 --> 00:39:51.010 And the transmission coefficient is actually 00:39:51.010 --> 00:39:58.630 T equal to 2n1 divided by n1 plus n2. 00:39:58.630 --> 00:40:01.870 So basically, what I am actually talking about here 00:40:01.870 --> 00:40:04.150 is a conclusion from the exercise 00:40:04.150 --> 00:40:10.960 we have done in the last lecture, just a quick reminder. 00:40:10.960 --> 00:40:17.500 So I would like to discuss with you various situation related 00:40:17.500 --> 00:40:19.970 to R value. 00:40:19.970 --> 00:40:24.730 So the n1 and the n2 are related to the property 00:40:24.730 --> 00:40:27.790 of the first medium and the second medium. 00:40:27.790 --> 00:40:36.700 So it could be that n1 is actually greater than n2. 00:40:36.700 --> 00:40:42.760 So if n1 is greater than n2 in the experimental setup, 00:40:42.760 --> 00:40:48.370 that means that R will be greater than 0. 00:40:51.050 --> 00:40:52.940 Because R is actually n1 minus n2 00:40:52.940 --> 00:40:57.090 divided by the sum of n1 and n2. 00:40:57.090 --> 00:41:01.700 Therefore, what I'm going to get is something like this. 00:41:01.700 --> 00:41:04.750 So basically, I'm going to have an incident 00:41:04.750 --> 00:41:09.580 wave like this, where, say, I use the notation pointing 00:41:09.580 --> 00:41:12.250 upwards. 00:41:12.250 --> 00:41:19.690 Once they got reflected, it is actually still like this, 00:41:19.690 --> 00:41:25.070 pointing upward, because the R is actually greater than 0. 00:41:25.070 --> 00:41:29.830 There's no changing sign in the amplitude. 00:41:29.830 --> 00:41:41.582 Therefore, there's an no flip in amplitude 00:41:41.582 --> 00:41:46.960 if n1 is actually greater than n2. 00:41:46.960 --> 00:41:50.080 On the other hand, the transmitted wave, 00:41:50.080 --> 00:41:54.530 if you look at the functional form of the transmitted wave, 00:41:54.530 --> 00:41:57.400 and transmission coefficient, T is actually 00:41:57.400 --> 00:42:02.050 equal to 2n1 divided by n1 plus n2. 00:42:02.050 --> 00:42:05.900 It's always positive. 00:42:05.900 --> 00:42:11.370 Therefore, will there be any possibility to flip the sign? 00:42:11.370 --> 00:42:12.400 No. 00:42:12.400 --> 00:42:14.110 You are absolutely right. 00:42:14.110 --> 00:42:16.750 So therefore, what is going to happen 00:42:16.750 --> 00:42:20.020 is that I will use this little arrow to keep 00:42:20.020 --> 00:42:23.260 track of the sign change. 00:42:23.260 --> 00:42:27.340 Basically, you'll see that after it pass through the boundary, 00:42:27.340 --> 00:42:32.290 there will be no change in sign in amplitude no matter 00:42:32.290 --> 00:42:33.910 what happens. 00:42:33.910 --> 00:42:41.230 On the other hand, if I have the situation n1 smaller than n2, 00:42:41.230 --> 00:42:43.140 what is going to happen? 00:42:43.140 --> 00:42:51.520 If you calculate the R value, it will be negative, right? 00:42:51.520 --> 00:42:54.890 In this case, R will be smaller than 0. 00:42:54.890 --> 00:43:03.380 So what is going to happen is that, initially, the incident 00:43:03.380 --> 00:43:06.130 wave has positive amplitude. 00:43:06.130 --> 00:43:08.710 And I keep track of the sign of this amplitude 00:43:08.710 --> 00:43:13.120 by this arrow pointing up. 00:43:13.120 --> 00:43:17.080 Because the R is actually smaller than 0, 00:43:17.080 --> 00:43:23.080 therefore, there is flip in sign in the amplitude. 00:43:23.080 --> 00:43:24.770 So what is going to happen? 00:43:24.770 --> 00:43:26.620 So the reflective wave will look like this. 00:43:26.620 --> 00:43:32.470 And I use this arrow to keep track of the flip in amplitude. 00:43:32.470 --> 00:43:35.540 And finally, as I mentioned before, 00:43:35.540 --> 00:43:39.890 the transmitted wave, the T, is always positive. 00:43:39.890 --> 00:43:46.780 Therefore, there will be no change in sign in amplitude. 00:43:46.780 --> 00:43:49.390 Finally, the third example is, if I 00:43:49.390 --> 00:43:51.690 have somehow two different materials, 00:43:51.690 --> 00:43:55.220 but they have the same refractive index, what 00:43:55.220 --> 00:43:58.540 is going to happen is that there will be no reflection, 00:43:58.540 --> 00:44:01.240 and everything goes through. 00:44:01.240 --> 00:44:05.560 Even if you have two different kinds of material, but if they 00:44:05.560 --> 00:44:09.430 have the same refractive index, then what is going to happen 00:44:09.430 --> 00:44:11.240 is that everything will pass through. 00:44:11.240 --> 00:44:13.450 And what you are going to get is that you 00:44:13.450 --> 00:44:16.230 will have no reflected light. 00:44:16.230 --> 00:44:20.610 Meaning R is actually equal to 0. 00:44:20.610 --> 00:44:21.850 Any questions so far? 00:44:26.300 --> 00:44:29.840 I would like to make sure that everybody 00:44:29.840 --> 00:44:34.440 understands the consequence of this calculation. 00:44:34.440 --> 00:44:39.590 So if I introduce no flip in amplitude, 00:44:39.590 --> 00:44:42.350 this means that this contribution will 00:44:42.350 --> 00:44:46.310 introduce a filter equal to 0. 00:44:46.310 --> 00:44:51.140 So basically, there will be no change in the phase, 00:44:51.140 --> 00:44:54.830 because there's no flip in amplitude. 00:44:54.830 --> 00:45:02.120 On the other hand, if there's a changing sign in amplitude, 00:45:02.120 --> 00:45:05.050 what would be the resulting filter value? 00:45:05.050 --> 00:45:06.870 Can somebody actually tell me? 00:45:06.870 --> 00:45:07.810 AUDIENCE: Pi. 00:45:07.810 --> 00:45:09.240 YEN-JIE LEE: It would be pi. 00:45:09.240 --> 00:45:10.030 Very good. 00:45:10.030 --> 00:45:13.600 So that means you are getting hit 00:45:13.600 --> 00:45:16.660 by a phase difference of pi. 00:45:16.660 --> 00:45:20.650 Therefore, the amplitude changes by a factor 00:45:20.650 --> 00:45:25.300 of cosine pi, which is minus 1. 00:45:25.300 --> 00:45:28.450 So that is actually something pretty important 00:45:28.450 --> 00:45:38.990 when we have the discussion of the soap bubble reflection. 00:45:38.990 --> 00:45:43.450 So let me give you a quick example 00:45:43.450 --> 00:45:48.160 about why is actually the amount of the reflected light and also 00:45:48.160 --> 00:45:51.670 what is the amount of transmitted light. 00:45:51.670 --> 00:45:53.980 Let me give you a concrete example. 00:45:53.980 --> 00:46:01.660 For example, if I have n1 equal to 1, which is actually 00:46:01.660 --> 00:46:12.070 the refractive index of the air, and n2 equal to 1.5 00:46:12.070 --> 00:46:19.210 If that happens what is the resulting intensity? 00:46:19.210 --> 00:46:25.360 Just a quick reminder, average I, the average intensity 00:46:25.360 --> 00:46:33.680 will be equal to c times n times epsilon 0 A 00:46:33.680 --> 00:46:38.470 squared divided by 2, where A is the amplitude 00:46:38.470 --> 00:46:42.190 of the electric field. 00:46:42.190 --> 00:46:43.420 Just a quick reminder. 00:46:43.420 --> 00:46:48.370 And this 1/2 is coming from the time average, just a reminder. 00:46:48.370 --> 00:46:53.470 So now I can go ahead and use these two formula, R and T, 00:46:53.470 --> 00:46:57.790 to calculate the reflection coefficient 00:46:57.790 --> 00:46:59.650 and this transmission coefficient. 00:46:59.650 --> 00:47:07.510 So R will equal to 1 minus 1.5 divided by 1 plus 1.5. 00:47:07.510 --> 00:47:15.100 So basically, you get minus 0.5 divided by 2.5. 00:47:15.100 --> 00:47:23.560 And that is actually going to give you minus 0.2. 00:47:23.560 --> 00:47:25.410 Of course, I can also calculate what 00:47:25.410 --> 00:47:33.220 will be the T, which would be a 2 divided 2.5. 00:47:33.220 --> 00:47:39.970 So basically what you are getting is 0.8. 00:47:39.970 --> 00:47:47.020 So I can now calculate what will be the resulting intensity 00:47:47.020 --> 00:47:49.190 of the reflected light. 00:47:49.190 --> 00:47:52.150 Everybody's following? 00:47:52.150 --> 00:47:58.340 So what would be the intensity of the reflected light? 00:47:58.340 --> 00:48:06.090 This will be equal to minus 0.2 squared, right? 00:48:06.090 --> 00:48:12.330 Because the average intensity is proportional to A squared. 00:48:12.330 --> 00:48:16.410 A is actually the amplitude of the electric field. 00:48:16.410 --> 00:48:19.020 R should tell you what is actually 00:48:19.020 --> 00:48:22.390 the relative amplitude between the reflected light 00:48:22.390 --> 00:48:25.660 and the incident light. 00:48:25.660 --> 00:48:31.480 Therefore, you are getting hit by 0.2 squared multiplied 00:48:31.480 --> 00:48:37.130 by the initial intensity. 00:48:37.130 --> 00:48:39.190 Basically what you are going to get 00:48:39.190 --> 00:48:47.210 is 0.04 I, initial intensity. 00:48:47.210 --> 00:48:55.040 So basically 4% of the light is reflected. 00:48:55.040 --> 00:48:57.260 That may surprise you a bit, right? 00:48:57.260 --> 00:49:01.610 Because when you see, for example, the soap bubble, 00:49:01.610 --> 00:49:04.040 you see that it is still pretty bright, right? 00:49:04.040 --> 00:49:08.615 But in reality, only 4% of the light or 4% of the intensity 00:49:08.615 --> 00:49:10.610 got reflected. 00:49:10.610 --> 00:49:15.090 That is because your eyes is actually having nonlinear. 00:49:15.090 --> 00:49:17.315 Your eye responds to the-- 00:49:17.315 --> 00:49:22.130 or, say, receiving or interpreting the intensity 00:49:22.130 --> 00:49:24.380 is really highly nonlinear. 00:49:24.380 --> 00:49:28.210 So basically, you get 4% reflected. 00:49:28.210 --> 00:49:31.500 And the rest actually goes through. 00:49:31.500 --> 00:49:37.450 And just to convince you that the total intensity is 100%, 00:49:37.450 --> 00:49:42.320 we can calculate what would be the intensity 00:49:42.320 --> 00:49:43.890 of the transmitted light. 00:49:43.890 --> 00:49:47.050 This will be equal to 1.5-- 00:49:47.050 --> 00:49:54.470 this is actually related to n2, because the intensity 00:49:54.470 --> 00:50:00.470 is proportional to c n epsilon 0 A squared over 2-- 00:50:00.470 --> 00:50:03.110 times T squared. 00:50:03.110 --> 00:50:09.530 So basically you have a 0.8 squared 00:50:09.530 --> 00:50:13.680 and the I initial intensity. 00:50:13.680 --> 00:50:15.860 And if I calculate this value, basically you 00:50:15.860 --> 00:50:21.450 are going to get 96% of the initial intensity. 00:50:21.450 --> 00:50:25.490 So 96% of the initial intensity actually passes 00:50:25.490 --> 00:50:30.470 through the boundary and continues and propagates 00:50:30.470 --> 00:50:34.150 in the second medium, which, actually, in this case, 00:50:34.150 --> 00:50:35.214 is the soap. 00:50:42.770 --> 00:50:45.530 So the picture is the following. 00:50:45.530 --> 00:50:53.970 When 100% of light intensity going 00:50:53.970 --> 00:50:58.280 towards the boundary, what is going to happen 00:50:58.280 --> 00:51:06.220 is that 4% of the light got reflected. 00:51:06.220 --> 00:51:09.030 4% of intensity got reflected. 00:51:09.030 --> 00:51:14.140 And also, because n1 is smaller than n2, 00:51:14.140 --> 00:51:21.750 therefore, there is a flip in sign in the amplitude. 00:51:21.750 --> 00:51:27.680 And the rest continues, 96% of them. 00:51:27.680 --> 00:51:34.400 And there is no flipping sign in the amplitude. 00:51:34.400 --> 00:51:36.180 Any questions so far? 00:51:36.180 --> 00:51:37.515 We're really pretty close. 00:51:41.040 --> 00:51:45.480 So now we are in a position to discuss what is actually really 00:51:45.480 --> 00:51:50.290 happening to this soap bubble. 00:51:50.290 --> 00:51:55.070 So I'm going to keep this result here. 00:51:58.740 --> 00:52:04.330 And I will now discuss a situation in which 00:52:04.330 --> 00:52:09.002 you have two interfaces. 00:52:12.860 --> 00:52:19.150 So suppose I zoom in, zoom, and zoom in this soap bubble 00:52:19.150 --> 00:52:21.100 and put it on the board. 00:52:21.100 --> 00:52:29.990 So this is actually the soap film. 00:52:29.990 --> 00:52:35.090 And I have now an incident wave, which is actually 00:52:35.090 --> 00:52:40.310 going into this bubble. 00:52:40.310 --> 00:52:43.700 So now I have 100%, which is actually 00:52:43.700 --> 00:52:48.110 going toward this film. 00:52:48.110 --> 00:52:54.170 So after this light, this plane wave 00:52:54.170 --> 00:52:58.370 hits the film, what is going to happen? 00:52:58.370 --> 00:53:01.750 The first thing which happens is that there will be 00:53:01.750 --> 00:53:06.306 4% of the light got reflected. 00:53:09.420 --> 00:53:11.050 n2 is equal to 1.5. 00:53:11.050 --> 00:53:12.704 It's the same setup, just a reminder, 00:53:12.704 --> 00:53:17.060 just to make sure everybody is on the same page. 00:53:17.060 --> 00:53:20.350 So 4% of the light got reflected. 00:53:20.350 --> 00:53:23.225 Of course, the sign changed. 00:53:27.880 --> 00:53:33.880 96% of the intensity actually continue. 00:53:33.880 --> 00:53:37.570 And what is actually happening is 00:53:37.570 --> 00:53:45.010 that there will be no change in amplitude in sign. 00:53:45.010 --> 00:53:47.830 And this is actually not the end of the story, right? 00:53:47.830 --> 00:53:51.850 Because the light will continue and continue to propagate. 00:53:51.850 --> 00:53:53.680 What is going to happen is that it 00:53:53.680 --> 00:54:02.120 will reach another boundary, where the incident light is 00:54:02.120 --> 00:54:06.630 you're traveling, from n2 refractive index material, 00:54:06.630 --> 00:54:09.520 to n1, which is actually the air. 00:54:09.520 --> 00:54:13.600 Now I have a situation where the light is actually 00:54:13.600 --> 00:54:18.080 going through the boundary and going out of the air. 00:54:18.080 --> 00:54:22.690 That means the light is actually going into the bubble. 00:54:22.690 --> 00:54:25.280 So this is actually inside the bubble. 00:54:31.130 --> 00:54:33.910 So what is going to happen is the following. 00:54:33.910 --> 00:54:40.130 Basically, the calculation is the same, except that now 00:54:40.130 --> 00:54:46.340 the R is actually 0.2 instead of minus 0.2, right? 00:54:46.340 --> 00:54:51.110 Because now n2 minus n2 is actually 0.5. 00:54:51.110 --> 00:54:55.280 0.5 divided by 2.5 is positive 0.2. 00:54:55.280 --> 00:54:57.200 So basically, what you are going to get 00:54:57.200 --> 00:55:03.486 is a reflected light, which actually doesn't change. 00:55:03.486 --> 00:55:08.240 It doesn't change the sign of the amplitude. 00:55:08.240 --> 00:55:11.270 And what is actually the intensity? 00:55:11.270 --> 00:55:18.280 The intensity will be 96% times 4%, because only 4% 00:55:18.280 --> 00:55:20.650 of the light got reflected. 00:55:20.650 --> 00:55:23.680 And of course, a large fraction of the light 00:55:23.680 --> 00:55:28.580 actually pass through the bubble, 96% times 96%. 00:55:28.580 --> 00:55:32.350 And this would be, again, pointing upward, 00:55:32.350 --> 00:55:36.460 because T is always positive. 00:55:36.460 --> 00:55:39.437 Any questions so far. 00:55:39.437 --> 00:55:41.520 You can see that this actually really interesting, 00:55:41.520 --> 00:55:43.670 because most of the light actually 00:55:43.670 --> 00:55:47.990 pass through the bubble. 00:55:47.990 --> 00:55:49.490 So that's actually already one thing 00:55:49.490 --> 00:55:52.010 we've learned from this exercise. 00:55:52.010 --> 00:55:55.860 Now, what is going to happen to this light if I continue 00:55:55.860 --> 00:55:58.220 and increase the time? 00:55:58.220 --> 00:56:00.640 What is going to happen is that this reflected 00:56:00.640 --> 00:56:06.650 light, from the second surface or second boundary, 00:56:06.650 --> 00:56:10.580 will go backward and pass through the first boundary 00:56:10.580 --> 00:56:13.070 again. 00:56:13.070 --> 00:56:14.930 What is going to happen is the following. 00:56:14.930 --> 00:56:20.150 So basically, we're going to get, again, transmitted light 00:56:20.150 --> 00:56:21.393 and the reflected light. 00:56:26.090 --> 00:56:29.360 What will be the sign of the reflected light? 00:56:29.360 --> 00:56:31.769 Will the arrow be pointing up or down? 00:56:31.769 --> 00:56:32.310 AUDIENCE: Up. 00:56:32.310 --> 00:56:34.055 YEN-JIE LEE: Up, yeah, very good. 00:56:34.055 --> 00:56:36.470 Right now, if you are bored, then that 00:56:36.470 --> 00:56:39.560 means I am very successful. 00:56:39.560 --> 00:56:44.840 So that means I'm getting 4% times 96% times 4%. 00:56:47.870 --> 00:56:51.080 What would be the sign for the transmitted light? 00:56:51.080 --> 00:56:52.019 Pointing up or down? 00:56:52.019 --> 00:56:52.560 AUDIENCE: Up. 00:56:52.560 --> 00:56:53.226 YEN-JIE LEE: Up. 00:56:53.226 --> 00:56:56.060 Very good, so everybody gets it. 00:56:56.060 --> 00:57:06.151 And 96% pass through, 96% times 96% times 4% will pass. 00:57:09.330 --> 00:57:11.820 And of course, I can now continue and continue. 00:57:11.820 --> 00:57:15.580 What is going to happen is that now you have learned 8.03. 00:57:15.580 --> 00:57:20.350 You will see that this is a crazy phenomenon. 00:57:20.350 --> 00:57:22.050 What is going to happen is that there 00:57:22.050 --> 00:57:28.710 will be a tiny fraction of the light which is trapped forever 00:57:28.710 --> 00:57:30.620 between the two surfaces. 00:57:30.620 --> 00:57:33.630 They are going to be bouncing back and forth, 00:57:33.630 --> 00:57:36.870 boo, boo, boo, boo, boo, boo, boo, boo, forever. 00:57:36.870 --> 00:57:40.290 Of course, the fraction of the intensity 00:57:40.290 --> 00:57:44.010 is really, really small. 00:57:44.010 --> 00:57:47.130 Because every time you've got the reflection actually 00:57:47.130 --> 00:57:50.660 happening, you take a hit of 4%. 00:57:50.660 --> 00:57:55.800 But since we are talking about theoretical physics, 00:57:55.800 --> 00:57:58.650 so, theoretically, that would continue forever. 00:57:58.650 --> 00:58:01.900 That's actually pretty interesting. 00:58:01.900 --> 00:58:06.030 And going back to practical situation, 00:58:06.030 --> 00:58:11.490 basically, I can safely ignore any further reflection, 00:58:11.490 --> 00:58:13.410 because they are hitting so hard, 00:58:13.410 --> 00:58:16.830 because every time I get a 4% hit, right? 00:58:16.830 --> 00:58:20.200 Therefore, I can ignore all the other contribution. 00:58:20.200 --> 00:58:25.710 And what we are actually seeing is what? 00:58:25.710 --> 00:58:29.200 Our eye is here. 00:58:29.200 --> 00:58:34.050 We see the contribution of the first pass, 00:58:34.050 --> 00:58:38.430 which is actually reflected from the first surface. 00:58:38.430 --> 00:58:41.560 The second pass, OK, it pass through the first surface, 00:58:41.560 --> 00:58:46.200 got reflected from the second boundary, 00:58:46.200 --> 00:58:54.090 and pass through the first boundary, in the second round, 00:58:54.090 --> 00:58:58.500 and then, also, reaching your eye. 00:58:58.500 --> 00:59:00.960 So what are we looking at ? 00:59:00.960 --> 00:59:04.590 We are looking at the superposition 00:59:04.590 --> 00:59:10.776 of two electromagnetic waves coming from one, 00:59:10.776 --> 00:59:15.935 which is like this, and two, which is actually like this. 00:59:23.310 --> 00:59:31.200 The question now is what is the thickness of the film? 00:59:31.200 --> 00:59:34.580 Now I can define the thickness or, say, 00:59:34.580 --> 00:59:39.060 the width of this film to be d. 00:59:39.060 --> 00:59:42.720 Now the question we are actually asking 00:59:42.720 --> 00:59:46.970 is, what would be the thickness, d, 00:59:46.970 --> 00:59:53.820 which is needed such that I can have constructive interference? 00:59:53.820 --> 00:59:58.010 Now the question becomes really clear. 00:59:58.010 --> 01:00:01.770 And we can actually calculate that 01:00:01.770 --> 01:00:06.840 by evaluating the phase difference between the path 01:00:06.840 --> 01:00:09.780 number one and the path number two. 01:00:09.780 --> 01:00:15.030 So now, in order to have constructive interference, 01:00:15.030 --> 01:00:18.420 I need a specific phase difference. 01:00:18.420 --> 01:00:22.310 But before that, I need to calculate the phase difference 01:00:22.310 --> 01:00:32.100 first between path number one and path number two. 01:00:32.100 --> 01:00:35.430 What will be the phase difference? 01:00:35.430 --> 01:00:42.190 The phase difference delta will be equal to, of course, pi. 01:00:42.190 --> 01:00:49.560 This pi contribution is coming from the flip of the amplitude. 01:00:49.560 --> 01:00:53.810 That will actually give you an pi phase difference. 01:00:53.810 --> 01:00:58.140 The second phase difference is coming from the difference 01:00:58.140 --> 01:01:01.900 in the optical path length. 01:01:01.900 --> 01:01:07.140 You can see that the first path, it doesn't go into the film. 01:01:07.140 --> 01:01:10.270 It got reflected, directly. 01:01:10.270 --> 01:01:15.300 And the second path, which is path number two, 01:01:15.300 --> 01:01:20.120 it takes more effort or more time 01:01:20.120 --> 01:01:23.740 for the light to go back and reach your eye. 01:01:23.740 --> 01:01:27.320 How big is the path length difference? 01:01:27.320 --> 01:01:33.030 The size of the path length difference is 2 times d, right? 01:01:36.540 --> 01:01:40.500 Of course, I need to actually translate that back 01:01:40.500 --> 01:01:45.330 to the phase. 01:01:45.330 --> 01:01:49.020 So first, I need to actually calculate how many period. 01:01:49.020 --> 01:01:54.080 So the length divided by lambda will be the period. 01:01:54.080 --> 01:02:00.510 So lambda is actually the wavelength 01:02:00.510 --> 01:02:03.780 of the incident light. 01:02:03.780 --> 01:02:06.100 But I am missing a factor here. 01:02:06.100 --> 01:02:07.410 And can somebody help me? 01:02:07.410 --> 01:02:10.110 Because this lambda is actually inside the material, right? 01:02:10.110 --> 01:02:11.948 So which factor, I'm missing? 01:02:11.948 --> 01:02:13.232 AUDIENCE: n2. 01:02:13.232 --> 01:02:14.230 YEN-JIE LEE: n2, right? 01:02:14.230 --> 01:02:15.580 Yeah, thank you very much. 01:02:15.580 --> 01:02:19.060 So basically, inside the material, 01:02:19.060 --> 01:02:23.440 since the speed of light is 1.5 times smaller 01:02:23.440 --> 01:02:25.870 than the speed of light in vacuum, 01:02:25.870 --> 01:02:34.840 therefore, the wavelength is actually lambda divided by n2. 01:02:34.840 --> 01:02:37.030 And this is actually the number of period. 01:02:37.030 --> 01:02:40.780 And now, I need to translate that to phase difference. 01:02:40.780 --> 01:02:44.620 Therefore, I multiply this by 2pi. 01:02:44.620 --> 01:02:49.080 So you can see that now I have successfully evaluated 01:02:49.080 --> 01:02:52.510 or quantified the phase difference between path number 01:02:52.510 --> 01:02:54.370 one and two. 01:02:54.370 --> 01:02:56.830 That is there are two contributions. 01:02:56.830 --> 01:02:59.170 The first one is pi. 01:02:59.170 --> 01:03:04.750 It's related to the flip in amplitude. 01:03:04.750 --> 01:03:08.280 The second contribution, the blue one, 01:03:08.280 --> 01:03:13.720 is actually coming from the optical path length difference. 01:03:13.720 --> 01:03:19.600 And of course, we can evaluate that really precisely. 01:03:19.600 --> 01:03:22.240 Therefore, we can now quickly conclude 01:03:22.240 --> 01:03:36.170 that, in order to have constructive interference, 01:03:36.170 --> 01:03:43.190 I need to have filter equal to 2N pi, where N is an integer. 01:03:48.080 --> 01:03:51.410 And in order to have destructive interference, 01:03:51.410 --> 01:04:02.870 I need to have filter equal to 2N 01:04:02.870 --> 01:04:07.420 plus 1, pi, which is actually the result 01:04:07.420 --> 01:04:12.070 of the calculation which we have done, I think, before. 01:04:12.070 --> 01:04:12.880 Yeah, there. 01:04:12.880 --> 01:04:15.250 So this is actually based on the calculation 01:04:15.250 --> 01:04:18.730 we have done in the beginning. 01:04:18.730 --> 01:04:22.120 So we are really close. 01:04:22.120 --> 01:04:28.660 So now we have this result, delta is equal to pi plus 2d, 01:04:28.660 --> 01:04:32.670 times 2 pi divided by lambda divided by n2, right, 01:04:32.670 --> 01:04:35.680 so this complicated formula? 01:04:35.680 --> 01:04:39.010 Now we are in the position to evaluate what 01:04:39.010 --> 01:04:43.030 would be the phase difference. 01:04:43.030 --> 01:04:45.310 So the first thing which I would like to discuss 01:04:45.310 --> 01:04:51.140 is that, when d goes to 0, what does is actually the limit? 01:04:51.140 --> 01:04:57.780 The limit is when the width of the film 01:04:57.780 --> 01:05:01.030 is really, really small, it goes to 0. 01:05:01.030 --> 01:05:02.770 What is going to happen? 01:05:02.770 --> 01:05:10.621 You are going to have destructive interference. 01:05:14.310 --> 01:05:15.530 Why is that? 01:05:15.530 --> 01:05:19.450 That is because, even when you have d equal to 0, 01:05:19.450 --> 01:05:27.120 the filter is pi because of the flip in sign in path number 01:05:27.120 --> 01:05:29.640 one. 01:05:29.640 --> 01:05:31.490 The second thing is that now I can 01:05:31.490 --> 01:05:35.316 calculate what would be the constructive interference 01:05:35.316 --> 01:05:35.816 width. 01:05:44.650 --> 01:05:50.805 So this will happen when d is equal to 2N minus 1 lambda, 01:05:50.805 --> 01:05:53.620 divided by 4 n2. 01:05:53.620 --> 01:06:01.430 So basically, you can use that formula there and solve d. 01:06:01.430 --> 01:06:04.390 Then basically that's the formula we are going to get. 01:06:04.390 --> 01:06:07.750 And I will not go into detail with this. 01:06:10.420 --> 01:06:11.310 Any questions so far? 01:06:15.600 --> 01:06:25.750 So now, the third conclusion is that, if I 01:06:25.750 --> 01:06:35.890 fix d and the change in lambda, that is actually 01:06:35.890 --> 01:06:39.100 the more practical situation. 01:06:39.100 --> 01:06:40.900 Because I have the soap bubble. 01:06:40.900 --> 01:06:44.120 And it have a well-defined width, which is d. 01:06:44.120 --> 01:06:46.180 And what is happening is that I am trying 01:06:46.180 --> 01:06:50.285 to shine this soap bubble with light 01:06:50.285 --> 01:06:53.140 with different wavelengths, right? 01:06:53.140 --> 01:06:55.900 So that is actually the third situation. 01:06:55.900 --> 01:07:02.050 If I fix the width of the film, and the change the wavelength, 01:07:02.050 --> 01:07:04.640 lambda, what I am going to get is 01:07:04.640 --> 01:07:11.170 that the lambda max, which is the wavelength needed 01:07:11.170 --> 01:07:29.720 to have constructive interference, 01:07:29.720 --> 01:07:37.710 will be equal to 4d n2 divided by 2N minus 1. 01:07:37.710 --> 01:07:42.170 So basically, I can solve the lambda if I am given a d value. 01:07:45.180 --> 01:07:47.550 So actually, we already get the answer 01:07:47.550 --> 01:07:50.520 we are asking in the beginning. 01:07:50.520 --> 01:07:54.990 The first question is, why do we see color? 01:07:54.990 --> 01:07:57.650 The second question is, when I see color, 01:07:57.650 --> 01:08:01.080 what is actually the width of the soap film? 01:08:01.080 --> 01:08:03.945 We are going to know the result in a moment. 01:08:07.860 --> 01:08:11.170 So now, I have this formula in hand. 01:08:11.170 --> 01:08:24.979 If I have d roughly equal to 100 nanometer, which 01:08:24.979 --> 01:08:37.790 is the third option we were discussing, 01:08:37.790 --> 01:08:45.892 that is going to give you lambda maxima equal to 4 times 100 01:08:45.892 --> 01:08:49.910 nanometer times 1.5-- 01:08:49.910 --> 01:08:56.180 n2 is 1.5-- divided by 2N minus 1. 01:08:56.180 --> 01:09:00.450 So that is actually 600 nanometer 01:09:00.450 --> 01:09:03.000 divided by 2N minus 1. 01:09:06.050 --> 01:09:10.489 Suppose I have N equal to 1, basically I 01:09:10.489 --> 01:09:15.319 am getting 600 nanometer. 01:09:15.319 --> 01:09:22.220 Suppose I have N equal to 2, 2N minus 1 is actually 4 minus 1 01:09:22.220 --> 01:09:23.029 is 3. 01:09:23.029 --> 01:09:29.330 Therefore, you get 200 nanometer and 120 nanometer, 01:09:29.330 --> 01:09:32.050 et cetera, et cetera, which are the required 01:09:32.050 --> 01:09:38.870 wavelengths in order to have constructive interference 01:09:38.870 --> 01:09:42.930 between path number one and path number two. 01:09:42.930 --> 01:09:46.010 Everybody is following? 01:09:46.010 --> 01:09:50.450 If I plot the spectra of this lambda max, 01:09:50.450 --> 01:09:56.720 assuming d is 100 nanometer, what I am getting is like this. 01:10:00.740 --> 01:10:06.380 So this is a situation of very thin film. 01:10:06.380 --> 01:10:09.320 So this is the lambda. 01:10:09.320 --> 01:10:13.865 What I am getting is that there will be a maxima here, which 01:10:13.865 --> 01:10:17.700 is actually 600 nanometer. 01:10:20.580 --> 01:10:27.350 Red color is actually roughly 650 nanometer. 01:10:27.350 --> 01:10:31.130 This is red light. 01:10:31.130 --> 01:10:35.680 And this is actually roughly the range 01:10:35.680 --> 01:10:39.780 of the visible light, which is actually between lambda 01:10:39.780 --> 01:10:44.290 equal to lambda violet-- 01:10:44.290 --> 01:10:48.270 violet is equal to 400 nanometer. 01:10:48.270 --> 01:10:52.640 So you can see that the first maxima, lambda maxima, 01:10:52.640 --> 01:10:55.580 where you have constructive interference 01:10:55.580 --> 01:10:58.640 is at 600 nanometer. 01:10:58.640 --> 01:11:02.400 So that means you are going to see what kind of color 01:11:02.400 --> 01:11:03.623 in your soap bubble? 01:11:03.623 --> 01:11:04.490 AUDIENCE: Red. 01:11:04.490 --> 01:11:07.400 YEN-JIE LEE: You are going to see red, right? 01:11:07.400 --> 01:11:10.760 And then the next wavelength which 01:11:10.760 --> 01:11:15.190 you can have constructive interference is 200 nanometer. 01:11:15.190 --> 01:11:22.130 That is actually shorter than the wavelength of the violet 01:11:22.130 --> 01:11:23.390 light. 01:11:23.390 --> 01:11:26.690 It's out of the range of the visible light. 01:11:26.690 --> 01:11:27.820 What is going to happen? 01:11:27.820 --> 01:11:30.050 Your eye will not see it. 01:11:32.900 --> 01:11:36.680 So the next one would be here, whatever, blah, blah, blah, 01:11:36.680 --> 01:11:38.980 blah, which I don't care, because they 01:11:38.980 --> 01:11:41.070 are so short in wavelength. 01:11:41.070 --> 01:11:43.280 And you cannot see them. 01:11:43.280 --> 01:11:47.770 So you can see that, if I have a width which 01:11:47.770 --> 01:11:54.560 is roughly 100 nanometer, very same situation, what 01:11:54.560 --> 01:11:55.580 is going to happen? 01:11:55.580 --> 01:11:58.430 What is going to happen is that you 01:11:58.430 --> 01:12:04.540 are going to get only one maxima in the visible light range. 01:12:04.540 --> 01:12:06.205 And therefore, you can see color. 01:12:09.550 --> 01:12:10.580 Any questions so far? 01:12:14.120 --> 01:12:20.090 Now, what I'm going to do is take the same formula here, 01:12:20.090 --> 01:12:28.580 but now I would like to change this d. 01:12:28.580 --> 01:12:30.530 So now I would like to change the d 01:12:30.530 --> 01:12:36.930 to consider a situation where you have a very thick layer. 01:12:36.930 --> 01:12:40.280 So now I would like to change the situation to a very thick 01:12:40.280 --> 01:12:46.165 layer, so maybe I need to erase this part of the board 01:12:46.165 --> 01:12:49.320 to make some space. 01:12:49.320 --> 01:12:58.130 So now if I have d equal to 100 micron, what 01:12:58.130 --> 01:12:59.930 is going to happen? 01:12:59.930 --> 01:13:03.590 So I can now still use this formula 01:13:03.590 --> 01:13:07.320 to calculate what would be the lambda maxima. 01:13:07.320 --> 01:13:15.020 So lambda maxima will be equal to 600 micron, which 01:13:15.020 --> 01:13:18.190 is when you have N equal to 1. 01:13:18.190 --> 01:13:22.400 But this wavelength is way, way larger, much, much larger 01:13:22.400 --> 01:13:26.840 than the wavelength of the visible light. 01:13:26.840 --> 01:13:29.510 So it's not going to work. 01:13:29.510 --> 01:13:32.940 Therefore, you have to be patient. 01:13:32.940 --> 01:13:41.780 You have to increase the N value until N is equal to 500. 01:13:41.780 --> 01:13:46.520 So I am calculation 1, 2, 3, 4, 5, 6, until 500. 01:13:46.520 --> 01:13:51.360 Ahh, we are in the visible light range, right? 01:13:51.360 --> 01:13:57.147 Now, 5000 will give you 600.6 nanometer. 01:13:57.147 --> 01:13:58.430 Phew. 01:13:58.430 --> 01:14:02.210 Suddenly, your eye can see it. 01:14:02.210 --> 01:14:03.070 Very good. 01:14:03.070 --> 01:14:04.520 That's very nice, right? 01:14:04.520 --> 01:14:07.610 So I can now put it in my diagram. 01:14:07.610 --> 01:14:10.130 This is actually the wavelength, again. 01:14:10.130 --> 01:14:14.480 And, ah, I get one line here. 01:14:14.480 --> 01:14:18.980 How about the next one, N equal to 501? 01:14:18.980 --> 01:14:23.770 I'm going to get 599.4 nanometer. 01:14:23.770 --> 01:14:26.210 It's pretty close to this one. 01:14:26.210 --> 01:14:37.010 And the next one would be 598.2 nanometer if N is equal to 502. 01:14:37.010 --> 01:14:38.540 And what you are getting is that you 01:14:38.540 --> 01:14:42.380 can see that, no, things are not going very well. 01:14:42.380 --> 01:14:49.962 They are full spectra, all very, very narrow. 01:14:49.962 --> 01:14:53.600 Very, very large number of wavelengths 01:14:53.600 --> 01:14:58.150 can give you constructive interference. 01:14:58.150 --> 01:15:00.600 So what is going to happen? 01:15:00.600 --> 01:15:06.540 What is going to happen is that your eye will see reflection 01:15:06.540 --> 01:15:10.210 with all kinds of different wavelengths. 01:15:10.210 --> 01:15:11.884 And what color is that? 01:15:11.884 --> 01:15:12.800 AUDIENCE: White light. 01:15:12.800 --> 01:15:13.430 YEN-JIE LEE: White! 01:15:13.430 --> 01:15:15.346 You are going to see something which is white. 01:15:18.590 --> 01:15:22.620 So that is actually the answer to our question. 01:15:22.620 --> 01:15:26.690 So what would be the required thickness? 01:15:26.690 --> 01:15:32.410 The required thickness is something like 100 nanometer. 01:15:32.410 --> 01:15:37.190 So we can see how thin is the water bubble. 01:15:37.190 --> 01:15:39.080 That may surprise some of you, right? 01:15:39.080 --> 01:15:41.230 Most of you actually didn't think 01:15:41.230 --> 01:15:44.120 that's actually that thin. 01:15:44.120 --> 01:15:47.510 Secondly, if d equal to 0, you are 01:15:47.510 --> 01:15:49.710 going to have destructive interference. 01:15:49.710 --> 01:15:52.150 That means there will be no reflected light. 01:15:52.150 --> 01:15:54.400 Everything is going to pass through. 01:15:54.400 --> 01:15:58.610 And the bubble is like transparent. 01:15:58.610 --> 01:16:03.710 Finally, when the bubble is really thick, 01:16:03.710 --> 01:16:06.920 you are going to see white. 01:16:06.920 --> 01:16:13.670 So let me finish this lecture with a demonstration here. 01:16:13.670 --> 01:16:15.462 First, before I turn off the light, 01:16:15.462 --> 01:16:16.670 I would like to turn this on. 01:16:20.330 --> 01:16:28.340 So what I have here is a very complicated machine. 01:16:28.340 --> 01:16:31.220 It's not that complicated, actually. 01:16:31.220 --> 01:16:33.920 So basically, I have a light source here, 01:16:33.920 --> 01:16:41.540 which emits light with all kinds of different wavelengths. 01:16:41.540 --> 01:16:43.810 And I have this little device here. 01:16:43.810 --> 01:16:46.340 There's soap solution inside. 01:16:46.340 --> 01:16:48.830 And I can actually rotate from outside. 01:16:48.830 --> 01:16:52.790 You see how sweet is this setup. 01:16:52.790 --> 01:16:57.560 And I can actually create a soap film out of this. 01:16:57.560 --> 01:17:02.670 You can see that I am rotating and trying to actually project 01:17:02.670 --> 01:17:06.190 the result on the wall. 01:17:06.190 --> 01:17:14.480 You can see that, initially, there's nothing really 01:17:14.480 --> 01:17:17.160 striking in the beginning. 01:17:17.160 --> 01:17:21.584 You can see that the light is which color? 01:17:21.584 --> 01:17:22.390 AUDIENCE: White. 01:17:22.390 --> 01:17:24.750 YEN-JIE LEE: It's white, right? 01:17:24.750 --> 01:17:29.610 Remember, because of the optical setup, 01:17:29.610 --> 01:17:33.440 this image is actually upside down. 01:17:33.440 --> 01:17:39.190 So the upper edge of that image is actually 01:17:39.190 --> 01:17:43.990 the lower edge of my setup, which is 01:17:43.990 --> 01:17:46.295 the lower edge of my soap film. 01:17:51.830 --> 01:17:55.390 We have gravity, right, so that I can walk around. 01:17:55.390 --> 01:18:02.514 Due to gravity, you can see that it 01:18:02.514 --> 01:18:06.160 will form a thicker and thicker layer 01:18:06.160 --> 01:18:09.700 in the bottom of my experimental setup 01:18:09.700 --> 01:18:13.720 or in the upper edge of the image. 01:18:13.720 --> 01:18:16.490 On the other hand, due to gravity, 01:18:16.490 --> 01:18:21.700 the upper edge or the lower edge of the image 01:18:21.700 --> 01:18:26.060 will become thinner and thinner as a function of time. 01:18:26.060 --> 01:18:31.840 At some point, the color will start to show up. 01:18:31.840 --> 01:18:39.270 As you can see now, since we wait long enough, 01:18:39.270 --> 01:18:41.950 the soap film becomes thinner and thinner. 01:18:41.950 --> 01:18:46.011 And you can see that there are colors popping up. 01:18:46.011 --> 01:18:47.910 It's like a rainbow. 01:18:47.910 --> 01:18:48.610 Why is that? 01:18:48.610 --> 01:18:54.020 Because I am varying the thickness of the film 01:18:54.020 --> 01:18:58.180 as a function of the vertical distance. 01:18:58.180 --> 01:19:01.160 So therefore, you can see that this is actually showing you 01:19:01.160 --> 01:19:07.760 that different d value will give you very different colors. 01:19:07.760 --> 01:19:10.900 And if we wait long enough, basically what 01:19:10.900 --> 01:19:15.730 we are going to get is that the whole film will 01:19:15.730 --> 01:19:18.850 become more and more colorful. 01:19:18.850 --> 01:19:21.520 And I am sure that after this class, 01:19:21.520 --> 01:19:23.140 you can walk out of the classroom 01:19:23.140 --> 01:19:29.320 and explain to your friend why the soap bubble is colorful. 01:19:29.320 --> 01:19:30.530 Thank you very much. 01:19:30.530 --> 01:19:34.480 And if you have any questions, I will be around. 01:19:34.480 --> 01:19:37.750 And of course, if you want to make your own soap bubble, 01:19:37.750 --> 01:19:40.610 you can actually go ahead and play the demo here. 01:19:48.310 --> 01:19:49.870 So hello, everybody. 01:19:49.870 --> 01:19:54.250 So we are going to show you a demonstration, which 01:19:54.250 --> 01:20:00.060 we can see colorful interference pattern from a soap film. 01:20:00.060 --> 01:20:04.270 So basically, the experimental setup up is like this. 01:20:04.270 --> 01:20:06.040 Basically, we have light, which is 01:20:06.040 --> 01:20:11.710 trying to shine this thin layer of soap film. 01:20:11.710 --> 01:20:17.030 And the result is actually projected on the screen. 01:20:17.030 --> 01:20:20.170 And you can see, at first, you don't really 01:20:20.170 --> 01:20:26.640 see a lot of colorful pattern, because the thickness 01:20:26.640 --> 01:20:33.430 of the film is still rather large, rather thick. 01:20:33.430 --> 01:20:36.060 Therefore, you don't really see a lot of pattern. 01:20:36.060 --> 01:20:38.560 But as a function of time, you can 01:20:38.560 --> 01:20:40.960 see that this pattern is actually changing. 01:20:40.960 --> 01:20:46.220 Because of gravitational force, you 01:20:46.220 --> 01:20:51.940 will be able to see that the lower part of the film 01:20:51.940 --> 01:20:53.740 becomes thicker and thicker. 01:20:53.740 --> 01:20:58.540 And the upper part of the film, which you see that upside down 01:20:58.540 --> 01:21:03.490 on the screen, is actually becoming thinner and thinner. 01:21:03.490 --> 01:21:08.530 As we actually discussed during the class, when soap film is 01:21:08.530 --> 01:21:16.375 thin enough, there will be only one or only a few maximas 01:21:16.375 --> 01:21:23.500 in the interference pattern as a function of wavelengths, 01:21:23.500 --> 01:21:28.600 which happened to be inside the visible light range. 01:21:28.600 --> 01:21:32.910 And you can see it now, already, this colorful pattern 01:21:32.910 --> 01:21:33.820 really develops. 01:21:33.820 --> 01:21:35.470 It's really beautiful. 01:21:35.470 --> 01:21:42.620 And you can see that, in the lower part of the experiment, 01:21:42.620 --> 01:21:45.280 really have authentic color, because there 01:21:45.280 --> 01:21:51.260 are multiple maximas in the visible light range. 01:21:51.260 --> 01:21:55.400 On the other hand, in the upper part of the film, 01:21:55.400 --> 01:21:59.800 you typically have very little number of maximas 01:21:59.800 --> 01:22:04.430 or only have one maxima, in the visible light range, 01:22:04.430 --> 01:22:06.110 as a function of wavelength. 01:22:06.110 --> 01:22:08.570 Therefore, you see really, really dramatic 01:22:08.570 --> 01:22:14.310 and very, very colorful pattern develop from this experiment.