WEBVTT 00:00:02.490 --> 00:00:04.860 The following content is provided under a Creative 00:00:04.860 --> 00:00:06.280 Commons license. 00:00:06.280 --> 00:00:08.490 Your support will help MIT OpenCourseWare 00:00:08.490 --> 00:00:12.580 continue to offer high quality educational resources for free. 00:00:12.580 --> 00:00:15.120 To make a donation or to view additional materials 00:00:15.120 --> 00:00:17.610 from hundreds of MIT courses, visit 00:00:17.610 --> 00:00:19.490 MITOpenCourseWare at ocw.mit.edu. 00:00:23.890 --> 00:00:27.980 PROFESSOR: Welcome back, everybody to 8.03. 00:00:27.980 --> 00:00:30.080 Very happy to see you again. 00:00:30.080 --> 00:00:32.610 So as you can see from the slides, 00:00:32.610 --> 00:00:34.960 we will continue the discussion from last time. 00:00:34.960 --> 00:00:38.770 We were talking about interference phenomena, 00:00:38.770 --> 00:00:44.170 which involve two or multiple point light source. 00:00:44.170 --> 00:00:46.990 And they actually interact with each other 00:00:46.990 --> 00:00:49.780 and produce interesting phenomenon, which 00:00:49.780 --> 00:00:53.820 we see with laser, with water ripples, 00:00:53.820 --> 00:01:00.400 and also we discussed how to design a phased radar together. 00:01:00.400 --> 00:01:04.280 And one thing which we learned is that if you, for example, 00:01:04.280 --> 00:01:07.420 have two slit interference, OK. 00:01:07.420 --> 00:01:12.280 And if you look at the intensity of the resulting interference 00:01:12.280 --> 00:01:16.390 pattern as a function of angle, you 00:01:16.390 --> 00:01:21.350 will see that there are peaks, periodic peaks 00:01:21.350 --> 00:01:23.020 as a function of angle. 00:01:23.020 --> 00:01:26.650 And we also know how to calculate 00:01:26.650 --> 00:01:29.775 where would be the principal maxima, what 00:01:29.775 --> 00:01:31.930 would be the minima, which will have 00:01:31.930 --> 00:01:35.871 destructive interference between the two point light source. 00:01:35.871 --> 00:01:36.370 OK? 00:01:36.370 --> 00:01:40.810 So as usual, we go from one electromagnetic wave 00:01:40.810 --> 00:01:45.160 to two electromagnetic waves and two unelectromagnetic waves. 00:01:45.160 --> 00:01:49.360 And today we are going to do infinite number 00:01:49.360 --> 00:01:51.010 of electromagnetic waves and they 00:01:51.010 --> 00:01:54.210 are going to interact with each other 00:01:54.210 --> 00:01:57.880 or superpose an infinite number of electromagnetic waves 00:01:57.880 --> 00:01:59.080 all together. 00:01:59.080 --> 00:02:04.360 And that brings us to that discussion of diffraction. 00:02:04.360 --> 00:02:05.260 OK? 00:02:05.260 --> 00:02:08.550 So what are we going to talk about today is, 00:02:08.550 --> 00:02:12.790 for example, a point light source, a laser pointer. 00:02:12.790 --> 00:02:17.410 And what would the image of a laser pointer look like? 00:02:17.410 --> 00:02:23.420 When these lasers pass through a single slit or just the laser 00:02:23.420 --> 00:02:23.920 itself. 00:02:23.920 --> 00:02:27.510 The laser beam itself, what will happen to this laser beam? 00:02:27.510 --> 00:02:32.320 And also we will make some comments on the Star Trek, 00:02:32.320 --> 00:02:33.006 for example. 00:02:33.006 --> 00:02:33.505 Right? 00:02:33.505 --> 00:02:38.250 They have this super weapon which they shoot enemy 00:02:38.250 --> 00:02:39.840 with this laser beam. 00:02:39.840 --> 00:02:44.710 And we'll see how practical that is by the end of this course. 00:02:44.710 --> 00:02:48.380 And the third thing is that it's related to resolution. 00:02:48.380 --> 00:02:51.630 So we are going to design a phone, screen 00:02:51.630 --> 00:02:56.230 of your mobile phone together to see what is actually practical, 00:02:56.230 --> 00:02:59.470 what is actually not practical. 00:02:59.470 --> 00:03:03.370 If Yen-Jie is opening a new company to develop jPhone, 00:03:03.370 --> 00:03:07.030 what should be the requirement for the screen, for example? 00:03:07.030 --> 00:03:08.290 Which, I'm not going to do it. 00:03:11.140 --> 00:03:16.030 So this is actually what we are going to discuss today. 00:03:16.030 --> 00:03:20.470 So we are interested in a situation where 00:03:20.470 --> 00:03:24.700 you have plane waves, and those plane waves are approaching 00:03:24.700 --> 00:03:29.140 from the left-hand side of the screen toward a single slit. 00:03:29.140 --> 00:03:33.010 So basically, the setup is like this. 00:03:33.010 --> 00:03:38.290 So you have those wavefront basically 00:03:38.290 --> 00:03:41.560 is traveling to the right-hand side, the plane waves. 00:03:41.560 --> 00:03:48.910 And on the wall, there's a slit or hole, which is actually 00:03:48.910 --> 00:03:51.730 a opening, and the waves can actually 00:03:51.730 --> 00:03:53.980 penetrate through this hole. 00:03:53.980 --> 00:04:02.140 The width of this hole is denoted by, or essentially 00:04:02.140 --> 00:04:05.080 given to you, which is actually D. 00:04:05.080 --> 00:04:09.305 And we were wondering what is going to happen to-- 00:04:12.210 --> 00:04:16.720 what are we going to observe on the screen, which is actually 00:04:16.720 --> 00:04:19.089 pretty far away from the wall. 00:04:19.089 --> 00:04:29.950 And this screen is actually used to observe 00:04:29.950 --> 00:04:32.560 the pattern of the interference pattern 00:04:32.560 --> 00:04:39.110 of the electromagnetic wave passing this hole. 00:04:39.110 --> 00:04:45.600 So, as we discussed last time, due to Huygens' principle, 00:04:45.600 --> 00:04:52.730 every point is actually like a point-like source 00:04:52.730 --> 00:04:55.190 of spherical waves. 00:04:55.190 --> 00:04:57.560 So, as you can see now, we actually 00:04:57.560 --> 00:05:00.620 consider the size of our slit. 00:05:00.620 --> 00:05:07.740 Therefore, there must be a lot of point-like source inside, 00:05:07.740 --> 00:05:09.470 in this slit. 00:05:09.470 --> 00:05:12.950 When this wavefront actually pass through this wall, 00:05:12.950 --> 00:05:17.240 there should be infinite number of point-like source. 00:05:17.240 --> 00:05:20.680 And all of them, due to Huygens' principle, 00:05:20.680 --> 00:05:25.520 is going to be like point-like source of spherical waves. 00:05:25.520 --> 00:05:30.280 And they are all emitting from all those possible location, 00:05:30.280 --> 00:05:32.810 and that they are overlapping each other 00:05:32.810 --> 00:05:36.450 and they have constructive or destructive interference 00:05:36.450 --> 00:05:37.650 with each other. 00:05:37.650 --> 00:05:39.510 So that is actually what is happening 00:05:39.510 --> 00:05:43.970 with this single-slit experiment. 00:05:43.970 --> 00:05:47.670 And we call that diffraction. 00:05:47.670 --> 00:05:50.930 So you may be wondering, why do I call it diffraction? 00:05:50.930 --> 00:05:53.280 Why not interference? 00:05:53.280 --> 00:05:57.110 Because it's basically the same phenomenon, right? 00:05:57.110 --> 00:05:59.810 I think it's just a matter of wording. 00:05:59.810 --> 00:06:02.300 Feynman actually commented on this, 00:06:02.300 --> 00:06:07.340 and he said that, nobody was able to define 00:06:07.340 --> 00:06:10.960 the difference between diffraction and interference 00:06:10.960 --> 00:06:13.040 in a satisfactory way. 00:06:13.040 --> 00:06:14.060 Which is actually true. 00:06:14.060 --> 00:06:15.780 So it's just a matter of wording. 00:06:15.780 --> 00:06:20.780 So we are looking at exactly the same phenomena when we actually 00:06:20.780 --> 00:06:23.960 discuss this experiment. 00:06:23.960 --> 00:06:26.570 So what I am going to do today is now 00:06:26.570 --> 00:06:29.840 to introduce to you the way we can deal with this. 00:06:29.840 --> 00:06:34.010 I'm sure you have seen this experiment before, maybe 00:06:34.010 --> 00:06:37.130 in 8.02 or in high school days. 00:06:37.130 --> 00:06:39.350 On the other hand, what we are going to do today 00:06:39.350 --> 00:06:43.160 is to really make use of the mathematics which 00:06:43.160 --> 00:06:48.440 we have learned from 18.03 or from the previous lectures 00:06:48.440 --> 00:06:50.300 to attack this problem. 00:06:50.300 --> 00:06:51.950 So what is actually the mathematics 00:06:51.950 --> 00:06:54.560 I am going to use today? 00:06:54.560 --> 00:06:57.680 So the mathematics which I would like 00:06:57.680 --> 00:07:01.040 to use to attack this problem is to use 00:07:01.040 --> 00:07:05.240 a two-dimensional Fourier transform. 00:07:05.240 --> 00:07:08.690 I think by now you should not be afraid of Fourier 00:07:08.690 --> 00:07:09.904 transform any more. 00:07:09.904 --> 00:07:11.070 It should be pretty natural. 00:07:11.070 --> 00:07:14.540 It's just integration, and you evaluate, and then 00:07:14.540 --> 00:07:17.940 you are going to get the corresponding number 00:07:17.940 --> 00:07:19.160 whatsoever. 00:07:19.160 --> 00:07:22.790 But the cool thing is that 18.03 give them 00:07:22.790 --> 00:07:25.340 physical meaning of those numbers, 00:07:25.340 --> 00:07:27.950 and I'm going to talk about that. 00:07:27.950 --> 00:07:32.020 So what is actually the Fourier transform I'm going to use? 00:07:32.020 --> 00:07:35.570 So I am going to evaluate C, which 00:07:35.570 --> 00:07:41.060 is a function of kx and ky. 00:07:41.060 --> 00:07:43.520 And what is actually this C function? 00:07:43.520 --> 00:07:48.650 This C function is equal to 1 over 4 pi squared, which 00:07:48.650 --> 00:07:50.180 I really don't care too much. 00:07:50.180 --> 00:07:52.400 It's just a constant. 00:07:52.400 --> 00:07:55.730 And I do the integration from minus infinity 00:07:55.730 --> 00:08:01.910 to infinity for dx, and I do a integration from minus infinity 00:08:01.910 --> 00:08:08.270 to infinity dy, a small letter scale, dy. 00:08:08.270 --> 00:08:11.210 And I have a f function, which I will introduce you 00:08:11.210 --> 00:08:15.420 what the f function mean, what does the f function actually 00:08:15.420 --> 00:08:16.820 represent. 00:08:16.820 --> 00:08:26.420 And exponential minus i, k is the vector which is actually 00:08:26.420 --> 00:08:29.330 telling you the direction of the propagation 00:08:29.330 --> 00:08:35.400 of the spherical wave, times r, which is actually 00:08:35.400 --> 00:08:39.440 a function of x and y. 00:08:39.440 --> 00:08:41.630 And this is actually the kind of integration 00:08:41.630 --> 00:08:45.530 which we will employ in order to attack the problem we 00:08:45.530 --> 00:08:50.570 are interested in this lecture. 00:08:50.570 --> 00:08:53.740 So, what does this integration mean? 00:08:53.740 --> 00:09:03.160 So we have basically some kind of this two-dimensional Fourier 00:09:03.160 --> 00:09:04.290 transform. 00:09:04.290 --> 00:09:08.200 The f function is actually telling you 00:09:08.200 --> 00:09:11.710 the shape of the source. 00:09:11.710 --> 00:09:14.170 So basically, this is actually telling you 00:09:14.170 --> 00:09:19.840 about the shape of the source. 00:09:24.920 --> 00:09:27.620 As we discussed before, the shape 00:09:27.620 --> 00:09:34.700 of the source, every point on this shape 00:09:34.700 --> 00:09:41.360 is a source of spherical wave, by Huygens' principle. 00:09:41.360 --> 00:09:44.740 So that is actually telling you where should I 00:09:44.740 --> 00:09:48.410 do the integration. 00:09:48.410 --> 00:09:53.690 This one, exponential i k dot r, what is that? 00:09:53.690 --> 00:10:00.000 This is actually telling you about the spherical wave. 00:10:00.000 --> 00:10:05.290 So remember, we were doing two-slit interference before, 00:10:05.290 --> 00:10:08.270 and we have actually two exponential function, if you 00:10:08.270 --> 00:10:11.030 remember from last lecture. 00:10:11.030 --> 00:10:13.190 So now, this is actually put there 00:10:13.190 --> 00:10:15.350 because each source you are going 00:10:15.350 --> 00:10:18.400 to get exponential function, which is actually 00:10:18.400 --> 00:10:23.060 presenting the propagation of the electromagnetic wave. 00:10:23.060 --> 00:10:25.420 You can say that, oh, wait, wait, wait. 00:10:25.420 --> 00:10:28.340 The omega t disappeared, right? 00:10:28.340 --> 00:10:30.770 There's no omega t here, right? 00:10:30.770 --> 00:10:33.980 But I don't really care because everybody is actually 00:10:33.980 --> 00:10:38.000 oscillating at the same frequency, the same phase. 00:10:38.000 --> 00:10:40.670 Therefore, I factorize out. 00:10:40.670 --> 00:10:43.700 After I have done all the calculation, 00:10:43.700 --> 00:10:47.520 I can multiply the whole thing by cosine omega t, and probably 00:10:47.520 --> 00:10:48.500 some phi. 00:10:48.500 --> 00:10:53.998 Then that is actually modulating and oscillating up and down 00:10:53.998 --> 00:10:59.140 as the plane wave, as you approach the wall. 00:10:59.140 --> 00:11:03.950 So, therefore, I actually already factorize it out. 00:11:03.950 --> 00:11:08.510 So this is actually telling you about the electric field. 00:11:08.510 --> 00:11:11.120 And what is actually here? 00:11:11.120 --> 00:11:16.530 This is actually the unit area you are 00:11:16.530 --> 00:11:20.110 performing this integration. 00:11:20.110 --> 00:11:22.470 And you can actually do integration 00:11:22.470 --> 00:11:25.500 over the full universe. 00:11:25.500 --> 00:11:27.930 So you have a plane which actually extend 00:11:27.930 --> 00:11:29.820 to the whole full universe. 00:11:29.820 --> 00:11:32.190 But what is actually really contributing 00:11:32.190 --> 00:11:35.310 is defined by this f function, which is actually 00:11:35.310 --> 00:11:37.470 the shape of the source. 00:11:37.470 --> 00:11:41.160 And some normalization factor, which I don't really 00:11:41.160 --> 00:11:44.430 care too much. 00:11:44.430 --> 00:11:49.760 So this looks really fancy, but is actually not that fancy. 00:11:49.760 --> 00:11:53.910 And what product you are getting here is our C function, 00:11:53.910 --> 00:11:58.120 is that C is actually a function of kx and ky. 00:11:58.120 --> 00:12:00.210 What is kx and ky? 00:12:00.210 --> 00:12:05.110 It's actually telling you the direction of propagation. 00:12:05.110 --> 00:12:07.590 The k vector is actually telling you the direction 00:12:07.590 --> 00:12:08.970 of the propagation. 00:12:08.970 --> 00:12:18.660 If you evaluate C with a specific given kx and ky, 00:12:18.660 --> 00:12:27.600 basically you are evaluating the total electric field 00:12:27.600 --> 00:12:32.640 going some direction, which is actually defined by kx and ky. 00:12:32.640 --> 00:12:34.320 So the big picture is the following. 00:12:34.320 --> 00:12:37.115 So basically you have some source. 00:12:37.115 --> 00:12:41.610 It can look like this in the xy plane. 00:12:41.610 --> 00:12:44.000 This is x and y plane. 00:12:49.990 --> 00:12:54.310 All those things, all those points, 00:12:54.310 --> 00:12:59.610 all those little areas inside this hole 00:12:59.610 --> 00:13:04.210 is spherical wave source. 00:13:04.210 --> 00:13:11.230 And the f function actually define the shape of this hole. 00:13:11.230 --> 00:13:16.090 And this integration is actually integrating over 00:13:16.090 --> 00:13:19.020 all those little areas. 00:13:19.020 --> 00:13:21.030 And then calculate the contribution 00:13:21.030 --> 00:13:26.720 from each small area, sum them together. 00:13:26.720 --> 00:13:30.780 Then, finally, you are getting something which is actually 00:13:30.780 --> 00:13:32.760 a function of kx and ky. 00:13:32.760 --> 00:13:34.700 What is kx and ky? 00:13:34.700 --> 00:13:39.480 It's actually giving you the direction of propagation 00:13:39.480 --> 00:13:46.680 from this point-like source to observer P. 00:13:46.680 --> 00:13:50.280 And this C function is actually proportional 00:13:50.280 --> 00:13:54.450 to the total electric field. 00:13:54.450 --> 00:13:57.330 So you can see that, hah, we have learned 00:13:57.330 --> 00:14:01.440 this Fourier transform from the math department, 00:14:01.440 --> 00:14:04.860 and we give life to this function. 00:14:04.860 --> 00:14:08.250 Now actually we understand what we are doing now. 00:14:08.250 --> 00:14:12.430 We are actually really summing all over the, 00:14:12.430 --> 00:14:17.910 summing over the available point-like source. 00:14:17.910 --> 00:14:21.000 And add all the contribution of the electromagnetic wave 00:14:21.000 --> 00:14:21.850 together. 00:14:21.850 --> 00:14:24.980 Then what we are getting, the C function 00:14:24.980 --> 00:14:34.034 is actually proportional to the total electric field. 00:14:36.790 --> 00:14:40.060 So that is actually the big picture. 00:14:40.060 --> 00:14:43.640 Any questions so far? 00:14:43.640 --> 00:14:46.400 I hope you can actually understand what we are doing. 00:14:46.400 --> 00:14:52.160 So now what I am going to do is to really use this formula 00:14:52.160 --> 00:14:56.120 and attack the problem which we are actually 00:14:56.120 --> 00:15:00.050 trying to understand, the single-slit problem. 00:15:00.050 --> 00:15:05.900 So suppose I have a single slit which looks like this. 00:15:05.900 --> 00:15:12.950 I'm zooming in this thing maybe 100 times, 1,000 times. 00:15:12.950 --> 00:15:15.725 And this is actually a wall. 00:15:19.880 --> 00:15:21.170 It's very, very long. 00:15:23.910 --> 00:15:28.894 I would like to define first my coordinate system. 00:15:28.894 --> 00:15:32.260 The x direction, as I actually drew from there, 00:15:32.260 --> 00:15:34.540 is actually pointing upward. 00:15:34.540 --> 00:15:41.560 The y direction is actually parallel to the wall. 00:15:41.560 --> 00:15:44.560 And the z direction is actually going 00:15:44.560 --> 00:15:47.080 to where the screen which I am trying 00:15:47.080 --> 00:15:52.540 to display the outcome of this experiment. 00:15:52.540 --> 00:15:56.285 And the distance between these two walls 00:15:56.285 --> 00:16:01.380 is actually D, which is actually given there. 00:16:01.380 --> 00:16:04.830 So I would like to actually understand 00:16:04.830 --> 00:16:09.010 what is going to happen when the plane waves pass through 00:16:09.010 --> 00:16:12.420 with this single slit. 00:16:12.420 --> 00:16:17.520 Therefore, before I calculate the C function proportional 00:16:17.520 --> 00:16:21.960 to the total electric field, what I really need 00:16:21.960 --> 00:16:29.010 is a functional form, f, which describe this single slit. 00:16:29.010 --> 00:16:33.830 And just to make sure that everybody is on the same page, 00:16:33.830 --> 00:16:40.680 this wall is actually infinitely long, from minus infinity in y 00:16:40.680 --> 00:16:43.570 to positive infinity y. 00:16:43.570 --> 00:16:45.870 So it's actually a super long wall. 00:16:45.870 --> 00:16:53.010 And these two edge is actually-- the distance 00:16:53.010 --> 00:16:57.180 between the edges is actually D. 00:16:57.180 --> 00:17:00.570 So what would be the f function which describes 00:17:00.570 --> 00:17:05.780 the shape of the light source? 00:17:05.780 --> 00:17:09.030 f function is a function of x and y. 00:17:09.030 --> 00:17:13.710 And I define a f function, and I give you 00:17:13.710 --> 00:17:17.160 this function to describe the experimental setup. 00:17:17.160 --> 00:17:21.420 So the f function can be either 1, which actually shows 00:17:21.420 --> 00:17:24.329 that there are point-like source there, 00:17:24.329 --> 00:17:29.150 or 0 when I am talking about things on the wall, 00:17:29.150 --> 00:17:32.060 because there's no point-like source there. 00:17:32.060 --> 00:17:38.780 Because the wall is actually blocking the light. 00:17:38.780 --> 00:17:41.130 So it can be either 1 or 0. 00:17:41.130 --> 00:17:43.200 When is that equal to 1? 00:17:43.200 --> 00:17:47.460 When minus D over 2. 00:17:47.460 --> 00:17:51.660 If I define-- this is actually x equal to 0. 00:17:51.660 --> 00:17:56.310 The middle of the slit is actually x equal to 0. 00:17:56.310 --> 00:18:00.780 Then it is actually equal to 1 when minus D over 2 00:18:00.780 --> 00:18:08.200 smaller than or equal to x, smaller or equal to D over 2. 00:18:08.200 --> 00:18:13.710 So that will give you a slit with width of capital D. 00:18:13.710 --> 00:18:17.760 On the other hand, if the absolute value of x 00:18:17.760 --> 00:18:22.740 is greater than D over 2, then I get 0. 00:18:22.740 --> 00:18:24.940 So now you can see that is actually 00:18:24.940 --> 00:18:26.880 the meaning of f function. 00:18:26.880 --> 00:18:29.130 f function is actually giving you 00:18:29.130 --> 00:18:34.530 a map of the point-like source. 00:18:34.530 --> 00:18:38.850 And what I am going to do now is to really do the integration 00:18:38.850 --> 00:18:45.180 to sum over all the spherical electromagnetic waves coming 00:18:45.180 --> 00:18:46.950 from all those point-like source, 00:18:46.950 --> 00:18:50.670 and to calculate the total electric field. 00:18:50.670 --> 00:18:55.140 So now I can go ahead and calculate C function, which 00:18:55.140 --> 00:18:59.220 is actually a function of kx and ky, related 00:18:59.220 --> 00:19:03.390 to the direction of propagation, or, say, 00:19:03.390 --> 00:19:09.030 the relative position of the observer 00:19:09.030 --> 00:19:12.630 and the overall point-like source. 00:19:12.630 --> 00:19:16.540 And this is actually equal to 1 over 4 pi squared, 00:19:16.540 --> 00:19:18.690 according to my formula. 00:19:18.690 --> 00:19:23.040 And now I'm going to do an integration from minus infinity 00:19:23.040 --> 00:19:24.510 to infinity. 00:19:24.510 --> 00:19:28.490 But I found that there's a shortcut I can take. 00:19:28.490 --> 00:19:35.160 f x y is only nonzero between minus D over 2 and plus 00:19:35.160 --> 00:19:38.010 D over 2 in the x direction. 00:19:38.010 --> 00:19:41.260 Therefore, this integration becomes integration 00:19:41.260 --> 00:19:48.094 from minus D over 2 to positive D over 2 dx, 00:19:48.094 --> 00:19:53.830 exponential minus i kx times x. 00:19:53.830 --> 00:20:00.990 I'm taking part of the k vector dot r out of this formula. 00:20:00.990 --> 00:20:06.290 The relevant part related to x direction integration 00:20:06.290 --> 00:20:11.430 is exponential i kx times x. 00:20:11.430 --> 00:20:16.320 And now I can actually do the integration in the y direction. 00:20:16.320 --> 00:20:19.230 So you can see that, in the y direction, 00:20:19.230 --> 00:20:24.230 this slit is infinity long, covering 00:20:24.230 --> 00:20:27.977 the from the left-hand side edge of the universe 00:20:27.977 --> 00:20:29.810 to the right-hand side edge of the universe. 00:20:29.810 --> 00:20:31.650 Really long. 00:20:31.650 --> 00:20:33.060 Super long. 00:20:33.060 --> 00:20:41.110 Minus infinity to infinity in the y direction. 00:20:41.110 --> 00:20:45.600 The relevant part of the exponential minus k 00:20:45.600 --> 00:20:51.530 dot r is exponential minus i ky times y. 00:20:56.600 --> 00:21:00.470 So, before I do this integration, 00:21:00.470 --> 00:21:04.820 I would like to remind you one thing which is actually 00:21:04.820 --> 00:21:08.180 we have learned from the past, from the help of math 00:21:08.180 --> 00:21:10.030 department. 00:21:10.030 --> 00:21:15.860 So we know delta function x minus a is actually equal 00:21:15.860 --> 00:21:22.950 to 1 over 2 pi, integration from minus infinity to infinity, 00:21:22.950 --> 00:21:30.950 exponential i p x minus a dp. 00:21:30.950 --> 00:21:35.890 So we know about this formula. 00:21:35.890 --> 00:21:43.180 So that means I can easily evaluate this function. 00:21:43.180 --> 00:21:47.940 So this function, I'm actually doing the integration over y. 00:21:47.940 --> 00:21:50.460 Therefore, what I'm going to get is, 00:21:50.460 --> 00:21:54.060 I take 1 over 2 pi out of this. 00:21:54.060 --> 00:21:56.610 I take 2 pi out of this. 00:21:56.610 --> 00:21:59.040 Then, basically, I can actually arrive 00:21:59.040 --> 00:22:02.710 expression, which is actually delta function 00:22:02.710 --> 00:22:05.520 is a function of ky. 00:22:05.520 --> 00:22:10.960 After you do this integration using this formula here. 00:22:10.960 --> 00:22:16.710 So p here is actually y in my integration I'm doing. 00:22:16.710 --> 00:22:18.880 And what I'm going to get is actually 00:22:18.880 --> 00:22:23.600 ky equal to 0, minus 0, and I simplify 00:22:23.600 --> 00:22:27.660 that to be delta function of ky. 00:22:27.660 --> 00:22:29.320 So basically, you are going to get 00:22:29.320 --> 00:22:33.150 the ky contribution is going to give you a delta function. 00:22:35.880 --> 00:22:41.490 So how about the integration which 00:22:41.490 --> 00:22:44.590 is the other part of the integration? 00:22:44.590 --> 00:22:46.780 The other part of the integration 00:22:46.780 --> 00:22:52.660 is related to x direction, is here. 00:22:52.660 --> 00:22:55.090 So, basically, what is actually left over? 00:22:55.090 --> 00:22:58.750 I took already 1 over 2 pi from here. 00:22:58.750 --> 00:23:05.800 Therefore, I have 1 over 2 pi, and I do the integration. 00:23:05.800 --> 00:23:07.500 It's just a exponential function. 00:23:07.500 --> 00:23:09.580 I'm not super worried. 00:23:09.580 --> 00:23:17.910 Basically, I get 1 over minus i kx. 00:23:17.910 --> 00:23:23.470 Exponential minus i kx x. 00:23:23.470 --> 00:23:29.870 And evaluated at D over 2, x equal to D over 2, 00:23:29.870 --> 00:23:33.740 and x equal to minus D over 2. 00:23:37.460 --> 00:23:40.210 I hope this part is straightforward enough. 00:23:40.210 --> 00:23:41.670 Any questions so far? 00:23:41.670 --> 00:23:44.710 Everybody's following? 00:23:44.710 --> 00:23:45.530 All right. 00:23:45.530 --> 00:23:46.490 Very good. 00:23:46.490 --> 00:23:48.620 So I will continue the red part. 00:23:48.620 --> 00:23:51.010 So I will just look at the red part 00:23:51.010 --> 00:23:53.280 and then continue on this board. 00:23:53.280 --> 00:23:56.260 I'm using the red pen, right. 00:23:56.260 --> 00:23:58.420 So, basically, what I am going to get 00:23:58.420 --> 00:24:10.790 is, basically you have 1 over 2 pi, 1 over minus i kx, 00:24:10.790 --> 00:24:23.680 exponential minus i kx D over 2, minus exponential i kx D 00:24:23.680 --> 00:24:26.270 over 2. 00:24:26.270 --> 00:24:30.770 So the red part of the left function become this. 00:24:30.770 --> 00:24:33.620 And you can actually easily realize 00:24:33.620 --> 00:24:36.740 that this is actually proportional 00:24:36.740 --> 00:24:39.110 to a sine function, right? 00:24:39.110 --> 00:24:49.750 So basically I'm going to get 1 over 2 pi minus 2i sine kx 00:24:49.750 --> 00:24:58.110 D over 2, divided by i kx. 00:25:02.010 --> 00:25:04.210 This is actually coming from there. 00:25:04.210 --> 00:25:06.640 And this is actually coming from-- this minus 00:25:06.640 --> 00:25:12.000 2i sine function is coming from the exponential function 00:25:12.000 --> 00:25:15.500 I can cancel this 2, and basically I 00:25:15.500 --> 00:25:23.670 get 1 over pi kx sine kx D divided by 2. 00:25:27.360 --> 00:25:31.260 So if I put everything together, so basically 00:25:31.260 --> 00:25:45.270 what you are getting is delta function of ky, 1 over pi kx 00:25:45.270 --> 00:25:49.040 sine kx D over 2. 00:25:53.300 --> 00:25:54.990 Am I going too fast? 00:25:54.990 --> 00:25:56.627 Everybody is following? 00:25:59.340 --> 00:26:02.230 So I hope this mathematics is straightforward enough. 00:26:02.230 --> 00:26:05.970 And don't forget what we are doing. 00:26:05.970 --> 00:26:08.530 So what we are doing is the following. 00:26:08.530 --> 00:26:11.900 So we have this two-dimensional Fourier transform. 00:26:11.900 --> 00:26:16.840 And the goal is to sum over all the waves coming 00:26:16.840 --> 00:26:20.340 from a shape defined by f function. 00:26:20.340 --> 00:26:22.980 And I'm going to evaluate the C function, 00:26:22.980 --> 00:26:25.410 and the C function is proportional 00:26:25.410 --> 00:26:29.650 to the total electric field. 00:26:29.650 --> 00:26:33.244 C is a function of kx and ky. 00:26:33.244 --> 00:26:36.850 kx and ky give you the information 00:26:36.850 --> 00:26:41.530 about the direction, relative position 00:26:41.530 --> 00:26:48.660 of the source and the observer P. And from this exercise, 00:26:48.660 --> 00:26:50.590 what we actually learn from here is 00:26:50.590 --> 00:26:55.810 that the C function is a function of y, 00:26:55.810 --> 00:27:02.410 but essentially only nonzero when ky is equal to what? 00:27:05.880 --> 00:27:06.380 AUDIENCE: 0. 00:27:06.380 --> 00:27:07.990 PROFESSOR: 0, right? 00:27:07.990 --> 00:27:10.890 Does that surprise you? 00:27:10.890 --> 00:27:12.950 No, probably not. 00:27:12.950 --> 00:27:14.300 Why is that? 00:27:14.300 --> 00:27:16.470 Why should we expect that? 00:27:16.470 --> 00:27:23.810 Because in the y direction this slit is infinitely long. 00:27:23.810 --> 00:27:30.410 So if you have contribution of many, many spherical wave, 00:27:30.410 --> 00:27:34.760 and this slit is infinity long, the sum 00:27:34.760 --> 00:27:37.040 of all those spherical wave is going 00:27:37.040 --> 00:27:40.940 to be still like a wavefront. 00:27:40.940 --> 00:27:43.160 You can do this in your head. 00:27:43.160 --> 00:27:50.690 So that means the direction, if I choose a direction which 00:27:50.690 --> 00:27:56.320 is actually pointing to somewhere which is actually 00:27:56.320 --> 00:28:00.020 with a ky not equal to 0-- 00:28:00.020 --> 00:28:03.140 so that means I have a specific direction-- 00:28:03.140 --> 00:28:06.980 what I'm going to get is that the electric field, 00:28:06.980 --> 00:28:10.410 the total electric field, will be equal to 0. 00:28:14.030 --> 00:28:18.320 And, of course, you can actually also 00:28:18.320 --> 00:28:23.070 talk about what will happen in the x direction. 00:28:23.070 --> 00:28:26.750 So that is actually the dependence of the C function 00:28:26.750 --> 00:28:29.700 to the kx. 00:28:29.700 --> 00:28:32.690 And we found interesting dependence. 00:28:32.690 --> 00:28:37.620 It's actually sine kx D over 2 divided by kx. 00:28:40.760 --> 00:28:45.190 So what I'm going to do is to make our life slightly easier 00:28:45.190 --> 00:28:53.020 by defining something which is actually easier to understand. 00:28:53.020 --> 00:28:55.270 But before that, I would like to say 00:28:55.270 --> 00:29:00.310 that the electric field, as I mentioned before, 00:29:00.310 --> 00:29:04.860 is going to be proportional to the C function. 00:29:04.860 --> 00:29:08.980 And now I would like to drop the y direction, 00:29:08.980 --> 00:29:11.270 because it's just a delta function. 00:29:11.270 --> 00:29:14.150 Therefore, I can actually drop it in the discussion. 00:29:14.150 --> 00:29:17.410 Then I will say that this electric field 00:29:17.410 --> 00:29:25.150 is going to be proportional to the sine kx D over 2, divided 00:29:25.150 --> 00:29:26.550 by kx. 00:29:29.900 --> 00:29:32.230 Since we have the electric field, 00:29:32.230 --> 00:29:34.550 the magnitude of the electric field, 00:29:34.550 --> 00:29:38.600 then I can actually calculate what will be the intensity. 00:29:38.600 --> 00:29:40.880 Intensity is actually what we care. 00:29:40.880 --> 00:29:44.420 It's going to be proportional to E square, 00:29:44.420 --> 00:29:49.040 and that is actually proportional to C square. 00:29:49.040 --> 00:29:51.310 And what is actually that value? 00:29:51.310 --> 00:29:57.920 That is going to be proportional to sine square kx D divided 00:29:57.920 --> 00:30:04.250 by 2, divided by kx squared. 00:30:04.250 --> 00:30:05.250 Any questions so far? 00:30:09.820 --> 00:30:16.610 So remember what is actually we are discussing. 00:30:16.610 --> 00:30:20.960 So we are discussing about a single slit, 00:30:20.960 --> 00:30:28.610 and we were wondering what will happen to observer point P 00:30:28.610 --> 00:30:34.050 when they actually do get, when these observer do get 00:30:34.050 --> 00:30:39.200 the interference pattern of all the point-like source 00:30:39.200 --> 00:30:41.010 between these two walls. 00:30:43.680 --> 00:30:47.250 We can actually make it much more understandable 00:30:47.250 --> 00:30:50.250 by using angle, which is actually 00:30:50.250 --> 00:30:54.060 theta, which is the measure of AP, 00:30:54.060 --> 00:30:58.170 which is the direction of the-- 00:30:58.170 --> 00:31:01.980 which is a vector connecting the slit to the observer-- 00:31:01.980 --> 00:31:04.960 to the horizontal direction. 00:31:04.960 --> 00:31:09.390 And I can define the displacement with respect 00:31:09.390 --> 00:31:13.490 to the center to be x. 00:31:13.490 --> 00:31:18.840 And I can actually also express AP 00:31:18.840 --> 00:31:21.060 by a vector which is r vector. 00:31:24.410 --> 00:31:27.190 Basically, after this definition, 00:31:27.190 --> 00:31:33.800 we can actually calculate or express sine theta. 00:31:33.800 --> 00:31:40.970 Since the distance between the screen and the wall 00:31:40.970 --> 00:31:47.060 is very, very large, therefore the theta angle is very small. 00:31:47.060 --> 00:31:52.250 Therefore, I can safely assume that sine theta is actually 00:31:52.250 --> 00:31:53.720 x divided by r. 00:31:56.450 --> 00:31:59.050 And also, at the same time, this is actually 00:31:59.050 --> 00:32:02.940 equal to kx divided by k. 00:32:05.480 --> 00:32:08.660 Because the k vector is actually telling you 00:32:08.660 --> 00:32:12.260 the direction of propagation. 00:32:12.260 --> 00:32:14.790 So, therefore, I can actually rewrite this. 00:32:14.790 --> 00:32:18.780 This will become kx. 00:32:18.780 --> 00:32:23.270 The magnitude of k vector is actually 00:32:23.270 --> 00:32:25.760 basically 2 pi over lambda. 00:32:25.760 --> 00:32:28.330 So, therefore, you can actually calculate that, 00:32:28.330 --> 00:32:35.010 and you will get kx times lambda divided by 2 pi. 00:32:35.010 --> 00:32:38.850 Therefore, the goal is to rewrite kx 00:32:38.850 --> 00:32:42.660 in a form which we understand, which is theta. 00:32:42.660 --> 00:32:44.960 So now we have achieved that. 00:32:44.960 --> 00:32:46.593 What is actually kx? 00:32:46.593 --> 00:32:54.070 kx is actually equal to 2 pi sine theta divided by lambda. 00:32:59.700 --> 00:33:04.940 And this means that my intensity, which 00:33:04.940 --> 00:33:13.401 I appended there, will be proportional to sine square pi 00:33:13.401 --> 00:33:21.995 D divided by lambda sine theta, divided by 2 pi sine 00:33:21.995 --> 00:33:26.300 theta divided by lambda, squared. 00:33:26.300 --> 00:33:31.440 So basically what I'm doing is to replace kx and then 00:33:31.440 --> 00:33:35.660 write it in terms of theta. 00:33:35.660 --> 00:33:44.090 If I define beta to be equal to pi D sine theta 00:33:44.090 --> 00:33:49.610 over lambda, if I define this, basically you 00:33:49.610 --> 00:33:54.290 are getting sine square beta, this 00:33:54.290 --> 00:33:57.670 will be proportional to sine squared beta divided 00:33:57.670 --> 00:34:01.070 by beta squared. 00:34:01.070 --> 00:34:09.830 And this beta is actually proportional to theta and D. 00:34:09.830 --> 00:34:12.000 Any questions so far? 00:34:12.000 --> 00:34:17.239 I'm just doing a replace, I'm just replacing the variables 00:34:17.239 --> 00:34:20.060 so that it's actually in terms of theta 00:34:20.060 --> 00:34:23.594 and in terms of some variable which actually simplify 00:34:23.594 --> 00:34:25.870 the expression dramatically. 00:34:28.570 --> 00:34:31.130 So, that's very good. 00:34:31.130 --> 00:34:37.090 So we have actually evaluated the intensity, 00:34:37.090 --> 00:34:44.719 the resulting intensity which will show up on the screen. 00:34:44.719 --> 00:34:46.310 And then we found that essentially 00:34:46.310 --> 00:34:51.690 proportional to sine square pi D divided by lambda sine theta, 00:34:51.690 --> 00:34:53.659 divided by something squared. 00:34:53.659 --> 00:34:56.560 And then I called this constant, sorry, 00:34:56.560 --> 00:35:00.110 I called this expression, I defined this expression 00:35:00.110 --> 00:35:01.310 to be beta. 00:35:01.310 --> 00:35:04.340 Then the functional form become much simpler. 00:35:04.340 --> 00:35:07.460 It's become sine square beta divided by beta square. 00:35:10.790 --> 00:35:17.330 So what I am going to do now is to visualize this result. 00:35:17.330 --> 00:35:24.050 So what I'm trying to do now is to plot the intensity 00:35:24.050 --> 00:35:30.290 I as a function of sine theta, for example, using 00:35:30.290 --> 00:35:32.610 this expression. 00:35:32.610 --> 00:35:35.960 So what I'm going to get is something which is actually 00:35:35.960 --> 00:35:39.830 going to be decreasing. 00:35:39.830 --> 00:35:41.720 Something is going to be decreasing 00:35:41.720 --> 00:35:43.640 as a function of beta. 00:35:43.640 --> 00:35:45.740 So that's the dashed line. 00:35:45.740 --> 00:35:49.460 This dashed line is actually proportional to 1 00:35:49.460 --> 00:35:50.990 over beta squared. 00:35:53.760 --> 00:36:00.810 And sine theta very small, you actually 00:36:00.810 --> 00:36:03.630 reach a maximum value of I0. 00:36:06.630 --> 00:36:14.460 When you move away from theta equal to 0, 00:36:14.460 --> 00:36:20.460 you actually will hit a minimum when the sine theta 00:36:20.460 --> 00:36:27.790 is equal to lambda over D. Because if sine theta is 00:36:27.790 --> 00:36:33.200 equal to lambda over D, then this expression become what? 00:36:33.200 --> 00:36:35.860 Become what value when sine theta is actually 00:36:35.860 --> 00:36:36.550 lambda over D? 00:36:39.590 --> 00:36:40.540 Pi. 00:36:40.540 --> 00:36:43.130 Sine pi is 0, right? 00:36:43.130 --> 00:36:49.160 Therefore, you have a destructive interference. 00:36:49.160 --> 00:36:50.830 This point is really interesting. 00:36:50.830 --> 00:36:52.070 Why? 00:36:52.070 --> 00:36:57.430 Because that means all the point-like source, 00:36:57.430 --> 00:37:02.120 all of them between these two walls, 00:37:02.120 --> 00:37:09.480 are working together so nicely such that the total field is 00:37:09.480 --> 00:37:12.270 completely cancelled. 00:37:12.270 --> 00:37:14.290 Isn't that remarkable? 00:37:14.290 --> 00:37:19.070 That's really, really crazy when this happens. 00:37:19.070 --> 00:37:22.970 Takes a lot of work, infinite number of source, to do that. 00:37:22.970 --> 00:37:26.960 Then, if you actually increase further the sine theta, 00:37:26.960 --> 00:37:32.090 move away from the center of the screen, 00:37:32.090 --> 00:37:35.930 basically you see that this will increase again and reach 00:37:35.930 --> 00:37:41.960 a smaller maxima, and again reach 0 00:37:41.960 --> 00:37:46.340 when this is actually equal to 2 lambda over D. 00:37:46.340 --> 00:37:48.530 And this pattern continues. 00:37:48.530 --> 00:37:53.740 And, of course, because of the symmetry 00:37:53.740 --> 00:37:56.780 we observe in this expression, everything 00:37:56.780 --> 00:38:00.950 is actually proportional to sine squared something. 00:38:00.950 --> 00:38:04.720 Therefore, this distribution is actually symmetric. 00:38:04.720 --> 00:38:09.430 So you have minus lambda over D, minus 2 lambda over D, 00:38:09.430 --> 00:38:12.915 et cetera, et cetera. 00:38:12.915 --> 00:38:14.400 Any questions so far? 00:38:20.340 --> 00:38:26.620 So what you can see here is something really interesting. 00:38:26.620 --> 00:38:30.880 Sine theta, if you multiply that by r, 00:38:30.880 --> 00:38:36.620 is telling you the position which 00:38:36.620 --> 00:38:39.470 you will see on the screen. 00:38:39.470 --> 00:38:41.360 So this is actually-- 00:38:41.360 --> 00:38:45.660 if you are interested in some place, point of interest P, 00:38:45.660 --> 00:38:50.570 and this actually just r times sine theta. 00:38:50.570 --> 00:38:52.310 And this is actually the slit. 00:38:55.420 --> 00:38:58.580 And I will move this thing closer here. 00:38:58.580 --> 00:39:04.360 And the size of this slit is called D. 00:39:04.360 --> 00:39:07.990 So one thing which is actually very interesting in this result 00:39:07.990 --> 00:39:18.080 is that, if we look at the width of the central principal 00:39:18.080 --> 00:39:19.780 maxima. 00:39:19.780 --> 00:39:21.770 The width is actually the measure 00:39:21.770 --> 00:39:24.610 between the center and the first minima, 00:39:24.610 --> 00:39:29.430 where you have complete destructive interference. 00:39:29.430 --> 00:39:34.600 What you actually see here is that this is actually something 00:39:34.600 --> 00:39:35.860 very interesting is happening. 00:39:35.860 --> 00:39:39.260 When you increase D, if you increase 00:39:39.260 --> 00:39:42.400 D, what is going to happen to the position 00:39:42.400 --> 00:39:46.590 of the first principal minima, of our first minima? 00:39:46.590 --> 00:39:49.800 It's going to what? 00:39:49.800 --> 00:39:53.360 Going to become smaller. 00:39:53.360 --> 00:39:54.370 Right? 00:39:54.370 --> 00:40:01.180 So suppose I have a gap here and I'm shooting a gun like crazy, 00:40:01.180 --> 00:40:03.250 boo-boo-boo-boo boo-boo-boo-boo. 00:40:03.250 --> 00:40:10.090 And I produce huge amount of bullet, which I don't 00:40:10.090 --> 00:40:13.960 recommend to do that, for sure. 00:40:13.960 --> 00:40:16.140 What I'm going to do, what I'm going to get 00:40:16.140 --> 00:40:21.230 is a distribution like this, which are the bullets passing 00:40:21.230 --> 00:40:23.630 through this wall. 00:40:23.630 --> 00:40:27.230 If I increase the size of the wall, 00:40:27.230 --> 00:40:31.150 the distribution I'm getting is becoming what? 00:40:31.150 --> 00:40:32.490 Wider. 00:40:32.490 --> 00:40:33.290 Right? 00:40:33.290 --> 00:40:36.200 But the result here is actually surprising. 00:40:36.200 --> 00:40:38.030 Why? 00:40:38.030 --> 00:40:43.020 When you increase the width, when 00:40:43.020 --> 00:40:48.360 you increase the width of the D, this function becomes smaller. 00:40:48.360 --> 00:40:53.240 That means the central maxima will become narrower, 00:40:53.240 --> 00:40:56.680 as you can see from this demonstration. 00:40:56.680 --> 00:41:01.170 So the left-hand side is an experimental setup which you 00:41:01.170 --> 00:41:05.010 have a very, very narrow slit. 00:41:05.010 --> 00:41:09.120 And basically you get a very wide distribution 00:41:09.120 --> 00:41:15.600 in the intensity as a function of position on the screen. 00:41:15.600 --> 00:41:19.800 Right-hand side is another situation where 00:41:19.800 --> 00:41:23.640 you have wider distribution. 00:41:23.640 --> 00:41:26.220 I'm sorry, wider slit, and you are 00:41:26.220 --> 00:41:29.950 going to get a narrower central maxima. 00:41:29.950 --> 00:41:34.950 Which is actually different from the other experiment 00:41:34.950 --> 00:41:39.000 which we were actually doing. 00:41:39.000 --> 00:41:42.900 So that's the first thing which we learn from here. 00:41:42.900 --> 00:41:46.650 And, also, the distance, the distance 00:41:46.650 --> 00:41:52.920 between the maxima and the minima 00:41:52.920 --> 00:41:56.430 is proportional to wavelength. 00:41:56.430 --> 00:42:01.230 So that means I can measure wavelength 00:42:01.230 --> 00:42:05.340 by using the position of the minima. 00:42:05.340 --> 00:42:10.020 And we are going to do that to measure the wavelength 00:42:10.020 --> 00:42:13.210 of the laser beam. 00:42:13.210 --> 00:42:15.330 And, finally, the last thing which we learn 00:42:15.330 --> 00:42:23.280 is that, in the central region, you have a maxima of I0, 00:42:23.280 --> 00:42:27.570 and this intensity is going to be going down, 00:42:27.570 --> 00:42:31.950 proportional to 1 over beta squared, where beta is actually 00:42:31.950 --> 00:42:32.670 defined here. 00:42:32.670 --> 00:42:35.340 It's proportional to D sine theta 00:42:35.340 --> 00:42:41.500 and inversely proportional to wavelength. 00:42:41.500 --> 00:42:48.190 So now what I'm going to do is experiment 00:42:48.190 --> 00:42:51.040 which I would like to measure what would 00:42:51.040 --> 00:42:55.510 be the wavelength of my laser. 00:42:55.510 --> 00:42:58.040 So I have a laser here. 00:42:58.040 --> 00:42:58.540 Oh. 00:42:58.540 --> 00:43:01.720 OK, I don't want to hurt anybody. 00:43:01.720 --> 00:43:04.300 So I have a laser here. 00:43:04.300 --> 00:43:08.770 And I have a slit, which you cannot see, unfortunately. 00:43:08.770 --> 00:43:11.780 And I can read off the width of the slit for you. 00:43:11.780 --> 00:43:14.020 The width of the slit is carefully 00:43:14.020 --> 00:43:21.620 designed to be really small, is 0.16 millimeter. 00:43:21.620 --> 00:43:23.900 This is my width. 00:43:23.900 --> 00:43:30.800 The D is actually equal to 0.16 millimeter. 00:43:30.800 --> 00:43:36.440 And on the screen, you can see that there's a pattern formed 00:43:36.440 --> 00:43:40.220 here, which you probably cannot see very, very clearly, 00:43:40.220 --> 00:43:46.450 so I will try to lower the intensity of the other source. 00:43:46.450 --> 00:43:49.450 So you can see, then, see that there 00:43:49.450 --> 00:43:53.840 is an interference pattern or diffraction pattern which 00:43:53.840 --> 00:43:54.920 is actually showing here. 00:43:57.680 --> 00:44:05.660 So what I really need in order to calculate the wavelength 00:44:05.660 --> 00:44:08.450 is the sine theta angle. 00:44:08.450 --> 00:44:10.370 Which I will really need the sine theta angle. 00:44:10.370 --> 00:44:11.786 Then I can actually calculate what 00:44:11.786 --> 00:44:15.110 will be the wavelength of this laser. 00:44:15.110 --> 00:44:19.580 So that means I will need help from a volunteer. 00:44:19.580 --> 00:44:22.370 Who volunteer to help me to measure 00:44:22.370 --> 00:44:26.090 the distance between this slit and the large screen? 00:44:26.090 --> 00:44:27.590 Can somebody volunteer? 00:44:27.590 --> 00:44:28.210 Yes, please. 00:44:32.320 --> 00:44:36.220 So we are going to measure the distance. 00:44:36.220 --> 00:44:36.970 Can you hold this? 00:44:36.970 --> 00:44:39.720 And can you actually put it? 00:44:39.720 --> 00:44:43.330 OK, try to pull this thing, and we will try our best 00:44:43.330 --> 00:44:47.330 to make it straight. 00:44:47.330 --> 00:44:50.130 Thank you very much. 00:44:50.130 --> 00:44:52.345 We don't want to destroy the experiment as well. 00:44:56.360 --> 00:44:57.360 This is not working? 00:45:01.050 --> 00:45:02.790 Let me do this in the other way. 00:45:02.790 --> 00:45:06.310 So how about-- trial and error, right? 00:45:08.910 --> 00:45:10.240 How about this. 00:45:10.240 --> 00:45:14.690 You hold that thing, and I'm going to actually measure 00:45:14.690 --> 00:45:16.036 the distance from here. 00:45:18.980 --> 00:45:24.520 And I need to really make it really carefully, measure this 00:45:24.520 --> 00:45:25.810 very carefully. 00:45:25.810 --> 00:45:29.890 And I don't want to destroy anything, 00:45:29.890 --> 00:45:30.980 which is very possible. 00:45:33.910 --> 00:45:37.670 So what I'm getting? 00:45:37.670 --> 00:45:42.230 I get 7.5 meter. 00:45:42.230 --> 00:45:47.720 So that's actually the distance between the screen, the screen 00:45:47.720 --> 00:45:49.260 and the source. 00:45:49.260 --> 00:45:50.600 Hold that for a second. 00:45:50.600 --> 00:45:55.820 I am going to measure the width of, the distance between two 00:45:55.820 --> 00:45:56.750 minima. 00:45:56.750 --> 00:46:04.170 The distance between two minima is 7 centimeter. 00:46:04.170 --> 00:46:06.490 Thank you very much. 00:46:06.490 --> 00:46:08.620 Thank you for your help. 00:46:08.620 --> 00:46:12.170 So we have now everything we need 00:46:12.170 --> 00:46:14.070 to calculate the wavelength. 00:46:17.276 --> 00:46:19.200 I'm going to clean this up first. 00:46:24.380 --> 00:46:27.560 We have what? 00:46:27.560 --> 00:46:29.290 We have the distance now. 00:46:29.290 --> 00:46:34.970 The distance between the source and the screen is 7.6 meter. 00:46:38.190 --> 00:46:41.860 So now I would like to calculate what will be the lambda. 00:46:44.700 --> 00:46:47.600 And also I know the distance-- 00:46:47.600 --> 00:46:55.830 the distance between these two minima is 7 centimeter. 00:46:55.830 --> 00:47:01.303 So that means this will be 3.5 centimeter. 00:47:04.270 --> 00:47:10.770 So lambda divided by D is actually equal to sine theta. 00:47:10.770 --> 00:47:17.040 Which is actually small d, which is the distance 00:47:17.040 --> 00:47:18.225 between the minima. 00:47:18.225 --> 00:47:20.550 The small d is here. 00:47:20.550 --> 00:47:24.860 The small d is the distance between the minima 00:47:24.860 --> 00:47:26.580 and the center. 00:47:26.580 --> 00:47:31.190 Divided by r, which is the distance between the source 00:47:31.190 --> 00:47:33.620 and the screen. 00:47:33.620 --> 00:47:36.300 Therefore, I can have lambda will 00:47:36.300 --> 00:47:42.800 be equal to capital D times small d divided by r. 00:47:42.800 --> 00:47:44.450 So what is actually the answer? 00:47:44.450 --> 00:47:48.140 So basically I have capital D, which 00:47:48.140 --> 00:47:53.420 is actually 0.16 millimeter. 00:47:53.420 --> 00:47:57.080 So that is actually shown there but you cannot see it. 00:47:59.720 --> 00:48:05.750 So I will use a different board for this calculation. 00:48:05.750 --> 00:48:09.710 So, basically, we will actually get 00:48:09.710 --> 00:48:17.825 lambda is equal to capital D times small d divided by r. 00:48:17.825 --> 00:48:24.120 Capital D is 0.16 times 10 to the minus 3 meter. 00:48:27.710 --> 00:48:29.780 And what is actually the small d? 00:48:29.780 --> 00:48:34.834 The small d is actually 3.5 centimeters. 00:48:40.620 --> 00:48:48.030 And, finally, I have 7.6 meter, which is actually the small r. 00:48:48.030 --> 00:48:51.220 Divided by 7.6 meter. 00:48:56.140 --> 00:48:58.180 Can somebody actually calculate this for me? 00:49:00.930 --> 00:49:02.460 Anybody have a smartphone? 00:49:07.320 --> 00:49:10.480 This means that I haven't done this experiment myself, 00:49:10.480 --> 00:49:12.830 and we will see what is going to happen. 00:49:12.830 --> 00:49:13.700 I hope it will work. 00:49:17.390 --> 00:49:20.337 What is actually the value? 00:49:20.337 --> 00:49:24.330 AUDIENCE: 7.368 times 10 to the negative 7. 00:49:24.330 --> 00:49:29.330 PROFESSOR: 7.368 times 10 to the minus 7. 00:49:29.330 --> 00:49:34.280 This is actually equal to 7.37 times 10-- 00:49:34.280 --> 00:49:34.790 oh, wait. 00:49:34.790 --> 00:49:40.050 This is actually 737 nanometer. 00:49:40.050 --> 00:49:43.880 Actually, the wavelength of the red 00:49:43.880 --> 00:49:48.180 is actually between 620 and 750. 00:49:48.180 --> 00:49:51.791 And actually we are actually getting the correct value. 00:49:51.791 --> 00:49:52.290 You see? 00:49:52.290 --> 00:49:55.770 So, actually, now you can actually tell your friends 00:49:55.770 --> 00:49:59.400 that, although the wavelength is so small, 00:49:59.400 --> 00:50:04.490 but I can't measure it with such a square feet 00:50:04.490 --> 00:50:07.700 experimental setup. 00:50:07.700 --> 00:50:10.220 So that's a successful experiment. 00:50:10.220 --> 00:50:12.550 So that is actually telling you that it's 00:50:12.550 --> 00:50:16.400 a proof that this formula, which we actually 00:50:16.400 --> 00:50:19.110 do all the crazy work of this Fourier 00:50:19.110 --> 00:50:22.355 transform in two-dimensional integration, 00:50:22.355 --> 00:50:24.470 it should really work. 00:50:24.470 --> 00:50:29.750 And the result is actually not really far from what 00:50:29.750 --> 00:50:33.770 you can get from Wikipedia. 00:50:33.770 --> 00:50:37.190 So, at this point, I would like to take a five-minute break 00:50:37.190 --> 00:50:38.750 to take some questions. 00:50:38.750 --> 00:50:43.790 And then we are going to come back in, at 31, 00:50:43.790 --> 00:50:47.340 and we are going to discuss another very interesting issue, 00:50:47.340 --> 00:50:48.471 resolution. 00:50:53.770 --> 00:50:55.540 So welcome back. 00:50:55.540 --> 00:50:59.680 So there are a few questions about-- 00:51:02.590 --> 00:51:08.420 there were a few questions about the pattern here, 00:51:08.420 --> 00:51:09.790 which is interesting. 00:51:09.790 --> 00:51:12.970 So you can see that what we actually concluded from here 00:51:12.970 --> 00:51:18.700 is that the width of the central principal maxima 00:51:18.700 --> 00:51:23.950 is actually two times of the width of the secondary maxima. 00:51:23.950 --> 00:51:27.330 So you can see that the width here between these two points 00:51:27.330 --> 00:51:31.840 is actually lambda over D. But the width between these two 00:51:31.840 --> 00:51:34.070 points, which actually give you the width 00:51:34.070 --> 00:51:38.690 of the central principal maxima, is actually 2 times 00:51:38.690 --> 00:51:40.900 of lambda over D. 00:51:40.900 --> 00:51:44.190 And now this actually can be seen from the experiment there. 00:51:44.190 --> 00:51:47.090 Maybe not easy for the moment. 00:51:47.090 --> 00:51:50.340 But this is actually the width, and the smaller structure 00:51:50.340 --> 00:51:58.090 is actually having a width half of the central peak. 00:51:58.090 --> 00:52:00.950 So that is actually something which is interesting, 00:52:00.950 --> 00:52:04.550 and I would like to share that with everybody. 00:52:04.550 --> 00:52:07.870 So now we actually come back to the original question 00:52:07.870 --> 00:52:11.800 we were actually discussing last time. 00:52:11.800 --> 00:52:13.960 So one interesting thing we observed 00:52:13.960 --> 00:52:17.770 in this two-slit interference experiment 00:52:17.770 --> 00:52:22.330 is that you not only see all those little structures, which 00:52:22.330 --> 00:52:24.850 is actually kind of periodic structure, 00:52:24.850 --> 00:52:28.300 and that they are coming from the two-slit interference. 00:52:28.300 --> 00:52:31.000 And you also see this larger structure 00:52:31.000 --> 00:52:34.750 which is showing up there, which is actually going up and down, 00:52:34.750 --> 00:52:38.530 and also it produce minima at some specific point. 00:52:38.530 --> 00:52:42.890 Now we understand what is actually happening. 00:52:42.890 --> 00:52:49.960 Suppose I have two-slit interference experiment, where 00:52:49.960 --> 00:52:57.200 I have the width of the slits to be capital D, to be very small. 00:52:57.200 --> 00:52:59.185 D is very, very small. 00:53:01.890 --> 00:53:04.500 And the distance between the slit 00:53:04.500 --> 00:53:06.460 is actually called small d. 00:53:06.460 --> 00:53:09.870 Which is kind of weird, but you have 00:53:09.870 --> 00:53:13.260 to accept that because it's on my note. 00:53:13.260 --> 00:53:15.830 And you can see that, interestingly, 00:53:15.830 --> 00:53:21.240 if this is the situation, then you have this periodic pattern 00:53:21.240 --> 00:53:25.630 and you will see no decrease in amplitude 00:53:25.630 --> 00:53:30.150 as a function of distance with respect to the central point 00:53:30.150 --> 00:53:33.390 of the screen. 00:53:33.390 --> 00:53:35.490 So that's actually very nice. 00:53:35.490 --> 00:53:42.570 However, if you consider a realistic situation, 00:53:42.570 --> 00:53:50.360 where the size, or say the width of the slit is not negligible, 00:53:50.360 --> 00:53:52.040 is sizable. 00:53:52.040 --> 00:53:54.710 And what is going to happen is that-- 00:53:54.710 --> 00:54:00.170 OK, let's forget about the second one for a moment. 00:54:00.170 --> 00:54:05.060 We already learned that the output intensity 00:54:05.060 --> 00:54:10.250 of a single slit is already varying as a function of angle. 00:54:10.250 --> 00:54:12.300 So I have this pattern. 00:54:12.300 --> 00:54:20.570 Therefore, if you have these two realistic slit interacting 00:54:20.570 --> 00:54:23.570 with each other, have interference pattern, 00:54:23.570 --> 00:54:29.120 what you are going to expect is that you are going 00:54:29.120 --> 00:54:35.810 to have the two-slit interference pattern modulated 00:54:35.810 --> 00:54:39.670 by diffraction pattern. 00:54:39.670 --> 00:54:42.850 Because, originally, coming from a single slit, 00:54:42.850 --> 00:54:47.740 you already have a varying intensity 00:54:47.740 --> 00:54:54.500 as a function of sine theta, as we already discussed there. 00:54:54.500 --> 00:55:00.130 So, if we put all those information together, 00:55:00.130 --> 00:55:03.940 we are going to get I. The intensity is going 00:55:03.940 --> 00:55:13.390 to be equal to I0, which is some maxima, sine beta divided 00:55:13.390 --> 00:55:22.790 by beta, square of that, sine N delta divided by 2, 00:55:22.790 --> 00:55:29.610 divided by sine delta divided by 2, squared. 00:55:29.610 --> 00:55:32.130 So basically what I'm talking about 00:55:32.130 --> 00:55:36.210 is that, if you have N-slit experiment, 00:55:36.210 --> 00:55:40.950 each slit have the same width. 00:55:40.950 --> 00:55:42.540 And what you are going to get is-- 00:55:42.540 --> 00:55:47.720 this is actually the N-slit interference pattern. 00:55:56.510 --> 00:56:05.288 And that is actually modulated by diffraction pattern. 00:56:09.680 --> 00:56:13.760 Where beta, just a reminder, in this summary 00:56:13.760 --> 00:56:19.850 is pi capital D divided by lambda sine theta. 00:56:19.850 --> 00:56:22.700 And the delta, which is the optical path length 00:56:22.700 --> 00:56:28.880 difference we defined before, is k times d sine theta, 00:56:28.880 --> 00:56:35.160 and that is actually equal to 2 pi times d sine theta 00:56:35.160 --> 00:56:38.390 divided by lambda. 00:56:38.390 --> 00:56:42.290 So that is actually why, when we perform 00:56:42.290 --> 00:56:50.510 the experiment of a double-slit experiment in the last lecture, 00:56:50.510 --> 00:56:55.520 we get complicated interference pattern like this, 00:56:55.520 --> 00:56:58.400 and it has a very complicated structure. 00:56:58.400 --> 00:57:02.340 And now we actually understand why the structure is like this. 00:57:02.340 --> 00:57:06.710 The small structure in this case is actually 00:57:06.710 --> 00:57:10.970 coming from interference, two-slit interference. 00:57:10.970 --> 00:57:15.270 And the additional structure, larger-scale structure, 00:57:15.270 --> 00:57:18.300 is actually coming from diffraction, 00:57:18.300 --> 00:57:22.350 is coming from the varying intensity of a single slit 00:57:22.350 --> 00:57:26.180 as a function of sine theta. 00:57:26.180 --> 00:57:27.180 Any questions so far? 00:57:30.920 --> 00:57:33.500 We are making a lot of progress. 00:57:33.500 --> 00:57:36.710 So what I would like to move on is 00:57:36.710 --> 00:57:39.390 to discuss with you something really interesting. 00:57:39.390 --> 00:57:48.590 So we discussed and learned how to explain why we have actually 00:57:48.590 --> 00:57:52.130 colorful soap bubble. 00:57:52.130 --> 00:57:55.100 So I have something totally unrelated. 00:57:55.100 --> 00:58:00.160 So we have a soap bubble also in the space, which 00:58:00.160 --> 00:58:03.500 is the Soap Bubble nebula. 00:58:03.500 --> 00:58:05.720 Which is really interesting, and you can actually 00:58:05.720 --> 00:58:08.580 Google it and see what is actually happening there. 00:58:08.580 --> 00:58:11.290 But, actually, that's actually not my point. 00:58:11.290 --> 00:58:16.340 Then my point is that you really need very good resolution 00:58:16.340 --> 00:58:21.050 telescope so that you can actually observe those really 00:58:21.050 --> 00:58:26.030 beautiful objects which are already there and cannot be 00:58:26.030 --> 00:58:28.250 made by human. 00:58:28.250 --> 00:58:29.450 Made by somebody else. 00:58:32.130 --> 00:58:36.840 So this is actually what I'm getting into. 00:58:36.840 --> 00:58:41.220 So the resolution is really something important. 00:58:41.220 --> 00:58:45.000 So when you take a look at this picture, 00:58:45.000 --> 00:58:47.670 the resolution is not very good. 00:58:47.670 --> 00:58:52.470 So as you can see, now the peak position of two nearby peak 00:58:52.470 --> 00:58:55.060 is actually connecting to each other. 00:58:55.060 --> 00:58:58.370 Then what do we see from this picture? 00:58:58.370 --> 00:59:00.320 You see maybe a lion? 00:59:00.320 --> 00:59:01.190 I don't know. 00:59:01.190 --> 00:59:04.010 Maybe, maybe not. 00:59:04.010 --> 00:59:10.000 But if you improve the resolution, what do you see? 00:59:10.000 --> 00:59:12.800 It's actually zebra. 00:59:12.800 --> 00:59:14.750 So this is actually the kind of thing 00:59:14.750 --> 00:59:18.590 which we would like to discuss with you. 00:59:18.590 --> 00:59:22.790 We are actually touching this important phenomenon, 00:59:22.790 --> 00:59:25.475 which is actually needed for observing 00:59:25.475 --> 00:59:31.850 an interesting phenomena which is actually happening really 00:59:31.850 --> 00:59:35.600 far away from the Earth. 00:59:35.600 --> 00:59:39.480 What is actually the resolution? 00:59:39.480 --> 00:59:41.740 And we are going to talk about that as well. 00:59:41.740 --> 00:59:46.120 And I would like to show you another interesting example. 00:59:46.120 --> 00:59:52.200 So this is a comparison between not so serious picture 00:59:52.200 --> 00:59:55.440 and the picture from Hubble telescope. 00:59:55.440 --> 01:00:01.360 So I was using some telescope with D equal to 40 centimeter. 01:00:01.360 --> 01:00:02.850 And that's actually the best thing 01:00:02.850 --> 01:00:09.720 which I can achieve, shooting the same planetary nebula M57. 01:00:09.720 --> 01:00:13.650 That object is actually 2,500 light year away. 01:00:13.650 --> 01:00:17.490 And you can see that I cannot get really a lot of detail from 01:00:17.490 --> 01:00:19.350 this image. 01:00:19.350 --> 01:00:23.310 And now, if you compare that to D equal 01:00:23.310 --> 01:00:27.810 to 240 centimeter Hubble telescope, 01:00:27.810 --> 01:00:29.400 and also, at the same time, this thing 01:00:29.400 --> 01:00:33.324 is actually above the atmosphere. 01:00:33.324 --> 01:00:34.990 So that's actually very, very important. 01:00:34.990 --> 01:00:39.430 And you can see that you do get a much, much better resolution, 01:00:39.430 --> 01:00:43.090 and you can actually see all the fine detail, very, very 01:00:43.090 --> 01:00:46.690 fine detail of this image. 01:00:46.690 --> 01:00:50.630 And we are in the position to understand 01:00:50.630 --> 01:00:59.075 the resolution and the limit which we can have 01:00:59.075 --> 01:01:01.540 due to diffraction, actually. 01:01:01.540 --> 01:01:12.220 So if I consider now a pinhole with diameter equal 01:01:12.220 --> 01:01:18.170 to D. So right now what we are actually doing 01:01:18.170 --> 01:01:24.250 is not a single slit any more, but a hole with radius 01:01:24.250 --> 01:01:27.220 D over 2. 01:01:27.220 --> 01:01:34.150 And we can do the same, exactly the same calculation 01:01:34.150 --> 01:01:36.220 using this formula. 01:01:36.220 --> 01:01:39.800 But I'm not going to do that for the sake of time. 01:01:39.800 --> 01:01:44.680 So we can do exactly that same C function calculation. 01:01:44.680 --> 01:01:50.920 And what we are going to get is I as a function of theta 01:01:50.920 --> 01:02:01.120 is equal to I0 J1 beta divided by beta, squared. 01:02:01.120 --> 01:02:07.870 Where J1 is the Bessel function of the first kind. 01:02:07.870 --> 01:02:11.200 Sounds really scary, but it's actually not. 01:02:11.200 --> 01:02:20.440 So what I really need is the zeros of the Bessel function, 01:02:20.440 --> 01:02:24.160 so that I can actually extract the interference 01:02:24.160 --> 01:02:29.170 pattern and the width of the central maximum. 01:02:29.170 --> 01:02:31.880 So now, since we are having a pinhole, 01:02:31.880 --> 01:02:33.940 basically all of those things are, 01:02:33.940 --> 01:02:37.720 all those patterns are actually two-dimensional. 01:02:37.720 --> 01:02:46.660 And I was wondering what will be the needed beta value so that I 01:02:46.660 --> 01:02:50.490 can actually reach the minima. 01:02:50.490 --> 01:02:52.610 Why is that important? 01:02:52.610 --> 01:02:55.800 That is actually telling you the limit 01:02:55.800 --> 01:02:58.810 of the optical resolution. 01:02:58.810 --> 01:03:02.780 If I have two peaks which are actually 01:03:02.780 --> 01:03:06.960 placed too close to each other, like what we actually 01:03:06.960 --> 01:03:09.000 see in the previous slide, then we 01:03:09.000 --> 01:03:15.510 can actually not separate very well these two light source. 01:03:15.510 --> 01:03:19.590 On the other hand, if the distance between these two peak 01:03:19.590 --> 01:03:23.430 is larger than the first minima, then I 01:03:23.430 --> 01:03:25.221 can actually be very safe. 01:03:25.221 --> 01:03:26.220 I can actually separate. 01:03:26.220 --> 01:03:28.770 I can say that, ha, this is really two peaks. 01:03:28.770 --> 01:03:31.080 Two stars, two light source. 01:03:31.080 --> 01:03:33.200 I can tell. 01:03:33.200 --> 01:03:36.060 So that is actually why this is actually important. 01:03:36.060 --> 01:03:39.750 And where is actually the minima? 01:03:39.750 --> 01:03:42.990 And I can already solve that for you. 01:03:42.990 --> 01:03:47.130 And that is actually when x is equal to roughly, 01:03:47.130 --> 01:03:53.790 the numerical value is roughly 3.83. 01:03:53.790 --> 01:03:55.290 So that's actually not important. 01:03:55.290 --> 01:03:57.180 Those numbers are not important. 01:03:57.180 --> 01:04:01.210 The important result is really the conclusion. 01:04:01.210 --> 01:04:06.810 So beta is equal to 3.83, and that is actually 01:04:06.810 --> 01:04:13.470 equal to pi D sine theta divided by lambda. 01:04:13.470 --> 01:04:16.280 So that is actually our original definition. 01:04:16.280 --> 01:04:21.390 And I can solve what will be the sine theta, which is actually 01:04:21.390 --> 01:04:25.560 telling you the position of the minima. 01:04:25.560 --> 01:04:33.580 So sine theta will be equal to actually 1.22 lambda divided 01:04:33.580 --> 01:04:36.290 by D. 01:04:36.290 --> 01:04:38.040 So what does that mean? 01:04:38.040 --> 01:04:44.160 That means the position where you have the first minima 01:04:44.160 --> 01:04:47.610 is actually happening when you have sine theta-- 01:04:52.480 --> 01:04:54.250 this is the theta-- 01:04:54.250 --> 01:04:58.980 when you have sine theta equal to 1.22 times lambda divided 01:04:58.980 --> 01:05:04.280 by D. 01:05:04.280 --> 01:05:07.130 So that is actually very nice. 01:05:07.130 --> 01:05:09.830 Doing exactly the same exercise, and we now 01:05:09.830 --> 01:05:14.680 understand where my minima is. 01:05:14.680 --> 01:05:16.790 Then that is actually telling you 01:05:16.790 --> 01:05:19.994 something about the resolution. 01:05:26.910 --> 01:05:29.480 So what I'm going to try to get into 01:05:29.480 --> 01:05:34.910 is that, now, let's design a phone together, a mobile phone 01:05:34.910 --> 01:05:36.470 together. 01:05:36.470 --> 01:05:41.820 So what is actually the width of the human pupil? 01:05:44.780 --> 01:05:49.410 The width is actually roughly 2 to 4 millimeter-- 01:05:49.410 --> 01:05:54.480 when narrow, when you see a lot of light all over the place-- 01:05:54.480 --> 01:05:58.000 or 3 to 8 millimeter. 01:05:58.000 --> 01:06:03.860 So that is actually the typical length when wide. 01:06:03.860 --> 01:06:06.910 So that is actually the width of the pinhole. 01:06:09.590 --> 01:06:14.960 So, typically, the visible light, as we calculated, 01:06:14.960 --> 01:06:18.610 is something like 500 nanometer. 01:06:21.820 --> 01:06:25.940 And the width of the human pupil, we 01:06:25.940 --> 01:06:31.430 can actually take a number of 5 millimeter. 01:06:31.430 --> 01:06:37.700 And now we can actually try to give input to the phone design. 01:06:37.700 --> 01:06:42.650 So what will be the resolution if we take these two 01:06:42.650 --> 01:06:44.510 parameter together? 01:06:44.510 --> 01:06:47.750 So, basically, the resolution of your eye, we 01:06:47.750 --> 01:06:49.910 can now calculate that. 01:06:49.910 --> 01:06:51.520 So what is that? 01:06:51.520 --> 01:07:01.540 That is actually 1.22 times 500 nanometer divided by-- 01:07:01.540 --> 01:07:03.890 OK, my function is D-- 01:07:03.890 --> 01:07:06.980 so divided by D is equal to 5 millimeter. 01:07:06.980 --> 01:07:10.700 5 millimeter. 01:07:10.700 --> 01:07:13.070 Basically, what you are going to get 01:07:13.070 --> 01:07:17.900 is 1.22 times 10 to the minus 4. 01:07:17.900 --> 01:07:20.080 This is actually the resolution. 01:07:20.080 --> 01:07:23.670 Sine theta, roughly equal to theta, 01:07:23.670 --> 01:07:28.430 is actually equal to 1.22 times 10 to the minus 4. 01:07:33.020 --> 01:07:40.240 I have a iPhone 6 or 7, whatever you have. 01:07:40.240 --> 01:07:44.850 Basically is 401 ppi. 01:07:44.850 --> 01:07:48.280 401 ppi is actually what is that? 01:07:48.280 --> 01:07:51.250 Pixel per inch. 01:07:51.250 --> 01:07:53.200 So what is actually the delta x? 01:07:53.200 --> 01:07:57.680 So if I have a phone, OK, it has a camera there. 01:07:57.680 --> 01:08:00.640 That is my phone. 01:08:00.640 --> 01:08:03.010 And this is my eye. 01:08:03.010 --> 01:08:05.740 Looks like an eye. 01:08:05.740 --> 01:08:09.670 The distance is 20 centimeter. 01:08:09.670 --> 01:08:11.500 I do this, which is unusual. 01:08:17.200 --> 01:08:18.850 We have 400 ppi. 01:08:18.850 --> 01:08:24.189 So what is actually the delta x, the delta x between the pixels? 01:08:24.189 --> 01:08:32.109 The delta x is equal to 2.54 centimeter divided by 401, 01:08:32.109 --> 01:08:36.220 and that will give you something like 6.3 times 10 01:08:36.220 --> 01:08:38.098 to the minus 3 centimeter. 01:08:40.970 --> 01:08:45.930 If I am trying to be healthy and I do this, then 01:08:45.930 --> 01:08:47.649 what is actually the delta theta? 01:08:47.649 --> 01:08:52.540 The delta theta is delta x divided by 20 centimeter, 01:08:52.540 --> 01:08:55.660 and that is 3 times 10 to be minus 4. 01:08:59.420 --> 01:09:04.939 If you compare this value to the limit 01:09:04.939 --> 01:09:11.090 I calculated here, you can see that, what is the conclusion? 01:09:11.090 --> 01:09:16.010 Can I resolve the pixels on the phone? 01:09:16.010 --> 01:09:19.640 The answer is yes. 01:09:19.640 --> 01:09:22.950 So that means this phone is not good enough. 01:09:22.950 --> 01:09:24.080 They have to do more work. 01:09:26.600 --> 01:09:29.689 And now I'm going to design a jPhone. 01:09:34.330 --> 01:09:38.630 Maybe at some point I got really crazy 01:09:38.630 --> 01:09:41.779 and I decided to open a company, which 01:09:41.779 --> 01:09:44.450 is Yen-Jie's phone company. 01:09:44.450 --> 01:09:50.990 And, of course, I will say this is jPhone because it's Yen-Jie. 01:09:50.990 --> 01:10:01.600 And I'm going to put 40,000 ppi in this phone. 01:10:01.600 --> 01:10:03.424 Will you buy it? 01:10:03.424 --> 01:10:03.965 AUDIENCE: No. 01:10:03.965 --> 01:10:04.590 AUDIENCE: Sure. 01:10:04.590 --> 01:10:05.950 AUDIENCE: How much? 01:10:05.950 --> 01:10:07.257 PROFESSOR: $1. 01:10:07.257 --> 01:10:07.840 You'll buy it? 01:10:07.840 --> 01:10:10.440 We'll see. 01:10:10.440 --> 01:10:14.760 Maybe you will buy it because you are my student. 01:10:14.760 --> 01:10:16.420 But it's not worth it. 01:10:16.420 --> 01:10:18.080 Why is that? 01:10:18.080 --> 01:10:26.520 Because you cannot resolve this kind of fine or small distance 01:10:26.520 --> 01:10:27.780 between pixels. 01:10:27.780 --> 01:10:29.850 So it's actually useless. 01:10:29.850 --> 01:10:31.100 So what is actually the limit? 01:10:31.100 --> 01:10:34.590 You can also probably give that to your friends. 01:10:34.590 --> 01:10:40.720 2,000 pixel per inch is roughly the limit. 01:10:40.720 --> 01:10:46.140 Beyond that, maybe the next generation of our students 01:10:46.140 --> 01:10:49.790 will be using this like this. 01:10:49.790 --> 01:10:52.580 Then it works, and it is worth it. 01:10:52.580 --> 01:10:56.230 You can actually read this distance. 01:10:56.230 --> 01:11:00.980 It doesn't work for old people like me, but for young people 01:11:00.980 --> 01:11:03.260 it works. 01:11:03.260 --> 01:11:04.340 So very good. 01:11:04.340 --> 01:11:06.800 So that's another thing which you have learned. 01:11:06.800 --> 01:11:10.670 So, finally, as I promised you, we 01:11:10.670 --> 01:11:15.650 are going to go back to this business of designing 01:11:15.650 --> 01:11:18.860 the Enterprise for Star Trek. 01:11:18.860 --> 01:11:22.770 So what does Enterprise do to their friends? 01:11:22.770 --> 01:11:26.120 They shoot laser beam. 01:11:26.120 --> 01:11:29.780 And they try to attack the other ships. 01:11:29.780 --> 01:11:33.836 And what I'm going to do now is to calculate for you 01:11:33.836 --> 01:11:35.294 what is going to happen. 01:11:35.294 --> 01:11:38.700 OK, now I have this laser beam here. 01:11:38.700 --> 01:11:41.690 And, in principle, before you take 8.03, 01:11:41.690 --> 01:11:47.010 you are going to say, aha, I can shoot the moon. 01:11:47.010 --> 01:11:50.360 And this light is going to be really narrow 01:11:50.360 --> 01:11:53.800 and it's going to hit the moon, a very small area on the moon. 01:11:56.930 --> 01:11:59.750 Do you believe that now? 01:11:59.750 --> 01:12:02.780 I hope the answer is not. 01:12:02.780 --> 01:12:05.660 How crazy is this idea? 01:12:05.660 --> 01:12:07.950 What is the size of the spot? 01:12:07.950 --> 01:12:09.560 Can you guess? 01:12:09.560 --> 01:12:12.980 Is that 1 millimeter? 01:12:12.980 --> 01:12:15.500 10 meter? 01:12:15.500 --> 01:12:18.880 Or 200 kilometer? 01:12:18.880 --> 01:12:23.860 How many of you think by now is 1 millimeter? 01:12:23.860 --> 01:12:24.980 Nobody? 01:12:24.980 --> 01:12:26.830 Fortunately. 01:12:26.830 --> 01:12:30.330 How about 10 meter? 01:12:30.330 --> 01:12:34.660 One, two, three. 01:12:34.660 --> 01:12:35.320 OK. 01:12:35.320 --> 01:12:36.490 Three of you. 01:12:36.490 --> 01:12:39.305 How about 200 kilometer? 01:12:39.305 --> 01:12:41.100 You believe that? 01:12:41.100 --> 01:12:42.980 Really? 01:12:42.980 --> 01:12:49.680 The answer is really 200 kilometer. 01:12:49.680 --> 01:12:53.130 It's the size of Missouri state. 01:12:56.590 --> 01:13:00.310 So now you can see that this is not practical at all, 01:13:00.310 --> 01:13:04.120 and you have to really do what? 01:13:04.120 --> 01:13:07.216 Increase or decrease the radius? 01:13:07.216 --> 01:13:08.070 AUDIENCE: Increase. 01:13:08.070 --> 01:13:09.960 PROFESSOR: Increase. 01:13:09.960 --> 01:13:13.490 By the end of this lecture everybody get this idea. 01:13:13.490 --> 01:13:18.820 Thank you very much for the attention. 01:13:18.820 --> 01:13:21.800 And I hope you enjoyed this lecture.