1 00:00:02,125 --> 00:00:04,920 SPEAKER: The following content is provided under a Creative 2 00:00:04,920 --> 00:00:06,310 Commons license. 3 00:00:06,310 --> 00:00:08,520 Your support will help MIT OpenCourseWare 4 00:00:08,520 --> 00:00:12,610 continue to offer high quality educational resources for free. 5 00:00:12,610 --> 00:00:15,150 To make a donation or to view additional materials 6 00:00:15,150 --> 00:00:19,110 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:19,110 --> 00:00:20,330 at ocw.mit.edu. 8 00:00:23,180 --> 00:00:27,630 YEN-JIE LEE: All right, so let's get started. 9 00:00:27,630 --> 00:00:31,130 So today, very happy to have you here. 10 00:00:31,130 --> 00:00:33,550 We are going to talk about continuing 11 00:00:33,550 --> 00:00:36,260 our discussion on the two-dimensional and 12 00:00:36,260 --> 00:00:37,940 three-dimensional, waves. 13 00:00:37,940 --> 00:00:42,260 So as I mentioned before, there are two interesting situation 14 00:00:42,260 --> 00:00:44,750 which we can actually increase the dimensional. 15 00:00:44,750 --> 00:00:49,100 So for example, I can have all the objects oscillating 16 00:00:49,100 --> 00:00:55,410 in just one direction, but I changed the way 17 00:00:55,410 --> 00:00:59,450 that I placed all those objects in the space. 18 00:00:59,450 --> 00:01:01,820 For example, I can have particles 19 00:01:01,820 --> 00:01:05,690 which are arranged in two or three-dimensional arrays. 20 00:01:05,690 --> 00:01:09,740 And we were talking about how to understand this kind of system. 21 00:01:09,740 --> 00:01:12,170 And all those objects are oscillating 22 00:01:12,170 --> 00:01:14,510 in just one direction-- for example, 23 00:01:14,510 --> 00:01:16,580 up and down in one direction. 24 00:01:16,580 --> 00:01:22,460 That's actually one measure we can increase the dimension. 25 00:01:22,460 --> 00:01:24,710 There's another interesting example 26 00:01:24,710 --> 00:01:26,660 which we will talk about today is 27 00:01:26,660 --> 00:01:29,930 to change the direction of the oscillation, 28 00:01:29,930 --> 00:01:36,370 even rotate the natural electromagnetic waves, 29 00:01:36,370 --> 00:01:37,370 for example. 30 00:01:37,370 --> 00:01:39,770 And how can we achieve that and how 31 00:01:39,770 --> 00:01:42,770 do we understand this phenomena, that's actually going to be 32 00:01:42,770 --> 00:01:45,990 covered by the lecture today. 33 00:01:45,990 --> 00:01:50,180 So before we move forward, we will 34 00:01:50,180 --> 00:01:51,980 have a short review on what we have 35 00:01:51,980 --> 00:01:54,020 learned from the last lecture. 36 00:01:54,020 --> 00:01:57,980 And this is actually what we discussed. 37 00:01:57,980 --> 00:02:01,450 If I have two materials, the left-hand side 38 00:02:01,450 --> 00:02:05,060 and the right-hand side are two different kinds of materials, 39 00:02:05,060 --> 00:02:08,250 or very thin membranes. 40 00:02:08,250 --> 00:02:16,120 And if I have an incident wave coming 41 00:02:16,120 --> 00:02:20,300 into the boundary of these two materials, which is just 42 00:02:20,300 --> 00:02:23,850 x equal to 0, as we discussed last time, 43 00:02:23,850 --> 00:02:27,470 basically the boundary condition require 44 00:02:27,470 --> 00:02:34,760 that in order to make sure that the membrane doesn't break, 45 00:02:34,760 --> 00:02:38,670 that means the k vector, the projection 46 00:02:38,670 --> 00:02:43,650 in the y direction of the k vectors has to be the same. 47 00:02:43,650 --> 00:02:48,740 That means the projection of the wave in the y direction, which 48 00:02:48,740 --> 00:02:53,220 is along this line, the wavelength of all 49 00:02:53,220 --> 00:02:58,190 the incidents refracted and transmitted wave, 50 00:02:58,190 --> 00:03:01,130 the wavelengths should be all the same. 51 00:03:01,130 --> 00:03:05,210 Otherwise, I can change y and make the membrane break, 52 00:03:05,210 --> 00:03:07,280 or break the boundary condition. 53 00:03:07,280 --> 00:03:08,870 So that's actually the first thing 54 00:03:08,870 --> 00:03:13,070 which we learned from the math we were doing last time. 55 00:03:13,070 --> 00:03:18,500 And also, the k value are not arbitrary, right? 56 00:03:18,500 --> 00:03:25,140 So we were already assuming that all three waves, all the three 57 00:03:25,140 --> 00:03:28,490 plane waves are oscillating at the same frequency. 58 00:03:28,490 --> 00:03:30,890 If they are oscillating at different frequencies, 59 00:03:30,890 --> 00:03:33,580 it doesn't work because I can change time, then 60 00:03:33,580 --> 00:03:35,600 make the membranes break. 61 00:03:35,600 --> 00:03:38,290 So therefore, all the three plane waves 62 00:03:38,290 --> 00:03:43,040 will be oscillating at the same frequency-- omega. 63 00:03:43,040 --> 00:03:47,380 And according to this formula, so we have defined 64 00:03:47,380 --> 00:03:50,210 and which is actually the refraction index, 65 00:03:50,210 --> 00:03:53,090 which is equal to some constant c divided 66 00:03:53,090 --> 00:03:58,310 by the speed of the propagation, the phase velocity 67 00:03:58,310 --> 00:04:00,380 of the left-hand side material. 68 00:04:00,380 --> 00:04:03,340 And we also can define n prime, which is actually 69 00:04:03,340 --> 00:04:09,350 equal to c, the same constant, divided by v prime, which 70 00:04:09,350 --> 00:04:12,860 is the speed of propagation of the right-hand side material. 71 00:04:12,860 --> 00:04:17,209 And according to the dispersion relation, 72 00:04:17,209 --> 00:04:20,959 we can calculate what will be the length of the k 73 00:04:20,959 --> 00:04:22,940 if omega is given. 74 00:04:22,940 --> 00:04:25,930 The omega for the left-hand side and right-hand, 75 00:04:25,930 --> 00:04:28,620 refracted, transmitted, and incident waves 76 00:04:28,620 --> 00:04:30,030 should be the same. 77 00:04:30,030 --> 00:04:32,500 Therefore, I can immediately write down 78 00:04:32,500 --> 00:04:34,610 what would be the length of the k vector. 79 00:04:34,610 --> 00:04:38,360 And that means the length of the k vector 80 00:04:38,360 --> 00:04:42,350 would be equal to n times omega over c. 81 00:04:42,350 --> 00:04:46,190 So c over n is actually just v. So basically it's 82 00:04:46,190 --> 00:04:52,300 just the non-dispersive median dispersion relation. 83 00:04:52,300 --> 00:04:54,490 And also you can go ahead and write down 84 00:04:54,490 --> 00:04:56,530 what would be the length of the k prime. 85 00:04:56,530 --> 00:05:02,030 And that is actually determined by n prime omega over c. 86 00:05:02,030 --> 00:05:06,755 So based on these two interesting formations, 87 00:05:06,755 --> 00:05:12,890 what we can conclude is that that means k prime and kr 88 00:05:12,890 --> 00:05:14,930 cannot be arbitrary. 89 00:05:14,930 --> 00:05:19,670 They have to be aligned to form a specific pattern. 90 00:05:19,670 --> 00:05:21,800 The pattern is that the projection 91 00:05:21,800 --> 00:05:24,940 in the right direction should be the same. 92 00:05:24,940 --> 00:05:29,680 And also at the same time, the length of the k vector 93 00:05:29,680 --> 00:05:34,450 is determined by the refraction index. 94 00:05:34,450 --> 00:05:38,560 So once we fix all that and put them all together, 95 00:05:38,560 --> 00:05:42,410 we can conclude that based on the projection in the y 96 00:05:42,410 --> 00:05:45,500 direction, we will conclude that theta 97 00:05:45,500 --> 00:05:49,780 will be equal to theta r, where theta is actually that incident 98 00:05:49,780 --> 00:05:54,160 angle and theta r is the refraction angle, which 99 00:05:54,160 --> 00:05:58,060 is describing the direction of the refractive wave. 100 00:05:58,060 --> 00:06:01,210 And also the second very interesting information 101 00:06:01,210 --> 00:06:06,010 we learned is that based on the boundary condition, 102 00:06:06,010 --> 00:06:11,140 we can conclude n sine theta will be equal to n prime sine 103 00:06:11,140 --> 00:06:12,700 theta prime. 104 00:06:12,700 --> 00:06:15,610 So that's actually Snell's law, which some of you 105 00:06:15,610 --> 00:06:17,680 actually learned from the high school days 106 00:06:17,680 --> 00:06:21,640 or from the earlier lectures in physics. 107 00:06:21,640 --> 00:06:27,280 And these two interesting results 108 00:06:27,280 --> 00:06:32,760 form the basis of geometrical optics or laws. 109 00:06:32,760 --> 00:06:35,380 That's two of the most important laws we 110 00:06:35,380 --> 00:06:38,230 learn from geometrical optics. 111 00:06:38,230 --> 00:06:43,000 So that actually gives you some examples 112 00:06:43,000 --> 00:06:45,970 and gives you some more feeling about what 113 00:06:45,970 --> 00:06:47,650 we are talking about. 114 00:06:47,650 --> 00:06:50,650 So what we are talking about is that if I have an incident 115 00:06:50,650 --> 00:06:59,410 wave coming into this boundary, and that incident angle is 116 00:06:59,410 --> 00:07:04,900 theta 1, what I would expect based on what we have just 117 00:07:04,900 --> 00:07:10,690 derived is that theta r, the refracted wave 118 00:07:10,690 --> 00:07:15,080 direction, the refractive wave angle, 119 00:07:15,080 --> 00:07:18,160 the refraction angle would be equal to the incident angle, 120 00:07:18,160 --> 00:07:20,710 according to what we have just derived. 121 00:07:20,710 --> 00:07:24,280 I was just rotating this by 90 degrees. 122 00:07:24,280 --> 00:07:28,870 And also we can calculate what will be theta 2 according 123 00:07:28,870 --> 00:07:30,170 to Snell's law-- 124 00:07:30,170 --> 00:07:34,200 n1 sine theta 1 equal to n2 sine theta 2. 125 00:07:34,200 --> 00:07:40,240 And if I continue and propagate the incident wave, 126 00:07:40,240 --> 00:07:44,275 here I assume that n2 is larger. 127 00:07:50,260 --> 00:07:52,210 If you have a larger refraction index, 128 00:07:52,210 --> 00:07:57,340 that means the speed of propagation is smaller. 129 00:07:57,340 --> 00:08:01,870 Larger n value give you a smaller speed of propagation. 130 00:08:01,870 --> 00:08:06,430 Therefore, you can see that the wavefront, which is actually 131 00:08:06,430 --> 00:08:08,990 the position of the peak, actually 132 00:08:08,990 --> 00:08:14,020 got delayed compared to the original projection. 133 00:08:14,020 --> 00:08:17,550 And you can see that the red line is actually really 134 00:08:17,550 --> 00:08:20,990 what you would see in the second median. 135 00:08:20,990 --> 00:08:23,170 And then we can also continue to propagate 136 00:08:23,170 --> 00:08:25,600 and you will see that, interesting, 137 00:08:25,600 --> 00:08:32,049 that means the plane wave would change direction because 138 00:08:32,049 --> 00:08:34,409 of the matching boundary condition, 139 00:08:34,409 --> 00:08:36,520 the membranes doesn't break. 140 00:08:36,520 --> 00:08:39,730 You can see that the peak position 141 00:08:39,730 --> 00:08:44,260 match from the median number 1 and median number 2. 142 00:08:44,260 --> 00:08:49,810 The peak position, which is the position of the line, match. 143 00:08:49,810 --> 00:08:55,490 And also, due to the slower speed of propagation, 144 00:08:55,490 --> 00:08:57,700 the plane wave actually change direction. 145 00:08:57,700 --> 00:09:00,316 So that's actually how we can understand 146 00:09:00,316 --> 00:09:06,410 the mathematical result which we derived last time 147 00:09:06,410 --> 00:09:09,970 by this interesting example. 148 00:09:09,970 --> 00:09:13,150 So that's considered a situation which 149 00:09:13,150 --> 00:09:17,000 is maybe a little bit interesting to you. 150 00:09:17,000 --> 00:09:21,430 So what will happen if I now shoot light 151 00:09:21,430 --> 00:09:26,260 from inside some material which delay the light slightly-- 152 00:09:26,260 --> 00:09:30,250 for example, n1 is equal to 1.5. 153 00:09:30,250 --> 00:09:35,290 I shoot something through the material, 154 00:09:35,290 --> 00:09:39,830 and then the second material I have has higher speed of light, 155 00:09:39,830 --> 00:09:42,340 which is actually having the speed of light 156 00:09:42,340 --> 00:09:46,720 equal to light in a vacuum case. 157 00:09:46,720 --> 00:09:51,890 So in this example, I have n1 equal to n2. 158 00:09:51,890 --> 00:09:59,920 Since n1 sine theta 1 is equal to n2 sine theta 2, 159 00:09:59,920 --> 00:10:03,240 so now I can calculate what will be the resulting theta 160 00:10:03,240 --> 00:10:05,710 2 according to this equation. 161 00:10:05,710 --> 00:10:09,340 So the resulting theta 2 will be sine theta 2 will 162 00:10:09,340 --> 00:10:15,610 be equal to n1 over n2 sine theta 1. 163 00:10:15,610 --> 00:10:19,600 In this case, n1 is greater than n2, therefore, 164 00:10:19,600 --> 00:10:21,440 this term is greater than 1. 165 00:10:24,250 --> 00:10:29,830 In this case, actually, this ratio is equal to 1.5. 166 00:10:29,830 --> 00:10:31,480 So this is actually pretty interesting. 167 00:10:31,480 --> 00:10:36,910 That means this factor is actually greater than 1. 168 00:10:36,910 --> 00:10:41,770 I have sine theta 1, which is multiplied in this vector. 169 00:10:41,770 --> 00:10:48,530 So that means if I increase theta 1, if I increase 170 00:10:48,530 --> 00:10:52,630 theta 1, at some point I will not 171 00:10:52,630 --> 00:10:56,170 be able to get theta 2 because theta 2 will 172 00:10:56,170 --> 00:11:01,680 be arcsine 1.5 sine 1. 173 00:11:01,680 --> 00:11:07,540 And at some point, 1.5 sine theta 1 will be greater than 1, 174 00:11:07,540 --> 00:11:09,880 then I don't have a theta 2, which 175 00:11:09,880 --> 00:11:12,300 can satisfy this equation. 176 00:11:12,300 --> 00:11:15,090 So what will happen? 177 00:11:15,090 --> 00:11:19,340 Maybe the whole system explodes, I don't know. 178 00:11:19,340 --> 00:11:23,350 So we are going to do a simple experiment 179 00:11:23,350 --> 00:11:25,370 to see what is going to happen. 180 00:11:25,370 --> 00:11:30,310 So before we do this kind of dangerous experiment, 181 00:11:30,310 --> 00:11:37,480 we are going to turn this light off and be prepared. 182 00:11:37,480 --> 00:11:38,920 Hope everybody will survive. 183 00:11:42,110 --> 00:11:50,020 So here I have a laser, which is actually shooting a laser beam 184 00:11:50,020 --> 00:11:53,570 through this tank of water. 185 00:11:53,570 --> 00:11:57,970 So this is water, with n value roughly 1.33. 186 00:11:57,970 --> 00:12:05,170 And outside of water is air, so the speed of the light 187 00:12:05,170 --> 00:12:08,255 is roughly equal to c. 188 00:12:08,255 --> 00:12:11,180 So n value is actually roughly 1. 189 00:12:11,180 --> 00:12:14,830 So that's actually exactly the situation we are looking for. 190 00:12:14,830 --> 00:12:19,970 And let's turn on this experiment, careful. 191 00:12:19,970 --> 00:12:23,060 Wow, look at this. 192 00:12:23,060 --> 00:12:30,630 You can see that here there was no light coming out. 193 00:12:30,630 --> 00:12:31,860 Why is that? 194 00:12:31,860 --> 00:12:33,540 Because of the mathematics-- 195 00:12:33,540 --> 00:12:36,980 the mathematics say that, OK, sorry, guys, 196 00:12:36,980 --> 00:12:39,210 theta 2 doesn't work. 197 00:12:39,210 --> 00:12:41,700 Therefore, there will be no light really coming out. 198 00:12:41,700 --> 00:12:45,960 You can see my hand, you can see the light here. 199 00:12:45,960 --> 00:12:48,870 When I put my hand here, it's not burning my hand. 200 00:12:48,870 --> 00:12:50,230 But that's OK. 201 00:12:50,230 --> 00:12:53,730 All right, and you can see that nothing really comes out. 202 00:12:53,730 --> 00:12:59,070 And all the energy are bounced back into the water. 203 00:12:59,070 --> 00:13:01,050 The same thing also happened here. 204 00:13:01,050 --> 00:13:03,090 You can see that the light is actually 205 00:13:03,090 --> 00:13:07,620 bouncing back and forth and moving into the left-hand side 206 00:13:07,620 --> 00:13:11,610 direction until it passed through here, maybe into there, 207 00:13:11,610 --> 00:13:16,090 because it's actually still bouncing around 208 00:13:16,090 --> 00:13:19,320 inside this tank. 209 00:13:19,320 --> 00:13:23,580 So the good news is that no explosion, like what we 210 00:13:23,580 --> 00:13:25,320 have here. 211 00:13:25,320 --> 00:13:30,240 The interesting thing we find is that all the energy 212 00:13:30,240 --> 00:13:35,550 will be bounced back if you have large enough incident 213 00:13:35,550 --> 00:13:37,828 angle in this situation. 214 00:13:41,180 --> 00:13:48,530 As you can see from here that all the light's energy are 215 00:13:48,530 --> 00:13:50,680 bounced back into water. 216 00:13:53,610 --> 00:13:56,580 So that's actually very, very interesting. 217 00:13:56,580 --> 00:13:58,790 And that means we can probably make 218 00:13:58,790 --> 00:14:07,650 use of this property to send light through some large n 219 00:14:07,650 --> 00:14:09,060 material. 220 00:14:09,060 --> 00:14:16,310 So that's actually how optical fiber works. 221 00:14:16,310 --> 00:14:19,320 So basically you can actually shoot a light 222 00:14:19,320 --> 00:14:21,100 into the optical fiber. 223 00:14:21,100 --> 00:14:24,150 And the light is going to be bouncing back and forth 224 00:14:24,150 --> 00:14:26,880 between the boundaries inside the fiber. 225 00:14:26,880 --> 00:14:30,740 And then you can actually send those information through light 226 00:14:30,740 --> 00:14:33,300 by this kind of mechanism. 227 00:14:33,300 --> 00:14:37,350 So that's how optical fiber actually works. 228 00:14:37,350 --> 00:14:45,150 And we have a setup, which I have a light here, 229 00:14:45,150 --> 00:14:49,320 shooting light into the optical fiber. 230 00:14:49,320 --> 00:14:52,620 And I have paper here which tried 231 00:14:52,620 --> 00:14:55,950 to block part of the light. 232 00:14:55,950 --> 00:15:02,310 So from here, you can see the text which I put on the paper, 233 00:15:02,310 --> 00:15:06,360 because some of the light is blocked 234 00:15:06,360 --> 00:15:09,430 by this paper, the text on the paper. 235 00:15:09,430 --> 00:15:15,630 And this light goes through this optical fiber 236 00:15:15,630 --> 00:15:20,760 and continues and propagates and got captured by the camera. 237 00:15:20,760 --> 00:15:23,550 So you can see, can you read out loud 238 00:15:23,550 --> 00:15:25,470 what is actually written there? 239 00:15:25,470 --> 00:15:27,540 Can you see it? 240 00:15:27,540 --> 00:15:30,040 The-- 241 00:15:30,040 --> 00:15:39,540 AUDIENCE: Last question on the 242 00:15:39,540 --> 00:15:47,060 YEN-JIE LEE: The exam is-- 243 00:15:47,060 --> 00:15:50,750 oh, no, no, no, no, transmission interrupted. 244 00:15:50,750 --> 00:15:51,380 Oh my god. 245 00:15:54,080 --> 00:15:57,850 All right, so I'm sorry it didn't work, 246 00:15:57,850 --> 00:16:01,910 but it worked in the beginning, right? 247 00:16:01,910 --> 00:16:05,180 So you can see that this is a wonderful way 248 00:16:05,180 --> 00:16:08,140 to send optical signal. 249 00:16:08,140 --> 00:16:09,890 And we actually do that. 250 00:16:09,890 --> 00:16:15,730 We send signal from US to Asia through all those crazy lines 251 00:16:15,730 --> 00:16:16,710 under the sea. 252 00:16:16,710 --> 00:16:18,610 So that's really cool. 253 00:16:18,610 --> 00:16:22,220 And finally before we change the topic, 254 00:16:22,220 --> 00:16:27,160 I'm sure that you will enjoy doing the practice in your PSET 255 00:16:27,160 --> 00:16:28,120 number 8. 256 00:16:28,120 --> 00:16:32,510 We are going to learn that this is actually highly related 257 00:16:32,510 --> 00:16:37,310 to a beautiful phenomena we see in daily life-- 258 00:16:37,310 --> 00:16:41,440 the rainbow is actually really related to Snell's law, 259 00:16:41,440 --> 00:16:46,970 and that you are going to solve this problem in the PSET. 260 00:16:46,970 --> 00:16:49,750 All right, so any questions so far related 261 00:16:49,750 --> 00:16:55,800 to Snell's law and refraction index, et cetera? 262 00:16:55,800 --> 00:17:00,240 All right, so if not, we'll go ahead and talk 263 00:17:00,240 --> 00:17:03,460 about the second example. 264 00:17:03,460 --> 00:17:06,780 The second example is that instead 265 00:17:06,780 --> 00:17:11,339 introducing more objects in this array, 266 00:17:11,339 --> 00:17:15,690 we could change the direction of the oscillation 267 00:17:15,690 --> 00:17:19,890 as a function of time and see what is going to happen. 268 00:17:19,890 --> 00:17:25,050 For example, the direction of the electric field, 269 00:17:25,050 --> 00:17:28,020 I can make it rotate or change as a function of time. 270 00:17:28,020 --> 00:17:32,380 And that's actually called circularly polarized wave. 271 00:17:32,380 --> 00:17:37,860 And that means we are going to talk about polarization today. 272 00:17:37,860 --> 00:17:42,090 Before we start the real lecture, 273 00:17:42,090 --> 00:17:44,550 what we are going to do is that I 274 00:17:44,550 --> 00:17:47,120 will quote words from Feynman. 275 00:17:47,120 --> 00:17:50,280 "It doesn't matter how beautiful your theory is, 276 00:17:50,280 --> 00:17:53,480 it doesn't matter how smart you are, 277 00:17:53,480 --> 00:17:57,390 but if it doesn't agree with experiment, is wrong." 278 00:17:57,390 --> 00:17:59,880 So that's actually a very important lesson. 279 00:17:59,880 --> 00:18:04,950 And I have been telling you that electromagnetic wave is 280 00:18:04,950 --> 00:18:09,025 predicted to be oscillating in the transverse direction, 281 00:18:09,025 --> 00:18:11,640 as you can see from that little figure. 282 00:18:11,640 --> 00:18:13,500 So if the directional of propagation 283 00:18:13,500 --> 00:18:17,040 is to the right-hand side, so that means electric field 284 00:18:17,040 --> 00:18:20,280 and the magnetic field can only be oscillating 285 00:18:20,280 --> 00:18:23,890 in a plane which is perpendicular to the direction 286 00:18:23,890 --> 00:18:24,760 of propagation. 287 00:18:24,760 --> 00:18:26,420 That's actually what we have learned 288 00:18:26,420 --> 00:18:28,590 from Maxwell's equation. 289 00:18:28,590 --> 00:18:31,220 But all those things are theory, right? 290 00:18:31,220 --> 00:18:32,845 Do you believe those theory? 291 00:18:35,350 --> 00:18:37,240 I'm not sure. 292 00:18:37,240 --> 00:18:39,750 Maybe light is actually oscillating 293 00:18:39,750 --> 00:18:42,400 in the longitudinal direction, right? 294 00:18:42,400 --> 00:18:44,656 If you are a physicist, you should ask this question. 295 00:18:44,656 --> 00:18:46,030 Maybe you'll find something which 296 00:18:46,030 --> 00:18:51,940 is inconsistent with the theory, then you'll feel really happy. 297 00:18:51,940 --> 00:18:54,400 So the question we are trying to answer 298 00:18:54,400 --> 00:18:58,480 is how do we know the electromagnetic waves are 299 00:18:58,480 --> 00:19:00,110 transverse waves. 300 00:19:00,110 --> 00:19:03,080 And we are going to work through a few examples 301 00:19:03,080 --> 00:19:06,290 to convince ourselves, maybe that's the case, 302 00:19:06,290 --> 00:19:07,390 maybe we have some hint. 303 00:19:07,390 --> 00:19:12,580 Of course, we cannot prove 100% but very likely this is 304 00:19:12,580 --> 00:19:14,500 probably the case. 305 00:19:14,500 --> 00:19:20,360 So let's go ahead and start the discussion today. 306 00:19:20,360 --> 00:19:25,500 So instead of adding one more dimension 307 00:19:25,500 --> 00:19:29,590 by arranging materials, we can actually 308 00:19:29,590 --> 00:19:42,070 discuss how the pointing direction of the electric field 309 00:19:42,070 --> 00:19:44,700 depends as a function of time. 310 00:19:44,700 --> 00:19:47,710 So in order to do that, I need to remind you 311 00:19:47,710 --> 00:19:52,225 how to write down electric field for the electromagnetic wave. 312 00:19:52,225 --> 00:19:57,340 So in a previous lecture, if I organize myself 313 00:19:57,340 --> 00:20:01,030 so that the electromagnetic field is 314 00:20:01,030 --> 00:20:06,600 going in the z direction-- the electromagnetic field 315 00:20:06,600 --> 00:20:09,600 is going in the z direction. 316 00:20:09,600 --> 00:20:14,320 All right, if I choose that, then basically electric field 317 00:20:14,320 --> 00:20:18,730 as a function of z and time will be 318 00:20:18,730 --> 00:20:26,050 equal to the real part of some vector psi 0 times 319 00:20:26,050 --> 00:20:31,840 exponential i kz minus omega t. 320 00:20:31,840 --> 00:20:35,980 If I assume that this electric field, 321 00:20:35,980 --> 00:20:39,490 the electromagnetic field is propagating to where-- 322 00:20:39,490 --> 00:20:40,930 the z direction, right? 323 00:20:40,930 --> 00:20:44,320 So by now, you should get used to this already. 324 00:20:44,320 --> 00:20:47,740 This is going to the positive z direction. 325 00:20:47,740 --> 00:20:52,220 Where I intentionally write psi 0 326 00:20:52,220 --> 00:20:57,940 there is actually a vector which contains two components-- psi 1 327 00:20:57,940 --> 00:21:05,170 in the x direction and psi 2 in the y direction. 328 00:21:05,170 --> 00:21:09,940 And you can see from here that you can immediately 329 00:21:09,940 --> 00:21:13,370 recognize that this can be returned 330 00:21:13,370 --> 00:21:18,300 as a superposition of two waves. 331 00:21:18,300 --> 00:21:22,300 So one is actually electromagnetic wave 332 00:21:22,300 --> 00:21:25,660 with electric field in the x direction. 333 00:21:25,660 --> 00:21:29,320 And the other wave, the other electromagnetic wave, 334 00:21:29,320 --> 00:21:32,080 is actually oscillating with electric field in the y 335 00:21:32,080 --> 00:21:36,520 direction and with amplitude psi 2. 336 00:21:36,520 --> 00:21:42,580 So that is one you can get immediately because 337 00:21:42,580 --> 00:21:46,780 of the superposition principle. 338 00:21:46,780 --> 00:21:50,860 So it's a superposition of two electromagnetic waves. 339 00:21:50,860 --> 00:21:53,200 And of course, in this case, I take 340 00:21:53,200 --> 00:21:54,800 the real part of this vector. 341 00:21:54,800 --> 00:22:03,160 So in general, psi 1 can be a1 exponential i phi 1. 342 00:22:03,160 --> 00:22:09,256 And in general psi 2 can be a2 exponential i phi 2. 343 00:22:12,750 --> 00:22:18,550 So in this notation basically, we write everything 344 00:22:18,550 --> 00:22:21,370 in terms of vector. 345 00:22:21,370 --> 00:22:27,490 And sometimes we write those things in terms of matrix, 346 00:22:27,490 --> 00:22:32,950 and that sometimes serve a better purpose for calculation. 347 00:22:32,950 --> 00:22:37,570 So we can also rewrite this thing in matrix form. 348 00:22:37,570 --> 00:22:44,230 So I have a matrix E which have two components, E x and E y. 349 00:22:44,230 --> 00:22:52,300 And this is actually equal to a real part of some Z matrix, 350 00:22:52,300 --> 00:22:57,430 which also contain two components, times a scalar, 351 00:22:57,430 --> 00:23:02,390 which is exponent i kz minus omega t. 352 00:23:02,390 --> 00:23:06,250 So as can see, I am now just setting up 353 00:23:06,250 --> 00:23:10,400 the language we want to speak so that we communicate. 354 00:23:10,400 --> 00:23:17,230 So in this case, the z matrix is written as psi 1 and psi 2, 355 00:23:17,230 --> 00:23:18,790 which are the two components. 356 00:23:18,790 --> 00:23:21,680 One is in the x direction, the other one 357 00:23:21,680 --> 00:23:23,995 is in the y direction. 358 00:23:27,030 --> 00:23:29,610 And we are going to use this language 359 00:23:29,610 --> 00:23:32,490 and see what we learn from it. 360 00:23:38,280 --> 00:23:43,770 So you can see that we have been discussing 361 00:23:43,770 --> 00:23:46,760 electromagnetic wave, which is propagating 362 00:23:46,760 --> 00:23:48,810 toward the positive z direction. 363 00:23:48,810 --> 00:23:51,210 And I have two component which I can have-- 364 00:23:51,210 --> 00:23:56,450 x direction electric field and y direction electric field. 365 00:23:56,450 --> 00:24:02,240 So let me try to go through some example 366 00:24:02,240 --> 00:24:06,260 and see what we can actually learn from this. 367 00:24:06,260 --> 00:24:09,290 So the first example I would like to talk about 368 00:24:09,290 --> 00:24:15,550 is that if I have two waves, E1. 369 00:24:15,550 --> 00:24:20,460 The first wave is actually E0 cosine kz 370 00:24:20,460 --> 00:24:24,260 minus omega t in the x direction. 371 00:24:24,260 --> 00:24:30,545 And then the second wave is E2 equal to E0 cosine 372 00:24:30,545 --> 00:24:34,620 kz minus omega t, in the y direction. 373 00:24:34,620 --> 00:24:38,210 Be careful-- the direction of the electric field 374 00:24:38,210 --> 00:24:43,700 is different for the first and the second plane wave. 375 00:24:43,700 --> 00:24:49,260 And if you actually notice from this expression, 376 00:24:49,260 --> 00:24:53,390 these two waves are in phase. 377 00:24:53,390 --> 00:24:59,030 So in-phase means that they reach maxima at the same time. 378 00:24:59,030 --> 00:25:05,190 So in this case, if I add two waves with no phase difference, 379 00:25:05,190 --> 00:25:06,660 what is going to happen? 380 00:25:06,660 --> 00:25:15,950 So I am going to have a E vector, which is E1 plus E2. 381 00:25:15,950 --> 00:25:24,050 And if I plug the locus of this kind of electric field 382 00:25:24,050 --> 00:25:30,920 in the two-dimensional space, x and y, so now I am plotting, 383 00:25:30,920 --> 00:25:36,020 so I am fixing my z position for example at 0. 384 00:25:36,020 --> 00:25:40,700 And I would like to see how the locus of the E 385 00:25:40,700 --> 00:25:43,640 vary as a function of time. 386 00:25:43,640 --> 00:25:50,490 In this case, since E1 and E2 have the same amplitude, which 387 00:25:50,490 --> 00:25:56,280 is E0, and also they have no phase difference, 388 00:25:56,280 --> 00:25:59,570 that means they reach maxima simultaneously. 389 00:25:59,570 --> 00:26:02,990 So that means originally, for example, in the beginning 390 00:26:02,990 --> 00:26:04,700 the electric field is 0. 391 00:26:04,700 --> 00:26:09,000 And this electric field projection to the xy plane 392 00:26:09,000 --> 00:26:12,730 will increase until some maximum. 393 00:26:12,730 --> 00:26:17,960 At the maxima, this will be E0, the x and y position 394 00:26:17,960 --> 00:26:19,320 will be E0. 395 00:26:19,320 --> 00:26:26,690 Then it goes back to minus E0, minus E0, 396 00:26:26,690 --> 00:26:29,330 in the x and y direction position. 397 00:26:29,330 --> 00:26:30,970 And it goes back and forth. 398 00:26:34,840 --> 00:26:39,670 So in this case, this is actually still not 399 00:26:39,670 --> 00:26:42,250 very different from what we had discussed 400 00:26:42,250 --> 00:26:45,940 before because the electric field is still 401 00:26:45,940 --> 00:26:47,800 oscillating up and down. 402 00:26:47,800 --> 00:26:50,050 But the difference is that it's not 403 00:26:50,050 --> 00:26:53,920 oscillating at the x-axis or y-axis, 404 00:26:53,920 --> 00:26:59,260 but in the axis which is actually 45 degree with respect 405 00:26:59,260 --> 00:27:00,310 to the x or y-axis. 406 00:27:02,890 --> 00:27:06,900 And now we can do a exercise to write it down 407 00:27:06,900 --> 00:27:11,340 in the matrix notation. 408 00:27:11,340 --> 00:27:21,670 Now I have this E matrix will be equal to the real part of E0, 409 00:27:21,670 --> 00:27:22,790 E0. 410 00:27:22,790 --> 00:27:30,510 And both E0's are real, exponential i kz minus omega t. 411 00:27:30,510 --> 00:27:37,180 And basically I can conclude that z will be equal to E0, E0, 412 00:27:37,180 --> 00:27:40,110 and this will be E0, 1, 1. 413 00:27:42,670 --> 00:27:47,950 In this kind of situation, the electric field 414 00:27:47,950 --> 00:27:50,600 is varying as a function of time. 415 00:27:50,600 --> 00:27:55,810 The position to xy plane is a line. 416 00:27:55,810 --> 00:28:05,963 When I happens, we call it linearly polarized. 417 00:28:12,340 --> 00:28:13,870 So this is actually just a name. 418 00:28:13,870 --> 00:28:19,960 But indeed the locus of the electric field on the xy plane 419 00:28:19,960 --> 00:28:26,410 is a linear line, so that we call it linearly polarized. 420 00:28:26,410 --> 00:28:30,700 And of course you can say, OK, this is just one example. 421 00:28:30,700 --> 00:28:34,360 I can have many, many examples which you also create 422 00:28:34,360 --> 00:28:37,030 a line when you plot the locus. 423 00:28:37,030 --> 00:28:44,050 So for example, I can have z equal to E0, 1, 0. 424 00:28:44,050 --> 00:28:47,600 Can somebody tell me what does that notation mean? 425 00:28:47,600 --> 00:28:51,030 AUDIENCE: That just means one wave electromagnetic wave 426 00:28:51,030 --> 00:28:52,447 in one direction. 427 00:28:52,447 --> 00:28:53,530 YEN-JIE LEE: That's right. 428 00:28:53,530 --> 00:28:57,220 So that means you only have electric field 429 00:28:57,220 --> 00:28:58,660 in the x direction. 430 00:28:58,660 --> 00:29:01,210 So this is the x direction, this is y direction. 431 00:29:01,210 --> 00:29:05,320 And this is also linearly polarized. 432 00:29:05,320 --> 00:29:11,420 Of course you can have very similarly E0, 0, 1. 433 00:29:11,420 --> 00:29:14,830 And in this case, you only have electric field 434 00:29:14,830 --> 00:29:16,090 in the y direction. 435 00:29:16,090 --> 00:29:19,160 Just want to tell you my language 436 00:29:19,160 --> 00:29:22,600 I'm introducing here so that we communicate. 437 00:29:22,600 --> 00:29:28,770 And of course, you can have z equal to E0 cosine theta 438 00:29:28,770 --> 00:29:31,150 sine theta. 439 00:29:31,150 --> 00:29:32,810 What does that mean? 440 00:29:32,810 --> 00:29:38,850 This means that when I plot the locus on the xy plane, 441 00:29:38,850 --> 00:29:41,690 I'm going to have a straight line still, 442 00:29:41,690 --> 00:29:46,630 but now the angle between this line and the x-axis 443 00:29:46,630 --> 00:29:48,910 is going to be theta. 444 00:29:48,910 --> 00:29:54,110 So all those examples, you can see that in the first case 445 00:29:54,110 --> 00:29:57,100 the oscillation is in the x direction. 446 00:29:57,100 --> 00:29:59,310 And in the second case the oscillation 447 00:29:59,310 --> 00:30:00,630 is in the y direction. 448 00:30:00,630 --> 00:30:04,840 And then in the third case it can be in an arbitrary line-- 449 00:30:04,840 --> 00:30:08,470 theta angle away from the x-axis. 450 00:30:08,470 --> 00:30:12,520 So all those things are linearly polarized. 451 00:30:12,520 --> 00:30:17,020 So so far, there is nothing to surprise you. 452 00:30:17,020 --> 00:30:19,311 So that's what we have been talking about. 453 00:30:19,311 --> 00:30:20,810 And you can say that, Professor Lee, 454 00:30:20,810 --> 00:30:23,080 you were just not doing a very good job. 455 00:30:23,080 --> 00:30:28,600 You didn't rotate the axis right so that all the electric field 456 00:30:28,600 --> 00:30:30,940 is in the x direction. 457 00:30:30,940 --> 00:30:34,510 So in principle for the discussion of the linearly 458 00:30:34,510 --> 00:30:37,090 polarized wave, you can do a good job 459 00:30:37,090 --> 00:30:40,570 by rotating your axis so that you only 460 00:30:40,570 --> 00:30:45,170 have one component in the x direction. 461 00:30:45,170 --> 00:30:51,010 So I say, OK, yes, I agree, but this is useful discussion. 462 00:30:51,010 --> 00:30:55,600 All right, so now maybe you're bored. 463 00:30:55,600 --> 00:30:57,790 They have no phase difference, right? 464 00:30:57,790 --> 00:31:02,830 So how about we introduce some phase difference and see what 465 00:31:02,830 --> 00:31:05,170 will happen. 466 00:31:05,170 --> 00:31:09,450 So now if I consider the second situation-- 467 00:31:09,450 --> 00:31:17,460 I have E1 is equal to E0 cosine kz minus omega 468 00:31:17,460 --> 00:31:20,890 t in the x direction. 469 00:31:20,890 --> 00:31:30,000 Now I consider E2, this would be E0 sine kz minus omega t 470 00:31:30,000 --> 00:31:33,040 in the y direction. 471 00:31:33,040 --> 00:31:34,690 I hope you can see it. 472 00:31:34,690 --> 00:31:39,770 And then you can see that now they reach the x component 473 00:31:39,770 --> 00:31:42,820 and y component electric field. 474 00:31:42,820 --> 00:31:47,080 They reach maxima at different times because of the phase 475 00:31:47,080 --> 00:31:48,550 difference. 476 00:31:48,550 --> 00:31:50,320 How big is the phase difference? 477 00:31:50,320 --> 00:31:52,030 Can somebody actually remind me? 478 00:31:52,030 --> 00:31:53,710 AUDIENCE: Pi over 2. 479 00:31:53,710 --> 00:31:55,400 YEN-JIE LEE: Pi over 2, very good. 480 00:31:55,400 --> 00:31:57,730 All right, so this I can write it 481 00:31:57,730 --> 00:32:04,690 down as E0 cosine kz minus omega t minus pi 482 00:32:04,690 --> 00:32:08,242 over 2, as you already figured out. 483 00:32:10,870 --> 00:32:16,630 And now I can also again write it in the language I like, 484 00:32:16,630 --> 00:32:17,470 the matrix format. 485 00:32:21,670 --> 00:32:27,630 Before I achieve that, I can write it as a real part of E0 486 00:32:27,630 --> 00:32:36,220 in the x direction plus E0 exponential minus i pi over 2, 487 00:32:36,220 --> 00:32:39,390 because this minus pi over 2 sign, 488 00:32:39,390 --> 00:32:43,690 the factor here for the y direction. 489 00:32:43,690 --> 00:32:49,780 And all those things are multiplied by exponential i kz 490 00:32:49,780 --> 00:32:53,140 minus omega t. 491 00:32:53,140 --> 00:32:58,090 Both of them are actually oscillating at the same omega 492 00:32:58,090 --> 00:33:00,430 angular frequency. 493 00:33:00,430 --> 00:33:04,580 And you can see that now I can collect this phase difference 494 00:33:04,580 --> 00:33:12,300 back into a complex factor here, exponential minus i pi over 2. 495 00:33:12,300 --> 00:33:19,050 And exponential minus i pi over 2 is actually just minus i. 496 00:33:22,890 --> 00:33:26,082 So basically you can figure out that it's just minus i. 497 00:33:28,740 --> 00:33:34,560 So then I can write down my expression for the E matrix. 498 00:33:34,560 --> 00:33:40,630 This one look like real part E0, 1, 499 00:33:40,630 --> 00:33:49,770 minus i, exponential i kz minus omega t. 500 00:33:49,770 --> 00:33:56,500 And in this case, z vector will be 1 minus i 501 00:33:56,500 --> 00:34:01,530 in the language which I introduced today. 502 00:34:01,530 --> 00:34:05,220 Everybody is following? 503 00:34:05,220 --> 00:34:07,050 Am I going too fast? 504 00:34:07,050 --> 00:34:07,820 Very good. 505 00:34:07,820 --> 00:34:10,210 Thank you for the feedback. 506 00:34:10,210 --> 00:34:14,300 So now I'm going to do the same thing 507 00:34:14,300 --> 00:34:18,540 to plot the locus of the electric field 508 00:34:18,540 --> 00:34:22,550 as a function of time on the xy plane. 509 00:34:32,270 --> 00:34:36,280 So now you can see that it's pretty interesting. 510 00:34:36,280 --> 00:34:41,500 Since I have a cosine and sine, I assume that t is equal to 0, 511 00:34:41,500 --> 00:34:45,159 z is equal to 0, then I will only 512 00:34:45,159 --> 00:34:49,969 have electric field in the x direction. 513 00:34:49,969 --> 00:34:53,480 So this is actually when t is equal to 0. 514 00:34:53,480 --> 00:34:57,490 All right, and if I increase time, 515 00:34:57,490 --> 00:35:00,850 I fix the z to be equal to 0. 516 00:35:00,850 --> 00:35:07,910 If I increase the time, this locus 517 00:35:07,910 --> 00:35:10,970 is going to do an interesting thing. 518 00:35:10,970 --> 00:35:21,250 It's going to be rotating until it finish one period. 519 00:35:21,250 --> 00:35:22,520 Why is that happening? 520 00:35:22,520 --> 00:35:28,750 Because what I'm plotting is the locus of the electric field. 521 00:35:28,750 --> 00:35:33,920 And you can see that if I set z equal to 0, when 522 00:35:33,920 --> 00:35:37,190 z is equal to 0, this will give you 523 00:35:37,190 --> 00:35:42,350 E0 cosine omega t in the x direction 524 00:35:42,350 --> 00:35:49,720 and E0 sine omega t in the y direction. 525 00:35:49,720 --> 00:35:54,250 So E0 cosine omega t, E0 sine omega t-- wow, 526 00:35:54,250 --> 00:35:58,240 this reminds you about the previous discussion. 527 00:35:58,240 --> 00:36:01,390 If you have a cosine and a sine, they 528 00:36:01,390 --> 00:36:04,660 work together in different directions, that's 529 00:36:04,660 --> 00:36:06,960 going to get you a circle. 530 00:36:09,810 --> 00:36:14,520 So that's why in this locus you see a circle. 531 00:36:14,520 --> 00:36:23,640 And it's rotating in a counter-clockwise direction. 532 00:36:23,640 --> 00:36:31,110 And the speed of the rotation is related to omega t. 533 00:36:31,110 --> 00:36:37,710 And now if I increase the time, then this point 534 00:36:37,710 --> 00:36:43,080 is going to be going in a counter-clockwise direction. 535 00:36:43,080 --> 00:36:48,870 And you may be super surprised because what I have been doing 536 00:36:48,870 --> 00:36:53,700 is to add two linearly polarized waves together. 537 00:36:53,700 --> 00:36:55,320 The only thing which I always say, 538 00:36:55,320 --> 00:36:58,590 the magic I have been doing, is to introduce a phase 539 00:36:58,590 --> 00:37:00,630 difference. 540 00:37:00,630 --> 00:37:04,070 And you can see that instead of going up and down, 541 00:37:04,070 --> 00:37:06,900 up and down as a function of time, now 542 00:37:06,900 --> 00:37:09,780 it's actually doing rotation. 543 00:37:09,780 --> 00:37:13,980 So what is going to happen is that as a function of the time 544 00:37:13,980 --> 00:37:18,510 the direction of the electric field 545 00:37:18,510 --> 00:37:23,680 is going to rotate as a function of time. 546 00:37:23,680 --> 00:37:30,750 So we call this situation circularly polarized. 547 00:37:35,120 --> 00:37:41,090 By the way, because of the initial two components 548 00:37:41,090 --> 00:37:47,520 I put in, both of them have amplitude E0. 549 00:37:47,520 --> 00:37:50,170 Therefore, it's circularly polarized. 550 00:37:53,080 --> 00:37:56,440 I can also try to do something different. 551 00:37:56,440 --> 00:38:02,400 For example, I can change the x direction. 552 00:38:02,400 --> 00:38:04,800 So the third situation is that I can 553 00:38:04,800 --> 00:38:13,680 change the first x direction amplitude from E0 to E0 over 2. 554 00:38:13,680 --> 00:38:16,560 Then what I'm going to get is something like this. 555 00:38:23,370 --> 00:38:27,960 The only thing which I change with respect to 2 556 00:38:27,960 --> 00:38:33,690 is that I changed the amplitude of the electric field 557 00:38:33,690 --> 00:38:38,700 in the x direction by a factor of 2. 558 00:38:41,490 --> 00:38:43,680 Can somebody actually tell me what 559 00:38:43,680 --> 00:38:46,360 is going to happen in this situation? 560 00:38:46,360 --> 00:38:49,000 What will happen to the locus? 561 00:38:49,000 --> 00:38:50,013 AUDIENCE: Gets squished. 562 00:38:50,013 --> 00:38:52,430 YEN-JIE LEE: Yeah, it gets squished in, 563 00:38:52,430 --> 00:38:54,460 so in the x direction. 564 00:38:54,460 --> 00:38:56,460 Yeah, that's right, very good. 565 00:38:56,460 --> 00:39:02,790 So what you are going to get is that instead of a circle 566 00:39:02,790 --> 00:39:04,920 you get something like this. 567 00:39:04,920 --> 00:39:08,480 So this is the x direction and this is the y direction. 568 00:39:08,480 --> 00:39:15,510 And this will give you a maxima x equal to E0 over 2. 569 00:39:15,510 --> 00:39:21,300 And of course the original amplitude in the y direction 570 00:39:21,300 --> 00:39:25,200 didn't change, and that gives you a maximum value of E0. 571 00:39:25,200 --> 00:39:31,370 And this becomes an elliptically polarized wave. 572 00:39:33,990 --> 00:39:42,795 So this kind of situation is called elliptically polarized. 573 00:39:47,450 --> 00:39:52,100 How do we actually visualize this situation? 574 00:39:52,100 --> 00:39:56,240 So that means, as in the beginning, 575 00:39:56,240 --> 00:40:01,170 when the electric field is aligned with the x-axis, 576 00:40:01,170 --> 00:40:03,670 it's pointing in the x-axis. 577 00:40:03,670 --> 00:40:07,490 It has a shorter length. 578 00:40:07,490 --> 00:40:12,620 And when it rotate, rotate as a function of time to the y-axis, 579 00:40:12,620 --> 00:40:15,260 the amplitude becomes bigger. 580 00:40:15,260 --> 00:40:19,670 And it gets smaller again, and increase again, 581 00:40:19,670 --> 00:40:20,830 it becomes bigger. 582 00:40:20,830 --> 00:40:24,080 So you can see the amplitude is actually 583 00:40:24,080 --> 00:40:27,620 changing as a function of time. 584 00:40:27,620 --> 00:40:31,220 When you have this kind of situation, 585 00:40:31,220 --> 00:40:35,210 they have different amplitudes, although they have a fixed 586 00:40:35,210 --> 00:40:39,850 phase difference, so pi over 2. 587 00:40:39,850 --> 00:40:40,920 Any questions so far? 588 00:40:44,890 --> 00:40:51,560 OK, so if you noticed that there's actually 589 00:40:51,560 --> 00:40:57,910 another way to produce elliptically polarized wave. 590 00:40:57,910 --> 00:41:04,630 What we could do is that instead of changing the phase 591 00:41:04,630 --> 00:41:13,380 by a factor of pi over 2, we can change the phase difference. 592 00:41:13,380 --> 00:41:15,770 The phase difference can be delta 593 00:41:15,770 --> 00:41:18,910 phi, which is the phase difference, 594 00:41:18,910 --> 00:41:22,750 can be a different value, arbitrary value, not equal 595 00:41:22,750 --> 00:41:27,700 to pi over 2 or 3 pi over 2. 596 00:41:27,700 --> 00:41:40,520 If the delta phi, which is the phase difference between x 597 00:41:40,520 --> 00:41:44,705 and the y direction, EM waves-- 598 00:41:49,720 --> 00:41:50,590 electric field. 599 00:41:54,910 --> 00:42:02,420 If the phase difference is not pi over 2 or not 2 pi over 2, 600 00:42:02,420 --> 00:42:07,070 then you can also create an elliptically polarized wave. 601 00:42:07,070 --> 00:42:14,160 So that starts from the original figure, 602 00:42:14,160 --> 00:42:16,640 which we actually discussed-- 603 00:42:16,640 --> 00:42:18,860 situation number 1. 604 00:42:18,860 --> 00:42:23,100 Situation number 1, I have delta phi, 605 00:42:23,100 --> 00:42:26,750 the phase difference equal to 0. 606 00:42:26,750 --> 00:42:32,450 When that happened, you have a linearly polarized wave. 607 00:42:35,690 --> 00:42:37,580 Basically what you are going to get 608 00:42:37,580 --> 00:42:45,440 is a line in this two-dimensional space. 609 00:42:45,440 --> 00:42:50,100 If I increase, so now I slightly delayed. 610 00:42:50,100 --> 00:42:57,440 If I slightly delay the electric field in the y direction. 611 00:42:57,440 --> 00:43:00,750 So if I slightly delay electric field it in the y direction 612 00:43:00,750 --> 00:43:05,180 so that delta phi now is greater than 1, what is going to happen 613 00:43:05,180 --> 00:43:11,560 is that it's going to look like this. 614 00:43:11,560 --> 00:43:16,340 This means that they will not reach maxima really 615 00:43:16,340 --> 00:43:17,860 simultaneously. 616 00:43:17,860 --> 00:43:20,540 There will be because of the phase difference. 617 00:43:20,540 --> 00:43:23,800 And you'll see that originally when 618 00:43:23,800 --> 00:43:25,910 there were no phase difference, you 619 00:43:25,910 --> 00:43:30,320 would be oscillating back and forth in this blue line. 620 00:43:30,320 --> 00:43:33,720 When you increase the delta phi slightly, 621 00:43:33,720 --> 00:43:38,380 then you are going to get also a elliptical shape, 622 00:43:38,380 --> 00:43:43,560 but now is tilted with some degree, which 623 00:43:43,560 --> 00:43:48,950 is 45 degree in this case. 624 00:43:48,950 --> 00:43:52,630 So therefore you can see that now I can also 625 00:43:52,630 --> 00:43:57,980 create this is also elliptically polarized wave, 626 00:43:57,980 --> 00:44:01,100 I can also create elliptically polarized wave 627 00:44:01,100 --> 00:44:05,930 by adding two components, which they have some slight phase 628 00:44:05,930 --> 00:44:06,950 difference. 629 00:44:10,600 --> 00:44:16,690 So that is actually how I can create a something which 630 00:44:16,690 --> 00:44:22,620 is called circularly polarized or elliptically polarized. 631 00:44:22,620 --> 00:44:25,940 Originally before you come into this class, 632 00:44:25,940 --> 00:44:28,640 it may look really completely bizarre 633 00:44:28,640 --> 00:44:32,225 that, oh, I can have electric field going up 634 00:44:32,225 --> 00:44:33,890 and down as a function of time. 635 00:44:33,890 --> 00:44:36,990 But how could I rotate this thing, right? 636 00:44:36,990 --> 00:44:38,180 Looks really strange. 637 00:44:38,180 --> 00:44:41,300 How can I see this from the Maxwell's equation? 638 00:44:41,300 --> 00:44:43,390 But now you get the idea. 639 00:44:43,390 --> 00:44:48,639 Basically, that's because I can now overlap two components. 640 00:44:48,639 --> 00:44:54,890 Both components individually are linearly polarized. 641 00:44:54,890 --> 00:44:57,140 But if I introduce a phase difference, 642 00:44:57,140 --> 00:45:00,190 then the superposition of these two components 643 00:45:00,190 --> 00:45:02,240 become something which is actually 644 00:45:02,240 --> 00:45:03,870 rotating as a function of time. 645 00:45:03,870 --> 00:45:06,380 And that's pretty interesting. 646 00:45:06,380 --> 00:45:10,200 So let's visualize what we have learned so far. 647 00:45:10,200 --> 00:45:12,710 The first one is linearly polarized. 648 00:45:12,710 --> 00:45:14,270 Actually, it doesn't surprise you-- 649 00:45:14,270 --> 00:45:16,550 that's the example which we have been 650 00:45:16,550 --> 00:45:20,300 using in the previous lectures. 651 00:45:20,300 --> 00:45:24,740 And this is the situation of circularly polarized wave. 652 00:45:24,740 --> 00:45:33,740 So let's focus on the figure at z equal to 0. 653 00:45:33,740 --> 00:45:38,900 So you can see that in this case the direction 654 00:45:38,900 --> 00:45:40,550 of the electric field is actually 655 00:45:40,550 --> 00:45:43,530 rotating as a function of time. 656 00:45:43,530 --> 00:45:47,600 And of course as we discussed, I can actually 657 00:45:47,600 --> 00:45:50,990 add two electric fields with different amplitudes 658 00:45:50,990 --> 00:45:55,220 or introduce slightly different phase, 659 00:45:55,220 --> 00:45:59,360 then you will see that not only that the direction is changing, 660 00:45:59,360 --> 00:46:02,500 but also the amplitude is changing. 661 00:46:02,500 --> 00:46:09,550 And in this case we call it elliptically polarized wave. 662 00:46:09,550 --> 00:46:13,030 Any questions so far? 663 00:46:13,030 --> 00:46:13,716 Yes? 664 00:46:13,716 --> 00:46:18,380 AUDIENCE: So the magnitude always is constant with time 665 00:46:18,380 --> 00:46:19,314 now? 666 00:46:19,314 --> 00:46:24,640 YEN-JIE LEE: Yes, so the magnitude, 667 00:46:24,640 --> 00:46:28,131 or say the intensity, is proportional to E squared, 668 00:46:28,131 --> 00:46:28,630 right? 669 00:46:28,630 --> 00:46:30,085 So it's actually a constant. 670 00:46:34,540 --> 00:46:40,030 So now let me add more excitement here. 671 00:46:40,030 --> 00:46:46,930 So now suppose I have a perfect conductor, all right. 672 00:46:46,930 --> 00:46:49,060 So what I am going to do is that I'm 673 00:46:49,060 --> 00:46:53,710 going to introduce some more excitement by shooting 674 00:46:53,710 --> 00:46:58,780 this linearly polarized wave through some material. 675 00:46:58,780 --> 00:47:04,930 If I have a perfect conductor where all those strips, 676 00:47:04,930 --> 00:47:09,110 I have only strips of perfect conductor instead of a plate-- 677 00:47:09,110 --> 00:47:11,830 we were talking about plate before. 678 00:47:11,830 --> 00:47:15,250 And the lesson we learned is that the plate 679 00:47:15,250 --> 00:47:18,430 is going to refract the electromagnetic wave 680 00:47:18,430 --> 00:47:22,240 because that electrons on the plate is so busy, 681 00:47:22,240 --> 00:47:26,080 it's trying to make sure that the electric field 682 00:47:26,080 --> 00:47:29,530 in the surface of the plate is equal to 0 683 00:47:29,530 --> 00:47:32,240 because that all those electrons are really moving around. 684 00:47:32,240 --> 00:47:34,900 So that's making sure the electric field is canceled. 685 00:47:34,900 --> 00:47:37,700 And therefore, it's going to refract 686 00:47:37,700 --> 00:47:40,060 the electromagnetic wave. 687 00:47:40,060 --> 00:47:45,670 So how about I restrict the direction 688 00:47:45,670 --> 00:47:50,350 of the movement of the electron so that it can only move 689 00:47:50,350 --> 00:47:55,870 in the horizontal direction? 690 00:47:55,870 --> 00:48:01,040 What is going to happen is that in this case, 691 00:48:01,040 --> 00:48:06,730 the electric field is actually oscillating up and down. 692 00:48:06,730 --> 00:48:09,550 And the electrons see this field and they will say, 693 00:48:09,550 --> 00:48:11,730 no, no, no, this is not why we work. 694 00:48:11,730 --> 00:48:14,710 This is not we are going to vote for. 695 00:48:14,710 --> 00:48:18,720 And I'm going to rearrange ourselves to compensate that, 696 00:48:18,720 --> 00:48:20,670 all right. 697 00:48:20,670 --> 00:48:22,860 And they were asking, can I move. 698 00:48:22,860 --> 00:48:26,270 Oh yes, I can move in the horizontal direction. 699 00:48:26,270 --> 00:48:28,660 So they jump up and down, then they can actually 700 00:48:28,660 --> 00:48:33,100 cancel this electric field. 701 00:48:33,100 --> 00:48:36,010 Therefore, what is going to happen 702 00:48:36,010 --> 00:48:38,170 when we have this perfect conductor is 703 00:48:38,170 --> 00:48:42,370 that the electromagnetic field is going to be bounced back 704 00:48:42,370 --> 00:48:46,900 like what we had before when we talked about a metal plate, 705 00:48:46,900 --> 00:48:47,680 all right. 706 00:48:47,680 --> 00:48:52,210 So that's actually pretty nice. 707 00:48:52,210 --> 00:48:56,870 Now suppose I have another perfect conductor. 708 00:48:59,520 --> 00:49:04,460 If I have another perfect conductor, 709 00:49:04,460 --> 00:49:08,680 this time the perfect conductor is arranged such 710 00:49:08,680 --> 00:49:14,560 that all those charges can only move inside or outside 711 00:49:14,560 --> 00:49:19,310 of the board instead of going up and down. 712 00:49:19,310 --> 00:49:25,210 Now I have the incident wave, which is actually polarized 713 00:49:25,210 --> 00:49:29,140 in the horizontal direction. 714 00:49:29,140 --> 00:49:31,210 So you can see that this time the electrons 715 00:49:31,210 --> 00:49:34,010 are really nervous about this. 716 00:49:34,010 --> 00:49:35,810 And oh no, I have to do something, 717 00:49:35,810 --> 00:49:39,070 but they cannot move up and down. 718 00:49:39,070 --> 00:49:40,540 Therefore, what is going to happen 719 00:49:40,540 --> 00:49:45,430 is that there will be no cancellation of electric field 720 00:49:45,430 --> 00:49:47,200 at this boundary. 721 00:49:47,200 --> 00:49:51,400 And this polarized wave can pass through 722 00:49:51,400 --> 00:49:57,300 this so-called polarizer or perfect conductor 723 00:49:57,300 --> 00:50:03,520 without getting stopped or get refracted. 724 00:50:03,520 --> 00:50:09,130 And as you can see that in these two examples, 725 00:50:09,130 --> 00:50:13,250 I have so called the easy axis. 726 00:50:13,250 --> 00:50:17,680 So you can see that the easy axis, as the name 727 00:50:17,680 --> 00:50:21,820 should tell you, so easy axis is the axis which is easy. 728 00:50:25,360 --> 00:50:26,890 All right, so what does that mean? 729 00:50:26,890 --> 00:50:31,540 That means if I have electric field aligned 730 00:50:31,540 --> 00:50:37,230 with the easy axis, it will pass through. 731 00:50:37,230 --> 00:50:42,480 And the easy axis is perpendicular to the direction 732 00:50:42,480 --> 00:50:48,540 of all those streets I have in the perfect conductor. 733 00:50:48,540 --> 00:50:51,370 Let's look at the first example. 734 00:50:51,370 --> 00:50:54,990 It tells you that the electric field is not 735 00:50:54,990 --> 00:50:56,430 aligned with the easy axis. 736 00:50:56,430 --> 00:51:00,590 That means it's not easy, so it got 737 00:51:00,590 --> 00:51:03,270 refracted-- oh, life is hard. 738 00:51:03,270 --> 00:51:08,110 And the second electric field is smarter. 739 00:51:08,110 --> 00:51:10,770 It got aligned with the easy axis. 740 00:51:10,770 --> 00:51:13,230 So life is easy, it passed through. 741 00:51:13,230 --> 00:51:18,310 That's how I remember this so-called easy axis. 742 00:51:18,310 --> 00:51:19,362 Any questions? 743 00:51:24,820 --> 00:51:29,770 So we have been talking about all those crazy names 744 00:51:29,770 --> 00:51:33,310 of polarized light. 745 00:51:33,310 --> 00:51:39,610 And I hope that you also have learned about unpolarized light 746 00:51:39,610 --> 00:51:40,510 before. 747 00:51:40,510 --> 00:51:43,120 Most of the lights are unpolarized. 748 00:51:43,120 --> 00:51:46,735 For example, light emitted from those light bulbs 749 00:51:46,735 --> 00:51:51,670 are not polarized because every emission of a photon 750 00:51:51,670 --> 00:51:55,330 can be aligned in different direction. 751 00:51:55,330 --> 00:51:57,760 Therefore, as I was talking about, 752 00:51:57,760 --> 00:52:00,150 there are an infinite number of photons hitting my face 753 00:52:00,150 --> 00:52:01,950 and bouncing back to your eye. 754 00:52:01,950 --> 00:52:05,930 All those things are not aligned to the same direction 755 00:52:05,930 --> 00:52:09,010 as what we have been discussing before. 756 00:52:09,010 --> 00:52:13,610 In the case of linearly polarized wave, 757 00:52:13,610 --> 00:52:16,930 all the electric fields are pointing to the same direction. 758 00:52:16,930 --> 00:52:22,540 And apparently those lights hitting my face is not aligned 759 00:52:22,540 --> 00:52:25,840 and it's actually pointing to random directions. 760 00:52:25,840 --> 00:52:32,530 So that brings me to the fifth situation, which 761 00:52:32,530 --> 00:52:34,630 I would like to talk about. 762 00:52:34,630 --> 00:52:47,800 The fifth situation is that I could have unpolarized light. 763 00:52:47,800 --> 00:52:50,890 So what do I mean by unpolarized light? 764 00:52:50,890 --> 00:52:54,340 That means electromagnetic waves which 765 00:52:54,340 --> 00:52:59,780 were produced independently by a large number 766 00:52:59,780 --> 00:53:03,850 of uncorrelated source. 767 00:53:03,850 --> 00:53:05,750 You may ask, OK, wait, wait, wait-- 768 00:53:05,750 --> 00:53:09,500 unpolarized means that we have a lot of different light source 769 00:53:09,500 --> 00:53:11,060 pointing to different angles. 770 00:53:11,060 --> 00:53:13,700 Shouldn't it just look like this? 771 00:53:13,700 --> 00:53:16,880 So you have pointing this direction, all the directions, 772 00:53:16,880 --> 00:53:20,150 all the possible directions, pointing 773 00:53:20,150 --> 00:53:24,130 to all different possible directions, like this. 774 00:53:24,130 --> 00:53:29,530 Shouldn't that diagram actually tell you 775 00:53:29,530 --> 00:53:33,690 something about unpolarized light? 776 00:53:33,690 --> 00:53:35,310 Life is hard. 777 00:53:35,310 --> 00:53:38,670 If everything is aligned and also 778 00:53:38,670 --> 00:53:44,350 arriving at the same time like this, what is going to happen? 779 00:53:44,350 --> 00:53:46,660 They cancel, right? 780 00:53:46,660 --> 00:53:48,660 Because all those things are vectors. 781 00:53:48,660 --> 00:53:51,890 If they reach maxima at the same time, 782 00:53:51,890 --> 00:53:53,825 this means that they are emitting always 783 00:53:53,825 --> 00:53:56,950 at the same time, but in random direction. 784 00:53:56,950 --> 00:54:01,690 And emitting at the same wavelength, 785 00:54:01,690 --> 00:54:04,030 then they are going to cancel each other. 786 00:54:04,030 --> 00:54:08,560 So apparently that's not what I mean by unpolarized light. 787 00:54:08,560 --> 00:54:12,670 What I mean is that they can be pointing 788 00:54:12,670 --> 00:54:15,750 to different directions, but that emission 789 00:54:15,750 --> 00:54:20,860 time and the wavelength of all those emitted electromagnetic 790 00:54:20,860 --> 00:54:24,070 waves can be slightly different, otherwise they 791 00:54:24,070 --> 00:54:29,330 will just cancel each other completely. 792 00:54:29,330 --> 00:54:31,440 Any questions about what I just said? 793 00:54:34,800 --> 00:54:36,240 So very good. 794 00:54:36,240 --> 00:54:42,360 So now what I am going to do is to show you some examples 795 00:54:42,360 --> 00:54:44,880 related to polarizer. 796 00:54:44,880 --> 00:54:47,790 Before that, I will take a five minute break before we 797 00:54:47,790 --> 00:54:50,220 come back to the demonstration. 798 00:54:50,220 --> 00:54:52,090 So we will come back at 36. 799 00:54:58,670 --> 00:55:00,920 OK, so welcome back. 800 00:55:00,920 --> 00:55:06,020 So we would continue the discussion of the polarization 801 00:55:06,020 --> 00:55:10,080 and the polarized wave and the polarizers. 802 00:55:10,080 --> 00:55:16,070 So before that, I already introduced the polarizer 803 00:55:16,070 --> 00:55:19,850 by the perfect conductor arranged in the x direction 804 00:55:19,850 --> 00:55:22,290 or y direction, many of them are strips, 805 00:55:22,290 --> 00:55:25,500 aligned in the x direction and the y direction. 806 00:55:25,500 --> 00:55:28,430 And usually we use a simple diagram 807 00:55:28,430 --> 00:55:32,540 like this, a circle and an arrow, 808 00:55:32,540 --> 00:55:37,490 to indicate the easy axis to tell you 809 00:55:37,490 --> 00:55:40,430 about the property of the polarizer. 810 00:55:40,430 --> 00:55:43,520 So in this case, if I have a circle 811 00:55:43,520 --> 00:55:49,700 and I have an arrow, which is pointing toward the x-axis, 812 00:55:49,700 --> 00:55:52,490 in this case my coordinate system is here, 813 00:55:52,490 --> 00:55:55,280 horizontal is x, vertical is y, that 814 00:55:55,280 --> 00:56:01,590 means my easy axis is in the x direction. 815 00:56:01,590 --> 00:56:04,220 And I can actually write down what 816 00:56:04,220 --> 00:56:09,140 would be the matrix presentation of p0, which would present 817 00:56:09,140 --> 00:56:11,450 the effect of this polarizer. 818 00:56:11,450 --> 00:56:18,740 The p0 will be a matrix in this form, 1, 0, 0, 0. 819 00:56:18,740 --> 00:56:25,280 If I have this P0 acting on the z, 820 00:56:25,280 --> 00:56:29,120 for example, I can have the p0 acting on the z, then 821 00:56:29,120 --> 00:56:34,220 basically what would happen after the light passed 822 00:56:34,220 --> 00:56:41,960 through this polarizer will become z multiplied by p0. 823 00:56:41,960 --> 00:56:47,900 So basically all this p0 matrix does 824 00:56:47,900 --> 00:56:56,530 is to extract the position of the field in the x-axis. 825 00:56:56,530 --> 00:57:01,130 But I'm just writing it down in the matrix format. 826 00:57:01,130 --> 00:57:03,950 And on the other hand, for example, I 827 00:57:03,950 --> 00:57:09,850 can have this easy axis, which is 90 degrees, which 828 00:57:09,850 --> 00:57:12,920 affects the x-axis, which is in the y direction, 829 00:57:12,920 --> 00:57:17,960 then I write this down in the form of p pi over 2. 830 00:57:17,960 --> 00:57:23,330 This pi over 2 is the angle between the easy axis 831 00:57:23,330 --> 00:57:25,940 and the x direction. 832 00:57:25,940 --> 00:57:29,250 And in this case, you are going to get 0, 0, 0, 833 00:57:29,250 --> 00:57:32,570 1 because all this matrix set notation 834 00:57:32,570 --> 00:57:36,440 does is to extract the component which is actually 835 00:57:36,440 --> 00:57:38,840 in the y direction. 836 00:57:38,840 --> 00:57:41,630 And you can of course multiply-- 837 00:57:41,630 --> 00:57:45,440 when you have an incident electromagnetic wave 838 00:57:45,440 --> 00:57:50,450 which you can extract the polarization z matrix, you just 839 00:57:50,450 --> 00:57:54,770 multiply p pi over 2 and z. 840 00:57:54,770 --> 00:57:59,270 Basically you can actually get the resulting polarization 841 00:57:59,270 --> 00:58:03,080 of the electromagnetic field after passing 842 00:58:03,080 --> 00:58:06,530 through this polarizer. 843 00:58:06,530 --> 00:58:11,000 What is going to happen is that only the y component 844 00:58:11,000 --> 00:58:12,110 will survive. 845 00:58:12,110 --> 00:58:15,820 In the case of easy axis aligned in the x direction, 846 00:58:15,820 --> 00:58:20,330 only the x component will survive. 847 00:58:20,330 --> 00:58:25,170 So now let me go through a short example here. 848 00:58:25,170 --> 00:58:31,010 So suppose initially I have an electromagnetic field 849 00:58:31,010 --> 00:58:36,110 polarized, it's linearly polarized in the y direction. 850 00:58:36,110 --> 00:58:38,630 And I have that electromagnetic field 851 00:58:38,630 --> 00:58:43,130 passing through a polarizer, which is actually theta 852 00:58:43,130 --> 00:58:47,000 degree away from the y-axis. 853 00:58:47,000 --> 00:58:56,240 So what is going to happen is that all the components which 854 00:58:56,240 --> 00:58:59,550 is parallel to the axis survive. 855 00:59:04,890 --> 00:59:09,940 All the projected components in the axis perpendicular 856 00:59:09,940 --> 00:59:14,800 to the easy axis didn't make it because the electrons are going 857 00:59:14,800 --> 00:59:17,860 to be oscillating up and down like crazy 858 00:59:17,860 --> 00:59:21,670 to compensate and then refract that component. 859 00:59:21,670 --> 00:59:24,880 So what is going to happen is that after passing 860 00:59:24,880 --> 00:59:30,190 through the easy axis, the direction of the polarization 861 00:59:30,190 --> 00:59:33,040 will be altered such that it's actually 862 00:59:33,040 --> 00:59:38,560 in line with the easy axis of the polarizer. 863 00:59:38,560 --> 00:59:40,500 So you can see that, after passing 864 00:59:40,500 --> 00:59:45,310 through this polarizer, the direction of the polarization, 865 00:59:45,310 --> 00:59:48,940 this is still linearly polarized, 866 00:59:48,940 --> 00:59:52,690 the direction is changed by theta degree. 867 00:59:52,690 --> 00:59:55,810 And also the amplitude also changed 868 00:59:55,810 --> 01:00:00,880 because only the component parallel to the easy axis 869 01:00:00,880 --> 01:00:01,770 survived. 870 01:00:01,770 --> 01:00:07,880 Therefore, the magnitude of the E vector 871 01:00:07,880 --> 01:00:13,090 becomes E0 cosine of cosine theta. 872 01:00:13,090 --> 01:00:17,550 And therefore, the amplitude of the light got reduced. 873 01:00:17,550 --> 01:00:21,190 It becomes I0 cosine theta squared 874 01:00:21,190 --> 01:00:24,460 because the intensity of the light 875 01:00:24,460 --> 01:00:28,660 is proportional to E squared. 876 01:00:28,660 --> 01:00:33,070 So after all this, this is time to take a look 877 01:00:33,070 --> 01:00:36,280 at some demonstration here. 878 01:00:36,280 --> 01:00:40,090 So here I have a setup. 879 01:00:40,090 --> 01:00:50,650 I have emission of linearly polarized microwave 880 01:00:50,650 --> 01:00:52,990 from the left-hand side, which is the source. 881 01:00:52,990 --> 01:00:56,800 And I have a receiver which is connected to this scope. 882 01:00:56,800 --> 01:01:00,980 You can see the result on the scope. 883 01:01:00,980 --> 01:01:04,540 You can see that indeed they are energy 884 01:01:04,540 --> 01:01:07,600 passing from this source. 885 01:01:07,600 --> 01:01:13,030 And it got accepted and then recorded by the scope. 886 01:01:13,030 --> 01:01:15,240 Now I have all of those-- 887 01:01:15,240 --> 01:01:17,160 OK, all are not so perfect, but those 888 01:01:17,160 --> 01:01:23,320 are metals which have many, many strips 889 01:01:23,320 --> 01:01:26,480 or many, many little rods here. 890 01:01:26,480 --> 01:01:32,210 And my initial linearly polarized wave 891 01:01:32,210 --> 01:01:40,270 is actually pointing in the vertical direction. 892 01:01:40,270 --> 01:01:48,500 So if I have my polarizer arranged in this direction, 893 01:01:48,500 --> 01:01:52,820 can you predict what will happen to the readout on the scope? 894 01:01:52,820 --> 01:01:55,070 Will I see signal or not? 895 01:01:55,070 --> 01:01:58,700 How many of you think we will see signal 896 01:01:58,700 --> 01:02:03,050 if I arrange that the electromagnetic wave is 897 01:02:03,050 --> 01:02:05,480 polarized in the up and down direction? 898 01:02:05,480 --> 01:02:09,970 OK, one, two, three-- 899 01:02:09,970 --> 01:02:12,180 only four people think so. 900 01:02:12,180 --> 01:02:20,660 How many of you think nothing will be observed by the scope? 901 01:02:20,660 --> 01:02:22,350 Most of you actually think so. 902 01:02:22,350 --> 01:02:26,080 So let's really do this experiment. 903 01:02:26,080 --> 01:02:27,600 Look at what is happening. 904 01:02:27,600 --> 01:02:28,890 Can you see it? 905 01:02:28,890 --> 01:02:31,030 Do I see a signal? 906 01:02:31,030 --> 01:02:32,070 No. 907 01:02:32,070 --> 01:02:34,700 All the signals are canceled, right? 908 01:02:34,700 --> 01:02:35,240 Why? 909 01:02:35,240 --> 01:02:39,810 Because all the hard work of all those electrons in the metal, 910 01:02:39,810 --> 01:02:42,050 they are like crazy, oh my god, this is a disaster. 911 01:02:42,050 --> 01:02:45,020 I am going to oscillate up and down-- cancel, cancel, cancel. 912 01:02:45,020 --> 01:02:47,240 Then it got cancelled. 913 01:02:47,240 --> 01:02:50,720 On the other hand, I can save all those electrons. 914 01:02:50,720 --> 01:02:55,070 So now I'm going to rotate this thing by 90 degrees 915 01:02:55,070 --> 01:02:56,420 and see what is going to happen. 916 01:03:00,140 --> 01:03:01,590 Can you see it? 917 01:03:01,590 --> 01:03:02,730 Nothing happened, right? 918 01:03:02,730 --> 01:03:05,460 Because those electrons-- oh my god, crisis coming, 919 01:03:05,460 --> 01:03:08,500 but now there's nothing I can do because I cannot move up 920 01:03:08,500 --> 01:03:09,370 and down. 921 01:03:09,370 --> 01:03:13,050 Therefore, there will be no refraction. 922 01:03:13,050 --> 01:03:15,720 So they have just to accept the fact 923 01:03:15,720 --> 01:03:21,300 that this electromagnetic field went through. 924 01:03:21,300 --> 01:03:23,370 All right, and that actually does 925 01:03:23,370 --> 01:03:26,100 something really interesting. 926 01:03:26,100 --> 01:03:32,810 Look at this-- so now I can actually 927 01:03:32,810 --> 01:03:36,890 make electromagnetic field completely destroyed, 928 01:03:36,890 --> 01:03:40,490 because originally the electromagnetic field is 929 01:03:40,490 --> 01:03:44,780 linearly polarized, up and down in this direction. 930 01:03:56,790 --> 01:04:00,070 So if I have a plate which is 45 degrees 931 01:04:00,070 --> 01:04:02,580 and I put in another one here-- 932 01:04:06,190 --> 01:04:09,520 it doesn't really work very well. 933 01:04:09,520 --> 01:04:12,950 The signal probably is too small. 934 01:04:12,950 --> 01:04:17,180 How about we do this demo in a different way. 935 01:04:17,180 --> 01:04:22,670 So right now here I have my computer here. 936 01:04:22,670 --> 01:04:24,060 Can you see my computer? 937 01:04:24,060 --> 01:04:25,700 You can see the screen, right? 938 01:04:25,700 --> 01:04:30,650 The screen essentially is made of LED. 939 01:04:30,650 --> 01:04:34,910 Those LED screens emit polarized light. 940 01:04:34,910 --> 01:04:36,590 And those are the polarizer, which 941 01:04:36,590 --> 01:04:39,290 is the equivalent version of those metals 942 01:04:39,290 --> 01:04:42,890 but arranged in a really fine grain-- 943 01:04:45,440 --> 01:04:47,880 so you cannot really see all those strips. 944 01:04:47,880 --> 01:04:50,810 And they look pretty transparent. 945 01:04:50,810 --> 01:04:53,100 But the idea is pretty similar. 946 01:04:53,100 --> 01:05:00,080 And the polarizing axis is in the horizontal direction. 947 01:05:00,080 --> 01:05:02,870 The easy axis is in horizontal direction. 948 01:05:02,870 --> 01:05:05,000 So let me put this here. 949 01:05:05,000 --> 01:05:07,640 Can you still see the screen? 950 01:05:07,640 --> 01:05:09,130 You cannot, right. 951 01:05:09,130 --> 01:05:14,900 But if I rotate this by 90 degrees, you can see it. 952 01:05:14,900 --> 01:05:20,330 That is because all those light emitted 953 01:05:20,330 --> 01:05:24,170 from the screen it linearly polarized 954 01:05:24,170 --> 01:05:26,840 up and down direction. 955 01:05:26,840 --> 01:05:36,260 So now if I have another identical polarizer which I 956 01:05:36,260 --> 01:05:39,740 insert between these two, so now you cannot see anything 957 01:05:39,740 --> 01:05:44,920 because all the light is pointing up and down, 958 01:05:44,920 --> 01:05:47,570 or the electric field is pointing up and down. 959 01:05:47,570 --> 01:05:54,870 And now if I insert another one, which is 45 degrees, 960 01:05:54,870 --> 01:05:56,730 can you see something? 961 01:05:56,730 --> 01:05:58,470 You can see it-- why? 962 01:05:58,470 --> 01:06:07,650 That is because if I insert this additional polarizer, like what 963 01:06:07,650 --> 01:06:12,510 I was discussing, it's going to take 964 01:06:12,510 --> 01:06:19,890 the position to the direction of the easy axis. 965 01:06:19,890 --> 01:06:25,170 And if I have another one, which is the easy axis pointing 966 01:06:25,170 --> 01:06:29,640 toward the x direction, then again since this vector is 967 01:06:29,640 --> 01:06:32,350 tilted already by some theta degree, 968 01:06:32,350 --> 01:06:36,510 then you can see some residual component 969 01:06:36,510 --> 01:06:42,030 which can pass through the second polarizer. 970 01:06:42,030 --> 01:06:46,290 On the other hand, if I remove this polarizer in between, 971 01:06:46,290 --> 01:06:48,840 then what is going to happen is that all the components 972 01:06:48,840 --> 01:06:51,790 is pointing to the y direction. 973 01:06:51,790 --> 01:06:56,230 Then you cannot see any component which pass through 974 01:06:56,230 --> 01:06:57,247 this polarizer. 975 01:06:57,247 --> 01:06:58,830 So that's actually pretty interesting. 976 01:06:58,830 --> 01:07:01,269 And I can rotate this, and you can see 977 01:07:01,269 --> 01:07:02,560 that the magnitude is changing. 978 01:07:06,930 --> 01:07:12,120 Only when I have 45 degrees, I see a maximum intensity. 979 01:07:14,890 --> 01:07:16,112 Any questions? 980 01:07:21,370 --> 01:07:22,030 Very good. 981 01:07:25,640 --> 01:07:31,405 So now what I'm going to do is to discuss with you 982 01:07:31,405 --> 01:07:35,640 an interesting question, which was posted by Einstein. 983 01:07:38,190 --> 01:07:46,790 Einstein said that, as I said, "God doesn't play dice 984 01:07:46,790 --> 01:07:48,340 with the world." 985 01:07:48,340 --> 01:07:53,200 So that's actually what he believed. 986 01:07:53,200 --> 01:07:58,120 So we can do an interesting experiment, which 987 01:07:58,120 --> 01:08:01,000 I have single photon source. 988 01:08:01,000 --> 01:08:03,160 So what do I mean by single photon source? 989 01:08:03,160 --> 01:08:05,820 I can emit one photon at a time-- 990 01:08:05,820 --> 01:08:09,490 just one, and then the second one, and the third one-- 991 01:08:09,490 --> 01:08:15,340 and have then pass through some imaginary polarizer. 992 01:08:15,340 --> 01:08:17,979 So let's suppose I have unpolarized 993 01:08:17,979 --> 01:08:21,880 light with intensity at 0. 994 01:08:21,880 --> 01:08:25,010 So basically, after it passed through this polarizer 995 01:08:25,010 --> 01:08:27,910 with a axis pointing to the x direction, 996 01:08:27,910 --> 01:08:29,810 in this case the x direction is pointing 997 01:08:29,810 --> 01:08:33,460 to this direction, what you are going to get 998 01:08:33,460 --> 01:08:37,149 is that you are going to filter from all 999 01:08:37,149 --> 01:08:40,270 those unpolarized direction, you are 1000 01:08:40,270 --> 01:08:46,300 going to filter only the light which is actually parallel 1001 01:08:46,300 --> 01:08:48,220 to the easy axis. 1002 01:08:48,220 --> 01:08:52,630 So therefore, after the unpolarized light source 1003 01:08:52,630 --> 01:08:55,060 passes through this polarizer, you 1004 01:08:55,060 --> 01:08:57,460 are going to have all the electric field 1005 01:08:57,460 --> 01:09:01,160 pointing to the x direction. 1006 01:09:01,160 --> 01:09:02,840 Now everybody can accept. 1007 01:09:02,840 --> 01:09:05,140 And of course if I have a second one, which 1008 01:09:05,140 --> 01:09:07,939 is in the y direction, since there will 1009 01:09:07,939 --> 01:09:10,720 be no component pointing to the y direction, 1010 01:09:10,720 --> 01:09:14,100 you get zero electric field. 1011 01:09:14,100 --> 01:09:16,359 That's very nice. 1012 01:09:16,359 --> 01:09:18,540 So Einstein was really happy that, oh, 1013 01:09:18,540 --> 01:09:23,920 that means unpolarized light maybe is just 50% 1014 01:09:23,920 --> 01:09:27,160 of the polarized light in the x direction 1015 01:09:27,160 --> 01:09:30,660 and 50% of the polarized light in the y direction. 1016 01:09:30,660 --> 01:09:42,170 Because each time I emit one photon, the first half of them 1017 01:09:42,170 --> 01:09:48,970 got stopped by the x direction easy axis polarizer. 1018 01:09:48,970 --> 01:09:50,529 And then the second half got stopped 1019 01:09:50,529 --> 01:09:52,180 by the second polarizer. 1020 01:09:52,180 --> 01:09:55,720 So that probably makes sense. 1021 01:09:55,720 --> 01:10:00,490 But how about-- as we did in the demonstration-- 1022 01:10:00,490 --> 01:10:06,880 how about we rotate the second polarizer by 45 degrees? 1023 01:10:06,880 --> 01:10:10,105 What will be the intensity? 1024 01:10:10,105 --> 01:10:15,900 But now, hey, you cannot split a single photon because a photon 1025 01:10:15,900 --> 01:10:16,440 is a photon. 1026 01:10:16,440 --> 01:10:17,790 How can I split, right? 1027 01:10:17,790 --> 01:10:20,415 Because if I describe my unpolarized light 1028 01:10:20,415 --> 01:10:26,010 as 50% linearly polarized in x, the other half is linearly 1029 01:10:26,010 --> 01:10:29,520 polarized in y, then I am in trouble 1030 01:10:29,520 --> 01:10:33,010 because I don't know how to calculate 1031 01:10:33,010 --> 01:10:36,290 what is this intensity. 1032 01:10:36,290 --> 01:10:44,490 So basically maybe all of them pass through or all of them 1033 01:10:44,490 --> 01:10:47,430 doesn't pass through. 1034 01:10:47,430 --> 01:10:50,670 Then we can do this experiment. 1035 01:10:50,670 --> 01:10:52,810 And this is the result-- 1036 01:10:52,810 --> 01:10:55,590 what you are going to get is that the intensity 1037 01:10:55,590 --> 01:10:59,680 is going to be I0 over 4. 1038 01:10:59,680 --> 01:11:04,500 That is actually not 0 or I over 2. 1039 01:11:04,500 --> 01:11:08,100 So that means really the nature of plate 1040 01:11:08,100 --> 01:11:16,740 dies because a single photon is equivalent to something which 1041 01:11:16,740 --> 01:11:20,100 can be described by a wave. 1042 01:11:20,100 --> 01:11:23,130 So that gives us some possible connection 1043 01:11:23,130 --> 01:11:25,840 to quantum mechanics, because that 1044 01:11:25,840 --> 01:11:29,970 means a single photon is not like a single object, which 1045 01:11:29,970 --> 01:11:33,480 is actually passing through all those polarizers. 1046 01:11:33,480 --> 01:11:42,630 But they actually act also like waves. 1047 01:11:42,630 --> 01:11:48,540 So that is something which we'll follow up with later lectures. 1048 01:11:48,540 --> 01:11:51,660 When we talk about interference, et cetera, 1049 01:11:51,660 --> 01:11:54,720 we are going to also discuss related issues 1050 01:11:54,720 --> 01:12:00,240 about the connection to quantum mechanics. 1051 01:12:00,240 --> 01:12:03,870 So today, we actually have learned 1052 01:12:03,870 --> 01:12:08,580 about polarization, linearly polarized, circularly 1053 01:12:08,580 --> 01:12:11,010 polarized, and elliptically polarized 1054 01:12:11,010 --> 01:12:15,930 electromagnetic waves, and also unpolarized light source. 1055 01:12:15,930 --> 01:12:18,550 And we have learned about how to produce 1056 01:12:18,550 --> 01:12:22,170 polarized electromagnetic wave with polarizer. 1057 01:12:22,170 --> 01:12:25,010 Basically have the unpolarized light 1058 01:12:25,010 --> 01:12:30,330 source pass through a polarizer, you'll have a polarized light. 1059 01:12:30,330 --> 01:12:33,390 We didn't cover quarter wave plate yet. 1060 01:12:33,390 --> 01:12:35,610 We are going to cover that next time. 1061 01:12:35,610 --> 01:12:40,950 And also, next time we will talk about how we actually can 1062 01:12:40,950 --> 01:12:43,260 generate electromagnetic field. 1063 01:12:43,260 --> 01:12:45,420 We have been talking about electromagnetic field 1064 01:12:45,420 --> 01:12:47,600 for a long time, but how are they actually 1065 01:12:47,600 --> 01:12:51,750 generated is an issue which we have actually touched. 1066 01:12:51,750 --> 01:12:55,620 And that is actually going to happen after the midterm 1067 01:12:55,620 --> 01:12:56,710 next week. 1068 01:12:56,710 --> 01:12:57,790 Thank you very much. 1069 01:12:57,790 --> 01:13:02,280 And if you have any questions, please let me know. 1070 01:13:02,280 --> 01:13:05,930 AUDIENCE: [INAUDIBLE]