1 00:00:00 --> 00:00:00,84 2 00:00:00,84 --> 00:00:04,001 Earlier in this course, we discussed linear 3 00:00:04,001 --> 00:00:08,443 polarization of electromagnetic radiation, and I demonstrate 4 00:00:08,443 --> 00:00:12,282 this at seventy-five megaHertz and at ten gigaHertz. 5 00:00:12,282 --> 00:00:16,498 Today, I will concentrate exclusively on the polarization 6 00:00:16,498 --> 00:00:19,886 of light, which is at a much higher frequency. 7 00:00:19,886 --> 00:00:23,951 The light from the sun or light from light bulbs is not 8 00:00:23,951 --> 00:00:26,435 polarized. So I can ask myself the 9 00:00:26,435 --> 00:00:31,028 question, now, what does it mean when light is 10 00:00:31,028 --> 00:00:34,553 not polarized? Let's think of individual light 11 00:00:34,553 --> 00:00:38,626 photons as plane waves, with a well-defined direction 12 00:00:38,626 --> 00:00:41,682 of polarization. So each one is linearly 13 00:00:41,682 --> 00:00:44,737 polarized. A beam is coming straight out 14 00:00:44,737 --> 00:00:48,105 of the blackboard. The first photon arrives, 15 00:00:48,105 --> 00:00:51,317 it's linearly polarized in this direction. 16 00:00:51,317 --> 00:00:55,469 This second photon arrives, linearly polarized in this 17 00:00:55,469 --> 00:01:00,483 direction, so the electric field vector is oscillating like that. 18 00:01:00,483 --> 00:01:04,183 Another photon, another photon, 19 00:01:04,183 --> 00:01:07,445 and another photon. And what you see here, 20 00:01:07,445 --> 00:01:10,547 very clearly, that there is no preferred 21 00:01:10,547 --> 00:01:15,639 direction which you average over time, and that's what we call -- 22 00:01:15,639 --> 00:01:19,219 call unpolarized light. It was Edwin Land who, 23 00:01:19,219 --> 00:01:23,515 in nineteen thirty eight, developed a material that can 24 00:01:23,515 --> 00:01:26,618 turn this into linearly polarized light, 25 00:01:26,618 --> 00:01:30,675 for which he became very famous, in addition to this 26 00:01:30,675 --> 00:01:35,29 demonstration that I showed you last time. 27 00:01:35,29 --> 00:01:40,302 If I take one of Edwin Land's sheets, which will turn light 28 00:01:40,302 --> 00:01:45,142 into polarization in this direction, and I first take one 29 00:01:45,142 --> 00:01:47,735 photon, for instance, this one. 30 00:01:47,735 --> 00:01:51,97 That one comes in from the blackboard towards you, 31 00:01:51,97 --> 00:01:56,032 and so here it is. Oscillating the E vector like 32 00:01:56,032 --> 00:02:00,699 this, E zero is the maximum value of the electric field 33 00:02:00,699 --> 00:02:05,626 strength in that plane electromagnetic wave. 34 00:02:05,626 --> 00:02:11 And this is the direction of the polarizer that I have 35 00:02:11 --> 00:02:16,475 through which this photon goes. I can now make a simple 36 00:02:16,475 --> 00:02:22,356 calculation, by projecting this E-vector onto the preferred 37 00:02:22,356 --> 00:02:27,831 direction of polarization, and this new E-vector is now 38 00:02:27,831 --> 00:02:33,003 down by the cosine of theta, if this angle is theta, 39 00:02:33,003 --> 00:02:40,101 this E-vector is now E -- E zero times the cosine of theta. 40 00:02:40,101 --> 00:02:43,887 If you ask me now, whether the light is reduced in 41 00:02:43,887 --> 00:02:48,677 intensity, I would have to say, "Yes, of course," because light 42 00:02:48,677 --> 00:02:51,768 intensity depends on the pointing vector, 43 00:02:51,768 --> 00:02:56,018 and the pointing vector is always proportional to E zero 44 00:02:56,018 --> 00:03:00,345 squared, because the pointing vector is the cross-product 45 00:03:00,345 --> 00:03:03,126 between E and B. And if E is reduced, 46 00:03:03,126 --> 00:03:06,758 B is also reduced. And so we get a cosine square 47 00:03:06,758 --> 00:03:09,617 reduction. If, now, I average over all 48 00:03:09,617 --> 00:03:14,387 incoming photons -- so I take all of these, 49 00:03:14,387 --> 00:03:19,489 which represent an unpolarized beam -- so I get not only one 50 00:03:19,489 --> 00:03:23,553 like so, but I get one like so, and one like so, 51 00:03:23,553 --> 00:03:26,925 and one like so, and one like so -- then 52 00:03:26,925 --> 00:03:31,94 clearly, I have to calculate, now, the mean value of cosine 53 00:03:31,94 --> 00:03:35,485 square theta. And the mean value of cosine 54 00:03:35,485 --> 00:03:40,241 square theta is one-half, and so if the intensity of the 55 00:03:40,241 --> 00:03:43,526 unpolarized beam, unpolarized light was 56 00:03:43,526 --> 00:03:48,303 originally I zero, once it comes through 57 00:03:48,303 --> 00:03:53,538 this polarizer that Edwin Land gave me, then I get one-half I 58 00:03:53,538 --> 00:03:57,639 zero, but that is now hundred percent polarized. 59 00:03:57,639 --> 00:04:02,263 And it is hundred percent polarized in this direction. 60 00:04:02,263 --> 00:04:07,498 And the one-half is the result of the average value of cosine 61 00:04:07,498 --> 00:04:10,552 square theta. If this were the case, 62 00:04:10,552 --> 00:04:15,962 it would be an extremely ideal polarizer, we would call this an 63 00:04:15,962 --> 00:04:20,002 H N fifty polarizer -- they don't exist, 64 00:04:20,002 --> 00:04:23,866 it's only in your head -- and the fifty refers to the fact 65 00:04:23,866 --> 00:04:26,577 that fifty percent get through polarized. 66 00:04:26,577 --> 00:04:30,508 In the optic skits that we hand out today that we will need 67 00:04:30,508 --> 00:04:33,694 throughout this course, you don't have H N fifty 68 00:04:33,694 --> 00:04:37,625 polarizers, they don't exist. I don't quite know what yours 69 00:04:37,625 --> 00:04:41,014 is, I didn't measure it, yours may be an H N twenty 70 00:04:41,014 --> 00:04:45,014 five, or maybe an H N thirty, which would then mean that the 71 00:04:45,014 --> 00:04:48,538 I zero strength of an unpolarized light of beam would 72 00:04:48,538 --> 00:04:53,524 not be half of I zero, but maybe only point two five, 73 00:04:53,524 --> 00:04:55,874 or point three. But in any case, 74 00:04:55,874 --> 00:04:59,969 the light that will come through your linear polarizers 75 00:04:59,969 --> 00:05:03,76 will be very closely two hundred percent polarized. 76 00:05:03,76 --> 00:05:07,702 So what I will do now, I will take unpolarized light, 77 00:05:07,702 --> 00:05:11,645 and I will have this light coming straight out of the 78 00:05:11,645 --> 00:05:15,663 blackboard perpendicular to you, with strength I zero, 79 00:05:15,663 --> 00:05:19,53 and here is one of my polarizers, and the light that 80 00:05:19,53 --> 00:05:23,809 comes through here is linearly polarized in 81 00:05:23,809 --> 00:05:26,823 this direction. And so we already know that 82 00:05:26,823 --> 00:05:30,41 one-half I zero will come through if it is an ideal 83 00:05:30,41 --> 00:05:33,854 polarizer, and it is polarized in this direction. 84 00:05:33,854 --> 00:05:36,652 I take a second sheet, an identical one, 85 00:05:36,652 --> 00:05:39,809 I put it also in the plane of the blackboard, 86 00:05:39,809 --> 00:05:42,32 but I rotate it over an angle theta. 87 00:05:42,32 --> 00:05:46,697 So here is now a second sheet, which has a preferred direction 88 00:05:46,697 --> 00:05:50,715 of polarization in -- in this direction, and the angle is 89 00:05:50,715 --> 00:05:53,894 rotated over an angle theta. 90 00:05:53,894 --> 00:05:57,278 So between this one and this one is an angle theta. 91 00:05:57,278 --> 00:06:01,337 And so you can now immediately tell what the intensity of the 92 00:06:01,337 --> 00:06:04,653 light is that comes through this second polarizer. 93 00:06:04,653 --> 00:06:08,036 It must, of course, be polarized in this direction, 94 00:06:08,036 --> 00:06:11,961 because that is the allowed direction polarization for that 95 00:06:11,961 --> 00:06:15,682 second sheet -- and the intensity must now be one-half I 96 00:06:15,682 --> 00:06:19,877 zero, because that's what comes in, and then I have to multiply 97 00:06:19,877 --> 00:06:22,042 it by the cosine square of theta. 98 00:06:22,042 --> 00:06:27,086 I don't have to average it now over all angles, 99 00:06:27,086 --> 00:06:33,492 because there is only one value of theta between this sheet and 100 00:06:33,492 --> 00:06:38,038 this sheet, so this is now the new intensity, 101 00:06:38,038 --> 00:06:42,171 and it's all polarized in this direction. 102 00:06:42,171 --> 00:06:46,614 And this law, whereby the light intensity is 103 00:06:46,614 --> 00:06:50,85 reduced by the factor cosine square theta, 104 00:06:50,85 --> 00:06:54,363 is known as Malus' Law. Malus' Law. 105 00:06:54,363 --> 00:06:57,359 If theta were thirty degrees, 106 00:06:57,359 --> 00:06:59,632 107 00:06:59,632 --> 00:07:04,103 the light intensity here would be one-half I zero ties the 108 00:07:04,103 --> 00:07:08,888 cosine square of thirty degrees, which is oh point seven five. 109 00:07:08,888 --> 00:07:13,28 If theta were zero degrees, that means that this sheet is 110 00:07:13,28 --> 00:07:17,908 in the same direction as this one, if everything were ideal, 111 00:07:17,908 --> 00:07:21,829 one-half I zero would get through the second sheet. 112 00:07:21,829 --> 00:07:26,3 If theta is ninety degrees, then nothing will get through, 113 00:07:26,3 --> 00:07:29,751 because the cosine of ninety degrees is zero. 114 00:07:29,751 --> 00:07:33,317 We call that crossed polarizers. 115 00:07:33,317 --> 00:07:37,344 If you cross them like this, no light will get through. 116 00:07:37,344 --> 00:07:41,818 Now, before I demonstrate this, I have to be honest with you, 117 00:07:41,818 --> 00:07:46,293 because the idea of reducing the energy of individual photons 118 00:07:46,293 --> 00:07:50,096 by reducing their electric field strength, as I did, 119 00:07:50,096 --> 00:07:52,408 is a cheat. A light photon has a 120 00:07:52,408 --> 00:07:55,988 well-defined energy which depends uniquely on the 121 00:07:55,988 --> 00:07:59,418 frequency of the light. Blue light has a higher 122 00:07:59,418 --> 00:08:03,222 frequency than red light, so blue light has a higher 123 00:08:03,222 --> 00:08:05,893 energy than red light. 124 00:08:05,893 --> 00:08:08,716 And when you send blue light through a polarizer, 125 00:08:08,716 --> 00:08:11,538 the way I did here, it either comes through or it 126 00:08:11,538 --> 00:08:14,419 doesn't come through. But if it does come through, 127 00:08:14,419 --> 00:08:17,359 it is still blue light, there is no such thing as a 128 00:08:17,359 --> 00:08:19,888 reduction in energy. Whereas this reduction, 129 00:08:19,888 --> 00:08:22,416 by cosine theta, would imply that the energy 130 00:08:22,416 --> 00:08:26,062 goes down, and that moo- would imply, then, that there would be 131 00:08:26,062 --> 00:08:28,473 a color change, that it would no longer be 132 00:08:28,473 --> 00:08:30,178 blue. And that's not the case. 133 00:08:30,178 --> 00:08:33,529 If you want to treat this properly, you have to do it in a 134 00:08:33,529 --> 00:08:36,356 quantum mechanical way. 135 00:08:36,356 --> 00:08:40,695 The interesting thing is that if you use quantum mechanics, 136 00:08:40,695 --> 00:08:44,809 you find exactly the same law, you find also Malus' Law. 137 00:08:44,809 --> 00:08:48,249 So the law is OK, even though the derivation is 138 00:08:48,249 --> 00:08:51,241 not kosher. Now, I want you to get out of 139 00:08:51,241 --> 00:08:55,43 your envelope one of your green plates, which is a linear 140 00:08:55,43 --> 00:08:58,422 polarizer. This is the kind of plate that 141 00:08:58,422 --> 00:09:00,89 you have, you have three in there. 142 00:09:00,89 --> 0. Only take one out. 143 0. --> 00:09:02,311 If this light that comes in is Ah, ready for this? I want you to see two things. 144 00:09:02,311 --> 00:09:08,09 These two lights shining on me, unpolarized light. 145 00:09:08,09 --> 00:09:12,231 So the light that comes to you now is unpolarized. 146 00:09:12,231 --> 00:09:16,965 I'm now going to hold in front of my face this polarizer. 147 00:09:16,965 --> 00:09:22,036 So the light that comes through is linearly polarized in this 148 00:09:22,036 --> 00:09:25,333 direction. And you are going to play the 149 00:09:25,333 --> 00:09:28,968 role of the second polarizer. Close one eye, 150 00:09:28,968 --> 00:09:33,448 put the polarizer in front of your eye, and rotate it. 151 00:09:33,448 --> 00:09:38,265 And you will see a huge difference in light 152 00:09:38,265 --> 00:09:41,525 intensity. If you cross-polarize with me, 153 00:09:41,525 --> 00:09:45,68 then you can't see me. That may make you very happy. 154 00:09:45,68 --> 00:09:48,695 But keep in mind, if you can't see me, 155 00:09:48,695 --> 00:09:50,977 then I can't see you, either. 156 00:09:50,977 --> 00:09:55,214 So rotate it around and convince that this light that 157 00:09:55,214 --> 00:09:58,962 reaches you is now, indeed, linearly polarized, 158 00:09:58,962 --> 00:10:03,688 and when you rotate around your polarimeters -- we call the 159 00:10:03,688 --> 00:10:07,192 polarimeters, we call them polarizers -- you 160 00:10:07,192 --> 00:10:09,953 can see me either, 161 00:10:09,953 --> 00:10:14,238 or you cannot see me at all, and there is anything and 162 00:10:14,238 --> 00:10:16,825 everything in between. Very well. 163 00:10:16,825 --> 00:10:21,352 There is a second way that we can produce hundred percent 164 00:10:21,352 --> 00:10:25,071 linearly polarized light, and we can do that by 165 00:10:25,071 --> 00:10:28,709 reflecting unpolarized light off a dielectric. 166 00:10:28,709 --> 00:10:30,972 For instance, water or glass. 167 00:10:30,972 --> 00:10:35,58 None of this follows from Snell's Law, Snell's Law was two 168 00:10:35,58 --> 00:10:39,946 hundred fifty years before Maxwell, polarization wasn't 169 00:10:39,946 --> 00:10:43,708 even known in the days of Snell. 170 00:10:43,708 --> 00:10:47,675 But Maxwell's equations allow you to properly deal with 171 00:10:47,675 --> 00:10:51,568 refraction and reflection, including the polarization. 172 00:10:51,568 --> 00:10:55,902 And I will make no attempt to derive this for you in detail, 173 00:10:55,902 --> 00:11:00,016 that's really part of eight oh three if you ever take it, 174 00:11:00,016 --> 00:11:04,276 but I will present you with some results so that you can at 175 00:11:04,276 --> 00:11:08,096 least appreciate the far-reaching consequences of the 176 00:11:08,096 --> 00:11:11,843 reflection in which we can produce a hundred percent 177 00:11:11,843 --> 00:11:16,257 polarized light. Suppose I have unpolarized 178 00:11:16,257 --> 00:11:20,383 light coming in here, medium one, index of refraction 179 00:11:20,383 --> 00:11:23,874 N one, medium two, index of refraction N two. 180 00:11:23,874 --> 00:11:27,683 It's coming in at an incident angle of theta one, 181 00:11:27,683 --> 00:11:31,413 it's unpolarized. Some of it is reflected -- and 182 00:11:31,413 --> 00:11:36,412 this angle is also theta one as we discussed earlier -- and some 183 00:11:36,412 --> 00:11:40,856 of it is refracted into this medium, and this angle we'll 184 00:11:40,856 --> 00:11:43,157 call theta two. So this light, 185 00:11:43,157 --> 00:11:47,998 unpolarized comes in, reflects, and refracts. 186 00:11:47,998 --> 00:11:52,177 If you want to use Maxwell's equations, at the action, 187 00:11:52,177 --> 00:11:56,435 where everything is happening right here at the surface 188 00:11:56,435 --> 00:12:00,063 between the two, you will have to decompose the 189 00:12:00,063 --> 00:12:03,454 incoming light, the electric field vector in 190 00:12:03,454 --> 00:12:06,214 two directions. And one direction is 191 00:12:06,214 --> 00:12:10,63 perpendicular to the blackboard, so this is the E-vector, 192 00:12:10,63 --> 00:12:14,099 and the other direction is in the blackboard. 193 00:12:14,099 --> 00:12:18,515 You will have to do the same here, and you have to do the 194 00:12:18,515 --> 00:12:22,307 same here. And if you look at this 195 00:12:22,307 --> 00:12:24,836 decomposition, then notice that both 196 00:12:24,836 --> 00:12:28,81 components -- this component, as well as this component, 197 00:12:28,81 --> 00:12:32,279 is perpendicular to the direction of propagation. 198 00:12:32,279 --> 00:12:36,325 That is always a must with traveling electric -- magnetic 199 00:12:36,325 --> 00:12:38,349 waves. You see the same here. 200 00:12:38,349 --> 00:12:42,106 This component and this are both perpendicular to the 201 00:12:42,106 --> 00:12:45,141 reflected beam, this component and this are 202 00:12:45,141 --> 00:12:48,032 both perpendicular to the refracted beam. 203 00:12:48,032 --> 00:12:51,789 This one, the one perpendicular to 204 00:12:51,789 --> 00:12:54,615 the blackboard, we normally give a symbol 205 00:12:54,615 --> 00:12:57,44 perpendicular, we call this the incidence 206 00:12:57,44 --> 00:13:01,537 plane, the plane through the incident light and then normal 207 00:13:01,537 --> 00:13:04,362 to the surface, we call that the incident 208 00:13:04,362 --> 00:13:08,6 plane, in this place -- in this case, that is the blackboard. 209 00:13:08,6 --> 00:13:12,908 We call this the perpendicular component, and we call this the 210 00:13:12,908 --> 00:13:15,098 parallel component. And this is, 211 00:13:15,098 --> 00:13:18,771 of course, the incident beam. So this one we call the 212 00:13:18,771 --> 00:13:23,785 perpendicular component, this one we call the parallel 213 00:13:23,785 --> 00:13:27,562 component, and this is now of the refracted beam. 214 00:13:27,562 --> 00:13:31,969 This is the parallel component, this is the perpendicular 215 00:13:31,969 --> 00:13:36,454 component of the reflected beam. The incident light is not 216 00:13:36,454 --> 00:13:39,759 polarized in this sense. So if you average, 217 00:13:39,759 --> 00:13:43,929 there is no preferred direction of the electric field. 218 00:13:43,929 --> 00:13:47,47 That means, then, that in this representation, 219 00:13:47,47 --> 00:13:51,719 the strength of this component and the strength of this 220 00:13:51,719 --> 00:13:58,006 component must be exactly equal, because if one were stronger 221 00:13:58,006 --> 00:14:00,385 than the other, then it wouldn't be 222 00:14:00,385 --> 00:14:03,463 unpolarized, then there would be, on average, 223 00:14:03,463 --> 00:14:07,102 a preferred direction. So this component has the same 224 00:14:07,102 --> 00:14:10,461 strength a that component for the incoming light. 225 00:14:10,461 --> 00:14:14,519 What Maxwell's equations now can do for us -- it's a lot of 226 00:14:14,519 --> 00:14:18,577 work, but you may see it in eight oh three -- it can relate 227 00:14:18,577 --> 00:14:22,286 the parallel component in reflection with the parallel 228 00:14:22,286 --> 00:14:25,644 component of incidence, the parallel component of 229 00:14:25,644 --> 00:14:29,761 refraction with the parallel component of 230 00:14:29,761 --> 00:14:34,005 incidence, it gives you two relations, two -- two equations. 231 00:14:34,005 --> 00:14:37,457 It can also relate the perpendicular component of 232 00:14:37,457 --> 00:14:40,621 reflection with this perpendicular component, 233 00:14:40,621 --> 00:14:44,505 and the perpendicular component in refraction with this 234 00:14:44,505 --> 00:14:47,095 component. So you get four equations. 235 00:14:47,095 --> 0. 236 0. --> 00:14:49,324 If this light that comes in is Ah, ready for this? I want you to see two things. 237 00:14:49,324 --> 00:14:53,424 unpolarized, the strength of this component is the same as 238 00:14:53,424 --> 00:14:55,869 this one. In general, you will see, 239 00:14:55,869 --> 00:15:00,113 if you apply these four equations, 240 00:15:00,113 --> 00:15:02,448 that that's no longer the case here. 241 00:15:02,448 --> 00:15:06,319 They're no longer the same intensity, and they're no longer 242 00:15:06,319 --> 00:15:09,989 the same intensity here. That means this reflected light 243 00:15:09,989 --> 00:15:13,86 and the refracted light has now become partially polarized. 244 00:15:13,86 --> 00:15:17,263 I will only give you the relation, one of those four 245 00:15:17,263 --> 00:15:19,866 equations. I will only address today the 246 00:15:19,866 --> 00:15:22,468 parallel component of the incident beam, 247 00:15:22,468 --> 00:15:25,671 and the parallel component of the reflected beam. 248 00:15:25,671 --> 00:15:29,609 And I don't imagine that any one of you will try to remember 249 00:15:29,609 --> 00:15:31,922 that equation. 250 00:15:31,922 --> 00:15:35,9 I don't either, I have to look it up every time 251 00:15:35,9 --> 00:15:39,791 that I deal with this. So E zero, which always 252 00:15:39,791 --> 00:15:43,769 represents, then, the maximum possible value of 253 00:15:43,769 --> 00:15:47,66 the electric field, of the parallel component, 254 00:15:47,66 --> 00:15:52,157 of the reflected beam, that's the one that I'm after, 255 00:15:52,157 --> 00:15:56,221 I'm going to make polarized light by reflecting, 256 00:15:56,221 --> 00:16:00,805 is E zero, the parallel component of the incident beam 257 00:16:00,805 --> 00:16:06,08 times -- and this depends now on the angle of 258 00:16:06,08 --> 00:16:11,665 incidence and on the indices of refraction -- is going to be N 259 00:16:11,665 --> 00:16:17,25 one times the cosine of theta two minus N two times the cosine 260 00:16:17,25 --> 00:16:20,088 of theta one. Believe it or not, 261 00:16:20,088 --> 00:16:24,758 all of this follows from Maxwell -- divided by N one 262 00:16:24,758 --> 00:16:30,16 times the cosine of theta two plus N two times the cosine of 263 00:16:30,16 --> 00:16:33,731 theta one. And if you apply Snell's Law, 264 00:16:33,731 --> 00:16:37,942 you can simplify this equation -- 265 00:16:37,942 --> 00:16:43,537 in this case it will help us -- and so you get minus E zero 266 00:16:43,537 --> 00:16:47,684 parallel incidence, and now you get here the 267 00:16:47,684 --> 00:16:51,253 tangent of theta one, minus theta two, 268 00:16:51,253 --> 00:16:56,076 divided by the tangent of theta one plus theta two. 269 00:16:56,076 --> 00:16:59,548 So these two equations are identical. 270 00:16:59,548 --> 00:17:05,528 Though that may not be obvious to you, certainly not obvious to 271 00:17:05,528 --> 00:17:09,094 me, either, but if you substitute 272 00:17:09,094 --> 00:17:12,651 Snell's Law in here, you can show that these two are 273 00:17:12,651 --> 00:17:15,161 the same. Keep in mind if you're ever 274 00:17:15,161 --> 00:17:19,485 interested in the intensity of this light, then you must always 275 00:17:19,485 --> 00:17:23,53 remember that the pointing vector is proportional to E zero 276 00:17:23,53 --> 00:17:27,854 squared, so you always have to square these numbers when you're 277 00:17:27,854 --> 00:17:31,899 interested in light intensity. This is just the strength of 278 00:17:31,899 --> 00:17:34,968 the E-vector. There is something very special 279 00:17:34,968 --> 00:17:38,855 hidden in this equation, and that is, 280 00:17:38,855 --> 00:17:43,153 when theta one plus theta two is ninety degrees, 281 00:17:43,153 --> 00:17:47,177 then the downstairs here is infinitely large. 282 00:17:47,177 --> 00:17:52,482 And so that means that the parallel component in reflection 283 00:17:52,482 --> 00:17:56,049 is zero. So E parallel in refection goes 284 00:17:56,049 --> 00:17:59,432 to zero. And if that one goes to zero, 285 00:17:59,432 --> 00:18:03,548 there is only this one left which is not zero, 286 00:18:03,548 --> 00:18:09,309 and that means the reflected light is now hundred 287 00:18:09,309 --> 00:18:13,182 percent polarized in this direction, because I have killed 288 00:18:13,182 --> 00:18:16,851 this component completely. But it only works if this is 289 00:18:16,851 --> 00:18:19,908 met, this condition. If this condition is met, 290 00:18:19,908 --> 00:18:23,101 that theta one plus theta two is ninety degrees, 291 00:18:23,101 --> 00:18:26,702 then it follows from high school math that the sine of 292 00:18:26,702 --> 00:18:29,487 theta two is then the cosine of theta one. 293 00:18:29,487 --> 00:18:31,729 That's immediately obvious, right? 294 00:18:31,729 --> 00:18:35,33 You remember the triangle, theta one plus theta two is 295 00:18:35,33 --> 00:18:38,251 ninety degrees, the sine of one angle is the 296 00:18:38,251 --> 00:18:41,097 cosine of the other. 297 00:18:41,097 --> 00:18:46,569 And if now, I remember Snell's Law, which says that the sine of 298 00:18:46,569 --> 00:18:52,217 theta one divided by the sine of theta two, if N two divided by N 299 00:18:52,217 --> 00:18:56,63 one, I can replace now, only for this special case, 300 00:18:56,63 --> 00:19:02,014 the sine of -- sine theta I can replace by the cosine of theta 301 00:19:02,014 --> 00:19:03,955 one. And so I get here, 302 00:19:03,955 --> 00:19:09,162 now, the tangent of theta one. And so if this is the tangent 303 00:19:09,162 --> 00:19:13,113 of theta one, that is under these 304 00:19:13,113 --> 00:19:18,189 conditions, then we have met the condition that I was looking 305 00:19:18,189 --> 00:19:23,011 for, that I end up with hundred percent linearly polarized 306 00:19:23,011 --> 00:19:25,887 light. And so this is the secret to 307 00:19:25,887 --> 00:19:29,186 getting hundred percent polarized light. 308 00:19:29,186 --> 00:19:32,824 And this angle is called the Brewster angle. 309 00:19:32,824 --> 00:19:35,024 And so if we, for instance, 310 00:19:35,024 --> 00:19:40,184 look at the transition from air to glass, glass has a index of 311 00:19:40,184 --> 00:19:44,414 refraction approximately one point 312 00:19:44,414 --> 00:19:49,144 five -- depends on the kind of glass that you have -- if I go 313 00:19:49,144 --> 00:19:52,848 from air to glass, which is what I will do in my 314 00:19:52,848 --> 00:19:56,317 demonstration, then the tangent of this angle 315 00:19:56,317 --> 00:19:59,47 is N two divided by N one, this is glass, 316 00:19:59,47 --> 00:20:02,701 so this is one point five, and one is one, 317 00:20:02,701 --> 00:20:06,012 then you will find that the Brewster angle, 318 00:20:06,012 --> 00:20:10,268 theta Brewster turns out to be about fifty six degrees. 319 00:20:10,268 --> 00:20:14,131 I can also make linearly polarized 320 00:20:14,131 --> 00:20:18,641 light by going from glass to air, by bouncing it off this 321 00:20:18,641 --> 00:20:20,251 way. Then, of course, 322 00:20:20,251 --> 00:20:23,876 I have to invert this, and then you will get a 323 00:20:23,876 --> 00:20:28,789 Brewster angle which is smaller, which is thirty-four degrees. 324 00:20:28,789 --> 00:20:32,655 But since I will do it with -- from air to glass, 325 00:20:32,655 --> 00:20:37,084 I want you to concentrate on the fifty-six degree angle. 326 00:20:37,084 --> 00:20:40,467 The way I'm going to do this demonstration, 327 00:20:40,467 --> 00:20:44,736 it's right set up here, we have light, 328 00:20:44,736 --> 00:20:48,684 a light beam that strikes a piece of plane parallel glass. 329 00:20:48,684 --> 00:20:51,871 That's all it is, there's nothing special about 330 00:20:51,871 --> 00:20:55,335 this piece of glass. So the light comes in like so, 331 00:20:55,335 --> 00:20:57,621 and here, I have a piece of glass. 332 00:20:57,621 --> 00:21:00,461 This is the angle of incidence, theta one. 333 00:21:00,461 --> 00:21:03,925 And so it's going to be reflected in this direction 334 00:21:03,925 --> 00:21:08,15 whereby this angle is also theta one, and something will go in 335 00:21:08,15 --> 00:21:12,376 here, that is that angle theta two, which I don't worry about, 336 00:21:12,376 --> 00:21:15,979 because I want you to see that this 337 00:21:15,979 --> 00:21:18,348 can become hundred percent polarized. 338 00:21:18,348 --> 00:21:21,046 As this light comes in, it is unpolarized, 339 00:21:21,046 --> 00:21:24,535 this component and this component have equal strength, 340 00:21:24,535 --> 00:21:28,023 if theta one is fifty-six degrees or somewhere in that 341 00:21:28,023 --> 00:21:31,511 vicinity, this light is now hundred percent polarized. 342 00:21:31,511 --> 00:21:34,473 And I'm going to project that onto the screen, 343 00:21:34,473 --> 00:21:37,632 and I'm going to convince you that it is, indeed, 344 00:21:37,632 --> 00:21:39,804 polarized. You cannot use your own 345 00:21:39,804 --> 00:21:43,292 polarizers to see that, because the light in this beam 346 00:21:43,292 --> 00:21:47,241 is going to be hundred percent polarized. 347 00:21:47,241 --> 00:21:51,233 However, once it reflects off the screen, it no longer is, 348 00:21:51,233 --> 00:21:55,015 so you cannot use your polarimeter, so I have to use my 349 00:21:55,015 --> 00:21:57,256 own polarimeter to show you this. 350 00:21:57,256 --> 00:22:01,318 So if you can turn the light off there, off the overhead -- 351 00:22:01,318 --> 00:22:05,45 thank you very much -- I will turn on the -- the light of my 352 00:22:05,45 --> 00:22:09,162 light beam, there it is, and I'm going to make it very 353 00:22:09,162 --> 00:22:12,383 dark for you so that we can see that very well. 354 00:22:12,383 --> 00:22:16,025 So the light comes in this direction, hits the glass, 355 00:22:16,025 --> 00:22:19,946 and the angle of incidence is now about 356 00:22:19,946 --> 00:22:22,849 forty-five degrees. I purposely didn't make it 357 00:22:22,849 --> 00:22:25,429 fifty-six yet. I have here a large set of 358 00:22:25,429 --> 00:22:28,525 polarizer, one of Edwin Land's polar- polarizers, 359 00:22:28,525 --> 00:22:30,782 and I will rotate that in this beam. 360 00:22:30,782 --> 00:22:33,555 You will see that it is partially polarized, 361 00:22:33,555 --> 00:22:36,78 not yet hundred percent, but it's already partially 362 00:22:36,78 --> 00:22:38,844 polarized. So there is already an 363 00:22:38,844 --> 00:22:42,133 imbalance between the perpendicular and the parallel 364 00:22:42,133 --> 00:22:44,584 component. So if I hold it in the beam, 365 00:22:44,584 --> 00:22:47,487 and I rotate it, you will clearly see now that 366 00:22:47,487 --> 00:22:51,747 it is much fainter than -- than it is now. 367 00:22:51,747 --> 00:22:54,987 And now I will go for the fifty-six angle, 368 00:22:54,987 --> 00:22:58,386 fifty-six degree angle, roughly, and so now, 369 00:22:58,386 --> 00:23:01,389 I rotate my polarizer, and now, notice, 370 00:23:01,389 --> 00:23:03,918 I can kill that light completely. 371 00:23:03,918 --> 00:23:06,605 Hundred percent linearly polarized. 372 00:23:06,605 --> 00:23:10,32 I may not have the angle perfect, but that's OK, 373 00:23:10,32 --> 00:23:13,797 you get the idea. It is very close to totally 374 00:23:13,797 --> 00:23:15,931 dark. Now you see the light, 375 00:23:15,931 --> 00:23:19,962 it's polarized in this direction, and now I can kill 376 00:23:19,962 --> 00:23:22,412 it. So that's quite a remarkable 377 00:23:22,412 --> 00:23:27,133 thing, that if we reflect light off a 378 00:23:27,133 --> 00:23:32,066 dielectric, that we can -- at one angle, and one angle only, 379 00:23:32,066 --> 00:23:36,414 which is the Brewster angle, that we can turn it into 380 00:23:36,414 --> 00:23:39,759 hundred percent linearly polarized light. 381 00:23:39,759 --> 00:23:42,518 This does not apply to conductors. 382 00:23:42,518 --> 00:23:47,535 The behavior of conductors is very different from dielectrics 383 00:23:47,535 --> 00:23:50,378 such as -- such as water and glass. 384 00:23:50,378 --> 00:23:54,224 You can use Maxwell's equations, 385 00:23:54,224 --> 00:23:57,7 of course, to study the reflection of electromagnetic 386 00:23:57,7 --> 00:24:00,708 waves off metals, but you get a very different 387 00:24:00,708 --> 00:24:02,914 result. And so never expect to get 388 00:24:02,914 --> 00:24:06,19 linearly polarized light which bounces off metals. 389 00:24:06,19 --> 00:24:08,195 I have here, for your pleasure, 390 00:24:08,195 --> 00:24:10,936 a metal sphere, and I have a glass sphere, 391 00:24:10,936 --> 00:24:14,412 and if you still have your linear polarizers at hand, 392 00:24:14,412 --> 00:24:18,556 uh, you can now -- or a little later -- just hold them in front 393 00:24:18,556 --> 00:24:21,297 of your eyes, and rotate them around -- of 394 00:24:21,297 --> 00:24:26,176 course, you're not seeing light at the Brewster angle, 395 00:24:26,176 --> 00:24:30,47 the chances are that some of the light that's reflected off 396 00:24:30,47 --> 00:24:34,541 this glass that you can clearly see that it is partially 397 00:24:34,541 --> 00:24:37,28 polarized. You can see a difference in 398 00:24:37,28 --> 00:24:39,649 light intensity as you rotate it. 399 00:24:39,649 --> 00:24:42,832 You're not going to see that off this metal. 400 00:24:42,832 --> 00:24:45,942 Now, I come to a third way of polarization. 401 00:24:45,942 --> 00:24:49,791 There is a third way that we can make hundred percent 402 00:24:49,791 --> 00:24:53,788 polarized light by the scattering of unpolarized light, 403 00:24:53,788 --> 00:24:58,609 and we have to scatter it off very fine particles, 404 00:24:58,609 --> 00:25:02,928 preferably a tenth of a micron. Dust particles would work very 405 00:25:02,928 --> 00:25:04,981 well. Now, the theory of light 406 00:25:04,981 --> 00:25:08,804 scattering is extremely complicated, but I will be able 407 00:25:08,804 --> 00:25:12,769 to convince you that if I scatter the light over an angle 408 00:25:12,769 --> 00:25:15,955 of ninety degrees -- so it comes in like this, 409 00:25:15,955 --> 00:25:19,99 and it scatters over an angle of ninety degrees -- that it 410 00:25:19,99 --> 00:25:22,964 becomes hundred percent linearly polarized. 411 00:25:22,964 --> 00:25:25,158 I will stay on the center board. 412 00:25:25,158 --> 00:25:28,911 Suppose I have light coming in like 413 00:25:28,911 --> 00:25:31,246 so. And I have one light photon, 414 00:25:31,246 --> 00:25:35,766 I concentrate on one to start with, and it happens to be that 415 00:25:35,766 --> 00:25:40,061 that light photon is linearly polarized in this direction, 416 00:25:40,061 --> 00:25:43,375 and so the E-vector is oscillating like this. 417 00:25:43,375 --> 00:25:47,368 Later, we're going to add all directions that we want. 418 00:25:47,368 --> 00:25:50,909 I just picked one now. And here are my find dust 419 00:25:50,909 --> 00:25:54,676 particles, and these dust particles have electrons, 420 00:25:54,676 --> 00:25:58,292 and the electric field passes by, 421 00:25:58,292 --> 00:26:00,656 and these electrons, which are charged, 422 00:26:00,656 --> 00:26:03,144 are going to oscillate in this direction. 423 00:26:03,144 --> 00:26:05,818 They're going to experience an acceleration, 424 00:26:05,818 --> 00:26:09,301 which is the force that they experience, divided by their 425 00:26:09,301 --> 00:26:12,287 mass, and therefore, that is the charge that they 426 00:26:12,287 --> 00:26:15,583 have, times the electric field, divided by their mass. 427 00:26:15,583 --> 00:26:19,066 And so as this electric field vector, this electric field 428 00:26:19,066 --> 00:26:21,865 component, oscillating with frequencies omega, 429 00:26:21,865 --> 00:26:25,473 is passing by these electrons, they themselves are going to 430 00:26:25,473 --> 00:26:30,519 oscillate with frequency omega, and this is the force that they 431 00:26:30,519 --> 00:26:33,267 will experience. Uh, this is the acceleration 432 00:26:33,267 --> 00:26:36,452 they will experience. Notice that the electrons will 433 00:26:36,452 --> 00:26:39,763 experience a way higher acceleration than the protons, 434 00:26:39,763 --> 00:26:43,385 because the protons have a mass which is more than eighteen 435 00:26:43,385 --> 00:26:45,759 hundred times larger than the electron. 436 00:26:45,759 --> 00:26:48,882 So whatever follows, it's really the electrons that 437 00:26:48,882 --> 00:26:52,629 do the job and not the protons. So, we have charges that move 438 00:26:52,629 --> 00:26:56,502 up and down, and now comes the question which we have discussed 439 00:26:56,502 --> 00:26:59,959 earlier, so this is just simply to 440 00:26:59,959 --> 00:27:03,333 refresh your memory, if you're here at point P, 441 00:27:03,333 --> 00:27:07,66 in what direction will you now see electromagnetic radiation 442 00:27:07,66 --> 00:27:11,621 that is produced by charges that are being accelerated? 443 00:27:11,621 --> 00:27:15,655 We discussed that earlier, and we even had a movie about 444 00:27:15,655 --> 00:27:18,148 that. And perhaps you will remember 445 00:27:18,148 --> 00:27:22,109 that the electric field, at point P, is now oscillating 446 00:27:22,109 --> 00:27:25,336 in this direction. A spherical wave goes out, 447 00:27:25,336 --> 00:27:30,699 if I accelerate charges here. And the rules about this 448 00:27:30,699 --> 00:27:34,737 direction of the electric field are very simple. 449 00:27:34,737 --> 00:27:39,635 E is always perpendicular to the direction of propagation, 450 00:27:39,635 --> 00:27:44,962 which is the position vector -- I call this the position vector 451 00:27:44,962 --> 00:27:50,031 R -- and the second rule is that A R and E are in one plane, 452 00:27:50,031 --> 00:27:53,125 and that happens to be, in this case, 453 00:27:53,125 --> 00:27:56,647 the plane of my blackboard. But of course, 454 00:27:56,647 --> 00:28:01,892 that doesn't have to be the plane of the blackboard, 455 00:28:01,892 --> 00:28:06,318 because I can choose my point P in space here and these rules 456 00:28:06,318 --> 00:28:08,679 still apply. The incident photon, 457 00:28:08,679 --> 00:28:11,408 which in this case, I picked only one, 458 00:28:11,408 --> 00:28:15,17 is completely destroyed. It is absorbed by the dust. 459 00:28:15,17 --> 00:28:18,416 And the electrons which are going to radiate, 460 00:28:18,416 --> 00:28:21,957 reradiate a photon at exactly the same frequency, 461 00:28:21,957 --> 00:28:26,309 because if this is oscillating with angular frequency omega, 462 00:28:26,309 --> 00:28:31,915 then the acceleration would be with angular frequency omega, 463 00:28:31,915 --> 00:28:35,658 and so this E field will have angular frequency omega. 464 00:28:35,658 --> 00:28:39,683 So it really is as if the photon comes in and takes off in 465 00:28:39,683 --> 00:28:42,861 a different direction, that's why we call this 466 00:28:42,861 --> 00:28:45,615 scattering. So the frequency remains the 467 00:28:45,615 --> 00:28:49,711 same, but the direction changes. And the probably that that 468 00:28:49,711 --> 00:28:53,807 photon will go in this direction or this direction is zero, 469 00:28:53,807 --> 00:28:57,408 because you remember that no electromagnetic wave is 470 00:28:57,408 --> 00:29:00,727 propagated in the direction of the acceleration, 471 00:29:00,727 --> 00:29:04,328 there's a high probability that it 472 00:29:04,328 --> 00:29:07,814 goes out in this plane, perpendicular to A, 473 00:29:07,814 --> 00:29:12,047 and at this angle theta, the probability is a little 474 00:29:12,047 --> 00:29:14,702 lower. We discussed that earlier. 475 00:29:14,702 --> 00:29:19,516 Now, I'm going to convince you why light that scatters over 476 00:29:19,516 --> 00:29:24,329 ninety degrees that comes in like this, and then [wssshhht] 477 00:29:24,329 --> 00:29:27,649 comes to you, why that is hundred percent 478 00:29:27,649 --> 00:29:31,217 linearly polarized. Here is a beam of light. 479 00:29:31,217 --> 00:29:35,615 And this beam of light is unpolarized. 480 00:29:35,615 --> 00:29:39,564 That means, if I look down on this beam, you have individual 481 00:29:39,564 --> 00:29:43,512 photons, and I described through those -- maybe a little bit 482 00:29:43,512 --> 00:29:45,653 artificially, but I will do that, 483 00:29:45,653 --> 00:29:49,668 it was successful with Malus' Law -- I describe through them, 484 00:29:49,668 --> 00:29:52,145 individual directions of polarization. 485 00:29:52,145 --> 00:29:54,888 So each photon, on its own, has a uniquely 486 00:29:54,888 --> 00:29:57,097 defined direction of polarization. 487 00:29:57,097 --> 00:30:00,978 It is a picture that is not kosher, you really need quantum 488 00:30:00,978 --> 00:30:04,257 mechanics to do this right, but on the other hand, 489 00:30:04,257 --> 00:30:08,054 the result that you find is probably 490 00:30:08,054 --> 00:30:10,802 correct. You will find the same result 491 00:30:10,802 --> 00:30:13,847 if you did it in a quantum mechanical way. 492 00:30:13,847 --> 00:30:18,229 So here, you have these dust particles, and so what is going 493 00:30:18,229 --> 00:30:20,68 to happen is, this light comes in, 494 00:30:20,68 --> 00:30:24,245 and then [wssshhht], one may be scattered in this 495 00:30:24,245 --> 00:30:28,107 direction, [wssshhht], another one in this direction, 496 00:30:28,107 --> 00:30:30,483 another one in forward direction. 497 00:30:30,483 --> 00:30:33,9 And you can go in all directions as you please. 498 00:30:33,9 --> 00:30:37,761 I'm now going to look in the plane 499 00:30:37,761 --> 00:30:40,899 perpendicular to the blackboard, because in the plane 500 00:30:40,899 --> 00:30:44,699 perpendicular to the blackboard, every photon that ends up there 501 00:30:44,699 --> 00:30:47,715 is at ninety degrees angles. It comes in like this, 502 00:30:47,715 --> 00:30:50,491 that is ninety degrees, this is ninety degrees, 503 00:30:50,491 --> 00:30:53,567 this is also ninety degrees. And here is that plane. 504 00:30:53,567 --> 00:30:56,523 I just draw a circle, but there is nothing special 505 00:30:56,523 --> 00:30:59,178 about the circle. And so the photons come now 506 00:30:59,178 --> 00:31:01,349 straight to you, perpendicular to the 507 00:31:01,349 --> 00:31:03,823 blackboard. That's the picture that I have 508 00:31:03,823 --> 00:31:07,08 now in mind. And let's say you were 509 00:31:07,08 --> 00:31:09,557 looking here. So you were sitting there, 510 00:31:09,557 --> 00:31:13,303 so you have to lift this up, so you're really sitting there. 511 00:31:13,303 --> 00:31:16,796 So this is where you are. And let's take one photon that 512 00:31:16,796 --> 00:31:20,288 comes in, which happens to be linearly polarized in this 513 00:31:20,288 --> 00:31:22,13 direction. We pick one photon, 514 00:31:22,13 --> 00:31:25,622 but it's unpolarized light because we're going to add so 515 00:31:25,622 --> 00:31:28,226 many photons that, on average, there is no 516 00:31:28,226 --> 00:31:31,337 preferred direction. But I pick one to start with. 517 00:31:31,337 --> 00:31:34,576 And now I ask myself the question, if that photon is 518 00:31:34,576 --> 00:31:37,497 scattered in your direction, in this direction, 519 00:31:37,497 --> 00:31:41,056 how is the E-vector oscillating here? 520 00:31:41,056 --> 00:31:44,175 So this is now the position vector R, and this is the 521 00:31:44,175 --> 00:31:47,354 direction A in which the electrons are going to shake, 522 00:31:47,354 --> 00:31:50,712 because the photon comes in, with an E field shaking like 523 00:31:50,712 --> 00:31:53,771 this, so the electrons are going to shake like this. 524 00:31:53,771 --> 00:31:56,77 And you will immediately conclude that the electric 525 00:31:56,77 --> 00:31:59,229 vector here must be oscillating like this. 526 00:31:59,229 --> 00:32:00,669 Why? Because it has to be 527 00:32:00,669 --> 00:32:02,528 perpendicular to R, which it is, 528 00:32:02,528 --> 00:32:04,927 and it has to be in the plane of A and R. 529 00:32:04,927 --> 00:32:08,166 There's only one solution, and then this is the correct 530 00:32:08,166 --> 00:32:11,224 solution. Now there is a next 531 00:32:11,224 --> 00:32:14,314 photon coming in. And the next photon -- I will 532 00:32:14,314 --> 00:32:17,268 give it color, just to distinguish the two -- 533 00:32:17,268 --> 00:32:20,156 happens to be oscillating in this direction. 534 00:32:20,156 --> 00:32:24,185 And I ask myself the question, if that photon is scattered in 535 00:32:24,185 --> 00:32:26,805 your direction, in what direction is the 536 00:32:26,805 --> 00:32:30,162 E-field oscillating? And you'll come to exactly the 537 00:32:30,162 --> 00:32:32,446 same conclusion, in this direction. 538 00:32:32,446 --> 00:32:34,057 Why? Because it has to be 539 00:32:34,057 --> 00:32:36,139 perpendicular to R, which it is, 540 00:32:36,139 --> 00:32:40,37 and it has to be in the plane A and R, that's the only solution. 541 00:32:40,37 --> 00:32:44,734 A little later, there is another photon that 542 00:32:44,734 --> 00:32:47,166 comes in. How is that electric field 543 00:32:47,166 --> 00:32:50,572 being observed here? Of course, in this direction. 544 00:32:50,572 --> 00:32:54,256 And so, no matter how they come in, unpolarized light, 545 00:32:54,256 --> 00:32:58,565 you will always see that if the photon is scattered over ninety 546 00:32:58,565 --> 00:33:02,666 degrees, you will always see it polarized in this direction, 547 00:33:02,666 --> 00:33:05,377 and therefore, you have created linearly 548 00:33:05,377 --> 00:33:07,948 polarized light. So if you watch here, 549 00:33:07,948 --> 00:33:11,91 that is at ninety degrees angle -- or if you watch here at 550 00:33:11,91 --> 00:33:15,795 ninety degrees angles -- but of course, 551 00:33:15,795 --> 00:33:19,725 it's a whole plane -- then you end up with hundred percent 552 00:33:19,725 --> 00:33:23,242 linearly polarized light. If you go through the same 553 00:33:23,242 --> 00:33:25,379 exercise, say, at an angle here, 554 00:33:25,379 --> 00:33:28,137 of forty five degrees, and you look here, 555 00:33:28,137 --> 00:33:32,275 it's only partially polarized. Indeed, if you rotate in front 556 00:33:32,275 --> 00:33:35,998 of your eye the polarimeter, you will see clearly light 557 00:33:35,998 --> 00:33:38,825 intensity changes. But not hundred percent 558 00:33:38,825 --> 00:33:41,238 polarized. You couldn't turn it into 559 00:33:41,238 --> 00:33:43,859 darkness. And if you look from above -- 560 00:33:43,859 --> 00:33:47,276 and you may want to go through that 561 00:33:47,276 --> 00:33:50,171 exercise for yourself, you will see that the light 562 00:33:50,171 --> 00:33:53,833 remains completely unpolarized. So it is only the ninety-degree 563 00:33:53,833 --> 00:33:57,083 angle that is very special. And that's what I'm going to 564 00:33:57,083 --> 00:33:59,977 demonstrate to you. But before I demonstrate this, 565 00:33:59,977 --> 00:34:03,581 there is something that I have to tell you, that I cannot hide 566 00:34:03,581 --> 00:34:05,767 from you, I wish I could, but I can't. 567 00:34:05,767 --> 00:34:08,957 It's not something that is my goal during this lecture, 568 00:34:08,957 --> 00:34:11,202 it has nothing to do with polarization. 569 00:34:11,202 --> 00:34:13,624 But that's the fact, that the probability, 570 00:34:13,624 --> 00:34:17,077 when you scatter light off very small 571 00:34:17,077 --> 00:34:20,416 particles, a tenth of a micron or so, dust particles, 572 00:34:20,416 --> 00:34:23,691 that the probability of scattering is way higher for 573 00:34:23,691 --> 00:34:26,003 blue light than it is for blue light. 574 00:34:26,003 --> 00:34:29,47 The shorter the wavelength, the higher the probability. 575 00:34:29,47 --> 00:34:32,874 If you ever take eight of three, you're going to see a 576 00:34:32,874 --> 00:34:35,764 derivation of that, a quantitative derivation. 577 00:34:35,764 --> 00:34:38,974 Blue light has a ten times higher probability to be 578 00:34:38,974 --> 00:34:42,571 scattered than red light. And so whenever I'm going to do 579 00:34:42,571 --> 00:34:46,681 my scatter experiments on very fine particles, 580 00:34:46,681 --> 00:34:49,99 you will see that that light is going to be bluish. 581 00:34:49,99 --> 00:34:53,367 You can't miss that. Now, if the particles off which 582 00:34:53,367 --> 00:34:56,346 I scatter are larger than a tenth of a micron, 583 00:34:56,346 --> 00:34:58,861 say one micron, this effect of color on 584 00:34:58,861 --> 00:35:01,708 probability of scattering is highly reduced, 585 00:35:01,708 --> 00:35:05,746 and if I scatter off very large particles -- ten microns or so 586 00:35:05,746 --> 00:35:08,791 -- then there is no dependence any more at all. 587 00:35:08,791 --> 00:35:12,763 And in this lies the secret why the sky is blue -- I will get 588 00:35:12,763 --> 00:35:17,992 back to that later today -- the reason why cigarette smoke can 589 00:35:17,992 --> 00:35:20,918 be blue, if the smoke particles are very small, 590 00:35:20,918 --> 00:35:23,461 and it's the reason why clouds are white. 591 00:35:23,461 --> 00:35:27,022 Because the sunlight hits the clouds, the light scatters, 592 00:35:27,022 --> 00:35:30,392 but the water drops in the clouds are not oh point one 593 00:35:30,392 --> 00:35:33,953 microns, but they are much larger, they are more like ten 594 00:35:33,953 --> 00:35:36,688 microns and up, and so there is no preferred 595 00:35:36,688 --> 00:35:40,121 wavelength that scatters, and so the clouds look white. 596 00:35:40,121 --> 00:35:43,937 And so the first demonstration that I want to do is very much 597 00:35:43,937 --> 00:35:47,243 like what you see here. I'm going to send unpolarized 598 00:35:47,243 --> 00:35:50,168 light up here, straight up. 599 00:35:50,168 --> 00:35:53,926 We have bright spotlights, and the light goes straight up. 600 00:35:53,926 --> 00:35:57,289 In here, I'm going to put very small dust particles. 601 00:35:57,289 --> 00:36:01,179 And I have decided to do that smoke, simply cigarette smoke. 602 00:36:01,179 --> 00:36:04,541 So I'm going to hold cigarette smoke in these beams, 603 00:36:04,541 --> 00:36:08,497 and the light that will come to you, no matter where you sit, 604 00:36:08,497 --> 00:36:11,596 must have scattered closely over ninety degrees, 605 00:36:11,596 --> 00:36:13,442 right? It comes up like this, 606 00:36:13,442 --> 00:36:16,343 but if you see it, almost for everyone in the 607 00:36:16,343 --> 00:36:20,166 audience, ninety degree angle scattering. 608 00:36:20,166 --> 00:36:27,689 So with your linear polarizers, you will be able to see that 609 00:36:27,689 --> 00:36:34,574 that light is polarized, and it's going to be polarized 610 00:36:34,574 --> 00:36:40,567 in this direction, which is the direction that I 611 00:36:40,567 --> 00:36:45,667 have here. And that's the first thing I'm 612 00:36:45,667 --> 00:36:50,129 going to do with this demonstration. 613 00:36:50,129 --> 00:36:55,484 And so I need cigarettes, and I need smoke. 614 00:36:55,484 --> 0. As much as I hate this. 615 0. --> 00:36:58,544 If this light that comes in is Ah, ready for this? I want you to see two things. 616 00:36:58,544 --> 00:37:00,329 [laughter]. OK. 617 00:37:00,329 --> 0. That should do. 618 0. --> 00:37:06,449 If this light that comes in is Ah, ready for this? I want you to see two things. 619 00:37:06,449 --> 00:37:10,191 [laughter]. OK. 620 00:37:10,191 --> 00:37:22,754 So, you need a lot of light, and as light comes, 621 00:37:22,754 --> 00:37:32,911 presumably from here yes, there it is. 622 00:37:32,911 --> 0. 623 0. --> 00:37:44,137 If this light that comes in is Ah, ready for this? I want you to see two things. 624 00:37:44,137 --> 00:37:48,27 OK, so have your polarizers ready. 625 00:37:48,27 --> 0. 626 0. --> 00:37:52,027 If this light that comes in is Ah, ready for this? I want you to see two things. 627 00:37:52,027 --> 00:37:58,539 Number one, that the light is bluish, and number two, 628 00:37:58,539 --> 00:38:04,175 that it is polarized. Take your time for that. 629 00:38:04,175 --> 00:38:11,064 If you don't see it as blue, then the reason for that is 630 00:38:11,064 --> 00:38:19,565 that at low-light intensities, your eyes are not very 631 00:38:19,565 --> 00:38:26,016 sensitive for color any more. It looks quite bluish to me, 632 00:38:26,016 --> 0. though. 633 0. --> 00:38:26,922 If this light that comes in is Ah, ready for this? I want you to see two things. 634 00:38:26,922 --> 00:38:31,223 Now I want to do something in addition. 635 00:38:31,223 --> 00:38:36,429 I mentioned that if the particles grow in size, 636 00:38:36,429 --> 00:38:42,54 that the scattering is no longer preferred in the blue. 637 00:38:42,54 --> 00:38:50,424 And I can demonstrate that. I can kill two birds with one 638 00:38:50,424 --> 00:38:53,84 stone. What I can do is I can hold the 639 00:38:53,84 --> 00:38:58,365 smoke in my lungs for a while, and when I do that, 640 00:38:58,365 --> 00:39:03,72 the the water vapor in my lungs will precipitate on these 641 00:39:03,72 --> 00:39:07,875 dust particles, and they will form small water 642 00:39:07,875 --> 00:39:10,737 drops. And when I puff that out, 643 00:39:10,737 --> 00:39:15,815 you will see a distinct difference in color between what 644 00:39:15,815 --> 00:39:21,686 you see now, and the smoke that comes out of my 645 00:39:21,686 --> 00:39:25,061 lungs when the particles are ten microns and even larger. 646 00:39:25,061 --> 00:39:27,23 You will see, then, that the light is 647 00:39:27,23 --> 00:39:29,581 whitish. So this is -- this comes extra, 648 00:39:29,581 --> 00:39:31,57 over and above, it comes for free. 649 00:39:31,57 --> 00:39:34,945 In order to make you see the difference, shortly before I 650 00:39:34,945 --> 00:39:38,139 puff out the smoke in here, I will again show you this 651 00:39:38,139 --> 00:39:41,092 smoke as it is now, so you can compare the colors, 652 00:39:41,092 --> 00:39:43,684 and you will see that there is a difference. 653 00:39:43,684 --> 00:39:47,24 Even though this doesn't -- this may not look very bluish to 654 00:39:47,24 --> 00:39:50,494 you, for reasons that I mentioned, 655 00:39:50,494 --> 00:39:57,54 in darkness, you don't have a very good 656 00:39:57,54 --> 00:40:06,996 sensitive for color. So I'm going to hold this smoke 657 00:40:06,996 --> 00:40:16,637 now -- as much as I hate it, this is one of the worst 658 00:40:16,637 --> 00:40:28,133 demonstrations that I have to do -- I hold it in my lungs for a 659 00:40:28,133 --> 0. while. 660 0. --> 00:40:32,583 If this light that comes in is Ah, ready for this? I want you to see two things. 661 00:40:32,583 --> 00:40:37,605 I see a huge difference, but I'm very close. 662 00:40:37,605 --> 00:40:44,263 The second puff [huh] [whew] was very wide compared to the 663 00:40:44,263 --> 00:40:48,702 first one. So we were able to catch two 664 00:40:48,702 --> 00:40:55,36 birds with one stone here. The sky is blue because of this 665 00:40:55,36 --> 00:40:59,215 phenomenon. Here, you are standing 666 00:40:59,215 --> 00:41:06,139 innocently on the Earth. And sunlight is coming in, 667 00:41:06,139 --> 00:41:09,43 onto the Earth's atmosphere. The sun is there. 668 00:41:09,43 --> 00:41:12,428 Sunlight comes in, and the light scatters. 669 00:41:12,428 --> 00:41:16,377 And the light that reaches you, that scatters off these 670 00:41:16,377 --> 00:41:20,618 extremely fine dust particles, and it also scatters off the 671 00:41:20,618 --> 00:41:24,64 air molecules themselves. There are thermal fluctuations 672 00:41:24,64 --> 00:41:27,858 that go on all the time, which causes density 673 00:41:27,858 --> 00:41:31,807 fluctuations in the air, and they are sufficient to act 674 00:41:31,807 --> 00:41:36,062 as scatterers. And so if light from here comes 675 00:41:36,062 --> 00:41:40,253 to you, the chances are that it is blue, because it has a higher 676 00:41:40,253 --> 00:41:43,578 probability than red. And this is also likely to be 677 00:41:43,578 --> 00:41:45,707 blue. And so when you look at the 678 00:41:45,707 --> 00:41:48,5 sky, the sky looks blue, that's the reason, 679 00:41:48,5 --> 00:41:52,158 it has to do with this strong preference for color to be 680 00:41:52,158 --> 00:41:56,348 scattered when it is blue light. If you look in the direction of 681 00:41:56,348 --> 00:42:00,339 the sky at an angle of ninety degrees to the direction of the 682 00:42:00,339 --> 00:42:04,263 sun, the sky is also linearly polarized for the reasons that 683 00:42:04,263 --> 00:42:07,603 we now understand, because there is 684 00:42:07,603 --> 00:42:11,151 scattering over ninety degrees. And so when the sun is out 685 00:42:11,151 --> 00:42:14,948 there, there is always a whole plane, great circle in the sky, 686 00:42:14,948 --> 00:42:17,501 which is ninety degrees away from the sun. 687 00:42:17,501 --> 00:42:20,426 So when you come out with your linear polarizes, 688 00:42:20,426 --> 00:42:23,29 as soon as the weather clears, look at the sky, 689 00:42:23,29 --> 00:42:26,029 at the blue sky, and look ninety degrees away 690 00:42:26,029 --> 00:42:28,644 from the sun, and you will see that the sky 691 00:42:28,644 --> 00:42:32,254 is very strongly polarized. If you look at angles different 692 00:42:32,254 --> 00:42:36,239 from ninety degrees, it's partially polarized. 693 00:42:36,239 --> 00:42:40,093 But not as strongly polarized as it is at ninety degrees. 694 00:42:40,093 --> 00:42:42,778 So this explains, in a very natural way, 695 00:42:42,778 --> 00:42:44,567 why the sun, when it rises, 696 00:42:44,567 --> 00:42:46,563 and why the sun, when it sets, 697 00:42:46,563 --> 00:42:50,005 why the sun is red. Because if the sun rises or the 698 00:42:50,005 --> 00:42:52,552 sun sets, the sun is near the horizon. 699 00:42:52,552 --> 00:42:55,168 So now the sunlight comes in like this. 700 00:42:55,168 --> 00:42:58,747 Imagine how much atmosphere it has to travel through, 701 00:42:58,747 --> 00:43:02,739 how many scattering particles it will encounter on the way. 702 00:43:02,739 --> 00:43:06,731 And so, right here there is scattering. 703 00:43:06,731 --> 00:43:08,77 That is blue light, that is blue light, 704 00:43:08,77 --> 00:43:10,809 that is blue light, that is blue light, 705 00:43:10,809 --> 00:43:13,438 that has a higher probability, that is blue light. 706 00:43:13,438 --> 00:43:15,638 So what do you think is left over for you? 707 00:43:15,638 --> 00:43:17,301 There isn't very much left over. 708 00:43:17,301 --> 00:43:19,179 What is left over, the blue is gone, 709 00:43:19,179 --> 00:43:21,593 the green is gone, so if there's anything left 710 00:43:21,593 --> 00:43:24,007 over, it is red. And it is that red light that 711 00:43:24,007 --> 00:43:25,939 you see. That's why the sun looks red 712 00:43:25,939 --> 00:43:28,3 when it sets, and it looks red when it rises, 713 00:43:28,3 --> 00:43:30,768 the same is true for planets, and bright stars, 714 00:43:30,768 --> 00:43:32,914 and the moon. When they're just above the 715 00:43:32,914 --> 00:43:36,347 horizon, they look very reddish. And if 716 00:43:36,347 --> 00:43:39,288 there happens to be a cloud here in the sky, 717 00:43:39,288 --> 00:43:42,296 well, the cloud will also see that red light, 718 00:43:42,296 --> 00:43:45,989 so you get a red cloud. And that's exactly what you see 719 00:43:45,989 --> 00:43:48,45 at sunset, all these clouds turn red. 720 00:43:48,45 --> 00:43:51,459 And so that is, again, the consequence of the 721 00:43:51,459 --> 00:43:55,493 fact that the probability for light scattering of blue light 722 00:43:55,493 --> 00:43:58,775 is ten times larger, roughly, that the scattering 723 00:43:58,775 --> 00:44:01,647 for red light. So it explains both the blue 724 00:44:01,647 --> 00:44:05,613 skies and the reason why the sky -- why the sunrise and the 725 00:44:05,613 --> 00:44:10,331 sunsets are -- are red. I can show you two slides, 726 00:44:10,331 --> 00:44:14,424 whereby you do see this phenomenon, that light that 727 00:44:14,424 --> 00:44:19,172 scatters to you becomes bluish. We can see it in astronomy, 728 00:44:19,172 --> 00:44:23,101 and the first slide is a picture of the Pleiades, 729 00:44:23,101 --> 00:44:26,375 called the Seven Sisters, very hot stars, 730 00:44:26,375 --> 00:44:29,977 they are surrounded by very small, fine dust, 731 00:44:29,977 --> 00:44:34,889 and the light that reaches you is not only polarized -- which 732 00:44:34,889 --> 00:44:38,573 you cannot see, of course, on the slide -- but 733 00:44:38,573 --> 00:44:42,903 it's also bluish. And so let's take a look at 734 00:44:42,903 --> 00:44:45,437 that first. So here you see the Pleiades, 735 00:44:45,437 --> 00:44:49,175 it's called the Seven Sisters -- some people think there are 736 00:44:49,175 --> 00:44:52,659 only seven stars in there, but there are hundreds -- and 737 00:44:52,659 --> 00:44:56,587 you see here these very bright stars, and here you see the dust 738 00:44:56,587 --> 00:44:59,881 surrounding these stars, and this is distinctly blue. 739 00:44:59,881 --> 00:45:02,985 That's the effect, the fact that short wavelengths 740 00:45:02,985 --> 00:45:06,976 have a higher probability to be scattered than long wavelengths. 741 00:45:06,976 --> 00:45:10,081 So the blue has a higher probability than the red. 742 00:45:10,081 --> 00:45:14,199 And the next slide shows you a man on the moon. 743 00:45:14,199 --> 00:45:18,844 And this person is walking on the moon, but as he walks on the 744 00:45:18,844 --> 00:45:21,814 moon, he produces dust, by just walking, 745 00:45:21,814 --> 00:45:26,003 very fine dust particles that come from the soil that he 746 00:45:26,003 --> 00:45:28,364 brings up. And what he is doing, 747 00:45:28,364 --> 00:45:32,552 he is creating around himself sort of a blue atmosphere, 748 00:45:32,552 --> 00:45:36,589 a blue sky, because the sunlight comes from the right, 749 00:45:36,589 --> 00:45:41,158 and so the light that you see that comes in your direction is 750 00:45:41,158 --> 00:45:43,138 being scattered, of course. 751 00:45:43,138 --> 00:45:47,175 Because there is no air on the moon, 752 00:45:47,175 --> 00:45:50,812 so that can only be the dust that he has produced, 753 00:45:50,812 --> 00:45:53,336 and you see that that dust is blue. 754 00:45:53,336 --> 00:45:56,603 It would probably also be strongly polarized, 755 00:45:56,603 --> 00:45:59,721 because if it changes ninety degrees angle, 756 00:45:59,721 --> 00:46:02,765 then, of course, it would also be strongly 757 00:46:02,765 --> 00:46:04,843 polarized. Now I want to do a 758 00:46:04,843 --> 00:46:06,848 demonstration, the last one, 759 00:46:06,848 --> 00:46:10,411 which catches more than two birds with one stone, 760 00:46:10,411 --> 00:46:14,123 it's going to kill three. I have here a bucket with 761 00:46:14,123 --> 00:46:17,241 thiosulfate. And we have lights coming from 762 00:46:17,241 --> 00:46:22,006 this side which we s- shine through that bucket. 763 00:46:22,006 --> 00:46:25,869 And we're going to put a little bit of sulfuric acid in there, 764 00:46:25,869 --> 00:46:28,655 and when we do that, sulfur will precipitate, 765 00:46:28,655 --> 00:46:32,264 small particles of sulfur. These small particles of sulfur 766 00:46:32,264 --> 00:46:35,114 will become the scatterers, and so this light, 767 00:46:35,114 --> 00:46:38,597 which is unpolarized white light, will begin to scatter. 768 00:46:38,597 --> 00:46:41,953 And you're going to see it. If you're sitting at right 769 00:46:41,953 --> 00:46:45,246 angles, then you will see that that light is linearly 770 00:46:45,246 --> 00:46:48,665 polarized, you can enjoy that. If you're sitting there, 771 00:46:48,665 --> 00:46:52,973 you're not so well off, because then the light is not 772 00:46:52,973 --> 00:46:55,746 at ninety degrees. But you will see it partially 773 00:46:55,746 --> 00:46:57,752 polarized. If you're sitting there, 774 00:46:57,752 --> 00:47:00,643 it's not at ninety degrees either, you will see it 775 00:47:00,643 --> 00:47:03,593 partially polarized. But you will also see it blue, 776 00:47:03,593 --> 00:47:07,016 because the probability that blue light scattered is higher 777 00:47:07,016 --> 00:47:09,258 than red light. So you're going to see, 778 00:47:09,258 --> 00:47:12,739 slowly in front of your eye, a blue sky is going to develop. 779 00:47:12,739 --> 00:47:16,338 It's going to be polarized in the vertical direction for those 780 00:47:16,338 --> 00:47:19,996 people who are sitting at right angles, partially polarized for 781 00:47:19,996 --> 00:47:23,03 the rest of the audience, 782 00:47:23,03 --> 00:47:28,867 and then we're going to look at the light that remains after the 783 00:47:28,867 --> 00:47:32,11 light had penetrated the atmosphere. 784 00:47:32,11 --> 00:47:35,817 After the blue light is slowly exhausted. 785 00:47:35,817 --> 00:47:39,338 All right. I will also take a polarizer 786 00:47:39,338 --> 00:47:44,434 with me, so I can also show you, then, the polarization. 787 00:47:44,434 --> 00:47:47,955 Let me first put that sulfuric acid in. 788 00:47:47,955 --> 00:47:54,163 I will do that now, so that I know that I OK. 789 00:47:54,163 --> 00:47:58,57 So very slowly am I creating, now, my own atmosphere. 790 00:47:58,57 --> 00:48:01,961 And I'm going to make it completely dark. 791 00:48:01,961 --> 00:48:06,623 And so you see white light, keep an eye on the -- on the 792 00:48:06,623 --> 00:48:11,2 bucket, the bucket cannot be seen by all of us so well, 793 00:48:11,2 --> 00:48:14,93 depending upon where you sit in the audience, 794 00:48:14,93 --> 00:48:20,1 but I can already begin to see that it turned slightly bluish. 795 00:48:20,1 --> 00:48:24,253 Of course, I have -- I'm very close. 796 00:48:24,253 --> 00:48:27,393 I admit I'm very close, which is very good. 797 00:48:27,393 --> 00:48:31,132 It's slightly bluish, more and more sulfur is going 798 00:48:31,132 --> 00:48:35,094 to precipitate as time goes on, we have to be patient. 799 00:48:35,094 --> 00:48:38,533 Oh, boy, it's bluish. I will show you that it's 800 00:48:38,533 --> 00:48:41,523 polarized. I will rotate in front of it a 801 00:48:41,523 --> 00:48:45,71 polarimeter, and you see, there's a distinct polarization 802 00:48:45,71 --> 00:48:48,626 of this light, if it comes out at ninety 803 00:48:48,626 --> 00:48:50,943 degrees. More and more sulfur is 804 00:48:50,943 --> 00:48:54,158 precipitating, and look at the sun -- if you 805 00:48:54,158 --> 00:48:58,504 think of this as the sun. The sun is, um, 806 00:48:58,504 --> 00:49:02,518 is getting a little yellowish, it's not so white any more. 807 00:49:02,518 --> 00:49:05,757 And you wonder why. Well, you should be able to 808 00:49:05,757 --> 00:49:08,714 answer that, now. Because the light that is 809 00:49:08,714 --> 00:49:13,08 scattered in the atmosphere -- I call this the atmosphere -- is 810 00:49:13,08 --> 00:49:15,333 blue. It has a higher probability 811 00:49:15,333 --> 00:49:18,15 than red. And so what is left over in the 812 00:49:18,15 --> 00:49:22,234 direction that you see on the screen there is the remaining 813 00:49:22,234 --> 00:49:24,417 light. And the more blue leaves, 814 00:49:24,417 --> 00:49:29,671 and the more green leaves, the redder the sun is going to 815 00:49:29,671 --> 00:49:31,754 be. And if we're going to be a 816 00:49:31,754 --> 00:49:36,064 little bit more patient -- and we certainly have the time for 817 00:49:36,064 --> 00:49:40,303 that -- you're going to see the sun getting pretty -- pretty 818 00:49:40,303 --> 00:49:42,602 bloody red. So keep an eye on it, 819 00:49:42,602 --> 00:49:46,337 also have your linear polarizers, and try to see that 820 00:49:46,337 --> 00:49:50,863 the light that comes to you from the atmosphere -- if you are at 821 00:49:50,863 --> 00:49:54,527 ninety degree angles, I will once more do it with my 822 00:49:54,527 --> 00:49:58,813 polarizers -- that that's strongly polarized. 823 00:49:58,813 --> 00:50:01,987 Oh, boy, look at the sun. We're getting close, 824 00:50:01,987 --> 00:50:03,75 we're getting close. Whew. 825 00:50:03,75 --> 00:50:06,994 Imagine you're now on the beach, very romantic, 826 00:50:06,994 --> 00:50:09,957 and there is the sun, you're with a friend, 827 00:50:09,957 --> 00:50:13,131 you don't have your polari- well, from now on, 828 00:50:13,131 --> 00:50:16,375 you should always have your polarizer with you, 829 00:50:16,375 --> 00:50:18,985 of course. You can really impress your 830 00:50:18,985 --> 00:50:21,313 friend, believe me. For one thing, 831 00:50:21,313 --> 00:50:25,686 you can point out that the sky at ninety-degree angles from the 832 00:50:25,686 --> 00:50:29,365 sun is nearly a hundred percent 833 00:50:29,365 --> 00:50:32,364 polarized. You can even tell your friend 834 00:50:32,364 --> 00:50:35,901 why the sky is blue. And as you experience this 835 00:50:35,901 --> 00:50:39,284 romantic sunset, you can also explain why the 836 00:50:39,284 --> 00:50:43,052 sun is getting red, because all that blue light -- 837 00:50:43,052 --> 00:50:46,897 and the green light -- gets out first, and this is, 838 00:50:46,897 --> 00:50:51,126 then, what is left over. And the sun is really beautiful 839 00:50:51,126 --> 00:50:55,047 now, I already feel these butterflies in my stomach, 840 00:50:55,047 --> 00:50:58,584 and ants in my pants, this is sunset all right? 841 00:50:58,584 --> 00:51:01,766 [sniffles] This is very, 842 00:51:01,766 --> 00:51:03,745 very nice sunset. Oh, man. 843 00:51:03,745 --> 00:51:06,594 What a beautiful sunset. Yes, indeed, 844 00:51:06,594 --> 00:51:09,205 indeed! We are approaching sunset! 845 00:51:09,205 --> 51:14 OK, see you Wednesday. [applause]