1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high-quality educational resources for free. 5 00:00:10,120 --> 00:00:12,690 To make a donation, or to view additional materials 6 00:00:12,690 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,536 at ocw.MIT.edu. 8 00:00:21,086 --> 00:00:26,020 COLIN SHEPPARD: Right, everyone, we're going to start now. 9 00:00:26,020 --> 00:00:29,420 So my name is Colin Sheppard, and I'm going 10 00:00:29,420 --> 00:00:31,510 to be giving the lecture today. 11 00:00:31,510 --> 00:00:36,880 George is here to keep me in order, 12 00:00:36,880 --> 00:00:39,580 and I think, probably, he'll come up 13 00:00:39,580 --> 00:00:42,570 with some comments at times, maybe-- 14 00:00:42,570 --> 00:00:45,070 I hope, anyway. 15 00:00:45,070 --> 00:00:51,880 So he started off by saying that there's no real announcements 16 00:00:51,880 --> 00:00:53,770 except that he's changed-- 17 00:00:53,770 --> 00:00:56,710 he's uploaded some revised notes, apparently. 18 00:01:02,557 --> 00:01:05,140 GEORGE BARBASTATHIS: There were some minor error in the notes, 19 00:01:05,140 --> 00:01:05,980 so I have-- 20 00:01:05,980 --> 00:01:07,570 I have fixed them. 21 00:01:07,570 --> 00:01:09,500 And the current versions should be-- 22 00:01:09,500 --> 00:01:13,810 the one that's in the website now is the corrected ones. 23 00:01:13,810 --> 00:01:17,680 COLIN SHEPPARD: All right, so only minor changes, I think. 24 00:01:17,680 --> 00:01:20,590 OK, so I'm going to try and take off 25 00:01:20,590 --> 00:01:24,040 from where George left it, which was 26 00:01:24,040 --> 00:01:27,510 with Maxwell's equations and the derivation of the wave 27 00:01:27,510 --> 00:01:28,010 equation. 28 00:01:33,820 --> 00:01:37,870 It's quite nice to see this derivation of the wave equation 29 00:01:37,870 --> 00:01:39,730 starting from Maxwell's equations, 30 00:01:39,730 --> 00:01:42,940 because it brings everything together and allows 31 00:01:42,940 --> 00:01:45,250 you to see how the-- 32 00:01:45,250 --> 00:01:49,570 these different areas of physics are interrelated. 33 00:01:49,570 --> 00:01:52,750 But normally, of course, in optics, we 34 00:01:52,750 --> 00:01:56,710 don't usually go to the electromagnetic-type theory, 35 00:01:56,710 --> 00:01:59,380 so we usually get as far as the wave equation. 36 00:01:59,380 --> 00:02:02,230 We show the existence of simple forms 37 00:02:02,230 --> 00:02:07,030 like plane waves and spherical waves. 38 00:02:07,030 --> 00:02:10,180 We carry on then using those plane waves and spherical 39 00:02:10,180 --> 00:02:14,070 waves in a simplified form. 40 00:02:14,070 --> 00:02:16,900 But anyway, so back to Maxwell's equations. 41 00:02:16,900 --> 00:02:19,900 These were the Maxwell's equations. 42 00:02:19,900 --> 00:02:27,400 George, before, explained these and tried to-- 43 00:02:27,400 --> 00:02:30,610 I'm not going to go through the meaning of them, 44 00:02:30,610 --> 00:02:33,400 again, because I think you've got that. 45 00:02:33,400 --> 00:02:35,410 But in the differential form, you've 46 00:02:35,410 --> 00:02:40,150 just got these four equations connecting the electric field 47 00:02:40,150 --> 00:02:42,160 and the magnetic field, right? 48 00:02:42,160 --> 00:02:46,300 So you'll see that there's two E's there. 49 00:02:46,300 --> 00:02:51,940 Sorry, there's three E's, and then there's three B's. 50 00:02:51,940 --> 00:02:55,955 And then there's a charge row and a current that's CJ. 51 00:03:02,740 --> 00:03:06,110 So how we get the wave equation from that is quite simple. 52 00:03:06,110 --> 00:03:10,270 It's just a case of doing a bit of vector manipulation. 53 00:03:15,820 --> 00:03:20,230 Curl E, you can see, is minus the Bdt 54 00:03:20,230 --> 00:03:21,890 from the Maxwell's equations. 55 00:03:21,890 --> 00:03:25,060 So if you take the curl of both sides of that, 56 00:03:25,060 --> 00:03:27,580 you get this equation. 57 00:03:27,580 --> 00:03:33,430 And curl B is another of the Maxwell's equations, 58 00:03:33,430 --> 00:03:37,480 so you can substitute in from the Maxwell's equation 59 00:03:37,480 --> 00:03:42,670 into this one, and then you'll get another equation. 60 00:03:42,670 --> 00:03:46,900 What you get after that George hasn't actually written here, 61 00:03:46,900 --> 00:03:49,660 but basically, what you'd obviously get then, 62 00:03:49,660 --> 00:03:54,100 this is curl of this, so it's going to be curl of curl. 63 00:03:57,630 --> 00:04:00,540 So this is what you actually end up with, is curl, curl, 64 00:04:00,540 --> 00:04:04,260 curl E is going to be-- 65 00:04:04,260 --> 00:04:06,620 and then the other term is going to be something-- 66 00:04:06,620 --> 00:04:13,790 a second derivative in the E. So actually, that equation, maybe 67 00:04:13,790 --> 00:04:15,530 I'll write it down. 68 00:04:15,530 --> 00:04:18,019 It's not actually written there, but after you've just 69 00:04:18,019 --> 00:04:21,050 done that substitution, you'll get something 70 00:04:21,050 --> 00:04:24,250 like-- is this thing working, or-- yeah, it is. 71 00:04:24,250 --> 00:04:34,820 You'd get something like curl of curl E. There we are. 72 00:04:34,820 --> 00:04:36,210 Now you can see it. 73 00:04:39,380 --> 00:04:42,590 A lot of people right cross product like that, as a little, 74 00:04:42,590 --> 00:04:49,380 sort of upside down V. Curl curl E plus mu naught 75 00:04:49,380 --> 00:04:57,080 epsilon naught, d2, edt squared equals naught, 76 00:04:57,080 --> 00:04:59,720 is I think what you get after you've just 77 00:04:59,720 --> 00:05:01,940 substituted that in there. 78 00:05:05,420 --> 00:05:07,490 And then finally-- so this is-- this 79 00:05:07,490 --> 00:05:10,250 has still got this thing, curl of curl E, which 80 00:05:10,250 --> 00:05:12,290 is quite complicated if you try and work out 81 00:05:12,290 --> 00:05:16,400 what that is in spherical polars or something, 82 00:05:16,400 --> 00:05:19,790 especially if you're expressing E as something which is 83 00:05:19,790 --> 00:05:22,130 spatially varying or whatever. 84 00:05:22,130 --> 00:05:24,920 This could be quite a complicated thing 85 00:05:24,920 --> 00:05:27,140 to have to work out. 86 00:05:27,140 --> 00:05:32,210 But anyway, but the-- you can get it into a slightly more 87 00:05:32,210 --> 00:05:34,430 usable form-- 88 00:05:34,430 --> 00:05:36,020 often we do this-- 89 00:05:36,020 --> 00:05:39,890 by using this identity, curl curl E 90 00:05:39,890 --> 00:05:46,760 is equal to grad of div E minus Del squared E. 91 00:05:46,760 --> 00:05:49,400 It calls it here an identity. 92 00:05:49,400 --> 00:05:52,460 Really, it's actually the definition 93 00:05:52,460 --> 00:05:54,620 of what this thing means. 94 00:05:54,620 --> 00:05:58,685 And this is not obvious, what this means, actually. 95 00:06:01,730 --> 00:06:04,940 But these other things, OK, they're 96 00:06:04,940 --> 00:06:06,800 well defined, what they mean. 97 00:06:06,800 --> 00:06:09,530 This thing is actually just what it 98 00:06:09,530 --> 00:06:13,880 says in Cartesian coordinates, so you 99 00:06:13,880 --> 00:06:21,050 know you've got Del is equal to ddx in the I 100 00:06:21,050 --> 00:06:25,730 direction, plus ddy in the J direction, 101 00:06:25,730 --> 00:06:28,440 plus dd zed in the K direction. 102 00:06:28,440 --> 00:06:32,130 So Del squared is going to be this dot this, 103 00:06:32,130 --> 00:06:38,360 which is going to be d2 Del squared equals d2, dx squared, 104 00:06:38,360 --> 00:06:42,600 plus d2, dy squared, plus d2, d zed squared. 105 00:06:42,600 --> 00:06:46,250 And you'll notice that that's exactly the same form 106 00:06:46,250 --> 00:06:48,880 as the normal Laplacian, right? 107 00:06:48,880 --> 00:06:51,820 So the lower Laplacian you've had 108 00:06:51,820 --> 00:06:55,000 before, Del squared of a scalar, but this is different, 109 00:06:55,000 --> 00:06:58,990 because this is Del squared of a vector. 110 00:06:58,990 --> 00:07:03,070 So what we're saying is that, for Cartesian coordinates, 111 00:07:03,070 --> 00:07:06,610 if we define Del squared of a vector by this thing, 112 00:07:06,610 --> 00:07:10,090 we find that it's got exactly the same form 113 00:07:10,090 --> 00:07:11,830 as Del squared of a scalar. 114 00:07:11,830 --> 00:07:13,480 Now, that doesn't actually work with 115 00:07:13,480 --> 00:07:16,840 any other coordinate system, so that's a bit of a warning. 116 00:07:16,840 --> 00:07:21,760 If you go into sphericals or cylindrical coordinates, 117 00:07:21,760 --> 00:07:26,530 you'll find that this Del squared of a vector 118 00:07:26,530 --> 00:07:29,200 is not the same as Del squared that you 119 00:07:29,200 --> 00:07:30,190 would use for a scalar. 120 00:07:34,020 --> 00:07:36,170 But nevertheless, let's do that. 121 00:07:36,170 --> 00:07:39,320 We stick in this as being that. 122 00:07:39,320 --> 00:07:42,100 And then we look at this term here, 123 00:07:42,100 --> 00:07:46,880 and mostly, we're going to be interested in, for example, 124 00:07:46,880 --> 00:07:50,630 regions where there's no charges. 125 00:07:50,630 --> 00:07:54,610 And very often, also, we're going 126 00:07:54,610 --> 00:07:57,450 to be interested in isotropic media. 127 00:07:57,450 --> 00:08:03,950 If these things are true, you know that div D equals row, 128 00:08:03,950 --> 00:08:10,070 so Div d is equal to naught if there's no charges, which 129 00:08:10,070 --> 00:08:13,790 means that Div E is equal to naught if there's no charges, 130 00:08:13,790 --> 00:08:16,130 and it's also isotropic. 131 00:08:16,130 --> 00:08:20,000 So I'm saying this all out in length 132 00:08:20,000 --> 00:08:24,310 because you have to be very careful about this, sometimes. 133 00:08:24,310 --> 00:08:29,180 Div E equals naught is only true if you've got no charges, 134 00:08:29,180 --> 00:08:31,620 and it's an isotropic medium. 135 00:08:31,620 --> 00:08:36,140 So if you've got, for example, let's say a diffraction 136 00:08:36,140 --> 00:08:41,120 grating, where you've got some variations in permittivity, 137 00:08:41,120 --> 00:08:44,360 then in general, you can't actually assume this, 138 00:08:44,360 --> 00:08:47,720 because epsilon is changing. 139 00:08:47,720 --> 00:08:49,770 And therefore, with div D equals naught, 140 00:08:49,770 --> 00:08:52,520 div epsilon doesn't equal naught. 141 00:08:52,520 --> 00:08:58,370 But if you're in free space or some constant isotopic medium, 142 00:08:58,370 --> 00:08:59,520 this is true. 143 00:08:59,520 --> 00:09:02,600 So this term goes, and so this is just 144 00:09:02,600 --> 00:09:04,710 equal to minus Del squared. 145 00:09:04,710 --> 00:09:07,460 And so you can see then, this Dell squared then 146 00:09:07,460 --> 00:09:11,270 is just replaced by this, and we got a sign change there. 147 00:09:11,270 --> 00:09:13,880 So this is the final expression. 148 00:09:13,880 --> 00:09:15,950 And you'll notice then that this is exactly 149 00:09:15,950 --> 00:09:19,790 the same form as the normal wave equation 150 00:09:19,790 --> 00:09:23,660 that we're very used to in this scalar form, 151 00:09:23,660 --> 00:09:26,550 except that the scalar is replaced by a vector. 152 00:09:26,550 --> 00:09:29,300 So that's really very nice. 153 00:09:34,940 --> 00:09:39,480 Yeah, so comparing with that wave equation, 154 00:09:39,480 --> 00:09:42,330 this is the normal wave equation we've had, 155 00:09:42,330 --> 00:09:45,750 the scalar wave equation before, and you can see then they're 156 00:09:45,750 --> 00:09:48,420 exactly the same, except that the scalar is 157 00:09:48,420 --> 00:09:50,250 replaced by a vector. 158 00:09:50,250 --> 00:09:52,980 And the one over c squared, where 159 00:09:52,980 --> 00:09:55,470 c is the velocity of the wave, is 160 00:09:55,470 --> 00:09:57,900 replaced by this thing, so we can say then 161 00:09:57,900 --> 00:10:03,030 that 1 over c squared is equal to mu naught epsilon naught, 162 00:10:03,030 --> 00:10:06,770 and that gives us then c is 1 over the square root of mu 163 00:10:06,770 --> 00:10:08,910 naught epsilon naught. 164 00:10:08,910 --> 00:10:12,330 And so here, there are some figures put in, 165 00:10:12,330 --> 00:10:21,610 and we get the expression for the speed of light in vacuum. 166 00:10:21,610 --> 00:10:24,360 So that's very neat. 167 00:10:24,360 --> 00:10:26,910 I guess Maxwell must have been really amazed 168 00:10:26,910 --> 00:10:27,855 when he did this sum. 169 00:10:30,880 --> 00:10:33,690 You don't know, of course, the actual history of the thing, 170 00:10:33,690 --> 00:10:36,840 but he played around with these equations, didn't he? 171 00:10:36,840 --> 00:10:41,040 And he came up with this form which looked nicely 172 00:10:41,040 --> 00:10:42,810 symmetrical to him. 173 00:10:42,810 --> 00:10:46,290 And then he comes out with an expression, comes out 174 00:10:46,290 --> 00:10:49,800 with a value, for the speed of light that it predicts, 175 00:10:49,800 --> 00:10:52,260 which is what they already knew was true. 176 00:10:52,260 --> 00:10:55,500 So he must have felt really as though he 177 00:10:55,500 --> 00:10:58,350 was going to get the Nobel Prize or something. 178 00:10:58,350 --> 00:11:00,090 But probably, I don't know-- maybe 179 00:11:00,090 --> 00:11:06,060 it was before the Nobel Prize, so quite interesting. 180 00:11:06,060 --> 00:11:08,670 And actually, Naveen was telling me 181 00:11:08,670 --> 00:11:13,200 that he'd been looking at the original paper by Maxwell. 182 00:11:13,200 --> 00:11:17,250 Maxwell's equations, if you look in the original, are horrible, 183 00:11:17,250 --> 00:11:19,860 because he doesn't-- 184 00:11:19,860 --> 00:11:23,640 this terminology for vectors wasn't invented then. 185 00:11:23,640 --> 00:11:26,640 So it's done all in terms of components, 186 00:11:26,640 --> 00:11:28,980 horribly complicated. 187 00:11:28,980 --> 00:11:31,980 And I think Naveen said it was the-- 188 00:11:31,980 --> 00:11:37,290 Heaviside who actually really derived the Maxwell's equations 189 00:11:37,290 --> 00:11:43,410 in the form that we know and love so well nowadays. 190 00:11:43,410 --> 00:11:46,925 Heaviside you might remember, he's famous for a few things, 191 00:11:46,925 --> 00:11:48,300 but one of them is a thing called 192 00:11:48,300 --> 00:11:53,150 the Heaviside function, which is basically a step function. 193 00:11:53,150 --> 00:11:55,820 And the other thing is the Heaviside layer, 194 00:11:55,820 --> 00:12:02,150 which is an ionic layer in the ionosphere 195 00:12:02,150 --> 00:12:04,160 that he reflects radio waves from. 196 00:12:09,830 --> 00:12:14,880 OK, so that's all about free space. 197 00:12:14,880 --> 00:12:18,690 So how can we now deal with matter? 198 00:12:18,690 --> 00:12:21,320 Well, it turns out actually there's lots of different ways 199 00:12:21,320 --> 00:12:23,360 you can deal with-- 200 00:12:23,360 --> 00:12:26,870 lots of levels you can deal with propagation 201 00:12:26,870 --> 00:12:30,710 of electromagnetic waves in matter. 202 00:12:30,710 --> 00:12:34,370 You can, for example, treat it really 203 00:12:34,370 --> 00:12:36,740 from a proper atomic point of view. 204 00:12:36,740 --> 00:12:40,160 You can think of the material being made up 205 00:12:40,160 --> 00:12:48,500 of atoms which have got nuclei and electrons clouds and so on. 206 00:12:48,500 --> 00:12:51,290 Or you can think of it in terms of matter 207 00:12:51,290 --> 00:12:53,510 just being like an isotropic material, where 208 00:12:53,510 --> 00:12:56,460 you don't really go into the microscopic view of it, 209 00:12:56,460 --> 00:13:01,270 but just think of it in a macroscopic way. 210 00:13:01,270 --> 00:13:05,880 And actually, to some degree, I think that, very often, 211 00:13:05,880 --> 00:13:10,270 the second of those is just as well, because normally, we're 212 00:13:10,270 --> 00:13:14,800 not really interested in the actual atomic nature of where 213 00:13:14,800 --> 00:13:16,540 these properties come from. 214 00:13:16,540 --> 00:13:20,020 But this is going back to how you can think of it, really, 215 00:13:20,020 --> 00:13:22,060 in terms of atoms. 216 00:13:22,060 --> 00:13:23,770 So as you know, atoms are made up 217 00:13:23,770 --> 00:13:28,010 of a nucleus with an electron cloud. 218 00:13:28,010 --> 00:13:31,300 And the nucleus is relatively fixed, 219 00:13:31,300 --> 00:13:35,830 of course, because it's heavy, but the electron cloud, 220 00:13:35,830 --> 00:13:38,260 because the electrons are much lighter, 221 00:13:38,260 --> 00:13:43,300 can be moved around by the field that's supplied. 222 00:13:43,300 --> 00:13:46,520 So you get the distortion of the electron cloud. 223 00:13:46,520 --> 00:13:48,940 So this is what this is showing here. 224 00:13:48,940 --> 00:13:53,180 It's showing here the nucleus here, 225 00:13:53,180 --> 00:13:55,840 which is pretty well fixed, because it's so heavy. 226 00:13:55,840 --> 00:13:58,900 And then this electron cloud, you 227 00:13:58,900 --> 00:14:01,180 can think of it like being joined to the nucleus 228 00:14:01,180 --> 00:14:02,980 by a lot of springs. 229 00:14:02,980 --> 00:14:05,730 And then the electrostatic-- 230 00:14:05,730 --> 00:14:09,580 the forces on that electric cloud 231 00:14:09,580 --> 00:14:13,010 caused by the electric field will tend to, for example, 232 00:14:13,010 --> 00:14:17,500 displace the electron cloud relative to the nucleus. 233 00:14:17,500 --> 00:14:19,870 So it gets sort of distorted. 234 00:14:19,870 --> 00:14:25,960 And the way that's described is in terms of a property called 235 00:14:25,960 --> 00:14:29,200 the polarization then. 236 00:14:29,200 --> 00:14:31,000 So this is showing how-- 237 00:14:31,000 --> 00:14:34,060 you can see that if you move the negative charges 238 00:14:34,060 --> 00:14:36,280 relative to the positive charges, 239 00:14:36,280 --> 00:14:38,680 you'll set up a sort of dipole moment. 240 00:14:38,680 --> 00:14:41,620 Separate the charges, you've got a plus and a minus. 241 00:14:41,620 --> 00:14:45,430 They're separated in distance, so that acts like a dipole. 242 00:14:45,430 --> 00:14:47,590 So that acts like a-- 243 00:14:47,590 --> 00:14:50,020 gives you a dipole moment, and then 244 00:14:50,020 --> 00:14:53,170 if you sum over all those dipole moments, 245 00:14:53,170 --> 00:14:55,530 you'll get the total polarization 246 00:14:55,530 --> 00:14:57,550 effects of applying that field. 247 00:15:01,560 --> 00:15:08,450 So what that does then is introduces these charge 248 00:15:08,450 --> 00:15:17,450 variations, which distort the charge that you really 249 00:15:17,450 --> 00:15:21,120 have there, which is caused by the bound charges. 250 00:15:21,120 --> 00:15:23,600 So this is what we get. 251 00:15:23,600 --> 00:15:26,240 You can say that div D equals row, 252 00:15:26,240 --> 00:15:33,440 so div E is equal to row over epsilon naught, assuming the-- 253 00:15:33,440 --> 00:15:35,450 yeah, well, this is-- epsilon naught, of course, 254 00:15:35,450 --> 00:15:37,060 is really a constant. 255 00:15:37,060 --> 00:15:43,250 And we are breaking this up into it's bound-and-free components, 256 00:15:43,250 --> 00:15:48,620 and we're saying that the free part-- 257 00:15:48,620 --> 00:15:51,870 we've got something wrong with this again, haven't we? 258 00:15:51,870 --> 00:15:53,260 Sorry, that should say-- 259 00:15:53,260 --> 00:15:55,040 that should say free there, shouldn't it? 260 00:15:55,040 --> 00:15:55,800 Yeah. 261 00:15:55,800 --> 00:16:01,360 The free part and the bound part is this part, 262 00:16:01,360 --> 00:16:03,880 and so this is it, the final result then. 263 00:16:03,880 --> 00:16:06,760 We can now take the epsilon naught the other side, 264 00:16:06,760 --> 00:16:08,920 take this the other side, and we get 265 00:16:08,920 --> 00:16:17,170 the div of epsilon naught E plus P is equal to the free charges, 266 00:16:17,170 --> 00:16:17,770 right? 267 00:16:17,770 --> 00:16:21,290 And then this thing in brackets here 268 00:16:21,290 --> 00:16:26,420 is what we call D, the displacement in the medium. 269 00:16:26,420 --> 00:16:28,660 So it's like if you've got this medium, 270 00:16:28,660 --> 00:16:32,520 div D equals row is what we know is 271 00:16:32,520 --> 00:16:38,140 the normal expression that we write from Maxwell's equations. 272 00:16:38,140 --> 00:16:42,670 And now, we've derived that that D, in the medium, 273 00:16:42,670 --> 00:16:47,350 can be written in terms of the polarization of the molecules 274 00:16:47,350 --> 00:16:49,700 that make up the medium. 275 00:16:49,700 --> 00:16:52,500 So how do these things connect? 276 00:16:52,500 --> 00:16:54,570 This shows us how we do this then. 277 00:16:54,570 --> 00:17:00,010 So we've got D equals epsilon naught E plus P. Now, 278 00:17:00,010 --> 00:17:02,110 in general, of course, we don't really 279 00:17:02,110 --> 00:17:07,150 know what this does at the moment. 280 00:17:07,150 --> 00:17:12,550 And in general, it can be very complicated, actually. 281 00:17:12,550 --> 00:17:14,500 It can be a non-linear relationship 282 00:17:14,500 --> 00:17:16,369 with the electric field. 283 00:17:16,369 --> 00:17:19,760 But if I go back again to the diagram, 284 00:17:19,760 --> 00:17:23,109 you'll see that this is a bit like-- 285 00:17:23,109 --> 00:17:25,050 you know, you've got springs here. 286 00:17:25,050 --> 00:17:28,750 Hooke's law tells us, in mechanics, 287 00:17:28,750 --> 00:17:34,390 if a system obeys Hooke's law, then the extension 288 00:17:34,390 --> 00:17:36,242 is proportional to the-- 289 00:17:36,242 --> 00:17:39,760 the tension is-- what is it? 290 00:17:39,760 --> 00:17:42,838 Extension is proportional to the attention. 291 00:17:42,838 --> 00:17:44,380 Sorry, I'm not a mechanical engineer, 292 00:17:44,380 --> 00:17:46,450 so I don't know these things. 293 00:17:46,450 --> 00:17:50,680 So you'd expect the same would be true-- could be the same-- 294 00:17:50,680 --> 00:17:52,000 could be true here. 295 00:17:52,000 --> 00:17:55,480 If you're only applying a very weak field on here, 296 00:17:55,480 --> 00:17:59,200 then you might think that this relationship might be linear. 297 00:17:59,200 --> 00:18:03,490 You'll get these springs behaving like Hooke's law, 298 00:18:03,490 --> 00:18:08,770 so you'll get a linear relationship between P and D. 299 00:18:08,770 --> 00:18:13,540 Now, also, in analogy with the mechanical situation-- yeah? 300 00:18:13,540 --> 00:18:16,900 STUDENT: So if we're in a mat-- like in a material, 301 00:18:16,900 --> 00:18:19,570 why is it the permitivity of free space and not just 302 00:18:19,570 --> 00:18:23,220 the permittivity of the material? 303 00:18:23,220 --> 00:18:25,260 Like why is it epsilon naught and not epsilon? 304 00:18:29,305 --> 00:18:31,180 COLIN SHEPPARD: Why is it epsilon naught here 305 00:18:31,180 --> 00:18:32,230 that you can write this? 306 00:18:32,230 --> 00:18:35,350 This is either you can think of as epsilon naught-- 307 00:18:38,050 --> 00:18:41,140 epsilon E, or you can think of it 308 00:18:41,140 --> 00:18:43,570 as epsilon naught E plus P. So these 309 00:18:43,570 --> 00:18:46,638 are two alternative ways of looking 310 00:18:46,638 --> 00:18:47,680 at the same thing, right? 311 00:18:47,680 --> 00:18:57,190 So we've got D equals epsilon naught E plus P, 312 00:18:57,190 --> 00:19:01,660 and you can think of this as being epsilon E. 313 00:19:01,660 --> 00:19:04,670 So we're just about to carry onto this on the next slide, 314 00:19:04,670 --> 00:19:05,170 actually. 315 00:19:05,170 --> 00:19:08,770 So that, you can see, tells us that there is a relationship 316 00:19:08,770 --> 00:19:15,160 now between epsilon and P, right? 317 00:19:15,160 --> 00:19:17,620 We're going to look what that relationship is. 318 00:19:17,620 --> 00:19:21,370 But in general, it's actually quite complicated, 319 00:19:21,370 --> 00:19:24,100 because we don't know the relationship between P and E 320 00:19:24,100 --> 00:19:24,970 yet, right? 321 00:19:24,970 --> 00:19:29,350 So what I'm trying to establish at the moment is 322 00:19:29,350 --> 00:19:32,080 that if this thing-- 323 00:19:32,080 --> 00:19:35,770 if the fields are very weak, then you'd expect, 324 00:19:35,770 --> 00:19:38,020 maybe, a linear relationship. 325 00:19:38,020 --> 00:19:43,150 P will be proportional to E. So this term here 326 00:19:43,150 --> 00:19:46,900 will also be something times E, and so 327 00:19:46,900 --> 00:19:50,620 you'll be able to take out E as a factor from this. 328 00:19:50,620 --> 00:19:55,900 Now, this is only going to be true if this electric field is 329 00:19:55,900 --> 00:19:58,870 weak compared with the-- 330 00:19:58,870 --> 00:20:01,840 you know, if the force due to the electric field 331 00:20:01,840 --> 00:20:06,850 is weak compared with the forces involved in the binding 332 00:20:06,850 --> 00:20:09,430 the electrons to the nucleus. 333 00:20:09,430 --> 00:20:14,410 And I suppose it was probably not until the invention 334 00:20:14,410 --> 00:20:18,230 of the laser that probably even people thought 335 00:20:18,230 --> 00:20:20,950 you could ever possibly get to a case 336 00:20:20,950 --> 00:20:22,390 where that might not be true. 337 00:20:22,390 --> 00:20:25,420 But now, of course, we know, if you've got a laser, 338 00:20:25,420 --> 00:20:28,600 you can make really big fields. 339 00:20:28,600 --> 00:20:31,660 You can get, very easily, into this rate regime where 340 00:20:31,660 --> 00:20:34,050 this polarization changes. 341 00:20:34,050 --> 00:20:39,100 And so you'll get non-linear-type terms coming. 342 00:20:39,100 --> 00:20:42,593 You're not going to do this at all in the course. 343 00:20:42,593 --> 00:20:44,260 But there is this whole area, of course, 344 00:20:44,260 --> 00:20:49,300 of non-linear optics, which is when this linear approximation 345 00:20:49,300 --> 00:20:50,110 breaks down. 346 00:20:55,030 --> 00:20:57,870 So here we are. 347 00:20:57,870 --> 00:21:02,850 So this is back to this, what I said, D equals epsilon E. OK, 348 00:21:02,850 --> 00:21:06,270 so that's really the answer to your question. 349 00:21:06,270 --> 00:21:13,350 And the D equals epsilon E, but D equals this other expression, 350 00:21:13,350 --> 00:21:16,620 epsilon naught E plus P. And now, we say 351 00:21:16,620 --> 00:21:19,080 that P is proportional to E. 352 00:21:19,080 --> 00:21:24,770 This Chi is called the electrics susceptibility, 353 00:21:24,770 --> 00:21:28,550 which I think is a confusing term because it's often, 354 00:21:28,550 --> 00:21:30,950 in practice, just called susceptibility. 355 00:21:30,950 --> 00:21:33,260 People just neglect the electric bit. 356 00:21:33,260 --> 00:21:36,542 But of course, there is also a magnetic susceptibility 357 00:21:36,542 --> 00:21:38,000 which is very important when you're 358 00:21:38,000 --> 00:21:40,760 doing magnetic materials. 359 00:21:40,760 --> 00:21:43,040 And unfortunately, people use Chi for that 360 00:21:43,040 --> 00:21:47,220 too, so that really gets people confused if you're not careful. 361 00:21:47,220 --> 00:21:50,330 So we're not going to be doing anything with magnetism, 362 00:21:50,330 --> 00:21:53,520 so maybe it's no problem for us. 363 00:21:53,520 --> 00:21:57,950 But anyway, this is the electric susceptibility. 364 00:21:57,950 --> 00:22:04,110 So we can put in our P as Chi E, and therefore, D 365 00:22:04,110 --> 00:22:07,820 equals epsilon naught 1 plus Chi times E. 366 00:22:07,820 --> 00:22:12,080 And this is equal to epsilon E, as we've said. 367 00:22:12,080 --> 00:22:16,370 And you can see then, we have that epsilon 368 00:22:16,370 --> 00:22:21,110 must equal this thing, epsilon equals epsilon naught 1 369 00:22:21,110 --> 00:22:21,665 plus Chi. 370 00:22:24,440 --> 00:22:27,200 And this is a another expression we've had before, 371 00:22:27,200 --> 00:22:30,830 in terms of the refractive index, so you can write 372 00:22:30,830 --> 00:22:35,510 the permittivity, the relative permittivity, 373 00:22:35,510 --> 00:22:38,640 in terms of refractive index, right? 374 00:22:38,640 --> 00:22:41,940 So this is the refractive index coming in here. 375 00:22:41,940 --> 00:22:45,710 So under refractive index then is the square root of 1 376 00:22:45,710 --> 00:22:47,480 plus Chi. 377 00:22:47,480 --> 00:22:52,130 And so you see how all these things are connected. 378 00:22:52,130 --> 00:22:54,050 You could also say, one more, of course. 379 00:22:57,690 --> 00:22:58,270 Where are we? 380 00:22:58,270 --> 00:23:01,960 Yeah, we could say that we've got 381 00:23:01,960 --> 00:23:06,010 n squared equals 1 plus Chi, so Chi 382 00:23:06,010 --> 00:23:08,155 is equal to n squared minus 1. 383 00:23:08,155 --> 00:23:13,880 So this is another sort of form you might come across at times. 384 00:23:13,880 --> 00:23:17,110 You can see then, we know in free space-- 385 00:23:17,110 --> 00:23:21,100 free space, n is 1, so this thing is 0. 386 00:23:21,100 --> 00:23:23,282 So there is no polarization of free space, 387 00:23:23,282 --> 00:23:25,490 which is what we know, because there's nothing in it. 388 00:23:25,490 --> 00:23:30,310 So there's nothing to polarize. 389 00:23:30,310 --> 00:23:31,610 OK? 390 00:23:31,610 --> 00:23:35,450 So, yeah, this bottom bit just says that you 391 00:23:35,450 --> 00:23:40,460 can do the same for magnetism. 392 00:23:40,460 --> 00:23:43,460 You can do virtually the same sort of expansions 393 00:23:43,460 --> 00:23:46,040 for magnetism, but we're not going 394 00:23:46,040 --> 00:23:48,200 to deal with this, because we're not going to deal 395 00:23:48,200 --> 00:23:50,570 with magnetic materials at all. 396 00:23:50,570 --> 00:23:53,600 But there are some quite interesting, 397 00:23:53,600 --> 00:23:56,300 you know, magneto optic materials and so on, 398 00:23:56,300 --> 00:23:59,060 which are quite important actually, 399 00:23:59,060 --> 00:24:03,680 nowadays, in terms of things like optical data storage 400 00:24:03,680 --> 00:24:06,050 and so on. 401 00:24:06,050 --> 00:24:11,000 We're just going to be assuming that, from the magnetic point 402 00:24:11,000 --> 00:24:12,830 of view, the material just behaves 403 00:24:12,830 --> 00:24:18,680 exactly the same as free space, so B equals mu naught H. Yeah? 404 00:24:23,280 --> 00:24:25,330 STUDENT: The electric polarization subtracts 405 00:24:25,330 --> 00:24:29,560 the field out of the original. 406 00:24:29,560 --> 00:24:31,600 Polarization adds the field, magnetization 407 00:24:31,600 --> 00:24:33,320 subtracts the field. 408 00:24:33,320 --> 00:24:34,195 COLIN SHEPPARD: Yeah. 409 00:24:37,060 --> 00:24:40,600 [INAUDIBLE] has asked this question of why this is a plus, 410 00:24:40,600 --> 00:24:42,940 and this is a minus. 411 00:24:42,940 --> 00:24:46,510 And the answer is, I think it's all 412 00:24:46,510 --> 00:24:48,730 to do with all these horrible things you 413 00:24:48,730 --> 00:24:52,780 learn about in magnetism, dire magnetism and paramagnetism. 414 00:24:52,780 --> 00:24:56,560 And sometimes, it's whether they oppose 415 00:24:56,560 --> 00:25:00,590 the change to which they were due and all that sort of stuff. 416 00:25:00,590 --> 00:25:02,650 So I think it's just a case of whether-- 417 00:25:02,650 --> 00:25:06,490 if you define magnetization in the way 418 00:25:06,490 --> 00:25:09,350 that it's normally done, then it turns out there's a minus sign, 419 00:25:09,350 --> 00:25:12,250 but I think you could equally well have defined it as being 420 00:25:12,250 --> 00:25:14,740 the opposite side, actually. 421 00:25:14,740 --> 00:25:16,510 Yeah. 422 00:25:16,510 --> 00:25:18,670 I don't really know much about magnetism, 423 00:25:18,670 --> 00:25:20,680 so don't ask me too much about that. 424 00:25:23,902 --> 00:25:25,360 STUDENT: Also, I think it has to do 425 00:25:25,360 --> 00:25:27,220 with the part of the stuff that-- 426 00:25:27,220 --> 00:25:32,500 B is actually the conjugate of the electric displacement. 427 00:25:32,500 --> 00:25:33,700 COLIN SHEPPARD: Ah. 428 00:25:33,700 --> 00:25:35,942 STUDENT: So it includes the effect of magnetization. 429 00:25:35,942 --> 00:25:37,900 So another way to put it is if you could please 430 00:25:37,900 --> 00:25:43,180 go back one, if you solve this, if you 431 00:25:43,180 --> 00:25:46,450 take M to the other side, then multiply it by mu naught, 432 00:25:46,450 --> 00:25:48,408 then you'll get exactly the question as before. 433 00:25:48,408 --> 00:25:50,033 COLIN SHEPPARD: Of course that's right. 434 00:25:50,033 --> 00:25:51,550 If you just expand this out, you get 435 00:25:51,550 --> 00:25:55,180 B equals mu naught H plus M, so it would 436 00:25:55,180 --> 00:25:57,850 be exactly the same form then. 437 00:25:57,850 --> 00:26:01,420 STUDENT: So the field is actually H, not B. 438 00:26:01,420 --> 00:26:04,950 The conjugate of the electric field, E, 439 00:26:04,950 --> 00:26:08,390 is the magnetic field, H, not B. B is the induction. 440 00:26:08,390 --> 00:26:10,590 COLIN SHEPPARD: Yeah. 441 00:26:10,590 --> 00:26:11,090 OK. 442 00:26:16,490 --> 00:26:20,990 So now then, let's write down those Maxwell's equations 443 00:26:20,990 --> 00:26:23,470 for these various different cases. 444 00:26:23,470 --> 00:26:26,060 So this is how you can write them in vacuum. 445 00:26:26,060 --> 00:26:28,370 That's how we first came across them. 446 00:26:28,370 --> 00:26:33,620 And if you've got vacuum, and in addition, you've got no charges 447 00:26:33,620 --> 00:26:38,210 and no currents, then of course, this term, this will be 0, 448 00:26:38,210 --> 00:26:40,500 and this J will also be 0. 449 00:26:40,500 --> 00:26:44,540 And now, you can see that they look nicely symmetrical. 450 00:26:44,540 --> 00:26:48,920 These two are both equal to 0, and these things are both equal 451 00:26:48,920 --> 00:26:56,810 to a first-- a time derivative, but there is a difference 452 00:26:56,810 --> 00:26:58,130 of a sine-- 453 00:26:58,130 --> 00:27:01,910 but apart from that, nicely symmetrical. 454 00:27:01,910 --> 00:27:06,680 And then the middle row here says, 455 00:27:06,680 --> 00:27:12,230 if you've got some matter, and it's still 456 00:27:12,230 --> 00:27:15,020 allowed to have free charges and currents, 457 00:27:15,020 --> 00:27:17,250 then we can write these things. 458 00:27:17,250 --> 00:27:20,320 So this is what we just said a minute ago. 459 00:27:20,320 --> 00:27:25,490 Of course, if you've got no free charges, then div D is 0. 460 00:27:25,490 --> 00:27:29,510 And then if you've got a material, matter, 461 00:27:29,510 --> 00:27:32,270 without the free charges, then again, we put the rows 462 00:27:32,270 --> 00:27:37,670 in the J's, 0, and we end up with these quite nice, simple, 463 00:27:37,670 --> 00:27:42,540 and symmetrical sort of relationships. 464 00:27:42,540 --> 00:27:45,350 And in any of these cases, of course, 465 00:27:45,350 --> 00:27:49,440 you could derive the wave equation 466 00:27:49,440 --> 00:27:53,950 and write it in that same form we had before, OK? 467 00:27:53,950 --> 00:27:59,200 And as we said, this term here is 468 00:27:59,200 --> 00:28:04,660 related to the speed of light in the medium, and what we find 469 00:28:04,660 --> 00:28:07,510 is that the speed of light in the medium 470 00:28:07,510 --> 00:28:10,690 is equal to the speed of light, your free space, 471 00:28:10,690 --> 00:28:12,550 divided by the refractive index. 472 00:28:12,550 --> 00:28:15,790 All of those things are things we've had before. 473 00:28:15,790 --> 00:28:18,970 So I'll just stress again, one thing that I did 474 00:28:18,970 --> 00:28:20,890 say earlier, this was originally, 475 00:28:20,890 --> 00:28:23,860 if you remember, curl curl E. And then 476 00:28:23,860 --> 00:28:26,080 when we got rid of the curl curl E, 477 00:28:26,080 --> 00:28:29,680 we actually had to get rid of a-- 478 00:28:29,680 --> 00:28:33,670 what was it-- grad of div E. And we 479 00:28:33,670 --> 00:28:36,190 assumed that grad div E is 0. 480 00:28:36,190 --> 00:28:38,650 I'm just reminding you, again, that is not 481 00:28:38,650 --> 00:28:42,072 always true, so beware. 482 00:28:42,072 --> 00:28:44,530 And you probably won't come across anything in this course, 483 00:28:44,530 --> 00:28:48,440 but in the general wide world, it's not always true. 484 00:28:48,440 --> 00:28:50,890 And the second thing to remember is 485 00:28:50,890 --> 00:28:56,890 that this thing, Del squared of a vector, that Del squared is 486 00:28:56,890 --> 00:28:59,770 not equal to the same as the Laplacian 487 00:28:59,770 --> 00:29:02,670 except if you're in Cartesian coordinates. 488 00:29:07,380 --> 00:29:11,760 OK, so now we've got the wave equation, 489 00:29:11,760 --> 00:29:16,380 we go back just like we did for scalar waves. 490 00:29:16,380 --> 00:29:19,830 We could look at different solutions of that wave 491 00:29:19,830 --> 00:29:23,400 equation, and the simplest possible one 492 00:29:23,400 --> 00:29:26,650 is the equation of a plane wave. 493 00:29:26,650 --> 00:29:30,270 And so this is a picture of what it looks like. 494 00:29:30,270 --> 00:29:33,240 I presume these pictures are taken from [INAUDIBLE],, 495 00:29:33,240 --> 00:29:36,330 but to me, it seems very perverse to have the wave 496 00:29:36,330 --> 00:29:38,920 propagating in the x direction. 497 00:29:38,920 --> 00:29:42,480 He's a bit strange there, but anyway, virtually every book 498 00:29:42,480 --> 00:29:45,670 has the wave propagating in the zed direction. 499 00:29:45,670 --> 00:29:48,420 But nevermind, we won't worry too much. 500 00:29:48,420 --> 00:29:50,450 It obviously doesn't make any difference. 501 00:29:50,450 --> 00:29:55,140 But this is the important thing, is we get the electric field 502 00:29:55,140 --> 00:29:59,146 vector and the magnetic field vector, 503 00:29:59,146 --> 00:30:02,770 E and B he's drawing here. 504 00:30:02,770 --> 00:30:05,920 It could equally well be B and H, because the two-- 505 00:30:05,920 --> 00:30:09,440 the B and H are going to be proportional to each other, 506 00:30:09,440 --> 00:30:12,790 or at right angles to each other. 507 00:30:12,790 --> 00:30:14,530 I'm just telling you the results first. 508 00:30:14,530 --> 00:30:16,390 I haven't proved this yet. 509 00:30:16,390 --> 00:30:19,990 And the wave is propagating in a direction 510 00:30:19,990 --> 00:30:22,790 which is at right angles to both of those. 511 00:30:22,790 --> 00:30:27,730 And so these three form a triad, and it's 512 00:30:27,730 --> 00:30:32,800 a right-handed triad such that E cross B is 513 00:30:32,800 --> 00:30:34,480 in the direction the wave goes. 514 00:30:34,480 --> 00:30:37,980 So that's the way to always remember it. 515 00:30:37,980 --> 00:30:39,610 Well, that's the way I remember it, 516 00:30:39,610 --> 00:30:44,775 is that E cross B is in the direction the field travels, 517 00:30:44,775 --> 00:30:46,150 which means, of course, if you've 518 00:30:46,150 --> 00:30:49,840 got a right-hand coordinate system, 519 00:30:49,840 --> 00:30:53,830 normally, we would take E in the x direction 520 00:30:53,830 --> 00:30:56,020 and B in the y direction, and then 521 00:30:56,020 --> 00:30:57,710 the wave moves in the zed direction. 522 00:30:57,710 --> 00:31:03,130 But he's taken it with E in the y direction like this, 523 00:31:03,130 --> 00:31:05,320 in order to get it so that it's still 524 00:31:05,320 --> 00:31:08,940 a right-handed coordinate system. 525 00:31:08,940 --> 00:31:10,870 And so how do we get that? 526 00:31:10,870 --> 00:31:13,600 Well, first of all, of course, we 527 00:31:13,600 --> 00:31:18,100 can say it's quite straightforward for the E. 528 00:31:18,100 --> 00:31:22,870 We can just solve this thing for the-- 529 00:31:22,870 --> 00:31:28,630 just as a-- it's a scalar and get an expression for E. E 530 00:31:28,630 --> 00:31:32,800 is going to be then polarized in a particular direction. 531 00:31:32,800 --> 00:31:38,470 We've then got to work out how B is polarized 532 00:31:38,470 --> 00:31:42,790 for this particular value of E. Well, 533 00:31:42,790 --> 00:31:47,860 George has said something here which is-- 534 00:31:47,860 --> 00:31:51,580 I guess it's correct, but it's quite sort of condensed, 535 00:31:51,580 --> 00:31:54,498 and I went through how I would really derive it. 536 00:31:54,498 --> 00:31:56,290 So perhaps, I'll say a bit more about that. 537 00:31:59,560 --> 00:32:01,330 Well, this first part is all right. 538 00:32:01,330 --> 00:32:05,410 We've got curl E is minus db dt. 539 00:32:05,410 --> 00:32:08,800 And we know that the time dependence is this e 540 00:32:08,800 --> 00:32:10,360 to the minus i omega t. 541 00:32:10,360 --> 00:32:14,750 So ddt of e to the minus i omega t is very straightforward. 542 00:32:14,750 --> 00:32:20,040 That's just-- the minus i omega comes out the front. 543 00:32:20,040 --> 00:32:21,070 So that's what we get. 544 00:32:24,140 --> 00:32:35,520 Curl e is equal to minus db, dt, is equal to i omega b. 545 00:32:35,520 --> 00:32:38,910 And so we can say then that b is equal to minus 546 00:32:38,910 --> 00:32:44,060 i over omega curl e, all right? 547 00:32:44,060 --> 00:32:50,360 So if we know what e is, we can work out what b is from that. 548 00:32:50,360 --> 00:32:55,730 And so we take e as being this thing, all right? 549 00:32:55,730 --> 00:32:59,330 So in general, this might not be traveling 550 00:32:59,330 --> 00:33:02,720 in the x direction or any other particular direction. 551 00:33:02,720 --> 00:33:07,650 It might be just pointing in some direction in space. 552 00:33:07,650 --> 00:33:09,860 So really, I guess, you ought to do it 553 00:33:09,860 --> 00:33:12,620 for that general case, which means 554 00:33:12,620 --> 00:33:14,330 that effectively what you've got to do 555 00:33:14,330 --> 00:33:20,650 is to work out what curl of the e is. 556 00:33:20,650 --> 00:33:26,800 And e is, effectively, as you can see, a scalar quantity. 557 00:33:26,800 --> 00:33:28,930 Well, let's forget about the time dependence. 558 00:33:28,930 --> 00:33:31,550 We're only looking at the space now. 559 00:33:31,550 --> 00:33:34,900 There's this scalar quantity here. 560 00:33:34,900 --> 00:33:37,010 Time is a constant vector. 561 00:33:37,010 --> 00:33:41,200 So we then have to use another of our identities, which 562 00:33:41,200 --> 00:33:45,490 is the curl of a scalar times a vector. 563 00:33:48,060 --> 00:33:49,960 So let's write it like that. 564 00:33:49,960 --> 00:33:53,740 And you might remember that that is equal to-- and I had to look 565 00:33:53,740 --> 00:33:56,780 it up because I never know these things-- 566 00:33:56,780 --> 00:34:06,890 5 times curl a plus grad phi across a, all right? 567 00:34:06,890 --> 00:34:11,250 So that is a general vector identity 568 00:34:11,250 --> 00:34:14,820 which allows you to work out the curl of this product of two 569 00:34:14,820 --> 00:34:16,010 things. 570 00:34:16,010 --> 00:34:19,050 And in our case, this is a plane wave. 571 00:34:19,050 --> 00:34:23,739 So this direction of the e vector is constant. 572 00:34:23,739 --> 00:34:27,210 So our a is a constant. 573 00:34:27,210 --> 00:34:30,840 So curl of a is going to be 0 because it's constant. 574 00:34:30,840 --> 00:34:32,400 So that goes. 575 00:34:32,400 --> 00:34:39,090 And we're then left with the curl of phi times a 576 00:34:39,090 --> 00:34:41,449 is grad phi cross a. 577 00:34:41,449 --> 00:34:49,830 And phi is equal to-- phi equals e to the i k dot r. 578 00:34:49,830 --> 00:34:55,770 And a is equal to this direction of the e vector, all right? 579 00:34:55,770 --> 00:35:00,310 So if phi equals e to the i k dot r, 580 00:35:00,310 --> 00:35:07,570 then grad phi is equal to ik times e to the i k dot r. 581 00:35:07,570 --> 00:35:10,520 And so we stick that into here. 582 00:35:10,520 --> 00:35:14,260 And after we've done that, we can derive this expression 583 00:35:14,260 --> 00:35:14,760 here. 584 00:35:14,760 --> 00:35:19,620 We find that finally the magnetic vector 585 00:35:19,620 --> 00:35:22,470 is equal to the cross products of these two. 586 00:35:22,470 --> 00:35:24,600 So that's why, of course, all these three things 587 00:35:24,600 --> 00:35:29,740 have to be right angles to each other, all right? 588 00:35:29,740 --> 00:35:32,340 So the next thing, this is a picture of it 589 00:35:32,340 --> 00:35:34,670 moving along then. 590 00:35:34,670 --> 00:35:39,510 And we'll stress, very strongly, something. 591 00:35:39,510 --> 00:35:45,090 And that is that these two, the electric and magnetic field, 592 00:35:45,090 --> 00:35:47,680 are in phase with each other. 593 00:35:47,680 --> 00:35:51,060 I don't know what it is, but I've often 594 00:35:51,060 --> 00:35:54,780 found that students don't seem to pick this up. 595 00:35:54,780 --> 00:35:57,180 And they come up with these funny ideas 596 00:35:57,180 --> 00:35:59,120 about them being in quadrature. 597 00:35:59,120 --> 00:36:02,070 And I think they get it mixed up with some other things. 598 00:36:02,070 --> 00:36:06,990 But for a plane polarized wave, the electric field 599 00:36:06,990 --> 00:36:08,730 and the magnetic field, you can see, 600 00:36:08,730 --> 00:36:10,110 are in phase with each other. 601 00:36:10,110 --> 00:36:13,890 When the e is a maximum, the b is a maximum. 602 00:36:13,890 --> 00:36:16,950 And then they will have this wave form. 603 00:36:16,950 --> 00:36:22,140 Here this is showing the wave as a function of distance. 604 00:36:22,140 --> 00:36:26,070 This is like a snapshot of what happens at a particular time. 605 00:36:26,070 --> 00:36:29,940 But you could equally well get a very similar sort of thing 606 00:36:29,940 --> 00:36:32,760 by looking at a particular position 607 00:36:32,760 --> 00:36:35,130 and seeing how it changes with time. 608 00:36:35,130 --> 00:36:37,920 You'd also get sine waves like this. 609 00:36:37,920 --> 00:36:41,490 And you would also see that these two, 610 00:36:41,490 --> 00:36:43,950 the electric and magnetic fields, 611 00:36:43,950 --> 00:36:48,670 have to be in phase with each other for that case, too. 612 00:36:48,670 --> 00:36:49,170 OK? 613 00:36:49,170 --> 00:36:52,800 This is expressing-- showing you, again, 614 00:36:52,800 --> 00:36:57,090 this right-hand rule or whatever it is for-- 615 00:36:57,090 --> 00:36:59,040 I don't know how you do these things. 616 00:36:59,040 --> 00:37:01,320 But anyway, I always like the-- 617 00:37:01,320 --> 00:37:04,120 I can never do these, partly because I'm left-handed, 618 00:37:04,120 --> 00:37:04,890 I think. 619 00:37:04,890 --> 00:37:06,520 It ruins things. 620 00:37:06,520 --> 00:37:10,650 But this corkscrew rule is the one 621 00:37:10,650 --> 00:37:12,210 that I always know that one. 622 00:37:12,210 --> 00:37:14,160 And my students will know it's because I'm 623 00:37:14,160 --> 00:37:16,974 very handy with a corkscrew. 624 00:37:16,974 --> 00:37:18,890 [LAUGHS] 625 00:37:18,890 --> 00:37:22,970 So this is showing, then, how this electric field varies 626 00:37:22,970 --> 00:37:26,360 with space, the fixed time. 627 00:37:26,360 --> 00:37:31,490 And the distance, then, between these maxima 628 00:37:31,490 --> 00:37:35,330 in the electric field is the wavelength, all right? 629 00:37:35,330 --> 00:37:38,270 So this is going to propagate. 630 00:37:38,270 --> 00:37:40,100 It's a traveling wave. 631 00:37:40,100 --> 00:37:43,700 So this structure moves bodily, doesn't it, 632 00:37:43,700 --> 00:37:48,590 in time along this direction of propagation. 633 00:37:53,360 --> 00:37:55,460 Ah, and now we get onto this thing 634 00:37:55,460 --> 00:37:58,550 that I was talking with George about on the way, the Poynting 635 00:37:58,550 --> 00:37:59,990 vector. 636 00:37:59,990 --> 00:38:05,620 And I don't think you actually really sort of prove it, 637 00:38:05,620 --> 00:38:06,940 do you, what it really does. 638 00:38:06,940 --> 00:38:13,720 But basically the Poynting vector, it turns out, 639 00:38:13,720 --> 00:38:17,170 is a measure of the energy flow. 640 00:38:17,170 --> 00:38:19,780 So it's a way of looking at the energy that's 641 00:38:19,780 --> 00:38:21,790 carried by a wave. 642 00:38:21,790 --> 00:38:25,570 And this is the definition of it. 643 00:38:25,570 --> 00:38:29,800 So s is equal to 1 over mu 0, e cross b. 644 00:38:29,800 --> 00:38:31,860 Actually, you can see b equals-- 645 00:38:31,860 --> 00:38:34,930 actually, I think this should really be mu, shouldn't it, 646 00:38:34,930 --> 00:38:39,550 not mu 0, really, if it was in a general medium. 647 00:38:39,550 --> 00:38:42,250 And b equals mu h, of course. 648 00:38:42,250 --> 00:38:46,110 So this is really cross h. 649 00:38:46,110 --> 00:38:52,780 So that's the way it's sometimes derived, sometimes defined. 650 00:38:52,780 --> 00:38:56,950 But anyway, but if you're in free space, then 651 00:38:56,950 --> 00:38:59,530 you can write it with mu 0 here like that. 652 00:38:59,530 --> 00:39:06,550 And then, using our expression for the velocity of light, 653 00:39:06,550 --> 00:39:08,330 then we can write it like this. 654 00:39:08,330 --> 00:39:13,570 But I think this is also true for free space only. 655 00:39:13,570 --> 00:39:15,970 I think that's right. 656 00:39:15,970 --> 00:39:19,440 Presumably this is from Hecht again. 657 00:39:19,440 --> 00:39:24,520 OK, so this is what our waves look like. 658 00:39:24,520 --> 00:39:30,460 Each of electric and magnetic fields is a cosine-type wave. 659 00:39:30,460 --> 00:39:33,550 So this represents, then, a wave which 660 00:39:33,550 --> 00:39:38,520 is moving in this direction, all right? 661 00:39:38,520 --> 00:39:42,610 So this thing in here basically is the phase, isn't it? 662 00:39:42,610 --> 00:39:45,280 You got cos of a phase term. 663 00:39:45,280 --> 00:39:48,930 So you can see that as omega t varies, 664 00:39:48,930 --> 00:39:50,940 so the shape of the wave is going 665 00:39:50,940 --> 00:39:54,420 to sort of move in a particular direction. 666 00:39:54,420 --> 00:39:57,880 And you'll notice that-- oh, this is a vector. 667 00:39:57,880 --> 00:40:00,480 This is a vector, but this is a constant vector. 668 00:40:00,480 --> 00:40:03,600 So this is a vector which tells you 669 00:40:03,600 --> 00:40:08,550 the amplitude of that wave in magnitude and direction. 670 00:40:08,550 --> 00:40:11,190 And you'll notice that these cosine bits 671 00:40:11,190 --> 00:40:14,060 are exactly the same. 672 00:40:14,060 --> 00:40:16,850 That's because, as we said, the e and the b 673 00:40:16,850 --> 00:40:19,510 are in phase with each other. 674 00:40:19,510 --> 00:40:21,830 And so all that's very simple. 675 00:40:21,830 --> 00:40:25,020 And so what we can say is that if we represent 676 00:40:25,020 --> 00:40:28,240 the fields like this, we can say, according 677 00:40:28,240 --> 00:40:30,640 to our definition of the Poynting vector, 678 00:40:30,640 --> 00:40:32,900 we can write it like this. 679 00:40:32,900 --> 00:40:38,020 And so we end up, then, cos times cos is cos squared. 680 00:40:38,020 --> 00:40:45,580 So notice that, although cos of course can go negative, 681 00:40:45,580 --> 00:40:46,660 it's periodic, isn't it? 682 00:40:46,660 --> 00:40:50,830 So it's negative half as long as it's positive. 683 00:40:50,830 --> 00:40:53,780 Cos squared is always positive. 684 00:40:53,780 --> 00:40:57,610 So the way this is defined, the Poynting vector 685 00:40:57,610 --> 00:41:00,320 is always going in the same direction. 686 00:41:00,320 --> 00:41:03,560 It's always, when you've got a plane wave, 687 00:41:03,560 --> 00:41:06,400 the energy is going in the direction of the direction 688 00:41:06,400 --> 00:41:07,720 of propagation of the wave. 689 00:41:13,700 --> 00:41:20,330 And OK, so this shows us how we get all this. 690 00:41:23,330 --> 00:41:27,570 Sorry, let me-- this shows us-- 691 00:41:27,570 --> 00:41:28,350 can I go back? 692 00:41:28,350 --> 00:41:30,350 Aw, yeah. 693 00:41:30,350 --> 00:41:32,830 Oh, I've gone back too far now. 694 00:41:32,830 --> 00:41:34,690 Here we are. 695 00:41:34,690 --> 00:41:38,290 This shows us how this Poynting vector varies 696 00:41:38,290 --> 00:41:40,120 in space and time, all right? 697 00:41:40,120 --> 00:41:45,730 So this Poynting vector has also got a sort of wave-type nature. 698 00:41:45,730 --> 00:41:47,410 But normally what we're interested in 699 00:41:47,410 --> 00:41:50,920 is not how it varies in space and time, 700 00:41:50,920 --> 00:41:54,950 but what the time-averaged form of that is going to look like. 701 00:41:54,950 --> 00:41:57,280 And so we can do a time average of it. 702 00:42:02,010 --> 00:42:05,810 And so that's what this thing here is. 703 00:42:05,810 --> 00:42:12,090 This s with the lines on either side is the time average of s. 704 00:42:12,090 --> 00:42:20,490 And so we've got b is k times e over omega. 705 00:42:20,490 --> 00:42:24,620 And so therefore we've got here e cross b. 706 00:42:24,620 --> 00:42:26,150 And we know what b is. 707 00:42:26,150 --> 00:42:27,680 So we can put that in. 708 00:42:27,680 --> 00:42:30,086 And we're going to get now e squared. 709 00:42:30,086 --> 00:42:32,820 All right, so finally what we get 710 00:42:32,820 --> 00:42:38,880 is that the Poynting vector is proportional to the time 711 00:42:38,880 --> 00:42:41,580 average of the electric field squared. 712 00:42:41,580 --> 00:42:43,333 AUDIENCE: [INAUDIBLE] 713 00:42:43,333 --> 00:42:44,250 COLIN SHEPPARD: Sorry? 714 00:42:44,250 --> 00:42:45,132 AUDIENCE: Let's take the average. 715 00:42:45,132 --> 00:42:46,455 We haven't done the average. 716 00:42:46,455 --> 00:42:48,622 COLIN SHEPPARD: Oh, we haven't done any average yet. 717 00:42:48,622 --> 00:42:49,950 Sorry, but this is the modulus. 718 00:42:49,950 --> 00:42:50,450 So-- 719 00:42:50,450 --> 00:42:52,170 AUDIENCE: Anyway [INAUDIBLE]. 720 00:42:52,170 --> 00:42:53,298 COLIN SHEPPARD: It's the-- 721 00:42:53,298 --> 00:42:54,630 AUDIENCE: So right now-- 722 00:42:54,630 --> 00:42:56,440 COLIN SHEPPARD: What do the two lines mean? 723 00:42:56,440 --> 00:42:58,946 AUDIENCE: The magnitude of the vector. 724 00:42:58,946 --> 00:43:01,154 COLIN SHEPPARD: OK, it's the magnitude of the vector. 725 00:43:01,154 --> 00:43:02,363 AUDIENCE: A function of time. 726 00:43:02,363 --> 00:43:04,470 COLIN SHEPPARD: It's still a function of time. 727 00:43:04,470 --> 00:43:08,800 OK, we'll take it as that, yeah. 728 00:43:08,800 --> 00:43:12,980 And yeah, OK. 729 00:43:16,120 --> 00:43:19,620 And I think, really, if it's in a medium, 730 00:43:19,620 --> 00:43:21,400 this is epsilon, not epsilon 0. 731 00:43:21,400 --> 00:43:22,010 Is that right? 732 00:43:22,010 --> 00:43:22,507 AUDIENCE: Yeah. 733 00:43:22,507 --> 00:43:23,382 COLIN SHEPPARD: Yeah. 734 00:43:26,980 --> 00:43:29,830 OK, so now we're going to do the time averaging. 735 00:43:29,830 --> 00:43:33,880 So this e, e is a time-varying quantity. 736 00:43:33,880 --> 00:43:37,140 e varies with both space and time. 737 00:43:37,140 --> 00:43:45,870 And so this is the modular square of it, of the vector. 738 00:43:45,870 --> 00:43:47,940 So we put that in. 739 00:43:47,940 --> 00:43:53,655 And we get, then, the s is equal to something we cos squared. 740 00:43:53,655 --> 00:43:55,780 So it's not much different from what we had before. 741 00:43:58,440 --> 00:44:02,760 OK, and then the next thing we do is to do the time averaging. 742 00:44:02,760 --> 00:44:10,770 So this is something which is positive but oscillating, 743 00:44:10,770 --> 00:44:12,270 time-varying. 744 00:44:12,270 --> 00:44:16,480 And the speed of these variations is very, very fast. 745 00:44:16,480 --> 00:44:21,600 We've said before the frequency of light is around 10 746 00:44:21,600 --> 00:44:27,300 to the 15 Hertz, which is faster than any detector 747 00:44:27,300 --> 00:44:30,640 that we have to measure the fields directly, all right? 748 00:44:30,640 --> 00:44:33,990 So you can't actually measure the electric field 749 00:44:33,990 --> 00:44:39,930 or the power of an optical wave because the frequency's 750 00:44:39,930 --> 00:44:41,580 so high. 751 00:44:41,580 --> 00:44:48,120 So a normal detector, like a photo diode or whatever, 752 00:44:48,120 --> 00:44:51,180 would be measuring some time average of this 753 00:44:51,180 --> 00:44:52,920 because there's no way it can respond 754 00:44:52,920 --> 00:44:55,150 to this sort of frequency. 755 00:44:55,150 --> 00:44:58,440 So we do the time average. 756 00:44:58,440 --> 00:45:02,590 And so this is how we can do the time average, right? 757 00:45:02,590 --> 00:45:06,600 The square things mean time average, the caret signs. 758 00:45:06,600 --> 00:45:11,040 And the time average, you integrate this over a long time 759 00:45:11,040 --> 00:45:15,440 and divide by the time to get the time average. 760 00:45:15,440 --> 00:45:20,140 And so this is called-- 761 00:45:20,140 --> 00:45:22,070 well, usually called intensity. 762 00:45:22,070 --> 00:45:27,110 I think that "intensity" is now frowned upon by purists. 763 00:45:27,110 --> 00:45:31,220 And they come up with new names every now and then. 764 00:45:31,220 --> 00:45:36,770 So "irradiance" is a word that you often read nowadays 765 00:45:36,770 --> 00:45:38,460 in the literature. 766 00:45:38,460 --> 00:45:40,970 And yeah, so here it's measured in watts 767 00:45:40,970 --> 00:45:47,240 per square meter, which is obviously a unit of power 768 00:45:47,240 --> 00:45:49,700 density, all right? 769 00:45:49,700 --> 00:45:52,130 There's one other thing that's worth mentioning here. 770 00:45:52,130 --> 00:45:57,200 And that is actually that normal detectors actually really 771 00:45:57,200 --> 00:46:00,250 don't measure the Poynting vector. 772 00:46:00,250 --> 00:46:04,880 They actually usually measure the electric energy density. 773 00:46:04,880 --> 00:46:09,830 And there's a thing called Poynting's theorem, which, 774 00:46:09,830 --> 00:46:12,080 if you study electromagnetism, will 775 00:46:12,080 --> 00:46:16,490 show that the power flow out of a volume 776 00:46:16,490 --> 00:46:20,090 is equal to the rate of change of the stored energy. 777 00:46:20,090 --> 00:46:21,770 And the stored energy is made up of 778 00:46:21,770 --> 00:46:23,330 electric and magnetic energy. 779 00:46:23,330 --> 00:46:25,760 So all these things are related. 780 00:46:25,760 --> 00:46:31,100 And it turns out that for all practical purposes normally 781 00:46:31,100 --> 00:46:35,270 it doesn't really matter whether you're measuring the Poynting 782 00:46:35,270 --> 00:46:40,700 vector or the electric energy density or whatever. 783 00:46:40,700 --> 00:46:43,160 It's going to give the same sort of answer. 784 00:46:43,160 --> 00:46:46,100 But there are cases where you have to be careful. 785 00:46:50,070 --> 00:46:53,850 OK, and then finally we've got to do this time average of cos 786 00:46:53,850 --> 00:46:55,450 squared. 787 00:46:55,450 --> 00:47:00,270 And the time average of cos squared is just a half. 788 00:47:00,270 --> 00:47:02,660 You remember how you do that? 789 00:47:02,660 --> 00:47:06,900 Cos squared, you go into double angles. 790 00:47:06,900 --> 00:47:09,300 And then you get a cosine that cancels out. 791 00:47:09,300 --> 00:47:11,760 And you're just left with the constant bit. 792 00:47:11,760 --> 00:47:14,970 And then, finally then, you put that half in. 793 00:47:14,970 --> 00:47:18,090 And we've now got an expression for the intensity, 794 00:47:18,090 --> 00:47:23,100 defined as being the time average of the Poynting vector 795 00:47:23,100 --> 00:47:30,080 as being equal to the amplitude of the electric field squared, 796 00:47:30,080 --> 00:47:34,520 multiplied by some constant, which, 797 00:47:34,520 --> 00:47:37,680 in probably 99 cases out of 100, we don't even 798 00:47:37,680 --> 00:47:39,840 bother to even think what that constant 799 00:47:39,840 --> 00:47:43,390 is, let alone thinking of whether there's a half there. 800 00:47:43,390 --> 00:47:46,890 So you'll see loads of books or papers 801 00:47:46,890 --> 00:47:48,840 where they just say that the intensity is 802 00:47:48,840 --> 00:47:53,420 equal to e squared, which is strictly not true, I guess, 803 00:47:53,420 --> 00:47:54,650 but people do say. 804 00:47:58,690 --> 00:48:00,400 So how are we going? 805 00:48:00,400 --> 00:48:06,570 We got one hour, yeah. 806 00:48:06,570 --> 00:48:11,490 OK, so that's how you can calculate the power 807 00:48:11,490 --> 00:48:16,060 flow for a plane wave. 808 00:48:16,060 --> 00:48:18,470 I guess the method is not only going 809 00:48:18,470 --> 00:48:19,760 to be true for plane waves. 810 00:48:19,760 --> 00:48:23,350 But you can apply it for other sorts of waves, too. 811 00:48:23,350 --> 00:48:27,220 But the next thing is you might think back to what 812 00:48:27,220 --> 00:48:29,380 we were doing with phases. 813 00:48:29,380 --> 00:48:31,330 And there are a few things that are 814 00:48:31,330 --> 00:48:34,030 important to realize here, what you can do 815 00:48:34,030 --> 00:48:37,280 and what you can't do. 816 00:48:37,280 --> 00:48:39,470 You remember this is what we said from our phases. 817 00:48:39,470 --> 00:48:44,990 We said that what we actually measure in the real world 818 00:48:44,990 --> 00:48:46,730 is a wave that's like this. 819 00:48:46,730 --> 00:48:52,030 It's a cosine dependence in space and time. 820 00:48:52,030 --> 00:48:57,250 And we say that we use this complex representative 821 00:48:57,250 --> 00:49:00,460 where we say that this is the real part 822 00:49:00,460 --> 00:49:03,940 of some complex exponential. 823 00:49:03,940 --> 00:49:06,370 And if you remember, the reason for doing that 824 00:49:06,370 --> 00:49:08,560 is that these complex exponentials 825 00:49:08,560 --> 00:49:11,770 are much easier to manipulate. 826 00:49:11,770 --> 00:49:15,160 It means that you don't have to remember all those horrible cos 827 00:49:15,160 --> 00:49:17,900 of a plus b things and so on. 828 00:49:17,900 --> 00:49:23,220 So it's much easier to do the algebra using these. 829 00:49:23,220 --> 00:49:27,100 So what you do normally, as you remember, 830 00:49:27,100 --> 00:49:32,700 is you do all your algebra with each of the i-somethings. 831 00:49:32,700 --> 00:49:35,910 And then right at the very end normally, 832 00:49:35,910 --> 00:49:39,420 you find the real part to find what 833 00:49:39,420 --> 00:49:45,060 the real field is in real space, as we're used to. 834 00:49:45,060 --> 00:49:48,990 But here, now you've got to be very careful because you 835 00:49:48,990 --> 00:49:53,480 saw that the Poynting vector actually was cos squared, 836 00:49:53,480 --> 00:49:54,120 all right? 837 00:49:54,120 --> 00:49:57,450 So this is a nonlinear. 838 00:49:57,450 --> 00:50:01,170 We performed a nonlinear operation. 839 00:50:01,170 --> 00:50:04,050 And if you do nonlinear operations, 840 00:50:04,050 --> 00:50:06,300 the same would be true for nonlinear optics, 841 00:50:06,300 --> 00:50:07,350 as well, actually. 842 00:50:07,350 --> 00:50:11,310 You can't do it in this complex notation 843 00:50:11,310 --> 00:50:13,830 directly because you just get the wrong answer. 844 00:50:13,830 --> 00:50:20,400 If you squared this, you get an e to the 2i kz minus 5, 845 00:50:20,400 --> 00:50:23,010 wouldn't you? 846 00:50:23,010 --> 00:50:25,980 Now the real part of that is not what we want. 847 00:50:29,100 --> 00:50:35,600 So what you've really got to do is to work through, I guess-- 848 00:50:35,600 --> 00:50:39,120 well, the surest way of doing it is to work through in terms 849 00:50:39,120 --> 00:50:46,440 of the cosines and so on rather than to go into e to the i's. 850 00:50:46,440 --> 00:50:48,600 So this is saying a bit more about this. 851 00:50:51,590 --> 00:50:56,010 So what we've got, then, is this is showing how you can do it 852 00:50:56,010 --> 00:50:58,530 in terms of the cosines. 853 00:50:58,530 --> 00:51:02,460 So you can see that, for example, here we've 854 00:51:02,460 --> 00:51:05,090 got two fields. 855 00:51:05,090 --> 00:51:07,870 And let's say that you see the point is 856 00:51:07,870 --> 00:51:11,450 that, although the fields are additive, 857 00:51:11,450 --> 00:51:18,700 the power of those two waves is not 858 00:51:18,700 --> 00:51:21,400 going to be equal to the sum of the powers of the two waves, 859 00:51:21,400 --> 00:51:22,120 all right? 860 00:51:22,120 --> 00:51:28,000 So we can say that the e's would add coherently. 861 00:51:28,000 --> 00:51:30,160 And then we can work out the power flow 862 00:51:30,160 --> 00:51:32,530 of the sum of these two fields. 863 00:51:32,530 --> 00:51:35,190 And you can see of course, expanding this square, 864 00:51:35,190 --> 00:51:37,750 you're going to have the power of the two. 865 00:51:37,750 --> 00:51:42,610 But you're also going to get this cross term, 866 00:51:42,610 --> 00:51:45,280 the cross product term, which depends-- 867 00:51:45,280 --> 00:51:46,390 you can see here-- 868 00:51:46,390 --> 00:51:48,730 on the relative phase of the two waves. 869 00:51:48,730 --> 00:51:50,830 If they're out of phase by 90 degrees, 870 00:51:50,830 --> 00:51:53,000 then that will go to 0. 871 00:51:53,000 --> 00:51:59,020 But if they're in phase, this of course is going to be strong. 872 00:51:59,020 --> 00:52:04,720 And so that's one way of doing it. 873 00:52:04,720 --> 00:52:08,630 But in terms of phases, then, you 874 00:52:08,630 --> 00:52:10,280 can do it in terms of phases. 875 00:52:10,280 --> 00:52:13,170 I'm just saying you have to be careful how you do it. 876 00:52:13,170 --> 00:52:19,610 And so you have to add the phases first and then find 877 00:52:19,610 --> 00:52:21,380 the modulus square. 878 00:52:21,380 --> 00:52:25,130 You can't actually do the squaring 879 00:52:25,130 --> 00:52:27,810 and then do the real power, all right? 880 00:52:27,810 --> 00:52:30,330 So you add these two phases together, 881 00:52:30,330 --> 00:52:33,110 which gives the total field. 882 00:52:33,110 --> 00:52:36,650 And then the intensity is going to be 883 00:52:36,650 --> 00:52:38,350 the modulus square of this. 884 00:52:38,350 --> 00:52:39,500 And you expand it. 885 00:52:39,500 --> 00:52:43,860 And you can see we've got exactly the same answer. 886 00:52:43,860 --> 00:52:46,190 So it doesn't matter which way you do it, 887 00:52:46,190 --> 00:52:48,470 as long as you do it one of those ways. 888 00:52:48,470 --> 00:52:52,340 What you don't want to do is to try squaring the e-to-the-i 889 00:52:52,340 --> 00:52:55,670 things and then taking the real power because you'll get 890 00:52:55,670 --> 00:52:57,700 something different then. 891 00:52:57,700 --> 00:53:00,700 And that is the very last slide. 892 00:53:00,700 --> 00:53:03,250 And that is dead on time. 893 00:53:06,360 --> 00:53:08,300 So, well, whoa, we've got a couple 894 00:53:08,300 --> 00:53:09,550 of minutes for some questions. 895 00:53:09,550 --> 00:53:11,850 So any questions? 896 00:53:11,850 --> 00:53:13,460 Yes, Sharon? 897 00:53:13,460 --> 00:53:14,930 AUDIENCE: [INAUDIBLE]. 898 00:53:18,360 --> 00:53:19,340 Hello? 899 00:53:19,340 --> 00:53:21,820 I'm quite curious about the Poynting vector. 900 00:53:21,820 --> 00:53:24,200 And you mentioned that it's actually a cross term, 901 00:53:24,200 --> 00:53:26,880 and it's similar like propagation and direction, 902 00:53:26,880 --> 00:53:28,640 like where the energy flows. 903 00:53:28,640 --> 00:53:32,900 But what about the peculiar case in evanescent waves? 904 00:53:32,900 --> 00:53:36,042 Does it mean the energy is not propagating there? 905 00:53:36,042 --> 00:53:37,250 COLIN SHEPPARD: Oh, oh, dear. 906 00:53:37,250 --> 00:53:38,510 Oh, yes. 907 00:53:38,510 --> 00:53:41,810 Evanescent waves, you're getting onto something 908 00:53:41,810 --> 00:53:42,880 really controversial-- 909 00:53:42,880 --> 00:53:43,640 AUDIENCE: No, I'm thinking-- 910 00:53:43,640 --> 00:53:44,960 COLIN SHEPPARD: --here, I think-- 911 00:53:44,960 --> 00:53:45,470 AUDIENCE: [LAUGHS] 912 00:53:45,470 --> 00:53:46,040 COLIN SHEPPARD: --actually. 913 00:53:46,040 --> 00:53:47,852 AUDIENCE: Well, where has the energy gone then? 914 00:53:47,852 --> 00:53:48,769 COLIN SHEPPARD: Sorry? 915 00:53:48,769 --> 00:53:50,810 AUDIENCE: I mean, if the Poynting vector is 916 00:53:50,810 --> 00:53:52,880 the direction of the energy flow, 917 00:53:52,880 --> 00:53:55,830 then it seems like there's no propagation in energy 918 00:53:55,830 --> 00:53:57,935 in the evanescent wave then-- 919 00:53:57,935 --> 00:53:59,450 COLIN SHEPPARD: Well, it is normally 920 00:53:59,450 --> 00:54:02,310 said that there is no energy propagation 921 00:54:02,310 --> 00:54:03,490 in an evanescent wave. 922 00:54:03,490 --> 00:54:06,790 But you know, in a way, there is. 923 00:54:06,790 --> 00:54:10,280 There's sort of transverse propagation of energy. 924 00:54:10,280 --> 00:54:14,420 But there's no energy flow across-- 925 00:54:14,420 --> 00:54:17,390 if you're doing total internal reflection 926 00:54:17,390 --> 00:54:19,460 and you're looking at the steady state, 927 00:54:19,460 --> 00:54:22,310 there's no energy flow across the boundary, 928 00:54:22,310 --> 00:54:24,300 I think is true to say. 929 00:54:24,300 --> 00:54:27,350 Is that true, or-- 930 00:54:27,350 --> 00:54:28,100 yeah, this is-- 931 00:54:28,100 --> 00:54:28,340 GEORGE BARBASTATHIS: Yes. 932 00:54:28,340 --> 00:54:30,673 COLIN SHEPPARD: --getting a bit controversial, isn't it? 933 00:54:30,673 --> 00:54:32,650 GEORGE BARBASTATHIS: Yes, you can-- 934 00:54:32,650 --> 00:54:33,950 if you have the interface-- 935 00:54:33,950 --> 00:54:35,700 I don't know if anyone can see, but if you 936 00:54:35,700 --> 00:54:39,740 have the interface of the TAR, if you compute the Poynting 937 00:54:39,740 --> 00:54:42,800 vector in the vertical direction, those have to be 0. 938 00:54:42,800 --> 00:54:44,140 The time average is 0. 939 00:54:44,140 --> 00:54:44,360 COLIN SHEPPARD: Yeah, yeah. 940 00:54:44,360 --> 00:54:46,610 GEORGE BARBASTATHIS: So you don't have energy flow. 941 00:54:46,610 --> 00:54:48,440 But you do have energy density. 942 00:54:48,440 --> 00:54:52,160 COLIN SHEPPARD: Yeah, yeah. 943 00:54:52,160 --> 00:54:53,690 And the other thing is, you know, 944 00:54:53,690 --> 00:54:56,060 the question of if you've got-- and this is 945 00:54:56,060 --> 00:54:57,600 your total internal reflection. 946 00:54:57,600 --> 00:55:00,230 So here you get this evanescent wave. 947 00:55:00,230 --> 00:55:04,130 Now, of course, all these questions 948 00:55:04,130 --> 00:55:08,210 of these evanescent waves and so on, they all 949 00:55:08,210 --> 00:55:13,760 rely on assuming infinite plane waves and infinite surfaces 950 00:55:13,760 --> 00:55:17,180 and things like that, which of course none of these things 951 00:55:17,180 --> 00:55:20,670 can really be strictly true in practice. 952 00:55:20,670 --> 00:55:23,690 But if we do this, then we'd expect 953 00:55:23,690 --> 00:55:25,340 to get something which is actually 954 00:55:25,340 --> 00:55:27,420 traveling in this direction. 955 00:55:27,420 --> 00:55:31,820 And so in some sense, I think that there 956 00:55:31,820 --> 00:55:36,660 is some sort of energy flow in the transverse direction. 957 00:55:36,660 --> 00:55:40,460 But I'm just having a big argument about this, not 958 00:55:40,460 --> 00:55:44,090 for this case, but to do with some other project 959 00:55:44,090 --> 00:55:47,360 we're doing at the moment where it's 960 00:55:47,360 --> 00:55:52,130 to do with when the h vector is 0. 961 00:55:52,130 --> 00:55:56,097 And so the question is, is there energy flow? 962 00:55:56,097 --> 00:55:57,180 Well, I guess there isn't. 963 00:55:57,180 --> 00:56:00,380 But I think there's also this question 964 00:56:00,380 --> 00:56:06,860 of whether Poynting vector really always means the energy 965 00:56:06,860 --> 00:56:09,470 flow. 966 00:56:09,470 --> 00:56:12,200 In fact, I remember having a conversation with Emil Wolf 967 00:56:12,200 --> 00:56:14,240 once about this. 968 00:56:14,240 --> 00:56:19,170 And, well, he said, let me give you an example. 969 00:56:19,170 --> 00:56:25,520 Imagine you have a DC planar electric field 970 00:56:25,520 --> 00:56:31,040 in this direction, and you have a DC planar magnetic field 971 00:56:31,040 --> 00:56:32,720 in this direction. 972 00:56:32,720 --> 00:56:34,580 Is there any energy flow? 973 00:56:34,580 --> 00:56:36,320 No. 974 00:56:36,320 --> 00:56:39,380 But you can work out what e cross h is. 975 00:56:39,380 --> 00:56:44,510 And you know, so I think that he put this forward 976 00:56:44,510 --> 00:56:51,800 as an example which makes you wonder whether it necessarily 977 00:56:51,800 --> 00:56:54,110 really always means energy flow. 978 00:56:54,110 --> 00:56:55,820 We were talking about this in the lab 979 00:56:55,820 --> 00:56:58,730 today because people have come up 980 00:56:58,730 --> 00:57:04,520 with these solutions where, for example, energy flow sometimes 981 00:57:04,520 --> 00:57:06,340 does this. 982 00:57:06,340 --> 00:57:10,030 So you might get the lines of-- 983 00:57:10,030 --> 00:57:14,140 the Poynting vector might do something like this. 984 00:57:18,513 --> 00:57:19,930 So here you can see that there has 985 00:57:19,930 --> 00:57:24,920 to be some sort of 0 around here because this is going this way, 986 00:57:24,920 --> 00:57:27,020 and this is going this way. 987 00:57:27,020 --> 00:57:30,730 And here you can actually get these, sort of, 988 00:57:30,730 --> 00:57:36,730 closed eddies of Poynting vector that's going round in circles. 989 00:57:36,730 --> 00:57:39,430 Now, whether this really represents 990 00:57:39,430 --> 00:57:42,370 something that's physical or not, I don't really know. 991 00:57:42,370 --> 00:57:44,920 You know, in water flow, I guess you could get this. 992 00:57:44,920 --> 00:57:47,410 There's no reason why you couldn't get eddies 993 00:57:47,410 --> 00:57:49,840 of water going in circles. 994 00:57:49,840 --> 00:57:53,080 But you know, so this is what you get. 995 00:57:53,080 --> 00:57:54,800 This is what you get if you've just 996 00:57:54,800 --> 00:58:00,450 got a very simple example of a lens focusing the light. 997 00:58:00,450 --> 00:58:06,530 You get this sort of pattern if I look at this in some detail. 998 00:58:06,530 --> 00:58:11,760 You get this sort of behavior in this region here. 999 00:58:11,760 --> 00:58:15,090 And there've been papers that have described this. 1000 00:58:15,090 --> 00:58:18,180 But what it really means in practice-- 1001 00:58:18,180 --> 00:58:24,000 well, I guess nothing really means anything 1002 00:58:24,000 --> 00:58:25,770 unless you can measure it. 1003 00:58:25,770 --> 00:58:27,420 So the question is, can you come up 1004 00:58:27,420 --> 00:58:30,630 with a detector that measures Poynting vector? 1005 00:58:30,630 --> 00:58:32,130 And I don't know the answer to that. 1006 00:58:32,130 --> 00:58:36,810 As I said earlier, I think normally detectors measure 1007 00:58:36,810 --> 00:58:38,670 electric energy density. 1008 00:58:38,670 --> 00:58:42,510 So people have looked at what happens in this focused 1009 00:58:42,510 --> 00:58:43,920 region of a lens. 1010 00:58:43,920 --> 00:58:47,160 And there are various ways you can look at that. 1011 00:58:47,160 --> 00:58:50,290 People have used near-field optics, 1012 00:58:50,290 --> 00:58:53,190 you know, a tapered fiber to look at it. 1013 00:58:53,190 --> 00:58:58,290 Or you can do some sort of tomography-type experiments 1014 00:58:58,290 --> 00:59:01,860 with a detector that you rotate and things like that. 1015 00:59:01,860 --> 00:59:07,820 But to actually measure the Poynting vector, I don't know. 1016 00:59:07,820 --> 00:59:10,500 Right, any questions from over there?