1 00:00:00,135 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,850 Commons license. 3 00:00:03,850 --> 00:00:06,060 Your support will help MIT OpenCourseWare 4 00:00:06,060 --> 00:00:10,150 continue to offer high quality educational resources for free. 5 00:00:10,150 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,650 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,650 --> 00:00:18,012 at ocu.mit.edu. 8 00:00:21,347 --> 00:00:21,930 PROFESSOR: OK. 9 00:00:21,930 --> 00:00:26,700 First of all, I was in touch by email with George, 10 00:00:26,700 --> 00:00:31,890 and he asked me to flash this up again, which 11 00:00:31,890 --> 00:00:34,530 was from a couple of lectures ago, 12 00:00:34,530 --> 00:00:38,910 because he thought this was very critical and very important 13 00:00:38,910 --> 00:00:42,220 that you sort of understood what it says. 14 00:00:42,220 --> 00:00:44,790 So I'm going to just say a few things about it. 15 00:00:44,790 --> 00:00:47,490 I guess the first thing to note is 16 00:00:47,490 --> 00:00:52,080 that it compares these two different cases, 17 00:00:52,080 --> 00:00:55,710 coherent imaging and incoherent imaging. 18 00:00:55,710 --> 00:00:59,520 And you see that, apart from that, these two, the top half 19 00:00:59,520 --> 00:01:02,670 and the bottom half, are really identical. 20 00:01:02,670 --> 00:01:07,440 So you use the same method in either case, 21 00:01:07,440 --> 00:01:10,260 except that, of course, in the coherent case you're 22 00:01:10,260 --> 00:01:13,770 working with amplitudes, whereas in the incoherent case 23 00:01:13,770 --> 00:01:15,900 you're working with intensities. 24 00:01:15,900 --> 00:01:19,360 But apart from that, it's identical. 25 00:01:19,360 --> 00:01:24,300 So what it's saying is, if you illuminate a thin transparency, 26 00:01:24,300 --> 00:01:30,390 then you get a field on the far side of this transparency which 27 00:01:30,390 --> 00:01:33,310 goes into your optical system. 28 00:01:33,310 --> 00:01:36,010 So this is his GN. 29 00:01:36,010 --> 00:01:39,640 And to calculate the image, you can either 30 00:01:39,640 --> 00:01:45,670 do it by convolving that with a point spread function 31 00:01:45,670 --> 00:01:51,280 to get the amplitude out, or you can do it in the Fourier domain 32 00:01:51,280 --> 00:01:55,480 by Fourier transforming to get the object spectra, 33 00:01:55,480 --> 00:02:00,430 and then multiplying it by a transfer function 34 00:02:00,430 --> 00:02:04,093 and then getting the field out. 35 00:02:04,093 --> 00:02:06,010 And then once you've got the field, of course, 36 00:02:06,010 --> 00:02:08,590 you can calculate the intensity from that 37 00:02:08,590 --> 00:02:10,930 just by doing the modulus square. 38 00:02:10,930 --> 00:02:13,810 So the difference between these two things 39 00:02:13,810 --> 00:02:16,900 is here we working with aptitudes, 40 00:02:16,900 --> 00:02:20,290 so we have a coherent point spread function 41 00:02:20,290 --> 00:02:26,470 and an amplitude transfer function, 42 00:02:26,470 --> 00:02:30,070 whereas for this one, the incoherent case, 43 00:02:30,070 --> 00:02:33,640 we have an intensity point spread function, which is just 44 00:02:33,640 --> 00:02:36,400 the modulus square of this. 45 00:02:36,400 --> 00:02:40,180 And here we have an optical transfer function, which 46 00:02:40,180 --> 00:02:44,570 is the autocorrelation of this. 47 00:02:44,570 --> 00:02:47,930 So that's really all there is to do, 48 00:02:47,930 --> 00:02:54,670 all there is that you need to know about calculating images. 49 00:02:54,670 --> 00:02:57,250 Any questions about that? 50 00:02:57,250 --> 00:03:00,040 So I'll just-- as I say, I'll just 51 00:03:00,040 --> 00:03:04,880 flash that because George said he wanted me to. 52 00:03:04,880 --> 00:03:12,310 And now we'll go on to where we got up to last time 53 00:03:12,310 --> 00:03:14,450 in the lecture. 54 00:03:14,450 --> 00:03:20,870 So George suggested-- he sent me some more material for today, 55 00:03:20,870 --> 00:03:27,590 and he also asked me to add some extra to it of my own material. 56 00:03:27,590 --> 00:03:30,860 But I don't know how far we get with this. 57 00:03:30,860 --> 00:03:34,640 So we've still got some left over from last time, actually, 58 00:03:34,640 --> 00:03:38,300 and then the bit that George sent. 59 00:03:38,300 --> 00:03:40,100 You might be fed up by then. 60 00:03:40,100 --> 00:03:42,020 But anyway, if we've still got some time 61 00:03:42,020 --> 00:03:44,510 and you're not fed up, I'll carry on with some stuff 62 00:03:44,510 --> 00:03:49,440 that I put in there which is to do with focusing. 63 00:03:49,440 --> 00:03:53,820 But it might be-- you might think 64 00:03:53,820 --> 00:03:56,820 it's sort of rather higher level than the rest of the course. 65 00:03:56,820 --> 00:03:59,210 So this is not for-- 66 00:03:59,210 --> 00:04:02,810 this bit that I've added is not for examining or anything. 67 00:04:02,810 --> 00:04:05,940 It's meant just for your interest. 68 00:04:05,940 --> 00:04:12,710 Anyway, this is where we got up to, the defocus ATF. 69 00:04:12,710 --> 00:04:15,855 So this, again, you remember-- 70 00:04:15,855 --> 00:04:16,730 let's make it bigger. 71 00:04:22,078 --> 00:04:24,300 Now, what's going on? 72 00:04:24,300 --> 00:04:26,555 I've gone the wrong thing. 73 00:04:26,555 --> 00:04:27,430 I should [INAUDIBLE]. 74 00:04:27,430 --> 00:04:27,580 Sorry. 75 00:04:27,580 --> 00:04:28,747 I haven't got my glasses on. 76 00:04:28,747 --> 00:04:31,290 I can't see what I'm doing. 77 00:04:31,290 --> 00:04:32,870 And it's all in a different place 78 00:04:32,870 --> 00:04:37,730 from across the version of PDF that-- 79 00:04:37,730 --> 00:04:39,500 PowerPoint that I use. 80 00:04:39,500 --> 00:04:43,520 Anyway, here we are, the defocus ATF. 81 00:04:43,520 --> 00:04:46,760 So you remember, we introduced this last time. 82 00:04:46,760 --> 00:04:50,150 We said that if you've got a defocused object, 83 00:04:50,150 --> 00:04:54,800 then you can work out the imaging of it 84 00:04:54,800 --> 00:04:59,280 by using an amplitude transfer function just like before, 85 00:04:59,280 --> 00:05:01,640 but it's not just the normal one. 86 00:05:01,640 --> 00:05:07,550 It's modified by the effects of defocus. 87 00:05:07,550 --> 00:05:11,600 What happens is that you have to multiply the transfer 88 00:05:11,600 --> 00:05:17,180 function by complex exponential, e to the i, this thing. 89 00:05:17,180 --> 00:05:20,300 It therefore becomes complex. 90 00:05:20,300 --> 00:05:24,050 And you can divide that up into two parts, a real part 91 00:05:24,050 --> 00:05:26,090 and an imaginary part. 92 00:05:26,090 --> 00:05:29,300 The real part will image the real information 93 00:05:29,300 --> 00:05:31,190 of the objects, which is basically 94 00:05:31,190 --> 00:05:33,930 the absorption of the object. 95 00:05:33,930 --> 00:05:37,280 And that's what this shows here, the real part of that transfer 96 00:05:37,280 --> 00:05:41,210 function, which is cosine of this thing. 97 00:05:41,210 --> 00:05:47,400 And so this is the cutoff of the optical system here. 98 00:05:47,400 --> 00:05:47,900 Sorry. 99 00:05:47,900 --> 00:05:49,700 No, it isn't, is it? 100 00:05:49,700 --> 00:05:50,240 No, sorry. 101 00:05:50,240 --> 00:05:53,420 The cutoff, I guess, is some arbitrary place 102 00:05:53,420 --> 00:05:55,760 I guess it's not actually shown here. 103 00:05:55,760 --> 00:05:58,430 But anyway, the fact is that as you 104 00:05:58,430 --> 00:06:01,340 change the amount of defocus, this 105 00:06:01,340 --> 00:06:05,690 will be scaled accordingly relative to the cutoff 106 00:06:05,690 --> 00:06:08,490 of the transfer function. 107 00:06:08,490 --> 00:06:10,700 So you can imagine if the transfer function was 108 00:06:10,700 --> 00:06:14,300 right in here, it would be almost as though it was 109 00:06:14,300 --> 00:06:18,350 the same as an in-focus system. 110 00:06:18,350 --> 00:06:25,010 And then as you make the cutoff wider and wider 111 00:06:25,010 --> 00:06:28,340 relative to this cos curve, you see you getting 112 00:06:28,340 --> 00:06:30,050 more and more of these wiggles. 113 00:06:30,050 --> 00:06:36,390 So this was a case where it's not changing very much. 114 00:06:36,390 --> 00:06:37,230 So there we are. 115 00:06:37,230 --> 00:06:41,050 There's the region where it's non-zero, 116 00:06:41,050 --> 00:06:44,380 and so it's not changing very much in that region. 117 00:06:44,380 --> 00:06:48,680 And so we said that corresponds to multifocus. 118 00:06:48,680 --> 00:06:55,110 And then the next case we looked at 119 00:06:55,110 --> 00:06:58,890 was now when it's got a lot of defocus. 120 00:06:58,890 --> 00:07:02,520 So you can see here, it starts really wiggling. 121 00:07:02,520 --> 00:07:04,410 Remember-- I said this last time-- 122 00:07:04,410 --> 00:07:07,960 this is actually cos of this thing squared. 123 00:07:07,960 --> 00:07:11,510 So that's why it doesn't look like an ordinary cos function, 124 00:07:11,510 --> 00:07:13,130 because of the x squared. 125 00:07:13,130 --> 00:07:17,850 So it scales it in this nonlinear way. 126 00:07:17,850 --> 00:07:20,610 And so here's the and it cut off again in this case. 127 00:07:20,610 --> 00:07:23,280 Now, in this case you've got some wiggles, 128 00:07:23,280 --> 00:07:30,660 and those wiggles are going to lead to artifacts in the image. 129 00:07:30,660 --> 00:07:34,660 Firstly, because of the fact this goes negative, 130 00:07:34,660 --> 00:07:37,300 you can see there's some spatial frequencies will 131 00:07:37,300 --> 00:07:39,800 be imaged with the wrong sign. 132 00:07:39,800 --> 00:07:43,210 So it's just like if you've got a Fourier 133 00:07:43,210 --> 00:07:47,020 series for, let's say, a square wave, if you take the Fourier 134 00:07:47,020 --> 00:07:50,590 series for a square wave and you change the sign of some 135 00:07:50,590 --> 00:07:53,920 of those components, it doesn't look like a square wave 136 00:07:53,920 --> 00:07:55,510 anymore. 137 00:07:55,510 --> 00:07:58,610 So you get a very poor image in that case. 138 00:07:58,610 --> 00:08:01,860 So this is really bad news for the image. 139 00:08:01,860 --> 00:08:06,370 And the other thing that George mentions up there 140 00:08:06,370 --> 00:08:07,930 is that, of course, there's regions 141 00:08:07,930 --> 00:08:11,390 where the value of the transfer function is very small. 142 00:08:11,390 --> 00:08:15,100 So this spatial frequency here, for example, is not 143 00:08:15,100 --> 00:08:16,600 going to be imaged at all. 144 00:08:16,600 --> 00:08:21,260 So you'll get some gaps in the response 145 00:08:21,260 --> 00:08:26,321 of the different transfer of spatial frequencies. 146 00:08:29,520 --> 00:08:34,500 So the significance of depth of focus in imaging-- 147 00:08:34,500 --> 00:08:37,530 so here he shows a 4F system. 148 00:08:37,530 --> 00:08:41,700 It shows here the NA in and the NA out. 149 00:08:41,700 --> 00:08:44,520 And you remember, of course, that as you changed the NA, 150 00:08:44,520 --> 00:08:46,650 you also change the magnification. 151 00:08:46,650 --> 00:08:49,540 You change the lateral magnification, 152 00:08:49,540 --> 00:08:54,210 and you also change the longitudinal or axial 153 00:08:54,210 --> 00:08:55,410 magnification. 154 00:08:55,410 --> 00:08:58,360 That goes as the NA squared. 155 00:08:58,360 --> 00:09:01,710 And so we define these two terms. 156 00:09:01,710 --> 00:09:05,910 Depth of field is basically that the defocus tolerance 157 00:09:05,910 --> 00:09:09,810 in the object space, so far you can move the objects 158 00:09:09,810 --> 00:09:14,030 without it going too much out of focus. 159 00:09:14,030 --> 00:09:21,760 And so this gives some idea about the value of that, then. 160 00:09:21,760 --> 00:09:24,490 The depth that depth of a field is 161 00:09:24,490 --> 00:09:28,975 given by this lambda over 2NA squared. 162 00:09:31,990 --> 00:09:37,300 You get this NA squared because the longitudinal magnification 163 00:09:37,300 --> 00:09:40,320 also goes as NA squared. 164 00:09:40,320 --> 00:09:42,980 And then on the other side, in the image side, 165 00:09:42,980 --> 00:09:46,160 you have a depth of focus rather than a depth of field. 166 00:09:46,160 --> 00:09:50,300 Notice in the title, he calls it DoF, which can mean either, 167 00:09:50,300 --> 00:09:52,290 so Depth of Field, Depth of Focus. 168 00:09:52,290 --> 00:09:57,200 You'll find that, of course, in general terminology people 169 00:09:57,200 --> 00:10:00,520 often use these phrases almost interchangeably 170 00:10:00,520 --> 00:10:05,250 and are not very careful about using the right one 171 00:10:05,250 --> 00:10:07,070 for the right space. 172 00:10:07,070 --> 00:10:08,870 But this is what it really means. 173 00:10:08,870 --> 00:10:12,050 Depth of field is how much you can move the object. 174 00:10:12,050 --> 00:10:17,120 Depth of focus is how much you can move the screen here 175 00:10:17,120 --> 00:10:19,220 without it going out of focus. 176 00:10:19,220 --> 00:10:27,420 And you can see that this is expressing the depth of focus 177 00:10:27,420 --> 00:10:30,120 now in terms of numerical aperture. 178 00:10:30,120 --> 00:10:31,830 You get the same expression, but it's now 179 00:10:31,830 --> 00:10:34,540 this NA rather than that one. 180 00:10:34,540 --> 00:10:37,440 But of course, the magnification also 181 00:10:37,440 --> 00:10:40,680 goes as NA squared as well. 182 00:10:40,680 --> 00:10:45,210 So the image of the bit that's in focus, here, of the object 183 00:10:45,210 --> 00:10:53,220 is, of course, the same as the depth of focus of the system. 184 00:10:53,220 --> 00:10:55,500 Is that all clear? 185 00:10:55,500 --> 00:10:56,280 Yeah? 186 00:10:56,280 --> 00:10:57,840 Good. 187 00:10:57,840 --> 00:11:00,390 OK. 188 00:11:00,390 --> 00:11:03,330 Even though our derivations were carried out 189 00:11:03,330 --> 00:11:06,870 for spatially coherent imaging, exactly the same arguments 190 00:11:06,870 --> 00:11:11,730 you can use for the incoherent case. 191 00:11:11,730 --> 00:11:15,030 So the idea of a-- 192 00:11:15,030 --> 00:11:19,470 we showed the defocused amplitude transfer function, 193 00:11:19,470 --> 00:11:22,740 in the same way you can also have a defocused OTF. 194 00:11:27,760 --> 00:11:31,090 So let's think of now what happens 195 00:11:31,090 --> 00:11:36,490 if we've got some objects which has got some height variations. 196 00:11:36,490 --> 00:11:39,820 Here he's taking the case of a surface, 197 00:11:39,820 --> 00:11:42,760 and he's called this a 2 and 1/2 D object. 198 00:11:42,760 --> 00:11:46,060 I think he's probably called it that because I sometimes 199 00:11:46,060 --> 00:11:48,122 use that terminology. 200 00:11:48,122 --> 00:11:49,330 I'm not the only one, really. 201 00:11:49,330 --> 00:11:50,440 I didn't invent it. 202 00:11:50,440 --> 00:11:53,860 But people often use this 2 and 1/2 D 203 00:11:53,860 --> 00:11:57,640 to represent something which is not a 2D object. 204 00:11:57,640 --> 00:11:59,530 It's not flat, it's got some depth. 205 00:11:59,530 --> 00:12:02,830 But it's not a true 3D object in that 206 00:12:02,830 --> 00:12:05,090 it's got no internal structure. 207 00:12:05,090 --> 00:12:07,090 It's just a surface. 208 00:12:07,090 --> 00:12:10,720 So anyway, in this case, this is the object 209 00:12:10,720 --> 00:12:12,730 and this is its image. 210 00:12:12,730 --> 00:12:16,510 And you can see that if this object is quite deep, 211 00:12:16,510 --> 00:12:19,420 it's going to be deeper than that depth of field 212 00:12:19,420 --> 00:12:20,660 of the system. 213 00:12:20,660 --> 00:12:24,910 And so some parts of that are going to be out of focus. 214 00:12:24,910 --> 00:12:28,600 And so this is a region, for example, 215 00:12:28,600 --> 00:12:31,890 which is a long way away from the focal plane, 216 00:12:31,890 --> 00:12:36,090 and therefore it's going to be imaged out of focus. 217 00:12:36,090 --> 00:12:40,400 And here it corresponds to this region here. 218 00:12:40,400 --> 00:12:48,390 Notice, if you look at this image, it's inverted in x. 219 00:12:48,390 --> 00:12:52,770 But you see that in the z direction, 220 00:12:52,770 --> 00:12:56,760 a distance which is moved this way here 221 00:12:56,760 --> 00:12:58,930 is also moved this way here. 222 00:12:58,930 --> 00:13:02,880 So there's no inversion in the z direction. 223 00:13:02,880 --> 00:13:05,760 Did I hear a question over there or was it just someone 224 00:13:05,760 --> 00:13:07,790 shuffling around? 225 00:13:07,790 --> 00:13:08,290 No. 226 00:13:08,290 --> 00:13:08,790 OK. 227 00:13:12,110 --> 00:13:17,890 So as George points out here, what this means then 228 00:13:17,890 --> 00:13:21,190 for this particular object is the point spread function 229 00:13:21,190 --> 00:13:24,220 is not now spatially invariant. 230 00:13:24,220 --> 00:13:29,440 It's a spatially shift variance PSF 231 00:13:29,440 --> 00:13:32,920 because different parts of this surface 232 00:13:32,920 --> 00:13:35,440 are going to experience a different point spread 233 00:13:35,440 --> 00:13:37,615 function because they might be in or out of focus. 234 00:13:43,520 --> 00:13:48,380 So yeah, and the part, just like before, the part which 235 00:13:48,380 --> 00:13:53,270 is in focus is given by this depth of field, which we just 236 00:13:53,270 --> 00:13:53,790 had before. 237 00:14:01,000 --> 00:14:03,360 So you can either think of it as a depth of field 238 00:14:03,360 --> 00:14:06,208 or a depth of focus, the same expression. 239 00:14:08,720 --> 00:14:11,160 So this is an example of this. 240 00:14:11,160 --> 00:14:13,860 Let's take an object which is a letter 241 00:14:13,860 --> 00:14:18,650 M. I guess he didn't use MIT this time because he wants 242 00:14:18,650 --> 00:14:23,390 to use the same for each so that you can see how each of them 243 00:14:23,390 --> 00:14:27,210 is affected by the different point spread function. 244 00:14:27,210 --> 00:14:30,320 So this is in the focal plane, and then this 245 00:14:30,320 --> 00:14:33,560 is to depth of fields out of the focal plane. 246 00:14:33,560 --> 00:14:36,810 This is for depth the fields out of the focal plane. 247 00:14:36,810 --> 00:14:39,260 So when you image that you can see 248 00:14:39,260 --> 00:14:41,780 that it's going to become-- this one's going to be in focus. 249 00:14:41,780 --> 00:14:43,280 It's still a bit blurred, of course, 250 00:14:43,280 --> 00:14:47,250 because of the refraction effect in the imaging. 251 00:14:47,250 --> 00:14:50,900 So this is convolved with the point spread function 252 00:14:50,900 --> 00:14:53,570 of the coherent optical system. 253 00:14:53,570 --> 00:15:00,350 And then this is for the two defocused ones, getting 254 00:15:00,350 --> 00:15:01,520 more and more out of focus. 255 00:15:05,830 --> 00:15:08,470 So right now, once we've got that blur, 256 00:15:08,470 --> 00:15:10,610 what can we do about it? 257 00:15:10,610 --> 00:15:14,710 And there's been quite a lot of papers 258 00:15:14,710 --> 00:15:17,080 that have looked at this question of whether you 259 00:15:17,080 --> 00:15:20,450 can actually undo the blur. 260 00:15:20,450 --> 00:15:22,670 Deblurring, they call it, or sometimes 261 00:15:22,670 --> 00:15:26,020 deconvolution because if you think of the blurring 262 00:15:26,020 --> 00:15:28,000 as being a convolution, you wanted 263 00:15:28,000 --> 00:15:31,690 to undo that, you want to do a deconvolution. 264 00:15:31,690 --> 00:15:36,730 So what we actually get then, in the Fourier domain 265 00:15:36,730 --> 00:15:40,420 the spectrum of the image is equal to the spectrum 266 00:15:40,420 --> 00:15:45,340 of the object multiplied by some transfer function. 267 00:15:45,340 --> 00:15:46,810 And this would be true whether it's 268 00:15:46,810 --> 00:15:48,840 coherent or incoherent, in general. 269 00:15:48,840 --> 00:15:51,280 They'd be the same sort of effects, 270 00:15:51,280 --> 00:15:53,230 but you'd have to work with intensities 271 00:15:53,230 --> 00:15:55,210 rather than amplitudes. 272 00:15:55,210 --> 00:15:59,950 And so what we say then is that if we-- 273 00:15:59,950 --> 00:16:02,950 this is the blurred thing, so we can actually 274 00:16:02,950 --> 00:16:07,490 blur it by just dividing by that to get rid of it. 275 00:16:07,490 --> 00:16:11,110 So if we get our image, we multiply by 1 276 00:16:11,110 --> 00:16:13,840 over the transfer function, and that gives us 277 00:16:13,840 --> 00:16:19,750 our deblurred version of the image. 278 00:16:19,750 --> 00:16:24,100 So as he calls it here, you can call it inverse filtering, 279 00:16:24,100 --> 00:16:27,910 or sometimes called deconvolution. 280 00:16:27,910 --> 00:16:32,040 But that is actually-- 281 00:16:32,040 --> 00:16:34,810 probably you all thought of, that can't work. 282 00:16:34,810 --> 00:16:37,040 It doesn't sound as like it ought to work. 283 00:16:37,040 --> 00:16:40,300 And of course, it doesn't work very well. 284 00:16:40,300 --> 00:16:44,050 We're going to show a bit more about this in a minute. 285 00:16:44,050 --> 00:16:46,720 In particular, it really goes wrong 286 00:16:46,720 --> 00:16:50,620 if this thing goes to 0, obviously. 287 00:16:50,620 --> 00:16:54,100 So you can't know very much about that. 288 00:16:54,100 --> 00:16:59,440 And then also, if it goes to something 289 00:16:59,440 --> 00:17:04,130 which is even very small, you have to amplify it by a lot. 290 00:17:04,130 --> 00:17:07,839 And when you amplify it, you're going to amplify the noise. 291 00:17:07,839 --> 00:17:10,900 So you'll end up with a noisy reconstruction 292 00:17:10,900 --> 00:17:13,270 after you've done that. 293 00:17:13,270 --> 00:17:15,369 So this sort of inverse filtering 294 00:17:15,369 --> 00:17:19,660 invariably increases the noise in the image. 295 00:17:19,660 --> 00:17:24,980 And so how can we get around that? 296 00:17:24,980 --> 00:17:30,160 Well, we just get around the 0 by making it so 297 00:17:30,160 --> 00:17:32,300 that there isn't a 0 anymore. 298 00:17:32,300 --> 00:17:34,120 So this is our new version. 299 00:17:34,120 --> 00:17:38,350 You can see that if mu were 0, this would just 300 00:17:38,350 --> 00:17:39,890 become the same as before. 301 00:17:39,890 --> 00:17:44,050 This is H times H star, so the H stars would cancel, 302 00:17:44,050 --> 00:17:47,030 and this is 1 over H. 303 00:17:47,030 --> 00:17:50,920 So if this is being compared with that, 304 00:17:50,920 --> 00:17:54,250 then it becomes the same as before. 305 00:17:54,250 --> 00:17:59,050 But if this becomes small, then we're just left with this, 306 00:17:59,050 --> 00:18:05,830 and it avoids it blowing up and giving infinity. 307 00:18:05,830 --> 00:18:08,950 So if mu is 0, it reduces to the direct filter, 308 00:18:08,950 --> 00:18:12,040 and we just described that that's not a good thing to do. 309 00:18:12,040 --> 00:18:15,650 So the more noise there is there, 310 00:18:15,650 --> 00:18:20,920 the bigger you need to make this to avoid getting 311 00:18:20,920 --> 00:18:23,350 too much noise in the image. 312 00:18:26,630 --> 00:18:28,640 So as the noise increases, you would tend 313 00:18:28,640 --> 00:18:30,500 to increase the value of mu. 314 00:18:35,600 --> 00:18:42,330 Yeah, so here it's saying that if the signal and noise 315 00:18:42,330 --> 00:18:46,430 both obey particular type of statistics, 316 00:18:46,430 --> 00:18:52,310 then you can find what the optimum value of mu is. 317 00:18:52,310 --> 00:18:56,570 And we find then that it's equal to 1 318 00:18:56,570 --> 00:19:00,650 over the signal-to-noise ratio, and this 319 00:19:00,650 --> 00:19:03,200 is what's called a vena filter. 320 00:19:03,200 --> 00:19:08,420 So this N comes down to be the same as a vena 321 00:19:08,420 --> 00:19:11,950 filter, which you might have come across, some of you, 322 00:19:11,950 --> 00:19:13,850 in other things. 323 00:19:13,850 --> 00:19:18,230 So what we are doing, really, is you're 324 00:19:18,230 --> 00:19:22,490 amplifying the spatial frequencies that 325 00:19:22,490 --> 00:19:27,540 have been pushed down by the imaging process. 326 00:19:27,540 --> 00:19:31,220 But if there is if they become smaller than the noise, 327 00:19:31,220 --> 00:19:35,460 then you don't do that anymore. 328 00:19:35,460 --> 00:19:39,510 So he's going to give an example of this. 329 00:19:39,510 --> 00:19:44,070 So this is the convolution using this second 330 00:19:44,070 --> 00:19:48,370 of regularized inverse filter. 331 00:19:48,370 --> 00:19:58,520 And yeah, OK, he's saying here that he's 332 00:19:58,520 --> 00:20:02,210 assuming he knows that depth before he does this. 333 00:20:02,210 --> 00:20:08,280 So obviously, that is something you wouldn't always necessarily 334 00:20:08,280 --> 00:20:09,190 know. 335 00:20:09,190 --> 00:20:12,700 But he knows the depth, therefore 336 00:20:12,700 --> 00:20:16,530 he can calculate what the transfer function was 337 00:20:16,530 --> 00:20:17,940 on the imaging side. 338 00:20:17,940 --> 00:20:21,030 And therefore he knows what filter inverse 339 00:20:21,030 --> 00:20:24,870 filter to apply in order to get the best reconstruction. 340 00:20:24,870 --> 00:20:28,170 And you can see that it works extremely well for this one. 341 00:20:28,170 --> 00:20:33,390 This one is slightly blurred, slightly a bit more blurred. 342 00:20:33,390 --> 00:20:35,940 And you can see also that the background here-- 343 00:20:35,940 --> 00:20:37,630 or at least, I can see on this screen, 344 00:20:37,630 --> 00:20:39,930 but this one doesn't show very much-- 345 00:20:39,930 --> 00:20:43,157 the background is becoming not black anymore. 346 00:20:49,580 --> 00:20:51,470 And then this is what happens if you've 347 00:20:51,470 --> 00:20:56,420 got a noisy image, if you add noise to the original-- 348 00:20:56,420 --> 00:20:59,780 to the image before you do the deconvolution. 349 00:20:59,780 --> 00:21:02,690 Now you see that that actually it's 350 00:21:02,690 --> 00:21:04,220 not working nearly so well. 351 00:21:04,220 --> 00:21:06,080 And this one here, you can't really 352 00:21:06,080 --> 00:21:08,860 make out what there is there. 353 00:21:08,860 --> 00:21:14,310 So this is showing that these methods can work, 354 00:21:14,310 --> 00:21:19,260 but you can see that there are also 355 00:21:19,260 --> 00:21:21,240 limits to how well they can work, 356 00:21:21,240 --> 00:21:26,140 and especially in the presence of noise. 357 00:21:26,140 --> 00:21:29,040 Actually, I should mention that, of course, 358 00:21:29,040 --> 00:21:31,870 at MIT in the Media Lab, there's a lot 359 00:21:31,870 --> 00:21:35,050 of work going on in this area of what 360 00:21:35,050 --> 00:21:38,380 they call computational photography, which is all 361 00:21:38,380 --> 00:21:46,390 to do with changing the properties of the camera 362 00:21:46,390 --> 00:21:50,560 in order to make it so that this sort of reconstruction 363 00:21:50,560 --> 00:21:52,340 is more efficient. 364 00:21:52,340 --> 00:21:55,480 So it turns out that if you're clever about the design 365 00:21:55,480 --> 00:21:57,520 of the imaging system, you can actually 366 00:21:57,520 --> 00:22:01,440 make it so that this all works a lot better. 367 00:22:01,440 --> 00:22:04,920 But we weren't going to go into all that. 368 00:22:04,920 --> 00:22:05,680 Oh, sorry. 369 00:22:05,680 --> 00:22:06,640 I've got this wrong. 370 00:22:06,640 --> 00:22:08,050 This one was the-- 371 00:22:08,050 --> 00:22:09,702 sorry, this was the-- 372 00:22:09,702 --> 00:22:10,660 oh, yeah, that's right. 373 00:22:10,660 --> 00:22:16,360 This is the image before you've done the inverse filtering, 374 00:22:16,360 --> 00:22:17,950 and this is the image after you've 375 00:22:17,950 --> 00:22:19,750 done the inverse filtering. 376 00:22:19,750 --> 00:22:21,250 So it's actually doing quite well. 377 00:22:21,250 --> 00:22:26,090 This one here is pretty well not very different from that. 378 00:22:26,090 --> 00:22:32,320 And this one, as before, is not as good, but doing pretty well. 379 00:22:37,390 --> 00:22:40,840 Of course if, you didn't know what the height of these things 380 00:22:40,840 --> 00:22:45,910 is, then you might have to use some iterative algorithm 381 00:22:45,910 --> 00:22:51,250 in order to come up with the best reconstruction. 382 00:22:51,250 --> 00:22:53,350 You can imagine also that-- 383 00:22:53,350 --> 00:22:56,860 so this is for this so-called 2 and 1/2 D object. 384 00:22:56,860 --> 00:23:01,450 You could imagine an object which really did have structure 385 00:23:01,450 --> 00:23:04,390 inside itself, so maybe these three Ms 386 00:23:04,390 --> 00:23:09,940 would be on top of each other and semi-transparent. 387 00:23:09,940 --> 00:23:13,990 And then the trick would be to try to actually 388 00:23:13,990 --> 00:23:21,270 produce a reconstruction of the 3D object from the 3D image, 389 00:23:21,270 --> 00:23:24,990 and there are actually computer packages 390 00:23:24,990 --> 00:23:29,360 that you can get to do just that very sort of thing. 391 00:23:29,360 --> 00:23:32,250 So that was what we should have done, 392 00:23:32,250 --> 00:23:34,690 and we'll now go on to today's lecture. 393 00:23:38,630 --> 00:23:43,160 And so today's lecture is, first of all, 394 00:23:43,160 --> 00:23:48,210 about polarization, which we've been talking about in the lab 395 00:23:48,210 --> 00:23:49,070 today-- 396 00:23:49,070 --> 00:23:51,330 or yesterday, wasn't it? 397 00:23:51,330 --> 00:23:54,660 So that's a bit topical. 398 00:23:54,660 --> 00:23:59,950 And then George wrote these. 399 00:23:59,950 --> 00:24:03,910 So the bit that I've added extra is basically 400 00:24:03,910 --> 00:24:07,300 addressing these points here, effects of polarization 401 00:24:07,300 --> 00:24:12,580 on imaging in a microscope objective or a lens 402 00:24:12,580 --> 00:24:16,310 with a high numerical aperture. 403 00:24:16,310 --> 00:24:19,400 So we'll start off with the polarization. 404 00:24:19,400 --> 00:24:23,900 I'm not sure what he said about polarization, first of all. 405 00:24:23,900 --> 00:24:28,310 But basically, then, the wave equation 406 00:24:28,310 --> 00:24:31,400 for an electromagnetic field is basically a vector wave 407 00:24:31,400 --> 00:24:32,470 equation. 408 00:24:32,470 --> 00:24:37,390 It's a function of electric field here. 409 00:24:37,390 --> 00:24:41,230 And we had then that the polarization 410 00:24:41,230 --> 00:24:46,430 is some function of the electric field 411 00:24:46,430 --> 00:24:50,600 and how the relationship between these 412 00:24:50,600 --> 00:24:52,710 depends on the sort of material. 413 00:24:52,710 --> 00:24:55,010 And so this is giving some different examples. 414 00:24:55,010 --> 00:24:59,990 The simple case is this one, which we did look at earlier, 415 00:24:59,990 --> 00:25:02,690 is where the polarization is proportional 416 00:25:02,690 --> 00:25:04,790 to the electric field. 417 00:25:04,790 --> 00:25:10,340 And you remember that you get D by adding the P to the E. 418 00:25:10,340 --> 00:25:16,220 And so here, this is called the electric susceptibility, 419 00:25:16,220 --> 00:25:21,920 and P equals Chi times E. So if this is a constant, 420 00:25:21,920 --> 00:25:23,840 this is just a linear relationship, which 421 00:25:23,840 --> 00:25:26,510 is how we described it before. 422 00:25:26,510 --> 00:25:30,680 But there are two ways that that can break down in practice. 423 00:25:30,680 --> 00:25:36,350 And usually, actually, for materials where it breaks down, 424 00:25:36,350 --> 00:25:39,800 it breaks down for both at the same time, unfortunately. 425 00:25:39,800 --> 00:25:41,360 So it becomes very complicated. 426 00:25:41,360 --> 00:25:45,650 But here we've written it with these two effects separately. 427 00:25:45,650 --> 00:25:46,580 this. 428 00:25:46,580 --> 00:25:51,980 Is where actually the relationship is still linear, 429 00:25:51,980 --> 00:25:56,480 but P is related to e not by a simple scalar 430 00:25:56,480 --> 00:25:59,610 relationship like this, but by a tensor relationship. 431 00:25:59,610 --> 00:26:02,630 So the direction of P is not in the same direction 432 00:26:02,630 --> 00:26:05,780 as the direction of E. And so you've 433 00:26:05,780 --> 00:26:10,130 got this 3 by 3 matrix, which describes 434 00:26:10,130 --> 00:26:12,680 the properties of the material. 435 00:26:12,680 --> 00:26:15,440 And this will depend on-- 436 00:26:15,440 --> 00:26:22,250 well, for a isotopic material, like a crystal or something 437 00:26:22,250 --> 00:26:26,180 like that, it would depend on the crystal structure. 438 00:26:26,180 --> 00:26:32,150 Of course, if you've got some complicated varying objects, 439 00:26:32,150 --> 00:26:37,760 then it can be all sorts of even more complicated things. 440 00:26:37,760 --> 00:26:41,540 So that's the first case. 441 00:26:41,540 --> 00:26:45,260 So this is what's going to lead to some interesting 442 00:26:45,260 --> 00:26:49,070 polarization effects as you put that electric field 443 00:26:49,070 --> 00:26:51,540 into a crystal, for example. 444 00:26:51,540 --> 00:26:53,510 And then this is the other one which 445 00:26:53,510 --> 00:26:57,750 is now looking at isotopic material, but nonlinear. 446 00:26:57,750 --> 00:27:00,810 So this is now no longer a constant. 447 00:27:00,810 --> 00:27:02,780 And so what that means, you can you 448 00:27:02,780 --> 00:27:05,550 could expand it as a power series. 449 00:27:05,550 --> 00:27:10,950 And you can see that there might be some extra terms here. 450 00:27:10,950 --> 00:27:13,700 This is what's called the Kerr effect. 451 00:27:13,700 --> 00:27:18,590 There's effectively a susceptibility 452 00:27:18,590 --> 00:27:24,220 which is dependent on the intensity of the light which 453 00:27:24,220 --> 00:27:24,960 shines on it. 454 00:27:27,630 --> 00:27:29,850 And of course, you might expect that there 455 00:27:29,850 --> 00:27:34,380 will be higher orders of this as well, of course. 456 00:27:34,380 --> 00:27:35,910 These sorts of effects, you're only 457 00:27:35,910 --> 00:27:43,350 going to excite these if you have a large value of E. It's 458 00:27:43,350 --> 00:27:46,110 a bit like, if you imagine how this all comes about, 459 00:27:46,110 --> 00:27:48,750 this comes about because your material 460 00:27:48,750 --> 00:27:52,400 is made up of atoms which are all bonded together 461 00:27:52,400 --> 00:27:55,200 and joined to each other with springs. 462 00:27:55,200 --> 00:27:57,580 And when you apply the electric field, 463 00:27:57,580 --> 00:28:02,400 it makes these atoms vibrate against the spring. 464 00:28:02,400 --> 00:28:05,670 And if the vibrations are small, then it's 465 00:28:05,670 --> 00:28:07,680 all going to satisfy Hooke's law, 466 00:28:07,680 --> 00:28:09,430 and it's going to be linear. 467 00:28:09,430 --> 00:28:13,170 But if you increase the forcing term, 468 00:28:13,170 --> 00:28:17,100 then the vibrations get bigger, Hooke's law breaks down, 469 00:28:17,100 --> 00:28:20,190 and the vibrations become nonlinear. 470 00:28:20,190 --> 00:28:26,090 That's the basic physics of what's going on here. 471 00:28:26,090 --> 00:28:28,530 So this is what's called the Kerr effect. 472 00:28:28,530 --> 00:28:30,510 There is actually another effect which 473 00:28:30,510 --> 00:28:33,370 George hasn't mentioned here. 474 00:28:33,370 --> 00:28:37,350 There's another-- you see this is actually E cubed here. 475 00:28:37,350 --> 00:28:39,150 But you can have an effect where this 476 00:28:39,150 --> 00:28:45,240 is E squared, which is called the-- what's it called? 477 00:28:45,240 --> 00:28:47,130 The Pockels effect. 478 00:28:47,130 --> 00:28:49,140 I was really only asking because I'd forgotten 479 00:28:49,140 --> 00:28:50,760 the word for a minute. 480 00:28:50,760 --> 00:28:54,150 But my students here know that I'm always forgetting names, 481 00:28:54,150 --> 00:29:00,920 and they usually come back eventually. 482 00:29:00,920 --> 00:29:03,250 Did you see what that did? 483 00:29:03,250 --> 00:29:05,930 Oh, yeah, it did this, the relationship 484 00:29:05,930 --> 00:29:10,235 between the susceptibility and the refractive index. 485 00:29:13,190 --> 00:29:16,480 And this is saying that the phase delay 486 00:29:16,480 --> 00:29:20,240 depends on the polarization. 487 00:29:20,240 --> 00:29:23,300 And in this one, the index of refraction 488 00:29:23,300 --> 00:29:25,910 depends on the intensity. 489 00:29:25,910 --> 00:29:28,160 So this is nonlinear optics. 490 00:29:28,160 --> 00:29:32,597 And as I say, those last two effects very often 491 00:29:32,597 --> 00:29:33,180 come together. 492 00:29:36,090 --> 00:29:40,090 So this is now talking about polarization of light. 493 00:29:40,090 --> 00:29:42,210 I think that probably George must have mentioned 494 00:29:42,210 --> 00:29:45,210 this to some degree earlier on. 495 00:29:45,210 --> 00:29:49,980 But what this is showing, then, is a plane wave of light. 496 00:29:49,980 --> 00:29:54,660 It's part linearly polarized in the x direction-- 497 00:29:54,660 --> 00:29:55,890 sorry, the y direction. 498 00:29:55,890 --> 00:29:58,110 This is taken as y. 499 00:29:58,110 --> 00:30:01,360 So the electric field is in the y direction, 500 00:30:01,360 --> 00:30:03,700 which actually means that the magnetic field is going 501 00:30:03,700 --> 00:30:06,240 to be in the x direction because they're 502 00:30:06,240 --> 00:30:09,270 at right angles with each other, if you remember. 503 00:30:09,270 --> 00:30:15,270 And you remember that you can think of the wave propagating 504 00:30:15,270 --> 00:30:16,810 in two different ways. 505 00:30:16,810 --> 00:30:22,080 One is to think of it other at a certain instant of time, 506 00:30:22,080 --> 00:30:25,440 so what we call here it's a snapshot, 507 00:30:25,440 --> 00:30:29,610 and then we plot what happens as a function of z. 508 00:30:29,610 --> 00:30:34,110 So of course, we have to do this because it's 509 00:30:34,110 --> 00:30:38,310 difficult to plot it against space and time at the same time 510 00:30:38,310 --> 00:30:42,550 because it's difficult to visualize what's going on. 511 00:30:42,550 --> 00:30:45,660 So you can see here this electric field is always 512 00:30:45,660 --> 00:30:49,770 in the same plane, and it's a maximum here. 513 00:30:49,770 --> 00:30:52,740 It reduces, it becomes 0. 514 00:30:52,740 --> 00:30:55,890 It then increases in the negative direction, 515 00:30:55,890 --> 00:30:59,280 comes back to 0 again, and so on and so on. 516 00:30:59,280 --> 00:31:06,170 So this is linearly polarized light as a function of space. 517 00:31:06,170 --> 00:31:11,340 And then this one shows the same thing as a function of time. 518 00:31:11,340 --> 00:31:15,950 So we're looking at now one particular value of positions 519 00:31:15,950 --> 00:31:18,800 that equals 0, and we're looking at how this 520 00:31:18,800 --> 00:31:21,410 varies as a function of time. 521 00:31:21,410 --> 00:31:23,120 Exactly the same diagram, of course. 522 00:31:23,120 --> 00:31:28,080 You probably saw when I flicked to the next slide, 523 00:31:28,080 --> 00:31:29,280 it didn't change at all. 524 00:31:29,280 --> 00:31:32,225 It just changed the caption. 525 00:31:34,780 --> 00:31:38,560 And this is what circular polarization does. 526 00:31:38,560 --> 00:31:42,490 Circular polarization, you can see what this is trying to-- 527 00:31:42,490 --> 00:31:44,000 it looks a bit complicated here. 528 00:31:44,000 --> 00:31:47,950 But what it's really doing is actually quite straightforward. 529 00:31:47,950 --> 00:31:49,420 It's actually rotating. 530 00:31:49,420 --> 00:31:54,280 This electric vector is rotating either with distance 531 00:31:54,280 --> 00:31:58,090 or with time, according to which way you do the plot. 532 00:31:58,090 --> 00:32:01,310 And the length of this line stays the same. 533 00:32:01,310 --> 00:32:04,870 So this is what we mean by circular polarization. 534 00:32:08,580 --> 00:32:11,610 And this is showing how, actually, you 535 00:32:11,610 --> 00:32:14,040 can think of circular polarization 536 00:32:14,040 --> 00:32:17,150 as being made up of two components-- 537 00:32:17,150 --> 00:32:19,440 one linearly polarized in this side 538 00:32:19,440 --> 00:32:24,850 in the y direction, one linearly polarized in the x direction. 539 00:32:24,850 --> 00:32:30,500 And so you can see that there are also out of phase 540 00:32:30,500 --> 00:32:32,040 by 90 degrees. 541 00:32:32,040 --> 00:32:39,090 So you can see now that in this distance, position, 542 00:32:39,090 --> 00:32:41,050 this is plotted against z again. 543 00:32:41,050 --> 00:32:44,880 Here this one is a maximum, this one is 0. 544 00:32:44,880 --> 00:32:47,550 So the electric field is in the x direction. 545 00:32:47,550 --> 00:32:50,010 And then as you go, as you travel along, 546 00:32:50,010 --> 00:32:53,190 this gets smaller, but this gets bigger. 547 00:32:53,190 --> 00:32:57,150 So this position here, the electric field 548 00:32:57,150 --> 00:32:59,520 is now polarized in the y direction. 549 00:32:59,520 --> 00:33:02,010 And then this one goes back down to 0 again, 550 00:33:02,010 --> 00:33:04,620 and this one gets bigger. 551 00:33:04,620 --> 00:33:10,140 So this is exactly equivalent to the case 552 00:33:10,140 --> 00:33:13,350 that I showed before, the circular polarized case. 553 00:33:13,350 --> 00:33:17,730 I might add that electrical polarization 554 00:33:17,730 --> 00:33:22,020 is where the amplitude of this component and this component 555 00:33:22,020 --> 00:33:23,590 are not equal. 556 00:33:23,590 --> 00:33:26,888 So if that was the case, it would-- again, 557 00:33:26,888 --> 00:33:28,680 you that you'd think of this electric field 558 00:33:28,680 --> 00:33:32,280 vector as rotating, but the length of the vector 559 00:33:32,280 --> 00:33:34,860 would also change as it rotates. 560 00:33:34,860 --> 00:33:37,260 And of course, you can have two cases. 561 00:33:37,260 --> 00:33:39,780 You can have it rotating clockwise, 562 00:33:39,780 --> 00:33:42,880 or you can have it rotating anti-clockwise. 563 00:33:42,880 --> 00:33:46,410 So these are called left- and right-hand 564 00:33:46,410 --> 00:33:47,850 circular polarization. 565 00:33:51,310 --> 00:33:55,100 So this is showing an end-on view. 566 00:33:55,100 --> 00:33:59,080 So this is x and y. 567 00:33:59,080 --> 00:34:02,920 So this thing is pointing out towards you, isn't it, 568 00:34:02,920 --> 00:34:05,480 that means. 569 00:34:05,480 --> 00:34:09,850 No, it means it's going away, isn't it? 570 00:34:09,850 --> 00:34:11,050 But is it right? 571 00:34:11,050 --> 00:34:14,540 Because it's left-handed coordinate system. 572 00:34:14,540 --> 00:34:19,870 If that's the case because x-- 573 00:34:19,870 --> 00:34:21,199 oh, no, it's not. 574 00:34:21,199 --> 00:34:21,960 No, it's right. 575 00:34:21,960 --> 00:34:22,650 It's right. 576 00:34:22,650 --> 00:34:25,050 It's the right-handed coordinate system. 577 00:34:25,050 --> 00:34:29,670 Yeah, x cross y goes away from you. 578 00:34:29,670 --> 00:34:32,940 And so this is showing, then, end-on, 579 00:34:32,940 --> 00:34:35,489 what happens to this electric vector. 580 00:34:35,489 --> 00:34:37,590 And the wonders of modern science 581 00:34:37,590 --> 00:34:41,340 have in fact changed the circular polarization 582 00:34:41,340 --> 00:34:45,690 to electrical polarization without me doing anything. 583 00:34:49,929 --> 00:34:53,860 Now, so next is to introduce some-- 584 00:34:53,860 --> 00:34:55,179 sorry. 585 00:34:55,179 --> 00:34:56,530 I missed that one. 586 00:34:56,530 --> 00:35:01,960 There were lots of these things that my computer misinterpreted 587 00:35:01,960 --> 00:35:04,180 from George's computer. 588 00:35:04,180 --> 00:35:07,030 And I changed most of them, but I missed one there. 589 00:35:07,030 --> 00:35:10,960 That should say left lambda over 4 plate. 590 00:35:10,960 --> 00:35:14,020 I don't know what the difference is what he put in there. 591 00:35:14,020 --> 00:35:17,310 Do you know what he's put in there to make it go like that? 592 00:35:17,310 --> 00:35:20,440 Anyway, he did this on his Mac. 593 00:35:20,440 --> 00:35:21,503 AUDIENCE: [INAUDIBLE] 594 00:35:21,503 --> 00:35:22,170 PROFESSOR: Yeah. 595 00:35:22,170 --> 00:35:25,540 Yeah, but it's a funny font as well. 596 00:35:25,540 --> 00:35:30,700 Anyway, so this is showing what would happen 597 00:35:30,700 --> 00:35:32,590 if you put some birefringent-- 598 00:35:32,590 --> 00:35:35,830 a special birefringent material here, 599 00:35:35,830 --> 00:35:39,550 which is what's called a lambda by 4 plate. 600 00:35:39,550 --> 00:35:45,310 And what this does is it changes the relative phase 601 00:35:45,310 --> 00:35:50,770 of the components of electric field in two directions 602 00:35:50,770 --> 00:35:57,850 by a different amount as it transmits through the crystal. 603 00:35:57,850 --> 00:36:02,740 And so if you arrange the orientation of it 604 00:36:02,740 --> 00:36:04,810 correctly relative to the light, it 605 00:36:04,810 --> 00:36:09,650 has this effect of changing the linear polarization 606 00:36:09,650 --> 00:36:11,980 into circular polarization. 607 00:36:11,980 --> 00:36:15,400 The way it really does that is that this-- 608 00:36:15,400 --> 00:36:17,380 you would say that this you can think 609 00:36:17,380 --> 00:36:21,760 of as being made up of two linearly polarized components, 610 00:36:21,760 --> 00:36:24,190 and this you can always think of as being made up 611 00:36:24,190 --> 00:36:26,650 of two linearly polarized components 612 00:36:26,650 --> 00:36:30,180 as well at 45 degrees to that. 613 00:36:30,180 --> 00:36:33,460 So you just resolve that into two components. 614 00:36:33,460 --> 00:36:36,850 So the circular polarized, you get that 615 00:36:36,850 --> 00:36:41,380 very simply by just changing the relative phase of those two 616 00:36:41,380 --> 00:36:43,510 linear polarized components. 617 00:36:43,510 --> 00:36:46,600 That's how it works, but I think here he's just trying 618 00:36:46,600 --> 00:36:49,530 to show you what it does. 619 00:36:49,530 --> 00:36:55,050 And then the other important device 620 00:36:55,050 --> 00:36:59,820 that people use is a is a lambda by 2 plate 621 00:36:59,820 --> 00:37:03,480 rather than a lambda by 4 plate, and that 622 00:37:03,480 --> 00:37:10,080 has the property of changing x polarized, linearly polarized 623 00:37:10,080 --> 00:37:13,560 light into y polarized light. 624 00:37:13,560 --> 00:37:16,530 And again, you can see how this is working 625 00:37:16,530 --> 00:37:20,580 by thinking in terms of resolving this in so two 626 00:37:20,580 --> 00:37:26,010 components at 45 degrees to it, and then changing 627 00:37:26,010 --> 00:37:27,420 the relative sign. 628 00:37:27,420 --> 00:37:30,200 Maybe I should do that. 629 00:37:30,200 --> 00:37:31,420 This doesn't [INAUDIBLE]. 630 00:37:31,420 --> 00:37:32,190 Is it switched on? 631 00:37:32,190 --> 00:37:34,180 AUDIENCE: [INAUDIBLE] 632 00:37:34,180 --> 00:37:36,610 PROFESSOR: So maybe I can just do that quickly. 633 00:37:36,610 --> 00:37:39,480 So what I'm saying is that this, you can think of it-- 634 00:37:39,480 --> 00:37:41,845 AUDIENCE: [INAUDIBLE] 635 00:37:43,740 --> 00:37:45,980 PROFESSOR: This one you can think 636 00:37:45,980 --> 00:37:49,320 of as being resolved into these two components. 637 00:37:49,320 --> 00:37:51,590 And then the crystal is then changing 638 00:37:51,590 --> 00:37:54,110 the relative phase of one of these relative 639 00:37:54,110 --> 00:37:57,130 to that, making that. 640 00:37:57,130 --> 00:38:00,825 And so now the polarization is like this. 641 00:38:00,825 --> 00:38:01,950 So that's how it's working. 642 00:38:01,950 --> 00:38:06,600 For the lambda by 4 plate, what it was doing was 643 00:38:06,600 --> 00:38:11,010 it was changing the relative phase of this and this 644 00:38:11,010 --> 00:38:14,380 by 90 degrees, not by 180 degrees, which would give you 645 00:38:14,380 --> 00:38:15,760 a circularly polarized light. 646 00:38:15,760 --> 00:38:18,720 But in order to get this effect, you 647 00:38:18,720 --> 00:38:23,520 have to get the right orientation of the polarization 648 00:38:23,520 --> 00:38:27,960 relative to the crystal axes. 649 00:38:27,960 --> 00:38:33,340 So that part he hasn't actually said anything about here. 650 00:38:33,340 --> 00:38:37,120 The other thing-- maybe you'd be interested to know this. 651 00:38:37,120 --> 00:38:41,230 Of course, if you put linearly polarized light in and get 652 00:38:41,230 --> 00:38:43,660 circularly polarized light out, what 653 00:38:43,660 --> 00:38:45,440 would happen if you went the other way? 654 00:38:45,440 --> 00:38:48,490 Well, in fact, it does work the other way. 655 00:38:48,490 --> 00:38:51,070 You can put it in circularly polarized light 656 00:38:51,070 --> 00:38:53,050 and get out linearly polarized light. 657 00:38:55,930 --> 00:39:01,690 So you can do either of those two things. 658 00:39:01,690 --> 00:39:03,510 The other thing that you might worry about 659 00:39:03,510 --> 00:39:17,400 is it's interesting that if we change the phase of this light, 660 00:39:17,400 --> 00:39:21,840 then it's going to change the phase of that light, isn't it? 661 00:39:21,840 --> 00:39:28,910 And yeah, maybe that's not leading anywhere. 662 00:39:28,910 --> 00:39:31,210 I won't say any more about. 663 00:39:31,210 --> 00:39:33,830 OK, think about this. 664 00:39:33,830 --> 00:39:34,330 Yeah. 665 00:39:34,330 --> 00:39:35,650 Oh, well, there we are. 666 00:39:35,650 --> 00:39:39,860 I've already-- I've jumped the gun a bit. 667 00:39:39,860 --> 00:39:43,120 So he's saying, what happens here then? 668 00:39:43,120 --> 00:39:48,530 So we've got incoming linearly polarized light, the lambda 669 00:39:48,530 --> 00:39:49,400 by 4 plate. 670 00:39:49,400 --> 00:39:52,770 What does the lambda by 4 plate do? 671 00:39:52,770 --> 00:39:54,190 Circular polarized. 672 00:39:54,190 --> 00:39:56,980 So we've got circular polarized light on here. 673 00:39:56,980 --> 00:40:00,640 It strikes a mirror and it reflects. 674 00:40:00,640 --> 00:40:03,310 What do we get after it's reflected? 675 00:40:03,310 --> 00:40:06,420 That's a bit hard now, isn't it? 676 00:40:06,420 --> 00:40:08,750 Opposite circular, right. 677 00:40:08,750 --> 00:40:11,670 And then this gets back to here and it goes back 678 00:40:11,670 --> 00:40:15,360 through this birefringent plate. 679 00:40:15,360 --> 00:40:16,470 So what does it do? 680 00:40:16,470 --> 00:40:17,300 AUDIENCE: Same. 681 00:40:17,300 --> 00:40:18,600 PROFESSOR: Same polarization. 682 00:40:18,600 --> 00:40:19,190 Is that right? 683 00:40:21,990 --> 00:40:25,158 It rotates by 180 degrees. 684 00:40:25,158 --> 00:40:26,200 This is slower, actually. 685 00:40:26,200 --> 00:40:30,760 It's gone through a lambda by 2 plate then, isn't it? 686 00:40:30,760 --> 00:40:31,840 Well, I mean-- 687 00:40:31,840 --> 00:40:35,298 AUDIENCE: [INAUDIBLE] 688 00:40:40,977 --> 00:40:41,810 PROFESSOR: Oh, yeah. 689 00:40:41,810 --> 00:40:42,310 Yeah, sorry. 690 00:40:42,310 --> 00:40:43,352 Yeah, you're quite right. 691 00:40:43,352 --> 00:40:43,960 Yeah. 692 00:40:43,960 --> 00:40:44,460 Yeah. 693 00:40:46,810 --> 00:40:48,670 OK everyone, about that? 694 00:40:48,670 --> 00:40:50,560 I guess the next thing is to say what 695 00:40:50,560 --> 00:40:53,310 would happen if you change that mirror by a phase conjugate 696 00:40:53,310 --> 00:40:53,810 mirror. 697 00:40:53,810 --> 00:40:59,710 But maybe that's too difficult for us to think about. 698 00:40:59,710 --> 00:41:03,640 And then remember that-- 699 00:41:03,640 --> 00:41:05,800 I think we had this before as well-- 700 00:41:05,800 --> 00:41:09,760 that when you have lights interfering, 701 00:41:09,760 --> 00:41:13,960 the electric field, it's only the parallel components 702 00:41:13,960 --> 00:41:15,880 of electric field that interfere. 703 00:41:15,880 --> 00:41:20,560 So if you've got two waves where the electric fields are 704 00:41:20,560 --> 00:41:24,310 at right angles to each other, then they don't interfere. 705 00:41:24,310 --> 00:41:26,210 So this is showing this case here. 706 00:41:26,210 --> 00:41:29,230 We've got two plane polarized waves. 707 00:41:29,230 --> 00:41:32,590 They're polarized at right angles to each other, 708 00:41:32,590 --> 00:41:35,890 and so they won't interfere. 709 00:41:35,890 --> 00:41:38,602 And therefore, if you look at the trying 710 00:41:38,602 --> 00:41:40,060 to get an interference pattern, you 711 00:41:40,060 --> 00:41:41,393 won't see anything in this case. 712 00:41:44,010 --> 00:41:48,080 And this is showing them in the same plane. 713 00:41:48,080 --> 00:41:51,430 And then they would interfere , and you would get some 714 00:41:51,430 --> 00:41:56,840 interference pattern with maxima a minima like this. 715 00:41:56,840 --> 00:42:00,590 Another interesting one I guess I might draw over here 716 00:42:00,590 --> 00:42:02,600 is what happens if you-- 717 00:42:02,600 --> 00:42:04,790 this is our screen. 718 00:42:04,790 --> 00:42:07,450 And let's say we've got two waves coming in. 719 00:42:07,450 --> 00:42:13,190 One, let's say, is polarized like this, 720 00:42:13,190 --> 00:42:18,190 and the other one is coming in here 721 00:42:18,190 --> 00:42:21,660 and it's polarized like this. 722 00:42:21,660 --> 00:42:22,160 You see? 723 00:42:22,160 --> 00:42:24,070 So we've got these two waves coming in. 724 00:42:27,190 --> 00:42:31,030 And you can see that their electric field vectors are 725 00:42:31,030 --> 00:42:33,520 at right angles to each other, and so this 726 00:42:33,520 --> 00:42:36,620 is another example where they're not going to interfere. 727 00:42:36,620 --> 00:42:38,890 So you wouldn't see any interference fringes 728 00:42:38,890 --> 00:42:40,120 in this case. 729 00:42:40,120 --> 00:42:43,900 So this corresponds to what's called p-polarization. 730 00:42:43,900 --> 00:42:47,020 So if we did this the opposite, if we did it 731 00:42:47,020 --> 00:42:53,760 with the electric field in this direction, and the other one's 732 00:42:53,760 --> 00:42:58,900 coming in also with the actually field in this direction, 733 00:42:58,900 --> 00:43:03,720 then you can see these are now in parallel directions. 734 00:43:03,720 --> 00:43:07,710 and so you would see fringes where these two plane 735 00:43:07,710 --> 00:43:09,930 waves interfere. 736 00:43:09,930 --> 00:43:11,980 So I've drawn this for the particular case. 737 00:43:11,980 --> 00:43:13,710 This is supposed to be 90 degrees. 738 00:43:23,040 --> 00:43:25,440 Right, OK. 739 00:43:25,440 --> 00:43:27,000 That's all George's material. 740 00:43:27,000 --> 00:43:28,440 That was the end of George. 741 00:43:28,440 --> 00:43:34,930 It didn't end with a particularly startling thing. 742 00:43:34,930 --> 00:43:43,920 So that's really the end of the course as George wrote it. 743 00:43:43,920 --> 00:43:47,530 But he did ask me, as I say, to do a bit more. 744 00:43:47,530 --> 00:43:49,920 So what I think I'll do, why don't we have a break 745 00:43:49,920 --> 00:43:51,960 now for five minutes? 746 00:43:51,960 --> 00:43:55,232 And then I'll start again in five minutes' time. 747 00:43:55,232 --> 00:43:56,940 So this is-- now it's going to be talking 748 00:43:56,940 --> 00:43:59,430 about focusing of light. 749 00:43:59,430 --> 00:44:02,940 And as I said before, this material I'm going to do now 750 00:44:02,940 --> 00:44:05,010 is just for interest. 751 00:44:05,010 --> 00:44:07,770 It's not going to be examinable. 752 00:44:07,770 --> 00:44:09,810 And if you're not interested, you 753 00:44:09,810 --> 00:44:11,580 can go and have a cup of coffee instead. 754 00:44:11,580 --> 00:44:12,840 I don't really mind. 755 00:44:12,840 --> 00:44:15,960 But some people might be interested in learning 756 00:44:15,960 --> 00:44:17,940 a bit more about focusing of light. 757 00:44:26,250 --> 00:44:31,350 So I was going to say a bit more about focusing of light, 758 00:44:31,350 --> 00:44:34,680 and in particular some of the funny things that happen when 759 00:44:34,680 --> 00:44:39,960 you look at focusing of light with a microscope objective, 760 00:44:39,960 --> 00:44:42,870 so the tight focusing of light. 761 00:44:42,870 --> 00:44:47,880 And this is actually a picture of the intensity 762 00:44:47,880 --> 00:44:55,410 in the focal region of a lens, not high numerical aperture 763 00:44:55,410 --> 00:44:55,980 in this case. 764 00:44:55,980 --> 00:44:58,170 This is according to paraxial optics. 765 00:44:58,170 --> 00:45:01,350 This is a picture taken from [INAUDIBLE] Wolf. 766 00:45:01,350 --> 00:45:06,660 And it's similar but not identical to the one 767 00:45:06,660 --> 00:45:13,770 that George showed on the last lecture or the lecture before. 768 00:45:13,770 --> 00:45:16,510 So this is the focal plane. 769 00:45:16,510 --> 00:45:18,392 This is the optic axis. 770 00:45:18,392 --> 00:45:20,100 And of course, what you're really seeing, 771 00:45:20,100 --> 00:45:23,820 this is actually going to be radially symmetric. 772 00:45:23,820 --> 00:45:29,490 So if you imagine this was being revolved around the axis, 773 00:45:29,490 --> 00:45:33,240 you'd have the shape of this complete surface. 774 00:45:33,240 --> 00:45:36,300 And you can see what happens is so this 775 00:45:36,300 --> 00:45:40,380 is the focal plane where you got the Airy disk. 776 00:45:40,380 --> 00:45:43,170 And then as you go out of focus, then you 777 00:45:43,170 --> 00:45:48,750 start getting this strange sort of almost 778 00:45:48,750 --> 00:45:50,310 like a sort of butterfly shape. 779 00:45:54,740 --> 00:45:59,090 So this calculation that's given in [INAUDIBLE] Wolf, 780 00:45:59,090 --> 00:46:02,030 it first of all, as I've said, it's 781 00:46:02,030 --> 00:46:04,430 using this small angle approximation. 782 00:46:04,430 --> 00:46:08,720 You assume that sine theta is the same as theta. 783 00:46:08,720 --> 00:46:13,460 And it also makes an assumption, which is called the Debye 784 00:46:13,460 --> 00:46:16,820 approximation. 785 00:46:16,820 --> 00:46:20,780 So I'll say a bit more about the Debye approximation. 786 00:46:20,780 --> 00:46:23,360 But before I carry on, I'll say another thing. 787 00:46:23,360 --> 00:46:25,680 I'd forgotten I'd put this in. 788 00:46:25,680 --> 00:46:30,140 And that is that there was a really neat paper published 789 00:46:30,140 --> 00:46:36,410 way back in 1964 by McCutcheon, where 790 00:46:36,410 --> 00:46:42,350 he showed that you can postulate the idea, the concept of what 791 00:46:42,350 --> 00:46:45,050 he calls the 3D pupil. 792 00:46:45,050 --> 00:46:49,730 And this is the cap of a sphere where 793 00:46:49,730 --> 00:46:54,290 it's cut off at this angle alpha, the angle of the lens. 794 00:46:54,290 --> 00:46:59,290 And if you do love the 3D Fourier transform of this-- 795 00:46:59,290 --> 00:47:03,430 ta-da-- you get back to that. 796 00:47:03,430 --> 00:47:05,290 So isn't that neat? 797 00:47:05,290 --> 00:47:11,470 So I guess, if you think a bit more about it, it's probably-- 798 00:47:11,470 --> 00:47:15,700 I think you can satisfy yourself that perhaps it's right. 799 00:47:15,700 --> 00:47:18,700 This is in k space, effectively, then. 800 00:47:18,700 --> 00:47:25,540 And what it's really saying is that because the light 801 00:47:25,540 --> 00:47:31,000 satisfies the wave equation, the modulus of k is constant, 802 00:47:31,000 --> 00:47:39,990 and therefore the locus of this is a sphere of radius k, 803 00:47:39,990 --> 00:47:42,300 so very simple. 804 00:47:42,300 --> 00:47:48,240 So light propagating-- this convergent light propagating 805 00:47:48,240 --> 00:47:53,080 will have a k vector which lies on the surface of this sphere. 806 00:47:53,080 --> 00:47:57,750 So I just added that just because I thought people 807 00:47:57,750 --> 00:47:59,790 might find it a bit intriguing. 808 00:47:59,790 --> 00:48:04,050 You can sort of generalize a lot of imaging theory 809 00:48:04,050 --> 00:48:06,910 into 3D using this same sort of approach. 810 00:48:06,910 --> 00:48:09,810 But I'm not going to say anything more about that now. 811 00:48:09,810 --> 00:48:14,010 What I'm going to do now is talk about-- 812 00:48:14,010 --> 00:48:16,800 so this is this Debye approximation again. 813 00:48:16,800 --> 00:48:21,160 You remember, we said there's two ways of calculating 814 00:48:21,160 --> 00:48:23,370 to two different optical systems, which 815 00:48:23,370 --> 00:48:26,610 give virtually the same or similar sort of results. 816 00:48:26,610 --> 00:48:28,470 This one I'll look at first of all. 817 00:48:28,470 --> 00:48:30,570 This is where you got the aperture stop 818 00:48:30,570 --> 00:48:32,920 in the front focal plane of the lens. 819 00:48:32,920 --> 00:48:34,350 This is a distance f. 820 00:48:34,350 --> 00:48:35,880 This is a distance f. 821 00:48:35,880 --> 00:48:40,350 And then in this plane here, you will get the Airy disk. 822 00:48:40,350 --> 00:48:44,100 So the pattern I described a couple of slides 823 00:48:44,100 --> 00:48:50,530 ago for the field is in 3D space would be in this region here. 824 00:48:50,530 --> 00:48:54,510 So in this plane here, you get Fraunhofer diffraction, 825 00:48:54,510 --> 00:48:59,430 and then in the regions around it you get Fresnel diffraction. 826 00:48:59,430 --> 00:49:03,570 But you can get quite similar results, if you remember, 827 00:49:03,570 --> 00:49:05,980 by using this sort of system. 828 00:49:05,980 --> 00:49:09,420 So this is where the aperture stop 829 00:49:09,420 --> 00:49:13,200 is not actually in the front focal plane of the lens, but it 830 00:49:13,200 --> 00:49:15,210 actually in the plane of the lens. 831 00:49:15,210 --> 00:49:18,960 Or you could think another way of doing this experiment would 832 00:49:18,960 --> 00:49:23,520 be to get a convergent beam and place a circular aperture 833 00:49:23,520 --> 00:49:25,110 in that convergent beam. 834 00:49:25,110 --> 00:49:27,540 So the question is, what do you get in the focal region 835 00:49:27,540 --> 00:49:29,190 in that case? 836 00:49:29,190 --> 00:49:37,950 And it turns out that if you satisfy this condition, 837 00:49:37,950 --> 00:49:43,080 that a squared over lambda z is very large, where 838 00:49:43,080 --> 00:49:47,040 is a is this radius here-- 839 00:49:47,040 --> 00:49:49,590 this is called the Fresnel number. 840 00:49:49,590 --> 00:49:55,000 And if that quantity is very large, 841 00:49:55,000 --> 00:49:59,490 it means that what you'll get in this focal region 842 00:49:59,490 --> 00:50:01,080 is very similar to this. 843 00:50:01,080 --> 00:50:03,900 Actually, there's a parabolic phase term 844 00:50:03,900 --> 00:50:06,010 that we described when we did the Fourier 845 00:50:06,010 --> 00:50:08,220 optics that makes it different. 846 00:50:08,220 --> 00:50:12,510 But the intensity is going to be the same as for this case here. 847 00:50:12,510 --> 00:50:14,670 So these are two cases where you can 848 00:50:14,670 --> 00:50:18,040 get this Debye approximation. 849 00:50:18,040 --> 00:50:23,280 So in this case, it's exact for this particular geometry. 850 00:50:23,280 --> 00:50:26,440 In this case, it's an approximation 851 00:50:26,440 --> 00:50:31,320 which is obeyed if this Fresnel number is very large. 852 00:50:31,320 --> 00:50:36,540 And then, not that long ago, people 853 00:50:36,540 --> 00:50:41,050 realized that this was an approximation, 854 00:50:41,050 --> 00:50:43,200 an assumption that was made. 855 00:50:43,200 --> 00:50:49,410 And there was a very nice paper published by Li and Wolf, 856 00:50:49,410 --> 00:50:50,770 so this paper here-- 857 00:50:50,770 --> 00:50:57,780 1984, quite a long time ago now, but not that long, but anyway, 858 00:50:57,780 --> 00:51:00,810 as history goes, Li and Wolf. 859 00:51:00,810 --> 00:51:03,420 And they showed what would happen 860 00:51:03,420 --> 00:51:06,030 if this Fresnel number was not large 861 00:51:06,030 --> 00:51:08,640 but was of some intermediate value. 862 00:51:08,640 --> 00:51:10,065 In this case, N equals 10. 863 00:51:12,670 --> 00:51:17,830 So you see what happens is that this pattern has got distorted. 864 00:51:17,830 --> 00:51:21,190 You can see that it's tending to sort of fan out. 865 00:51:21,190 --> 00:51:25,150 And the physical explanation of that 866 00:51:25,150 --> 00:51:27,850 is that, effectively, the light is 867 00:51:27,850 --> 00:51:30,790 being distracted by the aperture and spreading 868 00:51:30,790 --> 00:51:34,300 because of diffraction effects. 869 00:51:34,300 --> 00:51:37,690 And that is superimposed on the focusing effect. 870 00:51:37,690 --> 00:51:40,900 So the lens is trying to focus the light down, 871 00:51:40,900 --> 00:51:45,080 and the diffraction is tending to focus the light out. 872 00:51:45,080 --> 00:51:50,860 And so you get a sort of balance between these two things. 873 00:51:50,860 --> 00:51:57,740 So in this focal plane here, you get the same intensity 874 00:51:57,740 --> 00:52:02,250 as you would if the Debye approximation was valid. 875 00:52:02,250 --> 00:52:05,940 But in the region elsewhere, it's a bit different. 876 00:52:05,940 --> 00:52:11,952 It's scaled according to some sort of affine transformation, 877 00:52:11,952 --> 00:52:13,160 I think it probably would be. 878 00:52:13,160 --> 00:52:15,080 What do you think? 879 00:52:15,080 --> 00:52:18,320 What do you think, [INAUDIBLE]? 880 00:52:18,320 --> 00:52:22,700 But one very interesting thing that happens 881 00:52:22,700 --> 00:52:24,380 is that it turns out-- you can't really 882 00:52:24,380 --> 00:52:26,300 see it very well from this diagram, 883 00:52:26,300 --> 00:52:32,360 but the maximum intensity is no longer in the focal plane, no 884 00:52:32,360 --> 00:52:35,210 longer in the geometrical focal plane. 885 00:52:35,210 --> 00:52:37,700 It's shifted towards the aperture. 886 00:52:37,700 --> 00:52:40,670 And that is called the focus shift effect, 887 00:52:40,670 --> 00:52:42,800 and there's been a lot of papers that 888 00:52:42,800 --> 00:52:47,210 have been concerned with that and calculating what happens 889 00:52:47,210 --> 00:52:49,760 for lots of different cases. 890 00:52:49,760 --> 00:52:52,740 So that's the Debye approximation. 891 00:52:52,740 --> 00:52:55,370 That shows what happens if it isn't true. 892 00:52:55,370 --> 00:52:57,530 But now I'm going to go back. 893 00:52:57,530 --> 00:52:59,750 And all the rest I'm going to carry on now 894 00:52:59,750 --> 00:53:02,810 is talking about what happens if the paraxial approximation 895 00:53:02,810 --> 00:53:08,960 breaks down, but the Debye approximation is valid. 896 00:53:08,960 --> 00:53:13,520 The case when the Debye approximation is not valid 897 00:53:13,520 --> 00:53:17,270 and it's a high numerical aperture is very complicated, 898 00:53:17,270 --> 00:53:18,890 and there have been very few papers 899 00:53:18,890 --> 00:53:20,900 that I've treated that case. 900 00:53:20,900 --> 00:53:23,390 Anyway, tight focusing of light. 901 00:53:23,390 --> 00:53:27,540 We get light, we put it into a microscope objective. 902 00:53:27,540 --> 00:53:30,680 The light then-- a microscope objective, 903 00:53:30,680 --> 00:53:35,570 a dry microscope objective might have a numerical aperture 904 00:53:35,570 --> 00:53:41,060 of 0.95, which means that the light is converging something 905 00:53:41,060 --> 00:53:45,080 maybe over 70 degrees relative to the axis, so very-- 906 00:53:45,080 --> 00:53:48,020 you can't really think of 70 degrees as being a small angle, 907 00:53:48,020 --> 00:53:49,460 I guess. 908 00:53:49,460 --> 00:53:52,810 So the normal paraxial approximations break down. 909 00:53:52,810 --> 00:53:57,300 And this is some sort of areas where we might want to do this. 910 00:53:57,300 --> 00:54:00,920 The one, of course, that I'm most interested in, microscopy, 911 00:54:00,920 --> 00:54:06,320 but also laser micromachining, optical data storage, 912 00:54:06,320 --> 00:54:11,360 optical lithography, laser trapping and cooling, looking 913 00:54:11,360 --> 00:54:14,630 at the physics of light-atom interactions, 914 00:54:14,630 --> 00:54:16,850 and cavity quantum electrodynamics. 915 00:54:16,850 --> 00:54:19,220 So there's a lot of basic physics 916 00:54:19,220 --> 00:54:25,190 that's being done nowadays that uses these sort of principles, 917 00:54:25,190 --> 00:54:27,370 too. 918 00:54:27,370 --> 00:54:30,410 So this is how we model the light. 919 00:54:30,410 --> 00:54:33,050 Again, we've got-- just like before, 920 00:54:33,050 --> 00:54:35,240 we think of the aperture as being 921 00:54:35,240 --> 00:54:38,720 placed in the front focal plane of the lens. 922 00:54:38,720 --> 00:54:40,620 Of course, the lens is very complicated. 923 00:54:40,620 --> 00:54:43,070 It's got loads of pieces of glass in. 924 00:54:43,070 --> 00:54:46,970 But we think of it just as like a black box, 925 00:54:46,970 --> 00:54:51,440 and we describe it just by this surface here, 926 00:54:51,440 --> 00:54:54,320 which is called the equivalent refractive locus. 927 00:54:54,320 --> 00:54:58,100 And what that is, is the locus of points 928 00:54:58,100 --> 00:55:02,980 where this ray and this ray intersect. 929 00:55:02,980 --> 00:55:08,870 And actually, if you've got a well-designed system that 930 00:55:08,870 --> 00:55:12,410 is satisfies what's called the sine condition, 931 00:55:12,410 --> 00:55:15,088 this surface will be a sphere. 932 00:55:15,088 --> 00:55:17,130 But I'm not going to go into the details of that. 933 00:55:17,130 --> 00:55:22,230 I'm just going to assume it's a sphere in what comes next. 934 00:55:22,230 --> 00:55:25,550 So in this case, you've effectively 935 00:55:25,550 --> 00:55:30,380 got lots of plane waves incident in the focus. 936 00:55:30,380 --> 00:55:34,940 But unlike the paraxial case, we really 937 00:55:34,940 --> 00:55:37,580 have to take into account now the polarization 938 00:55:37,580 --> 00:55:41,750 of these electric fields and add together the electric fields 939 00:55:41,750 --> 00:55:43,580 in the focal region. 940 00:55:43,580 --> 00:55:47,570 And so this is what happens after the light's 941 00:55:47,570 --> 00:55:50,660 gone through the lens. 942 00:55:50,660 --> 00:55:54,140 It starts off as being plane polarized. 943 00:55:54,140 --> 00:56:01,080 So in my diagram here the red represents the electric field, 944 00:56:01,080 --> 00:56:01,760 say. 945 00:56:01,760 --> 00:56:04,490 So we start off with the electric field 946 00:56:04,490 --> 00:56:08,210 being plane polarized in the vertical direction. 947 00:56:08,210 --> 00:56:10,990 And then after it's gone through the lens, 948 00:56:10,990 --> 00:56:17,510 the rays are focused on the focal point, which is 949 00:56:17,510 --> 00:56:19,700 at the center of this sphere. 950 00:56:19,700 --> 00:56:22,280 But if you actually calculate what 951 00:56:22,280 --> 00:56:26,720 happens to the electric field vector, this is what it does. 952 00:56:26,720 --> 00:56:29,510 After you've focused it, the electric field 953 00:56:29,510 --> 00:56:33,500 lies on the surface of the sphere with this geometry 954 00:56:33,500 --> 00:56:36,170 that-- you see that these lines all 955 00:56:36,170 --> 00:56:40,460 go through a point on the far side of the sphere. 956 00:56:40,460 --> 00:56:45,080 And the same is true for the magnetic field as well. 957 00:56:45,080 --> 00:56:48,470 If you calculate what happens, this is what you find happens. 958 00:56:48,470 --> 00:56:54,980 And you can show that this is actually exactly the same field 959 00:56:54,980 --> 00:56:58,220 as you would get if you placed at the center 960 00:56:58,220 --> 00:57:01,670 an electric dipole along this axis 961 00:57:01,670 --> 00:57:05,010 and a magnetic dipole along the y-axis. 962 00:57:05,010 --> 00:57:09,280 So if you add together those fields, you would get this. 963 00:57:09,280 --> 00:57:13,900 And one more thing to say is, you 964 00:57:13,900 --> 00:57:19,360 can see that if you've got a small aperture system, low NA, 965 00:57:19,360 --> 00:57:21,010 you're in this region here. 966 00:57:21,010 --> 00:57:23,500 And you can see now it's becoming very close to being 967 00:57:23,500 --> 00:57:25,060 linearly polarized again. 968 00:57:25,060 --> 00:57:27,460 It's only when you get to big angles 969 00:57:27,460 --> 00:57:32,550 that it starts becoming a bit more complicated. 970 00:57:32,550 --> 00:57:37,410 So Richards and Wolf in 1959-- 971 00:57:37,410 --> 00:57:42,110 so this is the same Emil Wolf who wrote the book. 972 00:57:42,110 --> 00:57:46,160 And in their paper-- 973 00:57:46,160 --> 00:57:48,020 I haven't got the reference here, I'm sorry. 974 00:57:48,020 --> 00:57:52,722 But anyone remember what it was published in? 975 00:57:52,722 --> 00:57:54,300 I've forgotten. 976 00:57:54,300 --> 00:57:57,140 No, it doesn't matter. 977 00:57:57,140 --> 00:57:59,890 Proceedings of the Physical Society, I think, maybe. 978 00:57:59,890 --> 00:58:01,140 But anyway, it doesn't matter. 979 00:58:01,140 --> 00:58:04,710 You can easily find it on the web if you're after it. 980 00:58:04,710 --> 00:58:08,792 They showed-- you found it? 981 00:58:08,792 --> 00:58:10,780 AUDIENCE: [INAUDIBLE] 982 00:58:10,780 --> 00:58:12,800 PROFESSOR: OK, Proceedings of the Royal Society 983 00:58:12,800 --> 00:58:15,290 of London in 1959. 984 00:58:15,290 --> 00:58:19,220 They showed that the field in the focal region 985 00:58:19,220 --> 00:58:20,840 could be written in this form. 986 00:58:20,840 --> 00:58:23,960 So rather than just have one diffraction integral, 987 00:58:23,960 --> 00:58:28,190 we now end up with three for the three-- 988 00:58:28,190 --> 00:58:30,590 defining these three integrals. 989 00:58:30,590 --> 00:58:33,650 And then these are the electric fields 990 00:58:33,650 --> 00:58:37,390 in the focal region, written in terms of these integrals-- 991 00:58:37,390 --> 00:58:39,110 i0, i1, and i2. 992 00:58:39,110 --> 00:58:44,100 And these things are very similar to the paraxial things. 993 00:58:44,100 --> 00:58:49,400 The normal paraxial one is a bit similar to this, with the j0. 994 00:58:49,400 --> 00:58:53,300 And this is an extra bit which comes about 995 00:58:53,300 --> 00:58:59,190 because of the system satisfying the sine condition. 996 00:58:59,190 --> 00:59:02,150 And I'll draw it for you, what's happening. 997 00:59:02,150 --> 00:59:06,650 Basically, if this is this aplanatic sphere surface, 998 00:59:06,650 --> 00:59:12,860 if you put in light, plane polarized light, 999 00:59:12,860 --> 00:59:16,190 into this system, you can see, if this 1000 00:59:16,190 --> 00:59:20,690 is uniform intensity when it goes in, 1001 00:59:20,690 --> 00:59:24,950 you can see that the amount of energy that's gone into here 1002 00:59:24,950 --> 00:59:29,100 and the amount of energy that's going to go into this angle, 1003 00:59:29,100 --> 00:59:32,000 which is very much bigger, is going to be the same. 1004 00:59:32,000 --> 00:59:33,210 So you see what I'm saying? 1005 00:59:33,210 --> 00:59:35,620 So this amount of energy is going 1006 00:59:35,620 --> 00:59:39,650 to be squeezed into this angle, whereas this amount of energy 1007 00:59:39,650 --> 00:59:43,350 is going to be spread out over this angle. 1008 00:59:43,350 --> 00:59:48,560 So this produces this factor here. 1009 00:59:52,710 --> 00:59:56,220 And then you can calculate what happens. 1010 00:59:56,220 --> 01:00:00,300 So this is-- yeah, I'm sorry about that. 1011 01:00:00,300 --> 01:00:02,394 You see, you write things like that not so ti 1012 01:00:02,394 --> 01:00:03,405 can be symmetric. 1013 01:00:03,405 --> 01:00:07,530 But of course, what I mean is that this is not 1014 01:00:07,530 --> 01:00:09,120 circularly symmetric. 1015 01:00:09,120 --> 01:00:14,730 This really should be a circle before the modern technology 1016 01:00:14,730 --> 01:00:17,030 distorted it. 1017 01:00:17,030 --> 01:00:21,240 And so rather than the normal Airy disk, what you get 1018 01:00:21,240 --> 01:00:26,370 is this elliptical focal spot like this. 1019 01:00:30,470 --> 01:00:33,540 And this is another calculation. 1020 01:00:33,540 --> 01:00:35,630 These are both from a couple of my papers. 1021 01:00:35,630 --> 01:00:42,830 This one is a really old paper, 1977, really old. 1022 01:00:42,830 --> 01:00:47,270 And this one was from a bit more recent, 1997. 1023 01:00:47,270 --> 01:00:49,290 So this was written with Peter Torok. 1024 01:00:49,290 --> 01:00:53,930 And we came up here with a new way of calculating it 1025 01:00:53,930 --> 01:00:59,030 where, basically, we look at the multimodal expansion 1026 01:00:59,030 --> 01:01:03,110 of the field over the sphere and express it 1027 01:01:03,110 --> 01:01:06,740 as a sum of these spherical harmonics, 1028 01:01:06,740 --> 01:01:08,750 the modes of the sphere. 1029 01:01:08,750 --> 01:01:13,550 And so this becomes a very actually quite computationally 1030 01:01:13,550 --> 01:01:15,620 efficient way of calculating things. 1031 01:01:15,620 --> 01:01:17,540 And this is for two systems, then. 1032 01:01:17,540 --> 01:01:23,790 This is for an angle of pi by 3, so 60 degrees, 1033 01:01:23,790 --> 01:01:26,030 and this is for 90 degrees. 1034 01:01:26,030 --> 01:01:30,080 So this is if you're actually the limiting case, when 1035 01:01:30,080 --> 01:01:33,670 you're actually focusing a whole hemisphere of light 1036 01:01:33,670 --> 01:01:34,840 onto the focus. 1037 01:01:34,840 --> 01:01:38,530 And you can see that the sort of general shape 1038 01:01:38,530 --> 01:01:41,440 stays much the same, but it changes a bit 1039 01:01:41,440 --> 01:01:44,923 as you change the aperture. 1040 01:01:44,923 --> 01:01:46,840 The other thing I was going to say a bit about 1041 01:01:46,840 --> 01:01:49,390 is Bessel beams. 1042 01:01:49,390 --> 01:01:52,960 I don't think George did anything about Bessel beams 1043 01:01:52,960 --> 01:01:56,770 in the end, but I did see it as a title on some 1044 01:01:56,770 --> 01:02:00,910 on the contents of some of the lectures, 1045 01:02:00,910 --> 01:02:02,410 as though it was just about to come. 1046 01:02:02,410 --> 01:02:05,140 But maybe I just missed it. 1047 01:02:05,140 --> 01:02:07,600 But anyway, this is a concept that 1048 01:02:07,600 --> 01:02:13,780 is quite interesting at the moment. 1049 01:02:13,780 --> 01:02:17,440 Basically, there are three ways I'm showing here of doing it. 1050 01:02:19,970 --> 01:02:23,720 One is what happens if you put in your-- 1051 01:02:23,720 --> 01:02:26,960 this is the aperture stop of this lens, 1052 01:02:26,960 --> 01:02:30,920 and we've put this central obstruction so that it becomes 1053 01:02:30,920 --> 01:02:32,600 just that annular region. 1054 01:02:32,600 --> 01:02:37,050 And we make the width of this annulus very, very narrow. 1055 01:02:37,050 --> 01:02:40,170 Then, you remember, just in the last lecture, 1056 01:02:40,170 --> 01:02:43,550 we looked at what happens for this annular lens. 1057 01:02:43,550 --> 01:02:46,220 And we know that what happens is, 1058 01:02:46,220 --> 01:02:50,960 compared with the Airy disk, the central lobe gets narrower, 1059 01:02:50,960 --> 01:02:53,040 but the side lobes get bigger. 1060 01:02:53,040 --> 01:02:55,670 Well, it turns out that in the limiting cases, 1061 01:02:55,670 --> 01:03:01,520 this becomes very narrow, you get the Airy disk becomes, 1062 01:03:01,520 --> 01:03:05,240 given by this very simple expression, just j0 squared, 1063 01:03:05,240 --> 01:03:08,310 where j0 is a Bessel function. 1064 01:03:08,310 --> 01:03:12,130 So that's why it's called a Bessel beam. 1065 01:03:12,130 --> 01:03:17,390 And it has this very nice property 1066 01:03:17,390 --> 01:03:22,460 that actually this is it this satisfies the wave equation. 1067 01:03:22,460 --> 01:03:25,290 So it's like a sort of mode of free space. 1068 01:03:25,290 --> 01:03:30,440 So this Bessel beam propagates with an infinite depth 1069 01:03:30,440 --> 01:03:32,240 of focus. 1070 01:03:32,240 --> 01:03:34,850 It just propagates without spreading, 1071 01:03:34,850 --> 01:03:37,960 and so that's why these beams have become 1072 01:03:37,960 --> 01:03:40,160 of quite a lot of interest. 1073 01:03:40,160 --> 01:03:43,910 Actually, the concept is really quite old. 1074 01:03:43,910 --> 01:03:48,980 There was a device proposed by MacLeod in 1954 1075 01:03:48,980 --> 01:03:52,380 for producing something rather similar, 1076 01:03:52,380 --> 01:03:53,990 which is called the Axicon. 1077 01:03:53,990 --> 01:03:57,960 An Axicon is like a conical piece of glass. 1078 01:03:57,960 --> 01:03:59,840 It's like a conical prism. 1079 01:03:59,840 --> 01:04:03,260 And you can see, then, that a prism refracts 1080 01:04:03,260 --> 01:04:06,350 the light parallel, like this. 1081 01:04:06,350 --> 01:04:09,620 And so you can see wherever you are along this axis 1082 01:04:09,620 --> 01:04:12,140 you can see that what you see is light 1083 01:04:12,140 --> 01:04:16,880 that's coming at the same angle, and so you 1084 01:04:16,880 --> 01:04:21,800 can see from this how you get this increased depth of focus 1085 01:04:21,800 --> 01:04:23,210 of this system. 1086 01:04:23,210 --> 01:04:27,110 And another system was proposed, again, a long time ago, 1087 01:04:27,110 --> 01:04:31,100 by Dyson, to do this not with a prism 1088 01:04:31,100 --> 01:04:33,740 but with a diffractive optical structure. 1089 01:04:33,740 --> 01:04:35,810 So you make a sort of grating, which 1090 01:04:35,810 --> 01:04:40,190 is a bit like a zone plate except the zone plate-- 1091 01:04:40,190 --> 01:04:44,900 the rings on a zone plate go as R squared, whereas-- 1092 01:04:44,900 --> 01:04:48,410 sorry, the width of them, the spacing goes as 1 over R 1093 01:04:48,410 --> 01:04:54,110 squared, whereas this one, the rings are linearly spaced. 1094 01:04:54,110 --> 01:04:59,390 And in this case, because the grating is linearly spaced, 1095 01:04:59,390 --> 01:05:01,760 all these rays go at the same angle, 1096 01:05:01,760 --> 01:05:03,260 and you get the same effect as this. 1097 01:05:06,070 --> 01:05:08,720 So that's how you make Bessel beams. 1098 01:05:08,720 --> 01:05:13,870 So I've been interested in them for a very long time. 1099 01:05:13,870 --> 01:05:18,590 And this is from another paper of mine going back to 1978, 1100 01:05:18,590 --> 01:05:21,140 very long time ago. 1101 01:05:21,140 --> 01:05:24,290 And this here was one of the lines of that paper, 1102 01:05:24,290 --> 01:05:25,160 as it says. 1103 01:05:25,160 --> 01:05:29,690 "A wave with zero-order Bessel function radial distribution 1104 01:05:29,690 --> 01:05:32,770 propagates without change." 1105 01:05:32,770 --> 01:05:36,230 And so what this is showing is what 1106 01:05:36,230 --> 01:05:38,780 happens if you, instead of thinking 1107 01:05:38,780 --> 01:05:43,490 of the paraxial theory, what happens to this Bessel beam 1108 01:05:43,490 --> 01:05:48,000 if it's not a paraxial case, if you put, say, 1109 01:05:48,000 --> 01:05:54,410 this annular mask in front of a high numerical aperture 1110 01:05:54,410 --> 01:05:55,910 objective lens. 1111 01:05:55,910 --> 01:05:57,960 And this is what you get. 1112 01:05:57,960 --> 01:05:59,690 So this is for different angles-- 1113 01:05:59,690 --> 01:06:02,780 30, 60, 90, as before. 1114 01:06:02,780 --> 01:06:04,130 I've even gone further here. 1115 01:06:04,130 --> 01:06:05,880 I've gone to 120. 1116 01:06:05,880 --> 01:06:08,660 And you might wonder what that could possibly mean, 1117 01:06:08,660 --> 01:06:12,260 except it means you can see that you could do it 1118 01:06:12,260 --> 01:06:17,840 by using a mirror, the parabolic mirror that George spoke 1119 01:06:17,840 --> 01:06:20,340 about spoke about earlier. 1120 01:06:20,340 --> 01:06:26,340 So you can have light coming in this way, 1121 01:06:26,340 --> 01:06:29,410 say, a narrow annular beam coming in here, 1122 01:06:29,410 --> 01:06:32,340 so this might correspond, say, to this case. 1123 01:06:32,340 --> 01:06:37,920 And then the 120 case would be when the light's coming in 1124 01:06:37,920 --> 01:06:40,540 like this. 1125 01:06:40,540 --> 01:06:43,780 So you can actually get a bigger than 90 degrees 1126 01:06:43,780 --> 01:06:47,110 by using a mirror sort of structure. 1127 01:06:47,110 --> 01:06:50,950 But anyway, what you see is what happens, 1128 01:06:50,950 --> 01:06:54,680 if the angle of the focusing is small, 1129 01:06:54,680 --> 01:06:57,250 then it's pretty similar to what I showed earlier 1130 01:06:57,250 --> 01:07:00,520 for the full lens, full aperture. 1131 01:07:00,520 --> 01:07:02,890 But as you make this angle bigger and bigger, 1132 01:07:02,890 --> 01:07:05,920 eventually very nasty things start happening. 1133 01:07:05,920 --> 01:07:10,480 This thing starts splitting into two spots rather than one, 1134 01:07:10,480 --> 01:07:12,580 and you can see that it does more and more of that 1135 01:07:12,580 --> 01:07:15,220 as you make it even bigger. 1136 01:07:15,220 --> 01:07:17,620 And so this is all assuming that we're 1137 01:07:17,620 --> 01:07:22,600 putting plane polarized light into the lens. 1138 01:07:22,600 --> 01:07:26,980 So the conclusion we come so is that putting plane polarized 1139 01:07:26,980 --> 01:07:29,500 light into that lens is not really 1140 01:07:29,500 --> 01:07:32,770 a good idea, not good news. 1141 01:07:32,770 --> 01:07:36,610 And so this is explaining things a bit more. 1142 01:07:36,610 --> 01:07:40,660 First of all, we look at the black curves. 1143 01:07:40,660 --> 01:07:45,490 So these correspond, then-- so a normal Airy disk is this one, 1144 01:07:45,490 --> 01:07:48,700 and this other one here corresponds 1145 01:07:48,700 --> 01:07:50,590 to the narrow annulus. 1146 01:07:50,590 --> 01:07:52,690 So you can see this is the Bessel beam. 1147 01:07:52,690 --> 01:07:54,850 You can see it sharpened up quite 1148 01:07:54,850 --> 01:08:00,760 considerably but with some rather larger side lobes. 1149 01:08:00,760 --> 01:08:05,430 But then if you go to the high NA case-- 1150 01:08:05,430 --> 01:08:09,390 so this is calculating it for a numerical aperture of 1.4, 1151 01:08:09,390 --> 01:08:12,480 so for an [INAUDIBLE] immersion objective. 1152 01:08:12,480 --> 01:08:16,050 And you'll find that for the annulus now, 1153 01:08:16,050 --> 01:08:20,470 it's actually no narrower than for the full lens. 1154 01:08:20,470 --> 01:08:26,580 So the polarization effects are really basically getting 1155 01:08:26,580 --> 01:08:31,680 rid of any advantage in this sharpening of the spot 1156 01:08:31,680 --> 01:08:34,720 that we get in this case. 1157 01:08:34,720 --> 01:08:38,600 But it turns out there is a way of getting round that, 1158 01:08:38,600 --> 01:08:42,149 and that is to not use plane polarized light going in. 1159 01:08:42,149 --> 01:08:45,359 And what we use instead is putting 1160 01:08:45,359 --> 01:08:47,640 radially polarized light. 1161 01:08:47,640 --> 01:08:52,050 If we put in radially polarized light-- 1162 01:08:52,050 --> 01:08:56,670 this corresponds to this magenta color, 1163 01:08:56,670 --> 01:08:58,260 so as it says here, radial. 1164 01:08:58,260 --> 01:09:03,550 I've also called it here TM0 because it corresponds 1165 01:09:03,550 --> 01:09:06,910 to a transverse magnetic mode. 1166 01:09:06,910 --> 01:09:10,029 I'll show you how that comes about in a minute. 1167 01:09:10,029 --> 01:09:12,840 And now you can see for that case 1168 01:09:12,840 --> 01:09:15,330 we're back again to getting something 1169 01:09:15,330 --> 01:09:19,470 very close to what you would get for the paraxial case. 1170 01:09:19,470 --> 01:09:22,680 So there's been a number of papers 1171 01:09:22,680 --> 01:09:28,260 which have exploited using radially polarized input 1172 01:09:28,260 --> 01:09:33,330 into your lens in order to get a smaller focal spot. 1173 01:09:33,330 --> 01:09:35,370 So yeah, this is how it comes about, 1174 01:09:35,370 --> 01:09:38,010 why it's called transverse magnetic. 1175 01:09:38,010 --> 01:09:43,590 So this is radially polarized light. 1176 01:09:43,590 --> 01:09:46,420 Sorry, yeah, the red is the electric field. 1177 01:09:46,420 --> 01:09:48,439 So that's in the radial direction. 1178 01:09:48,439 --> 01:09:51,060 If we had a small NA, it would just 1179 01:09:51,060 --> 01:09:53,979 be still radially polarized. 1180 01:09:53,979 --> 01:09:58,470 But you can see that if you look on the complete sphere, 1181 01:09:58,470 --> 01:10:05,860 it starts becoming like lines of longitude-- 1182 01:10:05,860 --> 01:10:08,500 latitude on the sphere. 1183 01:10:08,500 --> 01:10:13,270 And you see the magnetic field are these lines, which 1184 01:10:13,270 --> 01:10:15,730 are purely transverse, so that's why 1185 01:10:15,730 --> 01:10:18,400 we call it transverse magnetic. 1186 01:10:18,400 --> 01:10:22,600 And the 0 corresponds to the fact it's the lowest order 1187 01:10:22,600 --> 01:10:24,510 mode that's like that. 1188 01:10:24,510 --> 01:10:30,298 There is another case which is azimuthal polarization. 1189 01:10:30,298 --> 01:10:31,840 This is where the electric field goes 1190 01:10:31,840 --> 01:10:35,540 like this and the magnetic field is like this. 1191 01:10:35,540 --> 01:10:38,260 So this corresponds to TE, but I'm not 1192 01:10:38,260 --> 01:10:41,500 going to say anything more about that because that doesn't 1193 01:10:41,500 --> 01:10:42,715 produce anything useful. 1194 01:10:45,630 --> 01:10:50,340 And this is-- actually, I said there were three ways 1195 01:10:50,340 --> 01:10:51,840 of making these Bessel beams. 1196 01:10:51,840 --> 01:10:59,160 There's actually a fourth way, which some people together 1197 01:10:59,160 --> 01:11:03,720 with me at the Data Storage Institute here in Singapore 1198 01:11:03,720 --> 01:11:06,120 have been working on, and this is 1199 01:11:06,120 --> 01:11:10,710 to use a lens with some sort of binary optic structure. 1200 01:11:10,710 --> 01:11:12,540 And actually, this was from a paper 1201 01:11:12,540 --> 01:11:15,930 where we were simulating what happens 1202 01:11:15,930 --> 01:11:20,580 if you put radially polarized light into this sort of system 1203 01:11:20,580 --> 01:11:23,970 and you optimize this binary mask in order 1204 01:11:23,970 --> 01:11:26,250 to get close to a Bessel beam. 1205 01:11:26,250 --> 01:11:28,260 And these are the sort of results you get. 1206 01:11:28,260 --> 01:11:31,050 This is along the axis you can see that you get something 1207 01:11:31,050 --> 01:11:34,800 which really propagates without spreading over 1208 01:11:34,800 --> 01:11:35,745 quite a long distance. 1209 01:11:42,840 --> 01:11:50,250 This is showing how there are various ways of specifying how 1210 01:11:50,250 --> 01:11:53,280 the performance of a focusing system. 1211 01:11:53,280 --> 01:11:59,880 And one way is to calculate or measure the electric field 1212 01:11:59,880 --> 01:12:03,540 at the focal point for a given amount of power going in. 1213 01:12:03,540 --> 01:12:06,690 And what you want, of course, is for the power of the focus 1214 01:12:06,690 --> 01:12:07,860 to be-- 1215 01:12:07,860 --> 01:12:10,960 the electric field at the focus to be as big as possible. 1216 01:12:10,960 --> 01:12:15,430 And so this ratio is this thing called F here. 1217 01:12:15,430 --> 01:12:20,580 And what this is showing is that this one here is actually 1218 01:12:20,580 --> 01:12:24,930 for our plane polarized light going in and showing what 1219 01:12:24,930 --> 01:12:28,170 happens to this ratio as you increase 1220 01:12:28,170 --> 01:12:30,280 the aperture of the lens. 1221 01:12:30,280 --> 01:12:32,340 And so you can see that as you increase 1222 01:12:32,340 --> 01:12:35,160 the aperture of the lens, so, of course, 1223 01:12:35,160 --> 01:12:37,470 it focuses better and better. 1224 01:12:37,470 --> 01:12:39,330 And therefore for the same amount of power 1225 01:12:39,330 --> 01:12:43,110 get getting going in, you get a higher field at the focus. 1226 01:12:43,110 --> 01:12:45,870 So that's why it rises like this. 1227 01:12:45,870 --> 01:12:50,130 But it, you see, only gets to eventually-- 1228 01:12:50,130 --> 01:12:52,920 the maximum it ever gets to is a half. 1229 01:12:52,920 --> 01:12:56,730 And the reason it gets to a half is because-- 1230 01:12:56,730 --> 01:12:59,220 I showed that this polarization is actually 1231 01:12:59,220 --> 01:13:04,830 equivalent to an electric dipole plus a magnetic dipole. 1232 01:13:04,830 --> 01:13:11,830 And so only half of the power is going into the electric dipole, 1233 01:13:11,830 --> 01:13:14,490 and it's the electric dipole that gives an electric field 1234 01:13:14,490 --> 01:13:15,420 at the center. 1235 01:13:15,420 --> 01:13:18,150 The magnetic dipole gives a magnetic field 1236 01:13:18,150 --> 01:13:19,930 at the center, which we don't want. 1237 01:13:19,930 --> 01:13:23,400 So only half of the power is going into the right thing. 1238 01:13:23,400 --> 01:13:25,620 So that's why that does that. 1239 01:13:25,620 --> 01:13:29,250 And then you can see there another two curves here. 1240 01:13:29,250 --> 01:13:32,980 This one is our radially polarized light. 1241 01:13:32,980 --> 01:13:36,970 So you can see that it's very slow to get started here, 1242 01:13:36,970 --> 01:13:41,560 and that's because with the radial polarized distribution, 1243 01:13:41,560 --> 01:13:44,680 you've got very little power near the axis anyway. 1244 01:13:44,680 --> 01:13:48,100 So it eventually starts taking off, 1245 01:13:48,100 --> 01:13:51,230 and you see that at some angle here, 1246 01:13:51,230 --> 01:13:53,090 which is really quite big-- 1247 01:13:53,090 --> 01:13:55,690 so this is 90 degrees down here somewhere, 1248 01:13:55,690 --> 01:13:58,510 so this is going to be maybe 80 degrees or something 1249 01:13:58,510 --> 01:13:59,290 like that-- 1250 01:13:59,290 --> 01:14:03,690 it overtakes the plane polarized one and becomes better. 1251 01:14:03,690 --> 01:14:05,660 But you can see there's another one, 1252 01:14:05,660 --> 01:14:10,660 another curve here, which is for just 1253 01:14:10,660 --> 01:14:13,810 an electric dipole on its own, so 1254 01:14:13,810 --> 01:14:16,060 a transverse electric dipole. 1255 01:14:16,060 --> 01:14:19,600 So rather than try and give this field, 1256 01:14:19,600 --> 01:14:22,150 we just might try to get this polarization. 1257 01:14:22,150 --> 01:14:26,090 And if we do that, then we see we do even better. 1258 01:14:26,090 --> 01:14:28,810 So in this region here, you can see this is doing better 1259 01:14:28,810 --> 01:14:31,120 than either of these other two. 1260 01:14:31,120 --> 01:14:35,740 So this is what we call an electric dipole wave. 1261 01:14:35,740 --> 01:14:38,600 And this is what this looks like, then. 1262 01:14:38,600 --> 01:14:40,610 This mixed dipole, which is the electric 1263 01:14:40,610 --> 01:14:45,050 plus a magnetic dipole, is equal to an electric dipole 1264 01:14:45,050 --> 01:14:47,360 plus a magnetic dipole. 1265 01:14:47,360 --> 01:14:49,820 So this is the electric dipole field. 1266 01:14:49,820 --> 01:14:52,820 So what we need to do is to make it 1267 01:14:52,820 --> 01:14:55,070 so that after we focus the light, 1268 01:14:55,070 --> 01:14:57,620 the electric field of the focused light 1269 01:14:57,620 --> 01:15:01,160 is on the surface along these lines, like this, basically. 1270 01:15:01,160 --> 01:15:03,940 You see that if you look in the region near the axis, 1271 01:15:03,940 --> 01:15:05,750 all these behave the same. 1272 01:15:05,750 --> 01:15:08,210 They're all we're all going to be virtually linearly 1273 01:15:08,210 --> 01:15:08,930 polarized. 1274 01:15:08,930 --> 01:15:11,750 It's only when you get so big apertures 1275 01:15:11,750 --> 01:15:14,260 that these become different. 1276 01:15:17,550 --> 01:15:24,275 And so this is now looking at what happens with Bessel beams. 1277 01:15:27,970 --> 01:15:30,890 So this the so-called mixed dipole, 1278 01:15:30,890 --> 01:15:33,890 which is the plain polarization case, 1279 01:15:33,890 --> 01:15:42,800 for 30 degrees, 60 degrees, 90 degrees 150 degrees. 1280 01:15:42,800 --> 01:15:48,790 This is for the electric dipole case, 30 degrees, 60 degrees. 1281 01:15:48,790 --> 01:15:54,030 Now, you see here this is a bit better than this one. 1282 01:15:54,030 --> 01:15:58,200 And this one, this spot is divided into two, 1283 01:15:58,200 --> 01:16:00,450 but this one hasn't divided into two. 1284 01:16:00,450 --> 01:16:03,200 But it has got these rather large side 1285 01:16:03,200 --> 01:16:05,350 lobes developed here. 1286 01:16:05,350 --> 01:16:10,350 And then if you look at this other curve, these other ones 1287 01:16:10,350 --> 01:16:13,020 here, this is what I call a T1. 1288 01:16:13,020 --> 01:16:15,450 So this is another mode I haven't mentioned before. 1289 01:16:15,450 --> 01:16:21,240 It's another TE mode, but it's the TE1 rather than the TE0. 1290 01:16:21,240 --> 01:16:22,950 And you can see this one-- actually, 1291 01:16:22,950 --> 01:16:24,450 ti's the same in all these diagrams. 1292 01:16:24,450 --> 01:16:26,880 It's independent of the angle now. 1293 01:16:26,880 --> 01:16:29,400 So you can see, actually, this one turns out 1294 01:16:29,400 --> 01:16:31,260 to be slightly smaller than this one even. 1295 01:16:34,140 --> 01:16:37,180 And this shows what the TE1 looks like. 1296 01:16:37,180 --> 01:16:39,280 This is TE1. 1297 01:16:39,280 --> 01:16:44,770 So again, the electric field is transverse. 1298 01:16:44,770 --> 01:16:47,710 But it's not going all the way around like TE0. 1299 01:16:47,710 --> 01:16:50,860 It's going this way on this side and this way on this side. 1300 01:16:50,860 --> 01:16:53,680 So if you could make this polarization, 1301 01:16:53,680 --> 01:16:58,360 this would produce that results I've just described there. 1302 01:16:58,360 --> 01:17:01,210 So this shows all these different types. 1303 01:17:01,210 --> 01:17:05,580 So what do you have to put into the lens to get that? 1304 01:17:05,580 --> 01:17:08,410 So this is what you have to put into the lens. 1305 01:17:08,410 --> 01:17:12,120 So if you put it in plane polarized light, 1306 01:17:12,120 --> 01:17:16,260 you get this what I call the mixed dipole case. 1307 01:17:16,260 --> 01:17:19,680 In order to get electric dipole polarization out, 1308 01:17:19,680 --> 01:17:22,290 you have to put in light that looks like this. 1309 01:17:22,290 --> 01:17:25,740 For the magnetic dipole, it has to look like this. 1310 01:17:25,740 --> 01:17:27,220 See, here it's going-- 1311 01:17:27,220 --> 01:17:30,580 it's pointing in, and here it's pointing out. 1312 01:17:30,580 --> 01:17:35,970 And for the TE1, it's like azimuthal. 1313 01:17:35,970 --> 01:17:40,380 And for the TM1, it's radial, like this. 1314 01:17:40,380 --> 01:17:42,630 So by altering the polarization of the light that 1315 01:17:42,630 --> 01:17:46,015 goes into the lens, you can generate all 1316 01:17:46,015 --> 01:17:47,265 these different mode patterns. 1317 01:17:50,030 --> 01:17:52,010 And this is what it does. 1318 01:17:52,010 --> 01:17:55,250 This is calculating the area of the focal spot 1319 01:17:55,250 --> 01:17:59,310 as a function of angle for all of these different cases. 1320 01:17:59,310 --> 01:18:02,870 So here there's a whole range of them 1321 01:18:02,870 --> 01:18:06,170 for a full lens, full circular aperture. 1322 01:18:06,170 --> 01:18:10,850 And this is for a few different cases for the Bessel beam. 1323 01:18:10,850 --> 01:18:13,640 And you can see what happens is we want this area, of course, 1324 01:18:13,640 --> 01:18:15,600 to be as small as possible. 1325 01:18:15,600 --> 01:18:18,440 And you can see that actually in this region 1326 01:18:18,440 --> 01:18:21,530 here, the electric dipole one is the smallest, 1327 01:18:21,530 --> 01:18:23,840 and then the TE1 is the smallest, 1328 01:18:23,840 --> 01:18:27,890 and then eventually the radial one becomes the smallest. 1329 01:18:27,890 --> 01:18:30,740 But this doesn't become bigger until you 1330 01:18:30,740 --> 01:18:35,230 get over a numerical aperture of 0.89. 1331 01:18:35,230 --> 01:18:38,260 And then this is for the full circular case. 1332 01:18:38,260 --> 01:18:40,960 I've got lots of different cases here. 1333 01:18:40,960 --> 01:18:44,290 This one corresponds to a paraboloid mirror. 1334 01:18:44,290 --> 01:18:48,070 This one corresponds to basically 1335 01:18:48,070 --> 01:18:54,130 a different apertization of the light, which you can see it 1336 01:18:54,130 --> 01:18:57,970 actually is doing better here than the others, 1337 01:18:57,970 --> 01:19:01,690 but eventually runs out of steam when you get to big angles. 1338 01:19:01,690 --> 01:19:05,770 And you can see here, the TE case is the smallest, 1339 01:19:05,770 --> 01:19:07,920 and then eventually the radial case 1340 01:19:07,920 --> 01:19:10,770 is the smallest after you get above a numerical aperture 1341 01:19:10,770 --> 01:19:16,158 0.91. 1342 01:19:16,158 --> 01:19:17,950 Now, the other thing you'll notice, though, 1343 01:19:17,950 --> 01:19:20,470 about all those spots I was showing 1344 01:19:20,470 --> 01:19:22,240 is they're all not symmetrical. 1345 01:19:22,240 --> 01:19:25,180 So what can we do to make them symmetrical? 1346 01:19:25,180 --> 01:19:30,220 Well, this fits in a bit with what we were talking about 1347 01:19:30,220 --> 01:19:34,000 before, the polarization case, the circularly polarized. 1348 01:19:34,000 --> 01:19:39,670 The TE and the TM0 are already circularly symmetric, 1349 01:19:39,670 --> 01:19:41,080 rotationally symmetric. 1350 01:19:41,080 --> 01:19:44,350 This one corresponds to our normal radial polarization. 1351 01:19:44,350 --> 01:19:47,140 But these are the ones, we know that x 1352 01:19:47,140 --> 01:19:52,330 polarized plus iy polarized equals circular polarized. 1353 01:19:52,330 --> 01:19:55,150 So we can do similar sorts of things 1354 01:19:55,150 --> 01:20:00,880 with these TEs or the electric dipole polarizations. 1355 01:20:00,880 --> 01:20:05,170 We can add we can add one of them along the x with one 1356 01:20:05,170 --> 01:20:08,120 of them along the y with an i there, 1357 01:20:08,120 --> 01:20:10,630 and that actually is now going to give something which 1358 01:20:10,630 --> 01:20:13,090 is rotationally symmetric. 1359 01:20:13,090 --> 01:20:15,610 And it actually, if you think a lot more about it, 1360 01:20:15,610 --> 01:20:20,620 it corresponds to azimuthal polarization 1361 01:20:20,620 --> 01:20:23,680 with a phase singularity, with a vortex, a phase 1362 01:20:23,680 --> 01:20:26,620 vortex superimposed on it. 1363 01:20:26,620 --> 01:20:31,210 And this one corresponds to some elliptical polarization 1364 01:20:31,210 --> 01:20:33,580 with a phase singularity. 1365 01:20:33,580 --> 01:20:34,960 So what I mean by this? 1366 01:20:34,960 --> 01:20:40,680 This one, azimuthal polarization with a phase singularity, 1367 01:20:40,680 --> 01:20:50,360 what I mean by that is at anywhere around here, 1368 01:20:50,360 --> 01:20:54,320 it's polarized in the azimuthal direction. 1369 01:20:54,320 --> 01:20:57,530 But as you go round here, the phase 1370 01:20:57,530 --> 01:21:01,595 changes from 0 right around to 2 pi. 1371 01:21:04,710 --> 01:21:06,080 [INAUDIBLE]? 1372 01:21:06,080 --> 01:21:06,630 No? 1373 01:21:06,630 --> 01:21:07,780 OK. 1374 01:21:07,780 --> 01:21:10,480 So it changes. 1375 01:21:10,480 --> 01:21:13,550 As you go round here, it changes from 0 to 2 pi. 1376 01:21:13,550 --> 01:21:16,040 So where it gets back to here, it matches up again. 1377 01:21:16,040 --> 01:21:18,830 But here it's actually changed by pi, 1378 01:21:18,830 --> 01:21:22,250 so it's effectively in the opposite direction. 1379 01:21:22,250 --> 01:21:23,510 So that's what I mean by that. 1380 01:21:23,510 --> 01:21:25,100 There's a continuous phase variation 1381 01:21:25,100 --> 01:21:27,070 as you go round there. 1382 01:21:27,070 --> 01:21:31,400 And on this one, elliptical polarization 1383 01:21:31,400 --> 01:21:35,870 with a phase singularity, what it means is that the light is-- 1384 01:21:35,870 --> 01:21:40,550 anywhere on here, it's elliptically polarized, 1385 01:21:40,550 --> 01:21:45,320 like this, say, but at any angle it will 1386 01:21:45,320 --> 01:21:47,600 be a similar sort of ellipse. 1387 01:21:47,600 --> 01:21:51,320 But the phase of this ellipse will 1388 01:21:51,320 --> 01:21:56,690 change with a face singularity as you go around in the circle. 1389 01:21:56,690 --> 01:22:02,620 And so what happens if you do that? 1390 01:22:02,620 --> 01:22:05,370 Well, this is what you get. 1391 01:22:05,370 --> 01:22:07,070 So this is looking at the cross section 1392 01:22:07,070 --> 01:22:10,070 for three different angles for all these different types. 1393 01:22:10,070 --> 01:22:14,660 And so this is for Bessel beams again. 1394 01:22:14,660 --> 01:22:16,760 They're the simplest to calculate. 1395 01:22:16,760 --> 01:22:20,550 So what we find is that for small angles, 1396 01:22:20,550 --> 01:22:23,030 the radial polarized is no good. 1397 01:22:23,030 --> 01:22:26,510 It turns out that the reason why radial polarization works, 1398 01:22:26,510 --> 01:22:29,330 actually, is because-- 1399 01:22:29,330 --> 01:22:29,830 sorry. 1400 01:22:29,830 --> 01:22:31,960 I should have said this earlier. 1401 01:22:31,960 --> 01:22:34,600 I don't know why I didn't, really. 1402 01:22:34,600 --> 01:22:39,620 Why it works is because the light that comes down here 1403 01:22:39,620 --> 01:22:42,600 has got some polarization in this direction here, 1404 01:22:42,600 --> 01:22:47,050 which means it's got a component in this longitudinal direction. 1405 01:22:47,050 --> 01:22:51,280 So actually, the radially polarized light 1406 01:22:51,280 --> 01:22:55,360 put into the lens produces a longitudinal electric field 1407 01:22:55,360 --> 01:22:56,680 in the focal region. 1408 01:22:56,680 --> 01:23:00,010 And that might seem very strange to people, first of all. 1409 01:23:00,010 --> 01:23:01,900 You can actually generate a wave that's 1410 01:23:01,900 --> 01:23:07,720 moving along which has got a longitudinal electric field. 1411 01:23:07,720 --> 01:23:11,120 But that seems to be what is true. 1412 01:23:11,120 --> 01:23:15,710 But the point is, of course, if this angle is small, 1413 01:23:15,710 --> 01:23:19,280 then this longitudinal component is going to be very weak. 1414 01:23:19,280 --> 01:23:22,090 And that is why this racially polarized one here 1415 01:23:22,090 --> 01:23:27,220 doesn't work well, because the transverse field is dominating 1416 01:23:27,220 --> 01:23:29,020 over the longitudinal field. 1417 01:23:29,020 --> 01:23:31,000 But as you increase the aperture, 1418 01:23:31,000 --> 01:23:35,170 you see it gets better, like that. 1419 01:23:35,170 --> 01:23:40,380 And then I also show here the electric dipole. 1420 01:23:40,380 --> 01:23:43,600 All these are all the same on this, for this small angle, 1421 01:23:43,600 --> 01:23:48,290 but they for the larger anguish they become different. 1422 01:23:48,290 --> 01:23:51,240 And this is showing how-- 1423 01:23:51,240 --> 01:23:52,410 sorry. 1424 01:23:52,410 --> 01:23:54,770 This arrow has moved itself for some reason. 1425 01:23:54,770 --> 01:23:57,400 I don't know why it's moved itself, 1426 01:23:57,400 --> 01:24:00,210 but that this was supposed to be pointing to this blue line 1427 01:24:00,210 --> 01:24:03,980 here and showing how, in this case, 1428 01:24:03,980 --> 01:24:08,030 the TE1 case is the narrowest. 1429 01:24:08,030 --> 01:24:11,300 It's actually even narrower than the radial in this case 1430 01:24:11,300 --> 01:24:15,740 here, but with some rather slightly-- 1431 01:24:15,740 --> 01:24:17,330 some bigger side lobes here. 1432 01:24:17,330 --> 01:24:21,550 And this shows also the electric dipole case as well. 1433 01:24:21,550 --> 01:24:25,850 And this one's showing how the plane polarized case is really 1434 01:24:25,850 --> 01:24:28,730 very bad, as we showed earlier, once you 1435 01:24:28,730 --> 01:24:30,050 get to really big angles. 1436 01:24:33,280 --> 01:24:39,240 And then this is comparing them a bit more. 1437 01:24:39,240 --> 01:24:42,290 This is actually plotting the normalized width 1438 01:24:42,290 --> 01:24:44,270 of that rotationally symmetric spot 1439 01:24:44,270 --> 01:24:47,710 that you get, again, for all these different cases, 1440 01:24:47,710 --> 01:24:49,430 and showing-- 1441 01:24:49,430 --> 01:24:52,310 so these are the cases of a full aperture. 1442 01:24:52,310 --> 01:24:55,370 These are the cases of a Bessel beam. 1443 01:24:55,370 --> 01:24:57,650 And you can see that, actually, it turns out 1444 01:24:57,650 --> 01:25:04,390 that the TE annulus is the narrowest, the TE Bessel beam, 1445 01:25:04,390 --> 01:25:06,650 all the time, for any angle. 1446 01:25:06,650 --> 01:25:10,420 It's always narrower than the radially polarized one. 1447 01:25:10,420 --> 01:25:14,020 And for these cases here, for the full circular case, 1448 01:25:14,020 --> 01:25:17,890 the TE one is always narrower than the green one 1449 01:25:17,890 --> 01:25:20,210 until you get to this point here, 1450 01:25:20,210 --> 01:25:24,100 which corresponds to a numerical aperture of 0.98, which 1451 01:25:24,100 --> 01:25:28,880 is higher than you would ever get to in a practical system. 1452 01:25:28,880 --> 01:25:33,610 So what this is all showing is that, as I say, 1453 01:25:33,610 --> 01:25:37,240 putting plane polarized light in is not the optimum. 1454 01:25:37,240 --> 01:25:41,920 It turns out that very often the optimum can be different 1455 01:25:41,920 --> 01:25:43,990 according to the circumstances, but it 1456 01:25:43,990 --> 01:25:46,960 can be this electric dipole polarization 1457 01:25:46,960 --> 01:25:50,890 or it can be the radially polarized input 1458 01:25:50,890 --> 01:25:54,100 or it can be the transverse electric. 1459 01:25:54,100 --> 01:25:57,150 And I think I-- 1460 01:25:57,150 --> 01:26:01,490 yeah, this is showing the strength of the side lobes 1461 01:26:01,490 --> 01:26:06,830 and showing how the transverse electric might be narrower, 1462 01:26:06,830 --> 01:26:08,480 but it might have bigger side lobes. 1463 01:26:08,480 --> 01:26:11,540 So here the radially polarized has smaller side lobes, 1464 01:26:11,540 --> 01:26:14,900 weaker side lobes, than the TE. 1465 01:26:18,810 --> 01:26:22,800 So and there's our conclusions about this, then. 1466 01:26:22,800 --> 01:26:25,890 So if you focus plane polarized light, 1467 01:26:25,890 --> 01:26:27,960 you'll get a big focal spot. 1468 01:26:27,960 --> 01:26:32,562 So you can improve that by using radially polarized light, 1469 01:26:32,562 --> 01:26:34,020 and that comes about because you've 1470 01:26:34,020 --> 01:26:37,920 got this strong longitudinal field on the axis. 1471 01:26:37,920 --> 01:26:41,280 But we find that electric dipole polarization actually 1472 01:26:41,280 --> 01:26:45,100 gives a higher electric energy density at the focus, 1473 01:26:45,100 --> 01:26:47,250 as I showed earlier, higher than you get 1474 01:26:47,250 --> 01:26:50,340 with radial polarized light. 1475 01:26:50,340 --> 01:26:56,100 And then I showed that the TE1 mode polarization gives 1476 01:26:56,100 --> 01:26:59,400 the smallest central lobe, and it's smaller 1477 01:26:59,400 --> 01:27:04,320 even than you get with radially polarized light. 1478 01:27:04,320 --> 01:27:07,830 But the TE on its own is asymmetric, 1479 01:27:07,830 --> 01:27:11,200 but you can make a symmetric version of it, 1480 01:27:11,200 --> 01:27:14,430 which is equivalent so azimuthal polarization 1481 01:27:14,430 --> 01:27:17,100 with a phase singularity. 1482 01:27:17,100 --> 01:27:23,120 So that's about it for what I was going to say about this. 1483 01:27:23,120 --> 01:27:25,790 So this is finishing up the course 1484 01:27:25,790 --> 01:27:29,810 with something which is really sort of ongoing research 1485 01:27:29,810 --> 01:27:33,423 and something that, I think, there's 1486 01:27:33,423 --> 01:27:35,090 a lot of people around the world working 1487 01:27:35,090 --> 01:27:38,180 on these different problems and trying to optimize 1488 01:27:38,180 --> 01:27:43,280 these systems to get the best possible focusing properties 1489 01:27:43,280 --> 01:27:45,940 that you can get with a high NA lens. 1490 01:27:45,940 --> 01:27:50,090 And I mentioned before all the many different applications 1491 01:27:50,090 --> 01:27:54,240 that there's this could be used for. 1492 01:27:54,240 --> 01:27:56,490 So I'm going to stop, then. 1493 01:27:56,490 --> 01:27:57,310 Questions? 1494 01:27:57,310 --> 01:27:58,650 Any questions about this? 1495 01:27:58,650 --> 01:28:01,150 Any questions about earlier? 1496 01:28:01,150 --> 01:28:03,130 There's lots of questions about this are there? 1497 01:28:03,130 --> 01:28:04,698 Yeah, maybe. 1498 01:28:04,698 --> 01:28:06,490 There's a lot of information there, I know. 1499 01:28:06,490 --> 01:28:11,260 I think you're probably all suffering from overload. 1500 01:28:11,260 --> 01:28:17,650 But it's all very complicated, actually, but very beautiful 1501 01:28:17,650 --> 01:28:18,580 as well, I think. 1502 01:28:18,580 --> 01:28:22,500 It's nice physics, I think.