1 00:00:00,120 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,550 at ocw.mit.edu. 8 00:00:20,805 --> 00:00:22,040 PROFESSOR: Right, OK. 9 00:00:22,040 --> 00:00:23,750 So I think this was where we were up to. 10 00:00:23,750 --> 00:00:26,030 I wasn't here last time, so I don't really know. 11 00:00:26,030 --> 00:00:30,320 But George told me that this was where you're up to, 12 00:00:30,320 --> 00:00:33,150 so that's where I'm going to carry on from. 13 00:00:33,150 --> 00:00:40,340 And so this section is all about imaging and resolution, 14 00:00:40,340 --> 00:00:43,190 and it starts off here looking in 15 00:00:43,190 --> 00:00:49,250 some strange American dictionary at the definition 16 00:00:49,250 --> 00:00:51,890 of a "resolution." 17 00:00:51,890 --> 00:00:54,530 Of course, myself, coming from Oxford, 18 00:00:54,530 --> 00:00:56,660 I would always use the Oxford Dictionary. 19 00:00:56,660 --> 00:00:58,700 I don't know what that says. 20 00:00:58,700 --> 00:01:03,090 But anyway, this is quite interesting. 21 00:01:03,090 --> 00:01:08,480 The verb here, resolve, "to break up its constituent parts, 22 00:01:08,480 --> 00:01:12,545 to analyze, to find an answer to, 23 00:01:12,545 --> 00:01:15,440 to solve, determine, and decide," blah, blah, blah. 24 00:01:15,440 --> 00:01:18,500 But anyway, I think this first part is-- the very first thing 25 00:01:18,500 --> 00:01:21,380 is a good one, isn't it? 26 00:01:21,380 --> 00:01:25,400 So break up into its constituent parts. 27 00:01:25,400 --> 00:01:28,610 So you look at an image, and you've 28 00:01:28,610 --> 00:01:32,990 got to actually determine what the different parts 29 00:01:32,990 --> 00:01:34,220 of the image really mean. 30 00:01:34,220 --> 00:01:36,860 And if they sort of merge together, 31 00:01:36,860 --> 00:01:39,290 you won't be able to resolve what 32 00:01:39,290 --> 00:01:42,200 the image is trying to say. 33 00:01:42,200 --> 00:01:43,580 So that's what it means. 34 00:01:48,300 --> 00:01:52,170 And yeah, now, there are lots of different ways 35 00:01:52,170 --> 00:01:55,500 of determining resolution, different definitions 36 00:01:55,500 --> 00:01:57,170 that people apply. 37 00:01:57,170 --> 00:02:03,330 The most used one, perhaps, is the Rayleigh resolution limit, 38 00:02:03,330 --> 00:02:05,670 which was proposed by Lord Rayleigh. 39 00:02:05,670 --> 00:02:09,479 Lord Rayleigh was very interested in astronomy, 40 00:02:09,479 --> 00:02:15,450 and so he was interested in trying to quantify 41 00:02:15,450 --> 00:02:19,480 the imaging of stars. 42 00:02:19,480 --> 00:02:22,370 So if you look up in the telescope at the sky 43 00:02:22,370 --> 00:02:25,890 and you've got two stars very close together, 44 00:02:25,890 --> 00:02:30,810 then you'll see, like the two images of the stars, 45 00:02:30,810 --> 00:02:33,720 this shows to two different separations of those two 46 00:02:33,720 --> 00:02:35,370 points. 47 00:02:35,370 --> 00:02:40,320 And now, it's important to note that stars, of course, 48 00:02:40,320 --> 00:02:42,720 two independent stars would be completely 49 00:02:42,720 --> 00:02:45,240 incoherent with respect to each other 50 00:02:45,240 --> 00:02:47,700 because they obviously don't know-- the one doesn't 51 00:02:47,700 --> 00:02:49,390 know what the other's doing. 52 00:02:49,390 --> 00:02:54,210 So they emit light incoherently with respect to each other. 53 00:02:54,210 --> 00:02:57,190 So to calculate the image of these two points, 54 00:02:57,190 --> 00:03:02,640 you have to add together the intensities of the two images, 55 00:03:02,640 --> 00:03:05,310 and that's what this shows here. 56 00:03:05,310 --> 00:03:08,940 And so this shows two different spaces. 57 00:03:08,940 --> 00:03:11,100 And this one here, this is what I 58 00:03:11,100 --> 00:03:17,520 think is really normally taken as the Rayleigh resolution 59 00:03:17,520 --> 00:03:19,660 limit. 60 00:03:19,660 --> 00:03:21,313 I don't know about this other one. 61 00:03:21,313 --> 00:03:22,980 I don't where Georgia has got this from, 62 00:03:22,980 --> 00:03:26,790 but he talks about these two different limits 63 00:03:26,790 --> 00:03:28,330 in these notes. 64 00:03:28,330 --> 00:03:33,870 So in this one here, you'll see that this peak here 65 00:03:33,870 --> 00:03:39,000 is placed exactly over the 0 of the other, 66 00:03:39,000 --> 00:03:42,900 the first 0 of the other image. 67 00:03:42,900 --> 00:03:46,530 So that's what defines this separation shown 68 00:03:46,530 --> 00:03:48,280 in this picture. 69 00:03:48,280 --> 00:03:54,720 And we find that if they're separated by that amount, 70 00:03:54,720 --> 00:03:57,600 the total image is given by this blue line. 71 00:03:57,600 --> 00:04:01,170 We get that just by adding together the two intensities 72 00:04:01,170 --> 00:04:03,960 of the two points together. 73 00:04:03,960 --> 00:04:12,130 And you can see that this drops a bit at the center. 74 00:04:12,130 --> 00:04:13,210 What is it? 75 00:04:13,210 --> 00:04:19,995 I think the intensity there is 0.888-- 76 00:04:19,995 --> 00:04:20,870 I can't remember now. 77 00:04:20,870 --> 00:04:22,460 No, it doesn't matter. 78 00:04:22,460 --> 00:04:25,640 But anyway, it's about 0.8, isn't it? 79 00:04:25,640 --> 00:04:29,810 0.835, so that figure sounds-- 80 00:04:29,810 --> 00:04:32,310 But anyway, the other interesting thing, of course, 81 00:04:32,310 --> 00:04:35,960 is that because this one is 0 at this point, 82 00:04:35,960 --> 00:04:39,470 it means that this maximum is 1. 83 00:04:39,470 --> 00:04:40,460 It's exactly 1. 84 00:04:40,460 --> 00:04:44,360 It's just the one point with no contribution 85 00:04:44,360 --> 00:04:46,840 from the other one. 86 00:04:46,840 --> 00:04:51,220 Anyway, so that is arbitrarily taken 87 00:04:51,220 --> 00:04:55,810 as being the definition of the point being just resolved. 88 00:04:55,810 --> 00:04:58,750 So we say that if there are any closer together than that, 89 00:04:58,750 --> 00:05:00,520 you can't resolve them. 90 00:05:00,520 --> 00:05:04,540 You would see a little bit of a dip still, 91 00:05:04,540 --> 00:05:08,200 but we say that the dips not really big enough for you 92 00:05:08,200 --> 00:05:11,890 to really be confident that there are two objects there. 93 00:05:11,890 --> 00:05:15,040 And if you make them further apart, 94 00:05:15,040 --> 00:05:17,620 then, of course, they'll become more distinguished. 95 00:05:17,620 --> 00:05:22,640 And this shows an example here where you can see now, 96 00:05:22,640 --> 00:05:26,230 this is twice the separation. 97 00:05:26,230 --> 00:05:29,940 So this is where the PSF diameter equals the point 98 00:05:29,940 --> 00:05:30,565 source spacing. 99 00:05:33,280 --> 00:05:38,790 So the-- what does he mean by that? 100 00:05:38,790 --> 00:05:41,010 Sorry. 101 00:05:41,010 --> 00:05:45,100 This one, the PSF radius equals the point source spacing. 102 00:05:45,100 --> 00:05:46,000 Yeah, OK. 103 00:05:46,000 --> 00:05:51,060 So that is the distance between the center of the image 104 00:05:51,060 --> 00:05:53,160 And the first dark ring. 105 00:05:53,160 --> 00:05:59,790 In this one, the PSF diameter equals the point source 106 00:05:59,790 --> 00:06:02,073 spacing. 107 00:06:02,073 --> 00:06:02,990 Has he got this right? 108 00:06:07,640 --> 00:06:09,590 No, it's not the second zero. 109 00:06:09,590 --> 00:06:14,620 It's not at the second zero exactly, anymore, is it? 110 00:06:14,620 --> 00:06:17,480 So the spacing here is twice the spacing there. 111 00:06:17,480 --> 00:06:20,150 Anyway, I think that's what I'm taking at this meaning. 112 00:06:20,150 --> 00:06:23,600 But as Charlene just pointed out, 113 00:06:23,600 --> 00:06:27,620 that means this is not actually at a zero. 114 00:06:27,620 --> 00:06:31,340 This is not placed at a zero of this other one 115 00:06:31,340 --> 00:06:36,170 because, if you remember, the zeros of the Airy disk 116 00:06:36,170 --> 00:06:38,720 are irregularly spaced, aren't they? 117 00:06:38,720 --> 00:06:42,470 So they're not equally spaced, like the light for a sink. 118 00:06:42,470 --> 00:06:47,480 And so because this isn't 0 at this point here, 119 00:06:47,480 --> 00:06:50,960 this maximum here will be not exactly 1. 120 00:06:50,960 --> 00:06:54,010 It will be slightly more than 1. 121 00:06:54,010 --> 00:06:54,540 OK. 122 00:06:54,540 --> 00:07:03,070 And if this condition is satisfied, then, in this case, 123 00:07:03,070 --> 00:07:08,820 the delta R, the-- 124 00:07:08,820 --> 00:07:10,290 where does he define that? 125 00:07:10,290 --> 00:07:15,226 That's the-- I guess it's this distance here, isn't it? 126 00:07:15,226 --> 00:07:17,540 AUDIENCE: [INAUDIBLE] 127 00:07:17,540 --> 00:07:21,470 PROFESSOR: It's the separation of the peaks. 128 00:07:21,470 --> 00:07:24,190 Yeah, OK, it's the separation of the peaks. 129 00:07:24,190 --> 00:07:28,190 It's 0.61 lambda over NA. 130 00:07:28,190 --> 00:07:28,700 Oh, yeah. 131 00:07:28,700 --> 00:07:29,200 OK. 132 00:07:29,200 --> 00:07:30,280 So here he's-- 133 00:07:30,280 --> 00:07:33,760 R and R dashed are obviously coordinates in the object 134 00:07:33,760 --> 00:07:35,690 played in the image plane, effectively. 135 00:07:35,690 --> 00:07:36,190 Yeah. 136 00:07:36,190 --> 00:07:38,530 So, of course, it depends on-- 137 00:07:38,530 --> 00:07:42,430 the numerical aperture in those two spaces 138 00:07:42,430 --> 00:07:43,843 won't be the same, of course. 139 00:07:43,843 --> 00:07:45,760 But there will be magnification as well, which 140 00:07:45,760 --> 00:07:47,800 is going to mean that you get the same answer 141 00:07:47,800 --> 00:07:50,020 for the resolution in both planes. 142 00:07:50,020 --> 00:07:54,940 So this 0.61 is this magic figure 143 00:07:54,940 --> 00:07:58,570 that comes about basically because of the-- 144 00:07:58,570 --> 00:08:02,800 it's related to the first zero of the Bessel function, j1. 145 00:08:02,800 --> 00:08:07,150 And the first zero of the Bessel function j1 is 3.83, 146 00:08:07,150 --> 00:08:11,620 and this is 3.83 divided by 2 pi. 147 00:08:11,620 --> 00:08:14,360 Does that sound right? 148 00:08:14,360 --> 00:08:14,980 Yeah. 149 00:08:14,980 --> 00:08:17,560 Yeah, I think that sounds right. 150 00:08:17,560 --> 00:08:21,240 This one here is twice the spacing, 151 00:08:21,240 --> 00:08:25,670 so 1.22 lambda over NA. 152 00:08:25,670 --> 00:08:28,710 So this is 3.83 divided by pi. 153 00:08:28,710 --> 00:08:29,210 Yeah. 154 00:08:29,210 --> 00:08:31,100 So here, actually, these are obviously 155 00:08:31,100 --> 00:08:34,370 extremely well resolved. 156 00:08:34,370 --> 00:08:37,020 You wouldn't have any problem in seeing 157 00:08:37,020 --> 00:08:38,270 that there's two points there. 158 00:08:43,780 --> 00:08:50,530 And so here he calls these two definitions the safe definition 159 00:08:50,530 --> 00:08:53,440 and the pushy definition. 160 00:08:53,440 --> 00:08:56,110 My experience is that the pushy definition 161 00:08:56,110 --> 00:09:00,670 is the one that is normally quoted in places, 162 00:09:00,670 --> 00:09:04,090 and so this is this one where you can only just see 163 00:09:04,090 --> 00:09:07,390 a bit of a dip in between them. 164 00:09:07,390 --> 00:09:12,760 Note that if we did the same thing for the 1D system, i.e. 165 00:09:12,760 --> 00:09:17,560 for a slit aperture rather than a circular aperture, then 166 00:09:17,560 --> 00:09:20,110 you'd be looking at the first zero of a sink rather 167 00:09:20,110 --> 00:09:22,760 than a Bessel function. 168 00:09:22,760 --> 00:09:27,310 And so you get these expressions rather than these, so 1.22 169 00:09:27,310 --> 00:09:32,360 rather than 1 or 0.61 rather than 0.5. 170 00:09:32,360 --> 00:09:33,940 So in this case you actually you can 171 00:09:33,940 --> 00:09:38,800 see you can resolve slightly smaller spacing 172 00:09:38,800 --> 00:09:44,390 with a slit aperture than with a circular aperture. 173 00:09:44,390 --> 00:09:48,550 And this is a true effect that sometimes people 174 00:09:48,550 --> 00:09:52,980 try to take advantage of. 175 00:09:52,980 --> 00:09:55,710 You will see that different authors 176 00:09:55,710 --> 00:09:58,320 give different definitions. 177 00:09:58,320 --> 00:10:02,190 So I'm not going to go into all that because it's not 178 00:10:02,190 --> 00:10:04,560 really important at this stage. 179 00:10:04,560 --> 00:10:09,170 Rayleigh, in his original paper, noted the issue of noise 180 00:10:09,170 --> 00:10:13,800 and warned that the definition of just resolvable points 181 00:10:13,800 --> 00:10:18,120 is system or application dependent. 182 00:10:18,120 --> 00:10:19,510 Actually, what you find-- 183 00:10:19,510 --> 00:10:21,870 well, a lot of that is true, of course. 184 00:10:21,870 --> 00:10:26,398 Here we're-- it very much depends on the actual system 185 00:10:26,398 --> 00:10:28,440 and what you're trying to measure all these sorts 186 00:10:28,440 --> 00:10:29,070 of things. 187 00:10:29,070 --> 00:10:32,460 And you can come up with lots of different resolution 188 00:10:32,460 --> 00:10:35,820 criteria for different shaped objects, 189 00:10:35,820 --> 00:10:38,460 and the comparison of different systems 190 00:10:38,460 --> 00:10:40,770 doesn't always give the same answers 191 00:10:40,770 --> 00:10:42,970 according to what you chose. 192 00:10:42,970 --> 00:10:43,480 So beware. 193 00:10:43,480 --> 00:10:47,950 There is not one answer to this question of resolution. 194 00:10:47,950 --> 00:10:49,680 The other thing is, as he says here, 195 00:10:49,680 --> 00:10:52,860 is noise is an important thing, of course. 196 00:10:52,860 --> 00:10:54,620 If you add noise to these image-- 197 00:10:54,620 --> 00:10:57,720 we've got some examples later where 198 00:10:57,720 --> 00:10:59,850 there's noise added to the images 199 00:10:59,850 --> 00:11:02,320 so that you can see what it does to it. 200 00:11:02,320 --> 00:11:06,420 But one thing that is perhaps not quite what George is saying 201 00:11:06,420 --> 00:11:08,640 there-- actually, you do find-- 202 00:11:08,640 --> 00:11:10,770 if you look at the image of these two points, 203 00:11:10,770 --> 00:11:13,560 as you bring them together, you actually 204 00:11:13,560 --> 00:11:19,590 do find that the degradation in the image 205 00:11:19,590 --> 00:11:22,200 is really rather sudden. 206 00:11:22,200 --> 00:11:27,390 So although, OK, it's not a specific point, 207 00:11:27,390 --> 00:11:29,850 it is quite so sudden. 208 00:11:29,850 --> 00:11:38,140 And so you'll find that if you change the separation 209 00:11:38,140 --> 00:11:42,440 around slightly more or slightly less than that, 210 00:11:42,440 --> 00:11:46,220 you'd find that this central bit would go up and down quite 211 00:11:46,220 --> 00:11:47,960 a lot, actually. 212 00:11:47,960 --> 00:11:55,860 So it is quite a sensitive change around that spacing. 213 00:11:55,860 --> 00:12:01,990 So that gives us some idea that maybe this resolution 214 00:12:01,990 --> 00:12:07,280 limit as it's defined does mean something, anyway. 215 00:12:07,280 --> 00:12:11,210 Now, that's for a perfect aberration-free system. 216 00:12:11,210 --> 00:12:13,130 But very often, you've got systems 217 00:12:13,130 --> 00:12:16,730 where you've got aberrations, and that's obviously 218 00:12:16,730 --> 00:12:18,500 going to mean that the resolution is not 219 00:12:18,500 --> 00:12:20,160 going to be as good. 220 00:12:20,160 --> 00:12:23,600 So this picture shows an example with coma. 221 00:12:23,600 --> 00:12:28,250 So this is with an off-axis ray, off-axis beam going 222 00:12:28,250 --> 00:12:34,650 through a lens, and you get this coma spot here. 223 00:12:34,650 --> 00:12:37,100 So now, of course, this spot is bigger 224 00:12:37,100 --> 00:12:40,130 than it would be if there were no aberrations, 225 00:12:40,130 --> 00:12:44,070 and therefore you'd expect that the resolution would be worse. 226 00:12:44,070 --> 00:12:50,740 So this is another thing you have to take into account. 227 00:12:50,740 --> 00:12:54,080 I don't think there's going to be much more about that. 228 00:12:54,080 --> 00:12:55,800 But did that do something there? 229 00:12:55,800 --> 00:12:59,040 Or perhaps this didn't push it fast enough. 230 00:12:59,040 --> 00:13:01,770 Now, the other way of looking at the effects-- 231 00:13:01,770 --> 00:13:05,100 resolution and the effects of aberrations 232 00:13:05,100 --> 00:13:08,620 is looking at the transfer function approach. 233 00:13:08,620 --> 00:13:16,230 And so this is giving results for a one-dimensional system. 234 00:13:16,230 --> 00:13:21,010 And it you see it's an MTF, Modulation Transfer Function. 235 00:13:21,010 --> 00:13:24,050 So that means it's an incoherent system. 236 00:13:24,050 --> 00:13:31,800 The MTF is the Fourier transform of the point spread function. 237 00:13:31,800 --> 00:13:34,470 So this is for the aberration-free case. 238 00:13:34,470 --> 00:13:36,450 This is a sink squared, the point spread 239 00:13:36,450 --> 00:13:39,020 function for the 1D case. 240 00:13:39,020 --> 00:13:41,010 And if you take the Fourier transform, 241 00:13:41,010 --> 00:13:44,850 you get the modulation transfer function, 242 00:13:44,850 --> 00:13:47,520 as he calls it, and vise versa, of course. 243 00:13:47,520 --> 00:13:49,290 These are Fourier transform pairs. 244 00:13:49,290 --> 00:13:52,420 This is just a triangle function. 245 00:13:52,420 --> 00:13:55,390 And this shows what would happen if you 246 00:13:55,390 --> 00:13:57,270 got some aberrations there. 247 00:13:57,270 --> 00:14:01,200 In general, what would happen is that this would be broader, 248 00:14:01,200 --> 00:14:06,150 and this would be deformed so that you've got a lower 249 00:14:06,150 --> 00:14:08,430 spatial frequency response. 250 00:14:08,430 --> 00:14:10,770 Here, actually, he seems to be showing something which 251 00:14:10,770 --> 00:14:15,910 has got an improved response at the very high frequencies, 252 00:14:15,910 --> 00:14:19,060 which is quite interesting because what that would mean, 253 00:14:19,060 --> 00:14:22,740 actually, is that if your object only 254 00:14:22,740 --> 00:14:27,050 had those high spatial frequencies, 255 00:14:27,050 --> 00:14:29,520 the system with aberrations would actually give a better 256 00:14:29,520 --> 00:14:33,120 image than the one without aberrations. 257 00:14:33,120 --> 00:14:35,520 So maybe that's a bit anomalous. 258 00:14:39,760 --> 00:14:44,970 So now some common misinterpretations, 259 00:14:44,970 --> 00:14:47,940 attempting to resolve object feature smaller 260 00:14:47,940 --> 00:14:50,380 than the resolution limit, i.e. 261 00:14:50,380 --> 00:14:54,210 1.22 lambda over NA, is hopeless. 262 00:14:54,210 --> 00:14:57,880 So he's saying that that's not strictly true 263 00:14:57,880 --> 00:14:59,790 because this is not, for a start, 264 00:14:59,790 --> 00:15:04,030 it's not a sudden cutoff. 265 00:15:04,030 --> 00:15:08,920 And it depends on the presence of noise and other features. 266 00:15:08,920 --> 00:15:14,310 So there is a very big no to say that, yes, you can sometimes 267 00:15:14,310 --> 00:15:19,560 see things which are smaller than the resolution limit. 268 00:15:19,560 --> 00:15:23,520 And then also another point is that there 269 00:15:23,520 --> 00:15:27,840 are ways you can actually improve the performance using, 270 00:15:27,840 --> 00:15:30,240 for example, the CLEAN algorithm. 271 00:15:30,240 --> 00:15:32,820 The CLEAN algorithm is a thing they use in astronomy 272 00:15:32,820 --> 00:15:36,090 for doing deconvolution. 273 00:15:36,090 --> 00:15:38,280 It works particularly well-- 274 00:15:38,280 --> 00:15:41,550 it would be very good, actually, for some of these microscope 275 00:15:41,550 --> 00:15:46,140 things that people do nowadays with these very sparse 276 00:15:46,140 --> 00:15:48,780 molecules because it's designed to work 277 00:15:48,780 --> 00:15:53,220 with you know in astronomy with isolated point images. 278 00:15:53,220 --> 00:15:57,630 So if it looks at a continuous object 279 00:15:57,630 --> 00:16:01,210 like a motorcar or something like that, it wouldn't work. 280 00:16:01,210 --> 00:16:08,670 It would actually only work on point-type images. 281 00:16:08,670 --> 00:16:13,290 Then vena filtering-- we'll say something more about that-- 282 00:16:13,290 --> 00:16:15,960 expectation, maximization, et cetera. 283 00:16:15,960 --> 00:16:20,880 So there are various digital ways you can improve the image 284 00:16:20,880 --> 00:16:22,830 and get information out of the image, 285 00:16:22,830 --> 00:16:27,180 even if you can't see it by [INAUDIBLE].. 286 00:16:27,180 --> 00:16:32,250 And then there's this other word, super-resolution. 287 00:16:32,250 --> 00:16:35,430 Super-resolution, it means, basically, 288 00:16:35,430 --> 00:16:38,970 getting better resolution than you are suppose to, i.e. 289 00:16:38,970 --> 00:16:42,210 beating the Rayleigh diffraction limit. 290 00:16:42,210 --> 00:16:46,260 And various ways have been proposed for doing this. 291 00:16:46,260 --> 00:16:49,200 And until recently, I guess, most of them 292 00:16:49,200 --> 00:16:52,950 were pretty unsuccessful. 293 00:16:52,950 --> 00:16:56,340 But more recently, nowadays, some of these techniques 294 00:16:56,340 --> 00:16:57,490 are doing really well. 295 00:16:57,490 --> 00:16:59,850 I think the most noticeable one probably 296 00:16:59,850 --> 00:17:05,910 is STED, Stimulated Emission Depletion, microscopy, 297 00:17:05,910 --> 00:17:10,079 where Stefan Hell is getting resolutions which, 298 00:17:10,079 --> 00:17:14,190 I don't know, more than a factor of 10 better than you should 299 00:17:14,190 --> 00:17:17,490 according to the Rayleigh diffraction limit. 300 00:17:17,490 --> 00:17:20,550 But anyway, here he's talking about what are 301 00:17:20,550 --> 00:17:23,339 called super-resolving filters. 302 00:17:23,339 --> 00:17:29,220 What you do is you alter the pupil function of the lens 303 00:17:29,220 --> 00:17:34,440 by some clever trick in order to sharpen up the point spread 304 00:17:34,440 --> 00:17:35,200 function. 305 00:17:35,200 --> 00:17:38,280 And I think that maybe George did mention that earlier 306 00:17:38,280 --> 00:17:40,630 on at some point. 307 00:17:40,630 --> 00:17:43,890 But the problem is what always happens 308 00:17:43,890 --> 00:17:47,310 is that it doesn't actually-- 309 00:17:47,310 --> 00:17:48,810 you never get anything for nothing. 310 00:17:48,810 --> 00:17:50,970 That's the problem. 311 00:17:50,970 --> 00:17:57,360 You make a central lobe narrower, 312 00:17:57,360 --> 00:18:01,440 but you find that the side lobes become stronger. 313 00:18:01,440 --> 00:18:04,710 And the other thing is that very often the power transmitted 314 00:18:04,710 --> 00:18:08,530 through the system is much reduced. 315 00:18:08,530 --> 00:18:11,160 So you can imagine if you've got some filter that 316 00:18:11,160 --> 00:18:17,630 absorbs part of the power, then you're wasting power. 317 00:18:17,630 --> 00:18:20,430 So both of these can be bad. 318 00:18:20,430 --> 00:18:27,120 And so, consequently, this idea of super-resolving filters 319 00:18:27,120 --> 00:18:31,440 in practice has never really, up until now, 320 00:18:31,440 --> 00:18:36,350 been very successful from a practical point of view. 321 00:18:36,350 --> 00:18:39,100 But anyway, this shows one example. 322 00:18:39,100 --> 00:18:42,330 This shows a circular pupil. 323 00:18:42,330 --> 00:18:49,410 And yeah, and then this one is an annual pupil, 324 00:18:49,410 --> 00:18:53,550 which you can model as being one circular function 325 00:18:53,550 --> 00:18:55,320 minus another one. 326 00:18:55,320 --> 00:18:59,190 And of course, once you write it in that simple way, 327 00:18:59,190 --> 00:19:03,580 as a circ function, as the difference between two 328 00:19:03,580 --> 00:19:06,450 circ functions, you can do the Fourier transform with that 329 00:19:06,450 --> 00:19:09,870 very easily and get the point spread function of these two 330 00:19:09,870 --> 00:19:11,950 cases very easily. 331 00:19:11,950 --> 00:19:17,730 And so he's working it out for a focal length of 20 centimeters 332 00:19:17,730 --> 00:19:22,320 and a wavelength of 0.5 microns. 333 00:19:22,320 --> 00:19:26,220 And the width of that aperture is-- 334 00:19:26,220 --> 00:19:27,700 what is it? 335 00:19:27,700 --> 00:19:35,120 It looks about 4 millimeters diameter, something like that. 336 00:19:35,120 --> 00:19:36,100 Yeah. 337 00:19:36,100 --> 00:19:40,830 Anyway, so this is-- and he works out what it looks like. 338 00:19:40,830 --> 00:19:42,530 So this is the PSF. 339 00:19:42,530 --> 00:19:46,640 This is supposed to be an Airy disk. 340 00:19:46,640 --> 00:19:49,700 It's been sampled a bit coarsely, 341 00:19:49,700 --> 00:19:52,530 so it looks a bit ragged. 342 00:19:52,530 --> 00:19:58,640 And so what you see is when you look at the annular case, 343 00:19:58,640 --> 00:20:02,720 you put this central obscuration in the center 344 00:20:02,720 --> 00:20:08,720 there, what it does is it narrows up that central lobe. 345 00:20:08,720 --> 00:20:12,380 And you can see, though, it also increases 346 00:20:12,380 --> 00:20:14,260 the strength of these side lobes, 347 00:20:14,260 --> 00:20:17,630 that these are much higher than these ones. 348 00:20:17,630 --> 00:20:22,720 So although in some cases this can give a good 349 00:20:22,720 --> 00:20:26,300 an improved image compared with this, for example, 350 00:20:26,300 --> 00:20:29,320 for point objects it would do well, 351 00:20:29,320 --> 00:20:31,250 but if you've got an extended object 352 00:20:31,250 --> 00:20:34,170 like an image of a car or something, 353 00:20:34,170 --> 00:20:36,080 it might not work so well. 354 00:20:36,080 --> 00:20:40,850 The other way of thinking on it is in terms of the MTF. 355 00:20:40,850 --> 00:20:46,190 And this is our normal MTF for a circular pupil. 356 00:20:46,190 --> 00:20:49,750 It's this so-called Chinese hat function. 357 00:20:49,750 --> 00:20:52,550 And then for the annular pupil, this is what you get. 358 00:20:52,550 --> 00:20:54,800 You remember, you calculate these. 359 00:20:54,800 --> 00:21:00,230 This is the convolution, the autocorrelation of a circle. 360 00:21:00,230 --> 00:21:02,050 So you have to-- 361 00:21:02,050 --> 00:21:04,490 you get two circles and you displace 362 00:21:04,490 --> 00:21:06,740 it one relative to the other. 363 00:21:06,740 --> 00:21:09,650 You calculate the area of overlap, 364 00:21:09,650 --> 00:21:12,740 and you plot the area of overlap as a function of the distance 365 00:21:12,740 --> 00:21:14,720 between the two center. 366 00:21:14,720 --> 00:21:17,000 So that's how you actually calculate this. 367 00:21:17,000 --> 00:21:20,030 And you do the same with this one, 368 00:21:20,030 --> 00:21:23,840 and you can see what you get is you see that it drops 369 00:21:23,840 --> 00:21:27,090 more quickly, first of all. 370 00:21:27,090 --> 00:21:29,460 This is normalized to 1 because you normally 371 00:21:29,460 --> 00:21:35,220 normalize transfer functions to 1 for 0 spatial frequencies. 372 00:21:35,220 --> 00:21:36,930 That's the way we normally do it. 373 00:21:36,930 --> 00:21:42,730 But I guess maybe it's not the only way of doing it. 374 00:21:42,730 --> 00:21:45,690 This drops down quickly because you can see, 375 00:21:45,690 --> 00:21:48,570 if you're working out the autocorrelation at this angular 376 00:21:48,570 --> 00:21:50,790 aperture, you can see that [INAUDIBLE] 377 00:21:50,790 --> 00:21:56,700 you move it a small amount, the one the one bright region 378 00:21:56,700 --> 00:21:59,610 is not going to overlap with the other one anymore. 379 00:21:59,610 --> 00:22:02,100 So that's why it drops off quickly there. 380 00:22:02,100 --> 00:22:05,400 And then it has another blip out here, 381 00:22:05,400 --> 00:22:10,900 which basically comes about from when the toe annularly-- 382 00:22:10,900 --> 00:22:15,020 where they both got their bright bits on top of each other. 383 00:22:15,020 --> 00:22:19,510 And so if you compare this one with this one, 384 00:22:19,510 --> 00:22:23,290 you can see that here the spatial frequency response 385 00:22:23,290 --> 00:22:25,420 is much lower. 386 00:22:25,420 --> 00:22:29,153 Here, actually, it can be higher. 387 00:22:29,153 --> 00:22:30,820 The cutoff of course, is still the same. 388 00:22:34,790 --> 00:22:36,940 So you could calculate the image of any object 389 00:22:36,940 --> 00:22:40,540 you fancied from knowing that. 390 00:22:40,540 --> 00:22:43,090 And this is another example. 391 00:22:43,090 --> 00:22:47,770 This is the case of a Gaussian apodization. 392 00:22:47,770 --> 00:22:52,040 So what we're now doing is instead of-- 393 00:22:52,040 --> 00:22:54,550 the annulus is effectively boosting up 394 00:22:54,550 --> 00:22:56,500 the edges of the pupil, which is why 395 00:22:56,500 --> 00:23:00,880 you get this improved high spatial frequency response. 396 00:23:00,880 --> 00:23:03,040 Here we're doing the opposite of that. 397 00:23:03,040 --> 00:23:07,900 apodization-- if George was here, he would tell us-- 398 00:23:07,900 --> 00:23:11,050 in fact, he already did it one of the previous lectures-- 399 00:23:11,050 --> 00:23:16,780 apodization means to cut off the feet in Greek, I believe. 400 00:23:16,780 --> 00:23:19,690 He'll probably be listening to my lecture and tell me 401 00:23:19,690 --> 00:23:20,950 I've got my Greek all wrong. 402 00:23:20,950 --> 00:23:23,240 But anyway, it's to do-- 403 00:23:23,240 --> 00:23:25,450 it means to cut off the feet. 404 00:23:25,450 --> 00:23:28,120 And what it means, why it's given that name, 405 00:23:28,120 --> 00:23:30,560 is because it cuts off the side lobes. 406 00:23:30,560 --> 00:23:35,350 So you see the side lobes of the Airy disk have disappeared now. 407 00:23:35,350 --> 00:23:37,900 So by making this shaded aperture-- in this case, 408 00:23:37,900 --> 00:23:40,030 is a Gaussian function-- 409 00:23:40,030 --> 00:23:42,250 you make something which is a bit broader. 410 00:23:42,250 --> 00:23:45,340 Actually, in this case, doesn't look very much broader at all. 411 00:23:45,340 --> 00:23:49,060 But the side lobes have gone, and this 412 00:23:49,060 --> 00:23:54,350 is what it's done to the transfer function. 413 00:23:54,350 --> 00:24:00,100 You see now it's slower to fall off here than this one. 414 00:24:00,100 --> 00:24:03,890 But the high spatial frequencies are now reduced relative 415 00:24:03,890 --> 00:24:04,640 to this other one. 416 00:24:10,550 --> 00:24:13,600 So overview on this, then. 417 00:24:13,600 --> 00:24:15,340 This is what we call-- 418 00:24:15,340 --> 00:24:19,220 or what George is calling pupil engineering. 419 00:24:19,220 --> 00:24:25,330 And so if you try to make the main, the central lobe 420 00:24:25,330 --> 00:24:29,620 of the PSF smaller, then you're going to get bigger side lobes. 421 00:24:29,620 --> 00:24:32,830 And that seems to be something that you can't avoid. 422 00:24:32,830 --> 00:24:37,360 Nobody's come up with a way of doing something clever that 423 00:24:37,360 --> 00:24:39,160 avoids that. 424 00:24:39,160 --> 00:24:41,110 And the opposite, of course, if you 425 00:24:41,110 --> 00:24:47,740 make the main side lobe bigger, the side lobes get smaller. 426 00:24:47,740 --> 00:24:52,420 No doubt, that is maybe not so necessarily going to happen. 427 00:24:52,420 --> 00:24:54,010 I suspect that if you really wanted, 428 00:24:54,010 --> 00:24:55,385 you could probably make something 429 00:24:55,385 --> 00:24:58,120 that had a big central lobe and big side lobes. 430 00:24:58,120 --> 00:25:02,860 But nobody would really want that anyway. 431 00:25:02,860 --> 00:25:06,190 And then also this power loss problem, 432 00:25:06,190 --> 00:25:10,120 that there is going to be some worse signal-to-noise sort 433 00:25:10,120 --> 00:25:11,890 of performance. 434 00:25:11,890 --> 00:25:16,800 So annular-type pupils typically narrow the main lobe 435 00:25:16,800 --> 00:25:20,020 at the point spread function at the expense of high side lobes. 436 00:25:20,020 --> 00:25:22,660 Gaussian-type pupil functions typically 437 00:25:22,660 --> 00:25:26,760 suppress the side lobes but broaden the main lobe. 438 00:25:26,760 --> 00:25:30,700 And so, as it says here, different sorts 439 00:25:30,700 --> 00:25:33,580 of filters like this might possibly 440 00:25:33,580 --> 00:25:41,497 be worth considering for certain particular applications. 441 00:25:41,497 --> 00:25:43,080 Yeah, and then he gives another couple 442 00:25:43,080 --> 00:25:47,080 of problems with these things. 443 00:25:47,080 --> 00:25:52,570 Making a filter which is of varying amplitude is not easy. 444 00:25:52,570 --> 00:25:56,040 And I guess nowadays you might do it with some liquid crystal 445 00:25:56,040 --> 00:25:58,220 device, hopefully. 446 00:25:58,220 --> 00:26:00,630 We might be doing that so, won't we, Charlene? 447 00:26:03,810 --> 00:26:07,350 But until recently, this would have 448 00:26:07,350 --> 00:26:12,150 to have been done by some sort of photolithographic process. 449 00:26:12,150 --> 00:26:18,870 And actually getting controlling the absorption of the filter 450 00:26:18,870 --> 00:26:24,940 to accurately fit what you designed is not trivial. 451 00:26:24,940 --> 00:26:28,950 And there's also this energy loss problem, then. 452 00:26:28,950 --> 00:26:34,645 Yeah, nowadays people are very interested in phase filters. 453 00:26:34,645 --> 00:26:36,270 You can do a lot of these clever things 454 00:26:36,270 --> 00:26:43,480 also by using phase masks rather than amplitude masks. 455 00:26:43,480 --> 00:26:49,180 And of course, a phase apodizer is going to be lossless. 456 00:26:49,180 --> 00:26:52,640 So that might make it sound attractive. 457 00:26:52,640 --> 00:26:58,180 And so George says that maybe it is attractive. 458 00:26:58,180 --> 00:27:02,590 Actually, it turns out, I think, it's not quite as attractive 459 00:27:02,590 --> 00:27:05,080 as it might sound because, actually, 460 00:27:05,080 --> 00:27:10,810 very often what happens is if you use an absorbing filter, 461 00:27:10,810 --> 00:27:13,720 the absorption that you can introduce actually 462 00:27:13,720 --> 00:27:17,080 can reduce the size of the side lobes. 463 00:27:17,080 --> 00:27:20,830 So if you do it cleverly, you might 464 00:27:20,830 --> 00:27:26,110 be able to make the side lobes lower and also 465 00:27:26,110 --> 00:27:29,200 still get as much energy into the central lobe with it 466 00:27:29,200 --> 00:27:32,410 with an absorbing mask as with a pure phase mask. 467 00:27:37,020 --> 00:27:40,450 Yeah, and then we get on to all these terminologies 468 00:27:40,450 --> 00:27:44,980 that is obviously annoyed by as much as me. 469 00:27:44,980 --> 00:27:46,360 This is one. 470 00:27:46,360 --> 00:27:49,570 This super cool digital camera has a resolution 471 00:27:49,570 --> 00:27:51,820 of eight megapixels. 472 00:27:51,820 --> 00:27:55,750 And so what's he going to say about this? 473 00:27:55,750 --> 00:27:57,880 Big no again. 474 00:27:57,880 --> 00:28:03,670 So this is using the word wrongly. 475 00:28:03,670 --> 00:28:06,980 This has nothing to do with resolution. 476 00:28:06,980 --> 00:28:10,150 And so what they are actually referring 477 00:28:10,150 --> 00:28:15,400 to is the space bandwidth product of the camera, how 478 00:28:15,400 --> 00:28:17,500 many pixels there are, which is not 479 00:28:17,500 --> 00:28:20,980 really measured as resolution. 480 00:28:20,980 --> 00:28:23,890 The other one, as all the people in my group 481 00:28:23,890 --> 00:28:26,500 would know that I always get very cross about, 482 00:28:26,500 --> 00:28:29,230 is these interferometry people who are always 483 00:28:29,230 --> 00:28:31,570 talking about inerferometers being 484 00:28:31,570 --> 00:28:36,370 able to resolve one Angstrom, which is also wrong 485 00:28:36,370 --> 00:28:40,080 because it's nothing about resolving. 486 00:28:40,080 --> 00:28:44,100 So resolving, resolution, we've said what resolution means. 487 00:28:46,850 --> 00:28:49,790 So this question of the number of pixels, 488 00:28:49,790 --> 00:28:53,330 is there a connection between the two? 489 00:28:53,330 --> 00:28:57,750 Well, I guess the answer is, there is and there isn't. 490 00:28:57,750 --> 00:29:01,010 There is a relationship if you design the optical system 491 00:29:01,010 --> 00:29:02,270 properly. 492 00:29:02,270 --> 00:29:05,810 But what George is pointing out here 493 00:29:05,810 --> 00:29:10,730 is that, actually, this is an example where the pixels are 494 00:29:10,730 --> 00:29:13,040 very much smaller than the point spread 495 00:29:13,040 --> 00:29:16,200 function of the optical system. 496 00:29:16,200 --> 00:29:20,460 So actually, here you're very much over sampling this image. 497 00:29:20,460 --> 00:29:24,290 So all you're measuring-- you're measuring very accurately 498 00:29:24,290 --> 00:29:26,480 a blurred picture. 499 00:29:26,480 --> 00:29:30,440 So that's not really giving you very good resolution 500 00:29:30,440 --> 00:29:32,760 and not really very-- 501 00:29:32,760 --> 00:29:39,297 the number of resolution elements in your image 502 00:29:39,297 --> 00:29:41,630 is actually much smaller then than the number of pixels. 503 00:29:47,110 --> 00:29:49,510 Sometimes they use this word resels, 504 00:29:49,510 --> 00:29:51,070 don't they, resolution elements. 505 00:29:51,070 --> 00:29:52,760 Have you seen that? 506 00:29:52,760 --> 00:29:57,250 They spell it R-E-S-E-L. It's an abbreviation for resolution 507 00:29:57,250 --> 00:30:02,190 elements, so it's the number of resolved spots you've got 508 00:30:02,190 --> 00:30:07,160 in your device rather than the number of pixels. 509 00:30:07,160 --> 00:30:11,090 Some more misstatements-- it is pointless to attempt 510 00:30:11,090 --> 00:30:15,350 to resolve beyond the Rayleigh criterion, however defined. 511 00:30:15,350 --> 00:30:18,440 No, the difficulty increases gradually as 512 00:30:18,440 --> 00:30:22,050 feature sizes shrink, and difficulty is noise dependent. 513 00:30:22,050 --> 00:30:26,390 So it's not a sharp hard and fast line. 514 00:30:26,390 --> 00:30:30,440 But that said, I think you I think it probably 515 00:30:30,440 --> 00:30:34,760 would be pointless to attempt to resolve 516 00:30:34,760 --> 00:30:38,930 10 times the Rayleigh limit with an ordinary optical system 517 00:30:38,930 --> 00:30:42,170 because you know you're not going to be able to do that. 518 00:30:42,170 --> 00:30:47,180 But STED, that we were talking about earlier, does do that. 519 00:30:47,180 --> 00:30:50,420 So this is [INAUDIBLE] well, OK, so maybe it 520 00:30:50,420 --> 00:30:54,260 is worth attempting to do it because people have come up 521 00:30:54,260 --> 00:30:56,600 with ways of doing it now. 522 00:30:56,600 --> 00:31:00,320 Apodization can be used to beat the resolution limit imposed 523 00:31:00,320 --> 00:31:02,760 by the numerical aperture. 524 00:31:02,760 --> 00:31:05,390 And there's a big no there again. 525 00:31:05,390 --> 00:31:11,540 Watch the side lobe growth and poor efficiency loss. 526 00:31:11,540 --> 00:31:16,240 I don't know if you've noticed, but I spoke to one of-- 527 00:31:16,240 --> 00:31:18,830 maybe Charlene-- I can't remember now. 528 00:31:18,830 --> 00:31:25,460 There was a paper in Nature Methods recently by Nikolai 529 00:31:25,460 --> 00:31:28,070 [? Segilev, ?] where he gave an example 530 00:31:28,070 --> 00:31:30,980 of a super-resolving mask, and he gave the pic 531 00:31:30,980 --> 00:31:35,540 this picture of a very narrow point spread function, 532 00:31:35,540 --> 00:31:39,830 improving on the Rayleigh rate resolution limit 533 00:31:39,830 --> 00:31:45,570 by a factor of, I don't know, a few, I think. 534 00:31:45,570 --> 00:31:49,430 But I did some calculations on the design he gave. 535 00:31:49,430 --> 00:31:53,120 It turned out that this little central peak 536 00:31:53,120 --> 00:31:57,020 was surrounded by side lobes that went up 537 00:31:57,020 --> 00:32:00,350 to something like 10 to the power of 40-something 538 00:32:00,350 --> 00:32:03,880 or something really ridiculous. 539 00:32:03,880 --> 00:32:07,340 And the amount of energy that went into this central spot 540 00:32:07,340 --> 00:32:09,500 was, like, nothing. 541 00:32:09,500 --> 00:32:13,340 And what made it even worse was that if you actually 542 00:32:13,340 --> 00:32:17,360 calculated what happened as you went out of focus, 543 00:32:17,360 --> 00:32:21,770 you found that you also got these big walls around it, very 544 00:32:21,770 --> 00:32:23,780 high all around it, actually. 545 00:32:23,780 --> 00:32:29,390 So it was this like this very small bright spot completely 546 00:32:29,390 --> 00:32:32,390 surrounded by something which was, like, 547 00:32:32,390 --> 00:32:34,880 10 to the 40 times higher. 548 00:32:34,880 --> 00:32:37,720 So how you would ever use that in practice, you 549 00:32:37,720 --> 00:32:41,900 can think it really obviously wouldn't 550 00:32:41,900 --> 00:32:43,430 be a very practical device. 551 00:32:47,500 --> 00:32:51,820 Then the number of the rest of the pixels in your camera 552 00:32:51,820 --> 00:32:55,050 is not the resolution. 553 00:32:55,050 --> 00:32:56,730 So what is resolution? 554 00:32:59,240 --> 00:33:05,490 Our ability to resolve to point objects based on the image. 555 00:33:05,490 --> 00:33:09,420 However, this may be difficult to quantify. 556 00:33:09,420 --> 00:33:11,800 Resolution is related to the NA, i.e. 557 00:33:11,800 --> 00:33:14,880 it's proportional to the NA. 558 00:33:14,880 --> 00:33:18,950 The distance is inversely proportional to the NA. 559 00:33:18,950 --> 00:33:20,700 That's another thing that people sometimes 560 00:33:20,700 --> 00:33:24,960 get into knots about of course because resolution 561 00:33:24,960 --> 00:33:28,060 means that the bigger the number, 562 00:33:28,060 --> 00:33:30,180 the more is the resolution. 563 00:33:30,180 --> 00:33:33,870 But actually, the smaller the size, the more the resolution, 564 00:33:33,870 --> 00:33:35,400 so beware. 565 00:33:35,400 --> 00:33:41,520 So sometimes, actually, people use the term "resolving power" 566 00:33:41,520 --> 00:33:45,560 to get around that problem because otherwise-- people 567 00:33:45,560 --> 00:33:49,940 say things which they don't really mean sometimes. 568 00:33:49,940 --> 00:33:54,000 Yeah, so other factors that can affect 569 00:33:54,000 --> 00:33:59,800 resolution-- aberrations, apodization, and noise. 570 00:33:59,800 --> 00:34:03,370 So is there an easy answer? 571 00:34:03,370 --> 00:34:07,410 When in doubt, quote, 0.61 lambda over NA. 572 00:34:07,410 --> 00:34:09,100 And then you get the marks in the exam. 573 00:34:11,659 --> 00:34:12,739 So that's that one done. 574 00:34:16,260 --> 00:34:18,320 So how are we going? 575 00:34:18,320 --> 00:34:21,330 Got a bit more time yet. 576 00:34:21,330 --> 00:34:24,170 That was last week's lecture I just gave then, anyway. 577 00:34:27,570 --> 00:34:31,260 So I think it must be this one. 578 00:34:34,710 --> 00:34:37,690 So I don't know how far we're going to get with this one, 579 00:34:37,690 --> 00:34:39,030 but we can at least start it. 580 00:34:45,469 --> 00:34:48,350 So what we're supposed to be doing today, 581 00:34:48,350 --> 00:34:50,030 more applications of the transfer 582 00:34:50,030 --> 00:34:52,670 function, a bit about depth of focus, 583 00:34:52,670 --> 00:34:54,230 and a bit about deconvolution. 584 00:34:54,230 --> 00:34:59,270 And there's some nice simulations 585 00:34:59,270 --> 00:35:02,250 that George is done to show that. 586 00:35:02,250 --> 00:35:04,940 And then he's got down for Wednesday, which 587 00:35:04,940 --> 00:35:06,230 I'm also going to be giving. 588 00:35:06,230 --> 00:35:07,710 I think George is not back. 589 00:35:07,710 --> 00:35:10,760 Do you know when George comes back? 590 00:35:10,760 --> 00:35:12,587 AUDIENCE: No. 591 00:35:12,587 --> 00:35:13,170 PROFESSOR: No. 592 00:35:13,170 --> 00:35:16,910 Anyway, so I shall be giving the lecture on Wednesday. 593 00:35:16,910 --> 00:35:24,920 And so polarization-- what we don't finish from today 594 00:35:24,920 --> 00:35:27,620 will obviously also be on Wednesday, 595 00:35:27,620 --> 00:35:31,610 and then there's polarization, intensity distribution 596 00:35:31,610 --> 00:35:36,410 near the focus of high-NA imaging systems. 597 00:35:36,410 --> 00:35:39,950 And yeah, I don't know what else we're going to do, actually, 598 00:35:39,950 --> 00:35:42,573 because George keeps changing his mind about what we ought 599 00:35:42,573 --> 00:35:43,740 to put in that last lecture. 600 00:35:47,640 --> 00:35:48,960 So defocus. 601 00:35:48,960 --> 00:35:50,758 So this is an example of defocus. 602 00:35:50,758 --> 00:35:52,050 Anyone know what this movie is? 603 00:35:52,050 --> 00:35:53,670 I don't. 604 00:35:53,670 --> 00:35:57,330 I'm out of touch with cinema. 605 00:35:57,330 --> 00:36:00,460 Recognize who the other stars are? 606 00:36:00,460 --> 00:36:01,750 No? 607 00:36:01,750 --> 00:36:02,510 No. 608 00:36:02,510 --> 00:36:04,010 I'm only asking because I just did-- 609 00:36:04,010 --> 00:36:06,180 AUDIENCE: It's Fight Club. 610 00:36:06,180 --> 00:36:07,360 PROFESSOR: Is that right? 611 00:36:07,360 --> 00:36:08,832 AUDIENCE: Yeah. 612 00:36:08,832 --> 00:36:09,540 PROFESSOR: Right. 613 00:36:09,540 --> 00:36:12,990 Anyway, what you'll notice is this guy here-- 614 00:36:12,990 --> 00:36:13,620 is it a guy? 615 00:36:13,620 --> 00:36:17,040 I can't really recognize what it is, actually. 616 00:36:17,040 --> 00:36:19,320 But it's out of focus. 617 00:36:19,320 --> 00:36:20,910 AUDIENCE: [INAUDIBLE] 618 00:36:20,910 --> 00:36:22,320 PROFESSOR: Yeah, it could be. 619 00:36:22,320 --> 00:36:25,050 OK, so anyway, so that's an example of defocus. 620 00:36:32,270 --> 00:36:35,910 Then this is what happens when you 621 00:36:35,910 --> 00:36:40,100 get a lens in the proximal approximation 622 00:36:40,100 --> 00:36:43,660 and look at the intensity in the focal region. 623 00:36:43,660 --> 00:36:46,670 You get something that looks a bit like that. 624 00:36:46,670 --> 00:36:48,930 It doesn't look quite the same as [INAUDIBLE].. 625 00:36:48,930 --> 00:36:52,120 I don't know how George did this calculation. 626 00:36:52,120 --> 00:36:56,380 But you'll notice that this is the focal plane, 627 00:36:56,380 --> 00:36:58,920 so this is a cross section through the Airy disk. 628 00:36:58,920 --> 00:37:00,465 So you see the central-- 629 00:37:00,465 --> 00:37:03,180 ah, well, I can tell you why it's different, then. 630 00:37:03,180 --> 00:37:06,420 You can see these have got its regularly spaced zeros. 631 00:37:06,420 --> 00:37:10,570 So he's done this through a slit aperture, hasn't he? 632 00:37:10,570 --> 00:37:14,800 And so that might explain why it looks a bit different. 633 00:37:14,800 --> 00:37:18,240 But what you'll see-- 634 00:37:18,240 --> 00:37:22,300 excuse me-- is that as you go out of focus-- 635 00:37:22,300 --> 00:37:23,340 well, note one thing. 636 00:37:23,340 --> 00:37:26,340 It's symmetrical about the focal plan. 637 00:37:26,340 --> 00:37:29,670 And this is-- 638 00:37:29,670 --> 00:37:32,430 I think we probably had that somewhere before, actually, 639 00:37:32,430 --> 00:37:34,620 that it's symmetrical. 640 00:37:34,620 --> 00:37:41,770 If it's calculated using the usual Debye approximation, 641 00:37:41,770 --> 00:37:43,630 you get a result like that. 642 00:37:43,630 --> 00:37:50,110 So if you've got, for example, the aperture stop 643 00:37:50,110 --> 00:37:53,600 is in the front focal plane of the lens 644 00:37:53,600 --> 00:37:57,760 and you look at what happens in the back focal plane, then 645 00:37:57,760 --> 00:37:58,810 that would be the case. 646 00:37:58,810 --> 00:38:02,460 You get this symmetrical behavior like this. 647 00:38:02,460 --> 00:38:08,070 And you can see also these sort of bright bands 648 00:38:08,070 --> 00:38:12,150 that seem to go off at diagonal lines here. 649 00:38:12,150 --> 00:38:15,660 I guess, yeah, so that corresponds roughly 650 00:38:15,660 --> 00:38:17,610 to the edge of the aperture then, of course. 651 00:38:21,840 --> 00:38:26,280 And yeah, so the width of the central spot in this direction, 652 00:38:26,280 --> 00:38:31,260 then, we've said is this 0.61 lambda over NA, 653 00:38:31,260 --> 00:38:35,970 and this distance from the focal plane where 654 00:38:35,970 --> 00:38:38,310 you get to this first zero is given 655 00:38:38,310 --> 00:38:40,860 by some expression like this. 656 00:38:40,860 --> 00:38:43,380 It goes as if-- this is really only 657 00:38:43,380 --> 00:38:45,150 true for the paraxial case. 658 00:38:45,150 --> 00:38:49,200 But for the paraxial case, this NA 659 00:38:49,200 --> 00:38:53,940 here, NA squared here, if it's the high aperture case, which 660 00:38:53,940 --> 00:38:56,730 we might consider on Wednesday, you'll 661 00:38:56,730 --> 00:38:59,955 find that this is not quite right anymore. 662 00:39:05,790 --> 00:39:08,080 Yeah, so delta x is the radial resolution. 663 00:39:08,080 --> 00:39:13,060 Delta z is the depth of focus, today's topics, depth of focus 664 00:39:13,060 --> 00:39:14,500 or depth of field. 665 00:39:14,500 --> 00:39:17,650 And both of those he calls DoF. 666 00:39:17,650 --> 00:39:19,540 Note the very high numerical apertures. 667 00:39:19,540 --> 00:39:22,150 The scalar approximation is no longer good. 668 00:39:22,150 --> 00:39:24,880 The vectorial nature of the electromagnetic field 669 00:39:24,880 --> 00:39:26,650 becomes important. 670 00:39:26,650 --> 00:39:27,520 That is true. 671 00:39:27,520 --> 00:39:29,920 There's other things as well, though. 672 00:39:29,920 --> 00:39:33,520 When you do all this paraxial stuff, if you remember, 673 00:39:33,520 --> 00:39:36,760 all the time you're replacing sine theta by theta and things 674 00:39:36,760 --> 00:39:39,680 like that, making approximations like that. 675 00:39:39,680 --> 00:39:44,590 So there are actually lots of approximations you make. 676 00:39:44,590 --> 00:39:48,610 One is the fact that the angles are small. 677 00:39:48,610 --> 00:39:51,970 The other is that you neglect the polarization effects. 678 00:39:51,970 --> 00:39:56,260 When the apertures become big, then the polarization effects 679 00:39:56,260 --> 00:39:57,760 become very important. 680 00:40:02,670 --> 00:40:05,880 So we're going to look at this sort of system. 681 00:40:05,880 --> 00:40:12,710 And so this is a 4F system with magnification, 682 00:40:12,710 --> 00:40:14,940 so the two the focal length of the two lenses 683 00:40:14,940 --> 00:40:17,040 are different in this case. 684 00:40:17,040 --> 00:40:20,790 And so this shows what happens if you're 685 00:40:20,790 --> 00:40:26,790 looking at an object which is placed a distance F in front 686 00:40:26,790 --> 00:40:31,350 of this lens and your screen is placed the difference F2 687 00:40:31,350 --> 00:40:34,170 behind this lens, and your aperture 688 00:40:34,170 --> 00:40:36,880 is F1 from this one and F2 from this one. 689 00:40:36,880 --> 00:40:41,760 So everything there is set up properly 690 00:40:41,760 --> 00:40:44,070 as defined for a 4F system. 691 00:40:44,070 --> 00:40:46,880 And this is the case where you'd expect, 692 00:40:46,880 --> 00:40:51,540 as I was just describing, the defocus point spread function 693 00:40:51,540 --> 00:40:54,660 is going to be symmetric. 694 00:40:54,660 --> 00:41:02,580 So we place our mask there, and we've got our pupil. 695 00:41:02,580 --> 00:41:05,410 So this is our object function. 696 00:41:05,410 --> 00:41:11,330 This is the angular spectrum, the spectrum 697 00:41:11,330 --> 00:41:14,660 of the object in this plane here, 698 00:41:14,660 --> 00:41:18,980 which is then filtered by the aperture stop. 699 00:41:18,980 --> 00:41:26,120 And then that gives some sort of amplitude in this plane 700 00:41:26,120 --> 00:41:28,460 here, which is going to be inverted 701 00:41:28,460 --> 00:41:32,660 and also magnified or otherwise according 702 00:41:32,660 --> 00:41:36,810 to the ratio of F1 over F2. 703 00:41:36,810 --> 00:41:38,440 And there's the rays going through it. 704 00:41:41,120 --> 00:41:43,220 And this is showing how-- 705 00:41:43,220 --> 00:41:46,040 yeah, this plane, this is his terminology. 706 00:41:46,040 --> 00:41:47,510 This is x. 707 00:41:47,510 --> 00:41:48,980 This is x prime. 708 00:41:48,980 --> 00:41:51,590 This is x double prime. 709 00:41:51,590 --> 00:41:55,370 And you can see here x double prime maximum is 710 00:41:55,370 --> 00:42:00,440 equal to the and the radius of the aperture. 711 00:42:00,440 --> 00:42:01,700 So this is his terminology. 712 00:42:01,700 --> 00:42:05,420 We are going to get that in a minute coming [INAUDIBLE].. 713 00:42:05,420 --> 00:42:09,860 This is a function of x This is a function of x double prime. 714 00:42:09,860 --> 00:42:11,390 This is a function of x prime. 715 00:42:14,900 --> 00:42:17,180 This is the numerical aperture of the lens 716 00:42:17,180 --> 00:42:19,530 that's looking at it. 717 00:42:19,530 --> 00:42:24,540 And yeah, so the numerical aperture 718 00:42:24,540 --> 00:42:29,160 is equal to x double prime maximum, which 719 00:42:29,160 --> 00:42:38,740 is equal to A over F1, again assuming that the sine theta 720 00:42:38,740 --> 00:42:41,260 or tan theta are both theta. 721 00:42:41,260 --> 00:42:42,760 Now, what we're going to look at now 722 00:42:42,760 --> 00:42:45,470 is what happens if we defocus this system. 723 00:42:45,470 --> 00:42:49,090 So now our object has been displaced the distance delta 724 00:42:49,090 --> 00:42:52,870 away, and we want to calculate what the image of that 725 00:42:52,870 --> 00:42:55,370 looks like now. 726 00:42:55,370 --> 00:42:59,060 And so what we can say is that after we've 727 00:42:59,060 --> 00:43:01,370 eliminated this object, we've got 728 00:43:01,370 --> 00:43:04,400 this GT of x, and then that light is going 729 00:43:04,400 --> 00:43:06,750 to propagate from here to here. 730 00:43:06,750 --> 00:43:09,290 And in order to calculate that, we 731 00:43:09,290 --> 00:43:14,750 have to convolve it with the propagation kernel. 732 00:43:14,750 --> 00:43:16,430 So you convolve it with this thing. 733 00:43:20,800 --> 00:43:24,670 For some reason, when George converted this from his 734 00:43:24,670 --> 00:43:28,150 whatever his other thing's called, keynotes-- 735 00:43:28,150 --> 00:43:29,140 it's a PowerPoint. 736 00:43:29,140 --> 00:43:32,560 All the equations are like as though someone's 737 00:43:32,560 --> 00:43:34,390 had their martinis before the lecture. 738 00:43:38,000 --> 00:43:40,940 So we then look at the Fourier transform of that in this plane 739 00:43:40,940 --> 00:43:41,970 here. 740 00:43:41,970 --> 00:43:43,890 So this thing convolves-- 741 00:43:43,890 --> 00:43:46,080 Fourier transforms to this. 742 00:43:46,080 --> 00:43:50,240 The convolution transforms to a product, 743 00:43:50,240 --> 00:43:54,110 and this Gaussian transforms to another Gaussian. 744 00:43:54,110 --> 00:43:56,360 And notice that, as usual, of course, 745 00:43:56,360 --> 00:43:58,730 you remember that the scaling thing for Fourier 746 00:43:58,730 --> 00:44:01,460 transforms, the bigger a function is, 747 00:44:01,460 --> 00:44:04,380 the smaller it's Fourier transform and vice versa. 748 00:44:04,380 --> 00:44:10,010 So this lambda delta at the bottom appears now at the top. 749 00:44:10,010 --> 00:44:13,760 So the bigger delta is, the bigger 750 00:44:13,760 --> 00:44:16,940 this thing's going to be here. 751 00:44:16,940 --> 00:44:19,218 Yeah? 752 00:44:19,218 --> 00:44:22,200 AUDIENCE: [INAUDIBLE] 753 00:44:24,973 --> 00:44:25,640 PROFESSOR: Yeah. 754 00:44:25,640 --> 00:44:28,620 AUDIENCE: Is E to the imaginary x squared a Gaussian? 755 00:44:28,620 --> 00:44:31,267 Or only the E to the real x squared a Gaussian? 756 00:44:31,267 --> 00:44:31,850 PROFESSOR: OK. 757 00:44:31,850 --> 00:44:37,300 Yeah, the real Gaussian is E to the minus x squared, isn't it? 758 00:44:37,300 --> 00:44:39,910 E to the minus x squared does that. 759 00:44:39,910 --> 00:44:45,650 This is what you might call an imaginary Gaussian. 760 00:44:45,650 --> 00:44:49,280 But it all comes out in the wash, actually. 761 00:44:49,280 --> 00:44:53,000 So just the same as the Fourier transform 762 00:44:53,000 --> 00:44:56,100 of the normal Gaussian is another Gaussian, 763 00:44:56,100 --> 00:45:00,020 so the Fourier transform of this imaginary Gaussian 764 00:45:00,020 --> 00:45:02,750 is another imaginary Gaussian. 765 00:45:02,750 --> 00:45:06,830 So what's really converting is a parabolic phase front 766 00:45:06,830 --> 00:45:11,340 here to a parabolic phase front here. 767 00:45:11,340 --> 00:45:16,040 So you can get it just by getting the normal expression. 768 00:45:16,040 --> 00:45:20,810 I mean, it sometimes amazes me having how much liberties 769 00:45:20,810 --> 00:45:22,130 you can take with these things. 770 00:45:22,130 --> 00:45:25,370 But you just take the expressions for the Gaussian, 771 00:45:25,370 --> 00:45:27,290 and you put in this complex number, 772 00:45:27,290 --> 00:45:28,590 and it gives the right answer. 773 00:45:28,590 --> 00:45:31,800 So it's very nice. 774 00:45:31,800 --> 00:45:33,330 I guess occasionally there might be 775 00:45:33,330 --> 00:45:34,650 occasions when it doesn't work. 776 00:45:37,720 --> 00:45:45,070 Yeah, so what he's then saying is that you can think of this-- 777 00:45:45,070 --> 00:45:47,290 this is the object for our-- 778 00:45:47,290 --> 00:45:50,500 this is the spectrum for our defocused objects, 779 00:45:50,500 --> 00:45:59,290 which is multiplied by the pupil function of the system. 780 00:45:59,290 --> 00:46:02,200 But that is exactly equivalent of course 781 00:46:02,200 --> 00:46:05,560 to thinking of it as the ordinary object 782 00:46:05,560 --> 00:46:13,690 spectrum multiplied by a modified pupil function. 783 00:46:13,690 --> 00:46:18,040 So it doesn't matter whether you associate this with this 784 00:46:18,040 --> 00:46:21,020 or with the pupil function. 785 00:46:21,020 --> 00:46:22,330 So that's what we can think of. 786 00:46:22,330 --> 00:46:26,770 We can think of the objects as being the same as where 787 00:46:26,770 --> 00:46:29,260 it is if it wasn't defocused, and we 788 00:46:29,260 --> 00:46:33,310 can think of the pupil function of the thing being 789 00:46:33,310 --> 00:46:37,150 modified to take into account of defocus. 790 00:46:37,150 --> 00:46:41,110 We sometimes called that the defocus transfer function. 791 00:46:41,110 --> 00:46:44,720 And so he's now going to calculate that. 792 00:46:44,720 --> 00:46:46,660 Yeah, so the defocus-- 793 00:46:46,660 --> 00:46:51,510 he calls it here the defocus amplitude transfer function. 794 00:46:51,510 --> 00:46:53,440 So remember, here we're dealing-- 795 00:46:53,440 --> 00:46:58,140 this is dealing now with coherent system. 796 00:46:58,140 --> 00:47:00,970 Before, when we were doing the resolution, 797 00:47:00,970 --> 00:47:03,340 we were looking at incoherent systems. 798 00:47:03,340 --> 00:47:06,870 So now we've gone back to coherent systems. 799 00:47:06,870 --> 00:47:11,020 So you have to be wary of that because sometimes there's 800 00:47:11,020 --> 00:47:13,670 some differences. 801 00:47:13,670 --> 00:47:15,680 So all this is saying then-- 802 00:47:15,680 --> 00:47:20,710 yeah, you remember, to go from the front 803 00:47:20,710 --> 00:47:24,500 the pupil to the transfer function, all you have to do 804 00:47:24,500 --> 00:47:27,890 is a coordinate scaling. 805 00:47:27,890 --> 00:47:31,570 So the amplitude transfer function 806 00:47:31,570 --> 00:47:36,930 is just the scaled version of this pupil function. 807 00:47:36,930 --> 00:47:39,960 And again, it comes out to be now, of course-- 808 00:47:39,960 --> 00:47:41,280 it's going to be-- 809 00:47:41,280 --> 00:47:45,030 it's multiplied, of course, still by the circular aperture. 810 00:47:45,030 --> 00:47:47,450 But inside that circular aperture 811 00:47:47,450 --> 00:47:49,880 where it's got some value, its value 812 00:47:49,880 --> 00:47:55,040 is given by this complex exponential-- 813 00:47:55,040 --> 00:48:00,920 a complex Gaussian, like we were just saying before. 814 00:48:00,920 --> 00:48:03,550 And that's what this looks like. 815 00:48:03,550 --> 00:48:05,490 Now, notice, so this is cosine. 816 00:48:05,490 --> 00:48:07,630 This is looking at just the real part. 817 00:48:07,630 --> 00:48:09,480 So actually, it was a complex-- 818 00:48:09,480 --> 00:48:12,630 it was an E to the minus I something. 819 00:48:12,630 --> 00:48:16,470 So you can break that up into cos plus I sine. 820 00:48:16,470 --> 00:48:19,960 And the cos part is the real part, 821 00:48:19,960 --> 00:48:23,370 and the sine part is the imaginary part. 822 00:48:23,370 --> 00:48:27,120 And as when we looked at imaging earlier, 823 00:48:27,120 --> 00:48:30,150 the real part images real objects. 824 00:48:30,150 --> 00:48:34,010 So an amplitude object will be imaged by the real part. 825 00:48:34,010 --> 00:48:38,442 A phase object would be imaged by the imaginary part. 826 00:48:38,442 --> 00:48:40,400 But we're not going to say anything about that. 827 00:48:40,400 --> 00:48:42,370 We'll just keep with the real part. 828 00:48:42,370 --> 00:48:45,120 So this is cos, but it's not a normal cos. 829 00:48:45,120 --> 00:48:46,990 It's cos of an x squared. 830 00:48:46,990 --> 00:48:50,640 So that's why it looks a rather strange shape compared 831 00:48:50,640 --> 00:48:51,870 with the normal cosine. 832 00:48:51,870 --> 00:48:55,410 You can see it's become very flat there, 833 00:48:55,410 --> 00:48:58,920 and then it becomes more oscillating later on. 834 00:48:58,920 --> 00:49:01,590 And so this is what it looks like. 835 00:49:01,590 --> 00:49:04,650 Of course, the actual scaling of this 836 00:49:04,650 --> 00:49:12,600 is going to depend on the value of this delta. 837 00:49:12,600 --> 00:49:16,005 So here he looks at where this crosses the axis. 838 00:49:18,570 --> 00:49:22,600 So this parameter, x double prime over lambda F1, 839 00:49:22,600 --> 00:49:26,450 is equal to u, the spatial frequency. 840 00:49:26,450 --> 00:49:29,930 So this is just u squared here. 841 00:49:29,930 --> 00:49:33,800 And he says that this crosses the axis where this argument is 842 00:49:33,800 --> 00:49:36,200 equal to pi over 2. 843 00:49:36,200 --> 00:49:41,960 And if you solve that, you'll get this. 844 00:49:41,960 --> 00:49:46,880 And so that's where it crosses the axis. 845 00:49:46,880 --> 00:49:50,750 And then you have to multiply this, of course, 846 00:49:50,750 --> 00:49:54,290 by the circular aperture. 847 00:49:54,290 --> 00:49:56,900 So this is the circular aperture. 848 00:49:56,900 --> 00:50:01,760 This is the radium where the object spectrum is non-zero. 849 00:50:01,760 --> 00:50:08,310 And so you can see that only the parts 850 00:50:08,310 --> 00:50:11,970 of this blue curve up to here are going to have any meaning. 851 00:50:11,970 --> 00:50:14,380 The bits out here are not going to do anything. 852 00:50:14,380 --> 00:50:16,380 So this is obviously looking at a case 853 00:50:16,380 --> 00:50:18,810 where this is almost flat. 854 00:50:18,810 --> 00:50:20,640 If you made it a bit smaller, even, it 855 00:50:20,640 --> 00:50:22,740 would look even flatter. 856 00:50:22,740 --> 00:50:27,800 So this is a not a lot of defocus. 857 00:50:27,800 --> 00:50:31,270 And so here he's got, then-- yeah, 858 00:50:31,270 --> 00:50:35,110 so here now is labeling where that cutoff is. 859 00:50:35,110 --> 00:50:38,140 So this is equal to NA over lambda, as we know. 860 00:50:38,140 --> 00:50:40,990 And in terms of the x prime coordinate, 861 00:50:40,990 --> 00:50:42,610 it's given right by this thing here. 862 00:50:46,290 --> 00:50:50,330 So this is talking about what he calls mild defocus, 863 00:50:50,330 --> 00:50:52,910 and the condition for that to occur 864 00:50:52,910 --> 00:50:55,790 is that this has got to be much smaller than this. 865 00:51:00,030 --> 00:51:06,480 And then-- yeah, and that corresponds, then, 866 00:51:06,480 --> 00:51:08,640 in terms of delta. 867 00:51:08,640 --> 00:51:11,500 So we've now got a condition for this to happen. 868 00:51:11,500 --> 00:51:14,100 The value of delta has to be small compared 869 00:51:14,100 --> 00:51:17,040 with lambda over 2 NA squared. 870 00:51:17,040 --> 00:51:18,810 So you remember, this is what we said 871 00:51:18,810 --> 00:51:23,450 was the depth of field or depth of focus of the system. 872 00:51:26,830 --> 00:51:31,870 Depth of field is measured in the object space. 873 00:51:31,870 --> 00:51:34,120 It doesn't say anything about the defocus on that one. 874 00:51:34,120 --> 00:51:35,880 I thought I was going to [INAUDIBLE].. 875 00:51:35,880 --> 00:51:37,410 He says a bit about that later. 876 00:51:37,410 --> 00:51:39,210 This is showing another case, then. 877 00:51:39,210 --> 00:51:42,030 This is with much more defocus now. 878 00:51:42,030 --> 00:51:45,210 So now it's oscillating a lot more wildly, 879 00:51:45,210 --> 00:51:48,900 so now this cutoff is much smaller than this. 880 00:51:48,900 --> 00:51:52,550 And so this is where you've got a lot of defocus. 881 00:51:52,550 --> 00:51:55,710 And the fact that this goes negative 882 00:51:55,710 --> 00:51:57,540 is probably the worst thing. 883 00:51:57,540 --> 00:51:59,910 It means that some spatial frequencies 884 00:51:59,910 --> 00:52:02,820 are going to be imaged with the wrong contrast. 885 00:52:02,820 --> 00:52:06,060 So when you try doing your Fourier synthesis 886 00:52:06,060 --> 00:52:10,980 to add up to make the image, it's not going to work right. 887 00:52:10,980 --> 00:52:14,070 And the other point he makes is that the regions around here, 888 00:52:14,070 --> 00:52:17,850 of course, these spatial frequencies around these zeros 889 00:52:17,850 --> 00:52:19,440 are not imaged at all. 890 00:52:19,440 --> 00:52:24,630 So that's another reason why it doesn't give a very good image. 891 00:52:24,630 --> 00:52:27,540 So this is the case of strongly focus where delta 892 00:52:27,540 --> 00:52:31,350 is much bigger than this thing, not necessarily much bigger, 893 00:52:31,350 --> 00:52:32,134 I suppose. 894 00:52:34,878 --> 00:52:36,170 I think we ought to stop there. 895 00:52:36,170 --> 00:52:38,220 We've run over time anyway. 896 00:52:38,220 --> 00:52:43,020 So let's stop at this point and we'll pick that up next time.