1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,660 To make a donation or to view additional materials 6 00:00:12,660 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,640 at ocw.mit.edu. 8 00:00:21,415 --> 00:00:22,290 PROFESSOR: Right, OK. 9 00:00:22,290 --> 00:00:24,690 So I guess we ought to start. 10 00:00:24,690 --> 00:00:26,625 We seem to be a bit depleted. 11 00:00:30,630 --> 00:00:39,210 Anyway, so today last time we spoke about 4f imaging systems 12 00:00:39,210 --> 00:00:41,040 and a bit about spatial filtering. 13 00:00:41,040 --> 00:00:43,890 And I guess we're going to do quite a lot more on that today, 14 00:00:43,890 --> 00:00:45,150 really. 15 00:00:45,150 --> 00:00:51,780 So I mean this is really the most basic part of Fourier 16 00:00:51,780 --> 00:00:57,480 optics, really, I think so you need to get lots of practice 17 00:00:57,480 --> 00:01:06,240 seeing how it all works until it comes second nature really. 18 00:01:06,240 --> 00:01:09,950 So this is what we said last time. 19 00:01:09,950 --> 00:01:13,760 We went through this slide last time 20 00:01:13,760 --> 00:01:18,570 showing the significance of the point spread 21 00:01:18,570 --> 00:01:21,210 function of a low pass filter. 22 00:01:21,210 --> 00:01:27,310 The idea is that if you put in a point object-- 23 00:01:27,310 --> 00:01:27,810 i.e. 24 00:01:27,810 --> 00:01:30,830 You put in an opaque screen with a small hole 25 00:01:30,830 --> 00:01:33,750 in, illuminate that with light, this is going 26 00:01:33,750 --> 00:01:38,460 to then give a spherical wave. 27 00:01:38,460 --> 00:01:42,060 And then this lens is going to collimate that spherical wave 28 00:01:42,060 --> 00:01:44,910 to make a plane wave. 29 00:01:44,910 --> 00:01:48,680 And in this back focal plane of that lens, 30 00:01:48,680 --> 00:01:53,180 you will get a nice smooth variation here-- 31 00:01:53,180 --> 00:01:58,530 no variation, uniform intensity, uniform amplitude, 32 00:01:58,530 --> 00:02:01,130 which is basically the Fourier transform 33 00:02:01,130 --> 00:02:02,630 of this delta function. 34 00:02:02,630 --> 00:02:06,050 Fourier transform of a delta function is a constant, 35 00:02:06,050 --> 00:02:07,680 isn't it? 36 00:02:07,680 --> 00:02:13,010 And then this light then goes through a pupil mask. 37 00:02:13,010 --> 00:02:20,480 The plane wave after the mask then gets to this second lens. 38 00:02:20,480 --> 00:02:23,390 And the plane wave is then focused down 39 00:02:23,390 --> 00:02:27,830 and produces a focused spot at some image plane, right? 40 00:02:27,830 --> 00:02:30,050 So that's how it all works. 41 00:02:30,050 --> 00:02:33,800 And so now what do you actually see here? 42 00:02:33,800 --> 00:02:36,050 What's the image of a single point object? 43 00:02:36,050 --> 00:02:41,930 What you see is the Fourier transform of this pupil mask. 44 00:02:41,930 --> 00:02:45,600 Because again, this is also producing a Fourier transform. 45 00:02:45,600 --> 00:02:48,140 Right, so the point spread function-- 46 00:02:48,140 --> 00:02:51,000 what's called the amplitude point spread function 47 00:02:51,000 --> 00:02:56,000 is just the Fourier transform of this pupil mask. 48 00:02:56,000 --> 00:02:59,390 And we sometimes talk about an intensity point spread 49 00:02:59,390 --> 00:03:02,300 function, which is actually the model of the square of that, 50 00:03:02,300 --> 00:03:06,380 the intensity in that image of a point object. 51 00:03:06,380 --> 00:03:10,010 OK, so and this is reminding you where 52 00:03:10,010 --> 00:03:14,030 you might have come across this sort of concept before. 53 00:03:14,030 --> 00:03:18,080 This is like-- we spoke in the last lecture about 54 00:03:18,080 --> 00:03:20,270 if in electronics, you might think 55 00:03:20,270 --> 00:03:24,350 of an electronic system like an amplifier or something 56 00:03:24,350 --> 00:03:26,090 as a sort of black box. 57 00:03:26,090 --> 00:03:29,810 You put in a signal, and you get out a signal. 58 00:03:29,810 --> 00:03:34,730 And you can characterize that black box 59 00:03:34,730 --> 00:03:37,850 by what's called the impulse response, which 60 00:03:37,850 --> 00:03:43,280 is the response that it would give to a sharp spike going in. 61 00:03:43,280 --> 00:03:46,580 All right, so ideally you'd expect-- 62 00:03:46,580 --> 00:03:49,880 you'd want a sharp spike to come out as well. 63 00:03:49,880 --> 00:03:51,500 But in practice, you don't get that. 64 00:03:51,500 --> 00:03:55,820 You get some sort of broader function 65 00:03:55,820 --> 00:04:00,920 because of the imperfections of the transfer system. 66 00:04:00,920 --> 00:04:04,250 OK, so this is our point spread function. 67 00:04:04,250 --> 00:04:09,080 And it's denoted-- little h of x dash, y dash. 68 00:04:09,080 --> 00:04:11,390 This the x dash y dash plane. 69 00:04:11,390 --> 00:04:16,459 And I guess this terminology little h 70 00:04:16,459 --> 00:04:18,440 seems to be pretty universal. 71 00:04:18,440 --> 00:04:21,620 Goodman, I think, uses that in other places. 72 00:04:21,620 --> 00:04:27,920 All right, so this is just now describing again 73 00:04:27,920 --> 00:04:31,460 how you how you calculate this 4 pi-- 74 00:04:31,460 --> 00:04:34,190 this 4f system. 75 00:04:34,190 --> 00:04:40,310 We notice everywhere that the input transparency 76 00:04:40,310 --> 00:04:43,220 of a delta function gives one here. 77 00:04:43,220 --> 00:04:45,890 You multiply by the pupil mask, and then you 78 00:04:45,890 --> 00:04:50,520 do another Fourier transform to find the image. 79 00:04:50,520 --> 00:04:53,610 All right, so that's-- 80 00:04:53,610 --> 00:04:57,080 I ought to say something bit more about the scaling of this. 81 00:05:07,650 --> 00:05:10,080 Yeah, so you've got of course, this. 82 00:05:10,080 --> 00:05:14,798 The output point spread function is a function of x dashed. 83 00:05:14,798 --> 00:05:17,340 But, of course, when you do a Fourier transform of something, 84 00:05:17,340 --> 00:05:22,770 it's going to be a function of frequency, not of distance, 85 00:05:22,770 --> 00:05:23,760 isn't it? 86 00:05:23,760 --> 00:05:28,890 So really this h here, h of x dashed, 87 00:05:28,890 --> 00:05:32,400 we get by doing the Fourier transform of this, 88 00:05:32,400 --> 00:05:35,910 and then transforming the coordinates-- scaling 89 00:05:35,910 --> 00:05:41,050 the coordinates in order to get it into the coordinates x dash. 90 00:05:41,050 --> 00:05:43,450 OK, so and then last time, we also-- 91 00:05:43,450 --> 00:05:45,640 I'll go through this very quickly, what 92 00:05:45,640 --> 00:05:52,570 we said last time, if you've got a low pass filter in 1D, 93 00:05:52,570 --> 00:06:02,870 like a slip pupil, then the pupil mask is a rect function. 94 00:06:02,870 --> 00:06:07,420 And so if we're trying to calculate the image of a point 95 00:06:07,420 --> 00:06:11,890 object again, the field when it gets to this pupil plane 96 00:06:11,890 --> 00:06:13,460 is just a constant. 97 00:06:13,460 --> 00:06:16,660 And so after it's gone through the aperture, 98 00:06:16,660 --> 00:06:21,390 it will have this sort of amplitude variation. 99 00:06:21,390 --> 00:06:26,260 And then so this is what this is showing. 100 00:06:26,260 --> 00:06:28,870 On the positive side is what's on the negative side, which 101 00:06:28,870 --> 00:06:33,780 is just 1 times this, which is this. 102 00:06:33,780 --> 00:06:38,230 And then if you then Fourier transform that, 103 00:06:38,230 --> 00:06:41,470 you find that the point spread function 104 00:06:41,470 --> 00:06:43,812 is going to be the Fourier transform of this, 105 00:06:43,812 --> 00:06:44,770 which is just the sync. 106 00:06:49,660 --> 00:06:53,540 And then the point spread function 107 00:06:53,540 --> 00:06:57,920 is the Fourier transform of this. 108 00:06:57,920 --> 00:07:01,013 And then this is doing the scaling of the coordinates 109 00:07:01,013 --> 00:07:02,680 and getting it into the right hand side. 110 00:07:02,680 --> 00:07:04,055 I'm not going to go through that. 111 00:07:04,055 --> 00:07:10,300 We went through it last time, but hopefully we're going to-- 112 00:07:10,300 --> 00:07:11,220 yeah, here we are. 113 00:07:11,220 --> 00:07:12,580 And there's the sync. 114 00:07:12,580 --> 00:07:14,680 Right, so this is that the amplitude point 115 00:07:14,680 --> 00:07:18,010 spread function or just normally called point spread function. 116 00:07:18,010 --> 00:07:21,670 Note that it goes negative. 117 00:07:21,670 --> 00:07:23,060 This is 0 here. 118 00:07:23,060 --> 00:07:25,180 So note that it goes negative, right. 119 00:07:25,180 --> 00:07:28,260 So you've got to convolve that with your objects 120 00:07:28,260 --> 00:07:32,140 to find the image of some arbitrary objects, which we'll 121 00:07:32,140 --> 00:07:34,750 carry on to talk about in a minute. 122 00:07:34,750 --> 00:07:39,580 This was just another example of a phase filter 123 00:07:39,580 --> 00:07:41,260 and the Fourier transform of that. 124 00:07:41,260 --> 00:07:43,635 But I don't think there's point going through that again. 125 00:07:43,635 --> 00:07:47,470 We'll just carry on now where we've got up too, 126 00:07:47,470 --> 00:07:51,520 which was to do with shift invariance of the 4f system. 127 00:07:51,520 --> 00:07:52,570 Now what we find-- 128 00:07:52,570 --> 00:07:56,170 what we've looked at is if we've got a point object that we get 129 00:07:56,170 --> 00:07:58,840 from this hole in the screen, we get 130 00:07:58,840 --> 00:08:02,080 an image which is this amplitude point spread function. 131 00:08:02,080 --> 00:08:06,360 The amplitude here would be this Fourier transform 132 00:08:06,360 --> 00:08:09,470 of this pupil mask. 133 00:08:09,470 --> 00:08:13,370 Then, yeah I'd just mention, there is this constant. 134 00:08:13,370 --> 00:08:15,810 This is the point spread function. 135 00:08:15,810 --> 00:08:21,080 It's the Fourier transform of this pupil mass. 136 00:08:21,080 --> 00:08:23,840 But there's this constant out at the front. 137 00:08:23,840 --> 00:08:27,370 And this constant out at the front is written like this. 138 00:08:27,370 --> 00:08:31,940 And you can see that it's made up of two parts. 139 00:08:31,940 --> 00:08:33,890 First of all, this phase part-- 140 00:08:33,890 --> 00:08:37,700 this just comes about because of the distance 141 00:08:37,700 --> 00:08:40,429 the wave is traveled through the optical system. 142 00:08:40,429 --> 00:08:43,789 Right, so if it starts off with zero phase here, 143 00:08:43,789 --> 00:08:47,660 it's going to come up with some particular finite phase 144 00:08:47,660 --> 00:08:50,940 when it travels all this distance through the system. 145 00:08:50,940 --> 00:08:55,070 And then the other bet is this F2 146 00:08:55,070 --> 00:08:59,420 over F1, which is a scaling factor, 147 00:08:59,420 --> 00:09:02,030 the ratio of the focal lengths. 148 00:09:02,030 --> 00:09:08,365 And basically you need that to satisfy conservation of energy. 149 00:09:08,365 --> 00:09:14,030 All right, so you can imagine that as the point spread 150 00:09:14,030 --> 00:09:19,100 function gets broader, then it has to also get lower 151 00:09:19,100 --> 00:09:22,880 because it has to contain the right amount of energy 152 00:09:22,880 --> 00:09:25,620 equal to the energy going into the system. 153 00:09:25,620 --> 00:09:28,670 So when you work out the intensity 154 00:09:28,670 --> 00:09:30,410 of this image of a point, this would 155 00:09:30,410 --> 00:09:34,880 be F2 over F1 squared, which is the ratio of the heights-- 156 00:09:34,880 --> 00:09:39,200 the maximum intensities of the center 157 00:09:39,200 --> 00:09:41,420 of that image of a point. 158 00:09:41,420 --> 00:09:43,820 Right, now so what we'll be carrying on to look at now 159 00:09:43,820 --> 00:09:49,040 is what happens if we displace our point object. 160 00:09:49,040 --> 00:09:52,850 We move this delta function sideways. 161 00:09:52,850 --> 00:09:54,600 And then let's look, first of all, 162 00:09:54,600 --> 00:09:59,150 at just the physical situation of what's happening. 163 00:09:59,150 --> 00:10:01,700 This produces a spherical wave. 164 00:10:01,700 --> 00:10:04,400 And then when it goes through the lens, 165 00:10:04,400 --> 00:10:07,030 you can see that it's going to produce again-- it's 166 00:10:07,030 --> 00:10:09,770 going to be collimated into a plane wave. 167 00:10:09,770 --> 00:10:16,040 But now this plane wave is not actually moving horizontally 168 00:10:16,040 --> 00:10:18,200 but is moving at an angle. 169 00:10:18,200 --> 00:10:21,920 And this comes about because when you Fourier transform 170 00:10:21,920 --> 00:10:28,990 this, you will get some exponential phase factor, which 171 00:10:28,990 --> 00:10:33,400 actually represents a wave which is moving at an angle. 172 00:10:33,400 --> 00:10:37,060 And then when this inclined plane wave 173 00:10:37,060 --> 00:10:38,920 goes through the pupil mask, of course, 174 00:10:38,920 --> 00:10:40,870 its angle doesn't change anymore. 175 00:10:40,870 --> 00:10:44,270 So it's still moving at the same angle here. 176 00:10:44,270 --> 00:10:48,640 And then again, it's collimated and produces an image. 177 00:10:48,640 --> 00:10:52,900 And the image-- what we find-- 178 00:10:52,900 --> 00:10:55,990 is the same as it was before. 179 00:10:55,990 --> 00:10:59,540 The amplitude is exactly the same as it was before. 180 00:10:59,540 --> 00:11:01,350 But it's shifted. 181 00:11:01,350 --> 00:11:05,800 So this is what we found before when we looked at the image. 182 00:11:05,800 --> 00:11:11,200 You got this-- you remember-- this inverted magnified image. 183 00:11:11,200 --> 00:11:16,470 The inversion means that it's the opposite side of the axis. 184 00:11:16,470 --> 00:11:20,200 The scaling is this F2 over F1, which comes about from the fact 185 00:11:20,200 --> 00:11:24,130 that the focal lengths are not equal. 186 00:11:24,130 --> 00:11:26,500 But the important thing to notice 187 00:11:26,500 --> 00:11:30,790 is that apart from this shift, this function is the same 188 00:11:30,790 --> 00:11:31,950 as the function was before. 189 00:11:35,880 --> 00:11:40,140 So what that means is that this is 190 00:11:40,140 --> 00:11:42,450 what's called shift invariance. 191 00:11:42,450 --> 00:11:44,790 It means that the image of a point 192 00:11:44,790 --> 00:11:47,430 is independent of where the point is. 193 00:11:47,430 --> 00:11:51,510 Right, so that's a very important property, which 194 00:11:51,510 --> 00:11:53,640 simplifies things tremendously. 195 00:11:56,980 --> 00:11:58,810 Yeah, and here's a note just to say 196 00:11:58,810 --> 00:12:03,370 that that is the lateral magnification of this system 197 00:12:03,370 --> 00:12:04,460 as we described before. 198 00:12:09,880 --> 00:12:15,560 OK, so now this is just showing this diagram again. 199 00:12:15,560 --> 00:12:18,570 These are not meant as sort of rays. 200 00:12:18,570 --> 00:12:22,140 These are just sort of showing regions where the light is 201 00:12:22,140 --> 00:12:24,270 as it goes through. 202 00:12:24,270 --> 00:12:31,890 But we've got this amplitude here, 203 00:12:31,890 --> 00:12:34,230 which is Fourier transformed by a lens 204 00:12:34,230 --> 00:12:39,726 and then goes through a mask, Fourier transformed again 205 00:12:39,726 --> 00:12:40,600 [AUDIO OUT] field. 206 00:12:43,920 --> 00:12:48,120 So what this slide is trying to show 207 00:12:48,120 --> 00:12:51,020 is the significance of the point spread function. 208 00:12:51,020 --> 00:12:54,030 At some arbitrary objects-- 209 00:12:54,030 --> 00:12:57,060 we can think of an arbitrary object g of t, 210 00:12:57,060 --> 00:12:59,250 an arbitrary transparency. 211 00:12:59,250 --> 00:13:00,750 You can you can think of-- 212 00:13:03,340 --> 00:13:03,840 sorry. 213 00:13:03,840 --> 00:13:06,930 You're illuminating that with some field. 214 00:13:06,930 --> 00:13:09,990 So what actually you get after the transparency 215 00:13:09,990 --> 00:13:11,610 is the products of those two. 216 00:13:11,610 --> 00:13:15,110 So is the g in that I really meant to talk about. 217 00:13:15,110 --> 00:13:17,940 The g in you can think of as being made up 218 00:13:17,940 --> 00:13:19,680 of lots of delta functions. 219 00:13:19,680 --> 00:13:22,830 Any arbitrary function, you can divide it up 220 00:13:22,830 --> 00:13:26,280 into lots of delta functions where 221 00:13:26,280 --> 00:13:29,160 the strength of each of them corresponds to the strength 222 00:13:29,160 --> 00:13:30,938 of that function at that point. 223 00:13:30,938 --> 00:13:32,730 Right, so this is what this expression here 224 00:13:32,730 --> 00:13:34,210 is writing down. 225 00:13:34,210 --> 00:13:38,970 So the g in is written as-- 226 00:13:38,970 --> 00:13:42,720 you can write it as an integral over delta functions 227 00:13:42,720 --> 00:13:44,415 with the strength given by the function. 228 00:13:46,940 --> 00:13:49,890 But each of those delta functions 229 00:13:49,890 --> 00:13:52,170 is going to propagate through this system 230 00:13:52,170 --> 00:13:54,840 and produce a point spread function exactly 231 00:13:54,840 --> 00:13:57,430 like we described before. 232 00:13:57,430 --> 00:14:00,120 And then the image of the total object 233 00:14:00,120 --> 00:14:03,300 is going to be the sum of the images of all 234 00:14:03,300 --> 00:14:04,990 these delta functions. 235 00:14:04,990 --> 00:14:09,960 So all of these delta functions produce point spread functions. 236 00:14:09,960 --> 00:14:12,930 So you integrate over those point spread functions 237 00:14:12,930 --> 00:14:17,070 with the relative strengths that you need from here 238 00:14:17,070 --> 00:14:20,580 and you'll get the final image. 239 00:14:20,580 --> 00:14:24,600 And yeah, you can see that this is 240 00:14:24,600 --> 00:14:27,600 a bit like Huygens principle, that you are breaking up 241 00:14:27,600 --> 00:14:34,560 this wavefront into lots of components that are all 242 00:14:34,560 --> 00:14:36,300 represented as delta functions. 243 00:14:46,010 --> 00:14:52,630 OK, so therefore what we get out is going to be this integral. 244 00:14:52,630 --> 00:14:56,050 Each of these delta functions is going 245 00:14:56,050 --> 00:14:59,530 to produce an image given by this point spread function. 246 00:14:59,530 --> 00:15:04,150 The point spread function is shifted a distance 247 00:15:04,150 --> 00:15:07,830 which depends on the original distance 248 00:15:07,830 --> 00:15:10,240 but is scaled according to this factor we just 249 00:15:10,240 --> 00:15:12,080 mentioned earlier. 250 00:15:12,080 --> 00:15:14,560 And the strength of that component 251 00:15:14,560 --> 00:15:19,740 is given by this value g in at this point as we said there. 252 00:15:19,740 --> 00:15:25,230 So our final expression says that the output amplitude, 253 00:15:25,230 --> 00:15:29,160 but amplitude convolved with the point spread function. 254 00:15:29,160 --> 00:15:33,360 Right, so this expression-- you can see-- 255 00:15:33,360 --> 00:15:37,140 is exactly the same as an ordinary convolution 256 00:15:37,140 --> 00:15:45,460 expression, except [AUDIO OUT] But you can see it's 257 00:15:45,460 --> 00:15:55,730 like [AUDIO OUT] 258 00:15:55,730 --> 00:16:01,060 OK, yeah so because of this scaling, 259 00:16:01,060 --> 00:16:06,060 we can-- and also the inversion as well, 260 00:16:06,060 --> 00:16:10,110 we sometimes express things in terms 261 00:16:10,110 --> 00:16:13,680 of a reduced optical output coordinates. 262 00:16:13,680 --> 00:16:16,230 So we think of the output coordinate 263 00:16:16,230 --> 00:16:19,990 as being [? measured ?] [AUDIO OUT] to get rid 264 00:16:19,990 --> 00:16:23,560 of the sine and also scaled it to get rid 265 00:16:23,560 --> 00:16:25,510 of the scaling factor. 266 00:16:25,510 --> 00:16:31,450 And if we do it in terms of these reduced quantities, 267 00:16:31,450 --> 00:16:34,300 then you can write this as are just a normal convolution 268 00:16:34,300 --> 00:16:34,830 expression. 269 00:16:34,830 --> 00:16:37,240 So all we've done really is scaled 270 00:16:37,240 --> 00:16:40,750 the output coordinates in order to make it 271 00:16:40,750 --> 00:16:46,350 so that you can use the normal convolution expression. 272 00:16:46,350 --> 00:16:50,280 And then we think back to our case 273 00:16:50,280 --> 00:16:55,270 I was mentioning before of the electronic black box. 274 00:16:55,270 --> 00:16:58,530 So an electronic black box, you can either 275 00:16:58,530 --> 00:17:02,310 characterize in terms of an impulse function 276 00:17:02,310 --> 00:17:04,800 or in terms of a transfer function. 277 00:17:04,800 --> 00:17:07,020 So an impulse function-- you look 278 00:17:07,020 --> 00:17:11,890 at a delta function in time and look at what comes out. 279 00:17:11,890 --> 00:17:16,500 In terms of a transfer function, you put in a pure frequency 280 00:17:16,500 --> 00:17:19,230 and you look at the output that you 281 00:17:19,230 --> 00:17:21,630 get from that pure frequency. 282 00:17:21,630 --> 00:17:26,550 Right, so we can do exactly the same for the optical case, 283 00:17:26,550 --> 00:17:30,180 instead of thinking in terms of a point spread function, which 284 00:17:30,180 --> 00:17:39,830 is the same as this impulse response, 285 00:17:39,830 --> 00:17:42,740 we can think in terms of a transfer function. 286 00:17:42,740 --> 00:17:46,310 And now one thing to be very clear 287 00:17:46,310 --> 00:17:49,070 here is of course, this whole thing, 288 00:17:49,070 --> 00:17:52,100 we're talking about coherent systems. 289 00:17:52,100 --> 00:17:55,560 So we're working with amplitudes all the time. 290 00:17:55,560 --> 00:17:59,600 And so it's the amplitudes that are 291 00:17:59,600 --> 00:18:04,500 going to be affected by the optical system. 292 00:18:04,500 --> 00:18:06,860 And that's why we have to think of what's called 293 00:18:06,860 --> 00:18:09,750 an amplitude transfer function. 294 00:18:09,750 --> 00:18:13,550 All right, and we've written that as ATF, amplitude transfer 295 00:18:13,550 --> 00:18:14,430 function. 296 00:18:14,430 --> 00:18:19,490 So this-- if you do have just a Fourier 297 00:18:19,490 --> 00:18:23,750 transform of this convolution, you get a product. 298 00:18:23,750 --> 00:18:28,360 And so this is saying that the spectrum out-- 299 00:18:28,360 --> 00:18:31,340 the spatial frequency content of the image 300 00:18:31,340 --> 00:18:33,770 is equal to the spatial frequency 301 00:18:33,770 --> 00:18:38,390 content of the input multiplied by some transfer 302 00:18:38,390 --> 00:18:44,300 function, which is the Fourier transform of the point spread 303 00:18:44,300 --> 00:18:45,870 function. 304 00:18:45,870 --> 00:18:49,140 Right, so this thing here is called our amplitude transfer 305 00:18:49,140 --> 00:18:50,140 function. 306 00:18:50,140 --> 00:18:56,370 So we have to multiply the spectral content of the object 307 00:18:56,370 --> 00:18:58,290 by the transfer function. 308 00:18:58,290 --> 00:19:01,350 And then we could do an inverse Fourier transfer 309 00:19:01,350 --> 00:19:05,400 and we'd end up now with the image, 310 00:19:05,400 --> 00:19:06,690 the amplitude in the image. 311 00:19:10,400 --> 00:19:11,685 That there's something. 312 00:19:17,330 --> 00:19:18,680 It wrote something, did it? 313 00:19:21,390 --> 00:19:22,318 Yeah, OK. 314 00:19:22,318 --> 00:19:24,110 Right, so we've already [? reached ?] that. 315 00:19:28,260 --> 00:19:32,670 OK, so let's look at these examples we said again. 316 00:19:32,670 --> 00:19:37,170 So these are two pupil masks that we looked at before. 317 00:19:37,170 --> 00:19:39,890 The first one is just a low pass filter, 318 00:19:39,890 --> 00:19:42,120 lets through the low spatial frequencies 319 00:19:42,120 --> 00:19:45,760 but doesn't let through the high spatial frequencies. 320 00:19:45,760 --> 00:19:47,880 And this was another-- this was a phase 321 00:19:47,880 --> 00:19:51,240 filter we looked at before where we only 322 00:19:51,240 --> 00:19:56,400 let through the spatial frequencies within this region. 323 00:19:56,400 --> 00:20:00,990 But also within that region, we changed the phase-- 324 00:20:00,990 --> 00:20:05,040 the relative phase of the components 325 00:20:05,040 --> 00:20:06,660 which are in this region here. 326 00:20:06,660 --> 00:20:09,120 Right, so you can think of this as being 327 00:20:09,120 --> 00:20:11,567 the modulus of the filter. 328 00:20:11,567 --> 00:20:12,900 This is the phase of the filter. 329 00:20:16,010 --> 00:20:20,380 And outside here, of course, it's zero anyway. 330 00:20:23,668 --> 00:20:31,670 So [AUDIO OUT] Right and then, this 331 00:20:31,670 --> 00:20:38,720 is how we calculate the transfer function from this. 332 00:20:38,720 --> 00:20:40,430 This is a distance, right? 333 00:20:40,430 --> 00:20:43,610 The pupil is actually a real physical thing. 334 00:20:43,610 --> 00:20:45,470 So it's got a physical size. 335 00:20:45,470 --> 00:20:47,900 So this is measured in centimeters here. 336 00:20:47,900 --> 00:20:51,030 It cuts off at 1.5 centimeters. 337 00:20:51,030 --> 00:20:55,010 And in order to convert that into spatial frequency, 338 00:20:55,010 --> 00:20:58,700 we have to divide that by lambda f. 339 00:20:58,700 --> 00:21:00,350 And this is putting in some figures. 340 00:21:00,350 --> 00:21:07,010 That says that u is equal to x double dash in centimeters 341 00:21:07,010 --> 00:21:11,810 times 10 to the minus 3 centimeters 342 00:21:11,810 --> 00:21:19,370 to the minus [AUDIO OUT] [? Scaling ?] of our transfer 343 00:21:19,370 --> 00:21:25,040 function, and then finally this is what the ATF is. 344 00:21:31,120 --> 00:21:34,714 5 is going to be 1.5 times 10 to the 3 cent-- 345 00:21:34,714 --> 00:21:38,740 [AUDIO OUT] which is what this shows here in this diagram. 346 00:21:41,420 --> 00:21:44,015 It's identical, just scaled. 347 00:21:47,950 --> 00:21:50,020 Of course, to the other example-- 348 00:21:50,020 --> 00:21:56,237 or any [AUDIO OUT] [INAUDIBLE] [AUDIO OUT] 349 00:21:56,237 --> 00:21:57,070 AUDIENCE: Professor? 350 00:21:57,070 --> 00:22:00,190 PROFESSOR: [INAUDIBLE] is going to have this modulus of phase 351 00:22:00,190 --> 00:22:00,930 components. 352 00:22:00,930 --> 00:22:01,690 Yeah? 353 00:22:01,690 --> 00:22:04,060 AUDIENCE: Your sound is breaking up a little bit. 354 00:22:04,060 --> 00:22:08,830 And I wanted to ask how is the point spread function related 355 00:22:08,830 --> 00:22:09,610 to ATF? 356 00:22:09,610 --> 00:22:12,130 Is it just the-- 357 00:22:12,130 --> 00:22:16,360 ATF is the Fourier transform of the point spread function. 358 00:22:16,360 --> 00:22:18,770 PROFESSOR: Yeah, they're a Fourier transform pair. 359 00:22:18,770 --> 00:22:21,925 Yeah, but you have to be careful with the scalings. 360 00:22:26,200 --> 00:22:28,790 OK, so I don't know why the sound. 361 00:22:28,790 --> 00:22:31,290 I moved that a bit closer. 362 00:22:31,290 --> 00:22:31,920 Is it? 363 00:22:31,920 --> 00:22:33,850 TA: Maybe clip it on the other side. 364 00:22:33,850 --> 00:22:35,975 PROFESSOR: I'll clip it on the other side, perhaps. 365 00:22:40,240 --> 00:22:40,990 Right, OK. 366 00:22:40,990 --> 00:22:42,770 Let's see if that's better. 367 00:22:42,770 --> 00:22:43,900 Maybe just walking around. 368 00:22:47,950 --> 00:22:50,910 OK, so that's that one finished. 369 00:22:50,910 --> 00:22:53,010 Where am I going to find something, George? 370 00:22:53,010 --> 00:23:12,060 TA: [INAUDIBLE] That's proper. 371 00:23:12,060 --> 00:23:14,770 PROFESSOR: OK, so now we're up to today. 372 00:23:19,670 --> 00:23:23,530 OK, so really today we're going to carry on talking 373 00:23:23,530 --> 00:23:24,880 about the same sort of stuff. 374 00:23:24,880 --> 00:23:28,850 I think this is the simplest way to describe it. 375 00:23:28,850 --> 00:23:31,240 And so what we've been looking at-- this 376 00:23:31,240 --> 00:23:33,760 is just revising again with the fact 377 00:23:33,760 --> 00:23:36,640 that we're thinking of the 4f system 378 00:23:36,640 --> 00:23:40,090 as being a linear shifting variance system. 379 00:23:40,090 --> 00:23:45,190 Right, so if you remember the shift invariance means 380 00:23:45,190 --> 00:23:50,210 that the point spread function is independent of position? 381 00:23:50,210 --> 00:23:56,590 Right, so there in that case, you can [? right ?] the image 382 00:23:56,590 --> 00:23:57,850 as a convolution. 383 00:23:57,850 --> 00:23:58,900 Yeah? 384 00:23:58,900 --> 00:24:11,780 AUDIENCE: Mr. [INAUDIBLE] There's [? nothing ?] with 385 00:24:11,780 --> 00:24:19,620 [INAUDIBLE] 386 00:24:19,620 --> 00:24:21,890 PROFESSOR: OK. 387 00:24:21,890 --> 00:24:23,500 So I hope that this is better. 388 00:24:26,070 --> 00:24:31,040 So we can take about a 4f system as being an linear shift 389 00:24:31,040 --> 00:24:33,580 invariant imaging system. 390 00:24:33,580 --> 00:24:40,080 And so this is going through then 391 00:24:40,080 --> 00:24:42,790 how you calculate that sort of system. 392 00:24:42,790 --> 00:24:45,720 So we start off with some illuminating field 393 00:24:45,720 --> 00:24:47,860 that comes in here. 394 00:24:47,860 --> 00:24:50,510 It illuminates some transparency. 395 00:24:50,510 --> 00:24:52,890 And so what you get on the far side of that 396 00:24:52,890 --> 00:24:56,220 is going to be the product of illumination 397 00:24:56,220 --> 00:24:57,420 and the transparency. 398 00:25:00,750 --> 00:25:03,650 All right, so this is basically the object 399 00:25:03,650 --> 00:25:07,480 of the imaging system, 400 00:25:07,480 --> 00:25:10,070 And then it goes through an imaging system 401 00:25:10,070 --> 00:25:15,410 with a certain pupil mask, which is related 402 00:25:15,410 --> 00:25:19,280 to an amplitude transfer function, which 403 00:25:19,280 --> 00:25:27,780 is just a scaled version of that pupil mask. 404 00:25:27,780 --> 00:25:33,150 And by doing the Fourier transform of that, 405 00:25:33,150 --> 00:25:36,650 you will get the point spread function of the system. 406 00:25:36,650 --> 00:25:42,280 So the Fourier transform of the pupil mask 407 00:25:42,280 --> 00:25:46,160 gives this point spread function like this. 408 00:25:52,540 --> 00:25:56,380 And then your output field is going 409 00:25:56,380 --> 00:26:01,840 to be your input field convolved with the point spread 410 00:26:01,840 --> 00:26:04,750 function of the system. 411 00:26:04,750 --> 00:26:08,920 So I noticed that-- 412 00:26:08,920 --> 00:26:13,030 so there are two versions of this that are described here, 413 00:26:13,030 --> 00:26:17,000 one in terms of x dashed, and one in terms of x naught 414 00:26:17,000 --> 00:26:18,130 dashed. 415 00:26:18,130 --> 00:26:23,980 This x naught dashed one is the rescaled one 416 00:26:23,980 --> 00:26:26,480 to make it look like a proper convolution. 417 00:26:26,480 --> 00:26:32,170 So we've got rid of the scaling factor F2 over F1. 418 00:26:32,170 --> 00:26:34,000 And you see what that does back here 419 00:26:34,000 --> 00:26:40,840 in the point spread function is it just changes an F2-- 420 00:26:40,840 --> 00:26:44,950 well, one over F2 to one over F1. 421 00:26:44,950 --> 00:26:46,562 So you can work in either of those. 422 00:26:46,562 --> 00:26:47,770 There's no difference really. 423 00:26:50,780 --> 00:26:56,640 OK, so let's now think about how these concepts fit in 424 00:26:56,640 --> 00:26:59,950 with what we learn from geometrical optics. 425 00:26:59,950 --> 00:27:04,290 And first of all, then we look at natural magnification. 426 00:27:04,290 --> 00:27:12,480 What we find then is that if we have a point object here-- 427 00:27:12,480 --> 00:27:14,700 we just mentioned this before-- 428 00:27:14,700 --> 00:27:16,920 what it would give is an image which 429 00:27:16,920 --> 00:27:21,840 is going to be a blurred image of a point object. 430 00:27:21,840 --> 00:27:25,090 It's going to be this amplitude point spread function. 431 00:27:25,090 --> 00:27:27,270 But let's for the moment-- 432 00:27:27,270 --> 00:27:29,220 we're trying to find out how this fits in 433 00:27:29,220 --> 00:27:31,740 with the geometric optics limit. 434 00:27:31,740 --> 00:27:34,950 Of course, in geometrical optics there is no diffraction. 435 00:27:34,950 --> 00:27:39,650 So you can look at the sort of limiting case 436 00:27:39,650 --> 00:27:42,450 as when that point spread function becomes 437 00:27:42,450 --> 00:27:43,960 a delta function. 438 00:27:43,960 --> 00:27:46,710 So in this case, this is not physical 439 00:27:46,710 --> 00:27:50,450 because you couldn't do this either very easily. 440 00:27:50,450 --> 00:27:54,540 But you could do it only by making the wavelength 10 441 00:27:54,540 --> 00:27:58,050 to zero or you could make the aperture 10 442 00:27:58,050 --> 00:28:00,840 to infinity, which of course, both things 443 00:28:00,840 --> 00:28:02,610 you can't really do. 444 00:28:02,610 --> 00:28:06,000 But just think of it like this just to see what's going on. 445 00:28:06,000 --> 00:28:08,790 If you could think of the point spread function 446 00:28:08,790 --> 00:28:11,940 as being a delta function, the image of our point 447 00:28:11,940 --> 00:28:14,720 here would be a point here. 448 00:28:14,720 --> 00:28:17,940 And we've already found where that point is. 449 00:28:17,940 --> 00:28:20,640 We've found that it's a point which 450 00:28:20,640 --> 00:28:23,160 is on the opposite side of the axis. 451 00:28:23,160 --> 00:28:28,140 And the distance here is scaled relative to this one 452 00:28:28,140 --> 00:28:30,880 by the ratio of the focal lengths. 453 00:28:30,880 --> 00:28:37,680 And this is exactly what we found from geometrical optics. 454 00:28:37,680 --> 00:28:43,620 So the lateral magnification is this n here. 455 00:28:43,620 --> 00:28:46,440 nt is the lateral magnification which 456 00:28:46,440 --> 00:28:52,410 is given by minus the ratio of the focal length in F2 over F1. 457 00:29:07,400 --> 00:29:11,600 OK, so if we assume that our amplitude point spread 458 00:29:11,600 --> 00:29:14,210 function is just a delta function, 459 00:29:14,210 --> 00:29:18,200 then the output amplitude is going 460 00:29:18,200 --> 00:29:20,060 to be equal to the input amplitude 461 00:29:20,060 --> 00:29:22,160 but with this scaling. 462 00:29:28,200 --> 00:29:30,540 OK, so that's the result we get. 463 00:29:30,540 --> 00:29:33,895 It's exactly the same as you get from geometrical optics. 464 00:29:37,370 --> 00:29:40,790 Analyzing the same system would also give you 465 00:29:40,790 --> 00:29:44,220 this same magnification. 466 00:29:44,220 --> 00:29:46,890 Right, so this is another example. 467 00:29:46,890 --> 00:29:50,820 This is looking at the angular magnification of this system. 468 00:29:50,820 --> 00:29:52,980 So before we looked at what happens 469 00:29:52,980 --> 00:29:55,770 for a point in the input plane. 470 00:29:55,770 --> 00:29:59,400 Now we're going to look at what happens to a plane wave 471 00:29:59,400 --> 00:30:03,030 at a certain angle in the input plane. 472 00:30:03,030 --> 00:30:06,090 And you can see what that plane wave does as it goes through. 473 00:30:06,090 --> 00:30:09,420 Of course, the plane wave when it's Fourier transformed now 474 00:30:09,420 --> 00:30:13,380 becomes a converging wave, which converges on this point. 475 00:30:13,380 --> 00:30:17,310 It then spreads again and then is 476 00:30:17,310 --> 00:30:18,840 coming is collimated by this lens 477 00:30:18,840 --> 00:30:25,210 to produce an another parallel wave there. 478 00:30:25,210 --> 00:30:28,500 But you'll notice that the angle of this wave 479 00:30:28,500 --> 00:30:31,250 is not the same as the angle of this wave. 480 00:30:31,250 --> 00:30:34,230 For a start, you'll notice they've got opposite signs. 481 00:30:34,230 --> 00:30:35,460 That one's going up. 482 00:30:35,460 --> 00:30:37,080 That one's coming down. 483 00:30:37,080 --> 00:30:39,660 But that's what you'd expect of course, because 484 00:30:39,660 --> 00:30:43,810 of this same inversion that we've been talking about. 485 00:30:43,810 --> 00:30:50,130 So this input field then is just an exponential face factor. 486 00:30:58,770 --> 00:31:02,680 OK, so still assuming this ideal geometrical point spread 487 00:31:02,680 --> 00:31:07,750 function, the output field is going to be given by this. 488 00:31:07,750 --> 00:31:12,990 And you can then express that in terms of phases. 489 00:31:12,990 --> 00:31:15,310 And it comes out to be this. 490 00:31:15,310 --> 00:31:18,660 So this is the angle-- 491 00:31:18,660 --> 00:31:24,240 the angle in was 3 to 1 and the angle out is now minus F1 492 00:31:24,240 --> 00:31:26,200 over F2 times 3 to 1. 493 00:31:26,200 --> 00:31:31,030 So again, it's scaled and got an inversion relative 494 00:31:31,030 --> 00:31:32,920 to the input angle. 495 00:31:32,920 --> 00:31:36,550 So you can see the angle of magnification 496 00:31:36,550 --> 00:31:40,720 then is given by this expression, minus F1 over F2. 497 00:31:40,720 --> 00:31:44,170 And again, that is the same as you get 498 00:31:44,170 --> 00:31:45,690 from your geometrical optics. 499 00:31:49,040 --> 00:31:54,710 OK, so based on the interpretation of propagation 500 00:31:54,710 --> 00:31:58,520 of spatial frequency, the magnification results 501 00:31:58,520 --> 00:32:02,030 are also in agreement with the scaling similarity 502 00:32:02,030 --> 00:32:04,520 theorem of Fourier transforms. 503 00:32:04,520 --> 00:32:09,410 Right, so you've got an input amplitude, 504 00:32:09,410 --> 00:32:14,570 an output amplitude related to an input amplitude, an output 505 00:32:14,570 --> 00:32:20,000 spatial frequency related to an input spatial frequency. 506 00:32:20,000 --> 00:32:22,850 And you notice this one's got an F1 over F2. 507 00:32:22,850 --> 00:32:24,940 And this one's got an F2 over F1. 508 00:32:24,940 --> 00:32:28,220 So this is back to where idea is that if you make 509 00:32:28,220 --> 00:32:31,640 the functions smaller, the Fourier transform is 510 00:32:31,640 --> 00:32:33,110 broader and vise versa. 511 00:32:37,400 --> 00:32:41,240 OK, so now this geometry I guess you could see-- 512 00:32:41,240 --> 00:32:43,570 maybe I could just draw one. 513 00:32:50,430 --> 00:32:51,800 So this is our system. 514 00:32:57,730 --> 00:33:00,930 And what I was going to draw, this is F1. 515 00:33:00,930 --> 00:33:04,110 F1, F1, F2, F2. 516 00:33:04,110 --> 00:33:08,040 And you can see this wave is coming in like this. 517 00:33:08,040 --> 00:33:11,430 So we could have drawn a ray like this. 518 00:33:11,430 --> 00:33:21,440 And this would of course, have been refracted so that it's 519 00:33:21,440 --> 00:33:24,400 traveling parallel to the axis. 520 00:33:24,400 --> 00:33:26,890 And then this would come down like that. 521 00:33:26,890 --> 00:33:29,640 So you can see here, this is our phase one. 522 00:33:29,640 --> 00:33:32,130 So f phase one is this distance. 523 00:33:32,130 --> 00:33:33,060 And that is also-- 524 00:33:33,060 --> 00:33:37,390 if you think of this as being theta out, 525 00:33:37,390 --> 00:33:41,370 we can see that this distance is the same as this distance. 526 00:33:41,370 --> 00:33:46,180 And so F2 times theta out is equal to F1 times theta in. 527 00:33:50,500 --> 00:33:54,900 OK, so everything agrees nicely with geometrical optics. 528 00:33:54,900 --> 00:33:56,760 So that's good. 529 00:33:56,760 --> 00:34:00,110 So let's look at some examples now. 530 00:34:00,110 --> 00:34:03,650 Let's look at some examples of some apertures. 531 00:34:03,650 --> 00:34:08,500 So first of all, the case of let's put it as our pupil mask 532 00:34:08,500 --> 00:34:12,960 a rectangular aperture, which has got a distance a and b 533 00:34:12,960 --> 00:34:14,520 in the two directions. 534 00:34:14,520 --> 00:34:17,869 So we can think of this as being rect, x over a times directive 535 00:34:17,869 --> 00:34:21,260 rect of y over b. 536 00:34:21,260 --> 00:34:26,880 And we can then work out what the transfer function of that 537 00:34:26,880 --> 00:34:27,570 is. 538 00:34:27,570 --> 00:34:29,280 The transfer function-- we said-- 539 00:34:29,280 --> 00:34:34,710 is just equal to a scaled version of the pupil mask. 540 00:34:34,710 --> 00:34:38,969 So if this is a rectangle, this is also a rectangle. 541 00:34:38,969 --> 00:34:45,400 But the scaling has got this lambda F1 coming in. 542 00:34:45,400 --> 00:34:47,074 Right, so this is now-- 543 00:34:47,074 --> 00:34:48,449 because it has to have something. 544 00:34:48,449 --> 00:34:50,695 This is a function of distance. 545 00:34:55,219 --> 00:34:56,960 Sorry, x is distance. 546 00:34:56,960 --> 00:34:58,070 a is distance. 547 00:34:58,070 --> 00:35:01,640 Right, so this is the dimension and its quantity. 548 00:35:01,640 --> 00:35:04,580 And in here, this is distance. 549 00:35:04,580 --> 00:35:05,960 This is distance. 550 00:35:05,960 --> 00:35:07,740 And this is distance. 551 00:35:07,740 --> 00:35:11,600 So this has to be, of course, inverse distance, doesn't it-- 552 00:35:11,600 --> 00:35:15,730 to make this whole thing dimensionless. 553 00:35:15,730 --> 00:35:20,090 So this is why you need to have something in there 554 00:35:20,090 --> 00:35:22,680 to get it the right dimensions. 555 00:35:22,680 --> 00:35:24,410 And then the point spread function 556 00:35:24,410 --> 00:35:27,950 is given by the Fourier transform of that. 557 00:35:27,950 --> 00:35:29,630 And that's what this looks like. 558 00:35:29,630 --> 00:35:32,440 Fourier transform of rect is sinc. 559 00:35:32,440 --> 00:35:34,290 So this is separable. 560 00:35:34,290 --> 00:35:38,180 So you just do the two Fourier transforms separately. 561 00:35:38,180 --> 00:35:42,830 And you end up with this pattern that you've seen before. 562 00:35:42,830 --> 00:35:48,980 It looks like a central spot and then a lot of extra spots 563 00:35:48,980 --> 00:35:53,390 that decrease in amplitude or intensity 564 00:35:53,390 --> 00:35:56,050 as you get away from that. 565 00:35:56,050 --> 00:35:57,170 Yeah? 566 00:35:57,170 --> 00:35:58,970 TA: Maybe we mentioned that in the notes 567 00:35:58,970 --> 00:36:02,510 that they posted the ATF and PSF were mislabeled. 568 00:36:02,510 --> 00:36:03,860 PROFESSOR: Yeah, OK. 569 00:36:03,860 --> 00:36:06,680 In the notes that were on the web, 570 00:36:06,680 --> 00:36:11,480 this said PSF and this said ATF. 571 00:36:11,480 --> 00:36:13,970 So it's pretty obvious when you see them. 572 00:36:13,970 --> 00:36:15,470 It's obviously the wrong way around. 573 00:36:15,470 --> 00:36:17,500 TA: And we posted corrections. 574 00:36:17,500 --> 00:36:19,830 PROFESSOR: Yeah, OK. 575 00:36:19,830 --> 00:36:22,050 [? George ?] posted a correction. 576 00:36:22,050 --> 00:36:23,930 Now I was going to ask one question. 577 00:36:23,930 --> 00:36:26,810 This is presumably the intensity of the point spread function 578 00:36:26,810 --> 00:36:28,500 that your plotting here-- 579 00:36:28,500 --> 00:36:29,788 TA: Is the square. 580 00:36:29,788 --> 00:36:30,830 PROFESSOR: Is the square. 581 00:36:30,830 --> 00:36:33,470 Yeah, the square of the amplitude point spread 582 00:36:33,470 --> 00:36:35,960 function. 583 00:36:35,960 --> 00:36:38,330 Right, So that's our first example, very important 584 00:36:38,330 --> 00:36:39,950 example. 585 00:36:39,950 --> 00:36:43,320 The second one is probably even more important. 586 00:36:43,320 --> 00:36:46,350 So this is a circular aperture. 587 00:36:46,350 --> 00:36:49,780 And of course, most optical systems 588 00:36:49,780 --> 00:36:51,770 do have circular apertures. 589 00:36:51,770 --> 00:36:54,950 So this is why this is of great importance. 590 00:36:54,950 --> 00:36:56,420 It seems to have turned out being 591 00:36:56,420 --> 00:37:01,160 bit like American football for some reason, but never mind. 592 00:37:01,160 --> 00:37:05,510 And practice because it's going to over to the US, I suppose. 593 00:37:05,510 --> 00:37:07,760 TA: This would have been OK except the perception 594 00:37:07,760 --> 00:37:09,250 [INAUDIBLE] the other way around. 595 00:37:09,250 --> 00:37:10,310 PROFESSOR: Oh yeah. 596 00:37:10,310 --> 00:37:11,150 That's true. 597 00:37:11,150 --> 00:37:13,160 Yes. 598 00:37:13,160 --> 00:37:16,740 It doesn't know how to do Fourier transforms obviously. 599 00:37:16,740 --> 00:37:19,860 Anyway, this is supposed to be a circle. 600 00:37:19,860 --> 00:37:25,070 And if this is the pupil mask, the transfer 601 00:37:25,070 --> 00:37:28,190 function-- the amplitude transfer function is just 602 00:37:28,190 --> 00:37:30,200 a scaled version of this. 603 00:37:30,200 --> 00:37:34,320 So it's again just a circular thing. 604 00:37:34,320 --> 00:37:37,250 And sorry, perhaps I just ought to stress this. 605 00:37:37,250 --> 00:37:41,130 It's obvious to you now, I'm sure, by this stage. 606 00:37:41,130 --> 00:37:44,120 So this is plotted here as black or white. 607 00:37:44,120 --> 00:37:47,690 Of course, white means one-- black means zero in this case. 608 00:37:47,690 --> 00:37:52,520 So that means if spatial frequencies land on this, those 609 00:37:52,520 --> 00:37:55,340 inside this get through and those outside this 610 00:37:55,340 --> 00:37:56,540 don't get through, right? 611 00:37:56,540 --> 00:38:00,830 So this is what we have to multiply our frequency 612 00:38:00,830 --> 00:38:03,770 response of our object by in order to get the frequency 613 00:38:03,770 --> 00:38:06,370 response of the image. 614 00:38:06,370 --> 00:38:09,530 And that's just like before. 615 00:38:09,530 --> 00:38:15,380 We have to scale this with these lambda F1s. 616 00:38:15,380 --> 00:38:18,560 And then the Fourier transform of this 617 00:38:18,560 --> 00:38:20,990 will give the point spread function. 618 00:38:20,990 --> 00:38:26,070 And this one has now gone into glorious technicolor. 619 00:38:26,070 --> 00:38:30,110 And so this is what we call the airy disc. 620 00:38:30,110 --> 00:38:33,530 The equation for best is given in terms of Bessel functions. 621 00:38:33,530 --> 00:38:35,930 You've defined this at some point, I presume. 622 00:38:35,930 --> 00:38:39,110 I'm not sure I was in the lecture where we defined 623 00:38:39,110 --> 00:38:43,700 this [INAUDIBLE] function, which is basically 2j 1 of something 624 00:38:43,700 --> 00:38:45,460 over something. 625 00:38:45,460 --> 00:38:48,910 And so this is our normal airy disc, 626 00:38:48,910 --> 00:38:51,410 the image of a point object, which 627 00:38:51,410 --> 00:38:54,560 has got this central bright spot, and then 628 00:38:54,560 --> 00:39:02,120 a dark ring, and then some bright rings outside a not so 629 00:39:02,120 --> 00:39:06,230 bright ring, and then another dark ring, and so on and so on. 630 00:39:06,230 --> 00:39:10,790 All right, so what we find then is 631 00:39:10,790 --> 00:39:17,600 the size of this has got this magic factor 1.22 that comes up 632 00:39:17,600 --> 00:39:21,290 all the time in terms of when you look at Rayleigh resolution 633 00:39:21,290 --> 00:39:24,470 and things, you always get this magic factor, 634 00:39:24,470 --> 00:39:30,930 which is actually 3.83 which is the first zero of J1 divided 635 00:39:30,930 --> 00:39:34,490 by pi it turns out. 636 00:39:34,490 --> 00:39:36,290 So that's why you get this magic figure. 637 00:39:38,930 --> 00:39:42,470 OK, and this is just to remind you of course, 638 00:39:42,470 --> 00:39:44,980 when it comes to resolution, we've 639 00:39:44,980 --> 00:39:47,380 mentioned before how resolution is 640 00:39:47,380 --> 00:39:50,180 related to numerical aperture. 641 00:39:50,180 --> 00:39:54,430 And so what we're doing here is producing an image 642 00:39:54,430 --> 00:39:55,810 of an object, aren't we? 643 00:39:55,810 --> 00:39:59,320 So if we've got an object, it's imaged 644 00:39:59,320 --> 00:40:03,100 with this lens, which has got a certain numerical aperture. 645 00:40:03,100 --> 00:40:08,230 Right, so this aperture here is what 646 00:40:08,230 --> 00:40:11,380 decides the resolution of the image 647 00:40:11,380 --> 00:40:13,810 that you will get in this system. 648 00:40:13,810 --> 00:40:16,020 This aperture here, you see is a ray that 649 00:40:16,020 --> 00:40:21,050 was drawn so it goes to the edge of this pupil mask. 650 00:40:21,050 --> 00:40:23,650 And if you trace this strobe, you'll 651 00:40:23,650 --> 00:40:26,230 see that it comes back down here. 652 00:40:26,230 --> 00:40:29,590 Sorry, of course, the numerical aperture is actually-- 653 00:40:29,590 --> 00:40:32,750 the sine of this angle, if you remember. 654 00:40:32,750 --> 00:40:33,790 That's what this means. 655 00:40:33,790 --> 00:40:35,270 It's not the angle itself. 656 00:40:35,270 --> 00:40:37,000 NA is not the angle. 657 00:40:37,000 --> 00:40:42,050 NA is equal to N sine alpha, if you remember. 658 00:40:42,050 --> 00:40:46,540 The refractive index of the material 659 00:40:46,540 --> 00:40:49,310 times the sine of this angle. 660 00:40:49,310 --> 00:40:53,530 But what we find is that if F1 and F2 are not equal, 661 00:40:53,530 --> 00:40:55,240 this angle here, of course, is not 662 00:40:55,240 --> 00:40:56,720 going to be equal to this one. 663 00:41:04,000 --> 00:41:11,830 You can see that if NA is small, if NA is small, 664 00:41:11,830 --> 00:41:13,840 you can see that these are going to be 665 00:41:13,840 --> 00:41:19,970 scaled in the same way as your F1 and your F2. 666 00:41:19,970 --> 00:41:21,970 It [INAUDIBLE] exactly be that. 667 00:41:21,970 --> 00:41:26,160 Of course, this NA would have to be the tan of that. 668 00:41:26,160 --> 00:41:30,940 But the sine-- if NA is small, then this distance 669 00:41:30,940 --> 00:41:34,310 is the same here and here. 670 00:41:34,310 --> 00:41:38,890 So this distance over this distance 671 00:41:38,890 --> 00:41:44,310 divided by this distance over this distance 672 00:41:44,310 --> 00:41:47,500 is pretty close to being the ratio of the focal length. 673 00:41:53,870 --> 00:41:58,220 OK, so we've got a certain NA on the input side. 674 00:41:58,220 --> 00:42:02,110 And we've got a certain NA on the output side. 675 00:42:02,110 --> 00:42:05,090 And the ratio between those names 676 00:42:05,090 --> 00:42:10,690 is just equal to the magnification of the system. 677 00:42:10,690 --> 00:42:14,290 So actually, of course, we're well used to the fact 678 00:42:14,290 --> 00:42:22,760 that rays of light go equally well in both directions. 679 00:42:22,760 --> 00:42:26,990 So we could use this same ray diagram 680 00:42:26,990 --> 00:42:30,890 to look at what happens if this was the object 681 00:42:30,890 --> 00:42:33,620 and this was the image. 682 00:42:33,620 --> 00:42:37,180 But in this case, of course, all these things are now inverted. 683 00:42:37,180 --> 00:42:40,540 So if it was producing-- 684 00:42:40,540 --> 00:42:43,750 as it stands here, it's producing a demagnified image 685 00:42:43,750 --> 00:42:44,800 in this way. 686 00:42:44,800 --> 00:42:48,400 So it would produce a magnified image going this way. 687 00:42:48,400 --> 00:42:55,300 And so you see that the object that you'd have to be imaging 688 00:42:55,300 --> 00:42:57,840 is scaled relative to this one. 689 00:42:57,840 --> 00:43:00,310 So the resolution would be the same basically 690 00:43:00,310 --> 00:43:01,540 in two directions. 691 00:43:01,540 --> 00:43:04,560 I mean, sorry-- that's not quite true. 692 00:43:04,560 --> 00:43:06,730 What am I trying to say? 693 00:43:06,730 --> 00:43:11,530 I'm trying to say that the NA relative to the size 694 00:43:11,530 --> 00:43:14,080 of our object that we're imaging here 695 00:43:14,080 --> 00:43:16,840 will be the same when we think of it as going 696 00:43:16,840 --> 00:43:21,820 this way or that way, because in real terms, how 697 00:43:21,820 --> 00:43:26,380 small a thing you can actually see with an optical system 698 00:43:26,380 --> 00:43:29,380 depends on the numerical aperture of the lens 699 00:43:29,380 --> 00:43:33,700 that you're looking at with, not the lens at the far end. 700 00:43:38,800 --> 00:43:43,590 OK, and then for our slit aperture, 701 00:43:43,590 --> 00:43:47,170 this is what the point spread function looks like. 702 00:43:47,170 --> 00:43:52,350 And then you can express that then very simply just 703 00:43:52,350 --> 00:43:55,800 replacing our a over F1 by numerical aperture, 704 00:43:55,800 --> 00:43:59,550 we've got now that the width at this spot 705 00:43:59,550 --> 00:44:01,770 is given by lambda over NA, which 706 00:44:01,770 --> 00:44:04,920 is exactly what we came about-- 707 00:44:04,920 --> 00:44:09,015 I guess you said that before somewhere. 708 00:44:09,015 --> 00:44:09,890 TA: I can't remember. 709 00:44:09,890 --> 00:44:12,330 PROFESSOR: But the resolution of a microscope 710 00:44:12,330 --> 00:44:15,690 or whatever and of an imaging system, 711 00:44:15,690 --> 00:44:17,040 you can write in this form. 712 00:44:17,040 --> 00:44:20,190 So of course, this distance will be 713 00:44:20,190 --> 00:44:22,890 smallest when this is biggest. 714 00:44:22,890 --> 00:44:30,510 So the biggest this could be is if NA was one 715 00:44:30,510 --> 00:44:34,620 and you made n, the refractive index nice and big. 716 00:44:37,960 --> 00:44:41,290 And this one, these distances are in terms-- 717 00:44:41,290 --> 00:44:43,690 this is distance in the image in terms 718 00:44:43,690 --> 00:44:47,570 of the NA in the output plane. 719 00:44:47,570 --> 00:44:49,990 So you could have converted this so this 720 00:44:49,990 --> 00:44:52,150 is distance in the objects. 721 00:44:52,150 --> 00:44:58,930 And then this would become NA in the input plane. 722 00:44:58,930 --> 00:45:04,060 And right and then circular aperture, 723 00:45:04,060 --> 00:45:07,070 our point spread function looks like this. 724 00:45:07,070 --> 00:45:10,090 And so this is what we get. 725 00:45:10,090 --> 00:45:15,790 And this delta r dashed here given by this expression, 726 00:45:15,790 --> 00:45:22,000 this 0.61 is half of the 1.22 we had before. 727 00:45:22,000 --> 00:45:26,170 But notice this is delta r dashed 728 00:45:26,170 --> 00:45:29,980 is the radius of the PSF [INAUDIBLE] 729 00:45:29,980 --> 00:45:32,260 So it's this distance, not the whole distance. 730 00:45:32,260 --> 00:45:33,940 It's this distance, right? 731 00:45:33,940 --> 00:45:36,170 So this is why we got half-- 732 00:45:36,170 --> 00:45:44,950 why the 1.22 is divided by 2. 733 00:45:44,950 --> 00:45:45,510 OK. 734 00:45:45,510 --> 00:45:49,740 Now I didn't mislead you also by saying from now 735 00:45:49,740 --> 00:45:53,220 to the end of the lecture's a breeze because it 736 00:45:53,220 --> 00:45:56,730 isn't, because the next bit is this thing about sampling. 737 00:45:56,730 --> 00:46:03,450 So that one more bit which is a bit difficult to take in. 738 00:46:03,450 --> 00:46:08,640 And this is really pointing out about effectively the number 739 00:46:08,640 --> 00:46:13,110 of degrees of freedom in an image. 740 00:46:13,110 --> 00:46:17,370 So imagine first of all, you've got an object 741 00:46:17,370 --> 00:46:18,660 that you're imaging. 742 00:46:18,660 --> 00:46:20,460 So this is this input field. 743 00:46:20,460 --> 00:46:25,140 It's got a certain size which is called here delta x. 744 00:46:25,140 --> 00:46:27,660 And of course, in any real practical systems, 745 00:46:27,660 --> 00:46:31,800 you don't actually measure it as a continuous function. 746 00:46:31,800 --> 00:46:33,570 It's actually going to be measured 747 00:46:33,570 --> 00:46:35,830 with a certain pixel size. 748 00:46:35,830 --> 00:46:39,350 So here we're calling the pixel size delta x. 749 00:46:39,350 --> 00:46:42,860 Right, now so what we've got here then 750 00:46:42,860 --> 00:46:48,320 is this function is like an array of spikes 751 00:46:48,320 --> 00:46:50,150 in this direction. 752 00:46:50,150 --> 00:46:53,270 An array of spikes delta functions in this direction, 753 00:46:53,270 --> 00:46:54,060 isn't it-- 754 00:46:54,060 --> 00:46:56,460 so in the products of those. 755 00:46:56,460 --> 00:47:01,120 And if you do the Fourier transform of that, 756 00:47:01,120 --> 00:47:04,890 you'll get something which is very similar again an array 757 00:47:04,890 --> 00:47:05,390 of spikes. 758 00:47:08,120 --> 00:47:12,440 If you look in the standard books on Fourier transforms 759 00:47:12,440 --> 00:47:16,940 like Bracewell or whatever, he calls this a cone function. 760 00:47:16,940 --> 00:47:19,970 And the Fourier transform of a cone function 761 00:47:19,970 --> 00:47:22,130 is another cone function. 762 00:47:22,130 --> 00:47:27,650 Right, so this is a cone array of delta functions multiplied 763 00:47:27,650 --> 00:47:29,820 by a rectangle. 764 00:47:29,820 --> 00:47:33,140 And the Fourier transform of a product 765 00:47:33,140 --> 00:47:37,580 is the convolution of the Fourier transforms. 766 00:47:37,580 --> 00:47:42,900 Right, so the Fourier transform of the rect is actually-- 767 00:47:42,900 --> 00:47:45,570 so this is really-- 768 00:47:45,570 --> 00:47:46,880 I ought to get into that. 769 00:47:46,880 --> 00:47:50,970 So just think of this then as being in the spatial frequency 770 00:47:50,970 --> 00:47:51,960 domain. 771 00:47:51,960 --> 00:47:56,220 It's got a certain range of spatial frequencies, 772 00:47:56,220 --> 00:48:04,600 and it's made up of certain resolution of frequencies. 773 00:48:04,600 --> 00:48:09,030 So what we're going to say is something about Nyquist now. 774 00:48:09,030 --> 00:48:11,870 So we can say that-- 775 00:48:11,870 --> 00:48:18,920 we know that in order to image this input field, knowing 776 00:48:18,920 --> 00:48:22,430 what this bandwidth is here, we have 777 00:48:22,430 --> 00:48:26,000 to sample with this distance. 778 00:48:32,330 --> 00:48:34,940 Is that what I'm trying to say? 779 00:48:34,940 --> 00:48:37,640 Yeah. 780 00:48:37,640 --> 00:48:40,610 The bandwidth is equal to 1 over twice the sampling distance. 781 00:48:40,610 --> 00:48:41,770 That's right. 782 00:48:41,770 --> 00:48:44,280 Yeah, and this is in the x direction. 783 00:48:44,280 --> 00:48:46,220 This is in the y direction. 784 00:48:46,220 --> 00:48:47,810 But we can also-- 785 00:48:47,810 --> 00:48:52,550 you can see by the similarity of these diagrams and the fact 786 00:48:52,550 --> 00:48:55,760 that Fourier transforms go either way, 787 00:48:55,760 --> 00:48:58,790 you can see that there will be a similar sort of Nyquist type 788 00:48:58,790 --> 00:49:02,900 criterion going from the other way around. 789 00:49:02,900 --> 00:49:05,670 So because this has got a finite field, 790 00:49:05,670 --> 00:49:09,410 this would mean that you've got a certain frequency 791 00:49:09,410 --> 00:49:10,400 resolution there. 792 00:49:10,400 --> 00:49:14,990 So there's a relationship between delta u and delta x. 793 00:49:14,990 --> 00:49:17,930 The fact that this has got a 2 here, 794 00:49:17,930 --> 00:49:21,140 of course, you know from the normal definition of Nyquist. 795 00:49:21,140 --> 00:49:23,530 And the fact that this one doesn't have a 2 in, 796 00:49:23,530 --> 00:49:25,400 it just comes about from the fact 797 00:49:25,400 --> 00:49:31,350 that this is defined here as being two times this. 798 00:49:31,350 --> 00:49:39,210 And right, so just like before, the spatial frequency domain-- 799 00:49:39,210 --> 00:49:41,400 those spatial frequencies are then 800 00:49:41,400 --> 00:49:45,480 multiplied by the pupil mask to find the spatial frequencies 801 00:49:45,480 --> 00:49:47,000 in the image, aren't they? 802 00:49:47,000 --> 00:49:51,220 Right, so this is looking now in the plane of the pupil mask. 803 00:49:51,220 --> 00:49:55,200 So this is actually the same as equal to this but just 804 00:49:55,200 --> 00:49:57,270 with a rescaling. 805 00:49:57,270 --> 00:50:03,120 And so these are the expressions for that. 806 00:50:03,120 --> 00:50:06,750 So this is just applying that rescaling, putting these lambda 807 00:50:06,750 --> 00:50:09,810 F1s in these expressions. 808 00:50:09,810 --> 00:50:13,080 So you'll get then what you need, 809 00:50:13,080 --> 00:50:20,120 how you need to sample in this plane here. 810 00:50:20,120 --> 00:50:27,880 And what you can notice from that is if you 811 00:50:27,880 --> 00:50:32,950 you'll notice that these things are all equal and equal 812 00:50:32,950 --> 00:50:35,890 to some constant, therefore. 813 00:50:35,890 --> 00:50:40,270 And this thing is called the Space bandwidth product. 814 00:50:40,270 --> 00:50:42,040 So you look at what this means. 815 00:50:42,040 --> 00:50:46,450 Delta x over delta x is the field size 816 00:50:46,450 --> 00:50:49,690 divided by the pixel size. 817 00:50:49,690 --> 00:50:55,360 So it's basically the number of pixels in the field, isn't it? 818 00:50:55,360 --> 00:50:59,200 Right, so your original object here 819 00:50:59,200 --> 00:51:05,335 has got the number of pixels equal to n x times n y. 820 00:51:05,335 --> 00:51:09,850 So that's the total amount of the number of pixels. 821 00:51:09,850 --> 00:51:13,750 And therefore if you think of it as being a binary thing, 822 00:51:13,750 --> 00:51:15,730 it's the amount of information that you've 823 00:51:15,730 --> 00:51:19,300 got in that original object. 824 00:51:19,300 --> 00:51:22,150 And of course, that information is 825 00:51:22,150 --> 00:51:28,700 going to be transferred through the system. 826 00:51:28,700 --> 00:51:33,610 So if you look at now the ratio in the next plane in terms 827 00:51:33,610 --> 00:51:38,590 of use, this distance over this distance 828 00:51:38,590 --> 00:51:41,170 is equal to again, the number of pixels. 829 00:51:41,170 --> 00:51:46,720 And as you know, when you do a fast Fourier transform 830 00:51:46,720 --> 00:51:48,430 of something which has got n elements, 831 00:51:48,430 --> 00:51:50,020 you get something which is n elements. 832 00:51:50,020 --> 00:51:52,720 So it all works right. 833 00:51:52,720 --> 00:51:58,240 So effectively, you can think of this as being a bit like-- 834 00:51:58,240 --> 00:52:00,430 of course, you can't really-- 835 00:52:00,430 --> 00:52:04,600 if this has actually got some finite bandwidth, then 836 00:52:04,600 --> 00:52:07,750 of course you can't sample on points. 837 00:52:07,750 --> 00:52:11,560 So it's a bit of an idealization to thinking this for the terms 838 00:52:11,560 --> 00:52:12,800 really. 839 00:52:12,800 --> 00:52:16,270 But nevertheless, it gives you a good idea. 840 00:52:16,270 --> 00:52:20,950 So of course, for a repetitive type function like this, 841 00:52:20,950 --> 00:52:24,190 also you can think in terms of a Fourier series 842 00:52:24,190 --> 00:52:25,870 becomes a Fourier transfer. 843 00:52:25,870 --> 00:52:28,990 Right, so all we're doing here is really 844 00:52:28,990 --> 00:52:35,350 resolving our function here into a Fourier series. 845 00:52:35,350 --> 00:52:38,390 And the Fourier series consists of a lot of sine terms, 846 00:52:38,390 --> 00:52:38,890 doesn't it? 847 00:52:38,890 --> 00:52:47,290 Sine of n, and sine of x, and sine of 2x, 848 00:52:47,290 --> 00:52:52,390 and so on and so on, and these correspond in the Fourier 849 00:52:52,390 --> 00:52:55,210 domain to these points on this grid. 850 00:52:55,210 --> 00:52:58,390 Right, so these each of these points 851 00:52:58,390 --> 00:53:03,970 represents the particular Fourier series component 852 00:53:03,970 --> 00:53:08,050 in the original object if you think of it like that. 853 00:53:08,050 --> 00:53:11,080 So the idea of a space bandwidth product 854 00:53:11,080 --> 00:53:13,270 is quite a powerful way of thinking 855 00:53:13,270 --> 00:53:18,670 about the fundamental limits of an optical system. 856 00:53:18,670 --> 00:53:24,920 And people have gone on to study this sort of approach 857 00:53:24,920 --> 00:53:28,520 in much more detail, actually trying 858 00:53:28,520 --> 00:53:31,790 to get around the limitations that I just described. 859 00:53:31,790 --> 00:53:35,450 The fact that, of course, that we've truncated this 860 00:53:35,450 --> 00:53:38,980 means that this is no longer a repetitive function. 861 00:53:38,980 --> 00:53:41,030 And similarly for here, right? 862 00:53:41,030 --> 00:53:44,060 So that actually you know this idea of thinking 863 00:53:44,060 --> 00:53:47,780 of truncated repetitive functions 864 00:53:47,780 --> 00:53:50,330 is a bit of a anomaly, really. 865 00:53:50,330 --> 00:53:54,530 But people have come up with mathematical ways 866 00:53:54,530 --> 00:53:58,700 of dealing with these sorts of systems which are probably 867 00:53:58,700 --> 00:54:01,170 way beyond what we are going to get to in this course. 868 00:54:04,480 --> 00:54:07,360 OK, did something come up then? 869 00:54:07,360 --> 00:54:08,080 Oh, yeah. 870 00:54:08,080 --> 00:54:09,510 Next slide. 871 00:54:09,510 --> 00:54:13,120 So OK, this is just pointing out then 872 00:54:13,120 --> 00:54:18,370 we said that you put a mask in here 873 00:54:18,370 --> 00:54:22,850 and that mask produces a certain transfer function-- 874 00:54:22,850 --> 00:54:27,130 amplitude transfer function, and the Fourier transform of that 875 00:54:27,130 --> 00:54:29,000 gives the pupil function. 876 00:54:29,000 --> 00:54:30,610 And we've shown a couple of examples 877 00:54:30,610 --> 00:54:33,700 of what this pupil function looks like 878 00:54:33,700 --> 00:54:37,130 and what the image would look like. 879 00:54:37,130 --> 00:54:39,670 But this is just pointing out that actually, 880 00:54:39,670 --> 00:54:45,710 of course, that means that we can choose whatever 881 00:54:45,710 --> 00:54:50,340 we want to put in there in order to produce the results we want. 882 00:54:50,340 --> 00:54:52,580 So this is what George has called here 883 00:54:52,580 --> 00:54:56,600 pupil engineering, which Ernst Stelzer doesn't believe in, 884 00:54:56,600 --> 00:54:59,140 I think, from what we were saying. 885 00:54:59,140 --> 00:55:04,790 But anyway, so the idea is that by changing this mask, 886 00:55:04,790 --> 00:55:11,150 you can improve the properties of that point spread function 887 00:55:11,150 --> 00:55:16,610 in some way, which would suit the that particular application 888 00:55:16,610 --> 00:55:17,540 you're looking at. 889 00:55:17,540 --> 00:55:22,220 Actually, it turns out that you can't actually 890 00:55:22,220 --> 00:55:25,670 do a lot better normally than the very simple example 891 00:55:25,670 --> 00:55:28,040 of a circular aperture. 892 00:55:28,040 --> 00:55:35,960 And the circular aperture is actually very special 893 00:55:35,960 --> 00:55:40,910 because it does give according to some measures anyway, a very 894 00:55:40,910 --> 00:55:43,650 sharp point spread function. 895 00:55:43,650 --> 00:55:47,870 And you can come up with other pupil functions 896 00:55:47,870 --> 00:55:51,560 that make the central lobe narrower, but normally 897 00:55:51,560 --> 00:55:56,660 what happens is then the size of the side lobes get stronger. 898 00:55:56,660 --> 00:55:58,200 So it's a sort of compromise. 899 00:55:58,200 --> 00:56:02,180 You can't have a very small central lobe together 900 00:56:02,180 --> 00:56:05,540 with very weak side lobes. 901 00:56:05,540 --> 00:56:08,590 OK, so some examples then. 902 00:56:08,590 --> 00:56:13,200 So let's imagine first of all, our object 903 00:56:13,200 --> 00:56:16,170 is our intensity at the input plane 904 00:56:16,170 --> 00:56:21,360 is this rectangle with the stripy pattern. 905 00:56:21,360 --> 00:56:24,330 And so we're going to then look at what 906 00:56:24,330 --> 00:56:31,050 happens if we image that with some mask that we put in here. 907 00:56:31,050 --> 00:56:33,150 And what should we put in there, then? 908 00:56:33,150 --> 00:56:38,530 Well, the first one let's put in something like this. 909 00:56:38,530 --> 00:56:40,290 So now this is going to let through-- 910 00:56:40,290 --> 00:56:42,690 this is a low pass filter, isn't it? 911 00:56:42,690 --> 00:56:45,870 Right, so it's a very small low pass filter as well. 912 00:56:45,870 --> 00:56:50,220 So it only lets through the very slowly varying parts 913 00:56:50,220 --> 00:56:52,570 of that object. 914 00:56:52,570 --> 00:56:57,090 But it stops all the fast variations of the object. 915 00:56:57,090 --> 00:56:58,930 Right, so if you look at the object, 916 00:56:58,930 --> 00:57:02,130 you'll see that it's got this stripy pattern, which is 917 00:57:02,130 --> 00:57:04,800 actually a very fine structure. 918 00:57:04,800 --> 00:57:07,530 So these are actually really being 919 00:57:07,530 --> 00:57:13,250 calculated using Mat lab, not by me, by George. 920 00:57:13,250 --> 00:57:17,150 And so this is what you'd see in the output plane of that. 921 00:57:17,150 --> 00:57:18,110 So there you are. 922 00:57:18,110 --> 00:57:21,260 You can see that this stripy pattern 923 00:57:21,260 --> 00:57:23,720 has disappeared virtually. 924 00:57:23,720 --> 00:57:26,960 We've lost some sharpness at the edge 925 00:57:26,960 --> 00:57:32,630 though because we've also lost the high spatial frequencies, 926 00:57:32,630 --> 00:57:34,880 of course, that defined the edge. 927 00:57:34,880 --> 00:57:39,680 And a sharp cut off at the edge will include quite a lot 928 00:57:39,680 --> 00:57:42,090 of high spatial frequencies. 929 00:57:42,090 --> 00:57:47,830 But anyway, so that's a good example of spatial filtering. 930 00:57:47,830 --> 00:57:50,880 And then the next ones we're going 931 00:57:50,880 --> 00:57:54,070 to look at something which has got a bit more structure. 932 00:57:54,070 --> 00:57:59,500 And so this is MIT written as a binary mask. 933 00:57:59,500 --> 00:58:02,210 And so this is our original object. 934 00:58:02,210 --> 00:58:05,800 And we're going to image that in an optical system 935 00:58:05,800 --> 00:58:09,230 and look at what the image looks like in different cases. 936 00:58:09,230 --> 00:58:12,650 So the first one is quite straightforward. 937 00:58:12,650 --> 00:58:18,820 We've got now a circular aperture. 938 00:58:18,820 --> 00:58:21,770 So this is what the image looks like in this case. 939 00:58:21,770 --> 00:58:29,780 You see that there's already a bit of artifacts in that image. 940 00:58:29,780 --> 00:58:33,850 It's no longer-- you can still read it as MIT, 941 00:58:33,850 --> 00:58:36,963 but you see that inside the letters, 942 00:58:36,963 --> 00:58:38,630 you'll see there's some structure there, 943 00:58:38,630 --> 00:58:40,810 which is not really present. 944 00:58:40,810 --> 00:58:44,020 So this is some sort of edge ringing, I guess from the fact 945 00:58:44,020 --> 00:58:47,680 that this aperture has got this sharp cut off 946 00:58:47,680 --> 00:58:52,180 at some particular spatial frequency. 947 00:58:52,180 --> 00:58:55,420 So if you made it so that it tailed off gradually, 948 00:58:55,420 --> 00:58:58,930 you could probably get rid of some of that-- 949 00:58:58,930 --> 00:59:01,300 those fringes there. 950 00:59:01,300 --> 00:59:03,580 But OK, so that's an aperture. 951 00:59:03,580 --> 00:59:07,550 So we've got rid of some high spatial frequencies. 952 00:59:07,550 --> 00:59:13,900 Let's look what happens if we reduce the spatial frequencies 953 00:59:13,900 --> 00:59:15,520 even more. 954 00:59:15,520 --> 00:59:18,290 So this is making the aperture smaller. 955 00:59:18,290 --> 00:59:23,070 And now you can see that the resolution has degraded. 956 00:59:23,070 --> 00:59:27,010 It looks rather blurry now. 957 00:59:27,010 --> 00:59:29,020 You can you can't really see whether it 958 00:59:29,020 --> 00:59:33,670 says MIT of HIT or maybe HIY. 959 00:59:33,670 --> 00:59:36,930 It's all getting a bit vague, isn't it? 960 00:59:36,930 --> 00:59:41,430 And also you can see that it gets a bit sort of blurry 961 00:59:41,430 --> 00:59:43,060 around the edges. 962 00:59:43,060 --> 00:59:45,118 Yeah, sorry if we make that-- 963 00:59:45,118 --> 00:59:45,910 AUDIENCE: Question? 964 00:59:45,910 --> 00:59:48,770 One thing to point out here which is kind of obvious, 965 00:59:48,770 --> 00:59:51,820 but it's worth pointing is that if you do this also, 966 00:59:51,820 --> 00:59:54,220 the average brightness of the image 967 00:59:54,220 --> 00:59:57,010 would decrease as you stop down the aperture. 968 00:59:57,010 --> 00:59:58,420 But in this case, they cheated. 969 00:59:58,420 --> 01:00:00,580 They normalized the intensity to the full level 970 01:00:00,580 --> 01:00:02,290 so it is visible in the projector. 971 01:00:02,290 --> 01:00:05,590 But in a real system, it would also become less bright. 972 01:00:05,590 --> 01:00:08,290 PROFESSOR: Right, OK so I didn't notice that because I 973 01:00:08,290 --> 01:00:09,570 didn't do the calculations. 974 01:00:09,570 --> 01:00:10,570 But yeah, OK. 975 01:00:10,570 --> 01:00:14,235 So of course, if this aperture gets smaller and smaller, 976 01:00:14,235 --> 01:00:15,610 the amount of light going through 977 01:00:15,610 --> 01:00:17,320 gets smaller or smaller. 978 01:00:17,320 --> 01:00:21,250 And so in the end, you'd end up with black. 979 01:00:21,250 --> 01:00:26,390 Right, so anyway we're now going to make this even smaller. 980 01:00:26,390 --> 01:00:29,020 So what would you expect? 981 01:00:29,020 --> 01:00:31,830 Well, you'd expect that again the thing 982 01:00:31,830 --> 01:00:34,070 to become more blurry looking. 983 01:00:34,070 --> 01:00:36,730 And in fact, it really does become. 984 01:00:36,730 --> 01:00:40,240 Just before we carry on, just take this fully in. 985 01:00:40,240 --> 01:00:44,860 You notice that there are four legs here, four straight lines. 986 01:00:44,860 --> 01:00:46,630 And these two are closer together 987 01:00:46,630 --> 01:00:49,960 than these other spacings, right? 988 01:00:49,960 --> 01:00:55,740 So as we decrease the size of the pupil, 989 01:00:55,740 --> 01:00:57,540 now you see what's happened. 990 01:00:57,540 --> 01:00:58,600 This is quite amazing. 991 01:00:58,600 --> 01:01:00,030 I was quite surprised at this. 992 01:01:00,030 --> 01:01:03,130 The four legs have become three legs. 993 01:01:03,130 --> 01:01:07,470 So I guess these two central legs are now not resolved. 994 01:01:07,470 --> 01:01:10,770 They're below the resolution limit. 995 01:01:10,770 --> 01:01:14,400 And you'd have no idea of course what that said now. 996 01:01:14,400 --> 01:01:16,860 It looks something really blurred. 997 01:01:16,860 --> 01:01:19,590 So we're not getting enough information obviously there 998 01:01:19,590 --> 01:01:27,350 to be able to see the image properly. 999 01:01:27,350 --> 01:01:32,520 OK, then let's try doing some other funny things. 1000 01:01:32,520 --> 01:01:36,870 So this one is a high pass filter. 1001 01:01:36,870 --> 01:01:39,400 Now, so what we're doing here is we're 1002 01:01:39,400 --> 01:01:42,190 blocking out this central part and letting 1003 01:01:42,190 --> 01:01:45,510 through all the other filters, all the other frequencies. 1004 01:01:45,510 --> 01:01:47,140 Now there's one other thing that we 1005 01:01:47,140 --> 01:01:51,520 ought to mention here is that this, of course, is done-- 1006 01:01:51,520 --> 01:01:54,790 these calculations are done using FFTs, right? 1007 01:01:54,790 --> 01:02:00,280 So we're actually working on a field of a certain size 1008 01:02:00,280 --> 01:02:04,210 when we calculate the FFTs. 1009 01:02:04,210 --> 01:02:06,570 The fact that this-- 1010 01:02:06,570 --> 01:02:11,980 as you know FFTs do like a sort of repeating, don't they? 1011 01:02:11,980 --> 01:02:16,600 Right, so this is not really quite the same as this being 1012 01:02:16,600 --> 01:02:19,180 going out to infinity in spatial frequency. 1013 01:02:19,180 --> 01:02:22,750 So I think that the result in the image you might get 1014 01:02:22,750 --> 01:02:28,000 could depend a little on the properties of the zero padding 1015 01:02:28,000 --> 01:02:30,640 and things like that in this case. 1016 01:02:30,640 --> 01:02:33,880 But anyway, this is the image in this case. 1017 01:02:33,880 --> 01:02:38,930 So you'll notice that we've got the high spatial frequencies. 1018 01:02:38,930 --> 01:02:42,550 So we actually get some sort of enhancement of the edges. 1019 01:02:42,550 --> 01:02:46,870 You see the edges come out really very crisply now. 1020 01:02:46,870 --> 01:02:49,510 So you get this enhancement of the edges. 1021 01:02:49,510 --> 01:02:52,870 It's like an edge enhancement filter. 1022 01:02:52,870 --> 01:02:59,360 But on top of that, there's a rather strange sort 1023 01:02:59,360 --> 01:03:01,670 of glowing patterns, which I'm not really quite 1024 01:03:01,670 --> 01:03:04,280 sure what they refer to at all. 1025 01:03:04,280 --> 01:03:06,020 But you could-- I guess-- think of them 1026 01:03:06,020 --> 01:03:08,330 as being some sort of artifact. 1027 01:03:08,330 --> 01:03:11,300 It looks a lot better on this screen than that one. 1028 01:03:11,300 --> 01:03:12,361 Interesting. 1029 01:03:15,310 --> 01:03:19,000 But if we do another of these high pass filtering 1030 01:03:19,000 --> 01:03:23,200 but with the central block even bigger, 1031 01:03:23,200 --> 01:03:27,790 this effect of edge enhancement shows really nicely now. 1032 01:03:27,790 --> 01:03:30,190 So all we're now seeing is the edges. 1033 01:03:30,190 --> 01:03:35,080 And the bits inside the letters are not showing up or outside 1034 01:03:35,080 --> 01:03:37,040 the letters, so neither. 1035 01:03:37,040 --> 01:03:39,370 I can't remember what the object looked lie now. 1036 01:03:39,370 --> 01:03:41,350 It's interesting. 1037 01:03:41,350 --> 01:03:44,080 The object, whether it was black writing 1038 01:03:44,080 --> 01:03:45,940 on a white background or white writing 1039 01:03:45,940 --> 01:03:48,070 on the black background, probably this 1040 01:03:48,070 --> 01:03:49,952 would look very similar. 1041 01:03:49,952 --> 01:03:51,610 It wouldn't make a lot of difference-- 1042 01:03:51,610 --> 01:03:52,810 I don't think. 1043 01:03:52,810 --> 01:03:58,930 OK, and so all these examples have been circularly symmetric 1044 01:03:58,930 --> 01:04:00,080 type patterns. 1045 01:04:00,080 --> 01:04:02,080 But you can also, of course, put in things which 1046 01:04:02,080 --> 01:04:04,180 are not certainly symmetric. 1047 01:04:04,180 --> 01:04:10,310 So this is throwing out these frequencies here and here. 1048 01:04:10,310 --> 01:04:11,900 So we haven't done anything here. 1049 01:04:11,900 --> 01:04:14,900 So we're going to have a high resolution in that direction. 1050 01:04:14,900 --> 01:04:17,800 But in this direction here, we've reduced the resolution. 1051 01:04:17,800 --> 01:04:23,190 So we'd expect it to be blurred in the vertical direction. 1052 01:04:23,190 --> 01:04:28,900 And of course, just to prove that we can do the opposite, 1053 01:04:28,900 --> 01:04:32,090 and this one now is going to be blurred 1054 01:04:32,090 --> 01:04:33,590 in the horizontal direction. 1055 01:04:33,590 --> 01:04:36,410 So you see here again, you're almost 1056 01:04:36,410 --> 01:04:41,740 losing the resolution of those four legs. 1057 01:04:44,460 --> 01:04:52,230 OK, and then probably one of the most important applications 1058 01:04:52,230 --> 01:04:57,220 of pupil filters is in phase imaging. 1059 01:04:57,220 --> 01:05:02,760 And so this leads on to a Zernike face contrast, 1060 01:05:02,760 --> 01:05:08,000 which Zernike got the Nobel Prize for. 1061 01:05:08,000 --> 01:05:09,170 It's quite interesting. 1062 01:05:09,170 --> 01:05:12,120 It was a long time after he invented it though, I believe. 1063 01:05:12,120 --> 01:05:14,550 He invented it sometime in the '30s. 1064 01:05:14,550 --> 01:05:18,755 And it was commercialized-- 1065 01:05:18,755 --> 01:05:20,130 actually he seems to have managed 1066 01:05:20,130 --> 01:05:25,200 to get Zeiss and [? Leicher ?] and everyone to make it. 1067 01:05:25,200 --> 01:05:30,780 So he must have sold his patent to lots of people. 1068 01:05:30,780 --> 01:05:33,930 But anyway, eventually it became very popular. 1069 01:05:33,930 --> 01:05:36,930 But it didn't get it didn't get the Nobel Prize until the '50s 1070 01:05:36,930 --> 01:05:38,130 some time. 1071 01:05:38,130 --> 01:05:43,640 And very useful for a long time, I 1072 01:05:43,640 --> 01:05:48,090 think eventually it got to a certain degree 1073 01:05:48,090 --> 01:05:53,240 anyway, outclassed by some later techniques, 1074 01:05:53,240 --> 01:05:56,540 but still used a fair bit in the lab. 1075 01:05:56,540 --> 01:05:59,690 He has one advantage of being quite cheap as well. 1076 01:05:59,690 --> 01:06:04,220 Anyway, what you're trying to do with [? phase-phase ?] contrast 1077 01:06:04,220 --> 01:06:06,130 is image of phase objects. 1078 01:06:06,130 --> 01:06:10,550 And this is an example of what a phase object might look like. 1079 01:06:10,550 --> 01:06:16,910 Imagine, for example, you've got some glass slide-- 1080 01:06:16,910 --> 01:06:20,540 well, one thing which would approximate quite well to this 1081 01:06:20,540 --> 01:06:23,690 is if you had a biological cell sitting 1082 01:06:23,690 --> 01:06:25,720 on top of a glass slide. 1083 01:06:25,720 --> 01:06:29,390 Right, so this shows that something like that. 1084 01:06:29,390 --> 01:06:32,520 But actually what we're thinking-- 1085 01:06:32,520 --> 01:06:34,070 well, if it was a cell, of course, 1086 01:06:34,070 --> 01:06:35,780 the refractive index of this would be 1087 01:06:35,780 --> 01:06:37,850 slightly different from this. 1088 01:06:37,850 --> 01:06:44,450 But that's really not an important difference. 1089 01:06:44,450 --> 01:06:46,600 So this is looking on the top of it. 1090 01:06:46,600 --> 01:06:49,150 So here we've got this object, which 1091 01:06:49,150 --> 01:06:51,340 has got a certain thickness. 1092 01:06:51,340 --> 01:06:54,160 And what we'd like to be able to do 1093 01:06:54,160 --> 01:06:57,850 is to produce an image which actually shows up 1094 01:06:57,850 --> 01:06:59,830 the shape of this object, right, which 1095 01:06:59,830 --> 01:07:02,450 might be a cell or something. 1096 01:07:02,450 --> 01:07:06,110 And so this phase object-- 1097 01:07:06,110 --> 01:07:10,010 we can think of the light propagating through this 1098 01:07:10,010 --> 01:07:12,770 and producing a phase shift then. 1099 01:07:12,770 --> 01:07:16,910 So the phase shift is given by this expression. 1100 01:07:16,910 --> 01:07:21,680 So n here is the refractive index of this cell. 1101 01:07:21,680 --> 01:07:25,850 We have to have n minus 1, of course, because the cell is 1102 01:07:25,850 --> 01:07:27,260 displacing air. 1103 01:07:27,260 --> 01:07:31,130 So the cell wasn't there, there would be air there. 1104 01:07:31,130 --> 01:07:35,420 So we have to subtract away the air that would have been there. 1105 01:07:35,420 --> 01:07:39,570 And so the phase is 2 pi times n minus 1 times 1106 01:07:39,570 --> 01:07:43,520 the thickness, which is a function of x or of x of y 1107 01:07:43,520 --> 01:07:47,340 in this case, divided by the wavelength. 1108 01:07:47,340 --> 01:07:53,240 And so if we illuminate this with some certain illumination 1109 01:07:53,240 --> 01:07:57,890 field this, then what we'll get on the output side 1110 01:07:57,890 --> 01:08:00,430 is just this times this. 1111 01:08:00,430 --> 01:08:04,340 There's no-- each of the i phi, our object 1112 01:08:04,340 --> 01:08:08,930 just behaves here as a pure phase object. 1113 01:08:08,930 --> 01:08:11,570 There's no modulus change, only a phase change. 1114 01:08:15,640 --> 01:08:19,080 Yeah, so useful for imaging biological objects 1115 01:08:19,080 --> 01:08:21,640 such as cells, but not only that. 1116 01:08:21,640 --> 01:08:27,560 It's got applications in material science, and so on. 1117 01:08:27,560 --> 01:08:33,529 And yeah, so if we looked at the-- 1118 01:08:33,529 --> 01:08:38,149 image this illumination is just unity. 1119 01:08:38,149 --> 01:08:41,689 And we look at the modulus squared 1120 01:08:41,689 --> 01:08:44,479 of this object function. 1121 01:08:44,479 --> 01:08:47,210 You see, because it's just only a phase object, 1122 01:08:47,210 --> 01:08:49,310 when you look at the modulus square, 1123 01:08:49,310 --> 01:08:51,060 you'll see nothing at all. 1124 01:08:51,060 --> 01:08:54,290 So if you look at this in an ordinary microscope, 1125 01:08:54,290 --> 01:08:55,760 you wouldn't see anything. 1126 01:08:55,760 --> 01:08:59,930 You'd just see completely uniformly white. 1127 01:08:59,930 --> 01:09:02,720 And so you wouldn't see the shape of this thing 1128 01:09:02,720 --> 01:09:04,890 that we're trying to see. 1129 01:09:04,890 --> 01:09:07,680 So the question is, how can we go about that? 1130 01:09:07,680 --> 01:09:10,189 What can we do to solve that? 1131 01:09:10,189 --> 01:09:14,600 So yeah, what we're now going to look at 1132 01:09:14,600 --> 01:09:18,130 is back to this MIT pattern. 1133 01:09:18,130 --> 01:09:20,250 And we're going to think of an MIT 1134 01:09:20,250 --> 01:09:24,540 pattern, which is now not a binary object but a phase 1135 01:09:24,540 --> 01:09:25,500 object. 1136 01:09:25,500 --> 01:09:27,210 So imagine that this is our object. 1137 01:09:29,760 --> 01:09:32,779 So this represents phase. 1138 01:09:32,779 --> 01:09:40,360 So red represents a phase change of 0.1 of a radian. 1139 01:09:44,310 --> 01:09:49,590 And this one shows the modulus of this object, which we're 1140 01:09:49,590 --> 01:09:53,160 assuming is completely uniform. 1141 01:09:53,160 --> 01:09:57,850 And so it's a pure phase object. 1142 01:09:57,850 --> 01:10:00,510 And if we looked at this in our ordinary optical imaging 1143 01:10:00,510 --> 01:10:05,430 system, all we'd see is the modular square of this thing. 1144 01:10:05,430 --> 01:10:09,280 So we'd see basically nothing. 1145 01:10:09,280 --> 01:10:14,180 So the question is what can we do to try and improve that. 1146 01:10:14,180 --> 01:10:17,080 Well, Zernike came up with the answer. 1147 01:10:17,080 --> 01:10:20,270 And actually, this is not quite the way he does it. 1148 01:10:20,270 --> 01:10:22,270 We're not going to really do the way he does it. 1149 01:10:22,270 --> 01:10:26,260 But the principle is here anyway. 1150 01:10:26,260 --> 01:10:31,360 What you do is you introduce a pupil mask, which has also 1151 01:10:31,360 --> 01:10:34,000 got a phase objects in it. 1152 01:10:34,000 --> 01:10:38,320 It's got basically a small region 1153 01:10:38,320 --> 01:10:43,090 at the center, i.e. where the low spatial frequencies are, 1154 01:10:43,090 --> 01:10:46,450 which we change the relative phase 1155 01:10:46,450 --> 01:10:50,290 of those spectral components relative to the other spatial 1156 01:10:50,290 --> 01:10:54,760 frequencies by pi over 2. 1157 01:10:54,760 --> 01:10:58,330 And so we put that in our in our system. 1158 01:11:05,990 --> 01:11:08,120 And what we find, as it says there, 1159 01:11:08,120 --> 01:11:13,070 is we can actually get contrast of the phase. 1160 01:11:13,070 --> 01:11:16,080 We actually see the phase now. 1161 01:11:16,080 --> 01:11:18,030 And this will explain why that works. 1162 01:11:18,030 --> 01:11:19,230 So this is what we're doing. 1163 01:11:19,230 --> 01:11:22,440 We're putting in our phase MIT thing here. 1164 01:11:22,440 --> 01:11:26,090 And we're putting in this pi over 2 phase mask here. 1165 01:11:31,900 --> 01:11:34,740 So that's what it looks like. 1166 01:11:34,740 --> 01:11:37,530 It's some bit of dielectric, which 1167 01:11:37,530 --> 01:11:39,600 has got this central region which 1168 01:11:39,600 --> 01:11:43,200 changes the phase by pi over 2. 1169 01:11:43,200 --> 01:11:46,680 Ideally, what we want to do, as it says here, 1170 01:11:46,680 --> 01:11:48,930 is for this to be very small. 1171 01:11:48,930 --> 01:11:50,610 What we're trying to do is to use 1172 01:11:50,610 --> 01:11:54,600 it to change the phase of just the DC component, 1173 01:11:54,600 --> 01:11:59,400 the unscattered light component relative to all the rest. 1174 01:11:59,400 --> 01:12:02,220 But of course, you can never do that because it's 1175 01:12:02,220 --> 01:12:06,840 got to have some definite real size. 1176 01:12:06,840 --> 01:12:11,070 So whatever size it is, there might 1177 01:12:11,070 --> 01:12:15,270 be some low spatial frequencies of the object, which 1178 01:12:15,270 --> 01:12:17,105 are going to be-- 1179 01:12:17,105 --> 01:12:18,480 the phase is going to be changing 1180 01:12:18,480 --> 01:12:22,320 in the same way as the DC term. 1181 01:12:22,320 --> 01:12:25,870 So let's look at the mass of how that works. 1182 01:12:25,870 --> 01:12:30,240 So what we do is we assume our phase object is what 1183 01:12:30,240 --> 01:12:33,080 we call a weak phase object. 1184 01:12:33,080 --> 01:12:37,250 It's got the form e to the i phi. 1185 01:12:37,250 --> 01:12:42,720 But we think of that as being approximately one plus i phi. 1186 01:12:42,720 --> 01:12:44,990 And we neglect all the other higher order 1187 01:12:44,990 --> 01:12:53,660 terms in the Haus's series expansion for the exponential. 1188 01:12:53,660 --> 01:12:58,670 And you can see then that in an ordinary system, what we'd see 1189 01:12:58,670 --> 01:13:02,810 is you can think of it-- 1190 01:13:02,810 --> 01:13:05,630 keeping all the terms, the modular square of this 1191 01:13:05,630 --> 01:13:07,660 is obviously one. 1192 01:13:07,660 --> 01:13:10,130 And this also, the modular square 1193 01:13:10,130 --> 01:13:14,210 of this to the first order of small quantities 1194 01:13:14,210 --> 01:13:15,610 is also going to be one. 1195 01:13:15,610 --> 01:13:19,550 Of course, if you square this, you'll get phi squared terms. 1196 01:13:19,550 --> 01:13:21,260 But actually, there are also really 1197 01:13:21,260 --> 01:13:25,980 some extra phi squared terms in this series, which actually, 1198 01:13:25,980 --> 01:13:28,740 if you do it properly, all cancel out. 1199 01:13:28,740 --> 01:13:29,930 So you still must get one. 1200 01:13:35,360 --> 01:13:39,440 OK, so what we're now going to say, this is our object. 1201 01:13:39,440 --> 01:13:44,210 But now, this object is going to go through this mask here, 1202 01:13:44,210 --> 01:13:48,200 which is going to change the relative phase of this relative 1203 01:13:48,200 --> 01:13:49,770 to this. 1204 01:13:49,770 --> 01:13:54,920 And so if we change the one to i, 1205 01:13:54,920 --> 01:13:58,580 we're not going to get i plus i phi. 1206 01:13:58,580 --> 01:14:01,520 And therefore now, the output that we get 1207 01:14:01,520 --> 01:14:02,780 is going to be given by-- 1208 01:14:05,720 --> 01:14:08,520 this should be into i, intensity actually. 1209 01:14:08,520 --> 01:14:12,740 Yeah, the intensity that you see is the modular square 1210 01:14:12,740 --> 01:14:17,840 of this, which is 1 plus phi squared, which to small-- 1211 01:14:17,840 --> 01:14:21,100 Because phi is small, we can expand that 1212 01:14:21,100 --> 01:14:24,560 to first two terms of the series. 1213 01:14:24,560 --> 01:14:26,900 And we get one plus two phi. 1214 01:14:26,900 --> 01:14:31,340 Right, so now we're getting an image where the phase is 1215 01:14:31,340 --> 01:14:34,730 visible to the order of this-- 1216 01:14:34,730 --> 01:14:39,200 order of the weak phase object phi. 1217 01:14:39,200 --> 01:14:42,230 In this case here, we weren't seeing anything 1218 01:14:42,230 --> 01:14:43,660 to the order of phi. 1219 01:14:43,660 --> 01:14:46,340 The lowest things that were there 1220 01:14:46,340 --> 01:14:49,280 would be of the order of phi squared, which 1221 01:14:49,280 --> 01:14:54,050 is going to be negligible if phi is going to be small enough. 1222 01:14:54,050 --> 01:14:55,050 Right, so there we are. 1223 01:14:55,050 --> 01:14:58,010 So by putting this in, we've managed 1224 01:14:58,010 --> 01:15:02,030 to make this phase object visible. 1225 01:15:02,030 --> 01:15:04,910 So this is the demonstration of that. 1226 01:15:04,910 --> 01:15:09,830 So this is our looking out at our phase MIT. 1227 01:15:09,830 --> 01:15:14,010 And this is the size of our phase mask. 1228 01:15:14,010 --> 01:15:16,490 So it's quite big here actually. 1229 01:15:16,490 --> 01:15:25,970 And so I guess, it's changing the phase of the central DC 1230 01:15:25,970 --> 01:15:26,810 term. 1231 01:15:26,810 --> 01:15:30,770 But it's also changing the phase of some low spatial frequency 1232 01:15:30,770 --> 01:15:35,210 terms of the MIT, which I guess accounts 1233 01:15:35,210 --> 01:15:39,770 for some sort of imperfections in this image. 1234 01:15:39,770 --> 01:15:41,960 And I think a lot of the ripples-- 1235 01:15:41,960 --> 01:15:44,870 again, are coming from the fact that this 1236 01:15:44,870 --> 01:15:48,290 has got a sharp edge to it. 1237 01:15:48,290 --> 01:15:50,120 Probably if you smoothed this, you 1238 01:15:50,120 --> 01:15:53,870 could make it so that you didn't get such a strong ringing 1239 01:15:53,870 --> 01:15:54,740 in this thing here. 1240 01:15:58,480 --> 01:16:01,510 But this is the next example. 1241 01:16:01,510 --> 01:16:04,150 We've now made this a lot smaller. 1242 01:16:04,150 --> 01:16:08,800 So it's now changing the phase of the DC term. 1243 01:16:08,800 --> 01:16:10,870 But it seems that the object itself probably 1244 01:16:10,870 --> 01:16:14,830 has got very, very weak components in that region. 1245 01:16:14,830 --> 01:16:19,790 And you end up then with this really wonderful image. 1246 01:16:19,790 --> 01:16:28,450 So I guess that really shows the power of the Zernike phase 1247 01:16:28,450 --> 01:16:29,370 contrast method. 1248 01:16:36,030 --> 01:16:37,240 That's the end. 1249 01:16:37,240 --> 01:16:39,530 How are we going? 1250 01:16:39,530 --> 01:16:41,040 Good, we're up to time. 1251 01:16:41,040 --> 01:16:42,910 I've been fast. 1252 01:16:42,910 --> 01:16:45,348 Right, any questions anywhere? 1253 01:16:45,348 --> 01:16:47,390 Anyone want to look at these nice pictures again? 1254 01:16:54,788 --> 01:16:56,330 AUDIENCE: We had a question actually. 1255 01:16:56,330 --> 01:16:57,840 PROFESSOR: Yes good. 1256 01:16:57,840 --> 01:17:00,920 AUDIENCE: It seemed like the last analysis you're doing, 1257 01:17:00,920 --> 01:17:02,780 phi was much, much less than 1. 1258 01:17:02,780 --> 01:17:05,820 But then you're saying that phi was pi over 2. 1259 01:17:05,820 --> 01:17:06,570 PROFESSOR: No, no. 1260 01:17:06,570 --> 01:17:07,720 Two different phi's. 1261 01:17:11,027 --> 01:17:12,110 How do I get back in this? 1262 01:17:14,720 --> 01:17:17,400 TA: You click the play button. 1263 01:17:17,400 --> 01:17:18,962 PROFESSOR: The play button. 1264 01:17:18,962 --> 01:17:20,220 Ah, the play button. 1265 01:17:20,220 --> 01:17:22,470 It's right there. 1266 01:17:22,470 --> 01:17:24,530 Yeah, OK. 1267 01:17:24,530 --> 01:17:26,390 So two different phi's. 1268 01:17:26,390 --> 01:17:32,240 The object is the one that has to be weak. 1269 01:17:32,240 --> 01:17:38,880 So here we said the object has got this phi. 1270 01:17:42,030 --> 01:17:44,790 So we got e to the i phi as the object. 1271 01:17:44,790 --> 01:17:47,820 And this phi has to be small. 1272 01:17:47,820 --> 01:17:52,270 And somewhere it said-- 1273 01:17:52,270 --> 01:17:55,700 maybe it was after. 1274 01:17:55,700 --> 01:17:58,696 The phase MIT objects-- 1275 01:17:58,696 --> 01:17:59,760 here we are. 1276 01:17:59,760 --> 01:18:02,780 The phase MIT objects-- 1277 01:18:02,780 --> 01:18:05,940 the phrase change is 0.1 of a radian. 1278 01:18:05,940 --> 01:18:09,820 So 0.1 of a radian we're taking as being small. 1279 01:18:09,820 --> 01:18:15,310 But so red here represents 0.1 of a radian. 1280 01:18:15,310 --> 01:18:20,580 Red there doesn't represent 0.1 of a radian. 1281 01:18:20,580 --> 01:18:22,660 So this is our mask. 1282 01:18:22,660 --> 01:18:25,230 So this has got a phase change of pi by 2. 1283 01:18:25,230 --> 01:18:30,700 So this is much bigger than 0.1 of a radian. 1284 01:18:30,700 --> 01:18:31,720 OK? 1285 01:18:31,720 --> 01:18:34,900 Yeah, so that's-- it's the object that has to be a weak 1286 01:18:34,900 --> 01:18:36,320 phase object, not the mask. 1287 01:18:40,980 --> 01:18:45,000 TA: I think the principle still works for strong phase objects. 1288 01:18:45,000 --> 01:18:48,970 But the explanation is easier if you assume a weak phase object. 1289 01:18:48,970 --> 01:18:50,190 PROFESSOR: Yeah. 1290 01:18:50,190 --> 01:18:53,640 I think I agree with that. 1291 01:18:53,640 --> 01:18:56,910 Yeah, but actually for a strong phase object, 1292 01:18:56,910 --> 01:19:00,610 you do also get an image in your ordinary imaging system. 1293 01:19:00,610 --> 01:19:05,400 But it's very complicated to actually relate it 1294 01:19:05,400 --> 01:19:06,360 to the object. 1295 01:19:06,360 --> 01:19:08,040 It's a bit like frequency modulation. 1296 01:19:08,040 --> 01:19:10,590 TA: Or another example of a strong phrase object 1297 01:19:10,590 --> 01:19:13,420 is the waves in a swimming pool. 1298 01:19:13,420 --> 01:19:16,140 They create an image at the bottom which 1299 01:19:16,140 --> 01:19:18,660 looks like very bright lines. 1300 01:19:18,660 --> 01:19:20,580 In a way these lines are an image 1301 01:19:20,580 --> 01:19:24,720 of the index of reflection of the water. 1302 01:19:24,720 --> 01:19:26,430 And these lines are known as caustics. 1303 01:19:26,430 --> 01:19:29,490 But it is very difficult to interpret them and relate them 1304 01:19:29,490 --> 01:19:32,360 to the actual shape of the water. 1305 01:19:32,360 --> 01:19:34,110 PROFESSOR: Yeah. 1306 01:19:34,110 --> 01:19:38,220 OK, any more questions? 1307 01:19:38,220 --> 01:19:41,320 Not many from this end, today, either. 1308 01:19:41,320 --> 01:19:45,030 I usually rely on [INAUDIBLE] to come up with some questions. 1309 01:19:45,030 --> 01:19:47,560 You know, at FOM, he was in FOM, and he 1310 01:19:47,560 --> 01:19:50,110 asked the very first question of the whole conference. 1311 01:19:50,110 --> 01:19:55,233 TA: Oh, that is quite the distinction. 1312 01:19:55,233 --> 01:19:56,650 To the plenary speaker, I imagine? 1313 01:19:56,650 --> 01:19:58,317 PROFESSOR: Yeah, to the plenary speaker. 1314 01:20:02,470 --> 01:20:03,520 All very clear? 1315 01:20:03,520 --> 01:20:06,970 I think these demonstrations are really 1316 01:20:06,970 --> 01:20:11,470 great because I think that you get a very nice feel for what 1317 01:20:11,470 --> 01:20:13,580 happens. 1318 01:20:13,580 --> 01:20:15,100 TA: We have the quiz coming up. 1319 01:20:15,100 --> 01:20:17,627 But we have enough time. 1320 01:20:17,627 --> 01:20:19,335 Maybe we can introduce temporal coherence 1321 01:20:19,335 --> 01:20:21,060 at the beginning of the next lecture 1322 01:20:21,060 --> 01:20:23,225 or should we just stop here? 1323 01:20:23,225 --> 01:20:24,100 PROFESSOR: Up to you. 1324 01:20:26,680 --> 01:20:30,460 TA: If anybody has no questions at all, we have quiz coming up. 1325 01:20:30,460 --> 01:20:33,470 This is your chance to ask questions, I suppose. 1326 01:20:33,470 --> 01:20:34,910 PROFESSOR: So when's the quiz? 1327 01:20:34,910 --> 01:20:35,565 TA: On Monday. 1328 01:20:35,565 --> 01:20:36,440 PROFESSOR: Oh, right. 1329 01:20:36,440 --> 01:20:36,940 OK. 1330 01:20:39,530 --> 01:20:41,530 TA: That's why he could not answer the question. 1331 01:20:41,530 --> 01:20:43,125 He was expecting a phone call. 1332 01:20:43,125 --> 01:20:49,200 AUDIENCE: When you have a phase grading, 1333 01:20:49,200 --> 01:20:54,792 normally you try to solve for a system that has the 4f system, 1334 01:20:54,792 --> 01:20:56,250 you normally play different tricks. 1335 01:20:56,250 --> 01:21:03,690 One of them is you try to expand part of the grading in Fourier 1336 01:21:03,690 --> 01:21:04,730 series. 1337 01:21:04,730 --> 01:21:06,420 And after that, you somehow managed 1338 01:21:06,420 --> 01:21:09,240 to decompose that into something easy, 1339 01:21:09,240 --> 01:21:15,600 or you actually change the e to the exponent to something 1340 01:21:15,600 --> 01:21:19,890 easier to handle in terms of you look at the phase-- 1341 01:21:19,890 --> 01:21:21,060 and you look at it. 1342 01:21:21,060 --> 01:21:24,660 And if it is like pi over two or it's pi, 1343 01:21:24,660 --> 01:21:28,080 you change it to something different 1344 01:21:28,080 --> 01:21:32,790 that it's easier to handle, any advice on how to solve those? 1345 01:21:32,790 --> 01:21:34,560 I found those particularly tricky 1346 01:21:34,560 --> 01:21:39,635 to actually get a grasp of how to solve things. 1347 01:21:39,635 --> 01:21:42,130 PROFESSOR: I'm not quite sure you're saying. 1348 01:21:42,130 --> 01:21:43,300 I mean if you've got-- 1349 01:21:43,300 --> 01:21:46,090 I mean, you start off by saying you've got 1350 01:21:46,090 --> 01:21:49,270 some sort of gradient object. 1351 01:21:49,270 --> 01:21:51,310 And then if you illuminate this, you'll 1352 01:21:51,310 --> 01:21:56,040 get a series of diffraction orders. 1353 01:21:56,040 --> 01:22:00,720 And of course, you know, the relative strengths of these 1354 01:22:00,720 --> 01:22:04,510 might be complex to account for the phase changes. 1355 01:22:04,510 --> 01:22:11,610 And so what you're saying is that by changing the phrase 1356 01:22:11,610 --> 01:22:16,560 or whatever of these components, you can improve the image. 1357 01:22:16,560 --> 01:22:18,570 But is that what you're saying? 1358 01:22:18,570 --> 01:22:19,980 Something like that? 1359 01:22:19,980 --> 01:22:21,510 AUDIENCE: Yeah, my question was more 1360 01:22:21,510 --> 01:22:25,440 in the order of like how to handle the math? 1361 01:22:25,440 --> 01:22:28,950 Normally if you get something in terms of Fourier series, 1362 01:22:28,950 --> 01:22:32,020 you get a bunch of terms. 1363 01:22:32,020 --> 01:22:37,583 And it's kind of like it ends up being pretty messy. 1364 01:22:37,583 --> 01:22:38,250 PROFESSOR: Yeah. 1365 01:22:38,250 --> 01:22:41,590 But you know, this is like going through the mask. 1366 01:22:41,590 --> 01:22:48,440 So if we've got some mask which has got some phase variation. 1367 01:22:48,440 --> 01:22:50,850 So the relative strength of these gradient orders, 1368 01:22:50,850 --> 01:22:56,550 it will just be changed by the phase of the phase mask. 1369 01:22:56,550 --> 01:22:57,790 So the forward problem-- 1370 01:22:57,790 --> 01:23:00,580 I've got the feeling that maybe you 1371 01:23:00,580 --> 01:23:03,820 were trying to solve the inverse problem, you know. 1372 01:23:07,030 --> 01:23:09,490 If you know what the image looks like, 1373 01:23:09,490 --> 01:23:11,650 how do you know how to improve the image 1374 01:23:11,650 --> 01:23:13,010 or something like that? 1375 01:23:13,010 --> 01:23:18,100 But normally, of course, if you knew what the object was, 1376 01:23:18,100 --> 01:23:21,402 then it probably wouldn't be a lot of point in imaging it. 1377 01:23:21,402 --> 01:23:24,280 But anyway, to calculate this, this is all you have to do. 1378 01:23:24,280 --> 01:23:28,540 Just multiply the strength of these gradient components 1379 01:23:28,540 --> 01:23:34,120 by the strength of your mask, both in amplitude and phase. 1380 01:23:34,120 --> 01:23:36,730 So it's very straightforward really. 1381 01:23:36,730 --> 01:23:41,620 TA: And I don't think there's a way out of the Fourier series 1382 01:23:41,620 --> 01:23:42,790 expansion. 1383 01:23:42,790 --> 01:23:45,400 Like if you really want to solve this kind of a problem 1384 01:23:45,400 --> 01:23:50,257 with a phase gradient, you must compute the Fourier series 1385 01:23:50,257 --> 01:23:51,340 expansion of the gradient. 1386 01:23:54,560 --> 01:23:56,760 Unless, for example, if the phase is weak, 1387 01:23:56,760 --> 01:23:59,150 then you can simplify it as we just saw in the example. 1388 01:23:59,150 --> 01:24:01,430 But otherwise, yeah, you have to go 1389 01:24:01,430 --> 01:24:04,960 through the pain of the Fourier series expansion. 1390 01:24:04,960 --> 01:24:06,080 OK, thanks. 1391 01:24:06,080 --> 01:24:07,480 PROFESSOR: Yeah. 1392 01:24:07,480 --> 01:24:09,360 But you know, there are some times of course, 1393 01:24:09,360 --> 01:24:11,527 for lots of other objects, you can just look them up 1394 01:24:11,527 --> 01:24:13,710 in the tables of Fourier transforms. 1395 01:24:13,710 --> 01:24:17,580 And even the Fourier series, you can sometimes 1396 01:24:17,580 --> 01:24:19,500 get from that, of course. 1397 01:24:19,500 --> 01:24:22,830 I mean, for example, you know, these square wave 1398 01:24:22,830 --> 01:24:24,810 gradients like this. 1399 01:24:24,810 --> 01:24:32,040 You can think of this as being this convolved with this. 1400 01:24:32,040 --> 01:24:33,740 So the Fourier transform of this is 1401 01:24:33,740 --> 01:24:36,140 going to be the Fourier transform of this, which 1402 01:24:36,140 --> 01:24:40,640 is a sinc times the Fourier transform of this, 1403 01:24:40,640 --> 01:24:43,380 which is another one of these. 1404 01:24:43,380 --> 01:24:48,590 Right, so you end up with basically an envelope, 1405 01:24:48,590 --> 01:24:53,800 which is this and then some components within here. 1406 01:24:53,800 --> 01:24:57,320 And the height of these gives you 1407 01:24:57,320 --> 01:25:01,440 the strengths of these different gradient components. 1408 01:25:01,440 --> 01:25:03,360 So I think there was a sketch like that in one 1409 01:25:03,360 --> 01:25:07,350 of the earlier slides showing how you can think of that. 1410 01:25:07,350 --> 01:25:09,840 So these are the strengths of the Fourier 1411 01:25:09,840 --> 01:25:12,990 series for this object. 1412 01:25:12,990 --> 01:25:14,490 They're just given by the product 1413 01:25:14,490 --> 01:25:20,418 of a sinc and an appropriately scaled sampling function. 1414 01:25:20,418 --> 01:25:22,710 TA: I think you need to push the paper a little bit up. 1415 01:25:22,710 --> 01:25:23,880 PROFESSOR: Sorry. 1416 01:25:23,880 --> 01:25:25,207 Yeah. 1417 01:25:25,207 --> 01:25:25,790 Can I do that? 1418 01:25:25,790 --> 01:25:26,500 Oh, yeah. 1419 01:25:26,500 --> 01:25:27,000 Sorry. 1420 01:25:27,000 --> 01:25:29,550 There you can see what I'm saying. 1421 01:25:29,550 --> 01:25:34,920 Yeah, so some sometimes you can find 1422 01:25:34,920 --> 01:25:39,798 shortcuts to be able to deal with some 1423 01:25:39,798 --> 01:25:41,340 of these various areas without having 1424 01:25:41,340 --> 01:25:44,520 to go through that the hard work of doing what you did when 1425 01:25:44,520 --> 01:25:45,990 you were a first year student. 1426 01:25:49,960 --> 01:25:53,620 TA: There was a homework like that actually where it was not 1427 01:25:53,620 --> 01:25:55,720 quite phrased like this-- but yeah, 1428 01:25:55,720 --> 01:26:00,270 I had you guys compute the Fourier 1429 01:26:00,270 --> 01:26:03,200 series in this clever way. 1430 01:26:07,580 --> 01:26:09,410 PROFESSOR: OK no more questions. 1431 01:26:09,410 --> 01:26:11,270 Yes, [INAUDIBLE] got one. 1432 01:26:11,270 --> 01:26:14,066 Yeah? 1433 01:26:14,066 --> 01:26:20,890 AUDIENCE: How did the idea of numerical aperture come about? 1434 01:26:20,890 --> 01:26:25,000 And why is it defined as a sine of an angle? 1435 01:26:25,000 --> 01:26:27,150 PROFESSOR: Rather the tangent of an angle. 1436 01:26:27,150 --> 01:26:28,440 Yeah, good point. 1437 01:26:28,440 --> 01:26:31,780 Well, I guess it's because George in one of the earlier 1438 01:26:31,780 --> 01:26:36,130 lectures pointed out that according to Snell's law, 1439 01:26:36,130 --> 01:26:38,800 you've got n sine alpha is the invariant. 1440 01:26:38,800 --> 01:26:44,840 So as rays goes through a series of interfaces, 1441 01:26:44,840 --> 01:26:46,445 then that will be an invariant. 1442 01:26:46,445 --> 01:26:48,110 It won't change. 1443 01:26:48,110 --> 01:26:52,150 So I think that was the justification for it. 1444 01:26:52,150 --> 01:26:55,440 But yeah, you've hit something which is quite-- 1445 01:26:55,440 --> 01:27:00,340 you know, the fact that sine theta isn't tan theta 1446 01:27:00,340 --> 01:27:05,540 is basically why paraxial optics eventually breaks down. 1447 01:27:05,540 --> 01:27:10,790 And so a lot of the stuff that we've presented 1448 01:27:10,790 --> 01:27:14,680 has been based on assuming that these angles are all small. 1449 01:27:14,680 --> 01:27:17,330 And a lot of these things just fail 1450 01:27:17,330 --> 01:27:23,090 if you can't assume that these angles are small. 1451 01:27:23,090 --> 01:27:25,310 TA: For example, aberrations are another type 1452 01:27:25,310 --> 01:27:31,090 of not shift invariant behavior with the exception 1453 01:27:31,090 --> 01:27:31,990 of spherical. 1454 01:27:31,990 --> 01:27:35,290 But the aberrations are not shift invariant 1455 01:27:35,290 --> 01:27:38,800 so our approximations fail. 1456 01:27:38,800 --> 01:27:41,100 For example, astigmatism if you recall, 1457 01:27:41,100 --> 01:27:45,810 astigmatism occurs if you have a plane wave entering the lens-- 1458 01:27:45,810 --> 01:27:48,240 I'm sorry, I should've said coma. 1459 01:27:48,240 --> 01:27:50,220 Coma occurs if you have a plane wave entering 1460 01:27:50,220 --> 01:27:52,050 the lens at an angle. 1461 01:27:52,050 --> 01:27:53,810 So of course, as you increase the angle, 1462 01:27:53,810 --> 01:27:54,900 the coma becomes worse. 1463 01:27:54,900 --> 01:28:02,050 So that is shift invariant 1464 01:28:02,050 --> 01:28:03,770 PROFESSOR: OK, do you want to stop or do 1465 01:28:03,770 --> 01:28:05,250 you want to say something? 1466 01:28:05,250 --> 01:28:07,310 TA: We could talk about the Talbot effect 1467 01:28:07,310 --> 01:28:10,430 that we skipped in the previous lecture. 1468 01:28:10,430 --> 01:28:14,288 I can have to do to find it. 1469 01:28:14,288 --> 01:28:16,580 PROFESSOR: I think that might not be interested in that 1470 01:28:16,580 --> 01:28:18,372 because I don't think my students have ever 1471 01:28:18,372 --> 01:28:21,926 come across the Talbot effect. 1472 01:28:21,926 --> 01:28:40,585 [INAUDIBLE] I was just going to say-- 1473 01:28:46,372 --> 01:28:47,580 [INAUDIBLE] that's all right. 1474 01:28:47,580 --> 01:28:48,810 I'll come and say it here. 1475 01:28:48,810 --> 01:28:52,690 I was just going to say that this Talbot effect. 1476 01:28:52,690 --> 01:28:54,720 He was actually a very long time ago. 1477 01:28:54,720 --> 01:28:57,150 I can't quite remember the history, 1478 01:28:57,150 --> 01:29:01,350 but it's a very old effect. 1479 01:29:01,350 --> 01:29:04,260 TA: And I think Talbot was also another professor of mechanics 1480 01:29:04,260 --> 01:29:09,180 who ended up doing contributions in optics if I'm not mistaken, 1481 01:29:09,180 --> 01:29:13,380 very similar to Maxwell and a bunch of other people. 1482 01:29:13,380 --> 01:29:14,300 No, no. 1483 01:29:14,300 --> 01:29:15,795 I went the other way around there. 1484 01:29:15,795 --> 01:29:18,420 I shouldn't be expected to make the contributions in mechanics. 1485 01:29:18,420 --> 01:29:23,360 [INAUDIBLE] OK, oh, I'm not showing anything. 1486 01:29:23,360 --> 01:29:24,183 I'm sorry. 1487 01:29:27,810 --> 01:29:32,310 So the Talbot effect is a phenomenon 1488 01:29:32,310 --> 01:29:37,320 that happens when we have a periodic pattern, for example 1489 01:29:37,320 --> 01:29:40,830 a gradient and you illuminate it with a plane wave. 1490 01:29:40,830 --> 01:29:45,670 And then what you do is you observe the intensity pattern 1491 01:29:45,670 --> 01:29:50,010 forming after the field diffracts, 1492 01:29:50,010 --> 01:29:52,890 after it has gone through the transparency. 1493 01:29:52,890 --> 01:29:55,350 So I don't when the movie will show possibly 1494 01:29:55,350 --> 01:29:58,020 because of something [INAUDIBLE] the projector, 1495 01:29:58,020 --> 01:29:59,412 it might not show. 1496 01:29:59,412 --> 01:30:00,870 But if you look carefully, you will 1497 01:30:00,870 --> 01:30:05,700 see that as the field diffracts, the original grate and pattern 1498 01:30:05,700 --> 01:30:08,250 becomes visible at certain distances. 1499 01:30:08,250 --> 01:30:09,270 Then it disappears. 1500 01:30:09,270 --> 01:30:11,050 Then it becomes visible again. 1501 01:30:11,050 --> 01:30:14,420 It will play it in the case of diagonal gradient 1502 01:30:14,420 --> 01:30:16,920 and then I will play both of them again. 1503 01:30:16,920 --> 01:30:21,270 So you can see, it diffracts, then the sinusoid reappears 1504 01:30:21,270 --> 01:30:22,225 and so on. 1505 01:30:22,225 --> 01:30:23,850 In this case, it is actually very clear 1506 01:30:23,850 --> 01:30:27,480 when it will stop happening because as we said, 1507 01:30:27,480 --> 01:30:30,300 in the case of a gradient, in the field 1508 01:30:30,300 --> 01:30:34,440 after the gradient splits up into diffraction orders. 1509 01:30:34,440 --> 01:30:37,540 Now this gradient is finite, so as you can see the diffraction 1510 01:30:37,540 --> 01:30:39,140 orders separate. 1511 01:30:39,140 --> 01:30:40,710 And by the time it reached here, you 1512 01:30:40,710 --> 01:30:44,220 can see that you have basically three rectangles that have 1513 01:30:44,220 --> 01:30:46,520 split up out of the gradient. 1514 01:30:46,520 --> 01:30:49,860 So the Talbot effect will stop happening when the rectangles 1515 01:30:49,860 --> 01:30:51,030 completely separate. 1516 01:30:51,030 --> 01:30:52,560 It hasn't quite stopped yet. 1517 01:30:52,560 --> 01:30:54,240 But it can only happen that region 1518 01:30:54,240 --> 01:30:58,250 of overlap between gradients. 1519 01:30:58,250 --> 01:30:59,473 OK, so the next-- 1520 01:30:59,473 --> 01:31:01,390 let me play this once again so you can see it. 1521 01:31:04,760 --> 01:31:09,880 So again, look out for the repetition of the periodicity 1522 01:31:09,880 --> 01:31:11,320 of this pattern now. 1523 01:31:16,490 --> 01:31:18,190 And again for the next one. 1524 01:31:21,833 --> 01:31:23,750 I hope with this is showing up also in Boston. 1525 01:31:23,750 --> 01:31:27,960 Here you can see it very clearly on the projector. 1526 01:31:27,960 --> 01:31:31,570 OK, so yes? 1527 01:31:31,570 --> 01:31:35,820 OK, so the next slide is cross sections? 1528 01:31:35,820 --> 01:31:37,180 AUDIENCE: George? 1529 01:31:37,180 --> 01:31:39,910 Did you see that this effect occurs only 1530 01:31:39,910 --> 01:31:43,960 when there is an overlap between the diffraction orders of the-- 1531 01:31:43,960 --> 01:31:44,930 TA: Yes. 1532 01:31:44,930 --> 01:31:46,930 So if the gradient is infinite, then, of course, 1533 01:31:46,930 --> 01:31:48,243 it occurs forever. 1534 01:31:48,243 --> 01:31:50,410 But if the gradient is finite, eventually the orders 1535 01:31:50,410 --> 01:31:54,410 will separate and then it doesn't happen. 1536 01:31:54,410 --> 01:31:58,282 So this is cross sections of the intensity pattern 1537 01:31:58,282 --> 01:31:59,990 at different distances, which of course I 1538 01:31:59,990 --> 01:32:02,960 chose strategically because I know the formula 1539 01:32:02,960 --> 01:32:05,800 that this effect follows. 1540 01:32:05,800 --> 01:32:07,550 Of course, you don't have the formula yet. 1541 01:32:07,550 --> 01:32:10,220 But anyway, this is what you see. 1542 01:32:10,220 --> 01:32:13,640 Now what is really interesting and you may not 1543 01:32:13,640 --> 01:32:16,040 have seen in the previous one, but there's also 1544 01:32:16,040 --> 01:32:20,790 a plane where you see a periodic pattern at twice the frequency. 1545 01:32:20,790 --> 01:32:23,880 So this like the second harmonic of the original gradient. 1546 01:32:23,880 --> 01:32:26,650 So the question is why does that happen. 1547 01:32:26,650 --> 01:32:29,630 Well, there is the physical explanation, which is probably 1548 01:32:29,630 --> 01:32:33,410 the most interesting. 1549 01:32:33,410 --> 01:32:35,450 And that has to do with the following. 1550 01:32:35,450 --> 01:32:39,320 So this really goes to the heart of diffraction, 1551 01:32:39,320 --> 01:32:41,690 why really diffraction happens and why 1552 01:32:41,690 --> 01:32:43,950 we see all of this phenomenon. 1553 01:32:43,950 --> 01:32:47,550 So imagine that you have a plane wave going 1554 01:32:47,550 --> 01:32:51,390 into a sinusoid amplitude gradient. 1555 01:32:51,390 --> 01:32:53,860 And of course, in this case you only get three diffraction 1556 01:32:53,860 --> 01:32:56,900 orders, as we discussed, the plus 1, the minus 1, 1557 01:32:56,900 --> 01:32:58,860 and the [INAUDIBLE] 1558 01:32:58,860 --> 01:33:01,680 And these diffraction orders, each one of them 1559 01:33:01,680 --> 01:33:03,300 is really a plane wave. 1560 01:33:03,300 --> 01:33:04,520 So you have now-- 1561 01:33:04,520 --> 01:33:07,790 you started with one coming in, but after the gradient, 1562 01:33:07,790 --> 01:33:11,930 you have three plane waves propagating out. 1563 01:33:11,930 --> 01:33:15,270 And imagine that you pick any point of the gradient really, 1564 01:33:15,270 --> 01:33:16,230 it doesn't matter. 1565 01:33:16,230 --> 01:33:18,090 But let's pick this one. 1566 01:33:18,090 --> 01:33:22,820 And we'll draw a sphere over radius z. 1567 01:33:22,820 --> 01:33:26,190 we'll draw a sphere centered at this point. 1568 01:33:26,190 --> 01:33:28,170 Now if you compare the three plane waves 1569 01:33:28,170 --> 01:33:33,450 and you take rays really, centered 1570 01:33:33,450 --> 01:33:35,760 at the center of this sphere, and if you 1571 01:33:35,760 --> 01:33:41,130 compare what happens to these plane waves by the time 1572 01:33:41,130 --> 01:33:43,440 that each of the cell of this sphere, 1573 01:33:43,440 --> 01:33:47,340 then you realize that they all have the same phase delay 1574 01:33:47,340 --> 01:33:50,910 because they only propagated the same distance from the center 1575 01:33:50,910 --> 01:33:52,130 to the edge of this sphere. 1576 01:33:52,130 --> 01:33:53,980 They all propagated by the same amount. 1577 01:33:53,980 --> 01:33:55,920 So they have the same phase delay. 1578 01:33:55,920 --> 01:33:59,370 However, when we observe the field 1579 01:33:59,370 --> 01:34:01,740 in all of this cross-section that I showed before 1580 01:34:01,740 --> 01:34:04,290 in my movie and in the calculation, 1581 01:34:04,290 --> 01:34:07,140 I did not really observe the field on a sphere. 1582 01:34:07,140 --> 01:34:09,400 I observed it on a plane. 1583 01:34:09,400 --> 01:34:14,430 So if a center this plane at a distance z from the axis, 1584 01:34:14,430 --> 01:34:18,150 then the central order that propagates a axis, 1585 01:34:18,150 --> 01:34:20,430 it sustains itself in phase delay. 1586 01:34:20,430 --> 01:34:24,450 But you can see that the plus one and minus one order, 1587 01:34:24,450 --> 01:34:27,660 they propagate the longer distance from this sphere 1588 01:34:27,660 --> 01:34:28,920 to that gradient. 1589 01:34:28,920 --> 01:34:30,760 And you can calculate this distance. 1590 01:34:30,760 --> 01:34:34,090 It is the difference between the z itself 1591 01:34:34,090 --> 01:34:38,060 and an expansion that is that you get from its Pythagoras's 1592 01:34:38,060 --> 01:34:39,150 theorem. 1593 01:34:39,150 --> 01:34:43,890 So this expression is basically the hypotenuse of this triangle 1594 01:34:43,890 --> 01:34:47,780 all the way out to the plane. 1595 01:34:47,780 --> 01:34:50,090 And if you do after that the paraxial approximation, 1596 01:34:50,090 --> 01:34:51,882 then you find that is given by a quadratic. 1597 01:34:51,882 --> 01:34:54,500 That is this is our familiar quadratic term that 1598 01:34:54,500 --> 01:34:58,510 appears in the Fresnel kernel and in all of these expressions 1599 01:34:58,510 --> 01:35:01,040 of Fresnel diffraction. 1600 01:35:01,040 --> 01:35:06,930 So however, now what am I really observing? 1601 01:35:06,930 --> 01:35:10,520 Well, in this plane, I observe the interference pattern. 1602 01:35:10,520 --> 01:35:12,740 You can think of it as a Fresnel diffraction 1603 01:35:12,740 --> 01:35:15,140 pattern from the gradient or you can also 1604 01:35:15,140 --> 01:35:17,930 think of it as an interference pattern between these three 1605 01:35:17,930 --> 01:35:19,550 plane waves. 1606 01:35:19,550 --> 01:35:22,940 It's like an interferometer, right? 1607 01:35:22,940 --> 01:35:25,460 I mean, physically as I think Professor separately 1608 01:35:25,460 --> 01:35:28,520 mentioned and I also mentioned at some point or another, 1609 01:35:28,520 --> 01:35:31,198 diffraction is really no different than interference. 1610 01:35:31,198 --> 01:35:32,990 It is just that in the case of diffraction, 1611 01:35:32,990 --> 01:35:35,990 you have many, many, many waves interfering produced 1612 01:35:35,990 --> 01:35:39,210 by the Huygen's wavelengths originating 1613 01:35:39,210 --> 01:35:41,120 at your original object. 1614 01:35:41,120 --> 01:35:42,830 But in this case, instead of thinking 1615 01:35:42,830 --> 01:35:44,630 of all these Huygens wavelengths, 1616 01:35:44,630 --> 01:35:47,030 I'm better off just thinking about it 1617 01:35:47,030 --> 01:35:49,760 as plane waves interfering because the physics 1618 01:35:49,760 --> 01:35:53,330 of the gradient tell me that the diffracting field is really 1619 01:35:53,330 --> 01:35:56,210 composed of three plane waves. 1620 01:35:56,210 --> 01:36:01,460 So now there's a phase delay between this guy and this guy. 1621 01:36:01,460 --> 01:36:04,900 Actually, this guy and this guy, the plus 1 and minus 1 1622 01:36:04,900 --> 01:36:06,800 are always in phase. 1623 01:36:06,800 --> 01:36:10,790 But they can have a difference in phase delay 1624 01:36:10,790 --> 01:36:14,690 between the themselves and the 0-th order. 1625 01:36:14,690 --> 01:36:18,870 And of course, if that phase delay happens to be 2 pi, 1626 01:36:18,870 --> 01:36:20,640 then there's no phase delay. 1627 01:36:20,640 --> 01:36:25,470 And when did these things start having no phase delay? 1628 01:36:25,470 --> 01:36:27,720 Well, at the gradient, right? 1629 01:36:27,720 --> 01:36:30,840 So if I let this thing propagate enough, whatever 1630 01:36:30,840 --> 01:36:33,810 parameter I can vary here is z. 1631 01:36:33,810 --> 01:36:37,530 If I let z grow so that this phase delay becomes 1632 01:36:37,530 --> 01:36:40,860 equal to 2 pi, then I reproduced my original gradient 1633 01:36:40,860 --> 01:36:45,105 because of course, there was no phase delay here. 1634 01:36:47,870 --> 01:36:49,960 So when the phase delay becomes 2 pi, 1635 01:36:49,960 --> 01:36:53,420 then again I will see the same pattern. 1636 01:36:53,420 --> 01:37:04,480 It is not as obvious why you get the subharmonic, why 1637 01:37:04,480 --> 01:37:07,360 you get the second harmonic that I discussed before. 1638 01:37:07,360 --> 01:37:08,890 But there is a little bit fortunate 1639 01:37:08,890 --> 01:37:15,300 that the lectures got shuffled in order because by now, you 1640 01:37:15,300 --> 01:37:17,290 remember from one of the examples of Professor 1641 01:37:17,290 --> 01:37:21,520 Sheppard showed, when you actually get rid of the DC 1642 01:37:21,520 --> 01:37:23,440 component, you can get the second harmonic 1643 01:37:23,440 --> 01:37:26,330 to pop out all of a sudden. 1644 01:37:26,330 --> 01:37:29,200 A very similar thing happens if you get the pi phase shift. 1645 01:37:29,200 --> 01:37:32,268 This pi phase shift can produce a very distinct sub harmonic. 1646 01:37:32,268 --> 01:37:33,226 So this is that reason. 1647 01:37:38,860 --> 01:37:42,393 Now the rest, I will actually not go through. 1648 01:37:42,393 --> 01:37:43,810 But there's a mathematical-- well, 1649 01:37:43,810 --> 01:37:45,435 I can go through very quickly, I guess. 1650 01:37:45,435 --> 01:37:47,730 There's a mathematical way to derive it. 1651 01:37:47,730 --> 01:37:49,510 And it is also described in Goodman. 1652 01:37:49,510 --> 01:37:51,780 And I also did it in the slides. 1653 01:37:51,780 --> 01:37:55,620 So if you start with a grate with a field 1654 01:37:55,620 --> 01:37:57,880 after the gradient-- 1655 01:37:57,880 --> 01:38:00,130 of course, you can express it as a diffraction orders. 1656 01:38:00,130 --> 01:38:03,040 We did this before. 1657 01:38:03,040 --> 01:38:06,880 You can put it in a form the field after the gradient. 1658 01:38:06,880 --> 01:38:10,340 We've done this before ask in the context of the 4f system. 1659 01:38:10,340 --> 01:38:12,130 But mathematically, it is really the same. 1660 01:38:12,130 --> 01:38:14,800 Each one of these exponentials in the field 1661 01:38:14,800 --> 01:38:17,000 will produce a delta function. 1662 01:38:17,000 --> 01:38:18,760 Now, the question is what happens 1663 01:38:18,760 --> 01:38:23,060 after the field propagates by a certain distance. 1664 01:38:23,060 --> 01:38:28,870 So in order to do that, let me show the Fresnel diffraction 1665 01:38:28,870 --> 01:38:29,680 again. 1666 01:38:29,680 --> 01:38:31,290 These we're familiar with. 1667 01:38:31,290 --> 01:38:34,180 We said many times that Fresnel diffraction 1668 01:38:34,180 --> 01:38:36,400 is expressed as a convolution that 1669 01:38:36,400 --> 01:38:38,170 has to do with the Huygen's wavelengths 1670 01:38:38,170 --> 01:38:40,780 propagating and producing spherical waves that 1671 01:38:40,780 --> 01:38:42,100 again interfere. 1672 01:38:42,100 --> 01:38:45,520 So the interference is expressed as a convolution integral. 1673 01:38:45,520 --> 01:38:49,600 But because of the properties of the Fourier transform, 1674 01:38:49,600 --> 01:38:52,750 if I take a Fourier transform of the output field, 1675 01:38:52,750 --> 01:38:56,270 then it will be expressed as a product of the Fourier 1676 01:38:56,270 --> 01:39:00,790 transform of the input field times the Fourier transform 1677 01:39:00,790 --> 01:39:04,660 of this Fresnel kernel. 1678 01:39:04,660 --> 01:39:08,020 And I really am not going to prove this. 1679 01:39:08,020 --> 01:39:12,490 But you can look up in Goodman's book in the table of Fourier 1680 01:39:12,490 --> 01:39:13,390 integrals. 1681 01:39:13,390 --> 01:39:15,820 You can look up the Fourier transform 1682 01:39:15,820 --> 01:39:18,070 of an expression like this one. 1683 01:39:18,070 --> 01:39:20,770 And you'll find that it looks very similar. 1684 01:39:20,770 --> 01:39:24,158 This is a quadratic phase delay 1685 01:39:24,158 --> 01:39:25,200 It's a Fourier transform. 1686 01:39:25,200 --> 01:39:28,240 It also looks like a quadratic phase delay. 1687 01:39:28,240 --> 01:39:30,350 But it has two key differences. 1688 01:39:30,350 --> 01:39:32,020 One is that it has a minus sign. 1689 01:39:32,020 --> 01:39:35,070 So it is actually quadratic phase advance if you wish. 1690 01:39:35,070 --> 01:39:37,450 And the second is, of course, the [? scalene ?] theorem 1691 01:39:37,450 --> 01:39:41,690 that will take up this factor of lambda z from the denominator 1692 01:39:41,690 --> 01:39:43,750 and will pop it up in the numerator. 1693 01:39:43,750 --> 01:39:47,440 And of course, the happy outcome of this scale operation 1694 01:39:47,440 --> 01:39:50,260 is that also the units are preserved. 1695 01:39:50,260 --> 01:39:52,414 This quantity has to be dimensionless. 1696 01:39:52,414 --> 01:39:53,122 So indeed, it is. 1697 01:39:53,122 --> 01:39:56,560 I have distance square in the numerator 1698 01:39:56,560 --> 01:39:58,240 as well as the denominator. 1699 01:39:58,240 --> 01:40:00,320 In this case it uses spatial frequency which 1700 01:40:00,320 --> 01:40:02,980 is inverse distance square. 1701 01:40:02,980 --> 01:40:05,500 By multiplying by this distance squared, 1702 01:40:05,500 --> 01:40:07,490 I get again a one dimensional quantity. 1703 01:40:07,490 --> 01:40:10,030 So the [? scalene ?] theorem works quite nicely. 1704 01:40:10,030 --> 01:40:15,570 And you can also think then of-- 1705 01:40:15,570 --> 01:40:17,490 we talked about transfer functions 1706 01:40:17,490 --> 01:40:20,003 in the context over the 4f system. 1707 01:40:20,003 --> 01:40:22,170 But really you can also think of a transfer function 1708 01:40:22,170 --> 01:40:24,360 in the case of Fresnel propagation 1709 01:40:24,360 --> 01:40:26,070 because it is shift invariant. 1710 01:40:26,070 --> 01:40:29,430 Clearly if you have free space, that is it truly 1711 01:40:29,430 --> 01:40:30,930 a shift invariant system. 1712 01:40:30,930 --> 01:40:33,060 No matter where you start in free space, 1713 01:40:33,060 --> 01:40:37,110 you would expect the field to shift by the same amount. 1714 01:40:37,110 --> 01:40:40,800 It would be really odd if anything else happened. 1715 01:40:40,800 --> 01:40:43,470 Unless you believe in space and [INAUDIBLE] I guess. 1716 01:40:43,470 --> 01:40:48,390 Only close to the Big Bang, this property was not observed. 1717 01:40:48,390 --> 01:40:51,720 But since billions of years have lapsed since the Big Bang, 1718 01:40:51,720 --> 01:40:53,082 space is isotropic. 1719 01:40:53,082 --> 01:40:54,540 So we don't have to worry about it. 1720 01:40:54,540 --> 01:40:58,110 So of course, the Fourier integral approaches. 1721 01:40:58,110 --> 01:41:00,540 And of course, then this quantity that we derive 1722 01:41:00,540 --> 01:41:03,060 is simply the transfer function of free space. 1723 01:41:03,060 --> 01:41:07,190 It is another quadratic phase delay. 1724 01:41:07,190 --> 01:41:11,120 OK, so we can actually apply this formulation now 1725 01:41:11,120 --> 01:41:13,230 to the mathematics of the Talbot effect. 1726 01:41:27,990 --> 01:41:31,780 All I have to do is multiply the Fourier transform 1727 01:41:31,780 --> 01:41:34,840 of the gradient itself. 1728 01:41:34,840 --> 01:41:37,650 All I have to do is multiply it with a Fourier transform 1729 01:41:37,650 --> 01:41:40,250 of the Fresnel propagation kernel that is 1730 01:41:40,250 --> 01:41:42,355 another quadratic phase delay. 1731 01:41:42,355 --> 01:41:44,230 And this multiple multiplication is very easy 1732 01:41:44,230 --> 01:41:46,300 to do because I'm multiplying the function 1733 01:41:46,300 --> 01:41:49,130 that I know with a bunch of delta functions. 1734 01:41:49,130 --> 01:41:54,400 So all that will remain really is the value 1735 01:41:54,400 --> 01:42:00,760 of this quadratic phase delay computed at the spots 1736 01:42:00,760 --> 01:42:04,340 where the delta functions occurred. 1737 01:42:04,340 --> 01:42:06,200 And this is what I'm doing the next slide. 1738 01:42:08,970 --> 01:42:12,720 This looks a little bit nasty but really 1739 01:42:12,720 --> 01:42:15,890 that's all you need to know. 1740 01:42:15,890 --> 01:42:19,190 It looks like an expression very similar to the gradient. 1741 01:42:19,190 --> 01:42:20,820 This is the DC component. 1742 01:42:20,820 --> 01:42:22,610 This is the plus one diffracted order. 1743 01:42:22,610 --> 01:42:24,980 This is a minus one diffracted order. 1744 01:42:24,980 --> 01:42:29,000 But it picked up this additional quadratic phase delay, 1745 01:42:29,000 --> 01:42:31,210 which is really it came out of the math now. 1746 01:42:31,210 --> 01:42:33,350 I used Fourier transforms and convolutions 1747 01:42:33,350 --> 01:42:34,500 and everything else. 1748 01:42:34,500 --> 01:42:43,070 But if you go back to my picture here with the sphere 1749 01:42:43,070 --> 01:42:45,530 and the phase delays, you can really 1750 01:42:45,530 --> 01:42:49,520 see that this is really the same expression. 1751 01:42:49,520 --> 01:42:51,400 It came out of the math in one case. 1752 01:42:51,400 --> 01:42:54,140 It really came out of a physical picture 1753 01:42:54,140 --> 01:42:56,390 of phase delay of [? planes ?] as they propagate. 1754 01:42:56,390 --> 01:42:57,390 But it's all consistent. 1755 01:42:57,390 --> 01:43:04,010 It is nearly all the same answer. 1756 01:43:04,010 --> 01:43:06,950 And after a little bit more pain, 1757 01:43:06,950 --> 01:43:16,080 which involves really taking one Fourier transform, 1758 01:43:16,080 --> 01:43:22,970 and then combining these things to form a sinusoid, 1759 01:43:22,970 --> 01:43:26,210 you'll get this expression for the field. 1760 01:43:26,210 --> 01:43:31,160 And now you can see what I was saying before, if you-- 1761 01:43:31,160 --> 01:43:32,830 this looks like a nasty phase delay 1762 01:43:32,830 --> 01:43:34,940 that got appended to the diffracted field. 1763 01:43:34,940 --> 01:43:36,740 This is really diffracted field now. 1764 01:43:36,740 --> 01:43:38,540 So this is the Fresnel diffraction pattern 1765 01:43:38,540 --> 01:43:39,770 from a gradient. 1766 01:43:39,770 --> 01:43:45,780 But if this quantity and exponent happens to be 2 pi, 1767 01:43:45,780 --> 01:43:47,180 then it is not there anymore. 1768 01:43:47,180 --> 01:43:49,160 So you recover your original gradient. 1769 01:43:52,790 --> 01:43:54,350 OK, that's the intensity. 1770 01:43:54,350 --> 01:43:56,760 And you can also see from this-- you can also 1771 01:43:56,760 --> 01:44:00,600 get the condition for the second harmonic generation. 1772 01:44:00,600 --> 01:44:02,580 Basically, if you kill this term, which 1773 01:44:02,580 --> 01:44:05,250 means that this quantity becomes pi over 2, 1774 01:44:05,250 --> 01:44:06,230 then this disappears. 1775 01:44:06,230 --> 01:44:08,730 So all you're left is the cosine squared, which is of course 1776 01:44:08,730 --> 01:44:10,090 the second harmonic. 1777 01:44:10,090 --> 01:44:16,300 So this is really delineated in the next slide. 1778 01:44:16,300 --> 01:44:18,460 If the propagation distance-- 1779 01:44:18,460 --> 01:44:20,830 if this quantity, this is also non-dimensional. 1780 01:44:20,830 --> 01:44:23,440 It is distance squares over distance squared. 1781 01:44:23,440 --> 01:44:26,890 If you propagate far enough so that this equals 1782 01:44:26,890 --> 01:44:30,100 an even integer, then you get the replica 1783 01:44:30,100 --> 01:44:31,690 of the original gradient. 1784 01:44:31,690 --> 01:44:33,970 This is basically-- this quantity 1785 01:44:33,970 --> 01:44:37,000 becomes a phase delay of 2 pi. 1786 01:44:37,000 --> 01:44:42,690 If you get it to become equal to [? n, ?] you call this distance 1787 01:44:42,690 --> 01:44:44,580 a Talbot play. 1788 01:44:44,580 --> 01:44:49,320 If you're get it to propagate so that it is an odd integer, 1789 01:44:49,320 --> 01:44:53,940 then it is pi that shifts this term by pi. 1790 01:44:53,940 --> 01:44:55,550 Basically, you flip the sign. 1791 01:44:55,550 --> 01:44:56,860 Now the gradient has shifted. 1792 01:44:56,860 --> 01:45:00,680 If you really do work it out, you still get the same sinusoid 1793 01:45:00,680 --> 01:45:03,430 but shifted by half a period. 1794 01:45:03,430 --> 01:45:07,560 And finally, if you completely eliminate the linear term 1795 01:45:07,560 --> 01:45:09,210 by choosing-- 1796 01:45:09,210 --> 01:45:14,790 this tends to be pi over 2 or 3 pi over 2 or 5 pi over 2, 1797 01:45:14,790 --> 01:45:18,120 and so on and so forth, then you can actually get-- 1798 01:45:18,120 --> 01:45:20,380 I forgot to put to close my bracket here, 1799 01:45:20,380 --> 01:45:24,060 but anyway, then you get the period doubled. 1800 01:45:24,060 --> 01:45:27,270 And you'll get the second harmonic. 1801 01:45:27,270 --> 01:45:30,070 OK, and this the schematic. 1802 01:45:30,070 --> 01:45:32,380 These are basically now repeating periodically 1803 01:45:32,380 --> 01:45:37,860 every time you add this propagation distance-- 1804 01:45:37,860 --> 01:45:41,250 I'm sorry, these are flipped here. 1805 01:45:41,250 --> 01:45:44,940 This should have been 2 lambda z over lambda squared, 2 lambda 1806 01:45:44,940 --> 01:45:46,210 z over lambda squared. 1807 01:45:46,210 --> 01:45:49,410 So basically every time you propagate by this quantity, 1808 01:45:49,410 --> 01:45:51,540 you go from a Talbot sub plane where 1809 01:45:51,540 --> 01:45:54,580 you have a second harmonic to a half period 1810 01:45:54,580 --> 01:45:58,760 shift, to another second harmonic to a replica. 1811 01:45:58,760 --> 01:46:00,000 And then it repeats. 1812 01:46:00,000 --> 01:46:02,790 So these are the Talbot plains. 1813 01:46:02,790 --> 01:46:06,280 Now we're out of time so I will say 1814 01:46:06,280 --> 01:46:07,870 something that's interesting for those 1815 01:46:07,870 --> 01:46:12,150 of you who have time to stay. 1816 01:46:12,150 --> 01:46:15,670 This-- as Professor Sheppard mentioned 1817 01:46:15,670 --> 01:46:18,860 has been noticed since a very long time ago. 1818 01:46:18,860 --> 01:46:21,950 But more recently, it became actually kind of a hot topic 1819 01:46:21,950 --> 01:46:24,730 over [INAUDIBLE] because people noticed 1820 01:46:24,730 --> 01:46:29,740 that it happens not only in sinusoidal gradients, 1821 01:46:29,740 --> 01:46:34,180 but it can also happen in every periodic pattern. 1822 01:46:34,180 --> 01:46:36,700 If, for example, you have a square gradient, 1823 01:46:36,700 --> 01:46:39,562 it would also have its own Talbot planes. 1824 01:46:39,562 --> 01:46:41,020 And then, not only that, but people 1825 01:46:41,020 --> 01:46:47,570 noticed that it can happen in some strange patterns that 1826 01:46:47,570 --> 01:46:50,560 are called was quasi-crystals. 1827 01:46:50,560 --> 01:46:53,260 So I don't know how many of you have taken solid state physics. 1828 01:46:53,260 --> 01:46:56,020 But in solid state physics they teach you 1829 01:46:56,020 --> 01:47:02,132 that there are certain patterns in nature that-- 1830 01:47:06,640 --> 01:47:09,258 there's often symmetries in nature. 1831 01:47:09,258 --> 01:47:10,800 For example, a very simple symmetries 1832 01:47:10,800 --> 01:47:12,370 is rectangular symmetry. 1833 01:47:12,370 --> 01:47:17,620 If you take this as a unit cell and then you repeat it, 1834 01:47:17,620 --> 01:47:20,500 you can keep filling out the space like this periodically. 1835 01:47:20,500 --> 01:47:23,130 Another example is a hexagon. 1836 01:47:23,130 --> 01:47:26,253 Let's see if I know how to do the hexagon. 1837 01:47:26,253 --> 01:47:27,420 I ended up doing an octagon. 1838 01:47:27,420 --> 01:47:28,128 Sorry about that. 1839 01:47:34,840 --> 01:47:37,220 That's a pentagon. 1840 01:47:37,220 --> 01:47:38,410 I need one more, OK. 1841 01:47:38,410 --> 01:47:39,460 I got my hexagon, OK. 1842 01:47:39,460 --> 01:47:45,390 That's nice because this does repeat. 1843 01:47:45,390 --> 01:47:48,970 OK, never mind anyway. 1844 01:47:53,440 --> 01:47:56,570 OK, that will repeat. 1845 01:47:56,570 --> 01:47:57,070 Good. 1846 01:48:00,500 --> 01:48:03,770 OK, so fortunately I did a pentagon also. 1847 01:48:08,042 --> 01:48:09,500 So the question is what will happen 1848 01:48:09,500 --> 01:48:13,350 if you start sticking pentagons next to each other. 1849 01:48:13,350 --> 01:48:16,760 Now this will be really impossible for me to do, 1850 01:48:16,760 --> 01:48:18,890 but you can imagine if you tried to stick 1851 01:48:18,890 --> 01:48:22,730 another pentagon, identical pentagon next to here, 1852 01:48:22,730 --> 01:48:27,860 then you will get something like this, I guess. 1853 01:48:27,860 --> 01:48:29,680 And then you can keep doing it. 1854 01:48:29,680 --> 01:48:33,650 But you noticed when I did it, then I got to stick a pentagon 1855 01:48:33,650 --> 01:48:35,990 but it is slightly rotated. 1856 01:48:35,990 --> 01:48:38,150 It is still is difficult for me to draw 1857 01:48:38,150 --> 01:48:41,570 but if you google quasi-crystals. 1858 01:48:46,522 --> 01:48:48,730 I don't know if it's the first website that comes up. 1859 01:48:48,730 --> 01:48:51,250 There's a website at Caltech, in the physics department 1860 01:48:51,250 --> 01:48:51,910 at Caltech. 1861 01:48:51,910 --> 01:48:54,310 And you will see some very nice pictures 1862 01:48:54,310 --> 01:48:57,670 of what happens if you stick pentagons next to each other. 1863 01:48:57,670 --> 01:48:59,860 That's called a quasi-crystal. 1864 01:48:59,860 --> 01:49:01,610 OK, now why are people interested in this? 1865 01:49:01,610 --> 01:49:04,450 Physicists are interested because in solid state 1866 01:49:04,450 --> 01:49:07,870 they teach you that if you have atoms arranged 1867 01:49:07,870 --> 01:49:13,180 in one of the patterns that produce symmetries, then 1868 01:49:13,180 --> 01:49:18,240 if you pass X-rays through this symmetric lattice, 1869 01:49:18,240 --> 01:49:20,250 you get a diffraction pattern, very 1870 01:49:20,250 --> 01:49:23,250 similar to a gradient that we saw here, 1871 01:49:23,250 --> 01:49:25,920 except that we're in 3D because the symmetries are 1872 01:49:25,920 --> 01:49:27,030 very simple in 2D. 1873 01:49:27,030 --> 01:49:28,620 I think in 2D you only get two-- 1874 01:49:28,620 --> 01:49:31,188 you get the rectangular and the hexagonal symmetry. 1875 01:49:31,188 --> 01:49:32,730 In 3D, you get many more symmetries-- 1876 01:49:32,730 --> 01:49:34,110 I forget, how many do you get? 1877 01:49:34,110 --> 01:49:38,680 I think it is 7 groups or 12 or anyway. 1878 01:49:38,680 --> 01:49:40,430 I should refresh my solid state physics. 1879 01:49:40,430 --> 01:49:44,650 But anyway, each one of these gives rise to-- 1880 01:49:44,650 --> 01:49:48,350 each symmetry gives rise to a distinct diffraction pattern. 1881 01:49:48,350 --> 01:49:49,870 And then crystallographers basically 1882 01:49:49,870 --> 01:49:51,610 bypass the next stage to a crystal 1883 01:49:51,610 --> 01:49:53,560 and observe the diffraction pattern-- 1884 01:49:53,560 --> 01:49:56,170 they can guess what was the original symmetry that 1885 01:49:56,170 --> 01:49:58,220 produced it. 1886 01:49:58,220 --> 01:50:01,670 We also know that non symmetric patterns 1887 01:50:01,670 --> 01:50:03,290 can produce diffraction. 1888 01:50:03,290 --> 01:50:06,680 And that is what won Crick and Watson the Nobel Prize 1889 01:50:06,680 --> 01:50:09,210 because they figured out that you have a helix, which 1890 01:50:09,210 --> 01:50:11,620 not quite symmetric, it can still produce a diffraction 1891 01:50:11,620 --> 01:50:12,320 pattern. 1892 01:50:12,320 --> 01:50:14,850 And Crick was a very clever mathematician who solved it. 1893 01:50:14,850 --> 01:50:17,240 And he won the Nobel Prize for it. 1894 01:50:17,240 --> 01:50:21,020 Anyway, recently people have observed that quasi-crystals 1895 01:50:21,020 --> 01:50:22,730 like this one-- if you take pentagons 1896 01:50:22,730 --> 01:50:24,500 and you stick them next to each other, 1897 01:50:24,500 --> 01:50:26,400 they also produce a diffraction pattern, 1898 01:50:26,400 --> 01:50:28,400 which looks actually like a bunch 1899 01:50:28,400 --> 01:50:33,907 of bright dots in a pentagonal kind of symmetry. 1900 01:50:33,907 --> 01:50:34,740 It looks very funny. 1901 01:50:34,740 --> 01:50:38,760 If you go to this Caltech website, you will see it. 1902 01:50:38,760 --> 01:50:41,520 And then people also noticed that this kind of thing 1903 01:50:41,520 --> 01:50:44,010 also produces Talbot patterns. 1904 01:50:44,010 --> 01:50:50,550 If you you diffract a field from a pentagonal quasi-crystal, 1905 01:50:50,550 --> 01:50:54,390 it will produce Talbot patterns as you'll go behind it. 1906 01:50:54,390 --> 01:50:56,340 And that caused quite a bit of stir. 1907 01:50:56,340 --> 01:51:00,600 Actually there's a lot of papers analyzing this and looking 1908 01:51:00,600 --> 01:51:01,860 into this interest. 1909 01:51:01,860 --> 01:51:05,875 Anyway, so I don't know if this has any use-- 1910 01:51:05,875 --> 01:51:07,250 it's kind of interesting physics. 1911 01:51:07,250 --> 01:51:11,020 But yeah, people spend some time on that. 1912 01:51:20,340 --> 01:51:20,840 Oh, yeah. 1913 01:51:30,690 --> 01:51:33,480 Actually, Talbot effect found a use recently 1914 01:51:33,480 --> 01:51:38,350 in a very surprising way in lithography. 1915 01:51:38,350 --> 01:51:43,100 It is a very easy way to reproduce a pattern lenslessly, 1916 01:51:43,100 --> 01:51:46,590 as long as the pattern is periodic. 1917 01:51:46,590 --> 01:51:49,940 And so for planar lithography is not 1918 01:51:49,940 --> 01:51:52,100 that interesting, but nowadays many 1919 01:51:52,100 --> 01:51:55,780 people who want to make photonic crystals which are periodic. 1920 01:51:55,780 --> 01:51:58,710 So in the photonic crystals, you want to start with the volume. 1921 01:51:58,710 --> 01:52:00,380 So this is a cross-section, and you 1922 01:52:00,380 --> 01:52:05,280 want to write patterns that look like this. 1923 01:52:05,280 --> 01:52:08,630 So for example, what you do is you take a polymer-- 1924 01:52:08,630 --> 01:52:09,710 OK, I take it back. 1925 01:52:09,710 --> 01:52:14,000 You take a monomer and if you could expose some [INAUDIBLE] 1926 01:52:14,000 --> 01:52:19,018 these regions and then you develop it, 1927 01:52:19,018 --> 01:52:20,810 then you would end up with a structure that 1928 01:52:20,810 --> 01:52:24,530 has low index in the unexposed regions and high index 1929 01:52:24,530 --> 01:52:26,030 in exposed regions. 1930 01:52:26,030 --> 01:52:28,090 That is a photonic crystal. 1931 01:52:28,090 --> 01:52:29,820 So the question is how could you do this. 1932 01:52:29,820 --> 01:52:31,720 Well, one way is with a Talbot effect 1933 01:52:31,720 --> 01:52:34,520 because a Talbot effect will actually focus the light very 1934 01:52:34,520 --> 01:52:40,157 sharply in the Talbot planes and then you'll get your exposure. 1935 01:52:40,157 --> 01:52:42,240 But everywhere else, the light is kind of diffuse. 1936 01:52:42,240 --> 01:52:44,270 So it does not expose very much. 1937 01:52:44,270 --> 01:52:47,660 So if you play with the threshold of the photoresist, 1938 01:52:47,660 --> 01:52:50,240 you can actually get the pattern to work like this. 1939 01:52:50,240 --> 01:52:52,360 So there's a colleague of mine at MIT, Ned Thomas. 1940 01:52:52,360 --> 01:52:54,830 I don't know if any of Ned's students are in the class. 1941 01:52:54,830 --> 01:52:57,200 But he came up with a very clever system 1942 01:52:57,200 --> 01:53:02,120 that he can expose and create photonic crystals this way. 1943 01:53:02,120 --> 01:53:04,460 And then interestingly, a student of mine 1944 01:53:04,460 --> 01:53:12,550 discovered that if you vary the duty cycle of this thing, 1945 01:53:12,550 --> 01:53:14,450 I may be disclosing-- 1946 01:53:14,450 --> 01:53:16,550 everybody here is MIT, right? 1947 01:53:16,550 --> 01:53:19,100 And those of you who are NUS are sort of bound 1948 01:53:19,100 --> 01:53:21,150 by a vow of secrecy or whatever. 1949 01:53:21,150 --> 01:53:22,990 But anyway, I may be disclosing something 1950 01:53:22,990 --> 01:53:24,710 that we have a patent pending for. 1951 01:53:24,710 --> 01:53:27,380 But anyway, so it is also well known-- 1952 01:53:27,380 --> 01:53:34,360 people have analyzed patterns with a duty cycle 1953 01:53:34,360 --> 01:53:37,313 that these different [INAUDIBLE] point have. 1954 01:53:37,313 --> 01:53:39,730 So for example, if you have a gradient that is like this-- 1955 01:53:43,130 --> 01:53:45,850 this also produces a Talbot effect. 1956 01:53:45,850 --> 01:53:49,470 So it will also produce replicas of that pattern. 1957 01:53:49,470 --> 01:53:52,315 So but my student discovered that if you vary the duty 1958 01:53:52,315 --> 01:53:54,440 cycle, that's really interesting and we don't fully 1959 01:53:54,440 --> 01:53:56,660 understand why but it is true. 1960 01:53:56,660 --> 01:53:59,780 I mean, it is physically correct-- we don't know how 1961 01:53:59,780 --> 01:54:02,460 to explain it very intuitively. 1962 01:54:02,460 --> 01:54:04,210 But mathematically, you find that you have 1963 01:54:04,210 --> 01:54:07,380 a pattern with a fixed period. 1964 01:54:07,380 --> 01:54:16,810 But the variable duty cycle, for example, like this, 1965 01:54:16,810 --> 01:54:19,120 it also for this is a Talbot effect. 1966 01:54:19,120 --> 01:54:24,370 Not forever, but over a finite region maybe about-- 1967 01:54:24,370 --> 01:54:27,030 I don't know, it depends but maybe about 10 or so Talbot 1968 01:54:27,030 --> 01:54:29,750 planes, it reproduces itself. 1969 01:54:29,750 --> 01:54:32,772 So it is also possible to make non-periodic patterns, 1970 01:54:32,772 --> 01:54:35,230 which as [? Colin ?] knows, we're very interested in making 1971 01:54:35,230 --> 01:54:36,907 non-periodic kind of-- 1972 01:54:36,907 --> 01:54:38,740 they're not for [? tiny ?] crystals anymore, 1973 01:54:38,740 --> 01:54:42,340 but they're not periodic strong index modulations. 1974 01:54:42,340 --> 01:54:46,820 That is possibly one way to do it. 1975 01:54:46,820 --> 01:54:49,080 AUDIENCE: George, it seems like it 1976 01:54:49,080 --> 01:54:53,140 seems like that was still a periodic pattern though. 1977 01:54:53,140 --> 01:54:56,318 TA: No, because the duty cycle varies. 1978 01:54:56,318 --> 01:54:58,610 We call it periodic because you must have the same duty 1979 01:54:58,610 --> 01:55:00,500 cycle in every period. 1980 01:55:00,500 --> 01:55:02,950 AUDIENCE: So even though it repeats every period 1981 01:55:02,950 --> 01:55:07,190 with a different duty cycle, that's not periodic. 1982 01:55:07,190 --> 01:55:10,345 TA: I don't know, would you call this function periodic? 1983 01:55:10,345 --> 01:55:12,470 AUDIENCE: I understand what you mean about the duty 1984 01:55:12,470 --> 01:55:15,155 cycle changing, but it's got a definite period to it 1985 01:55:15,155 --> 01:55:16,470 where features are-- 1986 01:55:16,470 --> 01:55:17,420 TA: Yeah. 1987 01:55:17,420 --> 01:55:21,710 Yeah, but it's non-periodic. 1988 01:55:21,710 --> 01:55:25,860 OK, what you're saying is a possible explanation 1989 01:55:25,860 --> 01:55:28,020 for why it has a Talbot effect. 1990 01:55:28,020 --> 01:55:30,330 It has a Talbot effect because it 1991 01:55:30,330 --> 01:55:34,370 has this fundamental period underlined the variable duty 1992 01:55:34,370 --> 01:55:35,070 cycle. 1993 01:55:35,070 --> 01:55:36,778 AUDIENCE: Right, I look at the duty cycle 1994 01:55:36,778 --> 01:55:39,720 as changing the orders that are coming out 1995 01:55:39,720 --> 01:55:41,160 for that particular region. 1996 01:55:45,150 --> 01:55:47,700 TA: Well, in a slowly varying approximation, perhaps. 1997 01:55:47,700 --> 01:55:49,110 Yes. 1998 01:55:49,110 --> 01:55:50,990 I mean, this thing-- 1999 01:55:50,990 --> 01:55:52,880 I've not looking at the diffraction pattern 2000 01:55:52,880 --> 01:55:53,880 of this thing in detail. 2001 01:55:53,880 --> 01:55:57,810 But yes, it would produce diffracted orders. 2002 01:55:57,810 --> 01:56:01,860 But it is not clear how you would compute their strength 2003 01:56:01,860 --> 01:56:03,360 from a Fourier series. 2004 01:56:03,360 --> 01:56:05,390 At least, I don't know how to do it. 2005 01:56:05,390 --> 01:56:09,800 There may be a way with somebody adiabatic approximation. 2006 01:56:09,800 --> 01:56:12,840 Because it is true that in our simulation, 2007 01:56:12,840 --> 01:56:15,890 the duty cycle was very slowly varied. 2008 01:56:15,890 --> 01:56:17,250 And I believe it is also true. 2009 01:56:17,250 --> 01:56:18,542 In fact, I think Will-- 2010 01:56:18,542 --> 01:56:20,750 my student who did this-- he also did this simulation 2011 01:56:20,750 --> 01:56:23,690 with a fast, varying duty cycle, and he found out 2012 01:56:23,690 --> 01:56:26,120 that the Talbot effect actually does not happen 2013 01:56:26,120 --> 01:56:28,280 if the duty cycle varies fast. 2014 01:56:28,280 --> 01:56:29,210 So, yeah. 2015 01:56:29,210 --> 01:56:34,130 There's certainly an adiabatic approximation happening here. 2016 01:56:34,130 --> 01:56:38,480 But I guess we haven't really worked out all the details. 2017 01:56:38,480 --> 01:56:40,910 But it is sort of interesting still. 2018 01:56:46,230 --> 01:56:51,180 Anyway, yeah, it's one of those things that-- 2019 01:56:51,180 --> 01:56:53,710 the reason I'm bringing it up is because the Talbot effect 2020 01:56:53,710 --> 01:56:56,290 doesn't look like something particularly useful. 2021 01:56:56,290 --> 01:56:58,660 It looks more like a curiosity. 2022 01:56:58,660 --> 01:57:00,830 But yeah, I guess sometimes even curiosities 2023 01:57:00,830 --> 01:57:03,220 can find applications, right?