1 00:00:00,090 --> 00:00:02,430 The following content is provided under a Creative 2 00:00:02,430 --> 00:00:03,820 Commons license. 3 00:00:03,820 --> 00:00:06,030 Your support will help MIT OpenCourseWare 4 00:00:06,030 --> 00:00:10,120 continue to offer high quality educational resources for free. 5 00:00:10,120 --> 00:00:12,690 To make a donation or to view additional materials 6 00:00:12,690 --> 00:00:16,620 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:16,620 --> 00:00:17,536 at ocw.mit.edu. 8 00:00:22,000 --> 00:00:23,830 GUEST SPEAKER: Good morning. 9 00:00:23,830 --> 00:00:25,480 Today we're going to start with a demo, 10 00:00:25,480 --> 00:00:28,120 and we're going to show experimentally 11 00:00:28,120 --> 00:00:29,710 some of the principles we've learned 12 00:00:29,710 --> 00:00:32,439 about 4F systems in the past classes, 13 00:00:32,439 --> 00:00:36,820 and actually we're going to see a comparison of an incoherent 14 00:00:36,820 --> 00:00:42,670 versus a coherent 4F system when imaging resolution target. 15 00:00:42,670 --> 00:00:45,470 So but since now we have all the tools to understand, 16 00:00:45,470 --> 00:00:48,900 finally, the set up here, let me just describe it. 17 00:00:48,900 --> 00:00:50,650 I've been showing some of these components 18 00:00:50,650 --> 00:00:53,680 over and over the past demos, but I 19 00:00:53,680 --> 00:00:56,020 guess now we can understand it, so 20 00:00:56,020 --> 00:00:58,930 let me try to focus the camera. 21 00:00:58,930 --> 00:01:05,600 So let me start with this is a green laser. 22 00:01:05,600 --> 00:01:07,550 So the first component that we have here 23 00:01:07,550 --> 00:01:10,910 is [? anti ?] filter that its job-- 24 00:01:10,910 --> 00:01:14,910 thank you-- so this component, its job 25 00:01:14,910 --> 00:01:17,130 is to control the intensity of light 26 00:01:17,130 --> 00:01:19,650 so we can attenuate it in this case for experiment. 27 00:01:19,650 --> 00:01:22,410 This is an interesting component called a spatial filter 28 00:01:22,410 --> 00:01:26,130 that is composed of a microscope objective and a pinhole, 29 00:01:26,130 --> 00:01:30,240 and now that we know what Fourier transforms are, 30 00:01:30,240 --> 00:01:31,890 the microscope objective is essentially 31 00:01:31,890 --> 00:01:34,710 taking the Fourier transform of the laser beam, which 32 00:01:34,710 --> 00:01:37,830 ideally is a plane wave, so it should be a delta function. 33 00:01:37,830 --> 00:01:40,095 However, since this plane wave is not perfect, 34 00:01:40,095 --> 00:01:42,420 it has high frequency noise. 35 00:01:42,420 --> 00:01:44,070 So then the pinhole, what it does 36 00:01:44,070 --> 00:01:46,380 is it does a low-pass spatial filtering. 37 00:01:46,380 --> 00:01:49,680 So it basically removes all these high frequency signal, 38 00:01:49,680 --> 00:01:52,720 and after these we have a nice spherical wave coming out. 39 00:01:52,720 --> 00:01:55,380 This component here is an iris, control 40 00:01:55,380 --> 00:01:57,510 the diameter of this spherical wave, 41 00:01:57,510 --> 00:02:00,720 and this element here is my collimating lens 42 00:02:00,720 --> 00:02:03,720 that will transform this spherical wave into a plane 43 00:02:03,720 --> 00:02:04,500 wave. 44 00:02:04,500 --> 00:02:07,030 In this case, this is just a beam splitter 45 00:02:07,030 --> 00:02:09,930 that will allow me to couple from this side, where 46 00:02:09,930 --> 00:02:12,060 I have a white light source. 47 00:02:12,060 --> 00:02:16,000 In this white light source, we have a green filter, 48 00:02:16,000 --> 00:02:18,870 so then we only have green light coming out. 49 00:02:18,870 --> 00:02:20,850 We have an iris again to control the diameter. 50 00:02:20,850 --> 00:02:23,190 Right now I'm blocking it, so it is a beam block. 51 00:02:23,190 --> 00:02:24,960 And these are collimating lens. 52 00:02:24,960 --> 00:02:27,480 So here it should be the planar equivalent 53 00:02:27,480 --> 00:02:28,650 of the incoherent side. 54 00:02:28,650 --> 00:02:31,350 This is the planar equivalent of the coherent side. 55 00:02:31,350 --> 00:02:36,930 This mirror here bends the light 90 degrees to send it 56 00:02:36,930 --> 00:02:39,330 to the side from here. 57 00:02:39,330 --> 00:02:41,340 This is the one line of the incoherent line, 58 00:02:41,340 --> 00:02:43,680 and then should go into the common path 59 00:02:43,680 --> 00:02:46,500 of the coherent line. 60 00:02:46,500 --> 00:02:50,080 Then they basically bend again 90 degrees, 61 00:02:50,080 --> 00:02:52,660 and this is the 4F system section. 62 00:02:52,660 --> 00:02:55,710 So here we have first an element, which is the object 63 00:02:55,710 --> 00:02:57,390 that we want to image. 64 00:02:57,390 --> 00:02:59,640 The object is called the resolution target, 65 00:02:59,640 --> 00:03:03,150 and it's a clear substrate, so a piece of glass 66 00:03:03,150 --> 00:03:07,510 that has patterned chrome lines of smaller and smaller sizes 67 00:03:07,510 --> 00:03:09,090 that we're going to see. 68 00:03:09,090 --> 00:03:12,000 Then we have our first lens of the 4F system. 69 00:03:12,000 --> 00:03:14,620 This is a non-magnifying 4F system, 70 00:03:14,620 --> 00:03:17,950 so these two lenses are exactly the same focal length. 71 00:03:17,950 --> 00:03:19,170 So first lens. 72 00:03:19,170 --> 00:03:22,020 Then this is an aperture in the Fourier plane 73 00:03:22,020 --> 00:03:25,260 because we're going to start with low-pass filtering. 74 00:03:25,260 --> 00:03:28,200 The second lens, and then we're imaging this 75 00:03:28,200 --> 00:03:32,010 into our detector, which uses CCD seamless array. 76 00:03:37,260 --> 00:03:39,170 So now we're going to switch to the camera 77 00:03:39,170 --> 00:03:40,790 and start with a coherent case. 78 00:04:02,880 --> 00:04:07,440 So the first case, it's like what we've seen in class. 79 00:04:07,440 --> 00:04:11,980 We're going to illuminate this object with an inline plane 80 00:04:11,980 --> 00:04:12,480 wave. 81 00:04:16,430 --> 00:04:18,740 OK, let me just zoom in a little bit. 82 00:04:21,790 --> 00:04:23,210 So this is the resolution target, 83 00:04:23,210 --> 00:04:25,130 and it's in imaging condition, right? 84 00:04:25,130 --> 00:04:28,010 So the object is in the front focal plane of the first lens, 85 00:04:28,010 --> 00:04:31,580 and the detector is in the back focal plane of the second lens. 86 00:04:31,580 --> 00:04:34,940 So what you can see here is we are actually 87 00:04:34,940 --> 00:04:38,400 imaging all the way to groups two and three. 88 00:04:38,400 --> 00:04:40,900 This resolution target has lines even larger, 89 00:04:40,900 --> 00:04:43,400 which are one and two that we don't see here. 90 00:04:43,400 --> 00:04:46,650 And the first thing we see is the following. 91 00:04:46,650 --> 00:04:49,310 We see some fringes, which are produced 92 00:04:49,310 --> 00:04:51,860 by the interference from the glass, 93 00:04:51,860 --> 00:04:54,500 so it's basically the reflection of the plane waves interfering 94 00:04:54,500 --> 00:04:56,340 with each other. 95 00:04:56,340 --> 00:05:01,760 So this is very typical in coherent imaging. 96 00:05:01,760 --> 00:05:06,860 We see other rings that are produced by dust particles 97 00:05:06,860 --> 00:05:08,490 maybe along the way. 98 00:05:08,490 --> 00:05:11,360 So all these fringes right now look destructive to the image, 99 00:05:11,360 --> 00:05:13,770 and perhaps they are for certain applications. 100 00:05:13,770 --> 00:05:17,120 However, for those of you doing digital holography, 101 00:05:17,120 --> 00:05:20,240 you notice that these fringes contain a lot of information 102 00:05:20,240 --> 00:05:23,030 that is encoded about the amplitude and the phase 103 00:05:23,030 --> 00:05:25,760 of the signal that we can exploit. 104 00:05:25,760 --> 00:05:27,870 So just for comparison, I'm going 105 00:05:27,870 --> 00:05:34,030 to zoom in and see what is the maximum that we can see here. 106 00:05:34,030 --> 00:05:38,100 So you can see that we can have a resolution now that we know 107 00:05:38,100 --> 00:05:40,770 what the optimal resolution. 108 00:05:40,770 --> 00:05:42,690 With good accuracy we see maybe down 109 00:05:42,690 --> 00:05:46,330 to this side of the group number five. 110 00:05:46,330 --> 00:05:48,550 All right, so in the homework we learned 111 00:05:48,550 --> 00:05:53,673 about the low-pass filtering in Fourier domain. 112 00:05:53,673 --> 00:05:55,090 So just to remind you, essentially 113 00:05:55,090 --> 00:05:58,120 what we're going to do is that the first lens 114 00:05:58,120 --> 00:05:59,980 is taking the Fourier transform, and it 115 00:05:59,980 --> 00:06:02,110 happens in the Fourier plane. 116 00:06:02,110 --> 00:06:05,260 So then in that plane we put an aperture, 117 00:06:05,260 --> 00:06:07,630 centered, and then what we do is we're 118 00:06:07,630 --> 00:06:11,080 going to start reducing the diameter of the aperture 119 00:06:11,080 --> 00:06:14,930 to try to low-pass filter the signal more and more. 120 00:06:14,930 --> 00:06:17,500 So what do we expect to see here? 121 00:06:17,500 --> 00:06:19,720 Well, you did it in the p-set, so we 122 00:06:19,720 --> 00:06:22,480 expect to see some blurring, progressive blurring. 123 00:06:22,480 --> 00:06:24,160 So I'm going to do it, and hopefully you 124 00:06:24,160 --> 00:06:25,220 can see it from there. 125 00:06:25,220 --> 00:06:31,963 So right now it's all open, and I'm going to blur it. 126 00:06:31,963 --> 00:06:33,130 So I'm going to do it again. 127 00:06:33,130 --> 00:06:36,150 This is clear. 128 00:06:36,150 --> 00:06:36,650 Blurry. 129 00:06:40,470 --> 00:06:43,590 So I'm controlling with this iris. 130 00:06:43,590 --> 00:06:47,760 And you can see how the signal that contains the higher 131 00:06:47,760 --> 00:06:51,360 spatial frequency in this area goes first, 132 00:06:51,360 --> 00:06:55,200 then this one and this one. 133 00:06:55,200 --> 00:06:56,850 But all of them now-- 134 00:06:56,850 --> 00:07:03,230 in this blurry condition, if I try to zoom back again, 135 00:07:03,230 --> 00:07:06,220 now we see how the signal is really blurry. 136 00:07:06,220 --> 00:07:09,430 If I open the aperture, it becomes clear again. 137 00:07:12,268 --> 00:07:14,560 So now let's do the comparison of the exact same thing. 138 00:07:14,560 --> 00:07:17,800 What would you think it would happen if I switch illumination 139 00:07:17,800 --> 00:07:21,300 now to incoherent light? 140 00:07:21,300 --> 00:07:24,790 So I'm going to block the coherent light here, and then 141 00:07:24,790 --> 00:07:28,270 the first thing that we notice is that we don't have 142 00:07:28,270 --> 00:07:30,040 those fringes anymore, right? 143 00:07:30,040 --> 00:07:33,400 So it looks like a very clean image. 144 00:07:35,888 --> 00:07:37,680 So again, in this case, you can think about 145 00:07:37,680 --> 00:07:40,500 that for microscopy, this is very nice. 146 00:07:40,500 --> 00:07:43,350 However, from the digital holographic point of view, 147 00:07:43,350 --> 00:07:45,330 we don't have that extra information 148 00:07:45,330 --> 00:07:49,620 that we can use in encoding of the signal. 149 00:07:49,620 --> 00:07:52,890 Now let me zoom in to compare the resolution in this case. 150 00:07:58,230 --> 00:08:01,350 So you can see it's hard to tell if we don't 151 00:08:01,350 --> 00:08:06,900 do a quantitative study to see exactly what is the smallest 152 00:08:06,900 --> 00:08:08,500 line that we can see in each case, 153 00:08:08,500 --> 00:08:10,230 but we can see the comparison again 154 00:08:10,230 --> 00:08:14,400 versus the resolution in the coherent and incoherent cases. 155 00:08:14,400 --> 00:08:17,800 And now let's do the low-pass filtering. 156 00:08:17,800 --> 00:08:20,140 So again, this aperture right now is fully open, 157 00:08:20,140 --> 00:08:21,390 and then I started closing it. 158 00:08:26,040 --> 00:08:28,280 And I'm closing and closing and closing it. 159 00:08:31,130 --> 00:08:33,630 Open again. 160 00:08:33,630 --> 00:08:35,470 Close it. 161 00:08:35,470 --> 00:08:37,820 So we see two things. 162 00:08:37,820 --> 00:08:40,809 Remember that the incoherent system is basically regulated 163 00:08:40,809 --> 00:08:44,685 by the OTF, as opposed to the ATF in the coherent system. 164 00:08:44,685 --> 00:08:46,060 So now that you can think of it-- 165 00:08:46,060 --> 00:08:50,230 the easy terms is it's a [? combo ?] because the OTF is 166 00:08:50,230 --> 00:08:55,510 a normalized cross correlation of the ATF, which 167 00:08:55,510 --> 00:08:58,450 means that when I close this aperture, 168 00:08:58,450 --> 00:09:02,200 you can see how the signal becomes weaker 169 00:09:02,200 --> 00:09:05,950 and weaker because essentially I'm restricting the DC 170 00:09:05,950 --> 00:09:09,250 component that happens to be in all the signal there, 171 00:09:09,250 --> 00:09:10,600 but also gets blurry. 172 00:09:10,600 --> 00:09:14,370 It's hard to see, but it's getting blurry. 173 00:09:14,370 --> 00:09:16,780 It is getting blurry at a different rate 174 00:09:16,780 --> 00:09:17,710 as the other system. 175 00:09:23,560 --> 00:09:25,890 So now let's go to high-pass filter, 176 00:09:25,890 --> 00:09:30,490 and I'm going to switch back to the coherent case. 177 00:09:30,490 --> 00:09:31,900 So this is the coherent case. 178 00:09:31,900 --> 00:09:33,275 So right now what I'm going to do 179 00:09:33,275 --> 00:09:36,640 is that I'm going to remove the aperture in the Fourier plane, 180 00:09:36,640 --> 00:09:41,880 and I just did a very improvised high-pass filter, which 181 00:09:41,880 --> 00:09:45,150 basically is a microscope slide that has a little piece of tape 182 00:09:45,150 --> 00:09:49,020 in the center just to block the basic component, or kind of, 183 00:09:49,020 --> 00:09:49,980 but anyways, it works. 184 00:09:59,400 --> 00:10:03,690 Somewhere around here-- let me crank up the light. 185 00:10:09,080 --> 00:10:12,770 And we can see how basically we remove the DC term. 186 00:10:17,810 --> 00:10:20,090 And you see the high-pass filtering effect, 187 00:10:20,090 --> 00:10:23,480 which is essentially accentuating 188 00:10:23,480 --> 00:10:24,740 all the edges of the signal. 189 00:10:32,110 --> 00:10:34,070 Now I'm going to put the incoherent case 190 00:10:34,070 --> 00:10:39,697 and the coherent superimposed just to show edge detection. 191 00:10:39,697 --> 00:10:40,780 So this is a normal thing. 192 00:10:40,780 --> 00:10:42,442 We can accentuate the edges. 193 00:10:42,442 --> 00:10:43,900 So what is interesting also here is 194 00:10:43,900 --> 00:10:46,990 that if you move the filter in different places, 195 00:10:46,990 --> 00:10:50,890 you can, say, just accentuate the edges 196 00:10:50,890 --> 00:10:54,520 in the horizontal direction or in the vertical direction, et 197 00:10:54,520 --> 00:10:55,270 cetera, et cetera. 198 00:10:58,910 --> 00:11:01,270 We see that the incoherent case, we 199 00:11:01,270 --> 00:11:04,510 don't see the same effect, which is as expected. 200 00:11:04,510 --> 00:11:06,850 Again, it's a normalized cross correlation. 201 00:11:06,850 --> 00:11:10,780 But with a coherent case, we do see these drastic change. 202 00:11:29,188 --> 00:11:30,480 GEORGE BARBASTATHIS: Thank you. 203 00:11:33,840 --> 00:11:40,110 Any questions about the demo or about coherence, incoherence? 204 00:11:52,455 --> 00:11:54,330 AUDIENCE: How do you tell whether something's 205 00:11:54,330 --> 00:11:57,930 partially incoherent or completely incoherent? 206 00:11:57,930 --> 00:11:59,430 GEORGE BARBASTATHIS: OK, so the test 207 00:11:59,430 --> 00:12:00,990 is with one of these interferometers 208 00:12:00,990 --> 00:12:04,020 that Se Baek showed last time. 209 00:12:04,020 --> 00:12:06,820 You basically measure the fringe contrast. 210 00:12:06,820 --> 00:12:10,740 For example, if you're worried about spatial coherence, 211 00:12:10,740 --> 00:12:13,590 you set up a [INAUDIBLE] interferometer [INAUDIBLE],, 212 00:12:13,590 --> 00:12:16,380 and the fringe visibility will tell you 213 00:12:16,380 --> 00:12:18,000 the degree of coherence. 214 00:12:18,000 --> 00:12:20,518 If it is full contrast, it is fully coherent. 215 00:12:20,518 --> 00:12:22,310 And of course, you never see full contrast. 216 00:12:22,310 --> 00:12:25,290 You never see-- it's always somewhere in between. 217 00:12:34,230 --> 00:12:39,170 OK, so today we'll have a flashback 218 00:12:39,170 --> 00:12:40,250 of geometrical optics. 219 00:12:40,250 --> 00:12:45,570 I will go back to something we saw in geometrical optics, 220 00:12:45,570 --> 00:12:49,430 and that's the single lens imaging system. 221 00:12:49,430 --> 00:12:53,910 Oh, and before I forget, on Wednesday, 222 00:12:53,910 --> 00:12:56,900 it is the project presentation for the graduate version 223 00:12:56,900 --> 00:13:00,060 of the class, so I wanted to remind you. 224 00:13:00,060 --> 00:13:02,630 I will also send an email from the website, 225 00:13:02,630 --> 00:13:05,540 but if you haven't done so already, 226 00:13:05,540 --> 00:13:08,570 you should have a meeting with your project mentors, 227 00:13:08,570 --> 00:13:11,510 finalize your presentations, and then 228 00:13:11,510 --> 00:13:15,710 on Wednesday, what we'll do is it'll be a conference style. 229 00:13:15,710 --> 00:13:20,110 Each group will have about 15 minutes, approximately 12 230 00:13:20,110 --> 00:13:24,163 for the presentation, approximately 3 for questions. 231 00:13:24,163 --> 00:13:24,830 How many groups? 232 00:13:24,830 --> 00:13:26,330 I think we have six or seven groups. 233 00:13:28,890 --> 00:13:32,300 So typically, these things run over, 234 00:13:32,300 --> 00:13:34,080 so we'll take probably the full two 235 00:13:34,080 --> 00:13:36,750 hours with the presentations. 236 00:13:36,750 --> 00:13:41,790 So for those of you in the 2.71, you don't have to attend, 237 00:13:41,790 --> 00:13:44,407 but I think it would be very useful if you do because it's 238 00:13:44,407 --> 00:13:45,240 sort of interesting. 239 00:13:45,240 --> 00:13:47,270 It is more advanced material. 240 00:13:47,270 --> 00:13:49,440 The graduate students will present some topics 241 00:13:49,440 --> 00:13:51,390 from current literature, and you'll 242 00:13:51,390 --> 00:13:55,170 get an idea not only over the basic optics that we'll do here 243 00:13:55,170 --> 00:14:00,750 but also what kind of things happen in the research front. 244 00:14:00,750 --> 00:14:04,050 For example, last week several of us-- 245 00:14:04,050 --> 00:14:07,020 Pat, Yuan, and Sean-- 246 00:14:07,020 --> 00:14:09,560 were at the conference in Vancouver, 247 00:14:09,560 --> 00:14:12,810 so some of the topics that were represented on Wednesday 248 00:14:12,810 --> 00:14:17,650 were actually big items in the conference. 249 00:14:17,650 --> 00:14:19,500 The conference was actually called Advances 250 00:14:19,500 --> 00:14:22,580 in Imaging, this spring congress of the optical society. 251 00:14:22,580 --> 00:14:23,080 Anyway. 252 00:14:27,630 --> 00:14:29,150 So going back to the lecture, this 253 00:14:29,150 --> 00:14:31,340 is something that we've seen before, 254 00:14:31,340 --> 00:14:32,840 but with geometrical optics. 255 00:14:32,840 --> 00:14:38,000 Now we try to understand how the same system works 256 00:14:38,000 --> 00:14:40,160 but with the wave optics. 257 00:14:40,160 --> 00:14:46,560 And if I could have the tablet again, please. 258 00:14:46,560 --> 00:14:53,290 So the way we analyze the system, of course, is we'll 259 00:14:53,290 --> 00:14:57,070 basically have to propagate using Fresnel diffraction 260 00:14:57,070 --> 00:15:00,370 to propagate the field from one element to the next 261 00:15:00,370 --> 00:15:02,510 until we reach the end of the system. 262 00:15:02,510 --> 00:15:06,483 So for example, at the input we have the input field. 263 00:15:06,483 --> 00:15:07,900 I will write it in one [INAUDIBLE] 264 00:15:07,900 --> 00:15:10,070 so we don't write too much. 265 00:15:10,070 --> 00:15:12,190 The input field is simply the product 266 00:15:12,190 --> 00:15:22,920 of the illumination field times the complex transmittivity 267 00:15:22,920 --> 00:15:24,750 of the input transparency. 268 00:15:24,750 --> 00:15:30,380 I will do everything for a spatially coherent 269 00:15:30,380 --> 00:15:34,600 illumination, and we know now that if the illumination is 270 00:15:34,600 --> 00:15:37,805 spatially incoherent, if we have already 271 00:15:37,805 --> 00:15:39,680 solved the coherent case, we're in good shape 272 00:15:39,680 --> 00:15:42,735 because we can simply take the modular square of the point 273 00:15:42,735 --> 00:15:44,360 spread function and the autocorrelation 274 00:15:44,360 --> 00:15:48,240 of the coherent amplitude transfer function, 275 00:15:48,240 --> 00:15:50,520 and then we can compute the correspondent quantities 276 00:15:50,520 --> 00:15:51,710 for the incoherent case. 277 00:15:51,710 --> 00:15:55,350 So the coherent is the one that we'll always start with. 278 00:15:55,350 --> 00:15:58,540 OK, so doing that now, we can calculate, for example, 279 00:15:58,540 --> 00:16:02,280 the field immediately to the left of the lens. 280 00:16:09,310 --> 00:16:14,650 So let's call it g sub L minus for left, 281 00:16:14,650 --> 00:16:21,660 and I think I still use x double prime as my coordinate. 282 00:16:21,660 --> 00:16:25,940 So this is now a Fresnel propagated version 283 00:16:25,940 --> 00:16:30,560 of the input field, so we'll write out the big, bad Fresnel 284 00:16:30,560 --> 00:16:31,680 integral. 285 00:16:31,680 --> 00:16:36,320 e the i 2 pi z1 over lambda i lambda z1. 286 00:16:36,320 --> 00:16:38,080 So this is the [? prequel, ?] and then we 287 00:16:38,080 --> 00:16:40,400 write our big integral here, g sub 288 00:16:40,400 --> 00:16:43,340 in of x, and then the quadratic. 289 00:16:52,810 --> 00:16:57,250 And of course, two things happen at the lens plane. 290 00:16:57,250 --> 00:16:58,840 First of all is the lens itself. 291 00:16:58,840 --> 00:17:01,900 That will impose an additional quadratic phase delay 292 00:17:01,900 --> 00:17:03,970 upon the signal, but now it's a little bit 293 00:17:03,970 --> 00:17:06,339 different than the geometrical optics case. 294 00:17:06,339 --> 00:17:08,440 I forgot to mention earlier that I 295 00:17:08,440 --> 00:17:14,480 have added a thin transparency in front of the lens. 296 00:17:14,480 --> 00:17:16,450 So this is again a pupil mask. 297 00:17:16,450 --> 00:17:18,400 As we will see in a moment, it does 298 00:17:18,400 --> 00:17:21,339 pretty much the same job as the pupil mask 299 00:17:21,339 --> 00:17:25,640 did at the Fourier plane of the 4F system, but in this case, 300 00:17:25,640 --> 00:17:30,067 we actually put it sort of in contact with the lens. 301 00:17:30,067 --> 00:17:31,150 In fact, we don't have to. 302 00:17:31,150 --> 00:17:33,340 We can put the mask anywhere we like, 303 00:17:33,340 --> 00:17:36,690 but anyway, it makes the math a bit simpler in this case. 304 00:17:36,690 --> 00:17:39,580 And in most implementations, for practical reasons, 305 00:17:39,580 --> 00:17:41,950 they also do it this way because it is much better 306 00:17:41,950 --> 00:17:44,410 to have everything stacked together, 307 00:17:44,410 --> 00:17:46,450 as opposed to having different elements 308 00:17:46,450 --> 00:17:49,070 sort of floating around in an optical system. 309 00:17:49,070 --> 00:17:53,320 OK, so this means then that the field to the right hand 310 00:17:53,320 --> 00:17:57,310 side of the lens, so the plus now means to the right, 311 00:17:57,310 --> 00:17:59,500 still x double prime coordinates, 312 00:17:59,500 --> 00:18:03,040 is going to equal to the field that was 313 00:18:03,040 --> 00:18:07,480 on the left multiplied by 2. 314 00:18:07,480 --> 00:18:08,680 Two functions now. 315 00:18:08,680 --> 00:18:11,960 One is the thin transparency itself, 316 00:18:11,960 --> 00:18:14,800 which I denote there as g sub pm. 317 00:18:14,800 --> 00:18:16,720 pm, of course, stands for pupil mask, 318 00:18:16,720 --> 00:18:20,570 like in the previous cases. 319 00:18:20,570 --> 00:18:25,730 And then times another complex quadratic 320 00:18:25,730 --> 00:18:28,860 exponential that corresponds to the lens itself. 321 00:18:28,860 --> 00:18:31,880 So this would be e to the minus i pi x 322 00:18:31,880 --> 00:18:36,300 double prime square upon lambda f, 323 00:18:36,300 --> 00:18:40,610 where f is the focal length of the lens. 324 00:18:40,610 --> 00:18:43,565 And finally, the field at the output. 325 00:18:48,720 --> 00:18:51,410 Now the output coordinate is x prime. 326 00:18:51,410 --> 00:18:55,190 It's going to equal another Fresnel propagation 327 00:18:55,190 --> 00:18:58,250 from the pupil mask to the output plain, 328 00:18:58,250 --> 00:19:01,820 so I have to write another propagation term here. 329 00:19:01,820 --> 00:19:03,470 This would be z2 this time. 330 00:19:06,430 --> 00:19:12,980 And what I will put now is gL plus because this is the field 331 00:19:12,980 --> 00:19:17,340 to the right of the right of the composite lens and the pupil 332 00:19:17,340 --> 00:19:20,990 mask, so that would be x double prime and then 333 00:19:20,990 --> 00:19:23,015 another quadratic exponential. 334 00:19:35,400 --> 00:19:42,978 OK, so this is it, and when you plug in all this math-- 335 00:19:42,978 --> 00:19:44,520 this is something we've done already, 336 00:19:44,520 --> 00:19:47,380 so I will not do it in the class again, 337 00:19:47,380 --> 00:19:49,000 but there is a supplement. 338 00:19:49,000 --> 00:19:51,400 There is a scan of my handwritten notes 339 00:19:51,400 --> 00:19:54,310 in the website where you will see a little bit more detail 340 00:19:54,310 --> 00:19:56,257 on how I did the derivation here, 341 00:19:56,257 --> 00:19:58,090 but I will skip some of the immediate steps, 342 00:19:58,090 --> 00:20:00,110 and I will show the result here. 343 00:20:00,110 --> 00:20:02,590 So the way you handle this kind of-- as you imagine, 344 00:20:02,590 --> 00:20:04,360 when you combine all of these integrals, 345 00:20:04,360 --> 00:20:08,700 you will get a sort of integral that involves all 346 00:20:08,700 --> 00:20:13,990 of these functions, g sub in, g sub pm and so on, 347 00:20:13,990 --> 00:20:17,060 but also you will have a bunch of quadratic exponentials. 348 00:20:17,060 --> 00:20:19,480 So what you do is you expand the exponents 349 00:20:19,480 --> 00:20:26,620 of these exponentials, and you rearrange terms, 350 00:20:26,620 --> 00:20:29,120 and you write it out this way. 351 00:20:29,120 --> 00:20:30,670 So these are the output coordinates. 352 00:20:30,670 --> 00:20:32,720 They don't participate in the integration, 353 00:20:32,720 --> 00:20:35,440 so we can knock them out of the integral. 354 00:20:35,440 --> 00:20:38,020 And then of course, we cannot do much about this thing 355 00:20:38,020 --> 00:20:42,460 because, well, we don't know what these functions are yet, 356 00:20:42,460 --> 00:20:45,940 but we can certainly try to see what we can do with 357 00:20:45,940 --> 00:20:47,150 the remainder. 358 00:20:47,150 --> 00:20:52,600 So what is sort of glaring here is that in the exponent, 359 00:20:52,600 --> 00:20:56,650 you have quadratic terms, and you also have linear terms. 360 00:20:56,650 --> 00:20:58,630 The linear terms, we know from experience 361 00:20:58,630 --> 00:21:01,030 that they lead into Fourier transform, so they're nice. 362 00:21:01,030 --> 00:21:02,560 We'll know what to do with them. 363 00:21:02,560 --> 00:21:04,420 The quadratic terms, they usually 364 00:21:04,420 --> 00:21:08,960 correspond to nasty things like the focus, but in this case, 365 00:21:08,960 --> 00:21:11,380 you can see that the quadratic term is multiplied 366 00:21:11,380 --> 00:21:13,990 by a coefficient here, which allows 367 00:21:13,990 --> 00:21:16,030 us to knock it out if we select the distance 368 00:21:16,030 --> 00:21:20,740 as z1 and z2 judiciously, and of course that 369 00:21:20,740 --> 00:21:23,290 is the result of the lens law. 370 00:21:23,290 --> 00:21:25,330 This is the same equation that we 371 00:21:25,330 --> 00:21:27,070 derived using geometrical optics, 372 00:21:27,070 --> 00:21:29,180 and we'll call it the imaging condition. 373 00:21:29,180 --> 00:21:31,600 So you can see here that when you satisfied this imaging 374 00:21:31,600 --> 00:21:35,620 condition, then you essentially knock out the focus term 375 00:21:35,620 --> 00:21:37,450 from the diffraction integral. 376 00:21:37,450 --> 00:21:40,060 So that is gratifying because we actually 377 00:21:40,060 --> 00:21:43,390 derived a result from geometrical optics, 378 00:21:43,390 --> 00:21:46,510 but this time it is perhaps more rigorous because it 379 00:21:46,510 --> 00:21:48,328 is based on electromagnetics. 380 00:21:51,365 --> 00:21:52,240 What is the rest now? 381 00:21:52,240 --> 00:21:54,880 Well, we'll deal with the rest in a moment, 382 00:21:54,880 --> 00:21:59,710 but you see that there's another quadratic here that we cannot 383 00:21:59,710 --> 00:22:05,530 quite so easily get rid of, and this extra quadratic that 384 00:22:05,530 --> 00:22:10,750 popped up there is interesting because it is not predicted 385 00:22:10,750 --> 00:22:11,900 by geometrical optics. 386 00:22:11,900 --> 00:22:13,988 So this is something that-- 387 00:22:13,988 --> 00:22:16,030 well, it will happen only with spatially coherent 388 00:22:16,030 --> 00:22:21,750 illumination, and geometrical optics cannot predict it 389 00:22:21,750 --> 00:22:24,560 and actually cannot deal with it. 390 00:22:24,560 --> 00:22:26,830 And in fact, not so long ago, about the time 391 00:22:26,830 --> 00:22:29,793 I was born in the early '70s, this 392 00:22:29,793 --> 00:22:31,960 was a topic of research, what to do about this term. 393 00:22:31,960 --> 00:22:36,550 In fact, Professor Goodman, who wrote your textbook, one 394 00:22:36,550 --> 00:22:41,680 of his early results as a young scientist, 395 00:22:41,680 --> 00:22:45,160 he wrote the famous paper about this term, 396 00:22:45,160 --> 00:22:46,120 what this term means. 397 00:22:46,120 --> 00:22:46,828 And he wrote it-- 398 00:22:46,828 --> 00:22:50,140 I believe he wrote it in 1971, exactly the year I was born. 399 00:22:50,140 --> 00:22:52,750 I'm not that old, so I guess that you can still 400 00:22:52,750 --> 00:22:57,300 call it a relatively recent development. 401 00:22:57,300 --> 00:22:59,980 So again, we're talking about this term over here, 402 00:22:59,980 --> 00:23:02,290 what to do about this one. 403 00:23:02,290 --> 00:23:08,650 And then of course, there's an entire section in Goodman 404 00:23:08,650 --> 00:23:11,050 that talks about this term, and I would encourage 405 00:23:11,050 --> 00:23:12,480 you to read the section. 406 00:23:12,480 --> 00:23:16,120 I will show a summary here. 407 00:23:16,120 --> 00:23:18,705 Something got messed up that with my animations here. 408 00:23:18,705 --> 00:23:20,580 This was not supposed to be here, but anyway. 409 00:23:24,240 --> 00:23:27,750 So Goodman actually goes over three methods 410 00:23:27,750 --> 00:23:31,440 that you can use to eliminate this unwanted quadratic. 411 00:23:31,440 --> 00:23:36,180 One of them is you lay out the input transparency as a sphere, 412 00:23:36,180 --> 00:23:37,680 or in a spherical surface, instead 413 00:23:37,680 --> 00:23:39,210 of the typical planar surface. 414 00:23:39,210 --> 00:23:40,710 That may sound a little bit strange, 415 00:23:40,710 --> 00:23:43,992 and it's probably very difficult to implement in practice, 416 00:23:43,992 --> 00:23:45,450 but conceptually you can see what's 417 00:23:45,450 --> 00:23:49,440 going on because now these points in the transparency 418 00:23:49,440 --> 00:23:51,930 start with an original phase delay, 419 00:23:51,930 --> 00:23:53,700 you can work out the curvature right here. 420 00:23:53,700 --> 00:23:55,075 In fact, the curvature would have 421 00:23:55,075 --> 00:23:57,480 to be exactly z1, the areas of curvature, 422 00:23:57,480 --> 00:23:59,760 so that it cancels this unwanted quadratic. 423 00:23:59,760 --> 00:24:01,770 So that's one way. 424 00:24:01,770 --> 00:24:04,590 The second way is to actually stick 425 00:24:04,590 --> 00:24:07,830 a lens in front of the transparency. 426 00:24:07,830 --> 00:24:12,390 This is actually very often done in geometrical optics as well. 427 00:24:12,390 --> 00:24:13,890 This is called the condenser lens. 428 00:24:13,890 --> 00:24:17,130 So for example, we don't have this anymore, 429 00:24:17,130 --> 00:24:19,710 but actually, we do have one in the classroom. 430 00:24:19,710 --> 00:24:24,330 We used to use these projectors. 431 00:24:24,330 --> 00:24:28,650 Basically, if you look at the top surface of this projector-- 432 00:24:28,650 --> 00:24:30,450 I don't know if I can take it out here-- 433 00:24:30,450 --> 00:24:32,790 but if you look at it, it's actually a lens. 434 00:24:32,790 --> 00:24:34,570 After class, come over and look. 435 00:24:34,570 --> 00:24:39,240 There is grooves on it, and basically, 436 00:24:39,240 --> 00:24:43,590 sort of the grooves implement equivalent of a lens surface. 437 00:24:47,090 --> 00:24:56,730 If you do the operation modulo 2 pi on the surface of the lens, 438 00:24:56,730 --> 00:25:01,350 instead of having this surface that we typically 439 00:25:01,350 --> 00:25:03,867 think of as a lens, if you do a modulo 2 pi operation, 440 00:25:03,867 --> 00:25:05,700 you will get something like this, of course. 441 00:25:14,370 --> 00:25:18,800 OK, so that's a way to make a thin lens. 442 00:25:18,800 --> 00:25:21,050 That is called a condenser. 443 00:25:21,050 --> 00:25:22,520 The reason we put it in a projector 444 00:25:22,520 --> 00:25:24,530 is not to eliminate this quadratic factor, 445 00:25:24,530 --> 00:25:29,750 it is simply because it makes the illumination uniform. 446 00:25:29,750 --> 00:25:32,143 So when you project the slide on-- 447 00:25:32,143 --> 00:25:33,560 I don't know if you guys are maybe 448 00:25:33,560 --> 00:25:35,480 too young to have ever seen anything 449 00:25:35,480 --> 00:25:38,002 like this in operation, but the next time if I remember, 450 00:25:38,002 --> 00:25:39,710 I'll bring an old fashioned transparency, 451 00:25:39,710 --> 00:25:42,260 and I will show you how people of my age 452 00:25:42,260 --> 00:25:43,820 used to give talks at conferences 453 00:25:43,820 --> 00:25:47,040 when we didn't have PowerPoint. 454 00:25:47,040 --> 00:25:51,350 But anyway, this is done actually very commonly, 455 00:25:51,350 --> 00:25:53,300 to stick a lens in front of-- 456 00:25:53,300 --> 00:25:55,707 just attached to a thin object. 457 00:25:55,707 --> 00:25:57,540 But there are some cases you cannot do that. 458 00:25:57,540 --> 00:25:58,957 For example, in the microscope, it 459 00:25:58,957 --> 00:26:00,890 is very difficult to imagine sticking 460 00:26:00,890 --> 00:26:04,640 a lens behind the glass slide. 461 00:26:04,640 --> 00:26:06,890 So in some cases, this is applicable, 462 00:26:06,890 --> 00:26:07,960 and in some it is not. 463 00:26:07,960 --> 00:26:11,720 Anyway, a big application of this type of lens 464 00:26:11,720 --> 00:26:14,120 is of course in overhead projectors. 465 00:26:14,120 --> 00:26:15,830 Does anybody know another application 466 00:26:15,830 --> 00:26:17,630 where people use this kind of lenses, 467 00:26:17,630 --> 00:26:21,527 or what might motivate you to make this kind of lens? 468 00:26:21,527 --> 00:26:22,110 Can you think? 469 00:26:28,450 --> 00:26:30,170 These lenses are commonly used-- 470 00:26:30,170 --> 00:26:30,980 actually, you know? 471 00:26:33,887 --> 00:26:35,470 AUDIENCE: Do they sometimes stick them 472 00:26:35,470 --> 00:26:38,413 on the backs of vans or cars to make it wide angle? 473 00:26:38,413 --> 00:26:40,080 GEORGE BARBASTATHIS: That's right, yeah. 474 00:26:40,080 --> 00:26:42,160 That's another application. 475 00:26:42,160 --> 00:26:46,163 Exactly, to improve the rear view of the driver. 476 00:26:46,163 --> 00:26:47,080 And that's a good one. 477 00:26:47,080 --> 00:26:49,330 I wasn't thinking of that. 478 00:26:49,330 --> 00:26:54,970 Yet another one is in very high power illuminators, 479 00:26:54,970 --> 00:26:59,350 which are most commonly used in photography in shooting movies 480 00:26:59,350 --> 00:27:06,460 because the glass, as you know, is not a very good conductor, 481 00:27:06,460 --> 00:27:10,610 so it would hit up very badly if you illuminate 482 00:27:10,610 --> 00:27:13,590 with very high power. 483 00:27:13,590 --> 00:27:15,350 They're called, actually, babies, 484 00:27:15,350 --> 00:27:17,830 these really big light bulbs that they 485 00:27:17,830 --> 00:27:19,780 use when they shoot movies. 486 00:27:19,780 --> 00:27:21,490 So they actually use these. 487 00:27:21,490 --> 00:27:24,590 It is called a Fresnel lens, the same Fresnel 488 00:27:24,590 --> 00:27:28,550 lens in Fresnel propagation. 489 00:27:28,550 --> 00:27:34,280 So they use the Fresnel lenses in these so-called babies. 490 00:27:34,280 --> 00:27:36,170 All right. 491 00:27:36,170 --> 00:27:39,200 And lighthouses also, yeah. 492 00:27:39,200 --> 00:27:42,890 So again, every time that you have a really huge light 493 00:27:42,890 --> 00:27:43,390 source. 494 00:27:43,390 --> 00:27:45,598 Of course, it's a matter of practicality also, right? 495 00:27:45,598 --> 00:27:48,680 If you have a major lens, as you can imagine, 496 00:27:48,680 --> 00:27:51,470 if the diameter is big, then also this size 497 00:27:51,470 --> 00:27:52,750 would also be very big, right? 498 00:27:52,750 --> 00:27:54,170 So in a lighthouse, you might have 499 00:27:54,170 --> 00:27:57,980 a lens that is maybe 1 or 2 meters in diameter, 500 00:27:57,980 --> 00:28:00,947 then it would have to bulge over another half meter or so. 501 00:28:00,947 --> 00:28:02,030 That's not very practical. 502 00:28:02,030 --> 00:28:04,252 It is heavy, fragile, blah blah blah. 503 00:28:04,252 --> 00:28:06,710 So they make the Fresnel lenses to get around this problem. 504 00:28:09,805 --> 00:28:11,420 A good thing to remember about those 505 00:28:11,420 --> 00:28:15,330 is that they generally give very poor image quality. 506 00:28:15,330 --> 00:28:18,850 So if you try to use this as an imaging lens, 507 00:28:18,850 --> 00:28:23,180 you typically get a very bad blur and a very bad image 508 00:28:23,180 --> 00:28:27,530 quality, but obviously in the lighthouse you don't care. 509 00:28:27,530 --> 00:28:29,540 In an illuminator, you don't care. 510 00:28:29,540 --> 00:28:33,470 And also the projector, it is not used as an imaging lens. 511 00:28:33,470 --> 00:28:35,990 The imaging lens is a real lens, this one. 512 00:28:35,990 --> 00:28:38,270 This is simply collimating the illumination 513 00:28:38,270 --> 00:28:43,220 so you get a uniform image at the image plane. 514 00:28:43,220 --> 00:28:45,890 So for jobs like this one where you basically 515 00:28:45,890 --> 00:28:48,830 want to take a light bulb that is highly un-uniform. 516 00:28:48,830 --> 00:28:51,300 It has filaments, all kinds of crap. 517 00:28:51,300 --> 00:28:54,080 So this is a very nonuniform field. 518 00:28:54,080 --> 00:28:57,200 This type of lens is very good to uniformize 519 00:28:57,200 --> 00:28:58,940 the intensity, and then of course 520 00:28:58,940 --> 00:29:01,343 if you need to image a sharp object, 521 00:29:01,343 --> 00:29:02,510 then you need the real lens. 522 00:29:02,510 --> 00:29:05,550 You cannot get around that. 523 00:29:05,550 --> 00:29:09,540 OK, that's a bit about Fresnel lenses. 524 00:29:09,540 --> 00:29:12,180 So that's a bit of a detour, but I 525 00:29:12,180 --> 00:29:14,760 wanted to point out some practical ways 526 00:29:14,760 --> 00:29:18,030 that you might be able to do this kind of thing 527 00:29:18,030 --> 00:29:19,860 if you need to get rid of the lens. 528 00:29:19,860 --> 00:29:23,980 And finally, what John Goodman proved back when he was young 529 00:29:23,980 --> 00:29:28,710 is that there's a condition that you can 530 00:29:28,710 --> 00:29:31,453 neglect this nasty quadratic. 531 00:29:31,453 --> 00:29:33,120 Again, we're talking about the quadratic 532 00:29:33,120 --> 00:29:35,330 that you see at the top of the slide. 533 00:29:35,330 --> 00:29:41,280 So it turns out if the field of the object-- 534 00:29:41,280 --> 00:29:43,950 when we say the field, sometimes in geometrical optics 535 00:29:43,950 --> 00:29:46,000 we mean the size of this. 536 00:29:46,000 --> 00:29:49,950 So if the field is smaller than about a quarter 537 00:29:49,950 --> 00:29:53,220 of the imaging lens, then it turns out again 538 00:29:53,220 --> 00:30:01,380 that the effects of this term are negligible, 539 00:30:01,380 --> 00:30:04,680 and therefore we can get rid of it in clear conscience. 540 00:30:04,680 --> 00:30:07,580 So I will not go into the details. 541 00:30:07,580 --> 00:30:10,140 There is discussion in the book and also 542 00:30:10,140 --> 00:30:13,830 reference 303 from the book is an article 543 00:30:13,830 --> 00:30:17,040 that Goodman wrote back then, and he 544 00:30:17,040 --> 00:30:19,140 goes into this in a little bit more detail. 545 00:30:19,140 --> 00:30:21,622 It's a pretty interesting article. 546 00:30:21,622 --> 00:30:23,830 I'm going to leave a post on the website if you like. 547 00:30:23,830 --> 00:30:25,520 It's pretty interesting to read. 548 00:30:28,370 --> 00:30:32,580 So assuming then that will get rid of this extra quadratic-- 549 00:30:32,580 --> 00:30:33,450 let me go back one. 550 00:30:39,240 --> 00:30:42,820 So we got rid of one quadratic from the lens law, 551 00:30:42,820 --> 00:30:47,560 and I spent a lot of time arguing about this quadratic, 552 00:30:47,560 --> 00:30:49,200 that maybe you can get rid of that 553 00:30:49,200 --> 00:30:53,980 then as well, assuming you can get rid of that as well. 554 00:30:53,980 --> 00:30:55,930 Well, anyway, even if we don't get rid 555 00:30:55,930 --> 00:30:59,290 of that, what is left in here is actually Fourier transform. 556 00:30:59,290 --> 00:31:01,060 I think I pointed that out before. 557 00:31:01,060 --> 00:31:02,290 A Fourier transform-- why? 558 00:31:02,290 --> 00:31:04,480 Because you have linear now, linear 559 00:31:04,480 --> 00:31:08,520 exponents times some coefficient, 560 00:31:08,520 --> 00:31:10,580 and then here you have the pupil mask. 561 00:31:10,580 --> 00:31:12,330 So again you recognize something familiar. 562 00:31:12,330 --> 00:31:15,210 The same thing happened in the case of the 4F system. 563 00:31:15,210 --> 00:31:17,580 We again got a term like this one, 564 00:31:17,580 --> 00:31:19,690 but it was slightly different. 565 00:31:19,690 --> 00:31:23,660 Instead of z1 z2 in the denominators here, 566 00:31:23,660 --> 00:31:26,880 it had f1 and f2, the two focal lengths. 567 00:31:26,880 --> 00:31:30,210 Here we have the two distances, but it's pretty much the same, 568 00:31:30,210 --> 00:31:34,680 and again the pupil mask appeared in the kernel here. 569 00:31:34,680 --> 00:31:38,130 So this is a Fourier transform, and of course we 570 00:31:38,130 --> 00:31:41,890 get rid of this term, and let me jump through these animations 571 00:31:41,890 --> 00:31:42,390 now. 572 00:31:45,990 --> 00:31:50,010 OK, so if we're doing that, then what we finally get 573 00:31:50,010 --> 00:31:54,480 is this expression here, where, again, this 574 00:31:54,480 --> 00:31:57,690 is the Fourier transform of the pupil mask. 575 00:31:57,690 --> 00:32:00,540 I wrote it in a form that looks a bit more like a Fourier 576 00:32:00,540 --> 00:32:04,360 transform, and of course u and v are now arbitrary arguments. 577 00:32:04,360 --> 00:32:06,360 These are the spatial frequencies in the Fourier 578 00:32:06,360 --> 00:32:09,720 transform, but what actually goes in there 579 00:32:09,720 --> 00:32:13,390 is these parameters over here. 580 00:32:13,390 --> 00:32:18,550 So basically we get again a term that looks like a convolution. 581 00:32:18,550 --> 00:32:23,930 So we see that the image, the complex optical field 582 00:32:23,930 --> 00:32:32,260 at the image is a convolution of the input field times something 583 00:32:32,260 --> 00:32:35,555 that again we will call the point spread function. 584 00:32:35,555 --> 00:32:37,180 And this point spread function, just as 585 00:32:37,180 --> 00:32:41,210 in the case of the 4F system, again here the point spread 586 00:32:41,210 --> 00:32:44,050 function is given as the Fourier transform over 587 00:32:44,050 --> 00:32:47,830 the pupil of the pupil mask scaled 588 00:32:47,830 --> 00:32:51,200 with appropriate coordinates. 589 00:32:51,200 --> 00:32:55,520 OK, and there is various scaling forms and factors here. 590 00:32:55,520 --> 00:33:00,570 For example, this one again ensures energy conservation, 591 00:33:00,570 --> 00:33:03,140 which is kind of important, but very often 592 00:33:03,140 --> 00:33:06,270 in optics we don't care so much about the absolute intensities, 593 00:33:06,270 --> 00:33:08,850 which means that we don't really care about this term. 594 00:33:08,850 --> 00:33:11,720 What we really care about is about the spatial distribution, 595 00:33:11,720 --> 00:33:15,560 like the type we're solving, whether features make it 596 00:33:15,560 --> 00:33:18,830 to the image or they get lost due to spatial filtering 597 00:33:18,830 --> 00:33:19,860 and so on and so forth. 598 00:33:19,860 --> 00:33:23,360 So very often these multiplicative constants, 599 00:33:23,360 --> 00:33:25,880 we just drop them. 600 00:33:25,880 --> 00:33:28,560 That's sort of convention. 601 00:33:28,560 --> 00:33:31,520 Now, to get a little more insight out of this, 602 00:33:31,520 --> 00:33:38,000 imagine for a moment that the point spread function 603 00:33:38,000 --> 00:33:39,620 becomes extremely narrow. 604 00:33:39,620 --> 00:33:42,240 It becomes a delta function. 605 00:33:42,240 --> 00:33:44,330 So if you substitute the delta function 606 00:33:44,330 --> 00:33:46,130 into the expression we had before, 607 00:33:46,130 --> 00:33:47,780 then you get this expression, which 608 00:33:47,780 --> 00:33:50,780 is of course again very gratifying 609 00:33:50,780 --> 00:33:53,300 because now the output really looks like, 610 00:33:53,300 --> 00:33:56,720 again, some multiplicative term, but then it really looks 611 00:33:56,720 --> 00:33:59,500 like the input, except in the input 612 00:33:59,500 --> 00:34:02,870 now, the coordinates have been scaled, 613 00:34:02,870 --> 00:34:05,120 so therefore you can immediately see that this term is 614 00:34:05,120 --> 00:34:07,210 the magnification. 615 00:34:07,210 --> 00:34:11,239 In fact, the lateral magnification of the system 616 00:34:11,239 --> 00:34:12,960 is given by this expression. 617 00:34:12,960 --> 00:34:18,739 So we basically rederived the same result 618 00:34:18,739 --> 00:34:21,290 that we got from geometrical optics-- 619 00:34:21,290 --> 00:34:24,469 namely, the lens law that is the imaging condition, 620 00:34:24,469 --> 00:34:27,300 the magnification and so on and so forth. 621 00:34:27,300 --> 00:34:32,480 Now we've got them back from our wave optics approach, 622 00:34:32,480 --> 00:34:34,219 but we had to do some approximations. 623 00:34:34,219 --> 00:34:37,280 For example, we had to pretend that the point spread 624 00:34:37,280 --> 00:34:39,020 function is a delta function in order 625 00:34:39,020 --> 00:34:41,688 to get the geometrical optics result. If it's not, 626 00:34:41,688 --> 00:34:42,980 which is never the case, right? 627 00:34:42,980 --> 00:34:46,679 Delta function is a very good approximation. 628 00:34:46,679 --> 00:34:50,270 So if it's not a delta function, then the result 629 00:34:50,270 --> 00:34:52,940 is not exactly like this, but it is a convolution. 630 00:34:52,940 --> 00:34:55,550 That is, basically this will become blurry. 631 00:34:55,550 --> 00:35:00,650 Like [? Pepe ?] showed earlier, you will get a blurred vision-- 632 00:35:00,650 --> 00:35:04,040 or in general, an especially filtered version 633 00:35:04,040 --> 00:35:06,080 of the original input will actually 634 00:35:06,080 --> 00:35:08,270 survive through to the output. 635 00:35:13,800 --> 00:35:15,510 What I would like to point out now, 636 00:35:15,510 --> 00:35:17,940 this is something that I believe Se Baek also showed 637 00:35:17,940 --> 00:35:20,360 last time, the same slide. 638 00:35:20,360 --> 00:35:24,750 What I would like to do is basically pay 639 00:35:24,750 --> 00:35:28,080 a little bit of attention to the scaling factors here. 640 00:35:28,080 --> 00:35:30,450 These scaling factors are actually very important. 641 00:35:30,450 --> 00:35:34,530 So I mentioned that to drop multiplicative factors that 642 00:35:34,530 --> 00:35:36,150 appear in the front here. 643 00:35:36,150 --> 00:35:39,103 This, it is OK to drop, but these scaling factors 644 00:35:39,103 --> 00:35:40,520 that go inside the argument, these 645 00:35:40,520 --> 00:35:43,080 are very important because they determine 646 00:35:43,080 --> 00:35:48,240 the amount of spatial filtering that goes on inside the system. 647 00:35:48,240 --> 00:35:50,770 So what I will do actually for a while 648 00:35:50,770 --> 00:35:54,170 is I will compare the 4F system with a single lens 649 00:35:54,170 --> 00:35:55,180 similar system. 650 00:35:55,180 --> 00:35:57,180 So the scale factors, again, they are different. 651 00:35:57,180 --> 00:36:03,470 You see f1 is inside the transfer function, and z1 652 00:36:03,470 --> 00:36:05,160 in the case of the single lens. 653 00:36:05,160 --> 00:36:06,950 So what is the effect of that one? 654 00:36:06,950 --> 00:36:10,390 Also in the incoherent case. 655 00:36:10,390 --> 00:36:12,730 Of course, in the incoherent case, 656 00:36:12,730 --> 00:36:16,540 the transfer function is the autocorrelation 657 00:36:16,540 --> 00:36:19,220 of the coherent transfer function, 658 00:36:19,220 --> 00:36:23,380 and since the coherent transfer function is basically 659 00:36:23,380 --> 00:36:27,430 proportional to the pupil mask, then what 660 00:36:27,430 --> 00:36:31,060 we call the optical transfer function, the incoherent case, 661 00:36:31,060 --> 00:36:33,520 is an autocorrelation of the pupil mask itself. 662 00:36:33,520 --> 00:36:37,780 This is again a very basic result, but again, be careful. 663 00:36:37,780 --> 00:36:40,390 When we compute these autocorrelations, 664 00:36:40,390 --> 00:36:42,880 we have to apply different scaling factors 665 00:36:42,880 --> 00:36:44,210 in the argument. 666 00:36:44,210 --> 00:36:44,710 OK. 667 00:36:44,710 --> 00:36:47,650 So I will show in a second how this works, and I will show it 668 00:36:47,650 --> 00:36:49,360 in two cases. 669 00:36:49,360 --> 00:36:53,895 One is what we called before the Zernike phase mask. 670 00:36:53,895 --> 00:36:55,520 Actually, this is not exactly accurate. 671 00:36:55,520 --> 00:36:57,355 Zernike did not invent exactly this. 672 00:36:57,355 --> 00:37:00,395 He invented another one that has pi over 2 phase 673 00:37:00,395 --> 00:37:01,300 shift at the edges. 674 00:37:01,300 --> 00:37:02,290 It's like a ring. 675 00:37:02,290 --> 00:37:04,360 But anyway, the function is the same, 676 00:37:04,360 --> 00:37:09,220 so everybody refers to all of these types of masks as Zernike 677 00:37:09,220 --> 00:37:11,080 nowadays. 678 00:37:11,080 --> 00:37:11,980 So this is familiar. 679 00:37:11,980 --> 00:37:14,320 We saw a few examples in the past 680 00:37:14,320 --> 00:37:19,570 when it was in the middle in the focal plane of the 4F system, 681 00:37:19,570 --> 00:37:24,950 and this is the case when we put it as a pupil mask 682 00:37:24,950 --> 00:37:26,510 in a single lens system. 683 00:37:26,510 --> 00:37:28,100 It's the same mask. 684 00:37:28,100 --> 00:37:30,100 It will do a very similar thing, but there's 685 00:37:30,100 --> 00:37:33,110 a subtle difference, so this is what I want to point out. 686 00:37:33,110 --> 00:37:35,920 So this is the subtle difference. 687 00:37:35,920 --> 00:37:37,960 What you see here is, of course, the schematics, 688 00:37:37,960 --> 00:37:41,320 so you see it is opaque outside some amplitude. 689 00:37:41,320 --> 00:37:43,930 Then inside, you have this little extra phase 690 00:37:43,930 --> 00:37:48,730 delay near the optical access in a small region. 691 00:37:48,730 --> 00:37:51,430 And OK, this is the mathematical expression 692 00:37:51,430 --> 00:37:53,050 that I don't want to dwell upon. 693 00:37:53,050 --> 00:37:55,460 I'd rather focus on these plots here. 694 00:37:55,460 --> 00:38:01,450 So these two plots are the magnitude 695 00:38:01,450 --> 00:38:03,550 and the phase of the mask. 696 00:38:03,550 --> 00:38:05,200 So the magnitude is the top. 697 00:38:05,200 --> 00:38:08,680 The blue plot it goes from 0 to 1. 698 00:38:08,680 --> 00:38:11,800 So it is 1 within the amplitude, so let's see if that's correct. 699 00:38:11,800 --> 00:38:14,380 The amplitude has a size of 1 centimeter, 700 00:38:14,380 --> 00:38:19,060 so indeed this is 1 between minus 0.5 and 0.5. 701 00:38:19,060 --> 00:38:23,190 And the phase, well, the phase, nobody knows what the phase is. 702 00:38:23,190 --> 00:38:25,450 Where the magnitude is zero, we can not 703 00:38:25,450 --> 00:38:28,810 define the phase for a complex number that equals zero. 704 00:38:28,810 --> 00:38:31,810 Nevertheless, let's set it equal to zero. 705 00:38:31,810 --> 00:38:35,470 But what I want to emphasize is that within the past band 706 00:38:35,470 --> 00:38:40,360 of the system, the phase is 0, except for a small chunk 707 00:38:40,360 --> 00:38:42,730 of width of 0.2 centimeters where 708 00:38:42,730 --> 00:38:45,230 the phase jumps to pi over 2. 709 00:38:45,230 --> 00:38:49,570 Pi over 2 is of course i. 710 00:38:49,570 --> 00:38:51,490 And this is the real and imaginary part. 711 00:38:51,490 --> 00:38:53,350 These two are actually equivalent. 712 00:38:53,350 --> 00:38:58,030 So the real part, it is 1, except at 713 00:38:58,030 --> 00:39:02,500 the little extra phase protrusion there where 714 00:39:02,500 --> 00:39:05,350 the real part goes to 0, and the imaginary part 715 00:39:05,350 --> 00:39:12,010 becomes 1 because of course when the phase equals pi over 2, 716 00:39:12,010 --> 00:39:15,010 then the actual complex amplitude equals i. 717 00:39:15,010 --> 00:39:17,170 So that's what you see here. 718 00:39:17,170 --> 00:39:22,570 So these two pairs describe both of the phase mask. 719 00:39:22,570 --> 00:39:26,550 So the [INAUDIBLE] transfer functions of course 720 00:39:26,550 --> 00:39:28,250 are the same. 721 00:39:28,250 --> 00:39:31,730 I only plotted them here as a real and imaginary part, 722 00:39:31,730 --> 00:39:33,570 but they have the same shape. 723 00:39:33,570 --> 00:39:36,330 What is really different-- and unfortunately, 724 00:39:36,330 --> 00:39:39,240 my plot is a little bit too small here. 725 00:39:39,240 --> 00:39:40,620 You cannot see very well-- 726 00:39:40,620 --> 00:39:43,810 is that they actually have different sizes. 727 00:39:43,810 --> 00:39:47,430 This one, if you do the scaling factors here for the numbers 728 00:39:47,430 --> 00:39:49,815 that I used, I think I used-- 729 00:39:53,810 --> 00:39:56,870 so I used lambda equals 1 micron. 730 00:39:59,960 --> 00:40:07,880 Lambda equals 1 micron, f1 equals 10 centimeters, 731 00:40:07,880 --> 00:40:11,210 f2 equals 1 centimeter for the 4F system. 732 00:40:11,210 --> 00:40:15,630 And then z1 equals 11 centimeters, 733 00:40:15,630 --> 00:40:23,750 z2 equals 1.1 centimeters, and f1 equals still 10 centimeters 734 00:40:23,750 --> 00:40:25,060 for the single lens. 735 00:40:25,060 --> 00:40:27,050 And I worked those out so that in both cases, 736 00:40:27,050 --> 00:40:30,570 you get the demagnification of a factor of 10. 737 00:40:30,570 --> 00:40:33,100 That's why the numbers are a little bit strange there. 738 00:40:33,100 --> 00:40:36,057 So in both cases, you get a demagnification factor of 10, 739 00:40:36,057 --> 00:40:38,390 and I did that deliberately because the two systems give 740 00:40:38,390 --> 00:40:41,060 you the same effect kind of, but you 741 00:40:41,060 --> 00:40:43,940 can see in terms of geometrical optics, they are identical. 742 00:40:43,940 --> 00:40:46,070 They both give you a demagnification 743 00:40:46,070 --> 00:40:48,518 by a factor of 10, but you see here 744 00:40:48,518 --> 00:40:50,060 that in terms of wave optics, they're 745 00:40:50,060 --> 00:40:52,430 slightly different because one of them, 746 00:40:52,430 --> 00:40:55,160 the 4F system is scaled by the focal length. 747 00:40:55,160 --> 00:40:57,290 Its spatial frequency extends, goes 748 00:40:57,290 --> 00:41:00,650 from minus 50 to 50 inverse millimeters. 749 00:41:00,650 --> 00:41:02,450 This is what you see here. 750 00:41:02,450 --> 00:41:05,960 Whereas in this case, it goes from approximately minus 48 751 00:41:05,960 --> 00:41:09,020 to 48 inverse millimeters. 752 00:41:09,020 --> 00:41:12,610 So the single lens actually does a slightly more severe 753 00:41:12,610 --> 00:41:20,428 spatial filter than the case of the 4F system. 754 00:41:20,428 --> 00:41:22,720 That is to be expected, of course, because the distance 755 00:41:22,720 --> 00:41:23,240 z1-- 756 00:41:23,240 --> 00:41:25,880 you can see from here-- 757 00:41:25,880 --> 00:41:34,520 the distance z1 that you subtend from the object to the aperture 758 00:41:34,520 --> 00:41:36,710 is longer, so therefore this system 759 00:41:36,710 --> 00:41:39,830 is cutting off more angles, or more spatial 760 00:41:39,830 --> 00:41:43,100 frequencies than this system. 761 00:41:43,100 --> 00:41:44,980 We will see that in more detail in a second. 762 00:41:44,980 --> 00:41:46,730 The other thing I want to point out to you 763 00:41:46,730 --> 00:41:50,390 is that if you take that autocorrelation 764 00:41:50,390 --> 00:41:54,380 of this function, which is the coherent transfer 765 00:41:54,380 --> 00:41:58,070 function or the ATF, if you take its autocorrelation, 766 00:41:58,070 --> 00:42:03,930 you obtain the OTF, which is, as Se Baek demonstrated last time, 767 00:42:03,930 --> 00:42:06,230 it is the transfer function for spatially 768 00:42:06,230 --> 00:42:08,060 incoherent illumination. 769 00:42:08,060 --> 00:42:10,130 So again, this is a little bit of an exercise 770 00:42:10,130 --> 00:42:15,720 that I will not do here, but it is in the notes. 771 00:42:15,720 --> 00:42:19,230 I have posted another set of practice problems. 772 00:42:19,230 --> 00:42:23,180 So it is in pages 16 and 17 of the last set 773 00:42:23,180 --> 00:42:24,140 of practice problems. 774 00:42:24,140 --> 00:42:26,120 I have gone ahead and derived this one, 775 00:42:26,120 --> 00:42:28,360 and I would encourage you to go through. 776 00:42:28,360 --> 00:42:31,700 It's a little bit of algebra, but you 777 00:42:31,700 --> 00:42:34,610 can think of it as mental fitness right 778 00:42:34,610 --> 00:42:38,210 because you might say, why do I need to do all this algebra? 779 00:42:38,210 --> 00:42:41,540 When would I ever have to do so much algebra? 780 00:42:41,540 --> 00:42:43,670 I can just plug it into Matlab, and I get it. 781 00:42:43,670 --> 00:42:45,290 Well, I'm sure all of you do some kind of fitness. 782 00:42:45,290 --> 00:42:46,332 You go to the gym, right? 783 00:42:46,332 --> 00:42:48,860 When you do bench presses, what is the probability 784 00:42:48,860 --> 00:42:51,290 that you need to do a bench press motion in real life? 785 00:42:51,290 --> 00:42:52,940 It's actually very small. 786 00:42:52,940 --> 00:42:55,370 So you do bench presses in order to keep fit. 787 00:42:55,370 --> 00:42:57,650 So the reason we do these kind of calculations 788 00:42:57,650 --> 00:43:00,260 is it's the equivalent of mental bench presses, right, 789 00:43:00,260 --> 00:43:01,760 so I would encourage you to do it. 790 00:43:01,760 --> 00:43:04,010 Because presumably you are at MIT because you have you 791 00:43:04,010 --> 00:43:06,830 want to have mental fitness as well as physical fitness. 792 00:43:11,220 --> 00:43:16,110 Let me talk a little bit more about these scaling factors. 793 00:43:21,100 --> 00:43:25,580 So in this case, this is just a clear aperture that I put here, 794 00:43:25,580 --> 00:43:29,980 and you can see that in the case of the clear aperture, 795 00:43:29,980 --> 00:43:32,950 it's actually very easy to do sort of the ray diagrams, 796 00:43:32,950 --> 00:43:34,470 and you can see also here that-- 797 00:43:37,060 --> 00:43:38,810 well, let me back up for a second. 798 00:43:38,810 --> 00:43:40,810 So you remember from geometrical optics, 799 00:43:40,810 --> 00:43:43,850 we had the definition of the numerical aperture. 800 00:43:43,850 --> 00:43:47,140 So we defined the numerical aperture as the angle 801 00:43:47,140 --> 00:43:50,740 that we subtend towards the optical system 802 00:43:50,740 --> 00:43:54,460 if we place a point source on axis. 803 00:43:54,460 --> 00:43:56,570 So therefore, the numerical aperture 804 00:43:56,570 --> 00:43:59,680 is limited by one of the physical apertures that 805 00:43:59,680 --> 00:44:01,040 are in the system. 806 00:44:01,040 --> 00:44:04,450 So in the case of the 4F system, that would be the pupil mask. 807 00:44:04,450 --> 00:44:06,270 At some point in any physical system, 808 00:44:06,270 --> 00:44:08,920 the pupil mask cannot be infinite. 809 00:44:08,920 --> 00:44:11,890 It has to be finite, so the physical diameter of the pupil 810 00:44:11,890 --> 00:44:16,180 mask, assuming the lenses themselves are large enough, 811 00:44:16,180 --> 00:44:19,130 the pupil mask will become the limiting factor. 812 00:44:19,130 --> 00:44:22,040 And so we can compute the numerical aperture then. 813 00:44:22,040 --> 00:44:27,430 It is the ratio of the radius of the mask 814 00:44:27,430 --> 00:44:29,110 over the focal length, f1. 815 00:44:29,110 --> 00:44:33,460 You can see it very easily from this triangle over here. 816 00:44:37,840 --> 00:44:39,840 And of course this is an approximate expression. 817 00:44:39,840 --> 00:44:42,180 In reality, the numerical apertures 818 00:44:42,180 --> 00:44:47,250 would be the sine over the inverse tangent 819 00:44:47,250 --> 00:44:49,200 of this quantity, but of course we're 820 00:44:49,200 --> 00:44:51,730 doing [INAUDIBLE] approximations here, 821 00:44:51,730 --> 00:44:55,950 so we can drop the trigonometric functions. 822 00:44:55,950 --> 00:45:01,050 In the case of the single lens the numerical aperture is-- 823 00:45:01,050 --> 00:45:04,290 again, you can get it from this triangle over here, 824 00:45:04,290 --> 00:45:09,975 but now the one of the orthogonal sides is z1, not f1. 825 00:45:09,975 --> 00:45:11,850 So you can see that the numerical aperture is 826 00:45:11,850 --> 00:45:18,630 r over z1, and of course if this system is supposed to produce 827 00:45:18,630 --> 00:45:22,350 a real image, as opposed to a virtual image, 828 00:45:22,350 --> 00:45:25,290 then z1 must be longer than f1. 829 00:45:25,290 --> 00:45:29,010 You can see very clearly from the imaging condition, 830 00:45:29,010 --> 00:45:32,460 1 over z1 plus 1 over z2 equals 1 over f. 831 00:45:32,460 --> 00:45:35,170 If you want z2 to be positive-- 832 00:45:35,170 --> 00:45:40,200 so for z2 to be positive, it means I have a real image-- 833 00:45:40,200 --> 00:45:45,060 for this, it is required that z1 is bigger than f. 834 00:45:45,060 --> 00:45:48,780 So because of that, you can see that if you have the same size 835 00:45:48,780 --> 00:45:51,810 pupil mask, the single lens system would have a smaller 836 00:45:51,810 --> 00:45:53,400 numerical aperture. 837 00:45:53,400 --> 00:45:57,240 That is sort of an unfortunate fact of life. 838 00:45:59,760 --> 00:46:01,650 So another thing I want to point out here 839 00:46:01,650 --> 00:46:04,620 is that the numerical aperture actually 840 00:46:04,620 --> 00:46:07,350 changes when you go to the second leg of the system 841 00:46:07,350 --> 00:46:10,110 because, of course, the system has angular magnification. 842 00:46:10,110 --> 00:46:14,130 So what started as a numerical aperture at the input 843 00:46:14,130 --> 00:46:16,230 is of course an angle. 844 00:46:16,230 --> 00:46:18,570 If you take the marginal ray, it is 845 00:46:18,570 --> 00:46:20,700 propagating at a given angle with respect 846 00:46:20,700 --> 00:46:21,940 to the optical axis. 847 00:46:21,940 --> 00:46:25,410 By the time it comes out, this angle will have changed. 848 00:46:25,410 --> 00:46:26,100 By how much? 849 00:46:26,100 --> 00:46:29,070 By the angular magnification of the system. 850 00:46:29,070 --> 00:46:32,090 So therefore, the numerical aperture at the output 851 00:46:32,090 --> 00:46:32,970 is not the same. 852 00:46:32,970 --> 00:46:37,800 It equals the numerical aperture at the input times the angular 853 00:46:37,800 --> 00:46:39,910 magnification of the system. 854 00:46:39,910 --> 00:46:42,810 So this is not a cause for confusion. 855 00:46:42,810 --> 00:46:44,380 We can use either one. 856 00:46:44,380 --> 00:46:47,490 We'll get actually the same conclusions whether we 857 00:46:47,490 --> 00:46:50,010 use the numerical aperture or the input 858 00:46:50,010 --> 00:46:52,470 or the numerical aperture at the output, 859 00:46:52,470 --> 00:46:54,090 but we have to be a little bit careful 860 00:46:54,090 --> 00:46:56,340 that we don't confuse things, as long 861 00:46:56,340 --> 00:46:58,800 as we remember this simple relationship 862 00:46:58,800 --> 00:47:04,120 that the two are connected by the angular magnification. 863 00:47:04,120 --> 00:47:08,560 So the reason the numerical aperture is so important is 864 00:47:08,560 --> 00:47:13,088 because if you consider a circular-- of course, 865 00:47:13,088 --> 00:47:14,380 you cannot see the circle here. 866 00:47:14,380 --> 00:47:15,820 This is just the projection. 867 00:47:15,820 --> 00:47:20,590 But imagine that I have a circular pupil. 868 00:47:20,590 --> 00:47:23,290 Then recall that the point spread function of the system 869 00:47:23,290 --> 00:47:25,270 is actually the Fourier transform 870 00:47:25,270 --> 00:47:29,770 of that circular pupil because this is a clear aperture now. 871 00:47:29,770 --> 00:47:32,860 And we learned sometime ago-- 872 00:47:32,860 --> 00:47:34,120 I don't remember when-- 873 00:47:34,120 --> 00:47:36,130 but we learned that the Fourier transform 874 00:47:36,130 --> 00:47:41,180 of a circular function is this crazy [? zinc ?] 875 00:47:41,180 --> 00:47:45,310 so-called function that is given by a ratio of a Bessel function 876 00:47:45,310 --> 00:47:48,220 to its argument, and it's probably better 877 00:47:48,220 --> 00:47:50,560 to think of it as a plot like this one that 878 00:47:50,560 --> 00:47:53,790 has a main lobe and then a smaller side lobe. 879 00:47:53,790 --> 00:47:55,490 And actually, it has a lot of lobes. 880 00:47:55,490 --> 00:48:00,340 It continues on forever, but the size of the lobes decays away. 881 00:48:00,340 --> 00:48:04,810 How fast it decays away is 1 over the argument. 882 00:48:04,810 --> 00:48:08,988 So what I want to point out here is that in both cases, 883 00:48:08,988 --> 00:48:10,780 the point spread the function of the system 884 00:48:10,780 --> 00:48:13,390 looks like this [? zinc ?] functions. 885 00:48:13,390 --> 00:48:16,245 Sometimes it's also known as an Airy disk. 886 00:48:16,245 --> 00:48:17,620 Not an Airy function, by the way. 887 00:48:17,620 --> 00:48:20,380 Airy function is different. 888 00:48:20,380 --> 00:48:24,313 This one is referred to as Airy disk. 889 00:48:24,313 --> 00:48:25,980 If you're curious what Airy function, is 890 00:48:25,980 --> 00:48:27,930 you can open the table of formulas. 891 00:48:27,930 --> 00:48:30,810 Abramowitz and [INAUDIBLE],, and you'll see a monstrous thing. 892 00:48:30,810 --> 00:48:32,935 That's called the Airy function, and this turns out 893 00:48:32,935 --> 00:48:35,070 to be a special case of an Airy function. 894 00:48:35,070 --> 00:48:38,410 But anyway, this is the Airy disk, 895 00:48:38,410 --> 00:48:42,190 and what I want to emphasize is that in both cases, 896 00:48:42,190 --> 00:48:47,650 you get the same shape of Airy disk, but different in size. 897 00:48:47,650 --> 00:48:51,040 And you'll get different size because in the case of the 4F 898 00:48:51,040 --> 00:48:56,690 system, the size of the aperture is the same, 899 00:48:56,690 --> 00:48:59,260 but because the scaling factor is smaller-- 900 00:48:59,260 --> 00:49:04,090 it is f1-- then you actually get a bigger ATF. 901 00:49:04,090 --> 00:49:08,980 The size of the disk in the frequency domain is bigger. 902 00:49:08,980 --> 00:49:11,980 Therefore, it will give you a slightly narrower point spread 903 00:49:11,980 --> 00:49:13,750 function. 904 00:49:13,750 --> 00:49:17,260 If you work out these ratios over here, 905 00:49:17,260 --> 00:49:18,760 and you work out these coefficients, 906 00:49:18,760 --> 00:49:21,190 you will see that very clearly, but I 907 00:49:21,190 --> 00:49:23,500 want you to get it sort of intuitively 908 00:49:23,500 --> 00:49:27,910 by using the scaling theorem of Fourier transform. 909 00:49:27,910 --> 00:49:31,510 In both cases, you start with the same physical aperture, 910 00:49:31,510 --> 00:49:34,780 but what matters is not the physical size of the aperture, 911 00:49:34,780 --> 00:49:36,490 but the numerical aperture. 912 00:49:36,490 --> 00:49:40,860 So in one case, you have this numerical aperture. 913 00:49:40,860 --> 00:49:41,750 This is a f1. 914 00:49:41,750 --> 00:49:44,950 This is the physical size, and numerical aperture 915 00:49:44,950 --> 00:49:47,240 is R over f1. 916 00:49:47,240 --> 00:49:50,570 In the other case, you have again the same aperture, 917 00:49:50,570 --> 00:49:55,220 but now you have a longer distance here. 918 00:49:55,220 --> 00:49:58,500 This is still R. So now the numerical aperture 919 00:49:58,500 --> 00:50:01,430 would be R over z1. 920 00:50:01,430 --> 00:50:05,030 So the same physical aperture in the two cases 921 00:50:05,030 --> 00:50:09,020 will actually give you a different size in the ATF. 922 00:50:09,020 --> 00:50:10,475 This will have a bigger ATF. 923 00:50:16,900 --> 00:50:20,350 So the size of the ATF, if you work it out, 924 00:50:20,350 --> 00:50:26,680 it will be proportional to actually 1 over lambda f1, 925 00:50:26,680 --> 00:50:30,220 and the size of the ATF here-- 926 00:50:30,220 --> 00:50:32,230 I'm exaggerating, of course-- 927 00:50:32,230 --> 00:50:35,140 will be proportional to 1 over lambda z1. 928 00:50:38,970 --> 00:50:44,080 OK, so since we've got the smaller size of the ATF 929 00:50:44,080 --> 00:50:48,120 in this case, it means that we'll get a broader PSF. 930 00:50:48,120 --> 00:50:50,220 So which system is better? 931 00:50:50,220 --> 00:50:52,630 Well, obviously this one because this one 932 00:50:52,630 --> 00:50:54,610 will give you a narrower PSF. 933 00:50:54,610 --> 00:50:57,260 Therefore, it will give you a smaller blur. 934 00:50:57,260 --> 00:50:58,510 So of course there's a caveat. 935 00:50:58,510 --> 00:51:01,360 I sort of took it for granted that the desirable 936 00:51:01,360 --> 00:51:04,210 in an optical system is to minimize the blur, which 937 00:51:04,210 --> 00:51:05,800 in most cases it's true. 938 00:51:05,800 --> 00:51:07,990 If for some reason someone asked you deliberately 939 00:51:07,990 --> 00:51:10,750 to produce a system that causes a lot of blur, 940 00:51:10,750 --> 00:51:13,600 then of course you would go for this one, but in most cases, 941 00:51:13,600 --> 00:51:15,910 we try to minimize blur. 942 00:51:15,910 --> 00:51:19,300 So this means, given our resources-- 943 00:51:19,300 --> 00:51:23,410 that is, given our physical size of the numerical aperture-- 944 00:51:23,410 --> 00:51:27,008 we should try to maximize the numerical aperture, 945 00:51:27,008 --> 00:51:28,800 and that is what the photo system is doing. 946 00:51:34,702 --> 00:51:36,910 So there are a few comments about the resolution that 947 00:51:36,910 --> 00:51:38,200 are in the rest of the slides. 948 00:51:38,200 --> 00:51:40,242 I don't want to destroy the rest of your morning. 949 00:51:45,650 --> 00:51:47,930 Because the PSF is finite, you can 950 00:51:47,930 --> 00:51:49,520 imagine that if you have two point 951 00:51:49,520 --> 00:51:51,213 sources that are at distance. 952 00:51:51,213 --> 00:51:52,630 So now this is like an experiment. 953 00:51:52,630 --> 00:51:54,950 I'm moving two point sources together. 954 00:51:54,950 --> 00:51:57,410 There will come a point where the two point 955 00:51:57,410 --> 00:51:59,120 sources will merge. 956 00:51:59,120 --> 00:52:01,730 And if this is your image now, you 957 00:52:01,730 --> 00:52:05,067 don't really know whether you had one point source or two. 958 00:52:05,067 --> 00:52:06,650 Now, of course you say, well, you also 959 00:52:06,650 --> 00:52:08,030 get twice the intensity. 960 00:52:08,030 --> 00:52:10,520 That's true, but very often you don't know 961 00:52:10,520 --> 00:52:11,940 the intensity you started with. 962 00:52:11,940 --> 00:52:14,180 For example, if you're looking at the sky, 963 00:52:14,180 --> 00:52:15,680 and you're looking at the bright dot 964 00:52:15,680 --> 00:52:18,770 and you're trying to decide, is it one star or two stars? 965 00:52:18,770 --> 00:52:22,160 Well, so far we cannot yet go to the stars and measure 966 00:52:22,160 --> 00:52:24,020 their brightness, right? 967 00:52:24,020 --> 00:52:25,700 We can only measure the total brightness 968 00:52:25,700 --> 00:52:27,192 that we receive here. 969 00:52:27,192 --> 00:52:29,150 So therefore, if you're looking at a telescope, 970 00:52:29,150 --> 00:52:31,850 and you see this, you don't know if there's 971 00:52:31,850 --> 00:52:36,380 one star or two stars that are too close to be resolved 972 00:52:36,380 --> 00:52:38,480 by your telescope. 973 00:52:38,480 --> 00:52:41,000 So actually, this sort of situation 974 00:52:41,000 --> 00:52:47,550 was dealt with by a fellow called Rayleigh, and of course, 975 00:52:47,550 --> 00:52:49,280 the size of the PSF, as I worked out 976 00:52:49,280 --> 00:52:51,960 before, it depends on the numerical aperture. 977 00:52:51,960 --> 00:52:54,950 So I will go over this in more detail 978 00:52:54,950 --> 00:52:56,990 later, but this is just a preview 979 00:52:56,990 --> 00:53:00,605 that the numerical aperture actually gives you an idea-- 980 00:53:00,605 --> 00:53:02,480 I actually believe also Professor [INAUDIBLE] 981 00:53:02,480 --> 00:53:04,520 mentioned it in one of the past lectures. 982 00:53:04,520 --> 00:53:07,040 The numerical aperture gives you an idea 983 00:53:07,040 --> 00:53:12,980 about the capability of your system to resolve point images. 984 00:53:12,980 --> 00:53:15,890 So imagine that each one of these lobes 985 00:53:15,890 --> 00:53:19,190 here corresponds to the point spread function of one point 986 00:53:19,190 --> 00:53:20,360 object. 987 00:53:20,360 --> 00:53:23,300 So in this case, they were resolved because I spaced them 988 00:53:23,300 --> 00:53:28,580 so that the diameter of the main lobe 989 00:53:28,580 --> 00:53:32,570 falls exactly on one null of the other lobe. 990 00:53:32,570 --> 00:53:34,700 So if you work it out, and you take into care 991 00:53:34,700 --> 00:53:38,000 into where the zeros of the Bessel function 992 00:53:38,000 --> 00:53:39,830 are located and so on, you come up 993 00:53:39,830 --> 00:53:41,880 with an expression that looks like this. 994 00:53:41,880 --> 00:53:43,760 The space in between the two sources 995 00:53:43,760 --> 00:53:47,210 is 1.22 times the wavelength divided 996 00:53:47,210 --> 00:53:49,130 by the numerical aperture. 997 00:53:49,130 --> 00:53:53,510 And of course this would be at the input plane. 998 00:53:53,510 --> 00:53:57,980 So this would be the spacing at the input required 999 00:53:57,980 --> 00:54:00,740 so that the two sources can be resolved. 1000 00:54:00,740 --> 00:54:04,400 But when you look at the output, the distance of course 1001 00:54:04,400 --> 00:54:06,980 would be given by a similar expression, 1002 00:54:06,980 --> 00:54:09,680 but the numerical aperture at the output 1003 00:54:09,680 --> 00:54:11,330 would appear in this case. 1004 00:54:11,330 --> 00:54:14,720 And of course this is sort of the most common definition. 1005 00:54:14,720 --> 00:54:17,120 Some books define the resolution-- 1006 00:54:17,120 --> 00:54:19,430 instead of the diameter, they use the radius, 1007 00:54:19,430 --> 00:54:22,550 so they come up with an expression that is exactly 1/2. 1008 00:54:22,550 --> 00:54:23,960 Which one is correct, it actually 1009 00:54:23,960 --> 00:54:25,190 depends on your requirements. 1010 00:54:25,190 --> 00:54:28,700 If you require a very crisp image, 1011 00:54:28,700 --> 00:54:33,195 or if you're dealing with a very noisy situation, 1012 00:54:33,195 --> 00:54:34,820 then you go for this one because as you 1013 00:54:34,820 --> 00:54:37,380 can imagine if you organize noise into this one, 1014 00:54:37,380 --> 00:54:38,870 then it becomes progressively more 1015 00:54:38,870 --> 00:54:41,480 difficult to resolve the two spots. 1016 00:54:41,480 --> 00:54:43,190 So in a very noisy environment, when 1017 00:54:43,190 --> 00:54:44,840 you have a very little light, then 1018 00:54:44,840 --> 00:54:46,088 you go for this definition. 1019 00:54:46,088 --> 00:54:47,630 If you have plenty of light, then you 1020 00:54:47,630 --> 00:54:50,690 can possibly resolve two sources that are very close like this, 1021 00:54:50,690 --> 00:54:51,190 right. 1022 00:54:51,190 --> 00:54:53,210 Here you see you have very poor contrast. 1023 00:54:53,210 --> 00:54:54,800 You basically have to rely-- 1024 00:54:54,800 --> 00:54:56,300 in order to resolve the two sources, 1025 00:54:56,300 --> 00:54:59,610 you have to rely on this little dip in density. 1026 00:54:59,610 --> 00:55:02,120 So if your signal is noisy, this may be lost, 1027 00:55:02,120 --> 00:55:04,277 and then you cannot resolve anymore. 1028 00:55:04,277 --> 00:55:05,860 OK, so that's a preview of resolution. 1029 00:55:05,860 --> 00:55:08,400 We'll talk about this a little bit more.