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 00:00:21,900 --> 00:00:24,600 SE BAEK OH: So today's class, we are going to talk about-- 9 00:00:24,600 --> 00:00:25,670 called coherence. 10 00:00:25,670 --> 00:00:31,310 And basically Fourier system with incoherent imaging. 11 00:00:31,310 --> 00:00:34,490 So particularly, this topic is very important 12 00:00:34,490 --> 00:00:37,520 if you want to take our optics course because We're dealing 13 00:00:37,520 --> 00:00:39,125 with coherence and incoherence. 14 00:00:43,070 --> 00:00:45,590 So far, we have dealt with the Fourier system 15 00:00:45,590 --> 00:00:48,350 with a coherent illumination. 16 00:00:48,350 --> 00:00:52,340 Basically, laser light from a point source-- 17 00:00:52,340 --> 00:00:53,180 I mean, pinhole. 18 00:00:53,180 --> 00:00:59,360 So the coherent plane waves illuminating the Fourier 19 00:00:59,360 --> 00:01:04,480 system, and the frequency content appears at the plane. 20 00:01:04,480 --> 00:01:06,830 And you can put the [INAUDIBLE] mask 21 00:01:06,830 --> 00:01:10,910 to play with the spatial filtering, 22 00:01:10,910 --> 00:01:14,880 and you're going to get some image at the alpha plane. 23 00:01:14,880 --> 00:01:19,730 But in everyday life, if you use your digital camera, 24 00:01:19,730 --> 00:01:23,390 you're not using the really laser illumination. 25 00:01:23,390 --> 00:01:28,190 You mostly use this ambient light, or if it's too dark, 26 00:01:28,190 --> 00:01:32,460 then you're going to use the flashlight. 27 00:01:32,460 --> 00:01:36,720 So today, we are going to talk about more realistic case. 28 00:01:36,720 --> 00:01:38,600 We use the incoherent light. 29 00:01:38,600 --> 00:01:42,290 By the way, the difference between coherent and incoherent 30 00:01:42,290 --> 00:01:45,530 is basically, you can make the interference 31 00:01:45,530 --> 00:01:47,060 with the coherent light. 32 00:01:47,060 --> 00:01:49,130 So if you're not make interference-- 33 00:01:49,130 --> 00:01:52,190 you're using laser, but in most case, 34 00:01:52,190 --> 00:01:53,690 like this white light source, you're 35 00:01:53,690 --> 00:01:55,910 not going to see interference. 36 00:01:55,910 --> 00:02:00,590 So we're going to first define what coherence is. 37 00:02:00,590 --> 00:02:02,210 And actually, there are two kinds 38 00:02:02,210 --> 00:02:06,230 of coherence, temporal coherence and spatial coherence. 39 00:02:06,230 --> 00:02:08,570 And the rest of the topic is pretty straightforward. 40 00:02:08,570 --> 00:02:14,750 We will define incoherent point spread function and its Fourier 41 00:02:14,750 --> 00:02:17,300 transform which is optical transfer function 42 00:02:17,300 --> 00:02:19,370 and modulation transfer function, which 43 00:02:19,370 --> 00:02:22,850 is modulus of OTF. 44 00:02:22,850 --> 00:02:25,760 And we're going to see what MTF means. 45 00:02:25,760 --> 00:02:28,250 And we want to compare the difference 46 00:02:28,250 --> 00:02:32,861 between spatially coherent and incoherent images. 47 00:02:38,660 --> 00:02:40,400 So as I just described, the coherence 48 00:02:40,400 --> 00:02:45,090 is basically ability to make interference. 49 00:02:45,090 --> 00:02:47,750 So if you have incoherent light, you cannot make interference. 50 00:02:47,750 --> 00:02:49,850 And if you have coherent light, then you 51 00:02:49,850 --> 00:02:51,530 can make the interference. 52 00:02:51,530 --> 00:02:55,663 And I'm going to describe it later, but mathematically-- 53 00:02:55,663 --> 00:02:57,080 actually, the coherence is nothing 54 00:02:57,080 --> 00:03:03,050 but the correlation of wave at two time space points, which 55 00:03:03,050 --> 00:03:08,090 means we have some light, and you just sample two points. 56 00:03:08,090 --> 00:03:11,750 It could be different time points, but you combine them. 57 00:03:11,750 --> 00:03:14,450 And if you see interference, then you can say it's coherent. 58 00:03:14,450 --> 00:03:16,270 But if you don't see interference, 59 00:03:16,270 --> 00:03:19,160 then it is incoherent. 60 00:03:19,160 --> 00:03:21,930 So first, let me describe what's going 61 00:03:21,930 --> 00:03:25,100 to happen if I use the incoherent light in Michelson 62 00:03:25,100 --> 00:03:25,730 interferometer. 63 00:03:25,730 --> 00:03:28,850 So here, we have the Michelson interferometer. 64 00:03:28,850 --> 00:03:31,740 So from the point source, the light-- 65 00:03:31,740 --> 00:03:38,240 oops-- is coming in this direction, this direction. 66 00:03:38,240 --> 00:03:41,450 And at the beam splitter, half of the light 67 00:03:41,450 --> 00:03:44,090 just transmit the beam splitter, but the other half 68 00:03:44,090 --> 00:03:45,680 is reflected. 69 00:03:45,680 --> 00:03:48,470 So this path gets reflected again at the mirror, 70 00:03:48,470 --> 00:03:50,090 and passed through the beam splitter, 71 00:03:50,090 --> 00:03:52,010 and arrives at the detector. 72 00:03:52,010 --> 00:03:55,815 And the other path is reflected again, and reflected 73 00:03:55,815 --> 00:03:56,690 at the beam splitter. 74 00:03:56,690 --> 00:03:58,850 And again you get-- 75 00:03:58,850 --> 00:04:01,530 I mean, it arrives at the detector. 76 00:04:04,490 --> 00:04:10,055 So let's just suppose that we have the laser source here. 77 00:04:10,055 --> 00:04:11,930 Then what you're going to see at the detector 78 00:04:11,930 --> 00:04:15,940 is a nice sinusoidal pattern if I move one of the mirror. 79 00:04:15,940 --> 00:04:23,990 Because I can tune the optical path length that light travels. 80 00:04:23,990 --> 00:04:27,080 And if the distance match, then I 81 00:04:27,080 --> 00:04:28,550 get the constructive interference. 82 00:04:28,550 --> 00:04:33,590 And if the phase delay pi, then I get disruptive interference. 83 00:04:33,590 --> 00:04:35,690 So as I move the one with the mirror, 84 00:04:35,690 --> 00:04:39,015 I get nice sinusoidal pattern with the full contrast. 85 00:04:43,410 --> 00:04:48,400 And next, let's suppose instead of the laser, 86 00:04:48,400 --> 00:04:51,350 I have incoherent source, which means-- 87 00:04:51,350 --> 00:04:54,880 so the horizontal axis is time, and the vertical axis 88 00:04:54,880 --> 00:04:56,065 is amplitude. 89 00:04:56,065 --> 00:05:00,100 But the phase or amplitude is varying randomly. 90 00:05:00,100 --> 00:05:10,510 So over time, the amplitudes are varying randomly. 91 00:05:10,510 --> 00:05:15,730 And even though it's random, this wave train 92 00:05:15,730 --> 00:05:18,730 will travel through the interferometer. 93 00:05:18,730 --> 00:05:28,570 So in this path, the optical path length 94 00:05:28,570 --> 00:05:30,380 is actually constant, which is-- 95 00:05:30,380 --> 00:05:33,680 I mean, these paths, they're common in two path. 96 00:05:33,680 --> 00:05:37,510 So constants plus 2d1 over c, which is time-- 97 00:05:37,510 --> 00:05:39,110 speed of light. 98 00:05:39,110 --> 00:05:43,270 And in the second path, I have the same-- 99 00:05:43,270 --> 00:05:44,780 this wave train also travels. 100 00:05:44,780 --> 00:05:49,530 But due to the difference between d1 and d2, 101 00:05:49,530 --> 00:05:51,550 at the detector, essentially, I'm 102 00:05:51,550 --> 00:05:53,480 going to sample a different time point. 103 00:05:53,480 --> 00:05:57,460 So even though it's random, I basically 104 00:05:57,460 --> 00:06:01,210 choose two time points here and here, 105 00:06:01,210 --> 00:06:04,600 and combine them at the detector. 106 00:06:04,600 --> 00:06:10,720 So if the d1 and d2 are same, then basically, 107 00:06:10,720 --> 00:06:13,340 identical two waves are coming together, 108 00:06:13,340 --> 00:06:16,150 so I always get constructive interference. 109 00:06:16,150 --> 00:06:19,300 But what if d1 and d2 are not same? 110 00:06:19,300 --> 00:06:23,860 And then basically, they get different phase shift. 111 00:06:23,860 --> 00:06:28,720 And since it's random, they kind of average out. 112 00:06:28,720 --> 00:06:30,830 So d1 and d2 are getting bigger. 113 00:06:30,830 --> 00:06:34,930 Then basically, the degree of correlation 114 00:06:34,930 --> 00:06:39,700 is getting decreased, which means at the detector, 115 00:06:39,700 --> 00:06:43,780 the contrast will also get decreased. 116 00:06:43,780 --> 00:06:47,830 So if I plot the intensity at the detector with respect 117 00:06:47,830 --> 00:06:52,110 to d2 minus d1, which is the distance difference of two 118 00:06:52,110 --> 00:06:57,010 arms, then the intensity is going to be like this. 119 00:06:57,010 --> 00:07:02,800 So remember that if I have the laser at point source, then 120 00:07:02,800 --> 00:07:07,180 I will get the sinusoidal pattern with the full contrast. 121 00:07:07,180 --> 00:07:11,860 But since now I have the randomly varying source, 122 00:07:11,860 --> 00:07:14,060 this envelop is getting attenuated like this. 123 00:07:14,060 --> 00:07:18,490 So if I have the rapid decay curve, which means 124 00:07:18,490 --> 00:07:20,570 I have more incoherent light. 125 00:07:20,570 --> 00:07:23,390 And if I have the slowly decaying envelop, 126 00:07:23,390 --> 00:07:24,550 then it's more coherent. 127 00:07:29,490 --> 00:07:35,290 And there is another kind of coherence 128 00:07:35,290 --> 00:07:36,760 which is spatial coherence. 129 00:07:36,760 --> 00:07:38,510 So basically, the concept is the same, 130 00:07:38,510 --> 00:07:43,240 but we only define the correlation in space domain. 131 00:07:43,240 --> 00:07:46,060 So now I have the Young interferometer, which 132 00:07:46,060 --> 00:07:48,390 I have two pinholes, x1 and x2. 133 00:07:48,390 --> 00:07:51,850 And I put a detector at the center of the two pinholes. 134 00:07:51,850 --> 00:07:56,350 So from this pinhole, we see-- 135 00:07:56,350 --> 00:07:59,830 I mean, the spherical wave is going to be emanating, 136 00:07:59,830 --> 00:08:05,050 but since it is at the center, these paths are exactly same. 137 00:08:05,050 --> 00:08:08,770 And first, let's suppose we have laser illumination. 138 00:08:08,770 --> 00:08:11,650 So I have the plane wave coming in. 139 00:08:11,650 --> 00:08:15,010 Then this spherical wave will make the Young's interference. 140 00:08:15,010 --> 00:08:18,640 So I'm going to have the sinusoidal pattern 141 00:08:18,640 --> 00:08:21,250 with the full contrast at this plane, 142 00:08:21,250 --> 00:08:23,530 but I'm just seeing at one point. 143 00:08:23,530 --> 00:08:32,059 And again, if I have the randomly varying wave here-- 144 00:08:32,059 --> 00:08:36,280 so in this case, this horizontal axis amplitude 145 00:08:36,280 --> 00:08:39,820 and vertical axis is space instead of time. 146 00:08:39,820 --> 00:08:46,000 So this amplitude and phase are randomly varying over space. 147 00:08:46,000 --> 00:08:49,450 Since I have two pinholes, so I essentially 148 00:08:49,450 --> 00:08:54,490 sample at two points here and there, and combine them. 149 00:08:58,520 --> 00:09:02,140 So again, if these two pinholes are very close, 150 00:09:02,140 --> 00:09:04,990 then these sampling points are also very close. 151 00:09:04,990 --> 00:09:10,390 So the amplitude and phase are more likely similar. 152 00:09:10,390 --> 00:09:12,550 [INAUDIBLE] 153 00:09:12,550 --> 00:09:17,620 But if I move these two pinholes in the upward and downward 154 00:09:17,620 --> 00:09:20,760 like this, then the spacing between two pinholes 155 00:09:20,760 --> 00:09:22,660 are getting bigger, and basically, I 156 00:09:22,660 --> 00:09:25,840 lose the correlation, which means 157 00:09:25,840 --> 00:09:29,780 the spatial coherence is getting attenuated-- getting smaller. 158 00:09:29,780 --> 00:09:33,490 So again, if I plot the intensity 159 00:09:33,490 --> 00:09:37,900 at the detector with respect to x2 minus x1, which 160 00:09:37,900 --> 00:09:41,140 is the spacing between two pinholes, 161 00:09:41,140 --> 00:09:46,258 then intensity varying like this. 162 00:09:46,258 --> 00:09:47,800 Again, remember, if I have the laser, 163 00:09:47,800 --> 00:09:50,340 then it doesn't really matter. 164 00:09:50,340 --> 00:09:52,330 I mean, I always get a full contrast. 165 00:09:52,330 --> 00:09:54,590 Even though those frequencies-- 166 00:09:54,590 --> 00:09:58,130 I mean, [INAUDIBLE] plane-- the frequency of the interference 167 00:09:58,130 --> 00:09:58,690 is changing. 168 00:09:58,690 --> 00:10:00,760 But I always get the full contrast, 169 00:10:00,760 --> 00:10:04,980 but contrast getting smaller if you have the incoherent source. 170 00:10:14,020 --> 00:10:20,800 So let me describe mathematically 171 00:10:20,800 --> 00:10:22,780 why this happens. 172 00:10:22,780 --> 00:10:25,600 It doesn't really matter whether I 173 00:10:25,600 --> 00:10:28,630 use the Michelson interferometer or Young's interferometer. 174 00:10:28,630 --> 00:10:38,750 Basically, interferometer, we have two waves. 175 00:10:38,750 --> 00:10:43,610 So a1, e to the pi 1, and a2, d to the pi 2. 176 00:10:43,610 --> 00:10:47,450 Basically, we have two waves I could sample in space or time 177 00:10:47,450 --> 00:10:47,950 domain. 178 00:10:47,950 --> 00:10:50,000 But anyway, I'm going to have two waves 179 00:10:50,000 --> 00:10:52,880 and just add them together, and take the magnitude 180 00:10:52,880 --> 00:10:55,250 square, which is intensity. 181 00:10:55,250 --> 00:10:59,510 So Michelson interferometer, this is-- 182 00:10:59,510 --> 00:11:06,090 I mean, these two waves came from the different-- the path 183 00:11:06,090 --> 00:11:07,940 length, d1 and d2. 184 00:11:07,940 --> 00:11:11,270 And Young's interferometer, basically, they 185 00:11:11,270 --> 00:11:15,050 came from the two different pinholes, x1 and x2. 186 00:11:15,050 --> 00:11:17,750 But anyway, I can always write the interference 187 00:11:17,750 --> 00:11:19,940 of two waves like this. 188 00:11:19,940 --> 00:11:24,990 So wave one and wave two, it's negative square 189 00:11:24,990 --> 00:11:30,374 is going to be i0 and 1 plus m cosine, the phase difference. 190 00:11:30,374 --> 00:11:32,834 AUDIENCE: [INAUDIBLE] 191 00:11:32,834 --> 00:11:35,420 SE BAEK OH: We are assuming monochromatic, yes. 192 00:11:35,420 --> 00:11:37,050 I mean, you can write-- 193 00:11:37,050 --> 00:11:41,210 [INAUDIBLE] we use assuming monochromatic 194 00:11:41,210 --> 00:11:43,040 because we're using phase notation. 195 00:11:43,040 --> 00:11:47,510 So we write this and we write phaser, 196 00:11:47,510 --> 00:11:51,350 but actually, there is e to the i or negative i. 197 00:11:51,350 --> 00:11:55,490 So yeah, we assume it's monochromatic. 198 00:11:55,490 --> 00:11:58,100 But even though it's not chromatic, 199 00:11:58,100 --> 00:12:01,548 you can always write just addition of two waves. 200 00:12:01,548 --> 00:12:02,590 It really doesn't matter. 201 00:12:08,770 --> 00:12:12,460 So interference is basically the dc term which is i0. 202 00:12:12,460 --> 00:12:16,210 So i0 is actually a1 squared plus a2 squared, 203 00:12:16,210 --> 00:12:20,800 which is intensity of wave 1 and intensity of wave 2. 204 00:12:20,800 --> 00:12:25,090 So dc term is just addition of the intensity, 205 00:12:25,090 --> 00:12:28,400 and we have some cosine modulation term. 206 00:12:33,430 --> 00:12:36,890 So this is basically interference. 207 00:12:36,890 --> 00:12:43,000 So if we have coherent light, then we have some dc term, 208 00:12:43,000 --> 00:12:47,020 and have a sinusoidal modulation. 209 00:12:47,020 --> 00:12:52,660 And as the phase varies, we're going 210 00:12:52,660 --> 00:12:55,640 to see the sinusoidal pattern is changing. 211 00:12:55,640 --> 00:12:59,740 But remember that this optical frequency is very fast-- 212 00:12:59,740 --> 00:13:03,850 like tens of gigahertz [INAUDIBLE].. 213 00:13:03,850 --> 00:13:05,500 So actually, what we measure-- 214 00:13:05,500 --> 00:13:07,780 I mean, intensity is time average of this guy. 215 00:13:07,780 --> 00:13:14,560 So we're going to take the time average of intensity. 216 00:13:14,560 --> 00:13:17,680 And if you do the math-- 217 00:13:17,680 --> 00:13:20,580 actually, we have the dc term and this kind 218 00:13:20,580 --> 00:13:25,710 of modulation term, which is time average of cosine 219 00:13:25,710 --> 00:13:27,780 with a phase difference. 220 00:13:27,780 --> 00:13:30,380 But remember that if we have incoherent light, 221 00:13:30,380 --> 00:13:32,200 this phase difference is random. 222 00:13:32,200 --> 00:13:36,270 So what is the average of cosine of random phase? 223 00:13:39,870 --> 00:13:40,799 [INAUDIBLE] 224 00:13:49,550 --> 00:13:53,280 What is the average of cosine or random phase? 225 00:13:56,440 --> 00:13:56,940 AUDIENCE: Q. 226 00:13:56,940 --> 00:13:59,370 SE BAEK OH: Yeah, Q. Cosine is oscillating 227 00:13:59,370 --> 00:14:02,160 between plus 1 and to minus 1. 228 00:14:02,160 --> 00:14:06,720 So if you have random phase and average, then you get 0. 229 00:14:06,720 --> 00:14:11,100 So if you have incoherent light, then basically intensity 230 00:14:11,100 --> 00:14:13,125 is just addition of-- 231 00:14:13,125 --> 00:14:15,210 the intensity of 2 waves, and we don't 232 00:14:15,210 --> 00:14:17,070 have any interference term. 233 00:14:17,070 --> 00:14:20,520 So that's the difference between coherent and incoherent light. 234 00:14:20,520 --> 00:14:24,030 So coherent light, you always have a modulation term. 235 00:14:24,030 --> 00:14:26,950 I mean, it could be 0 depending on phase difference. 236 00:14:26,950 --> 00:14:32,130 But in [INAUDIBLE],, you always have just addition of dc term-- 237 00:14:32,130 --> 00:14:33,744 I mean addition of intensity. 238 00:14:41,590 --> 00:14:44,530 So if it's perfectly incoherent, then we 239 00:14:44,530 --> 00:14:46,570 have just addition of intensity. 240 00:14:46,570 --> 00:14:49,875 And actually, coherent and incoherent is very-- 241 00:14:49,875 --> 00:14:53,260 actually two opposite extreme cases. 242 00:14:53,260 --> 00:14:57,197 So it's just like monochromatic. 243 00:14:57,197 --> 00:14:58,780 In real case, we are not going to have 244 00:14:58,780 --> 00:15:02,080 really monochromatic source. 245 00:15:02,080 --> 00:15:05,070 Our source is always finite bandwidth. 246 00:15:05,070 --> 00:15:06,570 It's not going to be delta function. 247 00:15:06,570 --> 00:15:07,760 So it's same thing. 248 00:15:07,760 --> 00:15:11,670 We are not going to have the perfectly coherent or perfectly 249 00:15:11,670 --> 00:15:12,250 incoherent. 250 00:15:12,250 --> 00:15:15,190 Everything is partially coherent. 251 00:15:15,190 --> 00:15:19,990 So if you want to deal with it, then you have to apply 252 00:15:19,990 --> 00:15:25,720 basically statistics to all these Fresnel and [INAUDIBLE]---- 253 00:15:25,720 --> 00:15:27,770 I mean, this kind of [INAUDIBLE] optics. 254 00:15:27,770 --> 00:15:29,950 So it's called statistical optics, 255 00:15:29,950 --> 00:15:32,980 and it's covered in 2.707. 256 00:15:32,980 --> 00:15:36,230 So if you want to learn more about it, 257 00:15:36,230 --> 00:15:39,940 then you can take 2.7 or 1.7. 258 00:15:39,940 --> 00:15:42,760 But nevertheless, the coherent and incoherent case 259 00:15:42,760 --> 00:15:46,990 are enough to describe many, many phenomena. 260 00:15:46,990 --> 00:15:53,506 So we are focused on coherent and incoherent state detection. 261 00:15:57,350 --> 00:15:58,350 Going a little bit fast. 262 00:16:02,110 --> 00:16:07,728 So-- oops. 263 00:16:07,728 --> 00:16:08,228 Sorry. 264 00:16:15,400 --> 00:16:17,620 So since we have two kinds of coherence-- so 265 00:16:17,620 --> 00:16:20,310 temporal coherence and spatial coherence. 266 00:16:20,310 --> 00:16:24,230 So actually, we have four different combinations. 267 00:16:24,230 --> 00:16:26,320 So temporarily and spatially coherent, 268 00:16:26,320 --> 00:16:29,030 and temporarily/spatially incoherent. 269 00:16:29,030 --> 00:16:33,740 And temporal incoherent, spatially coherent, temporarily 270 00:16:33,740 --> 00:16:35,940 coherent, spatially incoherent. 271 00:16:35,940 --> 00:16:39,040 So the rule of thumb-- 272 00:16:39,040 --> 00:16:42,460 so temporal coherence is related to the bandwidth 273 00:16:42,460 --> 00:16:44,590 of your source-- 274 00:16:44,590 --> 00:16:47,300 basically, the color of your source. 275 00:16:47,300 --> 00:16:50,930 So if you have one color in monochromatic, then it's-- 276 00:16:50,930 --> 00:16:51,890 question? 277 00:16:51,890 --> 00:16:55,240 AUDIENCE: Yeah, where do diode lasers go in that chart? 278 00:16:57,750 --> 00:17:00,392 SE BAEK OH: Diode laser is more like temporally 279 00:17:00,392 --> 00:17:01,350 and spatially coherent. 280 00:17:05,780 --> 00:17:07,520 So yeah, let me just continue. 281 00:17:07,520 --> 00:17:10,500 So temporal coherence is related to color. 282 00:17:10,500 --> 00:17:13,760 So if you have laser, like single-color monochromatic, 283 00:17:13,760 --> 00:17:15,869 then it's more like coherent. 284 00:17:15,869 --> 00:17:18,710 But if you have white light, like this fluorescent lamp 285 00:17:18,710 --> 00:17:21,800 or halogen lamp, then it's broadband, 286 00:17:21,800 --> 00:17:24,349 so it's temporally incoherent. 287 00:17:24,349 --> 00:17:28,520 And spatial coherence is related to actually the size 288 00:17:28,520 --> 00:17:30,820 of the source and the distance. 289 00:17:30,820 --> 00:17:32,945 So everything coming from the point source 290 00:17:32,945 --> 00:17:36,890 is spatially coherent because they're basically 291 00:17:36,890 --> 00:17:38,720 coming from the same point. 292 00:17:38,720 --> 00:17:41,550 But if your source is extended like-- 293 00:17:41,550 --> 00:17:43,820 or you have the filament in a bulb, 294 00:17:43,820 --> 00:17:46,040 or this kind of fluorescent light, 295 00:17:46,040 --> 00:17:49,190 then it's spatially incoherent. 296 00:17:49,190 --> 00:17:53,240 So the monochromatic laser source, 297 00:17:53,240 --> 00:17:59,630 laser is very monochromatic, and the output is very small. 298 00:17:59,630 --> 00:18:02,870 So laser is typically very temporarily and spatially 299 00:18:02,870 --> 00:18:05,930 incoherent, and that's why people use it 300 00:18:05,930 --> 00:18:08,360 in interferometry. 301 00:18:08,360 --> 00:18:11,060 And white light source-- 302 00:18:11,060 --> 00:18:13,620 and if you are nearby the white light source, 303 00:18:13,620 --> 00:18:19,610 then it's more like temporally and spatially incoherent. 304 00:18:19,610 --> 00:18:23,110 And actually, these two are a little bit interesting. 305 00:18:23,110 --> 00:18:26,130 So temporally incoherent or spatially coherent. 306 00:18:26,130 --> 00:18:30,020 So if you put a small pinhole in front of the white light, 307 00:18:30,020 --> 00:18:33,530 then it's temporally incoherent because it's broadband, 308 00:18:33,530 --> 00:18:36,500 but it's coming from a pinhole, so it's spatially coherent. 309 00:18:36,500 --> 00:18:42,080 At the same reason, the sunlight is very spatially coherent, 310 00:18:42,080 --> 00:18:43,730 but temporally incoherent. 311 00:18:43,730 --> 00:18:47,150 But the sun is very huge, but also the distance is very huge. 312 00:18:47,150 --> 00:18:51,240 So if you see the sun, then it looks-- 313 00:18:51,240 --> 00:18:53,000 sunlight is coming from point source, 314 00:18:53,000 --> 00:18:57,854 so that's why it's spatially coherent. 315 00:18:57,854 --> 00:19:00,500 And yeah, the light from-- 316 00:19:00,500 --> 00:19:03,948 yeah, press the button, please. 317 00:19:03,948 --> 00:19:04,860 It doesn't work? 318 00:19:04,860 --> 00:19:06,010 AUDIENCE: Yeah. 319 00:19:06,010 --> 00:19:12,110 But anyway, so one of the things that [INAUDIBLE] 320 00:19:12,110 --> 00:19:15,290 to measure coherence, but what exactly 321 00:19:15,290 --> 00:19:17,130 is coherent [INAUDIBLE]? 322 00:19:21,760 --> 00:19:25,300 SE BAEK OH: Yeah, that is a good question. 323 00:19:25,300 --> 00:19:29,830 So actually, it's a little bit lengthy to explain, 324 00:19:29,830 --> 00:19:32,380 but what essentially the light-- 325 00:19:32,380 --> 00:19:35,620 I mean, this kind of light, it excites some atoms. 326 00:19:35,620 --> 00:19:39,720 And when they are going down, you get the energy. 327 00:19:39,720 --> 00:19:43,210 Basically, light energy by e to the-- 328 00:19:43,210 --> 00:19:45,280 I mean energy is H mu. 329 00:19:49,330 --> 00:19:50,590 But it's not-- 330 00:19:50,590 --> 00:19:53,470 I mean, this kind of spontaneous emission is not correlated. 331 00:19:53,470 --> 00:19:57,410 You have many, many atoms, but they are not synchronized. 332 00:19:57,410 --> 00:20:00,020 They are emitting randomly. 333 00:20:00,020 --> 00:20:03,180 So basically, you have some finite-- 334 00:20:03,180 --> 00:20:06,430 there are wave train. 335 00:20:06,430 --> 00:20:09,690 So in that wave train, you get basically coherent. 336 00:20:09,690 --> 00:20:12,620 But you have many, many of them with the random phase. 337 00:20:12,620 --> 00:20:14,490 That's why you add them together, 338 00:20:14,490 --> 00:20:15,870 then you get incoherent. 339 00:20:15,870 --> 00:20:18,370 So coherent length is essentially 340 00:20:18,370 --> 00:20:21,910 what is the length of a finite wave train 341 00:20:21,910 --> 00:20:25,120 that you can define deterministic phase. 342 00:20:25,120 --> 00:20:28,870 So d1-- so in just imagine the Michelson interferometer, 343 00:20:28,870 --> 00:20:33,160 you have the optical path first, d1 minus d2. 344 00:20:33,160 --> 00:20:37,760 But that difference is within the distance, 345 00:20:37,760 --> 00:20:40,270 then you get interference. 346 00:20:40,270 --> 00:20:42,540 That's why you call the coherence length and coherence 347 00:20:42,540 --> 00:20:44,290 time. 348 00:20:44,290 --> 00:20:49,540 And yeah, essentially, that's it. 349 00:20:49,540 --> 00:20:52,680 So remember that if we had-- 350 00:20:59,000 --> 00:20:59,923 AUDIENCE: Excuse me. 351 00:20:59,923 --> 00:21:00,690 SE BAEK OH: Yep? 352 00:21:00,690 --> 00:21:02,580 AUDIENCE: Here I also will have one question. 353 00:21:02,580 --> 00:21:05,510 So I want to check whether this understanding is correct. 354 00:21:05,510 --> 00:21:09,470 So let's say if the light comes at a perfect parallel 355 00:21:09,470 --> 00:21:12,350 direction, even though they have different wavelengths, 356 00:21:12,350 --> 00:21:15,210 it's considered especially coherent. 357 00:21:15,210 --> 00:21:18,230 And even if light go into different directions, 358 00:21:18,230 --> 00:21:19,490 but [INAUDIBLE] perfect-- 359 00:21:19,490 --> 00:21:23,370 the single wavelength, they're considered time coherent. 360 00:21:23,370 --> 00:21:25,260 Is this understanding correct? 361 00:21:25,260 --> 00:21:27,590 SE BAEK OH: So actually, if you have plain wave, 362 00:21:27,590 --> 00:21:31,950 then you already assume your wave is spatially coherent. 363 00:21:31,950 --> 00:21:36,410 Because plane wave, it doesn't really matter where you sample. 364 00:21:36,410 --> 00:21:38,090 They have always same phase. 365 00:21:38,090 --> 00:21:41,880 So your assumption is spatially coherent 366 00:21:41,880 --> 00:21:44,610 if you assume spatially coherent light. 367 00:21:47,260 --> 00:21:51,602 AUDIENCE: You mean even if they have different wavelengths? 368 00:21:51,602 --> 00:21:52,310 SE BAEK OH: Yeah. 369 00:21:52,310 --> 00:21:53,270 Even though you have different wavelengths, 370 00:21:53,270 --> 00:21:54,500 that's temporally incoherent. 371 00:21:54,500 --> 00:21:56,840 But in space domain, you have spatially coherent. 372 00:21:59,880 --> 00:22:02,610 Another way to think of spatially incoherent 373 00:22:02,610 --> 00:22:06,150 is you have many, many plane waves propagating 374 00:22:06,150 --> 00:22:10,140 all the different directions, but they are randomly-- 375 00:22:10,140 --> 00:22:12,060 they have random initial phase. 376 00:22:12,060 --> 00:22:16,570 So especially in coherent light, you cannot really define 377 00:22:16,570 --> 00:22:19,620 a unique propagation direction. 378 00:22:19,620 --> 00:22:21,870 But spatially coherent light, you 379 00:22:21,870 --> 00:22:24,090 can always define direction. 380 00:22:24,090 --> 00:22:26,970 Even though spherical wave it has all the directions, 381 00:22:26,970 --> 00:22:28,740 but you can always say-- 382 00:22:28,740 --> 00:22:32,520 if you put-- if you define the position, then that position, 383 00:22:32,520 --> 00:22:34,390 you have a k vector. 384 00:22:34,390 --> 00:22:36,778 So you can define the direction. 385 00:22:46,420 --> 00:22:49,780 So let me just finish this thing. 386 00:22:49,780 --> 00:22:53,740 So if d1 and d2 are within the coherence length or coherence 387 00:22:53,740 --> 00:22:56,700 time, then you get some interference. 388 00:22:56,700 --> 00:22:58,872 But after that-- so basically, this 389 00:22:58,872 --> 00:23:00,330 is essentially the coherence length 390 00:23:00,330 --> 00:23:03,300 of this particular source. 391 00:23:03,300 --> 00:23:04,500 And remember that-- 392 00:23:04,500 --> 00:23:05,970 I mean, this always happens. 393 00:23:05,970 --> 00:23:08,220 So even though you have a white light, 394 00:23:08,220 --> 00:23:12,000 you can get interference, but within a very limited range. 395 00:23:44,250 --> 00:23:45,000 Where did I stop? 396 00:23:51,580 --> 00:23:52,205 [CLEARS THROAT] 397 00:23:52,205 --> 00:23:53,048 Excuse me. 398 00:23:53,048 --> 00:23:54,590 So another interesting thing is there 399 00:23:54,590 --> 00:23:59,720 is a laser that has a broadband which is [INAUDIBLE] laser. 400 00:23:59,720 --> 00:24:02,810 So it's spatially coherent, but it's broadband, 401 00:24:02,810 --> 00:24:06,050 so you have temporally incoherence. 402 00:24:06,050 --> 00:24:08,600 And the other kind of this combination 403 00:24:08,600 --> 00:24:13,190 is temporally coherent, but it's spatially incoherent. 404 00:24:13,190 --> 00:24:19,170 And it's typically referred to as quasi-monochromatic-- 405 00:24:19,170 --> 00:24:23,390 temporally coherent and spatially incoherent. 406 00:24:23,390 --> 00:24:25,400 So if you put a rotating diffuser, which 407 00:24:25,400 --> 00:24:31,400 is when you rotate the ground glass or milky glass in front 408 00:24:31,400 --> 00:24:36,500 of the laser, then basically, it induces some random phase 409 00:24:36,500 --> 00:24:39,950 to the laser beam where we can say it destroyed 410 00:24:39,950 --> 00:24:42,120 the correlation of the related. 411 00:24:42,120 --> 00:24:45,770 So it's monochromatic, so it temporally coheres. 412 00:24:45,770 --> 00:24:48,270 But in space domain, it is a random phase, 413 00:24:48,270 --> 00:24:52,040 so you can have the spatially incoherent light. 414 00:24:56,430 --> 00:24:59,750 So these are a list of optical instruments using 415 00:24:59,750 --> 00:25:02,416 spatial coherence on imaging. 416 00:25:02,416 --> 00:25:05,420 The Michelson stellar interferometer and the radio 417 00:25:05,420 --> 00:25:09,920 telescope, they are basically measuring spatial coherence 418 00:25:09,920 --> 00:25:14,260 of the light coming from a distant star, or sun, or-- 419 00:25:14,260 --> 00:25:16,340 yeah, they're basically for astronomy. 420 00:25:16,340 --> 00:25:18,845 So they can measure, for example, 421 00:25:18,845 --> 00:25:22,860 the diameter of a distant star. 422 00:25:22,860 --> 00:25:24,330 Remember, the spatial coherence is 423 00:25:24,330 --> 00:25:26,410 related to the source size and the distance. 424 00:25:26,410 --> 00:25:29,080 So if you properly measure spatial coherence-- 425 00:25:29,080 --> 00:25:31,290 actually, the envelope of the spatial coherence, 426 00:25:31,290 --> 00:25:33,450 then you can estimate the diameter of the star. 427 00:25:37,072 --> 00:25:39,030 I believe Michelson got a Nobel Prize for this. 428 00:25:42,694 --> 00:25:46,220 And optical coherence tomography is actually 429 00:25:46,220 --> 00:25:48,310 utilizing temporal incoherence. 430 00:25:48,310 --> 00:25:51,210 So as I just described, even though you have white light, 431 00:25:51,210 --> 00:25:54,200 if you properly match the optical path difference, 432 00:25:54,200 --> 00:25:58,440 then you can get interference within a very small range. 433 00:25:58,440 --> 00:26:02,840 So let's say you have multiple layers like tissue, 434 00:26:02,840 --> 00:26:05,120 and you construct the Michelson interferometer 435 00:26:05,120 --> 00:26:08,370 with the white light, and as you move the other mirror, 436 00:26:08,370 --> 00:26:10,490 then if the optical path difference matches, 437 00:26:10,490 --> 00:26:11,870 then you get interference. 438 00:26:11,870 --> 00:26:14,120 But if you slightly off that point, 439 00:26:14,120 --> 00:26:16,130 then you don't get the interference. 440 00:26:16,130 --> 00:26:19,550 So if you plot the interference versus the distance 441 00:26:19,550 --> 00:26:22,550 of the mirror, then you can see kind of the peak 442 00:26:22,550 --> 00:26:30,440 whenever you have the right position at each layer. 443 00:26:30,440 --> 00:26:35,220 So you can kind of do the optical sectioning. 444 00:26:35,220 --> 00:26:39,210 And in the lithography replication technique, 445 00:26:39,210 --> 00:26:42,470 people use some spatial coherence 446 00:26:42,470 --> 00:26:49,640 to basically make some nice pattern. 447 00:26:49,640 --> 00:26:52,460 Because as we described spatial coherence 448 00:26:52,460 --> 00:26:57,110 is essentially affecting this resolution 449 00:26:57,110 --> 00:27:00,340 and the contrast of your image. 450 00:27:00,340 --> 00:27:14,240 So there are these kind of things 451 00:27:14,240 --> 00:27:16,720 Even though we deal with the spatially coherent 452 00:27:16,720 --> 00:27:20,200 with the first system, so what if we change the illumination 453 00:27:20,200 --> 00:27:24,450 to a spatially incoherent or temporally incoherent? 454 00:27:24,450 --> 00:27:30,100 And obviously, my image will be different depending 455 00:27:30,100 --> 00:27:33,055 on the coherent state. 456 00:27:33,055 --> 00:27:38,880 Actually, that's the bottom line of today's lecture. 457 00:27:38,880 --> 00:27:42,100 So first, if we have the temporal incoherent-- 458 00:27:42,100 --> 00:27:44,630 basically, if you have the broadband for many, 459 00:27:44,630 --> 00:27:45,730 many colors-- 460 00:27:45,730 --> 00:27:48,880 then the chromatic aberration is typical evidence 461 00:27:48,880 --> 00:27:53,410 that I have temporal incoherent source. 462 00:27:53,410 --> 00:27:58,420 Then how to deal with it if you have many, many colors? 463 00:27:58,420 --> 00:28:00,010 That's pretty straightforward. 464 00:28:00,010 --> 00:28:03,940 So you can define point spread function, or ATF, or out-- 465 00:28:03,940 --> 00:28:05,770 or whatever it is. 466 00:28:05,770 --> 00:28:08,720 You can define that thing for each wavelength. 467 00:28:08,720 --> 00:28:11,500 So for example, if I can define point spread function 468 00:28:11,500 --> 00:28:13,750 for red wavelength, I can define point spread function 469 00:28:13,750 --> 00:28:15,200 for blue wavelengths. 470 00:28:15,200 --> 00:28:16,000 So it's just-- 471 00:28:16,000 --> 00:28:19,840 I can do with a straightforward equation. 472 00:28:19,840 --> 00:28:23,010 And spatial coherence is a little bit-- 473 00:28:23,010 --> 00:28:25,090 and you need to be careful. 474 00:28:25,090 --> 00:28:28,720 So the bottom-- actually, this is the take-home message 475 00:28:28,720 --> 00:28:30,430 of the lecture. 476 00:28:30,430 --> 00:28:32,680 If the illumination is spatially coherent, 477 00:28:32,680 --> 00:28:37,950 then input field is conversion of the-- actually, 478 00:28:37,950 --> 00:28:43,090 output field is a convolution of input field 479 00:28:43,090 --> 00:28:48,130 with a coherent point spread function, as we all know. 480 00:28:48,130 --> 00:28:50,710 But if the illumination is spatially incoherent, 481 00:28:50,710 --> 00:28:54,280 then output intensity, not the output field. 482 00:28:54,280 --> 00:28:57,160 So output intensity is a convolution 483 00:28:57,160 --> 00:29:00,950 of the input intensity with the incoherent point spread 484 00:29:00,950 --> 00:29:01,450 function. 485 00:29:05,170 --> 00:29:12,450 So that's the-- which means if you have spatial coherent 486 00:29:12,450 --> 00:29:13,675 illumination, then-- 487 00:29:13,675 --> 00:29:15,270 fields are linear systems. 488 00:29:15,270 --> 00:29:17,710 You can do the convolution in field. 489 00:29:17,710 --> 00:29:20,530 But if the illumination is incoherent, 490 00:29:20,530 --> 00:29:22,730 then intensity is a linear system, 491 00:29:22,730 --> 00:29:27,640 so you can deal with the intensity with the convolution. 492 00:29:27,640 --> 00:29:29,140 So rest of the lecture, we are going 493 00:29:29,140 --> 00:29:33,200 to deal with the spatial incoherent illumination. 494 00:29:33,200 --> 00:29:37,060 So how we can define point spread function, and transfer 495 00:29:37,060 --> 00:29:38,260 function, and et cetera. 496 00:29:46,760 --> 00:29:51,720 So first, let's review what we have learned so far. 497 00:29:51,720 --> 00:29:55,040 So we have fourth system with a two length 498 00:29:55,040 --> 00:29:57,650 whose focal lengths are f1 and f2. 499 00:29:57,650 --> 00:30:00,890 And typically, we put our input transparency 500 00:30:00,890 --> 00:30:03,290 grating at the input plane. 501 00:30:03,290 --> 00:30:07,610 And we illuminate with the spatially coherent illumination 502 00:30:07,610 --> 00:30:11,560 and also temporally coherent. 503 00:30:11,560 --> 00:30:15,670 So the input field just right after the input 504 00:30:15,670 --> 00:30:20,560 transparency is going to be just a product of the illumination 505 00:30:20,560 --> 00:30:24,010 field and the complex transparency. 506 00:30:24,010 --> 00:30:25,990 That's the input field. 507 00:30:25,990 --> 00:30:29,260 And you're going to take the Fourier transform to get 508 00:30:29,260 --> 00:30:31,030 the field at the wave plane. 509 00:30:31,030 --> 00:30:35,090 And another Fourier transform, you get the final image. 510 00:30:35,090 --> 00:30:38,890 And from that, we all know that amplitude [INAUDIBLE] 511 00:30:38,890 --> 00:30:43,690 function is nothing but the [INAUDIBLE] function 512 00:30:43,690 --> 00:30:46,660 with the coordinate scaling. 513 00:30:46,660 --> 00:30:51,710 So ATF, H of u [INAUDIBLE] some constant 514 00:30:51,710 --> 00:30:57,310 and [INAUDIBLE] mask function, and lambda f1u and lambda f1v. 515 00:31:00,130 --> 00:31:03,550 And the point spread function in field 516 00:31:03,550 --> 00:31:09,320 is just Fourier transform of the ATF. 517 00:31:09,320 --> 00:31:12,800 Or you can say it's field transform [INAUDIBLE] function. 518 00:31:12,800 --> 00:31:16,000 So this is the coherent point spread function. 519 00:31:16,000 --> 00:31:19,840 And you can define [INAUDIBLE] function reduce coordinate. 520 00:31:19,840 --> 00:31:23,530 And the relation between input and output field 521 00:31:23,530 --> 00:31:25,180 is this convolution. 522 00:31:25,180 --> 00:31:28,180 So G output-- I mean, output field 523 00:31:28,180 --> 00:31:31,000 is convolution of input field, and this 524 00:31:31,000 --> 00:31:33,070 is coherent point spread function 525 00:31:33,070 --> 00:31:36,010 which is defined in field. 526 00:31:36,010 --> 00:31:41,210 So this is the stuff that we all know. 527 00:31:41,210 --> 00:31:47,140 So the next thing is let me put the two pinholes at the input 528 00:31:47,140 --> 00:31:47,640 plane. 529 00:31:50,760 --> 00:31:53,646 And illuminate with the spatially coherent light. 530 00:31:53,646 --> 00:31:56,730 So basically, we have Young interferometer here, 531 00:31:56,730 --> 00:31:59,640 and I'm going to image that pattern. 532 00:32:03,310 --> 00:32:06,480 And since it is coherent-- it is illuminated 533 00:32:06,480 --> 00:32:07,810 with the coherent light. 534 00:32:07,810 --> 00:32:12,360 So essentially, I have-- at the image plane, 535 00:32:12,360 --> 00:32:15,610 there's actually two point spread functions 536 00:32:15,610 --> 00:32:17,140 coming from two pinholes. 537 00:32:17,140 --> 00:32:19,440 So in the upper image is actually 538 00:32:19,440 --> 00:32:21,790 coming from the lower pinhole, and the lower image 539 00:32:21,790 --> 00:32:23,530 actually coming from the top pinhole. 540 00:32:23,530 --> 00:32:27,730 But I have the addition of two point spread 541 00:32:27,730 --> 00:32:31,310 functions which is shifted by the position of the two 542 00:32:31,310 --> 00:32:31,810 pinholes. 543 00:32:31,810 --> 00:32:36,520 And I may have some initial phase, a1, e to the phi 1, 544 00:32:36,520 --> 00:32:37,570 and a to-- 545 00:32:37,570 --> 00:32:38,710 e to the phi 2. 546 00:32:38,710 --> 00:32:42,230 But the bottom line is in coherent case, 547 00:32:42,230 --> 00:32:46,250 I just have coherent addition of two point spread 548 00:32:46,250 --> 00:32:48,640 functions like this. 549 00:32:48,640 --> 00:32:51,620 And to get the intensity, as you all know, 550 00:32:51,620 --> 00:32:55,450 we just take the negative square of the sum, 551 00:32:55,450 --> 00:32:59,300 and if we expand it, then we have [INAUDIBLE] 552 00:32:59,300 --> 00:33:01,810 which is the intensity of the first point spread 553 00:33:01,810 --> 00:33:05,170 function and intensity of the second point spread function. 554 00:33:05,170 --> 00:33:09,330 And we have some additional modulation term, which 555 00:33:09,330 --> 00:33:11,470 means I have some interference. 556 00:33:11,470 --> 00:33:14,920 So it's a1, a2, and rear path of this guy. 557 00:33:14,920 --> 00:33:17,860 But rear path is always cosine. 558 00:33:17,860 --> 00:33:21,890 So if you change the phase, you have some sinusoidal modulation 559 00:33:21,890 --> 00:33:22,390 term. 560 00:33:26,410 --> 00:33:29,470 So this is sometimes called interference term 561 00:33:29,470 --> 00:33:31,630 or [INAUDIBLE] term. 562 00:33:31,630 --> 00:33:35,440 It implies that you have interference. 563 00:33:35,440 --> 00:33:38,350 So with these parameters-- so initial phase 564 00:33:38,350 --> 00:33:41,920 is just d to the negative pi over 3, 565 00:33:41,920 --> 00:33:45,790 and the other one is the positive pi over 3. 566 00:33:45,790 --> 00:33:49,330 Then if I plot this intensity, then I'm 567 00:33:49,330 --> 00:33:51,760 going to have this blue line. 568 00:33:55,100 --> 00:34:00,410 So this is the case of spatially coherent illumination. 569 00:34:00,410 --> 00:34:03,755 And next, let's suppose I illuminate 570 00:34:03,755 --> 00:34:07,970 with the incoherent illumination. 571 00:34:07,970 --> 00:34:13,989 So as I described earlier, the difference 572 00:34:13,989 --> 00:34:17,350 between incoherent and coherent is 573 00:34:17,350 --> 00:34:19,480 I don't have interference term if I illuminate 574 00:34:19,480 --> 00:34:21,880 with the incoherent source. 575 00:34:21,880 --> 00:34:26,035 So at the intensity, I only have these two terms-- 576 00:34:26,035 --> 00:34:27,410 just intensity of the first point 577 00:34:27,410 --> 00:34:29,743 spread function and intensity of the second point spread 578 00:34:29,743 --> 00:34:30,340 function. 579 00:34:30,340 --> 00:34:34,870 And I don't have the extra cosine term. 580 00:34:34,870 --> 00:34:39,460 So this is just intensity. 581 00:34:39,460 --> 00:34:44,469 So if I plot the intensity with the same parameter, 582 00:34:44,469 --> 00:34:48,020 then this is how it looks. 583 00:34:48,020 --> 00:34:52,330 And if I put together, then the upper left 584 00:34:52,330 --> 00:35:01,670 is the two intensity profiles for coherent and incoherent 585 00:35:01,670 --> 00:35:03,050 illumination. 586 00:35:03,050 --> 00:35:05,460 So they are slightly different. 587 00:35:05,460 --> 00:35:07,270 And if I play with a different parameter, 588 00:35:07,270 --> 00:35:11,120 like the different initial phase. 589 00:35:11,120 --> 00:35:14,440 And in this case, my point spread function 590 00:35:14,440 --> 00:35:17,170 has extra linear phase term, then 591 00:35:17,170 --> 00:35:19,450 the difference is quite dramatic. 592 00:35:19,450 --> 00:35:23,140 So you always have to be careful what kind of coherence 593 00:35:23,140 --> 00:35:26,490 your illumination is. 594 00:35:26,490 --> 00:35:27,829 Any questions? 595 00:35:39,500 --> 00:35:44,370 So in previous case, which is spatially coherent, 596 00:35:44,370 --> 00:35:48,870 then the input field and output field is linear, 597 00:35:48,870 --> 00:35:51,020 so you can use the convolution. 598 00:35:51,020 --> 00:35:57,950 But in spatially incoherent case, 599 00:35:57,950 --> 00:36:00,590 we deal with the input intensity which is-- 600 00:36:06,920 --> 00:36:10,760 so we're going to deal with the input intensity and output 601 00:36:10,760 --> 00:36:11,480 intensity. 602 00:36:11,480 --> 00:36:14,570 And at the image plane, the intensity 603 00:36:14,570 --> 00:36:19,070 will be [INAUDIBLE] incoherently. 604 00:36:19,070 --> 00:36:26,940 So I illuminate with the intensity-- 605 00:36:26,940 --> 00:36:31,380 I mean, incoherent illumination with the intensity [INAUDIBLE],, 606 00:36:31,380 --> 00:36:34,130 and g sub t is the complex transparency. 607 00:36:34,130 --> 00:36:38,820 But this is field, so you have to take negative square. 608 00:36:38,820 --> 00:36:47,460 So product of this guy is basically your input intensity. 609 00:36:47,460 --> 00:36:50,870 So this is illumination intensity and-- 610 00:36:50,870 --> 00:36:53,130 intensity of your grating-- 611 00:36:53,130 --> 00:36:59,790 and this is point spread function-- actually, 612 00:36:59,790 --> 00:37:01,440 coherent point spread function where 613 00:37:01,440 --> 00:37:04,320 you take the negative square because you 614 00:37:04,320 --> 00:37:07,245 take the negative square to get intensity. 615 00:37:07,245 --> 00:37:09,370 [CLEARS THROAT] Excuse me. 616 00:37:09,370 --> 00:37:12,150 We define incoherent point spread function, 617 00:37:12,150 --> 00:37:13,770 which is ipsf-- 618 00:37:13,770 --> 00:37:16,540 it's actually negative square over the coherent point spread 619 00:37:16,540 --> 00:37:18,570 function. 620 00:37:18,570 --> 00:37:19,710 It's pretty simple, right? 621 00:37:19,710 --> 00:37:23,580 You have point spread function in field. 622 00:37:23,580 --> 00:37:25,170 And if you just take the intensity 623 00:37:25,170 --> 00:37:29,750 and test the intensity point spread function. 624 00:37:29,750 --> 00:37:33,220 But you probably already noticed the difference. 625 00:37:33,220 --> 00:37:36,870 So if you use the coherent illumination, 626 00:37:36,870 --> 00:37:40,450 then the fields are related to convolution, 627 00:37:40,450 --> 00:37:42,240 but in coherent case, intensities 628 00:37:42,240 --> 00:37:47,950 are related to convolution, and point spread function 629 00:37:47,950 --> 00:37:48,957 are different. 630 00:37:55,976 --> 00:37:57,870 Any questions? 631 00:37:57,870 --> 00:38:00,040 I'm going to [INAUDIBLE]. 632 00:38:11,300 --> 00:38:14,420 So we just defined the intensity point spread function, 633 00:38:14,420 --> 00:38:17,370 which is negative square coherent point spread function. 634 00:38:17,370 --> 00:38:23,100 And by analogy, I mean in the coherence case, 635 00:38:23,100 --> 00:38:25,120 we had the coherent point spread function. 636 00:38:25,120 --> 00:38:28,230 And if you take the Fourier transform of the coherent point 637 00:38:28,230 --> 00:38:32,220 spread function, then that is ATF-- 638 00:38:32,220 --> 00:38:34,150 amplitude transfer function. 639 00:38:34,150 --> 00:38:36,540 So we could define the same transform function 640 00:38:36,540 --> 00:38:42,490 for incoherent point spread function, which is OTF here. 641 00:38:42,490 --> 00:38:47,220 So [INAUDIBLE] says that-- 642 00:38:52,380 --> 00:38:58,080 so actually, the small h is the coherent point spread function, 643 00:38:58,080 --> 00:39:02,790 and the capital H is ATF. 644 00:39:02,790 --> 00:39:06,180 So they are a Fourier transform pair. 645 00:39:06,180 --> 00:39:08,820 And hI, which is intensity point spread function, 646 00:39:08,820 --> 00:39:12,750 is the negative square of the coherent point spread function. 647 00:39:12,750 --> 00:39:21,080 But hI and OTF which is this weird h, 648 00:39:21,080 --> 00:39:26,740 they are also a Fourier transform pair. 649 00:39:26,740 --> 00:39:28,900 So if you go through this derivation, 650 00:39:28,900 --> 00:39:33,630 then it will prove actually they are-- 651 00:39:33,630 --> 00:39:47,810 that OTF is actually this integration of ATF. 652 00:39:47,810 --> 00:39:53,930 So once you have ATF, and after you compute this information, 653 00:39:53,930 --> 00:39:57,590 then you get OTF which is auto correlation. 654 00:39:57,590 --> 00:40:01,860 And I'm going to derive in a different way here. 655 00:40:05,390 --> 00:40:06,680 [INAUDIBLE] please. 656 00:40:17,500 --> 00:40:23,260 So we start with hI of x, which is intensity point spread 657 00:40:23,260 --> 00:40:27,200 function is negative square of the coherent point spread 658 00:40:27,200 --> 00:40:29,280 function. 659 00:40:29,280 --> 00:40:30,172 And-- yes? 660 00:40:30,172 --> 00:40:31,672 AUDIENCE: Can you write much bigger? 661 00:40:31,672 --> 00:40:32,580 [INAUDIBLE] 662 00:40:32,580 --> 00:40:33,497 SE BAEK OH: All right. 663 00:40:35,780 --> 00:40:36,280 [LAUGHS] 664 00:40:36,280 --> 00:40:38,560 Let me try. 665 00:40:38,560 --> 00:40:41,018 That's too big, right? 666 00:40:41,018 --> 00:40:42,419 AUDIENCE: [INAUDIBLE] 667 00:40:42,419 --> 00:40:43,360 [LAUGHS] 668 00:40:43,360 --> 00:40:44,510 SE BAEK OH: It's too small? 669 00:40:44,510 --> 00:40:45,010 OK. 670 00:40:55,790 --> 00:40:58,790 A little bit smaller. 671 00:40:58,790 --> 00:41:01,970 We start with this thing, and we know 672 00:41:01,970 --> 00:41:10,310 that the OTF is Fourier transform of intensity point 673 00:41:10,310 --> 00:41:11,020 spread function. 674 00:41:17,020 --> 00:41:19,250 And I can just plug into here. 675 00:41:19,250 --> 00:41:25,020 So it's Fourier transform of negative square 676 00:41:25,020 --> 00:41:30,430 of the coherent point spread function and Fourier transform. 677 00:41:30,430 --> 00:41:33,640 And since it is complex conjugate, 678 00:41:33,640 --> 00:41:38,230 I can write as h of x. 679 00:41:38,230 --> 00:41:40,652 And it's complex conjugate. 680 00:41:45,660 --> 00:41:48,530 And since these two are products, 681 00:41:48,530 --> 00:41:53,570 then I can write as convolution of two guys. 682 00:41:53,570 --> 00:41:58,210 So it's Fourier transform of h of x, 683 00:41:58,210 --> 00:42:01,810 conversion with Fourier transform 684 00:42:01,810 --> 00:42:06,240 of a complex conjugate like this. 685 00:42:09,100 --> 00:42:11,980 [INAUDIBLE] 686 00:42:11,980 --> 00:42:20,920 And Fourier transform of the H is actually H of u-- 687 00:42:20,920 --> 00:42:25,650 I mean this guy. 688 00:42:25,650 --> 00:42:29,610 ATF is just nothing but Fourier transform of the coherent point 689 00:42:29,610 --> 00:42:30,860 spread function. 690 00:42:30,860 --> 00:42:32,980 And what about this guy? 691 00:42:32,980 --> 00:42:38,780 Fourier transform of complex conjugate of small h. 692 00:42:38,780 --> 00:42:41,720 So I can write this in integral form. 693 00:42:41,720 --> 00:42:46,964 So integral h, complex conjugate of x. 694 00:42:46,964 --> 00:42:53,030 e to the minus j2pi, xu, dx. 695 00:42:56,250 --> 00:42:58,320 And since it is complex conjugate, 696 00:42:58,320 --> 00:43:03,450 I can write as integral h of x, e 697 00:43:03,450 --> 00:43:16,531 to the I2pi xu, dx, x, and conjugate. 698 00:43:21,720 --> 00:43:34,670 So I can write as OTF is H of u convolution, 699 00:43:34,670 --> 00:43:46,010 and complex conjugate of Fourier transform of-- 700 00:43:46,010 --> 00:43:46,950 like this. 701 00:43:51,200 --> 00:43:53,483 Actually, [INAUDIBLE] transform. 702 00:44:00,110 --> 00:44:02,570 So actually, this inverse Fourier transform, 703 00:44:02,570 --> 00:44:05,540 if you remember, the difference of the Fourier transform 704 00:44:05,540 --> 00:44:15,390 is actually integral h of x, e to the I2pi xu, dx. 705 00:44:15,390 --> 00:44:15,890 But 706 00:44:15,890 --> 00:44:24,320 You can write as h of x, e to the minus 2pi x minus u, dx. 707 00:44:24,320 --> 00:44:29,390 So actually, it is Fourier transform, and h of x. 708 00:44:29,390 --> 00:44:31,850 But you change the spatial frequency variable 709 00:44:31,850 --> 00:44:35,360 as a minus u, So it's going to be H of minus u. 710 00:44:41,260 --> 00:44:47,530 So at the end, we have H of u convolution, 711 00:44:47,530 --> 00:44:49,450 and complex conjugate of-- 712 00:44:52,282 --> 00:44:53,730 ah, sorry. 713 00:44:56,300 --> 00:44:59,025 H complex conjugate of minus u. 714 00:45:04,850 --> 00:45:10,180 [INAUDIBLE] where I can write as an integral form. 715 00:45:10,180 --> 00:45:23,880 Integration H of u prime, H of minus u 716 00:45:23,880 --> 00:45:30,330 minus u prime, complex conjugate, and du prime. 717 00:45:30,330 --> 00:45:37,100 So essentially, integration H complex conjugate-- 718 00:45:37,100 --> 00:45:38,254 sorry. 719 00:45:38,254 --> 00:45:47,400 So I have H of u prime, complex conjugate u prime minus 720 00:45:47,400 --> 00:45:53,610 u, du prime, which is essentially 721 00:45:53,610 --> 00:45:57,750 this guy in [INAUDIBLE]. 722 00:46:04,400 --> 00:46:11,185 And sometimes, we denote by H of u [INAUDIBLE] H of u, 723 00:46:11,185 --> 00:46:14,332 which is auto-correlation. 724 00:46:28,340 --> 00:46:30,497 Any questions on the derivation? 725 00:46:42,880 --> 00:46:47,010 So before we move on, can anyone guess 726 00:46:47,010 --> 00:46:49,810 what this integration means? 727 00:46:49,810 --> 00:46:53,520 So you have-- you want to compute the OTF, which 728 00:46:53,520 --> 00:46:56,370 is auto-correlation of the ATF. 729 00:46:56,370 --> 00:47:01,920 But this integration is basically you have H, 730 00:47:01,920 --> 00:47:03,400 and you take the complex conjugate, 731 00:47:03,400 --> 00:47:06,350 but you shift and take the integral. 732 00:47:15,100 --> 00:47:21,220 So next slide, I'm going to explain how we can compute-- 733 00:47:21,220 --> 00:47:23,790 so bottom line of this slide is OTF 734 00:47:23,790 --> 00:47:27,130 is nothing but auto-correlation of your ATF. 735 00:47:27,130 --> 00:47:30,960 But next slide is how to compute when-- 736 00:47:30,960 --> 00:47:32,820 you can simply compute this integration, 737 00:47:32,820 --> 00:47:38,695 but there is another easy way to guess the OTF. 738 00:47:38,695 --> 00:47:44,950 AUDIENCE: [INAUDIBLE] 739 00:47:44,950 --> 00:47:46,290 SE BAEK OH: Conceptually, yes. 740 00:47:46,290 --> 00:47:48,120 I mean, actually, Fourier transform 741 00:47:48,120 --> 00:47:51,420 of the input intensity and Fourier transform of the output 742 00:47:51,420 --> 00:47:53,480 intensity, and the ratio of [INAUDIBLE].. 743 00:48:08,060 --> 00:48:12,290 So let's first compute the OTF for the simple case. 744 00:48:12,290 --> 00:48:17,930 I have the rectangular aperture in my [INAUDIBLE] plane. 745 00:48:17,930 --> 00:48:21,100 And from that, I know that the ATF-- 746 00:48:21,100 --> 00:48:25,280 H of u-- is just [INAUDIBLE] function, and u over umx. 747 00:48:25,280 --> 00:48:28,130 So umx is defined by the physical size 748 00:48:28,130 --> 00:48:29,780 of the [INAUDIBLE] function. 749 00:48:29,780 --> 00:48:32,640 And so if it's just x, double prime x, 750 00:48:32,640 --> 00:48:37,560 and with the scaling vector of lambda F1. 751 00:48:37,560 --> 00:48:40,520 So this is the ATF. 752 00:48:40,520 --> 00:48:46,670 And I just put that OTF is auto-correlation of ATF. 753 00:48:46,670 --> 00:48:49,970 In integral form, I have this integration. 754 00:48:49,970 --> 00:48:54,650 So let's just take a look at this [INAUDIBLE] actually. 755 00:48:54,650 --> 00:48:58,325 So OTF is function of u, but this integration-- 756 00:48:58,325 --> 00:49:00,340 I mean inside of this integration, 757 00:49:00,340 --> 00:49:04,020 I have function of H of u prime. 758 00:49:04,020 --> 00:49:07,220 And H complex conjugate, u prime minus u. 759 00:49:07,220 --> 00:49:09,470 So I have the first function, which 760 00:49:09,470 --> 00:49:13,330 is essentially same as my original function 761 00:49:13,330 --> 00:49:18,230 even though I change the variable from u to u prime. 762 00:49:18,230 --> 00:49:24,560 But this guy is function of u prime where you shift by u. 763 00:49:24,560 --> 00:49:25,920 And [INAUDIBLE] integration. 764 00:49:25,920 --> 00:49:28,490 And this whole thing is function of u, 765 00:49:28,490 --> 00:49:30,890 which means you're going to compute 766 00:49:30,890 --> 00:49:37,370 this integration for every u, and then you get OTF. 767 00:49:37,370 --> 00:49:39,410 And this integration is basically 768 00:49:39,410 --> 00:49:43,010 you have two functions, one is shifted by u, 769 00:49:43,010 --> 00:49:46,130 and multiply them, and take the integration, which 770 00:49:46,130 --> 00:49:53,060 is essentially you compute the overlapped area of the two 771 00:49:53,060 --> 00:49:53,850 functions. 772 00:49:57,050 --> 00:50:00,775 So graphically, if I have [INAUDIBLE] function and plug 773 00:50:00,775 --> 00:50:05,180 it in this integration, I have this integration. 774 00:50:05,180 --> 00:50:07,940 But I can imagine for a different situation, 775 00:50:07,940 --> 00:50:10,100 the first-- 776 00:50:10,100 --> 00:50:13,560 so this [INAUDIBLE] function is the coherent [INAUDIBLE] 777 00:50:13,560 --> 00:50:14,780 function. 778 00:50:14,780 --> 00:50:17,180 And the second one which is moving 779 00:50:17,180 --> 00:50:19,170 is a coherent [INAUDIBLE] function. 780 00:50:19,170 --> 00:50:21,650 So if they are not overlapped, then I 781 00:50:21,650 --> 00:50:25,490 get simply 0 the first situation. 782 00:50:25,490 --> 00:50:28,820 And as soon as they are overlapped, 783 00:50:28,820 --> 00:50:35,520 and then the overlapped area is getting linearly increased. 784 00:50:35,520 --> 00:50:40,430 So I get this u plus umx until they are completely overlapped. 785 00:50:40,430 --> 00:50:43,670 And then-- so this is [INAUDIBLE]---- 786 00:50:43,670 --> 00:50:50,660 and then once you move a little bit further, 787 00:50:50,660 --> 00:50:54,510 then basically this overlapped area is getting decreased. 788 00:50:54,510 --> 00:50:58,260 So this is third situation, which is this guy. 789 00:50:58,260 --> 00:51:01,370 And eventually, they are not going to overlap, 790 00:51:01,370 --> 00:51:04,850 so I get 0 again. 791 00:51:04,850 --> 00:51:07,100 Remember that this OTF is functional of u, 792 00:51:07,100 --> 00:51:11,000 which is the amount of shift of this [INAUDIBLE] function. 793 00:51:16,357 --> 00:51:18,752 Any questions? 794 00:51:18,752 --> 00:51:22,620 So if you plot this overlapped area-- 795 00:51:22,620 --> 00:51:24,840 I mean this guy and this guy-- 796 00:51:24,840 --> 00:51:28,070 with respect to u, then actually we 797 00:51:28,070 --> 00:51:36,130 are going to get a triangular function, which 798 00:51:36,130 --> 00:51:38,945 looks like this one. 799 00:51:41,920 --> 00:51:46,310 And of course, we normalize at the 0 frequency. 800 00:51:46,310 --> 00:51:50,390 So at the 0, we always have a 1. 801 00:51:50,390 --> 00:51:54,380 So you notice the difference between these two. 802 00:51:54,380 --> 00:51:58,210 So ATF-- my ATF is nothing but my [INAUDIBLE] function-- 803 00:51:58,210 --> 00:51:59,650 just left function. 804 00:51:59,650 --> 00:52:02,200 And the color frequency, umx, is defined 805 00:52:02,200 --> 00:52:04,990 by my physical size of the [INAUDIBLE] 806 00:52:04,990 --> 00:52:07,020 with the scaling factor. 807 00:52:07,020 --> 00:52:11,050 But OTF is auto-correlation, so the shape 808 00:52:11,050 --> 00:52:13,630 is now the triangular shape. 809 00:52:13,630 --> 00:52:19,330 But also the color frequency-- the umx is twice of the umx, 810 00:52:19,330 --> 00:52:25,460 because basically you go through all this process. 811 00:52:25,460 --> 00:52:29,362 So you get twice of the color frequency. 812 00:52:37,730 --> 00:52:41,700 So-- yeah? 813 00:52:41,700 --> 00:52:42,590 Oh. 814 00:52:42,590 --> 00:52:44,370 So whatever [INAUDIBLE] you have, 815 00:52:44,370 --> 00:52:48,050 if you compute this kind of overlapped area, 816 00:52:48,050 --> 00:52:51,060 then you can get OTF. 817 00:52:51,060 --> 00:52:56,900 So [INAUDIBLE] you have a rectangular square aperture, 818 00:52:56,900 --> 00:53:00,890 then you if you go through this auto-correlation, 819 00:53:00,890 --> 00:53:02,460 then you can get this kind of OTF. 820 00:53:02,460 --> 00:53:06,090 So this is just completely linear in u and [INAUDIBLE] 821 00:53:06,090 --> 00:53:08,190 direction. 822 00:53:08,190 --> 00:53:12,820 Again, the fx and fy, they are [INAUDIBLE].. 823 00:53:12,820 --> 00:53:16,370 And if you have the circular aperture, it's almost linear, 824 00:53:16,370 --> 00:53:18,610 but it's slightly curved at the edge. 825 00:53:22,270 --> 00:53:27,930 But essentially, they are the same thing. 826 00:53:27,930 --> 00:53:30,190 So if you move around two circles 827 00:53:30,190 --> 00:53:32,800 in completely overlapped area, then you 828 00:53:32,800 --> 00:53:37,070 end up with this function. 829 00:53:42,880 --> 00:53:45,970 I guess this is a good time to take a break. 830 00:53:52,820 --> 00:53:53,930 Any questions so far? 831 00:53:56,630 --> 00:53:59,930 So we discussed there is a lot of difference 832 00:53:59,930 --> 00:54:02,210 between temporal and spatial coherence, 833 00:54:02,210 --> 00:54:05,510 and defined the intensity point spread function, 834 00:54:05,510 --> 00:54:06,948 and also defined the OTF which is 835 00:54:06,948 --> 00:54:08,990 Fourier transform with the intensity point spread 836 00:54:08,990 --> 00:54:10,340 function. 837 00:54:10,340 --> 00:54:12,312 And at the last-- 838 00:54:12,312 --> 00:54:14,495 [CLEARS THROAT] excuse me-- 839 00:54:14,495 --> 00:54:17,780 OTF is auto-correlation of the ATF. 840 00:54:17,780 --> 00:54:21,420 And I visually showed how to compute the OTF. 841 00:54:31,240 --> 00:54:35,170 So let's just summarize what we discussed so far. 842 00:54:35,170 --> 00:54:41,250 So first, we all know that coherent point spread function 843 00:54:41,250 --> 00:54:44,917 is nothing but an optical field generated by a point 844 00:54:44,917 --> 00:54:47,240 source at [INAUDIBLE] plane. 845 00:54:47,240 --> 00:54:50,620 And that's the field. 846 00:54:50,620 --> 00:54:54,880 And also you know that coherent point spread function 847 00:54:54,880 --> 00:54:56,710 is Fourier transform [INAUDIBLE] function 848 00:54:56,710 --> 00:55:00,370 with a coordinate scaling. 849 00:55:00,370 --> 00:55:02,680 And incoherent point spread function 850 00:55:02,680 --> 00:55:07,400 is negative square of coherent point spread. 851 00:55:07,400 --> 00:55:10,680 Or another way to say is intensity point 852 00:55:10,680 --> 00:55:14,420 spread function is intensity of the field coherent point spread 853 00:55:14,420 --> 00:55:14,920 function. 854 00:55:21,820 --> 00:55:26,940 And we define the amplitude [INAUDIBLE] function, ATF, 855 00:55:26,940 --> 00:55:30,150 which is Fourier transform of the coherent point spread 856 00:55:30,150 --> 00:55:32,680 function. 857 00:55:32,680 --> 00:55:41,260 And it is nothing but the [INAUDIBLE] function with 858 00:55:41,260 --> 00:55:42,490 the scaling vector-- 859 00:55:42,490 --> 00:55:44,540 I mean coordinate scaling. 860 00:55:44,540 --> 00:55:49,330 And we just derive the OTF, which 861 00:55:49,330 --> 00:55:53,340 is Fourier transform of the intensity point spread function 862 00:55:53,340 --> 00:55:56,230 is actually auto-correlation with the ATF. 863 00:56:02,860 --> 00:56:06,310 In the rest of the lecture, we're 864 00:56:06,310 --> 00:56:08,920 going to mostly deal with the MTF which 865 00:56:08,920 --> 00:56:12,760 is negative modulus of the OTF. 866 00:56:12,760 --> 00:56:14,490 Actually, it should be OTF. 867 00:56:14,490 --> 00:56:17,740 So MTF is the negative component of OTF. 868 00:56:22,450 --> 00:56:25,870 In terms of input and output, we're 869 00:56:25,870 --> 00:56:27,590 dealing with a linear system approach. 870 00:56:27,590 --> 00:56:32,080 So if you have the spatially coherent illumination-- 871 00:56:32,080 --> 00:56:36,170 so [INAUDIBLE] is the illumination field. 872 00:56:36,170 --> 00:56:38,440 And I have the thin transparency-- 873 00:56:38,440 --> 00:56:41,280 the complex transparency which is sub t. 874 00:56:41,280 --> 00:56:44,820 Then my input field is just product of my illumination 875 00:56:44,820 --> 00:56:47,050 field and complex transparency. 876 00:56:47,050 --> 00:56:49,490 So g sub n, that's the input field. 877 00:56:49,490 --> 00:56:53,260 And if I convert with the coherent point spread function, 878 00:56:53,260 --> 00:56:58,140 that's going to be my output field, which is g sub f. 879 00:56:58,140 --> 00:57:01,970 And I kind of think this whole thing in frequency domain. 880 00:57:01,970 --> 00:57:04,280 So if I take a Fourier transform-- 881 00:57:04,280 --> 00:57:06,790 [INAUDIBLE] is actually Fourier transform 882 00:57:06,790 --> 00:57:15,040 of small gn, which is Fourier component of my input field. 883 00:57:15,040 --> 00:57:18,377 And convolution is multiplication 884 00:57:18,377 --> 00:57:19,210 in frequency domain. 885 00:57:19,210 --> 00:57:24,250 So I multiply by ATF, then I'm going to get g sub-- 886 00:57:24,250 --> 00:57:28,340 [INAUDIBLE] which is Fourier inform of my output field. 887 00:57:30,970 --> 00:57:35,370 And if I have the spatially incoherent illumination, 888 00:57:35,370 --> 00:57:38,000 then I deal with the intensity. 889 00:57:38,000 --> 00:57:43,830 So I have the intensity of illumination, and I-- 890 00:57:43,830 --> 00:57:45,180 still complex transparency. 891 00:57:45,180 --> 00:57:51,960 So actually, my input intensity is illumination intensity 892 00:57:51,960 --> 00:57:56,220 times negative square of g sub t. 893 00:57:56,220 --> 00:57:59,550 So remember that gt is complex transparency. 894 00:57:59,550 --> 00:58:02,850 So to make intensity, you have to take negative square. 895 00:58:02,850 --> 00:58:05,280 So this is my input intensity. 896 00:58:05,280 --> 00:58:09,640 And in current case, it's linear in intensity, 897 00:58:09,640 --> 00:58:12,300 so you have to convert with the intensity point spread 898 00:58:12,300 --> 00:58:16,870 function, which is negative square of the coherent point 899 00:58:16,870 --> 00:58:17,980 spread function. 900 00:58:17,980 --> 00:58:22,600 Then you're going to get in output intensity. 901 00:58:22,600 --> 00:58:25,300 And again, in the frequency domain, 902 00:58:25,300 --> 00:58:30,204 we can take the Fourier transform-- so g sub i-- 903 00:58:30,204 --> 00:58:31,410 what can I say? 904 00:58:31,410 --> 00:58:38,730 Character gi input-- character g sub i in, 905 00:58:38,730 --> 00:58:42,480 which is Fourier transform of the input intensity. 906 00:58:42,480 --> 00:58:45,450 And you're going to multiply by OTF, 907 00:58:45,450 --> 00:58:48,780 which is what auto-correlation of the ATF. 908 00:58:48,780 --> 00:58:52,380 And you'll get the Fourier transform 909 00:58:52,380 --> 00:58:54,057 of the output intensity. 910 00:58:58,350 --> 00:59:02,880 So actually, if you're familiar with the control theory, 911 00:59:02,880 --> 00:59:05,430 then you probably know the Bode plot. 912 00:59:05,430 --> 00:59:09,040 So Bode plot is nothing but you have-- 913 00:59:09,040 --> 00:59:11,490 in frequency domain, it describes the weighting 914 00:59:11,490 --> 00:59:14,280 function that is going to be multiplied by your input 915 00:59:14,280 --> 00:59:16,240 and produce your output. 916 00:59:16,240 --> 00:59:19,720 So this ATF, and OTF, and also MTF, 917 00:59:19,720 --> 00:59:22,650 they are basically the same thing. 918 00:59:22,650 --> 00:59:25,310 You multiply your-- so basically, 919 00:59:25,310 --> 00:59:29,745 this transform function in frequency domain 920 00:59:29,745 --> 00:59:32,200 describes the weighting function here 921 00:59:32,200 --> 00:59:34,600 that is going to be multiplied by your input. 922 00:59:34,600 --> 00:59:38,010 And you can just compare the-- 923 00:59:38,010 --> 00:59:41,050 and also another way to say it is the ratio between input 924 00:59:41,050 --> 00:59:41,550 and output. 925 00:59:47,110 --> 00:59:49,930 So MTF-- so, yeah. 926 00:59:49,930 --> 00:59:54,850 So as I described, this transform function 927 00:59:54,850 --> 00:59:58,870 is nothing but with weighting function in frequency domain. 928 00:59:58,870 --> 01:00:01,490 So let me show you how to use it. 929 01:00:01,490 --> 01:00:06,110 Because if you buy a lens for your digital camera-- 930 01:00:06,110 --> 01:00:08,925 I mean, typically the SLR camera-- 931 01:00:08,925 --> 01:00:11,740 so lets make us provide the MTF layer, 932 01:00:11,740 --> 01:00:16,690 which is this weighting function with the spatial frequency, 933 01:00:16,690 --> 01:00:19,020 and with the color frequency. 934 01:00:19,020 --> 01:00:21,500 It's cool-- you can say what kind of image quality 935 01:00:21,500 --> 01:00:24,670 I can expect with the lens. 936 01:00:24,670 --> 01:00:28,510 So first, let's consider our [INAUDIBLE] grating 937 01:00:28,510 --> 01:00:33,590 whose MTF transform function is given by this equation. 938 01:00:33,590 --> 01:00:39,040 So the input intensity is going to be-- 939 01:00:39,040 --> 01:00:42,460 I mean intensity of the grating is going to be negative square, 940 01:00:42,460 --> 01:00:44,020 so I add 1 over 2. 941 01:00:44,020 --> 01:00:48,250 Then 1 plus n modulation, and cosine term. 942 01:00:48,250 --> 01:00:51,207 And the phase [INAUDIBLE] it doesn't 943 01:00:51,207 --> 01:00:53,290 matter why, because if you take a negative square, 944 01:00:53,290 --> 01:00:54,010 it's always 1. 945 01:01:00,480 --> 01:01:05,820 The Fourier transform of this input intensity is-- 946 01:01:05,820 --> 01:01:10,220 we all know that it has three 3 orders. 947 01:01:10,220 --> 01:01:14,647 So 0-th order, and the 1 over capital lambda, and minus 1 948 01:01:14,647 --> 01:01:15,480 over capital lambda. 949 01:01:15,480 --> 01:01:18,480 We have three different orders. 950 01:01:18,480 --> 01:01:26,370 And it's going to be multiplied by the OTF, which is at this h. 951 01:01:26,370 --> 01:01:29,670 And since this is delta function, which means 952 01:01:29,670 --> 01:01:33,190 this one is 1 only at u equals 0, 953 01:01:33,190 --> 01:01:37,050 and when u equals 1 over capital lambda, 954 01:01:37,050 --> 01:01:39,700 and at minus 1 capital lambda. 955 01:01:39,700 --> 01:01:41,310 So if you multiply the OTF-- 956 01:01:41,310 --> 01:01:44,550 and actually, you still have the three components. 957 01:01:44,550 --> 01:01:50,940 So at 0 component, you have the OTF at the 0 frequency. 958 01:01:50,940 --> 01:01:54,430 And these two delta functions are multiplied by OTF 959 01:01:54,430 --> 01:01:58,320 at the 1 over capital lambda and minus 1 over capital lambda. 960 01:01:58,320 --> 01:02:00,810 We still have the three [INAUDIBLE] orders. 961 01:02:05,070 --> 01:02:15,320 And let me just go back. 962 01:02:15,320 --> 01:02:18,830 So since this is your Fourier transform of the output 963 01:02:18,830 --> 01:02:21,370 intensity, you can just take the inverse 964 01:02:21,370 --> 01:02:25,010 Fourier transform to compute the output intensity, right? 965 01:02:25,010 --> 01:02:26,520 But there is another-- 966 01:02:26,520 --> 01:02:29,970 there is an important property of the OTF. 967 01:02:29,970 --> 01:02:32,030 So this guy and this guy. 968 01:02:32,030 --> 01:02:41,220 So OTF is actually-- 969 01:02:41,220 --> 01:02:44,730 intensity point spread function, which is h sub i, 970 01:02:44,730 --> 01:02:48,180 is negative square of the coherent point spread function. 971 01:02:48,180 --> 01:02:50,520 So it is always positive-- 972 01:02:50,520 --> 01:02:53,340 it's non-negative I should say. 973 01:02:53,340 --> 01:02:55,320 So it's non-negative. 974 01:02:55,320 --> 01:02:57,170 So this Fourier transform is actually 975 01:02:57,170 --> 01:03:00,660 Hermitian, which mathematically, you 976 01:03:00,660 --> 01:03:04,290 can [INAUDIBLE] h of minus u-- is actually 977 01:03:04,290 --> 01:03:06,800 h complex conjugate of u. 978 01:03:06,800 --> 01:03:11,040 Or you can say its real path is [INAUDIBLE] function, 979 01:03:11,040 --> 01:03:12,750 but its simulated path is out function. 980 01:03:15,540 --> 01:03:18,520 So OTF of the physically realizable optical system 981 01:03:18,520 --> 01:03:19,600 must be Hermitian. 982 01:03:19,600 --> 01:03:22,950 So this condition should be satisfied always. 983 01:03:30,280 --> 01:03:33,666 So in the previous slide, I had-- 984 01:03:33,666 --> 01:03:34,280 let me just-- 985 01:03:49,010 --> 01:03:49,900 [INAUDIBLE] 986 01:03:52,420 --> 01:04:08,100 Capital G, I, out, was 1 over 2 OTF at 0, and delta u. 987 01:04:08,100 --> 01:04:19,760 Plus n over 2, H, 1 over capital lambda, and delta u, 988 01:04:19,760 --> 01:04:21,240 1 over capital lambda. 989 01:04:21,240 --> 01:04:26,400 Plus H minus 1 over capital lambda, 990 01:04:26,400 --> 01:04:29,731 delta u, 1 over capital lambda. 991 01:04:37,910 --> 01:04:47,510 Since the OTF is Hermitian, which means my OTF over minus 1 992 01:04:47,510 --> 01:04:51,860 over capital lambda is actually complex conjugate 993 01:04:51,860 --> 01:04:54,080 of H, 1 over capital lambda. 994 01:04:57,230 --> 01:04:59,161 This is Hermitian. 995 01:05:01,810 --> 01:05:03,190 So G out-- 996 01:05:03,190 --> 01:05:16,550 I mean, the capital G, I, out, I can write as 1 over 2 OTF at 0 997 01:05:16,550 --> 01:05:18,946 is still delta function. 998 01:05:18,946 --> 01:05:26,870 And m over 2, H, 1 over capital lambda, delta u, 999 01:05:26,870 --> 01:05:29,040 1 over capital lambda. 1000 01:05:29,040 --> 01:05:31,190 But the last delta function I can 1001 01:05:31,190 --> 01:05:43,560 write as H complex conjugate, 1 over capital lambda, decay u 1002 01:05:43,560 --> 01:05:45,010 plus 1 over capital lambda. 1003 01:05:48,142 --> 01:05:49,350 So the first delta function-- 1004 01:05:49,350 --> 01:05:51,892 I mean, if you take the inverse Fourier transform [INAUDIBLE] 1005 01:05:51,892 --> 01:05:56,010 transform, the first delta function [INAUDIBLE]---- 1006 01:05:56,010 --> 01:05:57,300 just 1 or something. 1007 01:05:57,300 --> 01:05:59,140 So it's pretty straightforward. 1008 01:05:59,140 --> 01:06:03,030 The question is still what happened in these two delta 1009 01:06:03,030 --> 01:06:08,070 functions, because this one has H of-- 1010 01:06:08,070 --> 01:06:11,220 I mean the OTF at one over capital lambda, 1011 01:06:11,220 --> 01:06:14,820 and the other one is actually complex conjugate. 1012 01:06:14,820 --> 01:06:21,600 So if I just write the two delta function part, 1013 01:06:21,600 --> 01:06:24,180 and I'm going to take the inverse Fourier transform. 1014 01:06:24,180 --> 01:06:32,380 So inverse Fourier transform, or m over 2, [INAUDIBLE] 1015 01:06:32,380 --> 01:06:39,060 OTF at 1 over capital lambda, delta u, 1 over capital lambda, 1016 01:06:39,060 --> 01:06:45,040 plus H complex conjugate, capital lambda, delta u plus 1 1017 01:06:45,040 --> 01:06:45,950 over capital lambda. 1018 01:06:50,550 --> 01:06:56,310 Then what is-- inverse Fourier transform [INAUDIBLE] 1019 01:06:56,310 --> 01:07:00,880 function is going to be exponential. 1020 01:07:00,880 --> 01:07:02,300 And [INAUDIBLE] do we scale it? 1021 01:07:02,300 --> 01:07:04,370 So [INAUDIBLE] just outside. 1022 01:07:04,370 --> 01:07:09,830 And I'm going to have H of 1 over capital lambda, 1023 01:07:09,830 --> 01:07:17,880 and exponent e to the I2pi, x over capital lambda, 1024 01:07:17,880 --> 01:07:22,910 plus H complex conjugate, 1 over capital lambda. 1025 01:07:22,910 --> 01:07:27,090 e to the minus I2pi, x over capital lambda. 1026 01:07:32,956 --> 01:07:40,820 And I can write as capital lambda-- 1027 01:07:40,820 --> 01:07:45,110 1 over capital lambda, e to the I2pi, x over lambda. 1028 01:07:45,110 --> 01:07:48,230 Plus-- actually the same thing-- 1029 01:07:48,230 --> 01:07:53,570 H, 1 over capital lambda, e to the I2pi, x over lambda. 1030 01:07:53,570 --> 01:07:56,600 And complex conjugate like this. 1031 01:08:00,860 --> 01:08:07,070 So what happens if you add A and its complex conjugate? 1032 01:08:07,070 --> 01:08:14,290 So for example, this is A, and this is A complex conjugate. 1033 01:08:14,290 --> 01:08:17,720 And if you add them together, then you 1034 01:08:17,720 --> 01:08:28,189 get only rear path, because two rear paths over A. 1035 01:08:28,189 --> 01:08:38,899 So then I'm going to have m, the rear path of OTF 1036 01:08:38,899 --> 01:08:47,560 at 1 over capital lambda, to the I2pi x over lambda, 1037 01:08:47,560 --> 01:08:53,040 which I can write as m, and negative-- 1038 01:08:53,040 --> 01:08:56,290 this H is just non-zero. 1039 01:08:56,290 --> 01:08:58,600 So it could have-- 1040 01:08:58,600 --> 01:09:03,310 I mean, it can have only 0 or minus [INAUDIBLE] basically. 1041 01:09:03,310 --> 01:09:04,930 So I can write it as negative here. 1042 01:09:04,930 --> 01:09:09,939 And this [INAUDIBLE] cosine 2pi, x over lambda. 1043 01:09:13,920 --> 01:09:20,010 So that's why the output intensity is just 1 1044 01:09:20,010 --> 01:09:23,520 plus m, which is original modulation. 1045 01:09:23,520 --> 01:09:27,630 But I add extra modulation due to OTF and cosine term. 1046 01:09:36,540 --> 01:09:40,229 Should I prove why OTF is Hermitian? 1047 01:09:51,479 --> 01:09:54,900 Yeah, let me prove why OTF is Hermitian. 1048 01:10:00,670 --> 01:10:07,540 So OTF is auto-correlation of ATF, which is like this. 1049 01:10:10,246 --> 01:10:14,450 And integration with the integral form I can write as H 1050 01:10:14,450 --> 01:10:22,003 of u prime, H complex conjugate, u prime minus u, 1051 01:10:22,003 --> 01:10:22,920 and [INAUDIBLE] prime. 1052 01:10:25,660 --> 01:10:29,290 And how to compute H-- 1053 01:10:29,290 --> 01:10:33,850 I mean OTF and complex conjugate of u. 1054 01:10:33,850 --> 01:10:37,280 So I can write as-- 1055 01:10:37,280 --> 01:10:39,020 the total thing-- 1056 01:10:39,020 --> 01:10:46,600 H of u prime, H complex conjugate u prime u, u prime, 1057 01:10:46,600 --> 01:10:47,720 and complex conjugate. 1058 01:10:47,720 --> 01:10:48,580 AUDIENCE: Excuse me. 1059 01:10:48,580 --> 01:10:49,510 SE BAEK OH: Yes? 1060 01:10:49,510 --> 01:10:52,700 AUDIENCE: We're not able to see what you are writing. 1061 01:10:52,700 --> 01:10:55,027 SE BAEK OH: Oh, it's too small? 1062 01:10:55,027 --> 01:10:55,735 AUDIENCE: OK now. 1063 01:10:59,400 --> 01:11:01,400 SE BAEK OH: So I just take the complex conjugate 1064 01:11:01,400 --> 01:11:02,110 on both sides. 1065 01:11:04,730 --> 01:11:09,050 And I can chain the complex conjugate. 1066 01:11:09,050 --> 01:11:15,020 So it's going to be H complex conjugate of u prime, H, 1067 01:11:15,020 --> 01:11:18,730 and u prime and u, and du prime. 1068 01:11:18,730 --> 01:11:21,870 So I just put the complex conjugate inside 1069 01:11:21,870 --> 01:11:24,640 of the integration. 1070 01:11:24,640 --> 01:11:27,110 And here, I define the new variable. 1071 01:11:27,110 --> 01:11:32,590 So u double prime is u prime minus u. 1072 01:11:32,590 --> 01:11:39,680 So u prime is u plus u double prime. 1073 01:11:39,680 --> 01:11:43,290 And du prime is actually du double prime. 1074 01:11:48,520 --> 01:11:53,190 So this H complex conjugate is actually 1075 01:11:53,190 --> 01:12:01,510 H complex conjugate, u plus u double prime, 1076 01:12:01,510 --> 01:12:05,430 h of u double prime, and du double 1077 01:12:05,430 --> 01:12:15,070 prime, which I can write as integration h u double prime. 1078 01:12:15,070 --> 01:12:23,110 A complex conjugate u double prime, minus u, 1079 01:12:23,110 --> 01:12:24,460 du double prime. 1080 01:12:24,460 --> 01:12:27,990 So it's H of minus u. 1081 01:12:30,880 --> 01:12:39,460 So this is complex conjugate Hermitian. 1082 01:12:48,050 --> 01:12:51,010 I just proved you can easily do that. 1083 01:12:51,010 --> 01:12:54,500 So then the bottom line is at the output plane-- 1084 01:12:54,500 --> 01:12:57,710 I mean output intensity, I have extra modulation 1085 01:12:57,710 --> 01:12:59,720 coming from the OTF. 1086 01:12:59,720 --> 01:13:01,640 I mean, you can say intensity is negative, 1087 01:13:01,640 --> 01:13:03,149 so you can say [INAUDIBLE]. 1088 01:13:07,550 --> 01:13:14,150 So if you compare in terms of the contrast or visibility, 1089 01:13:14,150 --> 01:13:17,750 so the contrast is defined by imx minus ime 1090 01:13:17,750 --> 01:13:19,550 over imx plus ime. 1091 01:13:19,550 --> 01:13:24,140 But at the output plane, we have the extra modulation term. 1092 01:13:24,140 --> 01:13:27,080 So if you compute the same thing, 1093 01:13:27,080 --> 01:13:29,460 then actually your contrast is going 1094 01:13:29,460 --> 01:13:34,700 to be at the original modulation multiplied by MTF 1095 01:13:34,700 --> 01:13:37,526 at the spatial frequency. 1096 01:13:43,640 --> 01:13:48,170 So MTF is just contrast at the spatial frequency 1097 01:13:48,170 --> 01:13:51,230 relative to the transmission that we're discussing. 1098 01:13:51,230 --> 01:13:53,280 So what does it mean? 1099 01:13:53,280 --> 01:13:59,900 So for example, I have MTF looks like this. 1100 01:13:59,900 --> 01:14:03,075 And I initially had the grating-- 1101 01:14:03,075 --> 01:14:05,870 the intensity of the grating is actually 1 over 2, 1102 01:14:05,870 --> 01:14:08,850 1 plus n, cosine, 2pi, x over capital lambda. 1103 01:14:08,850 --> 01:14:15,200 So if I draw in grayscale-- so the white is 1 and black is 0. 1104 01:14:15,200 --> 01:14:20,090 So I have nice sinusoid with a full contrast like this. 1105 01:14:20,090 --> 01:14:22,790 But the output intensity, I'm going 1106 01:14:22,790 --> 01:14:29,420 to have 1 plus m, and additional modulation, and cosine term. 1107 01:14:29,420 --> 01:14:33,450 And this factor, you can find [INAUDIBLE].. 1108 01:14:33,450 --> 01:14:36,170 So if you have the 1 over capital lambda, 1109 01:14:36,170 --> 01:14:40,300 then you can find the value at that point-- 1110 01:14:40,300 --> 01:14:43,910 I mean MTF at this point. 1111 01:14:43,910 --> 01:14:47,810 So if that value is 0.6 in this case, 1112 01:14:47,810 --> 01:14:52,070 this sinusoidal pattern will look like this. 1113 01:14:52,070 --> 01:14:58,340 But if the modulation is 0.2, then the black becomes 1114 01:14:58,340 --> 01:15:00,320 brighter and white becomes darker. 1115 01:15:00,320 --> 01:15:03,770 So you kind of get the grayscale-- 1116 01:15:03,770 --> 01:15:06,810 [INAUDIBLE] version of the gradient like this. 1117 01:15:06,810 --> 01:15:12,170 So basically, this MTF tells you what kind of contrast 1118 01:15:12,170 --> 01:15:16,820 I expect at this particular spatial frequency, 1119 01:15:16,820 --> 01:15:20,750 and also it specifies the maximum spatial frequency 1120 01:15:20,750 --> 01:15:21,867 you can resolve-- 1121 01:15:21,867 --> 01:15:22,700 I mean, you can see. 1122 01:15:31,960 --> 01:15:33,149 Any questions? 1123 01:15:42,000 --> 01:15:44,260 So so far, we just deal with the idea 1124 01:15:44,260 --> 01:15:46,730 of thin lens with a clear aperture, 1125 01:15:46,730 --> 01:15:49,550 like the rectangular aperture is a clear aperture. 1126 01:15:49,550 --> 01:15:52,400 So in the case of the rectangular aperture, 1127 01:15:52,400 --> 01:15:55,670 we know that the OTF is just auto-correlation of the two 1128 01:15:55,670 --> 01:15:58,940 [INAUDIBLE] functions which is just triangular function. 1129 01:15:58,940 --> 01:16:03,260 So it's linear from 1 to color frequency, 1130 01:16:03,260 --> 01:16:07,310 which is twice of the [INAUDIBLE] frequency at OTF-- 1131 01:16:07,310 --> 01:16:08,990 the ATF. 1132 01:16:08,990 --> 01:16:11,060 And if you have the circular aperture, 1133 01:16:11,060 --> 01:16:15,410 it's almost linear except at the [INAUDIBLE] 1134 01:16:15,410 --> 01:16:18,560 you have slightly curved line. 1135 01:16:18,560 --> 01:16:21,420 But it's the same thing. 1136 01:16:21,420 --> 01:16:27,750 So this blue line and the black line we call defraction 1137 01:16:27,750 --> 01:16:32,630 limited OTF, because we assume the idea of a thin lens 1138 01:16:32,630 --> 01:16:34,070 without any aberration. 1139 01:16:34,070 --> 01:16:40,170 So this is kind of best OTF you can achieve. 1140 01:16:40,170 --> 01:16:43,790 But in practice, you always have aberration-- 1141 01:16:43,790 --> 01:16:46,310 chromatic aberration, or any high aberration. 1142 01:16:46,310 --> 01:16:53,300 Or you can have a defocus, or some imperfect factor in lens. 1143 01:16:53,300 --> 01:16:57,140 So typically, the typical lens has 1144 01:16:57,140 --> 01:17:01,670 this kind of distorted or degraded OTF like this. 1145 01:17:01,670 --> 01:17:03,426 The red one right here or here. 1146 01:17:07,710 --> 01:17:10,550 So in terms of this defraction limited, 1147 01:17:10,550 --> 01:17:14,780 you want to buy a lens that OTF is similar to this blue line, 1148 01:17:14,780 --> 01:17:17,080 because this is defraction limited, 1149 01:17:17,080 --> 01:17:21,386 which means the best OTF we can achieve. 1150 01:17:21,386 --> 01:17:23,650 And [INAUDIBLE]. 1151 01:17:30,940 --> 01:17:35,260 So I describe-- I mean, this is slope is linear, 1152 01:17:35,260 --> 01:17:39,670 and circular aperture. 1153 01:17:42,730 --> 01:17:45,500 I mean, locally, it can have a higher value, 1154 01:17:45,500 --> 01:17:50,270 but the overall shape should be smaller than this blue line. 1155 01:18:01,690 --> 01:18:09,740 So let's just apply all these concepts in [INAUDIBLE] system. 1156 01:18:09,740 --> 01:18:13,260 So we have first system whose focal length f1, f2. 1157 01:18:13,260 --> 01:18:15,720 And I put the input transparency, 1158 01:18:15,720 --> 01:18:18,520 which is a binary [INAUDIBLE] grading right here. 1159 01:18:18,520 --> 01:18:21,860 And illuminate with the quasi-monochromatic which 1160 01:18:21,860 --> 01:18:24,650 is on a single color, and spatial [INAUDIBLE] 1161 01:18:24,650 --> 01:18:30,460 illumination, and [INAUDIBLE] we assume the intensity variation 1162 01:18:30,460 --> 01:18:32,240 is uniform. 1163 01:18:32,240 --> 01:18:37,120 So the possible question is what kind of image 1164 01:18:37,120 --> 01:18:44,170 we can expect at the image plane, or what is the OTF, 1165 01:18:44,170 --> 01:18:49,210 or what is ATF, and blah, blah, blah. 1166 01:18:49,210 --> 01:18:50,350 So let's go through it. 1167 01:18:56,460 --> 01:18:59,140 So this is the binary amplitude grating. 1168 01:18:59,140 --> 01:19:03,510 So it is still complex transparency in here. 1169 01:19:03,510 --> 01:19:06,690 So if you have [INAUDIBLE] illumination, 1170 01:19:06,690 --> 01:19:08,650 you want to compute the intensity. 1171 01:19:08,650 --> 01:19:11,470 So you have to take the negative scale. 1172 01:19:11,470 --> 01:19:15,120 So this is just 0 or 1, so basically you have 1173 01:19:15,120 --> 01:19:20,070 the same thing in intensity. 1174 01:19:20,070 --> 01:19:24,900 So that's the input intensity at the upper right. 1175 01:19:24,900 --> 01:19:30,210 And this is the phase mask located at the rear plane. 1176 01:19:30,210 --> 01:19:35,030 And we know that the ATF is basically the same function 1177 01:19:35,030 --> 01:19:41,470 with different coordinates. 1178 01:19:41,470 --> 01:19:45,580 So ATF is phase mask, but the function 1179 01:19:45,580 --> 01:19:47,710 is the x double prime-- well, I mean 1180 01:19:47,710 --> 01:19:49,710 you have this coordinate scaling vector. 1181 01:19:53,080 --> 01:19:56,830 So if you compute the OTF from this ATF, 1182 01:19:56,830 --> 01:20:03,330 basically, what you do is you have two [INAUDIBLE] functions. 1183 01:20:03,330 --> 01:20:06,610 So you will have actually 4 [INAUDIBLE] functions. 1184 01:20:06,610 --> 01:20:10,450 And you move toward them, so whenever they overlap, 1185 01:20:10,450 --> 01:20:13,030 you get some value. 1186 01:20:13,030 --> 01:20:16,270 So if could do the auto-correlation, 1187 01:20:16,270 --> 01:20:19,750 then actually you're going to have three triangular. 1188 01:20:19,750 --> 01:20:24,550 But remember, if you have the [INAUDIBLE] function, 1189 01:20:24,550 --> 01:20:26,330 then you move two [INAUDIBLE] function. 1190 01:20:26,330 --> 01:20:29,750 So you're going to have one triangular. 1191 01:20:29,750 --> 01:20:32,560 So since we have two [INAUDIBLE] functions, so whenever 1192 01:20:32,560 --> 01:20:36,670 they first [INAUDIBLE],, then we have the one triangle. 1193 01:20:36,670 --> 01:20:40,240 But the second and the two [INAUDIBLE] functions 1194 01:20:40,240 --> 01:20:42,880 overlap, then we have the center triangle. 1195 01:20:42,880 --> 01:20:46,480 And like this, we another triangle, so 1196 01:20:46,480 --> 01:20:49,140 we get three triangles. 1197 01:20:49,140 --> 01:20:50,750 So that's the OTF. 1198 01:20:53,790 --> 01:20:59,400 And if you expand with the Fourier series of your input 1199 01:20:59,400 --> 01:21:01,890 intensity, then we're going to have an infinite number 1200 01:21:01,890 --> 01:21:03,000 of diffraction. 1201 01:21:03,000 --> 01:21:08,910 But in ATF case, we have only first order 1202 01:21:08,910 --> 01:21:11,640 and the minus first order, and minus 1203 01:21:11,640 --> 01:21:14,270 first order going to be transmitted. 1204 01:21:14,270 --> 01:21:17,320 But in OTF, actually, they are blocked. 1205 01:21:17,320 --> 01:21:19,294 [INAUDIBLE] 1206 01:21:19,294 --> 01:21:23,340 Actually, plus 2, and 0, and minus 2, 1207 01:21:23,340 --> 01:21:25,412 they are going to be transmitted. 1208 01:21:29,480 --> 01:21:34,420 So you just multiply these three delta functions 1209 01:21:34,420 --> 01:21:37,930 by the value of the ATF-- 1210 01:21:37,930 --> 01:21:40,000 this, this, this. 1211 01:21:40,000 --> 01:21:46,260 And the Fourier transform will be your output intensity, 1212 01:21:46,260 --> 01:21:48,950 which is this. 1213 01:21:48,950 --> 01:21:53,200 So 1 over 3 just came from the central [INAUDIBLE] function, 1214 01:21:53,200 --> 01:21:54,430 because this is 1-- 1215 01:21:54,430 --> 01:21:56,530 so just 1 over 3. 1216 01:21:56,530 --> 01:22:04,450 And for the first order, at the 0.1 and minus 0.1 1217 01:22:04,450 --> 01:22:08,500 micrometer [INAUDIBLE],, actually they have 0 value. 1218 01:22:08,500 --> 01:22:12,100 So I multiply by 0. 1219 01:22:12,100 --> 01:22:16,450 And for the second order, I have one over 2 which 1220 01:22:16,450 --> 01:22:21,400 is this value here and here. 1221 01:22:21,400 --> 01:22:25,672 So that is the [INAUDIBLE] of the output intensity. 1222 01:22:28,450 --> 01:22:32,140 So if you compute the inverse Fourier transform 1223 01:22:32,140 --> 01:22:35,770 of this equation, then we are going to have 1 over 3, 1224 01:22:35,770 --> 01:22:39,730 and 1 over 2, and two exponent here 1225 01:22:39,730 --> 01:22:45,970 which is actually cosine like this. 1226 01:22:45,970 --> 01:22:48,160 So in terms of grayscale image, we 1227 01:22:48,160 --> 01:22:50,570 have the binary amplitude grating, 1228 01:22:50,570 --> 01:22:54,260 which varies just black or white like this. 1229 01:22:54,260 --> 01:22:56,950 But if you go through this [INAUDIBLE] system 1230 01:22:56,950 --> 01:23:00,520 at the image, we're going to have this kind of grayscale 1231 01:23:00,520 --> 01:23:06,910 because contrast is just 0.1034. 1232 01:23:06,910 --> 01:23:09,460 So it's very little. 1233 01:23:09,460 --> 01:23:14,770 And the average value is about 0.33 or something. 1234 01:23:14,770 --> 01:23:19,120 So basically, it looks gray with a little wiggling. 1235 01:23:21,910 --> 01:23:26,830 So ATF basically tells you the contrast 1236 01:23:26,830 --> 01:23:29,311 of the particular spatial frequency. 1237 01:23:38,680 --> 01:23:41,760 So, yeah. 1238 01:23:41,760 --> 01:23:44,390 What happened-- I mean, the orders at this side. 1239 01:23:44,390 --> 01:23:51,926 So if my OTF is 0, then what kind of image I can expect? 1240 01:23:51,926 --> 01:23:52,854 It's black? 1241 01:23:55,353 --> 01:23:56,270 It's not light, right? 1242 01:23:58,870 --> 01:24:00,910 So if it is 0-- 1243 01:24:00,910 --> 01:24:03,535 I mean if the values are 0, then-- 1244 01:24:08,830 --> 01:24:13,130 so I still have [INAUDIBLE] so some kind of grayscale. 1245 01:24:13,130 --> 01:24:16,150 So I don't see any [INAUDIBLE]. 1246 01:24:18,900 --> 01:24:20,880 OTF is auto-correlation, so you always 1247 01:24:20,880 --> 01:24:26,020 have 1 at the dc frequency. 1248 01:24:26,020 --> 01:24:28,460 Because if you move the two functions, 1249 01:24:28,460 --> 01:24:30,810 so when they exactly overlap, you always 1250 01:24:30,810 --> 01:24:31,770 have the maximum value. 1251 01:24:31,770 --> 01:24:38,810 So you always have [INAUDIBLE] system. 1252 01:24:38,810 --> 01:24:41,840 And actually, that is very different in current case. 1253 01:24:41,840 --> 01:24:44,770 So in ATF case, you can actually cancel [INAUDIBLE] system, 1254 01:24:44,770 --> 01:24:45,820 right? 1255 01:24:45,820 --> 01:24:47,860 But OTF, you can not cancel the system, 1256 01:24:47,860 --> 01:24:49,400 because it's auto-correlation. 1257 01:24:55,670 --> 01:25:01,370 So again, we have some relation with the [INAUDIBLE].. 1258 01:25:01,370 --> 01:25:06,870 So this particular circular aperture, 1259 01:25:06,870 --> 01:25:09,650 if I eliminate the coherent illumination, 1260 01:25:09,650 --> 01:25:10,910 then this is the image-- 1261 01:25:10,910 --> 01:25:11,940 AUDIENCE: Excuse me. 1262 01:25:11,940 --> 01:25:13,550 SE BAEK OH: Yes? 1263 01:25:13,550 --> 01:25:16,870 AUDIENCE: In Singapore slide, the [INAUDIBLE] slides 1264 01:25:16,870 --> 01:25:18,500 are not [INAUDIBLE]. 1265 01:25:25,630 --> 01:25:29,440 SE BAEK OH: Maybe it takes time or-- 1266 01:25:29,440 --> 01:25:35,750 AUDIENCE: No, it even didn't show that input and output 1267 01:25:35,750 --> 01:25:38,885 image when you explained about that contrast. 1268 01:25:42,286 --> 01:25:45,170 SE BAEK OH: Just let me ask [INAUDIBLE].. 1269 01:25:45,170 --> 01:25:49,030 Hello, [INAUDIBLE]? 1270 01:25:49,030 --> 01:25:50,010 AUDIENCE: Yes? 1271 01:25:50,010 --> 01:25:51,330 SE BAEK OH: Yes, [INAUDIBLE]. 1272 01:25:54,610 --> 01:25:55,960 It's frozen. 1273 01:25:55,960 --> 01:26:00,080 AUDIENCE: [INAUDIBLE] 1274 01:26:00,080 --> 01:26:02,330 SE BAEK OH: I think they're going to reset the system. 1275 01:26:04,980 --> 01:26:09,150 So let me just take a few moments. 1276 01:26:09,150 --> 01:26:10,060 Any questions so far? 1277 01:26:27,000 --> 01:26:28,450 [INAUDIBLE] 1278 01:26:28,450 --> 01:26:34,851 AUDIENCE: [INAUDIBLE] 1279 01:26:34,851 --> 01:26:36,540 SE BAEK OH: So actually-- 1280 01:26:36,540 --> 01:26:37,505 yeah? 1281 01:26:37,505 --> 01:26:39,380 AUDIENCE: What happens when you [INAUDIBLE]?? 1282 01:26:41,888 --> 01:26:43,930 SE BAEK OH: Actually, I'm going to talk about it, 1283 01:26:43,930 --> 01:26:48,760 but in general, if you use the incoherent light, then 1284 01:26:48,760 --> 01:26:52,337 you don't see any phase effects because [INAUDIBLE] 1285 01:26:52,337 --> 01:26:53,170 the negative square. 1286 01:26:53,170 --> 01:26:55,673 So it doesn't really matter. 1287 01:26:55,673 --> 01:26:56,590 So you don't see any-- 1288 01:26:56,590 --> 01:26:58,510 I mean, you barely see some variation, 1289 01:26:58,510 --> 01:27:02,573 but usually it's not dramatic as [INAUDIBLE].. 1290 01:27:05,680 --> 01:27:08,697 And what was I about to say? 1291 01:27:15,170 --> 01:27:20,220 Oh, yeah-- so MTF is just magnitude of the OTF. 1292 01:27:20,220 --> 01:27:24,390 So we haven't talk about any phase of the OTF. 1293 01:27:24,390 --> 01:27:26,790 So what's the effect of the-- 1294 01:27:26,790 --> 01:27:31,230 I mean, what does it mean if I have 1295 01:27:31,230 --> 01:27:32,540 some phase variation in OTF? 1296 01:27:35,600 --> 01:27:37,660 Can you guess? 1297 01:27:37,660 --> 01:27:40,630 So for example, OTF is always non-zero. 1298 01:27:40,630 --> 01:27:44,540 It's always 0 or positive. 1299 01:27:44,540 --> 01:27:49,390 But its phase can be 0 or pi. 1300 01:27:49,390 --> 01:27:52,130 So what if I have pinhole phase shift in OTF? 1301 01:28:00,780 --> 01:28:04,950 For example, I have sinusoidal grating, and at the output 1302 01:28:04,950 --> 01:28:06,430 I have a sinusoidal grating. 1303 01:28:06,430 --> 01:28:09,430 But what does it mean I have the pi phase shift? 1304 01:28:27,560 --> 01:28:33,810 AUDIENCE: [INAUDIBLE] 1305 01:28:33,810 --> 01:28:35,160 SE BAEK OH: Not really. 1306 01:28:35,160 --> 01:28:39,840 What happens if you shift by pi-- 1307 01:28:39,840 --> 01:28:43,260 I mean if you move to a cosine by pi? 1308 01:28:43,260 --> 01:28:44,940 You get sine. 1309 01:28:44,940 --> 01:28:47,593 So you have cosine pattern here. 1310 01:28:47,593 --> 01:28:49,260 And if you have pi phase shift, then you 1311 01:28:49,260 --> 01:28:50,790 will have the sine pattern, which 1312 01:28:50,790 --> 01:28:53,640 means you flip the black and white of the image. 1313 01:29:04,720 --> 01:29:07,400 And I flip the [INAUDIBLE]. 1314 01:29:07,400 --> 01:29:09,260 So I flip the-- but still I [INAUDIBLE] 1315 01:29:09,260 --> 01:29:12,430 between the black and white, not the sine/cosine. 1316 01:29:16,760 --> 01:29:21,447 [INAUDIBLE] can anybody see the slide right now, or? 1317 01:29:21,447 --> 01:29:22,030 AUDIENCE: Yes. 1318 01:29:24,760 --> 01:29:26,220 I have a question. 1319 01:29:26,220 --> 01:29:30,771 How do you introduce this pi phase shift in OTF? 1320 01:29:30,771 --> 01:29:33,400 SE BAEK OH: So if you have pi phase shift in OTF, then 1321 01:29:33,400 --> 01:29:36,250 basically you flip the black and white. 1322 01:29:36,250 --> 01:29:40,090 So your input image is black-- 1323 01:29:40,090 --> 01:29:42,280 let's say the black, white, black, white. 1324 01:29:42,280 --> 01:29:45,450 And your image is going to be white, black, white, black. 1325 01:29:45,450 --> 01:29:50,470 AUDIENCE: No, my question is physically, 1326 01:29:50,470 --> 01:29:54,400 how do you introduce that controllable phase shift 1327 01:29:54,400 --> 01:29:55,030 in OTF? 1328 01:29:57,950 --> 01:30:02,020 SE BAEK OH: So this ATF can be complex. 1329 01:30:02,020 --> 01:30:06,100 So if you compute the auto-correlation, 1330 01:30:06,100 --> 01:30:10,240 then actually, you can have negative value of OTF. 1331 01:30:13,090 --> 01:30:14,860 And if you have the negative value 1332 01:30:14,860 --> 01:30:18,200 in OTF, which means you have still positive and negative, 1333 01:30:18,200 --> 01:30:20,356 but you have some negative-- 1334 01:30:20,356 --> 01:30:23,680 I mean, basically pi phase. 1335 01:30:23,680 --> 01:30:25,660 So if you want to do that kind of thing, 1336 01:30:25,660 --> 01:30:28,150 basically, you have some complex transparency 1337 01:30:28,150 --> 01:30:30,720 at your [INAUDIBLE] mask. 1338 01:30:30,720 --> 01:30:34,690 But if you have clear aperture, then you don't get that effect. 1339 01:30:44,970 --> 01:30:48,440 So let me continue this slide. 1340 01:30:48,440 --> 01:30:53,450 I mean, these three images we already seen maybe two or three 1341 01:30:53,450 --> 01:30:54,520 lectures ago. 1342 01:30:54,520 --> 01:30:56,000 So this is MIT lecture-- 1343 01:30:56,000 --> 01:31:00,590 I mean the image of MIT letters with the coherent emission. 1344 01:31:00,590 --> 01:31:06,980 And the right two images obtained 1345 01:31:06,980 --> 01:31:08,930 with the incoherent emission. 1346 01:31:08,930 --> 01:31:13,740 So I can notice two main differences here. 1347 01:31:13,740 --> 01:31:17,090 So if I use the [INAUDIBLE] emission, 1348 01:31:17,090 --> 01:31:19,310 basically, the image looks sharper 1349 01:31:19,310 --> 01:31:22,130 than the one with the coherent. 1350 01:31:22,130 --> 01:31:26,330 And that's because OTF has a higher-- 1351 01:31:26,330 --> 01:31:29,270 twice of higher color frequency even 1352 01:31:29,270 --> 01:31:30,590 though they get attenuated. 1353 01:31:30,590 --> 01:31:33,450 But basically, it can pass through the higher spatial 1354 01:31:33,450 --> 01:31:38,630 frequency, so that's why it preserves more 1355 01:31:38,630 --> 01:31:40,240 higher frequency-- 1356 01:31:40,240 --> 01:31:43,865 preserves higher frequency. 1357 01:31:43,865 --> 01:31:45,090 And it's same thing here. 1358 01:31:45,090 --> 01:31:50,090 So with the coherent light, you barely see the letter "I", 1359 01:31:50,090 --> 01:31:53,450 but you can still see the MIT. 1360 01:31:53,450 --> 01:31:57,800 And another difference is actually if you see "M" or "T", 1361 01:31:57,800 --> 01:32:01,070 there are some variations, like black or white. 1362 01:32:01,070 --> 01:32:03,620 Some wiggling in MIT here. 1363 01:32:03,620 --> 01:32:08,480 But in coherent case, the intensity is more like uniform. 1364 01:32:08,480 --> 01:32:11,650 You don't see some weird variation. 1365 01:32:11,650 --> 01:32:14,430 So that's another difference. 1366 01:32:21,260 --> 01:32:23,690 So here's the incoherent light. 1367 01:32:23,690 --> 01:32:26,640 So generally, you don't have the ringing 1368 01:32:26,640 --> 01:32:30,840 artifact as I described, and also no [INAUDIBLE].. 1369 01:32:30,840 --> 01:32:35,340 So you're not going to see any interference. 1370 01:32:35,340 --> 01:32:38,940 And it is higher bandwidth because the color frequency 1371 01:32:38,940 --> 01:32:43,480 is twice higher than the color frequency in ATF. 1372 01:32:43,480 --> 01:32:48,060 So you can see the smaller features-- 1373 01:32:48,060 --> 01:32:54,020 I mean the two points closer. 1374 01:32:54,020 --> 01:32:55,650 You can see the smaller feature. 1375 01:32:58,590 --> 01:33:04,060 But as I answered to [INAUDIBLE] question, 1376 01:33:04,060 --> 01:33:07,690 incoherent image is insensitive to phase variation. 1377 01:33:11,280 --> 01:33:14,610 So mostly, we just deal with the monochromatic, and especially 1378 01:33:14,610 --> 01:33:15,330 incoherent. 1379 01:33:15,330 --> 01:33:18,340 But if you have polychromatic, which is broadband, 1380 01:33:18,340 --> 01:33:22,110 then you're going to have some chromatic aberrations. 1381 01:33:22,110 --> 01:33:26,490 But again, you can define all these OTF, or MTF, 1382 01:33:26,490 --> 01:33:29,410 or point spread functions for each different wavelength. 1383 01:33:29,410 --> 01:33:32,730 So it just extends this in a straightforward fashion. 1384 01:33:38,120 --> 01:33:44,900 It's the end of show, so any questions? 1385 01:33:44,900 --> 01:33:47,450 So the take-home message of this [INAUDIBLE] lecture 1386 01:33:47,450 --> 01:33:51,450 is if you have coherent illumination, 1387 01:33:51,450 --> 01:33:55,510 then you have linear convolution in field. 1388 01:33:55,510 --> 01:33:58,050 But if you have incoherent case, and you 1389 01:33:58,050 --> 01:34:01,710 have the linear relation in intensity, 1390 01:34:01,710 --> 01:34:03,720 and for incoherent light, you have 1391 01:34:03,720 --> 01:34:07,400 to deal with the OTF, which is auto-correlation of ATF. 1392 01:34:07,400 --> 01:34:09,865 And that's pretty much about it. 1393 01:34:13,260 --> 01:34:19,440 If you don't have questions, then let's see next Monday.