1 00:00:00,120 --> 00:00:02,661 ANNOUNCER: The following content is provided under a Creative 2 00:00:02,661 --> 00:00:04,059 Commons license. 3 00:00:04,059 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,750 continue to offer high quality educational resources for free. 5 00:00:10,750 --> 00:00:13,350 To make a donation or view additional materials 6 00:00:13,350 --> 00:00:17,310 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,310 --> 00:00:18,480 at ocw.mit.edu. 8 00:00:22,970 --> 00:00:27,180 DENNIS FREEMAN: Hello Welcome. 9 00:00:27,180 --> 00:00:30,120 The good news, we're basically done. 10 00:00:30,120 --> 00:00:32,189 We've covered all of the theory. 11 00:00:32,189 --> 00:00:33,960 The only thing we're going to do, 12 00:00:33,960 --> 00:00:35,670 the remaining part of the course, 13 00:00:35,670 --> 00:00:40,380 is to think through several applications of Fourier. 14 00:00:40,380 --> 00:00:43,530 Today I'll do some just really easy motivational things. 15 00:00:43,530 --> 00:00:47,130 I'll do another example from filtering, 16 00:00:47,130 --> 00:00:49,980 just because that's such an important idea. 17 00:00:49,980 --> 00:00:53,880 I'll talk about how Fourier transforms show up in physics. 18 00:00:53,880 --> 00:00:56,940 And then, starting next week, we'll 19 00:00:56,940 --> 00:01:00,270 talk about sampling, which is just a simple application 20 00:01:00,270 --> 00:01:02,110 of Fourier. 21 00:01:02,110 --> 00:01:05,530 You've already sort of seen it in the last lecture. 22 00:01:05,530 --> 00:01:07,710 And then we'll talk about modulation, which is yet 23 00:01:07,710 --> 00:01:10,020 another application of Fourier. 24 00:01:10,020 --> 00:01:11,730 So basically you have the theory, 25 00:01:11,730 --> 00:01:15,446 and all we're going to do for the remainder of the course 26 00:01:15,446 --> 00:01:17,070 is think about applications of Fourier. 27 00:01:19,830 --> 00:01:22,600 So we already talked about a filter. 28 00:01:22,600 --> 00:01:25,080 I motivated a filter by thinking about Fourier series 29 00:01:25,080 --> 00:01:26,880 because it's easy. 30 00:01:26,880 --> 00:01:29,910 If you think about the system that's 31 00:01:29,910 --> 00:01:32,610 comprised of a resistor and a capacitor, 32 00:01:32,610 --> 00:01:35,200 and you have as an input vi and an output vo, 33 00:01:35,200 --> 00:01:38,490 then you can think about that system being a filter. 34 00:01:38,490 --> 00:01:41,700 The filter can be characterized by a frequency response. 35 00:01:41,700 --> 00:01:44,220 You're all experts at that. 36 00:01:44,220 --> 00:01:48,840 And then if you can break the signal into Fourier components, 37 00:01:48,840 --> 00:01:51,816 which was easy when we had a Fourier series, 38 00:01:51,816 --> 00:01:53,940 if you can break the signal into Fourier components 39 00:01:53,940 --> 00:02:00,180 then it's easy to calculate the output as the sum of weighted 40 00:02:00,180 --> 00:02:02,580 and possibly time shifted versions 41 00:02:02,580 --> 00:02:04,230 of the components of the input. 42 00:02:04,230 --> 00:02:07,890 So we can decompose the signal, the square wave, 43 00:02:07,890 --> 00:02:11,610 into a bunch of Fourier components. 44 00:02:11,610 --> 00:02:14,550 And then think about how they pass through the filter. 45 00:02:14,550 --> 00:02:17,100 Where if they all pass through the low frequency 46 00:02:17,100 --> 00:02:21,540 part of the filter, so that the gain is one and the phase is 0, 47 00:02:21,540 --> 00:02:23,470 the output looks just like the input. 48 00:02:23,470 --> 00:02:26,550 And if they all pass through the part of the filter that's 49 00:02:26,550 --> 00:02:28,535 sloping down with a slope of minus 1, 50 00:02:28,535 --> 00:02:31,230 with the phase lagging by pi over 2, 51 00:02:31,230 --> 00:02:33,130 you turn a square wave into a triangle wave. 52 00:02:33,130 --> 00:02:35,340 That was the simplest example of a filter. 53 00:02:35,340 --> 00:02:40,440 We saw it again when we thought about speech production. 54 00:02:40,440 --> 00:02:44,220 Because there the idea was that the different kinds of sounds 55 00:02:44,220 --> 00:02:47,820 that we make are partially generated by the larynx, 56 00:02:47,820 --> 00:02:51,270 and partially generated by the throat. 57 00:02:51,270 --> 00:02:53,950 And it's all the musculature in your face 58 00:02:53,950 --> 00:02:59,610 that enables you to make the precise different sounds. 59 00:02:59,610 --> 00:03:01,620 And that was what we called the source filter 60 00:03:01,620 --> 00:03:03,120 model of speech production. 61 00:03:03,120 --> 00:03:06,270 The sources down here, the filter is here. 62 00:03:06,270 --> 00:03:10,410 And we again, we can think about it by thinking about filtering. 63 00:03:10,410 --> 00:03:12,270 Filtering comes up all over the place, 64 00:03:12,270 --> 00:03:14,700 and it's one of the most important applications 65 00:03:14,700 --> 00:03:17,340 of Fourier techniques generally. 66 00:03:17,340 --> 00:03:19,860 And it's for all the reasons you already know. 67 00:03:19,860 --> 00:03:21,870 If you can think of a way of breaking down 68 00:03:21,870 --> 00:03:24,000 a signal into Fourier components, 69 00:03:24,000 --> 00:03:28,440 then you can thing about an LTI system as a filter. 70 00:03:28,440 --> 00:03:33,930 And in filtering applications, in signal processing filtering 71 00:03:33,930 --> 00:03:35,550 applications, we usually try to think 72 00:03:35,550 --> 00:03:37,810 about high frequencies, low frequencies, 73 00:03:37,810 --> 00:03:39,510 designing systems that pass the lows, 74 00:03:39,510 --> 00:03:41,260 pass the highs, that sort of thing. 75 00:03:41,260 --> 00:03:43,090 But the key is breaking down an input, 76 00:03:43,090 --> 00:03:46,200 which might be complicated in the pieces. 77 00:03:46,200 --> 00:03:49,350 That's where the Fourier transform is so good. 78 00:03:49,350 --> 00:03:50,910 And so, I want to illustrate that 79 00:03:50,910 --> 00:03:53,940 by thinking about a hard problem in signal processing, 80 00:03:53,940 --> 00:03:57,180 and that is an electrocardiogram. 81 00:03:57,180 --> 00:03:59,220 So first off, it's completely amazing 82 00:03:59,220 --> 00:04:02,920 that you can measure an electrocardiogram at all. 83 00:04:02,920 --> 00:04:05,280 The voltages produced by cells are 84 00:04:05,280 --> 00:04:07,930 on the order of 100 millivolts. 85 00:04:07,930 --> 00:04:11,040 That's true for all cells for extremely fundamental reasons, 86 00:04:11,040 --> 00:04:14,310 which if you're interested in, 6021 has a lot of information 87 00:04:14,310 --> 00:04:15,390 about why. 88 00:04:15,390 --> 00:04:17,880 There are fundamental physical limits 89 00:04:17,880 --> 00:04:21,750 on what kinds of a voltage a cell can make. 90 00:04:21,750 --> 00:04:28,170 So the voltages are small, but they're much worse than that, 91 00:04:28,170 --> 00:04:30,360 they're constrained to the inside of a cell. 92 00:04:30,360 --> 00:04:33,207 You can't generally get access to the inside of a cell. 93 00:04:33,207 --> 00:04:35,290 In fact, there's a big technological breakthrough, 94 00:04:35,290 --> 00:04:37,140 and people figured out how to draw-- 95 00:04:37,140 --> 00:04:39,030 cells are little. 96 00:04:39,030 --> 00:04:41,820 It's kind of hard to imagine how little they are. 97 00:04:41,820 --> 00:04:48,870 About two to three million cells would fit in the length, 98 00:04:48,870 --> 00:04:52,032 in a one inch length of a human hair. 99 00:04:52,032 --> 00:04:53,490 And that's a skinny hair like mine. 100 00:04:53,490 --> 00:04:57,240 So blonde skinny hair. 101 00:04:57,240 --> 00:05:00,480 The rest is not necessarily skinny, but they're skinny. 102 00:05:00,480 --> 00:05:02,400 So you can get about a million cells, 103 00:05:02,400 --> 00:05:07,200 maybe two million cells in one length, one inch of-- 104 00:05:07,200 --> 00:05:08,757 length of a human hair. 105 00:05:08,757 --> 00:05:10,590 To give you some idea of how small they are. 106 00:05:10,590 --> 00:05:12,110 These tiny, tiny, tiny little things. 107 00:05:12,110 --> 00:05:13,020 So it was a big deal when people were 108 00:05:13,020 --> 00:05:14,552 able to measure inside a cell. 109 00:05:14,552 --> 00:05:16,260 And when you put electrodes on your chest 110 00:05:16,260 --> 00:05:19,920 to measure an electrocardiogram, you're not inside a cell. 111 00:05:19,920 --> 00:05:22,510 Cells are surrounded by an insulator. 112 00:05:22,510 --> 00:05:26,760 So not much of that electrical current is available to you. 113 00:05:26,760 --> 00:05:31,710 Furthermore, the cells in the heart are surrounded by saline. 114 00:05:31,710 --> 00:05:34,600 What's important about saline? 115 00:05:34,600 --> 00:05:37,400 What is saline? 116 00:05:37,400 --> 00:05:38,621 Salt water. 117 00:05:38,621 --> 00:05:39,870 What's important about saline? 118 00:05:42,804 --> 00:05:44,220 What would be different if we were 119 00:05:44,220 --> 00:05:47,160 filled with distilled water? 120 00:05:47,160 --> 00:05:50,810 Besides the fact that we would die. 121 00:05:50,810 --> 00:05:52,590 Yeah. 122 00:05:52,590 --> 00:05:54,310 Conducts electricity. 123 00:05:54,310 --> 00:05:57,810 So the fluid that bathes the cells 124 00:05:57,810 --> 00:05:59,950 has a very high ionic concentration. 125 00:05:59,950 --> 00:06:02,784 And that means that it conducts electricity, 126 00:06:02,784 --> 00:06:04,200 which means most of the potentials 127 00:06:04,200 --> 00:06:06,866 are shorted out before they ever get to the surface of the skin. 128 00:06:06,866 --> 00:06:09,900 Then there's another 20 layers of insulation, 129 00:06:09,900 --> 00:06:12,540 called your skin. 130 00:06:12,540 --> 00:06:15,760 So it's astonishing that you can even measure these things. 131 00:06:15,760 --> 00:06:17,310 And when you do, it's not surprising 132 00:06:17,310 --> 00:06:20,850 that there's a lot of signal in the waveform, 133 00:06:20,850 --> 00:06:22,750 other than the signal that you intend. 134 00:06:22,750 --> 00:06:27,300 There's about 10 to the 13th neurons in your body, 135 00:06:27,300 --> 00:06:29,460 and they're all chattering away. 136 00:06:29,460 --> 00:06:30,850 So you see those. 137 00:06:30,850 --> 00:06:34,950 So that the idea of filtering out the part of the signal 138 00:06:34,950 --> 00:06:37,862 that results from the EKG is completely non-trivial. 139 00:06:37,862 --> 00:06:40,320 And it's very useful to think about it in a Fourier domain. 140 00:06:40,320 --> 00:06:42,870 And absolutely everybody does it that way. 141 00:06:42,870 --> 00:06:47,220 If you were to take the Fourier transform of this waveform, 142 00:06:47,220 --> 00:06:48,720 you would get a wave form that looks 143 00:06:48,720 --> 00:06:51,420 like this, which looks perhaps more complicated. 144 00:06:51,420 --> 00:06:53,820 except that you can make some sense out of it. 145 00:06:53,820 --> 00:06:56,130 Hearts beat about 60 or 70 times a minute, 146 00:06:56,130 --> 00:06:59,340 depending on how athletic and how old you are, 147 00:06:59,340 --> 00:07:03,480 which means that it's the components around 1 148 00:07:03,480 --> 00:07:07,020 or bigger that are coming from the heart. 149 00:07:07,020 --> 00:07:09,007 Things significantly lower than hertz 150 00:07:09,007 --> 00:07:10,590 probably aren't coming from the heart. 151 00:07:10,590 --> 00:07:13,170 Things that are up in the 10 kilohertz region probably 152 00:07:13,170 --> 00:07:16,390 are not from the heart. 153 00:07:16,390 --> 00:07:21,040 This enormous spike, what's that? 154 00:07:21,040 --> 00:07:22,510 That's the lights. 155 00:07:22,510 --> 00:07:23,680 That's the power. 156 00:07:23,680 --> 00:07:28,450 That's the-- we distribute electrical power by modulating 157 00:07:28,450 --> 00:07:30,670 it at 60 hertz because that distributes better, 158 00:07:30,670 --> 00:07:32,380 but then it radiates. 159 00:07:32,380 --> 00:07:37,300 And so some of that can be coupled into everything, 160 00:07:37,300 --> 00:07:39,430 including me. 161 00:07:39,430 --> 00:07:42,520 So the big line is the 60 hertz that's 162 00:07:42,520 --> 00:07:46,720 being coupled from the power distribution network 163 00:07:46,720 --> 00:07:50,260 into the person whose EKG this is. 164 00:07:50,260 --> 00:07:52,450 So what we'd like to do then is generate 165 00:07:52,450 --> 00:07:56,620 a filter that takes out the stuff that isn't EKG. 166 00:07:56,620 --> 00:07:58,840 So we'd like to eliminate this low frequency stuff, 167 00:07:58,840 --> 00:08:02,380 and we'd like to eliminate this high frequency stuff, right? 168 00:08:02,380 --> 00:08:05,710 So what we do is we design a filter. 169 00:08:05,710 --> 00:08:08,870 Filter design Bode of course. 170 00:08:08,870 --> 00:08:09,370 Right? 171 00:08:09,370 --> 00:08:10,060 Smile? 172 00:08:10,060 --> 00:08:11,640 Everybody smile, you know? 173 00:08:11,640 --> 00:08:16,270 There's nothing on a quiz for weeks to come, right? 174 00:08:16,270 --> 00:08:17,440 So Bode. 175 00:08:17,440 --> 00:08:21,580 So we would think about passing the high frequencies. 176 00:08:25,730 --> 00:08:27,080 I didn't say that right. 177 00:08:27,080 --> 00:08:29,330 So we would like to cut off the very lows, 178 00:08:29,330 --> 00:08:32,210 we'd like to cut off the very highs. 179 00:08:32,210 --> 00:08:36,830 And we'd like to get rid of that one glitch at 60 hertz. 180 00:08:36,830 --> 00:08:40,280 So the 60 hertz thing, that's what this is intended to do. 181 00:08:40,280 --> 00:08:42,950 It's intended to be a very narrow filter that 182 00:08:42,950 --> 00:08:46,130 just wipes out the 60. 183 00:08:46,130 --> 00:08:49,220 So this is a kind of filter that I'd like to design. 184 00:08:49,220 --> 00:08:51,410 And here is the design. 185 00:08:51,410 --> 00:08:54,110 Handful of poles and 0s. 186 00:08:54,110 --> 00:08:59,450 Your task, talk to your neighbor and figure out which poles 187 00:08:59,450 --> 00:09:01,550 and which 0s go with the high pass, the low pass, 188 00:09:01,550 --> 00:09:02,133 and the notch. 189 00:09:07,760 --> 00:09:09,745 Look at your neighbor, say hi, smile. 190 00:10:53,340 --> 00:10:57,090 OK, so which of the poles and 0s contribute to the high pass? 191 00:11:00,295 --> 00:11:01,170 Rught, you can point. 192 00:11:01,170 --> 00:11:04,300 Like, there's up, and there's left, and there's center. 193 00:11:04,300 --> 00:11:05,859 So which ones are high pass? 194 00:11:11,130 --> 00:11:12,000 The ones near 0. 195 00:11:14,816 --> 00:11:16,986 The 0 or the poles? 196 00:11:16,986 --> 00:11:18,360 Or one of each, or both, or what? 197 00:11:24,270 --> 00:11:29,464 So what would the Bode plot look like if you only had the 0? 198 00:11:29,464 --> 00:11:30,340 Yeah. 199 00:11:30,340 --> 00:11:36,740 So the 0 alone would cancel out frequency components at 0. 200 00:11:36,740 --> 00:11:41,092 Which our Bode plot is a way over there, right? 201 00:11:41,092 --> 00:11:42,050 What about these poles? 202 00:11:42,050 --> 00:11:42,924 What's the poles for? 203 00:11:46,470 --> 00:11:48,450 They flatten that it out. 204 00:11:48,450 --> 00:11:50,390 So if you think about reading off 205 00:11:50,390 --> 00:11:53,330 the effect of the poles and 0s starting at 0 206 00:11:53,330 --> 00:11:57,047 and going to bigger and bigger radiuses, 207 00:11:57,047 --> 00:11:58,630 then the first thing you have to worry 208 00:11:58,630 --> 00:12:01,300 about going left to right on the Bode plot, 209 00:12:01,300 --> 00:12:04,090 is the 0s doing this. 210 00:12:04,090 --> 00:12:06,040 And then these poles flatten out the 0s. 211 00:12:10,030 --> 00:12:13,310 So what's this doing? 212 00:12:13,310 --> 00:12:16,570 That's the thing that's attenuating 213 00:12:16,570 --> 00:12:19,510 the high frequencies, which leaves us with this. 214 00:12:19,510 --> 00:12:21,286 What's that? 215 00:12:21,286 --> 00:12:22,660 And how do you know it's a notch? 216 00:12:25,366 --> 00:12:27,110 There's a 0. 217 00:12:27,110 --> 00:12:29,960 So the 0 on the j-omega axis means 218 00:12:29,960 --> 00:12:33,370 there's a frequency that it completely wipes out. 219 00:12:33,370 --> 00:12:35,710 Why do I need to have this stupid pole here? 220 00:12:35,710 --> 00:12:37,860 Can't I just put the 0 there? 221 00:12:37,860 --> 00:12:39,640 AUDIENCE: [INAUDIBLE]. 222 00:12:39,640 --> 00:12:41,330 DENNIS FREEMAN: Bring it back up. 223 00:12:41,330 --> 00:12:46,990 So the idea is that if you put the pole close to the 0, 224 00:12:46,990 --> 00:12:49,820 if you get to a frequency that's pretty far away from those, 225 00:12:49,820 --> 00:12:51,770 their effects cancel, and there's 226 00:12:51,770 --> 00:12:54,200 no effect of either of them. 227 00:12:54,200 --> 00:12:57,080 So their effect is constrained to frequencies 228 00:12:57,080 --> 00:13:03,620 that are very close measured in terms of this distance. 229 00:13:03,620 --> 00:13:06,680 So you have to be at a frequency that's very close in order 230 00:13:06,680 --> 00:13:09,830 to have an effect. 231 00:13:09,830 --> 00:13:13,160 And when you do that, when you design such a filter, 232 00:13:13,160 --> 00:13:14,570 you clean up the wave form a lot, 233 00:13:14,570 --> 00:13:19,580 and no one in their right mind would generate an EKG machine 234 00:13:19,580 --> 00:13:20,720 that didn't do this. 235 00:13:20,720 --> 00:13:23,660 I mean, that's just built into the preamplifier that's 236 00:13:23,660 --> 00:13:25,070 attached to the electrodes. 237 00:13:25,070 --> 00:13:28,970 So the point is that filtering is very important application 238 00:13:28,970 --> 00:13:30,590 of Fourier transforms. 239 00:13:30,590 --> 00:13:33,980 We can take an arbitrary signal and often get a lot of insight 240 00:13:33,980 --> 00:13:38,060 into what we would like to preserve and remove by thinking 241 00:13:38,060 --> 00:13:39,770 about the Fourier transform, insights 242 00:13:39,770 --> 00:13:42,061 that you wouldn't get by looking at the time wave form. 243 00:13:44,780 --> 00:13:47,650 Next thing I want to think about is a little bit different. 244 00:13:47,650 --> 00:13:50,080 I want to think about physics. 245 00:13:50,080 --> 00:13:52,960 This isn't, of course, in physics, nor am I a physicist, 246 00:13:52,960 --> 00:13:56,410 but I do do optics. 247 00:13:56,410 --> 00:13:59,267 And the thing I want to think about is diffraction. 248 00:13:59,267 --> 00:14:00,850 You can't understand optics unless you 249 00:14:00,850 --> 00:14:03,370 think about diffraction, and honestly the easiest way 250 00:14:03,370 --> 00:14:11,100 to think about diffraction is Fourier transform. 251 00:14:11,100 --> 00:14:13,510 So a very simple example of diffraction. 252 00:14:13,510 --> 00:14:15,870 So here is a diffraction grating. 253 00:14:15,870 --> 00:14:19,230 So if I pass a coherent beam through there, which I just 254 00:14:19,230 --> 00:14:22,560 happen to have, if I pass a coherent beam 255 00:14:22,560 --> 00:14:24,650 through a diffraction grating, what do you see? 256 00:14:29,840 --> 00:14:33,490 The diffraction grating split one beam. 257 00:14:33,490 --> 00:14:36,350 All term I've been pointing with one. 258 00:14:36,350 --> 00:14:41,130 Well, if I put this in front of it, now I get three. 259 00:14:41,130 --> 00:14:45,700 So somehow there's something about this that's 260 00:14:45,700 --> 00:14:47,365 breaking one beam into three. 261 00:14:49,980 --> 00:14:52,960 Can you think why? 262 00:14:52,960 --> 00:14:58,750 Can you remember 802 where they probably mentioned this? 263 00:15:02,440 --> 00:15:03,515 What is this? 264 00:15:07,170 --> 00:15:08,318 What's in here? 265 00:15:08,318 --> 00:15:10,070 AUDIENCE: Slits. 266 00:15:10,070 --> 00:15:11,170 DENNIS FREEMAN: Slits. 267 00:15:11,170 --> 00:15:11,770 Close. 268 00:15:11,770 --> 00:15:13,420 Yeah. 269 00:15:13,420 --> 00:15:15,430 So the simplest experiment where you 270 00:15:15,430 --> 00:15:18,385 could illustrate this kind of a diffraction phenomenon is-- 271 00:15:21,570 --> 00:15:22,425 goes by the name-- 272 00:15:26,460 --> 00:15:27,020 yes. 273 00:15:27,020 --> 00:15:27,650 Shout the name. 274 00:15:31,140 --> 00:15:33,775 Young. 275 00:15:33,775 --> 00:15:35,619 AUDIENCE: Double slit experiment. 276 00:15:35,619 --> 00:15:37,940 DENNIS FREEMAN: Double slit experiment. 277 00:15:37,940 --> 00:15:41,090 So if you pass light through two slits, 278 00:15:41,090 --> 00:15:42,470 something phenomenal happens. 279 00:15:42,470 --> 00:15:46,370 And it's very closely related to this. 280 00:15:46,370 --> 00:15:47,120 What's in this? 281 00:15:49,862 --> 00:15:51,690 AUDIENCE: [INAUDIBLE]. 282 00:15:51,690 --> 00:15:54,470 DENNIS FREEMAN: Something like a slit. 283 00:15:54,470 --> 00:15:57,790 It's actually something like a whole bunch of slits. 284 00:15:57,790 --> 00:15:58,310 So, yeah? 285 00:15:58,310 --> 00:15:59,196 AUDIENCE: Polarizer. 286 00:15:59,196 --> 00:16:00,370 DENNIS FREEMAN: Polarizer. 287 00:16:00,370 --> 00:16:01,560 Close. 288 00:16:01,560 --> 00:16:06,286 It's one of those neat words from physics, but not quite. 289 00:16:06,286 --> 00:16:07,660 Polarized has to do with the fact 290 00:16:07,660 --> 00:16:10,540 that the light has an e field and an m field, 291 00:16:10,540 --> 00:16:12,280 and they are perpendicular to each other, 292 00:16:12,280 --> 00:16:16,580 and sources like this one actually keep them separate. 293 00:16:16,580 --> 00:16:19,120 So this actually does generate polarized light, 294 00:16:19,120 --> 00:16:23,230 but you can't really tell because if I spin the laser 295 00:16:23,230 --> 00:16:24,940 pointer, you can't really see anything. 296 00:16:24,940 --> 00:16:26,065 But if I spin the grating-- 297 00:16:30,760 --> 00:16:31,840 I just said something. 298 00:16:31,840 --> 00:16:33,311 So what did I just call this? 299 00:16:33,311 --> 00:16:34,060 AUDIENCE: Grating. 300 00:16:34,060 --> 00:16:35,380 DENNIS FREEMAN: Grating. 301 00:16:35,380 --> 00:16:37,136 So what is it? 302 00:16:37,136 --> 00:16:38,260 It's a diffraction grating. 303 00:16:38,260 --> 00:16:43,880 It's got a bunch of little horizontal structure. 304 00:16:43,880 --> 00:16:47,770 So the idea is here. 305 00:16:47,770 --> 00:16:51,460 The grating, you can think about it as an array, 306 00:16:51,460 --> 00:16:53,410 big array, big compared to the size 307 00:16:53,410 --> 00:16:55,360 of the point that I was eliminating, 308 00:16:55,360 --> 00:17:00,040 big array of scatterers. 309 00:17:00,040 --> 00:17:03,010 So you think about having a material that 310 00:17:03,010 --> 00:17:05,200 transmits light and a material that 311 00:17:05,200 --> 00:17:08,109 scatters light and a material that scatters light is arranged 312 00:17:08,109 --> 00:17:08,890 in lines. 313 00:17:08,890 --> 00:17:11,619 And the lines are separated from each other by some kind 314 00:17:11,619 --> 00:17:16,329 of a distance, here represented by a capital D. 315 00:17:16,329 --> 00:17:20,440 And the reason that you see the far-field pattern, 316 00:17:20,440 --> 00:17:22,329 we call that the far-field pattern as opposed 317 00:17:22,329 --> 00:17:23,440 to the near-field pattern. 318 00:17:23,440 --> 00:17:30,601 If I were to do this and put it very close, 319 00:17:30,601 --> 00:17:32,350 then I would think about the pattern being 320 00:17:32,350 --> 00:17:36,830 the near-field pattern if I get close compared to D. 321 00:17:36,830 --> 00:17:40,940 So if the slits are D apart, the near-field is D close, 322 00:17:40,940 --> 00:17:44,920 and the far field is when I'm far away compared to D. 323 00:17:44,920 --> 00:17:47,450 So if I'm far away compared to D, 324 00:17:47,450 --> 00:17:52,050 then you can imagine that the light is coming in this way, 325 00:17:52,050 --> 00:17:55,680 and each one of these generates scatter. 326 00:17:55,680 --> 00:17:58,230 So there is a spherical wave coming way out 327 00:17:58,230 --> 00:18:00,207 from each of these. 328 00:18:00,207 --> 00:18:01,290 That's Huygens' principle. 329 00:18:01,290 --> 00:18:03,300 Right 330 00:18:03,300 --> 00:18:06,870 So the idea then is that there are contributions 331 00:18:06,870 --> 00:18:08,430 from each one of these scatterers 332 00:18:08,430 --> 00:18:10,620 out here in the far field. 333 00:18:10,620 --> 00:18:12,450 Each point in the far field picks up 334 00:18:12,450 --> 00:18:16,340 a little bit of light from each of the scatterers, 335 00:18:16,340 --> 00:18:20,070 but they're in different phase relationships. 336 00:18:20,070 --> 00:18:26,430 They were in phase when the coherent light struck these. 337 00:18:26,430 --> 00:18:30,230 But now if I were to think about a point here, 338 00:18:30,230 --> 00:18:34,505 the phase that you get is different from the point here. 339 00:18:34,505 --> 00:18:41,150 So here the phase from two points would be about the same. 340 00:18:41,150 --> 00:18:46,220 Here they would be different because the distance traveled 341 00:18:46,220 --> 00:18:49,220 from this scatterer is that much longer. 342 00:18:53,020 --> 00:18:57,280 So if I look at a particular angle, 343 00:18:57,280 --> 00:19:02,560 then each of these scatterers is a different optical path length 344 00:19:02,560 --> 00:19:06,290 from the source. 345 00:19:06,290 --> 00:19:14,020 That means that if I arrange the angle so that this is lambda, 346 00:19:14,020 --> 00:19:16,480 the light that scatters off of here 347 00:19:16,480 --> 00:19:18,610 will constructively interfere with the light that 348 00:19:18,610 --> 00:19:22,540 scatters off of here and here. 349 00:19:22,540 --> 00:19:25,080 Everybody see that? 350 00:19:25,080 --> 00:19:29,130 So that means that there's a funny angle, the sign of which 351 00:19:29,130 --> 00:19:32,090 is lambda over D in which you get constructive interference, 352 00:19:32,090 --> 00:19:34,800 and that's what's going on with the diffraction grating. 353 00:19:37,800 --> 00:19:40,640 And you can see that. 354 00:19:40,640 --> 00:19:43,380 There are several illustrations of that all over the place. 355 00:19:43,380 --> 00:19:45,930 Here I have a CD and a DVD. 356 00:19:45,930 --> 00:19:48,300 Which one is which? 357 00:19:48,300 --> 00:19:51,380 I'm showing you the backside of a CD and a DVD. 358 00:19:51,380 --> 00:19:52,890 AUDIENCE: DVD is in the left side. 359 00:19:52,890 --> 00:19:54,536 DENNIS FREEMAN: DVDs is this one. 360 00:19:54,536 --> 00:19:55,410 How do you know that? 361 00:19:55,410 --> 00:19:56,710 AUDIENCE: Because it looks blue. 362 00:19:56,710 --> 00:19:58,293 DENNIS FREEMAN: Because it looks blue. 363 00:19:58,293 --> 00:19:59,700 Why does it look blue? 364 00:19:59,700 --> 00:20:00,740 Not a clue. 365 00:20:00,740 --> 00:20:03,752 It's just they always do. 366 00:20:03,752 --> 00:20:05,150 AUDIENCE: [INAUDIBLE]. 367 00:20:05,150 --> 00:20:07,050 DENNIS FREEMAN: Smaller. 368 00:20:07,050 --> 00:20:11,340 So somehow this one reflects better, the bluer lights, 369 00:20:11,340 --> 00:20:13,140 so you see more of that, and this one 370 00:20:13,140 --> 00:20:16,790 reflects better longer lights. 371 00:20:16,790 --> 00:20:19,090 This one has got a greenish twinge maybe. 372 00:20:22,660 --> 00:20:24,130 Kind of like all colors. 373 00:20:24,130 --> 00:20:26,110 I like to call it green. 374 00:20:26,110 --> 00:20:29,390 Maybe it's because I know the answer. 375 00:20:29,390 --> 00:20:32,500 So if I take these guys, I can get the same sort of thing, 376 00:20:32,500 --> 00:20:35,110 so I'll take the CD first. 377 00:20:35,110 --> 00:20:37,629 Now, of course, this has a coding on it, 378 00:20:37,629 --> 00:20:39,670 so if I try to shine a laser straight through it, 379 00:20:39,670 --> 00:20:41,950 not much will happen. 380 00:20:41,950 --> 00:20:46,180 I don't really want to blind somebody. 381 00:20:46,180 --> 00:20:47,680 So I'm going to take this and I'm 382 00:20:47,680 --> 00:20:50,050 going to attempt to hit this while watching back here, 383 00:20:50,050 --> 00:20:52,330 which for somebody my age it's pretty hard to do. 384 00:20:52,330 --> 00:20:55,720 But the idea is going to be that instead of using this 385 00:20:55,720 --> 00:20:58,690 as a transmission grating, I'm going to use it as a reflection 386 00:20:58,690 --> 00:20:59,810 grating. 387 00:20:59,810 --> 00:21:00,925 So if I do this-- 388 00:21:05,200 --> 00:21:06,880 so that's kind of cute. 389 00:21:06,880 --> 00:21:11,680 So my laser is green. 390 00:21:11,680 --> 00:21:13,795 What's lambda? 391 00:21:13,795 --> 00:21:14,785 AUDIENCE: 530. 392 00:21:14,785 --> 00:21:16,030 DENNIS FREEMAN: 530. 393 00:21:16,030 --> 00:21:16,930 So let's round off. 394 00:21:16,930 --> 00:21:17,941 Let's say 500. 395 00:21:17,941 --> 00:21:18,440 Yes. 396 00:21:21,310 --> 00:21:25,900 So I've got a green laser, and what 397 00:21:25,900 --> 00:21:29,954 will happen if I move the laser back and forth? 398 00:21:29,954 --> 00:21:31,120 How will the pattern change? 399 00:21:35,072 --> 00:21:37,050 It won't change. 400 00:21:37,050 --> 00:21:41,960 So if I do this, which I can't do while I'm watching it, 401 00:21:41,960 --> 00:21:43,730 no change. 402 00:21:43,730 --> 00:21:45,669 What will happen if I move this way this way? 403 00:21:50,122 --> 00:21:51,455 Well, the angle is what matters. 404 00:21:54,230 --> 00:21:58,760 So as I move it farther away, the pattern gets bigger. 405 00:21:58,760 --> 00:22:08,720 So if I do this, if I were at all coordinated, which I'm not, 406 00:22:08,720 --> 00:22:10,190 I'm trusting that it's doing what 407 00:22:10,190 --> 00:22:14,150 I'm telling it to because I can't look at the same time. 408 00:22:14,150 --> 00:22:21,110 So let me put this at a precise 1 meter away, 3 feet. 409 00:22:21,110 --> 00:22:24,560 And let me shine this in here, and let me 410 00:22:24,560 --> 00:22:26,690 do a very precise measurement. 411 00:22:26,690 --> 00:22:30,255 And those are now one-foot apart by my precise measurement 412 00:22:30,255 --> 00:22:30,755 skills. 413 00:22:34,570 --> 00:22:39,420 So what I'd like you to do is figure out the pitch. 414 00:22:39,420 --> 00:22:43,860 The information on a CD is written will spirally. 415 00:22:43,860 --> 00:22:47,700 That's the reason there is an apparent color. 416 00:22:47,700 --> 00:22:50,220 I just measured the pitch. 417 00:22:50,220 --> 00:22:52,290 How far apart are the tracks? 418 00:22:52,290 --> 00:22:54,780 So take the data. 419 00:22:54,780 --> 00:22:57,330 So if I'm three feet away from the screen, 420 00:22:57,330 --> 00:22:59,100 the dots are separated by one foot 421 00:22:59,100 --> 00:23:02,640 and figure out how closely spaced the tracks are on a CD. 422 00:24:25,045 --> 00:24:25,920 So what's the answer? 423 00:24:25,920 --> 00:24:27,560 1, 2, 3, 4. 424 00:24:27,560 --> 00:24:28,310 None of the above. 425 00:24:34,580 --> 00:24:41,210 A very small number of votes, but about 90% correct. 426 00:24:41,210 --> 00:24:41,900 So what do I do? 427 00:24:41,900 --> 00:24:42,858 How do I figure it out? 428 00:24:50,100 --> 00:24:50,600 Yeah. 429 00:24:50,600 --> 00:24:54,830 So we just made a big deal out of how 430 00:24:54,830 --> 00:24:59,270 the distance, which was D sine theta 431 00:24:59,270 --> 00:25:02,281 had to be an integer multiple of lambda. 432 00:25:02,281 --> 00:25:04,280 So all we need to do is think through some trig. 433 00:25:14,080 --> 00:25:15,520 So if I think about the dots being 434 00:25:15,520 --> 00:25:18,320 a foot apart when I'm three feet away, then the tangent is 1/3. 435 00:25:21,528 --> 00:25:26,749 So the small angle approximation is about 1/3, 436 00:25:26,749 --> 00:25:27,790 so the sine is about 1/3. 437 00:25:31,450 --> 00:25:37,960 So lambda is about 500 nanometers divided by 1/3. 438 00:25:37,960 --> 00:25:41,410 And if I carry that out a little more precisely, I get 1,613, 439 00:25:41,410 --> 00:25:43,900 and the manufacturer's specification for CDs 440 00:25:43,900 --> 00:25:46,670 is 1,600 nanometers. 441 00:25:46,670 --> 00:25:51,070 So it's 1.6 micrometers. 442 00:25:51,070 --> 00:25:51,729 Yeah. 443 00:25:51,729 --> 00:25:55,082 AUDIENCE: [? What ?] is the precision on [INAUDIBLE]?? 444 00:25:55,082 --> 00:25:58,670 DENNIS FREEMAN: Oh, they're pretty good. 445 00:25:58,670 --> 00:26:01,410 In fact, we'll talk about this a little later. 446 00:26:01,410 --> 00:26:03,690 They do this by an embossing process. 447 00:26:03,690 --> 00:26:09,450 So you make a master out of aluminum, 448 00:26:09,450 --> 00:26:11,090 and then you stamp it. 449 00:26:11,090 --> 00:26:14,370 And the stamping is actually very good. 450 00:26:14,370 --> 00:26:16,680 So probably the biggest manufacturing problem 451 00:26:16,680 --> 00:26:18,090 is the thermals. 452 00:26:18,090 --> 00:26:24,360 So you stamp it in heated polystyrene, so there's 453 00:26:24,360 --> 00:26:27,314 a little bit of a coefficient of shrinkage as it cools, 454 00:26:27,314 --> 00:26:28,230 but it's not very big. 455 00:26:28,230 --> 00:26:30,300 It's fractions of a percent. 456 00:26:33,330 --> 00:26:38,140 So what will happen then if I switch to the DVD, 457 00:26:38,140 --> 00:26:41,330 how many tracks are on a DVD? 458 00:26:41,330 --> 00:26:43,540 So we can do the same kind of an experiment. 459 00:26:43,540 --> 00:26:48,174 Now if I try to do the DVD, in order 460 00:26:48,174 --> 00:26:50,590 to get them to be a foot apart, I have to get much closer. 461 00:26:53,674 --> 00:26:56,340 So now to get D space at about a foot, I have to be about a foot 462 00:26:56,340 --> 00:26:57,420 away. 463 00:26:57,420 --> 00:27:03,140 So what's the spacing on the DVD, smaller or bigger? 464 00:27:06,000 --> 00:27:09,760 So it's smaller because the angle got bigger. 465 00:27:09,760 --> 00:27:13,510 So it's the same sort of deal. 466 00:27:13,510 --> 00:27:20,140 If you think about the one foot, so you get one-foot spacing 467 00:27:20,140 --> 00:27:20,995 with about one foot. 468 00:27:23,820 --> 00:27:25,830 So then the tangent is about 1. 469 00:27:25,830 --> 00:27:29,910 It's not quite a small angle anymore. 470 00:27:29,910 --> 00:27:35,980 So theta is 0.78. 471 00:27:35,980 --> 00:27:41,410 And so the experimental value comes out about 704 nanometers, 472 00:27:41,410 --> 00:27:46,730 so it's about a factor of 2. 473 00:27:46,730 --> 00:27:48,710 Smaller. 474 00:27:48,710 --> 00:27:50,540 So we got about 700 nano, so it's 475 00:27:50,540 --> 00:27:53,837 a fraction of a micron, rather than 1.6 microns, 476 00:27:53,837 --> 00:27:55,670 and that's why they look a little different. 477 00:27:55,670 --> 00:27:59,930 So we get about a factor of 2 in terms of information density 478 00:27:59,930 --> 00:28:01,941 in terms of track count. 479 00:28:01,941 --> 00:28:04,190 Of course, the information is also stored more closely 480 00:28:04,190 --> 00:28:06,660 within a track as well. 481 00:28:06,660 --> 00:28:08,680 So instead of being about a factor of 2 better, 482 00:28:08,680 --> 00:28:12,020 it's maybe a factor of 4 better because the track 483 00:28:12,020 --> 00:28:13,290 is based about a factor of 2. 484 00:28:17,260 --> 00:28:18,850 You can do more interesting things. 485 00:28:18,850 --> 00:28:20,474 That's kind of the most degenerate case 486 00:28:20,474 --> 00:28:26,800 because that's one D. So here is a fancier 487 00:28:26,800 --> 00:28:30,970 grating from my lab, which has the property that if I shine 488 00:28:30,970 --> 00:28:37,070 the light through it, I get more dots. 489 00:28:37,070 --> 00:28:38,660 So what's inside this? 490 00:28:44,516 --> 00:28:46,470 AUDIENCE: A 2D grating. 491 00:28:46,470 --> 00:28:47,900 DENNIS FREEMAN: It's a 2D grating. 492 00:28:47,900 --> 00:28:49,858 So instead of having just lines going this way, 493 00:28:49,858 --> 00:28:54,050 it has lines going this way and lines going that way. 494 00:28:54,050 --> 00:28:59,150 And in fact, back about 5 or 10 years ago, 495 00:28:59,150 --> 00:29:01,250 when laser pointers were a novelty, 496 00:29:01,250 --> 00:29:02,780 they became all the rage to put them 497 00:29:02,780 --> 00:29:05,960 right into your laser pointer. 498 00:29:05,960 --> 00:29:09,950 So here's another laser pointer. 499 00:29:09,950 --> 00:29:13,640 So here is a thing that you screw 500 00:29:13,640 --> 00:29:17,660 in the end, which has a fraction grating built right into it. 501 00:29:17,660 --> 00:29:23,390 And so now if I mean to say some graphics reading off 502 00:29:23,390 --> 00:29:25,910 the cursor, so there is a cursor now in my laser pointer 503 00:29:25,910 --> 00:29:28,550 rather than being a dot. 504 00:29:28,550 --> 00:29:36,990 Or if I'm more chintzy, now I can circle things. 505 00:29:36,990 --> 00:29:40,200 I'm not sure why I want to do that, but in case I wanted to, 506 00:29:40,200 --> 00:29:40,740 I can do it. 507 00:29:47,090 --> 00:29:48,859 If I wanted to underline things, I guess. 508 00:29:48,859 --> 00:29:49,400 I'm not sure. 509 00:29:53,480 --> 00:29:56,660 The point is that these things are just a diffraction grating. 510 00:29:56,660 --> 00:29:59,321 All that's in there is a little piece of glass that's got-- 511 00:29:59,321 --> 00:30:00,320 it's not actually glass. 512 00:30:00,320 --> 00:30:04,970 It's actually plastic that was made with an embossing process, 513 00:30:04,970 --> 00:30:07,620 same as they make CDs with. 514 00:30:07,620 --> 00:30:09,054 So if you're into-- 515 00:30:09,054 --> 00:30:09,970 AUDIENCE: [INAUDIBLE]. 516 00:30:09,970 --> 00:30:10,490 DENNIS FREEMAN: Yeah. 517 00:30:10,490 --> 00:30:10,989 Who knows. 518 00:30:13,520 --> 00:30:19,950 And if you go backwards in time, they get even chintzier. 519 00:30:19,950 --> 00:30:23,930 So this is an even older laser pointer. 520 00:30:33,391 --> 00:30:33,890 They're fun. 521 00:30:42,340 --> 00:30:46,090 You can tell it's older because it's red and it's dimmer. 522 00:30:52,560 --> 00:30:55,560 If you think the click art is kind of still-- 523 00:30:59,820 --> 00:31:00,990 this is my favorite. 524 00:31:00,990 --> 00:31:02,662 Save the favorite for the end. 525 00:31:05,760 --> 00:31:08,360 Dark side. 526 00:31:08,360 --> 00:31:09,780 I don't know. 527 00:31:09,780 --> 00:31:15,150 Anyway, so that's kind of the idea. 528 00:31:15,150 --> 00:31:17,350 The way you can think about diffraction grating, 529 00:31:17,350 --> 00:31:19,380 so now I want think about a more general theory. 530 00:31:19,380 --> 00:31:21,240 I've worked out a specific theory 531 00:31:21,240 --> 00:31:24,510 for a one-dimensional diffraction grating, 532 00:31:24,510 --> 00:31:27,540 now I want to think about the general theory. 533 00:31:27,540 --> 00:31:31,440 What if I simply told you some pattern in space? 534 00:31:31,440 --> 00:31:36,190 What if I told you your job is to make the diffraction grating 535 00:31:36,190 --> 00:31:39,890 to project a dollar sign. 536 00:31:39,890 --> 00:31:42,410 How would you do it? 537 00:31:42,410 --> 00:31:44,690 Take a wild guess. 538 00:31:44,690 --> 00:31:46,175 AUDIENCE: [INAUDIBLE]. 539 00:31:46,175 --> 00:31:48,470 DENNIS FREEMAN: Take the Fourier transform, of course. 540 00:31:52,870 --> 00:31:53,451 So the idea. 541 00:31:53,451 --> 00:31:55,700 So what would happen if the target is more complicated 542 00:31:55,700 --> 00:31:58,460 than a grating, the way to think about this 543 00:31:58,460 --> 00:32:01,760 is the think about just like in the case of the grating, 544 00:32:01,760 --> 00:32:07,130 if you collect the light that hits the far field, 545 00:32:07,130 --> 00:32:10,050 the far field bit has a point in the far field, 546 00:32:10,050 --> 00:32:14,780 came from all over the target just like in the grating. 547 00:32:14,780 --> 00:32:18,230 In the grating, the point in the far field 548 00:32:18,230 --> 00:32:20,300 came from every one of the lines and there 549 00:32:20,300 --> 00:32:22,605 was a simple relationship among the phases that 550 00:32:22,605 --> 00:32:24,230 came from each of the lines, and that's 551 00:32:24,230 --> 00:32:27,410 what gave rise to the pattern in the far field. 552 00:32:27,410 --> 00:32:29,280 Well, the same thing is always true. 553 00:32:29,280 --> 00:32:33,780 All the points in the image in the far field came from-- 554 00:32:33,780 --> 00:32:37,580 each point, had contributions from every point in the target. 555 00:32:40,270 --> 00:32:44,030 And if you think about points in the target, 556 00:32:44,030 --> 00:32:46,240 now I'm going to think about just one point 557 00:32:46,240 --> 00:32:48,880 so this is the target space. 558 00:32:48,880 --> 00:32:52,870 Way out there is the far field. 559 00:32:52,870 --> 00:32:56,560 So in the target space, there's an x-coordinate, 560 00:32:56,560 --> 00:32:59,770 there's a y-coordinate, there's a z-coordinate. 561 00:32:59,770 --> 00:33:03,130 Let me just worry about x for the moment. 562 00:33:03,130 --> 00:33:08,440 This point that is x displaced in this direction 563 00:33:08,440 --> 00:33:13,090 contributes a different phase to the far field 564 00:33:13,090 --> 00:33:14,980 than a point that's at 0. 565 00:33:14,980 --> 00:33:18,420 Because the point is at x, you get a different phase. 566 00:33:18,420 --> 00:33:21,610 And that phase relationship depends, not just 567 00:33:21,610 --> 00:33:25,600 on x, but also on the angle. 568 00:33:25,600 --> 00:33:29,290 And this is the relationship, the phase generated 569 00:33:29,290 --> 00:33:31,840 by a scatterer at the point, x. 570 00:33:34,370 --> 00:33:36,470 Just like the one-dimensional grating, 571 00:33:36,470 --> 00:33:39,560 if you take x and multiply by the sine, 572 00:33:39,560 --> 00:33:43,460 that tells you the delay in meters, 573 00:33:43,460 --> 00:33:45,710 but then you can convert that into radians 574 00:33:45,710 --> 00:33:51,630 by dividing by the wavelength and multiplying by 2 pi. 575 00:33:51,630 --> 00:33:57,020 So the phase that is generated by scattering 576 00:33:57,020 --> 00:34:00,170 from this point displaced a distance, x, 577 00:34:00,170 --> 00:34:04,264 when measured in angle, theta, in the far field, 578 00:34:04,264 --> 00:34:04,930 looks like that. 579 00:34:07,790 --> 00:34:11,179 That's supposed to be just a simple reiteration of what 580 00:34:11,179 --> 00:34:13,420 we just did. 581 00:34:13,420 --> 00:34:17,050 Now the tricky part is that all of the light that 582 00:34:17,050 --> 00:34:20,409 hits a particular point in the far field, 583 00:34:20,409 --> 00:34:24,010 a point in the far field is characterized as theta. 584 00:34:24,010 --> 00:34:26,320 Where is my laser pointer? 585 00:34:26,320 --> 00:34:28,810 So if you go to the far field at a place, 586 00:34:28,810 --> 00:34:34,620 theta, the total amount of light that hits the point, theta, 587 00:34:34,620 --> 00:34:39,150 is the sum, the integral, of all the different scatterers. 588 00:34:39,150 --> 00:34:41,159 So f is x represents how much light gets 589 00:34:41,159 --> 00:34:43,859 scattered as a function of x. 590 00:34:43,859 --> 00:34:46,900 x is the target. 591 00:34:46,900 --> 00:34:50,040 So this is integrating over how much 592 00:34:50,040 --> 00:34:53,850 light gets scattered at each x, and at a particular x, 593 00:34:53,850 --> 00:34:55,179 you get this phase delay. 594 00:34:57,680 --> 00:35:01,650 So there's a coherent sum then. 595 00:35:01,650 --> 00:35:04,030 So you add up all of these complex numbers 596 00:35:04,030 --> 00:35:06,870 by the integral. 597 00:35:06,870 --> 00:35:10,660 Now imagine that the sine of theta is about theta. 598 00:35:10,660 --> 00:35:13,730 That was true all of the examples I've done so far. 599 00:35:13,730 --> 00:35:17,780 So here I'm what, 15 feet away, and it's a foot, 600 00:35:17,780 --> 00:35:21,250 so that's small angle approximation. 601 00:35:21,250 --> 00:35:25,800 So now let's replace sine of theta with theta, 602 00:35:25,800 --> 00:35:29,190 and let's think about this number, 2 pi theta over lambda. 603 00:35:29,190 --> 00:35:32,730 That's just frequency omega. 604 00:35:32,730 --> 00:35:34,920 So I'm thinking about now the far field, 605 00:35:34,920 --> 00:35:38,560 which we had previously called dependent on theta. 606 00:35:38,560 --> 00:35:41,890 Theta is just omega. 607 00:35:41,890 --> 00:35:45,520 So I've got a relationship now between how much scattering 608 00:35:45,520 --> 00:35:51,430 happens at each x in the target and what does the picture 609 00:35:51,430 --> 00:35:53,650 look like in the far field. 610 00:35:53,650 --> 00:35:56,970 And that's a Fourier transform. 611 00:35:56,970 --> 00:35:59,360 So there is an exact Fourier transform relationship. 612 00:35:59,360 --> 00:36:03,080 Well, exact is a little bit of an approximation 613 00:36:03,080 --> 00:36:05,270 because I'm using a small angle approximation. 614 00:36:05,270 --> 00:36:07,380 I'm ignoring a few things. 615 00:36:07,380 --> 00:36:10,250 Not only am I making a small angle approximation, 616 00:36:10,250 --> 00:36:12,380 but I'm assuming that if the light goes straight 617 00:36:12,380 --> 00:36:15,950 and if the light goes up, they have the same intensity when 618 00:36:15,950 --> 00:36:17,780 they hit the board. 619 00:36:17,780 --> 00:36:18,770 That's not quite true. 620 00:36:18,770 --> 00:36:21,950 That's with Fraunhofer approximation blah, blah, blah. 621 00:36:21,950 --> 00:36:23,840 Fancy names for a bunch of approximations. 622 00:36:23,840 --> 00:36:27,170 The point is that if you make reasonable approximations, 623 00:36:27,170 --> 00:36:29,150 you get a Fourier transform relationship 624 00:36:29,150 --> 00:36:33,320 between the scattering in the target and far field image. 625 00:36:40,110 --> 00:36:42,600 So now we have a very convenient way 626 00:36:42,600 --> 00:36:46,380 of thinking about what happened whenever I shot 627 00:36:46,380 --> 00:36:49,050 my laser through this thing. 628 00:36:49,050 --> 00:36:54,390 I get a bunch of spots in the far field because-- 629 00:36:59,380 --> 00:37:01,210 everybody shout. 630 00:37:01,210 --> 00:37:02,000 This is the aha. 631 00:37:05,460 --> 00:37:11,610 So we've got an impulse train in the Fourier transform, which 632 00:37:11,610 --> 00:37:17,670 means that the thing that was down here was an impulse train. 633 00:37:17,670 --> 00:37:20,400 because we know that the Fourier transform of an impulse train 634 00:37:20,400 --> 00:37:23,010 is an impulse train. 635 00:37:23,010 --> 00:37:26,620 So this is an impulse train in space. 636 00:37:26,620 --> 00:37:29,920 The scatterers represent an impulse train. 637 00:37:29,920 --> 00:37:32,810 So whenever I illuminate it, I getting impulse train 638 00:37:32,810 --> 00:37:35,470 in the free transform because the Fourier 639 00:37:35,470 --> 00:37:37,519 transfer on an impulse train is an impulse train. 640 00:37:37,519 --> 00:37:38,560 Well, that's pretty cool. 641 00:37:42,790 --> 00:37:47,620 So you all know why an impulse train is an impulse train. 642 00:37:47,620 --> 00:37:51,940 So this two dimensional grating then, the interesting thing 643 00:37:51,940 --> 00:37:54,460 about that then is that it must be 644 00:37:54,460 --> 00:38:02,470 the case that the Fourier transform of a 2D impulse train 645 00:38:02,470 --> 00:38:04,390 is a 2D impulse train. 646 00:38:04,390 --> 00:38:07,450 So I want to think just a minute about that. 647 00:38:07,450 --> 00:38:10,900 What will we mean by a two-dimensional Fourier 648 00:38:10,900 --> 00:38:12,557 transform. 649 00:38:12,557 --> 00:38:14,140 So two-dimensional Fourier transforms. 650 00:38:14,140 --> 00:38:20,440 I want x j omega 1, j omega 2. 651 00:38:20,440 --> 00:38:25,150 So it's a lot like a 1D transform. 652 00:38:25,150 --> 00:38:29,410 I am going to think about having x except now there's two time 653 00:38:29,410 --> 00:38:33,046 variables, time 1 and time 2. 654 00:38:33,046 --> 00:38:34,420 And I'm going to have to have two 655 00:38:34,420 --> 00:38:36,730 of these funny exponential things. 656 00:38:36,730 --> 00:38:38,500 So I'm going to have e to the minus j 657 00:38:38,500 --> 00:38:49,980 omega 1 t1 plus omega 2 t2 d t1 d t2. 658 00:38:49,980 --> 00:38:53,040 It's almost like a 1D transform except I have two d's now. 659 00:38:56,700 --> 00:38:59,400 And the way to think about that, probably 660 00:38:59,400 --> 00:39:00,780 shouldn't distract you with this. 661 00:39:00,780 --> 00:39:01,696 That's the next slide. 662 00:39:06,550 --> 00:39:12,130 So the way to think about this is this separates. 663 00:39:12,130 --> 00:39:16,750 e to the j sum is e to the j1 plus e to the j2. 664 00:39:16,750 --> 00:39:24,550 So I can write this more simply as x of t1, t2 e to the minus j 665 00:39:24,550 --> 00:39:28,960 omega 1 t1 d t1. 666 00:39:28,960 --> 00:39:34,080 So then I can integrate that, e to the minus j omega d t2. 667 00:39:38,230 --> 00:39:41,940 This is the integral over the t1 variable. 668 00:39:44,820 --> 00:39:47,930 So if I think about doing the transform in steps, 669 00:39:47,930 --> 00:39:49,580 this is the integral I might say-- 670 00:39:49,580 --> 00:39:57,250 so let's say that I had my original 2D thing 671 00:39:57,250 --> 00:39:59,090 was a square that I'd like to characterize 672 00:39:59,090 --> 00:40:04,470 as a t1 dependence and a t2 dependence. 673 00:40:04,470 --> 00:40:06,590 The way I think about the 2D transform, 674 00:40:06,590 --> 00:40:14,170 this thing says, for each t2, treat t2 as a constant, 675 00:40:14,170 --> 00:40:16,870 for each t2, that's for each row. 676 00:40:16,870 --> 00:40:20,080 Treat t2 as a constant and just take the Fourier transform 677 00:40:20,080 --> 00:40:23,470 of the t1 direction. 678 00:40:23,470 --> 00:40:32,190 So what I do is I take the Fourier transform of this, 679 00:40:32,190 --> 00:40:35,460 and I put it here, but when I've done that, 680 00:40:35,460 --> 00:40:39,810 I've changed the t1 axis into an omega 1 axis 681 00:40:39,810 --> 00:40:42,210 because the result of integrating over t1 682 00:40:42,210 --> 00:40:44,640 throws away the t1, but I'm left with an omega 1. 683 00:40:47,940 --> 00:40:52,980 If I repeat that for all the different lines, 684 00:40:52,980 --> 00:40:56,160 I didn't really change this axis, which was t2, 685 00:40:56,160 --> 00:40:57,310 so it's still t2. 686 00:40:59,914 --> 00:41:01,330 All I've done now is taken a bunch 687 00:41:01,330 --> 00:41:02,620 of integrals in the middle. 688 00:41:02,620 --> 00:41:06,670 Now, if I take the outer integral, what I need to do 689 00:41:06,670 --> 00:41:09,340 is integrate over t2. 690 00:41:09,340 --> 00:41:19,481 So if I integrate over t2, now I want to integrate this way. 691 00:41:19,481 --> 00:41:20,980 Now, I want to integrate out the t2. 692 00:41:20,980 --> 00:41:23,930 The t2 goes away. 693 00:41:23,930 --> 00:41:26,290 So I take Fourier transforms this way. 694 00:41:26,290 --> 00:41:29,530 For each column in this space, I generate 695 00:41:29,530 --> 00:41:33,520 a new Fourier transform over here, and I repeat. 696 00:41:33,520 --> 00:41:34,780 That doesn't change this one. 697 00:41:34,780 --> 00:41:36,640 This one is still omega 1. 698 00:41:36,640 --> 00:41:39,460 But then when I take the transform this way, I had a t2. 699 00:41:39,460 --> 00:41:42,820 That turns it into an omega 2. 700 00:41:42,820 --> 00:41:48,580 So I started with t1-t2 space, take all the transforms row 701 00:41:48,580 --> 00:41:53,440 wise, make a new array, take all the transforms column wise, 702 00:41:53,440 --> 00:41:55,840 make a new array, and I end up with omega 1, omega 2. 703 00:41:59,160 --> 00:42:10,360 So if I do that, what's the two-dimensional transform? 704 00:42:10,360 --> 00:42:12,010 What if I started with a vertical line? 705 00:42:14,920 --> 00:42:17,380 What if my spatial dependence had a vertical line in it? 706 00:42:17,380 --> 00:42:19,960 What would be the two-dimensional transform 707 00:42:19,960 --> 00:42:20,770 of a vertical line? 708 00:42:25,730 --> 00:42:31,730 Well, the rule says transform all the rows. 709 00:42:31,730 --> 00:42:33,942 What's the transform? 710 00:42:33,942 --> 00:42:35,150 What do I call that function? 711 00:42:39,380 --> 00:42:40,161 Shout. 712 00:42:40,161 --> 00:42:40,660 I'm deaf. 713 00:42:40,660 --> 00:42:42,308 AUDIENCE: [INAUDIBLE]. 714 00:42:42,308 --> 00:42:45,890 DENNIS FREEMAN: So let's say the line has a height of 1, 715 00:42:45,890 --> 00:42:49,610 and the background has a height of 0. 716 00:42:49,610 --> 00:42:53,620 So as I read across here it's, 0 0, 0, 1, 0, 0, 0. 717 00:42:53,620 --> 00:42:55,021 AUDIENCE: [INAUDIBLE]. 718 00:42:57,823 --> 00:43:00,390 DENNIS FREEMAN: Exponential. 719 00:43:00,390 --> 00:43:01,260 Impulse. 720 00:43:01,260 --> 00:43:03,450 Impulse in left. 721 00:43:03,450 --> 00:43:06,660 So when I take the transform, I get 1. 722 00:43:06,660 --> 00:43:07,860 I get constant. 723 00:43:07,860 --> 00:43:13,920 So I get a constant over here, and then I take this one. 724 00:43:13,920 --> 00:43:15,240 Same thing. 725 00:43:15,240 --> 00:43:17,250 Same constant, same constant, same constant. 726 00:43:17,250 --> 00:43:19,990 I end up with a constant over here. 727 00:43:19,990 --> 00:43:24,790 So this started out being delta of t1. 728 00:43:24,790 --> 00:43:27,070 This is a constant everywhere. 729 00:43:27,070 --> 00:43:34,550 So now if I take the column-wise transforms, 730 00:43:34,550 --> 00:43:36,474 what's the transform of a constant? 731 00:43:40,694 --> 00:43:41,360 It's an impulse. 732 00:43:41,360 --> 00:43:42,526 Where should the impulse be? 733 00:43:44,860 --> 00:43:46,670 0. 734 00:43:46,670 --> 00:43:52,790 So I end up when I do the omega 2, the t2 transforms. 735 00:43:52,790 --> 00:43:54,890 I get a delta in omega 2. 736 00:43:54,890 --> 00:43:59,270 So I have omega 1 here, and I have omega 2 here. 737 00:43:59,270 --> 00:44:04,320 So I end up with a delta function like that. 738 00:44:04,320 --> 00:44:06,030 Then I do it again, and I get one here. 739 00:44:06,030 --> 00:44:07,840 And then I do it again, and I get one here. 740 00:44:07,840 --> 00:44:10,000 So I end up with a line. 741 00:44:10,000 --> 00:44:13,070 So vertical line in t1-t2 space turns into a horizontal line 742 00:44:13,070 --> 00:44:16,270 in omega 1-omega 2 space. 743 00:44:16,270 --> 00:44:17,470 Got it? 744 00:44:17,470 --> 00:44:20,110 It's all very simple. 745 00:44:20,110 --> 00:44:23,680 And if I do the same thing, if I did a horizontal line, 746 00:44:23,680 --> 00:44:26,060 it will come out vertical. 747 00:44:26,060 --> 00:44:28,530 And if I do a diagonal line, it will rotate 90. 748 00:44:31,590 --> 00:44:33,240 Very cute. 749 00:44:33,240 --> 00:44:34,710 Y'all got it? 750 00:44:34,710 --> 00:44:39,660 So the idea then is that there's a very natural generalization 751 00:44:39,660 --> 00:44:42,390 from a 1D transform to a 2D transform, 752 00:44:42,390 --> 00:44:46,180 and that's kind of fun, but it's also kind of important. 753 00:44:46,180 --> 00:44:47,820 So now for the important part. 754 00:44:47,820 --> 00:44:57,570 So this is a picture, a diffraction photomicrograph 755 00:44:57,570 --> 00:45:02,880 done by Rosalind Franklin back about 50 years ago. 756 00:45:02,880 --> 00:45:04,192 She had taken DNA. 757 00:45:04,192 --> 00:45:04,900 I can't remember. 758 00:45:04,900 --> 00:45:07,410 I think it was from Drosophila, but I don't really remember. 759 00:45:07,410 --> 00:45:09,660 She had purified DNA. 760 00:45:09,660 --> 00:45:14,700 And at that time, lasers were not this size. 761 00:45:14,700 --> 00:45:15,900 Actually, she used x-rays. 762 00:45:15,900 --> 00:45:18,170 She used a source of coherent light from an x-ray, 763 00:45:18,170 --> 00:45:22,140 so it's not even a laser, it was x-rays. 764 00:45:22,140 --> 00:45:27,960 She fired coherent x-rays at DNA-- 765 00:45:27,960 --> 00:45:30,990 so she had just a big glob of DNA-- 766 00:45:30,990 --> 00:45:34,330 and took this picture. 767 00:45:34,330 --> 00:45:37,870 And it was pretty and pretty confusing. 768 00:45:37,870 --> 00:45:40,270 By the way, just so you know, this is an artifact. 769 00:45:40,270 --> 00:45:45,100 This is a hole in the photographic film. 770 00:45:45,100 --> 00:45:49,660 So the idea was coherent source of x-rays, sample of DNA, 771 00:45:49,660 --> 00:45:51,720 into a photographic emotion. 772 00:45:51,720 --> 00:45:53,334 Develop it, you get this. 773 00:45:53,334 --> 00:45:55,750 So it's very similar to the diffraction grating experiment 774 00:45:55,750 --> 00:45:59,050 that I did except that the target was not a diffraction 775 00:45:59,050 --> 00:46:04,030 grating, it was not a CD, it was not a DVD, it was DNA 776 00:46:04,030 --> 00:46:06,220 That cute thing is that she showed 777 00:46:06,220 --> 00:46:09,460 this to Watson and Crick, who were 778 00:46:09,460 --> 00:46:13,150 geniuses at so many different levels it's absurd, 779 00:46:13,150 --> 00:46:17,440 and especially Watson knew all about Fourier transforms. 780 00:46:17,440 --> 00:46:19,600 And so he saw that and immediately 781 00:46:19,600 --> 00:46:22,240 was able to interpret it as telling him 782 00:46:22,240 --> 00:46:24,950 something about the three-dimensional structure 783 00:46:24,950 --> 00:46:26,920 of DNA. 784 00:46:26,920 --> 00:46:32,200 In particular, if you think about our modern view of DNA, 785 00:46:32,200 --> 00:46:34,540 you can make an association between the structure 786 00:46:34,540 --> 00:46:39,150 of the DNA and this picture. 787 00:46:39,150 --> 00:46:41,950 So there's a high-frequency band. 788 00:46:41,950 --> 00:46:44,710 So if you imagine that what you're seeing 789 00:46:44,710 --> 00:46:49,390 is three bright points from a diffraction pattern 790 00:46:49,390 --> 00:46:52,660 like I showed previously, this and this 791 00:46:52,660 --> 00:46:54,880 represents a high frequency. 792 00:46:54,880 --> 00:46:57,100 This represents DC. 793 00:46:57,100 --> 00:47:00,010 So the distance between DC and the high frequency 794 00:47:00,010 --> 00:47:03,970 is telling you something about the inverse distance 795 00:47:03,970 --> 00:47:06,710 between base pairs. 796 00:47:06,710 --> 00:47:16,530 The highest frequency in the x-ray picture 797 00:47:16,530 --> 00:47:18,740 is inversely related to the distance between the base 798 00:47:18,740 --> 00:47:20,960 pairs. 799 00:47:20,960 --> 00:47:26,675 The fact that there is these closer-spaced frequencies, 800 00:47:26,675 --> 00:47:28,670 those are actually lower frequencies 801 00:47:28,670 --> 00:47:31,160 because they're closer to 0-- 802 00:47:31,160 --> 00:47:34,550 we're in the far field, so we're in the frequency domain. 803 00:47:34,550 --> 00:47:37,550 Big frequency, small frequency. 804 00:47:37,550 --> 00:47:41,450 The fact that we're seeing these smaller frequency things here 805 00:47:41,450 --> 00:47:45,000 and there are periodic, that means there some structure 806 00:47:45,000 --> 00:47:50,610 at bigger spacing, which Watson and Crick interpreted 807 00:47:50,610 --> 00:47:55,450 as the pitch of the double helix. 808 00:47:55,450 --> 00:47:58,220 So the helix is twisting. 809 00:47:58,220 --> 00:48:00,230 This is the modern picture, and it 810 00:48:00,230 --> 00:48:03,770 fits very nicely with the picture 811 00:48:03,770 --> 00:48:09,140 that Rosalind Franklin made. 812 00:48:09,140 --> 00:48:15,700 And the angles, the fact that there is an axis here, 813 00:48:15,700 --> 00:48:17,450 the reason for going through all this junk 814 00:48:17,450 --> 00:48:19,130 was to motivate the idea that there's 815 00:48:19,130 --> 00:48:22,870 an angle transformation in a two-dimensional transform too. 816 00:48:22,870 --> 00:48:28,040 In fact, angles in space get rotated 90 degrees 817 00:48:28,040 --> 00:48:30,780 when they're in frequency. 818 00:48:30,780 --> 00:48:33,740 So the angle between these solid lines 819 00:48:33,740 --> 00:48:35,630 tells you something about the angle 820 00:48:35,630 --> 00:48:39,750 that the helix makes as it's wrapping around. 821 00:48:39,750 --> 00:48:44,150 So you can easily make 003 model of this experiment, 822 00:48:44,150 --> 00:48:45,560 and that's what this is. 823 00:48:45,560 --> 00:48:49,400 Here all I've taken is a trivial model. 824 00:48:49,400 --> 00:48:53,990 I just made a wire frame that had 825 00:48:53,990 --> 00:48:59,540 base pairs arranged as a rotating spiral, just 826 00:48:59,540 --> 00:49:02,130 like a spiral staircase. 827 00:49:02,130 --> 00:49:04,380 Then that was a three-dimensional picture. 828 00:49:04,380 --> 00:49:08,180 Then I just flattened it, and then I 829 00:49:08,180 --> 00:49:11,180 took the two-dimensional Fourier transform and got that. 830 00:49:14,810 --> 00:49:15,670 Got it? 831 00:49:15,670 --> 00:49:20,010 So the idea was I made a trivial model for what DNA should look 832 00:49:20,010 --> 00:49:22,110 like according to a modern conception, 833 00:49:22,110 --> 00:49:23,910 took the two-dimensional Fourier transform, 834 00:49:23,910 --> 00:49:26,190 and what you can see there is all of the features 835 00:49:26,190 --> 00:49:30,750 that you see in Rosalind Franklin's picture. 836 00:49:30,750 --> 00:49:35,250 You can see the big band that corresponds to the base pairs, 837 00:49:35,250 --> 00:49:38,100 you can see the smaller bands that correspond to the twist, 838 00:49:38,100 --> 00:49:39,990 and you can see the angle that corresponds 839 00:49:39,990 --> 00:49:42,780 to the angle of the double helix. 840 00:49:42,780 --> 00:49:45,030 So the point then is just that there's 841 00:49:45,030 --> 00:49:47,040 a lot of physical phenomenon. 842 00:49:47,040 --> 00:49:48,690 Besides the signal processing things 843 00:49:48,690 --> 00:49:50,370 which we're very interested in, there's actually 844 00:49:50,370 --> 00:49:51,745 a lot of physical phenomenon that 845 00:49:51,745 --> 00:49:55,082 also have intrinsic meaning within a Fourier domain, 846 00:49:55,082 --> 00:49:57,040 and optics is a very important example of that. 847 00:50:00,180 --> 00:50:00,840 See you later. 848 00:50:00,840 --> 00:50:02,720 Have a good day.