WEBVTT 00:00:00.090 --> 00:00:02.430 The following content is provided under a Creative 00:00:02.430 --> 00:00:03.820 Commons license. 00:00:03.820 --> 00:00:06.030 Your support will help MIT OpenCourseWare 00:00:06.030 --> 00:00:10.120 continue to offer high quality educational resources for free. 00:00:10.120 --> 00:00:12.660 To make a donation or to view additional materials 00:00:12.660 --> 00:00:16.620 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:16.620 --> 00:00:17.640 at ocw.mit.edu. 00:00:21.900 --> 00:00:24.600 SE BAEK OH: So today's class, we are going to talk about-- 00:00:24.600 --> 00:00:25.670 called coherence. 00:00:25.670 --> 00:00:31.310 And basically Fourier system with incoherent imaging. 00:00:31.310 --> 00:00:34.490 So particularly, this topic is very important 00:00:34.490 --> 00:00:37.520 if you want to take our optics course because We're dealing 00:00:37.520 --> 00:00:39.125 with coherence and incoherence. 00:00:43.070 --> 00:00:45.590 So far, we have dealt with the Fourier system 00:00:45.590 --> 00:00:48.350 with a coherent illumination. 00:00:48.350 --> 00:00:52.340 Basically, laser light from a point source-- 00:00:52.340 --> 00:00:53.180 I mean, pinhole. 00:00:53.180 --> 00:00:59.360 So the coherent plane waves illuminating the Fourier 00:00:59.360 --> 00:01:04.480 system, and the frequency content appears at the plane. 00:01:04.480 --> 00:01:06.830 And you can put the [INAUDIBLE] mask 00:01:06.830 --> 00:01:10.910 to play with the spatial filtering, 00:01:10.910 --> 00:01:14.880 and you're going to get some image at the alpha plane. 00:01:14.880 --> 00:01:19.730 But in everyday life, if you use your digital camera, 00:01:19.730 --> 00:01:23.390 you're not using the really laser illumination. 00:01:23.390 --> 00:01:28.190 You mostly use this ambient light, or if it's too dark, 00:01:28.190 --> 00:01:32.460 then you're going to use the flashlight. 00:01:32.460 --> 00:01:36.720 So today, we are going to talk about more realistic case. 00:01:36.720 --> 00:01:38.600 We use the incoherent light. 00:01:38.600 --> 00:01:42.290 By the way, the difference between coherent and incoherent 00:01:42.290 --> 00:01:45.530 is basically, you can make the interference 00:01:45.530 --> 00:01:47.060 with the coherent light. 00:01:47.060 --> 00:01:49.130 So if you're not make interference-- 00:01:49.130 --> 00:01:52.190 you're using laser, but in most case, 00:01:52.190 --> 00:01:53.690 like this white light source, you're 00:01:53.690 --> 00:01:55.910 not going to see interference. 00:01:55.910 --> 00:02:00.590 So we're going to first define what coherence is. 00:02:00.590 --> 00:02:02.210 And actually, there are two kinds 00:02:02.210 --> 00:02:06.230 of coherence, temporal coherence and spatial coherence. 00:02:06.230 --> 00:02:08.570 And the rest of the topic is pretty straightforward. 00:02:08.570 --> 00:02:14.750 We will define incoherent point spread function and its Fourier 00:02:14.750 --> 00:02:17.300 transform which is optical transfer function 00:02:17.300 --> 00:02:19.370 and modulation transfer function, which 00:02:19.370 --> 00:02:22.850 is modulus of OTF. 00:02:22.850 --> 00:02:25.760 And we're going to see what MTF means. 00:02:25.760 --> 00:02:28.250 And we want to compare the difference 00:02:28.250 --> 00:02:32.861 between spatially coherent and incoherent images. 00:02:38.660 --> 00:02:40.400 So as I just described, the coherence 00:02:40.400 --> 00:02:45.090 is basically ability to make interference. 00:02:45.090 --> 00:02:47.750 So if you have incoherent light, you cannot make interference. 00:02:47.750 --> 00:02:49.850 And if you have coherent light, then you 00:02:49.850 --> 00:02:51.530 can make the interference. 00:02:51.530 --> 00:02:55.663 And I'm going to describe it later, but mathematically-- 00:02:55.663 --> 00:02:57.080 actually, the coherence is nothing 00:02:57.080 --> 00:03:03.050 but the correlation of wave at two time space points, which 00:03:03.050 --> 00:03:08.090 means we have some light, and you just sample two points. 00:03:08.090 --> 00:03:11.750 It could be different time points, but you combine them. 00:03:11.750 --> 00:03:14.450 And if you see interference, then you can say it's coherent. 00:03:14.450 --> 00:03:16.270 But if you don't see interference, 00:03:16.270 --> 00:03:19.160 then it is incoherent. 00:03:19.160 --> 00:03:21.930 So first, let me describe what's going 00:03:21.930 --> 00:03:25.100 to happen if I use the incoherent light in Michelson 00:03:25.100 --> 00:03:25.730 interferometer. 00:03:25.730 --> 00:03:28.850 So here, we have the Michelson interferometer. 00:03:28.850 --> 00:03:31.740 So from the point source, the light-- 00:03:31.740 --> 00:03:38.240 oops-- is coming in this direction, this direction. 00:03:38.240 --> 00:03:41.450 And at the beam splitter, half of the light 00:03:41.450 --> 00:03:44.090 just transmit the beam splitter, but the other half 00:03:44.090 --> 00:03:45.680 is reflected. 00:03:45.680 --> 00:03:48.470 So this path gets reflected again at the mirror, 00:03:48.470 --> 00:03:50.090 and passed through the beam splitter, 00:03:50.090 --> 00:03:52.010 and arrives at the detector. 00:03:52.010 --> 00:03:55.815 And the other path is reflected again, and reflected 00:03:55.815 --> 00:03:56.690 at the beam splitter. 00:03:56.690 --> 00:03:58.850 And again you get-- 00:03:58.850 --> 00:04:01.530 I mean, it arrives at the detector. 00:04:04.490 --> 00:04:10.055 So let's just suppose that we have the laser source here. 00:04:10.055 --> 00:04:11.930 Then what you're going to see at the detector 00:04:11.930 --> 00:04:15.940 is a nice sinusoidal pattern if I move one of the mirror. 00:04:15.940 --> 00:04:23.990 Because I can tune the optical path length that light travels. 00:04:23.990 --> 00:04:27.080 And if the distance match, then I 00:04:27.080 --> 00:04:28.550 get the constructive interference. 00:04:28.550 --> 00:04:33.590 And if the phase delay pi, then I get disruptive interference. 00:04:33.590 --> 00:04:35.690 So as I move the one with the mirror, 00:04:35.690 --> 00:04:39.015 I get nice sinusoidal pattern with the full contrast. 00:04:43.410 --> 00:04:48.400 And next, let's suppose instead of the laser, 00:04:48.400 --> 00:04:51.350 I have incoherent source, which means-- 00:04:51.350 --> 00:04:54.880 so the horizontal axis is time, and the vertical axis 00:04:54.880 --> 00:04:56.065 is amplitude. 00:04:56.065 --> 00:05:00.100 But the phase or amplitude is varying randomly. 00:05:00.100 --> 00:05:10.510 So over time, the amplitudes are varying randomly. 00:05:10.510 --> 00:05:15.730 And even though it's random, this wave train 00:05:15.730 --> 00:05:18.730 will travel through the interferometer. 00:05:18.730 --> 00:05:28.570 So in this path, the optical path length 00:05:28.570 --> 00:05:30.380 is actually constant, which is-- 00:05:30.380 --> 00:05:33.680 I mean, these paths, they're common in two path. 00:05:33.680 --> 00:05:37.510 So constants plus 2d1 over c, which is time-- 00:05:37.510 --> 00:05:39.110 speed of light. 00:05:39.110 --> 00:05:43.270 And in the second path, I have the same-- 00:05:43.270 --> 00:05:44.780 this wave train also travels. 00:05:44.780 --> 00:05:49.530 But due to the difference between d1 and d2, 00:05:49.530 --> 00:05:51.550 at the detector, essentially, I'm 00:05:51.550 --> 00:05:53.480 going to sample a different time point. 00:05:53.480 --> 00:05:57.460 So even though it's random, I basically 00:05:57.460 --> 00:06:01.210 choose two time points here and here, 00:06:01.210 --> 00:06:04.600 and combine them at the detector. 00:06:04.600 --> 00:06:10.720 So if the d1 and d2 are same, then basically, 00:06:10.720 --> 00:06:13.340 identical two waves are coming together, 00:06:13.340 --> 00:06:16.150 so I always get constructive interference. 00:06:16.150 --> 00:06:19.300 But what if d1 and d2 are not same? 00:06:19.300 --> 00:06:23.860 And then basically, they get different phase shift. 00:06:23.860 --> 00:06:28.720 And since it's random, they kind of average out. 00:06:28.720 --> 00:06:30.830 So d1 and d2 are getting bigger. 00:06:30.830 --> 00:06:34.930 Then basically, the degree of correlation 00:06:34.930 --> 00:06:39.700 is getting decreased, which means at the detector, 00:06:39.700 --> 00:06:43.780 the contrast will also get decreased. 00:06:43.780 --> 00:06:47.830 So if I plot the intensity at the detector with respect 00:06:47.830 --> 00:06:52.110 to d2 minus d1, which is the distance difference of two 00:06:52.110 --> 00:06:57.010 arms, then the intensity is going to be like this. 00:06:57.010 --> 00:07:02.800 So remember that if I have the laser at point source, then 00:07:02.800 --> 00:07:07.180 I will get the sinusoidal pattern with the full contrast. 00:07:07.180 --> 00:07:11.860 But since now I have the randomly varying source, 00:07:11.860 --> 00:07:14.060 this envelop is getting attenuated like this. 00:07:14.060 --> 00:07:18.490 So if I have the rapid decay curve, which means 00:07:18.490 --> 00:07:20.570 I have more incoherent light. 00:07:20.570 --> 00:07:23.390 And if I have the slowly decaying envelop, 00:07:23.390 --> 00:07:24.550 then it's more coherent. 00:07:29.490 --> 00:07:35.290 And there is another kind of coherence 00:07:35.290 --> 00:07:36.760 which is spatial coherence. 00:07:36.760 --> 00:07:38.510 So basically, the concept is the same, 00:07:38.510 --> 00:07:43.240 but we only define the correlation in space domain. 00:07:43.240 --> 00:07:46.060 So now I have the Young interferometer, which 00:07:46.060 --> 00:07:48.390 I have two pinholes, x1 and x2. 00:07:48.390 --> 00:07:51.850 And I put a detector at the center of the two pinholes. 00:07:51.850 --> 00:07:56.350 So from this pinhole, we see-- 00:07:56.350 --> 00:07:59.830 I mean, the spherical wave is going to be emanating, 00:07:59.830 --> 00:08:05.050 but since it is at the center, these paths are exactly same. 00:08:05.050 --> 00:08:08.770 And first, let's suppose we have laser illumination. 00:08:08.770 --> 00:08:11.650 So I have the plane wave coming in. 00:08:11.650 --> 00:08:15.010 Then this spherical wave will make the Young's interference. 00:08:15.010 --> 00:08:18.640 So I'm going to have the sinusoidal pattern 00:08:18.640 --> 00:08:21.250 with the full contrast at this plane, 00:08:21.250 --> 00:08:23.530 but I'm just seeing at one point. 00:08:23.530 --> 00:08:32.059 And again, if I have the randomly varying wave here-- 00:08:32.059 --> 00:08:36.280 so in this case, this horizontal axis amplitude 00:08:36.280 --> 00:08:39.820 and vertical axis is space instead of time. 00:08:39.820 --> 00:08:46.000 So this amplitude and phase are randomly varying over space. 00:08:46.000 --> 00:08:49.450 Since I have two pinholes, so I essentially 00:08:49.450 --> 00:08:54.490 sample at two points here and there, and combine them. 00:08:58.520 --> 00:09:02.140 So again, if these two pinholes are very close, 00:09:02.140 --> 00:09:04.990 then these sampling points are also very close. 00:09:04.990 --> 00:09:10.390 So the amplitude and phase are more likely similar. 00:09:10.390 --> 00:09:12.550 [INAUDIBLE] 00:09:12.550 --> 00:09:17.620 But if I move these two pinholes in the upward and downward 00:09:17.620 --> 00:09:20.760 like this, then the spacing between two pinholes 00:09:20.760 --> 00:09:22.660 are getting bigger, and basically, I 00:09:22.660 --> 00:09:25.840 lose the correlation, which means 00:09:25.840 --> 00:09:29.780 the spatial coherence is getting attenuated-- getting smaller. 00:09:29.780 --> 00:09:33.490 So again, if I plot the intensity 00:09:33.490 --> 00:09:37.900 at the detector with respect to x2 minus x1, which 00:09:37.900 --> 00:09:41.140 is the spacing between two pinholes, 00:09:41.140 --> 00:09:46.258 then intensity varying like this. 00:09:46.258 --> 00:09:47.800 Again, remember, if I have the laser, 00:09:47.800 --> 00:09:50.340 then it doesn't really matter. 00:09:50.340 --> 00:09:52.330 I mean, I always get a full contrast. 00:09:52.330 --> 00:09:54.590 Even though those frequencies-- 00:09:54.590 --> 00:09:58.130 I mean, [INAUDIBLE] plane-- the frequency of the interference 00:09:58.130 --> 00:09:58.690 is changing. 00:09:58.690 --> 00:10:00.760 But I always get the full contrast, 00:10:00.760 --> 00:10:04.980 but contrast getting smaller if you have the incoherent source. 00:10:14.020 --> 00:10:20.800 So let me describe mathematically 00:10:20.800 --> 00:10:22.780 why this happens. 00:10:22.780 --> 00:10:25.600 It doesn't really matter whether I 00:10:25.600 --> 00:10:28.630 use the Michelson interferometer or Young's interferometer. 00:10:28.630 --> 00:10:38.750 Basically, interferometer, we have two waves. 00:10:38.750 --> 00:10:43.610 So a1, e to the pi 1, and a2, d to the pi 2. 00:10:43.610 --> 00:10:47.450 Basically, we have two waves I could sample in space or time 00:10:47.450 --> 00:10:47.950 domain. 00:10:47.950 --> 00:10:50.000 But anyway, I'm going to have two waves 00:10:50.000 --> 00:10:52.880 and just add them together, and take the magnitude 00:10:52.880 --> 00:10:55.250 square, which is intensity. 00:10:55.250 --> 00:10:59.510 So Michelson interferometer, this is-- 00:10:59.510 --> 00:11:06.090 I mean, these two waves came from the different-- the path 00:11:06.090 --> 00:11:07.940 length, d1 and d2. 00:11:07.940 --> 00:11:11.270 And Young's interferometer, basically, they 00:11:11.270 --> 00:11:15.050 came from the two different pinholes, x1 and x2. 00:11:15.050 --> 00:11:17.750 But anyway, I can always write the interference 00:11:17.750 --> 00:11:19.940 of two waves like this. 00:11:19.940 --> 00:11:24.990 So wave one and wave two, it's negative square 00:11:24.990 --> 00:11:30.374 is going to be i0 and 1 plus m cosine, the phase difference. 00:11:30.374 --> 00:11:32.834 AUDIENCE: [INAUDIBLE] 00:11:32.834 --> 00:11:35.420 SE BAEK OH: We are assuming monochromatic, yes. 00:11:35.420 --> 00:11:37.050 I mean, you can write-- 00:11:37.050 --> 00:11:41.210 [INAUDIBLE] we use assuming monochromatic 00:11:41.210 --> 00:11:43.040 because we're using phase notation. 00:11:43.040 --> 00:11:47.510 So we write this and we write phaser, 00:11:47.510 --> 00:11:51.350 but actually, there is e to the i or negative i. 00:11:51.350 --> 00:11:55.490 So yeah, we assume it's monochromatic. 00:11:55.490 --> 00:11:58.100 But even though it's not chromatic, 00:11:58.100 --> 00:12:01.548 you can always write just addition of two waves. 00:12:01.548 --> 00:12:02.590 It really doesn't matter. 00:12:08.770 --> 00:12:12.460 So interference is basically the dc term which is i0. 00:12:12.460 --> 00:12:16.210 So i0 is actually a1 squared plus a2 squared, 00:12:16.210 --> 00:12:20.800 which is intensity of wave 1 and intensity of wave 2. 00:12:20.800 --> 00:12:25.090 So dc term is just addition of the intensity, 00:12:25.090 --> 00:12:28.400 and we have some cosine modulation term. 00:12:33.430 --> 00:12:36.890 So this is basically interference. 00:12:36.890 --> 00:12:43.000 So if we have coherent light, then we have some dc term, 00:12:43.000 --> 00:12:47.020 and have a sinusoidal modulation. 00:12:47.020 --> 00:12:52.660 And as the phase varies, we're going 00:12:52.660 --> 00:12:55.640 to see the sinusoidal pattern is changing. 00:12:55.640 --> 00:12:59.740 But remember that this optical frequency is very fast-- 00:12:59.740 --> 00:13:03.850 like tens of gigahertz [INAUDIBLE].. 00:13:03.850 --> 00:13:05.500 So actually, what we measure-- 00:13:05.500 --> 00:13:07.780 I mean, intensity is time average of this guy. 00:13:07.780 --> 00:13:14.560 So we're going to take the time average of intensity. 00:13:14.560 --> 00:13:17.680 And if you do the math-- 00:13:17.680 --> 00:13:20.580 actually, we have the dc term and this kind 00:13:20.580 --> 00:13:25.710 of modulation term, which is time average of cosine 00:13:25.710 --> 00:13:27.780 with a phase difference. 00:13:27.780 --> 00:13:30.380 But remember that if we have incoherent light, 00:13:30.380 --> 00:13:32.200 this phase difference is random. 00:13:32.200 --> 00:13:36.270 So what is the average of cosine of random phase? 00:13:39.870 --> 00:13:40.799 [INAUDIBLE] 00:13:49.550 --> 00:13:53.280 What is the average of cosine or random phase? 00:13:56.440 --> 00:13:56.940 AUDIENCE: Q. 00:13:56.940 --> 00:13:59.370 SE BAEK OH: Yeah, Q. Cosine is oscillating 00:13:59.370 --> 00:14:02.160 between plus 1 and to minus 1. 00:14:02.160 --> 00:14:06.720 So if you have random phase and average, then you get 0. 00:14:06.720 --> 00:14:11.100 So if you have incoherent light, then basically intensity 00:14:11.100 --> 00:14:13.125 is just addition of-- 00:14:13.125 --> 00:14:15.210 the intensity of 2 waves, and we don't 00:14:15.210 --> 00:14:17.070 have any interference term. 00:14:17.070 --> 00:14:20.520 So that's the difference between coherent and incoherent light. 00:14:20.520 --> 00:14:24.030 So coherent light, you always have a modulation term. 00:14:24.030 --> 00:14:26.950 I mean, it could be 0 depending on phase difference. 00:14:26.950 --> 00:14:32.130 But in [INAUDIBLE],, you always have just addition of dc term-- 00:14:32.130 --> 00:14:33.744 I mean addition of intensity. 00:14:41.590 --> 00:14:44.530 So if it's perfectly incoherent, then we 00:14:44.530 --> 00:14:46.570 have just addition of intensity. 00:14:46.570 --> 00:14:49.875 And actually, coherent and incoherent is very-- 00:14:49.875 --> 00:14:53.260 actually two opposite extreme cases. 00:14:53.260 --> 00:14:57.197 So it's just like monochromatic. 00:14:57.197 --> 00:14:58.780 In real case, we are not going to have 00:14:58.780 --> 00:15:02.080 really monochromatic source. 00:15:02.080 --> 00:15:05.070 Our source is always finite bandwidth. 00:15:05.070 --> 00:15:06.570 It's not going to be delta function. 00:15:06.570 --> 00:15:07.760 So it's same thing. 00:15:07.760 --> 00:15:11.670 We are not going to have the perfectly coherent or perfectly 00:15:11.670 --> 00:15:12.250 incoherent. 00:15:12.250 --> 00:15:15.190 Everything is partially coherent. 00:15:15.190 --> 00:15:19.990 So if you want to deal with it, then you have to apply 00:15:19.990 --> 00:15:25.720 basically statistics to all these Fresnel and [INAUDIBLE]---- 00:15:25.720 --> 00:15:27.770 I mean, this kind of [INAUDIBLE] optics. 00:15:27.770 --> 00:15:29.950 So it's called statistical optics, 00:15:29.950 --> 00:15:32.980 and it's covered in 2.707. 00:15:32.980 --> 00:15:36.230 So if you want to learn more about it, 00:15:36.230 --> 00:15:39.940 then you can take 2.7 or 1.7. 00:15:39.940 --> 00:15:42.760 But nevertheless, the coherent and incoherent case 00:15:42.760 --> 00:15:46.990 are enough to describe many, many phenomena. 00:15:46.990 --> 00:15:53.506 So we are focused on coherent and incoherent state detection. 00:15:57.350 --> 00:15:58.350 Going a little bit fast. 00:16:02.110 --> 00:16:07.728 So-- oops. 00:16:07.728 --> 00:16:08.228 Sorry. 00:16:15.400 --> 00:16:17.620 So since we have two kinds of coherence-- so 00:16:17.620 --> 00:16:20.310 temporal coherence and spatial coherence. 00:16:20.310 --> 00:16:24.230 So actually, we have four different combinations. 00:16:24.230 --> 00:16:26.320 So temporarily and spatially coherent, 00:16:26.320 --> 00:16:29.030 and temporarily/spatially incoherent. 00:16:29.030 --> 00:16:33.740 And temporal incoherent, spatially coherent, temporarily 00:16:33.740 --> 00:16:35.940 coherent, spatially incoherent. 00:16:35.940 --> 00:16:39.040 So the rule of thumb-- 00:16:39.040 --> 00:16:42.460 so temporal coherence is related to the bandwidth 00:16:42.460 --> 00:16:44.590 of your source-- 00:16:44.590 --> 00:16:47.300 basically, the color of your source. 00:16:47.300 --> 00:16:50.930 So if you have one color in monochromatic, then it's-- 00:16:50.930 --> 00:16:51.890 question? 00:16:51.890 --> 00:16:55.240 AUDIENCE: Yeah, where do diode lasers go in that chart? 00:16:57.750 --> 00:17:00.392 SE BAEK OH: Diode laser is more like temporally 00:17:00.392 --> 00:17:01.350 and spatially coherent. 00:17:05.780 --> 00:17:07.520 So yeah, let me just continue. 00:17:07.520 --> 00:17:10.500 So temporal coherence is related to color. 00:17:10.500 --> 00:17:13.760 So if you have laser, like single-color monochromatic, 00:17:13.760 --> 00:17:15.869 then it's more like coherent. 00:17:15.869 --> 00:17:18.710 But if you have white light, like this fluorescent lamp 00:17:18.710 --> 00:17:21.800 or halogen lamp, then it's broadband, 00:17:21.800 --> 00:17:24.349 so it's temporally incoherent. 00:17:24.349 --> 00:17:28.520 And spatial coherence is related to actually the size 00:17:28.520 --> 00:17:30.820 of the source and the distance. 00:17:30.820 --> 00:17:32.945 So everything coming from the point source 00:17:32.945 --> 00:17:36.890 is spatially coherent because they're basically 00:17:36.890 --> 00:17:38.720 coming from the same point. 00:17:38.720 --> 00:17:41.550 But if your source is extended like-- 00:17:41.550 --> 00:17:43.820 or you have the filament in a bulb, 00:17:43.820 --> 00:17:46.040 or this kind of fluorescent light, 00:17:46.040 --> 00:17:49.190 then it's spatially incoherent. 00:17:49.190 --> 00:17:53.240 So the monochromatic laser source, 00:17:53.240 --> 00:17:59.630 laser is very monochromatic, and the output is very small. 00:17:59.630 --> 00:18:02.870 So laser is typically very temporarily and spatially 00:18:02.870 --> 00:18:05.930 incoherent, and that's why people use it 00:18:05.930 --> 00:18:08.360 in interferometry. 00:18:08.360 --> 00:18:11.060 And white light source-- 00:18:11.060 --> 00:18:13.620 and if you are nearby the white light source, 00:18:13.620 --> 00:18:19.610 then it's more like temporally and spatially incoherent. 00:18:19.610 --> 00:18:23.110 And actually, these two are a little bit interesting. 00:18:23.110 --> 00:18:26.130 So temporally incoherent or spatially coherent. 00:18:26.130 --> 00:18:30.020 So if you put a small pinhole in front of the white light, 00:18:30.020 --> 00:18:33.530 then it's temporally incoherent because it's broadband, 00:18:33.530 --> 00:18:36.500 but it's coming from a pinhole, so it's spatially coherent. 00:18:36.500 --> 00:18:42.080 At the same reason, the sunlight is very spatially coherent, 00:18:42.080 --> 00:18:43.730 but temporally incoherent. 00:18:43.730 --> 00:18:47.150 But the sun is very huge, but also the distance is very huge. 00:18:47.150 --> 00:18:51.240 So if you see the sun, then it looks-- 00:18:51.240 --> 00:18:53.000 sunlight is coming from point source, 00:18:53.000 --> 00:18:57.854 so that's why it's spatially coherent. 00:18:57.854 --> 00:19:00.500 And yeah, the light from-- 00:19:00.500 --> 00:19:03.948 yeah, press the button, please. 00:19:03.948 --> 00:19:04.860 It doesn't work? 00:19:04.860 --> 00:19:06.010 AUDIENCE: Yeah. 00:19:06.010 --> 00:19:12.110 But anyway, so one of the things that [INAUDIBLE] 00:19:12.110 --> 00:19:15.290 to measure coherence, but what exactly 00:19:15.290 --> 00:19:17.130 is coherent [INAUDIBLE]? 00:19:21.760 --> 00:19:25.300 SE BAEK OH: Yeah, that is a good question. 00:19:25.300 --> 00:19:29.830 So actually, it's a little bit lengthy to explain, 00:19:29.830 --> 00:19:32.380 but what essentially the light-- 00:19:32.380 --> 00:19:35.620 I mean, this kind of light, it excites some atoms. 00:19:35.620 --> 00:19:39.720 And when they are going down, you get the energy. 00:19:39.720 --> 00:19:43.210 Basically, light energy by e to the-- 00:19:43.210 --> 00:19:45.280 I mean energy is H mu. 00:19:49.330 --> 00:19:50.590 But it's not-- 00:19:50.590 --> 00:19:53.470 I mean, this kind of spontaneous emission is not correlated. 00:19:53.470 --> 00:19:57.410 You have many, many atoms, but they are not synchronized. 00:19:57.410 --> 00:20:00.020 They are emitting randomly. 00:20:00.020 --> 00:20:03.180 So basically, you have some finite-- 00:20:03.180 --> 00:20:06.430 there are wave train. 00:20:06.430 --> 00:20:09.690 So in that wave train, you get basically coherent. 00:20:09.690 --> 00:20:12.620 But you have many, many of them with the random phase. 00:20:12.620 --> 00:20:14.490 That's why you add them together, 00:20:14.490 --> 00:20:15.870 then you get incoherent. 00:20:15.870 --> 00:20:18.370 So coherent length is essentially 00:20:18.370 --> 00:20:21.910 what is the length of a finite wave train 00:20:21.910 --> 00:20:25.120 that you can define deterministic phase. 00:20:25.120 --> 00:20:28.870 So d1-- so in just imagine the Michelson interferometer, 00:20:28.870 --> 00:20:33.160 you have the optical path first, d1 minus d2. 00:20:33.160 --> 00:20:37.760 But that difference is within the distance, 00:20:37.760 --> 00:20:40.270 then you get interference. 00:20:40.270 --> 00:20:42.540 That's why you call the coherence length and coherence 00:20:42.540 --> 00:20:44.290 time. 00:20:44.290 --> 00:20:49.540 And yeah, essentially, that's it. 00:20:49.540 --> 00:20:52.680 So remember that if we had-- 00:20:59.000 --> 00:20:59.923 AUDIENCE: Excuse me. 00:20:59.923 --> 00:21:00.690 SE BAEK OH: Yep? 00:21:00.690 --> 00:21:02.580 AUDIENCE: Here I also will have one question. 00:21:02.580 --> 00:21:05.510 So I want to check whether this understanding is correct. 00:21:05.510 --> 00:21:09.470 So let's say if the light comes at a perfect parallel 00:21:09.470 --> 00:21:12.350 direction, even though they have different wavelengths, 00:21:12.350 --> 00:21:15.210 it's considered especially coherent. 00:21:15.210 --> 00:21:18.230 And even if light go into different directions, 00:21:18.230 --> 00:21:19.490 but [INAUDIBLE] perfect-- 00:21:19.490 --> 00:21:23.370 the single wavelength, they're considered time coherent. 00:21:23.370 --> 00:21:25.260 Is this understanding correct? 00:21:25.260 --> 00:21:27.590 SE BAEK OH: So actually, if you have plain wave, 00:21:27.590 --> 00:21:31.950 then you already assume your wave is spatially coherent. 00:21:31.950 --> 00:21:36.410 Because plane wave, it doesn't really matter where you sample. 00:21:36.410 --> 00:21:38.090 They have always same phase. 00:21:38.090 --> 00:21:41.880 So your assumption is spatially coherent 00:21:41.880 --> 00:21:44.610 if you assume spatially coherent light. 00:21:47.260 --> 00:21:51.602 AUDIENCE: You mean even if they have different wavelengths? 00:21:51.602 --> 00:21:52.310 SE BAEK OH: Yeah. 00:21:52.310 --> 00:21:53.270 Even though you have different wavelengths, 00:21:53.270 --> 00:21:54.500 that's temporally incoherent. 00:21:54.500 --> 00:21:56.840 But in space domain, you have spatially coherent. 00:21:59.880 --> 00:22:02.610 Another way to think of spatially incoherent 00:22:02.610 --> 00:22:06.150 is you have many, many plane waves propagating 00:22:06.150 --> 00:22:10.140 all the different directions, but they are randomly-- 00:22:10.140 --> 00:22:12.060 they have random initial phase. 00:22:12.060 --> 00:22:16.570 So especially in coherent light, you cannot really define 00:22:16.570 --> 00:22:19.620 a unique propagation direction. 00:22:19.620 --> 00:22:21.870 But spatially coherent light, you 00:22:21.870 --> 00:22:24.090 can always define direction. 00:22:24.090 --> 00:22:26.970 Even though spherical wave it has all the directions, 00:22:26.970 --> 00:22:28.740 but you can always say-- 00:22:28.740 --> 00:22:32.520 if you put-- if you define the position, then that position, 00:22:32.520 --> 00:22:34.390 you have a k vector. 00:22:34.390 --> 00:22:36.778 So you can define the direction. 00:22:46.420 --> 00:22:49.780 So let me just finish this thing. 00:22:49.780 --> 00:22:53.740 So if d1 and d2 are within the coherence length or coherence 00:22:53.740 --> 00:22:56.700 time, then you get some interference. 00:22:56.700 --> 00:22:58.872 But after that-- so basically, this 00:22:58.872 --> 00:23:00.330 is essentially the coherence length 00:23:00.330 --> 00:23:03.300 of this particular source. 00:23:03.300 --> 00:23:04.500 And remember that-- 00:23:04.500 --> 00:23:05.970 I mean, this always happens. 00:23:05.970 --> 00:23:08.220 So even though you have a white light, 00:23:08.220 --> 00:23:12.000 you can get interference, but within a very limited range. 00:23:44.250 --> 00:23:45.000 Where did I stop? 00:23:51.580 --> 00:23:52.205 [CLEARS THROAT] 00:23:52.205 --> 00:23:53.048 Excuse me. 00:23:53.048 --> 00:23:54.590 So another interesting thing is there 00:23:54.590 --> 00:23:59.720 is a laser that has a broadband which is [INAUDIBLE] laser. 00:23:59.720 --> 00:24:02.810 So it's spatially coherent, but it's broadband, 00:24:02.810 --> 00:24:06.050 so you have temporally incoherence. 00:24:06.050 --> 00:24:08.600 And the other kind of this combination 00:24:08.600 --> 00:24:13.190 is temporally coherent, but it's spatially incoherent. 00:24:13.190 --> 00:24:19.170 And it's typically referred to as quasi-monochromatic-- 00:24:19.170 --> 00:24:23.390 temporally coherent and spatially incoherent. 00:24:23.390 --> 00:24:25.400 So if you put a rotating diffuser, which 00:24:25.400 --> 00:24:31.400 is when you rotate the ground glass or milky glass in front 00:24:31.400 --> 00:24:36.500 of the laser, then basically, it induces some random phase 00:24:36.500 --> 00:24:39.950 to the laser beam where we can say it destroyed 00:24:39.950 --> 00:24:42.120 the correlation of the related. 00:24:42.120 --> 00:24:45.770 So it's monochromatic, so it temporally coheres. 00:24:45.770 --> 00:24:48.270 But in space domain, it is a random phase, 00:24:48.270 --> 00:24:52.040 so you can have the spatially incoherent light. 00:24:56.430 --> 00:24:59.750 So these are a list of optical instruments using 00:24:59.750 --> 00:25:02.416 spatial coherence on imaging. 00:25:02.416 --> 00:25:05.420 The Michelson stellar interferometer and the radio 00:25:05.420 --> 00:25:09.920 telescope, they are basically measuring spatial coherence 00:25:09.920 --> 00:25:14.260 of the light coming from a distant star, or sun, or-- 00:25:14.260 --> 00:25:16.340 yeah, they're basically for astronomy. 00:25:16.340 --> 00:25:18.845 So they can measure, for example, 00:25:18.845 --> 00:25:22.860 the diameter of a distant star. 00:25:22.860 --> 00:25:24.330 Remember, the spatial coherence is 00:25:24.330 --> 00:25:26.410 related to the source size and the distance. 00:25:26.410 --> 00:25:29.080 So if you properly measure spatial coherence-- 00:25:29.080 --> 00:25:31.290 actually, the envelope of the spatial coherence, 00:25:31.290 --> 00:25:33.450 then you can estimate the diameter of the star. 00:25:37.072 --> 00:25:39.030 I believe Michelson got a Nobel Prize for this. 00:25:42.694 --> 00:25:46.220 And optical coherence tomography is actually 00:25:46.220 --> 00:25:48.310 utilizing temporal incoherence. 00:25:48.310 --> 00:25:51.210 So as I just described, even though you have white light, 00:25:51.210 --> 00:25:54.200 if you properly match the optical path difference, 00:25:54.200 --> 00:25:58.440 then you can get interference within a very small range. 00:25:58.440 --> 00:26:02.840 So let's say you have multiple layers like tissue, 00:26:02.840 --> 00:26:05.120 and you construct the Michelson interferometer 00:26:05.120 --> 00:26:08.370 with the white light, and as you move the other mirror, 00:26:08.370 --> 00:26:10.490 then if the optical path difference matches, 00:26:10.490 --> 00:26:11.870 then you get interference. 00:26:11.870 --> 00:26:14.120 But if you slightly off that point, 00:26:14.120 --> 00:26:16.130 then you don't get the interference. 00:26:16.130 --> 00:26:19.550 So if you plot the interference versus the distance 00:26:19.550 --> 00:26:22.550 of the mirror, then you can see kind of the peak 00:26:22.550 --> 00:26:30.440 whenever you have the right position at each layer. 00:26:30.440 --> 00:26:35.220 So you can kind of do the optical sectioning. 00:26:35.220 --> 00:26:39.210 And in the lithography replication technique, 00:26:39.210 --> 00:26:42.470 people use some spatial coherence 00:26:42.470 --> 00:26:49.640 to basically make some nice pattern. 00:26:49.640 --> 00:26:52.460 Because as we described spatial coherence 00:26:52.460 --> 00:26:57.110 is essentially affecting this resolution 00:26:57.110 --> 00:27:00.340 and the contrast of your image. 00:27:00.340 --> 00:27:14.240 So there are these kind of things 00:27:14.240 --> 00:27:16.720 Even though we deal with the spatially coherent 00:27:16.720 --> 00:27:20.200 with the first system, so what if we change the illumination 00:27:20.200 --> 00:27:24.450 to a spatially incoherent or temporally incoherent? 00:27:24.450 --> 00:27:30.100 And obviously, my image will be different depending 00:27:30.100 --> 00:27:33.055 on the coherent state. 00:27:33.055 --> 00:27:38.880 Actually, that's the bottom line of today's lecture. 00:27:38.880 --> 00:27:42.100 So first, if we have the temporal incoherent-- 00:27:42.100 --> 00:27:44.630 basically, if you have the broadband for many, 00:27:44.630 --> 00:27:45.730 many colors-- 00:27:45.730 --> 00:27:48.880 then the chromatic aberration is typical evidence 00:27:48.880 --> 00:27:53.410 that I have temporal incoherent source. 00:27:53.410 --> 00:27:58.420 Then how to deal with it if you have many, many colors? 00:27:58.420 --> 00:28:00.010 That's pretty straightforward. 00:28:00.010 --> 00:28:03.940 So you can define point spread function, or ATF, or out-- 00:28:03.940 --> 00:28:05.770 or whatever it is. 00:28:05.770 --> 00:28:08.720 You can define that thing for each wavelength. 00:28:08.720 --> 00:28:11.500 So for example, if I can define point spread function 00:28:11.500 --> 00:28:13.750 for red wavelength, I can define point spread function 00:28:13.750 --> 00:28:15.200 for blue wavelengths. 00:28:15.200 --> 00:28:16.000 So it's just-- 00:28:16.000 --> 00:28:19.840 I can do with a straightforward equation. 00:28:19.840 --> 00:28:23.010 And spatial coherence is a little bit-- 00:28:23.010 --> 00:28:25.090 and you need to be careful. 00:28:25.090 --> 00:28:28.720 So the bottom-- actually, this is the take-home message 00:28:28.720 --> 00:28:30.430 of the lecture. 00:28:30.430 --> 00:28:32.680 If the illumination is spatially coherent, 00:28:32.680 --> 00:28:37.950 then input field is conversion of the-- actually, 00:28:37.950 --> 00:28:43.090 output field is a convolution of input field 00:28:43.090 --> 00:28:48.130 with a coherent point spread function, as we all know. 00:28:48.130 --> 00:28:50.710 But if the illumination is spatially incoherent, 00:28:50.710 --> 00:28:54.280 then output intensity, not the output field. 00:28:54.280 --> 00:28:57.160 So output intensity is a convolution 00:28:57.160 --> 00:29:00.950 of the input intensity with the incoherent point spread 00:29:00.950 --> 00:29:01.450 function. 00:29:05.170 --> 00:29:12.450 So that's the-- which means if you have spatial coherent 00:29:12.450 --> 00:29:13.675 illumination, then-- 00:29:13.675 --> 00:29:15.270 fields are linear systems. 00:29:15.270 --> 00:29:17.710 You can do the convolution in field. 00:29:17.710 --> 00:29:20.530 But if the illumination is incoherent, 00:29:20.530 --> 00:29:22.730 then intensity is a linear system, 00:29:22.730 --> 00:29:27.640 so you can deal with the intensity with the convolution. 00:29:27.640 --> 00:29:29.140 So rest of the lecture, we are going 00:29:29.140 --> 00:29:33.200 to deal with the spatial incoherent illumination. 00:29:33.200 --> 00:29:37.060 So how we can define point spread function, and transfer 00:29:37.060 --> 00:29:38.260 function, and et cetera. 00:29:46.760 --> 00:29:51.720 So first, let's review what we have learned so far. 00:29:51.720 --> 00:29:55.040 So we have fourth system with a two length 00:29:55.040 --> 00:29:57.650 whose focal lengths are f1 and f2. 00:29:57.650 --> 00:30:00.890 And typically, we put our input transparency 00:30:00.890 --> 00:30:03.290 grating at the input plane. 00:30:03.290 --> 00:30:07.610 And we illuminate with the spatially coherent illumination 00:30:07.610 --> 00:30:11.560 and also temporally coherent. 00:30:11.560 --> 00:30:15.670 So the input field just right after the input 00:30:15.670 --> 00:30:20.560 transparency is going to be just a product of the illumination 00:30:20.560 --> 00:30:24.010 field and the complex transparency. 00:30:24.010 --> 00:30:25.990 That's the input field. 00:30:25.990 --> 00:30:29.260 And you're going to take the Fourier transform to get 00:30:29.260 --> 00:30:31.030 the field at the wave plane. 00:30:31.030 --> 00:30:35.090 And another Fourier transform, you get the final image. 00:30:35.090 --> 00:30:38.890 And from that, we all know that amplitude [INAUDIBLE] 00:30:38.890 --> 00:30:43.690 function is nothing but the [INAUDIBLE] function 00:30:43.690 --> 00:30:46.660 with the coordinate scaling. 00:30:46.660 --> 00:30:51.710 So ATF, H of u [INAUDIBLE] some constant 00:30:51.710 --> 00:30:57.310 and [INAUDIBLE] mask function, and lambda f1u and lambda f1v. 00:31:00.130 --> 00:31:03.550 And the point spread function in field 00:31:03.550 --> 00:31:09.320 is just Fourier transform of the ATF. 00:31:09.320 --> 00:31:12.800 Or you can say it's field transform [INAUDIBLE] function. 00:31:12.800 --> 00:31:16.000 So this is the coherent point spread function. 00:31:16.000 --> 00:31:19.840 And you can define [INAUDIBLE] function reduce coordinate. 00:31:19.840 --> 00:31:23.530 And the relation between input and output field 00:31:23.530 --> 00:31:25.180 is this convolution. 00:31:25.180 --> 00:31:28.180 So G output-- I mean, output field 00:31:28.180 --> 00:31:31.000 is convolution of input field, and this 00:31:31.000 --> 00:31:33.070 is coherent point spread function 00:31:33.070 --> 00:31:36.010 which is defined in field. 00:31:36.010 --> 00:31:41.210 So this is the stuff that we all know. 00:31:41.210 --> 00:31:47.140 So the next thing is let me put the two pinholes at the input 00:31:47.140 --> 00:31:47.640 plane. 00:31:50.760 --> 00:31:53.646 And illuminate with the spatially coherent light. 00:31:53.646 --> 00:31:56.730 So basically, we have Young interferometer here, 00:31:56.730 --> 00:31:59.640 and I'm going to image that pattern. 00:32:03.310 --> 00:32:06.480 And since it is coherent-- it is illuminated 00:32:06.480 --> 00:32:07.810 with the coherent light. 00:32:07.810 --> 00:32:12.360 So essentially, I have-- at the image plane, 00:32:12.360 --> 00:32:15.610 there's actually two point spread functions 00:32:15.610 --> 00:32:17.140 coming from two pinholes. 00:32:17.140 --> 00:32:19.440 So in the upper image is actually 00:32:19.440 --> 00:32:21.790 coming from the lower pinhole, and the lower image 00:32:21.790 --> 00:32:23.530 actually coming from the top pinhole. 00:32:23.530 --> 00:32:27.730 But I have the addition of two point spread 00:32:27.730 --> 00:32:31.310 functions which is shifted by the position of the two 00:32:31.310 --> 00:32:31.810 pinholes. 00:32:31.810 --> 00:32:36.520 And I may have some initial phase, a1, e to the phi 1, 00:32:36.520 --> 00:32:37.570 and a to-- 00:32:37.570 --> 00:32:38.710 e to the phi 2. 00:32:38.710 --> 00:32:42.230 But the bottom line is in coherent case, 00:32:42.230 --> 00:32:46.250 I just have coherent addition of two point spread 00:32:46.250 --> 00:32:48.640 functions like this. 00:32:48.640 --> 00:32:51.620 And to get the intensity, as you all know, 00:32:51.620 --> 00:32:55.450 we just take the negative square of the sum, 00:32:55.450 --> 00:32:59.300 and if we expand it, then we have [INAUDIBLE] 00:32:59.300 --> 00:33:01.810 which is the intensity of the first point spread 00:33:01.810 --> 00:33:05.170 function and intensity of the second point spread function. 00:33:05.170 --> 00:33:09.330 And we have some additional modulation term, which 00:33:09.330 --> 00:33:11.470 means I have some interference. 00:33:11.470 --> 00:33:14.920 So it's a1, a2, and rear path of this guy. 00:33:14.920 --> 00:33:17.860 But rear path is always cosine. 00:33:17.860 --> 00:33:21.890 So if you change the phase, you have some sinusoidal modulation 00:33:21.890 --> 00:33:22.390 term. 00:33:26.410 --> 00:33:29.470 So this is sometimes called interference term 00:33:29.470 --> 00:33:31.630 or [INAUDIBLE] term. 00:33:31.630 --> 00:33:35.440 It implies that you have interference. 00:33:35.440 --> 00:33:38.350 So with these parameters-- so initial phase 00:33:38.350 --> 00:33:41.920 is just d to the negative pi over 3, 00:33:41.920 --> 00:33:45.790 and the other one is the positive pi over 3. 00:33:45.790 --> 00:33:49.330 Then if I plot this intensity, then I'm 00:33:49.330 --> 00:33:51.760 going to have this blue line. 00:33:55.100 --> 00:34:00.410 So this is the case of spatially coherent illumination. 00:34:00.410 --> 00:34:03.755 And next, let's suppose I illuminate 00:34:03.755 --> 00:34:07.970 with the incoherent illumination. 00:34:07.970 --> 00:34:13.989 So as I described earlier, the difference 00:34:13.989 --> 00:34:17.350 between incoherent and coherent is 00:34:17.350 --> 00:34:19.480 I don't have interference term if I illuminate 00:34:19.480 --> 00:34:21.880 with the incoherent source. 00:34:21.880 --> 00:34:26.035 So at the intensity, I only have these two terms-- 00:34:26.035 --> 00:34:27.410 just intensity of the first point 00:34:27.410 --> 00:34:29.743 spread function and intensity of the second point spread 00:34:29.743 --> 00:34:30.340 function. 00:34:30.340 --> 00:34:34.870 And I don't have the extra cosine term. 00:34:34.870 --> 00:34:39.460 So this is just intensity. 00:34:39.460 --> 00:34:44.469 So if I plot the intensity with the same parameter, 00:34:44.469 --> 00:34:48.020 then this is how it looks. 00:34:48.020 --> 00:34:52.330 And if I put together, then the upper left 00:34:52.330 --> 00:35:01.670 is the two intensity profiles for coherent and incoherent 00:35:01.670 --> 00:35:03.050 illumination. 00:35:03.050 --> 00:35:05.460 So they are slightly different. 00:35:05.460 --> 00:35:07.270 And if I play with a different parameter, 00:35:07.270 --> 00:35:11.120 like the different initial phase. 00:35:11.120 --> 00:35:14.440 And in this case, my point spread function 00:35:14.440 --> 00:35:17.170 has extra linear phase term, then 00:35:17.170 --> 00:35:19.450 the difference is quite dramatic. 00:35:19.450 --> 00:35:23.140 So you always have to be careful what kind of coherence 00:35:23.140 --> 00:35:26.490 your illumination is. 00:35:26.490 --> 00:35:27.829 Any questions? 00:35:39.500 --> 00:35:44.370 So in previous case, which is spatially coherent, 00:35:44.370 --> 00:35:48.870 then the input field and output field is linear, 00:35:48.870 --> 00:35:51.020 so you can use the convolution. 00:35:51.020 --> 00:35:57.950 But in spatially incoherent case, 00:35:57.950 --> 00:36:00.590 we deal with the input intensity which is-- 00:36:06.920 --> 00:36:10.760 so we're going to deal with the input intensity and output 00:36:10.760 --> 00:36:11.480 intensity. 00:36:11.480 --> 00:36:14.570 And at the image plane, the intensity 00:36:14.570 --> 00:36:19.070 will be [INAUDIBLE] incoherently. 00:36:19.070 --> 00:36:26.940 So I illuminate with the intensity-- 00:36:26.940 --> 00:36:31.380 I mean, incoherent illumination with the intensity [INAUDIBLE],, 00:36:31.380 --> 00:36:34.130 and g sub t is the complex transparency. 00:36:34.130 --> 00:36:38.820 But this is field, so you have to take negative square. 00:36:38.820 --> 00:36:47.460 So product of this guy is basically your input intensity. 00:36:47.460 --> 00:36:50.870 So this is illumination intensity and-- 00:36:50.870 --> 00:36:53.130 intensity of your grating-- 00:36:53.130 --> 00:36:59.790 and this is point spread function-- actually, 00:36:59.790 --> 00:37:01.440 coherent point spread function where 00:37:01.440 --> 00:37:04.320 you take the negative square because you 00:37:04.320 --> 00:37:07.245 take the negative square to get intensity. 00:37:07.245 --> 00:37:09.370 [CLEARS THROAT] Excuse me. 00:37:09.370 --> 00:37:12.150 We define incoherent point spread function, 00:37:12.150 --> 00:37:13.770 which is ipsf-- 00:37:13.770 --> 00:37:16.540 it's actually negative square over the coherent point spread 00:37:16.540 --> 00:37:18.570 function. 00:37:18.570 --> 00:37:19.710 It's pretty simple, right? 00:37:19.710 --> 00:37:23.580 You have point spread function in field. 00:37:23.580 --> 00:37:25.170 And if you just take the intensity 00:37:25.170 --> 00:37:29.750 and test the intensity point spread function. 00:37:29.750 --> 00:37:33.220 But you probably already noticed the difference. 00:37:33.220 --> 00:37:36.870 So if you use the coherent illumination, 00:37:36.870 --> 00:37:40.450 then the fields are related to convolution, 00:37:40.450 --> 00:37:42.240 but in coherent case, intensities 00:37:42.240 --> 00:37:47.950 are related to convolution, and point spread function 00:37:47.950 --> 00:37:48.957 are different. 00:37:55.976 --> 00:37:57.870 Any questions? 00:37:57.870 --> 00:38:00.040 I'm going to [INAUDIBLE]. 00:38:11.300 --> 00:38:14.420 So we just defined the intensity point spread function, 00:38:14.420 --> 00:38:17.370 which is negative square coherent point spread function. 00:38:17.370 --> 00:38:23.100 And by analogy, I mean in the coherence case, 00:38:23.100 --> 00:38:25.120 we had the coherent point spread function. 00:38:25.120 --> 00:38:28.230 And if you take the Fourier transform of the coherent point 00:38:28.230 --> 00:38:32.220 spread function, then that is ATF-- 00:38:32.220 --> 00:38:34.150 amplitude transfer function. 00:38:34.150 --> 00:38:36.540 So we could define the same transform function 00:38:36.540 --> 00:38:42.490 for incoherent point spread function, which is OTF here. 00:38:42.490 --> 00:38:47.220 So [INAUDIBLE] says that-- 00:38:52.380 --> 00:38:58.080 so actually, the small h is the coherent point spread function, 00:38:58.080 --> 00:39:02.790 and the capital H is ATF. 00:39:02.790 --> 00:39:06.180 So they are a Fourier transform pair. 00:39:06.180 --> 00:39:08.820 And hI, which is intensity point spread function, 00:39:08.820 --> 00:39:12.750 is the negative square of the coherent point spread function. 00:39:12.750 --> 00:39:21.080 But hI and OTF which is this weird h, 00:39:21.080 --> 00:39:26.740 they are also a Fourier transform pair. 00:39:26.740 --> 00:39:28.900 So if you go through this derivation, 00:39:28.900 --> 00:39:33.630 then it will prove actually they are-- 00:39:33.630 --> 00:39:47.810 that OTF is actually this integration of ATF. 00:39:47.810 --> 00:39:53.930 So once you have ATF, and after you compute this information, 00:39:53.930 --> 00:39:57.590 then you get OTF which is auto correlation. 00:39:57.590 --> 00:40:01.860 And I'm going to derive in a different way here. 00:40:05.390 --> 00:40:06.680 [INAUDIBLE] please. 00:40:17.500 --> 00:40:23.260 So we start with hI of x, which is intensity point spread 00:40:23.260 --> 00:40:27.200 function is negative square of the coherent point spread 00:40:27.200 --> 00:40:29.280 function. 00:40:29.280 --> 00:40:30.172 And-- yes? 00:40:30.172 --> 00:40:31.672 AUDIENCE: Can you write much bigger? 00:40:31.672 --> 00:40:32.580 [INAUDIBLE] 00:40:32.580 --> 00:40:33.497 SE BAEK OH: All right. 00:40:35.780 --> 00:40:36.280 [LAUGHS] 00:40:36.280 --> 00:40:38.560 Let me try. 00:40:38.560 --> 00:40:41.018 That's too big, right? 00:40:41.018 --> 00:40:42.419 AUDIENCE: [INAUDIBLE] 00:40:42.419 --> 00:40:43.360 [LAUGHS] 00:40:43.360 --> 00:40:44.510 SE BAEK OH: It's too small? 00:40:44.510 --> 00:40:45.010 OK. 00:40:55.790 --> 00:40:58.790 A little bit smaller. 00:40:58.790 --> 00:41:01.970 We start with this thing, and we know 00:41:01.970 --> 00:41:10.310 that the OTF is Fourier transform of intensity point 00:41:10.310 --> 00:41:11.020 spread function. 00:41:17.020 --> 00:41:19.250 And I can just plug into here. 00:41:19.250 --> 00:41:25.020 So it's Fourier transform of negative square 00:41:25.020 --> 00:41:30.430 of the coherent point spread function and Fourier transform. 00:41:30.430 --> 00:41:33.640 And since it is complex conjugate, 00:41:33.640 --> 00:41:38.230 I can write as h of x. 00:41:38.230 --> 00:41:40.652 And it's complex conjugate. 00:41:45.660 --> 00:41:48.530 And since these two are products, 00:41:48.530 --> 00:41:53.570 then I can write as convolution of two guys. 00:41:53.570 --> 00:41:58.210 So it's Fourier transform of h of x, 00:41:58.210 --> 00:42:01.810 conversion with Fourier transform 00:42:01.810 --> 00:42:06.240 of a complex conjugate like this. 00:42:09.100 --> 00:42:11.980 [INAUDIBLE] 00:42:11.980 --> 00:42:20.920 And Fourier transform of the H is actually H of u-- 00:42:20.920 --> 00:42:25.650 I mean this guy. 00:42:25.650 --> 00:42:29.610 ATF is just nothing but Fourier transform of the coherent point 00:42:29.610 --> 00:42:30.860 spread function. 00:42:30.860 --> 00:42:32.980 And what about this guy? 00:42:32.980 --> 00:42:38.780 Fourier transform of complex conjugate of small h. 00:42:38.780 --> 00:42:41.720 So I can write this in integral form. 00:42:41.720 --> 00:42:46.964 So integral h, complex conjugate of x. 00:42:46.964 --> 00:42:53.030 e to the minus j2pi, xu, dx. 00:42:56.250 --> 00:42:58.320 And since it is complex conjugate, 00:42:58.320 --> 00:43:03.450 I can write as integral h of x, e 00:43:03.450 --> 00:43:16.531 to the I2pi xu, dx, x, and conjugate. 00:43:21.720 --> 00:43:34.670 So I can write as OTF is H of u convolution, 00:43:34.670 --> 00:43:46.010 and complex conjugate of Fourier transform of-- 00:43:46.010 --> 00:43:46.950 like this. 00:43:51.200 --> 00:43:53.483 Actually, [INAUDIBLE] transform. 00:44:00.110 --> 00:44:02.570 So actually, this inverse Fourier transform, 00:44:02.570 --> 00:44:05.540 if you remember, the difference of the Fourier transform 00:44:05.540 --> 00:44:15.390 is actually integral h of x, e to the I2pi xu, dx. 00:44:15.390 --> 00:44:15.890 But 00:44:15.890 --> 00:44:24.320 You can write as h of x, e to the minus 2pi x minus u, dx. 00:44:24.320 --> 00:44:29.390 So actually, it is Fourier transform, and h of x. 00:44:29.390 --> 00:44:31.850 But you change the spatial frequency variable 00:44:31.850 --> 00:44:35.360 as a minus u, So it's going to be H of minus u. 00:44:41.260 --> 00:44:47.530 So at the end, we have H of u convolution, 00:44:47.530 --> 00:44:49.450 and complex conjugate of-- 00:44:52.282 --> 00:44:53.730 ah, sorry. 00:44:56.300 --> 00:44:59.025 H complex conjugate of minus u. 00:45:04.850 --> 00:45:10.180 [INAUDIBLE] where I can write as an integral form. 00:45:10.180 --> 00:45:23.880 Integration H of u prime, H of minus u 00:45:23.880 --> 00:45:30.330 minus u prime, complex conjugate, and du prime. 00:45:30.330 --> 00:45:37.100 So essentially, integration H complex conjugate-- 00:45:37.100 --> 00:45:38.254 sorry. 00:45:38.254 --> 00:45:47.400 So I have H of u prime, complex conjugate u prime minus 00:45:47.400 --> 00:45:53.610 u, du prime, which is essentially 00:45:53.610 --> 00:45:57.750 this guy in [INAUDIBLE]. 00:46:04.400 --> 00:46:11.185 And sometimes, we denote by H of u [INAUDIBLE] H of u, 00:46:11.185 --> 00:46:14.332 which is auto-correlation. 00:46:28.340 --> 00:46:30.497 Any questions on the derivation? 00:46:42.880 --> 00:46:47.010 So before we move on, can anyone guess 00:46:47.010 --> 00:46:49.810 what this integration means? 00:46:49.810 --> 00:46:53.520 So you have-- you want to compute the OTF, which 00:46:53.520 --> 00:46:56.370 is auto-correlation of the ATF. 00:46:56.370 --> 00:47:01.920 But this integration is basically you have H, 00:47:01.920 --> 00:47:03.400 and you take the complex conjugate, 00:47:03.400 --> 00:47:06.350 but you shift and take the integral. 00:47:15.100 --> 00:47:21.220 So next slide, I'm going to explain how we can compute-- 00:47:21.220 --> 00:47:23.790 so bottom line of this slide is OTF 00:47:23.790 --> 00:47:27.130 is nothing but auto-correlation of your ATF. 00:47:27.130 --> 00:47:30.960 But next slide is how to compute when-- 00:47:30.960 --> 00:47:32.820 you can simply compute this integration, 00:47:32.820 --> 00:47:38.695 but there is another easy way to guess the OTF. 00:47:38.695 --> 00:47:44.950 AUDIENCE: [INAUDIBLE] 00:47:44.950 --> 00:47:46.290 SE BAEK OH: Conceptually, yes. 00:47:46.290 --> 00:47:48.120 I mean, actually, Fourier transform 00:47:48.120 --> 00:47:51.420 of the input intensity and Fourier transform of the output 00:47:51.420 --> 00:47:53.480 intensity, and the ratio of [INAUDIBLE].. 00:48:08.060 --> 00:48:12.290 So let's first compute the OTF for the simple case. 00:48:12.290 --> 00:48:17.930 I have the rectangular aperture in my [INAUDIBLE] plane. 00:48:17.930 --> 00:48:21.100 And from that, I know that the ATF-- 00:48:21.100 --> 00:48:25.280 H of u-- is just [INAUDIBLE] function, and u over umx. 00:48:25.280 --> 00:48:28.130 So umx is defined by the physical size 00:48:28.130 --> 00:48:29.780 of the [INAUDIBLE] function. 00:48:29.780 --> 00:48:32.640 And so if it's just x, double prime x, 00:48:32.640 --> 00:48:37.560 and with the scaling vector of lambda F1. 00:48:37.560 --> 00:48:40.520 So this is the ATF. 00:48:40.520 --> 00:48:46.670 And I just put that OTF is auto-correlation of ATF. 00:48:46.670 --> 00:48:49.970 In integral form, I have this integration. 00:48:49.970 --> 00:48:54.650 So let's just take a look at this [INAUDIBLE] actually. 00:48:54.650 --> 00:48:58.325 So OTF is function of u, but this integration-- 00:48:58.325 --> 00:49:00.340 I mean inside of this integration, 00:49:00.340 --> 00:49:04.020 I have function of H of u prime. 00:49:04.020 --> 00:49:07.220 And H complex conjugate, u prime minus u. 00:49:07.220 --> 00:49:09.470 So I have the first function, which 00:49:09.470 --> 00:49:13.330 is essentially same as my original function 00:49:13.330 --> 00:49:18.230 even though I change the variable from u to u prime. 00:49:18.230 --> 00:49:24.560 But this guy is function of u prime where you shift by u. 00:49:24.560 --> 00:49:25.920 And [INAUDIBLE] integration. 00:49:25.920 --> 00:49:28.490 And this whole thing is function of u, 00:49:28.490 --> 00:49:30.890 which means you're going to compute 00:49:30.890 --> 00:49:37.370 this integration for every u, and then you get OTF. 00:49:37.370 --> 00:49:39.410 And this integration is basically 00:49:39.410 --> 00:49:43.010 you have two functions, one is shifted by u, 00:49:43.010 --> 00:49:46.130 and multiply them, and take the integration, which 00:49:46.130 --> 00:49:53.060 is essentially you compute the overlapped area of the two 00:49:53.060 --> 00:49:53.850 functions. 00:49:57.050 --> 00:50:00.775 So graphically, if I have [INAUDIBLE] function and plug 00:50:00.775 --> 00:50:05.180 it in this integration, I have this integration. 00:50:05.180 --> 00:50:07.940 But I can imagine for a different situation, 00:50:07.940 --> 00:50:10.100 the first-- 00:50:10.100 --> 00:50:13.560 so this [INAUDIBLE] function is the coherent [INAUDIBLE] 00:50:13.560 --> 00:50:14.780 function. 00:50:14.780 --> 00:50:17.180 And the second one which is moving 00:50:17.180 --> 00:50:19.170 is a coherent [INAUDIBLE] function. 00:50:19.170 --> 00:50:21.650 So if they are not overlapped, then I 00:50:21.650 --> 00:50:25.490 get simply 0 the first situation. 00:50:25.490 --> 00:50:28.820 And as soon as they are overlapped, 00:50:28.820 --> 00:50:35.520 and then the overlapped area is getting linearly increased. 00:50:35.520 --> 00:50:40.430 So I get this u plus umx until they are completely overlapped. 00:50:40.430 --> 00:50:43.670 And then-- so this is [INAUDIBLE]---- 00:50:43.670 --> 00:50:50.660 and then once you move a little bit further, 00:50:50.660 --> 00:50:54.510 then basically this overlapped area is getting decreased. 00:50:54.510 --> 00:50:58.260 So this is third situation, which is this guy. 00:50:58.260 --> 00:51:01.370 And eventually, they are not going to overlap, 00:51:01.370 --> 00:51:04.850 so I get 0 again. 00:51:04.850 --> 00:51:07.100 Remember that this OTF is functional of u, 00:51:07.100 --> 00:51:11.000 which is the amount of shift of this [INAUDIBLE] function. 00:51:16.357 --> 00:51:18.752 Any questions? 00:51:18.752 --> 00:51:22.620 So if you plot this overlapped area-- 00:51:22.620 --> 00:51:24.840 I mean this guy and this guy-- 00:51:24.840 --> 00:51:28.070 with respect to u, then actually we 00:51:28.070 --> 00:51:36.130 are going to get a triangular function, which 00:51:36.130 --> 00:51:38.945 looks like this one. 00:51:41.920 --> 00:51:46.310 And of course, we normalize at the 0 frequency. 00:51:46.310 --> 00:51:50.390 So at the 0, we always have a 1. 00:51:50.390 --> 00:51:54.380 So you notice the difference between these two. 00:51:54.380 --> 00:51:58.210 So ATF-- my ATF is nothing but my [INAUDIBLE] function-- 00:51:58.210 --> 00:51:59.650 just left function. 00:51:59.650 --> 00:52:02.200 And the color frequency, umx, is defined 00:52:02.200 --> 00:52:04.990 by my physical size of the [INAUDIBLE] 00:52:04.990 --> 00:52:07.020 with the scaling factor. 00:52:07.020 --> 00:52:11.050 But OTF is auto-correlation, so the shape 00:52:11.050 --> 00:52:13.630 is now the triangular shape. 00:52:13.630 --> 00:52:19.330 But also the color frequency-- the umx is twice of the umx, 00:52:19.330 --> 00:52:25.460 because basically you go through all this process. 00:52:25.460 --> 00:52:29.362 So you get twice of the color frequency. 00:52:37.730 --> 00:52:41.700 So-- yeah? 00:52:41.700 --> 00:52:42.590 Oh. 00:52:42.590 --> 00:52:44.370 So whatever [INAUDIBLE] you have, 00:52:44.370 --> 00:52:48.050 if you compute this kind of overlapped area, 00:52:48.050 --> 00:52:51.060 then you can get OTF. 00:52:51.060 --> 00:52:56.900 So [INAUDIBLE] you have a rectangular square aperture, 00:52:56.900 --> 00:53:00.890 then you if you go through this auto-correlation, 00:53:00.890 --> 00:53:02.460 then you can get this kind of OTF. 00:53:02.460 --> 00:53:06.090 So this is just completely linear in u and [INAUDIBLE] 00:53:06.090 --> 00:53:08.190 direction. 00:53:08.190 --> 00:53:12.820 Again, the fx and fy, they are [INAUDIBLE].. 00:53:12.820 --> 00:53:16.370 And if you have the circular aperture, it's almost linear, 00:53:16.370 --> 00:53:18.610 but it's slightly curved at the edge. 00:53:22.270 --> 00:53:27.930 But essentially, they are the same thing. 00:53:27.930 --> 00:53:30.190 So if you move around two circles 00:53:30.190 --> 00:53:32.800 in completely overlapped area, then you 00:53:32.800 --> 00:53:37.070 end up with this function. 00:53:42.880 --> 00:53:45.970 I guess this is a good time to take a break. 00:53:52.820 --> 00:53:53.930 Any questions so far? 00:53:56.630 --> 00:53:59.930 So we discussed there is a lot of difference 00:53:59.930 --> 00:54:02.210 between temporal and spatial coherence, 00:54:02.210 --> 00:54:05.510 and defined the intensity point spread function, 00:54:05.510 --> 00:54:06.948 and also defined the OTF which is 00:54:06.948 --> 00:54:08.990 Fourier transform with the intensity point spread 00:54:08.990 --> 00:54:10.340 function. 00:54:10.340 --> 00:54:12.312 And at the last-- 00:54:12.312 --> 00:54:14.495 [CLEARS THROAT] excuse me-- 00:54:14.495 --> 00:54:17.780 OTF is auto-correlation of the ATF. 00:54:17.780 --> 00:54:21.420 And I visually showed how to compute the OTF. 00:54:31.240 --> 00:54:35.170 So let's just summarize what we discussed so far. 00:54:35.170 --> 00:54:41.250 So first, we all know that coherent point spread function 00:54:41.250 --> 00:54:44.917 is nothing but an optical field generated by a point 00:54:44.917 --> 00:54:47.240 source at [INAUDIBLE] plane. 00:54:47.240 --> 00:54:50.620 And that's the field. 00:54:50.620 --> 00:54:54.880 And also you know that coherent point spread function 00:54:54.880 --> 00:54:56.710 is Fourier transform [INAUDIBLE] function 00:54:56.710 --> 00:55:00.370 with a coordinate scaling. 00:55:00.370 --> 00:55:02.680 And incoherent point spread function 00:55:02.680 --> 00:55:07.400 is negative square of coherent point spread. 00:55:07.400 --> 00:55:10.680 Or another way to say is intensity point 00:55:10.680 --> 00:55:14.420 spread function is intensity of the field coherent point spread 00:55:14.420 --> 00:55:14.920 function. 00:55:21.820 --> 00:55:26.940 And we define the amplitude [INAUDIBLE] function, ATF, 00:55:26.940 --> 00:55:30.150 which is Fourier transform of the coherent point spread 00:55:30.150 --> 00:55:32.680 function. 00:55:32.680 --> 00:55:41.260 And it is nothing but the [INAUDIBLE] function with 00:55:41.260 --> 00:55:42.490 the scaling vector-- 00:55:42.490 --> 00:55:44.540 I mean coordinate scaling. 00:55:44.540 --> 00:55:49.330 And we just derive the OTF, which 00:55:49.330 --> 00:55:53.340 is Fourier transform of the intensity point spread function 00:55:53.340 --> 00:55:56.230 is actually auto-correlation with the ATF. 00:56:02.860 --> 00:56:06.310 In the rest of the lecture, we're 00:56:06.310 --> 00:56:08.920 going to mostly deal with the MTF which 00:56:08.920 --> 00:56:12.760 is negative modulus of the OTF. 00:56:12.760 --> 00:56:14.490 Actually, it should be OTF. 00:56:14.490 --> 00:56:17.740 So MTF is the negative component of OTF. 00:56:22.450 --> 00:56:25.870 In terms of input and output, we're 00:56:25.870 --> 00:56:27.590 dealing with a linear system approach. 00:56:27.590 --> 00:56:32.080 So if you have the spatially coherent illumination-- 00:56:32.080 --> 00:56:36.170 so [INAUDIBLE] is the illumination field. 00:56:36.170 --> 00:56:38.440 And I have the thin transparency-- 00:56:38.440 --> 00:56:41.280 the complex transparency which is sub t. 00:56:41.280 --> 00:56:44.820 Then my input field is just product of my illumination 00:56:44.820 --> 00:56:47.050 field and complex transparency. 00:56:47.050 --> 00:56:49.490 So g sub n, that's the input field. 00:56:49.490 --> 00:56:53.260 And if I convert with the coherent point spread function, 00:56:53.260 --> 00:56:58.140 that's going to be my output field, which is g sub f. 00:56:58.140 --> 00:57:01.970 And I kind of think this whole thing in frequency domain. 00:57:01.970 --> 00:57:04.280 So if I take a Fourier transform-- 00:57:04.280 --> 00:57:06.790 [INAUDIBLE] is actually Fourier transform 00:57:06.790 --> 00:57:15.040 of small gn, which is Fourier component of my input field. 00:57:15.040 --> 00:57:18.377 And convolution is multiplication 00:57:18.377 --> 00:57:19.210 in frequency domain. 00:57:19.210 --> 00:57:24.250 So I multiply by ATF, then I'm going to get g sub-- 00:57:24.250 --> 00:57:28.340 [INAUDIBLE] which is Fourier inform of my output field. 00:57:30.970 --> 00:57:35.370 And if I have the spatially incoherent illumination, 00:57:35.370 --> 00:57:38.000 then I deal with the intensity. 00:57:38.000 --> 00:57:43.830 So I have the intensity of illumination, and I-- 00:57:43.830 --> 00:57:45.180 still complex transparency. 00:57:45.180 --> 00:57:51.960 So actually, my input intensity is illumination intensity 00:57:51.960 --> 00:57:56.220 times negative square of g sub t. 00:57:56.220 --> 00:57:59.550 So remember that gt is complex transparency. 00:57:59.550 --> 00:58:02.850 So to make intensity, you have to take negative square. 00:58:02.850 --> 00:58:05.280 So this is my input intensity. 00:58:05.280 --> 00:58:09.640 And in current case, it's linear in intensity, 00:58:09.640 --> 00:58:12.300 so you have to convert with the intensity point spread 00:58:12.300 --> 00:58:16.870 function, which is negative square of the coherent point 00:58:16.870 --> 00:58:17.980 spread function. 00:58:17.980 --> 00:58:22.600 Then you're going to get in output intensity. 00:58:22.600 --> 00:58:25.300 And again, in the frequency domain, 00:58:25.300 --> 00:58:30.204 we can take the Fourier transform-- so g sub i-- 00:58:30.204 --> 00:58:31.410 what can I say? 00:58:31.410 --> 00:58:38.730 Character gi input-- character g sub i in, 00:58:38.730 --> 00:58:42.480 which is Fourier transform of the input intensity. 00:58:42.480 --> 00:58:45.450 And you're going to multiply by OTF, 00:58:45.450 --> 00:58:48.780 which is what auto-correlation of the ATF. 00:58:48.780 --> 00:58:52.380 And you'll get the Fourier transform 00:58:52.380 --> 00:58:54.057 of the output intensity. 00:58:58.350 --> 00:59:02.880 So actually, if you're familiar with the control theory, 00:59:02.880 --> 00:59:05.430 then you probably know the Bode plot. 00:59:05.430 --> 00:59:09.040 So Bode plot is nothing but you have-- 00:59:09.040 --> 00:59:11.490 in frequency domain, it describes the weighting 00:59:11.490 --> 00:59:14.280 function that is going to be multiplied by your input 00:59:14.280 --> 00:59:16.240 and produce your output. 00:59:16.240 --> 00:59:19.720 So this ATF, and OTF, and also MTF, 00:59:19.720 --> 00:59:22.650 they are basically the same thing. 00:59:22.650 --> 00:59:25.310 You multiply your-- so basically, 00:59:25.310 --> 00:59:29.745 this transform function in frequency domain 00:59:29.745 --> 00:59:32.200 describes the weighting function here 00:59:32.200 --> 00:59:34.600 that is going to be multiplied by your input. 00:59:34.600 --> 00:59:38.010 And you can just compare the-- 00:59:38.010 --> 00:59:41.050 and also another way to say it is the ratio between input 00:59:41.050 --> 00:59:41.550 and output. 00:59:47.110 --> 00:59:49.930 So MTF-- so, yeah. 00:59:49.930 --> 00:59:54.850 So as I described, this transform function 00:59:54.850 --> 00:59:58.870 is nothing but with weighting function in frequency domain. 00:59:58.870 --> 01:00:01.490 So let me show you how to use it. 01:00:01.490 --> 01:00:06.110 Because if you buy a lens for your digital camera-- 01:00:06.110 --> 01:00:08.925 I mean, typically the SLR camera-- 01:00:08.925 --> 01:00:11.740 so lets make us provide the MTF layer, 01:00:11.740 --> 01:00:16.690 which is this weighting function with the spatial frequency, 01:00:16.690 --> 01:00:19.020 and with the color frequency. 01:00:19.020 --> 01:00:21.500 It's cool-- you can say what kind of image quality 01:00:21.500 --> 01:00:24.670 I can expect with the lens. 01:00:24.670 --> 01:00:28.510 So first, let's consider our [INAUDIBLE] grating 01:00:28.510 --> 01:00:33.590 whose MTF transform function is given by this equation. 01:00:33.590 --> 01:00:39.040 So the input intensity is going to be-- 01:00:39.040 --> 01:00:42.460 I mean intensity of the grating is going to be negative square, 01:00:42.460 --> 01:00:44.020 so I add 1 over 2. 01:00:44.020 --> 01:00:48.250 Then 1 plus n modulation, and cosine term. 01:00:48.250 --> 01:00:51.207 And the phase [INAUDIBLE] it doesn't 01:00:51.207 --> 01:00:53.290 matter why, because if you take a negative square, 01:00:53.290 --> 01:00:54.010 it's always 1. 01:01:00.480 --> 01:01:05.820 The Fourier transform of this input intensity is-- 01:01:05.820 --> 01:01:10.220 we all know that it has three 3 orders. 01:01:10.220 --> 01:01:14.647 So 0-th order, and the 1 over capital lambda, and minus 1 01:01:14.647 --> 01:01:15.480 over capital lambda. 01:01:15.480 --> 01:01:18.480 We have three different orders. 01:01:18.480 --> 01:01:26.370 And it's going to be multiplied by the OTF, which is at this h. 01:01:26.370 --> 01:01:29.670 And since this is delta function, which means 01:01:29.670 --> 01:01:33.190 this one is 1 only at u equals 0, 01:01:33.190 --> 01:01:37.050 and when u equals 1 over capital lambda, 01:01:37.050 --> 01:01:39.700 and at minus 1 capital lambda. 01:01:39.700 --> 01:01:41.310 So if you multiply the OTF-- 01:01:41.310 --> 01:01:44.550 and actually, you still have the three components. 01:01:44.550 --> 01:01:50.940 So at 0 component, you have the OTF at the 0 frequency. 01:01:50.940 --> 01:01:54.430 And these two delta functions are multiplied by OTF 01:01:54.430 --> 01:01:58.320 at the 1 over capital lambda and minus 1 over capital lambda. 01:01:58.320 --> 01:02:00.810 We still have the three [INAUDIBLE] orders. 01:02:05.070 --> 01:02:15.320 And let me just go back. 01:02:15.320 --> 01:02:18.830 So since this is your Fourier transform of the output 01:02:18.830 --> 01:02:21.370 intensity, you can just take the inverse 01:02:21.370 --> 01:02:25.010 Fourier transform to compute the output intensity, right? 01:02:25.010 --> 01:02:26.520 But there is another-- 01:02:26.520 --> 01:02:29.970 there is an important property of the OTF. 01:02:29.970 --> 01:02:32.030 So this guy and this guy. 01:02:32.030 --> 01:02:41.220 So OTF is actually-- 01:02:41.220 --> 01:02:44.730 intensity point spread function, which is h sub i, 01:02:44.730 --> 01:02:48.180 is negative square of the coherent point spread function. 01:02:48.180 --> 01:02:50.520 So it is always positive-- 01:02:50.520 --> 01:02:53.340 it's non-negative I should say. 01:02:53.340 --> 01:02:55.320 So it's non-negative. 01:02:55.320 --> 01:02:57.170 So this Fourier transform is actually 01:02:57.170 --> 01:03:00.660 Hermitian, which mathematically, you 01:03:00.660 --> 01:03:04.290 can [INAUDIBLE] h of minus u-- is actually 01:03:04.290 --> 01:03:06.800 h complex conjugate of u. 01:03:06.800 --> 01:03:11.040 Or you can say its real path is [INAUDIBLE] function, 01:03:11.040 --> 01:03:12.750 but its simulated path is out function. 01:03:15.540 --> 01:03:18.520 So OTF of the physically realizable optical system 01:03:18.520 --> 01:03:19.600 must be Hermitian. 01:03:19.600 --> 01:03:22.950 So this condition should be satisfied always. 01:03:30.280 --> 01:03:33.666 So in the previous slide, I had-- 01:03:33.666 --> 01:03:34.280 let me just-- 01:03:49.010 --> 01:03:49.900 [INAUDIBLE] 01:03:52.420 --> 01:04:08.100 Capital G, I, out, was 1 over 2 OTF at 0, and delta u. 01:04:08.100 --> 01:04:19.760 Plus n over 2, H, 1 over capital lambda, and delta u, 01:04:19.760 --> 01:04:21.240 1 over capital lambda. 01:04:21.240 --> 01:04:26.400 Plus H minus 1 over capital lambda, 01:04:26.400 --> 01:04:29.731 delta u, 1 over capital lambda. 01:04:37.910 --> 01:04:47.510 Since the OTF is Hermitian, which means my OTF over minus 1 01:04:47.510 --> 01:04:51.860 over capital lambda is actually complex conjugate 01:04:51.860 --> 01:04:54.080 of H, 1 over capital lambda. 01:04:57.230 --> 01:04:59.161 This is Hermitian. 01:05:01.810 --> 01:05:03.190 So G out-- 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 01:05:16.550 --> 01:05:18.946 is still delta function. 01:05:18.946 --> 01:05:26.870 And m over 2, H, 1 over capital lambda, delta u, 01:05:26.870 --> 01:05:29.040 1 over capital lambda. 01:05:29.040 --> 01:05:31.190 But the last delta function I can 01:05:31.190 --> 01:05:43.560 write as H complex conjugate, 1 over capital lambda, decay u 01:05:43.560 --> 01:05:45.010 plus 1 over capital lambda. 01:05:48.142 --> 01:05:49.350 So the first delta function-- 01:05:49.350 --> 01:05:51.892 I mean, if you take the inverse Fourier transform [INAUDIBLE] 01:05:51.892 --> 01:05:56.010 transform, the first delta function [INAUDIBLE]---- 01:05:56.010 --> 01:05:57.300 just 1 or something. 01:05:57.300 --> 01:05:59.140 So it's pretty straightforward. 01:05:59.140 --> 01:06:03.030 The question is still what happened in these two delta 01:06:03.030 --> 01:06:08.070 functions, because this one has H of-- 01:06:08.070 --> 01:06:11.220 I mean the OTF at one over capital lambda, 01:06:11.220 --> 01:06:14.820 and the other one is actually complex conjugate. 01:06:14.820 --> 01:06:21.600 So if I just write the two delta function part, 01:06:21.600 --> 01:06:24.180 and I'm going to take the inverse Fourier transform. 01:06:24.180 --> 01:06:32.380 So inverse Fourier transform, or m over 2, [INAUDIBLE] 01:06:32.380 --> 01:06:39.060 OTF at 1 over capital lambda, delta u, 1 over capital lambda, 01:06:39.060 --> 01:06:45.040 plus H complex conjugate, capital lambda, delta u plus 1 01:06:45.040 --> 01:06:45.950 over capital lambda. 01:06:50.550 --> 01:06:56.310 Then what is-- inverse Fourier transform [INAUDIBLE] 01:06:56.310 --> 01:07:00.880 function is going to be exponential. 01:07:00.880 --> 01:07:02.300 And [INAUDIBLE] do we scale it? 01:07:02.300 --> 01:07:04.370 So [INAUDIBLE] just outside. 01:07:04.370 --> 01:07:09.830 And I'm going to have H of 1 over capital lambda, 01:07:09.830 --> 01:07:17.880 and exponent e to the I2pi, x over capital lambda, 01:07:17.880 --> 01:07:22.910 plus H complex conjugate, 1 over capital lambda. 01:07:22.910 --> 01:07:27.090 e to the minus I2pi, x over capital lambda. 01:07:32.956 --> 01:07:40.820 And I can write as capital lambda-- 01:07:40.820 --> 01:07:45.110 1 over capital lambda, e to the I2pi, x over lambda. 01:07:45.110 --> 01:07:48.230 Plus-- actually the same thing-- 01:07:48.230 --> 01:07:53.570 H, 1 over capital lambda, e to the I2pi, x over lambda. 01:07:53.570 --> 01:07:56.600 And complex conjugate like this. 01:08:00.860 --> 01:08:07.070 So what happens if you add A and its complex conjugate? 01:08:07.070 --> 01:08:14.290 So for example, this is A, and this is A complex conjugate. 01:08:14.290 --> 01:08:17.720 And if you add them together, then you 01:08:17.720 --> 01:08:28.189 get only rear path, because two rear paths over A. 01:08:28.189 --> 01:08:38.899 So then I'm going to have m, the rear path of OTF 01:08:38.899 --> 01:08:47.560 at 1 over capital lambda, to the I2pi x over lambda, 01:08:47.560 --> 01:08:53.040 which I can write as m, and negative-- 01:08:53.040 --> 01:08:56.290 this H is just non-zero. 01:08:56.290 --> 01:08:58.600 So it could have-- 01:08:58.600 --> 01:09:03.310 I mean, it can have only 0 or minus [INAUDIBLE] basically. 01:09:03.310 --> 01:09:04.930 So I can write it as negative here. 01:09:04.930 --> 01:09:09.939 And this [INAUDIBLE] cosine 2pi, x over lambda. 01:09:13.920 --> 01:09:20.010 So that's why the output intensity is just 1 01:09:20.010 --> 01:09:23.520 plus m, which is original modulation. 01:09:23.520 --> 01:09:27.630 But I add extra modulation due to OTF and cosine term. 01:09:36.540 --> 01:09:40.229 Should I prove why OTF is Hermitian? 01:09:51.479 --> 01:09:54.900 Yeah, let me prove why OTF is Hermitian. 01:10:00.670 --> 01:10:07.540 So OTF is auto-correlation of ATF, which is like this. 01:10:10.246 --> 01:10:14.450 And integration with the integral form I can write as H 01:10:14.450 --> 01:10:22.003 of u prime, H complex conjugate, u prime minus u, 01:10:22.003 --> 01:10:22.920 and [INAUDIBLE] prime. 01:10:25.660 --> 01:10:29.290 And how to compute H-- 01:10:29.290 --> 01:10:33.850 I mean OTF and complex conjugate of u. 01:10:33.850 --> 01:10:37.280 So I can write as-- 01:10:37.280 --> 01:10:39.020 the total thing-- 01:10:39.020 --> 01:10:46.600 H of u prime, H complex conjugate u prime u, u prime, 01:10:46.600 --> 01:10:47.720 and complex conjugate. 01:10:47.720 --> 01:10:48.580 AUDIENCE: Excuse me. 01:10:48.580 --> 01:10:49.510 SE BAEK OH: Yes? 01:10:49.510 --> 01:10:52.700 AUDIENCE: We're not able to see what you are writing. 01:10:52.700 --> 01:10:55.027 SE BAEK OH: Oh, it's too small? 01:10:55.027 --> 01:10:55.735 AUDIENCE: OK now. 01:10:59.400 --> 01:11:01.400 SE BAEK OH: So I just take the complex conjugate 01:11:01.400 --> 01:11:02.110 on both sides. 01:11:04.730 --> 01:11:09.050 And I can chain the complex conjugate. 01:11:09.050 --> 01:11:15.020 So it's going to be H complex conjugate of u prime, H, 01:11:15.020 --> 01:11:18.730 and u prime and u, and du prime. 01:11:18.730 --> 01:11:21.870 So I just put the complex conjugate inside 01:11:21.870 --> 01:11:24.640 of the integration. 01:11:24.640 --> 01:11:27.110 And here, I define the new variable. 01:11:27.110 --> 01:11:32.590 So u double prime is u prime minus u. 01:11:32.590 --> 01:11:39.680 So u prime is u plus u double prime. 01:11:39.680 --> 01:11:43.290 And du prime is actually du double prime. 01:11:48.520 --> 01:11:53.190 So this H complex conjugate is actually 01:11:53.190 --> 01:12:01.510 H complex conjugate, u plus u double prime, 01:12:01.510 --> 01:12:05.430 h of u double prime, and du double 01:12:05.430 --> 01:12:15.070 prime, which I can write as integration h u double prime. 01:12:15.070 --> 01:12:23.110 A complex conjugate u double prime, minus u, 01:12:23.110 --> 01:12:24.460 du double prime. 01:12:24.460 --> 01:12:27.990 So it's H of minus u. 01:12:30.880 --> 01:12:39.460 So this is complex conjugate Hermitian. 01:12:48.050 --> 01:12:51.010 I just proved you can easily do that. 01:12:51.010 --> 01:12:54.500 So then the bottom line is at the output plane-- 01:12:54.500 --> 01:12:57.710 I mean output intensity, I have extra modulation 01:12:57.710 --> 01:12:59.720 coming from the OTF. 01:12:59.720 --> 01:13:01.640 I mean, you can say intensity is negative, 01:13:01.640 --> 01:13:03.149 so you can say [INAUDIBLE]. 01:13:07.550 --> 01:13:14.150 So if you compare in terms of the contrast or visibility, 01:13:14.150 --> 01:13:17.750 so the contrast is defined by imx minus ime 01:13:17.750 --> 01:13:19.550 over imx plus ime. 01:13:19.550 --> 01:13:24.140 But at the output plane, we have the extra modulation term. 01:13:24.140 --> 01:13:27.080 So if you compute the same thing, 01:13:27.080 --> 01:13:29.460 then actually your contrast is going 01:13:29.460 --> 01:13:34.700 to be at the original modulation multiplied by MTF 01:13:34.700 --> 01:13:37.526 at the spatial frequency. 01:13:43.640 --> 01:13:48.170 So MTF is just contrast at the spatial frequency 01:13:48.170 --> 01:13:51.230 relative to the transmission that we're discussing. 01:13:51.230 --> 01:13:53.280 So what does it mean? 01:13:53.280 --> 01:13:59.900 So for example, I have MTF looks like this. 01:13:59.900 --> 01:14:03.075 And I initially had the grating-- 01:14:03.075 --> 01:14:05.870 the intensity of the grating is actually 1 over 2, 01:14:05.870 --> 01:14:08.850 1 plus n, cosine, 2pi, x over capital lambda. 01:14:08.850 --> 01:14:15.200 So if I draw in grayscale-- so the white is 1 and black is 0. 01:14:15.200 --> 01:14:20.090 So I have nice sinusoid with a full contrast like this. 01:14:20.090 --> 01:14:22.790 But the output intensity, I'm going 01:14:22.790 --> 01:14:29.420 to have 1 plus m, and additional modulation, and cosine term. 01:14:29.420 --> 01:14:33.450 And this factor, you can find [INAUDIBLE].. 01:14:33.450 --> 01:14:36.170 So if you have the 1 over capital lambda, 01:14:36.170 --> 01:14:40.300 then you can find the value at that point-- 01:14:40.300 --> 01:14:43.910 I mean MTF at this point. 01:14:43.910 --> 01:14:47.810 So if that value is 0.6 in this case, 01:14:47.810 --> 01:14:52.070 this sinusoidal pattern will look like this. 01:14:52.070 --> 01:14:58.340 But if the modulation is 0.2, then the black becomes 01:14:58.340 --> 01:15:00.320 brighter and white becomes darker. 01:15:00.320 --> 01:15:03.770 So you kind of get the grayscale-- 01:15:03.770 --> 01:15:06.810 [INAUDIBLE] version of the gradient like this. 01:15:06.810 --> 01:15:12.170 So basically, this MTF tells you what kind of contrast 01:15:12.170 --> 01:15:16.820 I expect at this particular spatial frequency, 01:15:16.820 --> 01:15:20.750 and also it specifies the maximum spatial frequency 01:15:20.750 --> 01:15:21.867 you can resolve-- 01:15:21.867 --> 01:15:22.700 I mean, you can see. 01:15:31.960 --> 01:15:33.149 Any questions? 01:15:42.000 --> 01:15:44.260 So so far, we just deal with the idea 01:15:44.260 --> 01:15:46.730 of thin lens with a clear aperture, 01:15:46.730 --> 01:15:49.550 like the rectangular aperture is a clear aperture. 01:15:49.550 --> 01:15:52.400 So in the case of the rectangular aperture, 01:15:52.400 --> 01:15:55.670 we know that the OTF is just auto-correlation of the two 01:15:55.670 --> 01:15:58.940 [INAUDIBLE] functions which is just triangular function. 01:15:58.940 --> 01:16:03.260 So it's linear from 1 to color frequency, 01:16:03.260 --> 01:16:07.310 which is twice of the [INAUDIBLE] frequency at OTF-- 01:16:07.310 --> 01:16:08.990 the ATF. 01:16:08.990 --> 01:16:11.060 And if you have the circular aperture, 01:16:11.060 --> 01:16:15.410 it's almost linear except at the [INAUDIBLE] 01:16:15.410 --> 01:16:18.560 you have slightly curved line. 01:16:18.560 --> 01:16:21.420 But it's the same thing. 01:16:21.420 --> 01:16:27.750 So this blue line and the black line we call defraction 01:16:27.750 --> 01:16:32.630 limited OTF, because we assume the idea of a thin lens 01:16:32.630 --> 01:16:34.070 without any aberration. 01:16:34.070 --> 01:16:40.170 So this is kind of best OTF you can achieve. 01:16:40.170 --> 01:16:43.790 But in practice, you always have aberration-- 01:16:43.790 --> 01:16:46.310 chromatic aberration, or any high aberration. 01:16:46.310 --> 01:16:53.300 Or you can have a defocus, or some imperfect factor in lens. 01:16:53.300 --> 01:16:57.140 So typically, the typical lens has 01:16:57.140 --> 01:17:01.670 this kind of distorted or degraded OTF like this. 01:17:01.670 --> 01:17:03.426 The red one right here or here. 01:17:07.710 --> 01:17:10.550 So in terms of this defraction limited, 01:17:10.550 --> 01:17:14.780 you want to buy a lens that OTF is similar to this blue line, 01:17:14.780 --> 01:17:17.080 because this is defraction limited, 01:17:17.080 --> 01:17:21.386 which means the best OTF we can achieve. 01:17:21.386 --> 01:17:23.650 And [INAUDIBLE]. 01:17:30.940 --> 01:17:35.260 So I describe-- I mean, this is slope is linear, 01:17:35.260 --> 01:17:39.670 and circular aperture. 01:17:42.730 --> 01:17:45.500 I mean, locally, it can have a higher value, 01:17:45.500 --> 01:17:50.270 but the overall shape should be smaller than this blue line. 01:18:01.690 --> 01:18:09.740 So let's just apply all these concepts in [INAUDIBLE] system. 01:18:09.740 --> 01:18:13.260 So we have first system whose focal length f1, f2. 01:18:13.260 --> 01:18:15.720 And I put the input transparency, 01:18:15.720 --> 01:18:18.520 which is a binary [INAUDIBLE] grading right here. 01:18:18.520 --> 01:18:21.860 And illuminate with the quasi-monochromatic which 01:18:21.860 --> 01:18:24.650 is on a single color, and spatial [INAUDIBLE] 01:18:24.650 --> 01:18:30.460 illumination, and [INAUDIBLE] we assume the intensity variation 01:18:30.460 --> 01:18:32.240 is uniform. 01:18:32.240 --> 01:18:37.120 So the possible question is what kind of image 01:18:37.120 --> 01:18:44.170 we can expect at the image plane, or what is the OTF, 01:18:44.170 --> 01:18:49.210 or what is ATF, and blah, blah, blah. 01:18:49.210 --> 01:18:50.350 So let's go through it. 01:18:56.460 --> 01:18:59.140 So this is the binary amplitude grating. 01:18:59.140 --> 01:19:03.510 So it is still complex transparency in here. 01:19:03.510 --> 01:19:06.690 So if you have [INAUDIBLE] illumination, 01:19:06.690 --> 01:19:08.650 you want to compute the intensity. 01:19:08.650 --> 01:19:11.470 So you have to take the negative scale. 01:19:11.470 --> 01:19:15.120 So this is just 0 or 1, so basically you have 01:19:15.120 --> 01:19:20.070 the same thing in intensity. 01:19:20.070 --> 01:19:24.900 So that's the input intensity at the upper right. 01:19:24.900 --> 01:19:30.210 And this is the phase mask located at the rear plane. 01:19:30.210 --> 01:19:35.030 And we know that the ATF is basically the same function 01:19:35.030 --> 01:19:41.470 with different coordinates. 01:19:41.470 --> 01:19:45.580 So ATF is phase mask, but the function 01:19:45.580 --> 01:19:47.710 is the x double prime-- well, I mean 01:19:47.710 --> 01:19:49.710 you have this coordinate scaling vector. 01:19:53.080 --> 01:19:56.830 So if you compute the OTF from this ATF, 01:19:56.830 --> 01:20:03.330 basically, what you do is you have two [INAUDIBLE] functions. 01:20:03.330 --> 01:20:06.610 So you will have actually 4 [INAUDIBLE] functions. 01:20:06.610 --> 01:20:10.450 And you move toward them, so whenever they overlap, 01:20:10.450 --> 01:20:13.030 you get some value. 01:20:13.030 --> 01:20:16.270 So if could do the auto-correlation, 01:20:16.270 --> 01:20:19.750 then actually you're going to have three triangular. 01:20:19.750 --> 01:20:24.550 But remember, if you have the [INAUDIBLE] function, 01:20:24.550 --> 01:20:26.330 then you move two [INAUDIBLE] function. 01:20:26.330 --> 01:20:29.750 So you're going to have one triangular. 01:20:29.750 --> 01:20:32.560 So since we have two [INAUDIBLE] functions, so whenever 01:20:32.560 --> 01:20:36.670 they first [INAUDIBLE],, then we have the one triangle. 01:20:36.670 --> 01:20:40.240 But the second and the two [INAUDIBLE] functions 01:20:40.240 --> 01:20:42.880 overlap, then we have the center triangle. 01:20:42.880 --> 01:20:46.480 And like this, we another triangle, so 01:20:46.480 --> 01:20:49.140 we get three triangles. 01:20:49.140 --> 01:20:50.750 So that's the OTF. 01:20:53.790 --> 01:20:59.400 And if you expand with the Fourier series of your input 01:20:59.400 --> 01:21:01.890 intensity, then we're going to have an infinite number 01:21:01.890 --> 01:21:03.000 of diffraction. 01:21:03.000 --> 01:21:08.910 But in ATF case, we have only first order 01:21:08.910 --> 01:21:11.640 and the minus first order, and minus 01:21:11.640 --> 01:21:14.270 first order going to be transmitted. 01:21:14.270 --> 01:21:17.320 But in OTF, actually, they are blocked. 01:21:17.320 --> 01:21:19.294 [INAUDIBLE] 01:21:19.294 --> 01:21:23.340 Actually, plus 2, and 0, and minus 2, 01:21:23.340 --> 01:21:25.412 they are going to be transmitted. 01:21:29.480 --> 01:21:34.420 So you just multiply these three delta functions 01:21:34.420 --> 01:21:37.930 by the value of the ATF-- 01:21:37.930 --> 01:21:40.000 this, this, this. 01:21:40.000 --> 01:21:46.260 And the Fourier transform will be your output intensity, 01:21:46.260 --> 01:21:48.950 which is this. 01:21:48.950 --> 01:21:53.200 So 1 over 3 just came from the central [INAUDIBLE] function, 01:21:53.200 --> 01:21:54.430 because this is 1-- 01:21:54.430 --> 01:21:56.530 so just 1 over 3. 01:21:56.530 --> 01:22:04.450 And for the first order, at the 0.1 and minus 0.1 01:22:04.450 --> 01:22:08.500 micrometer [INAUDIBLE],, actually they have 0 value. 01:22:08.500 --> 01:22:12.100 So I multiply by 0. 01:22:12.100 --> 01:22:16.450 And for the second order, I have one over 2 which 01:22:16.450 --> 01:22:21.400 is this value here and here. 01:22:21.400 --> 01:22:25.672 So that is the [INAUDIBLE] of the output intensity. 01:22:28.450 --> 01:22:32.140 So if you compute the inverse Fourier transform 01:22:32.140 --> 01:22:35.770 of this equation, then we are going to have 1 over 3, 01:22:35.770 --> 01:22:39.730 and 1 over 2, and two exponent here 01:22:39.730 --> 01:22:45.970 which is actually cosine like this. 01:22:45.970 --> 01:22:48.160 So in terms of grayscale image, we 01:22:48.160 --> 01:22:50.570 have the binary amplitude grating, 01:22:50.570 --> 01:22:54.260 which varies just black or white like this. 01:22:54.260 --> 01:22:56.950 But if you go through this [INAUDIBLE] system 01:22:56.950 --> 01:23:00.520 at the image, we're going to have this kind of grayscale 01:23:00.520 --> 01:23:06.910 because contrast is just 0.1034. 01:23:06.910 --> 01:23:09.460 So it's very little. 01:23:09.460 --> 01:23:14.770 And the average value is about 0.33 or something. 01:23:14.770 --> 01:23:19.120 So basically, it looks gray with a little wiggling. 01:23:21.910 --> 01:23:26.830 So ATF basically tells you the contrast 01:23:26.830 --> 01:23:29.311 of the particular spatial frequency. 01:23:38.680 --> 01:23:41.760 So, yeah. 01:23:41.760 --> 01:23:44.390 What happened-- I mean, the orders at this side. 01:23:44.390 --> 01:23:51.926 So if my OTF is 0, then what kind of image I can expect? 01:23:51.926 --> 01:23:52.854 It's black? 01:23:55.353 --> 01:23:56.270 It's not light, right? 01:23:58.870 --> 01:24:00.910 So if it is 0-- 01:24:00.910 --> 01:24:03.535 I mean if the values are 0, then-- 01:24:08.830 --> 01:24:13.130 so I still have [INAUDIBLE] so some kind of grayscale. 01:24:13.130 --> 01:24:16.150 So I don't see any [INAUDIBLE]. 01:24:18.900 --> 01:24:20.880 OTF is auto-correlation, so you always 01:24:20.880 --> 01:24:26.020 have 1 at the dc frequency. 01:24:26.020 --> 01:24:28.460 Because if you move the two functions, 01:24:28.460 --> 01:24:30.810 so when they exactly overlap, you always 01:24:30.810 --> 01:24:31.770 have the maximum value. 01:24:31.770 --> 01:24:38.810 So you always have [INAUDIBLE] system. 01:24:38.810 --> 01:24:41.840 And actually, that is very different in current case. 01:24:41.840 --> 01:24:44.770 So in ATF case, you can actually cancel [INAUDIBLE] system, 01:24:44.770 --> 01:24:45.820 right? 01:24:45.820 --> 01:24:47.860 But OTF, you can not cancel the system, 01:24:47.860 --> 01:24:49.400 because it's auto-correlation. 01:24:55.670 --> 01:25:01.370 So again, we have some relation with the [INAUDIBLE].. 01:25:01.370 --> 01:25:06.870 So this particular circular aperture, 01:25:06.870 --> 01:25:09.650 if I eliminate the coherent illumination, 01:25:09.650 --> 01:25:10.910 then this is the image-- 01:25:10.910 --> 01:25:11.940 AUDIENCE: Excuse me. 01:25:11.940 --> 01:25:13.550 SE BAEK OH: Yes? 01:25:13.550 --> 01:25:16.870 AUDIENCE: In Singapore slide, the [INAUDIBLE] slides 01:25:16.870 --> 01:25:18.500 are not [INAUDIBLE]. 01:25:25.630 --> 01:25:29.440 SE BAEK OH: Maybe it takes time or-- 01:25:29.440 --> 01:25:35.750 AUDIENCE: No, it even didn't show that input and output 01:25:35.750 --> 01:25:38.885 image when you explained about that contrast. 01:25:42.286 --> 01:25:45.170 SE BAEK OH: Just let me ask [INAUDIBLE].. 01:25:45.170 --> 01:25:49.030 Hello, [INAUDIBLE]? 01:25:49.030 --> 01:25:50.010 AUDIENCE: Yes? 01:25:50.010 --> 01:25:51.330 SE BAEK OH: Yes, [INAUDIBLE]. 01:25:54.610 --> 01:25:55.960 It's frozen. 01:25:55.960 --> 01:26:00.080 AUDIENCE: [INAUDIBLE] 01:26:00.080 --> 01:26:02.330 SE BAEK OH: I think they're going to reset the system. 01:26:04.980 --> 01:26:09.150 So let me just take a few moments. 01:26:09.150 --> 01:26:10.060 Any questions so far? 01:26:27.000 --> 01:26:28.450 [INAUDIBLE] 01:26:28.450 --> 01:26:34.851 AUDIENCE: [INAUDIBLE] 01:26:34.851 --> 01:26:36.540 SE BAEK OH: So actually-- 01:26:36.540 --> 01:26:37.505 yeah? 01:26:37.505 --> 01:26:39.380 AUDIENCE: What happens when you [INAUDIBLE]?? 01:26:41.888 --> 01:26:43.930 SE BAEK OH: Actually, I'm going to talk about it, 01:26:43.930 --> 01:26:48.760 but in general, if you use the incoherent light, then 01:26:48.760 --> 01:26:52.337 you don't see any phase effects because [INAUDIBLE] 01:26:52.337 --> 01:26:53.170 the negative square. 01:26:53.170 --> 01:26:55.673 So it doesn't really matter. 01:26:55.673 --> 01:26:56.590 So you don't see any-- 01:26:56.590 --> 01:26:58.510 I mean, you barely see some variation, 01:26:58.510 --> 01:27:02.573 but usually it's not dramatic as [INAUDIBLE].. 01:27:05.680 --> 01:27:08.697 And what was I about to say? 01:27:15.170 --> 01:27:20.220 Oh, yeah-- so MTF is just magnitude of the OTF. 01:27:20.220 --> 01:27:24.390 So we haven't talk about any phase of the OTF. 01:27:24.390 --> 01:27:26.790 So what's the effect of the-- 01:27:26.790 --> 01:27:31.230 I mean, what does it mean if I have 01:27:31.230 --> 01:27:32.540 some phase variation in OTF? 01:27:35.600 --> 01:27:37.660 Can you guess? 01:27:37.660 --> 01:27:40.630 So for example, OTF is always non-zero. 01:27:40.630 --> 01:27:44.540 It's always 0 or positive. 01:27:44.540 --> 01:27:49.390 But its phase can be 0 or pi. 01:27:49.390 --> 01:27:52.130 So what if I have pinhole phase shift in OTF? 01:28:00.780 --> 01:28:04.950 For example, I have sinusoidal grating, and at the output 01:28:04.950 --> 01:28:06.430 I have a sinusoidal grating. 01:28:06.430 --> 01:28:09.430 But what does it mean I have the pi phase shift? 01:28:27.560 --> 01:28:33.810 AUDIENCE: [INAUDIBLE] 01:28:33.810 --> 01:28:35.160 SE BAEK OH: Not really. 01:28:35.160 --> 01:28:39.840 What happens if you shift by pi-- 01:28:39.840 --> 01:28:43.260 I mean if you move to a cosine by pi? 01:28:43.260 --> 01:28:44.940 You get sine. 01:28:44.940 --> 01:28:47.593 So you have cosine pattern here. 01:28:47.593 --> 01:28:49.260 And if you have pi phase shift, then you 01:28:49.260 --> 01:28:50.790 will have the sine pattern, which 01:28:50.790 --> 01:28:53.640 means you flip the black and white of the image. 01:29:04.720 --> 01:29:07.400 And I flip the [INAUDIBLE]. 01:29:07.400 --> 01:29:09.260 So I flip the-- but still I [INAUDIBLE] 01:29:09.260 --> 01:29:12.430 between the black and white, not the sine/cosine. 01:29:16.760 --> 01:29:21.447 [INAUDIBLE] can anybody see the slide right now, or? 01:29:21.447 --> 01:29:22.030 AUDIENCE: Yes. 01:29:24.760 --> 01:29:26.220 I have a question. 01:29:26.220 --> 01:29:30.771 How do you introduce this pi phase shift in OTF? 01:29:30.771 --> 01:29:33.400 SE BAEK OH: So if you have pi phase shift in OTF, then 01:29:33.400 --> 01:29:36.250 basically you flip the black and white. 01:29:36.250 --> 01:29:40.090 So your input image is black-- 01:29:40.090 --> 01:29:42.280 let's say the black, white, black, white. 01:29:42.280 --> 01:29:45.450 And your image is going to be white, black, white, black. 01:29:45.450 --> 01:29:50.470 AUDIENCE: No, my question is physically, 01:29:50.470 --> 01:29:54.400 how do you introduce that controllable phase shift 01:29:54.400 --> 01:29:55.030 in OTF? 01:29:57.950 --> 01:30:02.020 SE BAEK OH: So this ATF can be complex. 01:30:02.020 --> 01:30:06.100 So if you compute the auto-correlation, 01:30:06.100 --> 01:30:10.240 then actually, you can have negative value of OTF. 01:30:13.090 --> 01:30:14.860 And if you have the negative value 01:30:14.860 --> 01:30:18.200 in OTF, which means you have still positive and negative, 01:30:18.200 --> 01:30:20.356 but you have some negative-- 01:30:20.356 --> 01:30:23.680 I mean, basically pi phase. 01:30:23.680 --> 01:30:25.660 So if you want to do that kind of thing, 01:30:25.660 --> 01:30:28.150 basically, you have some complex transparency 01:30:28.150 --> 01:30:30.720 at your [INAUDIBLE] mask. 01:30:30.720 --> 01:30:34.690 But if you have clear aperture, then you don't get that effect. 01:30:44.970 --> 01:30:48.440 So let me continue this slide. 01:30:48.440 --> 01:30:53.450 I mean, these three images we already seen maybe two or three 01:30:53.450 --> 01:30:54.520 lectures ago. 01:30:54.520 --> 01:30:56.000 So this is MIT lecture-- 01:30:56.000 --> 01:31:00.590 I mean the image of MIT letters with the coherent emission. 01:31:00.590 --> 01:31:06.980 And the right two images obtained 01:31:06.980 --> 01:31:08.930 with the incoherent emission. 01:31:08.930 --> 01:31:13.740 So I can notice two main differences here. 01:31:13.740 --> 01:31:17.090 So if I use the [INAUDIBLE] emission, 01:31:17.090 --> 01:31:19.310 basically, the image looks sharper 01:31:19.310 --> 01:31:22.130 than the one with the coherent. 01:31:22.130 --> 01:31:26.330 And that's because OTF has a higher-- 01:31:26.330 --> 01:31:29.270 twice of higher color frequency even 01:31:29.270 --> 01:31:30.590 though they get attenuated. 01:31:30.590 --> 01:31:33.450 But basically, it can pass through the higher spatial 01:31:33.450 --> 01:31:38.630 frequency, so that's why it preserves more 01:31:38.630 --> 01:31:40.240 higher frequency-- 01:31:40.240 --> 01:31:43.865 preserves higher frequency. 01:31:43.865 --> 01:31:45.090 And it's same thing here. 01:31:45.090 --> 01:31:50.090 So with the coherent light, you barely see the letter "I", 01:31:50.090 --> 01:31:53.450 but you can still see the MIT. 01:31:53.450 --> 01:31:57.800 And another difference is actually if you see "M" or "T", 01:31:57.800 --> 01:32:01.070 there are some variations, like black or white. 01:32:01.070 --> 01:32:03.620 Some wiggling in MIT here. 01:32:03.620 --> 01:32:08.480 But in coherent case, the intensity is more like uniform. 01:32:08.480 --> 01:32:11.650 You don't see some weird variation. 01:32:11.650 --> 01:32:14.430 So that's another difference. 01:32:21.260 --> 01:32:23.690 So here's the incoherent light. 01:32:23.690 --> 01:32:26.640 So generally, you don't have the ringing 01:32:26.640 --> 01:32:30.840 artifact as I described, and also no [INAUDIBLE].. 01:32:30.840 --> 01:32:35.340 So you're not going to see any interference. 01:32:35.340 --> 01:32:38.940 And it is higher bandwidth because the color frequency 01:32:38.940 --> 01:32:43.480 is twice higher than the color frequency in ATF. 01:32:43.480 --> 01:32:48.060 So you can see the smaller features-- 01:32:48.060 --> 01:32:54.020 I mean the two points closer. 01:32:54.020 --> 01:32:55.650 You can see the smaller feature. 01:32:58.590 --> 01:33:04.060 But as I answered to [INAUDIBLE] question, 01:33:04.060 --> 01:33:07.690 incoherent image is insensitive to phase variation. 01:33:11.280 --> 01:33:14.610 So mostly, we just deal with the monochromatic, and especially 01:33:14.610 --> 01:33:15.330 incoherent. 01:33:15.330 --> 01:33:18.340 But if you have polychromatic, which is broadband, 01:33:18.340 --> 01:33:22.110 then you're going to have some chromatic aberrations. 01:33:22.110 --> 01:33:26.490 But again, you can define all these OTF, or MTF, 01:33:26.490 --> 01:33:29.410 or point spread functions for each different wavelength. 01:33:29.410 --> 01:33:32.730 So it just extends this in a straightforward fashion. 01:33:38.120 --> 01:33:44.900 It's the end of show, so any questions? 01:33:44.900 --> 01:33:47.450 So the take-home message of this [INAUDIBLE] lecture 01:33:47.450 --> 01:33:51.450 is if you have coherent illumination, 01:33:51.450 --> 01:33:55.510 then you have linear convolution in field. 01:33:55.510 --> 01:33:58.050 But if you have incoherent case, and you 01:33:58.050 --> 01:34:01.710 have the linear relation in intensity, 01:34:01.710 --> 01:34:03.720 and for incoherent light, you have 01:34:03.720 --> 01:34:07.400 to deal with the OTF, which is auto-correlation of ATF. 01:34:07.400 --> 01:34:09.865 And that's pretty much about it. 01:34:13.260 --> 01:34:19.440 If you don't have questions, then let's see next Monday.