WEBVTT 00:00:00.000 --> 00:00:02.976 align:middle line:90% [SQUEAKING] 00:00:02.976 --> 00:00:04.464 align:middle line:90% [RUSTLING] 00:00:04.464 --> 00:00:05.456 align:middle line:90% [CLICKING] 00:00:05.456 --> 00:00:12.478 align:middle line:90% 00:00:12.478 --> 00:00:14.770 align:middle line:84% JACK HARE: OK, so you'll remember, in the last lecture, 00:00:14.770 --> 00:00:18.707 align:middle line:84% we discussed the technique called schlieren. 00:00:18.707 --> 00:00:21.330 align:middle line:90% 00:00:21.330 --> 00:00:23.430 align:middle line:84% And schlieren, we were relying on the fact 00:00:23.430 --> 00:00:30.210 align:middle line:84% that the plasma has refractive index gradients inside it which 00:00:30.210 --> 00:00:33.660 align:middle line:84% cause a deflection, and that deflection angle 00:00:33.660 --> 00:00:36.300 align:middle line:84% of the rays going through the plasma 00:00:36.300 --> 00:00:37.980 align:middle line:84% is going to be something like 1 over 2 00:00:37.980 --> 00:00:42.390 align:middle line:84% times the critical density times the integral of the gradient 00:00:42.390 --> 00:00:46.740 align:middle line:84% of the electron density along the path of r probing b. 00:00:46.740 --> 00:00:48.840 align:middle line:84% And what we did with schlieren is 00:00:48.840 --> 00:00:52.860 align:middle line:84% we filtered the rays using a lens 00:00:52.860 --> 00:00:55.290 align:middle line:84% and a stop at the focal plane, and we 00:00:55.290 --> 00:00:57.420 align:middle line:84% we're able to filter out rays with certain angles. 00:00:57.420 --> 00:00:59.010 align:middle line:84% We could let through undeflected rays. 00:00:59.010 --> 00:01:00.630 align:middle line:90% We could block undeflected rays. 00:01:00.630 --> 00:01:02.375 align:middle line:84% We could do it in the isotropic fashion. 00:01:02.375 --> 00:01:03.750 align:middle line:84% We could do it with a knife edge. 00:01:03.750 --> 00:01:07.410 align:middle line:84% And this enabled us to image effectively the gradients 00:01:07.410 --> 00:01:10.292 align:middle line:84% in the electron density inside our plasma, 00:01:10.292 --> 00:01:12.000 align:middle line:84% and this is particularly useful if you've 00:01:12.000 --> 00:01:15.570 align:middle line:84% got something with sharp density gradients such as shocks. 00:01:15.570 --> 00:01:19.770 align:middle line:90% 00:01:19.770 --> 00:01:21.810 align:middle line:84% But when we were doing this, we actually made 00:01:21.810 --> 00:01:24.270 align:middle line:84% a pretty big assumption that I kind of buried here. 00:01:24.270 --> 00:01:32.170 align:middle line:84% We assumed that we have no displacement of the ray, 00:01:32.170 --> 00:01:34.930 align:middle line:84% that the rays pick up some angle as they 00:01:34.930 --> 00:01:36.490 align:middle line:90% travel through our plasma. 00:01:36.490 --> 00:01:40.000 align:middle line:84% So I'll draw a little plasma here. 00:01:40.000 --> 00:01:43.670 align:middle line:84% We assume that the rays picked up some sort of angle 00:01:43.670 --> 00:01:45.580 align:middle line:84% but they didn't themselves get displaced. 00:01:45.580 --> 00:01:48.910 align:middle line:84% Whereas, for some realistic extended plasma, 00:01:48.910 --> 00:01:51.440 align:middle line:84% they would be displaced inside the plasma 00:01:51.440 --> 00:01:53.270 align:middle line:84% as well as picking up some angle. 00:01:53.270 --> 00:01:56.493 align:middle line:84% And so the rays themselves would not only have an angle, 00:01:56.493 --> 00:01:57.910 align:middle line:84% they would have some displacement, 00:01:57.910 --> 00:02:00.730 align:middle line:84% and when we use our imaging system with a lens, 00:02:00.730 --> 00:02:04.750 align:middle line:84% the rays should end up somewhere else on our object plane, 00:02:04.750 --> 00:02:07.000 align:middle line:84% not in the same place that they started with. 00:02:07.000 --> 00:02:11.050 align:middle line:84% For schlieren, we assumed that there was no ray displacement. 00:02:11.050 --> 00:02:16.920 align:middle line:90% 00:02:16.920 --> 00:02:20.990 align:middle line:84% And this is actually equivalent to assuming that our object was 00:02:20.990 --> 00:02:21.930 align:middle line:90% very, very thin. 00:02:21.930 --> 00:02:24.330 align:middle line:84% So if instead of having this extended object like this, 00:02:24.330 --> 00:02:27.770 align:middle line:84% we have an object that's incredibly thin like that, 00:02:27.770 --> 00:02:30.980 align:middle line:84% and you have rays of light coming in that then just 00:02:30.980 --> 00:02:35.630 align:middle line:84% get instantaneously deflected like that-- this works out OK 00:02:35.630 --> 00:02:39.230 align:middle line:84% because there's no sort of path along inside the plasma 00:02:39.230 --> 00:02:41.720 align:middle line:84% for these rays to become displaced. 00:02:41.720 --> 00:02:43.560 align:middle line:84% Whereas, they are in an extended object. 00:02:43.560 --> 00:02:45.290 align:middle line:84% So we made this assumption, but we're now 00:02:45.290 --> 00:02:49.010 align:middle line:84% going to relax that and have a look at what happens when 00:02:49.010 --> 00:02:51.980 align:middle line:84% we do true shadowgraphy, which is where we're interested 00:02:51.980 --> 00:02:54.560 align:middle line:84% not just in the angles of the rays but their displacements 00:02:54.560 --> 00:02:55.530 align:middle line:90% as well. 00:02:55.530 --> 00:02:57.830 align:middle line:84% So I'm going to try and draw a slightly better version 00:02:57.830 --> 00:03:00.870 align:middle line:84% of this image, hopefully with some nice, straight lines. 00:03:00.870 --> 00:03:02.220 align:middle line:90% Let's see how it goes. 00:03:02.220 --> 00:03:05.000 align:middle line:90% Here's my plasma again. 00:03:05.000 --> 00:03:10.620 align:middle line:90% I'm going to have a lens here. 00:03:10.620 --> 00:03:13.440 align:middle line:84% I'm going to set up my lens such that it brings 00:03:13.440 --> 00:03:24.050 align:middle line:84% into focus rays from this object plane onto some image plane 00:03:24.050 --> 00:03:24.770 align:middle line:90% back here. 00:03:24.770 --> 00:03:32.540 align:middle line:90% 00:03:32.540 --> 00:03:35.680 align:middle line:84% And what I'm going to have is the undeflected rays. 00:03:35.680 --> 00:03:37.810 align:middle line:84% So imagine there was no plasma here. 00:03:37.810 --> 00:03:39.310 align:middle line:84% The rays would just go through here. 00:03:39.310 --> 00:03:43.290 align:middle line:90% 00:03:43.290 --> 00:03:45.870 align:middle line:84% And they would all go through a focal point back here. 00:03:45.870 --> 00:03:49.640 align:middle line:90% 00:03:49.640 --> 00:03:52.140 align:middle line:84% So this again looks like the setup we had for our schlieren, 00:03:52.140 --> 00:03:54.450 align:middle line:84% but the difference is now if we consider the rays being 00:03:54.450 --> 00:03:56.670 align:middle line:84% deflected inside the plasma, they're 00:03:56.670 --> 00:04:01.335 align:middle line:84% going to pick up some sort of displacement. 00:04:01.335 --> 00:04:02.710 align:middle line:84% So for example, this ray is going 00:04:02.710 --> 00:04:04.795 align:middle line:90% to be displaced down like this. 00:04:04.795 --> 00:04:07.420 align:middle line:84% This ray maybe is also going to be displaced down a little bit, 00:04:07.420 --> 00:04:10.690 align:middle line:84% and this ray is going to be displaced upwards such 00:04:10.690 --> 00:04:12.430 align:middle line:84% that when they get to the object plane, 00:04:12.430 --> 00:04:15.130 align:middle line:84% they now have some distance that they 00:04:15.130 --> 00:04:20.589 align:middle line:84% have been translated compared to the undeflected one. 00:04:20.589 --> 00:04:22.780 align:middle line:84% And then, all that the lens does is 00:04:22.780 --> 00:04:24.910 align:middle line:84% it puts light back on the image plane 00:04:24.910 --> 00:04:26.920 align:middle line:84% where it came from on the object plane, 00:04:26.920 --> 00:04:28.910 align:middle line:84% with some inversion, other things like that. 00:04:28.910 --> 00:04:30.550 align:middle line:84% But that means because these rays have actually 00:04:30.550 --> 00:04:32.740 align:middle line:84% been moved, because the light has been moved around, 00:04:32.740 --> 00:04:34.930 align:middle line:84% they're going to show up in different places on the image 00:04:34.930 --> 00:04:35.140 align:middle line:90% plane. 00:04:35.140 --> 00:04:37.150 align:middle line:84% And this is where I really hope I can draw this properly, 00:04:37.150 --> 00:04:39.550 align:middle line:84% because this is quite a complicated diagram, so give me 00:04:39.550 --> 00:04:42.860 align:middle line:90% a moment to focus on it. 00:04:42.860 --> 00:04:45.080 align:middle line:84% So let's start with this ray here. 00:04:45.080 --> 00:04:46.660 align:middle line:90% It's going to go down. 00:04:46.660 --> 00:04:48.640 align:middle line:84% It's going to go down below the focal point 00:04:48.640 --> 00:04:50.720 align:middle line:84% because it's being deflected downwards. 00:04:50.720 --> 00:04:57.200 align:middle line:84% And so it's going to end up here, so slightly above 00:04:57.200 --> 00:04:59.990 align:middle line:84% where the original ray was going to go. 00:04:59.990 --> 00:05:03.860 align:middle line:84% This ray here is going to get deflected downwards, 00:05:03.860 --> 00:05:06.410 align:middle line:84% and then the lens is going to push it up. 00:05:06.410 --> 00:05:11.210 align:middle line:84% And it's also going to go below the focal point 00:05:11.210 --> 00:05:13.790 align:middle line:84% here as well, so it's going to be deflected up 00:05:13.790 --> 00:05:15.140 align:middle line:90% from its original position. 00:05:15.140 --> 00:05:17.120 align:middle line:84% And this one here is going to go up. 00:05:17.120 --> 00:05:21.640 align:middle line:90% 00:05:21.640 --> 00:05:23.140 align:middle line:84% I'm going to bend that one slightly. 00:05:23.140 --> 00:05:24.890 align:middle line:84% Oh, no, I didn't want it to go over there. 00:05:24.890 --> 00:05:28.090 align:middle line:84% Ah, I got it right the first time. 00:05:28.090 --> 00:05:28.590 align:middle line:90% There we go. 00:05:28.590 --> 00:05:30.690 align:middle line:84% And this one is deflected downwards 00:05:30.690 --> 00:05:32.890 align:middle line:84% because it was deflected off of the object here. 00:05:32.890 --> 00:05:36.780 align:middle line:84% So what we're seeing is that this lens is now 00:05:36.780 --> 00:05:39.390 align:middle line:84% imaging the light into slightly different places 00:05:39.390 --> 00:05:42.435 align:middle line:84% than we had before, so we have the displacement of the rays. 00:05:42.435 --> 00:05:48.590 align:middle line:90% 00:05:48.590 --> 00:05:50.590 align:middle line:84% And you might want to think of that displacement 00:05:50.590 --> 00:05:54.110 align:middle line:84% as an intensity modulation, because, of course, 00:05:54.110 --> 00:05:55.360 align:middle line:90% there aren't just three rays. 00:05:55.360 --> 00:05:58.250 align:middle line:84% There are a large number of rays, 00:05:58.250 --> 00:06:00.500 align:middle line:90% and these all make up an image. 00:06:00.500 --> 00:06:04.120 align:middle line:84% And so initially, maybe you had a nice uniform pattern. 00:06:04.120 --> 00:06:08.050 align:middle line:84% These yellow rays would have made a nice uniform 00:06:08.050 --> 00:06:11.060 align:middle line:90% circular beam like this. 00:06:11.060 --> 00:06:15.910 align:middle line:84% And now your blue rays are going to be distorted, 00:06:15.910 --> 00:06:19.000 align:middle line:84% and there's going to be regions of very bright intensity 00:06:19.000 --> 00:06:21.400 align:middle line:84% and regions of lesser intensity and regions 00:06:21.400 --> 00:06:23.360 align:middle line:90% that are about the same. 00:06:23.360 --> 00:06:26.140 align:middle line:84% So we'll have all sorts of interesting features 00:06:26.140 --> 00:06:28.090 align:middle line:90% inside this. 00:06:28.090 --> 00:06:32.230 align:middle line:84% And one way to think about this, imagine we took our beam-- 00:06:32.230 --> 00:06:33.970 align:middle line:84% and now we're looking at the laser beam. 00:06:33.970 --> 00:06:36.250 align:middle line:84% It's coming directly towards us through the plasma. 00:06:36.250 --> 00:06:39.400 align:middle line:84% Imagine we arranged it so we have nine little dots. 00:06:39.400 --> 00:06:42.700 align:middle line:84% Maybe we've got nine little laser pointers pointing 00:06:42.700 --> 00:06:46.790 align:middle line:90% through our plasma like this. 00:06:46.790 --> 00:06:49.490 align:middle line:84% Now, as these rays come through the plasma 00:06:49.490 --> 00:06:51.920 align:middle line:84% and they're deflected onto our camera here, 00:06:51.920 --> 00:06:54.330 align:middle line:84% we would end up with the dots in different places. 00:06:54.330 --> 00:06:57.480 align:middle line:84% So we could have this one is moved up here, 00:06:57.480 --> 00:07:01.740 align:middle line:84% this one is moved like this, this one is moved like this, 00:07:01.740 --> 00:07:02.240 align:middle line:90% and so on. 00:07:02.240 --> 00:07:04.910 align:middle line:84% And you can just think there's a series of displacements 00:07:04.910 --> 00:07:07.910 align:middle line:84% for these dots, and if you're able to measure 00:07:07.910 --> 00:07:10.250 align:middle line:84% all these displacements, perhaps you 00:07:10.250 --> 00:07:12.778 align:middle line:84% could work out something about the plasma itself. 00:07:12.778 --> 00:07:14.570 align:middle line:84% And we'll talk a little bit about some more 00:07:14.570 --> 00:07:16.490 align:middle line:84% quantitative measurements for doing this, 00:07:16.490 --> 00:07:19.110 align:middle line:84% but this is just qualitatively the picture here. 00:07:19.110 --> 00:07:21.183 align:middle line:84% Another way to think about this, if you prefer-- 00:07:21.183 --> 00:07:22.850 align:middle line:84% these are just different conceptual ways 00:07:22.850 --> 00:07:24.558 align:middle line:84% of trying to think about the same thing-- 00:07:24.558 --> 00:07:28.400 align:middle line:84% imagine we initially put through this sort of tic-tac-toe grid 00:07:28.400 --> 00:07:29.180 align:middle line:90% of light here. 00:07:29.180 --> 00:07:30.770 align:middle line:84% We've sort of blocked it off so we've 00:07:30.770 --> 00:07:33.510 align:middle line:84% got two vertical lines, two horizontal lines. 00:07:33.510 --> 00:07:35.960 align:middle line:84% These lines, as they go through the plasma, 00:07:35.960 --> 00:07:38.270 align:middle line:84% the light will be deflected by different amounts. 00:07:38.270 --> 00:07:41.960 align:middle line:84% And so for example, this line here could be deflected like. 00:07:41.960 --> 00:07:45.030 align:middle line:84% This one could be deflected like this. 00:07:45.030 --> 00:07:48.360 align:middle line:84% And so you sort of see we have some sort of distortion 00:07:48.360 --> 00:07:50.980 align:middle line:84% to the grid that we're getting out of this, 00:07:50.980 --> 00:07:53.550 align:middle line:84% and our initial grid with all these nice, straight, parallel 00:07:53.550 --> 00:07:56.590 align:middle line:84% lines is now a distorted grid instead. 00:07:56.590 --> 00:07:59.070 align:middle line:84% And that's very important because shadowgraphy does not 00:07:59.070 --> 00:08:00.750 align:middle line:90% actually produce an image. 00:08:00.750 --> 00:08:02.400 align:middle line:84% I've called this the image plane here, 00:08:02.400 --> 00:08:05.010 align:middle line:84% but we don't actually get an image out, 00:08:05.010 --> 00:08:07.995 align:middle line:84% because we have something which is significantly distorted 00:08:07.995 --> 00:08:09.870 align:middle line:84% and it no longer preserves the things that we 00:08:09.870 --> 00:08:10.920 align:middle line:90% like to preserve in images. 00:08:10.920 --> 00:08:12.510 align:middle line:84% We take a picture with a camera, we'd 00:08:12.510 --> 00:08:14.250 align:middle line:84% like straight lines to remain straight. 00:08:14.250 --> 00:08:16.395 align:middle line:84% Here, the straight lines do not remain straight. 00:08:16.395 --> 00:08:17.520 align:middle line:90% They don't remain parallel. 00:08:17.520 --> 00:08:19.680 align:middle line:84% We don't preserve length, anything like that. 00:08:19.680 --> 00:08:22.020 align:middle line:84% But there's still clearly information 00:08:22.020 --> 00:08:25.740 align:middle line:84% stored inside these images, pictures, 00:08:25.740 --> 00:08:27.930 align:middle line:84% that correspond to the density inside the plasma, 00:08:27.930 --> 00:08:30.090 align:middle line:84% and we'll talk a little bit more about how 00:08:30.090 --> 00:08:31.810 align:middle line:90% to do this quantitatively. 00:08:31.810 --> 00:08:33.510 align:middle line:90% So any questions on this? 00:08:33.510 --> 00:08:34.770 align:middle line:90% Yeah? 00:08:34.770 --> 00:08:37.320 align:middle line:84% AUDIENCE: So if the plasma here is acting like a lens, 00:08:37.320 --> 00:08:38.820 align:middle line:84% what do we need the second lens for? 00:08:38.820 --> 00:08:39.653 align:middle line:90% Just as a reference? 00:08:39.653 --> 00:08:41.549 align:middle line:84% JACK HARE: Absolutely, yeah, great question. 00:08:41.549 --> 00:08:45.940 align:middle line:84% Yeah, so I put the second lens in here for a couple of reasons. 00:08:45.940 --> 00:08:48.810 align:middle line:84% First of all, I find it conceptually simpler 00:08:48.810 --> 00:08:50.238 align:middle line:84% because you can see the deflection 00:08:50.238 --> 00:08:52.530 align:middle line:84% angles at the focal point here, because when I read off 00:08:52.530 --> 00:08:54.405 align:middle line:84% the position of the rays at this focal plane, 00:08:54.405 --> 00:08:56.910 align:middle line:84% I can tell whether they've been deflected up or down, which 00:08:56.910 --> 00:08:58.590 align:middle line:90% is slightly harder to do here. 00:08:58.590 --> 00:09:01.110 align:middle line:84% You could absolutely put your detector 00:09:01.110 --> 00:09:04.150 align:middle line:84% right here, OK, so you could put a bit of film or something 00:09:04.150 --> 00:09:04.650 align:middle line:90% out here. 00:09:04.650 --> 00:09:07.233 align:middle line:84% In practice, you can't, because it's right next to the plasma. 00:09:07.233 --> 00:09:09.470 align:middle line:84% So you need some sort of lens optic. 00:09:09.470 --> 00:09:11.220 align:middle line:84% This technique actually looks a great deal 00:09:11.220 --> 00:09:14.190 align:middle line:84% like proton radiography, which we'll talk about later on here. 00:09:14.190 --> 00:09:16.320 align:middle line:84% In proton radiography, you don't have any lenses. 00:09:16.320 --> 00:09:19.350 align:middle line:84% You can't-- it's very difficult to make lenses for charged 00:09:19.350 --> 00:09:20.280 align:middle line:90% particles. 00:09:20.280 --> 00:09:23.070 align:middle line:84% And so you do, in fact, put your bit of paper, 00:09:23.070 --> 00:09:25.748 align:middle line:84% like your film in this case, very close to your plasma. 00:09:25.748 --> 00:09:27.540 align:middle line:84% You might put it a little bit further back, 00:09:27.540 --> 00:09:30.660 align:middle line:84% and we'll talk in a moment about actually how important is 00:09:30.660 --> 00:09:34.020 align:middle line:84% your choice of exactly where you put your detector, because I 00:09:34.020 --> 00:09:36.750 align:middle line:84% could put my detector here or here or here, 00:09:36.750 --> 00:09:38.650 align:middle line:84% and I will still get an image forming, 00:09:38.650 --> 00:09:40.260 align:middle line:84% but the image will look different. 00:09:40.260 --> 00:09:42.760 align:middle line:84% But yeah, we do not need the lens to do shadowgraphy, 00:09:42.760 --> 00:09:44.710 align:middle line:84% but in almost all realistic setups 00:09:44.710 --> 00:09:47.080 align:middle line:84% where we're using some probing beam through a plasma, 00:09:47.080 --> 00:09:48.700 align:middle line:84% we're going to have a lens that allows 00:09:48.700 --> 00:09:50.800 align:middle line:84% us to put our detector nice and far away 00:09:50.800 --> 00:09:52.240 align:middle line:90% and safe from the plasma. 00:09:52.240 --> 00:09:52.780 align:middle line:90% So, yeah. 00:09:52.780 --> 00:09:52.930 align:middle line:90% AUDIENCE: All right. 00:09:52.930 --> 00:09:53.430 align:middle line:90% Thank you. 00:09:53.430 --> 00:09:54.385 align:middle line:90% JACK HARE: Cool. 00:09:54.385 --> 00:09:55.240 align:middle line:90% Other questions? 00:09:55.240 --> 00:09:58.200 align:middle line:90% 00:09:58.200 --> 00:09:59.075 align:middle line:90% Anyone from Columbia? 00:09:59.075 --> 00:10:04.330 align:middle line:90% 00:10:04.330 --> 00:10:06.960 align:middle line:84% I don't see anyone, so I'm going to keep going. 00:10:06.960 --> 00:10:08.940 align:middle line:90% Cool. 00:10:08.940 --> 00:10:10.875 align:middle line:90% So let me erase this side. 00:10:10.875 --> 00:10:21.900 align:middle line:90% 00:10:21.900 --> 00:10:24.240 align:middle line:84% So as I just mentioned, then, we can actually 00:10:24.240 --> 00:10:26.040 align:middle line:84% change the shadowgraphic image we 00:10:26.040 --> 00:10:28.950 align:middle line:84% get by moving the position of our object plane. 00:10:28.950 --> 00:10:31.008 align:middle line:84% So that could be moving our detector, 00:10:31.008 --> 00:10:32.550 align:middle line:84% if we've just placed a detector here, 00:10:32.550 --> 00:10:34.140 align:middle line:84% or it could be by moving our lens, 00:10:34.140 --> 00:10:35.973 align:middle line:84% because as we move the lens, we change 00:10:35.973 --> 00:10:37.140 align:middle line:90% the place we're focusing on. 00:10:37.140 --> 00:10:40.270 align:middle line:84% Or we could move other-- anyway, you get the idea. 00:10:40.270 --> 00:10:42.480 align:middle line:84% There's lots of different ways of moving this 00:10:42.480 --> 00:10:43.690 align:middle line:90% to some other place. 00:10:43.690 --> 00:10:45.130 align:middle line:90% So let's have a look here. 00:10:45.130 --> 00:10:46.635 align:middle line:90% We've got, again, our plasma. 00:10:46.635 --> 00:10:49.940 align:middle line:90% 00:10:49.940 --> 00:10:51.920 align:middle line:84% And we got our rays coming through it. 00:10:51.920 --> 00:10:57.470 align:middle line:90% 00:10:57.470 --> 00:10:59.660 align:middle line:84% We've got a ray that gets deflected downwards 00:10:59.660 --> 00:11:02.000 align:middle line:84% and a ray that gets deflected upwards 00:11:02.000 --> 00:11:05.880 align:middle line:84% and another one that gets deflected down like this. 00:11:05.880 --> 00:11:09.200 align:middle line:84% I can draw some different planes in which 00:11:09.200 --> 00:11:10.762 align:middle line:90% we can put our detector. 00:11:10.762 --> 00:11:12.470 align:middle line:84% Let me just get them in the right places. 00:11:12.470 --> 00:11:18.890 align:middle line:90% 00:11:18.890 --> 00:11:25.963 align:middle line:84% Or maybe I can call these 1, 2, 3, and 4, like that. 00:11:25.963 --> 00:11:27.380 align:middle line:84% And so then we can have a look at, 00:11:27.380 --> 00:11:30.320 align:middle line:84% What would we see for this very simple system where we've 00:11:30.320 --> 00:11:32.810 align:middle line:84% got some sort of focusing density gradients 00:11:32.810 --> 00:11:35.520 align:middle line:84% and some defocusing density gradients here? 00:11:35.520 --> 00:11:40.085 align:middle line:84% So this would be a maxima in electron density, 00:11:40.085 --> 00:11:43.010 align:middle line:84% and this would be a minima here, right, 00:11:43.010 --> 00:11:45.830 align:middle line:84% so that we would get focusing and defocusing. 00:11:45.830 --> 00:11:49.290 align:middle line:84% So let's draw what those patterns would look like. 00:11:49.290 --> 00:11:53.860 align:middle line:84% So if I have intensity like this-- 00:11:53.860 --> 00:11:56.990 align:middle line:84% and let's say that this is the y-coordinate, 00:11:56.990 --> 00:12:03.720 align:middle line:84% so I'll call this y here, and I'll do the first one here-- 00:12:03.720 --> 00:12:05.880 align:middle line:84% we'll get something-- we're going 00:12:05.880 --> 00:12:09.695 align:middle line:84% to have a little bit of deficit of intensity around about here 00:12:09.695 --> 00:12:11.320 align:middle line:84% because the ray's being deflected away. 00:12:11.320 --> 00:12:14.280 align:middle line:84% So if this is the background intensity of our probing beam, 00:12:14.280 --> 00:12:16.620 align:middle line:84% we'll have some sort of drop, and then we'll 00:12:16.620 --> 00:12:20.700 align:middle line:84% have an increase because the light is being focused together 00:12:20.700 --> 00:12:22.140 align:middle line:90% in the middle here. 00:12:22.140 --> 00:12:25.800 align:middle line:84% We'll have another deficit in this region, 00:12:25.800 --> 00:12:28.630 align:middle line:84% and we'll have another little increase over there. 00:12:28.630 --> 00:12:31.190 align:middle line:84% So we get some sort of nice little modulation, 00:12:31.190 --> 00:12:32.490 align:middle line:90% what it's looking like. 00:12:32.490 --> 00:12:35.160 align:middle line:84% If we go to 2, we can see that the rays are now 00:12:35.160 --> 00:12:37.210 align:middle line:84% getting closer together and further apart, 00:12:37.210 --> 00:12:41.900 align:middle line:84% so you'd expect for this pattern to be even more exaggerated. 00:12:41.900 --> 00:12:44.400 align:middle line:84% Maybe I should've drawn this less exaggerated to give myself 00:12:44.400 --> 00:12:44.900 align:middle line:90% more space. 00:12:44.900 --> 00:12:52.390 align:middle line:84% But let's say it looks like this, and this is bigger. 00:12:52.390 --> 00:12:55.850 align:middle line:84% If we get over to 3 now, we see something interesting. 00:12:55.850 --> 00:12:57.405 align:middle line:84% We actually have rays crossing here, 00:12:57.405 --> 00:12:58.780 align:middle line:84% so this means that a lot of light 00:12:58.780 --> 00:13:01.127 align:middle line:84% is all being piled into the same place. 00:13:01.127 --> 00:13:03.460 align:middle line:84% So we're still going to have a little bit of defocusing, 00:13:03.460 --> 00:13:07.270 align:middle line:84% but we're going to have a very sharp spike here, which goes off 00:13:07.270 --> 00:13:08.380 align:middle line:90% my page. 00:13:08.380 --> 00:13:11.860 align:middle line:84% And then we don't really have so much precise focusing down here, 00:13:11.860 --> 00:13:15.320 align:middle line:84% but maybe there's a little bit of light, so like that. 00:13:15.320 --> 00:13:20.900 align:middle line:84% And then for number 4, we've got away 00:13:20.900 --> 00:13:24.260 align:middle line:84% from having this crossing here, so we won't see it so strongly. 00:13:24.260 --> 00:13:27.320 align:middle line:84% It's going to look a little bit more like this, 00:13:27.320 --> 00:13:33.010 align:middle line:84% but something maybe a little bit like that. 00:13:33.010 --> 00:13:35.700 align:middle line:84% Now, one thing I haven't really exaggerated enough here 00:13:35.700 --> 00:13:38.130 align:middle line:84% is the fact that as these rays move outwards, 00:13:38.130 --> 00:13:41.380 align:middle line:84% the positions of these maxima and minima are changing. 00:13:41.380 --> 00:13:46.500 align:middle line:84% So in reality, maybe this one is a little bit closer in, 00:13:46.500 --> 00:13:49.710 align:middle line:84% and then I'll make these ones go further out. 00:13:49.710 --> 00:13:54.390 align:middle line:84% So you can see that actually the position of where 00:13:54.390 --> 00:13:57.777 align:middle line:84% we get peak intensity is going to change depending on how far 00:13:57.777 --> 00:14:00.360 align:middle line:84% away we are from the detector, because the rays have traveled, 00:14:00.360 --> 00:14:04.120 align:middle line:84% and so their displacement has changed as well. 00:14:04.120 --> 00:14:05.940 align:middle line:84% So there's a lot going on inside here. 00:14:05.940 --> 00:14:08.280 align:middle line:84% You can see that there's a lot of richness, 00:14:08.280 --> 00:14:11.370 align:middle line:84% and you can see that although we can identify why there 00:14:11.370 --> 00:14:14.550 align:middle line:84% are light and dark regions inside our image, 00:14:14.550 --> 00:14:17.790 align:middle line:84% it's hard to map them directly back onto whatever's 00:14:17.790 --> 00:14:19.500 align:middle line:90% going on inside the plasma. 00:14:19.500 --> 00:14:24.060 align:middle line:84% And again, that's because, as I said, this is not an image. 00:14:24.060 --> 00:14:26.400 align:middle line:84% You can tell it's not an image because if you 00:14:26.400 --> 00:14:28.680 align:middle line:84% move your detector or if you move 00:14:28.680 --> 00:14:30.630 align:middle line:84% where you're taking the data, you'd 00:14:30.630 --> 00:14:32.560 align:middle line:84% get a very different picture out here. 00:14:32.560 --> 00:14:37.530 align:middle line:84% So although this maxima here corresponds 00:14:37.530 --> 00:14:42.060 align:middle line:84% to this region of the plasma, it shows up 00:14:42.060 --> 00:14:44.950 align:middle line:84% at different positions on your detector at different places. 00:14:44.950 --> 00:14:47.670 align:middle line:84% So you can't really say this is a direct image. 00:14:47.670 --> 00:14:52.690 align:middle line:84% It is bijective, so this is means 00:14:52.690 --> 00:14:57.980 align:middle line:84% that at least for these smaller deflections, we have 00:14:57.980 --> 00:14:59.360 align:middle line:90% what's a one-to-one mapping. 00:14:59.360 --> 00:15:03.990 align:middle line:90% 00:15:03.990 --> 00:15:09.210 align:middle line:84% That's certainly true for the small deflections, 1 and 2, 00:15:09.210 --> 00:15:12.120 align:middle line:84% like that, for these two slices here. 00:15:12.120 --> 00:15:15.060 align:middle line:84% Once we get to 3 and we have the rays crossing, 00:15:15.060 --> 00:15:17.880 align:middle line:84% you can see that at this point, the light which 00:15:17.880 --> 00:15:23.730 align:middle line:84% is coming from these two regions now maps into the same place 00:15:23.730 --> 00:15:27.040 align:middle line:84% on our image-- or on our detector, 00:15:27.040 --> 00:15:29.230 align:middle line:84% and so we no longer have that one-to-one mapping. 00:15:29.230 --> 00:15:33.130 align:middle line:84% And indeed, after the rays have crossed, these two cross over, 00:15:33.130 --> 00:15:34.980 align:middle line:84% and so it's very confusing looking at this 00:15:34.980 --> 00:15:38.560 align:middle line:84% and trying to work out where all the light has come from. 00:15:38.560 --> 00:15:42.520 align:middle line:84% Although we do have this one-to-one map for 1 and 2-- 00:15:42.520 --> 00:15:45.335 align:middle line:84% so in principle, we can work out where everything 00:15:45.335 --> 00:15:47.710 align:middle line:84% came from-- it doesn't have properties that we'd normally 00:15:47.710 --> 00:15:49.310 align:middle line:90% like to have from an image. 00:15:49.310 --> 00:15:52.120 align:middle line:84% So for example, parallel lines do not 00:15:52.120 --> 00:15:56.590 align:middle line:84% go to parallel lines from the object plane to the image plane, 00:15:56.590 --> 00:15:58.583 align:middle line:84% as we sort of discussed over here, 00:15:58.583 --> 00:16:01.000 align:middle line:84% and so you might want that in a normal imaging diagnostic. 00:16:01.000 --> 00:16:02.170 align:middle line:84% If you take a picture of a square 00:16:02.170 --> 00:16:04.390 align:middle line:84% and it comes out looking like something spaghetti, 00:16:04.390 --> 00:16:06.370 align:middle line:90% that's not really an image. 00:16:06.370 --> 00:16:08.396 align:middle line:84% And also, the lengths are not preserved. 00:16:08.396 --> 00:16:16.840 align:middle line:90% 00:16:16.840 --> 00:16:19.150 align:middle line:90% Things get even worse-- 00:16:19.150 --> 00:16:27.630 align:middle line:84% preserved-- things get even worse for 3 and 4 00:16:27.630 --> 00:16:30.420 align:middle line:84% because now we don't even have this bijective property. 00:16:30.420 --> 00:16:35.160 align:middle line:84% We end up in a regime which is called the caustic regime, which 00:16:35.160 --> 00:16:36.285 align:middle line:90% I'll talk more about later. 00:16:36.285 --> 00:16:40.070 align:middle line:90% 00:16:40.070 --> 00:16:43.460 align:middle line:84% Caustics come up all the time in other fields. 00:16:43.460 --> 00:16:46.250 align:middle line:84% They are sometimes called optical catastrophes, 00:16:46.250 --> 00:16:48.500 align:middle line:84% and although they're very, very pretty, 00:16:48.500 --> 00:16:51.673 align:middle line:84% they do make analysis of this data very, very difficult. 00:16:51.673 --> 00:16:53.090 align:middle line:84% So what I'm trying to show here is 00:16:53.090 --> 00:16:55.370 align:middle line:84% something as simple as moving exactly where you 00:16:55.370 --> 00:16:57.860 align:middle line:84% make this measurement makes a big difference to the data 00:16:57.860 --> 00:16:59.030 align:middle line:90% that you have. 00:16:59.030 --> 00:17:02.750 align:middle line:84% It turns out that the easiest place theoretically to analyze 00:17:02.750 --> 00:17:05.486 align:middle line:84% your data is nice and close to your plasma, 00:17:05.486 --> 00:17:07.069 align:middle line:84% but the easiest place to actually make 00:17:07.069 --> 00:17:08.861 align:middle line:84% the measurement with decent signal-to-noise 00:17:08.861 --> 00:17:10.160 align:middle line:90% is somewhere around about 3. 00:17:10.160 --> 00:17:11.810 align:middle line:90% So you can't have it all. 00:17:11.810 --> 00:17:14.550 align:middle line:84% You can't end up in a regime where you have the absolute best 00:17:14.550 --> 00:17:15.050 align:middle line:90% data. 00:17:15.050 --> 00:17:17.670 align:middle line:90% 00:17:17.670 --> 00:17:19.420 align:middle line:84% Again, this has all been very qualitative. 00:17:19.420 --> 00:17:22.359 align:middle line:84% We're about to make it quantitative by showing, 00:17:22.359 --> 00:17:24.160 align:middle line:84% at least in this small deflection 00:17:24.160 --> 00:17:26.285 align:middle line:90% regime, what you can measure. 00:17:26.285 --> 00:17:28.660 align:middle line:84% But does anyone have any questions on this general schema 00:17:28.660 --> 00:17:29.830 align:middle line:90% before we keep going? 00:17:29.830 --> 00:17:30.850 align:middle line:90% Yeah? 00:17:30.850 --> 00:17:32.392 align:middle line:84% AUDIENCE: Are you saying that there's 00:17:32.392 --> 00:17:35.470 align:middle line:84% a bijective map between your measured intensity 00:17:35.470 --> 00:17:38.620 align:middle line:84% distribution and the incident intensity distribution, 00:17:38.620 --> 00:17:40.090 align:middle line:84% or between the measured intensity 00:17:40.090 --> 00:17:44.287 align:middle line:84% distribution and the plasma density profile? 00:17:44.287 --> 00:17:45.495 align:middle line:90% JACK HARE: Really the former. 00:17:45.495 --> 00:17:46.080 align:middle line:90% AUDIENCE: OK. 00:17:46.080 --> 00:17:48.400 align:middle line:84% JACK HARE: So between the incident laser, 00:17:48.400 --> 00:17:51.060 align:middle line:84% the laser on this side, and on this side. 00:17:51.060 --> 00:17:53.170 align:middle line:84% But I would argue that if you have that, 00:17:53.170 --> 00:17:55.650 align:middle line:84% you can then infer something about the plasma 00:17:55.650 --> 00:17:58.200 align:middle line:84% that you've gone through, because you have 00:17:58.200 --> 00:18:00.720 align:middle line:84% some idea of-- if you're measuring here 00:18:00.720 --> 00:18:02.220 align:middle line:84% and you know it corresponds to here, 00:18:02.220 --> 00:18:04.595 align:middle line:84% then you know that the chord that you took for the plasma 00:18:04.595 --> 00:18:05.325 align:middle line:90% was roughly that. 00:18:05.325 --> 00:18:06.450 align:middle line:90% AUDIENCE: Right, you have-- 00:18:06.450 --> 00:18:08.580 align:middle line:84% JACK HARE: So you have some idea of what sort of plasma 00:18:08.580 --> 00:18:09.840 align:middle line:90% properties you were sensing. 00:18:09.840 --> 00:18:12.270 align:middle line:84% Or, alternative way around, you have some idea 00:18:12.270 --> 00:18:14.930 align:middle line:84% that if you ended up here, that you 00:18:14.930 --> 00:18:16.680 align:middle line:84% must have gone through this bit of plasma, 00:18:16.680 --> 00:18:19.920 align:middle line:84% and to get that displacement, that bit of plasma 00:18:19.920 --> 00:18:23.003 align:middle line:84% must have had some certain density gradients within it. 00:18:23.003 --> 00:18:25.545 align:middle line:84% AUDIENCE: Right, like, some sort of average density gradient. 00:18:25.545 --> 00:18:25.710 align:middle line:90% OK. 00:18:25.710 --> 00:18:26.790 align:middle line:90% JACK HARE: Yes, exactly. 00:18:26.790 --> 00:18:28.110 align:middle line:84% This is all line integrated, and we'll 00:18:28.110 --> 00:18:29.652 align:middle line:84% talk about that in a moment when we-- 00:18:29.652 --> 00:18:31.320 align:middle line:90% we'll make this displacement. 00:18:31.320 --> 00:18:32.670 align:middle line:90% We will work out-- 00:18:32.670 --> 00:18:35.160 align:middle line:84% I guess it's a y, isn't it, because I did choose a y-axis. 00:18:35.160 --> 00:18:37.260 align:middle line:84% We will work out what that displacement is 00:18:37.260 --> 00:18:38.830 align:middle line:84% in terms of the plasma parameters, 00:18:38.830 --> 00:18:40.170 align:middle line:90% so we'll make that quantitative. 00:18:40.170 --> 00:18:42.570 align:middle line:84% But yeah, this is kind of what I'm getting at, 00:18:42.570 --> 00:18:45.527 align:middle line:84% is have some idea of what you're actually measuring. 00:18:45.527 --> 00:18:47.110 align:middle line:84% Once you get into this caustic regime, 00:18:47.110 --> 00:18:49.027 align:middle line:84% you don't really know exactly where everything 00:18:49.027 --> 00:18:50.450 align:middle line:90% has come from anymore. 00:18:50.450 --> 00:18:52.240 align:middle line:90% Yeah, so I saw a hand first. 00:18:52.240 --> 00:18:54.730 align:middle line:84% AUDIENCE: Yeah, is the intensity still proportional 00:18:54.730 --> 00:18:57.790 align:middle line:84% to the density gradient, as it was previously? 00:18:57.790 --> 00:18:58.417 align:middle line:90% Or is it-- 00:18:58.417 --> 00:19:01.000 align:middle line:84% JACK HARE: I'm sad that you took away from my previous lecture 00:19:01.000 --> 00:19:03.160 align:middle line:84% that initially the intensity is proportional to the density 00:19:03.160 --> 00:19:03.660 align:middle line:90% gradient. 00:19:03.660 --> 00:19:06.020 align:middle line:84% It almost never is in any realistic situation. 00:19:06.020 --> 00:19:08.590 align:middle line:84% So that's not true, and it's still not true here. 00:19:08.590 --> 00:19:10.450 align:middle line:84% And we will derive that it's proportional 00:19:10.450 --> 00:19:12.670 align:middle line:84% to the second derivative of the density gradient, 00:19:12.670 --> 00:19:14.350 align:middle line:84% and we'll also show that is also not 00:19:14.350 --> 00:19:16.340 align:middle line:90% true in most reasonable cases. 00:19:16.340 --> 00:19:18.670 align:middle line:84% So please take away from this that neither 00:19:18.670 --> 00:19:20.650 align:middle line:84% of these diagnostics are easy to interpret, 00:19:20.650 --> 00:19:22.783 align:middle line:84% but you will read in the textbooks or online 00:19:22.783 --> 00:19:25.450 align:middle line:84% that they're proportional to the first or the second derivative. 00:19:25.450 --> 00:19:28.150 align:middle line:84% It's almost impossible to set up your diagnostics such 00:19:28.150 --> 00:19:29.510 align:middle line:90% that that's actually true. 00:19:29.510 --> 00:19:31.990 align:middle line:90% So, cool. 00:19:31.990 --> 00:19:35.020 align:middle line:84% AUDIENCE: So you say that if we are at, like, 2, 00:19:35.020 --> 00:19:39.700 align:middle line:84% we can guess that it's coming from the first ray, but how can 00:19:39.700 --> 00:19:40.210 align:middle line:90% we say that? 00:19:40.210 --> 00:19:43.570 align:middle line:84% Or why can't we say that it's like the lower ray, incredibly 00:19:43.570 --> 00:19:44.170 align:middle line:90% deflected off? 00:19:44.170 --> 00:19:46.807 align:middle line:84% So how can we know what the caustic regime is? 00:19:46.807 --> 00:19:47.890 align:middle line:90% JACK HARE: We will-- yeah. 00:19:47.890 --> 00:19:49.183 align:middle line:90% Yeah, so that's a great point. 00:19:49.183 --> 00:19:51.100 align:middle line:84% So it is very hard to find the caustic regime, 00:19:51.100 --> 00:19:54.400 align:middle line:84% but the caustic regime is defined by extreme intensity 00:19:54.400 --> 00:19:55.300 align:middle line:90% variations. 00:19:55.300 --> 00:19:57.370 align:middle line:84% So if you see only small intensity variations, 00:19:57.370 --> 00:20:00.290 align:middle line:84% you can't be in the caustic regime. 00:20:00.290 --> 00:20:04.570 align:middle line:84% Theoretically, this is a spike to infinity, right. 00:20:04.570 --> 00:20:06.558 align:middle line:84% Fortunately, the universe doesn't 00:20:06.558 --> 00:20:08.850 align:middle line:84% allow that to happen because your optics aren't perfect 00:20:08.850 --> 00:20:11.380 align:middle line:84% and your detector isn't perfect, but this is very, very bright. 00:20:11.380 --> 00:20:13.005 align:middle line:84% So you can tell by looking at an image, 00:20:13.005 --> 00:20:14.710 align:middle line:84% if it's got no hugely bright regions, 00:20:14.710 --> 00:20:16.330 align:middle line:90% you're not in this regime. 00:20:16.330 --> 00:20:18.175 align:middle line:84% And then your second question was, yeah, 00:20:18.175 --> 00:20:20.050 align:middle line:84% but still, it could come from somewhere else. 00:20:20.050 --> 00:20:20.830 align:middle line:90% You're right. 00:20:20.830 --> 00:20:26.200 align:middle line:84% When we talk about some of the advanced methods for, I guess, 00:20:26.200 --> 00:20:28.180 align:middle line:84% deconvolving or processing this data, 00:20:28.180 --> 00:20:31.990 align:middle line:84% we will come across some long, exciting-sounding phrases 00:20:31.990 --> 00:20:34.460 align:middle line:84% like optimal transport and Voronoi diagrams, 00:20:34.460 --> 00:20:36.670 align:middle line:84% and there you're trying to minimize 00:20:36.670 --> 00:20:38.590 align:middle line:84% how ridiculous your density distribution has 00:20:38.590 --> 00:20:41.030 align:middle line:90% to be to give you this result. 00:20:41.030 --> 00:20:43.510 align:middle line:84% And so there are very mathematically grounded 00:20:43.510 --> 00:20:45.880 align:middle line:84% ways of trying to put this back, but if you're just 00:20:45.880 --> 00:20:48.565 align:middle line:84% staring at an image and it's got small intensity perturbations, 00:20:48.565 --> 00:20:49.940 align:middle line:84% you're probably going to be like, 00:20:49.940 --> 00:20:52.630 align:middle line:84% hey, it's most likely to have come from up here. 00:20:52.630 --> 00:20:55.100 align:middle line:84% It's unlikely that it went, whoop, like that. 00:20:55.100 --> 00:20:58.870 align:middle line:84% AUDIENCE: So if we image way far to the right, 00:20:58.870 --> 00:21:01.958 align:middle line:84% then it would start to look more reasonable, right? 00:21:01.958 --> 00:21:03.250 align:middle line:90% But then the image would have-- 00:21:03.250 --> 00:21:05.140 align:middle line:90% JACK HARE: Well, yes. 00:21:05.140 --> 00:21:07.930 align:middle line:84% What I haven't drawn here is, if I draw in more rays, 00:21:07.930 --> 00:21:10.320 align:middle line:90% they will actually be more-- 00:21:10.320 --> 00:21:12.742 align:middle line:90% oh, can do it easily here? 00:21:12.742 --> 00:21:14.950 align:middle line:84% We'll find out that there are more caustics coming in 00:21:14.950 --> 00:21:15.740 align:middle line:90% later on. 00:21:15.740 --> 00:21:17.290 align:middle line:84% So once you've gone past this point, 00:21:17.290 --> 00:21:18.873 align:middle line:84% you will always have caustics in here. 00:21:18.873 --> 00:21:21.490 align:middle line:84% I just haven't drawn enough rays to make that point really 00:21:21.490 --> 00:21:22.040 align:middle line:90% clearly. 00:21:22.040 --> 00:21:23.665 align:middle line:84% But if you can imagine drawing more in, 00:21:23.665 --> 00:21:24.940 align:middle line:90% you'll see that they will-- 00:21:24.940 --> 00:21:27.855 align:middle line:84% like, maybe there's one that doesn't cross this one here, 00:21:27.855 --> 00:21:29.230 align:middle line:84% but eventually-- no, that doesn't 00:21:29.230 --> 00:21:30.813 align:middle line:84% work because it's not a straight line. 00:21:30.813 --> 00:21:35.130 align:middle line:90% 00:21:35.130 --> 00:21:37.710 align:middle line:84% Like that, OK, and then this one, crop this here, 00:21:37.710 --> 00:21:39.260 align:middle line:84% we'd have a caustic in that region. 00:21:39.260 --> 00:21:39.490 align:middle line:90% AUDIENCE: Right. 00:21:39.490 --> 00:21:40.890 align:middle line:84% So they eventually just start processing 00:21:40.890 --> 00:21:41.490 align:middle line:90% more and more and more. 00:21:41.490 --> 00:21:43.073 align:middle line:84% JACK HARE: They will eventually cross. 00:21:43.073 --> 00:21:45.420 align:middle line:84% And so I guess what I'd say is there is always 00:21:45.420 --> 00:21:48.180 align:middle line:84% a point, some place where you can 00:21:48.180 --> 00:21:50.233 align:middle line:84% put your detector back here where you will end up 00:21:50.233 --> 00:21:51.150 align:middle line:90% in the caustic regime. 00:21:51.150 --> 00:21:53.130 align:middle line:84% And there's indeed a dimensionless parameter 00:21:53.130 --> 00:21:55.200 align:middle line:84% that tells you whether you're in the caustic regime or not, 00:21:55.200 --> 00:21:56.850 align:middle line:84% and it's to do with the deflection 00:21:56.850 --> 00:21:58.470 align:middle line:90% angle and this distance. 00:21:58.470 --> 00:21:59.340 align:middle line:90% Thank you. 00:21:59.340 --> 00:21:59.880 align:middle line:90% Cool. 00:21:59.880 --> 00:22:00.720 align:middle line:90% Any other questions? 00:22:00.720 --> 00:22:01.678 align:middle line:90% Anything from Columbia? 00:22:01.678 --> 00:22:05.550 align:middle line:90% 00:22:05.550 --> 00:22:06.660 align:middle line:90% OK. 00:22:06.660 --> 00:22:08.610 align:middle line:84% So let's try and make this a bit quantitative, 00:22:08.610 --> 00:22:10.290 align:middle line:84% because I can see that folks want 00:22:10.290 --> 00:22:12.075 align:middle line:90% to get some numbers into this. 00:22:12.075 --> 00:22:19.010 align:middle line:90% 00:22:19.010 --> 00:22:21.240 align:middle line:84% I think here I'm basically following Hutchinson, 00:22:21.240 --> 00:22:24.480 align:middle line:84% so if you need to look up the equations in more detail, 00:22:24.480 --> 00:22:26.100 align:middle line:90% this is where you want to head. 00:22:26.100 --> 00:22:31.850 align:middle line:84% We're going to be working in a regime with small angles, so 00:22:31.850 --> 00:22:34.050 align:middle line:90% small values of theta. 00:22:34.050 --> 00:22:40.315 align:middle line:84% And remember that theta is going to be equal to, I guess-- 00:22:40.315 --> 00:22:43.330 align:middle line:84% I put d dy here, though I want to make this kind of two 00:22:43.330 --> 00:22:46.550 align:middle line:84% dimensional, so I'm just going to write "gradient" here. 00:22:46.550 --> 00:22:49.270 align:middle line:84% And you can think of theta as a vector which 00:22:49.270 --> 00:22:52.240 align:middle line:84% contains the angle with respect to the x-axis 00:22:52.240 --> 00:22:54.370 align:middle line:84% and the angle with respect to the y-axis here. 00:22:54.370 --> 00:22:55.180 align:middle line:90% OK. 00:22:55.180 --> 00:22:59.380 align:middle line:84% This is the gradient of the line-integrated refractive 00:22:59.380 --> 00:23:00.110 align:middle line:90% index. 00:23:00.110 --> 00:23:01.420 align:middle line:84% So I'm still going to work in refractive index 00:23:01.420 --> 00:23:03.253 align:middle line:84% here because it's a little bit more compact, 00:23:03.253 --> 00:23:06.493 align:middle line:84% and also because this applies to any inhomogeneous medium, not 00:23:06.493 --> 00:23:08.410 align:middle line:84% just the plasma, so you could use this for air 00:23:08.410 --> 00:23:10.237 align:middle line:90% and other things like that. 00:23:10.237 --> 00:23:12.820 align:middle line:84% So we're going to write it just in terms of N, and at the end, 00:23:12.820 --> 00:23:14.720 align:middle line:84% I'll turn this into plasma density 00:23:14.720 --> 00:23:17.090 align:middle line:90% so you can see the final result. 00:23:17.090 --> 00:23:22.030 align:middle line:84% So we're going to assume that we've got our rays of light, 00:23:22.030 --> 00:23:30.602 align:middle line:84% again, incident like this onto some plane 00:23:30.602 --> 00:23:31.685 align:middle line:90% that we're going to call-- 00:23:31.685 --> 00:23:34.350 align:middle line:90% 00:23:34.350 --> 00:23:37.230 align:middle line:90% that has coordinates x and y. 00:23:37.230 --> 00:23:44.710 align:middle line:84% We're going to have some initial intensity profile that's 00:23:44.710 --> 00:23:47.740 align:middle line:90% incident on this plasma here. 00:23:47.740 --> 00:23:54.658 align:middle line:84% So our plasma is just past the plane we're doing this. 00:23:54.658 --> 00:23:56.950 align:middle line:84% And of course, this could be something like in uniform, 00:23:56.950 --> 00:23:57.800 align:middle line:90% or it could be Gaussian. 00:23:57.800 --> 00:24:00.258 align:middle line:84% It could be whatever you want, so whatever you can actually 00:24:00.258 --> 00:24:02.380 align:middle line:84% come up with for your laser probing. 00:24:02.380 --> 00:24:14.310 align:middle line:84% And our rays are going to get deflected by some angle theta, 00:24:14.310 --> 00:24:15.510 align:middle line:90% like this. 00:24:15.510 --> 00:24:20.800 align:middle line:84% And then we're going to have our detector, 00:24:20.800 --> 00:24:23.610 align:middle line:84% and that's going to be in a prime coordinate system 00:24:23.610 --> 00:24:25.660 align:middle line:90% x prime, y prime. 00:24:25.660 --> 00:24:28.470 align:middle line:84% And of course, if don't want to put my detector just here, 00:24:28.470 --> 00:24:32.235 align:middle line:90% I can always put the lens-- 00:24:32.235 --> 00:24:34.360 align:middle line:84% no, I can put my detector here, and that would also 00:24:34.360 --> 00:24:38.352 align:middle line:84% have x prime, y prime, with maybe any magnification 00:24:38.352 --> 00:24:40.810 align:middle line:84% that the lens does, but that's not really relevant to this. 00:24:40.810 --> 00:24:42.440 align:middle line:90% That's just optics. 00:24:42.440 --> 00:24:48.350 align:middle line:84% So we're trying to work out how we get from the intensity 00:24:48.350 --> 00:24:51.230 align:middle line:90% initial to the intensity-- 00:24:51.230 --> 00:24:53.100 align:middle line:90% what am I going to call it-- 00:24:53.100 --> 00:25:00.230 align:middle line:84% I detector, x prime, y prime, like this. 00:25:00.230 --> 00:25:01.070 align:middle line:90% OK. 00:25:01.070 --> 00:25:05.540 align:middle line:84% So we can just stare at this and do some simple geometry. 00:25:05.540 --> 00:25:08.120 align:middle line:84% We can say that if we're just talking about coordinates, 00:25:08.120 --> 00:25:13.640 align:middle line:84% x prime, y prime, is going to equal wherever we started out, 00:25:13.640 --> 00:25:15.230 align:middle line:90% x plus-- 00:25:15.230 --> 00:25:16.432 align:middle line:90% ah, this is important. 00:25:16.432 --> 00:25:17.390 align:middle line:90% We need a length scale. 00:25:17.390 --> 00:25:21.250 align:middle line:90% 00:25:21.250 --> 00:25:24.350 align:middle line:84% We're going to put our detector at distance L from the plasma 00:25:24.350 --> 00:25:24.850 align:middle line:90% here. 00:25:24.850 --> 00:25:27.580 align:middle line:84% We're assuming the plasma is pretty thin still. 00:25:27.580 --> 00:25:41.880 align:middle line:84% So this is going to be Ld dx of the integral of N dl, 00:25:41.880 --> 00:25:43.530 align:middle line:84% and this coordinate in y is going 00:25:43.530 --> 00:25:52.430 align:middle line:84% to be y plus L times d dy integral of N dl. 00:25:52.430 --> 00:25:55.410 align:middle line:84% So again, I said we're using a small-angle approximation, 00:25:55.410 --> 00:25:58.580 align:middle line:84% so we've taken the approximation that tan theta is approximately 00:25:58.580 --> 00:26:02.380 align:middle line:84% sine theta is approximately theta, 00:26:02.380 --> 00:26:06.030 align:middle line:84% so this is just a simple linear relationship here, 00:26:06.030 --> 00:26:11.250 align:middle line:84% where this is L times the angle in x and this 00:26:11.250 --> 00:26:19.272 align:middle line:84% is L times the angle of the line, the theta line. 00:26:19.272 --> 00:26:20.980 align:middle line:84% And so we could write this more compactly 00:26:20.980 --> 00:26:24.390 align:middle line:84% in a sort of vector notation as some vector x prime 00:26:24.390 --> 00:26:33.970 align:middle line:84% is equal to some vector x plus L, gradient operator on N dl, 00:26:33.970 --> 00:26:34.830 align:middle line:90% like that. 00:26:34.830 --> 00:26:39.980 align:middle line:84% So this is just another representation of this. 00:26:39.980 --> 00:26:40.590 align:middle line:90% OK. 00:26:40.590 --> 00:26:44.100 align:middle line:84% Now, one thing that we need in order to make some progress here 00:26:44.100 --> 00:26:47.550 align:middle line:84% is we need to assume that the overall intensity, so 00:26:47.550 --> 00:26:49.920 align:middle line:84% the integral of this over x and y, 00:26:49.920 --> 00:26:52.230 align:middle line:84% is equal to the integral of this over x and y. 00:26:52.230 --> 00:26:54.460 align:middle line:84% So we're sort of conserving our intensity, 00:26:54.460 --> 00:26:59.070 align:middle line:84% so we could write that down as I on the detector 00:26:59.070 --> 00:27:07.040 align:middle line:84% d x prime d y prime is equal to I incident dx dy. 00:27:07.040 --> 00:27:09.200 align:middle line:84% So the plasma is not absorbing, and I 00:27:09.200 --> 00:27:11.600 align:middle line:84% guess it's also not emitting any light 00:27:11.600 --> 00:27:13.680 align:middle line:90% in this wavelength equation. 00:27:13.680 --> 00:27:17.160 align:middle line:84% Yeah, that's a pretty reasonable approximation. 00:27:17.160 --> 00:27:21.470 align:middle line:84% And then we can skip ahead, and we can say that the light which 00:27:21.470 --> 00:27:24.800 align:middle line:84% is incident divided-- the intensity which is incident 00:27:24.800 --> 00:27:27.680 align:middle line:84% divided by the intensity on our detector is going to be equal 00:27:27.680 --> 00:27:33.540 align:middle line:90% to 1 plus gradient squared-- 00:27:33.540 --> 00:27:39.420 align:middle line:84% sorry, 1 plus L times gradient squared 00:27:39.420 --> 00:27:44.350 align:middle line:84% of the integral of the refractive index along the path. 00:27:44.350 --> 00:27:50.230 align:middle line:84% And if we work with the relatively small value of this, 00:27:50.230 --> 00:27:54.450 align:middle line:84% so if we assume that this is much, much less than 1, 00:27:54.450 --> 00:27:57.700 align:middle line:84% we can rewrite this in terms of a change in intensity. 00:27:57.700 --> 00:28:02.740 align:middle line:84% So this would be the change in intensity in our image delta 00:28:02.740 --> 00:28:07.940 align:middle line:84% I normalized to our initial intensity, 00:28:07.940 --> 00:28:12.370 align:middle line:84% and that is going to be equal to 1 minus L 00:28:12.370 --> 00:28:18.280 align:middle line:84% over 2 critical density, delta squared integral 00:28:18.280 --> 00:28:21.400 align:middle line:84% of any dl, where in this last step 00:28:21.400 --> 00:28:24.100 align:middle line:84% I've substituted out the refractive index 00:28:24.100 --> 00:28:29.510 align:middle line:84% for the expression that we had for Ne much less than N 00:28:29.510 --> 00:28:30.010 align:middle line:90% critical. 00:28:30.010 --> 00:28:31.810 align:middle line:84% So again, we've made an assumption 00:28:31.810 --> 00:28:34.060 align:middle line:84% that we have small intensity variations. 00:28:34.060 --> 00:28:37.280 align:middle line:84% We've also made the assumption that Ne is much, 00:28:37.280 --> 00:28:39.010 align:middle line:84% much less than N critical, so we can 00:28:39.010 --> 00:28:44.800 align:middle line:84% use this nice, linear formula for the refractive index 00:28:44.800 --> 00:28:45.640 align:middle line:90% of the plasma. 00:28:45.640 --> 00:28:47.150 align:middle line:90% That simplifies it. 00:28:47.150 --> 00:28:50.830 align:middle line:84% So this is kind of like our final nice result 00:28:50.830 --> 00:28:53.560 align:middle line:84% here, so for very small intensity variations, 00:28:53.560 --> 00:28:57.250 align:middle line:84% you do indeed get an intensity variation which 00:28:57.250 --> 00:29:01.360 align:middle line:84% is proportional to 1 minus gradient squared 00:29:01.360 --> 00:29:02.960 align:middle line:90% of your electron density. 00:29:02.960 --> 00:29:05.290 align:middle line:84% So using this formula, where do we 00:29:05.290 --> 00:29:08.110 align:middle line:84% expect to have bright regions in our plasma? 00:29:08.110 --> 00:29:09.565 align:middle line:84% So where-- or, sorry, where do we 00:29:09.565 --> 00:29:11.690 align:middle line:84% expect to have bright regions in our interferogram? 00:29:11.690 --> 00:29:12.940 align:middle line:90% What do they correspond to? 00:29:12.940 --> 00:29:17.570 align:middle line:90% 00:29:17.570 --> 00:29:18.070 align:middle line:90% Yeah? 00:29:18.070 --> 00:29:19.043 align:middle line:90% Anyone? 00:29:19.043 --> 00:29:20.460 align:middle line:84% AUDIENCE: The bright regions would 00:29:20.460 --> 00:29:22.600 align:middle line:84% be places where the second term is small, 00:29:22.600 --> 00:29:25.980 align:middle line:84% so the second derivative being small, the point of inflection 00:29:25.980 --> 00:29:26.820 align:middle line:90% of the density. 00:29:26.820 --> 00:29:28.980 align:middle line:84% JACK HARE: OK, so only where it's small? 00:29:28.980 --> 00:29:35.680 align:middle line:90% 00:29:35.680 --> 00:29:38.236 align:middle line:84% AUDIENCE: Like, where it's negative and big? 00:29:38.236 --> 00:29:39.460 align:middle line:90% JACK HARE: Beg pardon? 00:29:39.460 --> 00:29:42.232 align:middle line:84% AUDIENCE: Like, where it's negative and big? 00:29:42.232 --> 00:29:42.940 align:middle line:90% JACK HARE: Right. 00:29:42.940 --> 00:29:43.910 align:middle line:90% Yeah, exactly. 00:29:43.910 --> 00:29:45.520 align:middle line:84% So the bright regions here are going 00:29:45.520 --> 00:29:51.870 align:middle line:84% to correspond to minima in the density, 00:29:51.870 --> 00:29:55.755 align:middle line:84% and the dark regions are going to correspond to maxima. 00:29:55.755 --> 00:29:59.496 align:middle line:90% 00:29:59.496 --> 00:30:02.790 align:middle line:84% And that's because, very roughly, 00:30:02.790 --> 00:30:05.290 align:middle line:84% when we think about what the plasma is doing, 00:30:05.290 --> 00:30:07.530 align:middle line:84% we see that minima in the electron density act 00:30:07.530 --> 00:30:10.930 align:middle line:84% as focusing lenses and maxima act as diverging regions. 00:30:10.930 --> 00:30:14.400 align:middle line:84% And so that's kind of what we saw around about here. 00:30:14.400 --> 00:30:15.720 align:middle line:90% OK. 00:30:15.720 --> 00:30:17.400 align:middle line:90% AUDIENCE: How did you get the-- 00:30:17.400 --> 00:30:22.350 align:middle line:84% how did you get the I initial over Id formula, 00:30:22.350 --> 00:30:23.280 align:middle line:90% with the Laplace-- 00:30:23.280 --> 00:30:24.480 align:middle line:84% JACK HARE: I skipped a few of the steps. 00:30:24.480 --> 00:30:25.022 align:middle line:90% AUDIENCE: OK. 00:30:25.022 --> 00:30:26.940 align:middle line:84% JACK HARE: Yeah, it's not obvious, 00:30:26.940 --> 00:30:28.620 align:middle line:84% but you can go to Hutchinson's book 00:30:28.620 --> 00:30:30.578 align:middle line:84% and see if you can follow the derivation there. 00:30:30.578 --> 00:30:33.870 align:middle line:84% But yeah, there's a little bit of magic to do that step there. 00:30:33.870 --> 00:30:35.370 align:middle line:84% The thing that you want to recognize 00:30:35.370 --> 00:30:36.960 align:middle line:84% is that we're doing something that 00:30:36.960 --> 00:30:39.630 align:middle line:84% looks like it's got a Jacobian or something like that involved 00:30:39.630 --> 00:30:44.178 align:middle line:84% there, so it all eventually ends up working out. 00:30:44.178 --> 00:30:46.470 align:middle line:84% There's also a paper by Kugland that I'll mention later 00:30:46.470 --> 00:30:47.520 align:middle line:84% that's really good for this stuff, 00:30:47.520 --> 00:30:49.540 align:middle line:84% if you want to see an alternative derivation. 00:30:49.540 --> 00:30:50.240 align:middle line:90% AUDIENCE: OK. 00:30:50.240 --> 00:30:51.990 align:middle line:84% I'm wondering, how do we-- what is 00:30:51.990 --> 00:30:54.570 align:middle line:84% the precise meaning of the gradient 00:30:54.570 --> 00:30:56.555 align:middle line:90% of a line-integrated quantity? 00:30:56.555 --> 00:30:58.680 align:middle line:84% Is it like we're changing the limits of integration 00:30:58.680 --> 00:31:00.700 align:middle line:90% by an infinitesimal amount? 00:31:00.700 --> 00:31:02.215 align:middle line:90% Like, what does that mean? 00:31:02.215 --> 00:31:05.157 align:middle line:90% 00:31:05.157 --> 00:31:06.740 align:middle line:84% JACK HARE: I mean, mathematically, you 00:31:06.740 --> 00:31:08.300 align:middle line:90% can just work it out, right. 00:31:08.300 --> 00:31:10.740 align:middle line:84% There's nothing wrong with this, because, for example, 00:31:10.740 --> 00:31:18.740 align:middle line:84% if we are doing any of x, y, and z integrated dz, like that, 00:31:18.740 --> 00:31:23.225 align:middle line:84% then you can still take the derivative of this. 00:31:23.225 --> 00:31:25.100 align:middle line:84% You just won't have any z components anymore. 00:31:25.100 --> 00:31:27.350 align:middle line:84% You'll just have components in x and y. 00:31:27.350 --> 00:31:29.210 align:middle line:90% AUDIENCE: OK. 00:31:29.210 --> 00:31:31.700 align:middle line:84% JACK HARE: In reality, when I see this written down, 00:31:31.700 --> 00:31:32.930 align:middle line:90% I often see people-- 00:31:32.930 --> 00:31:36.200 align:middle line:84% and I do the same thing-- who sort of sneakily move this 00:31:36.200 --> 00:31:41.200 align:middle line:84% integration sign to here, and then it makes a little bit more 00:31:41.200 --> 00:31:43.090 align:middle line:84% sense because you're actually looking-- 00:31:43.090 --> 00:31:46.720 align:middle line:84% for each step, dl, that the ray goes through the plasma, 00:31:46.720 --> 00:31:49.880 align:middle line:84% you look at what the local density gradient is. 00:31:49.880 --> 00:31:53.890 align:middle line:84% And I think these two things are different, kind of obviously, 00:31:53.890 --> 00:31:56.500 align:middle line:84% but they are pretty close for thin plasmas. 00:31:56.500 --> 00:31:58.630 align:middle line:84% And what most of the time people are doing 00:31:58.630 --> 00:32:02.590 align:middle line:84% is making approximations that the width of this plasma, which 00:32:02.590 --> 00:32:06.490 align:middle line:84% might be a or something like that, is much, much less than L. 00:32:06.490 --> 00:32:11.110 align:middle line:84% So this is effectively assuming that the actual path 00:32:11.110 --> 00:32:14.050 align:middle line:84% that the ray takes through the plasma is unimportant. 00:32:14.050 --> 00:32:17.500 align:middle line:84% We only just care about what path it takes through free space 00:32:17.500 --> 00:32:20.020 align:middle line:84% afterwards, which is going to be in a straight line. 00:32:20.020 --> 00:32:22.030 align:middle line:84% If you don't have this condition, 00:32:22.030 --> 00:32:24.637 align:middle line:84% you actually end up in the caustic regime more easily, 00:32:24.637 --> 00:32:26.470 align:middle line:84% and again, I'll point you to some references 00:32:26.470 --> 00:32:29.335 align:middle line:84% later which talk about this in a bit more detail. 00:32:29.335 --> 00:32:33.758 align:middle line:84% For the-- in the case where the thickness of the plasma 00:32:33.758 --> 00:32:35.800 align:middle line:84% is much less than the distance between the plasma 00:32:35.800 --> 00:32:37.630 align:middle line:84% and the detector, it doesn't really 00:32:37.630 --> 00:32:39.930 align:middle line:84% matter which way around you do this operation. 00:32:39.930 --> 00:32:41.630 align:middle line:90% So, yeah, good question. 00:32:41.630 --> 00:32:42.350 align:middle line:90% Yeah? 00:32:42.350 --> 00:32:44.450 align:middle line:84% AUDIENCE: When you say "the thin plasma regime," 00:32:44.450 --> 00:32:46.970 align:middle line:84% we're sort of saying that the path inside the plasma 00:32:46.970 --> 00:32:48.050 align:middle line:90% doesn't matter. 00:32:48.050 --> 00:32:51.490 align:middle line:84% Isn't that just like the schlieren? 00:32:51.490 --> 00:32:53.290 align:middle line:84% JACK HARE: Yeah, and indeed, this effect 00:32:53.290 --> 00:32:54.798 align:middle line:90% pops up in schlieren as well. 00:32:54.798 --> 00:32:56.590 align:middle line:84% But in schlieren, the bigger effect you get 00:32:56.590 --> 00:32:58.780 align:middle line:84% is by putting the stop and blocking out the rays, 00:32:58.780 --> 00:33:00.405 align:middle line:84% but you will have shadowgraphic effects 00:33:00.405 --> 00:33:02.155 align:middle line:84% inside your schlieren imaging system, too. 00:33:02.155 --> 00:33:02.988 align:middle line:90% AUDIENCE: OK, I see. 00:33:02.988 --> 00:33:05.650 align:middle line:84% JACK HARE: So, yeah, I just introduced the schlieren first, 00:33:05.650 --> 00:33:07.090 align:middle line:84% and now we have the shadowgraphy. 00:33:07.090 --> 00:33:08.950 align:middle line:84% But these are both present here, and they're also 00:33:08.950 --> 00:33:10.742 align:middle line:84% present in some of the interferometry we'll 00:33:10.742 --> 00:33:11.810 align:middle line:90% talk about later. 00:33:11.810 --> 00:33:14.490 align:middle line:84% But in each of these, there's something 00:33:14.490 --> 00:33:16.650 align:middle line:84% that causes the biggest intensity modulations. 00:33:16.650 --> 00:33:19.050 align:middle line:84% In the schlieren, it's the schlieren stop, clearly, 00:33:19.050 --> 00:33:21.300 align:middle line:84% but here, again, we're only looking 00:33:21.300 --> 00:33:23.250 align:middle line:84% in this regime with small intensity 00:33:23.250 --> 00:33:25.220 align:middle line:90% modulation for the moment. 00:33:25.220 --> 00:33:26.720 align:middle line:84% But without the schlieren stop, this 00:33:26.720 --> 00:33:29.380 align:middle line:90% is the effect that shows up. 00:33:29.380 --> 00:33:30.130 align:middle line:90% Mm-hmm? 00:33:30.130 --> 00:33:32.650 align:middle line:84% AUDIENCE: Our signal is proportional to the Laplacian 00:33:32.650 --> 00:33:36.610 align:middle line:84% density, and so why is it that maxima and minima of the signal 00:33:36.610 --> 00:33:39.100 align:middle line:84% correspond to maxima and minima of the density 00:33:39.100 --> 00:33:42.947 align:middle line:84% rather than maxima and minima of the density gradient? 00:33:42.947 --> 00:33:44.114 align:middle line:90% JACK HARE: Hm-hm, hm-hm, hm. 00:33:44.114 --> 00:33:48.973 align:middle line:90% 00:33:48.973 --> 00:33:50.265 align:middle line:90% Yeah, I see what you're saying. 00:33:50.265 --> 00:33:53.280 align:middle line:84% AUDIENCE: I think it's because it's the Laplacian of the line 00:33:53.280 --> 00:33:56.190 align:middle line:84% integral of the density as opposed to the Laplacian 00:33:56.190 --> 00:33:57.860 align:middle line:90% density. 00:33:57.860 --> 00:33:59.930 align:middle line:84% AUDIENCE: Oh, like, the integration over space 00:33:59.930 --> 00:34:01.220 align:middle line:90% brings you back a level? 00:34:01.220 --> 00:34:02.273 align:middle line:90% AUDIENCE: Yeah. 00:34:02.273 --> 00:34:04.190 align:middle line:84% JACK HARE: I'll have a look at that and check. 00:34:04.190 --> 00:34:06.110 align:middle line:84% I do see what you're saying, and I see what you're saying. 00:34:06.110 --> 00:34:07.735 align:middle line:84% And I don't know what the resolution is 00:34:07.735 --> 00:34:10.190 align:middle line:84% right at the moment, but yeah, I'll have a look 00:34:10.190 --> 00:34:11.838 align:middle line:90% and see if I can work that out. 00:34:11.838 --> 00:34:12.380 align:middle line:90% AUDIENCE: OK. 00:34:12.380 --> 00:34:12.710 align:middle line:90% Thank you. 00:34:12.710 --> 00:34:13.400 align:middle line:90% JACK HARE: Cool. 00:34:13.400 --> 00:34:18.020 align:middle line:84% So again, this looks really nice because it 00:34:18.020 --> 00:34:21.050 align:middle line:84% looks like you can get out the second derivative with Nz, 00:34:21.050 --> 00:34:23.715 align:middle line:84% and then maybe you could double-integrate that 00:34:23.715 --> 00:34:26.090 align:middle line:84% and you could get out the actual line-integrated electron 00:34:26.090 --> 00:34:27.000 align:middle line:90% density. 00:34:27.000 --> 00:34:29.929 align:middle line:84% But in reality, the assumptions we'd have to make to get here 00:34:29.929 --> 00:34:32.750 align:middle line:84% mean this is really hard because the actual signal 00:34:32.750 --> 00:34:35.030 align:middle line:84% term, we have assumed, is much, much smaller 00:34:35.030 --> 00:34:38.045 align:middle line:84% than 1, which means that it's really, really hard to measure. 00:34:38.045 --> 00:34:46.670 align:middle line:90% 00:34:46.670 --> 00:34:50.800 align:middle line:84% So for any realistic system with a realistic signal-to-noise, 00:34:50.800 --> 00:34:52.969 align:middle line:84% we don't want to have this limitation. 00:34:52.969 --> 00:34:56.750 align:middle line:84% We don't want to be working at position 1 or even position 2. 00:34:56.750 --> 00:34:58.750 align:middle line:84% We're actually going to get the best signal when 00:34:58.750 --> 00:35:02.000 align:middle line:84% we go to position 3, where all of this no longer applies 00:35:02.000 --> 00:35:04.160 align:middle line:84% and we can no longer get this nice result. 00:35:04.160 --> 00:35:06.490 align:middle line:84% So this is what I'm saying where you 00:35:06.490 --> 00:35:08.050 align:middle line:84% will find people saying shadowgraphy 00:35:08.050 --> 00:35:10.383 align:middle line:84% is proportional to the second derivative of the electron 00:35:10.383 --> 00:35:12.520 align:middle line:84% density, and that's sort of true. 00:35:12.520 --> 00:35:14.110 align:middle line:84% But I've never seen anyone do it. 00:35:14.110 --> 00:35:16.420 align:middle line:84% Like, it's not actually possible to do that measurement 00:35:16.420 --> 00:35:17.505 align:middle line:90% in a meaningful way. 00:35:17.505 --> 00:35:19.630 align:middle line:84% You have to work in a regime where you can actually 00:35:19.630 --> 00:35:20.740 align:middle line:90% measure the modulation. 00:35:20.740 --> 00:35:23.080 align:middle line:84% And I'll show you some example pictures of schlieren 00:35:23.080 --> 00:35:25.460 align:middle line:84% and shadowgraphy towards the end of this lecture 00:35:25.460 --> 00:35:28.505 align:middle line:84% so you get an idea of what it looks like in a plasma. 00:35:28.505 --> 00:35:29.005 align:middle line:90% OK. 00:35:29.005 --> 00:35:32.520 align:middle line:90% 00:35:32.520 --> 00:35:34.550 align:middle line:84% I think I've kind of said a lot of this already. 00:35:34.550 --> 00:35:37.790 align:middle line:84% If we end up in this regime here, 00:35:37.790 --> 00:35:42.110 align:middle line:84% where we have these caustics, we've clearly lost information. 00:35:42.110 --> 00:35:45.290 align:middle line:84% We can no longer do this mapping here 00:35:45.290 --> 00:35:46.880 align:middle line:90% because it's no longer unique. 00:35:46.880 --> 00:35:50.240 align:middle line:84% We've got raised crossings, so different bits of x prime 00:35:50.240 --> 00:35:52.250 align:middle line:90% and y prime are mapping onto-- 00:35:52.250 --> 00:35:54.440 align:middle line:84% or, one place in x prime, y prime, 00:35:54.440 --> 00:35:57.590 align:middle line:84% might map onto multiple x and y, and so we 00:35:57.590 --> 00:35:59.882 align:middle line:90% can't do this simple thing. 00:35:59.882 --> 00:36:01.340 align:middle line:84% And it's kind of obvious, actually, 00:36:01.340 --> 00:36:03.530 align:middle line:84% that you'll be losing information, 00:36:03.530 --> 00:36:05.180 align:middle line:84% and even a more advanced technique 00:36:05.180 --> 00:36:07.567 align:middle line:84% isn't going to be able to do the reconstruction. 00:36:07.567 --> 00:36:09.650 align:middle line:84% So I want to talk just now a little bit about some 00:36:09.650 --> 00:36:12.350 align:middle line:84% of these advanced reconstruction techniques which go beyond this, 00:36:12.350 --> 00:36:13.933 align:middle line:84% and then I'll show you these examples. 00:36:13.933 --> 00:36:31.020 align:middle line:90% 00:36:31.020 --> 00:36:33.440 align:middle line:84% So the first paper I've seen that really tackles this 00:36:33.440 --> 00:36:42.660 align:middle line:84% very nicely is Kugland, et al., in RSI, 2012. 00:36:42.660 --> 00:36:45.420 align:middle line:84% So Kugland points out pretty quickly 00:36:45.420 --> 00:36:51.990 align:middle line:84% that shadowgraphy is a direct equivalent 00:36:51.990 --> 00:36:54.690 align:middle line:90% to proton radiography. 00:36:54.690 --> 00:36:56.903 align:middle line:84% So mathematically, they both deal 00:36:56.903 --> 00:36:58.320 align:middle line:84% with the same quantities, which is 00:36:58.320 --> 00:37:00.120 align:middle line:90% a sort of deflection potential. 00:37:00.120 --> 00:37:03.090 align:middle line:84% In proton radiography, this deflection potential is to do 00:37:03.090 --> 00:37:05.250 align:middle line:84% with electric and magnetic fields-- 00:37:05.250 --> 00:37:07.740 align:middle line:84% and we'll get on to proton radiography later-- 00:37:07.740 --> 00:37:11.490 align:middle line:84% and in shadowgraphy, the deflection potential 00:37:11.490 --> 00:37:13.650 align:middle line:90% is to do with density gradients. 00:37:13.650 --> 00:37:16.770 align:middle line:84% But once you work in terms of this deflection potential 00:37:16.770 --> 00:37:19.770 align:middle line:84% and forget where it came from, you get exactly the same results 00:37:19.770 --> 00:37:23.830 align:middle line:84% mathematically, and he has this nice geometric approach. 00:37:23.830 --> 00:37:26.730 align:middle line:84% And so if you've found the derivation I just 00:37:26.730 --> 00:37:29.920 align:middle line:84% did not very convincing, you can go and have a look at this. 00:37:29.920 --> 00:37:33.030 align:middle line:84% It's a little bit more rigorous, and also, he then 00:37:33.030 --> 00:37:35.640 align:middle line:84% extends it into the caustic regime and beyond, 00:37:35.640 --> 00:37:38.490 align:middle line:84% and shows what you would expect to get from a plasma 00:37:38.490 --> 00:37:39.960 align:middle line:90% where you have caustics. 00:37:39.960 --> 00:37:42.675 align:middle line:84% So this is nice, but he still doesn't really 00:37:42.675 --> 00:37:43.800 align:middle line:90% tell you how to analyze it. 00:37:43.800 --> 00:37:45.600 align:middle line:84% It's really talking about the problem 00:37:45.600 --> 00:37:47.160 align:middle line:84% where you go from knowing the density 00:37:47.160 --> 00:37:49.920 align:middle line:84% of the plasma to predicting what you're going to get out. 00:37:49.920 --> 00:37:52.350 align:middle line:84% Going the opposite direction to going from your intensity 00:37:52.350 --> 00:37:56.770 align:middle line:84% variations to the density isn't particularly well developed, 00:37:56.770 --> 00:37:59.670 align:middle line:84% so this is what some people call the forward problem. 00:37:59.670 --> 00:38:05.630 align:middle line:90% 00:38:05.630 --> 00:38:11.210 align:middle line:84% And the forward problem is going from Ne of x, y, 00:38:11.210 --> 00:38:16.010 align:middle line:84% and z, to intensity on your detector 00:38:16.010 --> 00:38:18.980 align:middle line:90% in x prime and y prime-- 00:38:18.980 --> 00:38:23.870 align:middle line:84% useful, but not exactly the solution. 00:38:23.870 --> 00:38:27.580 align:middle line:84% So then in 2017, there were two almost competing papers 00:38:27.580 --> 00:38:28.720 align:middle line:90% that came out about this. 00:38:28.720 --> 00:38:37.820 align:middle line:84% There was a paper by Kasim, et al., in Physical Review E, 2017. 00:38:37.820 --> 00:38:41.450 align:middle line:84% What Kasim did is he tried to reconstruct this deflection 00:38:41.450 --> 00:38:46.495 align:middle line:84% potential that Kugland had come up with. 00:38:46.495 --> 00:38:49.850 align:middle line:90% 00:38:49.850 --> 00:38:54.650 align:middle line:84% And he did this using a technique-- 00:38:54.650 --> 00:38:57.970 align:middle line:84% that was borrowed from, some would say, 00:38:57.970 --> 00:38:59.720 align:middle line:84% computer graphics, or at least some fields 00:38:59.720 --> 00:39:01.520 align:middle line:90% of applied mathematics-- 00:39:01.520 --> 00:39:05.690 align:middle line:90% which used a Voronoi diagram. 00:39:05.690 --> 00:39:08.692 align:middle line:84% Has anyone come across Voronoi diagrams before? 00:39:08.692 --> 00:39:10.150 align:middle line:84% AUDIENCE: They're huge in robotics. 00:39:10.150 --> 00:39:10.600 align:middle line:90% JACK HARE: Sorry? 00:39:10.600 --> 00:39:11.500 align:middle line:84% AUDIENCE: They're huge in robotics. 00:39:11.500 --> 00:39:12.070 align:middle line:90% JACK HARE: OK, cool. 00:39:12.070 --> 00:39:13.735 align:middle line:84% Right, so there's like-- they basically-- 00:39:13.735 --> 00:39:15.250 align:middle line:84% I think this worked well because they 00:39:15.250 --> 00:39:16.420 align:middle line:84% took some work that other people have 00:39:16.420 --> 00:39:18.545 align:middle line:84% been doing in different fields and applied it here. 00:39:18.545 --> 00:39:19.690 align:middle line:90% So a Voronoi diagram-- 00:39:19.690 --> 00:39:21.310 align:middle line:84% ha-ha, please don't shout at me if I 00:39:21.310 --> 00:39:25.420 align:middle line:84% get this wrong-- is roughly, if you have a series of points 00:39:25.420 --> 00:39:30.460 align:middle line:84% which are randomly distributed, how do you draw quadrilaterals 00:39:30.460 --> 00:39:34.690 align:middle line:84% around them such that every point inside the quadrilateral 00:39:34.690 --> 00:39:38.450 align:middle line:84% is closest to this point and not to any of the other points? 00:39:38.450 --> 00:39:41.005 align:middle line:84% So it's a way of tiling up a space. 00:39:41.005 --> 00:39:44.350 align:middle line:90% 00:39:44.350 --> 00:39:49.090 align:middle line:84% And from this, you can imagine that these tiles that you've 00:39:49.090 --> 00:39:53.680 align:middle line:84% produced in your shadowgraphy can then be related back 00:39:53.680 --> 00:39:58.820 align:middle line:84% to a more uniform grid of tiles which all have 00:39:58.820 --> 00:40:01.980 align:middle line:84% the same shape and the same intensity, 00:40:01.980 --> 00:40:05.070 align:middle line:84% which is your intensity beforehand, 00:40:05.070 --> 00:40:07.880 align:middle line:84% and this is your intensity at the detector. 00:40:07.880 --> 00:40:10.370 align:middle line:84% Again, this is just a very hand-wavy sketch 00:40:10.370 --> 00:40:11.100 align:middle line:90% of what they did. 00:40:11.100 --> 00:40:14.100 align:middle line:84% And they came up with an algorithm to do this, 00:40:14.100 --> 00:40:18.260 align:middle line:84% and this enabled them to do the inverse problem, which 00:40:18.260 --> 00:40:22.700 align:middle line:84% is intensity at our detector in x prime, y prime, 00:40:22.700 --> 00:40:27.285 align:middle line:84% going back towards density in x, y, and z. 00:40:27.285 --> 00:40:29.660 align:middle line:84% And as we discussed before, this is an ill-posed problem. 00:40:29.660 --> 00:40:32.930 align:middle line:84% There are a large family of possible density structures 00:40:32.930 --> 00:40:34.620 align:middle line:90% that produce the same intensity. 00:40:34.620 --> 00:40:38.030 align:middle line:84% But this Voronoi diagram is making conservative assumptions 00:40:38.030 --> 00:40:40.040 align:middle line:84% about where light has come from in order 00:40:40.040 --> 00:40:42.600 align:middle line:90% to be able to put stuff back. 00:40:42.600 --> 00:40:46.030 align:middle line:84% At almost exactly the same time, Bott-- 00:40:46.030 --> 00:40:47.380 align:middle line:90% both these groups are at Oxford. 00:40:47.380 --> 00:40:47.880 align:middle line:90% Yes? 00:40:47.880 --> 00:40:50.370 align:middle line:84% AUDIENCE: So would this be applicable to shooting 00:40:50.370 --> 00:40:53.115 align:middle line:84% an array of lasers through, or shooting in a grid? 00:40:53.115 --> 00:40:56.448 align:middle line:84% JACK HARE: Oh, so no one does that, but they should. 00:40:56.448 --> 00:40:58.740 align:middle line:84% So they did it in proton radiography in the early days. 00:40:58.740 --> 00:41:00.313 align:middle line:84% They actually had little beamlets, 00:41:00.313 --> 00:41:01.980 align:middle line:84% and then you can uniquely work out where 00:41:01.980 --> 00:41:03.300 align:middle line:90% each beamlet is deflected. 00:41:03.300 --> 00:41:04.860 align:middle line:84% But of course, you only get, like, 00:41:04.860 --> 00:41:06.993 align:middle line:84% n beamlets of points of data, and it's 00:41:06.993 --> 00:41:08.910 align:middle line:84% like, it doesn't look pretty, you can't put it 00:41:08.910 --> 00:41:10.160 align:middle line:90% in nature, that sort of thing. 00:41:10.160 --> 00:41:12.410 align:middle line:84% So people moved very quickly away from that technique, 00:41:12.410 --> 00:41:14.410 align:middle line:84% so now we have data that's impossible to analyze 00:41:14.410 --> 00:41:15.330 align:middle line:90% but is very beautiful. 00:41:15.330 --> 00:41:16.747 align:middle line:84% Whereas, we used to have data that 00:41:16.747 --> 00:41:18.760 align:middle line:84% was entirely analyzable but not very beautiful. 00:41:18.760 --> 00:41:19.552 align:middle line:90% So I have opinions. 00:41:19.552 --> 00:41:23.100 align:middle line:84% AUDIENCE: So this actually would be just a finite number of rays 00:41:23.100 --> 00:41:23.910 align:middle line:90% that should be-- 00:41:23.910 --> 00:41:27.210 align:middle line:84% JACK HARE: No, I mean, this is entirely to do with-- 00:41:27.210 --> 00:41:29.320 align:middle line:84% this is not to do with the finite number of rays. 00:41:29.320 --> 00:41:32.340 align:middle line:84% This is still to do with our nice, initially uniform beam. 00:41:32.340 --> 00:41:35.460 align:middle line:84% It's just the way that we segment up the final image. 00:41:35.460 --> 00:41:37.800 align:middle line:90% These dots don't really exist. 00:41:37.800 --> 00:41:41.380 align:middle line:84% You're actually-- in reality, you're trying to find-- 00:41:41.380 --> 00:41:43.630 align:middle line:84% ha, this is where I'm probably going to get it wrong-- 00:41:43.630 --> 00:41:46.738 align:middle line:84% you're trying to find regions which contain the same intensity 00:41:46.738 --> 00:41:49.155 align:middle line:84% as one of these initial squares, and you're trying to sort 00:41:49.155 --> 00:41:51.793 align:middle line:84% of tile them together in such a way that you don't have to have 00:41:51.793 --> 00:41:54.210 align:middle line:84% something that looks like a congressional district, right, 00:41:54.210 --> 00:41:54.750 align:middle line:90% where it-- 00:41:54.750 --> 00:41:56.550 align:middle line:90% [LAUGHTER] 00:41:56.550 --> 00:41:58.170 align:middle line:84% OK, because that's obviously silly. 00:41:58.170 --> 00:41:59.380 align:middle line:90% That's unlikely to happen. 00:41:59.380 --> 00:42:00.960 align:middle line:84% So we're trying to-- this is-- you could maybe 00:42:00.960 --> 00:42:03.335 align:middle line:84% use this algorithm to sort out a lot of problems in this. 00:42:03.335 --> 00:42:06.870 align:middle line:84% Anyway, so at the same time, Bott 00:42:06.870 --> 00:42:09.960 align:middle line:84% was working on an algorithm that ended up doing the same thing, 00:42:09.960 --> 00:42:13.410 align:middle line:90% and this is in JPP in 2017. 00:42:13.410 --> 00:42:15.840 align:middle line:84% This paper is something like 120 pages long. 00:42:15.840 --> 00:42:18.090 align:middle line:84% It very much helps to have your thesis advisor be 00:42:18.090 --> 00:42:19.690 align:middle line:84% the editor in chief of JPP if you want 00:42:19.690 --> 00:42:21.150 align:middle line:90% to publish a paper with them. 00:42:21.150 --> 00:42:25.800 align:middle line:84% And they use a very interesting technique called the-- 00:42:25.800 --> 00:42:27.480 align:middle line:84% I'm probably going to say this wrong, 00:42:27.480 --> 00:42:30.150 align:middle line:84% and I certainly handled all the accents in my notes here-- 00:42:30.150 --> 00:42:34.320 align:middle line:90% Monge-Ampere optimal transport. 00:42:34.320 --> 00:42:37.510 align:middle line:84% And this optimal transport algorithm had actually 00:42:37.510 --> 00:42:42.130 align:middle line:84% won someone the Fields Medal only a few years before, 00:42:42.130 --> 00:42:45.863 align:middle line:84% Cédric Villani, who wears these incredibly huge bow ties, 00:42:45.863 --> 00:42:48.280 align:middle line:84% and he has a wonderful book called The Life of the Theorem 00:42:48.280 --> 00:42:51.130 align:middle line:84% where he discusses how great it is to be Cédric Villani. 00:42:51.130 --> 00:42:53.890 align:middle line:84% But, so this algorithm was derived not at all 00:42:53.890 --> 00:42:56.930 align:middle line:84% to do with proton radiography, but it is to do with how we-- 00:42:56.930 --> 00:42:59.800 align:middle line:84% what is the most conservative way 00:42:59.800 --> 00:43:04.910 align:middle line:84% to map one function into another function like this. 00:43:04.910 --> 00:43:07.150 align:middle line:84% And once you derive that, you can then put it back 00:43:07.150 --> 00:43:08.920 align:middle line:84% where you started from, so this also 00:43:08.920 --> 00:43:14.720 align:middle line:84% enables you to do this same inverse problem like this. 00:43:14.720 --> 00:43:18.350 align:middle line:84% And I'm not even going to slightly go through my guess 00:43:18.350 --> 00:43:19.940 align:middle line:84% at what the Monge-Ampere equation does 00:43:19.940 --> 00:43:22.050 align:middle line:90% because I don't have one. 00:43:22.050 --> 00:43:23.780 align:middle line:90% But it seems to work. 00:43:23.780 --> 00:43:25.050 align:middle line:90% They give similar results. 00:43:25.050 --> 00:43:26.490 align:middle line:84% It turns out this is much faster. 00:43:26.490 --> 00:43:30.020 align:middle line:84% I think most people use some version of this at the moment, 00:43:30.020 --> 00:43:33.740 align:middle line:84% but this one is maybe easier to understand. 00:43:33.740 --> 00:43:36.263 align:middle line:84% So both of these give similar results 00:43:36.263 --> 00:43:37.680 align:middle line:84% with slightly different techniques 00:43:37.680 --> 00:43:39.722 align:middle line:84% And I think, in the end, Kasim wrote a code based 00:43:39.722 --> 00:43:42.180 align:middle line:84% on Bott's paper that was faster than what Bott had done, 00:43:42.180 --> 00:43:44.963 align:middle line:84% so I think people have converged on using something 00:43:44.963 --> 00:43:45.630 align:middle line:90% to do with this. 00:43:45.630 --> 00:43:47.255 align:middle line:84% And again, these techniques were mostly 00:43:47.255 --> 00:43:50.675 align:middle line:84% invented for proton radiography, but now, as we said, 00:43:50.675 --> 00:43:52.550 align:middle line:84% the mathematics is the same for shadowgraphy. 00:43:52.550 --> 00:43:55.160 align:middle line:84% I have not seen anyone use either technique 00:43:55.160 --> 00:43:56.727 align:middle line:84% to properly analyze shadowgraphy, 00:43:56.727 --> 00:43:57.810 align:middle line:90% but it should be possible. 00:43:57.810 --> 00:43:58.310 align:middle line:90% Yes? 00:43:58.310 --> 00:43:59.893 align:middle line:84% AUDIENCE: How does this problem differ 00:43:59.893 --> 00:44:02.310 align:middle line:84% from more general tomography problems we encounter? 00:44:02.310 --> 00:44:05.052 align:middle line:84% Right, like, tomography or when people do 00:44:05.052 --> 00:44:06.260 align:middle line:90% tomographic reconstructions-- 00:44:06.260 --> 00:44:07.160 align:middle line:84% JACK HARE: There's no tomography here. 00:44:07.160 --> 00:44:08.120 align:middle line:90% We only have one line of sight. 00:44:08.120 --> 00:44:09.470 align:middle line:84% You can't do a tomographic reconstruction-- 00:44:09.470 --> 00:44:10.980 align:middle line:84% AUDIENCE: OK, so there's only one line of sight. 00:44:10.980 --> 00:44:11.880 align:middle line:84% JACK HARE: Yeah, I mean you can-- 00:44:11.880 --> 00:44:13.350 align:middle line:84% OK, so now you can ask yourself, If I 00:44:13.350 --> 00:44:14.808 align:middle line:84% have multiple lines of sight, can I 00:44:14.808 --> 00:44:16.680 align:middle line:90% do tomographic reconstruction? 00:44:16.680 --> 00:44:20.310 align:middle line:84% Which, like, yes, obviously, but it's also hard. 00:44:20.310 --> 00:44:21.810 align:middle line:84% But, you know, it might be possible. 00:44:21.810 --> 00:44:25.620 align:middle line:84% But this is a single line of sight, so we're not trying to-- 00:44:25.620 --> 00:44:27.000 align:middle line:90% ah, thank you. 00:44:27.000 --> 00:44:28.560 align:middle line:84% This is the mistake I've been making. 00:44:28.560 --> 00:44:31.320 align:middle line:84% We are not actually getting this out. 00:44:31.320 --> 00:44:36.660 align:middle line:84% We are getting our best guess at this out. 00:44:36.660 --> 00:44:38.640 align:middle line:84% Right, so this is not a full three-dimensional 00:44:38.640 --> 00:44:39.880 align:middle line:90% reconstruction. 00:44:39.880 --> 00:44:45.600 align:middle line:84% This is still a reconstruction of the line-integrated electron 00:44:45.600 --> 00:44:46.540 align:middle line:90% density here. 00:44:46.540 --> 00:44:48.677 align:middle line:84% So, yeah, yeah, that's a good point. 00:44:48.677 --> 00:44:49.510 align:middle line:90% I forgot about that. 00:44:49.510 --> 00:44:52.010 align:middle line:84% And it's sort of obvious that you should be able to do that. 00:44:52.010 --> 00:44:54.690 align:middle line:84% But there still could be multiple profiles that 00:44:54.690 --> 00:44:56.860 align:middle line:84% still produce the same intensity distribution, 00:44:56.860 --> 00:44:59.070 align:middle line:84% so it's still not particularly well posed. 00:44:59.070 --> 00:45:02.970 align:middle line:84% If you do have some caustics inside your protected image, 00:45:02.970 --> 00:45:04.950 align:middle line:84% none of these work, right, so we no longer 00:45:04.950 --> 00:45:06.840 align:middle line:84% are able to do the reconstruction. 00:45:06.840 --> 00:45:09.328 align:middle line:84% You can actually have a go at doing the reconstruction 00:45:09.328 --> 00:45:10.620 align:middle line:90% if you have some strong priors. 00:45:10.620 --> 00:45:13.620 align:middle line:84% So if you put-- you have some optimization algorithm that 00:45:13.620 --> 00:45:15.090 align:middle line:84% thinks, like, there's a shot here 00:45:15.090 --> 00:45:16.975 align:middle line:84% and that shot is going to cause caustics, 00:45:16.975 --> 00:45:18.850 align:middle line:84% and I think the caustics will look like this, 00:45:18.850 --> 00:45:19.980 align:middle line:90% you might be able to do it. 00:45:19.980 --> 00:45:22.272 align:middle line:84% But of course, you'd obviously have very strong priors. 00:45:22.272 --> 00:45:24.390 align:middle line:84% The techniques are very line integrated, 00:45:24.390 --> 00:45:26.760 align:middle line:84% as I just remember them pointed out here, 00:45:26.760 --> 00:45:29.220 align:middle line:84% so we're not getting a full 3D structure. 00:45:29.220 --> 00:45:30.690 align:middle line:84% That's maybe a bit too much to ask 00:45:30.690 --> 00:45:32.815 align:middle line:84% from a diagnostic which is clearly line integrated, 00:45:32.815 --> 00:45:34.830 align:middle line:90% but it's still a limitation. 00:45:34.830 --> 00:45:38.490 align:middle line:84% And then the final problem that in proton radiography 00:45:38.490 --> 00:45:41.190 align:middle line:84% is particularly profound is actually 00:45:41.190 --> 00:45:47.443 align:middle line:84% how reproducible this is, how reproducible your initial-- 00:45:47.443 --> 00:45:48.735 align:middle line:90% I'm going to run out of space-- 00:45:48.735 --> 00:45:51.620 align:middle line:90% 00:45:51.620 --> 00:45:54.680 align:middle line:84% your initial intensity is, because they-- 00:45:54.680 --> 00:45:56.930 align:middle line:84% before you do the experiment, you fire your laser beam 00:45:56.930 --> 00:46:00.185 align:middle line:84% through the chamber, and you measure that beam profile. 00:46:00.185 --> 00:46:02.060 align:middle line:84% But then when you actually do the experiment, 00:46:02.060 --> 00:46:03.143 align:middle line:90% that beam profile changes. 00:46:03.143 --> 00:46:05.138 align:middle line:84% You know, lasers are not completely stable. 00:46:05.138 --> 00:46:06.930 align:middle line:84% The beam profile changes from time to time. 00:46:06.930 --> 00:46:09.560 align:middle line:84% And so that means what you think you're mapping from here to here 00:46:09.560 --> 00:46:10.910 align:middle line:84% is actually slightly different, and that's 00:46:10.910 --> 00:46:12.160 align:middle line:90% going to introduce some noise. 00:46:12.160 --> 00:46:14.660 align:middle line:84% This is very important for proton 00:46:14.660 --> 00:46:17.780 align:middle line:84% radiography, where it's very hard to measure the beam. 00:46:17.780 --> 00:46:19.863 align:middle line:84% For a laser shadowgraphy setup, you actually 00:46:19.863 --> 00:46:20.780 align:middle line:90% have more of a chance. 00:46:20.780 --> 00:46:24.050 align:middle line:84% You can put a beam splitter for the plasma and sample the beam 00:46:24.050 --> 00:46:26.330 align:middle line:84% itself, so you actually simultaneously measure 00:46:26.330 --> 00:46:28.783 align:middle line:90% this quantity and this quantity. 00:46:28.783 --> 00:46:31.200 align:middle line:84% So there's a lot of scope for doing some really cool stuff 00:46:31.200 --> 00:46:33.370 align:middle line:90% with shadowgraphy. 00:46:33.370 --> 00:46:35.090 align:middle line:90% Questions? 00:46:35.090 --> 00:46:36.030 align:middle line:90% Yeah? 00:46:36.030 --> 00:46:38.450 align:middle line:84% AUDIENCE: Are there obvious practical reasons 00:46:38.450 --> 00:46:40.820 align:middle line:84% why you wouldn't be measuring the dynamically 00:46:40.820 --> 00:46:42.818 align:middle line:84% helpful distances, or even if you have-- 00:46:42.818 --> 00:46:44.360 align:middle line:84% JACK HARE: I think it's a great idea. 00:46:44.360 --> 00:46:44.825 align:middle line:90% No one's done it. 00:46:44.825 --> 00:46:45.470 align:middle line:90% I want to do it. 00:46:45.470 --> 00:46:45.970 align:middle line:90% [LAUGHTER] 00:46:45.970 --> 00:46:46.970 align:middle line:90% Yeah, absolutely. 00:46:46.970 --> 00:46:51.305 align:middle line:84% So it seems to me like if you have 00:46:51.305 --> 00:46:52.680 align:middle line:84% these images of different places, 00:46:52.680 --> 00:46:54.055 align:middle line:84% you should be able to reconstruct 00:46:54.055 --> 00:46:56.370 align:middle line:84% the trajectories of the rays, and that would give you 00:46:56.370 --> 00:46:57.330 align:middle line:90% more information. 00:46:57.330 --> 00:46:58.200 align:middle line:90% AUDIENCE: Yeah. 00:46:58.200 --> 00:47:00.825 align:middle line:84% JACK HARE: And in fact, in some of the first proton radiography 00:47:00.825 --> 00:47:04.160 align:middle line:84% papers, this is discussed, from looking at the position 00:47:04.160 --> 00:47:05.160 align:middle line:90% through multiple stacks. 00:47:05.160 --> 00:47:07.080 align:middle line:84% But I haven't seen it actually done in practice. 00:47:07.080 --> 00:47:07.800 align:middle line:84% AUDIENCE: You could kind of do it. 00:47:07.800 --> 00:47:08.383 align:middle line:90% JACK HARE: OK. 00:47:08.383 --> 00:47:12.493 align:middle line:84% AUDIENCE: But not fully, because it's-- the proton-- 00:47:12.493 --> 00:47:14.660 align:middle line:84% typically, all of your energy gets deposited first-- 00:47:14.660 --> 00:47:15.900 align:middle line:84% JACK HARE: Yeah, so I think that's the problem, 00:47:15.900 --> 00:47:17.250 align:middle line:90% but I like the idea. 00:47:17.250 --> 00:47:18.240 align:middle line:90% Like, it's a cool-- 00:47:18.240 --> 00:47:19.918 align:middle line:84% if it works, if people used this, yeah. 00:47:19.918 --> 00:47:21.960 align:middle line:84% But you could definitely do it with shadowgraphy, 00:47:21.960 --> 00:47:23.190 align:middle line:90% relatively straightforward. 00:47:23.190 --> 00:47:23.730 align:middle line:90% Yeah. 00:47:23.730 --> 00:47:26.615 align:middle line:84% The other thing you could do is you could put multiple lasers 00:47:26.615 --> 00:47:27.990 align:middle line:84% at different wavelengths through, 00:47:27.990 --> 00:47:29.830 align:middle line:84% and they'd be deflected by different angles. 00:47:29.830 --> 00:47:32.475 align:middle line:84% And then you could use those different deflections, like when 00:47:32.475 --> 00:47:35.100 align:middle line:84% you have your proton radiography and you use different particle 00:47:35.100 --> 00:47:38.040 align:middle line:84% energies to determine between electric and magnetic fields. 00:47:38.040 --> 00:47:40.640 align:middle line:84% Here, we don't have electric and magnetic fields. 00:47:40.640 --> 00:47:43.140 align:middle line:84% There's only one thing that can cause the deflections, which 00:47:43.140 --> 00:47:45.150 align:middle line:84% is density gradients, but those different colors 00:47:45.150 --> 00:47:46.200 align:middle line:90% would enable you to. 00:47:46.200 --> 00:47:48.630 align:middle line:84% So you could imagine that one of your other rays 00:47:48.630 --> 00:47:51.982 align:middle line:84% would get deflected less if it had a shorter wavelength, 00:47:51.982 --> 00:47:53.690 align:middle line:84% and it would take a trajectory like that. 00:47:53.690 --> 00:47:56.242 align:middle line:84% So by comparing where it ends up in one wavelength to where 00:47:56.242 --> 00:47:57.700 align:middle line:84% it ends up in the other wavelength, 00:47:57.700 --> 00:47:59.980 align:middle line:84% you should be able to actually just precisely 00:47:59.980 --> 00:48:01.960 align:middle line:84% measure the angle that's made and therefore 00:48:01.960 --> 00:48:03.543 align:middle line:84% what the density gradients are inside, 00:48:03.543 --> 00:48:06.170 align:middle line:84% but I haven't seen anyone try that yet. 00:48:06.170 --> 00:48:06.930 align:middle line:90% Yeah? 00:48:06.930 --> 00:48:08.930 align:middle line:84% AUDIENCE: When you responded to John's question, 00:48:08.930 --> 00:48:11.550 align:middle line:84% you mentioned that you only have one line of sight here. 00:48:11.550 --> 00:48:14.740 align:middle line:84% But if your detector is able to do 00:48:14.740 --> 00:48:18.180 align:middle line:84% x and y positions of your intensity, 00:48:18.180 --> 00:48:20.930 align:middle line:84% could you consider each pixel as a different line of sight? 00:48:20.930 --> 00:48:23.467 align:middle line:84% JACK HARE: But it's only a chord through a certain bit 00:48:23.467 --> 00:48:24.050 align:middle line:90% of the plasma. 00:48:24.050 --> 00:48:26.660 align:middle line:84% It's not a line of sight through the same bit of plasma. 00:48:26.660 --> 00:48:28.282 align:middle line:84% When we do tomography, you know, you 00:48:28.282 --> 00:48:29.990 align:middle line:84% imagine you've got some cloud, and you're 00:48:29.990 --> 00:48:32.510 align:middle line:84% looking through the same bit of plasma from multiple angles. 00:48:32.510 --> 00:48:34.190 align:middle line:84% Here, you're looking at different bits of plasma, 00:48:34.190 --> 00:48:36.470 align:middle line:84% so you can't topographically reconstruct the density there 00:48:36.470 --> 00:48:38.780 align:middle line:84% because it's literally a different place in the plasma. 00:48:38.780 --> 00:48:41.320 align:middle line:90% 00:48:41.320 --> 00:48:43.580 align:middle line:90% AUDIENCE: Sure, OK, yes. 00:48:43.580 --> 00:48:44.630 align:middle line:90% JACK HARE: Yeah? 00:48:44.630 --> 00:48:46.520 align:middle line:84% AUDIENCE: I would've expected a smaller wave 00:48:46.520 --> 00:48:49.460 align:middle line:84% to be reflected more because the length 00:48:49.460 --> 00:48:52.022 align:middle line:84% scale of the plasma this large wavelength-- 00:48:52.022 --> 00:48:53.730 align:middle line:84% JACK HARE: We are doing geometric optics, 00:48:53.730 --> 00:48:54.950 align:middle line:84% so we don't actually care about the wavelength. 00:48:54.950 --> 00:48:55.825 align:middle line:90% AUDIENCE: Oh, oh, OK. 00:48:55.825 --> 00:48:57.822 align:middle line:84% JACK HARE: Yeah, so in systems where you're not 00:48:57.822 --> 00:48:59.780 align:middle line:84% doing geometric optics, where the wavelength is 00:48:59.780 --> 00:49:01.322 align:middle line:84% comparable to the size of the plasma, 00:49:01.322 --> 00:49:02.900 align:middle line:84% that would be a diffraction effect, 00:49:02.900 --> 00:49:03.710 align:middle line:84% and that would be more important. 00:49:03.710 --> 00:49:04.010 align:middle line:90% AUDIENCE: Got you. 00:49:04.010 --> 00:49:04.380 align:middle line:90% OK. 00:49:04.380 --> 00:49:05.780 align:middle line:84% JACK HARE: But we are actually doing geometric optics 00:49:05.780 --> 00:49:07.160 align:middle line:84% where that isn't important, but you're right. 00:49:07.160 --> 00:49:08.330 align:middle line:84% So all of the stuff I've been doing, 00:49:08.330 --> 00:49:10.330 align:middle line:84% the reason I'm drawing straight lines everywhere 00:49:10.330 --> 00:49:12.570 align:middle line:84% is I'm doing geometric optics and doing ray optics. 00:49:12.570 --> 00:49:15.680 align:middle line:84% So in that case, a different deflection angle 00:49:15.680 --> 00:49:17.780 align:middle line:84% comes about because for shorter wavelengths, 00:49:17.780 --> 00:49:20.840 align:middle line:84% the critical density is higher, and so this quantity becomes 00:49:20.840 --> 00:49:23.370 align:middle line:90% smaller. 00:49:23.370 --> 00:49:26.630 align:middle line:84% This is just a little bit of a lighthearted little picture 00:49:26.630 --> 00:49:27.347 align:middle line:90% show. 00:49:27.347 --> 00:49:29.180 align:middle line:84% And we may finish off the lecture with this, 00:49:29.180 --> 00:49:30.930 align:middle line:84% or we might get started on interferometry, 00:49:30.930 --> 00:49:32.152 align:middle line:90% depending on how I feel. 00:49:32.152 --> 00:49:33.485 align:middle line:90% But here are some nice pictures. 00:49:33.485 --> 00:49:36.100 align:middle line:90% 00:49:36.100 --> 00:49:37.690 align:middle line:84% Shadowgraphy is absolutely everywhere. 00:49:37.690 --> 00:49:40.030 align:middle line:84% You have already seen many shadowgraphs before. 00:49:40.030 --> 00:49:43.120 align:middle line:84% If you've ever seen a mirage, that is a shadowgraph. 00:49:43.120 --> 00:49:47.740 align:middle line:84% That is the natural focusing of light by refractive index 00:49:47.740 --> 00:49:50.643 align:middle line:84% gradients in hot air, right, and you see that shimmering. 00:49:50.643 --> 00:49:52.060 align:middle line:84% You see the fact that there appear 00:49:52.060 --> 00:49:54.367 align:middle line:84% to be mountains below where the mountains actually are. 00:49:54.367 --> 00:49:56.950 align:middle line:84% That's because the rays of light have been bent by the hot air 00:49:56.950 --> 00:49:58.507 align:middle line:90% back upwards into your eye. 00:49:58.507 --> 00:50:01.090 align:middle line:84% And so this is what I mean when I say that shadowgraphy is not 00:50:01.090 --> 00:50:01.690 align:middle line:90% an image. 00:50:01.690 --> 00:50:04.398 align:middle line:84% Whenever you take a shadowgraphy thing, what you're really seeing 00:50:04.398 --> 00:50:06.590 align:middle line:90% is some sort of mirage. 00:50:06.590 --> 00:50:11.360 align:middle line:84% So the first person to ever, we know, study geography 00:50:11.360 --> 00:50:13.520 align:middle line:90% was this guy Jean-Paul Marat. 00:50:13.520 --> 00:50:14.630 align:middle line:90% That's a portrait of him. 00:50:14.630 --> 00:50:16.160 align:middle line:90% Here's his shadowgrams. 00:50:16.160 --> 00:50:19.250 align:middle line:84% He actually drew these by hand because he didn't have cameras 00:50:19.250 --> 00:50:20.263 align:middle line:90% back in the day. 00:50:20.263 --> 00:50:20.930 align:middle line:90% This is the guy. 00:50:20.930 --> 00:50:21.973 align:middle line:90% Here he is in 1793. 00:50:21.973 --> 00:50:23.390 align:middle line:84% He was actually deeply unpleasant. 00:50:23.390 --> 00:50:24.080 align:middle line:90% He was a Jacobin. 00:50:24.080 --> 00:50:26.420 align:middle line:84% He was responsible for the deaths of hundreds of thousands 00:50:26.420 --> 00:50:27.890 align:middle line:84% of people in the French Revolution, 00:50:27.890 --> 00:50:30.175 align:middle line:84% and he was eventually murdered in the bathtub. 00:50:30.175 --> 00:50:32.200 align:middle line:90% [LAUGHTER] 00:50:32.200 --> 00:50:35.540 align:middle line:84% But before he was murdered, he had a very famous guest. 00:50:35.540 --> 00:50:38.180 align:middle line:84% He had a very famous guest who came and actually sat, 00:50:38.180 --> 00:50:43.760 align:middle line:84% and he sketched the shadowgraphic effect of this-- 00:50:43.760 --> 00:50:47.930 align:middle line:84% the effect of this guest's bald head on the air around. 00:50:47.930 --> 00:50:49.780 align:middle line:90% So does anyone know who this is? 00:50:49.780 --> 00:50:52.670 align:middle line:84% That is, of course, Ben Franklin-- so there we go-- 00:50:52.670 --> 00:50:57.290 align:middle line:84% who had made a habit of hanging around in France. 00:50:57.290 --> 00:50:59.570 align:middle line:84% But these days, we usually do things 00:50:59.570 --> 00:51:01.920 align:middle line:84% which are much more exciting than Ben Franklin's head. 00:51:01.920 --> 00:51:06.080 align:middle line:84% So here are several different models. 00:51:06.080 --> 00:51:10.430 align:middle line:84% These are models for the Gemini capsule 00:51:10.430 --> 00:51:12.410 align:middle line:84% that was part of the American space program, 00:51:12.410 --> 00:51:15.035 align:middle line:84% and they want to understand what the shockwaves were around it. 00:51:15.035 --> 00:51:18.330 align:middle line:84% So you can see that this capsule is coming from right to left. 00:51:18.330 --> 00:51:20.000 align:middle line:84% It's got this blunt end here, and we 00:51:20.000 --> 00:51:23.443 align:middle line:84% have this very well-defined bow-shock structure here. 00:51:23.443 --> 00:51:24.860 align:middle line:84% So we think that that's a caustic? 00:51:24.860 --> 00:51:27.627 align:middle line:90% 00:51:27.627 --> 00:51:29.210 align:middle line:84% It's a big intensity variation, right? 00:51:29.210 --> 00:51:30.950 align:middle line:84% It's extremely black here, and it's 00:51:30.950 --> 00:51:32.735 align:middle line:84% very bright around the outside here. 00:51:32.735 --> 00:51:33.860 align:middle line:90% Behind it, what do we have? 00:51:33.860 --> 00:51:36.380 align:middle line:84% We've got a set of shocks there, a shock here. 00:51:36.380 --> 00:51:38.820 align:middle line:84% There's another set of shocks coming off here and here, 00:51:38.820 --> 00:51:41.000 align:middle line:84% which interact with the outer shocks, 00:51:41.000 --> 00:51:44.240 align:middle line:84% and then we have this beautifully turbulent flow 00:51:44.240 --> 00:51:46.010 align:middle line:84% behind it, right, and the same thing 00:51:46.010 --> 00:51:50.288 align:middle line:84% here on this more hemispherical object. 00:51:50.288 --> 00:51:51.830 align:middle line:84% And so don't know how big these were, 00:51:51.830 --> 00:51:54.012 align:middle line:84% but you can do this in a wind tunnel. 00:51:54.012 --> 00:51:55.970 align:middle line:84% In this case, it was probably not a wind tunnel 00:51:55.970 --> 00:51:58.160 align:middle line:84% but a static tube of gas that they fired these 00:51:58.160 --> 00:51:59.115 align:middle line:90% through with a cannon. 00:51:59.115 --> 00:52:00.740 align:middle line:84% And then they would have used some sort 00:52:00.740 --> 00:52:03.200 align:middle line:84% of bright light source, probably not a laser, 00:52:03.200 --> 00:52:04.085 align:middle line:90% in order to do this. 00:52:04.085 --> 00:52:06.772 align:middle line:90% 00:52:06.772 --> 00:52:08.730 align:middle line:84% Also, when you start looking at the literature, 00:52:08.730 --> 00:52:10.980 align:middle line:84% you get lots of beautiful pictures of bullets. 00:52:10.980 --> 00:52:12.610 align:middle line:84% Bullets are particularly good sources. 00:52:12.610 --> 00:52:15.970 align:middle line:84% So I believe, in this picture, the gun is just here, 00:52:15.970 --> 00:52:19.620 align:middle line:84% and so you can see the cloud of exhaust vapor 00:52:19.620 --> 00:52:20.910 align:middle line:90% coming out of the gun barrel. 00:52:20.910 --> 00:52:24.570 align:middle line:84% You can see this is the sound wave of the shot going off, 00:52:24.570 --> 00:52:28.380 align:middle line:84% and then supersonically moving away from the gun is the bullet. 00:52:28.380 --> 00:52:32.080 align:middle line:84% We can see the trail with the defined structure behind it 00:52:32.080 --> 00:52:34.808 align:middle line:84% and these very clearly defined shockwaves here. 00:52:34.808 --> 00:52:37.350 align:middle line:84% And again, these shockwaves have these light and dark regions 00:52:37.350 --> 00:52:39.700 align:middle line:84% corresponding to changes in refractive index. 00:52:39.700 --> 00:52:42.640 align:middle line:84% And this is, I think, a zoom-in of that photograph, 00:52:42.640 --> 00:52:43.770 align:middle line:90% but I can't be sure-- 00:52:43.770 --> 00:52:46.590 align:middle line:90% it looks like it is-- 00:52:46.590 --> 00:52:47.090 align:middle line:90% of that. 00:52:47.090 --> 00:52:49.173 align:middle line:84% So you can see you can get an extraordinary amount 00:52:49.173 --> 00:52:51.065 align:middle line:90% of qualitative detail. 00:52:51.065 --> 00:52:52.690 align:middle line:84% What you can measure from this straight 00:52:52.690 --> 00:52:55.090 align:middle line:84% away is the shock opening angle, so you 00:52:55.090 --> 00:52:56.407 align:middle line:90% can measure the Mach number. 00:52:56.407 --> 00:52:58.990 align:middle line:84% What's going to be a lot more tricky looking at these pictures 00:52:58.990 --> 00:53:01.060 align:middle line:84% is to work out what the density and temperature of the air 00:53:01.060 --> 00:53:02.560 align:middle line:84% is everywhere inside this picture. 00:53:02.560 --> 00:53:04.930 align:middle line:84% That's not really going to be doable 00:53:04.930 --> 00:53:07.730 align:middle line:84% because we're not in this small-intensity-variation 00:53:07.730 --> 00:53:08.230 align:middle line:90% regime. 00:53:08.230 --> 00:53:10.780 align:middle line:84% We've deliberately gone into a regime where we get caustics, 00:53:10.780 --> 00:53:12.890 align:middle line:84% which means we also get a strong intensity variation, 00:53:12.890 --> 00:53:13.930 align:middle line:90% so we can actually measure. 00:53:13.930 --> 00:53:16.360 align:middle line:84% If we were looking at one of those small-intensity-variation 00:53:16.360 --> 00:53:17.943 align:middle line:84% shadowgraphs, it would be very boring. 00:53:17.943 --> 00:53:20.770 align:middle line:84% It would mostly be gray with very, very tiny modulations 00:53:20.770 --> 00:53:21.640 align:middle line:90% to it, so. 00:53:21.640 --> 00:53:22.780 align:middle line:90% Yeah? 00:53:22.780 --> 00:53:25.900 align:middle line:84% AUDIENCE: On figure B, why are the light and dark regions 00:53:25.900 --> 00:53:28.024 align:middle line:90% above and below swaths? 00:53:28.024 --> 00:53:32.290 align:middle line:90% 00:53:32.290 --> 00:53:34.600 align:middle line:90% JACK HARE: Pass. 00:53:34.600 --> 00:53:36.316 align:middle line:84% I'll put it on a little problem set. 00:53:36.316 --> 00:53:38.020 align:middle line:90% [LAUGHTER] 00:53:38.020 --> 00:53:40.960 align:middle line:84% I don't know immediately why that is. 00:53:40.960 --> 00:53:44.080 align:middle line:84% The thing is also to remember, if you end up in a regime 00:53:44.080 --> 00:53:47.740 align:middle line:84% where you have caustics and you have strong deflections 00:53:47.740 --> 00:53:50.440 align:middle line:84% of your rays, you may actually end up in a regime 00:53:50.440 --> 00:53:52.720 align:middle line:84% where the rays don't go through your first optics, 00:53:52.720 --> 00:53:54.460 align:middle line:84% that they may be deflected out of the deflection 00:53:54.460 --> 00:53:55.585 align:middle line:90% volume of your first optic. 00:53:55.585 --> 00:53:57.770 align:middle line:84% And then they would just show up as dark regions. 00:53:57.770 --> 00:54:00.970 align:middle line:84% So you can also have, overlaid with the shadowgraphy effects, 00:54:00.970 --> 00:54:03.760 align:middle line:84% what are effectively schlieren-type effects, where 00:54:03.760 --> 00:54:05.380 align:middle line:84% we are rejecting rays by their angle, 00:54:05.380 --> 00:54:07.240 align:middle line:84% but that's just due to the physical size of the optics 00:54:07.240 --> 00:54:07.953 align:middle line:90% we use. 00:54:07.953 --> 00:54:09.620 align:middle line:84% So I don't know if that's the case here, 00:54:09.620 --> 00:54:11.703 align:middle line:84% but it does complicate the interpretation further. 00:54:11.703 --> 00:54:12.850 align:middle line:90% Yes? 00:54:12.850 --> 00:54:15.010 align:middle line:84% AUDIENCE: In the previous pictures with the Gemini 00:54:15.010 --> 00:54:22.060 align:middle line:84% capsule, there's the effect of pressure waves, shockwaves, 00:54:22.060 --> 00:54:25.690 align:middle line:84% and there's also supposedly intense 00:54:25.690 --> 00:54:30.820 align:middle line:84% heating that should be, like, on the surface of the sphere, 00:54:30.820 --> 00:54:33.290 align:middle line:84% and that's also going to change the refraction index. 00:54:33.290 --> 00:54:34.610 align:middle line:90% JACK HARE: Yes. 00:54:34.610 --> 00:54:36.860 align:middle line:84% AUDIENCE: Which effect of those-- 00:54:36.860 --> 00:54:39.410 align:middle line:84% like, how could we distinguish if something 00:54:39.410 --> 00:54:44.150 align:middle line:84% is a thermal refractive index change or a pressure-- 00:54:44.150 --> 00:54:45.152 align:middle line:90% JACK HARE: In air. 00:54:45.152 --> 00:54:45.860 align:middle line:90% AUDIENCE: In air? 00:54:45.860 --> 00:54:47.150 align:middle line:84% JACK HARE: Yeah, I think that's very difficult 00:54:47.150 --> 00:54:49.610 align:middle line:84% because they both are just the change in refractive index. 00:54:49.610 --> 00:54:50.920 align:middle line:90% AUDIENCE: Yeah? 00:54:50.920 --> 00:54:51.587 align:middle line:90% JACK HARE: Yeah. 00:54:51.587 --> 00:54:53.670 align:middle line:84% I don't think it's possible to tell the difference 00:54:53.670 --> 00:54:54.890 align:middle line:90% between those two, yeah. 00:54:54.890 --> 00:54:58.710 align:middle line:90% 00:54:58.710 --> 00:55:00.810 align:middle line:84% There's also some funky pictures in this book. 00:55:00.810 --> 00:55:02.230 align:middle line:84% This looks pretty straightforward 00:55:02.230 --> 00:55:04.850 align:middle line:84% until you realize the bullet's actually flying backwards. 00:55:04.850 --> 00:55:06.850 align:middle line:84% I don't know why they did that, but there we go. 00:55:06.850 --> 00:55:09.290 align:middle line:84% This is a great book, by the way, Settles' Schlieren 00:55:09.290 --> 00:55:10.500 align:middle line:90% and Shadowgraph Techniques. 00:55:10.500 --> 00:55:11.890 align:middle line:84% It's got wonderful pictures inside it 00:55:11.890 --> 00:55:13.390 align:middle line:84% if you want to know more about this. 00:55:13.390 --> 00:55:16.320 align:middle line:84% It's in the bibliography on the syllabus, and it's a great read. 00:55:16.320 --> 00:55:18.300 align:middle line:84% But I appreciate most people are not 00:55:18.300 --> 00:55:19.530 align:middle line:84% going to be using shadowgraphy and schlieren, 00:55:19.530 --> 00:55:20.700 align:middle line:90% but it's still good stuff. 00:55:20.700 --> 00:55:23.480 align:middle line:90% 00:55:23.480 --> 00:55:25.410 align:middle line:84% But you don't have to fire bullets at things. 00:55:25.410 --> 00:55:27.077 align:middle line:84% This is actually a picture of the author 00:55:27.077 --> 00:55:30.650 align:middle line:84% of this book writing his book next to his heating unit. 00:55:30.650 --> 00:55:31.250 align:middle line:90% There he is. 00:55:31.250 --> 00:55:34.550 align:middle line:84% His head is not quite as impressive as Ben Franklin, 00:55:34.550 --> 00:55:37.560 align:middle line:84% but there's his computer as he types away. 00:55:37.560 --> 00:55:40.160 align:middle line:84% And you can see that you can actually make these measurements 00:55:40.160 --> 00:55:41.780 align:middle line:84% even in relatively benign conditions, 00:55:41.780 --> 00:55:45.170 align:middle line:84% and that's because, again, if we put our detector further 00:55:45.170 --> 00:55:49.070 align:middle line:84% and further back, even small variations in the angle going 00:55:49.070 --> 00:55:51.170 align:middle line:84% through the refractive medium are 00:55:51.170 --> 00:55:53.750 align:middle line:84% going to be mapped into large intensity variations. 00:55:53.750 --> 00:55:56.750 align:middle line:84% So people use this technique to look for flows of air. 00:55:56.750 --> 00:55:59.522 align:middle line:84% You can look for flows of air for all sorts of reasons. 00:55:59.522 --> 00:56:01.730 align:middle line:84% A pretty benign reason would be in the HVAC industry, 00:56:01.730 --> 00:56:04.170 align:middle line:84% if you want to see whether these things are working. 00:56:04.170 --> 00:56:06.530 align:middle line:84% So there are applications for shadowgraphic techniques 00:56:06.530 --> 00:56:09.800 align:middle line:84% and schlieren techniques just in very benign conditions 00:56:09.800 --> 00:56:15.480 align:middle line:84% like this, but we, of course, are interested in plasmas. 00:56:15.480 --> 00:56:18.875 align:middle line:84% So we talked about this a little bit more on the last lecture, 00:56:18.875 --> 00:56:20.750 align:middle line:84% but remember that we need a very bright light 00:56:20.750 --> 00:56:22.760 align:middle line:84% source to overcome the self-emission from a plasma. 00:56:22.760 --> 00:56:24.677 align:middle line:84% So that means we really have to go to a laser. 00:56:24.677 --> 00:56:27.440 align:middle line:84% We just don't have any light sources that are great. 00:56:27.440 --> 00:56:30.050 align:middle line:84% Lasers are actually not ideal for any of these techniques. 00:56:30.050 --> 00:56:32.125 align:middle line:84% You really want to have a nice, large focal spot. 00:56:32.125 --> 00:56:34.250 align:middle line:84% That turns out to be true for shadowgraphy as well, 00:56:34.250 --> 00:56:35.840 align:middle line:84% but we needn't really go into why. 00:56:35.840 --> 00:56:37.580 align:middle line:84% And so this small focal spot gives us 00:56:37.580 --> 00:56:41.390 align:middle line:84% quite limited dynamic range, not to mention the fact 00:56:41.390 --> 00:56:42.957 align:middle line:84% that we are assuming in all of this 00:56:42.957 --> 00:56:45.290 align:middle line:84% that we don't have any coherence effect so we don't have 00:56:45.290 --> 00:56:48.410 align:middle line:84% interference, and we'll talk a lot about interference 00:56:48.410 --> 00:56:49.890 align:middle line:90% in a little bit. 00:56:49.890 --> 00:56:53.125 align:middle line:84% And so really, we don't want to have that coherence when we're 00:56:53.125 --> 00:56:54.500 align:middle line:84% doing schlieren and shadowgraphy, 00:56:54.500 --> 00:56:56.667 align:middle line:84% but if you've got a laser, we tend to have coherence 00:56:56.667 --> 00:56:57.540 align:middle line:90% that we don't want. 00:56:57.540 --> 00:56:59.360 align:middle line:84% So these are not great, but you can still 00:56:59.360 --> 00:57:02.010 align:middle line:90% get some nice images out of it. 00:57:02.010 --> 00:57:05.180 align:middle line:84% Here's a device called an X-pinch. 00:57:05.180 --> 00:57:08.437 align:middle line:84% It consists of two wires that are crossed here. 00:57:08.437 --> 00:57:10.020 align:middle line:84% This is only 1 millimeter across here, 00:57:10.020 --> 00:57:11.395 align:middle line:84% so this is a pretty small object. 00:57:11.395 --> 00:57:14.880 align:middle line:84% And these are X-ray, gated X-ray images of the X-pinch. 00:57:14.880 --> 00:57:18.540 align:middle line:84% We put, in this case, 200 kiloamps through each wire, 00:57:18.540 --> 00:57:21.240 align:middle line:84% and it forms a plasma here, which pinches. 00:57:21.240 --> 00:57:24.540 align:middle line:84% The fields here are maybe a hundred or a thousand tesla. 00:57:24.540 --> 00:57:26.640 align:middle line:84% And it pinches the plasma inwards 00:57:26.640 --> 00:57:29.970 align:middle line:84% like this, compressing it up so it becomes extremely hot, 00:57:29.970 --> 00:57:32.880 align:middle line:84% and it emits a burst of X-rays, which you can then 00:57:32.880 --> 00:57:35.020 align:middle line:90% use for imaging things. 00:57:35.020 --> 00:57:38.150 align:middle line:84% So when we get on to self-emission diagnostics, X-ray 00:57:38.150 --> 00:57:40.150 align:middle line:84% diagnostics, we'll talk a little bit about this. 00:57:40.150 --> 00:57:42.840 align:middle line:84% So these are the X-ray images, but this rather beautiful 00:57:42.840 --> 00:57:47.640 align:middle line:84% schlieren image was captured of this X-pinch in 2008. 00:57:47.640 --> 00:57:50.400 align:middle line:84% Here, they used a dark-field schlieren system 00:57:50.400 --> 00:57:51.630 align:middle line:90% with a circular stop. 00:57:51.630 --> 00:57:54.570 align:middle line:84% You can tell that because it looks up-down symmetric, 00:57:54.570 --> 00:57:55.343 align:middle line:90% so we're not-- 00:57:55.343 --> 00:57:56.760 align:middle line:84% we don't have a knife edge, but we 00:57:56.760 --> 00:57:59.940 align:middle line:84% have some distinction between the different directions. 00:57:59.940 --> 00:58:02.970 align:middle line:84% And you can tell it's dark-field because outside, 00:58:02.970 --> 00:58:05.070 align:middle line:84% where there is no plasma, it's dark here. 00:58:05.070 --> 00:58:06.665 align:middle line:84% If it was light-field, this region 00:58:06.665 --> 00:58:08.040 align:middle line:84% would be filled with laser light, 00:58:08.040 --> 00:58:11.010 align:middle line:84% and we'd have darkness wherever we have lightness here. 00:58:11.010 --> 00:58:13.500 align:middle line:84% And you can see a beautiful amount of detail. 00:58:13.500 --> 00:58:16.020 align:middle line:84% This projector isn't really doing it justice. 00:58:16.020 --> 00:58:18.840 align:middle line:84% You can see in the center here, this is the pinching region. 00:58:18.840 --> 00:58:20.748 align:middle line:84% It's far too dense for all of the laser light 00:58:20.748 --> 00:58:21.540 align:middle line:90% to make it through. 00:58:21.540 --> 00:58:23.200 align:middle line:84% It's much above the critical density, 00:58:23.200 --> 00:58:25.620 align:middle line:84% and so there's very strong refraction of the laser light 00:58:25.620 --> 00:58:27.270 align:middle line:90% outwards, dark in the center. 00:58:27.270 --> 00:58:30.750 align:middle line:84% There's jets of plasma going up and down out 00:58:30.750 --> 00:58:31.990 align:middle line:90% of this compressed region. 00:58:31.990 --> 00:58:34.440 align:middle line:84% And there's also ablation streams coming off 00:58:34.440 --> 00:58:37.350 align:middle line:84% each of the wires which have this beautiful modulated 00:58:37.350 --> 00:58:40.080 align:middle line:84% pattern, which is actually due to a instability in the wire 00:58:40.080 --> 00:58:40.930 align:middle line:90% ablation process. 00:58:40.930 --> 00:58:42.322 align:middle line:90% So this is extremely rich. 00:58:42.322 --> 00:58:44.530 align:middle line:84% There's a lot of information you can get out of this, 00:58:44.530 --> 00:58:48.600 align:middle line:84% even though it's not quantitative. 00:58:48.600 --> 00:58:51.600 align:middle line:84% Another thing that's been done is using schlieren imaging 00:58:51.600 --> 00:58:55.140 align:middle line:84% to image shocks in what's initially a gas, 00:58:55.140 --> 00:58:56.650 align:middle line:90% but quickly becomes a plasma. 00:58:56.650 --> 00:59:00.040 align:middle line:84% So what we had in these experiments was a metal liner. 00:59:00.040 --> 00:59:02.800 align:middle line:84% This is only about 5 millimeters across, 00:59:02.800 --> 00:59:04.030 align:middle line:90% so it's still pretty small. 00:59:04.030 --> 00:59:07.170 align:middle line:84% And it was filled with a gas, 8 millibars of argon, 00:59:07.170 --> 00:59:09.450 align:middle line:90% 15 millibars of nitrogen. 00:59:09.450 --> 00:59:12.030 align:middle line:84% A current was put up through this metal cylinder. 00:59:12.030 --> 00:59:14.730 align:middle line:84% Again, the cylinder is only about that tall and that wide. 00:59:14.730 --> 00:59:18.060 align:middle line:84% And as it does so, it heats the outside of the cylinder 00:59:18.060 --> 00:59:19.590 align:middle line:84% and launches a shockwave inwards. 00:59:19.590 --> 00:59:21.120 align:middle line:84% And that shockwave couples the gas, 00:59:21.120 --> 00:59:23.748 align:middle line:84% and it launches this first shockwave in. 00:59:23.748 --> 00:59:25.290 align:middle line:84% And as the current continues to rise, 00:59:25.290 --> 00:59:28.020 align:middle line:84% you actually get above the melt point of the metal, 00:59:28.020 --> 00:59:30.510 align:middle line:84% and that launches another shockwave due to material 00:59:30.510 --> 00:59:31.990 align:middle line:90% strength that starts coming in. 00:59:31.990 --> 00:59:33.660 align:middle line:84% So we get these converging shockwaves. 00:59:33.660 --> 00:59:36.510 align:middle line:84% And the beautiful thing here is that in argon, we 00:59:36.510 --> 00:59:38.880 align:middle line:84% had a beautifully circular shock, 00:59:38.880 --> 00:59:41.430 align:middle line:84% but in nitrogen, for some reason, 00:59:41.430 --> 00:59:44.465 align:middle line:84% we got this hexagonal shape of shock which has never 00:59:44.465 --> 00:59:46.590 align:middle line:84% been explained, absolutely bizarre, because there's 00:59:46.590 --> 00:59:48.397 align:middle line:84% no sixfold symmetry in this system. 00:59:48.397 --> 00:59:49.230 align:middle line:90% It shouldn't happen. 00:59:49.230 --> 00:59:51.690 align:middle line:84% So there's some instability which is giving it 00:59:51.690 --> 00:59:53.620 align:middle line:90% this really bizarre shape. 00:59:53.620 --> 00:59:55.920 align:middle line:84% And again, this was dark-field with a circular stop. 00:59:55.920 --> 00:59:57.665 align:middle line:84% That's a pretty standard configuration. 00:59:57.665 --> 00:59:59.790 align:middle line:84% Dark-field is a bit more sensitive than light-field 00:59:59.790 --> 01:00:01.410 align:middle line:84% because if you see any light at all, 01:00:01.410 --> 01:00:02.730 align:middle line:84% you know that it's being deflected, 01:00:02.730 --> 01:00:04.688 align:middle line:84% and that's really what you're looking for here. 01:00:04.688 --> 01:00:07.730 align:middle line:90% 01:00:07.730 --> 01:00:11.470 align:middle line:84% And then, shadowgraphy, this is an image 01:00:11.470 --> 01:00:14.200 align:middle line:84% that I took of an imploding wire array. 01:00:14.200 --> 01:00:17.410 align:middle line:84% So again, this thing is only about 16 millimeters tall, 01:00:17.410 --> 01:00:19.150 align:middle line:90% 16 millimeters in diameter. 01:00:19.150 --> 01:00:21.250 align:middle line:84% We've got eight carbon rods here. 01:00:21.250 --> 01:00:24.715 align:middle line:84% Current goes up through the rods, ablates plasma off them. 01:00:24.715 --> 01:00:26.740 align:middle line:84% J-cross-B force accelerates the plasma inwards, 01:00:26.740 --> 01:00:29.710 align:middle line:84% and you get a Z-pinch column in the center here. 01:00:29.710 --> 01:00:32.830 align:middle line:84% And looking from the side using a green laser beam, 01:00:32.830 --> 01:00:35.260 align:middle line:84% we can see that we've got shadows corresponding 01:00:35.260 --> 01:00:36.832 align:middle line:90% to the four wires. 01:00:36.832 --> 01:00:38.290 align:middle line:84% So there's four wires on this side, 01:00:38.290 --> 01:00:40.790 align:middle line:84% and they're blocking the four wires on the other side. 01:00:40.790 --> 01:00:43.818 align:middle line:84% And you can see the column of plasma in the middle here, 01:00:43.818 --> 01:00:46.360 align:middle line:84% and what you can see is there's these very strong modulations 01:00:46.360 --> 01:00:47.068 align:middle line:90% to the intensity. 01:00:47.068 --> 01:00:48.730 align:middle line:90% These are caustics, OK. 01:00:48.730 --> 01:00:52.930 align:middle line:84% We also see modulations sort of flow, as we saw in the X-pinch. 01:00:52.930 --> 01:00:55.540 align:middle line:84% And these caustics mean that this data, while very pretty, 01:00:55.540 --> 01:00:56.540 align:middle line:90% is pretty useless. 01:00:56.540 --> 01:00:58.790 align:middle line:84% There's not much we can actually do with it because we 01:00:58.790 --> 01:01:00.040 align:middle line:90% can't do any decent analysis. 01:01:00.040 --> 01:01:01.958 align:middle line:84% But it does tell us, because the caustics 01:01:01.958 --> 01:01:04.000 align:middle line:84% are on a large range of different spatial scales, 01:01:04.000 --> 01:01:06.657 align:middle line:84% that we must have density perturbations inside the plasma, 01:01:06.657 --> 01:01:08.240 align:middle line:84% and a lot of different spatial scales. 01:01:08.240 --> 01:01:11.090 align:middle line:84% And so this means that plasma is likely to be turbulent. 01:01:11.090 --> 01:01:14.210 align:middle line:84% So this is a little turbulent Z-pinch inside a pulsed power 01:01:14.210 --> 01:01:16.100 align:middle line:90% machine. 01:01:16.100 --> 01:01:17.210 align:middle line:90% So is that it? 01:01:17.210 --> 01:01:21.890 align:middle line:84% Oh, and you can also do some pretty good 3D simulations 01:01:21.890 --> 01:01:22.640 align:middle line:90% of these things. 01:01:22.640 --> 01:01:24.680 align:middle line:84% And then you can spend a lot of time 01:01:24.680 --> 01:01:27.110 align:middle line:84% doing Monte Carlo ray tracing, tracking rays through them, 01:01:27.110 --> 01:01:28.950 align:middle line:84% and seeing what the shadowgraphy looks like. 01:01:28.950 --> 01:01:30.950 align:middle line:84% And don't think this is a particularly bad match 01:01:30.950 --> 01:01:33.170 align:middle line:84% between what we saw from our simulations 01:01:33.170 --> 01:01:35.100 align:middle line:84% and what we saw in our actual data here. 01:01:35.100 --> 01:01:38.690 align:middle line:84% So it is possible to use computational tools to work out 01:01:38.690 --> 01:01:40.260 align:middle line:90% what we would predict. 01:01:40.260 --> 01:01:40.837 align:middle line:90% So that's it. 01:01:40.837 --> 01:01:42.920 align:middle line:84% That's all I've got on shadowgraphy and schlieren. 01:01:42.920 --> 01:01:43.940 align:middle line:90% Any questions on that? 01:01:43.940 --> 01:01:44.890 align:middle line:90% Yeah? 01:01:44.890 --> 01:01:47.300 align:middle line:84% AUDIENCE: Even if you have to worry about caustics, 01:01:47.300 --> 01:01:49.820 align:middle line:84% and you don't feel like you can get all the way back 01:01:49.820 --> 01:01:53.060 align:middle line:84% to initial distribution, can you at least analyze it 01:01:53.060 --> 01:01:57.170 align:middle line:84% for frequency distribution or something to get, like, 01:01:57.170 --> 01:01:59.750 align:middle line:84% oh, I must have had this many spatial scales 01:01:59.750 --> 01:02:02.270 align:middle line:84% in my original plasma or something, or-- 01:02:02.270 --> 01:02:04.910 align:middle line:84% JACK HARE: Yeah, so it's very tempting, when you have an image 01:02:04.910 --> 01:02:07.730 align:middle line:84% or when you have a time-series bit of data, 01:02:07.730 --> 01:02:11.288 align:middle line:84% to Fourier transform it and look for spectral content. 01:02:11.288 --> 01:02:13.580 align:middle line:84% And in particular, when we're talking about turbulence, 01:02:13.580 --> 01:02:16.920 align:middle line:84% we might do that, and we might look for power spectra 01:02:16.920 --> 01:02:20.770 align:middle line:84% corresponding to some turbulent density fluctuation spectrum, 01:02:20.770 --> 01:02:24.640 align:middle line:84% so, like, Kolmogorov "K to the minus 5/3" distribution. 01:02:24.640 --> 01:02:25.990 align:middle line:90% And so I did this. 01:02:25.990 --> 01:02:29.350 align:middle line:84% I did this, and of course, you get a really nice K to the minus 01:02:29.350 --> 01:02:30.255 align:middle line:90% 5/3 on this. 01:02:30.255 --> 01:02:31.960 align:middle line:84% And then I took the background image, 01:02:31.960 --> 01:02:34.990 align:middle line:84% the one without the plasma, and you also get K to the minus 5/3. 01:02:34.990 --> 01:02:37.540 align:middle line:84% And then I took a photograph of the experimental apparatus 01:02:37.540 --> 01:02:39.550 align:middle line:84% and Fourier-transformed the photograph, 01:02:39.550 --> 01:02:42.255 align:middle line:84% and you also get something like K to the minus 5/3. 01:02:42.255 --> 01:02:43.630 align:middle line:84% The trouble is, when you're doing 01:02:43.630 --> 01:02:46.450 align:middle line:84% Fourier transforms on images, you've got to think, 01:02:46.450 --> 01:02:47.620 align:middle line:90% How many pixels have I got? 01:02:47.620 --> 01:02:50.710 align:middle line:84% You've probably got, like, a thousand by a thousand pixels. 01:02:50.710 --> 01:02:53.230 align:middle line:84% And so when you're doing your Fourier transform, 01:02:53.230 --> 01:02:55.840 align:middle line:84% your dynamic range is only going to be about 10 plus 3. 01:02:55.840 --> 01:03:00.217 align:middle line:84% But at those large scales, at the smallest K, 01:03:00.217 --> 01:03:02.050 align:middle line:84% it's going to be like large-scale structure, 01:03:02.050 --> 01:03:03.680 align:middle line:84% so you wouldn't fit a power spectrum there. 01:03:03.680 --> 01:03:05.680 align:middle line:84% And at small scales, it's going to be down at pixel noise, 01:03:05.680 --> 01:03:06.460 align:middle line:90% so you wouldn't fit it there. 01:03:06.460 --> 01:03:08.835 align:middle line:84% So actually, you've only got maybe an order of magnitude, 01:03:08.835 --> 01:03:12.610 align:middle line:84% and you can fit any straight line you want to a curve 01:03:12.610 --> 01:03:15.700 align:middle line:84% and claim that you've got K to the minus 5/3 or K to the minus 01:03:15.700 --> 01:03:18.980 align:middle line:84% 3/2 or whatever, because when you do turbulence theory, 01:03:18.980 --> 01:03:20.990 align:middle line:84% they all turn out to be roughly the same. 01:03:20.990 --> 01:03:23.680 align:middle line:84% So that's one reason it's really hard just 01:03:23.680 --> 01:03:24.820 align:middle line:90% to Fourier-transform these. 01:03:24.820 --> 01:03:26.945 align:middle line:84% The second reason is, as we've talked about before, 01:03:26.945 --> 01:03:27.890 align:middle line:90% this is a mirage. 01:03:27.890 --> 01:03:29.660 align:middle line:90% It's not an image. 01:03:29.660 --> 01:03:32.230 align:middle line:84% So if I see a region like this black region 01:03:32.230 --> 01:03:34.660 align:middle line:84% here, or another region-- maybe it's 01:03:34.660 --> 01:03:36.010 align:middle line:90% easier to point out on this one. 01:03:36.010 --> 01:03:37.720 align:middle line:84% You see these sort of black voids here, 01:03:37.720 --> 01:03:39.810 align:middle line:84% and you think, OK, I could just be like, hey, 01:03:39.810 --> 01:03:42.310 align:middle line:84% this is about 2 millimeters long, this is 1 millimeter long, 01:03:42.310 --> 01:03:44.890 align:middle line:84% make a histogram, fit a power law or something to it. 01:03:44.890 --> 01:03:46.870 align:middle line:84% But these don't represent an object 01:03:46.870 --> 01:03:48.250 align:middle line:90% that is 2 millimeters long. 01:03:48.250 --> 01:03:50.380 align:middle line:90% They are a defocusing mechanism. 01:03:50.380 --> 01:03:53.110 align:middle line:84% That could be a really tiny region of very high density, 01:03:53.110 --> 01:03:55.630 align:middle line:84% that defocus, and then it's projected out 01:03:55.630 --> 01:03:57.020 align:middle line:90% into this larger region. 01:03:57.020 --> 01:04:01.100 align:middle line:84% So we can't-- there's no spatial information properly left inside 01:04:01.100 --> 01:04:01.765 align:middle line:90% this image. 01:04:01.765 --> 01:04:03.140 align:middle line:84% There's some spatial information. 01:04:03.140 --> 01:04:05.323 align:middle line:84% This sort of structure here corresponds 01:04:05.323 --> 01:04:06.740 align:middle line:84% to this sort of structure, and you 01:04:06.740 --> 01:04:08.990 align:middle line:84% think, it's about 5 millimeters across, it's probably 01:04:08.990 --> 01:04:11.210 align:middle line:84% slightly de-magnified, because it will actually 01:04:11.210 --> 01:04:13.400 align:middle line:90% make a larger image. 01:04:13.400 --> 01:04:15.350 align:middle line:84% But each of these individual voids 01:04:15.350 --> 01:04:16.883 align:middle line:84% no longer has the same spatial size 01:04:16.883 --> 01:04:18.300 align:middle line:84% as the structure that produced it, 01:04:18.300 --> 01:04:20.383 align:middle line:84% so it's very hard to infer things about turbulence 01:04:20.383 --> 01:04:21.450 align:middle line:90% from them. 01:04:21.450 --> 01:04:24.290 align:middle line:90% But, yeah, it's a good question. 01:04:24.290 --> 01:04:26.680 align:middle line:90% Any other questions? 01:04:26.680 --> 01:04:27.870 align:middle line:90% Mm-hmm? 01:04:27.870 --> 01:04:29.870 align:middle line:84% AUDIENCE: Can you get back into the difficulties 01:04:29.870 --> 01:04:31.450 align:middle line:90% of using Bott's base? 01:04:31.450 --> 01:04:34.330 align:middle line:84% If you reconstruct Bott's example, 01:04:34.330 --> 01:04:36.380 align:middle line:84% even though there's the usual caustics, 01:04:36.380 --> 01:04:41.183 align:middle line:84% you still get some information where one-- 01:04:41.183 --> 01:04:42.600 align:middle line:84% JACK HARE: So don't think you can. 01:04:42.600 --> 01:04:45.510 align:middle line:84% I mean, when you run this algorithm on your data, 01:04:45.510 --> 01:04:47.435 align:middle line:84% shadowgraphy or proton radiography, one 01:04:47.435 --> 01:04:49.560 align:middle line:84% of these Monge-Ampere optimal transport algorithms, 01:04:49.560 --> 01:04:52.130 align:middle line:84% it will always give you an answer, right. 01:04:52.130 --> 01:04:54.920 align:middle line:84% So it returns it returns a solution, right. 01:04:54.920 --> 01:04:58.580 align:middle line:84% But we know that these reconstruction algorithms don't 01:04:58.580 --> 01:05:00.930 align:middle line:84% work when we have the caustic regime, 01:05:00.930 --> 01:05:02.840 align:middle line:84% and so that solution is very suspect. 01:05:02.840 --> 01:05:04.882 align:middle line:84% Right, like, we don't believe we should trust it. 01:05:04.882 --> 01:05:06.257 align:middle line:84% So I don't think there's anything 01:05:06.257 --> 01:05:08.570 align:middle line:84% you can use from that solution to help you reconstruct 01:05:08.570 --> 01:05:09.305 align:middle line:90% the actual thing. 01:05:09.305 --> 01:05:11.263 align:middle line:84% I think, at that point, if you've got caustics, 01:05:11.263 --> 01:05:13.880 align:middle line:84% your best bet is having some very strong priors as 01:05:13.880 --> 01:05:15.740 align:middle line:84% to the sort of plasma you think you've got, 01:05:15.740 --> 01:05:19.330 align:middle line:84% and then pushing Monte Carlo rays or protons through it, 01:05:19.330 --> 01:05:21.830 align:middle line:84% doing the forward problem, and adjusting the forward problem 01:05:21.830 --> 01:05:23.372 align:middle line:84% until it matches some of the features 01:05:23.372 --> 01:05:24.673 align:middle line:90% you see on your actual data. 01:05:24.673 --> 01:05:26.840 align:middle line:84% I don't think you can do the inverse problem easily. 01:05:26.840 --> 01:05:28.490 align:middle line:84% But there have been a few papers where 01:05:28.490 --> 01:05:30.470 align:middle line:84% people have tried to do this, because we almost always end up 01:05:30.470 --> 01:05:31.440 align:middle line:90% in the caustic regime. 01:05:31.440 --> 01:05:33.710 align:middle line:84% So people have all this data, and they want to use it. 01:05:33.710 --> 01:05:36.252 align:middle line:84% Like, it's reasonable to try and do something with that data. 01:05:36.252 --> 01:05:39.760 align:middle line:90% It's just very hard, so, yeah. 01:05:39.760 --> 01:05:40.260 align:middle line:90% Yeah? 01:05:40.260 --> 01:05:42.420 align:middle line:84% AUDIENCE: So for those images of, like, 01:05:42.420 --> 01:05:46.170 align:middle line:84% the heads and the room, those are not 01:05:46.170 --> 01:05:47.620 align:middle line:90% super small-scale effects. 01:05:47.620 --> 01:05:51.330 align:middle line:84% So to see those, do you just push your imaging surface really 01:05:51.330 --> 01:05:52.170 align:middle line:90% far away? 01:05:52.170 --> 01:05:55.980 align:middle line:84% JACK HARE: If you take a laser pointer 01:05:55.980 --> 01:05:58.095 align:middle line:84% and make it diverge by taking off 01:05:58.095 --> 01:06:00.720 align:middle line:84% the little lens at the front of it, and you take a candle flame 01:06:00.720 --> 01:06:02.803 align:middle line:84% and you project it onto a wall, you will see this. 01:06:02.803 --> 01:06:06.120 align:middle line:84% Like, again, you see mirages just using sunlight, so, yeah, 01:06:06.120 --> 01:06:07.630 align:middle line:84% this is not a hard thing to observe. 01:06:07.630 --> 01:06:07.770 align:middle line:90% Yeah. 01:06:07.770 --> 01:06:09.610 align:middle line:84% AUDIENCE: OK, so if you see something with your naked eye, 01:06:09.610 --> 01:06:12.060 align:middle line:84% it's either a really small object or something really 01:06:12.060 --> 01:06:12.600 align:middle line:90% far away? 01:06:12.600 --> 01:06:12.990 align:middle line:90% JACK HARE: Yes. 01:06:12.990 --> 01:06:13.650 align:middle line:90% AUDIENCE: OK. 01:06:13.650 --> 01:06:15.180 align:middle line:84% JACK HARE: Yes, exactly, and usually, it's 01:06:15.180 --> 01:06:16.097 align:middle line:90% quite far away, right. 01:06:16.097 --> 01:06:18.450 align:middle line:84% I mean, if we-- you can often see even the heat rising 01:06:18.450 --> 01:06:19.908 align:middle line:84% from a vent or something like that, 01:06:19.908 --> 01:06:21.640 align:middle line:84% but not when you're right up close to it. 01:06:21.640 --> 01:06:24.000 align:middle line:84% Give it a go next time you see something, if it's safe. 01:06:24.000 --> 01:06:25.395 align:middle line:84% [LAUGHS] Put your eye right up next to it 01:06:25.395 --> 01:06:27.060 align:middle line:84% and see if it disappears, so, yeah. 01:06:27.060 --> 01:06:27.602 align:middle line:90% AUDIENCE: OK. 01:06:27.602 --> 01:06:28.680 align:middle line:90% Cool. 01:06:28.680 --> 01:06:30.186 align:middle line:90% JACK HARE: Please don't, anyway. 01:06:30.186 --> 01:06:32.010 align:middle line:90% [LAUGHTER] 01:06:32.010 --> 01:06:34.905 align:middle line:90% Any other questions? 01:06:34.905 --> 01:06:38.670 align:middle line:84% AUDIENCE: Can you give one more example of when you can see-- 01:06:38.670 --> 01:06:42.990 align:middle line:84% like, in this exact duration, is like 01:06:42.990 --> 01:06:46.170 align:middle line:84% if you have sunlight coming through a window 01:06:46.170 --> 01:06:48.163 align:middle line:84% and over like a heater or something like that. 01:06:48.163 --> 01:06:49.580 align:middle line:84% JACK HARE: Yes, onto the far wall. 01:06:49.580 --> 01:06:50.750 align:middle line:90% AUDIENCE: Onto the far wall. 01:06:50.750 --> 01:06:51.100 align:middle line:90% OK. 01:06:51.100 --> 01:06:51.725 align:middle line:90% JACK HARE: Yes. 01:06:51.725 --> 01:06:52.590 align:middle line:90% Yeah. 01:06:52.590 --> 01:06:53.590 align:middle line:90% AUDIENCE: Yeah, exactly. 01:06:53.590 --> 01:06:54.010 align:middle line:90% That's what I'm saying. 01:06:54.010 --> 01:06:55.093 align:middle line:90% JACK HARE: Another place-- 01:06:55.093 --> 01:06:56.390 align:middle line:90% the bottom of a swimming pool. 01:06:56.390 --> 01:06:59.232 align:middle line:84% So when you have waves on the top surface of a swimming pool, 01:06:59.232 --> 01:07:00.940 align:middle line:84% you get those bright lines on the bottom. 01:07:00.940 --> 01:07:03.820 align:middle line:84% Those are caustics, right, so exactly the same sort of physics 01:07:03.820 --> 01:07:05.020 align:middle line:90% is produced as these. 01:07:05.020 --> 01:07:07.180 align:middle line:84% So there is a beautiful book called 01:07:07.180 --> 01:07:10.600 align:middle line:84% The Natural Focusing of Light which tries to analyze caustics 01:07:10.600 --> 01:07:13.270 align:middle line:84% but mostly does it in a way that I don't think is practical. 01:07:13.270 --> 01:07:15.410 align:middle line:84% It's theoretically beautiful, and one 01:07:15.410 --> 01:07:17.410 align:middle line:84% of the things that they point out about caustics 01:07:17.410 --> 01:07:19.202 align:middle line:84% is that every time you get a bright region, 01:07:19.202 --> 01:07:23.110 align:middle line:84% you get a dark region on one side but not on the other side, 01:07:23.110 --> 01:07:26.060 align:middle line:84% and that tells you what direction the caustic came from. 01:07:26.060 --> 01:07:29.230 align:middle line:84% And so you might be able to trace back a series of arrows 01:07:29.230 --> 01:07:31.762 align:middle line:84% around one of these caustics here and work out 01:07:31.762 --> 01:07:33.970 align:middle line:84% where the point was that the caustic originated from, 01:07:33.970 --> 01:07:36.950 align:middle line:84% but it's very tricky to do the analysis of it. 01:07:36.950 --> 01:07:38.560 align:middle line:84% But the point is the book is called 01:07:38.560 --> 01:07:40.360 align:middle line:84% The Natural Focusing of Light, so people 01:07:40.360 --> 01:07:41.710 align:middle line:84% refer to this field as natural focusing. 01:07:41.710 --> 01:07:43.180 align:middle line:90% No one has tried to make a lens. 01:07:43.180 --> 01:07:44.950 align:middle line:84% No one is trying to do any focusing. 01:07:44.950 --> 01:07:47.950 align:middle line:84% Our medium, with its inhomogeneous refractive index, 01:07:47.950 --> 01:07:49.540 align:middle line:84% has just done it for us, and then we 01:07:49.540 --> 01:07:52.150 align:middle line:84% might try and work out what we can learn about the medium 01:07:52.150 --> 01:07:54.192 align:middle line:84% from looking at the light that's gone through it. 01:07:54.192 --> 01:07:58.050 align:middle line:90% 01:07:58.050 --> 01:08:00.377 align:middle line:90% Yes? 01:08:00.377 --> 01:08:01.710 align:middle line:90% AUDIENCE: One more thing, that-- 01:08:01.710 --> 01:08:04.510 align:middle line:84% to make sure I don't try this later. 01:08:04.510 --> 01:08:07.530 align:middle line:84% So if you have an image like the one with the bullet, 01:08:07.530 --> 01:08:10.880 align:middle line:84% for instance, there are some really clear caustics, 01:08:10.880 --> 01:08:13.010 align:middle line:84% then parts of the rest of the image 01:08:13.010 --> 01:08:16.220 align:middle line:84% look like they might be fine, if you 01:08:16.220 --> 01:08:18.555 align:middle line:84% want to analyze the rest of your image, is that-- 01:08:18.555 --> 01:08:21.180 align:middle line:84% JACK HARE: Oh, yeah, you can cut out the bit with the caustics. 01:08:21.180 --> 01:08:21.390 align:middle line:90% AUDIENCE: OK. 01:08:21.390 --> 01:08:22.250 align:middle line:84% JACK HARE: Yeah, yeah, that's fine. 01:08:22.250 --> 01:08:25.279 align:middle line:84% So, I mean, almost none of this image is actually suitable, 01:08:25.279 --> 01:08:28.217 align:middle line:84% but maybe this bit would be more-- 01:08:28.217 --> 01:08:30.050 align:middle line:84% because it's not small intensity variations. 01:08:30.050 --> 01:08:33.260 align:middle line:84% You can see, if you think about this gray as, like, 0.5 01:08:33.260 --> 01:08:35.937 align:middle line:84% and you think about the white as 1 and the black as 0, 01:08:35.937 --> 01:08:37.729 align:middle line:84% you can see that you're getting modulations 01:08:37.729 --> 01:08:39.600 align:middle line:84% on the order of 0.5 inside this image, 01:08:39.600 --> 01:08:42.290 align:middle line:84% so it's clearly not in that small regime. 01:08:42.290 --> 01:08:44.312 align:middle line:84% But it's not clear that these are caustics, 01:08:44.312 --> 01:08:45.770 align:middle line:84% so you may still be able to use one 01:08:45.770 --> 01:08:49.790 align:middle line:84% of the complicated Monge-Ampere style reconstruction techniques. 01:08:49.790 --> 01:08:52.160 align:middle line:84% You just won't be able to use the nice formula 01:08:52.160 --> 01:08:54.009 align:middle line:84% that we wrote down analytically, so. 01:08:54.009 --> 01:08:57.140 align:middle line:90% 01:08:57.140 --> 01:08:58.770 align:middle line:84% OK, you've successfully timed me out. 01:08:58.770 --> 01:09:00.140 align:middle line:84% I was going to start talking on interferometry. 01:09:00.140 --> 01:09:01.010 align:middle line:90% Well done, everyone. 01:09:01.010 --> 01:09:03.859 align:middle line:84% So we'll leave it there, and we will pick up on interferometry 01:09:03.859 --> 01:09:06.350 align:middle line:90% on Thursday. 01:09:06.350 --> 01:09:08.200 align:middle line:90% Sounds good. 01:09:08.200 --> 01:09:10.000 align:middle line:90%