WEBVTT 00:00:00.000 --> 00:00:00.960 align:middle line:90% 00:00:00.960 --> 00:00:02.400 align:middle line:90% [SQUEAKING] 00:00:02.400 --> 00:00:03.840 align:middle line:90% [RUSTLING] 00:00:03.840 --> 00:00:10.100 align:middle line:90% [CLICKING] 00:00:10.100 --> 00:00:12.780 align:middle line:84% JACK HARE: So we're going to go on to a slightly new topic 00:00:12.780 --> 00:00:13.280 align:middle line:90% today. 00:00:13.280 --> 00:00:15.590 align:middle line:84% We're going to be studying Abel inversion 00:00:15.590 --> 00:00:19.310 align:middle line:84% and then talking about Faraday rotation imaging. 00:00:19.310 --> 00:00:22.040 align:middle line:84% Does anyone have any questions on all the interferometry stuff 00:00:22.040 --> 00:00:25.340 align:middle line:84% we've covered so far before we leave that part alone? 00:00:25.340 --> 00:00:30.940 align:middle line:90% 00:00:30.940 --> 00:00:33.040 align:middle line:84% Everyone seems very happy with interferometry. 00:00:33.040 --> 00:00:33.790 align:middle line:90% All right. 00:00:33.790 --> 00:00:39.032 align:middle line:84% So interferometry-- this technique 00:00:39.032 --> 00:00:41.490 align:middle line:84% that we're going to be talking about today, Abel inversion, 00:00:41.490 --> 00:00:43.860 align:middle line:90% is actually quite general. 00:00:43.860 --> 00:00:46.620 align:middle line:84% And I'm introducing it in the context of interferometry 00:00:46.620 --> 00:00:50.650 align:middle line:84% because we often use it with this technique, 00:00:50.650 --> 00:00:55.590 align:middle line:84% but it could also be used with emission from plasma, 00:00:55.590 --> 00:00:57.000 align:middle line:90% from individuals. 00:00:57.000 --> 00:00:59.220 align:middle line:84% We'll talk a little bit about what Abel inversion is 00:00:59.220 --> 00:01:00.600 align:middle line:84% and what symmetry requirements we have, 00:01:00.600 --> 00:01:02.880 align:middle line:84% and you'll see quite quickly it's actually quite general. 00:01:02.880 --> 00:01:05.005 align:middle line:84% And it can be used in lots of different cases here. 00:01:05.005 --> 00:01:07.980 align:middle line:84% So what we have from interferometry-- 00:01:07.980 --> 00:01:10.530 align:middle line:84% I'll just write as IF like that-- 00:01:10.530 --> 00:01:15.750 align:middle line:84% is we've got some integration of any dl. 00:01:15.750 --> 00:01:18.150 align:middle line:84% So we have some line integrated quantity. 00:01:18.150 --> 00:01:20.910 align:middle line:84% And we might have that line integrated quantity 00:01:20.910 --> 00:01:22.830 align:middle line:90% along a very specific chord. 00:01:22.830 --> 00:01:26.500 align:middle line:84% So we might be integrating along the z direction, 00:01:26.500 --> 00:01:33.150 align:middle line:84% and we might be at x equals x 0, y equals y 0. 00:01:33.150 --> 00:01:35.700 align:middle line:84% And we might be resolving it as a function of time. 00:01:35.700 --> 00:01:39.320 align:middle line:84% That would be our temporally resolved interferometry. 00:01:39.320 --> 00:01:43.060 align:middle line:84% Or we might have an image of x and y, 00:01:43.060 --> 00:01:45.860 align:middle line:84% and we would have it at some specific time. 00:01:45.860 --> 00:01:48.920 align:middle line:84% This would be our spatially resolved interferometry. 00:01:48.920 --> 00:01:49.930 align:middle line:90% So this is what we have. 00:01:49.930 --> 00:01:52.720 align:middle line:90% 00:01:52.720 --> 00:01:56.110 align:middle line:90% And of course, what we want-- 00:01:56.110 --> 00:01:59.610 align:middle line:84% that's what we can't have, which is the electron density 00:01:59.610 --> 00:02:05.070 align:middle line:84% as a function of position everywhere in space, 00:02:05.070 --> 00:02:07.181 align:middle line:84% preferably as a function of position of time. 00:02:07.181 --> 00:02:08.639 align:middle line:84% This is what you'd like if you want 00:02:08.639 --> 00:02:11.760 align:middle line:84% to compare a simulation or a theory or something like that. 00:02:11.760 --> 00:02:15.270 align:middle line:84% And instead, we have these line integrated measurements. 00:02:15.270 --> 00:02:16.380 align:middle line:90% OK. 00:02:16.380 --> 00:02:22.080 align:middle line:84% So if you want to calculate this density from some 00:02:22.080 --> 00:02:24.870 align:middle line:84% of these limited reduced data sets, 00:02:24.870 --> 00:02:29.170 align:middle line:84% you could do a technique like tomography. 00:02:29.170 --> 00:02:34.060 align:middle line:84% So if you've ever gone for an MRI or something similar-- 00:02:34.060 --> 00:02:40.310 align:middle line:84% tomography-- you'll know that what they do 00:02:40.310 --> 00:02:42.500 align:middle line:84% is they'll take lots and lots of different images, 00:02:42.500 --> 00:02:45.050 align:middle line:84% slowly scanning around your head or the injured part 00:02:45.050 --> 00:02:48.110 align:middle line:84% of your body, and then they'll do some very fancy computer 00:02:48.110 --> 00:02:50.960 align:middle line:84% techniques to reconstruct the three-dimensional structure 00:02:50.960 --> 00:02:52.820 align:middle line:84% of whatever it is they're scanning. 00:02:52.820 --> 00:02:54.740 align:middle line:84% And this works extremely well because people 00:02:54.740 --> 00:02:56.150 align:middle line:90% have lifetimes of years. 00:02:56.150 --> 00:03:00.260 align:middle line:84% But plasmas only have lifetimes of seconds or milliseconds 00:03:00.260 --> 00:03:02.120 align:middle line:84% or nanoseconds, and so it's very hard 00:03:02.120 --> 00:03:04.382 align:middle line:84% to just slowly rotate your plasma in place 00:03:04.382 --> 00:03:06.090 align:middle line:84% when you've taken lots of pictures of it. 00:03:06.090 --> 00:03:09.500 align:middle line:84% The alternative, if you want to do single-shot tomography, 00:03:09.500 --> 00:03:12.175 align:middle line:84% will be to surround your plasma with lots and lots of cameras 00:03:12.175 --> 00:03:14.050 align:middle line:84% and look at it from lots of different angles. 00:03:14.050 --> 00:03:20.340 align:middle line:84% So for example, if we've got a circular cross-section plasma 00:03:20.340 --> 00:03:23.400 align:middle line:84% like this, maybe this is some sort of automat type thing. 00:03:23.400 --> 00:03:29.170 align:middle line:84% We could just have lots of different lines of sight. 00:03:29.170 --> 00:03:31.960 align:middle line:84% And we can do our tomographic reconstruction like this. 00:03:31.960 --> 00:03:37.340 align:middle line:84% But of course, lines of sight, LOS, are expensive. 00:03:37.340 --> 00:03:40.330 align:middle line:84% So we don't tend to be able to just have a very, very 00:03:40.330 --> 00:03:41.540 align:middle line:90% large number of them. 00:03:41.540 --> 00:03:43.520 align:middle line:84% But of course, for some applications, 00:03:43.520 --> 00:03:45.190 align:middle line:84% this might be justifiable on each 00:03:45.190 --> 00:03:48.003 align:middle line:84% so they've got 550 kilometer lines of sight 00:03:48.003 --> 00:03:49.670 align:middle line:84% for reconstructing the emission from it. 00:03:49.670 --> 00:03:53.170 align:middle line:84% So they made a choice to have a lot of imaging lines of sight. 00:03:53.170 --> 00:03:56.330 align:middle line:84% In general, for interferometry, this is too expensive, 00:03:56.330 --> 00:03:58.400 align:middle line:90% so we don't do it. 00:03:58.400 --> 00:04:01.150 align:middle line:84% But what we can do is a version of this 00:04:01.150 --> 00:04:03.970 align:middle line:84% where we make strong arguments about the symmetry 00:04:03.970 --> 00:04:04.660 align:middle line:90% of our plasma. 00:04:04.660 --> 00:04:07.120 align:middle line:84% Because if our plasma has some underlying symmetry, 00:04:07.120 --> 00:04:09.790 align:middle line:84% it helps us need fewer lines of sight, 00:04:09.790 --> 00:04:12.760 align:middle line:84% and we can still start getting out an approximation 00:04:12.760 --> 00:04:14.950 align:middle line:90% of the full density profile. 00:04:14.950 --> 00:04:17.560 align:middle line:84% And the symmetry we're going to talk about today with Abel 00:04:17.560 --> 00:04:22.890 align:middle line:84% inversion is an assumption of cylindrical symmetry. 00:04:22.890 --> 00:04:28.350 align:middle line:84% So we're going to assume cylindrical symmetry. 00:04:28.350 --> 00:04:36.740 align:middle line:90% 00:04:36.740 --> 00:04:38.590 align:middle line:84% And so that could be, again, something 00:04:38.590 --> 00:04:43.910 align:middle line:84% like this plasma here with a circular cross-section. 00:04:43.910 --> 00:04:46.640 align:middle line:90% 00:04:46.640 --> 00:04:49.098 align:middle line:84% And we're going to assume that maybe there's variation 00:04:49.098 --> 00:04:50.390 align:middle line:90% in this direction out of plane. 00:04:50.390 --> 00:04:51.020 align:middle line:90% That doesn't matter. 00:04:51.020 --> 00:04:53.000 align:middle line:84% We're only trying to measure it in one plane. 00:04:53.000 --> 00:04:56.480 align:middle line:84% But we're going to assume that our electron density, any 00:04:56.480 --> 00:05:04.910 align:middle line:84% of xy and z, is just a function, any of R and z, 00:05:04.910 --> 00:05:10.020 align:middle line:84% where R squared is equal to x squared plus y squared. 00:05:10.020 --> 00:05:13.160 align:middle line:84% So that's the same as saying that the density is 00:05:13.160 --> 00:05:18.120 align:middle line:84% constant nested circular surfaces. 00:05:18.120 --> 00:05:20.810 align:middle line:84% And of course, if this is something like a tokamak, 00:05:20.810 --> 00:05:24.260 align:middle line:84% that's quite nice because we know that our flux services 00:05:24.260 --> 00:05:26.420 align:middle line:90% quantities tend to be constant. 00:05:26.420 --> 00:05:29.360 align:middle line:84% And this could also be true for a cylindrical z-pinch plasma 00:05:29.360 --> 00:05:31.838 align:middle line:84% and the sorts of experiments I do. 00:05:31.838 --> 00:05:33.380 align:middle line:84% And you can think of other situations 00:05:33.380 --> 00:05:36.530 align:middle line:84% where you think things are approximately symmetric. 00:05:36.530 --> 00:05:39.120 align:middle line:84% And if you have a system like this, 00:05:39.120 --> 00:05:42.710 align:middle line:84% you could have a set of interferometers 00:05:42.710 --> 00:05:47.410 align:middle line:84% looking along parallel chords like this. 00:05:47.410 --> 00:05:51.610 align:middle line:84% Or if you are working with a tokamak and you can do imaging, 00:05:51.610 --> 00:05:57.610 align:middle line:84% you could have a camera and expanded laser beam, 00:05:57.610 --> 00:05:59.080 align:middle line:90% as we discussed before. 00:05:59.080 --> 00:06:05.530 align:middle line:84% And that camera is now measuring any of-- 00:06:05.530 --> 00:06:08.352 align:middle line:90% let's see, xy. 00:06:08.352 --> 00:06:10.820 align:middle line:90% That t moves to 0. 00:06:10.820 --> 00:06:13.850 align:middle line:84% Or these could be a series of chords which are measuring any 00:06:13.850 --> 00:06:17.752 align:middle line:90% at x equals x0 times like that. 00:06:17.752 --> 00:06:20.210 align:middle line:84% So this is just two different ways to look at this problem. 00:06:20.210 --> 00:06:22.970 align:middle line:84% I can either build up my data from multiple time resolved 00:06:22.970 --> 00:06:26.180 align:middle line:84% interferometers, looking along parallel chords, 00:06:26.180 --> 00:06:30.020 align:middle line:84% or I can have an imaging system looking at a single time. 00:06:30.020 --> 00:06:32.000 align:middle line:84% In both these cases, we can do this thing 00:06:32.000 --> 00:06:34.100 align:middle line:90% called the Abel inversion. 00:06:34.100 --> 00:06:35.830 align:middle line:90% I'll just write down-- 00:06:35.830 --> 00:06:37.790 align:middle line:90% oh I wrote it down. 00:06:37.790 --> 00:06:38.690 align:middle line:90% OK, good. 00:06:38.690 --> 00:06:40.460 align:middle line:84% Abel is one of these guys who's depressing 00:06:40.460 --> 00:06:41.630 align:middle line:90% when you read his biography. 00:06:41.630 --> 00:06:44.000 align:middle line:90% He was a Danish mathematician. 00:06:44.000 --> 00:06:46.040 align:middle line:84% He invented all sorts of wonderful things 00:06:46.040 --> 00:06:48.200 align:middle line:84% in mathematics, as well as the Abel transformation. 00:06:48.200 --> 00:06:50.640 align:middle line:84% And he died of consumption at the age of 26. 00:06:50.640 --> 00:06:53.147 align:middle line:84% So I don't know how many of you are still younger than 26, 00:06:53.147 --> 00:06:54.980 align:middle line:84% but you've still got maybe a couple of years 00:06:54.980 --> 00:06:56.510 align:middle line:84% to make such groundbreaking discoveries. 00:06:56.510 --> 00:06:57.455 align:middle line:90% I'm already past it. 00:06:57.455 --> 00:06:59.060 align:middle line:90% I don't have a chance. 00:06:59.060 --> 00:07:01.010 align:middle line:84% So you look at his biography and you're like, 00:07:01.010 --> 00:07:03.050 align:middle line:84% damn, Abel got a lot of work done. 00:07:03.050 --> 00:07:04.590 align:middle line:90% OK. 00:07:04.590 --> 00:07:05.090 align:middle line:90% Cool. 00:07:05.090 --> 00:07:08.570 align:middle line:84% So what we have, say, from either 00:07:08.570 --> 00:07:21.150 align:middle line:84% of these two systems is a map of the line-integrated electron 00:07:21.150 --> 00:07:22.948 align:middle line:90% density as a function of y. 00:07:22.948 --> 00:07:24.990 align:middle line:84% And I'm just going to draw this very suggestively 00:07:24.990 --> 00:07:27.490 align:middle line:90% as this very blocky setup here. 00:07:27.490 --> 00:07:31.410 align:middle line:84% So each of these densities could be the density 00:07:31.410 --> 00:07:34.080 align:middle line:84% that we've measured at a single pixel on our image 00:07:34.080 --> 00:07:36.150 align:middle line:84% or it could be the density measured 00:07:36.150 --> 00:07:41.290 align:middle line:84% by our n time-resolved interferometers here. 00:07:41.290 --> 00:07:46.990 align:middle line:84% So we've got some density value at each of these points, 00:07:46.990 --> 00:07:48.410 align:middle line:90% like this here. 00:07:48.410 --> 00:08:05.000 align:middle line:84% And what we want is, of course, our plasma density 00:08:05.000 --> 00:08:07.040 align:middle line:84% as a function of R, not as a function 00:08:07.040 --> 00:08:10.370 align:middle line:90% of y, which is a coordinate. 00:08:10.370 --> 00:08:14.145 align:middle line:84% For example, it could be one of these two. 00:08:14.145 --> 00:08:16.270 align:middle line:84% I'm not really distinguishing between y and x here. 00:08:16.270 --> 00:08:18.280 align:middle line:84% As long as it's perpendicular to the probing direction, 00:08:18.280 --> 00:08:19.190 align:middle line:90% it doesn't matter. 00:08:19.190 --> 00:08:21.730 align:middle line:84% But what we want is this as a function of R. 00:08:21.730 --> 00:08:25.555 align:middle line:84% And we'd like to have some nice, smooth function. 00:08:25.555 --> 00:08:27.180 align:middle line:84% And it doesn't necessarily have to have 00:08:27.180 --> 00:08:29.435 align:middle line:84% the same shape, of course, because it's clear 00:08:29.435 --> 00:08:30.810 align:middle line:84% if you look at this and you think 00:08:30.810 --> 00:08:34.140 align:middle line:84% about it for a little while that your profile in dl 00:08:34.140 --> 00:08:38.049 align:middle line:84% doesn't have to be the same as your profile in any like this. 00:08:38.049 --> 00:08:40.230 align:middle line:84% So what we want is some mathematical formalism 00:08:40.230 --> 00:08:45.840 align:middle line:84% to allow us to take this data and produce this. 00:08:45.840 --> 00:08:47.760 align:middle line:84% It's clear how to go back the other way. 00:08:47.760 --> 00:08:50.940 align:middle line:84% You can certainly do it numerically very easily. 00:08:50.940 --> 00:08:53.420 align:middle line:90% You can just make this up. 00:08:53.420 --> 00:08:55.142 align:middle line:84% You can make up some profile like this, 00:08:55.142 --> 00:08:57.600 align:middle line:84% and then you can just calculate the line-integrated density 00:08:57.600 --> 00:08:58.980 align:middle line:90% along each line of sight. 00:08:58.980 --> 00:09:01.920 align:middle line:84% What's less clear is how to go back the other way from the data 00:09:01.920 --> 00:09:03.460 align:middle line:90% we have to the data we want. 00:09:03.460 --> 00:09:05.320 align:middle line:84% So this is the setup of the problem. 00:09:05.320 --> 00:09:08.340 align:middle line:84% So does anyone have any questions about this so far? 00:09:08.340 --> 00:09:12.045 align:middle line:90% 00:09:12.045 --> 00:09:12.920 align:middle line:90% Any questions online? 00:09:12.920 --> 00:09:31.090 align:middle line:90% 00:09:31.090 --> 00:09:32.300 align:middle line:90% This is what we measure. 00:09:32.300 --> 00:09:37.150 align:middle line:84% And I'm going to call this some function, F of y. 00:09:37.150 --> 00:09:41.065 align:middle line:84% And I'm going to call this some function f of R. 00:09:41.065 --> 00:09:42.940 align:middle line:84% And the reason is because it doesn't actually 00:09:42.940 --> 00:09:46.000 align:middle line:84% matter whether this is density or brightness or whatever else. 00:09:46.000 --> 00:09:47.540 align:middle line:84% The mathematics are all the same. 00:09:47.540 --> 00:09:49.600 align:middle line:84% So I'm just going to refer to them as these two 00:09:49.600 --> 00:09:50.470 align:middle line:90% different functions. 00:09:50.470 --> 00:09:52.420 align:middle line:84% And we're trying to convert one to the other. 00:09:52.420 --> 00:09:55.710 align:middle line:90% 00:09:55.710 --> 00:09:59.630 align:middle line:84% So mathematically, we have our line-integrated function, 00:09:59.630 --> 00:10:06.410 align:middle line:84% F of y, which is equal to the integral of any 00:10:06.410 --> 00:10:08.350 align:middle line:90% exerted like this. 00:10:08.350 --> 00:10:11.230 align:middle line:84% And this is equal to the integral 00:10:11.230 --> 00:10:18.130 align:middle line:84% from minus a squared minus y squared square rooted 00:10:18.130 --> 00:10:24.812 align:middle line:84% to plus a squared plus y squared like this-- 00:10:24.812 --> 00:10:25.856 align:middle line:90% f of r. 00:10:25.856 --> 00:10:30.720 align:middle line:90% 00:10:30.720 --> 00:10:32.390 align:middle line:90% Exactly. 00:10:32.390 --> 00:10:35.030 align:middle line:90% [INAUDIBLE] very quickly. 00:10:35.030 --> 00:10:48.765 align:middle line:90% 00:10:48.765 --> 00:10:49.265 align:middle line:90% OK. 00:10:49.265 --> 00:10:57.190 align:middle line:90% 00:10:57.190 --> 00:10:59.900 align:middle line:84% I had it right in my notes, besides the change on the fly. 00:10:59.900 --> 00:11:00.980 align:middle line:90% I hate it. 00:11:00.980 --> 00:11:02.930 align:middle line:90% We're going back to the notes. 00:11:02.930 --> 00:11:03.500 align:middle line:90% OK. 00:11:03.500 --> 00:11:05.465 align:middle line:84% And I'll draw you a diagram of the geometry here. 00:11:05.465 --> 00:11:07.160 align:middle line:84% It might be slightly different from the background 00:11:07.160 --> 00:11:09.210 align:middle line:84% of the geometry I had just previously here. 00:11:09.210 --> 00:11:20.090 align:middle line:84% So we have some plasma which has an approximately circular shape. 00:11:20.090 --> 00:11:25.580 align:middle line:84% It's bounded at a, so we can say that the pressure at a 00:11:25.580 --> 00:11:27.680 align:middle line:90% is equal to 0. 00:11:27.680 --> 00:11:29.722 align:middle line:84% This just stops us having to integrate out 00:11:29.722 --> 00:11:32.180 align:middle line:84% to infinity, which is very inconvenient when you try and do 00:11:32.180 --> 00:11:33.150 align:middle line:90% it in reality. 00:11:33.150 --> 00:11:35.715 align:middle line:84% So we're going to stop our plasma at some distance. 00:11:35.715 --> 00:11:37.340 align:middle line:84% So we only need to make measurements up 00:11:37.340 --> 00:11:40.640 align:middle line:84% to the boundary, a, in order to solve this problem. 00:11:40.640 --> 00:11:46.350 align:middle line:84% We've got some chords going through the plasma. 00:11:46.350 --> 00:11:49.430 align:middle line:84% We've got our coordinate system, where 00:11:49.430 --> 00:11:53.150 align:middle line:84% y is transverse to the direction that we're probing 00:11:53.150 --> 00:11:55.430 align:middle line:84% and the x now is the direction that we're probing. 00:11:55.430 --> 00:11:58.100 align:middle line:84% I was using z previously, but the way I've got it written 00:11:58.100 --> 00:11:58.730 align:middle line:90% is x. 00:11:58.730 --> 00:12:01.820 align:middle line:84% I got myself getting confused with how to do that. 00:12:01.820 --> 00:12:06.190 align:middle line:84% And so any point inside the plasma, 00:12:06.190 --> 00:12:09.280 align:middle line:84% you can say that there is some distance, y, 00:12:09.280 --> 00:12:12.280 align:middle line:84% that it sits at from the origin here 00:12:12.280 --> 00:12:16.690 align:middle line:84% and some distance x along that it sits out. 00:12:16.690 --> 00:12:19.610 align:middle line:90% And so there's some distance, r. 00:12:19.610 --> 00:12:21.280 align:middle line:84% And this is the radial coordinate. 00:12:21.280 --> 00:12:23.110 align:middle line:84% We're assuming that we have symmetry 00:12:23.110 --> 00:12:25.850 align:middle line:84% in the azimuthal direction, in the angular direction. 00:12:25.850 --> 00:12:28.810 align:middle line:84% So we're only interested in the size of this radial coordinate 00:12:28.810 --> 00:12:30.260 align:middle line:90% here. 00:12:30.260 --> 00:12:32.840 align:middle line:84% And then if you look at this and stare at it for long enough, 00:12:32.840 --> 00:12:35.298 align:middle line:84% you can see that this is indeed the procedure I was talking 00:12:35.298 --> 00:12:36.910 align:middle line:84% about before, where you can easily 00:12:36.910 --> 00:12:41.200 align:middle line:84% go from your-- if you come up with some calculated profile, 00:12:41.200 --> 00:12:43.270 align:middle line:84% some guess at what you think the distribution is, 00:12:43.270 --> 00:12:45.130 align:middle line:84% like a Gaussian-- how you go from that 00:12:45.130 --> 00:12:47.650 align:middle line:84% to your prediction of what you're actually going 00:12:47.650 --> 00:12:48.830 align:middle line:90% to get from your detector. 00:12:48.830 --> 00:12:50.230 align:middle line:90% So this is the easy direction. 00:12:50.230 --> 00:12:52.510 align:middle line:84% This is actually the Abel transformation. 00:12:52.510 --> 00:12:55.720 align:middle line:84% And what we're really interested in is the inverse Abel 00:12:55.720 --> 00:12:56.621 align:middle line:90% transform. 00:12:56.621 --> 00:13:05.290 align:middle line:90% 00:13:05.290 --> 00:13:09.310 align:middle line:84% Now, we're technically requiring this condition here. 00:13:09.310 --> 00:13:12.121 align:middle line:84% I've written as pressure, but why don't we, 00:13:12.121 --> 00:13:15.390 align:middle line:84% now that we're talking in terms of these functions, 00:13:15.390 --> 00:13:19.680 align:middle line:84% I've written that I want f of a to be equal to 0. 00:13:19.680 --> 00:13:21.090 align:middle line:90% This isn't actually quite true. 00:13:21.090 --> 00:13:27.400 align:middle line:84% The rigorous requirement is that f of r as r tends to infinity. 00:13:27.400 --> 00:13:36.010 align:middle line:84% This needs to drop off or fall faster than 1 over r. 00:13:36.010 --> 00:13:38.167 align:middle line:84% So you can get away with a Gaussian type function 00:13:38.167 --> 00:13:40.750 align:middle line:84% or something like that, as long as it falls sufficiently fast. 00:13:40.750 --> 00:13:42.417 align:middle line:84% You can't get away with something that's 00:13:42.417 --> 00:13:43.900 align:middle line:90% uniform across all space. 00:13:43.900 --> 00:13:46.690 align:middle line:84% But as long as your function falls off nice and quickly, 00:13:46.690 --> 00:13:48.360 align:middle line:90% you can use this technique here. 00:13:48.360 --> 00:13:54.510 align:middle line:90% 00:13:54.510 --> 00:13:58.490 align:middle line:84% But this is actually quite the Abel transformation. 00:13:58.490 --> 00:14:00.230 align:middle line:84% What we do at this point here is we 00:14:00.230 --> 00:14:04.425 align:middle line:84% realize that this is a horrific mess of r's and y's and x's 00:14:04.425 --> 00:14:05.300 align:middle line:90% and things like that. 00:14:05.300 --> 00:14:07.490 align:middle line:84% And we decide that we want to rewrite everything. 00:14:07.490 --> 00:14:08.930 align:middle line:84% And we already said that r squared 00:14:08.930 --> 00:14:12.420 align:middle line:84% is equal to y squared plus x squared. 00:14:12.420 --> 00:14:15.260 align:middle line:84% And so we want to have a go at substituting this x out 00:14:15.260 --> 00:14:18.020 align:middle line:84% for something that's in terms of r instead. 00:14:18.020 --> 00:14:20.340 align:middle line:84% And this gives us a real transformation, 00:14:20.340 --> 00:14:26.090 align:middle line:84% which is f of y is equal to 2 times the integral from y 00:14:26.090 --> 00:14:39.292 align:middle line:84% to a, f of r r dr, square root of r squared minus y squared. 00:14:39.292 --> 00:14:41.000 align:middle line:84% And if you stare at this for long enough, 00:14:41.000 --> 00:14:43.250 align:middle line:84% you can convince yourself that doing this substitution 00:14:43.250 --> 00:14:46.630 align:middle line:84% into integration will work out, and that we've correctly dealt 00:14:46.630 --> 00:14:48.320 align:middle line:90% with the limits here as well. 00:14:48.320 --> 00:14:51.929 align:middle line:84% And so this is the thing which is called the Abel transform. 00:14:51.929 --> 00:14:59.920 align:middle line:90% 00:14:59.920 --> 00:15:00.460 align:middle line:90% OK. 00:15:00.460 --> 00:15:04.540 align:middle line:84% But we, as I mentioned, don't want the Abel transformation. 00:15:04.540 --> 00:15:06.040 align:middle line:90% That's relatively easy to do. 00:15:06.040 --> 00:15:10.180 align:middle line:84% What we have is f of y when we want f of r. 00:15:10.180 --> 00:15:13.270 align:middle line:84% So what we want to be able to do is the inverse. 00:15:13.270 --> 00:15:17.910 align:middle line:90% 00:15:17.910 --> 00:15:20.650 align:middle line:84% And I'm not going to derive this. 00:15:20.650 --> 00:15:22.350 align:middle line:90% I'm not even sure I know how to. 00:15:22.350 --> 00:15:24.820 align:middle line:84% But if you stare at what I'm about to write down 00:15:24.820 --> 00:15:26.430 align:middle line:84% in this for long enough, you can convince yourself 00:15:26.430 --> 00:15:27.750 align:middle line:84% there's enough shared features to it 00:15:27.750 --> 00:15:28.980 align:middle line:90% that it's probably correct. 00:15:28.980 --> 00:15:32.040 align:middle line:84% And you can go look it up if you want to [INAUDIBLE] 00:15:32.040 --> 00:15:40.200 align:middle line:84% So this gives us that f of r is equal to minus 1 upon pi times 00:15:40.200 --> 00:15:48.690 align:middle line:84% the integral from r of a of dF dy. 00:15:48.690 --> 00:15:52.300 align:middle line:90% That's our capital F here-- 00:15:52.300 --> 00:15:58.900 align:middle line:90% dy y squared minus r squared. 00:15:58.900 --> 00:16:02.460 align:middle line:90% I'll just [INAUDIBLE]. 00:16:02.460 --> 00:16:06.770 align:middle line:84% Remember, this capital F is something 00:16:06.770 --> 00:16:09.440 align:middle line:90% we have as a function of y. 00:16:09.440 --> 00:16:11.660 align:middle line:84% So this is our line incentive measurement, 00:16:11.660 --> 00:16:16.208 align:middle line:84% and this is as we move to the symmetric radial dependence 00:16:16.208 --> 00:16:17.000 align:middle line:90% on the measurement. 00:16:17.000 --> 00:16:20.950 align:middle line:84% This is the thing that we're actually trying to get at. 00:16:20.950 --> 00:16:21.550 align:middle line:90% OK. 00:16:21.550 --> 00:16:25.070 align:middle line:84% So this looks like a complete solution to the problem. 00:16:25.070 --> 00:16:28.060 align:middle line:84% If I have some measurement from my detector 00:16:28.060 --> 00:16:32.110 align:middle line:84% which is line integrated and I have enough samples in y, 00:16:32.110 --> 00:16:34.570 align:middle line:84% I should be able to work out what that involves. 00:16:34.570 --> 00:16:37.720 align:middle line:84% Can anyone spot any limitations for this procedure? 00:16:37.720 --> 00:16:40.990 align:middle line:90% There are two obvious ones. 00:16:40.990 --> 00:16:45.130 align:middle line:84% AUDIENCE: It's not clear where the plasma ends all the time. 00:16:45.130 --> 00:16:47.560 align:middle line:84% JACK HARE: Can you just say that again, please? 00:16:47.560 --> 00:16:48.220 align:middle line:90% AUDIENCE: Yeah. 00:16:48.220 --> 00:16:51.447 align:middle line:84% It's not always clear what a you should choose. 00:16:51.447 --> 00:16:52.030 align:middle line:90% JACK HARE: OK. 00:16:52.030 --> 00:16:53.950 align:middle line:84% So that's a reasonable one, actually. 00:16:53.950 --> 00:16:56.660 align:middle line:90% So where is a? 00:16:56.660 --> 00:17:01.640 align:middle line:90% 00:17:01.640 --> 00:17:03.450 align:middle line:90% That is actually a problem. 00:17:03.450 --> 00:17:05.310 align:middle line:84% I would definitely agree it's a problem. 00:17:05.310 --> 00:17:08.280 align:middle line:84% It's less of a problem as long as a drops 00:17:08.280 --> 00:17:10.750 align:middle line:84% off rapidly enough, which is related to this. 00:17:10.750 --> 00:17:13.822 align:middle line:84% But you're right, the edge of our plasma is fuzzy. 00:17:13.822 --> 00:17:15.780 align:middle line:84% You know where the density definitely goes to 0 00:17:15.780 --> 00:17:17.337 align:middle line:90% if you have a vacuum chamber-- 00:17:17.337 --> 00:17:19.170 align:middle line:84% the hard metal walls of your vacuum chamber. 00:17:19.170 --> 00:17:20.837 align:middle line:84% So maybe that would be good enough. 00:17:20.837 --> 00:17:22.920 align:middle line:84% But yeah, you certainly want to do this experiment 00:17:22.920 --> 00:17:24.270 align:middle line:90% to know where a is. 00:17:24.270 --> 00:17:25.702 align:middle line:90% Other limitations? 00:17:25.702 --> 00:17:27.160 align:middle line:84% AUDIENCE: If you do the derivative. 00:17:27.160 --> 00:17:29.880 align:middle line:84% So that's going to be limited by detector resolution and then 00:17:29.880 --> 00:17:31.140 align:middle line:90% experimental noise. 00:17:31.140 --> 00:17:31.807 align:middle line:90% JACK HARE: Yeah. 00:17:31.807 --> 00:17:35.820 align:middle line:90% So dF dy is noisy. 00:17:35.820 --> 00:17:41.760 align:middle line:84% If we go back to this very suggestive picture 00:17:41.760 --> 00:17:43.680 align:middle line:84% that I put in here deliberately like this, 00:17:43.680 --> 00:17:46.873 align:middle line:84% for any realistic system, you have discrete measurements 00:17:46.873 --> 00:17:47.790 align:middle line:90% at discrete locations. 00:17:47.790 --> 00:17:50.790 align:middle line:84% And we all know that doing derivatives of discrete data 00:17:50.790 --> 00:17:53.952 align:middle line:84% is a nightmare because you're taking something that's noisy, 00:17:53.952 --> 00:17:55.660 align:middle line:84% and you're dividing it by a small number, 00:17:55.660 --> 00:17:57.930 align:middle line:84% so any noise here gets really amplified up. 00:17:57.930 --> 00:18:02.130 align:middle line:84% So this straight away looks problematic. 00:18:02.130 --> 00:18:05.020 align:middle line:90% Third thing. 00:18:05.020 --> 00:18:06.010 align:middle line:90% Yes. 00:18:06.010 --> 00:18:08.010 align:middle line:84% AUDIENCE: I don't know if we want to count this, 00:18:08.010 --> 00:18:10.480 align:middle line:84% but once you start getting close to the edge 00:18:10.480 --> 00:18:14.840 align:middle line:84% and do that correctly, it'll be able to have a much 00:18:14.840 --> 00:18:17.150 align:middle line:90% smaller number in the bottom. 00:18:17.150 --> 00:18:18.927 align:middle line:90% JACK HARE: Close to the edge? 00:18:18.927 --> 00:18:20.510 align:middle line:84% AUDIENCE: [INAUDIBLE] we're closer to. 00:18:20.510 --> 00:18:23.800 align:middle line:90% 00:18:23.800 --> 00:18:26.983 align:middle line:90% [INAUDIBLE] I was thinking-- 00:18:26.983 --> 00:18:29.150 align:middle line:84% JACK HARE: You're close, but you're not quite right. 00:18:29.150 --> 00:18:31.660 align:middle line:90% 00:18:31.660 --> 00:18:33.408 align:middle line:90% Anyone else know? 00:18:33.408 --> 00:18:34.700 align:middle line:90% What are we talking about here? 00:18:34.700 --> 00:18:37.330 align:middle line:84% We're talking about the fact that there's something 00:18:37.330 --> 00:18:38.590 align:middle line:90% interesting going on here. 00:18:38.590 --> 00:18:42.040 align:middle line:84% And your physicist eyes have seen this and gone, a-ha. 00:18:42.040 --> 00:18:45.430 align:middle line:84% Whenever we start having numbers minus other numbers 00:18:45.430 --> 00:18:48.010 align:middle line:84% in the denominator here, there's some chance 00:18:48.010 --> 00:18:49.690 align:middle line:90% that this thing will go to 0. 00:18:49.690 --> 00:18:53.710 align:middle line:84% And it will actually go to 0 for y close to r equals 0. 00:18:53.710 --> 00:18:58.960 align:middle line:84% So near the center here, this thing will have a singularity. 00:18:58.960 --> 00:19:01.003 align:middle line:84% Now in reality, we won't have a singularity 00:19:01.003 --> 00:19:02.920 align:middle line:84% because we'll never have a detector bin that's 00:19:02.920 --> 00:19:04.250 align:middle line:90% exactly at that position. 00:19:04.250 --> 00:19:08.170 align:middle line:84% But what we will do is for the bins which 00:19:08.170 --> 00:19:09.700 align:middle line:84% are close to the center here, we'll 00:19:09.700 --> 00:19:12.020 align:middle line:84% have a very small number on the bottom. 00:19:12.020 --> 00:19:14.810 align:middle line:84% And so we will amplify the value of this big number. 00:19:14.810 --> 00:19:17.860 align:middle line:84% So if this is noisy, if there's some noise near the center here, 00:19:17.860 --> 00:19:19.540 align:middle line:84% that noise will be massively amplified 00:19:19.540 --> 00:19:21.800 align:middle line:84% and will appear everywhere else in our solution. 00:19:21.800 --> 00:19:24.470 align:middle line:84% And it will cause problems for the rest of our solution. 00:19:24.470 --> 00:19:34.630 align:middle line:84% So I'll just write here that we've got a singularity near y 00:19:34.630 --> 00:19:37.300 align:middle line:84% equals 0, and that this amplifies 00:19:37.300 --> 00:19:40.645 align:middle line:90% the noise of these points. 00:19:40.645 --> 00:19:45.880 align:middle line:90% 00:19:45.880 --> 00:19:47.450 align:middle line:90% OK. 00:19:47.450 --> 00:19:49.140 align:middle line:90% Any questions on any of this? 00:19:49.140 --> 00:19:49.640 align:middle line:90% Yes. 00:19:49.640 --> 00:19:55.460 align:middle line:84% AUDIENCE: Why are we collecting for the data where y is minus? 00:19:55.460 --> 00:19:58.520 align:middle line:84% y can only be positive in this picture. 00:19:58.520 --> 00:20:01.458 align:middle line:84% But numerical, you can have my y be negative. 00:20:01.458 --> 00:20:03.125 align:middle line:84% JACK HARE: That's an excellent question. 00:20:03.125 --> 00:20:06.680 align:middle line:84% And does anyone know why we are neglecting the data 00:20:06.680 --> 00:20:08.750 align:middle line:90% that we have for y less than 0? 00:20:08.750 --> 00:20:12.050 align:middle line:90% 00:20:12.050 --> 00:20:14.660 align:middle line:90% We have assumed symmetry. 00:20:14.660 --> 00:20:17.720 align:middle line:84% In order to do this calculation, we 00:20:17.720 --> 00:20:19.560 align:middle line:84% have assumed as a mutual symmetry. 00:20:19.560 --> 00:20:21.770 align:middle line:84% And so the data must be identical for y 00:20:21.770 --> 00:20:24.440 align:middle line:84% less than 0 than y greater than 0. 00:20:24.440 --> 00:20:26.090 align:middle line:90% Now, in reality it won't be. 00:20:26.090 --> 00:20:28.370 align:middle line:84% We never have a system which is perfectly symmetric. 00:20:28.370 --> 00:20:31.340 align:middle line:84% So the good way to present your data 00:20:31.340 --> 00:20:34.352 align:middle line:84% is to do the Abel inversion on one half of your data 00:20:34.352 --> 00:20:36.560 align:middle line:84% and the Abel inversion on the other half of your data 00:20:36.560 --> 00:20:40.710 align:middle line:84% separately, and then see whether those two match. 00:20:40.710 --> 00:20:43.190 align:middle line:84% And if they match close enough with an experimental error, 00:20:43.190 --> 00:20:44.030 align:middle line:90% great. 00:20:44.030 --> 00:20:45.500 align:middle line:90% You've got a good inversion. 00:20:45.500 --> 00:20:47.120 align:middle line:84% If they don't match at all, then you 00:20:47.120 --> 00:20:49.495 align:middle line:84% shouldn't have used an Abel inversion in the first place. 00:20:49.495 --> 00:20:52.040 align:middle line:84% Your prior that you have this cylindrical symmetry 00:20:52.040 --> 00:20:52.800 align:middle line:90% is incorrect. 00:20:52.800 --> 00:20:54.120 align:middle line:90% So you can't use this method. 00:20:54.120 --> 00:20:56.370 align:middle line:84% So it's a good check, actually, on your data. 00:20:56.370 --> 00:20:59.090 align:middle line:84% The other reason is because of this symmetry, if you're 00:20:59.090 --> 00:21:02.897 align:middle line:84% trying to save money, you might only put detectors in one half. 00:21:02.897 --> 00:21:04.730 align:middle line:84% If you really sure that you've got symmetry, 00:21:04.730 --> 00:21:06.870 align:middle line:90% then you don't need to check. 00:21:06.870 --> 00:21:08.580 align:middle line:84% Maybe you do the experiment a few times 00:21:08.580 --> 00:21:10.570 align:middle line:84% with all your detectors spread out, and then you're like, hey, 00:21:10.570 --> 00:21:11.153 align:middle line:90% this is great. 00:21:11.153 --> 00:21:14.320 align:middle line:84% Now I can get higher resolution by moving half of my detectors 00:21:14.320 --> 00:21:15.070 align:middle line:90% to the first half. 00:21:15.070 --> 00:21:16.778 align:middle line:84% So there may be some reasons why you only 00:21:16.778 --> 00:21:18.000 align:middle line:90% need half the data here. 00:21:18.000 --> 00:21:20.433 align:middle line:84% But yeah, that's a really good point. 00:21:20.433 --> 00:21:21.600 align:middle line:90% Any other questions on this? 00:21:21.600 --> 00:21:22.267 align:middle line:90% Anything online? 00:21:22.267 --> 00:21:28.580 align:middle line:90% 00:21:28.580 --> 00:21:29.080 align:middle line:90% OK. 00:21:29.080 --> 00:21:31.370 align:middle line:84% Like I said, this is very generic. 00:21:31.370 --> 00:21:34.150 align:middle line:84% This could be interferometry on a tokamak. 00:21:34.150 --> 00:21:36.550 align:middle line:84% This could be interferometry on the sorts 00:21:36.550 --> 00:21:37.780 align:middle line:90% of plasmas I work with. 00:21:37.780 --> 00:21:40.750 align:middle line:84% This could be used for unfolding a mission 00:21:40.750 --> 00:21:44.540 align:middle line:84% on a hotspot from an X-ray image or something like that. 00:21:44.540 --> 00:21:46.720 align:middle line:84% So we're just introducing it here because it's 00:21:46.720 --> 00:21:48.950 align:middle line:90% a useful technique to know. 00:21:48.950 --> 00:21:49.490 align:middle line:90% OK. 00:21:49.490 --> 00:21:51.115 align:middle line:84% How do we actually do this in practice? 00:21:51.115 --> 00:21:59.000 align:middle line:90% 00:21:59.000 --> 00:22:01.610 align:middle line:84% Can anyone think of some way to overcome 00:22:01.610 --> 00:22:04.150 align:middle line:84% some of these limitations, particularly this one here? 00:22:04.150 --> 00:22:07.860 align:middle line:90% 00:22:07.860 --> 00:22:14.330 align:middle line:84% So we've got, again, our data, which is 00:22:14.330 --> 00:22:16.940 align:middle line:90% discrete and potentially noisy. 00:22:16.940 --> 00:22:26.220 align:middle line:90% 00:22:26.220 --> 00:22:26.930 align:middle line:90% Yeah. 00:22:26.930 --> 00:22:29.150 align:middle line:84% AUDIENCE: We mentioned before, just take data 00:22:29.150 --> 00:22:32.060 align:middle line:84% exactly as you [INAUDIBLE] always 00:22:32.060 --> 00:22:34.283 align:middle line:90% going very close to [INAUDIBLE]. 00:22:34.283 --> 00:22:34.950 align:middle line:90% JACK HARE: Sure. 00:22:34.950 --> 00:22:36.975 align:middle line:90% That could be a problem, yes. 00:22:36.975 --> 00:22:39.510 align:middle line:84% So you could deliberately shift your data 00:22:39.510 --> 00:22:43.140 align:middle line:84% so that y equals 0 is on one side of one of your bins 00:22:43.140 --> 00:22:44.768 align:middle line:84% or something like that so it's close. 00:22:44.768 --> 00:22:47.310 align:middle line:84% Hard to get in practice because the plasma might move around, 00:22:47.310 --> 00:22:49.352 align:middle line:84% so you're probably not going to be able to do it. 00:22:49.352 --> 00:22:50.340 align:middle line:90% Yeah, other ideas. 00:22:50.340 --> 00:22:52.590 align:middle line:84% AUDIENCE: I guess that also means that this idea might 00:22:52.590 --> 00:22:53.250 align:middle line:90% have issues. 00:22:53.250 --> 00:22:55.320 align:middle line:84% But if you have prior information about where 00:22:55.320 --> 00:22:56.910 align:middle line:84% high grades are in your plasma, you 00:22:56.910 --> 00:22:58.930 align:middle line:84% concentrate your measurements in those regions 00:22:58.930 --> 00:23:00.638 align:middle line:84% so that you're better resolving hydrating 00:23:00.638 --> 00:23:01.800 align:middle line:90% ingredients can go for it. 00:23:01.800 --> 00:23:02.467 align:middle line:90% JACK HARE: Yeah. 00:23:02.467 --> 00:23:04.680 align:middle line:84% So certainly these high gradients 00:23:04.680 --> 00:23:06.390 align:middle line:84% are going to be dominating this integral. 00:23:06.390 --> 00:23:09.330 align:middle line:84% But of course, also any gradients near the center 00:23:09.330 --> 00:23:10.410 align:middle line:90% is going to be dominated. 00:23:10.410 --> 00:23:12.827 align:middle line:84% So you might want to put more measurements near the middle 00:23:12.827 --> 00:23:15.150 align:middle line:84% so you'd have higher fidelity there. 00:23:15.150 --> 00:23:16.110 align:middle line:90% Other techniques? 00:23:16.110 --> 00:23:18.330 align:middle line:84% AUDIENCE: Is it possible to integrate it by parts? 00:23:18.330 --> 00:23:19.860 align:middle line:84% Therefore, you don't differentiate 00:23:19.860 --> 00:23:23.530 align:middle line:84% by inside your different [INAUDIBLE] the other part. 00:23:23.530 --> 00:23:25.770 align:middle line:84% JACK HARE: I don't think you can do this by parts, 00:23:25.770 --> 00:23:28.690 align:middle line:84% but I haven't seen the analytical version of this, 00:23:28.690 --> 00:23:29.940 align:middle line:90% which looks like it does that. 00:23:29.940 --> 00:23:32.250 align:middle line:84% But I can't immediately check this and tell you that it 00:23:32.250 --> 00:23:32.910 align:middle line:90% doesn't. 00:23:32.910 --> 00:23:34.300 align:middle line:90% I suspect it doesn't work. 00:23:34.300 --> 00:23:35.260 align:middle line:90% Yeah. 00:23:35.260 --> 00:23:36.510 align:middle line:90% Any other ideas? 00:23:36.510 --> 00:23:37.720 align:middle line:84% AUDIENCE: You might have some reasonable idea 00:23:37.720 --> 00:23:39.095 align:middle line:84% of the distribution ahead of time 00:23:39.095 --> 00:23:42.197 align:middle line:84% and form an expected interpolation. 00:23:42.197 --> 00:23:43.030 align:middle line:90% JACK HARE: OK, yeah. 00:23:43.030 --> 00:23:45.280 align:middle line:84% So maybe we've got some priors about the distribution, 00:23:45.280 --> 00:23:46.600 align:middle line:90% and we fit to this. 00:23:46.600 --> 00:23:47.440 align:middle line:90% That would be good. 00:23:47.440 --> 00:23:49.780 align:middle line:84% A similar version is we could fit this noisy data 00:23:49.780 --> 00:23:51.400 align:middle line:84% with a set of basis functions that we 00:23:51.400 --> 00:23:54.430 align:middle line:84% think has the sort of information in it 00:23:54.430 --> 00:23:55.352 align:middle line:90% that describes this. 00:23:55.352 --> 00:23:57.310 align:middle line:84% And those basis functions won't have any noise, 00:23:57.310 --> 00:23:59.470 align:middle line:84% and they'll be nice and differentiable 00:23:59.470 --> 00:24:02.560 align:middle line:84% because we'll use some nice analytical basis functions. 00:24:02.560 --> 00:24:08.020 align:middle line:84% So we could have a sum of m Gaussian functions, 00:24:08.020 --> 00:24:11.902 align:middle line:84% where the Gaussians have some position and sigma and width 00:24:11.902 --> 00:24:12.860 align:middle line:90% or something like that. 00:24:12.860 --> 00:24:15.277 align:middle line:84% And then we know what the derivatives of all of those are, 00:24:15.277 --> 00:24:15.850 align:middle line:90% so yeah. 00:24:15.850 --> 00:24:19.090 align:middle line:84% AUDIENCE: So that's putting a set of basis functions 00:24:19.090 --> 00:24:22.870 align:middle line:90% to the brightness of emissivity? 00:24:22.870 --> 00:24:24.078 align:middle line:90% Big F or small? 00:24:24.078 --> 00:24:25.870 align:middle line:84% JACK HARE: We'd still be fitting into this. 00:24:25.870 --> 00:24:26.880 align:middle line:90% This is the only thing we know. 00:24:26.880 --> 00:24:27.070 align:middle line:90% AUDIENCE: OK. 00:24:27.070 --> 00:24:27.737 align:middle line:90% JACK HARE: Yeah. 00:24:27.737 --> 00:24:32.800 align:middle line:84% So you could say F of y is equal to some weighted set of basis 00:24:32.800 --> 00:24:34.270 align:middle line:90% functions. 00:24:34.270 --> 00:24:35.740 align:middle line:84% And there are some basis functions 00:24:35.740 --> 00:24:38.680 align:middle line:84% that work really well because they've got analytical Abel 00:24:38.680 --> 00:24:39.880 align:middle line:90% transformations. 00:24:39.880 --> 00:24:41.830 align:middle line:84% Gaussians are one of them, unsurprisingly. 00:24:41.830 --> 00:24:44.680 align:middle line:84% But some functions have a nice analytical Abel version and some 00:24:44.680 --> 00:24:45.180 align:middle line:90% don't. 00:24:45.180 --> 00:24:47.080 align:middle line:84% So you'd want to use a set of functions 00:24:47.080 --> 00:24:49.550 align:middle line:84% that have nice analytical algorithm versions. 00:24:49.550 --> 00:24:51.708 align:middle line:90% So that works pretty well. 00:24:51.708 --> 00:24:53.000 align:middle line:90% And that's what most people do. 00:24:53.000 --> 00:24:55.960 align:middle line:84% So if you go online, Python has a nice Abel inversion package, 00:24:55.960 --> 00:24:57.730 align:middle line:84% and they've got different basis functions. 00:24:57.730 --> 00:24:59.272 align:middle line:84% And depending on your exact problem, 00:24:59.272 --> 00:25:00.730 align:middle line:84% you might want basis functions that 00:25:00.730 --> 00:25:03.907 align:middle line:84% have got more spiky features at the edge or smooth features 00:25:03.907 --> 00:25:04.490 align:middle line:90% in the middle. 00:25:04.490 --> 00:25:06.670 align:middle line:84% And so just like all of these sorts of problems, 00:25:06.670 --> 00:25:08.600 align:middle line:90% there's no one size fits all. 00:25:08.600 --> 00:25:10.840 align:middle line:84% You have to tailor it to what you're doing. 00:25:10.840 --> 00:25:13.420 align:middle line:90% Did I see another question? 00:25:13.420 --> 00:25:17.020 align:middle line:84% AUDIENCE: Can we fold it for a low pass or something 00:25:17.020 --> 00:25:18.663 align:middle line:90% to get rid of 5 [INAUDIBLE]? 00:25:18.663 --> 00:25:19.330 align:middle line:90% JACK HARE: Yeah. 00:25:19.330 --> 00:25:22.160 align:middle line:84% So as always with our data, we can smooth it out, 00:25:22.160 --> 00:25:24.650 align:middle line:84% but then we lose spatial resolution. 00:25:24.650 --> 00:25:27.860 align:middle line:84% So that could help us, but at some cost. 00:25:27.860 --> 00:25:31.490 align:middle line:84% And so you have to balance those things. 00:25:31.490 --> 00:25:32.750 align:middle line:90% AUDIENCE: My first thought-- 00:25:32.750 --> 00:25:34.850 align:middle line:84% I actually think how to do this is 00:25:34.850 --> 00:25:39.122 align:middle line:84% the ideal would be get that derivative in analog directly. 00:25:39.122 --> 00:25:39.830 align:middle line:90% JACK HARE: Right. 00:25:39.830 --> 00:25:44.000 align:middle line:84% So the idea was how to get this derivative in analog directly. 00:25:44.000 --> 00:25:45.440 align:middle line:84% I don't know how to do it either, 00:25:45.440 --> 00:25:48.595 align:middle line:84% but I think this is just a fundamental limitation when 00:25:48.595 --> 00:25:50.220 align:middle line:84% you're making one of these measurements 00:25:50.220 --> 00:25:53.070 align:middle line:90% that you can't overcome. 00:25:53.070 --> 00:25:54.810 align:middle line:84% There are some really interesting links 00:25:54.810 --> 00:25:58.180 align:middle line:84% between the Abel inversion; the radon transform, 00:25:58.180 --> 00:26:01.740 align:middle line:84% which is also used in tomography; 00:26:01.740 --> 00:26:03.000 align:middle line:90% and Fourier transform. 00:26:03.000 --> 00:26:04.800 align:middle line:84% They form some sort of weird, little cycle. 00:26:04.800 --> 00:26:05.880 align:middle line:84% If you do all of them in a row, you 00:26:05.880 --> 00:26:07.210 align:middle line:90% get back to where you started. 00:26:07.210 --> 00:26:09.960 align:middle line:84% So there's really fun mathematics going on inside this 00:26:09.960 --> 00:26:12.747 align:middle line:84% as well, which I'm not intelligent enough 00:26:12.747 --> 00:26:13.330 align:middle line:90% to know about. 00:26:13.330 --> 00:26:14.700 align:middle line:84% But if you like that sort of thing, 00:26:14.700 --> 00:26:16.050 align:middle line:84% you should go look on the Wikipedia page. 00:26:16.050 --> 00:26:17.350 align:middle line:90% There's a lot of good stuff. 00:26:17.350 --> 00:26:19.900 align:middle line:84% So any other questions or thoughts on Abel inversion? 00:26:19.900 --> 00:26:21.150 align:middle line:90% This is just a little aside. 00:26:21.150 --> 00:26:23.040 align:middle line:84% We're going to go on and do Faraday rotation after this, 00:26:23.040 --> 00:26:24.730 align:middle line:84% so we'll completely change topics. 00:26:24.730 --> 00:26:27.460 align:middle line:84% So if you've got any more questions, speak now. 00:26:27.460 --> 00:26:30.360 align:middle line:84% AUDIENCE: So this condition of cylindrical symmetry 00:26:30.360 --> 00:26:31.660 align:middle line:90% is very strict. 00:26:31.660 --> 00:26:35.530 align:middle line:84% So for instance, if we had a highly shaped [INAUDIBLE] flask 00:26:35.530 --> 00:26:38.710 align:middle line:84% with diverters, et cetera, it technically doesn't-- 00:26:38.710 --> 00:26:41.260 align:middle line:84% it possesses a symmetry on flex surfaces, 00:26:41.260 --> 00:26:43.270 align:middle line:84% but those flex surfaces aren't cylindrical. 00:26:43.270 --> 00:26:46.563 align:middle line:84% So there would be no way to incorporate that information. 00:26:46.563 --> 00:26:47.230 align:middle line:90% JACK HARE: Yeah. 00:26:47.230 --> 00:26:50.740 align:middle line:84% So the question was how do we deal with non-circular flux 00:26:50.740 --> 00:26:52.690 align:middle line:84% surfaces, like in most modern tokamaks? 00:26:52.690 --> 00:26:55.850 align:middle line:84% And yes, you can no longer use an Abel inversion in that case. 00:26:55.850 --> 00:26:57.790 align:middle line:84% But there is still sufficient symmetry, 00:26:57.790 --> 00:26:59.080 align:middle line:90% and there's a lot of symmetry. 00:26:59.080 --> 00:27:00.940 align:middle line:84% So what you would do for a tokamak 00:27:00.940 --> 00:27:03.460 align:middle line:84% is I believe you'd do a gradual runoff 00:27:03.460 --> 00:27:05.650 align:middle line:84% reconstruction of the flux surfaces 00:27:05.650 --> 00:27:07.390 align:middle line:90% from your magnetic diagnostics. 00:27:07.390 --> 00:27:09.550 align:middle line:84% You would then know that the density is 00:27:09.550 --> 00:27:13.755 align:middle line:84% constant along the flux surface because they are surfaces 00:27:13.755 --> 00:27:15.130 align:middle line:84% of constant pressure, and there's 00:27:15.130 --> 00:27:17.320 align:middle line:84% enough motion in the toroidal direction 00:27:17.320 --> 00:27:19.540 align:middle line:84% to smooth out any density perturbations very quickly. 00:27:19.540 --> 00:27:21.640 align:middle line:84% This is roughly-- obviously there's fluctuations and stuff 00:27:21.640 --> 00:27:22.040 align:middle line:90% like that. 00:27:22.040 --> 00:27:23.623 align:middle line:84% But in general, there's this constant. 00:27:23.623 --> 00:27:25.690 align:middle line:84% And then you would use that as a prior. 00:27:25.690 --> 00:27:28.883 align:middle line:84% And you wouldn't do this Abel inversion analytically or even 00:27:28.883 --> 00:27:30.425 align:middle line:84% semi-analytically, but you would have 00:27:30.425 --> 00:27:33.090 align:middle line:84% to feed into your tomographic reconstruction algorithm. 00:27:33.090 --> 00:27:36.230 align:middle line:84% And for any tomographic reconstruction algorithm, 00:27:36.230 --> 00:27:38.720 align:middle line:84% the more data you have, the better it is. 00:27:38.720 --> 00:27:39.472 align:middle line:90% So it's obvious. 00:27:39.472 --> 00:27:41.180 align:middle line:84% And it's obvious here as well-- if I only 00:27:41.180 --> 00:27:42.932 align:middle line:84% have four chords of interferometry, 00:27:42.932 --> 00:27:44.390 align:middle line:84% my data is so sparse that I'm going 00:27:44.390 --> 00:27:46.080 align:middle line:90% to have a bad Abel inversion. 00:27:46.080 --> 00:27:49.737 align:middle line:84% And so you can have four chords of-- 00:27:49.737 --> 00:27:52.070 align:middle line:84% you can have your chords of interferometry crisscrossing 00:27:52.070 --> 00:27:53.743 align:middle line:84% the plasma like this, or you could still 00:27:53.743 --> 00:27:55.410 align:middle line:84% have them crossing the plasma like this. 00:27:55.410 --> 00:27:57.320 align:middle line:84% And even if your plasma was strongly 00:27:57.320 --> 00:28:04.540 align:middle line:84% shaped-- so if it was some classic single null no x point 00:28:04.540 --> 00:28:07.930 align:middle line:84% type thing like this, you could have your chords 00:28:07.930 --> 00:28:09.310 align:middle line:90% frothing in this fashion. 00:28:09.310 --> 00:28:14.020 align:middle line:84% And that would be good enough to be 00:28:14.020 --> 00:28:16.362 align:middle line:84% able to do some sort of inversion in the middle here. 00:28:16.362 --> 00:28:18.820 align:middle line:84% And there's actually-- I think one of Anne White's students 00:28:18.820 --> 00:28:21.487 align:middle line:84% who's about to graduate who has been working on this for X-rays. 00:28:21.487 --> 00:28:23.230 align:middle line:84% And he came up with some cool ideas 00:28:23.230 --> 00:28:27.760 align:middle line:84% about what if you have some sparse chords of data 00:28:27.760 --> 00:28:30.610 align:middle line:84% and then in some section here, you have really, really 00:28:30.610 --> 00:28:31.300 align:middle line:90% fine chords? 00:28:31.300 --> 00:28:33.070 align:middle line:84% Can you combine those and measure 00:28:33.070 --> 00:28:35.500 align:middle line:84% very small turbulent fluctuations with this? 00:28:35.500 --> 00:28:37.030 align:middle line:84% And apparently, the answer is yes. 00:28:37.030 --> 00:28:40.980 align:middle line:84% So there's lots of cool things you can do with [INAUDIBLE].. 00:28:40.980 --> 00:28:43.670 align:middle line:90% Yeah. 00:28:43.670 --> 00:28:46.460 align:middle line:84% AUDIENCE: Do people commonly [INAUDIBLE] integral formulation 00:28:46.460 --> 00:28:48.650 align:middle line:84% of the algorithm or is it more a problem 00:28:48.650 --> 00:28:54.590 align:middle line:84% to do a matrix formula where you know the width of your sight 00:28:54.590 --> 00:28:56.540 align:middle line:84% line or all of your bins in space, 00:28:56.540 --> 00:28:58.640 align:middle line:84% and then you can take the inverse of the matrix 00:28:58.640 --> 00:29:02.288 align:middle line:90% with some math massaging? 00:29:02.288 --> 00:29:04.580 align:middle line:84% JACK HARE: I don't know what technique is more popular. 00:29:04.580 --> 00:29:07.880 align:middle line:84% When I've done this, I've tended to use fitting 00:29:07.880 --> 00:29:13.160 align:middle line:84% with basis functions and then do the inversion analytically 00:29:13.160 --> 00:29:14.870 align:middle line:84% to the fits of the basis function. 00:29:14.870 --> 00:29:17.210 align:middle line:84% But I imagine there might be good reasons for doing 00:29:17.210 --> 00:29:19.520 align:middle line:84% that matrix formulation, especially 00:29:19.520 --> 00:29:21.840 align:middle line:84% if you're trying to use this for real time control. 00:29:21.840 --> 00:29:24.680 align:middle line:84% So the bolometers are going to be 00:29:24.680 --> 00:29:27.275 align:middle line:84% used for showing where the glowy bit of the plasma is. 00:29:27.275 --> 00:29:28.900 align:middle line:84% And we want that to stay in the middle. 00:29:28.900 --> 00:29:30.320 align:middle line:84% If it starts going somewhere else, 00:29:30.320 --> 00:29:31.867 align:middle line:84% then you want to feed back on that. 00:29:31.867 --> 00:29:33.450 align:middle line:84% And so then you need to do it quickly. 00:29:33.450 --> 00:29:35.242 align:middle line:84% And so having some sort of matrix technique 00:29:35.242 --> 00:29:37.430 align:middle line:84% will be beneficial compared to, oh, I'll 00:29:37.430 --> 00:29:38.900 align:middle line:84% carefully tune my fitting function. 00:29:38.900 --> 00:29:41.420 align:middle line:84% So you don't have time to do that. 00:29:41.420 --> 00:29:41.920 align:middle line:90% OK. 00:29:41.920 --> 00:29:42.795 align:middle line:90% Any questions online? 00:29:42.795 --> 00:29:45.890 align:middle line:90% 00:29:45.890 --> 00:29:46.390 align:middle line:90% All right. 00:29:46.390 --> 00:29:47.380 align:middle line:90% Let's do some Faraday-- 00:29:47.380 --> 00:29:50.110 align:middle line:84% AUDIENCE: Are there ways of handling systems-- 00:29:50.110 --> 00:29:54.197 align:middle line:84% are there ways of handling systems that don't have-- 00:29:54.197 --> 00:29:55.780 align:middle line:84% I guess you already talked about this. 00:29:55.780 --> 00:29:58.270 align:middle line:84% But are there ways of handling systems that 00:29:58.270 --> 00:30:02.407 align:middle line:90% don't have any symmetry priors? 00:30:02.407 --> 00:30:03.490 align:middle line:90% JACK HARE: Yeah, actually. 00:30:03.490 --> 00:30:04.948 align:middle line:84% I have a grad student of mine who's 00:30:04.948 --> 00:30:07.760 align:middle line:84% been working on this recently for tomographic reconstruction. 00:30:07.760 --> 00:30:10.750 align:middle line:84% So you can always do a reconstruction, 00:30:10.750 --> 00:30:12.820 align:middle line:90% but it's always poorly posed. 00:30:12.820 --> 00:30:14.530 align:middle line:84% You don't have enough information 00:30:14.530 --> 00:30:20.170 align:middle line:84% to fully reconstruct the [INAUDIBLE] points in space. 00:30:20.170 --> 00:30:22.660 align:middle line:84% But if you have some priors, you can make some guesses. 00:30:22.660 --> 00:30:24.457 align:middle line:84% Even with just a single line of sight, 00:30:24.457 --> 00:30:25.540 align:middle line:90% you can make some guesses. 00:30:25.540 --> 00:30:26.780 align:middle line:84% And the more lines of sight you have, 00:30:26.780 --> 00:30:28.030 align:middle line:90% the more you can constrain it. 00:30:28.030 --> 00:30:29.742 align:middle line:84% And he was working on a technique, 00:30:29.742 --> 00:30:30.700 align:middle line:90% which I can't remember. 00:30:30.700 --> 00:30:33.520 align:middle line:84% The acronym is ART, and I can't remember what it stands for. 00:30:33.520 --> 00:30:35.590 align:middle line:84% But with two orthogonal lines of sight 00:30:35.590 --> 00:30:38.050 align:middle line:84% and just a flat uniform prior-- so 00:30:38.050 --> 00:30:39.505 align:middle line:84% no information about what it looks 00:30:39.505 --> 00:30:42.130 align:middle line:84% like to start with-- he was able to reconstruct some relatively 00:30:42.130 --> 00:30:46.330 align:middle line:90% complicated shapes out of this. 00:30:46.330 --> 00:30:48.995 align:middle line:84% So there are some clever things that you can do. 00:30:48.995 --> 00:30:50.620 align:middle line:84% And I think there's a lot of cool stuff 00:30:50.620 --> 00:30:53.092 align:middle line:84% that we can take in plasma physics from other fields. 00:30:53.092 --> 00:30:55.300 align:middle line:84% So a lot of stuff in computer vision from tomography, 00:30:55.300 --> 00:30:57.752 align:middle line:84% from medical imaging, which can be used 00:30:57.752 --> 00:30:58.960 align:middle line:90% for understanding this stuff. 00:30:58.960 --> 00:31:02.347 align:middle line:84% And people do already, so there's lots of nice things 00:31:02.347 --> 00:31:03.430 align:middle line:90% out there that you can do. 00:31:03.430 --> 00:31:05.180 align:middle line:84% At the end of the day, the more data you have, 00:31:05.180 --> 00:31:06.010 align:middle line:90% the easier this is. 00:31:06.010 --> 00:31:07.570 align:middle line:84% So if you have very little data, you 00:31:07.570 --> 00:31:10.028 align:middle line:84% have to provide that information from somewhere else, which 00:31:10.028 --> 00:31:12.250 align:middle line:84% is your intuition, your guesses about the plasma. 00:31:12.250 --> 00:31:14.170 align:middle line:90% So you can't win. 00:31:14.170 --> 00:31:18.000 align:middle line:84% You can't get just get this information for free. 00:31:18.000 --> 00:31:18.500 align:middle line:90% OK. 00:31:18.500 --> 00:31:21.800 align:middle line:84% I'm going to go on now so you have time to cover all of this. 00:31:21.800 --> 00:31:36.860 align:middle line:90% 00:31:36.860 --> 00:31:39.925 align:middle line:84% So we're going to be looking at Faraday rotation. 00:31:39.925 --> 00:31:42.830 align:middle line:90% 00:31:42.830 --> 00:31:45.950 align:middle line:84% But we're actually just going to take a little side 00:31:45.950 --> 00:31:50.900 align:middle line:84% track for maybe most of the rest of this lecture to look again 00:31:50.900 --> 00:31:57.680 align:middle line:84% at waves in magnetized plasmas, of which Faraday rotation is 00:31:57.680 --> 00:31:58.820 align:middle line:90% one. 00:31:58.820 --> 00:32:01.250 align:middle line:84% The reason is although you hopefully 00:32:01.250 --> 00:32:03.908 align:middle line:84% have seen some of this in earlier plasma classes, 00:32:03.908 --> 00:32:06.200 align:middle line:84% I think there are lots of different ways of looking it. 00:32:06.200 --> 00:32:08.150 align:middle line:84% And I think the way that Hutchinson has 00:32:08.150 --> 00:32:10.890 align:middle line:84% and that I've adapted is quite a nice way of looking at it. 00:32:10.890 --> 00:32:13.220 align:middle line:84% And we're also going to need lots of these results, 00:32:13.220 --> 00:32:17.360 align:middle line:84% not only for Faraday rotation, but also for reflectometry 00:32:17.360 --> 00:32:19.167 align:middle line:90% and electron cyclotron emission. 00:32:19.167 --> 00:32:20.750 align:middle line:84% So we need to know how waves propagate 00:32:20.750 --> 00:32:22.132 align:middle line:90% in a magnetized plasma. 00:32:22.132 --> 00:32:24.090 align:middle line:84% And so we may as well just review this quickly. 00:32:24.090 --> 00:32:26.423 align:middle line:84% So if you've seen this before and you're very confident, 00:32:26.423 --> 00:32:28.050 align:middle line:90% feel free to relax. 00:32:28.050 --> 00:32:31.640 align:middle line:84% And if you haven't seen it, maybe pay attention. 00:32:31.640 --> 00:32:35.150 align:middle line:84% So remember, we had before some assumptions 00:32:35.150 --> 00:32:39.073 align:middle line:84% that our plasma was cold, and we quantified that 00:32:39.073 --> 00:32:41.240 align:middle line:84% by saying that the thermal velocity of the electrons 00:32:41.240 --> 00:32:43.700 align:middle line:84% is much less than the speed of light. 00:32:43.700 --> 00:32:48.320 align:middle line:84% We said that our frequency was high, 00:32:48.320 --> 00:32:52.970 align:middle line:84% and we quantified that by saying that our ion plasma 00:32:52.970 --> 00:32:59.258 align:middle line:84% frequency was much less than a frequency of the waves ascending 00:32:59.258 --> 00:33:00.050 align:middle line:90% through the plasma. 00:33:00.050 --> 00:33:02.250 align:middle line:84% And that meant that the ions are effectively stationary. 00:33:02.250 --> 00:33:04.792 align:middle line:84% So we can just neglect them and just deal with the electrons. 00:33:04.792 --> 00:33:07.250 align:middle line:90% That made life much easier. 00:33:07.250 --> 00:33:09.380 align:middle line:84% And we also made this following restriction, 00:33:09.380 --> 00:33:13.620 align:middle line:90% that k dot e was equal to 0. 00:33:13.620 --> 00:33:16.980 align:middle line:84% And this was the restriction that the waves were transverse. 00:33:16.980 --> 00:33:21.470 align:middle line:90% 00:33:21.470 --> 00:33:23.300 align:middle line:84% Now, this restriction-- the final one, 00:33:23.300 --> 00:33:27.440 align:middle line:84% the transverse waves-- made our life very simple, algebraically. 00:33:27.440 --> 00:33:31.190 align:middle line:84% But it turns out that if you try and find 00:33:31.190 --> 00:33:34.340 align:middle line:84% the waves in a magnetized plasma while keeping this restriction, 00:33:34.340 --> 00:33:35.870 align:middle line:90% you don't get all of the waves. 00:33:35.870 --> 00:33:37.970 align:middle line:84% You, in fact, have explicitly ruled out 00:33:37.970 --> 00:33:40.380 align:middle line:84% one of the most important waves-- the extraordinary mode, 00:33:40.380 --> 00:33:41.880 align:middle line:90% which is extraordinarily useful. 00:33:41.880 --> 00:33:45.890 align:middle line:84% And so we have to drop this restriction here and then 00:33:45.890 --> 00:33:48.710 align:middle line:84% deal with all the horrible consequences of that. 00:33:48.710 --> 00:33:55.428 align:middle line:84% So now we have k dot e not equal to 0. 00:33:55.428 --> 00:33:57.720 align:middle line:84% And if you go back and you look through the derivation, 00:33:57.720 --> 00:34:01.750 align:middle line:84% and you start rederiving bits, you end up with an equation 00:34:01.750 --> 00:34:04.500 align:middle line:84% now that looks like omega squared 00:34:04.500 --> 00:34:08.940 align:middle line:90% minus c squared k squared. 00:34:08.940 --> 00:34:10.550 align:middle line:84% We had something like this before, 00:34:10.550 --> 00:34:13.980 align:middle line:84% but our equation previously was a scalar equation. 00:34:13.980 --> 00:34:19.820 align:middle line:84% Our equation now is going to be a matrix equation with these 3 00:34:19.820 --> 00:34:20.918 align:middle line:90% by 3 matrices. 00:34:20.918 --> 00:34:22.460 align:middle line:84% And this is just the identity matrix. 00:34:22.460 --> 00:34:26.030 align:middle line:90% 00:34:26.030 --> 00:34:30.170 align:middle line:84% And this is an even more odd looking 00:34:30.170 --> 00:34:34.280 align:middle line:84% object, which is kk, which is a dyad, which 00:34:34.280 --> 00:34:37.520 align:middle line:90% is also a 3 by 3 matrix here. 00:34:37.520 --> 00:34:39.110 align:middle line:84% You can look up more of the details 00:34:39.110 --> 00:34:40.850 align:middle line:84% of this in Hutchison's book if you 00:34:40.850 --> 00:34:42.530 align:middle line:90% haven't seen this in a while. 00:34:42.530 --> 00:34:46.790 align:middle line:84% All of this is now dotted with the electric field. 00:34:46.790 --> 00:34:55.230 align:middle line:84% And that is equal to minus I omega J over epsilon. 00:34:55.230 --> 00:34:59.150 align:middle line:84% Now, once again, we realized that J is equal 00:34:59.150 --> 00:35:03.950 align:middle line:90% to minus e and eVe. 00:35:03.950 --> 00:35:06.980 align:middle line:84% And what we want to try and do is write this entire equation 00:35:06.980 --> 00:35:08.600 align:middle line:90% in terms of e. 00:35:08.600 --> 00:35:10.670 align:middle line:84% We want to get rid of J completely. 00:35:10.670 --> 00:35:12.680 align:middle line:84% And then, of course, this is an equation 00:35:12.680 --> 00:35:15.320 align:middle line:84% that looks like a matrix times a vector, 0. 00:35:15.320 --> 00:35:16.632 align:middle line:90% And we know how to do this. 00:35:16.632 --> 00:35:18.590 align:middle line:84% This is what we've trained our whole lives for. 00:35:18.590 --> 00:35:19.800 align:middle line:90% We take the determinant of this. 00:35:19.800 --> 00:35:21.710 align:middle line:84% We find the mode to search by [INAUDIBLE].. 00:35:21.710 --> 00:35:22.280 align:middle line:90% That's great. 00:35:22.280 --> 00:35:24.330 align:middle line:84% So we really love these sorts of equations. 00:35:24.330 --> 00:35:28.110 align:middle line:84% And so we're trying to make this equation look like one of those. 00:35:28.110 --> 00:35:31.730 align:middle line:84% So now we need an equation of motion for Ve. 00:35:31.730 --> 00:35:33.230 align:middle line:84% Previously, remember, we just looked 00:35:33.230 --> 00:35:36.020 align:middle line:84% at the response of the electrons to the electric field. 00:35:36.020 --> 00:35:38.190 align:middle line:84% But now we want to have the magnetic field as well. 00:35:38.190 --> 00:35:45.830 align:middle line:84% So we have any d Ve et is equal to minus 00:35:45.830 --> 00:35:51.390 align:middle line:84% e times the electric field plus V cross b. 00:35:51.390 --> 00:35:54.240 align:middle line:84% So this is just the Lorentz force, 00:35:54.240 --> 00:35:57.060 align:middle line:84% but we've now got the magnetic field in here. 00:35:57.060 --> 00:36:00.360 align:middle line:84% And we're going to assume that our magnetic field is 00:36:00.360 --> 00:36:03.630 align:middle line:84% the first order, is just some static field. 00:36:03.630 --> 00:36:06.030 align:middle line:84% And we're going to point it in the z direction. 00:36:06.030 --> 00:36:08.190 align:middle line:84% Of course, I can point in any direction we want. 00:36:08.190 --> 00:36:11.400 align:middle line:90% I've chosen z in this case here. 00:36:11.400 --> 00:36:14.580 align:middle line:84% And then from this, you get out a series 00:36:14.580 --> 00:36:16.685 align:middle line:90% of equations for the velocity. 00:36:16.685 --> 00:36:18.060 align:middle line:84% And I think you've all seen this, 00:36:18.060 --> 00:36:21.060 align:middle line:84% so I'm actually not going to go through this line by line. 00:36:21.060 --> 00:36:25.430 align:middle line:90% 00:36:25.430 --> 00:36:26.730 align:middle line:90% Well, I might go through. 00:36:26.730 --> 00:36:28.350 align:middle line:90% OK. 00:36:28.350 --> 00:36:30.288 align:middle line:84% I'm not going to write this equation out 00:36:30.288 --> 00:36:32.580 align:middle line:84% in terms of vector components, like I had in the notes. 00:36:32.580 --> 00:36:34.920 align:middle line:84% I will make the point that, as before, 00:36:34.920 --> 00:36:36.450 align:middle line:84% we're going to say that we're going 00:36:36.450 --> 00:36:45.490 align:middle line:84% to assume that V equals Ve 0 exponential of I k 00:36:45.490 --> 00:36:48.610 align:middle line:90% dot x minus omega t. 00:36:48.610 --> 00:36:57.260 align:middle line:84% And that allows us to replace d dt with minus I omega. 00:36:57.260 --> 00:36:59.450 align:middle line:84% So we turn to this differential equation 00:36:59.450 --> 00:37:01.540 align:middle line:90% into an algebraic equation. 00:37:01.540 --> 00:37:06.740 align:middle line:84% And then we find that the velocity of our single electron 00:37:06.740 --> 00:37:09.290 align:middle line:84% here is going to have a structure 00:37:09.290 --> 00:37:11.850 align:middle line:84% with some nice symmetry due to the magnetic field. 00:37:11.850 --> 00:37:14.220 align:middle line:84% So dx and dy are going to look very similar. 00:37:14.220 --> 00:37:23.330 align:middle line:84% They're going to look like minus Ie over omega me times 1 over 1 00:37:23.330 --> 00:37:26.660 align:middle line:84% minus omega squared over omega squared. 00:37:26.660 --> 00:37:28.660 align:middle line:84% It doesn't work so well when I say it like that. 00:37:28.660 --> 00:37:32.270 align:middle line:84% Capital omega squared over lowercase omega squared. 00:37:32.270 --> 00:37:35.950 align:middle line:84% And this is going to be equal to the electric field 00:37:35.950 --> 00:37:38.800 align:middle line:84% in the x direction minus I capital 00:37:38.800 --> 00:37:42.070 align:middle line:84% omega over lowercase omega electric field 00:37:42.070 --> 00:37:45.760 align:middle line:84% in the y direction, where we've defined here 00:37:45.760 --> 00:37:52.550 align:middle line:84% capital omega as the cyclotron frequency, e B0 over me. 00:37:52.550 --> 00:37:58.510 align:middle line:84% So dy, we have all of these terms again. 00:37:58.510 --> 00:38:00.400 align:middle line:84% But in the brackets, we have something 00:38:00.400 --> 00:38:02.740 align:middle line:84% that looks similar, but slightly different. 00:38:02.740 --> 00:38:10.303 align:middle line:84% I capital omega over lowercase omega ex plus ey. 00:38:10.303 --> 00:38:12.720 align:middle line:84% And when you look at this and you squint and you calculate 00:38:12.720 --> 00:38:16.818 align:middle line:84% dx squared plus dy squared, you find out that this is just-- 00:38:16.818 --> 00:38:18.360 align:middle line:84% when you just look at how these work, 00:38:18.360 --> 00:38:20.910 align:middle line:84% this is, of course, how particles are circulating 00:38:20.910 --> 00:38:23.170 align:middle line:90% around the magnetic field line. 00:38:23.170 --> 00:38:27.340 align:middle line:84% Remember, the field line is going in z direction. 00:38:27.340 --> 00:38:33.780 align:middle line:84% And so these two components are just the spiraling dx and dy 00:38:33.780 --> 00:38:34.740 align:middle line:90% here. 00:38:34.740 --> 00:38:39.612 align:middle line:84% And finally, we have the z components. 00:38:39.612 --> 00:38:40.570 align:middle line:90% And that's very simple. 00:38:40.570 --> 00:38:47.218 align:middle line:84% That's minus Ie over omega me [INAUDIBLE] like that. 00:38:47.218 --> 00:38:50.400 align:middle line:90% 00:38:50.400 --> 00:38:50.910 align:middle line:90% OK. 00:38:50.910 --> 00:38:52.670 align:middle line:90% You've seen all this before. 00:38:52.670 --> 00:38:57.500 align:middle line:84% You can then take this, make it into a nice vector, 00:38:57.500 --> 00:39:00.320 align:middle line:84% substitute it back into this equation for J, 00:39:00.320 --> 00:39:02.330 align:middle line:90% substitute J back into here. 00:39:02.330 --> 00:39:05.750 align:middle line:84% And you see very quickly that all you're going to have left 00:39:05.750 --> 00:39:08.600 align:middle line:84% are things like omegas and c's and k's and capital 00:39:08.600 --> 00:39:10.588 align:middle line:90% omegas and this electric field. 00:39:10.588 --> 00:39:12.380 align:middle line:84% So we're going to have some matrix equation 00:39:12.380 --> 00:39:16.610 align:middle line:90% dot e is equal to. 00:39:16.610 --> 00:39:20.910 align:middle line:84% And that matrix-- well, we can write 00:39:20.910 --> 00:39:27.270 align:middle line:84% J is equal to minus e and b times the velocity. 00:39:27.270 --> 00:39:29.400 align:middle line:84% And we can write that in terms of some sort 00:39:29.400 --> 00:39:33.420 align:middle line:84% of conductivity tensor times the electric field, 00:39:33.420 --> 00:39:43.040 align:middle line:84% where that conductivity is this monster I, any e squared over me 00:39:43.040 --> 00:39:48.170 align:middle line:84% omega 1 over 1 minus capital omega 00:39:48.170 --> 00:39:52.310 align:middle line:84% squared lowercase omega squared times pi on a nice, big 3 00:39:52.310 --> 00:39:55.300 align:middle line:90% by 3 matrix. 00:39:55.300 --> 00:39:57.310 align:middle line:90% If you want, down the diagonal. 00:39:57.310 --> 00:40:00.270 align:middle line:90% 00:40:00.270 --> 00:40:02.440 align:middle line:90% [INAUDIBLE] here. 00:40:02.440 --> 00:40:12.030 align:middle line:84% And minus I omega in here, pointing to off-diagonal 00:40:12.030 --> 00:40:15.730 align:middle line:90% elements, then 0s elsewhere. 00:40:15.730 --> 00:40:18.340 align:middle line:84% If you're surprised that seeing something interesting 00:40:18.340 --> 00:40:20.260 align:middle line:84% involving the magnetic field show up 00:40:20.260 --> 00:40:22.240 align:middle line:84% in the zz component of this tensor, 00:40:22.240 --> 00:40:24.410 align:middle line:84% it's literally only there to cancel it out here. 00:40:24.410 --> 00:40:26.028 align:middle line:84% So in fact, it doesn't exist at all. 00:40:26.028 --> 00:40:27.820 align:middle line:84% It's just it's more convenient than writing 00:40:27.820 --> 00:40:30.410 align:middle line:84% this factor underneath all four of these terms. 00:40:30.410 --> 00:40:33.280 align:middle line:90% So this is just a symbol. 00:40:33.280 --> 00:40:35.602 align:middle line:84% So this is the conductivity for our particle. 00:40:35.602 --> 00:40:50.690 align:middle line:90% 00:40:50.690 --> 00:40:53.330 align:middle line:84% Which means we can then write out 00:40:53.330 --> 00:40:58.860 align:middle line:84% in short form that omega squared minus c squared 00:40:58.860 --> 00:41:03.800 align:middle line:84% k squared times the identity matrix 00:41:03.800 --> 00:41:15.610 align:middle line:84% minus this dyad, ak, plus I omega over epsilon null sigma 00:41:15.610 --> 00:41:18.490 align:middle line:90% dot e equals [INAUDIBLE]. 00:41:18.490 --> 00:41:23.350 align:middle line:90% 00:41:23.350 --> 00:41:26.300 align:middle line:84% And I like writing things in terms of this conductivity term. 00:41:26.300 --> 00:41:29.050 align:middle line:84% But if you like things in terms of e-- 00:41:29.050 --> 00:41:33.950 align:middle line:84% I'm sorry, in terms of epsilon, the dielectric tensor, 00:41:33.950 --> 00:41:35.840 align:middle line:90% you could rewrite this. 00:41:35.840 --> 00:41:41.720 align:middle line:84% And you will get Hutchinson's equation, 4.1.2. 00:41:41.720 --> 00:41:43.950 align:middle line:90% OK. 00:41:43.950 --> 00:41:48.660 align:middle line:84% So the magnetic field in the system breaks the symmetry. 00:41:48.660 --> 00:41:51.900 align:middle line:84% So we have to treat that differently from dx and dy. 00:41:51.900 --> 00:41:54.960 align:middle line:84% But we don't have to treat dx and dy the same. 00:41:54.960 --> 00:41:58.650 align:middle line:84% We can pick our orientation of our x and y axes 00:41:58.650 --> 00:42:01.200 align:middle line:84% to make our life simple in what's going to follow. 00:42:01.200 --> 00:42:05.600 align:middle line:90% And we're going to do that now. 00:42:05.600 --> 00:42:07.540 align:middle line:84% I've got just about enough space here. 00:42:07.540 --> 00:42:34.520 align:middle line:90% 00:42:34.520 --> 00:42:39.310 align:middle line:84% So we're going to have a coordinate system like this, 00:42:39.310 --> 00:42:41.300 align:middle line:90% with z pointing upwards. 00:42:41.300 --> 00:42:44.640 align:middle line:84% And that's the direction of the magnetic field. 00:42:44.640 --> 00:42:47.300 align:middle line:84% We're going to have this coordinate, y, 00:42:47.300 --> 00:42:50.810 align:middle line:90% and a coordinate, x, like this. 00:42:50.810 --> 00:42:55.390 align:middle line:84% And I'm going to choose our k vector to always 00:42:55.390 --> 00:43:03.280 align:middle line:84% be in the yz plane, with some angle of theta to the z-axis 00:43:03.280 --> 00:43:03.800 align:middle line:90% here. 00:43:03.800 --> 00:43:07.150 align:middle line:84% So we're effectively choosing our coordinate system such 00:43:07.150 --> 00:43:12.550 align:middle line:84% that we can write k is equal to the size of the vector, k, 00:43:12.550 --> 00:43:20.360 align:middle line:84% times 0 sine theta, cosine of theta. 00:43:20.360 --> 00:43:23.710 align:middle line:84% This makes our life much, much easier and more flawless. 00:43:23.710 --> 00:43:26.380 align:middle line:84% But even if you do that, when you 00:43:26.380 --> 00:43:29.218 align:middle line:84% go to solve this equation in full generality, 00:43:29.218 --> 00:43:30.760 align:middle line:84% you have all these sines and cosines. 00:43:30.760 --> 00:43:32.170 align:middle line:90% It's an ungodly mess. 00:43:32.170 --> 00:43:33.730 align:middle line:90% So this is absolutely horrible. 00:43:33.730 --> 00:43:36.970 align:middle line:84% So you still get something terrible, 00:43:36.970 --> 00:43:40.510 align:middle line:84% which is equation 4.1.24, which I think is called 00:43:40.510 --> 00:43:44.090 align:middle line:84% the Appleton-Hartree dispersion relationship. 00:43:44.090 --> 00:43:46.420 align:middle line:84% And it's a mess, so it's very, very hard to work with. 00:43:46.420 --> 00:43:48.340 align:middle line:84% And the trick is that no one actually 00:43:48.340 --> 00:43:49.660 align:middle line:90% works with it most of the time. 00:43:49.660 --> 00:43:52.180 align:middle line:84% We just work in the case where we 00:43:52.180 --> 00:43:58.620 align:middle line:84% have the theta equal to 0 or theta equal to pi upon 2. 00:43:58.620 --> 00:44:00.078 align:middle line:84% So those are the two cases that I'm 00:44:00.078 --> 00:44:02.495 align:middle line:84% going to tell you about-- when we've got waves propagating 00:44:02.495 --> 00:44:04.423 align:middle line:84% along the magnetic field or perpendicular 00:44:04.423 --> 00:44:05.340 align:middle line:90% to the magnetic field. 00:44:05.340 --> 00:44:06.830 align:middle line:84% If you're ever in the unfortunate 00:44:06.830 --> 00:44:08.580 align:middle line:84% case of having to do something in between, 00:44:08.580 --> 00:44:10.320 align:middle line:84% you'll have to go back to this equation 00:44:10.320 --> 00:44:11.770 align:middle line:90% and work it out yourself. 00:44:11.770 --> 00:44:14.010 align:middle line:84% But at least in the limiting cases, 00:44:14.010 --> 00:44:16.087 align:middle line:84% the behavior is slightly easier to understand. 00:44:16.087 --> 00:44:18.420 align:middle line:84% So those are the cases that I'm going to go through now. 00:44:18.420 --> 00:44:20.610 align:middle line:84% So that was a bit of a whistlestop tour. 00:44:20.610 --> 00:44:21.540 align:middle line:90% Any questions? 00:44:21.540 --> 00:44:43.260 align:middle line:90% 00:44:43.260 --> 00:44:43.760 align:middle line:90% Good. 00:44:43.760 --> 00:44:46.330 align:middle line:84% Everyone loves waves [INAUDIBLE].. 00:44:46.330 --> 00:45:16.490 align:middle line:90% 00:45:16.490 --> 00:45:22.770 align:middle line:84% So we're going to start with the theta equals 0 case. 00:45:22.770 --> 00:45:33.300 align:middle line:90% 00:45:33.300 --> 00:45:33.800 align:middle line:90% Am I? 00:45:33.800 --> 00:45:34.300 align:middle line:90% Why? 00:45:34.300 --> 00:45:43.870 align:middle line:90% 00:45:43.870 --> 00:45:46.200 align:middle line:90% No chance. 00:45:46.200 --> 00:45:48.240 align:middle line:90% Probably makes sense. 00:45:48.240 --> 00:45:51.750 align:middle line:84% That was a z equals pi over 2 case. 00:45:51.750 --> 00:45:55.220 align:middle line:84% So this case is maybe particularly relevant 00:45:55.220 --> 00:45:58.310 align:middle line:84% if you're trying to diagnose something like a tokamak, which 00:45:58.310 --> 00:45:59.310 align:middle line:90% many of you are. 00:45:59.310 --> 00:46:03.050 align:middle line:84% So if we draw our tokamak looking from above, 00:46:03.050 --> 00:46:06.920 align:middle line:84% we have magnetic field lines going around like this. 00:46:06.920 --> 00:46:09.800 align:middle line:84% And remember, the toroidal magnetic field of the tokamak 00:46:09.800 --> 00:46:10.950 align:middle line:90% is very strong. 00:46:10.950 --> 00:46:13.670 align:middle line:84% And so really, although the magnetic field lines 00:46:13.670 --> 00:46:17.450 align:middle line:84% are slightly not like this, they really are very much just 00:46:17.450 --> 00:46:19.140 align:middle line:90% circles around the top. 00:46:19.140 --> 00:46:21.140 align:middle line:84% And where do we put our diagnostics? 00:46:21.140 --> 00:46:25.010 align:middle line:84% Well, we can't put them on the inside, usually. 00:46:25.010 --> 00:46:28.250 align:middle line:84% We don't want to put them at some weird line of sight 00:46:28.250 --> 00:46:29.780 align:middle line:84% like this because that would mean 00:46:29.780 --> 00:46:31.820 align:middle line:84% integrating through things where we're not 00:46:31.820 --> 00:46:34.370 align:middle line:90% really sure about the symmetry. 00:46:34.370 --> 00:46:38.750 align:middle line:84% We're very likely to put our diagnostics like this 00:46:38.750 --> 00:46:40.730 align:middle line:84% on a line of sight, which is indeed 00:46:40.730 --> 00:46:44.540 align:middle line:84% perpendicular at 90 degrees at the local magnetic field. 00:46:44.540 --> 00:46:46.310 align:middle line:84% That might be because you've got to fit 00:46:46.310 --> 00:46:47.930 align:middle line:84% through some gaps between the magnets 00:46:47.930 --> 00:46:49.305 align:middle line:84% or simply because it's very easy. 00:46:49.305 --> 00:46:50.780 align:middle line:84% And almost every tokamak I've seen 00:46:50.780 --> 00:46:53.647 align:middle line:84% has diagnostic designed to look along this line of sight. 00:46:53.647 --> 00:46:55.730 align:middle line:84% If you have something that looks at a weird angle, 00:46:55.730 --> 00:46:57.000 align:middle line:90% that's very unusual. 00:46:57.000 --> 00:47:00.290 align:middle line:84% So it's very relevant to ask how the waves propagate 00:47:00.290 --> 00:47:03.470 align:middle line:84% in a magnetized plasma like a tokamak perpendicular 00:47:03.470 --> 00:47:05.870 align:middle line:90% to the magnetic field. 00:47:05.870 --> 00:47:09.680 align:middle line:84% And what you get are two different modes here. 00:47:09.680 --> 00:47:13.490 align:middle line:84% We have one mode where the dispersion relationship 00:47:13.490 --> 00:47:14.780 align:middle line:90% is very familiar. 00:47:14.780 --> 00:47:23.081 align:middle line:84% n squared is equal to 1 minus omega p squared upon c squared-- 00:47:23.081 --> 00:47:23.990 align:middle line:90% oh, omega squared. 00:47:23.990 --> 00:47:27.000 align:middle line:90% 00:47:27.000 --> 00:47:30.240 align:middle line:84% So this is just the wave that we found in unmagnetized plasmas. 00:47:30.240 --> 00:47:32.430 align:middle line:84% What's interesting is that although we've 00:47:32.430 --> 00:47:34.260 align:middle line:84% done all of that mathematics, there 00:47:34.260 --> 00:47:37.570 align:middle line:84% is one wave which propagates the perpendicular magnetic field, 00:47:37.570 --> 00:47:40.630 align:middle line:84% which looks like the magnetic field doesn't matter at all. 00:47:40.630 --> 00:47:47.070 align:middle line:84% And so we call this wave the O mode, where O stands for Order. 00:47:47.070 --> 00:47:53.420 align:middle line:90% 00:47:53.420 --> 00:47:54.040 align:middle line:90% All right. 00:47:54.040 --> 00:47:58.670 align:middle line:84% So in the ordinary mode, remember, we can then go back. 00:47:58.670 --> 00:48:00.280 align:middle line:90% This is our eigenvalue. 00:48:00.280 --> 00:48:01.930 align:middle line:84% We can work out what the eigenmode is 00:48:01.930 --> 00:48:05.290 align:middle line:90% in terms of dx and dy and dz. 00:48:05.290 --> 00:48:12.550 align:middle line:84% And we find that dx is equal to dy is equal to 0 here. 00:48:12.550 --> 00:48:15.540 align:middle line:84% And so all of our electric field is in the z direction. 00:48:15.540 --> 00:48:18.660 align:middle line:84% Just remember we have this diagram where 00:48:18.660 --> 00:48:27.330 align:middle line:84% we have z like this, y like this, and x like this. 00:48:27.330 --> 00:48:30.230 align:middle line:84% I've restricted my k vector to be in the yz plane. 00:48:30.230 --> 00:48:32.870 align:middle line:84% I've now set theta to be pi upon 2, 00:48:32.870 --> 00:48:39.630 align:middle line:84% and so therefore, k is pointing in this direction, which 00:48:39.630 --> 00:48:41.910 align:middle line:84% means that our electric field is pointing 00:48:41.910 --> 00:48:44.280 align:middle line:90% purely in the z direction. 00:48:44.280 --> 00:48:46.925 align:middle line:90% 00:48:46.925 --> 00:48:48.300 align:middle line:84% And this actually gives us a hint 00:48:48.300 --> 00:48:50.788 align:middle line:84% why the mode dispersion relationship 00:48:50.788 --> 00:48:53.080 align:middle line:84% doesn't seem to know anything about the magnetic field. 00:48:53.080 --> 00:48:56.370 align:middle line:84% And that's because the electrons are traveling along 00:48:56.370 --> 00:48:57.810 align:middle line:90% in the z direction. 00:48:57.810 --> 00:49:00.990 align:middle line:84% And the electric field, as it oscillates up and down, 00:49:00.990 --> 00:49:03.750 align:middle line:84% is simply accelerating or decelerating them 00:49:03.750 --> 00:49:05.470 align:middle line:90% along the magnetic field. 00:49:05.470 --> 00:49:08.460 align:middle line:84% And so it doesn't have any effect from the gyrating 00:49:08.460 --> 00:49:09.390 align:middle line:90% particle orbits. 00:49:09.390 --> 00:49:13.187 align:middle line:84% You just have particles which are going like this. 00:49:13.187 --> 00:49:15.520 align:middle line:84% And maybe they're being slightly accelerated or slightly 00:49:15.520 --> 00:49:18.500 align:middle line:84% decelerated, but it's all in the d direction, 00:49:18.500 --> 00:49:20.020 align:middle line:84% so it doesn't have an interaction 00:49:20.020 --> 00:49:24.170 align:middle line:84% with the magnetic field whatsoever. 00:49:24.170 --> 00:49:25.408 align:middle line:90% So these are nice and easy. 00:49:25.408 --> 00:49:26.950 align:middle line:84% And the nice thing about these modes, 00:49:26.950 --> 00:49:29.980 align:middle line:84% actually, is that they are transverse. 00:49:29.980 --> 00:49:33.130 align:middle line:90% So k dot e is equal to 0. 00:49:33.130 --> 00:49:35.140 align:middle line:84% So although we relax that condition, 00:49:35.140 --> 00:49:36.790 align:middle line:84% we obviously didn't need it in order 00:49:36.790 --> 00:49:38.517 align:middle line:84% to get this mode, as you can see, 00:49:38.517 --> 00:49:40.600 align:middle line:84% because we already got the mode before when we did 00:49:40.600 --> 00:49:42.440 align:middle line:90% have that transverse condition. 00:49:42.440 --> 00:49:45.590 align:middle line:90% So this is our nice, easy wave. 00:49:45.590 --> 00:49:47.990 align:middle line:90% The next one is not ordered. 00:49:47.990 --> 00:49:55.220 align:middle line:84% So the next one has the dispersion relationship 00:49:55.220 --> 00:49:56.520 align:middle line:90% that looks like this. 00:49:56.520 --> 00:49:59.870 align:middle line:84% Now, in Hutchinson's book, he rewrites 00:49:59.870 --> 00:50:02.660 align:middle line:84% some of these terms as x and y and things 00:50:02.660 --> 00:50:04.077 align:middle line:84% like that to make it more compact, 00:50:04.077 --> 00:50:06.452 align:middle line:84% which I think is great if you're going to write it a lot. 00:50:06.452 --> 00:50:08.180 align:middle line:84% But just in this case, I want to write it 00:50:08.180 --> 00:50:11.648 align:middle line:84% out in its full generality, in terms of things like the plasma 00:50:11.648 --> 00:50:13.940 align:middle line:84% frequency and stuff like that so that you can see where 00:50:13.940 --> 00:50:15.080 align:middle line:90% all of those terms come from. 00:50:15.080 --> 00:50:16.220 align:middle line:84% So it's going to look a little bit more 00:50:16.220 --> 00:50:18.560 align:middle line:84% complicated than what you get in Hutchinson's book, 00:50:18.560 --> 00:50:19.950 align:middle line:90% but I think it's more useful. 00:50:19.950 --> 00:50:26.620 align:middle line:84% So it's 1 minus omega p squared over omega squared. 00:50:26.620 --> 00:50:28.680 align:middle line:84% Looks good, but then it's actually times 1 00:50:28.680 --> 00:50:33.910 align:middle line:84% minus omega p squared over omega squared. 00:50:33.910 --> 00:50:36.910 align:middle line:90% And all of those-- 00:50:36.910 --> 00:50:37.840 align:middle line:90% not the 1. 00:50:37.840 --> 00:50:42.910 align:middle line:84% All the rest of these are over 1 minus omega p 00:50:42.910 --> 00:50:47.950 align:middle line:84% squared over omega squared minus capital omega squared 00:50:47.950 --> 00:50:49.858 align:middle line:90% over lowercase omega squared. 00:50:49.858 --> 00:50:51.400 align:middle line:84% So this is a little bit Escher-esque. 00:50:51.400 --> 00:50:55.148 align:middle line:84% They're bits of the same thing repeated in a fractal pattern. 00:50:55.148 --> 00:50:57.190 align:middle line:84% And the more you stare at it, the more you think, 00:50:57.190 --> 00:50:59.350 align:middle line:84% I wonder what plasma is playing at going with that. 00:50:59.350 --> 00:51:00.460 align:middle line:90% It seems very complicated. 00:51:00.460 --> 00:51:02.590 align:middle line:84% And remember, this is the simplified version, 00:51:02.590 --> 00:51:04.960 align:middle line:84% where we've already taken theta equals pi upon 2. 00:51:04.960 --> 00:51:07.570 align:middle line:84% If you want to put in all the cosines and sines, 00:51:07.570 --> 00:51:09.680 align:middle line:84% it becomes much more complicated. 00:51:09.680 --> 00:51:13.460 align:middle line:84% So straight away, we can see that this is more than ordinary. 00:51:13.460 --> 00:51:15.340 align:middle line:84% And so this is, of course, the x mode, 00:51:15.340 --> 00:51:17.510 align:middle line:90% and it is the extraordinary. 00:51:17.510 --> 00:51:27.820 align:middle line:90% 00:51:27.820 --> 00:51:30.850 align:middle line:84% Now, you can probably guess just by looking at this 00:51:30.850 --> 00:51:34.120 align:middle line:84% that when we substitute this back in to our equation 00:51:34.120 --> 00:51:37.230 align:middle line:84% and we try and get out the eigenmodes of the system, 00:51:37.230 --> 00:51:39.230 align:middle line:84% they're not going to be quite as simple as this. 00:51:39.230 --> 00:51:40.430 align:middle line:90% And you are quite right. 00:51:40.430 --> 00:51:45.240 align:middle line:84% So what we find out is that dx over dy-- 00:51:45.240 --> 00:51:47.980 align:middle line:90% 00:51:47.980 --> 00:51:50.590 align:middle line:84% so the x and the y components of the electric field 00:51:50.590 --> 00:51:51.850 align:middle line:90% are related to each other. 00:51:51.850 --> 00:51:57.330 align:middle line:84% And the relationship between them is minus I times 1 00:51:57.330 --> 00:52:02.920 align:middle line:84% minus omega p squared over omega squared minus capital omega 00:52:02.920 --> 00:52:06.260 align:middle line:90% squared over omega squared. 00:52:06.260 --> 00:52:12.300 align:middle line:84% All of that is over omega p squared over omega squared, 00:52:12.300 --> 00:52:13.565 align:middle line:90% capital omega. 00:52:13.565 --> 00:52:15.630 align:middle line:90% Then I've got omega like this. 00:52:15.630 --> 00:52:18.270 align:middle line:90% 00:52:18.270 --> 00:52:21.540 align:middle line:84% Although that looks complicated, fortunately for us, 00:52:21.540 --> 00:52:25.320 align:middle line:84% the z component of the electric field is, in fact, 0. 00:52:25.320 --> 00:52:28.360 align:middle line:84% So that saves us a little bit of hassle. 00:52:28.360 --> 00:52:38.300 align:middle line:84% So if I plot out now y and z, x like this, 00:52:38.300 --> 00:52:42.190 align:middle line:84% I still get my magnetic field in this direction. 00:52:42.190 --> 00:52:45.580 align:middle line:84% And I've still got my k vector in this direction. 00:52:45.580 --> 00:52:47.560 align:middle line:90% It's got exactly the same k. 00:52:47.560 --> 00:52:50.550 align:middle line:90% 00:52:50.550 --> 00:52:54.292 align:middle line:84% Can anyone tell me what the electric field looks like here? 00:52:54.292 --> 00:52:56.250 align:middle line:84% Previously, we said the electric field was just 00:52:56.250 --> 00:52:59.010 align:middle line:84% pointing in the z direction because it was. 00:52:59.010 --> 00:53:01.650 align:middle line:84% But now it looks a little bit more complicated. 00:53:01.650 --> 00:53:04.675 align:middle line:90% Does anyone [INAUDIBLE]? 00:53:04.675 --> 00:53:05.175 align:middle line:90% Yes. 00:53:05.175 --> 00:53:06.673 align:middle line:90% AUDIENCE: An ellipse? 00:53:06.673 --> 00:53:08.340 align:middle line:84% JACK HARE: So the answer was an ellipse. 00:53:08.340 --> 00:53:09.820 align:middle line:90% Yes. 00:53:09.820 --> 00:53:11.620 align:middle line:90% Do you want to be more specific? 00:53:11.620 --> 00:53:14.170 align:middle line:90% AUDIENCE: In the xy plane. 00:53:14.170 --> 00:53:17.200 align:middle line:84% They'll be out of phase by [INAUDIBLE].. 00:53:17.200 --> 00:53:17.830 align:middle line:90% JACK HARE: OK. 00:53:17.830 --> 00:53:21.240 align:middle line:84% So we're talking about an ellipse in the xy plane. 00:53:21.240 --> 00:53:23.865 align:middle line:84% When we have an ellipse, we have a minor axis and a major axis. 00:53:23.865 --> 00:53:26.795 align:middle line:90% 00:53:26.795 --> 00:53:29.650 align:middle line:84% What is the orientation of the major axis, with respect 00:53:29.650 --> 00:53:33.310 align:middle line:90% to this xy coordinate system? 00:53:33.310 --> 00:53:35.830 align:middle line:84% That's effectively asking you, is ex bigger than ey 00:53:35.830 --> 00:53:38.250 align:middle line:90% or is ey bigger than ex? 00:53:38.250 --> 00:53:40.240 align:middle line:84% And we made some very strong assumptions 00:53:40.240 --> 00:53:43.660 align:middle line:84% in deriving this that will help you work that out. 00:53:43.660 --> 00:53:47.990 align:middle line:84% In particular, we assumed that this was a high frequency wave. 00:53:47.990 --> 00:53:51.748 align:middle line:90% 00:53:51.748 --> 00:53:52.540 align:middle line:90% Any answers online? 00:53:52.540 --> 00:53:56.320 align:middle line:90% 00:53:56.320 --> 00:53:58.187 align:middle line:84% You've got a 50/50 chance of being right. 00:53:58.187 --> 00:53:59.770 align:middle line:84% I hope everyone realizes that it's not 00:53:59.770 --> 00:54:02.615 align:middle line:84% going to be wildly at some random angle. 00:54:02.615 --> 00:54:04.990 align:middle line:84% AUDIENCE: If we make the frequency really large in there, 00:54:04.990 --> 00:54:08.830 align:middle line:84% then it would be basically 1 divided by-- 00:54:08.830 --> 00:54:10.033 align:middle line:90% so ex is way bigger. 00:54:10.033 --> 00:54:11.450 align:middle line:84% JACK HARE: Yeah, ex is way bigger. 00:54:11.450 --> 00:54:14.050 align:middle line:84% So if we have a high frequency, this term 00:54:14.050 --> 00:54:16.240 align:middle line:84% is going to be very, very small, which means 00:54:16.240 --> 00:54:18.505 align:middle line:90% ex is much bigger than ey. 00:54:18.505 --> 00:54:25.360 align:middle line:84% And so we have an ellipse that is extended like this. 00:54:25.360 --> 00:54:30.480 align:middle line:90% So ex, much bigger than ey. 00:54:30.480 --> 00:54:33.020 align:middle line:84% And because there's this I inside here, 00:54:33.020 --> 00:54:35.520 align:middle line:84% they're actually related to each other in a complex fashion. 00:54:35.520 --> 00:54:37.170 align:middle line:84% So it means the electric field vector 00:54:37.170 --> 00:54:41.610 align:middle line:84% is going to be sweeping out this ellipse as the wave propagates. 00:54:41.610 --> 00:54:42.450 align:middle line:90% OK. 00:54:42.450 --> 00:54:46.200 align:middle line:84% Now, the main thing to note here is that this wave is not 00:54:46.200 --> 00:54:47.190 align:middle line:90% transverse. 00:54:47.190 --> 00:54:53.460 align:middle line:84% So k dot e is not equal to 0 because of course, ky 00:54:53.460 --> 00:54:58.200 align:middle line:84% is parallel to our small but existent ey. 00:54:58.200 --> 00:55:00.210 align:middle line:84% So this is the wave we would not have 00:55:00.210 --> 00:55:02.580 align:middle line:84% got if we insisted on only looking 00:55:02.580 --> 00:55:04.200 align:middle line:84% for transverse waves, which is why 00:55:04.200 --> 00:55:07.420 align:middle line:84% you have to go back and rederive it all with this. 00:55:07.420 --> 00:55:09.570 align:middle line:84% Of course, we also put in the equation of motion 00:55:09.570 --> 00:55:11.430 align:middle line:84% for particles in the magnetic field. 00:55:11.430 --> 00:55:14.010 align:middle line:84% But you can see why it's much easier to go and derive 00:55:14.010 --> 00:55:16.170 align:middle line:84% the ordinary mode in an unmagnetized plasma 00:55:16.170 --> 00:55:19.740 align:middle line:84% than it is to get these two modes in a magnetized plasma 00:55:19.740 --> 00:55:21.670 align:middle line:90% straight away. 00:55:21.670 --> 00:55:24.550 align:middle line:84% So this is a very fun way indeed. 00:55:24.550 --> 00:55:28.320 align:middle line:90% 00:55:28.320 --> 00:55:33.060 align:middle line:84% There is a question in Hutchinson's book 00:55:33.060 --> 00:55:37.560 align:middle line:84% which caused me a lot of thought as a grad student 00:55:37.560 --> 00:55:41.010 align:middle line:84% because I didn't understand what the hell he was getting at. 00:55:41.010 --> 00:55:42.890 align:middle line:84% And so I'll put it to you now, and we 00:55:42.890 --> 00:55:45.530 align:middle line:84% shall see whether you can spot it straight away or not. 00:55:45.530 --> 00:55:47.090 align:middle line:90% The question is in the books. 00:55:47.090 --> 00:55:49.790 align:middle line:84% You have set up an interferometer 00:55:49.790 --> 00:55:54.410 align:middle line:84% looking across the plasma like this. 00:55:54.410 --> 00:55:57.410 align:middle line:84% If you have accidentally set up your interferometer 00:55:57.410 --> 00:56:01.220 align:middle line:84% to measure the x mode rather than the o mode, 00:56:01.220 --> 00:56:03.422 align:middle line:84% what would the error be in your measurements? 00:56:03.422 --> 00:56:04.880 align:middle line:84% And you give some plasma parameters 00:56:04.880 --> 00:56:06.330 align:middle line:90% so you can calculate it. 00:56:06.330 --> 00:56:07.910 align:middle line:84% So remember, in our interferometer, 00:56:07.910 --> 00:56:11.510 align:middle line:84% what we measure is not density, but we measure changes 00:56:11.510 --> 00:56:13.040 align:middle line:90% in refractive index. 00:56:13.040 --> 00:56:16.250 align:middle line:84% And so you're saying, if you set up your interferometer such it 00:56:16.250 --> 00:56:19.310 align:middle line:84% measures this refractive index, how different would your result 00:56:19.310 --> 00:56:23.520 align:middle line:84% be than if you were measuring this refractive index? 00:56:23.520 --> 00:56:25.380 align:middle line:84% And my question I was always asking-- 00:56:25.380 --> 00:56:28.030 align:middle line:84% how the hell do you even set it up to do one or the other? 00:56:28.030 --> 00:56:31.180 align:middle line:84% So perhaps we can work it out together. 00:56:31.180 --> 00:56:33.150 align:middle line:84% How can I get to choose whether I'm 00:56:33.150 --> 00:56:34.880 align:middle line:90% using the o mode or the x mode? 00:56:34.880 --> 00:56:40.320 align:middle line:90% 00:56:40.320 --> 00:56:43.920 align:middle line:84% AUDIENCE: Well, for o mode, you have a z chord, 00:56:43.920 --> 00:56:48.463 align:middle line:84% so you could polarize your light coming in, right? 00:56:48.463 --> 00:56:49.130 align:middle line:90% JACK HARE: Yeah. 00:56:49.130 --> 00:56:52.670 align:middle line:84% So if I'm injecting light into my interferometer-- 00:56:52.670 --> 00:56:55.170 align:middle line:84% and maybe it's collecting here and bounces back to that. 00:56:55.170 --> 00:57:00.350 align:middle line:84% So if I have this polarization, that's 00:57:00.350 --> 00:57:04.610 align:middle line:84% the o mode because the polarization, e, is parallel 00:57:04.610 --> 00:57:06.140 align:middle line:90% to b. 00:57:06.140 --> 00:57:14.560 align:middle line:84% And if I have this polarization out of the page like that, 00:57:14.560 --> 00:57:17.280 align:middle line:90% that's the x mode. 00:57:17.280 --> 00:57:19.050 align:middle line:84% So you get to choose which refractive 00:57:19.050 --> 00:57:23.893 align:middle line:84% index you probe by your choice of the polarization 00:57:23.893 --> 00:57:24.810 align:middle line:90% that you're injecting. 00:57:24.810 --> 00:57:27.060 align:middle line:84% And that was the bit that I missed as a grad student. 00:57:27.060 --> 00:57:29.400 align:middle line:84% It took me until I was teaching this class for the first time 00:57:29.400 --> 00:57:30.870 align:middle line:84% to finally work out what's going on-- 00:57:30.870 --> 00:57:32.350 align:middle line:84% is that you actually do have a choice. 00:57:32.350 --> 00:57:33.970 align:middle line:84% It's not like the plasma decides for you, 00:57:33.970 --> 00:57:36.345 align:middle line:84% which is why I was like, what will the plasma want to do? 00:57:36.345 --> 00:57:38.507 align:middle line:84% But that doesn't matter in this case. 00:57:38.507 --> 00:57:39.840 align:middle line:90% In this case, you have a choice. 00:57:39.840 --> 00:57:41.970 align:middle line:84% The plasma is not in control of you. 00:57:41.970 --> 00:57:44.490 align:middle line:90% 00:57:44.490 --> 00:57:46.260 align:middle line:84% And in general, you could launch a wave 00:57:46.260 --> 00:57:48.660 align:middle line:84% through the plasma at some arbitrary 00:57:48.660 --> 00:57:50.580 align:middle line:90% polarization at 45 degrees. 00:57:50.580 --> 00:57:54.900 align:middle line:84% And then because you can always decompose your wave 00:57:54.900 --> 00:57:57.660 align:middle line:84% into various sets of modes-- but this 00:57:57.660 --> 00:58:01.110 align:middle line:84% is a good set of modes which is valid for theta 00:58:01.110 --> 00:58:02.550 align:middle line:90% equals pi upon 2. 00:58:02.550 --> 00:58:05.520 align:middle line:84% Then you would see that one of your polarizations 00:58:05.520 --> 00:58:07.950 align:middle line:84% would travel faster than the other polarization 00:58:07.950 --> 00:58:10.440 align:middle line:84% because it would have a different refractive index. 00:58:10.440 --> 00:58:12.990 align:middle line:84% And so you'd see all sorts of funky effects going on that 00:58:12.990 --> 00:58:14.350 align:middle line:90% may be very hard to interpret. 00:58:14.350 --> 00:58:17.860 align:middle line:84% So it's definitely worth thinking about the polarization. 00:58:17.860 --> 00:58:20.800 align:middle line:84% And when I went looking recently for some papers on how people 00:58:20.800 --> 00:58:23.020 align:middle line:84% actually do this in tokamaks, they 00:58:23.020 --> 00:58:25.555 align:middle line:84% spend an awful lot of time thinking about the polarization. 00:58:25.555 --> 00:58:27.430 align:middle line:84% And often, they have the ability to switch it 00:58:27.430 --> 00:58:29.080 align:middle line:84% between x mode and o mode so they 00:58:29.080 --> 00:58:30.640 align:middle line:90% can do different measurements. 00:58:30.640 --> 00:58:31.715 align:middle line:90% Yes. 00:58:31.715 --> 00:58:34.720 align:middle line:84% AUDIENCE: The x mode should be polarized. 00:58:34.720 --> 00:58:36.490 align:middle line:84% JACK HARE: The x mode should be polarized. 00:58:36.490 --> 00:58:42.460 align:middle line:90% So yeah, out of the page. 00:58:42.460 --> 00:58:45.400 align:middle line:84% Because you're primarily going to be exciting this large ex. 00:58:45.400 --> 00:58:48.220 align:middle line:84% And the plasma will do the work of giving you the ey, 00:58:48.220 --> 00:58:50.437 align:middle line:84% and it will do that because of the electrons 00:58:50.437 --> 00:58:51.770 align:middle line:90% gyrating around the field lines. 00:58:51.770 --> 00:58:53.270 align:middle line:84% So you don't have to worry about it. 00:58:53.270 --> 00:58:55.490 align:middle line:84% ey is truly fantastically small here. 00:58:55.490 --> 00:58:56.410 align:middle line:90% So it's OK. 00:58:56.410 --> 00:58:58.420 align:middle line:84% You don't have to give it that yourself, which 00:58:58.420 --> 00:59:00.340 align:middle line:84% would be impossible because in free space, 00:59:00.340 --> 00:59:01.450 align:middle line:90% waves are only transverse. 00:59:01.450 --> 00:59:03.580 align:middle line:84% You can't launch a wave in free space 00:59:03.580 --> 00:59:07.060 align:middle line:84% that has a polarization along the direction of propagation. 00:59:07.060 --> 00:59:07.600 align:middle line:90% But it's OK. 00:59:07.600 --> 00:59:09.820 align:middle line:84% The plasma has got your back there. 00:59:09.820 --> 00:59:12.430 align:middle line:84% And these are the only two waves-- 00:59:12.430 --> 00:59:14.230 align:middle line:84% the only two electromagnetic high frequency 00:59:14.230 --> 00:59:17.990 align:middle line:84% cold magnetized waves which can propagate in a plasma. 00:59:17.990 --> 00:59:20.140 align:middle line:84% So if you hit the plasma of some wave, 00:59:20.140 --> 00:59:22.960 align:middle line:84% it will instantly convert into some mixture of these two, 00:59:22.960 --> 00:59:24.627 align:middle line:84% because they're the only modes which are 00:59:24.627 --> 00:59:26.230 align:middle line:90% supported by the plasma media. 00:59:26.230 --> 00:59:27.370 align:middle line:90% Yeah. 00:59:27.370 --> 00:59:30.270 align:middle line:84% AUDIENCE: When that converting step is happening, 00:59:30.270 --> 00:59:32.340 align:middle line:84% are you at all worried about reflected power 00:59:32.340 --> 00:59:35.790 align:middle line:90% than being different-- 00:59:35.790 --> 00:59:37.980 align:middle line:84% because if you're injecting a wave into the plasma, 00:59:37.980 --> 00:59:38.940 align:middle line:90% that's a vacuum wave. 00:59:38.940 --> 00:59:41.400 align:middle line:90% And so [INAUDIBLE] x. 00:59:41.400 --> 00:59:44.100 align:middle line:84% I'd imagine there's some chance of that vacuum wave reflecting 00:59:44.100 --> 00:59:44.640 align:middle line:90% back on you. 00:59:44.640 --> 00:59:48.510 align:middle line:84% Would that hurt your signal to noise ratio in some way? 00:59:48.510 --> 00:59:51.220 align:middle line:84% Or is there a very small amount reflects back 00:59:51.220 --> 00:59:52.770 align:middle line:84% so it doesn't really [INAUDIBLE]?? 00:59:52.770 --> 00:59:53.880 align:middle line:84% JACK HARE: So the question is when 00:59:53.880 --> 00:59:55.800 align:middle line:84% you have this conversion from the vacuum wave 00:59:55.800 --> 00:59:58.770 align:middle line:84% into the plasma waves, is there any power which is reflected? 00:59:58.770 --> 01:00:00.220 align:middle line:90% And the answer is I don't know. 01:00:00.220 --> 01:00:02.640 align:middle line:84% I would imagine that there could be some power reflected 01:00:02.640 --> 01:00:04.740 align:middle line:90% in that interface there. 01:00:04.740 --> 01:00:08.550 align:middle line:84% And yeah, also an important thing to notice 01:00:08.550 --> 01:00:10.440 align:middle line:84% is that on a tokamak, the magnetic field 01:00:10.440 --> 01:00:11.950 align:middle line:90% stays in the same direction. 01:00:11.950 --> 01:00:13.698 align:middle line:84% But if you're in something like an RFP-- 01:00:13.698 --> 01:00:15.990 align:middle line:84% Reversed Field Pinch, where the magnetic field rotates, 01:00:15.990 --> 01:00:17.868 align:middle line:90% your wave will invert. 01:00:17.868 --> 01:00:19.410 align:middle line:84% You may launch in o mode and then may 01:00:19.410 --> 01:00:21.340 align:middle line:90% convert yourself to x mode. 01:00:21.340 --> 01:00:22.830 align:middle line:84% And intermediate, you'll have all 01:00:22.830 --> 01:00:26.263 align:middle line:84% those intermediate polarizations and k vectors, with respect 01:00:26.263 --> 01:00:27.180 align:middle line:90% to the magnetic field. 01:00:27.180 --> 01:00:28.555 align:middle line:84% And you'll have to go back and do 01:00:28.555 --> 01:00:31.003 align:middle line:90% the Appleton-Hartree formalism. 01:00:31.003 --> 01:00:33.420 align:middle line:84% And that's probably why people don't work on RFPs anymore, 01:00:33.420 --> 01:00:35.700 align:middle line:84% because they're extremely difficult to work with. 01:00:35.700 --> 01:00:39.180 align:middle line:84% Now, one thing I will note is you might ask know, Jack, you 01:00:39.180 --> 01:00:41.838 align:middle line:84% work on z-pinch and they've got strong magnetic fields. 01:00:41.838 --> 01:00:43.380 align:middle line:84% And yet, you don't seem to be worried 01:00:43.380 --> 01:00:46.710 align:middle line:84% at all about the magnetic field or polarizing your beam 01:00:46.710 --> 01:00:47.862 align:middle line:90% or anything like that. 01:00:47.862 --> 01:00:49.320 align:middle line:84% And the reason for that is when you 01:00:49.320 --> 01:00:52.410 align:middle line:84% start looking at the frequency orderings 01:00:52.410 --> 01:00:56.190 align:middle line:84% in the sorts of plasmas I work with, this term here 01:00:56.190 --> 01:00:59.440 align:middle line:84% is very, very, very, very, very, very, very, very small. 01:00:59.440 --> 01:01:01.470 align:middle line:90% And so this cancels out that. 01:01:01.470 --> 01:01:03.533 align:middle line:84% And huzzah, you just have two o modes, 01:01:03.533 --> 01:01:05.700 align:middle line:84% so you're basically back to the unmagnetized plasma. 01:01:05.700 --> 01:01:07.980 align:middle line:84% So depending on your plasma regime, 01:01:07.980 --> 01:01:10.512 align:middle line:84% you may not be sensitive to the field anyway. 01:01:10.512 --> 01:01:11.970 align:middle line:84% And basically, that's a requirement 01:01:11.970 --> 01:01:16.410 align:middle line:84% that the probing frequency of your wave is much, much higher 01:01:16.410 --> 01:01:19.890 align:middle line:84% than the gyro frequency, which is the unmagnetized condition we 01:01:19.890 --> 01:01:22.260 align:middle line:84% wrote down all the way back when we 01:01:22.260 --> 01:01:23.700 align:middle line:90% derived the unmagnetized waves. 01:01:23.700 --> 01:01:25.200 align:middle line:84% So you may not be sensitive to this. 01:01:25.200 --> 01:01:26.610 align:middle line:84% It just turns out in the tokamak, 01:01:26.610 --> 01:01:29.440 align:middle line:84% you tend to be in a regime where this is important. 01:01:29.440 --> 01:01:31.890 align:middle line:84% So we'll talk a little bit about that 01:01:31.890 --> 01:01:35.160 align:middle line:84% when we talk about electron cyclotron emission. 01:01:35.160 --> 01:01:37.020 align:middle line:90% OK. 01:01:37.020 --> 01:01:38.370 align:middle line:90% Any questions on this? 01:01:38.370 --> 01:01:41.878 align:middle line:90% 01:01:41.878 --> 01:01:43.420 align:middle line:84% AUDIENCE: You noted just a minute ago 01:01:43.420 --> 01:01:44.920 align:middle line:84% there's a number of assumptions that 01:01:44.920 --> 01:01:46.750 align:middle line:84% have gone into derivation, chief among them 01:01:46.750 --> 01:01:48.860 align:middle line:90% being low temperature. 01:01:48.860 --> 01:01:52.210 align:middle line:84% Most of today's modern tokamaks are operating 1 01:01:52.210 --> 01:01:54.790 align:middle line:90% to 10 keV ion temperature. 01:01:54.790 --> 01:01:56.830 align:middle line:84% It does not seem to me like something that needs 01:01:56.830 --> 01:01:58.220 align:middle line:90% a low temperature requirement. 01:01:58.220 --> 01:01:59.820 align:middle line:90% So what is the-- 01:01:59.820 --> 01:02:02.460 align:middle line:90% 01:02:02.460 --> 01:02:07.380 align:middle line:84% maximum sequence theory yet applied to a diagnostic system, 01:02:07.380 --> 01:02:09.522 align:middle line:90% is this valid? 01:02:09.522 --> 01:02:11.210 align:middle line:90% Do we still meet the thermal? 01:02:11.210 --> 01:02:12.787 align:middle line:84% JACK HARE: Yeah, so when we say cold, 01:02:12.787 --> 01:02:15.120 align:middle line:84% we're just talking about compared to the speed of light. 01:02:15.120 --> 01:02:16.290 align:middle line:90% And that's pretty fast. 01:02:16.290 --> 01:02:18.260 align:middle line:90% So even when you've got-- 01:02:18.260 --> 01:02:22.430 align:middle line:84% so you can't neglect relativistic effects completely 01:02:22.430 --> 01:02:24.140 align:middle line:90% for electrons at 10 keV. 01:02:24.140 --> 01:02:27.260 align:middle line:90% They've got over 20%-- 01:02:27.260 --> 01:02:30.480 align:middle line:84% some decent fraction of their rest mass at 500 keV. 01:02:30.480 --> 01:02:31.920 align:middle line:90% It's not less than that. 01:02:31.920 --> 01:02:34.940 align:middle line:84% So in some cases, relativistic effects will be important. 01:02:34.940 --> 01:02:37.430 align:middle line:84% And we'll come across those when we do cyclotron radiation. 01:02:37.430 --> 01:02:40.220 align:middle line:84% I believe that for interferometry, we 01:02:40.220 --> 01:02:42.650 align:middle line:84% don't have to worry about relativistic effects here, 01:02:42.650 --> 01:02:44.318 align:middle line:90% and this holds well enough. 01:02:44.318 --> 01:02:45.860 align:middle line:84% And the corrections that you get will 01:02:45.860 --> 01:02:48.590 align:middle line:84% be on the order of the ratio between the thermal velocity 01:02:48.590 --> 01:02:51.015 align:middle line:84% and the speed of light, and that is a small number. 01:02:51.015 --> 01:02:52.640 align:middle line:84% So I think there's a really good point. 01:02:52.640 --> 01:02:56.420 align:middle line:84% But this actually-- this holds very, very well. 01:02:56.420 --> 01:02:58.970 align:middle line:84% We're not dealing with plasma waves, 01:02:58.970 --> 01:03:01.490 align:middle line:84% the waves inside a plasma which are being generated, 01:03:01.490 --> 01:03:03.987 align:middle line:84% where thermal effects are important, like Langmuir waves. 01:03:03.987 --> 01:03:06.570 align:middle line:84% We're dealing with these high frequency electromagnetic waves, 01:03:06.570 --> 01:03:09.003 align:middle line:84% which are traveling so fast that their phase velocity is 01:03:09.003 --> 01:03:10.170 align:middle line:90% close to the speed of light. 01:03:10.170 --> 01:03:12.810 align:middle line:84% Really, sorry-- I shouldn't say c here. 01:03:12.810 --> 01:03:14.343 align:middle line:90% This is the phase velocity. 01:03:14.343 --> 01:03:17.010 align:middle line:84% It's just the phase velocity is very close to the speed of light 01:03:17.010 --> 01:03:20.110 align:middle line:84% for these electromagnetic waves, so c is close enough. 01:03:20.110 --> 01:03:22.020 align:middle line:90% So this is the actual condition. 01:03:22.020 --> 01:03:23.520 align:middle line:84% For the waves inside your plasma, 01:03:23.520 --> 01:03:26.490 align:middle line:84% where you deal with hot plasma effects and Landau damping 01:03:26.490 --> 01:03:28.250 align:middle line:84% and all sorts of fun things like that, 01:03:28.250 --> 01:03:29.667 align:middle line:84% that phase velocity is much lower. 01:03:29.667 --> 01:03:32.125 align:middle line:84% And so the phase velocity is close to the thermal velocity. 01:03:32.125 --> 01:03:33.540 align:middle line:84% And then you have the interaction 01:03:33.540 --> 01:03:36.387 align:middle line:84% between the wave and the distribution function of plasma 01:03:36.387 --> 01:03:38.470 align:middle line:84% that gives you Landau damping, all that fun stuff. 01:03:38.470 --> 01:03:40.050 align:middle line:90% We're nowhere near that. 01:03:40.050 --> 01:03:41.970 align:middle line:84% But that's a really good question. 01:03:41.970 --> 01:03:43.130 align:middle line:90% OK. 01:03:43.130 --> 01:03:44.240 align:middle line:90% Any other questions? 01:03:44.240 --> 01:03:51.720 align:middle line:90% 01:03:51.720 --> 01:03:52.220 align:middle line:90% OK. 01:03:52.220 --> 01:03:53.603 align:middle line:90% Now we'll do the interface. 01:03:53.603 --> 01:03:57.230 align:middle line:90% 01:03:57.230 --> 01:04:00.390 align:middle line:84% This has nothing to do with the diagram. 01:04:00.390 --> 01:04:04.690 align:middle line:84% But it's interesting, and we'll use it twice again. 01:04:04.690 --> 01:04:06.430 align:middle line:84% So now we're going to do the case 01:04:06.430 --> 01:04:11.800 align:middle line:84% where the wave is propagating along the magnetic field here. 01:04:11.800 --> 01:04:12.300 align:middle line:90% OK. 01:04:12.300 --> 01:04:16.560 align:middle line:84% So this was maybe a harder case to get inside the tokamak, 01:04:16.560 --> 01:04:18.600 align:middle line:84% but it's a very easy case to get inside, 01:04:18.600 --> 01:04:21.520 align:middle line:84% for example, the z-pinch or many other systems like that. 01:04:21.520 --> 01:04:26.810 align:middle line:84% So for example, if you have a z-pinch, some wobbly plasma 01:04:26.810 --> 01:04:30.690 align:middle line:84% like this, it's got current like this. 01:04:30.690 --> 01:04:37.160 align:middle line:84% And then it's got magnetic field like this. 01:04:37.160 --> 01:04:39.063 align:middle line:84% If I fire a laser beam through this, 01:04:39.063 --> 01:04:41.480 align:middle line:84% there's going to be at least parts of the laser beam which 01:04:41.480 --> 01:04:47.470 align:middle line:84% are parallel or anti-parallel-- you get the same result-- 01:04:47.470 --> 01:04:48.850 align:middle line:90% to the magnetic field. 01:04:48.850 --> 01:04:52.270 align:middle line:84% And in fact, we'll show that this result that we derived 01:04:52.270 --> 01:04:56.680 align:middle line:84% for theta equals 0 applies to almost every angle between 0 01:04:56.680 --> 01:04:57.820 align:middle line:90% and pi over 2. 01:04:57.820 --> 01:05:00.790 align:middle line:84% And the theory only breaks down very, very close to pi over 2. 01:05:00.790 --> 01:05:03.860 align:middle line:84% So in fact, this still works even when, 01:05:03.860 --> 01:05:06.280 align:middle line:84% for example, here, my k is in the same direction 01:05:06.280 --> 01:05:07.840 align:middle line:90% but my magnetic field is bent. 01:05:07.840 --> 01:05:10.940 align:middle line:84% And there's quite a large theta between here. 01:05:10.940 --> 01:05:13.840 align:middle line:90% So this is very useful. 01:05:13.840 --> 01:05:15.430 align:middle line:90% OK. 01:05:15.430 --> 01:05:20.500 align:middle line:84% We go and we plug theta equals 0 into the Appleton-Hartree 01:05:20.500 --> 01:05:24.160 align:middle line:84% relationship, and then we solve the determinant 01:05:24.160 --> 01:05:27.070 align:middle line:84% and we get out our eigenvalues and our eigenmodes. 01:05:27.070 --> 01:05:28.900 align:middle line:90% And we get, first of all-- 01:05:28.900 --> 01:05:33.710 align:middle line:90% 01:05:33.710 --> 01:05:35.730 align:middle line:90% well, we've got two nodes. 01:05:35.730 --> 01:05:38.022 align:middle line:84% And these nodes I tend to write with a plus and a minus 01:05:38.022 --> 01:05:40.105 align:middle line:84% and write the two of them together because they're 01:05:40.105 --> 01:05:41.020 align:middle line:90% very, very similar. 01:05:41.020 --> 01:05:43.590 align:middle line:84% And so this mode plus and minus-- 01:05:43.590 --> 01:05:45.240 align:middle line:84% that's the names of the two modes-- 01:05:45.240 --> 01:05:50.490 align:middle line:84% is equal to 1 minus omega p squared upon omega 01:05:50.490 --> 01:05:54.270 align:middle line:90% squared over 1 plus and minus-- 01:05:54.270 --> 01:05:56.040 align:middle line:84% this is the difference between the two-- 01:05:56.040 --> 01:05:59.850 align:middle line:84% capital omega over lowercase omega. 01:05:59.850 --> 01:06:02.680 align:middle line:84% So the first thing that we notice, of course, 01:06:02.680 --> 01:06:07.130 align:middle line:84% is that for capital omega and the lowercase omega 01:06:07.130 --> 01:06:10.640 align:middle line:84% much less than 1, which was our unmagnetized condition, 01:06:10.640 --> 01:06:14.970 align:middle line:84% we just reduce back to our magnetized dispersion relation. 01:06:14.970 --> 01:06:15.980 align:middle line:90% So that's good. 01:06:15.980 --> 01:06:19.090 align:middle line:84% We haven't introduced anything funky in the math. 01:06:19.090 --> 01:06:21.660 align:middle line:90% 01:06:21.660 --> 01:06:22.620 align:middle line:90% OK. 01:06:22.620 --> 01:06:25.740 align:middle line:84% And then when we solve to get the eigenmodes, 01:06:25.740 --> 01:06:33.040 align:middle line:84% we find out that we have ex over ey is equal to plus or minus I 01:06:33.040 --> 01:06:33.540 align:middle line:90% here. 01:06:33.540 --> 01:06:38.320 align:middle line:84% Let me just draw again the geometry of the system. 01:06:38.320 --> 01:06:43.180 align:middle line:84% That's upwards y like this, x like this. 01:06:43.180 --> 01:06:46.810 align:middle line:84% Our magnetic field is in the z direction. 01:06:46.810 --> 01:06:50.770 align:middle line:84% And now our k vector, theta equals 0, 01:06:50.770 --> 01:06:54.280 align:middle line:90% is also in the z direction. 01:06:54.280 --> 01:06:58.350 align:middle line:84% And now we have the ex and ey lying in [? k. ?] 01:06:58.350 --> 01:07:00.240 align:middle line:90% And ez is equal to 0. 01:07:00.240 --> 01:07:03.800 align:middle line:84% So this is, again, a transverse wave. 01:07:03.800 --> 01:07:06.160 align:middle line:90% k dot equals 0, like that. 01:07:06.160 --> 01:07:09.260 align:middle line:90% 01:07:09.260 --> 01:07:13.390 align:middle line:84% So does anyone know what this wave is? 01:07:13.390 --> 01:07:16.780 align:middle line:84% The electric and the two components of the electric field 01:07:16.780 --> 01:07:19.780 align:middle line:84% appear to be out of phase from each other by a factor of pi 01:07:19.780 --> 01:07:21.270 align:middle line:90% upon 2, plus or minus sign. 01:07:21.270 --> 01:07:25.733 align:middle line:90% 01:07:25.733 --> 01:07:27.025 align:middle line:90% AUDIENCE: Circularly polarized? 01:07:27.025 --> 01:07:28.660 align:middle line:84% JACK HARE: Circularly polarized, yes. 01:07:28.660 --> 01:07:32.020 align:middle line:84% Everyone in the room is doing this. 01:07:32.020 --> 01:07:34.730 align:middle line:84% I guess they wanted to say circularly polarized. 01:07:34.730 --> 01:07:35.230 align:middle line:90% OK. 01:07:35.230 --> 01:07:40.030 align:middle line:84% So what we can do is we can say ex is going to go 01:07:40.030 --> 01:07:47.170 align:middle line:90% to exponential of I k dot x. 01:07:47.170 --> 01:07:51.420 align:middle line:84% In reality, this is just [INAUDIBLE] 01:07:51.420 --> 01:07:54.150 align:middle line:84% because we know which way our [INAUDIBLE] here. 01:07:54.150 --> 01:08:00.570 align:middle line:84% Minus omega t and ey is exactly the same, 01:08:00.570 --> 01:08:05.265 align:middle line:84% plus an extra factor of pi upon 2 inside the brackets. 01:08:05.265 --> 01:08:09.280 align:middle line:90% 01:08:09.280 --> 01:08:09.780 align:middle line:90% OK. 01:08:09.780 --> 01:08:15.802 align:middle line:84% So if we plot in either space or time-- 01:08:15.802 --> 01:08:17.760 align:middle line:84% it doesn't matter-- we can either fix ourselves 01:08:17.760 --> 01:08:19.590 align:middle line:84% in one place and watch a wave go by 01:08:19.590 --> 01:08:21.479 align:middle line:84% or we can attach ourselves to the wave 01:08:21.479 --> 01:08:23.279 align:middle line:90% and see how it changes in space. 01:08:23.279 --> 01:08:27.210 align:middle line:84% We're going to get at two oscillating fields. 01:08:27.210 --> 01:08:32.250 align:middle line:84% The electric field in x will look like this. 01:08:32.250 --> 01:08:36.100 align:middle line:84% The electric field in y will look like-- 01:08:36.100 --> 01:08:39.880 align:middle line:90% 01:08:39.880 --> 01:08:42.049 align:middle line:84% I'm trying very hard to do this properly-- 01:08:42.049 --> 01:08:42.549 align:middle line:90% that. 01:08:42.549 --> 01:08:45.750 align:middle line:90% 01:08:45.750 --> 01:08:51.220 align:middle line:84% And if you then look at what in the xy plane 01:08:51.220 --> 01:08:54.100 align:middle line:84% the electric field is doing, and you 01:08:54.100 --> 01:08:58.840 align:middle line:84% stop at this point here and then at this point here and then 01:08:58.840 --> 01:09:01.630 align:middle line:84% at this point here and then at this point here, 01:09:01.630 --> 01:09:05.560 align:middle line:84% and ask, which direction is the electric field pointing? 01:09:05.560 --> 01:09:09.229 align:middle line:84% Well, first of all, it's going to be pointing entirely 01:09:09.229 --> 01:09:10.670 align:middle line:90% in the y direction. 01:09:10.670 --> 01:09:13.430 align:middle line:84% And then it's going to be pointing entirely 01:09:13.430 --> 01:09:14.779 align:middle line:90% in the x direction. 01:09:14.779 --> 01:09:17.370 align:middle line:90% 01:09:17.370 --> 01:09:21.015 align:middle line:84% And so we can see that the electric field vector traces out 01:09:21.015 --> 01:09:21.515 align:middle line:90% a circle. 01:09:21.515 --> 01:09:25.250 align:middle line:84% And this is what we call circular polarization. 01:09:25.250 --> 01:09:28.100 align:middle line:84% And often, these two waves, which I've 01:09:28.100 --> 01:09:31.020 align:middle line:84% been calling plus or minus here-- 01:09:31.020 --> 01:09:33.750 align:middle line:84% you might call them the right hand and the left hand 01:09:33.750 --> 01:09:36.312 align:middle line:90% circularly polarized wave. 01:09:36.312 --> 01:09:37.979 align:middle line:84% And I can never remember the convention, 01:09:37.979 --> 01:09:41.412 align:middle line:84% but I think it's right hand rule with a k vector. 01:09:41.412 --> 01:09:42.870 align:middle line:84% And it's like, is it going this way 01:09:42.870 --> 01:09:45.524 align:middle line:84% or is it going the other way like that? 01:09:45.524 --> 01:09:48.120 align:middle line:84% I never really mind too much about which way around it is. 01:09:48.120 --> 01:09:51.390 align:middle line:84% But there is a convention about whether it's right or left, 01:09:51.390 --> 01:09:54.540 align:middle line:84% and that's what this plus and negative sign really mean here. 01:09:54.540 --> 01:09:57.600 align:middle line:84% We've got two modes, one of which is going clockwise 01:09:57.600 --> 01:10:01.290 align:middle line:84% and one of which is going counterclockwise. 01:10:01.290 --> 01:10:02.400 align:middle line:90% OK. 01:10:02.400 --> 01:10:03.520 align:middle line:90% Any questions on that? 01:10:03.520 --> 01:10:05.110 align:middle line:84% We're going to use that in a moment. 01:10:05.110 --> 01:10:06.450 align:middle line:84% There's not really anything particularly profound 01:10:06.450 --> 01:10:08.742 align:middle line:84% at this point, but we will get on to something profound 01:10:08.742 --> 01:10:09.382 align:middle line:90% in a moment. 01:10:09.382 --> 01:10:10.590 align:middle line:90% Any questions on that before? 01:10:10.590 --> 01:10:37.720 align:middle line:90% 01:10:37.720 --> 01:10:38.350 align:middle line:90% OK. 01:10:38.350 --> 01:10:40.240 align:middle line:84% Just so I can keep that up on the board, 01:10:40.240 --> 01:10:41.860 align:middle line:84% we're going to get into this stuff. 01:10:41.860 --> 01:10:46.239 align:middle line:84% Forget about that until we get on to reflectometry. 01:10:46.239 --> 01:11:05.320 align:middle line:90% 01:11:05.320 --> 01:11:07.343 align:middle line:84% So now, finally, we are fulfilling the promise 01:11:07.343 --> 01:11:09.010 align:middle line:84% we started the lecture with when we were 01:11:09.010 --> 01:11:10.302 align:middle line:90% talking about Faraday rotation. 01:11:10.302 --> 01:11:17.130 align:middle line:90% 01:11:17.130 --> 01:11:19.830 align:middle line:84% And the main point about Faraday rotation 01:11:19.830 --> 01:11:24.270 align:middle line:84% is that your magnetic fields cause a phenomena called 01:11:24.270 --> 01:11:25.140 align:middle line:90% birefringence. 01:11:25.140 --> 01:11:28.400 align:middle line:90% 01:11:28.400 --> 01:11:31.100 align:middle line:84% Has anyone come across birefringence 01:11:31.100 --> 01:11:34.940 align:middle line:84% studying optics before and can give us a concise definition? 01:11:34.940 --> 01:11:40.960 align:middle line:90% 01:11:40.960 --> 01:11:41.860 align:middle line:90% Yes. 01:11:41.860 --> 01:11:42.610 align:middle line:90% Either one of you. 01:11:42.610 --> 01:11:46.270 align:middle line:90% You can say it in unison. 01:11:46.270 --> 01:11:51.430 align:middle line:84% AUDIENCE: 2, 3-- [INAUDIBLE] refraction [INAUDIBLE],, 01:11:51.430 --> 01:11:53.020 align:middle line:90% which [INAUDIBLE]. 01:11:53.020 --> 01:11:54.857 align:middle line:84% JACK HARE: Its object's index of refraction 01:11:54.857 --> 01:11:57.190 align:middle line:84% is different, depending on the direction of propagation. 01:11:57.190 --> 01:11:57.815 align:middle line:90% AUDIENCE: Yeah. 01:11:57.815 --> 01:11:58.450 align:middle line:90% JACK HARE: No. 01:11:58.450 --> 01:11:59.230 align:middle line:90% AUDIENCE: Darn. 01:11:59.230 --> 01:12:00.805 align:middle line:90% OK. 01:12:00.805 --> 01:12:02.305 align:middle line:84% JACK HARE: Was that your guess, too? 01:12:02.305 --> 01:12:03.057 align:middle line:90% AUDIENCE: No. 01:12:03.057 --> 01:12:05.140 align:middle line:84% JACK HARE: That is a really interesting phenomena, 01:12:05.140 --> 01:12:07.125 align:middle line:84% and it definitely happens, but it's not this. 01:12:07.125 --> 01:12:09.250 align:middle line:84% AUDIENCE: I'm trying to make sure these words don't 01:12:09.250 --> 01:12:10.240 align:middle line:90% mean the same thing. 01:12:10.240 --> 01:12:13.820 align:middle line:84% That the angle of deflection depends on your frequency, 01:12:13.820 --> 01:12:15.100 align:middle line:90% which [INAUDIBLE]. 01:12:15.100 --> 01:12:15.760 align:middle line:90% JACK HARE: OK. 01:12:15.760 --> 01:12:17.928 align:middle line:84% So that was the angle of deflection 01:12:17.928 --> 01:12:18.970 align:middle line:90% depends on the frequency. 01:12:18.970 --> 01:12:20.260 align:middle line:90% No, that's also not true. 01:12:20.260 --> 01:12:24.850 align:middle line:84% But that's not a good definition of birefringence. 01:12:24.850 --> 01:12:28.000 align:middle line:84% Anyone got any thoughts about what birefringence 01:12:28.000 --> 01:12:30.130 align:middle line:90% could be related to? 01:12:30.130 --> 01:12:33.250 align:middle line:84% Anything on this board here that makes you think? 01:12:33.250 --> 01:12:35.080 align:middle line:90% Anyone online? 01:12:35.080 --> 01:12:36.610 align:middle line:90% AUDIENCE: I'll give it a try. 01:12:36.610 --> 01:12:37.390 align:middle line:90% JACK HARE: Sorry? 01:12:37.390 --> 01:12:38.598 align:middle line:90% AUDIENCE: I'll give it a try. 01:12:38.598 --> 01:12:39.825 align:middle line:90% JACK HARE: Yeah, please. 01:12:39.825 --> 01:12:41.950 align:middle line:84% AUDIENCE: Birefringence is that there are different 01:12:41.950 --> 01:12:44.293 align:middle line:84% refractive indexes for different directions-- 01:12:44.293 --> 01:12:46.210 align:middle line:84% not different directions, different components 01:12:46.210 --> 01:12:48.710 align:middle line:90% of the material. 01:12:48.710 --> 01:12:51.640 align:middle line:84% So it's like a tensor, and then the different components 01:12:51.640 --> 01:12:53.140 align:middle line:84% have different refractive indexes. 01:12:53.140 --> 01:12:54.890 align:middle line:84% JACK HARE: Yeah, this is very, very close. 01:12:54.890 --> 01:12:56.680 align:middle line:84% So it's different refractive indexes 01:12:56.680 --> 01:12:59.230 align:middle line:84% for different polarizations for different directions 01:12:59.230 --> 01:13:00.410 align:middle line:90% of the electric field. 01:13:00.410 --> 01:13:02.660 align:middle line:84% So everyone had some thought about direction in there, 01:13:02.660 --> 01:13:03.800 align:middle line:90% and that was all good. 01:13:03.800 --> 01:13:06.190 align:middle line:84% It's not to do with frequency, though of course, this 01:13:06.190 --> 01:13:07.390 align:middle line:90% is frequency dependent. 01:13:07.390 --> 01:13:09.040 align:middle line:84% But of course, the refractive index for plasma 01:13:09.040 --> 01:13:10.290 align:middle line:90% is always frequency dependent. 01:13:10.290 --> 01:13:13.300 align:middle line:84% So that's not a unique thing for this solution here. 01:13:13.300 --> 01:13:15.880 align:middle line:84% The unique thing about it is that these waves 01:13:15.880 --> 01:13:16.970 align:middle line:90% have got different speeds. 01:13:16.970 --> 01:13:19.270 align:middle line:84% So the other thing, the x mode and the o mode-- those 01:13:19.270 --> 01:13:21.030 align:middle line:90% are also birefringence as well. 01:13:21.030 --> 01:13:23.530 align:middle line:84% They're just not birefringence in a particularly useful way. 01:13:23.530 --> 01:13:28.010 align:middle line:84% This is birefringence in a way that we can exploit. 01:13:28.010 --> 01:13:29.750 align:middle line:84% And if you ever want to go down a rabbit 01:13:29.750 --> 01:13:32.840 align:middle line:84% hole of interesting stuff, there were several Viking variants 01:13:32.840 --> 01:13:34.790 align:middle line:84% where they found, from these Viking warlords, 01:13:34.790 --> 01:13:36.480 align:middle line:90% buried with lumps of calcite. 01:13:36.480 --> 01:13:38.480 align:middle line:84% And people were like, why have they got calcite? 01:13:38.480 --> 01:13:40.250 align:middle line:84% It's not a particularly pretty crystal. 01:13:40.250 --> 01:13:41.477 align:middle line:90% It's transparent. 01:13:41.477 --> 01:13:43.310 align:middle line:84% And you can't really make jewelry out of it. 01:13:43.310 --> 01:13:45.260 align:middle line:90% It's quartzy. 01:13:45.260 --> 01:13:46.705 align:middle line:84% And it wasn't carved in any case. 01:13:46.705 --> 01:13:48.080 align:middle line:84% It was just this lump of calcite. 01:13:48.080 --> 01:13:51.230 align:middle line:84% And there is a belief unproven amongst archeologists 01:13:51.230 --> 01:13:54.140 align:middle line:84% that this calcite, which is a birefringent material, 01:13:54.140 --> 01:13:56.182 align:middle line:84% could be used to navigate on a cloudy day. 01:13:56.182 --> 01:13:58.640 align:middle line:84% So when you're sailing across the Atlantic or the North Sea 01:13:58.640 --> 01:14:02.330 align:middle line:84% to raid some poor monastery and it's all cloudy-- like, damn. 01:14:02.330 --> 01:14:04.415 align:middle line:84% I don't know how to get to this monastery. 01:14:04.415 --> 01:14:06.290 align:middle line:84% And they didn't have compasses in those days, 01:14:06.290 --> 01:14:07.457 align:middle line:90% at least the Vikings didn't. 01:14:07.457 --> 01:14:08.837 align:middle line:90% The Chinese did. 01:14:08.837 --> 01:14:10.670 align:middle line:84% So with compasses, this birefringent crystal 01:14:10.670 --> 01:14:12.253 align:middle line:84% is very interesting, because you might 01:14:12.253 --> 01:14:15.860 align:middle line:84% know that the light from the sun is polarized by scattering. 01:14:15.860 --> 01:14:19.130 align:middle line:84% And so although that light isn't directly 01:14:19.130 --> 01:14:21.408 align:middle line:84% hitting your ship in the fog, some of that light 01:14:21.408 --> 01:14:22.200 align:middle line:90% is getting through. 01:14:22.200 --> 01:14:24.950 align:middle line:84% And although the fog is messing with that polarization, 01:14:24.950 --> 01:14:27.410 align:middle line:84% there's still going to be some overall polarization there. 01:14:27.410 --> 01:14:29.720 align:middle line:84% And by rotating this birefringent crystal 01:14:29.720 --> 01:14:31.760 align:middle line:84% in the light that you're getting that's slightly 01:14:31.760 --> 01:14:33.350 align:middle line:84% polarized from the sun and looking 01:14:33.350 --> 01:14:35.960 align:middle line:84% at markings on a bit of stone, when you rotate the crystal 01:14:35.960 --> 01:14:38.420 align:middle line:84% just so, the light is going to come through 01:14:38.420 --> 01:14:39.890 align:middle line:84% and the two markings will line up. 01:14:39.890 --> 01:14:42.080 align:middle line:84% And then you will know roughly where north is, 01:14:42.080 --> 01:14:44.480 align:middle line:84% and then you can go and sail and raid your monastery. 01:14:44.480 --> 01:14:46.335 align:middle line:90% So I'm not saying it's true. 01:14:46.335 --> 01:14:47.960 align:middle line:84% You can go read some really cool papers 01:14:47.960 --> 01:14:49.400 align:middle line:90% on people trying to do this. 01:14:49.400 --> 01:14:51.608 align:middle line:84% People have tried to go out in a boat on a cloudy day 01:14:51.608 --> 01:14:54.800 align:middle line:84% and navigate like this, and it did not go well. 01:14:54.800 --> 01:14:56.417 align:middle line:84% But they didn't have as much practice 01:14:56.417 --> 01:14:58.500 align:middle line:84% as the Vikings did because we have GPS these days. 01:14:58.500 --> 01:15:00.167 align:middle line:84% And so we don't worry about such things. 01:15:00.167 --> 01:15:01.110 align:middle line:90% But it's really cool. 01:15:01.110 --> 01:15:03.805 align:middle line:84% So if you like optics, go look up Viking sunstones, 01:15:03.805 --> 01:15:04.430 align:middle line:90% they call them. 01:15:04.430 --> 01:15:09.256 align:middle line:90% 01:15:09.256 --> 01:15:13.000 align:middle line:90% I always like rings. 01:15:13.000 --> 01:15:14.430 align:middle line:90% OK. 01:15:14.430 --> 01:15:17.380 align:middle line:90% Back to plasmas. 01:15:17.380 --> 01:15:23.550 align:middle line:84% So we said we have these two modes inside the plasma. 01:15:23.550 --> 01:15:26.610 align:middle line:84% These are the right-hand and left-hand circularly polarized 01:15:26.610 --> 01:15:29.750 align:middle line:84% modes, which I'm going to refer to as plus and minus like this. 01:15:29.750 --> 01:15:32.280 align:middle line:84% So plus is the clockwise going mode, 01:15:32.280 --> 01:15:34.560 align:middle line:84% and minus is the anti-clockwise going mode. 01:15:34.560 --> 01:15:37.920 align:middle line:84% If you're confused about why I'm confused about this convention, 01:15:37.920 --> 01:15:41.280 align:middle line:84% it's because is it clockwise as you look down the ray of light? 01:15:41.280 --> 01:15:43.650 align:middle line:84% Or is it clockwise as you look towards the ray of light? 01:15:43.650 --> 01:15:44.483 align:middle line:90% Those are different. 01:15:44.483 --> 01:15:46.858 align:middle line:84% And I can never remember which one the convention is for. 01:15:46.858 --> 01:15:49.525 align:middle line:84% I think it probably should be as you look down the ray of light, 01:15:49.525 --> 01:15:50.160 align:middle line:90% but who knows? 01:15:50.160 --> 01:15:52.410 align:middle line:84% And so maybe I've got this the wrong way around. 01:15:52.410 --> 01:15:54.240 align:middle line:84% If I got the wrong way around, you just 01:15:54.240 --> 01:15:57.140 align:middle line:84% swap where I'm looking down the red light or at the red light 01:15:57.140 --> 01:15:58.890 align:middle line:84% or whether I'm looking down the red light. 01:15:58.890 --> 01:16:01.830 align:middle line:84% And then I'll be writing it like so. 01:16:01.830 --> 01:16:06.235 align:middle line:84% Note, by the way, that this does depend on the direction 01:16:06.235 --> 01:16:07.520 align:middle line:90% of the magnetic field. 01:16:07.520 --> 01:16:09.910 align:middle line:84% This is not capital omega squared. 01:16:09.910 --> 01:16:10.840 align:middle line:90% It's just omega. 01:16:10.840 --> 01:16:13.660 align:middle line:84% So the direction of the magnetic field changes. 01:16:13.660 --> 01:16:16.210 align:middle line:84% Which of these modes propagates faster? 01:16:16.210 --> 01:16:19.548 align:middle line:84% If you're propagating along the magnetic field, 01:16:19.548 --> 01:16:21.340 align:middle line:84% one of your modes is faster than the other. 01:16:21.340 --> 01:16:22.840 align:middle line:84% If you're propagating against the magnet field, 01:16:22.840 --> 01:16:23.990 align:middle line:90% the other mode is faster. 01:16:23.990 --> 01:16:25.000 align:middle line:90% That's very important. 01:16:25.000 --> 01:16:27.400 align:middle line:84% That's going to be what we use to measure 01:16:27.400 --> 01:16:31.090 align:middle line:84% both the magnetic field's amplitude and its direction. 01:16:31.090 --> 01:16:33.970 align:middle line:84% And we can use this neat technique called 01:16:33.970 --> 01:16:38.980 align:middle line:84% Stokes vectors, where a Stokes vector is just 01:16:38.980 --> 01:16:46.550 align:middle line:84% a vector of the ex ey, normalized by the sum of ex 01:16:46.550 --> 01:16:49.910 align:middle line:90% squared plus ey squared. 01:16:49.910 --> 01:16:52.430 align:middle line:84% These Stokes vectors make our life very easy 01:16:52.430 --> 01:16:56.250 align:middle line:84% when we want to do the calculations-- 01:16:56.250 --> 01:16:58.680 align:middle line:90% same guy as Stokes' theorem. 01:16:58.680 --> 01:17:04.950 align:middle line:84% And we can say, looking at this relation between these two, 01:17:04.950 --> 01:17:08.520 align:middle line:84% that the e plus is going to have something triangular called 01:17:08.520 --> 01:17:09.600 align:middle line:90% the vector r. 01:17:09.600 --> 01:17:13.885 align:middle line:84% And that's going to be 1 over I. Oh, I lied. 01:17:13.885 --> 01:17:15.510 align:middle line:84% I'm not going to normalize all of them. 01:17:15.510 --> 01:17:17.920 align:middle line:84% I can't be bothered to put a square root 2 in front of this. 01:17:17.920 --> 01:17:19.878 align:middle line:84% So just remember there should be square root 2. 01:17:19.878 --> 01:17:21.340 align:middle line:90% It doesn't really matter. 01:17:21.340 --> 01:17:27.370 align:middle line:84% And then the left vector is going to be 1 minus I. 01:17:27.370 --> 01:17:30.640 align:middle line:84% And you can see that these have the same relationships 01:17:30.640 --> 01:17:32.530 align:middle line:90% between ex and ey. 01:17:32.530 --> 01:17:34.660 align:middle line:84% And so these are the Stokes vectors for the right 01:17:34.660 --> 01:17:36.640 align:middle line:84% and left circularly polarized light. 01:17:36.640 --> 01:17:39.670 align:middle line:84% We also can write some other polarizations 01:17:39.670 --> 01:17:41.690 align:middle line:90% in this Stokes vector notation. 01:17:41.690 --> 01:17:45.100 align:middle line:84% So if we were polarized entirely in the x direction 01:17:45.100 --> 01:17:52.480 align:middle line:84% here so y equals 0, this would be the vector, x. 01:17:52.480 --> 01:17:56.200 align:middle line:90% And that would be equal to 1, 0. 01:17:56.200 --> 01:17:58.870 align:middle line:84% And if we have the x equal to 0, this 01:17:58.870 --> 01:18:03.190 align:middle line:84% would be the vector capital Y. This would be equal to 0, 1, 01:18:03.190 --> 01:18:04.360 align:middle line:90% like that. 01:18:04.360 --> 01:18:06.730 align:middle line:84% Now, there are two modes inside the plasma. 01:18:06.730 --> 01:18:09.790 align:middle line:84% We can write any arbitrary polarization of our wave 01:18:09.790 --> 01:18:13.880 align:middle line:84% as a sum of these two basis vectors, effectively. 01:18:13.880 --> 01:18:16.330 align:middle line:84% And so we can switch basis vectors if we want to. 01:18:16.330 --> 01:18:18.580 align:middle line:84% So generally, when we're launching light, 01:18:18.580 --> 01:18:20.530 align:middle line:84% we don't launch it as a circular polarization. 01:18:20.530 --> 01:18:21.860 align:middle line:90% That's quite hard to come by. 01:18:21.860 --> 01:18:24.730 align:middle line:84% We launch it with some linear polarization. 01:18:24.730 --> 01:18:28.480 align:middle line:84% But this linear polarization is made up of these two 01:18:28.480 --> 01:18:29.930 align:middle line:90% circular polarizations. 01:18:29.930 --> 01:18:34.360 align:middle line:84% So for example, this x linear polarization is r. 01:18:34.360 --> 01:18:37.320 align:middle line:90% What's l over 2? 01:18:37.320 --> 01:18:42.600 align:middle line:84% The y polarization is r minus l over 2. 01:18:42.600 --> 01:18:48.910 align:middle line:90% 01:18:48.910 --> 01:18:50.410 align:middle line:84% I don't know why I got this again. 01:18:50.410 --> 01:18:53.320 align:middle line:90% 01:18:53.320 --> 01:18:53.820 align:middle line:90% Oh, well. 01:18:53.820 --> 01:18:54.030 align:middle line:90% OK. 01:18:54.030 --> 01:18:55.270 align:middle line:90% Maybe it's a good point. 01:18:55.270 --> 01:18:59.250 align:middle line:84% So just to draw our geometry another time like this, 01:18:59.250 --> 01:19:01.890 align:middle line:84% we've got some k vector, which is 01:19:01.890 --> 01:19:08.640 align:middle line:84% an angle, theta, to the magnetic field, b, like that. 01:19:08.640 --> 01:19:12.820 align:middle line:84% And again, for theta equals 0, this 01:19:12.820 --> 01:19:15.170 align:middle line:90% is our dispersion relationship. 01:19:15.170 --> 01:19:17.747 align:middle line:84% I just want to point out one slightly funny-- 01:19:17.747 --> 01:19:19.330 align:middle line:84% I think it's funny-- thing about this. 01:19:19.330 --> 01:19:23.220 align:middle line:84% So if we have k in this direction, 01:19:23.220 --> 01:19:29.320 align:middle line:84% that means that this wave is a parallel propagation. 01:19:29.320 --> 01:19:31.510 align:middle line:84% But of course, it's still a transverse wind. 01:19:31.510 --> 01:19:34.180 align:middle line:84% I think people often get this very confused because the words 01:19:34.180 --> 01:19:37.030 align:middle line:84% parallel and perpendicular and transverse and longitudinal 01:19:37.030 --> 01:19:38.090 align:middle line:90% have similar meanings. 01:19:38.090 --> 01:19:40.690 align:middle line:84% So this is a parallel and transverse wave. 01:19:40.690 --> 01:19:43.600 align:middle line:90% 01:19:43.600 --> 01:19:45.170 align:middle line:84% And there's one more word as well 01:19:45.170 --> 01:19:45.980 align:middle line:90% which means something similar. 01:19:45.980 --> 01:19:46.930 align:middle line:84% I can't remember which one it is. 01:19:46.930 --> 01:19:48.070 align:middle line:84% But yeah, there's lots of different ways. 01:19:48.070 --> 01:19:48.570 align:middle line:90% Yeah. 01:19:48.570 --> 01:19:50.320 align:middle line:84% AUDIENCE: For the r minus l over 2, 01:19:50.320 --> 01:19:53.057 align:middle line:84% wouldn't that be the I in the bottom? 01:19:53.057 --> 01:19:53.890 align:middle line:90% JACK HARE: Oh, yeah. 01:19:53.890 --> 01:19:54.430 align:middle line:90% I had meant to. 01:19:54.430 --> 01:19:55.805 align:middle line:84% There's meant to be an I in here. 01:19:55.805 --> 01:20:00.970 align:middle line:90% And it's going to be like that. 01:20:00.970 --> 01:20:04.300 align:middle line:84% Or maybe minus that, but you get the idea. 01:20:04.300 --> 01:20:06.400 align:middle line:90% No, it's that. 01:20:06.400 --> 01:20:07.270 align:middle line:90% Cool. 01:20:07.270 --> 01:20:08.440 align:middle line:90% OK. 01:20:08.440 --> 01:20:12.208 align:middle line:84% It turns out I derived this for theta equals pi over 2. 01:20:12.208 --> 01:20:13.000 align:middle line:90% I didn't derive it. 01:20:13.000 --> 01:20:16.343 align:middle line:84% I just showed you it for pi over 2 and theta equals 0. 01:20:16.343 --> 01:20:18.010 align:middle line:84% But it actually turns out that the theta 01:20:18.010 --> 01:20:20.740 align:middle line:84% equals 0 case applies over a huge range 01:20:20.740 --> 01:20:23.120 align:middle line:90% of different conditions here. 01:20:23.120 --> 01:20:26.980 align:middle line:84% So in fact, this theta equals 0, which 01:20:26.980 --> 01:20:34.520 align:middle line:84% we call the quasi-parallel case, it isn't good 01:20:34.520 --> 01:20:36.320 align:middle line:90% simply for theta equals 0. 01:20:36.320 --> 01:20:40.130 align:middle line:84% It's good for capital omega over omega. 01:20:40.130 --> 01:20:46.420 align:middle line:84% The secant of theta is much, much less than 1. 01:20:46.420 --> 01:20:52.270 align:middle line:84% And it turns out that for some reasonable values of omega 01:20:52.270 --> 01:20:57.670 align:middle line:84% here and here, this can apply for theta almost to pi upon 2. 01:20:57.670 --> 01:21:00.310 align:middle line:84% So you can use this dispersion relationship 01:21:00.310 --> 01:21:04.090 align:middle line:84% as I said up here, even for relatively large angles. 01:21:04.090 --> 01:21:05.925 align:middle line:84% The waves will propagate as if they 01:21:05.925 --> 01:21:07.300 align:middle line:84% had this dispersion relationship, 01:21:07.300 --> 01:21:09.250 align:middle line:84% or at least extremely close to it. 01:21:09.250 --> 01:21:12.020 align:middle line:84% And that's very, very convenient. 01:21:12.020 --> 01:21:14.510 align:middle line:84% They will propagate without dispersion relationship 01:21:14.510 --> 01:21:19.640 align:middle line:84% as long as you write your b to be the component 01:21:19.640 --> 01:21:21.030 align:middle line:90% parallel to propagation. 01:21:21.030 --> 01:21:27.030 align:middle line:84% So you replace b with b 0 cosine of theta. 01:21:27.030 --> 01:21:29.280 align:middle line:84% So now the dispersion relationship here, 01:21:29.280 --> 01:21:32.390 align:middle line:84% which has capital omega in, is to do 01:21:32.390 --> 01:21:35.330 align:middle line:84% with the projection of the magnetic field along your wave. 01:21:35.330 --> 01:21:38.600 align:middle line:84% So effectively, the wave fields components of the magnetic field 01:21:38.600 --> 01:21:41.900 align:middle line:84% in the direction it's traveling and ignores that other component 01:21:41.900 --> 01:21:44.390 align:middle line:84% until it gets very, very close to pi upon 2, 01:21:44.390 --> 01:21:48.870 align:middle line:84% when the components along the direction of travel is almost 0. 01:21:48.870 --> 01:21:53.130 align:middle line:84% And then we suddenly see this pi over 2 condition here. 01:21:53.130 --> 01:21:56.840 align:middle line:84% So this is very, very useful because it means that we only 01:21:56.840 --> 01:21:59.510 align:middle line:84% need to have-- for some cylindrical object like this, 01:21:59.510 --> 01:22:04.520 align:middle line:84% we can use this dispersion relationship 01:22:04.520 --> 01:22:06.860 align:middle line:84% for almost the entire region that we're 01:22:06.860 --> 01:22:08.900 align:middle line:90% probing for Faraday rotation. 01:22:08.900 --> 01:22:09.440 align:middle line:90% OK. 01:22:09.440 --> 01:22:11.250 align:middle line:84% I'm well aware that I've gone over time, 01:22:11.250 --> 01:22:13.410 align:middle line:90% so I'm going to leave it here. 01:22:13.410 --> 01:22:16.640 align:middle line:84% And we will get on to exactly how we exploit 01:22:16.640 --> 01:22:18.710 align:middle line:84% this interesting mathematics and these Stokes 01:22:18.710 --> 01:22:21.170 align:middle line:84% vectors in order to measure magnetic fields 01:22:21.170 --> 01:22:22.200 align:middle line:90% in the next lecture. 01:22:22.200 --> 01:22:23.340 align:middle line:90% So thank you very much. 01:22:23.340 --> 01:22:26.870 align:middle line:84% I'll see you-- oh, not on Tuesday because you 01:22:26.870 --> 01:22:28.130 align:middle line:90% all have a student holiday. 01:22:28.130 --> 01:22:29.540 align:middle line:84% So for the people in Columbia who 01:22:29.540 --> 01:22:32.750 align:middle line:84% are unfamiliar with this idea, we have Monday off. 01:22:32.750 --> 01:22:36.080 align:middle line:84% And then to recover from the three-day weekend, 01:22:36.080 --> 01:22:38.810 align:middle line:84% the students have a second day off from classes. 01:22:38.810 --> 01:22:41.725 align:middle line:90% So I will see you on Thursday. 01:22:41.725 --> 01:22:46.000 align:middle line:90%