WEBVTT 00:00:00.000 --> 00:00:02.460 align:middle line:90% [SQUEAKING] 00:00:02.460 --> 00:00:04.428 align:middle line:90% [RUSTLING] 00:00:04.428 --> 00:00:07.380 align:middle line:90% [CLICKING] 00:00:07.380 --> 00:00:10.340 align:middle line:90% 00:00:10.340 --> 00:00:12.350 align:middle line:84% JACK HARE: Right, so today, we are 00:00:12.350 --> 00:00:15.920 align:middle line:84% going to start a series of several lectures 00:00:15.920 --> 00:00:18.890 align:middle line:90% on refractive index diagnostics. 00:00:18.890 --> 00:00:21.830 align:middle line:84% So these are diagnostics which use the fact 00:00:21.830 --> 00:00:23.540 align:middle line:84% that the refractive index of a plasma 00:00:23.540 --> 00:00:27.710 align:middle line:84% is not 1 in order to make measurements about the plasma. 00:00:27.710 --> 00:00:29.480 align:middle line:84% So first of all, what we're going to do 00:00:29.480 --> 00:00:35.050 align:middle line:84% is a very quick recap of electromagnetic waves 00:00:35.050 --> 00:00:35.860 align:middle line:90% in a plasma. 00:00:35.860 --> 00:00:39.010 align:middle line:90% 00:00:39.010 --> 00:00:42.150 align:middle line:84% And you have doubtless seen this before in some courses 00:00:42.150 --> 00:00:44.790 align:middle line:84% that you've done, like it's in Chen. 00:00:44.790 --> 00:00:48.330 align:middle line:84% There's also a very good explanation of it in Hutchinson. 00:00:48.330 --> 00:00:51.030 align:middle line:84% And I'm going to give a little taster of how 00:00:51.030 --> 00:00:53.640 align:middle line:84% this derivation goes, just to remind 00:00:53.640 --> 00:00:57.810 align:middle line:84% folks of how this result actually comes about. 00:00:57.810 --> 00:00:59.910 align:middle line:84% And I'm going to do it in a rather restrictive way 00:00:59.910 --> 00:01:01.860 align:middle line:84% here so that we can make some rapid progress. 00:01:01.860 --> 00:01:03.870 align:middle line:84% And we'll go back and add in more bits of theory 00:01:03.870 --> 00:01:07.080 align:middle line:90% later on as we need it. 00:01:07.080 --> 00:01:11.760 align:middle line:84% So for our electromagnetic waves in a plasma, we have two fields. 00:01:11.760 --> 00:01:14.880 align:middle line:90% We have E and B, like this. 00:01:14.880 --> 00:01:17.130 align:middle line:90% And we have Maxwell's equations. 00:01:17.130 --> 00:01:20.010 align:middle line:90% 00:01:20.010 --> 00:01:27.020 align:middle line:84% So we have that the curl of E is equal to minus B dot, 00:01:27.020 --> 00:01:31.970 align:middle line:84% and we have C squared times the curl of B 00:01:31.970 --> 00:01:40.590 align:middle line:84% equals E dot plus J over epsilon 0. 00:01:40.590 --> 00:01:43.108 align:middle line:84% You might be more used to seeing this equation with a mu 0. 00:01:43.108 --> 00:01:45.150 align:middle line:84% I've just moved the C squared over the other side 00:01:45.150 --> 00:01:47.340 align:middle line:84% because it makes the math a little bit easier 00:01:47.340 --> 00:01:50.088 align:middle line:90% on the next step here. 00:01:50.088 --> 00:01:51.630 align:middle line:84% And so these equations are just true. 00:01:51.630 --> 00:01:53.680 align:middle line:90% They're true in any medium. 00:01:53.680 --> 00:01:56.790 align:middle line:84% And so what we want to do is try and reduce them 00:01:56.790 --> 00:01:59.285 align:middle line:84% down a little bit and then put some plasma physics in. 00:01:59.285 --> 00:02:00.660 align:middle line:84% So the first thing we normally do 00:02:00.660 --> 00:02:03.270 align:middle line:84% when we're deriving electromagnetic waves is we say, 00:02:03.270 --> 00:02:07.440 align:middle line:84% what if all of our vectors had some sort of time and space 00:02:07.440 --> 00:02:11.990 align:middle line:84% variation that looked like exponential of i, 00:02:11.990 --> 00:02:14.990 align:middle line:84% some wave vector k dot, some position 00:02:14.990 --> 00:02:21.660 align:middle line:84% vector x minus omega the frequency times time, like that. 00:02:21.660 --> 00:02:24.590 align:middle line:84% And so this is a little bit like Fourier transforming 00:02:24.590 --> 00:02:25.760 align:middle line:90% our equations here. 00:02:25.760 --> 00:02:38.660 align:middle line:84% And we end up with k cross E equals i omega B, 00:02:38.660 --> 00:02:51.160 align:middle line:84% and we have C squared k cross B equals minus i omega E plus J 00:02:51.160 --> 00:02:53.200 align:middle line:90% upon epsilon 0. 00:02:53.200 --> 00:02:54.850 align:middle line:84% And the astute amongst you have noticed 00:02:54.850 --> 00:02:57.680 align:middle line:84% I made a sign error in the first equation here. 00:02:57.680 --> 00:03:02.930 align:middle line:84% So this is plus i omega B, not minus i omega B. Very good. 00:03:02.930 --> 00:03:06.460 align:middle line:84% And so we could call these equations 1 and 2. 00:03:06.460 --> 00:03:13.330 align:middle line:84% And we can note that if we do k cross with equation 1, 00:03:13.330 --> 00:03:19.847 align:middle line:84% that is equal to i omega upon C squared of equation 2. 00:03:19.847 --> 00:03:21.430 align:middle line:84% So what we're doing here is we're just 00:03:21.430 --> 00:03:24.392 align:middle line:84% looking at this term and this term and being like, 00:03:24.392 --> 00:03:25.600 align:middle line:90% hey, they both got B in them. 00:03:25.600 --> 00:03:27.267 align:middle line:84% We can probably make them look the same. 00:03:27.267 --> 00:03:28.450 align:middle line:90% So then we can equate. 00:03:28.450 --> 00:03:30.940 align:middle line:84% We can do this calculation and we can equate 00:03:30.940 --> 00:03:32.630 align:middle line:90% the two sides of the equation. 00:03:32.630 --> 00:03:36.340 align:middle line:84% And with a little bit of vector magic, 00:03:36.340 --> 00:03:38.290 align:middle line:84% calculus magic, we would end up with something 00:03:38.290 --> 00:03:47.530 align:middle line:84% that looks like k k dot E plus k squared 00:03:47.530 --> 00:03:56.570 align:middle line:84% E is equal to i omega J over epsilon 0 C squared 00:03:56.570 --> 00:04:03.590 align:middle line:84% plus omega squared upon C squared E, like that. 00:04:03.590 --> 00:04:07.670 align:middle line:84% Now, we are searching for transverse waves here, 00:04:07.670 --> 00:04:09.890 align:middle line:84% electromagnetic waves that tend to be transverse, 00:04:09.890 --> 00:04:13.400 align:middle line:84% and that means that k dot E is 0. 00:04:13.400 --> 00:04:18.670 align:middle line:84% So we're just going to look for transverse wave solutions here. 00:04:18.670 --> 00:04:19.760 align:middle line:90% We can drop that. 00:04:19.760 --> 00:04:21.279 align:middle line:84% And that means that our equation can 00:04:21.279 --> 00:04:25.120 align:middle line:84% be rewritten as omega squared minus C squared 00:04:25.120 --> 00:04:38.000 align:middle line:84% k squared times E equals minus i omega J upon epsilon 0, 00:04:38.000 --> 00:04:38.708 align:middle line:90% like that. 00:04:38.708 --> 00:04:40.250 align:middle line:84% And I just want to point out, there's 00:04:40.250 --> 00:04:42.738 align:middle line:84% absolutely no plasma physics in this at the moment. 00:04:42.738 --> 00:04:44.780 align:middle line:84% All we've done is manipulate Maxwell's equations. 00:04:44.780 --> 00:04:47.820 align:middle line:84% We haven't said anything about the plasma. 00:04:47.820 --> 00:04:51.140 align:middle line:84% So if I take a standard limit here, 00:04:51.140 --> 00:04:54.040 align:middle line:84% say I let the current equal 0, which 00:04:54.040 --> 00:04:55.790 align:middle line:84% is what it would be in the vacuum of space 00:04:55.790 --> 00:04:58.160 align:middle line:84% where there's no particles to carry any current, 00:04:58.160 --> 00:05:02.750 align:middle line:84% then we would simply end up with an equation for light 00:05:02.750 --> 00:05:05.600 align:middle line:84% in free space and you would have a dispersion relationship 00:05:05.600 --> 00:05:09.540 align:middle line:84% omega squared equals C squared k squared, like that. 00:05:09.540 --> 00:05:10.800 align:middle line:90% So those are light waves. 00:05:10.800 --> 00:05:14.522 align:middle line:90% So life is good. 00:05:14.522 --> 00:05:16.980 align:middle line:84% Any questions on that before we put some plasma physics in? 00:05:16.980 --> 00:05:21.257 align:middle line:90% 00:05:21.257 --> 00:05:23.590 align:middle line:84% Hopefully, you're dredging up your memories of Griffiths 00:05:23.590 --> 00:05:25.510 align:middle line:84% and Jackson and all sorts of wonderful things like this, 00:05:25.510 --> 00:05:26.780 align:middle line:90% and this is all making sense. 00:05:26.780 --> 00:05:28.480 align:middle line:90% So let's keep going. 00:05:28.480 --> 00:05:32.760 align:middle line:90% 00:05:32.760 --> 00:05:34.280 align:middle line:84% The next thing we want to do then 00:05:34.280 --> 00:05:35.705 align:middle line:90% is add in some plasma physics. 00:05:35.705 --> 00:05:37.580 align:middle line:84% And we're going to add in some plasma physics 00:05:37.580 --> 00:05:41.450 align:middle line:84% with some serious assumptions which 00:05:41.450 --> 00:05:44.810 align:middle line:84% let us make significant progress quickly. 00:05:44.810 --> 00:05:48.620 align:middle line:84% And these assumptions can all be justified for most plasmas. 00:05:48.620 --> 00:05:50.600 align:middle line:84% And if you can't justify it for your plasma, 00:05:50.600 --> 00:05:53.240 align:middle line:84% you may want to revisit these a little bit. 00:05:53.240 --> 00:05:55.790 align:middle line:84% So the first assumption we're going to make 00:05:55.790 --> 00:05:59.960 align:middle line:84% is that we're using high-frequency waves here. 00:05:59.960 --> 00:06:03.320 align:middle line:84% And high frequency, this is kind of a wishy-washy term. 00:06:03.320 --> 00:06:05.930 align:middle line:84% We want to make that more precise by having 00:06:05.930 --> 00:06:07.310 align:middle line:90% a dimensionless parameter. 00:06:07.310 --> 00:06:10.850 align:middle line:84% And so we're going to say that omega is much, much larger 00:06:10.850 --> 00:06:13.080 align:middle line:90% than omega pi here. 00:06:13.080 --> 00:06:18.440 align:middle line:84% So the frequency of our waves is much higher than the ion plasma 00:06:18.440 --> 00:06:19.580 align:middle line:90% frequency. 00:06:19.580 --> 00:06:22.010 align:middle line:84% What does that condition physically respond to? 00:06:22.010 --> 00:06:24.320 align:middle line:84% What are we saying about the ions and their interaction 00:06:24.320 --> 00:06:26.153 align:middle line:84% with this wave by putting this condition in? 00:06:26.153 --> 00:06:31.205 align:middle line:90% 00:06:31.205 --> 00:06:32.830 align:middle line:84% STUDENT: You're saying they essentially 00:06:32.830 --> 00:06:34.803 align:middle line:90% don't interact at that point. 00:06:34.803 --> 00:06:35.470 align:middle line:90% JACK HARE: Yeah. 00:06:35.470 --> 00:06:39.458 align:middle line:90% So the ions are frozen in place. 00:06:39.458 --> 00:06:41.000 align:middle line:84% And they are not going to participate 00:06:41.000 --> 00:06:43.380 align:middle line:84% in any of the physics that we're interested in here. 00:06:43.380 --> 00:06:45.380 align:middle line:84% And again, you can work out the ion plasma 00:06:45.380 --> 00:06:49.800 align:middle line:84% frequency for some density that you're interested in, 00:06:49.800 --> 00:06:51.300 align:middle line:84% and you'll find out it's pretty low. 00:06:51.300 --> 00:06:53.180 align:middle line:84% So this is pretty reasonable, but if you 00:06:53.180 --> 00:06:56.250 align:middle line:84% start using very low-frequency waves, it won't be reasonable. 00:06:56.250 --> 00:06:59.330 align:middle line:84% So another thing that we're going to do 00:06:59.330 --> 00:07:04.870 align:middle line:84% is make the cold plasma approximation. 00:07:04.870 --> 00:07:06.940 align:middle line:84% And again, we can't just use the word cold. 00:07:06.940 --> 00:07:09.100 align:middle line:84% We have to say what we mean by cold. 00:07:09.100 --> 00:07:11.890 align:middle line:84% And we're going to say the thermal of the electrons 00:07:11.890 --> 00:07:15.310 align:middle line:84% is much, much less than the speed of light here. 00:07:15.310 --> 00:07:17.890 align:middle line:84% And this condition is equivalent to us 00:07:17.890 --> 00:07:20.440 align:middle line:84% not worrying about the Maxwellian distribution 00:07:20.440 --> 00:07:21.202 align:middle line:90% of the electrons. 00:07:21.202 --> 00:07:22.660 align:middle line:84% So all the electrons are just going 00:07:22.660 --> 00:07:25.630 align:middle line:84% to be moving with no-- they will be moving, 00:07:25.630 --> 00:07:27.200 align:middle line:84% unlike the ions, which are frozen, 00:07:27.200 --> 00:07:29.980 align:middle line:84% but they will be moving all at the same speeds. 00:07:29.980 --> 00:07:31.970 align:middle line:84% There's no spread of velocities here. 00:07:31.970 --> 00:07:36.970 align:middle line:84% So we have a delta function of velocities, 00:07:36.970 --> 00:07:42.980 align:middle line:84% and we do not have our Maxwellian distribution that we 00:07:42.980 --> 00:07:45.850 align:middle line:90% might normally think about. 00:07:45.850 --> 00:07:48.180 align:middle line:84% And the final condition we're going to write down 00:07:48.180 --> 00:07:50.830 align:middle line:90% is unmagnetized. 00:07:50.830 --> 00:07:53.710 align:middle line:84% Now, this is the one which I think is most complicated. 00:07:53.710 --> 00:07:56.527 align:middle line:84% Because, in fact, there are a dozen different ways 00:07:56.527 --> 00:07:58.360 align:middle line:84% you can write down a dimensionless parameter 00:07:58.360 --> 00:08:01.030 align:middle line:84% for unmagnetized, and they all mean slightly different things. 00:08:01.030 --> 00:08:02.440 align:middle line:84% We could think about collisionality. 00:08:02.440 --> 00:08:03.880 align:middle line:84% We could think about pressure balance, 00:08:03.880 --> 00:08:05.130 align:middle line:90% all sorts of things like that. 00:08:05.130 --> 00:08:07.360 align:middle line:84% For the purposes of this derivation, 00:08:07.360 --> 00:08:10.810 align:middle line:84% unmagnetized means omega is much, much larger 00:08:10.810 --> 00:08:13.800 align:middle line:90% than capital omega. 00:08:13.800 --> 00:08:15.060 align:middle line:90% I know it's confusing. 00:08:15.060 --> 00:08:17.050 align:middle line:84% And I'll put a little subscript e here 00:08:17.050 --> 00:08:18.300 align:middle line:90% because we've frozen the ions. 00:08:18.300 --> 00:08:21.250 align:middle line:84% But this is the gyro motion of the electrons. 00:08:21.250 --> 00:08:24.840 align:middle line:84% So the electrons may be gyrating around field lines. 00:08:24.840 --> 00:08:26.340 align:middle line:84% There may be some magnetic field. 00:08:26.340 --> 00:08:27.450 align:middle line:90% It doesn't have to be 0. 00:08:27.450 --> 00:08:30.690 align:middle line:84% But on the time scale that the wave goes by and does its stuff, 00:08:30.690 --> 00:08:33.210 align:middle line:84% the electrons do not move appreciably 00:08:33.210 --> 00:08:34.380 align:middle line:90% around their gyro orbit. 00:08:34.380 --> 00:08:36.820 align:middle line:84% So we don't have to care about their gyro motion. 00:08:36.820 --> 00:08:38.700 align:middle line:84% This is one of the places we will definitely 00:08:38.700 --> 00:08:41.460 align:middle line:84% have to relax later on when we want to do Faraday rotation 00:08:41.460 --> 00:08:43.830 align:middle line:84% imaging which relies on that gyro motion 00:08:43.830 --> 00:08:46.050 align:middle line:90% to give us the effect. 00:08:46.050 --> 00:08:49.410 align:middle line:90% So this is the final condition. 00:08:49.410 --> 00:08:51.420 align:middle line:84% Any questions on those three assumptions? 00:08:51.420 --> 00:08:56.720 align:middle line:90% 00:08:56.720 --> 00:08:59.840 align:middle line:84% OK, so then, we can write down that the current 00:08:59.840 --> 00:09:01.670 align:middle line:84% inside our plasma is simply going 00:09:01.670 --> 00:09:04.610 align:middle line:84% to be equal to the charge on the electrons, the number 00:09:04.610 --> 00:09:07.710 align:middle line:84% of electrons, and how fast they're moving. 00:09:07.710 --> 00:09:09.950 align:middle line:84% And so we've simply transferred our lack of knowledge 00:09:09.950 --> 00:09:12.050 align:middle line:84% about J into our lack of knowledge about V. 00:09:12.050 --> 00:09:13.910 align:middle line:84% So we better do something about that. 00:09:13.910 --> 00:09:18.920 align:middle line:84% And we're going to do that using the electron equation of motion. 00:09:18.920 --> 00:09:20.690 align:middle line:84% And that electron equation of motion 00:09:20.690 --> 00:09:29.030 align:middle line:84% looks like m dVe dt is equal to minus e the charge times capital 00:09:29.030 --> 00:09:34.540 align:middle line:84% E, the vector electric field, like that. 00:09:34.540 --> 00:09:37.540 align:middle line:84% So we could rearrange that so that we have Ve 00:09:37.540 --> 00:09:43.740 align:middle line:84% is equal to e vector E over im omega. 00:09:43.740 --> 00:09:45.960 align:middle line:84% We've done the same Fourier transform trick 00:09:45.960 --> 00:09:47.250 align:middle line:84% that we did with all the other quantities. 00:09:47.250 --> 00:09:49.375 align:middle line:84% We assume it's going to be oscillating in some way. 00:09:49.375 --> 00:09:52.380 align:middle line:84% Substitute that back in to our equation for E, 00:09:52.380 --> 00:09:54.840 align:middle line:84% and we'll get omega squared minus C squared 00:09:54.840 --> 00:09:58.290 align:middle line:84% k squared E, as we had before, is 00:09:58.290 --> 00:10:06.873 align:middle line:84% equal to nee squared over epsilon 0 and E, capital E. 00:10:06.873 --> 00:10:08.040 align:middle line:90% Sorry, was there a question? 00:10:08.040 --> 00:10:11.230 align:middle line:90% I heard someone speak. 00:10:11.230 --> 00:10:11.730 align:middle line:90% STUDENT: No. 00:10:11.730 --> 00:10:12.480 align:middle line:90% That was an accident. 00:10:12.480 --> 00:10:13.050 align:middle line:90% Sorry. 00:10:13.050 --> 00:10:13.967 align:middle line:90% JACK HARE: No worries. 00:10:13.967 --> 00:10:14.730 align:middle line:90% All good. 00:10:14.730 --> 00:10:17.220 align:middle line:84% And just to be clear here, this equation of motion 00:10:17.220 --> 00:10:20.620 align:middle line:84% doesn't have the V cross B term in it, 00:10:20.620 --> 00:10:22.560 align:middle line:84% which would be like the magnetic field term, 00:10:22.560 --> 00:10:24.750 align:middle line:84% because we've dropped it because we're 00:10:24.750 --> 00:10:27.090 align:middle line:84% making this unmagnetized approximation. 00:10:27.090 --> 00:10:28.590 align:middle line:84% But that's where you put it back in. 00:10:28.590 --> 00:10:30.540 align:middle line:84% You put in a little term that was like, 00:10:30.540 --> 00:10:34.330 align:middle line:84% cross V cross B in here and put little brackets around. 00:10:34.330 --> 00:10:36.990 align:middle line:84% But for now, we're just setting that equal to 0. 00:10:36.990 --> 00:10:38.160 align:middle line:90% OK. 00:10:38.160 --> 00:10:39.980 align:middle line:90% Good. 00:10:39.980 --> 00:10:43.070 align:middle line:84% And the solution to this equation that we've got now 00:10:43.070 --> 00:10:45.830 align:middle line:84% is omega squared equals omega p squared 00:10:45.830 --> 00:10:48.870 align:middle line:90% plus C squared K squared. 00:10:48.870 --> 00:10:51.630 align:middle line:84% So this looks an awful lot like what we had before, 00:10:51.630 --> 00:10:53.480 align:middle line:90% which was just these two terms. 00:10:53.480 --> 00:10:55.490 align:middle line:84% But now, we've got this additional term, which 00:10:55.490 --> 00:10:57.080 align:middle line:90% includes some plasma physics. 00:10:57.080 --> 00:11:01.280 align:middle line:84% And I could write that this is omega pe squared 00:11:01.280 --> 00:11:03.680 align:middle line:84% and make it clear that it's the electrons, but remember, 00:11:03.680 --> 00:11:05.222 align:middle line:84% we've dropped the arm motion already. 00:11:05.222 --> 00:11:07.280 align:middle line:84% So there's only one plasma frequency 00:11:07.280 --> 00:11:09.890 align:middle line:84% that's very interesting, the electron plasma frequency 00:11:09.890 --> 00:11:13.560 align:middle line:90% for this derivation here. 00:11:13.560 --> 00:11:16.170 align:middle line:84% And then, we can go ahead and do all the standard things 00:11:16.170 --> 00:11:21.710 align:middle line:84% we do with one of these functions, which 00:11:21.710 --> 00:11:24.265 align:middle line:84% is to try and write down the phase velocity. 00:11:24.265 --> 00:11:25.640 align:middle line:84% And the phase velocity squared is 00:11:25.640 --> 00:11:28.590 align:middle line:84% just equal to omega squared upon k squared. 00:11:28.590 --> 00:11:34.750 align:middle line:84% And so that means that our phase velocity is C upon 1 00:11:34.750 --> 00:11:39.680 align:middle line:84% minus omega p squared upon omega squared. 00:11:39.680 --> 00:11:42.400 align:middle line:90% It's a half. 00:11:42.400 --> 00:11:44.650 align:middle line:84% And then, we can write down the refractive index, 00:11:44.650 --> 00:11:46.900 align:middle line:84% because our refractive index, capital N, 00:11:46.900 --> 00:11:51.020 align:middle line:84% is just equal to C over the phase velocity. 00:11:51.020 --> 00:11:56.050 align:middle line:84% And that is going to be equal to 1 minus omega p squared 00:11:56.050 --> 00:11:58.860 align:middle line:90% over omega squared. 00:11:58.860 --> 00:12:01.540 align:middle line:84% And we often write this not in terms of frequencies, 00:12:01.540 --> 00:12:05.610 align:middle line:84% but in terms of densities, because the plasma frequency has 00:12:05.610 --> 00:12:07.370 align:middle line:90% inside it an electron density. 00:12:07.370 --> 00:12:11.880 align:middle line:84% We write 1 minus ne over n critical, 00:12:11.880 --> 00:12:16.672 align:middle line:84% like this, where n critical is some critical density. 00:12:16.672 --> 00:12:18.880 align:middle line:84% And we'll get on to why it's so critical in a moment. 00:12:18.880 --> 00:12:20.400 align:middle line:84% But if you want to approximate it, 00:12:20.400 --> 00:12:24.840 align:middle line:84% then n critical in centimeters to the minus 3, 00:12:24.840 --> 00:12:28.200 align:middle line:84% so particles per cubic centimeter, is roughly 10 00:12:28.200 --> 00:12:33.610 align:middle line:84% to the 21 over lambda when lambda is in microns here. 00:12:33.610 --> 00:12:36.480 align:middle line:84% So if you're using a laser beam at 1 micron, 00:12:36.480 --> 00:12:38.100 align:middle line:84% that would be a very standard laser 00:12:38.100 --> 00:12:41.130 align:middle line:84% wavelength from a neodymium YAG or a neodymium glass laser, 00:12:41.130 --> 00:12:44.110 align:middle line:84% then you have a critical density of about 10 to the 21. 00:12:44.110 --> 00:12:46.110 align:middle line:84% Which depending on which field you're working in 00:12:46.110 --> 00:12:49.680 align:middle line:84% is either hilariously high and unreachable or crazy low 00:12:49.680 --> 00:12:50.727 align:middle line:90% and happens all the time. 00:12:50.727 --> 00:12:52.560 align:middle line:84% So again, this is one of the exciting things 00:12:52.560 --> 00:12:55.740 align:middle line:84% about doing plasma diagnostics course where we span 16 00:12:55.740 --> 00:12:57.585 align:middle line:90% orders of magnitude in density. 00:12:57.585 --> 00:13:00.130 align:middle line:90% 00:13:00.130 --> 00:13:02.560 align:middle line:90% Any questions on that? 00:13:02.560 --> 00:13:06.740 align:middle line:84% STUDENT: Yeah, what's the critical density point again? 00:13:06.740 --> 00:13:09.247 align:middle line:84% JACK HARE: As in, what is its physical significance? 00:13:09.247 --> 00:13:09.830 align:middle line:90% STUDENT: Yeah. 00:13:09.830 --> 00:13:12.152 align:middle line:84% Like, why did you choose that number? 00:13:12.152 --> 00:13:14.360 align:middle line:84% JACK HARE: We're going to look into that in a moment. 00:13:14.360 --> 00:13:15.235 align:middle line:90% It's a good question. 00:13:15.235 --> 00:13:16.190 align:middle line:90% It's a solid question. 00:13:16.190 --> 00:13:20.930 align:middle line:84% But just to be clear, the reason I've got there is I've 00:13:20.930 --> 00:13:25.370 align:middle line:84% taken this omega p squared and this omega squared 00:13:25.370 --> 00:13:27.290 align:middle line:84% and I've noticed that omega p squared 00:13:27.290 --> 00:13:30.830 align:middle line:84% has inside it the electron density, 00:13:30.830 --> 00:13:32.550 align:middle line:84% and I've rewritten all the other terms. 00:13:32.550 --> 00:13:36.230 align:middle line:84% So this n critical now has inside it 00:13:36.230 --> 00:13:39.680 align:middle line:90% things like omega squared. 00:13:39.680 --> 00:13:41.780 align:middle line:84% I made a critical mistake in my equation up here. 00:13:41.780 --> 00:13:45.740 align:middle line:84% This is lambda squared here for approximating this 00:13:45.740 --> 00:13:48.990 align:middle line:90% in terms of simple quantities. 00:13:48.990 --> 00:13:53.060 align:middle line:84% So the n critical now depends on the laser wavelength 00:13:53.060 --> 00:13:55.590 align:middle line:84% or the frequency of your probing electromagnetic radiation. 00:13:55.590 --> 00:13:58.053 align:middle line:84% So it's different for every frequency. 00:13:58.053 --> 00:13:59.720 align:middle line:84% But that's the only thing it depends on. 00:13:59.720 --> 00:14:01.803 align:middle line:84% The rest of it is all like, fundamental constants, 00:14:01.803 --> 00:14:03.990 align:middle line:84% like E and the electron mass that don't change. 00:14:03.990 --> 00:14:06.560 align:middle line:84% So this is a parameter there's a critical density 00:14:06.560 --> 00:14:09.200 align:middle line:84% for every single laser wavelength 00:14:09.200 --> 00:14:11.480 align:middle line:84% or electromagnetic wave frequency. 00:14:11.480 --> 00:14:13.412 align:middle line:84% I'm going to talk about lasers a lot. 00:14:13.412 --> 00:14:15.620 align:middle line:84% Of course, this also applies to microwaves and things 00:14:15.620 --> 00:14:16.370 align:middle line:90% like that as well. 00:14:16.370 --> 00:14:19.050 align:middle line:84% But some sort of source of radiation. 00:14:19.050 --> 00:14:22.382 align:middle line:84% So I'm going to get on to what n critical is in just a moment. 00:14:22.382 --> 00:14:24.590 align:middle line:84% But any other questions on this before we keep going? 00:14:24.590 --> 00:14:30.820 align:middle line:90% 00:14:30.820 --> 00:14:35.290 align:middle line:84% OK, so we just said that n is equal to the square root of 1 00:14:35.290 --> 00:14:38.260 align:middle line:90% minus ne over n critical. 00:14:38.260 --> 00:14:42.940 align:middle line:84% If we work in a regime where ne is much, much less than n 00:14:42.940 --> 00:14:45.370 align:middle line:84% critical, so we work far from the critical density, 00:14:45.370 --> 00:14:47.590 align:middle line:84% we can do a Taylor expansion, which 00:14:47.590 --> 00:14:52.750 align:middle line:84% is what we often end up doing, and we write 1 minus ne over 2 n 00:14:52.750 --> 00:14:54.740 align:middle line:90% critical, like this. 00:14:54.740 --> 00:14:57.460 align:middle line:90% So my questions for you now-- 00:14:57.460 --> 00:14:58.990 align:middle line:84% and we will try and work out what 00:14:58.990 --> 00:15:00.760 align:middle line:90% n critical is doing together-- 00:15:00.760 --> 00:15:07.290 align:middle line:84% for n, for a density which is greater 00:15:07.290 --> 00:15:13.390 align:middle line:84% than the critical density, what happens 00:15:13.390 --> 00:15:20.660 align:middle line:90% to n, the refractive index? 00:15:20.660 --> 00:15:27.670 align:middle line:90% 00:15:27.670 --> 00:15:30.063 align:middle line:84% STUDENT: The wave becomes evanescent, right? 00:15:30.063 --> 00:15:30.730 align:middle line:90% JACK HARE: Yeah. 00:15:30.730 --> 00:15:36.120 align:middle line:84% Well, that's the result. So what happens to n itself? 00:15:36.120 --> 00:15:37.680 align:middle line:90% STUDENT: Big N? 00:15:37.680 --> 00:15:38.982 align:middle line:90% It's imaginary. 00:15:38.982 --> 00:15:39.690 align:middle line:90% JACK HARE: Right. 00:15:39.690 --> 00:15:40.170 align:middle line:90% Exactly. 00:15:40.170 --> 00:15:40.380 align:middle line:90% Yeah. 00:15:40.380 --> 00:15:40.880 align:middle line:90% Yeah. 00:15:40.880 --> 00:15:42.240 align:middle line:90% You've got the right answer. 00:15:42.240 --> 00:15:43.990 align:middle line:84% I just wanted to take in a few more steps. 00:15:43.990 --> 00:15:47.590 align:middle line:84% So n is imaginary because it's going 00:15:47.590 --> 00:15:50.140 align:middle line:84% to be the square root of a negative number, which 00:15:50.140 --> 00:15:53.300 align:middle line:90% we can see here. 00:15:53.300 --> 00:15:56.020 align:middle line:84% And so that means our wave becomes evanescent. 00:15:56.020 --> 00:15:59.540 align:middle line:84% So it's going to have properties which decay in time. 00:15:59.540 --> 00:16:04.675 align:middle line:84% So it's going to look like e to the minus alpha x 00:16:04.675 --> 00:16:07.880 align:middle line:84% e to the minus gamma t, like this. 00:16:07.880 --> 00:16:09.820 align:middle line:90% So it dies off. 00:16:09.820 --> 00:16:12.070 align:middle line:84% So that means that we can't propagate 00:16:12.070 --> 00:16:15.910 align:middle line:84% a wave at densities greater than the critical density. 00:16:15.910 --> 00:16:17.860 align:middle line:84% What happens to all the energy, then? 00:16:17.860 --> 00:16:21.263 align:middle line:90% 00:16:21.263 --> 00:16:22.805 align:middle line:84% Because this wave is carrying energy. 00:16:22.805 --> 00:16:27.760 align:middle line:90% 00:16:27.760 --> 00:16:31.290 align:middle line:84% STUDENT: It's absorbed by the plasma. 00:16:31.290 --> 00:16:32.760 align:middle line:84% JACK HARE: No absorption mechanism 00:16:32.760 --> 00:16:34.140 align:middle line:90% in our equations, actually. 00:16:34.140 --> 00:16:38.200 align:middle line:90% 00:16:38.200 --> 00:16:39.200 align:middle line:90% STUDENT: Yeah, so that-- 00:16:39.200 --> 00:16:43.580 align:middle line:84% I mean, from the math we have, the only option's reflection, 00:16:43.580 --> 00:16:44.210 align:middle line:90% right? 00:16:44.210 --> 00:16:46.380 align:middle line:90% JACK HARE: Yeah. 00:16:46.380 --> 00:16:48.475 align:middle line:84% So reflection is indeed the answer. 00:16:48.475 --> 00:16:50.100 align:middle line:84% So we will get reflection of this wave. 00:16:50.100 --> 00:16:52.905 align:middle line:84% It will bounce off the critical surface and go somewhere else. 00:16:52.905 --> 00:16:56.660 align:middle line:90% 00:16:56.660 --> 00:16:58.160 align:middle line:84% We'd only be able to have absorption 00:16:58.160 --> 00:17:00.403 align:middle line:84% if we put, for example, collisions into the equation 00:17:00.403 --> 00:17:01.820 align:middle line:84% of motion, all the way back there. 00:17:01.820 --> 00:17:04.362 align:middle line:84% And then, we could have what's called inverse Bremsstrahlung, 00:17:04.362 --> 00:17:07.880 align:middle line:84% which is, effectively, the electrons get oscillated 00:17:07.880 --> 00:17:09.950 align:middle line:84% by the wave, and then they collide with some ions 00:17:09.950 --> 00:17:11.450 align:middle line:84% and transfer the energy to the ions. 00:17:11.450 --> 00:17:12.829 align:middle line:90% So that's a damping mechanism. 00:17:12.829 --> 00:17:15.050 align:middle line:84% There could be other damping mechanisms like lambda, 00:17:15.050 --> 00:17:16.849 align:middle line:84% but in the equations we've got so far, 00:17:16.849 --> 00:17:18.170 align:middle line:90% we don't actually have those. 00:17:18.170 --> 00:17:21.119 align:middle line:84% So reflection is the only thing that can happen. 00:17:21.119 --> 00:17:25.960 align:middle line:84% Now, we also had this equation for the phase velocity, 00:17:25.960 --> 00:17:33.510 align:middle line:84% which was that Vp is equal to c over 1 minus omega p squared 00:17:33.510 --> 00:17:36.420 align:middle line:90% upon omega squared to the 1/2. 00:17:36.420 --> 00:17:38.190 align:middle line:84% Does anyone want to comment on what 00:17:38.190 --> 00:17:40.630 align:middle line:84% happens to the phase velocity as we 00:17:40.630 --> 00:17:41.880 align:middle line:90% go above the critical density? 00:17:41.880 --> 00:17:49.810 align:middle line:90% 00:17:49.810 --> 00:17:52.798 align:middle line:90% STUDENT: Does it go to infinity? 00:17:52.798 --> 00:17:54.840 align:middle line:84% JACK HARE: I mean, we'll do that eventually, yes. 00:17:54.840 --> 00:17:55.513 align:middle line:90% Yes. 00:17:55.513 --> 00:17:56.680 align:middle line:90% Right there, at that point-- 00:17:56.680 --> 00:17:57.595 align:middle line:90% STUDENT: It diverges? 00:17:57.595 --> 00:17:58.720 align:middle line:90% JACK HARE: Extremely large. 00:17:58.720 --> 00:18:00.780 align:middle line:90% Yes. 00:18:00.780 --> 00:18:01.280 align:middle line:90% Yeah. 00:18:01.280 --> 00:18:06.360 align:middle line:84% So this is going to start doing very silly things here. 00:18:06.360 --> 00:18:08.110 align:middle line:84% And those things don't seem very physical, 00:18:08.110 --> 00:18:10.443 align:middle line:84% of course, because we don't want things traveling faster 00:18:10.443 --> 00:18:11.590 align:middle line:90% than the speed of light. 00:18:11.590 --> 00:18:14.500 align:middle line:84% Fortunately, this isn't a big problem 00:18:14.500 --> 00:18:16.912 align:middle line:84% because the phase velocity doesn't carry any information, 00:18:16.912 --> 00:18:18.370 align:middle line:84% and there are limitations on things 00:18:18.370 --> 00:18:19.870 align:middle line:84% going faster than the speed of light 00:18:19.870 --> 00:18:21.190 align:middle line:90% have to do with information. 00:18:21.190 --> 00:18:24.010 align:middle line:84% And the information is encoded in a quantity called the group 00:18:24.010 --> 00:18:26.530 align:middle line:84% velocity, and the group velocity just 00:18:26.530 --> 00:18:31.860 align:middle line:84% looks like this, c times the square root of 1 00:18:31.860 --> 00:18:34.620 align:middle line:84% minus omega p squared upon omega squared. 00:18:34.620 --> 00:18:36.960 align:middle line:84% And so as we get close to the critical density here, 00:18:36.960 --> 00:18:39.450 align:middle line:84% all that happens is the group velocity goes to 0, 00:18:39.450 --> 00:18:41.730 align:middle line:84% and effectively, we transmit no information 00:18:41.730 --> 00:18:43.090 align:middle line:90% through that evanescent region. 00:18:43.090 --> 00:18:44.910 align:middle line:84% So we don't have to worry about the fact that phase velocity. 00:18:44.910 --> 00:18:47.640 align:middle line:84% It's superluminal because our group velocity is still 00:18:47.640 --> 00:18:49.830 align:middle line:90% subluminal, as we'd like. 00:18:49.830 --> 00:18:52.150 align:middle line:90% Sean? 00:18:52.150 --> 00:18:54.160 align:middle line:84% STUDENT: So if the wave's being reflected here, 00:18:54.160 --> 00:18:57.280 align:middle line:84% and the group velocity is going to 0, to me, 00:18:57.280 --> 00:18:59.830 align:middle line:84% that seems like the wave information 00:18:59.830 --> 00:19:02.800 align:middle line:84% is sort of stagnating at the reflection point. 00:19:02.800 --> 00:19:04.780 align:middle line:84% How do we see from these equations 00:19:04.780 --> 00:19:06.580 align:middle line:84% that the wave information is actually 00:19:06.580 --> 00:19:09.655 align:middle line:90% being reflected back out. 00:19:09.655 --> 00:19:10.905 align:middle line:90% JACK HARE: Very good question. 00:19:10.905 --> 00:19:13.550 align:middle line:90% 00:19:13.550 --> 00:19:15.260 align:middle line:90% STUDENT: I think-- excuse me. 00:19:15.260 --> 00:19:19.120 align:middle line:84% I think you need to impose boundary conditions to do that. 00:19:19.120 --> 00:19:22.203 align:middle line:84% Because there would be some sort of discontinuity, right? 00:19:22.203 --> 00:19:22.870 align:middle line:90% JACK HARE: Yeah. 00:19:22.870 --> 00:19:24.287 align:middle line:84% I suspect the model I'm presenting 00:19:24.287 --> 00:19:27.012 align:middle line:84% is a little bit too simplistic to handle this stuff. 00:19:27.012 --> 00:19:29.470 align:middle line:84% STUDENT: So we would need to bake in some more information. 00:19:29.470 --> 00:19:30.220 align:middle line:90% JACK HARE: I think so. 00:19:30.220 --> 00:19:31.720 align:middle line:84% There's certainly-- as you get very, 00:19:31.720 --> 00:19:36.740 align:middle line:84% very slow group velocities, you're going to start-- 00:19:36.740 --> 00:19:38.360 align:middle line:84% we've been making some assumptions 00:19:38.360 --> 00:19:42.047 align:middle line:84% about the homogeneity here, and so, reflectively, there's 00:19:42.047 --> 00:19:43.880 align:middle line:84% going to be some length scale in here, which 00:19:43.880 --> 00:19:48.440 align:middle line:84% is going to be like, k, the size of the wave vector of the light 00:19:48.440 --> 00:19:50.960 align:middle line:84% at a given point, and we're going to be comparing that 00:19:50.960 --> 00:19:54.500 align:middle line:84% to the length scale associated with how quickly the electron 00:19:54.500 --> 00:19:57.390 align:middle line:84% density changes, this gradient length scale here. 00:19:57.390 --> 00:20:00.950 align:middle line:84% And as the group velocity gets very, very low 00:20:00.950 --> 00:20:04.280 align:middle line:84% that k is going to get very, very long, 00:20:04.280 --> 00:20:08.930 align:middle line:84% and we're going to start violating our assumption 00:20:08.930 --> 00:20:10.640 align:middle line:90% that k is-- 00:20:10.640 --> 00:20:11.750 align:middle line:90% let's see. 00:20:11.750 --> 00:20:14.060 align:middle line:84% Maybe I can write this as lambda. 00:20:14.060 --> 00:20:16.460 align:middle line:84% That lambda is going to be much, much less 00:20:16.460 --> 00:20:19.740 align:middle line:84% than this change in the gradient. 00:20:19.740 --> 00:20:23.270 align:middle line:84% So for any realistic system, our density has to ramp up. 00:20:23.270 --> 00:20:28.233 align:middle line:84% It can't just immediately get up to the critical density. 00:20:28.233 --> 00:20:29.900 align:middle line:84% This could be the critical density here. 00:20:29.900 --> 00:20:31.880 align:middle line:84% And we're going to find out that there's 00:20:31.880 --> 00:20:33.920 align:middle line:84% some region where some of the approximations 00:20:33.920 --> 00:20:36.050 align:middle line:84% we've implicitly been making break down. 00:20:36.050 --> 00:20:38.300 align:middle line:84% And then, you need to start doing 00:20:38.300 --> 00:20:41.160 align:middle line:84% wbk, and all that sort of stuff and doing everything properly. 00:20:41.160 --> 00:20:44.100 align:middle line:84% So I think, effectively, this simple model breaks down. 00:20:44.100 --> 00:20:46.280 align:middle line:84% But if you do it properly-- and I think 00:20:46.280 --> 00:20:48.410 align:middle line:84% Hutchinson does this in the reflectometry section. 00:20:48.410 --> 00:20:50.330 align:middle line:90% So we may end up doing it. 00:20:50.330 --> 00:20:53.540 align:middle line:84% You can get the answer about the reflection there. 00:20:53.540 --> 00:20:54.440 align:middle line:90% It's a good question. 00:20:54.440 --> 00:20:56.090 align:middle line:84% This is very hand-wavy at this point. 00:20:56.090 --> 00:20:57.350 align:middle line:90% I agree. 00:20:57.350 --> 00:20:58.183 align:middle line:90% STUDENT: Thanks. 00:20:58.183 --> 00:20:58.850 align:middle line:90% JACK HARE: Cool. 00:20:58.850 --> 00:21:00.080 align:middle line:90% Any other questions on this? 00:21:00.080 --> 00:21:02.750 align:middle line:84% Because this equation here, we are now 00:21:02.750 --> 00:21:04.170 align:middle line:90% going to use an awful lot. 00:21:04.170 --> 00:21:06.110 align:middle line:84% So I'd like you to agree that it's 00:21:06.110 --> 00:21:08.040 align:middle line:84% valid within the assumptions that we've made. 00:21:08.040 --> 00:21:10.332 align:middle line:84% And if you don't agree, we should have a chat about it. 00:21:10.332 --> 00:21:19.540 align:middle line:90% 00:21:19.540 --> 00:21:20.470 align:middle line:90% OK, good. 00:21:20.470 --> 00:21:22.690 align:middle line:90% So let's keep going. 00:21:22.690 --> 00:21:25.740 align:middle line:84% So there is a series of different measurements 00:21:25.740 --> 00:21:26.650 align:middle line:90% that we can make. 00:21:26.650 --> 00:21:28.710 align:middle line:84% And these are the refractive index diagnostics. 00:21:28.710 --> 00:21:34.270 align:middle line:84% So I'm going to just call these n measurements or n diagnostics. 00:21:34.270 --> 00:21:37.370 align:middle line:84% Because they rely on the change in refractive index here. 00:21:37.370 --> 00:21:41.350 align:middle line:84% So one type is when the refractive index is not 00:21:41.350 --> 00:21:43.507 align:middle line:90% equal to 1. 00:21:43.507 --> 00:21:45.340 align:middle line:84% There's a refractive index inside our plasma 00:21:45.340 --> 00:21:47.410 align:middle line:84% is not equal to 1, which is true anytime there's 00:21:47.410 --> 00:21:49.810 align:middle line:90% any density inside the plasma. 00:21:49.810 --> 00:21:53.570 align:middle line:84% This sort of diagnostic causes a phase shift. 00:21:53.570 --> 00:21:56.567 align:middle line:84% So the plasma ends up-- the laser beam going through 00:21:56.567 --> 00:21:58.150 align:middle line:84% or the electromagnetic radiation going 00:21:58.150 --> 00:22:00.310 align:middle line:84% through the plasma ends up with a different phase 00:22:00.310 --> 00:22:02.540 align:middle line:84% than it would have done in the absence of the plasma. 00:22:02.540 --> 00:22:05.350 align:middle line:84% And we can measure that phase shift using a technique 00:22:05.350 --> 00:22:08.880 align:middle line:90% called interferometry. 00:22:08.880 --> 00:22:10.980 align:middle line:84% And with interferometry, we can therefore 00:22:10.980 --> 00:22:15.300 align:middle line:84% say something about the density inside the plasma. 00:22:15.300 --> 00:22:20.040 align:middle line:84% Another technique is when the gradient of the refractive index 00:22:20.040 --> 00:22:22.530 align:middle line:90% is not equal to 0. 00:22:22.530 --> 00:22:25.680 align:middle line:84% So this is when there is any change in refractive index. 00:22:25.680 --> 00:22:28.860 align:middle line:84% And in a plasma, that corresponds very clearly to just 00:22:28.860 --> 00:22:32.370 align:middle line:84% changes in the electron density, so gradients in the electron 00:22:32.370 --> 00:22:33.032 align:middle line:90% density. 00:22:33.032 --> 00:22:34.740 align:middle line:84% But of course, in general, this technique 00:22:34.740 --> 00:22:37.323 align:middle line:84% can be used for any medium where the refractive index changes. 00:22:37.323 --> 00:22:40.620 align:middle line:84% So air, if you heat it up, the refractive index changes, 00:22:40.620 --> 00:22:42.700 align:middle line:84% and so you could use these techniques. 00:22:42.700 --> 00:22:44.670 align:middle line:84% These are not specific to plasma physics. 00:22:44.670 --> 00:22:48.570 align:middle line:84% And these diagnostics tend to be called refraction diagnostics, 00:22:48.570 --> 00:22:51.990 align:middle line:84% because the light refracts and it bends. 00:22:51.990 --> 00:22:57.800 align:middle line:84% And we end up doing techniques such as schlieren 00:22:57.800 --> 00:22:59.300 align:middle line:90% and sonography. 00:22:59.300 --> 00:23:04.550 align:middle line:90% 00:23:04.550 --> 00:23:07.520 align:middle line:84% And then, the final type that I'm going to talk about 00:23:07.520 --> 00:23:12.380 align:middle line:84% are ones where, actually, the polarization, the medium 00:23:12.380 --> 00:23:13.280 align:middle line:90% is birefringent. 00:23:13.280 --> 00:23:15.750 align:middle line:84% It treats different polarizations differently. 00:23:15.750 --> 00:23:18.620 align:middle line:84% And so we can have polarizations of light which are circular. 00:23:18.620 --> 00:23:23.540 align:middle line:84% We can have the left-handed and the right-handed polarizations, 00:23:23.540 --> 00:23:26.540 align:middle line:84% which we sometimes refer to as plus and minus. 00:23:26.540 --> 00:23:29.000 align:middle line:84% And here, we would say that the refractive index 00:23:29.000 --> 00:23:32.840 align:middle line:84% for the plus wave is not equal to the refractive index 00:23:32.840 --> 00:23:35.070 align:middle line:90% to the minus wave here. 00:23:35.070 --> 00:23:38.790 align:middle line:84% And so here, we measure the polarization. 00:23:38.790 --> 00:23:41.880 align:middle line:90% 00:23:41.880 --> 00:23:45.200 align:middle line:84% And this is using a technique called 00:23:45.200 --> 00:23:51.255 align:middle line:90% Faraday or Faraday rotation. 00:23:51.255 --> 00:23:53.920 align:middle line:90% 00:23:53.920 --> 00:23:56.580 align:middle line:84% Which we briefly discussed in the context of magnetic field 00:23:56.580 --> 00:23:59.280 align:middle line:90% measurements using Verdet glass. 00:23:59.280 --> 00:24:01.140 align:middle line:84% And in fact, it turns out that you 00:24:01.140 --> 00:24:05.040 align:middle line:84% need to have magnetic fields that are non-zero, 00:24:05.040 --> 00:24:10.650 align:middle line:84% and we also need to relax our assumption that the plasma is 00:24:10.650 --> 00:24:14.880 align:middle line:84% unmagnetized in the sense that the light frequency is 00:24:14.880 --> 00:24:18.280 align:middle line:84% much larger than the electron cyclotron frequency. 00:24:18.280 --> 00:24:21.210 align:middle line:84% So those are three different types of refractive index 00:24:21.210 --> 00:24:22.770 align:middle line:84% diagnostic, and we're going to start 00:24:22.770 --> 00:24:26.310 align:middle line:84% with what I think is conceptually the simplest, 00:24:26.310 --> 00:24:28.680 align:middle line:84% but still often causes us lots of problems, 00:24:28.680 --> 00:24:31.485 align:middle line:84% which are the refraction diagnostics here. 00:24:31.485 --> 00:24:37.712 align:middle line:90% 00:24:37.712 --> 00:24:39.920 align:middle line:84% I see some people writing, so I'm just going to pause 00:24:39.920 --> 00:24:41.045 align:middle line:90% on this slide for a moment. 00:24:41.045 --> 00:24:49.050 align:middle line:90% 00:24:49.050 --> 00:24:49.890 align:middle line:90% Okey doke. 00:24:49.890 --> 00:24:55.400 align:middle line:84% Now, I just want to have a little aside. 00:24:55.400 --> 00:25:06.450 align:middle line:84% And this is on conceptual models for electromagnetic propagation. 00:25:06.450 --> 00:25:09.305 align:middle line:90% 00:25:09.305 --> 00:25:10.680 align:middle line:84% Because I'm going to be switching 00:25:10.680 --> 00:25:12.630 align:middle line:84% quite a lot between different ways of thinking 00:25:12.630 --> 00:25:15.180 align:middle line:84% about electromagnetic radiation and how 00:25:15.180 --> 00:25:16.290 align:middle line:90% it moves through a plasma. 00:25:16.290 --> 00:25:18.360 align:middle line:84% Because sometimes, some models are easier 00:25:18.360 --> 00:25:20.070 align:middle line:90% to work with than others. 00:25:20.070 --> 00:25:22.110 align:middle line:84% Sometimes, models are simplifications 00:25:22.110 --> 00:25:24.810 align:middle line:84% and they throw away physics, but they make 00:25:24.810 --> 00:25:26.110 align:middle line:90% the intuition much simpler. 00:25:26.110 --> 00:25:28.110 align:middle line:84% So I just want to show you two different models 00:25:28.110 --> 00:25:30.880 align:middle line:84% that we're going to be using in these next few lectures 00:25:30.880 --> 00:25:32.950 align:middle line:84% so that you have an idea of what's going on. 00:25:32.950 --> 00:25:38.130 align:middle line:84% One model would be a model of wavefronts. 00:25:38.130 --> 00:25:40.790 align:middle line:84% So this is based on the idea that, as we said, 00:25:40.790 --> 00:25:46.570 align:middle line:84% our electric and magnetic fields can be written as just 00:25:46.570 --> 00:25:47.982 align:middle line:90% a single Fourier component. 00:25:47.982 --> 00:25:49.690 align:middle line:84% So there's some strength and polarization 00:25:49.690 --> 00:25:52.180 align:middle line:84% of the electric field here, and this 00:25:52.180 --> 00:25:54.670 align:middle line:84% is multiplied by the exponential of what 00:25:54.670 --> 00:25:56.230 align:middle line:90% we call the phase factor. 00:25:56.230 --> 00:26:02.070 align:middle line:84% So i k dot x minus omega t, like this, 00:26:02.070 --> 00:26:07.940 align:middle line:84% which we could write as E0 exponential of i times 00:26:07.940 --> 00:26:12.330 align:middle line:84% some scalar quantity, which is the phase here. 00:26:12.330 --> 00:26:14.960 align:middle line:84% And so if we think of our electromagnetic wave 00:26:14.960 --> 00:26:17.720 align:middle line:84% as having a phase, and the electromagnetic wave still 00:26:17.720 --> 00:26:21.120 align:middle line:84% exists in all places, all points in time, blah, blah, blah, blah, 00:26:21.120 --> 00:26:23.870 align:middle line:84% blah, but that's a very difficult thing for me 00:26:23.870 --> 00:26:27.720 align:middle line:84% to sketch on my iPad here or on the board in front of you. 00:26:27.720 --> 00:26:29.600 align:middle line:84% So what I'll probably end up sketching 00:26:29.600 --> 00:26:33.020 align:middle line:84% are what we call isophase contours. 00:26:33.020 --> 00:26:39.720 align:middle line:84% So these are controls along which the phase is constant. 00:26:39.720 --> 00:26:44.700 align:middle line:84% And so, for example, it could be at some integer multiple of pi, 00:26:44.700 --> 00:26:45.950 align:middle line:90% right? 00:26:45.950 --> 00:26:47.090 align:middle line:90% Yes. 00:26:47.090 --> 00:26:47.930 align:middle line:90% And belonging to z. 00:26:47.930 --> 00:26:52.430 align:middle line:84% So this might look like some waves like this. 00:26:52.430 --> 00:26:55.465 align:middle line:84% This would be an electromagnetic wave which is diverging here. 00:26:55.465 --> 00:26:57.590 align:middle line:84% Actually, it could be an electromagnetic wave which 00:26:57.590 --> 00:26:59.630 align:middle line:90% is converging to the left. 00:26:59.630 --> 00:27:03.470 align:middle line:84% But at the moment, it looks like it's diverging to the right. 00:27:03.470 --> 00:27:07.840 align:middle line:84% So that's one way I could draw a wave here. 00:27:07.840 --> 00:27:09.885 align:middle line:84% Another way I could do it is with a ray model. 00:27:09.885 --> 00:27:13.270 align:middle line:90% 00:27:13.270 --> 00:27:16.690 align:middle line:84% And this gets into a topic which is called geometric optics. 00:27:16.690 --> 00:27:22.070 align:middle line:90% 00:27:22.070 --> 00:27:23.750 align:middle line:84% And it turns out what you can do, 00:27:23.750 --> 00:27:27.230 align:middle line:84% if you have some isophase contours like the ones 00:27:27.230 --> 00:27:29.900 align:middle line:84% I just drew, say, these contours here, 00:27:29.900 --> 00:27:32.240 align:middle line:84% they're doing something slightly strange, 00:27:32.240 --> 00:27:34.020 align:middle line:84% but perhaps there's a plasma there, 00:27:34.020 --> 00:27:36.753 align:middle line:84% which is like moving the phase contours around. 00:27:36.753 --> 00:27:38.420 align:middle line:84% If there's a change of refractive index, 00:27:38.420 --> 00:27:40.130 align:middle line:90% will affect the phase. 00:27:40.130 --> 00:27:42.290 align:middle line:84% The rays that we draw here, I can just 00:27:42.290 --> 00:27:45.290 align:middle line:84% take these phase contours and I can draw rays such 00:27:45.290 --> 00:27:49.910 align:middle line:84% that they are everywhere normal to the isophase contours. 00:27:49.910 --> 00:27:53.430 align:middle line:84% So this ray would look like this. 00:27:53.430 --> 00:27:56.220 align:middle line:90% This one would look like this. 00:27:56.220 --> 00:27:59.370 align:middle line:84% And this one would look like that. 00:27:59.370 --> 00:28:02.810 align:middle line:84% So they are perpendicular to the wavefronts. 00:28:02.810 --> 00:28:05.470 align:middle line:90% 00:28:05.470 --> 00:28:11.306 align:middle line:84% Conveniently, they are also parallel to the Poynting vector. 00:28:11.306 --> 00:28:13.380 align:middle line:84% At least, I'm pretty convinced they are. 00:28:13.380 --> 00:28:15.150 align:middle line:84% If someone knows more about geometric optics and thinks 00:28:15.150 --> 00:28:16.900 align:middle line:84% I'm wrong, please, shout out, because this 00:28:16.900 --> 00:28:19.095 align:middle line:84% was a very hard fact to check in like, 10 00:28:19.095 --> 00:28:20.220 align:middle line:90% minutes before the lecture. 00:28:20.220 --> 00:28:22.350 align:middle line:84% But I'm pretty certain they represent 00:28:22.350 --> 00:28:25.180 align:middle line:84% the direction of the energy flux in electromagnetic waves. 00:28:25.180 --> 00:28:27.480 align:middle line:84% So they're quite conceptually useful as well. 00:28:27.480 --> 00:28:29.820 align:middle line:84% They tell us where the power is flowing. 00:28:29.820 --> 00:28:31.830 align:middle line:84% Now, when we think about these rays here, 00:28:31.830 --> 00:28:33.420 align:middle line:84% we can start thinking a little bit 00:28:33.420 --> 00:28:37.730 align:middle line:90% like it's a particle trajectory. 00:28:37.730 --> 00:28:42.887 align:middle line:84% And I put particle here in speech marks. 00:28:42.887 --> 00:28:45.470 align:middle line:84% I don't think you really need to think about these as photons, 00:28:45.470 --> 00:28:48.980 align:middle line:84% but you can think about them as little point particles that 00:28:48.980 --> 00:28:51.500 align:middle line:90% move around inside a plasma. 00:28:51.500 --> 00:28:53.900 align:middle line:84% And we'll find out some rules for how they move 00:28:53.900 --> 00:28:55.280 align:middle line:90% inside the plasma in a moment. 00:28:55.280 --> 00:28:57.948 align:middle line:84% And if you track their trajectory, 00:28:57.948 --> 00:28:59.240 align:middle line:90% that's where the ray's gone on. 00:28:59.240 --> 00:29:01.280 align:middle line:84% And then, you also know some places 00:29:01.280 --> 00:29:03.990 align:middle line:84% where you have lines which are normal to the wavefronts. 00:29:03.990 --> 00:29:07.220 align:middle line:84% So maybe you could reconstruct the wavefronts later on. 00:29:07.220 --> 00:29:09.380 align:middle line:84% But it's important that when we're doing this, 00:29:09.380 --> 00:29:12.850 align:middle line:90% we ignore the wave effects. 00:29:12.850 --> 00:29:16.410 align:middle line:84% So we no longer track the phase of each particle. 00:29:16.410 --> 00:29:18.150 align:middle line:84% It's now just a little billiard ball. 00:29:18.150 --> 00:29:20.140 align:middle line:84% And billiard balls don't have phase. 00:29:20.140 --> 00:29:25.660 align:middle line:84% And so we're going to get rid of effects like interference 00:29:25.660 --> 00:29:32.040 align:middle line:84% and diffraction, and we're going to keep effects only 00:29:32.040 --> 00:29:34.570 align:middle line:90% like refraction here. 00:29:34.570 --> 00:29:36.160 align:middle line:90% So no interference. 00:29:36.160 --> 00:29:37.270 align:middle line:90% No diffraction. 00:29:37.270 --> 00:29:38.980 align:middle line:90% Just refraction. 00:29:38.980 --> 00:29:41.180 align:middle line:90% So this is our ray model. 00:29:41.180 --> 00:29:43.540 align:middle line:84% So does anyone have any questions on these models 00:29:43.540 --> 00:29:45.700 align:middle line:84% before we start trying to use them? 00:29:45.700 --> 00:29:49.257 align:middle line:90% 00:29:49.257 --> 00:29:51.590 align:middle line:84% STUDENT: I was just wondering, so if the Poynting vector 00:29:51.590 --> 00:29:54.020 align:middle line:84% right is E cross B, Does that mean that 00:29:54.020 --> 00:29:58.610 align:middle line:84% if we have any parallel electric field to k-- 00:29:58.610 --> 00:30:03.020 align:middle line:84% I'm just wondering, your point about the Poynting vector, 00:30:03.020 --> 00:30:06.680 align:middle line:84% would that break if there was like a parallel e 00:30:06.680 --> 00:30:09.500 align:middle line:84% to the main background magnetic field, 00:30:09.500 --> 00:30:12.005 align:middle line:84% or is that just the oscillating B there? 00:30:12.005 --> 00:30:14.635 align:middle line:90% 00:30:14.635 --> 00:30:16.052 align:middle line:90% If that question makes sense. 00:30:16.052 --> 00:30:17.260 align:middle line:90% JACK HARE: It does, actually. 00:30:17.260 --> 00:30:18.427 align:middle line:90% And I know the answer to it. 00:30:18.427 --> 00:30:19.090 align:middle line:90% That's good. 00:30:19.090 --> 00:30:21.130 align:middle line:84% So if you had some background magnetic field, 00:30:21.130 --> 00:30:22.180 align:middle line:90% like in a tokamak. 00:30:22.180 --> 00:30:24.550 align:middle line:90% And then E was parallel to it. 00:30:24.550 --> 00:30:27.410 align:middle line:90% 00:30:27.410 --> 00:30:29.140 align:middle line:90% Well, let's put it this way. 00:30:29.140 --> 00:30:31.510 align:middle line:90% The Poynting flux oscillates. 00:30:31.510 --> 00:30:34.132 align:middle line:84% And so, when you're averaging it, time averaging it, 00:30:34.132 --> 00:30:36.340 align:middle line:84% that's what gives you the actual power that's moving. 00:30:36.340 --> 00:30:38.290 align:middle line:84% If you've got a static magnetic field, 00:30:38.290 --> 00:30:40.670 align:middle line:90% your average power will go to 0. 00:30:40.670 --> 00:30:41.170 align:middle line:90% STUDENT: OK. 00:30:41.170 --> 00:30:41.500 align:middle line:90% Yeah. 00:30:41.500 --> 00:30:41.920 align:middle line:90% Thank you. 00:30:41.920 --> 00:30:43.592 align:middle line:84% JACK HARE: So you'll only get power flow 00:30:43.592 --> 00:30:45.550 align:middle line:84% from oscillating components here because that's 00:30:45.550 --> 00:30:47.508 align:middle line:84% what's transporting the electromagnetic energy. 00:30:47.508 --> 00:30:49.810 align:middle line:90% But it's a really good question. 00:30:49.810 --> 00:30:50.620 align:middle line:90% STUDENT: Thank you. 00:30:50.620 --> 00:30:52.330 align:middle line:84% JACK HARE: Like I said, I'm not completely 100% 00:30:52.330 --> 00:30:54.550 align:middle line:84% sure that rays follow the trajectory of the Poynting 00:30:54.550 --> 00:30:56.320 align:middle line:84% vector, but I'm pretty certain, after thinking 00:30:56.320 --> 00:30:58.160 align:middle line:84% about it for about 10 minutes, that they do. 00:30:58.160 --> 00:31:00.285 align:middle line:84% So if someone finds out that's wrong, please let me 00:31:00.285 --> 00:31:02.610 align:middle line:90% know and I'll take it out. 00:31:02.610 --> 00:31:04.610 align:middle line:84% OK, so everyone is going to be pretty happy if I 00:31:04.610 --> 00:31:05.990 align:middle line:90% start drawing ray diagrams. 00:31:05.990 --> 00:31:08.330 align:middle line:84% And they'll understand that these ray diagrams represent 00:31:08.330 --> 00:31:10.700 align:middle line:84% the trajectory of little beamlets of light, 00:31:10.700 --> 00:31:14.000 align:middle line:84% and you can also reconstruct the wavefronts from them, 00:31:14.000 --> 00:31:16.820 align:middle line:84% and therefore, you could reconstruct visually 00:31:16.820 --> 00:31:19.460 align:middle line:84% what the entire electromagnetic field looks like. 00:31:19.460 --> 00:31:21.530 align:middle line:84% And we're implicitly assuming everywhere here 00:31:21.530 --> 00:31:24.500 align:middle line:84% that our magnetic field is perpendicular 00:31:24.500 --> 00:31:27.050 align:middle line:84% to our electric field, which is a pretty good approximation 00:31:27.050 --> 00:31:28.640 align:middle line:84% to the assumptions we've made so far. 00:31:28.640 --> 00:31:31.850 align:middle line:90% 00:31:31.850 --> 00:31:34.100 align:middle line:84% OK, so now, let's try putting an electromagnetic wave 00:31:34.100 --> 00:31:34.808 align:middle line:90% through a plasma. 00:31:34.808 --> 00:31:45.170 align:middle line:90% 00:31:45.170 --> 00:31:47.210 align:middle line:84% And it's not going to be any old plasma here. 00:31:47.210 --> 00:31:50.750 align:middle line:84% I'm going to choose a slab of plasma like this. 00:31:50.750 --> 00:31:54.860 align:middle line:84% And this slab is going to be much denser at the top 00:31:54.860 --> 00:31:58.010 align:middle line:84% than it is at the bottom, which I've tried to really clumsily do 00:31:58.010 --> 00:31:59.570 align:middle line:90% with some shading here. 00:31:59.570 --> 00:32:04.250 align:middle line:84% So it's going to have a gradient of electron density going up, 00:32:04.250 --> 00:32:05.180 align:middle line:90% like that. 00:32:05.180 --> 00:32:07.670 align:middle line:84% And of course, you remember our formula, 00:32:07.670 --> 00:32:13.663 align:middle line:84% 1 minus ne over 2n critical for our refractive index. 00:32:13.663 --> 00:32:15.080 align:middle line:84% We're going to work in this regime 00:32:15.080 --> 00:32:17.130 align:middle line:84% where the density is much less than the critical density, 00:32:17.130 --> 00:32:19.380 align:middle line:84% so we don't have to worry about what happens if we get 00:32:19.380 --> 00:32:21.270 align:middle line:90% close to the critical density. 00:32:21.270 --> 00:32:24.380 align:middle line:84% And so you can see, then, that if the gradient in the electron 00:32:24.380 --> 00:32:27.260 align:middle line:84% density is in this direction, then the gradient 00:32:27.260 --> 00:32:30.950 align:middle line:84% in the refractive index is in the opposite direction, 00:32:30.950 --> 00:32:33.490 align:middle line:90% like that. 00:32:33.490 --> 00:32:36.300 align:middle line:84% And we're going to start by putting through some phase 00:32:36.300 --> 00:32:37.888 align:middle line:90% fronts. 00:32:37.888 --> 00:32:39.680 align:middle line:84% And we're going to start with a plane wave. 00:32:39.680 --> 00:32:41.450 align:middle line:84% So this is a wave in which the phase 00:32:41.450 --> 00:32:46.640 align:middle line:84% fronts are flat and parallel and uniformly spaced. 00:32:46.640 --> 00:32:50.090 align:middle line:90% So those are my wavefronts. 00:32:50.090 --> 00:32:51.950 align:middle line:84% I'll put a little coordinate system in here. 00:32:51.950 --> 00:32:54.620 align:middle line:84% I'm going to tend to put the z-coordinate in the direction 00:32:54.620 --> 00:32:55.740 align:middle line:90% waves are going. 00:32:55.740 --> 00:32:58.580 align:middle line:84% And so there'll be two transverse coordinates, y and x. 00:32:58.580 --> 00:33:00.830 align:middle line:84% And I'll probably just write y on most of these. 00:33:00.830 --> 00:33:03.110 align:middle line:84% I'll try and do things in a one-dimensional sense. 00:33:03.110 --> 00:33:04.910 align:middle line:84% But everything I say you can imagine 00:33:04.910 --> 00:33:08.440 align:middle line:84% could be applied to a three-dimensional picture here. 00:33:08.440 --> 00:33:12.970 align:middle line:84% I'm going to say that this plasma slab has some length L, 00:33:12.970 --> 00:33:16.270 align:middle line:84% and it's homogeneous within that length apart from the gradient 00:33:16.270 --> 00:33:21.860 align:middle line:90% in the density here. 00:33:21.860 --> 00:33:24.890 align:middle line:84% Does anyone know what happens to the phase fronts 00:33:24.890 --> 00:33:29.920 align:middle line:84% as they emerge out from this plasma? 00:33:29.920 --> 00:33:31.225 align:middle line:90% The wavefronts or the rays. 00:33:31.225 --> 00:33:32.170 align:middle line:90% I don't mind. 00:33:32.170 --> 00:33:39.020 align:middle line:90% 00:33:39.020 --> 00:33:45.682 align:middle line:84% STUDENT: They bend in the up or down, right? 00:33:45.682 --> 00:33:46.390 align:middle line:90% JACK HARE: Sorry? 00:33:46.390 --> 00:33:50.010 align:middle line:90% 00:33:50.010 --> 00:33:50.625 align:middle line:90% Yeah, Daniel? 00:33:50.625 --> 00:33:53.868 align:middle line:90% 00:33:53.868 --> 00:33:54.370 align:middle line:90% STUDENT: Oh. 00:33:54.370 --> 00:33:55.780 align:middle line:90% Yeah. 00:33:55.780 --> 00:33:58.810 align:middle line:90% They're bent downward, right? 00:33:58.810 --> 00:34:01.900 align:middle line:84% Because you've got a lower refractive index 00:34:01.900 --> 00:34:04.293 align:middle line:90% in the upper half. 00:34:04.293 --> 00:34:04.960 align:middle line:90% JACK HARE: Yeah. 00:34:04.960 --> 00:34:05.980 align:middle line:90% You're absolutely right. 00:34:05.980 --> 00:34:08.920 align:middle line:84% So these rays will emerge or these wavefronts 00:34:08.920 --> 00:34:11.610 align:middle line:90% will emerge bent, like this. 00:34:11.610 --> 00:34:14.929 align:middle line:84% And so the rays-- which I didn't draw on before, but I meant to. 00:34:14.929 --> 00:34:17.060 align:middle line:84% So here are some rays for you here. 00:34:17.060 --> 00:34:19.159 align:middle line:84% You see how they're all normal to the wavefronts. 00:34:19.159 --> 00:34:20.495 align:middle line:90% Here are some rays for you here. 00:34:20.495 --> 00:34:23.679 align:middle line:90% 00:34:23.679 --> 00:34:26.590 align:middle line:84% And they're going to be bent by some angle, which 00:34:26.590 --> 00:34:28.219 align:middle line:90% we'll call theta here. 00:34:28.219 --> 00:34:31.449 align:middle line:84% And it turns out, if you go and look how to do this, 00:34:31.449 --> 00:34:39.940 align:middle line:84% theta is going to be equal to d phase dy times lambda over 2pi. 00:34:39.940 --> 00:34:42.989 align:middle line:84% And so we can actually put that all together and we can say it's 00:34:42.989 --> 00:34:49.980 align:middle line:84% going to be equal to d dy times the integral of capital N dz, 00:34:49.980 --> 00:34:50.790 align:middle line:90% like that. 00:34:50.790 --> 00:34:57.090 align:middle line:84% Which, for our plasma, is minus 1 over 2 n critical integral 00:34:57.090 --> 00:34:59.865 align:middle line:84% of gradient of the electron density dz. 00:34:59.865 --> 00:35:03.620 align:middle line:90% 00:35:03.620 --> 00:35:14.410 align:middle line:84% And this dz here is going to be running from 0 to L. Now, 00:35:14.410 --> 00:35:16.510 align:middle line:84% I don't know how clear this is to everyone 00:35:16.510 --> 00:35:19.090 align:middle line:84% that the rays should bend or that they bend downwards 00:35:19.090 --> 00:35:19.840 align:middle line:90% or why they bend. 00:35:19.840 --> 00:35:22.007 align:middle line:84% There's lots of different ways of thinking about it. 00:35:22.007 --> 00:35:25.090 align:middle line:84% You can go and just solve a load of equations, if you want to. 00:35:25.090 --> 00:35:26.630 align:middle line:90% I like to think of it-- 00:35:26.630 --> 00:35:28.600 align:middle line:84% and you may laugh at me for this-- 00:35:28.600 --> 00:35:32.170 align:middle line:84% as a bunch of soldiers marching arm in arm 00:35:32.170 --> 00:35:34.400 align:middle line:90% through some mixed terrain here. 00:35:34.400 --> 00:35:35.350 align:middle line:90% So here's my soldiers. 00:35:35.350 --> 00:35:37.528 align:middle line:90% I'm looking at them from above. 00:35:37.528 --> 00:35:40.070 align:middle line:84% You can see how I'm lining them up nicely with the wavefronts 00:35:40.070 --> 00:35:41.060 align:middle line:90% very suggestively. 00:35:41.060 --> 00:35:43.720 align:middle line:90% 00:35:43.720 --> 00:35:45.700 align:middle line:84% And maybe some of the soldiers over here 00:35:45.700 --> 00:35:48.520 align:middle line:84% have got some sort of marsh that they've got to walk through, 00:35:48.520 --> 00:35:50.590 align:middle line:84% and these soldiers are going to fall behind. 00:35:50.590 --> 00:35:54.850 align:middle line:90% 00:35:54.850 --> 00:35:56.350 align:middle line:84% And because they've all linked arms, 00:35:56.350 --> 00:35:58.970 align:middle line:84% they've still got to stay in a straight line with each other. 00:35:58.970 --> 00:36:03.680 align:middle line:84% And so, as they go, they turn more and more round like this, 00:36:03.680 --> 00:36:05.990 align:middle line:84% and this is what leads to our bending here. 00:36:05.990 --> 00:36:09.230 align:middle line:84% And you can make this a bit more rigorous 00:36:09.230 --> 00:36:11.510 align:middle line:84% if you start thinking about the rays as particles, 00:36:11.510 --> 00:36:13.030 align:middle line:84% and you think about their velocity, 00:36:13.030 --> 00:36:14.780 align:middle line:84% and you think about the speed that they're 00:36:14.780 --> 00:36:16.600 align:middle line:84% going at inside the plasma, and you 00:36:16.600 --> 00:36:18.350 align:middle line:84% realize that they're actually going slower 00:36:18.350 --> 00:36:20.000 align:middle line:84% in the denser regions, and that's going 00:36:20.000 --> 00:36:21.893 align:middle line:90% to start giving you a twist. 00:36:21.893 --> 00:36:23.810 align:middle line:84% So they're going faster in the denser regions. 00:36:23.810 --> 00:36:26.990 align:middle line:84% They're going slower in the regions with high refractive 00:36:26.990 --> 00:36:27.530 align:middle line:90% index. 00:36:27.530 --> 00:36:29.322 align:middle line:84% And that's what gives you the bending here. 00:36:29.322 --> 00:36:31.345 align:middle line:84% So this is just like a little mental model 00:36:31.345 --> 00:36:32.720 align:middle line:84% to think about when you're trying 00:36:32.720 --> 00:36:35.240 align:middle line:84% to work out why it is that the rays of light are turning. 00:36:35.240 --> 00:36:38.570 align:middle line:84% But there's many, many different ways to get this. 00:36:38.570 --> 00:36:41.000 align:middle line:84% STUDENT: Is there a great density over all 00:36:41.000 --> 00:36:44.070 align:middle line:90% you've got use this? 00:36:44.070 --> 00:36:49.900 align:middle line:84% Hi, is there a gradient and density along the z-axis? 00:36:49.900 --> 00:36:53.940 align:middle line:84% The way it's drawn it looks like it's only within the y-axis. 00:36:53.940 --> 00:36:55.830 align:middle line:84% JACK HARE: It isn't only in the y-axis. 00:36:55.830 --> 00:36:57.520 align:middle line:90% Yes. 00:36:57.520 --> 00:37:03.170 align:middle line:84% STUDENT: So an integral of the gradient of density along z 00:37:03.170 --> 00:37:08.940 align:middle line:84% from that constant, then, was it-- 00:37:08.940 --> 00:37:12.055 align:middle line:84% is there no-- is there a density change in z, si what I'm asking. 00:37:12.055 --> 00:37:14.430 align:middle line:84% JACK HARE: Not in this really simple model I'm proposing. 00:37:14.430 --> 00:37:17.610 align:middle line:84% Of course, in general, there can be a density change. 00:37:17.610 --> 00:37:27.460 align:middle line:84% Really, this should read gradient of density dot dl, 00:37:27.460 --> 00:37:30.880 align:middle line:90% where L is an infinitesimal. 00:37:30.880 --> 00:37:31.450 align:middle line:90% No. 00:37:31.450 --> 00:37:31.720 align:middle line:90% Sorry. 00:37:31.720 --> 00:37:32.275 align:middle line:90% Ignore that. 00:37:32.275 --> 00:37:36.620 align:middle line:90% 00:37:36.620 --> 00:37:38.962 align:middle line:84% Yeah, there is no gradient in density in z. 00:37:38.962 --> 00:37:40.670 align:middle line:84% And we don't need one to get any bending. 00:37:40.670 --> 00:37:42.878 align:middle line:84% And in fact, if there was a gradient of density in z, 00:37:42.878 --> 00:37:46.340 align:middle line:84% it wouldn't have any effect on the light. 00:37:46.340 --> 00:37:49.100 align:middle line:84% It would just go forwards at a different speed, 00:37:49.100 --> 00:37:52.188 align:middle line:90% but it wouldn't get bent. 00:37:52.188 --> 00:37:52.980 align:middle line:90% STUDENT: All right. 00:37:52.980 --> 00:37:58.945 align:middle line:84% So then that integral is just gradient of ne times L. 00:37:58.945 --> 00:37:59.570 align:middle line:90% JACK HARE: Yes. 00:37:59.570 --> 00:38:01.940 align:middle line:84% For this very simple model, you're absolutely right. 00:38:01.940 --> 00:38:03.560 align:middle line:84% I'm just introducing the generality 00:38:03.560 --> 00:38:04.910 align:middle line:84% because we may have something different. 00:38:04.910 --> 00:38:06.020 align:middle line:90% But you're quite right. 00:38:06.020 --> 00:38:10.100 align:middle line:84% We could write this as minus 2 ncr times L. 00:38:10.100 --> 00:38:11.960 align:middle line:84% And maybe I'll put a subscript z so 00:38:11.960 --> 00:38:14.630 align:middle line:84% that I know that it's my length scale z and times 00:38:14.630 --> 00:38:16.820 align:middle line:90% by the gradient in ne. 00:38:16.820 --> 00:38:22.150 align:middle line:84% And if this is some simple density ramp, 00:38:22.150 --> 00:38:28.300 align:middle line:84% so I would have any 0 times 1 minus x upon Ly, 00:38:28.300 --> 00:38:30.430 align:middle line:84% or something like that, I can simply 00:38:30.430 --> 00:38:33.640 align:middle line:84% put this in and say that the entire beam is now 00:38:33.640 --> 00:38:39.340 align:middle line:84% twisted by a nice linear angle, which has an Lz inside it, 00:38:39.340 --> 00:38:49.650 align:middle line:84% an ne0, and minus 1 upon Ly, like this. 00:38:49.650 --> 00:38:52.532 align:middle line:84% So this, is if I give you some analytical result, 00:38:52.532 --> 00:38:54.740 align:middle line:84% you can then go and work out what the angle would be. 00:38:54.740 --> 00:38:57.950 align:middle line:84% And that's a super useful thing to be able to do. 00:38:57.950 --> 00:39:00.710 align:middle line:84% As I segue perfectly into my next remark, 00:39:00.710 --> 00:39:03.780 align:middle line:84% which is to do with the first problem that this causes. 00:39:03.780 --> 00:39:05.420 align:middle line:84% So this is issues with deflection. 00:39:05.420 --> 00:39:10.170 align:middle line:90% 00:39:10.170 --> 00:39:13.950 align:middle line:84% The first issue is if you've got some plasma 00:39:13.950 --> 00:39:18.780 align:middle line:84% and you're trying to put some electromagnetic radiation in it, 00:39:18.780 --> 00:39:20.910 align:middle line:84% you want to collect that radiation. 00:39:20.910 --> 00:39:22.560 align:middle line:84% You want to put it onto a detector. 00:39:22.560 --> 00:39:24.390 align:middle line:84% Maybe that detector is a camera, or it 00:39:24.390 --> 00:39:27.970 align:middle line:84% might be a waveguide that you're collecting microwaves with. 00:39:27.970 --> 00:39:31.140 align:middle line:84% And so that camera has some physical size. 00:39:31.140 --> 00:39:34.740 align:middle line:84% And so maybe the camera is represented by this lens here. 00:39:34.740 --> 00:39:37.860 align:middle line:90% It's got some physical size D. 00:39:37.860 --> 00:39:42.230 align:middle line:90% And if your rays get deflected-- 00:39:42.230 --> 00:39:45.590 align:middle line:84% that was a terrible straight line. 00:39:45.590 --> 00:39:47.170 align:middle line:90% OK. 00:39:47.170 --> 00:39:50.680 align:middle line:84% If your rays get deflected by an angle greater than theta max, 00:39:50.680 --> 00:39:59.980 align:middle line:84% where tangent of theta max is equal to d over 2 times L-- 00:39:59.980 --> 00:40:01.700 align:middle line:90% I forgot to put in this L here. 00:40:01.700 --> 00:40:03.010 align:middle line:90% There we go. 00:40:03.010 --> 00:40:05.120 align:middle line:84% Then your ray is going to be lost. 00:40:05.120 --> 00:40:13.428 align:middle line:84% So for theta greater than theta max, you lose your rays. 00:40:13.428 --> 00:40:14.970 align:middle line:84% So that means you can't collect them. 00:40:14.970 --> 00:40:16.420 align:middle line:90% You can't detect them anymore. 00:40:16.420 --> 00:40:18.360 align:middle line:84% So this causes big problems because it 00:40:18.360 --> 00:40:20.580 align:middle line:84% means that we're going to start losing light here. 00:40:20.580 --> 00:40:24.870 align:middle line:84% And for most situations, we can use the small angle paraxial 00:40:24.870 --> 00:40:30.130 align:middle line:84% approximation and just replace the tan theta with theta here. 00:40:30.130 --> 00:40:34.500 align:middle line:84% So that means you want to keep your deflection angle 00:40:34.500 --> 00:40:38.400 align:middle line:84% theta, which is equal to, as we said, 00:40:38.400 --> 00:40:41.730 align:middle line:90% ddy of the integral of Ndl. 00:40:41.730 --> 00:40:44.760 align:middle line:84% That wants to be less than theta max. 00:40:44.760 --> 00:40:48.120 align:middle line:84% So there's a few things that you can do to try and do this. 00:40:48.120 --> 00:40:49.425 align:middle line:90% You can have a nice big lens. 00:40:49.425 --> 00:40:53.417 align:middle line:90% 00:40:53.417 --> 00:40:54.500 align:middle line:90% You can have a close lens. 00:40:54.500 --> 00:40:57.000 align:middle line:84% You can put it nice and close in. 00:40:57.000 --> 00:41:01.250 align:middle line:84% Or you can use a shorter wavelength. 00:41:01.250 --> 00:41:03.320 align:middle line:84% Because if you go to a shorter wavelength, 00:41:03.320 --> 00:41:06.120 align:middle line:84% you get a smaller deflection angle, 00:41:06.120 --> 00:41:12.860 align:middle line:84% which you can see if you go back to maybe this formula here. 00:41:12.860 --> 00:41:16.050 align:middle line:84% A shorter wavelength corresponds to a larger n critical, 00:41:16.050 --> 00:41:18.080 align:middle line:84% and so you'll get a smaller angle. 00:41:18.080 --> 00:41:19.790 align:middle line:84% Now, not all of these things are possible 00:41:19.790 --> 00:41:20.832 align:middle line:90% in a standard experiment. 00:41:20.832 --> 00:41:22.670 align:middle line:84% If you've got a tokamak, you may have 00:41:22.670 --> 00:41:24.980 align:middle line:84% a limit on how big your detector can be, 00:41:24.980 --> 00:41:27.315 align:middle line:84% because it's got to fit in a gap between some magnets. 00:41:27.315 --> 00:41:29.690 align:middle line:84% You'll certainly have a limit on how close you can put it 00:41:29.690 --> 00:41:32.810 align:middle line:84% to the plasma because you don't want to stick it right inside. 00:41:32.810 --> 00:41:34.370 align:middle line:84% And you may not be able to choose 00:41:34.370 --> 00:41:35.578 align:middle line:90% whatever wavelength you want. 00:41:35.578 --> 00:41:37.815 align:middle line:84% Perhaps you're looking at electron cyclotron emission 00:41:37.815 --> 00:41:40.190 align:middle line:84% and you've got no choice but to use the wavelength that's 00:41:40.190 --> 00:41:40.910 align:middle line:90% emitted at. 00:41:40.910 --> 00:41:42.990 align:middle line:90% So this can cause big problems. 00:41:42.990 --> 00:41:45.407 align:middle line:84% And so if you're doing some electromagnetic probing 00:41:45.407 --> 00:41:47.990 align:middle line:84% of your plasma, one of the first things you should probably do 00:41:47.990 --> 00:41:49.573 align:middle line:84% is check whether the density gradients 00:41:49.573 --> 00:41:52.610 align:middle line:84% are going to make it hard to actually measure anything. 00:41:52.610 --> 00:41:53.840 align:middle line:90% Any questions on this? 00:41:53.840 --> 00:42:05.845 align:middle line:90% 00:42:05.845 --> 00:42:07.470 align:middle line:84% All right, so we're now going to plunge 00:42:07.470 --> 00:42:10.605 align:middle line:90% in to our first diagnostic. 00:42:10.605 --> 00:42:11.980 align:middle line:84% And the point I want to make here 00:42:11.980 --> 00:42:14.590 align:middle line:84% is although deflection can be frustrating, 00:42:14.590 --> 00:42:17.870 align:middle line:90% it can also be useful. 00:42:17.870 --> 00:42:20.555 align:middle line:84% Because we can use it to measure something about the plasma. 00:42:20.555 --> 00:42:24.160 align:middle line:90% 00:42:24.160 --> 00:42:26.785 align:middle line:84% The first thing we're going to talk about is schlieren imaging. 00:42:26.785 --> 00:42:32.830 align:middle line:90% 00:42:32.830 --> 00:42:37.670 align:middle line:84% This word, schlieren, people often assume refers to a person. 00:42:37.670 --> 00:42:39.110 align:middle line:90% It does not. 00:42:39.110 --> 00:42:40.990 align:middle line:84% So it doesn't have a capital, despite what 00:42:40.990 --> 00:42:42.430 align:middle line:90% Overleaf will tell you. 00:42:42.430 --> 00:42:44.950 align:middle line:84% And it's actually after a German word, schlierer, 00:42:44.950 --> 00:42:47.290 align:middle line:84% which is like streaks, because this was first 00:42:47.290 --> 00:42:50.467 align:middle line:84% used for looking at small imperfections in optics. 00:42:50.467 --> 00:42:52.300 align:middle line:84% And so looking at these little streaks here. 00:42:52.300 --> 00:42:55.300 align:middle line:84% So it's a way of imaging things which would otherwise 00:42:55.300 --> 00:42:57.550 align:middle line:84% be impossible to see because they 00:42:57.550 --> 00:43:00.680 align:middle line:84% cause small gradients in refractive index. 00:43:00.680 --> 00:43:03.670 align:middle line:84% So let's have a little example, building up 00:43:03.670 --> 00:43:05.147 align:middle line:90% towards schlieren imaging. 00:43:05.147 --> 00:43:06.730 align:middle line:84% This first thing I'm going to show you 00:43:06.730 --> 00:43:07.940 align:middle line:90% is not schlieren imaging. 00:43:07.940 --> 00:43:09.160 align:middle line:90% This is just imaging. 00:43:09.160 --> 00:43:12.040 align:middle line:84% But my impression is that some folks need 00:43:12.040 --> 00:43:13.507 align:middle line:90% a refresher with some optics. 00:43:13.507 --> 00:43:15.340 align:middle line:84% So we're going to start with a solid object. 00:43:15.340 --> 00:43:19.250 align:middle line:84% It's going to be this nice little chalice here. 00:43:19.250 --> 00:43:22.600 align:middle line:90% 00:43:22.600 --> 00:43:25.060 align:middle line:84% And we're going to put in some rays of light. 00:43:25.060 --> 00:43:27.590 align:middle line:90% 00:43:27.590 --> 00:43:28.580 align:middle line:90% Like this. 00:43:28.580 --> 00:43:30.800 align:middle line:84% Now, this object is solid, and so 00:43:30.800 --> 00:43:33.050 align:middle line:84% it blocks any rays of light which hit it, 00:43:33.050 --> 00:43:36.590 align:middle line:84% these two centers ones, and allows through rays of light 00:43:36.590 --> 00:43:39.910 align:middle line:90% going past it. 00:43:39.910 --> 00:43:41.740 align:middle line:84% Allows through rays of light going past it. 00:43:41.740 --> 00:43:42.250 align:middle line:90% Very good. 00:43:42.250 --> 00:43:46.140 align:middle line:90% 00:43:46.140 --> 00:43:48.793 align:middle line:84% And what we would probably do here 00:43:48.793 --> 00:43:50.460 align:middle line:84% if we're doing a standard imaging system 00:43:50.460 --> 00:43:52.200 align:middle line:90% is we would have a lens. 00:43:52.200 --> 00:43:53.950 align:middle line:84% So this is how you form an image. 00:43:53.950 --> 00:43:55.800 align:middle line:90% So we'll put our lens here. 00:43:55.800 --> 00:43:57.630 align:middle line:84% It's going to have a focal length F. 00:43:57.630 --> 00:44:00.340 align:middle line:84% And we're going to place it at a distance, 00:44:00.340 --> 00:44:06.610 align:middle line:84% which is 2F away from the object we're trying to image. 00:44:06.610 --> 00:44:09.890 align:middle line:90% I'll just put that F up there. 00:44:09.890 --> 00:44:13.430 align:middle line:84% Now, behind this lens, if we're doing a standard 2F imaging 00:44:13.430 --> 00:44:15.800 align:middle line:84% system, we're going to have a focal point, 00:44:15.800 --> 00:44:17.660 align:middle line:84% and that's going to be at F away. 00:44:17.660 --> 00:44:20.180 align:middle line:84% And then, we're going to have an object 00:44:20.180 --> 00:44:24.913 align:middle line:90% plane, which is also at F away. 00:44:24.913 --> 00:44:25.830 align:middle line:90% Sorry, an image plane. 00:44:25.830 --> 00:44:29.478 align:middle line:90% 00:44:29.478 --> 00:44:30.520 align:middle line:90% This is the object plane. 00:44:30.520 --> 00:44:35.050 align:middle line:90% 00:44:35.050 --> 00:44:37.650 align:middle line:84% And this is the lens with focal length F. 00:44:37.650 --> 00:44:40.380 align:middle line:84% So hopefully, some of you have seen this sort of thing before. 00:44:40.380 --> 00:44:45.960 align:middle line:84% You know that the rays will pass through the focal point here. 00:44:45.960 --> 00:44:48.510 align:middle line:90% He says, drawing them carefully. 00:44:48.510 --> 00:44:53.668 align:middle line:90% 00:44:53.668 --> 00:44:54.835 align:middle line:90% And what we'll end up with-- 00:44:54.835 --> 00:45:00.910 align:middle line:90% 00:45:00.910 --> 00:45:04.820 align:middle line:90% can't do this on a chalkboard-- 00:45:04.820 --> 00:45:09.260 align:middle line:84% is a copy of our image, of our object here. 00:45:09.260 --> 00:45:10.552 align:middle line:90% But it's going to be inverted. 00:45:10.552 --> 00:45:12.260 align:middle line:84% And you can tell that because you can see 00:45:12.260 --> 00:45:13.920 align:middle line:90% the rays have changed place. 00:45:13.920 --> 00:45:18.250 align:middle line:90% So this is a nice 1 to 1 image. 00:45:18.250 --> 00:45:25.880 align:middle line:84% It's at magnification 1, and it is inverted. 00:45:25.880 --> 00:45:27.700 align:middle line:90% So this is the simplest-- 00:45:27.700 --> 00:45:29.740 align:middle line:84% I think, the simplest possible imaging system 00:45:29.740 --> 00:45:31.390 align:middle line:90% you could possibly develop. 00:45:31.390 --> 00:45:34.600 align:middle line:84% It simply takes whatever is at the object plane 00:45:34.600 --> 00:45:37.615 align:middle line:84% and puts it at the image plane some distance away. 00:45:37.615 --> 00:45:38.740 align:middle line:90% This could be a microscope. 00:45:38.740 --> 00:45:40.130 align:middle line:90% This could be a camera. 00:45:40.130 --> 00:45:41.380 align:middle line:90% All sorts of things like that. 00:45:41.380 --> 00:45:44.152 align:middle line:90% Yes, Vincent? 00:45:44.152 --> 00:45:45.360 align:middle line:90% STUDENT: I think I missed it. 00:45:45.360 --> 00:45:46.632 align:middle line:90% What was F again? 00:45:46.632 --> 00:45:47.840 align:middle line:90% JACK HARE: I beg your pardon. 00:45:47.840 --> 00:45:50.000 align:middle line:84% STUDENT: What was F, like, in the diagram? 00:45:50.000 --> 00:45:51.687 align:middle line:84% JACK HARE: The focal length of the lens. 00:45:51.687 --> 00:45:52.880 align:middle line:90% STUDENT: Oh. 00:45:52.880 --> 00:45:54.103 align:middle line:90% Thank you. 00:45:54.103 --> 00:45:54.770 align:middle line:90% JACK HARE: Cool. 00:45:54.770 --> 00:45:55.940 align:middle line:90% Any other questions? 00:45:55.940 --> 00:46:01.388 align:middle line:90% 00:46:01.388 --> 00:46:02.930 align:middle line:84% OK, let's make this more interesting. 00:46:02.930 --> 00:46:04.595 align:middle line:90% Let's put a plasma here instead. 00:46:04.595 --> 00:46:09.320 align:middle line:90% 00:46:09.320 --> 00:46:11.240 align:middle line:84% And we're still going to have our lens. 00:46:11.240 --> 00:46:14.240 align:middle line:90% 00:46:14.240 --> 00:46:16.770 align:middle line:84% There still can be a focal point. 00:46:16.770 --> 00:46:19.950 align:middle line:84% And there's still going to be an image plane here. 00:46:19.950 --> 00:46:24.540 align:middle line:84% But the plasma doesn't block the rays of light, 00:46:24.540 --> 00:46:28.050 align:middle line:84% as long as we've got, for example, ne much, much less 00:46:28.050 --> 00:46:30.840 align:middle line:84% than the critical density here so 00:46:30.840 --> 00:46:32.760 align:middle line:84% that the rays can pass through easily. 00:46:32.760 --> 00:46:35.460 align:middle line:84% Instead, what we're going to have is rays that come in. 00:46:35.460 --> 00:46:41.590 align:middle line:90% 00:46:41.590 --> 00:46:43.590 align:middle line:84% And then, they're going to be deflected slightly 00:46:43.590 --> 00:46:44.623 align:middle line:90% inside this plasma. 00:46:44.623 --> 00:46:46.290 align:middle line:84% So I haven't drawn the density gradient. 00:46:46.290 --> 00:46:48.957 align:middle line:84% We can imagine, we've just got a whole range of exciting density 00:46:48.957 --> 00:46:50.560 align:middle line:84% gradients that cause some deflections. 00:46:50.560 --> 00:46:53.790 align:middle line:84% So they deflect this ray slightly downwards. 00:46:53.790 --> 00:46:57.540 align:middle line:84% They deflect this ray slightly upwards. 00:46:57.540 --> 00:47:00.692 align:middle line:90% They deflect this bottom ray-- 00:47:00.692 --> 00:47:01.900 align:middle line:90% what have I done to this one? 00:47:01.900 --> 00:47:03.492 align:middle line:90% Let's have this one go straight. 00:47:03.492 --> 00:47:05.950 align:middle line:84% For some reason, there's no density gradient exactly there, 00:47:05.950 --> 00:47:07.540 align:middle line:90% so the ray just goes through. 00:47:07.540 --> 00:47:11.810 align:middle line:84% And this bottom one gets deflected downwards as well. 00:47:11.810 --> 00:47:17.710 align:middle line:84% And let's say it just about makes it onto the lens. 00:47:17.710 --> 00:47:21.320 align:middle line:84% And I'll move the lens downwards to make that true. 00:47:21.320 --> 00:47:23.450 align:middle line:84% Can't do that on a chalkboard either. 00:47:23.450 --> 00:47:24.890 align:middle line:90% OK, good. 00:47:24.890 --> 00:47:28.400 align:middle line:84% So what will the lens do to these rays now? 00:47:28.400 --> 00:47:45.640 align:middle line:90% 00:47:45.640 --> 00:47:47.470 align:middle line:84% Well, it's still going to reflect the rays, 00:47:47.470 --> 00:47:49.647 align:middle line:90% and it's going to reflect them-- 00:47:49.647 --> 00:47:51.730 align:middle line:84% let's start with this one that actually didn't get 00:47:51.730 --> 00:47:52.640 align:middle line:90% deflected at all. 00:47:52.640 --> 00:47:54.110 align:middle line:90% So its angle hasn't changed. 00:47:54.110 --> 00:47:56.440 align:middle line:84% It's going to go straight through the focal point, 00:47:56.440 --> 00:47:57.760 align:middle line:90% as you'd expect. 00:47:57.760 --> 00:48:00.310 align:middle line:84% This one that was deflected upwards 00:48:00.310 --> 00:48:02.380 align:middle line:84% is going to be deflected down, but it's 00:48:02.380 --> 00:48:04.370 align:middle line:84% going to slightly miss the focal point. 00:48:04.370 --> 00:48:07.000 align:middle line:84% It's going to be slightly above it, like that. 00:48:07.000 --> 00:48:08.947 align:middle line:84% This one that was deflected downwards 00:48:08.947 --> 00:48:10.280 align:middle line:90% is going to be the opposite way. 00:48:10.280 --> 00:48:12.280 align:middle line:84% It's going to be slightly below the focal point. 00:48:12.280 --> 00:48:14.680 align:middle line:84% And this one was deflected downwards. 00:48:14.680 --> 00:48:18.220 align:middle line:84% It's also going to be slightly below the focal point. 00:48:18.220 --> 00:48:20.410 align:middle line:90% And that was a mistake. 00:48:20.410 --> 00:48:20.995 align:middle line:90% There we go. 00:48:20.995 --> 00:48:25.580 align:middle line:90% 00:48:25.580 --> 00:48:26.290 align:middle line:90% There we go. 00:48:26.290 --> 00:48:28.150 align:middle line:84% It's not a very good image compared 00:48:28.150 --> 00:48:30.760 align:middle line:84% to the one I was hoping to draw, but there we go. 00:48:30.760 --> 00:48:31.870 align:middle line:90% Nothing quite works out. 00:48:31.870 --> 00:48:35.170 align:middle line:84% So we should have an image of our plasma here. 00:48:35.170 --> 00:48:39.900 align:middle line:84% This image of the plasma should still be 1 to 1 00:48:39.900 --> 00:48:42.930 align:middle line:90% mag 1 and inverted. 00:48:42.930 --> 00:48:46.338 align:middle line:90% 00:48:46.338 --> 00:48:48.630 align:middle line:84% The fact that I haven't quite managed to get it to work 00:48:48.630 --> 00:48:50.047 align:middle line:84% is probably just a flaw with how I 00:48:50.047 --> 00:48:52.870 align:middle line:84% managed to draw the rays this time round. 00:48:52.870 --> 00:48:55.060 align:middle line:84% Not quite sure what went wrong there. 00:48:55.060 --> 00:49:01.150 align:middle line:90% 00:49:01.150 --> 00:49:03.160 align:middle line:90% Looks good on my notes, anyway. 00:49:03.160 --> 00:49:04.850 align:middle line:84% This stuff gets a little bit tricky. 00:49:04.850 --> 00:49:06.520 align:middle line:84% The point is, although the rays here 00:49:06.520 --> 00:49:08.560 align:middle line:84% look like they've all gone upwards slightly, 00:49:08.560 --> 00:49:11.890 align:middle line:84% they should actually still end up in the same places 00:49:11.890 --> 00:49:13.318 align:middle line:90% that they did before. 00:49:13.318 --> 00:49:15.610 align:middle line:84% And the fact that I can't get it to work right now just 00:49:15.610 --> 00:49:18.040 align:middle line:84% means that I've made a mistake while drawing it live. 00:49:18.040 --> 00:49:19.645 align:middle line:90% OK, good. 00:49:19.645 --> 00:49:21.520 align:middle line:84% STUDENT: Jack, this is a very basic question, 00:49:21.520 --> 00:49:23.650 align:middle line:84% but what's the point of having all the rays go 00:49:23.650 --> 00:49:26.530 align:middle line:84% through the trouble of going through the lens when we could 00:49:26.530 --> 00:49:29.380 align:middle line:84% just have them go straight through and hit our image plane? 00:49:29.380 --> 00:49:31.210 align:middle line:90% Guess I missed that. 00:49:31.210 --> 00:49:32.800 align:middle line:90% You know what I mean? 00:49:32.800 --> 00:49:34.540 align:middle line:90% JACK HARE: Yeah, absolutely. 00:49:34.540 --> 00:49:39.843 align:middle line:84% So in the top case, the solid object, the rays could go-- 00:49:39.843 --> 00:49:42.260 align:middle line:84% if the rays went straight through and hit the image plane, 00:49:42.260 --> 00:49:44.220 align:middle line:84% they will be deflected slightly at the edges. 00:49:44.220 --> 00:49:45.970 align:middle line:84% And so you'll end up with something fuzzy, 00:49:45.970 --> 00:49:48.160 align:middle line:90% so it'll be out of focus. 00:49:48.160 --> 00:49:50.500 align:middle line:84% So you need a lens to bring it to focus, 00:49:50.500 --> 00:49:52.630 align:middle line:84% which effectively is mapping the rays 00:49:52.630 --> 00:49:56.950 align:middle line:84% from where they came from back to the same place on the object 00:49:56.950 --> 00:49:57.490 align:middle line:90% plane. 00:49:57.490 --> 00:50:00.288 align:middle line:84% If in the case of the plasma, if you don't have the lens 00:50:00.288 --> 00:50:02.080 align:middle line:84% and you just put-- if you don't have a lens 00:50:02.080 --> 00:50:04.085 align:middle line:84% and you just let the rays propagate to a screen, 00:50:04.085 --> 00:50:05.710 align:middle line:84% that's a technique called shadowgraphy, 00:50:05.710 --> 00:50:08.200 align:middle line:84% which we'll talk about next lecture, which I actually think 00:50:08.200 --> 00:50:10.940 align:middle line:84% is more difficult even though it's simpler to draw. 00:50:10.940 --> 00:50:13.970 align:middle line:84% And so I want to talk about it after this one. 00:50:13.970 --> 00:50:18.030 align:middle line:84% So we haven't done anything here at the moment. 00:50:18.030 --> 00:50:19.980 align:middle line:84% And in fact, if you do this with a plasma, 00:50:19.980 --> 00:50:22.260 align:middle line:90% you won't see anything at all. 00:50:22.260 --> 00:50:24.360 align:middle line:84% Because all of the rays are mapped back 00:50:24.360 --> 00:50:25.870 align:middle line:90% to where they started from. 00:50:25.870 --> 00:50:28.860 align:middle line:84% And that means that you are going 00:50:28.860 --> 00:50:32.417 align:middle line:84% to end up with just the same laser beam that you originally 00:50:32.417 --> 00:50:34.500 align:middle line:84% started with or the same microwaves you originally 00:50:34.500 --> 00:50:35.110 align:middle line:90% started with. 00:50:35.110 --> 00:50:37.080 align:middle line:90% So this will be invisible. 00:50:37.080 --> 00:50:39.300 align:middle line:84% So the only way we can make this visible 00:50:39.300 --> 00:50:41.910 align:middle line:84% is to notice that the rays do not all 00:50:41.910 --> 00:50:43.545 align:middle line:90% pass through the focal point. 00:50:43.545 --> 00:50:54.410 align:middle line:90% 00:50:54.410 --> 00:50:57.200 align:middle line:84% Now, you saw in the case where we did the imaging 00:50:57.200 --> 00:50:59.255 align:middle line:84% that all the rays did, indeed, still 00:50:59.255 --> 00:51:00.380 align:middle line:90% pass through a focal point. 00:51:00.380 --> 00:51:02.450 align:middle line:84% But here, some of them have gone above and some 00:51:02.450 --> 00:51:03.380 align:middle line:90% of them gone below. 00:51:03.380 --> 00:51:05.270 align:middle line:84% And in fact, the distance they've gone above 00:51:05.270 --> 00:51:06.728 align:middle line:84% and the distance they've gone below 00:51:06.728 --> 00:51:10.220 align:middle line:84% is directly proportional to the angle with which they 00:51:10.220 --> 00:51:12.270 align:middle line:90% exited the plasma here. 00:51:12.270 --> 00:51:14.270 align:middle line:84% And so we can learn something about the angle 00:51:14.270 --> 00:51:16.310 align:middle line:84% they can select exited the plasma 00:51:16.310 --> 00:51:19.220 align:middle line:84% by placing a filter at this focal plane. 00:51:19.220 --> 00:51:25.454 align:middle line:84% And this filter maybe looks a little bit like this. 00:51:25.454 --> 00:51:29.680 align:middle line:84% This filter, for example, here is like a little aperture. 00:51:29.680 --> 00:51:32.830 align:middle line:84% And it lets through these two rays. 00:51:32.830 --> 00:51:34.150 align:middle line:90% Let me color code them. 00:51:34.150 --> 00:51:36.670 align:middle line:90% This one and this one. 00:51:36.670 --> 00:51:40.750 align:middle line:84% And it blocks off these two rays. 00:51:40.750 --> 00:51:43.000 align:middle line:84% And so light is coming from that bit of the plasma, 00:51:43.000 --> 00:51:45.410 align:middle line:84% where the density gradients were large will be blocked 00:51:45.410 --> 00:51:48.230 align:middle line:84% and it will no longer appear on our final image. 00:51:48.230 --> 00:51:50.260 align:middle line:84% And this is what Schlieren imaging is. 00:51:50.260 --> 00:52:04.085 align:middle line:84% So we place a stop at the focal plane and we filter by angle. 00:52:04.085 --> 00:52:06.870 align:middle line:90% 00:52:06.870 --> 00:52:09.690 align:middle line:84% I'm going to do some exhaustive examples of this 00:52:09.690 --> 00:52:11.490 align:middle line:84% to try and build some intuition for what's 00:52:11.490 --> 00:52:13.943 align:middle line:84% going on if you're a little bit confused right now. 00:52:13.943 --> 00:52:15.735 align:middle line:84% Any questions on this before we keep going? 00:52:15.735 --> 00:52:27.610 align:middle line:90% 00:52:27.610 --> 00:52:30.640 align:middle line:84% STUDENT: I have sort of an overall conceptual question. 00:52:30.640 --> 00:52:33.820 align:middle line:84% I feel like the highest gradients, a lot of the time, 00:52:33.820 --> 00:52:34.533 align:middle line:90% or-- 00:52:34.533 --> 00:52:35.950 align:middle line:84% well, maybe this isn't quite true. 00:52:35.950 --> 00:52:37.810 align:middle line:84% But I can imagine, in a lot of cases, 00:52:37.810 --> 00:52:40.420 align:middle line:84% this is going to be affected most by the edges where 00:52:40.420 --> 00:52:43.700 align:middle line:84% it's entering and leaving the plasma because you're-- 00:52:43.700 --> 00:52:45.680 align:middle line:84% yeah, depending on how uniform things are. 00:52:45.680 --> 00:52:48.673 align:middle line:84% So just curious what you actually get an image of. 00:52:48.673 --> 00:52:50.090 align:middle line:84% I mean, if you-- especially if you 00:52:50.090 --> 00:52:54.440 align:middle line:84% don't have a great sense of where the gradients are 00:52:54.440 --> 00:52:56.450 align:middle line:84% or if you have gradients inside you 00:52:56.450 --> 00:52:58.500 align:middle line:90% don't know about or something. 00:52:58.500 --> 00:52:59.910 align:middle line:90% JACK HARE: Yeah, absolutely. 00:52:59.910 --> 00:53:01.750 align:middle line:84% These images are difficult to interpret. 00:53:01.750 --> 00:53:04.020 align:middle line:84% So this is not a generic diagnostic technique 00:53:04.020 --> 00:53:06.300 align:middle line:84% that will immediately tell you what's going on. 00:53:06.300 --> 00:53:08.430 align:middle line:84% You need to know something about your plasma. 00:53:08.430 --> 00:53:11.030 align:middle line:84% Maybe you have a simulation and you do a synthetic diagnostic 00:53:11.030 --> 00:53:11.530 align:middle line:90% on it. 00:53:11.530 --> 00:53:13.540 align:middle line:84% Or maybe you've set up your experiment such 00:53:13.540 --> 00:53:14.790 align:middle line:90% that it's particularly simple. 00:53:14.790 --> 00:53:17.100 align:middle line:84% We'll talk a little bit about some simple distributions 00:53:17.100 --> 00:53:18.600 align:middle line:84% and what patterns they make and that 00:53:18.600 --> 00:53:20.550 align:middle line:84% will give us an idea for what sorts of things 00:53:20.550 --> 00:53:24.100 align:middle line:84% we might be able to measure with this. 00:53:24.100 --> 00:53:26.100 align:middle line:84% Turbulence in the edge of a tokamak or something 00:53:26.100 --> 00:53:29.475 align:middle line:84% like that, this is maybe not the ideal diagnostic for it. 00:53:29.475 --> 00:53:30.350 align:middle line:90% STUDENT: Fair enough. 00:53:30.350 --> 00:53:31.800 align:middle line:90% JACK HARE: Yeah. 00:53:31.800 --> 00:53:33.840 align:middle line:84% I want to make very clear because I 00:53:33.840 --> 00:53:36.720 align:middle line:84% don't know if it came across when we 00:53:36.720 --> 00:53:37.960 align:middle line:90% were talking about it before. 00:53:37.960 --> 00:53:41.470 align:middle line:84% But we only get deflections from density gradients, 00:53:41.470 --> 00:53:44.050 align:middle line:84% which are perpendicular to the direction here. 00:53:44.050 --> 00:53:48.000 align:middle line:84% So if our ray is going in this direction, in the z direction, 00:53:48.000 --> 00:53:56.530 align:middle line:84% we sense ddy of ne and d dx sub ne, 00:53:56.530 --> 00:54:03.710 align:middle line:90% but we do not sense ddz of any. 00:54:03.710 --> 00:54:06.230 align:middle line:84% So if there's density gradients in the direction the ray is 00:54:06.230 --> 00:54:09.950 align:middle line:84% propagating, the ray will slow down or speed up, 00:54:09.950 --> 00:54:12.212 align:middle line:84% but it won't actually deflect from that. 00:54:12.212 --> 00:54:13.670 align:middle line:84% And so that helps you a little bit. 00:54:13.670 --> 00:54:15.800 align:middle line:84% You're only sensitive to gradients perpendicular 00:54:15.800 --> 00:54:16.910 align:middle line:90% to the probing direction. 00:54:16.910 --> 00:54:19.313 align:middle line:84% That might also tell you if you've got a plasma, which 00:54:19.313 --> 00:54:20.480 align:middle line:90% you think has some geometry. 00:54:20.480 --> 00:54:22.580 align:middle line:84% There may be a good direction to send the probing 00:54:22.580 --> 00:54:24.510 align:middle line:84% beam through there maybe a bad direction. 00:54:24.510 --> 00:54:26.385 align:middle line:84% So you want to think about that a little bit. 00:54:26.385 --> 00:54:28.890 align:middle line:90% 00:54:28.890 --> 00:54:32.120 align:middle line:84% So let's have a talk about some of these stops, right? 00:54:32.120 --> 00:54:34.910 align:middle line:84% So I've said that we can place these stops here. 00:54:34.910 --> 00:54:37.130 align:middle line:84% Let's have a chat about what sort of different stops 00:54:37.130 --> 00:54:40.580 align:middle line:90% are available to us. 00:54:40.580 --> 00:54:43.450 align:middle line:90% So types of stop. 00:54:43.450 --> 00:54:46.220 align:middle line:90% 00:54:46.220 --> 00:54:49.340 align:middle line:84% The first type is to decide whether our stop is going to be 00:54:49.340 --> 00:54:52.555 align:middle line:90% dark field or light field. 00:54:52.555 --> 00:54:53.930 align:middle line:84% So I'm putting darken in brackets 00:54:53.930 --> 00:54:56.800 align:middle line:84% because it will save me writing in a moment. 00:54:56.800 --> 00:54:59.110 align:middle line:84% The difference between dark field and light field 00:54:59.110 --> 00:55:06.010 align:middle line:84% is that the dark field blocks undeflected rays. 00:55:06.010 --> 00:55:09.190 align:middle line:84% So ones that do pass through the focal point. 00:55:09.190 --> 00:55:11.290 align:middle line:84% And the light field blocks deflected 00:55:11.290 --> 00:55:14.360 align:middle line:84% rays, ones which do not pass through the focal point. 00:55:14.360 --> 00:55:16.690 align:middle line:84% So you can either look for regions 00:55:16.690 --> 00:55:18.160 align:middle line:84% where there are density gradients 00:55:18.160 --> 00:55:20.320 align:middle line:84% or where there aren't density gradients. 00:55:20.320 --> 00:55:22.810 align:middle line:84% We can also choose the shape of our stop 00:55:22.810 --> 00:55:26.680 align:middle line:84% because our stop is a two dimensional 00:55:26.680 --> 00:55:28.190 align:middle line:90% plane at the focal plane here. 00:55:28.190 --> 00:55:32.230 align:middle line:90% So we can have a circular stop. 00:55:32.230 --> 00:55:35.140 align:middle line:84% And that doesn't care what direction 00:55:35.140 --> 00:55:38.330 align:middle line:84% the ray is deflected in, it only cares on the size of the angle. 00:55:38.330 --> 00:55:39.890 align:middle line:90% So the size of theta. 00:55:39.890 --> 00:55:44.890 align:middle line:84% So that is basically, are there any large density gradients? 00:55:44.890 --> 00:55:48.770 align:middle line:84% Or we can have what's called a knife edge, which 00:55:48.770 --> 00:55:55.300 align:middle line:90% is linear like this. 00:55:55.300 --> 00:55:59.710 align:middle line:84% And that is sensitive to density gradients in only one direction 00:55:59.710 --> 00:56:03.160 align:middle line:84% and it still cares about the size of the density gradient. 00:56:03.160 --> 00:56:07.850 align:middle line:90% That is like x hat here. 00:56:07.850 --> 00:56:15.320 align:middle line:84% So we could, for example, have a stop at the focal plane. 00:56:15.320 --> 00:56:16.900 align:middle line:90% No, it's not going to do it. 00:56:16.900 --> 00:56:21.890 align:middle line:90% 00:56:21.890 --> 00:56:22.700 align:middle line:90% OK, fine. 00:56:22.700 --> 00:56:24.530 align:middle line:84% I can't get a nice, round circle. 00:56:24.530 --> 00:56:29.900 align:middle line:84% We can have a stop which is an opening inside an opaque sheet 00:56:29.900 --> 00:56:31.670 align:middle line:90% of material here. 00:56:31.670 --> 00:56:34.640 align:middle line:84% And this opening could be positioned such 00:56:34.640 --> 00:56:39.690 align:middle line:84% that the focal spot in the absence of any plasma 00:56:39.690 --> 00:56:41.160 align:middle line:90% sits inside it. 00:56:41.160 --> 00:56:42.885 align:middle line:90% What sort of stop would this be? 00:56:42.885 --> 00:56:50.070 align:middle line:90% 00:56:50.070 --> 00:56:51.980 align:middle line:90% STUDENT: Light field? 00:56:51.980 --> 00:56:54.760 align:middle line:84% JACK HARE: So this is a light field stop. 00:56:54.760 --> 00:56:55.960 align:middle line:90% And what shape is it? 00:56:55.960 --> 00:56:58.670 align:middle line:90% 00:56:58.670 --> 00:57:00.085 align:middle line:90% [INTERPOSING VOICES] 00:57:00.085 --> 00:57:01.270 align:middle line:90% Yes, OK, circle. 00:57:01.270 --> 00:57:01.870 align:middle line:90% Thank you. 00:57:01.870 --> 00:57:05.360 align:middle line:90% 00:57:05.360 --> 00:57:10.835 align:middle line:84% We could also have a stop that looks like this. 00:57:10.835 --> 00:57:14.010 align:middle line:90% 00:57:14.010 --> 00:57:17.330 align:middle line:84% And we can position it such that the focal spot is actually 00:57:17.330 --> 00:57:19.760 align:middle line:90% within the opaque region. 00:57:19.760 --> 00:57:21.320 align:middle line:84% And what sort of stop would this be? 00:57:21.320 --> 00:57:32.040 align:middle line:90% 00:57:32.040 --> 00:57:32.925 align:middle line:90% STUDENT: Dark field. 00:57:32.925 --> 00:57:34.230 align:middle line:90% STUDENT: Dark field. 00:57:34.230 --> 00:57:35.610 align:middle line:90% JACK HARE: OK, dark field. 00:57:35.610 --> 00:57:39.040 align:middle line:90% And what shape is it? 00:57:39.040 --> 00:57:40.543 align:middle line:90% STUDENT: Linear. 00:57:40.543 --> 00:57:41.960 align:middle line:84% JACK HARE: So the knife edge here. 00:57:41.960 --> 00:57:43.680 align:middle line:90% Yeah. 00:57:43.680 --> 00:57:46.180 align:middle line:84% We call it a knife edge because actually using a razor blade 00:57:46.180 --> 00:57:48.730 align:middle line:84% is a pretty good thing to have because you get a very nice, 00:57:48.730 --> 00:57:51.070 align:middle line:90% sharp, uniform edge to it. 00:57:51.070 --> 00:57:54.310 align:middle line:84% OK, and so depending on these stops 00:57:54.310 --> 00:57:56.470 align:middle line:84% you can think of as filters in angle space, right? 00:57:56.470 --> 00:57:58.060 align:middle line:84% So they allow through certain angles. 00:57:58.060 --> 00:58:00.950 align:middle line:84% You can think of arbitrarily complicated versions of this. 00:58:00.950 --> 00:58:04.990 align:middle line:84% There's a technique called angular-- 00:58:04.990 --> 00:58:06.670 align:middle line:90% not fringe. 00:58:06.670 --> 00:58:08.530 align:middle line:84% Angular filter refractometry, which 00:58:08.530 --> 00:58:12.970 align:middle line:84% has a set of nested annuli, which 00:58:12.970 --> 00:58:16.690 align:middle line:84% let through light which has been deflected 00:58:16.690 --> 00:58:19.240 align:middle line:90% by certain specific angles. 00:58:19.240 --> 00:58:20.573 align:middle line:90% So the world is your oyster. 00:58:20.573 --> 00:58:22.990 align:middle line:84% You can come up with all sorts of exciting different stops 00:58:22.990 --> 00:58:24.190 align:middle line:90% if you want to. 00:58:24.190 --> 00:58:27.220 align:middle line:84% One thing I will note is that the dynamic range 00:58:27.220 --> 00:58:30.880 align:middle line:84% of your diagnostic, which we'll talk about more later, 00:58:30.880 --> 00:58:35.870 align:middle line:84% depends a great deal on your focal spot size. 00:58:35.870 --> 00:58:40.880 align:middle line:84% So I've shown these focal spots to be relatively small here. 00:58:40.880 --> 00:58:42.920 align:middle line:84% But that focal spot size, at least 00:58:42.920 --> 00:58:44.840 align:middle line:84% in a diffraction limited sense, it 00:58:44.840 --> 00:58:47.450 align:middle line:84% covers an angle, which is equal to the wavelength 00:58:47.450 --> 00:58:51.380 align:middle line:84% of your light over the size of your lens, the diameter 00:58:51.380 --> 00:58:52.560 align:middle line:90% of your lens here. 00:58:52.560 --> 00:58:56.030 align:middle line:84% And so you might end up having focal spots which 00:58:56.030 --> 00:59:00.180 align:middle line:84% are not small, but actually could be rather large. 00:59:00.180 --> 00:59:03.320 align:middle line:84% And then, part of the focal spot could be obscured 00:59:03.320 --> 00:59:06.880 align:middle line:84% and part of the focal spot could be clear for some given 00:59:06.880 --> 00:59:09.600 align:middle line:90% deflection angle. 00:59:09.600 --> 00:59:12.610 align:middle line:84% And in general, when we've got a plasma here, 00:59:12.610 --> 00:59:15.370 align:middle line:84% we have, say, our small focal spot 00:59:15.370 --> 00:59:18.132 align:middle line:84% before we put any plasma in the way. 00:59:18.132 --> 00:59:20.590 align:middle line:84% When the plasma is gone in the way, different rays of light 00:59:20.590 --> 00:59:22.298 align:middle line:84% have been deflected by different amounts. 00:59:22.298 --> 00:59:27.577 align:middle line:84% So this thing may take on some complicated shape here. 00:59:27.577 --> 00:59:29.410 align:middle line:84% And this is the shape that you're filtering. 00:59:29.410 --> 00:59:33.670 align:middle line:84% You might be filtering it with your knife edge like this, 00:59:33.670 --> 00:59:37.440 align:middle line:84% or you might be filtering it with your circular stop 00:59:37.440 --> 00:59:38.050 align:middle line:90% like this. 00:59:38.050 --> 00:59:39.630 align:middle line:84% So we're basically filtering the rays 00:59:39.630 --> 00:59:41.940 align:middle line:84% based on how far they've been deflected. 00:59:41.940 --> 00:59:43.770 align:middle line:84% At the focal plane, there's no information 00:59:43.770 --> 00:59:46.080 align:middle line:84% about where the rays came from inside their plasma. 00:59:46.080 --> 00:59:48.340 align:middle line:84% So their spatial information has been lost. 00:59:48.340 --> 00:59:52.210 align:middle line:84% The only thing we know about them is their angular position. 00:59:52.210 --> 00:59:54.600 align:middle line:84% So rays, which are deflected by an angle theta 1 00:59:54.600 --> 00:59:56.190 align:middle line:84% from at the top of the plasma, are 00:59:56.190 --> 00:59:58.200 align:middle line:84% rays which are deflected by the same angle 00:59:58.200 --> 00:59:59.670 align:middle line:90% from the bottom of plasma. 00:59:59.670 --> 01:00:02.725 align:middle line:84% It ends up at the same place in the focal plane, 01:00:02.725 --> 01:00:05.100 align:middle line:84% even though they came from different parts of the plasma. 01:00:05.100 --> 01:00:08.860 align:middle line:84% This is the magic of geometric optics. 01:00:08.860 --> 01:00:12.510 align:middle line:84% So any questions on this or should we do a little example? 01:00:12.510 --> 01:00:18.990 align:middle line:90% 01:00:18.990 --> 01:00:21.680 align:middle line:90% OK, let's do our example. 01:00:21.680 --> 01:00:27.070 align:middle line:84% So let us consider a very simple plasma. 01:00:27.070 --> 01:00:34.300 align:middle line:84% This plasma-- we'll have the coordinate system y vertically 01:00:34.300 --> 01:00:37.300 align:middle line:84% and we'll have z in the direction of propagation 01:00:37.300 --> 01:00:39.370 align:middle line:90% as we discussed before. 01:00:39.370 --> 01:00:41.445 align:middle line:90% We'll have rays. 01:00:41.445 --> 01:00:42.820 align:middle line:84% Well, I'll draw the plasma first. 01:00:42.820 --> 01:00:46.690 align:middle line:84% So the plasma is going to have a density distribution that 01:00:46.690 --> 01:00:51.385 align:middle line:84% sort of looks Gaussian ish, some sort of nice peaked function. 01:00:51.385 --> 01:00:54.180 align:middle line:90% 01:00:54.180 --> 01:00:56.840 align:middle line:90% So this is density, ne. 01:00:56.840 --> 01:01:00.510 align:middle line:84% So you can think about this, for example, as like a cylinder. 01:01:00.510 --> 01:01:02.660 align:middle line:84% So you've got a cylinder of plasma, like a z pinch, 01:01:02.660 --> 01:01:04.970 align:middle line:84% and you're probing down the axis of this z pinch 01:01:04.970 --> 01:01:08.780 align:middle line:84% and it's got a Gaussian distribution of density to it. 01:01:08.780 --> 01:01:10.650 align:middle line:90% Anything like that. 01:01:10.650 --> 01:01:13.340 align:middle line:84% And then, we'll have the rays of light coming through. 01:01:13.340 --> 01:01:15.890 align:middle line:84% So we'll have rays of light, which 01:01:15.890 --> 01:01:19.910 align:middle line:84% are sampling very small density gradients at the edges here. 01:01:19.910 --> 01:01:23.600 align:middle line:84% And these rays e will just go straight through. 01:01:23.600 --> 01:01:26.930 align:middle line:90% 01:01:26.930 --> 01:01:29.650 align:middle line:84% There's also be a ray that goes through the center, which 01:01:29.650 --> 01:01:32.062 align:middle line:84% also sees a very small density gradient here, right? 01:01:32.062 --> 01:01:33.520 align:middle line:84% At the center of this distribution, 01:01:33.520 --> 01:01:35.620 align:middle line:90% the density gradient is zero. 01:01:35.620 --> 01:01:39.280 align:middle line:84% But the edges here where the density gradient is large 01:01:39.280 --> 01:01:40.750 align:middle line:90% will have some deflection. 01:01:40.750 --> 01:01:49.020 align:middle line:90% 01:01:49.020 --> 01:01:54.900 align:middle line:84% And if we place our lens, as we did before, some distance away, 01:01:54.900 --> 01:02:01.180 align:middle line:84% it's going to focus those rays onto our focal plane. 01:02:01.180 --> 01:02:05.260 align:middle line:84% And in our focal plane, we're going to put some sort of stop 01:02:05.260 --> 01:02:05.960 align:middle line:90% here. 01:02:05.960 --> 01:02:13.250 align:middle line:84% So the undeflected rays are just going to go straight 01:02:13.250 --> 01:02:14.270 align:middle line:90% through the focal point. 01:02:14.270 --> 01:02:29.320 align:middle line:90% 01:02:29.320 --> 01:02:30.970 align:middle line:84% But the deflected rays are not going 01:02:30.970 --> 01:02:32.830 align:middle line:90% to go through the focal point. 01:02:32.830 --> 01:02:38.020 align:middle line:84% The deflected rays are going to go above and below. 01:02:38.020 --> 01:02:47.170 align:middle line:90% 01:02:47.170 --> 01:02:49.940 align:middle line:84% So now we can put a series of stops inside here. 01:02:49.940 --> 01:02:56.210 align:middle line:84% So we can have a stop on that blue dashed line that looks 01:02:56.210 --> 01:03:04.400 align:middle line:90% like a light field knife edge. 01:03:04.400 --> 01:03:15.810 align:middle line:84% We can have a stop that looks like dark field knife edge. 01:03:15.810 --> 01:03:25.980 align:middle line:84% We can have a stop that looks like a light field circle. 01:03:25.980 --> 01:03:35.120 align:middle line:84% And we could have a stop that looks like a dark field circle. 01:03:35.120 --> 01:03:37.460 align:middle line:84% And what we're going to do is sketch out what 01:03:37.460 --> 01:03:41.390 align:middle line:84% we expect the intensity to look like from each 01:03:41.390 --> 01:03:47.030 align:middle line:84% of these different knife edges here. 01:03:47.030 --> 01:03:53.510 align:middle line:90% So if I plot this one-- 01:03:53.510 --> 01:03:58.410 align:middle line:90% 01:03:58.410 --> 01:04:03.920 align:middle line:84% I'll just draw the density distribution again like that. 01:04:03.920 --> 01:04:05.360 align:middle line:90% So that's our density. 01:04:05.360 --> 01:04:09.160 align:middle line:84% Now we want to know what our intensity distribution looks 01:04:09.160 --> 01:04:09.660 align:middle line:90% like. 01:04:09.660 --> 01:04:12.900 align:middle line:90% So this is now intensity. 01:04:12.900 --> 01:04:15.660 align:middle line:84% We have our initial intensity 1 and we 01:04:15.660 --> 01:04:19.630 align:middle line:84% have 0 intensity corresponding to all the rays being blocked. 01:04:19.630 --> 01:04:22.470 align:middle line:84% So where for the light field knife edge 01:04:22.470 --> 01:04:25.110 align:middle line:90% do we see no intensity? 01:04:25.110 --> 01:04:34.988 align:middle line:90% 01:04:34.988 --> 01:04:37.530 align:middle line:84% STUDENT: In the upper side where there is the largest density 01:04:37.530 --> 01:04:37.702 align:middle line:90% gradient. 01:04:37.702 --> 01:04:39.150 align:middle line:84% JACK HARE: I'm sorry, I didn't hear that properly. 01:04:39.150 --> 01:04:40.025 align:middle line:90% Can you say it again? 01:04:40.025 --> 01:04:42.330 align:middle line:84% STUDENT: I think the upper portion 01:04:42.330 --> 01:04:45.570 align:middle line:84% where there is the largest density gradient. 01:04:45.570 --> 01:04:48.450 align:middle line:84% JACK HARE: So I think what you said was in the upper side 01:04:48.450 --> 01:04:50.550 align:middle line:84% where there's the largest density gradient, right? 01:04:50.550 --> 01:04:53.560 align:middle line:84% So this is where the density gradient is very large. 01:04:53.560 --> 01:04:56.130 align:middle line:84% So we expect outside of that region 01:04:56.130 --> 01:04:58.560 align:middle line:84% our intensity would be pretty much constant. 01:04:58.560 --> 01:05:00.840 align:middle line:84% But inside that region, we'd expect 01:05:00.840 --> 01:05:02.760 align:middle line:90% the intensity would drop to 0. 01:05:02.760 --> 01:05:05.670 align:middle line:84% Because we're blocking the rays which have a large deflection 01:05:05.670 --> 01:05:09.540 align:middle line:84% angle in one direction upwards and the rays which 01:05:09.540 --> 01:05:11.970 align:middle line:84% have a large deflection angle in one direction upwards 01:05:11.970 --> 01:05:15.630 align:middle line:84% corresponds to that specific density gradient there. 01:05:15.630 --> 01:05:20.860 align:middle line:90% 01:05:20.860 --> 01:05:24.470 align:middle line:84% OK, anyone want to have a go at telling me 01:05:24.470 --> 01:05:26.165 align:middle line:90% what happens with this one? 01:05:26.165 --> 01:05:39.240 align:middle line:90% 01:05:39.240 --> 01:05:40.620 align:middle line:90% What's happening at the edges? 01:05:40.620 --> 01:05:43.920 align:middle line:84% What's happening out in these regions 01:05:43.920 --> 01:05:45.960 align:middle line:84% here where the density gradients are small? 01:05:45.960 --> 01:05:56.700 align:middle line:90% 01:05:56.700 --> 01:05:59.420 align:middle line:84% STUDENT: They're cropped out because they mostly 01:05:59.420 --> 01:06:02.462 align:middle line:90% go through the focal spot. 01:06:02.462 --> 01:06:03.170 align:middle line:90% JACK HARE: Right. 01:06:03.170 --> 01:06:05.810 align:middle line:84% Yeah, so these are going to be 0, right? 01:06:05.810 --> 01:06:12.290 align:middle line:84% So 0, 0, is there any region where it's not zero? 01:06:12.290 --> 01:06:17.870 align:middle line:84% STUDENT: I think it's not 0 for the high gradient 01:06:17.870 --> 01:06:20.660 align:middle line:90% region that's lower in y. 01:06:20.660 --> 01:06:21.860 align:middle line:90% Because it's deflected. 01:06:21.860 --> 01:06:25.100 align:middle line:84% JACK HARE: So you think it's not 0 for this region here? 01:06:25.100 --> 01:06:26.067 align:middle line:90% STUDENT: Yes. 01:06:26.067 --> 01:06:27.650 align:middle line:84% JACK HARE: Does anyone agree with Sara 01:06:27.650 --> 01:06:28.650 align:middle line:90% or does anyone disagree? 01:06:28.650 --> 01:06:31.003 align:middle line:90% 01:06:31.003 --> 01:06:32.420 align:middle line:84% STUDENT: I think that looks right. 01:06:32.420 --> 01:06:35.080 align:middle line:90% 01:06:35.080 --> 01:06:39.335 align:middle line:84% JACK HARE: OK, so we're talking about this ray here. 01:06:39.335 --> 01:06:39.835 align:middle line:90% Yeah. 01:06:39.835 --> 01:06:46.730 align:middle line:90% 01:06:46.730 --> 01:06:49.880 align:middle line:84% So this ray is, indeed, passing low down 01:06:49.880 --> 01:06:51.170 align:middle line:90% compared to the knife edge. 01:06:51.170 --> 01:06:55.360 align:middle line:84% So we'd expect this to show some intensity here. 01:06:55.360 --> 01:06:59.480 align:middle line:84% But not for the upper one, which passes higher up. 01:06:59.480 --> 01:07:00.980 align:middle line:90% OK, good. 01:07:00.980 --> 01:07:01.970 align:middle line:90% We're getting there. 01:07:01.970 --> 01:07:07.753 align:middle line:84% What about for the circular light field? 01:07:07.753 --> 01:07:10.170 align:middle line:84% Anyone want to tell me what the intensity looks like here? 01:07:10.170 --> 01:07:12.860 align:middle line:90% 01:07:12.860 --> 01:07:14.700 align:middle line:84% STUDENT: You'd probably get three peaks. 01:07:14.700 --> 01:07:18.530 align:middle line:84% So on the edges where the density isn't really-- 01:07:18.530 --> 01:07:20.113 align:middle line:84% where the density is just low and then 01:07:20.113 --> 01:07:22.030 align:middle line:84% right in the middle where the density is high, 01:07:22.030 --> 01:07:23.683 align:middle line:84% but not really changing too much. 01:07:23.683 --> 01:07:24.350 align:middle line:90% JACK HARE: Yeah. 01:07:24.350 --> 01:07:26.132 align:middle line:84% So you say three peaks, I'm going 01:07:26.132 --> 01:07:27.590 align:middle line:84% to think about them as two notches, 01:07:27.590 --> 01:07:29.640 align:middle line:84% but I agree with what you're saying. 01:07:29.640 --> 01:07:33.260 align:middle line:84% So these notches here correspond to the points 01:07:33.260 --> 01:07:36.660 align:middle line:84% where the density gradient is largest here. 01:07:36.660 --> 01:07:39.620 align:middle line:84% And so those rays got a big deflection angle there being 01:07:39.620 --> 01:07:40.550 align:middle line:90% blocked out. 01:07:40.550 --> 01:07:43.940 align:middle line:84% What about for the final one here, 01:07:43.940 --> 01:07:45.831 align:middle line:90% dark field circular aperture? 01:07:45.831 --> 01:07:53.540 align:middle line:90% 01:07:53.540 --> 01:07:56.600 align:middle line:84% STUDENT: Well, there you're basically notching out 01:07:56.600 --> 01:08:01.070 align:middle line:84% all the minimally deflected rays, 01:08:01.070 --> 01:08:04.055 align:middle line:84% so you lose the ones on the edges and center. 01:08:04.055 --> 01:08:05.895 align:middle line:90% 01:08:05.895 --> 01:08:07.520 align:middle line:84% JACK HARE: Yeah, so we'd have something 01:08:07.520 --> 01:08:09.380 align:middle line:90% that looks like this, right? 01:08:09.380 --> 01:08:11.840 align:middle line:84% So we would just see light where there was 01:08:11.840 --> 01:08:13.295 align:middle line:90% a significant deflection angle. 01:08:13.295 --> 01:08:16.222 align:middle line:90% 01:08:16.222 --> 01:08:19.149 align:middle line:84% OK, this may still seem a little bit abstract, 01:08:19.149 --> 01:08:21.635 align:middle line:84% so we're going to try one more thing. 01:08:21.635 --> 01:08:23.010 align:middle line:84% And I hope that you don't hate me 01:08:23.010 --> 01:08:24.635 align:middle line:84% for spending so much time on Schlieren, 01:08:24.635 --> 01:08:27.640 align:middle line:84% but I absolutely love it, so that's your loss. 01:08:27.640 --> 01:08:30.029 align:middle line:84% Which is going to be a two dimensional example here. 01:08:30.029 --> 01:08:32.970 align:middle line:84% I'll just draw on this distribution function. 01:08:32.970 --> 01:08:35.760 align:middle line:84% There we go, density function now on that last one 01:08:35.760 --> 01:08:37.050 align:middle line:90% just so you've got it. 01:08:37.050 --> 01:08:40.100 align:middle line:90% 01:08:40.100 --> 01:08:44.439 align:middle line:84% So this actually goes back to a sketch 01:08:44.439 --> 01:08:46.510 align:middle line:84% that I did during the b dot lecture 01:08:46.510 --> 01:08:50.810 align:middle line:84% where we stuck a little b dot inside the plasma. 01:08:50.810 --> 01:08:54.430 align:middle line:84% And we have plasma flow coming from left to right. 01:08:54.430 --> 01:08:56.800 align:middle line:84% And because it collides with this, 01:08:56.800 --> 01:09:00.819 align:middle line:84% we get some sort of bow shock like this. 01:09:00.819 --> 01:09:03.340 align:middle line:90% 01:09:03.340 --> 01:09:07.000 align:middle line:84% And as we all know from our shop physics, at the bow shock, 01:09:07.000 --> 01:09:13.840 align:middle line:84% we have strong density gradients like this where the density 01:09:13.840 --> 01:09:17.560 align:middle line:90% jumps across the shock here. 01:09:17.560 --> 01:09:21.870 align:middle line:84% So if we look at this system and we have our probing laser coming 01:09:21.870 --> 01:09:26.370 align:middle line:84% towards us, so our laser is looking towards us like this, 01:09:26.370 --> 01:09:27.490 align:middle line:90% we are the camera. 01:09:27.490 --> 01:09:30.000 align:middle line:84% What would we see about this bow shock? 01:09:30.000 --> 01:09:31.359 align:middle line:90% What could we tell about it? 01:09:31.359 --> 01:09:34.380 align:middle line:84% So maybe the first thing we could do 01:09:34.380 --> 01:09:38.580 align:middle line:84% is ask for a light field circular aperture, what 01:09:38.580 --> 01:09:40.290 align:middle line:90% do we think we would see here? 01:09:40.290 --> 01:09:43.939 align:middle line:90% 01:09:43.939 --> 01:09:46.960 align:middle line:84% And if you're not quite sure, at each of these places where I've 01:09:46.960 --> 01:09:48.760 align:middle line:84% drawn this little arrow, you could 01:09:48.760 --> 01:09:54.680 align:middle line:84% say that the density maybe as a simple model 01:09:54.680 --> 01:09:58.640 align:middle line:84% looks like a sort of hyperbolic tangent type thing like that. 01:09:58.640 --> 01:10:02.630 align:middle line:84% It's got some region where the density is ne0 some region where 01:10:02.630 --> 01:10:06.530 align:middle line:84% the density is ne1, and then some region where 01:10:06.530 --> 01:10:07.910 align:middle line:90% the density changes rapidly. 01:10:07.910 --> 01:10:10.292 align:middle line:84% That's not true for a shock, but it's 01:10:10.292 --> 01:10:12.750 align:middle line:84% a model just to get us thinking about what this looks like. 01:10:12.750 --> 01:10:14.300 align:middle line:90% So what would I see on my image? 01:10:14.300 --> 01:10:17.660 align:middle line:84% I've got this expanded laser beam going through this shock 01:10:17.660 --> 01:10:23.380 align:middle line:84% and I've decided to use a circular light field stop. 01:10:23.380 --> 01:10:26.030 align:middle line:84% STUDENT: I might be doing this backwards, but from-- 01:10:26.030 --> 01:10:28.580 align:middle line:84% it looks like what we drew above. 01:10:28.580 --> 01:10:31.430 align:middle line:84% That would mean that you're not getting your steep gradient 01:10:31.430 --> 01:10:35.338 align:middle line:84% sections, so it should be dark where the bow shock is. 01:10:35.338 --> 01:10:37.130 align:middle line:84% JACK HARE: So you'd actually have an image. 01:10:37.130 --> 01:10:40.042 align:middle line:84% Yeah, you'd have an image, which if I was on a chalkboard, 01:10:40.042 --> 01:10:42.500 align:middle line:84% I could do as an eraser, but it's actually quite hard here. 01:10:42.500 --> 01:10:46.160 align:middle line:84% If you imagine this is filled, a green laser 01:10:46.160 --> 01:10:48.740 align:middle line:84% beam beautiful nice laser beam image. 01:10:48.740 --> 01:10:52.280 align:middle line:84% Then you would have a region where it-- no, the arrays is not 01:10:52.280 --> 01:10:52.880 align:middle line:90% very good. 01:10:52.880 --> 01:10:55.250 align:middle line:84% If I can make the eraser smaller that would work much better 01:10:55.250 --> 01:10:57.708 align:middle line:84% you would have a region where there was no light whatsoever 01:10:57.708 --> 01:10:59.330 align:middle line:84% and there was just darkness, right? 01:10:59.330 --> 01:11:01.790 align:middle line:84% So you just have this dark region here 01:11:01.790 --> 01:11:03.300 align:middle line:90% corresponding to the bow shock. 01:11:03.300 --> 01:11:06.830 align:middle line:84% And if you did this with a dark field circular aperture, 01:11:06.830 --> 01:11:09.170 align:middle line:90% then you'd have the opposite. 01:11:09.170 --> 01:11:12.380 align:middle line:84% You'd have complete darkness and you would just 01:11:12.380 --> 01:11:15.710 align:middle line:84% have a region where the bow shock shows up 01:11:15.710 --> 01:11:18.325 align:middle line:90% very nicely like this. 01:11:18.325 --> 01:11:19.700 align:middle line:84% And so this is actually typically 01:11:19.700 --> 01:11:23.030 align:middle line:84% what we use for shock measurements is dark field 01:11:23.030 --> 01:11:23.990 align:middle line:90% circular aperture. 01:11:23.990 --> 01:11:27.860 align:middle line:90% 01:11:27.860 --> 01:11:31.730 align:middle line:84% If you do happen to do something like dark field 01:11:31.730 --> 01:11:38.300 align:middle line:84% with a knife edge and you put your knife edge like this, 01:11:38.300 --> 01:11:43.510 align:middle line:84% so you were measuring density gradients that 01:11:43.510 --> 01:11:47.770 align:middle line:84% were in this direction, you would end up 01:11:47.770 --> 01:11:51.580 align:middle line:84% seeing something a little bit like just one half 01:11:51.580 --> 01:11:53.570 align:middle line:90% of the bow shock like that. 01:11:53.570 --> 01:11:56.408 align:middle line:84% So if you set-- your probe was sitting here, 01:11:56.408 --> 01:11:58.450 align:middle line:84% you would just see a little bit of the bow shock. 01:11:58.450 --> 01:12:01.748 align:middle line:84% You wouldn't see this section because the density gradients 01:12:01.748 --> 01:12:03.290 align:middle line:84% wouldn't be in the correct direction. 01:12:03.290 --> 01:12:05.350 align:middle line:84% So you wouldn't be able to observe those. 01:12:05.350 --> 01:12:07.840 align:middle line:84% But the knife edge might be a better choice for some shock 01:12:07.840 --> 01:12:10.420 align:middle line:84% geometries if you're only interested in gradients 01:12:10.420 --> 01:12:12.100 align:middle line:90% in one direction. 01:12:12.100 --> 01:12:14.880 align:middle line:90% Aidan, I see your hand. 01:12:14.880 --> 01:12:22.610 align:middle line:84% STUDENT: Yeah, I'm curious how distinct the actual image 01:12:22.610 --> 01:12:26.340 align:middle line:84% gradients would be given that this 01:12:26.340 --> 01:12:30.160 align:middle line:84% is like a cylindrical phenomenon, right? 01:12:30.160 --> 01:12:32.770 align:middle line:90% The shock. 01:12:32.770 --> 01:12:36.210 align:middle line:84% So I assume there's some density gradients that's 01:12:36.210 --> 01:12:41.880 align:middle line:84% they slowly become parallel to your propagation direction 01:12:41.880 --> 01:12:43.033 align:middle line:90% as you rotate to the-- 01:12:43.033 --> 01:12:45.450 align:middle line:84% JACK HARE: Yeah, so you won't see those density gradients, 01:12:45.450 --> 01:12:47.617 align:middle line:84% but you will see the ones on the edges of the shock. 01:12:47.617 --> 01:12:50.190 align:middle line:84% Yeah, it depends on the exact shock morphology, 01:12:50.190 --> 01:12:52.770 align:middle line:84% whether it's some extended like a bow shock that's 01:12:52.770 --> 01:12:54.540 align:middle line:84% extended like this or whether it's 01:12:54.540 --> 01:12:57.810 align:middle line:84% a bow shock that's sort of rotated like the tip of a nose 01:12:57.810 --> 01:13:00.480 align:middle line:84% cone on an aircraft or something like that. 01:13:00.480 --> 01:13:01.780 align:middle line:90% So it will make a difference. 01:13:01.780 --> 01:13:04.197 align:middle line:84% But this is just to try and get a feel for what this looks 01:13:04.197 --> 01:13:06.460 align:middle line:90% like in terms of imaging here. 01:13:06.460 --> 01:13:09.100 align:middle line:84% I can send around some papers later, 01:13:09.100 --> 01:13:12.490 align:middle line:84% which have some very nice images of Schlieren of shock structures 01:13:12.490 --> 01:13:14.587 align:middle line:84% that show what sort of quality of data 01:13:14.587 --> 01:13:15.670 align:middle line:90% you can get out from this. 01:13:15.670 --> 01:13:20.200 align:middle line:90% 01:13:20.200 --> 01:13:22.440 align:middle line:84% So any other questions on Schlieren? 01:13:22.440 --> 01:13:25.710 align:middle line:84% I will then just summarize exactly how to use it 01:13:25.710 --> 01:13:28.263 align:middle line:84% and when not to use it and things like that. 01:13:28.263 --> 01:13:30.930 align:middle line:84% But if anyone has any questions on these sort of worked examples 01:13:30.930 --> 01:13:33.180 align:middle line:84% we've done, please go ahead and shout out. 01:13:33.180 --> 01:13:49.060 align:middle line:90% 01:13:49.060 --> 01:13:56.590 align:middle line:84% So Schlieren is good for visualizing density gradients, 01:13:56.590 --> 01:13:57.220 align:middle line:90% right? 01:13:57.220 --> 01:14:00.520 align:middle line:84% And in particular, it's good for visualizing strong density 01:14:00.520 --> 01:14:01.150 align:middle line:90% gradients. 01:14:01.150 --> 01:14:04.240 align:middle line:84% So in particular, it's good at visualizing shocks. 01:14:04.240 --> 01:14:06.460 align:middle line:84% So if you're looking at shocks in plasmas, 01:14:06.460 --> 01:14:07.775 align:middle line:90% this is a nice diagnostic. 01:14:07.775 --> 01:14:09.400 align:middle line:84% And it's particularly nice because it's 01:14:09.400 --> 01:14:12.010 align:middle line:90% very simple to set up. 01:14:12.010 --> 01:14:16.510 align:middle line:84% As I showed you, it's just got at minimum a single optic. 01:14:16.510 --> 01:14:19.510 align:middle line:84% You need to form a focal point, so you need to have an optic. 01:14:19.510 --> 01:14:21.522 align:middle line:84% You need a focal point and you need a stop. 01:14:21.522 --> 01:14:23.230 align:middle line:84% So this is a very simple thing to set up. 01:14:23.230 --> 01:14:24.700 align:middle line:84% You can do it very, very quickly. 01:14:24.700 --> 01:14:26.900 align:middle line:84% The trouble is, although it's simple, 01:14:26.900 --> 01:14:29.710 align:middle line:84% it's very difficult to be quantitative. 01:14:29.710 --> 01:14:32.770 align:middle line:90% 01:14:32.770 --> 01:14:36.520 align:middle line:84% We can say there is some sort of density gradient there. 01:14:36.520 --> 01:14:38.080 align:middle line:84% We think the density gradient has 01:14:38.080 --> 01:14:40.960 align:middle line:84% to be larger than some certain limiting value. 01:14:40.960 --> 01:14:43.940 align:middle line:84% But other than that, we can't really say much more than that. 01:14:43.940 --> 01:14:46.840 align:middle line:84% So it's useful for seeing where shocks are and their morphology, 01:14:46.840 --> 01:14:50.440 align:middle line:84% but is not useful for measuring gradient of any. 01:14:50.440 --> 01:14:56.970 align:middle line:90% So we can't measure gradient Ne. 01:14:56.970 --> 01:15:05.030 align:middle line:84% We can measure shape and location 01:15:05.030 --> 01:15:08.990 align:middle line:90% of the density gradients. 01:15:08.990 --> 01:15:10.910 align:middle line:84% Now, there are some ways where you can 01:15:10.910 --> 01:15:12.720 align:middle line:90% make this more quantitative. 01:15:12.720 --> 01:15:18.390 align:middle line:84% So imagine you've got a little beam of light coming in 01:15:18.390 --> 01:15:19.080 align:middle line:90% like this. 01:15:19.080 --> 01:15:21.180 align:middle line:90% This is going in this direction. 01:15:21.180 --> 01:15:25.320 align:middle line:84% And you have a knife edge that looks like this. 01:15:25.320 --> 01:15:28.650 align:middle line:84% That knife edge is going to be entirely blocking 01:15:28.650 --> 01:15:31.390 align:middle line:84% what's coming through, we'll have 0 light coming through. 01:15:31.390 --> 01:15:33.600 align:middle line:84% But if we have a beam that comes through 01:15:33.600 --> 01:15:37.720 align:middle line:84% and it's lined up like this, you can 01:15:37.720 --> 01:15:40.880 align:middle line:84% see that now about half the light is going to get through. 01:15:40.880 --> 01:15:42.700 align:middle line:84% And if we have a beam coming through that's 01:15:42.700 --> 01:15:45.220 align:middle line:84% lined up above the knife edge, we'll 01:15:45.220 --> 01:15:47.690 align:middle line:84% have all of the light coming through. 01:15:47.690 --> 01:15:51.400 align:middle line:84% And it turns out that if you have a large enough spot 01:15:51.400 --> 01:15:55.360 align:middle line:84% that you can actually see this partial obscuration 01:15:55.360 --> 01:15:58.510 align:middle line:84% of the focal point, you end up in a regime where 01:15:58.510 --> 01:16:01.120 align:middle line:84% the intensity that you see on your detector eye 01:16:01.120 --> 01:16:03.820 align:middle line:84% is, in fact, proportional directly 01:16:03.820 --> 01:16:06.790 align:middle line:84% to the gradient of the electron density. 01:16:06.790 --> 01:16:08.980 align:middle line:84% I'm putting it in brackets because you need 01:16:08.980 --> 01:16:15.155 align:middle line:90% a large uniform focal spot. 01:16:15.155 --> 01:16:17.280 align:middle line:84% And I'll talk in a moment about why that's actually 01:16:17.280 --> 01:16:19.950 align:middle line:84% extremely difficult using a laser, which 01:16:19.950 --> 01:16:23.210 align:middle line:90% is what we normally use here. 01:16:23.210 --> 01:16:27.170 align:middle line:84% OK, if you can't guarantee that you have a nice large uniform 01:16:27.170 --> 01:16:33.550 align:middle line:84% focal spot, you could also use a graded neutral density filter. 01:16:33.550 --> 01:16:37.420 align:middle line:84% So a neutral density filter is sort of smoky glass 01:16:37.420 --> 01:16:38.740 align:middle line:90% that blocks light. 01:16:38.740 --> 01:16:41.020 align:middle line:84% And you can change the amount of impurities in it 01:16:41.020 --> 01:16:42.070 align:middle line:90% to block more light. 01:16:42.070 --> 01:16:45.040 align:middle line:84% So we could carefully fabricate for ourselves 01:16:45.040 --> 01:16:50.050 align:middle line:84% a neutral density filter that has a changing absorption. 01:16:50.050 --> 01:16:52.260 align:middle line:84% So we can have a gradient in, say, we'll 01:16:52.260 --> 01:16:54.370 align:middle line:90% call the absorption alpha here. 01:16:54.370 --> 01:16:56.910 align:middle line:84% And then, the idea is that different rays 01:16:56.910 --> 01:16:58.800 align:middle line:84% of light at different points of your stop 01:16:58.800 --> 01:17:07.370 align:middle line:84% will either come through not at all or very attenuated. 01:17:07.370 --> 01:17:13.060 align:middle line:84% Or they will come through partially attenuated 01:17:13.060 --> 01:17:20.380 align:middle line:84% or they will come through lazy not attenuated at all. 01:17:20.380 --> 01:17:23.760 align:middle line:84% And so if you have a really good graded neutral density filter, 01:17:23.760 --> 01:17:26.340 align:middle line:84% you can, again, end up in this regime 01:17:26.340 --> 01:17:31.940 align:middle line:84% where you actually get some sensitivity. 01:17:31.940 --> 01:17:35.030 align:middle line:90% To the position of the beam. 01:17:35.030 --> 01:17:39.140 align:middle line:84% And that can get you back into this nice regime 01:17:39.140 --> 01:17:41.060 align:middle line:84% up here where the intensity of your signal 01:17:41.060 --> 01:17:43.570 align:middle line:84% is actually proportional to the gradient of the electron 01:17:43.570 --> 01:17:44.070 align:middle line:90% density. 01:17:44.070 --> 01:17:48.680 align:middle line:84% So that'd be a very nice place to be, but this is hard to make. 01:17:48.680 --> 01:17:51.260 align:middle line:90% 01:17:51.260 --> 01:17:55.220 align:middle line:84% And you still need a uniform beam to start with, 01:17:55.220 --> 01:17:58.490 align:middle line:90% which is also hard to make. 01:17:58.490 --> 01:18:01.150 align:middle line:84% So these techniques, a lot of Schlieren 01:18:01.150 --> 01:18:03.792 align:middle line:84% was actually developed using light sources, which 01:18:03.792 --> 01:18:05.500 align:middle line:84% are very different from the light sources 01:18:05.500 --> 01:18:11.740 align:middle line:84% that we have to end up using in our experiments with plasmas. 01:18:11.740 --> 01:18:13.750 align:middle line:84% And so you end up, yeah, so just looking 01:18:13.750 --> 01:18:17.290 align:middle line:84% at the beautiful pictures that Matthew put in the chat here. 01:18:17.290 --> 01:18:19.490 align:middle line:84% These pictures here are absolutely glorious. 01:18:19.490 --> 01:18:22.640 align:middle line:84% And you can see there's a huge amount of detail on them. 01:18:22.640 --> 01:18:25.077 align:middle line:84% And this detail is due to the fact 01:18:25.077 --> 01:18:27.160 align:middle line:84% that actually I think for these ones they're using 01:18:27.160 --> 01:18:29.200 align:middle line:90% the sun as the back lighter. 01:18:29.200 --> 01:18:30.970 align:middle line:84% I may have forgotten this exactly, 01:18:30.970 --> 01:18:32.410 align:middle line:90% but light source is the sun. 01:18:32.410 --> 01:18:34.460 align:middle line:84% And the sun is actually quite large. 01:18:34.460 --> 01:18:37.180 align:middle line:84% It's an extended object, you may have noticed, in the sky. 01:18:37.180 --> 01:18:39.940 align:middle line:84% And this gives you really nice Schlieren imaging. 01:18:39.940 --> 01:18:43.390 align:middle line:84% But we don't tend to use nice large objects 01:18:43.390 --> 01:18:48.520 align:middle line:84% for our experiments with plasmas. 01:18:48.520 --> 01:18:50.680 align:middle line:84% We tend to use things like lasers. 01:18:50.680 --> 01:18:54.980 align:middle line:90% 01:18:54.980 --> 01:18:58.525 align:middle line:84% And lasers are not a good Schlieren source. 01:18:58.525 --> 01:19:03.247 align:middle line:90% 01:19:03.247 --> 01:19:05.830 align:middle line:84% And if you want to read in more detail why lasers are actually 01:19:05.830 --> 01:19:07.880 align:middle line:84% a really bad idea, you should go read 01:19:07.880 --> 01:19:09.630 align:middle line:84% the book that's listed in the bibliography 01:19:09.630 --> 01:19:13.000 align:middle line:84% by Settles, which is an absolutely cracking book. 01:19:13.000 --> 01:19:15.010 align:middle line:84% Has lots of lovely pictures, like the ones which 01:19:15.010 --> 01:19:16.600 align:middle line:90% were just posted in the chat. 01:19:16.600 --> 01:19:20.380 align:middle line:84% And also, a very detailed description of this stuff. 01:19:20.380 --> 01:19:22.810 align:middle line:90% But can anyone tell me why-- 01:19:22.810 --> 01:19:24.550 align:middle line:84% if lasers aren't very good, why do we 01:19:24.550 --> 01:19:27.700 align:middle line:84% end up using lasers as the light source for Schlieren 01:19:27.700 --> 01:19:29.050 align:middle line:90% imaging in plasma physics? 01:19:29.050 --> 01:19:33.450 align:middle line:90% 01:19:33.450 --> 01:19:35.835 align:middle line:84% What property of a laser is it that we particularly want? 01:19:35.835 --> 01:19:38.730 align:middle line:90% 01:19:38.730 --> 01:19:40.470 align:middle line:90% STUDENT: Monochromatic light. 01:19:40.470 --> 01:19:41.700 align:middle line:90% STUDENT: Monochromatic. 01:19:41.700 --> 01:19:45.365 align:middle line:84% JACK HARE: So monochromatic is somewhat useful. 01:19:45.365 --> 01:19:46.740 align:middle line:84% Actually, it turns out you can do 01:19:46.740 --> 01:19:49.410 align:middle line:84% really cool Schlieren techniques with a broadband light source 01:19:49.410 --> 01:19:52.560 align:middle line:84% as well and filters because the different wavelengths will be 01:19:52.560 --> 01:19:54.370 align:middle line:90% reflected by different amounts. 01:19:54.370 --> 01:19:57.090 align:middle line:84% And so if you have multiple cameras with different filters, 01:19:57.090 --> 01:19:59.070 align:middle line:84% you can really carefully reconstruct 01:19:59.070 --> 01:20:00.370 align:middle line:90% the deflection angles. 01:20:00.370 --> 01:20:01.980 align:middle line:84% So monochromatic is a good guess. 01:20:01.980 --> 01:20:04.320 align:middle line:84% But actually, we'll be OK with broadband. 01:20:04.320 --> 01:20:08.100 align:middle line:84% What else-- when you think laser, what do you think? 01:20:08.100 --> 01:20:09.485 align:middle line:90% STUDENT: Coherence. 01:20:09.485 --> 01:20:10.360 align:middle line:90% JACK HARE: Coherence. 01:20:10.360 --> 01:20:12.690 align:middle line:84% Actually, coherence isn't good for this. 01:20:12.690 --> 01:20:16.620 align:middle line:84% And we've been using a picture and the coherence screws 01:20:16.620 --> 01:20:17.762 align:middle line:90% up our nice simple picture. 01:20:17.762 --> 01:20:18.970 align:middle line:90% It's going to cause problems. 01:20:18.970 --> 01:20:20.887 align:middle line:84% So we really want coherence for interferometry 01:20:20.887 --> 01:20:23.578 align:middle line:84% and we absolutely hate it for Schlieren. 01:20:23.578 --> 01:20:25.870 align:middle line:84% And that's another thing that Settles says in his book, 01:20:25.870 --> 01:20:27.370 align:middle line:90% so coherence is a bad thing. 01:20:27.370 --> 01:20:30.120 align:middle line:90% So I'm going to put coherence. 01:20:30.120 --> 01:20:32.097 align:middle line:90% No, we actually don't want it. 01:20:32.097 --> 01:20:33.555 align:middle line:84% What else do we think about lasers? 01:20:33.555 --> 01:20:36.490 align:middle line:90% 01:20:36.490 --> 01:20:39.560 align:middle line:84% STUDENT: Well, they tend to produce nice round spots. 01:20:39.560 --> 01:20:45.730 align:middle line:84% So it's also easy to deflect them without changing them that. 01:20:45.730 --> 01:20:46.810 align:middle line:90% JACK HARE: Nice beam. 01:20:46.810 --> 01:20:48.850 align:middle line:84% Yeah, you can do that with other light sources. 01:20:48.850 --> 01:20:52.220 align:middle line:90% But fine, it is a nice beam. 01:20:52.220 --> 01:20:56.340 align:middle line:84% Anyone ever shone a laser pointer in their eye? 01:20:56.340 --> 01:20:59.430 align:middle line:90% If not, why not? 01:20:59.430 --> 01:21:00.868 align:middle line:90% STUDENT: High energy density. 01:21:00.868 --> 01:21:02.160 align:middle line:90% JACK HARE: They're very bright. 01:21:02.160 --> 01:21:03.420 align:middle line:90% Should we put it that way? 01:21:03.420 --> 01:21:05.520 align:middle line:84% OK, so lasers are extremely bright. 01:21:05.520 --> 01:21:07.830 align:middle line:84% Why do we want a bright light source 01:21:07.830 --> 01:21:10.860 align:middle line:90% when we're dealing with plasmas? 01:21:10.860 --> 01:21:12.430 align:middle line:90% STUDENT: The plasmas glow. 01:21:12.430 --> 01:21:15.790 align:middle line:84% JACK HARE: Yes, so we need to overcome 01:21:15.790 --> 01:21:19.130 align:middle line:84% the plasma glow, should we call it, 01:21:19.130 --> 01:21:21.140 align:middle line:90% or should I say emission here. 01:21:21.140 --> 01:21:24.530 align:middle line:84% So when you're working with jet planes flying around 01:21:24.530 --> 01:21:26.723 align:middle line:84% and you're taking Schlieren images of them using 01:21:26.723 --> 01:21:28.640 align:middle line:84% the sun as your background, you don't actually 01:21:28.640 --> 01:21:33.922 align:middle line:84% have to worry about the plasma or the shocked 01:21:33.922 --> 01:21:36.380 align:middle line:84% air around the plane glowing and ruining your measurements. 01:21:36.380 --> 01:21:37.755 align:middle line:84% Whereas, a plasma, you really do. 01:21:37.755 --> 01:21:39.180 align:middle line:90% It makes an awful lot of light. 01:21:39.180 --> 01:21:41.383 align:middle line:84% So you need an incredibly bright light source. 01:21:41.383 --> 01:21:42.800 align:middle line:84% And so despite the fact lasers are 01:21:42.800 --> 01:21:45.383 align:middle line:84% terrible for Schlieren it's the brightest light source that we 01:21:45.383 --> 01:21:47.480 align:middle line:84% have available so we end up using that. 01:21:47.480 --> 01:21:49.820 align:middle line:84% But the trouble with lasers is that they 01:21:49.820 --> 01:21:53.050 align:middle line:90% have a very small focal spot. 01:21:53.050 --> 01:21:56.460 align:middle line:84% So there are, in fact, incredibly easy to focus down 01:21:56.460 --> 01:21:57.750 align:middle line:90% to a point. 01:21:57.750 --> 01:21:58.750 align:middle line:90% And we don't want that. 01:21:58.750 --> 01:22:00.510 align:middle line:90% We want a nice large focal spot. 01:22:00.510 --> 01:22:02.940 align:middle line:84% And it's actually very difficult to get a laser 01:22:02.940 --> 01:22:05.790 align:middle line:84% to decohere enough to get a nice big focal spot. 01:22:05.790 --> 01:22:08.430 align:middle line:84% But maybe there are some techniques we can do. 01:22:08.430 --> 01:22:10.860 align:middle line:84% And because we've got a small focal spot, 01:22:10.860 --> 01:22:15.390 align:middle line:84% laser Schlieren is effectively binary. 01:22:15.390 --> 01:22:17.610 align:middle line:84% And I'll explain what I mean by that when I remember 01:22:17.610 --> 01:22:18.882 align:middle line:90% how to spell Schlieren. 01:22:18.882 --> 01:22:23.040 align:middle line:90% 01:22:23.040 --> 01:22:26.190 align:middle line:84% What I mean by that is the spot is so small, 01:22:26.190 --> 01:22:30.510 align:middle line:84% the laser Schlieren, it is either completely blocked 01:22:30.510 --> 01:22:37.070 align:middle line:84% by the knife edge or it is completely visible, 01:22:37.070 --> 01:22:38.870 align:middle line:90% unblocked, by the knife edge. 01:22:38.870 --> 01:22:41.390 align:middle line:84% And so when you do a laser Schlieren image, 01:22:41.390 --> 01:22:45.050 align:middle line:84% you get an image which is either dark or light, 01:22:45.050 --> 01:22:50.300 align:middle line:84% 0 or 1, no intensity or full intensity. 01:22:50.300 --> 01:22:53.540 align:middle line:84% And so you don't get those nice images 01:22:53.540 --> 01:22:55.520 align:middle line:84% that we were just looking at that were linked. 01:22:55.520 --> 01:22:58.880 align:middle line:84% And we also don't have the ability 01:22:58.880 --> 01:23:03.020 align:middle line:84% to use the change in intensity to measure the gradients 01:23:03.020 --> 01:23:05.910 align:middle line:84% in the electron density because the intensity is either 0 or 1. 01:23:05.910 --> 01:23:08.527 align:middle line:84% So we have very, very limited dynamic range. 01:23:08.527 --> 01:23:09.860 align:middle line:90% You can think about it that way. 01:23:09.860 --> 01:23:11.030 align:middle line:90% We have no dynamic range. 01:23:11.030 --> 01:23:13.100 align:middle line:84% We even know the density gradient is larger 01:23:13.100 --> 01:23:16.310 align:middle line:84% than the number or it's smaller than a number and that's it. 01:23:16.310 --> 01:23:18.800 align:middle line:84% So it makes very good shock pictures, 01:23:18.800 --> 01:23:20.930 align:middle line:84% but you can't do all the beautiful techniques 01:23:20.930 --> 01:23:24.860 align:middle line:84% that people use Schlieren for in standard fluid dynamics where 01:23:24.860 --> 01:23:26.850 align:middle line:84% they have access to different sources. 01:23:26.850 --> 01:23:32.270 align:middle line:84% So someone came up with a nice, bright, incoherent large area 01:23:32.270 --> 01:23:34.128 align:middle line:90% laser that we could use. 01:23:34.128 --> 01:23:35.420 align:middle line:90% That would be absolutely great. 01:23:35.420 --> 01:23:37.820 align:middle line:84% I have a few ideas along this direction, 01:23:37.820 --> 01:23:40.160 align:middle line:84% but haven't had a chance to try them out yet. 01:23:40.160 --> 01:23:42.852 align:middle line:84% So yeah, they have some serious limitations. 01:23:42.852 --> 01:23:44.810 align:middle line:84% So Schlieren is a lovely technique for plasmas. 01:23:44.810 --> 01:23:45.960 align:middle line:90% It's very limited. 01:23:45.960 --> 01:23:47.840 align:middle line:84% It's obviously useful when you have shocks, 01:23:47.840 --> 01:23:50.110 align:middle line:84% so you need to be having fast moving plasmas. 01:23:50.110 --> 01:23:52.610 align:middle line:84% And so this is not really a technique that we would normally 01:23:52.610 --> 01:23:55.790 align:middle line:84% use inside say a tokamak or a magnetically confined device, 01:23:55.790 --> 01:23:59.570 align:middle line:84% but it is very good when you have larger densities. 01:23:59.570 --> 01:24:02.120 align:middle line:90% 01:24:02.120 --> 01:24:04.535 align:middle line:84% All right, so any questions on this? 01:24:04.535 --> 01:24:10.580 align:middle line:90% 01:24:10.580 --> 01:24:12.520 align:middle line:90% Nigel, yeah. 01:24:12.520 --> 01:24:16.660 align:middle line:84% STUDENT: So did we ever say how exactly the critical density was 01:24:16.660 --> 01:24:17.830 align:middle line:90% determined? 01:24:17.830 --> 01:24:19.215 align:middle line:90% Is that for a later lecture? 01:24:19.215 --> 01:24:20.590 align:middle line:84% JACK HARE: Do you mean like, what 01:24:20.590 --> 01:24:22.698 align:middle line:84% is a numeric-- what is an analytical result 01:24:22.698 --> 01:24:23.740 align:middle line:90% for the critical density? 01:24:23.740 --> 01:24:27.310 align:middle line:84% STUDENT: Yeah, when we had like 1 minus n over n critical. 01:24:27.310 --> 01:24:29.350 align:middle line:90% Where that was ever determined? 01:24:29.350 --> 01:24:31.670 align:middle line:84% Or is it just kind of guess and check. 01:24:31.670 --> 01:24:32.920 align:middle line:90% JACK HARE: No, absolutely not. 01:24:32.920 --> 01:24:34.180 align:middle line:90% You can write it down. 01:24:34.180 --> 01:24:35.350 align:middle line:90% It's a function. 01:24:35.350 --> 01:24:38.090 align:middle line:84% And I neglected to write it down here. 01:24:38.090 --> 01:24:40.270 align:middle line:84% And I am not going to try and remember what it is. 01:24:40.270 --> 01:24:44.170 align:middle line:84% But it is a function which has inside it 01:24:44.170 --> 01:24:48.700 align:middle line:84% the frequency of the laser light, epsilon naught, e 01:24:48.700 --> 01:24:52.720 align:middle line:90% and me and the-- 01:24:52.720 --> 01:24:53.623 align:middle line:90% not the density. 01:24:53.623 --> 01:24:55.540 align:middle line:84% Definitely doesn't have the density inside it. 01:24:55.540 --> 01:24:58.870 align:middle line:84% So this effectively comes from rearranging the plasma frequency 01:24:58.870 --> 01:24:59.590 align:middle line:90% here. 01:24:59.590 --> 01:25:01.450 align:middle line:84% And you can find this in Hutchinson's book, 01:25:01.450 --> 01:25:03.325 align:middle line:84% and I really should have written in my notes, 01:25:03.325 --> 01:25:05.568 align:middle line:90% but I just didn't, OK? 01:25:05.568 --> 01:25:08.110 align:middle line:84% And I'm not going to try and bullshit you and look it up now. 01:25:08.110 --> 01:25:10.870 align:middle line:84% So yeah, the critical density is a number that we can find. 01:25:10.870 --> 01:25:15.710 align:middle line:84% It is uniquely defined for every electromagnetic wavelength 01:25:15.710 --> 01:25:18.275 align:middle line:84% and that is the density, which you cannot go above. 01:25:18.275 --> 01:25:20.150 align:middle line:84% And if you're sensible, you'll work well away 01:25:20.150 --> 01:25:21.320 align:middle line:90% from the critical density. 01:25:21.320 --> 01:25:22.790 align:middle line:84% Because if you get anywhere close to it, 01:25:22.790 --> 01:25:24.748 align:middle line:84% it really screws up most of these calculations. 01:25:24.748 --> 01:25:28.160 align:middle line:84% So most of the time, we're relying on this approximation 01:25:28.160 --> 01:25:30.980 align:middle line:84% that the density we're using is much 01:25:30.980 --> 01:25:32.370 align:middle line:90% less than the critical density. 01:25:32.370 --> 01:25:34.037 align:middle line:84% And we'll come across that approximation 01:25:34.037 --> 01:25:38.100 align:middle line:84% very strongly when we deal with interferometry in a little bit. 01:25:38.100 --> 01:25:40.230 align:middle line:84% But we did also use it already when we were 01:25:40.230 --> 01:25:43.870 align:middle line:90% deriving the Schlieren angle. 01:25:43.870 --> 01:25:46.150 align:middle line:84% When we got to this point here, we 01:25:46.150 --> 01:25:48.610 align:middle line:84% used this linear approximation where 01:25:48.610 --> 01:25:52.410 align:middle line:84% the density is much less than the critical density. 01:25:52.410 --> 01:25:53.580 align:middle line:90% Any other questions? 01:25:53.580 --> 01:25:57.560 align:middle line:90% 01:25:57.560 --> 01:25:58.070 align:middle line:90% Yeah? 01:25:58.070 --> 01:25:58.570 align:middle line:90% No? 01:25:58.570 --> 01:26:01.200 align:middle line:90% 01:26:01.200 --> 01:26:02.697 align:middle line:90% STUDENT: So can you hear me? 01:26:02.697 --> 01:26:03.780 align:middle line:90% JACK HARE: I can hear you. 01:26:03.780 --> 01:26:04.990 align:middle line:90% Yeah. 01:26:04.990 --> 01:26:05.740 align:middle line:90% STUDENT: OK, cool. 01:26:05.740 --> 01:26:10.630 align:middle line:84% So would it not be possible to use the quote unquote glow 01:26:10.630 --> 01:26:14.683 align:middle line:84% of the plasma itself to use in some other part of the plasma 01:26:14.683 --> 01:26:16.600 align:middle line:84% that isn't glowing itself or that it's glowing 01:26:16.600 --> 01:26:18.048 align:middle line:90% at some other frequency? 01:26:18.048 --> 01:26:19.090 align:middle line:90% Say, once we're putting-- 01:26:19.090 --> 01:26:21.760 align:middle line:90% 01:26:21.760 --> 01:26:23.230 align:middle line:90% JACK HARE: I thought about this. 01:26:23.230 --> 01:26:25.465 align:middle line:90% I think it's a really cool idea. 01:26:25.465 --> 01:26:28.030 align:middle line:84% You definitely have to arrange your plasma so there's a bit 01:26:28.030 --> 01:26:29.530 align:middle line:84% that's glowing and a bit that isn't. 01:26:29.530 --> 01:26:31.447 align:middle line:84% It's a bit you want to measure as also glowing 01:26:31.447 --> 01:26:34.010 align:middle line:84% that's going to make things very, very difficult. 01:26:34.010 --> 01:26:36.520 align:middle line:84% So yeah, you could potentially have something like that, 01:26:36.520 --> 01:26:40.930 align:middle line:84% like a ICF hotspot backlighting the whole plasma. 01:26:40.930 --> 01:26:42.730 align:middle line:90% That sort of thing could work. 01:26:42.730 --> 01:26:43.510 align:middle line:90% Yeah. 01:26:43.510 --> 01:26:47.020 align:middle line:84% But I don't know of anyone who's done it. 01:26:47.020 --> 01:26:49.163 align:middle line:90% STUDENT: OK, Thank you. 01:26:49.163 --> 01:26:51.080 align:middle line:84% JACK HARE: All right, we're past the hour now. 01:26:51.080 --> 01:26:52.465 align:middle line:90% So I think we'll leave it here. 01:26:52.465 --> 01:26:54.340 align:middle line:84% I will be back in the classroom next Tuesday. 01:26:54.340 --> 01:26:55.798 align:middle line:84% I look forward to seeing all of you 01:26:55.798 --> 01:26:57.730 align:middle line:84% there and all of the Columbia folks online. 01:26:57.730 --> 01:27:02.460 align:middle line:84% And yeah, enjoy your weekend and bye for now. 01:27:02.460 --> 01:27:05.000 align:middle line:90%