An Imaging Refractometer for Density Fluctuation Measurements in High Energy Density Plasmas
J. D. Hare, G. C. Burdiak, S. Merlini, J. P. Chittenden, T. Clayson, A. J. Crilly, J. W. D. Halliday, D. R. Russell, R. A. Smith, N. Stuart, L. G. Suttle, S. V. Lebedev
AAn Imaging Refractometer for Density Fluctuation Measurements in HighEnergy Density Plasmas
J. D. Hare, a) G. C. Burdiak, S. Merlini, J. P. Chittenden, T. Clayson, A. J. Crilly, J. W. D. Halliday, D.R. Russell, R. A. Smith, N. Stuart, L. G. Suttle, and S. V. Lebedev Blackett Laboratory, Imperial College, London, SW7 2AZ, UK First Light Fusion Ltd, 10 Oxford Industrial Park, Yarnton, Kidlington OX5 1QU, UK (Dated: 10 July 2020)
We report on a recently developed laser-based diagnostic which allows direct measurements of ray-deflection anglesin one axis, whilst retaining imaging capabilities in the other axis. This allows us to measure the spectrum of angulardeflections from a laser beam which passes though a turbulent high-energy-density plasma. This spectrum containsinformation about the density fluctuations within the plasma, which deflect the probing laser over a range of angles.The principle of this diagnostic is described, along with our specific experimental realisation. We create syntheticdiagnostics using ray-tracing to compare this new diagnostic with standard shadowgraphy and schlieren imaging ap-proaches, which demonstrates the enhanced sensitivity of this new diagnostic over standard techniques. We presentexperimental data from turbulence behind a reverse shock in a plasma and demonstrate that this technique can measureangular deflections between 0.05 and 34 mrad, corresponding to a dynamic range of over 500.
I. INTRODUCTION
Turbulence drives inhomogeneities in fluids, creating a cas-cade of fluctuations from the driving scale down to the scaleat which viscous dissipation dominates. These fluctuationscreate density gradients over a wide range of scales, whichbroaden a probing laser beam. This broadening has attractedwidespread interest in aero-optics and plasma physics.
Two techniques for measuring these density fluctuations areoften used in high-energy-density (HED) plasmas: shadowg-raphy and schlieren imaging. In shadowgraphy, density fluc-tuations produce intensity variations (light and dark regions)in the laser beam which are proportional to the second deriva-tive of the electron density. However, as the rays are spatiallyredistributed, there is no one-to-one relation between the im-age on the detector and the object, and so it is not possible tomeasure spatial scales directly from a shadowgram. In schlieren imaging, a stop is placed at a focal plane tocut off rays with specific deflection angles. With a finite sizedsource, the schlieren image contains information on the gra-dients sampled by the probe beam, but with a laser source themeasurement is often only sensitive to a very narrow rangeof gradients. Therefore laser schlieren images are almost bi-nary, showing the location of density gradients but not theirmagnitude. With both shadowgraphy and schlieren, the anal-ysis of turbulence is often similar: the image is Fourier trans-formed, the spectrum of intensity variations is linked to thespectrum of density fluctuations, and a power-law fit is madeand compared with theoretical predictions.
These imaging techniques have two significant drawbacks.First, large density gradients result in ray crossings, leading tocaustics which prevent a unique reconstruction of the den-sity fluctuations. Second, the smallest resolvable length scaleis at least a few times the detector pixel size, which means thatin a realistic setup the difference between the driving length a) Corresponding author: [email protected] scale and the smallest resolvable length scale is often only asingle order of magnitude. It is difficult to fit a power-lawspectra over such a limited range, and it is difficult to probethe full inertial range down to the dissipation scale.The spectrum of deflection angles within a probing laserbeam is directly related to the spectrum of density fluctua-tions within the turbulent plasma. This deflection angle spec-trum can be simply measured by placing a detector at the fo-cal plane of a lens, which contains a Fourier transform of theprobing beam, which is the two-dimensional spectrum of de-flection angles. However, this spectrum lacks informationabout spatial variations in the properties of the turbulence.
FIG. 1. A shadowgraphy image of a turbulent plasma column formedat the centre of an imploding, eight-wire, carbon z-pinch. In this paper we present a new hybrid diagnostic, an imag-ing refractometer, which has spatial resolution along one axisand angular resolution along the perpendicular axis. This is apowerful technique when the properties of turbulence withinan HED plasma are homogeneous in one direction, eg. an ex-periment with cylindrical or planar geometry. In this case, in-tegrating over this spatial dimension can provide angular res-olution with an increased signal-to-noise ratio. For example,in Fig. 1 the turbulence in a plasma column formed inside awire array z-pinch appears homogeneous in z , but varies in x .This new diagnostic uses physical optics to image a one-dimensional Fourier transform of a probing beam. This diag-nostic’s dynamic range is over 500, far in excess of techniquesusing digital Fourier transforms, and is not limited by the for-mation of caustics. We give an example from an experimentin which density fluctuations form behind a reverse shock. a r X i v : . [ phy s i c s . p l a s m - ph ] J u l FIG. 2. A ray diagram of the imaging refractometer, from two orthogonal directions. An initially collimated laser propagates through a plasmaof length D from the left. The first lens (cyan) forms a Fourier plane and an Image plane, which are imaged by a composite optic consisting ofa spherical lens (cyan) and a cylindrical lens (pale purple). This results in a composite image, with b) one axis of spatial resolution, and a) oneaxis of angular resolution. The diagram shows three example rays: green; a collimated ray, and red and orange; deflected, collimated rays.
II. OPTICAL CONFIGURATION
We will illustrate the operation of this diagnostic for thespecific focal length relations shown in Fig. 2, with a nearlycollimated laser beam travelling from left to right, whichpasses through a turbulent plasma of length D . After the firstlens (spherical, f = L /
2, cyan) there are two planes of inter-est, the image plane and the Fourier plane. At the Image plane,there is an image of the probing laser beam at the exit of theplasma, containing the spatial distribution of the rays. At theFourier plane, there is the Fourier transform of the probinglaser beam, containing the angular distribution of the rays.The second optic consists of a spherical (cyan) lens and acylindrical (pale purple) lens placed next to each other, whichfocuses light differently along two orthogonal axes. In theImaging axis (Fig. 2a), the combination of the cylindrical andspherical lenses focus the rays with f = L /
3, reproducingthe Image plane at the detector. In the Fourier axis (Fig. 2b)the spherical lens focuses the rays with f = L /
2, forming animage of the Fourier plane at the detector. Hence the x-axis (y-axis) of the image at the detector shows the spatial (angular)distribution of rays at the object plane.In Fig. 2, the green line represents a collimated beam prop-agating parallel to the optical axis. This ray crosses the opti-cal axis at the Fourier Plane, and crosses the Image plane atthe same distance from the optical axis as it originated. Af-ter the second lens, this ray arrives at the optical axis in theFourier direction ( φ = M = M =
1. The second lens images both rays to thesame off-axis location in the Fourier axis, indicating that theray exited the plasma with φ (cid:54) =
0. The rays are imaged with M = Wetreat all of the lenses as thin, and calculate the position of aray X f at the detector as: X = x θ y φ → X f = x θ / + x / L − L φ / y / L (1)where X is the ray at the exit of the plasma, and for x , y , θ and φ see Fig. 2. The x-direction is imaging with a magnifi-cation of 2, and has no dependence on the initial angle. They-direction only depends on the angle of the initial ray ( φ )from the z axis in the y-z plane. The y location of the ray onthe CCD is L φ /
2, which defines the angular sensitivity of thedetector. The resolution of the detector is set by any imper-fections in the optics, the initial divergence of the beam, thepixel size of the detector, and limitations due to diffraction.
III. IMPLEMENTATION FOR HED EXPERIMENTS
In an inhomogeneous medium, rays are deflected by refrac-tive index gradients normal to the direction of propagation: α = (cid:90) ∇ Ndl = (cid:90) ∇ n e n cr dl , (2)where α is in radians and the gradient is taken perpendic-ular to the path dl . In a plasma the refractive index ( N )is related to the electron density n e and the critical density n cr = ω ε m e / e = . × λ , where λ is the freespacewavelength in µ m, and n cr is in units of cm − . It is thesedeflections which are measured by the imaging refractometer.In order to measure the deflection angle with a high dy-namic range, we require a laser which a) has a small angulardivergence, b) has a smooth beam profile, c) is highly repro-ducible, d) has a short coherence length, and e) is sufficientlyintense to overcome the self-emission of the plasma.The smoothness of the laser beam ensures that any varia-tions in intensity are due to deflected light, rather than varia-tions in the initial source profile. The high collimation ensuresthat the incoming laser beam can be well described by a singleFourier component, and hence focuses to a very narrow line,which contributes to the resolution of this diagnostic. Thehigh reproducibility is necessary to compare the results ob-tained in the absence of the plasma and with the plasma, todetermine how the intensity of the laser beam is redistributed.The short coherence length is necessary to minimise interfer-ence effects which degrade laser-based schlieren and shad-owgraphy techniques. These can cause variations in the mea-sured intensity which could be mistaken for fluctuations in themeasured deflection angle. The intensity of the laser must bemuch larger than the measurable self-emission of the plasma,which degrades the signal-to-noise ratio.For our experiments we use the long-pulse arm of theNd:Glass CERBERUS laser (1053 nm, 100 mJ, 1 ns). Thislaser passes through several vacuum spatial filters, resultingin a divergence < .
05 mrad, and the reproducibility of theintensity profile is better than 1%. The laser has a low coher-ence length ( < α ∼ / n cr ∼ λ .We implemented this diagnostic with L =
400 mm (see Fig.2), which is determined by the radius of our vacuum chamberand the available optics. Therefore f =
200 mm for the twospherical lenses (50 mm diameter Thorlabs Achromatic dou-blet AC508-200-C-ML), and the cylindrical lens has f = ×
30 mm Thorlabs plano-convex LJ1363L2-C).The second optic consists of the cylindrical lens and sphericallens placed back to back, with f = / λ = ≈
22 mm.For the shadowgraphy arm, we use a 15 MP DSLR (Canon500D), with the IR filter removed and no lens mounted to thebody. All the optics are mounted to a breadboard outside thevacuum chamber. The exposure time of the camera is 1.3 s,so the effective exposure time of the image is instead set to 1ns by the duration of the laser pulse.For the imaging refractometer arm, we use an ATIK 383L+camera, which has a cooled 8 MP (3448 × × . IV. COMPARISON WITH SHADOWGRAPHY ANDSCHLIEREN
To compare this system with existing laser-imaging sys-tems, we have created synthetic diagnostics representing theimaging refractometer, shadowgraphy and schlieren imagingsystems. The imaging refractometer is as shown in Fig. 2, andthe shadowgraphy and schlieren systems use the optics shownin Fig. 2a. For the schlieren system, a knife edge is placedat the Fourier plane at L / φ > . z ). We generate a density field described by n e ( x , y ) = n e x / s [ + cos ( π y / L y )] . We use n e = × cm − , s = L y = x , y , z = [ − , ] mm).We generate 9.6 × test rays randomly located within abeam of diameter 10 mm. The initial ray angles are drawnfrom a normal distribution with width 0 .
05 mrad, correspond-ing to the experimentally measured response function. Thisaccounts for the imperfect collimation of the input beam, andany optical aberrations. These rays are traced through the den-sity cube using a ray tracer, which interpolates the refractiveindex gradient at the ray location. After the rays have ex-ited the plasma, they are propagated through the three opticalsystems using a ray transfer matrix technique.The results are shown in Fig. 3. The schlieren image (Fig.3a) shows a series of bright horizontal bands, centred on themaximum value of ∂ n e / ∂ y , which increase in width from leftto right, with caustics forming at the far-right of the image.These bands are where the gradients deflect rays in the + φ direction. The schlieren effect results in a binary image, be-cause laser-based schlieren imaging is only sensitive to a nar-row range of density gradients - for larger density gradients,all of the rays either pass the knife edge, or they are blocked. Intensity variations in laser-schlieren imaging are often dueto shadowgraphy effects (proportional to the second derivative
FIG. 3. Synthetic images for a) Schlieren, b) Shadowgraphy and c) Imaging Refractometry, for 10 rays passing through a sinusoidal densityperturbation in y (shown on the left), which increase in strength in x . The ray locations at the detector binned into 3448 × × . φ = ±
34 mrad. of the electron density) rather than schlieren effects (propor-tional to the first derivative). This is particularly clear insidethe caustic forming region on the right of the schlieren image,which is identical to the intensity profile in the shadowgraphyimage. Therefore digital Fourier transforms of schlieren im-ages may in fact be measuring shadowgraphy effects, whichaffects the interpretation of the power spectrum and how itrelates to the spectrum of density fluctuations.The synthetic shadowgraphy diagnostic (Fig. 3b) showshorizontal bands in the same locations as the schlieren im-age, but the dynamic range is not binary. The contrast of thebands increases as the density gradients increase from left toright. Towards the right of the image, the sinusoidal perturba-tions focus the rays, causing bright central bands surroundedby darker bands. At the far-right of the image, the bright bandsbroaden again as the rays cross, and caustics form.The imaging refractometer (Fig. 3c) shows a bright bound-ing curve which increases exponentially from φ =
0, reflect-ing the exponential density ramp from left to right. This brightcurve represents the regions of highest deflection, or steepestelectron density gradient, and bounds a region of lower inten-sity, corresponding to smaller deflection angles.It is clear that the imaging refractometer is more sensitiveto angular deflections than shadowgraphy and schlieren tech-niques. Indeed, the schlieren technique is really only appro-priate for showing the location of shocks, and does not pro-vide any information about their properties. The shadowgra-phy technique does provide more information, but in practicethe double-integration necessary to retrieve the properties ofthe density perturbation is very sensitive to numerical noise.Other statistical techniques may be more appropriate for re-trieving the density modulations, providing the input laserbeam is well characterised.
Of course, the imaging refractometer gives no informationabout the location of the density perturbations in the y direc-tion, and so is ideally complemented by an inline shadowgra-phy arm, as discussed earlier. V. EXPERIMENTAL RESULTS
To demonstrate the capabilities of this diagnostic, we car-ried out an experiment on the MAGPIE pulsed-power genera- tor (1.4 MA peak current, 250 ns rise time) . We drove an ex-ploding wire array to produce a super-sonic, magnetiseddiverging aluminium plasma outflow. This outflow collidedwith a planar target and created a reverse shock, which slowlypropagated back (left) towards the array. FIG. 4. Experimental data from a pulsed-power driven supersonicplasma colliding with a planar target. The two spatial scale barsare located at the same location in both a) shadowgraphy and b) theimaging refractometer, with their right end at x = The results are shown in Fig. 4. The flow propagates fromleft to right and the target is on the right of the shadowgra-phy image Fig. 4a. The shock is very clear, an intense nar-row band which represents a strong, focusing density gradient.The intensity variations are small in the upstream and imme-diate downstream flow, but close to the planar target there aresmall scale intensity fluctuations.We can gain further insight from Fig. 4b, which showsdata from the imaging refractometer. Here we see that theupstream flow is structured, with horizontal lines indicatingdistinct deflection angles. This is due to the well-understoodaxial-modulation of the ablation in exploding and implodingwire arrays.
Immediately post-shock the imaging refrac-tometer records no intensity due to the strong shadowgraphyeffect which deflects the rays horizontally. Further right, thereis a region with smaller deflection angles than in the upstreamregion, suggesting that the axial modulations are damped af-ter the shock. In this region, the shadowgraphy shows no in-tensity variations, but they are clearly visible in the imagingrefractometer, indicating the higher sensitivity of this new di-agnostic to small deflections. Close to the target we observean abrupt transition, to a wide spread of deflection angles, cor-responding to a region with a density fluctuations over a broadrange of spatial scales, which may indicate turbulence.The spatial variations in the deflection angle spectrum areapparent in Fig. 4c and d, which shows two lineouts fromFig. 4b (averaged over 0.5 mm in x ), as well as the detectorresponse functions measured in a shot without plasma. The re-sponse function is well approximated by a normal distributionwith a width of 0.06 mrad, which is over 500 times smallerthan the maximum angle of 34 mrad (shown in green in Fig.4c and d). This response function is set by the initial qualityof the probing laser beam, the optics, diffraction effects andthe resolution of the CCD.The pre-shock spectrum is significantly broader than the re-sponse function, consistent with the axial modulation of theincoming flow as discussed above. The spectrum at the tar-get is much broader still, by a factor of 200 greater than theresponse function, which again provides evidence for a broadrange of density fluctuations consistent with turbulence. VI. CONCLUSIONS
In this paper we have outlined the theory and application ofa new hybrid diagnostic, which combines angular and spatialresolution with a very high dynamic range. When used with ashort optical pulse, we also achieve high temporal resolution.We explored the theory of this diagnostic using a ray-transfermatrix based approach, and we used a ray-tracing techniqueto compare this imaging refractometer with the familiar shad-owgraphy and schlieren imaging diagnostics.The spectrum of deflection angles is directly linked to thespectrum of density fluctuations, and the imaging refractome-ter directly measures this spectrum of angular deflections inone direction. In contrast, schlieren and shadowgraphy imag-ing require the spectrum to be inferred from digital Fouriertransforms, which suffer from limited resolution, and arestrongly affected by ray crossing which form caustics. Neitherlimitation applies to the imaging refractometer, which offers anew way to study turbulence in HED plasmas. We demonstrated the real-world capabilities of the imagingrefractometer in a pulsed-power driven high-energy-densityplasma experiment, using a supersonic outflow from an ex-ploding wire array to create turbulence behind a reverse shock,which we simultaneously imaged using shadowgraphy and theimaging refractometer.In future, we will use ray-tracing to calculate the propaga-tion of a laser beam through a turbulent medium with a givenspectral index and intermittency properties. The output rayswill then be transferred through our optical system, so thatwe can link the measured spectrum of angular deflections di-rectly to the spectrum of density fluctuations within a turbu-lent plasma.
ACKNOWLEDGEMENTS
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