Simulated Refraction-Enhanced X-Ray Radiography of Laser-Driven Shocks
Arnab Kar, T. R. Boehly, P. B. Radha, D. H. Edgell, S. X. Hu, P. M. Nilson, A. Shvydky, W. Theobald, D. Cao, K. S. Anderson, V. N. Goncharov, S. P. Regan
SSimulated Refraction-Enhanced X-Ray Radiography of Laser-Driven Shocks
Arnab Kar, a) T. R. Boehly, P. B. Radha, D. H. Edgell, S. X. Hu, P. M. Nilson, A. Shvydky, W. Theobald, D. Cao, K. S. Anderson, V. N. Goncharov, and S. P.Regan Laboratory for Laser Energetics, University of Rochester, Rochester, New York 14623,USA
Refraction-enhanced x-ray radiography (REXR) is used to infer shock-wave positions ofmore than one shock wave, launched by a multiple-picket pulse in a planar plastic foil.This includes locating shock waves before the shocks merge, during the early time and themain drive of the laser pulse that is not possible with the velocity interferometer systemfor any reflector. Simulations presented in this paper of REXR show that it is necessaryto incorporate refraction and attenuation of x rays along with the appropriate opacity andrefractive-index tables to interpret experimental images. Simulated REXR shows goodagreement with an experiment done on the OMEGA laser facility to image a shock wave.REXR can be applied to design multiple-picket pulses with a better understanding of theshock locations. This will be beneficial to obtain the required adiabats for inertial confine-ment fusion implosions. a) Electronic mail: [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] M a r . INTRODUCTION In direct-drive inertial confinement fusion (ICF) implosions, a spherical target (plastic shell)containing the fusion fuel [deuterium (D) and tritium (T)] is irradiated directly with nominallyidentical laser beams with the goal of achieving sufficient compression needed for ignition. Ig-nition refers to a high gain when the fusion energy from the target is greater than the incidentlaser energy after the onset on a thermonuclear burn wave. To achieve ignition, the minimumlaser energy ( E L ) and areal density ( ρ R ) required scale with the adiabat or isentrope parameter ( α ) as E L ∝ α . and ρ R ∝ α − . . The adiabat is a measure of compressibility, defined as theratio of the electron pressure to the Fermi pressure of the degenerate electron gas at absolute zerotemperature. In the early stage of the implosion, laser light is absorbed by the target and plasmaformation occurs through laser–matter interaction. Typically, the laser pulse comprises a sequenceof one to three low-intensity pulses known as “pickets." These pickets trigger a series of pressurepulses in the plasma that launch a sequence of shock waves to compress the target. This sequence,characterized by the individual shock speeds and temporal spacing, creates the desired adiabatprofile in the shell necessary for the expected compression. Adiabat shaping is a technique used inICF implosions to increase shell compressibility and target yield by increasing the entropy on theouter portion of the shell and preserving a low entropy inside the shell.
Knowing the positionof the shock waves launched over time is crucial to designing the desired adiabat.Adiabat shaping in implosions is achieved through hydrodynamic code validation based onshock-timing results from the velocity interferometer system for any reflector (VISAR).
VISAR provides the speed of the first shock along the axis of propagation and the time at whichthe other shock waves catch up with the first shock. Discrepancies still exist between simulationsand experiment for shock velocities and shock-merger time measurements at low adiabats α ∼
1. This diagnostic does not provide the longitudinal spatial position of the shock waves over time.Moreover, VISAR does not provide any information about the shock wave early in time becauseof a time lag associated with the critical surface formation for the diagnostic to work. During themain drive, the high intensity of the laser leads to x-ray photoionization of the target ahead of theshock front. This blanks out the VISAR signal, preventing it from determining the shock wave’slocation.
X-ray radiography can be a complementary diagnostic in ICF implosions to obtain shock po-sitions by imaging shock waves.
In this paper, simulated refraction-enhanced x-ray radiogra-2hy (REXR) has been developed to locate multiple shock waves launched from a multiple-picketpulse. Phase-contrast imaging, which is widely used in biological and medical imaging, iscalled refraction-enhanced radiography for Fresnel numbers ( F ) significantly larger than 1.Here, the Fresnel number is defined as F = L / d λ , where L is the spatial size of the object and d denotes the distance of propagation of the x rays of wavelength λ . For F (cid:29)
1, geometrical opticsis a good approximation for wave optics that are applicable for imaging typical ICF implosionswith soft x rays. The benefit of this technique over existing x-ray postprocessors when viewingICF implosions is that it includes x-ray refraction. For example,
SPECT3D accounts only forthe attenuation of x rays. Other potential applications of this method are the study of hydrody-namic mixing at the ablator/fuel interface and the study of mixing of dopants in materials alongwith the effect of the stalk on the spherical targets during implosions. More recently, it has beenshown that inflight density profiles can be inferred for ICF implosions using REXR. Through an experimental design, we show that REXR can view shock waves both early intime and during the main drive of the laser pulse, unlike the VISAR diagnostic. This is possiblesince REXR does not rely on the formation of the critical surface in early time. During the maindrive, the x-ray photoionization of the target ahead of the shock front does not affect REXR sinceit relies on the density gradient of the shock front, which is not opaque to x rays. Additionally,REXR is used to interpret the image of a shock wave for an experiment that was done on OMEGA.In this context, the importance of the appropriate choice of the opacity and refractive-index tablesis discussed in Appendix A.This paper has been organized as follows: In Sec. II, an overview of the REXR is presentedalong with a discussion of the experimental design parameters. In Sec. III, the results from anOMEGA experiment and simulated REXR are analyzed and compared. In the experiment, a shockwave was launched inside a planar plastic foil by irradiating it with a laser. In Sec. IV, an experi-mental design to image multiple shock waves launched by a picket pulse with a main drive pulseis presented. Finally, in Sec. V, the capabilities of this diagnostic are summarized.
II. OVERVIEW
Here we present an overview of REXR, which generates images of shock waves. Figure 1shows a schematic of the different components of this method. To generate a shock wave, a laserbeam characterized by a pulse shape is incident on a planar foil (ablator) at room temperature,3
IG. 1. A schematic showing a planar foil being irradiated with a laser pulse that generates a shock waveinto the target. This process is simulated with the hydrodynamic code
DRACO , which gives a spatial densityprofile with the shock features. Point-projected radiography of this profile with x rays of specific energygenerates the radiograph on the detector. generating shock waves inside the foil. This process is simulated with the hydrodynamic code
DRACO . Next, x rays sidelight the target at a specific energy and a ray-tracing code tracks thetrajectory of the rays as they travel through the density profile obtained from
DRACO to generatesimulated radiographs. The x-ray energy is chosen such that sufficient fluence to resolve the spatialfeatures is possible. The Henke tables provide the refractive indices and opacities corresponding tothe density profiles at this x-ray energy. The simulated radiograph shows the x-ray flux (analogousto the photon count of a detector) by tracking the position of the x rays to account for refractionand their intensity to estimate attenuation. The details of the ray-tracing code along with a modelsystem to illustrate this idea have been included in Appendix B. The distances labeled in theschematic determine the magnification of the imaging system.The experimental design parameters to image a shock wave include the target, the pulse shapethat generates the shock waves, and the spatial resolution of the imaging system. The relevantablator materials used in ICF targets include CH and beryllium, which have sufficient x-ray trans-parency in their compressed states. The laser drive instrumental in launching (one or more) shockwaves determines the degree of compression of the target; then these shock waves are diagnosedwith an x-ray framing camera. The effective spatial resolution of the system is given by the pinhole4ize.
III. COMPARISON OF SIMULATED REXR AND OMEGA EXPERIMENTSA. Experimental setup n m × 3 mm × 3 mm Al20 n m × 3 mm × 3 mm CH0.25 mm × 0.4 mm × 1.5 mm CH FIG. 2. The target consisted of a 250- µ m-wide planar plastic foil shielded by an aluminum foil and a thinplastic foil in front of it. The laser beam on the target was incident on the right, driving the shock wave fromthe thin plastic foil into the aluminum pusher, followed by the thicker plastic foil.FIG. 3. A ∼ ∼
350 J of energy was used for shot 38808. Aluminum also serves as a good shield for the coronal xrays that are generated from the laser drive to enhance the image contrast in the shocked region.Elements with a low atomic number such as the plastic are used as an overcoat in front of thealuminum foil to provide a higher exhaust velocity and reduce x-ray generation in the corona. Asingle laser with a ∼ µ m spot size was incident normally on the target. The laser drive (shownin Fig. 3) was comprised of a square pulse with ∼
350 J of energy that generated the shock wave inthe foil. For the x-ray radiography, x-rays of 5.2-keV energy corresponding to the He- α emissionsof Vanadium were projected from a 10- µ m pinhole to image the shock wave onto an x-ray framingcamera. A single strip x-ray framing camera with an integration time of ∼
300 ps and pixel size of18 µ m was used. The framing camera started to acquire the image at 8 . ± . B. Simulated REXR
FIG. 4. Density profile at 8.62 ns obtained from a
DRACO simulation modeling an experiment on OMEGA.Along the center of the beam axis, the highest-density point at –320 µ m marks the shock front in plasticthat was launched from the point of the incident laser energy at 0 µ m on the right. The solid white linecorresponds to the plastic/aluminum interface. RACO simulations for this experimental setup were performed. A Eulerian version of thecode modeled the laser energy deposition on the ablator through ray-tracing and included processessuch as heat conduction and radiation transport. DRACO is an axially symmetric code by nature,and the shock waves propagated along the center of the beam axis in these simulations. Figure 4shows the density profile that was obtained from the simulation at 8.62 ns when the target was nolonger being irradiated by the laser beam. The interface between the vacuum and rear end of theplastic is 360 µ m from the initial surface upon which the laser was incident. Near the surface ofthe laser deposition, the figure shows plastic and aluminum of low density as they were ablated.This was followed by a gradual rise in the density of the aluminum caused by the transition ofthe shock wave. The density gradually increased in space as it approached the shock front in theplastic region. The highest-density point in plastic corresponding to the shock front occurs around − µ m. The shock profile in plastic shows a prominent bowing effect because of the smallerlaser spot size relative to the target. The shock wave was not being supported by then as the laserdrive was turned off. FIG. 5. Simulated radiograph for an OMEGA experiment shows the shock profile in a plastic ablator. Thex-ray flux is representative of the degree of transmission of the x rays through the different areas: vacuum(in red), unshocked plastic (in yellow), shocked plastic (in cyan), and shocked aluminum (in blue). Thecolor contrast in the figure shows the shock profile distinctly.
Subsequently, x rays of 5.2-keV energy were used to generate the simulated radiograph inFig. 5 using refraction-enhanced radiography. The image at the detector is a magnified version ofthe figure shown here in the target plane, where the magnification is determined by the ratio ofthe distance between the detector and pinhole to the distance between the target and pinhole. For7imulated radiography, the Henke tables determine the refractive indices and the opacity of themedium corresponding to the density profiles obtained from the hydrodynamic code. The reasonbehind the particular choice of this refractive-index/opacity table is provided in Appendix A. Thehighest x-ray flux in the vacuum region of the image shows that the x rays do not get attenuatedhere. The x-ray flux represents the degree of transmission of the x rays in these regions thatproduces the contrast in the radiograph so that the shock profile is visible. The transmission thatis dependent on the density conditions and the material opacities gradually increases as the shockwave transits. The bowing effect of the shock wave is also prominently visible in the radiograph.
FIG. 6. The effect of refraction on the x-ray flux is shown here by taking the normalized difference in tworadiographs: one that includes refraction of x rays ( I R + A ) and one that does not ( I A ). The figure showsthat the refraction of x rays leads to changes in the x-ray flux near the shock front and the vacuum/plasticinterface. Due to the bowing effect of the shock front, it is important to account for the refraction ofthe x rays. Moreover, the sharp density gradient across the shock front leads to the deflection ofthe x rays because of the steep gradient in the refractive indices. To see the contribution of therefractive effects specifically, the density profile from
DRACO was radiographed by two methods:In the first method, both refraction and attenuation ( I R + A ) of the x rays were included, as seen inFig. 5; in the second method, only attenuation ( I A ) of the x rays was included. Figure 6 shows thedifference in the normalized flux ( | − I R + A / I A | ) between the two radiographs. The difference inthe x-ray flux in the bow-shaped shocked region and the material interfaces is evident, suggestingthe deflection of the x rays. In the aluminum region, the x rays have very low intensities and8he relative differences are not noticeable. This deflection of the x rays across the vacuum/plasticinterface was seen in the experiment, as discussed later. C. OMEGA experiment
FIG. 7. Image obtained from an x-ray framing camera on OMEGA. The image was captured around 8.63ns after the target was irradiated with a square pulse that launched the shock wave. The image shows thebowing effect of the shock wave in plastic with the main features labeled. The region marked with a blackbox shows the vacuum/unshocked plastic interface. Lineouts from this region shown in Fig. 8 are used toinfer the opacity of unshocked plastic as a consistency check.
Figure 7 shows the experimental image of the shock wave inside a planar plastic foil thatwas obtained on OMEGA. The unshocked CH, shocked CH, and aluminum pusher are easilydistinguishable in the image because of the difference in the degree of transmission of the x-raysin these regions. The bowing effect of the shock front is also prominent. It should be pointedout that the unshocked plastic region in the image extends over ∼ µ m in the direction of theshock-wave propagation; however, it should have been 250 µ m as per the experimental design.One possible reason why we do not see the full extent of the plastic in the image is an error intarget alignment. It is possible that the x rays were not incident normally on the target and wereattenuated by the large aluminum pusher foil in the target. This would also account for the sharpjump in the flux count of the shock profile at the plastic/aluminum interface. Instead, a smoothbow-shaped shock front was expected at the aluminum/plastic interface as we see in the DRACO simulations (see Fig. 5). Overall, this experiment shows that dynamic shock waves can be imaged9sing this technique.
FIG. 8. Lineouts across the vacuum/plastic interface from Fig. 7 are shown here. The red and blue linesrepresent the linear fits to the x-ray flux in the vacuum and unshocked CH respectively. The peak and valleyin the lineout show the effect of the refraction of the x rays across the CH/vacuum interface.
For verification purposes, a simple check to infer the opacity of the unshocked plastic regionwas performed. Lineouts from the interface of the vacuum and unshocked CH region marked witha black box in Fig. 7 are shown in Fig. 8. The x-ray flux in the vacuum is high since there is noattenuation. At the interface, the x-ray flux fluctuates because of refraction of x rays across theplastic/vacuum interface, making the x rays deflect from their trajectory of propagation. In theplastic, the x-ray flux decreases because of its higher opacity compared to the vacuum. In fact,these x-ray flux can be used to quantify the opacity of plastic using the relation I = I e − κ CH L , (1)where I , I are the x-ray flux in plastic and vacuum, respectively, L is the distance of propagationof the x rays through the plastic foil, and κ CH denotes the opacity of unshocked plastic. In theexperiment, L = µ m, i.e., the depth of the plastic foil. From the intercepts of the linear fits, I = . × and I = . × are obtained. Using these values in Eq. (1), the opacity of theunshocked plastic is inferred to be 16.5 cm − . This matches well against the opacity of 16.3 cm − for plastic of 1.05-gm/cm density as per the Henke tables at a photon energy of 5.2 keV.10 IG. 9. The transmission curves along the center of the beam axis were obtained from the simulated ra-diograph and the experimental image which show good agreement between them for plastic and aluminum.For reference, the transmission in the vacuum region is set to 1 since there is no attenuation.
D. Comparison of simulation to experiment
To compare the experimental and the simulation results, the transmission curves have beencompared in Fig. 9. The lineouts have been taken about the center of the beam axis of Figs. 5 and7. Both the lineouts have been scaled so that the transmission through the vacuum region is ∼
1. Webegin by inferring the opacity of the unshocked plastic obtained from simulation. By using Eq. (1)with I = .
51 and I = . κ CH = 16.7 cm − . This inferred opacity matches the experimental dataand atomic physics calculations of the Henke tables for x rays of 5.2-keV energy. The plot showsthat the relative degree of transmission in the unshocked plastic, shocked plastic, and shockedaluminum with respect to the vacuum is in good agreement between the experiment and refraction-enhanced radiography. The actual position of the shock front (adjacent to unshocked CH on theright in the figure) and the spatial extent of the shocked regions are in reasonable agreement witheach other. A plausible explanation for the lack of an exact match in the spatial extent of theshocked regions is caused by lateral heat flow from the decaying shock since the image was takenvery late in time after the laser pulse had been turned off. Typically, experimental investigations oflaser-driven shocks occur within a few hundred picoseconds at the end of the laser pulse, unlike thisspecific experiment. Therefore, any disagreement in the drive (that determines the shock strength)after the pulse was turned off will be amplified over time. Near the ablation front, the transmissionprofiles do not agree well. The target alignment error is the likely reason for this mismatch.11 small tilt in the large aluminum foil, when projected onto the 2-D image, can blank out thesignal over a significant region. However, in the ablation region, the results from the simulatedradiograph and the hydrodynamic simulations are consistent with each other (comparing Figs. 4and 5). The discrepancy in the ablation region could also occur from a lower mass ablation ratein simulations compared to the experiment. Overall, the good agreement between the relativedegree of transmission between refraction-enhanced radiograph and the experiment gives us theconfidence to rely on our modeling capability. The target alignment error and the imaging at a latetime after the pulse is turned off discussed above affect the spatial extent of the shocked/unshockedregions of a material but do not alter the relative degree of transmission in a material with respectto the vacuum.It should be mentioned that for refraction-enhanced x-ray radiography, it is not possible to infera 1-D density profile (lineout) accurately from the transmission profile. This is because of the x-ray refraction and the bowing of the shock wave, which is inherently a 3-D phenomenon. Even ifthe shock curvature is assumed to be axially symmetric and thereby account for a variable density(therefore, opacity) along the path length, the specific shape of the post-shock density profile atdifferent radii from the beam axis will determine the deflection of the x rays caused by refraction.In regular absorption radiography that infers a path-integrated density for a straight line ray path,x-ray refraction is neglected. However, a recent paper has shown that inflight density gradientscan be inferred for spherical implosions. IV. APPLICATION OF REXR TO PROPOSED EXPERIMENTS
In this section, we propose an experimental design to image multiple shock waves withrefraction-enhanced radiography. The objective is to view and locate the shock position dur-ing the main drive pulse when the experimental diagnostics such as VISAR fail to locate the shockposition. Due to the high intensity during the main drive, x-ray photoionization of the target aheadof the shock front leads to absorption of the optical diagnostic of the VISAR and no signal isgenerated.
This analysis will be useful for understanding the dynamics of shock coalescingand designing pulse shapes with multiple-picket pulses to achieve the desired adiabats.
Suchan analysis will also provide additional information about shock locations before they merge. Thisinformation will improve the way pulse shapes are currently designed based on VISAR data ofshock-merger experiments. This may also shed some light on why the shock-merger experimental12ata do not agree well with theoretical estimates for low adiabats. A. Picket pulse preceding the main target drive pulse P o w e r ( T W ) FIG. 10. A single-picket pulse is shown with a main drive pulse of 190-J energy. A pulse shape of this kindlaunches two shock waves whose positions are analyzed in this section.
Vacuum y ( n m ) –200 0–4004000 x ( n m) –200 0 x ( n m)CHCH 04020 F l ux c oun t t ( g / c m ) FIG. 11. The density profile from a
DRACO simulation (on left) and the corresponding radiograph (onright) are shown. The white boxes mark the region of interest where the two shock waves caused by thepicket and the main drive pulse are labeled. To see the spatial features clearly, only this region of interestwill be shown in Fig. 12 for five time steps.
13n this design, a picket with a main drive pulse as shown in Fig. 10 is used to drive shock wavesthrough a square planar plastic foil of 1.2-mm height and 200- µ m width. This setup is modeledwith a DRACO simulation that provides the density profiles over time. One such
DRACO profilealong with the corresponding simulated radiograph is shown in Fig. 11. The two shock fronts inplastic due to the picket pulse and main drive are distinctly visible. The degree of attenuationchanges with the density conditions in the plastic, which is captured by the radiograph.
FIG. 12. The spatial profiles obtained from
DRACO simulations and the corresponding simulated radio-graphs are shown here over time. With increasing time (going from left to right), the second shock wavelaunched during the main drive catches up with the slow-moving first shock wave launched from the singlepicket. The bowing effect of the first shock becomes more prominent over time compared to the secondshock wave, which is launched later.
Figure 12 shows the evolution of the spatial profiles and the corresponding simulated radio-graphs over time. These profiles highlight the regions of interest, although the radiographs weregenerated over a larger area. The figure shows the second shock wave that was triggered late intime catching up with the slow-moving first shock wave. This analysis provides the spatial infor-mation about the distance between the shock waves before they merge. It shows that by the timethe second shock wave is detectable around 3.1 ns, half of the energy from the main drive hasalready irradiated the target. The shocks merge around 4 ns, which is ∼ –80–120 –40 0 x ( n m) T r a n s m i ss i on ( a r b it r a r y un it s ) D e n s it y ( g / c m ) x ( n m)0.40.00.81.2 1023Time = 3.10 ns –80–120 –40 0 x ( n m)0.40.00.81.2 10234Time = 3.40 ns2046–80–120 –40 0 x ( n m) T r a n s m i ss i on ( a r b it r a r y un it s ) Time = 3.60 ns D e n s it y ( g / c m ) –80–120 –40 0 x ( n m)0.40.00.81.2 Time = 4.00 ns 20460.40.00.81.21st shock 1st shock 1st shock2nd shock 2nd shock2nd shock2nd shock Shocksmerge1st shock FIG. 13. The transmission (red) and density profiles (blue) across the center of the beam axis obtained fromFig. 12 are shown here. The transmission has been scaled so that the intensity in the vacuum region is 1.The spikes followed by the dip (local minimum) in the transmission curve correspond to the shock fronts aslabeled. The density profile also spikes at those points to illustrate this fact.
The density profile and degree of transmission of the x rays through these shock profiles aboutthe center of the beam axis are shown in Fig. 13. As expected, the transmission of the x rays reflectsthe density conditions in the shock waves and the post-shock density conditions. The spike anddip in the transmission correspond to the position of the peak densities of the shock waves. Thespike and dip occur due to the refraction of the x rays across the shock front. The position andshape of these features are dependent on the density profiles and are not affected by the finite sizeof the pinhole in the imaging system. The local minimum (or drop) in the transmission occurdue to the lowest transmission of the x rays through the relatively highest-density condition (localmaximum) across the shock front. The position of the shock front can be inferred from thesedrops by locating the local minimum in the transmission curves. The inferred shock positionsfrom the simulated radiographs were found to be in good agreement with shock locations definedby the density profiles. The fluctuations near the ablation front show that the x-ray trajectories getdeflected significantly in this region. This happens due to the changes in the density profile in the15ateral direction at different radii from the beam axis as a result of the ablation. In the experimentalimage shown previously, this was absent due to absorption in aluminum. To summarize, the shock-wave transit during the main drive is noticeable before it catches up with the first shock launchedfrom the picket pulse around 4.0 ns.
V. CONCLUSION
We have developed a method to generate images of shock waves that are relevant for ICFimplosions through refraction-enhanced x-ray radiography. This method is capable of inferringshock-wave position, both during the early stage of the laser pulse and the main drive, which isnot possible with the current experimental diagnostics such as VISAR. The point-projection back-lighting described in this paper provides a transverse 2-D image of the shock waves from whichtheir positions can be inferred for more than one shock wave before they coalesce. The usefulnessof the technique was illustrated by generating images of multiple shock waves and determiningtheir locations. The calculations indicate that it is important to account for the refraction of x raysacross density gradients of the shock fronts and material interfaces along with their attenuation andto choose appropriate opacity tables to interpret experimental observation. Efforts are underway touse REXR as a diagnostic for OMEGA experiments to image shock positions and mergers. Thispost-processor will be used to identify observables for the experiments that will be enhanced dueto the refraction in shock profiles.
ACKNOWLEDGMENTS
We would like to thank T. J. B. Collins, L. Antonelli and F. Barbato for useful discussions. Thismaterial is based upon work supported by the Department of Energy National Nuclear SecurityAdministration under Award Number de-na0003856, the University of Rochester, and the NewYork State Energy Research and Development Authority.This report was prepared as an account of work sponsored by an agency of the U.S. Govern-ment. Neither the U.S. Government nor any agency thereof, nor any of their employees, makesany warranty, express or implied, or assumes any legal liability or responsibility for the accuracy,completeness, or usefulness of any information, apparatus, product, or process disclosed, or rep-resents that its use would not infringe privately owned rights. Reference herein to any specific16ommercial product, process, or service by trade name, trademark, manufacturer, or otherwisedoes not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S.Government or any agency thereof. The views and opinions of authors expressed herein do notnecessarily state or reflect those of the U.S. Government or any agency thereof.
Appendix A: Sensitivity of REXR to opacities and refractive indices
REXR is sensitive to the refractive indices and opacities of the materials and the x-ray photonenergy. X-rays traveling through a medium are refracted and attenuated. The refractive index of amaterial ( n ) has a real and imaginary part. If we denote n = ( − δ ) + i β , then Re ( n ) = ( − δ ) leads to refraction and Im ( n ) = β controls attenuation. Specifically, the opacity of a material ( κ ) is related to the imaginary part of the refractive index through the relation κ = πβ / λ , where λ isthe x-ray wavelength.To incorporate the material properties accurately, the choice of the refractive indices and theopacity tables play a crucial role. The first-principles opacity tables (FPOT’s) and Henke ta-bles are two databases that can be used [by inferring opacity from Im ( n ) ] since they both providethe Re ( n ) and Im ( n ) of the refractive indices. Some of the standard choices for the opacity tables,such as the astrophysical tables, PROPACEOS tables, and NIST tables, do not provide thereal part of the refractive index. It is worth mentioning that exact quantum mechanical calculationsthat have been done for x-ray energies of less than 500 eV for the FPOT’s are in good agreementwith the standard NIST opacities for cold materials. Beyond that for higher-energy x rays, thevalues in the FPOT’s are extrapolated using the Drude model. It is noticed that the agreement be-tween the extrapolated values from FPOT’s and the standard NIST cold opacities differ by a largeramount (not shown here). Taking these into consideration, the Henke tables are the only avail-able choice to obtain the density-dependent Re ( n ) and Im ( n ) for different materials. To check thereliability of the Henke tables, the Re ( n ) is compared to the refractive indices obtained from theFPOT’s. The opacity corresponding to Im ( n ) of the Henke tables is compared to the NIST opacity.Figure 14 shows good agreement between the Henke tables and the above-mentioned databases.Since plastic (CH) of 1.05- gm/cm density has a plasma frequency of ω p <
30 eV, the refractiveindex increases beyond that and approaches 1 for x-ray energies (cid:29) ω p . Around photon energiesof 280 eV, both the refractive index and the opacity are discontinuous because of the K edge ofcarbon in CH when the 1 s -core electrons become accessible, which increases the electrical con-17 IG. 14. (a) The real part of the refractive index Re ( n ) obtained from the first-principles opacity tables(FPOT’s) and the Henke tables is compared for x-ray photon energy between 40 eV and 1 keV. (b) Theopacity ( κ ) , which is related to the imaginary part of the refractive index ( κ = πβ / λ ) , is comparedbetween the Henke tables and the NIST tables for x-ray photon energy between 30 eV and 6 keV. In thisplot, the NIST data and Henke table data correspond to room temperature values of 1.05-gm/cm CH whilethe FPOT values are at 5000 K. ductivity. It should be noted that the discontinuity in the Re ( n ) around photon energy of 280 eVdoes not coincide exactly between the FPOT’s and the Henke tables. This happens because of theunderestimation of the band gap (or correspondingly the K edge) in the density functional theorycalculation that generates the FPOT’s. Since Re ( n ) and κ are related to electrical conductivity,their values are discontinuous; but overall, the opacity of CH monotonically decreases after the Kedge.Intuitively, material properties such as refractive index and opacity under shocked conditionsshould be different from room-temperature conditions. To check this, we compared the Re ( n ) from the FPOT’s and opacity from the astrophysical tables for CH at solid density for different18emperatures. It was observed that temperature does not influence the refractive indices and opac-ities significantly so its dependence can be neglected. Nevertheless, it should be noted that thereis a K edge shift in the opacity values with a rise in temperature, but the K edge for C in plasticis at 284 eV while the x-ray imaging for shock-wave radiography is performed around 5 keV.The x-ray photon energy is another parameter that affects the refractive indices. Specifically,the real and imaginary parts of the refractive index scale as δ ∝ ( h ν ) − , β ∝ ( h ν ) − , (A1)with the x-ray photon energy h ν . This leads to a competition between the refractive and attenuativeeffects based on the x-ray energy. As the x-ray photon energy increases, the refractive effectsbecome less dominant as the refractive indices approach the refractive index of vacuum, i.e., 1(since δ → Appendix B: Refraction-Enhanced X-Ray Radiography
Here, we discuss the details of the refraction-enhanced x-ray radiography (REXR). X rays arelaunched in a point-projection radiography setup at a specific energy, and their trajectories aretracked as they pass through the material profiles obtained from
DRACO . The x rays are modeledas rays and their trajectories are determined through the ray equation described below. The ray-tracing approach or geometrical optics is applicable in our setup and a wave-based treatment isnot necessary. This is justified by the fact that the Fresnel number ( F ) of the setup is significantlylarger than 1. Fresnel number F is defined as F = L / d λ , where L is the characteristic size ofthe spatial features and d is the propagation distance of the x rays of wavelength λ . For F (cid:29) dds (cid:20) n ( r ) d r ds (cid:21) = ∇ n ( r ) , (B1)where r denotes the position vector of a point on the ray, ds is an element of the arc length onthe ray, and n ( r ) represents spatial dependence of the refractive index of the medium. Through a19hange of variables dt = ds / n , Eq. (B1) can be written as d r dt = n ( r ) ∇ n ( r ) . (B2)This second-order equation can be further simplified into two first-order differential equations bydefining an optical ray vector T : T = d r dt , d T dt = ∇ n ( r ) . (B3)This pair of ray-tracing equations is solved through a Runge–Kutta method described in Ref. 41.The refractive index in Eq. (B3) is obtained from the Henke tables based on the material densities.The method also accounts for attenuation of the x rays by tracking the cumulative decrease in theintensity of the x-ray along its path length through exp (cid:20) − (cid:90) κ ( r ) ds (cid:21) where κ ( r ) is the opacitydistribution of the medium. Henke tables provide the imaginary part of the refractive index, i.e., β and the opacities are determined from β by using κ = πβ / λ . The final position of the x raysin the detector plane and their intensities after they pass through the spatial profile are determinedfrom the ray-tracing equations. The final positions account for the amount of refraction or theextent of deflection of the x rays and their intensity determines the attenuation.Here, we have outlined a model system to show how this information is used to create the radio-graphs. Figure 15 shows a target partitioned into two regions of refractive index n , n and opac-ities κ , κ . When x rays are incident on the target normally, they travel along the direction fromwhere they were launched until they are refracted because of a change in the refractive index. Also,the rays are launched with an arbitrary intensity of 1; their final intensity of exp (cid:20) − (cid:90) L κ ( r ) ds (cid:21) is then determined based on their total path length L . In this model system, randomly launchedrays are incident normally on the rectangular slab shown in Fig. 15. Three such representativeray trajectories through the three regions are shown. The rays in regions I and III do not get de-flected, while the rays in region II get refracted by an angle of π / θ plotted in the figure shows the degree ofdeflection from its launching point and not from the interface. Due to the geometry of the setup, θ is maximum near the interface of regions I and II where the rays have the largest path length afterrefraction. This path length is decreased as we move from the interface of regions I and II to theinterface of regions II and III and the angle θ decreases linearly. Figure 15 also shows the intensityrepresenting the phenomena of refraction and attenuation. When only refraction is considered, themodel assigns a unit intensity to all the x-rays. There are no rays accumulated near the region I20 I n t e n s it y ( × a r b it r a r y un it s ) D e fl ec ti on a ng l e ( i ) ° x Region I Region II Region III2 m2 m2 m2 m n , l i r r r RefractionRefractionand attenuation , n
23 2 l (a)(b)(c) FIG. 15. (a) A rectangular slab with two materials having different refractive indices and opacities is shownalong with the path of representative rays (in red) in the three regions of the slab. (b) The angle of deflectionof the rays from their initial direction of propagation is plotted as a function of their launch position. (c) Theintensity of the rays shows the effect of refraction (or refraction and attenuation) by plotting the number ofrays with unit intensity {or intensity exp (cid:20) − (cid:90) κ ( r ) ds (cid:21) } as a function of their launch position. and II interface as they deflected and get accumulated towards the end of region II, increasing theflux in that region. If both refraction and attenuation are considered, the x-ray flux is weightedby the decrease in the intensity of the rays due to attenuation. If we choose κ =
10 cm − , thefinal intensity of the rays in regions I and III are exp ( − . ) and exp ( − . ) respectively. Through21 derivation, the intensity in region II is found to be exp (cid:110) − . (cid:104) x − ( − x ) cos ( π / ) (cid:105)(cid:111) , where x is thedistance between the launch position of the ray and region I, II interface. This result can also bederived by using Eq. (10) of Ref. 42. Since the intensity is exponentially decreasing with x , thisexplains the increase in attenuation from the left to right in region II. The analysis shown here forthe model system was used to construct the simulated radiographs shown in this paper. REFERENCES J. Nuckolls, L. Wood, A. Thiessen, and G. Zimmerman, Nature , 139 (1972). R. S. Craxton, K. S. Anderson, T. R. Boehly, V. N. Goncharov, D. R. Harding, J. P. Knauer,R. L. McCrory, P. W. McKenty, D. D. Meyerhofer, J. F. Myatt, A. J. Schmitt, J. D. Sethian,R. W. Short, S. Skupsky, W. Theobald, W. L. Kruer, K. Tanaka, R. Betti, T. J. B. Collins, J. A.Delettrez, S. X. Hu, J. A. Marozas, A. V. Maximov, D. T. Michel, P. B. Radha, S. P. Regan, T. C.Sangster, W. Seka, A. A. Solodov, J. M. Soures, C. Stoeckl, and J. D. Zuegel, Phys. Plasmas ,110501 (2015). A. R. Christopherson, R. Betti, J. Howard, K. M. Woo, A. Bose, E. M. Campbell, andV. Gopalaswamy, Phys. Plasmas , 072704 (2018). M. Herrmann, M. Tabak, and J. Lindl, Nucl. Fusion , 99 (2001). A. Kemp, J. Meyer-ter Vehn, and S. Atzeni, Phys. Rev. Lett. , 3336 (2001). R. Betti, C. D. Zhou, K. S. Anderson, L. J. Perkins, W. Theobald, and A. A. Solodov, Phys. Rev.Lett. , 155001 (2007). S. Atzeni and J. Meyer-ter Vehn,
The Physics of Inertial Fusion: Beam Plasma Interaction,Hydrodynamics, Hot Dense Matter , 1st ed., International Series of Monographs on Physics, Vol.125 (Oxford University Press, Oxford, 2004). J. H. Gardner, S. E. Bodner, and J. P. Dahlburg, Phys. Fluids B , 1070 (1991). V. N. Goncharov, T. C. Sangster, T. R. Boehly, S. X. Hu, I. V. Igumenshchev, F. J. Marshall,R. L. McCrory, D. D. Meyerhofer, P. B. Radha, W. Seka, S. Skupsky, C. Stoeckl, D. T. Casey,J. A. Frenje, and R. D. Petrasso, Phys. Rev. Lett. , 165001 (2010). K. Anderson and R. Betti, Phys. Plasmas , 5 (2004). L. M. Barker and R. E. Hollenbach, J. Appl. Phys. , 4669 (1972). T. R. Boehly, V. N. Goncharov, W. Seka, M. A. Barrios, P. M. Celliers, D. G. Hicks, G. W.Collins, S. X. Hu, J. A. Marozas, and D. D. Meyerhofer, Phys. Rev. Lett. , 195005 (2011).22 D. Cao, T. R. Boehly, M. C. Gregor, D. N. Polsin, A. K. Davis, P. B. Radha, S. P. Regan, andV. N. Goncharov, Phys. Plasmas , 052705 (2018). T. R. Boehly, E. Vianello, J. E. Miller, R. S. Craxton, T. J. B. Collins, V. N. Goncharov, I. V.Igumenshchev, D. D. Meyerhofer, D. G. Hicks, P. M. Celliers, and G. W. Collins, Phys. Plasmas , 056303 (2006). W. Theobald, J. E. Miller, T. R. Boehly, E. Vianello, D. D. Meyerhofer, T. C. Sangster, J. Eggert,and P. M. Celliers, Phys. Plasmas , 122702 (2006). S. X. Hu, V. A. Smalyuk, V. N. Goncharov, J. P. Knauer, P. B. Radha, I. V. Igumenshchev, J. A.Marozas, C. Stoeckl, B. Yaakobi, D. Shvarts, T. C. Sangster, P. W. McKenty, D. D. Meyerhofer,S. Skupsky, and R. L. McCrory, Phys. Rev. Lett. , 185003 (2008). D. S. Montgomery, A. Nobile, and P. J. Walsh, Rev. Sci. Instrum. , 3986 (2004). J. A. Koch, O. L. Landen, B. J. Kozioziemski, N. Izumi, E. L. Dewald, J. D. Salmonson, andB. A. Hammel, J. Appl. Phys. , 113112 (2009). L. Antonelli, S. Atzeni, A. Schiavi, S. D. Baton, E. Brambrink, M. Koenig, C. Rousseaux,M. Richetta, D. Batani, P. Forestier-Colleoni, E. Le Bel, Y. Maheut, T. Nguyen-Bui, X. Ribeyre,and J. Trela, Phys. Rev. E , 063205 (2017). T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, Nature , 595 (1995). A. Momose and J. Fukuda, Med. Phys. , 375 (1995). S. C. Mayo, P. R. Miller, S. W. Wilkins, T. J. Davis, D. Gao, T. E. Gureyev, D. Paganin, D. J.Parry, A. Pogany, and A. W. Stevenson, J. of Microsc. , 79 (2002). F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. , 258 (2006). J. A. Koch, O. L. Landen, L. J. Suter, L. P. Masse, D. S. Clark, J. S. Ross, A. J. Mackinnon,N. B. Meezan, C. A. Thomas, and Y. Ping, Appl. Opt. , 3538 (2013). J. MacFarlane, I. Golovkin, P. Wang, P. Woodruff, and N. Pereyra, High Energy Density Phys. , 181 (2007). R. Epstein, C. Stoeckl, V. N. Goncharov, P. W. McKenty, F. J. Marshall, S. P. Regan, R. Betti,W. Bittle, D. R. Harding, S. X. Hu, I. V. Igumenshchev, D. Jacobs-Perkins, R. T. Janezic, J. H.Kelly, T. Z. Kosc, C. Mileham, S. F. B. Morse, P. B. Radha, B. Rice, T. C. Sangster, M. J.Shoup III, W. T. Shmayda, C. Sorce, J. Ulreich, and M. D. Wittman, High Energy Density Phys. , 167 (2017). I. V. Igumenshchev, F. J. Marshall, J. A. Marozas, V. A. Smalyuk, R. Epstein, V. N. Goncharov,T. J. B. Collins, T. C. Sangster, and S. Skupsky, Phys. Plasmas , 082701 (2009).23 E. L. Dewald, O. L. Landen, L. Masse, D. Ho, Y. Ping, D. Thorn, N. Izumi, L. Berzak Hopkins,J. Kroll, A. Nikroo, and J. A. Koch, Rev. Sci. Instrum. , 10G108 (2018). P. B. Radha, V. N. Goncharov, T. J. B. Collins, J. A. Delettrez, Y. Elbaz, V. Yu. Glebov, R. L.Keck, D. E. Keller, J. P. Knauer, J. A. Marozas, F. J. Marshall, P. W. McKenty, D. D. Meyerhofer,S. P. Regan, T. C. Sangster, D. Shvarts, S. Skupsky, Y. Srebro, R. P. J. Town, and C. Stoeckl,Phys. Plasmas , 032702 (2005). D. G. Hicks, T. R. Boehly, P. M. Celliers, J. H. Eggert, S. J. Moon, D. D. Meyerhofer, and G. W.Collins, Phys. Rev. B , 014112 (2009). N. Ozaki, T. Sano, M. Ikoma, K. Shigemori, T. Kimura, K. Miyanishi, T. Vinci, F. H. Ree,H. Azechi, T. Endo, Y. Hironaka, Y. Hori, A. Iwamoto, T. Kadono, H. Nagatomo, M. Nakai,T. Norimatsu, T. Okuchi, K. Otani, T. Sakaiya, K. Shimizu, A. Shiroshita, A. Sunahara, H. Taka-hashi, and R. Kodama, Phys. Plasmas , 062702 (2009). S. X. Hu, L. A. Collins, J. P. Colgan, V. N. Goncharov, and D. P. Kilcrease, Phys. Rev. B ,144203 (2017). S. X. Hu, L. A. Collins, T. R. Boehly, Y. H. Ding, P. B. Radha, V. N. Goncharov, V. V. Karasiev,G. W. Collins, S. P. Regan, and E. M. Campbell, Phys. Plasmas , 056306 (2018). B. Henke, E. Gullikson, and J. Davis, At. Data and Nucl. Data Tables , 181 (1993). W. F. Huebner, A. L. Merts, N. H. Magee, Jr., and M. F. Argo, Los Alamos National Laboratory,Los Alamos, NM, Report No. LA-6760-M (1977). J. MacFarlane, I. Golovkin, and P. Woodruff, J. of Quant. Spectrosc. and Radiat. Transf. , 381(2006). J. H. Hubbell and S. M. Seltzer,
Tables of X-Ray Mass Attenuation Coefficients and Mass Energy-Absorption Coefficients from 1 keV to 20 MeV for Elements Z = 1 to 92 and 48 AdditionalSubstances of Dosimetric Interest S. X. Hu, L. A. Collins, V. N. Goncharov, T. R. Boehly, R. Epstein, R. L. McCrory, and S. Skup-sky, Phys. Rev. E , 033111 (2014). R. Fitzgerald, Phys. Today , 23 (2000). M. Born and E. Wolf,
Principles of Optics: Electromagnetic Theory of Propagation, Interferenceand Diffraction of Light , 7th ed. (Cambridge University Press, Cambridge, England, 1999). A. Sharma, D. V. Kumar, and A. K. Ghatak, Appl. Opt. , 984 (1982). J. A. Koch, O. L. Landen, L. J. Suter, and L. P. Masse, J. Opt. Soc. Am. A30