1. Introduction

The measurement of time- and space-resolving X-ray flux on hohlraum wall is of most importance in indirect drive inertial fusion, because it is essential in the ignition hohlraum design that needs to control the time-dependent capsule asymmetry [1] via beam phasing inside a cylindrical hohlraum. For instance, in the National Ignition Facility (NIF) [2], the time-varying flux symmetry on capsule is controlled by using two laser rings on each side of hohlraum and varying the power ratio between the two rings, and the power ratio is decided by the time-dependent radiation distribution on hohlraum wall.

Usually, the X-ray flux emitted from hohlraum can be directly measured with a Soft X-ray Spectrometer (SXS) outside one laser entrance hole (LEH), which views the emission from the hot laser source regions as well as the cooler re-emitting hohlraum wall and even part of LEH at the other end of hohlraum. Consequently, the measured flux sensitively depends on the direction of observation and can not distinguish the X-rays emitted by hot laser spot from those of cooler hohlraum wall. In addition, the observed radiation flux is usually affected by the cold plasmas outside the hohlraum as well as the shrinking of LEH. To distinguish the X-ray flux emitted by hot laser spot from that of cooler hohlraum wall, recently, it was tried to open additional diagnostic holes at the hohlraum wall [3]. Nevertheless, the hole closure effect, the change of hohlraum geometrical structure by digging the diagnostic holes and the laser spot movement could seriously affect the experimental results. The ViewFactor experiments allowing for the direct diagnosis of the x-ray drive just from the capsule point of view were done by MacLaren et al [4], but the time- and space-resolving flux accurate detection on hohlraum wall could still not be realized through their method. On the other hand, even though the X-ray streak and X-ray framing cameras [5–8] can help the time- and space-resolving X-ray diagnostics in ICF, it is hard for them to obtain the X-ray flux message quantitatively.

In this letter, we report a novel time- and space-resolving flux detector (SRFD) which can quantitatively detect temporal X-ray flux with a high space resolution of 0.11 mm. For the first time, we implement the simultaneous measurement using two SRFDs for the respective X-ray fluxes emitted from the laser hot spot and re-emitting area in hohlraum experiments on SGIII prototype laser facility [9]. According to our experiments, the ratio of laser spot flux to re-emitted flux is strongly time-dependent, and the area-weighted flux post-processed from the measured laser spot flux and re-emitting wall flux agrees well with that measured by using flat-response X-ray detector (F-XRD) [10, 11]. The observations are simulated by our two-dimensional radiation hydrodynamic code LARED [12] and understood with the power balance relationship.

2. Principle of SRFD

Our SRFD combines the X-ray flux detection technique of F-XRD [10, 11] with the pinhole imaging technique, so that the X-ray flux detection with a high space resolution can be achieved. A schematic drawing of SRFD is shown in Fig. 1. A pinhole of 0.1 mm diameter d₁ is used to create X-ray images of the hohlraum LEH, where the defining aperture (space-resolving aperture) of 2 mm size allows for two-dimensional observation of the hohlraum LEH image. This is defined as an imaging plane. An imaging plate also with a 2 mm diameter hole is pasted on the imaging plane, which can record the X-ray images of the hohlraum LEH created by the pinhole and the position relation between the X-ray images and the defining aperture. The F-XRD is a time-dependent and absolutely calibrated X-ray detector with a flat response in the photon energy range of 0.1–4 keV [15], placed behind the defining aperture. Hence, only the specified portion of the X-rays, exiting the hohlraum from LEH and streaming sequentially through the pinhole, the imaging plate and the defining aperture, can collide with the recording plane in the F-XRD. Following the F-XRD’s response to these collisions, the voltage signal recorded by the oscilloscope is output. The detected X-ray component will only correspond to the X-rays from a specified area of 0.2 mm diameter in the hohlraum. With this, high space resolution can be realized by targeting SRFD to any pre-specified region of hohlraum.

 

Fig. 1 Schematic diagram of the time- and space-resolving flux detector. In the figure, D is diameter of the defining aperture, d is diameter of the target area, and u and v are object distance and image distance, respectively.

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The pinhole-lens is the key component of SRFD. As we know, the spatial resolution of apparatus is mainly determined by the pinhole diameter, which is limited by machining accuracy. However, the pinhole diameter can also not be very small because the noise might affect the voltage signal recorded by the oscilloscope. In this experiment, we choose the optimized pinhole size as 0.1mm. The 0.1 mm diameter pinhole is positioned on a 0.1 mm thick tantalum substrate located at the center of the annular lens. The magnification and the object distance of the annular lens imaging are almost identical to those of the pinhole imaging, respectively. This makes images of LEH created by the annular lens nearly the same in position and size as the ones created by the pinhole. So when the LEH is illuminated by visible light, with a camera monitoring the position relation between the defining aperture and the images created by the lens, the lens can make auxiliary targeting in real time. In front of the pinhole-lens component there is also an adjustable diaphragm which is used to protect the lens and prevent X-rays from passing through the lens and causing signal interference. As a result, the images are magnified 10 times and the space resolution of the pinhole imaging is 0.11 mm, which can be determined by the pinhole diameter and imaging magnification. This also verifies the feasibility of using SRFD to detect X-ray flux from a region of 0.2 mm in size can also be verified.

In the SRFD, the X-ray flux F(t) at time t from a specified small region in hohlraum can be calculated as follows. From geometrical relations shown in Fig. 1, we have: $F(t)S\mathrm{\Delta }\mathrm{\Omega }/\pi =F(t)S{d}_1^2/4u^2$, where S is the size of the detecting area, ΔΩ is the solid angle of the pinhole, d₁ is the pinhole diameter and u is the object distance from hohlraum to lens. By denoting V(t) as the voltage signal recorded by the oscilloscope, then the detected X-ray energy is NV(t)/η_reR_os, where N is the attenuation value of signal, η_re is the average response of the F-XRD and R_os is the oscilloscope resistance. Hence, we have $F(t)=4u^{2}NV(t)/S{d}_1^2{\eta }_{re}{R}_{os}$.

3. Experiment

As the first application of SRFD, we use it to measure the respective fluxes of hot laser spot and cooler re-emitting wall region in the same shot of hohlraum experiments on SGIII prototype laser facility. A schematic diagram of the experimental setup is shown in Fig. 2. In this experiment, we use a relatively large hohlraum in order to avoid the influence of plasma filling or jetting on observations. The hohlraum is a 1.2-mm-diameter and 2.4-mm-long gold cylinder with two 0.85-mm-diameter LEHs. According to the plasma-filling model [16], n_e in this hohlraum is about 0.07 at the end of laser drive, far smaller than the criterion of n_e = 0.1 used for ignition hohlraum design. Here, n_e is the electron density in unit of the critical density. Eight laser beams, with 800J per beam at 0.35 μm, enter from two LEHs and irradiate the hohlraum wall at a 45° incident angle. The laser pulse has a 1.5 ns flat top with 100 ps rising and falling edges. The four laser beams from one LEH have 45° difference in azimuthal direction with those from the other LEH. The eight employed lasers are almost synchronously emitting light, with a synchronization uncertainty of 10 ps. Two SRFDs were arranged at 20° relative to the hohlraum axis, with 180° separation in the azimuthal direction. Thus, one SRFD can view the laser spot region while the other views the re-emitted wall. For comparison, an F-XRD was installed at 55° relative to the hohlraum axis. Theoretically, the observations from F-XRD should agree with the area-weighted flux post-processed from the laser spot flux and re-emitting wall flux which are measured by using SRFD. The time resolutions of all the three detectors are the same.

 

Fig. 2 Experimental setup for the first measurement of respective X-ray fluxes from hot laser spot and cooler re-emitting wall of hohlraum on SGIII prototype laser facility.

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Figure 3 shows the target regions of No. 1 SRFD and No. 2 SRFD, observed on the imaging plate pasted on the defining aperture of 2 mm diameter. To make it clearer, the scenography of the target regions of the SRFDs and the view field of F-XRD are also presented in Fig. 3, which are plotted according to the experiments by using a view-factor code. As shown, No. 1 SRFD well targets the cooler re-emitting wall, while No. 2 SRFD well targets the hot laser spot. However, for F-XRD, it views part of laser spot and part of re-emitting wall, respectively, with 0.101 mm² and 0.226 mm² in area.

 

Fig. 3 (a) The target region of No. 1 SRFD is a part of re-emitted wall, observed on the imagine plate (up) and drawn by view factor code (down); (b) The target region of No. 2 SRFD is a laser spot, observed on the imagine plate (up) and drawn by view factor code (down); (c) The view of F-XRD includes both re-emitted wall and laser spot, drawn by view factor code.

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Figure 4 shows the measured X-ray fluxes from No. 1 SRFD, No. 2 SRFD and F-XRD, hereafter denoted as F_w(t), F_l(t) and F_x(t), respectively. As shown, the flux from hot spot is obviously higher than that from re-emitting wall, and the flux measured by F-XRD is in between. In addition, F_l(t) rises much earlier and faster than F_w(t) and F_x(t), and also reaches its peak a little earlier. As presented, F_l(t) reaches peak at 1.505 ns, while F_w(t) and F_x(t) reach at 1.53 ns and 1.535 ns, respectively. Assuming the area of laser spot viewed by F-XRD is A_l and that of re-emitting wall is A_w, then post-processed area-weighted flux F_*(t) = [A_lF_l(t) + A_wF_w(t)]/(A_l + A_w). Taking A_l = 0.101 mm² and A_w = 0.226 mm², we can have F_*(t) from F_w(t) and F_l(t), which should agree with F_x(t) theoretically. As shown in Fig. 4, the post-processed result F_*(t) agrees very well with that observed by F-XRD. Quantitatively, the disagreement between the area-weighted flux F_*(t) and the flux F_x(t) from F-XRD is around 1.9 % at the peak value. This comparison partly verifies the validity of SRFD experimentally.

 

Fig. 4 Temporal evolution of observed fluxes F_w(t) (black line) by No. 1 SRFD, F_l(t)(red line) by No. 2 SRFD, F_x(t) (violet) by F-XRD and the area-weighted flux F_*(t) (blue), with experimental error bars of 13.9% for No. 1 SRFD, 12.7% for No. 2 SRFD and 10% for F-XRD. Inset is the enhanced view of the region near the flux peaks.

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We further give the ratio of F_l(t) to F_w(t) in Fig. 5. As shown, the ratio is strongly time-dependent and decreases monotonously from almost infinity at beginning, reaching 2 at 1ns and 1.5 at 1.5 ns when the laser pulse stops, and then gradually tends to 1 after the laser pulse stops. This is reasonable, because the re-emitting wall is triggered by emission from the laser spot, and therefore F_w(t) is generated not only later than F_l(t) but also also weaker during the laser pulse. After laser drive stops, F_l(t) decreases rapidly and gradually tends to be the same as F_w(t). It is interesting to compare the measurements of SFRD with that in Ref.[3]. The trend of F_l/F_w in Ref. [3] is similar to Fig. 5 in principle, but the details of their temporal histories are very different. In contrast to F_l/F_w in Fig. 5, which is strongly time-dependent during whole laser pulse, F_l/F_w in Ref. [3] drops rapidly in a short time and then keeps a nearly steady state with value of 2.3 until to laser ends. According to Ref. [3], their measurements are affected by the LEH closure and the laser spot movements.

 

Fig. 5 Ratios of fluxes emitted from hot laser spot to cooler re-emitting wall. The error bars of the experimental ratio is determined by the uncertainties of No. 1 SRFD and No. 2 SRFD.

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4. Hydrodynamic simulation

We compare above observations with our two-dimensional simulation code LARED-H and the post-processed calculations. LARED-H is a 2D radiation hydrodynamics code for hohlraum physics study and is widely used to design and study the hohlraum experiments on SG laser facilities [3, 12–14]. In LARED-H, the three-temperature approximation with the flux-limited model for electron heat transport is coupled with the averaged atomic model, the thermodynamic quantities are derived either from the ideal gas model or from data of realistic equation of state, and the mean opacity is calculated with our relativistic HFS self-consistent average atom model OPINCH. Although LARED-H has been used successfully to design and study many hohlraum experiments, it is hard to give accurate radiation spectrum and plasma status inside the hohlraums. In fact, the physics models in LARED-H, including atomic model, electron heat transport model, opacities and equation of states, are to be improved in order to increase the quality of the comparison with data.

In 2D simulations, the laser spots are given as annular rings on hohlraum wall, rather than separate round spots in experiment. To compare with observation, we consider two models in simulation. In the first, we use a narrow laser ring to keep the same spot area as in experiment, in order to assure the same laser intensity on wall. In the second, we use a wide laser ring which width is the same as that of the axial length of laser spot on wall in experiment. Of course, the laser intensity in the second model is lower than in experiment. Based on the simulation results, we use post-processor to obtain the X-ray fluxes emitting, respectively, in the same directions of two SRFDs and F-XRD installed in experiment. Shown in Fig. 6 are the corresponding fluxes of two models. We can see that the simulations reestablish the scenario of the experimental observations, i.e., the laser spot flux rises earlier and increases faster than the re-emitting wall flux, while the calculated flux at the direction of F-XRD lies in between. Also from simulations, the area-weighted flux F_*(t) = [A_lF_l(t) + A_wF_w(t)]/(A_l + A_w) agrees very well with F_x(t), the calculated flux at the direction of F-XRD.

 

Fig. 6 Corresponding fluxes from 2D simulation and post-processor: (a) with narrow laser ring, (b) with wide laser ring.

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Notice that there are some differences between observation and simulation. From detailed comparison of X-ray flux from hot spot, the simulated F_l(t) rises obviously more slowly than observation. Especially, F_l(t) with a wide laser ring rises most slowly due to its lower laser intensity. For the narrow ring model, F_l(t) reaches its peak at 1.3 ns and then rolls over, which happens 0.2 ns before the laser pulse ends because the laser deposited place moves out of viewed region at this time. For the wide ring model, F_l(t) reaches its plateau at 1.3 ns, much later than observation which reaches at 0.5 ns. As to the X-ray flux emitted from wall, the simulated F_w(t) with narrow ring is far from observation, while the simulated F_w(t) with wide ring agrees very well with observation except rolling over a little earlier. Furthermore, we compare the simulated ratio of F_l(t) to F_w(t) with observation in Fig. 5. For the narrow ring model, the ratio agrees with observation during most part of the laser pulse, but with remarkable deviations before 0.3 ns and after 1.3 ns. For the wide ring model, the simulated ratio rapidly drops to 2.4 at about 0.2 ns, and then keeps at around 2 during most part of the laser pulse. No matter how, it is hard to find any model in 2D simulation can agree with the observations during whole laser drive, which indicates the limitation of 2D code on 3D hohlraum simulation.

5. Power balance relationship and some scalings

Approximately, the ratio of F_l(t) to F_w(t) during laser drive can be estimated by using the power balance [2]. We denote the input laser power as P_L, the absorbed laser efficiency as η and the laser-to-X-ray coupling efficiency as ξ, then the X-ray power in hohlraum is ηξ P_L and the

X-ray flux from wall is F_w = ηξ P_L/[(1 −α)A_W + A_LEH]. Here, A_W is wall area, A_LEH is LEH area and α is the albedo of wall. Under a radiation that is proportional to time t, wall radiation temperature T_r can be expressed as T_r0(t/τ)^p and α can be expressed as $1-H/(T_r^{\gamma }{t}^{\beta })$, where T_r0 is radiation temperature at t = τ and τ is radiation pulse width. The parameters p, H, γ, and β are related to the temporal profile of radiation. On the other hand, the laser is deposited on the spots mainly via inverse bremsstrahlung and converted to X-ray mainly through atomic processes. We define ζ as the conversion efficiency from laser to hot spot emission towards hohlraum, and then the X-ray flux from hot spot is F_l = ηζP_L/A_L. Here, A_L is total area of the eight laser spots on wall. Thus, we have F_l/F_w = ζ/ξ × [(1 − α)A_W + A_LEH]/A_L. For the hohlraum used in our experiment, we have A W = 10.17 mm², A_LEH = 1.135 mm², and A_L = 1.57 mm². From 2D simulations for this model, we have ξ ≈ 81% and ζ ≈ 55%. By fitting the 2D simulation results, we have T_r0 =184 eV, p = 0.21, H= 9.44 for T_r in eV, γ = 0.68, β = 0.4. As shown in Fig. 5, the ratio obtained from power balance agrees approximately with that obtained from experiment and simulation.

Lastly, let us discuss the uncertainties in the SRFD measurement that might arise from four sources. The first set is produced in the calibration of the average response R_e, the measurement of the oscilloscope voltage signal V(t), the determination of the attenuation value of signal N and the oscilloscope resistance R₀, and it is 10% for both SRFDs. The second set is created in the measurement of diameter d₁ of the pinhole, and it is 4.70% for No. 1 SRFD and 3.75% for No. 2 SRFD. The third set includes the ones deriving from the determination of the size of the detecting area S, which is 1.08% for No. 1 SRFD and 1.6% for No. 2 SRFD. The fourth set is stemmed from the measurement of the object distance u, and it is 0.79% for No. 1 SRFD and 0.73% for No. 2 SRFD. As a result, the total relative uncertainty is 13.9% for No. 1 SRFD and 12.7% for No. 2 SRFD, as presented in Fig. 4. The relative uncertainty of the flux given by the F-XRD installed at 55° relative to the hohlraum axis is 10%.

In summary, we develop a novel time- and space-resolving flux detector, i.e., SRFD, which can measure the X-ray flux emitted from a specified region with very small size. For the first time, SRFD is applied to measure the respective X-ray fluxes emitted from hot laser spot and cooler re-emitting region in the hohlraum experiments on SGIII prototype laser facility. The observations are simulated by our two-dimensional radiation hydrodynamic code LARED, and understood with the power balance relationship in which the scalings associated with the albedo of hohlraum wall can be calibrated by the experimental measurements. We expect the SRFD will have versatile applications in ICF diagnosis, such as the M-band fraction of laser spot, the re-emitting flux from the capsule, and so on. Related works are undergoing. In addition, to further improve the spatial resolution, we plan to optimize the pinhole scale and increase the imaging magnification in future.