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Abstract
To precisely measure and evaluate X-ray generation and evolution in a hohlraum during an implosion process, we present a two-dimensional (2D) time- and space-resolved diagnostic method by combining a compressed ultrafast photography (CUP) system and a simplified version of space-resolving flux detector (SSRFD). Numerical experiment results showed that the reconstruction quality of the conventional CUP significantly improved owing to the addition of the external SSRFD, especially when a coded mask with a large pixel size was used in the CUP. Further, the performance of the CUP cooperation with the SSRFD was better than that of adding an external charge-coupled device or streak camera. Compared with existing ultrafast imaging techniques in laser fusion, the proposed method has a prominent advantage of measuring the 2D evolution of implosion by combining high temporal resolution of streak camera and high spatial resolution of SSRFD; moreover, it can provide guidance for designing diagnostic experiments in laser fusion research.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
In the study of inertial confinement fusion (ICF), the precise measurement of X-ray generation and evolution plays a crucial role in controlling physical experiment parameters and verifying simulation programs. Two principal approaches are used with lasers to drive an ICF implosion [1]. In the direct-drive approach, the laser directly irradiates the target filled with fuel, while in the indirect-drive approach, the laser is injected into the hohlraum of specific configuration and interacts with the cavity wall to excite X-ray radiation and then irradiates the target. Under the ablation of laser or X-ray, the surface material of the capsule is ablated and sprayed away, and a huge recoil pressure is generated, which compresses the fuel to the extremely high temperature and high-pressure state and achieves the ignition condition. The implosion experiment mainly focuses on the whole process of the implosion compression and the final implosion efficiency. Recording the self-emitting X-ray of the core during the implosion process with high temporal and spatial resolution is essential to evaluate the final implosion symmetry and bang time. With the development of X-ray streak and X-ray framing cameras [2–5] specifically designed for ICF measurements, these devices can achieve ultrahigh temporal resolution (TR) and spatial resolution (SR) of X-ray diagnosis [6–9]. However, the high TR of X-ray streak cameras is achieved at the expense of imaging dimension. The streak cameras used to obtain streaked radiographic image of the imploded capsule limit their imaging to only one spatial dimension. Multi-imaging X-ray streak camera can obtain ultrafast two-dimensional (2D) X-ray images [5]. However, its size of the viewing field is only 200 $\mu$m $\times$ 200 $\mu$m [10], and not suitable for imaging targets larger than the field of view due to overlapping. The X-ray framing cameras can simultaneously provide time- and space-resolving X-ray diagnostics; however, owing to its limited frame number, it cannot achieve ultrahigh TR in the implosion process. Up to 8 hot-spot images with sub-30$ps$ temporal resolution were obtained in the experiments [11–13]. In 2015, a new 2D space-resolving flux detection technique was developed to measure X-ray flux inside a hohlraum using a time- and space-resolving flux detector (SRFD) [14] and successfully conducted in the China laser facility [15]. Based on the radiation fluxes measured accurately by SRFD, radiation temperature distributions within a hohlraum was first explored in Ref. [15].
In the last few decades, high-speed imaging has made significant progress, especially with the invention of high-performance image sensors based on charge-coupled device (CCD) and complementary metal-oxide semiconductor, high-speed imaging speed has been able to achieve $10^7$ frames per second (fps) [16]. Such frame rates have been unable to meet the increasing demand for ultrafast imaging. Recently, the pump-probe technique has become the dominant method for capturing transient events [17]; however, it requires that the captured events are repeatable. Capturing nonrepeating time evolution events at ultrahigh speed is essential, especially in ICF. Compressed ultrafast photography (CUP) [18], as a receiving-only ultrahigh-speed imaging technology, was proposed in 2014. Such a method realized 2D computational ultrafast imaging on the basis of streak camera and compressed sensing. Related work has been proposed and effective results have been obtained, continuously improving the CUP—from monochromatic to multispectral imaging [19–22], from visible to ultraviolet light [23], from three dimensions to higher dimensions [24]; the reported frame rates have been able to achieve $7 \times 10^{13}$ fps [25] and $1.8 \times 10^{14}$ fps [26] by using different techniques. While achieving higher TR, SR is insufficient owing to the existence of encoding and decoding steps. To achieve better quality of the reconstructed dynamic image, except the original used TwIST [27] algorithm , a series of algorithm-based methods [28–30] and machine learning methods [31–33] have been adopted in the reconstruction process. Using different coded mask designs [34,35] and external CCD [17] also improve the reconstruction results. However, these methods are still not sufficient for the high SR diagnosis of implosion process.
In this paper, we present a 2D time- and space-resolved diagnostic method for the integrated implosion process by combining a CUP system and a simplified version of SRFD (SSRFD). Figure 1 shows the schematic diagram of the proposed method. Measurement via the CUP system is imaged through a pinhole on the left side of the hohlraum, whereas the SSRFD measurement is obtained through a pinhole on the right side of the hohlraum. Finally, the dynamic scenes of the integrated implosion process are reconstructed from these measurements by adopting a compressive sensing algorithm. To the best of our knowledge, no 2D time- and space-resolved diagnostic method with high resolution in the entire implosion process exists. Numerical experiment results show that the proposed method achieves better reconstruction quality than the conventional CUP. Compared with the CUP, the high TR of the streak camera was retained, and insufficient SR of CUP significantly improved by adding the SSRFD.
2. Method
2.1 CUP
In the conventional CUP system, dynamic scenes $I\left (x,y,t\right )$ are imaged via an objective lens and a 4f imaging system, spatially modulated by a digital micromirror device (DMD), and then measured using a streak camera. In the simulation, the pixel size of DMD needs to match with CCD. However, in the experiment, since the pixel size of DMD is not equal to that of CCD, the two need to bin pixels to match each other. The measured image $E\left (m,n\right )$ using the conventional CUP system can be mathematically expressed as follows [18]:
where $C$ is the spatial-encoding operator which means the dynamic screens are encoded by the coded mask, $S$ is the temporal-shearing operator which means the dynamic screens are sheared along one spatial direction by streak camera with time, and $T$ is the spatiotemporal-integrating operator which means the temporally integrating of the sheared dynamic screens. Given the information on the coded mask, the inverse problem of Eq. (1) is solved to reconstruct the dynamic scenes from the CUP measurement. This process can be formulated as follows:2.2 SSRFD
The concept of SRFD was first proposed in Ref. [14]. The SSRFD comprises an X-ray CCD at the periphery and a flat-response X-ray detector (FXRD) [36] at the center (Fig. 1). The FXRD is an absolutely calibrated flux detector for the X-ray energy range of 0.1–4 keV [14]. The size of the SSRFD is the same as that of CCD in the streak camera which is total 100$\times$100 pixels with pixel size of 10 $\mu$m $\times$ 10 $\mu$m. The peripheral CCD provides the time-integrated image of dynamic scenes with high SR, whereas the internal FXRD provides the X-ray flux information with TR. The diameter of internal FXRD is one-tenth that of the SSRFD. The internal FXRD can be customized and placed in an area where the dynamic scenes change the most with time. In this study, the internal FXRD was placed at the center of the SSRFD and and the temporal resolutions of FXRD were chosen range from 25$ps$ to 2$ns$.
2.3 Combination of the CUP and SSRFD
To reconstruct a 2D dynamic scene with high TR and SR, the measurement system is composed of a CUP and an SSRFD (Fig. 1). The data $E$ using the proposed method can be mathematically expressed as follows:
2.4 Evaluation metrics
To quantitatively assess the quality of recovery, the peak signal-to-noise ratio (PSNR) is defined. The PSNR between the source $x$ and the restored $y$ is defined as follows:
where $MSE(x,y)$ is the mean square error between $x$ and $y$. the $peakval$ is the maximum value of the image data type. The larger the value of PSNR, the higher is the reconstruction quality. Moreover, to evaluate the similarity of the reconstructed dynamic scene, the structural similarity (SSIM) [40] is defined as follows:3. Numerical experiments setup
X-ray self-emission images of hot-spot have been obtained experimentally in Ref. [12] and Ref. [13] from different laser facilities. Regardless of the inhomogeneity of the image intensity in some directions, the distributions of targets could be approximated as a Gaussian distribution. To make the simulation closer to the real data, the physical parameters of the hohlraum and capsule are derived from experiments at laser facilities in China. So, in these simulations, a 100 $\times$ 100 image with two Gaussian distributions was used as a base image, corresponding to the physical size of 1 mm $\times$ 1 mm, which means the CCD’s pixel size is 10 $\mu$m $\times$ 10 $\mu$m. We consider a dynamic scene set with 80 frames and a total time of 2 $ns$ with a TR of 25 $ps$. Similar to the result in Ref. [41], the maximum intensity of each frame with respect to time is shown in Fig. 2(a). Every pixel of each frame follows the relative change over time. The means of the Gaussian distribution along the x and y-axis are both 0 and the variances of the Gaussian distribution along the x and y-axis are both 0.03. Under the consideration of inhomogeneity in each frame, One of the Gaussian distributions with half intensity is shifted 40 pixels from the center. The correlation coefficient between the x and y-axis of each frame is slightly different ranging from −0.01 to 0.01 so that the difference in the radiation field distribution among frames is not only the intensity but also the outline. (Figs. 2(b)-2(d)). In ICF experiments, the laser interacts with the target to produce strong electromagnetic and ionizing radiation, including neutrons, gamma rays, X-rays and electromagnetic pulses. The sources of noise in experimental measurement are various and the radiation flux is usually very large, so we simply choose Gaussian noise as the noise model. In these simulations, the mean of noise was 0, and the variance of noise was 1$\%$ of the mean measured values of CUP and SSRFD, respectively.
Fig. 2. Numerical experiments setup of intensity: (a) maximum intensity of each frame; (b-d) different intensity distributions of three frames.
3.1 Validation test
In the conventional CUP, the coded mask is usually a DMD displayed at a pseudorandom binary pattern. In laser fusion, DMD is unsuitable as a coded mask for X-ray. We consider using a transmission mask with a large pixel size because of manufacturing difficulties. In this test, we use a mask of 10 times CCD unit pixel, that is, a 10 $\times$ 10 mask with a pixel size of 0.1 mm $\times$ 0.1 mm. The TR of SSRFD is 100 $ps$.
3.2 Comparison tests
To illustrate the advantages of our method, we set up four groups of numerical experiments, the first was the conventional CUP (marked as CUP), the second was two CUP systems using complementary masks (marked as Two_SC), the third was CUP with an external CCD camera placed at the right of a hohlraum (marked as SC_CCD), and the last was the proposed method with different TR for the SSRFD. The contribution to reconstruction quality from the pixel size of coded mask was also investigated in three cases: 10 $\times$ 10, 20 $\times$ 20, and 100 $\times$ 100 coded masks (pixel sizes of 0.1 mm $\times$ 0.1 mm, 50 $\mu$m $\times$ 50 $\mu$m, and 10 $\mu$m $\times$ 10 $\mu$m, respectively).
4. Results and discussion
4.1 Validation test
Figure 3 shows three frames of the ground truth and the reconstructed intensity profile; the general shape and intensity of distributions are all maintained using the proposed method. Focusing on the absolute pixel-wise error in the figure, the intensities of the proposed method were closer to the ground truth. The reconstructed intensity distribution of the conventional CUP distorted owing to noise and the large pixel size of the mask, but the proposed method does not. The proposed method achieves better smoothness.
Fig. 3. Comparison of three frames: (a) simulated ground truth, reconstructed frames of the conventional CUP system and the proposed method. (b) Absolute pixel-wise error of the conventional CUP system and the proposed method, respectively.
The PSNR between ground truth and reconstructed intensity distribution was calculated for the conventional CUP and the proposed method as 30.518 and 36.633, respectively. The PSNRs of all frames were calculated to evaluate the reconstruction difference during the implosion (Fig. 4(a)). The proposed method was superior to the conventional CUP at each point. Further, the SSIM of the proposed method was 0.9513, whereas that of the conventional CUP was 0.8958. Figure 4(b) shows the SSIMs of all frames, which has the same trend as Fig. 4(a). That is, the proposed method was superior to the conventional CUP. In indirectly driven laser fusion, the measurement of the radiation flux crucial in the diagnostic process. To compare the reconstruction quality of the two methods across frames, we plotted the normalized total intensity against the frame index (Fig. 4(c)). The reconstruction of the proposed method got closer to the ground truth than the conventional CUP, demonstrating a better reconstruction performance in the time domain.
Fig. 4. Comparisons of the conventional CUP with the proposed method for different frames: (a) PSNRs of different frames; (b) SSIMs of different frames. (c) Normalized total intensity of different frames.
4.2 Comparison tests
To quantitatively evaluate the reconstruction quality of the four methods, we calculate PSNR and SSIM in different cases, and the results are shown in Fig. 5. The PSNRs of the first three methods were 30.518, 32.640, and 32.389, respectively. From Fig. 5(a), when the TR of SSRFD exceeded 500 $ps$, the PSNR of the proposed method was higher than that of the first three methods. The SSIMs of the first three methods were 0.8958, 0.9439, and 0.9296, respectively. From Fig. 5(b), when the TR of SSRFD exceeded 200 $ps$, the SSIM of the proposed method was higher than that of the first three methods. The above result implies that when the TR of SSRFD was more than 200 $ps$, the proposed method was better than the other three methods in terms of SSIM and PSNR. This requirement for the TR of SSRFD can be easily achieved.
Fig. 5. Comparison of different SSRFD settings: (a) PSNRs of the proposed method under different SSRFD settings; (b) SSIMs of the proposed method under different SSRFD settings.
The mask size used in the above experimental results was 10 times the pixel size of CCD. To evaluate the effect of mask unit size on reconstruction quality, we tested two other mask sizes: one and five times the pixel size of CCD. We calculated the SSIMs and PSNRs of the four methods in different mask settings. The results of our method also showed the performance of the SSRFD with different TRs.
As shown in Table 1, the PSNRs and SSIMs of the reconstructed dynamic scenes improved with the decrease in the minimum pixel size of the mask. The improvements of the first three methods were larger than the last, implying that the first three methods were more dependent on the pixel size of the mask than the proposed method. With the increase in the pixel size of coded masks, the advantages of the proposed method were more. In fusion experiment diagnosis, the pixel size of coded masks is unlikely to be as fine as that of the conventional CUP system for visible light. When the mask size was small, the proposed method was better than the first three methods in terms of both SSIM and PSNR. Moreover, the reconstructed results improved by elevating the TR of the SSRFD.
Table 1. PSNRs and SSIMs of the reconstructed dynamic scenes using different methods
To illustrate the relationship between the number of FXRD and the reconstruction of dynamic images, a case was studied where one FXRD was located in the center of the CCD and there were some FXRDs in the adjacent regions. In this case, we did not consider the effect of noise, and the temporal resolution of FXRD was fixed at 100$ps$. The PSNR and SSIM of the reconstructed dynamic image were calculated under different mask sizes as shown in Fig. 6. With the increased number of FXRD, the reconstructed quality of dynamic images showed a trend of decreasing at first and then increased slightly. The reconstructed results of multiple FXRD were not as good as those of single FXRD. One possible reason was that FXRD had a poor spatial resolution, and more FXRD would reduce the high spatial resolution characteristics of SSRFD. In addition, the results were affected by the size of FXRD which was not considered yet and could be investigated in future work.
Fig. 6. Comparison of different number of FXRD: (a) PSNRs of the proposed method under different number of FXRD; (b) SSIMs of the proposed method under number of FXRD.
5. Conclusion
In summary, based on the SSRFD and CUP, a 2D space- and time-resolved diagnostic method for integrated implosion targets was developed. Compared with the CUP, the high TR of the streak camera was retained, and insufficient SR of CUP was significantly improved by adding an SSRFD. Situations of coded masks with different pixel size were also investigated. Further, the performance of CUP cooperation with the SSFRD is better than that of adding an external CCD or streak camera. Numerical experiments showed that when the TR of the entire system was limited by a TR of the streak camera, a smaller pixel size of coded masks or an SSFRD with higher TR significantly improved the SR of reconstruction. In future studies, the proposed method can be further combined with pinhole array imaging, which has potential applications in three-dimensional space- and time-resolved diagnosis.
Funding
National Natural Science Foundation of China (11675014, 11675157, 11805180).
Acknowledgments
The authors thank Prof. Molin Ge for the valuable discussions.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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