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Abstract
The X-ray sources for Compton radiography of ICF experiments are generated by using intense picosecond lasers to irradiate wire targets. The wire diameter must be designed thin enough, for example ∼ 10 µm in many published works, to comply a high spatial resolution. This results in a low laser-target interception, which limits the photon yield. We investigated a technique of coded-source radiography based on laser-driven annular sources via Monte Carlo and PIC simulations. The annular X-ray source is formed by laser irradiating tube target in which the effect of electron recirculation plays an important role. We proved that this technique has an increased spatial resolution and contrast than that using the Gaussian source produced by wire targets. Therefore, the diameter of the backlighter target can be significantly increased to uplift laser-target interception without compromising on spatial resolution. This contributes towards a reconciliation between the spatial resolution and photon yield for Compton radiography. The results predict the possibility of improving source photon yield by several times in future experiments.
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1. Introduction
The advent of the chirped-pulse amplification (CPA) technology [1] has led to a rapid development in high-power lasers [2], which unlocked new physics regimes of novel particle accelerators based on the ultra-short intense laser [3–7]. Among these, picosecond petawatt laser-driven hard X-rays motivated by the application in backlighting for inertial confined fusion (ICF) [8] and high energy density (HED) physics have garnered widespread attention [9–15].
In ICF experiments, in order to diagnose the fuel density near peak compression, wire targets are irradiated with intense picosecond lasers, generating quasi-point X-ray sources to backlight the capsule. This technique, namely Compton radiography [16–19], can provide highly temporally (∼ 10 ps) and spatially (∼ 10 µm) resolved images. However, under ignition conditions (Te ∼ 10 keV), self-emissions of the imploded capsule are so intense that even above the lower limit of the high-pass-filtered diagnostic photons, i.e., 70 keV, the photon yield approaches 1012 photons/sr. On the other hand, previous experiments of laser-driven X-rays using wire targets thinner than 10 µm measured the energy conversion efficiencies (CE) from laser to X-rays of 70 - 200 keV to be 1 - 6 × 10−4 [17,20]. Assuming a 1-kJ laser input and isotropic emissions of 100 keV photons, it corresponds to a yield of 0.5 - 3 × 1012 photons/sr, which is inadequate for high signal-to-noise ratio (SNR).
CE is directly proportional to the laser-target interception. Picosecond laser facilities constructed around the world, such as NIF-ARC [21], PETA [22], and SG-II UP [23], deliver laser beams with foci of ∼100 µm. The backlighter wire applied in current experiments are commonly 5 - 30 µm in diameter, which means the interception is less than 30%. However, an increased diameter is detrimental to the spatial resolution, as the resolvable spatial size of a point-projection imaging system is determined by:
where, d is the detector pixel size, M is the magnification of the imaging system, and a is the backlighter size. When M is much greater than 1 and d is small enough, $Bw$ is roughly equal to the backligher size a.Improving spatial resolution and source photon yield simultaneously is crucial for Compton radiography. Tommasini et al. used 25-µm-diamter wires and deblurred the images by deconvolving the backlighter. As a result, the resolution of about 10 µm was achievable for the reconstructed images [14,19]. However, this method doesn’t work well for those even larger backlighter sources as described in the following context. Variation of the source distribution between shot to shot will also diminish the accuracy of reconstruction. Zhang et al. proposed a coded-source radiography based on wire array target, in which the single wire is replaced with an array of very fine, regularly arranged wires to generate a point-array X-ray source [24]. They used it to backlight the object, forming encoded images and then obtained reconstructed images by decoding. This technique achieves higher spatial resolution, but has a minimal impact on the laser interception improvement since employing too many fine wires is very difficult in practice.
The aim of this work is to develop a technique of picosecond laser-driven annular coded-source X-ray radiography. It involves utilizing laser to irradiate a large-diameter, thin-walled tube target, as depicted in Fig. 1. The generated X-ray source has the same shape as the tube’s cross-section, which we use to backlight the object to form an convolved image. A high-resolution reconstructed image will be obtained through deconvolution. This technique realizes enhancing laser interception by increasing the tube diameter without compromising on spatial resolution. Thus, the diameter can be designed to make full use of the driving laser and maximize the source photon yield.
In this paper, we conducted simulation studies on this kind of improved radiography. The paper is structured as follows: Section 2 describes the advantages in image quality of the annular coded-source radiography and the influence of the backlighter’s structure parameters using Monte Carlo (MC) codes. In this section, we assumed that the intensity distribution of the annular source is completely uniform. Section 3 verified the assumption through demonstrating a generation of the laser-driven annular source using a combination of Particle-In-Cell (PIC), hydrodynamic and MC codes. Section 4 makes a conclusion.
2. Simulations of X-ray backlighting
2.1 Setup of Monte Carlo models
We conducted the X-ray backlighting simulations through MC platform Geant4 [25]. From number of ready to use physical processes suites, the “FTFP_BERT_EMZ” physics list class was selected. The physical processes related to photon absorption include photoelectric effect, Compton scattering, e-/e + pair production and Rayleigh scattering.
The backlighted object and IP were placed 10 mm and 500 mm away from the source respectively. The space distribution of energy deposition in the IP sensitive layer was used as original image, which was processed to reconstruct the real-life image of the object. The IP image was 12 mm × 12 mm with pixels of 50 µm × 50 µm. The X-ray source was isotropic and monoenergetic at 20 keV in order to maximize the IP sensitivity [26,27]. Figure 2(b) illustrates the cross-section of the backlighted object which was made of tantalum (Ta) and 105 µm × 70 µm × 50 µm in size. It contains three groups of uniform linearly spaced cutting lines. They are 10 µm, 5 µm and 2.5 µm in width respectively. The blue, red, and black dashed boxes represent the ‘knife-edge area’, ‘10 µm lines’ and ‘5 µm lines’ respectively (see below).
Fig. 2. (a) Scheme of the MC simulations of X-ray backlighting using Geant4. (b) Cross-section of the backlighted object. The red, black and blue dashed boxes indicate ‘10 µm lines’, ‘5 µm lines’, and ‘knife-edge area’, respectively. (c) IP recorded original image illuminated by an ideal point-like source.
Different types of backlighter sources were simulated, including annular, point-like, disc-shaped and Gaussian sources. The latter two are produced by wire targets which are commonly used in current experiments. Their source photon yield ranged from 1.3 × 109 photons/sr to 2.6 × 1010 photons/sr. Figure 2(c) present the IP recorded original image illuminated by an ideal point-like source with yield of 1.3 × 1010 photons/sr. It gives a clear shape of the object with very sharp edges, as point-like sources with infinitely small size is the most ideal for backlighting. In Fig. 2, as well as all the original and reconstructed IP images throughout the text, the intensity represents energy deposition in a pixel in unit of keV.
2.2 Image reconstruction
The images received by IP were processed using Richardson-Lucy (R-L) deconvolution algorithm [28]. The R-L algorithm, named after its creators William Richardson [29] and Leon Lucy [30], is an iterative algorithm commonly used in image deconvolution that have been blurred by a known point spread function (PSF). The algorithm's foundation lies in the concept of maximum likelihood estimation, where it iteratively estimates the real-life image by comparing the originally observed image with the reconstructed image. It involves alternating between two steps: 1. the current estimate of the real-life image is convolved with the known PSF to produce a convolved image; 2. the observed image is divided by the convolved image. This division step gradually improve the reconstruction by minimizing the difference between the observed and estimated images. The basic formula is:
To verify the deconvolution effect in our models, a 10-µm-diameter disc-shaped source (emitting photons from positions distributed homogeneously within a circle) was simulated. Figure 3(a) presents the original and reconstructed images obtained with 5 to 30 iterations, and Fig. 3(b) presents profiles of the ‘10 µm lines’ on them. With increased iteration number, the profiles evolve towards a square wave indicating effectiveness in restoring sharpness to images. However, it is sensitive to noise and can amplify noise during the iterative process. As shown in Fig. 3(a), speckles becomes more significant with more iterations, which do not represent any real structure but numerical noises of fitting the original noises in the image too closely.
Fig. 3. (a) Original and reconstructed images with different iteration numbers, illuminated by a disc-shaped source. The red dashed box indicates ‘bright area’. (b) Profiles of the ‘10 µm lines’ on images of (a). (c) Profile and derivative of the ‘knife-edge area’ on reconstructed image with 10 iterations. (d) The CNR and spatial resolution versus the iterations.
Contrast-to-noise ratio (CNR) were calculated using:
Additionally, we calculated the spatial resolution through knife-edge method as depicted in Fig. 3(c). As the iteration number increases from 1 to 30, the resolution increases from 50 to 400 pairs/mm and the CNR declines down to 1 as demonstrated in Fig. 3(d). Therefore, it is important to comprehensively factor in spatial resolution, CNR, and sharpness to determine the iteration number which achieves the finest reconstruction.
2.3 Improved spatial resolution and contrast
Annular and Gaussian sources with 25-µm diameter and yield of 1.3 × 1010 photons/sr were compared. The two-dimensional Gaussian source distribution follows $I(r )\propto \textrm{exp}({ - {r^2}/2{\sigma^2}} )$, where r is the position relative to the center, $\sigma $ is the std equal to 12 µm which was set based on the experimental results conducted on NIF-ARC [19]. The annular source has a 15-µm inner diameter with 5-µm ring width. Reconstructed images are displayed in Fig. 4(c) and (f). The Gaussian case appears more blurred, and the ‘bright area’ seem to have more non-convergent structures compared to the annular one. We can clearly resolve the 10-µm and 5-µm-spaced cutting lines on the reconstructed images of the annular source, but for the Gaussian case, the 10-µm-spaced cutting lines can barely be resolved whereas the 5 µm cannot completely. The spatial resolutions are 5.8 µm and 4.0 µm for the Gaussian and annular sources respectively, consistent with the resolvable cutting lines on their reconstructed images.
Fig. 4. Source distribution, along with their original and reconstructed images for Gaussian and annular sources.
Figure 5 presents the profiles of the ‘10 µm lines’ and ‘5 µm lines’ on the reconstructed images. The same data of an ideal point-like source (see Fig. 2(c)) are plotted for comparison, which exhibits a square wave shape. However, they degrade into sine-like waves for the Gaussian and annular sources attributed to a resolution degradation. Image contrast were calculated using:
Fig. 5. Profiles of the (a) ‘10 µm lines’ and (b) ‘5 µm lines’ on reconstructed images of Gaussian, annular, and ideal point-like sources.
2.4 Contrast enhancement with increased diameter
We proved that the annular coded-source radiography has a higher spatial resolution and contrast intrinsically than the Gaussian sources. In situations of even larger diameters, this advantage will become more noticeable. Figure 6(a) and (b) present the original images illuminated by Gaussian and annular sources with 100-µm diameter and yield of 1.3 × 1010 photons/sr. The $\sigma $ of the Gaussian distribution was set to 48 µm, and the annular source had a 2-µm ring width. Because of too large backlighter, the image from the Gaussian source degenerates into an obscured shadow which carries such little information that reconstruction cannot be performed. Figure 6(c) presents the reconstructed image of the annular source. A rough sketch can be discerned, though it has a poorer image quality than that of the 25-µm-diameter annular source, manifested by much more serious numerical noises. This is attributed to the effect of diameter, as will be discussed below. Nevertheless, annular coded-source scheme mitigates the damages of the large backlighter to the reconstructed image’s quality, providing a novel insight for increasing the contrast.
Fig. 6. Original images illuminated by (a) Gaussian and (b) annular sources with 100-µm diameter. (c) Reconstructed image from (b).
Next, two sources under conditions with self-emission from the object were simulated. One is classical disc-like with 5-µm diameter and yield of 1.3 × 109 photons/sr. Another is annular with 50-µm diameter, 5-µm ring width and yield of 1.3 × 1010 photons/sr. Here we simply assumed source photon yield to be proportional to the diameter. Self-emissions of the backlighted object is 1.3 × 109 photons/sr, which created uniform additive background noises on the IP. Note that to save computation time, we set the photon yields of the backlighter sources and the self-emission to be reduced by a factor of 103 compared to their actual range. Figure 7(a) and (b) present the original images of the two sources and (c) presents the reconstructed images of the annular source. The disc-shaped source produces a sharper image with a resolution of ∼5 µm, consistent with the source size. Figure 7(d) and (e) show the profiles of the ‘10 µm lines’ on (a) and (c). Note that the curve of the disc-shaped source has been normalized to an appropriate level. The image’s contrast is dramatically suppressed by the self-emissions in the disc-shaped case (∼ 3.04). However, due to 10 times brighter backlighter resulting from the increased diameter, both profile and contrast (∼ 12.76) in the annular case remain similar to the noise-free case discussed above.
Fig. 7. Original images illuminated by (a) disc-shaped and (b) annular sources with different sizes and yields, accompanied by uniform additive background noises. (c) Reconstructed image from (b). (d)-(e) Profiles of the ‘10 µm lines’ on (a) and (c).
2.5 Influence of structure parameters
Figure 8 shows the reconstructed images for annular sources with different structure parameters but the same photon yield of 1.3 × 1010 photons/sr, on which ‘D’ indicate the diameter, ‘W’ indicate the ring width, and ‘it’ indicate the iteration number. The finest resolvable cutting lines are 2 µm for D10W2, 5 µm for D50W5, and 10 µm for D50W10, which are consistent with their ring widths.
As demonstrated in Fig. 9, for annular sources with the same diameter, the thinner the ring width, the steeper the CNR decline with iterations. For a very thin width, numerical noises may develop too severely before sufficient iterations to converge to the real-life image. In other words, the reconstructed image’s quality will be limited by the rapid development of numerical noises. For D50W2, we chose 17 iterations to achieve the optimal reconstruction, which is only half of 30 iterations for D50W10. Therefore, it suffers from more pronounced non-convergent structures, and even so, has a lower CNR of 3 than that of 4 for D50W10, so cannot resolve the 2.5-µm-spaced cutting lines despite having a ring width of 2 µm. On the other hand, as displayed in the insert, D50W10 which has the thickest ring width reproduces profile of the ‘10 µm lines’ closest to that of the point-like source. This is attributed to an increased tolerance to iterations.
Fig. 9. CNR versus the iterations for 50-µm-diameter annular sources with different ring widths, marked are the chosen iteration numbers. Insert: Profiles of the ‘10 µm lines’ on reconstructed images
The diameter of annular source is a key parameter affecting the required iteration number. As demonstrated in Fig. 10, for the same ring width, the rate of CNR decline is roughly the same, but fewer iterations are needed when the diameter is smaller. Hence, one can use a small diameter to prevent the numerical noise development in reconstruction. For example, D10W2 achieves the optimal reconstruction after 11 iterations with 2.5-µm-spaced cutting lines resolved clearly, and the CNR is 4. But for D100W2, due to more iterations required to achieve the convergence, the images will suffer from either low fidelity or strong numerical noises during the reconstruction. Figure 8(e) represents the reconstructed image after 29 iterations with a CNR lower than 1.6.
Fig. 10. CNR versus the iterations for 2-µm-width annular sources with different diameters, marked are the chosen iteration numbers.
3. Picosecond laser-driven annular X-ray source
We presented the advantages in image quality of the proposed annular-source radiography relative to the one using classical backlighters in Section 2. In this technique, an annular X-ray source with uniform intensity is critical since it affects the reconstruction accuracy. Here we demonstrate the generation of such a desired annular source driven by ultra-intense laser.
The effect of electron recirculation in the interactions of ultra-intense laser with mass-limited target was well-studied [31–37] and plays an important role in the formation of a uniformly distributed annular source. When ultra-intense laser hitting a tube target, sheath fields formed on both surfaces of the tube and confine the laser-accelerated electrons to oscillate and transport in the target volume as depicted in the insert of Fig. 11(c). The electrons ultimately distribute uniformly within the target, resulting in the uniformity of the intensity distribution of the X-ray source on the target cross-section. Many effects contribute to the production of X-ray source including preplasma, LPI, electron transport and collision in the target. To include these effects, a variety of codes including FLASH [38], EPOCH [39,40] and Geant4 were used to verify the scenario.
Fig. 11. (a) Scheme of the 2D PIC simulation of laser interactions with tube target using EPOCH. (b) 2D Electron number density and (insert) the profile in the laser propagation region obtained from FLASH. (c) Scheme of the MC simulation of electron incident to the tube target using Geant4. Insert: Front view of a typical trajectory of electron recirculation in the target.
3.1 Setup of the models to simulate the generation of the annular X-ray source
First, we employed the PIC code EPOCH to model the interaction of a laser with a tube target containing preplasma. It provided the hot electron spectrum and the distribution of sheath fields during the laser interactions. The initial preplasma structure was determined using the hydrodynamics code FLASH (version 4.6.2). Next, we incorporated the PIC simulation results into a MC model to simulate electron interactions with the tube target for X-ray radiation using Geant4.
The EPOCH model was considered in 2D (X, Y) plane as illustrated in Fig. 11(a). The p-polarized laser pulse whose electric field is described as Gaussian profile, i.e., $E({y,t} )= {E_0}\textrm{exp}( - {y^2}/({{w^2}} )\textrm{exp}({ - {{({t - {t_0}} )}^2}/w_t^2} )$, is incident from the left boundary and propagates to impacts the gold (Au) tube target with preplasma. ${E_0}$ is the peak amplitude corresponding to a peak laser intensity ${I_0} = 4.4 \times {10^{18}}\; W/c{m^2}$. $w = 25\; \mu m$ is the beam waist radius. $2{w_t} = 500\; fs$ is the pulse duration. ${t_0} = 300\; fs$ is the reference time when the peak intensity occurs on the left boundary. The tube target has a 20-µm outer diameter and a 5-µm wall thickness. The electron density is 5.9 × 1022 /cm3 and the ionization degree Z* is 1.
As is well known, preplasma formed on the target surface by the prepulse is very important because it has a significant impact on the generation of hot electrons [41]. We conducted a 2D hydrodynamic calculation using FLASH to determine the structure of preplasma before the short laser pulse arrives. In this simulation, a 5-ns prepulse with contrast of 10−6 (4.4 × 1018 W/cm2) irradiating a semi-infinite gold planar target as illustrated by Fig. 11(b). The hydrodynamic model was based on the Eulerian grid, using unsplit Hydro solvers, and the Spitzer conductivity was implemented. Figure 11(b) gives the 2D electron number density and (insert) the profile of it in the laser propagation region as marked by the white dash line, just as the laser has just ended. It exhibits a density peak at critical density (∼1021 /cm3) extending 10 µm at a distance of 100 µm. This may results from localized laser absorption, leading to ion density accumulation and high ionization. Since hot electrons are mostly accelerated in this region before moving forward into the solid target [42–44], we simplified preplasma as a layer of 10-µm-thick preplasma of critical density on the outer surface of the tube as illustrated in Fig. 11(a).
Numerically, the PIC simulation domain was 80 µm × 80 µm, divided into 10000 × 10000 cells, corresponding to the cell size Δx = Δy = 8 nm which is considered to resolve the relativistic skin depth and plasma wave. 5 macroparticles for electrons and 0.5 macroparticles for ions were set in each cell. The boundary conditions were “simple_outflow” which means particles and electromagnetic waves impinging on it are fully transmitted. The ions were immobile. The collision among charged particles was included. The total simulation time was 2.5 ps.
Figure 11(c) illustrates the modeling of electrons hitting a tube target and producing X-rays using Geant4. The insert depicts a typical trajectory of electron recirculation under the constraint of the sheath fields. The tube target is made of Au, with a 20-µm outer diameter, a 5-µm wall thickness, and a 300-µm length. Sheath fields acquired from PIC (see Fig. 12(b)) were applied with axial symmetry on the tube surfaces. Note that we omitted the preplasma in the MC model, since we are focusing on hard X-rays with spectral range so high as tens to hundreds keV, that the emission from the preplasma can be neglected. Electrons acquired from PIC (see Fig. 12(c)) were shot at the end of the outer surface (shadow area), and enter the target along the x-axis without divergence. The emittance area extends 50 µm in z-direction, covering half circle of the surface.
Fig. 12. (a) Er distribution at 800 fs. (b) Er profile along the red dashed line of the central axis shown in (a). (c) Energy spectrum of the hot electrons inside the target volume at 500 fs.
3.2 Simulation results
Figure 12(a) shows the distribution of the electrostatic fields perpendicular to target surfaces (Er) from PIC model. It is a snapshot at 800 fs when the driving laser has just concluded and Er has reached a steady state. The fields exhibit an axially symmetric manner both outside the preplasma and inside the solid target. The maximum Er outside is about 3.2 TV/m, in good agreement with that of a planar target model we studied earlier [35]. Figure 12(b) displays the Er profile along the red dashed line of the central axis shown in (a). Generally, it decreases exponentially with the radial distance. The outer field extends about 7 µm, and the inner field extends about 2 µm when they decrease by one order of magnitude. The Er equals 0 at the central point.
Figure 12(c) shows the energy spectrum of the hot electrons inside the solid target at 500 fs, immediately following the peak laser-plasma interaction. It exhibits a double-temperature distribution, with the temperatures at 120 keV and 1.2 MeV. Note that it doesn’t contain all the accelerated electrons integrated over time which is difficulty to obtain in PIC simulation. But the spectrum snapshot at 500 fs reaches peak in both spectral range and particle number, hence it can reflect the general characteristics of the electrons when considering the radiation by them.
In the MC model, 107 electrons were simulated following the energy spectrum as described in Fig. 12(c). Figure 13(a) and (b) respectively show the energy spectrum of the generated photons and the distribution of their generation positions in the range of 70 - 200 keV. Viewed from the z-axis, the generation positions are distributed throughout the entire solid target, forming roughly uniform annular X-ray source. The slight non-uniformity may be caused by electron escape and absorption within the target. The differential range of source intensities was about 10%. In fact, the current challenge for this kind of coded-source radiography lies in consistently producing uniformly distributed annular X-ray sources. We hope to do the optimization studies in the near future.
Fig. 13. (a) Energy spectrum of the generated photons. (b) Distribution of photon generation positions of 70 - 200 keV.
4. Conclusion
We have investigated the technique of annular coded-source radiography via Monte Carlo and PIC simulations. Advantages in increased spatial resolution and contrast over the classical backlighter were demonstrated through visual and quantity comparison. First, Gaussian and annular sources with 25-µm diameter and yield of 1.3 × 1010 photons/sr were simulated. The former can only resolve 10 µm lines on the object, with a contrast of 7.49. While the latter can resolve 5 µm lines, with a contrast of 13.90. Moreover for resolving 10 µm lines, the contrast increased to 16.29, which is twice as high as the former. Second, in the case of using a large-diameter target, such as 100 µm, the Gaussian source is more likely to experience image degradation so that it cannot be deconvolved. On the other hand, an annular source can effectively reduce this risk, enabling the image reconstruction, which facilitates higher laser absorption and photon yield through increased diameter. Third, we calculated the impact on contrast of using a 50-µm-diameter annular source to increase the photon yield to 10 times that of a 5-µm-diameter classical backlighter, under conditions with self-emission from the object. The photon yields from the classical backlighter and self-emission are equal, both set to 1.3 × 109 photons/sr. The results showed that the contrast increased from 3.04 to 12.76, experiencing a fourfold improvement.
Additionally, based on the R-L algorithm, diameter and ring width of the annular source have effects on the reconstruction. A thinner width leads to more rapid development of numerical noises with iterations, and a larger diameter leads to more required iterations to achieve convergence. Although the actual source photon yield and detector response in experiments may differ from those we simulated, the results discussed above can provide broad conclusions. In practice, one need to comprehensively consider the laser conditions, background noises, and resolution requirements, etc., to determine the backlighter’ structural parameters when designing the imaging system.
An example of annular source generation has been demonstrated. A tube target with 20-µm diameter and 5-µm wall thickness was irradiated by a laser of 4.4 × 1018 W/cm2. We presented the formation of axially symmetric fields on the surfaces of solid target. The hot electrons recirculate under constraint of the fields and induce a relatively uniform annular source.
We believe that this technique of annular coded-source radiography has great prospects in scientific facilities equipped with multiple large-foci picosecond lasers, since it can not only fully utilize the laser energy, but also employ multi-angle lasers to drive the annular source to easily achieve uniform intensity.
Funding
National Natural Science Foundation of China (12235014); National Key Research and Development Program of China (2022YFA1603300).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated or analyzed in the presented research.
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