Radiography has been a powerful tool for exploring the world ever since the discovery of x-rays, enabling the internal structure and features of objects to be determined without causing damage.1,2 It has wide applications in medical, security, industrial, manufacturing, and scientific research fields. As a nondestructive testing (NDT) tool, radiography aims to achieve high resolution and thereby obtain more detailed information while penetrating materials. The spatial resolution is one of the key parameters of NDT and is mainly determined by the spot size of the x-ray or gamma-ray source.3 For photon energies in the keV to hundreds of keV range, a micro-focus x-ray tube can generate radiation with a spot size of the order of a micrometer, and the spatial resolution can reach micrometer to submicrometer scales.4 This enables refined imaging technologies, such as micro-computed tomography (μCT) and nanoscale computed tomography (nanoCT).

However, it is challenging to improve both the penetration ability and the spatial resolution simultaneously. For the internal detection of large and heavy objects, achieving high penetration ability requires gamma rays with photon energy reaching several MeV. In conventional industrial applications, MeV electron accelerators are commonly utilized as gamma-ray sources. High-energy electrons passing through a high-Z converter emit high-energy photons via the bremsstrahlung process. The spot size of the gamma-ray source is typically determined by the focusing capability of the electron beam.5,6 On the basis of extensive research on accelerator technology, various advanced techniques, such as high-performance radio-frequency (RF) accelerators, dielectric accelerators, and laser–plasma accelerators, have been shown to generate high-performance electron beams and thus provide micro-spot high-energy gamma-ray sources, thereby significantly improving spatial resolution.

Among the highly promising new techniques, laser–plasma accelerators have undergone rapid development, with ultrashort intense lasers7,8 being able to produce various particle beams and types of radiation. Laser wakefield accelerators (LWFAs)9,10 can generate electron beams with excellent qualities, including high energy (MeV to GeV), micro-spot size (∼10 µm), low emittance (∼mrad), and short duration (∼10 fs).11–13 LWFA-based secondary radiation is a significant area of research with great potential for applications in scientific, medical, and industrial fields.14–16 The development of compact micro-spot gamma-ray sources has been a subject of investigation for several years. In 2005, Glinec et al.17 demonstrated a gamma-ray source with a spot size of 450 µm using a 40 MeV electron beam with a divergence of ∼17 mrad and a 2.5 mm-thick tantalum converter. In 2011, Ben-Ismaïl et al.18 achieved a spot size of around 30 µm by employing a 1 mm-thick tantalum target and an electron beam with a divergence of about 3 mrad. In 2016, Dong et al.19 conducted a detailed experimental investigation into the properties and operating conditions of a micro-spot gamma-ray source and were able to achieve a spot size of 40 µm using a 5 mrad electron beam. In all of these studies, a reduction in converter thickness or an improvement in electron beam emittance was employed to achieve a smaller spot size. However, both of these methods lead to a decrease in the brightness of the gamma-ray source. Furthermore, there has been significant research into novel methods for laser acceleration and radiation generation. Although these methods have not yet been realized experimentally, they have tremendous potential. For instance, Zhu et al.20 have proposed a two-stage laser–plasma accelerator that numerical simulations have shown should be capable of generating high-energy and high-brightness gamma rays with an efficiency of over 10%.

Laser-driven micro-spot gamma-ray sources have also been utilized for high-penetration radiography. Previous studies have successfully obtained 2D radiographic images with total areal density ranging from ∼10 to 50 g/cm², achieving submillimeter spatial resolutions.17–25 In 2018, Wu et al.26,27 demonstrated a scheme for high-energy computed tomography using a laser-driven source and were able to achieve a spatial resolution of 100 µm. However, to date, no studies have demonstrated a spatial resolution better than 100 µm in high-penetration radiography within the MeV photon energy range.

The spot size of the gamma-ray source, which mainly determines the spatial resolution, is influenced by the experimental conditions, including electron source size, electron beam divergence, and converter thickness. Typically, the spot size and photon yield both increase with the converter thickness, necessitating a compromise in experimental conditions. Despite numerous studies on laser-driven micro-spot gamma source generation and demonstrations of high-penetration radiography, the intrinsic relationship between gamma-ray source size and experimental conditions remains incompletely understood. In the present study, a virtual source approach is proposed to optimize the spatial resolution of radiography. In radiography, high-energy gamma rays generated from bremsstrahlung radiation can be regarded as originating from a localized area within the converter. This specific area, considered as a virtual source, exhibits a spot size that is several tens of times smaller than the original emission area of the gamma rays. By utilizing this virtual source approach, the imaging parameters can be efficiently optimized to maximize the spatial resolution in radiography. Experimental investigations have demonstrated a spatial resolution of 91 µm under a penetration condition of 120 g/cm². Through extensive Monte Carlo simulations, we have developed a series of empirical formulas that describe the variations in the size and divergence angle of the gamma source with the experimental conditions. Under optimized conditions, the gamma source size for imaging can reach about 15 µm, which is about one-hundredth of the size achieved by conventional methods, thus greatly improving the technical capabilities of high-penetration high-resolution radiography. The virtual source approach is primarily based on the use of micro-spot and low-divergence high-energy electron beams. As a result, this method can be applied widely to various advanced acceleration techniques, such as high-performance RF accelerators and dielectric accelerators, in addition to LWFA.

An experimental study of high-penetration imaging based on a laser-driven micro-focus gamma source was performed under experimental conditions similar to those in our previous studies.26 As shown in Fig. 1(a), 0.8 J, 24 fs laser pulse was focused on a gas jet with an outlet diameter of 0.7 mm to generate high-energy electron beams (see Appendix A). The electron beams had divergence angles ranging from 5 to 20 mrad (full width at half maximum, FWHM) at different backing pressure points of the gas target. The electron energy spectra under different beam divergences showed little difference, covering a wide range with a central energy near 40 MeV. The beam charge was about 100–130 pC. A tungsten converter was placed behind the gas target, and when the high-energy electron beam passed through this converter, gamma-ray photons were produced by bremsstrahlung. In the experiment, the thicknesses of the converter varied from 1 to 3 mm, and the distances of the converter front side to the gas jet rear edge were also controlled from 1 to 3 mm.

To assess spatial resolution, an object was placed behind the gamma-ray source at a distance of 800 mm. The object was composed of W-steel alloy, with a thickness of 30 mm and an areal density of 42.6 g/cm². It was designed as a line pair plate, consisting of many slits forming periodic structures with line widths ranging from 500 to 100 µm. To increase the total areal density in radiography, some steel plates were added between the front collimator and the object for resolution. The point projection images had a geometric magnification of 2.5. The images were recorded using a gamma camera composed of a CsI scintillator and a fiber-coupled CCD camera and with a detector intrinsic resolution of 100 µm.28

During the experiment, the linear pair plate was used to measure the spatial resolution. Figure 1(b) shows a typical image of the linear pair plate, and from an analysis of the periodic structure profile shown in Fig. 1(c), it was possible to obtain the modulation transfer function (MTF), a commonly used tool for evaluating spatial resolution. The MTF value m for a periodic structure can be calculated using the formulam=Imax−IminImax+Imin,(1)where I_max is the peak value in the periodic structure and I_min is the valley value. Following the IEC evaluation criterion,29 the spatial frequency corresponding to the 10% MTF was used as the limit for distinguishing capability, while the line width of this cycle was employed as the spatial resolution.

In radiography, the spatial resolution is primarily determined by the x-ray spot size and detector aperture, but it is also affected by geometry. According to imaging theory, the spatial resolution can be evaluated using the equivalent beam width (BW),30,31 which can be calculated using the following formula:BW=d2+[aM−1]2M,(2)where a is the spot size, d is the detector aperture (for an area detector, the detector aperture d is considered as the intrinsic resolution of the gamma camera), and M is the geometric magnification.

As shown in Fig. 2, the spatial resolution under different experimental conditions was tested, with each test image being obtained by accumulating 20 shots. A 20 mm-thick stainless steel plate was placed in front of the object to shield low-energy photons. In Fig. 2(a), the MTF curves are compared for different electron beam divergence angles of 5 and 20 mrad, with the converter thickness and distance both being set at 1 mm. Additionally, different converter thicknesses and distances were investigated for each electron beam divergence. In Fig. 2(b), the MTF curves obtained with a 20 mrad electron beam divergence, a 2 mm converter thickness, and distances between 1 and 3 mm are presented. Figure 2(c) shows the MTF curves for a 5 mrad electron beam divergence, converter thicknesses of 1–3 mm, and a 2 mm distance. The MTF curves are nearly the same under all conditions, indicating similar spatial resolutions. These experimental findings suggest that spatial resolution is relatively unaffected by experimental conditions, thus indicating that gamma-ray spot sizes differ little under all conditions of the experiment.

To analyze and understand the experimental results, a large number of Monte Carlo simulations were carried out. These were performed using Geant4,32,33 which is a widely used simulation toolkit for modeling particle transport in matter. The interactions of an electron beam with a tungsten converter were simulated. In the simulations, the energy spectrum and divergence angle of the electron beam were set according to the experimental results, and the spot size of the electron beam was set to 10 µm (the typical scale in laser wakefield acceleration). To achieve a satisfactory statistical fluctuation, the incident electron number was set from 4 × 10¹¹ to 1.2 × 10¹². The standard electromagnetic physics process was applied, which includes all electromagnetic interactions of electrons, positrons, and gamma rays.

A point projection imaging geometry was used to evaluate the spot size for radiography. By collecting the position of gamma-photon emission on the back of the converter, the emission area of the gamma-ray source could be obtained. A knife-edge method was used to measure the spot size in radiography,34,35 in which an ideal knife edge (a thin knife edge and complete absorption) was used for point projection imaging. By analyzing the image formed by the knife edge, the point spread function of the imaging gamma-ray source could be obtained, and its FWHM was considered as the spot size (see Appendix B).

Typical results are shown in Fig. 3, where the electron beam divergence is 20 mrad, the converter has a thickness of 2 mm, and the distance from the electron beam source is 3 mm. The emission area on the back of the converter can reach 320 µm, while the equivalent source size measured by the knife-edge method is only 45.6 µm. Comparison results are presented in Table I. In radiography, the spot sizes are significantly smaller than the emission area sizes, typically on the scale of several tens of micrometers. In the worst-case scenario, the gamma-ray emission area, measured at the converter’s rear surface, had a diameter of 460 µm, while the imaging spot size was only 46.8 µm.

The trajectories of the gamma rays emitted from the back of the target and used for imaging were extracted. By extending these rays in the reverse direction, it was observed that the emitted gamma-ray photons originated effectively from a localized area within the target, shown in Fig. 1(a). We call this area the “virtual source.” The gamma rays generated by bremsstrahlung radiation could be considered as equivalent to being emitted from a virtual source inside the converter, which had a size closer to the initial electron source and much smaller than the emission area at the rear of the converter. According to the equivalent beam width method, if the spatial resolution is determined by the gamma-ray emission area, it will rapidly increase from ∼80 to 250 µm. On the other hand, if it is determined by the virtual source, the resolution will be limited to a small range of ∼40–50 µm. For multishot cumulative imaging, the spatial resolution also depends on the fluence of the source location fluctuation, which was measured to be 86 µm in our previous experiment.26 Therefore, the resolutions under different conditions are all around 100 µm, which is consistent with the experimental results.

To verify the high spatial resolution enabled by the virtual source approach for high-penetration imaging, we compared spatial resolution under different experimental conditions. Radiographic images with different total areal density are presented in Figs. 4(a)–4(c). These images were acquired by accumulating multiple laser shots: 60 shots for the object only, and 500 shots with the addition of 100 and 130 mm-thick steel plates. Even with the addition of 130 mm-thick steel plates, reaching a total area density of 145.3 g/cm², the periodic structures of the object can still be clearly distinguished. Profiles of the periodic structures after denoising are plotted in Figs. 4(d)–4(i) display the corresponding MTF curves used to evaluate spatial resolution. The experimental results indicate that spatial resolutions can reach about 85.3 µm for the object only, 90.7 µm for 121.6 g/cm² (with 100 mm-thick steel plates added), and ∼93.4 µm for 145.3 g/cm² (with 130 mm-thick steel plates added). In our view, degradation of imaging resolution is primarily induced by scattering of high-energy photons as they penetrate the object and steel plates.

A virtual source of high-energy gamma rays primarily benefits from the initial electron beam generated from LWFA, which typically has a spot size of several micrometers. Owing to bremsstrahlung, high-energy photons exhibit a significant feature of forward emission.36,37 Therefore, the spot size of the virtual source from high-energy bremsstrahlung tends toward the spot size of the initial electron beam. In practical experiments, the spot size is determined by the electron beam spot size D_e, energy E_e, divergence angle θ_e, converter thickness t, and distance from electron source to converter l. The relationships of the virtual source size and the photon emission angle to those parameters were investigated by performing a large number of Monte Carlo simulations.

First, a collimated point electron source (D_e = 0, θ_e = 0) is considered. The spot size of the virtual source D₁ is primarily determined by the electron energy. Figure 5(a) shows that the intrinsic spot sizes were less than 2.5 µm for electron energies of 20–100 MeV with tungsten converter thicknesses less than 10 mm. The following empirical power-law relationship is found to hold between spot size and converter thickness:D1(μm)=at(mm)b,a=12.2Ee(MeV)−0.63,b=0.046Ee(MeV)0.32,(3)where D₁ is the spot size (FWHM) of the virtual source for the collimated electron beam, t is the converter thickness, and the power-law coefficient a and exponent b themselves follow power laws with respect to the electron energy E_e. The gamma-ray emission angle also has a power-law dependence on the converter thickness, as shown in Fig. 5(b):θ1(mrad)=at(mm)b,a=1989Ee(MeV)−0.96,b=1989Ee(MeV)−0.96,(4)where θ₁ is the gamma-ray emission angle (FWHM) for the collimated electron beam, and the power-law coefficient and exponent again follow power laws with respect to the electron energy.

The results of these situations have revealed the characteristics of high-energy electrons passing through matter. Large numbers of secondary electrons and photons are generated by collision and radiation. The secondary radiation deviates gradually from the incident direction of the electron beam as the propagating distance increases, leading to increases in the spot size and emission angle of bremsstrahlung photons. The cross sections of electrons and photons indicate that the emission or scattering angles of secondary electrons and photons decrease as the electron energy increases. Consequently, a smaller spot size and emission angle can be achieved by using an electron beam with higher energy.

When considering radiographic applications, both the projection angle and photon yield are important factors. Figure 5(c) presents the photon yields integrated within a cone angle of 15°. A high-flux gamma-ray beam was achieved by choosing a converter thickness of 3–4 mm for the high-energy electron beam.

The-following simulations consider the electron divergence angle θ_e and the distance between the electron source and the converter l, with the converter thickness set as 3 and 4 mm. Unlike the power-law relationships observed for collimated beams, the properties of bremsstrahlung gamma rays exhibit linear dependences on the electron divergence angle and the converter distance. When a divergent point electron source is located at the front surface of the converter (l = 0), the size of the virtual source D₂ increases linearly with the electron beam divergence angle, as shown in Figs. 6(a) and 6(b):D2(μm)=kθe(mrad)+D1,k=aEe(MeV)b,(5)witha=0.0013,b=0.95fort=3mm,a=0.0031,b=0.82fort=4mm⋅

Figure 6(c) shows that the gamma-ray emission angle θ₂ also follows a linear relation:θ2(mrad)=kθe(mrad)+θ1,k=−780Ee(MeV)−2.5+1.15⋅(6)The coefficients k for both D₂ and θ₂ have power-law relationships with the incident electron energy. An interesting observation is that the growth rate of the spot size in the high-energy condition is greater than that in the low-energy condition. When the electron divergence angle increases to more than 20 mrad, the high-energy electron beam will result in a larger spot size, reversing the situation at low beam divergence. In our opinion, this phenomenon is mainly caused by self-absorption in the converter. The spot size is essentially determined by the dispersion of secondary photons and electrons generated in the converter. As the divergence angle increases, the electrons will follow a longer transmission path. Secondary photons and electrons generated from a low-energy electron beam in a long transmission path have a higher self-absorption rate, which decreases the dispersion to some extent and results in a smaller spot size. For the gamma-ray emission angle, there is almost no difference between the results for 3 and 4 mm converters. A high-energy electron beam will generate a low-emittance gamma-ray beam.

As the distance from the converter increases, the incident electron beam covers a larger area of the converter, which causes more intense dispersion and increases the spot size. We found that the spot size of the virtual source D₃ for different electron divergences and converter distances consistently follows a linear relationship, as shown in Fig. 6(d):D3(μm)−D2(μm)θe(mrad)=kl(mm),k=1.52.(7)Here, D₂ is the spot size for the divergent point source with l = 0, and θ_e is the divergence angle (FWHM) of the electron beam. The coefficient k represents the rate of increase. We can see that the converter distance significantly affects the spot size. For example, when an electron beam with E_e = 40 MeV and θ_e = 20 mrad is used, the spot size is about 30 µm for a converter distance of 1 mm and 150 µm for a distance of 5 mm. Therefore, to minimize the spot size, the electron source should be placed as close as possible to the converter. Figure 6(e) presents the gamma-ray emission angle for different converter distances, and it can be seen that this angle is unaffected by the distance.

When considering an actual electron beam with a beam size (FWHM) of D_e, the final gamma-ray source can be considered as a convolution of a gamma-ray source with a point electron source and the actual electron beam. As the convolution of two Gaussian functions, the final virtual source spot size D_γ can be expressed asDγ=D32+De2,(8)where D₃ is the spot size of the virtual source generated by a point electron source under given values of E_e, θ_e, t, and l.

For a broad-spectrum electron beam, the final virtual source size can be considered as a weighted average of individual monoenergetic electron beams that contribute to the virtual source, with each contribution weighted by its respective proportion:Dγave=∫E1E2NENtotDγ(E),(9)where Dγave is the spot size of the virtual source generated by an electron source with a spectrum covering a range of [E₁, E₂], N(E)/N_tot is the fraction of electrons with an energy of E, and D_γ(E) is the spot size of the virtual source corresponding to electrons with an energy of E.

From Eqs. (3)–(8), we can predict the properties of the bremsstrahlung gamma-ray source generated using a micro-spot and low-divergence high-energy electron beam, and determine the optimal conditions for creating a micro-spot gamma-ray source for radiographic applications. As shown in Figs. 7(a) and 7(c), a virtual source spot size of 15 µm can be achieved by controlling the electron divergence angle to about 5 mrad over a wide range of electron energies, from 20 to 100 MeV, with a 10 µm spot size electron beam passing through a 4 mm converter from a 1 mm distance. The gamma-ray emission angle under these conditions is shown in Figs. 7(b) and 7(d).

Figure 8 presents the results of imaging simulations conducted under the conditions E_e = 40 MeV, θ_e = 10 mrad, D_e = 10 µm, t = 3 mm, and l = 1 mm. The object used in the simulation was a line-pair plate consisting of four sets of periodic structures with line widths of 50, 75, 100, and 200 µm. We explored different materials, namely, polythene (C₂H₄), aluminum (Al), and tungsten (W), with object thicknesses of 10 mm. The simulation results demonstrate the successful differentiation of all the periodic structures, implying that the virtual source approach can provide a spatial resolution of better than 50 µm.

We have presented a high-penetration and high-resolution radiographic technique using a micro-spot gamma-ray source driven by a compact laser–plasma accelerator. We have introduced a virtual source approach to optimize bremsstrahlung radiography and have successfully demonstrated an image resolution of ∼90 µm even after passage of the radiation through a 100 mm steel object. Relationships of the virtual source properties to experimental conditions, namely, electron energy, electron divergence, converter thickness and distance, have been derived from the results of detailed simulations. Empirical formulas derived from these simulations indicate that achieving a small-sized virtual source relies on the existence of a micro-spot and a low-divergence electron beam. This enables the virtual source approach to be effectively applied in various advanced accelerator techniques. Under the conditions of this study, according to the empirical formulas, the spot size can be as small as 15 µm, which is almost 100 times smaller than the traditional bremsstrahlung source used in high-energy industrial computed tomography.

For different materials, the empirical formulas can be expressed in terms of the radiation length361RL=4aNAAZZ+1re2ln183Z−1/31+0.12Z822,(10)which is a dimensionless length unit that eliminates the dependence on the material used. Here, a is the electron fine structure constant, r_e is the classical radius of the electron, N_A is Avogadro’s number, and Z and A are respectively the charge and mass number of the material.

Further development of high-peak-power and high-repetition lasers should lead to substantial improvements in the average flux of laser-driven gamma rays. Moreover, there should be further enhancements to the stability of the electron beams from LWFA, which already have energy fluctuations of less than 3% and pointing uncertainties of about 2 mrad.16,38,39 Moving forward, laser-based high energy NDT equipment could be constructed with excellent spatial resolutions of less than 50 µm.

Acknowledgment. This work was supported by the National Natural Science Foundation of China (Grant Nos. 12175212, 11991071, 12004353, 11975214, and 11905202), the National Key R&D Program of China (Grant No. 2022YFA1603300), the Science Challenge Project (Project No. TZ2018005), and the Sciences and Technology on Plasma Physics Laboratory at CAEP (Grant No. 6142A04200103).