Introduction

Ever since its discovery in 1895 by Röntgen, x-ray radiation has always been pushing the frontiers of our knowledge as one of the most powerful tools for exploring the properties of matter1,2,3. State-of-the-art x-ray sources can produce ultrabright x-rays at keV photon energies4,5,6,7,8. They have become indispensable for many applications in fundamental research, industry and medicine. However, despite the extensive demand, only a few dedicated synchrotrons and x-ray free-electron lasers exist in the world, because limited by the weak acceleration field of radio-frequency accelerators, these facilities are usually very large and expensive. Furthermore, the natural narrowband spectrum from undulators imposes restrictions on their applications in several important diagnostics for material science and high-energy-density (HED) physics9.

In relativistic laser plasma-based accelerators, the excited accelerating field strength can be several orders of magnitude higher, and therefore, laser-plasma based radiation sources have been proposed to be an attractive alternative for modern x-ray applications. In the last two decades, x-ray sources based on laser wakefield accelerators (LWFA) have been widely investigated10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25. However, in LWFA, the accelerated electron beam is characterized by moderate charge (about hundreds of picocoulombs to nanocoulombs)26,27, which determines that the conversion efficiency of laser energy to photons and the photon number are relatively low. The total number of photons in full spectrum is usually on the order of 108, and the peak flux is about 104–105 photons eV−13,28. This is not sufficient to make high-quality measurements, especially in a single-shot. For instance, to achieve good statistics, x-ray absorption spectroscopy techniques, as essential tools for probing both electronic and atomic structural properties of matter, require the flux satisfying Nph > 106 eV−13,29,30,31,32, where Nph is the number of photons in the energy band of interest. This is simply determined by the requirement that the random statistic noise \(SN=1/\sqrt{{N}_{{{{{{{{\rm{ph}}}}}}}}}}\) should be smaller than 1/1000 of the signal, while recent experiments indicate that due to background noise, an even higher flux about 107 photons eV−1 is necessary to fulfill the required signal-to-noise ratio33. On the other hand, for many applications, especially in HED experiments3,33,34,35,36,37,38, single-shot measurements are crucial because of complex target designs and low repetition rate of powerful lasers. Though it is possible to make such measurements by utilizing implosion-based x-ray sources39,40,41, they are based on large, expansive and unique facilities and their brilliance is relatively low3. Therefore, it is critical to develop tabletop ultrabright high-flux x-ray sources.

Recently, there has been a growing focus on enhancing the photon flux of x-rays at keV range. Most of the works are based on delicate manipulation of the LWFA by making a trade-off between high brilliance and high flux33,42,43,44,45,46,47 due to the limitation of the beam loading effect48. Another approach to achieving higher flux involves using electrons from direct laser acceleration (DLA) mechanism with near-critical-density plasmas49,50,51,52,53 or microwires54,55, where compared to the LWFA, much larger number of electrons but with lower energy and larger angular spread are produced. The divergence of the emitted x-rays is usually around hundreds of mrad, leading to rather low brilliance56,57,58,59,60,61. Moreover, electron acceleration and coherent harmonics generation in laser-microtape target interaction at oblique incidence62,63,64 and MeV electron acceleration driven by THz surface waves in laser-microwire interaction65 have been investigated. Until now, in spite of the significant demand in diverse applications, how to achieve ultrabright femtosecond x-rays with a flux of up to or even beyond 107 photons eV−1 at keV range still remains an open question.

In this paper, we demonstrate a scheme for generation of high-flux high-brilliance x-ray radiations, where a linearly-polarized (LP) femtosecond laser pulse is incident on an edge of a submillimeter wide microtape, as illustrated in Fig. 1a. When the laser pulse impinges on the edge, a strong surface plasma wave (SPW) is easily excited and significant amounts of electrons are peeled off from the tape into vacuum. These electrons will be accelerated forward to superponderomotive energies, while undergoing betatron oscillations near the lateral plasma-vacuum interfaces, leading to intense x-ray bursts. Our three-dimensional (3D) particle-in-cell (PIC) simulations demonstrate that by utilizing a readily available tabletop 150 TW femtosecond laser, hard x-ray sources characterized by flux up to 107 photons eV−1 around 5 keV and peak brilliance about 1023 photons s−1 mm−2 mrad−2 0.1%BW−1 can be generated. The energy conversion efficiency is about 2.5 × 10−4. We have identified the existence of the SPW and demonstrated that the radiating electrons are primarily accelerated by the longitudinal electric field of the SPW and undergo betatron oscillations in the self-generated quasistatic transverse electric and magnetic fields at the vicinity of the interfaces. Our theoretical model fits well with the numerical results.

Fig. 1: Schematic and three-dimensional particle-in-cell simulation results.
figure 1

a An intense femtosecond laser pulse (red cone) is incident on an edge of a submillimeter wide microtape target (gray cuboid) to drive electron acceleration and betatron radiation. Distribution of energetic electrons at t = 40T0 is represented by the spheres where each individual electron is color-coded with its γ-factor. Note that the tape target can be infinitely long along z-direction and our scheme can tolerate the existence of a short-scale preplasma. b Transverse field Ey at t = 0 when the peak laser enters the simulation box. c, d Angular distributions of the electrons and photons, respectively. e Final photon spectrum at t = 120T0. Here \(d\Omega =\sin \theta d\theta d\phi\) with \(\theta =\arctan (\sqrt{{p}_{y}^{2}+{p}_{z}^{2}})/{p}_{x}\) and \(\phi =\arctan ({p}_{z}/{p}_{y})\), T0 = 2π/ωL and the field is normalized to E0 = mecωL/e.

Full size image

Results

Source characterization

Figure 1 illustrates the 3D simulation results. When the laser pulse impinges on the front edge of the relativistically overdense plasmas (gray cuboid), a strong SPW can be easily induced without requiring special resonance condition66. It propagates along the plasma-vacuum interfaces with a velocity close to the light speed c67,68,69,70. Subsequently, electrons are continuously peeled off by the transverse field Ey [Fig. 1b] and accelerated forward by the self-generated longitudinal field Ex [Fig. 2c], forming periodically distributed electron nanobunches on both sides of the target [Fig. 1a]. The maximum electron γ already reaches about 400 at t = 40T0, and most of high-energy electrons are confined close to the lateral interfaces.

Fig. 2: Identification of the electron injection, acceleration and betatron oscillation processes.
figure 2

a Electron density (green-yellow) and longitudinal field Ex (blue-red) at t = − 16T0, in which `A', `B' and `C' mark three different electron nanobunches in space, but can also be used to phenomenologically represent the compression, injection and acceleration stages of the same electron bunch over time. b Transverse field Ey at t = − 16T0. c Longitudinal field Ex at t = 12.5T0 and electron phase space (black-yellow dots), while (d) shows the transverse field Fy and the electron density distribution (green-yellow), where the electron density below y < 0.6λ and ne < 5nc are removed to clearly show the Fy. The inset is a zoom-in. In (c), the dashed cyan line demarcates the surface plasma wave (SPW)-dominated region (below) from the diffracted wave-dominated (above). e Comparison of the transverse profile of the SPW model (red dashed) and Ex (black) from the PIC simulation, extracted from the vertical black dashed line marked in (c). f The 1D profiles of ne (blue), Fy (black) and the potential (red) obtained from (d).

Full size image

The angular distribution of electrons is shown in Fig. 1c. The divergence angle is about 5∘ which is much smaller than that observed in DLA49,53,57. The effective electron temperature reaches about 25 MeV, about four times higher than the ponderomotive scaling of 6.7 MeV71. There are about 3.1 × 1011 high-energy electrons ( ~ 50 nC) with γ > 10 generated along the moving window. This is more than one order of magnitude higher than the typical LWFA results26,27. Such high charge, superponderomotive, collimated electrons produce ultrabright radiations near the lateral surfaces since there they experience transverse betatron oscillations in strong self-generated fields.

The obtained photon energy spectrum at t = 120T0 is shown in Fig. 1e. The peak brightness is up to about 4.5 × 107 photons 0.1%BW−1 with critical energy of 20 keV. The number of photons with energy > 10 keV (i.e., hard x-ray) reaches 8.6 × 1010. The x-ray flux is high up to 107 photons eV−1 around 5 keV region and the corresponding spectral yield efficiency is about 2.2 × 106 photons eV−1 J−1. Both are more than one order of magnitude higher than the LWFA results at the laser energy3,28,33. This is the key advantage of our scheme compared to other known regimes. Such a high photon flux is crucial for achieving single-shot high-quality measurements3,33. The total radiated energy is about 1.1 mJ, resulting in a laser-to-photon conversion efficiency of 2.5 × 10−4, about one order of magnitude higher than that in other betatron schemes56,58,72. Though comparable conversion efficiency was observed in the interaction of PW fs lasers with near-critical-density plasmas73, in our scheme, the obtained critical energy and hard x-ray photon flux are much higher and divergence is significantly lower.

Figure 1d presents the angular distribution of photons. The opening angles θy and θz (RMS) are about 50 mrad and 60 mrad, respectively. The source size dy × dz ≈ 5 μm2, where dy [estimated as the sum of ly and averaged transverse oscillation amplitudes, see Fig. 3e] and dz ( ~ dL) are diameters along y- and z-direction, respectively. Taking the x-ray pulse duration as 30 fs ( ~ τL), we have the peak brilliance about 1023 photons s−1 mm−2 mrad−2 0.1%BW−1. This is comparable to state-of-the-art third-generation synchrotrons. We highlight that in diagnostic applications, though the x-ray divergence is relatively larger than that obtained in the LWFA regimes33, it is feasible to ensure that the sample collects the entire photon beam by simply adjusting the source to sample (and sample to detector) distance.

Fig. 3: Energy spectra and typical trajectories of electrons.
figure 3

a Energy-gain plane (Γx, Γy) of electrons with their net γ-factor color coded. b Electron energy spectra, where the black, red and blue lines represent the energy spectra of the total electrons, the longitudinal field dominant acceleration (LFDA) and transverse field dominant acceleration (TFDA) electrons, respectively. c, d Temporal evolution of the γ-factor and py of the tracked LFDA electrons, respectively, demonstrating that the γ-factors of the electrons are continuously increasing while undergoing transverse betatron oscillations, highlighted with the red lines from a single electron. e Trajectories of representative electrons from the LFDA mechanisms, respectively. In (c, d), representative electrons from both sides of the plasma are included.

Full size image

Electron acceleration

To reveal the underlying physics, 2D simulations with particle tracer are performed. The laser and plasma parameters remain the same as described above.

When the laser pulse hits the front edge of an overdense tape target, not only is it cut into two parts [Fig. 1b], but also piles up electrons into a compressed overdense layer in front of the target. A strong SPW can be effectively excited since both the laser pulse and the SPW propagate along the target lateral surfaces at velocities close to the light speed c. The resonance conditions of ωL = ωSP and kL = kSP can be easily satisfied67,68,69, where ωL and kL are the laser frequency and wavenumber, respectively, while ωSP and kSP corresponding to the frequency and wavenumber of the SPW. After fast Fourier transformation, in (ω, k) space, by considering a target with a lower electron density of ne = 5nc to make the dispersion relations of the laser pulse and SPW discernible, we identified the existence of the SPW at the flat interface, see Supplementary Note I and Fig. S2.

Considering that the physics and phenomena are similar in both sides of the target except for a phase shift [see Fig. 1a, b], in Fig. 2, we elucidate the key physics by only showcasing the results above the target (i.e., y > 0). For a LP laser pulse, its Lorentz force has an oscillating component, and therefore, this dense layer will be dragged back at later time [marked as electron nanobunch ‘A’ in Fig. 2a]. If at the same time the local laser field Ey [Fig. 2b] is negative, the layer will be deflected upward whilst moving backward, forming the electron nanobunch ‘B’. Electrons located in the region of a negative longitudinal field Ex will be further injected into the accelerating phase and accelerated to high energies. This forms the accelerated electron nanobunches above the tape target, corresponding to the bunch ‘C’ in Fig. 2a and those displayed in Figs. 1a and 2d. In contrast, those experiencing a positive Ex field will be decelerated and debunched to very low density. On the other hand, if the backward electrons feel a positive Ey field, the dense layer will be turned downward, but part of them can also be injected and accelerated to high energies, forming the electron nanobunches below the target [see Fig. 1a].

Figure 2c illustrates the distribution of the Ex field (blue-red) and the phase space of electrons (black-yellow dots) at t = 12.5T0, where to show it clearly, only electrons with γ > 100 are shown. It is evident that most high-energy electrons are located in the negative components of the Ex field and the majority of them including those most energetic ones are confined very close to the plasma-vacuum interface [see Fig. 2d]. Since the phase velocity of the SPW \({v}_{ph}^{SPW}=\omega /{k}_{x} \, \approx \, c(1-{\omega }^{2}/{\omega }_{pe}^{2})\) is very close to the light speed c, electrons can stay in the accelerating phase for tens of laser periods and be accelerated to hundreds of MeV68.

Figure 3a shows the energy-gain plane (Γx, Γy) of every electron, while in Fig. 3b, the red and blue lines depict energy spectra of electrons from longitudinal field dominant acceleration (LFDA, Γx > Γy) and transverse field dominant acceleration (TFDA, Γx < Γy), respectively. Here Γx = − ∫ evxExdt and Γy = − ∫ evyEydt represent the accumulated work from the longitudinal and transverse fields, respectively. One can clearly see that most energetic electrons are predominantly accelerated by the longitudinal field and the number of LFDA electrons is much higher that of TFDA ones.

To further understand which group of electrons is responsible for the x-ray generation, in Fig. 3e, we show trajectories of some representative electrons from the LFDA. While they are accelerated forward, each of them undergoes multiple betatron oscillations near the lateral plasma-vacuum interface, where both the longitudinal accelerating and transverse wiggling fields are the strongest. Therefore, at each turning point, these electrons emit x-rays efficiently. Their oscillation strength which can be estimated as the amplitude of py is K ≈ 10 [see Fig. 3d], indicating the radiations occur in the wiggler regime2, in which the opening angle of the radiation cone is much smaller than the maximum angle between the electron velocity and the propagation. In comparison, the TFDA electrons almost move straightly, except at the turning points, but most of them are only reflected once and then never come back, see Supplementary Note V and Fig. S6. Considering that the number and energy of the TFDA electrons are both relatively small, their contributions to radiation are negligible. Hence, the transverse oscillating LFDA electrons are the main interest here.

In Fig. 3(c) and (d), we correspondingly show the temporal evolution of the γ and py of the representative electrons from the LFDA group. One can see that the values of the γ is continuously increasing even when py changes sign. This indicates that the electron acceleration and the recoil force of electron betatron motion cannot directly stem from the laser fields. Otherwise, the γ should be oscillating when py changes sign49. Therefore, we further identify the origin of the acceleration and wiggling fields. This is essential for optimizing the electron acceleration and betatron radiation.

In laser-plasma interactions, different modes can be excited especially in our scheme, including the SPW aforementioned, the diffracted wave, the high-order harmonics etc. As shown in Fig. 2(a), near the target edge, the profile of the Ex field is curved which indicates that it is combined by both the SPW and the diffracted wave. However, the diffracted wave diverges to all the directions in the propagation, leading to a rapid decrease of its amplitude. In contrast, the SPW mainly travels forward along the plasma-vacuum interface and its amplitude decreases very slowly. Figure 2(c) displays the distribution of the Ex field at t = 12.5T0, where one can clearly see a gap (highlighted with a cyan dashed line) which roughly demarcates the SPW-dominated region (below) from the diffracted wave-dominated region (above). In other words, the Ex field near the interface is mainly contributed by the longitudinal component of the SPW.

To further demonstrate the Ex field responsible for the acceleration of the radiating electrons (i.e., the LFDA electrons) indeed stemming from the SPW, we compared the transverse profile of the Ex field to the prediction of the SPW theory. By considering an SPW propagating along the interface between vacuum and a cold plasma and assuming the field having the form67,69

$${E}_{sw}={E}_{sw0}{e}^{-{k}_{y}y}{e}^{-i(\omega t-{k}_{x}x)}$$
(1)

where Esw0 is the field amplitude. For the field on the vacuum side,

$${k}_{xv}=\frac{\omega }{c}\sqrt{\frac{1-{n}_{{{{{{{{\rm{e}}}}}}}}}/{n}_{{{{{{{{\rm{c}}}}}}}}}}{2-{n}_{{{{{{{{\rm{e}}}}}}}}}/{n}_{{{{{{{{\rm{c}}}}}}}}}}},\,\,{k}_{yv}=\frac{\omega }{c}\sqrt{\frac{1}{{n}_{{{{{{{{\rm{e}}}}}}}}}/{n}_{{{{{{{{\rm{c}}}}}}}}}-2}}.$$
(2)

For overdense plasmas with ne/nc ≫ 2, kyv > 0, which means the SPW decays in y-direction, but very slowly. The characterize length for the field decaying to 1/eN is \(\lambda \sqrt{{n}_{{{{{{{{\rm{e}}}}}}}}}/{n}_{{{{{{{{\rm{c}}}}}}}}}-2}/2\pi \, > \, \lambda\) which is much longer than the amplitude of the betatron oscillations shown in Fig. 3e. Here eN = 2.71828... is the Euler’s number. For the field on the plasma side,

$${k}_{xp}={k}_{xv},\,\,{k}_{yp}=\frac{\omega }{c}\left(\frac{{n}_{{{{{{{{\rm{e}}}}}}}}}}{{n}_{{{{{{{{\rm{c}}}}}}}}}}-1\right)\sqrt{\frac{1}{{n}_{{{{{{{{\rm{e}}}}}}}}}/{n}_{{{{{{{{\rm{c}}}}}}}}}-2}}.$$
(3)

Therefore, inside the plasma the SPW fields decay much faster.

In order to compare with the above theory, we extracted the transverse profile of the Ex field at x = 17.35 λ as marked by the vertical black dashed line in Fig. 2c. Its profile is depicted by the black solid line in Fig. 2e, while the prediction of the SPW model according to Eqs. (1), (2) and (3) is presented by the red dashed line. One can clearly see that the simulation result fits quite well with the SPW model and the Ex field decreases slowly along y-direction. Here ne and Esw0 are determined from the simulation and we choose the part with positive Ex field to avoid the effects induced by the electron nanobunches located in the region of negative components.

Furthermore, to exclude the effects caused by charge-separation field, we also conducted a hybrid 2D simulation with the hybrid VLPL code74 where the target acts as a metal. The separation between the diffracted fields and the SPW is even more clear. The distribution of the Ex field near the interface fits almost perfectly with the SPW model after a relatively long propagation. Details can be found in the Supplementary Note I and Fig. S1.

The electron density at t = 12.5T0 is shown in Fig. 2d, and the 1D cut is depicted by the blue line in Fig. 2f, where one can see that the density is peaked near the interface. About 92% of electrons outside the initial plasma boundary are confined within ∣y∣ < 3λ. Therefore, in our scheme, it is clear that most of radiating electrons are accelerated by the longitudinal component of the SPW. Only a minor part of them can travel to farther distance along y-direction experiencing TFDA or being accelerated by the diffracted laser field, but their contribution to the x-ray generation is negligible.

Betatron radiation

In order to reveal the source of the recoil force, we present the distribution of the transverse field Fy = Ey − Bz at t = 12.5T0 in Fig. 2d and its 1D cut in Fig. 2f. One can see that Fy is positive above the plasma, but negative inside the skin depth of the plasma. This is because outside the plasma, there is a strong charge-separation field \({E}_{y}^{s}\) between the peeled-off electrons and those almost stationary ions. Thanks to these forward-moving electrons, a return current is formed inside the plasma to maintain the target neutral. This large return current in turn induces a strong localized magnetic field \({B}_{z}^{s}\) that will reflect electrons outward. Its amplitude is much stronger than the charge-separation field, but is constrained in a small region characterized by several skin depths. As a result, the quasistatic attracting \({E}_{y}^{s}\) field and the repelling \({B}_{z}^{s}\) field not only constitute the recoil force that drives the electron betatron motion, but also create a potential well [red line in Fig. 2f] in transverse direction that can confine most electrons at the vicinity of the target lateral surfaces.

So far we have clarified that the radiating electrons are predominantly accelerated by the longitudinal components of the SPW, whilst wiggled by the self-generated transverse fields along the forward propagation. Note that due to the overdense target used here, our wiggling field up to 4.5 × 1013 V/m [see Fig. 2f] is significantly stronger than that of the LWFA2,10 and DLA49,56. This leads to a very short oscillation wavelength λu which is about 10 μm. Considering that in the wiggler regime, the critical energy ℏωc = (3/2)γ3c/ρ = (3/2)Kγ2hc/λu, such a strong wiggling field is the key reason that with K ≈ 10, we can achieve x-rays with critical energy of tens of keV without requiring GeV-class electrons as in LWFA3,28. While the proposed scheme presents some differences in target parameters with respect to the widely investigated LWFA-based betatron x-ray sources, here the electrons still undergo betatron oscillations. Therefore, we retain the name of betatron x-ray source to characterize the fundamental physical process, as in other literature18,45,58,75.

Theoretical model and scaling

To establish a theoretical model, we consider single electron motion in presence of combined fields of a y-polarized laser pulse (\({E}_{y}^{l},{B}_{z}^{l}\)) and the self-generated electric (\({E}_{x,y}^{s}\)) and magnetic (\({B}_{z}^{s}\)) fields. For simplicity, we exploit 2D (x, y) planar geometry, and use dimensionless variables: t → ωLt, x → k0x, v → v/c, p → p/mec, E → eE/mecωL. The equations of single electron motion are dr/dt = v and dp/dt = − (E + v × B). Since in the comoving frame, \({E}_{x}^{s}\) varies very slowly, and therefore can be treated as a constant. As the red line shown in Fig. 2f, we have a skewed potential well. Near the point of the minimum potential energy, the transverse force is assumed to have the form of \({F}_{y}^{s}={E}_{y}^{s}-{v}_{x}{B}_{z}^{s}={\kappa }_{i,o}y\), see Supplementary Note II and Fig. S3. Here the subscripts i and o denote inside and outside of the minimum potential energy point. We calculate the coefficients as \({\kappa }_{i}={F}_{y0}^{s}{k}_{p}\) and \({\kappa }_{o}=-{k}_{h}^{2}/{e}_{N}\), where \({k}_{p,h}=\sqrt{{n}_{p,h}/{n}_{{{{{{{{\rm{c}}}}}}}}}}{k}_{0}\) is the wave number, np and nh represent the local plasma density and the hot electron density, respectively68,76.

Considering that the longitudinal field dominates the acceleration, and in relativistic limit, vx ≈ 1 ≫ vy, we have \({E}_{x}^{s} \, \gg \, {E}_{y}^{l}-{v}_{x}{B}_{z}^{l} \, \approx \, 0\). Then we can write the equation of the electron transverse motion as

$$\frac{{d}^{2}y}{d{t}^{2}}+\frac{{E}_{x}^{s}}{{\gamma }_{0}+{E}_{x}^{s}t}\frac{dy}{dt}+\frac{\kappa }{{\gamma }_{0}+{E}_{x}^{s}t}y=0.$$
(4)

This is an equation of a damped oscillator with eigenfrequency \({\omega }_{\beta }=\sqrt{\kappa /({\gamma }_{0}+{E}_{x}^{s}t)}\). On each side of the lowest potential point, the electron orbits can be approximately given as

$$x\simeq {x}_{0}+t,\,\,y\simeq {y}_{\beta }\cos \left(\frac{2{\kappa }_{i,o}}{{\omega }_{\beta }{E}_{x}^{s}}+\phi \right);$$
(5)
$${p}_{x} \, \approx \, {\gamma }_{t}\simeq {\gamma }_{0}+{E}_{x}^{s}t,\,\,{p}_{y}\simeq -{\gamma }_{t}{y}_{\beta }{\omega }_{\beta }\sin \left(\frac{2{\kappa }_{i,o}}{{\omega }_{\beta }{E}_{x}^{s}}+\phi \right)$$
(6)

where \({y}_{\beta }={C}_{1}\sqrt{{\omega }_{\beta }{E}_{x}^{s}/\pi {\kappa }_{i,o}}\) is the betatron oscillation amplitude, C1 and ϕ are constants which have to be determined by initial conditions.

According to our simulation results discussed above, we know \({F}_{y0}^{s}\simeq 3{F}_{0}\), np ≃ a0nc and nh ≃ 5nc, where F0 = mecωL. Then we calculate ∣κi/κo∣ ≈ 7.3 and the oscillation period \({\tau }_{i}/{\tau }_{o}=\sqrt{| {\kappa }_{o}/{\kappa }_{i}| } \, \approx \, 0.4\), which agrees well with the result in Fig. 3d. By taking the electron γ as 500, we have τo = 2.5τi = 16.5T0, leading to the betatron duration of τβ = 0.5(τi + τo) ≈ 11.5T0, which also fits well the simulation results [see Fig. 3d].

From the electron trajectory, we can further study the radiation features by assuming a relativistic electron in instantaneously circular motion with radius ρ. Expressions of the radiated power Pr, the critical frequency ωc and the radiation angle θ can be written as2,77

$${P}_{r}=\frac{2}{3}{\gamma }^{4}\frac{1}{{\rho }^{2}},\,{\omega }_{{{{{{{{\rm{c}}}}}}}}}=\frac{3}{2}{\gamma }^{3}\frac{1}{\rho },\,\theta =\frac{2\pi {y}_{\beta }}{{\lambda }_{\beta }},$$
(7)

where ρ = λβ/2πθ. The total radiated energy can be estimated as \({{{{{{{{\mathcal{E}}}}}}}}}_{r}={N}_{{{{{{{{\rm{e}}}}}}}}}{P}_{r}L/c\), where Ne is the electron number and L is the acceleration distance. One can estimate the critical energy10 \(\hslash {\omega }_{{{{{{{{\rm{c}}}}}}}}}=(3/2)\hslash {\gamma }^{2}| {F}_{y0}^{s}| \approx 17.5\) keV by substituting the electron temperature of \(\bar{\gamma }=50\). Besides, considering the electron initial position is within the skin depth, i.e., 1/kp, the amplitude of betatron motion is yβ ≃ 0.062λ, close to the simulation results 0.056 λ, shown in Fig. 3e. The opening angle estimated from Eq. (7) gives about 0.066 rad, also in good agreement with the simulation results [Fig. 1d].

In intense laser-solid-density target interactions, the self-generated fields are always mainly determined by the peak laser fields78,79,80, which means \({E}_{x}^{s},\,{E}_{y}^{s},\,{B}_{z}^{s}\propto {a}_{0}\), leading to γt ∝ a0. Therefore, \({y}_{\beta }\propto {(\kappa /{\gamma }_{t})}^{1/4}\propto {a}_{0}^{0}\), \({\omega }_{\beta }\propto \kappa /{E}_{x}^{s}\propto {a}_{0}^{0}\). Since the energy conversion efficiency to electrons is almost comparable for moderately relativistic laser intensities considered here, \({N}_{{{{{{{{\rm{e}}}}}}}}}\propto {{{{{{{{\mathcal{E}}}}}}}}}_{{{{{{{{\rm{L}}}}}}}}}/{T}_{{{{{{{{\rm{e}}}}}}}}}\propto {a}_{0}\). Finally, we have

$${\omega }_{{{{{{{{\rm{c}}}}}}}}}\propto {a}_{0}^{3},\,\,{{{{{{{{\mathcal{E}}}}}}}}}_{r}\propto {N}_{{{{{{{{\rm{e}}}}}}}}}{P}_{r}\propto {a}_{0}^{5},\,\,\theta \propto {a}_{0}^{0}.$$
(8)

To demonstrate these relations, we performed more 3D simulations by varying a0. The results are shown in Fig. 4a, where the black squares and red triangles represent the simulation results of ℏωc and \({{{{{{{{\mathcal{E}}}}}}}}}_{r}\), respectively, and the black and red lines show the corresponding fittings from Eq. (8). It is evident that the simulation results are in accordance with the theoretical scalings. Moreover, as predicted by our theory, the radiation divergence is almost independent of a0, see blue circles in Fig. 4a.

Fig. 4: Scaling and robustness.
figure 4

a Dependence of synchrotron radiated x-ray energy \({{{{{{{{\mathcal{E}}}}}}}}}_{r}\) (red triangles), critical energy ℏωc (black squares) and opening angle (blue circles) on a0. The black and red solid lines correspondingly show fittings of \({{{{{{{{\mathcal{E}}}}}}}}}_{r}\) and ℏωc according to Eq. (8). Here \(\theta =\max \{{\theta }_{y},{\theta }_{z}\}\). b Final photon spectra from the case with ne = 50nc (black), ne = 200nc (blue), ne = 200nc and an elliptical focal spot with larger ry = 3.53λ (red) and ne = 50nc,λ=400nm (dashed gray). Detailed parameters are introduced in the Discussion section.

Full size image

Discussion

The interaction of intense lasers with solid targets has been widely investigated in both theory and experiments81. The key difference in the experimental setup between our scheme and previous studies is the incident angle of the laser pulse. Previous studies81 focused on the schemes where the laser is normally or obliquely incident on a plane of a solid target, while here we demonstrate the physical advantages of the parallel incidence on an edge of a solid target. Compared to these known setups especially those employing normal incidence setups, no special considerations including the ionization of the solid target in the vacuum chamber are required68. Note that the ratio of laser energy reflected by the thin edge can be roughly estimated as dy/dL ≈ 20%, comparable to the 2D simulation result of about 14% though a short-scale preplasma is considered. The potential damage caused by the reflection can be avoided via utilizing a plasma mirror which can also enhance the laser pulse contrast82,83. Usually, the reflectivity of plasma mirrors is about 80%84 and it can be further improved to exceed 90% with a controlled prepulse85.

Considering that for laser wavelength of 800 nm, the electron density of solid targets is usually above 100nc, we did a new simulation with initial electron density of 200nc and the other parameters remaining unchanged to check the effects of the electron density. The obtained photon energy spectrum at t = 120T0 is shown by the blue line in Fig. 4b. The peak flux at 5 keV is about twenty percent lower, but still about 8 times higher than the record reported in the LWFA regime33. However, the divergences are even smaller (40 mrad and 30 mrad along y- and z-direction, respectively) benefited from narrower profile of the transverse fields [see detailed discussion in Supplementary Note III and Fig. S4d]. These together actually lead to two times higher brilliance compared to the standard case described in Fig. 1.

Further, to show that our scheme works for metal targets and different wavelength laser pulses, we consider the second-harmonic laser pulses which recently have attracted great interest in experiments due to its high contrast86,87,88. Since the normalized critical density \({n}_{{{{{{{{\rm{c}}}}}}}}}={m}_{{{{{{{{\rm{e}}}}}}}}}{\omega }_{{{{{{{{\rm{L}}}}}}}}}^{2}/4\pi {e}^{2}\) depends on the wave frequency, for different wavelength laser pulses, the same material is characterized by different normalized densities. For example, for second-harmonic laser pulses, ne = 50nc,λ=400nm already represents the realistic density of plastics and for ne = 200nc,λ=400nm, it is even higher than the realistic density of some common metals such as Aluminum and Copper.

We conducted another 3D PIC simulation with λ = 400 nm and the other normalized parameters remaining the same (i.e., a0 = 20, dL = 4.7λ, ne = 50nc,λ=400nm) except that the pulse duration is still 30 fs to deliver the same laser energy. The corresponding photon spectrum is shown by the dashed gray line in Fig. 4b. The critical energy is about 37 keV and the flux is about 1.25 × 107 photons eV−1 around 5 keV. The opening angles remain almost the same as that shown in Fig. 1d, but the spot size becomes about four times lower due to the shorter wavelength. These together result in an ultrabright x-ray pulse with brilliance of about 7.5 × 1023 photons s−1 mm−2 mrad−2 0.1%BW−1, which is much brighter than the record obtained in laser-based x-ray sources via the LWFA regime14. The reason for the higher critical energy can be attributed to the fact that the absolute values of the peak fields are twice larger than the case with wavelength of 800 nm, though their normalized value is the same. The stronger fields lead to stronger radiation. Note that the similarity theory is not applicable when the radiation reaction is considered.

Another feasible approach to further enhance the photon flux without consuming too many laser energies is using a laser pulse with an elliptical focal spot. An straightforward idea would be increasing the size of rz which may further linearly enhance the number of accelerated electrons and also the photon flux. However, we found that increasing the size of ry is a more efficient approach. For example, by increasing the size of ry by 50% to 3.53 λ and remaining rz and other parameters the same as the case with ne = 200nc described above (leading to a laser power of 225TW), the obtained photon flux reaches about 1.3 × 107 photons eV−1 around 5 keV [red line in Fig. 4b], slightly stronger than the linear scaling and about 15% higher than the case of increasing rz. Our analysis shows that the reasons can be attributed to that not only the local laser intensity felt by the electrons is slightly increased, but also the effects induced by the Rayleigh length can be mitigated. Moreover, with an appropriately longer pulse duration, more electron nanobunches are pulled out, accelerated forward and wiggled at the lateral surfaces, emitting more photons, see Supplementary Note IV and Fig. S5.

In summary, we have theoretically and numerically demonstrated an electron acceleration and radiation mechanism, where a femtosecond laser pulse is incident on a solid tape target. The underlying physics is identified, where electrons are accelerated efficiently by the longitudinal component of the SPW and emit photons continuously in the transverse self-generated quasistatic electric and magnetic fields. Our 3D simulations show that a high-flux high-brilliance x-ray source can be obtained with a tabletop 100 TW-class femtosecond laser. This would greatly advance the development of the laser-plasma based x-ray sources as a compact platform for radiography studies and ultrafast diagnostics.

Methods

PIC simulation

The 3D PIC simulations are conducted with the VLPL code89. Since the short wavelength radiation cannot be resolved by the simulation grid, in VLPL, the radiation spectrum is calculated by assuming that an angular distribution is peaked in the direction of the electron momentum and can be approximated with the δ function and the frequency spectrum S(ω/ωc), where \(S(x)=x\int\nolimits_{x}^{\infty }{K}_{5/3}(\xi )d\xi\) and ωc = (3/2)γ2∣F⊥∣ is the critical frequency determined by the instant electron γ and transverse force acting on the electron59,90. The recoil force on the electron exerted by the emitted photon is included into the equations of the electron motion. This model has been benchmarked against experimental results11,15,90.

A y-polarized Gaussian laser pulse is normally incident on an edge of a longitudinally orientated microtape [see Fig. 1a]. The pulse has a normalized intensity of a0 = 20 (I0 = 8.65 × 1020 W/cm2), wavelength of λ = 0.8 μm, duration of τL = 30fs (FWHM), and transverse spot size of dL = 4.7 λ, leading to a power of about 150 TW. Such laser systems have been widely constructed around the world with a compact size of ~ 10m291,92. A microtape target with relativistically overdense electron density ne = 50nc is used to save computational resources. Here \({n}_{{{{{{{{\rm{c}}}}}}}}}={m}_{{{{{{{{\rm{e}}}}}}}}}{\omega }_{{{{{{{{\rm{L}}}}}}}}}^{2}/4\pi {e}^{2}\) is the critical density, where me and e are the electron mass and charge, respectively. The plasma tape has dimensions of x × y × z = 140λ × 1λ × 24 λ to ensure large numbers of betatron oscillations and adapt to the simulation domain of 42λ × 24λ × 24 λ. It is sampled by 2800 × 960 × 240 cells with each cell containing 8 macroparticles. To account for possible finite contrast in experiment, we put some preplasmas around the tape with scale lengths that can be achieved in current experimental conditions81. The preplasma at the front has 1 λ linear density ramp and that at lateral surfaces has 0.25 λ linear ramp. A moving window is employed.

For the simulations in the discussions of scaling law and robustness, except the parameters specified in the text, the others remain unchanged. In the simulation with higher electron density of ne = 200nc, finer resolutions along x- and y-directions (i.e., hx = 0.01λ, hy = 0.0125 λ) are considered while the resolution along z-direction remains the same to reduce the simulation size. Collisions are neglected since the collisional time79 is orders of magnitude longer than the time scale of interest.