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Abstract
Electrons can be accelerated to GeV energies with high collimation via laser wakefield acceleration in the bubble regime and emit bright betatron radiation in a table-top size. However, the radiation brightness is usually limited to the third-generation synchrotron radiation facilities operating at similar photon energies. Using a two-stage plasma configuration, we propose a novel scheme for generating betatronlike radiation with an extremely high brilliance. In this scheme, the relativistic electrons inside the bubble injected from the first stage can catch up with the frequency-downshifted laser pulse formed in the second stage. The laser red shift originates from the phase modulation, together with the group velocity dispersion, which enables more energy to be transfered from the laser pulse to γ-photons, giving rise to ultra-brilliant betatronlike radiation. Multi-dimensional particle-in-cell simulations indicate that the radiated γ-photons have the cut-off energy of GeV and a peak brilliance of 1026 photons s−1 mm−2 mrad−2 per 0.1%BW at 1 MeV, which may have diverse applications in various fields.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Thanks to the rapid development of laser technology, laser-plasma interaction provides vast possibilities for fundamental researches. For example, it has been widely used for compact particle acceleration [1], Thomson scattering [2], free-electron laser (FEL) [3], and mid-IR laser pulse generation [4], resulting in highly collimated, ultra-brilliant femtosecond x/$\gamma$-ray sources [5,6]. It was first proposed in 1979 that the wake field created by an intense laser pulse could accelerate electrons in gases to relativistic energies in a much more efficient way as compared to conventional accelerators [7]. When an ultra-intense laser pulse injects into underdense plasmas, the ponderomotive force pushes the plasma electrons aside rapidly. However, the ions are much heavier than electrons, so that they can be regarded to be still during the short interaction time. Finally, a plasma wave with the frequency ${\omega _p} = {({e^{2}}{n_0}/{m_e}{\varepsilon _0})^{1/2}}$ would be excited behind the pulse, where $n_0$ denotes the electron density, $e$ and $m_e$ are the electron charge and mass respectively, and $\varepsilon _0$ is the dielectric constant. Meanwhile, the phase velocity of the wave equals to the group velocity of the drive pulse, $v_{ph}^{wave} = v_g^{laser}$. A number of experiments have shown that the laser wakefield acceleration (LWFA) in the bubble regime is one of the most efficient schemes for electron acceleration, which is valid when the spot waist of the laser pulse $r_0$ satisfies the condition $r_0 \approx {2}\sqrt {{a_0}} /k_p$ ($a_0$ is the laser normalized vector potential, and $k_p$ is the plasma wave number). This regime is characterized by an approximate sphere-shaped bubble with a radius of $R=r_0$ [8], when the laser pulse evacuates all the electrons out of the focal center, leaving an ion cavity, which is so-called bubble [9]. The electrons would self-inject from the end of the bubble, trapped and accelerated by the the longitudinal acceleration field. They would not be decelerated until reaching the decelerating field, and the distance of the preceding acceleration process is defined as dephasing length $L_d$ [10]. Usually, the acceleration length is determined by the dephasing length, which scales as $L_d \approx 2/3(\omega _0^{2}/\omega _p^{2})R$ ( $\omega _0$ is the laser frequency). In addition, the pulse duration should satisfy a matching condition $\tau \le r_0/c$, allowing for the laser pulse fitting the half ion cavity completely. The trapped electron bunch can be accelerated by the longitudinal focusing force, and be simultaneously wiggled transversally due to a restoring force from the transverse charge separation field, emitting a collimated x/$\gamma$-ray beam, which is the so-called betatron radiation. The wake electric field that provides the above forces for the electrons could achieve the maximum $E_{max}=m_e\omega _p c\sqrt {a_0}/e$ [8], which depends on the laser dimensionless parameter $a_0$.
It was proven that with the assistance of external guiding [11] or density gradient injection [12–15], or using the clustering gas target [16], or increasing the laser intensity [17,18], the efficiency of betatron radiation could be improved significantly. In addition, a longitudinal density depression or step of the plasma has been also used to increase the transverse oscillation amplitude in the bubble [19]. However, the transverse focusing force produced by the wake field in the bubble is still relatively weak as compared to the laser field, restricting the energy of x/$\gamma$-ray photons within keV or MeV range. Alternatively, another possible solution that has been considered is that the trapped electrons can be accelerated by the laser pulse directly, and gain more energy from the laser pulse by resonating with the laser frequency [20–26]. An experiment of such an alternative solution [27] demonstrated that the peak brilliance of $\gamma$-photons achieved at 450 keV is as high as $10^{23}\; \textrm{photons}\; \textrm{s}^{-1}\; \textrm{mm}^{-2} \;\textrm{mrad}^{-2}\; \textrm{per}\; 0.1\% \;\textrm{BW}$, and the total number of photons is estimated as $10^{8}$. Although the electron oscillation and photon emission get improved apparently in this scheme, both the electron and $\gamma$-ray photon energy are excessively restricted as a result of the overlong pulse duration [28].
Therefore, it is of significance to increase the radiation probabilities for electrons without breaking the matching condition of the bubble acceleration. During the past decades, the few-cycle mid-infrared (wavelength $\lambda$ < 5 $\mu$m) and long-wavelength infrared (LWIR, $\lambda \sim 10\mu$m) pulses have motivated new ways to produce bright x/$\gamma$-rays [29,30]. It has been shown that the harmonics photons per pulse generated with mid-infrared laser pulses of a peak intensity of $6\times 10^{16}\; \textrm{W cm}^{-2}$ is about 25 times that of 800 nm pulses, since the longer wavelength could accelerate the electrons much more efficiently [31]. In this paper, we put forward a new scheme of brilliant GeV $\gamma$-rays generation from a tailored two-stage plasma in the near QED regime. In the first stage, the electrons self-inject into the bubble and are accelerated to ultrahigh energies, while in the second one, the laser pulse propagates into a much denser plasma with its frequency redshift to infrared band, so that the accelerated electrons could catch up with the end of the infrared pulse, giving rise to ultra-brilliant radiation. In this regime, we achieved a dense electron bunch with the peak energy of 4.5 GeV, and the emitted $\gamma$-ray photons with photon number and brilliance 3-4 orders of magnitude higher than that of the Compton scattering case assisted by LWFA [32]. Such bright high-energy $\gamma$-ray sources have broad uses ranging from probing photo-nuclear spectroscopy [33,34] in fundamental science, through industrial radiography [35], to light-matter interaction [36–38] or $\gamma$-$\gamma$ collider [39–41] in the field of high-energy-density physics.
2. Laser wakefield acceleration in a tailored two-stage plasma
In the following, we first carry out 2D PIC simulations with the open source code EPOCH to investigate the LWFA and infrared laser involved synchrotron radiation process in a tailored two-stage plasma with a petawatt (PW) laser. In simulations, the QED BLOCK has been enabled to include the stochastic emission of photons via a Monte Carlo algorithm, the radiation reaction effect, and the feedback between the plasma and photon-emission processes, whereas the effect of spin polarization is ignored. Here, we take use of the similarity parameter $S=n_0/a_0n_c$ to scale the electron acceleration and radiaiton [28], where $n_c= {m_e}{\omega _0}^{2}/4\pi {e^{2}}$ is the critical plasma density. When $S \ll 1$, it corresponds to the case of relativistic underdense plasma. The plasma composes of two parts: the first part is the preplasma located from $5\lambda _0$ to $20\lambda _0$ along the x-axis and -$20\lambda _0$ to $20\lambda _0$ along the y-axis, accompanied by the uniform underdense one with the density of $n_1=1.5\times 10^{-3}a_0n_c=0.3{n_c}$, located between $20\lambda _0$ and $250\lambda _0$; the second part is the uniform plasma with $n_2=1.5\times 10^{-2}a_0n_c=3n_c$ from $250\lambda _0$ to $350\lambda _0$, where $\lambda _0=0.8\; \mu\textrm{m}$ is the wavelength of the incident laser pulse. A linearly polarized laser pulse propagates from the left boundary with the Gaussian envelope ${a_{t,r}} =a_0 \textrm{exp}(-r_\bot ^{2}/r_0^{2}-t^{2}/\tau ^{2})$. Here, $a_0=200$ corresponds to the peak intensity of $I_0=8.6\times 10^{22}\; \textrm{W cm}^{-2}$, which is polarized along the y axis. The laser pulse is focused on the first stage plasma at $x=5\lambda _0$ with the spot radius of ${r_0} = 8\lambda _0$ and the pulse duration $\tau = 20T_0$, where $T_0 = \lambda _0/c=8/3$ fs is the laser period. The moving window module is applied, with the window moving at the speed of light in vacuum. The simulation box has the size of $x\times y=60\lambda _0 \times 40\lambda _0$ with $0.02\lambda _0\times 0.1\lambda _0$ for each cell. Particles are represented by 16 macro-particles in each cell and ions are movable in all simulations. Absorbing boundary conditions are used for both electromagnetic fields and particles.
The two-stage acceleration and bright radiation emission processes are shown schematically in Fig. 1(a). At the first stage, the ultra-intense laser pulse excites a stable ion cavity and the injected electrons can be accelerated steadily as shown in Fig. 1(b). The radius of the cavity is approximately 8 $\mu$m, which equals to the focal waist of the incident laser pulse. At this instant, the phase velocity of plasma wave is approximately $0.9c$, while the wake field reaches $x=185\ \mu$m along the propagation direction. To describe the ion cavity quantitatively, we assume a cylindrical coordinate system ($r, \theta , x$) with the distance from the central propagation axis $r$, axial angle $\theta$, and propagation direction $x$. Thus, a concomitant variable $\zeta = x - {v_{ph}}t$ can be used to represent the cavity [42]. In this system, the radial electric field $E_{rs}/E_{wb}={k _p}r /4$ (corresponding to $E_{ys}/E_{wb}={k _p}y /4$ in our 2D simulations) is estimated to have the maximum $E_{ys}^{max}=8E_{wb}$, where ${E_{wb}} = {m_e}{\omega _p}c/e$ is the wave breaking field [43–45]. Figure 1(e) shows the electric field at $t=250T_0$, fitting well with the analytical result. Since $t=270T_0$, the laser pulse starts to propagate into the second stage plasma. In this stage, The LWFA is replaced by direct laser acceleration (DLA). The laser pulse continues to expel the electrons from the second plasma out of the central laser axis, developing a plasma channel. These electrons are recoiled back to the focal spot on account of the radiation reaction effect [46,47]. However, the electrons from this part of plasma contribute little radiation emission as compared to the relativistic electrons accelerated in the first stage, due to their much lower number and energy. As presented in Fig. 1(f), the laser field is compressed sharply along the y axis and dissipates enormously along the x axis as it propagates through the ten times denser plasma at $t=290T_0$. On the one hand, the local group velocity of the laser pulse ${v_g}(\zeta )/c \simeq 1 - k_p^{2}(\zeta )/(2{k^{2}})$ reduces, so does the phase velocity of the self-generated electric field. However, the electrons from the first part of plasma still move at approximately the former velocities during such short interaction time of the second stage (electrons from the second part of plasma are not shown in Fig. 1). On the other hand, the wave frequency of plasma rises along with the square root of electron density, $\omega _p^{2}(\zeta ) \propto n(\zeta )$, and the local phase velocity of the light changes as ${\beta _{ph}} = {v_{ph}}(\zeta )/c \simeq 1 + \omega _p^{2}(\zeta )/(2{\omega ^{2}})$ on the basis of dispersion relation $\omega ^{2} =c^{2}k^{2} + \omega _p^{2}(\zeta )$, resulting in the fact that the phase velocity of the laser wavefront is larger than that at the rear of the pulse, ${v_{ph}}(\tau /2) > {v_{ph}}( -\tau /2)$ [48,49]. Therefore, the wavefront moves faster than the rear of the pulse and the wavelength increases, which is the so-called phase modulation. As a result, the electron bunch catches up with the back of the frequency-downshifted laser pulse. The laser frequency redshift continues until $t=330T_0$ when the electron bunch gets into the pulse rear more deeply, as we can see in Figs. 1(d) and (g).
Fig. 1. (a) Concept of the two-stage acceleration and betatronlike radiation processes. A linearly polarized laser pulse propagates through a tailored plasma with a density step at $x=200\ \mu m$. The self-injected electrons at the first stage would be overlapped and accelerated by the frequency-downshifted infrared laser pulse after moving into the second stage. (b-d) Density distribution of electrons $n_e$ from the first stage plasma and (e-g) electric field $E_y$ distribution at $t=250T_0$, $290T_0$, and $330T_0$, respectively. Here, the electric fields are normalized by ${E_0} = {m_e}c{\omega _0}/e$.
During the enhancement of the radial electric field, the longitudinal field also increases enormously after the laser pules enters the second stage, which is apparently shown in Figs. 2(a) and (b). It can be well explained by the relation $E_x/E_{wb}=k_p\zeta /2$, from which we see the acceleration field would triple as the plasma density becomes ten times denser. Hence, the electrons could be further accelerated to higher energies driven by the longitudinal electric field, with the monoenergetic peak of the spectrum increased from 2.7 GeV at $t=250T_0$ to 3.8 GeV at $t=330T_0$ as indicated in Figs. 2(c) and (d). Here, the maximum energy that the trapped electrons could achieve in the bubble at the first stage is ${\gamma _{\max }} \simeq 2\gamma _p^{2}{\widehat E_{\max }^{2}}$ [49], where ${\gamma _p} = {(1 - v_p^{2}/{c^{2}})^{ - 1/2}}$ is the Lorentz factor of the plasma wave and ${\widehat E_{\max }} = {E_{\max }}/{E_{wb}}$, corresponding to $E_e^{max}=\gamma _{\max }m_e c^{2}=3.7\; \textrm{GeV}$, which agrees well with the simulations (Fig. 2(c)). The insets in Fig. 2(d) show that the energy divergence of the collimated electron bunch gets maintained in the formed plasma channel, and around $10^{11}$ electrons with high density ($10^{21} cm^{-3}$), high charge (tens-nC) and high energy (multi-GeV) are trapped and accelerated efficiently within a cone angle of ${7^ \circ }$. For more details, some typical electrons’ trajectories are also presented in Fig. 2(e), showing the whole process that the electrons are firstly expelled from the laser axis, then get kicked back to the focal center, and finally travel around the propagation axis steadily. It can be also observed that the trajectories of electrons appear to be planar sinusoidal in the 2D simulations.
Fig. 2. Electron acceleration. Distribution of longitudinal accelerating field $E_x$ at (a) $t=250T_0$ and (b) $270T_0$. The longitudinal electric field gets enhanced when the laser pulse interacts with the higher-density plasma at $270T_0$. Energy spectrum and energy divergence of electrons from the first stage at (c) $t=250T_0$ and (d) $330T_0$. (e) Trajectories of electrons from the very beginning to $t=360T_0$.
3. Bright betatronlike radiation in frequency-downshifted laser pulses
The dense, high-charge, low-divergence electron bunch is not only accelerated to relativistic energy by the longitudinal force, but also wiggled by the transverse electric field force, radiating high-energy $\gamma$-ray photons. The dynamic motion of an electron in such electric fields can be described as
Here, the radiation process is determined by the key QED parameter ${\chi _e} \approx \gamma _e{F_ \bot }/E_s=\gamma _e({E_r}{\widehat e_r} + \overrightarrow v \times ({B_\theta }{\widehat e_\theta }))/E_s$ [53], where $E_s= m_e^{2}{c^{3}}/e \hbar$ is the Schwinger electric field, and its distribution along the $x$ and $y$ axis in the two stages has been shown in Figs. 3(a) and (b). When ${\chi _e}\geq 1$, the photon emission would be nonlinearly intense. It is shown that the parameter at $t=250T_0$ approaches a maximum ${\chi _e=1.4}$, peaking on the $x$ axis, much larger than the one at $t=330T_0$ with ${\chi _e\le 0.17}$. As discussed above, the transverse force in the first stage mainly comes from the plasma wakefield, which is much weaker as compared to the laser field, and the betatron radiation is limited by the cavity radius and electron energy. By contrast, when the laser pulse propagates into a denser NCD plasma, the transverse force gets strengthened in two ways: (a) the electrons overlap and interact with the frequency-downshifted LWIR pulse, whose electric field is much stronger than the wakefield; (b) the transverse wakefield near the light propagation axis has the maximum $E_{ys}\approx 12E_0$, three times higher than the former field at the first stage. Hence, much more energetic photons are emitted at $t=330T_0$ than that at $t=250T_0$ as shown in Figs. 3(c) and (d). There are approximately $10^{14}$ photons emitted within a cone angle of ${10^ \circ }$, with a cross section of $2\ \mu$m and duration of 30 fs. Figure 3(e) shows the energy spectrum evolution of the $\gamma$-photons. The cut-off photon energy increases by three times after the laser pulse propagates 50 times initial laser periods in the second stage. Additionally, the $\gamma$-ray peak brilliance as a function of the photon energy is given in Fig. 3(f), assuming the emission angle to be 33 mrad, with a peak brilliance up to $10^{26}$ photons $ \textrm{s}^{-1} \textrm{mm}^{-2} \textrm{mrad}^{-2} \textrm{per}\ 0.1\%$BW at 1 MeV, which is 3-4 orders of magnitude higher than that of the Compton scattering case assisted by LWFA [32]. Such ultra-brilliant radiation sources could be broadly applied in various domains ranging from fundamental researches, to medical imaging in industry, medicine, and so on. Compared with the other two-stage ideas to generate intense $\gamma$-rays in Table 1, the conversion efficiency of our scheme is comparable to the case in [6] where the accelerated electron beam by LWFA radiates high-energy $\gamma$-photons in an electric dipole, 1-2 orders of magnitude higher than the two other schemes. However, the proposed scheme in [6] requires the pre-accelerated electrons from additional LWFA or traditional accelerators, which is relatively difficult in real experiments.
Fig. 3. $\gamma$-ray emission. QED parameter ${\chi _e}$ at $t=250T_0$ (a) and $330T_0$ (b). The kinetic energy density of the radiated $\gamma$-photons at $t=250T_0$ (c) and $330T_0$ (d). The energy spectrum of $\gamma$-photons at $t=250T_0$, $290T_0$ and $330T_0$ (e), and the angular spectrum at $t=330T_0$ (inset). The brilliance spectrum (photons $\textrm{s}^{-1} \textrm{mm}^{-2} \textrm{mrad}^{-2} \textrm{per}\ 0.1\%$BW) of the $\gamma$-ray as a function of the energy spectrum at the corresponding time (f).
Table 1. Comparison of $\gamma$-ray source from several proposed schemes.
The production of such bright radiation sources results from the fact that more laser energy can be transfered to the electrons from the first stage. To illustrate the underlying physics, we show the evolution of the laser field, the electron density distribution along the propagation axis, the kinetic energy density distribution of electrons and the evolution of laser intensity from $t=200T_0$, through $t=270T_0$ to $t=330T_0$. In Figs. 4(a)-(c), the black curves in the top row shows the combination density of electrons from the two stages, while the black ones below illustrate the electron density distribution from the first stage plasma. In Fig. 4(a), it is shown that the laser pulse propagates steadily inside the first half of the ion cavity with a small density peak ahead, followed by the trapped electrons behind. There is nearly no modulation of the pulse wavelength as we can see from Fig. 4(d). However, when the laser pulse starts to enter into the second stage plasma, the ponderomotive force presses the plasma electrons forwards quickly, forming a sharp density gradient at the front of the pulse as shown in Fig. 4(b), which plays a key role in the laser phase self-modulation. The electron density $n_e(\zeta , \tau )$ and plasma frequency $\omega _p(\zeta , \tau )$ ($\tau =t$) are different at the front and rear of the pulse in space and time, leading to the change of the local phase velocity of light, $d{v_p}/d\zeta \propto \partial {n_e}(\zeta ,\tau )/\partial \zeta$ [42,48]. Finally, the wavelength variation could be expressed as $d\lambda = \Delta {v_p}d\tau$ [49], and the modulated laser wavelength could be estimated by [4]
Fig. 4. Laser frequency self-modulation. The black and red curves in the top row represent the profiles of the electron density and transverse electric field on the laser axis at (a) $t=200T_0$, (b) $270T_0$ and (c) $330T_0$. Here, the density profile of electrons from the $0.3n_c$ plasmas is shown with the black curve in the bottom row, and the electron energy-density distribution is also projected in the x-y plane. The transverse laser field has been multiplied by four times (red curves) in (c). (d-f) The corresponding spectra of the laser pulse intensity with the same normalized unit, and the LWIR pulse at $t=330T_0$ is shown in the shaded region. The inset in (f) shows the temporal waveform of the developed LWIR pulse at the central wavelength $\lambda _c\approx 7\mu$m.
4. Discussion
In our scheme, both the tailored plasma and frequency-downshifted infrared pulse play a key role in the final ultra-bright $\gamma$-ray emission. Firstly, such low-density or near-critical-density (NCD) plasmas with density of $(0.1-10)n_c$ could be produced by focusing laser beams on gas, foam or cluster jets from ultra-low density plastic foam [54], which has been extensively investigated and applied in laser-plasma community to ion acceleration and x-rays emission.
Furthermore, as we know, when a laser pulse propagates into a uniform low-density plasma, the laser frequency would be downshifted and a density disturbance would be excited. However, the process usually lasts several and even tens of Rayleigh length (depending on the ratio of laser duration to wavelength) and the laser pulse also depletes as its energy is coupled into the plasma during the process. Therefore, the frequency-downshifted pulse is so weak that it could be ignored, when it falls behind to overlap with the electrons. For comparison, we perform additional 2D simulations with a uniform plasma density. All the other parameters keep unchanged. It has been shown in Fig. 5(a) that the accelerated electron beam and the drive laser remain separate in space, without direct overlap with each other. The electron density is mainly distributed behind the laser field even at $t=330T_0$. Therefore, although the electrons could be accelerated to higher energy, the total number and peak energy of $\gamma$ photons are a half lower as compared to our scheme above as we can see from Figs. 5(b) and (c). Eventually, the peak brilliance of the emitted $\gamma$ photons in this uniform plasma case is estimated as $10^{24}$ photons $\textrm{s}^{-1} \textrm{mm}^{-2} \textrm{mrad}^{-2} \textrm{per}\ 0.1\% \textrm{BW}$, which is two orders of magnitude lower than that of our scheme. In addition, the spectral evolution of the modulated laser pulse from $t=250T_0, 290T_0$, to $330T_0$ is given in Fig. 5(d), which proves that the red-shifting due to the rapid phase modulation is absent in the uniform medium. This is radically different from our scheme, where the down-shifted infrared laser pulse can be generated in the two-stage plasma above.
Fig. 5. Simulation results without tailored plasma. (a) Distribution of electron density and electric field at $t=330T_0$. Energy spectrum evolution of (b) electron and (c) $\gamma$-ray photons, and (d) the spectrum of modulated laser pulse from $t=250T_0, 290T_0$, to $330T_0$.
We also investigate the robustness and convergence of the regime by using different laser intensities and performing additional higher-resolution simulations. Figure 6 shows that the energy conversion efficiency from the laser pulse to $\gamma$-rays grows nonlinearly as the laser amplitude rises from $a_0=160$ to 220, producing more and brighter photons in a steady scaling. The number of emitted photons can be estimated by ${N_\gamma } \propto N_e N_\beta K$ and the spectral range for a synchrotron radiation is derived as $\hbar {\omega _c \propto K{\gamma ^{2}}/{\lambda _u}}$, where ${\omega _c}$ is the critical frequency of synchrotron spectrum [45]. Here, $\lambda _u$ is the electron oscillation period, $N_e$ and $N_\beta$ are the number of electrons and oscillation number respectively. In addition, $K = \gamma \theta _\gamma$ is the dimensionless strength parameter, where $\theta _\gamma$ is the maximum angle of the electron velocity with regard to the x axis and also the emission divergence of the $\gamma$ photons. The electron energy gain scales as $\gamma \propto a_0$ and the betatron oscillation period obeys $\lambda _u \propto a_0^{1/2}$. The bubble radius, which equals the laser focal waist, scales as $r_b\propto a_0^{1/2}$, so that the number of the trapped electrons scales as the size of the bubble $N_e \propto r_b^{3} \propto a_0^{3/2}$. Moreover, both the direction of electron motion $\theta _\gamma$, and the number of oscillation $N_\beta$ scale as $\theta _\gamma \propto n_e^{1/2}$. Thus, the number, cut-off energy, and conversion efficiency of the emitted photons have a scaling of ${N_\gamma } \propto a_0^{5/2}$, $\hbar {\omega _c}\propto a_0^{5/2}$, and ${\eta _\gamma } \propto a_0^{3/2}$, in an excellent agreement with the simulation results as shown in Fig. 6. In addition, the peak brilliance of the $\gamma$-photons at 1 MeV also increases with the laser intensity, of the order of $10^{26}$ photons $\textrm{s}^{-1} \textrm{mm}^{-2} \textrm{mrad}^{-2} \textrm{per}\ 0.1\% \textrm{BW}$ from $a_0=160$ to 220, corresponding to the laser intensity from $5.5\times 10^{22}$ to $10^{23}\; \textrm{W cm}^{2}$. The next generation of 10 PW laser facilities can provide such laser pulses, offering possibilities for our scheme to produce ultra-brilliant high-energy $\gamma$-ray sources. The obtained ultra-brilliant $\gamma$-rays are versatile tools for probing dense matter, homeland security, isotope production in astrophysics [33] and conducting pump-probe experiments in ultrafast optics [55]. Meanwhile, the convergence of simulations is also demonstrated by the higher-resolution simulations with the resolution of $0.002 \mu m\times 0.5 \mu m$ each cell for the two-stage plasma case and $0.01 \mu m\times 0.5 \mu m$ each cell for the uniform case. We find that the higher resolution would not significantly influence the $\gamma$-ray brilliance. Furthermore, the total number and peak energy of $\gamma$ photons in the uniform case are still a half lower as compared to the two-stage one, and the peak brilliance of the emitted $\gamma$ photons in the former case is still two orders of magnitude lower, in accordance with the our original results.
Fig. 6. Robustness of the scheme. (a) Influence of laser intensity on the peak brilliance, and cut-off energy of $\gamma$ photons. (b) Total photon number and energy conversion efficiency of laser pulse converted to $\gamma$-rays as a function of $a_0$. Here, the plasma density in the two stages are determined by $S_1= n_1/a_0n_c=1.5\times 10^{-3}$ and $S_2=n_2/a_0n_c=1.5\times 10^{-2}$ . All the other parameters are the same as those in the simulations above.
Additionally, we carry out a full-scale 3D simulation as well to ensure the reliability of our scheme. In Figs. 7(a) and (b), a time sequence of the isosurface contours of electron density and electric field is plotted, with $n_e=4n_c$ and $E_0=\pm 16E_0$. It is clearly shown that the electron bunch overlaps with the infrared laser pulse and oscillates inside at $t=330T_0$, resulting in an outburst of ultra-bright $\gamma$-ray radiation as presented in Figs. 7(c) and (d). The spectrum evolutions of the photon energy and laser pulse are also given in Figs. 7(e) and (f) from $t=250T_0$ to $330T_0$. When the laser pulse propagates into the density up-ramp at $t=250T_0$, its frequency gets downshifted gradually, and gets to a long-wavelength range (the red-shaded region) with a spectral peak at about ${\lambda _c} \approx 6\ \mu$m at $t=330T_0$ as shown in Fig. 7(f). In Table 2, we compare the $\gamma$-ray brilliances in 2D and 3D simulations. It is shown that the peak brilliances of $\gamma$-rays at 1 MeV and average brilliance in 3D simulations are as the same level as in the 2D simulations. Therefore, the results show a very similar performance for the electron acceleration, frequency downshift of laser pulses and photon emission to the 2D simulations, further demonstrating the validity of the simulations and robustness of our scheme.
Fig. 7. 3D snapshots of electron density and electric field at (a) $t=250T_0$ and (b) $330T_0$, where the isosurfaces of electron density (in green color) and electric field intensity (in red and blue colors) are taken at $4n_c$ and $\pm 16E_0$, respectively. The 2D rejections of photon density distributions in $x-y$ plane at (c) $t=250T_0$ and (d) $330T_0$. Evolution of the photon energy spectrum (e) and laser pulse (f) at $t=330T_0$, and the LWIR pulse is shown in the shaded region.
Table 2. Comparison of photon brilliances in 2D and 3D simulations.
5. Conclusion
In conclusion, high power laser pulse can accelerate electrons to a couple of dozens GeV to date, which offers more possibilities to produce bright high-energy $\gamma$-ray sources via laser driven betatron radiation. In this article, we introduce a novel scheme for ultra-brilliant $\gamma$-ray emission by use of a modulated LWIR laser pulse caused by a plasma density step in a tailed plasma. It strengthens the oscillation field for the trapped electrons from the LWFA. Finally, the multi-dimensional simulations demonstrate that ultra-brilliant $\gamma$-ray source is achieved with the cut-off energy of GeV and peak brilliance of $10^{26}$ photons $\textrm{s}^{-1} \textrm{mm}^{-2} \textrm{mrad}^{-2} \textrm{per}\ 0.1\% \textrm{BW}$ at 1 MeV in our scheme, which is two orders of magnitude higher than that in the uniform plasma case. Such bright $\gamma$-ray sources are serviceable for the applications in a broad range of fields, like fundamental researches in astrophysics, exploring elementary particles in high-energy physics, and imagery in chemistry, biology and medicine area, etc.
Funding
National Key Research and Development Program of China (2018YFA0404802); Science Challenge Project (TZ2016005); National Natural Science Foundation of China (11774430, 11875319, 12005297); Science and Technology Innovation Program of Hunan Province (2020RC4020); Natural Science Foundation of Hunan Province (2020JJ5651); Research Project of NUDT (ZK18-02-02); Fok Ying Tung Education Foundation (161007); Hunan Provincial Innovation Foundation for Postgraduate (CX20190017, CX20190018, CX20200002, CX20200038).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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