1. Introduction

High energy γ-rays have wide applications. Photons with MeV range can be used in manipulating nucleus materials [1,2]. In radiosurgery of oncology, bremsstrahlung γ-ray with the energy of ∼ 10MeV is the standard tool for for tumor treatment known as ’Gamma Knife’ [3]. For the studies in astrophysics and other fundamental science, one requires well collimated γ-ray beams with higher energy [4]. In relativistic laser-plasma interaction, photons are mainly emitted via the nonlinear Compton scattering regime when the energetic electrons are interacting with the strong laser pulse [5–7]. It requires both electron energy and the laser intensity to be sufficiently high. With the development of the laser facilities, multi-PW laser is available nowadays [8,9] which provides a unique possibility to realize the nonlinear Compton scattering photon emission. Various regimes have been proposed in recent years [10–13,15–23].In the pioneering experiment, hundreds keV X-ray radiation has been obtained with the total photon number of 10⁸ by the head- on collision between the back-reflected laser pulse and the wakefield accelerated electrons [13]. At the Texas PW Laser, Compton γ-ray with a tunable peak energy from 5 to 85 MeV is obtained [14]. Here we propose a self-organizing for high energy γ-photons emission by the collision between the relativistic electron beam and the high order harmonics attosecond pulse. The electrons are accelerated by the laser-wakefield and the high order harmonics are generated via the relativistic oscillating plasma mirror. The results are based on the kinetic simulations by the relativistic electromagnetic Particle-in-Cell code EPOCH [24,25]. The laser parameters employed here are achievable nowadays and the setup is simple for experiments.

2. Description of the interaction processes

The probability of photon emission via Compton scattering is characterised by the relativistic and gauge invariant parameter ${\chi }_e=\sqrt{{\left(F^{\mu \nu }{p}_{\nu }\right)}^2}/(E_s{m}_{e}c)$ [26], here F_μν = ∂_μA_ν − ∂_νA_μ is the 4-tensor of the electromagnetic wave, p_ν is the electron 4-momentum and $E_s=m_e^2{c}^3/e\hslash$ is the critical quantum electrodynamics (QED) electric field [27]. In the case of a plane electromagnetic (EM) wave propagating along x–direction, χ_e = (E/E_s)(γ − p_x/m_ec). In the ultra-relativistic limit, γ >> 1, the photon emission probability is maximized if the electrons and the laser pulse are counter-propagating, ${\chi }_e^{\downarrow \uparrow }\simeq 2\gamma (E/E_s)$. It is dramatically reduced to ${\chi }_e^{\uparrow \uparrow }\simeq {\left(2\gamma \right)}^{-1}(E/E_s)$ in the co-propagating case. Several configurations have been proposed based on multi-pulses interaction to satisfy the laser-electron counter-propagating condition. Here we add a solid density Aluminium target at the end of the hydrogen gas jet to reflect the laser pulse. When the laser propagates in the underdense hydrogen plasma, it induce a strong wakefield to trap and accelerate the electrons according to the typical laser-wakefield model (LWFA). The laser is also self-focused while propagating since the self-focusing threshold [28], P > P_cn_c/n₀ ($P_c=2m_e^2{c}^5/e^2=17\text{GW}$), is satisfied. The laser is reflected from the solid target before deletion becomes important. Since the high density of the aluminium layer provides a large charge separation field, the expelled electrons are pulled back to the ion layer by the restoring force. The interface of the critical electron density moves back and forth, which makes the solid target mirror becoming an oscillating mirror [29]. Due to the Doppler effect experienced by the incident laser, the frequency spectrum of the reflected laser field extends into the high frequency range and the wave breaks up into short wave packets. The schematic is shown in Fig. 1. The subsequent energetic LWFA electrons then collide with the high order harmonic fields, which provides a large probability for γ-ray emission.

 

Fig. 1 Schematic of the simulation setup.

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The maximum electron energy obtained by wakefield acceleration is estimated as $W\approx 0.22(c\tau /\lambda )\sqrt{P(\mathit{GW})m_e}{c}^2$ [30], therefore the corresponding electron Lorentz factor is ${\gamma }_e\approx 0.22(c\tau /\lambda )\sqrt{P(\mathit{GW})}$. Due to the head-on collision geometry of the relativistic electrons and the laser field, the obtained maximal quantum invariant is

The average χ_e can be evaluated as <χ_e >≃ 2(ħω_L/m_ec²) <a_f>², here <a_f> is the mean amplitude of the electric field [24]. In the case of Compton scattering, the emitted γ photon energy scales as ${\epsilon }_{k\gamma }=4{\gamma }_e^2\hslash {\omega }_L$, here ω_L is the frequency of the EM field and ħ is the reduced Planck constant. Then the maximum photon energy is obtained as

The photon energy is proportional to the incident laser power and the frequency of the EM field colliding with the electrons. In the Compton scattering regime with ω₀ laser and a GeV electron, one get the maximum photon energy about 16 MeV. However, in the case of a₀ >> 1, the nonlinear Compton scattering should be taken into account, in which the photon energy scales as ${\epsilon }_{k\gamma }=4{\gamma }_e^2\hslash {\omega }_L{a}_0$ [31]. Then the corresponding maximum photon energy becomes

The characteristic photon energy can be written as

Based on the above model and estimate, a bunch of GeV electrons accelerated by the laser wakefield is expected by using a PW class laser with the intensity of 10²² W/cm². With collisions of the GeV electrons and the reflected laser field, the emitted photon energy also reaches to the GeV order of magnitude. The characteristic photon energy is about 13.5 MeV. The simulations results in the next section show the well agreements with the estimate. The reflected EM field passes through the whole LWFA electron beam so that the emitted photon bunch has almost the same length of the LWFA electron beam. Therefore the photon beam length is also controllable by the LWFA electron adjusting methods such as [32]. Furthermore, the γ-ray flux is well collimated due to the directional electron beam.

3. PIC simulation results

In this section, we will show the results from the 2D kinetic simulations. A linear-polarized Gaussian pulse with the peak intensity of 10²² W/cm² propagates along the x-axis. The normalized amplitude is a₀ = eE₀/m_eω₀c ≈ 85.7, where $E_0=\sqrt{8\pi I_0/c}$ and ω₀ are the laser electric field strength and frequency, e and m_e are the electron charge and mass, respectively; and c is the speed of light in vacuum. The pulse duration is τ = 15 fs and the spot size (FWHM) is about r₀ = 5 λ. The laser wavelength is λ = 1 μm. The gas jet of hydrogen plasma has the peak density of n₀ = 0.3 n_c, where n_c = m_eω²/4πe² is the plasma critical density. The simulation box has the size of 200 λ and 40 λ in the x and y direction. The longitudinal density profile exponentially increases from 10⁻³ n_c to 0.3 n_c for 10 λ < x < 110 λ, then remains constant for 85 λ. At the end of the hydrogen plasma target, a warm expanded Aluminium plasma layer [33] is attached with the density of 50 n_c and the thickness of 1 λ. The solid density layer assumed to be pre-ionized as Al¹³⁺. The mesh size for the 2D simulation is δx = δy = λ/60. Ion motion is included in the simulations. All the quasiparticles (64 per cell) are initially at rest. Nonlinear Compton scattering process is included in the simulations [24,25]. The Bremsstrahlung radiation effect is ignored. A Monte-Carlo algorithm is implemented in the code for modeling the photon emission process. An optical depth is given to each electron. The photon is emitted when the optical depth reaches an extracted value from random-number-generator between 0 and 1. The local constant field approximation is used to treat the external electromagnetic field as a plane wave in the instantaneous rest frame of the charge. It is valid in describing the strong-field QED processes and the hundreds MeV photon emission in our case.

When the laser propagates in the underdense plasma, the ponderomotive force expels the electrons and leaves an ion background. Then the electron bubble and the wakefield structure are formed, which have been thoroughly discussed in [34–36] and the references quoted there. With the accumulating of the electrons in the tail of the bubble, the wave breaking condition is satisfied [37]. A bunch of electrons are injected into the wakefield being captured and accelerated with the strength on the order of 100 GV/cm as seen in Fig. 2(a) at t = 164 T₀. The electron density distribution of the bubble structure is shown in the (x,y) plane, in which the black line is the laser intensity profile (normalized to 0.1 × I₀) along y = 0 and the red line is the longitudinal electric field (normalized to 30GV/cm) profile. The injected electrons locate in the acceleration phase of the wakefield and experience stable acceleration. The energy spectra of the total electrons at 144 T₀, 164 T₀, 184 T₀ and 204 T₀ are plotted in the (y,z) plane. The maximum energy increases from 0.5 GeV to 1.1 GeV within about 60 μm. The corresponding energy spectra of the injected electrons, which means the electrons trapped and accelerated by the wakefield, are presented in the (x,z) plane. The injected electrons are quasi-mononenergetic with the peak energy increasing from 100 MeV to 700 MeV. Furthermore, the accelerated electrons are well collimated as shown in Fig. 2(b), which is the energy angular distribution of the electrons at t = 204 T₀. The energy and beam quality of the accelerated electrons will affect the emitted γ-photons.

 

Fig. 2 (a) The electron distributions at 164 T₀ is presented in the (x,y) plane. The black and red line represent the profiles of the laser intensity and longitudinal electric field on the laser axis (y = 0). The energy spectra of the total electrons at 144 T₀ (green), 164 T₀ (blue), 184 T₀ (yellow) and 204 T₀ (purple) are in the (y,z) plane. The corresponding spectra for the injected electrons are shown in the (x,z) plane. (b) The energy angular distribution of the injected electrons at t = 204 T₀. The colorbar represents the number of the particles on logarithm scale.

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The laser pulse is reflected from the solid density Aluminium at the end of the hydrogen plasma. The electrons on the solid surface are pushed inward. The heavier ions, Al¹³⁺, do not respond on the same time scale. A double-layer is formed which is similar to the laser piston model [38]. The corresponding piston velocity can be estimated by balancing the momentum flux of the radiation and those of the charged particles. One obtains $I/\left[\left(m_i/Z_i+m_e\right)n_e{c}^3\right](1-{\beta }_f)/(1+{\beta }_f)={\gamma }_f^2{\beta }_f^2$. Here Z_i is the ion charge state, β_f is the piston velocity normalized to c, and ${\gamma }_f={\left(1-{\beta }_f^2\right)}^{-1/2}$ is the corresponding Lorentz factor. The piston velocity is ∼ 0.12c in our case. It takes about 27fs to burnt out the target with the thickness of 1 μm. Therefore the pulse is completely reflected before the destroying of the mirror. The high density of the solid target provides a large charge separation field, which pulls the electrons back to the ion layer as a restoring force. The oscillations of the electrons and the ions are shown in the density evolution in Figs. 3(a) and (b). The critical density surface is moving back and forth which reflects the incident laser pulse serving as a relativistic oscillating plasma mirror (ROM). The ROM model and high order harmonics generation have been proposed in [29]. The reflected laser field contains high frequency components and the wave breaks up into packets due to the Doppler effect and the mirror movement [39]. In Fig. 3(c), we present the frequency spectrum of the incident (at 20 T₀) and reflected (at 203 T₀) laser field (analyzed along y = 0 profile). The intensity of the high order harmonics decrease based on the power law I_ω ≈ ω^−5/2 according to the predictions from [40]. However, due to the focusing and compression effect, the intensity decrease is partially balanced and the high frequency components have the intensities comparable to the incident laser pulse. The maximum resolvable harmonics based on our mesh size is 30ω₀, which corresponds to the wavelength of λ/30. The formation length can be estimated as l_f ∼ λ/a₀ [41], which means the local constant field approximation is generally applicable in the ultra-intense laser cases as a₀1 for the fundamental frequency ω₀. However, the intensity of the high order of harmonics decreases with the frequency. As shown in Fig. 3(c), the intensity drops three order of magnitude within 10ω₀, which means the components with higher frequency have very low probability for gamma-ray emission. The cutoff wavelength of 10ω₀ is much larger than the formation length, implying the local constant field approximation is still valid in our case.

 

Fig. 3 (a) and (b) are the electron and ion (Al¹³⁺) density evolution of the solid density target. (c) is the frequency power spectrum of the incident and reflected laser field at 20 T₀ (black) and 203 T₀ (red), respectively.

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As mentioned in the theoretical model part, the probability for γ-ray emission depends on the relative motion of the electrons and the laser field, which is maximized in the counter-propagating case. Therefore, the reflected high order harmonics provide the optimal conditions for the process of Compton scattering. In Fig. 4(a), the reflected laser field intensity (at 206 T₀) is shown by the black line on the (x,z) plane. One finds that intensities of the wave packets are higher than that of the incident intensity due to the focusing and compression effects. At this moment, the reflected laser starts colliding with the LWFA electrons, which is shown by the corresponding density distribution in the (x,y) plane. A large number of γ-photons are produced as seen in the photon density profile (along the laser axis) on the (x,z) plane. The γ-photon density, reaching 23 n_c (corresponding to the initial laser wavelength), is well consistent with the electron density profile (red). In this case, the pulse length of the γ-photons can be controlled by adjusting the LWFA electrons [32]. The energy spectrum evolution of the γ-photons are presented on the (y,z) plane. Before the collisions of the laser field and the LWFA electrons, the number and the energy of the photons are relatively low as shown by the lines represented for 186 T₀ (blue) and 204 T₀ (purple). Significant increase starts after the collisions as shown in the line for 206 T₀(cyan). The yellow line (at 212 T₀) is the final state of the γ-ray when the reflected laser has passed through the whole LWFA electron beam. The maximum energy reaches ε_kγ ∼ 0.97GeV. The corresponding energy angular distribution of the γ-photons is presented in Fig. 4(b). The γ-ray flux is well collimated with the opening angle of ∼ 18° and the transverse emittance as 3.18mm × mrad with the total photon number (ε_kγ > 1MeV) as 1.1 × 10¹¹/μm. The energy of the γ-ray flux is about 0.435J/μm, which is about 3.72% of the incident laser energy.

 

Fig. 4 (a) The electron density distribution at t = 206 T₀ is projected on the (x,y) plane. The corresponding density profiles (on the laser axis) of the electron and γ-photons (normalized to 10 n_c) are plotted on the (x,z) plane with red and green lines, respectively. The laser intensity profile is shown with the black line normalized to I₀. The negative value indicates that the laser field is reflected and propagating backwards. The energy spectrum evolutions of the γ-photons at 186 T₀ (blue), 204 T₀ (purple), 206 T₀ (cyan), and 212 T₀ (yellow) are displayed on the (y,z) plane. (b) The energy angular distribution of the emitted γ-photons at 206 T₀. The colorbar represents the number of the photons on logarithm scale.

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4. Conclusion

A high energy γ-ray flux is generated by the reflected high order harmonic fields via nonlinear Compton scattering. The laser parameters proposed here are available nowadays and the setup relies only on a single laser beam. This mechanism can be a potential experiment on 10PW-class laser facilities for the research of radiation effect and bright γ-ray production.