1. Introduction

The basic properties of atoms, molecules, and solids are governed by electron dynamics which usually take place on extremely short time scales. To measure and control these dynamics require ultrafast radiation sources combined with efficient detection techniques. Currently these researches are still limited to convenient high quality light sources. So far, successive radiation sources have been developed mainly by using conventional radio frequency-based accelerators or laser-driven plasmas-based accelerators [1]. These sources can deliver radiation with high brightness, and short wavelength and pulse duration [1–15]. They are therefore very useful to investigate ultrafast phenomena at short temporal scale and lead to crucial progress in our understanding of the fundamental processes in matter [16–19].

Although the generation of ultrafast light sources at femtosecond (fs) time scale have been proposed and experimentally investigated, the production, characterization and application of bright, attosecond (as) pulses are still under active investigations. Single isolated extreme-ultraviolet light pulse with duration of a few tens of attoseconds has been realized by using high-order harmonic generation from ultra-intense laser radiated gas medium due to nonlinear process [20]. However, attosecond pulses at subangstrom wavelengths have not been demonstrated experimentally so far [21,22]. Attosecond hard x-ray pulse bursts, either in the form of isolated pulses or trains of pulses can offer the opportunity to investigate unexplored research areas with unprecedented time resolution. Especially for the trains of attosecond hard x-ray pulse, their unique properties will uncover new understanding of fundamental processes in atoms, molecules, plasmas and materials, including the timescales on which electrons are emitted from atoms [23], the timescale for spin-spin and electron-electron interactions [24,25], the timescale that determines molecular dissociation and electron localization [26–28], the timescale and mechanisms for spin and energy transport in nanosystems [29–31], as well as new capabilities to radiograph medium and high-Z materials [32].

We have previously shown the possibility of producing attosecond hard x-rays by scattering a laser pulse off a counter-propagating relativistic attosecond electron bunch (e bunch) [33]. In this letter, we present a novel scheme to generate bright, attosecond pulse trains (APTs) for hard x-ray pulses. Figure 1 shows a schematic view of the experimental set-up, which is based on Thomson scattering (TS) of an electromagnetic wave (i.e. Thomson scattering laser pulse) off multi-bunch attosecond relativistic electron trains. The latter is resulted from the interaction between an intense laser pulse and a channel with sharp inner boundaries consisting of overdense plasmas. Compared with [33], a different target geometry (a channel instead of a wire) is employed to produce attosecond electron bunches within a train fine structure. Such e bunches have some potential to produce a similarly structured x-ray flashes. It would be intriguing indeed to demonstrate their characterization. We also note that another two theoretical models [34,35] have been used to study the Thomson or Compton scattering light source generation driven by laser wakefield acceleration. However, these efforts are mainly focused on the femtosecond x-ray pulse generation rather than bright attosecond x-ray pulse trains generation. To the best of our knowledge, it is the first time to propose and demonstrate attosecond hard x-ray bursts within a fined train structure.

 

Fig. 1 Attosecond electron bunches and the successive TS-based x-ray pulse train generation. An USUI laser pulse, denoted as a driving laser, is focused on the entrance of the empty channel. Such channel has an abrupt interface with the plasma walls. We use the driving laser to eject e bunches from the inner channel wall and accelerate them toward a counter-propagating Thomson scattering laser pulse. After injecting in the scattering laser, the laser-plasma accelerated electrons oscillate transversally and emit x-rays due to synchrotron radiation.

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2. Generation of attosecond electron bunches

Attosecond electron bunch generation driven by laser ponderomotive force acceleration (PFA) in vacuum and plasma have been demonstrated extensively [36–38]. To create a train of such short e bunches, we suggest using an USUI laser pulse tightly focused on the entrance of a vacuum channel inside an over dense plasma boundary, as demonstrated in Fig. 1. Such “bunching” laser pulse could eject e bunches from the channel inner wall transversely due to intense laser electric field and accelerate them longitudinally due to pondermotive force. The channel is suggested to be a hole with diameter of a few microns in a solid target that has already been experimentally realized recently [39].

To demonstrate the e bunch generation in this channel geometry, we perform two dimensional (2D) simulations with fully electromagnetic PIC code VLPL [40]. The simulation box is x × y = 20λ₀ × 6λ₀, with spatial resolution 100 cells per laser wavelength (λ₀). To resolve the density gradient, we take 100 electrons and 100 ions per cell. A p-polarized Gaussian laser pulse (with its electric field in the (x, y) plane) with a trapezoidal profile in time is incident from the left boundary and focused to a spot size of $\sigma =2.5{\lambda }_0$ at the entrance of the channel. We take the dimensionless laser amplitude $a_0=eE/m\omega c$ = 10, and the duration ${\tau }_L=10T_0$, thus the maximal intensity in the focus point corresponds to 2 × 10²⁰ W/cm² for a ${\lambda }_0$ = 0.8 μm laser. For this intensity and interaction time scale the ion motion can be neglected. The empty channel has an abrupt interface inside with an inner radius of $2.0{\lambda }_0$, and a wall plasma density of $n_0=15n_{cr}$, where $n_{cr}$ = 1.7 × 10²¹/cm³ is the critical density for the laser pulse. We choose $t=0$ to be the instant when the peak of the pulse envelope reaches the channel entrance at $x=5{\lambda }_0$. Spatial coordinates are normalized by laser wavelength (λ₀) and time is normalized by laser period (T₀). A shifting window is employed in our case to save the simulation time and the data size.

Figure 2 shows the typical electron distribution inside the channel. As shown in Fig. 2(a), e bunches with density ~4$n_{cr}$ are observed. They are pulled out from the plasma walls and enter into the vacuum channel. These bunches exhibit a remarkable micro-bunching structure with about 0.18${\lambda }_0$ thickness and 1.0${\lambda }_0$ longitudinal spacing. Accordingly, the e bunch duration is expected to be 200 as (rms). Figure 2(b) shows the spatial distributions of electrons whose energy greater than 10MeV. The bunched electrons show fine structure character and approach energies up to 40 MeV. Here we describe the formation process of the e bunch with “alternating” structure in the following way. Assuming a 1D configuration, the ponderomotive force from the incident waves acts on electrons within the skin depth. For a sufficiently large wave amplitude (e.g., peak of the laser pulse) this leads immediately to breaking of the stimulated Langmuir oscillations of electrons. Thus, electrons from the skin layer are thrown toward the bulk of the plasma. At the same time, a counterstream of electrons arises from the bulk plasma in the direction of the skin layer due to the attractive force toward ions that were left behind by the inward-driven electrons and by the repulsive force of those electrons. The velocity of the counterstreaming electrons is relativistic. This behavior is typical of normal incidence effects and will lead to the self-intersection of e trajectories, as shown in Fig. 2(b), i.e., alternating structure of the attosecond e bunches. We also note that after escaping from the channel such bunched electrons could maintain a similar attosecond train structure although part of their energy is lost [36].

 

Fig. 2 Attosecond electron bunches generated inside the channel at $t=\text{4}$. (a) Electron phase space distribution (x, p_x) and (b) Spatial distribution for electrons with energies greater than 10 MeV inside the channel.

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Figure 3 shows the electron spectral distribution inside the channel at $t=\text{4}$. The simulation results show that more than 6 × 10⁸ electrons with energies higher than 10 MeV are pulled out from the plasma wall and accelerated as a train of e bunches. For both cases of plasmas wall densities $n_0=25n_{cr}$ and $n_0=15n_{cr}$, the accelerated electrons have a similar spectral distribution whose peak energies are around 20 MeV. However, a lower density of the plasma walls will basically result in a higher peak of the electron bunch energy. This is to be expected, for a lower density plasma wall, the electrons are more easily pulled out and the ponderomotive force can accelerate them to a higher energy so that their peak energy are larger. On the contrary, for a higher density plasma wall, though their peak energy is smaller, the corresponding total particle numbers are a little bit more than the lower density case because their original density is higher, as demonstrated in Fig. 3. Such relativistic e bunches with attosecond train structure have some potential for generating ultrashort x-ray sources via linear or nonlinear TS scattering. In the following sections, we will focus on the radiation generation.

 

Fig. 3 Electron energy spectra for different plasma wall densities. For $n_0=25n_{cr}$ case, simulation parameters are laser duration ${\tau }_L=8T_0$ and focused size $\sigma =3.5{\lambda }_0$ while other parameters are the same as the case above.

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3. Generation and characteristics of intense APTs

From the radiation energy scaling of linear TS $E_x=4{\gamma }^2{E}_L$, where $E_x$ is the radiated x-ray photon energy, $\gamma$ is the electron’s relativistic factor and $E_L$ is the photon energy of the scattered laser, we expect to generate a 10 keV x-ray by scattering 1.5 eV photons off 20 MeV e beam. The temporal profile of the radiation from relativistic e bunch is given by the convolution between the e bunch temporal profile and the radiation profile from a single electron. Since the typical e bunch length is in the sub-micron range in our case, whereas the radiation length from a single electron is in the nanometer range for x-rays, the duration of the radiation via TS process from an e bunch is approximately equal to the e bunch duration [1]. Note that the effect of the photon pulse stretching due to the finite electron interaction time with the laser pulse should be considered in some cases. Then we elaborate the stretching effect acting on the photon pulse and find that it is much smaller than the electron micro-bunch thickness. We therefore suppose that it may not result in washing out the attosecond modulation of the produced radiation in this case.

A four-dimensional Monte Carlo laser-Compton scattering simulation code, MCLCSS [41–43], combined with the Geant4 toolkit [44], is employed to model the performance of x-ray pulse trains from the PFA e beams. The input to the code is directly provided by the output of the PIC code describing the PFA process. The main parameters of Thomson scattering laser and e bunches are summarized in Table 1. The spectral distribution of the x-rays simulated by the MCLCSS code is shown in Fig. 4. Since the normalized field strength a₀ that drives TS process is less than unity in our case, the nonlinear effects resulting from relativistic nonlinear TS process could be neglected [45] and have not been taken into account in our simulations. On the one hand, the x-ray spectrum shown in the inserted figure in Fig. 4 is quite broad. This is mainly due to a large energy spread of the PFA e bunch. Such broad x-ray spectrum is ideal for ultrafast Laue diffraction experiments. On the other hand, a thin metal foil could be employed to attenuate the low energy part of the Thomson-backscattered photons and then to obtain a quasi-monochromatic APTs for x-rays. It has been confirmed by the Geant4 simulation that when a 10 μm Ag filter is employed here, a filtered x-ray spectrum is peaked at 11 keV and its high energy range extends to 50 keV.

Table 1. Typical parameters of scattering laser (${\omega }_L$), electron (e) for TS process

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Fig. 4 Simulated energy spectrum of Thomson backscattered x-rays filtering by a 10 μm Ag foil. The inserted one corresponds to the total x-ray energy spectrum without the attenuation of the thin Ag foil.

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As demonstrated in [33,34], the e bunch temporal shape was faithfully reproduced in the x-ray pulse via TS process. Now, let us progress to investigate the possibility of producing multi-pulse attosecond x-ray train via multi-collision TS process, wherein the TS driving laser counterpropagates to the direction of the multi-bunch e train from the PFA process. Using the coordinate space distributions of electrons shown in Fig. 2(b) as input, the temporal x-ray profile of APTs projected on x-y plane is reproduced. It is shown in Fig. 5 that the driving PFA e beam with micro-bunch fine structure (see Fig. 2(b)) could generate an intense APTs of x-rays whose temporal pattern is superimposed on the temporal electron distribution indicating a close correlation between each other. This may open the possibilities of preparing specific e beam pulse shapes for different x-ray temporal patterns and special applications. In particular, double pulsing source on the attosecond time scale appears feasible. Further, the x-ray duration for APTs is around 200 as in the simulations and its longitudinal spacing is only 0.8 μm.

 

Fig. 5 Temporal x-ray profile for APTs. The color scale represents the intensity variation of temporal x-ray profile.

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With the e beam obtained above, the x-ray flux for APTs is expected to be the order of 2 × 10⁷ photons/shot. The x-ray divergence is about 0.12 × 0.12 rad², which is a little larger than the e beam divergence ${\theta }_e$, 0.1 × 0.1 rad². Here, the radiation angular dependence is determined as follows:

Finally, we also check the x-ray beam qualities in the far field, i.e. away from the 2-μm aperture where the x-ray is produced. An observation plate (plane), which is 0.2 meters downstream from the collision point (where the x-rays are produced) is used to record the x-ray beam parameters. It shows that the filtered x-ray flux and spectral distribution keep the same while its angular distribution decreases from 0.12 × 0.12 rad² to about 0.1 × 0.1 rad². This is mainly due to the fact that the scattered x-rays with low energies but large divergence angles has been attenuated by the thin Ag filter while those with high energies but comparatively small divergence angles penetrate through the filter and then arrive at the observation plate successfully. The 1/e² source radius of the x-ray bunch enlarges to about 2 × 2 cm². However, the pulse duration of x-rays when they arrive at the observation plate is not diagnosed since their arrival time can hardly be recorded with Geant4 toolkit.

4. Conclusions and perspectives

In summary, an all-optical approach for generating bright, attosecond hard x-ray pulse trains is proposed and demonstrated by PIC and Monte Carlo simulations. Our simulation results show that trains of bright, attosecond x-ray flashes at subangstrom wavelengths with peak brightness up to 10²⁰ photons/s/mm²/mrad²/0.1%BW could be generated directly from start-of-the-art laser-plasma accelerator using the proposed scheme. With the rapid progress of laser-plasma science, we believe it may pave the way for generating experimentally compact, high-quality attosecond hard x-ray sources with a tunable energy range at a reasonable cost using combined laser-plasma accelerator and TS scheme. Our results would also expand the field of attosecond science, making it possible to capture the attosecond motion of electrons in a broader range of atoms, molecules, liquids and materials.