1. Introduction

Laser wakefield acceleration (LWFA) [1] is a novel particle acceleration technique that can accelerate particles to very high energies at very short distances. In LWFA, an intense laser propagates through an initially neutral, homogeneous plasma, forming a bubble structure that can sustain orders of magnitude higher accelerating fields than conventional accelerators. Experiments have shown that LWFAs can produce electron energies over GeV [2–5] with unique properties, such as kiloampere peak current [6–8] with few-femtosecond (fs) bunch length and low emittance [9,10]. Recent experiments have used a laser-heated capillary discharge waveguide to guide a petawatt laser, an electron beam with multiple quasi-monoenergetic peaks up to 7.8 GeV was generated in a low-density plasma [11]. By modulating the longitudinal plasma density and guiding the self-focusing process of the laser pulse to control the electron injection, electron beams with relative energy spread (rms) of 2.4‰ – 4.1‰ and divergence (rms) of 0.1 – 0.4 mrad were obtained [12]. The reproducible and controllable producing of electron beams [13] provides a concrete basis for developing the laser-plasma accelerator from the laboratory to practical applications, especially in future accelerator-based light sources. Simulations and the proof-of-principle experiments [14–17] suggest that electron beams produced from such compact plasma wakefield accelerators either driven by an intense laser pulse or a relativistic electron beam may be good candidates for driving compact free-electron lasers (FEL).

Periodic longitudinal density modulation of relativistic electron beams at optical wavelengths (i.e. microbunching) gives rise to coherent light emission, which is essential to the free electron lasing process. Typically, a laser modulator comprised of a laser and an undulator is used as a bunching system, where the undulator makes the electrons undergo transverse oscillations while the co-propagating laser exchanges energy with the electrons [18]. In LWFA, when the accelerated electrons interact with the rear of the drive laser pulse, it leads to enhanced transverse oscillations and causes microbunching of the beam [19]. A detailed understanding of microbunching is critical to developing LWFAs and LWFA-driven FELs.

Experimental efforts are ongoing to characterize electron beams from LWFAs, especially their temporal structures [20–22]. In LWFA, the nonlinear trapping process is a key issue to the ultimate beam quality. A cold optical injection scheme is proposed to produce ultra-low energy spread beams as shown by two-dimensional simulations [23]. Experimental studies show that an electron bunch with a temporal feature as short as 1.4 fs [6] can be produced using controlled optical injection [24]. The ionization-induced injection is an attractive injection scheme in LWFA because of its controllability, which can generate a high-brightness, stable, and tunable electron beam. In this mechanism, the frontier of relativistic intense laser pulses can completely ionize the outer electrons of high Z atoms or the electrons of low Z atoms to excite plasma waves. In contrast, the inner electrons of the high Z atoms are ionized only near the peak intensity of the laser pulse and are born in the wakefield. Thus, the emittance of the electron beam can be limited to the residual momentum and initial radius of the electrons from the ionization process itself [25–28]. However, the continuous injection over long distances generates strong beam-loading effects [29] and leads to a large energy spread because of the different injection phases into the wake and different accelerating times for electrons. Some early experiments [30,31] showed that for mixed gas lengths greater than 1 mm, the larger injection length of electrons leads to a larger final energy spread, making the quality of the ionization-injected electron beam inferior to other injection schemes. Therefore, schemes have been proposed to reduce the injection distance and the energy spread by using two gas cells to separate the injection and acceleration stages [32–34].

The generation of electron bunch trains in the plasma wakefield has been studied previously. They originate either from electron trapping in the subsequent plasma periods with separation in scales of the plasma wavelength [35] or from multiple injections into the wake [36] or periodic triggering of ionization injections by employing modulated laser field [37]. However, the modulation of the electron bunching is highly dependent on laser propagation and evolution in the plasma. The subcycle interaction of laser and electrons holds great promise as a candidate for attosecond electron generation. Phase-dependent streaking of electrons has been demonstrated either in a surface plasma wave [38] or in the plasma accelerator [17,39]. In the plasma wakefield accelerator, electrons will have intrinsic discrete final phase space when they are trapped by the wake due to the phase-dependent tunnel ionization in the drive laser field. However, the ultrashort micro-bunched feature may not be maintained when the electrons leave the plasma because of the effects of phase mixing [26] and/or inverse energy chirp compensation from the accelerating gradient during the acceleration.

In this paper, we study the dynamics of a micro bunched beam generated under the near-threshold ionization scheme in laser wakefield acceleration through PIC simulations. Due to the phase-dependent tunneling ionization in the drive laser field, electrons with periodic modulation in longitudinal density are non-linearly mapped to a discrete final phase space when they are trapped by the wake. And they can preserve their initial bunching structure during the acceleration, leading to an attosecond electron bunch train after leaving the plasma with separations of the same time scale consequently. Such a pre-bunched beam with comb-like longitudinal density distribution has a high temporal coherence and can be a candidate for emitting coherent radiation.

2. Simulations and results

In the plasma wakefield, the relative longitudinal position $\xi \equiv x-{{v}_{\phi }}t$ of the trapped electrons can be expressed as [39]

To illustrate the concept, we consider ionization injection using a single laser pulse via two-dimensional (2D) PIC simulations using the EPOCH code [40]. A $\lambda =0.8$ $\mathrm{\mu}\textrm{m}$ drive laser pulse polarized in the $y$ direction with normalized amplitude ${{a}_{0}}=1.5$, spot size ${{w}_{0}}=20$ $\mathrm{\mu}\textrm{m}$, and a pulse duration ${{\tau }_{\text {FWHM}}}=30$ $\text {fs}$ (FWHM) propagates into a mixture of pre-ionized plasma with a density of ${{n}_{p}}=5\times {{10}^{18}}\text {c}{{\text {m}}^{-3}}$ and ${\text {N}}^{5+}$ ions with a concentration of ${{n}_{{{\text {N}}^{\text {5+}}}}}=0.1{{n}_{p}}$. ${\text {N}}^{5+}$ locates from $x=0$ $\mathrm{\mu}\textrm{m}$ to $x=300$ $\mathrm{\mu}\textrm{m}$, with an up-ramp of 250 $\mathrm{\mu}\textrm{m}$ and a down-ramp of 50 $\mathrm{\mu}\textrm{m}$ density profiles. The simulation window has a dimension of $40 \mathrm{\mu}\textrm{m} \times 80 \mathrm{\mu}\textrm{m}$ with $2000\times 1200$ cells in the $x$ and $y$ directions, respectively. This corresponds to cell sizes of $0.025k_{0}^{-1}$ in the $x$ direction and $0.08k_{0}^{-1}$ in the $y$ direction, respectively. Each cell contains 4 macroparticles. The Keldysh parameter [41] in our case is smaller than unity: ${{\gamma }_{k}}=\sqrt {{{I}_{p}}/2{{U}_{p}}}\ll 1$, where ${{I}_{p}}\approx$ 552 eV, 667 eV are the IPs of the sixth and seventh nitrogen electrons respectively, and ${{U}_{p}}$ is the ponderomotive potential. Therefore, tunneling or barrier suppression ionization is the main mechanism responsible for the electron injection, which is implemented in the code used for our simulations.

The pre-ionized electrons from the helium atoms and the $L$-shell of nitrogen form the plasma wake, and electrons from the $K$-shell of nitrogen are tunneling ionized near the peak of the laser pulse and trapped by the wakefield. The snapshots of the trapped charge near $x$ = 0.35 mm where the injected electrons exit the gas mixture stage are shown in Fig. 1(a), and the fine bunch structure and the on-axis current density profile of the injected electrons are shown in Fig. 1(b). The current density profile shows strong periodic modulation in the $({{\xi }_{f}},y)$ space and a comb-like density bunch profile is pronounced. The average interval between the peaks of the density bunch profile is $\sim$0.97 fs, and the average length (rms) of each micro bunch is $\sim$118 as.

 

Fig. 1. The electron microbunching of injected charge in an LWFA. (a) Snapshots of the charge density distributions of the background electrons, the trapped N$^{6+}$ electron of nitrogen, and the on-axis longitudinal electric field along the $x$ direction. (b) The density distribution of the trapped charge and the on-axis current density profile. (c) The initial phase space distribution of the injected electrons ($\xi _i$, $p_y$) when they are ionized in the electric field of the drive laser. (d) The ($\xi _f$, $y$) space of the electrons when they are trapped by the wake at $x$ = 0.35 mm. The plasma density of the gas mixture is $n_{p}=5\times 10^{18}$ cm$^{-3}$, and $n_{{\text {N}}^{5+}}=0.1n_p$.

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Figure 1(c) shows the initial phase space distribution ($\xi _i$, $p_y$) of the injected electrons when they are ionized in the electric field of the drive laser. As shown in Fig. 1(c), because the laser intensity is just above the threshold of N$^{5+}$, the electrons are ionized off-peak of the laser electric field and born with the initial coordinates ${{\xi }_{i}}<0$. When they are trapped behind the laser pulse, they will have transverse momenta ${{p}_{\bot }}\simeq {{a}_{\bot }}(\varphi )-{{a}_{\bot }}({{\varphi }_{i}})=-{{a}_{\bot }}({{\varphi }_{i}})$, where ${{a}_{\bot }}$ is the transverse laser amplitude and ${{\varphi }_{i}}={{k}_{p}}{{\xi }_{i}}={{k}_{p}}({{x}_{i}}-{{v}_{\phi }}t)$ is the initial phase of the electrons. These residual transverse momenta relative to the phases where the electrons are ionized contribute to the initial beam emittance. As discussed above, with a linear polarized laser pulse, the transverse momentum distribution of the electrons oscillates with the laser period, and the longitudinal momentum distribution oscillates twice per laser period. Due to the phase-dependent ionization, the initial electron distribution has a strong modulation at $2{{k}_{0}}$, where ${{k}_{0}}$ is the wavenumber of the laser pulse. Electrons with the same ${{\xi }_{i}}$ will be injected into the same position ${{\xi }_{f}}$, while any spread in ${{r}_{i}}$, $r$ and $\gamma -({{v}_{\phi }}/c){{p}_{x}}$ will broaden the $\xi$ distribution in the injected position.

In Fig. 1(d) we plot the position $({{\xi }_{f}},y)$ of each electron, with each color representing a different birth time. For the electrons that are ionized at the same time ${{t}_{i}}$ (the same ${{x}_{i}}$ in the lab frame) with the same ${{\xi }_{i}}$ (in the co-moving frame) and then injected to the same $r$ coordinates (especially $r\approx 0$), the spread of $\xi$ due to the different ${{r}_{i}}$ and the transverse motion is $\Delta {{\xi }_{{{r}_{i}}}}\approx 0.05$. Also, one can see that electrons ionized earlier (small ${{t}_{i}}$) but with the same ${{\xi }_{i}}$ will be injected closer to the end of the wake, while those ionized later are mapped to different $\xi$. The term of $\gamma -({{v}_{\phi }}/c){{p}_{x}}\approx (1-{{v}_{\phi }}/c)\gamma$ contributes to the spread of $\xi$ for electrons with the different energy (i.e. different born time ${{t}_{i}}$), and the spread in the final position is $\Delta {{\xi }_{\gamma }}\approx 0.12$, which could be limited by controlling the length of the gas mixture.

Figure 2(a-c) show us the dynamics of longitudinal phase spaces $(x,{{p}_{x}})$ of the injected beam during the acceleration, with each color representing a different micro bunch. As shown in Fig. 2(a), when the laser pulse enters the gas mixture, the $K$-shell electrons are ionized and the distribution of longitudinal velocity oscillates twice per laser period. Electrons are nonlinearly trapped by the wake, that is, electrons ionized near the peak of the laser electric field are mapped to the position closer to the end of the wake, resulting in discretized phase space structure as shown in Fig. 2(b). And the structure can be maintained, while the chirp of the injected beam will be reversed during the acceleration. After about 3 mm acceleration, the electron beam is accelerated to about 420 MeV, with a large relative energy spread of $4.06\%$ and a normalized transverse emittance of 0.41$\pi$ $\mathrm{\mu}\textrm{m}$ rad. Each micro bunch has different peak energy of 429.5 MeV, 419 MeV, 411.0 MeV, 400.2 MeV, 385.0 MeV and a smaller energy spread of $1.62\%$, $0.90\%$, $1.10\%$, $1.11\%$, $1.29\%$, respectively. We analyze the evolutions of energy spread and emittance for micro bunches, as shown in Fig. 2(e-f). The minimum energy spread during the acceleration can be down to $\sim$0.7$\%$, and all emittances are saturated to a level lower than $1.5$ $\mathrm{\mu}\textrm{m}$ rad.

 

Fig. 2. The dynamic of the micro bunched beam during the acceleration. (a-c) The longitudinal phase spaces $(x,{{p}_{x}})$ of electrons at different times: (a) $t=0.85$ ps when the laser enters the gas mixture and ionizes the N$^{5+}$ ions, (b) $t=1.07$ ps when the injected beam exits the gas mixture, (c) $t=6.74$ ps when the injected beam electrons with the lowest energy spread. (d) Multichromatic micro-bunches with different label numbers ($i$) are formed after acceleration of $3$ $\text {mm}$. (e) The evolutions of the relative energy spread ${{(\Delta E/E)}_{i}}$ for each micro bunch $i$, and the inset shows the energy $(E)$ and energy spread $(\Delta E/E)$ for the whole beam. (f) The evolutions of the normalized transverse emittance (${\varepsilon }_{ny}$) for the whole beam and each micro bunch. Electrons in the same micro bunch are labeled with the same color.

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The snapshots of the electron density after the injection and during the acceleration process are shown in Fig. 3(a-c). The density profile in panel (a) at $t=1.67$ $\text {ps}$ suggests that the electron bunch is micro-bunched immediately after the injection. The electrons are further accelerated and focused transversely in later times (6.67 ps and 10.0 ps in panels (b) and (c)). It is clear to identify the electron bunching structure, and one can see that the comb-like structure of the bunch is significantly enhanced from the on-axis current density profiles, as shown in Fig. 3(d). The intervals between the peaks of the density profile are about 1.40 fs, which is nearly half of the laser period $\sim$1.33 fs. Each bunch has an average rms length of $\sim$165 as.

 

Fig. 3. Injection beam evolution and electron microbunching. (a-c) Snapshots of the electron beam density after the injection ((a) at $t$ = 1.67 ps, $x$ = 0.5 mm) and during the acceleration process ((b) $t$ = 6.67 ps, $x$ = 2.00 mm and (c) $t$ = 10.0 ps, $x$ = 3.00 mm), respectively. (d-e) The on-axis current density profile and the bunching factor during the acceleration process correspond to (a-c) respectively.

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In Fig. 3(e), we present the bunching factor $b(k)$ of the electron beam at the corresponding moments. The bunching factor $b(k)$ is defined as $b(k)=\left |\int {dxf(x)\exp (ikx)}\right |$, where $f(x)$ is the normalized density distribution of the trapped electrons along the longitudinal direction. The modulation in the current profile is peaked at $k\approx 2{{k}_{0}}$ after the injection ($t=1.67$ ps), where ${{k}_{0}}$ is the wavenumber of the drive laser pulse. Also, the modulation and the bunching factor are enhanced during the acceleration process, which peaked at ${\sim }2.15{{k}_{0}}$ in the later times. Therefore, we can still confirm that electrons preserve the steady phase during the acceleration and keep the bunching structure at twice the laser frequency.

3. Discussions

Now we discuss the robustness of the attosecond micro bunched electron beam generation. Figure 4 shows the effect of nitrogen gas doping lengths and concentration in the gas mixture plasma on the generation of the attosecond micro bunch. For a fixed pre-ionized plasma density of ${{n}_{p}}=5\times {{10}^{18}}\text {c}{{\text {m}}^{\text {-3}}}$, we modulate the nitrogen gas doping length and the concentration of N$^{5+}$ in the gas mixture to preserve the same level of trapped charge, i.e., ${{n}_{p}}={{n}_{\text {He}}}+5{{n}_{{{\text {N}}^{\text {5+}}}}}$ where ${{n}_{{{\text {N}}^{\text {5+}}}}}={{C}_{{{\text {N}}^{5+}}}}{{n}_{p}}$. The snapshots of the charge density distribution are shown in Fig. 4(a-d). It is indicated that the electron bunching structure can be more obviously identified for shorter gas mixture lengths with higher concentrations. It can be easily understood because the injected beam is composed of electrons ionized at different longitudinal positions and at different times. When the injection stage with the gas mixture is long, the spread due to the trapped position $\Delta {{\xi }_{\gamma }}$ will be large, and the overlap between the micro bunches will eliminate the bunching factor. As a result, the longitudinal phase mixing process will flatten the longitudinal current profile of the injected beam. As shown in Fig. 4(e), the comb-like current profile distributions are emphatic in the cases with shorter injection gas lengths and higher concentrations. Bunches as short as $\sim$127 as (rms) can be obtained with a density profile of case IV with a short gas mixture length of 125 $\mathrm{\mu}\textrm{m}$ and higher concentration of ${{n}_{{{\text {N}}^{5+}}}}=0.08{{n}_{p}}$, and the peak current density is about $\sim$6 times of that in case I with a longer gas mixture length of 300 $\mathrm{\mu}\textrm{m}$ and a low concentration of ${{n}_{{{\text {N}}^{5+}}}}=0.01{{n}_{p}}$. Moreover, the modulation and the bunching factors are enhanced in the cases with shorter gas mixture lengths, which are found to be peaked at $\sim$2.87${{k}_{0}}$, $2.39{{k}_{0}}$, $2.54{{k}_{0}}$ and $2.23{{k}_{0}}$, respectively, as shown in Fig. 4(f). For a laser pulse with linear polarization and moderate laser intensity of ${{a}_{0}}=1.5$, the final longitudinal parameter $\xi =-\sqrt {4+\xi _{i}^{2}}$ from Eq. (1), which gives the modulation wavenumber of the trapped electron beam can be expressed as ${{k}_{\text {beam}}}=2{{k}_{0}}\sqrt {4+\xi _{i}^{2}}/\left |{{\xi }_{i}}\right |$. For the electrons with an average initial position $\left |{{\xi }_{i}}\right |\approx 2$, the modulation wavenumber ${{k}_{\text {beam}}}\approx 2.82{{k}_{0}}$, which is close to the simulation values.

 

Fig. 4. The attosecond electron bunching of injected charge for different gas mixture profiles. (a)-(d) Snapshots of charge density distributions of the injected $K$-shell electrons of Nitrogen at $x$ = 0.6 mm, with the N$^{5+}$ plateaus of $200$ $\mathrm{\mu}\textrm{m}$, $100$ $\mathrm{\mu}\textrm{m}$, $50$ $\mathrm{\mu}\textrm{m}$, $25$ $\mathrm{\mu}\textrm{m}$ and the concentrations of N$^{5+}$ ions are ${{n}_{{{\text {N}}^{\text {5+}}}}}=0.01{{n}_{p}}$, ${{n}_{{{\text {N}}^{\text {5+}}}}}=0.02{{n}_{p}}$, ${{n}_{{{\text {N}}^{\text {5+}}}}}=0.04{{n}_{p}}$, ${{n}_{{{\text {N}}^{\text {5+}}}}}=0.08{{n}_{p}}$, respectively. (e) and (f) the on-axis current density distributions and the bunching factors for the cases in fig. (a)-(d). In all cases, the nitrogen gases are bordered by two ramp-like density profiles of $50$ $\mathrm{\mu}\textrm{m}$ on both sides of the plateaus, i.e. case I: the N$^{5+}$ ions locate from $x$ = 0 mm to $x$ = 0.3 mm; case II: the N$^{5+}$ ions are distributed from $x$ = 0 mm to $x$ = 0.2 mm; case III: the N$^{5+}$ ions are distributed from $x$ = 0 mm to $x$ = 0.15 mm; case IV: the N$^{5+}$ ions are distributed from $x$ = 0 mm to $x$ = 0.125 mm; The laser pulse with parameters ${{a}_{0}}=1.5$, ${{w}_{0}}=20$ $\mathrm{\mu}\textrm{m}$, ${{\tau }_{\text {FWHM}}}=30$ $\text {fs}$ is focused at $x$ = $50$ $\mathrm{\mu}\textrm{m}$.

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Figure 5 indicates the result from the 3D PIC simulation with the same plasma profile (${{n}_{p}}=5\times {{10}^{18}} \text {c}{{\text {m}}^{\text {-3}}}$, ${{n}_{{{\text {N}}^{\text {5+}}}}}=0.1{{n}_{p}}$) and the same laser pulse $({{a}_{0}}=1.5$, ${{w}_{0}}=20$ $\mathrm{\mu}\textrm{m}$, ${{\tau }_{\text {FWHM}}}=30$ $\text {fs})$ as those in the 2D simulations in Fig. 1. The simulations have been carried out within a box that was $40 \mathrm{\mu}\textrm{m}\times 65 \mathrm{\mu}\textrm{m}\times 65 \mathrm{\mu}\textrm{m}$ with $1500\times 400\times 400$ number of cells with 2 particles per cell. The snapshots of the injected electron charge density at $x=500$ $\mathrm{\mu}\textrm{m}$ are shown in Fig. 5(a) with the central slices of the wakefield structure in/out the plane of laser polarization as shown in Fig. 5(b). One can still obviously see the discrete structure of the injected electrons. The total charge of the injected beam is about 12.4 pC, in which one of the brightest micro bunch is about $\sim$4.9 pC with a bunch length of $\sim$254 as (rms). In Fig. 5(b), the emittances of this micro bunch in the $y$ and $z$ plane are estimated to ${{\varepsilon }_{ny}}\approx 1.2\times {{10}^{-6}}\text {m rad}$ and ${{\varepsilon }_{nz}}\approx 2.7\times {{10}^{-7}}\text {m rad}$, the corresponding 5D-brightness ${{B}_{5D}}={2{{I}_{p}}}/{{{\varepsilon }_{ny}}{{\varepsilon }_{nz}}}\;$ reaches the level of ${\sim }1.0\times {{10}^{17}}\text {A }{{\text {m}}^{\text {2}}}\text {ra}{{\text {d}}^{\text {2}}}$. Combined with a further reduced energy spread of the micro-bunch to the level of around $\sim$1$\%$ and the further transverse focusing of the electron beam, one can get a micro bunched electron beam with unprecedented 6D-brightness values ${{B}_{6D}}\approx {{B}_{5D}}/0.1\%(\Delta {{E}_{rms}}/E)$, which may have a transformative impact on the realization of high brightness light source. Moreover, the interval between the peaks of the charge density as marked is $\sim$1.45 fs, and the modulation in the current profile indicated by the bunching factor is peaked at $k\approx 2{{k}_{0}}$, as shown in Fig. 5(c). We can predict that such a density-modulated beam could produce coherent X-rays with attosecond duration, which could enable novel applications.

 

Fig. 5. Micro bunching effect from a 3D simulation. (a) The snapshots of the injected electron charge density at $x=500$ $\mathrm{\mu}\textrm{m}$, (b) the central slices of the wakefield structure in/out the plane of the laser polarization, (c) the on-axis current profile and the bunching factor of the electrons at $x=500$ $\mathrm{\mu}\textrm{m}$. The laser and plasma parameters are the same as those used in the 2D simulation above.

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4. Conclusion

In summary, attosecond micro bunches with a few pC in charge are accessible in the LWFA by using near-threshold ionization injection. Due to the phase-dependent tunneling ionization in the driving laser field, electrons with periodic modulation in longitudinal density are non-linearly mapped to a discrete final phase space when they are trapped by the wake. The electrons preserve the steady phase during the acceleration, which consequently produces attosecond micro bunches with dispersive comb-like longitudinal density distribution and energy spectrum. A shorter gas mixture stage with a high concentration is favorable to obtain micro bunched beams with significant periodic modulations. Such ultra-short nature of the micro bunching also leads to ultra-high beam current, making secondary sources (X-rays, gamma rays, etc.) generated by the electron beam from LWFA have great applications in medical radiotherapy, material imaging applications, and the study of molecular and atomic ultrafast dynamics.