Enhanced laser wakefield acceleration using dual-color relativistic pulses

Figure 1 shows the simulation results of a LWFA driven by single 77 TW, 800 nm P-polarized laser pulse (with a corresponding critical density of n_c = 1.745 × 10²¹ cm⁻³) in helium of 8 × 10¹⁸ cm⁻³ gas density; electrons are self-injected into the plasma wave due to the transverse wave-breaking as the laser intensity evolves to a₀ > 4.5. Figure 1(a) shows a plasma wave (ions bubble) [21, 22] excited by the pondermotive force of the laser pulse which undergoes a strong self-focusing. The laser pulse front starts to steepen and shorten, as shown in figure 1(b), due to the self-phase modulation and an electron bunch is injected into the bubble. More electrons are continuously expelled by the laser's ponderomotive force, flow backwards due to the charge separation with the ions towards the rear of the bubble, and then are self-injected and accelerated, as seen in figure 1(c). During the process of acceleration, as the laser pulse propagates it is gradually eroded and depleted. One also can see that electrons start to perform the well-known betatron (transverse) oscillations due to the electromagnetic interaction with the stationary ions of the bubble, figures 1(c)-(e). Eventually, the laser pulse is depleted upon propagating ∼1.8 mm in the plasma, then the bubble is driven by high-energy electrons, as shown in figure 1(d). Due to continuous oscillations of the accelerated electrons and the obvious deterioration of its geometrical quality or emittance, the bubble gradually shrinks and finally disappears within 3 mm, figures 1(e) and (f). The electron energy spectrum is continuous, the maximum energy (i.e. highest energy) of some electrons reaches 400 MeV at ∼4 mm, whereas most of the electrons have energies from 0 to 200 MeV, figure 1(f).

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On the other hand, figure 2 shows the evolution of a LWFA driven by dual-color co-propagating FL (70 TW, 30 fs, P-polarized) and SH (7 TW, 30 fs, S-polarized) laser pulses with a time delay of 0 fs in helium gas having a density of 8 × 10¹⁸ cm⁻³. Figure 2(a) shows a bubble driven by the FL pulse which is strongly self-focused, as in figure 1(a). As the bubble is formed, electrons started to be injected and accelerated, figures 2(b) and (c), to high energy with a maximum of ∼400 MeV. Then, the FL pulse is almost depleted, as shown earlier in the single-laser driven LWFA case. The accelerated electrons enter into deceleration phase of the wakefield and the acceleration is almost terminated due to laser depletion. However, the accelerated electron bunch plays a key role in sustaining a bubble for acceleration, replacing the role of the FL pulse in accelerating a trailing electron bunch, as shown in figures 2(d) and (e), to ∼450 MeV (max) within 2 mm of propagation in the plasma. The synchronized SH laser pulse, being relativistically self-guided for 2 mm of propagation in the plasma, is still energetic to play a role in the remaining 2 mm of the plasma. It is capable of exciting a plasma wave, of a similar wakefield intensity [max E_x = −0.15 (in normalized units of ${m_{\rm{e}}}c{\omega _0}/e$)], which can be considered as a second-stage accelerating structure for the electrons, which are organized in two bunches separated by a plasma wavelength, as shown in figures 2(e) and (f). Finally, the maximum energy of the second electron bunch (labeled 2) in figure 2(f) reaches 770 MeV and the first electron bunch (labeled 1) reaches a maximum energy of 520 MeV over an acceleration distance of 4 mm.

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Figure 3(a) shows evolutions of the peak-electric field amplitudes of single FL pulse and dual-color pulses during the LWFA processes. One can see that the FL pulse undergoes strong self-focusing and spot size oscillations due to the fact the laser spot size is large and unmatched with the plasma wavevector (${k_{\text{p}}}{w_{\text{1}}}{\text{ = 13}}{\text{.5}}$≫${\text{2}}\sqrt {{a_{0\left( {{\omega _{\text{1}}}} \right)}}} {\text{ = 2}}{\text{.7}}$), and then it depletes quickly within ∼2 mm. The SH pulse, which is tightly focused and possesses a closer matching with the plasma wavevector (${k_{\text{p}}}{w_{\text{2}}}{\text{ = 1}}{\text{.35}}$ < ${\text{2}}\sqrt {{a_{0\left( {{\omega _{\text{2}}}} \right)}}} {\text{ = 3}}{\text{.3}}$), undergoes a modest evolution (to an intensity similar to the initial value) over the whole interaction length with some oscillations near the end of the laser-plasma interaction. Thus, the SH pulse has an ability to drive wakefield acceleration to further accelerate electrons from 450 MeV to 770 MeV (max). As plotted in figure 3(b), the maximum electron energy can be tuned by changing the time delay between the dual-color laser pulses. By fine-tuning the time delay to 0 fs, an electron energy of 770 MeV can be obtained, as seen in figure 3(c). Furthermore, the effect of polarization of the SH pulse on the electron energy gain is presented in figure 3(d). For a time delay of 0 fs, the maximum electron beam energy could reach 530 MeV for P-polarized SH pulse (as that of the FL pulse) and a similar electron energy of 536 MeV was also achieved when it was circularly (C-) polarized, respectively. Since the E-field direction of the S-polarized SH pulse is not in the x-z plane, the electron bunch 2 in figure 2(f), which is self-injected in the bubble driven by the electron bunch 1, could be easily captured by the second-stage accelerating wakefield driven by the SH pulse with less oscillations. Simultaneously, the electron bunch 2 experiences the accelerating phase while the electron bunch 1 already enters into decelerating phase, leading bunch 2 to gain a higher energy than that of bunch 1 at this position, see figure 2(f). As a result, there is a sharp increase in the maximum energy of the accelerated electrons, which refers to the maximum energy of electron bunch 2, in figure 3(d). However, for the P- and C-polarized SH pulses, the energy of electron bunch 2 is lower than that of electron bunch 1. Therefore, the boosting capability of the S-polarized SH pulse on the electron energy is stronger than that of P- and C-polarized SH pulses, consistent with published work [13].

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In addition, it is possible to have an estimate for the energy gain of a dual-color-laser driven LWFA according to the input parameters of the lasers in our simulations. In the LWFA, the effective acceleration length is determined by the laser's depletion length ${L_1} = {\left( {{\omega _{\rm{p}}}/{\omega _{\rm{0}}}} \right)^2}c\tau$ or the dephasing length ${L_{\rm{d}}} = \left( {{\rm{4}}\omega _{\rm{0}}^{\rm{2}}/{\rm{3}}\omega _{\rm{p}}^{\rm{2}}} \right)\sqrt {{a_0}} c/{\omega _{\rm{p}}}$, where ${\omega _{\rm{p}}}{\rm{ = }}\sqrt {4\pi {n_0}{e^2}/m}$ is the plasma frequency with n_0, m, and e are the plasma density, electron rest mass, and electron's elementary charge, respectively. The energy gain in the LWFA driven by the FL pulse is given by $\Delta {E_1} \cong m{c^2}\left( {{\rm{2}}\omega _{\rm{0}}^{\rm{2}}/{\rm{3}}\omega _{\rm{p}}^{\rm{2}}} \right){a_{0\left( {{\omega _1}} \right)}}$ [12, 23]. After the FL pulse has been depleted, the SH pulse took over the bubble excitation and the acceleration length is estimated as ${L_{\rm{2}}} = \left( {{\rm{8}}\omega _{\rm{0}}^{\rm{2}}/{\rm{3}}\omega _{\rm{p}}^{\rm{2}}} \right)R$, where $R = 2\sqrt {{a_{0\left( {{\omega _2}} \right)}}}$ is the bubble radius [23]. Thus, the energy gain in the wakefield driven by the SH pulse is given by $\Delta {E_{\rm{2}}} \cong m{c^2}\left( {{\rm{8}}\omega _{\rm{0}}^{\rm{2}}/{\rm{3}}\omega _{\rm{p}}^{\rm{2}}} \right){a_{0\left( {{\omega _{\rm{2}}}} \right)}}$ and the final total energy gain can be estimated as [12, 23],

$$\begin{equation}\Delta E \cong m{c^2}\left( {{\rm{2}}\omega _{\rm{0}}^{\rm{2}}/{\rm{3}}\omega _{\rm{p}}^{\rm{2}}} \right){a_{0\left( {{\omega _1}} \right)}}\left[ {1 + 4{a_{0\left( {{\omega _2}} \right)}}/{a_{0\left( {{\omega _1}} \right)}}} \right]\end{equation} \tag{ 1 }$$

For a total input laser power of $P = {P_{{\omega _1}}} + {P_{{\omega _2}}}$ with the FL power of ${P_{{\omega _1}}}$ and the SH power of ${P_{{\omega _2}}}$, if the power proportion of the SH pulse is defined as $\alpha = {P_{{\omega _2}}}/P$, then equation (1) can be rewritten as

$$\begin{align} \Delta E\left( {{\rm{GeV}}} \right) & = {\rm{1}}{\rm{.7}}G{\left[ {\frac{{P\left( {{\rm{TW}}} \right)}}{{100}}} \right]^{1{\rm{ }}/3}}{\left[ {\frac{{{{10}^{18}}}}{{{n_0}\left( {{\rm{c}}{{\rm{m}}^{ - 3}}} \right)}}} \right]^{2/3}}\nonumber\\ & \quad{\left[ {\frac{{0.8}}{{{\lambda _0}\left( {\mu {\rm{m}}} \right)}}} \right]^{4/3}}\end{align} \tag{ 2 }$$

where G = (1-α)^1/3 + 2^4/3α^1/3 is the extra energy gain factor for a LWFA driven by dual-color laser pulses. For α = 0 and G = 1, the energy gain will be same as that of Ref. [23]. According to the laser parameters used in our simulations, α= 0.091 and G is calculated to be around 2.1, which is close to the extra energy gain (G = 1.92) obtained by comparing the results of figures 1(f) and 2(f). Compared with the simulations conducted in Ref. [12], our results show that using a SH pulse of a relatively smaller waist and with only 10% of the full laser energy (sub-Joule, tightly focused) can still dramatically boost the electrons energy gain, and can be managed experimentally [19], as will be discussed in a later section.

According to the simulations of the self-injection regime shown in figures 1 and 2, one can see that the nonlinear evolution of the self-guided laser pulse will easily cause deformation of the bubble, resulting in a continuous injection, poorly collimated electron spectra, and polychromatic background [24–26]. Ionization injection, on the other hand, occurs at a relatively low plasma density but often results in electron beams with a continuous energy spectrum (large energy spread) [4]. The simulations, which employ co-propagating dual-color laser pulses to excite the ionization injection in the LWFA, are conducted for helium-nitrogen gas mixture with a gas density of 5 × 10¹⁸ cm⁻³ where 99.5% helium gas was doped with 0.5% nitrogen gas. The simulation box also had a size of 120 × 160 μm² along z and x directions, respectively, and was moving with the light speed c. The cell divided from the simulation box had a size of 1/64 μm × 1/4 μm and the average number of particles per cell was 14. The other laser-plasma parameters are the same as those used in the earlier self-injection simulations. Figure 4 shows the evolution of a LWFA process employing ionization injection driven by an 800 nm, 30 fs, 77 TW single FL pulse. Figure 4(a) shows that ionization injection has occurred and the beam-loading effect [27–29] is triggered as a result of the increased number of trapped electrons. The initially injected electrons entered into the deceleration phase due to the beam-loading effect, as shown in figure 4(b). The subsequently injected electrons entered the acceleration phase and are continuously accelerated by the plasma wakefield, as shown in figures 4(b)-(d). The bubble size is enlarged in both longitudinal and transverse directions due to the strong self-focusing and front-steepened effects of the laser, resulting in a stronger wakefield and acceleration gradient, as seen in figures 4(b)-(c). It is noticed that, in figure 4(d) the laser pulse is depleted and the bubble is driven cooperatively by the laser remanence and the accelerated electron beam. After propagating a distance of ∼3 mm, the laser pulse is almost disappeared and the bubble is solely driven by the electron beam, as shown in figure 4(e). The brown-dotted line in the lower-half panel of figure 4(e) is the final electron beam energy spectrum after an acceleration distance of 4 mm. Eventually, electrons with a maximum energy of 400 MeV, a peak energy (the energy at the highest monoenergetic peak in the energy spectrum) at 326 MeV, and a relative energy-spread of 24.4% in full-width at half-maximum (FWHM) could be obtained.

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Figure 5 shows the evolution of a LWFA driven by dual-color co-propagating FL (70 TW, 30 fs, P-polarized) and SH (7 TW, 30 fs, S-polarized) laser pulses with a time delay of 0 fs in a gas mixture of 99.5% helium and 0.5% nitrogen at a gas density of 5 × 10¹⁸ cm⁻³. When compared with the single-laser driven LWFA, figures 5(a)-(c) show a similar process: a lot of electrons are ionized from the inner-shell of nitrogen atoms and trapped by the wakefield of the bubble driven by the FL pulse and are rapidly accelerated up to a maximum energy of ∼300 MeV; meanwhile, some of the early injected electrons already entered the deceleration phase due to beam-loading effect. However, it is observed that the bubble structure could be sustained for a longer time due to the use of the SH pulse, as seen in figures 5(d) and (e). Similarly, over the entire 4 mm acceleration distance, the synchronized SH laser pulse has been self-guided in the plasma and helped in maintaining the bubble fields structure. Thus, the subsequently injected electrons can be accelerated to a higher energy as compared to the single-laser driven LWFA case. The brown-dotted plot in the lower-half panel of figure 5(e) is the final energy spectrum after an acceleration distance of 4 mm. Eventually, an electron beam with a maximum energy of 480 MeV, a peak energy of 444 MeV, and a relative energy spread of 9.2% in FWHM is obtained. Not only the maximum and peak energies are enhanced by 20% and 36% respectively, but also the electron beam's energy-spread is reduced significantly.

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Figure 6(a) shows evolutions of the laser electric-field peak amplitudes of the FL and SH laser pulses in the plasma of 99.5% helium and 0.5% nitrogen for the single-laser and dual-color-laser driven LWFA cases. As discussed earlier in figure 3, due to the mismatch between the FL laser spot sizes and the plasma wavevector during the LWFA process, the FL pulses (2.3 J and 2.1 J) are undergoing strong self-focusing and evolution (focusing and defocusing), and then almost depleted within ∼3 mm distance [33% longer than the case of figure 3(a)] for both of the two cases. However, the FL (2.1 J) pulse of the dual-color-laser LWFA case has depleted relatively slower as comparted with the case of the FL (2.3 J) single-laser LWFA case. The SH laser pulse evolution is modest, and it retained its intensity over most of the interaction length, as shown in blue curve. Figures 6(b) and (c) show the energy spectra of the accelerated electron beams over 4 mm distance for various time delays between the dual-color laser pulses. It is important that the maximum and peak energies of the electron beams as well as the beam's energy-spread are all controllable by tuning the time delay between the dual-color laser pulses. Comparing figures 6(b) and (c) with figures 3(b) and (c), it is obvious that a LWFA driven by dual-color pulses in the ionization-injection regime generates high-quality electron beams with a higher number of electrons at the highest energies and this is achievable over a relatively wider range of time delays as compared with the case of dual-color LWFA in helium plasma (self-injection regime). Additionally, the effect of the SH laser pulse's polarization on the final energy gain has been also investigated for the dual-color LWFA in ionization injection regime, figure 6(d). It is found that the maximum energy gain of the electron beam using C-polarized and S-polarized SH laser pulses is somewhat similar (500–550 MeV), but the energy spectrum is continuous in the former case and quasi-monoenergetic in the latter case (figure 6(b)). Using P-polarized SH pulse, the maximum energy of electrons could reach ∼700 MeV over the 4 mm acceleration distance and the energy spectrum is also continuous in this case.

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