⇦Chinese Physics Letters, 2016, Vol. 33, No. 9, Article code 095202⇨ Laser Wakefield Acceleration Using Mid-Infrared Laser Pulses * Guo-Bo Zhang(张国博)1,2, N. A. M. Hafz2,3**, Yan-Yun Ma(马燕云)1,3**, Lie-Jia Qian(钱列加)2,3, Fu-Qiu Shao(邵福球)1, Zheng-Ming Sheng(盛政明)2,3,4 Affiliations 1College of Science, National University of Defense Technology, Changsha 410073 2Key Laboratory for Laser Plasmas (MOE) and Department of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240 3Collaborative Innovation Center of IFSA, Shanghai Jiao Tong University, Shanghai 200240 4SUPA, Department of Physics, University of Strathclyde, Glasgow G4 0NG, UK Received 25 May 2016 ^*Supported by the National Basic Research Program of China under Grant Nos 2013CBA01504, the National Natural Science Foundation of China under Grant Nos 11475260, 11374209 and 11375265.
^**Corresponding author. Email: nasr@sjtu.edu.cn; plasim@163.com Citation Text: Zhang G B, Hafz N A M, Ma Y Y, Qian L J and Shao F Q et al 2016 Chin. Phys. Lett. 33 095202 Abstract We study a laser wakefield acceleration driven by mid-infrared (mid-IR) laser pulses through two-dimensional particle-in-cell simulations. Since a mid-IR laser pulse can deliver a larger ponderomotive force as compared with the usual 0.8 μm wavelength laser pulse, it is found that electron self-injection into the wake wave occurs at an earlier time, the plasma density threshold for injection becomes lower, and the electron beam charge is substantially enhanced. Meanwhile, our study also shows that quasimonoenergetic electron beams with a narrow energy-spread can be generated by using mid-IR laser pulses. Such a mid-IR laser pulse can provide a feasible method for obtaining a high quality and high charge electron beam. Therefore, the current efforts on constructing mid-IR terawatt laser systems can greatly benefit the laser wakefield acceleration research. DOI:10.1088/0256-307X/33/9/095202 PACS:52.38.Kd, 52.65.Rr, 52.35.Mw © 2016 Chinese Physics Society Article Text Laser wakefield acceleration (LWFA) is a promising scheme to realize table-top electron accelerators due to the extremely high acceleration gradients (1 GV/cm) of wakewaves and cm-scale guiding ability of driving laser pulses through plasma channels.[1,2] In LWFA, plasma electrons are radially expelled by the ponderomotive force of an ultraintense laser pulse that leads to the formation of an ion cavity (void of electrons) behind the laser pulse, and this is called the 'bubble' regime.[3-5] Electrons can be self-injected into the bubble and accelerated by its strong accelerating electric fields until they reach the center of the bubble over a propagation distance called 'dephasing length'. A great challenge is how to effectively inject electrons into the accelerating phase of wakefields. For this purpose, several schemes have been proposed, such as the optical pulse injection,[6,7] density gradient injection,[8,9] ionization injection,[10-13] electron bow-wave injection (EBWI),[14] and other mechanisms.[15-18] Most of these injection methods are focused on achieving electron beams of low energy-spread and small transverse emittance, which have potential applications to bright x-ray sources,[19] generation of attosecond electron pulses,[20] and high-energy positrons.[21] In recent LWFA experiments, high-quality GeV electron beams have been observed.[13,22] The ponderomotive force of the laser pulse plays the major role in exciting the plasma wave (wakefield) and the formation of the bubble structure. In a laser-plasma interaction, the ponderomotive force can be expressed as $F_{\rm p}=-(e^{2}/4m_{\rm e}{\omega}_0^{2}){\bigtriangledown}E^{2}$,[23] where $\omega_0=2{\pi}c/{\lambda}_0$ is the laser frequency, $\lambda_0$ is the laser wavelength, $c$ is the light speed in vacuum, $m_{\rm e}$ and $e$ are the electron mass and charge, respectively. Therefore, for a fixed laser power, the ponderomotive force of a laser pulse scale quadratically with the laser wavelength; that is, $F_{\rm p} {\propto} {\lambda}_0^{2}$. Nowadays, most laser-driven particle acceleration and radiation experiments are conducted by using 0.8 μm or 1.0 μm wavelength laser pulses available from Ti:sapphire and Nd:glass laser systems.[24-27] On the other hand, femtosecond intense mid-infrared lasers are recently produced by either the difference frequency generation or optical parametric amplification technologies.[28,29] Such laser pulses have been extensively used in studying high-harmonic generation,[30] terahertz emission,[31] x-rays.[32] In our laboratory, we are constructing such a mid-infrared (mid-IR) laser system at the power level of 50 TW for LWFA experiments. In this Letter, we investigate the laser wakefield acceleration using mid-IR laser pulses. Two-dimensional (2D) particle-in-cell (PIC) simulations by using the PLASIM code are conducted.[33] As is expected, due to the strong ponderomotive force which is delivered to the electrons by mid-IR laser pulses, we find that more intense plasma wave is excited. Thus the electron injection occurs earlier, and the plasma density threshold for self-injection is reduced. Trapped electrons can be accelerated to high energies over a short distance. In addition, we also observe an enhancement of the accelerated number of electrons by a factor of 60 by using 4.0 μm laser wavelength as compared with the electron number generated by using 0.8 μm lasers. In our 2D-PIC simulations, the laser pulse polarization is along the $x$-direction and it propagates in the $z$-direction with a spatial-temporal profile given by the following relationship $$ E={E_0}\exp(-(y-y_0)^{2}/{\sigma}^{2})\exp(-(t-t_0)^{2}/{\tau}^{2}),~~ \tag {1} $$ where $\tau=100$ fs is the laser pulse duration at full-width at half maximum (FWHM), and $\sigma=15$ μm is the laser spot radius, $E_0$ is the maximum electric field of the laser pulse. We consider fully ionized hydrogen plasma with a trapezoidal density profile: 50 μm linear up-ramp, 3 mm plateau, and 50 μm linear down-ramp. We consider the wavelength cases: (1) 0.8 μm, (2) 2.0 μm and (3) 4.0 μm. The simulation box sizes for cases (1) and (2) are $200\times100$ μm$^2$, and for case (3) it is larger, $300\times100$ μm$^2$, due to the large bubble size for the 4.0 μm wavelength. All the cell sizes are $0.05\times0.1$ μm$^2$, and the time-step interval is 0.094 fs.

Fig. 1. (Color online) Spatial distribution of electron energy density with the laser wavelength of (a) 0.8 μm, (b) 2.0 μm and (c) 4.0 μm after the laser has propagated for $t=2.2$ ps. The matching plasma densities for each case are $0.88\times10^{18}$ cm$^{-3}$, $2.23\times10^{18}$ cm$^{-3}$ and $4.48\times10^{18}$ cm$^{-3}$, respectively.

For a fixed laser power, a laser of longer wavelength possesses a larger normalized vector potential $a_0$. Figure 1 shows the spatial distributions of the electron energy density for the three cases at the time $t=2.2$ ps using the laser power of $P=50$ TW. Based on the matching condition $k_{\rm p}{\sigma}\approx2\sqrt{a_{0}}$,[34] we use the respective electron densities, as follows: case 1 $n_{\rm e}=0.88\times10^{18}$ cm$^{-3}$; case 2 $n_{\rm e}=2.23\times10^{18}$ cm$^{-3}$ and case 3 $n_{\rm e}=4.48\times10^{18}$ cm$^{-3}$, where $k_{\rm p}$ is the plasma wave number. For 0.8 μm laser wavelength at the above-mentioned plasma density, there were no trapped electrons in the bubble, as shown in Fig. 1(a). Only a periodic plasma wave is formed in the wake of the laser pulse with a plasma wavelength of ${\lambda}_{\rm p}$ (μm) ${\approx}\,3.3\times10^{10}/\sqrt{n_{\rm e}({\rm cm}^{-3})}=35.1\,μ$m. On the other hand, for the 2.0 μm laser pulse case (Fig. 1(b)), we can clearly see that self-injection of electrons occurred in the third and fourth plasma waves. These trapped electrons are accelerated to high energies. For the case of 4.0 μm wavelength pulse case (Fig. 1(c)), we can see a bubble with a huge size due to the evolution of the laser pulse. A large number of electrons are injected into the bubble structure leading to much higher energy density, and the bubble size is elongated gradually due to the intense self-focusing effect. Nevertheless, the laser with wavelength of 4.0 μm could only propagate a short distance if compared with the normal 0.8 μm pulse, this is due to the large laser energy depletion at longer wavelengths. The pump depletion length scales as $L_{\rm pd}={\omega}_0^{2}c\tau/{\omega}_{\rm p}^{2}=n_{\rm c}c{\tau}/n_{\rm e}$,[34] where $n_{\rm c}$ (cm$^{-3}$) = $1.11\times10^{21}/{\lambda}_0^{2}$ (μm$^2$) is the critical density for the incident laser pulse, and $n_{\rm e}$ is the background plasma density. In the simulation, the pump depletion length for the laser wavelength of 4.0 μm case is only about 465 μm for the matching density condition.

Fig. 2. (Color online) Trapped electron charge ($E\geq6.4$ MeV) as a function of the density keeping the laser spot radius and pulse duration while varying laser power and laser wavelength: (a) 30 TW, (b) 50 TW and (c) 100 TW.

The large electron charge has many applications, for example, the generation of energetic positrons where there is a direct relationship between the yield of positrons and the incident electron beam charge.[35] Theoretical prediction shows that the total trapped electron number scales as $N\approx2.5\times10^{9}(\lambda_0$( μm)/0.8)$\sqrt{P({\rm TW})/100}$[34] in the bubble regime. Therefore, a mid-infrared laser pulse is beneficial in enhancing the trapped electron charge. Recently, an electron beam of 10 nC charge has been studied by using a CO$_2$ laser system with the wavelength of 10.6 μm,[36] it also proves the feasibility that a laser of longer wavelength can drive a plasma wakefield which can accelerate high-charge electron beam with moderate laser power. Figures 2(a), 2(b) and 2(c) show the charge of trapped electrons as a function of the plasma density for different laser powers 30 TW, 50 TW and 100 TW. We count the trapped electron charge at different times for each laser power case due to the fact that the laser energy for a longer wavelength rapidly depletes. Thus we choose the time at which the bubble structure is stable and intact (before the bubble structure disappears) as the suitable time for counting the number of trapped electrons. We can see that there was no injection in the bubble for the laser wavelength case of 0.8 μm at the matched density that we mentioned earlier. The trapped electron charge for various laser wavelengths at the laser power of $P=30$ TW is shown in Fig. 2(a). We observe an increase in the threshold plasma density for trapping along with decreasing the laser wavelength when we keep laser spot radius and the pulse duration unchanged. The electron beam charge rises rapidly with the plasma density until eventually it reaches some sort of saturation, and the bubble driven by a longer wavelength laser traps a large number of self-injected electrons. In the matching condition, the trapped electron charge is 760 pC/μm for the laser wavelength of 4.0 μm, 51.3 pC/μm with laser wavelength of 2.0 μm, and no injection with laser wavelength of 0.8 μm. One can also see that there is enhancement of the total trapped charge for the same plasma density for the three cases. Based on the theory, the estimated electron charge is 685 pC for the 4.0 μm wavelength. The simulation value is greater than the estimated value. For $P=50$ TW, the total trapped electron beam charge shows a similar trend: a trapped electron charge of 943 pC/μm for the 4.0 μm laser and 366 pC/μm for the 2.0 μm laser at a fixed density of $4.48\times10^{18}$ cm$^{-3}$. The total trapped electron charge increases with the laser power. The maximum trapped electron charge up to 2.1 nC/ μm is observed by using 100 TW laser power at the plasma density of $6.25\times10^{18}$ cm$^{-3}$. Trapped electron charge is enhanced by 60 times as the laser wavelength increases from 0.8 μm to 4.0 μm. By comparing the trapped electron charge with different powers for the same laser wavelength of 0.8 μm (black square in Fig. 2), one can see that the electron injection occurs at density $1.2\times10^{19}$ cm$^{-3}$ for $P=30$ TW, $8\times10^{18}$ cm$^{-3}$ for $P=50$ TW, $6.35\times10^{18}$ cm$^{-3}$ for $P=100$ TW. A threshold plasma density is commonly observed, below which no electron beam is produced, and the self-injection density threshold is gradually reduced versus the laser power. Thus the longer wavelength laser not only enhances the trapped electron charge but it also reduces the threshold plasma density for self-injection.

Fig. 3. (Color online) Electron self-injection occurrence time as a function of the density for the same laser spot radius and pulse duration while varying laser power and laser wavelength: (a) 30 TW, (b) 50 TW and (c) 100 TW. The maximum electron energy for the three laser wavelengths with keeping the laser spot radius and pulse duration while varying laser power and laser wavelength: (d) 30 TW, (e) 50 TW and (f) 100 TW.

In LWFA, the maximum accelerating field can strongly affect the self-injection process. The maximum amplitude of the plasma wave in the nonlinear regime can be estimated as $E_{\max}=E_0\sqrt{a_0}$ according to the model in Ref. [34], where $E_0$ (V/m)$\approx$96$\sqrt{n_{\rm e}\,({\rm cm}^{-3})}$. The maximum accelerating field for the laser wavelength of 4.0 μm is estimated to be 869 GV/m for $n_{\rm e}=6.35\times10^{18}$ cm$^{-3}$ at $P=100$ TW. In our simulations, the observed accelerating field is 614 GV/m after the bubble formation. Such a large acceleration field can rapidly accelerate electrons, which leads to the electrons being injection into the bubble. Figures 3(a)–3(c) show the self-injection occurrence time as a function of the plasma density for various laser powers. We choose the self-injection time, for those electrons with energy $E\geq6.4$ MeV and are located in the bubbles accelerating phase, as the diagnostic of the self-injection onset. In previous studies, self-trapping occurs when the bubble radius is larger than a certain value[37] $$ {k_{\rm p}}{r_{\rm b}}>2{\sqrt{\ln{2({\gamma_{\rm p}}^2)}-1}},~~ \tag {2} $$ where $\gamma_{\rm p}{\approx}\sqrt{n_{\rm c}/3n_{\rm e}}$ is the Lorentz factor associated with the phase velocity of the bubble.[38] For $P=100$ TW, the critical densities $n_{\rm c}=1.74\times10^{21}$ cm$^{-3}$, $0.28\times10^{21}$ cm$^{-3}$ and $0.07\times10^{21}$ cm$^{-3}$ corresponding to the laser wavelengths of 0.8 μm, 2.0 μm and 4.0 μm, respectively. Thus we can obtain the bubble radius for self-trapping $r_{\rm b}$(0.8 μm)$\geq$8.55 μm, $r_{\rm b}$(2.0 μm)$\geq$6.43 μm, $r_{\rm b}$(4.0 μm)$\geq$4.16 μm for the plasma density $n_{\rm e}=6.35\times10^{18}$ cm$^{-3}$. That is, in the case of longer wavelength laser pulses, self-injection is highly likely to occur, which occurs earlier during the bubble evolution. Self-injection takes place as soon as the bubble formation is just completed. This is consistent with our results, as shown in Figs. 3(a)–3(c). We also studied the maximum electron energy for the three laser wavelengths at the laser powers of 30 TW (Fig. 3(d)), 50 TW (Fig. 3(e)) and 100 TW (Fig. 3(f)). As we mentioned earlier, although the injection time occurs very early for the laser pulse of longer wavelength case, the pump depletion length becomes very short. Thus the trapped electrons can only be accelerated over a short distance leading to modest electron energies. The scaling of the maximum electron energy can be expressed as $\Delta E{\sim}P/n_{\rm e}{\lambda}_0$. In our simulation, the maximum electron energy with laser wavelength of 2.0 μm is estimated to be 233 MeV for the matching density of various laser powers. However, the simulation results are 86 MeV for the laser power of 30 TW, 303 MeV for the laser power of 50 TW, and 459 MeV for the laser power of 100 TW. One can see that 303 MeV is close to the theory. However, for the case of $P=30$ TW, the maximum electron energy is seriously lower than the theory due to the fact that the trapped electrons are accelerated in the second and third bubbles, and the accelerating field and dephasing distance are overvalued. For the case of $P=100$ TW, the value is higher than the theory due to the rapid evolution of the bubble size. In general, the obtained maximum electron energy using the 2.0 μm laser pulse is relatively higher than electron energy using the 4.0 μm pulse. However, when compared with the 0.8 μm laser pulses, generally the mid-IR laser pulses generate higher electron energies at the same conditions. Therefore, we need to choose the suitable laser wavelength based on the desired application. In our laboratory, we are constructing a 50 TW laser system with wavelength of 2.0 μm, which we are going to use for LWFA experiments in the near future. Therefore, we have simulated such an experiment, and the simulation results are as follows. We used the plasma density of $n_{\rm e}=1.1\times10^{18}$ cm$^{-3}$ with the following profile: 200 μm linear up-ramp, 3 mm plateau, and 200 μm linear down-ramp to satisfy the experimental conditions of our gas jets. The spatial distribution of the electron energy density at $t=11$ ps is shown in Fig. 4(a). We can clearly see that the electrons are self-injected in the second and third bubbles. However, the trapped electron beam is spatially divided into two parts due to the dynamic evolution of the laser pulse and the plasma wave, and an ultrarelativistic electron ring might be formed in 3D geometry, which was experimentally reported recently.[39] We also plot the electron energy spectrum in Fig. 4(b) corresponding to those in Fig. 4(a). Two quasi-monoenergetic peaks are generated. The first high-energy peak corresponds to the electron beam in the second bubble, and the peak energy is 280 MeV with the energy spread of 4.3%. The second lower energy peak corresponds to an electron beam in the third bubble. The peak energy is 208 MeV with an energy spread of 2.4%. Therefore, we can use the longer wavelength laser driven laser wakefield acceleration to produce narrow energy-spread high-quality electron beams.

Fig. 4. (Color online) (a) Spatial distribution of electron energy density with the laser wavelength of 2.0 μm at $t=11$ ps, and the simulation electron density is $1.1\times10^{18}$ cm$^{-3}$. (b) The energy spectrum of the trapped electrons at the same time.

In summary, laser wakefield acceleration using mid-IR laser pulses has been studied by using the 2D-PIC simulations. Considering near-infrared and mid-infrared laser wavelengths with 0.8 μm, 2.0 μm and 4.0 μm, a laser of longer wavelength can provide a larger ponderomotive force for a fixed laser power condition. Such a strong ponderomotive force can drive an intense plasma wave, and the electron self-injection can occur more efficiently. As a result, electron self-injection occurs earlier and the plasma density threshold for injection is lowered as the laser wavelength becomes longer. Additionally, the trapped electron charge is enhanced by 60 times as the laser wavelength increases from 0.8 μm to 4.0 μm by using $P=100$ TW laser pulses. We also obtain two quasi-monoenergetic electron beams with narrow energy spreads of 2.4% and 4.3% using the laser wavelength of 2.0 μm corresponding to a planned experiment at our laboratory. We can also obtain a high-charge high-energy electron beam over a short distance by optimizing the laser wavelength for a fixed laser and plasma system. Our simulation results suggest that mid-IR laser pulses are beneficial for future experimental investigations of the laser wakefield acceleration. We appreciate helpful discussions with Chen M. References Laser Electron AcceleratorLaser plasma acceleratorsLaser wake field acceleration: the highly non-linear broken-wave regimePhenomenological theory of laser-plasma interaction in “bubble” regimeAcceleration of on-axis and ring-shaped electron beams in wakefields driven by Laguerre-Gaussian pulsesCold Optical Injection Producing Monoenergetic, Multi-GeV Electron BunchesEnhancement of electron injection in laser wakefield acceleration using auxiliary interfering pulsesLaser wakefield acceleration using wire produced double density rampsStudy of the plasma wave excited by intense femtosecond laser pulses in a dielectric capillaryElectron injection and trapping in a laser wakefield by field ionization to high-charge states of gasesTheory of ionization-induced trapping in laser-plasma acceleratorsInjection and Trapping of Tunnel-Ionized Electrons into Laser-Produced WakesThermal emittance from ionization-induced trapping in plasma acceleratorsDemonstration of self-truncated ionization injection for GeV electron beamsElectron bow-wave injection of electrons in laser-driven bubble accelerationMagnetic Control of Particle Injection in Plasma Based AcceleratorsElectron injection by a nanowire in the bubble regimeElectron Acceleration in the Bubble Regime with Dense-Plasma Wall Driven by an Ultraintense Laser PulseEnhanced electron injection in laser-driven bubble acceleration by ultra-intense laser irradiating foil-gas targetsLaser-wakefield accelerators as hard x-ray sources for 3D medical imaging of human boneDense quasi-monoenergetic attosecond electron bunches from laser interaction with wire and slice targetsHigh-flux low-divergence positron beam generation from ultra-intense laser irradiated a tapered hollow targetMulti-GeV Electron Beams from Capillary-Discharge-Guided Subpetawatt Laser Pulses in the Self-Trapping RegimePhysics of laser-driven plasma-based electron acceleratorsStable generation of GeV-class electron beams from self-guided laser–plasma channelsEnhanced single-stage laser-driven electron acceleration by self-controlled ionization injectionProton Acceleration with Nano-Scale Micro-Structured Target by Circularly Polarized Laser PulseEnhanced laser-radiation-pressure-driven proton acceleration by moving focusing electric-fields in a foil-in-cone targetControl of target-normal-sheath-accelerated protons from a guiding coneBright Betatronlike X Rays from Radiation Pressure Acceleration of a Mass-Limited Foil TargetBetatron-like resonance in ultra-intense laser mass-limited foil interactionUltrabroadband, highly flexible amplifier for ultrashort midinfrared laser pulses based on aperiodically poled Mg:LiNbO_3Carrier-envelope-phase-stable, 12 mJ, 15 cycle laser pulses at 21 μmAtoms in high intensity mid-infrared pulsesEfficient terahertz emission by mid-infrared laser pulses from gas targetsHigh-brightness table-top hard X-ray source driven by sub-100-femtosecond mid-infrared pulsesGenerating multi-GeV electron bunches using single stage laser wakefield acceleration in a 3D nonlinear regimeGenerating positrons with femtosecond-laser pulsesHigh quality electron bunch generation with CO2-laser-plasma interactionScalings for radiation from plasma bubblesThe evolution of ultra-intense, short-pulse lasers in underdense plasmasFormation of Ultrarelativistic Electron Rings from a Laser-Wakefield Accelerator

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