Electron self-injection threshold for the tandem-pulse laser wakefield accelerator

accelerator Zahra M. Chitgar,1, a) Paul Gibbon,1, 2 Jürgen Böker,3 Andreas Lehrach,3, 4 and Markus Büscher5, 6 1)Institute for Advanced Simulation, Jülich Supercomputing Centre, Forschungszentrum Jülich, D-52425 Jülich, Germany 2)Centre for Mathematical Plasma Astrophysics, Katholieke Universiteit Leuven, 3000 Leuven, Belgium 3)Institut für Kernphysik (IKP-4), Forschungszentrum Jülich, D-52425 Jülich, Germany 4)Institut für Experimentalphysik III B und III. Physikalisches Institut, RWTH Aachen, Germany 5)Peter Grünberg Institut (PGI-6), Forschungszentrum Jülich GmbH, D-52425 Jülich, Germany 6)Institut für Laserund Plasmaphysik Heinrich-Heine Universität Düsseldorf, D-40225 Düsseldorf, Germany


I. INTRODUCTION
Compact laser-plasma electron accelerators have made enormous strides over the past two decades in terms of beam energy, quality and reproducibility. A key factor in this advance has been the separation of acceleration and injection phases, offering more control over the beam dynamics 1 . Apart from providing a route to cheaper, more accessible GeV electron beams, laser-based accelerators are also becoming an important source of bright X-rays with femtosecond to picosecond pulse duration. These are of great importance in different branches of science for resolving atomic structure down to sub-nanometer range and for capturing ultrashort time scale events 2-4 . Until now synchrotrons and free electron lasers have provided such sources, but these have limited accessibility because of their costs and huge scale. Laser-driven betatron radiation is a relatively recent source of femtosecond X-rays which could potentially offer much wider availability [5][6][7][8] .
The most widely studied scheme is the laser wakefield accelerator (LWFA) 9 , in which plasma electrons are pushed away by the ponderomotive force of a focused laser pulse to regions with lower intensity. Ions remain immobile for such short interaction times, so that the electrons start to oscillate around their initial position due to the restoring force. This perturbation, or wake, is strongest for pulse durations matched to the electron plasma period, or τ L ∼ ω −1 p , and follows the laser pulse with a phase velocity equal to the group velocity of the laser pulse. In this way, injected electrons surfing on this wave can get accelerated.
Since the LWFA was first conceived, several techniques a) z.chitgar@fz-juelich. de have been proposed to increase the energy and flux of the injected electrons and their corresponding emitted radiation. Multi-pulse (MP-LWFA) schemes have been advocated before to successively increase the amplitude of the plasma wave and achieve higher electron energies [10][11][12] with greater efficiency, and have also been studied in the context of betatron radiation in the quasi-linear regime 13 . Tailored density profiles 14,15 and two-color driven ionization injection [16][17][18] have been explored as a means of reducing the energy spread and emittance of the electron beam. Recently, a combined cluster/gas-jet target was shown to yield higher betatron radiation flux and energy 19 than a standard gas target. In this paper we show how the injection process in the nonlinear bubble regime can be better controlled using two co-propagating pulses with the same wavelength and focal lengths but differing intensity. This scheme results in a lower injection threshold in terms of the laser intensity, while at the same time yielding improved beam qualities such as energy, charge and emittance compared to the single-pulse scheme, potentially offering advantages for TW laser-driven betatron radiation sources. These findings may also make it easier to extend the recent progress in generation of compact MeV electron beams to a new class of kHz laser facilities now coming into use 20,21 .
In order to focus the investigation on the role of the second pulse in the injection process, we modelled a homogeneous gas target where the lasers are focused far from the edges and a simulation time limited to around one Rayleigh length. This choice is designed to reduce the influence of other injectiontriggers such as density gradients and nonlinear pulse propagation effects such as self-focusing, which will likely come into play at later times.
The paper is organized as follows. In Sec. II, the fundamentals of electron injection and different means of achieving it are briefly introduced, including the nonlinear cavity regime of electron acceleration. In Sec. III the simulation methodology is described and the optimal conditions for the doublepulse scheme with respect to the electron beam properties are evaluated. In Sec. IV the injection threshold using the double pulse scheme is determined and compared to a single-pulse driver for experimentally relevant conditions. The paper concludes with a discussion on the merits and practicalities of this scheme.

II. ELECTRON SELF-INJECTION IN LASER-DRIVEN ACCELERATORS
The 'blowout' or 'bubble' regime was first predicted for electron-beam-driven 22 and laser-driven 23 plasma wakefields respectively, and later analyzed 24,25 as the highly non-linear regime of electron wakefield acceleration. Lu et al. 26 demonstrated that 'matched' cavity formation can be achieved by balancing the transverse ponderomotive force of the laser to the radial space-charge force created by the ion channel. This yields a condition for which the focal spot-size w 0 is comparable to the blowout radius, ie: k p w 0 k p r b 2 √ a 0 , for laser amplitudes a 0 ≥ 2. The pulse duration should also be roughly a plasma period, τ L = ω −1 p , where ω p = ck p is the cold plasma frequency. Here λ p = 2πc/k p denotes the plasma wavelength, a 0 = 8.5 × 10 −10 (I L [W /cm 2 ]) 1/2 λ L [µm] the dimensionless laser field strength for laser intensity I L , and τ L the pulse length.
Under certain conditions, some plasma electrons can get trapped at the back of this bubble and be accelerated in the strong longitudinal electric field of the ion cavity 27 . Moreover, its radial electric field causes electrons start to oscillate around the laser propagation axis. This motion results in betatron radiation; emission of bright X-rays directed according to the energy and trajectory of the oscillating electrons 28 . This implies that the radiation flux can be tuned through the electron dynamics, for which the injection process and the cavity structure both play crucial roles.
In the simplest one-dimensional picture, the minimum energy of the electrons to get injected into a laser-driven wakefield is required to be greater than the wake phase velocity, which to a first approximation is tied to the group velocity of the driver pulse 29 , p min /m e c ≈ γ p ω 0 /ω p . This condition can be met in different ways: either through modification of the (generally three-dimensional) wakefield structure so that some of the plasma wake electrons get injected, or by injecting additional background electrons into a pre-formed wake. Wave-breaking and formation of cavity provide the necessary conditions for betatron oscillations, but electron injection does not occur automatically. To inject the electrons at the right point in the wakefield, several techniques have been considered and implemented so far, which all have their advantages and drawbacks 30 .
In order to obtain high quality electron beams for applications, a low-emittance, narrow-energy-spread electron bunch is required, meaning that control over the electron injection process is essential. An attractive aspect of the cavity regime is that self-injection readily occurs provided that -in addition to the matching requirements introduced earlier -the laser power P satisfies P/P c 5, where P c = 17.4(ω 0 /ω p ) 2 GW is the critical power for self-focusing 26,31 . This rule of thumb has been largely confirmed in experiments, but these also show that the electron beam quality thus obtained is very erratic 30,32,33 . To improve reproducibility of beam properties therefore, various injection aids have been considered such as: ionization injection [34][35][36] , optical injection [37][38][39] , or modification of the plasma wave dispersion properties via density gradients [40][41][42][43][44] .
In this paper we investigate the threshold conditions and bunch quality for an all-optical, double-pulse injection scheme, in which the nonlinear cavity field is amplified in a controlled manner via a 2nd trailing pulse, and the injected electron bunch current is enhanced by charge accumulation at the rear of the second cavity thanks to recycling of electrons initially expelled by the first pulse. We note in passing that similar collinear pulse configurations have also been considered in the context of ionization injection 16 , two-color driven wakefield 45,46 and channel-guided wakefield acceleration 47 , where the latter experiment used a wider, lower-intensity leading pulse to create a plasma channel. By contrast, the scheme studied here assumes that the laser pulses share the same focal optics and enter a homogeneous gas target.

III. DOUBLE PULSE SCHEME
Such a collinear double-pulse scheme was recently examined by Horný et al. 48 , using a lower-intensity preceding laser pulse as a means of gaining more control over the self-injected bunch charge and dynamics, potentially leading to higher energy gain and/or bunch charge for electrons and enhanced betatron emission in the nonlinear blowout regime 49,50 . This tandem-pulse scheme, which relies on a double-cavity, has two advantages: first, the leading pulse ensures that the plasma encountered by the 2nd driver is fully pre-ionized; second, the accumulation and recycling of free electrons at the back of the first cavity provides a concentrated source for the second pulse to act on, ultimately enhancing the accelerating field behind it, resulting in higher electron beam energy and current. These observations naturally raise the question of when it becomes worthwhile to divide the available laser energy into such an 'injector-driver' configuration, and if so, how their relative intensities and timing need to be arranged. In the following sections we attempt to examine this question before making a more systematic assessment of the scheme compared to the conventional single-pulse drive mode.

A. Simulation method
To make a quantitative survey covering a range of realistic parameters we have performed both 2-dimensional and 3-dimensional particle-in-cell simulations using the EPOCH code 51 . All 2D (3D) simulations were performed using a 100 × 80 µm 2 (100 × 80 × 80 µm 3 ) box filled with an underdense helium gas (preionized helium plasma), discretised by a computational grid with dimensions n x × n y = 3125 × 400 (n x × n y × n z = 3125 × 400 × 400) and 2 (4) particles per cell. A number of higher-resolution simulations were also done to check statistical convergence with no significant change in the results. The target is preceded by a vacuum region of 5 µm followed by a 12 µm ramp in the gas density in order to avoid an overly steep gradient at the plasma edge. A 20 fs laser pulse with wavelength 800 nm was focused from the left hand boundary down to a 10 µm 1/e focal waist w 0 (FWHM= 16.6 µm ) at the box center. A moving window was deployed in order to follow the development of the ensuing plasma wake and electron injection, which is switched on 260 fs after the start of simulation, before the laser pulse reaches the right side of the boundary. For the 2D runs, the amount of charge fully injected into the cavity is estimated from the number of particles extrapolated into a 3D box surrounding the injected bunch using a fifth order (B-Spline) particle weighting.
Most simulations were run up to a time corresponding to the Rayleigh length of the laser pulse (z R = πw 2 0 /λ ), allowing a well-defined early-time assessment of the double-pulse scheme compared to its single-pulse equivalent. Limiting the simulation time in this way also minimizes additional propagation effects (refraction, focussing or etching) which might influence the injection process at later times 52 . We will return to this point later in Section V.
A representative example of the new scheme is displayed in Fig. 1, which compares the electron number density after t = 1300 fs for the single-and double-pulse schemes respectively, corresponding to approximately one Rayleigh length of laser propagation; in this example ∼ 400 µm. The target density is 1.9 × 10 18 cm −3 (γ p = 30) which is irradiated by a laser pulse corresponding to a 0 = 5 for both single-and double-pulse schemes, meaning that the same total energy of U total = 3.4 J is used for both cases, but shared between pulses in the double pulse scheme (U 2 /U total = 0.84, U 1 /U total = 0.16).
In the double-pulse scheme the second cavity is slightly longer and wider than the first cavity of the single-pulse case in Fig. 1(a,b). Consequently a longer acceleration length could be expected, as well as a potentially larger cross-section for injected particles. The latter is confirmed by the trajectories shown later in Fig. 6(a), which trace the origin of the electron bunches injected into the second cavity seen in Fig 1(b). With the same laser parameters and target characteristics, there is no electron injection in the first bubble in the single-pulse scheme - Fig. 1(a). This indicates that for the same total laser energy, the injection threshold appears to be markedly reduced for the tandem scheme compared to a single pulse driver. Figure 2 depicts the electron distribution in longitudinal (x, p x ) momentum phase space for (a) single-and (b) doublepulse scheme in Fig. 1, and the small inset in each figure shows the transverse (y, p y ) momentum phase space. In this example the injected electrons in the second cavity of the double-pulse scheme carry around 48 pC with a normalized emittance of 7.86π mm mrad, accelerated up to 140 MeV. In the single-pulse case, there is no injection in the first cavity. However, there is a bunch of electrons at the back of the first cavity carrying 228 pC with a normalized emittance of 23.28π mm mrad. This bunch is accelerated up to 50 MeV, within the same target length and the pulse energy. Although this latter bunch is carrying a much higher amount of charge, the quality of bunch is significantly degraded by the onset of cavity wall deformation.
A key characteristic of bubble acceleration is that a volume devoid of electrons is created by the first pulse. other hand, expelled electrons return back to the rear side of the bubble providing surplus concentrated charge for the second pulse to resonantly act on in creating a second, stronger cavity - Fig. 1(a,b). The advantage of double pulse scheme over single pulse is also apparent by the higher electric field strength in (d) compared to (c) extended over a longer distance: ∆(E.d) ∼ 1.0 MV and 0.7 MV, respectively.
An initial set of simulations was performed aimed at optimizing the double-pulse scheme, before comparing the injection threshold in single-and double-pulse schemes. These simulations were done for a specific laser pulse with amplitude a 0 = 5 (corresponding to a total energy of 3.4 J) and helium gas density of 1.9 × 10 18 cm −3 , the details of which follow in Sec. III B and III C. A parameter scan of injection threshold simulations for six laser amplitudes and 5 different target densities are described later in Sec. IV.

B. Optimization of Pulse delay
Given the apparent advantages of the double-pulse scheme just demonstrated above, it is natural to ask under which conditions the injected electron beam properties are optimal. First, the ideal delay between the pulses would be expected to correspond to the size of the first cavity, or (following the notation used in Ref. [26]) twice its longitudinal radius 2r b 3.8a 1/2 01 k −1 p . This is confirmed by a series of simulations with different delays but keeping the pulse energies equally divided, i.e. U 2 = U 1 , and a 02 = a 01 = 3.5- Fig. 3(a). Note that the scheme appears to be relatively robust with respect to a non-optimal pulse interval- Fig. 3(b). When choosing the optimum condition, we need to specify the desired electron bunch characteristics, and in this case we set the maximum achievable beam energy as the criterion. Based on these results, the maximum energy is attained if the center of the second laser pulse is delayed by ∆t = 2r b /c with respect to the first bubble. Within a tolerance of ±1.5 µm (±5 fs), the scheme still works advantageously and delivers even higher flux albeit with lower maximum energy - Fig. 3(b).
Repeating this exercise for the full 3D case, we see that the delay optimum is shifted to lower values probably because of stronger self-focusing/guiding effects and the shape of the wakefield in 3D, c.f. Sec. IV. On the other hand the slight anti-correlation of optimal charge and energy persists in 3D.

C. Optimization of pulse energy division
The optimal energy division was determined by performing a further series of five 2D simulations at the optimal pulse separation, dividing laser pulse energy between each pulses as follows: the original pulse amplitude is a 0 = 5. In the first simulation the first pulse carries a fraction of total pulse energy corresponding to the amplitude a 01 = 2, the remaining energy belongs to the second pulse. This was repeated in steps of ∆a 0 = 0.5, up to the case where the first pulse carries an energy corresponding to a 01 = 4 and the remaining energy is allocated to the second pulse, which was placed on the rear side of the cavity created by the first pulse as discussed in Section III B. As a result, it is found for these specific laser parameters that the most efficient energy division is when the second pulse carries three times the energy of first pulse - Fig. 4(a,b). Generally there is a range of relative energy fraction where the injected beam can be optimized for energy or charge, or a combination thereof. For a total energy U total = 3.4 J we take U 1 /U total = 0.25, or U 2 /U 1 = 3.
In order to find a more general rule for the energy fraction of each pulse, several other simulations are carried out with different laser intensity. The result is summarized in Table I.
In all simulations, both pulses are the same in focal size and pulse length. It is evident that the intensity of the first pulse must be sufficiently high to meet the usual condition for the cavity formation, a 01 > 2, supplying an optimal quantity of ionized electrons for the second laser pulse. The rest of the laser energy can then be invested in the second laser pulse such that a 2 0 = a 2 01 + a 2 02 , with a 02 /a 01 > 1. Although these  considerations for the pulse delay and amplitude ratio serve as a good initial guide to optimising the injection process, we will see later in Sec. V that propagation effects may complicate this choice.

IV. SELF-INJECTION THRESHOLD
The self-injection process depends both on the laser amplitude and wake phase velocity, which is tied to the group velocity of the laser pulse, 53 (v gr = c 1 − ω 2 p /ω 2 ), so to compare thresholds in the cavity regime for the double-and single-pulse cases, a set of simulations were done with different laser amplitudes and target densities and then following Ref. [53] mapped in the (γ p , a 0 ) plane, where γ p is the Lorentz factor corresponding to the plasma wake moving with a velocity v p = v gr . The results are collected in Fig. 5.
The red solid lines in Fig. 5(a) and (b) show the effective threshold when keeping the laser pulse length and waist (and therefore energy) the same while changing the target density (through γ p ω 0 /ω p ), for single-and double-pulse scheme respectively. Note that for this parameter scan we only ensure that the matching conditions (k p w 0 = 2 √ a 0 , cτ = λ p /4) are applied in some average sense but are not adjusted for each (γ p , a 0 ) pair; the procedure which would likely be followed in an experiment. As one might expect, there is a density for which the wake phase velocity (γ p = 30) seems to be optimal for electron injection: above and below this density, the cavity is driven non-resonantly with respect to injected bunch charge. Lower densities provide fewer electrons and correspondingly lower injected charge, and the wake has higher phase velocity. Therefore, it is more difficult for the electrons to become injected. Higher densities provide more charge but make the cavity regime more difficult to reach.
Comparison of these charts between the single-and doublepulse schemes reveals a 30% reduction of the threshold amplitude a 0 necessary for beam injection, corresponding to a 2× reduction in (total) required intensity (or power), bringing obvious practical advantages.
To shed light on how a twin driver alters the injection process, we analyze trajectories of particles either side of the injection threshold. The latter has often been examined for the cavity regime before, but often with assumptions about the exact shape and (non-)rigidity of the cavity 52,54-56 . Figure 6  shows the trajectory of electrons in a 2D simulation for the double-pulse scheme Fig. 6(a) and (b), as well as the singlepulse scheme Fig. 6(c).
Notice how electrons from different lateral positions relative to the laser axis can still end up phase-synchronised after injection -a feature studied in some detail in Ref. [48]. Here, electrons are injected and accelerated up to 140 MeV. The same injected particles are shown in 6(b), which follows the electrons' trajectory in phase space, (red dots), together with the non-injected ones (black). Comparing this case to its single-pulse counterpart 6(c), using the same total laser pulse energy, some of the electrons which fail to get injected in the first cavity follow an extended orbit which deepens the potential well of the 2nd. In other words, when using two laser pulses the trapping separatrix of the 2nd cavity is modified in such a way that it becomes more favorable for electron injection, in an analogous fashion to an evolving bubble 52 .
In order to compare our single-pulse simulation results to previous theoretical work 26,53,54 , we kept the point γ p = 30 as the optimum matching condition, where in fact cτ L = λ p /4. For each target density (or γ p ), the pulse duration and focal size were then adjusted to maintain a roughly spherical cavity shape - Fig. 7. The resulting red solid line (I) follows a similar trend to the dashed blue line (III) 53 , confirming that injection is easier in a higher density target, whereas higher thresholds can be expected with increasing wake phase velocity. The apparently higher laser amplitude threshold in our double-pulse case is likely because of the geometrical differences inherent in 2D and 3D simulation; a point we will examine in more detail shortly. Several independent theoretical and numerical studies on self-injection threshold scaling have been previously published, but for various reasons it is difficult to compare these results quantitatively. In our simulations we used a helium gas target with a short ramp of 12 µm, and focused the laser 38 µm downstream of the ramp to ensure that the self-injection took place in the flat part of the density profile. Lu et al. 26 used a plasma channel (dashed green line IV), but did not explicitly investigate the dependence of self-injection on γ p : they found that self-injection occurs when the normalized blowout radius k p r b ∼ 4 − 5. According to Ref. [54] (dashed yellow line II) and Ref. [53] (dashed blue line III), the injection condition is dependant on the wake velocity (or γ p ); however Benedetti et al. predict a stronger dependency of self-injection on γ p . In Ref. [54] a uniform electron density is assumed, whereas Benedetti et al. use a long target ramp with ionization enabled artificially only after a stable cavity is established. This deliberately suppresses the influence of the ramp on the injection process, allowing for a 'clean' evaluation of the threshold. On the other hand such idealised conditions are probably difficult to realise experimentally.
In fact, it turns out that the injected beam current can be further increased by tuning the density ramp, as already demonstrated in Ref. [57]. Our own 2D simulations to check the influence of the ramp size on electron injection at the point (a 0 , γ p ) = (5,30) in Fig. 7 for ramp sizes from 7 µm to 22 µm predict an 17% increase of injected charge over this range. These findings are not too sensitive to the placement of the laser focal spot as long as it is focused at least one cavity diameter beyond the top of the ramp.
It is well known that nonlinear laser pulse propagation, plasma wake evolution and injection can exhibit quantitative differences between 2D and 3D geometry 31,58 , so to get some idea of how geometrical effects might quantitatively alter our findings, several full 3D simulations were done at the points (a 0 , γ p ) = {(4, 26), (5,26), (3, 30), (4,30), (5,30), (4,34), (5,34)} and the results of single and double-pulse interactions compared in Tab. II. Based on our 3D results, injection occurs for lower intensities for the double-pulse scheme and leads to a larger amount of injected charge, confirming the general trend seen in the 2D simulations. However, depending on the chosen delay the maximum energy is not necessarily higher for double-pulse scheme. This can be seen in the Tab. II; the delay for the cases with γ p = 30 is chosen to be 66 fs, since for this density the delay was optimized- Fig. 3(b). In this case both energy and charge are higher than in the corresponding single-pulse case. For other target densities, γ p = 26, 34, the delay was not optimized and this may be the reason for the lower electron energy observed; however, the amount of injected charge is still higher than the single-pulse scheme. Finally, we observe that the emittance in the doublepulse scheme is generally comparable to the value from the single-pulse scheme despite higher injected charge and in some cases higher beam energy for the cases examined. The energy could probably be increased further by devising a proper matching condition for double-pulse drive.
As expected, the same general behaviour as in Fig. 5 is ob-  8. Comparison between single-(1p) and double-pulse (2p) for the same simulation parameters in 2D (solid lines) and 3D (dashed lines); The 3D results confirm the advantage of double-pulse over the single-pulse scheme in terms of the amount of injected charge and the maximum energy. In general, 3D simulations yield higher amount of charge than the 2D simulations. The upper axis shows the P/P c for each of the simulations, where the spot size is w 0 = 10 µm and the γ p = 30. served for the 3D simulations, where there is an optimal density (γ p = 30) for injection. Furthermore, the higher injection threshold seen in our 2D simulations in Fig. 7 compared to other works can also be accounted for by the geometry. This difference is displayed more explicitly in Fig. 8, which shows how the beam charge and energy increases with laser power at fixed plasma density (here γ p = 30, or n e = 1.9 × 10 18 cm −3 ). According to Fig. 8a), the single-pulse threshold P/P c > 5 previously observed in a number of experiments 32,33 is effectively reduced to a value P/P c ∼ 2 − 3 for the double pulse scheme.

V. DISCUSSION AND CONCLUSION
We have shown that a tandem-pulse wakefield accelerator in the nonlinear regime can offer significant advantages over the conventional single-pulse driver, yielding higher electron currents and energies thanks to an enhanced cavity size and a lower electron injection threshold for the same total laser energy. Full 3D simulations show that the injection threshold intensity or power can be reduced by at least a factor of two, confirming a wider 2D parameter scan for a set of laser intensities and target densities.
The latter simulations indicate that further optimisation of this scheme may be possible by adjusting the pulse separation and relative amplitudes to allow for dynamical evolution of the cavity over longer propagation distances. To explore this properly, a comprehensive 3D or quasi-3D 53 set of simulations would be needed in a multi-dimensional parameter space; a task beyond the scope of the present work.
Nevertheless, it is of interest to examine how robust the tandem scheme is over significantly longer propagation distances, so a proof-of-principle 3D simulation corresponding to two Rayleigh lengths has been carried out for (γ p , a 0 ) = (30,5). A time series of the electron number density is plotted in Fig. 9, in which the laser pulse evolution is also partially visible. Starting with the target density corresponding to γ p = 30, the cavity has a spherical shape at the beginning, Fig. 9-(a). For the chosen laser parameter with focal spot size of w 0 = 10 µm, two Rayleigh lengths corresponds to 785 µm, so the laser intensity should undergo a significant decrease unless there is some self-guiding 26 . Moreover, because of the aggregation of electrons at the back of the first cavity and change of the target density, the second laser pulse may also be affected by additional refraction, which causes the cavity size to decrease in the longitudinal direction. The latter effect eventually leads to loss of trapped electrons from the back of the second cavity in this case.
Despite these factors, acceleration of 0.29 nC electron bunches up to 536 MeV with a peak at 395 MeV and energy spread ∆E/E = 12.7% at FWHM within this distance is observed. By comparison, the matched single-pulse scheme yields a beam with 0.22 nC charge and maximum energy 450 MeV with FWHM bandwidth ∆E/E = 33% and a peak at 300 MeV. On the other hand, the cavity shape remains roughly spherical over this distance and exhibits no leakage of the trapped beam, raising the possibility of stabilizing the tandem-pulse propagation over many Rayleigh lengths in the blowout regime via a modified matching condition. For example, more control over the laser pulse evolution might be achieved by starting with a slightly mis-matched configuration such that the 2nd cavity is initially elongated compared to the first, or by setting a longer-than-optimal delay suggested by Fig.3(b).
Clearly further study is needed to mitigate and properly exploit the interplay between relativistic propagation effects and dynamical refraction to achieve a fully 'matched' tandempulse scheme. At this point we can conclude that this scheme has tangible advantages for producing electron beams with modest energies (10s to 100s of MeV) suited to x-ray generation with smaller TW lasers, perhaps with high repetition rate. Additional trial simulations with ∼ 100 mJ laser pulse energy, 10 fs pulse duration and focal spot size down to 5 µm confirm the advantage of double-pulse drive over single-pulse. Finally, it is possible that trains of 3 or more pulses might permit even greater long-term control over cavity dynamics and beam energies for the same total pulse energy, extending the original 1D pulse-train concept 10 to the nonlinear, three-dimensional regime.