6.2 aspects of the laser-driven accelerator stage 91 acceleration is decreased. Further, the computational costs to simulate such a scenario is lowered, since the longer the laser wavelength, the larger the cell size that can be used in the simulation. On the other hand, a larger simulation window is necessary because lower plasma densities and thus larger plasma waves need to be monitored, increasing the number of cells.
6.2.1 Laser guiding
Laser diffraction in vacuum is determined by the Rayleigh length zR = πw20/λl ≈ 1.57 mmforw0 =20µm atTi:Sawavelength. Hence, the acceleration would cease after a few millimeters if laser diffraction is not stopped. This can be done by either guiding the laser within a pre-formed plasma channel [7], or by choosing a laser intensity that is high enough for relativistic effects to set in, which can also lead to self-guiding of the laser pulse [210]. Applying the first method, the required laser energy can be reduced at the cost of having to provide for a plasma channel.
Pre-formed plasma channel
To demonstrate the guiding of a laser pulse by a pre-formed plasma channel, and to achieve the aforementioned reduction in computational time, a simulation with a CO2 laser with λl = 10 µm is implemented. To guide the laser, a hollow parabolic plasma density channel is implemented. For the long wavelength, the strength parame-tera0 ∝λlis increased and a longer cell size is used. This is shown in figure16, using a0 =6,w0 = 10 µm,τ =100 fs,λ= 10 µm, and a hydrogen density of1×1017cm−3, where the laser is stably guided over many Rayleigh lengths. The plasma channel was implemented by a parabolic plasma profile with a radius of100 µm. Because the plasma wakefield does not self-inject electrons for these conditions, a density down-ramp was implemented to inject electrons into the plasma wave (not shown here). In these simulations, only less than10 pCof charge is injected for different peak densities and widths of the downramp. To inject more charge, a higher laser energy is required.
In figure16, the laser focus is chosen very small and results in a tiny Rayleigh length in combination with the10 µm-long laser wavelength. However, this simulation nicely demonstrates the functionality of external guiding of the laser pulse. In addition, using a plasma channel to guide the laser pulse has the beneficial effect that the accelerating field can be flattened, or even reversed to reduce the energy spread of the accelerated bunch [183,220].
Others reported self-injected charge of several100 pCs, using simulation ofLWFAin a plasma channel [223]. Leemans et. al.[133] reported self-injected electrons with up to 4.2 GeV of energy and up to50 pC of charge, using a 16 J laser in a plasma channel
Figure16: Guiding of a CO2 laser with (λ = 10 µm, a0 = 6, τ = 100 fs, w0 = 10 µm, W≈82.3 mJ) innp=1×1017cm−3dense neutral hydrogen. The neutral hydrogen (orange-red colored) is shown to visualize the hollow plasma channel with100 µm radius over many Rayleigh lengths (zR≈31.4 µm).
generated by capillary discharge. This result is remarkable, as it uses significantly less laser energy than in other experiments [233], while achieving similar results.
External guiding becomes necessary for laser pulses with a power less than the critical power (79), e. g. at a low plasma density. Otherwise, a strongly relativistic laser pulse can be self-guided over a dephasing length with the relativistic self-focusing mecha-nism [43,64,118]. Either way, some form of laser guiding is essential to provide for a sufficiently long acceleration length inLWFA.
6.2.2 Plasma species and density
Neutral hydrogen was chosen as target gas for theLWFAstage, because it can only be ionized once, no higher-order states occur, and it is in gaseous phase under normal conditions. Furthermore, its ionization threshold is sufficiently low to allow the laser to ionize at its very edges and provide a stable and wide plasma for the acceleration.
Operating with gases of higher atomic number (Z) might lead to an unfortunate den-sity distribution, where the strong laser peak-electric-field strength required to drive the wakefield (on the order ofT V/m) is capable of ionizing many electrons from one nuclei. Therefore, fewer states are ionized at lower field strengths, giving the highest
6.2 aspects of the laser-driven accelerator stage 93 plasma density on axis. Such a plasma-density profile would strongly defocus a laser pulse (ionization defocusing). Therefore, high-Z gases are usually not beneficial for
LWFA(with few exceptions [248]).
In general, extending the dephasing length,Ldeph∝n−3/2by operation at low density is favorable to simultaneously increase the achievable energy gain∆W ∝ n−1. How-ever, when going to lower plasma densities, simulations become increasingly challeng-ing, as the required computational resources explode with the requirement of a larger simulation box to fully model the bigger plasma wavelength. In addition, the nec-essary simulation length is extended as well, quickly making a full three-dimensional standardPICsimulation very ineffective without enhancements in computational meth-ods [156, 230]. Therefore, the plasma density was chosen to be at the lowest level that could be used with the available computational resources, while simultaneously using a high laser energy to quickly accelerate the electrons to a high energy. With this, the laser pulse can also be self-guided and no external plasma channel is required.
A comparable restriction in minimal density applies to experiments as well, arising from the available laser power. So far, the cutting-edge high-power lasers utilized in
LWFAexperiments, with and without a plasma channel, are on the100 T W level, only allowing the usage of densities above np ∼ 1×1018cm−3, and restricting the gained energy to approximately the1 GeV mark [43,118,134]. Using a PW-level laser allows a lower plasma density to be used, resulting in a higher achievable energy of poten-tially up to10 GeV [233] and beyond [156, 230] if acceleration can be sustained over a sufficiently long distance.
6.2.3 LWFA injection
As described in section 2.2.5, electrons can enter the plasma wave from behind when its accelerating field is strong enough to suck them in (self injection). This self injection naturally occurs in nonlinear laser-driven plasma waves, because the plasma wave travels with a velocity below the vacuum speed of light. To implement other injection methods, self injection must therefore first be suppressed or avoided. This can be done by tuning the laser power down to operate in the linear regime, but then external guidance of the laser pulse is required.
Another way to suppress self-injection is to fully load the wakefield with electrons at the start to lower the accelerating field to a level where no electrons can be self-injected (beam loading). Because self-injection is a continuous process, a large amount of charge can be injected with a continuous energy spectrum, which is very bad for most applica-tions, except for driving a plasma wakefield. So, arising from the requirement for high charge and energy, operating at a high density and laser energy in the bubble regime using self injection is the most promising scenario to produce suitablePWFAdrivers. In a more advanced scenario, where an external guidance for the laser can be provided,
lower plasma densities would be better to increase the energy of the witness beam in
LWFA.
The high energy of the laser was also chosen due to computational reasons, so the self injected electrons are accelerated very rapidly and the total acceleration length can be kept short. Because the final energy is proportional to ∆W ∝ a0/nλl, and neither the laser wavelength nor the plasma density can be adjusted in the needed direction, tuning up the laser amplitude is the only parameter left to increase the final electron beam energy. Apart from that, a high accelerating field also promises to inject a large amount of charge, which is also crucial for a potentialPWFAdrive beam. When considering the total energy content of the neededPWFAdrive beam, a high demand on total laser energy on the100 J-level is required (see discussion in section2.5), especially when taking the expected low efficiency of theLWFAinto account.
The spatial dimensions of the laser pulse can be adjusted to the dimensions of the plasma wave to maximize the response and the efficiency of the accelerator. However, as the laser pulse undergoes rapid transformations during the acceleration, and the wakefield does not change much for small variations around the optimum, the spot size and pulse duration optimization is not crucial as long as they are within a reasonable range around the optimum value [61]. As one optimization goal is to also keep the total required laser energyWl∝w20a20σt/λ2l as low as possible, choosing a smallerw0 and σt than demanded by the plasma-response-maximization conditions might save some energy without loosing much of the possible witness energy. This correlation, however, would require a more detailed investigation aimed at enhancing the overall efficiency.
Finally, the density and laser parameters have been chosen such that, according to equation (85) and equation (86), a bunch of1.4 nCcharge can be accelerated to6.1 GeV in a self-guidedLWFA. This bunch would meet the requirements of aPWFAdrive bunch very well.