5.2 justification of applied approximations 85
The use of the ADK model for ionization is justified because the Keldysh parameter given in equation (52) indicates the dominance of tunnel ionization over MPI for the used parameters. For instance, in the main PWFA simulation displayed in Table 13, γK ≈ 0.14 < 1. For the same simulation, the peak electric field of the laser at focus E0 ≈ 482 GV/m is smaller than the maximum allowable field for ADK, Ec,ADK ≈ 987.13 GV/m (given in equation (54)), also indicating the applicability of the ADK
model. If the occurring field strength is beyond the critical value, the ionization pro-cess might still be modeled accurately when the tunnel ionization already provided for complete ionization before the critical field strength is exeeded. This, however, needs to be verified manually when usingVSim; in the case that the electric field strength rapidly increases beyond the criticalBSIfield, the appliedADK-model becomes inaccurate.
For hydrogen for example, the critical value is Ec,ADK ≈ 75.26 GV/m and WADK ≈ 11.59 /fs(compare table2). At a density of1023/m3, the corresponding linear charge density amounts to λ ≈ 1×1023/m31/3
≈ 4.64×107/m, or 46.4 particles per µm.
A laser is therefore passing 46.4 µm/c ≈14 particles per femto-second along the axis.
Hence, an ionization rate exceedingWADK '14/fscould fully ionize hydrogen with a density of1023/m3. The use of macro-particles underestimates the amount of charge that is ionized in general, which is especially visible for ionization near the threshold energy. Originating from the fact that only whole macro-particles can be ionized, the maximum error is less than the charge of one macro-particle. When considering the spatial volume that is ionized and its temporal development, this might add up to more than the charge of one macro-particle. In thePWFAsimulations shown in figure34 and35, 163 840electrons have been combined in every macro-particle, corresponding to 0.26 pC. Compared to the obtained charge in the generated bunch of 35.9 pC, this error affects at least the last shown digit.
Atoms and ions can also be ionized by collisions with an electron. The cross section for thisimpact ionizationhas a maximum shortly after the ionization energy, and decreases asymptotically thereafter. Furthermore, it takes more time than tunnel ionization. Thus it is negligible for electrons with energies much larger than the ionization energyξ≫ ξion. Especially the high-energy drive beam is very ineffective in ionizing via impact ionization. However, impact ionization by moderate-energy plasma electrons can be of importance as a second-order effect, e. g. from electrons that are transversely scattered outside the plasma by the driver of the wakefield. In addition, when operating with a gas mixture that includes HIT species, impact ionization of HIT species by plasma electrons must be considered as a source of dark current.
Part II
S TA R T- T O - E N D - S I M U L AT I O N S : F R O M H Y B R I D P L A S M A WA K E F I E L D A C C E L E R AT I O N T O
U N D U L AT O R R A D I AT I O N
1. production of a plasma wakefield drive beam in lwfaIn the first stage of the hybrid plasma wakefield accelerator, a high-power laser is utilized to accelerate a suitedPWFAdrive beam. In consideration of the available resources and methods, suitableLWFAsimulation parameters are investigated and implemented resulting in a long, fully three-dimensional simulation that will be shown and discussed. The optimal position for the extraction of the obtained electron beam is considered.
2. high-quality electron-bunch generation in lwfa-beam driven pwfa The preparation and injection of the obtained drive beam into the PWFA stage will be discussed and simulated. In the second stage of the hybrid plasma wakefield accelerator, the sustained excitation of the plasma wakefield without unintentionally injecting electrons will be dis-cussed. To show the influence of the different TH-laser parameters and to find the best witness-bunch, a parameter scan over the ionization laser pa-rameters is conducted and evaluated. Eventually, the acceleration of the high-quality witness bunch will be given and the optimal position to ex-tract this bunch from the plasma will be investigated.
3. generating high-power short-wavelength radiation in an undulator In the last stage of the simulation chain, the possibilities of the obtained high-quality bunch to drive aFEL will be considered and an according choice for the undulator design will be discussed. The capturing and matching of the bunch with a conventional beamline will be simulated and analyzed, followed by the simulation of the fullFELprocess.
4. conclusions and outlook In the last chapter of this work, the re-sults will be summarized, and important findings will be emphasized. Fi-nally, future improvements on the presented scheme will be discussed.
6
P R O D U C T I O N O F A P L A S M A WA K E F I E L D D R I V E B E A M I N LW FA
6.1 computational resources
LWFAsimulations are particularly computationally expensive, demanding big amounts of time and core hours. Therefore only one long LWFA simulation could be realized, which will be displayed and discussed in the following. The expenses of the simu-lations have been covered by the Jülich Supercomputing Centre (JSC), Norddeutsche Verbund für Hoch- und Höchstleistungsrechnen (HLRN), and the National Energy Research Scientific Computing Center (NERSC) that I hereby gratefully acknowledge.
The simulations ran on the Supercomputers Gottfried (named after Gottfried Willhelm Leibnitz) provided by HLRN, the Jülich Research on Exascale Cluster Architectures (JU-RECA) at JSC, and the supercomputer named in honor of Thomas Edison (EDISON) at NERSC.
Super computing performance is typically measured in floating point operations per second (flops), computed from the CPU frequency times the number of floating point operations per computing cycle times the number of cores.
The latest update from HLRN (HLRN III) runs a Cray XC30(Intel IvyBridge) and XC40 (Intel Haswell) system with2.7petaflops combined peak performance of85 248cores1. The JURECA installation comes with 2.2 petaflops per second peak performance on 45 216Intel Haswell cores2, and Edison, a Cray XC30with a peak performance of2.57 petaflops per second on133 824compute cores3.
The simulations are set up under the restrictions of available computational resources and methods and scenarios that would have been too expensive could therefore not be conducted. This effectively restrictsLWFA simulations to a minimum density beneath which, no simulations are possible without applying advanced computational methods.
For instance, using a frame of reference that is moving with a relativistic velocity along with the wakefield (boosted frame) can significantly reduce the computational load as the wavelength of the laser is stretched and therefore a larger cell size can be used.
However, this mixes up the order of time events within the simulation window, making a transformation of the gathered output back into the lab frame very cumbersome, which somewhat compensates the speedup gained during the simulation. This method has therefore not been utilized.
1 https://www.hlrn.de/home/view/NewsCenter/ArticleNov2013KonradAndGottfried
2 https://www.fz-juelich.de/ias/jsc/EN/Expertise/Supercomputers/JURECA/JURECA_node.html 3 https://www.nersc.gov/users/computational-systems/edison/
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