4.2. THE NONLINEAR REGIME OF PWFA
CHAPTER 4. INSTABILITIES AND NONLINEAR PWFA
Eq. (2.29). Plasma electrons in nonlinear PWFA can reach relativistic ve-locities. These high energy plasma electrons oscillate at different frequencies than the plasma frequency due to the relativistic mass increase [compare mass dependence in Eq. (2.11)]. This first leads to the change of the wake-field shape from sinusoidal to sawtooth-like and second to a dependence of the length of the plasma bubble on the amplitude of the wakefield and thus the driver density (driver current in the strong blowout). Another feature of the bubble regime is the so-called wave-breaking and trapping of plasma electrons in the plasma wake [125,126]. The acceleration of plasma electrons, which reach velocities close to the phase velocity of the wake, allows to form bunches from plasma electrons and production of short, high-brightness elec-tron beams [127] but can also lead to dark current [128]. In PWFA this is mainly an issue in cases of strong longitudinal plasma density inhomo-geneities due to the wake phase velocity, i.e. the driver velocity, being close to c.
Various analytical descriptions of the nonlinear regime have been presented [121,129,130] whereas many can only be solved numerically for arbitrary driver bunch shapes. A simple approach is to assume the plasma to be a multi-mode medium, with a frequency spectrum depending on the density perturbation nb/ne [131,132]:
E(ζ) = ne
e ǫ0
Z ζ 0
X∞
m=1
mm 2m−1m!
nb
ne
m
cos(mkpζ′)dζ′ . (4.4) From this point of view it is also evident, that the fundamental theorem of beamloading, derived in Sec. 3.1 under the assumption of a single-mode medium, does not apply to the nonlinear regime interaction. Hence, TRs above 2 are possible for symmetrical bunches in the nonlinear regime [54,121]
but have not been measured to date [133].
The blowout regime has been shown to provide very high acceleration gra-dients [41,108,134] and is considered the favourable regime for a PWFA for electrons, due to the linear focusing fields and transverse homogeneous ac-celeration.
If ˜Q . 1 but nb > ne the interaction is set in the quasi-nonlinear regime [80]. In this regime favourable characteristics of the bubble-regime (e.g. linear focusing) are maintained, while the fields behind the driver bunch do not exhibit a strong nonlinearity or wave-breaking. Therefore, resonant excitation of wakefields is possible, which is not the case in the blowout regime. Operation in the quasi-nonlinear regime was hence proposed to solve
4.2. THE NONLINEAR REGIME OF PWFA certain issues of bunch train driver scenarios for high transformer ratio PWFA discussed in Sec. 3.2 [80].
4.2.2 Beam transport in the nonlinear regime
Figure 4.3: Numerical solution of on-axis wakefields in the nonlinear regime for a HTR capable bunch. Particles are moving to the left. The small dashed line shows the particle density, the long dashed line the transverse fields and the solid line the longitudinal fields [120]
Figure 4.3 shows a calculation of wakefields in the nonlinear regime of PWFA. The maximum beam density is equal to the plasma density and the length of the driver beam is ∼ λp. Transverse fields Wr show that the full bunch is in the focusing phase of the wakefields and in the second half of the bunch the transverse fields are constant. This is different to the linear regime [compare Fig.3.2(b)] and prevents variations in the wakefield strength which can lead to instability (see Sec. 4.1.3). Therefore the bunch can be transported through the plasma stably even though its length exceeds the length of the focusing phase of the plasma wake, which is λp/2.
The linear character of the transverse fields allows to preserve the beam quality of a witness bunch, as no nonlinearities are introduced to its phase space. As already mentioned, the length of the wake can depend on the driver density. This can lead to phase variation at the witness beam position when e.g. betatron oscillations within the driver cause driver density variations along the plasma. As the maximum accelerating fields are at the back of the wake, close to the strongly defocusing phase of the transverse fields, this can quickly lead to loss of the witness bunch [135].
Chapter 5
Numerical simulations
Previous chapters have discussed analytical and phenomenological descrip-tions of classical accelerators and PWFA. As the complex particle dynamics in the longitudinal and transverse fields in accelerators can only be described analytically for certain ideal cases, usually numerical simulations are em-ployed. These allow to accurately calculate the dynamics of realistic field and particle distributions. Every simulation code is based on an idealised physical model and makes certain assumptions to solve the equations asso-ciated with this model. To achieve reliable results, choosing the right codes, i.e. not violating the assumptions the codes are based upon with the physics of the processes under study, is essential.
In classical accelerator physics various simulation codes have been devel-oped. Among the most commonly used are MAD-X [136], elegant [137] and ASTRA [138]. While the former ones are not considering collective, i.e. inter-particle effects like space charge forces between bunch inter-particles, the latter also models these effects and is therefore most suitable for the rather low beam energies and high bunch charge densities encountered in the present work.
An introduction to ASTRA will be given in the next section.
For modeling PWFA, a variety of simulation codes has evolved through-out the last years. Widely used codes are Warp [139], OSIRIS [140,141], LCODE [142,143], QuickPIC [144] and HiPACE [145]. Due to its fast per-formance and efficient parallelisation the latter was used for the PWFA sim-ulations conducted during the course of the presented studies. A brief in-troduction to the theoretical background of HiPACE will therefore also be provided in one of the following sections.
5.1. PARTICLE TRACKING WITH ASTRA
5.1 Particle tracking with ASTRA
A space charge tracking algorithm (ASTRA) [138] is a code developed for modeling low energy beam transport in photo-injector linacs (see also Ch.6).
It is a so-called particle tracking code, i.e. it numerically solves the differen-tial equations of motion for individual particles for given external transverse and longitudinal magnetic and electric field distributions. To reduce the amount of necessary numerical calculations beam particles are summarised in so-called macroparticles. Each macroparticle represents e.g. 103–105 elec-trons and their use is justified by the assumption that particles which are close to each other will experience similar forces and therefore have similar trajectories in the phase space.
In addition to external fields ASTRA also involves routines to calculate the space charge forces between the particles in a bunch. This is done by sub-division of the volume occupied by beam particles into a grid of sub-cells.
The grid can either be 2D or 3D, depending on whether only rotationally symmetric forces are expected or not. Fields induced by the space charge of the (macro-)particles are calculated on this grid for the sum of all particles in each grid cell: the system is Lorentz-transformed into the average rest frame of the bunch, fields on the grid are calculated by solving the Poisson equation and the system is transformed back to the laboratory frame. The resulting map of the space charge fields is then included into the equations of motion of the macroparticles.
As input for the simulation an initial particle distribution, field maps and parameters for all accelerating cavities and magnets, parameters of the calcu-lation (e.g. space charge grid parameters) and the parameters for data output are needed. In the present work, usually 5·105 macroparticles were used.
2D space charge calculation was used until after the last accelerating cavity, where the first quadrupole magnets are located (see also Ch. 6). As these have non-rotationally symmetric magnetic fields, 3D space charge calculation was used for modeling the rest of the linac. The grid size for space charge calculation was defined in convergence studies as 64×16×16 cells (z-x-y).