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).
CHAPTER 5. NUMERICAL SIMULATIONS
accelerators, plasma wakefield accelerators are much more difficult to ac-cess with diagnostics experimentally. To model beam-plasma interaction,
Figure 5.1: Illustration of quasi-static PIC time step: a slice of plasma (blue/yellow) is in-troduced to the beam (red) (a)].
Fields (black lines) are calculated and plasma particles evolved (b).
Plasma slices are moved (c) and process is repeated until the full field map is acquired (d). Finally the beam particles are pushed (green) (e) and the next beam time step can be calculated.
usually the PIC method is applied.
In this approach, the simulated sys-tem is divided into a 3D grid of sub-cells. The differential equations of motion of the simulated particles are solved for fields which are cal-culated on this grid. This is done for the beam macroparticles as well as for the plasma macroparticles. In plasma wakefield applications usu-ally no external fields are included in the calculation, as they are not present or negligible compared to the field strengths in the plasma wake.
In the calculation loop of one time step of the simulation first the cur-rent densities of the macroparticles are distributed between the clos-est grid points. After the cur-rent density on the grid is calcu-lated, conventional (also called fully-explicit) PIC codes numerically solve the time-dependent Maxwell equa-tions [Eqs. (2.3) and (2.4)]. Mag-netic and electric fields are calcu-lated on the faces and edges of the grid cells, respectively, for full and half integer time steps [146]. After-wards the fields at the particle po-sitions are interpolated between the field calculation points and the equa-tions of motion can be solved. The numerical stability of this method depends on the time step size via the Courant-Friedrich-Lewy
condi-5.2. HIPACE PWFA SIMULATION tion [146,147]: ∆t = CCF L
√∆x2+∆y2+∆z2
c where CCLF < 1, i.e. no particle may cross more than one grid cell diagonally. By this condition the maximum time step size is fundamentally linked to the grid cell size. In simulations of PWFA the features that have to be resolved (e.g. wakefield structure) can be very small, while the length of a PWFA can span up to meter sizes. Due to the maximum time step limitation this leads to PIC simulations which consume large amounts of computational resources.
To resolve this issue, the so-called quasi-static approach was introduced to PWFA simulations [148,149]. When the driver beam does not evolve sig-nificantly during the transit of a plasma electron, one can decouple the plasma and beam time scales. This is equal to the betatron wavelength λβ of the beam being much larger than the plasma wavelength λp, e.g.
λβ = √2γbλp [150] in the linear focusing of nonlinear PWFA with usually γb ≫ 10. This typically allows orders of magnitude longer simulation time steps and corresponding reduction of calculation time. Instead of evolving the fields from time step to time step by explicitly solving Maxwell’s equa-tions, the field maps are constructed self-consistently in each time step by moving a slice of plasma along the beam and evolving the particles and cor-responding fields in this slice. The process is illustrated in Fig. 5.1. On the one hand this means that the entry into a plasma and plasma density inho-mogeneities on length scales similar to the bunch length cannot be simulated (as in every time step the beam is initialised fully surrounded by plasma con-structed from one plasma slice). On the other hand it e.g. allows to import beam particle distributions from other PIC codes for further simulation.
HiPACE (Highly efficient Plasma ACelerator Emulation) is based on this quasi-static approach. It also employs fast Poisson-solvers for field calcu-lation to reduce simucalcu-lation time even further. Fields at the boundaries of the simulation box are assumed to be zero, i.e. perfectly conducting walls.
The size of the time steps can automatically be adjusted depending on the beam particle energies and maximum plasma density perturbation. Typical simulations in this work were conducted with a z-x-y grid of 512×256×256 cells, which was determined by checking the convergence of the simulation results with increasing grid cell number for the simulation boxes of around 11×1.5×1.5 mm3 size (depending on exact beam parameters and plasma den-sities). Such a simulation usually finished within a few hours using 32 or 64 parallel processors. This rather short run time even allowed for limited parameter scans to conduct error studies and to reconstruct measurement results (see also Sec. 8.4 and Ch. 10).
Part II
High transformer ratio plasma
wakefield acceleration at PITZ
A
fter introducing the theoretical basis of high transformer ratio PWFA in the first part, this second part will now give details about the experimental environment at PITZ and present the results that have been achieved in the course of this work.It is structured as follows:
• In Ch. 6 the PITZ facility, where experiments have been conducted, will be introduced.
• Chapter 7 describes the work that has been done on production of HTR-capable electron bunches via photocathode laser pulse shaping and prospects of the method.
• Gas discharge plasma physics and the gas discharge plasma cell, that was developed for the HTR PWFA experiments, are introduced in Ch. 8.
• The beam dynamics simulations, that have been carried out to identify favourable electron beam and plasma parameters for experiment are described in Ch. 9.
• Chapter10finally concludes with the description of the first observation of and further experimental results on HTR PWFA which have been achieved in the course of this work.
Chapter 6
The PITZ facility
The experimental work for the demonstration of high transformer ratio PWFA has been carried out at thePhoto-InjectorTest facility at DESY inZeuthen (PITZ) [151,152]. In the following sections, the general principle of photo-injectors will be introduced before a detailed description of the PITZ electron source and the beamline layout will be given.
6.1 Photoinjection radio frequency accelera-tors
Various applications of high energy electron beams like X-ray free-electron lasers and PWFA demand for high beam brightness. The brightness is defined as
B= I ǫn,xǫn,yǫn,z
, (6.1)
where I is the bunch current and ǫn,x/y/z are the normalised emittances of the particle bunch in all three dimensions [153]. To produce bunches with B ∼ 100(mm mrad)A 2 usually electron sources based on the extraction of elec-trons from a cathode via the photoelectric effect [154] are employed. The cathode is illuminated with a high intensity (∼ 10kWmm2 ), short (∼ps), ultravi-olet (UV) laser pulse. Due to their space charge, the extracted electrons repel each other, which leads to divergence and blow-up of the extracted bunch both transversely and longitudinally. To reduce the bunch quality degrada-tion due to the space charge forces, the photocathode is placed inside an RF accelerating cavity, also called electron gun. Bunch electrons are accelerated immediately after extraction by the cavity fields. The quicker the electrons