4.4 characteristic fel parameters 79
Slippage
Electrons perform a sinusoidal motion in the undulator (see equation (145)), and there-fore travel a longer distance compared to the straight path of the emitted photons.
Consequently, electrons have a smaller average velocity in the forward direction and slip back by one resonant wavelength with respect to the photons in every undulator period, as dictated by the resonance condition (153). The instantaneously accumulated phase slippage
Ls=Nuλr (173)
therefore increases with longer resonant wavelength, λr, and number of undulator periods,Nu = Lsat/λu ≈1/ρFEL. Within an undulator of lengthLsat = λu/ρFEL, the total accumulated slippage is simplyLs,max=λr/ρFEL. For the generated radiation to not move out of the electron bunch and thus to overlap within the undulator, the total accumulated slippage should not be larger than the bunch length
Ls,max= λu
ρFEL 6σz. (174)
This is an important condition, as electron bunches from plasma accelerators are typ-ically very short (few microns). If this condition is not fulfilled, the saturation of the lasing process is substantially delayed and the generated light pulse is stretched. In the transverse direction, an analogous condition must be fulfilled [116]
ǫn6 γλr
4π , (175)
which guarantees transverse overlap by ensuring that the bunch divergence is smaller than the divergence of the emitted photons. This condition also shows the energy and wavelength possibilities for a given bunch, and limits the attainable minimum wavelength through the emittance for a given energy.
Self-amplified spontaneous emission
The FEL process can be seen as an instability that needs an initial seed to start. This
Interestingly, spontaneous emission of radiation can be interpreted as emission that is stimulated by vacuum fluctuations [88].
seed can be provided by either a slightly pre-bunched electron distribution, or an external seed laser at the resonant wavelength. The other option is to let the process start from the incoherent shot noise of the initial undulator radiation. This process is called SASE[23] and requires a longer undulator and starting phase of the FEL. The radiation pulse in aSASE-FELdevelops phase correlation only within a short range, the cooperation length, that is determined by the phase slippage over one gain length
Lc= λr
√34πρFEL
. (176)
These regions develop temporal coherence and cause a spike in the spectrum, so a bunch of length σz will show approximatelyσz/(2πLc) spikes. This includes the pos-sibility to build a single spike SASE-FEL, where a very short bunch with σz ≈ 2πLc is
4.4 characteristic fel parameters 81 applied [190, 191]. Then, close attention must be paid to keep the phase slippage as short as the bunch length by using a short undulator and resonant wavelength.
The cooperation length, Lc, has an important role in the applicability of electron bunches with energy chirp in FELs. Because the coherence length in the SASE mode is≈2πLc, it is sufficient that the energy spread fulfills the condition (172) only within this range. This opens the possibility for electron bunches with an energy spread that in total violatesσγ/γ < ρFEL/2, to be used in aFELwhen the fractional (or slice) energy spread meets this condition. The length of the slices must therefore be about as long as the cooperation length of the considered undulator, and the energy chirp is limited by [204]
αc
σz,fwhm λr
ρ2FEL
≪1 (177)
withσz,fwhm = σz2√
2ln2theFWHMbunch length, ρFELthe Pierce parameter,λr the resonant wavelength of the undulator, and the slope of the energy chirpαc given by
∆γ γ =αc
ξ
σz,fwhm, (178)
whereξ=z−ctis zero in the center of the bunch.
5
C O M P U TAT I O N A L M E T H O D S
5.1 particle-in-cell algorithm
Physicists have largely profited from computational methods that are used to predict the development of physical systems. Especially in very complex systems like plasmas, this has led to large accomplishments, enabled by ever better computer systems. De-spite very large modern high-performance computing (HPC) machines, plasma wake-field acceleration is not—and does not need to be—modeled including all details of the system. Usually, a selection of relevant interactions and other simplifying assumptions are made that do not affect the physical processes of interest.
To simplify the computation of a macroscopic weakly coupled plasma, it is convenient to definemacro-particlesthat are composed of a large number of physical particles and thereby reduce the number of particles of the system. Another benefit of this technique is the use of finite size particles, instead of using point-like electrons that interact with the Coulomb potential that becomes infinite for zero distance. The macro-particles are designed to avoid this singularity by using an interaction that decreases again as soon as the macro-particles start to overlap and is the same as the Coulomb potential for distances larger than the particle radius. The thereby reduced potential energy in the system is compensated for by using less particles to reproduce the plasma parameter (40).
These macro-particles are used in thePICmethod [20,95], which allows the interaction of three-dimensional collisionless plasmas with relativistic particle beams and high-power lasers to be computed. The PIC method is widely applied in the computation of plasma-based accelerator research and is based on the Maxwell equations (2-5) in combination with the Vlassov equation (equation (48) with the right-hand side set to zero). To give a small error, the Maxwell equations are solved with the finite-difference time-domain (FDTD) method and advanced in time on a Yee-mesh [247]. In this mesh, the values of the magnetic field are defined at the center of the faces of the cells, whereas the values of the electric field and current are defined at the middle of the edges of the cells. In theFDTDmethod, the time is advanced in half steps, where the electric-field values are computed at half-integer time steps and the magnetic-field values are computed at integer time steps (leap-frog algorithm). Particles are advanced through free space [24].
The fields and the particles are alternately advanced within the PIC cycle. The charge and current densities are deposited onto the grid, followed by solving the Maxwell
83
equations to calculate the electric and magnetic field values at the respective grid points. Then, the fields are interpolated back onto the locations of the particles to calcu-late the Lorentz force and the particles are moved after the integration of the equations of motion using the leap-frog method.
Naturally, the spatial size of the cell of the grid must be chosen sufficiently small to resolve all important phenomena. To avoid numerical instability, the time step must fulfill theCourant-Friedrichs-Lewy condition [46,247]
c∆t <p
∆x2+∆y2+∆z2, (179)
which precludes propagating particles faster than one cell per time step. For thePWFA
simulations in this work, the time step c∆t = ∆z/2 is used to minimize the growth rate of numerical instabilities associated with a relativistic particle bunch [78]. Other restrictions can arise, e. g. when a high-power laser is used, a smaller time step is required that scales with the laser amplitude [11].
Further, the discretization of the electromagnetic field leads to a minimum wavelength that can be resolved with the chosen cell size. This alters the dispersion at small wave-lengths and leads to the artificial effect that particles may travel faster than the numer-ical speed of their own radiation. Hence, resonant interaction between the light and the particle can lead to the generation of an artificial source of radiation. This numeri-cal Cherenkovradiation was studied intensely [47,78,86, 135, 249] and can be reduced by a careful choice of the simulation parameters or alternative field solvers. However, the origin of this effect, the discrete nature of the electric field on the grid, cannot be avoided within thePIC method. The minimum wavelength that can be resolved by a grid with the resolution∆z isλmin = 2∆z, restricting the wave numbers tok < π/∆z.
In the applied standard Yee-mesh, electromagnetic waves propagate according to the dispersion relation [86]
1 c∆tsin
ω∆t 2
2
= 1
∆xsin
kx∆x 2
2
+ 1
∆ysin
ky∆y 2
2
+ 1
∆zsin
kz∆z 2
2
, (180)
whereas in vacuum the dispersion isω2/c2=k2x+k2y+k2z. This numerical dispersion error significantly slows down the speed of electromagnetic waves near the cut-off fre-quency. Note that the numerical dispersion is most accurate along the diagonal of the simulation cell and least accurate along the axes.
In this work, the high-performance multi-physics cross-platform computational simu-lation framework [169] (VSim)1 was used to compute full three-dimensional laser- and
1 Plasma acceleration package (PA), solely developed to perform large-scale simulations of laser-plasma and beam-plasma acceleration experiments by Tech-X
5.2 justification of applied approximations 85