Emittance Growth Study of an Electron Beam in a Double-Alpha Magnet Compressor Used in an Inverse Compton Scattering X-Ray Source

An Inverse Compton Scattering (ICS) X-ray source is under development at the ELSA (Electrons Laser X-Sources and Applications) electron RF linac of CEA DAM (Commissariat à l’Energie Atomique et aux Energies Alternatives, Direction des Applications Militaires). X-rays are emitted by the interaction of 30 MeV electron bunches with Nd:YAG laser pulses. The electron bunches duration is 30 ps rms before compression in two alpha magnets. In such a system, electron trajectories are curved with a short radius, resulting in a noticeable degradation of the beam emittance. In the specific case of strongly curved trajectories, the straight trajectory and beam shape assumptions used for space charge calculation in most simulation codes are questionable. Two different approaches to the simulation of electron beam dynamics within the alpha magnets are compared. A specific method to deal with changes from the reference particle frame to the laboratory reference frame, which does not imply any trajectory and beam shape assumptions, is proposed. Calculation results are presented. Along with an important emittance growth, they show that more physical effects can be taken into account in the latter simulation.


Introduction
In the Inverse Compton Scattering (ICS) X-ray source of the ELSA accelerator of CEA DAM [1], photons emitted by a Nd:YAG laser interact with electron bunches, compressed in a magnetic scheme involving two vertical alpha magnets (Fig. 1).The energy chirp along the bunch is achieved by shifting its phase in the upstream 433 MHz accelerating cavities.This vertical setup was implemented more than twenty years ago, primarily chosen for its compactness [2].Duration as short as 7 ps rms have been obtained in the case of low charge per bunch (typ.100 pC), with rms-normalized transverse emittance better than 5 mm.mrad.In order to increase the X-ray flux, it is now necessary to increase the charge up to 3 nC while preserving the beam quality.
Most accelerator simulation codes -such as Trace-Win/PARTRAN [3] -are based in the frame of the reference particle.For PIC simulation it is common to use the laboratory frame, as in CST Studio Suite [4].In this frame, the specific regime of short radius trajectory curvature is taken into account more accurately, since no assumptions about the beam shape and trajectory are made.ELSA alpha magnet compressor has thus been modelled in the PIC module of CST Particle Studio (PS).
Great attention must be paid to mesh sizing, code convergence and to the difficulty of dealing with different reference frames, before being able to compare numerical results.

Beam transport through double alpha magnet compressor
The alpha-dipole field structure is a 1 T/m constant gradient field, giving a 0.4 m depth of penetration of 17 MeV electron bunches inside the alpha-dipole.In this case the radius of curvature at its shortest is 0.1 m.The field is quadrupolar: an alpha-dipole can be thought of as a half quadrupole used off-axis.
For an injection angle of 40.71°, the entry and exit points coincide: the dipole is perfectly achromatic and the trajectories for different energies are homothetic, looking like the greek letter α, hence the name.Table 1 summarizes the beam parameters at the entrance of the first alpha magnet for 3 nC.For lower charges, parameters values were kept identical to ease comparison between simulation results.287 Field maps of our alpha magnet have been generated by the magnetostatic module of CST Electromagnetic Simulation Solvers (Fig. 2).However, the use of an analytical model was preferred in this study, removing the influence of magnetic field imperfections on beam dynamics, in order to focus on the space charge effect.

Setting simulations parameters in CST Studio
CST Particle Studio is an electromagnetic simulation software with several modules.The PIC (Particle In Cell) solver module allows calculation of charged particle bunch transport in an electromagnetic structure in a stationary mesh in the laboratory frame, while most of accelerator calculation codes rely on calculation in the moving frame of the bunch reference particle.The mesh size has to be small enough for correct space charge effect calculation, and the number of particles must be chosen accordingly to avoid numerical artefacts.If it is insufficient, computation may result in the apparition of non-physical artefacts in the computed fields as illustrated in Fig. 3.For a  rms-size beam profile, a 5 mesh size was chosen.Figure 4 shows that simulation convergence is obtained when exceeding 1M particles.Since the volume of the alpha magnet is large, and the number of meshcells cannot exceed 350 million with the current version of CST PS, the beam rms-size was limited to 1 mm at the focalisation point, and the rms-emittance to 5 mm.mrad at the entrance.A mesh size of 0.2 mm was used, which remains above 5 all along the beam trajectory.
Further studies with the upcoming version of CST PS will enable up to 2 billion mesh cells for smaller size and emittance.With such numbers of meshcells, calculations are done using Topaze supercalculator at CCRT (Computing Center for Research and Technology [5]).

Comparisons between CST PS and TraceWin/PARTRAN
Comparing results of codes making calculations in different frames is not straightforward.One difficulty lies in the definition of the reference particle.A striking example is shown in the exaggerated case of a 600 ps bunch in ELSA alpha magnet (Fig. 5).Because of the curved trajectory, the barycentre lies outside of the bunch so the reference trajectory cannot be computed using the successive mean positions.Besides, it shows that field computation and space charge effects cannot be taken into account properly in a simulation done in the bunch reference frame.The other pitfall lies in the fact that particle coordinates are calculated at fixed time steps in CST PS, while they are calculated at fixed curvilinear abscissa steps of the reference particle in routine codes like PARTRAN.
To solve these problems, a Python code was developed to transform results from the laboratory frame to the reference particle frame (Fig. 6), that allows for proper comparisons between results obtained from CST PS and most of accelerator calculation codes like Trace-Win/PARTRAN, PARMELA [6], ASTRA [7].In our study, TraceWin/PARTRAN and its ALPHA_MAGNET command was used to benchmark our Python code.
The first task consists in computing the trajectory of the reference particle.As stated before, calculating the barycentre of the bunch leads to erroneous results.Instead, the trajectory of each particle is computed by interpolating discrete calculation results of CST PS, with a much higher resolution.At a given step of the calculation, the barycentre is instead projected on these trajectories, allowing to find the actual bunch center.Then each particle trajectory is projected onto a plane perpendicular to the reference trajectory, enabling comparison of emittance calculated in CST PS and TraceWin/PARTRAN.The results obtained with CST PS and Trace-Win/PARTRAN are in close agreement all along the beam trajectory inside the alpha magnet, especially at low charge per bunch, which validates the process described above (Fig. 7).The energy dispersion and the geometrical effects inside the alpha magnet temporarily boost the transverse emittance in the X direction.This increase gets perfectly compensated along the second half of the alpha magnet, except for the part due to space charge.
With high charge per bunch, curves of Fig. 7 tend to separate, because space charge effects become prominent.In this case the different calculation methods and reference frames impact the simulation: both methods reveal an important net-growth of the X-axis emittance at high charge per bunch, as can be seen after the compensation phase, right at the output of the alpha magnet in Fig. 8.This effect is noticeably more important in calculations in the laboratory frame, demonstrating the interest of this method that takes more accurately into account the effects of short radius trajectory curvature.

Conclusion
The emittance growth within ELSA compressor primarily depends on the transport of electron bunches in its two alpha magnets.The behaviour of electron bunches in the first alpha magnet has been investigated with two different codes, showing in both cases an important growth of transverse emittance.CST PS calculations are done in the laboratory frame and TraceWin/PARTRAN in the reference particle frame.A numerical method was developed to deal with pitfalls related to change of frame between the two approaches allowing rightful comparison.Though differences can generally be overlooked in most accelerator setups, our numerical evaluation of emittance demonstrates that simulating beam transport directly in the laboratory frame allows taking into account effects related with short radius trajectory curvature, which is of paramount importance in the ELSA Inverse Compton Scattering (ICS) setup with high charge per bunch.This method will help us to understand emittance degradation in the case of the whole compressor with both alpha magnets and to optimize bean transport, or to design, if absolutely needed, a new beam line on a different spot of ELSA facility for its ICS.

Figure 1 .
Figure 1.Side view of the compressor with 2 alpha dipoles, 3 quadrupoles and diagnostics.The inset shows the trajectory of electrons in the first alpha magnet.

Figure 2 .
Figure 2. (zy) plane cut view of the alpha magnet, simulation with CST Magnetostatic module showing constant gradient along z.

Figure 3 .
Figure 3. Simulation with CST PS.Left: electric field map generated by the electron bunch at the focus point inside the alpha magnet, with a sufficient number of particles.Right: electric field calculated in a drift with too few particles.

Figure 4 .
Figure 4. Convergence study of CST PS calculations in a 30 cm drift.Ratio between transverse emittance at the input and at the output of the drift (theoretical value = 1) as a function of the number of meshcells per σ.

Figure 5 .
Figure 5. Exaggerated case of a 600-ps bunch half-way through the alpha magnet.Left: in the laboratory frame, the beam is curved and the mean position lies outside of the bunch.Right: in the reference particle frame.

Figure 6 .
Figure 6.Left: Particles positions at successive time steps in CST PS.Middle: Interpolation of trajectories, the dashed line corresponds to the intersection of the interpolated trajectories with the plane of the reference particle.Right: Particles positions in the reference particle frame at given values of curvilinear abscissa s.

Figure 7 .
Figure 7. Comparisons of phase-space for different charges.Simulations with TraceWin/PARTRAN (TW) and reconstruction along curvilinear abscissa s for CST PS.

Figure 8 .
Figure 8. Comparisons of X, Y and Z emittance along curvilinear abscissa s for TraceWin/PARTRAN and CST PS.For space charge calculation in PARTRAN, the PIC-NIC [8] (3D) routine was used with 10 M particles and 18x18 cells on +/-6 σ.

Table 1 . Beam Parameters of Electron Bunches at the Entrance of the Alpha Magnet for 3 nC.
RMS Energy spread [keV]