A FOUR-DIMENSIONAL VLASOV SOLVER FOR MICROBUNCHING INSTABILITY INTHE INJECTION SYSTEM FOR X-RAY FELS Page: 3 of 4
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Figure 1: Emittance comparison between the Vlasov solver
and the beam transport matrix.
If we consider, for example, the longitudinal phase space at
the entrance of the compressor, as shown in Fig. (2), we can
see that there is a strong correlation between the variables
z - 6. The same happens in the transverse plane. Using an
uniform orthogonal grid, a large fraction of the mesh points
(> 90%) corresponds to empty regions in the phase space,
and therefore represents a waste of memory and compu-
tational resources. Studying the MBI in 2D (longitudinal
phase space only) we have found that grids of the order
of 1000x500 nodes were needed in order to follow the in-
stability at short wavelenghts, say 10 pm and below. In
4D a very coarse grid with a hundred points per direction
correspond to 1004 mesh nodes, and requires typically 2
GB of memory space at runtime. Preliminary investiga-
tions show that high resolution in the longitudinal phase
space z - 6 calls for a comparable resolution in the trans-
verse space x - 0, because the two spaces are strongly cou-
pled by the dynamics. It is clear that increasing by a factor
10 the grid resolution is not feasible, since it would cor-
respond to a factor of 104 increase in the computational
resources. Presently we are considering two options for
addressing this problem: the distribution support could be
remapped to a uniform orthogonal space using a coordi-
nate transformation similar to that presented for a 2D solver
in , or the phase space could be divided into parallel
slices which closely follow the distribution support. The
first option would require a more complex redefinition of
the Hamiltonian operator, and could introduce numerical
noise due to the remapping of the longitudinal coordinate.
Therefore we are currently implementing the second option
which requires just a careful book keeping of grid, without
any changes to the computational kernel.
On the basis of the considerations just presented, it is
worthwhile to note that Vlasov solvers for particle acceler-
ator problems require large computational resource and, at
least with present computers, it seems very hard to address
the full 6D problem at a resolution high enough to resolve
the CSR dynamics at very short wavelengths.
We have discussed an accurate analysis of the micro-
bunching instability. Macroparticle codes suffer by intrin-
sic numerical noise problems, while the possible limita-
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Figure 2: Longitudinal phase space at the entrance of the
compressor. Linear color scheme.
tions of the V-S method is due to the grid number of points
which may become prohibitively large with many useless
points, since the phase space is strongly correlated. The
essence of the instability itself is such a correlation, which
gives rise to bunching and hence to coherent emission. The
MBI dynamics share strong analogies with the FEL dynam-
ics itself, whose final effect on the beam is that of creating a
large (uncorrelated) energy spread. The interplay between
FEL and saw-tooth type instabilities (a different flavour of
MBI) has shown that they are competing effects , there-
fore MBI dampers based on FEL heater devices have been
proposed . A quantitative understanding of the MBI will
therefore provide elements to design such a tool to inhibit
the growth of the instability.
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Beam tianspri- -
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Venturini, Marco; Migliorati, Mauro; Schiavi, Angelo; Dattoli, Giuseppe & Venturini, Marco. A FOUR-DIMENSIONAL VLASOV SOLVER FOR MICROBUNCHING INSTABILITY INTHE INJECTION SYSTEM FOR X-RAY FELS, article, June 23, 2008; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc894193/m1/3/: accessed January 20, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.