A FOUR-DIMENSIONAL VLASOV SOLVER FOR MICROBUNCHING INSTABILITY INTHE INJECTION SYSTEM FOR X-RAY FELS Page: 2 of 4
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O = HT
with T1o, =To
where s refers to the propagation coordinate.
The formal solution of eq. (1) can be written, for a small
propagation interval As and if H is not explicitly a time
dependent operator, as
I' exp (HAs) To (2)
representing the Vlasov evolution operator. Actually, in
writing eq. (2), we are neglecting any contribution due to
time ordering corrections, that arises whenever the operator
H is explicitly time dependent and does not commute with
itself at different times. With these assumptions we neglect
third order terms in the integration step As, the symplectic-
ity is however automatically preserved by the exponential
form of the evolution operator.
The operator H encloses the physical properties of the
propagation problem and contains the beam dynamics. In
absence of any collective effects generated by wake fields,
it is a differential operator, describing the beam propaga-
tion through magnetic lens systems . If we include the
wake fields effects, H becomes an integral operator and
eq. (2) becomes non linear . For example, in case of
beam transport through a bending magnet of radius R and
length L, in a 4D beam dynamics, we can write
Hr =- +(kix - -+
R tz 0x k1 R &
& e2N f
m6 E- W(z' - z)p(z')dz' (3)
with x and 0 the transverse coordinates, z and 3 the longitu-
dinal ones, k, the magnet focusing strength, N the number
of particles in the bunch, p(z) the projection of the density
distribution T on the longitudinal axis, and W (z) the lon-
gitudinal wake function, providing, for the problem we are
considering, the CSR wake field that we write as 
1 2 &
W(z) 47reo (3R2)1/3 z/3 3 z > 0 (4)
Accordingly, we neglect the screening effect of conducting
walls and we consider only the steady state radiation from
an ultra-relativistic particle in a long magnet .
Equations analogous to eq. (3) can be written for the
quadrupoles, drifts and RF cavities, which are the beam
transport devices used for our study of the MBI. Further-
more, the longitudinal space charge wake field, of the
1+ 2log - ' (5)
with 3' the derivative of the Dirac delta function, has been
used in the drift sections by considering a transversally uni-
form bunch density with circular cross section a and a cir-
cular beam pipe of radius b. This wake field is supposed to
be the main cause leading to the MBI .
Once the expression of the operator H is known for each
device of the bunch compressor, we can write the explicit
solution of eq. (2). In order to do that, we observe that any
expression of H of the kind of eq. (3) can be always written
as the sum of the beam transport matrix without collective
effects Ao, and the non linear term due to the wake field,
that is H Ao + F(z)&/&8.
With this assumption, the operator H consists of two
parts that we decouple by using the operator splitting tech-
EHs _ AsAo AsF(z) }-g2AsAo
Such a decoupling realizes a kind of interaction picture in
which the collective effects are separated from the ordinary
transport part whose action on the beam distribution func-
tion is known exactly. We have indeed
e2 Z sAo' =F (R X o)
with R(s) the transport matrix of the element and Xo the
initial coordinates vector. The action of the non linear part
can be evaluated in an analogous way and it is readily un-
derstood since the exponential operator accounting for its
effect is just a shift operator in the coordinate 3, provid-
ing a translation of the same coordinate by AsF(z) in the
distribution function, i. e.
eos/F(z)2 4T (z, s, x, 0) = (z, + AsF(z), x, 0) (8)
This is the way we transport the distribution function
through the compressor devices in our V-S.
The simulation code TEO  was upgraded to the 4D do-
main. The initial e-beam distribution 4 was sampled on a
uniform dense Cartesian grid (z,, 6j, Xk, 01). The beam was
then advanced by a discrete step As along the beam line us-
ing the operator method described in the previous section.
For each grid point the advanced distribution 4" was ob-
tained by numerical interpolation of the distribution 4' at
the beginning of the step using the method of local char-
acteristics. Namely, ''(z,, 6j, xk, 01) - (r,, p k, 61),
where the origin of the characteristic zi, j, k, l was ob-
tained by applying the exponential operator as in (8). La-
grange polynomials of the 5th order where used for the 4D
off-grid interpolation of the distribution.
The first tests of the Vlasov solver have been performed
in order to verify if the 4D dynamics of the simple trans-
port without wake fields is well represented. In Fig. (1) we
have used a magnetic bunch compressor with parameters
close to those of SPARX BC2  and have compared the
transverse emittance with that obtained with the transport
matrix method. The agreement is excellent. Also with the
longitudinal dynamics there is total agreement. One prob-
lem arising in the use of a Vlasov solver is related to the
high resolution of the grid necessary to evidence the MBI.
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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/m1/2/: accessed January 23, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.