# A FOUR-DIMENSIONAL VLASOV SOLVER FOR MICROBUNCHING INSTABILITY INTHE INJECTION SYSTEM FOR X-RAY FELS Page: 2 of 4

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equation

O = HT

aswith T1o, =To

(1)

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 [5]. If we include the

wake fields effects, H becomes an integral operator and

eq. (2) becomes non linear [3]. 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 [6]

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 [7].

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

kind [8]

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 [9].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-

niqueEHs _ AsAo AsF(z) }-g2AsAo

(6)

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 indeede2 Z sAo' =F (R X o)

(7)

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.

NUMERICAL RESULTS

The simulation code TEO [3] 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 [10] 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.