The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

PIC SIMULATIONS OF SASE FELs
Toshiyuki OZAKIHigh Energy Accelerator Research Organization (KEK), Accelerator Laboratory
Oho1-1, Tsukuba-shi, Ibaraki-ken, JapanAbstract
A one-dimensional PIC simulation of SASE FELs isdeveloped. In the code, the electron bunch is divided into
many cells with the ponderomotive wavelength. The
evolution of the radiation pulse is studied. The simulation
results are compared with steady-state FELs.

1 INTRODUCTION
The free electron laser (FEL) based on self-amplifiedspontaneous emission (SASE) is achieved in a single passof an electron bunch through a very long undulator. Theelectron beam amplifies exponentially an initially existingradiation field under the FEL resonance condition that theradiation wave must travel one radiation wavelength λfurther than the electron during the time that the electron
travels one wiggler period λW. Therefore the electron
slips a distance NWλ relative to the leading edge of theradiation pulse, where NW is the number of periods in theundulator. In SASE FELs, the slippage distance NWλ isnot negligibly small relative to the beam length. So theslippage between the radiation pulse and the electron
beam is important in a simulation of the SASE FEL.

2 BASIC EQUATIONS
The motion of the simulation electron (i=1,2,3…..n) is
expressed by the following first-order equations:∂φi (η ,τ )= pi (η , τ ),∂τ(1)∂pi (η ,τ )= −[ A(η ,τ ) exp(iφ (η , t )) + c.c.],∂τ

(2)where theintroduced:

following

dimensionlessvariables

areγi −γr;τ = 2 ρkW ct ;η = 2 ρkW (z − β Z ct );ργ r
ε0 2E ⋅ λΣ
12pi =4A =.
ρ Nγ r mc 2

The dynamical variables are the phase φ and therelative energy deviation p. φ is the conjugate variable top. τ and η are scaled coordinates. τ is the normalized
distance from the undulator entrance. η is the electron
coordinate with respect to the beam center. The radiation
field is represented by a complex amplitude A. Thequantity ρ is known as the FEL parameter.
The field equation under the slowly varying envelope
approximation can be written in complex form as follows:ª∂1 n∂ º()+vAη,τ=¦ exp(− iφl (η ,τ )),« ∂τ∂η »¼
n l =1¬(3)where the sum represent the contribution from allsimulation particles within a ponderomotive wavelength.

3 PIC CODE
In order to study the radiation profile, we divide theelectron bunch into many cells with dimensions given by
the ponderomotive wavelength. We use the PIC(particle-in-cell) simulation method.
We numerically solve the FEL equations (1) and (2) ineach cell and obtain the generated field in the cell byusing the following differential form:§1 n·dA[k ] = ¨ ¦ exp (− iφl (η ,τ ))¸dτ .
© n l =1¹(3a)The velocity slip between the radiation field and theelectron can be realized by using the following first-orderwave equation to represent the field transfer into the nextcell:

A[k ]New = A[k ] +14πρ

( A[k + 1] − A[k ]).

(3b)

Under the new field, the particles are accelerated ordecelerated. The field acting on the simulation particles
can be obtained to interpolate the field on the neighboring
grid:

A = A[k ] ± ( A[k # 1] − A[k ])∆ϕ iπ.

The movement of the particle generates a new field.
Such iterations are continued to the end of the undulator.

The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

4 INITIAL CONDITION

φi (η ) =0.0015

AIn SASE FELs, the initial power source is the shotnoise in the beam. The startup from shot noise is modeledby adding random fluctuations to the longitudinalcoordinates of the simulated particles. We place thesimulated particles near regularly-spaced positon:

0.0020.001

2πi+ xi .
n0.0005

The fluctuation xi has a normal distribution with width00∆:

50100150200250300200250300200250300

200

250300

Fig.1(a)P( xi ) =§ x2 ·1exp¨¨ − i 2 ¸¸.
2π ∆© 2∆ ¹

0.006

0.005

We choose n=50 and ∆=0.022 according to ref. (1).
A0.002

0.001

0050

100150

Fig.1(b)0.025

0.020.015

AThe FEL equations (1)~(3) are solved numericallyunder the assumption that the initial beam has no energy
spread. The shape of the initial beam bunch is taken to beparabolic. The charge of the simulated particle has an
assigned value which reflects the parabolic profile of thebeam. The number of particles in each cell is fixed fornumerical stability.
In our simulations, the following values are assumed:
the total number of cells is 300, and the particles areloaded in 200 successive cells. In order to savecomputation time, the number of the particles in each cellis 50. The FEL fundamental parameter ρ equals 0.0239.
As described above, every dτ step, the field equations
are solved without the slippage term in each bucket, andthen the field is transferred forward.
Figs.1(a)~(d) respectively show the normalized field A
at the normalized distance τ=0.1,1,5,12 from theundulator entrance. The spikes seeded by the noise
non-uniformity are distributed throughout the radiation
pulse around the entrance of the undulator. The envelopeof the generated field is approximately parabolic.
Gradually, sub-pulse structures appear as the field grow.
As can be seen, the generated pulse is composed of anumber of spikes which are spaced at a distance related tothe slippage length. The field saturates around τ=12.

0.003

0.01

0.005

0050

100150

Fig.1(c)1.51A5 SIMULATIONS

0.004

0.50050

100150

Fig.1(d)The 14th Symposium on Accelerator Science and Technology, Tsukuba, Japan, November 2003

6 COMPARISON WITH STEADY STATEFELs
Fig.2 shows the result of the steady state model FEL
simulation code with the initial field A=0.0005. The field
begins to grow slowly with distance, grows exponentiallyand saturates.
In the SASE FEL, the development of the peak value ofthe generated radiation is plotted by the squares on theFig.2. At the beginning of the undulator, the field grows
quickly with distance. In the exponential regime, anexcellent agreement is obtained between the steady-statemodel and the pulsed SASE model.

101

A

0.1

0.01

0.001

0.0001
02

4

6

8

10

12tauFig. 2

REFERENCES(1) R.Graham and S.Isermann: Phys. Rev. A 43(1991)
3982(2) Zhirong Huang and Kwang-Je Kim: Nucl. Instrum. &Methods A445(2000) 105