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A
LICE is a


L
asing


I
nvestigation


C
o
-

d
E

Igor Zagorodnov

BDGM, DESY

16.10.06

1D and 3D mathematical models are described in [SSY, 1999]
and coincide with those used in the code FAST of the same author
s
2
2
0
(,,) (..)
2
i
z
z
H P z CP P Ue CC eE d
c


 
 
    

Equations of motion
correspond to effective Hamiltonian
Field Equations
are used in parabolic approximation
2
1
2 4
s
c i E j
c z




 
   
 

 
0
4 ( )
z z
E j j
t


 



(/)
i z c t
x y
E E iE e

 
 
with simplified
space charge model
Mathematical model
[SSY, 1999]
E.L.Saldin
,
E.A.Schneidmiller
,
M.Y.Yurkov
, The Physics 0f Free Electron
Lasers, Springer, 1999
1D and 3D mathematical models are described in [SSY, 1999]
and coincide with those used in the code FAST of the same author
s
2
2
0
(,,) (..)
2
i
z
z
H P z CP P Ue CC eE d
c


 
 
    

Equations of motion
correspond to effective Hamiltonian
Field Equations
are used in parabolic approximation
2
1
2 4
s
c i E j
c z




 
   
 

 
0
4 ( )
z z
E j j
t


 



(/)
i z c t
x y
E E iE e

 
 
with simplified
space charge model
Mathematical model
[SSY, 1999]
E.L.Saldin
,
E.A.Schneidmiller
,
M.Y.Yurkov
, The Physics 0f Free Electron
Lasers, Springer, 1999
Why write a code with the same mathematical model


There are a lot of codes for Maxwell’s equations (wakefields), why do
not write
one more

for the FEL equations?


To
study the theory

through numerical modeling


To implement
simpler and faster

numerical methods without loss of
accuracy


To have
consistent and matched 1D, 2D and 3D models

in the same
code


To have a thoroughly
tested code

with full control and possibility of
future development

Motivation

Numerical methods

Equations of motion

FAST

ALICE

Runge
-
Kutta method

Leap
-
Frog method

Field Equation

Non
-
local integral representation

(two fold singular integral with

special functions)

Finite
-
Difference Solver with

Perfectly Matched Layer

Why other methods?



Leap
-
Frog is
faster

than Runge
-
Kutta and „symplectic“(?)



Finite
-
Difference solver is
local
: uses only information from one previous
„slice“; it should be faster than non
-
local retarded integral representation
which uses all slices in the slippage length



Like to the integral representation the Perfectly Matched Layer
approximates the
„open boundary“ condition accurately


Computer realization and testing



The code was initially developed in Matlab and then rewritten in C/C++



the numerical results are compared with the analytical ones when possible
(propogation of different Laguerre
-
Gaussian azimuthal modes, analytical
results for linear regime in 1D and 3D theories)


The figures from chapters 2, 3, 6 of [SSY, 1999] are reproduced with the new
code

[SSY, 1999] E.L.Saldin, E.A.Schneidmiller, M.Y.Yurkov, The Physics 0f Free Electron
Lasers, Springer, 1999

Tests.
Space
charge
algorithm
(1D)
-2
-1
0
1
2
3
4
0
0.2
0.4
0.6
0.8
1
analytical
ALICE
ˆ
C
ˆ
Re

Field
growth rate.
ˆ
1
p
 
ˆ
2
p
 
Tests.
Space
charge
algorithm
(1D)
-2
-1
0
1
2
3
4
0
0.2
0.4
0.6
0.8
1
analytical
ALICE
ˆ
C
ˆ
Re

Field
growth rate.
ˆ
1
p
 
ˆ
2
p
 
Tests.
Energy
spread
algorithm
(1D)
-2
0
2
4
6
0
0.2
0.4
0.6
0.8
1
Field
growth rate.
ˆ
C
ˆ
Re

ˆ
2
T
 
analytical
ALICE
Tests.
Energy
spread
algorithm
(1D)
-2
0
2
4
6
0
0.2
0.4
0.6
0.8
1
Field
growth rate.
ˆ
C
ˆ
Re

ˆ
2
T
 
analytical
ALICE
Tests. Tapering (3D)

5
7
9
11
13
15
0
2
4
6
0
1
2
3
4
5
0
20
40
60
80
100
*
[SSY, 1995] E.L.Saldin, E.A.Schneidmiller, M.Y.Yurkov, The Physics 0f Free Electron
Lasers, An Introduction, Physics Reports 260 ( 1995) 187
-
327

FAST
*

ALICE

Tests.
Field
Solver
,
Perfectly
Matched
Layer
(3D)
0
1
2
3
0
0.01
0.02
0.03
0.04
0.05
0
1
2
3
0
0.01
0.02
0.03
0.04
0.05
0
1
2
3
0
0.01
0.02
0.03
0.04
0.05
0
1
2
3
0
0.002
0.004
0.006
0.008
0.01
0.012
Dirichlet
BC
PML
analytical
numerical
PML
ˆ
z
ˆ
u
ˆ
z
ˆ
u
ˆ
z
ˆ
u
ˆ
z
ˆ
u
ˆ
1
z

ˆ
5
z

Tests.
Field
Solver
,
Perfectly
Matched
Layer
(3D)
0
1
2
3
0
0.01
0.02
0.03
0.04
0.05
0
1
2
3
0
0.01
0.02
0.03
0.04
0.05
0
1
2
3
0
0.01
0.02
0.03
0.04
0.05
0
1
2
3
0
0.002
0.004
0.006
0.008
0.01
0.012
Dirichlet
BC
PML
analytical
numerical
PML
ˆ
z
ˆ
u
ˆ
z
ˆ
u
ˆ
z
ˆ
u
ˆ
z
ˆ
u
ˆ
1
z

ˆ
5
z

Tests.
Field
Solver
,
Perfectly
Matched
Layer
(3D)
0
5
10
15
0
0.2
0.4
0.6
0.8
Dirichlet
BC vs. PML
ˆ
z
ˆ

max
ˆ
3.3
D
r

max
ˆ
5
D
r

(7)
max
ˆ
2
PML
r

max
ˆ
10
D
r

Tests.
Field
Solver
,
Perfectly
Matched
Layer
(3D)
0
5
10
15
0
0.2
0.4
0.6
0.8
Dirichlet
BC vs. PML
ˆ
z
ˆ

max
ˆ
3.3
D
r

max
ˆ
5
D
r

(7)
max
ˆ
2
PML
r

max
ˆ
10
D
r

Tests. Energy spread and diffraction (3D)

0
0.2
0.4
0.6
0.8
0
0.2
0.4
0.6
0.8
1
L
T
*L
T
-2
-1
0
1
0.4
0.5
0.6
0.7
0.8
*
[SSY, 1995] E.L.Saldin, E.A.Schneidmiller, M.Y.Yurkov, The Physics 0f Free Electron
Lasers, An Introduction, Physics Reports 260 ( 1995) 187
-
327

FAST
*

ALICE

Tests. Space charge (3D)

0
0.2
0.4
0.6
0.8
1
0.8
1
1.2
1.4
*
[SSY, 1995] E.L.Saldin, E.A.Schneidmiller, M.Y.Yurkov, The Physics 0f Free Electron
Lasers, An Introduction, Physics Reports 260 ( 1995) 187
-
327

FAST
*

ALICE

Tests. SASE (1D)
ˆ
s
ˆ
z
Normalized
power
in
the
radiation
pulse
ALICE
Tests. SASE (1D)
ˆ
s
ˆ
z
Normalized
power
in
the
radiation
pulse
ALICE
-1
0
1
2
3
0
1
2
3
4
x 10
-3
-1
0
1
2
3
0
1
2
3
4
x 10
-3
FAST
*
ALICE
*
[SSY, 1999]
E.L.Saldin
,
E.A.Schneidmiller
,
M.Y.Yurkov
, The Physics 0f Free Electron
Lasers, Springer, 1999
Tests. SASE,
Gaussian
axial
bunch
profile
(1D)
ˆ
0.5
b


-1
0
1
2
3
0
1
2
3
4
x 10
-3
-1
0
1
2
3
0
1
2
3
4
x 10
-3
FAST
*
ALICE
*
[SSY, 1999]
E.L.Saldin
,
E.A.Schneidmiller
,
M.Y.Yurkov
, The Physics 0f Free Electron
Lasers, Springer, 1999
Tests. SASE,
Gaussian
axial
bunch
profile
(1D)
ˆ
0.5
b


Tests. SASE,
Gaussian
axial
bunch
profile
(1D)
-10
0
10
0
0.02
0.04
0.06
-10
0
10
0
0.05
0.1
0.15
0.2
FAST
*
ALICE
*
[SSY, 1999]
E.L.Saldin
,
E.A.Schneidmiller
,
M.Y.Yurkov
, The Physics 0f Free Electron
Lasers, Springer, 1999
ˆ
8
b


Tests. SASE,
Gaussian
axial
bunch
profile
(1D)
-10
0
10
0
0.02
0.04
0.06
-10
0
10
0
0.05
0.1
0.15
0.2
FAST
*
ALICE
*
[SSY, 1999]
E.L.Saldin
,
E.A.Schneidmiller
,
M.Y.Yurkov
, The Physics 0f Free Electron
Lasers, Springer, 1999
ˆ
8
b


Longitudinal and transverse coherence (SASE, 3D)

Longitudinal and transverse coherence (SASE, 3D)

Longitudinal and transverse coherence (SASE, 3D)

Acknowledgements to

Martin Dohlus,

Torsten Limberg

for helpful discussions and interest

and to

E.L. Saldin,

E.A. Schneidmiller,

M.V.Yurkov

for the nice book on the FEL theory