ALICE is a Lasing Investiga tion Co- dE Igor Zagorodnov BDGM, DESY 16.10.06
Jan 22, 2016
ALICE is a Lasing Investigation Co-dE
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 authors
22
0
( , , ) ( . .)2
iz
z
H P z CP P Ue C C eE dc
Equations of motion correspond to effective Hamiltonian
Field Equations are used in parabolic approximation
212 4 sc i E j
c z
04 ( )z zE j jt
( / )i z c tx yE 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 authors
22
0
( , , ) ( . .)2
iz
z
H P z CP P Ue C C eE dc
Equations of motion correspond to effective Hamiltonian
Field Equations are used in parabolic approximation
212 4 sc i E j
c z
04 ( )z zE j jt
( / )i z c tx yE 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 40
0.2
0.4
0.6
0.8
1
analytical
ALICE
C
ˆRe
Field growth rate.
ˆ 1p ˆ 2p
Tests. Space charge algorithm (1D)
-2 -1 0 1 2 3 40
0.2
0.4
0.6
0.8
1
analytical
ALICE
C
ˆRe
Field growth rate.
ˆ 1p ˆ 2p
Tests. Energy spread algorithm (1D)
-2 0 2 4 60
0.2
0.4
0.6
0.8
1
Field growth rate.
C
ˆRe ˆ 2T
analytical
ALICE
Tests. Energy spread algorithm (1D)
-2 0 2 4 60
0.2
0.4
0.6
0.8
1
Field growth rate.
C
ˆRe ˆ 2T
analytical
ALICE
Tests. Tapering (3D)
5 7 9 11 13 150
2
4
6
0 1 2 3 4 50
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 30
0.01
0.02
0.03
0.04
0.05
0 1 2 30
0.01
0.02
0.03
0.04
0.05
0 1 2 30
0.01
0.02
0.03
0.04
0.05
0 1 2 30
0.002
0.004
0.006
0.008
0.01
0.012
Dirichlet BC PML
analytical
numerical
PML
z
u
z
u
z
u
z
u
ˆ 1z
ˆ 5z
Tests. Field Solver, Perfectly Matched Layer (3D)
0 1 2 30
0.01
0.02
0.03
0.04
0.05
0 1 2 30
0.01
0.02
0.03
0.04
0.05
0 1 2 30
0.01
0.02
0.03
0.04
0.05
0 1 2 30
0.002
0.004
0.006
0.008
0.01
0.012
Dirichlet BC PML
analytical
numerical
PML
z
u
z
u
z
u
z
u
ˆ 1z
ˆ 5z
Tests. Field Solver, Perfectly Matched Layer (3D)
0 5 10 150
0.2
0.4
0.6
0.8
Dirichlet BC vs. PML
z
maxˆ 3.3Dr
maxˆ 5Dr
(7)maxˆ 2PMLr
maxˆ 10Dr
Tests. Field Solver, Perfectly Matched Layer (3D)
0 5 10 150
0.2
0.4
0.6
0.8
Dirichlet BC vs. PML
z
maxˆ 3.3Dr
maxˆ 5Dr
(7)maxˆ 2PMLr
maxˆ 10Dr
Tests. Energy spread and diffraction (3D)
0 0.2 0.4 0.6 0.80
0.2
0.4
0.6
0.8
1
LT*L
T
-2 -1 0 10.4
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 10.8
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)
s
zNormalized power in the radiation pulse
ALICE
Tests. SASE (1D)
s
zNormalized power in the radiation pulse
ALICE
-1 0 1 2 30
1
2
3
4x 10
-3
-1 0 1 2 30
1
2
3
4x 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 0.5b -1 0 1 2 3
0
1
2
3
4x 10
-3
-1 0 1 2 30
1
2
3
4x 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 0.5b
Tests. SASE, Gaussian axial bunch profile (1D)
-10 0 100
0.02
0.04
0.06
-10 0 100
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 8b
Tests. SASE, Gaussian axial bunch profile (1D)
-10 0 100
0.02
0.04
0.06
-10 0 100
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 8b
Longitudinal and transverse coherence (SASE, 3D)
Longitudinal and transverse coherence (SASE, 3D)
Longitudinal and transverse coherence (SASE, 3D)
Acknowledgements to Martin Dohlus,
Torsten Limbergfor helpful discussions and interest
and toE.L. Saldin,
E.A. Schneidmiller, M.V.Yurkov
for the nice book on the FEL theory