A LICE is a L asing I nvestigation C o- d E

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


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


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


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


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


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


FAST*

ALICE

Tests. Space charge (3D)

0 0.2 0.4 0.6 0.8 10.8

1

1.2

1.4


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



ˆ 0.5b


-10 0 100

0.02

0.04

0.06

-10 0 100

0.05

0.1

0.15

0.2

FAST*

ALICE


ˆ 8b


-10 0 100

0.02

0.04

0.06

-10 0 100

0.05

0.1

0.15

0.2

FAST*

ALICE


ˆ 8b

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

A LICE is a L asing I nvestigation C o- d E

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