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