Lecture 4. High-gain FELs X-Ray Free Electron Lasers Igor Zagorodnov Deutsches Elektronen Synchrotron TU Darmstadt, Fachbereich 18 19. May 2014

Lecture 4. High-gain FELs

X-Ray Free Electron Lasers

Igor Zagorodnov

Deutsches Elektronen Synchrotron

TU Darmstadt, Fachbereich 1819. May 2014

PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 4 | 19. May 2014 | Seite 2

Contents

Review

Exponential growth and saturation

Bunching factor and space charge field

Linearized model

FEL parameters

FEL bandwidth

3D effects

Outlook


Review

undulator radiation

low gain FEL

high gain FEL

electrons

Interaction

EM field

EM field

EM field

Model

FEL radiation

electrons

electrons


Review

What is shown? (Exercise 7)


Review

( ) sin( )wu

ux t t

k

2

8( ) sin(2 )w

uz uk

z t v t t

2

22

14

zK

v c

u z u z uv k ck

Electron motion

wK

trajectory

z 0

e u

eBK

m ck

2u uk

2e

e

W

m c


Review

Exercise 7Exercise 4

, 0,( )

e , , 02

i

iz

RR

2

8

, ,2

( ) sin(2 )

sin( ) ,

w

u

w

u u z

z z uk

wz u

u

ik k

z k

i k elsek


Review

zv c

the electron should be slower by one wavelength

- the electron is slower than the light

Resonance condition

2

2122

u K


Review

020

[ ]cos

2 e

eK JJ Ed

dt m c

2 ud

k cdt

Low gain FEL model0 constE

( )uk k z t

0

0


Review

data from FLASH

W. Ackermann et al, Nature Photonics 1, 336 (2007)

rad ~ elP N 2rad ~ elP N

[μJ]E

[ ]z m [nm]The amplification is very high

Microbunching


Review

01

[ ]( )

4x zr

d cK JJE z j

dz

2 , 1,2,...n u nd

k n Ndz

2 2

[ ]( )

2ni

n xe r

d eK JJE e

dz m c

High-gain FEL model

2

1 010

1 2m

Nii

z z zm

j j e d j eN



(0) 0.1MV/mxE

0 5 10 15 20 250

2

4

-3 -2 -1 0 1 2 3-5

0

5

-3 -2 -1 0 1 2 30

10

20

0mz

r

[kA]I

[GW]P

[m]z

Exercise 6



0 5 10 15 20 250

2

4

-3 -2 -1 0 1 2 3-5

0

5

-3 -2 -1 0 1 2 30

10

20

saturation: beam fully modulated

20mz [GW]P

[m]z

r

[kA]I

Exercise 6



0 5 10 15 20 250

2

4

-3 -2 -1 0 1 2 3-5

0

5

-3 -2 -1 0 1 2 30

10

20

21mz [GW]P

[m]z

r

[kA]I

Exercise 6



0 5 10 15 20 250

2

4

-3 -2 -1 0 1 2 3-5

0

5

-3 -2 -1 0 1 2 30

10

20

22mz [GW]P

[m]z

r

[kA]I

Exercise 6



0 5 10 15 20 250

2

4

-3 -2 -1 0 1 2 3-5

0

5

-3 -2 -1 0 1 2 30

10

20

23mz [GW]P

[m]z

r

[kA]I

Exercise 6


0 5 10 15 20 25

10-5

100[GW]P

[m]z


linear

saturationExercise 6



( ) inn

n

f x f e

0

1( )

Tinx

nf f x e dxT

0 0( ) ( ) ( )x f x dx f x

*( ) ( )f x f x *n nf f

( ) ( )f x f x

Fourier series of real periodic function

Dirac delta-function

01

( )2

inn

n

ff x f e

*0 01 1

( )2

in in in innn n n

n n n

ff x f e f f e f e f e



01

( ) inz z zn

n

j j j e

2

1 0 10

1( ) 2i

z z zj j e d j b

2

00

1( )

2z zj j d

0

1

2( ) ( )

N

z z nn

j jN

11

1n

Ni

n

b eN



0 E

01

( ) inz z zn

n

E E E e

01

( ) inn

n

e

0

inin n

zne

E ez

0

( ) nu znin k k E

0

nznE i

kn

0

znzn

jE i

n

0

1

2m

Nin

zn zm

j j eN

( )uk k z t

uk k



01 1

0

0 01 1 1

0

0 1 1

0

0 1

( )

2 1Im Im

sin2

sgn

m

in inz z zn zn

n n

Nininzn z

n n m

Nmz

m n

Nz

m mm

E E E e E e

j je e

n N n

nj

N n

j

N

1

sgn( ) sin( )

2 n

x x nx

n



x x z zdW

e ev E ev Edt

v E

Revision of energy equation

2 2

[ ]( )

2ni

n xe r

d eK JJE e

dz m c

2 2 2

( )[ ]( )

2nin z n

xe r r e

d eEeK JJE e

dz m c m c

0

0 1

( ) sgnN

zz n n m n m

m

jE

N



01

[ ]( )

4x zr

cK JJdE z j

dz

2 , 1,2,...n u nd

k n Ndz

High-gain FEL model with space-charge

2 2 2

( )[ ]( )

2nin z n

xe r r e

d eEeK JJE e

dz m c m c

0

0 1

( ) sgnN

zz n n m n m

m

jE

N

1 01

2m

Ni

z zm

j j eN


Linearized model

0 1( , , ) ( ) Re ( , ) iF z F F z e

1 0F F

0( , ) ( , , )z F z d

1 0 1( ) ( , )F z d


Linearized model

0dF F F d F d

dz z dz dz

Vlasov-Maxwell equations

00 1

[ ]( ) ( , )

4x zr

cK JJdE z j F z d

dz

2 2 2

( )[ ]( )

2

i zx

e r r e

eEd eK JJE e

dt m c m c

2 u

dk

dz

0 ( ) ( )F - monoenergetic beam

According to the Liouville Theorem the phasespace volume occupied by an ensemble of particles is conserved along theparticle trajectory.


FEL parameters

2 23 2

ˆˆ ˆ2 0x x xp x

E E Ei iE

12 2 3

0 03

[ ]

4u

r e

eK JJ k j

m c

Gain parameter Detuning

22

2ˆ pp

Space charge parameter

ˆ

2 0 02 up

r e

k j e

m

2 uk

Pierce parameter


FEL parameters

ˆ , 1, 2,...ˆ n nd

n Ndz

Normalized high-gain FEL model with space-charge

2ˆ ˆ ˆ ˆ( ) ( )ˆ

ninx p z n

dE e E

dz

1

1ˆ ( ) sgnN

z n n m n mm

EN

11

2ˆ mN

iz

m

j eN

2 2

02

[ ]e rm c

EeK JJ

0

ˆ xx

EE

E

1ˆ ˆ( )ˆ x zdE z j

dz

z z


FEL parameters

Power gain length

0r

r

30x

xE

iE

zxE Ae 3 3i

1 3 2i

Im

Re

2 3 2i 3 i

12

31 21 2 3

zz zxE C e C e C e


FEL parameters

01

22 ( ) 3

0~

2g

z

Lx zES e e e

Z

31 21 2 3

zz zxE C e C e C e 1

1z

xE C ez

Power gain length

01

3gL

Field gain length

02

3FgL

1( )~ z

xE e


FEL parameters

Power amplification

(0)x inE E

( , ) cosseedx inE z t E kz t

(0) 0xE

(0) 0xE 31 2

3zz zin

xE

E e e e

beamA

P Sdxdy 0

9g

z

LinPP e S E H


FEL parameters

0

9g

z

LinPP e

“lethargy regime”


FEL parameters

?satP

( ) 1

9g

z

L

in

P ze

P

?satz

Saturation


0 5 10 15 20 25

2468

x 109 field amplitude vs undulator length

z[m]

|Ex|[V

/m]

-3 -2 -1 0 1 2 3

-4-202

x 10-3 phase space

psi[rad]

eta

-3 -2 -1 0 1 2 30

5000

current

psi[rad]

I[A

]

0FgL


Exercise 6( ) ?x

dE z

dz


FEL parameters

Saturation power

11

1n

Ni

n

b eN

2

1 0 10

1( ) 2i

z z zj j e d j b

1 1b

ˆ ( ) 2ˆ xdE z

dz

0

ˆ xx

EE

E

z z

0( ) 2xdE z E

dz

1ˆ ˆ( )ˆ x zdE z j

dz


FEL parameters

Saturation power

, 0 04

( )3

Fx sat x g

dE E z L E

dz

2, 2 0

00 0 0

1 1 8 4

2 3 3x sat

sat beam beam

E IP A E P

Z Z j

0( ) 2xdE z E

dz

2 0beam

IP mc

e

0 0 0 0( ) ( ) ( )f x f x f x x x

2 2

02

[ ]e rm c

EeK JJ


FEL parameters

4

3sat beamP P

0

( ) 1

9g

z

LP ze

P

020 usat gz L

Saturation


FEL Bandwidth

0

0

2

2122

u K

2020

122

uk K

k

0

0

02

0

Detuning for resonance frequency

Detuning for resonance energy


FEL Bandwidth

and are equivalent and present a “detuning” in the resonance condition

2

2

11

2 22u

u

k Kk

k

0 2

02

00

11 (2)

2 22( )u

u

k Kk O

k

0 2

02

00

11 (2)

2 2 22u

u

k k K kk O

k k


FEL Bandwidth

0 0

0 02

23 2

ˆ ˆ2 0x x xx

E E Ei iE

ˆ z

xE Ae

3 2 2ˆ ˆ ˆ ˆ ˆ2 0i i

2ˆ ˆ ˆi i

zxE Ae ˆ

z z


FEL Bandwidth

3Re 0

2

1 20

1Re ( ) 1

2 9gL


FEL Bandwidth

2

1 20

1Re ( ) 1

2 9gL

2 2

2 20 0 09 2( , )

( , ) ~ g g g

zz zL L Lin

in

P z PG z e e e e

P

202 9

2gL

z

0 0

0 02

00 02 3 2 gL

z 0 02


FEL Bandwidth

-2 -1 0 1 20

20

40

60

80

100

120Fig. 2.5. from SSY

02

(0)x

x

E

E

0~

11gz L

amplification at resonance energy small positive detuning increases amplification


3D Effects

Diffraction

Diffraction reduces the field amplitude on the axis

012 ( ) 2

4xKd

ik E z i jdz

Δ


Outlook

Electrons produce spontaneous undulators radiation

How to obtain a useful external field?

SASE

Self-Amplified Spontaneous Emission (SASE)

A. Kondratenko, E. Saldin, Part. Accelerators 10, 207 (1980)

R.Bonifacio et al, Opt. Comm.50, 373 (1984)

Documents

Lecture 4. High-gain FELs X-Ray Free Electron Lasers Igor Zagorodnov Deutsches Elektronen Synchrotron TU Darmstadt, Fachbereich 18 19. May 2014