Lecture 3. Low-gain and high-gain FELs X-Ray Free Electron Lasers Igor Zagorodnov Deutsches Elektronen Synchrotron TU Darmstadt, Fachbereich 18 12. May

Lecture 3. Low-gain and high-gain FELs

X-Ray Free Electron Lasers

Igor Zagorodnov

Deutsches Elektronen Synchrotron

TU Darmstadt, Fachbereich 1812. May 2014

PD Dr. Igor Zagorodnov| X-Ray Free Electron Lasers. Lecture 3 | 12. May2 2014 | Seite 2

Contents

Low-gain FEL

Energy exchange between electrons and EM wave

FEL pendulum equations

FEL gain and Madey theorem

Microbunching

High-gain FEL equations in 1D


Low-gain FEL

Estimated spectrum of spontaneous undulator radiation and FEL radiation in LCLS (Stanford)

FEL radiation

(H.-D. Nuhn, SLAC)


Low-gain FEL

undulator radiation

low gain FEL

high gain FEL

electrons

Interaction

EM field

EM field

EM field

Model

FEL radiation

electrons

electrons


Low-gain FEL

out in 1 NP P

Upon each passage the light intensity grows only by small gain factor


Energy exchange

22 2 2 20W m c c p 2dW d

W cdt dt

p

pd dW

dt dt

pv

L : q F v B E

Ld

dt

pF

0: mp v

2

0

22 2 20

:W m c

m c c

p

LdW

dt v F

dWe

dt v E


Energy exchange

0 sin( )y uB B k zField on the axis

0

e u

eBK

m ck

2u uk

Undulator parameter

Electron motion

wK

trajectory

z


1K 5K

*x

a

*z a

Energy exchange

( ) cos( ) cos( ( ))x u uK K

v t c t c k z t

2cos(2 )

4w

z z uc

v v t

( ) sin( )wu

ux t t

k

2

8( ) sin(2 )w

uz uk

z t v t t

in frame moving with averaged velocity

*( *) sin( * *)x t a t

*( *) sin(2 * *)z t b t

wK

trajectory 2

22

14

zK

v c

u z uck

Electron motion

z


Energy exchange

zv c - the electron is slower than the light

A steady energy transfer?

v

xE

0( , ) cos( )xE z t E kz t

Laser field is approximated by a plane EM wave

2k

c

x xdW

ev Edt


Energy exchange

zv cv

xE

- the electron is slower than the light

A steady energy transfer?x x

dWev E

dt


Energy exchange



dWev E

dt


Energy exchange



dWev E

dt


Energy exchange

zv c

the electron should be slower by one wavelength

- the electron is slower than the light


dWev E

dt


Energy exchange

the electron should be slower by one wavelengthu

z

tv

2

0 12 2u K


2

21

22u K

2

2

11 1

22u u

u u uz z

Ktc c

v

Slippages by odd number of wavelength is also possible (higher harmonics)

Spontaneous radiation has the same wavelength and can serve as a seed radiation


Energy exchangeA steady energy transfer?

0

0

cos( ) cos( )

cos( ) cos( )2

x x udW K

ev E ec k z E kz tdt

KEec

0( , ) cos( )xE z t E kz t ( ) cos( )x uK

v z c k z

( )uk k z t 2 uk z

The electron energy changes as described by the equation


Energy exchange


0 cos( ) cos( )2

KEdWec

dt

0( ) 0u zd

k k vdt

2

20

122

u K

( ) 0u zd

k k v kcdt

0?!zuz

vk k

v c

zz v t

The first term in the first equation provides a continuous energy transfer from the electron to the light wave, if the coherence condition is hold.


Energy exchange

Energy transfer equation

2 uk z - makes 2 oscilations per undulator period and cancels out

zz v t0 cos

2

cKEdW e

dt


Energy exchange

0[ ]cos

2

cK JJ EdW e

dt

0 cos( ) cos( )2

KEdEec

dt

2

8( ) sin(2 )w

uz uk

z t v t t zz v t

0 cos2

cKEdW e

dt

2 2

0 12 2[ ]

4 2 4 2

K KJJ J J

K K

Refinement (see, for example, P. Schmüser et al)


4 2 0 2 4


2s

0zs z v t

0 0

2( )u

z zk k z t z t s s

v v

0u

z

k kv

0( ) 0u z

dk k v

dt

Ponderomotive phase

where we have used the coherence condition



2

0 2 20 0 0 0

2 1 1

4u z z z zz z z

d Kk k v v v v

dt v v

2 20 0 04 2

00 0

( )2 ( )2 11 2

4 2 ud K K

k cdt

0

0

0u

z

k kv

2

22

14

zK

v c

Phase equation



020

[ ]cos

2 e

eK JJ Ed

dt m c

2 ud

k cdt

2 020

[ ]( , ) sin

2u

e

eK JJ EH k c

m c

020

[ ]2 cos

2u

e

eK JJ Ed dk c

dt dtm c

2 020

[ ]sin

2u

e

eK JJ Ek c H

m c

Hamiltonian



02 2

0

[ ]( ) cos

2 4sepu e

eK JJ En

k m c

Separatrix

0,2

020

[ ]

2sep

e

eK JJ EH

m c

2 020

[ ]sin

2sep u

e

eK JJ EH k c

m c

3 2 2 2



020

[ ]cos

2 e

eK JJ Ed

dt m c

2 ud

k cdt

02 2

0

[ ]cos

2 e

eK JJ Ed

dz m c

2 ud

kdz

zd dz d d d

v cdt dt dz dz dz

Change of the independent variable

zz v t



Asymptotic expansion

cosd

dz

da

dz

2 30 1 2 ( )O

2 30 1 2 ( )O

0 0d

adz

0 0d

dz

0 0 ia z

No impact of the EM field on the particles in the lowest order

0 (0)const

020

[ ]1

2 e

eK JJ E

m c

0

0

2 ua k



First order

1 0cosd

dz

1 1d

adz

01 0

00

sin ( ) sin( ) cos ( )

ziz

z z dza

2

1 10

1( ) ( ) 0

2iiz z d

01 1

0 00

cos ( ) cos1( ) sin

zi

iz

z a dz za

No mean energy exchange between the particles and the EM field, but there are energy modulation in the particle beam

1( ) ~ cos iz z

0 0 ia z



Second order

2 1 0( )sin ( )d

z zdz

22d

adz

2 1 00

( ) ( )sin ( )z

z z z dz

2 23

2 20

1 sin( ) ( )

2 4i

uik z d

z z dd

There is a mean energy exchange between the particles and the EM field in this order

002 u

az k z



The change in the electron energy density reads

2 22~e

z

jw mc n mc

ev

The net decrease in particle energy results in an increase in the EM energy

The energy density of the seed EM field (plane wave) reads

2002

u E

The energy gain is2

sin~

w dG

u d



2sin

~w d

Gu d

0u uk L

0 00

0 02

00 02u u u u uk L k L N

2

21

22u K

The energy deviation (of electron) is equivalent to the wavelength deviation (of EM wave)



20

( ) ~ sinc , uS N

( ) ~ ( )d

G Sd

J.M.J. Madey, Nuovo Cimento, B50, 64 (1979)

spontaneous radiation spectrum energy gain

0

energy deviation ηfrequency deviation Δω0



Low-gain FEL

no energy gain at the resonance energy the electron energy has to be higher


Microbunching

1.2K 6 nm

1.6 kApeakI

0.1mmbeamr

FLASH in low-gain model

27mmu

FLASH (Hamburg)


Microbunching

FLASH in low-gain model (Exercise 5)

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

0

5x 10

-4 phase space

psi[rad]

eta

-3 -2 -1 0 1 2 3-2000

-1000

0

1000

2000

eta*ku*L

und

gain

(0) 30MV/mxE

und 27mL


Microbunching

-3 -2 -1 0 1 2 3-0.5

0

0.5

-3 -2 -1 0 1 2 30

5

10

15

20

r

[kA]I

(0) 30MV/mxE Low-gain model0mz


Microbunching

-3 -2 -1 0 1 2 3-0.5

0

0.5

-3 -2 -1 0 1 2 30

5

10

15

20

r

[kA]I

Low-gain model (0) 30MV/mxE 15mz


Microbunching

-3 -2 -1 0 1 2 3-0.5

0

0.5

-3 -2 -1 0 1 2 30

5

10

15

20

r

[kA]I

(0) 30MV/mxE Low-gain model


s

Microbunching

0 1( ) ( ( ) )iz z z zj j j j z e

experimental evidence of microbunching in Stanford

K.N. Ricci ant T.I Smith, PR-STAB 3, 032801 (2000)

2

10

1 iz zj j e d



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



( , ) ( , )z t z tE E2

02 20

1 1

tc t

jE

The laser field can not be considered as constant

( , ) ( , )z t z tj j

2 2

02 2 2

1( , ) xx

jE z t

tz c t

1D model



Representation with slowly changing amplitude

2 2

02 2 2

1( , ) xx

jE z t

tz c t

( ) ( )x xE z k E z ( )( , ) ( ) i kz tx xE z t E z e

( )02 ( ) ( ) i kz t x

x xj

ikE z E z et

( )0( )2

i kz txx

i jE z e

k t


cos( )xzx x x z z u

z

vj Kj v v j j k z

v c

( )0 1 0 1( ) ( ) ui k z kz ti

z z z z zj j j z e j j z e

( )1( ) ui k z kz t

z zj i j z et

201( ) 1

4ui k z

x zd cKE z j e

dz


01( )

4x zd cKE z j

dz

2

10

1 iz zj j e d



numerical methods are required

very high number of electrons

calculations with the help of macroparticles.0

1( )4xcKd

E z jdz

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



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



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



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



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


0 5 10 15 20 25

10-5

100[GW]P

[m]z

Outlook

linear

saturation

Documents

Lecture 3. Low-gain and high-gain FELs X-Ray Free Electron Lasers Igor Zagorodnov Deutsches Elektronen Synchrotron TU Darmstadt, Fachbereich 18 12. May