Free-Electron Laser

A) Motivation and Introduction

C) Experimental Realization / Challenges

B) Theoretical Approach

v1

coherent radiation

micro-bunching

why SASE?

self amplifying spontaneous emission (SASE)

amplifier and oscillator

free electron ↔ wave interaction

why FELs?

need for short wavelengths

A) Motivation and Introduction

need for short wavelengths

state of the art:structure of biological macromolecule

needs ≈1015 samples

crystallized � not in life environment

reconstructed from diffractionpattern of protein crystal:

LYSOZYME MW=19,806the crystal lattice imposes restrictions on molecular motion

images courtesy Janos Hajdu, slide from Jörg Rossbach

resolution does not depend on sample qualityneeds very high radiation power @ λ ≈1Åcan see dynamics if pulse length < 100 fs

need for short wavelengths - 2

simulated image

we need a radiation source with

SINGLEMACROMOLECULE

• very high peak and average power• wavelengths down to atomic scale λ ~ 1Å• spatially coherent• monochromatic• fast tunability in wavelength & timing• sub-picosecond pulse length

courtesy Janos Hajdu

principle of a quantum laser

why FELs?

problem & solution:active medium → free electron – EM wave interaction

free electron ↔ wave Interaction

free electron in uniform motion + electric field lines

trajectory

undulator

(before undulator)

courtesy T. Shintakehttp://www.shintakelab.com/en/enEducationalSoft.htm

free electron is source of waves

cm∝uλÅ→lλ(in principle)

after undulator:waves ahead of particle

there are backward waves

uλλ 2≈

micro-bunching

longitudinal motion to 1st order is trivial, but

micro-bunching is a 2nd order effect → coupled theory of particle motion and

wave generation

transverse bunch structure is much largerthan longitudinal sub-structure→ 1d theory with plane waves

nm 1.0

nm 00005

amplifier and oscillator

amplifier:

in principle FEL

wave

electrons

oscillator:

amplified noise:

instability, driven by noise, growth until amplifier saturates

amplitude

frequency

amplitude

frequency

noise …

amplitude

frequency

saturation

self amplifying spontaneous emission (SASE)

bunch of electronswith increasing micro-

modulation

log(P/P0)

exponential gain

saturation

uniform random distribution of particles at entrance

incoherent emission of EM waves (noise, wide bandwidth)

amplification (→ resonant wavelength, micro-bunching)

saturation, full micro modulation, coherent radiation

why SASE?

oscillator needs resonator

alternative: seed laser + harmonic generation + amplifier

but there are no mirrors for wavelengths < 100 nm

seed laser

seedλmodulation ofelectron beam

micro bunching ofa particular harmonic

amplifier seedλλ nl =

bunch of electronswith increasing micro-

modulation

log(P/P0)

exponential gain

saturation

coherent radiation

no micro-bunching(only shot noise)

P0 ∝ N

saturation:full micro-bunching

P ∝ N2

with N = particles per λl

3ϕ1ϕ 2ϕ1ϕ 2ϕ 3ϕ

particle dynamics: undulator motion

about particles

independent parameter: z

particle dynamics: interaction with EM wave

longitudinal equation of motion

phase space and pendulum

FEL low gain theory

micro bunching

electrodynamics (1D)

FEL high gain theory (1D)

continuous phase space: Vlasov equation

FEL third order equation

B) Theoretical Approach

particle dynamics: undulator motion

field of planar undulator zkB uyu sin0eB −=

Bvv ×−= eme &γequation of motion

uλ uuk λπ2=

approach:

txx uωsinˆ≈

tztvz uω2cosˆ−≈ uu v λπω 2=with

uue k

KB

kmv

ex

βγγ== 02

ˆ with undulator parameter 10 ∝=ueckm

eBK

+−≈

−=2

122

2

2

2Kc

cK

vvvγβγ

uk

Kz

2

2

8ˆ

γ=

example: FLASH

mm 27=uλ

T 47.00 =B

2.1cmT

934.00 ≈≈ uBK λ

µm 6.2ˆ ==uk

Kx

γ

GeV 1≈W 1957≈→ γ

nm 2.08

ˆ2

2

==uk

Kz

γ

about particles

z

x

“reference particle”: only interaction with undulator fieldconstant energy, averaged velocity, …index “0” uλ

0γ L≈0z2

0000 2

−=

γK

vvv200 1 γ−= cv txx uωsinˆ00 ≈

ordinary particles:in interaction with undulator, external waves, self fields, …slowly variation of energy, velocity (compared to λu)index “ν” or skipped

z

x

wave( )t,rE lλ

νγ νv νv L

resonancecondition

new independent parameter: z

( ) zkxzx usinˆ00 =

( ) zkv

z

v

zzt u2cos

ˆ

0

0

00 +=

0γ 0v 0v

( ) ( ) zkv

z

zv

dztzt u

z

i 2cosˆ

0

0

0

, +⌡

⌠+≈ν

νν

( )zνγ ( )zvν ( )zvν approach: nearly constant on one period λu

slippage effects

( ) zkxxzx ui sinˆ0, +≈ νν

initial condition

energy parameter:0

0

γγγη ν

ν−=

“reference particle”:

ordinary particles:

( ) ( )ztzT 0+= ν

( )zxx i 0, += ν

particle dynamics: interaction with EM wave

lλwavelength of light

z

x

wave( )t,rE

( ) rrE dtedW ⋅−= ,

resonance condition (permanent energy transfer): ( ) →=− ul kvck 01

it is a condition for the energy of the reference particle or the wavelength λl

+=2

12 2

0

Kul γ

λλ

plane wave with kl=2π/λx polarization

( )( ){ } zkeK

zctzkEedz

dWul cos cos 0 γν

ν ⋅−−=

( ) ( )( ){ } zkzkzzcTkzvckKeE

dz

dWuull cos 2cosˆ 1 cos 00

0 ⋅+−−−= νν

γ

dx/dz

zkv

z

v

zTt u2cos

ˆ

0

0

0

++= νν

particle dynamics: interaction with EM wave

cos2

0ν

ν ψγKeE

dz

dW −=

averaged vs. undulator period ?=dz

dWν ( ) constzT ≈ν

estimation without longitudinal oscillation ( ) ψψ cos2

1cos2cosˆcos 0 =++ zkzkzzk uuu

( ) zcTkl ννψ = ponderomotive phase

with longitudinal oscillation: replace K by

+−

+=

2

2

12

2

0 2424ˆ

K

KJ

K

KJKK

(modified undulator parameter)

example: FLASH

mm 27=uλ

T 47.00 =B

2.1cmT

934.00 ≈≈ uBK λ

µm 6.2ˆ ==uk

Kx

γ

GeV 1≈W 1957≈→ γ

nm 2.08

ˆ2

2

==uk

Kz

γ

nm 62

12

2

2≈

+= Ku

l γλλ

+−

+=

2

2

12

2

0 2424ˆ

K

KJ

K

KJKK

06.1ˆ =K

νν ψ

γη

cos2

ˆ20

20

cm

KeE

dz

d

e

−≈

νν ηψ ukdz

d2≈

particle equations

longitudinal equation of motion

( ) ( ) ( ) ( ) ( )ztck

zztzTzt

l00 +=+= ν

ννψ

new longitudinal parameters

( ) ( )1

0

−=W

zWz ν

νη

ponderomotive phase

relative energy deviation

( )ν

ν

νν ηψu

lll k

v

ck

v

ck

z

zdTck

dz

d2

0

≈−==with and follow20

20 γcmW e=

equations are analog to mathematical pendulum

φ lm

two types of solution:

trajectories in phase space

15 particles with different initial conditions

“oscillation”“rotation”separatrix φ

v

0=t

0>t(end of undulator)

( ) 00 =νη

η

2πψ +separatrix

2πψ +

η

( ) 00 >νη

( ) 00 γγν >

or

phase space and pendulum

FEL low gain theory

neglect change of field amplitude

indirect gain calculation

( ) ( )energy field initial

outinenergy field initial

energy particle of lossenergy field initialenergy field of gain ΣΣ −=== WW

G

( ) ( )outin ΣΣ = WW

0=G

( ) ( )outin ΣΣ > WW

0>G

( ) 00 =νη

η

2πψ +( ) 00 γγν >

or

2πψ +

η

( ) 00 >νη

=uN periods of undulator

FEL low gain theory

uN

η

G

neglect change of field amplitude

micro-bunching

Iz(ψ)

Iz,0

Fourier analysis→ amplitude of micro modulation

( )∑ −∝ νψiI expˆ

(fundamental mode)

electrodynamics (1D)

Maxwell equations↓

wave equation↓

1D wave equation

xx Jt

Etcz ∂

∂=

∂∂−

∂∂

02

2

22

2 1 µ ( ) ( )( ) ( )tzJzv

zvtzJ z

z

xx ,, =

( ) ( ) ( )( ){ } L+− ctzikzItzJ lz expˆRe~, 1

( ) ( ) ( )( ){ } L+− ctzikzEtzE lxx expˆRe~,

( ) 10

0 ˆ4

ˆˆ I

KczE

dz

dx γ

µ−=

approach with slowly varying amplitude

zkkx uu cosˆ0

FEL high gain theory (1D)

( ){ }νν ψ

γη

iEcm

Ke

dz

dx

e

expˆRe2

ˆ20

2−≈

νν ηψ ukdz

d2≈

particle equations

( )∑ −∝ νψiI expˆ

10

0 ˆ4

ˆˆ I

KcE

dz

dx γ

µ−≈

micro modulation

FEL high gain theory (1D)

numerical solution(Mathcad)

3rd order equation

numerical

P(z)/P(0)

phase space

modulation

FEL high gain theory (1D)

continuous phase space, Vlasov equation

10 −= γγη

ψ

many point particles → continuous density distributionνν ηψ , ( )zF ,,ηψ

continuous phase space, Vlasov equation

( )zF ,,ηψ

( ) ( )zFz ,,,, ηψηψ

ηψ

′′

=J

( ) 0,, =∂∂+⋅∇

z

FzηψJ

( ) ( )0=

∂∂+

∂∂′+

∂′∂+

∂′∂

z

FFFF

ηη

ηη

ψψ

( )zgz

,ψηη =∂∂=′

( )zfz

,ηψψ =∂∂=′

0=∂∂+

∂∂′+

∂∂′

z

FFF

ηη

ψψ 0=

dz

dF

phase space density:

continuity equation:

with “current density”

therefore

with particle equations

Vlasov equation: or

FEL 3rd order equation

perturbation approach: ( ) ( ) ( ) ( ){ }ψηηηηψ izFFzF exp,ˆRe,, 1off0 +−≈

0ˆˆ

4ˆ

4ˆ

32off

22

2

off3

3

=Γ−−+ xx

ux

ux Ei

dz

Edk

dz

Edik

dz

Ed ηη

with energy offset0

0off γ

γγηη ν

ν−

==

gain parameter 330

20

ˆ

4

1

γµ Kk

A

I

cm

e u

e

=Γ beam current I

beam cross-section A

solution ( ) zAzAzAzEx 332211 expexpexpˆ ααα ++=

FEL 3rd order equation

no energy offset: 0γγν =0off =η or

( ) zAzAzAzEx 332211 expexpexpˆ ααα ++=

Γ+=2

31

iα

Γ−=2

32

iα

Γ−= i3α

positive real part → exponential growth !

3rd order equation

numerical

( )( )

2

0ˆ

ˆ

x

x

E

zE

power gain length: ( ) ( ) 3 exp2exp 121 Γ∝→ zzAzP α

1/Lg

C) Experimental Realization / Challenges

Linac Coherent Light Source - LCLS

scales

challenges

rf gun

European X-FEL

bunch compression

Linac Coherent Light Source- LCLS

SLAC mid-April 2009 – first lasing at 1.5 Å

injector

nC 25.0≈q

linac & bunch compression

m10 3∝L m 100 ∝L

undulator

GeV 6.13=E

kA 3=I

cm 3≈uλ

Å 5.1=lλ

m 3.3≈gL

saturation> 10 GW

exponential gain

incoherent

410≈=ec

IN lλ

particles per λl

scales

total length m10 3∝L

undulator length m 100 ∝uL

undulator period m10 2−∝uλ

cooperation length m10 8−∝lL

bunch length m 10 5−∝bL

power gain length m 10 .. 1≈gL

Rayleigh length RL (scale of widening of photon beam)

saturation length uggs LLLL <≈ 20 ..10

photon wavelength m10 10−∝lλ 2γλu∝

transverse oscillation m10ˆ 6−∝x (undulator trajectory)

bunch width Rlw Lλσ ∝wavem10 5bunch −∝wσ width of photon beam

overlap of particle beamwith photon beam

challenges

+=

21

2

2

2

Kul γ

λλ

312

2

34

3

1

=

IKe

mcL ru

g

σλγµ ● high peak current >~ kA

● λl → Å

● Energy → 10 .. 20 GeV

● gain length Lg <~ 10 m

● transverse beam size σr ∝ 10 µm

● energy spread

● overlap electron-photon beam

longitudinal: compression

(undulator parameter K ∝ 1)

glr Lλσ ∝2

transverse: generate low emittance beampreservation of emittance

diagnostic and steering

acceleration

undulator alignment

qc

L

Ibrr

22 σσ =volume

bunch charge

space charge forces:

22

1

rsq

qE

σγ∝

rf gun

nC 1.0∝q MeV 5∝E A 5∝I

typical parameters of FLASH & European XFEL:

10∝γlongitudinal compression 1 → 0.001 needed ! (5 kA)

magnetic bunch compression

magnetic compression: path length depends on energy

γ >>1 → velocity differences are too small for effective compression

accelleration “off crest” →head particle with less energy than tail

t

Ezz

r

B

r

B

r

B

r

B

FLASH:

gun accellarating module ~ 10 m 1st bunch compressor

beam dynamics with space charge and CSR effects

4 magnet chicane

2nd

com

pres

sor,

mor

e m

odul

esun

dula

tor

magnetic bunch compression - 2

LCLS (14 GeV, 0.15 nm)

FLASH (1.2 GeV, 4 nm)

European XFEL (0.1 nm)

BCs at 130 MeV, 500 MeV and 2 GeV

1.2 GeV450 MeV130 MeV

European XFEL

total length 3.4 kmaccelerator length 1.7 kmenergy 17.5 MeVminimal wavelength 1 Åbunches / second 30000bunch charge 0.01 .. 1 nCtypical peak current 5 kA

European X-FEL - 2

superconducting cavity, 1.3 GHz Eacc → 40 MeV/m23.5 MeV/m are needed

FLASH tunnel: cryo module undulator

European X-FEL - 3

beamlines

multi bunch operation