Electron beam proper.es and FEL Lecture I - - PowerPoint PPT Presentation

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¡Electron ¡beam ¡proper.es ¡and ¡FEL ¡– ¡Lecture ¡I ¡

Massimo.Ferrario@LNF.INFN.IT ¡

Fresnel diffraction pattern at the European XFEL (30 June 2017)

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X-FEL based on last 1-km of existing SLAC linac

LCLS at SLAC

1.5-15 Å

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Short Wavelength SASE FEL

LCLS FLASH XFEL SwissFel FERMI SACLA PAL SDUV

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Genera.ons ¡of ¡Synchrotron ¡Light ¡Sources ¡

I. Bending magnets in HEP rings

• II. Dedicated Undulators
• III. Optimized Rings
• IV. Short Wavelength FEL
• V. Compact Sources

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A Free Electron Laser is a device that converts a fraction of the electron kinetic energy into coherent radiation via a collective instability in a long undulator (Tunability - Harmonics)

λrad ≈ λu 2γ 2 1 + K 2 2 + γ 2ϑ 2 & ' ( ) * +

SPARX 12.4 1.24 0.124 λ (nm)

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Electron source and acceleration

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Magnetic bunch compressor (< 1 ps)

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Long undulators chain

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Beam separation

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Experimental hall (Single Protein Imaging)

http://lcls.slac.stanford.edu/AnimationViewLCLS.aspx

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Transverse electron motion in an Undulator:

β// = β 2 − β⊥

2 =

1− 1 γ 2 − β⊥

2 ≈ 1 − 1

2 1 γ 2 + β⊥

2

' ( ) * + ,

β

// = 1−

1 2γ 2 1+ K 2 2 % & ' ( ) *

vx c = β⊥ = K γ cos kuz

( )

SLIDE 15

θ = 1 γ The ¡electron ¡trajectory ¡ is ¡inside ¡ ¡the ¡radia.on ¡ cone ¡if: ¡

K ≤ 1

The ¡electron ¡trajectory ¡is ¡determined ¡ by ¡ the ¡ undulator ¡ field ¡ and ¡ the ¡ electron ¡energy ¡

β

// = 1 −

1 2γ 2 1 + K 2 2 % & ' ( ) *

Undulator Radiation

! x = K γ cos kuz

( )

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Relativistic Mirrors

λu

' = λu

γ //

λrad

'

= λu

'

Counter propagating pseudo-radiation Thompson back-scattered radiation in the mirror moving frame

λrad ≈ λu 2γ 2 1+ K 2 2 + γ 2ϑ 2 & ' ( ) * +

Tunability & Red Shift

Doppler effect in the laboratory frame

λrad = γ $ λ

rad 1− β cosϑ

( ) ≈ λu 1− β

// cosϑ

( )

β

// = 1 −

1 2γ 2 1 + K 2 2 % & ' ( ) *

cosϑ ≈ 1 − ϑ 2 2

SLIDE 17

Δλrad λrad = −2 Δγ γ + 2K 2 1+ K 2 ΔK K + γ 2ϑ 2 1+ K 2

λrad ≈ λu 2γ 2 1+ K 2 2 +γ 2ϑ 2 " # $ % & '

Resonant Wavelength Sensitivity to beam parameters

γ 2ϑ 2 =γ 2σ !

x 2 = γ 2ε 2

σ x

2 = εn 2

σ x

2

Energy spread Beam Emittance Undulator tolerances

SLIDE 18

P

1 =

e2 6πεoc 3 γ 4 ˙ v

⊥ 2

Peak power of one accelerated charge:

P

T = N e 2e2

6πεoc 3 γ 4 ˙ v

⊥ 2

Coherent Stimulated Radiation Power:

P

T = N e

e2 6πεoc 3 γ 4 ˙ v

⊥ 2

Different electrons radiate indepedently hence the total power depends linearly on the number Ne of electrons per bunch: Incoherent Spontaneous Radiation Power: Bunching on the scale of the wavelength:

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Spontaneous Emission ==> Random phases N

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Coherent Light ==> Stimulated Emission N2

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Radiation Simulator – T. Shintake, @ http://www-xfel.spring8.or.jp/Index.htm

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Lpulse = Nuλrad

Nu = 5

{ { { { {

λrad ∝ λu 2γ 2

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Letargy Spontaneous Emission Low Gain Slow Bunching Exponential Growth Stimulated emission High Gain Enhanced Bunching Saturation Absorption No Gain Debunching

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Free Electron Laser 1D Self Consistent Model

dγ dt = − e mc  E ⋅  β = − e mc E⊥β⊥

Energy exchange occurs only if there is transverse motion Consider“seeding”by an external light source with wavelength λr The light wave is co-propagating with the relativistic electron beam

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Newton Lorentz Equations Maxwell Equations

J⊥

E,B

Problem: electrons are slower than light Question: can there be a continuous energy transfer from electron beam to light wave? Answer: We need a Self Consistent Model

(R. Bonifacio, C.Pellegrini, L.Narducci, Opt. Comm., 50, 373 (1984))

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The relative slippage of the radiation envelope through the electron beam can be neglected, provided that lb>>Nuλr (Steady State Regime) After one wiggler period the electron sees the radiation with the same phase if the flight time delay is exactly one radiation period:

Δt = te − tph = Trad

Δt = λu cβ// − λu c = λrad c & → & λrad = 1− β

//

β

//

λu

β

// ≈1

& → & & λrad ≈ λu 2γ 2 1+ K 2 2 * + ,

• .

/

γ res ≈ λu 2λrad 1 + K 2 2 % & ' ( ) *

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In a resonant and randomly phased electron beam, nearly one half of the electrons absorbs energy and one half loses energy, with no net energy exchange.

dγ dt = − e mec Exβx = − eEoK 2γmec cosψ −cosψ

[ ]

Ex z,t

( ) = Eo cos klz − ωlt +ψo ( )

kl = ωl c = 2π λl

Plane wave with constant amplitude , co-propagating with the electron beam:

ψ z,t

( ) = kl + ku ( )z − ωlt +ψo

Ponderomotive phase: Fast oscillating phase (we can neglect it)

βx = K γ cos kuz

( )

Ponderomotive potential

SLIDE 30

dψ dt = kl + ku

( )vz − klc ≈

ku<<kl klc

2 1 γr

2 − 1

γ 2 # $ % & ' ( 1+ K 2 2 # $ % & ' ( dψ dt ≈ 2kucγ −γr γr

βz =1− 1 2γ 2 1+ K 2 2 " # $ % & '

γr ≈ λu 2λrad 1+ K 2 2 " # $ % & '

Electrons with energies above the resonant energy move faster, while energies below will make the electrons fall back

Off-resonance electrons motion

SLIDE 31

λ

t>0 t=0 Optical potential Ponderomotive Potential t=0 …………………………

If the undulator is sufficiently long the energy modulation becomes a phase modulation: the electrons self-bunch on the scale of a radiation wavelength. The particles bunch around a phase where there is weak coupling with the radiation:

ψr

dψ dt ≈ 2kucγ −γr γr dψ dt > 0 for γ >γr dψ dt < 0 for γ <γr # $ % % & % %

SLIDE 32

λ

t>0 t=0 Optical potential Ponderomotive Potential t=0 …………………………

b z,t

( ) = 1

N e−iψ j

j= 1 N

∑

= e−iψ j

b ≈ 0

b → 1

Spontaneous emission Stimulated emission Bunching Parameter:

The Bunching parameter:

SLIDE 33

For particles with off resonance energy, the ponderomotive phase is no longer constant

Motion in the potential well: the electron pendulum equations

dψ dt ≈ 2kucγ −γr γr = 2kucη dη dt = 1 γr dγ dt = − eEoK 2γr

2mec cosψ

# $ % % & % %

η = γ − γ r γ r << 1

Two coupled first order differential equations

SLIDE 34

Combining the two coupled first order differential equations:

dψ dt = 2kucη dη dt = − eEoK 2γ r

2mec

cosψ & ' ( ( ) ( (

d 2ψ dt 2 = − eEoKku γ r

2me

cosψ

d 2ψ dt 2 + Ω 2 cosψ = 0

Ω 2 = eEoKku γ r

2me

ηsep = ± eEK kumec 2γ r

2 cos ψ −ψr

2 & ' ( ) * +

η η

ψ ψ

Separatrix

SLIDE 35

Letargy Exponential Growth Stimulated emission High Gain Enhanced Bunching Saturation Absorption No Gain Debunching

dψ dt = 2kucη dη dt = − eEoK 2γ r

2mec

cosψ & ' ( ( ) ( (

SLIDE 36

High gain FEL regime (1D model )

˜ E

x z,t

( ) = ˜

E

x z

( )ei kl z−ωl t

( ) = Eo z

( )eiϕ

2 ei kl z−ωl t

( )

Test solution

∇⊥

2 + ∂ 2

∂z2 − 1 c2 ∂ 2 ∂t2 $ % & ' ( )  Ex z,t

( ) = µo

∂ jx ∂t

2ikl ˜ " E

x z

( ) + ˜ "

" E

x z

( )

[ ]ei kl z−ωl t

( ) = µo

∂jx ∂t

Slowly Varying Envelope Approximation (SVEA):

the amplitude variation within one undulator period is very small

˜ " E

x z

( ) <<

˜ E

x z

( )

λu ⇒ ˜ " " E

x z

( ) <<

˜ " E

x z

( )

λu

d˜ E

x z

( )

dz = − iµo 2kl ∂jx ∂t e−i kl z−ωl t

( )

2ikl ˜ " E

x = µo

1 T ∂ ˜ j

x

∂t

t t +T

∫

e−i kl z−ωl t

( )dt

To be consistent with SVEA we should average also the source term

• ver a time in which could be considered constant

T ≈ n λl c

˜ E

x z

( )

SLIDE 37

˜ j

x = e

S vxj

j= 1 N

∑

δ z − z j t

( )

( ) =

e Svz vxj

j= 1 N

∑

δ t − t j z

( )

( )

1 T ˜ j

xe−i kl z−ωl t

( )dt

t t +T

∫

= e SvzT vxj

j= 1 N

∑

δ t − t j z

( )

( )e−i kl z−ωl t

( )dt

t t +T

∫

= e V vxj

j= 1 N

∑

e

−i kl z−ωl t j

( ) where : V = SvzT

= e V Kc γ j cos kuz

( )e

−i kl z−ωl t j

( )

j= 1 N

∑

using vxj = ..... = eKc Vγ r e

−i kl +ku

( )z−ωl t j

( )

j= 1 N

∑

= eKc Vγ r e−iψ j

j= 1 N

∑

using γ j ≈ γ r = eKc Vγ r N e−iψ j = eKc γ r ne e−iψ j where ne = N V

1 T ∂  jx ∂t

t t+T

∫

e

−i klz−ωlt

( )dt = −iωl

T  jxe

−i klz−ωlt

( ) dt

t t+T

∫

Integration by parts with: Beam model S: transverse beam area Exercise: verify there are not misprints (~mistakes):

∂ 2  jx ∂t2 ≈ 0

SLIDE 38

d  Ex dz = µo 2 eK γr ne e

−iψ j

dψ j dz = 2kucη j dη j dz = − eK 2mec2γr

2 ℜe 

Exe

iψ j

( )

# $ % % % & % % % j = 1,N e

b = 1 N e−iψ j

j= 1 N

∑

= e−iψ j

Three coupled first order differential equations. They describe a collective instability of the system which leads to electron self- bunching and to exponential growth of the radiation until saturation effects set a limit on the conversion of electron kinetic energy into radiation energy. Saturation effects prevent the beam to radiate as N2, limting the radiated power scaling to N4/3, due to a competition between neighbours slices . When propagation effects and slippage are relevant, i.e. when the elctron beam is as short as a slippage length, the emitted radiation leaves the bunch before saturation occurs and the power scaling becomes N2 (Super-radiant or Single Spike regime) Bunching parameter

SLIDE 39

• Case 1: not interesting

b z,t = 0

( ) = 0

˜ E

x z,t = 0

( ) = 0

" # $ % $ ⇒ ˜ E

x z,t

( ) = 0 d  Ex dz = µo 2 eK γr neb(z,t) dψ j dz = 2kucη j dη j dz = − eK 2mec2γr

2 ℜe 

Exe

iψ j

( )

# $ % % % & % % %

• Case 3: Self Amplification of Spontaneous Emission (SASE)

b z,t = 0

( ) ≠ 0

˜ E

x z,t = 0

( ) = 0

# $ % & % ⇒ ˜ E

x z,t

( ) ≠ 0

b z,t = 0

( ) = 0

˜ E

x z,t = 0

( ) ≠ 0

# $ % & % ⇒ ˜ E

x z,t

( ) ≠ 0

• Case 2: Amplification of input signal (Seeding)

SLIDE 40

The particles within a micro-bunch radiate coherently. The resulting strong radiation field enhances the micro-bunching even further. Result: collective instability, exponential growth of radiation power.

Newton Lorentz Equations Maxwell Equations

J⊥

E,B

SLIDE 41

Letargy Exponential Growth Saturation Absorption No Gain Debunching

dψ dt = 2kucη dη dt = − eEoK 2γ r

2mec

cosψ & ' ( ( ) ( (

d  Ex dz = µo 2 eK γr ne e

−iψ j

dψ j dz = 2kucη j dη j dz = − eK 2mec2γr

2 ℜe 

Exe

iψ j

( )

# $ % % % & % % %

SLIDE 42

dψ j dz = 2kucη j

z=2kuρz

! → !!!!!! dψ j dz = cη j ρ = cη dη j dz = − eK 2mec2γr

2 ℜe 

Exe

iψ j

( )

A= eK 4kuρ2mec2γr

2 ℜe 

Ex

( )

! → !!!!!! dη j dz = −ℜe Ae

iψ j

( )

d  Ex dz = µo 2 eK γr ne e

−iψ j

! → !!!!!!! dA dz = 1 ρ3 µone γr

3mec2

eK 4ku % & ' ( ) *

2

+ ,

• .

/ e

−iψ j

1 2 3 3 3 3 4 3 3 3 3

ρ = 1 γr µone mec2 eK 4ku ! " # $ % &

2

' ( ) ) * + , ,

1/3 ene=I/2πcσ x

2

IA=4πmc/eµo

• →
• --- 1

γr 2I IA K 4ckuσ x ! " # $ % &

2

' ( ) ) * + , ,

1/3

Universal Scaling – Adimensional variables –only one free parameter ρ

SLIDE 43

dψ j dz = cη dη j dz = −ℜe Ae

iψ j

( )

dA dz = e

−iψ j

# $ % % % & % % %

A

2 + η = const

η

sat = A

• 2 + η
• − A

sat 2 = −1 ⇒ η sat = Δγ

ργ ≈1 ⇒ Psat = ρP

beam

SLIDE 44

SASE FEL at short wavelengths require a very intense, high quality e-beam

• FEL Parameter
• Exponential growth
• Gain Length
• Saturation power
• Constraint on emittance
• Constraint on energy spread
• Relative bandwidth

ρ = 0.136 1 γ r J 1/ 3Bu

2 / 3λu 4 / 3

ρ π λ 3 4

u G

L =

! ! " # $ $ % & =

G

L z P z P exp 9 ) (

ε = εn γ < λ0 4π

P

sat = ρPbeam ∝ Ne 4 / 3

ρ γ γ < Δ

Δω ω = ρ Nu

SLIDE 45

SASE

Courtesy L. Giannessi (Perseo in 1D mode http://www.perseo.enea.it)

SLIDE 46

Radiation Simulator – T. Shintake, @ http://www-xfel.spring8.or.jp/Index.htm

SLIDE 47

SASE Longitudinal coherence

The radiation “slips” over the electrons for a distance Nuλrad

ζ

independent processes

Nuλrad

Slippage length ≈

SLIDE 48

SEEDING

Courtesy L. Giannessi (Perseo in 1D mode http://www.perseo.enea.it)

SLIDE 49

• R. Saldin et al. in Conceptual Design of a 500 GeV e+e- Linear

Collider with Integrated X-ray Laser Facility, DESY-1997-048

SASE FEL Electron Beam Requirements: High Brightness Bn

Bn = 2I εn

2

Bn

λr

MIN ∝σδ

1+ K 2 2

( )

γBnK 2

γ Bn K2

Lg ∝ γ 3 2 K Bnn 1+ K 2 2

( )

Bn

energy spread undulator parameter minimum radiation wavelength gain length

SLIDE 50

Bunch compressors (RF & magnetic) Laser Pulse shaping Emittance compensation Cathode emittance

Bn ≈ 2I εn

2

Short Wavelength SASE FEL Electron Beam Requirement: High Brightness Bn > 1015 A/m2

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