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

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

Massimo.Ferrario@LNF.INFN.IT ¡

• 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

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

Longitudinal ¡Manipula.on ¡

The problem of relativistic bunch length

γ ≈1 Lb = 3mm ≈ " Lb γ =1000 Lb = ! Lb γ = 3µm

Length contraction? Low energy electron bunch injected in a linac:

Magnetic compressor (Chicane)

σz0 ΔΕ/Ε z σz

under- compression

V = V0sin(ωτ)

RF Accelerating Voltage

Δz = R56ΔΕ/Ε

Path Length-Energy Dependent Beamline …or over- compression

ΔΕ/Ε z σE/E ΔΕ/Ε z ‘chirp’

Magnetic compressor (Chicane)

Courtesy

• P. Emma

ΔΕ/Ε = 0

Δx = R16(s)ΔE/E bend-plane emittance growth e–

R

Coherent Synchrotron Radiation (CSR)

σz

coherent radiation for λ > σz

• vertaking length: L0 ≈ (24σzR2)1/3

ΔΕ/Ε < 0 s Δx

• Powerful radiation generates energy spread in bends
• Causes bend-plane emittance growth (short bunch worse)
• Energy spread breaks achromatic system

θ L0 λ

Courtesy

• P. Emma

Velocity bunching concept (RF Compressor)

If the beam injected in a long accelerating structure at the crossing field phase and it is slightly slower than the phase velocity of the RF wave , it will slip back to phases where the field is accelerating, but at the same time it will be chirped and compressed. The key point is that compression and acceleration take place at the same time within the same linac section, actually the first section following the gun, that typically accelerates the beam, under these conditions, from a few MeV (> 4) up to 25-35 MeV.

FEL Single Spike THz Radiation LWFA_ext LASER COMB Velocity Bunching PWFA Thomson C_Band injector DWFA Narrow THz Rad 2 pulses FEL

Average current vs RF compressor phase

100 200 300 400 500 600 700 800 900 1000 1100 1200 1300

• 95
• 90
• 85
• 80
• 75
• 70
• 65
• 60

RF compressor phase (deg) Average current (A)

LOW COMPRESSION MEDIUM COMPRESSION HIGH COMPRESSION OVER- COMPRESSION

Transverse ¡Beam ¡Dynamics ¡

https://arxiv.org/ftp/arxiv/papers/1705/1705.10564.pdf

X X’

Trace space of an ideal laminar beam

x ! x = dx dz = px pz " # $ % $ px << pz

X X’

Trace space of a laminar beam

X X’

Trace space of non laminar beam

In a system where all the forces acting on the particles are linear (i.e., proportional to the particle’s displacement x from the beam axis), it is useful to assume an elliptical shape for the area occupied by the beam in x-x‘ trace space.

! x! x x

 x + kx = 0

Twiss parameters:

1

2 =

−α βγ

Ellipse equation: Geometric emittance:

εg

γx 2 + 2αx $ x + β $ x 2 = εg

Ellipse area:

A = πεg ! β = −2α

Trace space evolution

With space charge => no cross over No space charge => cross over

X X’ X X’

rms emittance

x x’

σ x σ x'

σ x

2 = x 2 =

x 2

−∞ +∞

∫

−∞ +∞

∫

f x, & x

( )dxd &

x

rms beam envelope:

γx 2 + 2αx $ x + β $ x 2 = εrms

σ x = x 2 = βεrms σ x' = % x 2 = γεrms

Define rms emittance: such that:

εrms

f x, ! x

( )dxd !

x

−∞ +∞

∫

−∞ +∞

∫

=1

! f x, ! x

( ) = 0

γβ −α 2 = 1

σ x'

2

εrms σ x

2

εrms − σ xx' εrms " # $ % & '

2

=1

It holds also the relation: Substituting we get

α,β,γ εrms = σ x

2σ x' 2 −σ xx' 2 =

x2 " x 2 − x " x

2

( )

We end up with the definition of rms emittance in terms of the second moments of the distribution: σ x = x2 = βεrms σ x' = x'2 = γεrms σ xx' = x ! x = −αεrms

" x = px pz

α = − " β 2

x x’ a a’

εrms

2

= x 2 # x 2 − x # x

2

! x = Cxn

εrms

2

= C2 x2 x2n − xn+1

2

( )

When n = 1 ==> εrms = 0 When n = 1 ==> εrms = 0

x x’ a a’

What does rms emittance tell us about phase space distributions under linear or non-linear forces acting on the beam? Assuming a generic correlation of the type:

x, " x

εn,rms = σ x

2σ px 2 −σ xpx 2 = 1

moc x2 px

2 − xpx 2

( )

Normalized rms emittance:

px = pz ! x = mocβγ ! x

Canonical transverse momentum: Liouville theorem: the density of particles n, or the volume V

• ccupied by a given number of particles in phase space

(x,px,y,py,z,pz) remains invariant. It hold also in the projected phase spaces (x,px),(y,py)(,z,pz) provided that there are no couplings

pz ≈ p

εn,rms

Limit of single particle emittance

Limits are set by Quantum Mechanics on the knowledge of the two conjugate variables (x,px). According to Heisenberg: This limitation can be expressed by saying that the state of a particle is not exactly represented by a point, but by a small uncertainty volume of the order of in the 6D phase space. In particular for a single electron in 2D phase space it holds:

σ xσ px ≥ ! 2 !3

εn,rms = 1 moc σ x

2σ px 2 −σ xpx 2 ⇒

= 0 classical limit ≥ 1 2 ! moc = ! c 2 =1.9×10−13m quantum limit % & ' ( '

Where is the reduced Compton wavelength.

! c

εn,rms = 1 moc x2 px

2 − xpx 2

( ) =

x2 βγ " x

( )

2 − xβγ "

x

2

( ) = βγ εrms

Assuming small energy spread within the beam, the normalized and un-normalized emittances can be related by the above approximated relation.

px = pz ! x = mocβγ ! x

This approximation that is often used in conventional accelerators may be strongly misleading when adopted to describe beams with significant energy spread, as the one at present produced by plasma accelerators. Normalized and un-normalized emittances

When the correlations between the energy and transverse positions are negligible (as in a drift without collective effects) we can write:

εn,rms

2

= β 2γ 2 x2 ! x 2 − βγ

2 x !

x

2

Considering now the definition of relative energy spread:

σ γ

2 =

β 2γ 2 − βγ

2

βγ

2

which can be inserted in the emittance definition to give:

εn,rms

2

= β 2γ 2 σ γ

2 x2

! x 2 + βγ

2

x2 ! x 2 − x ! x

2

( )

Assuming relativistic electrons (β=1) we get:

εn,rms

2

= γ 2 σ γ

2σ x 2σ ! x 2 +εrms 2

( )

εn,rms

2

= γ 2 σ γ

2σ x 2σ ! x 2 +εrms 2

( ) = γ 2

σ γ

2σ o, ! x 4

z − zo

( )

2 +εrms 2

( )

showing that beams with large energy spread an divergence undergo a significant normalized emittance growth even in a drift

• Energy 350 MeV
• Beam divergence 1 mrad
• Energy spread 1%
• Beam spot-size 1 µm

Z <γ>ε

Simulation Formula

Phase space, slice emittance and longitudinal correlations x px Projected Phase Space Slice Phase Spaces FEL cooperation length

γβ −α 2 = 1

σ x'

2

εrms σ x

2

εrms − σ xx' εrms " # $ % & '

2

=1

It holds also the relation: Substituting we get

α,β,γ εrms = σ x

2σ x' 2 −σ xx' 2 =

x2 " x 2 − x " x

2

( )

We end up with the definition of rms emittance in terms of the second moments of the distribution:

σ x = x2 = βεrms ! σ x = x'2 = γεrms σ xx' = x ! x = −αεrms

" x = px pz

dσ x dz = d dz x2 = 1 2σ x d dz x2 = 1 2σ x 2 x ! x = σ x !

x

σ x d 2σ x dz2 = d dz σ x !

x

σ x = 1 σ x dσ x !

x

dz − σ x !

x 2

σ x

3 = 1

σ x ! x 2 + x ! x

( )− σ x !

x 2

σ x

3 = σ ! x 2 + x !!

x σ x − σ x !

x 2

σ x

3

Envelope Equation without Acceleration

Now take the derivatives:

!! σ x = σ x

2σ x' 2 −σ xx' 2

σ x

3

+ x !! x σ x = εrms

2

σ x

3 + x !!

x σ x

And simplify:

!! σ x − x !! x σ x = εrms

2

σ x

3

We obtain the rms envelope equation in which the rms emittance enters as defocusing pressure like term.

εrms

2

σ x

3 ≈ T

V ≈ P

kBTx = m vx

2 T = 1

3 Tx +Ty +Tz

( ) Ek = 1

2 m v2 = 3 2 kBT

Beam Thermodynamics

Kinetic theory of gases defines temperatures in each directions and global as: Definition of beam temperature in analogy:

kBTbeam,x =γmo vx

2

vx

2 = β 2c2

! x 2 = β 2c2σ x'

2 = β 2c2 εrms 2

σ x

2

We get:

kBTbeam,x =γmo vx

2 =γmoβ 2c2 εrms 2

σ x

2

P

beam,x = nkBTbeam,x = nγmoβ 2c2 εrms 2

σ x

2 = NTγmoβ 2c2 εrms 2

σ Lσ x

2

n = N πσ Lσ x

2 = NT

σ L

S = kN log πε

( )

kBTbeam,x =γmoβ 2c2 εrms βx

Assuming that each particle is subject only to a linear focusing force, without acceleration: take the average over the entire particle ensemble

!! x + kx

2x = 0

" " σ x + kx

2σ x = εrms 2

σ x

3

x !! x = −kx

2 x2

We obtain the rms envelope equation with a linear focusing force in which, unlike in the single particle equation of motion, the rms emittance enters as defocusing pressure like term.

!! σ x − x !! x σ x = εrms

2

σ x

3

Envelope Equation with Linear Focusing

Space Charge: What does it mean?

The net effect of the Coulomb interactions in a multi-particle system can be classified into two regimes: 1) Collisional Regime ==> dominated by binary collisions caused by close particle encounters ==> Single Particle Effects 2) Space Charge Regime ==> dominated by the self field produced by the particle distribution, which varies appreciably only over large distances compare to the average separation of the particles ==> Collective Effects

γ= 1 γ = 5 γ = 10 L(t) Rs(t) Δt

Er(r,s,γ ) = Ir 2πε0R2βc g s,γ

( )

Ez(0,s,γ ) = I 2πγε0R2βc h s,γ

( ) Fr = eEr γ 2 = eIr 2πγ 2ε0R2βc g s,γ

( )

Lorentz Force

F

r = e Er − βcBϑ

( ) = e 1− β 2

( )Er = eEr

γ 2

The attractive magnetic force , which becomes significant at high velocities, tends to compensate for the repulsive electric force. Therefore space charge defocusing is primarily a non-relativistic effect. is a linear function of the transverse coordinate

dpr dt = Fr = eEr γ 2 = eIr 2πγ 2ε0R2βc g s,γ

( ) B

ϑ = β

c Er

Fx = eIx 2πγ 2ε0σ x

2βc g s,γ

( )

Er(r,s,γ ) = Ir 2πε0R2βc g s,γ

( ) ksc s,γ

( ) =

2I IA βγ

( )

3 g s,γ

( )

Envelope Equation with Space Charge

!! x = ksc s,γ

( )

σ x

2

x

Space Charge de-focusing force Single particle transverse motion:

dpx dt = Fx px= p ! x = βγmoc ! x d dt p ! x

( ) = βc d

dz p ! x

( ) = Fx

!! x = Fx βcp

Generalized perveance

ksc s,γ

( ) =

2I IA βγ

( )

3 g s,γ

( )

IA = 4πεomoc3 e =17kA

x !! x = ksc σ x

2 x2 =ksc

!! σ x + k2σ x = εn

2

βγ

( )

2σ x 3 + ksc

σ x

External Focusing Forces Space Charge De-focusing Force Emittance Pressure Now we can calculate the term that enters in the envelope equation

x " " x

" " σ x = εrms

2

σ x

3 − x "

" x σ x

Including all the other terms the envelope equation reads:

ρ = βγ

( )

2 kscσ x 2

εn

2

Laminarity Parameter:

" " σ x + k 2σ x = εn

2

βγ

( )

2σ x 3 + ksc

σ x " " σ x + k 2σ x = εn

2

βγ

( )

2σ x 3 + ksc

σ x

ρ>>1 ρ<<1 Laminar Beam Thermal Beam The beam undergoes two regimes along the accelerator

Space Charge induced emittance oscillations in a laminar beam

Surface charge density Surface electric field Restoring force Plasma frequency Plasma oscillations

Neutral Plasma

Magnetic focusing Magnetic focusing

Single Component Cold Relativistic Plasma

• Oscillations
• Instabilities
• EM Wave propagation

Single Component Relativistic Plasma

" " σ + ks

2σ = ksc s,γ

( )

σ

ks = qB 2mcβγ

δ # # σ s

( ) + 2ks

2δσ s

( ) = 0

σ eq s,γ

( ) =

ksc s,γ

( )

ks

Equilibrium solution:

σ ζ

( ) = σ eq s ( ) +δσ s ( )

Small perturbation:

σ s

( ) = σ eq s ( ) +δσ o s ( )cos

2ksz

( )

Perturbed trajectories oscillate around the equilibrium with the same frequency but with different amplitudes:

δσ s

( ) = δσ o s ( )cos

2ksz

( )

σ s

( ) = σ eq s ( ) +δσ o s ( )cos

2ksz

( )

Perturbed trajectories oscillate around the equilibrium with the same frequency but with different amplitudes:

εrms = σ x

2σ x' 2 −σ xx' 2 ≈ sin

2ksz

( )

1 2 3 4 5 metri

• 0.5

0.5 1 1.5 2 envelopes

1 2 3 4 5 metri 20 40 60 80 emi

σ(z) ε(z) Envelope oscillations drive Emittance oscillations

δγ γ = 0

! σ = 0

εrms = σ x

2σ x' 2 −σ xx' 2 =

x 2 % x 2 − x % x

2

( ) ≈ sin

2ksz

( )

Emittance Oscillations are driven by space charge differential defocusing in core and tails of the beam x px Projected Phase Space Slice Phase Spaces

X X ’ Perturbed trajectories oscillate around the equilibrium with the same frequency but with different amplitudes

Plasma Accelerator

po =γomoβoc px << po p = po + ! p z ! p = βγ

( )! moc

dpx dt = d dt p ! x

( ) = βc d

dz p ! x

( ) = 0

!! x + ! p p ! x = 0

!! x = − βγ

( )!

βγ ! x

Envelope Equation with Longitudinal Acceleration

Envelope Equation with Longitudinal Acceleration

x !! x = − βγ

( )!

βγ x ! x = − βγ

( )!

βγ σ xx'

!! σ x + βγ

( )!

βγ ! σ x + k2σ x = εn

2

βγ

( )

2σ x 3 + ksc

σ x

Other External Focusing Forces Space Charge De-focusing Force Adiabatic Damping Emittance Pressure

εn = βγεrms dσ x dz = ! σ x = σ xx' σ x

" " σ x = εrms

2

σ x

3 − x "

" x σ x

!! σ x + ! γ γ ! σ x + kp

2

3γ σ x = εn

2

γ 2σ x

3 + ksc

• γ 3σ x

Envelope ¡equa.on ¡in ¡a ¡plasma ¡accelerator ¡

Er = en1 3εo r

Radial ¡field ¡

Rsphere ≈ λp 2

Bubble ¡radius ¡

ρ+

n1 ≈ ndrive

Bubble ¡density ¡

F

r = e Er − βcBϑ

( ) = e2n1

3ε0 r

x !! x = kp

2

γ x2 = kp

2

γ σ x

2

!! x = Fx βcp = e2n1x 3εoγmc2 = kp

2

3γ x

kp

2 = e2n1

εomc2

!! γ = 0 ! γ ≠ 0

When

η = 4γkp

2

3 ! γ 2 >>1

!! σ x

x + kp 2

3γ σ x = εn

2

γ 2σ x

3

ρ = ksc

0σ x 2

γoεn

2 <<1

Looking for an equilibrium solution of the form: σ ε =γ nσ o We get the matching condition with acceleration:

σ ε = 3 γ

4

εn kp

1¥1016 2¥1016 3¥1016 4¥1016 5¥1016 3.0 3.5 4.0 4.5 5.0

n @cm-3D sigma_r @umD

εn = 2µm