Monte Carlo Mean Field Treatment of Coherent Synchrotron...

Monte Carlo Mean Field Treatment of CSR Effects with Application to µBI in BCs / Gabriele Bassi

Monte Carlo Mean Field Treatment of Coherent

Synchrotron Radiation Effects with Application to

Microbunching Instability in Bunch Compressors

Gabriele BassiDepartment of Physics, University of Liverpool and the Cockcroft Institute, UK

Collaborators

Jim Ellison, Klaus Heinemann, Dept. of Math and Stats, University of New Mexico, Albuquerque, NM, USA

Robert Warnock, SLAC National Accelerator Laboratory, Menlo Park, CA, USA

1. Motivation

2. Coherent Synchrotron Radiation in Bunch Compressors

3. Self Consistent Vlasov-Maxwell Treatment in Lab Frame

4. Self Consistent Vlasov-Maxwell Treatment in Beam Frame

5. Beam to Lab Density Transformations

6. Field Calculation and Density Estimation

7. Microbunching Instability Studies for FERMI@Elettra

8. Discussion

Accelerator Physics Center, Fermilab, February 16, 2010, Chicago, USA


Motivation

• Collective effects, such as coherent synchrotron radiation (CSR), can play a

detrimental role on the beam quality and limit machine performance

• The need of high precision simulations of these effects requires state-of-the-art

numerical techniques and high performance computing

• A system to study collective effects: 6D Vlasov-Maxwell system (models the

particle beam as a non-neutral collisionless plasma)

• Its numerical integration is computationally too intensive

• A 4D Monte Carlo mean field approach has been developed and implemented on

parallel high performance computer clusters

• We discuss its application to the study of the microbunching instability for the

bunch compressor system of the FERMI@Elettra Free Electron Laser



FERMI@Elettra Free Electron Laser



ALICE Energy Recovery Linac

-



Coherent Synchrotron Radiation

Charged particles on a curved tra-

jectory (bending magnet) emit syn-

chrotron radiation

incoherent λ < σs

coherent λ > σs

Synchrotron radiation is coherent (CSR) at wavelenghts λ longer than the bunch

length σs.

If the bunch has microstructures, a coherent emission at wavelenghts shorter than

the bunch length can lead to an instability (microbunching instability).



Magnetic Bunch Compressor: 4-Dipole Chicane

• Particles at higher energy move along shorter trajectory

• A proper correlation (called chirp) between energy and long. position is created in

the linac before the chicane entrance, so that particles in the front of the bunch

have less energy than particles in the tail

• This leads to bunch compression



Magnetic Bunch Compressor System



Self Consistent Vlasov-Maxwell Treatment: Basic Lab Frame Setup

h



Self Consistent Vlasov-Maxwell Treatment for Sheet Beam in Lab Frame I

Sheet Beam model: f(Z, X, Y, PZ , PX , PY ; u0) = δ(Y )δ(PY )fL(Z, X, PZ , PX ; u0).

3D Wave equation (equivalent to Maxwell equations)

(∂2

Z + ∂2

X + ∂2

Y − ∂2

u)E = δ(Y )S(R, u), E(R, Y = ±g, u) = 0.

where u = ct, E(R, Y, u) = (EZ , EX , BY ), R = (Z, X)T and ˙= d/du.

Vlasov equation (equation for the evolution of the average number of particles per

unit volume for collisionless (uncorrelated) systems)

∂ufL + R · ∂RfL + P · ∂PfL = 0, fL(R,P; ui) = fL0(R,P)

where

R =P

mγ(P )c,

P =q

c

[

(

EZ(R; u)

EX(R; u)

)

+[Bext(Z) + BY (R; u)]

mγ(P)

(

PX

−PZ

)

]

.

and P = (PZ , PX)T and (EZ(R; u), EX(R; u), BY (R; u)) ≡ E(R, 0; u).



Self Consistent Vlasov-Maxwell Treatment for Sheet Beam in Lab Frame II

Field Formula:

FL(R; u) = − 1

4π

∞∑

k=−∞

(−1)k

∫

R2

dR′S(R′; u − [|R′ − R|2 + (kh)2]1/2)

[|R′ − R|2 + (kh)2]1/2.

where FL(R; u) = (EZ(R; u), EX(R; u), BY (R; u)) and the source is

S(R; u) = Z0QH(u − ui)

c∂ZρL + ∂uJL,Z

c∂XρL + ∂uJL,X

∂XJL,Z − ∂ZJL,X

,

where H is the unit step function.The Vlasov equation and the self fields are coupled by QρL and QJL

ρL(R; u) =

∫

R2

dPfL(R,P; u),

JL(R; u) =

∫

R2

dP(P/mγ(P))fL(R,P; u).



Self Consistent Vlasov-Maxwell Treatment for Sheet Beam in Beam Frame I

Beam frame Frenet-Serret coordinates defined in terms of the reference orbit

Rr(s) = (Zr(s), Xr(s))T in the Y = 0 plane.

Phase space transformation (R,P; u) → (s, x, ps, px; u)

R = Rr(s) + xn(s), P = Pr(pst(s) + pxn(s)).

Lab to beam transformation steps

(R,P; u) → (s, x, ps, px; u) → (u, x, ps, px; s) → (z, x, pz, px; s),

where z := s − βru and pz := (γ − γr)/γr.

Exact relation between lab, fL, and beam, fB, phase space densities

fB(r,p; s) =P 2

r

β2r

fL{Rr(s) + xn(s), Pr[ps(p)t(s) + pxn(s)]; (s − z)/βr}.



Self Consistent Vlasov-Maxwell Treatment for Sheet Beam in Beam Frame II

Approximate beam frame equations of motion

z′ = −κ(s)x, p′z = Fz1(z, x; s) + pzFz2(z, x; s),

x′ = px, p′x = κ(s)pz + Fx(z, x; s),

where the self forces are

Fz1 =q

PrcE‖(R(s, x);

s − z

βr) · t(s), Fz2

q

PrcE‖(R(s, x);

s − z

βr) · n(s),

Fx =q

Prc[E‖(R(s, x);

s−z

βr

) · n(s) − cBY (R(s, x);s − z

βr

)].

Associated Vlasov IVP for the evolution of the beam frame phase space density

∂sfB + r′ · ∇rfB + p′ · ∇pfB = 0, fB(r,p; 0) =: fB0(r,p).



Beam to Lab Density Transformations

• To solve Maxwell equations in lab frame must express lab frame charge/current

density in terms of beam frame phase space density

• To a very good approximation

ρB(r; s) ≈ ρL(Rr(s) + xn(s); (s − z)/βr),

thus

ρL(R; u) ≈ ρB(s(R) − βru, x(R); s(R)).

Replacing s by βru + z and expanding in z gives ρL(Rr(βru) + M(βru)r; u) ≈ρB(r; βru + z), where M(s) = [t(s),n(s)]. Finally, inverting (similarly for JL)

ρL(R; u) ≈ ρB( MT (βru)[R− Rr(βru)]; βru),

JL(R; u) ≈ βrc[ρB( MT (βru)[R − Rr(βru)]; βru)t(βru)

+τB( MT (βru)[R − Rr(βru)]; βru)n(βru)].

where τB(z, x; s) :=∫

pxfB(z, x, pz, px; s)dpzdpx.



Fields in Terms of Beam Frame Density and Causality Issue

To solve the beam frame equations of motion we need (ignoring shielding)

FL(Rr(s)+xn(s); (s−z)/βr)=−∫

R2

dR′S[R′; (s − z)/βr − |R′ − (Rr(s) + xn(s))|]4π|R′ − (Rr(s) + xn(s))| ,

To compute this we need ρL[R′; (s− z)/βr − |R′ − (Rr(s) + xn(s))|], as R′ varies

over the support of ρL in R2, given ρB(·; s′) for 0 ≤ s′ ≤ s.

There is a causality issue here, since the calculation of ρL requires values ρB for s′

slightly outside the range 0 ≤ s′ ≤ s.

This issue can be easily resolved with the following slowing varying approximation

fB(r,p; s) ≈ fB(r,p; s + ∆)

where ∆ is of the order of the bunch size.



Field Calculation: Polar Coordinates

• Transform to polar coordinates (χ, θ), and then take the temporal argument v in

place of the radial coordinate χ: make the transformation R′ → (θ, v) via

R′ = R + χe(θ), e(θ) = (cos θ, sin θ)T , v = u −√

χ2 + (kh)2

This conveniently removes the integrable singularity, giving the field simply as an

integral over the source (ignoring shielding)

FL(Rr(s) + M(s)r, s/βr) = − 1

4π

∫ s/βr

ui

dv

∫ θM

θm

dθ S[R(θ, v; r, s), v],

where R(θ, v; r, s) = Rr(s) + M(s)r + (s/βr − v)e(θ).

• θ integration: superconvergent trapezoidal rule (localization in θ for v ≪ s/βr)

• v integration: adaptive Gauss-Kronrod rule (non uniform behavior in v)

The computational effort is O(NzNxNvNθ), where Nz and Nx are the number of

grid points in z ans x respectively, Nv is the number of evaluations for the v

integration, and Nθ is the number of evaluations for the θ integration.

For Nz = 1000, Nx = 128, Nv = Nθ = 1000, O(NzNxNvNθ) = O(1012).



Density Estimation: Orthogonal Series Method

• From scattered beam frame points at s → smooth/global lab frame charge/current

density via a 2D Fourier method.

1D Example: 1D orthogonal series estimator of f(x), x ∈ [0, 1]

fJ(x) :=J∑

j=0

θjφj(x), θj =

∫

1

0

φj(x)f(x)dx, φ0(x) = 1, φj(x) =√

2 cos(jπx), j = 1, 2, ...

Since f(x) is a probability density (X,Xn random variables distributed via f)

θj = E{φj(X)}, thus from Monte Carlo a natural estimate is θj :=1

N

N∑

n=1

φj(Xn)

• The computational effort is O(NJzJx), where N is the number of simulated

particles, Jz and Jx the number of Fourier coefficients in z and x respectively.

For N = 5 × 108, Jz = 150 and Jx = 50, O(NJzJx) = O(1012).



Density Estimation: Search for Improvement

• Cloud in cell charge deposition followed by computation of the Fourier coefficients

of the truncated Fourier series by a simple quadrature (already implemented).

The computational effort is O(N ) + O(NzNxJzJx), where N is the number of

simulated particles, Nz and Nx are the number of grid points in z ans x

respectively, and Jz and Jx the number of Fourier coefficients in z and x

respectively.

For Nz = 1000, Nx = 128, Jz = 150 and Jx = 50, O(NzNxJzJx) = O(109).

• Kernel density estimation using standard kernels like bivariate Gaussians or bivariate

compact support kernels (e.g. Epanechnikov kernels).

The computational effort is O(N NzNx), where N is the number of simulated

particles and NzNx is the number of grid points inside the circle of radius h

(bandwidth) centered at the scattered particle position z, x.

For N = 5 × 108 and Nz = Nx = 4, O(NNzNx) = O(1010).

• Wavelets-denoising (G. Bassi, B. Terzic, PAC09)



Interaction Picture

• Interaction picture to isolate CSR dynamics.

From Fz = Fx = 0 =⇒ ζ = Φ(s|0)ζ0

∴ ζ ′0

= Φ(0|s)F, F = (0, Fz, 0, Fx).

• In component form

z′0

= −R56(s)Fz − D(s)Fx, p′z0= Fz,

x′0

= (sD′(s) − D(s))Fz − sFx, p′x0= −D′(s)Fz + Fx,

where D(s) =∫ s

0κ(τ)dτ and R56(s) = −

∫ s

0D(τ)κ(τ)dτ .

Here the principal solution matrix is

Φ(s|0) =

1 R56(s) −D′(s) D(s) − sD′(s)

0 1 0 0

0 D(s) 1 s

0 D′(s) 0 1

.



Microbunching in FERMI@Elettra First Bunch Compressor

Microbunching can cause an instability which degrades beam quality, i.e. can

cause an increase in energy spread and emittance

This is a major concern for free electron lasers where very bright electron beams are

required

We discuss numerical results for the FERMI@Elettra first bunch compressor system.

See G. Bassi, J.A. Ellison, K. Heinemann and R. Warnock, PRSTAB 12, 080704

(2009)

This system was proposed as a benchmark for testing codes at the first Workshop

on Microbunching Instability held in Trieste in 2007

Layout first bunch compressor system



FERMI@Elettra First Bunch Compressor Parameters

Table 1: Chicane parameters and beam parameters at first dipole

Parameter Symbol Value Unit

Energy reference particle Er 233 MeVPeak current I 120 ABunch charge Q 1 nCNorm. transverse emittance γǫ0 1 µmAlpha function α0 0Beta function β0 10 mLinear energy chirp h -12.6 1/mUncorrelated energy spread σE 2 KeVMomentum compaction R56 0.057 mRadius of curvature ρ0 5 mMagnetic length Lb 0.5 mDistance 1st-2nd, 3rd-4th bend L1 2.5 mDistance 2rd-3nd bend L2 1 m



Initial 2D Spatial Density

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6

-0.4

-0.2

0

0.2

0.4

0.6 0

0.5

1

1.5

2

2.5

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

Initial spatial density in grid coordinates for A=0.05, λ0 = 100µm.

Init. phase space density = (1 + A cos(2πz/λ0))µ(z)ρc(pz − hz)g(x, px).



Gain factor

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 100 200 300 400 500 600

Ga

in

λ0 (µm)

Numeric: ratio of values at k0 and Cfk0Analytic: CSR on

Numeric: ratio of peak values Analytic: CSR off

Gain := |ρ(kf , sf)/ρ(k0, 0)|, ρ(k, s) =∫

dz exp(−ikz)ρ(z, s) and kf = C(sf)k0

for λ0 = 2π/k0. Here C(sf) = 1/(1 + hR56(sf)) = 3.54, sf = 829m.H. Huang and K. Kim, PRSTAB 5, 074401, 129903 (2002); S. Heifets, G. Stupakov and S. Krinsky, PRSTAB 5,064401 (2009); G. Bassi, J.A. Ellison, K. Heinemann and R. Warnock, PRSTAB 12, 080704 (2009).



Spectra Longitudinal Density I

0

0.2

0.4

0.6

0.8

1

0 5000 10000 15000 20000 25000 30000 35000

˜ |ρ(k

,s)|

k (1/m)

λ0 = 300 µm (k0 = 20944 1/m), Gain = 0.0243/0.0240= 1.0128

(20944,0.0240)

s=0 m, A=0.05s=0 m, A=0

0

0.2

0.4

0.6

0.8

1

0 20000 40000 60000 80000 100000

˜ |ρ(k

,s)|

k (1/m)

λ0 = 300 µm (k0 = 20944 1/m), Gain = 0.0243/0.0240= 1.0128

(74247,0.0243)

s=8.028 m, A=0.05s=8.028 m, A=0

0

0.2

0.4

0.6

0.8

1

0 5000 10000 15000 20000

˜ |ρ(k

,s)|

k (1/m)

λ0 = 600 µm (k0 = 10472 1/m), Gain = 0.0236/0.0176 = 1.3409

(10472,0.0176)

s=0 m, A=0.05s=0 m, A=0

0

0.2

0.4

0.6

0.8

1

0 10000 20000 30000 40000 50000

˜ |ρ(k

,s)|

k (1/m)

λ0 = 600 µm (k0 = 10472 1/m), Gain = 0.0236/0.0176 = 1.3409

(37123,0.0236)

s=8.028 m, A=0.05s=8.028 m, A=0



Spectra Longitudinal Density II

0

0.05

0.1

0.15

0.2

100000 200000 300000 400000 500000 600000

˜ |ρ(k

,s)|

k (1/m)

λ0 = 100 µm (k0 = 62832 1/m), Gain = 0.0649/0.0248 =2.6169

(222741,0.0649)(62832,0.0248)

s=8.028 m, A=0.05s=0 m, A=0.05

0

0.05

0.1

0.15

0.2

100000 200000 300000 400000 500000 600000

˜ |ρ(k

,s)|

k (1/m)

λ0 = 80 µm (k0 = 78534 1/m), Gain = 0.0620/0.0248 = 2.5000

(278405,0.0620)(78534,0.0248)

s=8.028 m, A=0.05s=0 m, A=0.05

0

0.05

0.1

0.15

0.2

100000 200000 300000 400000 500000 600000

˜ |ρ(k

,s)|

k (1/m)

λ0 = 60 µm (k0 = 104720 1/m), Gain = 0.0593/0.0248 =2.3870

(371235,0.0593)(104720,0.0248)

s=8.028 m, A=0.05s=0 m, A=0.05

0

0.05

0.1

0.15

0.2

100000 200000 300000 400000 500000 600000

˜ |ρ(k

,s)|

k (1/m)

λ0 = 40 µm (k0 = 157070, 1/m), Gain = 0.0291/0.0248 = 1.1734

(556815,0.0291)(157070,0.0248)

s=8.028 m, A=0.05s=0 m, A=0.05



Longitudinal Density I

0

0.5

1

1.5

2

-1.5 -1 -0.5 0 0.5 1 1.5

ρ(z,s

) (1

/mm

)

z (mm)

s=sf: self fields OFFs=0

0

0.5

1

1.5

2

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

ρ(z,s

) (1

/mm

)

z (mm)

self fields OFFself fields ON

0

0.5

1

1.5

2

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

ρ(z,s

) (1

/mm

)

z (mm)


0

0.5

1

1.5

2

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

ρ(z,s

) (1

/mm

)

z (mm)


λ0 = 200µm (top left), λ0 = 200µm at s = sf (top right),λ0 = 100µm at s = sf (bottom left), λ0 = 80µm at s = sf (bottom right).



Longitudinal Density II

0

0.5

1

1.5

2

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

ρ(z,s

) (1

/mm

)

z (mm)

0

0.5

1

1.5

2

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

ρ(z,s

) (1

/mm

)

z (mm)

0

0.5

1

1.5

2

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

ρ(z,s

)(1/m

m)

z (mm)

0

0.5

1

1.5

2

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

ρ(z,s

) (1

/mm

)

z (mm)

λ0 = 200µm at s = sf (top left), λ0 = 100µm at s = sf (top right),λ0 = 60µm at s = sf (bottom left), λ0 = 40µm at s = sf (bottom right).



Stationary Grid

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6

-0.4

-0.2

0

0.2

0.4

0.6 0

0.5

1

1.5

2

2.5

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6

-0.4

-0.2

0

0.2

0.4

0.6 0

0.5

1

1.5

2

2.5

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6

-0.4

-0.2

0

0.2

0.4

0.6 0

0.5

1

1.5

2

2.5

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6

-0.4

-0.2

0

0.2

0.4

0.6 0

0.5

1

1.5

2

2.5

2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

λ0=200µm. s=0.25sf (top left), s=0.5sf (top right), s=0.75sf (bottom left), s=sf (bottom right).



2D spatial density and longitudinal force at s = sf

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6

-0.4

-0.2

0

0.2

0.4

0.6 0

0.5

1

1.5

2

2.5

2.2 2

1.8 1.6 1.4 1.2 1

0.8 0.6 0.4 0.2

z

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6

-0.4

-0.2

0

0.2

0.4

0.6 0

0.5

1

1.5

2

2.5

,xn,s)

2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2

z

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-0.0015

-0.001

-0.0005

0

0.0005

0.0003 0.0001 -1e-04 -0.0003 -0.0005 -0.0007

z

-0.6 -0.4 -0.2 0 0.2 0.4 0.6-0.6

-0.4

-0.2

0

0.2

0.4

0.6

-0.0015

-0.001

-0.0005

0

0.0005

0.0003 0.0001 -1e-04 -0.0003 -0.0005 -0.0007 -0.0009 -0.0011 -0.0013

z

λ0 = 200µm (top left), λ0 = 100µm (top right), λ0 = 200µm (bottom left), λ0 = 100µm (bottom right)



Mean Power, Transverse Emittance and 2D vs 1D longitudinal force

-0.0009

-0.0008

-0.0007

-0.0006

-0.0005

-0.0004

-0.0003

-0.0002

-1e-04

0

0.0001

0 1 2 3 4 5 6 7 8

mean

po

wer

path length (m)

A=0A=0.05, λ=100µm

0

1e-05

2e-05

3e-05

4e-05

5e-05

6e-05

7e-05

8e-05

9e-05

0 1 2 3 4 5 6 7 8

x-e

mit

tan

ce (

m-r

ad

)

path length (m)

A=0: 1.48 mm-mrad

A=0.05: λ=100µm: 1.50 mm-mrad

A=0A=0.05, λ=100µm

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.0012

-0.001

-0.0008

-0.0006

-0.0004

-0.0002

0

0.0002

0.0004

0.0006

-5e-05

-4e-05

-3e-05

-2e-05

-1e-05

0

1e-05

2e-05

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

Mean power (top left), transverse emittance (top right), 2D longitudinal force (section at x = 0)

at s = sf (bottom left), 1D longitudinal force (steady state CSR wake) (bottom right).



Discussion

• FERMI@Elettra microbunching studies at λ0 ≥ 40µm:

- Very small effect of µBI on mean power and transverse emittance

- Gain factor at long wavelengths shows breakdown coasting beam assumption

- Gain factor at short wavelengths indicates deviations from analytical gain formula

• Work in progress and future work:

- Study wavelengths shorter than λ0 = 40µm

- Study dependence on the amplitude of the initial modulation and on the

uncorrelated energy spread

- Study initial perturbation with more than one frequency

- Study energy modulations.



Computational Issues

• Intensive memory requirement and expensive computational cost:

- Typical simulations done on the parallel clusters ENCANTO in New Mexico and

NERSC at LBNL: N procs = 200-1000, N particles = 2×107-5×108, few hours

of CPU time

- Memory requirement: for λ0 = 50µm store 3D array of dimension

1500 × 128 × 200 on master processor (to avoid massive communications

between slave processors)

• To reduce storage/computational cost:

- Analytical work + state of the art numerical techniques: integration,

interpolation, density estimation

- Parallel computing


Monte Carlo Mean Field Treatment of Coherent Synchrotron...

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