Coherent Synchrotron Radiation and Longitudinal Beam Dynamics in Rings

July 2002 M. Venturini 1

Coherent Synchrotron Radiation and Longitudinal

Beam Dynamics in Rings

M. Venturini and R. WarnockStanford Linear Accelerator Center

ICFA Workshop on High Brightness Beams Sardinia, July 1-5, 2002


Outline• Review of recent observations of CSR in electron

storage rings. Radiation bursts.• Two case studies:

– Compact e-ring for a X-rays Compton Source.– Brookhaven NSLS VUV Storage Ring.

• Model of CSR impedance. • Modelling of beam dynamics with CSR in terms of

1D Vlasov and Vlasov-Fokker-Planck equation.– Linear theory CSR-driven instability. – Numerical solutions of VFP equation. Effect of nonlinearities. – Model reproduces main features of observed CSR.


Observations of CSR - NSLS VUV RingCarr et al. NIM-A 463 (2001) p. 387

Current Threshold for Detection of Coherent Signal

Spectrum of CSR Signal (wavelength ~ 7 mm)


Observations of CSR - NSLS VUV RingCarr et al. NIM-A 463 (2001) p. 387

•CSR is emitted in bursts.

•Duration of bursts is

•Separation of bursts is of the order of few ms but varies with current.

s100

Detector Signal vs. Time


NSLS VUV RingParameters

Energy 737 MeV

Average machine radius 8.1 mLocal radius of curvature 1.9 mVacuum chamber aperture 4.2 cm

Nominal bunch length (rms) 5 cmNominal energy spread (rms)

Synchrotron tuneLongitudinal damping time 10 ms

5 10 4

s 0 018.


X-Ring Parameters (R. Ruth et al.)

Energy 25 MeV

Circumference 6.3 mLocal radius of curv. R=25 cmPipe aperture h~ 1 cmBunch length (rms) cmEnergy spreadSynchrotron tune

Long. damping time ~ 1 secFilling rate 100 Hz# of particles/bunch

z 1 3 10 3

s 0 018.

N 6 25 109.RADIATION DAMPING UNIMPORTANT!

Can a CSR-driven instability limit performance?

When Can CSR Be Observed ?

• CSR emissions require overlap between (single particle) radiation spectrum and charge density spectrum:

• What causes the required modulation on top of the bunch charge density?

P N P N P e dnn

nin

n

z22

( )

incoherent coherent

Radiated Power:

Presence of modulation (microbunches) in bunch density

CSR may become significant

Collective forces associated with CSR induce instability

Instability feeds back, enhances microbunching

Dynamical Effects of CSR


Content of Dynamical Model

• CSR emission is sustained by a CSR driven instability [first suggested by Heifets and Stupakov]

• Self-consistent treatment of CSR and effects of CSR fields on beam distribution.

• No additional machine impedance.

• Radiation damping and excitations.


Model of CSR Impedance• Instability driven by CSR is similar to ordinary

microwave instability. Use familiar formalism, impedance, etc.

• Closed analytical expressions for CSR impedance in the presence of shielding exist only for simplified geometries (parallel plates, rectangular toroidal chamber, etc.)

• Choose model of parallel conducting plates.• Assume e-bunch follows circular trajectory.• Relevant expressions are already available in the

literature [Schwinger (1946), Nodvick & Saxon (1954), Warnock & Morton (1990)].


Parallel Plate Model for CSR: Geometry Outline

E

Rh


Analytic Expression for CSR Impedance(Parallel Plates)

By definition: 2 RE n Z n I n( , ) ( , ) ( , )

Z nn

Z Rh

Rnc

J H J Hpp

n np

pn n

( , )

, ,...

( ) ( )

LNMM

OQPP

2 20

1 3

12

21

Argument of Bessel functions R p

With p p h / p c p2 2 2 ( / ),

Impedance

p x x sin( ) / x p h h / 2

h

,

beam height


Collective Force due CSRFT of (normalized) charge density of bunch.

Assume charge distribution doesn’t change much over one turn (rigid bunch approx).

E eR

e d e Z n dt e tin i t i n tt

nn

z z0

220

( )( , ) ' ( ' )( ) '

E eR

e Z n n tin t

nn

002

0( ) ( , ) ( )


Parallel Plate Model : Two Examples

n mm 1000 12n mm 1000 1 6 .

R cm h cm E MeV 25 1 25, , R m h cm E MeV 1 9 4 2 737. , . ,

500 1000 1500 2000n

- 2

0

2

4

6

HWL

Re ZHnL€€€€€€€€€€€€€€€€n

ImZ HnL€€€€€€€€€€€€€€€€n

1000 2000 3000 4000 5000n

- 2

0

2

4

6

HWLRe ZHnL€€€€€€€€€€€€€€€€

n

ImZ HnL€€€€€€€€€€€€€€€€n

R =1.9 mh =4.2 cmE =737 MeV

X-Ring NSLS VUV RingZ n Z n n( ) ( , ) 0


Properties of CSR Impedance

• Shielding cut off

• Peak value

• Low frequency limit of impedance

n R hc ( / ) /3 2

Re ( )( )Z nn

hR

130

lim ( , )

, ,...n

p

p

Z n nn

i Zp

hpR

FHG IKJ L

NMMOQPP0

0 0

1 32

2 21 3 18

energy-dependent term curvature term


Longitudinal Dynamics• Zero transverse emittance but finite y-size. • Assume circular orbit (radius of curvature R).• External RF focusing + collective force due to CSR.• Equations of motion

( ) 1

dzdt

c ddt

c sR

z ceE

E t

FHG

IKJ

2

0( , )

RF focusing collective force

z t R ( ) 0 is distance from synchronous particle. ( ) /P P P0 0 is relative momentum (or energy) deviation.

:


Vlasov Equation

• Scaled variables • Scale time 1 sync. Period.

q z p Ez E / , / . 0 0 0

st ,

I e N Es E 202/ .

FHG IKJ

f p fq

fp

q I R Z n ez n

ninq Rz

0 0( ) /

n

inq Rdqe f q p dpz z z12

/ ( , )

2


Equilibrium Distribution in the Presence of CSR Impedance Only (Low Energy)

• Haissinski equilibria i.e.• Only low-frequency part of impedance affects

equilibrium distribution.• For small n impedance is purely capacitive

• If energy is not too high imaginary part of Z may be significant (space-charge term ).

Z Z n iZn lim / ,0

/Z 1 2

Z 0

f H0 exp( )


- 3 - 2 - 1 0 1 2 3 4q

0

0.1

0.2

0.3

0.4r 0HqL

•If potential-well distortion is small, Haissinski can be approximated as Gauss with modified rms-length:

q

n z

Zin

I R

FHGIKJ

140

02

2

Haissinski Equilibrium(close to Gaussian withrms length )=>Bunch Shortening.

I=0.844 pC/V corresponding to N part bnch 4 6 1010. ./

q 0 96.

Haissinski Equilibrium for X-Ring

2 cm


Linearized Vlasov Equation

f p fq

fp

I Z n en

ninq Rz1 1 0

0 0

( ) /

• Set and linearize about equilibrium:

• Equilibrium distribution:

•Equilibrium distribution for equivalent coasting beam (Boussard criterion):

fe ep q

q

q

0

2 22 2 2

2 2

/ /

f e p

q0

22

21

2

/

f f f 0 1


(Linear) Stability Analysis

• Ansatz

• Dispersion relation

with ,

and

• Look for for instability.

f f p i nq Rz1 1 0 ( ) exp[ ( / )]

II

Z nn iW0

1( )( )

12

1

0

02

02I

R

z q

R nz/ 0

W z iz w z( ) / ( / ) 1 2 2

Im 0

Error functionof complex arg


Keil-Schnell Stability Diagram for X-Ring

- 1.5 - 1 - 0.5 0 0.5 1 1.5 2Re@- IZHnL� I0 nD- 1

- 0.5

0

0.5

1

mI@-ZIHnL�I 0n

Dn=702

n=1235

n=494

above threshold

on threshold

1 0iW

for( )

Im

(stability boundary)

Threshold (linear theory): I pC V part bnchth 0 8183 4 1010. / ./

Keil-Schnell criterion: II

Z nn0

1( ) I pC Vth 0 63. /

n mm 702 2 2 .Most unstable harmonic:

0 0.1 0.2 0.3 0.4 0.5 0.6q

.5

1

1.5

I=0.76

I=0.88

Amplitude of perturbation vs time (different currents)

pert mm 2 2.

Numerical Solution of Vlasov Equation coasting beam –linear regime

Initial wave-like perturbationgrows exponentially.

Wavelength of perturbation:

pert mm 2 2.

#grid pts 0.8241 0.6 0.8202 0.3 0.8189 0.2 -- 0.8183 --

res mm( )2400280021200

Growth rate vs. currentfor 3 different mesh sizes

I th

Validation of Code Against Linear Theory (coasting beam)

0.76 0.78 0.8 0.82 0.84 0.86 0.88IHpC � VL- 2

- 1.5- 1

- 0.50

0.51

1.5

htworg

etarmI

n

dq=

500.0,m

=45

4002 grid

8002 grid

12002 grid

Theory


- 0.2 0 0.2q

- 2

0

2

p

q=1.2

- 0.2 0 0.2q

- 2

0

2

p

q=1.6

- 0.2 0 0.2q

- 2

0

2

p

q=2.0

- 2 0 2p

0.5

1.0

1.5

p-birtsid.

q=1.2

- 2 0 2p

0.5

1.0

1.5

p-birtsid.

q=1.6

- 2 0 2p

0.5

1.0

1.5

p-birtsid.

q=2.0

Coasting Beam: Nonlinear Regime (I is 25% > threshold).

Density Contours in Phase space

Energy Spread Distribution

2 mm

Coasting Beam: Asymptotic Solution

- 2 0 2p

0.5

1.0

1.5

p-birtsid.

q=9.6

- 0.2 0 0.2q

- 2

0

2

p

q=9.6

Large scale structures have disappeared. Distribution approaches some kind of steady state.

Energy Spread vs. TimeDensity Contours in Phase Space

Energy Spread Distribution

0 2 4 6 8q

1

1.1

1.2

1.3

1.4sp I=0.98


Bunched Beam Numerical Solutions of Vlasov Eq. - Linear Regime.

0 0.1 0.2 0.3 0.4 0.5 0.6q

.5

1

1.5

edom

edutilpma

I=0.76

I=0.88

RF focusing spoils exponential growth.Current threshold (5% larger than predicted by Boussard).

pert mm 2 2.

Amplitude of perturbation vs. time

I th 0 836.

Wavelength of initial perturbation:


- 1.5 0 1.5q

- 1.5

0

1.5

p

q=1.2

- 3 - 2 - 1 0 1 2 3 4q

0.1

0.2

0.3

0.4

egrahcytisned

q=1.2

- 1.5 0 1.5q

- 1.5

0

1.5

p

q=3.2

- 3 - 2 - 1 0 1 2 3 4q

0.1

0.2

0.3

0.4 q=3.2

- 1.5 0 1.5q

- 1.5

0

1.5

p

q=9.6

- 3 - 2 - 1 0 1 2 3 4q

0.1

0.2

0.3

0.4 q=9.6

Bunched Beam: Nonlinear Regime (I is 25% > threshold).

Density Plots in Phase space

Charge Distribution

2 cm


Bunched Beam: Asymptotic Solutions

2 4 6 8 10 12q

0.951

1.051.1

1.151.2

1.25

s q

s p

Bunch Length and Energy Spread vs. Time

Quadrupole-like modeoscillations continue indefinitely.

Microbunching disappears within 1-2 synchr. oscillations


Charge Density Bunch Length (rms)

bnchpartVpCI /108.5/02.1 10 (25% above instability threshold)

X-Ring: Evolution of Charge Density and Bunch Length


Inclusion of Radiation Damping and Quantum Excitations

• Add Fokker-Planck term to Vlasov equation

FHG IKJ

f p fq

fq

q I R Z n ez n

ninq Rz

0 ( ) /2

s d

fp

pf fp

FHG IKJ

Case study NSLS VUV Ring

s sT kHz 2 2 12/

d ms synch periods 10 106 .

Synch. Oscill. frequency

Longitudinal damping time

dampingquantum excit.

G. Carr et al., NIM-A 463 (2001) p. 387

- 1.5 - 1 - 0.5 0 0.5 1 1.5 2Re@- IZHnL� I0 nD- 1

- 0.5

0

0.5

1

mI@-ZIHnL�I 0n

Dn=1764

n=2664

n=1224

above threshold

on threshold

Measurements Model

Current threshold 100 mA 168 mA

CSR wavelength 7 mm 6.7 mm

Keil-Schnell Diagram for NSLS VUV Ring

Most unstable harmonic:n mm 1764 6 7 .

I pC V

part bnchmA

th

6 2

18 10168

11

. /

. ./

Current theshold:

*

*

0 50 100 150 200n. of synch. periods

1

1.2

1.4

1.6

1.8

s q


1.´ 10- 7

0.0001

0.1

100

100000.hoC�hocnI

rewoPP N e Z ncoh

nn

20

2 2 b g Re ( )

P N e Z nincoh

n

02b g Re ( )

P Pcoh incoh/ vs. Time

CSR Power:

Incoherent SR Power:

Bunch Length vs. Time

NSLS VUV Model

current =338 mA, (I=12.5 pC/V)

10 ms


105 110 115 120 125 130n. of synch. periods

1.15

1.2

1.25

1.3

1.35

s q


1

1.2

1.4

1.6

1.8

s q


1.´ 10- 7

0.0001

0.1

100

100000.

hoC�hocnI

rewoP

105 110 115 120 125 130n. of synch. periods

1.´ 10- 7

0.00001

0.001

0.1

10

1000

100000.

hoC�hocnI

rewoP

Bunch Length vs. Time P Pcoh incoh/ vs. Time

1.5 ms

Snapshots of Charge Density and CSR Power Spectrum

- 1.5 - 1 - 0.5 0 0.5 1 1.5 2q

0.220.240.260.28

0.30.320.34

Charge Densityn.

sdoirep=

119.94

- 4 - 2 0 2 4q

00.050.1

0.150.2

0.250.3

Charge Density

n.sdoirep=

119.9413.2 6.63 4.42

l HmmL0.05

0.1

0.15Ha.u.L

CSR Spectrum

n.sdoirep=

119.94

5 cmcurrent =311 mA, (I=11.5 pC/V)


Charge Density Bunch Length (rms)

Radiation SpectrumRadiation Power

NSLS VUV Storage Ring


Conclusions• Numerical model gives results consistent with

linear theory, when this applies.

• CSR instability saturates quickly• Saturation removes microbunching, enlarges

bunch distribution in phase space.

• Relaxation due to radiation damping gradually restores conditions for CSR instability.

• In combination with CSR instability, radiation damping gives rise to a sawtooth-like behavior and a CSR bursting pattern that seems consistent with observations.

Coherent Synchrotron Radiation and Longitudinal Beam Dynamics in Rings

Documents

Transcript of Coherent Synchrotron Radiation and Longitudinal Beam Dynamics in Rings