Coherent Synchrotron Radiation and Longitudinal Beam Dynamics

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July 2002


M. Venturini


1

Coherent Synchrotron
Radiation and Longitudinal
Beam Dynamics in Rings

M. Venturini and R. Warnock

Stanford Linear Accelerator Center


ICFA Workshop on High Brightness Beams


Sardinia, July 1
-
5, 2002

July 2002


M. Venturini


2

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
.

July 2002


M. Venturini


3

Observations of CSR
-

NSLS VUV Ring

Carr
et al
. NIM
-
A
463

(2001) p. 387

Current Threshold for Detection


of Coherent Signal

Spectrum of CSR Signal


(
wavelength ~ 7 mm
)

July 2002


M. Venturini


4

Observations of CSR
-

NSLS VUV Ring

Carr
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.


s

100

Detector Signal vs. Time

July 2002


M. Venturini


5

NSLS VUV Ring

Parameters

Energy 737 MeV


Average machine radius 8.1 m

Local radius

of curvature 1.9 m

Vacuum chamber
aperture

4.2 cm


Nominal bunch length (rms) 5 cm

Nominal energy spread (rms)


Synchrotron tune

Longitudinal damping time 10 ms







5
1
0
4

s

0
018
.
July 2002


M. Venturini


6


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


Energy 25 MeV


Circumference 6.3 m

Local radius of curv. R=25 cm

Pipe aperture h~ 1 cm

Bunch length (rms) cm

Energy spread

Synchrotron tune


Long. damping time ~ 1 sec

Filling rate 100 Hz

# of particles/bunch



z

1





3
1
0
3

s

0
018
.
N


6
25
10
9
.
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
d
n
n
n
in
n



z

2
2




(
)
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

July 2002


M. Venturini


9

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
.

July 2002


M. Venturini


10

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)].

July 2002


M. Venturini


11

Parallel Plate Model for CSR
:
Geometry Outline


E


R
h
July 2002


M. Venturini


12

Analytic

Expression for CSR Impedance

(Parallel Plates)

By definition:



2






RE
n
Z
n
I
n
(
,
)
(
,
)
(
,
)
Z
n
n
Z
R
h
R
nc
J
H
J
H
p
p
n
n
p
p
n
n
(
,
)
,
,
.
.
.
(
)
(
)










L
N
M
M
O
Q
P
P


2
2
0
1
3
1
2
2
1

Argument of Bessel functions

R
p

With



p
p
h

/



p
c
p
2
2
2


(
/
)
,

Impedance



p
x
x

sin(
)
/
x
p
h
h



/
2

h

,

beam height

July 2002


M. Venturini


13

Collective Force due CSR

FT of (normalized) charge


density of bunch.

Assume charge distribution doesn’t change much


over one turn (rigid bunch approx).

E
e
R
e
d
e
Z
n
dt
e
t
in
i
t
i
n
t
t
n
n

















z
z

0
2
2
0
(
)
(
,
)
'
(
'
)
(
)
'
E
e
R
e
Z
n
n
t
in
t
n
n











0
0
2
0
(
)
(
,
)
(
)
July 2002


M. Venturini


14

Parallel Plate Model : Two Examples

n
mm



1000
12

n
mm



1000
1
6

.
R
cm
h
cm
E
MeV



25
1
25
,
,
R
m
h
cm
E
MeV



1
9
4
2
737
.
,
.
,
X
-
Ring

NSLS VUV Ring

Z
n
Z
n
n
(
)
(
,
)


0
July 2002


M. Venturini


15

Properties of CSR Impedance


Shielding cut off



Peak value




Low frequency limit of impedance




n
R
h
c


(
/
)
/
3
2
Re
(
)
(
)
Z
n
n
h
R

130

lim
(
,
)
,
,
.
.
.
n
p
p
Z
n
n
n
i
Z
p
h
pR




F
H
G
I
K
J

L
N
M
M
O
Q
P
P

0
0
0
1
3
2
2
2
1
3
1
8







energy
-
dependent term

curvature term

July 2002


M. Venturini


16

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
dz
dt
c



d
dt
c
s
R
z
ce
E
E
t






F
H
G
I
K
J

2
0
(
,
)
RF focusing

collective force

z
t
R


(
)


0
is distance from synchronous particle.




(
)
/
P
P
P
0
0
is relative momentum (or energy) deviation.

:

July 2002


M. Venturini


17

Vlasov

Equation


Scaled variables




Scale time 1 sync. Period
.





q
z
p
E
z
E



/
,
/
.



0
0
0



s
t
,
I
e
N
E
s
E

2
0
2
/
.











F
H
G
I
K
J





f
p
f
q
f
p
q
I
R
Z
n
e
z
n
n
inq
R
z





0
0
(
)
/



n
inq
R
dq
e
f
q
p
dp
z

z
z

1
2
/
(
,
)





2
July 2002


M. Venturini


18

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
i
Z
n



lim
/

,
0

/
Z

1
2


Z

0
f
H
0


exp(
)
July 2002


M. Venturini


19


If potential
-
well distortion is small, Haissinski can be approximated


as Gauss with modified rms
-
length:





q
n
z
Z
i
n
I
R


F
H
G
I
K
J

1
4
0
0
2
2
Haissinski Equilibrium

(close to Gaussian with

rms length )

=>Bunch Shortening.



I=0.844 pC/V
corresponding
to


N
part
bnch


4
6
10
10
.
./

q

0
96
.
Haissinski Equilibrium for X
-
Ring

2 cm

July 2002


M. Venturini


20

Linearized Vlasov Equation














f
p
f
q
f
p
I
Z
n
e
n
n
inq
R
z
1
1
0
0
0




(
)
/



Set and linearize about equilibrium:








Equilibrium distribution:




Equilibrium distribution for


equivalent coasting beam


(Boussard criterion):

f
e
e
p
q
q
q
0
2
2
2
2
2
2
2



/
/




f
e
p
q
0
2
2
2
1
2


/



f
f
f


0
1
July 2002


M. Venturini


21

(Linear) Stability Analysis


Ansatz



Dispersion relation




with ,



and



Look for for instability.

f
f
p
i
nq
R
z
1
1
0




(
)
exp[
(
/
)]




I
I
Z
n
n
iW
0
1
(
)
(
)

1
2
1
0
0
2
0
2
I
R
z
q









R
n
z
/
0
W
z
iz
w
z
(
)
/
(
/
)


1
2
2

Im


0
Error function

of complex arg

July 2002


M. Venturini


22

Keil
-
Schnell Stability Diagram
for X
-
Ring

1
0
iW
for
(
)
Im



(stability boundary)

Threshold (linear theory):

I
pC
V
part
bnch
th



0
8183
4
10
10
.
/
./
Keil
-
Schnell criterion:

I
I
Z
n
n
0
1
(
)



I
pC
V
th
0
63
.
/
n
mm



702
2
2

.
Most unstable harmonic:

Amplitude of perturbation vs time


(different currents)


pert
mm

2
2
.
Numerical Solution of Vlasov Equation



coasting beam

linear regime

Initial wave
-
like perturbation

grows 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
(
)
2
400
2
800
2
1200

Growth rate vs. current

for 3 different mesh sizes

I
th
Validation of Code Against Linear Theory



(
coasting beam
)

Theory

July 2002


M. Venturini


25

Coasting Beam: Nonlinear Regime

(I is 25% > threshold).

Density Contours in Phase space

Energy Spread Distribution

2 mm

Coasting Beam: Asymptotic Solution


Large scale structures have


disappeared.




Distribution approaches some


kind of steady state.



Energy Spread vs. Time

Density Contours in Phase Space


Energy Spread Distribution

July 2002


M. Venturini


27

Bunched Beam


Numerical Solutions of Vlasov Eq
.
-

Linear Regime.

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:


July 2002


M. Venturini


28

Bunched Beam: Nonlinear Regime

(I is 25% > threshold).

Density Plots in Phase space

Charge Distribution

2 cm

July 2002


M. Venturini


29

Bunched Beam: Asymptotic Solutions


Bunch Length and Energy Spread vs. Time



Quadrupole
-
like mode

oscillations continue

indefinitely.

Microbunching disappears within 1
-
2 synchr. oscillations

July 2002


M. Venturini


30

Charge Density

Bunch Length (rms
)

bnch
part
V
pC
I
/
10
8
.
5
/
02
.
1
10



(25% above instability threshold)


X
-
Ring:


Evolution of Charge Density and Bunch Length

July 2002


M. Venturini


31

Inclusion of Radiation Damping and
Quantum Excitations


Add Fokker
-
Planck term to Vlasov equation










F
H
G
I
K
J





f
p
f
q
f
q
q
I
R
Z
n
e
z
n
n
inq
R
z





0
(
)
/
2


s
d
f
p
pf
f
p





F
H
G
I
K
J
Case study NSLS VUV Ring





s
s
T
kHz



2
2
12
/

d
ms
synch
periods


10
106
.

Synch. Oscill. frequency

Longitudinal damping time

damping

quantum excit
.


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


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
bnch
mA
th




6
2
1
8
10
168
11
.
/
.
./
Current theshold:

*
*
P
N
e
Z
n
coh
n
n


2
0
2
2


b
g
Re
(
)
P
N
e
Z
n
incoh
n



0
2
b
g
Re
(
)
P
P
coh
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

July 2002


M. Venturini


34

Bunch Length vs. Time


P
P
coh
incoh
/

vs. Time

1.5 ms

Snapshots of Charge Density and


CSR Power Spectrum

5 cm

current =311 mA,
(I=11.5 pC/V)

July 2002


M. Venturini


36

Charge Density

Bunch Length (rms)

Radiation Spectrum

Radiation Power

NSLS VUV Storage Ring

July 2002


M. Venturini


37

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.