Double RF system at IUCF

Shaoheng Wang

06/15/04

Contents

1. Introduction of Double RF System

2. Phase modulation Single cavity case Double cavity case

3. Voltage modulation Single cavity case Double cavity case

Introduction: Double RF system

: Harmonic number

: RF peak voltages

: RF phase of syn. particle

: Orbit angle

: Synchrotron tune at zero

amplitude for primary cavity

s

ss

VV

hh

21

21

21

,

,

,

)(

)(

222

11

ss

ss

h

h

h2

V2

1s22 s

Primary cavity

Secondary harmonic

cavityh1

V1

s

s

Synchronous particle

Other particles in the bunch

Particle bunch

Introduction: Why Double RF system

• By reducing the voltage gradient at the bunch position, it will also increase the bunch length. Hence, lower the space charge effect.

• There is an increase in the spread of synchrotron frequencies within the bunch. This spread can help in damping coherent instabilities such as the longitudinal coupled bunch instabilities through an effect known as Landau damping which come from the non-linearity of the voltage along the bunch .

)sin()sin()( 0 ns nnkVV The voltage seen by the beam with a double RF system is

0)(cos0

20

V

V s

The equation of synchrotron motion

HY

s

s

2

0

22

0

ˆ

cos

),(2

00

2 )(1

),( dVV

Y s

Make the integration

To maximize the bunch length, the first derivative of V should vanish at the center of the bunch.

To avoid having a second region of phase stability close by, the second derivative of V must also vanish.

sn

sn

nkn

nnk

sinsin

coscos2

The peak value along a given trajectory

Introduction: Working conditions

Computed distribution in synchrotron tune

t

t

dHT

2

1

2)(

In the phase space, along the H contour,

Period:

Single RF System

Zero Gradient

Qs

density

Reduced voltage slope

Shifting

Nonlinearity Spread

Introduction: Synchrotron tune spread

Introduction: IUCF cooler ring

Injection

Cavity 1

Cavity 2

Introduction: Experiments at IUCF

86.0 factor slip phase

679 frequency n synchrotro

6 ,3 number harmonic

03168.1 frequency revolution

protons MeV 45

21

0

Hzf

hh

MHzf

syn

The bunched beam intensity was found to increase by about a factor of 4 in comparison with that achieved in operating only the primary rf cavity at same rf voltage.

Introduction: Equations of motion

]}sin))([sin()sin{(sin 211

22

1

21 sssss

s

h

h

V

VP

P

0211

1

2 E

eVh

p

phP

s

s

: Normalized momentum deviation

: Synchrotron tune at zero amplitude for primary cavity

Contribution from primary cavity

Contribution from secondary cavity

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Introduction: Hamiltonian

]}sin)())(cos([cos

sin)()cos{(cos2

1

21122

111

2

sssss

ssss

s

hhh

r

U

UPH

1

2

1

2

h

hh

V

Vr

: Ratio of the amplitude of RF voltages

: Ratio of harmonic numbers

0 0.2 0.4 0.6 0.8

1 1.2 1.4 1.6

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

phi

shape of the potential

Primary cavity onlyDouble RF system

Flattened potentialPotential shape example

)2cos1(2

cos1 r

U

r = 0 r = 0.5

Flattened potentialPotential shape example

)2cos1(2

cos1 r

U

-0.5

0

0.5

1

1.5

2

-3 -2 -1 0 1 2 3

phi

r > 0.5

At the equilibrium state, the particle distribution follows the shape of the potential

)2cos1(2

cos12

as simplified is hamiltonia the,0 ,2When

2

21

rPH

h

ss

ss

Synchrotron tune: Hamiltonian

)2sin(sin

rH

P

PP

H

s

s

02sinsin2 rs

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

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

P

phi

r<0.5

r > 0.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

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

P

phi

r < 0.5

P

P

phi

Synchrotron tune: action-angle variables

ˆ

ˆ

1 :Action PdJ

1

:n tunesynchrotro

E

JQs

ˆˆ2

ˆ2

''

:becomes variableangle The

')'(),( :function generatingWith

P

dQd

E

P

J

E

J

F

dPJF

s

s

Synchrotron tune: synchrotron tune

ˆ and , 5.0hen w,

11

2

ˆ and , 5.0or

,5.0when ,

12

21221

222

120

40

20

b

lu

u

b

s

s

rkKtt

tr

r

r

kKt

trtr

Q

integral elliptical theof moduli are ,)21(2)21(

)21(1

kind.first theof integral elliptical complete theis )(),2

1arccos(2

ˆ,2

sin2

sinarcsin2,2

tan,2

ˆtan where,

22

240

20

200

1

22,,0

u

lu

b

uub

llu

lu

t

ttk

trtr

trtk

kKr

tt

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

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

P

phi

r<0.5

r > 0.5

r < 0.5

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

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

P

phi

P

P

phi

Synchrotron tune: graph

(deg) ̂

ss

Q/

Synchrotron tune variation w.r.t. phase amplitude for a double RF system.

Synchrotron tune compare with experiments

J.Y. Liu et al, Phys. Rev. E 50, 3349 (1994)

The synchrotron tune spread is maximized at r=0.5 for a given bunch area, which is provided to Landau damping.

JdJdQQ ss /

The effective tune spread is given by:

Synchrotron tuneCompare with one RF cavity system

Synchrotron tune measured as a function of phase amplitude at IUCF.

M. Ellison et al, PRL 70, 591, 1993

Synchrotron tune: Synchrotron phase space measurement

• Synchrotron phase is measured with phase detectors.– By comparing the bunch arrival time with the RF cavity wave.

• Momentum deviation is measured according to the dispersion relation:

• FFT on to get synchrotron tune

p

pDx xco

n

tinB

lB eeNlTteNtI )(

00),(

Synchrotron tune: Phase shifter

To have a certain phase amplitude, a phase shifter is used.

1. The RF signal is split into two channels, one of them is 90 degrees shifted.

2. Each of these two channels is multiplied with a signal proportional to sine or cosine of intended phase shift.

3. They are combined again. )ˆcos(

)ˆsin()2

cos(

)ˆcos()cos(

)2

cos(

)cos(

thA

thA

thA

thA

thA

Phase modulation Phase Modulation Signal

With this phase modulation, the phase variation will be given by :

)sin( ma

amplitude modulation theis

tune,modulation theis where0

a

mm

d

ad

d

d m ))sin(( 0

Consider a sinusoidal RF phase modulation:

cavity. thearrives particle when phase RF dunperturbe is where 0

Single cavity: Equations of motion

mms

s

p

PaP

UHH

cos2

sin22

22

0

sin

cos

s

mms

P

aP

Hence, the equations of motion are given by :

The corresponding Hamiltonian is :

Perturbation potential

Single cavity expressed with action-angle variables

can be transformed into Action-angle coordinates0H ),( J

In this transformation, old coordinates are expressed as function of ,J

,P

Further more, can be expanded in Fourier harmonics series of

),( JP

Perturbation potential can be expressed as:

)]3cos()3[cos(128

2

)cos()cos(2

2/3

mmm

mmmp

Ja

JaU

Note: since is an odd function of , only odd harmonics exist.

P

When is close to , stationary phase condition exists for a parametric resonance term. All non-resonance terms can be neglected.

sn 12 m

Single cavity: dipole mode

)cos(216

2/12

mss

seff JaJ

JH

When n=0 case, or dipole mode, is considered, the approximate effective Hamiltonian is:

The effective Hamiltonian is dependent. We can go to resonance rotating frame to find the independent Hamiltonian.

Single cavity: resonance rotating frame

IF m )(2

JIm ,

)cos(216

)(~ 2

IaI

IH ssms

In the phase space, the structure of resonant islands can be characterized by fixed points, which satisfy conditions:

0

0

I

With the generating function:

We can realize the transformation:

And get the new Hamiltonian in resonance rotating frame:

Single cavity: Poincare surface of section

)cos(2 mI

)si

n(2

m

I : Outer SFP

: Inner SFP

: UFP

: Seperatrix phase axis crossings

2,1g

g

g

g

c

b

a

])4(16

31[

:when

3/2asc

cm

Single cavity: bifurcation

)cos(2 mI

)si

n(2

m

I

cWhen goes to from below, the fixed points move as arrows showmWhen = , and coincide. This is the bifurcation point, beyond which, only exists

m c bg cgag

cs

ms

3/14/ agb

3/12 4/ ag

3/11 4/ ag

3/14/ aga

3/14/ agc

Single cavity: experiments

)cos(2 mI

)si

n(2

m

I

Double cavities: Equations of motion

mms

s

p

Par

P

UHH

cos)2cos1(2

cos12

2

0

)2sin(sin

cos

rP

aP

s

mms

Hence, the equations of motion are given by :

The corresponding Hamiltonian is :

Perturbation potential

Double cavities: Perturbation analysis

in

nn

inn PeJgeJgP

2

1)( with ,)(

can expressed with action-angle variables of the unperturbed . can be expanded in Fourier series on ,

motion. particle perturb

tocoherently scontribute termresonance the, when Obviously,

n.Hamiltonia dunperturbe theofenergy theis )( where

)]cos()[cos()()(

becomesn Hamiltonia the),,( of In terms

nonzero. isn odd with only , offunction oddan is Since

0n

n

JE

nnJgaJEH

J

gP

m

nmnmnm

n

pU

0H P

Double cavities: Parametric resonances

0)(')(

)12,,2,1,0(,

:solved becan points fixed here, From

cos)(n

)(

:isn Hamiltonia new average timeThe

and ,

:become variablesangle-action new The

)(),( :function generating

usingby frame rotating resonance ation to transformMake

m

2

Igan

IQ

nlln

nIgaIIEH

nnJI

nnIIW

nmm

s

nm

nm

nm

Double cavities: numerical simulation (1)

)2sin(sin2

)2cos(22

111

1

nnsnn

mmnnn

r

na

Simulations are based on the difference equations:

with:

4108

5.2

s

oa

Double cavities: numerical simulation (2)

5.0 and 5.0th section wi of surfaces Poincare rs

m

Double cavities: bifurcation

5.0 and 3 toclose th section wi of surfaces Poincare rs

m

30.2s

m

32.2s

m

35.2s

m

Double cavities: wave structure

Figure (a) shows two beamlets obtained about 15 ms after the phase modulation was turned on, and Fig (b) shows the final beam profile captured after 25 ms, showing a wave structure. The beam profiles were extended from a half length of about 10 ns to 50 ns without beam loss.

Voltage modulation: Single cavity

111

1

sinsin12

2

nnmsnn

nsnn

0sinsin12 ms With the dot corresponds to the time derivative wrt θ.

The equation of motion can be derived from the Hamiltonian:

cos1sin

cos12

1

1

20

10

ms

ss

H

H

HHH

Unperturbed Hamiltonian:

Perturbation:

Action of the Unperturbed Hamiltonian:

dJ2

1

2

0 256

1

16

11 JJJJEH s

Synchrotron tune:

256

3

81

2

2JJ

EKJ

EJQ s

s

ss

Complete elliptic integral of the first kind

Single cavity: Action-angle variable

ˆ

',', dJJGGenerating function:

n

nmnmns nnJgH sinsin2

11 0

choose

deJg inn cos1

2

1is zero except for n=even with *

nn gg

RF voltage modulation contributes only even-order harmonics to the perturbation H1

J

JG

,

Single cavity: rotating frameJ

nnnF m ~

22

Generating function:

,~,

~~~cos~~~~

JHnJgJn

JEH nsm

~cos~~~~

nJgJn

JEH nsm

Including both ±n terms, the resonance Hamiltonian:

The time-averaged Hamiltonian:

~2cos4

~~

16

~

2

~ 2 JJJH ssm

s

2cos482

2sin2

ssms

s

J

JJ

When n=2:

For simplicity, the tilde notation is dropped:

22 if0

22 if2

218

ssm

ssm

s

m

SFPJ

22

22 if0

22 if2

218

ssm

ss

ssm

s

m

UFPJ

Fixed points: and 0with

2

3 and

2with

Single cavity: Fixed point

Single cavity: experiments

• The bunch was kicked longitudinally, all particles then were captured and dampened into one attractor, see fig.

• At the same time, rf voltage modulation was applied.

• A total of 16000 points at intervals of 50 revolutions, i.e. 800000 orbital revolutions, was recorded.

• Poincare surfaces in resonance processing frame, see fig, the particle damping paths and the island structure were clearly observed

Single cavity: experiments

Single cavity: beam profile

The profile of the beam in a single pass.

Double cavity: Hamiltonian

2sinsinsin1 rms

s

0

whereV

Vm

msm

mss

E

rH

cos2

1cos1

2cos12

cos1sin12

1

2

2

l

lnlin

nn

inn JgJgeJGeJG

22

2

1)( with ,)(

Gn is not zero only for even harmonics.

0

coscos2

1cos

nnmnmnsm nnJGJEJEH

Inn

IW nm

,2

nnIJ nm and ,

,,cos

2

1IHnGI

nEΗ ns

m

Time dependent part

Double cavity: Fixed point

2cos

2cos

2

ˆsin

2sin

21

4 0

02

2

2

n

K

un

K

nKun

q

qKQG

n

n

s

sn

KKeqk

,kk

kFuK

'2

2

0

and , 2

ˆsin1

2

1 modulus with kindfirst

theoffunction elliptical incomplete theand complete the

ly respective are 21

arccos and where

0'2

1

,,3,2,1,0 ,

:aren Hamiltonia averaged time theof points fixed The

FPnsm

FPs

FP

IGn

IQ

lln

Double cavity: numerical simulation

nnnmsnn

nsnn

rn

111

1

2sinsin2sin12

2

:equations differencefor performed weressimulation numerical

Double cavity: Bifurcation point

sm 4.1sm 3.1 sm 55.1

Bifurcation point

Conclusion

• The benefits of double RF system– Longer bunch, less space charge effect– Landau damping from synchrotron tune

spread

• Resonance structure when system is under phase and voltage modulation

reference

A. Hoffman,

SY Lee,

M. Ellison

JY Liu,

D. Li,

H. Huang

…