ILC Damping Ring Wolski (2007)

Dynamics with Transverse Coupled-Bunch Wake Fields in the ILC Damping Rings

Kai Hock and Andy Wolski

Objective: 1. Effect of varying beta function in macroparticle model 2. Analytic solution for constant beta function.

Previous work – for constant beta function :

1. Equation of motion decouple into Fourier modes (Chao 1993).2. Modes decay / grown exponentially, analytic formula available.3. Analytic formula for bunch trajectories available but incomplete (Thompson and Ruth 1991).

Current work:

1. Varying beta function – modes remain coupled, decay modes grow.2. Analytic formula for bunch trajectories completed with error bounds.

Example of an application

v

A Macroparticle Modely

No wake field

)(11 tyb n

Resistive wake force

n n+1

n+2

Equation of motion

Sands (1969),Thompson and Ruth (1991), …

History

Normal Modes

)(~)(~)(~2

1

2

2

2

ntyebtytydt

d M

ni

nn

1

0

2

)()(~M

m

M

mi

m etyty

where

coupled

decoupled

(Fourier modes)

DFT

Chao (1993)

If beta function is constant

OCS6 Damping Ring: Uniform Fill

A few surprises …

Mode Frequency Spectra

Turn 90

Parametric force

High frequency oscillation comes from …

Mode Coupling

Decay modes grow because …

Growth mode

Decay mode

Decay mode

?

Bunch Trajectories

Equation of motion

Analytic solution (Thompson, Ruth 1991)

Incomplete – Zero wake field limit is SHM, need – Else cannot have arbitrary

ti le

)0(),0( xx

tin

n

titi nba

eCBeAey)()()(~

Characteristic Equation

tieyty

)0(~)(~

iM

ni

n

N

n

eeb2

1

22

Elementary solution

22

1

22

iM

ni

n

N

n

eeb

Contour plot

General solution

If beta function is constant:

Integration of

Depends on initial history

tin

n

titi nba

eCBeAey)()()(~

titi ba

BeAey)()(~

If wake field very small

Conjecture

Depends on initial conditionat t = 0

Depends on initial history

Just like SHM

1

0

2)()(1

)(M

M

mititi

m eeBeAM

tyba

An analytic solution

Conjecture validated

H()

Rogers (2006)

Feedback system bandwidth

Frequency mode number Bandwidth must cover growth modesIf decay modes grow unexpectedly …