Nonlinear Effects of Phase Modulation in an Electron ... - CiteSeerX

SLAC-PUB-7727 December 1997

Nonlinear Effects of Phase Modulation in an Electron Storage Ring* J.M. t Lawrence

Byrd+, W.-H.

Berkeley

National

Chcngi,

F. Zimmermann*

Laboratory,

Berkeley,

California

94720

$ Dept. of Physics, University of California, Berkeley: California 94720 * Stanford Linear Accclcrator Center, Stanford, California 94309

We present results of experimental

studies of the nonlinear

dynamics

of syn-

chrotron oscillations in the presence of phase modulation in an electron stor_- age ring. A streak camera is used to directly observe the longitudinal distribution of an electron bunch as it forms two stable resonant islands. The positions of the fixed points as a function of modulation frequency agree well with theory. We also present rneasurernents of the diffusion rate from one stable island to the other for a fixed modulation frequency which show agreernent with the diffusion rates expected from large-angle intrabcam (Touschek) scattering. These results can explain anornalous features in bearn-transfer function diagnostic mcasurernents at other electron storage rings.

*Work supported by the U.S. 7GSF00098, DE-FG-03%95ER40936,

Department of Energy under and DE-AC03-7GSF00515.

contracts

DE-ACOS-

I

Nonlinear effects of phase modulation in an electron storage ring *

t Lawrence $Dept.

Berkeley

J. M. Byrd+, W.-H.

Chengt,

National

One Cyclotron

of Physics, §Stanford

Laboratory,

University

of California

Linear Accelerator

F. Zimmermanns Road, Berkeley,

Berkeley,

Center,

Berkeley,

Stanford,

California

California

California

9,$720

94 720

94309

Abstract We present chrotron ring.

results

oscillations

A streak

of experimental

studies

in the presence

camera

of the nonlinear

of phase modulation

is used to directly

observe

dynamics

of syn-

in an electron

the longitudinal

storage

distribution

_of an electron

bunch as it forms two stable

the fixed points

as a function

We also present

of modulation

measurements

for a fixed modulation

the diffusion

rates

ing.

These

diagnostic

results

also explain

measurements

frequency

from large-angle anomalous

obtained

islands.

frequency

of the diffusion

to the other

expected

resonant

The positions

agree well with theory.

rate

from one stable

island

which

show agreement

intrabeam

(Touschek)

results

at other electron

of beam storage

transfer

DE-FG-03-95ER40936, --

by the U.S. Dept.

of Energy

and DE-AC03-76SF00515. I

under Contract

with

scatterfunction

rings.

Typeset

*This work was supported

of

using REVW

Nos. DE-AC03-76SF00098,

I. INTRODUCTION

The

longitudinal

storage

charge

ring operation.

density

of an electron

In collider rings, maximum

length is adjusted

to be approximately

interaction

In a synchrotron

point.

ture of the radiation. storage

synchrotron

radiation

resulting

and intrabeam

charge density.

potential

The nonlinear presence

longitudinal

from a balance

ally from the main radiofrequency longitudinal

(RF)

Because

lifetime,

understand

impact

system,

longitudinal

synchrotron

oscillations

on the longitudinal

coherent

As performance

camera,

the effect of the nonlinear

on the longitudinal stable

distribution

fixed points of the system.

are mapped

as a function

dynamics

increase

increasingly

frequency

in the

with the rate expected

from large-angle

these results to explain an anomalous

--

important

to

have been studied experimenstudy of nonlinear

longitu-

storage ring. Using a dual axis streak can be directly

bunch as the electrons

and compared

intrabeam

effect of the nonlinear

z

or pe-

on beam

observed

populate

with calculated

Real time diffusion is observed from one stable island to the other and the measured compared

the

motion.

The stable fixed points as viewed on the streak

of modulation

usu-

sinusoidal,

synchrotron

demands

from

potential,

[1,2] and random

of single electrons

of the entire electron

instabilities,

rings.

in hadron accelerators

in an electron

in an electron

charge distribution

beam instabilities

effects in electron

with phase modulation

the time struc-

the corresponding

RF voltage is typically

tally in the past [336]. In this paper, we present an experimental dinal dynamics

focussing

gives rise to nonlinear

at the

and excitation

determines

in bunch length, it is becoming

the nonlinear

Nonlinear

beta function

damping

in

when the bunch

Neglecting

The longitudinal

oscillate

of the RF voltage.

and reductions

vertical

factors.

radiation

accelerating

of both single and multibunch

intensity,

between

the accelerating

in which electrons

modulations

is reached

parameter

phase space distribution

of different

scattering.

motion has a dramatic

riodic external

luminosity

light source the bunch length determines

by a combination

spread results

is a fundamental

equal to the minimum

The equilibrium

ring is determined

the energy

bunch

scattering.

the

camera values. rate is

As an aside, we use

motion on a common diagnostic

measurement Section

technique, II summarizes

scribes the calculation surements

islands,

beam transfer

a Hamiltonian

formulation

function

(BTF).

of the longitudinal

of the diffusion between stable fixed points.

of the longitudinal

rate between Section

the longitudinal

response

as a function

and the effect on longitudinal

IV provides a discussion

II. SYNCHROTRON

dynamics

Section

and de-

III presents mea-

of modulation

frequency,

beam transfer

function

the diffusion

measurements.

and summary.

OSCILLATIONS

WITH

PHASE

MODULATION

A. Single particle dynamics

In this section, The equations

we present a summary

of motion

for an electron

v,

= fmT,

momentum

per turn.

Eo is particle’s Ignoring

(2)

[sin(cp + qL> - sin &] - X6,

The synchrotron

--

phase relative

relative

pm is the amplitude

h is the harmonic

number,

energy, X is the radiation

+ (p,v,$

the Hamiltonian

period,

phase 4,, S = (p - pe)/pe is

momentum

~0, ’ = d/dQ, 19 = 2m,

7 is the slip factor,

damping

of rf phase

To is the revolution

to the synchronous

to the synchronous

rate,

n

$’ is the peak rf

and Ue is the energy loss

for this system can be written

as

(3)

cos u,Q + u,e [l - cos cp + (sin cp - cp) tan &] ,

tune is us0 = (-hqe?

Near a resonance in a coordinate

as (Pm& = pm sinu,Q,

is the tune of rf phase modulation,

damping,

= Fs2

as

6’ = &

is the number of revolution, voltage,

following ref. [4].

(1)

cp = q5- qSs is th e particle’s the particle’s

can be written

formulation

p’ = hrj4 + pmum cos u,Q,

where the rf phase is modulated modulation,

of the Hamilitonian

cos $s/27rEe)1/2, and 8 = (hq/uso)6.

(urn = use), it is most useful to examine

frame with the phase relative

3

to the modulated

the averaged

Hamiltonian

phase and rotating

at the

modulation

frequency.

Hamiltonian,

including

tonian expressed

where m

This can be done via a series of canonical the nonzero synchronous

in action-angle

(4) can be approximated

and by assuming

The expression

implicitly

expressed

sin