In this chapter we will first concentrate on manipulation of the zeroth (I) and second moments ( z=c/ω·, σ ) of the beam distribution. Afterwards methods of emittance control (longitudinal and transverse) in linacs is presented. All applications to be dis-cussed have in common the use of rf systems to manipulate the beam properties.

z , I

I

x,y(frf)

bunch length compression

bunch length pre-compression

bunch lengthening using harmonic cavities

bunch coalescingbunch splitting

energy compression(single and multibunch)

changing the transverse beam emittancesusing the rf system (rf frequency shifts)

linear accelerators and linac FELs

transfer between circular accelerators

circular accelerators

linear accelerators and transfer lines

circular accelerators

parameter of interest

manipulation method application in

synchrotron light sources

(phase space matching)

Longitudinal phase space manipulation

z, σ

z, I

Introduction

single-particle equation of motion

harmonic oscillator equations

synchrotron frequency

synchrotron tune

e+/-: =momentum compaction factorp: -1/γt

22πfrf

energy deviation

phase deviation

syn|dt

dV

=t

E=beam energyT=revolution period

ellipses in phasespace (for linear rf;i.e. small amplitudeparticle motion)

(=s,=0)

derivatives wrt time, t

Analogy between transverse and longitudinal motiontransverse longitudinal

equations of motion

phase space

second moments

beam matrix

emittance

x’

x

kx

(=s,=0)

=t

(xco+x=0, x’co+x’=0)

(taking <x>=0, <x’>=0) (taking <>=0, <>=0)

k (/c)^2

px + kpx = 0..

Bunch length compressionmotivation: produce short bunches (at expense of increased energy spread) in order to minimize the projected energy spread along bunch maximize peak intensity (for linac-based FELs) in a collider: minimize luminosity reduction due to the hour-glass effect allow for smallest possible β* (β-function at the interaction point)

concept: introduce an E-z correlation within the bunch combined with an energy-dependent path length

example: two-stage bunch compression scheme for the NLC

E-z correlation

(R56≠0) using wiggler

BC1: σz = 5mm500 μm BC2: (100-150) μm

2 rf “sections”

18

0° a

rc

magnet chicane

(compressor cavity)

for E-Z correlation πn (=360˚, NLC) to minimize sensitivity to phase errors

(16 ps -> 1.6 ps -> 330 fs (!))

example: SLC bunch compressor

damping ring compressor cavity, V

main linac

E=eV

s1

2

3

R56≠0

1 upstream of compressor

z

downstream of cavity

z

at injection into linac

z

2 3

I(z)I(z)

H

H H

T

T T

=-z/c

Again,

combining the equations:

in the limit of a linear rf; i.e. bunch length short compared to rf wavelength, so sin ~ =-z1/c

final bunch length: ( assuming < z1 > = 0 )

if the compressor amplitude is adjusted so that

z,f = |R56|,0then the final bunch length is independent of the initial bunch length:

Phase errors in bunch compressors

for the single-stage compressor just described, letting =-z/c

at the SLC the tolerance on the phase of the beam injected into the linac was <0.1 with frf=2856 MHz (barely measurable!)

here we consider the sources of phase error

compressor phase error(deviation of beam from thezero crossing of the rf)

assume: the errors in the injected beam phase i and the compressor phase c are independent the initial momentum deviations are independent of these phases; i.e. d1/d1=d1/dc=0

then with

combining the two contributions in quadrature:

in particular, for =-1 (full compression), the final phase error is independent of theinitial phase error

Bunch length pre-compression

motivation: produce short bunches (at “expense” of increased energy spread) for longitudinal matching from one accelerator to a downstream accelerator for reducing beam loss in downstream transfer line with high dispersion

concept: induce longitudinal quadrupole-mode oscillation by variation of rf voltage amplitude

initial state:bunches matchedin longitudinalphase space

mismatch inlongitudinalphase spaceafter raisingcavity voltage

longitudinalphase spaceafter 1/4synchrotronoscillationperiod

matched phasespace ellipse inreceiving accel-erator

Equation of motion for the bunch length (pp. 178-179)

notation: equations of motion:

a few equalities: for example:

seek

then

first result [A]

omit derivative, p. 178, before “using (7.1)”

omit minus signs(2 occasions) inEq. 8.12

express <2> in terms of and the longitudinal emittance

here

next result [B]

combine [A] and [B] using

gives the result [C]

noting that

combining (iv) and [C] while noting the cancellation of the terms

omit minus sign in Eq. 8.15

equation of motion for bunch length

equation of motion for energy spread

measured cavityvoltage Vc

measured peakcurrent I~z

-1

measured beamcentroid energy

50 kV (800 kV max),10 μs per division

10%, 5μsper division

with =50μm,0.77%, 2.3 μsper division

example: bunch pre-compression at the SLC (variant when higher V not an option)

(not shown: cancellation of dipole-mode oscillation)

t

Vc

side-benefit: reduced energy “tails” in the final focus (chromatic aberrations…)

energy aperture given by trans-verse aperture (=x/) avoided

effect in downstream compressor section:

I(z) I(z)

less particle loss in downstream transfer line (~25% more Ie-Ie+ at SLC IP)

also, eliminate “tails” by short-ening bunch

phase space for case of long bunches wrt compressor cavity wavelength:

shorten the bunch at extraction, energy apertures are avoided

Bunch splitting (for the LHC)motivation: using existing accelerators, produce multiple high-current bunches produce ~40 bunch trains of 72 bunches with 1011 protons and 25 ns bunch spacing (LHC)history: debunching of 6-7 high intensity bunches in the CERN PS + capture in higher-f rf system (microwave instability observed in the process leading to non-uniform beam distributions)

concept: application of higher-harmonic rf cavities

one bunch from the PS boostergets split into twelve bunchesin the CERN PS

layout of the LHC including the preinjectors

example: simulation of bunch triple-splitting in the CERN PS (courtesy R. Garoby, 1999)

example: measurement of bunch triple-splitting in the CERN PS (courtesy R. Garoby, 2001)

tim

eti

meone of 6 bunches from the

booster in the CERN PS

Split factor 1 3

Issues:

preservation of longitudinal beam emittance stability of initial conditions complicated (then) by B-field drift requires careful synchronization control of longitudinal coupled-bunch instabilities bunch intensity fluctuations stability of initial conditions

example: simulation of bunch quadrupole-splitting in the CERN PS (courtesy R. Garoby, 1999)

example: measurement of bunch triple-splitting in the CERN PS (courtesy R. Garoby, 2001)

tim

eti

me

Split factor 1 4(cumulative split factor: 112)

Bunch splitting arising from modulation of the beam near the synchrotron frequency(from a study of the effects of ground motion for the SSC)

Example (IUCF): bunch deformationsresulting from modulation of a trans-verse dipole (at nonzero dispersion)near the synchrotron frequency

Bunch coalescing

motivation: combine many bunches into 1 bunch for high peak intensity (and luminosity)

concept:

1) initial condition with multiple bunches in different high frequency rf buckets

2) lower (vector sum) of cavity voltages

bunches “shear”due to longitudinalmismatch

3) turn on a subharmonic rf system

4) restore initial rf (with appro- priate phase), turn off the lower frequency rf system

bunches rotate with newsynchrotron frequency

example: bunch coalescing in the Fermilab Main Ring (courtesy P. Martin, 1999)

initial condition: 11 bunchescaptured in 53 MHz rf buckets

“paraphrasing” – adiabatic reduc-tion of the vector sum rf voltage by shift of the relative phases between rf cavities

application of higher voltage2.5 MHz rf system (in practice,a 5 MHz rf system was used to help linearize the rotation)

capture of bunches in a single 53 MHz rf bucket

peak intensity monitor with successive traces spaced by 6.8 ms intervals

“snap coalescing” – fast change in voltage amplitude applied (instead of adiabatic voltage reduction) observed advantage: avoidance of high-current beam instabilities during paraphrasing observed disadvantage: reduced capture efficiency (~10%)

time

motivation: increase beam lifetime (reduce the loss rate) by reducing the probability for Touschek scattering (large angle intrabeam scattering)

concept: increase the bunch length (i.e. reduce the volume density) by adding a higher harmonic rf system so that the vector sum of the voltages seen by the beam is constant

Bunch lengthening using harmonic cavities

example: harmonic cavity design for the ALS (courtesy J. Byrd, 1999)

cross section for scattering beyondthe energy accep-tance (given byrf or by physicalapertures, which-ever is smaller)no. particles

per bunch

particle bunch density

k=amplitude ratio=|Vh|/|Vc|

n=ratio of rf frequencies=h/c

relative phase of the two rf systems

nominal rf

rf of thethird har-nomic rfsystem

vector sum of thetwo rf systems

phase of primary rf(wrt zero crossing)

Again,

“Boundary conditions”:

i.e. energy loss per turn (due to radiation) is compensated

voltage profile across bunch is flat without curvature

optimum amplitude

optimum relative phase

1 rad ~ 9.5 mm @ 500 MHz

optimum phasing of the primary rf

potential withprimary rf

example: harmonic cavity design for the ALS (courtesy J. Byrd, 1999)

potential withboth rf systems

bunch profilewith primary rf

bunch profilewith bothrf systems

Experience with the ALS harmonic (5 single-cell, passively driven) cavities:

factor of 2 increase in beam lifetime using uniform current distribution (with minimum ~2% gap for clearing ions and for allowing for dump kicker rise and fall times)50% increase in beam lifetime during normal operation with 20% gap

Example: streak camera images from the ALS (courtesy J. Byrd, 2000)

17% gap

2.4% gap

15 mm

rapid variations in the beamcurrent produce transient loading in the h.c. variable harmonic voltage across the bunch train and variation in synchrotron phase (increased Landau damping)

=(2πf)t=18º (t=100 ps)@500 MHz; ~54º at 1.5 GHz

Energy spreadcircular accelerators

e+/- naturally damped to limit of quantum fluctuationsp, pbar given by accelerating rf or controlled using electron and stochastic cooling

transport lines

linear accelerators

can be modified using an energy compressor (“backwards” bunch length compressor)

given by accelerating rf and controlled using compressors and bunch shaping

Example: energy spread in the SLC damping ring vs beam current

wire scanner data made inthe downstream transfer line after adjustment of the optics to make high dispersion at the wire

evidence ofcurrent-dependentenergy spreadincrease attributedto a microwaveinstability

compared to (x , x’ , y , y’ , ), is perhaps the most difficult to measure and control

particle energy in a linear accelerator

injected beam energy

energy gain from each klystronlongitudinal wake function [eV/Cm]

longitudinaldensity distribution

phase of beam wrt rf crest

energy gain energy loss

phase of particle wrt i

Energy spread E is given by averaging over the particle distribution after subtracting out the mean energy <E>. Normalized to this mean energy of the bunch

mean energy of the bunch distribution:

spacing between successive klystrons

low current limit:

beam does not takeaway energy

In pictures:

Again,

effectiveenergy gain

s

energy spread (obtained byprojecting onto the energy axis)

minimum energyspread (and maximumenergy) obtained byplacing beam at crest

acceleratingrf voltage

a low current, mis-phased beam hashigher energy spread

high current limit:

beam takes away energy and the twoterms in [ ] aboveshould be balanced

acceleratingrf voltage

beam-inducedvoltage

minimum energyspread obtained byplacing beam off-crest

a high current beamplaced on crest hashigher energy spread

Bunch shaping using bunch compressors (upstream of linac)

Recall the resulting beam distribution following the bunch compressor for the case of long bunches

We have seen that the energy spread in a linac is given not by the incoming beam energy spread but rather by the incoming beam bunch length

Minimizing the bunch length therefore results (with appropriate linac phasing) in the smallest energy spread

Example: bunch “over-compression” (courtesy F.-J. Decker, 1999)

bunchlength

normal compression(z,f=R56 ,0)

over- compression

s

long. phase space:

s

I

measured beam profile at end of linac withunder-compression and over-compression

y

x=

Bunch shaping in the linac proper (G. Loew and J.W. Wang)

energy gain of particle within a bunch

same equation asbefore expressedhere in terms of the phase of the beam wrt crest k

integration over all preceeding bunches

0 phase of the head of the bunch wrt rf crest

Minimum energy spread when V(k) is independent of k:

Taking the derivative, it has been shown that an optimal bunch charge distribution () exists which can be found by numerically solving

()

please replace x with in Eq. (8.27), p. 194

general trend: the higher the bunch charge,the more forward-peaked the charge distri-bution for minimum energy spread

Example: conceptual illustration for optimizing the relative phase of the beam for the case of very long (or high current) bunches -- trade-off between beam energy and energy spread in the SLC linac (courtesy J. Seeman, 1999)

acceleratingrf voltage

longitudinalwakefield

loading fromfront of bunchcompensatesthe curvatureof the rfperfectly

energy spread of core smallbut large energy “tails” arepresent

energy spread of core slightlylarger with fewer particles at extreme high or low beam energy

remark: visualizing such projections is very helpful in interpreting data next slide

energy spread of core large, many particles at extreme high or low beam energy, mean energy also lower(worst case)

undercompen-sation of the rf

overcompen-sation of thelongitudinalwake

Example: bunch energy spread measurements at the linac-based SASE FEL (data courtesy F. Stulle, 2003)

y

Beam loading and long-range wakefields, single pass

Fundamental theorem of beam loading (P. Wilson)

wakepotential

loss parameter

filltime

t0 (au)

two-particle model: V is the wake potential seen by trailing particle due to drive bunch

with multiple particles: the contributions from all leading bunches are added

t0 is time between the driving and the trailing particle

Example: influence of long-range wakefields in the SLC linac

se+e-

60 ns(~ 170 rf)

orbit of driving (e+)bunch (dashed line)

difference orbit of trailing (e-) bunch takenbefore and after displacingthe drive bunch by 1 rfwavelength

e-

Consequence: the “split-tune” lattice (R. Ruth)

Observation: the pulse-to-pulse variation of the e- orbit was significantly less if the e+ current was reduced (by about a factor of 2 for the FFTB experiments)

Experiment:

Until the observation, the horizontal and vertical phase advance of the simple FODO latticewere equal. Therefore, long-range wakefields generated by the e+ beam acted resonantlyon the e- beam. By splitting the tunes (making the phase advance in each plane unequal),the pulse-to-pulse jitter was reduced from 0.4x to 0.3x (15%) and from 0.75y to 0.50y

(30%).

Aside:LR-wakesin HERA-P

The loss factor k is often calculated for each cavity mode using numerical programs; e.g. MAFIA

Example: accelerator structure R&D for the Next Linear Collider (courtesy C. Adolphsen, 2002)

sstructure under test

e-

e+

damping rings

first sector of main linac

bunch spacing between drive (e+) andtest (e-) beam varied, differenceorbits analyzed to determine deflectionseen by trailing bunch and used to inferthe amplitude of the transverse wake-fields

model includes5 MHz rmsfrequency errors

model includes12 MHz rmsfrequency errors

crosses: data in xdiamonds: data in ytheory (line)

Beam loading and long-range wakefields, multiple passes (t>>f):

Driving the cavity on resonance (=rf)

Summing the contributions from all previous turns:

Neglect the small self-loading factor and use

gives

Example: phase variations along the bunchtrains of the PEP-II B-Factory

8º

beam currents

beam phases (variation affects max feedback gain)

relative phase difference between the beams (the relevant parameter for luminosity)

(slow oscillation dependent on cavity tuning)

one revolution period

Multi-bunch energy compensation1. t-method (for the NLC)

concept: change (advance) the relative time of arrival of the bunch train with respect to the accelerating voltage from the pulsed power source so that the vector sum Vk+Vb is constant

note: if the compensation is not perfect, while the projected energy spread is minimized, each bunch could have a different energy spread

Vk

Vb

Vk+Vb

2. f-method (for the JLC and ATF)

concept: detune some structures positively, some later structures negatively so that the position-energy correlations introduced by the slope of the rf cancels V0 = cost

V1 = sin(t+t) sin(t)cos(t) + tcos(t)V2 = sin(t-t) -sin(t)cos(t) + tcos(t)

compensatingvoltages add

single-bunchenergy spreadcancelled

bunch energy spread and projected energy spreadof train both minimized

Next we describe the commonly used method (SLC, LEP, HERA,…)for changing the transverse beam emittance by changing the accelerating frequency of the rf cavities in a storage ring.

We begin with

introductory remarks on equilibrium emittance including the definitions of the partition numbers, damping times, and a statement of Robinson’s theorem

then, changes to the transverse emittance by changing the accelerator circumference will be presented (somewhat analogous to the case in point)

lastly we describe emittance control via change of the rf frequency

(inadvertentlyomitted in lastweek’s lecture)

Equilibrium emittance (reference: famous paper by M. Sands)

limit to phase focussing in e+/e- storage rings is given by the quantum excitation (Sands, 1955):

bending radius

with

emittance growth due to quantum excitation:

emittance decreasefrom radiation damping

partition number

equilibrium emittance reached when the quantum excitation equals the damping:

The equilibrium emittance can be changed by changing D (either by changing thering circumference or the accelerating rf frequency) or by adding wiggler magnets

(eq. 8.35)

(~ 1/Ju)

(Dx is dispersion function)

Radiation damping rates, the partition numbers, and Robinson’s theorem

The damping times may be expressed in convenient form in terms of the partition numbers, Ji (i=x,y,z)

which depends on the beam energy E0 and on the average rate of energy loss <Pγ>

We may also write here <Pγ>=U0/T, where U0 is the energy loss per particle per turn andT is the revolution period. Hence <Pγ> may be easily calculated. Since particle tracking codes evaluate the Ji, it is then easy to calculate the change in damping times.

Robinson’s theorem: Jx+Jy+Jz=4

Jx=1-DJy=1Jz=2+D

(horizontal ring with no vertical bends)

again

so to evaluate the influences of the applied change (C, f) one needs to calculate D

Here and (isomagnetic ring)

(~ 1/Ju)

Jx=1-DJy=1Jz=2+D

(horizontal ring with no vertical bends)

Again,

The following methods rely on manipulating D

Simplified expressions (compared to text):

Consider GK=0 (no combined-function magnets). Then D|GK=0 ~ Dx/ ~ R/2

We estimate then D:

static case - “stretched” ring (on-energy orbits is offset in quadrupoles): G = kx then

dynamic case – frequency shift (off-energy orbit is offset in quadrupoles): G = kDx (E/E)

Δxρ2π

DLN2kΔD

xqqq2

(E/E)=-(1/)(f/f)

E

ΔE

ρ2π

DLN2k~ΔD

xq2

qq2

Circumference changechange in D after changing the magnetic circumference of the acceleratorwhile holding the rf frequency fixed, G=Kxmag :

on-energy orbits are offset in quadrupoles

example from the SLC:

measured damping timesand equilibrium beam sizebefore circumferencechange

same as above after increasing the circum-ference by 9 mm

radial displacement

~ 1/Jx, Jx=1-D

RF frequency change

change in D after shifting the rf frequency (while the magnetic circumference of the accelerator is unchanged), G=KE/E:

off-energy orbits are offset in quadrupoles

example from the SLC (details of implementation will be covered later in chapter 8):

beam emittancewithoutfrequencyshift

beamemittancewith 62.5 kHz(f/f~10-4)frequencyshift

Emittance control in lepton acc.’s via rf frequency shift (LEP, HERA, SLC)

x,xinf ~ 1/JxJx=1-D

Recall from chapter 4

By reducing Jx therefore, the beam damps faster and the equilibrium beam emittance is smaller.

Example: simulated reduction in normalized horizontal beam emittance for different amplitude frequency shifts in the SLC damping rings

We have already seen an example where Jx is modified by “stretching” the accelerator.Here we consider an equivalent method using an rf frequency shift f, where

beam size (squared), proportional to emittance,atextraction

injection time infinity

damping time

f=0

f=2E-4

extraction

In storage rings and colliders, there is no tight tolerance on maintaining the desired rffrequency. In a damping ring, the time required to reset the frequency and to relock the beam phase to the desired extraction phase is critical.

Practical issues associated with the rf system: 1. changing the rf frequency changes the cavity tuning 2. voltage feedback acts to maintain the desired voltage 3. if the detuning is large, there may be insufficient klystron power in the case of no beam

cavity tuning angle:

new tuning angle with fixed (or very slow) tuners:

tuning angle for minimum reflected power

using

Example: beam emittance measured before (left) and after (right) implementation of an rf frequency shift in the SLC damping rings

Emittance control in linear accelerators

E=50 GeVL=3 km frf=2856 MHzfrep=120 HzNe+ = Ne- =41010ppbE/E~0.1%x/y=4.5/0.9 10-5 m-rad

layout of the SLC

sources of emittance dilution in linacs optical errors gradient errors β-mismatch -mismatch (including T166, U1666,…)trajectory errors BPM and quad misalignments generating dispersive errors structure misalignments generating rf deflections “klystron complement”stability issues: injection errors/launch variations offsets (x,x’,y,y’) phase offsets () component vibration phase reference stability current jitter

It seems appropriate to discuss this topic here having already covered the material onlongitudinal dynamics. As we have seen, the longitudinal properties of a beam in a linac influence strongly the projected transverse beam emittances. Specific examples include dilutions arising from nonzero dispersion and bunch energy spread, and dilutions arising from trajectory distortions resulting from long-range wakefields. The examplesgiven here are from the SLC, however the beam dynamics is of interest to any futurelinear collider (NLC,GLC, Tesla, CLIC), future linac-based SASE FEL’s, and possiblyfor future ERLs.

review x=transverse displacement

s = emittance ~ 2/

= beam size = <x2>1/2

x = xco+x+xxco

x

x

Equation of motion (Chao, Richter, Yao, 1980)

longitudinalcoordinate,s

beamenergy

lattice strength

relative longitudinalcoordinate, z

longitudinal distributionfunction

transverse wakefield

In the following, we assume that k(s) is smoothly varying (as opposed to consisting of discrete quadrupoles)

In the following, we consider these limiting cases:

transverse wakefield W

caseBeam energy

latticefocussing k

(i)

(ii)

(iii)

(iv)

(v)

concept of interest

0

0

W’ z

W’ z

W’ z

E0

E0(1+Gs)

E0

E0(1+Gs)

E0(1+Gs)

k0

k0

k0

k(s) –special cases

k(s) - general

simplest case

adiabatic damping

motivate tail vshead oscillation

beam breakup

BNS & autophasing

Again,

(i) W=0 zero current limit k=k0 constant focusing E=E0 no acceleration

(ii) W=0 zero current limit k=k0 constant focusing E=E0(1+Gs) linear acceleration with gradient G

betatron oscillations damp as 1/sqrt(E) --- “adiabatic damping”

equation ofmotion

solution:

equation ofmotion

solution:

betatron oscillation of peak amplitude x about a reference trajectory xco+x

<

initial conditions: x(0)=x, x’(0)=0

initial conditions: x(0)=x, x’(0)=0

<

<

in pictures:

emittance damps as 1/E (~ 1/)in practice, often express emittance as (“normalized emittance”) along linac

p is unchangedby acceleration

differentiating,

i.e. the equation of motion with k=0

horizontal emittance:

Example: measured horizontal oscillations along the SLC linac after application of an initial displacement versus number particles per bunch

So far, only single particle motion has been considered. For a bunch consisting of multiple particles, particles of different energy are focused differently. The motion of the bunch centroid (the position of the mean of the bunch charge) may therefore not damp as 1/sqrt(E); the macroparticle approximation breaks down if the bunch has an internal energy spread.

dependence of the lattice focussing on energy:

centroid motiondecays faster than1/sqrt(E) due to spread in phaseadvance of particleswithin the bunch

longitudinal profileoptimized and centroid motion decays initiallyas 1/sqrt(E)

at higher particledensities, the effectof the wakefieldsneed to be considered

(iii) W≠ W’z k=k0 E=E0(1+s)

equation ofmotion

3-particle model (J. Seeman, 1991)

equations of motion for each macroparticle:

effect of head on tail at z=2z

effect of core on tail at 2(N/4)

effect of head on core

solutions:

sT C H

zmacroparticlespacing

macroparticlechargeN/4 N/4N/2

comments: 1. solutions for xh, xc, and xt are all linear in x 2. each slice adds a power of B (~NW) and s growth in amplitude of tail of bunch due to transverse wakefields

initial conditions: x(0)=x, x’(0)=k0x

< <

<

Example: profile monitor measurements showing that the bunch head drives the tail to larger amplitudes during a betatron oscillation due to W

linac

=0

0

screen

e-

y

x~E

projected beamemittance largerthan the “sliceemittances”

Example: measured images demonstrating emittance growth due to wakefields for different initial displacements at start of linac (courtesy J. Seeman, 2000)

x=0 x=0.2 x=0.5 x=1.0 mm

Ne-=21010 ppb

x

y

(fast kickers used to deflect beam onto screens)

(iv) W= W’z dE/ds≠0 k(s) – special casesa) k adiabatic

In this case, the solution to the equation of motion has no closed form expression. A solutionmay be obtained (Chao, Richter, Yao) by expanding the solution x(z,s) in a power series and solvingrecursively. In the asymptotic limit of strong wakefields (|}>>1), the peak-to-initial amplitudegiven at the end of the linac of length L is

The trajectory of the lagging particle increases exponentially with the transverse wakefield. This phenomenon has come to be referred to as “beam breakup”.

Example: measured transverse profile of a long bunch showing large transverse displacements of the bunch tail (courtesy J. Seeman, 2000)

x

y

(fast kickers used to deflect beam onto screens)

bunch intentionallyelongated to about2.5 times normallength, 2.5 mm oscil-lations introduced inboth transverse planesearly in linac

bunch head

b) k tailored (Balakin, Novokhatsky, Smirnov - BNS - damping)

Motivation: combat the beam breakup instability by providing a different focussing function across the bunch (with the same focussing function experienced by all the particles in the bunch, a perturbation at the head of the bunch may resonantly drive particles in the tail of the bunch). Expressed another way, by appropriate adjustment of the beam phase relative to the rf, compensate the defocusing due to wakefields with stronger focusing for lower-energy tail particles

Using the 3-slice model, let the focussing be k=k0+ at the head of the bunch k=k0 for the bunch core k=k0- in the tail of the bunch

Requiring that the head and the core follow the same trajectories, =4 and

Example: estimated energy spread at the end of the linac for different complements of klystron phases (courtesy F.-J. Decker, 2000)

(%)

s

klystrons “back-phased”(bunch preceeds more in time the rf wave) in beginning of linac whereeffects are strongest

energy spreadrestored in remainderof linac by forward-phasing the remainingklystrons

(v) W= W’z autophasing dE/ds≠0 k(s) – exact

Example: measured horizontal trajectories obtained under nominally identical conditions without BNS damping (top) and with BNS damping (bottom) with Ne-=21010 ppb (courtesy J. Seeman, 2000)

measured centroid displacementnormalized to initial kick amplitudewas reduced by ~10 with BNSdamping

With BNS damping alone, the projected 6-D emittance may assume large valuesalong the linac where is large. Emittance dilutions could result if the dispersionis not perfectly corrected. Alternatively, substituting x(s)=x0 cos(k0s+0) into the equation of motion, leads to

<

If k(z,s) can be adjusted to exactly compensate the second term for all particles within the bunch, then all particles would follow the same trajectory and experience the same focussing so that the projected 6-D emittance would be maintained over the entire linac.This condition is hard to realize in practice. Of the adjustable parameters k, , and, the longitudinal density distribution is perhaps most easy to control.

Summary

Many techniques using rf systems to manipulate the LPS were presented:

The (linearized) equations of motion in longitudinal phase space (LPS) were reviewed

Small-amplitude motion in LPS obeys that of a simple harmonic oscillator

bunch length compression for shortening the bunch length in linear accelerators (to subsequently minimize the energy spread and hence the projected transverse beam emittances) for producing maximum peak intensities (e.g. as required for SASE-FELs)

bunch length precompression for better matching in LPS (to avoid longitudinal emittance dilution, HERA) for minimizing beam loss in downstream transfer lines (SLC)

bunch lengthening using harmonic cavities in synchrotron light sources for minimizing the particle density (to improve the beam lifetime)

bunch coalescing (FNAL) and bunch splitting (LHC) to change the current distributions (i.e. number of bunches)

energy compression single-bunch for avoiding “longitudinal apertures” multi-bunch as a method for compensating beam loading in linacs

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RF systems may also be used to control parameters in transverse phase space

Two cases were reviewed: transverse emittance control in a) circular and b) linear acc.’s

In circular accelerators with radiation damping changing the rf frequency changes the partition number Jx (and Je by Robinson’s theorem) and reduces both x and x

a similar technique (“stretching” the ring) achieving the same was also discussed

For the case of linear accelerators, in the absence of optical errors, the transverse emittances are dominated by the incoming bunch length and beam loading. The equation of motion was considered in 5 cases (from most easy to most general) illustrating the concepts of simple harmonic motion (constant focussing and energy) adiabatic damping (recall ) motion in the front of the bunch influencing particles in the tail of the bunch beam breakup BNS-damping and autophasing

In the most general case with BNS-damping and autophasing (which has never beenachieved), the variable parameters of interest are k – the particle focusing - the particle energy - the longitudinal density distribution (which is perhaps the easiest to control)