ERL2005

BBU Codes OverviewMasaru Sawamura and Ryoichi Hajima

JAERI

Outline• Introduction• Beam transport Equation• How to solve (BBU-R, TDBBUU, bi, MATBBU etc.)

• Comparison of BBU codes

ERL2005

IntroductionERL current limited by • Beam Breakup

– Transversedeflect beam

– Longitudinalarrival time differenceHOM-induced energy spread

HOM

T12 or T34

HOM

T56

ERL2005

Beam Transport Equation

∑ ∑=

−−+

=−−− +−+−+−=

pn

r

MrpM

knkrn

ppnnp

ppnnp skMrpMnUIZGTMnUTMnU

1

1)(

1011,1,

0

)())(,1(),1(),( τω

n-th Cavity (n-1)-th Cavity

(p+1)-th pass

p-th pass

(p-1)-th pass

M

0

M-

M+

M

MM-

Transport

Kick due to Induceded HOMsby Previous Bunches

Induced HOM Decay&

Phase shift

•transport •kick by HOM

currentaverageIsitescavityofnumbern

passesnkes

lengthcavitylionrecirculatoneinbunchesofnumberM

impedancetransverseZ

G

matrixtransferT

p

nQ

k

nk

n

pqmn

n

::

:)sin()(

::

:

:

0

2

0

,

0100

τωτωτω

−

=

⎟⎠⎞⎜

⎝⎛=

ERL2005

How to Solve

• Beam Tracking (Beam position vs. Time)– BBU-R (JAERI)– TDBBU (JLab)– bi (Cornell Univ.)– new code (JLab)

• Eigenvalue Solution (Current vs. Frequency)– MATBBU (JLab)

ERL2005

BBU-Rfeature• Transverse BBU• Beam position vs. time (time increment Trf/2)• Point-like bunch• X,Y HOM orientation• X- & Y-axis independently• Impulse of HOM kick• Using 2X2 Transfer Matrix• Two-pass recirculation

ERL2005

BBU-R Algorism

0 n m nloc-1

0

Trf/2

k Trf/2

nloc Trf/2

Cavity Cavity

Expand beam line into a consecutive time-step array

Set initial bunch

Move Trf/2 step

Bunch in cavity? Update bunch position to next cavity

Update HOM power Decay & Phase shift

Bunch in cavity? Bunch kicked by HOM

Last location? Create new bunch

Y

Y

Y

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Threshold current of JAERI ERL-FEL• layout

-4

-2

0

2

4

0 500 1000 1500 2000

x (m

m)

Time ( sec)µ

3.42A

-50

-25

0

25

50

0 500 1000 1500 2000

x (m

m)

Time ( sec)

6.11A

µ

Beam Dump

Main 5-cell SCAElectron Gun

SHB

1-cell SCA

Injection Merger

2nd Arc1st ArcUndulator

Half Chicane

ERL2005

Example for future ERL design

β-function in the linac

Threshold current vs. HOM randomization

Threshold current vs. Cavity gradient

6GeV 1.3GHz 15MV/m

48 cryomodules 47 external QT

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TDBBU

• Transverse BBU• (a) Move bunch

(b) calculate position of bunch entering the cavity(c) update all HOM excitation level

• Turn the current up/down to find instability• X- &Y-axis entirely independent• Suitable for large HOM numbers or

short characteristic time (2Q/ω)

ERL2005

Example of TDBBU results

[K.Beard et al., PAC2003 Proc. 332 (2003)]

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bi

• Transverse or longitudinal BBU• Allow any ERL topology• transient effect for arbitrary bunch pattern• Arbitrary HOM orientation

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Example of bi results

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 50000 100000 150000

-3

-2

-1

0

1

2

3

0 50000 100000 150000

-10-8-6-4-202468

10

0 50000 100000 150000

bunch #

arriv

al ti

me

diff

eren

ce [p

s]

I = 22.0 mA

I = 23.0 mA

I = 22.1 mA

Longitudinal instability

ERL2005

New code at JLab

• Transverse BBU• Two-path machine• Full 2D particle tracking (4X4) or 1D (2X2)• Arbitrary HOM angle• Decoupled transverse motion and coupled motion

effect of rotated HOMeffects of rotated HOMs and of rotated optics

• Include FB for BBU suppression

ERL2005

Example of new code results

[C.Tennant and E.Pozdeyev, JLAB-TN-02-020 (2004)]

Beam displacement as a function of time for Ib>Ith

FFTs of the indicated”slices”of the beam displacement

Threshold current versus HOM orientation for a single cavity with one HOM and a decoupled recirculation matrix

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MATBBU

• Solve eigenvalue of matrix representing system

• Transverse BBU• X,Y axis treated sequentially,

entirely independently

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Algorism of MATBBU

Ω is also unknown Scan Ω and find Im(I)=0 & I>0

∑∑∑∑∑=

−

== < =

−Ω Ω+Ω=pp n

p

i

llll

ppli

n

p pr

n

llll

rpMprlii DhZTIDhZeTID

1

1

112,

2 1

)(12, )()()()(

00 τ

0&)(),( >>= Ω MnVeMnU pMi

pτ

∑=

Ω−−−− −Ω+−=

pn

rrn

Mrpippnnnp

ppnnp nVheGTIZnVTnV

1

)(1,11, )1()()1()( 0 τ

∑ ∑=

−−+

=−−− +−+−+−=

pn

r

MrpM

knkrn

ppnnp

ppnnp rskMrpMnUIZGTMnUTMnU

1

1)(

1011,1,

0

)())(,1(),1(),( ω

∑=

=pn

kk

ikM nVenV0

)()( 0τ

assume a steady state solution

sum over all passes

)(iVofcomponentxDi −

DI

DM 1)( =ΩEigenvalue problem

Ω : coherent frequency

)cos()(2)(1)sin()()( 2 τω

τω

nnn

nnn HH

HhΩ−Ω+

Ω=Ω

ττω

Ω−−

=Ω iQn eeH

n

2)(

ERL2005

Example of MATBBU results

[K.Beard et al., PAC2003 Proc. 332 (2003)]

[J.J.Bisognano et al., 1987 PAC Proc. 1078 (1987)]

Stability plot of complex threshold current (amperes) for one site, two-pass configuration

Sweep coherent frequency and search Positive real current

ERL2005

Comparison of BBU codes

JLabJLabCornell Univ.JLabJAERIDeveloper

Fortran/CC++C++Fortran/CCProgrammingLanguage

Arbitrary2ArbitraryArbitrary2No. of Recirculation

X,YArbitraryArbitraryX,YX,YHOM direction

EigenvalueTrackingTrackingTrackingTrackingSolve

1D1D/2D2D1D1DDimension

TTT/LTTTransverse/Longitudinal

MATBBUNew code @JLabbiTDBBUBBU-RName

ERL2005

Other problems

• Thin cavityImpulse kickWhere does a kick work -entrance, middle or exit? Single- or multi- kick for low energy?

• Faster methods to determine whether stable or instablebeam displace or HOM voltage?