Accelerator Physics Topic VII Coupled Bunch Effects

J. J. Bisognano

Topic Seven: Coupled Bunch Effects

1

UW Spring 2008

Accelerator Physics

Accelerator PhysicsTopic VII

Coupled Bunch Effects

Joseph Bisognano

Engineering Physics &

Synchrotron Radiation Center

University of Wisconsin-Madison

J. J. Bisognano


2

UW Spring 2008

Accelerator Physics

Coupled Bunch Instabilities• We have discussed instabilities internal to a single bunch of charged

particles• Typically in a storage ring or linear accelerator there are trains (finite or

cw) of bunches separated by nanoseconds to maybe milliseconds• Say we have a resonant structure at 300 MHz, with an angular frequency

of 2(300) 2 GHz• If it has a Q of 20,000 (typical of Cu), its fields ring for 20,000/2 GHz=10

microsecond; if the Q were 2 109 more typical of superconducting RF, the ringing would last a full second

• So a sequence of bunches can talk to each other through resonant structures

• Whereas low Q impedances have a large bandwidth and can “see” the peak current, these high Q structures have a narrow bandwith and only see the average current.

• In other words, broadband impedances generate peak current limitations in accelerators, narrowband impedances generate average current limitations

J. J. Bisognano


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Accelerator Physics

Bunch Spectrum

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J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Robinson Instability

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J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Robinson/cont.

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J. J. Bisognano


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Accelerator Physics

Robinson/cont.

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J. J. Bisognano


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Accelerator Physics

Robinson/cont.

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J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Robinson/cont.

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J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Robinson Conclusions

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Accelerator Physics

Robinson Stability Condition

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- + - +

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Coupled Bunch Instabilities

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J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Coupled Bunch/cont.

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J. J. Bisognano


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Accelerator Physics

General Phase Relationship

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J. J. Bisognano


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Accelerator Physics

Normal Modes N=4

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Accelerator Physics

Spectrum/cont.

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Growth Rates

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Accelerator Physics

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Accelerator Physics

Mode Coupling

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J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Mode Coupling at SRC

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Accelerator Physics

Transverse Phenomena

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J. J. Bisognano


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UW Spring 2008

Accelerator Physics

Transverse Coupling

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Accelerator Physics

Deflecting Modes

:pattern field withcan dipole

but deflect, tdon' modes 0 So

1

Particle on axis doesn’t see Ez , doesn’t deposit energy

Particle off axis can excite mode through Ez

But deflection is constant through derivative of Ez

J. J. Bisognano


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Accelerator Physics

Resonant Wakefield

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UW Spring 2008

Accelerator Physics

Beam Breakup in Linear Accelerators• In a linac there the higher order cavity modes produce the same

basic resonant self-interaction, both longitudinal and transverse• For relativistic linacs, the longitudinal motion is more “frozen” than

in a storage ring, which has bending. So transverse effects are often the limiting factor in linacs

• For transverse effects, the primary difference in the dynamics is number of times the same bunch sees a given cavity HOMs– Straight linac: once, amplification– Recirculated linac: several times, instability with finite threshold– Storage ring: infinite times, zero threshold unless some form of

damping present• In linacs, these effects are call Beam Breakup

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UW Spring 2008

Accelerator Physics

Regenerative Beam Breakup• Basic mechanism: a train of bunches excites a transverse deflecting mode of a

single cavity

• Feedback loop

– Say, HOM has small excitation

– Even a bunch perfectly aligned on axis will receive a transverse kick

– If energy is low and structure long, a significant deflection will occur while the bunch is in the cavity

– The offset bunch is now in a region of longitudinal electric field and can deposit energy into mode

– Go to next bunch

– We have a feedback loop that can go unstable unless the cavity losses (more with lower Q) exceed the gain of the loop

– An honest instability

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Accelerator Physics

Regenerative Beam Breakup

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Accelerator Physics

Threshold Condition

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yields whichequal, whenthreshold out,energy is Pl/c

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2

2

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Accelerator Physics

Cumulative BBU Amplification

cavity first in offset be beam let and cavities, of series

a through passing beam bunched a Consider

1 2 3 4 5

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Accelerator Physics

Cumulative BBU/cont.• Cavity 1: Bunch will coherently excite cavity, later

bunches will receive transverse kick• Cavity 2: Bunch will enter cavity 2 with an extra offset;

cavity 2 experiences an enhanced excitation• Cavity N: DITTO• Overall, initial offset causes growing excitation of

subsequent cavities which can increase offset downstream: Amplification

• Since there is no closure of loop, there is no instability as such

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UW Spring 2008

Accelerator Physics

Cumulative Beam Breakup

• Typically bunching frequency and transverse HOM frequency are not harmonically related

• So, there can be a large transient, but the equilibrium excitation can be rather small. For a pulsed linac, however, the transient can cause beam loss, limiting currents to ~100 mA

• For CW operation with equally spaced bunches, the excitation settles down to a DC value that can be steered away

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Accelerator Physics

Multipass Beam Breakup

• A “new” feature of SRF linacs is the possibility of recirculation, and even energy recovery

• SRC allows CW operation and the beam can pass through the linac several times

• The “cumulative” beam breakup amplifier now has its feedback loop closed and at high enough gain there can be instability

• Limited the first generation of SRF linaces to 10 microamps average currents when HOM Q’s were in the 10,000,000 range

• In some ways it’s a combination of cumulative and regenerative BBU

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UW Spring 2008

Accelerator Physics

Multipass BBU Mechanism

• Displaced bunch excites a HOM

• Following bunches deflected

• Recirculation optics transforms kick into a displacement

• Displaced bunch further excites HOM in same cavity

• Again threshold occurs when excitation rate exceeds damping rate

J. J. Bisognano


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Accelerator Physics

Beam Breakup Mechanism

Initial noise excitation of cavity mode kicks particle bunch

On subsequent pass,bunch enters off axis and coherently excitescavity mode

cavity

Beam on pass n+1

beam on pass n

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Accelerator Physics

CEBAF

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Jlab FEL

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Multipass BBU Theory

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Multipass BBU Theory/cont.

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Simulation: transient and steady state below threshold (cumulative-like)

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Simulation: instability

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Longitudinal Multipass BBU Theory

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General Scaling with Q• For a single cavity, the threshold scales like 1/Q• For several cavities at the same resonance frequency,

the threshold scales like 1/Q times weighted sum over the transport optics

• But HOMs have a distribution in frequency from construction errors which– Decreases the peak value of the weighted sum– Changes the 1/Q dependence to something more

typically like 1/Q until frequency spread is so large that the cavity modes don’t overlap significantly

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Accelerator Physics

Typical Frequency Distribution Scaling

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Accelerator Physics

Typical Q Scaling

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Energy Recovery Linacs

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Accelerator Physics

Current Limits• Typically, a storage ring light source will store a

few hundred milliamps• Progress in electron source development make

CW guns at the 100 milliamp level reasonable to talk about with emittance performance comparable and even better than storage rings

• Progress in HOM damping has made current limits at the 100 milliamp level obtainable

• So, an energy recovery linac should be able to produce storage ring levels of current with better emittance

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Accelerator Physics

Is It Worth Recirculating an ERL DriverMore Than Once

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Energy Recovery Linac (ERL)

superconducting

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Accelerator Physics

Issues• Good

– Save money: SRF, RF, cryo– Beams of different energy right there

• Bad– Costs of more magnets– Beam breakup– Coherent synchrotron radiation (CSR)

• emittance growth• energy spread growth

– CSR instability– Space charge – Weaker focusing: head tail could be worse– RF constraints: off phase choices– Layout of compressors

• Just different– Site constraints: shorter and wider

• Multipass harder, but are limits limiting and are the savings significant

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Accelerator Physics

One up/one down Current Limits with Arc Optics Variation

Threshold vs. Phase Advance

672.12

528.13

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425.63

750.21

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401.23389.03

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Phase Advance [2*pi]

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A)

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Accelerator Physics

Two up/two down Current Limits with Arc Optics Variation

300

200

100

mA

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Homework Problems Topic VII• Topic VII-1 Consider a storage ring with 4 equally spaced bunches.

– a) Derive an eigenvalue problem for longitudinal coupled bunch motion for 4 point-like bunches undergoing small oscillations. Assume the instability is excited by a single higher order mode at some frequency. (Proceed as follows: calculate the voltage produced by an individually oscillating bunch. Second, calculate the perturbatin induced by this voltage on any of the four bunches.)

– b) Show that the eigenmodes reduce to those discussed in lecture

– c)Discuss what happens if one of the bunches is missing; i.e., there is a gap.

• Topic VII-2 Consider a charge Q passing rQ off axis through a cavity and a charge e passing time behind

– For a transverse deflecting mode, write an expression for the energy loss of the charge Q and the transverse deflection of charge e if it were on axis, and also if it were at rQ

– In physical terms, why would isochronous transport in a two-total-pass recirculated linac prevent longitudinal beam breakup? What would be the analogous transport constraint to prevent transverse beam breakup? What are the limitations of such solutions?

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Accelerator Physics Topic VII Coupled Bunch Effects

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