I PhysicsP I llinois George Gollin, Cornell LC 7/031 A Fourier Series Kicker for the TESLA Damping...

George Gollin, Cornell LC 7/031 IPhysicsPIllinois

A Fourier Series Kicker for the TESLA Damping Rings

George Gollin

Department of Physics

University of Illinois at Urbana-Champaign

LCRD 2.22


Introduction

•The TESLA damping ring fast kicker must inject/eject every nth bunch, leaving adjacent bunches undisturbed.

•The minimum bunch separation inside the damping rings (which determines the size of the damping rings) is limited by the kicker design.

•We are investigating a novel extraction technique which might permit smaller bunch spacing: a “Fourier series kicker” in which a series of rf kicking cavities is used to build up the Fourier representation of a periodic function.

•Various issues such as finite bunch size, cavity geometry, and tune-related effects are under investigation.


Outline

Overview

• TESLA damping rings and kickers

• how a “Fourier series kicker” might work

Phasor representation of pT and dpT/dt

Flattening the kicker’s dpT/dt

Some of the other points:• finite separation of the kicker elements• timing errors at injection/extraction

Conclusions

4 IPhysicsPIllinois

Illinois participants in LCRD 2.22

Guy Bresler (REU student, from Princeton)Keri Dixon (senior thesis student, from UIUC)George Gollin (professor)Mike Haney (engineer, runs HEP electronics group)Tom Junk (professor)

We benefit from good advice from people at Fermilab and Cornell. In particular: Dave Finley, Vladimir Shiltsev, Gerry Dugan, and Joe Rogers.


Overview: linac and damping ring beams

Linac beam (TESLA TDR):

•One pulse: 2820 bunches, 337 nsec spacing (five pulses/second)

•length of one pulse in linac ~300 kilometers

•Cool an entire pulse in the damping rings before linac injection

Damping ring beam (TESLA TDR):

•One pulse: 2820 bunches, ~20 nsec spacing

•length of one pulse in damping ring ~17 kilometers

•Eject every nth bunch into linac (leave adjacent bunches undisturbed)

17 km damping ring circumference is set by the minimum bunch spacing in the damping ring: Kicker speed is the limiting factor.


Overview: TESLA TDR fast kickerFast kicker specs (à la TDR): B dl = 100 Gauss-meter = 3 MeV/c

• stability/ripple/precision ~.07 Gauss-meter = 0.07%

• ability to generate, then quench a magnetic field rapidly determines the minimum achievable bunch spacing in the damping ring

TDR design: bunch “collides” with electromagnetic pulses traveling in the opposite direction inside a series of traveling wave structures.

TDR Kicker element length ~50 cm; impulse ~ 3 Gauss-meter. (Need 20-40 elements.)

Structures dump each electromagnetic pulse into a load.


Something new: a “Fourier series kicker”

Fourier series kicker would be located in a bypass section.

While damping, beam follows the dog bone-shaped path (solid line).

During injection/extraction, deflectors route beam through bypass (straight) section. Bunches are kicked onto/off orbit by kicker.

injection path extraction path

kicker rf cavities

injection/extraction deflecting magnet

injection/extraction deflecting magnet

pT


Fourier series kicker

Kicker would be a series of N “rf cavities” oscillating at harmonics of the linac bunch frequency 1/(337 nsec) = 2.97 MHz:

1

0

2cos ;

337 ns

cavitiesj N

T j high low lowj

p A A j t

injection path extraction path

kicker rf cavities

fhigh fhigh + 3 MHz

fhigh + 6 MHz

fhigh + (N-1)3 MHz

...


Fourier series kicker

Run them at 3 MHz, 6 MHz, 9 MHz,… (original idea) or perhaps at higher frequencies, with 3 MHz separation: fhigh, fhigh+3 MHz, fhigh+6MHz,... (Shiltsev’s suggestion)

Cavities oscillate in phase, possibly with equal amplitudes.

They are always on so fast filling/draining is not an issue.

Kick could be transverse, or longitudinal, followed by a dispersive (bend) section (Dugan’s idea).

High-Q: perhaps amplitude and phase stability aren’t too hard to manage?

fhigh fhigh + 3 MHz

fhigh + 6 MHz

fhigh + (N-1)3 MHz

...


Kicker pT

A ten-cavity system might look like this (fhigh = 300 MHz):

0 50 100 150 200 250 300 350

10

5

0

5

10

Kick vs. time, 10cavity system, 300MHz lowest frequency, f 3MHz

(actually, one would want to use more than ten cavities)

1

0

~ coscavitiesj N

T high lowj

p j t

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Bunch timing

0 50 100 150 200 250 300 350

10

5

0

5

10


Kicked bunches are here…

…undisturbed bunches are here (call these “major zeroes”)

1 2 3 4 5 6 7 8 9 10

Interval between kick and adjacent “major zeroes” is uniform.

2 1 0 1 210

5

0

5

10Kicking pulse, 10cavity system, 300MHz lowest frequency, f 3MHz

32 33 34 35

0.4

0.2

0

0.2

0.4

Kick vs. time, 10cavity system, around first major zero


Extraction cycle timing

Assume bunch train contains a gap between last and first bunch while orbiting inside the damping ring.

first bunch

last bunch

1. First deflecting magnet is energized.



bunches 1, N+1, 2N+1,...

2. Second deflecting magnet is energized; bunches 1, N+1, 2N+1,… are extracted during first orbit through the bypass.



3. Bunches 2, N+2, 2N+2,… are extracted during second orbit through the bypass.

4. Bunches 3, N+3, 2N+3,… are extracted during third orbit through the bypass.

5. Etc. (entire beam is extracted in N orbits)


Injection cycle timingJust run the movie backwards…

With a second set of cavities, it should work to extract and inject simultaneously.


Sometimes pT can be summed analytically…

Here are some plots for a kicker system using frequencies 300 MHz, 303 MHz, 306 MHz,… and equal amplitudes Aj :

Define 2 300MHz; 2 3MHz; high low high lowK

1

0

coscavitiesj N

T high lowj

p j t

1 12 2

12

sin sin

2sincavities low low

low

K N t K t

t

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Lots of algebra. Visualizing this…

1

0

~ coscavitiesj N

T high lowj

p j t

Represent each cavity’s kick as a “phasor” (vector) whose x component is the kick, and whose y component is not...

cos high lowj t

j

sin high lowj t j high lowj t

Each cavity’s phasor spins around counterclockwise…


pT

Phasors: visualizing the pT kick

The horizontal component of the phasor (vector) sum indicates pT.

Here’s a four-phasor sum as an example:


Phasors: visualizing the pT kickHere’s a 10-cavity phasor diagram for equal-amplitude cavities…

start here

end here

pT

…and 30-cavity animations (30, A, B, C).

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Both the x and y components of the phasor sum…

1

0

1 12 2

12

: cos

sin sin

2sin

cavitiesj N

high lowj

cavities low low

low

x j t

K N t K t

t

1

0

1 12 2

12

: sin

cos cos

2sin

cavitiesj N

high lowj

cavities low low

low

y j t

K N t K t

t

…are zero when (m is integral)2cavities lowN t m



One kicker cycle comprises a kick followed by (Ncavities - 1) major zeroes.

Kicks occur when the denominator is zero:2 4

0, , ,low low

t

“Major zeroes” between two kicks are evenly spaced and occur at

1 22 4, , cavities

cavities low cavities low cavities low

Nt

N N N


Phasors when pT = 0 (30 cavities)

1.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 11.1111

scaled kick 7.21645 1017

Zero crossing 1

x,y 6.37275 1015, 1.43387 1014vx,vy 1.1002, 2.471091.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 22.2222


Zero crossing 2

x,y 1.06798 1014, 1.17477 1014vx,vy 0.909964, 1.01062

Phasor diagrams for major zeroes of one specific example:

• 30 cavity system

• 300 MHz lowest frequency, 3 MHz spacing

zero #1 zero #2

green dot:

tip of first phasor

larger red dot:

tip of last phasor


1.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 33.3333


Zero crossing 3

x,y 1.16765 1015, 3.59276 1016vx,vy 0.870195, 0.2827431.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 44.4444


Zero crossing 4

x,y 1.77423 1015, 1.61728 1014vx,vy 0.0726631, 0.691343


zero #3 zero #4


1.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 55.5556


Zero crossing 5

x,y 5.10703 1015, 2.88658 1015vx,vy 0.489726, 0.2827431.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 66.6667


Zero crossing 6

x,y 1.03885 1015, 7.5553 1016vx,vy 0.389163, 0.282743


zero #5 zero #6



1.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 77.7778


Zero crossing 7

x,y 1.32736 1015, 6.45015 1015vx,vy 0.0878538, 0.4133191.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 88.8889


Zero crossing 8

x,y 1.74232 1014, 3.66018 1015vx,vy 0.372155, 0.0791039

zero #7 zero #8



1.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 100.


Zero crossing 9

x,y 1.784 1015, 2.47016 1015vx,vy 0.205425, 0.2827431.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 111.111


Zero crossing 10

x,y 3.39723 1015, 5.89709 1015vx,vy 0.163242, 0.282743

zero #9 zero #10



1.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 122.222


Zero crossing 11

x,y 1.01401 1014, 1.09376 1015vx,vy 0.307806, 0.03235171.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 133.333


Zero crossing 12

x,y 5.25311 1016, 1.48838 1015vx,vy 0.0918689, 0.282743

zero #11 zero #12



1.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 144.444


Zero crossing 13

x,y 1.47553 1015, 1.36203 1015vx,vy 0.214813, 0.1934191.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 155.556


Zero crossing 14

x,y 8.12136 1015, 3.6281 1015vx,vy 0.259722, 0.115636

zero #13 zero #14



1.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 166.667

scaled kick 0.

Zero crossing 15

x,y 0., 8.82123 1015vx,vy 7.54029 1015, 0.2827431.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 177.778


Zero crossing 16

x,y 2.43083 1014, 1.07727 1014vx,vy 0.259722, 0.115636

zero #15 zero #16



1.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 188.889


Zero crossing 17

x,y 9.70447 1015, 8.68295 1015vx,vy 0.214813, 0.1934191.5 1 0.5 0 0.5 1 1.5 2

1.5

1

0.5

0

0.5

1

1.5

2Phasor plot

tnsec 200.


Zero crossing 18

x,y 1.57593 1015, 4.9029 1015vx,vy 0.0918689, 0.282743

…and so forth. (There are 29 major zeroes in all.)

zero #17 zero #18

IPhysicsPIllinois

dpT/dt considerationsWe’d like the slopes of the pT curves when not-to-be-kicked bunches pass through the kicker to be as small as possible so that the head, center, and tail of a (20 ps rms) bunch will experience about the same field integral.

33.2 33.4 33.6 33.8

0.1

0.05

0.05

0.1Kick vs. time, 10cavity system, around first major zero

66.2 66.4 66.6 66.8

0.1

0.05

0.05

0.1Kick vs. time, 10cavity system, around second major zero

0 50 100 150 200 250 300 350

10

5

0

5

10


1 nsec

1% of kick

pT in the vicinity of two zeroes

IPhysicsPIllinois

Sometimes dpT/dt at zeroes can be calculated…

1

0

coscavitiesj N

Thigh low

j

dp dj t

dt dt

12cos 2

2sin

cavities lowcavitiesT

cavities

KN m

Ndp

dt mN

At the mth “major zero” the expression evaluates to

Note that magnitude of the slope does not depend strongly on high. (It does for the other zeroes, however.)

high lowK


Flattening out d pT/d t

Endpoint is moving this way at the zero in pT

(80 psec intervals)

pT = 0



In terms of phasor sums: we want the endpoint of the phasor sum to have as small an x component of “velocity” as possible.

Endpoint velocity components (m ranges from 1 to Ncavities – 1):

12

sin 2

2sin

cavities lowy

cavities

cavities

KNv m

NmN

12

cos 2

2sin

cavities lowx

cavities

cavities

KNv m

NmN



How large a value for vx is acceptable?

• Size of kick: Ncavities

• rms bunch length: 20 psec (6 mm)

• maximum allowable kick error: ~.07%

2.020 nsec 0.07 10x cavitiesv N

Work in units of nsec and GHz… for 30 cavities: vx < 1.05 nsec-1.

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Phasor plots for d pT/d t

Phasor magnitude for mth zero’s dpT/dt:

2sin

cavities low

cavities

Nv

mN

1, 2, , 1cavitiesm N

122m

cavities

Km

N

Phasor angle:

5 10 15 20 25

0.5

1

1.5

2

2.5

Phasor sum endpoint velocity magnitude vs. which major zero

~maximum allowable value

cosx mv v

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Phasor plots for d pT/d t

Phasor plot for dpT/dt, including the phasor angles…

2 1 0 1 2

2

1

0

1

2

kicker phasor endpoint velocity at the various major zeroes

1

2

3

4

29

28

122m

cavities

Km

N

high lowK

Ncavities = 30

K=100 (300MHz, 3 MHz)

There are lots of parameters to play with.

Between the blue lines is

good.

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Changing parameters

2 1 0 1 2

2

1

0

1

2


Ncavities = 30, K=1002 1 0 1 2

2

1

0

1

2


Ncavities = 30, K=1012 1 0 1 2

2

1

0

1

2


Ncavities = 30, K=99

2 1 0 1 2

2

1

0

1

2


Ncavities = 29, K=1003 2 1 0 1 2 3

3

2

1

0

1

2

3


Ncavities = 31, K=100


More dramatic d pT/d t reduction…

…is possible with different amplitudes Aj in each of the cavities.

We (in particular Guy Bresler) are investigating this right now. It looks very promising!

Guy has constructed an algorithm to find sets of amplitudes which have dpT/dt = 0 at evenly-spaced “major zeroes” in pT.

There are lots of different possible sets of amplitudes which will work.


More dramatic d pT/d t reduction…Here’s one set for a 29-cavity system (which makes 28 zeroes in pT and dpT/dt in between kicks), with 300 MHz, 303 MHz,…:

0 5 10 15 20 25

0.01

0.02

0.03

0.04Cavity amplitudes


Kick corresponding to those amplitudes

The “major zeroes” aren’t quite at the obvious symmetry points.

0 50 100 150 200 250 300 350

0.5

0

0.5

1



Kick corresponding to those phasors

100 120 140 160 180 200

-0.05

0

0.05

0.1

Kick vs. time , Guy 's amplitudes , 300 MHz lowest frequency , f 3MHz

Here’s where some of them are.


Phasors with amplitudes chosen to give dpT/dt = 0 and pT = 0 (29 cavities)

zero #1 zero #2

0.1 0 0.1 0.2

0.1

0

0.1

0.2Phasor plot

tnsec 11.4943


Zero crossing 1

x,y 2.44179, 8.79454vx,vy 18.0926, 3.968320.1 0 0.1 0.2

0.1

0

0.1

0.2Phasor plot

tnsec 22.9885


Zero crossing 2

x,y 1.27271, 0.765766vx,vy 2.0668, 2.26901

The phasor sums show less geometrical symmetry.


Phasors with amplitudes chosen to give dpT/dt = 0 and pT = 0 (29 cavities)

zero #3 zero #4

0.1 0 0.1 0.2

0.1

0

0.1

0.2Phasor plot

tnsec 34.4828


Zero crossing 3

x,y 2.00884, 1.90287vx,vy 4.07264, 3.865210.1 0 0.1 0.2

0.1

0

0.1

0.2Phasor plot

tnsec 45.977


Zero crossing 4

x,y 0.675104, 1.27338vx,vy 2.70045, 1.14132

etc.

IPhysicsPIllinois

How well do we do with these amplitudes?

Old, equal-amplitudes scheme:

5 10 15 20 25

0.5

1

1.5

2

2.5

Phasor sum endpoint velocity magnitude vs. which major zero

~maximum allowable value

New, intelligently-selected-amplitudes scheme:

0 5 10 15 20 25

0.0002

0.0004

0.0006

0.0008

0.001Head of bunch kick when center is zero

~maximum allowable valueWow!

bunch number

bunch number


Multiple passes through the kicker

Previous plots were for a single pass through the kicker.

Most bunches make multiple passes through the kicker.

Modeling of effects associated with multiple passes must take into account damping ring’s

• synchrotron tune (0.10 in TESLA TDR)

• horizontal tune (72.28 in TESLA TDR)

We (in particular, Keri Dixon) are working on this now.

With equal-amplitude cavities some sort of compensating gizmo on the injection/extraction line (or immediately after the kicker) is probably necessary. However…


Multiple passes through the kicker…selecting amplitudes to zero out pT slopes fixes the problem! Here’s a worst-case plot for 300 MHz,… (assumes tune effects always work against us).

0 5 10 15 20 25

0.0002

0.0004

0.0006

0.0008

0.001Worstcase cumulative head of bunch kick when center is zero

maximum allowed value

bunch number


Some of our other concerns

1. Effect of finite separation of the kicker cavities along the beam direction (George)

2. Arrival time error at the kicker for a bunch that is being injected or extracted (Keri)

3. Inhomogeneities in field integrals for real cavities (Keri)

4. What is the optimal choice of cavity frequencies and amplitudes? (Guy)


Finite separation of the kicker cavities

...

Even though net pT is zero there can be a small displacement away from the centerline by the end of an N-element kicker.

For N = 16; 50 cm cavity spacing; 6.5 Gauss-meter per cavity:

Non-kicked bunches only (1, 2, 4, … 32)


Finite separation of the kicker cavities

... ...

Compensating for this: insert a second set of cavities in phase with the first set, but with the order of oscillation frequencies reversed: 3 MHz, 6 MHz, 9MHz,… followed by …, 9 MHz, 6 MHz, 3 MHz.

Non-kicked bunches only (N = 1, 2, 4, … 32)


Arrival time error at the kicker for a bunch that is being injected or extracted

What happens if a bunch about to be kicked passes through the kicker cavities slightly out of time?

It depends on the details of the kicker system:

• 16-cavity system with 3MHz, 6MHz,… cavities:

pT/pT~ 6 10-6

• 30-cavity system with 3.000GHz, 3.003GHz,… cavities:

pT/pT~ 9 10-4 so an extraction line corrector is probably necessary


What we’re working on now

• Lately we’ve been working with models using high-frequency cavities, split in frequency by multiples of the linac bunch frequency.

• We want to better understand how to select the best set of cavity frequencies an geometries.

• We are in the process of incorporating tune effects into our models.

• We will investigate the kinds of corrections necessary to compensate for tune and cavity-related effects.

• We will look into the relative merits of horizontal and longitudinal kicks.


Comments on doing this at a university• Participation by talented undergraduate students makes LCRD

2.22 work as well as it does. The project is well-suited to undergraduate involvement.

• We get most of our work done during the summer: we’re all free of academic constraints (teaching/taking courses). The schedule for evaluating our progress must take this into account.

• Support for students comes from (NSF-sponsored) REU program. We have borrowed PC’s from the UIUC Physics Department instructional resources pool for them this summer.

• LCRD 2.22 requested $2,362 in support from DOE (mostly for travel). In spite of a favorable review by the Holtkamp committee, DOE has rejected the proposal. (We don’t know why.) We’re continuing with the work, in spite of this.


Conclusions

• We haven’t found any obvious show-stoppers yet.

• It seems likely that intelligent selection of cavity amplitudes will provide us with a useful way to null out some of the problems present in a more naïve scheme.

• We haven’t studied issues relating to precision and stability yet. later this summer…

• This is a lot of fun.

I PhysicsP I llinois George Gollin, Cornell LC 7/031 A Fourier Series Kicker for the TESLA Damping...

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Transcript of I PhysicsP I llinois George Gollin, Cornell LC 7/031 A Fourier Series Kicker for the TESLA Damping...