Scaling Law of Coherent Synchrotron Radiation in a Rectangular Chamber

Yunhai CaiFEL & Beam Physics Department

SLAC National Accelerator Laboratory

TWIICE 2014, Synchrotron SOLEIL, 16-17 January 2014

CSR Impedance of Parallel-Plates

3/43/2

22ˆ

2 kpu

2/3

2/3ˆ

hnk

where h is the distance between two plates, n=k, Ai and Bi are Airy functions, and their argument u is defined as

An impedance with scaling property is given by

Dependence of n is all through

In fact, this scaling property holds for the CSR impedance in free space, formally

3/22/33/1 ])()[

223(

3)3/2()(4()(

hnicn

nZh free

1/17/2014 2Yunhai Cai, SLAC

/

1)(

3/42 ))]()()(())(')(')(('[]2

ˆ)[4)(2()( kh

oddppara

uiBiuAiuuAiuiBiuAiuAikcn

nZh

Scaling and Asymptotic Properties

The scale defines the strength of impedance and the location of the peakdefines where the shielding effects start.

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(/2)3/2

Threshold of Instabilityfor CSR of Parallel Metal Plates

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4

Threshold xth becomes a function of the shielding parameter c =sz1/2/h3/2. Simulation was carried out by Bane, Cai, and Stupakov, PRSTAB 13,104402 (2010). For a long bunch, the coasting beam theory agrees wellwith the VFP simulation.

A dip seen near sz1/2/h3/2 = 0.25.

3/22/3

23 c

x coasting beam theory:

Summary of the ComparisonsMachine sz

[mm]Radius [m]

Height h [cm]

c xth

(theory)xth

(meas.)BESSY II 2.6 4.23 5.0 0.48 0.67 0.89MLS 2.6 1.53 5.0 0.29 0.60 0.39ANKA 1.0 5.56 3.2 0.42 0.64 0.50SSRL 1.0 8.14 3.4 0.46 0.66 ?Diamond 0.7 7.13 3.8 0.25 0.17 ? 0.33

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ccx 34.05.0)( thWe have used

where c = sz1/2/h3/2 is the shielding parameter. This simple relationwas first obtained by fitting to the result of simulations (Bane, Cai, and Stupakov, PRSTAB 13, 104402 (2010)).

Since the MLS’s shielding parameter is very close to the dip. That may be a reason of its lower threshold.

Statements of Problem and Solution• Solve the Maxwell equation

with a circulating charge inside a perfect conducting chamber

• Express the longitudinal impedance in terms of Bessel functions

• Approximate Bessel functions with Airy functions under the uniform asymptotic expansion

A section of vacuum chamber

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For detail read: SLAC-PUB-15875, January 2014

CSR Impedance of Rectangular Chamber

/

1)(

3/42 ]),(ˆ),(ˆ),(ˆ

),(ˆ),(ˆ),(ˆ]

2

ˆ)[4)(2()( kh

oddprect uupuupuupu

uusuusuusk

ci

nnZ

h

3/43/2

22ˆ

2 kpu

where w is the width of chamber and h the height, n=k, p and s hats are products of Airy functions and their derivatives. Similar to the parallelplates, one of arguments u is

An impedance with scaling property is given by

Dependence of n is all through

1/17/2014 7Yunhai Cai, SLAC

In addition, for aspect ratio of A = w/h, the other two arguments are

)ˆˆ(21 3/23/4223/2 kAkpu

2/3

2/3ˆ

hnk

The Cross Products

)(')(')(')('),(ˆ),()()()(),(ˆyAixBiyBixAiyxs

yAixBiyBixAiyxp

)(')(')(')('),(),()()()(),(yJxYyYxJyxs

yJxYyYxJyxp

nnnnn

nnnnn

of the Bessel functions of the Airy functions

The uniform asymptotic expansion:

1/17/2014 Yunhai Cai, SLAC 8

))2((')2()1('

),)2((')2()1('

),)2(()2()1(

),)2(()2()1(

23/23/22

23/23/22

23/23/12

23/23/12

xn

Bin

xnY

xn

Ain

xnJ

xn

Bin

xnY

xn

Ain

xnJ

n

n

n

n

Scaling Property of Point-Charge Wakefield

)]ˆˆexp(2)ˆsgn()ˆˆcos()ˆ([)ˆ(ˆ]ˆ[ 2 zkzzkzk

QRkzW jj

jj

sjh

The integrated wake can be written as

where is bending radius and h height of chamber. We have defined adimensionless longitudinal position:

1/17/2014 Yunhai Cai, SLAC 9

2/32/1 /ˆ hzz which naturally leads to the shielding parameter

2/32/1 / hzsc if we introduce the normalized coordinate q=z/sz.

• An LRC resonance with infinite quality factor• Decay term is due to the curvature

Dimensionless Rs/Q (A=1)

kduusd

uusuusk

jj

jjjjj ˆ/),(ˆ

),(ˆ),(ˆ]ˆ[232 3/43/13

kduupd

uupuupuk

jj

jjjjjj ˆ/),(ˆ

),(ˆ),(ˆ]ˆ[232 3/43/13

It can be computed by

or

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p=1,3,5,746 modes

Point-Charge Wakefield (A=1)

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It is extremely hard to simulate a long bunch.

zoomed in

Comparisons of Bunch Wake

c=0.1

Agoh, PRSTAB, 12, 094402 (2009) Stupakov & Kotelnikov, PRSTAB, 12 104401 (2009)

c=0.5

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excellent agreements

Fig 4.cFig 11

Longitudinal Beam Dynamics

)'"()'('")(21 22 qqWqdqdqIpqH

q

n

)(2},{p

pp

H

Hamiltonian is given as

q=z/sz, p=-d/sd and W(q) is the integrated wake per turn (conventionused in Alex Chao’s book). The independent variable is =wst.Vlasov-Fokker-Planck equation is written as

where (q,p;) is the beam density in the phase space and =1/wstd. A robust numerical solver was developed by Warnock and Ellison (2000) .

ds s

ben

NrI2

where In is the normalized current introduced by Oide and Yokoya (1990)

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Scaling Law of the Threshold

In the normalized coordinate, it can be shown easily

)]ˆexp(2)sgn()ˆcos()([)ˆ(ˆ][ 3/4 qkqqkqk

QRkqWI jj

jj

sjn ccxc

where is the dimensionless current and the shielding parameter. We redefine a dimensionless wake as

Clearly, this wake depends on c and aspect ratio A through the values ofBased on the VFP equation. we conclude that the dimensionless current

is a function of other three dimensionless parameters. 1/17/2014 Yunhai Cai, SLAC 14

)]ˆexp(2)sgn()ˆcos()([)ˆ(ˆ][ 3/4 qkqqkqk

QRkqW jj

jj

sj cccx

jk̂

),,( cxx Ath

3/43/1 / znI sx 2/32/1 / hzsc

Rs/Q for Rectangular Chamber

Envelopes are very similar, also similar to the real part of Z(n)/nof the parallel model shown previously.

A=w/h=2p=1,3,5112 modes

A=w/h=3p=1,3,5204 modes

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VFP Simulation

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A square chamber has a much lower threshold: xth=1/4.

cx 34.05.0 th

Mitigation of CSR Effects

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3/10

2

3/72 cos)(8

sxZc

ffVI srevrfrfz

th

b

Rewrite the threshold in terms of some practical parameters. The beam becomes unstable if

• Extremely unfavorable scaling in terms of shortening bunch

• Stronger longitudinal focusing is very helpful

• Superconducting RF at higher frequency

Shorten the bunch to 1 ps in PEP-X

Conclusion• The scaling law found in the parallel-plate mode is

extended to the rectangular chamber by adding another parameter: A (aspect ratio of the chamber)

• The threshold of a square chamber is lower by a factor than the one of a rectangular chamber with A > 2

• More effective mean to shorten a bunch to a ps scale is to use superconducting RF at higher frequency

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Important and Relevant References• R. Warnock and P. Morton, “Fields Excited by a Beam in a Smooth

Toroidal Chamber,” SLAC-PUB-5462 March (1988); Also Pat. Accel. • J. Murphy, S. Krinsky, and R. Gluckstern, “Longitudinal Wakefield for an

Electron Moving on a Circular Orbit,” Pat. Accel. 57, pp. 9-64 (1997)• T. Agoh and K. Yokoya, “Calculation of coherent synchrotron radiation

using mesh,” Phys. Rev. ST Accel. Beams 7, 0544032 (2004)• T. Agoh, “Steady fields of coherent synchrotron radiation in a rectangular

pipe,” Phys. Rev. ST Accel. Beams 12, 094402 (2009)• G. Stupakov and I. Kotelnikov, “Calculation of coherent synchrotron

radiation impedance using mode expansion method,” Phys. Rev. ST Accel. Beams 12, 104401 (2009)

• K. Bane, Y. Cai, and G. Stupakov, “Threshold studies of microwave instability in electron storage rings,” Phys. Rev. ST Accel. Beams 13, 104402 (2010)

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Acknowledgements• My colleagues: Karl Bane, Alex Chao, Gennady

Stupakov, and Bob Warnock for many helpful and stimulating discussions, their insights, collaborations, and encouragements

• Marit Klein (ANKA), A.-S. Muller (ANKA), G. Wustefeld (BESSY, MLS), J. Corbett (SSRL), F. Sannibale (LBNL), I. Martin (Diamond) for many

helpful email exchanges and providing their experimental data and plots

1/17/2014 20Yunhai Cai, SLAC

Transverse Impedance

/

2)(

2

]),(ˆ),(ˆ),(ˆ

),(ˆ),(ˆ),(ˆ[)4)(8()(

kh

evenp uupuupuupu

uusuusuusu

chikZ

3/43/2

22ˆ

2 kpu

where w is the width of chamber and h the height, n=k, p and s hats are products of Airy functions and their derivatives. Similar to the parallelplates, one of arguments u is

An impedance with scaling property is given by

Dependence of n is all through

1/17/2014 21Yunhai Cai, SLAC

In addition, for aspect ratio of A = w/h, the other two arguments are

)ˆˆ(21 3/23/4223/2 kAkpu

2/3

2/3ˆ

hnk

Transverse Wakefield

)]ˆˆexp(21)ˆˆsin()ˆ([)ˆ(]ˆ[ 2/5

2/1

1 zkzkzkQcRzW jj

jj

sh

The integrated wake can be written as

where is bending radius and h height of chamber. We have defined adimensionless longitudinal position:

1/17/2014 Yunhai Cai, SLAC 22

2/32/1 /ˆ hzz which naturally leads to the shielding parameter

2/32/1 / hzsc if we introduce the normalized coordinate q=z/sz.

• An LRC resonance with infinite quality factor• Decay term is due to the curvature

Dimensionless cRs/Q (A=1)

kduusd

uusuusu

jj

jjjjj ˆ/),(ˆ

),(ˆ),(ˆ64 3

kduupd

uupuupu

jj

jjjjj ˆ/),(ˆ

),(ˆ),(ˆ64 23

It can be computed by

or

1/17/2014 Yunhai Cai, SLAC 23

p=1,3,5,746 modes

Transverse Instability

,),(ˆˆ

)(21)ˆˆ)((

21

0

2222

s

yy

s kkEqFy

pqpykk

H

In the normalized coordinates, Hamiltonian is given by

)2

)'()('(ˆ)'('ˆ

1

0

qqWqDqdqNr

EF ey

where the force is written as

If we define a dimensionless current )1)(1)(( 2/12/5

s

e

kkhNrI

),,(s

th

kk

AII cwe derive a scaling law

1/17/2014 Yunhai Cai, SLAC 24