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Scaling Law of Coherent Synchrotron Radiation in a Rectangular C hamber. Yunhai Cai FEL & Beam Physics Department SLAC National Accelerator Laboratory TWIICE 2014, Synchrotron SOLEIL, 16-17 January 2014. CSR Impedance of Parallel-Plates. An impedance with scaling property is given by. - PowerPoint PPT Presentation
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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.
1/17/2014 3Yunhai Cai, SLAC
(/2)3/2
Threshold of Instabilityfor CSR of Parallel Metal Plates
1/17/2014Yunhai Cai, SLAC
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
1/17/2014 Yunhai Cai, SLAC 5
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
1/17/2014 Yunhai Cai, SLAC 6
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
1/17/2014 Yunhai Cai, SLAC 10
p=1,3,5,746 modes
Point-Charge Wakefield (A=1)
1/17/2014 Yunhai Cai, SLAC 11
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
1/17/2014 Yunhai Cai, SLAC 12
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)
1/17/2014 13Yunhai Cai, SLAC
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
1/17/2014 Yunhai Cai, SLAC 15
VFP Simulation
1/17/2014 Yunhai Cai, SLAC 16
A square chamber has a much lower threshold: xth=1/4.
cx 34.05.0 th
Mitigation of CSR Effects
1/17/2014 Yunhai Cai, SLAC 17
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
1/17/2014 Yunhai Cai, SLAC 18
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)
1/17/2014 Yunhai Cai, SLAC 19
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