Longitudinal Stability of Short Bunches at BESSY
Peter Kuske,
M. Abo-Bakr, W. Anders, J. Feikes, K. Holldack, U. Schade, G. Wüstefeld
(BESSY)
H.-W. Hübers (DLR)
ICFA Mini-Workshop on
„Frontiers of Short Bunches in Storage Rings“
INFN-LNF, Frascati, 7- 8 November 2005
Content
1. Introduction
2. Experimental Techniques
2.1 Streak Camera
2.2 Observation of CSR
3. Theoretical Approaches
3.1 Impedance Model
3.2 Haissinski Equation
3.3 Vlasov-Fokker-Planck Equation
3.4 Instability Thresholds
4. Comparison of Experiment and Theory
4.1 Bunch Length
4.2 Turbulent Instability
5. Other “Instabilities”
5.1 Timing Jitter
5.2 Multi Bunch Instability at Small Negative Alpha
6. Summary
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
1. Introduction
BESSY:
3rd generation light source, in operation since 1998
Importance of short bunches:
Investigation of fast phenomena
Coherent synchrotron radiation (CSR)
Accademic interest
History of short bunches at BESSY:
1984 Isochronous SR based FEL project at BESSY I
(D. Deacon, A. Gaupp)
since 1999 more seriously persued at BESSY II
(G. Wüstefeld, ...)
Table: BESSY parameter
Energy

1.72 GeV
Natural energy spread
/
810-4
Longitudinal damping time
lon
7.7 ms
Momentum compaction factor 
510-5 .. + 10-4
Bunch length
o
0.5 … 15 ps
Accellerating voltage
Vrf
1.4 MV
RF-frequency
rf
5002 MHz
Gradient of RF-Voltage Vrf/t
4.63 kV/ps
Circumference
C
240 m
Revolution time
To
800 ns
Number of electrons
5106 per µA
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
2. Experimental Techniques
2.1 Streak Camera:
Dual sweep SC model C5680 with fast sweep=250 MHz, ±1mrad bending magnet radiation, for “direct” bunch
length measurements - PAC ’03: “Bunch Length Measurements at BESSY”, M. Abo-Bakr, et al.
ph.noise  2.4 ps
stat.res.  1.5 ps
 act 
 meas  ph.noise  stat.res.
2
2
2
Results of bunch length measurements as a
function of the synchrotron frequency Fsyn
Bunch length as a function of beam current for three
different momentum compaction factors
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
2. Experimental Techniques
2.2 Observation of CSR
Suppression due to shielding and finite acceptance angles
Detector: InSb-FIR detector HDL-5 (QMC Instrum. Ltd.)
most sensitive around 20 cm-1, very fast – revolution frequency resolved
Time-dependence of CSR-signal indicates instability, CSR-spectra (Martin-Puplett-spectrometer)
 bunch shape (K. Holldack, et al., Phys. Rev ST-AB 8, 040704 (2005))
Current dependence of the 1.25 MHz- CSRcomponent as a function of single bunch current
Appearance of CSR-bursts measured in time domain (left)
and the corresponding Fourier transformation (right)
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
3. Theoretical Approaches
3.1 Impedance Model
Observations:
Inductive impedance:
(I) prop. I1/3
Streak Camera and CSR
Assumption for chamber:
Z﴾﴿ ≈ R - iL
with R = 850 Ω
Lo = 0.2 ... 0.35 Ω
o ≤ 2 ps ... 13 ps
Consequences:
Current below threshold
Current above threshold
Long. particle distribution
Stationary (nonGaussian)
Non-Gaussian, time dependent
Momentum distribution
Stationary, Gaussian
with natural spread
Time dependent, non-Gaussian with increasing
spread
Theory
Potential well distortion
Turbulent bunch lengthening (energy widening),
longitudinal mode mixing-, µ-wave-, CSR-instability
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
3. Theoretical Approaches
3.1 Impedance Model
CSR-wake (J. B. Murphy, et al. Part. Acc. 1997, Vol. 57, pp 9-64)
Radiation wake field
Short bunch – 1 ps rms-bunch length
Assumption: R = 850 Ω, 0L ≈ 0.2 Ω valid for  ≈ 1 ps, unshielded CSR-wake can be added
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
3. Theoretical Approaches
3.2 Haissinski Equation:
 t2
1
I (t )  K exp 

2
2
 2 0 Vrf  0

Vind ( )d 

t

 I (t )dt  1


induced voltage per turn:
Vind (t )  I 0T0 R  I (t )  L  I(t )   WCSR (t   )  I ( )d 

Analytical solutions only in some cases  numerical approaches required
Solutions exist in most cases, if none can be found use relaxation technique (N. Towne, Phys. Rev.
ST-AB Vol 4, 114401 (2001)) - usually numerical difficulties have nothing to do with instability
>0
<0
Potential well distortion due
to CSR-wake in comparison
with results of K. Bane, et al.
AIP Conf. Proc. 367, p. 191
Exception is the purely inductive impedance with negative momentum compaction:
Potential well distortion due to
CSR- and vacuum chamber
impedance (<0)
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
3. Theoretical Approaches
3.2 Solution of the Haissinski Equation and CSR-Spectra
stationary distribution function  stable coherent SR spectrum
S CSR ( )   e
i 2t
 I (t )dt
2
Distortorted shapes of short bunches just below instability thresholds and their “free space” CSR-spectra.
The strong enhancement of CSR at frequencies>3 THz with <0 could not be observed ( J. Lee, G. Wüstefeld)
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
3. Theoretical Approaches
3.3 Vlasov-Fokker-Planck (VFP) Equation
q  z / z
p  E /  E
f
f
f
2  
f 
 pf  
 p  q  Fc (q, , f ) 

q
p  s t d p 
p 
  st
RF focusing
Collective Force
Damping
(M. Venturini)
Quantum Excitation
Numerical solution based on R.L. Warnock, J.A. Ellison, SLAC-PUB-8404, March 2000
S. Novokhatski, EPAC 2000 and SLAC-PUB-11251, May 2005
My code:
limited to 127x127 mesh points
and CPU-time
500-2000 time steps per ωs
simulation of 200 Tsyn
Bunch shape at end of simulation: 1ps bunch
with R, L-impedance and CSR-wake.
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
3. Theoretical Approaches
3.3 VFP- Results for 1ps Bunch and  > 0 - only CSR-wake
Results of the VFP-calculations in comparison with
solution of Haissinski equation. Threshold for energy
widening is at 7 A. Instability starts with bunch shape
oscillations at 1.8·Fsyn. Shown are the moments of the
momentum distribution as a function of time at the end
of the numerical calculation.
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
3. Theoretical Approaches
3.3 Bursting VFP - Results for 1ps Bunch and  < 0 – only CSR-wake
Some moments of the particle
distribution as a function of time
Results of the VFP-calculations in comparison with
solution of Haissinski equation. Threshold for energy
widening is at 25 A. -wave-type instability with
density modulation accompanied by a small increase
of the momentum spread. Well above threshold
random bursts at a rate small compared to Fsyn.
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
3. Theoretical Approaches
3.3 Bursting VFP - Results for 1ps Bunch – Vacuum chamber and CSR-wake
Results of the VFP-calculations in comparison
with solution of Haissinski equation.
Burst rate in units of the synchrotron
frequency as a function of intensity
Unstable longitudinal particle distribution and their projections
just above threshold - periodic bursts of coherent radiation.
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
3. Theoretical Approaches
3.4 Instability Thresholds
Stupakov & Heifets applied coasting beam instability analysis with CSR-impedance to bunched beams
(Phys. Rev. ST-AB 5, 054402)
Results are in agreement with observations if the wavelength of the perturbation = 0 chosen
Good agreement despite:
●Vacuum chamber ignored - except for
perfectly conducting infinite par. plates
●Discrepancy for <0 between this theory and
the VFP-calculations and
●observed thresholds < thresholds with >0
Comparison of observed and simulated bursting thresholds.
At BESSY a 1 ps-long bunch would have Fsyn~500 Hz.
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
4. Comparison of Experimental and Theoretical Results
4.1 Bunch Length:
Chosen vacuum chamber impedance leads to:
●very good agreement with the observations in the region of potential well distortion
●VFP-simulation give too high thresholds and too small energy widening
Comparison of measured and calculated bunch lengths
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
4. Comparison of Experimental and Theoretical Results
4.2 Turbulent Instability
time dependent CSR-bursts
observed in frequency domain:
0=14 ps, nom. optics, with 7T-WLS

CSR-bursting threshold
Stable, time independent CSR
Spectrum of the CSR-signal:
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
4. Comparison of Experimental and Theoretical Results
4.2 Turbulent Instability
Fourier spectra of the time dependent CSR-signals as a function of single bunch current
●With short bunches strong signal at 3·Fsyn
●appearance of additional sidebands
●with <0 1st and 2nd synchrotron sidebands at small Isb
CSR-signal as a function of time at Isb=160 A
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
5. Other „Instabilities“
5.1 Timing Jitter- Technical Imperfections?
Streak Camera measurements at Fsyn=1.7 kHz and Isb=0.85 mA
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
5. Other „Instabilities“
5.2 Multi Bunch Instability at Small Negative Alpha
experimental conditions: Fsyn ~ 300 Hz, 2 mA in 200 buckets
Horizontal beam position in the straight section (BPM 1) and in the center of the bending region (BPM 2). There are large energy
oscillations at Fsyn and much larger sporadic and slower energy variations. In this case the energy increases at a rate of ~12 Hz.
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005
6. Summary
●Rather short, intensity limited bunches can be produced in storage rings
●Potential well distortion can lead to enhanced emission of stable CSR
●Threshold for energy widening usually accompanied by non-stationary bunch shapes
and time dependent CSR emission
●Observation of CSR leads to information on small scale variations and fluctuations of
the particle density
●Diagnostic power of CSR
●Problems with threshold predictions – inclusion of vacuum chamber effects
●In the region of turbulence our understanding is limited and further studies are required
●Numerical solution of the VFP equation in combination with more realistic wakes or
impedances is certainly a way to go
●There remain technical and intellectual challenges for the production of short bunches
in storage rings – timing jitter and “orbit” stability
Longitudinal Stability of Short Bunches at BESSY, Peter Kuske, 7 November 2005