1Laurent S. Nadolski FFAG Workshop, Grenoble, 2007
Application of Frequency Map Analysis for Studying
Beam Transverse Dynamics
Laurent S. NadolskiAccelerator Physics Group
ALS frequency maps
Beam dataSimulation data
2Laurent S. Nadolski FFAG Workshop, Grenoble, 2007
Contents
• Introduction to FMA and motivations
• Application for the SOLEIL lattice– On momentum dynamics– Off momentum dynamics
• Experimental frequency maps (ALS)
• Discussion– How to use this method for FFAG?
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Frequency Map Analysis
Motivations– Global view of the
beam dynamics– Beam Lifetime– Injection Efficiency– Short and Long
term stability– Particle losses– Effect of insertion
devices– …
Selection of a good working point
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Frequency Map AnalysisLaskar A&A1988, Icarus1990
Quasi-periodic approximation through NAFF algorithm
of a complex phase space function
for each degree of freedom
with
defined over
and
Numerical Analysis of Fundamental Frequency
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I. Very accurate representation of the “signal” (if quasi-periodic) and thus of the amplitudes
II. b) Determination of frequency vector
with high precision for Hanning Filter Laskar NATO-ASI 1996
Long term prediction Accuracy gain (simulation, beam based experiments) Diffusion coefficient related to particle diffusion
Advantages of NAFF
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Rigid pendulum
Sampling effect
Hyperbolic Elliptic
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Accelerator 4D Dynamics
Accelerator
PoincaréSurface ofsection
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z
z’
x
x’
x
z
x0
z0
x0’= 0z0’= 0
Frequency map
Configuration space Phase space
Phase space
Tracking T
NAFF
FT : (x0,z0) (x,z)
resonance
Frequency map:
NAFFTracking T
Computing a frequency map
x
z
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Tools• Tracking codes (symplectic integrators)
– Simulation: Tracy II, Despot, MAD, AT, …– Nature: beam signal collected on BPM electrodes
• NAFF package (C, fortran, matlab)
• Turn number Selections– Choice dictated by
• Allows a good convergence near resonances• Beam damping times (electrons, protons)• 4D/6D
– AMD Opteron 2 GHz (Soleil lattice)• 0.7 s for tracking a particle over 2 x 1026 turns
– 1h00 for 100x50 (enough for getting main characteristics)– s 6h45 for 400x100
• Step size following a square root law (cf. Action)
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z
x
Regular areas
Resonances
Nonlinear or chaotic regionsFold
Reading a FMA
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4th order5th order7th order9th order
Resonance network: a x + b z = c order = |a| + |b|
Higher orderresonance
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Diffusion D = (1/N)*log10(||Dn||)
Color code:
||Dn||< 10-10
||Dn||> 10-2
Diffusion reveals as well slightly excited resonances
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Bare lattice(no errors)
WP sitting on
Resonance node
x + 6z = 80
5x = 91
x - 4z = -23
2x + 2z = 57
9x=164 x-4z=-234x=73
x+6z=80x+5z=88
x+4z=96
5x=91
x+2z=57
x+z=65
On-momentum Dynamics --Working point: (18.2,10.3)
x
z
4x=73x-4z=-23 9x=164
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Randomly rotating 160
Quads
•Map fold Destroyed
•Coupling strongly impacts
3x + z = 65
•Resonance node excited
PhysicalAperture
On-momentum dynamics w/ 1.9% coupling (18.2,10.3)
x+z=65
Resonance islandx+z=65
x-4z=-234x=73
x+6z=80x+5z=88
x+4z=96
5x=91
x+2z=57
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Off-momentum dynamics
Several approaches:
– Off-momentum frequency maps
– Energy/betatron-amplitude frequency maps
– Touschek lifetime• 4D tracking• 6D tracking
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Chromatic orbit
Chromatic orbit
Closed orbit0
1
x
x Ax
2'
00
'
000
2
02
0
xxx xA
ALS Example
WP
WP
Particle behavior after
Touschek scattering
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Off momentum dynamics
4x=73excited
4x=73
3x+z=65
3x- 2z=34
3z=31
3z=313z=313x+z=65
3x- 2z=34
>0 <0
z0 = 0.3mm
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Measured versus Calculated Frequency Map
Modeled Measured
See resonance excitation of unallowed 5th order resonancesNo strong beam loss isolated resonances are benign
D. Robin et al., PRL (85) 3
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Frequency Maps for Different Working Points
Region of strong beam lossDangerous intersection of excited resonances
D. Robin et al., PRL (85) 3
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FMA and FFAG
Light sources: 4D tracking useful since
– 4D dynamics + slow longitudinal dynamics• Still valid for proton FFAG? Resonant phenomena?• x-y fmap at a given energy (slices during acceleration ramping up)• x- fmap
– 6D tracking + FMA to investigate• Not very much used for 3GLS because not so important • Here not synchrotron oscillation but constant acceleration
– Tracking over 512 turns to get a good determination of the tunes
• Good tracking code with almost symplectic integrators• Resonances need time to build up• Definition of Dynamics aperture versus number of turns
– Investigation of dynamics for large amplitude• Injection efficiency• FFAG are very non linear by construction• Multipole errors, coupling errors
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Conclusions
FMA techniques
– Gives us a global view (footprint of the dynamics)– Reveals the dynamics sensitiveness to quads,
sextupoles and IDs – Reveals nicely effect of coupled resonances, specially
cross term z(x)– Enables us to modify the working point to avoid
resonances or regions in frequency space– Is suitable both for simulation and online data– 4D tracking: on- and off- momentum dynamics
Applications to FFAG ?
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References• Tracking Codes
– BETA (Loulergue – SOLEIL)– Tracy II (Nadolski – SOLEIL, Boege – SLS, Bengtsson – BNL)– AT (Terebilo http://www-ssrl.slac.stanford.edu/at/welcome.html)
• Papers– H. Dumas and J. Laskar, Phys. Rev. Lett. 70, 2975-2979 – J. Laskar and D. Robin, “Application of Frequency Map Analysis to the ALS”,
Particle Accelerators, 1996, Vol 54 pp. 183-192 – D. Robin and J. Laskar, “Understanding the Nonlinear Beam Dynamics of
the Advanced Light Source”, Proceedings of the 1997 Computational Particle Accelerator Conference
– J. Laskar, Frequency map analysis and quasiperiodic decompositions, Proceedings of Porquerolles School, sept. 01
– D. Robin et al., Global Dynamics of the Advanced Light Source Revealed through Experimental Frequency Map Analysis, PRL (85) 3
– Measuring and optimizing the momentum aperture in a particle accelerator, C. Steier et al., Phys. Rev. E (65) 056506
– L. Nadolski and J. Laskar, Review of single particle dynamics of third generation light sources through frequency map analysis, Phys. Rev. AB (6) 114801
– J. Laskar, Frequency map Analysis and Particle Accelerator, PAC03, Portland
– FMA Workshop’04 proceedings, Synchrotron SOLEIL, 2004 http://www.synchrotron-soleil.fr/images/File/soleil/ToutesActualites/Archives-Workshops/2004/frequency-map/index_fma.html
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Annexes
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Particle Computation Frame
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Decoherence of a particle bunch
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1 = 4.38 10-04
2 = 4.49 10-03
Non-linear synchrotron motion
Tracking 6D required
ds
1
2
2 2
ds1
+3.8% -6%