Application of Frequency Map
Analysis for Studying
Beam Transverse Dynamics
Laurent S. Nadolski
Accelerator Physics Group
Simulation data
Beam data
ALS frequency maps
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
1
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?
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
2
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
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
3
Frequency Map Analysis
Laskar A&A1988, Icarus1990
Numerical Analysis of Fundamental Frequency
Quasi-periodic approximation through NAFF
algorithm
of a complex phase space function
defined over
for each degree of freedom
with
and
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
4
Advantages of NAFF
I.
Very accurate representation of the “signal”
and thus of the amplitudes
(if quasi-periodic)
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
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
5
Rigid pendulum
Sampling effect
Hyperbolic
Laurent S. Nadolski
Elliptic
FFAG Workshop, Grenoble, 2007
6
Accelerator 4D Dynamics
Accelerator
Poincaré
Surface ofsection
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
7
Computing a frequency map
Frequency map:
Configuration space
z0
FT : (x0,z0)
(x,z)
Phase space
z’
x0’= 0
z0’= 0
z
x0
Tracking T
z
Tracking T
Frequency map
Phase space
x’
NAFF
x
Laurent S. Nadolski
NAFF
z
resonance
x
FFAG Workshop, Grenoble, 2007
x
8
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)
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
9
Reading a FMA
Resonances
x
z
Regular areas
Fold
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
Nonlinear or
chaotic regions
10
Resonance network: a x + b z = c
order = |a| + |b|
4th order
5th order
7th order
9th order
Higher order
resonance
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
11
Diffusion D = (1/N)*log10(||Dn||)
Color code:
||Dn||< 10-10
||Dn||> 10-2
Diffusion reveals as well slightly excited resonances
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
12
On-momentum Dynamics --Working point: (18.2,10.3)
9x=164 3x+z=65
4x=73
x-4z=-23
5x=91
Bare lattice
(no errors)
x
3x+4z=96
z
2x+2z=57
x+6z=80
2x+5z=88
WP sitting on
Resonance node
x + 6z = 80
5x = 91
x - 4z = -23
2x + 2z = 57
x-4z=-23
Laurent S. Nadolski
9x=164
4x=73
FFAG Workshop, Grenoble, 2007
13
On-momentum dynamics w/ 1.9% coupling (18.2,10.3)
Randomly rotating
160 Quads
•Map fold
Destroyed
3x+z=65
4x=73
x-4z=-23
5x=91
3x+4z=96
x+6z=80
2x+2z=57
2x+5z=88
•Coupling strongly
impacts
3x + z = 65
Physical
Aperture
•Resonance node
excited
Resonance island
3x+z=65
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
14
Off-momentum dynamics
Several approaches:
– Off-momentum frequency maps
– Energy/betatron-amplitude frequency maps
– Touschek lifetime
• 4D tracking
• 6D tracking
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
15
Particle behavior after
Touschek scattering
x
1
Ax 
0
 x     2          
2
0
0
Closed orbit
Chromatic orbit
A  x  
x
'
x0
0
'
0
x0
Chromatic orbit
2
0
WP
WP
ALS Example
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
16
Off momentum dynamics
3x- 2z=34
4x=73
 <0
3x+z=65
 >0
3z=31
3z=31
3x+z=65
3z=31
z0 = 0.3mm
3x- 2z=34
4x=73
excited
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
17
Measured versus
Calculated Frequency Map
Modeled
Measured
D. Robin et al., PRL (85) 3
See resonance excitation of unallowed 5th order resonances
No strong beam loss  isolated resonances are benign
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
18
Frequency Maps
for Different Working Points
D. Robin et al., PRL (85) 3
Region of strong beam loss
Dangerous intersection of excited resonances
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
19
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
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
20
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 ?
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
21
References
•
Tracking Codes
•
Papers
–
–
–
–
–
–
–
–
–
–
–
–
BETA (Loulergue – SOLEIL)
Tracy II (Nadolski – SOLEIL, Boege – SLS, Bengtsson – BNL)
AT (Terebilo http://www-ssrl.slac.stanford.edu/at/welcome.html)
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/ArchivesWorkshops/2004/frequency-map/index_fma.html
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
22
Annexes
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
23
Particle Computation Frame
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
24
Decoherence of a particle bunch
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
25
Non-linear synchrotron motion
+3.8%  -6%
1 = 4.38 10-04
2 = 4.49 10-03

1   ds

  2 1 
 2     ds
 2 
Tracking 6D required
Laurent S. Nadolski
FFAG Workshop, Grenoble, 2007
26