1 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Application of Frequency Map Analysis for Studying Beam Transverse Dynamics Laurent S. Nadolski Accelerator.

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


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?


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


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


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


Rigid pendulum

Sampling effect

Hyperbolic Elliptic


Accelerator 4D Dynamics

Accelerator

PoincaréSurface ofsection


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


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)


z

x

Regular areas

Resonances

Nonlinear or chaotic regionsFold

Reading a FMA


4th order5th order7th order9th order

Resonance network: a x + b z = c order = |a| + |b|

Higher orderresonance


Diffusion D = (1/N)*log10(||Dn||)

Color code:

||Dn||< 10-10

||Dn||> 10-2

Diffusion reveals as well slightly excited resonances


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


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


Off-momentum dynamics

Several approaches:

– Off-momentum frequency maps

– Energy/betatron-amplitude frequency maps

– Touschek lifetime• 4D tracking• 6D tracking


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


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


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


Frequency Maps for Different Working Points

Region of strong beam lossDangerous intersection of excited resonances

D. Robin et al., PRL (85) 3


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


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 ?


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


Annexes


Particle Computation Frame


Decoherence of a particle bunch


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%

1 Laurent S. Nadolski FFAG Workshop, Grenoble, 2007 Application of Frequency Map Analysis for Studying Beam Transverse Dynamics Laurent S. Nadolski Accelerator.

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