FMA workshop April 1st and 2nd, 2004 L. Nadolski 1

Guiding Lines for Computing and Reading a

Frequency Map

New FMA Calculations for SOLEIL including Insertion

Device EffectsLaurent S. Nadolski

Pascale Brunelle and Amor Nadji

FMA workshop April 1st and 2nd, 2004 L. Nadolski 2

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

FMA workshop April 1st and 2nd, 2004 L. Nadolski 3

Tools• Tracking codes

– 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, protrons)• 4D/6D

– AMD Opteron 2 GHz• 0.7 s for tracking a particle over 2 x 1026 turns

– 1h00 for 100x50 (enough for getting main caracteristics)– 6h45 for 400x100 (next fmap)

• Step size following a square root law (cf. Action)

FMA workshop April 1st and 2nd, 2004 L. Nadolski 4

z

x

Regular areas

Resonances

Nonlinear or chaotic regionsFold

Reading a FMA

FMA workshop April 1st and 2nd, 2004 L. Nadolski 5

5 10 15 201

5

10

1520

25

z (mm)

x (mm)

Mapping

FMA workshop April 1st and 2nd, 2004 L. Nadolski 6

4th order5th order7th order9th order

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

Higher orderresonance

FMA workshop April 1st and 2nd, 2004 L. Nadolski 7

Rigid pendulum

Sampling effect

Hyperbolic Elliptic

FMA workshop April 1st and 2nd, 2004 L. Nadolski 8

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

Color code:

||||< 10-10

||||> 10-2

Diffusion reveals as well slighted excited resonances

FMA workshop April 1st and 2nd, 2004 L. Nadolski 9

Soleil Beam Dynamics investigations using FMA

• Working point 18.20 10.30– Lattice: bare, errors, IDs– Optimization schemes

• Design of a new working point taking into account what was discovered through FMA

• Conclusions

FMA workshop April 1st and 2nd, 2004 L. Nadolski 10

Optimization Method• Tuneshift w/ amplitude• Tuneshift w/ energy• Robustness to errors

multipoles couplingIDs

Lattice designFine tuning

ImprovementNeeded

TrackingNAFF

NAFF suggestions

Yes

•(x-z) fmap injection eff. •(x-) fmap Lifetime•Touschek computation

Resonance identification

• 4D tracking• 6D tracking

Knobs10 quadrupole families 10 sextupole families

Dynamics analysis

Good WPNo

FMA workshop April 1st and 2nd, 2004 L. Nadolski 11

x

z

x|z=1m

z| x=1m

X or z (m)

Flat vertical tune

No coupling resonance crossingx- z = 8 ( = 0.1).

Reference working point (18.2, 10.3)

See M. Belgroune’s talk

Just looking at these curves, it seems very clean …

FMA workshop April 1st and 2nd, 2004 L. Nadolski 12

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

FMA workshop April 1st and 2nd, 2004 L. Nadolski 13

Randomly rotating 160

Quads

•Map fold Destroyed

•Coupling strongly impacts

3x + z = 65

•Resonance node excited

PhysicalAperture

On-momentum dynamics with 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

FMA workshop April 1st and 2nd, 2004 L. Nadolski 14

Importance of including vaccum chamber

Injection @ 14mm

x+z=65 4x=73

x

z

Skew resonance excited by coupling

FMA workshop April 1st and 2nd, 2004 L. Nadolski 15

Adding effect of 3 in-vacuum IDs

x+z=65

z = 4.5 10- 3

IDOctupole term

deeper

Injection trouble if stronger

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Particle behavior after

Touschek scattering Chromatic orbit

Chromatic orbit

Closed orbit0

1

x

x Ax

2'

00

'

000

2

02

0

xxx xA

ALS Example

WP

WP

FMA workshop April 1st and 2nd, 2004 L. Nadolski 17

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%

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Off momentum dynamics w/o IDs

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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Off momentum dynamics w/ 3 x U20

U20

B-Roll offx = 18 mg = 5mm

4x=73

3x+z=65

What’s about Effect of synchrotron radiation and damping?

Very narrow resonances

Synchrotron140 turns

Damping5600 turns

Loss over >400 turnsStable in 6D

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Coupling reduction by a factor 2 with 3 x

U20

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Optimization of a New PointEnhanced philosophy

• On momentum – 3 x + z = 65 to be avoided (not shown w/o fmap)– WP to be shifted from resonance node: locus of most

particles– Control of tune shift with amplitude using sextupole knobs

• x (Jx, Jz) = a Jx + b Jz

• z (Jx, Jz) = b Jx + c Jz

• Off momentum x()• Large energy acceptance • Control of the tune shift with energy using sextupoles• The 4 x = 73 resonance has to be avoided for insertion devices

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Energy tune shift for the new WP 18.19 / 10.29

x

z

dp/px

z

dp/p

18.19 / 10.2918.20 / 10.30

Tune shift w/ energy optimised with sextupoles to avoid in

addition the 4x = 73 resonance for negative energy offset

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On momentum fmap for the WP 18.19 / 10.29

3x- 2z=34

3x- 2z=34

WP to be slightly shifted

Clean DA

1% coupling

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On momentum fmap for the WP 18.19 / 10.29 with 3 x U20

Map Torsion

Strong diffusion related to ID roll off

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Off momentum fmap for the WP 18.19 / 10.29

Off-momentum

map

comparable

to the

nominal WP

2x+2z=57

2x+2z=57

3x-z=26

3x-z=26

1% coupling

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Off momentum fmap for the WP 18.19 / 10.29 with 3 x U20

Effect of the

2x+2z=57

resonance

becomes

« dangerous »

with the 3xU20

2x+2z=57

2x-z=26

2x-z=26

2x+2z=57

Smoothing by synchrotron oscillations

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Conclusions

FMA at design stage for the SOLEIL lattice

– Gives us a global view (footprint of the dynamics)– 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– Importance of coupling correction to small values (below

1%)– 4D/6D …

FMA workshop April 1st and 2nd, 2004 L. Nadolski 28

Aknowledgements and references

• Institutes– IMCCE, ALS, SOLEIL

• Codes– BETA (Loulergue -- SOLEIL)– Tracy II (Nadolski -- SOLEIL, Boege -- SLS)– AT (Terebilo http://www-ssrl.slac.stanford.edu/at/welcome.html)

• Papers– Frequency map analysis and quasiperiodic decompositions, J. Laskar,

Proceedings of Porquerolles School, sept. 01– Global Dynamics of the Advanced Light Source Revealed through

Experimental Frequency Map Analysis, D. Robin et al., PRL (85) 3– Measuring and optimizing the momentum aperture in a particle

accelerator, C. Steier et al., Phys. Rev. E (65) 056506– Review of single particle dynamics of third generation light sources

through frequency map analysis, L. Nadolski and J. Laskar, Phys. Rev. AB (6) 114801