Application of Frequency Map Analysis to Beam Beam
Application of Frequency Map Analysis to Beam -Beam Interaction in Crab Waist Scheme E. Levichev, D. Shatilov, E. Simonov BINP, Novosibirsk XI Super. B Workshop Frascati, 1 -5 December 2009
OUTLINE § Introduction § Description of the method § Example of the output data § Crab scan for DAΦNE § Crab scan for Super. B LER § Conclusions
Introduction The main goal of this presentation is to demonstrate how Crab Waist suppresses betatron coupling resonances. We found the FMA technique is very useful and suitable for that. Here the DAΦNE betatron tunes footprints are shown. All working resonances can be clearly identified. One can see how their strengths and widths depend on the Crab value (waist rotation). More pictures and explanations will be given in the next slides. y Crab = 0. 0 y x Crab = 0. 4 y x Crab = 0. 8 x
Frequency Map Analysis FMA was introduced by J. Laskar in 1990, and applied to beam-beam study in 2003 (reported on PAC’ 03). For any given particle a tracking code is used to get its coordinates for N sequential turns. An improved Fourier analysis is used to determine the betatron tunes with high accuracy within a “window” of M turns (for example M=1024, and M < N). If we shift the window by some number of turns, the tunes will NOT change for a regular trajectory, but they DO change for a chaotic trajectory. So, the spread of tunes along a trajectory can be used to determine its “degree of stochasticity”, or “diffusion index”. In the following plots the time-frequency dependence is shown for two different trajectories. On the horizontal axis – the “window” shift in turns (bold points correspond to 50 -turns intervals). On the vertical axis – the tune deviation from the average in logarithmic scale. The calculated diffusion indexes (numbers at the bottom) almost do not depend on the time scale. -3. 77 -6. 76 -3. 25 -6. 39 -3. 51 -6. 35
Brief description of the technique We track a particle for 2024 turns, use rectangular window of 1024 turns, and shift the window 20 times by 50 turns. So, we get 21 sets of 1024 turns. For each set an improved Fourier analysis is used to determine betatron tunes with accuracy of about 10 -8. The “diffusion index” is defined as Log 10( ), where is the spread of tunes for the given trajectory. Each trajectory is shown on the planes of betatron tunes (footprint) and betatron amplitudes by a point, colored according to the diffusion index and colormap shown on the right. How the resonances look like on the footprint? All the points within resonance islands have the same tunes. This makes a “gap” in tunes (white color), which is somehow filled by the points close to separatrix. y Points within the resonance islands are attracted to the resonance line Red points came from around separatrix x Small part of footprint
Parameters for FMA (DAΦNE) We used the DAΦNE parameters of weak-strong experiment, dated 28. 05. 2009. Agreement between experimental results and simulations is very good here. List of Main Parameters ex (cm) 2. 50· 10 -5 ey (cm) 1. 25· 10 -7 bx (cm) 26 by (cm) 0. 95 sz (cm) 1. 9 / 1. 28 Ne 4. 075· 1010 Np 0. 342· 1010 (mrad) 50 x (e+) 0. 1065 y (e+) 0. 1753 s 0. 01 xx (tracking) 0. 0154 xy (tracking) 0. 0894
Beam-Beam resonances in the tune and amplitude planes Ay y 4 y=1 2 x- y=0 x+4 y=1 5 x+2 y=1 2 x+4 y=1 6 y- x=1 x 9 x=1 Ax 6 x+2 y=1
Here the diffusion index is calculated from the spread of horizontal betatron tunes only Ay y 4 y=1 2 x- y=0 x+4 y=1 5 x+2 y=1 2 x+4 y=1 6 y- x=1 x 9 x=1 Ax 6 x+2 y=1
Transformations between tunes and betatron amplitudes y=0. 18 Ay y 0 1σx 1σy 2σx 3σx 4σx 5σx 2σy 3σy y=0. 19 4σy 5σy y=0. 20 10σx 15σx 20σx 10σy 15σy 20σy y=0. 22 x y=0. 25 Ax x=0. 120 x=0. 109
In the next 11 slides the “Crab” value will be scanned from 0 to 1 with a step of 0. 1. All the other parameters remain the same as in the table. Pay attention to the following: • Shape and size of the footprint depend on the strength of resonances. • The tune shifts ξx, y depend on the Crab value – due to crabbing of the opposite beam. • Synchro-betatron resonances are clearly seen around strong betatron ones. • Optimum Crab value is not the same for different resonances. This is explained by two mechanisms of resonance’s excitation: vertical betatron phase modulation (suppressed when Crab=1) and amplitude of beam-beam kick modulation (only partially suppressed, optimum at Crab<1). When considering both mechanisms, the optimum Crab must be < 1, but the actual value depends on particular resonances. For example, see the resonances x+4 y=1 and 5 x+2 y=1 for Crab=0. 6÷ 0. 8. • Good illustration on the Chirikov’s criterion of stochasticity (overlap of resonances) can be observed. See the resonances 2 x+4 y=1 and 6 y- x=1 for Crab=0. 2÷ 0. 4.
Crab Scan (11 slides): Crab = 0. 0 Ay y x Ax
Crab = 0. 1 Ay y x Ax
Crab = 0. 2 Ay y x Ax
Crab = 0. 3 Ay y x Ax
Crab = 0. 4 Ay y x Ax
Crab = 0. 5 Ay y x Ax
Crab = 0. 6 Ay y x Ax
Crab = 0. 7 Ay y x Ax
Crab = 0. 8 Ay y x Ax
Crab = 0. 9 Ay y x Ax
Crab = 1. 0 Ay y x Ax
Parameters for FMA (Super. B LER) List of Parameters LER/HER ex (cm) (2. 56 / 1. 6) · 10 -7 ey (cm) (6. 4 / 4. 0) · 10 -10 bx (cm) 3. 2 / 2. 0 by (cm) 0. 02 / 0. 032 sz (cm) 0. 5 Ne, p (mrad) 5. 74· 1010 60 x 0. 542 y 0. 580 The tune shift ξy is far below the limit (which is about 0. 2). As a result, the luminosity dependence on the Crab value is weak in vicinity of the optimum. s 0. 01 xx (tracking) 0. 0047 In contrast, when looking at FMA plots (see the next slides), the difference is very significant! xy (tracking) 0. 1063 Note that the actual beam tails are within 7σx and 10σy
Crab Scan (3 slides) The scales are much larger than the ones occupied by the beam – in order to see more complete plot. Note the shape of the footprint and the actual spread of x within 10σx – it is about 10 times smaller than ξx ! Crab = 0. 8 y Ay 10σx 10σy x Ax
Crab = 0. 9 y Ay x Ax
Crab = 1. 0 y Ay x Ax
FMA vs. “Old Style” Tracking § Both methods use the same tracking code, so they are inter-consistent and cannot cross-check each other. § “Old style” tracking produces the luminosity and equilibrium density distribution. For that, damping and noise must be taken into account. § On the “old style” contour plots only strong isolated resonances can be recognized. In general, identification of resonances there is difficult. § Tracking for FMA must be without noise and damping, that results in a very high resolution of resonances. Even resonances of high orders can be clearly identified. § FMA is very useful for investigating particular resonances, their strengths and widths – where they are located and how they depend on different conditions. § Tracking for FMA is much more time-consuming (more particle-turns required), and it cannot give the numbers for luminosity and beam tails (lifetime). Conclusion: These two techniques are mutually complementary. Together they give more complete understanding, in some sense “stereoscopic view” on the nonlinear beam dynamics. We should use both, as they answer different questions.
Conclusions § We found that FMA is very useful for beam-beam interaction study. Demonstration of how Crab Waist works is clear and impressive, it can be helpful for better understanding. § For those who still do not believe that Crab Waist works fine – this can be a decisive argument to reconvince. § For the present Super. B parameters the beam-beam tune shift is far below the limit. So, in linear lattice there are no problems at all. § Beam-beam simulations in nonlinear lattice would be much more interesting. We have a tool ready for that and waiting. But first of all the lattice must be finalized and DA optimized. As soon as this is done, we will proceed with beam-beam.
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