LOCALIZED CHAOS GENERATED BY RF-PHASE MODULATIONS / 25 PHASE-SPACE DILUTION

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1 APPLICATION OF A LOCALIZED CHAOS GENERATED BY RF-PHASE MODULATIONS IN PHASE-SPACE DILUTION 1 and S.Y. Lee 2 1 Fermilab 2 Physics Department, Indiana University HB2010, Morschach, Switzerland September 27 October 1, 2010

2 Introduction ALPHA is a 20-m electron storage ring under construction at IU CEEM. It calls for storing a 1000-nC bunch to be extended to 40 ns with uniform linear density, so that the slowly extracted power load on the target is not as high. Barrier bucket should be the best way to do this. Unfortunately, this ring is only 66.6 ns in length, so that the width of the barrier must be of the order of 10 ns or less. The risetime will therefore be of order of few ns, or the rf generating the barrier will require frequencies 100 MHz. 1. Ferrite is very lossy at such high frequency. 2. Barrier voltage will be very high, because of narrow barrier width. Such a barrier rf system will be very costly.

3 Diffusion by Phase Modulation A possibility is to do phase modulation of the rf so as to produce a large chaotic region at the center of the rf bucket, but bounded by well-behaved tori. The tiny beam at bucket center will be blown up to the chaotic region. If this region is truly chaotic, particle distribution will be uniform. We have performed two experiments on such an idea to generate a chaotic region at the bucket center. References: 1. J.Y. Liu, etal, Analytic Solution of Particle Motion in a Double RF System, Particle Accelerators 49, 221 (1995) 2. D. Jeon, etal, A Mechanism of Anomalous Diffusion in Particle Beams, Phys. Rev. Lett. 80, 2314 (1998) 3. C.M. Chu, etal, Effects of Overlapping Parametric Resonances on the Particle Diffusion Process, Phy. Ref. E60, 6051 (1999) 4., Particle Diffusion in Overlapping Resonances, Advanced IFCA Workshop on Beam Dynamics Issues for e + e Factories, Frascati, Oct , 1997 (Fermilab-Conf-98/001)

4 Former Simulation & Experimental Results { Hamiltonian: H = 1 2 ν sδ 2 + ν s (1 cos φ) r [ ] } 1 cos(hφ + φ) h φ = φ 0 + aν s sin ν m θ φ 0 is a sensitive parameter. No diffusion at φ 0 = 0, 180. φ 0 =180 o φ 0 =245 o Waterfall plot Beam Diffusion Experiment δ φ

5 Compare With Experiment Experiment Simulation f m =1400 Hz, f s0 =667 Hz, a=100 o, r=0.11 σ 2 (ns 2 ) f m =1400 Hz, f s =667 Hz, A=100 o, r=0.11 σ 2 (ns 2 ) φ 0 =245 o φ 0 =180 o t(µs) t(ms) Simulation agrees with experimental results. σ 2 φ time process is diffusion.

6 The Model In this paper, 1. We will further the investigation. 2. To determine the best φ 0 to use. 3. To understand the diffusion mechanism. The model is H = { H 0 + H 1, with H 0 = 1 2 ν sδ 2 + ν s (1 cos φ) r [ 1 cos(hφ + φ0 ) ]} h H 1 = aν s δ sin(ν m θ + η). dφ so that dθ = ν sδ + aν s sin(ν m θ + η) Or both phases are modulated. Model slightly different from above, but easier to analyze. Here, we study the situation: h = 1 r = 2, ν m = 2, η = 0. ν s Only parameters left are: rf phase difference φ 0 and modulation amplitude a.

7 Why 2-rf System? Action J = 0 at bottom of potential well. Resonance strengths g n (J) J and vanish at potential well bottom. Tiny bunch at well bottom is with J = 0, will not be driven into parametric resonance. We need to shift well bottom away from phase-space center. This explains why we need a 2-rf system with φ 0 0 or π. In other words, needs a right-left asymmetric potential. We also require synchrotron tune Q s to be larger at central phase space than the edge, so that central region well be in resonance and chaos, while the edge is well behaved.

8 Choice of φ 0 Well bottom is at φ 0 with V (φ 0 ) = 0. φ 0 is maximum when V (φ 0 ) = 0 = sin φ 0 = r. Corresponding φ 0 = π 2 h sin 1 r. Well Bottom Offset φ 0 (deg) RF: r=1/2 h= Phase Difference φ 0 (deg) For h = 1 r = 2, Well bottom at φ 0 = 30. Rf phase diff. φ 0 = 30. Well bottom not too sensitive to φ 0. Diffusion does occur when 20 φ However, no diffusion seen outside this range.

9 Synchrotron Tune Q s Q s /ν s or RF Potential Y. Ng J = 1 δ(φ)dφ, ψ = Q φ s dφ 2π ν s φ 1 δ(φ ), Q s is obtained by integrating once around while ψ advances by 2π. Here we first study the case of φ 0 = 30. 4x V(φ 1,2 ) 1x 2x 3x 5/2x h=1/r=2 ν m /ν s φ 0 =30 o Ends of Torus φ 1 or φ 2 (rad) Q s /ν s x 2x position of origin 1x 5/2x ν m /ν s V(J) h=1/r=2 φ 0 =30 o Action J 3:1 resonance will be excited, but fixed points and separatrices are far from bunch original position. 5:2 will also excited but not as strong. Need large modulation amplitude a to produce diffusion.

10 Resonance Strength Functions g n (J) Resonance Strength g n (J) H 1 = aν s δ sin ν m θ (η = 0) δ = g n (J)e inψ, g n (J)= g n (J) e iχn(j) = 1 π δe inψ dψ 2π n= π First-order terms: [ ] H = E(J) + aν s g n (J) sin(ν m θ+χ n nψ) + sin(ν m θ+χ n +nψ) n>0 + aν s g 0 (J) sin ν m θ Strength fcn g n measures ability to generate 1st-order n : 1 resonance. h=1/r=2 φ 0 =30 o Potential asymmetric n=1 δ(φ) asymmetric n even is possible. position of origin n=2 n=4 n=3 Note: g 2 (J) g 3 (J). Much easier to drive 2:1 resonance than 3:1 resonance Action J.Y. Ng

11 Phase-Space Pattern with φ 0 = 30.Y. Ng First look at phase-space structure at a = 8. Bunch in stable area enclosed by 5:2 resonance. Chaotic region between 8:3 and 25:8 resonances. For bunch to diffuse requires 1. bunch out of central stable area 2. 5:2 and 8:3 resonances collapse to join chaotic region. No. 2 happens when a is raised from 8 by not much. No. 1 happens when a 46. Note phase-space structure shifts left as a increases because of detuning.

12 Diffusion with φ 0 = particles tracked for 1M turns. Modulation period is 515 turns. Certainly 3:1 resonance excited. Bunch diffuses into thick chaotic layers surrounding separatrices. Larger modulation amplitude thicker chaotic layers larger area of diffusion. Red dots form the bounding good torus. σφ 2 and σ2 δ time diffusion..y. Ng

13 Variation of Modulation Amplitude a 1.0 σ φ and σ δ After Diffusion σ φ (rad) σ δ φ 0 =30 o Modulation Amplitude a (deg) Diffusion occurs only for limited ranges of modulation amplitude a. When diffusion occurs, final bunch distribution is roughly similar. σ φ and σ δ are not much different. When a = 80, phase-space center inside a 3:1 island no diffusion. Resonance pattern continues to shift left as a increases. At a 85, phase-space center moves out of 3:1 island, or the island gets filled up. The bunch diffuses again.

14 Variation of Modulation Amplitude a (cont.) Left: As a > 94, resonance pattern contracts and moves to left so much that chaotic region clears the bunch completely no more diffusion. When a > 110, beam loss occurs. Right: strong diffusion at a = 58. Particles streaming (not jumping) from one island to another through stochastic layers surrounding separatrices..y. Ng

15 Other Choice of φ Q s /ν s x 2x position of origin 1x 5/2x ν m /ν s V(J) h=1/r=2 φ 0 =30 o Well Bottom Offset φ 0 (deg) RF: r=1/2 h= Action J Phase Difference φ 0 (deg) The intercept of ν m /ν s = 2 with 3 Q s /ν s is far from phase space center (roughly J 0.05). Bunch initially is far from 3:1 resonance fixed points. Need very large modulation amplitude to generate thick stochastic layer to include the bunch. We can choose φ 0 = 45. Offset of well bottom is φ 0 = 29.12, slightly less than the best 30.

16 Q s /ν s Choice of φ 0 = x 2x position of origin 1x ν m /ν s V(J) h=1/r=2 5/2x φ 0 =45 o Action J Resonance Strength g n (J) φ 0 =45 o position of origin h=1/r=2 n=1 n=2 n=4 n= Action J Now initial bunch position is close to unstable fixed point of 2:1 resonance. 2:1 resonance strength function much larger than that of 3:1 resonance. Much smaller modulation amplitude will be necessary. there is good reason to pursue φ 0 = 45..Y. Ng

17 Simulations with φ 0 = σ φ (rad) or σ δ σ φ (rad) σ δ φ 0 =45 o beam loss Phase-space pattern at φ 0 = 45. Bunch initial position close to UFP of 2:1 resonance Modulation Amplitude a (deg) But not connected to chaotic region flanked by 7:3 and 8:5 resonances. For bunch to diffuse, chains of higher-order islands enclosing 2:1 resonance must collapse. This happens when a 9. There is another jump of bunch spread near a = 20.

18 Simulations with φ 0 = x ν m /ν s 8/3x 2.0 Q s /ν s x position of origin 1x V(J) h=1/r=2 5/2x φ 0 =45 o Action J Note the hump of Q s around J = 0.7 and 2 intercepts with ν m /ν s = 2. There are 2 sets of 8:3 resonances: one going out and one coming in as a increases. The one going out was broken when a 9. The one coming in enclosing the chaotic region (right plot). Around a 20, this chain of 8 islands starts to collapse, and chaotic region increases.

19 Simulations with φ 0 = 45 Typical bunch distribution at a = 28. The two 2:1 resonance islands almost filled up. Compared with φ 0 = 30, 1. Modulation amplitude is about 1/2 smaller. 2. Bunch spread is larger by 15%. Bunch length 38 ns. 3. Bunch distribution more rectangular, thus more uniform. Both σφ 2 and σ2 δ turn number process is diffusion and much faster.

20 Conclusions A method is devised to enlarge a bunch via phase modulation of rf. The enlarged bunch distribution is uniform and is close to rectangular. Total bunch length 38 ns as required. The chaotic region is well bounded by tori. To accomplish this, a 2-rf system is required to shift potential-well bottom off phase-space center. The optimum shift of well bottom is computed. The corresponding rf phase difference φ 0 required is also given.

21 Effects of Extra Phase η Recall H 1 = aδν s sin(ν m θ + η), where η = 0 was used before. We now try various η s with a = 58, φ 0 = 30 Note rotations: 3:1 islands by 1/3 2:1 center by 1/2 Extracted slice of beam will be of different intensity according to beam position in phase space.

22 Effects of Extra Phase η With H 1 =aδν s sin(ν m θ+η), and a = 116, φ 0 = 30, Note rotations: 8:3 islands by 3/8 5:2 islands by 2/5 2:1 center by 1/2 How to compute offset of central region as function of φ 0? How to compute the detuning, or the shrinking of resonance structure?

23 Detuning: One Resonance Consider H = ν s [ Q s I αi2 + ai 3/2 sin(3ψ + ν m θ) + H 2 ], where H 2 = 1 2 bi2 sin ν m θ or 1 2 bi2 cos ν m θ. Track using 1/ν m = 3333, ν m /ν s = , a = 0.05, b = 0.75 Q s = 1.1, α = 0.1. b = 0 b sin ν m θ b cos ν m θ With b-detuning term, resonance structure shrinks. There is a clockwise rotation with b sin but not b cos. There appears to be a string of 6 islands now, possibly 6:2 resonance. If resonance-driving term changed to ai 3/2 cos(3ψ+ν m θ), there is no change in stucture. But for the b-term, it does matter.

24 Detuning: One Resonance Actually, there is no detuning term 1 2 αi2 in original Hamiltonian. we delete this term, but keep 1 2 bi2 sin ν m θ or 1 2 bi2 cos ν m θ. Track with Q s = 1.05, b = 1.00, a = 0.05, ν m /ν s = When b = 0, only u.f.p. s, since Q s ν s ν m 3ν s = 0.05 > 0. b = 0 b sin ν m θ b cos ν m θ There is detuning from the b-term. There is curling if sine is used instead of cosine. Resonance expands instead of contracts, independent of sign of b.