A PHASE MODULATION* 2Fermilab,Box 500, Batavia, IL Brookhaven National Laboratory, Upton, N Y 11973

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1 A PHASE MODULATION* J.Y. Liu, M. Ball, B. Brabson, J. Budnick, D.D. Caussynl, P. Colestock,V. Derenchuk G.East, M. Ellison, D. Friesel, B. Hamilton, W.P.Jones, X. Kang, S.Y.Lee, D. Li K.Y. Ng, A. Pei, A. Riabko, T.Sloan, M. Syphers3,L. Wang Indiana University Cyclotron Facility, Bloomington, IN 7 a 'Physics Department, University of Michigan, Ann Arbor, MI 89 AU6 Fermilab,Box 5, Batavia, IL 65 3Brookhaven National Laboratory, Upton, N Y 973 EC EI V ED I996 OSTI Abstract instabilitiesand reduce the beamloss during the transitioncros: ing. In this report, we present the study of using the secondar The Hamiltonian system with phase modulation in a higher rf cavity with a phase modulation as a perturbation to the pr harmonic rfcavity is experimentally studied on the IUCF cooler mary rf cavity. We analysed parametric resonances and stochar ring. The Poin& maps in the resonant rotating frame are obtic motions, based on experiment data from the beam extained &om experimental data and compared with numerical tracking. The formation of the stochastic layer due to the over- CE37F at IUD. The controlledbeam evolutionwill be discussa in another paper []. lap of parametricresonances is discussed. The dependence of the stochastic layer on the voltage of the higher harmonic rfcavity, II. HAMILTONIAN ANALYSIS amplitude-and frequency of the phase modulation is studied. With a sinusoidal phase modulation to the secondary rf sys tem, the Hamiltonian can be written as I. INTRODUCTION The doublerf system, i.e. a primary rf system plus a secondary rf system working at a higher harmonic. can be used to overcome the space charge effect in low and median energy proton accelerators by reducing the peak current, and provide strong Landau damping against instabilitiesin high energy accelerators. It has been widely used to enhance the beam intensity in synchrotrons H = 6' iv,( - cos ) ~vae [ ccs(h - m(f3))],(t where, (6)= a,,, sin vm6, a, and v, are amplitude and tune (Wuency) of the phase modulation respectively. For a small c/h. wetreatthesecondary dsystemas aperturbationtotheprimary rf cavity, and therefore we are able to expand the timedependent Hamiltonian in action-angle variables { J,} of the unui. For particles in a double rf system, the synchrotronequations permbedhamiltonian [5I. Rewrittingthe term c a ( hq5,,) = of motion with respect to the orbiting angle f3 are generally given cos h&cos 6, -sin h sin,. we can expand sin h and cos h in the Fourier series, by * is the P -.-_. fractional momentum deviation from the synctxonous JW particle, 7 is the phase slip factor, vs = is the synchrotron tune determined by the primary rf system, h = and e = are harmonic and voltage ratios of the primary and the secondary rf cavities. In previous reports,we have systematically studied the double rf system with h = and discussed the stability of particle motion under the influence of parametric resonances by applying external phase and voltage modulations to both rf cavities [Z]. We recently studied the controlled beam emittance diiution using the double rf system with higher hamonic ratio by modulating either the primary rf cavity or the secondary one 3. The controlled beam blow-up is necessary in a high intensity accelerator with a small longitudinal emittance to avoid synchrotron *Work supported in pan from NSF Grant, No. PHY /96/$5. *996 IEEE -f sin[h tan-'(tan C,,(J) = s /cw[h tan-'(tan S n ( J ) = n ~cn)]e-"'"d, %n)le-'"*d, () and cn$ is the elliptical function. Hence, the texm cos hgl gives rise only to even harnlonics and sin hq5 to odd harmonia in the first order perturbation. Figure shows the resonance strengths 5' and C which drive the lowest harmonics of parametric resonances. The small amplitude approximations are compared and found to be good for a range < 5'. In terms of action-angle variables { J, $}, the Hamiltonian of Eq. () becomes 33

2 " * 9 (md) (rrd) Figure. Resonance strengths Sand CZ (solid lines) for m n g e s t parametric resonances, compared with the smaii amfigure 3. Attractors obtained from tracking x partcles :plitude approximations (dotted lines). with Q = 5 s-', us = 6.3 x c =., h = 9. %/v, = b k and a, = 7'. (a) The final distribution and (b) the initial is: dismbution showing basins of attractors. x cm(n$ - (k ) v d Xn) Cn(J) 6. k -I n k=l where E( J) is the-energy of the unperturbed Hamiltonian, and.jk(a,) is the Bessel function. In Eq.(5), only terms which con- -tributeto parametric resonances are kept. The funher analysis of,-anisolated parametric resonance can be easily accomplished by a canonical transformation. When the modulation tune is near ' one of the p-mc resonances, i.e. kv, nv,, the perturbation coherently acts on the particle motion. = ~ The parametric resonances ye numerically studied in a basis 'of turn by nun tracking. Figure shows parametric resonances at v, =.6v, and v, = v,. The overfap of higher harmonic lcsonances is responsible to the stochasticity near the boundary of the bucket However, in a dissipative dynamicd system such ' as the IUCF cooler ring, the stochastic motion of particles will not lead a significant beam loss. Instead, particles are damped mto the central region of the potential well and form a beam proae with waves on the top. In such a way, the beam emittance is blow-up, depending on the modulation amplitude and fi-equency. figure 3 shows the tracking results at v,/v, = with a phase.damping. Thedampingratecr = 5s-I isused. Thefixedpoints of resonance islands become attractors as observed in previous experiments r (- 9 (rad) Egure. Parametricresonances withv, = 6.3 x c =., h = 9 anda, = 7'. In(a). v,/v, =.6,andin(b), u,/v, = The IUCF cooler ring was operated with a single proton beam bunch at the energy of 5 MeV and the intensity about PAThe cycling time of the proton beam is about seconds with injection and electron cooling being accomplished in about 5 seconds.the electron cooling time is about 3 m. The beam emittance of the proton beam is electron-cooied to less than.3 ;f mm-mrad in about 3 seconds. The momentum spread A p / p is of order and the typical bunch length is about a = ns. The revolution frequency is fo=.368 MHz. The primary rf cavity and the secondary rf cavity were operated at harmonics hl = and h = 9 respectively. The voltage of the primary rf cavity was set at 6 = 85 v to achieve the synchrtron frequency off, = 63 Hz (or v, = 6.3 x and the secondary rf cavity was varied to obtain a proper voltage ratio to the primary rf cavity. When the experiment was started. the beam bunch was Iongitudinally kicked to drive the synchrotron oscillation by phase shifting the control signals for both rf cavities. The phase modulation with controllable ampiitude and frequency was added onto the phase shift to the secondary rf cavity. Once the beam is phase kicked, the beam closed orbit zc was measured from the ratio of the difference (A) and sum (C) signals of a BPM at a high dispersion location with an accuracy of. mm. Then the off-momentum variable was dculated &om h p / p = x c / D t, where, 3.9 m. The signal from this BPM was lead to a phase detector with a range of 7' which generated the phase coordinate by comparing the signal from a pickup loop in the primaryrfcavity witharesolutionofo.'. APoincaremap thencan be constructed h m the digitized A p / p and qj data. Figure shows a set of a typical measured data with E =., h = 9,a, = 7' and fm = 65 Hz which gave v,/v, =. In Fig. (a) the phase space is plottedexh turns for 5 turns. and in Fig. (b) the P o i n c h phase map shows a resonance island after data being manformed to the resonance rotating fr-ame. Because of the weak dissipative damping force of the electron cooling, the motion of the beam centroid is damped into the outer attractor as predicted in Fig. 3. The wiggling of the damping path is due to the time dependent effect. Figure 5 displays the beam profiles reconstructed from a fast sampling oscilloscope with c =., h = 9. u, = 5' and fm = 6 Hz, which 3 3 =

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4 A -:I i 3 L. l n a W P 8 -. e L # (ma) td) 5 a m (ded 5 Figure. Measured phase space with E =., h = 9, a,,, = Figure 6. Stochastic boundary as a function of the modulation 7' and fm = 65 Hz (v,/y, = ). (a) original data, and (b) amplitude for given frequencies (symbols), compared with nup o i n d phase map after transformation. merical simulations (solid lines). i shows the evidence of the parametric resonances. Figure 5(a) shows two beamlets obtained about 5 ms aftex the phase modulation was turned on,and Fig 5(b) shows the final beam profile captured after 5 ms, showinga wave structureresulted from the phase modulation. The beam profiles were extended from a half length of about ns to 5 ns without beam loss $ A. w j!i -. *. -_.. the paramemc resonances in the beam evolution process. In the measurement of the stochastic layer, we found that this dynamical system was complicated to detect. The diagnosticmethod of 3 5 t (n 3 5 t (n Figure 5. Beam profiles. In (a), two beamlets were due to the first harmonicresonance, and in (b), the beamprofile with a wave structure was resulted from the phase modulation. The initial beam profile is plotted as dotted line. For a given modulation frequency, the stochastic layer exists near the separatrix. As the amplitude of the phase modulation is increased, the stochastic layer increases as well. Numerical simulations indicate that when the beam is kicked inside the stochastic boundary,paxticle motions in the bunch decohere more rapidly. The change of the damping rates was experimentally observed to depend on the phase modulation. The measurements were done by fixing the phase kicks at = 6O, looo, ' and ' and varying the modulation amplitude in a step Aam = 8' for given modulation frequencies fm = 6 Hz,9 Hz and Hz. Because the experiment was time-consuming, very c o r n steps of the phase kick and the phase modulation were used. Figure 6 shows the measured results of the stochastic boundary versus the modulation amplitude, compared with numerical simulations. the stochastic layer is expected to work better in a simple system such as the double rf system reported in reference [] : References R. Averill et al., Roc. 8th Int. Conf. on High Energy Accelerators, CERN (97) p.3; l? Bramham et&., IEEE Trans.Nucl. Sci. NS-.9 (977); J.M.Baillod et d., IEEE T ~ snucl.. Sci. NS-3, 399 (983); G. Gelato et al., Roc. IEEE Part. Acc. Conf., Washington (987). p.98. [] J.Y. Liu, et ai., published on Particle Accelerators; J.Y. Liu, et al., Phys. Rev. B O,R339, (99) 3 R. Cappi, R. Garoby and E.N. Shaposhnikova,CERNPS 9- (RF) [] L. Wang er al.,this procedings. [5] H. Huang et ul., Phys. Rev. E 8, 678 (99); V.V. Balandin, M.B.Dyachkov and E.N.Shapostmikova, Particle Accelerators, 35 (99). N.CONCLUSION In summary, we have studied the parametric resonances due to a phase modulation in the secondary rf cavity. The resonance island was experimentally obtained,which agrees with the theoretical analysis. The beam profiles were evidently related to i ; -: -

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