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Nonlinear dynamics and nonlinear dynamical systems Controlled Hopf Bifurcation of a storage-ring free-electron laser Giovanni De Ninno1, Duccio Fanelli2 1. Sincrotrone Trieste, 34012 Trieste, Italy 2. Cell and Molecular Biology Department, Karolinska Institute, SE-171 77 Stockholm, Sweden (Dated: February 2, 2008) 4 Localbifurcationcontrolisatopicoffundamentalimportanceinthefieldofnonlineardynamical 0 systems. Wediscussanoriginalexamplewithinthecontextofstorage-ringfree-electronlaserphysics 0 bypresentinganewmodelthatenablesanalyticalinsightintothesystemdynamics. Thetransition 2 between the stable and the unstable regimes, depending on the temporal overlapping between the n light stored in the optical cavity and the electrons circulating into the ring, is found to be a Hopf a bifurcation. A feedback procedure is implemented and shown to provide an effective stabilization J of theunstable steady state. 7 2 PACSnumbers: 05.45.-a ] s Transition from stability to chaos is a common char- pass, n, inside the optical cavity according to: c acteristic of many physical and biological systems [1, 2]. ti Within this context, local bifurcation control is a topic yn+1(τ)=R2yn(τ −ǫ)[1+gn(τ)]+is(τ), (1) p of paramount importance especially for those systems o where τ is the temporal position of the electron bunch . in which stability is a crucial issue. This is, for ex- s distribution with respect to the centroid; R is the mir- c ample, the case for conventional and non conventional rorreflectivity;thedetuningparameterǫisthedifference i light sources, such as Storage-Ring Free-Electron Lasers s between the electrons revolution period (divided by the y (SRFELs), commonly employed in various scientific ap- numberofbunches)andthe periodofthe photonsinside h plications [3]. In a SRFEL [4], the physical mechanism the cavity; i (τ) accounts for the profile of the sponta- p responsiblefor lightemissionandamplificationis the in- s neousemissionoftheopticalklystron[9]. Assumingthat [ teraction between a relativistic electron beam and the the saturationis achievedwhenthe peakgainisequalto magnetostatic periodic field of an undulator. Due to the 1 thecavitylosses,P,theFELgaing (τ)isgivenby[7,8]: v effectofthemagneticfieldtheelectronsemitsynchrotron n 8 radiation,knownasspontaneousemission. Thelightpro- 13 danudceadmbpylitfiheededleucrtirnognsbuecacmessisivsetoturerdnsinofanthoepptiacratlicclaevsitiny g (τ)=g σ0 Pσe σn2−γσ02 exp − τ2 (2) 01 thering. Agiventemporaldetuning,i.e. adifferencebe- n iσn (cid:20)giσ0(cid:21) (cid:20) 2στ2,n(cid:21) tweentheelectronbeamrevolutionperiodandtheround 4 0 tripofthephotonsinsidetheopticalcavity,leadstoacu- where gi and σ0 are the initial (laser-off) peak gain and beam energy spread, σ and σ are the energy spread / mulativedelaybetweentheelectronsandthelaserpulses: n τ,n s and the bunch length after the nth light-electron beam the laser intensity may then appear as “continuouswave c interaction, and γ = σ2 −σ2. Note that equation (2) i (cw)” (for a weak or strong detuning) or show a stable e 0 s refers to the case of SRFELs implemented on an optical y pulsed behavior (for an intermediate detuning amount) klystron. The evolution of the laser-induced electron- h [5, 6]. The achievement of a large and stable “cw” zone p is a crucial issue, of fundamental importance for experi- beam energy spread is governed by the following equa- : mental applications. In this Letter, we characterize the tion: v i transitionbetween stablethe unstable regimesasa Hopf 2∆T X bifurcation. This result allows one to establish a formal σn2+1 =σn2 + τ (γIn+σ02−σn2). (3) s r bridge with the field of conventional lasers and to adopt a the universal techniques of control theory to enlarge the Here σ is the equilibrium value (i.e. thatreachedatthe e stablesignalregion. We developthisideabyintroducing lasersaturation)ofthe energyspreadatthe perfect tun- anew modelwhichrevealsto be particularlysuitablefor ingand∆T isthebouncingperiodofthelaserinsidethe ∞ analytic investigations. optical cavity; I = y (τ)dτ is the laser intensity n −∞ n The longitudinaldynamicsofa SRFELcanbe described normalized to its equilibrium value (i.e. the saturation R by a system ofrate equationsaccounting for the coupled value for ǫ=0) and τ stands for the characteristic time s evolutionof the electromagnetic field and of the longitu- of the damped oscillation of electrons in their longitudi- dinal parameters of the electron bunch [7, 8]. The tem- nal phase-space. Equations (1), (2) and (3) are shown poralprofile ofthe laserintensity, yn, is updated ateach to reproduce quantitatively the experimental results [8]. In particular, the laser intensity displays a stable “cw” behavior for small amount of detuning, while a pulsed 2 regimeisfoundforǫlargerthanacertaincriticalthresh- energyspread,intensity, centroidposition andrms value old, ǫ . This model represents the starting point of our of the laser distribution, to the light-electron beam de- c analysis. tuning. Througha stability analysisit is alsopossible to Equation (1) characterizes the evolution of the statis- determine the threshold value ǫ . To our knowledge this c tical parameters of the laser distribution: by assuming study represents the first attempt to fully characterize a specific form for the profile, it is in principle possible the detuned SRFEL dynamics. to make explicit the evolution of each quantity. For this purpose, we put forward the assumption of a Gaussian laser profile and compute the first three moments. The Laser intensity Electrons' energy spread details of the calculations are given elsewhere [10]. In dI ds dt dt addition,itisshownthatforǫspanningthecentral“cw” zone, the quantities (σ /σ )2 and [(τ +ǫ)/σ ]2 are l,n τ,n n τ,n small. Hence,aTaylorseriesexpansionisperformedand secondorderterms neglected. Finally, by approximating finite differences with differentials, the following contin- uous system is found: s I dI Laser intensity ds Electrons' energy spread dt dt dσ = α1 1 α2I+1−σ2 dt ∆T 2σ  (cid:2) (cid:3)  ddddIτtt ==−R∆2∆TIτT(cid:20)−+RP∆2τˆT+ g12iσ−α33gασi4σα2α3−2α14σ(cid:18)2α−22α1σ3σσ2l22−σl2−τˆ2(cid:19)(cid:21)+ ∆IsT I s (cid:20) (cid:21)  ddσtl =−∆1T g2iα3α4σα2−21σσl33 + ∆1T IIs21σl (cid:18)ασ23 +τ2(cid:19)(4,) FsǫlFauEIt=sGiLo.In0a.1s.1s:hfRrasPeivfghee<hartbeseǫnecec-co,nselpu.tpahmeTcerenhfb:oeporotvσm˙rtaoterlvmudaeeirtusosssu.noisnefLsgteσhftt.teohecrTǫoecllha=ueesvmea1tnno.o:3tpffIpt˙sphaaiers>naSepmlulǫsoecptt.erteererSd-fsAeirmvaCerutrOeo--: where τˆ=τ +ǫ and ∆T =120ns,τs=8.5ms,σ0 =5·10−4,σe/σ0 =1.5,Ω=14 kHz, gi =2%, P =0.8%, Is=1.4·10−8. 2∆T σ2−σ2 α1 = τs , α2 = eσ02 0, (5) The fixed points I, σ, τ, σl are found by imposing dI = dσ = dτ = dσl = 0 in (4), and solving the cor- dt dt dt d(cid:0)t (cid:1) responding system. Assume hereon ǫ > 0, being the 2 Ω Pσe scenario for ǫ < 0 completely equivalent. After some α3 = , α4 = . (6) (cid:18)σ0α(cid:19) giσ0 algebraiccalculations,the followingrelationsarefound: Here Ω represents the oscillation frequency of the elec- σ2−1 trons in their longitudinal phase-space and α, the mo- I = , (7) α2 mentumcompactionfactor,isacharacteristicparameter of the storage ring. Note the redefinition of σ which is from hereonnormalized to σ0. Although in approximate 1 form, system (4) still captures the main features of the 1 σ2 σ2 2 2 longitudinal SRFEL dynamics. In particular,the transi- τ = − + +4ǫ2A , (8) tionfromthe“cw”regimetotheunstable(pulsed)steady 2 α3 s(cid:18)α3(cid:19)  state occurs for a temporal detuning which is close to   theonefoundintheframeworkoftheexactformulation,   hence to the experimental value. However, system (4) 1 fs1aitipiloshnainsteor-setpphareocedlauptceoirnratglrat“ihctews ”cfoozrrorbencoetthibsetahhpaepvrliaoosarecrwhehindet.nenItnshietFyitgrauannrde- σl =2gIisα3α41−ασ22α2σ2σ−3 1ασ23 +s(cid:18)ασ23(cid:19)2+4ǫ2A4 ,  (9) the beam energy spread are plotted for different values where  of ǫ. Limit cycles are observed when ǫ>ǫ . For smaller c values of ǫ, the variables converge asymptotically to a 1−σ2 stable fixed point. The latter can be analytically char- σ3 σ2−1 α α2 A= 4 . (10) acterized, thus allowing one to relate the electron-beam (cid:0)α2Is (cid:1) giα3 3 These relations link the equilibrium values of I,τ,σ to Laser intensity Electrons' energy spread l 1 1.5 σ. The quantity σ is found from the following implicit equation: giασ2α−21 1− 1α3 σ 2+(τ +ǫ)2 = P , (11) σ 4 2σ2 l R2 where σ and(cid:20)τ are res(cid:0)pectively given(cid:1)(cid:21)by (9) and (8). 0.70 e (fs) 1 1.30 e (fs) 1 For anylgiven value of the detuning ǫ, equation (11) 0.7 Laser centroid 4.5 x10-2 Laser rms can be solved numerically, by using a standard bisection method. The estimates of σ are then inserted in equations (7), (8), (9), to compute the corresponding values of I, τ, σ . Results of the calculations (solid line) and l direct numerical simulations using the system (4) (symbols) are compared in Figure 2, displaying remarkably 0 e 4.3 e 0 (fs) 1 0 (fs) 1 good agreement. It is worth stressing that, by means of FIG. 2: The fixed points are plotted as function of the de- aperturbativeanalysis,aclosedanalyticalexpressionfor tuning parameter ǫ. Top left panel: Normalized laser in- σ asafunctionofǫisalsofound. Thedetailsofthequite tensity. Top right panel: Normalized electron-beam energy cumbersome calculations are given elsewhere [10]. spread. Bottom left panel: Laser centroid. Bottom right As a validation of the preceding analysis, we consider panel: rms value of the laser distribution. Symbols refer to the simulations, while the solid line stands for the analytic the case of perfect tuning, i.e. ǫ = 0, and compare our calculation. The list of parameters is enclosed in the caption estimate for the laser induced energy spread σ to the l of Figure 1. value (σ ) , derived in the context of the widely used l sm super-modes approach [11]. Both theoretical predictions are then compared to experiments performed on the Su- partofthe eigenvaluesasafunctionofǫis showninFig- perACO and Elettra FELs. Results are givenin Table I: ure 3. The system is by definition stable when all the theimprovementofthecalculationbasedonequation(9) real parts are negative. The transition to an unstable isclearlyshown. TheresultsofTableIindicatethatboth regime occurs when at least one of them becomes posi- σ and (σ ) are smaller than the experimental values. l l sm tive. Ingeneral,thelossofstabilitytakesplaceaccording This is probably due to the fact that the models neglect to different modalities. Consider the case of a Jacobian the effect ofthe microwaveinstability [12]resulting from matrix with a pair of complex conjugate eigenvaluesand the electronbeam interactionwith the ring environment assumetherealpartsofalltheeigenvaluestobenegative. (e.g. the metallic wall of the vacuum chamber). In the A Hopf bifurcation occurs when the real part of the two case of Elettra the situation is complicated by the pres- complexeigenvaluesbecomespositive,providedtheother ence of a “kick-like” instability (having a characteristic keep their signs unchanged [14]. This situation is clearly frequency of 50 Hz) which periodically switches off the displayedin Figure 3, thus allowingto conclude that the laser preventing the attainment of a stable “cw” regime transition between the “cw” and the pulsed regime in a [13]. SRFEL is a Hopf bifurcation. The critical detuning, ǫ , c canbecalculated(opencircleinFigure(3))anddisplays good agreement with both the simulated data and the SuperACO Elettra experimentalvalue. A closedrelationfor ǫ is also found c σl (ps) 5 2 [10], by making use of the analytic expressions for the fixed points. (σl)sm (ps) 3 1 Experimental values (ps) 10±2 5±2 Having characterizedthe transition from the stable to the unstable steady state in term of Hopf bifurcation opens up interesting perspectives to stabilize the signal TABLE I: Theoretical widths of the laser pulse compared to anddramaticallyimprovethesystemperformance. Inor- experimentalvaluesforthecaseoftheSuper-ACOandElettra dertomaintainthelaser-electronbeamsynchronismand FELs. The experimental setting for the case of Super ACO avoid the migration towards one of the unstable pulsed (operated at a beam energy of 800 MeV and at a laser wave- zones of the detuning curve, existing second-generation length of 350 nm) is that specified in the caption of Figure SRFELs,suchlikeSuper-ACOandUVSOR[15,16],have 1. The analogous parameters for ELETTRA (operated at a beamenergyof900MeVandatalaserwavelengthof250nm) implemented dedicated control systems. The idea is to are the following: ∆T = 216 ns, τs = 87 ms, σ0 = 1·10−3, re-adjust periodically the radio-frequency, thus dynami- σe/σ0 =1.5, Ω=16 kHz, gi=15%, P =7%, Is=4.3·10−7. cally confining the laser in the central “cw” zone. Even thoughgenerallysuitableforsecond-generationSRFELs, The stability of the fixed point I(ǫ),σ(ǫ),τ(ǫ),σ (ǫ) these systems are inappropriate for more recent devices, l canbedeterminedbystudyingthe eigenvaluesoftheJa- such as ELETTRA and DUKE. The latters are indeed (cid:2) (cid:3) cobian matrix associated with the system (4). The real characterizedby a much narrowerregionof stable signal 4 2x10-5 imental tests [10]. In this respect, a significant and re- producible extensionofthe stable “cw”regionusing this 0 techniquehasbeenrecentlyachievedatSuperACO[18]. s This result fully confirms our theoretical predictions. Part -2 Inconclusion,inthisLetterweproposeanewapproxi- eal -4 matemodelofaSRFEL.Thisformulationenablesadeep R analyticalinsightintothesystemdynamics,allowingone -6 to derive the explicit dependence of the main laser pa- rametersonthetemporaldetuning. Resultsarefullycon- -8 0 0.4 e (0f.s8) 1.2 1.6 firmed by numerical simulations and show satisfactory (cid:13) agreement with available experimental data. Further, FIG. 3: Real part of the eigenvalues of the Jacobian ma- the transition between the stable and unstable regimes trixassociated to thesystem (4)asafunctionof thedetuning is found to be a Hopf bifurcation, and the critical de- parameter ǫ. The solid line refers to the complex conjugate tuning ǫ is calculated explicitly. Finally, we introduced eigenvalues. Thecirclerepresents the transitionfrom thesta- c in the model a derivative feedback that is shown to stable to the pulsed regime, i.e. the Hopf bifurcation. bilize the laser intensity well beyond the threshold ǫ . c Successful experiments carried out at Super ACO con- 3 onlyoccasionallyexperimentallyobserved[13],makinga y priori impossible to pursuit the former strategy. On the sit n contrary,theapproachherediscussedexploitsanuniver- nte sal property of SRFELs, thus allowing to overcome the er i s limitations of other schemes. The procedure consists in La introducing a specific self-controlled (closed loop) feed- 0 0 200 backtosuppresslocallytheHopfbifurcationandenlarge 2 the zone of stable signal. This is achieved by replacing y sit the constantdetuning with the time-dependent quantity en nt [17]: er i s a ǫ(t)=ǫ0+β∆TI˙ , (12) L0 0 200 Time (ms) which is added to system (4). Here ǫ0 is assumed to be largerthatǫc: whenthecontrolisswitchedoff,i.e. β =0, FIG. 4: Behavior of the FEL (normalized) intensity in the laser is unstable and displays periodic oscillations. absence (upper panel) and in presence (lower panel) of the Forβ largerthana certainthreshold,β ,the oscillations derivative control system. The simulations refer to the case c aredampedandthelaserbehavesasifitwereinthe“cw” of Super ACO (see caption of Figure 1 for the list of the pa- region. Notethat,assoonassaturationisreached,I˙=0 rameters). Here ǫ0 = 1.3 fs > ǫc. The stabilization has been and,thus,thestableregimeismaintainedasymptotically achieved using β =6·10−3. Here, βc ≃5·10−4. for ǫ=ǫ0 >ǫc, i.e. well inside the former unstable zone. The results of the simulations are represented in Figure 4. firmed our predictions. Preliminary experiments carried This new theoreticalinsightsets the groundforexper- out at ELETTRA have also given encouraging results. [1] J. Guckenheimer, P. Holmes, Nonlinear Oscillation, [8] G.DeNinno,D.Fanelli,C.Bruni,M.E.Couprie,Europ. Dynamical systems and Bifurcation of Vector Fields Phys. Journ. D 22 267 (2003). (Springer-Verlag, Berlin, 1983). [9] N.A. Vinokurov et al., Preprint INP77.59 Novossibirsk [2] J. Keener, J. Sneyd, Mathematical Physiology Springer- (1977). Verlag, New York,(1998). [10] G. De Ninno, D. Fanelli, Elettra Technical Report [3] G. S. Edwards et al., Rev. of Scient. Instrum. 74 3207 ST/SL-03/03 (2003). (2003). [11] G. Dattoli et al., Phys. Rev. A 37, 4326 (1988). [4] W.B. Colson, Laser Handbook Vol.6, (North Holland [12] G.DattoliandA.Renieri,Nucl.Instr. and Meth. A375 1990). 1 (1996). [5] M E. Couprie et al. Nucl. Instr. and Meth. A 331 37 [13] G. De Ninno et al., Nucl. Instr. and Meth. A 507 274 (1993). (2003). [6] H.Hama et al, Nucl. Instr. and Meth. A 375 32 (1996). [14] R. C. Hilborn, Chaos and Nonlinear dynamics, Oxford [7] M.Billardon,D.Garzella,M.E.Couprie,Phys.Rev.Lett. University Press (1994). 69 2368 (1992). [15] M. E. Couprie et al., Nucl. Instr. and Meth. A 358 374 5 (1995). [18] M.E. Couprie et al., Proceedings of the FEL conference [16] S.Kodaetal.,Nucl. Instr. andMeth. A475211(2001). 2003, submitted to Nucl. Instr. and Meth. A. [17] S. Bielawski, M. Bouazaoui, D. Derozier, Phys. Rev A 47, 3276 (1993).