Chaotic behaviour of the storage ring free electron laser

Nuclear Instruments and Methods in Physics Research A304 (1991) 37-39 North-Holland

37

Section 11. Storage ring FEL experiments

Chaotic behaviour of the storage ring free electron laser Michel Billardon 1

LURE CNRS/CEA/MENJS, Bât. 209D, Université de Paris-Sud, 91405 Cedex Orsay, France Invited paper The macrotemporal structure of the storage ring free electron laser is studied in relation to deterministic chaos laws . Comparison between experimental and theoretical results shows that the laser obeys such laws. This can have some important consequences for the future operation of free electron lasers. Recently it has been shown that the macrotemporal structure (long time scale compared to the micropulse spacing) of a storage ring free electron laser (SRFEL) obeys the laws of the deterministic chaos [1]. Prior to the first SRFEL experiments made on the storage ring ACID, the old experiment at Orsay, it was commonly believed that the FEL would have a continuous structure. In fact it has been experimentally established that the FEL spontaneously adopts a more or less stable pulsed structure with a repetition period of several milliseconds . Similar results have been obtained on the three differents projects successively operated : ACO [2,3], Novosibirsk [4] and Super-ACO [5]. However, in the Super-ACO experiments cw laser operation has also been observed, though in an unpredictable manner, the temporal structure being determined in a complicated way by the electron bunch itself in the absence of an external gain modulation . This has made it difficult for experimentalists to exactly control the characteristics of the temporal structure . The best way to study the dynamics of the system is a controlled gain modulation in order to see how the system responds . Such experiments first require a relatively stable laser as the Super-Acv FEL. The gain modulation was performed by a periodical longitudinal detuning of the optical cavity due to a modulation of the radiofrequency f(t) = FRF + Of sin Qt . For SuperACO, FRF = 100 Mhz and a jump 0f = 300 Hz (corresponding to a longitudinal detuning of 50 lim) can stop the laser. Then, the modulation rate of the gain can be easily adjusted by varying Of from 0 to 300 Hz . The frequency d of the modulation is generally chosen close to the laser resonance frequency in the range 50 Hz-1 kHz. Fig. 1 is an example of such an experiment . without modulation the laser was continuous. By changing the i Permanent address: Lab. Optique Physique, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France. 0168-9002/91/$03 .50

0

gain modulation rate from 0 to 100%, several successive regimes are observed, which are summarized by the diagram at the bottom of fig. 1 . For a very low level of modulation, the laser is simply modulated at the Tl frequency (fig. la). For a slightly higher modulation the laser adopts a stable structure pulsed at Q/2 (fig . 1b), a regime which is known as period doubling (or 2T regime) in chaotic systems . The third regime (fig. lc) is a chaotic behaviour with a fluctuating intensity and a frequency not very well defined. Note that these different regimes were obtained with a modulation rate lower than 8% . For a higher modulation rate the laser exhibits a succession of relatively stable 2T and chaotic regimes

GAIN MODULATION

k-~~G~1nn~ ~nnnnnnnnnn~ nnn~ 5ms 0 VMS

0

LASER INTENSITY

bC 2TJ .J'

a I 0

~C a0s I 50%

--_

9 I 100%

0

700 Hz

Fig. 1. Macrotemporal structure of the Super-ACO FEL, for a modulation of the radiofrequency at 700 Hz. Records (a) to (g) show the evolution of the laser with respect to the gain modulation rate as summarized in the diagram.

1991 - Elsevier Science Publishers B.V . (North-Holland)

II . STORAGE RING EXPERIMENTS

38

M. Billardon / Chaotic behaviour

Fig. 2. Examples of higher subharmonics for the modulated Super-ACO FEL. (a) 4T regime, (b) 3T regime . illustrated by figs. Id-Ig. In all cases the transition between two successive regimes is very sharp and appears as a dump . This experiment was conducted at a frequency d2 = 700 Hz. Following the experimental conditions, or by changing the frequency S2, the laser can also exhibit oscillations at other subharmonics of S2 . Fig. 2 is an example of period quadrupling (4T) and period tripling (3T), but these regimes always appear through a sequence as for example 1T, 2T, 4T, chaos, 3T, chaos etc., when the gain modulation rate varies from 0 to 100% . This behaviour is very similar to the deterministic chaos observed for classical lasers [6-9] or other dynamic systems. In order to test this hypothesis we have made simulations with a simple theoretical model. As a first approximation, the main features of the macrotemporal structure can be described by the following model [10] :

of storage ring

FEL

term gives the gain reduction due to the effect of the induced energy spread . The last term, a sm2 (Sdt), represents an imposed modulation of the gain . Fig. 3 is an example of this theoretical simulation . Following the gain modulation the laser exhibits the successive regimes 1T, 2T, 4T, chaos, 3T and chaos. The change of the frequency or the initial conditions (the laser intensity and the energy spread at t = 0 when the modulation takes place) gives rise to other sequences. For these simulations the modulation was a pure sine . Because of the absence of noise and random function to our model, this chaotic behaviour can only be attributed to a deterministic chaos, that is to say the impossibility for the system to adopt a stable structure. This similarity between the simulations, the experimental results and also the chaotic dynamics observed in classical lasers [6-9], shows that the macrotemporal structure of a SRFEL has a deterministic evolution. The small differences between the simulations and the experiments can be explained by the real gain modulation resulting from the sinusoidal modulation of the radiofrequency, and by the actual initial conditions which cannot be imposed by the experimentalist, and probably by some residual fluctuations or modulations of the storage ring parameters. 12n,

~~'~~JIUUU v ~ . v

.

I

w'771~

dI I(g - p) +e 5 , ât 0 da 2 dt

2 TS(v2-aô)+al,

g= go exp(-k(a 2 - aô))(1 + a sm2 S2t) . The first equation describes the amplification of the laser intensity I. The parameters g, p and 0 are the gain, losses and transit time for each pass, respectively, and i s represents the spontaneous emission by an electron bunch. The second equation describes the evolution of the electron energy dispersion, a = DE/Eo, where Eo is the average energy of the electrons and 0 E the rms deviation. Without the FEL operating, the energy spread has an equilibrium value ao and for a system away from equilibrium, the difference (a2 - vâ ) evolves with a characteristic damping time Ts. The last term, al represents an additional energy spread that results from the interaction between the optical field and the electron bunch . The third equation describes the evolution of the optical gain . go is the maximum gain without the laser operation and the exponential

3T

c

4T 3T 2T ~b 1T

e --chaosrf

_

Gmax

9 1chaos -

a 0

h

I 50%

0

100%

Fig. 3. Simulation with the theoretical model (eq. (1)) for a gam modulation at 460 Hz, with a maximum gain-over-losses ratio equal to 3. For these calculations the laser was initially in a cw state and the records (a) to (h) present the laser state after a long time in order to eliminate the transient regime .

M. Billardon / Chaotic behaviour of storage ring FEL

(f>=

-~~

308 H z II

~ d

II ~L I~

_ Chaos (f)= __ 349 H z

.102

W

sec

350 Hz

39

that the average frequencies (308, 349, 350 Hz and 25 Hz) are harmonics and subharmonics of the 50 Hz main supply voltage. Similar situations were also observed on ACO (the first FEL at Orsay) . A great number of records exhibits the frequencies 350, 300, 150, 100 Hz and sometimes 75, 50 and 25 Hz . Then generally the natural macrotemporal structure is determined by a residual modulation of the storage ring supplies . The frequencies 75 and 25 Hz clearly indicate a 2T or 4T process and the stable or chaotic natural pulsed laser must be explained in terms of deterministic evolution. This behaviour has some consequences : 1) A stable cw laser requires an extremely stable storage ring without noise and without residual modulation . On this point, see also the effects of the coherent synchrotron oscillations discussed elsewhere [11]. 2) A modulated or Q-switched gain can give unpredictable results. A better understanding is necessary to enable efficient modulation and Q-switch methods . 3) Many interesting aspects of the deterministic chaos can be studied with a SRFEL since, as the classical lasers, they are relatively simple dynamic systems . Acknowledgements

0

51

VUV" I m sec

Fig. 4. Examples of "natural" macrotemporal structure observed on Super-Acv. In this article we only give a comparison concerning the macrotemporal structure. The chaotic regime can also be characterised by an attractor as made in ref. [1]. The similarity between the experimental and theoretical attractors is another proof of the deterministic chaos. Such a behaviour probably explains the "natural" macrotemporal structure of the FEL. Here the term "natural" is used to designate the properties of the system resulting from the unknown and uncontrollable gain modulation induced by the storage ring, in contrast with the case of a deliberate external gain modulation. Fig. 4 shows different examples of this structure for the Super-ACO FEL. Fig. 4a is a record of an approximately stable cw laser (fluctuations lower than a few percent can be obtained depending on the stability of the stored beam). Fig. 4b is an example of a stable pulsed laser. Fig. 4c exhibits a transition between a cw laser and a chaotic laser. Fig. 4d shows a particular structure with two frequencies (350 and 25 Hz). Simulations made with a random function instead of a sinusoidal modulation lead to an extremely noisy laser, without a very well defined frequency. Then a stable pulsed laser as to fig. 4b can be explained only with a stable periodical modulation of a parameter of the storage ring. From the records in fig. 4 it can be seen

The results discussed in this article were derived from several series of experiments conducted on the ACO and Super-ACO storage rings and have been supported by the LURE, the CNRS the CEA and the DRET (contract no . 85/179). References

[4]

[6] [7] [8] [9] [10] [11]

M. Billardon, Phys . Rev. Lett . 65 (1990) 713. M. Billardon, P. Elleaume, J M. Ortega, C. Bazin, M. Bergher, M. Velghe and Y. Petroff, Phys. Rev. Lett . 51 (1983) 1652 . M. Billardon, P. Elleaume, J.M . Ortega, C. Bazin, M. Bergher, M. Velghe, D.A .G . Deacon and Y. Petroff, IEEE J. Quantum Electron . QE-21 (1985) 805 . I.B . Drobyazko, G.N . Kultpanov, V.N . Litvtnenko, I.V. Pmayev, V.M . Popik, I.G. Silvestrov, A.N . Skrinsky, A.S. Sokolov and N.A . Vmokurov, in : Free Electron Lasers II, ed. Y. Petroff, SPIE vol 1133 (1989) 2. M.E. Couprie, C. Bazin, M. Billardon and M. Velghe, ibid ., p. 11 . C.O . Weiss and H. King, Opt. Commun . 44 (1982) 59. F.T . Arrecht, R. Meucci, G. Puccioni and J. Tredicce, Phys. Rev. Lett . 49 (1982) 1217 . D. Dangoisse, P. Glorieux and D. Hennequm, Phys . Rev. A36 (1987) 4775 . M.J . Feigenbaum, J. Stat . Phys . 19 (1978) 25 . P. Elleaume, J. Phys . 45 (1984) 997. M.E . Couprie, M. Billardon, M. Velghe and D. Jaroszynski, these Proceedings (12th Int . FEL Conf. Paris, France, 1990) Nucl. Instr. and Meth . A304 (1991) 58 . II . STORAGE RING EXPERIMENTS