HAL Id: jpa-00226919
https://hal.archives-ouvertes.fr/jpa-00226919
Submitted on 1 Jan 1987
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
CHAOS IN CO2 LASERS
P. Glorieux
To cite this version:
P. Glorieux. CHAOS IN CO2 LASERS. Journal de Physique Colloques, 1987, 48 (C7), pp.C7-433-
C7-438. �10.1051/jphyscol:19877103�. �jpa-00226919�
JOURNAL DE PHYSIQUE
Colloque C7, suppl6ment au n 0 1 2 , Tome 48, dbcembre 1987
CHAOS IN CO, LASERS
P. GLORIEUX
Laboratoire de Spectroscopie Hertzienne (associ6 au CNRS), Universit6 de Lille I , F-59655 Villeneuve-d'Ascq Cedex, France
ABSTRACT
Chaotic regimes have been observed experimentally in a C02 laser with modulated losses and in a C02 laser containing a saturable absorber. This chaos is a purely deterministic phenomenon and the transition from periodic to chaotic regimes follows well-known scenarios : the Feigenbeum cascade or Shilnikov-type chaos. Simple rate equation models provide a good description of most chaotic phenomena encountered in these C02 laser systems.
To understand the behavior of lasers o r to use them in different systems, it is interesting to consider first the simplest lasers, namely the single mode, single- line lasers. Even when they are pumped in a cw regime, the intensity they emit may present irregularities. This apparent disorder is not due to fluctuations or to noise-induced phenomena but it is rather a direct consequence of the nonlinear dyna- mics of the system itself. In the chaotic regimes which are described here, the lascr remains characterized by a set of dcterministic equations. Due to the non linear coupling of different variables such as for instance the electric field and the population inversion, the steady state (cw) regime becomes unstable and the laser emits irregular bursts of radiation although all the parameters determining its behavior (pumping rate, losses, cavity length
...
) are kept constant.CO laser are good candidates for the study of these irregular deterministic behaviors because gas lasers in which individual atoms or molecules interact with 2 a common radiation field are much easier to modelize than solid or liquid-.state lasers. Therefore the conditions are met for an efficient check of the many theore- tical predictions, and the strong interplay with theory is one of themotivations of this work. Moreover since CO lasers are extensively used in industry and laborato-
2 .
ries, their wide field of applications provides an additional impetus to understa2d some particuliarities of their behavior.
Chaotic phenomena have been observed in different laser systems which include lasers with modulated losses p - 6 2
,
lasers with an injected signal[7], lasers with feedback[8],lasers containing a saturable absorber191
and bidirectional ring lasers &0y.~ote that, in most conditions, these systems are stable but for some ran- ge of parameters, each of these systems present deterministic irregular behaviors of different kinds.To illustrate the chaotic phenomena which can occur in such C02 lasers, we have chosen two systems which have been extensively studied at the Laboratoire de Spectroscopie Hertzienne de Lille : (i) the laser with loss modulation for its simplicity and (ii) the laser containing a saturable a b ~ o r b e r for the variety of
its chaotic regimes.
The C02
- laser with internal modulation of los.ses or frequency
By inserting an elastooptic crystal inside the cavity of the laser, an effi- cient modulation of the cavity losses or frequency may be achieved. At very low modulation rates, the laser output intensity follows linearly the driving voltage V applied to the modulator. As this voltage is increased, the laser responds at a period equal to twice the modulation one T. If V is further increased, the period of the output intensity doub1.e~ again, and so on... until it reaches a domain in vhich any regularity is lost. Such a sequence of period doublines is also observed
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19877103
C7-434 JOURNAL
DE
PHYSIQUEin the evolution of other nonlinear systems, in numerical series and in the solu- tions of nonlinear differential equations. It is known as the "Fel enbaum cascade"
which is one of the most common scenarios of transition to chaos
&g .
A graphic display of that evolution towards chaos, may be provided by a perio- dic sampling of the laser output and by plotting the output of the sampler versus the amplitude of the driving voltage. By taking one sample at each period of the modulation, a.single value is obtained in the linear regime. In that region of linear response, the modulated intensity is proportional to the driving voltage, this gives a single line in the display. After the first period doubling, the perio- dicity of the intensity is twice that of the excitation and one usually obtains two different samples. In a domain of 2 " ~ periodicity, 2" different samples are obtai- ned. Eventually in the chaotic regime where the system presents no periodicity, all samples are different. Figure 1 gives an example of the bifurcation diagram (BD) obtained when the sampler output is displayed versus V. The subharmonic cascade, the periodic regimes and the peculiar statistics inside the chaotic domain are clearly visible. The BD1s are extremely convenient to study the transition to chaos and the evolution inside the chaotic domain since they visualize in one diagram the whole evolution of the laser system.TheBDobtained usingthisstroboscopic method
presents strong similarities with that of the Logistic map, in which limits of the series x = 1
-
u x2 are plotted for different values of r~ PI]. Among the common f8Qkures are tge period-doubling cascade and the inverse cascade, i.e. the evolution from 2 " ~ "noisy" signals to 4T, 2T and T "noisy" signals when entering the chaotic domain. However, there remain significant differences as far as the periodic windows are concerned. For instance, in the experimental BD the 3T regime appears much wider and at a position different from what is obtained in the logistic map. The origin of that difference is cleared out when the BDs obtained for increa- sing and decreasing-Vs are compared. While the corresponding BDs are obviously iden- tical in the logistic map, there exists a region in which different solutions are obtained in the BD of the laser for the same modulation amplitude. This is the so- called "generalized multistability", in which different laser regimes either perio- dic or chaotic can coexist (3,6).Figme
1 :B.i6uhcation d i a g m
( a )0 6 t h e .tWm
w L t h6teyuency rnoobktion when t h e moduhtion amptitude
b e h i e n Ma co&oL
pcuameXa ( b loh -.the CogDfic map xn+,
= 1 - p x 2n
Numerical simulatio,ns of the C02 laser with loss modulation have been performed in a rate equation model in which a time dependent loss term has been introduced in the field equation
In these equations, G is ;he gain, I and D are the intensity and the population inversion respectively Y and 2 K the corresponding damping rates, 6 is the cavi- ty detuning in units of the polarization relaxation rate. To account for modulation, K is taken as
K = K ( 1
+
m sin (2T f t+ a )
) where m is the modulation index.Generalized bistability together with other properties of the experimental BDs are well given bythese simple models. This is illustrated on Figure 2 which dis- plays examples of BDs obtained for increasing and decreasing values of m.
TIME
TIME
Figwre
2 :Calculated bidutcation
d i a g mdoh .the Labm luith loon modulation.
The conthal
pa/uunetmt h e modueatian voLtage which
.LA ( a )incheaned oh ( b ] decheabed. Note t h e change i n t h e width 06 t h e 3T-pehiodic hegion which c U 6 . i ~ t h e e&tence
0 6g e n U z e d b.LAtabXLty.
T k i nohimLLecLtion &a & & d ~ t h e dynamicat de6omuncLtion
0 6t h e bi6uhccLtion
diaghamdue t o t h e non zeho sweep rcate.
For some values of the control parameter (V in the experiments, m in the simu- lations), abrupt qualitative changes are observed in the chaotic regions. These phenomena which are called "crises" correspond to either a sudden expansion (inter- nal crisis) of the attractor or to a destruction of the original attractor together with its basin of attraction (boundary crisis). Both kinds of crises were observed in the C02 laser with modulated parameters and in its numerical simulation [ 6 ] . A boundary crlsis is observed for instance when the chaotic regime is followed by a 3T-periodic regime with no apparent continuity between them. The strong similarities between the experimental results and the simulation back up the idea that the sim- ple rate equation model given above provides excellent predictions of the dynamical behavior of the CO laser.
2
Other chaotic phenomena have also been experimentally observed in this system and compared with numerical simulation.
They are mostly related to the competition of noise and chaos, the dynamical deformation of BDs or the calculation of the dimension of the strange attractor
JOURNAL DE PHYSIQUE
representing the laser evolution in its phase state. Details of this work may be found elsewhere
[la.
The C02 laser contining a saturable absorber.
-
When a saturable absorber is placed inside the cavity of a standard C02 laser, the dynamics of this laser are strongly modified because of the coupling of laser radiation with the passive (and saturable) molecules. More precisely, optical bis- tability (OB) and passive Q-switching (PQS) are often observed. The former is due to the coexistence of several steady states of the system for the same operating conditions whi1.e the latter corresponds to spontaneous self-pulsing. Note that PQS and OB can simultaneously occur for some C02 lasers containing saturable absorbers
(LSA) [13-161
.
Because of the occurence of self-pulsing in the LSA, this system is a good candidate for the observation of optical chaos. However there were only few theoretical works in which this possibility was discussed and chaos was predic- ted only in ranges of parameter far away from those which can he explored in exi.s- ting C02 LSA'S [17, 187.
A peculiar scenario of transition towards chaos has recently been observed in a C02,1aser operating on the P 32 line and containing CH 1 as a saturable absorber, described in details elsewhere P 3 7 . Here the contro? parameter is the cavity detuning. Note that three different hyper-fine components of the CH I absorption line fall within the cavity mode width E9_1 Hence it is impossible
20
assign a single value to the detuning between the laser and the molecular lines. However the laser detuning appears to strongly influence the PQS lineshapes. As it is varied inside the mode profile, the laser intensity evolves differently depending of the absorber pressure p . For high values of j5(P >
500 mTorr) the laser is blocked off by the absorber, while at low pressures ( p<
10 mTorr), it operates in a continuous way.For intermediate pressures, PQS may be observed. As the cavity is swept from one side of the mode to the other, the PQS lineshape changes. Figure 3 gives an example of such an evolution at a CH 1 pressure of 26 mTorr whereavery rich scenario was observed. For central t~ning,~ttie PUS appears as a sinusoidal oscillation whose period doubles as the detuning is increased. After a region of completely irregular beha- vior, a new regime appears with a large peak followed by two smaller ones (denoted as T I 2), "chaos" again and another periodic regime going out of chaos through an inverHe cascade. In this regime, the initial large peak is followed by only one smaller peak. Therefore, the corresponding behaviors are called 2 T I 1 and TI1.
The first part of the scenario closely follows the Feigenbaum period-doubling cas- cade while the second part is completely different. An extensive investigation of the LSA in that region of the parameter space indicates that a succession of other regimes T12, T13
...
together with the corresponding period douhlings can be ohser- ved. Moreover there exist some range of parameters in which the LSA seems to "hesi- tate" between adjacent regimes e.g T 1 2 and T13'
The succession of Tln regimes, with period-doublingand"hesitation" are strong indications of a Shilnikov-type chaos
Ed.
A reconstruction of the attractor from a time series of the intensity shows that in the PQS regime, there exists a point in the phase space which acts a s an attractor point in one di-rection and as an unstable focus in two other directions of the phase space. This situation is just the symme- tric of that ofShilnikovD?.
Thus by time-reversal Shilnikov's results may be extended to the LSA under consideration here, i.e. chaos may by obtained if the divergence rate of the focus exceeds the attracting rate in the other direction.All these facts support the attribution of the chaotic regime observed for large laser detunings ( ?, 20 MHz) to "inverse Shilnikov chaos
".
Recent numerical simulations made o n a simple model of the LSA have already shown a good qualitative agreement with the experimental results and confirm that chaos observed in the LSA may be of the purely deterministic "inverse Shilnikov"
type and are definitely not noise-induced effects
Figwe 3 : Pannive a-nwLtcking
pubw
doh di66ehent detuningn06
Rhe
h e n .
cavLty. COTl a m
fine : I O P 3 2 , CH31 p h u - n w e 26 mTom. The .time hange explohed i n Rhw e xecolrdingnLb 5500
m.
CONCLUSION
Two examples of deterministic chaos in CO laser systems have been presented here. Different kinds of chaos and transition $0 chaos have been obtained, depending not only on the system but also on its range of operation. Simple rate equation models usually provide a good description of the dynamical properties of this class of laser systems.
It should be noticed that C02 lasers provide a very interesting tool for the investigation of such effects because (i) these systems are relatively well defined and known, (ii) large signal-to noise ratios are easily obtained and (iii) the time scale of the experiments in the microsecond region allows an easy separation of the transient response and also to scan wide ranges of parameters in a quasi adiabatic way.
The very rich phenomenology encountered in the C02 laser with loss modulation and in the LSA as wee1 as in other CO laser systems not presented here, makes it likely that the characterization of tgeir dynamical behavior will provide a large amount of interesting nonlinear phenomena.
The financial support of La R6gion Nord
-
Pas-de-Calais, D.R.E.T. are grate- fully acknowledged. The author also thanks Michel OUHAYOUN from la Soci6t6 Anonyme de TQlbcommunications for a generous loan of equipment and Claude BAESENS and Paul MANDEL for enlightening discussions.JOURNAL DE PHYSIQUE
REFERENCES
1 1 1 F.T. ARECCHI, R. MEUCCI, G.P. PUCCIONI and J.R. TREDICCE, Phys. Rev. Lett.
2,
1217 (1982).
[a
J.K. TREDICCE, N.B. ABRAHAM, G.P. PUCCIONI and F.T. ARECCHI, Opt. Comm.55,
131 (1985).
[a
G.P. PUCCIONI, A. POGGI, W. GADOMSKI, J.R. TREDICCE and F.T. ARECCHI, Phys. Rev.Lett.
2 ,
339 (1985).[g
T. MIDAVAINE, D. DANGOISSE and P. GLORIEUX, Phys. Rev. Lett.2 ,
1989 (1985).[a
J.R. TREDICCE, F.T. ARECCHI, G.P. PUCCIONI, A. POGGI and N. GADOMSKI, Phys.Rev.
m,
2073 (1986).[a
D. DANGOISSE, P. GLORIEUX and D. HENNEQUIN, Phys. Rev. Lett.57,
2657 (1986).[a
A. POLITI, G.L. OPPO and R. BADII, is Optical Instabilities,R.W. BOYD, M.G. RAYMER and L.M. NARDUCCI, Eds, Cambridge Univ. Press, Cambridge, (1986)
[a
F.T. ARECCHI, R. MEUCCI and W. GADOMSKI, Phys. Rev. Lett.3 ,
2205 (1987).[ g
A. BEKKALI, F. PAPOFF, D. DANGOISSE, P. GLORIEUX. Proc. Int. Conf. Laser Instab., I1 Ciocco, (1987).b g
F.T. AKECCHI in Optical Instabilities,R.W. BOYD, M.G. RAYMER and L.M. NARDUCCI, Eds. Cambridge Univ. Press, Cambridge, (1986).
ba
P. BERGE, Y. POMEAU and C. VIDAL, L'ordre dans le chaos, Herman, Paris (1984).ba
D. HENNEQUIN, Thsse Universite de Lille I.ba
A. JACQUES and P. GLORIEUX, Opt. Comm.40,
455 (1982).E g
E. ARIMONDO and E. MENCHI, Appl. Phys.m,
55 (1985).bg
J. HEPPNER, Z. SOLAJIC and G. MERKLE, Appl. Phys.m,
77 (1984)b a
E. ARIMONDO, D. HENNEQUIN, E. MENCHI and B. ZAMBON, Proc. Int. Conf. Laser Instab., I1 Ciocco (1987).ba
F. MRUGALA and P. PEPLOWSKI, Z. Phys.m,
359 (1980).ba
J.C. ANTORANZ, J. GEA and M.G. VELARDE, Phys. Rev. Lett.5,
1985 (1981) M. VELARDE and J.C. ANTORANZ, Prog. Theor. Phys.66,
717 (1981)Dg
E. ARIMONDO and P. GLORIEUX, Phys. Rev.g ,
1067 (1979).Bg
C. BAESENS, Private communication.Dg
L.P. SHILNIKOV, Sov. Math. Dokl.6,
163 (1965) and8,
54 (1967), Maths USSR SG, 10 (I), 91 (1970).-
