Types I and II intermittencies in a cascade laser model

16 October

1995

PHYSICS LETTERS A

ELSEVIER

Physics Letters A 206

(I 995) 359-364

Types I and II intermittencies in a cascade laser model Germ&n J. de Valck-eel a3132, Eugenio Roldh a,2, Victor Espinosa b-3, Ramon Vilaseca c,4 ’ Departament h Departament ’ Delwrtament

d’6ptica.

de Fisica Aplicadu,

de Fkica

i Enginyeria

Unicersitut

de Vahcia,

Unitiersitat

Polit~cnica

Nuclear.

Universitat

Dr. Moliner de Valhcia,

Politknicu

50, 46100 Burjussot,

de Catalunyu.

Received 22 May 1995; accepted for publication Communicated by C.R. Doering

Spain

Cami de Veru. 46071 VulZncitr, Spain Colom I I, 08222 Terraw,

Spain

28 July 1995

Abstract We report on types I and II intermittencies found in a cascade laser model. type of intermittency, which involves the coexistence of both types of laminar Type II intermittency has special characteristics such as its origin at a frequency this torus bifurcates to a three-torus, further giving rise to a type II intermittent laminar phases.

Intermittency

is among

the universal

mechanisms

that produce chaos from a periodic

orbit in a continuous way [l]. The intermittency scenario is characterized by the existence of regular (laminar) phases along the evolution of a system’s variable, interrupted by bursts of irregular behaviour, the distribution of the laminar phases duration verifying certain universal statistical properties [I]. In a classical paper Pomeau and Manneville [2] established the existence of three types of intermit-

I On leave of absence from Departament de Fkica Aplicada, Universitat Politkcnica de Valtncia, Cami de Vera, 46071 Valkcia, Spain. ’ E-mail: [email protected]. ’ E-mail: [email protected]. 4 E-mail: [email protected] 03759601/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SD1 037%9601(95)00628-l

A continuous transition from one to another phases within the same time series, is found. locked two-torus. When frequency unlocked like behaviour with new features during the

tency depending on the way a periodic orbit can lose its stability. The classification is made according to the way the eigenvalues of the differentiable PoincarC map cross the unit circle at the point of instability. Types I and III are associated with one real eigenvalue (+ 1, tangent bifurcation, and - 1, subcritical period doubling bifurcation, respectively). Type II is associated with a pair of complex conjugate eigenvalues (related to a subcritical Hopf bifurcation) inrroducing an additional incommensurate frequency with respect to that of the unstable periodic orbit at the onset of instability. In all three cases a reinjection mechanism must exist in order to approach the trajectory to the unstable periodic orbit after every irregular burst. Since the paper of Pomeau and Manneville the catalogue of intermittencies has become largely broadened: type X (which extends type I to cover

cases which exhibit intermittency near hysteretic transitions) [3]; type V (for discontinuous maps) [4]; and on-off (which is induced by an irregular variation of a control parameter around the bifurcation point of a steady state) [5]. Regarding the three classical types of intermittency, there are a number of observations both experimental and from physical models of types I (for experimental observations, see Ref. [6], for theoretical predictions, see Ref. [7]) and III (for experimental observations, see Ref. [8], for theoretical predictions, see Ref. [9]). The situation is quite different with respect to type II intermittency. There is an experimental observation by Huang and Kim [lo] in an electronic oscillator. Also Sacher et al. [I 11 found an her-ted version of type II intermittency in a feedback semiconductor laser. Nevertheless, it is not clear whether inverted type II intermittency can exist since it involves a spiral falling down of the oscillations towards the fixed point of the Poincark map, in apparent contradiction with the repulsive character of the periodic orbit assumed by the theory [l]. As far as we know the only clear theoretical prediction of type II intermittency is due to Richetti et al. [ 121 in a periodically driven nonlinear oscillator model. In this Letter we report a new theoretical prediction of the type II intermittency route to chaos. This behaviour is accompanied by outstanding peculiarities such as its quasiperiodic origin and its coexistence with type I intermittency, both types of laminar phases alternating in a same time series for a certain parameter set. We have found such a behaviour in a cascade laser model [ 131. In this laser model two fields of different frequencies, coupled to the two consecutive transitions of a three-level ladder medium, can be generated inside the laser cavity. When the cavity modes are resonant with the two transitions the model equations read [I41 di= -Dj+r,-4E,Pi+2f;(,y)EkPi, F$= -Pj+EiDj+(-l)jf,(fi)eE,, d=

-Q+W’,/+diW’,~

k, = a( P, -E,),

version of the j-transition (j = I (lower), 2 (upper)), respectively, Q is the two-photon coherence and f,(x) = x and f2( x) = l/.x. c stands for the (common) relaxation constant of the fields normalized to the relaxation rate of the material variables (absence of dephasing collisions is assumed). The transitions are incoherently pumped at a normalized rate r,. Finally x represents the lower to upper transitions gain ratio determined by the transition electric dipole moments and frequencies ( x > I means that the lower transition is structurally more favoured for light amplification than the upper one and vice versa). In Ref. [14] the dynamics of system (1) has been studied for x < 1 and r, = 0 (which corresponds to pumping the laser only in its upper level, which is the more reasonable situation from the experimental point of view), revealing the existence of periodic and chaotic attractors always connected by the quasiperiodic scenario [15]. In this situation the upper transition is clearly more favoured than the lower one. Here we also study the case with r, = 0, and consider x > 1 in order to favour the lower transition and increase the degree of competition between both fields. In the following we take x = 3, leaving r2 and u as the main control parameters. Eqs. (1) have three possible stationary solutions apart from the’ trivial one which is stable up to r2 = 1. Two of them are single-field solutions (only one field is amplified) and the other one is the two-field solution. A linear stability analysis shows that the stationary solution in which only the lower field E, is emitted is always unstable whilst the one in which only the upper mode E2 is emitted is stable 5 up to r2 = 1.5. The two-field solution appears at this pump value, and is destabilized further increasing pumping at r2 = rHB where a Hopf bifurcation appears. The dependence of this threshold on the cavity losses u is shown in Fig. I (dashed line). We describe next the dynamic behaviour appearing in this case. In Fig. 1 the solid lines mark the approximate limits of the different emission regimes (see caption). We have observed that the transition from P, to CH occurs via type I intermittency whilst the transition

(1)

where j, k = 1, 2 (j # k). ( Ej, Pj, 0,) are the field amplitude, induced polarization, and population in-

~ngeneral single-mode

expressionsof the linear stability analysis solutions are in Ref. [ 141.

of the

G.J. de ValcLircel

r2

35

et al. /Physics

Letters

I t

A 206 (I 99.5) 359-364

2.45

-

2.40

-

361

25

Fig.

I.

Domains of dynamic behaviour of the cascade laser model

for x = 3 on the cavity losses-pump

2.25

parameter plane. Dashed

line: result of the linear stability analysis (Hopf bifurcation).

2.30

2.35

2.40

2.45

ST:

stationary emission. P, and P,,: periodic behaviours associated to

0.12

two different attractors. T: torus. CH: chaos.

P(L) 0.06

from P,, to CH (through T) occurs via type II intermittency. As an example of a type I scenario Fig. 2 shows the time traces corresponding to the field E2 (similar traces are displayed by the field E,) as the pumping rz is increased for a fixed value of the cavity losses (T= 6. The periodic state (Fig. 2a) destabilizes at r2 = 9.587 and laminar phases alternate with irregular bursts (Fig. 2b), which become more and more frequent (Fig. 2c), until fully developed chaos is eventually reached. In order to characterize the laminar phases, Fig. 3a shows the PoincarC map constructed from the

0.00

Ez

200

150

100

I

3

2

-9

3

50

0

-a

-7

-6

-5

-4

13

Fig. 3. (a) Poincark map corresponding to one of the laminar

2

phases in Fig. 2c. (b) Histogram of the laminar phase length for

1

rz = 9.6. (c) Dependence

0

phases on the standard parameter E.

of the mean duration of the laminar

3 E,

2 1

maxima of one field amplitude (z,) in a laminar phase. It has the parabolic shape characteristic of type I intermittency, which can always be reduced to the standard form [l]

0 4

E2 3 2 1 ”

t 0

1 I

I “’ I

50

X

150

100

n+ I =&+X,+X;,

(2)

T

Fig. 2. Time evolution of the field amplitude E, increasing values of r2: r2 = 9.58 type

I

intermittency; and IO.50 (c).

for (T=6

and

(a): periodic state; 9.90 (b):

where x, is a shifted and scaled version of z,, and E is a universal control parameter (intermittency appears for E > 0). Fig. 3b shows the histogram of the

362

G.J. de Vakcircel et al. / Phyks

3

E.?

2 1

4

E

2

3

2

1 0

’

I 150

100

50

0

7

Fig.

4. Evolution

of

r2 = IO.74 (a) (type intermittencies, II

I

the

for CT= 7 and (type 1and type II reman numerals), and 13.IO (c) (type field

intermittency),

denoted by

amplitude

E,

12.45 (b)

intermittency).

duration of laminar phases 6 for r2 = 9.6. As expected it exhibits two large peaks at minimum and maximum durations. From histograms like this, the dependence of the mean duration of a laminar phase (1) on E can be found, as shown in Fig. 3~). The theory predicts the scaling law (I) a E-~ with (Y= l/2 (this value of cy is obtained assuming a white reinjection mechanism [l]). In our case we obtain a = 0.71 + 0.05. Presumably the origin of this discrepancy lies in the particular reinjection mechanism, which is not quite uniform in our case, as also occurs in other places [ 121. Here we also point out that scaling laws with cy # l/2 and even of logarithmic form have been recently determined in type I intermittency [ 161. Let us consider now a cavity losses value u = 7. In this case the bifurcation diagram for increasing r2 is much richer, presenting type II intermittent behaviour. Fig. 4 shows the evolution of one field amplitude for three increasing values of the pump parameter. As before, the periodic attractor P, undergoes type I intermittency (Fig. 4a) but, different from the case CT= 6, increasing pumping does not lead to chaos but lo the appearance of an intermittent be-

Letters A 206 (1995) 359-364

haviour with two kinds of laminar phases (Fig. 4b), one of them being type I and the other type II. Further increasing pump makes the type I laminar phases disappear and only type II laminar phases remain (Fig. 4~). In order to characterize these new laminar phases Fig. 5a shows the sequence of maxima of the field E2 along one of such phases. For the sake of clarity the maxima appear connected by three lines. Two features are evident: (i> there is a triple structure and (ii) the behaviour of the maxima in each of these structures shows an oscillation of increasing amplitude. Fig. 5b shows the Poincart! map of each structure (it is actually a third return map) and three spirals are observed. This is the clearest signature of type II intermittency [I]. The origin of the triple structure lies in the presence of a period-three attractor that is found for slightly higher pumping. Nevertheless this attractor is a special case of the quasiperiodic attractor T in Fig. 1 (which surrounds the strictly periodic attractor P,,), which becomes frequency locked (I : 3) for this parameter set. A slight increase of the pumping leads the system to the periodic attractor P,, (see Fig. 61, and hence these intermittencies do not end in chaos.

3.20 2, 3.10

3.00

\ 2.90

8 3240

3160

3.2 %+3 31

3320

7

(b)

k

1

3

3.0

2

2.9 2.9

3.0

3.1

3.2

%a

’ The duration by fitting

of any particular

successive points of

was previously

constructed

laminar

phase was calculated

the Poincti map to EQ.(2). which

from three selected laminar

phases.

Fig. 5. (a) Evolution

of the maxima

of the field amplitude

the case of Fig. 4c; each maximum

z,, is connected

maximum

Poincare

z,,,+ 3. (b) Corresponding

map.

E, for

by a line with

G.J. de Vulccircel et ul. / Physics Letters A 206 (1995) 359-364

The transition to chaos via type II intermittency occurs for a higher value of the pump parameter. The periodic attractor P,, is stable up to r2 = 19 (upper bound of P,, in Fig. 1) where it becomes the torus T of Fig. I. In this case the torus is again frequency locked but now with a ratio (1 : 5) (Fig. 6). This locked torus is stable in a small domain in r2 and is destabilized via type II intermittency. Thus the Poincarc map of the laminar phases shows five defined spirals similar to those in Fig. 5b. These intermittencies do eventually lead to fully developed chaos. The strategy adopted when characterizing type I laminar phases (see footnote 5) is not applicable now due to the numerical difficulties of constructing the standard spiral map associated to type II intermittency [I]. Alternatively the duration of a type II laminar phase has been assigned according to how many consecutive field maxima do not exceed a given threshold 8 (e.g. Z = 3.22, according to Fig. 5a). Nevertheless, this automatic criterion presents a large error, specially due to the small difference between the values of the field maxima along a laminar phase and those reached along the irregular bursts (see Fig. 4~). Anyway coarse grain estimations show the existence of a long tail in the laminar-phases length distribution, with only one maximum for short durations, as expected for type II intermittcncy [ 11. Finally, and for the sake of completeness, we report on the observation of a different type of intermittent like behaviour. It occurs for larger values of cavity losses u, for which the locking of the torus is not produced. In these cases the bifurcation diagram is more complex since first the torus T is replaced by a three-torus (through the appearance of a third incommensurate frequency), and finally this three-torus destabilizes giving rise to quite complicated behaviours reminiscent of type II intermittency.

T Fig. 7. Time evolution of the maxima of the field amplitude

This behaviour comprises the alternation between “regular” phases in which the field maxima exhibit oscillations which grow in time, and bursts of complete irregular behaviour. Nevertheless, the quasiperiodic nature of the underlying motion gives rise to laminar phases with an unusual appearance. Fig. 7 shows the field maxima evolution corresponding to one of these “quasi periodic type II intennittency” laminar phases computed at a cavity losses value u = 8.2. Different from Fig. 5a, no substructures can be identified. This special type of behaviour deserves special attention, and we leave its characterization to future work. In conclusion, we have shown that a cascade laser model can exhibit intermittent behaviour of types I and II. Moreover, it is possible to observe intermittent behaviour in which laminar phases of both types alternate in the same time series. Interestingly enough a behaviour closely related to type II intermittency has been observed from the destabilization of threetori. This work has been supported by the Direction General de Investigation Cientifica y TCcnica (Spain) through project no. PB-92-0600-C03-02.

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