Type-III intermittency in a four-level coherently pumped laser

We study a homogeneously broadened four-level model for a coherently pumped laser with pump and laser fields having crossed linear polarizations. For a parameter range of the type explored in the experiments by Tang et al. [Phys. Rev. A 44, R35 (1991)]the system exhibits a family of type-III-intermittency transitions to chaos in which the onset of intermittency is preceded by period-2, period-3, or period-4 states. We find similarities but also differences between the results of our theory and their experimental results.


I. INTRODUCTION
The occurrence of nonlinear dynamics and chaotic behavior are well-established  phenomena   that affect many systems belonging to different fields such as phys- ics, chemistry, or biology.Moreover, systems seemingly very different may exhibit some kind of universal behavior.Thus, it is well known that there are at least three universal routes from regular to chaotic motion: The Ruelle-Takens-Newhouse scenario, the Feigenbaum scenario or period-doubling route, and the Pomeau- Manneville scenario or intermittency route.The inter- mittency route is characterized by chaotic bursts ("turbulent" phases) that appear at random times, interrupt- ing the otherwise nearly regular oscillations ("laminar" phases).Three types of intermittency are known, which depend on the way the destabilization of the limit cycle associated with the regular oscillations takes place [1].
Lasers are among the most interesting physical systems for the study of the dynamic behavior of nonlinear dissi- pative systems and their transitions to chaos [2].Particu-  larly relevant in this respect are coherently pumped lasers, which have allowed the first experimental observa- tion [3] of a behavior remarkably similar to the predic- tions of the paradigmatic Lorenz-Haken model [4], in- cluding period-doubling transitions to chaos, and also the first observation in an optical system of a family of transi- tions to chaos by type-III intermittency [5] and, more re- cently, by type-I intermittency also [6].These observa- tions were all performed on the same coherently pumped NH3 far-infrared laser, operating in different domains of the control-parameter space.Sacher, Elsaser, and Gobel reported recently the first experimental observation of type-II intermittency in an optical system using a GaAs/Al Ga, "As semiconductor laser with external feedback [7].We focus in this paper in the modeling of laser dynamics of intermittency type III.
In almost all the dynamical models the laser field is considered as a scalar quantity coupled to a medium tran- sition between two nondegenerated levels whose popula- tion inverts an incoherent pumping mechanism.These models are therefore unable to describe two important physical aspects of the ammonia laser used in the experi- ments [3,5,6].On the one hand, the 81-pm laser transi- tion in NH3 is optically pumped with the P(13) line of the N30 laser via an adjacent transition sharing the upper level with the lasing transition.It is therefore coherently excited and associated effects such as Autler-Townes splitting or Raman two-photon processes may result in new dynamic features [8].On the other hand, the am- monia laser operated with pump and generated fields with crossed linear polarizations [5].This is also an im- portant aspect since it is known [9] that NH3 laser transi- tions with linear polarizations parallel to that of the pump laser exhibit dramatically different dynamics.To incorporate the vectorial character of the light field into the laser model one has to consider also the level degen- eracy related to the angular momentum of the molecular levels.
We developed recently a model that takes into account both the coherent optical pumping and the inAuence of light polarization on the laser dynamics [10].This model has proven its capacity to reproduce the Lorenz-like dy- namics of the NH3-laser experiments [11].Here we use this model in an attempt to reproduce the type-III inter- mittency observed on the same NH3 laser [5].In our nu- merical results we find qualitatively some of the main signatures of the dynamics observed by Tang, Pujol, and Weiss, but a complete understanding of the experimental behavior is not achieved.

II. MODEL
Figure 1 illustrates the J, =O~Jb=1~J, =O transi- tion scheme assumed for the homogeneously broadened gain medium of an optically pumped ring laser.It is the simplest level scheme that enables one to consider in- teraction with pump E2(z, t) and generated E&(z, t) fields composed by arbitrary superpositions of right and left circularly polarized components.
Both fields are con- sidered as uniform plane monochromatic waves propaga- E (z, t)= g e" A~"(t)expIi[k, z v-, "t -P"(t)]I/2+c.c. (j =1, 2), where p " represents the population of level j, pbb =-p++ =-p, NJ -= (p;; - pjj ) is a population inver- sion, and p, -is the slowly varying complex amplitude of the coherence associated with the transition i ~j (i, j =a, +, -, c),x;.=Re(p; ),and y, " =Im(p, ").Dissipa- tive processes are described by means of population relax- ation rates y;, coherence relaxation rates y;, and cavity where the unit vectors e -= +(e +ie )/v 2. The z axis is also chosen to be the quantization axis, so that only four levels are involved [the sublevel (Jb = I, m =0) does not couple to the fields], which we shall simply denote as a, +, -, and c.In the following we restrict to the case of M-degenerate levels (v=0, see Fig. 1) and fully resonant pump field (vz+ = v2 = Qz).The amplitudes A ~z and phases P~z of the pump beam are considered as constant in time control parameters.For the reference frequencies of the generated field we choose v, +=v& =co"being the closest empty cavity resonance frequency.The unknown amplitudes A II' and phases /II' of the generated beam are slowly varying functions of time.We assume that the pump beam drives the a-b transition only, and the gen- erated field the b-c transition only.The field-matter cou- pling is characterized by four real Rabi frequencies (see Fig. 1) defined as /3"=dz A2 /2A' and a"=d, A", /2A' with d (j =1,2) being molecular transition electric-dipole mo- ments.As in the experiment [5] we will restrict ourselves here to studying the case of pump and laser fields with crossed linear polarizations.

III. NUMERICAL RESULTS AND DISCUSSION
We have performed a numerical integration of Eqs. ( 2) using a seventh-to eighth-order Runge-Kutta method.
As mentioned in the Introduction, it is known [10,11] that our equations reveal the existence of periodic and chaotic regimes, with transitions to chaos occurring, for increasing pump field strength P, and for increasing cavi- ty detuning 6&, through an inverse sequence of period- doubling bifurcations as in the Lorenz-Haken model [4].A more detailed scanning of the parameter space has al- lowed us to find also type-III intermittencies close to periodic windows in the chaotic domain.It is interesting to note that in the Lorenz-Haken laser model intermit- tencies appear also close to periodic windows, but they are of type I [2,4] while in our model they are of type III.
Figure 2 shows the sequence of time dependences of the laser output that we obtain as the pump field intensity is increased from Fig. 2(a) to Fig. 2(e), for a ' NH3 gas pressure of 8 Pa and a fixed resonator tuning.At P/yi=0. 17the system is in a period-3 window [Fig. 2(c)].b) and 2(a)] or an intermittency road [Figs.2(d) and 2(e)].The time evolution shown in Fig. 2(d) has the characteristics of type-III intermittency.An inspection of this figure shows that during the laminar phases the intensity of one component of the period-3 pulses decreases while the in- tensity of the other two components grow.This behavior is time inverted in comparison with the one experimental- ly observed [5], where the largest peak of the period-3 pulses grows and the two smaller ones decrease.The laminar phases have a random time duration and are separated by chaotic bursts of fairly regular duration.As the pump strength P is further increased the average duration of the laminar phases decreases and finally the system reaches the chaotic state shown in Fig. 2(e).
The same behavior is also obtained with fixed pump in- tensity varying the resonator detuning as shown in Fig. 3, which corresponds to a gas pressure p = 6 Pa and a pump 3(a) to Fig. 3(c).The chaotic states preceding the period- 6 state [Fig. 3(a)] and following the intermittency pulsing state [Fig.3(c)] are not shown for brevity.
As in the experiment we have also found intermitten- cies preceded by period-2 and period-4 states, but not the ones preceded by a period-5 state which were also ob- served [5].For a gas pressure of 10 Pa and a cavity tuned to resonance, increasing the pump strength P the system exhibits an inverse period-doubling route to chaos [Fig.4(a)], with period-4 and period-2 windows in the chaotic domain.By slightly increasing P from the center of the period-4 window [Fig.4(b)] one observes a transition to a chaotic state (not reported) preceded by the intermittency pulsing shown in Fig. 4(c).In this intermittent regime the two larger peaks of the period-4 pulses increase dur- ing the laminar phases while the two smaller peaks de- crease.This is the type of behavior observed experimen- tally, however, while in the numerical results the larger peaks of the period-4 pulses are the first and the third; they are the first and the second in the experiment.By slightly decreasing P from the center of the period-4 win-
dow one again obtains a period-doubling route to chaos (not reported in Fig. 4).Similar results can be obtained with diQ'erent sets of parameters.Figure 5 shows the se- quence of time variations of the laser output for a gas pressure of 4 Pa and a fixed pump intensity, using the resonator tuning as control parameter.Note that the pulsing in this period-4 window [Fig.5(b)] is different from the one previously discussed.
Close to the period-2 window indicated in Fig. 4(a), de-  creasing or increasing P we again obtain transitions to chaos either through a period-doubling sequence or through the type-III-intermittency pulsing shown in Fig. 4(d), respectively.Starting from this intermittency pulsing and increasing the cavity detuning 6& from zero the system follows the sequence shown in Figs.4(e) and 4(f).e)].A further increase of b, ', leads to a new simplification of the dynamics of the laser, which now pulses periodically [Fig.4(f)].A sequence of this type was also found in the experiment, but with the inter- mittent state preceded by a chaotic state [5].In the nu- merical results we cannot find the chaotic state because the dynamic behavior of the resonantly pumped laser de- pends only on !5&! and not on the sign of the detuning, therefore we cannot further reduce 6; from zero, the value at which appears at the intermittency, in search of a more complex behavior.
In order to reinforce the evidence that the behavior predicted by our model represents type-III intermittency, we have analyzed the statistical distribution P(t) of the duration of the laminar phases corresponding to the in- termittent states shown in Figs.2(d), 4(d), and 5(c).In making this statistical analysis we have used data sets containing 950 consecutive laminar periods.Figures 2(d), 4(d), and 5(c) are samples of these large data sets.Ac- cording to [13],for type-III intermittency the fraction X of laminar phases lasting longer than to is given by X(t & to) = J dt P(t) 0 o-[ e/[exp(4@to ) -1 ] ] ' (3 where e is the bifurcation parameter.The dependence of N on to is completely characterized by the bifurcation pa- rameter e. Figure 6 shows that the distribution of the laminarphase durations predicted by our model exhibits the expected feature of the type-III intermittency: the points, calculated from the large data sets mentioned above, agree very well with the line obtained from Eq. (3) for E=0.060+3% [Fig.6(a)], E=O 027+4.%[Fig.6(b)], and @=0.019+3.5% [Fig.6(c)], that correspond to the inter- mittent states preceded by period-3, period-2, and period-4 states shown in Figs.2(d), 4(d), and 5(c), respec- tively.Notice that in Eq. (3) e and to are dimensionless quantities.Therefore the value of e obtained from the fitting of Eq. ( 3) to the numerical points strongly depends on the units used to normalize the time.As in the rest of the paper, for Fig. 6 we have measured it in units of (y~) '.If, as in the experiment [5], we measure the time 0.8- tor tuning is varied.Gas pressure is 4 Pa, g = 18 000y J, P=O.ly~, and y=5.2y~. (a) Period-8 state.  in ps the new values of the bifurcation parameter are, re- spectively, e' =0.326+3%%uo' , e' =0.184+4%%uo, and e'=0.051+3.5%%uo.The first two of these values are of the same order of magnitude as the corresponding ones ob- tained in [5] by fitting Eq. (3) to points calculated from experimental data.The distribution of the laminar-phase durations for the intermittent state preceded by a period-4 state was not measured in Ref. [5].It is probably dificult to get a better quantitative agreement between theory and experiment, taking into account that some ex- perimental parameters (e.g. , the resonator detuning) are not exactly known, and also that the real laser is a much more complex system than the one considered in our model.
IV. CONCLUSIONS AND OUTLOOK cases.We have not found the transition involving the period-5 state which appeared also in the experiment.
Numerically these transitions originate in a period-X window in the chaotic domain.This results in two dis- tinctive features of our results with respect to the experimental findings.First, our transitions to chaos occur over intervals of the control parameters narrower than the experimental ones.Second, our period-N state is not preceded generally by a period-1 state, as in the experi- ment, but by an inverse sequence of period-doubling bi- furcations leading to chaos.A closer fitting of theory and experiment would require both a lengthier exploration of the parameter space and the inclusion in the model of the inhomogeneous Doppler broadening, which is another important physical factor for the NH3 laser.
We have presented a theoretical study of the dynamics of type-III intermittency in a coherently pumped four- level laser, with pump and laser fields having crossed linear polarizations.Our model shows a family of transi- tions to chaos by type-III intermittencies preceded by period-N states (N =2, 3,4) of the same type as the ones observed experimentally, though time inverted in some ting along the optical (z) axis of the ring resonator 1050-2947/93/48(3)/2251(5)/$06.00 48 2251

FIG. 1 .
FIG. 1. Schematic representation of the level system and laser fields considered in this work.2P~represent the pump Rabi frequencies and 2athe generated ones.