Dynamics of coherently pumped lasers with linearly polarized pump and generated fields

The inhuence of light polarization on the dynamics of an optically pumped single-mode laser with a homogeneously broadened four-level medium is theoretically investigated in detail. Pump and laser fields with either parallel or crossed linear polarizations are considered, as are typical in far-infraredlaser experiments. Numerical simulations reveal dramatically different dynamic behaviors for these two polarization configurations. The analysis of the model equations allows us to find the physical origin of both behaviors. In particular, the crossed-polarization configuration is shown to be effective in decoupling the pump and laser fields, thus allowing for the appearance of Lorenz-type dynamics.


I. INTRODUCTION
Extensive research over the last fifteen years has shown that lasers are among the most versatile physical systems to study a variety of temporal phenomena characteristic of nonlinear dissipative systems such as, for example, the onset of spontaneous pulsations in the output intensity that may be regular, quasiperiodic, or chaotic [1].In par- ticular, optically pumped molecular lasers (OPML) in ad- dition to their practical interest as a source of stable coherent radiation in different regions of mid-and far- infrared [2], are very interesting systems from the viewpoint of nonlinear dynamics.In fact, OPML have allowed the experimental realization of a rich variety of dynamical behaviors that include (i) full amplitude periodic spiking associated with sustained relaxation os- cillations, directly from the onset of laser threshold [3,4]; (ii) high-frequency periodic pulsing at higher pump powers, associated with pump-induced Rabi sideband os- cillation [4]; (iii) a behavior remarkably similar [5 -10] to the predictions of the paradigmatic Lorenz-Haken model   for a plane-wave single-mode homogeneously broadened two-level laser [ll -13]; and (iv) transitions to chaos by type-I [14] and type-III [15]intermittencies.
From the theoretical point of view OPML have been extensively studied in the framework of single-mode three-level models [16 -21] with the aim to capture the a priori most salient feature of these lasers, namely, the coherent interaction of pump and lasing emission fields, which distinguishes them from ordinary incoherently pumped two-level lasers and can lead to characteristic dynamic behavior as discussed by Harrison and coworkers [19,20].Indeed, for specific domains of parame- ter values three-level laser models with either homogene- ous broadening [4] or inhomogeneous (Doppler) broaden- ing [22 -26] have shown reasonably good qualitative agreement [4] and even a surprising similarity [22 -26]  with corresponding experimental findings.These coincidences suggest that these models take into account physical factors playing an important role in OPML dynamics.However, at this moment they can be only con- sidered as a first step towards the understanding of the behavior of OPML since they do not take into account additional physical factors that are also present in the real laser, namely, longitudinal and transverse spatial dependence of pump and laser fields, M-level degeneracy of the radiatively coupled levels, and polarizations of the two laser fields.In principle any of these factors could have an influence in the laser dynamics that is worth in- vestigating.
To take into account simultaneously all the physical factors above would represent a formidable task practi- cally unaffordable with commonly available computers.
In this paper we analyze the influence of light polariza- tion and level degeneracy using a simple homogeneously broadened four-level OPML model.
One can expect to find a clear influence of these factors based on several previous experimental observations.
The possibility of pump-intensity-driven polarization control has been demonstrated by Petukov, Tochitsky,  1. Schematic representation of the level system and laser fields considered in this work.2Prepresent the pump Rabi frequencies and 2a + -the generated ones.28 is the Zeeman splitting which is taken to be zero along this paper.and Churakov in an optically pumped CO2 laser oscillating in the 4.3-pm region with a polarization-insensitive cavity [27].Weiss and co-workers observed on the 81.5pm [6 -8] and the 153-pm [7,10] lines of the NH~laser, whose linear polarization was orthogonal to the pump field polarization, a dynamic behavior remarkably diFerent from that of the 374-pm NH3 line [28,29] whose polarization was parallel to that of the pump field.In or- der to mimic the experimental conditions of Weiss and co-workers, pump and laser fields with fixed linear polar- izations are considered through this paper.The interest- ing case of an OPML with a polarization-insensitive cavi- ty in which the polarization state of the laser light can change in addition to the amplitude and phase changes that are usually investigated remains to be addressed in a future work.Very recently Bielawski, Derozier, and Glorieux [30] have shown theoretically and experimental- ly that the laser light polarization also plays an important role on the dynamics of a class of lasers different from the OPML considered here.They studied an Nd-doped opti- cal fiber laser which is an incoherently pumped class-8 laser.
The paper is organized as follows.In Sec.II we refer to a homogeneously broadened four-level model proposed previously by some of us [31],which allows us to consider arbitrary polarizations for pump and laser fields and the effect of level degeneracy in the simplest configuration: level scheme J, =0~J b = 1~J , =0 (see Fig. 1).Here we study this model in detail both analytically and numeri- cally for the case of linear polarizations for pump and laser fields and compare its predictions with those of two- and three-level models.The case of crossed polarizations is studied in Sec.III.For this configuration the model predicts Lorenz-like dynamics.For parallel polarizations the model describes a markedly different dynamics which is studied in Sec.IV.These predictions for the two polar- ization configurations are consistent with previous exper- imental observations [5 -10,28,29].A closer fitting of theory and experiment requires the inclusion in the mod- el of the Doppler broadening, the real degeneracy of lev- els (much larger than the one considered here), and probably other physical factors mentioned above and is there- fore beyond the aim of this paper.Section V is devoted to conclusions and outlook.

II. HOMOGENEOUSLY BROADENED
FOUR-LEVEL MODEL FOR AN OPML WITH LINEARLY POLARIZED FIELDS E", (z, t) = -, ' A", (t)exp[i[k, z Ht --P", (t)]], 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 = l, mJ =0) does not couple to the fields], which we shall simply denote as a, +, -, and c.In the fo11owing we restrict ourselves to the case of M-degenerated levels (s =0) and fully resonant pump field vz+ =vz -= Qz (see Fig. 1).The amplitudes A'z and phases Piz of the pump beam are considered as timeindependent control parameters.For the reference fre- quencies of the generated fields we choose v&+ =v& = -cu" co, being the closest empty cavity resonance frequency.
The unknown amplitudes A", (r) and phases p", (t) of the generated beam are slowly varying functions of time.We assume that the pump beam (lasing beam) drives the a b-(b c) transition -only.Therefore the field-matter coupling is characterized by four real Rabi frequencies (see Fig. 1) defined as P"=dz A~z/2iil and a"=di Aii'/2itt with d&.
As is typical in far-infrared-laser experiments we only consider in this paper pump and laser fields with fixed linear polarizations either parallel or crossed.Without loss of generality we take the generated field to be e~p o- larized, thus we have a+(t)=a (t):-a(t) and Pi+(t) =Pi (t) =P(t).For the pump beam we have Pz+ =Pz =Pz and either P =-P-: P for the Parallel case, or p+ = -p: -p for the orthogonal case.Making use of these relations in Eqs. ( 2), (3), and (4) of Ref. [31],and as- suming equal losses for both components of the generated To take into account the vectorial character of the laser fields in an OPML one has to consider also the an- gular momentum quantum number of the radiatively coupled molecular states.Usually the total angular momentum quantum number J of the three coupled lev- els is high (e.g. , J=7 in Ref. [9]; J=25 in Ref. [27]) and therefore the total number N = 3(2J + 1) of magnetic sublevels involved is too large to be considered in a sim- ple laser model.Instead, we study here a homogeneously broadened ring OPML that operates with the transition scheme J, =0~J b = 1~J , =0 shown in Fig.
1 [31].It is the simplest level scheme that enables one to consider in- teraction with pump Ez(z, t) and generated E,(z, t) fields composed by arbitrary superpositions of right and left circularly polarized components.
:-m, -QI is the cavity detuning and g= -Q&d&Xo/2coA represents the unsaturated gain parameter, No being the number of active molecules per unit volume.Note that Eq. (3.5) corresponds to the symmetrized expression p,b =(p, ++p, )/2, since (2.1) is verified.
As indicated above, Eqs. (3)are valid for both parallel and crossed linear polarizations.Next we wi11 elaborate on each one of these cases separately.

B. Stationary solutions
Equations (5) provide two difFerent classes of stationary solutions.One of them corresponds to the nonlasing state with a=0.A linear stability analysis (LSA) of this trivial solution shows that by increasing the pump inten- sity P it destablizes through a pitchfork bifurcation occurring at [33] where Nb, -= (pbb -p").Note that Nb, involves only half the total population of the upper lasing levels (p+++ p --). Relation (4.2) is of remarkable relevance since it means that in the orthogonal configuration there is no coherent coupling between the pump field and the laser field, which occurs in general via the two-photon coherence p".Therefore, no Raman processes occur at all, nor pump-induced Rabi splitting of the laser gain.There remains only an incoherent coupling between pump and laser fields through the populations of levels + andand the coherence x+ between them.This absence of Raman processes can be interpreted [32] as a destructive quantum interference between the two Raman channels a -++ -+c and a -+ -~c.On the other hand, relation (4.3) implies that the refractive index at the pump fre- quency is not modified for any cavity detuning, at vari- ance of what happens in the simpler three-level OPML model [21].
A. Model P =P =B/A, (6) In this section we will consider the interaction between a linearly polarized pump beam (e"polarized) and an or- thogonally polarized generated field (e polarized).For this case one has P+ = -P -=P.By introducing this con- In our case y', "'=0.004 and I, h, =0.285.The variation of the pump intensity threshold for cw laser emission P»" with the relaxation rate I as a function of the cavity de- tuning 6&, is shown in Fig. 2, where one can observe the rapid increase of Pi"as I approaches l, b, .
The stationary lasing solution that bifurcates from the nonlasing state when P ) P", is given by Since P is a real quantity Eq. ( 6) can only be fulfilled if the three following conditions are met: 20- consequence of the lack of coherent pumping effects in the orthogonal configuration.For the sake of brevity the stationary value of the rest of variables is not given here.
[Only the stationary value of x+ is given below in Eq. ( 15).] Figure 3 shows the variation of the steady laser intensi- ty a as a function of the pump intensity P, given by Eq. ( 9), for the case hi =0 and several values of I in Fig. 3(a), and for I =0.5 and different values of the detuning 5& in Fig. 3(b).In both figures it is clearly apparent the non- linear variation of the laser intensity with P at variance of what happens for an ordinary incoherently pumped two-level laser.Although the optical pumping does not introduce coherent effects in this orthogonal configuration [see Eq. ( 4.2) and subsequent discussion], there remains a nonlinear dependence due to pump field absorption saturation.
On the other hand, and at difference with the behavior of the usual three-level OPML, now there is not an upper limit for the pump in- and A and B given by Eqs.(7).
The pulling effect corresponding to this lasing solution is given by P = -scb.;/(a+ 1), F&G. 2. Variation of the first laser threshold Pzi"with the re- laxation rate I as a function of the cavity detuning 6& for g/y&=3642.5, K/yj =2, and crossed polarizations for pump and generated fields.Different curves from top to bottom correspond to 6&/y& = 7, 5, and 0. Other parameters are given in the text.FIG. 3. Variation of the steady laser intensity a as a func- tion of the pump intensity P~for crossed polarizations.For {a) 6&=0 and different curves from top to bottom correspond to I /y&= 1, 0.75, 0.5, and 0.3.For (b) I /yj =0.5 and curves from top to bottom correspond to 6&/@~=0, 10, and 20.(Other pa- rameters are as in Fig.    e a sence of Rabi splitting of the laser A LSA of thee stationary solution 9 h h h ldf ~~o or soft-mode laser in bihty threshold or s d instabilities (insta-  or second laser threshold (P " /f3", ) is shown ' F' n in ig. 4 as a fun b obs ree main feature i") tends to infinity for I =1 a, h, ig.c .ii nstabiiities appear i.e. , the "bad cav't " d  er instability threshold is fixed point and the other corresponds to the dynamical solutions.For the range of parameters explored in Fig. 5(b) the LSA indicates that the fixed point is always stable.Therefore, the generalized bistability occurs now for all the dynamic regimes shown in Fig. 5(b).As can be seen in this Fig. 5 there are periodic and chaotic regimes, with transitions to chaos occurring for increasing pump field strength P and for increasing cavity detuning b, ; through an inverse sequence of period-doubling bifurca- tions as in the Lorenz-Haken model [13].
Representative chaotic time series of the laser intensity are shown in Fig. 6 for three decreasing values of the coherence relaxation rate I =2, 0.5, and 0.295, approach- ing I, h" together with a phase portrait on the field- polarization plane and the corresponding intensity map.
As seen the attractor appears with the butterfly shape characteristic of the Lorenz-Haken model, and a (t) shows the typical "spiralings" of this model [12].This type of motion produces the "cusp"-shaped laser intensi- ty map shown in Figs. 6(c), 6(f), and 6(i) [12,8].
It is worth pointing out that all these features found in our model are well-known characteristics of the Lorenz- Haken model [12].Note also that the similarity between both models increases as I gets closer to I, h, .For in- stance, for I =2 the "cusp" map of Fig. 6(c) presents a well-defined double "cusp" which does not appear in the Lorenz model for pump parameters close to the instabili- ty threshold.As I decreases [Figs.6(f) and 6(i)] this dou- ble "cusp" tends to disappear, approaching the Lorenz behavior.It is interesting to comment that in some inten- sity maps obtained from the experimental recordings [8,9] a double "cusp" also appears.But it must be stressed that the origin of this second "cusp" is not the same as in our case.In the experiments, the second "cusp" was identified with the presence of an extra intermediate-height pulse at the beginning of each spiral [9].On the contrary, in our case the first pulse at the be- ginning of each spiral seems to be too short as compared to what one should expect from the Lorenz model.The origin of the similarities and differences between our model and the Lorenz-Haken model is further discussed in the next subsection.
Then Eqs. (5) Although neither the stationary solution given in Eq. ( 9) nor the LSA [11] behave as in a Lorenz-Haken laser (see Figs. 3 and 4, in particular, the stationary solution is nonlinear with respect to the pump parameter P ), in Sec. which is plotted in Fig. 7 as a function of p for I =yl, where Do -= (pbbp"), and use has been made of the simplified notations x(r)-: x+ (r), y (r) -= yb, (t).
Equations (13.1) -( 13.3) would be isomorphic to those of the complex Lorenz-Haken laser model [11]if the last term in the right-hand side of Eq. ( 13.3), i.e. , [ -(Iyl)x ], were null.This is the only term that cou- ples Eqs.(13.1) - (13.3) and Eqs. (13. 4) -(13.7).Since the coherence decay rate I may be written as I =(y+++y )/2+yah=y~i+y~", where the first con- tribution yll is related to the decay of molecular states and y h accounts for phase interrupting collisions, it fol- lows that when I =y~~i (i.e. , the lowest value I can take), the two-level variables (E,P, D) become unaffected by the evolution of the rest of variables (x,y, p",pbb ), although the former influence the latter as can be seen in Eqs.(13).
Moreover, since the optical pumping acts only directly on the variables (x,y, p",pbb ), the effective variables (E,P, D) are blind to this pump, and are only determined by the initial population inversion Do.Since Do is negative in the absence of an external incoherent pumping mecha- nism, the laser will not lase independently of the values of the pump strength il and the gain parameter g.This re- sult is in agreement with condition (8.2), which gives the minimum value of I that permits lasing operation, since in the particular case we are discussing (I =yii), I is below I, h, .
Nevertheless, if one looks at the stationary population inversion corresponding to the lasing transition FIG. 8. Flow diagram showing schematically the coupling [see Eqs.(13)] between two-level laser variables (E,P, D), the rest of variables (x, p», y, p") and the pump field P. 0 0 ~ll r "(p"pbb)+ 1+ one observes that there is a net population inversion.This inversion would give rise to emission for an appropriate value of the gain parameter g in a normal two- level laser, but in our case the laser remains o6' for any value of this parameter.This phenomenon of population inversion without lasing can be clearly interpreted in the context of the dressed-atom formalism [32,36].Inversion between bare states does not guarantee inversion between states dressed by the pump beam.In Ref.
[32] it is shown that population inversion between the dressed states cou- pled to the generated field is only possible if I ) l,h, .For I & I, h, the population is trapped in dressed states not coupled to the laser field [32].
Up to now, we have shown that our crossed- polarization OPML model exactly reduces to the Lorenz-Haken model in the particular case in which I takes its minimum value (I'=yii) and that in this situa- tion our laser will not emit if the optical pumping of the adjacent transitions a~b is the only external pumping mechanism.
For I ) I,h"however, our model does not reduce to the Lorenz-Haken model, but rather to a modified one with the extra term [ -(Iyl)x] in Eq. (13.3), which connects the dynamics of all the variables of the system.Nevertheless, it clearly appears in Fig. 8 that the role of the variable x is basically that of an incoherent pumping mechanism on the lasing transition.This pumping, how- ever, is coupled with the dynamics of the whole system.Figure 9 shows a typical time evolution of x in the chaot- ic domain.It consists of a nonzero dc contribution xd, and a superimposed modulation.
For Fig. 9 xd, = -0.0307,which is close to its (unstable) stationary value x= -0.0318 given by Eq. ( 15) below.The varia- tion of x around xd, is less than 6%%uo.When one considers periodic behaviors, the value xd, is closer to x than in the chaotic regime.
All these general features of the variable x permit a first approximate treatment which reduces the complete model of Eqs. ( 13) to an effective Lorenz-Haken model.
This can be done by substituting the actual instantaneous value of x by its stationary value x, which is given by -0.033 c2I, +c&I, +c0=0, where A, B, C, and H are given as in Eqs. ( 7) and (10).
With this approximation the model reduces to Eqs.
(13.1) and (13.2), and Eq. ( 13.3), which may be written as D=yi Do D+4 E-P*+PE*) D0 is the effective incoherent pump parameter of the equivalent Lorenz-Haken model, which is nonlinear with respect to the optical pump intensity p, as can be seen in Fig. 10, where the normalized pump parameter r =Do gla (as defined for the Lorenz model) is plotted for different cavity relaxation rates ~.Note that r satu- rates for p ~1 due to pump absorption saturation.
In order to test the degree of accuracy of this approxi- FIG. 9. Time evolution of the variable x in the chaotic domain for I /y~=0.5 and P/y~=0. 09.xd, denotes the aver- age value of this variable, and x the (unstable) stationary value which is marked by an arrow.
Substitution of Eq. ( 9) into Eq.( 18) enables one to ob- tain the pump strength p =pz"d at the threshold for in- stabilities.The results of this analysis are summarized in Fig. 11.A comparison of Figs. 4 and 11 shows that the approx roximate treatment reproduces fairly well the dependence of the ratio between threshold pump intensities on the cavity losses [Figs. 4(c) and 11(c)] for I =0.3.For increas- ing I one observes a monotonic variation of the curves in Figs.11(a) and 11(c) at variance with what happens in Figs.4(a) and 4(c) although the corresponding curves have qualitatively similar shapes.The largest discrepancies between the exact and the approximated results appear when one compares Figs.4(b) and 11(b).Note the different vertical scales and also that corresponding curves have different shapes in both figures.This confirms that the truncation is a good approximation for values of I close to I,h" in agreement with the numerical results of Sec.III C.
A better approximate treatment should also include the modulation of the variable x and not only its station- ry value.This can be done along the lines described in ary va ue.~~~e fF Ref. [25] by introducing different effective values yl and D0.We have tried also this additional approximate treatment and studied the corresponding LSA.In spite of a greatly increased numerical complexity the agree- ment with the exact treatment has not improved significantly.In this section we will consider the interaction between pump and generated fields with parallel linear polarizations.For this case one has p = p: -p, which substitut- ed into Eq. (2.2) leads to P++(t) =P (t) =Pbb(t), y+ (t)=0 .On the other hand making use of Eqs. ( 2) and (19) and of the equations of evolution for D + ( t ) =p++(t) p(t) and -y+ (t), we obtain FICx.10.Normalized.pump parameter r vs pump tntenssty p for 5'/y = 0 g /y2 = 3642.5, I /y, = 0.5, and di6'erent values of i yj. - (20) Finally, substitution of Eqs.(19)  allows us to write the equations which govern the dynam- ics of our system in the parallel configuration as &b, = r ~~[(pbb -p'") -iiib, ]+6m, b -2pyb.

15-
Let us point out that at variance with the case of or- thogonal polarizations, in the present case Raman processes and other coherent coupling processes between pump and generated fields governed by the two-photon coherence p"are present.Also since x,b&0, there are now dispersive effects at the pump frequency.These fundamental differences between both cases make them dramatically different.
Owing to the complexity of system (21) it is impossible to obtain general analytical results.Moreover, in the few particular cases in which this can be done the obtained expressions are not easy to handle, so we will study only numerically the dynamics of our system.The phase diagrams of Fig. 12 show in the parameter plane (p, 6;) the different emission regimes of the laser intensity a (t), obtained by hard-mode excitation.All the dynamic regimes shown here coexist with a steady lasing state, obtained by soft-mode excitation.A comparison of Figs. 5 and 12 shows that the laser is much more stable operating with parallel polarizations than with the orthogonal configuration [31].In fact, while the instability pump threshold is p=0.0743 at b, ', =0 for chaotic emis- sion in Fig. 5(b), it is increased to p=1.54 at El=0 f' or regular period-1 pulsing (P ') in Fig. 12(b).A global feature of Figs.er, our results indicate that the laser with crossed polar- izations is less sensitive than the one with parallel polarization to the stabilizing effect of detuning.Note, in fact, that the range of 5; values considered in Fig. 5 is about one order of magnitude larger then in Fig. 12.Note also in Fig. 12 that the instability pump threshold increases by increasing I .This again indicates that the coherence be- tween the upper degenerate levels has a strong inAuence on the laser dynamics.This point is made clearer in Fig. 13, which corresponds to the dynamic behavior observed for the minimum value of I (=yII) for which the parallel configuration laser is most unstable.While with crossed polarizations the system would not lase for I =yII, it exhibits now a chaotic pulsing for P=0.099, b, ;=0 [Fig.13(a)].Keeping b, ; fixed and in- creasing p the chaotic attractor transforms into a period- ic one.The period-4 pulsing regime is shown in Fig. 13(c).The chaotic attractor found here does not have the butterfly shape [Fig.13(b)] characteristic of the Lorenz model but closely resembles the ones found in three-level OPML models [21].

C. Discussion
The dynamics just reported is quite different from that found for the orthogonal case.On the other hand, a cer- tain resemblance exists between the results shown in Sec.
IV B and those presented in Ref. [21].This similarity is quite surprising since as was indicated in Sec.IIA the model of Ref. [21] corresponds to pump and generated fields with equal circular polarization.
Since the greatest similarity between the parallel case and the circular case [21] occurs when I'=yII let us make this particularization into Eqs. (21).One obtains In Eqs. (22)  Q bb =pbb+&+ -and g)b g bb p This closed system is isomorphic to that of Refs.[19] and [21] [38].This fact, which is in agreement with the numerical results of Sec.IVB, confirms the surprising conclusion that in the limit condition I = y II, the behavior of the three-level OPML model (circular polar- ization) is identical to that of the parallel-polarization OPML four-level model reported in this section.Thus, all the analytical and numerical results given in Refs.[19] and [21] apply also to the present case.Moreover, the numerical results indicate that the effect of increasing I is to make the behavior of the four-level system progres- sively different from that of the three-level model.

V. CONCLUSIONS AND OUTI.OOK
In this paper we have analyzed in detail the inhuence of light polarization on the dynamics of an optically pumped single-mode laser with a homogeneously broadened four-level medium.As typical in far-infrared- laser experiments we have considered pump and laser fields with either parallel or crossed linear polarizations.
Dramatically different dynamic behaviors have been found for these two polarization configurations, which suggest that light polarization and level degeneracy are key ingredients of the OPML dynamics.The dynamics strongly depends on the newly introduced relaxation rate I of the coherence between the two degenerate levels shared by the pump and lasing transitions.When I takes its lowest possible value (I = y ~~), our model for crossed polarizations exactly reduces to the Lorenz-Haken one, while for parallel polarization it is isomorphic with the coherently pumped three-level mod- el.This result enables one to get some insight into the influence the Doppler broadening (not considered in this paper) should have in our four-level model, since the cases of two-level and three-level lasers with Doppler broadening have been previously investigated in Refs.[39] and [22 -26], respectively.
In the configuration with crossed polarizations it has been shown that there is no coherent coupling between pump and laser fields for any set of parameter values, at variance with what happens for the three-level model.This allows for the persistence of Lorenz-type behavior over a wide domain of laser parameters.Distinctive features have also been observed, however, that are due to incoherent optical pumping effects.These features in- clude the possibility to have population inversion without lasing; a nonlinear dependence of the steady laser intensi- ty with the pump intensity; the dependence on various laser parameters of the ratio between pump intensities for second and first laser thresholds; or the structure of in- tensity return maps.
In the parallel-polarization configuration there is coherent coupling between the pump and laser fields and the OPML exhibits a rich and varied dynamics, including periodic and chaotic regimes, very different from the Lorenz-Haken ones, and already extensively studied in the case I = y ~~ [19,21].For this configuration an in- crease of I, by means of phase interrupting collisions for instance, is found to have a strong stabilizing inhuence on the OPML.
In the light of the above conclusions let us finally em- phasize that a completely acceptable theoretical descrip- tion of the dynamic behavior of real OPMI, is not yet available.By way of example let us consider the experi- mental observation by Weiss and co-workers [6] of Lorenz-type dynamics on the 81.5-pm crossed- polarization line of the NH3 laser.The relaxation rates considered in the present work are appropriate for that line, and our four-level model effectively predicts Lorenz-type dynamics in the case of crossed polarizations (although with some restrictions with respect to the values of I ).The real laser [6], however, is a Dopplerbroadened system, so that Doppler effects should be taken into account.This represents a long numerical task which will be undertaken in subsequent work.Thus, at present we do not know exactly the result of including these effects in the model with crossed polarizations, but since this configuration is effective in decoupling pump and laser fields, one can speculate that Doppler effects should not destroy the Lorenz-like behavior already present in the homogeneous system (on the contrary, they should make the predicted behavior even closer to that observed).
On the other hand, Lorenz-type behavior appears also in the present parallel-polarization four-level model, if we take I =y~~a nd include Doppler effects, since we then re- cover the three-level Doppler model of Refs. [22 -26].Thus if the above conjecture is true and this type of behavior occurs for both crossed and parallel polariza- tions one would come to the conclusion that light polar- ization does not strongly inhuence the laser dynamics.Then one should explain the different behavior observed on the 374-pm line of the NH3 laser [28,29], which is a Doppler-broadened system with parallel polarizations, on the basis only of the different parameter settings (gain, re- laxation rates, wave number, pump detuning, etc.) for these two lines.On the contrary, if Doppler effects des- troy the Lorenz-type dynamics in the four-level model with crossed polarizations, then the four-level model would be in contradiction with experiments (predicts the correct behavior for the wrong polarization configuration).Then one should again improve the mod- el considering, e.g. , higher J values (as occurs in the ex- periments).In our opinion all these are very challenging questions that deserve future work.

F 2 FIG. 4 .
FIG.4.R Ratio between threshold um mode laser instabilities ~~es o pump intensities for soft- i i ies I »d and for cw laser e crossed-polarization fi emission P&" in the ' n con guration, as a fun

FIG. 7 .
FIG. 7. Population inversion X&& as a function of pump in- tensity P' for I = rii.Other parameters as in Fig. 4.

FIG. 13 .
FIG.13.Intensity pulsing e (t) and two-dimensional (a, Imp, +) projections of the dynamic attractor found with hard-mode exci- tation in the parallel polarizations model for I =yII and 5;=0.The pump field strength in the chaotic regime (a) and (b) is p= 0.099yl; in the period-4 pulsing regime (c) and (d) it is p= 0. Syl (other parameters as in Fig.2).