Lyapunov Potential Description for Laser Dynamics

arXiv:cond-mat/9905423v1 28 May 1999 As 26.f 25.h 26.i 42.55.Px 42.60.Mi, 42.55.Ah, 42.65.Sf, PACs: pdb ab[,]adidpnetyb ae [4–7]. Haken by independently devel- fact and theory, This [2,3] laser Lamb semiclassical by equations. the oped inside Maxwell in field introduced the of semi- was electromagnetic through number a the cavity allowing the thus the of where one, treatment system than classical larger a much cavity. is is dynam- laser photons laser the description laser the quantum-mechanical within However, of fully interaction a matter-radiation understanding on of complete based mech- is pumping A a ics pro- iii) that and cavity feedback, anism. optical necessary an the ii) cavity, vides the inside prop- radiation agating electromagnetic the amplifying dy- of capable a by as laser then, the start, of We system. features namical main systems. the Lyapunov reviewing construct laser briefly to some is density work for probability this potentials of known aim a The to function. be respect can with averages and performed be potential condi- also Lyapunov the the can general by in distribution some governed probability under stationary i.e. and the terms, tions, dynamics, noise non–deterministic of of presence case landscape. potential the the trajectories in In particles stationary test and of movements transient as un- the intuitive the cases, an of allows some potential derstanding In determin- Lyapunov points. a as fixed of well those existence of the as stability function of the Lyapunov the ing determination as the system easy of (deterministic) an dynamical extrema on a allow of lies they points functions fixed Lyapunov that of fact usefulness trajec- the The along decreases func- [1]. that Lyapunov tories potential) (called Lyapunov function or a tion construct to possible ae a he ai nrdet:i anmedium gain a i) ingredients: basic three has laser A occasionally is it systems, non-mechanical for Even yais ntesohsi aei sfudta h hs of phase analyze the to that potential found is the it use state. case steady we stochastic the lasers the In A class dynamics. For potential. rmtesainr tt spooe n hce gis nu against p checked the and giving proposed is relation state semi-empirical stationary a the and from pre discussed considered also not g were is the oscillati which to power, relaxation corresponding emitted terms the spontaneously the expression of this in period included the relating expression edsrb h yaia eairo ohcasAadcasB class and A class both of behavior dynamical the describe We o ls aes h oeta bandi ai nteabs the in valid is obtained potential the lasers, B class For .INTRODUCTION I. ypnvPtnilDsrpinfrLsrDynamics Laser for Description Potential Lyapunov ( ‡ nttt eierae eEtdo vnao IEE,U (IMEDEA, Avanzados Estudios Mediterráneo de Instituto ) aaiaMayol Catalina ( † eatmn eFıia nvria elsIlsBalears Illes les de F´ısica, Universitat de Departament ) e ae http://www.imedea.uib.es Page: Web -77 am eMloc,Spain. Mallorca, de Palma E-07071 † , ‡ Raúl Toral , 1 † , innie aebe xesvl sdfrsemiconductor for used extensively emis- lasers. been spontaneous have equations, for noise, Rate accounting sion [8]. terms lasers) stochastic semiconductor inside with of number number) case (carriers the photons inversion in population (or the field and temporal cavity the electric the for the equations of can rate change description deriving simpler by A obtained be equations. atoms Bloch two-level the of through collection modeled a is elec- as medium quantum-mechanically the amplifying the of and nature field, quantum-mechanical tromagnetic the- laser the semiclassical ignores The ory medium. two-level a interact- the with field from ing single-mode constructed a was for equations dynamics Maxwell-Bloch laser for model This ubr n h lcrcfil.Aohreape fclass of CO examples are Another lasers B rate field. two electric carriers the just (or and inversion by number) population described atomic then the for are equations example, lasers an semiconductor adiabatically which are be of [10] can lasers, faster B it Class and much variables, eliminated. state two other steady po- the the the than lasers, towards B decays class For inversion satisfactory larization population nonlin- polarization. a material the three the field, for of and electric required set the of a is description therefore equations rates clas- and decay differential a order, the ear use all same to have the first lasers the of C po- were class material [9] scheme: and al. sification carriers et photons, Arecchi the larization. of rate decay the fcasAlsr,lk y ae,aesuidi [12,13]. in studied are laser, dye a like describe properties lasers, Some to A enough system. class the is of of field evolution electric dynamical and Both the the eliminated, for field. adiabatically equation electric be the the can than variables faster material polar- much material and decay inversion ization population lasers A class in ‡ n lui .Mirasso R. Claudio and ieettpso aescnb lsie codn to according classified be can lasers of types Different n oteptnili on.W have We found. is potential the to ons i auainadtema au fthe of value mean the and saturation ain ro fterlxto siltosfar oscillations relaxation the of expression this eriod of validity The viously. eia simulations. merical neo os.I hscs,ageneral a case, this In noise. of ence h lcrcfil rfswt iein time with drifts field electric the ohdtriitcadstochastic and deterministic both 2 aesi em faLyapunov a of terms in lasers aesadsldsaelsr 1] Finally, [11]. lasers state solid and lasers IB-CSIC) † In this paper we interpret the dynamics of both class A A non-potential flow, on the other hand, is one for which and class B lasers by using a Lyapunov potential. the splitting (2), satisfying (3), admits only the trivial The paper is organized as follows. In Section II we solution V (x) = constant, vi(x)= fi(x). present a brief review of the relation of Lyapunov poten- Since the above (sufficient) conditions for a potential tials to the dynamical equations and the splitting of those flow lead to dV/dt 0, one concludes that V (x) (when it into conservative and dissipative parts. We consider the satisfies the additional≤ condition of being bounded from example of class A lasers. In this case, the Lyapunov below) is a Lyapunov potential for the dynamical system. potential gives an intuitive understanding of the dynam- In this case, one can get an intuitive understanding of the ics observed in the numerical simulations. In the pres- dynamics: the fixed points are given by the extrema of ence of noise, the probability density function obtained V (x) and the trajectories relax asymptotically towards from the potential allows the calculation of stationary the surface of minima of V (x). This decay is produced mean values of interest as, for example, the mean value by the only effect of the terms containing the matrix S in of the number of photons. We will show that the mean Eq. (2), since the dynamics induced by vi conserves the value of the phase of the electric field in the steady state potential, and vi(x) represents the residual dynamics on varies linearly with time only when noise is present, in this minima surface. A particular case of potential flow is a phenomenon reminiscent of the noise–sustained flows. given when vi(x) can also be derived from the potential, In Section III, the dynamics of rate equations for class namely: B lasers is presented in terms of the intensity and the carriers number (we will restrict ourselves to the semi- N dxi ∂V conductor laser). In this case we have found a potential = Dij (4) dt − ∂x which helps to analyze the corresponding dynamics in the j=1 j X absence of noise. By using the conservative part of the equations, one can obtain an expression for the period of where the matrix D(x)= S(x)+ A(x), splits into a posi- the oscillations in the transient regime following the laser tive definite symmetric matrix, S, and an antisymmetric switch-on. This expression extends the one obtained in one, A. In this case, the residual dynamics also ceases a simpler case by an identification of the laser dynam- after the surface of minima of V (x) has been reached. ics with a Toda oscillator in [14]. Here, we have added We now describe the effect of noise on the dynamics of in the expression for the period the corresponding mod- the above systems. The stochastic equations (considered ifications for the gain saturation term and spontaneous in the Itôsense) are: emission noise. Finally, in section IV, we summarize the N main results obtained. dx i = f (x)+ g (x)ξ (t) (5) dt i ij j j=1 II. POTENTIALS AND LYAPUNOV X FUNCTIONS: CLASS A LASERS where gij (x) are given functions and ξj (t) are white noise: Gaussian random processes of zero mean and correla- The evolution of a system (dynamical flow) can be clas- tions: sified into different categories according to the relation ′ ′ of the Lyapunov potential to the actual equations of mo- ξi(t)ξj (t ) =2ǫδij δ(t t ) (6) h i − tion [15,16]. We first consider a deterministic dynamical flow in which the real variables (x1,...,xN ) x satisfy ǫ is the intensity of the noise. the general evolution equations: ≡ In the presence of noise terms, it is not adequate to talk about fixed points of the dynamics, but rather dxi = fi(x), i =1,...,N (1) consider instead the maxima of the probability density dt function P (x,t), which satisfies the multivariate Fokker- In the so–called potential flow, there exists a non– Planck equation [17,18] whose general solution is un- constant function V (x) (the potential) in terms of which known. When the deterministic part of (5) is a potential the above equations can be written as: flow, however, a closed form for the stationary distribution Pst(x) can be given in terms of the potential V (x) N dx ∂V if the following (sufficient) conditions are satisfied: i = S + v (2) dt − ij ∂x i j=1 j fluctuation–dissipation X 1. The condition, relating the symmetric matrix S to the noise matrix g: where S(x) is a symmetric and positive definite matrix, x and vi( ) satisfy the orthogonality condition: N S = g g , S = g gT (7) N ij ik jk · ∂V k=1 vi =0. (3) X ∂x i=1 i X

2 2. Sij satisfies: Where a = Γ/(κβ) and b = 1/β. ξ1(t) and ξ2(t) are white noise terms with zero mean and correlations given N ∂Sij by equation (6) with ǫ = ∆/κ. =0, i (8) In the deterministic case (ǫ = 0), these dynamical ∂x ∀ j=1 j X equations constitute a potential flow of the form (4) where the potential V (x) is [7] This condition is satisfied, for instance, for a con- 1 2 2 2 2 stant matrix S. V (x1, x2)= [x + x a ln(b + x + x )] (15) 2 1 2 − 1 2

3. vi is divergence free: and the matrix D(x) (split into symmetric and antisymmetric parts) is: N ∂v i = 0 (9) 1 0 0 α ∂x D = S + A = + − . (16) i=1 i 0 1 α 0 X A simpler expression for the potential is given in [6] and this third condition is automatically satisfied for poten- [17] valid for the case in which the gain term is expanded tial flows of the form (4) with a constant matrix A. in Taylor series. Under those circumstances, the stationary probability According to our discussion above, the fixed points of density function is: the deterministic dynamics are the extrema of the potential V (x): for a>b there is a maximum at (x1, x2)=0 x x −1 V ( ) (corresponding to the laser in the off state) and a line of Pst( )= Z exp (10) 2 2 − ǫ minima given by x1 +x2 = a b (see Fig. 1). The asymptotic stable situation, then,− is that the laser switches to 2 2 2 where Z is a normalization constant. Graham [19] has the on state with an intensity I E = x1 +x2 = a b. shown that if conditions 2 and 3 are not satisfied, then For a

∗ ′ ′ is only the ordinary phase diﬀusion around the circum- ζ(t)ζ (t ) = 4∆δ(t t ) , (12) 2 2 h i − ference x1 + x2 = a b that represents the set of all possible deterministic− equilibrium states [12]. Therefore, where ∆ measures the strength of the noise. for α = 0 the real and imaginary parts of E oscillate By writing the complex variable E as E = x1 +ix2 and not only6 in the transient dynamics but also in the steady introducing a new dimensionless time such that t κt, → state, and while the frequency of the oscillations still de- the evolution equations become: pends on α (as well as ǫ), their amplitude depends on the noise strength ǫ. a x˙ = 1 (x αx )+ ξ (t) (13) We can understand these aforementioned features of 1 b + x2 + x2 − 1 − 2 1 1 2 the noisy dynamics using the deterministic Lyapunov po- a tential V (x , x ). Since conditions 1-3 above are satis- x˙ = 1 (αx + x )+ ξ (t) (14) 1 2 2 b + x2 + x2 − 1 2 2 ﬁed, the stationary probability distribution is given by 1 2

3 (10) with V (x1, x2) given by (15). By changing vari- and, by using the distribution (17), one obtains the ables to intensity and phase, we find that the probability stochastic frequency shift: density functions for I and φ are independent functions, a 2ǫ Pst(φ)=1/(2π) is a constant and, exp( b/2ǫ)(b/2ǫ) φ˙ st = α − (21) h i − Γ a +1, b −1 −I/(2ǫ) a/(2ǫ) 2ǫ 2ǫ Pst(I)= Z e (b + I) (17) Notice that this average rotation speed is zero in the case where the normalization constant is Z = of no detuning (α = 0) or for the deterministic dynamics a b 2ǫ +1 2ǫ a b (2ǫ) e Γ 2ǫ +1, 2ǫ and Γ(x, y) is the incomplete (ǫ = 0) and that, due to the minus sign, the rotation Gamma function. From this expression, we see that, speed is opposite to that of the deterministic transient independently of the value for ǫ, Pst(I) has its max- dynamics when starting from the off state. These results ima at the deterministic stationary value Im = a b. are in excellent agreement with numerical simulations of Starting from a given initial condition corresponding,− for the rate equations in the presence of noise (see Fig. 2). instance, to the laser in the off state, the intensity fluctuates around a mean value that increases monotonically with time. In the stationary state, the intensity fluctu- III. CLASS B LASERS ates around the deterministic value I = a b but, since m − the distribution (17) is not symmetric around Im, the The dynamics of a typical class B laser, for instance mean value I st is larger than the deterministic value. a single mode semiconductor laser, can be described in h i By using (17) one can easily find that terms of two evolution equations, one for the slowly- varying complex amplitude E of the electric field inside a exp( b/2ǫ)(b/2ǫ) 2ǫ +1 the laser cavity and the other for the carriers number I = (a b)+2ǫ 1+ (18) st − a b N (or electron-hole pairs) [10]. These equations include h i − " Γ +1, # 2ǫ 2ǫ noise terms accounting for the stochastic nature of spon- An expression for the mean value of the intensity in the taneous emission and random non-radiative carrier re- steady state was also given in [17] in the simpler case combination due to thermal fluctuations. Both noise of an expansion of the saturation-term parameter in the sources are usually assumed to be white Gaussian noise. dynamical equations. The equation for the electric field can be written in As mentioned before, in the steady state of the stochas- terms of the optical intensity I and the phase φ by defin- iφ tic dynamics, the phase φ of the electric field fluctuates ing E = √Ie . For simplicity, we neglect the explicit around a mean value that changes linearly with time. Of random fluctuations terms and retain, as usual [10], the course, since any value of φ can be mapped into the in- mean power of the spontaneous emission. The equations terval [0, 2π), this is not inconsistent with the fact that are: the stationary distribution for φ is a uniform one. We dI noise sustained = (G(N, I) γ)I +4βN (22) can easily understand the origin of this dt − flow [20]: the rotation inducing terms, those proportional to α in the equations of motion, are zero at the line of dφ 1 minima of the potential V and, hence, do not act in the = (G(N, I) γ)α (23) steady deterministic state. Fluctuations allow the sys- dt 2 − tem to explore regions of the configuration space (x1, x2) where the potential is not at its minimum value. Since, dN according to Eq. (18), the mean value of I is not at the = C γeN G(N, I)I (24) dt − − minimum of the potential, there is, on average, a non-zero contribution of the rotation terms producing the phase G(N, I) is the material gain given by: drift observed. g(N N ) The rotation speed can be calculated by writing the G(N, I)= − o (25) evolution equation for the phase of the electric field as: 1+ sI

a 1 The definitions and typical values of the parameters for φ˙ = 1 α + ξ(t) (19) semiconductor lasers are given in Table 1. The first b + I − √ I term of Eq. (22) accounts for the stimulated emission where ξ(t) is a white noise term with zero mean value while the second accounts for the mean value of the and correlations given by (6). By taking the average spontaneous emission power. Eqs. (22 - 24) are writ- value and using the rules of the Itôcalculus, one arrives ten in the reference frame in which the frequency of the at: on state is zero when spontaneous emission noise is neglected. The threshold condition is obtained by setting a G(N, I)= γ, I = 0 and neglecting spontaneous emission, φ˙ = α 1 (20) γ h i b + I − i.e. Nth = No+ . The threshold carrier injected per unit g

4 time to turn the laser on is given by Cth = γeNth. Eq. the observed dynamics. This study was done in [14] with- (23) shows that φ˙ is linear with N and slightly (due to out considering neither the saturation term, nor the mean the smallness of the saturation parameter s, see Table 1) value of the spontaneously emission power, and under nonlinear with I. those conditions an expression for the period of the tran- Since in the deterministic case considered henceforth, sient oscillations was obtained. In our work, we calculate the evolution equations for I and N do not depend on the period of the oscillations by taking into account these the phase φ, we can concentrate only in the evolution two eﬀects. The period is obtained in terms of the po- of I and N. One can obtain a set of simpler dimen- tential, by assuming that the latter has a constant value sionless equations by performing the following change of during one period. It will be shown that this assumption variables: works reasonably well and gives a good agreement with numerical calculations. Near the steady state, the relax- 2g g γ y = I, z = (N No), τ = t (26) ation oscillations can also be calculated in this form, but γ γ − 2 the potential is almost constant and consequently so is The equations become then: the period. The evolutions equations (27), (28) can be casted in dy z the form of Eq. (4) with the following Lyapunov poten- =2 1 y + cz + d (27) dτ 1+¯sy − tial: dz z y A 1 = a bz , (28) V (y,z)= A y + A y2 + A ln(y)+ 4 + B2(y,z) dτ − − 1+¯sy 1 2 3 y 2

2g 2γe (32) where we have defined a = 2 (C γ N ), b = , γ − e o γ 16β 16βgNo sγ c = γ , d = γ2 ands ¯ = 2g . These equations form where the basis of our subsequent analysis. The steady states 1 1 1 1 are obtained by setting (27) and (28) equal to zero, i.e.: A = as¯ + bs¯ sd¯ (1 + bs¯) as¯2c 1 2 − 2 − 4 − 4 1 s¯ y = [2(a b)+ d(1 + bs¯)+ ca s¯ + √v] (29) A2 = (1 + bs¯) st 4(1 + bs¯) − 4 1 d a(1+¯syst) A3 = [a b + (ac + bd)s ¯ + ] zst = (30) −2 − 2 b + y (1 + bs¯) st (ac + bd) A4 = where the constant v is given by: 4 (d + cz) v = 4(a b)2 +4d(a + b)(1 + bs¯)+ d2(1 + bs¯)2 B(y,z)= z 1 sy¯ + (1+¯sy) . − − − 2y + c(8a +4as¯(a + b)+2das¯(1 + bs¯)) + c2a2s¯2 (31) The corresponding (non-constant) matrix D is given by: There is another steady state solution for yst given by Eq. (29) (with a minus sign in front of √v) which, however, 0 d12 D = − . (33) does not correspond to any possible physical situation, d12 d22 since yst < 0. For a value of the injected carriers per unit being time below threshold (C < Cth, equivalent to a b< 0) y is very small. This corresponds to the off −solution st 4y2 in which the only emitted light corresponds to the spon- d12 = (34) taneous emission. Above threshold, stimulated emission (1+¯sy) [2y + c (1+¯sy)] occurs and the laser operates in the on state with large 4y [(1 + 2¯s + bs¯) y2 + by + d + cz] d22 = (35) yst. In what follows, we will concentrate in the evolution (1+¯sy) [2y + c (1+¯sy)]2 following the laser switch-on to the on state. It is known that the dynamical evolution of y and z This potential reduces to the one obtained in ref. [14] is such that they both reach the steady state by per- when setting c = d =s ¯ = 0 (which corresponds to set the forming damped oscillations [10] whose period decreases laser parameters β = s = 0). As expected, non-vanishing with time. This fact is different from the usual relaxation values for the parameters s and β increase the dissipative oscillations that are calculated near the steady state by part of the potential (d22), associated with the damping linearizing the dynamical equations. The time evolution term. This result was pointed out in [21] when linearizing of y and z is shown in Fig. 3a for some parameters (for the rate equations around the steady state. another values of the parameters equivalent results are The equipotential lines of (32) are also plotted in Fig. obtained), while the corresponding projection in the y, 4. It is observed that there is only one minimum for V z phase-plane is shown in Fig. 4. We are interested in and hence the only stable solution (for this range of pa- obtaining a Lyapunov potential that can helps to explain rameters) is that the laser switches to the on state and

5 relaxes to the minimum of V . The movement towards where: the minimum of V has two components: a conservative 2 one that produces closed equipotential trajectories and a 1 A3 A4 1 (d + czst) E =2 A + + s¯ + damping that decreases the value of the potential. The 2 2 3 2 − 2 yst yst 2 2yst ! combined effects drives the system to the minimum fol- (1+¯sy ) 2 lowing a spiral movement, best observed in Fig. 4. F = 1+ c st The time evolution of the potential is also plotted in 2yst Fig. 3b. In this figure it can be seen that the Lyapunov c(1+¯syst) (d + czst) potential is approximately constant between two consec- H = 1+ s¯ + 2 − 2yst 2y utive peaks of the relaxation oscillations (This fact can st be also observed with the equipotential lines of Fig. 4). and d12,st is the coefficient d12 calculated in the steady This fact allows us to estimate the relaxation oscillation state. The period of the relaxation oscillations near the period by approximating V (y,z) = V , constant, during steady state can be obtained by linearizing eqs. (27) and this time interval. When the potential is considered to (28) after a small perturbation is applied. The frequency be constant, the period can be evaluated by the stan- of the oscillations in the steady state is the imaginary dard method of elementary Mechanics: z is replaced by part of the eigenvalues of the linearized equations. This its expression obtained from (27) in terms of y andy ˙ yields a period: (the dot stands for the time derivative) in V (y,z). Using −1/2 the condition that V (y,z) = V = constant, we obtain 2 2 2 π d22,st F an equation for y of the form: F (y, y˙) = V . From this Tst = 1 2 2 2 d12,st √EF H " − d12,st 4(EF H ) # equation, we can calculate the relaxation oscillation pe- − − riod (T ) by integrating over a cycle. This leads to the (38) expression:

y1 The difference between (37) and (38) vanishes with 1+¯sy dy 2 T = d22,st (i.e. d22 in the stationary state). Since EF H is 2 −1 1/2 − y0 y [2(V A y A y A ln(y) A y )] always a positive quantity, our approximation will give, Z − 1 − 2 − 3 − 4 (36) at least asymptotically, a smaller value for the period. In order to have a complete understanding of the variation of the period with time, we need to compute the where y and y are the values of y that cancel the de- 0 1 time variation of the potential V (τ) between two con- nominator. We stress the fact that the only one approx- secutive intensity peaks. This variation is induced by imation used in the derivation of this expression is that the dissipative terms in the equations of motion. We the Lyapunov potential is constant during two maxima have not been able to derive an expression for the vari- of the intensity oscillations. The previous equation for ation of the potential (see [14] for an approximate ex- the period reduces, in the case c = d =¯s = 0, to the one pression in a simpler case). However, we have found that previously obtained by using the relation between the a semi-empirical argument can yield a very simple law laser dynamics and the Toda oscillator derived in [14]. which is well reproduced by the simulations. We start Evaluation of the above integral shows that the period by studying the decay to the stationary state in the lin- T decreases as the potential V decreases. Since the Lya- earized equations. By expanding around the steady state: punov potential decreases with time, this explains the y = y +δy, z = z +δz, the dynamical equations imply fact that the period of the oscillations in the transient st st that the variables decay to the steady state as: δy(τ), regime decreases with time. In Fig. 5 we compare the δz(τ) exp( ρ τ), where: results obtained with the above expression for the period ∝ − 2 with the one obtained from numerical simulations of the ρ = d F (39) rate equations (27), (28). In the simulations we compute 22,st the period as the time between two peaks in the evolution Expanding V (y,z) around the steady state and taking of the variable y. As can be seen in this figure, the above an initial condition at τ0 we find an expression for the expression for the period, when using the numerical value decay of the potential: of the potential V , accurately reproduces the simulation results although it is systematically lower than the nu- ln [V (τ) Vst] = ln [V (τ0) Vst] ρ (τ τ0) (40) merical result. The discrepancy is less than one percent − − − − over the whole range of times. In Fig. 6 we plot ln[V (τ) Vst] versus time and compare it It is possible to quantify the difference between the ap- with the approximation− (40). One can see that the latter proximate expression (36) and the exact values near the fits ln[V (τ) Vst] not only near the steady state (where stationary state. In this case expression (36) reduces to : it was derived),− but also during the transient dynamics. The value of τ0, being a free parameter, was chosen at 2 π T = (37) the time at which the first peak of the intensity appears. d √EF H2 Although other values of τ0 might produce a better fit, 12,st −

6 the one chosen here has the advantage that it can be ACKNOWLEDGMENTS calculated analytically by following the technique of ref. [22]. It can be derived from Eq. (36) that the period T is We wish to thank Professor M. San Miguel and Pro- linearly related to the potential V . This, combined with fessor G.L. Oppo for a careful reading of this manuscript the result of Eq. (40), suggests the semi-empirical law and for useful comments. We acknowledge financial sup- for the evolution of the period of the form: port from DGES (Spain) under Project Nos. PB94-1167 and PB97-0141-C02-01. ln [T (τ) T ] = ln [T (τ ) T ] ρ (τ τ ) (41) − st 0 − st − − 0 This simple expression fits well the calculated period not only near the steady state, but also in the transient regime, see Figs. 5 and 7. The tiny differences observed near the steady state are due to the fact that the semi- empirical law, Eq. (41), is based on the validity of rela- [1] J. Guckenheimer and P. Holmes, Nonlinear Oscillations, tion Eq. (36) between the period and the potential. As Dynamical Systems and Bifurcations of Vector Fields, it was already discussed above, that expansion slightly Springer–Verlag (1983). underestimates the asymptotic (stationary) value of the [2] M. Sargent III, M.O. Scully and W.E. Lamb, Jr., Laser period. By complementing this study with the procedure Physics. Addison-Wesley Publ. Comp., Reading Mass. given in [22] to describe the switch-on process of a laser, (1974). 134 A and valid until the first intensity peak is reached, we can [3] W.E. Lamb, Jr. Phys Rev. , 1429 (1964). obtain a complete description of the variation of the os- [4] H. Haken, Light Vol. 2, Laser Light Dynamics, North- cillations period in the dynamical evolution following the Holland, Amsterdam (1985). [5] H. Haken ahd H. Sauermann, Z. Phys. 173, 261 (1963). laser switch-on. [6] H. Haken, Laser Theory. Springer-Verlag (1984). [7] H. Haken, Synergetics. Springer-Verlag (1983). [8] H. Statz and G.A. deMars, Quantum Electromics, Ed. IV. SUMMARY C.H. Townes, Colombia University Press, New York (1960). C.L. Tang, H. Statz and G.A. deMars, J. Appl. Phys., 34, 2289 (1963). In this work we have used Lyapunov potentials in the [9] F.T. Arecchi, G.L. Lippi, G.P. Puccioni and J.R. context of laser dynamics. For class A lasers, we have Tredicce, Opt. Comm. 51, 308 (1984); J. Opt Soc. Am. explained qualitatively the observed features of the de- B, 2, 173 (1985). terministic dynamics by the movement on the potential [10] G. P. Agrawal and N. K. Dutta, Long-Wavelength Semi- landscape. We have identified the relaxational and con- conductor Lasers, New York: Van Nostrand Reinhold servative terms in the dynamical equations of motion. In (1986). the stochastic dynamics (when additive noise is added [11] C. O. Weiss and R. Vilaseca, Dynamics of lasers, (VCH to the equations), we have explained the presence of a Publishers, Weinhemim, 1991). “noise sustained flow” for the phase of the electric field [12] S. Ciuchi, F. de Pasquale, M. San Miguel and N.B. Abra- as the interaction of the conservative terms with the noise ham, Phys. Rev. A 44, 7657 (1991). terms. An analytical expression allows the calculation of [13] E. Hernández-Garc´ıa, R. Toral and M. San Miguel, Phys. the phase drift. Rev. A 42, 6823 (1990). In the case of class B lasers, we have obtained a Lya- [14] G.L. Oppo and A. Politi, Z. Phys. B - Condensed Matter punov potential valid only in the deterministic case, when 59, 111 (1985). noise fluctuations are neglected. We have found that the [15] M. San Miguel, R. Montagne, A. Amengual, E. dynamics is non-relaxational with a non-constant matrix Hernández-Garc´ıa, in Instabilities and Nonequilibrium D. The fixed point corresponding to the laser in the on Structures V, E. Tirapegui and W. Zeller, eds. Kluwer state is interpreted as a minimum in the potential land- Acad. Pub. (1996). [16] R. Montagne, E. Hernández-Garc´ıa and M. San Miguel, scape. By observing that the potential is nearly constant Physica D 96, 47 (1996). between two consecutive intensity peaks during the re- [17] H. Risken, The Fokker-Plank equation, Springer-Verlag laxation process towards the steady state, but still in (1989). a highly non-linear regime, we were able to obtain an [18] M. San Miguel and R. Toral, Stochastic Effects in Phys- approximate expression for the period of the oscillations. ical Systems in Instabilities and Nonequilibrium Struc- Moreover, we have derived a simple exponential approach tures VI, E. Tirapegui and W. Zeller, eds. Kluwer Acad. of the period of the oscillations with time towards the pe- Pub. (1997). riod of the relaxation oscillations near the steady state. [19] R. Graham in Instabilities and Nonequilibrium Struc- This dependence appears to be valid after the first inten- tures, E. Tirapegui and D. Villaroel, eds. Reidel, Dor- sity peak following the switch-on of the laser. A possible drecht (1987). R. Graham in Instabilities and Nonequi- extension of our work could be to consider the presence librium Structures III, E. Tirapegui and W. Zeller, eds. of an external field, which is numerically studied in [23]. Reidel, Dordrecht (1991).

7 [20] M.O. Magnasco, Phys. Rev. Lett. 71, 1477 (1993). [22] S. Balle, P. Colet and M. San Miguel, Phys. Rev. A 43, [21] C.-H. Lee and S.-Y. Shin, IEEE J. Quantum Electronics 498 (1991). 25, 878 (1989). [23] A. Politi and G. L. Oppo, Phys. Rev. A 33, 4055 (1986).

Table 1

PARAMETERS VALUES C Carriers injected per unit time. > threshold γ Cavity decay rate. 0.5 ps−1 −1 γe Carrier decay rate. 0.001 ps 8 No Number of carriers at transparency. 1.5 10 g Diﬀerential gain parameter. 1.5 10×−8 ps−1 s Saturation parameter. 10×−8 10−7 β Spontaneous emission rate. 10−8−ps−1 α Linewidth enhancement factor. 3-6

8 FIGURE CAPTIONS

Fig. 1: Potential for a class A laser, Eq. (15) with the parameters: a = 2, b = 1. Dimensionless units.

Fig. 2: Time evolution of the mean value of the phase φ in a class A laser, in the case a = 2, b = 1, ǫ =0.1. For α =0 (dashed line) there is only phase diﬀusion and the average value is 0 for all times. When α = 5 (solid line) there is a linear variation of the mean value of the phase at late times. Error bars are incluted for some values. The dot-dashed line has the slope given by the theoretical prediction Eq. (21). The initial condition is taken as x1 = x2 = 0 and the results were averaged over 10000 trajectories with diﬀerent realizations of the noise. Dimensionless units.

Fig. 3: a) Normalized intensity, y (solid line) and normalized carriers number, z/40 (dot-dashed line) versus time in a class B laser obtained by numerical solution of Eqs. (27) and (28). b) Plot of the potential (32). Parameters: a = 0.009, b = 0.004,s ¯ = 0.5, c = 3.2 10−9, d = 1.44 10−8 which correspond to physical parameters in Table 1 with C =1.2C . The initial conditions× are taken as y =5× 10−8 and z =0.993. Dimensionless units. th ×

Fig. 4: Number of carriers versus intensity (scaled variables). The vector ﬁeld and contour plot (thick lines) are also represented. Same parameters than in Fig. 3. Dimensionless units.

Fig. 5: Period versus time in a class B laser. Solid line has been calculated as the distance between two peaks of intensity, with triangles plotted at the begining of each period; dashed line has been calculated using the expression (36), with the value of the potential V obtained also from the simulation; dotted line corresponds to the semi-empirical expression (41). Same parameters than in Fig. 3. We have used τ0 = 55.55, coinciding with the position of the ﬁrst intensity peak. Dimensionless units.

Fig. 6: Logarithm of the potential diﬀerence versus time in a class B laser (solid line), compared with the theoretical expression in the steady state (40) (dashed line). Same parameters than in Fig. 3 and τ0 as in Fig. 5. Dimensionless units.

Fig. 7: Logarithm of the period diﬀerence versus time in a class B laser. Triangles correspond to the period calculated from the simulations as the distance between two consecutive intensity peaks, at the same position than in Fig. 6. The dashed line is the semiempirical expression Eq. (41). Same parameters than in Fig. 3 and τ0 as in Fig. 5. Dimensionless units.

9 10 11 12 13