Lyapunov Potential Description for Laser Dynamics

Lyapunov Potential Description for Laser Dynamics Catalina Mayol†,‡, Raúl Toral†,‡ and Claudio R. Mirasso† (†) Departament de F´ısica, Universitat de les Illes Balears (‡) Instituto Mediterráneo de Estudios Avanzados (IMEDEA, UIB-CSIC) E-07071 Palma de Mallorca, Spain. Web Page: http://www.imedea.uib.es We describe the dynamical behavior of both class A and class B lasers in terms of a Lyapunov potential. For class A lasers we use the potential to analyze both deterministic and stochastic dynamics. In the stochastic case it is found that the phase of the electric field drifts with time in the steady state. For class B lasers, the potential obtained is valid in the absence of noise. In this case, a general expression relating the period of the relaxation oscillations to the potential is found. We have included in this expression the terms corresponding to the gain saturation and the mean value of the spontaneously emitted power, which were not considered previously. The validity of this expression is also discussed and a semi-empirical relation giving the period of the relaxation oscillations far from the stationary state is proposed and checked against numerical simulations. PACs: 42.65.Sf, 42.55.Ah, 42.60.Mi, 42.55.Px I. INTRODUCTION This model for laser dynamics was constructed from the Maxwell-Bloch equations for a single-mode field interact- Even for non-mechanical systems, it is occasionally ing with a two-level medium. The semiclassical laser the- possible to construct a function (called Lyapunov func- ory ignores the quantum-mechanical nature of the elec- tion or Lyapunov potential) that decreases along trajec- tromagnetic field, and the amplifying medium is modeled tories [1]. The usefulness of Lyapunov functions lies on quantum-mechanically as a collection of two-level atoms the fact that they allow an easy determination of the through the Bloch equations. A simpler description can fixed points of a dynamical (deterministic) system as the be obtained by deriving rate equations for the temporal extrema of the Lyapunov function as well as determin- change of the electric field (or photons number) inside ing the stability of those fixed points. In some cases, the the cavity and the population inversion (carriers number existence of a Lyapunov potential allows an intuitive un- in the case of semiconductor lasers) [8]. Rate equations, derstanding of the transient and stationary trajectories with stochastic terms accounting for spontaneous emis- as movements of test particles in the potential landscape. sion noise, have been extensively used for semiconductor In the case of non–deterministic dynamics, i.e. in the lasers. presence of noise terms, and under some general condi- Different types of lasers can be classified according to tions, the stationary probability distribution can also be the decay rate of the photons, carriers and material po- governed by the Lyapunov potential and averages can be larization. Arecchi et al. [9] were the first to use a clas- performed with respect to a known probability density sification scheme: class C lasers have all the decay rates arXiv:cond-mat/9905423v1 28 May 1999 function. The aim of this work is to construct Lyapunov of the same order, and therefore a set of three nonlin- potentials for some laser systems. We start, then, by ear differential equations is required for a satisfactory briefly reviewing the main features of the laser as a dy- description of the electric field, the population inversion namical system. and the material polarization. For class B lasers, the po- A laser has three basic ingredients: i) a gain medium larization decays towards the steady state much faster capable of amplifying the electromagnetic radiation prop- than the other two variables, and it can be adiabatically agating inside the cavity, ii) an optical cavity that pro- eliminated. Class B lasers, of which semiconductor lasers vides the necessary feedback, and iii) a pumping mech- [10] are an example, are then described by just two rate anism. A complete understanding of laser dynam- equations for the atomic population inversion (or carriers ics is based on a fully quantum-mechanical description number) and the electric field. Another examples of class of matter-radiation interaction within the laser cavity. B lasers are CO2 lasers and solid state lasers [11]. Finally, However, the laser is a system where the number of in class A lasers population inversion and material polar- photons is much larger than one, thus allowing a semi- ization decay much faster than the electric field. Both classical treatment of the electromagnetic field inside material variables can be adiabatically eliminated, and the cavity through the Maxwell equations. This fact the equation for the electric field is enough to describe was introduced in the semiclassical laser theory, devel- the dynamical evolution of the system. Some properties oped by Lamb [2,3] and independently by Haken [4–7]. of class A lasers, like a dye laser, are studied in [12,13]. 1 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.

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