ion of Shannon ' s Entropy to Classify

K e y w o r d s C o m p u t a t i o n a l physics, Parallel modelling and simulation, Cellular automata, Complex systems, Laser physics. 1. I N T R O D U C T I O N Laser dynamics has been t rad i t iona l ly analyzed by solving a set of coupled differential rate equat ions which describe the in terre la t ionships and t r ans i t ion rates among the electronic states in the laser active m e d i u m and the laser photons [1,2]. Bu t recently, an a l te rna t ive approach based Partially supported by the Junta de Extremadura (Consejer{a de Educacidn, Ciencia y Tecnolog{a) and the European Social Fund, Grant MOV03B170. 0895-7177/05/$ see h'ont matter (~) 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2005.09.012 Typeset by AMS-TEX 848 J.L. GUISADO et al. on a cellular automata (CA) model has been proposed [3], in which the laser properties arise as emergent phenomena from the collective effect of many locally interacting, simple components. In this paper we focus on the application of Shannon's entropy to recognize and classify the different types of behavior shown by the CA model. Cellular automata are a class of spatially and temporally discrete mathematical systems, characterized by local interaction and synchronous dynamical evolution. They were first introduced in the 1950s by von Neumann [4] to investigate self-reproduction and since then have at tracted much interest because of its capability to generate complex behavior from sets of components which interact locally with relatively simple rules [5,6]. In the last two decades, CA have been extensively used to build models of a wide variety of physical systems [7,8], for example reactiondiffusion processes [9], fluid dynamics [10], magnetization in solids [11], growth phenomena [12], molecular excited-state dynamics [13], etc. Recently, CA have become more attractive because the inherent parallelism makes them very suitable to be naturally and efficiently implemented in parallel computers. High performance simulations of physical systems [14,15] can be carried out in this way. A laser cellular automata model can be useful for two kinds of application: first, for cases in which the standard t reatment cannot be applied--as happens in lasers governed by stiff differential equations, with convergence problems--second, to take advantage of the cellular automata intrinsic parallel nature in order to efficiently implement three-dimensional simulations of laser devices in parallel computers. In this work, we study the following problem. In the laser cellular automata model, laser field and population inversion may perform different kinds of temporal behavior, depending on the values of the pumping probability, the life time of laser photons, and the upper laser level life time. The problem is how to find a magnitude to characterize all these possible outcomes of the system to be compared with the predictions of the traditional approach for laser dynamics (the laser rate equations) and with the experimental results. In order to address this problem, we choose the Shannon's entropy as such magnitude. In the laser CA model, two main kinds of temporal behavior are found: an overdamped dynamics with almost null Shannon's entropy and an oscillating one with larger Shannon's entropy. With the Shannon's entropy, we locate these different kinds of behavior and check the agreement between the laser CA model results and the predictions of the laser rate equations. This paper is organized as follows. Section 2 describes the cellular automata model that has been used for the simulations. In Section 3, the results of the simulations are presented and analyzed using the Shannon's entropy. Finally, the conclusions are summarized in Section 4. 2. C E L L U L A R A U T O M A T A M O D E L The model that has been used is a cellular automaton defined on a two-dimensional square lattice of Arc -200 × 200 cells with periodic boundary conditions. Two variables ai(t) and ci(t) are associated to each node of this lattice. The first one, ai(t), represents the state of the electron in node i at a given time t. An electron in the laser ground state takes the value a~(t) = 0, while an electron in the upper laser state takes the value ai(t) -1. The second variable ci(t) C {0, 1 , 2 , . . . , M} represents the number of photons in node i at t ime t. A large enough upper value of M is taken to avoid saturation of the system. The neighborhood considered is the Moore neighborhood, each cell having nine neighbors: the ceil itself, its four nearest neighbors (situated in the positions north, south, east, and west) and the four next neighbors (in the positions northeast, southeast, northwest and southwest). The time evolution of the CA is given by a set of transition rules which determine the state of any particular cell of the system at time t + 1 depending on the state of the cells included in its neighborhood at time t. These rules represent the different physical processes that work at the microscopic level in a laser system. Application of Shannon's Entropy 849 R1. Pumping: If the electronic s tate of a cell has a value of ai(t) = 0 in t ime t, then in t ime t + 1 tha t s tate will have a value of ai(t + 1) = 1 with a probabil i ty ,k. R2. Stimulated Emission: If, in t ime t, the electronic state of a cell has a value of ai(t) = 1 and the sum of the values of the laser photons states in the nine neighbor ceils is greater than a certain threshold (which in our simulations has been taken to be 1), then in t ime t + 1 a new photon will be created in tha t cell: ci(t + 1) = ci(t) + 1 and the electron will decay to the ground leveh ai(t + 1) = 0. R3. Photon Decay: A finite life t ime ~-~ is assigned to each photon when it is created. The photon will be destroyed ~-~ t ime steps after it is created. R4. Electron Decay: A finite life t ime ~-a is assigned to each electron tha t is promoted from the ground level to the upper laser level. Tha t electron will decay to the ground level again ~-~ t ime steps after it was promoted, if it has not yet decayed by st imulated emission. To simplify the model as much as possible, we consider this decay is entirely nonradiative, i.e., we do not take into account spontaneous emission. Also, as in an ideal four level laser the population of level E1 is negligible, st imulated absorption has not been considered. In addition, a small continuous noise level of random photons in the laser mode is introduced at every t ime step, in order to represent the experimentally observed noise level, responsible of the initial laser start-up. This is done by making c~(t + 1) = c~(t) + 1 for a small number of cells (< 0.01% of total) with randomly chosen positions. . S I M U L A T I O N R E S U L T S A N D S H A N N O N ' S E N T R O P Y A N A L Y S I S Three parameters determine the response of the system: the pumping probabil i ty (A), the life t ime of photons (~-c) and the life t ime of excited electrons (Ta). Initially, ai(O) = O, ci(O) = O, Vi, except a small fraction 0.01% of noise photons. We let the system evolve for 500 t ime steps. In each t ime step, the total number of laser photons n(t) N = ~ i 2 1 ci(t), and the total number of electrons in the upper laser state (population inversion) N(t) = ~N~ 1 ai(t) are measured. Running test simulations for different values of the three parameters of the system, two main types of behavior are observed: after a transient time, n(t) and N(t) show either a constant value or correlated damped oscillations. We are interested in making a more systematic exploration of the behavior of the laser CA model in the whole parameters space, and comparing it with the response of the laser rate equations. To this end, it should be interesting to find a magnitude to characterize the type of behavior shown by the system for each particular tr iad of values of the parameters . As such a magnitude, the Shannon's entropy S of the distribution of values taken by n(t) and N(t) , after running the simulation for a t ime interval, is calculated. This Shannon's entropy is computed by dividing the range of values taken by n or N in 10 a equally spaced bins, and computing the frequency (fi) at which this value lies inside every particular nonvoid bin i. Then, S is calculated as s ( a , = l o g s (1) i = 1 where rn is the number of nonvoid bins. The Shannon's entropy S measures the dispersion of the distribution of values taken by the nmnber of laser photons ~ (or the population inversion N). If this number is approximately constant (i.e., all the bins except one are void), S will tend to zero; if oscillations appear, the probabili ty distribution becomes wider and S takes higher values; and finally the max imum value of S would result from an equiprobable distribution. Therefore, S can be used as a good indicator of the presence of oscillations in the system. The dependence of the Shannon's entropy on two of