Chaos in lasers

After an overview of the main papers devoted to chaos in lasers (Section I.A) and in nonlinear optical processes (Section I.B), we present a more detailed analysis of dynamics in a process of second-harmonic generation of light (Section II) as well as in Kerr oscillators (Section III). The last case we consider particularly in the context of coupled nonlinear systems. Finally, we present a cumulant approach to the problem of quantum corrections to the classical dynamics in second-harmonic generation and Kerr processes (Section IV). 

P. W. Milonni, M.L. Shih, and J.R. Ackerhalt, Chaos in Laser-Matter Interactions, World Scientific, Singapore, 1987. 

Firth, W. J. (1986) Instabilities and chaos in lasers and optical resonators. In A. V. Holden, ed. Chaos (Princeton University Press, Princeton, NJ). 

It should be noted that there is a limited number of works on classical relativistic dynamical chaos (Chernikov et.al., 1989 Drake and et.al., 1996 Matrasulov, 2001). However, the study of the relativistic systems is important both from fundamental as well as from practical viewpoints. Such systems as electrons accelerating in laser-plasma accelerators (Mora, 1993), heavy and superheavy atoms (Matrasulov, 2001) and many other systems in nuclear and particle physics are essentially relativistic systems which can exhibit chaotic dynamics and need to be treated by taking into account relativistic dynamics. Besides that interaction with magnetic field can also strengthen the role of the relativistic effects since the electron gains additional velocity in a magnetic field. 

A major development reported in 1964 was the first numerical solution of the laser equations by Buley and Cummings [15]. They predicted the possibility of undamped chaotic oscillations far above a gain threshold in lasers. Precisely, they numerically found almost random spikes in systems of equations adopted to a model of a single-mode laser with a bad cavity. Thus optical chaos became a subject soon after the appearance Lorenz paper [2]. 

To achieve the instability of homogeneous broadened line lasers, a satisfaction of much more difficult conditions is required large gain and the so-called bad-cavity properties. This special regime for damping constants and mode intensity is fulfilled in the far-infrared lasers [36]. In 1985 Weiss et al. [37,38] experimentally found a period doubling route to chaos in the NH3 laser. Further experimental investigation of chaotic dynamics in such lasers was reported later [39]. 

Another class of good candidates for a study of chaos in nonlinear optics are wave-mixing processes in which chaos appears in the propagation of laser light through passive nonlinear media [93]. A chaotic behavior was observed in three-wave mixing [94] and in four-wave mixing [95]. 

The case of a frequency mismatch between laser pumps and cavity modes was investigated [83], and for the first time, chaos in SHG was found. When the pump intensity is increased, we observe a period doubling route to chaos for Ai = 2 = 1. Now, for/i = 5.5, Eq. (3) give aperiodic solutions and we have a chaotic evolution in intensities (Fig. 5a) and a chaotic attractor in phase plane (Imaj, Reai) (Fig. 5b). 

Here, we also include the contributions related to quantum mechanics The chapter by Takami et al. discusses control of quantum chaos using coarsegrained laser fields, and the contribution of Takahashi and Ikeda deals with tunnehng phenomena involving chaos. Both discuss how chaos in classical behavior manifests itself in the quantum counterpart, and what role it will play in reaction dynamics. 

For the experienced practitioner of atomic physics there appears to be an enigma right at this point. What does nonlinear chaos theory have to do with linear quantum mechanics, so successful in the classification of atomic states and the description of atomic dynamics The answer, interestingly, is the enormous advances in atomic physics itself. Modern day experiments are able to control essentially isolated atoms and molecules to unprecedented precision at very high quantum numbers. Key elements here are the development of atomic beam techniques and the revolutionary effect of lasers. Given the high quantum numbers, Bohr s correspondence principle tells us that atoms are best understood on the basis of classical mechanics. The classical counterpart of most atoms and molecules, however, is chaotic. Hence the importance of understanding chaos in atomic physics. 

Figure 10.4.5 shows an experimental example of the intermittency route to chaos in a laser. 

H. G. Winful and L. Rahman. Synchronized chaos and spatiotemporal chaos in arrays of coupled lasers. Phys. Rev. Lett, 65 1575, 1990. 

R. Y. Chio, P. G. Kwiai, and A. M. Steinberg Classical and Quantum Chaos in Atomic Systems, Dominique Delande and Andrease Buchleitner Measurements of Collisions between Laser-Cooled Atoms, Thad Walker and Paul Feng The Measurement and Analysis of Electric Fields in Glow Discharge Plasma,... 

On the other hand, the rapid advancement of laser spectroscopy has made it possible to produce atoms in highly exited states. It has been realized, that these Rydberg atoms in uniform external fields are gateway systems for studying various aspects of chaos in... 

It has been known since 1975 that chaotic behavior is possible in some lasers under specific conditions, due to the similarity between the Lorenz equations (which predict chaos in fluids) and the semiclassical Maxwell-Bloch equations describing single mode lasers including the ef-... 

Controlling chaos, even in the rather limited fashion described here, is a powerful and attractive notion. Showalter (1995) compares the process to the instinctive, apparently random motions of a clown as he makes precise corrections aimed at stabilizing his precarious perch on the seat of a unicycle. A small, but carefully chosen, variation in the parameters of a system can convert that system s aperiodic behavior not only into periodic behavior, but also into any of a large (in principle, infinite) number of possible choices. In view of the appearance of chaos in systems ranging from chemical reactions in the laboratory, to lasers, to flames, to water faucets, to hearts, to brains, our ability first to understand and then to control this ubiquitous phenomenon is likely to have major consequences. 

Kuz min, M. V., and Stuchebrukhov, A. A. (1989). Dynamical chaos and intramolecular vibrational relaxation in polyatomic molecules. In Laser spectroscopy of highly vihrationally excited molecules (ed. V. S. Letokhov), pp. 178-264. Adam Hilger,... 

Laser instabilities were experimentally investigated in many kinds of lasers (see an overview of early papers [14]), but the first experimental observation of the optical chaos was performed by Arecchi et al. [30] in 1982. They used a stabilized CO2 laser with modulated cavity loss = y(l + a cos fit) and by changing the frequency of modulation 2, they found a few period doubling oscillations of the output intensity, both numerically and experimentally. 

The CO2 lasers were also investigated in connection with chaotic behavior, and here we mention the most important papers in the field. The chaotic behavior associated with a transverse mode structure in a cw CO2 laser was observed in 1985 [40]. In the CO2 laser with elastooptically modulated cavity length, a period doubling route to chaos was also found [41]. 

Chaos was also investigated in solid-state lasers, and the important role of a pump nonuniformity leading to a chaotic lasing was pointed out [42]. A modulation of pump of a solid-state NdPsOn laser leads to period doubling route to chaos [43]. The same phenomenon was observed in the case of laser diodes with modulated currents [44,45]. Also a chaotic dynamics of outputs in Nd YAG lasers was also discovered [46 18]. In semiconductor lasers a period doubling route to chaos was found experimentally and theoretically in 1993 [49]. 

Many kinds of molecular systems pumped by a strong laser light show chaotic dynamics. Indeed, in a semiclassical model of a multiphoton excitation on molecular vibration, chaos was discovered by Ackerhalt et al. [85] and theoretically and numerically investigated in detail [86,87]. Moreover, the equations of motion that describe a rotating molecule in a laser field can exhibit a chaotic behavior and have been applied in the classical case of a rigid-rotator approximation [87,88]. 

The need to be able to control chaos has attracted considerable attention. Methods already available include a variety of minimal forms of interaction [150-155] and methods of strong control [156,157] that necessarily require a large modification of the system s dynamics, for at least a limited period of time. For example, in Refs. 158 and 159, the procedure of controlling chaos by means of minimal forms of interaction (saddle cycle stabilization) is realized for different laser systems. 

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