Chaotic dynamics in semiconductor lasers with optical feedback

Junji Ohtsubo    Faculty of Engineering, Shizuoka University, 3-5-1 Johoku, Hamamatsu, 432-8561, Japan

§ 1 Introduction

It was in 1963 that Lorenz [1963], investigating the behavior of convective fluid flow as a model for the atmospheric flow, showed that nonlinear systems described by three variables could exhibit chaotic dynamics. Chaos is the phenomenon of irregular variations of system output derived from models described by deterministic equations. In spite of the models being deterministic, we cannot foresee the future of the output since chaos is very sensitive to the initial conditions: each system behaves completely different every time, even if the difference in the initial state is very small. Chaos can be observed in various fields of engineering, physics, chemistry, economics, and biology. Although the fields are different, some of the chaotic systems can be characterized by similar differential equations. Since lasers are nonlinear systems and are typically characterized by three variables: the field, the polarization of matter, and the population inversion, they are candidates for chaotic systems. Indeed, it was proved in the mid-1970s by Haken [1975] that lasers are nonlinear systems similar to the Lorenz model and that they show chaotic behavior in their output power. Haken assumed a ring laser model and considered two-level atoms in a laser medium. Although lasers are not always described by his model, the approximations are reasonable for most lasers. Therefore, nonlinear laser rate equations with three variables and involving chaotic dynamics are called Lorenz–Haken equations (Haken [1985]). However, ordinary lasers do not exhibit chaotic behavior, and only few of the lasers with bad cavity conditions show chaotic dynamics. In the meantime, chaotic behavior were theoretically demonstrated in a ring laser system (Ikeda [1979]). Weiss and Brock [1986] were the first to observe Lorenz–Haken chaos in infrared NH3 lasers.

Contrary to Haken’s prediction, ordinary lasers are stable systems and only a few infrared lasers systems show chaotic behavior in their output power. Arecchi, Lippi, Puccioni and Tredicce [1984] investigated the laser systems from the viewpoint of the characteristic relaxation times of the three variables, and categorized lasers in three classes. According to their classifications, one or two of the relaxation times are very fast compared with the other time scales and most lasers are described by rate equations with one or two variables. These are, therefore, stable systems, categorized into class-A and class-B lasers. Only class-C lasers are described by the full set of rate equations with three variables and can show chaotic dynamics. However, class-A and class-B lasers can show chaotic dynamics when one or more degrees of freedom are introduced to the laser systems. Class-B lasers are characterized by rate equations for the field and the population inversion, and they are easily destabilized by an additional degree of freedom applied as an external perturbation. For example, solid-state lasers, fiber lasers, and CO2 lasers that are categorized as class-B lasers show unstable oscillation upon external optical injection or modulation of accessible laser parameters. Semiconductor lasers, the main topic of this review, are also classified as class-B lasers; they are very sensitive to optical injection, self-induced optical feedback, optical injection from other lasers, opto-electronic feedback, and injection current modulation. They show chaotic dynamics in the presence of external perturbations. An overview of chaotic instabilities in lasers has been given by Abraham, Mandel and Narducci [1988].

Since the early 1980s (Lang and Kobayashi [1980]), feedback-induced instabilities and chaos in semiconductor lasers have been closely examined. In a semiconductor laser, when the light reflected from an external reflector couples with the original field in the laser cavity, the laser oscillation is affected considerably. A variety of dynamical behaviors can be observed in semiconductor lasers with optical feedback and they have been investigated by many researchers for two decades. One of the main differences between semiconductor lasers and other lasers is the low reflectivity of the internal mirrors in the semiconductor laser cavity. It ranges typically only from 10 to 30% of the intensity in Fabry–Perot semiconductor lasers. This makes the feedback effects significant in semiconductor lasers. In the case of the Vertical-Cavity Surface-Emitting Lasers (VCSELs), the reflectivity of the internal mirror is very high at more than 99%, however they are also sensitive to external optical feedback because of the small number of photons in the internal cavity. Therefore, semiconductor lasers of all types are essentially very sensitive to external optical feedback. Another difference is a large absolute value of the linewidth enhancement factor α of the laser media. Values for the linewidth enhancement factor α of 2–7 have been reported in semiconductor lasers depending on the laser materials, while this value is almost zero for other lasers. Then, the coupling between the phase and the carrier density (equivalent to population inversion) is encountered in the laser dynamics. Interestingly, these factors lead to a variety of dynamics quite different from any other lasers. At weak to moderate optical feedback reflectivity, the laser output power shows interesting dynamical behaviors such as a stable state, periodic and quasi-periodic oscillations, and chaos with changes of the system parameters. These ranges of the external reflectivity are not only interesting from a viewpoint of fundamental physics, but also very important in practical applications of semiconductor lasers such as in optical data storage systems and optical communications. Extensive lists of recent literature for the dynamic characteristics in semiconductor lasers with optical feedback can be found in articles by van Tartwijk and Agrawal [1998] and Ohtsubo [1999].

Until now, many semiconductor laser devices with different structures have been proposed and fabricated. In spite of the different device structures, the dynamics of semiconductor lasers are the same as long as the laser rate equations are written in the same or similar forms. The dynamics of edge-emitting singlemode semiconductor lasers have been studied extensively for a long time, and many fruitful results have been obtained. However, there are still important issues on the fundamental physics of optical chaos and practical applications to be discussed. On the other hand, little investigation on the dynamics of other laser structures has been done, for example VCSELs, self-pulsating lasers, broad-area lasers. In the meantime, important breakthroughs in the applications of chaos were achieved in the early 1990s. The ideas of chaos control (Ott, Grebogi and Yorke [1990]) and chaos synchronization (Pecora and Carroll [1990]) have been proposed and developed in that decade. The idea of chaos control has been applied immediately to the stabilization of chaotic lasers (Roy, Murphy, Maier, Gills and Hunt [1992]). The possibility of chaos communications has been discussed based on chaos synchronization in systems with two chaotic lasers (Colet and Roy [1994]).

In this article, we focus on the dynamics and applications in semiconductor lasers with optical feedback. In § 2, we first introduce general laser rate equations which reduce to the Lorenz equations, and the classifications of the lasers are given. The instabilities intrinsically involved in the rate equations are studied. Next, semiconductor lasers as class-B lasers are described and the possibility for unstable oscillations of lasers by the introduction of external perturbations is discussed. A solitary semiconductor laser is characterized by two equations, for field and carrier density (population inversion). We then derive the forms of the rate equations for edge-emitting semiconductor lasers. After that, the rate equations of the semiconductor lasers for various laser structures are introduced. In § 3, we present the theory of semiconductor lasers with optical feedback and discuss the various...