Dynamics of a Semiconductor Laser with Two External Cavities

Abstract

We study numerically complex dynamics of a semiconductor with one and two external cavities. We show that dynamical regimes in the laser with one external cavity can be controlled by adding another external cavity and properly adjusting its length and feedback strength. We demonstrate the existence of difierent stable periodic, quasiperiodic, chaotic, and steady-state regimes which form Arnold's tongues in bi-dimensional parameter spaces of the length and feedback strength of the external cavities and the pump parameter. The dynamics of a semiconductor laser subjected to external optical feedback has been studied by many researchers, since this laser has many applications in optical communications, interferometric sensors, frequency stability, etc. (see, for example, Ref. 1 and references therein). For moderate and strong feedback strengths this laser can display a very rich dynamical behavior, from periodic and quasiperiodic oscillations to chaos and coherence collapse (2,3). When the laser injection current is close to the solitary laser threshold, the laser operates in a stable steady-state regime. As the pump current is increased, intermittent drops of the laser intensity appear, that gives rise to low- frequency ∞uctuations (LFF). At higher currents, the laser optical bandwidth broadens that is known as coherence collapse (4,5). The dynamical behavior of a semiconductor lasers with a single feedback was studied extensively and the basic mechanism for LFF is well understood (6,7). The possibility for controlling LFF in a semiconductor laser with an external cavity by using a second delayed optical feedback was initially deduced by Liu and Ohtsubo (8). Later their idea was developed by Rogister, etal., (9,10), who realized the suppression of antimodes, responsible for LFF crises by properly adjusting the second feedback strength. Recently Mendez, etal., (11) have demonstrated experimentally that the frequency of LFF can be locked by external periodic modulation applied to the feedback strength of the external cavity. They have found difierent periodic and quasiperiodic regimes which formed Arnold's tongues in the space of the amplitude and frequency of the external modulation. In this paper we demonstrate that the locking efiect can be achieved without any external forcing. We show how LFF in a semiconductor laser with external cavity can be adequately controlled by adjusting properly both the length and the feedback strength of the second external cavity. The feedback time and the feedback strength of the second external cavity act in a similar manner as the period and amplitude of external modulation. We demonstrate that the variation of the parameters of the second external cavity allows one to obtain difierent dynamical regimes of the laser operation without any modiflcations in the solitary laser with a single external cavity. A single-mode semiconductor laser with delayed feedback is usually modeled by the Lang- Kobayashi rate equations (12). Due to the inflnite dimension of the system, an analytical study is very di-cult. Therefore, to study the dynamics of a semiconductor laser with two external cavities shown in Figure 1, we make numerical calculations of the modifled equations similar to those ex- plored previously by Sivapakrasam, etal., (13) and Carr (14). For weak or moderate feedback the modifled equations can be written as follows d ¢ E0(t) = (1=2)G(N (t) i Nth)E0 (t) + •1E0 (t i ?1)cos(ˆ1 + `(t) i `(t i ?1))