Le Thai Son - Post-graduate Student. E-mail: lethaison1110@gmail.com
Yu.I. Alekseev - Dr.Sc. (Eng.), professor. E-mail: lethaison1110@gmail.com

Semiconductor lasers subjected to optical injection from another laser is known as nonlinear optical systems that can generate chaotic intensity fluctuations and lead to various nonlinear dynamics. Such dynamics have gained a lot of consideration due to their fundamental physical properties and potential for technological applications, including for example, chaotic RADAR, chaotic LIDAR, and secure communications. Several theoretical and experimental investigations of semiconductor lasers in the terms of chaos, transition routes to chaos, nonlinear dynamics, bifurcation and chaotic synchronization have been performed. The nonlinear dynamics of semiconductor lasers with injected optical signals are commonly investigated by using Lang and Kobayashi equations for the amplitude and the phase of the electric field and the average density in the active region. Numerical method is often used due to its simplicity. The stability maps of the system can be obtained by numerically integrating the Lang and Kobayashi equations using Rungge-Kutta method. However, the more investigations are needed in order to get the deeper understanding about the physical phenomena behind the system. In this article we show a simple analytical method for investigating stability of locking regime in an optically injected semiconductor laser. The proposed method is based on well-known oscillation theory and Routh-Hurwitz stability criterion. We first obtained the condition of stable locking regime, and then investigated its stability. As the results of this, we successfully found the mathematical expression for the Hopf boundary that can be easily built using a simple computer program. At the end of the article we showed the instability map of the system that was obtained using numerical method. The hopf boundary obtained this was agreed well with the result of analytical proposed method.