Asymptotic approach to the analysis of mode-hopping in semiconductor ring lasers

We investigate the directional mode-hopping in semiconductor ring lasers in the limit of small noise intensity. We show how only long time scales are involved in the hopping process. In such limit, the initial stochastic rate equations can be reduced to an auxiliary Hamiltonian system, and the dependence of the hopping on the laser parameters can be investigated. The predictions made from the reduced model agree well with the results obtained by simulating the full rate-equation system.

Figure 1

A schematic view of the semiconductor ring laser. SRL: semiconductor ring laser cavity; WG: bus waveguide; CW and CCW: clockwise and counterclockwise modes propagating in the cavity; OUT CW and CCW: output beams.

Figure 2

Example of a stochastic trajectory solution of Eqs. (1, 2) for the following parameter’s choice: $K=0.44{\text{ns}}^{-1}$, ${\varphi }_k=1.5$, $\mu =1.72$, $\alpha =3.5$. and $D^{\prime }=7.7\times {10}^{-4}{\text{ns}}^{-1}$. The intensities of the CW mode (black) and the CCW mode (gray) are shown together with the total intensity $P_1+P_2$ (light gray). The total power is shifted upward for clarity. The arrow indicates an example of first passage time. A short excursion to the opposite mode is shown in the inset.

Figure 3

Phase space portrait of Eqs. (5, 6). The two stationary points are marked by circles and correspond to CW and CCW unidirectional operations. The SP is marked by a cross and the stable manifold of SP is marked as a solid line. The unstable manifold of SP is not shown. The parameters are as follow $k=0.44$, ${\varphi }_k=1.5$, $\mu =1.65$, and $\alpha =3.5$.

Figure 4

MPEP paths for different values of the backscattering phase: (a) ${\varphi }_k=1.5$; (b) ${\varphi }_k=1.08$. The other parameters are as in Fig. 3.

Figure 5

Activation energy as a function of the current $\mu$ for different values of the backscattering phase. Markers are used to indicate the results of the integration of the full rate-equation model [Eqs. (1, 2)], whereas the solid lines represent the solution of auxiliary Hamiltonian systems (13, 14, 15, 16).

Figure 6

Activation energy as a function of the phase of the backscattering ${\varphi }_K$ for different values of the bias current $\mu$.

Figure 7

Activation energy $S$ as a function of the inverse backscattering $1/\left|K\right|$ obtained as a solution of Eqs. (13, 14, 15, 16). The current is set at $\mu =1.8$ and the phase at ${\varphi }_k=1.3$.

Figure 8

Activation energy $S$ as a function of the linewidth enhancement factor $\alpha$ when calculated (a) by fitting the MFPT obtained from Monte Carlo simulations of Eqs. (1, 2) and (b) from the Hamiltonian formalism. Both approaches reveal the “knee.” The current is set at $\mu =1.8$, the backscattering amplitude to $k=0.44$ and the phase at ${\varphi }_k=1.2$. (Bottom) Qualitatively different MPEP for (c) $\alpha =5$ and (d) $\alpha =6$ as calculated with Hamiltonian systems (13, 14, 15, 16).