Self-sustained pulsations in a quantum-dot laser

We analyze a delay differential equation for the amplitude of the electrical field in order to understand recent experimental observations of low-frequency oscillations in a QD laser. The laser contains no saturable absorber section and exhibits no relaxation oscillations. We investigate the problem both analytically and numerically. We show that there exists a homoclinic bifurcation from a cavity mode that is responsible for the generation of low-frequency pulsating oscillations. We discuss the role of optothermal effects in the formation of the pulsed dynamics.

Figure 1

Bifurcation diagram of the CMs $n=0$ and $n=2$ in the interval $0<{\varphi }_0<2\pi .$ ${\varphi }_H$ and ${\varphi }_c$ denote a Hopf bifurcation point and a homoclinic bifurcation point, respectively. As ${\varphi }_0-{\varphi }_H$ progressively decreases from zero, stable oscillations appear and the figure represents the extrema determined numerically. As ${\varphi }_0-{\varphi }_c$ increases from zero, a branch of periodic solutions emerges with a large period. The figure represents the time-average of $R^2$ obtained numerically. The values of the fixed parameters are $\Gamma =0.375,$ $J=7,$ $\kappa =0.48,$ and $\alpha =2.5.$

Figure 2

Numerical time traces: (a) bursts of spiking oscillations for ${\varphi }_0=1.9$ (b) large amplitude periodic oscillations for ${\varphi }_0=-0.5$ (or $\varphi =5.78).$

Figure 3

Time trace of low-frequency large-amplitude pulsations for ${\varphi }_0=5.748.$ $R_c^2$ corresponds to the CM limit point described in the legend of Fig. 1.

Figure 4

Unfiltered (a) and filtered (b) time traces of low-frequency large-amplitude pulsations. ${\gamma }_{\text{th}}=0.001;{\varphi }_0=4;\beta =0.8$ and $\Gamma =3,$ $J=5,$ ${\tau }_f=10.$ The other parameters remain the same as described in the legend of Fig. 1.