Bifurcation study of regular pulse packages in laser diodes subject to optical feedback

We study the influence of delayed optical feedback from a short external cavity on the emission dynamics of semiconductor lasers using the Lang and Kobayashi rate equation model. A combination of numerical integration and continuation techniques allows us to bring new light into the bifurcation scenario leading to the regular pulse packages (RPP) regime. We give examples of bistability between RPP and time-periodic or steady state solutions. Our bifurcation study of RPP regime is complemented by an analysis of the dependency of the RPP period on the laser and feedback parameters. We qualitatively study this new dynamical regime by plotting a two-dimensional map in the feedback parameters space. The occurrence of RPP is furthermore associated with a topological change in the bifurcation diagram and accompanied by the creation of a new type of bifurcation bridge between a mode and an antimode.

Figure 1

Numerically computed trajectory in the RPP dynamics using the LK model. We have plotted in (a) the time trace of intensity $P$ variations in time measured in units of the delay $\theta$ and in (b) phase portrait in the $\left[\varphi \left(s\right)-\varphi (s-\theta ),N\right]$ space. Modes are represented by triangles and antimodes by circles. Numbers indicate distinct moment in time during RPP. The parameters are $J=1.155$, $\eta =0.135$, $\theta =70$, $\alpha =5$, and $\omega \theta =-\arctan \alpha$.

Figure 2

Time traces of intensity $P$ variations in time [panels (1)] measured in units of the delay $\theta$ and phase portraits [panels (2)] in the $\left[\varphi \left(s\right)-\varphi (s-\theta ),N\right]$ space showing the development and disappearing of RPP. Modes are represented by triangles and antimodes by circles. In (a) $\eta =0.061$, (b) and (c) 0.064, (d) 0.0688, (e) 0.0707, (f) 0.0715, (g) and (h) 0.073, respectively. The parameters are $J=1.15$, $\theta =70$, $\alpha =5$, and $\omega \theta =-\arctan \alpha$.

Figure 3

Extrema of the laser intensity as a function of the feedback rate $\eta$. The figure shows a representation of different dynamical regimes in the form of the bifurcation diagram. The feedback rate was first (a) increased and then (b) decreased. Arrows indicate the dynamical situations from Fig. 2. In (a) $\eta =0.061$, (b) and (c) 0.064, (d) 0.0688, (e) 0.0707, (f) 0.0715, (g) and (h) 0.073, respectively. The parameters are $J=1.15$, $\theta =70$, $\alpha =5$, and $\omega \theta =-\arctan \alpha$.

Figure 4

Bifurcation diagrams of the laser intensity $P$ as a function of the feedback rate $\eta$ for different delay $\theta$. In (a) $\theta =50$, (b) 70, and (c) 90. The parameters are $J=1.155$, $\theta =70$, $\alpha =5$, and $\omega \theta =-\arctan \alpha$.

Figure 5

Period (a) and frequency (b) of envelope of RPP as a function of injection current. Indices $\left(\mathrm{RPP}1\right)$, $\left(\mathrm{RPP}2\right)$, $\left(\mathrm{RPP}3\right)$ and $\left(\mathrm{RPP}4\right)$ indicate the order of the period of RPP. The parameters are $\eta =0.2$, $\theta =85$, $\alpha =5$, and $\omega \theta =-\arctan \alpha$.

Figure 6

Map of the RPP dynamics in the $(\theta ,\eta )$ space. The parameters are $J=1.155$, $\alpha =5$, and $\omega \theta =-\arctan \alpha +\pi$. The inset illustrates the phase dependency of RPP: in black the map is plotted for $\omega \theta =-\arctan \alpha$ and in gray $\omega \theta =-\arctan \alpha +\pi$.

Figure 7

Period of RPP in units of ${\tau }_p$ versus feedback strength $\eta$. In (a) $\theta =50$ and in (b) $\theta =70$. Vertical lines represent the value of $\eta$ for which creation on new pair of ECMs takes place. The parameters are $J=1.155$, $\alpha =5$ and $\omega \theta =-\arctan \alpha$.

Figure 8

An enlargement of Fig. 7a in the vicinity of the creation of ECMs together with a phase trajectory in the $\left[\varphi \left(s\right)-\varphi (s-\theta ),N\right]$ space for different values of the feedback strength $\eta$. In (b) $\eta =0.2154$, (c) 0.2168 (d) 0.2182, and (e) 0.2213. The parameters are $J=1.155$, $\theta =50$, $\alpha =5$, and $\omega \theta =-\arctan \alpha$.

Figure 9

Period of RPP in units of ${\tau }_p$ versus delay $\theta$ in units of ${\tau }_p$ for different values of the injection current. Vertical lines represent the value of $\theta$ for which creation on new pair of ECMs takes place. The parameters are $\eta =0.3$, $\alpha =5$, and $\omega \theta =-\arctan \alpha$.

Figure 10

Stability of ECMs and branches of time-periodic solutions in the route to RPP dynamics. Stable (unstable) branches of steady and time-periodic solutions are indicated in solid (dashed) lines. Diamonds indicate Hopf bifurcations, triangles are used for period doubling bifurcation and torus bifurcations are indicated by stars. The parameters are $J=1.155$, $\theta =70$, $\alpha =4$, and $\omega \theta =-\arctan \alpha$.

Figure 11

Bifurcation diagram of $P$ in the route to RPP dynamics, for the same parameters than in Fig. 10. This figure complements Fig. 10 in that it shows the complex dynamics that emerge after the time-periodic solutions have bifurcated. A window of RPP dynamics is shown with the arrow.