Hopf bifurcation to square-wave switching in mutually coupled semiconductor lasers

Using advanced continuation techniques for dynamical systems, we elucidate the bifurcations leading to asymptotically stable square-wave pulsing and polarization mode switching in semiconductor lasers with mutual time-delayed and polarization rotating coupling. We find that the increase of coupling strength leads to a cascade of Hopf bifurcations on a mixed-mode steady state up to a transcritical bifurcation on a so-called pure-mode steady state where both lasers emit with the injected polarization state. From these successive Hopf bifurcations emerge time-periodic solutions that have a period close to the laser relaxation oscillation for weak coupling but a period close to twice the time delay for large coupling strength. The wave form of the time-periodic solutions also evolves from harmonic pulsing up to square-wave pulsing as has been observed recently in experiments.

Figure 1

We plot as a function of the coupling strength $\eta$ (a) the intensity of the $I_y$ polarization of the pure-mode solution, (b) the function determining the transcritical bifurcation $g_{1,y}-1$, and (c) the function determining the Hopf bifurcation $f_H$ in Eq. (18). The red (dark gray) square in (a) and (b) is for the transcritical bifurcation point. The green (light gray) dot in (a) and (c) is for the Hopf bifurcation.

Figure 2

Bifurcation diagram of $I_{y,1}$ as a function of the coupling strength $\eta$, as computed using a continuation method. Branches of mixed-mode (black) and pure-mode (red) steady states are plotted together with the Hopf bifurcations labeled H1 to H13 on the mixed-mode steady state. The branches of time-periodic solutions emerging from the different Hopf points are plotted with different colors. The stable parts of the branches of time-periodic solutions are shown in bold. Parameters are defined in the text.

Figure 3

Period of the time-periodic solutions along the different branches shown in Fig. 2 and emerging from the Hopf points labeled H1 to H13. The stable parts of the branches of time-periodic solutions are shown in bold. The dashed lines indicate a time period equal to $\tau$ or $2\tau$, where $\tau$ is the time delay in the laser mutual coupling.

Figure 4

Samples of the time-periodic pulsing computed along the different branches of time-periodic dynamics in Fig. 2. Only the $x$-mode [red (light gray)] and $y$-mode [blue (dark gray)] intensities of laser 1 are plotted, but the solutions are the same for laser 2. (a) Branch 1, $\eta =49.98$ ns${}^{-1}$, (b) branch 6, $\eta =50.14$ ns${}^{-1}$, (c) branch 10, $\eta =49.95$ ns${}^{-1}$, (d) branch 11, $\eta =50.02$ ns${}^{-1}$, and (e) branch 12, $\eta =50.00$ ns${}^{-1}$. Branch $j$ means that it emerges from the Hopf bifurcation labeled $j$ in Fig. 2.

Figure 5

Square-wave switching with a period close to the delay time $\tau$, obtained by direct integration of the rate equations. The coupling strength is $\eta =49.60$ ns${}^{-1}$. Only the $x$-mode [red (light gray)] and $y$-mode [blue (dark gray)] intensities of laser 1 are plotted, but the solutions are the same for laser 2.

Figure 6

Comparison between stable and unstable square-wave dynamics at the $2\tau$ period, along the branch emerging from Hopf bifurcation H12. (a) Stable solution, $\eta =50.00$ ns${}^{-1}$ and (b) unstable solution, $\eta =51.74$ ns${}^{-1}$. Only the $x$-mode [red (light gray)] and $y$-mode [blue (dark gray)] intensities of laser 1 are plotted, but the solutions are the same for laser 2.