Pulse-timing symmetry breaking in an excitable optical system with delay

Excitable systems with delayed feedback are important in areas from biology to neuroscience and optics. They sustain multistable pulsing regimes with different numbers of equidistant pulses in the feedback loop. Experimentally and theoretically, we report on the pulse-timing symmetry breaking of these regimes in an optical system. A bifurcation analysis unveils that this originates in a resonance phenomenon and that symmetry-broken states are stable in large regions of the parameter space. These results have impact in photonics for, e.g., optical computing and versatile sources of optical pulses.

Figure 1

(a) Bifurcation diagram of (1), showing the pulse intensity $I_p$ with respect to $\tau$, with the number of pulses per feedback loop along each stable periodic solution branch. (b1) Enlargement of the framed area in (a), with further enlargments around point (b2) $T_3$ and (b3) $S_3$. Stable equidistant ($E$) and nonequidistant ($N$) pulse solutions are represented in dark and light blue, respectively, and unstable $E$ and $N$ solutions in dark and light orange, respectively. The dots indicate Hopf ($H$), torus ($T$), period doubling ($P$), saddle-node ($S$), and homoclinic ($L$) bifurcations. (c1)–(c4) Floquet multipliers at points $P$, $T_3$, $T_4$, and $T_5$, with critical multipliers highlighted in red.

Figure 2

(a1) Maximum $I_p$ of pulse intensity and (a2) relative interpule timings $t_p$ along the branches of two equidistant and two nonequidistant pulses, with respect to $\tau$. Stable and unstable solutions are represented in blue and red, respectively. (b) Simulation of (1) for $\tau =1000$ [gray lines in panel (a)] with initial condition very near the (unstable) two-pulse solution, showing the long-term evolution of (b1) $I_p$ and of (b2) $t_p$. The subpanels in (b1) show the intensity time series during the two first and two last roundtrips through the feedback loop; the dots and arrows indicate the amplitudes and relative timings as represented in (b1) and (b2), respectively.

Figure 3

(a) Intensity profiles of coexisting periodic solutions of (1), for $\tau =1000$ and $\kappa =0.2$. (b) Regions of stability, in the $(\tau ,\kappa )$-plane of feedback parameters, of the families of equidistant ($E$) and nonequidistant ($N$) periodic solutions with one to eight pulses per feedback loop. The number of pulses is indicated in the colored regions, and the star indicates the parameter point $(\tau ,\kappa )=(1000,0.2)$ of the time series in panels (a).

Figure 4

Schematic of the experimental setup showing the optically pumped excitable micropillar laser with delayed optical feedback from an external mirror. DM: Dichroic mirror; BS: Beam splitter with 70/30 power split between reflected and transmitted path; MO: Microscope objective; APD: Avalanche photodiode; L: Lens with $f=5$ cm, M: High reflectivity feedback mirror; BD: Beam dump; $\mu \mathrm{Pillar}$: Micropillar laser; $\tau$: External cavity round trip time.

Figure 5

Experimental intensity pulse trains shortly after (a1) two and (b1) three external perturbations, and after a large number of roundtrips in the external cavity with associated calibration of timings between the respective (a2) two and (b2) three sustained pulses; colored arrows indicate the interpulse timings as represented in Fig. 6. The feedback delay is 8.2 ns.

Figure 6

Evolution over the roundtrip number of the relative interpulse timing $t_p$ (as shown by arrows in Fig. 5) of experimental pulse trains following (a) two and (b) three external perturbations, for a feedback delay of $\tau =8.2\text{ns}$: just after the external perturbation [panels (1)], during the convergence towards nonequidistant pulsing patterns [panels (2)], and in the long-term [panels (3)].