Attractor hopping between polarization dynamical states in a vertical-cavity surface-emitting laser subject to parallel optical injection

We present experimental and theoretical results of noise-induced attractor hopping between dynamical states found in a single transverse mode vertical-cavity surface-emitting laser (VCSEL) subject to parallel optical injection. These transitions involve dynamical states with different polarizations of the light emitted by the VCSEL. We report an experimental map identifying, in the injected power-frequency detuning plane, regions where attractor hopping between two, or even three, different states occur. The transition between these behaviors is characterized by using residence time distributions. We find multistability regions that are characterized by heavy-tailed residence time distributions. These distributions are characterized by a $-1.83\pm 0.17$ power law. Between these regions we find coherence enhancement of noise-induced attractor hopping in which transitions between states occur regularly. Simulation results show that frequency detuning variations and spontaneous emission noise play a role in causing switching between attractors. We also find attractor hopping between chaotic states with different polarization properties. In this case, simulation results show that spontaneous emission noise inherent to the VCSEL is enough to induce this hopping.

Figure 1

(a) Experimental setup. TL, tunable laser; PC1 and PC2, polarization controllers; VA, variable attenuator; OC, optical coupler; PBS, polarization beam splitter; PM, power meter; BOSA: Optical spectrum analyzer. Experimental optical spectra of the free-running VCSEL at various values of bias current. (b) $I_{\mathrm{bias}}=5.0$ mA and (c) $I_{\mathrm{bias}}=3.0$ mA.

Figure 2

Experimental optical spectra (left) and corresponding time traces (right) of the fundamental states involved in hopping dynamics: P1, P1-both, IL+PS, CH, and CH-both. Black (red, lower) traces correspond to the parallel (orthogonal) polarization signal. The injection parameters for (a)–(f) are $I_{\mathrm{bias}}=5.0$ mA, ${\nu }_i=0.9$ GHz, and, respectively, (a),(b) P1: $P_i=303.8\mu \mathrm{W}$, (c),(d) P1-both: $P_i=144.8\mu \mathrm{W}$, and (e),(f) IL+PS: $P_i=31.62\mu \mathrm{W}$. (g)–(j) are obtained for $I_{\mathrm{bias}}=3.0$ mA, ${\nu }_i=-2.2$ GHz, and, respectively, (g),(h) CH: $P_i=19.87\mu \mathrm{W}$, and (i),(j) CH-both: $P_i=14.21\mu \mathrm{W}$, obtained when decreasing $P_i$.

Figure 3

Experimental optical spectrum (a) and corresponding time trace (b) of the (P1 & IL+PS) state, resulting from an aperiodic switching between P1 and IL+PS. Black (red, lower) trace corresponds to the parallel (orthogonal) polarization signal. The injection parameters are $I_{\mathrm{bias}}=5.0$ mA, ${\nu }_i=0.9$ GHz, and $P_i=53.83\mu \mathrm{W}$.

Figure 4

(a) Experimental mapping of the nonlinear dynamics in which hopping is observed in the injected power-frequency detuning plane. Time series of the (P1 & P1-both) state (b) and the (P1 & P1-both & IL+PS) state (c) are obtained, respectively, for ${\nu }_i=0.7$ GHz, $P_i=120.6\mu \mathrm{W}$ and ${\nu }_i=0.9$ GHz, $P_i=106.2\mu \mathrm{W}$. Black (red, lower) traces correspond to the parallel (orthogonal) polarization signal. (d) Representation in the orthogonal-parallel polarization plane of the P1 & P1-both & IL+PS time series. The three attractors are P1-both, P1, and IL+PS, and the arrows show the order of the sequence. Cases illustrated in Figs. 5, 5 and 5 are indicated in the diagram of (a) by using circle, square, and star, respectively.

Figure 5

Experimental time traces of the orthogonal polarization signal corresponding to the P1 & IL+PS dynamics [(a)–(c)], zoomed parts of the time series [(d)–(f)], and residence time distributions in log-log scale [(g)–(i)]. The system is in the P1 (IL+PS) state when the signal is small (large). The bias current is $I_{\mathrm{bias}}=5$ mA. Left panel: ${\nu }_i=0.8$ GHz, $P_i=48.01\mu \mathrm{W}$. Center panel: ${\nu }_i=0.9$ GHz, $P_i=53.83\mu \mathrm{W}$. Right panel: ${\nu }_i=1.1$ GHz, $P_i=64.71\mu \mathrm{W}$.

Figure 6

Residence time distribution of ${\tau }_2$ for $I_{\mathrm{bias}}=5.0$ mA, ${\nu }_i=1.1$ GHz, and $P_i=64.71\mu \mathrm{W}$ (open dots) and data fit from ${\tau }_2={10}^{-5}$ to ${10}^{-2}$ s giving a decay characterized by a $-1.85$ power law (thick dashed line). Individual distributions obtained with fewer data are plotted with colored dashed lines. Decays characterized by $-3/2$ and $-2$ power laws (thin dashed lines) are also plotted.

Figure 7

Experimental time series of attractor hopping between (a) CH and CH-both, and (b) CH, CH-both, and IL, obtained for $I_{\mathrm{bias}}=3.0$ mA, ${\nu }_i=-2.2$ GHz, and, respectively, (a) $P_i=13.56\mu \mathrm{W}$, and (b) $P_i=15.08\mu \mathrm{W}$. Black (red, lower) trace is the parallel (orthogonal) polarization mode signal.

Figure 8

Simulated time series of attractor hopping between CH and CH-both, obtained for $I_{\mathrm{bias}}=3.0$ mA, ${\nu }_i=-2.2$ GHz, $P_i=0.024$ (a.u), and a noise strength of (a) ${\beta }_{\text{SF}}=2.7\times {10}^{-4}$, and (b) ${\beta }_{\text{SF}}=6.5\times {10}^{-4}$. (c) and (d) are zooms of (a) on a CH and a CH-both time window, respectively, such that (c) has been shifted in $50\mu \mathrm{s}$. (e) and (f) are results obtained with ${\beta }_{\text{SF}}=1\times {10}^{-30}$ for two different sets of initial conditions. Black (red, lower) trace is the parallel (orthogonal) polarization mode signal.

Figure 9

Simulated residence time distributions in log-log scale obtained for $I_{\mathrm{bias}}=3.0$ mA, ${\nu }_i=-2.2$ GHz, $P_i=0.024$ (a.u), and a noise strength of ${\beta }_{\text{SF}}=6.5\times {10}^{-4}$. Black (red, dotted) trace is the parallel (orthogonal) polarization mode signal.

Figure 10

Simulated bifurcation diagram showing the maximum and minimum of the power of both linear polarizations when increasing (squares), and decreasing (open dots) ${\nu }_i$. Results are obtained for $I_{\mathrm{bias}}=5.0$ mA, $P_i=0.022$ (a.u), and ${\beta }_{\text{SF}}={10}^{-30}$. Black (red, lower) symbols correspond to the parallel (orthogonal) polarization mode.

Figure 11

Simulated time series of (a) power of both linear polarizations, and (b) the total population inversion. Results are obtained for $I_{\mathrm{bias}}=5.0$ mA, ${\nu }_i=0.774\mathrm{GHz},P_i=0.022$ (a.u), and ${\beta }_{\text{SF}}=2.7\times {10}^{-4}$. Black (red, lower) lines correspond to the parallel (orthogonal) polarization mode.

Figure 12

Simulated time series of (a) frequency detuning, (b) power of both linear polarizations, and (c) total population inversion. Results are obtained for $I_{\mathrm{bias}}=5.0\mathrm{mA},P_i=0.022$ (a.u), ${\beta }_{\text{SF}}=2.7\times {10}^{-4}$, and ${\tau }_{\text{th}}=1\mu \mathrm{s}$. Black (red, lower) lines correspond to the parallel (orthogonal) polarization mode.