Excitability in semiconductor microring lasers: Experimental and theoretical pulse characterization

We characterize the operation of semiconductor microring lasers in an excitable regime. Our experiments reveal a statistical distribution of the characteristics of noise-triggered optical pulses that is not observed in other excitable systems. In particular, an inverse correlation exists between the pulse amplitude and duration. Numerical simulations and an interpretation in an asymptotic phase space confirm and explain these experimentally observed pulse characteristics.

Figure 1

Experimental MRL setup. (a) The actual MRL device with its contacting. The ring itself is pumped with a current $I_p$. Four waveguide contacts are depicted, of which only port A is biased with $I_w$. (b) The overall setup where the readout and the SOA are separated from each other by a circulator C.

Figure 2

(a) Experimental time series of port A demonstrating the excitable behavior of the MRL operating at $I_p=47.45$ mA. The current on the waveguide of port A is $I_w=13$ mA, and the current in the external SOA is $I_{\mathrm{SOA}}=700$ mA. Examples of (b) a single and (c) a double excited pulse taken from the same time series.

Figure 3

ISI distribution for pulses emitted by the symmetry-broken MRL for device parameters as given in Fig. 2. The logarithm of the number of events $N$ in each ISI bin is shown. (a) The distribution of long ISIs is well fitted by an exponential curve (see gray solid line). The gradient $1/T_1=1.6$ $\mu$s${}^{-1}$ indicates a characteristic time scale of $~0.63$ $\mu$s. (b) The distribution of short ISIs reveals a different time constant $T_2~18.4$ ns. The inset in (a) shows the dependence of the long ISI time scale $T_1$ (with error bars given by the standard deviation from the mean value $T_1$) on the current on the external SOA ($I_{\mathrm{SOA}}$), which is used as noise source.

Figure 4

Distribution of (a) the pulse amplitude and (b) the pulse width for $I_p=47.45$ mA, $I_{\mathrm{SOA}}=700$ mA, and $I_w=13.0$ mA. The number of events $N$ in each bin is plotted. The gray solid line shows the corresponding Gaussian distribution as predicted by the theory presented in Sec. 4.

Figure 5

Measured width and amplitude of the excited pulses for two different values of the bias current $I_p$ on the ring. $I_{\mathrm{SOA}}=0$ mA and $I_w=13.0$ mA. (a) $I_p=42.82$ mA and (b) $I_p=44.49$ mA. A linear fit of the data points is shown by the gray solid line. The correlation coefficient $C$ is approximately 0.7 in both cases.

Figure 6

The phase-space topology of an asymmetric MRL. CW, CCW, and S are two quasiunidirectional states and a saddle state whose stable and unstable manifolds are displayed, respectively. The basin of attraction of the CW (CCW) quasiunidirectional state is white (gray). Parameters are as in Table .

Figure 7

Results of simulations of Eqs. (3, 4, 5). In (a) and (b) $D=1.5\times {10}^{-4}$ ns${}^{-1}$, $t_{\mathrm{obs}}=100$ ms, and in (c) and (d) $D=2.5\times {10}^{-4}$ ns${}^{-1}$, $t_{\mathrm{obs}}=10$ ms, where $D$ is the noise strength and $t_{\mathrm{obs}}$ is the observation time. (a) and (c) Histogram of the trajectories in asymptotically reduced two-dimensional phase space. The solid (dashed) white lines indicate the stable (unstable) manifolds of the saddle. (b) and (d) Pulse width versus pulse amplitude. The white line indicates the prediction from the deterministic reduced model. Parameters are as in Table .

Figure 8

Results of simulations of Eqs. (3, 4, 5). The histogram of events $N/N_{\mathrm{tot}}$ is shown (a) and (c) as a function of the pulse amplitudes and (b) and (d) as a function of the pulse widths. $D=1.5\times {10}^{-4}$ ns${}^{-1}$ for (a) and (b), and $D=2.5\times {10}^{-4}$ ns${}^{-1}$ for (c) and (d). The histogram of the pulse amplitudes can be fitted by a Gaussian distribution: $A\exp \{-[(x-B)/C]{}^2\}$. In (a), $N_{\mathrm{tot}}=$ $15\hspace{0.5em}052$ events, $A=0.0575,B=0.9762,C=0.1237$. In (c), $N_{\mathrm{tot}}=$ $23\hspace{0.5em}748$ events, $A=0.0503,B=1.019,C=0.1508$. The vertical gray lines show the maxima of the histograms: A width of $\approx 9.20$ ns is found in (b), and a width of $\approx 7.99$ ns is found in (d). Parameters are as in Table. .

Figure 9

Simulation of Eqs. (3, 4, 5). Typical pulse shapes corresponding to different pulse widths are shown. (left) Single pulses are collected from the time traces, and (right) the corresponding trajectory in the $(\theta ,\psi )$ phase space is shown in gray. Stable (solid lines) and unstable (dashed lines) manifolds are in black. From (a) to (e), pulses are shown with widths in the following intervals: 5–6, 8–9, 11–12, 14–15, and 17–18 ns. $D=2.5\times {10}^{-4}$ ns${}^{-1}$, and the other parameters are as in Table. .

Figure 10

Simulation of Eqs. (3, 4, 5) for 10 ms, showing the interspike-interval diagram. $\ln (N/N_{\mathrm{tot}})$ is plotted as a function of the interspike interval. In (a) $\mu =1.65$, $D=2.5\times {10}^{-4}$ ns${}^{-1}$, and the bin size is chosen to be 1 ns. In (b)–(d), a zoom of the peak at shorter ISI is shown with a bin size of $0.05$ ns. (b) and (c) show the variation of the ISI with $\mu$ for $D=2.5\times {10}^{-4}$ ns${}^{-1}$ and $D=1.5\times {10}^{-4}$ ns${}^{-1}$, respectively. (d) demonstrates the change in ISI for asymmetric noise contributions to the CW ($D_{\mathrm{cw}}=D$) and the CCW ($D_{\mathrm{ccw}}$) modes for $\mu =1.65$. The other parameters are as in Table. .