Spike latency and response properties of an excitable micropillar laser

We present experimental measurements concerning the response of an excitable micropillar laser with saturable absorber to incoherent as well as coherent perturbations. The excitable response is similar to the behavior of spiking neurons but with much faster time scales. It is accompanied by a subnanosecond nonlinear delay that is measured for different bias pump values. This mechanism provides a natural scheme for encoding the strength of an ultrafast stimulus in the response delay of excitable spikes (temporal coding). Moreover, we demonstrate coherent and incoherent perturbations techniques applied to the micropillar with perturbation thresholds in the range of a few femtojoules. Responses to coherent perturbations assess the cascadability of the system. We discuss the physical origin of the responses to single and double perturbations with the help of numerical simulations of the Yamada model and, in particular, unveil possibilities to control the relative refractory period that we recently evidenced in this system. Experimental measurements are compared to both numerical simulations of the Yamada model and analytic expressions obtained in the framework of singular perturbation techniques. This system is thus a good candidate to perform photonic spike processing tasks in the framework of novel neuroinspired computing systems.

Figure 1

Median of the response amplitude $R$ (a) and median of the pulse response delay (b) versus normalized perturbation energy $P$ for different bias pumps with respect to the self-pulsing pump threshold. The perturbation and the response are normalized respectively to the excitable threshold ($P_{\mathrm{th}}^{99\%}$) and to the response at excitable threshold ($R_{\mathrm{th}}^{99\%}$) for a bias pump $P$ equal to $99\%$ of the pump at the self-pulsing threshold.

Figure 2

(a) Response pulse delay ${\tau }_d$ from Yamada model with spontaneous emission (full line) versus incoherent perturbation initial condition $G_0$ for different pump values ${\mu }_1$ from ${\mu }_1=2.8$ to ${\mu }_1=-18.2$ in steps of 1.5. (b) Numerical delay ${\tau }_d$ (circles) and analytical expression ${\tau }_d^{*}$ given by (2) for ${\mu }_1=2.9$ (dashed). The full line corresponds to a fit of the numerical delays (dashed orange line) with expression (2). (c) Numerical maximum response pulse for ${\mu }_1=2.9$ (full line) and analytical approximation. In (b) and (c) ${\beta }_{\mathrm{sp}}=0$ and $I\left(0\right)=0.01$.

Figure 3

(a): Experimental measurements of the amplitude of the response $R_2$ to a second perturbation pulse after an excitable response has been triggered by a first perturbation pulse at $t=0$ for different delays between the two perturbations. The perturbation amplitude $P$ of the second pulse is scaled to the perturbation amplitude at the excitable threshold $P_{\mathrm{th},\infty }$ for the delay 1.51 ns for which the system has almost completely recovered. [(b)–(d)] Same with the model [Eqs. (1)] and parameters: (b) $b_1=0.001$, $b_2=0.002$; (c) $b_1=0.001$, $b_2=0.001$; and (d) $b_1=0.0001$, $b_2=0.001$. In the simulations, a first excitable response is triggered at $t=0$ and is followed by a second $\delta$-like incoherent perturbation at a variable delay $\tau$. The perturbation is characterized by a gain carrier increase $G\left({\tau }^{+}\right)\equiv G_1$ with $G\left({\tau }^{-}\right)\equiv G_{\tau }$. The pump parameter is ${\mu }_1=2.8$ and the other parameters are listed in the text.

Figure 4

Net gain $R\left(t\right)=G\left(t\right)-Q\left(t\right)-1$ for the same set of parameters as Fig. 3 but different gain and SA recombination rates $b_{1,2}$ and for an initial excitable pulse triggered at $t=0$. Black vertical thick lines materialize the perturbations on the gain for $b_1=0.001$.

Figure 5

Left panel: Response amplitude for a coherent perturbation at $\lambda =980.47\mathrm{nm}$ and different bias pumping with respect to the self-pulsing threshold $P_{\mathrm{SP}}$ pumping value. The response and perturbation amplitudes are scaled to their maximum value for $P/P_{\mathrm{SP}}=94.3$. Upper left panel: Excitable threshold dependence for coherent perturbations versus bias pump.

Figure 6

Maximum intensity $I_M$ of the response pulse to a mixed perturbation characterized by initial conditions at $0^{+}$ for $\{I,G,Q\}$:$I_0,G_0,Q_0={\mu }_2$. Left: ${\mu }_1=1.5$. Right: Zoom for $I_0$ small and $G_0$ close to the threshold value. Parameters are the one given in the text except ${\beta }_{\mathrm{sp}}=0$. Dashed lines: Analytic formula for the excitable threshold Eq. (4). Sections of the surface are marked by black dotted lines and shown in the side panels for $G_0=1.5$, $G_0=3.5$, $I_0=1$, $I_0=40$.