Deterministic Optical Rogue Waves

Experimental observations of rare giant pulses or rogue waves were done in the output intensity of an optically injected semiconductor laser. The long-tailed probability distribution function of the pulse amplitude displays clear non-Gaussian features that confirm the rogue wave character of the intensity pulsations. Simulations of a simple rate equation model show good qualitative agreement with the experiments and provide a framework for understanding the observed extreme amplitude events as the result of a deterministic nonlinear process.

Figure 1

(a) Experimental bifurcation diagram displaying the optical spectra of the slave laser versus the bias current. The dashed line denotes the zero detuning condition between the master and the slave lasers. (b) Typical time series of the slave laser intensity, where rare large pulses can be observed. The experimental parameters are $J=0.976\mathrm{mA}$, $\Delta \nu =-1.34\mathrm{GHz}$, and $P_{\mathrm{inj}}=1.1\mathrm{mW}$.

Figure 2

Experimental histograms, without (a) and with (b) rogue waves. $N$ is the number of pulses detected on the time traces of the slave laser intensity. The vertical dashed lines denote the mean value plus 8 standard deviations and thus mark the limit for a pulse to be considered a rogue wave. The histogram in (b) corresponds to the time series displayed in Fig. 1b ($J=0.976\mathrm{mA}$), while the histogram in (a) was obtained by slightly decreasing the current ($J=0.972\mathrm{mA}$).

Figure 3

(a) Lyapunov exponents as function of slave laser pump current and the master-slave frequency detuning. The color code indicates negative exponents as gray scale and positive exponents as black-white (yellow-red) scale. The solid line indicates a simultaneous variation of $\mu$ and $\Delta \nu$ that results in a sequence of bifurcations very similar to that observed experimentally. (b) Detail of box shown in (a). The points labeled $A$ and $B$ indicate the parameters corresponding to Fig. 4. (c) Number of RWs, $N$, vs the slave laser pump current and the frequency detuning. $N$ was calculated over time series spanning $10\mu \mathrm{s}$. The model parameters are $\alpha =3$, $\kappa =300{\mathrm{ns}}^{-1}$, ${\gamma }_n=1{\mathrm{ns}}^{-1}$, $P_{\mathrm{inj}}=60{\mathrm{ns}}^{-2}$, and $D=0$.

Figure 4

Numerically calculated laser intensity (a),(d) and the PDFs of the pulse amplitude (b),(e) and intensity (c),(f), for parameters corresponding to point $A$ (a)–(c) in Fig. 3 ($\mu =1.8$, $\Delta \nu =0\mathrm{GHz}$), and to point $B$ (d)–(f) in Fig. 3 ($\mu =1.96$, $\Delta \nu =-1.86\mathrm{GHz}$).

Figure 5

(a) Time series of the laser intensity obtained by simulating the stochastic rate equations, without filter (dashed line) and with filter (solid line). PDFs of the pulse amplitude (b) and intensity (c). The noise strength is $D={10}^{-5}{\mathrm{ns}}^{-1}$; other parameters are as in Fig. 4, right-hand column.