Coarse Grained Variables and Deterministic Chaos in an Excitable System

Temporal coarse graining was applied to the dynamical variables of a semiconductor laser with optical feedback. The chaotic low frequency fluctuations obtained in numerical and experimental data are shown to have properties of a self-excitable deterministic system. External exciting noise is replaced by the ultrafast chaotic oscillations of the system. A low dimensional coarse-grained phase space is defined and time constants are introduced and measured for the exponential drop and recovery of the randomly excited equally shaped spikes.

Figure 1

Scheme of fixed points and manifolds for a bidimensional excitable dynamical system as proposed in 2.

Figure 2

Calculated time series for the laser power: (a) with weak feedback, $\kappa =1.2{\mathrm{ns}}^{-1}$, chaos appears as fast pulsation, in black, but there is no LFF. (b) Stronger feedback, $\kappa =15{\mathrm{ns}}^{-1}$, puts the system above threshold for excitation of LFF events. In gray (color) are the coarse-grained series.

Figure 3

One LFF drop and recovery from Fig. 2b.

Figure 4

Phase portrait: Laser intensity versus the carrier number, with ps time resolution (black), and after coarse graining (red online).

Figure 5

Phase portraits of the laser intensity versus its derivative. (a) Dark dots with the original ps time resolution. (b) An expanded scale of the coarse-grained portrait.

Figure 6

(a) Segment of experimental time series. (b) Phase portrait of the experimental laser intensity with LFF.