Similarity properties in the dynamics of delayed-feedback semiconductor lasers

Semiconductor lasers with delayed feedback exhibit two fundamentally different dynamical states: weak and strong chaos. We characterize experimentally the mechanism for the emergence of strong chaos. Based on these insights, we demonstrate similarity properties for long delays, i.e., similar dynamics for different pump currents when adjusting the feedback strength. For different delay times, even the same time- and amplitude-rescaled version of the dynamics can be generated. Using a simple rate-equation model, these properties can be corroborated. The results have major consequences for the characterization and tailoring of the dynamics for applications.

Figure 1

Scheme of the experimental setup. LD: laser diode; Circ: optical circulator; PC: polarization controller; Att: optical attenuator; 95/05: one by two splitter with 95% and 5% coupling ratios; $\to$: optical isolator; and PD: photodiode.

Figure 2

Maximum of the first delay echo as a function of the feedback attenuation for the DM laser for four different operating currents: $1.1I_{\mathrm{th}}$ (circle), $1.25I_{\mathrm{th}}$ (square), $1.5I_{\mathrm{th}}$ (triangle), and $2I_{\mathrm{th}}$ (cross), respectively. For clarity, we indicate on the top abscissa the feedback strength, i.e., the estimated fraction of emitted power reinjected into the laser.

Figure 3

Measured optical spectra vs attenuation for a current of $1.5I_{\mathrm{th}}$. (a) Vertical dashed-dotted and dotted lines indicate the solitary laser frequency ($f_{\mathrm{SOL}}$) and the corresponding relaxation oscillation frequency ($f_{\mathrm{RO}}^S$), respectively. The dashed curve denotes the position of the spectrally shifted feedback-induced high-gain region. The horizontal solid line indicates the attenuation for which we observed the smallest first delay echo. (b) is a zoom of the region close to the solitary frequency for weak feedback conditions.

Figure 4

Maximum of the first delay echo as a function of the feedback attenuation for (a) four different operating currents of the DM laser and (b) three different operating currents of the DFB laser. The pump currents are $1.1I_{\mathrm{th}}$ (circle), $1.25I_{\mathrm{th}}$ (square), $1.5I_{\mathrm{th}}$ (triangle), and $2I_{\mathrm{th}}$ (cross), respectively.

Figure 5

(a) Maximum of the first delay echo and (b) full first delay echo for two different dynamical conditions. Crosses (a) and the dashed line (b) correspond to $2I_{\mathrm{th}}$, ${\tau }_1=75.18\mathrm{ns}$. Triangles (a) and the solid line (b) correspond to $1.566I_{\mathrm{th}}$, ${\tau }_2=99.95\mathrm{ns}$. Both curves in (b) correspond to attenuations at the minimum of each respective AC curve in (a). The time axis is rescaled by $\Delta t^{\prime }=\frac{{\tau }_1}{{\tau }_2}\Delta t$ for the solid line in (b).

Figure 6

Maximum of the first autocorrelation peak as a function of the feedback attenuation for four different operating currents: $1.1I_{\mathrm{th}}$ (circle), $1.25I_{\mathrm{th}}$ (square), $1.5I_{\mathrm{th}}$ (triangle), and $2I_{\mathrm{th}}$ (cross), respectively. In (b) the feedback rate has been scaled as $\sqrt{p_{\mathrm{exc}}}$.

Figure 7

Color-coded amplitude of the numerically calculated optical spectra vs feedback rate for a pump of $p_{\mathrm{exc}}=0.5$ ($I=1.5I_{\mathrm{th}}$). The vertical dashed-dotted and dotted lines indicate the solitary laser frequency and its relaxation oscillation frequency, respectively. The horizontal solid line indicates the feedback for which we have the minimum of the AC, and the oblique dashed line depicts the position of $\Delta f_{\mathrm{fb}}$. (b) is a zoom of the region close to the solitary laser frequency for low feedback strengths of (a).