Coherence Resonance Near a Hopf Bifurcation

We report on the observation of coherence resonance for a semiconductor laser with short optical feedback close to Hopf bifurcations. Noise-induced self-pulsations are documented by distinct Lorentzian-like features in the power spectrum. The character of coherence is critically related to the type of the bifurcation. In the supercritical case, spectral width and height of the peak are monotonic functions of the noise level. In contrast, for the subcritical bifurcation, the width exhibits a minimum, translating into resonance behavior of the correlation time in the pulsation transients. A theoretical analysis based on the generic model of a self-sustained oscillator demonstrates that these observations are of general nature and are related to the fact that the damping depends qualitatively different on the noise intensity for the subcritical and supercritical case.

Figure 1

Upper left panel: Scheme of the multisection laser device; see text for explanations. Upper right panel: Power spectrum of electrical noise. Lower panel: Power spectra of the device under the influence of noise. (a) Before the supercritical Hopf bifurcation without external noise (lower plot) and at optimum noise level (upper plot), solid lines are fits with a Lorentzian shape ($I_{\mathrm{DFB}}=70\mathrm{mA}$, $I_{\mathrm{PH}}=11.8\mathrm{mA}$, $I_{\mathrm{A}}=5.62\mathrm{mA}$). (b) The same for the subcritical case ($I_{\mathrm{DFB}}=70\mathrm{mA}$, $I_{\mathrm{PH}}=98.2\mathrm{mA}$, $I_{\mathrm{A}}=30.6\mathrm{mA}$).

Figure 2

Peak height (gray dots), normalized width (open dots), and signal-to-noise ratio (black dots) versus noise intensity as deduced from the power spectra by Lorentzian fits. Note the log scale. (a),(c) Supercritical bifurcation, (b), (d) Subcritical bifurcation. Maxima of curves are normalized to unity.

Figure 3

Width $({\Delta }_{\omega })$, peak height $(H)$, and signal-to-noise ratio $(\beta )$ versus noise intensity $(D)$ as obtained from the theoretical analysis. Left column (a)–(c) Supercritical bifurcation $(F=F_1)$. Right column (d)–(f) Subcritical bifurcation $(F=F_2)$. Symbols “◯” represent the numerical solution of (2), lines are obtained from the analytical treatment. Insets: Shape of the potentials $U_1(r)$ and $U_2(r)$. Dashed lines in (d) show the asymptotic behavior $D^{-1}$ and $D^{2/3}$, respectively (see text for details).