Figure 1

(a) Image of the double wheel oscillator. (b) FEM calculation of the fundamental mode. (c) Basic scheme of the experimental apparatus. O.I.: optical isolator; H: half-wave plate; PD: photodiode; PBS: polarizing beam splitter; BS: beam splitter.Reuse & Permissions

Figure 2

Normalized reflected signal as the laser frequency is (a) increased and (b) decreased (scanning rate 6 MHz/ms) for $P_{\mathrm{in}}=40$ mW. Red (gray) trace in (a) shows the Lorentzian reflection dip obtained for $P_{\mathrm{in}}=0.1$ mW. (c) State diagram reconstructed from reflected signal spectra as $P_{\mathrm{in}}$ is varied: In black regions, the system has one stable fixed point. Gray regions refer to the coexistence of a fixed point and a nonstationary state and white regions to the coexistence of oscillating solutions. Normalized reflected intensity $I={\left|\varepsilon \right|}^2$ as ${\delta }_0$ is linearly (d) increased and (e) decreased in time (rate of 6 MHz/ms), during the numerical integration of Eqs. (1, 2, 3). Parameters: $\omega =1.2$, $Q=5000$, $\alpha =4.{910}^{-2}{P}_{\mathrm{in}}$, ${\omega }_{PH}=2.{410}^{-4}$, and $\beta =3.{510}^{-1}{P}_{\mathrm{in}}$ with $P_{\mathrm{in}}=40$ mW. Red (gray) trace in (d) shows the Lorentzian reflection dip obtained for $P_{\mathrm{in}}=0.1$ mW. (f) State diagram obtained from numerical reflection spectra.Reuse & Permissions

Figure 3

Experimental time series of the reflected intensity for $P_{\mathrm{in}}=50$ mW: (a) $\Delta \nu =-3.9$ MHz, (b) $\Delta \nu =-3.6$ MHz, (c) $\Delta \nu =-3.2$ MHz, (d) $\Delta \nu =-2.6$ MHz, (e) $\Delta \nu =0.8$ MHz. Time series of the reflected intensity $I={\left|\varepsilon \right|}^2$ as obtained by numerical integration of Eqs. (1, 2, 3), with $P_{\mathrm{in}}=50$ mW and initial conditions [$I=1,\varphi =(\alpha /{\omega }^2)I,\theta =-\beta I$]: (f) ${\delta }_0=4$, (g) ${\delta }_0=4.22$, (h) ${\delta }_0=4.25$, (i) ${\delta }_0=4.29$, (j) ${\delta }_0$ = 14.9. Other parameters as in Fig. 2.Reuse & Permissions

Figure 4

Experimental time series of the reflected intensity for $P_{\mathrm{in}}=50$ mW: (a) $\Delta \nu =-1.1$ MHz, (b) $\Delta \nu =-3.4$ MHz, (c) $\Delta \nu =-3.7$ MHz. Time series of the reflected intensity $I={\left|\varepsilon \right|}^2$ as obtained by numerical integration of Eqs. (1, 2, 3), with $P_{\mathrm{in}}=50$ mW and initial conditions [$I=0.1,\varphi =(\alpha /{\omega }^2)I,\theta =-\beta I$]: (d) ${\delta }_0=6.5$, (e) ${\delta }_0=4$, (f) ${\delta }_0=3$. Other parameters as in Fig. 2.Reuse & Permissions

Figure 5

(a) Bistable regimes (shaded regions) calculated by means of Eq. (5), together with the Hopf bifurcation boundaries [Eq. (8)] of the high-intensity steady state (solid lines) and low-intensity stable state (dashed line), implicitly defined by Eq. (4). LL: coexistence of nonstationary solutions. LF: coexistence of a fixed point and a nonstationary solution. FF: coexistence of fixed points. (b) Hopf instability domain [Eq. (8)] and the stationary solutions (solid line) in the $(I_s,{\delta }_0)$ plane, for $P_{\mathrm{in}}=50$ mW (other parameters as in Fig. 2). The intersections between the boundaries of the shaded region and the $I_s$ curve determine the Hopf bifurcation points. In white regions, the fixed points are stable.Reuse & Permissions

Figure 6

Left panels: three-dimensional reconstruction of the phase portrait through the Ruelle-Takens embedding technique with time delay $\tau =1.74$ $\mu$s, from experimental time series of the optical intensity. (a) Hopf quasiharmonic cycle [see Fig. 3a], (b) small-amplitude chaotic attractor [see Fig. 3b], and chaotic spiking regime [see Fig. 3c]. Right panels: the corresponding numerical phase-space trajectories (light curves) together with the slow manifold (black curve) (4) in the $(I,\theta )$ plane. Parameters as in Figs. 3f, 3g, 3h.Reuse & Permissions