Resonant self-pulsations in coupled nonlinear microcavities

A different point of view on the phenomenon of self-pulsations is presented, which shows that they are a balanced state formed by two counteracting processes: beating of modes and bistable switching. A structure based on two coupled nonlinear microcavities provides a generic example of a system with enhanced ability to support this phenomenon. The specific design of such a structure in the form of multilayered media is proposed, and the coupled-mode theory is applied to describe its dynamical properties. It is emphasized that the frequency of self-pulsations is related to the frequency splitting between resonant modes and can be adjusted over a broad range.

Figure 1

(a) Linear transmission spectrum of the Bragg structure consisting of 64 layers. For convenience, the re-sonances are counted from the upper band edge and marked by the red triangles with numbers. The positive Kerr nonlinearity can bend the linear resonances to the lower frequencies. (b) A typical hysteresis curve for signals with carrier frequency taken inside the band-gap region. The loops created by the resonances for the increasing and decreasing input intensity are shown by arrows. (c) The linear field profile of the second resonance with one of the localization centers highlighted by the red rectangle. The background shows alternation of layers inside the structure.

Figure 2

(a) Example of multilayered structure obtained from the symbolic formula (1), with various functional parts highlighted. (b) Transmission spectrum of the single microcavity for $m=3$ (left inset) that was computed with the transfer-matrix method (solid blue line) and the coupled mode theory (dashed red line). The resonance of perfect transmission caused by the point defect is in the middle of the band-gap region, where the two spectra overlap. (c) Transmission spectrum of two coupled microcavities for $m=3$ and $p=0$. Due to the coupling, the degenerate resonance splits into a doublet of even and odd modes.

Figure 3

(a) A sketch of the model used to describe coupling between microcavities. The response of each element is approximated by a scattering matrix. (b) The hysteresis of output intensity for a single microcavity. The real and imaginary parts of eigenfrequencies obtained from the linear stability analysis are superimposed on the right side. Switching occurs when $\text{Im}({\omega }_{\mathrm{p}})>0$ and $\text{Re}({\omega }_{\mathrm{p}})=0$. (c) The hysteresis of output intensity for two coupled microcavities. Self-pulsing occurs when $\mathit{Im}({\omega }_{\mathrm{p}})>0$ and $\mathit{Re}({\omega }_{\mathrm{p}})\ne 0$.

Figure 4

(a) The intensities of incident and transmitted signals as functions of time during the switching between stable states and after the transition to the self-pulsations. (b) A phase portrait of the switching, which is obtained from the real and imaginary parts of transmission amplitude. The final state corresponds to a stable spiral point. (c) A similar phase portrait for one period of self-pulsations. A limit cycle is formed due to the Hopf bifurcation [12].