Nonlinear Oscillations and Bifurcations in Silicon Photonic Microresonators

Silicon microdisks are optical resonators that can exhibit surprising nonlinear behavior. We present a new analysis of the dynamics of these resonators elucidating the mathematical origin of spontaneous oscillations and deriving predictions for observed phenomena such as a frequency comb spectrum with MHz-scale repetition rate. We test predictions through laboratory experiment and numerical simulation.

Figure 1

(a) Scanning electron microscope image of a silicon microdisk resonator. (b) Schematic of the physical system, in which a continuous-wave input field results in a periodically oscillating output field. (c)–(f) Steady-state resonance curves for the intracavity field amplitude as a function of normalized detuning. (c) At low power, the curve is stable and single valued. (d) As power increases, nonlinear effects grow and the resonance curve bends over, leading to an unstable middle branch (dashed red). (e) Further increase in pump power leads to two simultaneous Hopf bifurcations (blue dots) and the birth of a stable limit cycle (envelope shown in green). (f) When the limit cycle grows sufficiently large, it collides with the middle branch and is destroyed via a homoclinic bifurcation (this collision occurs in four dimensions and is not visible in this projection). Pump powers: 0.71, 45, 86, and $120\mu \mathrm{W}$.

Figure 2

Limit cycle oscillation. The panels show (a) field amplitude $|a|$, (b) free carrier population $N$, and (c) temperature change $\mathrm{\Delta }T$ vs time. The colors indicate different stages of the limit cycle. The pump power is 1 mW ($S=56$), and the pump detuning is 0.84 nm above resonance ($\delta {\omega }_0/{\gamma }_0=168$).

Figure 3

Phase space diagram of the system in a parameter space of power ($P_{\mathrm{in}}$) and detuning ($\delta {\omega }_0/{\gamma }_0$). Region I (gray): monostable (one stable equilibrium). Region II (blue): bistable (two stable, one unstable equilibria). Region III (pink): stable oscillations (one stable, two unstable equilibria). Region IV (green): monostable (one stable, two unstable equilibria). Blue boundary: saddle-node bifurcation. Red boundary: Hopf bifurcation. Dashed green boundary: homoclinic bifurcation. Black asterisk indicates the point where the Hopf bifurcation goes from subcritical to supercritical. Power ranges from 0 to 1 mW and detuning ranges from 0 to 1.5 nm above resonance.

Figure 4

Limit cycle for 2D and 1D reduction in the space of nondimensional free carrier population $\eta$ and temperature change $\theta$ [Eq. (3) and Eqs. (4) and (5), respectively]. Top panel: stable periodic solution to 2D model (solid), nullcline $d\eta /d\tau =0$ (dashed), unstable fixed points (filled diamonds), and points of interest in the 1D reduction (filled and open circles). Lower panel: same labeling scheme, solid lines represent branches of nullcline corresponding to the 1D limit cycle, arrows indicate instantaneous jumps. Input power is 1 mW ($S=56$) and pump detuning is 0.84 nm above resonance ($x=168$). The color coding indicates the portion of the cycle with the same scheme as Fig. 2.

Figure 5

The existence and period of the limit cycle. Comparison of 4D (blue), 2D (red), and 1D (green) models from Eqs. (2, 3, 4, 5, 6) to experiment (black points; the vertical error bars stem from Lorentzian fits to determine frequency peak locations and represent 1 standard deviation in the comb spacing) for $P_{\mathrm{in}}=400\mu \mathrm{W}$, ${\lambda }_0=1609\mathrm{nm}$, $Q=6\times {10}^5$, $V_{\text{eff}}=60{({\lambda }_0/n_{\mathrm{Si}})}^3$ (fit), ${\gamma }_{\mathrm{lin}}/{\gamma }_0=0.53$ (fit), ${\gamma }_{\mathrm{Th}}=1.4\times {10}^5\mathrm{Hz}$ (fit), and ${\gamma }_e/{\gamma }_0=0.08$. The solid line is the numerical solution, and the dashed line is the analytical approximation.

Figure 6

Frequency comb envelope. The frequency comb’s decay for both experimental data (connected black circles) and 4D numerics (connected green triangles) is compared with a best fit sawtooth pulse wave (solid lines, see inset). The primary spike height is arbitrarily scaled for visual purposes. The red reference line shows $n^{-2}$. The error bars indicate 90% bootstrap confidence interval derived from a Lorentzian fit.