Thermal instabilities, frequency-comb formation, and temporal oscillations in Kerr microresonators

We analyze the consequences of dissipative heating in driven Kerr microresonators theoretically and numerically, using a thermal Lugiato-Lefever model. We show that thermal sensitivity modifies the stability range of continuous waves in a way that blocks direct access to broadband frequency-comb-forming waveforms, and we propose a deterministic access path that bypasses the thermal instability barrier. We describe a thermal instability that leads to thermo-optical oscillations via a Hopf bifurcation.

Figure 1

Stability of continuous-wave solutions of the Lugiato-Lefever equation including thermal effects, Eqs. (5) and (6), as a function of pump detuning $\alpha$ and amplitude $F$ for four sets of thermal parameters. Each point in the $\alpha \text{−}F$ plane corresponds to three continuous-wave solutions, of which at most two are stable, inside the blue wedge, and to a unique solution (stable or unstable) outside the wedge. The yellow curve marks the lower boundary of the region of the finite-wavelength modulational instability, which affects only the upper branch of continuous waves for parameter values in the coexistence wedge in the three panels with $C>0$. The green curve, where present, marks the instability threshold of the uniform mode with complex growth eigenvalues, which affects only the upper branch where it overlaps with the coexistence wedge. The dark circles mark the points on the modulational instability boundaries where the fastest growing mode has the largest wavelength. The red dashed line in the upper right panel shows a deterministic path to access perfect soliton crystals.

Figure 2

Stability of continuous-wave solutions of the Lugiato-Lefever equation including thermal effects, Eqs. (5) and (6), as a function of pump detuning $\alpha$ and mean cavity power $P$ for the four sets of thermal parameters of Fig. 1. Each point in the $\alpha \text{−}P$ plane corresponds to a unique continuous-wave solution. The solid blue curve is the boundary of the region where three branches exist for the same pump power, and the dashed blue curve marks the boundary between the branches. The solutions in the shaded regions of the graphs are unstable, with shading color indicating the instability modes as follows: Blue, uniform instability with a real growth eigenvalue; red, long-wave modulational instability; yellow, finite-wavelength modulational instability; and green, uniform instability with complex growth eigenvalues. The significance of the dark circle is the same as in Fig. 1.

Figure 3

Left: Solid lines show deterministic access paths in the pump parameter space leading from the lower tip of the modulational instability curve to soliton crystals for several choices of the thermal parameters $A=C=0,2,4,6,8,10$ from left to right. The access paths are related by the exact mapping in Eq. (12), so that cnoidal waveforms with the same $F$ are identical. Stability properties, however, depend on the thermal parameters, and the cnoidal waves on the access paths for $C=8$ and $C=10$ become unstable before reaching $F=2$. The horizontal dashed lines show the values of $F$ for which the intensity profiles are shown in the right-hand panel. Dashed curves show a small part of the modulational instability threshold for the respective values of $C$. Right: The intensity profile $|{\psi }_s{\left(x\right)|}^2$ for stable cnoidal waves at several points on the access paths shown in the left-hand panel, located at $F=1.01,1.2,1.5,2.0$ (where applicable).

Figure 4

Steady-state thermal oscillations of a spatially uniform wave envelope field in a Kerr resonator. The deviations $\mathrm{\Delta }$ of the field envelope $\psi -{\psi }_c$ (real part in red, imaginary part in blue) and of the thermal detuning $\mathrm{\Theta }-{\mathrm{\Theta }}_c$ (green) from the constant values are shown as a function of time $t$. The left-hand column shows oscillation with thermal parameters $B=2,C=5$, and detuning $\alpha =8$, and the right-hand column shows oscillation with $B=0.6,C=10$, and $\alpha =15$. The pump amplitude is $F=F_h+F_{\mathrm{\Delta }}$, where $F_h$ is the instability threshold, and $F_{\mathrm{\Delta }}=0.0005,0.1,\text{and}2.0$ in the top, middle, and bottom rows, respectively. The quasiharmonic oscillations of the top row are well described by weakly nonlinear analysis. (Note different vertical axis scales.)

Figure 5

The nonlinear response parameter $\left|K\right|$ of the uniform instability Hopf bifurcation as a function of the pump detuning $\alpha$ and cooling rate $B$ for thermal sensitivity ratio $C=5$ (left) and $C=10$ (right). The plots are colored in parameter regions where the bifurcation is supercritical. Uncolored regions correspond either to points where the bifurcation is subcritical (large-$\alpha$ regions) or parameter values where the Hopf bifurcation does not occur at all (small-$\alpha$ regions).