Modulational instability and frequency combs in whispering-gallery-mode microresonators with backscattering

We introduce the first-principle model describing frequency comb generation in a whispering-gallery-mode microresonator with the backscattering-induced coupling between the counterpropagating waves. The elaborated model provides deep insights and an accurate description of the complex dynamics of nonlinear processes in such systems. We analyze the backscattering impact on the splitting and reshaping of the nonlinear resonances and demonstrate backscattering-induced modulational instability in the normal dispersion regime and subsequent frequency comb generation. We present and discuss features of the soliton comb dynamics induced by the backward wave.

Figure 1

Forward wave linear (red dash-dotted line) and nonlinear (blue solid line) resonance curves for different $\beta$ at $f=4$. Dashed arrow in the right panel shows that the loop crosses the main resonance region and the system will jump from the first to the second branch while tuning the frequency. Plotted variables are dimensionless.

Figure 2

(a) The root map of (13). The blue (dark) area provides stable solutions, yellow (light) and green (gray) oscillatory and nonoscillatory unstable ones. The red (black) line corresponds to $c_1+c_2=-1$, brown (dark gray) $-c_2={(c_1+4)}^2/4$, and magenta (gray) $c_2=c_1^2/4$; the critical point is $(1;-2)$. (b) The $d_1$ values for which the deviation between maximal real parts of eigenvalues is less than 5%. Plotted variables are dimensionless.

Figure 3

Evolution of the initial soliton for different linear coupling coefficients $\beta$ and finesse values $d_1/2$ at fixed $\alpha =12$, $f=4.11$, and $d_2=0.01$. The bottom figures coincide with the high-finesse case. Plotted variables are dimensionless.

Figure 4

Left: Nonlinear double resonance at anomalous GVD: thick blue, stationary analytical solution (solid, stable; dashed, unstable, high-finesse regime), thin solid red (dark) and yellow (light), solutions of (8) with different initial conditions. Right: Evolution of the spectrum upon frequency scan at anomalous GVD. Plotted variables are dimensionless.

Figure 5

Left: Nonlinear double resonance at normal GVD: thick blue, stationary analytical solution (solid, stable; dashed, unstable, high-finesse regime), thin solid red (dark) and yellow (light), solutions of (8) with different initial conditions. Right: Evolution of the spectrum upon frequency scan at normal GVD (second branch). Plotted variables are dimensionless.

Figure 6

The first sideband number ${\mu }_{\mathrm{sb}}$ and the detuning ${\alpha }_{\mathrm{sb}}$ after which it appears vs linear coupling coefficient $\beta$. Plotted variables are dimensionless.