Finesse and four-wave mixing in microresonators

We elaborate a comprehensive theory of the sharp variations of the four-wave-mixing (FWM) threshold happening with the tuning of the pump frequency along the dispersive tails of the nonlinear resonance in the driven Kerr microresonators with high finesses and high finesse dispersions. Our theory leads to the explicit estimates for the difference in the pump powers required for the excitation of one sideband pair with momenta $\pm \mu$ and for the simultaneous excitation of the two neighboring pairs with momenta $\pm \mu , \pm (\mu +1)$. This power difference also measures the depth of the Arnold tongues in the pump-frequency and pump-power parameter space, associated with the aforementioned power variations. A set of select pump frequencies and powers, where the instabilities of the two sideband pairs come first, forms the threshold of complexity, which is followed by a sequence of conditions specifying critical powers for the simultaneous instabilities of three, four and so on pairs. We formally link finesse dispersion and the density of states notion borrowed from the condensed matter context, and report a value of the finesse dispersion and a critical mode number signaling the transition from the low- to the high-contrast tongue structure. We also demonstrate that the large finesse dispersion makes possible a surprising for the multimode resonators regime of the bistability without FWM.

Figure 1

Normalised intra-resonator cw-state power computed from Eq. (12): $\mathcal{W}/{\mathcal{W}}_{*}=5$ (full line), $\mathcal{W}/{\mathcal{W}}_{*}=100$ (dashed line). Horizontal blue line shows the FWM threshold as per Eq. (22b). Magenta points are given by Eqs. (16) for $\mu =0$. $g/\gamma$ equals the intraresonator cw power in Watts, see Table 1.

Figure 2

An illustration of how the quasicontinuous residual spectrum, $|{\mathcal{F}}_d|=|D_2|/\kappa \ll 1$, leads to the coinciding FWM and complexity thresholds. (a) $g/\kappa =g_{\mu }^{\left(i\right)}/\kappa$ vs. ${\delta }_0/\kappa$ for $\mu =0,1,\cdots 64$ and ${\mathcal{F}}_d=0.005$ (anomalous dispersion). This is an example of the quasicontinuous residual spectrum forming the quasismooth line converging to $g/\kappa =1/2$, see the last column in Table 1 for scaling of $g/\kappa$ to the cw-state power, $g/\gamma$. Yellow shading shows the range corresponding to the middle branch of the cw-state. Magenta line is the cw-state for $\mathcal{W}/{\mathcal{W}}_{*}=2$. (b) Normalized threshold laser power, see Eq. (24), vs. ${\delta }_0/\kappa$ computed using the $g_{\mu }^{\left(i\right)}$ data from (a) taken outside the yellow shading, so that the plot applies to the upper branch of the cw-resonance. The color bar shows the number of the simultaneously growing sideband pairs. Magenta line is drawn at $\mathcal{W}/{\mathcal{W}}_{*}=2$.

Figure 3

An illustration of how the sparse residual spectrum, $|{\mathcal{F}}_d|=|D_2|/\kappa \sim 1$, leads to the diverging FWM (blue points) and complexity (red points) thresholds. (a) $g=g_{\mu }^{\left(i\right)}$ vs. ${\delta }_0$ for $\mu =0,1,\cdots 13$ and ${\mathcal{F}}_d=0.5$ (anomalous dispersion). Yellow shading shows the range of parameters corresponding to the middle branch of the cw-state, $g_0^{\left(2\right)}<g<g_0^{\left(1\right)}$. Gray shading shows the range of $\mu \ne 0$ instabilities. Magenta line shows the ${\delta }_0<0$ tail of the upper branch of the cw-state for $\mathcal{W}/{\mathcal{W}}_{*}=2000$. The in-figure numbers show $\left|\mu \right|$ for the respective tongues. For the chosen $\mathcal{W}/{\mathcal{W}}_{*}$, the $\left|\mu \right|=9,10,11$ tongues make isolated in ${\delta }_0$ instability intervals. One sideband pair, $\pm \mu$, is unstable inside the quasirhombs starting at the blue points, i.e., FWM threshold. Two pairs, $\pm \mu , \pm (\mu +1)$, are simultaneously unstable inside the quasirhombs extending upwards from the red points, i.e., threshold of complexity. (b) Normalized threshold laser power, see Eq. (24), vs. ${\delta }_0/\kappa$ computed using the $g_{\mu }^{\left(i\right)}$ data from panel (a) taken outside the yellow shading. The color bar shows the number of the simultaneously growing sideband pairs. Magenta line is drawn at $\mathcal{W}/{\mathcal{W}}_{*}=2000$. Panel (c) is like panel (a), but for ${\mathcal{F}}_d=-0.5$ (normal dispersion). Magenta line shows the lower and middle branches of the cw-state, $\mathcal{W}/{\mathcal{W}}_{*}=50$. The $\left|\mu \right|=4$ tongue makes an isolated instability interval around ${\delta }_0/\kappa \approx 5$. For scaling of $g/\kappa$ in panels (a) and (c) to the intraresonator power $g/\gamma$, see the last column in Table 1.

Figure 4

(a) ${\mathcal{F}}_d=0.1$: Transition from the red-red crossing of the $\mu$ and $\mu +1$ instability lines to the red-black ones, $\widehat{\mu }\approx 5.3, |{\widehat{\delta }}_0/\kappa |\approx 0.4$, see Sec. 4. The last red-red crossing is $\mu =5$ to $\mu =6$. (b) ${\mathcal{F}}_d=1.5$: In this case, all the crossings are the red-black ones, since ${\mathcal{F}}_d>{\tilde{\mathcal{F}}}_d$. Blue lines show the minima of the tongues, i.e., FWM threshold, $g/\kappa =1/2$, and the $g_{\mu }^{\left(1\right)}=g_{\mu }^{\left(2\right)}$ points, where $g/\kappa =1/\sqrt{3}$, i.e., bistability threshold. Black and red lines correspond to $g_{\mu }^{\left(1\right)}$ and $g_{\mu }^{\left(2\right)}$, respectively.

Figure 5

(a) Functions ${\mathcal{R}}_{j{j}^{\prime }}$ vs. residual finesse, ${\alpha }_{\mu }$. ${\mathcal{R}}_{j{j}^{\prime }}=0$ constitute equations for the threshold of complexity. ${\mathcal{R}}_{22}$ correspond to the red-red intersections, and ${\mathcal{R}}_{12}, {\mathcal{R}}_{21}$ to the red-black ones, see Fig. 4. (b) Bistability without FWM for ${\mathcal{F}}_d=6$. First three blue lines show the bistable cw-states, when both the upper and lower branches are stable relative to the sideband excitations. Two top blue lines have the upper branch entering the $\left|\mu \right|=1$ instability range. The pump values are $\mathcal{W}/{\mathcal{W}}_{*}=1$, 1.2, 1.4, 1.6, 1.8 for the blue lines from bottom to top.

Figure 6

FWM (blue dashed line) and complexity (red full line) thresholds in the sideband coupling strength, $g$, and the residual finesse, ${\alpha }_{\mu }\approx \mu {\mathcal{F}}_d$, parameter space. For scaling of $g/\kappa$ to the intraresonator power $g/\gamma$, see the last column in Table 1. Green shading corresponds to the stable cw-state. Yellow shading corresponds to the $\left|\mu \right|$-specific instability tongues, i.e., Arnold tongues, intermitting with the intervals of stability. Blue shading corresponds to the simultaneous instabilities with different $\left|\mu \right|$, and with the cw stability being fully lost. Dashed red lines show the higher-order thresholds for 3 and 4 sideband pairs to be unstable simultaneously.