Optimal frequency combs from cnoidal waves in Kerr microresonators

We present a thorough computational study of the existence, stability, and comb properties of cnoidal waves—dissipative periodic patterns—in Kerr microresonators. We show that cnoidal waves comprise a large set with multiple periods. Optimal comb power efficiency and bandwidth are obtained for highly red-detuned, intermediate-strength pump, and short-period waves, that are similar to a train of cavity solitons. We demonstrate a deterministic access path for optimal waves that yields combs of soliton-class bandwidth with a much higher power efficiency.

Figure 1

Constant-$\alpha$ sections of the CnW stability domain in the $\alpha \text{−}f\text{−}p$ parameter space. The section boundaries are shown in thick (green) curves. The dark-, medium-, and light-hued parts of the boundary curves correspond to type-$\mathsf{E}$, $\mathsf{H}$, and $\mathsf{F}$ instabilities (respectively). Unstable CnWs may exist outside the section boundaries. Dashed horizontal lines, where present, are the upper stability limits of continuous waves. In other sections the upper limit of continuous-wave stability is at the bottom of the CnW stability region. The interiors of the sections are colored by the comb power efficiency ${\eta }_c$, according to the color key shown on the right.

Figure 2

Stability sections of Fig. 1, with interiors colored by the rms bandwidth ${\mathrm{\Omega }}_b$, according to the color key shown on the right.

Figure 3

Projection of the principal-branch CnW stability region on the $\alpha \text{−}f$ pump-parameter plane, colored by optimal comb power efficiency ${\eta }_c$ (top) and rms bandwidth ${\mathrm{\Omega }}_b$ (bottom). The light-hued part of the lower CnW stability boundary is at the same time the upper stability boundary of continuous waves for $\alpha <{\alpha }_c$. For $\alpha >{\alpha }_c$, the upper continuous-wave stability boundary is shown by the dashed line, of which the compatible part (see text) is colored in gold (light hue).

Figure 4

(top) Intensity ${\left|\psi \right|}^2$ as a function of propagation length $x$ of the CnW with period $p=1.87\pi$ and pump parameters $\alpha =6.01,f=4.07$ (dark red) and of the $p=2.35\pi$-period CnW for $\alpha =6.01,f=5.88$ (light red). (bottom) Same as for top panel but for $p=1.87\pi$-period CnW (full dark red) and soliton (dashed black), with pump parameters $\alpha =6.01$, $f=4.07$.

Figure 5

The real part of the waveform obtained by numerical simulation of Eq. (1) with pump parameters $\alpha =1.6$ (top) and $\alpha =1.7$ (bottom) in the threshold-compatible interval, and $f=f_m\left(\alpha \right)+{10}^{-4}$, slightly above the modulational instability threshold, in a cavity length of $5p_m\left(\alpha \right)$, where $p_m$ is the period of the critical mode. Initial conditions (light blue) are randomly perturbed continuous waves. In both cases, like for others not shown here, the critical mode initially grows harmonically (medium green), before settling on the period-$p_m$ principal branch CnW (dark brown).

Figure 6

A deterministic access path in pump-parameter space starting with compatible subcritical modulational instability (green dot) and following the green arrow to the comb power efficiency optimal CnW (marked with a red dot). If pump detuning $\alpha$ and amplitude $f$ can only be controlled separately, the diagonal path should be replaced by a step-shaped path (not shown) where $f$ is first increased to 2.2, then $\alpha$ is increased from 1.7 to 6, and then $f$ is increased again to its final value of 4.07.

Figure 7

Snapshots showing the growth of a CnW along the access path of Fig. 6. Legend shows the detuning values $\alpha$ of the CnW.

Figure 8

Fundamental domain of stability spectra near instabilities of type $\mathsf{E}$ (left, detail, inset showing full domain, $\alpha =1.84$, $f=1.375$, $p=1.14\pi$), $\mathsf{H}$ (middle, $\alpha =1.84$, $f=1.375$, $p=2.92\pi$), and $\mathsf{F}$ (right, $\alpha =1.84$, $f=1.6$, $p=1.54\pi$). Shown are the real parts of the hydrodynamic band (thick blue) and the least stable nonhydrodynamic band (amber, for the type-$\mathsf{H}$ and -$\mathsf{F}$ cases) as a function of stability wave number $k$ in units of $\pi /p$, $p$ being the CnW period. The two curves join for $k$ values where the two least stable eigenvalues are a complex-conjugate pair. See text for further explanation.