Analysis of parameter combinations for optimal soliton microcomb generation efficiency in a simple single-cavity scheme

Dissipative Kerr solitons generated in high-$Q$ optical microresonators provide unique opportunities for different up-to-date applications. Increasing the generation efficiency of such signals is a problem of paramount importance. We perform a comprehensive analytical and numerical analysis using a simple single-cavity scheme. It is revealed that in order to obtain high pump-to-comb conversion efficiency such parameters as coupling rate, pump amplitude, detuning, and microresonator second-order dispersion should not be considered individually, only in the aggregate. The dependence of the optimal coupling rate on the pump power is shown, in addition to the trade-off relations balancing the efficiency versus the number of comb lines. Combining analytical predictions and numerical simulations, we find optimal conditions for the maximal pump-to-comb conversion efficiency (up to 100%) in the cases of free-running and self-injection-locked pump lasers. The discrepancy between numerical and analytical solutions and methods to increase the total comb power are also discussed.

Figure 1

Pump-to-comb conversion efficiency vs normalized pump amplitude for different values of effective pump frequency detuning and normalized dispersion coefficient. Solid lines show the numerical modeling, and dashed lines show the analytical expression (6). All quantities are plotted in dimensionless units.

Figure 2

Pump-to-comb conversion efficiency vs coupling rate and internal losses calculated via (9) for fixed pump power. $\zeta ={\zeta }^{\max }$, and $D_2/{\kappa }_{\max }=0.0143$. The dotted and dash-dotted lines correspond to the critical and minimal threshold couplings. The red solid curve shows $f=2$ [the isocontour for this is ${\overline{\kappa }}_0=({\overline{\kappa }}_c^{1/3}-{\overline{\kappa }}_c){\kappa }_{\max }/f$]. All quantities are plotted in dimensionless units.

Figure 3

Coupling efficiency needed to maintain fixed $f$ for the given pump power calculated from (10) (solid line). The dash-dotted line shows its asymptote for high pump power. All quantities are plotted in dimensionless units.

Figure 4

Pump-to-comb conversion efficiency vs intrinsic-loss-normalized coupling ${\kappa }_c/{\kappa }_0$ and dispersion $D_2/{\kappa }_0$ via (6) and (10) for $\zeta ={\zeta }^{\max }$ and fixed $f$. The left panel shows $f=1$, and the right panel shows $f=2$. The thick vertical line is the minimal $P/P_0=1$, the dark red dashed line is $d_2=1$, and the thin pink solid curve is the tuning of the intrinsic loss for $D_2/{\kappa }_{\max }=0.0143$. The light green solid line is the optimal coupling for fixed $D_2/{\kappa }_0$, and the dash-dotted curve is the optimum (7). All quantities are plotted in dimensionless units.

Figure 5

Pump-to comb-efficiency vs $f_0$ and the number of comb lines calculated via (6) and (10). Additional lines and parameters are the same as in the right panel of Fig. 4. All quantities are plotted in dimensionless units.

Figure 6

Total comb power without the pumped mode, normalized to $P_0$, vs the normalized pump power for different coupling rates. Solid lines correspond to maximum detuning, and dashed lines correspond to locked detuning. Lines 1 and 2 correspond to the threshold pumping case $f=1$ according to (10). Lines 3, 4, and 5 correspond to different fixed couplings. Blue lines 1 correspond to the case with $f=1$ and 2 times less internal losses. The black dotted line shows $P=P_{\mathrm{out}}$, e.g., the 100% efficiency limit. System parameters are ${\kappa }_0/2\pi =188$ MHz and $D_2/2\pi =14.3$ MHz. All quantities are plotted in dimensionless units.

Figure 7

Simulated soliton profiles (solid lines) and their analytical approximations (dashed lines) for the different values of the normalized pump amplitude $f$. The left panel shows the results for the near-cutoff ($\approx 0.95{\zeta }_{\mathrm{eff}}^{\max }$) detunings and optimal dispersion coefficient (7). The right panel shows the results for the SIL detuning and optimal dispersion coefficient (7). All quantities are plotted in dimensionless units.

Figure 8

Simulated pump-to-comb conversion efficiency (left) and its difference from the analytical approximation (6) (right). The blue and orange lines correspond to the optimal dispersion value (7) for the near-cutoff detuning and for the locked detuning. All quantities are plotted in dimensionless units.

Figure 9

Simulated pump-to-comb conversion efficiency (solid lines) and the analytical approximation (6) (dashed lines) for fixed dispersion. The left panel shows the pump sweep for near-cutoff ($\approx 0.98{\zeta }_{\mathrm{eff}}^{\max }$) detunings and different dispersion coefficients $d_2$. The right panel shows the detuning sweep for different normalized pump amplitudes $f$ (lines inside corresponding detuning boundaries) and different dispersion coefficients $d_2$ (line colors); the dotted vertical lines show the SIL (left) and cutoff (right) detunings for the corresponding pump amplitude. Note that while the cutoff detuning is strict for soliton existence, the SIL detuning is generally higher than the lower existence boundary. All quantities are plotted in dimensionless units.

Figure 10

Pump-to-comb dependence on the normalized backscattering coefficient $\beta$. SIL and CO stand for SIL detuning and cutoff detuning. Dispersion is chosen to be optimal (7). Vertical dashed lines show the backscattering-related cutoff. All quantities are plotted in dimensionless units.