Modulation instability of Kerr optical frequency combs in dual-coupled optical cavities

Kerr optical frequency combs generated in a coherently driven Kerr nonlinear resonator have the potential for a wide range of applications. However, in a single cavity which is a widely adopted configuration for Kerr optical frequency-comb generation, modulation instability is suppressed in the normal dispersion regime, and the pump-to-comb conversion efficiency is extremely low for a single dissipative Kerr soliton (DKS) in the anomalous dispersion regime. Dual-coupled cavities have been proposed to generate Kerr optical frequency combs in the normal dispersion regime, and have the potential to remarkably increase conversion efficiency for Kerr optical frequency combs. Here, we investigate modulation instability and Kerr optical frequency-comb formation in dual-coupled cavities. Based on solutions of the continuous-wave steady state, we obtain a quadric algebraic equation describing the modulation instability gain, and we find that it is intensely influenced by the scaled free spectral range mismatch between the two cavities. Our numerical simulations demonstrate that platicons can be generated via a pump scanning scheme for the case that both the two cavities possess normal dispersion, and a single DKS can be generated in the cavity with anomalous dispersion while the dispersion of the other cavity is normal. The conversion efficiency of a single DKS is determined by the pump power and detunings as well as intrinsic quality factors of two cavities. Our analysis of modulation instability provides a powerful tool for Kerr optical frequency-comb generation via pump modulation and cavity detuning tuning scheme in dual-coupled cavities.

Figure 1

Schematic of the OFC generation in dual-coupled cavities. The continuous-wave pump $A_{\mathrm{in}}$ is coupled into cavity 1 through a bus waveguide. The Kerr nonlinearity of the system gives rise to optical frequency combs $A_1$ in cavity 1 and $A_2$ in cavity 2.

Figure 2

Intracavity temporal profiles and corresponding comb spectra. (a) and (c) The DKS in cavity 2 and the temporal profile in cavity 1. (b) and (d) Corresponding comb spectra. (e) and (f) The evolution of the DKS in cavity 2 and the temporal profile in cavity 1, respectively.

Figure 3

Continuous-wave steady-state intracavity power as a function of detunings. The parameters $\alpha =1.17, \mathrm{\Gamma }=1, \kappa =12.05$, and $S=10$ are used for computing. (a) $Y_2$ and (b) $Y_1$ versus ${\mathrm{\Delta }}_2$, with ${\mathrm{\Delta }}_1=0$. The orange (green) arrows show the intracavity power path assuming MI does not occur, when ${\mathrm{\Delta }}_2$ is tuned from blue (red) detuning to red (blue) detuning. (c) $Y_2$ and (d) $Y_1$ versus ${\mathrm{\Delta }}_1$, with ${\mathrm{\Delta }}_2=0$. (e) $Y_2$ and (f) $Y_1$ versus ${\mathrm{\Delta }}_1$, with ${\mathrm{\Delta }}_2={\mathrm{\Delta }}_1$. The intracavity power shows tilted curves caused by the Kerr nonlinearity.

Figure 4

MI gains calculated with parameters as follows: $d=-300, {\mathrm{\Delta }}_1=5$, and ${\mathrm{\Delta }}_2=5$, and other parameters are the same as in Fig. 3. Note that the cw steady state is multistable under these parameters, and the modulation instability gains are computed on the upper branch. (a) ${\eta }_1=1$ and ${\eta }_2=1$, i.e., both cavity 1 and cavity 2 have normal dispersion. (b) ${\eta }_1=1$ but ${\eta }_2=-1$, i.e., cavity 1 has normal dispersion but cavity 2 has anomalous dispersion. The inset in (b) is zoom-in MI gain identical to the one in (a). (c) MI gain spectrum for a wide range of $d$, with dispersion parameters the same as in (a).

Figure 5

The simulated comb for the “$+/+$” case. (a) Evolution of the comb spectrum in cavity 2 with the normalized detuning ${\mathrm{\Delta }}_1$. (b) Normalized average intracavity power in cavity 1 (red line) and in cavity 2 (blue line). Orange dots denote the cw steady-state power in cavity 2 calculated by Eqs. (7) and (8). (c)–(e) The MI gain at ${\mathrm{\Delta }}_1=-5, {\mathrm{\Delta }}_1=-0.5$, and ${\mathrm{\Delta }}_1=8$, respectively. (f) and (g) Temporal profiles and corresponding comb spectra in cavity 2 for the different positions i–iii denoted by white lines in (a).

Figure 6

The simulated comb for the “$+/-$” case. (a) The MI gain. (b) Evolution of the temporal profile in cavity 2 with roundtrips. (c) and (e) A single DKS and the corresponding comb spectrum. (d) and (f) Pump modulation and detuning tuning scheme.

Figure 7

Pump-to-comb conversion efficiency of a single DKS with different input pump power and detunings. (a) The comb power (blue) and conversion efficiency (red) as a function of the pump power with detunings ${\delta }_1=0.0555{\mathrm{mm}}^{-1}$ and ${\delta }_2=0.776{\mathrm{mm}}^{-1}$. The blue shadow area represents no DKSs existing. (b) The conversion efficiency with different detunings for pump power $P_{\mathrm{in}}={\left|A_{\mathrm{in}}\right|}^2=180$ mW. DKSs cannot exist in the white area. The black “$\times$” marks the highest conversion efficiency point.