Free-carrier-driven Kerr frequency comb in optical microcavities: Steady state, bistability, self-pulsation, and modulation instability

Continuous-wave pumped optical microresonators have been vastly exploited to generate a frequency comb (FC) utilizing the Kerr nonlinearity. Most of the nonlinear materials used to build photonic platforms exhibit nonlinear losses such as multiphoton absorption, free-carrier absorption, and free-carrier dispersion which can strongly affect their nonlinear performances. In this work, we model the Kerr FC based on a modified Lugiato-Lefever equation (LLE) along with the rate equation and develop analytical formulations to make a quick estimation of the steady state, bistability, self-pulsation, and modulation instability (MI) gain and bandwidth in the presence of nonlinear losses. The analytical model is valid over a broad wavelength range as it includes the effects of all nonlinear losses. Higher-order $\left(>3\right)$ characteristic polynomials of intracavity power describing the steady-state homogeneous solution of the modified LLE are discussed in detail. We derive the generalized analytical expressions for the threshold (normalized) pump detuning that initiates the optical bistability when nonlinear losses are present. Free-carrier dispersion-led nonlinear cavity detuning is observed through the reverse Kerr tilt of the resonant peaks. We further deduce the expressions of the threshold pump intensity and the range of possible cavity detuning for the initiation of the MI considering the presence of nonlinear losses. The proposed model will be helpful in explaining several numerical and experimental results which have been previously reported and thereby will be able to provide a better understanding of the comb dynamics.

Figure 1

Effect of 2PA on optical bistability. (a) Normalized detuning Δ vs normalized 2PA coefficient $Q_2$. (b) Saddle-node positions $X_{\pm }$ and $X_{2\mathrm{PA}\pm }$ with $Q_2$ when Δ is fixed at 2.216. Intracavity power $Y$ with the change in input pump power; $X$ for (c) $\mathrm{\Delta }=\sqrt{3}$, (d) $\mathrm{\Delta }=2.216$, (e) $\mathrm{\Delta }=3.496$, and (f) $\mathrm{\Delta }=10$, with a set of five different values of $Q_2$, in each case $\left(Q_2=0,0.2,0.5,1.0,\mathrm{and}1.5\right)$. Bistable behavior initiates at $\mathrm{\Delta }=\sqrt{3}$, 2.216, and 3.496 when the $Q_2$ is 0, 0.2, and 0.5, respectively. For $Q_2>2\sqrt{3}$, bistability does not occur even if Δ is as high as 10.0 (a,f) with positive and realistic values of $X$ and $Y$.

Figure 2

The possible range of 2PA coefficients for which bistability can occur. Intracavity power $Y$ with respect to normalized detuning Δ at (a) $Q_2=0$, (c) $Q_2=4$. Input pump power $X$ vs Δ, at (b) $Q_2=0$, (d) $Q_2=4.0$. Bistability occurs at a positive value of $Y$ and $X$ if $Q_2=0$ ($<2\sqrt{3}$), whereas bistability can be obtained for a very small value of Δ, with negative values of $X$ and $Y$ when Δ = 4 ($>2\sqrt{3}$).

Figure 3

Effect of 2PA on Kerr tilt. (a) The ratio of maximum normalized intracavity power $\left(Y_{max}\right)$ and the normalized pump power ($X$) with 2PA parameters, and (b) normalized detuning (Δ) with 2PA coefficient ($Q_2$) where the maxima occur. Kerr tilts with different 2PA coefficients when normalized input power is (c) $X=1$ and (d) $X=1.5$.

Figure 4

(a) Bistability curve at $\mathrm{\Delta }=2.216$, and (b) Kerr tilt at $X=1$, for a set of six different values of 2PA-induced FCA-FCD ($C_2$ and $K_2$, respectively). (c) Dependence of Kerr tilt on input pump power $X$ when $C_2=K_2=5$. For all the cases, the 2PA coefficient $Q_2$ is taken as 0.2.

Figure 5

Effective detuning ${\mathrm{\Delta }}_{\mathrm{eff}}$ [diagonal, red (light gray) curve] and FCD-induced detuning ${\mathrm{\Delta }}_{\mathrm{FCD}}$ [horizontal, green (light gray) curve], with the variation in pump detuning when (a) $C_2=K_2=2$, and (b) $C_2=K_2=5$ at input pump power; $X=1$ in both cases. Input pump detuning is also plotted [blue (deep gray) curve] in (a,b) to indicate the reference level in the absence of FCA-FCD. ${\mathrm{\Delta }}_{\mathrm{FCD}}$ can possess multiple values if FCA-FCD coefficients are high.

Figure 6

Kerr tilt ($Y/X$) with the change in effective detuning, ${\mathrm{\Delta }}_{\mathrm{eff}}$, instead of pump detuning, Δ for two different pump powers: (a) $X=1$ and (b) $X=2.5$. FCD-induced steady-state cavity detuning, ${\mathrm{\Delta }}_{\mathrm{FCD}}$ with the intracavity power, $Y$ when the free carriers are generated due to the (c) 2PA [blue (deep gray) solid curve], (d) 3PA [red (light gray) solid curve], and 4PA [green (light gray) dotted curve].

Figure 7

(a) MI-gain lobes ${\lambda }_{\pm }$ in the absence and presence of 2PA for $\mathrm{\Omega }>0$, (b) ${\lambda }_{+}$ in the presence and absence of 3PA. Simulation parameters are $s=1, Y=2.5, D=7.5, Q_2=1, K_2=0.05, C_2=0.1, {\theta }_{c2}=0.05, {\theta }_{c3}=6.31\times {10}^{-7}, Q_3=8.73\times {10}^{-4}, C_3=0.049, K_3=4.9$. Effect of third-order dispersion $\left(d_3=0.1\right)$ is also considered in (b) [red (light gray) dashed curve]. Third-order dispersion induces asymmetry in the gain lobe with respect to $\mathrm{\Omega }=0$.

Figure 8

Real part of the MI gain λ vs (a) Ω and Δ (at $Y=2.5$), and (b) Ω and $Y(\mathrm{at}\mathrm{\Delta }=7.5)$ in the absence of all nonlinear losses. Variation of λ with (c) Ω and $\mathrm{\Delta }\left(\mathrm{at}Y=2.5\right)$, (d) Ω and $Y(\mathrm{at}\mathrm{\Delta }=7.5)$ while 2PA and FCA-FCD are present. Change in λ with the change in (e) Ω and $\mathrm{\Delta }\left(\mathrm{at}Y=2.5\right)$, and (f) Ω and $Y(\mathrm{at}\mathrm{\Delta }=7.5)$ when 3PA and FCA-FCD are present. λ with (g) Ω and $\mathrm{\Delta }\left(\mathrm{at}Y=2.5\right)$, and (h) Ω and $Y(\mathrm{at}\mathrm{\Delta }=7.5)$ in the presence of 4PA and FCA-FCD. For all the cases, third-order dispersion $\left(d_3=0.1\right)$ is taken into consideration. Other simulation parameters are $s=1, {\theta }_{c2}=0.0005, Q_2=0.93, C_2=29.81, K_2=7.5 (\mathrm{at}{\lambda }_0\sim 1.56\mu \mathrm{m})$; ${\theta }_{c3}=6.31\times {10}^{-7}, Q_3=8.73\times {10}^{-4}, C_3=0.049, K_3=4.9$ (at ${\lambda }_0\sim 2.4\mu \mathrm{m}$); ${\theta }_{c4}=6.018\times {10}^{-8}, Q_4=6.16\times {10}^{-6}, C_4=4.66\times {10}^{-3}, K_4=2.9$ (at ${\lambda }_0\sim 4.0\mu \mathrm{m}$). The cyan and magenta lines overlaid on the 2D plots indicate the corresponding values of λ for $\mathrm{\Delta }=7.5$ and $Y=2.5$, respectively.

Figure 9

MI gain λ with respect to Ω and Δ when $Y$ is fixed at 2.5 in (a) the absence of all nonlinear losses, (b) the presence of only 3PA and no FCA-FCD, and (c) the presence of 3PA and FCA-FCD. MI gain λ with respect to Ω and Δ when at $Y=10.5$ in (d) the absence of all nonlinear losses, (e) the presence of only 3PA and no FCA-FCD, and (f) the presence of 3PA, FCA-FCD. It is observed that if input power $X$ is high (high intracavity power, $Y$) parametric oscillation ceases to occur in the presence of FCA-FCD along with 3PA.

Figure 10

(a) Maximum MI gain with the change in intracavity power $Y$ with a set of four 2PA coefficients $\left(Q_2=0,0.2,0.5,0.9\right)$. (b) Range of Δ for which MI can initiate with different 2PA coefficients $\left(Q_2=0,0.2,0.5,0.9\right)$.