Stability and variational analysis of cavity solitons under various perturbations

We theoretically investigate the dynamics and stability of a temporal cavity soliton (CS) excited inside a silicon-based microresonator that exhibits free-carrier generation as a result of two-photon absorption (TPA). The optical propagation of the CS is modeled through a mean-field Lugiato-Lefever equation (LLE) coupled with an ordinary differential equation accounting for the generation of free carriers owing to TPA. The CS experiences several perturbations [like intrapulse Raman scattering (IRS), TPA, free-carrier absorption (FCA), free-carrier dispersion (FCD), etc.] during its round-trip evolution inside the cavity. We develop a full variational analysis based on a Ritz optimization principle which is useful in deriving simple analytical expressions describing the dynamics of individual pulse parameters of the CS under perturbation. TPA and FCA limit the efficient comb generation and modify the stability condition of the CS. We determine the critical condition of stability modified due to TPA and derive closed-form expressions of the saturated amplitude and width of the stable CS. We perform detailed modulation-instability analysis and obtain stability conditions against perturbations of the steady-state solution of LLE. The CS experiences FCD, which leads to a temporal acceleration resulting in spectral blueshift. Exploiting the variational analysis, we estimate these temporal and spectral shifts. We also include IRS in our perturbation theory and analytically estimate the frequency redshifting. Finally, we study the effect of pump-phase modulation on a stable CS. All our analytical results are found to be in good agreement with the data obtained from the full numerical solution of LLE.

Figure 1

(a) Formation of the CS in presence ($K=0.04$) and in absence ($K=0$) of TPA at $t=20$ is represented by dashed-gray and dashed-blue lines, respectively, when the sech pulse is used as input. The solid gray and blue lines give the variational predictions of the pulse shape for $K=0.04$ and 0, respectively. (b) Temporal evolution of the CS over time $t$ for $K=0.03$ with $\mathrm{\Delta }=3,S=\sqrt{3.5}$.

Figure 2

(a) Kerr bistability in $(X,Y)$ parameter space in the case of a homogeneous field for three different values of $K$. The turning points are shown by $X_{\pm }$ (for $K=0$) and $X_{\pm }^{\mathrm{TPA}}$ (for $K\ne 0$). (b) Kerr bistability in $(X,Y)$ parameter space, where the dashed parts are unstable. (c) The region of the intracavity MI for anomalous dispersion. The modulationally unstable region is indicated by the shaded area. (d) The steady-state CW intracavity power (black curve), peak intensity of the MI patterns (dashed blue curve), and peak intensity of CSs (solid green curve) as a function of $\mathrm{\Delta }$. The variation of (e) temporal pulse width $\left({\tau }_w=2{\eta }^{-1}\right)$ and (f) peak intensity $\left(|u_0{|}^2=E\eta /2\right)$ over the round-trip time $t$ for $K=0.03$. Saturated peak intensity (g) and temporal pulse width (h) as a function of the TPA coefficient. Equations (16) and (17) give the closed-form saturated peak intensity and temporal pulse width, respectively.

Figure 3

Different dynamical regimes of operation and the transition lines in the LLE in the case of TPA.

Figure 4

Kerr bistability in (a) $(\mathrm{\Delta },Y)$ parameter space for two values of ${\varphi }_c$ and in (b) $(X,Y)$ parameter space. (c) The region of the intracavity MI for anomalous dispersion. The modulationally unstable region is indicated by the shaded area. (d) Temporal evolution of the CS profile for realistic $\theta =0.0011$ and $\mu =3.7741$ with the parameters $\mathrm{\Delta }=3$ and $X=3.5$ for a single round trip. The inset gives the same evolution for multiple round trips. The unperturbed (dotted line) and perturbed (solid line) temporal CSs are also shown on the overhead; also the normalized ${\varphi }_c$ is shown at $t=100$ for ${\tau }_c={10}^{-3}$ by the light-green-shaded area on the top. (e) The CW response (black curve) and the peak intensity of CSs (solid green curve) as a function of $\mathrm{\Delta }$. (f) The variation of temporal delay of the CS as a function of $t$ is shown for a single round trip. The red circles give numerical data, whereas the solid blue line gives variational prediction. The closed form of the temporal delay [Eq. (28)] is also plotted by a green dot-dashed line. (g) Comparison between the numerical (red circles) and variational (solid blue line) results of the temporal delay of the CS as a function of $\theta$. (h) Different dynamical regimes of operation in the LLE for $\mu =7.7741$ and $\theta =0.0011$.

Figure 5

(a) Peak intensity of the intracavity field as a function of cavity detuning $\mathrm{\Delta }$ for $X=3.5$. The black curve represents the CW solution, while red c-d and blue a-b curves show CS solutions with and without IRS (${\tau }_R=0.04$), respectively. The dashed curves correspond to unstable solutions. The vertical dashed line represents the point where $\mathrm{\Delta }=3$. (b) Temporal ($\tau$) and (c) spectral $[\mathrm{\Omega }=(\omega -{\omega }_0){\tau }_s]$ evolution of CS profiles for ${\tau }_R=0.04$ with parameters $\mathrm{\Delta }=3$ and $X=3.5$. The unperturbed (dotted trace) and perturbed (solid line) CSs in temporal and spectral domains are also shown on the top of each panel. The inset of (b) gives the spectrogram plot at $t=30$. The variation of (d) frequency shift and (e) temporal delay of the CS over the round-trip time $t$. The saturated frequency (f) and corresponding temporal delay at $t=20$ (g) as a function of ${\tau }_R$.

Figure 6

(a) The evolution of the CS over round-trip time in three dimensions for $\rho =0.005$. (b) Kerr bistability in $(\mathrm{\Delta },Y)$ parameter space. The variation of peak intensity $\left(|u_0{|}^2=E\eta /2\right)$ over the round-trip time $t$ for (c) different values of $\rho$ and for (d) different values of $\sigma$. (e) The variation of output peak intensity of the CS with respect to $\sigma$ for three different sets of $\mathrm{\Delta }$ and $|S_0{|}^2$. For each of these three cases we find three critical values of $\sigma$ [${\left({\sigma }_c\right)}_1\approx 0.378,{\left({\sigma }_c\right)}_2\approx 0.679$, and ${\left({\sigma }_c\right)}_3\approx 0.879$] indicated by solid circles. (f) The variation of numerically simulated peak intensity over the round-trip time $t$ for different values of $\sigma$ at a fixed $\rho$.