Microresonator dynamics with frequency-dependent Kerr nonlinearity

We investigate the ultrafast nonlinear dynamics in a silica ringresonator composed of a silver-nanoparticle-doped core that exhibits rapidly varying frequency-dependent Kerr nonlinearity that even changes sign across a specific frequency, defined as zero-nonlinearity frequency. We model optical propagation in such a resonator through a modified Lugiato-Lefever equation, revealing that the formation of bright cavity solitons can be possible in both positive and negative nonlinearity domains. The intrapulse Raman scattering along with the frequency-dependent Kerr nonlinearity modify the stability and the dynamics of cavity solitons, which we analyze using a bistability analysis, intracavity modulation-instability analysis, and semianalytical variational approach. Our derived analytical results agree well with the direct numerical solutions of the Lugiato-Lefever equation. The generation of a dispersive wave from a cavity soliton encountering higher-order dispersion is also influenced by the location of the zero-nonlinearity frequency, which we describe by providing a modified phase matching equation. Our results and analysis provide a framework for better understanding the dynamics of cavity solitons and their interactions in two different bright-solitonic domains with possible applications in dual-pump spectroscopy, formation of soliton molecules, and counterpropagating solitons to other fields.

Figure 1

(a) Schematic diagram of a silica ringresonator with a single bus waveguide with power reflection and transmission coefficients of $\rho$ and $\mathrm{\Theta }$. The zoomed cross section of the fiber ring is depicted at the top. (b) The GVD and nonlinearity profiles of the fundamental mode of the doped fiber are shown by the solid blue and red lines, respectively, for a filling fraction of $f=6\times {10}^{-3}$ [21]. The GVD and nonlinearity profiles of the undoped fiber with the same geometry are also depicted by light dashed lines. The locations of the ZNF and ZDF are denoted by circles. The location of two inputs are indicated by $①$  (for ${\lambda }_0=1.08\mu \mathrm{m}$) and $②$  (for ${\lambda }_0=1.55\mu \mathrm{m}$).

Figure 2

(a) Temporal ($\tau$) and (b) spectral $[\mathrm{\Omega }=(\omega -{\omega }_0){\tau }_s]$ evolution of a CS over $t$ for $①$  with $X=4.5,\mathrm{\Delta }=-3$. (c) and (d) describe the same for $②$  with $X=4.5,\mathrm{\Delta }=3$. The other realistic parameters are defined in the main text. The temporal and spectral profiles of the unperturbed (dotted trace) and perturbed (solid line) CSs at the output are also depicted on the overhead of each panel.

Figure 3

(a) Kerr bistability curves in ($\mathrm{\Delta },Y$) parameter space from Eq. (5) and (b) cw bistability S curves in ($X,Y$) from Eq. (2) for two launching conditions. The dashed portions indicate the cw-unstable solutions. The turning points in (b) are indicated by $X_{\pm }^{\mathrm{f}}$. (c),(e) Kerr bistability curves are mapped with (d),(f) the corresponding intracavity MI regions. The dashed curves in (c) and (e) are MI-unstable solutions, and the shaded areas in (d) and (f) indicate the MI-unstable regions. (g) MI gain spectrum ${\mathrm{\Lambda }}^{\mathrm{f}}\left(\mathrm{\Omega }\right)$ in the presence of FDKN for two launching conditions with $Y=3.5$. (h) The intracavity cw solutions (black curve), peak intensity of the MI branch (dashed blue curve), and peak intensity of the CSs (solid green curve) as a function of $\mathrm{\Delta }$ under FDKN. The transition points from cw to MI to CS are indicated by ${\mathrm{\Delta }}_{\mathrm{MI}}^{\mathrm{f}}, {\mathrm{\Delta }}_{\uparrow }^{\mathrm{f}}$, and ${\mathrm{\Delta }}_{\max }^{\mathrm{f}}$, respectively.

Figure 4

The variation of (a) peak intensity ($|u_0{|}^2$), (b) temporal pulse width (${\tau }_w=2{\eta }^{-1}$), (c) frequency shift (${\mathrm{\Omega }}_p$), and (d) temporal delay (${\tau }_p$) of a CS over the round-trip time t for $①$ . Similarly, (e)–(h) represent the same for $②$ . The closed form of the saturated frequency [Eq. (15)] and corresponding temporal shift [Eq. (16)] are also plotted by dot-dashed lines.

Figure 5

Temporal evolution of the variational ansatz [Eq. (9)] for (a) $①$  and (b) $②$  under the influence of perturbations, whose pulse parameters are governed by Eqs. (10, 11, 12, 13, 14). The output pulse shapes from the variational ansatz (solid blue curve) and the numerical CS profile (red dotted curve) are shown on the top of each panel.

Figure 6

(a),(b) Spectral evolution of CSs under the influence of a strong TOD (${\delta }_3=0.2$) for two launching conditions. The upper panel corresponds to the output spectrum. The lower panel in each plot represents the graphical solution of the PM equation [Eq. (17)] with zero-crossing frequency that gives the location of the DW. Variation of the radiation frequency with (c) ${\mu }_{1\mathrm{eff}}$ and (d) ${\delta }_3$ for two launching conditions. All the other parameters are the same as in Fig. 2.