Coexistence of dark vector soliton Kerr combs in normal dispersion resonators

We investigate the formation of dark vector dissipative solitons in the presence of nonlinear polarization mode coupling in optical resonators subject to a coherent optical injection in the normal dispersion regime. This simple device is described by coupled Lugiato-Lefever equations. The stabilization of dark dissipative solitons is attributed to a front locking mechanism in the bistable regime as shown in a recent communication [B. Kostet et al., OSA Continuum 4, 1564 (2021)]. Here, we focus on a tristable homogeneous steady state regime. We show that two branches of dark dissipative solitons can coexist for a fixed value of the system parameters. These coexisting solutions possess different polarization states and different peak powers in the microresonator. We characterize their formation by drawing their bifurcation diagrams in regimes far from any modulational instability. It is shown that both branches of localized structures exhibit a heteroclinic collapsed snaking type of behavior. The coexistence of two vectorial branches of dark localized states is not possible without taking into account polarization degrees of freedom.

Figure 1

Schematics of two typical optical resonators: (left) macroscopic fiber resonator and (right) microscopic toroidal resonator. Both devices are injected with linearly polarized light.

Figure 2

Stability regions in the parameter space $E_i-{\theta }_x$. Parameters are ${\theta }_y=5$ and ${\beta }_2=-1$. Region I corresponds to monostability, with the presence of only one stable state. Region II corresponds to bistability between two stable states. Region III corresponds to bistability between one stable state and one modulationally unstable state. Region IV corresponds to tristability between three stable states. Region V corresponds to tristability between two stable states and one modulationally unstable state. Finally, region VI corresponds to tristability between one stable state and two modulationally unstable states. Examples of two consecutive bistable curves and of a tristable curve along the dashed lines (a) and (b) are shown in Figs. 3 and 3 respectively.

Figure 3

Homogeneous steady states. Bistable curve (a) and tristable (b) obtained for ${\theta }_y=5$. These curves are taken along the dashed lines (a) and (b) in Fig. 2 corresponding to ${\theta }_x=2.7$ and ${\theta }_x=6.5$, respectively. Full lines correspond to stable states, dashed lines correspond to unstable states, and dotted lines correspond to modulationally unstable states. Highlighted regions in (a) correspond to zones where DSs can be stabilized.

Figure 4

Normalized Stokes parameters. (a) CW solutions plotted in Fig. 3. (b) CW solutions plotted in Fig. 3. The meaning of the dashed, dotted, and solid lines is the same as in Fig. 3. The parameters are the same as in Fig. 3. The injected light is linearly polarized, with $S_1=S_3=0$ and $S_2=1$.

Figure 5

Profiles of the Stokes parameters $S_0, S_1, S_2$, and $S_3$ as a function of the fast time $\tau$ for the stable DS solutions created in the highlighted regions of Fig. 3. The size of the system was taken as $L=100$, which corresponds to 83.6 $\mu \text{m}$ in physical units. Profiles (a)–(c) correspond to the region highlighted on the left, while (${\mathrm{a}}^{\prime }$)–(${\mathrm{c}}^{\prime }$) correspond to the region highlighted on the right. Injection amplitude values are $E_i=$ (a) 2.2917, (b) 2.2922, (c) 2.2926, (${\mathrm{a}}^{\prime }$) 2.9394, (${\mathrm{b}}^{\prime }$) 2.9406, (${\mathrm{c}}^{\prime }$) 2.9406.

Figure 6

Vector Kerr combs corresponding to Fourier transform of the stable DS solutions shown in Fig. 5. The free spectral range is equal to 2.45 THz.

Figure 7

Double heteroclinic snaking. (a) Bifurcation diagram showing the L2-norm $\mathcal{N}$ as a function of the injected field intensity $E_i$. Stable (unstable) states are denoted by solid (dashed) lines. MI states are denoted by dotted lines. Parameters are ${\theta }_x=2.7$ and ${\theta }_y=5$. (b) Close-up on the snaking curves collapsing onto the Maxwell point of the left bistability. (c) Close-up on the snaking curves collapsing onto the Maxwell point of the right bistability.

Figure 8

Close-up showing the MI branch emerging from the end of the upper snaking curve from Fig. 7.

Figure 9

Tristable heteroclinic snaking. Left panel: Bifurcation diagram showing the L2-norm $\mathcal{N}$ as a function of the injected field intensity $E_i$. Stable (unstable) states are denoted by solid (dashed) lines. MI states are denoted with dotted lines. Parameters are ${\theta }_x=6.5,{\theta }_y=4.5$. The size of the system was taken as $L=200$, which corresponds to 334.4 $\mu \text{m}$ in physical units. Right panel: Close-up on the snaking curves collapsing onto the Maxwell point of each respective bistability showing the coexistence region $C$.

Figure 10

Profiles of the Stokes parameters $S_0, S_1, S_2$, and $S_3$ as a function of the fast time $\tau$ for the stable solutions indicated in Fig. 9. Parameters are the same as in Fig. 9. Injection amplitude values are $E_i=$ (a) 3.2195, (b) 3.217, (c) 3.2209, (${\mathrm{a}}^{\prime }$) 3.3094, (${\mathrm{b}}^{\prime }$) 3.3087, (${\mathrm{c}}^{\prime }$) 3.3085.

Figure 11

Vector Kerr combs corresponding to the stable solutions shown in Fig. 10. The free spectral range is equal to 610 GHz.

Figure 12

Moving vector DS under the effect of group velocity mismatch. (a) Profile of an asymmetric DS due to nonzero group velocity mismatch. (b) Space-time map of the same DS showing the drifting motion. (c) Drift velocity $v$ as a function of the group velocity mismatch coefficient $\mathrm{\Delta }{\beta }_1$.