Dissipative soliton interaction in Kerr resonators with high-order dispersion

We consider an optical resonator containing a photonic crystal fiber and driven coherently by an injected beam. This device is described by a generalized Lugiato-Lefever equation with fourth-order dispersion. We use an asymptotic approach to derive interaction equations governing the slow time evolution of the coordinates of two interacting dissipative solitons. We show that Cherenkov radiation induced by positive fourth-order dispersion leads to a strong increase of the interaction force between the solitons. As a consequence, a large number of equidistant soliton bound states in the phase space of the interaction equations can be stabilized. We show that the presence of even small spectral filtering not only dampens the Cherenkov radiation at the soliton tails and reduces the interaction strength, but can also affect the bound state stability.

Figure 1

Phase velocity $V$ of small dispersive waves with (a) positive and (b) negative fourth-order dispersion coefficient ${\beta }_4$, and ${\beta }_3=0$. Solid line corresponds to (a) ${\beta }_4=0.025$ and (b) ${\beta }_4=-0.025$. Dashed line corresponds to ${\beta }_4=0$. The parameter values are $S=1.8, \theta =3.5$, and $\delta =0.02$.

Figure 2

(a) Soliton solution of the Lugiato-Lefever equation (1) with ${\beta }_3={\beta }_4=\delta =0$ and (b) eigenvalue spectrum obtained by numerical linear stability analysis of this solution. Other parameters are the same as in Fig. 1.

Figure 3

(a) Soliton solution of the Lugiato-Lefever equation (1) with ${\beta }_4=0.025$ and $\delta =0.02$; (b) eigenvalue spectrum obtained by numerical linear stability analysis of this solution. Other parameters are the same as in Fig. 1.

Figure 4

Right-hand side of Eq. (16) as a function of the soliton separation ${\tau }_2-{\tau }_1$. Black (red) dots indicate the separations of the two solitons in stable (unstable) bound states calculated numerically. Parameter values: $S=2.0, \theta =3.5, \delta =0.02$, and ${\beta }_4=0.025$.

Figure 5

A stable bound state of two dissipative solitons corresponding to a black point in Fig. 4, $\delta =0.02$. (a) Intensity distribution and (b) eigenvalue spectrum. Other parameter values are the same as for Fig. 4.

Figure 6

Unstable bound state of two dissipative solitons corresponding to a red point in Fig. 4, $\delta =0.02$. (a) Intensity distribution and (b) eigenvalue spectrum. Other parameter values are the same as for Fig. 4.

Figure 7

The same bound state as shown in Fig. 5, but calculated for $\delta =0$. (a) Intensity distribution and (b) eigenvalue spectrum. The bound state is unstable with respect to an Andronov-Hopf bifurcation.