Effect of Cherenkov radiation on localized-state interaction

We study theoretically the interaction of temporal localized states in all fiber cavities and microresonator-based optical frequency comb generators. We show that Cherenkov radiation emitted in the presence of third-order dispersion breaks the symmetry of the localized structures interaction and greatly enlarges their interaction range thus facilitating the experimental observation of the dissipative soliton bound states. Analytical derivation of the reduced equations governing slow time evolution of the positions of two interacting localized states in a generalized Lugiato-Lefever model with the third-order dispersion term is performed. Numerical solutions of the model equation are in close agreement with analytical predictions.

Figure 1

The amplitude $\left|E\right|$ of a CS calculated by numerical solution of Eq. (1) in linear scale (left) and the deviation $A\left(\xi \right)$ of the CS amplitude from the background in logarithmic scale (right). Top: CS formation in the $(t,T)$ plane ($d_3=0.2$). Bottom: CS moving uniformly with the velocity $v=0.50679$ ($d_3=0.1$). Other parameters are $\theta =3.5$ and $E_{\mathrm{in}}=2.0$.

Figure 2

Soliton velocity $v$ vs. third-order dispersion coefficient $d_3$. Solid line corresponds to the plot of the asymptotic formula Eq. (2) with numerically calculated $s=3.895$. Dots indicate soliton velocities obtained by means of direct numerical integration of Eq. (1) and by calculation of stationary soliton solutions in the comoving frame with the help of the Newtons iteration method. Parameter values are the same as in Fig. 1.

Figure 3

Neutral mode $\left|\mathbf{u}\right|$ (gray) and adjoint neutral mode $\left|\mathbf{w}\right|$ (black) in logarithmic scale calculated for $d_3=0.1$. Other parameters are the same as for Fig. 1.

Figure 4

Top: The dependence of CS velocities on their time separation. Unlike the velocity of the first CS (black line), which is positive and fast decaying with the increase separation $\tau ={\tau }_2-{\tau }_1$, the velocity of the second CS (gray line) decays very slowly and oscillates fast as $\tau$ changes. Bottom: Difference of CS velocities as a function of their time separation. Zeros of this difference correspond to bound CS states. Numerically calculated CS time separations in the bound states are indicated by dots. Stable (unstable) bound states are shown by filled (empty) dots and correspond to decreasing (increasing) CS velocity difference. $d_3=0.1$, other parameters are the same as in Fig. 1.

Figure 5

Top: Stable bound state of two CSs calculated for $d_3=0.2$. Left CS is almost unaffected by the interaction while the right one has larger peak power and is much stronger modified by the interaction force. Note that for unstable bound states the peak power of the right CS is smaller than that of the left one. Bottom: Frequency comb envelope for a solitary pulse (black) and pulse bound state shown in left panel (gray). The envelope modulation period of the bound state comb is determined by the time separation of the two pulses. Other parameters are the same as in Fig. 1.

Figure 6

Formation of bound states of two (left) and five (right) CSs calculated numerically for $d_3=0.2$. Other parameters are the same as in Fig. 1.