Inverse Cherenkov effect and control of optical solitons by resonant dispersive waves

The interactions of optical solitons with dispersive waves of low intensity in optical fibers with high-order dispersion are considered numerically and analytically. It is shown that Cherenkov phase matching between the solitons and the dispersive waves makes it possible to achieve an efficient interaction between the solitary and the incident dispersive waves. The inverse Cherenkov effect (resonant absorption of dispersive wave by the soliton) and the effect of phase locking of the solitons to the dispersive waves are demonstrated numerically. To explain the phenomena observed in numerical simulations, a simple analytical theory is developed and compared against the numerical data.

Figure 1

The dependence of the square root of the field amplitude on time is shown in panel (a) for the propagation distance $z=0.5$. The green arrow shows the direction of the incident radiation velocity. The dynamics of $Q_s$ and ${\delta }_s$ are illustrated in panels (b) and (c), respectively, for the different initial phases of the incident field ${\theta }_p=m\pi /3$, $m=0,1,2,3,4,5$. Thick black curves correspond to dependencies extracted from direct numerical modeling, and the thinner red curves show the solution of the quasiparticle model. The vertical dashed lines show the propagation distances where the soliton and the incident radiation pulse overlap. The dependencies of the soliton parameters $Q_s$ and ${\delta }_s$ on the initial phase ${\theta }_p$ at propagation distance $z=7$ are shown in panels (d) and (e). The black dots show the data extracted from numerical simulations. The parameters are $\beta =-0.015$, ${\delta }_s=0$, $Q_s=11.9$, $a_p=0.04$, ${\delta }_p=-35.8$, $w_p=75$, $t_p=65$, and initially the soliton is situated at $t=-40$.

Figure 2

XFROG diagrams for propagation distance $z=0.5$ as shown in panel (a). This diagram looks very similar independently of the initial phase of the incident radiation. Panels (b) and (c) show the XFROG diagrams at $z=7$ for the initial phase ${\theta }_p=0$ and ${\theta }_p=\pi$. The width of the reference pulse is $w_r=2$, the other parameters as for Fig. 1.

Figure 3

The families of the stable and unstable equilibrium points of the quasiparticle model are shown in panel (a) by solid and dashed curves, respectively. The thinner blue curve shows the phase trajectory coming to a stable equilibrium point. The dependencies of the real and imaginary parts of the eigenvalues governing the stability of the equilibrium points on the mutual phase between the soliton and the dispersive wave are shown in panel (b). The third-order dispersion and the parameters of the dispersive radiation are the same as for Fig. 1.

Figure 4

Panel (a) shows the evolution of the field distribution for the case of nearly perfect absorption of the radiation by the soliton. The initial distribution of the field was obtained by numerical simulation of the propagation of a soliton and the transformation of the resulting field (see text for details). The amplitude of the initial soliton is 5.97. The high-order dispersion is the same as in the previous simulations. Panel (b) illustrates propagation of the field when the simulation started, with similar initial conditions as for panel (a), the only difference being that the initial field is set to zero in a short interval $7<t<10$. The evolution of the soliton parameters $Q_s$ and ${\delta }_s$ are shown in panels (c) and (d) for the propagation regimes shown in panel (a) (black curves) and in panel (b) (red curves).

Figure 5

Panel (a) shows the distribution of the square root of the absolute value of the initial field distribution (left vertical axis) and the distribution of the instantaneous frequency of the dispersive wave (right vertical axis.) The dynamics of the soliton parameters $Q_s$ and ${\delta }_s$ are shown in panels (b) and (c) for different initial phases of the incident field ${\theta }_p=m\pi /3$, $m=0,1,2,3,4,5$. The black curves show the results of the direct numerical simulations, and thinner red curves are obtained by solving the quasiparticle model. The parameters are $\beta =-0.015$, ${\delta }_s=0$, $Q_s=11.85$, $a_p=0.04$, ${\delta }_p=-36$, $w_p=95$, $t_p=100$, $\xi =0.0049$, $\eta =0.000344$, and initially the soliton is situated at $t=-20$.