Vector-soliton collision dynamics in nonlinear optical fibers

We consider the interactions of two identical, orthogonally polarized vector solitons in a nonlinear optical fiber with two polarization directions, described by a coupled pair of nonlinear Schrödinger equations. We study a low-dimensional model system of Hamiltonian ordinary differential equations (ODEs) derived by Ueda and Kath and also studied by Tan and Yang. We derive a further simplified model which has similar dynamics but is more amenable to analysis. Sufficiently fast solitons move by each other without much interaction, but below a critical velocity the solitons may be captured. In certain bands of initial velocities the solitons are initially captured, but separate after passing each other twice, a phenomenon known as the two-bounce or two-pass resonance. We derive an analytic formula for the critical velocity. Using matched asymptotic expansions for separatrix crossing, we determine the location of these “resonance windows.” Numerical simulations of the ODE models show they compare quite well with the asymptotic theory.

Figure 1

The exit velocity as a function of the input velocity for $\beta =0.05$, $\beta =0.2$, and $\beta =0.6$, from Tan and Yang [16], original authors’ annotations removed.

Figure 2

The input vs output velocity of a pair of orthogonally polarized solitons with $\beta =0.05$.

Figure 3

Top: The exit velocity graph for Eq. (4) for $\beta =0.2$, showing reflection windows and a variety of more complex fractal-like structures. Bottom: The same figure with all but the main resonant reflections removed.

Figure 4

Plots of the $w\left(t\right)$ component of Eq. (4) with initial velocity $v=0.09988$ (top) and $v=0.13464$ (bottom) and $\beta =0.05$, showing the 2-5 and 2-6 resonances.

Figure 5

The critical velocity for capture as a function of the coupling $\beta$ for the fully nonlinear system (4) (solid) and the simplified system (6) (dashed).

Figure 6

The exit velocity graph for the simplified system (6), showing qualitative agreement with Fig. 2.

Figure 7

The $X$ phase plane, showing trapped (dashed), untrapped (dash-dot), and separatrix (thin solid) orbits, corresponding to level sets of Eq. (10). Superimposed is the $X-\dot{X}$ projection of the 2-6 resonant solution to the fully nonlinear Eqs. (4) with $\beta =0.05$ (thick solid line).

Figure 8

The critical velocity of the ODE system (6) computed via direct numerical simulation (solid line), the first-order asymptotic approximation (dots), and the second-order asymptotic approximation (dashed line) from Eq. (24), and via numerical evaluation of integral (15).

Figure 9

The resonant velocities, indexed by the number of complete oscillations of $w\left(t\right)$, with $\beta =0.05$. The thick solid curve at the bottom is the result of direct numerical simulation. From top to bottom, the other curves are the asymptotic results of Eqs. (34, 40) and the value involving numerical calculation of the energy level, given the resonant period (43).

Figure 10

The solutions components of the 2-5 resonance of (a) $1+W\left(t\right)$ of the simplified equation (6) and (b) $w\left(t\right)$ of the full ODE (4).