Role of stationary invariant manifolds in the spatiotemporal dynamics of a nonlinear-wave system of finite extension: Application to polarization attraction in optical fibers

The study of the spatiotemporal dynamics of two counterpropagating beams in optical fibers has recently been the subject of a growing renewed interest. This system has been shown to exhibit a phenomenon of polarization attraction which can be used to achieve a complete polarization of an initially unpolarized beam, almost without any loss of energy. In previous works, a theoretical description of this phenomenon has been developed in the particular case where the underlying stationary system exhibits the important property of integrability. Our aim here is to provide a generalization of the theoretical description to nonintegrable stationary systems. The analysis reveals that the spatiotemporal dynamics of the system relaxes towards a stationary trajectory whose geometric structure is revealed by the stable and unstable manifolds of some singular fixed points of the stationary system. We illustrate the theory by considering the representative and concrete example of a weakly birefringent optical fiber system.

Figure 1

Results of the numerical simulations of the spatiotemporal PDE system on the Poincaré sphere with $\rho =5$. The red dots denote the states of the signal at $z=L$ when the stationary dynamics is reached. The fiber lengths are set to $L=1$ (a), $L=2$ (b), $L=3$ (c), and $L=4$ (d). The yellow dot displays the boundary condition of the pump SOP, $\vec{J}\left(L\right)=(1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})$, and the green dots the fixed points of the stationary ODE system. The blue solid line depicts the curve of points (line of polarization attraction) with the same energy $H$ as the fixed point (see Sec. 3 for details). We recall that $S_3=\pm 1$ denotes the right or left circular polarization states. The different quantities are dimensionless.

Figure 2

Stationary solutions that the PDE system governed by Eqs. (1) and (2) reaches after a transient spatiotemporal relaxation process. The figure illustrates the particular coordinate $S_3$ vs $z$. The fiber lengths are set to $L=1$ (a), $L=2$ (b), $L=4$ (c). The parameter $\rho$ is fixed to 5. The boundary condition of the pump SOP is fixed to $\vec{J}(z=L)=(1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})$. 64 different boundary conditions of the signal SOP, $\vec{S}(z=0)$, are considered (see Fig. 1). We recall that $S_3=\pm 1$ denotes the right or left circular polarization states. The different quantities are dimensionless.

Figure 3

Evolution of the distance $d$, defined in Eq. (13), as a function of the fiber length $L$ for the input pump SOP $\vec{J}(z=L)=(1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})$, while the signal boundary conditions are uniformly distributed on the surface of the Poincaré sphere. The distance is normalized with respect to its value for $L=1$ $[\tilde{d}\left(L\right)=d\left(L\right)/d(L=1)]$ to improve the visualization of the convergence toward the singular line of attraction. The parameter $\rho$ is set to 5. When $L$ varies from 1 to 6, the range of variation of $\mathrm{\Delta }$ is from $2\pi /5$ to $2\pi /30$. The different quantities are dimensionless.

Figure 4

Same as in Fig. 3, except that now the birefringence parameter $\mathrm{\Delta }$ is varied while keeping the fiber length constant, $L=4$.

Figure 5

Sequential Poincaré sphere evolution illustrating the two-stage process of polarization attraction. The initial 64 signal SOPs [red (dark gray) dots] are uniformly distributed on the Poincaré sphere (from left to right and top to bottom). The eight panels show the evolution of the SOPs for different values of the longitudinal variable $z$. In the first stage of polarization attraction (panels 1–4, $z=0$–2), all signal SOPs are contracted in the neighborhood of the two fixed points [green (light gray) points]. In the second stage (panels 5–8, $z=2.5$ to $z=4$), the SOPs follow the unstable manifolds to reach the final polarization attraction point. The blue (black), red (dark gray), and yellow (light gray) lines respectively depict the attraction line and the projection of a trajectory of the unstable manifold onto the Poincaré sphere. (The red color is associated with the $\vec{S}$ dynamics and the yellow one with the $\vec{J}$ dynamics.) The parameters are $\rho =5$, $\vec{J}\left(L\right)=(1/\sqrt{3},1/\sqrt{3},1/\sqrt{3})$, and $L=4$. A movie corresponding to this evolution is reported in the Supplemental Material [37]. The different quantities are dimensionless.