Polarization dynamics in nonlinear anisotropic fibers

We give an extensive study of polarization dynamics in anisotropic fibers exhibiting a third-order index nonlinearity. The study is performed in the framework of the Stokes parameters with the help of the Poincaré sphere. Stationary states are determined, and their stability is investigated. The number of fixed points and their stability depend on the respective magnitude of the linear and nonlinear birefringence. A conservation relation analogous to the energy conservation in mechanics allows evidencing a close analogy between the movement of the polarization in the Poincaré sphere and the motion of a particle in a potential well. Two distinct potentials are found, leading to the existence of two families of solutions, according to the sign of the total energy of the equivalent mechanical system. The mechanical analogy allows us to fully characterize the solutions and also to determine analytically the associated beat lengths. General analytical solutions are given for the two families in terms of Jacobi’s functions. The intensity-dependent transmission of a fiber placed between two crossed polarizers is calculated. Optimal conditions for efficient nonlinear switching compatible with mode-locking applications are determined. The general case of a nonlinear fiber ring with an intracavity polarizer placed between two polarization controllers is also considered.

Figure 1

The Poincaré sphere.

Figure 2

Evolution of the polarization state with no natural birefringence. The points in the poles are stationary states.

Figure 3

Stationary solutions in the general case.

Figure 4

Trajectory around the stationary point $P$.

Figure 5

Trajectories around the stationary point $P^{\prime }$. (a) Elliptic motion when $k>\Omega$. (b) Hyperbolic motion when $k⩽\Omega$. The dashed lines are the asymptotes.

Figure 6

Potential associated with the evolution of the $S_3$ component of the polarization state during the propagation along the fiber. (a) The initial conditions are such that $J<-k^2/\Omega$. Depending on the total energy, the trajectory in the Poincaré sphere is periodic around stationary points $Q,Q^{\prime }$ or $P,P^{\prime }$. A, B, C, D, E, F, and $P^{\prime }$ are turning points. (b) The initial conditions are such that $J⩾-k^2/\Omega$. The total energy is positive, and the trajectories in the Poincaré sphere are periodic around point $P,P^{\prime }$.

Figure 7

Evolution of the polarization state for the rotating solution. A and B are the turning points and $L_B$ is the beat length.

Figure 8

Approximate evolution of the beat length for the $(Q,Q^{\prime })$ family of solutions as a function of $\kappa$.

Figure 9

Evolution of the polarization state for the oscillating solution.

Figure 10

(a) Example of evolution around the point $P^{\prime }$ with points of interest. (b) Electric field polarization in the $x$-$y$ plane corresponding to the relevant points G (linear polarization), E (right-handed elliptic polarization), H (linear polarization), and F (left-handed elliptic polarization).

Figure 11

Evolution of the state of polarization in the Poincaré sphere. (a) Evolution around $P$ when $k>\Omega$. (b) Evolution around $P^{\prime }$ when $k>\Omega$. (c) Case where $k<\Omega$. The double-loop curve is the boundary between $Q$-$Q^{\prime }$ solutions and $P$-$P^{\prime }$ solutions.

Figure 12

Evolution of the nonlinear transmission as a function of $\Omega$ for increasing values of the input polarization angle $\theta$.

Figure 13

Evolution of the polarization state at the exit of the fiber in the Poincaré sphere as a function of $\Omega$ for an incident polarization close to the unstable fast axis, $\theta ={89}^{°}$

Figure 14

Evolution of the nonlinear transmission as a function of $\theta$ for increasing values of $\Omega$.

Figure 15

Evolution of the switching power parameter as a function of the incident polarization angle.

Figure 16

Nonlinear birefringent fiber ring cavity with an intracavity polarizer placed between two polarization controllers.