Stationary solitonlike pulses in birefringent optical fibers.

Propagation of solitonlike pulses in birefringent nonlinear fibers has attracted much attention in recent years [1—14]. The equations that describe pulse propagation in these fibers have been derived by Menyuk [1]. These equations are quite complicated and can be solved only in an approximate way for certain specific cases. Two main cases have been studied in depth: high and low birefringent fibers, for which two separate approximations have been developed. The case of high birefringence has been studied in detail in [1—5]. In this regime, one considers that the two linearly polarized components of the field have different phase velocities and different group velocities. Due to the nonlinearity, the pulses in these two components can capture each other, but their central frequencies become different [5] to make their group velocities equal. As a result of averaging, the fast oscillatory terms which relate the phases of the two components can be ignored and usually only trapping effects are considered in this approach [4,5]. On the other hand, the approximation of low birefringence takes into account the difference in phase velocities between the two linearly polarized components, but neglects their difference in group velocities, as this is assumed to be a higher-order effect. The two components of the soliton travel with the same group velocity and phase locking of these two components can occur. This approach has been considered numerically by Blow, Doran, and Wood [7]. In particular, polarization instabilities were first found in [7] and studied in more detail by Wright, Stegeman, and Wabnitz [9]. The full polarization dynamics of solitons in polarization-preserving fibers, in the approximation of low birefringement, has been considered in [14). It is interesting to know what happens if both effects, viz. , pulse trapping and phase locking, act together. In this paper we make a first step in trying to solve this problem. In particular, we are extending the results of [14], but now are taking into account the difference between the group velocities of the components. Specifically, we study numerically the stationary soliton states in a birefringent fiber, considering simultaneously the differences in phase and in group velocities between the components. In doing this we are not averaging over the fast oscillatory terms, which was done in [1]. Moreover, we show that, in certain regimes of propagation, these terms play an essential role in producing stationary solutions, e.g. , coupled soliton states. Stationary solutions play an essential role in the propagation dynamics of nonlinear pulses in optical fibers. In Hamiltonian dynamical systems, they determine, to some extent, the overall dynamics of solitonlike pulses. In the low birefringence approximation, it has been shown that two different regimes of propagation of solitonlike pulses exist [14]. They are related to the linearly polarized slow and fast soliton states and also to the elliptically polarized soliton states which bifurcate from the fast soliton branch. When taking into account the different group velocities of each component, we see that even the stationary solutions become different. However, the propagation dynamics changes greatly only at quite high values of the difference in group velocities. In this work we study numerically, and using the Poincare sphere formalism, the stationary solitonlike solutions when polarization group velocity dispersion is taken into account. We find that when the difference between the group velocities is small, stationary solutions are similar to those in the approximation of low birefringence, i.e., they consist of slow and fast solitons. When the difference in group velocities becomes high, the slow soliton splits into two other solutions. We find that in many aspects, these solutions possess the same features as the gap solitons considered by Aceves and Wabnitz [15] and Christodoulides and Joseph [16]. In particular, the velocity of the soliton depends on the relative amplitudes of the two components. The remainder of this paper is organized as follows. In Sec. II we formulate the problem, recalling some wellknown solutions, viz. , the so-called "slow" and "fast"