Inelastic collisions of polarized solitons in a birefringent optical fiber

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Abstract
The change of polarization in a collision of orthogonally polarized solitons [i.e., generation of the solitons’ shadows as defined in Opt. Lett. 12, 202 ( 1987)] is investigated analytically within the framework of the two-component nonlinear Schrödinger model of a birefringent fiber. The analysis can be performed explicitly in three cases: if the relative frequency of the colliding solitons, 2c, is much larger than the solitons’ amplitudes η1 and η2, with an arbitrary cross-phase-modulation coefficient B; if B ≪ 1 (the weakly coupled system); and if 0 < B−1≪ 1 (when the system is close to the exactly integrable Manakov system). In the first case the result (amplitudes of the collision-generated shadows) is exponentially small with respect to the large parameter c/η1,2. In the second case the amplitude is ∼B2, and the energy of the shadow is ∼B2, while the energy emitted during the collision in the form of radiation is ∼B2. In the third case (B − 1 ≪ 1) both the amplitudes of the shadows and their energies are ∼(B − 1), while the radiative energy losses vanish, being ∼(B − 1)2. Using conservation laws, I also find collision-induced changes of the solitons’ amplitudes and central frequencies. Finally, the possibility of a resonant soliton–soliton collision when one of the shadow’s spatial eigenfrequencies (wave numbers) is close to the wave number of the second soliton is analyzed qualitatively. In this case the collision may give rise to anomalously strong effects. An explicit calculation is performed for the particular case B ≪ 1, η2 ≪ η1.