The Raman effect and solitons in an optical fibre with general ellipticity

We derive equations for the quasi-instantaneous Raman effect in a nonlinear bimodal optical fibre with arbitrary (linear, circular or mixed) ellipticity. A balance-equations approach using an appropriate ansatz for the soliton's shape yields a set of ordinary differential equations which approximate the dynamics of the soliton in such an arbitrary ellipticity fibre with group-velocity birefringence, phase-velocity birefringence and linear cross-coupling. The latter two effects are the most general Hamiltonian linear terms without derivatives in the underlying partial differential equations and are identical in form to the terms which describe a periodic twist. When group-velocity birefringence and linear cross-coupling are both nonzero, the soliton cannot hold a constant polarization. If linear cross-coupling is zero, there are solitons with constant polarizations 0°, 45° and 90°. As the only term which couples the inter-mode phase difference to the other parameters, linear cross-coupling is a singular perturbation. With zero group-velocity birefringence and nonzero linear cross-coupling, a soliton's polarization may be constant at up to four different values, depending on the soliton and fibre parameters. Regarding stability of the polarization angle: without linear cross-coupling, at ellipticity angles from 0° (linear ellipticity) to tan^-1(1/√2) ≈ 35.3°, the 0° and 90° polarization solitons are stable and the 45° soliton is unstable and vice versa for ellipticity angles between 35.3° and 90° (circular ellipticity). With nonzero linear cross-coupling, the behaviour is analogous, but only approximately; the solitons which transform continuously to polarizations 0° and 90° are stable when the ellipticity is 0° to 35.3°, but at more than 35.3° the polarization cannot be constant in the first place. The soliton with polarization 45° tends to be but is not always unstable for ellipticity angles of 0° to 35.3° and at greater than 35.3° there is always a solution with a stable 45° polarization. This implies the possible coexistence of two different types of solitons, both stable, thus opening a new avenue to bistability of optical solitons.