Applicability of theoretical approaches for dispersion managed solitons

摘要

We present here recent results on dispersion management for non-linear return-to-zero pulse propagation in optical fibers. The main purpose of dispersion management is to reduce several effects such as radiation from the pulse due to lumped amplifiers compensating dispersion loss, modulational instability, jitters caused by the collisions between signals in different channels of wavelength-division-multiplexed (WDM) systems, the Gordon-Haus effect resulting from the interaction with noise, and to set a desired average value of dispersion. Additionally, DM solitons present a possibility of upgrading installed systems (see for example the UPGRADE project). The so-called dispersion managed soliton can be studied by different methods, which are compared and extended in the present paper. Those methods can generally be applied to general dispersion maps, but we particularize here to the case of a two-step dispersion map, of average dispersion D_(av), dispersion difference ΔD and of period Z_d. The Z = 0 point is at the center of the positive (or higher) dispersion fiber. Our model equation is iu_Z + D(Z)u_(TT/2 + S(Z)|u|~2u = 0, the nonlinear Schroedinger equation (NLSE). It can incorporate loss and amplification through S(Z) (see for example). This equation can be rescaled by three parameters acting on T,Z,u,D or S, but we do not impose conditions on them yet as they will be useful later. This NLSE admits some conservation laws: three trivial ones being the energy ∫ |u|~2dT, the moment i ∫ uu~*_TdT and the "Hamiltonian" H = 1/2 ∫ [S|u|~4 - D|u_T|~2]dT, this last one being only valid when both the dispersion and the non-linearities are constants (in which case there are an infinite number of conserved quantities). A test of validity of the different methods exposed here is to see how they conserve, or don't, those quantities.