A generalized nonlinear Schrödinger equation and optical soliton in a gradient index cylindrical media

A generalized form of nonlinear Schrödinger equation is deduced for the propagation of an optical pulse in a fiber with a cylindrical geometry having a gradient in refractive index in the radial direction. The configuration gives a simple model for a fiber with a cladding or multicore fiber. To begin with we have analyzed in detail the modulational instability in terms of Stokes and anti-Stokes side band amplitudes which shows a significant change with respect to the depth parameter L and dispersion constant. Next we have deduced the equations governing the modulation of parameters of a Gaussian pulse as it propagates through it. The moment method is used for the derivation. The gradient of the refractive index leads to the trapping of the pulse, whereas the balance between nonlinearity (Kerr type) and dispersion in the longitudinal direction guides the propagation. Instead of a constant dispersion profile we have considered the standard dispersion map which helps in shaping of the pulse. The numerical simulation of these derived equations shows how the chirp, width, amplitude of the pulse change with type of gradient and the distance travelled.