|
We use the method of moments to calculate the propagation of an arbitrarily shaped pulse in a nonlinear
dispersive fiber. By assuming that the pulse is linearly chirped, we are able to determine analytically the
evolution of the second order moments (representing the duration, bandwidth and chirp of the pulse) along
propagation regardless of the initial pulse shape. The evolution of the moments is given by an implicit equation
and several invariants. These invariants allow an easy estimation of the different pulse parameters. The linear
chirp approximation implies that the arbitrary pulse shape remains invariant along propagation but allows to
calculate the propagation in both dispersion regimes from the same solution. The solution show an oscillatory
behavior in the anomalous dispersion regime and a monotonic behavior in the normal dispersion regime. In both
regimes the calculations are compared to numerical split-step simulations and are shown to agree for propagation
over many dispersion and nonlinear lengths.
While this method describes well the evolution of the pulse duration, bandwidth and chirp, we need to proceed
differently to find the evolution of the pulse shape. From these propagation equations for the moments, we
derive an approximate implicit solution describing the propagation of a Gaussian pulse in the normal dispersion
regime. This approximate solution describes the pulse shaping toward a parabola that the pulse undergoes
along propagation. A good agreement is found between the pulse obtained from numerically solving the implicit
equation and the split-step propagation of the same pulse. Numerically solving the implicit analytical function
describing the pulse is much faster than using purely numerical simulations, which becomes time consuming for
highly chirped pulses with large bandwidths over long propagation distances. These and other results suggest
that pulse shaping along propagation is only adequately modeled by implicit functions.
|
