Numerical Solution of the Extended Nonlinear Schrodinger Equation

摘要

High-resolution mathematical models of the extended nonlinear Schroedinger equation have been designed which include diffraction combined with non-zero second-order group-velocity dispersion (GVD). These models follow a Gaussian pulse as it propagates in air to a large distance (several meters). With diffraction disabled, a pulse quickly collapses to a single singularity on the propagation axis. Alternatively, with diffraction included, a pulse will collapse into a pair of fins off the propagation axis. If the GVD is disabled, the fins eventually collapse to singularities. However, if the GVD is set an appropriate non-zero value, the fins can be propagated out to several meters (propagation distance) without singularities forming. In test cases with diffraction plus GVD, we see (A) an initial drop in intensity, followed by (B) a rise at about 2 to 3 meters, and then (C) a gradual drop thereafter. This pattern is most pronounced in our energy pattern depictions where we model the distribution of the total energy seen by a target plane as the pulse quickly passes through it. When viewed on a target plane at an optimal distance (roughly 2.5 meters), the energy pattern appears as a bright ring indicating that the initial Gaussian pulse has collapsed to a very thin cylindrical shape. Our results are based solely on mathematical formulations without any experimental verification. Additionally, these formulations do not attempt to completely ensure energy conservation.