Nonlinear temporal-spatial surface plasmon polaritons

The propagation of temporal-spatial surface plasmon polaritons propagating along a flat dielectric/metal interface is investigated. The governing envelope equation for these surface plasmons is found to be the damped nonlinear Schrödinger (NLS) equation with two spatial-like dimensions and one evolution dimension. Depending on whether the dispersion is anomalous or normal the dispersion of this multidimensional nonlinear NLS equation can be elliptic or hyperbolic. In the elliptic case a localized initial mode is found to focus before damping effects begin to act. In the hyperbolic case the solution is found to be self-similar which also eventually decays due to damping.