Numerical simulations of light bullets using the full-vector time-dependent nonlinear Maxwell equations

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Abstract
The vector Maxwell’s equations are solved to simulate propagating and colliding optical pulses in planar waveguides. The slowly varying envelope approximation is not made, and therefore the dynamics of the optical carrier are retained in the calculations. Some short optical pulses are found to be essentially unchanged over rather long propagation distances and stable over some energy variation in a manner characteristic of light bullets. With greater energy increase the pulses do not collapse, as predicted by the nonlinear Schrödinger equation; rather, after initially compressing, they undergo unlimited expansion. Because the optical pulses that are employed in these simulations are extremely short, they are beyond the limitations of the slowly varying envelope approximation that is used in the derivation of the nonlinear Schrödinger equation. The dispersive effects are modeled by a single Lorentzian resonance, and the nonlinear refraction is modeled by a Kerr-like instantaneous nonlinearity. The procedure for obtaining numerical solutions to the nonlinear Maxwell equations is described, and solutions are obtained by a finite-difference algorithm.