  The finite-difference time-domain (FDTD) method is a flexible and powerful
technique for rigorously solving Maxwell's equations. However,
three-dimensional optical nonlinearity in current commercial and research FDTD
softwares requires solving iteratively an implicit form of Maxwell's equations
over the entire numerical space and at each time step. Reaching numerical
convergence demands significant computational resources and practical
implementation often requires major modifications to the core FDTD engine. In
this paper, we present an explicit method to include second and third order
optical nonlinearity in the FDTD framework based on a nonlinear generalization
of the Lorentz dispersion model. A formal derivation of the nonlinear Lorentz
dispersion equation is equally provided, starting from the quantum mechanical
equations describing nonlinear optics in the two-level approximation. With the
proposed approach, numerical integration of optical nonlinearity and dispersion
in FDTD is intuitive, transparent, and fully explicit. A strong-field
formulation is also proposed, which opens an interesting avenue for FDTD-based
modelling of the extreme nonlinear optics phenomena involved in laser
filamentation and femtosecond micromachining of dielectrics.
