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The finite-difference time-domain (FDTD) method is a flexible and powerfultechnique for rigorously solving Maxwell's equations. However,three-dimensional optical nonlinearity in current commercial and research FDTDsoftwares requires solving iteratively an implicit form of Maxwell's equationsover the entire numerical space and at each time step. Reaching numericalconvergence demands significant computational resources and practicalimplementation often requires major modifications to the core FDTD engine. Inthis paper, we present an explicit method to include second and third orderoptical nonlinearity in the FDTD framework based on a nonlinear generalizationof the Lorentz dispersion model. A formal derivation of the nonlinear Lorentzdispersion equation is equally provided, starting from the quantum mechanicalequations describing nonlinear optics in the two-level approximation. With theproposed approach, numerical integration of optical nonlinearity and dispersionin FDTD is intuitive, transparent, and fully explicit. A strong-fieldformulation is also proposed, which opens an interesting avenue for FDTD-basedmodelling of the extreme nonlinear optics phenomena involved in laserfilamentation and femtosecond micromachining of dielectrics.

Explicit formulation of second and third order optical nonlinearity in the FDTD framework