Abstract
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This chapter describes a rigorous and efficient approach to model dispersive active materials in the Finite difference-time domain (FDTD) framework. It introduces an optimized unconditionally stable second order algorithm, which is able to deal with any type of dispersion. The chapter explores the model a fully three-dimensional quantum-mechanical active material with the aid of the Maxwell-Bloch (MB) formalism and shows that the solution of MB equations can be simplified using concepts from group and graph theories. It describes the numerical approach to the description of dispersive materials in the FDTD domain. The chapter explains the MB approach to active media in the density-matrix formalism and also shows how to use concepts from group theory to reduce the number of equations to be solved. The MB theory is fully derived from the quantum mechanical description of the atomic system, hence it allows for the simulation of realistic materials and direct comparison with experimental results event at the nanoscale.