In this research we discuss the design and implementation of high-order finite-difference methods to analyze Maxwell's equations in the context of meta-materials and non-linear optics, which are new and emerging material science research avenues. We first describe Maxwell's equation with Lorentz-Drude polarization, and derive an exact solution in the presence of an incident wave. Thereafter, using the modified equation method a fourth-order (in space and time) methods using finite-difference is developed. These schemes are then implemented for 1D as $(4,4)$ numerical method.
The second part of this paper we describe a dynamical systems based analysis technique for analyzing non-linear optics. We implement a fully explicit numerical scheme which is used to calculate the stable points and eigenvalues under varying system conditions (wavelength of incident light), and this can be used for design optimization.
Key Words: Maxwell's equation, meta-materials, Lorentz-Drude model, non-linear optics, high-order finite-difference methods.