When a light field interacts with structures that have complex geometric features comparable in size to the wavelength of the light, it is not permissible to invoke the assumptions of the classical diffraction theory, which simplify the problem and allow for approximate solutions. For such cases, direct numerical solutions of the governing equations are sought through approximating the continuous time and space derivatives by the appropriate difference operators. The Finite Difference Time Domain (FDTD) method discretizes Maxwell's equations by using a central difference operator in both the time and space variables. The E- and B-fields are then represented by their discrete values on the spatial grid, and are advanced in time in steps of Δt. The numerical solution thus obtained to Maxwell's equations (in conjunction with the relevant constitutive relations) provides a highly reliable representation of the electromagnetic field distribution in the space-time region under consideration.

This chapter presents examples of application of the FDTD method to problems involving the interaction between a focused beam of light and certain subwavelength structures of practical interest. A few general remarks concerning the nature of the FDTD method appear in the next section. This is followed by a description of the simulated system and two examples in which comparison is possible between the FDTD method and an alternative method of calculation. We then present simulation results for the case of a focused beam interacting with small pits and apertures in a thin film supported by a transparent substrate.