Recent studies in the field of EM optics are presented in terms of the methods used and the potential and limits of the field of research. Problems are addressed which can be reduced to the scalar level, where Maxwell's equation (ME) is reduced to Helmholtz's equation (HE) and often requiring the use of ME to amplify problems encountered in HE. Consideration is given uniquely to time-harmonic fields represented systematically by complex functions of space and with attention given to time-dependence in exp(-i omega t). The closed-form solutions are given, and the integral, differential, and modal expansion methods are set forth. Possible applications for the methods are discussed, with emphasis given to the properties of gratings. Diffraction is described for rough surfaces, for a slit, and for integrated optics. The EM theory of anisotropic gratings, homogenization techniques, and asymptotic analysis are shown to be significant concepts that can have practical applications in optics.