Validation of the effective-medium approximation for the dielectric permittivity of oriented nanoparticle-filled materials: effective permittivity for dielectric nanoparticles in multilayer photonic composites

A formula for the effective permittivity for two-dimensional particles embedded in a host matrix is derived and a method for its numerical evaluation is described. The method is applied to specific cases of circular, square, rectangular and triangular particles. Shapes are assumed for the inclusion particles. Data for obtaining the effective permittivity is provided for a wide range of filling fractions, geometries and dielectric contrasts between the particles and the matrix under the assumption of the quasi-static approximation, that is, the wavelength of the electric field is assumed to be much larger than the particle size. Metallic particles with complex and frequency-dependent dielectric constants are treated, as well as no-loss dielectric inclusions. Calculations are validated by comparing the results of the reflectivity obtained for a composite layer using the transfer-matrix method, assuming the layer to be an effective medium, to those using the finite-element method and accounting for the heterogeneous material.