The rapid development of micro-nano fabrication technologies enables the synthesis of various optical metamaterials with complex microstructures. The size or component ratio of optical metamaterials can be tuned to control the dielectric properties of optical metamaterials in a wide range. Therefore, analyzing the effective optical parameters of metamaterials has always been a core issue in electromagnetic computing. It is known that the traditional analysis methods for effective optical parameters of composite materials include Maxwell-Garnet effective medium theory and Bruggeman effective medium theory. However, they can only analyze the cases with a small difference between the dielectric constants of the doped and background crystals. When such a difference is large, the error of effective permittivity given by the effective medium theory becomes large as well. This work uses the transfer-matrix method to analyze the effective permittivity of optical metamaterials and discusses the application scope of the effective medium theory.

For convenience, we assume that the nanocrystals are periodically doped into the background crystal to form a composite system and the nanocrystals and background crystal are isotropic. These allow us to use the transfer-matrix method to calculate the transmission spectrum of the composite system, and then obtain the effective dielectric constant of the composite system by comparing the transmission spectrum of the original structure with that of the equivalent crystal. The matching degree between equivalent transmission spectrum and original transmission spectrum is measured by the coefficient of determination R2. A larger value of R2 means the effective dielectric constant is more accurate.

This work analyzes the dependence of the effective dielectric constant of composite crystals on the dielectric constant of each component, the geometry parameters of composite crystals, and the light wavelength. The comparison of the effective dielectric constants obtained from the transfer-matrix method and the Bruggeman effective medium theory separately indicates that the Bruggeman effective medium theory is indeed only applicable to the case with a small difference between the dielectric constants of the doped and background crystals (Fig. 3). The effective permittivity given by Bruggeman effective medium theory only relates to the volume fraction of nanocrystals and is independent of the light wavelength (Fig. 2 and Fig. 4). However, it is found that the effective dielectric constants rely on the light wavelength after the comparison of the equivalent transmission spectrum and the original transmission spectrum by the transfer-matrix method (Fig. 5). In particular, there is no effective dielectric constant at the optical band gap of the composite system due to the scattering of optical lattices.

When the difference between the dielectric constants of nanocrystals and background crystal is small, the effective dielectric constant of the composite system is consistent with the result of Bruggeman effective medium theory. However, when the wavelength dependence of the effective dielectric constant is considered, the effective medium theory can be extended to the case with a large difference. This can broaden the application scope of the effective medium theory. It is also found that there is no effective dielectric constant at the optical band gap of the composite system as a homogeneous material has no band gap. These conclusions can provide theoretical guidance for research on the optical properties of composite materials and help determine whether the effective medium theory is applicable.