Ability and limitations of the effective medium theory in terms of the filling fraction and number of layers for hyperbolic metamaterials

Hyperbolic metamaterials (HMMs) exhibit unusual electromagnetic properties, attracting significant interest from both fundamental physics and applied engineering perspectives. At subwavelength dimensions, the effective medium theory (EMT) is commonly used to simulate and describe HMMs, which are typically composed of metal-dielectric multilayers, due to their simplicity in design. In this context, the EMT approximation must align with exact electromagnetic solutions. However, significant discrepancies in the predicted properties sometimes arise due to these approximate methods. This article compares the EMT with the exact solution (i.e., the transfer matrix method) and demonstrates that the transmission and reflection in the HMM multilayer structure are sensitive to variations in the number of layers and the filling fraction, even when the layer thicknesses are much smaller than the operating wavelength. Understanding this dependence of EMT on the number of layers and the filling fraction is crucial for reducing errors and improving the design of plasmonics and photonics devices.