Optical parameter extraction for metamaterials via robust effective and equivalent medium models

Metamaterials are complex structured mixed-material systems with tailored physical properties that have found applications in a variety of optical and electronic technologies. New methods for homogenizing the optical properties of metamaterials are of increasing importance, both to study their exotic properties and because the simulation of these complex structures is computationally expensive. We propose a method to extract a homogeneous refractive index and wave impedance for inhomogeneous materials. We examine effective medium models, where inhomogeneities are subwavelength, and equivalent models where features are larger. Homogenization is only physically justified in the former; however, it is still useful in the latter if only the reflection, transmission, and absorption are of interest. We introduce a resolution of the branching problem in the Nicolson-Ross-Weir method that involves starting from the branch of the complex logarithm beginning with the minimum absolute mean derivative and then enforcing continuity, and also determine an effective thickness. We demonstrate the proposed method on patterned PbS colloidal quantum dot films in the form of disks and birefringent gratings. We conclude that effective models are Kramers-Kronig compliant, whereas equivalent models may not be. This work illuminates the difference between the two types of models, allowing for better analysis and interpretation of the optical properties of complex metamaterials.