On the Physical Significance of Strong Spatial Dispersion

Optical metamaterials consist of subwavelength inclusions that possess unconventional optical properties that are unavailable in natural materials. The specific shape, composition, and arrangement of these inclusions determine the optical response of the metamaterials. However, designing them for specific applications with traditional simulations and experimental tests is impractical due to their high degree of complexity. To address this, the effective medium theory provides an efficient approach by linking the actual metamaterial to a homogeneous material with specific constitutive relations, allowing it to interact with light in the same manner.

Local material laws have been frequently used to model metamaterials as a homogeneous medium, assuming that the electromagnetic response at a point depends solely on the fields at that point. However, the accuracy of these models depends on the characteristic length scale of the metamaterial, which is the ratio between the lattice period and the operational wavelength. When this ratio is not much smaller than one, as is the case with optical metamaterials, spatial nonlocality becomes dominant, and the electromagnetic response of a point is influenced by the fields at many other points within the material. ... mehrAs a result, traditional local material laws cannot accurately describe the behavior of optical metamaterials. To address this challenge, we propose a new approach for modeling the behavior of optical metamaterials using nonlocal material laws. Specifically, we use a Taylor expansion in the Fourier space to approximate a general and exact nonlocal response function of the electric field, allowing us to derive a set of effective material parameters. Our approach accurately captures the spatial nonlocality of optical metamaterials and can be used to design novel and unique optical properties for various applications.

In this thesis, firstly, we present two nonlocal models that account for significant spatial dispersion effects and analyze the dispersion behavior of eigenmodes in homogenized metamaterials. We then derive the interface conditions that facilitate the calculation of reflection and transmission coefficients for a homogeneous slab with an incident field.

Secondly, to conduct the actual homogenization of a MM, we discuss two methods. The first approach treats the metamaterial as a bulk material and utilizes computational parameter retrieval techniques to assign effective material parameters to the bulk. This technique employs a least-square fitting algorithm to determine the optimal values for the effective material parameters by comparing the reflection and transmission coefficients of the bulk with those of the actual metamaterial. Further, we use this method to study three artificial structures with predetermined scattering properties that are quantified in terms of multipole moments they sustain and reveal that the effective permittivity and permeability are linked to the electric and magnetic dipole moments of the scatterers. Additionally, nonlocal material parameters are related to higher-order multipolar moments and their interaction with dipolar terms. By understanding the significance of each material parameter, we can decide the truncation order for the Taylor expansion of the considered constitutive relations for a given metamaterial.

We also investigate the role of the period-to-operational wavelength ratio in homogenizing a metamaterial, specifically an electric dipolar lattice. Surprisingly, we observe a breakdown in homogenization at shorter lattice constants due to near-field interactions among the particles forming the lattice. This suggests that the period should not only be much smaller than the operational wavelength to homogenize a metamaterial but there exists an optimal period for a given inclusion size.

The second method introduced in this thesis is a novel homogenization approach using the "effective transition matrix" or $T_{eff}$-matrix, which provides an exact description of the linear interaction between light and the bulk material without the need for any computational retrieval processes. By homogenizing an isotropic 3D metamaterial made of gold nanospheres, we detail the calculation of the corresponding effective material parameters and further obtain the reflection and transmission coefficient for the homogeneous material. The highlight of this approach is the promise to homogenize a 3D metamaterial without requiring a target object as opposed to the case of any parameter retrieval methods.

Finally, we summarize all the analytical and numerical results and discuss possible future research endeavors as an extension of the results obtained in the thesis. Overall, the thesis contributes to a deeper understanding of the behavior of optical metamaterials at the effective level and offers valuable insights for future research in this field.