Green Function Formulism for Electromagnetic Wave Generated in Nanostructured Metamaterial of Finite Thickness: Isotropy and Anisotropy

A Green function formulism is developed to calculate the electromagnetic fields generated by sources embedded in nanostructured medium which could be represented by an effective electric permittivity tensor with finite thicknesses. The method begins with the decomposition of the generated mode into the eigenmodes in the medium, which have definite dispersions. To account the interface effect at boundaries, especially the mode conversion at the interface between anisotropic media, mode expansion method is combined into the theoretical framework. Thus, the electromagnetic wave in any given position can be gotten clearly. The formulism can provide conveniences of studying the novel properties of nanostructured metamaterials. 1. Introduction Artificially constructed metamaterials [1, 2] have unique electromagnetic properties which cannot be obtained in natural materials, such as artificial magnetism, negative refraction [3], and near-field focusing [4]. Recently, active research areas related to metamaterials attracted much attention [5, 6], for instance, high/low epsilon metamaterials, designer dispersion, transformation optics metamaterials, switchable metamaterials, amplifying metamaterials, sensor metamaterials, nonlinear metamaterials, and quantum metamaterials [7]. The origin of novel property of structured metamaterial comes from the modification of effective permittivity and permeability. There are many developed methods to determine the effective permittivity and permeability [8, 9]. Nanostructured metamaterials have wide applications in real world, such as manipulation of thermal radiation [10], plasmonic biosensor [11], artificial magnetic material [12], and superlens imaging [13]. Patterned metal at the interface between two media will generate surface-plasmon polaritons which alter surface waves and subsequently change radiative heat transfer, coherence properties, and Casimir forces [14]. To obtain the electromagnetic property of a nanostructured metamaterial, one would have to calculate Green function of the given system [15, 16]. The conventional formulism [17] can handle homogeneous isotropic media very well, and the interface effects can be trivially accounted by introducing Fresnel coefficients. However, a lot of metamaterials exhibit anisotropy properties due to the special arrangement of structure [18]. Thus, the story becomes quite complex, when anisotropy of media is introduced. One distinguishing phenomenon of anisotropic media is the mode conversion [19, 20] happening at the interface between anisotropic media. Under the situation,