Two-orders fast multipole analysis of meta-atoms

Meta-materials are complex structured media that are designed in order to get unusual physical properties that often cannot be reached in nature. In the field of electromagnetic, many interesting research topics of the past years are based on our ability to create these specific materials that would behave as no preexisting natural materials. Meta-materials are designed in order to mimic a needed medium that we can call the “targeted material”: most of the time, targeted materials are regular theoretical magneto-dielectrics, whose electromagnetic behavior is encoded into two property tensors: a permittivity tensor that describes the response to an electric field, and a permeability tensor that describes the response to a magnetic field. One of the most popular application involving meta-materials is the super-lens, that requires a negative refraction index: both negative permittivity and permeability are needed to get a negative refraction in the lens. Invisibility cloaking and routing are others well known applications, that need a rigorous control of gradient properties of both permittivity tensor and permeability tensor in the medium. Theoretically, a magneto-dielectric is composed of an assembly of small dipoles: the use of resonant devices playing the role of the dipoles allow researchers to reach almost every targeted values of permittivity and permeability. However, these resonances are complex and homogenization of arrays of such inclusions is challenging: spatial dispersion, bianisotropy, high-order multipolar moments and coupling effects contribute to move the reality away from the model of a hypothetical assembly of dipoles. In our researches, we try to develop meta-materials that would behave as there target for any incoming electromagnetic field and environment. Multipole decomposition is a powerful tool that helps in the study of scatterers, and especially for meta-atoms in meta-materials. We developed an algorithm that performs a fast two-orders multipole expansion based on the extraction of the far field in six different directions. A comparison of the scattered field of the simulated inclusion with the scattered field of the two-orders multipole model is performed, and informs us about the degree of precision of the model. This approach appears to be efficient in the design of meta-materials intended for cloaking applications.