نظریه چند قطبی الکترومغناطیسی برای نانومواد نوری Electromagnetic multipole theory for optical nanomaterials

The classical electromagnetic multipole expansion [1] is a powerful tool for analyzing the electric and magnetic fields created by spatially localized electric charges and currents. Irrespective of the complexity of the charge and current distributions, the fields produced by them can be represented as a superposition of the fields created by a corresponding set of point multipoles. This correspondence provides a common basis for characterizing the fields radiated by localized charge and current excitations in arbitrary configurations. In optics, the multipole expansion is well suited to describe the scattering of optical fields by small objects. Usually, if the wavelength of the field is large compared to the size of the object, the scattering is described mainly by the lowestorder multipole, the electric dipole, while the contributions from all higher-order multipoles are considered as mere perturbations. Recently, it has been shown that in specifically designed optical nanomaterials [2], such as metamaterials, the contribution of the magnetic dipole [3] and the electric quadrupole [4] excitation to the scattering by the material’s constituents can be made significant, which substantially affects the optical properties of the material and lead to extraordinary phenomena, such as negative refraction [5]. In certain materials, higher-order multipoles can even completely overshadow the electric dipole contribution [6]. Thus, it is clear that higher-order multipoles have to be taken into account when evaluating the macroscopic electromagnetic characteristics of such materials [7]. In order to create a material with prescribed optical properties, one should select the elementary unit of the material (often called the meta-atom) and optimize its scattering characteristics through adjusting the design. For an individual particle this can be done by numerically solving the Maxwell equations for the scattered field and applying the multipole expansion to determine the multipoles contributing to the scattering [8]. However, in a material composed of a large number of such elementary units, each particle interacts with the fields scattered by the other particles, which can significantly modify the amplitudes and phases of the excited multipoles. Since at each point in the material the field is a superposition that contains the fields scattered by all the particles, the approach developed for individual particles [8] can no longer be used, and one should perform the multipole decomposition on the excited electric currents in each of the particles individually. Previously, this decomposition has been introduced only for a single localized electric current distribution in vacuum [1] and, therefore, a multipole theory suitable for analysis of nanoscatterers in an array and in an arbitrary homogeneous dielectric host medium was missing. In this work, we introduce such a theory. Our multipole expansion approach is particularly suitable for direct numerical implementation. The geometry of the scatterer determines the types of electric current modes that can be excited in it by light. For the description of these modes, we propose a set of orthogonal electric current multipoles, which are composed of elementary point currents in simple configurations. Each element of the resulting current multipole tensor reflects the strength of one of these configurations, which enables one to visualize the real electric current modes that will be excited in a scatterer. We complete our theory by deriving expressions that relate the elements of the proposed electric current multipole tensor to the classical multipole expansion coefficients. In particular, these expressions reveal i) perfectly dark multipole modes that do not create any electromagnetic field, and ii) electric dipole radiation produced by electric currents with zero net electric dipole moment. These findings provide us with additional freedom in the choice of the particle geometry. In section 2, the classical multipole expansion is adjusted to describe the electromagnetic field scattered by individual nanoparticles and nanoparticle arrays embedded in a dielectric host medium. In section 3, we map the coefficients in the multipole expansion to the electromagnetic fields created by sets of sub-wavelength current elements. Explicit mapping relations are derived up to the orders of electric octupole and magnetic quadrupole, which both describe third-order excitations in the multipole hierarchy [9]. In section 4, we summarize our results.