Metasurfaces, optically thin structures with engineered diffraction, have gained attention in the past few years as new platform for ultra-compact photonics. Metasurfaces are typically based on arrays of planar optical resonators, fabricated at the interface between two dielectrics, arranged to produce diffracted waves of pre-designed amplitude and direction. The quasi-two dimensional nature of metasurfaces results in the majority of light-matter interaction occurring in the near-field proximity of the interfaces. Conventional techniques for calculating light interaction with composite systems, i.e. metamaterials, have been developed with volumetric materials in mind, and typically are inefficient in understanding the optics of metasurfaces. Moreover, since the optics of metasurfaces are more akin to the two dimensional structure graphene than to the traditional optics of volumetric composites, traditional effective medium theories cannot be used to describe the diffractive optics by characterizing the metasurface as an effective index or an effective permittivity/permeability. Although finite-difference and finite-element tools can in principle be used to characterize the optics of metasurface arrays, these techniques require volumetric meshes of the large structures with subwavelength resolution, which can be computationally intractable.
Here we propose an analog of effective medium description for metasurfaces by characterizing the metasurface by a two-dimensional polarizability. Since the typical spatial profile of a metasurface is inhomogeneous on the wavelength scale, polarizability becomes strongly nonlocal (dependent on the wavevector). Nonlocality is known to enable the propagation of multiple waves in volumetric structures; similarly, nonlocal polarization yields diffraction in two-dimensional metasurfaces. We show that adequate treatment of optics of metasurfaces requires discontinuity of both normal and tangential components of electric fields, and present comparison of the predictions of the developed formalism to full-wave solutions of Maxwell equations. The proposed formalism does not require any subwavelength volumetric meshing, or the solving of an eigenvalue problem, drastically reducing the total calculation time and required computational resources. The developed formalism can be used as a new convenient tool for understanding, design, and optimization of metasurface optics.