  Optical metasurfaces consist of a 2D arrangement of scatterers, and they
control the amplitude, phase, and polarization of an incidence field on demand.
Optical metasurfaces are the cornerstone for a future generation of flat
optical devices in a wide range of applications. The rapidly growing advances
in nanofabrication have made the versatile design and analysis of these
ultra-thin surfaces an ever-growing necessity. However, despite their
importance, a comprehensive theory to describe the optical response of periodic
metasurfaces in closed-form and analytical expressions has not been formulated,
and prior attempts were frequently approximate. Here, we develop a theory that
analytically links the properties of the scatterer, from which a periodic
metasurface is made, to its optical response via the lattice coupling matrix.
The scatterers are represented by their polarizability or T matrix, and our
theory works for normal and oblique incidence. We provide explicit expressions
for the optical response up to octupolar order in both spherical and Cartesian
coordinates. Several examples demonstrate that our analytical tool constitutes
a paradigm shift in designing and understanding optical metasurfaces. Novel
fully-diffracting metagratings and particle-independent polarization filters
are proposed, and novel insights into the response of Huygens' metasurfaces
under oblique incidence are provided. Our analytical expressions are a powerful
tool for exploring the physics of metasurfaces and designing novel flat optics
devices.
