  We consider the design and modeling of metasurfaces that couple energy from
guided waves to propagating wavefronts. This is a first step towards a
comprehensive, multiscale modeling platform for metasurface antennas-large
arrays of metamaterial elements embedded in a waveguide structure that radiates
into free-space--in which the detailed electromagnetic responses of
metamaterial elements are replaced by polarizable dipoles. We present two
methods to extract the effective polarizability of a metamaterial element
embedded in a one- or two-dimensional waveguide. The first method invokes
surface equivalence principles, averaging over the effective surface currents
and charges within an element to obtain the effective dipole moments; the
second method is based on computing the coefficients of the scattered waves
within the waveguide, from which the effective polarizability can be inferred.
We demonstrate these methods on several variants of waveguide-fed metasurface
elements, finding excellent agreement between the two, as well as with
analytical expressions derived for irises with simpler geometries. Extending
the polarizability extraction technique to higher order multipoles, we confirm
the validity of the dipole approximation for common metamaterial elements. With
the effective polarizabilities of the metamaterial elements accurately
determined, the radiated fields generated by a metasurface antenna (inside and
outside the antenna) can be found self-consistently by including the
interactions between polarizable dipoles. The dipole description provides an
alternative language and computational framework for engineering metasurface
antennas, holograms, lenses, beam-forming arrays, and other electrically large,
waveguide-fed metasurface structures.
