  We present a design strategy for selecting the effective polarizability
distribution for a metasurface aperture needed to form a desired radiation
pattern. A metasurface aperture consists of an array of subwavelength
metamaterial elements, each of which can be conceptualized as a radiating,
polarizable dipole. An ideal polarizability distribution can be determined by
using a holographic approach to first obtain the necessary aperture fields,
which can then be converted to a polarizability distribution using equivalence
principles. To achieve this ideal distribution, the polarizability of each
element would need to have unconstrained magnitude and phase; however, for a
single, passive, metamaterial resonator the magnitude and phase of the
effective polarizability are inextricably linked through the properties of the
Lorentzian resonance, with the range of phase values restricted to a span of at
most 180 degrees. Here, we introduce a family of mappings from the ideal to the
available polarizability distributions, easily visualized by plotting both
polarizabilities in the complex plane. Using one of these mappings it is
possible to achieve highly optimized beam patterns from a metasurface antenna,
despite the inherent resonator limitations. We introduce the mapping technique
and provide several specific examples, with numerical simulations used to
confirm the design approach.
