  Non-imaging optical lenses can shape the light intensity from incoherent
sources to a desired target intensity profile, which is important for
applications in lighting, solar light concentration, and optical beam shaping.
Their surface curvatures are designed to ensure optimal transfer of energy from
the light source to the target. The performance of such lenses is directly
linked to their asymmetric freeform surface curvature, which is challenging to
manufacture. Metasurfaces can mimic any surface curvature without additional
fabrication difficulty by imparting a spatially-dependent phase delay using
optical antennas. As a result, metasurfaces are uniquely suited to realize
non-imaging optics, but non-imaging design principles have not yet been
established for metasurfaces. Here, we take an important step in connecting
non-imaging optics and metasurface optics, by presenting a phase-design method
for beam shaping based on the concept of optimal transport. We establish a
theoretical framework that enables a collimated beam to be redistributed by a
metasurface to a desired output intensity profile. The optimal transport
formulation leads to metasurface phase profiles that transmit all energy from
the incident beam to the output beam, resulting in an efficient beam shaping
process. Through a variety of examples, we show that our approach accommodates
a diverse range of different input and output intensity profiles. Last but not
least, a full field simulation of a metasurface has been done to verify our
phase-design framework.
