  A fair beam allocation framework through reconfigurable intelligent surfaces
(RISs) is proposed, incorporating the Max-min criterion. This framework focuses
on designing explicit beamforming functionalities through optimization.
Firstly, realistic models, grounded in geometrical optics, are introduced to
characterize the input/output behaviors of RISs, effectively bridging the gap
between the requirements on explicit beamforming operations and their practical
implementations. Then, a highly efficient algorithm is developed for Max-min
optimizations involving quadratic forms. Leveraging the Moreau-Yosida
approximation, we successfully reformulate the original problem and propose
iterations to attain the optimal solution. A comprehensive analysis of the
algorithm's convergence is provided. Importantly, this approach exhibits
excellent extensibility, making it readily applicable to address a broader
class of Max-min optimization problems. Finally, numerical and prototype
experiments are conducted to validate the effectiveness of the framework. With
the proposed beam allocation framework and algorithm, we clarify that several
crucial redistribution functionalities of RISs, such as explicit
beam-splitting, fair beam allocation, and wide-beam generation, can be
effectively implemented. These explicit beamforming functionalities have not
been thoroughly examined previously.
