  This paper studies an extremely large-scale reconfigurable intelligent
surface (XL-RIS) empowered covert communication system in the near-field
region. Alice covertly transmits messages to Bob with the assistance of the
XL-RIS, while evading detection by Willie. To enhance the covert communication
performance, we maximize the achievable covert rate by jointly optimizing the
hybrid analog and digital beamformers at Alice, as well as the reflection
coefficient matrix at the XL-RIS. An alternating optimization algorithm is
proposed to solve the joint beamforming design problem. For the hybrid
beamformer design, a semi-closed-form solution for fully digital beamformer is
first obtained by a weighted minimum mean-square error based algorithm, then
the baseband digital and analog beamformers at Alice are designed by
approximating the fully digital beamformer via manifold optimization. For the
XL-RIS's reflection coefficient matrix design, a low-complexity alternating
direction method of multipliers based algorithm is proposed to address the
challenge of large-scale variables and unit-modulus constraints. Numerical
results unveil that i) the near-field communications can achieve a higher
covert rate than the far-field covert communications in general, and still
realize covert transmission even if Willie is located at the same direction as
Bob and closer to the XL-RIS; ii) the proposed algorithm can enhance the covert
rate significantly compared to the benchmark schemes; iii) the proposed
algorithm leads to a beam diffraction pattern that can bypass Willie and
achieve high-rate covert transmission to Bob.
