  Modulation of metasurfaces in time gives rise to several exotic space-time
scattering phenomena by violating the reciprocity and generation of
higher-order frequency harmonics. We introduce a new design paradigm for
time-modulated metasurfaces, offering electrically tunable engineering of the
generated frequency harmonics and their emerging wavefronts by controlling the
phase delay in modulation. It is demonstrated that the light acquires a
dispersionless phase shift regardless of incident angle and polarization, upon
undergoing frequency conversion in a time-modulated metasurface which is
linearly proportional to the modulation phase delay and the order of generated
frequency harmonic. The conversion efficiency to the frequency harmonics is
independent of modulation phase delay and only depends on the modulation depth
and resonant characteristics of the metasurface element, with the highest
efficiency occurring in the vicinity of resonance, and decreasing away from the
resonant regime. The design approach allows for creating tunable spatially
varying phase discontinuties with 2{\pi} span in the wavefronts of generated
frequency harmonics for a wide range of frequencies and incident angles.
Specifically, we apply this approach to a time-modulated metasurface in the
Teraherz regime consisted of graphene-wrapped silicon microwires. For this
purpose, we use an accurate and efficient semi-analytical framework based on
multipole scattering. We demonstrate the utility of the design for tunable beam
steering and focusing of the generated frequency harmonics. Furthermore, we
rigorously verify the broadband and wide-angle performance of the metasurface
in manipulation of the generated frequency harmonics. The proposed design
approach enables a new class of high-efficiency tunable metasurfaces with wide
angular and frequency bandwidth, wavefront engineering capabilities and
multi-functionality.
