  This work theoretically and experimentally studies metasurfaces with
spatially-discrete, traveling-wave modulation (SD-TWM). A representative
metasurface is considered consisting of columns of time-modulated subwavelength
unit cells, referred to as stixels. SD-TWM is achieved by enforcing a time
delay between temporal waveforms applied to adjacent columns. In contrast to
the continuous traveling-wave modulation commonly assumed in studies of
space-time metasurfaces, here the modulation is spatially discretized. In order
to account for the discretized spatial modulation, a modified Floquet analysis
is introduced based on a new boundary condition that has been derived for
SD-TWM structures. The modified Floquet analysis separates the scattered field
into its macroscopic and microscopic variations. The reported theoretical and
experimental results reveal that the electromagnetic behavior of an SD-TWM
metasurface can be categorized into three regimes. For electrically-large
spatial modulation periods, the microscopic field variation across each stixel
can be neglected. In this regime, the space-time metasurface allows
simultaneous frequency translation and angular deflection. When the spatial
modulation period on the metasurface is electrically small, the microscopic
variation results in new metasurface capabilities such as subharmonic mixing.
When the spatial modulation period of the metasurface is wavelength-scale, the
metasurface allows both subharmonic mixing and angular deflection to be
achieved simultaneously. To verify our analysis, a dual-polarized,
spatio-temporally modulated metasurface, is developed and measured at X-band
frequencies.
