  In the quest to realize analog signal processing using sub-wavelength
metasurfaces, in this paper, we demonstrate the first experimental
demonstration of programmable time-modulated metasurface processors based on
the key properties of spatial Fourier transformation. Exploiting space-time
coding strategy enables local, independent, and real-time engineering of not
only amplitude but also phase profile of the contributing reflective digital
meta-atoms at both central and harmonic frequencies. Several illustrative
examples are demonstrated to show that the proposed multifunctional calculus
metasurface is capable of implementing a large class of useful mathematical
operators, including 1st- and 2nd-order spatial differentiation, 1st-order
spatial integration, and integro-differential equation solving accompanied by
frequency conversions. Unlike the recent proposals, the designed time-modulated
signal processor effectively operates for input signals containing wide spatial
frequency bandwidths with an acceptable gain level. Proof-of-principle
simulations are also reported along with the successful realization of image
processing functions like edge detection. This time-varying wave-based
computing system can set the direction for future developments of programmable
metasurfaces with highly promising applications in ultrafast equation solving,
real-time and continuous signal processing, and imaging.
