  We study the capabilities of time-modulated diffraction gratings as imaging
devices. It is shown that a time-dependent but transversally homogeneous slab
can be used to make a perfect image of an object in the far-field, since all
the evanescent modes couple to propagative time-diffracted orders. It is found
that, if the image to be obtained is axially symmetric, it can be recovered by
measuring the time-signal at a single point, without the need of performing a
spatial scan, so that time gratings can act as well as single-pixel imaging
devices. In the case of having an object without axial symmetry, the
time-grating can be combined with a spatial grating, and then the full image
can be recovered again with a measurement at a single point. We apply the
theory of compressive sensing to optimize the recovery method and numerical
examples are provided. We show therefore that time-modulated gratings can be
used to perfectly recover the image of an object in the far field and after
measuring at a single point in the space, being therefore a promising approach
to superesoloution and ultra-fast imaging.
