  Optical imaging and sensing systems based on diffractive elements have seen
massive advances over the last several decades. Earlier generations of
diffractive optical processors were, in general, designed to deliver
information to an independent system that was separately optimized, primarily
driven by human vision or perception. With the recent advances in deep learning
and digital neural networks, there have been efforts to establish diffractive
processors that are jointly optimized with digital neural networks serving as
their back-end. These jointly optimized hybrid (optical+digital) processors
establish a new "diffractive language" between input electromagnetic waves that
carry analog information and neural networks that process the digitized
information at the back-end, providing the best of both worlds. Such hybrid
designs can process spatially and temporally coherent, partially coherent, or
incoherent input waves, providing universal coverage for any spatially varying
set of point spread functions that can be optimized for a given task, executed
in collaboration with digital neural networks. In this article, we highlight
the utility of this exciting collaboration between engineered and programmed
diffraction and digital neural networks for a diverse range of applications. We
survey some of the major innovations enabled by the push-pull relationship
between analog wave processing and digital neural networks, also covering the
significant benefits that could be reaped through the synergy between these two
complementary paradigms.
