  Wave-based signal processing has witnessed a significant expansion of
interest in a variety of science and engineering disciplines, as it provides
new opportunities for achieving high-speed and low-power operations. Although
flat optics desires integrable components to perform multiple missions, yet,
the current wave-based analog computers can engineer only the spatial content
of the input signal where the processed signal obeys the traditional version of
Snell's law. In this paper, we propose a multi-functional metagrating to
modulate both spatial and angular properties of the input signal whereby both
symmetric and asymmetric optical transfer functions are realized using
high-order space harmonics. The performance of the designed compound metallic
grating is validated through several investigations where closed-form
expressions are suggested to extract the phase and amplitude information of the
diffractive modes. Several illustrative examples are demonstrated to show that
the proposed metagrating allows for simultaneous parallel analog computing
tasks such as first- and second-order spatial differentiation through a single
multi-channel structured surface. It is anticipated that the designed platform
brings a new twist to the field of optical signal processing and opens up large
perspectives for simple integrated image processing systems.
