  We theoretically formulate and experimentally demonstrate an analytical
formalism for the design of printed circuit board (PCB) metagratings (MGs)
exercising individual control over the amplitude and phase of numerous
diffracted modes, in both reflection and transmission. Lately, these periodic
arrangements of subwavelength polarizable particles (meta-atoms) were shown to
deflect an incoming plane wave to prescribed angles with very high
efficiencies, despite their sparsity with respect to conventional metasurfaces.
Nonetheless, most reported MGs were designed based on full-wave optimization of
the meta-atoms, with the scarce analytical schemes leading directly to
realizable devices were restricted to single-layer reflecting structures,
controlling only the partition of power. In this paper, we present an
analytical model for plane-wave interaction with a general multilayered
multielement MG, composed of an arbitrary number of meta-atoms distributed
across an arbitrary number of layers in a given stratified media configuration.
For a desired (forward and backward) diffraction pattern, we formulate suitable
constraints, identify the required number of degrees of freedom, and
correspondingly set them to yield a detailed MG configuration implementing the
prescribed functionality; no full-wave optimization is involved. To verify and
demonstrate the versatility of this systematic approach, fabrication-ready
multilayer PCB MGs for perfect anomalous refraction and non-local focusing are
synthesized, fabricated, and experimentally characterized, for the first time
to the best of our knowledge, indicating very good correspondence with
theoretical predictions. This appealing semianalytical methodology is expected
to accelerate the development of MGs and extend the relevant range of
applications, yielding practical MG designs on demand for arbitrary beam
manipulation.
