Abstract-A general method, based on susceptibility tensors, is proposed for the synthesis of metasurfaces transforming arbitrary incident waves into arbitrary reflected and transmitted waves. The proposed method exhibits two advantages: 1) it is inherently vectorial, and therefore better suited for full vectorial (beyond paraxial) electromagnetic problems, 2) it provides closedform solutions, and is therefore extremely fast. Incidentally, the method reveals that a metasurface is fundamentally capable to transform up to four independent wave triplets (incident, reflected and refracted waves). In addition, the paper provides the closed-form expressions relating the synthesized susceptibilities and the scattering parameters simulated within periodic boundary conditions, which allows one to design the scattering particles realizing the desired susceptibilities. The versatility of the method is illustrated by examples of metasurfaces achieving the following transformations: generalized refraction, reciprocal and non-reciprocal polarization rotation, Bessel vortex beam generation, and orbital angular momentum multiplexing.

The paper presents partial overview of the mathematical synthesis and the physical realization of metasurfaces, and related illustrative examples. The synthesis consists in determining the exact tensorial surface susceptibility functions of the metasurface, based on generalized sheet transition conditions, while the realization deals with both metallic and dielectric scattering particle structures. The examples demonstrate the capabilities of the synthesis and realization techniques, thereby showing the plethora of possible metasurface field transmission and subsequent applications. The first example is the design of two diffraction engineering birefringent metasurfaces performing polarization beam splitting and orbital angular momentum multiplexing, respectively. Next, we discuss the concept of the "transistor" metasurface, which is an electromagnetic linear switch based on destructive interferences. Then, we introduce a non-reciprocal non-gyrotropic metasurface using a pick-up circuit radiator (PCR) architecture. Finally, the implementation of all-dielectric metasurfaces for spatial dispersion engineering is discussed.

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Abstract-We introduce the concept of metasurface spatial processor, whose transmission is remotely and coherently controlled by the superposition of an incident wave and a control wave through the metasurface. The conceptual operation of this device is analogous to both that of a transistor and a MachZehnder interferometer, while offering much more diversity in terms of electromagnetic transformations. We demonstrate two metasurfaces, that perform the operation of electromagnetic switching and amplification.

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