A simple waveguide-Floquet mapping is proposed and theoretically investigated that links the scattering response of a sub-wavelength metasurface unit cell under an infinite periodic array configuration and when measured inside a shielded rectangular waveguide environment. This mapping is based on extracting the effective surface susceptibilities as the fundamental constitutive parameters of the cells, via the Generalized Sheet Transition Conditions (GSTCs) of the unit cell using the waveguide measurements, and then reconstructing the unit cell response at an arbitrary incidence angles. Using two examples of a transmissive Huygens' metasurface and a purely reflective metasurface, the proposed waveguide-Floquet mapping is numerically confirmed to determine the complete angular scattering of the unit cells. The proposed waveguide-Floquet mapping thus represents a simple technique to experimentally determine the angular scattering characteristics of the unit cell using a lowcost effective waveguide measurements.

Higher-order boundary conditions are a useful tool for simplification of complex electromagnetic problems, such as spatially dispersive metasurfaces. A technique known as the Extended Generalized Sheet Transition Conditions (GSTCs) method has recently been proposed that allows for the systematic simulation of spatially dispersive metasurfaces using higherorder boundary conditions. A general metasurface, in general requires a boundary condition of infinite order spatial derivatives to accurately model the surface, making them unsuitable for conventional full-wave modeling techniques where higherorder derivatives are computed numerically. Here we propose a mixed spatial-spectral domain representation of the boundary conditions that avoids the explicit computation of higher-order derivatives while still being in a form suitable for integration into standard numerical methods. We integrate these boundary conditions into an integral equation solver which is then successfully validated through several numerical examples.

In this paper, we propose two different methods for time-domain finite-difference analysis of uniform temporally and spatially dispersive metasurfaces using their zero thickness sheet representations using the Generalized Sheet Transition Conditions (GSTCs). Metasurfaces are described here using their effective surface susceptibilities which are assumed to exhibit Lorentzian temporal dispersion characteristics. For both methods, the spatial dispersion of the, the surface susceptibilities (i.e., their dependence on angle of incidence) are represented using the extended GSTCs presented in [1]-[3]. However, the first method takes advantage of a polynomial expansion of the angle-dependent surface susceptibilities in terms of the transverse wavevector to implement spatial derivatives of the electric and magnetic polarization as well as the average field on the surface, leading to a coupled set of field equations encompassing the entire surface. Limitations for this method are presented in terms of poor conditioning for a coupled system of equations and an inconvenient extension to the higher-order expansion of the susceptibility terms. The second method lifts these limitations by solving the spatial dispersion problem in the spatial frequency domain at every time step. Both methods are validated for custom Lorentzian models and two canonical physical cells while comparing their transmission and reflection coefficients with analytical results.

A novel metasurface reflector unit cell based on two coupled resonators is proposed and demonstrated for real-time reconfigurable beamforming in the X-band (8-12 GHz). The unit cell is composed of a split-ring resonator (SRR) with tunable capacitance and a dipole-ring resonator (DRR) with tunable resistance, whose collective variations allow for control over the complex reflectance at a desired frequency. To gain physical insight into its working mechanism, the proposed unit cell is modeled as a coupled Lorentz oscillator using the surface susceptibility description of the unit cell. Thereafter, a metasurface reflector based on the proposed unit cell is demonstrated in full-wave simulations to achieve various beamforming examples, such as beam-steering, side-lobe level control, beam-steering with amplitude control, and multi-beam patterns, from a single normally incident plane wave excitation. Three metasurface reflectors are fabricated to experimentally demonstrate the proposed concept; the first is based on a SRR with a varactor diode, the second is based on a DRR with a PIN diode, and the third is based on the proposed coupled SRR-DRR configuration with both varactor and PIN diodes, for simultaneous magnitude and phase control. The metasurface reflector based on the coupled resonator unit cell is experimentally demonstrated to achieve versatile beam transformations including beam-steering with amplitude control and multi-beam patterns.

An analytical method is proposed to synthesize the angle-dependent surface susceptibilities, χ of spatially dispersive or non-local zero-thickness metasurfaces. The proposed method is based on the extended Generalized Sheet Transition Conditions (GSTCs), whereby spatially dispersive metasurfaces are modeled using angle-dependent surface susceptibilities that take the form of rational polynomial functions of the transverse wave-vector, k∥. The suggested method derives the rational polynomial form of χ(k∥), which can then be expressed in the space-domain using spatial derivatives of the fields, resulting in a corresponding higher-order spatial boundary condition to achieve the desired field operation. The proposed synthesis method is illustrated using variety of examples such as, a space-plate, spatial filters and field absorbers, which are then validated using an Integral Equation (IE) solver, in which the corresponding higher-order boundary conditions are integrated to predict the scattered fields. The proposed method thus not only represents a simple way to synthesize ideal zero-thickness metasurfaces, but helps establishes a way to define fundamental operational limits of spatially dis- persive metasurfaces. This is illustrated by considering the space- plate example, and deriving the fundamental trade-off between operation bandwidth and the achievable space-compression.

A novel shape-shifting mechatronic reflector system is proposed and demonstrated experimentally with a variety of real-time beam-forming examples. The reflector system consists of a 1D array of panels of sub-wavelength split ring resonators (SRR) loaded with PIN-diodes for reflection magnitude control, and a stepper motor array which de-embeds each panel to accomplish column-by-column phase control, thus offering excellent decoupling between the magnitudes and phases. A complete system prototype is built including a printed circuit board (PCB) based resonator array forming the reflector front end, a stepper motor array assembly, a custom built chassis using 3D printed parts and a fully programmable analog and digital electronic backend for PIN-diode biasing and motor control. The real-time beam-forming feature of the proposed reflector system is further experimentally demonstrated using the illustrative examples of single beam steering with gain control and generation of asymmetric dual beam radiation patterns where the two beams are independently controlled - a capability unlocked through achieving true decoupling between the reflection magnitudes and phases.

A metasurface reflector unit cell is proposed for achieving near-complete-range of complex reflectance (i.e., magnitude and phase) for versatile beamforming capabilities. The unit cell is based on coplanar coupled resonators: a split ring resonator (SRR) with a lumped capacitor and a lumped resistor, and a dipole ring resonator (DRR) with another lumped capacitor. The DRR inserted inside the SRR creates a coupled resonance configuration which results in an enhanced complex reflectance range at the desired frequency. To provide a physical insight and explain the operation principle of the structure, the response of the unit cell is modeled as a coupled Lorentz oscillator via the effective surface susceptibilities, where a unique plasma, damping constant, and resonant frequency can be attributed to each resonator. The proposed unit cell is demonstrated in an array configuration for linear-polarized beamforming, where full-wave simulations are used to demonstrate beam-steering, gain control, side-lobe level control and dual- and triple-beam generation, as illustrative examples. Finally experimental demonstration is performed to validate the full-wave results and obtain in-depth electrical characterization of the reflectors.

A simple method is proposed to characterize metasurface unit cells using rectangular waveguide. The method maps the S-parameters obtained using a dominant-mode waveguide to the S-parameters of fundamental Floquet mode for a plane wave incident at any desired angle. The mapping between the dominant waveguide mode and the fundamental Floquet mode is executed by extracting the equivalent surface susceptibilities of the metasurface unit cell. The results obtained from the proposed method show good agreement with the full-wave simulation results of the metasurface unit cell.

A semi-analytical technique to simulate spatially dispersive (SD) cylindrical metasurfaces is proposed and demon- strated. This method is based on the well-known Eigenfunction Expansion (EFE) method and extends the work of [1], [2] by integrating the Extended GSTCs into this pre-existing method specific to cylindrical surfaces. This method is then applied to several cylindrical metasurfaces composed of spatially dispersive wire dipole unit cells, and successfully verified against commercial full-wave simulations. Finally, using a case of a closed metasur- face shell, the usefulness of the proposed EFE-SD method is shown to illustrate the effect of metasurface thickness on the scattered fields inside and outside the shell.

A 1-bit reconfigurable metasurface reflector is designed. It can provide beam-steering of circularly polarized (CP) re- flected waves when a linearly polarized (LP) plane wave is incident upon it. The unit cell of the metasurface provides 1-bit coding (‘bit-0’ and ‘bit-1’) for both v- and u-polarized reflected waves when two distinct reverse bias voltages (3 V and 20 V) are applied to the varactor diodes. Using such unit elements, a 20×20 array is constructed. During the array construction, a quadrature phase difference is main- tained between the v- and u- polarized components of each element. Using both theory and full-wave simulation re- sults, it is shown that the proposed metasurface reflectarray can steer two symmetrically deflected CP beams with two opposite handedness for a LP plane wave incidence.

A zero thickness model of non-uniform spatially dispersive (SD) metasurfaces is developed and demonstrated numerically for practical structures. This is an extension of [1], [2], which proposed a method of modeling uniform spatially dispersive metasurfaces by expressing their surface susceptibilities as rational polynomial functions in the spatial frequency domain. This led to the extended generalized sheet transition conditions (GSTCs) forming a set of differential equations relating the spatial derivatives of both difference and average fields around the surface that were then integrated into an integral equation (IE) solver. Here, the extended GSTCs are further developed to model non-uniform metasurfaces by approximating them as locally linear space invariant (LSI). Using this model, a semi-analytical Floquet method is derived to predict scattered fields from periodically varying metasurfaces. The extended GSTCs, Floquet method, and IE method are tested on several nonuniform surfaces consisting of short metal dipole cells of varying lengths exhibiting strong spatial dispersion. The physical surfaces are simulated in Ansys FEM-HFSS while their zero thickness equivalents, both dispersive and non-dispersive, are simulated using the Floquet and IE methods. Excellent agreement is shown between the Floquet-SD and IE-GSTC-SD methods and HFSS, demonstrating the importance of spatial dispersion to model their scattered fields.

This paper presents a formulation of an eigenfunction expansion (EFE) technique for the analysis of multi-shell cylindrical metasurface configurations and tests the validity of zero thickness sheet models for simulating practical structures. The formulation homogenizes the metasurface, modeling the surface as a zero-thickness discontinuity characterized by surface susceptibilities and allows for the incorporation of Perfect Electric Conductor (PEC) cores and partial cylindrical surfaces (arcs/sectors). A primary advantage of the method is that it is a fully rigorous semi-analytical approach and is well suited for investigating the limits of the zero thickness sheet models model via the Generalized Sheet Transition Conditions (GSTCs). Comparison to full-wave simulations in commercial solvers show that, for a thin deep sub-wavelength unit cell, single shell arcs with increasing curvatures (up to at least half of a wavelength in radius) are accurately captured by the use of the model. The simulations also show that the edge diffraction off the finite length arcs are well modeled. This is an important result as it indicates that susceptibilities extracted from full-wave unit-cell simulations assuming a flat periodic infinite surface are appropriate for use in high curvature finite surfaces. A second set of simulations showed that the EFE predicted multi-shell field distributions accurately when compared to unit-cell based simulations. A final set of simulations for single and double shell configurations is used to demonstrate the limitations of the GSTC approach for unit cells of appreciable thickness. Using a relatively thick unit cell (?7% of the excitation wavelength) and two different extraction models (cell edge and center) comparison to full-wave simulation shows that for resonant structures errors in field prediction are present. This is a clear indication that the zero-thickness assumption will break down for thick cells for some surface configurations. This paper demonstrates some clear advantages of the EFE approach; it is fast, rigorous and can accommodate a large range of geometric structures.