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 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.

Nonlinear optical materials, such as transparent conductive oxides, have recently drawn a lot of attention when being integrated into metasurfaces and allowing full-optical control of the surface response. Although several methods for modeling the nonlinear materials have been proposed in the literature, most of them have the limitations on being non-dispersive and of instantaneous response. In this paper, we present a straightforward integration of an extended Drude-Lorentz model that captures the local intensity response of nonlinear materials while being dispersive and allowing for inertial response via a low-pass filtering process. This method is integrated into standard finite difference time-domain (FDTD) implementation of Maxwell’s equations and the auxiliary differential equations approach of the Drude-Lorentz model is extended via local intensity-dependent parameters. A numerical demonstration shows the response for a thin film of nonlinear material, where the parameters across the sample are time-varying with respect to the local intensity of the fields. Therefore, showing a direct feedback of the field profile to the nonlinear response of the material, which is critical when incorporating such films in resonating meta-atoms.

We present a rigorous semi-analytical formulation for the analysis of electromagnetic (EM) scattering from a cylinder constructed from a metasurface represented using surface susceptibilities. The formulation uses the Generalized Sheet Transition Conditions (GSTCs) to represent the surface and eigenfunction expansion (EFE) of the incident and scattered fields, by exploiting their angular periodicity. Incorporating a completely general non-uniform surface formulation of 36 susceptibility components, a matrix equation is formulated that can be solved for the field harmonic coefficients. The paper illustrates the methodology with a number of examples; including two formed from a finite sized practical unit-cell exhibiting a normally oriented magnetic resonance which is compared with commercial EM full-wave solver; other examples present surfaces that have a modulated gain/loss profile (i.e. amplitude modulation) and polarization conversion. It is found that for all cases the EFE solution very accurately captures the scattered fields in both the interior and exterior regions of the surface. Detailed convergence studies are further presented including the effect of susceptibility modulation on the number of terms needed in the EFEs, to reach a correct field solution. The proposed EFE framework is further compared with a second GSTC based simulator using an Integral Equations (IE) approach and it is found that the IE-GSTC simulated fields approach that of the EFE as the surface discretization is increased. The proposed EFE approach is thus a quick, rigorous methodology which, although limited in the geometry which it can model, has many advantages for investigation into the use of GSTCs, application development and providing a baseline for simulation studies.

A ray optical methodology based on the uniform theory of diffraction is proposed to model electromagnetic field scattering from curved metasurfaces. The problem addressed is the illumination of a purely reflective uniform cylindrical metasurface by a line source, models the surface with susceptibilities and employs a methodology previously used for cylinders coated in thin dielectric layers [1]. The approach is fundamentally based on a representation of the metasurface using the General Sheet Transition Conditions (GSTCs) which characterizes the surface in terms of susceptibility dyadics. An eigenfunction description of the metasurface problem is derived considering both tangential and normal surface susceptibilities, and used to develop a ray optics (RO) description of the scattered fields; including the specular geometrical optical field, surface diffraction described by creeping waves and a transition region over the shadow boundary. The specification of the fields in the transition region is dependent on the evaluation of the Pekeris caret function integral and the method follows [1]. The proposed RO-GSTC model is then successfully demonstrated for a variety of cases and is independently verified using a rigorous eigenfunction solution (EF-GSTC) and full-wave Integral Equation method (IE-GSTC), over the entire domain from the deep lit to deep shadow.

A static metasurface reflector based on a novel coupled resonator configuration is proposed to independently control the reflection phase and magnitude of linearly polarized incident fields, and is demonstrated experimentally in the millimeter-wave Ka-band around 30 GHz. The proposed concept is illustrated using a unit cell design consisting of a rectangular ring coupled with a rectangular slot resonator backed by a grounded dielectric slab. By geometrically tuning various dimensions of the two resonators, a near-perfect amplitude-phase coverage is achieved at a fixed design frequency of 30 GHz. To demonstrate the flexible beam-forming capability of the proposed metasurface reflectors, illustrative examples of fixed beam steering with varying reflection magnitudes, and asymmetric dual-beam patterns with specified reflection magnitude, reflection angles and beam-widths, are successfully shown. Compared to the standard method based on polarization rotation and resistive loadings with discrete values, the proposed technique does not generate undesired cross-polarization field reflection, and provides a continuous magnitude tuning including full absorption, along with wide phase coverage.

While metasurfaces (MSs) are constructed from deeply-subwavelength unit cells, they are generally electrically-large and full-wave simulations of the complete structure are computationally expensive. Thus, to reduce this high computational cost, non-uniform MSs can be modeled as zero-thickness boundaries, with sheets of electric and magnetic polarizations related to the fields by surface susceptibilities and the generalized sheet transition conditions (GSTCs). While these two-sided boundary conditions have been extensively studied for single sheets of resonant particles, it has not been shown if they can correctly model structures where the two sides are electrically isolated, such as a fully-reflective surface. In particular, we consider in this work whether the fields scattered from a fully reflective metasurface can be correctly predicted for arbitrary field illuminations, with the source placed on either side of the surface. In the process, we also show the mapping of a PEC sheet with a dielectric cover layer to bi-anisotropic susceptibilities. Finally, we demonstrate the use of the susceptibilities as compact models for use in various simulation techniques, with an illustrative example of a parabolic reflector, for which the scattered fields are correctly computed using a integral equation (IE) based solver.

An Integral Equation (IE) based field solver to compute the scattered fields from spatially dispersive metasurfaces is proposed and numerically confirmed using various examples involving physical unit cells. The work is a continuation of Part- 1 [1], which proposed the basic methodology of representing spatially dispersive metasurface structure in the spatial frequency domain, k. By representing the angular dependence of the surface susceptibilities in k as a ratio of two polynomials, the standard Generalized Sheet Transition Conditions (GSTCs) have been extended to include the spatial derivatives of both the difference and average fields around the metasurface. These extended boundary conditions are successfully integrated here into a standard IE-GSTC solver, which leads to the new IEGSTC-SD simulation framework presented here. The proposed IE-GSTC-SD platform is applied to various uniform metasurfaces, including a practical short conducting wire unit cell, as a representative practical example, for various cases of finite-sized flat and curvilinear surfaces. In all cases, computed field distributions are successfully validated, either against the semi-analytical Fourier decomposition method or the brute-force full-wave simulation of volumetric metasurfaces in the commercial Ansys FEM-HFSS simulator.

A simple method to describe spatially dispersive metasurfaces is proposed where the angle-dependent surface susceptibilities are explicitly used to formulate the zero thickness sheet model of practical metasurface structures. It is shown that if the surface susceptibilities of a given metasurface are expressed as a ratio of two polynomials of tangential spatial frequencies, k||, with complex coefficients, they can be conveniently expressed as spatial derivatives of the difference and average fields around the metasurface in the space domain, leading to extended forms of the standard Generalized Sheet Transition Conditions (GSTCs) accounting for the spatial dispersion. Using two simple examples of a short electric dipole and an all-dielectric cylindrical puck unit cells, which exhibit purely tangential surface susceptibilities and reciprocal/symmetric transmission and reflection characteristics, the proposed concept is numerically confirmed in 2D. A single Lorentzian has been found to describe the spatio-temporal frequency behavior of a short dipole unit cell, while a multi-Lorentzian description is developed to capture the complex multiple angular resonances of the dielectric puck. For both cases, the appropriate spatial boundary conditions are derived.

An accelerated Integral Equations (IE) field solver for determining scattered fields from electrically large electromagnetic metasurfaces utilizing Fast Multipole Method (FMM) is proposed and demonstrated in 2D. In the proposed method, practical general metasurfaces are expressed using an equivalent zero thickness sheet model described using surface susceptibilities, and where the total fields around it satisfy the Generalized Sheet Transition Conditions (GSTCs). While the standard IE-GSTC offers fast field computation compared to other numerical methods, it is still computationally demanding when solving electrically large problems, with a large number of unknowns. Here we accelerate the IE-GSTC method using the FMM technique by dividing the current elements on the metasurface into near- and far-groups, where either the rigorous or approximated Green’s function is used, respectively, to reduce the computation time without losing solution accuracy. Using numerical examples, the speed improvement of the FMM IE-GSTC method O(N1.5) over the standard IE-GSTC O(N3) method is confirmed. Finally, the usefulness of the FMM IE-GSTC is demonstrated by applying it to solve electromagnetic propagation inside an electrically large radio environment with strategically placed metasurfaces to improve signal coverage in blind areas, where a standard IE-GSTC solver would require prohibitively large computational resources and long simulation times.