Generalized Sheet Transition Conditions Research Articles

Eigenfunction Expansion (EFE) Analysis of Cylindrical Metasurfaces—Part II: Sectors and Multishells

This article presents a formulation of an eigenfunction expansion (EFE) technique for the analysis of multishell cylindrical metasurface (MS) configurations and tests the validity of zero-thickness sheet models for simulating practical structures. The formulation homogenizes the MS, modeling the surface as a zero-thickness discontinuity via the generalized sheet transition conditions (GSTCs) characterized by surface susceptibilities. Comparison with full-wave simulations in commercial solvers show that, for a thin deep subwavelength 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 ff 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 multishell field distributions accurately when compared with 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.

Time-Domain Meshless Method Simulation of Metasurfaces With Generalized Sheet Transition Conditions

This paper presents a new time-domain formulation of a meshless method based on the generalized sheet transition conditions (GSTCs). This formulation is applicable for modeling a kind of dispersive metasurface and a time-varying metasurface. This is the first time a meshless method is developed for analysis of the metasurfaces using GSTCs. The significance of time-domain meshless methods originates from their capabilities for multi-scale and conformal modeling in addition to analysis of problems in a broad frequency-range. Here, for simplicity, the meshless formulation is presented for analysis of 1-D problems consisting of monoisotropic metasurfaces. The extension of this method to 2-D and 3-D problems or bianisotropic metasurfaces is straightforward. The scheme is verified by numerical examples.

Zero Thickness Surface Susceptibilities and Extended GSTCs—Part I: Spatially Dispersive Metasurfaces

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 Lorentzian representation with angle-dependent parameters is further proposed and found to accurately model the complex temporal-spatial frequency response via their electric and magnetic susceptibilities of two unit cell examples - a metallic dipole and a dielectric puck resonator. For both cases, the appropriate spatial boundary conditions are derived.

Spatially Dispersive Metasurfaces—Part II: IE-GSTC-SD Field Solver With Extended GSTCs

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 IE-GSTC-SD simulation framework. 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.

Quadrupolar susceptibility modeling of substrated metasurfaces with application to the generalized Brewster effect.

We derive generalized sheet transition conditions (GSTCs) including dipoles and quadrupoles, using generalized functions (distributions). This derivation verifies that the GSTCs are valid for metasurfaces in non-homogeneous environments, such as for practical metasurfaces fabricated on a substrate. The inclusion of quadrupoles and modeling of spatial dispersion provides additional hyper-susceptibility components which serve as degrees of freedom for wave transformations. We leverage them to demonstrate a generalized Brewster effect with multiple angles of incidence at which reflection is suppressed, along with an "anti-Brewster" effect where transmission is suppressed.

A Full-Vectorial Spectral Element Method With Generalized Sheet Transition Conditions for High-Efficiency Metasurface/Metafilm Simulation

Recently, metasurfaces as 2-D metamaterials have been widely applied due to their advantages in low loss, ease of fabrication, and tunability in the phase, magnitude, and polarization of incident electromagnetic waves. However, due to the subwavelength-thickness properties of metasurfaces, their direct modeling and simulation are challenging because of the heavy computational burden. In this work, a high-efficiency spectral element method with generalized sheet transition conditions (SEM-GSTCs) is proposed by replacing finite thickness metasurfaces with zero-thickness equivalent boundary conditions to avoid dense elements and the associated enormous number of unknowns. Moreover, both the scalar and the full-vectorial SEM-GSTC are implemented. Some illustrative numerical examples are shown to verify that the proposed SEM-GSTC not only can obtain highly accurate numerical results but also is effective in addressing complex functions ranging from isotropic to fully bianisotropic metasurfaces.

A Locally-Implicit Discontinuous Galerkin Time-Domain Method to Simulate Metasurfaces Using Generalized Sheet Transition Conditions

The generalized sheet transition conditions (GSTCs) are incorporated into a discontinuous Galerkin time-domain (DGTD) method to efficiently simulate metasurfaces. The numerical flux for GSTCs is derived for the first time using the Rankine–Hugoniot jump conditions. This numerical flux includes the time derivatives of fields, and therefore, explicit time integration schemes, which are traditionally used with DGTD, do not yield a stable time marching method. To alleviate this bottleneck, a new time marching scheme, which solves a local matrix system for the unknowns of the elements touching the same GSTC face, is developed. This locally implicit method maintains its high-parallel efficiency just like the traditional explicit DGTD schemes. Numerical results, which validate the accuracy of the proposed method against analytical solutions and demonstrate its applicability to the simulation of curved and space/time-varying metasurfaces, are presented.

Surface Susceptibilities as Characteristic Models of Reflective Metasurfaces

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, nonuniform 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 (BCs) 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 MS 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 perfect electric conductor (PEC) sheet with a dielectric cover layer to bianisoptropic 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 an integral-equation (IE)-based solver.

Ray Optical Scattering From Uniform Reflective Cylindrical Metasurfaces Using Surface Susceptibility Tensors

A ray optical methodology based on the uniform theory of diffraction (UTD) is proposed to model electromagnetic (EM) field scattering from curved metasurfaces (MSs). The problem addressed is the illumination of a purely reflective uniform cylindrical MS by a line source, models the surface with susceptibilities and employs a methodology previously used for cylinders coated in thin dielectric layers [Kim and Wang (1989)]. The approach is fundamentally based on a representation of the MS using the generalized sheet transition conditions (GSTCs) which characterizes the surface in terms of susceptibility dyadics. An eigenfunction (EF) description of the MS 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 [Kim and Wang (1989)]. The proposed RO-GSTC model is then successfully demonstrated for a variety of cases and is independently verified using a rigorous EF solution (EF-GSTC) and full-wave Integral Equation method (IE-GSTC), over the entire domain from the deep lit to deep shadow.

Accelerated IE-GSTC Solver for Large-Scale Metasurface Field Scattering Problems Using Fast Multipole Method (FMM)

An accelerated integral equations (IE) field solver for determining scattered fields from electrically large electromagnetic metasurfaces using fast multipole method (FMM) is proposed and demonstrated in 2-D. 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 with 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 { <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N^{3/2})$ </tex-math></inline-formula> } over the standard IE-GSTC { <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$O(N^{3})$ </tex-math></inline-formula> } method is confirmed. Finally, the usefulness of 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.

Stable GSTC Formulation for Maxwell’s Equations

We revisit the classical zero-thickness generalized sheet transition conditions (GSTCs) that are a key tool for efficiently designing metafilms able to control the flow of light in the desired way. It is shown that it is more convenient to use an enlarged formulation of the GSTC in which the original metafilm is replaced by GSTCs that exclude the layer from the physical or computational domain. These new “layer” transition conditions have the same form as their “sheet” analogs; hence, they do not necessitate additional complications in their use, and their advantage is that they provide a well-posed problem and, hence, guarantee the stability of numerical schemes in the time domain. These assessments are demonstrated for an all-dielectric structure; the effective susceptibility tensors are derived due to asymptotic analysis combined with homogenization technique, and bounds for the susceptibilities entering the balance of energy are provided. While negative constant susceptibilities appear in the classical zero-thickness GSTCs, their values in the enlarged formulation are always positive, which ensures the stability of the effective problem. The validation of the effective model is provided by means of comparison with direct numerics in two and three dimensions.

Modeling of Metasurfaces Using Discontinuous Galerkin Time-Domain Method Based on Generalized Sheet Transition Conditions

A new discontinuous Galerkin time-domain (DGTD) method based on generalized sheet transition conditions (GSTCs) for analyzing the transient response of electromagnetic metasurfaces is presented. The discontinuities in electromagnetic fields around the vicinity of the metasurface are modeled by placing virtual edges on both sides of the metasurface. The GSTCs are incorporated in the updating equations of the standard DGTD algorithm by integrating the time-domain GSTCs formula with Galerkin’ s-weighted residual method. The DGTD-GSTC formulation can also simulate a bf piecewise linear approximation of a curved metasurfaces due to the unstructured mesh. This is the first time that DGTD is applied to the simulation of GSTCs. The implementation of this new method is verified by numerical cases.

A Vectorial Discontinuous Galerkin Time-Domain Method Incorporating Generalized Sheet Transition Conditions

A vectorial discontinuous Galerkin time-domain method incorporating generalized sheet transition conditions (DGTD-GSTC) is proposed for the simulation of curve metasurfaces. This scheme utilizes virtual fields to realize the discontinuity across the metasurface described by GSTCs, which is beyond the ability of conventional boundary conditions. Compared to the previous research on modeling approaches for metasurfaces based on GSTCs, this scheme is particularly suited for solving practical metasurface problems due to its ability for complex geometry modeling and broadband analysis. This scheme is also shown to provide a fundamental generalization of the conventional DGTD technique. Five 2-D representative examples are presented to validate the effectiveness of this newly proposed method.

A New High Aperture Efficiency Transmitarray Antenna Based on Huygens Metasurfaces

In this article, a new transmitarray (TA) antenna is designed and fabricated based on Huygens metasurfaces (HMSs). An analytical method has been used to achieve the distribution of surface parameters equivalent to HMS. In this method, the required equations for computing the surface parameters are derived using ray tracing, generalized sheet transition conditions, and two conditions of local power consumption and local impedance equalization. In order to implement these surface parameters, wire and loop unit cells are utilized, which are exposed to the radiation transversely. The array of these unit cells offers high aperture efficiency broadband TA antennas in low aperture profile. The fabricated TA antenna with a low aperture thickness of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0.2\lambda $ </tex-math></inline-formula> exhibits the bandwidth, sidelobe level (SLL), and cross-polarization level of 14%, −19 dB, and −19 dB, respectively. In addition, the high aperture efficiency of 65% is reached with this TA, which shows great improvement in comparison to previous works.

IE-GSTC Metasurface Field Solver Using Surface Susceptibility Tensors With Normal Polarizabilities

An Integral Equation (IE) based electromagnetic field solver using metasurface susceptibility tensors is proposed and validated using variety of numerical examples in 2D. The proposed method solves for fields generated by the metasurface which are represented as spatial discontinuities satisfying the Generalized Sheet Transition Conditions (GSTCs), and described using tensorial surface susceptibility densities, $\bar{\bar{\chi}}$. For the first time, the complete tensorial representation of susceptibilities is incorporated in this integrated IE-GSTC framework, where the normal surface polarizabilities and their spatial derivatives along the metasurface are rigorously taken into account. The proposed field equation formulation further utilizes a local co-ordinate system which enables modeling metasurfaces with arbitrary orientations and geometries. The proposed 2D BEM-GSTC framework is successfully tested using variety of examples including infinite and finite sized metasurfaces, periodic metasurfaces and complex shaped structures, showing comparisons with both analytical results and a commercial full-wave solver. It is shown that the zero-thickness sheet model with complete tensorial susceptibilities can very accurately reproduce the macroscopic fields, accounting for their angular field scattering response and the edge diffraction effects in finite-sized surfaces.

Optimization of Electromagnetic Metasurface Parameters Satisfying Far-Field Criteria

Electromagnetic metasurfaces offer the capability to realize arbitrary power-conserving field transformations. These field transformations are governed by the generalized sheet transition conditions, which relate the tangential fields on each side of the surface through the surface parameters. Ideally, designers would solve for the surface parameters based on their application-specific far-field criteria. However, determining the surface parameters for these criteria is challenging without knowledge of the tangential fields on each side of the surface, which are not unique for a given far-field pattern. Current designs are generally restricted to analytical examples where the tangential fields can be solved for, or determined via <i>ad hoc</i> methods, although there has been recent work to circumvent this. This article presents an optimization scheme, which determines surface parameters, such as electric impedance, magnetic admittance, and magnetoelectric coupling, satisfying far-field constraints, such as beam level, sidelobe level, and null locations. The optimization is performed using a method of moments-based model incorporating edge effects and mutual coupling. The surface parameters are optimized for using the alternating direction method of multipliers. Examples of this optimization scheme performing multicriteria pattern forming, extreme angle small surface refraction, and Chebyshev-like beamforming are presented.

Dynamic metasurface control using Deep Reinforcement Learning

Dynamic metasurface is an emerging concept for achieving a flexible control of electromagnetic waves. Generalized sheet transition conditions (GSTCs) can be used to model the relationship between the electromagnetic response and surface susceptibility parameters characterizing a metasurface. However, when it comes to the inverse problem of designing and controlling a metasurface in a space–time varying context based on GSTCs, the dynamic synthesis of the susceptibility parameters is a difficult and non-intuitive task. In this paper, we transform the inverse problem of solving dynamic susceptibility parameters into a sequence of control problems. Based on FDTD numerical simulations, a Deep Reinforcement Learning (DRL) framework using a proximal policy optimization (PPO) algorithm and a fully connected neural network is designed to control the susceptibility parameters intelligently and efficiently, promoting the further expansion of the application range of metasurface and thus helping realize more flexible and effective control of electromagnetic waves. We provide numerical results in a one-dimensional setting to show the applicability, correctness and effectiveness of the proposed approach.

Ray-Optical Evaluation of Scattering From Electrically Large Metasurfaces Characterized by Locally Periodic Surface Susceptibilities

This work continues the development of the ray-tracing method of de Jong (2021) for computing the scattered fields from metasurfaces characterized by locally periodic reflection and transmission coefficients. In this work, instead of describing the metasurface in terms of scattering coefficients that depend on the incidence direction, its scattering behavior is characterized by the surface susceptibility tensors that appear in the generalized sheet transition conditions (GSTCs). As the latter quantities are constitutive parameters, they do not depend on the incident field and, thus, enable a more compact and physically motivated description of the surface. The locally periodic susceptibility profile is expanded into a Fourier series subject to a spatially varying phase parameter, and the GSTCs are rewritten in a form that enables them to be numerically solved for the reflected and transmitted surface fields. The phase parameter can either be determined from a prescribed surface transformation or extracted from known surface susceptibilities. A method for the extraction of this phase parameter from a susceptibility profile using a spatial variant of the short-time Fourier transform (STFT) is proposed. The scattered field at arbitrary detector locations is constructed by evaluating critical-point contributions of the first and second kinds using a forward ray-tracing (FRT) scheme. The accuracy of the resulting framework has been verified with an integral equation-based boundary element method (BEM)-GSTC full-wave solver for a variety of examples, such as a periodically modulated metasurface, a metasurface diffuser, and a beam collimator.

Modeling of second-order nonlinear metasurfaces

We present a frequency-domain modeling technique for second-order nonlinear metasurfaces. The technique is derived from the generalized sheet transition conditions (GSTCs), which have been so far mostly used for modeling linear metasurfaces. In this work, we extend the GSTCs to include effective nonlinear polarizations. This allows retrieving the effective nonlinear susceptibilities of a given metasurface and predict its nonlinear scattering responses under arbitrary illumination conditions. We apply this modeling technique to the case of metasurfaces made of a periodic arrangement of T-shaped gold nanoparticles. For verification, several metasurfaces are fabricated and a fair agreement is found when comparing simulated data and experimental results. The proposed model may thus serve as a design platform to implement complex nonlinear metasurface based applications.

Wood's anomaly beyond the Meixner-Schäfke theorem: Analytical and experimental investigation of sinusoidally modulated metasurfaces with normal susceptibilities

We present a rigorous analysis of plane-wave-illuminated sinusoidally modulated metasurfaces (MSs) with normal susceptibility components, without limiting the modulation-index values. In contrast to tangentially polarizable scenarios, which are treated within the conventional continued-fraction framework of Meixner and Sch\"afke, the unconventional structure of the generalized sheet transition conditions with regard to normal components manifests exotic stability conditions, which were not addressed by this framework. By introducing small losses into the constituents of such MSs (inevitable in practice), we resolve these stability issues and establish a valid solution for the scattered fields. Such solutions reveal that for a certain range of modulations these surfaces feature a resonant absorptive notch at an angle related to the period, associated with strong coupling to the weakly evanescent first-order Floquet-Bloch harmonic. Based on our observations, we identify this phenomenon as Wood's anomaly for MSs with normal susceptibilities. We verify these observations numerically and experimentally using suitable, systematically designed, printed-circuit-board prototypes. This work highlights the fundamental intricacies involving theoretical analyses of inhomogeneous normally polarizable MSs and constitutes a firm ground for further exploration and utilization of nonuniform configurations with these mostly ignored degrees of freedom.