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This article investigates the dispersion properties of metasurfaces with hexagonal lattices, including potential higher symmetric configurations. We explore the relationships between the periodicity of hexagonal lattices and their dispersion properties, paying special attention to how hexagonal periodic structures can be analyzed with either a hexagonal primitive unit cell or a rectangular supercell. We also study the possibility of introducing higher symmetries into hexagonal periodic structures, including glide symmetry and mirrored half-turn symmetry. To complement and validate the analysis, we designed a graded-index Luneburg lens antenna with a dielectric-filled hexagonal holey structure working in the K$_{\mathrm{a}}$-band. The antenna generates steerable highly directive beams from 26 GHz to 30 GHz, which corroborates our analysis. Our findings provide valuable insight into the dispersion properties of hexagonal-lattice metasurfaces and demonstrate the feasibility of using such structures in practical applications.

In this paper, we present a general mode matching formulation to analyze the wave propagation in glide-symmetric metallic surfaces drilled with periodic holes with an arbitrary cross section. A generalized Floquet theorem is applied to reduce the computational cost by imposing boundary conditions only on one of the two surfaces. With a small modification, the formulation can be used also for dispersion analysis of a holey metallic surface with or without a metal plane above it. The method is fast and efficient, and it provides physical insight on the specific symmetry properties of Floquet harmonics in glide-symmetric structures. The formulation is applied to obtain dispersion diagrams of glide-symmetric structures with circular holes as this kind of hole is usually used in practical applications to realize gap waveguides or wideband planar lenses. The results agree well with the reference results from commercial software CST Microwave Studio.

We demonstrate the beneficial effects of introducing glide symmetry in a two-dimensional periodic structure. Specifically, we investigate dielectric parallel plate waveguides periodically loaded with Jerusalem cross slots in three configurations: conventional, mirror- and glide-symmetric. Out of these three configurations, it is demonstrated that the glide-symmetric structure is the least dispersive and has the most isotropic response. Furthermore, the glide-symmetric structure provides the highest effective refractive index, which enables the realization of a broader range of electromagnetic devices. To illustrate the potential of this glide-symmetric unit cell, a Maxwell fish-eye lens is designed to operate at 5 GHz. The lens is manufactured in printed circuit board technology. Simulations and measurements are in good agreement and a measured peak transmission coefficient of −0.5 dB is achieved. [ABSTRACT FROM AUTHOR]

A fully metallic glide-symmetric waveguide filter with transmission in $Ka$ -band and attenuation at its second harmonic is proposed. The filter is low-loss and cost-effective for high frequencies, and it can be easily integrated with an antenna. Glide symmetry and the possibility of breaking this symmetry provide an additional degree of freedom for passband and stopband control. A new kind of 2-D glide symmetry, referred to as braided glide symmetry, is presented, showing an increased attenuation per unit cell. [ABSTRACT FROM AUTHOR]

This work studies the propagation characteristics of a rectangular waveguide with aligned/misaligned double-sided dielectric-filled metallic corrugations. Two modes are found to propagate in the proposed double-sided configuration below the hollow-waveguide cutoff frequency: a quasi-resonant mode and a backward mode. This is in contrast to the single-sided configuration, which only allows for backward propagation. Moreover, the double-sided configuration can be of interest for waveguide miniaturization on account of the broader band of its backward mode. The width of the stopband between the quasi-resonant and backward modes can be controlled by the misalignment of the top and bottom corrugations, being null for the glide-symmetric case. The previous study is complemented with numerical results showing the impact of the height of the corrugations, as well as the filling dielectric permittivity, on the bandwidth and location of the appearing negative-effective-permeability band. The multi-modal transmission-matrix method has also been employed to estimate the rejection level and material losses in the structure and to determine which port modes are associated with the quasi-resonant and backward modes. Finally, it is shown that glide symmetry can advantageously be used to reduce the dispersion and broadens the operating band of the modes.

We propose a circular twist-symmetric dielectric waveguide that is polarization-selective. In the practical implementation of optical fibers, a selective circular polarization is more convenient than its linearly polarized counterpart where previous knowledge of the emitted polarization from the transmitter is unknown. The analysis of the waveguide was conducted with three methods: an eigenmode approach, simulation of a truncated structure, and the so-called multimodal transfer-matrix method (MMTMM). The presented simulations demonstrate that the operational band can be manipulated by tuning the parameters of the structure. Furthermore, the MMTMM allows for a direct and accurate calculation of the attenuation constant of the rejected circular polarization.

Recently, there has been an increased interest in exploring periodic structures with higher symmetry due to various possibilities of utilizing them in novel electromagnetic applications. The aim of this paper is to discuss design issues related to the implementation of holey glide-symmetric periodic structures in waveguide-based components. In particular, one can implement periodic structures with glide symmetry in one or two directions, which we differentiate as 1D and 2D glide symmetry, respectively. The key differences in the dispersion and bandgap properties of these two realizations are presented and design guidelines are indicated, with special care devoted to practical issues. Focusing on the design of gap waveguide-based components, we demonstrate using simulated and measured results that in practice it is often sufficient to use 1D glide symmetry, which is also simpler to mechanically realize, and if larger attenuation of lateral waves is needed, a diagonally directed 2D glide symmetric structure should be implemented. Finally, an analysis of realistic holes with conical endings is performed using a developed effective hole depth method, which combined with the presented analysis and results can serve as a valuable tool in the process of designing novel electrically-large waveguide-based components.

We demonstrate the beneficial effects of introducing glide symmetry in a two-dimensional periodic structure. Specifically, we investigate dielectric parallel plate waveguides periodically loaded with Jerusalem cross slots in three configurations: conventional, mirror- and glide-symmetric. Out of these three configurations, it is demonstrated that the glide-symmetric structure is the least dispersive and has the most isotropic response. Furthermore, the glide-symmetric structure provides the highest effective refractive index, which enables the realization of a broader range of electromagnetic devices. To illustrate the potential of this glide-symmetric unit cell, a Maxwell fish-eye lens is designed to operate at 5 GHz. The lens is manufactured in printed circuit board technology. Simulations and measurements are in good agreement and a measured peak transmission coefficient of −0.5 dB is achieved.

We study the wave propagation between two glide-symmetric metallic plates drilled with periodic rectangular holes. A mode-matching method is proposed in order to derive efficiently the dispersive properties of these periodic structures. The method takes advantage of the higher symmetry of the structure reducing the computational cost by enforcing boundary conditions on the field on only one of the two surfaces. Physical insight on specific symmetry properties of Floquet harmonics in glide-symmetric structures is also gained. The code is validated with commercial software assessing its accuracy when varying the most influential/critical parameters. We confirm the potential of glide-symmetric structures to tune the effective refractive index. Specifically, we demonstrate that glide-symmetric structures with rectangular shapes can be employed to synthesize anisotropic refractive indexes with a large band of operation, which makes such metasurface structures applicable for the realization of ultrawideband planar lenses. [ABSTRACT FROM AUTHOR]

We present a novel electromagnetic bandgap (EBG) structure, which can be used to manufacture low-cost waveguiding structures at high frequencies. The unit cell of the proposed EBG consists of glide-symmetric holes in parallel plate waveguide. Using this unit cell in groove gap waveguide technology has a number of advantages over pin-type EBG at high frequencies, such as acquiring higher accuracy because of larger periodicity as well as an easier and cheaper manufacturing process. The performance of the proposed waveguiding structure is demonstrated using both a straight and a double 90° bent lines through simulation and measurement. [ABSTRACT FROM PUBLISHER]

Recently, there has been an increased interest in exploring periodic structures having higher symmetry properties, primarily based on metallic realization. The design of dielectric glide-symmetric structures has many challenges, and this paper presents a systematic analysis approach based on Floquet mode decomposition and mode matching technique. The presented procedure connects the analysis of standard periodic structures and glide-symmetric realizations, thus giving insight into the wave propagation and interaction characteristics. The obtained results were verified in comparison with results from known references and using a commercial solver, proving that the proposed analysis technique is inherently accurate, and the degree of accuracy depends only on the number of modes used. The proposed analysis approach represents the first step in the design process of dielectric periodic structures with glide symmetry.

This work presents a new configuration to create glide-symmetric structures in a single plane, which facilitates fabrication and avoids alignment problems in the assembly process compared to traditional glide-symmetric structures based on several planes. The proposed structures can be printed on the metal face of a dielectric substrate, which acts as a support. The article includes a parametric study based on dispersion diagrams on the appearance of stop-bands and phase-shifting by breaking the symmetry. In addition, a procedure to regenerate symmetry is proposed that may be useful for reconfigurable devices. Finally, the measured and simulated S parameters of 10 × 10 unit-cell structures are presented to illustrate the attenuation in these stop-bands and the refractive index of the propagation modes. The attenuation obtained is greater than 30 dB in the stop-band for the symmetry-broken prototype.

Here we propose the use of twist and glide symmetries to increase the equivalent refractive index in a helical guiding structure. Twist- and glide-symmetrical distributions are created with corrugations placed at both sides of a helical strip. Combined twist-and glide-symmetrical helical unit cells are studied in terms of their constituent parameters. The increase of the propagation constant is mainly controlled by the length of the corrugations. In our proposed helix antenna, twist and glide symmetry cells are used to reduce significantly the operational frequency compared with conventional helix antenna. Equivalently, for a given frequency of operation, the dimensions of helix are reduced with the use of higher symmetries. The theoretical results obtained for our proposed helical structure based on higher symmetries show a reduction of 42.2% in the antenna size maintaining a similar antenna performance when compared to conventional helix antennas.

We study the interplay between the minimal representations of the orthogonal Lie algebra $\mathfrak{g}=\mathfrak{so}(n+2,\mathbb{C})$ and the algebra of symmetries$\mathscr{S}(\Box ^{r})$ of powers of the Laplacian $\Box$ on $\mathbb{C}^{n}$. The connection is made through the construction of a highest-weight representation of $\mathfrak{g}$ via the ring of differential operators ${\mathcal{D}}(X)$ on the singular scheme $X=(\mathtt{F}^{r}=0)\subset \mathbb{C}^{n}$, for $\mathtt{F}=\sum _{j=1}^{n}X_{i}^{2}\in \mathbb{C}[X_{1},\ldots ,X_{n}]$. In particular, we prove that $U(\mathfrak{g})/K_{r}\cong \mathscr{S}(\Box ^{r})\cong {\mathcal{D}}(X)$ for a certain primitive ideal $K_{r}$. Interestingly, if (and only if) $n$ is even with $r\geqslant n/2$, then both $\mathscr{S}(\Box ^{r})$ and its natural module ${\mathcal{A}}=\mathbb{C}[\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n},\ldots ,\unicode[STIX]{x2202}/\unicode[STIX]{x2202}X_{n}]/(\Box ^{r})$ have a finite-dimensional factor. The same holds for the ${\mathcal{D}}(X)$-module ${\mathcal{O}}(X)$. We also study higher-dimensional analogues $M_{r}=\{x\in A:\Box ^{r}(x)=0\}$ of the module of harmonic elements in $A=\mathbb{C}[X_{1},\ldots ,X_{n}]$ and of the space of ‘harmonic densities’. In both cases we obtain a minimal $\mathfrak{g}$-representation that is closely related to the $\mathfrak{g}$-modules ${\mathcal{O}}(X)$ and ${\mathcal{A}}$. Essentially all these results have real analogues, with the Laplacian replaced by the d’Alembertian $\Box _{p}$ on the pseudo-Euclidean space $\mathbb{R}^{p,q}$ and with $\mathfrak{g}$ replaced by the real Lie algebra $\mathfrak{so}(p+1,q+1)$. [ABSTRACT FROM AUTHOR]

We establish several relationships between the non-relativistic conformal symmetries of Newton–Cartan geometry and the Schrödinger equation. In particular we discuss the algebra s c h ( d ) of vector fields conformally-preserving a flat Newton–Cartan spacetime, and we prove that its curved generalisation generates the symmetry group of the covariant Schrödinger equation coupled to a Newtonian potential and generalised Coriolis force. We provide intrinsic Newton–Cartan definitions of Killing tensors and conformal Schrödinger–Killing tensors, and we discuss their respective links to conserved quantities and to the higher symmetries of the Schrödinger equation. Finally we consider the role of conformal symmetries in Newtonian twistor theory, where the infinite-dimensional algebra of holomorphic vector fields on twistor space corresponds to the symmetry algebra c n c ( 3 ) on the Newton–Cartan spacetime. [ABSTRACT FROM AUTHOR]

In this letter, we propose a new technique to tune the bandgap in gap-waveguide technology based on broken glide-symmetric holey structures. We demonstrate that breaking the glide symmetry in a proper manner provokes the presence of a passband within the bandgap due to the frequency sweep of the second propagating mode. This passband generates field leakage in the gap that is translated into a filtering property. This filtering effect may be used to reduce or eliminate filters in large complex devices. In order to avoid undesired coupling due to the leakage from the air gap between the plates, an absorbing sheet is proposed to dissipate the undesired fields. This idea has been numerically studied and experimentally validated with a specific design, a WR15-size gap-waveguide prototype with glide-symmetric holes with filtering properties. [ABSTRACT FROM AUTHOR]

Here, we study the properties of a new type of twist-symmetric configuration inside a waveguide. The structure consists of a periodic repetition of transversal metallic sheets with perforated holes. This twist-symmetric configuration permits the propagation of TM modes inside the waveguide and prevents the propagation of the first TE mode at the conventional band employed on those waveguides. The TM modes exhibit a broader band of operation as the periodicity is reduced, i.e., the structure is more compact. In addition, we demonstrate that the twist-symmetric structure has a broader band than that of the aligned holes in which the twist symmetry is absent. Furthermore, the number of folds of the twist symmetry determines the propagation constant of the propagating mode. Therefore, a reconfigurable structure based on a different number of folds creates future opportunities to produce fully metallic reconfigurable waveguides with different phase delays. [ABSTRACT FROM AUTHOR]

In this letter, we demonstrate that the dispersion properties of printed double-sided parallel-strip lines can be controlled by using glide symmetry. Glide symmetry is introduced in with corrugations in both strips of a double-sided line. We demonstrate that glide symmetry eliminates the stopband between first and second propagating modes and yields to a higher propagation constant, preserving its linearity, and the broadband nature of the underlying guiding technology. Thus, the glide-symmetric double-sided line can be designed to possess a high equivalent refractive index in an ultrawide range of frequencies. These exceptional properties have been numerically and experimentally validated. Finally, we demonstrate the possibilities of this technology with a specific design, a glide-symmetric double-sided parallel-strip line with filtering properties. Potential applications are low-dispersive leaky-wave antennas and electromechanical tunable phase shifters and filters. [ABSTRACT FROM AUTHOR]

We consider the Burgers-type system studied by Foursov, for which no recursion operator or master symmetry was known so far, and prove that this system admits infinitely many local higher symmetries that are constructed using a nonlocal two-term recursion relation rather than a recursion operator. [ABSTRACT FROM AUTHOR]

We consider two-dimensional relativistically invariant systems with a three-dimensional reducible configuration space and a chiral-type Lagrangian that admit higher symmetries given by polynomials in derivatives up to the fifth order. Nine such systems are known: two are Liouville-type systems, and zero-curvature representations for two others have previously been found. We here give zero-curvature representations for the remaining five systems. We show how infinite series of conservation laws can be derived from the established zero-curvature representations. We give the simplest higher symmetries; others can be constructed from the conserved densities using the Hamiltonian operator. We find scalar formulations of the spectral problems. [ABSTRACT FROM AUTHOR]

We study various Sp(2M)-invariant field equations corresponding to rank-r tensor products of the Fock (singleton) representation of Sp(2M). These equations describe localization on “branes” of different dimensions embedded in the generalized space-time $$\mathcal{M}_M$$ with matrix (i.e., “central charge”) coordinates. We consider the case of a bilinear tensor product in detail and show that the conserved currents built from bilinears of rank-1 fields in $$\mathcal{M}_M$$ satisfy the field equations for rank-2 fields in $$\mathcal{M}_M$$ . We also show that rank-2 fields in $$\mathcal{M}_M$$ are equivalent to rank-1 fields in $$\mathcal{M}_{2M}$$ . [ABSTRACT FROM AUTHOR]

We discuss a method for seeking integrable boundary conditions for nonlinear equations. For the Boussinesq equation, we find a new boundary condition that is compatible with the Lax pair and has an infinite set of higher symmetries and a Bäcklund transformation. We construct a class of explicit partial solutions of an equation satisfying this boundary condition. [ABSTRACT FROM AUTHOR]

We construct a 3 × 3 matrix zero-curvature representation for the system of three two-dimensional relativistically invariant scalar fields. This system belongs to the class described by the Lagrangian L = [gij (u)xiutj]/2 + f(u), where gij is the metric tensor of a three-dimensional reducible Riemannian space. We previously found all systems of this class that have higher polynomial symmetries of the orders 2, 3, 4, or 5. In this paper, we find a zero-curvature representation for one of these systems. The calculation is based on the analysis of an evolutionary system ut = S(u), where S is one of the higher symmetries. This approach can also be applied to other hyperbolic systems. We also find recursion relations for a sequence of conserved currents of the triplet of scalar fields under consideration. [ABSTRACT FROM AUTHOR]