All‐Dielectric Metasurface‐Based Gap Waveguides

Abstract

The development of modern optical communication systems requires specific waveguides, given that the widely used fiber‐optic components are poorly integrated with planar technologies. For planarization in the millimeter‐wave and subterahertz bands, so‐called gap waveguides are proposed, offering low‐loss performance and cost‐efficient manufacturing. Hence, the utilization of this technology in the optical range is very promising. Herein, a strategy for designing gap waveguides made of two metasurfaces composed of dielectric disk‐shaped resonators operated in hybrid HE (magnetic dipole) and EH (electric dipole) modes is proposed. The coupled dipole model is applied to the complex multiple‐scattering problem by substituting each resonator as an electric and magnetic dipole, providing equations for the efficient calculation of metasurface reflection and transmission properties. It is demonstrated that with the correct choice of metasurface geometry providing their resonant reflection conditions, a waveguide channel can be implemented between a pair of metasurfaces, which allows propagation of the transverse electric and transverse magnetic waves similar to those of a parallel plate waveguide with perfectly conducting either electric or magnetic walls. This approach may be seen as a novel metasurface‐based waveguide structure that uses flexibly mediated boundary conditions to control electromagnetic wave propagation.

Introduction

Metasurface-based waveguides utilize engineered metasurfaces to guide, manipulate, and confine electromagnetic waves efficiently.^[1] A metasurface is a 2D periodic or quasiperiodic array of subwavelength particles that exhibit tailored electromagnetic properties. They can provide control over transmission and reflection characteristics and introduce unique wave interactions, such as phase control, amplitude shaping, and polarization manipulation.^[2,3] By leveraging these properties, metasurface-based waveguides can confine and control wave propagation in unconventional ways, making them ideal for advanced electromagnetic and photonic applications. Moreover, metasurfaces allow for highly customizable waveguide designs, including bending, splitting, and merging wave paths.

Metasurface-based waveguides include several designs, the choice of which depends on the required operating range and applications. In particular, compared with conventional transmission lines such as metallic and microstrip waveguides, in millimeter-wave communications and terahertz devices, it is proposed to use gap waveguides, which benefit from low losses, flexible planar fabrication, and low cost.^[4] A gap waveguide uses a specific electromagnetic structure to guide waves without requiring direct physical contact between the guiding surfaces. Its operating principle relies on suppressing the propagation of electromagnetic waves in specific directions, utilizing two specially designed parallel artificial electromagnetic metasurfaces with periodic arrangements of conducting elements that serve as the waveguide's walls. These two metasurfaces can be plane or curved, separated by a small gap, and the waveguide is formed inside this gap between the two structures. The gap is usually filled with air, but can also be completely or partially filled with a dielectric. One or both walls of the waveguide are metasurfaces intended to implement high-impedance boundaries of some kind^[5] (a high-impedance surface is a boundary where the ratio of the electric field to the magnetic field is much higher than that of free space). They can be surfaces of a perfect electric conductor (PEC), a perfect magnetic conductor (PMC), or a PEC/PMC strip grid. In practice, the bandwidth of actual realizations of the gap waveguides can be different, where the cutoff bandwidth of the high-impedance surface can be found by modeling a unit cell of the PEC/PMC parallel plate part of the waveguide.

In gap waveguides, waves are confined and guided through an air gap between a ridge or strip and a flat conducting surface with a high-impedance metasurface. These structures impose specific boundary conditions that govern wave propagation,^[6] where the customized surface impedance is responsible for guiding waves. When targeted for optics and photonics, utilizing high-impedance dielectric metasurfaces in designs of gap waveguides is very promising, as they can allow confining waves with low losses in planar configurations.^[7–11] Similar to high-impedance metallic metasurfaces, the PEC/PMC conditions can also be realized in dielectric metasurfaces, which employ arrays of dielectric resonators that confine electromagnetic fields at the subwavelength scale. Such resonators support Mie-type modes,^[12–14] stemming from the Mie scattering theory.^[15] These modes are distinctive because they involve both electric and magnetic multipole resonances. Tuning mode conditions by changing the aspect ratio of dielectric particles or their symmetry properties makes it possible to flexibly manipulate the resonant states of dielectric metasurfaces. When illuminated with a plane wave, such a structure can behave as a perfect either electric or magnetic mirror depending on the type of the Mie resonance excited,^[16–19] thus resembling the PEC and PMC boundary conditions required for the implementation of gap waveguides. In dielectric metasurfaces, particles can be fabricated with various geometries (e.g., spheres, disks, cubes, or other shapes), which provides high flexibility in the design of gap waveguides while allowing impedance tuning and wave control.

The implementation of dielectric gap waveguides can be related to bilayer optical metasurfaces, which have attracted considerable interest recently.^[20] This interest comes from the fact that bilayer designs offer greater versatility in engineering optical responses, enabling a broader range of functionalities than single-layer metasurfaces. Each layer may have a unique pattern, thickness, and material to achieve specific needs. Such structures have already been proposed for use in the creation of novel devices such as holograms,^[21] multispectral achromatic optical components,^[22] metalenses,^[23] and high-quality lasers.^[24,25]

Here we suggest a general approach to the design of all-dielectric metasurface-based gap waveguides. Our approach is based on the concept of a bilayer dielectric metasurface composed of disk-shaped resonators. Each metasurface layer can behave as a PEC or PMC metamirror, providing the necessary conditions for reflection.

Design of a Metasurface-Based Waveguide

The all-dielectric metasurface-based waveguide proposed in our study is schematically shown in Figure 1. It is composed of two identical metasurfaces, separated from each other by a distance H_gap. In the following, unless otherwise stated, we assume, for simplicity, that the gap between metasurfaces is filled with air (${\epsilon }_{\text{gap}}=1$). This simplification does not violate the generality of our consideration and, nevertheless, allows us to exclude the influence of additional interfaces between media with different permittivities on the formation of the waveguide channel. In the waveguide, each metasurface is formed by an array of silicon disks (${\epsilon }_{\text{disk}}=12$) that are periodically arranged in both x- and y-directions. The radius and height of disks are R_disk and H_disk, respectively. The period of the array is P. The gap thickness, the disks’ radius and height, as well as the periodicity of the unit cell of the metasurfaces are subject to optimization to ensure the operation of the waveguide at a certain frequency. In this work, the waveguide is designed to operate in the telecommunications frequency range $f\in [\mathrm{140,190}]$ THz ($\lambda \in [\mathrm{1580,2140}]$ nm).

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As is known from the theory of microwave waveguides,^[26] each eigenwave of a parallel plate waveguide can be considered a plane electromagnetic wave propagating at a certain angle to the waveguide axis and receiving total reflection from its PEC walls. The wave propagation angle can be determined based on the values of γ and k_x, which are the propagation constant and the longitudinal component of the wave vector, respectively. They are dependent on the frequency f, the thickness H_gap, and the permittivity ε_gap of the material filling the waveguide channel (see Section S1, Supporting Information).

The situation changes when, instead of PEC walls, which totally reflect electromagnetic waves regardless of the angle of their incidence, dielectric metasurfaces are utilized to implement the metawaveguide. In this case, the following peculiarities of the metasurface-based waveguide should be accounted for: 1) a discrete structure of the metasurface imposes restrictions on the phase properties of electromagnetic radiation (spatial distribution along the metasurface), at which total reflection conditions are satisfied; 2) the resonant nature of the reflection is mediated by Mie modes of dielectric resonators forming the metasurface in contrast to broadband reflection from the infinitely extended PEC walls; 3) additional diffraction beams can appear in the waves reflected from the metasurface what is desirable to avoid.

To ensure waveguide operation, in an ideal loss-less case, the thickness of the gap H_gap must be such that the propagation constant of the waveguide mode is a real quantity. Using Table S1 given in Supporting Information, one can see that there is the following restriction on H_gap1$H_{\text{gap}}>\lambda /2=c/(2f)$where c is the speed of light, λ and f are the wavelength and frequency of the electromagnetic wave propagated inside the waveguide. For a waveguide to exist, the condition of total reflection of the waves from its walls must be met so that the wave is confined inside the channel. The propagation constant can be related to the angle of incidence of a plane wave on the wall of the waveguide. The angle of incidence on the metasurface can be calculated using a simple formula2$\theta =\arccos [c/(2f{H}_{\text{gap}})]$

To find the waveguide propagation conditions for the existence of the total reflection from a single metasurface behaving as a wall, a parametric problem must be solved concerning the frequency and angle of incidence for the wave of the given polarization. Knowing the angle of total reflection and frequency, using Equation (2), one can then estimate the distance between the metasurfaces, allowing the corresponding waveguide mode propagation.

Therefore, to implement a metasurface-based waveguide, it is necessary first to reveal conditions of total reflection for the case of oblique incidence of a plane electromagnetic wave on the metasurface. After the conditions for total reflection from the metasurface are found, at the second stage, the thickness of the waveguide channel ensuring the propagation of either TE or TM waveguide modes can be determined.

Reflection Conditions for a Dielectric Metasurface

The resonant reflection of a disk-shaped metasurface is associated with the excitation of the so-called Mie-type resonances supported by individual disks. When the disks are arranged into a 2D periodic array, the resonances associated with individual disks acquire a frequency shift that arises because of the electromagnetic interaction (coupling) between the resonators. Nevertheless, the configuration of the field distribution inside the disk preserves and can be associated with the corresponding eigenmode of the individual disk-shaped resonator. The eigenmodes of such a resonator are classified as the transverse electric (TE), transverse magnetic (TM), and hybrid (HE, EH) modes based on the orientation of the electromagnetic field components concerning the disk's symmetry axis (for the nomenclature of the eigenmodes, see Refs. [27,28]). In a disk-shaped resonator, the electromagnetic modes resemble Mie-type solutions, where the different eigenmodes of the resonator can be associated with electric and magnetic multipoles. In what follows, we are interested in the HE_11l and EH_11l eigenmodes of the disk, which appear as horizontal magnetic and electric dipoles, respectively, known as the lowest-order Mie resonances (see Table S1 in Ref. [29]; in the mode abbreviations, the first, second, and third indices denote the order of the azimuthal, radial, and vertical variations of the fields; here we do not fix the exact number for l). We have collected in Figure 2 parametric dependencies of the scattering cross section of a single disk used to implement the metasurface (see Ref. [30] for corresponding equations), the reflection characteristic of this metasurface, and typical distribution of the near field related to the manifestation of the HE_11l and EH_11l eigenmodes of the disk. It can be concluded that the dependencies of a single particle and a metasurface have similar patterns, but their resonant frequencies are somewhat shifted. Multipole analysis shows that in the chosen frequency range and structure parameter space, the main contribution to the scattering cross section appears from the electric dipole (ED) and magnetic dipole (MD) moments of the disk (see Section S2, Supporting Information, for separate contributions of the ED and MD moments to the scattering cross section of the single resonator).

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To further analyze these characteristics of the given metasurface operated on the lowest-order Mie resonances, the coupled dipole model can be applied. It simplifies the complex multiple-scattering problem by modeling each resonator as an electric and magnetic dipole, providing equations for the efficient calculation of metasurface reflection and transmission properties. In the coupled-dipole approximation, each disk in the metasurface is treated as a point dipole with an induced dipole moment that responds to both the incident field and the scattered fields from other dipoles. Thus, the method allows one to calculate the characteristics of the entire structure based on the properties of a single particle that forms a meta-atom of the metasurface.

Each resonator forming metasurface is characterized by its electric ${\widehat{\alpha }}^p$ and magnetic ${\widehat{\alpha }}^m$ polarizabilities, which are tensor quantities. Elements of these tensors are determined by the material and geometrical anisotropy of meta-atoms, and, in general, they can be calculated numerically. In our study, a linearly polarized plane wave excites the array of disks at oblique incidence. Hence, based on Ref. [31] and using Bloch's theorem as in Ref. [32] for a particle placed at the origin of the Cartesian coordinate frame and located in an infinitely extended array, the coupled-dipole equations for the ED moment p and MD moment m can be written as3$\mathbf{\text{p}}={\epsilon }_0{\epsilon }_s{\widehat{\alpha }}^p{\mathbf{\text{E}}}_0+{\widehat{\alpha }}^p\widehat{S}\mathbf{\text{p}}+{\widehat{\alpha }}^p\frac{\mathrm{i}}{v}[\mathbf{\text{g}}\times \mathbf{\text{m}}]$4$\mathbf{\text{m}}={\widehat{\alpha }}^m{\mathbf{\text{H}}}_0+{\widehat{\alpha }}^m\widehat{S}\mathbf{\text{m}}+{\widehat{\alpha }}^m\frac{v}{\mathrm{i}}[\mathbf{\text{g}}\times \mathbf{\text{p}}]$where ε₀ is the vacuum dielectric constant, ${\epsilon }_s={\epsilon }_{\text{gap}}$ is the relative permittivity of the surrounding medium, v is the speed of light in the surrounding medium, E₀ and H₀ are the electric and magnetic fields of the incident wave at the origin of the Cartesian coordinate frame, respectively, and ‘i’ is the imaginary unit. Expressions for the lattice sums $\widehat{S}$ and g, which account for the dipole coupling between particles in the metasurface, can be found in Ref. [33,34] The multipole moments of all other particles located at points with r_j in the array are ${\mathbf{\text{p}}}_{\mathit{\text{j}}}=\mathbf{\text{p}}\exp (\mathrm{i}{\mathbf{\text{k}}}_{\parallel }{\mathbf{\text{r}}}_j)$ and ${\mathbf{\text{m}}}_{\mathit{\text{j}}}=\mathbf{\text{m}}\exp (\mathrm{i}{\mathbf{\text{k}}}_{\parallel }{\mathbf{\text{r}}}_{\mathit{\text{j}}})$, where ${\mathbf{\text{k}}}_{\parallel }$ is the projection of the incident wave vector onto the array plane and j is the number of a particle in the array.

In the Cartesian coordinate frame presented in Figure 1, the electric field, magnetic field, and wave vectors of the incident wave are: ${\mathbf{E}}_{\text{inc}}={\mathbf{E}}_0\text{exp(i}\mathbf{\text{kr}}\text{)}$, ${\mathbf{H}}_{\text{inc}}={\mathbf{H}}_0\text{exp(i}\mathbf{\text{kr}}\text{)}$, and $\mathbf{\text{k}}=k_{\mathrm{s}}(\sin \theta ,0,\cos \theta )$, respectively, where $k_{\mathrm{s}}=\omega /v=2\pi f/v$, and θ is the angle of incidence of the primary wave given in the x-z plane. We define the components of E₀ and H₀ as5${\mathbf{E}}_0={\mathbf{E}}_0(0,1,0)$6${\mathbf{H}}_0={\mathbf{H}}_0(-\cos \theta ,0,\sin \theta )$for the TE-polarized incident wave, and7${\mathbf{E}}_0={\mathbf{E}}_0(\cos \theta ,0,-\sin \theta )$8${\mathbf{H}}_0={\mathbf{H}}_0(0,1,0)$for the TM-polarized one.

From solving the system of coupled-dipole Equations (3) and (4), the transmission (t) and reflection (r) coefficients of the metasurface for two orthogonal polarizations of the incident wave are expressed as follows^[33]

for the TE polarization9$r^{\text{TE}}=\frac{\mathrm{i}{k}_{\mathrm{s}}}{2S_L{\epsilon }_0{\epsilon }_{\mathrm{s}}{E}_0\cos \theta }\left(p_y+\frac{\cos \theta }{v}{m}_x+\frac{\sin \theta }{v}{m}_z\right)$10$t^{\text{TE}}=1+\frac{\mathrm{i}{k}_{\mathrm{s}}}{2S_{\mathit{\text{L}}}{\epsilon }_0{\epsilon }_{\mathrm{s}}{E}_0\cos \theta }(p_{\mathit{\text{y}}}-\frac{\cos \theta }{v}{m}_{\mathit{\text{x}}}+\frac{\sin \theta }{v}{m}_{\mathit{\text{z}}})$for the TM polarization11$r^{\text{TM}}=\frac{\mathrm{i}{k}_{\mathrm{s}}v}{2S_{\mathit{\text{L}}}{H}_0\cos \theta }\left(\frac{1}{v}{m}_{\mathit{\text{y}}}-p_{\mathit{\text{x}}}\cos \theta -p_{\mathit{\text{z}}}\sin \theta \right)$12$t^{\text{TM}}=1+\frac{\mathrm{i}{k}_{\mathrm{s}}v}{2S_{\mathit{\text{L}}}{H}_0\cos \theta }\left(\frac{1}{v}{m}_{\mathit{\text{y}}}+p_{\mathit{\text{x}}}\cos \theta -p_{\mathit{\text{z}}}\sin \theta \right)$where S_L is the area of the array unit cell. In the above equations, triples of components $\{m_{\mathit{\text{x}}},p_{\mathit{\text{y}}},m_{\mathit{\text{z}}}\}$ and $\{p_{\mathit{\text{x}}},m_{\mathit{\text{y}}},p_{\mathit{\text{z}}}\}$ correspond only to dipole moments located at the origin of the coordinate system. Then, reflectance R, transmittance T, and absorbance W are defined as $R=|r{|}^2$, $T=|t{|}^2$, and $W=1-R-T$, respectively. From the above equations, it follows that the out-of-plane components of the dipole moments (z-components) affect reflection and transmission only at oblique incidence, and their influence increases with increasing θ. At the same time, as shown in Ref. [33] under an oblique incidence of the external wave in the system, electromagnetic coupling occurs between in-plane and out-of-plane components of the ED and MD moments that depend on the polarization of the incident wave. For the TE polarization, coupling occurs between the p_y and m_z components, whereas, for the TM polarization, it is between the m_y and p_z components. Such coupling leads to a redistribution of energy between dipole moments, which affects reflection and transmission.

Representations for the transmission coefficient given by Equations (10) and (12) can be used to find the metasurface parameters that provide effective reflection in the required frequency band and for a wide range of the incidence angle θ. For example, let us consider the case of incidence of the TE-polarized wave related to Equation (10) (similar conclusions are also valid for the TM-polarized wave and Equation (12), by implementing the appropriate replacement of the components of the moments p and m). For systems with negligible material losses ($W\to 0$), which are typical for dielectric metasurfaces, the equality T = 0 corresponds to total reflection R = 1. Then from Equation (10) one can obtain two conditions13$\text{Re}\left[\frac{k_{\mathrm{s}}}{2S_{\mathit{\text{L}}}{\epsilon }_0{\epsilon }_s{E}_0\cos \theta }(p_{\mathit{\text{y}}}-\frac{\cos \theta }{v}{m}_{\mathit{\text{x}}}+\frac{\sin \theta }{v}{m}_{\mathit{\text{z}}})\right]=0$14$\text{Im}\left[\frac{k_{\mathrm{s}}}{2S_{\mathit{\text{L}}}{\epsilon }_0{\epsilon }_s{E}_0\cos \theta }(p_{\mathit{\text{y}}}-\frac{\cos \theta }{v}{m}_{\mathit{\text{x}}}+\frac{\sin \theta }{v}{m}_{\mathit{\text{z}}})\right]=1$which, by introducing the effective polarizabilities ${\alpha }_{\mathit{\text{y}}}^{\text{p,eff}}=p_{\mathit{\text{y}}}/({\epsilon }_0{\epsilon }_{\mathrm{s}}{E}_0)$, ${\alpha }_{\mathit{\text{x}}}^{\text{m,eff}}=-m_{\mathit{\text{x}}}/H_0$, and ${\alpha }_{\mathit{\text{z}}}^{\text{m,eff}}=m_{\mathit{\text{z}}}/H_0$, can be transformed to15$\text{Re}[{\alpha }_{\mathit{\text{y}}}^{\text{p,eff}}]+\text{Re}[{\alpha }_{\mathit{\text{x}}}^{\text{m,eff}}]\cos \theta +\text{Re}[{\alpha }_{\mathit{\text{z}}}^{\text{m,eff}}]\sin \theta =0$16$\text{Im}[{\alpha }_{\mathit{\text{y}}}^{\text{p,eff}}]+\text{Im}[{\alpha }_{\mathit{\text{x}}}^{\text{m,eff}}]\cos \theta +\text{Im}[{\alpha }_{\mathit{\text{z}}}^{\text{m,eff}}]\sin \theta =\frac{2S_{\mathit{\text{L}}}\cos \theta }{k_{\mathrm{s}}}$where the polarizabilities also depend on the incidence angle θ. One of the formal solutions to Equation (15) can be written out immediately17$\text{Re}[{\alpha }_{\mathit{\text{y}}}^{\text{p,eff}}]=\text{Re}[{\alpha }_{\mathit{\text{x}}}^{\text{m,eff}}]=\text{Re}[{\alpha }_{\mathit{\text{z}}}^{\text{m,eff}}]=0$

The existence of this solution is possible only at certain θ and depends in a complex way on many parameters of the system. Such an effect can be called “accidental” total reflection, which is usually a rather narrowband effect. This feature prevents the implementation of a waveguide system based on solution (17), so we next demonstrate alternative conditions related to the total reflection effect.

At normal incidence (θ = 0) of the primary plane wave, the contribution of the out-of-plane component m_z to reflection and transmission disappears. Moreover, due to the symmetry properties of the system, this component cannot be excited by a normally incident plane wave, and the optical response of the system is mediated only by the p_y and m_x components. Thus, when θ = 0, the following conditions for transmission suppression hold18$\text{Re}[{\alpha }_{\mathit{\text{y}}}^{\text{p,eff}}]+\text{Re}[{\alpha }_{\mathit{\text{x}}}^{\text{m,eff}}] =0$19$\text{Im}[{\alpha }_{\mathit{\text{y}}}^{\text{p,eff}}]+\text{Im}[{\alpha }_{\mathit{\text{x}}}^{\text{m,eff}}]=\frac{2S_{\mathit{\text{L}}}}{k_{\mathrm{s}}}$

Previously, this case was discussed in detail in Ref. [35] for metasurfaces composed of spherical particles. However, the conclusions drawn there are also applicable to the case of disk-shaped particles by replacing the corresponding polarizabilities, which in our case have a simple representation20${\alpha }_{\mathit{\text{y}}}^{\text{p,eff}}={\left(\frac{1}{{\alpha }_{\mathit{\text{y}}}^{\mathrm{p}}}-S_{\mathit{\text{y}}}\right)}^{-1} \text{and} {\alpha }_{\mathit{\text{y}}}^{\text{p,eff}}={\left(\frac{1}{{\alpha }_{\mathit{\text{x}}}^{\mathrm{m}}}-S_{\mathit{\text{x}}}\right)}^{-1}$where ${\alpha }_{\mathit{\text{y}}}^{\mathrm{p}}$ and ${\alpha }_{\mathit{\text{x}}}^{\mathrm{m}}$ are the ED and MD polarizabilities of a single particle, respectively, S_x and S_y are the dipole lattice sums, which for a square unit cell are equal to each other.^[33] Note that if the resonance of polarizabilities ${\alpha }_{\mathit{\text{y}}}^{\mathrm{p}}$, ${\alpha }_{\mathit{\text{x}}}^{\mathrm{m}}$ of single particles is realized in the diffraction-free (specular) region far from Rayleigh's anomaly (RA), then the influence of the lattice sums on effective polarizabilities (20) is insignificant. In this case, the spectral positions of the resonances of the metasurface are determined by the eigenmodes of a constitutive particle. Under these conditions, as was shown in another study,^[35] the fulfillment of conditions (18) and (19) for significant suppression of wave transmission in the metasurface can be achieved in separate resonances of effective polarizabilities (20), as well as in the spectral region between them in the area of their overlapping.

Therefore, our approach is as follows. Initially, it is necessary to select the parameters for which the resulting metasurface would have a maximal reflection at normal incidence of light in a given part of the spectrum in the diffraction-free region far from the RA (see Ref. [36] and Section S3, Supporting Information). The latter circumstance is important since it determines the interval of variation of the angle of incidence at which the maximal reflection of the metasurface is still realized. Such an interval always exists since the initial deviation of the angle of incidence from zero can be considered a small perturbation. With an additional increase in the angle of incidence, the reflection from the metasurface can change significantly due to the contribution of the out-of-plane dipole component (m_z or p_z) and their coupling with other in-plane dipole components.^[33,34] Moreover, with an increase in the angle of incidence, RA shifts toward long waves (to the red side), approaching the region of effective reflection. The closer RA is, the stronger its negative influence on the reflection effect will be. As has been demonstrated in many studies on the optical properties of metasurfaces with disk-shaped particles, with an increase in the angle of incidence, various interference effects appear in the system, leading to a significant change in reflection. Such effects include quasibound states in the continuum,^[34,37] the generalized Brewster effect,^[38,39] and surface lattice resonances.^[40,41]

The following sequence of actions can be proposed: 1) the sizes of individual particles are chosen based on the position of their dipole resonances in a given frequency range; 2) the period of the metasurface composed of these particles is selected so that it is noticeably smaller than the resonant wavelengths; 3) the dependence of the reflection on the angle of incidence is checked. For the waveguide implementation utilizing reflection resonances of a single metasurface, an important parameter is the maximal angle of incidence at which effective reflection is still observed, which is subject to an optimization problem.

Following this scenario, we first considered the scattering cross section in a given spectral range for particles with variable radius and fixed height (H_disk = 0.40 μm). It is evident from Figure 2A that particles with a radius given in the range $R_{\text{disk}}\in [\mathrm{0.30,0.36}]\text{μm}$ bear both the MD in ED resonances in the chosen frequency range. Therefore, following the second action, we select the metasurface period equal to 10 μm. The results of modeling the reflectivity at normal incidence of the primary wave on a disk-shaped metasurface are shown in Figure 2B. It can be seen that total reflection is realized in a relatively wide spectral region for particles with a radius around 0.34 μm. This spectral region arises because the MD and ED resonances overlap.

Figure 3A shows the dependence of the reflection coefficient on the frequency, disk radius, and angle of incidence for a metasurface specified in Figure 2B. As before, we fix the disk height to realize a region of total reflection for an oblique wave incidence in the required frequency range. The resulting color maps (Figure 3A for the TE- and TM-polarized waves) obtained for θ = 20° show the formation of such a region with high reflection, which shifts to the high-frequency range with decreasing disk radius. Similar to the case for θ = 0 given in Figure 2B, its existence is due to the manifestation of the closely spaced MD and ED resonances in the disks. It should be noted that to obtain a region of total reflection, the MD and ED resonances must not coincide; otherwise, the Kerker effect may occur,^[42] at which the metasurface becomes completely transparent to the incident wave. From the reflection distribution in Figure 3A, it is evident that effective (close to total) reflection is realized around the frequency of 160 THz for disks with a radius of 0.34 μm for both types of polarization. Therefore, in our subsequent simulation, we fix these geometric parameters as the basic ones.

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At an oblique incidence, the field distribution patterns for the case of hybrid HE_11l- and EH_11l-modes are no longer so clear due to a change in the phase distribution of the fields within the unit cell. However, these resonances can still be identified as Figure 3B suggests. As noted above, when the angle of incidence increases, the interaction between the MD and ED resonances increases, which leads to a more noticeable degradation of the region of total reflection for the TM-polarized wave. Thus, the angular dependencies show that the waveguide is easier to implement for the TE-polarized waves for which the phenomenon of total reflection is both broadband and stable for wide variations of the angle of incidence. For the TM-polarized waves, the waveguide can be implemented only in a quite small range of angles close to normal incidence.

In the case of oblique incidence, additional resonances also appear, which are associated with the excitation of “dark” modes of different topology due to the appearance of electromagnetic coupling between the disks’ eigenmodes and the field of the incident wave.^[43] The lowest-frequency high-Q resonance is most likely associated with the excitation of the trapped TE_01l-mode in the metasurface.^[44] Since these resonances are very narrow, the practical implementation of a waveguide operated on these modes is problematic.

We should note that the results presented in Figure 2 and 3 are calculated using the COMSOL Multiphysics solver, whereas those obtained within the framework of the coupled-dipole model given by Equation (9) and (11) are presented in Section S4, Supporting Information. Their excellent agreement implies the complete applicability of the dipole model for the analysis of the optical properties of the metasurface under consideration.

Using the dipole approximation allows us to select the values of the dimensional and configuration parameters, such as the size and aspect ratio of particles and the period of metasurfaces, to ensure suppression of transmission and confining of propagating fields only in the waveguide region. In this case, suppression of transmission guarantees concentration of decaying fields outside the waveguide at distances not exceeding the wavelength of the waveguide mode. With an increase in the particle size in the metasurface, the role of higher-order multipoles in the formation of fields increases, which can significantly change the dimensional parameters of the waveguide. In this case, it is necessary to separately adjust the effect of total reflection from metasurfaces at oblique wave incidence. Adding resonances of only the quadrupole moments of particles can significantly affect the process of angular reflection of light due to intermultipole coupling in metasurfaces.

Metawaveguide Characteristics

At the previous stage, the parameters of the metasurface were selected to ensure the conditions for total reflection of both TE-polarized and TM-polarized plane waves when they are incident obliquely on the metasurface at a certain angle θ. According to the hypothesis underlying this study, if two metasurfaces are placed at a certain distance H_gap from each other, they should form a waveguide channel within which waves can propagate. However, such a metawaveguide is a rather complex structure to be designed using only an analytical approach; therefore, the electromagnetic analysis of wave propagation in the waveguide should be performed numerically. In such a model, the key issue is the introduction of an exciting field, the configuration of which must coincide with or be as close as possible to the unknown field distribution of the eigenwaves of the metawaveguide. Fortunately, the configuration of this exciting field can be specified based on the assumption that the physics of the formation of a waveguide channel in a metawaveguide is identical to that in a parallel plate waveguide, from which it follows that the spectrum of their eigenwaves should be similar. Therefore, in our model, an ideal parallel plate waveguide with either PEC or PMC identical walls is used to excite waves propagating in the metawaveguide. With this approach, when combining a parallel plate waveguide and a metawaveguide together, it is impossible to achieve an ideal matching of their eigenwaves, so some reflection of waves can occur at the junctions. This leads to the formation of a mixed propagation mode in the metawaveguide, which combines the modes of the traveling and standing waves. Nevertheless, even this mode of operation allows us to evaluate the main features of the resulting metawaveguide.

Thus, in what follows, we consider a metawaveguide section containing N-periodic cells of two identical metasurfaces, which are separated by an air gap H_gap. This section is bounded on two sides by parallel plate waveguides with walls made of either PEC or PMC identical plates to excite the propagation of either TE_n or TM_n waves inside the metawaveguide, respectively (for the propagation conditions of these waves, see Table S1, Supporting Information). It should be noted that the junction of the metawaveguide with the external (input and output) parallel plate waveguides occurs at some distance d from the disks. If $d=P/2-R_{\text{disk}}$, then the section of the metawaveguide under consideration contains exactly N periods. Tuning d allows optimizing the matching conditions between the external parallel plate waveguides and metawaveguide to improve matching and reduce unwanted reflections at their junction.

The gap thickness H_gap for the waveguide system is estimated based on Equation (2) and the reference values for the desired operating frequency and corresponding incidence angles where the total reflection condition for the constitutive metasurfaces is met. Two different thicknesses were chosen to add variety to this study with target frequencies $f=160$ THz and $f=170$ THz to propagate TE₁- and TM₁-wave, respectively. Figure 4 shows the frequency dependencies of the transmission coefficient T and radiation losses W for such a complex waveguide system consisting of a metawaveguide and two sections of a parallel plate waveguide. These curves are presented for waves of two orthogonal polarizations. They are characterized by the presence of resonances with high transmission and low radiation losses. These resonances can be easily explained by the complex mode of wave propagation in the metawaveguide sustaining both traveling and standing waves. Accordingly, the maximal transmission is associated with the resonant conditions of the standing wave mode formation, while its minima correspond to the realization of the conditions of destructive interference of waves reflected at the junctions of the waveguides.

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One can also notice that the propagation of the TM₁ wave in Figure 4 is associated with significantly greater radiation losses compared to the propagation of the TE₁ wave. This is due to the peculiarity of the conditions of total reflection of the metasurface for waves of the corresponding type, discussed earlier. For given parameters of the structure, the region of total reflection rapidly degrades with the increasing angle of incidence for the TM-polarized wave. This leads to the fact that the corresponding waveguide can be realized only in a narrow range of parameters relative to the propagation constant. However, we hope that further optimization of the structure parameters can improve the conditions for TM-polarized wave propagation. This is a call for future study.

To confirm the resonant nature of wave propagation in the designed waveguides, which arises from the periodic distribution of resonators in the 2D arrays, we compared the transmission coefficients of the metawaveguide with those of a commensurate waveguide made of unstructured parallel dielectric plates (the characteristics of such a structure are presented in Section S5, Supporting Information). The comparison shows a significant difference between the two structures since, in an unstructured parallel plate waveguide with given parameters, the condition of total internal reflection is not satisfied, and the modes are not confined for both polarizations.

The formation of waveguide modes is also confirmed by distributions of electromagnetic fields calculated in the waveguide cross section (in the x-z plane) at frequencies where the transmission coefficient T reaches its maximal values. The corresponding results are collected in Figure 5 and 6 for the TE and TM waves, respectively. Here, each maximum in the transmission coefficient is associated with a corresponding number of electromagnetic field variations in the metawaveguide section. The amount of field variations along the height of the waveguide remains unchanged, indicating that the modes of first order (n = 1) are preserved for the selected heights H_gap. The obtained distributions also confirm good confinement of the TE₁ wave and moderate leakage of the TM₁ wave from the metawaveguide. Nevertheless, the formation of waves and their confinement in such a structure consisting of a discrete distribution of disk-shaped resonators is confirmed in both conditions of a metawaveguide with PEC and PMC walls.

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Finally, we explored an alternative metawaveguide design, more realistic from a practical perspective, which comprises germanium disks embedded in a silica host. Corresponding dependences of the reflection coefficient from a single constitutive metasurface and the transmission coefficient of the metawaveguide are collected in Section S6, Supporting Information. It can be concluded that for the alternative structure, the conditions for the propagation of waveguide waves are also satisfied. Thus, further implementation of the metawaveguide in practice is only a matter of optimizing the parameters of the structure based on available materials and manufacturing technologies.

Conclusion

In summary, we have employed a concept of an all-dielectric metawaveguide in the form of two metasurfaces composed of disk-shaped resonators. Such a waveguide can support the propagation of both TE and TM waves, providing its constitutive metasurfaces can behave as both PEC and PMC walls. Our approach provides a lossless alternative to microwave waveguides with metallic walls (approximated as PEC). It also provides a strategy for implementing PMC walls, where there is a lack of materials that can support such boundary conditions. More interestingly, in the visible, where the PEC approximation for metals no longer holds, our proposed concept provides a viable strategy to implement PEC/PMC-based gap waveguides.

The formation of conditions of total resonant reflection of a plane electromagnetic wave from a disk-shaped metasurface is due to the excitation of individual eigenmodes in the constitutive disks. The most promising from the points of view of creating artificial PEC and PMC walls for a metasurface-based waveguide are hybrid EH_11l and HE_11l eigenmodes of the disk. They are the lowest-frequency Mie-type resonances associated with electric and magnetic dipole modes, respectively. It was found that these two modes can interact constructively, forming a region of total reflection and destructively, significantly reducing the reflection level in the resonance region. The choice of metasurface parameters can control this interaction.

Further research may be aimed at finding conditions for better wave confinement in a metawaveguide with PMC walls, as well as specific metasurface designs for implementing PEC–PMC hybrid configurations. It can be done by optimizing the design of the metasurface unit cell to support corresponding Mie-type magnetic resonances at the operating frequency. In this case, the use of anisotropic or bianisotropic unit cells is promising, allowing control of the effective permeability tensor by adapting the artificial optical magnetic response at the interface. This enhances impedance mismatch for modes attempting to leave the waveguide, increasing the field confinement. It is also interesting to consider designs of a metawaveguide capable of supporting the TEM wave propagation condition.

Experimental Section

Numerical Modeling

To simulate reflection from the metasurface, we built its numerical model using COMSOL Multiphysics in 3D mode. The size of a tetrahedral spatial mesh was automatically chosen according to the COMSOL physics-controlled mesh preset. The permittivity of silicon was chosen to be dispersionless and equal to 12. The problems of polarized reflection were solved using the Electromagnetic Waves Frequency Domain module for one square unit cell with the periodic boundary conditions on the sides. The incident wave was generated on the top of the computation domain by subsequent excitation of the TE- and TM-polarized waves on the input periodic Port 1.

To confirm the propagation of the TE₁ and TM₁ waves in the metawaveguide and identify their characteristics (field distributions and the ability of the metawaveguide to hold the wave within the waveguide channel) for the optimized parameters of the metasurfaces, an auxiliary problem was solved in COMSOL. This problem consisted of modeling the transmission of the TE₁ and TM₁ waves of an ideal plane-parallel waveguide with PEC/PEC and PMC/PMC walls, respectively, through a section of the metawaveguide. The thicknesses of the parallel plate waveguides and metawaveguide were equal. Along the y axis, the structure had a thickness P and was limited by periodic boundary conditions; therefore, the field distributions were independent of y.

Note that the wall type was selected considering the expected symmetry type of the metawaveguide wave, which provides its excitation. The parallel plate waveguides were equipped with user-defined ports that allowed calculations of propagation of the TE₁ and TM₁ waves in the PEC/PEC and PMC/PMC waveguides for a given frequency and field distribution, which was specified through the y component of the electric and magnetic fields, respectively. The corresponding field distributions are selected from Table S1, Supporting Information.

To reduce reflection from the parallel plate waveguide and metawaveguide junctions, the problem of optimizing the distance between the open ends of the waveguides and the disks closest to them was solved. Since the TE₁ and TM₁ waves of parallel plate waveguides were able to penetrate the metasurface, there were perfectly matched layers in free space to provide absorption of waves leaking from the system.

Acknowledgements

V.R.T. and V.V.K. acknowledge funding from the European Union's Horizon 2020 Research and Innovation programme under grant agreement No. 871072. A.B.E., I.A., and A.C.L. acknowledge funding from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy within the Cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453).

Open Access funding enabled and organized by Projekt DEAL.

Conflict of Interest

The authors declare no conflict of interest.

Data Availability Statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

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