Extra high-Q resonances and extraordinary transparency in finite fragments of dielectric metasurfaces: Prospects for 5G applications

Fifth generation (5G) wireless communication systems are required to provide high-rate data transfer and robust connectivity for the drastically increasing number of mobile devices, which use data-consuming applications and services.¹ In comparison with 4G communication systems, 5G systems could utilize higher frequency bands from 24.25 to 52.6 GHz. The realization of 5G systems needs advanced electromagnetic devices, such as highly directive antennas, beam steerers, lenses, filters, sensors, and absorbers for the mm-wave range.² Metasurfaces (MSs), being planar lightweight structures, which operate due to resonances in constituent elements, were found to present a promising platform for implementing such devices.^3–10 However, most of recently developed 5G components based on MSs include conductive elements, which, at higher frequencies, cause increased losses and additional attenuation of electromagnetic signals on shorter distances. Low-loss dielectric MSs, known to outperform even plasmonic MSs, whose designs incorporate dielectric and metallic counterparts, promise significant advantages for 5G systems.

Similar to metamaterials, MSs consist of periodically arranged unit cells and represent, in fact, single planes of metamaterials. Therefore, MS responses are conventionally analyzed by modeling single unit cells with periodic boundary conditions (PBCs). This approach assumes identical responses from all unit cells. MS responses are defined by an interaction between waves incident on MS planes and resonances in meta-atoms. The resonances in dielectric MSs, i.e., in MS composed of dielectric resonators (DRs), can be typically considered analogous to Mie resonances, with the basic types being dipolar electric and magnetic ones. Since Mie-type dipolar resonances perform as the sources of radiation, they are expected to initiate the formation of surface waves (SWs), propagating in the MS plane. The interaction of SWs with DR lattices is controlled by diffraction processes, which could lead to collective responses in MSs, in particular, to the formation of lattice resonances (LRs). We have earlier reported about SWs and LRs in infinite dielectric MSs, investigated by numerical experiments using single cell models with periodic boundary conditions (PBCs).¹¹ In real applications, however, MSs should be represented by fragments with finite number of meta-atoms, when abrupt boundaries between MS and free space could affect propagation and reflection of SWs, modifying MS responses.

In this work, we investigate electromagnetic responses of MSs of finite size, composed of dielectric disks. We demonstrate that in addition to conventional dipolar resonances with identical responses of unit cells, observed in infinite MSs, MS fragments can exhibit in neighboring meta-atom's out-of-phase electric and magnetic responses, which have much higher intensity and very high-Q factors. Such resonances enable a significantly localized wave/matter interaction necessary for enhancing the performance of sensors and absorbers of 5G systems. We also report the detection of extraordinary transmission at electric and magnetic dipolar resonances in dielectric MSs that can be used to locally enhance mm-wave signals for 5G communications.

Full-wave simulations of electromagnetic responses from MSs, composed of identical dielectric disks, were performed by using COMSOL Multiphysics and CST Studio Suite software packages. Arrays of disks were arranged in free-space on the xy-plane, while their heights/thicknesses were extended along z-axis. Electric (E) and magnetic (H) components of incident wave were polarized, respectively, along x- and y-axes, while z-directed wave vector was normal to the MS plane. For simulating fragmented MSs, a finite set of disks was placed in the computational domain filled with free space and terminated by perfectly matching boundaries (PMBs). Such a model mimicked infinite free space around MS fragments under study. For analyzing infinite MSs, we simulated a single unit-cell model with PBCs applied at xz and yz boundaries of the cell.¹¹ The lattice constants marking the distance between centers of neighboring DRs in x- and y-directed rows were defined as Δ_X and Δ_Y, respectively. They were either equal, Δ = Δ_X = Δ_Y (square lattices), or different (rectangular lattices). Disks were made of ceramic material with the dielectric constant of 37.2 and had the diameter and the height of, respectively, 6 and 3 mm. At such parameters, dipolar resonances in DRs were excited within the microwave X-band for the convenience of experiments. However, we show that all features of the observed characteristics of these MSs can be reproduced in MSs, which operate in the 5G frequency range, being composed of resonators with properly re-scaled dimensions and made of a proper material.

Figure 1 allows for comparing 3D spectra of magnetic resonance responses for infinite MSs and 5 × 5 MS fragments at different lattice constants. Resonance responses are characterized by signal spectra from H-field probes placed either in centers of unit cells employed at modeling infinite MSs or in centers of central cells of MS fragments. For magnetic dipolar resonances, which are oriented along the y-direction at the above-described excitation type, primary radiation is expected along the x-direction that defines the choice of Δ_X as the changing lattice parameter in Fig. 1, which was varied between 7 and 35 mm. The lattice parameter Δy was kept constant and equal to 7 mm. Similar 3D spectra could be obtained for electric dipolar resonance responses at the changes of Δy lattice parameter from 7 up to 35 mm and fixing Δ_X = 7 mm.

As seen in Fig. 1(a), at increasing lattice constant Δ_X, magnetic response observed in infinite MS experiences red shifting (shifts to longer wavelengths) and asymptotically approaches the line marking the Rayleigh anomaly (RA), when the wavelength of incident wave becomes comparable to the lattice constant. We discussed such red shifting of dipolar resonances in Ref. 11. Similar effects were early noticed in Ref. 12 at the studies of planar arrays of spherical silicon resonators. According to Ref. 12, at lattice constants, comparable to the wavelengths of radiation, diffraction of surface waves, which are initiated by elementary electric and magnetic resonances, should lead to the formation of surface LRs, interplay of which with elementary dipolar resonances could cause their red shifting.

Figure 1(b), which presents H-field signal spectra for 5 × 5 MS fragment, shows that in addition to response with characteristic red shifting, MS of finite size demonstrates very strong response at Δ_X >12.5 mm, formed by coinciding branches, which are split at Δ_X <12.5 mm. New response forms an extremely narrow vertical line, indicating that its resonance frequency is fixed in a wide range of lattice constants. The Q-factor of the new narrow-band resonance appears to be much higher than Q-factor of the resonance, the characteristic for infinite MSs. Another vertical line of less strength, caused by an additional narrow band resonance, is seen in the longer wavelength part of the spectrum.

To characterize a variety of resonances observed in finite MSs, Fig. 2 presents the spectra of H- and E-probe signals, observed in the central DR of 5 × 5 fragment of MS with square lattice and lattice constant Δ = 10 mm. Figure 2 also presents H- and E-field patterns in the median xy-cross section of the fragment at peak frequencies of the spectra. As seen in Fig. 2(a), the H-field peak at 10.214 GHz [correlated with the response experiencing red shifting in Fig. 1(b)] corresponds to the formation of coherent magnetic dipolar resonances in all DRs of the fragment. (We can consider this case as the formation of even magnetic resonance mode.) Similar coherent oscillations are observed at magnetic dipolar resonances in infinite MSs [corresponding to the response shown in Fig. 1(a)]. Two sharp higher frequency H-field peaks in Fig. 2(a) [correlated with two responses with fixed resonance frequencies in Fig. 1(b)] and H-field peak at 9.93 GHz in Fig. 2(a) correspond to out-of-phase oscillations in neighboring DRs, i.e., to the formation of odd magnetic resonance modes. Odd modes apparently originate from partial reflections of SW from boundaries between fragmented MSs and free space. The strengths and high Q-factors of most prominent odd-mode resonances could be explained by mutual compensation of radiation generated by elementary dipolar resonances with opposite (shifted by π radians) phases of oscillations.

As shown in Fig. 2(b), the E-field signal spectrum for the same 5 × 5 MS fragment has the structure comparable with that of the magnetic response, seen in Fig. 2(a), except for the fact that even mode appears at the high frequency shoulder of the strongest peak in the spectrum. Coupled sharp peaks, seen in both E- and H-field responses, correspond to comparable arrangements of electric and magnetic dipolar resonances in neighboring DRs in the MS plane. The peaks in the middle of both spectra exhibit chess board-type alteration of dipolar resonances in the MS plane with opposite phases of oscillations in nearest DRs. However, while the high frequency H-field peak is defined by reflections of SWs propagating in the y-direction, its electric counterpart appears related to reflections of waves traveling in the x-direction.

To disclose the role of the boundaries between finite MSs and free space in the formation of resonance modes in fragments, we investigated responses from semi-infinite MSs, i.e., MSs with periodic boundaries (PBCs) applied at MS termination by either xz- or yz-planes, while at orthogonal planes, PMBs were terminating computational domain. Figures 3(a) and 3(b) present magnetic and electric responses of MSs, which are infinite in the y-direction and comprise only three cells in the x-direction. The simulation model was composed of 3 × 3 cells with non-periodic boundaries at MS termination by yz-planes and PBCs at xz boundaries of MS termination. Figures 3(c) and 3(d) present the responses for alternative case, i.e., for MS, infinite in the x-direction and comprising three cells in the y-direction, represented by the model comprised of 3 × 3 unit cells with PBCs at yz-boundaries and non-periodic boundaries at xz-planes. In difference from finite MS fragments, characterized in Figs. 1 and 2, semi-infinite MSs supported only two modes, one of even type, similar to that observed in infinite MSs, and another one of odd type, formed due to SW reflections from boundaries between MS fragment and free space. No field patterns with a chess-board type arrangement of dipoles could be observed in semi-infinite MSs, since reflections can be provided only for SWs traveling in either x- or y-directions, but not in both. It can be noticed in Fig. 3 that, while spectral positions of even electric and magnetic modes do not depend on whether MS is infinite in y- or in x-directions, the positions of odd modes in the spectra are changing significantly at changes in MS geometry. This can be explained by the fact that odd modes in MSs, infinite in the y-direction, have identical phases of resonance oscillations in DRs located in columns, while in MSs, infinite in the x-direction, identical phases of resonance oscillations are observed in rows of DRs. This makes configurations of field distributions for two odd modes quite different.

The specifics of field patterns presented in Fig. 3 allows for relating the observed modes to even and odd transmission modes, known in photonic crystals (PhCs). Although dipolar resonances are excited in constituent “atoms” of MS by normally incident wave, the presence of SWs, propagating in the MS plane, apparently leads to the formation of MS responses, which are similar to responses of PhCs. In MSs, infinite in both directions, PBCs, at plane wave incidence, allow for the formation of only coherent resonances in DRs. The presence of non-periodic boundaries in MSs, providing for multiple reflections of SWs, could make MSs performing similar to PhCs, even with only three unit cells between non-periodic boundaries (Fig. 3).

One more observation from Fig. 3 is that E-field intensities in the resonance spectra presented for MS fragments with non-periodic yz-boundary [Fig. 3(b)] are significantly stronger than E-field intensities in the spectra of MS fragments with non-periodic xz-boundary [Fig. 3(d)]. The Q-factor of odd mode resonance is also much higher in the former case. This could be explained by the fact that non-periodic boundaries of MSs, while providing for multiple partial reflections of SWs, allow energy to flow away, thus decreasing the strength of resonances. The revealed difference between Figs. 3(b) and 3(d) is in correspondence with the above-mentioned specifics of radiation from dipolar resonances: at electric response, radiation occurs primarily in the direction normal to the orientation of electric dipoles, i.e., along y-axis, while at magnetic response, radiation is expected primarily in the direction normal to the orientation of magnetic dipoles, i.e., along x-axis. As the result, non-periodic yz-boundaries at electric response should not affect the formation of electric resonances [Fig. 3(b)], while non-periodic xz-boundaries at electric response should provide a leakage of resonance energy from MS, decreasing the resonance strengths [Fig. 3(d)]. At magnetic response, non-periodic yz-boundaries should provide a leakage of resonance energy, decreasing the resonance strengths [Fig. 3(a)], while non-periodic xz-boundaries should not affect the formation of magnetic resonances [Fig. 3(c)].

Another effect, important for practical application of MSs, is extraordinary transmission, which can be detected in dielectric MSs at dipolar resonances. At the studies of electric responses from dense infinite dielectric MSs, irradiated by normally incident plane waves, we have revealed the phenomenon, analogous to the known in atomic physics electromagnetically induced transparency (EIT).¹³ The main specific of the new phenomenon was an occurrence of a narrow transparency window at the frequencies of electric dipolar resonances in disk-shaped DRs, when total reflection, instead of transmission, was generally expected. Investigation of Fano resonances, accompanying the new effect, allowed for relating it to destructive interference of background radiation and radiation produced by oscillations of resonance fields in MS atoms. Magnetic resonances in dense infinite MSs were very weak and could not produce similar effects. However, as it is seen from the presented above data, fragmented MSs demonstrate significant enhancement of the strength of magnetic resonances, so that accompanying unusual transparency can be expected.

For infinite MSs, modeled by using single unit cells with PBCs, transmission spectra (T) at plane wave incidence can be obtained using simulated scattering parameters T = |S21|. In the case of finite MS fragments, transmission spectra can be found based on the approach proposed in Ref. 14 by using the relation: $T=FSFS+BS$⁠, where FS is the power (W/m²) scattered by the fragment in the forward direction (+z) and BS is the power scattered by the fragment in the backward direction, with respect to the direction of incident wave flow (−z). Figure 4 presents transmission spectra for two MS fragments with different quantity of DRs in comparison with S21 spectrum calculated for infinite MS. All MSs have dense square lattices with lattice constant Δ = 7 mm. As seen in the figure, all obtained spectra coincide at frequencies f <10 GHz, where MS responses experience the Kerker's effect,¹⁵ and in the vicinity of electric dipolar resonance at about 11.6 GHz, where EIT is observed. It should be noticed, however, that transmission peaks at the electric resonance have higher Q-factors for MS fragments, in comparison with the peak obtained for infinite MS. As expected, S21 spectrum of infinite MS does not demonstrate any peaks in the vicinity of magnetic resonance, since dense infinite MSs support only very weak magnetic resonances.¹³ 

In difference from the spectrum of infinite MS, transmission spectra of finite MS fragments exhibit additional sharp peaks, with spectral positions corresponding to the magnetic dipolar resonance in DRs. These peaks are almost coinciding for two MS fragments; however, they appear on the background of high transmission. Since infinite MSs with square lattices leakage of the resonance energy could occur at all four boundaries of MS fragments, this could decrease the contrast of resonance effects.

To mitigate the above problem, we investigated the transmission spectra of fragments with small Δ_Y, but extended Δ_X, i.e., fragments with the geometry approaching that of MS, semi-infinite in the x-direction, for which stronger odd magnetic resonance mode was observed [Fig. 3(c)]. Figure 5(a) presents transmission spectra for MS fragments with Δ_Y = 7 mm and Δ_X increasing from 10 to 25 mm. As seen in the figure, increasing Δ_X leads to the formation of a very sharp and strongly pronounced transmission peak at expected frequency of dipolar magnetic resonance of about 10.7 GHz. The peak transparency is approaching 0.83. Although it does not achieve 1 and, so, does not confirm full transparency, as in the case of electric resonance, it demonstrates much higher Q-factors (400, 1087, and 1482 at, respectively, Δ_X = 10, 15, 25 mm) that is the characteristic for EIT. Field patterns, obtained in xy-cross sections of MS fragments at f = 10.7 GHz, demonstrated the formation of odd magnetic resonance modes. Additional investigations are required to ensure that the observed peak of MS transparency is defined by the phenomenon of EIT-type.

Figure 5(a) was obtained by using two different computational solvers (COMSOL and CST) to verify our simulation results. Figure 5(b) demonstrates that similar effects of extraordinary transparency can be reproduced in MSs, which operate in the 5G frequency range. MSs were composed of disks made of lossless ceramic material (barium zinc cobalt niobate¹⁶), having relative permittivity of 35. The diameter and height of the disks were equal to 2 and 1 mm, respectively. These results are promising for employing magnetic and electric resonances in finite dielectric MSs for 5G applications. Furthermore, we experimentally measured transmission spectrum for 5 × 5 fragment with Δ_Y = 7  and Δ_X = 25 mm. We used two X-band horn antennas and an Agilent vector network analyzer to measure the reflections (R) from the fragment as shown in the schematics of Fig. 5(c). Then, we calculated the measured transmission (T) as T = 1 − R. The fragmented MS sample was made by arranging ceramic disks on a thin and firm paper board by double-stick tapes. Figure 5(d) demonstrates that measured and simulated transmission spectra are in reasonable agreement, and expected peaks of MS transparency are clearly detected.

Table I presents Q-factors and amplitudes of electric and magnetic resonances for single DR and different MS geometries considered above. In all simulations, for illuminating the structures, we used identical incident plane waves with amplitude of 1 V/m for E-field and 1/120π = 0.0026 A/m for H-field. As seen from the table, odd-mode resonances in MS fragments provide much higher amplitudes and Q-factors in comparison with resonances in single DRs and infinite MS (with similar periodicity). Increasing the number of resonators in finite MSs leads to higher Q-factors and amplitudes of electric and magnetic responses. With such extraordinary characteristics demonstrated in Table I, fragments of dielectric MSs can be considered as a promising host for realizing very significant wave/matter interactions, which are requested in numerous applications, such as absorbing and sensing devices.^17–20 

The authors would like to thank Dr. G. Semouchkin for many useful discussions. This work was supported by the National Science Foundation (NSF) under Award No. ECCS-1709991.