Electromagnetic bandgap based on a compact three-hole double-layer periodic structure

IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 72, NO. 1, JANUARY 2024 1045 Communication Electromagnetic Bandgap Based on a Compact Three-Hole Double-Layer Periodic Structure Luis Fernando Herran , Qiao Chen , Francisco Mesa , and Oscar Quevedo-Teruel Abstract— We propose and study a new type of double-layer holey structure with a wide bandgap. The structure can have glide symmetry in two orthogonal directions but not 2-D glide symmetry. We report results in terms of dispersion diagrams calculated with the eigensolver of a commercial solver, as well as with a multimode transfer matrix approach that permits an accurate calculation of the attenuation constant. The results demonstrate that the bandgap of the proposed structure can provide a wider fractional bandwidth and a larger attenuation constant than those of a 2-D glide-symmetric holey configuration. Therefore, this new type of periodic structure can be advantageous in preventing leakage in gap waveguide technology or, in general, parallel plate configurations and filters. The operation of this new unit cell is experimentally demonstrated with a double-flange configuration between 40–60 GHz. Index Terms— Electromagnetic bandgap (EBG), flange transition, glide symmetry, holey periodic structure, multimodal analysis. I. INTRODUCTION Fully metallic waveguides are often used in millimeter-wave applications due to their low insertion losses. When used to design devices that integrate into high-frequency systems, waveguides are usually split into two pieces, so their internal details can be fabricated using the machining technique. In practice, this splitting introduces a small air gap between metal pieces due to manufacturing/assembly imperfection and/or metal deformation, which can cause significant energy leakage at millimeter-wave frequencies [1]. Leakage may also occur in the interconnection between the waveguides, where it is difficult to ensure perfect electrical contact [2]. Although conventional 1-D periodic corrugations/grooves only provide a high suppression of leakage in the direction of periodicity [3], the gap waveguide technology [4],[5] based on bed-of-nails structures can prohibit wave propagation in unwanted directions inside a parallel-plate waveguide (PPW). However, the manufacturing process of periodic pins can be fragile and costly when their physical dimensions are scaled down to a magnitude of submillimeters in millimeter-wave frequencies Manuscript received 16 July 2023; revised 19 October 2023; accepted 31 October 2023. Date of publication 15 November 2023; date of current version 9 February 2024. This work was supported in part by COST Action SyMat under Grant CA18223 and in part by COST (European Cooperation in Science and Technology). The work of Luis Fernando Herran was supported in part by under Grant PID2020-114172RB- C21/AEI/10.13039/501100011033 and in part by the Gobierno del Principado de Asturias under Grant IDI/2021/000097. The work of Francisco Mesa was supported by MCIN/AEI/10.13039/501100011033 under Grant PID2020- 116739GB-I00. (Corresponding author: Oscar Quevedo-Teruel.) Luis Fernando Herran is with the Department of Electrical Engineering, University of Oviedo, 33003 Oviedo, Spain (e-mail: [email protected]). Qiao Chen and Oscar Quevedo-Teruel are with the Division of Electromagnetic Engineering and Fusion Science, School of Electrical Engineering and Computer Science, KTH Royal Institute of Technology, SE 100 44 Stockholm, Sweden (e-mail: [email protected]; [email protected]). Francisco Mesa is with the Department of Applied Physics 1, ETS Ingeniería Informática, Universidad de Sevilla, 41012 Seville, Spain (e-mail: [email protected]). Color versions of one or more figures in this communication are available at https://doi.org/10.1109/TAP.2023.3331502. Digital Object Identifier 10.1109/TAP.2023.3331502 [6]. As an alternative, holey electromagnetic bandgap (EBG) structures have received increasing interest due to their robustness and cost-effectiveness [7]. It has recently been demonstrated that glide symmetry [8],[9] enables a number of advantages when applied to periodic structures, and in particular to holey EBG structures [10]. Two-dimensional glide-symmetric holey structures were first studied in [11], revealing that glide symmetry can be used to increase the operating bandwidth of a Luneburg lens. Furthermore, glide symmetry has been shown to improve EBG bandwidth [12],[13] and its attenuation [14], which are beneficial properties for gap waveguide technology [15], filters [16], leaky wave antennas [17],[18], and reduction of leakage in flanges [19],[20]. The characteristics of periodic EBG structures are studied primarily by means of dispersion analysis. The commonly used eigenmode solver (ES) of commercial full-wave simulators only computes the propagation constants of purely propagative modes, and hence only provides the frequency range of the stopband. However, knowledge of the nature of the stopband and its attenuation is crucial to fully characterize the behavior of the bandgap. This information can be obtained from the complete modal solutions offered by ad hoc analytical/numerical methods such as the equivalent circuit approach [21],[22] and the transverse resonance method [23],[24] widely used in single-sided metasurfaces. When a pair of periodic surfaces possess glide symmetry, higher order interactions between them cannot always be easily modeled with an equivalent circuit or homogenized using the transverse resonance method [25]. Although attempts to use the mode-matching technique [26],[27] have been reported for glide-symmetric corrugations [26] and holey surfaces [27], this technique is restricted to the analysis of some canonical geometries. The multimode transfer matrix approach [28],[29],[30],[31],[32], [33] combines the ability of full-wave simulators [34],[35],[36] to deal with complex geometry structures [14],[37] and inhomogeneous materials [38],[39],[40] with the advantage of ad hoc approaches to obtain complete modal solutions (both phase and attenuation constants) and provide physical insight [41].In[14], this approach proved to be capable of accurately characterizing the glide-symmetry holey EBG structure, including the rejection bandwidth, the level of attenuation, the nature of the mode (complex/evanescent/bound), parity of the mode, and the directional property of the stopband. In this work, we propose and study the EBG properties of a new type of glide-symmetric holey structure that has “broken” glide symmetry in two orthogonal directions rather than 2-D glide symmetry. The operation of this new unit cell is experimentally validated with a double-flange experimental set-up. II. DESCRIPTION OF THE PROPOSED PERIODIC UNIT CELL The common topology of holey periodic metasurfaces consists of two metallic layers separated by a narrow gap that forms a PPW. In both top and bottom layers, there is a periodic distribution of holes, and depending on its specific periodic configuration (or symmetry), © 2023 The Authors. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. For more information, see https://creativecommons.org/licenses/by-nc-nd/4.0/ 1046 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 72, NO. 1, JANUARY 2024 Fig. 1. Holey unit cells. (a) Mirror. (b) One edge displaced mirror. (c) Glide symmetric. (d) Proposed periodic unit cell. Top and bottom heights of the holes (h) are the same. the structure behaves differently. The most common configurations for a unit cell with holes in the upper and lower layers are: mirroring, mirroring with a displacement of one edge, and symmetric glide [12]. A schematic of these unit cells is illustrated in Fig. 1(a)–(c). In all cases, pis the periodicity of the unit cell, his the height of the top and bottom holes, gis the air gap between the bottom and top layers, and diis the diameter of the hole. For structures with different diameters in the top and bottom holes, d1will refer to the bottom holes and d2to the top holes. The novel unit cell proposed in this work is illustrated in Fig. 1(d). This unit cell has one hole of diameter d1in the bottom metallic layer and two holes with diameter d2in the top metallic layer. The top-layer holes are shifted in both plane directions for half the unit-cell period with respect to the mirror of the bottom hole. Unlike the other unit cells shown in Fig. 1, which have two holes in total, the proposed configuration has three holes per unit cell (p.u.c.). The diameter d2of the holes in the upper layer must be selected so that the adjacent holes do not overlap; that is, its maximum value is given by d2,max =p/√2. In the following, unit cells will also be denoted in terms of the number of holes p.u.c.; namely, the ones in Fig. 1(a)–(c) as “two-hole” unit cells, while the proposed novel unit cell will be denoted as a “threehole” unit cell. III. PARAMETRIC STUDY In this section, we study the behavior of the proposed three-hole unit cell, Fig. 1(d), and compare it with the other three two-hole unit cells illustrated in Fig. 1(a)–(c) through a detailed parametric study. A. Cells With the Same Period First, we show the dispersion diagrams of the four holey structures in Fig. 2when the size of their unit cells is the same. The remaining parameters of each unit cell are optimized to obtain the maximum stopband. As previously reported in [10] and [15], a glide-symmetric unit cell [Fig. 2(c)] exhibits a wider stopband bandwidth than its corresponding nonglide versions [Fig. 2(a) and (b)]. However, the proposed three-hole unit cell has a much lower frequency bound for the stopband with the same periodicity and similar bandwidth [Fig. 2(d)]. We can observe that the behavior of the fundamental mode of the mirrored two-hole unit cell and the one of the three-hole unit cell are qualitatively similar, although the three-hole unit cell limits the propagation in the 45◦direction (M0) at a significantly lower frequency, thus reducing the lower frequency bound of the stopband and increasing its bandwidth. This is a relevant feature, since it means that this newly proposed unit cell is more compact than other previously reported holey structures. An explanation of the above feature could be expected to be that the proposed three-hole unit cell has a higher filling factor than its two-hole counterparts. However, a further analysis of the stopband behavior reveals that the specific geometry of the three-hole unit cell also has a significant impact. For example, Fig. 3compares the behavior of two three-hole unit cells with the same filling factor: the Fig. 2. Dispersion diagrams for the units cells in Fig. 1with the same periodicity (p=3 mm). In all cases, h=2 mm and g=50 µm. (a) Mirror (d=2.8 mm). (b) One edge displaced mirror (d1=2.6 mm, d2=1.6 mm). (c) Glide symmetric (d=2.4 mm). (d) Proposed three-hole unit cell (d1= 2.6 mm, d2=1.6 mm). Fig. 3. Dispersion diagrams of two three-hole unit cells. (a) Proposed three-hole cell. (b) Three-hole unit cell with the top-layer holes shifted by p/2. In the inset of (b), the holes in the bottom layer are colored blue, while those in the top layer are colored red. one proposed in Fig. 1(d) and a variation of this cell with its top layer shifted by an additional length of p/2. The comparison of both plots in Fig. 3clearly shows that the value of the filling factor appears to be less relevant than the relative position of the bottom and top holes. It is somewhat unexpected that changing the relative position of the holes affects the stopband location and bandwidth so drastically. It can be inferred that the behavior of the shifted three-hole topology in Fig. 3(b) is a sort of hybridization between the two-hole mirror and glide two-hole cases shown in Fig. 2(a) and (c), in the sense that the behavior of the first mode in the M0region is similar to the two-hole mirror case in Fig. 2(a), while its stopband is confined between the second and third order of propagation as in the two-hole glide case in Fig. 2(c). A parametric study has also revealed that among the parameters that have more influence on the bandwidth performance of the proposed three-hole unit cell, both the top and bottom hole diameters have the greatest impact on it. The effect of the influence of these two parameters is plotted in Fig. 4, which shows the frequencies in the first and second modes, and thus the absolute bandwidth, for different combinations of d1and d2. This analysis was carried out using the same period as in the previous figures, p=3 mm. Both diameter values are taken to their practical size limits, the diameter of the bottom hole is less than the period of the unit cell, and the diameter of the top hole is chosen to avoid overlap [these limits are (d1/p)lim =1 and (d2/p)lim =0.7]. An important observation is IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 72, NO. 1, JANUARY 2024 1047 Fig. 4. Parametric analysis of the proposed three-hole cell for (a) frequency bounds of the first and second modes and (b) fractional bandwidths. All data have been obtained using a normalized height of h/p=0.66. Fig. 5. Holey periodic metasurfaces with the same lower frequency bound of the stopband. (a) Mirror. (b) Mirror with one displaced edge. (c) Glide symmetric. (d) Proposed periodic unit cell. that the stopband bandwidth increases as the diameter of the holes grows, suggesting that the larger the area of the surface occupied by the holes, the wider the bandwidth obtained. However, this increase shows a limit for an optimum combination of d1and d2. This fact is better observed in Fig. 4(b), where the fractional bandwidth is plotted versus the normalized hole diameter with respect to the unitcell period. The optimal fractional bandwidth is obtained for d1/p= 0.86 and d2/p=0.54. It has been observed that the variation in hole depth does not have a significant influence on fractional bandwidth for values greater than h/p=0.5. B. Cells With Different Periods Next, we study the behavior of the stopband for all holey structures when their unit-cell dimensions are adjusted to have the same lower frequency bound so the fractional bandwidth can be properly compared in all the structures. The corresponding dispersion diagrams are shown in Fig. 5, with the insets representing each holey unit cell with its actual aspect ratio to help compare the sizes of the different cells. All parameters used for this comparison have been selected to obtain the optimum stopband bandwidth [14]. These parameters are shown in Table I. The results of Fig. 5demonstrate that the three-hole unit cell proposed in this work exhibits a wider fractional stopband bandwidth and a smaller period size compared to the other three two-hole structures. Compared to the glide case, the three-hole unit cell has TABLE I HOLEY UNIT CELL PARAMETERS FOR FIG.5(DIMENSIONS IN mm) Fig. 6. Bloch’s analysis of the proposed unit cell [reproduced in (d)] with dimensions in Table Iusing MMTMM for its propagation and attenuation constants along (a) 0X and (b) 0M, in comparison with CST ES. (c) Root loci of the complex mode in (b) from 40.8 to 81 GHz. (d) Sketch of the proposed three-hole unit cell. 36% less size and 20% more fractional bandwidth. This result shows that the proposed three-hole unit cell has a very high potential for devices where a high stopband is required, with the additional benefit of a reduced size compared to other two-hole unit cells while maintaining its robustness and ease of manufacturing. Furthermore, the reduced size of the three-hole unit cell makes its use convenient at lower frequencies, where the larger size of the other two-hole unit cells could be impractical. However, it should be noted that size reduction can lead to a lower attenuation factor p.u.c. (e−2αp). In that case, the minimum number of unit cells required to achieve the target attenuation should be determined by investigating the characteristics of the stopband. IV. BLOCH ANALYSIS USING MMTMM In addition to the previous study, the multimode transfer matrix method (MMTMM) [41] is applied in this section to analyze an optimal three-hole p.u.c. holey metasurface (see dimensions in the fifth column of Table I). MMTMM allows us to obtain not only the phase constant but also the attenuation constant of any periodic structure. Although accurate values of the attenuation constant cannot be easily obtained using commercial electromagnetic software, MMTMM has already demonstrated being a reliable tool for this task [14],[41]. Fig. 6(a) shows both the phase shift and the normalized (to the wavenumber of the free space) attenuation constant along the edge 0X in the first Brillouin zone of the proposed three-hole unit cell with optimal dimensions using MMTMM. The figure also included the phase shift obtained with the ES of the CST Studio Suite using periodic boundaries in the propagation directions within the Brillouin zones and PEC boundaries on the rest. Both results are 1048 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 72, NO. 1, JANUARY 2024 Fig. 7. Frequency behavior of the attenuation for various normalized values of d2for the three-hole unit cell with dimensions given in Table I. (a) First Brillouin edge 0X. (b) Third Brillouin edge M0. perfectly correlated, showing both the validity of the MMTMM and the performance of the proposed cell. The results also demonstrate the high level of attenuation/rejection provided by the three-hole structure in the forbidden band, similar to the glide-symmetric holey EBG reported in [14]. As demonstrated in Fig. 6(b), when applying the MMTMM to the third edge 0M of the Brillouin zone, a good agreement in the phase constant with the CST-ES results is found again except for the appearance of a complex mode in the forbidden band. This mode cannot be detected with the CST-ES because it only computes propagative modes. A detailed discussion on the behavior of this kind of complex modes can be found in [14]. Similar to the complex mode in [14], the complex mode in Fig. 6(b) splits at about 77 GHz into two real propagative forward and backward modes. The lower branch of the attenuation constant corresponds to the evanescent modes in the stopband, while the upper branch corresponds to the complex mode. To obtain a better picture of the behavior of the complex mode, its root loci are presented in Fig. 6(c) from its onset frequency at 40.8 GHz until 81 GHz. It can be clearly observed that the complex mode splits at 77.4 GHz into two real propagative forward and backward modes. The effect of varying the top normalized diameter, d2(keeping d1fixed), on the attenuation constant is shown in Fig. 7. The value d1/pis taken as the optimal one that produces the highest limit of the stopband, as can be seen in Fig. 4(b). Fig. 7(a) shows that the effect of normalized d2on the first Brillouin edge is not very significant, except for its minimum value. The maximum level of attenuation is obtained at the optimal normalized value of d2/p=0.54, which is the same value that achieves the maximum fractional stopband bandwidth. The influence of d2/pis more significant at the third Brillouin edge, as observed in Fig. 7(a). In this region, it can be seen that the change from the minimum normalized value to its maximum significantly affects the attenuation factor. Again, the optimal value of d2/pfor attenuation matches the optimal value of the stopband. The results shown in Fig. 7(b) show that most of the influence on the propagation characteristics comes from the third Brillouin edge (kx=ky). This means that propagation in the 0M direction is the main responsible for the good performance of the proposed unit cell in terms of stopband. V. EXPERIMENTAL RESULTS In order to experimentally validate the previous analysis, two pairs of WR-19 waveguide spacers were manufactured and measured in a back-to-back configuration. All spacers are compatible with the UG-383/U flange, with a 2.5 mm length. One of the pairs is a regular spacer with no holes on it, as shown in Fig. 8(a), while the other uses the three-hole double-layer periodic structure proposed in this work, as shown in Fig. 8(b). The dimensions of the holes are given in Table Icorresponding to the dispersion diagram in Fig. 5(d). The measurement setup can be seen in Fig. 8(c), showing the back-to-back Fig. 8. Manufactured WR-19 waveguide spacers. (a) No holey regular spacer. (b) Three-hole double-layer spacer. (c) Spacers measurement setup in back-to-back configuration. Fig. 9. Measurements of the transmission coefficient. (a) Comparison of the response with and without holey structure for air gaps of 0 and 100 µm. (b) Response of the holey structure for different values of the air gap. case. To measure different air gaps, a separator is added between both spacers to achieve the desired air gap. Two different measurements have been performed for both pairs: one with zero air gap between both spacers and the other with an air gap of 100 µm. The transmission coefficients (S21) are illustrated in Fig. 9. Both structures perform as expected when no gap exists between each spacer; that is, minimal losses are obtained. However, when there is a 100 µm air gap, the standard structure IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 72, NO. 1, JANUARY 2024 1049 exhibits considerable losses, with a dip appearing around 47.5 GHz. In contrast, the holey structure has small losses with no peaks in the operating band, 40–60 GHz. Fig. 9(b) shows a comparison of transmission losses for the holey structure with varying air gaps. It is evident that the smaller the air gap, the lower the losses, although they remain almost insignificant in comparison to the nonholey case. Note that, in all cases, the measured reflection coefficients are below −20 dB. Therefore, the losses in the nonholey structure can only be explained by the leakage due to the air gap in the interface between both spacers. In conclusion, the three-hole structure drastically reduces leakage due to the spacer. These experimental results clearly demonstrate the potential of the proposed three-hole structure. This configuration is a compact and versatile candidate for different practical scenarios where leakage must be avoided. VI. CONCLUSION In this communication, a novel holey EBG is presented, which is based on a three-hole topology with different values for the top and bottom diameters. The performance of this unit cell has been compared with other similar holey EBGs, such as mirror, oneedge displaced mirror, and glide-symmetric topologies. In particular, we have studied the stopband behavior, maintaining the unit cell period equal in all structures and with the same lower frequency bound of the stopband. The results show that the proposed unit cell exhibits a reduction in size and an increase in the fractional bandwidth with respect to the glide-symmetric structure. These results suggest that the proposed unit cell could have potential uses in practical applications with limited space. With a parametric analysis, the optimum parameters for this unit cell have been found to achieve the maximum fractional bandwidth of the stopband. A MMTMM analysis was performed to determine the attenuation of the proposed unit cell, which shows a desired high level of attenuation in the stopband, similar to other glide-symmetric structures previously studied. To illustrate the potential of the proposed EBG structure in a practical scenario, it has been used to reduce leakage at the interface between two spacers. Our experimental results show a drastic improvement with respect to its nonholey counterpart. REFERENCES [1] P.-S. Kildal, S. Haasl, and P. Enoksson, “Gap waveguide structures for THz applications,” U.S. Patent 9806393, Oct. 31, 2017. [2] S. Carlred, E. A. Alós, and P.-S. Kildal, “Arrangement for interconnection of waveguide structures and a structure for a waveguide structure interconnecting arrangement,” U.S. Patent 17/156702, May 27, 2021. [3] B. Pyne, R. Naruse, H. Saito, J. Hirokawa, V. Ravindra, and P. R. Akbar, “Robust contactless noncircular choke flange for wideband waveguide applications,” IEEE Trans. Microw. Theory Techn., vol. 67, no. 3, pp. 861–867, Mar. 2019. [4] P.-S. Kildal, E. Alfonso, A. Valero-Nogueira, and E. Rajo-Iglesias, “Local metamaterial-based waveguides in gaps between parallel metal plates,” IEEE Antennas Wireless Propag. Lett., vol. 8, pp. 84–87, 2009. [5] P.-S. Kildal, A. U. Zaman, E. Rajo-Iglesias, E. Alfonso, and A. Valero-Nogueira, “Design and experimental verification of ridge gap waveguide in bed of nails for parallel-plate mode suppression,” IET Microw., Antennas Propag., vol. 5, no. 3, pp. 262–270, Mar. 2011. [6] F. Fan, J. Yang, V. Vassilev, and A. U. Zaman, “Bandwidth investigation on half-height pin in ridge gap waveguide,” IEEE Trans. Microw. Theory Techn., vol. 66, no. 1, pp. 100–108, Jan. 2018. [7] D. Dawn, Y. Ohashi, and T. Shimura, “A novel electromagnetic bandgap metal plate for parallel plate mode suppression in shielded structures,” IEEE Microw. Wireless Compon. Lett., vol. 12, no. 5, pp. 166–168, May 2002. [8] P. J. Crepeau and P. R. McIsaac, “Consequences of symmetry in periodic structures,” Proc. IEEE, vol. 52, no. 1, pp. 33–43, Jan. 1964. [9] A. Hessel, M. Hui Chen, R. C. M. Li, and A. A. Oliner, “Propagation in periodically loaded waveguides with higher symmetries,” Proc. IEEE, vol. 61, no. 2, pp. 183–195, Feb. 1973. [10] O. Quevedo-Teruel, Q. Chen, F. Mesa, N. J. G. Fonseca, and G. Valerio, “On the benefits of glide symmetries for microwave devices,” IEEE J. Microw., vol. 1, no. 1, pp. 457–469, Jan. 2021. [11] O. Quevedo-Teruel, M. Ebrahimpouri, and M. N. M. Kehn, “Ultrawideband metasurface lenses based on off-shifted opposite layers,” IEEE Antennas Wireless Propag. Lett., vol. 15, pp. 484–487, 2016. [12] M. Ebrahimpouri, O. Quevedo-Teruel, and E. Rajo-Iglesias, “Design guidelines for gap waveguide technology based on glide-symmetric holey structures,” IEEE Microw. Wireless Compon. Lett., vol. 27, no. 6, pp. 542–544, Jun. 2017. [13] A. Vosoogh, H. Zirath, and Z. S. He, “Novel air-filled waveguide transmission line based on multilayer thin metal plates,” IEEE Trans. THz Sci. Technol., vol. 9, no. 3, pp. 282–290, May 2019. [14] Q. Chen, F. Mesa, X. Yin, and O. Quevedo-Teruel, “Accurate characterization and design guidelines of glide-symmetric holey EBG,” IEEE Trans. Microw. Theory Techn., vol. 68, no. 12, pp. 4984–4994, Dec. 2020. [15] M. Ebrahimpouri, E. Rajo-Iglesias, Z. Sipus, and O. Quevedo-Teruel, “Cost-effective gap waveguide technology based on glide-symmetric holey EBG structures,” IEEE Trans. Microw. Theory Techn., vol. 66, no. 2, pp. 927–934, Feb. 2018. [16] A. Monje-Real, N. J. G. Fonseca, O. Zetterstrom, E. Pucci, and O. Quevedo-Teruel, “Holey glide-symmetric filters for 5G at millimeterwave frequencies,” IEEE Microw. Wireless Compon. Lett., vol. 30, no. 1, pp. 31–34, Jan. 2020. [17] Q. Chen, O. Zetterstrom, E. Pucci, A. Palomares-Caballero, P. Padilla, and O. Quevedo-Teruel, “Glide-symmetric holey leaky-wave antenna with low dispersion for 60 GHz point-to-point communications,” IEEE Trans. Antennas Propag., vol. 68, no. 3, pp. 1925–1936, Mar. 2020. [18] Q. Chen, F. Mesa, P. Padilla, X. Yin, and O. Quevedo-Teruel, “Efficient leaky-lens antenna at 60 GHz based on a substrate-integrated- holey metasurface,” IEEE Trans. Antennas Propag., vol. 68, no. 12, pp. 7777–7784, Dec. 2020. [19] M. Ebrahimpouri, A. Algaba Brazalez, L. Manholm, and O. Quevedo- Teruel, “Using glide-symmetric holes to reduce leakage between waveguide flanges,” IEEE Microw. Wireless Compon. Lett., vol. 28, no. 6, pp. 473–475, Jun. 2018. [20] Z. S. He, S. An, J. Liu, and C. Jin, “Variable high precision wide D-band phase shifter,” IEEE Access, vol. 8, pp. 140438–140444, 2020. [21] Q. Chen, F. Ghasemifard, G. Valerio, and O. Quevedo-Teruel, “Modeling and dispersion analysis of coaxial lines with higher symmetries,” IEEE Trans. Microw. Theory Techn., vol. 66, no. 10, pp. 4338–4345, Oct. 2018. [22] B. A. Mouris, A. Fernández-Prieto, R. Thobaben, J. Martel, F. Mesa, and O. Quevedo-Teruel, “On the increment of the bandwidth of mushroomtype EBG structures with glide symmetry,” IEEE Trans. Microw. Theory Techn., vol. 68, no. 4, pp. 1365–1375, Apr. 2020. [23] M. Bosiljevac, Z. Sipus, and P. S. Kildal, “Construction of green’s functions of parallel plates with periodic texture with application to gap waveguides—A plane-wave spectral-domain approach,” IET Microw., Antennas Propag., vol. 4, no. 11, pp. 1799–1810, Nov. 2010. [24] G. Valerio, D. R. Jackson, and A. Galli, “Fundamental properties of surface waves in lossless stratified structures,” Proc. Roy. Soc. A, Math., Phys. Eng. Sci., vol. 466, no. 2120, pp. 2447–2469, Mar. 2010. [25] G. Valerio, Z. Sipus, A. Grbic, and O. Quevedo-Teruel, “Accurate equivalent-circuit descriptions of thin glide-symmetric corrugated metasurfaces,” IEEE Trans. Antennas Propag., vol. 65, no. 5, pp. 2695–2700, May 2017. [26] F. Ghasemifard, M. Norgren, and O. Quevedo-Teruel, “Dispersion analysis of 2-D glide-symmetric corrugated metasurfaces using modematching technique,” IEEE Microw. Wireless Compon. Lett., vol. 28, no. 1, pp. 1–3, Jan. 2018. [27] G. Valerio, F. Ghasemifard, Z. Sipus, and O. Quevedo-Teruel, “Glidesymmetric all-metal holey metasurfaces for low-dispersive artificial materials: Modeling and properties,” IEEE Trans. Microw. Theory Techn., vol. 66, no. 7, pp. 3210–3223, Jul. 2018. [28] M. Tsuji, S. Matsumoto, H. Shigesawa, and K. Takiyama, “Guidedwave experiments with dielectric waveguides having finite periodic corrugation,” IEEE Trans. Microw. Theory Techn., vol. MTT-31, no. 4, pp. 337–344, Apr. 1983. [29] S. Amari, R. Vahldieck, J. Bornemann, and P. Leuchtmann, “Spectrum of corrugated and periodically loaded waveguides from classical matrix eigenvalues,” IEEE Trans. Microw. Theory Techn., vol. 48, no. 3, pp. 453–460, Mar. 2000. 1050 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 72, NO. 1, JANUARY 2024 [30] H. K. Liu and T. L. Dong, “Propagation characteristics for periodic waveguide based on generalized conservation of complex power technique,” IEEE Trans. Microw. Theory Techn., vol. 54, no. 9, pp. 3479–3485, Sep. 2006. [31] F. Bongard, J. Perruisseau-Carrier, and J. R. Mosig, “Enhanced periodic structure analysis based on a multiconductor transmission line model and application to metamaterials,” IEEE Trans. Microw. Theory Techn., vol. 57, no. 11, pp. 2715–2726, Nov. 2009. [32] R. Islam, M. Zedler, and G. V. Eleftheriades, “Modal analysis and wave propagation in finite 2D transmission-line metamaterials,” IEEE Trans. Antennas Propag., vol. 59, no. 5, pp. 1562–1570, May 2011. [33] J. Naqui et al., “Common-mode suppression in microstrip differential lines by means of complementary split ring resonators: Theory and applications,” IEEE Trans. Microw. Theory Techn., vol. 60, no. 10, pp. 3023–3034, Oct. 2012. [34] S. Marini, Á. Coves, V. E. Boria, and B. Gimeno, “Efficient modal analysis of periodic structures loaded with arbitrarily shaped waveguides,” IEEE Trans. Microw. Theory Techn., vol. 58, no. 3, pp. 529–536, Mar. 2010. [35] Á. Coves, S. Marini, B. Gimeno, and V. Boria, “Full-wave analysis of periodic dielectric frequency-selective surfaces under plane wave excitation,” IEEE Trans. Antennas Propag., vol. 60, no. 6, pp. 2760–2769, Jun. 2012. [36] Y. Weitsch and T. F. Eibert, “Modal series expansion of eigensolutions for closed and open periodic waveguides,” IEEE Trans. Antennas Propag., vol. 60, no. 12, pp. 5881–5889, Dec. 2012. [37] M. Bagheriasl, O. Quevedo-Teruel, and G. Valerio, “Bloch analysis of artificial lines and surfaces exhibiting glide symmetry,” IEEE Trans. Microw. Theory Techn., vol. 67, no. 7, pp. 2618–2628, Jul. 2019. [38] Q. Chen, F. Giusti, G. Valerio, F. Mesa, and O. Quevedo-Teruel, “Anisotropic glide-symmetric substrate-integrated-holey metasurface for a compressed ultrawideband Luneburg lens,” Appl. Phys. Lett., vol. 118, no. 8, Feb. 2021, Art. no. 084102. [39] F. Giusti, Q. Chen, F. Mesa, M. Albani, and O. Quevedo-Teruel, “Efficient Bloch analysis of general periodic structures with a linearized multimodal transfer-matrix approach,” IEEE Trans. Antennas Propag., vol. 70, no. 7, pp. 5555–5562, Jul. 2022. [40] P. Castillo-Tapia, K. Van Gassen, Q. Chen, F. Mesa, Z. Sipus, and O. Quevedo-Teruel, “Dispersion analysis of twist-symmetric dielectric waveguides,” Photonics, vol. 8, no. 6, p. 206, Jun. 2021. [41] F. Mesa, G. Valerio, R. Rodríguez-Berral, and O. Quevedo-Teruel, “Simulation-assisted efficient computation of the dispersion diagram of periodic structures: A comprehensive overview with applications to filters, leaky-wave antennas and metasurfaces,” IEEE Antennas Propag. Mag., vol. 63, no. 5, pp. 33–45, Oct. 2021.