Characterization and Suppression of Transmission Dips in Glide-Symmetric Holey Gap Waveguides

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES 1 Characterization and Suppression of Transmission Dips in Glide-Symmetric Holey Gap Waveguides Mingzheng Chen , Graduate Student Member, IEEE, Johan Bellbrant, Oskar Zetterstrom , Member, IEEE, Francisco Mesa , Fellow, IEEE, and Oscar Quevedo-Teruel , Fellow, IEEE Abstract—The spurious transmission dips that occur in glidesymmetric holey gap waveguides (GSHGWs) are systematically characterized in this work, and the obtained information is used to suppress them in the intended operating band of the gap waveguide. The analysis relies on the dispersion characteristics of the waveguide segment with electromagnetic bandgap (EBG) holes. These characteristics are explored through the multimodal transfer matrix approach, particularly focusing on identifying relevant edge and waveguide modes. We find four types of unwanted dips in the transmission coefficient within the intended operation frequency band of the gap waveguide under study. The first three types are all associated with the edge mode mostly concentrated in the small air-gap region between the waveguide and the EBG holes, whereas the fourth type is caused by a narrow stopband in the waveguide mode. Based on a thorough understanding of all dips, we propose three viable solutions: placing EBG holes away from the waveguide channel, intersecting EBG holes with the waveguide channel, and intersecting additional small holes with the waveguide channel and the EBG holes. After comparison, the last solution with two small holes per EBG hole along the waveguide channel was demonstrated to be the most advantageous in terms of transmission properties, compactness, and flexibility. This solution was also experimentally validated using a WR-19 GSHGW operating from 35 to 63 GHz. Index Terms—Electromagnetic bandgap (EBG), glide symmetry, holey gap waveguide, multimodal transfer matrix method (MMTMM), periodic structures, spurious transmission dips. I. INTRODUCTION METALLIC waveguides are widely used guiding structures in electromagnetic (EM) engineering due to their simplicity, low losses, and high power-handling capabilities [1]. At high frequencies (typically above 30 GHz), metallic waveguides are even more advantageous in terms of transmission losses and power handling compared to dielectric-based Received 9 February 2025; revised 16 April 2025 and 15 May 2025; accepted 17 May 2025. The work of Oscar Quevedo-Teruel was supported by the Vetenskapsr˚ adet (VR) Project through Call “Research Project Grant Within Natural and Engineering Sciences” under Grant 2022-03865. The work of Francisco Mesa was supported in part by MICIU/AEI/10.13039/501100011033 under Grant PID2023-148281NB-I00 and in part by ERDF/EU. (Corresponding author: Mingzheng Chen.) Mingzheng Chen, Johan Bellbrant, Oskar Zetterstrom, and Oscar Quevedo-Teruel are with the Division of Electromagnetic Engineering and Fusion Science, KTH Royal Institute of Technology, 100 44 Stockholm, Sweden (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Francisco Mesa is with the Department of Applied Physics 1, ETS Ingenier´ ıa Inform´ atica, Universidad de Seville, 41012 Seville, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2025.3572361 solutions, such as microstrip lines and substrate-integrated waveguides. However, due to the small dimension of the components at high frequencies, the manufacturing of high-quality metallic waveguides and waveguide-based feed networks becomes difficult and expensive [2],[3],[4]. In particular, it is important to ensure good electrical contact between waveguide plates because even a small gap of 50 µm can cause high transmission losses due to severe power leakage, as discussed in [5],[6], and [7]. In order to overcome the drawbacks of metallic waveguides at high frequencies, gap waveguide technology was proposed [8],[9]. In this technology, wave propagation is restricted to areas confined by electromagnetic bandgap (EBG) structures. As a result, electrical contact between the top and bottom plates is not required, which presents a great advantage for the manufacturing and assembly process. During the last decade, gap waveguide technology has found wide applications in packaging [10], waveguide array antennas [11],[12], waveguide filters [13],[14], and many other EM components [15],[16]. Pin-type EBG structures have been commonly used in gap waveguide technology to stop wave propagation in undesired directions and areas. However, at high frequencies, such structures require the manufacturing of thin and tall metallic pins, increasing the device cost and incurring fabrication difficulties with conventional production techniques, such as computerized numerical control (CNC) machining. Glide-symmetric holey EBG solutions were proposed as a cost-effective alternative to pin-type EBG structures in millimeter-wave (mmWave) bands [17],[18]. A glidesymmetric holey EBG structure consists of periodic holes in the upper and lower plates of a parallel-plate waveguide (PPW) separated by a small air gap. The EBG holes are shifted half the period in one or two orthogonal in-plane directions, causing a substantial increase in the stop bandwidth compared to its nonglide counterpart [19]. This EBG structure is made up of holes that are generally larger than conventional pins at the same operating frequency, with the depth of the holes less than the corresponding pin height [18]. Therefore, glidesymmetric holey EBG structures can result in higher accuracy and lower manufacturing costs at high frequencies and have found wide applications in waveguides [20], phase shifters [21],[22],[23], filters [23],[24],[25], flanges [26], and array antennas [5],[27]. However, spurious transmission dips have been reported to occur within the operating frequencies of gap ©2025 The Authors. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 2 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES waveguides implemented using such holey EBG structures, reducing their effective bandwidths [20],[22],[23],[28]. To avoid this problem, one proposed solution is to use small corrugations to connect the EBG holes and the waveguide [20],[22],[23]. Although effective in suppressing some of the observed transmission dips, the literature does not provide a satisfactory explanation for the cause of these dips. It is worth noting that Sipus et al. [28] related the detected transmission dip with the Bragg effect [29], explaining that reflections from periodic EBG holes constructively interfere in phase, which is only partially correct. Moreover, the addition of small corrugations increases the manufacturing cost when using CNC machining. In this work, we systematically characterize all transmission dips associated with gap waveguides implemented with glidesymmetric holey EBG structures using the multimodal transfer matrix method (MMTMM) [30],[31],[32]. This method allows for an accurate dispersion analysis of the gap waveguide structure implemented with EBG holes by representing its unit cell with a generalized multimodal transfer matrix. The inclusion of higher order modes in this method ensures that the relevant higher order mode coupling between adjacent unit cells can be properly accounted for. Moreover, the MMTMM also enables simultaneous computation of the phase and attenuation constants of the unit cell under study, offering valuable insight into the complex and evanescent modes present in the structure. Details about these complex/evanescent modes are not readily provided by commonly used commercial software. With the MMTMM tool, we identify four different types of transmission dips and give a detailed explanation of them. Moreover, we propose three viable solutions, aiming at giving guidelines for eliminating undesired transmission dips of gap waveguides implemented with glide-symmetric holey EBG structures. Among the proposed solutions, the introduction of additional small holes intersecting the waveguide channel and the EBG holes (two small holes per EBG hole) is primarily recommended and is also experimentally validated. Although this study primarily focuses on glide-symmetric holey gap waveguides (GSHGWs), similar transmission dips may also occur in other periodic guiding structures. Our methodology can be applied to understand and address these problems as well. This article is organized as follows. In Section II, a GSHGW and its corresponding unit cell are studied to investigate spurious transmission dips. Next, the sources of all the dips are analyzed in Section III. Then, in Section IV, three possible solutions are proposed and thoroughly discussed. In Section V, the proposed solution of intersecting additional small holes with the waveguide channel and the EBG holes is experimentally validated. Finally, the main conclusions are given in Section VI. II. SPURIOUS TRANSMISSION DIPS In this section, we use a Bloch analysis to investigate spurious transmission dips using a straight two-plate WR-19 gap waveguide implemented with glide-symmetric EBG holes in the side walls, as shown in Fig. 1. This structure is herein Fig. 1. Section of a GSHGW with a length of L=9p+2r. Cylindrical holes along the waveguide are added in the plates to form a glide-symmetric EBG. called a GSHGW, and the waveguide material, if not specified, is aluminum. A. Study of the Holey Gap-Waveguide Structure We first consider a finite section of the GSHGW shown in Fig. 1, where a part of the top plate is removed for clear illustration. The gap waveguide is separated into two aluminum plates by a small air gap (g=50 µm). This gap is selected to model the expected worst-case gap in the practical realization of a gap waveguide feeding system [21],[22]. The rectangular guiding channel has the same dimensions as the standard WR-19 waveguide with W=4.78 mm and H=2.39 mm. The WR-19 waveguide has a cutofffrequency of around 31.5 GHz for the fundamental TE10 mode, and the cutofffrequency for the next higher order mode is approximately 63 GHz. In order to study all spurious transmission dips occurring in the single-mode operating band of the waveguide, the frequency band of interest in this study is set to 35–63 GHz. Two rows of glide-symmetric EBG holes are placed along the lateral sides of the waveguide channel at a distance d(edge width). In order to obtain a stopband in the operational band of the waveguide, the depth hof the holes needs to be large enough, and the radius rof the holes is optimized for different p, where prepresents the periodic spacing of the holes along the z-axis. The total length of the GSHGW section is L=9p+2r, which is long enough to ensure a clear observation of transmission dips. Note that a short segment of length ris added on both sides of the waveguide to avoid truncating the EBG holes, an issue that will be discussed in detail in Sections III and IV. Before studying the transmission properties of the GSHGW, we need to characterize the stopband performance of the 2-D holey periodic EBG structure itself. For this purpose, we show in Fig. 2the dispersion diagram along the edges of the irreducible Brillouin zone (path ΓXMΓ) for periodic structures with p=5.6 and 6 mm and a small air gap of g=50 µm. Here, pis the diagonal length of the square unit cell, which is also shown in Fig. 2; the holes have a depth of h=1.5 mm and an optimized radius of r=0.274 p. We can observe that the stopband in all directions of the EBG covers the waveguide operational band from 35 to 63 GHz when pvaries between 5.6 and 6 mm. Note that when placed along the waveguide, the holes are repeated along the Γ→Mdirection. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. CHEN et al.: CHARACTERIZATION AND SUPPRESSION OF TRANSMISSION DIPS IN GSHGWs 3 Fig. 2. Dispersion diagram along the path ΓXMΓof the first Brillouin zone for the square glide-symmetric EBG unit cell of p=5.6 and 6 mm. The waveguide operational band is shaded yellow. Fig. 3. (a) Transmission and (b) reflection coefficients of the WR-19 GSHGW with different periods pand without (w.o.) EBG. Next, we study the transmission and reflection coefficients of the GSHGW of length Lwith different periods pusing the CST time-domain solver. The edge width is set to d=0.6 mm. The period pis varied from 5.6 to 6 mm in steps of 0.1 mm. The results of the GSHGW are reported in Fig. 3, together with a reference waveguide of length 55.38 mm without (w.o.) the EBG structure (the same length as the GSHGW with p=5.8 mm). We first look at the case of the GSHGW with p=6 mm, the results of which are the light purple curves in Fig. 3. The EBG holes, designed to minimize power leakage, are expected to allow efficient transmission across the working EBG. However, as demonstrated in Fig. 3(a), we note several transmission dips throughout the operational frequency band. In particular, effective transmission is only evident in the 41–46and 49–54-GHz frequency intervals. Fig. 4. (a) Unit cell of the GSHGW. (b) Equivalent network of the unit cell characterized by a 2N-port multimodal transfer matrix. Similar observations can also be made for other periods. In this work, we categorize the dips for any of the considered values of pinto four different types, namely, Types I-IV, from low to high frequencies. The reason for such classification is that the dips are found to have different properties and sources, which is discussed in more detail in later sections. Type I dips occur at the lower frequencies (below 45 GHz) where we can observe around four-five narrow dips. At the frequencies corresponding to the Type I dips, the transmission coefficients in Fig. 3(a) have values between −3 and −1 dB, and we observe that these dips are correlated with the peaks appearing in the reflection coefficients shown in Fig. 3(b). The Type II dips appear for different pbetween 46 and 52 GHz with values of |S21|around −3 dB. When moving higher in frequency, more severe Type III dips with values of the transmission coefficient around −10 dB and higher than −3 dB in the reflection coefficient are observed. Finally, Type IV dips are observed between 58 and 65 GHz with levels of approximately −5 dB for both |S21|and |S11|. An interesting observation in Fig. 3is that increasing the period pof the EBG holes from 5.6 to 6 mm results in dips of Types IIIV occurring at progressively lower frequencies, while the appearance of Type I dips is less predictable. This observation suggests a strong connection between the transmission dips and the periodicity of the EBG holes for the dips of Types II-IV. B. Analysis of the Unit Cell To understand the causes of these transmission dips, we note that the GSHGW analyzed in the previous section can be seen as a finite section of a 1-D periodic structure along the propagation direction z. To perform a Bloch analysis of this 1-D periodic structure, we apply the MMTMM to the unit cell of the GSHGW shown in Fig. 4(a). Ports are added at the boundaries of the unit cell along the z-axis, which is This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 4 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES the periodicity direction. To account for the coupling between adjacent cells through high-order mode interactions, the unit cell is studied using the MMTMM [32] and is modeled as a 2N-port equivalent network with input and output ports representing the first Nsignificant modes, as depicted in Fig. 4(b). The multimodal scattering parameters of the unit cell are first simulated using, for example, the CST frequencydomain solver with tetrahedral meshing and open boundaries set on the two physical ports, each excited by the Nmodes. From the multimodal scattering matrix, we obtain the 2N×2N multimodal transfer (ABCD) matrix [T(ω)] that characterizes the unit cell in the periodic environment. Finally, applying Bloch’s theorem, wave propagation along the periodic direction zcan be characterized as the following eigenvalue problem [32]: V2 I2=[T(ω)] V1 I1=e−jkp V1 I1(1) where V1,V2and I1,I2are the voltage and current vectors related to the N-port modes in the input and output ports, and k=β−jαis the Bloch wavenumber along the z-direction, with βand αbeing the phase and attenuation constants. As an example, the eigenproblem (1) is now solved for the unit cell with p=5.8 mm and other dimensions as defined in Section II-A. As shown in Fig. 5(a), a good agreement for the phase shift (βp/π) is obtained between the results of the MMTMM (solid lines) and the CST eigenmode solver (CST ES, circles). Fig. 5(b) shows the normalizedto-k0attenuation constants (k0is the wavenumber of the free space) calculated with the MMTMM with aluminum or perfect electric conductor (PEC) as the building material. Only MMTMM results are plotted since the CST ES cannot provide the attenuation constant in the stopband [32]. We can identify three significant modes in Fig. 5(a), which also shows the insets of their corresponding modal Eyprofiles given by CST ES (the top plate is omitted for better illustration). The first mode (blue curve) is the desired propagating waveguide mode. The second mode (red curve) is an edge mode that propagates in the small air-gap regions between the waveguide channel and the EBG holes. The third mode (orange curve) is another edge mode in the air-gap regions but now bounded to the external sides of them. The first relevant observation is that since Mode 3 is physically isolated from the waveguide mode by the EBG structures, it will not influence the propagation of the wave within the waveguide channel. In contrast, Mode 2 travels through the air gap that is directly connected to the waveguide channel. This mode has a field distribution that resembles that of the transverse electromagnetic (TEM) mode of a PPW. Hence, this quasi-TEM mode is expected to be coupled with the quasi-TE10 Mode 1 that travels along the waveguide channel. For p=5.8 mm, Fig. 5(a) shows that the edge Mode 2 has two passbands at 35–44 and 57–65 GHz, respectively, and a stopband from 44 to 57 GHz. This information is key to understanding the formation of spurious transmission dips. As plotted in Fig. 5(c), assuming that the metallic material is PEC, Mode 2 couples with Mode 1 starting at about 57 GHz to give rise to a pair of complex modes with a high attenuation coefficient in the range of Fig. 5. Bloch analysis of the GSHGW unit cell with p=5.8 mm. (a) Phase shifts (βp/π) of significant modes with the insets depicting the distribution of Eycomponents obtained with CST ES. (b) Normalized attenuation constant (α/k0) of Modes 1 and 2 with aluminum or PEC as the building material. (c) Root loci of the kzsolutions for Modes 1 and 2 from 56.5 to 60 GHz, where these modes couple to give rise to a complex mode. PEC is selected as the building material here to eliminate the influence of metallic losses in the analysis. 57–59 GHz. The appearance of these complex modes is what actually causes Type III transmission dips, as presented in Fig. 3. In fact, the sources of all the transmission dips can be revealed from the Bloch analysis of the GSHGW unit cell, which will be detailed in Section III. This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. CHEN et al.: CHARACTERIZATION AND SUPPRESSION OF TRANSMISSION DIPS IN GSHGWs 5 Fig. 6. (a) Transmission coefficients of the GSHGW with p=5.6 and 5.8 mm at 35–45 GHz, where Type I dips are indicated. (b) Phase shift (βL/π) of Mode 2 considering the total waveguide length L. (c) E-field distribution of the gap waveguide with p=5.8 mm at 38.3 GHz (the top plate is hidden for illustration). III. SOURCES OF SPURIOUS TRANSMISSION DIPS A. Type I Transmission Dips The transmission coefficients for the GSHGW segment with periods p=5.6 and 5.8 mm within the frequency range of 35–45 GHz are shown in Fig. 6(a), highlighting the frequencies where Type I dips appear. In Fig. 6(b), we show the phase shifts (βL/π) of Mode 2 along the z-direction in the same frequency range. It can be observed that Type I dips occur at frequencies that satisfy approximately βL/π =n,n=1,...,6.(2) To illustrate this phenomenon, the E-field distribution of the GSHGW with p=5.8 mm at the frequency of 38.3 GHz (one of the Type I resonances) is shown in Fig. 6(c). The waveguide is excited from Port 1 and the EM field propagates to Port 2 inside the guiding channel as a quasi-TE10 mode. Since this frequency falls within the passband of the edge Mode 2, part of the field travels along the thin air channel between the EBG holes and the waveguide channel. At the end of the structure Fig. 7. (a) Transmission coefficients of the GSHGW with different periods p at 46–52 GHz, where Type II dips occur. (b) Normalized attenuation constants α/k0of Mode 2 at 42–58 GHz. on the Port 2 side, the edge EM waves are reflected back with an amplitude given by Er=η0−η1 η0+η1 Ei(3) where Eiand Errepresent the amplitudes of the incident and reflected edge waves, respectively, while η0and η1denote the wave impedances in free space and within the narrow air-gap region, respectively. Given that this region can be considered as a PPW, its impedance is estimated to be η1=(g/d)η0, and because gd, we conclude that η1η0. This implies that Er≈Eifor the edge EM waves at the end of the waveguide. If we only consider one reflection, when the phase shift of the edge mode meets the condition βL/π =n, the electric field of the edge waves at any z-position can be obtained as E=Ei+Er≈Eie−jβzˆ y+Eiej(βz+nπ)ˆ y ≈Eie−jβz+(−1)nejβzˆ y.(4) Consequently, strong standing waves form along the thin edge next to the waveguide channel, leading to Type I dips, as illustrated in Fig. 6(c). It is also noted that the intended TE10 transmission mode within the waveguide channel is distorted by the intense standing edge waves. B. Type II Transmission Dips The GSHGW transmission coefficients corresponding to the frequency range where Type II transmission dips occur are shown in Fig. 7(a) for different periods pat 46–52 GHz. We observe that for each p, there exists only one dip of Type II, which shifts to lower frequencies as pincreases. Within the This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 6 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Fig. 8. E-field distribution of the gap waveguide with p=5.8 mm at 49 GHz when (a) L=9p+2r=55.38 mm and (b) L=9p=52.2 mm (truncated). (c) E-field magnitude along the middle of the waveguide channel and its edge without and with truncation (white dashed lines). frequency range where these dips occur, Fig. 5(a) indicates the presence of two relevant modes: a propagating Mode 1 and an evanescent Mode 2. The attenuation constants α/k0of Mode 2 for different pat 42–58 GHz are shown in Fig. 7(b), with the frequency range of Type II dips indicated. The first relevant observation is that the Type II dips occur approximately in the center of the stopbands of Mode 2, where the attenuation constants have a maximum. Taking p=5.8 mm as a reference, in Fig. 8(a), we show the E-field distribution of the GSHGW at 49 GHz, where the Type II dip occurs. It can be seen that the relevant edge Mode 2 is excited at the edges near the input and output ports, as highlighted in the red rectangles in Fig. 8(a). Due to the evanescent nature of this mode, after a few periods from the input port, only the propagating mode persists. When this propagating mode reaches the output port, the impedance mismatch at the end of the waveguide causes re-excitation of Mode 2, resulting in the high field concentration observed in the edge region at this end. Consequently, power transfer to Mode 2 results in the formation of the Type II dip. To further validate this hypothesis, we reduce the waveguide length by ron both port ends, effectively halving the edge regions where Mode 2 is mostly concentrated, as shown in Fig. 8(a). As clearly seen in Fig. 8(b), Mode 2 is hardly excited in these smaller edge regions, allowing us to achieve effective transmission of the quasi-TE10 mode through the waveguide channel. A comparison of the magnitude of the E-field along two straight lines in the middle of the waveguide channel and at its edge, without and with truncation, is shown in Fig. 8(c). Without truncation (L=9p+2r), strong fields are excited along the edge near the ports, which decay exponentially along the ±z-direction (see the blue dotted curve). The yellow curve in Fig. 8(c) shows that the attenuation fits well with the factor E0e−αz, where α=0.148 mm−1is the attenuation coefficient of Mode 2 obtained with MMTMM and shown in Fig. 7(b). Due to the excitation of these evanescent fields, the amplitude of the E-field along the center line of the waveguide significantly decreases near the ports, whereas the amplitude in the truncated waveguide remains nearly unchanged. In Fig. 7(a), we observed that there was no Type II dip for the case p=5.8 mm after truncation (blue dotted curve). In fact, it is recommended to always apply such a truncation to avoid the Type II dip. C. Type III Transmission Dips The transmission coefficients of the GSHGW with different periods pat 53–65 GHz are presented in Fig. 9(a). The Type III dips are those that occur at 55–61 GHz with a level lower than −6 dB. The frequency range aligns precisely with the location of the coupling of Mode 1 and Mode 2, resulting in the emergence of a pair of complex modes characterized approximately by the wavenumbers β±jα. Only in the lossless case are the βand αof the two complex modes equal. In practice, due to different metallic losses of the two modes, βand αof the complex modes will differ slightly. Due to the symmetric nature of the structure, an additional pair of complex modes with −β∓jαalso emerges. Detailed discussions on complex modes are given in [33],[34], and [35]. Importantly, Freire et al. [35] highlight that feeding Port 1 in the frequency range that supports only complex modes will excite a decaying field generated by the combination of two complex modes with kz=β−jαand −k∗ z=−β−jα. Assuming, for instance, that the field in the periodic structure is dominated by the zeroth harmonic, this combination would give rise to a so-called “modulated evanescent mode” with the following general form [35, eq. (10)]: E(x,y,z)≈A0(x,y) cos(βz)+B0(x,y) sin(βz)e−αz ≈C0(x,y) cos(βz+φ)e−αz.(5) In this case, the field profile corresponds to a standing field type with an amplitude function dependent on βthat exponentially decays with an attenuation constant α. The values of the phase and attenuation constants for Mode 1 within the range 55–61 GHz are plotted in Fig. 9(b) and (c). Due to the high attenuation constant αof the modulated evanescent mode, the magnitude of the E-field attenuates significantly along the propagating z-direction in the waveguide channel, as presented in Fig. 10(a) and (b). The attenuation of the E-field magnitude in Fig. 10(b) fits approximately with E0e−αz, where α=0.039 mm−1is the average of the attenuation coefficients of the two complex modes obtained with MMTMM when the metallic material is aluminum [see Fig. 5(b)]. Moreover, the This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. CHEN et al.: CHARACTERIZATION AND SUPPRESSION OF TRANSMISSION DIPS IN GSHGWs 7 Fig. 9. (a) Transmission coefficients of the GSHGW with different periods p at 53–65 GHz, where Types III and IV dips occur. (b) Phase shift βp/π and (c) attenuation constant α/k0of Mode 1. TABLE I FREQUENCIES OF THE TYPE IV TRANSMISSION DIPS frequency range for the occurrence of complex modes shifts with the change of period p, which is the reason for the shift of the Type III transmission dips. D. Type IV Transmission Dips To study the Type IV transmission dips, in Fig. 9(a)–(c), we can observe that the dips occurring in the frequency range of 58–64 GHz are related to stopbands of the GSHGW with β= 0. In these small stopbands, Fig. 9(c) shows that the attenuation constant αis much smaller than that causing the Type III dip, a fact that is consistent with the lower values observed for the transmission coefficients in Fig. 9(a). In Fig. 11, we can Fig. 10. (a) E-field distribution of the GSHGW p=5.8 mm at 57.7 GHz. (b) Amplitude of the E-field at 57.7 GHz along the central line of the guiding channel. Metallic parts are aluminum. Fig. 11. E-field distribution of the GSHGW with p=5.8 mm at 60.9 GHz. see that for p=5.8 mm, the attenuation of the E-field in the waveguide channel is much less severe at 60.9 GHz (Type IV dip) than that in Fig. 10(a) at 57.7 GHz (Type III dip). The Type IV transmission dips can be directly attributed to the Bragg effect [29],[36],[37], where consecutive reflections from the EBG holes constructively interfere in phase. As a result, the condition for the Type IV transmission dip is nλg=2p,n=1,2, . . . (6) where λgis the guiding wavelength. Approximating the GSHGW mode as the pure TE10 mode in normal rectangular waveguide, we obtain that the frequencies fB,nsatisfying (6) are given by fB,n=s2c np2 +fc2(7) where cis the speed of light in free space and fcis the cutoff frequency of TE10 mode. In our case, only fB,2falls within the operational frequency band. When pchanges from 5.6 to 6 mm, Table Ishows a good correlation between the values predicted by (7) and the frequencies of the attenuation peaks shown in Fig. 9(c). It should be noted that the origin of these This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. 8 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES TABLE II SOURCES OF ALL SPURIOUS TRANSMISSION DIPS Fig. 12. (a) Top view of the GSHGW unit cell (top plate is hidden here). Light/dark blue holes are in the top/bottom plate, and the violet rectangle represents the waveguide channel. (b) Transmission coefficients of the GSHGW with different edge widths dand periods p. transmission dips is very similar to the open stopband issue in leaky-wave antennas [38],[39],[40]. In summary, we have investigated the sources of four different spurious transmission dips in this section. As summarized in Table II, Type I dips are caused by strong standing edge waves, while Type II dips originate from attenuating edge waves near waveguide ports. Moreover, complex modes generated by the coupling of the propagating waveguide mode and the relevant edge mode give rise to Type III dips. Finally, unlike the Types I-III dips that are all related to the relevant edge mode, the Type IV dips are caused by the appearance of quasi-standing waves in the channel. IV. SOLUTIONS TO SPURIOUS TRANSMISSION DIPS In this section, we propose three possible solutions for eliminating spurious transmission dips in GSHGW. A. Place EBG Holes Away From Waveguide The first proposed solution might be better viewed as a mitigation strategy. As discussed in Sections II and III, the dips come from either the interaction between the waveguide mode and the undesired edge mode or in-phase reflections from the EBG holes. Therefore, the dips can be mitigated by reducing the coupling between the waveguide mode and the edge mode, and by physically increasing the isolation between the waveguide and the EBG structure. One way of simultaneously achieving these goals is placing the EBG holes further away from the waveguide channel (that is, increasing the Fig. 13. (a) Phase shift (βp/π) of significant modes for the GSHGW unit cell with p=5 mm and d=−0.1 mm. Negative values of dmean that the holes intersect the guiding channel. (b) Normalized attenuation constant (α/k0) of Mode 1 with the inset depicting the unit cell. Here, negative dmeans the EBG holes intersect with the waveguide channel. (c) Transmission coefficients of the GSHGW with different intersection distances dand periods p. width d), as the EM waves are in this case less confined to the edges. The transmission coefficients of the GSHGW (p=5.8 mm) with different widths dare presented in Fig. 12(b) (blue curves). We observe that when dincreases from 0.6 to 2 mm, all transmission dips are reduced. An additional observation is that the Types II-IV dips move to lower frequencies as dincreases, thereby decreasing the effective bandwidth of the GSHGW. Furthermore, the Type III dip remains significant, reaching more than −5 dB, even when d=2 mm. We can further reduce the period pof the EBG holes from 5.8 to 5 mm to shift the dips of Types II-IV to higher frequencies and reduce their magnitudes, as demonstrated in Section II-A. Finally, with a combination of p=5 mm and d=2 mm, we achieve good transmission in the frequency range from 40 to 60 GHz with a transmission coefficient higher than −0.5 dB, as shown by the red dotted curve in Fig. 12(b). When a user needs a specific operating bandwidth for the GSHGW, a potential approach is to enlarge the edge width dand decrease the period p, which will relocate unwanted dips beyond the desired band. However, mmWave systems require compactness, which means that dis generally kept to a minimum. Moreover, significantly reducing the period p could cause the stopband of the EBG structure to shift outside the waveguide operating band. Therefore, alternative methods for suppressing transmission dips are discussed Sections IV-B and IV-C. B. Intersecting EBG Holes Into the Waveguide Channel The second solution is to intersect the EBG holes directly in the waveguide channel, resulting in the GSHGW unit This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. CHEN et al.: CHARACTERIZATION AND SUPPRESSION OF TRANSMISSION DIPS IN GSHGWs 9 cell shown in the inset of Fig. 13(b). The benefit of this intersection is that the relevant edge mode (Mode 2) is eliminated, as illustrated in Fig. 13(a). The transmission coefficients of the GSHGW with different intersection distances dare presented in Fig. 13(c) (blue curves). In this figure, we can observe that using intersecting holes (d=−0.1 mm), the Types I-III dips are removed in the range of 35–55 GHz. Note that the disappearance of these three types of dips is due to the elimination of the relevant edge Mode 2. Nevertheless, this solution makes the Type IV dip even more prominent because of the increased reflections from the periodic EBG holes. If the intersection distance is slightly increased to −0.2 mm, the corresponding Type IV dip becomes much wider and deeper. To move this dip beyond our band of interest, we can reduce the period pto 5 mm, as illustrated by the red-dotted curve in Fig. 13(c). With a combination of p=5 mm and d=−0.1 mm, we achieve good transmission in the range of 40–63 GHz. However, as shown in the zoomed-in view inset of Fig. 13(c), we find significant dips in the range of 35–40 GHz. This is attributed to the fact that this frequency range is very close to the cutofffrequency of Mode 1, leading to potential leakage, as also revealed in the nonzero values of the attenuation constant in this frequency range observed in Fig. 13(b). We find that directly intersecting EBG holes in the waveguide channel and conveniently selecting the period pis a compromise solution in terms of operating bandwidth. However, it is still a better alternative compared to the first solution in at least two aspects. First, the problems of Types I-III dips are solved completely instead of just being mitigated. Second, the spaces needed for the EBG structures are significantly reduced, leading to more compact designs. C. Inserting Additional Small Holes To mitigate the Type IV dip observed in the second solution, we propose inserting a series of small holes at the boundary between the waveguide channel and the EBG holes. With this solution, the EBG holes do not need to intersect the waveguide channel (d>0). This strategy is expected to fully suppress the associated edge mode while avoiding the appearance of pronounced in-phase reflections. Our first test is to add an additional small hole for each EBG hole (denoted in the following as “1 small hole”), as illustrated in the unit cell shown in the inset of Fig. 14(e). In this case, the corresponding parameters are the radius of the small hole rs=0.6 mm, the height hs=h/2=1.195 mm, and the distance between the center of the hole and the waveguide channel ws=0.48 mm. The transmission coefficient for such GSHGW is plotted in Fig. 14(a) as a solid yellow curve. As expected, we successfully removed the Types I and III dips. However, the Type II dip remains present since Mode 2 can still be excited near the port regions in this structure. However, this dip is completely removed by implementing the truncation operation discussed in Section III-B, as confirmed by the yellow dotted curve in Fig. 14(a). Finally, a less pronounced Type IV dip is noticeable at 60 GHz compared to the case without small holes. This weaker dip is caused by reflections from the small holes. This effect is further corroborated by the normalized attenuation constant of the waveguide mode Fig. 14. (a) Transmission coefficients of the GSHGW without (w.o.) small holes, with (w.) one small hole, with two small holes, with one small hole and truncated, and with two small holes and truncated. p=5.8 mm and d=0.6 mm for all cases. (b) Implementation of the gap waveguide with two small holes (the top plate is hidden for illustration). (c) Illustration of a segment of the EBG structure with two additional small holes per EBG hole in the edge area. (d) Phase shifts (βp/π) of significant modes for the waveguide unit cell with one small hole. (e) Normalized attenuation constant (α/k0) of Mode 1 with the inset depicting the waveguide unit cell with one small hole. (f) Phase shift (βp/π) of significant modes for the waveguide unit cell with two small holes. (g) Normalized attenuation constant (α/k0) of Mode 1 with the inset depicting the waveguide unit cell with two small holes. in Fig. 14(e), where a distinct spike can be observed around 60 GHz. Once we identified the problem of inserting one small hole per EBG hole, we now increase the number of additional small holes to two per EBG hole in the edge area (denoted as “2 small holes”), as shown in Fig. 14(b). The detailed arrangement of the small holes is illustrated in Fig. 14(c), where the spacing of the adjacent small holes is ps=p/4. The other parameters are rs=0.5 mm, hs=h/2=1.195 mm, This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.