Introduction
As microwave systems increasingly operate at millimeter-wave frequencies to achieve higher bandwidths, there is an increasing need for filter topologies which allow for low-loss and simple manufacturing at these frequencies. For steerable antenna systems which require front-end filtering, a microwave filter is required for each antenna element in the array in both receive and transmit cases. Such filters for the antenna elements are known as array-element filters. For receivers, the sensitive Low-Noise Amplifiers need to be protected, and for transmitters, the out-of-band spurious signals and noise generated by each amplifier need to be filtered out. This requirement adds significant size constraints on the array-element filters, as one array-element filter typically needs to fit underneath one antenna element.
Gap waveguides, first presented in [Reference Kildal, Alfonso, Valero-Nogueira and Rajo-Iglesias1], have in recent years emerged as a low-loss alternative to conventional waveguides at very high frequencies, as they do not require electrical connections between the sidewalls and the roof of the waveguide, but rather create an effective sidewall by using periodic pin structures. This offers significant manufacturing advantages, as electrical connections at these interfaces require very accurate manufacturing in normal waveguides if excessive loss is to be prevented. Gap waveguide filters have been described in groove gap waveguide [Reference Shu, Xu, Wu, Guo, Chen and Zhang2, Reference Santiago, Tamayo-Domínguez, Laso, Lopetegi, Fernández-González, Martínez and Arregui5–Reference Zaman, Kildal and Kishk7, Reference Olmo-Olmeda, Baquero-Escudero, Boria-Esbert, Valero-Nogueira and Berenguer-Verdú12, Reference Al-Juboori, Zhou, Huang, Hussein, Alieldin, Otter, Klugmann and Lucyszyn14–Reference Shi, Zhang, Zhou, Feng, Cao and Che16], ridge gap waveguide [Reference Sorkherizi, Khaleghi and Kildal3, Reference Birgermajer, Janković, Radonić, Crnojević-Bengin and Bozzi9, Reference Ahmadi and Banai15], and printed ridge gap waveguide [Reference Deng, Li, Sun, Guo and Ma10, Reference Golboni, Arezoomand, Pirhadi and Asadi11]. Highly compact selective filters are presented in [Reference Rehman19–Reference Mushtaq, Khalid and Rehman21]. However, all these filters were designed as stand-alone components, without the constraints on size and input and output coupling imposed by the requirements of a filter for an antenna array. Currently, the only published array-element gap waveguide filter is a novel multi-layer stacked structure, utilizing one layer for each resonator [Reference Santiago, Fang, Zaman, Laso, Lopetegi and Arregui13]. This is an expensive solution, as each layer adds to the cost of the filter, and for high-order filters, many layers are required.
This letter presents a modified two-layer gap-waveguide array-element filter for a dense linear array of center-fed slot antennas, first presented in [Reference Fang, Yang, Emanuelsson, Andersson and Zaman18], as shown in Figure 1. For an array-element filter to fit underneath one slotted-waveguide antenna, each resonator is loaded to reduce the width and length and coupled together by coupling elements which allow for the reduced-size resonators. Both the loading elements and the coupling elements are designed to fit into the basic pattern of the gap-waveguide structure to reduce manufacturing costs. To incorporate a high-order filter within the same footprint as one slotted-waveguide antenna, the array-element filter is folded vertically, using a novel vertical coupling element. For improved manufacturing accuracy, one layer is flipped vertically to allow for both layers to be machined from one metal piece. A statistical analysis of the filter is done using methods presented in [Reference Klink, Meyer and Steyn17] to determine the required manufacturing tolerances. Finally, a novel ridge input coupling is presented. The filter presented in this letter is designed as a prototype to be later integrated with the antenna system and thus has the same input and output coupling for measurement purposes. A 7th order filter with a center frequency of 101 GHz and a bandwidth of 10% was built and measured, showing good performance, with a center band insertion loss of 0.5 dB.
Diagram showing the antenna array of the transmitter system the filter is being designed for as well as the stack-up of the individual layers of the system. Arrows show approximately where the layers couple to each other.
System specifications
The filter is designed to integrate with an antenna system being implemented in gap waveguide. The system consists of stacked layers of gap waveguide, a diagram of which is shown in Figure 1, where each layer has either single or multiple components for the transmitter system. The pertinent components of the transmitter for the design of the filter is the antenna layer which is immediately above the filter and the phase shifting and power amplification layer which feeds the filter. The antenna is a slotted waveguide antenna that is centrally fed by a ridged gap waveguide structure. To minimize grating lobes, the slotted waveguide arrays are spaced 
$\lambda_0/2$ apart, where λ 0 is the free space wavelength at the design frequency of the antenna. Including the width of the pins comprising the gap waveguide structure, this leaves the width of the waveguide to be less than 
$\lambda_0/2$. This implies that an empty gap waveguide cannot be used, and the waveguide must include a ridge to push the cutoff frequency down [Reference Shu, Xu, Wu, Guo, Chen and Zhang4]. Due to the spacing requirements of the slotted waveguide sub-arrays, each filter for the array must occupy the footprint of the antenna array above it. The specifications for the filter are center frequency = 101 GHz, bandwidth = 10 GHz, and return loss in passband = 20 dB. The gap waveguide comprises a rectangular lattice of square pins and is designed for maximum spurious free bandwidth to operate between 53 and 208 GHz, with dimensions and geometry shown in Figure 9. The ridged waveguide with geometry and dimensions are also shown in Figure 9.
Filter layout
The filter is required to be easily integrated into a pre-designed transmitter system, first presented in [Reference Fang, Yang, Emanuelsson, Andersson and Zaman18], shown in an exploded view in Figure 4. The system is composed of four layers, from top to bottom, the slot layer which contains the slots of the antenna in Figure 4(a), the feed layer which contains the ridged waveguides which feed the antenna slots in Figure 4(b), the connection layer which aligns the outputs from the signal layer to the center of the line in Figure 4(c), and the signal layer containing phase shifters and amplifiers in Figure 4(d). A section through a single column of the entire system is shown in Figure 2 showing the signal path from the power dividers to the antenna. The filter is proposed to occupy the layer between the signal layer and the antenna feed layer, the connection layer in Figure 4. It must integrate into the system without changing any of the preceding or successive layers. Examining the signal layer, the connection layer, and the feed layer in Figure 2 shows that the location of the layer transition between the signal layer and the centrally fed antennas is misaligned. The current connection layer consists of a section of ridged waveguide that is fed by a slot between layers and feeds another slot to the feed layer. The filter should occupy as little space as possible and not unduly extend the transmitter stack in the y-direction. If the filter extends only in the y-direction, using the coordinate system in Figure 2, another layer transition and another connection layer would be required in order to connect the signal from the end of the filter to the feed of the antenna. This would add an unnecessary length of transmission line and thus increase the costs and, more importantly, the losses of the system. To avoid this unnecessary transition and loss, the filter can be vertically folded in the transversal axis with a layer transition integrated in the design. This would allow the filter to extend in the y-direction and thus achieve the required filter order for the design specifications to be met while also removing the need for the connections layer which provides no signal conditioning and merely provides a connection between layers. The proposed filter configuration would increase the required layers of the transmitter stack up by one, as the filter would consist of two layers in the stack up and would replace the single layer of the connection layer, resulting in a net increase of one layer. As the feeds between the signal layer and the antenna feed layer are misaligned, and due to the odd order of the filter, the filter will have to be folded asymmetrically, as opposed to a traditional folded design in which the fold of the filter symmetrically divides the filter in two. In terms of manufacturing, the vertical folded structure has the advantage that no alignment issues within the two sections of the filter can occur in the assembly phase, as they are machined from one piece of metal. This, however, increases the manufacturing complexity, as a 5-axis CNC machine is required for such a step. During manufacturing, this will require the filter to be rotated during the machining process. As the filter is very sensitive to all the dimensions, this step can cause additional manufacturing errors if not done very accurately. The proposed 7th order filter will be implemented in a folded topology with the fold in the filter occurring between between resonators three and four, roughly dividing the filter between the two layers to minimize the space occupied by the filter.
Section through a column of the transmitter showing the signal path of a single channel.
Gap waveguides provide between 10 and 20 dB of attenuation of the electric field of the fundamental mode per row of pins [Reference Kildal, Alfonso, Valero-Nogueira and Rajo-Iglesias1]. Resonant structures have higher electric and magnetic field distributions due to their nature, and as such the strength of the fields coupled through the gap waveguides will be stronger. In order to prevent coupling between adjacent channels in the transmitter array which would interfere with the performance of the system, the filters must be arranged so that the resonant elements of each filter have more than one row of pins separating the channels. The filters for each channel will be arranged so that the filter for channel one will extend in the positive y-direction and the filter for channel two will extend in the negative y-direction with this configuration alternating for each channel of the transmitter, as shown in Figure 3. This arrangement will ensure that no resonators of adjacent filters will be separated by just one channel of pins as the only parts of the adjacent channels which overlap would be sections of ridged line, which have good isolation between them. The filter sections are physically separated by a large physical space which are filled with rows of gap waveguide pins, which would provide sufficient attenuation between the adjacent channels to minimize channel coupling. This method is suitable for large array configurations for slotted waveguide arrays as the filter will fit underneath each individual array element. The interleaved pattern allows the filters for each array element to be separated by at least more than one row of gap waveguide pins, regardless of the number of elements in the array. This method should also be applicable to high density arrays as long as they are linear.
Interleaved arrangement of filters to minimize coupling between channels.
Exploded view of the CAD model of the transmitter. (a) Antenna slot layer, (b) Antenna feed layer, (c) Connection layer and (d) Signal layer.
Filter design
The filter is designed as a coupled resonator filter with the coupling matrix calculated using the a center frequency of 101 GHz and a bandwidth of 10 GHz, as set by the system specifications [Reference Shu, Xu, Wu, Guo, Chen and Zhang8]. The filter is chosen to have an equiripple response with 20 dB of return loss in the passband. The elements of the coupling matrix are calculated to be 
$M_{01} = 0.995 = M_{78}$, 
$M_{12} = 0.830 = M_{67}$, and 
$M_{23} = 0.599 = M_{56}$, 
$M_{34} = 0.564 = M_{45}$. The resonators chosen for this design are loaded cavity resonators which comprise a resonant cavity with a capacitive load in the center. Due to the cavity operating below cut-off for an empty waveguide, the resonator requires a capacitive load to lower the resonant frequency to the design frequency of the filter, as shown in Figure 5. The cavity dimensions are width = 1.5 mm, height = 0.525 mm, and length = 1.55 mm. The fundamental mode of the unloaded cavity is a 
$\text{TE}_{101}$ mode resonating at 
$144 \text{ GHz}$. The use of the loads is required to fit the filter to the required physical dimensions prescribed by the antenna array. The capacitive load is implemented as a square pin in the center of a rectangular cavity. The load uses the same dimensions as the gap waveguide pins and is aligned with the gap waveguide to simplify manufacturing. A design graph of the resonant frequency of the first two modes of the resonator as well as the Q-factor of the fundamental mode as a function of the load height is plotted in Figure 6. The load height to ensure 
$\text{TE}{101}$ resonance at 101 GHz is 0.235 mm with an unloaded Q-factor simulated with a conductivity of 
$5.8\!\times\!10^7$ S/m of 1400.
CAD model of the loaded cavity resonator.
Resonant frequency of the first two modes of the loaded cavity resonator with the Q-factor of the first mode calculated with a conductance of 
$5.7\times10^7$ S/m. Line indicates where the resonant frequency of the first mode intercepts 101 GHz.
Coupling between resonators is accomplished through inductive irises, as seen in Figure 9. The input and output coupling in this filter requires a structure that offers both the correct coupling coefficient and changing from a ridged waveguide to an empty waveguide which cannot be achieved with a standard iris coupling. The coupling is achieved via protruding the input/output ridged waveguide line into the first/last resonator. This implements a type of electric probe coupling as the ridge is open circuited at the end, coupling the electric fields into the filter. The negative sign of the input/output coupling does not effect the performance of the filter. For the folded structure, the coupling between layers is a challenge. The coupling at the fold of the filter, between resonators three and four, is accomplished through a slot in the metal layer with a protrusion in it, seen in Figure 7. The slot couples the magnetic field of the resonators while the protrusion increases the coupling between the resonators by exciting higher order reactive modes, reducing the required size of the coupling slot. A design graph of the normalized coupling coefficient versus protrusion width and length at fixed length and width values, respectively, is shown in Figure 7. This shows that the normalized coupling coefficient is strongly dependent on the length of the protrusion and weakly dependent on the width of the protrusion.
Normalized coupling coefficient of the coupling slot with variations in length and width of the protrusion with a section view of the coupling slot.
The filter was designed and optimized using CST Studio Suite. The optimized parameters and filter are depicted in Figures 8 and 9. The simulated performance of the filter is shown in Figure 11(a).
Annotated vertical cross-section through the folded filter. Hatching denotes geometries through which the cross-section cuts through. All given dimensions are in millimeters: g = 0.025, h = 0.5, k01 = 0.676, k34 = 0.285, k78 = 0.658, r1 = 0.278, r2 = 0.1804, r3 = 0.182, r4 = 0.184, r5 = 0.192, r6 = 0.184, r7 = 0.272, t = 1. The pins forming the irises and the surrounding bed-of-nails have been omitted for clarity.
Annotated (a) top and (b) bottom view of the folded filter. Only the relevant sections of the filter are shown. All given dimensions are in millimeters: d = 0.45, k12 = 1.458, k23 = 1.409, k34w = 0.3, k45 = 1.393, k56 = 1.380, k67 = 1.449, lc = 1.5, p = 1, rd = 0.2, rh = 0.2, rw = 0.2, sl = 1, sw = 0.55, ϕr = 0.2.
Measurements
To determine the required manufacturing tolerance required for the filter, a statistical analysis of the yield was performed using non-linear partial least square (NLPLS)-based polynomial chaos expansion (PCE) which provides faster convergence to the mean when compared to Monte-Carlo analysis [Reference Klink, Meyer and Steyn17]. The filter was defined with 73 parameters of interest for the yield analysis. Three different manufacturing tolerances are investigated, namely, 
$5\ \mu$m, 
$2\ \mu$m, and 
$1\ \mu$m. For each tolerance level, 80 samples were simulated, which was deemed enough for the yield to converge. The yield was defined for this analysis as a return loss of less than 15 dB between 97 and 105 GHz. The yield criterion was adjusted from the filter specifications in order to provide meaningful yield results, as a 15 dB return loss was deemed acceptable and the bandwidth requirements were loosened to account for the inherent sensitivity of resonators to changes in dimensions. The simulated S-parameters for the yield analysis for the 
$5\ \mu$m, 
$2\ \mu$m, and 
$1\ \mu$m tolerances are plotted in Figure 10(a), 10(b), and 10(c) respectively. For the simulated tolerances of 
$5\ \mu$m, 
$2\ \mu$m, and 
$1\ \mu$m , the calculated percentage yields are 
$3.3\%$, 
$28.6\%$, and 
$95.6\%$ respectively. The low yields from the yield analysis indicate that this type of filter design is a prime candidate for optimising the design for yield in order to increase the yield rate for high-tolerance machining. Techniques presented in [Reference Klink, Meyer and Steyn17] are particularly suited for optimizing for yield. From the yield analysis, the filter was machined with a tolerance of 
$\pm2 \mu$m from a block of aluminium with no post processing treatments on the metal. The 
$2\ \mu$m tolerance was chosen as a compromise between yield and manufacturing cost. A 90∘ transition from ridge gap waveguide to WR-10 was included on both sides of the filter to facilitate measurements. These transitions are designed to be non-resonant so that the total loss is dominated by losses within the resonators of the filter. The manufactured filter is shown in Figure 11(b). The measured results are shown in Figure 11(a). The measured results show the filter is slightly detuned, with the center frequency slightly shifted up by ≈ 1 GHz, and has a large peak in the return loss at 98 GHz, but with performance in the rest of the passband that is comparable to filters presented in the literature, shown in Table 1. A parameter extraction was performed on the measured results to investigate the causes of the deviation from the measured results from the simulated. The extraction was done by creating an error function defined as the difference between the real and imaginary components of the measured S-parameters and the simulated S-parameters. The filter parameters were then optimized to minimize this error function. The parameter with the largest effect on filter performance was the gap height, g, of the gap waveguide. The nominal value specified was 25µm, while the extracted value was 45µm. This deviation in gap height caused the shift up in frequency as well as the large peak in the passband return loss at 98 GHz. A possible explanation for the large deviation from the nominal value of the gap height is the structure of the filter flexing when the WR-10 waveguide flanges are mounted to the filter assembly. The simulated S-parameters of the filter with extracted parameters is plotted in Figure 11(a) showing good agreement between the measured results and the extracted simulation.
NLPLS-based PCE simulation with (a) 
$5\ \mu$m, (b) 
$2\ \mu$m, and (c) 
$1\ \mu$m tolerances, 80 simulations plotted.
(a) 
$|S_{11}|\text{and } |S_{21}|$ of the measured and simulated filter with manufactured filter shown. Simulated 
$|S_{11}|$ with extracted parameters also shown. (b) Photograph of the manufacture filter with a €1 coin for scale.
Comparison of this filter with select published gap waveguide filters available in the literature
Conclusion
A folded W-bandgap waveguide filter is designed and manufactured with CNC milling and showed good matching between simulated and measured results with deviations accounted for due to manufacturing defects in the gap height. The filter is designed for a physically constrained environment and successfully implemented a vertically folded configuration to meet the imposed constraints.
Acknowledgements
This work was funded by the South African Department of Science and Innovation and part of the ENTRY100GHz research project.
Competing interests
The author(s) declare none.
Jack Fourie received his bachelors in Electrical and Electronic engineering from Stelllenbosch University in 2019. His Master’s work focused on tunable helical resonator filters and was achieved in 2021. He is currently pursuing his PhD focusing on gap waveguide filters.
Petrie Meyer received the Ph.D. degree in numerical analysis of microstrip circuits using the method-of-lines from the University of Stellenbosch, South Africa, in 1995, and the D.Eng. degree, in 2019. He is currently a Distinguished Professor in electromagnetics and microwaves at Stellenbosch University. He has worked actively in the fields of passive network design, optimization, and surrogate modeling and has authored or coauthored more than 100 technical journals and conference papers. Since 2010, he has been involved in the design of microwave filters and antenna elements for the square kilometer array radio astronomy antenna, the South-African precursor telescope, and MEERKAT. In 2004, he was awarded the South African THRIP Prize for human resource development, and the International CST Prize for a published journal articles making use of CST, in 2007.
Werner Steyn received his PhD degree researching multi-modal waveguides. He worked privately in industry developing radar systems. In 2020, he joined the academic staff at Stellenbosch University. His current research interests are in antenna design and radar systems.
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