Improved TM Dual-Mode Filters With Reduced Fabrication Complexity

This work presents an alternate and improved approach for the design of dual-mode filters that utilize transverse magnetic (TM) and nonresonating modes. The method that is proposed in this work allows for an improvement of the typical TM dual-mode filter design through the reformation of the coupling irises that connect the TM dual-mode cavity to the source/load waveguide ports, where the new interconnection means takes the form of resonate slot-irises. In this manner, a fourth-order quasi-elliptic response can be achieved within a very limited physical geometry and able to cover more than double the usable fractional bandwidth than previously reported. Furthermore, the selectivity and insertion loss of the filter response is significantly improved when compared to the typical single-cavity TM dual-mode filter response that uses coupling irises, while on the comparison of equal-order structures, a reduction in fabrication complexity and improved insertion loss is achieved. A characterization of the dimensional variations and effects of altering one of the source/load port positions in the proposed filter design is investigated in order to demonstrate notable effects on the rejection characteristics and positions of transmission zeros. A presentation on the design theory is given and formulations of various filter responses are examined. The fabrication of an experimental prototype with approximately 7.3% fractional bandwidth (FBW), centered at 90 GHz is conducted using high-precision computer numerical control (CNC) milling in order to demonstrate that the unique simplicity and overall compaction of this method can be easily applied at millimetre-wave frequencies without the need of tuning means. The results which are presented demonstrate highly accurate measurements throughout the W-band range, while an additional Q-factor analysis is provided in order to compare the improved filter scheme with other known design methodologies.


I. INTRODUCTION
As waveguide technology remains the dominant choice in industry due to its performance capabilities, high-end communication and space-bound systems require constant adaptation of both the electrical and physical properties to meet the stringent conditions imposed by industry as well as meet the needs of future applications that call for the use of higher frequency ranges. For decades, solutions for innate, yet critical properties such as size and weight have been an important area of research and have resulted in many compact multi-mode or multi-band solutions. In the case of multi-mode designs, many versatile methods, such as [1], [2], [3], [4], [5], have been proposed that allow for compact and light-weight microwave filters to be developed for high-end applications. One of the most resourceful methods of accomplishing this has been through the adoption of nonresonating nodes (NRN) and nonresonating modes (NRM) in order to connect resonators more flexibly or allocate new bypass couplings throughout the This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ structure; several advanced examples can be reviewed in [5], [6], [7], [8], [9], [10], [11], [12], where new classes of highlycompact filter components that utilize transverse magnetic (TM) mode cavities are demonstrated. The use of TM-mode filters with nonresonating modes has driven much innovation and research into modular building-block designs and have allowed for many new filter structures to be demonstrated, for example [13], [14], [15], [16], [17], [18], [19].
Maximizing the capabilities of filter structures that have highly stringent requirements is of great interest but also comes with great difficulty. In examples such as [6], [11], [20], [21], [22], [23], [24], [25], resonating irises (RIs) have been proposed as a means of maximizing the use of given filter structures for single-band or dual-band configurations. It is clear that the impact of these types of resonating irises can have a significant influence not only on the obtainable filtering characteristics, but also the resulting size and weight. In light of these results, the use of resonating iris configurations in the literature is relatively low. Pushing these types of filters to their maximum potential is of great interest and can grant design engineers the ability to utilize all of the cavities and interconnecting irises with a minimum of two purposes or modes during operation.
To this end, we propose an improved configuration to the TM dual-mode filters presented by [8], [9], [10], where the general operation and geometric simplicity of the filter is further maximized through the inclusion of resonating irises at the source/load positions of the dual-mode cavity. A description of the configuration and the subsequent impact on dimensional variations are exhibited for the first time. The impact of this work demonstrates: a more than doubling of the usable fractional bandwidth, an increase in selectivity, an increase in filter order, and improved Q-factor, all with minimal alterations to the original work proposed in [8], [9], [10] when comparing characteristics of designs with single resonators and iris/slot-irises. Furthermore, -notwithstanding the aforementioned attributes-when considering a comparison of equal-order designs, a substantial reduction in fabrication complexity can be obtained while attaining a significantly improved Q-factor. An experimental prototype of a filter is constructed for W-band (75-110 GHz) operation in order to demonstrate the geometrical simplicity and elegance for micro-scale design where each elementary structure of the filter operates with at least two purposes, (e.i., the resonant slot-irises for main-line coupling and producing passband poles, and the TM dual-mode cavity for producing two poles and a nonresonating mode for bypass coupling). The experimental prototype is specified for a fractional bandwidth (FBW) of approximately 7.3% and centered at 90 GHz in order to demonstrate the unique simplicity and low geometric complexity of the design for millimetre-wave operations where alignment becomes highly critical and tuning means become impractical or infeasible. A comparison of the proposed filter design and other TM-mode filter solutions is provided in order to highlight the benefits of the new design methodology.

II. RESONANT-IRIS APERTURES
Elementary resonating-irises can be configured from the general dimensions of a mixed-coupled (inductive and capacitive) slot-iris, or in the form of evanescent-mode partial-height posts (e.g., double ridge). Although evanescent-mode posttype designs have been utilized in work such as [11], the small and intricate details of partial-height posts make it difficult or even infeasible for the overall component to be scaled for millimetre and sub-millimetre wave applications. However, this partial-height post structure can be altered to form a synonymous configuration that avoids the use of partial-height posts and significantly eases fabrication. Fig. 1(a) depicts an evanescent-mode double-ridge resonant iris, while Fig. 1(b) depicts an equivalent form in the shape of a slot-type resonant iris. A comparison of the simulated S-parameter characteristics of both of these circuits is shown in Fig. 2 over the full W-band range, where the poles have been specified for 90 GHz operation. This demonstration instantiates a very similar behaviour over the full frequency band of interest for both the S 11 and S 21 response, and provides a suitable alternative to the manufacturing of partial-height post configurations in high-frequency applications. As a further demonstration, two interconnected resonating slot-irises are shown in Fig. 3 where the S-parameter responses for a variety of dimensional configurations are demonstrated in Fig. 4 for control over the bandwidth and center frequency. It can be clearly shown that the center frequency and bandwidth can be controlled with variations of either the S X or S Y dimensions from Fig. 3, respectively. This instance demonstrates the effect of a well controlled filtering response through the use of sequential resonant slot-irises and can be applied to more complex structures, as will be shown in the following sections.

III. TM DUAL-MODE FILTERS
Dual-mode filters using TM and nonresonating modes have been well defined and demonstrated in many works such as [10] and [19]. A short review of the theory and general operation will be discussed here as this work aims to be an improved and alternate approach that ultimately reduces the fabrication complexity and the need for cascading multiple TM-mode cavities as demonstrated in the works previously discussed.
Initially, the fundamental design of the TM dual-mode cavity follows the well-known design equations from [26], where w = width and h = height of the cavity. In general, the length of the cavity is kept smaller than a quarter of the guided wavelength due to the possible interference of the TE101 mode resonating near the passband of interest. The addition of stepped-corners allow for the formation of the TM120 and TM210 modes within the dual-mode resonant cavity while simultaneously allowing for the nonresonating TM11 mode to act as a source/load bypass coupling. However, one of the key differences in the structure that we propose in this work, is a modification to both of the coupling irises that are connected to the source/load. In this manner the typical thin coupling irises are replaced by resonating slot-irises that were previously described in Fig. 1(b). With this modification, the order of the filter is increased and effectively increases the rejection characteristics above and below the passband. Additionally, by reconfiguring one of the ports from its typical location (as shown in Fig. 5(a)) to the opposite side of TM-mode the cavity (as shown in Fig. 5(b)), and redefining the dimensions, a modified response can be expressed where, again, there is a significant change to the rejection characteristics. The configurations from Fig. 5 will be refereed to as configuration (a), and configuration (b) for simplicity. It is clear that in configuration (b), both of the source/load waveguide ports share a common corner of the TM-mode cavity that is not interrupted by one of the two necessary stepped-corners. A prospective view of this configuration is shown with integrated resonant slot-irises in Fig. 6.
It is important to note that a simple mirroring of the port in configuration (a) to configuration (b) will not satisfy the S-parameter goals, re-dimensioning is necessary to stimulate the transmission zero response. The evolution of the commonly used topology (e.g., [9]) to the new topology containing resonant slot-irises is described in Fig. 7(a) and (b), respectively. The addition of the two resonant slot-irises can facilitate two new poles within the passband while simultaneously acting as a typical coupling iris that allows for the nonresonating TM11 mode to act as a bypass between the first and fourth resonators (now resonating irises) and again on through to the source/load, effectively promoting the formation of two transmission zeros around the passband. Interestingly, the length of the slot irises have little effect on the position of the transmission zeros and simply allow for the bypass coupling to be passed over resonator 2 and 3; a demonstration on the effect of reducing the resonant slot-iris's length is further described in Section IV in order to highlight this effect. In light of the resonant slot-iris topology described in Fig. 7, it will be evident that for a comparison of single-cavity TM-mode cases (i.e., typical second-order versus the proposed fourth-order) that the proposed design achieves better rejection characteristics, better insertion loss, and can overcome narrow bandwidth limitations. Furthermore, a second dual-mode cavity and interconnecting coupling mechanism (such as a thick coupling  slot, quarter-wave waveguide section, or thin coupling slot) that is typically required in the cascaded fourth-order design [9], [10] can be forgone; ultimately allowing for reduced design complexity and a simpler milling procedure.
In previous works of this type, no equations have been outlined, nor is it evident how to easily evaluate this type of structure with resonant irises without the use of extraction tools. In regards to the structure shown in Fig. 6, we propose that a combination of coupling coefficient methods can be used to describe the coupling nature between the symmetric and asymmetric portions of the filter. In this manner, the structure can be evaluated based on the external quality factor Q e and coupling coefficient k 23 = k sym which follow from the standard equations [27] as where τ and f τ are the S 11 group delay and its associated center frequency with respect to the first resonator, and f i for i = 1, 2 are the eigenmode frequencies of two symmetric resonators, and the coupling coefficients k 12 = k 34 ≈ k asym follow from an asymmetric approximation of coupled resonators (in this case, resonator to resonant iris) as where f ix for i = 1, 2 and x = a, b are the eigenmode frequencies of two symmetric resonators formed from the mirroring of two decoupled asymmetric resonators [28], [29].
In the case of the TM11 bypass coupling, we propose a modified approach to the method described by [5] for TM110mode filters; this approach can be used to approximate the k 14 = k bypass of the proposed filter by removing the stepped corners of the TM-mode cavity and studying the transmission coefficient as where |S 21,SL | is the transmission coefficient of the source/load coupling through the TM-mode cavity [5], [30]. In this manner, the structures in Figs. 5 and 6 can be defined by coupling matrix theory.

IV. FILTER CONFIGURATIONS AND ATTRIBUTES
In order to demonstrate the functional nature of the proposed topology, a dimensional analysis is exhibited for simulated filters operating at 90 GHz with varying fractional bandwidths as well as a prototype filter to be fabricated for operation at 90 GHz with a fractional bandwidth of approximately 7.3%. Fig. 8 depicts the S-parameter responses of filter configuration (a) with and without resonant slot-irises. It can be shown that for similar specifications where the transmission zeros are held at the same locations, a significant improvement can be rendered over the full frequency band in favour of the resonant-iris type TM-mode filter structure. In a similar manner, configuration (a) and (b) are compared in Fig. 9 with similar dimensions and similar specified bandwidths.

FIGURE 9. The simulated S-parameters of the proposed TM dual-mode filter (Configuration (a) with resonant slot-irises) and its reconfigured version (Configuration (b) with resonant slot-irises). The 20 dB bandwidth is held in approximately the same position for a relative comparison between the return loss improvement and detuning of the transmission zero locations for filters with similar dimensions.
Observing the differences in transmission zero location, it is clear that the bypass coupling value is altered by changing from configuration (a) to (b), and the achievable rejection region characteristics are effected, particularly in the upper portion of the frequency band, and the return loss is further improved. Although comparisons generally maintain equivalent characteristics such as filter order and return loss ripple, this comparison highlights the evident improvements with minimal dimensional variation and similar bandwidth requirements.
Utilizing configuration (b) for its strong quasi-elliptic response, a filter operating at 90 GHz with a fractional bandwidth of approximately 7.3% is demonstrated and analyzed; first, with respect to the effects of parameter variations on the coupling values, and then the S-parameter characteristics with respect to dimensional variations in the lengths of both of the resonant slot-irises and the TM-mode resonator length itself. Fig. 10 depicts the simulated S-parameter response, where the coupling matrix response, described by (6), follows from equations (1)-(5) with regards to the filter structure defined in Fig. 6; note that the internal radii of corners is specified as 0.2 mm. Fig. 11 Control over the length of the resonant slot-iris dimension I Z is demonstrated in Fig. 12 for the structure and given dimensions outlined in Fig. 6. Studying Fig. 12, stimulation of two additional poles can be seen within the passband while varying the dimension I Z over the range of I Z = 0.1 mm to 0.868 mm. Inspection of this response sequence shows that a variation in the length of irises has negligible effect on the position of the transmission zeros. However, the bandwidth and selectivity of the filter is substantially increased, where on comparison of Fig. 12(a) and (f), the lowest rejection level at the edges of the plots can be shown to evolve from approximately 20 dB to 36 dB, while the return loss ripple evolves from approximately 1.8 GHz bandwidth at 36 dB to approximately 6.5 GHz bandwidth at 25 dB, demonstrating  a substantial variation of the characteristic response with respect to parameter I Z .
In regards to the simple mirroring of one of the source/load feeding ports to the opposite side of the TM-mode cavity (and with minimal re-dimensioning), a coupling sign change is invoked which results in an all-pole equalized filter response. This case is shown in Fig. 13, where a mirroring of one of the feed ports back to configuration (a) and a re-optimization of the dimensions is applied. In this scenario, the characteristics of a fourth-order Chebyshev response can be shown, where again, the stimulation of the two resonant slot-irises can be seen within the passband while varying the resonant slot-iris length from I Z = 0.1 mm to 0.868 mm. Again the bandwidth and selectivity of the filter is substantially increased with  .1 mm, 0.275 mm, 0.45 mm, 0.625 mm, 0.8 mm, and 0.868 mm  respectively. (re-optimized dimensions not shown).
the use of the resonant slot-irises, where on comparison of Fig. 13(a) and (f), the lowest rejection level at the edges of the plots can be shown to evolve from approximately 16.4 dB to 25.7 dB, while the return loss ripple evolves from approximately 2.1 GHz bandwidth at 30 dB to approximately 5.9 GHz bandwidth at 28 dB. Again, demonstrating a substantial variation of the characteristic response with respect to parameter I Z . In order to control the transmission zero response, a variation of the TM-mode cavity length (T Zi ) can be applied, where in Fig. 14, a demonstration of the transmission zero control is shown over the dimensional range of T Zi = 0.7 mm to 1.5 mm.  In [10], an upper limit of approximately 7% fractional bandwidth is suggested for the TM-mode filter. However, in light of the properties of the new topology proposed in Fig. 7(b), we expand this fractional bandwidth limit by more than double and demonstrate the simulated results to be suitable for at least 15% fractional bandwidth in both configuration (a) and (b). The S-parameter results are shown in Fig. 15 with a similar return loss ripple and fractional bandwidth over 60 GHz to 130 GHz. It can be noted that the achievable fractional bandwidth can be greater 15% and Fig. 15 is used as a wideband example. On the lower limit for a usable narrow fractional bandwidth, a practical limit of 2% is suggested. Notwithstanding, these bounds may be exceeded, however, can come at the cost of undesirable effects such as the introduction of spurious modes and increased losses. For bandwidth below 2%, the use of the classical structure is recommended. The simulated transmission response for the

V. FABRICATION AND RESULTS
As a demonstration of the compact and simple nature of this design, a prototype filter corresponding to the specifications outlined in the previous section for a 90 GHz filter operating with a 7.3% fractional bandwidth has been manufactured. Brass has been selected as the cutting material due to its machinability and high-quality surface finish in high-precision CNC milling. The dimensions of the structure are defined in Fig. 6 with internal radii of 0.2 mm. Images of the milled cavity and the assembled filter are shown in Fig. 17(a) and Fig. 17(b), respectively. The measured surface roughness of the milled cavity in Fig. 17(a) has been found to be approximately R a ≈ 1 μm, where R a describes the mean arithmetic height.
Once fabricated and assembled, the filter was tested using a Rohde & Schwarz ZVA67 with W-band up-converters. Fig. 18 presents a comparison of the simulated and measured results over 75 GHz to 110 GHz. This direct comparison demonstrates good measured results; the return loss is better than 20 dB, while the insertion loss is on range of 0.42 dB to 0.63 dB throughout the measured passband. Analyzing the response, the unloaded quality factor is found to be approximately Q u ≈ 550. The measured results demonstrate very high accuracy when compared to the simulated results in both the passband and rejection regions with negligible shift in center frequency. A close-up view of the simulated and measured insertion loss is given in Fig. 19 over the range of 83 GHz to 97 GHz. No gold (Au) or silver (Ag) coating has been applied to enhance the performance. The results of the prototype allow for the design approach to be demonstrated as a competitive option for ultra-compact structures, especially when selecting structures suitable for millimetre-wave design where tuning screws or the cascading and alignment of multiple cavities becomes increasingly difficult, if not infeasible.

VI. Q-FACTOR COMPARISON
As a verification of the design methodology, the Q-factor of a proposed configuration (b)-type filter is compared to several other TM-mode filter designs. Although it is evident that not all filtering characteristics can be matched, the two primary attributes in the comparison are in terms of equal return loss ripple and fractional bandwidth. In this comparison, the passband is set between 87.2 GHz to 92.8 GHz for a fractional bandwidth of 6.2%. In this manner, the typical dual-mode filter configuration - [9]-is within its suggested upper limit of 7% fractional bandwidth, while the proposed filter is closer to its lower limit.
Three filters are compared to the proposed filter design, two of which are variations of the typical dual-mode filter design -shown in Fig. 20(b) -and one of which is similar to the TM110 single-mode filters demonstrated in [31] -shown in Fig. 20(c). The simulated results of the four filters are shown in Fig. 21 over 70 GHz to 110 GHz. The equivalent conductivity of brass used for the measurement comparison provided in Section V has been applied to each of the structures for a realistic Q-factor comparison. The inset of Fig. 21 provides insight into the capabilities of each of these filters over 82 GHz to 98 GHz, where it can be clearly shown that the proposed filter in this work provides the lowest and flattest insertion loss of the four designs. Table 1 is provided for a quick comparison of values between the specified passband range. Although some argument can be made in regards to the variation of the S 21 parameters in terms of transmission zeros, spurious modes, and relative selectivity, it can be understood that it is difficult to perfectly match all characteristics, and the authors estimate the Q-factors as best as possible for simulated data for designs with equal return-loss ripple and fractional bandwidth. The proposed filter design reaches a Q u ≈ 660, the two typical dual-mode designs reach Q u ≈ 350, and the fourcavity single-mode design from Fig. 20(c) reaches Q u ≈ 350. It is note worthy to observe that with the use of resonant-irises, the Q-factor of the proposed design still overcomes that of the other simulated structures and provides a very flat insertion loss within the specified passband. Furthermore, the fabrication complexity is kept minimal by only requiring the use of one TM-mode cavity, opposed to two or four cavities that are required by the other design schemes to achieve fourth-order filter characteristics.  While it may be possible to increase the Q-factor of the structures shown in Fig. 20(b) and (c) by increasing their length, one must take into account that a simple lengthening of the cavities has intrinsic effects; primarily, the transmission zero positions will be altered due to a variation of the bypass coupling. In order to compensate this effect, one must try to vary the other dimensions of the filter, often resulting in other undesirable trade-offs that do not meet the required specifications. This becomes especially difficult if one considers that the length of the cavity should remain smaller than a quarter of the guided wavelength to keep the TE101 mode from resonating in the vicinity of the passband. Moreover, from a fabrication complexity point of view, the additional parts that are required to assemble the classical designs will cause more gaps, dimensional errors and misalignments that further degrade the Q-factor performance when applied in practice.

VII. CONCLUSION
In this work, an alternate and improved approach for the design of dual-mode filters that utilize TM and nonresonating modes has been investigated. A reconfiguration of the typical TM dual-mode design allows for the inclusion of resonant irises between the TM-mode cavity and the source/load connections. This approach demonstrates an improvement to the typical single-cavity design in several aspects, that being: increased filter order and suitability for wideband and millimetre-wave frequencies, all with minimal alterations to the original design, while in comparison to equal-order designs, a reduction in fabrication complexity and improved insertion loss is achieved. Furthermore, this method designates all of the filter's structural elements with a minimum of two operating purposes or modes in order to optimize the given geometry. In this manner, the novelty of the filter approach presented in this work allows for simple, compact and lightweight designs which do not require any tuning elements. A prototype of a W-band fourth-order quasi-elliptic filter has been manufactured using high-precision CNC milling in order to demonstrate the elegance and reduced complexity of the design. The prototype has been tested and demonstrates an exceptionally accurate response over the full frequency band of interest. A comparison of simulated fourth-order TM-mode filters is provided in order to highlight the obtainable Q-factor and flatness of the insertion loss with respect to other filter types, ultimately validating the design approach.