Compact Triangular-Cavity Singlet-Based Filters in Stackable Multilayer Technologies

In this letter, triangular-cavity bandpass filters are investigated in stackable multilayer technologies in order to achieve highly compact designs with reduced fabrication complexity. The triangular-shaped cavities are first introduced in the form of singlets and then expanded on as a novel method for achieving a quasi-triplet filter response, where the filter's input and output irises are utilized as resonating means for two additional passband poles. Exploitation of this advanced singlet scheme exemplifies innovative use of resonant irises for achieving highly compact filters that can be manufactured with simple multilayer fabrication steps for use in future terahertz applications.


I. INTRODUCTION
A S TRENDS for high-frequency applications rise, technologies must evolve to sustain current demands as well as prepare for future ones. For the next generation of satellite communications, radar, and deep-space exploration, filter designs must be continuously adapted to optimize performance while the allocation of frequency bands reaches beyond the radio-frequency spectrum. Although, different technologies, such as high-precision milling [1], SU-8 [2], silicon micromachining [3], and structured glass [4], have been able to demonstrate the ability to reach well into the terahertz and subterahertz regions, progressive design solutions must be continuously explored to coincide with technological advancements.
In order to facilitate future demands for the aforementioned applications, novel designs must be considered for very highfrequency regions where the effect of dimensional tolerances becomes critical, the ability to tune circuits diminishes, and the constraints imposed on size and weight become highly Manuscript   consequential. In the case of compact high-frequency designs, research works conducted, such as [3], [5], and [6], have been able to demonstrate the ability to minimize the overall footprint of structures, remove the need for lengthy input/output feeding transitions, and in some cases, employ resonant irises to enhance the filter response. Moreover, the introduction of filters with singlets have gained much attention in the literature due to their relatively small size, simple design, and their ability to easily control transmission zero locations [7]- [9]. Extension of this concept to a variety of different resonator types and filtering operations are exemplified in papers, such as [10]- [16]. In regards to simple manufacturing procedures for millimeter and submillimeter wave filter components, stackable technologies and multilayer designs have been demonstrated using methods, such as electronic-band-gap [17], diffusion bonding of laminated metal plates [18], deep reactive ion etching (DRIE) [3], [5], and structured-glass waveguide components [4]. miniaturization methods at the millimeter and submillimeter wave bands becomes increasingly difficult due to highly stringent dimensions combined with the need for precise alignment of multiple structural layers. In this letter, we propose a simple and elegant design scheme for compact bandpass filters with three stackable layers using an advanced singlet topology. The formulation utilizes a triangular cavity singlet that is combined with two resonant slot-type irises in order to create a third-order bandpass response with one transmission zero, resulting in a quasi-triplet filter topology. In order to verify the proposed filter scheme, a prototype is demonstrated using electrical-discharge machining (EDM) for W-band (75-110 GHz) operation, while a second prototype is demonstrated in DRIE for J-band (220-330 GHz) operation. Each of the bandpass filter iterations demonstrates a highly compact and simple fabrication scheme that is suitable for future terahertz applications.

II. FILTER DESIGN
The singlet is a first-order topology that is capable of generating a transmission zero either above or below the frequency location of the pole, where the position of the transmission zero depends on the sign of the bypass coupling. In this regard, the transmission zero can be modeled on the lower side of the stopband when M sl < 0, and on the upper side of the stopband when M sl > 0, [16].
Advancements in triangular waveguide structures, such as the analysis provided in [19]- [21], allow for the application of singlet theory to be extended to shapes beyond typical rectangular or circular cavities and formulated for the evolution of triangularshaped filter structures. As outlined in [19], the resonant frequency of an isosceles triangular cavity can be found from where m = 0, 1, 2,..., n = 0, 1, 2,..., and m + n = 0. Using (1), we can characterize triangular-waveguide singlets and quasitriplets operating with the TM120 mode for the first time in the literature as an alternative geometry-which is also convenient   for optimizing on-chip layout-and is capable of achieving an equivalent Q-factor via (1) when compared with rectangular or cylindrical cavities that have similar thicknesses and center frequencies, and operate with analogous electromagnetic field distributions (i.e., the TM110 and TM010 modes, respectively).
For the design at hand, we select the TM120 mode and formulate the singlet to be fed with slot-type irises in a position that can simultaneously allow for a bypass coupling to pass from the source to load; Fig. 1 depicts the magnetic field distribution for the 90 GHz triangular singlet example that follows in Fig. 2(a). In addition, the singlet topology is indicated in the image for reference. Fig. 2(a) and (b) shows two cases of a singlet, which is designed for operation at 90 GHz where the transmission zero position is selected relative to the positions of the source/load coupling; inline or offset. The structure proposed in Fig. 2(a) results in the transmission zero on the lower side (M sl < 0), while the structure proposed in Fig. 2(b) results in the transmission zero on the upper side (M sl > 0). This effect is demonstrated in Fig. 3 for the simulation of each structure over 80-100 GHz.
In order to extend this concept to bandpass filter design, the triangular singlet can be modified to include resonant irises similar to the formulations outlined in [22] and [23]. Fig. 4 shows the magnetic field distribution of a singlet with the two resonant irises and describes the basic interaction throughout the filter. The modified topology now includes the resonant irises and is indicated in the image for reference; it can be noted that the bypass coupling is now formed between the resonant slot irises (nodes 1 and 3) in a quasi-triplet fashion. The evanescent modes of the resonant slot irises can be treated as TE101 modes while the triangular cavity (node 2) utilizes the TM120 mode. Fig. 5 shows the vacuum shell of an inline quasi-triplet structure and outlines the dimensions for each of the upcoming prototypes when fed with their respective waveguide ports. Fig. 6 is provided as a demonstration of the transmission zero control when varying the port positions to be inline or offset while the triangular cavity and the resonant irises are optimized for an equivalent passband response.
After the initial design of the cavity from equation (1) for an isosceles cavity, the synthesis of the passband filter can be suitably approximated from the general equations outlined in [24] for external quality factors and synchronous coupling despite the resonant irises and cavity being asynchronous. This method is used as a good approximation; however, the structure can also be viewed in a transverse coupling matrix form for simple parameter extraction. In order to verify this concept, a third-order filter with a fractional bandwidth (FBW) and center frequency of approximately 7.2% and 91.7 GHz is specified for EDM wire erosion for WR-10 band operation, while another with a FBW and center frequency of approximately 5.4% and 268.9 GHz is specified for DRIE on silicon wafers for WR-3 band operation.

III. FABRICATION AND MEASUREMENTS
For the WR-10 version of the filter, wire erosion was selected for its ability to reduce the necessary corner radii within the structures. Brass was selected as the cutting material with an erosion wire of 60 μm radius. The filter is fabricated as three individual pieces, namely, brass plates that house the irises and the triangular cavity. For the WR-3 version of the filter, a standard silicon micromachining process based on DRIE has been employed for its ability to obtain micron-level detail and high repeatability. The fabrication has been performed on a silicon-on-insulator (SOI) wafer that consists of a 30 μm device    layer, a 275 μm handle layer, and a 3 μm buried oxide layer. The proposed filter was built using three stacked layers while gold (Au) sputtering was used to metallize the chips. Each of the layers is aligned on top of one another using Vernier scale alignment marks before thermocompression bonding. To minimize the effect of underetching and promote a high accuracy between the simulated and measured results, the middle layer is fabricated using a fallout technique [25], and a compensative side wall underetching effect has been applied to the design.
Once fabricated and assembled, both versions of the filters were tested using a Rohde & Schwarz ZVA67 with their respective up-converters. Fig. 7 shows the fabricated structure from EDM wire erosion, while Fig. 8 shows a comparison of the simulated and measured results over 75-110 GHz. The measured return loss is better than 20 dB in the measured passband while the measured insertion loss at the measured center frequency is approximately 0.31 dB with an estimated Q u of ≈ 500. Fig. 9 presents the fabricated structure from DRIE micromachining, while Fig. 10 presents a comparison of the simulated and measured results over 220-330 GHz. The measured return loss is better than 13.5 dB in the measured passband and the measured insertion loss at the measured center frequency is approximately 0.61 dB with an estimated Q u of ≈ 300. A tolerance analysis comparing 100 runs with Gaussian randomized points within ±2 μm is added to the figure. However, some mismatch may be attributed to the underetching effect caused by DRIE and may be compensated in further iterations. The WR-10 and WR-3 L·W·H internal dimensions are approximately 0.764·1.588·0.814 λ 3 and 0.864·1.821·0.815 λ 3 , respectively.

IV. CONCLUSION
In this work, the concept of triangular-cavity singlets combined with resonant irises is explored in order to achieve novel bandpass filters in stackable multilayer technologies. Prototypes of the filter concept are designed and measured for highfrequency applications in the WR-10 and WR-3 bands with wire erosion and silicon wafer technology, respectively. The introduction of singlets with slot-type resonant irises demonstrates a versatile approach for compact filter design suitable for future terahertz applications.