3-D Printed THz Waveguide Components

This paper presents the state-of-the-art in polymer-based 3-D printing of metal-pipe rectangular waveguides (MPRWGs) with the first reported terahertz filters, all operating within the WR-2.2 band (325 to 500 GHz): a 5 mm-long thru line, two 399 GHz single-cavity resonators and two 403 GHz bandpass filters (BPFs). Our thru line exhibits a measured average insertion loss of only 0.9 dB, with a worst-case return loss of 13.3 dB, across the band. The single-cavity resonators, without and with corner rounding compensation (CRC) are investigated with the use of an RLC equivalent circuit model. The uncompensated resonator exhibits a 2.3% frequency downshift and an increase of 10.8 GHz in its 3 dB bandwidth. The compensated resonator exhibits a 2.2% frequency upshift and an increase of only 2.2 GHz in its 3 dB bandwidth; clearly demonstrating that CRC helps to mitigate against increased coupling into the resonators, as a result of manufacturing limitations with low-cost 3-D printing. Finally, the $3^{rd}$ order Butterworth and Chebyshev MPRWG BPFs both have a measured passband insertion loss of only 1.0 dB. The Butterworth filter exhibits a 0.8% passband frequency upshift and worst-case return loss of 16.6 dB; while the Chebyshev filter exhibits a 1.2% passband frequency downshift and worst-case return loss of 10.4 dB. With our low-cost polymer-based 3-D printing technology, we have demonstrated measured performances that are better than those using metal-based 3-D printing in the WR-2.2 band and this may, in the not too distant future, challenge components manufactured using traditional machining technologies.


I. INTRODUCTION
3-D printing represents one form of additive manufacturing.Over the past four decades, interest in 3-D printing has increased exponentially within academia and industry; mainly because of its design flexibility, lightweight structures, rapid prototyping and low manufacturing cost.This emerging technology can be classified into two main categories: (i) polymer-based 3-D printing (e.g., fused deposition modeling (FDM), polymer jetting (PolyJet), stereolithography apparatus (SLA) and masked stereolithography apparatus (MSLA)); and (ii) metal-based 3-D printing (e.g., selective laser melting (SLM), micro laser sintering (MLS), as well The associate editor coordinating the review of this manuscript and approving it for publication was Chaker Larabi .as other derivatives).Both polymer-and metal-based 3-D printing have demonstrated applications at microwave and millimeter-wave frequencies, making them suitable candidates for future 5G+ mobile communications, radar and imaging systems.
When comparing these two categories, metal-based 3-D printing provides good structural strength.This makes it more suitable for applications requiring high mechanical tolerances.However, even when printing with copper powder, its surface roughness is the main limitation for realizing low-loss waveguide components at sub-terahertz/uppermillimeter-wave (ca. 100 to 300 GHz) [1], [2] and terahertz/submillimeter-wave (300 GHz to 3 THz) [3], [4] frequencies.In contrast, polymer-based 3-D printing exhibits limited physical strength and inherently requires additional electro(less)-plating to establish conducting walls.Nevertheless, this technology has greater potential for applications where smooth surface, low mass and low manufacturing cost are the main drivers.To this end, the authors have previously demonstrated examples of metal-pipe rectangular waveguide (MPRWG) components and subsystems, using a variety of polymer-based 3-D printing technologies, in different bands: X-band (8 to 12 GHz) using FDM [5]; Ku-band (12 to 18 GHz) using expensive PolyJet [6]; W-band (75 to 110 GHz) using SLA [5] and MSLA [7]; D-band (110 to 170 GHz) using MSLA [8]; G-band (140 to 220 GHz) using PolyJet [9] and MSLA [10], [11]; WR-2.2 band (325 to 500 GHz), also referred to as the WM-570 band by the IEEE, using PolyJet [12]; WR-1.5 band (500 to 750 GHz) and WR-1 band (750 GHz to 1.1 THz) using experimental RECILS [13].In addition, all within the last three years, a G-band multi-channel frontend subsystem [14] has been successfully demonstrated by combining MPRWG [11] and quasi-optical components [15]; using a mixture of polymer-based 3-D printing technologies.
Our previous research has shown that iris corner rounding associated with an inductively-coupled waveguide bandpass filter (BPF), can degrade performance (e.g., center frequency shifting and bandwidth increasing) significantly with the use of a low-cost 3-D printer [11].This becomes more significant when dealing with waveguide components as frequencies increase.To mitigate against this effect, iris corner rounding compensation (CRC) [11] is employed in this work.
For the first time, this paper demonstrates the design, manufacture and test of terahertz MPRWG filters, fabricated using an ultra-low-cost polymer-based MSLA 3-D printer having a pixel resolution of 22 µm in the x-y build plane.Three types of waveguide components (i.e., thru line, singlecavity resonators and bandpass filters), all having a flange-toflange length of 5 mm and operating within the WR-2.2 band are investigated.

II. LITERATURE REVIEW A. THRU LINES
The thru line is our most basic waveguide component.Terahertz (THz) commercial-off-the-shelf (COTS) MPRWG thru lines are normally manufactured using precision-machined processes.The submillimeter-wave seamless (single-block) waveguide product range from Flann Microwave Ltd. includes low loss COTS thru lines operating up to 1.1 THz [16].The nominal insertion loss, measured at the lower band edge frequency, is shown in Table 1.Either copper or nickel is electroformed (formative) to create the MPRWG, having a quoted internal mean profile surface roughness of R a < 0.2 µm [16].Information on the loss modeling of these commercial thru lines has been previously reported [11].
An exhaustive literature survey of THz 3-D printed MPRWG thru lines has been recently undertaken [11].To the best of our knowledge, Table 1 summarizes all the MPRWG thru lines that have been reported in the open literature.Table 1 is an updated version of our earlier survey [11].Only seven examples of 3-D printed thru lines operate at WR-3 (220 to 325 GHz) and higher frequency bands; all previously reported within the last decade (since 2014).Here, α ′ D and RL refer to the average dissipative attenuation per unit length and the worst-case return loss (RL) across the associated waveguide band, respectively.For the WR-3 band, there are two examples using polymer-based printing (i.e., SLA [17] and RECILS [18]) and three examples using expensive metal-based solutions (i.e., micro metal additive manufacturing (M-MAM) [19] and SLM [20], [21]).Note that, even though M-MAM does not involve 3-D printing (being formative), [19] is included in Table 1 for comparison against the more relevant additive manufacturing technologies.
In the WR-3 band, it can be seen that polymer-based thru lines show relatively good performance, with band-average dissipative attenuation being 13 dB/m [17] and 36 dB/m [18]; the COTS thru line has a worst-case value of 31 dB/m [16].In contrast, metal-based 3-D printed thru lines exhibit a higher band-average α ′ D > 90 dB/m [19], [20], [21].This is mainly due to the low effective bulk DC conductivity (dictated by the intrinsic conductivity of the metal powder and extrinsic surface roughness).
For the WR-2.2 band of interest, all three examples of 3-D printed thru lines exhibit high dissipative attenuation.Our earlier work resulted in a band-average α ′ D of 440 dB/m, using a PolyJet printer [12].It will be shown that our new MSLA printer can achieve 157 dB/m.By comparison, the COTS thru line has a worst-case value of 61 dB/m [16], while the bandaverage α ′ D is 199 dB/m for metal-based printing [19].

B. BANDPASS FILTERS
We have previously reported a detailed literature survey of sub-THz 3-D printed waveguide bandpass filters [11].At these frequencies, the range of all available manufacturing technologies is limited.The small internal dimensions associated with BPFs reach the limits for most low-cost manufacturing technologies.Moreover, the thin coupling irises are delicate.As a result, only a few examples of sub-THz MPRWG BPFs have been reported in open literature [2], [5], [10], [11], [14].In general, these filters show more significant center frequency shifting and changes in bandwidth, when compared to those operating at lowermillimeter-wave frequencies (30 to ca. 100 GHz).This highlights the significant challenge for (sub-)THz frequencies, due to the relatively poor manufacturing accuracy associated with today's low-cost 3-D printers.
An exhaustive literature survey has been recently undertaken for terahertz 3-D printed MPRWG bandpass filters.To the best of our knowledge, Table 2 summarizes all the MPRWG bandpass filters that have been reported in the open literature.Table 2 represents a frequency extension to the version in our earlier survey [11].Only two 3-D printed THz MPRWG BPF examples have been previously reported in open literature [3], [4]; both manufactured using metal-based printers (i.e., MLS [3] and 3-D screen printing [4]).Here, f c and 3 dB FBW represent the measured center frequency and 3 dB fractional bandwidth; IL is the measured minimum insertion loss; Q L corresponds to the measured loaded quality (Q-)factor defined by the 3 dB bandwidth.
With the 297 GHz filter, the metal powder is stainless steel (SS), which is subsequently electroless-plated with a 3 µm-thick layer of gold, to improve its effective conductivity.The in-band insertion loss varies between 1.1 dB and 2.7 dB, with a 1% frequency downshift, and return loss better than 10 dB.With the 314 GHz filter, this component is sliced into 4 layers, each individually screen printed, and then assembled using the precision alignment holes and additional dowel pins.The minimum insertion loss is 3.0 dB, with a 4.7% frequency upshift, and return loss better than 11 dB.The poor insertion loss is attributed to the tungsten-copper (W-Cu) paste used in the screen printing process, which exhibits a relatively very low effective conductivity, while the large frequency upshift is attributed to shrinkage.
Unlike these metal-based 5 th order Chebyshev 300 GHz filters, it will be shown that our polymer-based 3 rd order Butterworth 403 GHz filter has an insertion loss of only 1.0 dB, with a 0.25% frequency downshift, and return loss better than 16 dB.

III. DESIGNS A. THRU LINES
In this work, all the 3-D printed thru lines, resonators and bandpass filters are implemented in standard WR-2.2metalpipe rectangular waveguide, having internal cross-sectional dimensions a × b of 570 µm × 285 µm [22], and standard compatible flanges [23].All simulations are undertaken using Ansys High-Frequency Structure Simulator (HFSS) full-wave electromagnetic modeling software.The initial simulations for the thru line have no rounding, perfectly smooth walls and a textbook value of bulk DC conductivity σ 0 = 5.8×10 7 S/m, which is also the default value for copper in HFSS.Our 'trough-and-lid' split-block design is chosen to ensure that the plating can provide sufficient metallization at the corners.Conventional symmetric E-and H-plane split-block designs have been previously found to cause misalignment between the two parts [5].The effects caused by misalignment with 3-D printed MPRWG components are negligible at low frequencies and, therefore, such designs are widely adopted.However, adverse effects become significant at (sub-)THz frequencies.For this reason, an unorthodox H-plane a-edge split is employed here, which avoids misalignment and allows for easier removal of resin residues and visual inspection [11].As shown in Fig. 1, our split-block design comprises a flat lid top part and trough bottom part.The flat lid allows for slight misalignments in its plane; while two protrusive lips above the trough ensure a good transverse current path between the top and bottom parts, which mitigate against electromagnetic (EM) energy leakage.
Our previous 'trough-and-lid' design suffers from rectangular-to-trapezoidal waveguide cross-sectional deformation at the apertures, during initial assembly and flangeto-flange test mating [11].As a result, a design iteration was required here to ruggedize the lips.The new lip height H = 100 µm (cf., H = 200 µm in our previous work [11]).The previous work for G-band has 550 µm wide lips.However, with the smaller WR-2.2 band aperture, the internal lip width is now W 1 = 300 µm, which broadens out to W 2 = 500 µm at the aperture.This provides enhanced mechanical strength when the top and bottom parts are assembled and flange-to-flange test mating is undertaken.These empirical lip values seem to be (near-)optimal for the WR-2.2 band; with all values reduced for higher frequency bands.

B. SINGLE-CAVITY RESONATORS
In order to evaluate the adverse effects from manufacturing defects for MPRWG filters, single-cavity resonators are investigated.To this end, the 399 GHz cavity resonators at the center of our 3 rd order Butterworth and Chebyshev 403 GHz filters are chosen; having a 3 dB FBW of ∼14% (56 GHz) and ∼13% (54 GHz), respectively.Filter design details are given in Subsection III-C. Figure 2 illustrates the internal variable dimensions for an ideal (no rounding) MPRWG single-cavity resonator.
L FEED represents the feed length to/from the input and output ports; L R is the cavity resonator length; W R1 and W R2 correspond to the iris gap widths.With MPRWG filters, conventional diaphragm irises [11] require extremely-high dimensional tolerance manufacturing technologies and with high strength materials -not inherently compatible with lowcost polymer-based 3-D printing.In this work, all resonators are inductively coupled with transverse offset waveguides, while their width dimension has a constant value a [10].
As given in Subsection III-C, the corresponding center resonator design dimensions for the ideal (no rounding) Butterworth and Chebyshev filters are listed in Table 3. TABLE 3. 399 GHz center resonator design (HFSS simulated) dimensions associated with the ideal (no rounding) and CRC (rounding) Butterworth and Chebyshev 403 GHz filters (with smooth walls and σ 0 = 5.8 × 10 7 S/m).
Iris corner rounding, due to manufacturing imperfections (e.g. with 3-D printing [11]), has a significant effect on the resonance frequency and bandwidth for a single resonator.As a result, it may be necessary to apply appropriate compensation during the design phase in order to mitigate against this effect [11]. Figure 3 shows a plan view microphotograph of the bottom part for single-cavity resonator (after copper plating), showing significant iris corner rounding.Here, R in = 75 µm is the measured average inner corner rounding radius and R out = 34 µm is the measured average outer corner rounding radius.While the former is numerically larger, the latter has the dominant effect (on the coupling coefficients for the resonator).These values are subsequently used in the following corner rounding compensation process.Figure 4 shows the CRC simulation results (with corner rounding, smooth walls and σ 0 = 5.8×10 7 S/m) for the center resonators associated with the Butterworth and Chebyshev filters.The yellow solid line, red solid line, and blue dashed line represent the ideal single-cavity resonator without rounding, rounded before compensation and rounded after compensation, respectively.The yellow and blue arrows indicate frequency shifting of the transmission responses due to detuning and its compensation, respectively.It can be seen that the level of corner rounding, with our MSLA 3-D printer, causes a 1.3% frequency downshift and a 35% increase in the 3 dB bandwidth, due to an increase in both the cavity's effective electrical length and coupling coefficients.By finetuning the values of L R , W R1 and W R2 in HFSS (using our pre-determined measured values for R in and R out ), the compensated rounded resonator shows a very close-fit to the ideal non-rounded case.
Table 3 also gives the center resonator design dimensions for the compensated rounded Butterworth and Chebyshev filters (with smooth walls and σ 0 = 5.8 × 10 7 S/m).It can be seen that the designed dimensions for two center resonators are very similar.In practice, considering the 22 µm pixel resolution for our MSLA 3-D printer, these two resonators would be printed with exactly the same dimensions -defined by the level of quantization [10].As a result, only one resonator is needed and chosen as a reference benchmark for investigation.

C. BANDPASS FILTERS
To limit the number of unknowns, when investigating the adverse effects of manufacturing errors on MPRWG filters, we deliberately chose to investigate low (3 rd ) order filters.For the Chebyshev BPF, the worst-case passband return loss level is designed as 25 dB. Figure 5 shows the internal dimensions for an ideal (no rounding) filter.L Fi (i ∈ [1, 2, 3]) represents the cavity length for the i th cavity and W Fi (i ∈ [1, 2, 3, 4]) is the associated iris gap width.Following standard coupling matrix theory [24], with a filter order N = 3, two (N + 2) × (N + 2) = [5 × 5] prototype coupling matrices for our target Butterworth M B and Chebyshev M C filters, respectively, are extracted and given as: HFSS simulations are undertaken to optimize the filter geometry and the final design dimensions for the ideal (no rounding) 3 rd order Butterworth and Chebyshev filters (with smooth walls and σ 0 = 5.8 × 10 7 S/m) are given in Table 4.
For the final filter designs, CRC is applied (with smooth walls and σ 0 = 5.8 × 10 7 S/m), based on the same predetermined values for R in and R out .The corresponding EM simulation results are given in Fig. 6.
Table 4 also gives the design dimensions for the compensated rounded 3 rd order Butterworth and Chebyshev filters (with smooth walls and σ 0 = 5.8×10 7 S/m).By comparing the designed dimensions for the resonators (Table 3) and BPFs (Table 4), with and without CRC, with difference ≤ 11 µm, it is clear that the 22 µm pixel resolution of our 3-D printer is not capable of meeting the target CRC requirements and, thus, CRC is not applied to our BPFs.However, CRC is applied to the resonator, to highlight the problem of overcompensation.

IV. FABRICATION A. 3-D PRINTING
In this work, all 3-D drawings are undertaken using the computer-aided design (CAD) drawing package Autodesk Fusion 360 [25].The output drawing file, having the standard tessellation language (STL) format, is imported into the slicing software Chitubox [26].Here, the Phrozen Sonic Mini 8K MSLA 3-D printer is used, with a quoted print volume of 165 mm × 72 mm × 180 mm.As stated previously, the pixel resolution on the x-y build plane is quoted to be 22 µm, although this will degrade over time, and the default vertical layer thickness on the z-axis can range from 10 to 300 µm [27].
With our MSLA printer, the Elegoo water washable photopolymer resin (Ceramic Grey) was used; mainly for its high precision, low shrinkage and ease of post-processing.

B. METALIZATION
Visual inspection (under a microscope) of the 3-D printed parts is necessary to remove those with any signs of manufacturing defects.The remaining parts are then copper plated using a commercial process.Here, a thin layer of nickel is first electroless-plated onto the 3-D printed part surfaces; this is then followed by electroplating a 5 µm thick layer of copper.Finally, the plated parts are processed with a proprietary antitarnishing treatment.
Scanning electron microscope (SEM) images for a thru line and a 3 rd order BPF are shown in Figs.7(a After copper plating, the two parts are assembled for testing.Figure 8 shows photographs of an H-plane split-block WR-2.2 3 rd order Chebyshev MPRWG BPF before and after assembly.The black circular holes show where 1.5 mm diameter and 5.0 mm long stainless-steel dowel pins are inserted, to provide alignment.

V. MEASUREMENTS
Scattering (S-)parameter measurements were undertaken within the Department of Materials at Imperial College London, using their Rohde & Schwarz ZVA 67 vector network analyzer and ZVA-Z500 frequency extension heads.The typical measurement setup for 3-D printed MPRWG components is shown in Fig. 9.
For our two-port S-parameter measurements in waveguide, at submillimeter-wave frequencies, the Thru-Reflect-Line (TRL) calibration scheme was adopted [28].This choice offers the advantage of requiring only partial information regarding the calibration standards.Specifically, TRL requires identical Reflect standards at both waveguide test ports, and the Line standard with the correct waveguide aperture dimensions.
Traditionally, the TRL calibration scheme employs a Line standard having a 90 • electrical length (equivalent to a quarter-guided wavelength) at the midband frequency; this allows the transmission phase change to range between approximately 30 • and 150 • over the frequency range of interest within the band.However, at submillimeter-wave frequencies, the use of ¼-wave Line standards becomes problematic, due to their very short physical length (i.e., <250 µm for the WR-2.2 band), leading to mechanical integrity issues.
For this reason, as an alternative approach, our TRL calibration employed ¾-wave Line standards, which allow transmission phase changes to range between 210 • and 330 • over the frequency range of interest within the band.These waveguides are significantly more mechanically robust, due to their increased thickness [29].For this purpose, two Line standards were employed, both manufactured by SWISSto12.The first Line standard has a designed physical length of 951 µm, covering the lower frequency range of the WR-2.2 band (from 325 GHz to 394 GHz); while the second Line standard has a designed physical length of 651 µm, covering the higher frequency range (from 380 GHz to 500 GHz).
A flush short, also manufactured by SWISSto12, was used as the Reflect standard for both of the waveguide test ports.The calibration standards employed for these measurements are shown in Fig. 10.The S-parameter measurement results created using both ¾-wave Line standards were combined using the weighting scheme [29], to achieve a full-band calibration.

A. THRU LINE
Figure 11 shows the simulated, re-simulated and measured S-parameter results for the 5 mm-long thru line.As can be seen in Fig. 11(a), the measured return loss is better than 13.3 dB across the WR-2.2 band.The worst-case measured insertion loss is 1.2 dB at the lower band-edge frequency of 325 GHz, 0.8 dB at 400 GHz and 0.4 dB at the upper bandedge frequency of 500 GHz.The average insertion loss across the whole band is 0.9 dB.
According to initial HFSS simulations, where the default value of effective conductivity is used, there is no significant effect caused by ceiling rounding, with insertion loss degraded by less than 0.1 dB, which is well within experimental error at these frequencies.However, corner rounding will cause filter center frequency shifting and increased bandwidth, but without introducing significant insertion loss.
In all HFSS re-simulations, both ceiling and corner rounding are included to represent our practical scenario.The physical dimensions associated with our thru line, single-cavity resonators and filters are measured individually, and these dimensions are then incorporated into HFSS re-simulations.In addition, as will be discussed later in this Subsection, the effective conductivity is reduced from the default value to 0.36 × 10 7 S/m.From Fig. 11(a), the re-simulated S-parameter responses for the thru line match very closely with the measurements.
Assuming a perfect wave impedance match at both test ports (which is not a bad approximation, given the good measured return loss performance), the ideal transmittance |S 21 | 2 and its corresponding insertion loss IL| Ideal for an airfilled MPRWG thru line are given as: where, S 21 , α c , β = 2π/λ g , λ g , k 0 , η 0 and R S are the respective forward voltage-wave transmission coefficient, attenuation constant associated with the smooth conductor [30], phase constant, guided wavelength, modified wavenumber in free space, intrinsic impedance of free space and surface impedance of the conductor.The total power attenuation α T = α M + Lα ′ D = −10 log 10 |S 21 | 2 is equal to insertion loss, where, α M represents the contribution to the total attenuation due to the wave impedance mismatch reflection at the input port and α ′ D represents the dissipative attenuation per unit length due to ohmic losses and any leakage radiation [5], [11], [31]: where, S 11 is the input voltage-wave reflection coefficient.With general surface roughness models, roughness coefficient K is used to normalize either the power dissipated P DR or dissipative attenuation α ′ DR for rough conductors to those for smooth conductors (P DS and α ′ DS ), with [11] and [32]: In this work, two surface roughness models (i.e., the Extended-and Huray-Hemispherical models [11]) are used in the HFSS re-simulations.According to the SEM image in Fig. 7(d), the measured average hemispherical radius r base and separation distance between adjacent protrusions d peaks are 4.7 µm and 14.8 µm, respectively; this is commensurate with our respective previous measured values of 3.7 µm and 17 µm [11].The calculated roughness coefficients at 400 GHz are K (400GHz) = 1.66 and 1.62 for the Extended-and Huray-Hemispherical models, with the associated effective conductivity of 2.1 × 10 7 S/m and 2.2 × 10 7 S/m, respectively.As expected, these are greater than those of K (180GHz) = 1.32 and 1.28, previously found at G-band [11].
The HFSS re-simulated results that include the two surface roughness models are shown in Fig. 11(b), as solid and dashed green lines.It can be seen that there is a large discrepancy between the re-simulated and measured results.It is believed that, in addition to surface roughness, there may be a loss contribution due to contact resistance between the two parts of our H-plane a-edge split-block assembly.However, the anti-tarnishing coating applied to the copper surface is the dominant contributor to the measured dissipative attenuation (which further lowers the effective conductivity) at terahertz frequencies; this coating is not included in either of the surface roughness models.Figure 12 shows the simulated, re-simulated and measured dissipative attenuation for the thru line.For WR-2.2, our measured average dissipative attenuation is 157 dB/m and 0.16 dB/λ g .

B. SINGLE-CAVITY RESONATORS
Figures 13(a) and 13(b) show the simulated, re-simulated and measured S-parameter responses for the 3-D printed single-cavity resonators without and with CRC, respectively.In Fig. 13(a), the measured resonance frequency f 0c for the resonator without CRC is 390.0GHz, with 2.3% (9.2 GHz) downshift from the ideal simulated resonance frequency f 0cn = 399.2GHz.The measured 3 dB bandwidth f 3dB = 27.1 GHz, having an increase of 66.3% (10.8 GHz) from the ideal simulation of 16.3 GHz.
As discussed in Section III, corner rounding can cause resonance frequency downshifting and an increase in bandwidth.Therefore, the discrepancies found between the original simulated and measured results are mainly due to corner rounding.As shown in Fig. 13(b), the measured resonance frequency for the resonator with CRC is 407.9GHz, with 2.2% (8.7 GHz) upshift from the ideal simulated resonance frequency f 0cn = 399.2GHz.The measured 3 dB bandwidth f 3dB = 18.5 GHz, having an increase of 13.5% (2.2 GHz) from the ideal simulation of 16.3 GHz.Here, there is a significant reduction in the bandwidth error with the use of CRC.The upshift in resonance frequency is mainly due to Chitubox automatically reducing the width of the resonator by one pixel for our Phrozen 3-D printer, having a 22 µm pixel resolution, when compared to the required value ( ∼ 10 µm, given in Table 3) resulting in overcompensation in this case.
As we demonstrated previously [11], emulated timedomain reflectometry (TDR) is a useful tool.TDR measurements for the 5 mm-long thru line and single-cavity resonator, without CRC, have been undertaken and the results are shown in Fig. 14.The physical distance d between any two points on a TDR trace is given by [11] and [33]: where, v g is the group velocity of the MPRWG and t represents the round-trip time difference between any two points on the trace.For our thru line, the measured t between blue peaks #1 and #2 is 45.7 ps.With group velocity v g ∼ v g (400 GHz) ≃ 2.26×10 8 m/s, this corresponds to the waveguide flange-to-flange physical length d = 5,164 µm, with only 3.3% error, when compared to the design value for L = 5,000 µm.For the single-cavity resonator, the measured t between red peaks #1 and #2 is 20.6 ps.This corresponds to the waveguide feed length d = 2,328 µm, with only 1.4% error, which matches very closely with our design value for L FEED = 2,295 µm.
Figure 15 shows the RLC equivalent circuit model for any undriven resonator (i.e., uncoupled to any source or load impedances).With a lossy scenario, the associated complex natural angular (or eigen)frequency is related to the resonator unloaded Q-factor Q U (ω 0 ) as follows [34]: where, is the driven resonance frequency; ω ′ 0 is the damped (or undriven) natural angular resonance frequency; and ω ′′ 0 is the field (amplitude) decay rate or Napier frequency.
For any air-filled cavity resonator, the exact RLC parameter values can be extracted from [34]:  where, c = 1/ √ ε 0 µ 0 , ε 0 and µ 0 are the speed of light, permittivity and permeability, all in free space; V is the volume of the cavity resonator -for example, V = 0.06677 mm 3 for the center resonator of the ideal simulated Butterworth filter (V = 0.06742 mm 3 for the Chebyshev filter), using the values from Table 3.
Given that the cavity resonator is EM-coupled to both a source and load, both ideally having the same wave impedance (defined by the MPRWG), the new driven resonance frequency f 0c < f 0 .At this point, it is important to state that the unloaded . The extracted loaded quality factor Q L (f 0c ) and unloaded quality factor Q U (f 0c ) for the resonator are given as [5], [24], and [30]: Using ( 11)-( 15), the associated RLC elements for the cavity resonators can be extracted from measurements of f 0c , f 3dB and |S 21 (f 0c )|, with the results shown in Table 5.
Here, the ideal simulated cavity resonator represents the reference benchmark (with no rounding, smooth walls and σ 0 = 5.8 × 10 7 S/m), and all these parameters are depicted with subscript n.HFSS frequency-domain simulations give values for f 0cn = 399.2GHz, f 3dBn = 16.3GHz and, |S 21n (f 0cn )| = 0.97; giving Q Ln (f ocn ) = 24.5 using ( 14), and Q Un (f 0cn ) = 863.2using (15).Using the linear relationship When compared to the reference benchmark (ideal simulations), the measured cavity resonator without CRC exhibits a significant decrease of 41% in Q L (f 0c ).This is due to resonance frequency downshifting and an increase in its bandwidth (mainly caused by corner rounding).After applying CRC, there is an obvious improvement in Q L (f 0c ), with a decrease of only 10% from the ideal simulated value.Both resonators (without and with CRC) suffer from relatively low values of Q U (f 0 ), mainly due to the high resistance R that is attributed to the low effective conductivity.Moreover, when compared to the reference benchmark, both resonators exhibit ∼17% decrease in the inductance L and ∼20% increase in capacitance C; with negligible net change in f 0c and f 0 .

C. BANDPASS FILTERS
Figure 16 shows the simulated, re-simulated and measured S-parameter responses for the 3-D printed 3 rd order Butterworth and Chebyshev BPFs.Note that both MPRWG filters are designed without CRC; as previously stated, the resolution of our 3-D printer is not capable of meeting the target CRC requirements.
As shown in Fig. 16(a), the measured and re-simulated S-parameter responses for the 3 rd order Butterworth BPF do not attain the typical Butterworth approximation.This is mainly due to resonator detuning, caused by dimensional inaccuracies during manufacturing.The measured center frequency for this filter is 406.1 GHz, with its measured worstcase return loss of 16.6 dB and 3 dB bandwidth of 64.2 GHz.At center frequency, the measured insertion losses for the 5 mm filter and 5 mm reference thru line are 0.95 dB and 0.68 dB, respectively, having a discrepancy of only 0.27 dB at 406.1 GHz.When compared to the HFSS simulations, with ideal design dimensions given in Table 4, there is a slight center frequency upshift of 0.8% (3.4 GHz) and an increase of 14.2% (8 GHz) in 3 dB bandwidth, with a decrease in loaded quality factor from the simulated value of 7.2 to the measured value of 6.3.
As shown in Fig. 16(b), the measured and re-simulated S-parameter responses for the 3 rd order Chebyshev BPF attain the typical characteristic return loss zero distribution.The measured center frequency for this filter is 398.6 GHz, with its measured worst-case return loss of 10.4 dB and 3 dB bandwidth of 65.2 GHz.At center frequency, the measured insertion losses for the 5 mm filter and 5 mm reference thru line are 1.02 dB and 0.83 dB, respectively, having a discrepancy of only 0.19 dB at 398.6 GHz.When compared to the HFSS simulations, with ideal design dimensions given in Table 4, there is a slight center frequency downshift of 1.2% (4.8 GHz) and an increase of 20.7% (11.2 GHz) in 3 dB bandwidth, with a decrease in loaded quality factor from the simulated value of 7.5 to the measured value of 6.1.
With both filters, the increase in 3 dB bandwidth is mainly due to corner rounding and dimensional errors increasing the iris gap widths; both increasing the coupling coefficients associated with the cavity resonators.When compared to the ideal design simulation responses, the additional measured insertion loss is mainly attributed to the surface roughness and the anti-tarnishing coating on the copper surface, which reduce the total effective conductivity.When compared to the Chebyshev counterpart, it is clear that the Butterworth filter is more robust against manufacturing errors, in terms  of absolute frequency shift, insertion loss and worst-case return loss.
Emulated TDR measurements for the 5 mm-long 3-D printed Butterworth and Chebyshev filters have been undertaken and the results are shown in Fig. 17.For the Butterworth filter, the measured t between blue peaks #1 and #2 is 16.6 ps.This corresponds to the waveguide feed length d = 1,876 µm, with only 3.1% error, which matches closely with our design value for L FEED = 1,937 µm.For the Chebyshev filter, the measured t between red peaks #1 and #2 is 17.3 ps.This corresponds to the waveguide feed length d = 1,955 µm, with only 1.7% error, which matches very closely with our design value for L FEED = 1,923 µm.
The Chebyshev filter exhibits better defined reflections, when compared to the Butterworth filter.With reference to the blue and red #1 peaks, the weaker reflection demonstrated by the Butterworth filter is indicative of a better input impedance match, when compared to the Chebyshev filter.With reference to blue and red #2 peaks, the stronger reflection demonstrated by the Chebyshev filter is indicative of a narrower iris gap width, associated with a lower coupling coefficient from the feed into the first resonator, when compared to the Butterworth filter.

VI. DISCUSSION AND CONCLUSION
This paper presents the state-of-the-art in polymer-based 3-D printing of metal-pipe rectangular waveguides with the first reported terahertz filters.
Unlike metal-based 3-D printing, polymer-based 3-D printing suffers from relatively poor mechanical tolerances, which can result in rectangular-to-trapezoidal waveguide cross-sectional deformation at the apertures; being more pronounced at high frequencies.To this end, a design iteration for our 'trough-and-lid' solution was implemented.The more ruggedized lips are (near-)optimal for the WR-2.2 band, but will need to be redesigned for higher frequency bands.
Our 3-D printed thru line has the lowest reported dissipative attenuation of 157 dB/m, when compared to other 3-D printed thru lines at WR-2.2 band, with a measured worst-case return loss of 13.3 dB across the waveguide band.In addition, emulated TDR measurements have been investigated, and this shows negligible physical aberrations (e.g., aperture deformations and re-orientations) at the testport flange interfaces, insignificant surface defects within the waveguides and no obvious radiation leakage (indicated by excess losses and spurious resonances) from the split block.
Polymer-based 3-D printed waveguide bandpass filters have been demonstrated at terahertz frequencies.The measured insertion losses for these 3 rd order Butterworth and Chebyshev filters are only 1 dB at their center frequencies, with the worst-case return losses of 16.6 dB and 10.4 dB, respectively.This shows that the Butterworth filter is more robust against manufacturing errors than its Chebyshev counterpart.
With the single-cavity resonators, corner rounding compensation results in excess frequency shifting of the resonance frequency, indicating that the 22 µm pixel resolution of our Phrozen MSLA 3-D printer is not sufficient for the dimensional tolerance needed for the WR-2.2 band.For this reason, neither quantization predistortion [10] or corner rounding compensation [11] are employed here for our filters.
With quantization predistortion, dimensional measurement errors dominate when they become commensurate with pixel resolution, and so it is only effective with lower pixel resolution 3-D printers.However, with corner rounding compensation, higher pixel resolution is required (e.g., <10 µm for the WR-2.2 band).The Elegoo Mars 4 has recently come onto the market as a low-cost (currently retailing at $259) 9K 3-D printer, having an 18 µm pixel resolution on the x-y build plane [35].Using this 9K printer, it is believed that CRC can be applied to resonators and filters (without causing significant overcompensation) in the WR-3 band and lower frequencies.As pixel sizes continue to shrink, it is believed that THz waveguide components, employing CRC, can be 3-D printed in the near future.
Our latest work opens-up new opportunities for applications where high performance and low manufacturing cost are the main drivers (e.g., future 5G+ mobile communications, radar and imaging systems), which could one day compete with traditional machining technologies.

TABLE 1 .
Terahertz 3-D printed waveguide thru lines.( * Nominal worst-case insertion loss found at the lower band-edge frequency for COTS lines).

FIGURE 1 .
FIGURE 1. Ideal (no rounding) illustrations for our H-plane a-edge split-block WR-2.2 MPRWG thru line: (a) complete flange view; (b) close-in flange view; and (c) plan view of bottom trough.

FIGURE 2 .
FIGURE 2. Plan view illustration showing internal variable dimensions for an ideal (no rounding) MPRWG single-cavity resonator.

FIGURE 3 .
FIGURE 3. Plan view microphotograph of the bottom part for a single-cavity resonator, MSLA 3-D printed with a pixel resolution of 22 µm, showing iris corner rounding (after copper plating).

FIGURE 4 .
FIGURE 4. CRC simulations (with corner rounding, smooth walls and σ 0 = 5.8×10 7 S/m) for the single-cavity resonators, with yellow and blue arrows indicating frequency shifting due to detuning and its compensation, respectively, with the center resonators for the: (a) Butterworth filter; and (b) Chebyshev filter.

FIGURE 5 .
FIGURE 5. Plan view illustration showing internal variable dimensions for an ideal (no rounding) 3 rd order MPRWG bandpass filter.
) and 7(b), respectively.Figure 7(c) shows the close-in flange view of the bottom part of the BPF.The ceiling rounding radius has a measured average value R c = 34 µm.A close-in plan view of the bottom trough surface roughness is shown in Fig. 7(d).

FIGURE 7 .
FIGURE 7. SEM images for 3-D printed MPRWG components (after copper plating): (a) thru line; (b) 3 rd order Chebyshev filter; (c) close-in flange view of the bottom part of the filter; and (d) close-in plan view of the bottom trough surface.

FIGURE 15 .TABLE 5 .
FIGURE 15.RLC equivalent circuit model for an undriven resonator that is uncoupled to any source or load.

FIGURE 17 .
FIGURE 17. Emulated TDR measurement results for the 3-D printed Butterworth and Chebyshev 403 GHz filters.