Parasitic High Q-Factor Open-Box Modes With 3-D Printed Dielectric-Filled Metal Waveguides

High Q-factor open-box mode resonances have been found in the microwave measurements of several 3-D printed dielectric-filled metal-pipe rectangular waveguides (MPRWGs). These parasitic Fabry-Pérot eigenmodes are confined by the conductive walls in the transverse plane of the MPRWG and partially confined by the air-dielectric and dielectric-air boundaries in the longitudinal direction. The excitation of open-box modes was previously speculated to be due to the inhomogeneous and/or anisotropic nature of the 3-D printed dielectric-fillers. This has now been confirmed, by representing the inhomogeneous and anisotropic nature of the woodpile-like dielectric structure (physical realm), with an anisotropic dielectric constant tensor (simulation realm). Analytical and numerical eigenmode solvers, previously used by the authors with MPRWGs, are applied here to parallel-plate waveguides (PPWGs) and circular waveguides (CWGs); identifying all the individual parasitic open-box modes. With the former, its TM11 mode exhibits an ultra-high Q-factor of approximately 2,300 at X-band, which is considerably higher than those found with other modes and in other waveguide structures. Finally, a numerical full-wave frequency-domain simulator that employs the dielectric constant tensor is introduced in this paper. This new modeling technique independently confirms that open-box modes are excited in 3-D printed dielectric-filled MPRWG, PPWG and CWG structures. This paper provides the foundations for accurately modeling parasitic resonances associated with inhomogeneities and anisotropy in 3-D printed microwave components; not just the metal-walled waveguide structures considered here, but the methodology could also be extended to generic 3-D printed dielectric waveguides and substrate-based transmission lines.


I. INTRODUCTION
Over the past decade, 3-D printing of microwave components has gained increasing popularity, due to the availability of affordable 3-D printers that provide a cost-effective solution for prototyping microwave components [1]. This paper will only focus on the X-band (8 to 12 GHz) and Ku-band (12 to 18 GHz) parts of the microwave spectrum. 3-D printed air-filled metal-pipe rectangular waveguides (MPRWGs) were first demonstrated by D' Auria et al. in 2015 [2], operating at X-band and Wband (75 to 110 GHz). The following year, a 3-D printed mechanically-tunable X-band MPRWG phase shifter was reported by Gillatt et al., employing a moveable dielectric surrounded by conductive walls. With a MPRWG, the openbox modes are confined by the conductive walls in the transverse plane and partially confined by the air-dielectric and dielectric-air boundaries in the longitudinal direction.
When measuring a dielectric-filled MPRWG between a pair of traditional air-filled MPRWGs, associated with the vector network analyzer's (VNA's) test-set heads, the driven Fabry-Pérot resonance will be excited. For the MPRWG, the dominant transverse electric (TE) mode of propagation is the TE 10 mode. In addition to the expected spectral dips in |S 11 | and peaks in |S 21 |, associated with the driven Fabry-Pérot resonance, the measured scattering (S-)parameters show anomalous notches in |S 11 | and |S 21 | at fixed frequencies. The authors previously referred to these resonances as 'open-box' modes [5], [6]; they may also be referred to as 'parasitic Fabry-Pérot eigenmodes'. Excitation of parasitic open-box modes in 3-D printed woodpile-like cuboids were previously attributed to their inhomogeneous and/or anisotropic nature [5], [6].
High Q-factor open-box mode resonances have been found in microwave measurements of several 3-D printed dielectric-filled MPRWGs [5]- [9]. With conventional a priori frequency-domain modeling, these parasitic Fabry-Pérot eigenmodes cannot be predicted using analytical technique and may be difficult to predict using 'to-scale' numerical full-wave electromagnetic simulations; although an analytical or numerical eigenmode solver can be used.
In 2015, using an ultraviolet (UV)-cured ink-jet type 3-D printer, the U.S. Airforce Institute of Technology investigated both solid (homogenous) and periodic (inhomogeneous) polytetrafluoroethylene (PTFE) cuboids; the latter introduces periodic airgaps (rectangular inclusions) to create a biaxial anisotropic dielectric constant [7]. With this biaxial anisotropic material, X-band measurements of a dielectricfilled WR-90 MPRWG clearly show an open-box mode resonance at 10.4 GHz (not mentioned in their report). With their quoted value of effective dielectric constant of 2.1, using our analytical eigenmode solver, we found this to be the TE 201 open-box mode predicted at 10.45 GHz. Interestingly, using rectangular-to-square waveguide transitions, cubic sample measurements did not appear to show any open-box mode resonances [7].
Isakov et al. [8] demonstrated 3-D printed metamaterials by deliberately introducing inhomogeneity and anisotropy; either ABS or polypropylene (PP) was used for the low dielectric constant regions, while a mixed inorganic ceramic powder (BaTiO 3 ) and polymer (ABS) composite was used for high dielectric constant regions. Both striped and checkered cuboids (2 mm thick samples) were 3-D printed and inserted into Ku-band MPRWGs. With eight interleaving horizonal strips (having alternating dielectric constant values of 2.6 and 7.0), using the COMSOL Multiphysics R frequency-domain simulator, resonances at 15.19 GHz and 19.54 GHz are observed. The Oxford group attribute these resonances with the ''first TE 11 mode and second TE 12 mode of the Mie-resonances (magnetic and electric, respectively)''. Ku-band measurements revealed a very weak resonance at 12.7 GHz and a strong resonance at 15.9 GHz. However, by observing their simulated field plots [8] and using both our analytical and numerical eigenmode solvers, we found that these resonances are actually TE 111 and transverse magnetic (TM) TM 111 open-box modes. Moreover, the weaker resonance at 12.7 GHz, only observed in measurements, can be attributed to the TE 201 open-box mode. The subtle difference in mode interpretation is that Mie resonances are associated with unbound objects, while open-box modes are partially constrained by conducting sidewalls.
Yang et al. [9] reported the characterization of 3-D printed polylactic acid (PLA) cuboid samples at X-band. Dielectricfilled MPRWG measurements, having different filler lengths, indicate the presence of parasitic open-box mode resonances (not mentioned in their paper).
Parasitic Fabry-Pérot eigenmodes were previously identified and characterized in MPRWGs, having 3-D printed ABS dielectric fillers; at X-band by Sun et al. [5] and Ku-band by Márquez-Segura et al. [6]. In both cases, the ABS dielectric fillers were 3-D printed using fused deposition modeling (FDM). Interestingly, open-box modes were not found with conductive PLA cuboids, as they are heavily damped by the presence of carbon black powder [6].
These open-box modes were investigated using both analytical and numerical eigenmode solvers, but our previous analyses were restricted to just MPRWGs. Moreover, we used an irregular coarse tetrahedral meshing scheme within COM-SOL Multiphysics R frequency-domain simulator to excite the open-box modes; representing an artificial form of inhomogeneity and/or anisotropy (to demonstrate proof of concept). However, this coarse meshing scheme (simulation realm) is not a reliable way of accurately modeling an actual waveguide structure (physical realm).
In order to investigate the propagating and non-propagating modes associated with transmission lines and guided-wave structures, various analytical and numerical methods can be used. With the former, the variational method is very accurate above and below the cut-off frequency; while the more simplified textbook power-loss approximations can also be used for frequency-domain simulations above the cut-off frequency to good effect [5]. Moreover, analytical equations can be derived to give complex eigenfrequencies. With the latter, finite-element methods can be employed with both numerical full-wave frequency-domain simulators and eigenmode solvers. With numerical frequency-domain simulators, non-propagating modes can only be inferred from its spectral response; while eigenmode solvers give direct solutions as complex eigenfrequencies, even if the dielectric is represented as being homogenous and isotropic [5]. The COMSOL Multiphysics R RF Module has both a finite-element based numerical full-wave frequency-domain simulator and eigenmode solver [10]; the former provides a complex steady-state response for each swept frequency point, while the latter gives the complex eigenfrequencies associated with all possible eigenmodes for the structure.
In this paper, we introduce a reliable method for accurately predicting the excitation of open-box modes in MPRWGs by introducing controlled inhomogeneity and anisotropy in the dielectric filler, using the COMSOL Multiphysics R frequency-domain simulator. In addition, the excitation of open-box modes in dielectric-filled parallelplate waveguides (PPWGs) and circular waveguides (CWGs) are investigated. Finally, we demonstrate the accuracy of our predictions using both analytical and numerical eigenmode solvers.
Our previously derived analytical expressions for MPRWGs (Appendix B in [5]) have been adapted to also include PPWGs and CWGs. These are then used to verify the numerical results from COMSOL Multiphysics R . The generic analytical derivations, given in the Appendix, provide continuity between the different waveguide structures and completeness by including both TE and TM modes. Moreover, perfect electric conductor (PEC) walls are assumed throughout, to eliminate uncertainty in the extracted mode quality (Q-)factor, associated with the simulated boundary conditions used to approximate real metal walls. Also, throughout this paper, our analytical methods adopt the simplified textbook power-loss approximations.

II. PROPAGATION CONSTANT IN GUIDED MEDIA
In any medium (unbound or guided), the generic complex propagation constant is given by [5], where, α is the attenuation constant, β is the phase constant and k is the complex angular wavenumber. With a waveguide, a cut-off propagation constant can be defined as γ c = jk c , where k c ≡ β c is the cut-off angular wavenumber (real value); β c = ω c /v p is the phase constant associated with angular cut-off frequency ω c = 2π f c and phase velocity v p = c/Re{ñ} for a dielectric-filled waveguide, f c = k c c/2πRe{ñ} is the cut-off frequency; c = 1/ √ µ 0 ε 0 is the speed of light in free space, where µ 0 and ε 0 are the permeability and permittivity of free space, respectively; andñ is the complex refractive index for an unbounded dielectric medium. For a waveguide having PEC walls, operating in the TE mn or TM mn mode, with propagation along the longitudinal z-direction, the general textbook expression for guided-wave propagation constant γ g is, All variables have their usual meaning, where the subscripts m and n are the mode indices denoting the number of half-wavelength variations of the fields in the transverse plane; γ cmn is the cut-off propagation constant in the waveguide; γ ud is the propagation constant in an unbound dielectric medium with γ ud = jk ud = jω μ effεeff = ω μ reffεreff /c, where ω = 2πf is the angular frequency, f is the frequency; µ reff = µ reff −jµ reff (the dielectric is assumed non-magnetic FIGURE 1. Illustration of an ideal dielectric-filled metal-pipe rectangular waveguide having internal dimensions a and b. The length of the dielectric cuboid and air-filled sections are l and l air , respectively. In any section, As and Bs are the incident and reflected waves, respectively. This figure is redrawn from Sun et al. [5]. in this paper) andε reff = ε reff − jε reff are the effective complex relative permeability and permittivity, respectively, for the unbound dielectric; loss tangent is defined as tanδ = Im ñ 2 /Re ñ 2 , whereñ = μ reffεreff ; ε reff and tanδ e = ε reff /ε reff µ reff =0 are the dielectric constant and loss tangent of the unbound dielectric, respectively, since tanδ = tanδ e with µ reff = 0.

III. METAL-PIPE RECTANGULAR WAVEGUIDES A. BACKGROUND
With both the TE mn and TM mn modes for a MPRWG, the textbook cut-off propagation constant is, and, were, m and n correspond to the mode numbers in the horizontal x-and vertical y-directions in the transverse plane, respectively; k cmn = (mπ/a) 2 + (nπ/b) 2 is the cut-off angular wavenumber for either an air-or dielectric-filled waveguide; ω cmn = 2πf cmn and f cmn = k cmn v p /2π are the cut-off angular frequency and cut-off frequency, respectively.
Consider the MPRWG structure illustrated in Fig. 1, where a TE mn or TM mn mode is incident on the dielectricfilled section. For a WR-90/WR-62 waveguide operating at X-/Ku-band, in the dominant TE 10 mode, the ideal internal cross-sectional dimensions are a = 22.860/15.799 mm and b = 10.160/7.899 mm; while the length of the dielectricfilled section is l = 9.626/6.676 mm [5]/ [6].
An FDM 3-D printed sample is generally composed of dielectric layers, deposited layer-by-layer; even with a 100% infill printer setting, unintentional airgaps can effectively create a woodpile structure [5], [6]. Figure 2 illustrates layer-bylayer FDM 3-D printing of a dielectric cuboid, showing alternate layers having a 90 • angular offset in the printed strips and adjacent air gaps. As a result, the FDM 3-D printed MPRWG cuboid dielectric filler should be considered inhomogeneous and anisotropic in nature.

B. EXCITATION OF OPEN-BOX MODE RESONANCES WITH FREQUENCY-DOMAIN SIMULATOR
We previously speculated (without demonstrable proof) that the excitation of open-box modes is due to the inhomogeneous and/or anisotropic nature of our dielectric filler within MPRWGs [5], [6]. To-scale numerical electromagnetic modeling of our 3-D woodpile-like structure was not possible, given the limited computational resources available, as extremely fine meshing is required for the very small spatial features [5]. Up-scaling the spatial features or introducing minor defects to the cuboid geometry could not excite parasitic open-box modes associated with our woodpilelike structure. However, we were able to observe open-box mode resonances when an overly coarse meshing scheme was applied with purely homogenous and isotropic dielectric samples; using the COMSOL Multiphysics R frequency-domain simulator. With a finer meshing scheme, the open-box modes could not be observed.
A coarse meshing scheme can only give a qualitative prediction for the spectral locations of open-box modes. Moreover, this approach does not appear to work with other types of waveguides (e.g., parallel-plate and circular). For these reasons, an improved modeling scheme for predicting the excitation of open-box modes in dielectric-filled waveguides is needed.
The following method exploits the real part of the complex effective relative permittivity (dielectric constant) tensor ε reff within the COMSOL Multiphysics R frequencydomain simulator. This tensor represents an anisotropy for the dielectric-filler within any waveguide structure; allowing accurate predictions to be made for the parasitic open-box modes found with conventional microwave VNA measurements.
The generalized dielectric constant tensor can be expressed in (5), as shown at the bottom of the next page, where, ε reffx , ε reffy , ε reffz are bulk dielectric constant values in the respective x, y, z directions, with ε reffx , ε reffy , ε reffz = ε reff given an isotropic dielectric; c uv (u, v ∈ {x, y, z}) are scalar coefficients that represent the level of coupling between different field components due to anisotropy, with c uv = 0 given an isotropic dielectric; and f uv (x, y, z) are scalar functions that represent spatial inhomogeneity, with f uv = 0 (or any constant value) given a homogenous dielectric.
With FDM 3-D printed cuboids having a 100% infill printer setting, we previously demonstrated anisotropy for conducive PLA cuboids [6]. It was shown that the extracted bulk dielectric properties are different when TE 10 mode propagation within the dielectric-filled MPRWG is along build layers and against build layers [6]; for example, with the latter, ε reffx , ε reffy = ε reffz . However, for simplicity, when comparing numerical simulations with analytical modeling, ε reffx , ε reffy , ε reffz = ε reff . Moreover, since the dielectricfilled waveguide is both passive and reciprocal, to preserve its symmetrical S-parameter matrix, a symmetric dielectric constant tensor must be employed (with uv = vu). Note that a truly isotropic dielectric must satisfy all the following conditions: (i) ε reffx , ε reffy , ε reffz = ε reff ; (ii) c uv = 0; and (iii) f uv = 0 (or any constant value).
To ensure an effective bulk value for ε reff , f uv should be chosen to have a zero mean spatial value over the x, y, z directions of the dielectric. For simplicity, f uv is chosen to have the following linear spatial variation, represent the maximum normalized percentage variation in ε reff along x, y and z directions, respectively. For example, a 5% variation (corresponding to a ±2.5% change) in dielectric constant along the x-direction has, For simplicity, and throughout this paper, % can be chosen as a single parameter for the variation in ε reff along all axial directions. Moreover, with a symmetrical network, c uv = c vu and f uv = f vu . Within COMSOL Multiphysics R , using (5), the signs for the terms in (6) are arbitrarily chosen for our simulated MPRWG, as follows, with all the scalar coefficients c uv = 1 and a 2% variation of the dielectric constant in all axial directions. The choice of f uv assumes the origin (x, y, z = 0) is located at one of the corners of the dielectric filler.

C. SIMULATED SPECTRAL RESPONSES
Waveguides usually operate in a single mode. However, with mode conversion, normal ports in electromagnetic simulation software treat hybrid modes as single modes. The driven excitation mode (TE 10 in this case) must be chosen at the input port within the COMSOL Multiphysics R frequency-domain simulator. The S-parameters are calculated at each port from the driven excitation mode and a specified mode (which may be different to the driven mode, but not in this case); using their transverse field profiles (parallel to the port surfaces). Therefore, before simulations begin, the specified mode being analysed must be defined at each port. By default, within COMSOL Multiphysics R , the simulation space is terminated by a port boundary condition, which absorbs only the specified mode at that port; reflecting all other modes. Propagating (i.e., driven and spurious) and nonpropagating spurious (i.e., not driven or specified) modes can be excited within a waveguide having an embedded dielectric-filled section, due to inhomogeneity and/or anisotropy; these unmatched modes will be reflected from the ports -back into the simulation space. Indeed, using our chosen dielectric constant tensor, with (8)-(10), it was found that the ports reflected all non-TE 10 modes. As a result, the complete structure (which includes the two air-filled waveguide sections) acts as a closed-box resonator containing all spurious (including both closed-and open-box) modes. The open-box modes are excited within the central dielectricfilled waveguide section with decaying evanescent fields in the air-filled sections.
For multimode conditions, COMSOL Multiphysics R has the option of defining theoretical 'slit' ports, which mathematically allow all propagating modes to pass through without reflections or absorption. Therefore, a perfectly matched layer (PML) is also needed behind the slit port, to avoid any subsequent reflections, as illustrated in Fig. 3. These slit ports have no interaction with the waveguide fields, but can be used to excite or record fields (e.g., to calculate S-parameters) like normal ports.
All natural resonances are characterized by their complex eigenfrequency [11],f where, f 0 is the damped (undriven) resonance frequency and 2π f 0 is the Napier frequency (or field (amplitude) decay rate or loss rate). Conventional VNA measurements of MPRWGs, having FDM 3-D printed dielectric filler cuboids, can reveal unexpected spectral notches in the S-parameters.
With the VNA's steady-state frequency-domain responses, spectral notches represent the undamped (driven) resonance frequency f 0 = f 0 . For an ideal resonator without losses, Both the respective damped and undamped unloaded Q-factors for a resonance can be given as [11], and With reference to Fig. 1, the dominant TE 10 mode calibrated measurement reference planes are physically located at the air-dielectric (z = 0) and dielectric-air (z = l) boundaries within the MPRWG. However, the simulation reference planes are located at the slit ports. Therefore, the simulated Sparameters for the TE 10 mode must be de-embedded (shifting the measurement reference planes to the dielectric filler), to mathematically remove the air-filled sections. This can be achieved by simply multiplying S 11 and S 21 with e +2γ mn l air ; the brass-walled air-filled waveguide sections can be treated as lossless in practice. The lengths of the air-filled and PML sections are l air = l PML = 2l = 19.252 mm, used for the numerical frequency-domain simulations at X-band; the dielectric filler has ε reff 2 = 2.3284 and tanδ e 2 = 19.1480 × 10 −4 , corresponding to X-band values for ABS Sample #1 from Table 1 in [5].
Previously [5], the analytical S-parameter modeling and all numerical simulations used brass conductors (to model the physical realm), while PEC was used for the analytical eigenmode solver. Here, PEC is used in all simulations, to give a better Q-factor match between analytical and numerical eigenmode solver results. Moreover, finer meshing is introduced for the numerical frequency-domain simulator, to further increase accuracy. Finally, we employ the dielectric constant tensor and the use of slit ports with PML waveguide sections.
For the X-band MPRWG operating in the dominant TE 10 mode, the simulated |S 11 | 2 and |S 21 | 2 frequency responses are given in Fig. 4(a). The solid lines in Fig. 4(a) correspond to numerical frequency-domain simulations (using COM-SOL Multiphysics R ), while dotted lines correspond to our analytical textbook S-parameter modeling, using (A23) and (A26)-(A30) in the Appendix; both clearly predicting the driven Fabry-Pérot resonance at 11.14 GHz. Figure 4(b) shows the absorptance A calculated using [5], Note that (14) is only valid for passive, symmetrical and reciprocal two-port networks. The absorptance spectrum, given in Fig. 4(b) 14 GHz and f TE 111 = 12.05 GHz, respectively. The spectral location of these open-box modes can also be determined from local minima (dips) in |S 21 | 2 ; not clearly seen in Fig. 4(a). Using our analytical eigenmode solver, (A35)-(A36) in the Appendix, the dominant TE 101 open-box mode (loosely related to the driven Fabry-Pérot resonance, in terms of field patterns) was predicted to exist at f TE 101 = 5.6065 GHz and 5.6229 GHz for Samples #1 and #2, respectively, in [5]. These are below the f TE10 c = 6.5571 GHz cut-off frequency for an air-filled MPRWG and, therefore, outside the intended frequency range of operation for the WR-90 waveguide [5]. The lowest-order TM mode, TM 111 , has an eigenfrequency of f TM 111 = 12.8765 GHz, which is just above our frequency range of interest.
As seen in Table 1, unlike the previous numerical eigenmode solver Q-factor predictions [5], the new results from our analytical (using (A35) and (A36) in the Appendix) and numerical eigenmode solvers are almost identical. Moreover, the X-band numerical full-wave frequency-domain predictions for the undamped resonance frequencies are within 0.01 GHz of those from both the analytical and numerical eigenmode solvers; and within 0.1 GHz when compared to practical measurements (validating our simulations) [5].
Simulations were repeated for 3-D printed ABS at Ku-band. Here, the lengths of the air-filled and PML sections are l air = l PML = 2l = 13.352 mm, used for the numerical frequency-domain simulations at Ku-band; the band-average extracted values for dielectric constant and loss tangent are ε reff 2 = 2.3725 and tanδ e 2 = 16 × 10 −4 , respectively [6]. The S-parameters and absorptance spectra are given in Fig. 5   The Ku-band numerical full-wave frequency-domain predictions for the undamped resonance frequencies are within 0.02 GHz of those from both the analytical and numerical eigenmode solvers; and within approximately 0.2 GHz when compared to practical measurements (once again validating our simulations) [6].

D. IDENTIFYING OPEN-BOX MODE RESONANCES FROM FIELD PLOTS
A numerical eigenmode solver with fine meshing can also be used to create and study the 3-D field plots associated with  the parasitic open-box mode resonances. Here, the two airfilled waveguide sections have been increased in length from l air = 2l (used in the numerical frequency domain simulator) to l air = 11.5l (used in the numerical eigenmode solver), allowing the evanescent fields of the open-box modes below cut-off to sufficiently decay outside the dielectric cuboid [5]. Also, the extreme ends of the air-filled sections are terminated with absorbing 'second-order scattering boundary' surfaces, since a PML boundary has a significant spatial volume that has a large computational overhead [5], [10]. Figures 6 to 9 show E-field plots for the parasitic open-box TE mode resonances at X-band [5] and Fig. 10 for the additional TM mode at Ku-band. At X-band, Sun et al. [5] found that the parasitic TE 101 , TE 011 , TE 201 , and TE 111 open-box modes all have even longitudinal symmetry about the center of the sample. Similar field patterns will be found with the open-box modes at Ku-band [6] with the addition of the TM 111 mode, having odd longitudinal symmetry.

IV. PARALLEL-PLATE WAVEGUIDES A. BACKGROUND
Consider the vertical PEC-walled PPWG shown in Fig. 11, where the structure is assumed to be infinite in the y-direction. The length of the dielectric-filled section ( 2 ) is l and the air-filled sections ( 1 and 3 ) have length l air . Following the approach in Section IIIA, assume a TE m or TM m mode is incident on the dielectric-filled section.   With reference to the Appendix, the cut-off propagation constant is now γ cm = jk cm = jmπ/a for air-filled sections, with a representing the internal wall separation distance. The characteristic matrix equation for TE m modes is effectively the same as (A12) for TE mn modes with the MPRWG, but without the mode index n in the amplitude coefficients and propagation constant (i.e., A mn → A m , B mn → B m and γ mn → γ m ). Therefore, the amplitude coefficients correspond to those in (A23)-(A26); the generalized coefficients ζ and ς correspond to (A27) and (A28), respectively; and the S-parameters S 11 and S 21 to (A29) and (A30), respectively. In a similar way to that undertaken for the MPRWG, the associated even and odd modes can be obtained by numerically solving the analytical transcendental equations in (A35) and (A36), respectively, for the complex eigenfrequencies.

B. EXCITATION OF OPEN-BOX MODE RESONANCES WITH FREQUENCY-DOMAIN SIMULATOR
Following the approach outlined in Section IIIB, for the excitation of open-box modes using an anisotropic dielectric constant tensor, a 2-D PPWG was simulated using the COMSOL Multiphysics R frequency-domain simulator.
Within COMSOL Multiphysics R , using (5), the signs for the terms in (6) are arbitrarily chosen for our simulated PPWG, as follows, with all the scalar coefficients c uv = 5 and a 2% variation of the dielectric constant in all axial directions, giving % ε reff = 0.02. The choice of f uv in (15)-(17) assume the origin (x, z = 0) is located at one of the corners of the dielectric filler.

C. SIMULATED SPECTRAL RESPONSES
The S-parameters for the dielectric-filled PEC-walled X-band PPWG are simulated, assuming waveguide operation in the dominant TE 1 mode. The internal dimensions of the PPWG having wall separation distance a = 22.860 mm and dielectric-filled section length l = 9.626 mm; both corresponding to the previous X-band MPRWG dimensions [5]. For the numerical eigenmode solver only, the length of airfilled sections is now increased to l air = 17l (due to the extended evanescent fields of the TM 11 mode) and terminated at both ends of the air-filled waveguide sections with 'secondorder scattering boundary' surfaces. Moreover, the same X-band dielectric constant ε reff 2 = 2.3284 and loss tangent tanδ e 2 = 19.148 × 10 −4 values were used for ABS [5]. The cut-off frequency for an air-filled section of this PPWG is f TE1 c = 6.5571 GHz (identical to that for the dominant TE 10 mode with the X-band MPRWG).
For an X-band PPWG operating in the dominant TE 1 mode, the simulated |S 11 | 2 and |S 21 | 2 frequency responses are given in Fig. 12(a). The solid lines in Fig. 12(a) correspond to numerical full-wave frequency-domain simulations, while dotted lines correspond to our analytical textbook S-parameter modeling (using modified versions of (A29) and (A30) in the Appendix); both clearly predict the driven Fabry-Pérot resonance at 11.07 GHz.
The mode indices m and p inf mp refer to the number of half-wavelength variations along the x-and z-directions, respectively, for either the E-or H-fields. Open-box mode resonance frequencies (and associated mode Q-factors) calculated using analytical and numerical eigenmode solvers and numerical frequency-domain simulator for the PEC-walled X-band parallel-plate waveguide shown in Fig. 11.

FIGURE 13.
Numerically simulated E y -field plot for the even TE 11 Fabry-Pérot eigenmode below cut-off at f TE 11 = 5.6064 GHz.
The absorptance spectrum in Fig. 12(b) clearly shows parasitic TE 21 and TM 21 open-box mode resonances at f TE 21 = 10.07 and f TM 21 = 10.89 GHz, respectively; not clearly seen in Fig. 12(a).
The four lowest order parasitic Fabry-Pérot eigenmodes, along with the associated unloaded Q-factors, are predicted using our analytical and COMSOL Multiphysics R eigenmode solvers; the results are summarized in Table 3. From Table 3, it is clear that the first two parasitic Fabry-Pérot eigenmodes, TE 11 at f TE 11 = 5.6065 GHz and TM 11 at f TM 11 = 6.1403 GHz, are well below the f TE1 c = 6.5571 GHz cut-off frequency for an air-filled PPWG. As shown in Table 3, these numerical full-wave frequency-domain predictions are within 0.01 GHz of those from both the analytical and numerical eigenmode solvers.

D. IDENTIFYING OPEN-BOX MODE RESONANCES FROM FIELD PLOTS
E-field plots, from the numerical eigenmode solver, for the X-band PPWG (with l air = 17l) are shown in Figs. 13 to 16. It can be seen that for all the modes (TE 11 and TM 11 below cut-off and TE 21 and TM 21 above cut-off) the E-fields extend well outside the dielectric boundaries (represented by the two vertical black lines).

V. CIRCULAR WAVEGUIDES A. BACKGROUND
Consider the PEC-walled CWG shown in Fig. 17.    Subscripts m and n now refer to the number of halfwavelength radial (r) and circumferential (φ) variations, respectively, for either the E-or H-fields. The cut-off propagation constant for a CWG is [12], Here, r a is the internal radial distance from the axis of the circular waveguide to its PEC-walled boundary, p mn is the m th root of J n (x); such that J n (p mn ) = 0, where J n (x) are the Bessel functions of the first kind. Similarly, p mn is the m th root of J n (p mn ) = 0, where the operator J n (x) is the first derivative of J n (x). For both TE mn and TM mn propagating modes, the cut-off angular wavenumbers are k cmn = p mn /r a and k cmn = p mn /r a , respectively. The cut-off frequencies for the TE mn and TM mn propagating modes are then f cmn = k c /2π με = p mn /2πr a με and f cmn = k c /2π με = p mn /2πr a με, respectively. Selected values of p mn and p mn are given in Table 4 (rounded to three decimal places).
The characteristic matrix equations for the TE mn and TM mn modes are exactly the same as (A12) and (A22). Moreover, the amplitude coefficients correspond to those in (A23)-(A26); the generalized coefficients ζ and ς correspond to (A27) and (A28), respectively; and the S-parameters S 11 and S 21 to (A29) and (A30), respectively. In a similar way to that undertaken for the MPRWG and PPWG, the associated even and odd modes can be obtained by numerically solving the analytical transcendental equations in (A35) and (A36) for the complex eigenfrequencies.

B. EXCITATION OF OPEN-BOX MODE RESONANCES WITH FREQUENCY-DOMAIN SIMULATOR
Following the approach outlined in Sections IIIB and IVB, for the excitation of open-box modes using an anisotropic dielectric constant tensor, a 3-D CWG was simulated using the COMSOL Multiphysics R full-wave frequency domain simulator.
Within COMSOL Multiphysics R , using (5), the signs for the terms in (6) are arbitrarily chosen for our simulated CWG, as follows, where all the scalars coefficients c uv = 5 and a 2% variation of the dielectric constant in all axial directions, giving % ε reff = 0.02. The choice of f uv in (19)-(21) assumes that the z-axis lies in the middle of the circular waveguide with the origin (x, y, z = 0) located at the center of the waveguide at one of the air-dielectric boundaries.

C. SIMULATED SPECTRAL RESPONSES
For our example, the circular waveguide is based on Electronic Industries Alliance (EIA) WC-94 standards, having an internal diameter of 2r a = 23.825 mm [13]; the dominant TE 11 mode for an air-filled waveguide has a recommended frequency range of operation between 8.5 and 11.6 GHz within X-band. With an air-filled PEC circular waveguide, the lower-order cut-off frequencies f c for the TE and TM The S-parameters for the dielectric-filled PEC-walled Xband CWG are simulated, assuming waveguide operation in the dominant TE 11 mode. The internal dielectric-filled section length l = 9.626 mm corresponds to that used previously for the X-band MPRWG [5] and PPWG. Moreover, the same X-band dielectric constant ε reff 2 = 2.3284 and loss tangent tanδ e 2 = 19.148 × 10 −4 values are used for ABS [5].
For an X-band CWG operating in the dominant TE 11 mode, the simulated |S 11 | 2 and |S 21 | 2 frequency responses are given in Fig. 18(a).
The solid lines in Fig. 18(a) correspond to numerical full-wave frequency-domain simulations, while dotted lines correspond to our analytical textbook S-parameter modeling (using (A29) and (A30) in the Appendix); both clearly predict the driven Fabry-Pérot resonance at 11.29 GHz.
The mode index p inf mnp refers to the number of half-wavelength variations along the z-direction, for either the E-or H-fields. The absorptance spectrum in Fig. 18(b)   Open-box mode resonance frequencies (and associated mode Q-factors) calculated using analytical and numerical eigenmode solvers for the PEC-walled X-band circular waveguide shown in Fig. 17.  The resulting eigenfrequencies are given in Table 5. Here, the TE 111 open-box mode at f TE 111 = 6.1917 GHz is below the f TE11 c = 7.375 GHz cut-off frequency and not seen in the spectral responses. As shown in Table 5, these numerical fullwave frequency-domain predictions are within 0.03 GHz of those from our analytical and numerical eigenmode methods.

D. IDENTIFYING OPEN-BOX MODE RESONANCES FROM FIELD PLOTS
E-field plots, from the numerical eigenmode solver, for the X-band CWG are shown in Figs. 19 to 23. Here, isometric views are needed with cut-planes in order to uniquely identify the modes. For example, the TE 111 mode in Fig. 19(b) appears identical to the TM 111 mode in Fig. 23; the two different  modes can be differentiated by knowing the longitudinal position of their cuts (located at center and with quarterwavelength offset, respectively).

VI. DISCUSSION AND CONCLUSION
Open-box mode resonances have been found in a survey of microwave measurements of 3-D printed cuboid samples within MPRWGs. These parasitic Fabry-Pérot eigenmodes were either ignored or not rigorously investigated. These parasitic resonances can exhibit very high Q-factors (shown here to exceed 2,300 with a PEC-walled X-band PPWG), which may offer the potential for future exploitation. The authors of this paper previously studied open-box mode resonances within dielectric-filled MPRWGs, using both analytical and numerical eigenmode solvers. This preliminary work has now been extended in this paper, to cover three conventional types of dielectric-filled metal waveguides (MPRWGs, PPWGs and CWGs).
While previously speculating that open-box mode resonances can be excited within inhomogeneous and/or anisotropic media (3-D printed cuboids in this case), we have now proven that parasitic Fabry-Pérot eigenmodes exist with inhomogeneous and anisotropic dielectrics.
Complex eigenfrequencies for dielectric-filled waveguide structures were obtained by numerically solving analytical transcendental equations for the lowest-order even and odd eigenmodes. These results were compared against those obtained from the numerical COMSOL Multiphysics R eigenmode solver, showing excellent agreement. Moreover, from either the E-or H-field patterns, we have accurately identified the type of open-box modes, below and above the lowest cut-off frequencies associated with the corresponding air-filled waveguides.
Furthermore, we have demonstrated a new technique for studying open-box mode resonances that employs numerical full-wave frequency-domain simulations. Here, the inhomogeneous and anisotropic nature of the woodpile-like dielectric structure (physical realm) is represented by an anisotropic dielectric constant tensor (simulation realm).
The common scalar coefficient representing the level of coupling between different field components due to anisotropy was chosen to be c uv = 1 for the MPRWG and 5 for both PPWG and CWG. With the variation of the dielectric constant in all axial directions, a fixed value of % ε reff = 0.02 was adopted for all the waveguide structures. Both control parameters are perturbed just enough (with their lowest values) to reveal the presence of the parasitic open-box modes in the S-parameters, while being validated by practical measurements (in the case of the MPRWG).
The COMSOL Multiphysics R frequency-domain simulations accurately predicted the spectral location of the undamped resonance frequencies for all in-band parasitic open-box modes; to within 0.03 GHz of those obtained from the analytical and numerical eigenmode solvers for all waveguide structures (and within approximately 0.1 GHz and 0.2 GHz when compared to measurements at X-and Ku-bands, respectively, for the metal-pipe rectangular waveguide). Note that, additional simulations of a PPWG having a dielectric-filler that is inhomogeneous and isotropic does not excite all the open-box modes.
This paper provides the foundation for accurately modeling the parasitic resonances associated with inhomogeneity and anisotropy in 3-D printed microwave components; not just the metal-walled waveguide structures considered here, but the methodology could also be extended to generic 3-D printed dielectric waveguides and substrate-based transmission lines (e.g., printed microstrip, stripline and suspended microstrip [14]). In addition, our work can be applied to engineered metamaterials within waveguides and transmission lines. For example, periodically-loaded dielectric slabs could be inserted within a metal-pipe waveguide, such that the lowest order open-box mode below the air-filled cutoff frequency can result in wave propagation via evanescent mode coupling between slabs.

APPENDIX
With reference to Figs. 1, 11 and 17, under steady-state sinusoidal excitation, there will be forward and backward travelling waves, with As representing complex amplitude coefficients for the forward travelling waves and Bs representing the backward travelling waves. There are no reflected waves within Section 3 of the waveguide structure. It is assumed that Sections 1 and 3 have the same material parameters.
With the MPRWG, for the TE mn mode the electric (E-)field and magnetic (H-)field components within the three waveguide sections are [5], × sin(k cm0 x) cos(k c0n y) (A7) × sin(k cm0 x) cos(k c0n y) (A8) where the textbook cut-off propagation constant is, and, Here, m and n correspond to the mode numbers in the horizontal x-and vertical y-directions in the transverse plane, respectively; γ mn 1 and γ mn 2 are the generalized propagation constants for the TE mn mode within Sections 1 , 3 , and Section 2 , respectively. With respect to Fig. 1, γ mn 1 → γ air 10 and γ mn 2 → γ 10 are the propagation constants for the TE 10 mode within the air-filled (Sections 1 , 3 ) and dielectric-filled (Section 2 ) waveguide sections, respectively. All calculations assume a waveguide with PEC walls. The reference plane for fields with coefficient A mn 3 is z = l, as indicated by the z − l shift in the term A mn 3 e −γ mn 1 (z−l) . The reference plane for all other coefficients is z = 0. VOLUME 9, 2021 By applying TE mode continuity boundary conditions for the MPRWG (and PPWG): E y 1 (z = 0) = E y 2 (z = 0), E y 2 (z = l) = E y 3 (z = l), H x 1 (z = 0) = H x 2 (z = 0), H x 2 (z = l) = H x 3 (z = l), and after some algebraic manipulations, the following characteristic matrix equation is obtained for TE mn modes, For the TM mn mode, the E-field and H-field components within the three waveguide sections are [5], × cos(k cm0 x) sin(k c0n y) (A20) By applying TM mode continuity boundary conditions for the MPRWG (and PPWG): E x 1 (z = 0) = E x 2 (z = 0), E x 2 (z = l) = E x 3 (z = l), H y 1 (z = 0) = H y 2 (z = 0), H y 2 (z = l) = H y 3 (z = l), and after some algebraic manipulations, the following characteristic matrix equation is obtained for TM mn modes, While the MPRWG has been used to derive (A12) and (A22) for the TE and TM modes, respectively, they are generic and can also be used for the PPWG and CWG by the appropriate substitution of the cut-off propagation constant in (2).
For convenience, the waveguides in Figs. 1, 11 and 17 are excited by setting A mn 1 to unity and the rest of the amplitude coefficients can then be obtained from (A12) or (A22),  Setting A mn 1 = 1 corresponds to a driven excitation, while A mn 1 = 0 indicates undriven excitation. With the former, with no waveguide wall or dielectric losses, at the Fabry-Pérot resonance frequency, there is a return loss zero with B mn 1 = 0 and A mn 3 = 1. With the latter, the complex eigenfrequency response can be obtained (and as long as B mn 1 = 0) by setting the denominator of B mn 1 to zero in (A23).
Equations (35) for even modes and (36) for odd modes can be decomposed to give TE and TM solutions. For example, the following transcendental equations are obtained for the TE mn modes,