A novel two-port probe which uses dual-ridged waveguides for the nondestructive, broadband characterization of sheet materials is presented. The new probe is shown to possess approximately 2 to 3 times the bandwidth of traditional coaxial and rectangular/circular waveguide probe systems while maintaining the structural robustness characteristic of rectangular/circular waveguide probe systems. The theoretical development of the probe is presented, namely, by applying Love's equivalence theorem and enforcing the continuity of transverse fields at the dual-ridged waveguide apertures, a system of coupled magnetic field integral equations is derived. The system of coupled magnetic field integral equations is solved using the method of moments to yield theoretical expressions for the reflection and transmission coefficients. The complex permittivity and permeability of the unknown material under test are then found by minimizing the root-mean-square difference between the theoretical and measured reflection and transmission coefficients. Experimental results of two magnetic absorbing materials are presented to validate the new probe. The probe's sensitivity to measured scattering parameter, sample thickness, and flange-plate thickness errors is also investigated.
 Radio frequency waveguide probes have been the subject of extensive research for the past half century. Their applications are numerous with nondestructive inspection/evaluation and material characterization applications being the most popular. The most common types of waveguide probes are coaxial [Folgerø and Tjomsland, 1996; Wu et al., 2000; Boybay and Ramahi, 2011; Shin and Eom, 2003; Olmi et al., 2004; Li and Chen, 1995; De Langhe et al., 1993; Baker-Jarvis and Janezic, 1996; Bird, 2004] and rectangular/circular waveguide [Sanadiki and Mostafavi, 1991; Ganchev et al., 1992; Bois et al., 1999; Chang et al., 1997; Maode et al., 1998; Hyde et al., 2009b; Hyde and Havrilla, 2008; Tantot et al., 1997] probe systems. Coaxial probe systems provide the potential for broadband measurements; however, utilizing them for this purpose can be difficult. In general, small-aperture coaxial probes are most effective when used to characterize high-loss materials. Biological materials are often discussed in the literature, whose loss is driven by water content. On the other hand, large-aperture coaxial probes are required to measure low-loss materials [De Langhe et al., 1993, 1994]. This implies that broadband measurements using coaxial probes are only possible when the material under test (MUT) is known a priori to possess high loss. Rectangular/circular waveguide probes, generally, do not suffer from the coaxial probe “coupling” problem just described and are more structurally robust than their coaxial counterparts; however, they are naturally bandlimited (in some cases, severely).
1.1 One-Port Probe Systems
 Both coaxial and rectangular/circular waveguide probe one-port (single) and two-port (dual or clamped) systems have been developed. A great majority of the published waveguide probe research involves one-port probe devices [Folgerø and Tjomsland, 1996; Wu et al., 2000; Boybay and Ramahi, 2011; Shin and Eom, 2003; Olmi et al., 2004; Li and Chen, 1995; De Langhe et al., 1993, 1994; Sanadiki and Mostafavi, 1991; Ganchev et al., 1992; Bois et al., 1999; Chang et al., 1997; Maode et al., 1998; Tantot et al., 1997]. These systems are very well suited for determining the complex permittivity ϵr of a MUT yet suffer when one desires to fully characterize a material, i.e., determine unambiguously the complex permeability μr as well as ϵr, because of the lack of a second independent measurement.
 Several one-port-probe techniques have been developed to address this shortcoming by making a second “different” reflection S11 measurement, most notably, two-thickness method [Chen et al., 2005; Stewart and Havrilla, 2006; Maode et al., 1998; Chang et al., 1997], two-layer method [Baker-Jarvis et al., 1994; Dester et al., 2010; Maode et al., 1998; Tantot et al., 1997], frequency-varying method [Wang et al., 1998; Maode et al., 1998], short/free-space-backed method [Baker-Jarvis et al., 1994; Li and Chen, 1995; Tantot et al., 1997], and two-iris method [Dester et al., 2012]. However, these techniques typically require some prior knowledge of the MUT (two-thickness method and frequency-varying method), are numerically unstable (two-layer method), are physically difficult to measure (short/free-space-backed method), or require two waveguide probes (two-iris method).
1.2 Two-Port Probe Systems
 Two-port probe devices do not suffer from the aforementioned one-port-probe shortcoming because they are able to simultaneously collect two independent measurements, namely, S11 and the transmission measurement S21, from which ϵr and μr can be unambiguously determined. Two two-port probe geometries currently exist (in both coaxial and rectangular waveguide form)—the dual-probe geometry [Bird, 2004; Hyde et al., 2009b] and the clamped-probe geometry [Baker-Jarvis and Janezic, 1996; Hyde and Havrilla, 2008].
 In this paper, a novel two-port probe system (clamped-probe geometry) is presented, which uses dual-ridged waveguides (DRWGs) to provide broadband measurements (approximately 2 to 3 times the bandwidth of coaxial and rectangular/circular waveguide probe systems discussed in the literature) while maintaining the structural robustness characteristic of rectangular/circular waveguide probe systems. A schematic of the proposed two-port system (clamped DRWG probe) is shown in Figure 1. The structure consists of two DRWGs attached to Perfect Electric Conductor (PEC) flanges sandwiching the MUT.
 The derivation of the theoretical reflection and transmission coefficient expressions (theoretical S-parameters) necessary to determine ϵr and μr of the MUT is presented in section 2. Expressions for the theoretical S-parameters are derived using Love's equivalence theorem [Collin, 1991; Peterson et al., 1998], in which the DRWG apertures are replaced with equivalent magnetic currents. These currents excite modes in the DRWG and parallel-plate regions of the structure. Expressions for the magnetic field in the MUT are found using the parallel-plate Green's function. Enforcing transverse field continuity at the DRWG apertures produces a system of coupled magnetic field integral equations (MFIEs), which when solved using the method of moments (MoM) [Harrington, 1993; Peterson et al., 1998] yields the desired theoretical S-parameters. While the analytical approach is very similar to that of other published waveguide probe research, the complex nature of the DRWG fields greatly complicates the theoretical development and, combined with the substantial gain in measurement bandwidth, represents a significant contribution to waveguide probe research. Note that only the contribution from the dominant DRWG mode is considered here due to the theoretical and computational complexity of the problem.
 The system is experimentally validated in section 3, where measurement results of two magnetic absorbing materials obtained using the clamped DRWG probe are compared to results obtained using the traditional Nicolson-Ross-Weir technique [Nicolson and Ross, 1970; Weir, 1974]. Last, this paper is concluded with a summary of the work presented.
 In this section, theoretical expressions for the reflection and transmission coefficients are derived for the clamped DRWG probe depicted in Figure 1. The form of the transverse fields in the DRWG regions of the apparatus is detailed first, followed by the form of the transverse fields in the parallel-plate/MUT region. A system of coupled MFIEs is then derived by enforcing the continuity of the DRWG and parallel-plate region transverse fields at the DRWG apertures (z=0 and z=d in Figure 1). The solution of the coupled MFIEs via the MoM is then detailed with a final brief discussion on the computation of the impedance matrix elements.
2.1 DRWG Fields
 The DRWG field derivation follows the methodology used by Montgomery  and Elliot  because it is the most applicable approach to the problem of interest. Note that several other methods for analyzing the field behavior in DRWGs exist [Sun and Balanis, 1993; Cho and Eom, 2001; Hopfer, 1955; Pyle, 1966; Utsumi, 1985; Helszajn, 2000].
 To arrive at expressions for the fields in the DRWG regions of the clamped DRWG probe, each DRWG subregion (i.e., the gap subregion and the two trough subregions) is expanded in a set of TEz or TMz modes [Collin, 1991]. Note that since the DRWG dominant mode is a TEz mode, only the TEz mode development is relevant here. The mode-matching technique [Wexler, 1967] is then used to enforce the continuity of the transverse electric and magnetic fields at x=−Δx, Δx resulting in a coupled pair of matrix equations:
and are the y-directed DRWG wave numbers in the trough and gap subregions, respectively; and are the unknown x-directed DRWG wave numbers in the trough and gap subregions, respectively; and αn and βn are the unknown complex TEz modal amplitudes. Note that , where and ω is the radian frequency. Thus, at cutoff (kz=0), the above matrix equation is of the form
where A11, A12, A21, and A22 are N×N submatrices with A12 and A21 being diagonal.
 The cutoff wave number kc is found by forcing an eigenvalue of A(kc) to equal zero via numerical root search; is then the associated eigenvector of that zero eigenvalue. There are an infinite number of wave numbers which satisfy (3), each corresponding to a distinct TEz DRWG mode. The kc which corresponds to the first zero of (3) is the dominant DRWG mode cutoff wave number.
 After kc and have been found, the TEz transverse DRWG field expressions are given by
in the |x|<Δx (gap) subregion,
in the x>Δx (right) trough subregion, and
in the x<−Δx (left) trough subregion. In the above field expressions, ZTE=ωμ0/kz is the TEz wave impedance and
Note that modes which have even Ey and Hx distributions about the origin possess the relationship that αn=βn for n=0, 2, 4, ⋯ and αn=βn=0 for n=1, 3, 5, ⋯. This class of modes includes the DRWG dominant mode. A plot of the dominant DRWG mode is shown in Figure 2. The DRWG fields for z<0 are
and for z>d are
where and are the desired theoretical reflection and transmission coefficients.
2.2 Parallel-Plate Region Fields and Coupled MFIEs
 The transverse magnetic field in the parallel-plate region of the clamped DRWG structure is found by replacing the DRWG apertures with equivalent magnetic currents, and , which emanate in the presence of the background structure [Collin, 1991; Peterson et al., 1998]. The transverse magnetic field is given by
where the electric vector potential F is
G is the dyadic magnetic-current-excited parallel-plate Green's function [Hanson and Yakovlev, 2001], is the observation vector, is the source vector, and S represents the DRWG aperture cross sections.
 Based upon the expressions for the transverse fields in the DRWG and parallel-plate regions of the structure in Figure 1, a system of coupled MFIEs can be derived by enforcing the continuity of the transverse magnetic fields at z=0 and z=d:
where and ϵ and μ are the permittivity and permeability of the material filling the parallel-plate region (i.e., the MUT). The unknowns in the above MFIEs are , , Γ, and T.
2.3 MoM Solution and Computation of Impedance Matrix Elements
 Solving (12) for Γ and T is accomplished using the MoM [Harrington, 1993; Peterson et al., 1998]. The first step in the MoM is to choose suitable basis functions to represent the unknown currents, and . Since and are related to the transverse DRWG aperture electric fields, it makes physical and mathematical sense to expand and using the electric field distributions given in (4)–(6), namely,
where , the unit normal vector, points into the MUT region. Substitution of (13) into (12) and subsequent testing (the final step in the MoM) using the transverse magnetic field distributions given in (4)–(6) yields a 2 × 2 matrix equation:
where A is the impedance matrix and b is a vector containing the incident field excitation. Impedance matrix elements a11 and a22 are the “self” terms. They model how a source located at port 1 affects an observer located at port 1 and vice versa. Elements a12 and a21 are the “coupling” terms. They model how a source at port 1 affects an observer at port 2 and vice versa. Note that a11=a22 because of the symmetry of the clamped DRWG probe and a12=a21 because of reciprocity.
 Computing the elements of A can be accomplished by directly computing the convolution integrals in (11). This approach requires the numerical evaluation of four integrals (two basis and two testing integrals) for each integral in (11). It is numerically advantageous to apply the convolution theorem and perform the required integrations in the spectral domain [Hyde et al., 2012]. This approach permits all the basis and testing integrals to be computed in closed form yielding spectral domain integrals which, in the worst computational cases, are given by
where is the spectral domain wave number and
 The η integral can be evaluated using complex-plane analysis yielding a pole-series representation. The remaining ξ integral contains irremovable branch cuts and is therefore most easily computed numerically [Hyde and Havrilla, 2008; Hyde et al., 2009b; Stewart and Havrilla, 2006]. It should be noted that due to the summations in the basis and testing functions, special care must be taken when computing the spectral domain integrals. For optimal computational efficiency, it is best to bring the summations inside the ξ integral, evaluate the η integral via complex-plane analysis, evaluate the resulting summations, and lastly, calculate the ξ integral numerically. Note that there are a total of 32 distinct spectral domain integrals which must be evaluated.
 It should be stated that one could use a commercial EM solver to perform the theoretical work presented here. However, to yield theoretical S-parameters of sufficient accuracy to obtain accurate ϵr and μr values would require a very high-density mesh of the measurement structure, especially in the vicinity of the DRWG apertures and the parallel-plate/MUT region. This would significantly affect the computation time. The fact that iteration is required to find ϵr and μr only makes computation time more prohibitive. On the other hand, the method of analysis utilized here, being based on the equivalence theorem, only requires discretization of the unknown aperture surface currents. This produces a significant reduction in the size of the problem and computational burden when compared to finite element and finite difference time domain solvers. Also, by utilizing the approach presented here, significant physical insight into the field behavior in the structure is gained. One of the drawbacks of the proposed approach is that the effects of higher-order DRWG modes generated at the DRWG apertures are not included. These effects are included in commercial solvers. Future research will include the effects of higher-order DRWG modes in this technique.
3 Experimental Validation
 In this section, the technique introduced above is experimentally validated. Before progressing to the analysis of the experimental results, discussions of the experimental apparatus and procedure are warranted.
3.1 System Description and Measurement Procedure
 In this experiment, material characterization measurements were made at 6–18 GHz (as opposed to 8–12 GHz for traditional X-band rectangular waveguide probes) of two lossy silicon-based magnetic absorbing materials, ECCOSORB®; FGM-125 (d=3.12 mm) and ECCOSORB®; SF-3 (d=1.85 mm) manufactured by Emerson & Cuming Microwave Products [2012, 2007], using an Agilent E8362B vector network analyzer [Agilent Technologies, 2008]. The clamped DRWG apparatus consisted of two WRD650 DRWGs (a=18.29 mm, b=8.15 mm, Δx=2.20 mm, and Δy=1.28 mm) connected via screws to two aluminum flange plates. On the side of the flange plates which were connected to the DRWGs, 3.18 mm alignment holes were machined to ensure precision alignment between the DRWG and the flange-plate apertures. To ensure good alignment of the port 1 and port 2 apertures, i.e., the flange-plate apertures in contact with the MUT, 3.18 mm alignment holes were also machined on the outer edges of the flange plates. Locking pliers were used in the areas around the DRWG/flange-plate apertures to ensure good contact between the flange plates and the MUT. A photograph of the apparatus is shown in Figure 3. Note that square and circular aluminum flange plates of several different dimensions were machined: 15.24 cm × 15.24 cm × 0.95 cm, 10.16 cm × 10.16 cm × 0.95 cm, and 3.81 cm × 3.81 cm × 0.95 cm (square) and 15.24 cm diameter × 0.95 cm and 10.16 cm diameter × 0.95 cm (circular).
 Before the material measurements were made, the apparatus was calibrated using a thru-reflect-line (TRL) calibration [Engen and Hoer, 1979]. Two custom-made DRWG line standards were used in the calibration—one 6.98 mm thick to cover 6–12 GHz and one 3.40 mm thick to cover 12–18 GHz. These line standards can be seen in the photograph in Figure 3. The TRL calibration placed the port 1 and port 2 calibration planes at the DRWG apertures. These calibration planes were then phase shifted to their desired locations, i.e., the front (desired port 1 calibration plane location) and back (desired port 2 calibration plane location) faces of the MUT, by
where m, n=1, 2 and h=0.95 cm is the flange-plate thickness.
 The ϵr and μr of the MUTs were found by minimizing the root-mean-square difference between the theoretical Sthy and measured SmeasS-parameters using Broyden's method [Broyden, 1965], namely,
where and f is the frequency. Note that and because of measurement geometry symmetry and electromagnetic reciprocity, respectively.
3.2 Experimental Results
 Figures 4 and 5 show the relative ϵr (Figures 4a and 5a) and μr (Figures 4b and 5b) results for ECCOSORB®; FGM-125 and ECCOSORB®; SF-3, respectively. The clamped DRWG results are demarcated by the blue bars. The destructive Nicolson-Ross-Weir results, provided to serve as a reference, are demarcated by the solid black traces. Note that the Nicolson-Ross-Weir results were obtained by measuring FGM-125 and SF-3 samples which were carefully machined to uniformly fill the WRD650 DRWG aperture. Ensuring the samples uniformly fill the DRWG aperture is very difficult and paramount because the presence of air gaps, especially in the vicinity of the DRWG gap subregion where the dominant-mode fields are the strongest (see Figure 2), can cause very large variations in the extracted ϵr and μr values. To the authors' knowledge, this is the first time that DRWGs have been used to characterize materials via the Nicolson-Ross-Weir technique.
 The widths of the bars in the figures represent the errors in the clamped DRWG probe ϵr and μr measurements (±2Δϵr and ±2Δμr, respectively) considering the S-parameter measurement uncertainties specified in the vector network analyzer data sheet [Agilent Technologies, 2008] and ±0.1 mm measurement uncertainties in d and h (Δd and Δh) [Baker-Jarvis et al., 1990]:
where the superscripts “r” and “i” denote the real and imaginary parts, respectively. The values for , , and are calculated in a similar manner as above. Note that the partial derivatives in (19) were calculated numerically using the forward difference approximation. The ϵr and μr errors calculated using (19) are worst case estimates [Baker-Jarvis et al., 1990]. Because some of the bar widths in the figures are hard to discern, a summary of the Δϵr and Δμr values obtained for FGM-125 and SF-3 is provided in Table 1.
Table 1. Mean, Maximum, and Minimum of Δϵr and Δμr for FGM-125 and SF-3
 Overall, there is fairly good agreement between the clamped DRWG probe and reference Nicolson-Ross-Weir traces. The main discrepancy between the two occurs in the results for magnetic loss with the clamped DRWG probe generally over-predicting the amount of loss. This over-prediction of magnetic loss has precedent in the literature for similar probe geometries [Hyde et al., 2011; Hyde and Havrilla, 2008; Hyde et al., 2009b]. In all cases cited, the problem is alleviated when higher-order modes are included in the analytical model (recall that only the dominant DRWG mode is considered here). While including higher-order modes in rectangular waveguide probe systems is a relatively straightforward process and has been accomplished by many researchers [Stewart and Havrilla, 2006; Bois et al., 1999; Chang et al., 1997; Hyde and Havrilla, 2008; Hyde et al., 2009b; Maode et al., 1998; Seal et al., 2012; Hyde et al., 2012], because of the complexity of DRWG fields, including higher-order modes in the clamped DRWG probe analysis is a very difficult theoretical and computational problem. Including higher-order modes in the clamped DRWG probe is left to future work. Note that other lossy materials were measured as part of this analysis, in particular, resistive cards (R-cards) [Hyde et al., 2009a; Costa, 2013; Glover et al., 2008; Massman et al., 2010]. The results were similar to those just presented and therefore are not shown here for the sake of brevity.
3.3 Discussion on Characterizing Low-Loss MUTs
 While the presented experimental results focused on the characterization of lossy materials, the clamped DRWG apparatus can also be used to characterize low-loss MUTs. Characterization results of low-loss MUTs are not presented here for the sake of brevity; however, a brief discussion of the measurement procedure follows.
 For low-loss MUTs as opposed to lossy MUTs, reflections from the flange-plate edges can corrupt the measured S-parameters. These reflections must either be accounted for analytically (a very difficult problem) or removed from the measured S-parameters via time-domain gating [Hyde et al., 2011] to effectively model flange plates of infinite dimensions.
 To use the latter technique, the characterization measurement must possess sufficient bandwidth to resolve the flange-plate edge reflections. The criterion is
where B is the minimum required measurement bandwidth, is the speed of light in the MUT, and ρ is the radius of the flange plate [Hyde et al., 2011]. Since in most characterization experiments the measurement bandwidth is known a priori, it is more useful to solve (20) for ρ. In this context, ρ represents the minimum flange-plate radius in which the edge reflections will be resolvable and thus removable via time-domain gating. In other words, ρ is the probe's measurement footprint [Hyde et al., 2011]. Note that special care must be taken when using this relation. Common signal processing techniques, like frequency-domain windowing used to suppress side lobes in the time-domain response, will affect the resolution. It should also be mentioned that time-domain gating can introduce unwanted effects at the measurement frequency band edges thereby reducing the “usable” measurement bandwidth [Hyde et al., 2011]. For the DRWGs used in the research, the usable measurement bandwidth after time-domain gating will be reduced by 2–4 GHz.
 Nevertheless, the much broader bandwidth of the clamped DRWG probe compared to other waveguide probes means that much smaller flange plates (sizes ∝1/B) can be used to characterize materials. This is demonstrated in Figure 6. The figure shows measured time-domain |S11| plots of a 6.43 mm thick sample of 99.5% Al2O3 (ϵr≈9−j0, μr≈1−j0) using 15.24 cm × 15.24 cm (solid blue traces), 10.16 cm × 10.16 cm (dashed black traces), and 3.81 cm × 3.81 cm (dash-dotted red traces) square flange plates. Note that the S11 frequency-domain data were windowed using a Kaiser window (β=6) prior to Fourier transforming to arrive at the plotted time-domain responses [Oppenheim and Schafer, 2010]. Figure 6a shows the time-domain |S11| traces when the full 12 GHz WRD650 DRWG measurement bandwidth is used; Figure 6b shows the same traces when a 4 GHz bandwidth (representative of X-band rectangular waveguide probe systems) is used.
 It is clear from the results presented in the figure that the bandwidth of the clamped DRWG probe is sufficient to resolve the flange-plate edge reflections (labeled on the figure) for all three flange-plate sizes, while an X-band rectangular waveguide probe is only able to resolve the edge reflections for the 15.24 cm × 15.24 cm and 10.16 cm × 10.16 cm flange plates. Indeed, the bandwidth of the clamped DRWG probe is large enough to resolve the reflection from the port 2 flange plate, located on the backside of the MUT approximately 6.43 mm from the port 1 DRWG aperture. The additional bandwidth inherent in the clamped DRWG probe allows for the use of small flange plates making it especially attractive for nondestructive inspection/evaluation applications in the field or for the nondestructive characterization of small specimens.
 In this paper, a novel two-port probe system was presented which used DRWGs to provide nondestructive, broadband characterization of planar MUTs. The probe possessed approximately 2 to 3 times the bandwidth of traditional coaxial and rectangular/circular waveguide probe systems in the literature while maintaining the structural robustness characteristic of rectangular/circular waveguide probe systems. The theoretical development of the probe was discussed in section 2. Theoretical expressions for the reflection and transmission coefficients, necessary for extracting ϵr and μr of the MUT via numerical inversion, were derived by applying Love's equivalence theorem, enforcing transverse field continuity at the DRWG apertures, and solving the resulting coupled MFIEs via the MoM. Section 3 presented experimental characterization results of two magnetic absorbing materials using the new two-port probe. The probe's errors in determining ϵr and μr considering measured S-parameter, MUT thickness, and flange-plate thickness uncertainties were also examined. It was found that the results using the probe compared well with the destructive, traditional Nicolson-Ross-Weir results with the exception of magnetic loss; this discrepancy is due to considering only the dominant DRWG mode in the presented analysis. Last, measured reflection coefficient time-domain data of a low-loss material were analyzed. It was shown and discussed how the much larger bandwidth inherent with the new DRWG probe allows for the use of smaller flange plates making it an especially attractive option for nondestructive inspection/evaluation applications in the field or for the nondestructive characterization of small specimens.
 The views expressed in this paper are those of the authors and do not reflect the official policy or position of the U.S. Air Force, the Department of Defense, or the U.S. Government.