Broadband, nondestructive characterization of PEC-backed materials using a dual-ridged-waveguide probe

Summary form only given. A new single waveguide probe is introduced which utilizes a dual-ridged waveguide (DRWG) to provide nondestructive, broadband material characterization measurements of PEC-backed materials (much like existing coaxial probes, yet more broadly applicable) while maintaining the structural robustness of rectangular/circular waveguide probes. A schematic of the measurement geometry is shown. DRWG attached to an infinite PEC flange plate is placed in contact with a PEC-backed magnetic material of unknown εr and μr. The theoretical expression for the reflection coefficient Sthy11, necessary to characterize the material under test (MUT), is derived. This is achieved by replacing the DRWG aperture with an equivalent magnetic current which maintains the fields in the parallel-plate/MUT region in accordance with Love's equivalence theorem. Enforcing the continuity of the transverse magnetic fields at the DRWG aperture results in a magnetic field integral equation, which when solved using the Method of Moments, yields Sthy11. The εr and μr of the MUT are then found by minimizing the root-mean-square difference between the theoretical and measured reflection coefficients using nonlinear least squares. At a minimum, two independent reflection measurements are required to unambiguously characterize the MUT. In this research, two-thickness method is used for this purpose. To experimentally verify the new probe, broadband material characterization results of a magnetic absorbing material are presented and compared to those obtained using the traditional, destructive Nicolson-Ross-Weir technique. The new probe's sensitivity to sample thickness, flange-plate thickness, and measured S11 uncertainties is also presented.

In this paper, a new one-port waveguide probe is introduced which utilises a dual-ridged waveguide (DRWG) to provide ND, broadband (much like coaxial probes) material characterisation measurements of PEC-backed MUTs while maintaining the structural robustness of rectangular/circular waveguide probes. A schematic of the measurement geometry is shown in Fig. 1 a DRWG attached to an infinite PEC flange plate is placed in contact with a PEC-backed magnetic material of unknown complex relative permittivity ε r and permeability μ r .
In the next section, the theoretical expression for the reflection coefficient S thy 11 , necessary to characterise the MUT, is derived. This is achieved by replacing the DRWG aperture with an equivalent magnetic current which maintains the fields in the parallel-plate/MUT region in accordance with Love's equivalence theorem [31][32][33][34]. Enforcing the continuity of the transverse magnetic fields at the DRWG aperture results in a magnetic field integral equation (MFIE), which when solved using the method of moments (MoM) [34,35], yields S thy 11 . The ε r and μ r of the MUT are then found by minimising the root-mean-square difference between the theoretical S thy 11 and measured S meas 11 reflection coefficients using the trust-region-reflective method [36] subject to the constraints for passive materials, viz.
where S 11 = (S 11, 1 , S 11, 2 , …, S 11, n ) T and f is the frequency. At a minimum, n = 2 independent reflection measurements are required to unambiguously characterise the MUT. Several single-probe techniques have been developed to provide at least two independent reflection methodsmost notably, two-layer method [16,23,[37][38][39], two-iris method [40], frequency-varying method [16,41], short/free-space-backed method [15,23,37] and two-thickness method [9,16,22,42]. Of these, the two-layer and two-iris methods are the most universally applicable techniques for the ND characterisation of PEC backed MUTs. However, because of the analytical complexities involved in using the two-layer method (Green's function with off-diagonal elements combined with complex DRWG field expressions) and the additional hardware required to use the two-iris method, two-thickness method is utilised in this research. Note that this choice is only for convenience. Each of the techniques listed above can be used with the single DRWG probe presented in this paper.
Last, to experimentally verify the new probe, broadband material characterisation results of a magnetic absorbing material are presented and compared with those obtained using the traditional, destructive Nicolson-Ross-Weir (NRW) [43,44] technique. The new probe's sensitivity to sample thickness, flange-plate thickness, cutoff wavenumber and measured S 11 uncertainties is also presented.

Methodology
In this section, an expression for S thy 11 is derived for the single DRWG probe shown in Fig. 1. The forms of the fields in the DRWG and parallel-plate/MUT regions of the probe are detailed first. A MFIE is derived by enforcing the continuity of the transverse DRWG and parallel-plate/MUT region magnetic fields at the DRWG aperture, that is, z = 0. This MFIE is subsequently solved using the MoM.

DRWG and parallel-plate region field distributions
The fields in the DRWG region of Fig. 1 are found using the technique outlined by Montgomery [45] and later by Elliot [46]. First, the electric and magnetic fields in each DRWG subregion (i.e. the gap subregion, |x| < Δ x ∩ | y| < Δy, and the two trough subregions, Δ x < |x| < a/2 ∩ | y| < b/2) are expanded in a set of TE z and, if applicable, TM z modes [32]. Note that only the dominant DRWG mode, commonly termed a TE z 10 hybrid mode [45,47], is considered in this research. Thus, only the TE z mode development is reported here.
The mode-matching technique [48] is then used to enforce the continuity of the transverse electric and magnetic fields at x = − Δ x and Δ x producing the following homogeneous matrix equation where k c is the cutoff wavenumber and A 11 , A 12 , A 21 and A 22 are N × N submatrices whoseñth row and nth column entries are given by respectively. Here, n andñ represent basis and testing indices, respectively; δ ij is the Kronecker delta; k t ym = mp/b and k g yn = np/ 2Dy are the y-directed DRWG wavenumbers in the trough and gap subregions, respectively; k t xm = k 2 c − (k t ym ) 2 and k g xn = k 2 c − (k g yn ) 2 are the unknown x-directed DRWG wavenumbers in the trough and gap subregions, respectively; α n and β n are the unknown complex TE z modal amplitudes; and The cutoff wavenumber k c is found by forcing an eigenvalue of A(k c ) to zero via numerical root search. The vector containing the complex modal amplitudes, (α β) T , is the associated eigenvector of that zero eigenvalue. There are an infinite number of wavenumbers which satisfy (2) each corresponding to a distinct TE z DRWG mode. The k c which corresponds to the first zero of (2) is the dominant DRWG mode cutoff wavenumber. After k c and (α β) T have been found, expressions for the TE z transverse DRWG fields can be obtained. The transverse fields in the DRWG region (z < 0) are Here and h lt t are the DRWG dominant-mode transverse electric and magnetic field distributions in the gap and trough subregions, respectively. The analytical forms for these field distributions can be found in [4] and are not provided here for the sake of brevity.
The transverse magnetic field in the parallel-plate/MUT region of Fig. 1 is found by replacing the DRWG aperture with an equivalent transverse magnetic current M in accordance with Love's equivalence theorem [31][32][33][34]. The transverse magnetic field is given by where the electric vector potential F is where G is the dyadic magnetic-current-excited parallel-plate Green's function [49], I t =xx +ŷŷ, ∇ t =x(∂/∂x) +ŷ(∂/∂y), r =xx +ŷy is the observation vector, r ′ =xx ′ +ŷy ′ is the source vector, S represents the DRWG aperture cross section and k = 2pf 1m √ .

MFIE and MoM solution
A MFIE can be derived by enforcing the continuity of the transverse magnetic fields in the DRWG and parallel-plate/ MUT regions at z = 0, namely 1 jvm1 where the unknowns in the above MFIE are M and Γ.
Expanding M using the transverse DRWG electric field distribution given in (5) and testing the resulting expression with the transverse DRWG magnetic field distribution also given in (5) yields the desired S thy 11 .
It is possible to calculate the convolution integral in (7) directly. This approach requires the numerical evaluation of four integralstwo basis and two testing integrals. It is numerically advantageous to apply the convolution theorem and perform the required integrations in the spectral domain [39]. 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 p = j 2 + h 2 − k 2 is the spectral domain wavenumber and The η integral can be evaluated using complex-plane analysis yielding a pole-series representation. The remaining j integral contains irremovable branch cuts and is therefore most easily computed numerically [3,5,22]. It should be noted that because of 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 j integral, evaluate the η integral via complex-plane analysis, evaluate the resulting summations and lastly, calculate the j integral numerically. Note that there are a total of 16 distinct spectral domain integrals which must be evaluated.

Apparatus description and experimental procedure
To validate the new ND broadband probe, material measurements were made of ECCOSORB ® SF-3 (d = 1.85 mm) [50] using an Agilent E8362B vector network analyzer (VNA) [51]. The data were collected from 6 to 18 GHz using the apparatus shown in Fig. 2 Before the material measurements were made, the apparatus was calibrated using a thru-reflect-line (TRL) calibration [53]. Two custom made DRWG line standards were used in the calibrationone 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 Fig. 2. The TRL calibration placed the port 1 calibration plane at the DRWG aperture. This calibration plane was then phase shifted to the front face of the MUT by The ε r and μ r of SF-3 were found by solving (1) to within a tolerance of 10 −6 using the trust-region-reflective method [36]. As discussed above, two-thickness method [9,16,22,42] was utilised to provide the second independent S 11 measurement necessary to find ε r and μ r unambiguously. This method was chosen for analytical and computational convenience.
The new probe's sensitivity to measurement errors was also investigated. Uncertainty analysis was performed on the extracted ε r and μ r values taking into account errors in S meas 11 (s S 11 given in [51]), flange-plate thickness h (σ h = 0.05 mm), MUT thickness d (σ d1 = σ d2 = 0.05 mm) and cutoff wavenumber k c s k c = 0.03k c . Note that s k c arises mainly from uncertainties in DRWG aperture dimensions, in particular, errors because of gap width Δx and gap height Δy as well as errors because of rounded aperture corners [54]. Recent work has shown that the value of k c varies by approximately 3% when these factors are considered [54]. To the authors' knowledge, this is the first time that the effect of this error on the ε r and μ r extracted using a waveguide probe has been quantified.
One error source which was not considered in this analysis was probe lift-off or air-gap error [37]. To minimise the impact of this error, four vise-grip clamps were used to securely hold the flange plate, MUT and PEC-backing plate together. The clamps were placed surrounding and as close as possible to the DRWG aperture where the field concentration and therefore the impact of possible air gaps was the strongest. Quantifying probe lift-off error requires the two-layer magnetic-current-excited parallel-plate Green's function, which when combined with the complex analytical form of the DRWG fields, represents a significant theoretical and computational challenge beyond the scope of the work presented here. Quantifying probe lift-off error is therefore left to future work.
The following expression was used to calculate the measurement uncertainty in the real part of ε r [55] where the superscripts 'r' and 'i' denote the real and imaginary parts and '11, 1' and '11, 2' denote the first and second S 11 measurements (i.e. thickness 1 and thickness 2 mm), respectively. The partial derivatives in the above expression were estimated using the forward difference approximation. The values for s 1 i r , s m r r and s m i r were calculated in a similar manner as above. Note that the error values provided by (12) are worst case estimates [55]. Fig. 3 shows the SF-3 ε r (Fig. 3a) and μ r (Fig. 3b) results using the single DRWG probe (blue bars) introduced in this work. The solid black traces are the traditional, destructive NRW technique results using an SF-3 sample which uniformly fills the cross section of the DRWG. These results are provided to serve as a reference. Additional information about these DRWG NRW measurements, including descriptions of measurement procedures and sources of error, can be found in [54]. The widths of the blue bars in the plots represent the errors in ε r and μ r , that is, +2s 1 r and +2s m r , respectively, because of the measurement errors discussed above.

ECCOSORB ® SF-3 results
Overall, the SF-3 ε r and μ r reference results generally lie within the margins of error of the DRWG probe results. The major discrepancy between the two occurs in the m i r results, where the DRWG probe overestimates the amount of magnetic loss. Although the quality of the DRWG probe m i r estimate presented here is poor, it is consistent with those reported in the literature for similar ND single probe measurement geometries [15,16,22,23,38,[40][41][42].

Flange-plate size
Before concluding, it is worth discussing flange-plate size as this directly affects the probe's applicability for field measurements. Because of the possibility that reflections from the edges of the flange plate would corrupt the reflection measurements (recall that in the theoretical development, the flange plate is assumed to be infinite in extent), waveguide probes have traditionally been applied to lossy materials. In these cases, the flange plate needs to be large enough such that the unwanted edge reflected wave is attenuated to a degree less than the noise floor of the VNA. This condition is easily met for SF-3 and the 15.24 cm × 15.24 cm flange plate used in the experimental results discussed above. The traditional approach to determining flange size works well when one knows that the MUT is lossy; however, it stipulates an unrealistic flange size when the MUT is low loss. A simple technique to overcome this limitation is to remove the unwanted edge reflections via time-domain gating the measured reflection coefficient [56].
To use this approach, the characterisation measurement must possess sufficient bandwidth such that the edge reflections can be resolved. This criterion is where B is the minimum required bandwidth, c/ 1 r m r √ is the speed of light in the MUT and ρ is the radius of the flange plate [56]. Solving (13) for ρ yields the minimum flange-plate radius in which the edge reflections can be resolved and thus removed via time-domain gating. Since a vast majority of materials at microwave frequencies possess slower phase velocities than c, (13) can still be used to determine an acceptable flange-plate size even if ε r and μ r are unknown by assuming the MUT is free space. Applying this criterion for the DRWG probe presented here results in a minimum flange-plate radius ρ = 1.25 cm. This is significantly smaller than that stipulated by the traditional flange-size calculation, in which the minimum flange-plate radius is approximately 3 m.

Conclusion
In this paper, a new waveguide probe which utilised a DRWG to provide broadband, ND material characterisation results of PEC-backed materials was introduced. The new probe possessed two qualitiesbroad bandwidth similar to coaxial probes and structure robustness characteristic of rectangular waveguide probeswhich made it especially attractive for NDI/NDE applications in the field. The theoretical development of the DRWG probe was discussed in Section 2. This involved the derivation of the theoretical reflection coefficient necessary to characterise the MUT. This was achieved by utilising Love's equivalence theorem to replace the DRWG aperture with an equivalent magnetic current. The continuity of transverse magnetic fields at the DRWG aperture was then enforced yielding a MFIE, which was subsequently solved for the theoretical reflection coefficient using the MoM. The ε r and μ r of the MUT were then found by minimising the root-mean-square difference between the theoretical and measured reflection coefficients using the trust-region-reflective method. To experimentally verify the new probe, material characterisation results of ECCOSORB ® SF-3 were presented and compared with those obtained using the traditional, destructive NRW technique. The probe's sensitivity to sample thickness, flange-plate thickness, cutoff wavenumber and measured reflection coefficient uncertainties was also discussed. It was observed that the 1 r r , 1 i r and m r r values returned by the probe were consistent with the reference results; however, there was a significant discrepancy between the probe and reference m i r results. Although the new probe performed poorly in this regard, m i r results of this quality are reported elsewhere in the literature for similar ND single probe measurement geometries. It should be noted that in the analysis presented here, only the contribution from the dominant DRWG mode was considered because of the theoretical and computational complexity of the problem. Incorporating higher-order DRWG modes into the analysis is future work.

Acknowledgments
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. a Complex permittivity ε r and b Permeability μ r results for SF-3 using two-thickness method and the single DRWG probe (bars) analysed in this work The solid black traces are the traditional, destructive NRW technique results using an SF-3 sample which uniformly fills the cross section of the DRWG These results are provided to serve as a reference The widths of the bars in the plots represent the errors in ε r and μ r ( +2s 1 r and +2s m r ) due to uncertainties in S meas