A Formal Approach to the Extraction of Permittivity and Permeability of Isotropic and Anisotropic Media Using the TM₁₁ Mode in Rectangular Waveguides

Abstract

Based on our previous work on the propagation of electromagnetic waves in a layered medium placed in a rectangular waveguide, we present the theory of using the TE₁₀ and TM₁₁ modes to determine the complex parameters of isotropic and anisotropic media. The Nicolson–Ross–Weir method was used. The cases of isotropic, uniaxial, and biaxial materials were considered. It has been shown that the TM₁₁ mode can be used to extract parameters of non-magnetic uniaxial anisotropy media by a single measurement, without changing the sample position. This is not possible with the previously used TE₁₀ mode. It is also possible to use the TM₁₁ mode to quickly determine whether a material is isotropic or not. Experimental results are presented for some isotropic materials.

1. Introduction

One of the most important methods for determining the relative permittivity ε_r and relative permeability μ_r of isotropic media is the Nicolson–Ross–Weir (NRW) method. In 1970, Nicolson and Ross [1] gave the formulas for the retrieval of the electromagnetic (EM) parameters from measurements of the scattering matrix elements S₁₁ and S₂₁. The material under test (MUT) was placed in a coaxial transverse electromagnetic (TEM) waves transmission line. In 1974, Weir [2] obtained analogous relations for the MUT placed in a rectangular waveguide. In both works, the field reflection coefficient Γ at the interface, and the propagation coefficient P, were used. The algebraic inversion procedure used in the NRW method allows us to obtain ε_r and μ_r in terms of S₁₁ and S₂₁.

The overview of the NRW method and closed-form extraction equations for isotropic media can be found anywhere (see e.g., [3]). The NRW method is still being used and improved [4,5,6,7,8,9,10,11,12,13,14], in particular for the determination of the effective parameters of metamaterials (MM).

The above-mentioned methods use only a TEM mode (coaxial transmission line systems, and free-space systems), or TE modes (usually TE₁₀ in rectangular waveguide systems). To the best of our knowledge, TM modes have not been effectively used.

Relatively little work exists that relates to methods of retrieving EM properties of anisotropic media [15,16,17,18,19,20,21,22,23]. Isotropy and anisotropy relate to the directional properties of permittivity and permeability of the media (forming crystallographic systems). Consider the so-called linear media. Using the Einstein summation convention, the relationship between the components D_i (i = x, y, z) electric flux density, and E_k (k = x, y, z) electric field vectors is D_i = ε₀ε_ikE_k. The tensor ε_ik is the relative permittivity tensor, and ε₀ is the permittivity of vacuum. If the properties of the medium do not depend on the direction of the vector of the electric field, then such a medium is isotropic. Then, the permittivity tensor can be always written in diagonal form ε_ik = ε_rδ_ik, where δ_ik is the Kronecker delta.

In the general case, the properties of the medium may depend on the direction; then, such a medium is called anisotropic, and the permittivity tensor has nine components. One can find eigenvalues and eigenvectors of the coordinate system in which this tensor can be represented by a diagonal matrix. Normalized eigenvectors are unit vectors defining the directions of the main axes of the crystallographic system. For three different eigenvalues, such a medium is called biaxial. When two eigenvalues of the tensor are equal, such a medium is called uniaxial. Similar considerations can be carried out for the permittivity tensor μ_ik (see our previous work [24]). If μ_ik = δ_ik, then we say that the medium is non-magnetic and, in general, we are dealing only with electrical isotropy or anisotropy.

Many of the MMs realized so far are assumed to be biaxially or uniaxially anisotropic. Some conventional materials, i.e., double-positive MM, are also anisotropic.

Based on [24] on the propagation of electromagnetic waves in a layered medium placed in a rectangular waveguide, the theory of using TE₁₀ and TM₁₁ modes to determine the complex parameters of anisotropic and isotropic media is presented. The Nicolson–Ross–Weir method was used. The main aim of the work is to present the method. The cases of isotropic, uniaxial, and biaxial materials were considered.

The second section of the paper presents the above-mentioned theoretical basis of material parameter extraction in terms of the possibility of using the TM₁₁ mode. To facilitate the use of the formulas for specialists who are closer to the network description, rather than the field description (see e.g., our work [24]), a graph of the signal flow for TE and TM waves, and an anisotropic structure placed in a rectangular waveguide were used. A similar approach was used by Nicolson and Ross [1] for TEM waves in a coaxial line. Although the results for the anisotropic medium, when the main axes of the crystallographic system are oriented parallel to the edges of the waveguide, are identical to those of [1] for the isotropic medium, the physical basis is fundamentally different.

The third section contains a diagram of the proposed measuring system and its practical implementation. The advantage of this configuration is the constant position of the sample in the waveguide. At the end of the chapter, the S-parameter measurements for selected frequencies and the resulting calculations are presented. These results, for isotropic media with known electromagnetic parameters, confirm the theoretical assumptions presented in the second section.

The fourth section presents the advantages of the method used, compared to traditional methods using the TE₁₀ (or TEM) mode.

2. Extraction Procedure

2.1. Description of the Configurations

Consider the anisotropic slab of thickness d placed inside a rectangular waveguide with the cross-sectional dimensions a and b. The principal axes of the crystallographic system and the Cartesian system are oriented parallel to the waveguide edges (Figure 1), and the matrices of relative permeability ε and relative permittivity μ tensors of the medium are diagonal:

$$[\epsilon ]=\left[\begin{array}{ccc}{\epsilon }_x& 0& 0\\ 0& {\epsilon }_y& 0\\ 0& 0& {\epsilon }_z\end{array}\right]\hspace{1em}[\mu ]=\left[\begin{array}{ccc}{\mu }_x& 0& 0\\ 0& {\mu }_y& 0\\ 0& 0& {\mu }_z\end{array}\right]$$

(1)

Due to the possibility of using TM modes (in particular, the TM₁₁ mode) for the extraction of material parameters, it is convenient to distinguish two configurations: that which TM waves can propagate (A), and that for which this does not occur (B).

The first case occurs when the medium is isotropic (ε_x = ε_y = ε_z = ε_r, μ_x = μ_y = μ_z = μ_r) or uniaxially anisotropic with a special configuration of MUT (ε_x = ε_y ≠ ε_z, μ_x = μ_y ≠ μ_z), so-called transversely isotropic. In this case, TM and TE waves can propagate. For an isotropic sample, two material parameters (ε_r, μ_r) can be obtained, by using either the TE₁₀ or TM₁₁ modes. For the transversely isotropic configuration of MUT, it is possible to extract all four material parameters (ε_x = ε_y, ε_z, μ_x = μ_y, μ_z).

Configuration B is considered in the remaining cases. These include media with electrical anisotropy when all parameters ε_x (μ_x), ε_y (μ_y), and ε_z (μ_z) are different. In this configuration, only the TE_m0 or TE_0n mode (m, n = 1, 2, …) can be used to extract parameters, because other TE and TM modes do not propagate [15,24]. The same situation occurs for materials with uniaxial anisotropy, when ε_x = ε_z ≠ ε_y (μ_x = μ_z ≠ μ_y) or ε_y = ε_z ≠ ε_x (μ_y = μ_z ≠ μ_x).

In addition, for configuration A, parameter extraction is possible without changing the sample position, using a standard rectangular waveguide and two types of coax-to-waveguide (C-W) adapters (providing excitation of the TE₁₀ or TM₁₁ modes). If the sample is non-magnetic (i.e., μ_x = μ_z = 1) it is sufficient to use the TM₁₁ mode only to determine the parameters ε_x and ε_z. Figure 2 shows some representative examples of non-magnetic uniaxially anisotropic media. It is worth noticing that in the case of the most general biaxial anisotropy (configuration B), the extraction of the six material parameters (ε_x, ε_y, ε_z, μ_x, μ_y, μ_z) in the same way is not possible, even when the medium is non-magnetic. Detailed procedures for extracting material parameters are presented below.

First, we get Γ and P as a function of S₁₁ and S₂₁. The Rosetta Stone of understanding the problem is Figure 3, showing a network model of an air-filled rectangular waveguide with a MUT placed. Based on this model, a graph of signal flow was plotted. Based on the graph, the Mason’s rule formulas [25,26], used in the literature on the subject, are presented.

2.2. Determination of Γ and P as Functions of S₁₁ and S₂₁

The considered structure is shown in Figure 3. It is divided into five parts by four theoretical surfaces perpendicular to the waveguide’s propagation axis: 1, 1′, 2′, 2.

Air-filled parts of the waveguide have relative permittivity and permeability equal to 1, whereas layers of dielectric have their own specific values, and a specific complex propagation constant results from them. Respective impedances Z₀₁ for air-filled, and Z₀₂ for dielectric-filled waveguide can also be determined. The dielectric layer, 1′-2′, has length d, layers 1-1′ and 2-2′ have lengths tending to zero, the lengths of the outer layers are out of consideration. Additionally, incident C_s, a_x, and reflected b_x (x = {1, 1′, 2′, 2}) complex normalized wave-voltage amplitudes are marked for each connection surface. Last, there is no reflection from behind the 2 surface, nor from the 1 surface outside. This structure may be analyzed using a signal-flow graph approach.

Let us define the reflection coefficients at the interface between the layers as:

$${\mathrm{\Gamma }}_1=\frac{Z_{02}-Z_{01}}{Z_{01}+Z_{02}}, {\mathrm{\Gamma }}_2=\frac{Z_{01}-Z_{02}}{Z_{01}+Z_{02}}$$

(2)

and the propagation factor P (transmittance through the dielectric layer) as:

$$P=\exp (-\mathrm{j}{k}_{z}d)=\exp [-(\alpha +\mathrm{j}\beta )d]$$

(3)

where k_z = β − jα is the complex wavenumber, and β and α are the phase and attenuations coefficient, respectively. Figure 4 shows a graph of the signal flow adequate to the analyzed structure.

Based on this graph, the Mason rule [25,26] can be used to determine the transmittance T_ax-bx between the a_x and b_x nodes.

$$T=\frac{{\displaystyle \sum _i{T}_i\left[1+{\displaystyle \sum _j{\left(-1\right)}^j{\displaystyle \sum _k{L}_{k,i}^{\left(j\right)}}}\right]}}{1+{\displaystyle \sum _j{\left(-1\right)}^j{\displaystyle \sum _k{L}_k^{\left(j\right)}}}}$$

(4)

where

T—source-to-sink transmittance of the graph;

T_i—the ith forward-path transmittance between the initial and out nodes;

L_k^(j)—the kth loop transmittance; j—of this kind, occurring in the graph;

L_k,i^(j)—the kth loop transmittance not touching the ith path.

The transmittance between nodes a₁ and b₁ is the scattering matrix coefficient S₁₁

$$S_{11}=T_{a_1\to b_1}=\frac{{\mathrm{\Gamma }}_1\left(1-P^2{\mathrm{\Gamma }}_2^2\right)+P^2{\mathrm{\Gamma }}_2\left(1+{\mathrm{\Gamma }}_1\right)\left(1+{\mathrm{\Gamma }}_2\right)k_1{k}_2}{1-{\mathrm{\Gamma }}_2^2{P}^2}$$

(5)

where k₁ = 1/μ_z, k₂ = μ_z for TE modes, or k₁ = 1/ε_z, k₂ = ε_z for TM modes; moreover, Γ₁ = −Γ₂.

Taking this into account, and introducing the designation Γ = Γ₁, we get:

$$S_{11}=\frac{(1-P^2)\mathrm{\Gamma }}{1-{\mathrm{\Gamma }}^2{P}^2}$$

(6)

In the same way, the formula for S_21, which is the transmittance between nodes C_S and b₂, can be derived:

$$S_{21}=T_{C_S\to b_2}=\frac{\left(1+{\mathrm{\Gamma }}_1\right)\left(1+{\mathrm{\Gamma }}_2\right)P{k}_1{k}_2}{1-{\mathrm{\Gamma }}_2^2{P}^2}$$

(7)

$$S_{21}=\frac{(1-{\mathrm{\Gamma }}^2)P}{1-{\mathrm{\Gamma }}^2{P}^2}$$

(8)

The scattering-matrix elements S₁₁ and S₂₁ are expressed by the electric field reflection coefficient Γ and the propagation factor P, in the same form as for the isotropic medium. The system of Equations (6) and (8) will remain unchanged if Γ is mutually replaced with P, and S₂₁ with S₁₁. Moreover, the sum and difference of S₂₁ and S₁₁ can be expressed in a simple form:

$$S_{21}+S_{11}=\frac{P+\mathrm{\Gamma }}{1+\mathrm{\Gamma }P}, S_{21}-S_{11}=\frac{P-\mathrm{\Gamma }}{1-\mathrm{\Gamma }P}.$$

(9)

By eliminating P or Γ from the above equations, quadratic equations can be obtained:

$${\mathrm{\Gamma }}^2-2\mathrm{\Gamma }{X}_1+1=0 \mathrm{or} P^2-2P{X}_2+1=0$$

(10)

where

$$X_1=\frac{S_{11}^2-S_{21}^2+1}{2S_{11}}, X_2=\frac{S_{21}^2-S_{11}^2+1}{2S_{21}}$$

(11)

The solutions to the above equations are:

$$\mathrm{\Gamma }=X_1+\sqrt{X_1^2-1},\hspace{1em}P=X_2+\sqrt{X_2^2-1}$$

(12)

or

$$\mathrm{\Gamma }=X_1-\sqrt{X_1^2-1},\hspace{1em}P=X_2-\sqrt{X_2^2-1}$$

(13)

At this point, it is worth noting the ambiguity of the resulting pair of non-trivial solutions (Γ, P). Firstly, a quadratic equation has two solutions. Secondly, in the domain of complex numbers, the root in expressions (12) and (13) has two values.

This ambiguity is usually removed by adopting the solution for which |Γ| ≤ 1. It is true for the passive medium. Once Γ is determined, P is found as (cf. e.g., refs. [3,8]):

$$P=\frac{(S_{21}+S_{11})-\mathrm{\Gamma }}{1-(S_{21}+S_{11})\mathrm{\Gamma }} \mathrm{or} P=\frac{(S_{21}-S_{11})+\mathrm{\Gamma }}{1+(S_{21}-S_{11})\mathrm{\Gamma }}$$

(14)

The pairs (Γ, P) can be expressed in terms of the material parameters of the medium, and have a different form depending on the considered waveguide mode. The method is based on the use of two pairs of solutions, (Γ_TE, P_TE) and (Γ_TM, P_TM), determined in terms S₁₁ and S₂₁ for TE and TM polarization, respectively. This enables their direct application for the extraction of complex components of permittivity and permeability tensors.

For technical reasons, the frequency f_TE(TM) used for the TE₁₀ and TM₁₁ modes may be different. For instance, k_0TE(TM) and λ_0TE(TM) are wavenumbers and wavelengths in a vacuum, respectively.

$$k_{0\mathrm{TE}}=\frac{2\pi f_{\mathrm{TE}}}{c}=\frac{2\pi }{{\lambda }_{0\mathrm{TE}}}, k_{0\mathrm{TM}}=\frac{2\pi f_{\mathrm{TM}}}{c}=\frac{2\pi }{{\lambda }_{0\mathrm{TM}}}$$

(15)

where c is the velocity of light in a vacuum. Let us introduce the relative sample thickness d_TE(TM), and the transverse dimensions of the waveguide a_TE(TM) and b_TE(TM):

$$d_{\mathrm{TE}\left(\mathrm{TM}\right)}=\frac{d}{{\lambda }_{0\mathrm{TE}\left(\mathrm{TM}\right)}}, a_{\mathrm{TE}\left(\mathrm{TM}\right)}=\frac{a}{{\lambda }_{0\mathrm{TE}\left(\mathrm{TM}\right)}}, b_{\mathrm{TE}\left(\mathrm{TM}\right)}=\frac{b}{{\lambda }_{0\mathrm{TE}\left(\mathrm{TM}\right)}}$$

(16)

Next, let us define the dimensionless relative wavenumbers K_zTE(TM), and the relative cut-off wavenumbers K_TE(TM)

$$K_{z\mathrm{TE}\left(\mathrm{TM}\right)}=\frac{k_{z\mathrm{TE}\left(\mathrm{TM}\right)}}{k_{0\mathrm{TE}\left(\mathrm{TM}\right)}}$$

(17)

$$K_{\mathrm{TE}}=\frac{k_{10}}{k_{0\mathrm{TE}}}=\frac{1}{2a_{\mathrm{TE}}}$$

(18)

$$K_{\mathrm{TM}}=\frac{k_{11}}{k_{0\mathrm{TM}}}=\frac{1}{2}\sqrt{{\left(\frac{1}{a_{\mathrm{TM}}}\right)}^2+{\left(\frac{1}{b_{\mathrm{TM}}}\right)}^2}$$

(19)

where k₁₀ and k₁₁ are the cut-off wavenumbers for the TE₁₀ and TM₁₁ modes, respectively.

The propagation factors P_TE(TM) can be rewritten as

$$P_{\mathrm{TE}\left(\mathrm{TM}\right)}=\exp (-\mathrm{j}{d}_{\mathrm{TE}\left(\mathrm{TM}\right)}2\mathsf{\pi }{K}_{z\mathrm{TE}\left(\mathrm{TM}\right)})=\left|P_{\mathrm{TE}\left(\mathrm{TM}\right)}\right|\exp (\mathrm{j}{\varphi }_{\mathrm{TE}\left(\mathrm{TM}\right)})$$

(20)

where $-\pi <{\varphi }_{\mathrm{TE}},{\varphi }_{\mathrm{TM}}\le \pi$. Then

$$K_{z\mathrm{TE}\left(\mathrm{TM}\right)}=\frac{n_{\mathrm{TE}\left(\mathrm{TM}\right)}-\frac{{\varphi }_{\mathrm{TE}\left(\mathrm{TM}\right)}}{2\pi }}{d_{\mathrm{TE}\left(\mathrm{TM}\right)}}+\mathrm{j}\frac{\frac{\ln \left|P_{\mathrm{TE}\left(\mathrm{TM}\right)}\right|}{2\pi }}{d_{\mathrm{TE}\left(\mathrm{TM}\right)}}$$

(21)

where n_TE(TM) are the branch cut numbers of a complex logarithm.

A relative wave impedance z_TE(TM) (the ratio of the wave impedance in the medium, to the wave impedance in an air-filled waveguide) can be expressed in terms of the electric field reflection coefficient Γ_TE,TM, by the formula

$$z_{\mathrm{TE}\left(\mathrm{TM}\right)}=\frac{1+{\mathrm{\Gamma }}_{\mathrm{TE}\left(\mathrm{TM}\right)}}{1-{\mathrm{\Gamma }}_{\mathrm{TE}\left(\mathrm{TM}\right)}}.$$

(22)

Next, we will show how to obtain the complex material parameters ε_x, ε_y, ε_z, μ_x, μ_y, and μ_z using algebraic transformations.

2.3. Extraction of Material Parameters Using z and K_z—Configuration A

For transversely isotropic configuration, Equations (21) and (22) have the form

$$z_{\mathrm{TE}}=\frac{\sqrt{\frac{{\mu }_x}{{\epsilon }_x}}\sqrt{1-K_{\mathrm{TE}}^2}}{\sqrt{1-\frac{1}{{\epsilon }_x{\mu }_z}{K}_{\mathrm{TE}}^2}}$$

(23)

$$K_{z\mathrm{TE}}=\sqrt{{\mu }_x{\epsilon }_x}\sqrt{1-\frac{1}{{\epsilon }_x{\mu }_z}{K}_{\mathrm{TE}}^2}$$

(24)

$$z_{\mathrm{TM}}=\frac{\sqrt{\frac{{\mu }_x}{{\epsilon }_x}}\sqrt{1-\frac{1}{{\mu }_x{\epsilon }_z}{K}_{\mathrm{TM}}^2}}{\sqrt{1-K_{\mathrm{TM}}^2}}.$$

(25)

$$K_{z\mathrm{TM}}=\sqrt{{\mu }_x{\epsilon }_x}\sqrt{1-\frac{1}{{\mu }_x{\epsilon }_z}{K}_{\mathrm{TM}}^2}$$

(26)

We have obtained a system of four Equations (23)–(26), from which the complex material parameters ε_x, ε_z, μ_x, and μ_z should be determined. Equations (23) and (24) can be rewritten as:

$$\sqrt{\frac{{\mu }_x}{{\epsilon }_x}}=z_{\mathrm{TE}}\frac{\sqrt{1-\frac{1}{{\epsilon }_x{\mu }_z}{K}_{\mathrm{TE}}^2}}{\sqrt{1-K_{\mathrm{TE}}^2}}, \sqrt{{\mu }_x{\epsilon }_x}=\frac{K_{zTE}}{\sqrt{1-\frac{1}{{\epsilon }_x{\mu }_z}{K}_{\mathrm{TE}}^2}}$$

(27)

Multiplying the above expressions side by side, we get μ_x

$${\mu }_x=\sqrt{\frac{{\mu }_x}{{\epsilon }_x}}\sqrt{{\mu }_x{\epsilon }_x}=\frac{z_{\mathrm{TE}}{K}_{zTE}}{\sqrt{1-K_{\mathrm{TE}}^2}},$$

(28)

Similarly, from (25) and (26), we obtain ε_x

$$\sqrt{\frac{{\mu }_x}{{\epsilon }_x}}=z_{\mathrm{TM}}\frac{\sqrt{1-K_{\mathrm{TM}}^2}}{\sqrt{1-\frac{1}{{\mu }_x{\epsilon }_z}{K}_{\mathrm{TM}}^2}}, \sqrt{{\mu }_x{\epsilon }_x}=\frac{K_{zTM}}{\sqrt{1-\frac{1}{{\mu }_x{\epsilon }_z}{K}_{\mathrm{TM}}^2}}$$

(29)

$${\epsilon }_x=\frac{\sqrt{{\mu }_x{\epsilon }_x}}{\sqrt{\frac{{\mu }_x}{{\epsilon }_x}}}=\frac{K_{zTM}}{z_{\mathrm{TM}}\sqrt{1-K_{\mathrm{TM}}^2}}$$

(30)

Using (24) and (26) the remaining parameters, μ_z and ε_z can be expressed by μ_x and ε_x

$${\mu }_z=\frac{{\mu }_x{K}_{\mathrm{TE}}^2}{{\mu }_x{\epsilon }_x-K_{zTE}^2}$$

(31)

$${\epsilon }_z=\frac{{\epsilon }_x{K}_{\mathrm{TM}}^2}{{\mu }_x{\epsilon }_x-K_{zTM}^2}$$

(32)

Equation (32) shows that the TM₁₁ mode is necessary to determine the parameter ε_z. However, in the case of this mode and a non-magnetic medium (μ_x = μ_y = μ_z = 1), it is enough to use (30) and (32) to uniquely determine ε_x and ε_z. This is not possible using the TE₁₀ mode.

To extract parameters ε_r and μ_r of an isotropic MUT (ε_x = ε_y = ε_z = ε_r, μ_x = μ_y = μ_z = μ_r), we can use the results (28) and (31) for the TE₁₀ mode (see, e.g., ref. [3])

$${\mu }_r=\frac{z_{\mathrm{TE}}{K}_{zTE}}{\sqrt{1-K_{\mathrm{TE}}^2}},$$

(33)

$${\epsilon }_r=\frac{K_{zTE}^2+K_{\mathrm{TE}}^2}{{\mu }_r}$$

(34)

or (30) and (32) for the TM₁₁ mode

$${\epsilon }_r=\frac{K_{zTM}}{z_{\mathrm{TM}}\sqrt{1-K_{\mathrm{TM}}^2}}$$

(35)

$${\mu }_r=\frac{K_{zTM}^2+K_{\mathrm{TM}}^2}{{\epsilon }_r}$$

(36)

None among (28), (30), (31), and (32) for uniaxial anisotropy, or (35) and (36) for isotropy, has yet been reported in the scientific literature.

2.4. Extraction of Material Parameters Using z and K_z—Configuration B

In the case of the configuration B, only the TE₁₀ mode can be considered. For the most general biaxial medium, Equations (21) and (22) have the form:

$$z_{\mathrm{TE}}=\frac{\sqrt{\frac{{\mu }_x}{{\epsilon }_y}}\sqrt{1-K_{\mathrm{TE}}^2}}{\sqrt{1-\frac{1}{{\epsilon }_y{\mu }_z}{K}_{\mathrm{TE}}^2}},$$

(37)

$$K_{zTE}=\sqrt{{\mu }_x{\epsilon }_y}\sqrt{1-\frac{1}{{\epsilon }_y{\mu }_z}{K}_{\mathrm{TE}}^2}$$

(38)

By writing (37) and (38) as

$$\sqrt{\frac{{\mu }_x}{{\epsilon }_y}}=z_{\mathrm{TE}}\frac{\sqrt{1-\frac{1}{{\epsilon }_y{\mu }_z}{K}_{\mathrm{TE}}^2}}{\sqrt{1-K_{\mathrm{TE}}^2}},$$

(39)

$$\sqrt{{\mu }_x{\epsilon }_y}=\frac{K_{zTE}}{\sqrt{1-\frac{1}{{\epsilon }_y{\mu }_z}{K}_{\mathrm{TE}}^2}}$$

(40)

the parameter μ_x can be obtained

$${\mu }_x=\sqrt{\frac{{\mu }_x}{{\epsilon }_y}}\sqrt{{\mu }_x{\epsilon }_y}=z_{\mathrm{TE}}\frac{K_{zTE}}{\sqrt{1-K_{\mathrm{TE}}^2}}$$

(41)

Then, for non-magnetic media, using (38) can determin ε_y

$${\epsilon }_y=K_{zTE}^2+K_{\mathrm{TE}}^2$$

(42)

Since the TM₁₁ mode is disabled for biaxial anisotropy, this can be a simple test of its existence or non-existence.

3. Simulations and Measurements

3.1. Description of the Configuration

Measurements in a rectangular waveguide, especially using the TE₁₀ and TM₁₁ modes, have some known advantages over the use of the TEM mode in a coaxial line, or TEM waves in an open microwave path:

• the isolation of signals from the external environment in a closed microwave path, and the possibility of using a smaller sample and its more precise positioning, compared to the so-called free-space methods; and
• the easier preparation of the sample in the form of a hexahedron, than of a cylinder in a coaxial line.

For the experiments, one rectangular waveguide with dimensions a = 40 mm, b = 20 mm, and two C-W adapters, were used to excite the TE₁₀ and TM₁₁ modes in the waveguide. Measurements were made at 6 GHz and 10.55 GHz for the TE₁₀ and TM₁₁ modes, respectively. A C-W adapter with a magnetic loop was used to excite the TE₁₀ mode. To excite the TM₁₁ mode, a C-W adapter with electrical coupling in the form of an antenna was designed and manufactured.

The proposed measurement system is shown in Figure 5.

3.2. TM Mode Coax-to-Waveguide Adapter Design

Using the CST Studio Suite^® program, the C-W adapter was modeled for a rectangular waveguide in the 10–11 GHz band. The adapter matches the Z₀ = 50 Ω TEM coax line impedance to the wave impedance of the TM₁₁ mode, and ensures the excitation of this mode. Simulations of such a model were carried out. The simulation results are shown in Figure 6, Figure 7 and Figure 8.

The simulation results presented in Figure 7 and Figure 8 confirm the excitation of the TM₁₁ mode in the rectangular waveguide. The electric field E has the component along the direction of wave propagation. On the other hand, the magnetic field H, at any point in the waveguide space, is transverse to the direction of wave propagation. To ensure the excitation of the TM₁₁ mode with the longitudinal component of the electric field vector, a radiator located in the middle of the front and rear walls of the waveguide was used, as shown in Figure 6 and Figure 9. Figure 10 shows the result of the simulation of the waveguide-scattering matrix coefficients.

3.3. Practical Implementation

Figure 11 shows a view of the C-W adapter, designed according to the project.

A view of the waveguide structure is shown in Figure 12. This figure also shows the result of the measurements of the parameters S₁₁ and S₂₁ of the waveguide structure with the excited TM₁₁ mode. The measurements S₁₁ and S₂₁ fully confirmed the simulation results of the TM₁₁ mode C-W adapter.

3.4. Measurement Results

Measurements were made at two frequencies: f_TE = 6 GHz for TE₁₀ mode, and f_TM = 10.55 GHz for the TM₁₁ mode. In both cases, during the measurements, the tested material sample was in a rectangular waveguide, in which the type of C-W adapters was successively changed, depending on the analyzed mode. The measurements of the parameters S₁₁ and S₂₁ of the waveguide structure with the sample placed in it were carried out using a vector network analyzer (VNA). This is the most effective technique for measuring S-parameters. Using this instrument allowed us to measure S-parameters with high accuracy in relation to previously calibrated measurement gates. Measurements were made with a P9373B Streamline Vector Network Analyzer, 9 kHz to 14 GHz, 2-port. The VNA calibration was performed using the N7553A Electronic Calibration Module (ECal), DC-14 GHz, 2-port module that supports 3.5 mm 50 Ω connectors.

To reduce measurement errors, the procedure described in application note [27] was used. This application note described the techniques of de-embedding S-parameter networks with a device under test. Using the error-correcting algorithms of the vector network analyzer, the error coefficients can be modified, so that the process of de-embedding two port networks can be performed directly on the analyser in real time. This allowed us to use the de-embedding process, resulting in very accurate measurements.

Several isotropic and non-magnetic materials, whose electromagnetic parameters are described in the scientific literature, were selected for the measurements. The following materials were tested: PA-6 (Polyamide 6), FR4 (Epoxy Fiberglass Laminate), PVDF (Polyvinylidene Fluoride), and PTFE (Polytetrafluoroethylene).

The properties of the materials confirmed in the literature are presented in

Table 1. Properties of tested materials found in literature.

Material	f [GHz]	ε′	ε″	tanδ 1	μ′	μ″	Reference
PA-6	1.7–2.5	3.0723 ± 0.0095	0.0107 ± 0.0083	-	-	−	[28]
	8.2–12.4	2.999 ± 0.0059	0.0091 ± 0.0045	-	-	−	[28]
FR4	4–12	4.4	-	2 × 10−2	-	-	[29]
	nominal	4.2–5	-	-	-	-	[30]
	10	4.43	-	-	-	-	[30]
PVDF	8–12	2.5 ± 0.5		-	1.0 ± 1.5	0.3 ± 2	[31]
PTFE	8–13	2.046 ± 0.002	-	-	-	-	[5]
	5	2.1		3.1 × 10−4			[32]
	8–12	2.01 ± 0.05	0.003	-	-	-	[33]
	9.816	2.06	-	2.1 × 10−4	-	-	[34]
	8–12	2.07 ± 0.3	0.003 ± 0.0004		1.02 ± 0.02	0.0 ± 0.002	[35]
	70–114	2.12	-	3 × 10−4	-	-	[35]

¹ tanδ = ε″/ε′.

.

Table 2. Measurements and calculation results for the TM₁₁ mode (f_TM = 10.55 GHz).

Material	dTM [mm]	S11	S21	ε′	ε″	μ′	μ″
PA-6	3	0.171 < −152°	0.984 < −61.7°	3.23	0.006	0.999	0.0002
FR4	1.5	0.262 < −132.8°	0.956 < −77.9°	5.11	0.102	0.928	0.0023
PVDF	5	0.203 < −170.5°	0.909 < −63.6°	3.48	0.439	0.983	0.0220
PTFE	5	0.021 < 42.0°	0.999 < −48.9°	2.05	0.0028	1.110	0.00005

and

Table 3. Measurements and calculation results for the TE₁₀ mode (f_TE = 6 GHz).

Material	dTE [mm]	S11	S21	ε′	ε″	μ′	μ″
PA-6	3	0.449 < −134.2°	0.892 < −44°	3.23	0.008	0.999	0.0001
FR4	1.5	0.432 < −125.5°	0.892 < −34.1°	5.12	0.102	0.998	0.004
PVDF	5	0.459 < −143.8°	0.814 < −44.4°	3.47	0.438	1.001	0.002
PTFE	5	0.239 < −121°	0.970 < −30.9°	2.06	0.002	0.998	0.0019

shows the measurements of the S₁₁ and S₂₁, and calculation results of relative permittivity ε_r = ε′ − jε″ and relative permeability μ_r = μ′ − jμ″ for the TM₁₁ and TE₁₀ modes, respectively.

Measurements of the PA-6 sample for the TE₁₀ and TM₁₁ modes gave a consistent result of relative permittivity ε′ = 3.23. Similar values of parameters of this material are presented in [28]. For the remaining materials, there is an analogous agreement between the measurement results obtained using the TE₁₀ and TM₁₁ modes. For the FR4 sample, the obtained values were confirmed in [29,30], for the PVDF in [31,36], and for the PTFE in [5,32,33,34,35].

4. Discussion

The comparison of measurements presented in Table 2 (the novel method) and Table 3 (the traditional method) confirms the validity of using the TM₁₁ mode to extract the material parameters μ_r and ε_r of the isotropic medium.

The proposed method of extracting material parameters using the TM₁₁ mode has several advantages over the traditional method using the TE₁₀ mode.

• The first is due to the permittivity ε_r extraction resulting from (35). The traditional method requires first the determination of permeability μ_r, from (33), and then ε_r, from (34). The proposed method does not require the prior determination of permeability, so it does not introduce an additional error. In broadband studies, a known disadvantage of the NRW method is instability at frequencies corresponding to integer multiples of one-half wavelength in the MUT. An effective and widely used method of eliminating these instabilities is the iterative method, first described in [5]. This method applies to non-magnetic materials, i.e., with μ_r = 1. Using the TM₁₁ mode does not require this assumption.
• When the medium is non-magnetic (which is a common case) and uniaxial, both relative permittivity ε_x, and ε_z can be determined using the TM₁₁ mode without changing the position of the sample. This cannot be done with the TE₁₁ mode, because the latter does not has an electric field z-component.
• In turn, the use of the TM₁₁ and TE₁₀ modes allows us to determine all four parameters of the uniaxial medium, without changing the position of the sample.

An experimental confirmation of the presented theoretical results for uniaxial anisotropic materials will be the direction of further research.

The work fits into the broader context of verifying the properties of materials using various methods of microwave measurements [37,38]. In a narrower context, it provides an alternative to using TEM or the TE₁₀ modes in transmission/reflection measurement techniques.

At this point, it is necessary to mention the work [39], where the method of extracting the parameters of biaxial media in the rectangular waveguide, using the TM₁₁ mode, is presented. We questioned the theoretical basis of this work in comments [40] accepted for publication. Quite simply, the TM₁₁ mode is not propagated in the biaxial medium, as we have stated in [24], and clearly indicated in this paper.