1. Introduction

Since the first left-handed metamaterial (LHM) was realized based on split-ring resonators (SRR) and rods [1], a variety of designs as candidates to realize the LHMs has been experimentally realized and studied [2–7]. All of them present bulk electromagnetic properties, which can be characterized by constitutive parameter tensors [8–12].

Because the performance of metamaterial applications [14–18] depend much on the materials properties, it is very important that the constitutive parameters of these metamaterials can be well retrieved. In this paper, we aim to present an efficient experimental technique, which can be easily carried out to measure the permittivity and permeability tensors of the metamaterials. The material under test is assumed to be biaxially anisotropic, which, in general, is valid for the metamaterials realized so far. Various kinds of experimental approaches can be used for measuring the electromagnetic parameters of metamaterials, such as the resonator method [19], the open-ended coaxial method [20], the free-space method [21], and the rectangular waveguide method [22]. All of these methods have both advantages and disadvantages. The resonant method has a quite high accuracy and sensitivity, but it is narrowband, and requires the test specimen having a small electrical size and a specified geometrical shape [19], so it is unfit to measure metamaterials with frequency dispersive in nature. The coaxial line based reflection-transmission approach has an advantage in terms of bandwidth, but the test samples should be properly machined in the shape of circular cylinders [20], which is a very tough requirement for metamaterials. The free-space method has been reported to measure the parameters of the metamaterials where a plane wave was required to normally incident onto the material [23]. The thickness of the slab has to be very uniform along its transverse cross section because the undesirable reflection and diffraction from the incident wave over a big sample of material will cause the result less accurate. In the rectangular waveguide method [22], the matching of the test specimen is not so crucial as only slab-shaped samples with small cross sections are required, it has no other tough requirements, which are the advantages of this method. The method has been used to measure non-magnetic materials with uniaxial permittivity tensor [24] but, as the best of our knowledge, has not been applied to measure metamaterials with both complex permittivity and permeability tensors so far.

The waveguide-based retrieved algorithm for biaxial metamaterials is developed in this paper. Different with [22] where both TE₁₀ and TE₂₀ excitations are needed for dispersive material, here we only need TE₁₀ excitation. The cost is that four independent set of measurements with different orientations of the metamaterials are needed in order to retrieve six unknown parameters. The whole retrieval procedure is firstly verified by a slab of homogenous frequency dispersive material with known parameters, then to an anisotropic metamaterial composed of SRR structures. We confirmed that the effective permeability along the axis of the SRR µ ₁ is negative in a frequency band above the resonant frequency while the permeability along other directions is close to 1. The retrieved results using the waveguide-based method are compared with that obtained from normal incident measurement [23,25]. Good agreements are obtained.

2. Retrieve technique

 

Fig. 1. Experimental scheme for the S parameters measurement in waveguide.

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The experimental setup is shown in Fig. 1. It consists of two coaxial-to-waveguide adapters connected to a rectangular waveguide loaded with the slab of the material under test. The slab sample with a thickness of d completely fills the cross section of the waveguide. In the waveguide environment, the incident, reflected, and transmitted waves are stand waves. The distances between the adapters to the sample are large enough so that the higher order incident evanescent modes due to the coaxial-to-waveguide adapters are significantly attenuated prior to reaching the sample under test. With the origin coinciding with the first face of the slab, S ₁₁ is equal to the reflection coefficient, and S ₂₁ is related to the transmission coefficient T by $S_{21}={\mathit{Te}}^{i{k}_{0z}d}$ , where k_0z is the longitudinal wave number of the incident wave. In the experimental measurement, the recorded S parameters by the network analyzer are calibrated at the end of coaxial line, so they should be firstly calibrated to the two reference planes corresponding to the two interfaces between the material and the air, then we can retrieve the permittivity and permeability tensors using the calibrated S parameters. The constitutive relations [26–28] for the material are D̅=ε̿E̅, B̅=μ̿H̅, where the parameter tensors have the following forms in the principal system (e ₁, e ₂, e ₃): ε̿=ε ₀ diag[ε ₁ ε ₂ ε ₃], µ̿=µ ₀ diag[µ ₁ µ ₂ µ ₃]. The three coordinate axes e ₁, e ₂, e ₃ are called the principal axes [26]. Different slab samples of the material under test are required for measurement in order to get all the constitutive parameters. We first focus on three parameters: µ ₁, µ ₂ and ε ₃. In order to retrieve these three parameters, two independent measurements are necessary. In the first measurement (case a), the axes e ₁, e ₂, e ₃ of the slab sample are along the direction of x̂, - ẑ, and ŷ, respectively. In the second measurement (case b), the axes e ₁, e ₂, e ₃ of the slab sample are along with the direction of ẑ, x̂, and ŷ, respectively. From the measured S parameters, we can get the refractive index n and the impedance z [29]:

where X=(1-$S_{11}^2$+$S_{21}^2$)/2S ₂₁. Different with that in [29,30], here n and z are:

where k_x =π/a is the transverse wave number in the rectangular waveguide. The subscripts ‘a’ and ‘b’ denote that the results are calculated from the measurements of case (a) and case (b), respectively. Therefore, from Eq. (3–6) we get

ε ₃ can be calculated from Eqs. (3) and (7), or from Eqs. (5) and (7), respectively,

Similarly, we can achieve the other three constitutive parameters µ ₃, ε ₁, and ε ₂ based on two additional measurements: in the third measurement (case c), the principal axes (e ₁, e ₂, e ₃) of the material is along with the coordinate (-ẑ, ŷ, x̂), in this case we can get µ ₃ and ε ₂; in the fourth measurement (case d), the principal axes (e ₁, e ₂, e ₃) of the material is along with the coordinate (ŷ, ẑ, x̂), then we can get ε ₁.

3. Numerical validation

 

Fig. 2. Comparison of the analytical (markers) and the retrieved results (Solid lines: real part; dashed lines: imaginary part) for a loss homogeneous medium.

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For the purpose of assessing the accuracy of the measurement technique, the retrieval technique has been validated based on simulated scattering data measured from a slab of a homogeneous material. The constitutive parameters of the material are set to be: ε ₁=1, µ ₂=1, µ ₃=1, ε ₂=1-0.3/(f ²/3.66²+i0.00594f-1), ε ₃=1-1.2/(f ²/3.18²+i0.0785f-1), µ ₁=1-1/(f ²/2.01²+i0.0792f-1). A WR-430 (a=109.22 mm, b=54.61 mm) waveguide is used in the simulation. The retrieved constitutive parameters obtained from the measured S parameters are shown in Fig. 2. The retrieved ε ₁, µ ₂, and µ ₃ are equal to 1 and haven’t shown here. We see a perfect agreement between the retrieved parameters and the input ones.

4. Experimental results

We use two isotropic materials with known properties: a Teflon (ε _r=2.1) and a dielectric FR4 substrate (ε _r=4+i0.02) for calibration. A WR-430 waveguide is used in the experiment. The operating frequency range is 1.72~2.61 GHz. The two samples have a thickness of 10 mm and are loaded into the waveguide independently. The S parameters are recorded by an Agilent 8722ES network analyzer. The retrieved constitutive parameters of the two materials are shown in Fig. 3, where we see the relative permittivity of the Teflon sample is around 2.1 and the relative permeability is around 1. For the dielectric FR4 substrate, the relative permittivity is around 4.0, and the relative permeability is around 1. Both of the measured results are in good agreement with the known value, indicating the measurement method works very well.

 

Fig. 3. Measured ε _r and µ _r for the Teflon and FR4 substrate.

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The SRR structure as shown in Fig. 4(a) is fabricated for measurement. The sizes of the resonator are w=7.6 mm, c=1 mm, g=2 mm, d=2 mm. The metallic SRR strips are printed on the FR4 substrate. The structure has a periodicity of 4 mm along the e ₁ direction, 13.6 mm along both e ₂ and e ₃ direction. It should be noted that enough large number of unit cells should be used in the e ₁ directions in order to let the SRR exhibit the same bulk media properties in every measurement [31,32]. For example, here we focused on µ ₁, µ ₂, and ε ₃, and the two necessary measurements are shown in Fig. 4(b) and Fig. 4(c) corresponding respectively to the measurement of case (a) and case (b) indicated in section 2. In case (b), at least five layers of the ring in the e ₁ (or ẑ) direction is needed. While in case (a), the retrieved parameters are not sensitive to the number of layers used in e ₂ direction [29].

 

Fig. 4. (a) The dimensions of the SRR unit, (b) (c) two measurements corresponding to two different orientations of the SRR in the waveguide.

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The retrieved experimental results are shown in Fig. 5. We see that the real part of µ ₁ retrieved in the experimental measurement is negative in the frequency range from 1.96 GHz to 2.1 GHz [Fig. 5(a)]. The retrieved µ ₁ based on the two numerical simulations with different configurations are also shown. In the first simulation (Sim 1), we use the free-space method, modeling a normal incidence plane wave. In the second simulation (Sim 2), we model the real rectangular waveguide. The results are in good agreement with each other except the experimental result shows a smaller bandwidth. The retrieved µ ₂ shown in Fig. 5(a) is close to 1, indicating that the structure is non-magnetic in the e ₂ direction.

We present the retrieved ε ₃ in Fig. 5(b) from the calculation of Eq. (8). We show the one calculated from Eq. (8) exhibit an anti-resonant behavior around 1.96 GHz, which is accompanied by a negative imaginary part of permittivity. The unphysical phenomenon is very similar to that reported in Ref. [33], where it was pointed out that around the antiresonant frequency region, the wavelength in the medium is smaller than the periodicity of the SRRs and therefore, can not be characterized as a homogenous medium [33]. For the permittivity calculated from Eq. (9), we present the results in Fig. 5(c) for SRR with different layers in the ẑ (e ₁) direction. An interesting phenomenon can be seen that for SRR with less than five layers, the imaginary part of the permittivity is negative around the plasma frequency. The reason is that in the e ₁ direction, the magnetic plasma frequency of SRRs with a few layers, ω′_mp , is higher than that of a bulk SRR medium, ω_mp . When we use Eq. (9) to retrieve ε _3b, the µ ₁ we used is retrieved from measurement of case (a), where the sample is a bulk SRR medium, so µ ₁ has a plasma frequency of ω_mp . Since the real part of µ ₁ is close to zero near ω_mp , the second term in the numerator of Eq. (9) has a large negative imaginary part, which leads to a negative imaginary part of permittivity near ω_mp , as shown in Fig. 5(c) for the case of SRR with one or three layers. Similarly, the first term in the numerator of Eq. (9), $n_b^2$, is retrieved from measurement of case (b), which means µ ₁ in case (b) has a plasma frequency of ω′_mp. From Eq. (5), we can expect that $n_b^2$ has a large positive imaginary part near ω′_mp. As the number of layers increase in the test material, ω′_mp decreases to ω_mp , then the negative imaginary part of permittivity raised by the second term is canceled by the first term, which lead to more reasonable results, as shown in Fig. 5(c) for the case of SRR with five layers. Therefore, the principle backed in this unphysical phenomenon (the negative imaginary part of permittivity retrieved from SRR with less than 5 layers) is different with that in Fig. 5(b). It should also be noted that there is no anti-resonant behavior around the resonant frequency, the reason is that in the orientation of case (b), the wave number in the x̂ direction always equals to k_x =π/a, so the applied H _z field that is perpendicular to the SRR have a spatial variation on a scale significantly larger than the periodicity of SRR in the x̂ direction, which means in this orientation, SRR can form an effective medium over the resonant frequency regime.

 

Fig. 5. Measured constitutive parameters of the SRR structure.

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5. Conclusion

We presented a waveguide-based retrieval method for measuring complex permittivity and permeability tensors of metamaterials. Both the retrieval algorithm and experimental realizations are proposed. The whole retrieval procedure is verified both numerically and experimentally, using various kinds of materials with known properties. The successful retrieved results show the effectiveness and robustness of our method. In addition, the experiment measurement is easy to carry out, and there are no tough requirements for the material sample, indicating a good candidate in the homogenizations of metamaterials.