An analysis of a general system of resonators

An analysis of systems of NEP resonant elements was performed by the method presented in [Reference Machac3], rewritten to a description of the electric field instead of the magnetic field, i.e. by a dual approach. Two systems are studied. In the first material, resonant particles are located in a regular periodic 3D net with the period a, all aligned in the direction parallel to the z-axis, see Fig. 1(a). In the second system, the resonators are located in random positions dispersed around the lattice nodes in perturbations of Δx, Δy, Δz, and are oriented randomly around the z-axis in perturbation of Δθ. The variation of the second spherical angle φ is not relevant as the electric moment of the NEP element does not depend on this angle. The resonant element can be in the plane x – y oriented at any angle φ as the electric field is always parallel to its main axis, see Fig. 1. This corresponds to the TE ₁₀ mode in rectangular waveguide where the electric field is parallel to z-axis, see Fig. 1(b). A sketch of the i ^th resonator located in a nominal position in the coordinate system is shown in Fig. 1(c). The resonators are modeled by small electric dipoles. Their resonant frequencies differ from each other due to imperfect fabrication. The distribution of the resonant frequencies is assumed to be normal.

Fig. 1.

A 3D periodical system of elements aligned in the same direction (a), elements inserted into three polystyrene slices. The direction of z-axis is shown (b). Sketch of the i ^th NEP resonant element located in the coordinate system in the position defined by x _i, y _i, z _i, (the i ^th node – nominal position) and at orientation angle θ _i (c).

The total electric field averaged over the cell volume V = a ³ is aligned in the z direction ${ {\hat {\bi E}}} = {\hat {E}}\,{\bi z}_o$. The local electric field at the origin of the coordinate system that excites the reference dipole numbered as “0” is composed as the superposition of all fields E_i excited by dipoles located in positions i

(1)$${\bi E}_{0loc} = { {\hat {\bi E}}} + \sum\limits_{i \ne 0} {\left( {{\bi E}_{i{\kern 1pt}} - {{ {\hat {\bi E}}}}_{i{\kern 1pt}}} \right)} - { {\hat {\bi E}}}_0.$$

In fact, equation (1) compares the differences between the local fields and the averaged fields [Reference Tretyakov14]. Fields E_i are expressed by [Reference Jackson15]

(2)$${\bi E}_i = \displaystyle{1 \over {4\pi \varepsilon _o}}\left[ \matrix{k^2({\bi n}_i \times {\bi p}_i) \times {\bi n}_i\displaystyle{{e^{ - {\rm j}kr_i}} \over {r_i}} \hfill \cr + [3{\bi n}_i({\bi n}_i \cdot {\bi p}_i) - {\bi p}_i]{\kern 1pt} \left( {\displaystyle{1 \over {r_i^3}} + \displaystyle{{{\rm j}k} \over {r_i^2}}} \right)e^{ - {\rm j}kr_i} \hfill} \right],$$

where n_i and r _i are the direction vector and the distance between nodes 0 and i. In (1), the fields are summed over all dipoles with the exception of i = 0. The reference dipole at position i = 0 with electric moment p₀ excites a field with an averaged value [Reference Jackson15]

(3)$${ {\hat {\bi E}}}_0 = - \displaystyle{1 \over {3\varepsilon _oa^3}}{\bi p}_0.$$

The electric moments of dipoles oriented in directions defined by unit vectors p_oi are determined by the local electric fields E_iloc

(4)$${\bi p}_i = \alpha _i(\omega, \omega _{{\rm o}i})\,({\bi E}_{iloc} \cdot {\bi p}_{{\rm o}i})\,{\bi p}_{{\rm o}i},$$

where α _i(ω, ω _oi) is the electric polarizability of the resonator at frequency ω, and ω _oi is the resonant frequency.

The unknown components of vectors E_iloc are determined by solving a system of algebraic equations (1). The z component of the effective permittivity tensor is defined by the electric moment of the reference dipole, and assuming ${ {\hat {\bi E}}} = 1$ as a source quantity

(5)$$\varepsilon _{{\rm zz}}(\omega ,\omega _{\rm o}) = 1 + \displaystyle{{{\kern 1pt} \alpha _{\rm 0}(\omega ,\omega _{{\rm o0}})\,({\bi E}_{0loc} \cdot {\bi p}_{{\rm o}0})\,p_{{\rm o}0z}{\kern 1pt} } \over {\varepsilon _oa^3}},$$

where p _o0z is the z component of unit vector p_o0. Finally, the effective permittivity is determined by integrating (5) multiplied by the distribution function over the resonant frequencies. The electric polarizability of the dipole is taken in the Lorentz form

(6)$$\alpha _i(\omega, \omega _{{\rm o}i}) = \displaystyle{{A\omega ^2} \over {\omega _{{\rm o}i}^2 - \omega ^2 + j\omega \delta}}, $$

where δ represents losses and A is amplitude. These values were determined as proposed in [Reference Jelinek and Machac16] by simulating a single resonator in the R32 waveguide by the CST Microwave Studio and by measurements giving similar values.

In (1), the resonant frequencies are randomly chosen with normal probability of distribution. The perturbations of the resonator positions and of the orientations are randomly chosen independently for each dipole within the defined limits with a uniform distribution of probability. Finally, (5) is averaged over a number of realizations chosen high enough to obtain convergence. This averaging replaces N-dimensional integration. Generally, between 100 and 1000 realizations are taken. The number depends on the convergence of the simulation process. Convergence is ensured by getting smooth dependencies of permittivity on frequency. For most applications, it is sufficient to take into account one central cell surrounded by one layer of neighboring resonant particles from all sides (26 cells) to obtain smooth dependence of permittivity on frequency.

Selection of NEP resonant elements

There is a number of NEP resonant elements that have been presented in the literature. Some of them were taken here into consideration, those from ref. [Reference Tao, Padilla, Zhang; and Averitt11, Reference Imhof, Ziolkowski and Mosig13] and those already investigated by the author [Reference Machac, Rytir, Protiva and Zehentner2, Reference Machac, Protiva and Zehentner12]. These elements are shown in Fig. 2. Measurements, as described in [Reference Jelinek and Machac16], of all these resonators showed NEP. The amplitudes A for all resonators under consideration are shown in Table 1. This table shows that the NEP from Fig. 2(d) has an amplitude A almost one order higher than the other resonators. Therefore this resonant element is the most suitable for the application in a metamaterial showing negative permittivity between elements presented in Fig. 2.

Fig. 2.

View of particular NEP elements. An element is taken from [Reference Tao, Padilla, Zhang; and Averitt11] (a), an electric dipole terminated by an inductor [Reference Machac, Protiva and Zehentner12] (b), a double H-shaped resonator (DHR) [Reference Machac, Rytir, Protiva and Zehentner2] (c), a meander shaped resonator [Reference Imhof, Ziolkowski and Mosig13] (d).

Table 1.

Amplitudes A of the polarizabilities (6) of the resonant elements shown in Fig. 2.

The meander NEP resonators [Reference Imhof, Ziolkowski and Mosig13] were designed and fabricated using a Rogers RT Duroid 5870 substrate that was 0.254 mm in thickness with permittivity 2.33. The dimensions of the substrate were 6.7 × 6.7 mm², and the strips and slots were 0.15 mm in width. The resonant frequency measurements gave an average value of 2.965 GHz, with a standard deviation of σ = 0.015 GHz. The polarizability was measured as stated in [Reference Jelinek and Machac16] for a single resonant particle located at the center of an R32 waveguide. Fitting the resulting frequency dependence (averaged over six measurements of six resonators) determined values from (6) A = 1.28 × 10⁻¹⁸ m²F, δ = 0.355 GHz. The same measurements of the double H-shaped resonator (DHR) element [Reference Machac, Rytir, Protiva and Zehentner2] resulted in an average value of the resonant frequency 3.844 GHz and standard deviation of the resonant frequencies σ = 0.005 GHz. Measured values from (6) are A = 1.63 × 10⁻¹⁹ m²F, δ = 0.11 GHz. Similar values were determined for the resonators from Figs 2(a) and 2(b), see Table 1.

The suitability of the application of particular resonant elements to the design the metamaterial with negative permittivity has been proved by the presented homogenization method. The positions of the resonators were perturbed by spreading them randomly around the net nodes in all three directions by ±a/2, see Fig. 1(c). The two particular systems were analyzed. In the first system resonant elements were aligned in the same direction − Δθ  = 0°. In the second system, the resonators were randomly oriented in the perturbation of the full span that corresponds to Δθ = 180° taken with respect to the z-axis.

The dependence of the complex permittivity on frequency was calculated and plotted in Fig. 3. The metamaterials composed of both DHR and meander elements aligned in the same direction showed negative effective permittivity, see Fig. 3. The polarizability value of the DHR resonators is not high enough to ensure that a material composed of these resonant elements randomly located both in space and in orientation exhibits negative permittivity, Fig. 3(a). On the other hand, Fig. 3(b) proves that the real part of the complex effective permittivity of NEP in the shape of meander resonators calculated for randomly located resonators aligned in the full span corresponding to Δθ  = 180° reaches negative values.

Fig. 3.

Simulated complex effective permittivity of a system of DHRs [Reference Machac, Rytir, Protiva and Zehentner2] aligned in the same directions (Δθ =0°) and in a random distribution of directions (Δθ =180°) (a). The same plots for NEP [Reference Imhof, Ziolkowski and Mosig13] (b). Resonators are randomly spread in space in all three directions by ±a/2, the spread of resonant frequencies is defined for NEP by σ = 0.015 GHz, and for DHR by σ = 0.005 GHz.

An analysis of the resonators shown in Figs 2(a) and 2(b) proved in addition that they do not provide the negative real part of the effective permittivity if they are randomly distributed in space and in orientation.

Finally, this discussion showed that only meander-shaped resonators [Reference Imhof, Ziolkowski and Mosig13] out of the selected resonators can be used to design a metamaterial exhibiting negative effective permittivity.

Analysis and experiment – A metamaterial with negative permittivity

The scattering parameters of the designed metamaterial prisms were measured as inserted into a waveguide R32 with metal walls, and were compared with the results of the simulation done by the CST Microwave Studio. CST analyzes the prism of the metamaterial located in the waveguide as a homogeneous fictive material defined by the calculated complex permittivity, and permeability equal to 1.

A regular 3D metamaterial

The presented homogenization procedure was used to analyze the response of the regular 3D periodic system composed of NEP elements from Fig. 2(d) aligned in the same direction, Fig. 1(a). The experimental realization of this system is shown in Fig. 1(b). The resonant elements are inserted into three slices of polystyrene with a period of 11 mm. The calculated complex effective permittivity of this metamaterial is plotted in Fig. 4 with the standard deviation of the distribution of the resonant frequencies taken as a parameter. The system defined by σ = 0.015 GHz is a metamaterial that exhibits a negative real part of the effective permittivity in the interval from 2.97 to 3.13 GHz, with a maximum value of −1.4 at 3.0 GHz. With increasing σ, the frequency band of negative permittivity shrinks. Finally, for σ > 0.15 GHz this band disappears.

Fig. 4.

Calculated effective permittivity of the regular 3D periodic system of electric dipoles from Fig. 2d aligned in the same direction. Standard deviation σ is marked as “sig”.

Figure 5 shows the experimental verification of the simulation. The behavior of this metamaterial is determined by measuring the scattering parameters in an R32 (WG10) waveguide with metallic walls. The measured scattering parameters plotted in Fig. 5 were averaged over six permutations of three polystyrene slices, see Fig. 1(b). The agreement between the measured and calculated scattering parameters S ₂₁ is relatively good. For S ₁₁, there is a fit only in the frequency band of the resonance. The CST model used the block of a fictional homogeneous material defined by effective permittivity taken from Fig. 4 calculated for σ = 0.015 GHz.

Fig. 5.

Measured, simulated by CST scattering parameters of the metamaterial block composed by a 3D periodic system of electric dipoles aligned in the same direction. The metamaterial prism is located in waveguide R32.

An amorphous metamaterial

The amorphous metamaterial, see Fig. 6(a), was assembled by planar resonators from Fig. 1(d) put into spherical plastic shells 11 mm in outer diameter shown in Fig. 6(b). The diameter of the shells determines the system period a. The permittivity of the shell material is 3.3, and the losses are tanδ  = 0.02. The system consisting of 264 NEP elements in shells fills the sample holder that is R32 (WG10) waveguide with transverse dimensions 72 × 72 mm² and length 72 mm (a cube of edge equal to 72 mm). Now the resonators are fully randomly distributed and oriented. The scattering parameters were measured at reference planes at the faces of the holder.

Fig. 6.

The amorphous metamaterial in the partly disassembled measurement setup [Reference Zehentner and Machac1] (a). The two halves of the plastic shell with the fabricated NEP elements (b).

The model of the amorphous material for the purposes of the analysis presented here is composed of a system of resonators located in a 3D periodical net with the period a. This defines the basic geometry of the system. Now the positions of the resonators are perturbed by spreading them randomly around the net nodes in all three directions by ±a/2, see Fig. 1(c), with a uniform distribution of probability. The orientations of the resonators are randomly chosen with a uniform distribution of probability within the position angle measured with respect to the z-axis in the perturbation of the full span corresponding to Δθ = 180°. These conditions correspond to the real metamaterial implementation with no preference for any orientation in its realization, see Fig. 6(a).

The behavior of the resonators in this material is significantly influenced by the presence of the shells made of a plastic material with permittivity equal to 3.3. This material detunes the resonant frequency of the resonant particles downwards so that their average value is 2.727 GHz. The standard deviation of the distribution of the resonant frequencies was unchanged, and remained σ = 0.015 GHz. The losses of this material widen the particle polarizability response. The polarizability measurements [Reference Jelinek and Machac16] of NEP located in a plastic shell gave values for (6) A = 1.4 × 10⁻¹⁸ m²F, δ = 0.45 GHz. Now the polarizability (6) includes both the NEP resonator and the plastic shell. The presented homogenization takes this combination as a single resonator.

Figure 7 shows the results of the analysis of the amorphous material presented above, i.e. a metamaterial with resonators randomly located in space. The lines in Fig. 7(a) represent the real and imaginary parts of the effective permittivity (equal in all directions for the isotropic - amorphous – material studied here) calculated for different values of the standard deviation of resonant frequencies σ. The dispersion of the resonator orientations is described by the perturbation in the full span Δθ = 180°. It follows from the plot that the standard deviation of the resonant frequencies must be lower than or maximally equal to 0.015 GHz, in order to obtain a metamaterial with a negative real part of the permittivity. Therefore the fabricated resonant particles fulfill this requirement. This amorphous metamaterial exhibits the negative real part of the effective permittivity in the interval from 2.76 to 2.78 GHz, with a maximum value of −0.06 at 2.77 GHz. Better results would be obtained with the use of resonators with a smaller standard deviation value, see Fig. 7(a). Figure 7(b) shows the dependence of simulated effective permittivity on frequency, with the width of the perturbation of position angles Δθ as a parameter. The interval of the negative real part of the effective permittivity is naturally narrower as the perturbation of the angles increases. σ = 0.015 GHz is assumed in Fig. 7(b).

Fig. 7.

Calculated effective permittivity of the amorphous metamaterial for shown values of the standard deviation of the resonant frequency distribution marked as “sig”, Δθ = 180° is taken (a), for shown values of Δθ marked as “dth”, σ = 0.015 GHz is assumed (b).

An analysis of the magnetic metamaterial with negative permeability [Reference Machac3] has shown that a lower value of system period a must be used in order to obtain a higher negative effective permeability value. The resonators are more tightly coupled, which increases the value of the effective permeability. The same is valid for an electric metamaterial with negative permittivity.

A block of homogeneous metamaterial with the same dimensions as the fabricated specimen was analyzed by the CST Microwave Studio. The complex permittivity for this fictional material was taken from the analysis plotted in Fig. 7(a) for σ = 0.015 GHz. The simulated and measured scattering parameters are plotted in Fig. 8. The measured data in Fig. 8 were taken as the average value from 11 subsequent measurements for differently stirred resonators in the volume of the cube. They can be compared, but the comparison is not perfect. The reason is that fabricated resonators with dimensions of 6.7 mm are not negligibly small in comparison with the cell dimensions, which are equal to the system period of 11 mm. The cube inserted into the waveguide filled with a finite number of resonators located in shells has a limited volume. Additionally, the homogenization procedure used here models the resonators as point electric dipoles. This also reduces the preciseness of the results, because this condition is not exactly kept.

Fig. 8.

Measured, simulated by CST scattering parameters of the block of amorphous metamaterial located in a raised R32 (WG10) waveguide.