1. Introduction

V. G. Veselago [1] first postulated the existence of a negative index of refraction material (NIM) in 1968. But only in the last few years have NIMs been realized in practice by the appropriate combination of conductive or dielectric elements deposited on a substrate [2,3]. In current realizations in the microwave regime, NIMs are fabricated from metallic wires and rings assembled in a periodic cell structure. The rings are generally referred to as split ring resonators (SRR). The NIMs have the property that the effective permittivity ε_eff and permeability μ_eff are both negative. It has been shown that this results in a negative index of refraction, n=- (με)^1/2.

This paper presents numerical simulations and experimental investigations of NIM properties in the 10.0 to 15.0 GHz range for typical structures. The numerical simulations are performed using Microwave Studio (MWS). The simulated and measured results are compared for wires only, rings only and combinations of wires and rings. We show that the losses for wire and ring combinations can be reduced if careful attention is given to the materials used during construction. Next the effective permittivity ε⃡ _eff , permeability μ⃑ _eff and index of refraction n, are obtained from the scattering parameters obtained using the simulation tools. We compare these numerical values of n to experimental results obtained using a Snell’s Law experiment. For the structure examined the computed value of n is within 20% of the one experimentally measured from 13.6 to 14.8 GHz.

2. Numerical simulations using Maxwell’s equations solvers

The simulation of NIMs requires the ‘ab-initio’ numerical solution of Maxwell’s Equations in the time domain. Computational methods for reliable field simulations of NIM structures must meet a number of important criteria. Accurate geometric modeling of spatially intricate structures is required. The computational method must be able to model several kinds of materials, including metals with finite conductivity and dispersive dielectrics. Losses are a function of the inherent conductivity of the metals, the dielectric loss (loss tangent) of the substrate and of the other materials used to construct the NIM. The numerical method used for the simulation studies in this paper is the finite integration technique (FIT). Originally developed independently as a frequency domain approach, it can be regarded as a generalization of the FDTD method. A brief description of FIT can be found elsewhere [4]. We have used the commercial software packages MWS and Design Studio [5] (DS) for our simulations. Other authors [6,7] have used the transfer matrix method in their simulations. We have found that this approach is less desirable to study the full detail of the behavior of NIM’s, especially in the presence of ohmic losses.

The elementary cell structure of a one-dimensional (1D) NIM (labeled as 901) used for our numerical investigations is shown in Fig. 1. In a 1D NIM, the permittivity and permeability tensors, in the coordinate system of Fig. 1 where x is the direction of propagation, are given respectively by ε⃡=(1,1,ε_z) and μ⃡=(1, μ _y , 1). Here, the two concentric rectangular SRR generate the negative permeability μ _y , while the metal strip in the z-direction (the wire) generates the negative permittivity ε _z . The SRR and the wire are deposited on a dielectric substrate.

Similar to photonic band gap (PBG) structures, NIMs are formed from repeated unit cells containing scattering elements. A typical NIM contains a large number of unit cells as the one shown in Fig. 1. The cells are stacked along the direction of propagation x, the direction of the B-field, y and the direction of the E-field, z. The direct detailed simulation of a large number of unit cells is not possible due to computer memory and computational time requirements. The alternative efficient approach is to solve Maxwell’s Equations only in the unit cell with appropriate boundary conditions. The excitation of the cell is provided by a linearly z-polarized electromagnetic wave propagating in the x direction. The types of boundary conditions available are open, electrical, magnetic and periodic. Both electric / magnetic and periodic boundary conditions were used in this study.

 

Fig. 1. Unit cell showing electric / magnetic boundary conditions and the dimensions of the 901 structure used in the MWS simulations. Periodic boundary conditions were applied along the y-axis and z-axis when used. The direction of propagation of the electromagnetic field is along the x-axis, the electric field is oriented along the z-axis, the magnetic field along the y-axis. For the 901 structure C=0.025 cm, D=0.030 cm, G=0.046 cm, H=0.0254 cm, L=0.33cm, S=0.263 cm, T=17.0×10^-4 cm and W=0.025 cm.

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Once the solution of Maxwell’s Equations in the elementary cells has been found, the complex scattering parameters S₁₁ (reflection) and S₂₁ (transmission) in the direction of propagation, x, are computed. We then use DS to cascade multiple cells, thus obtaining the scattering parameters for an arbitrary number of cells in the direction of propagation. From the scattering parameters we can retrieve the effective ε_z, μ_y, n and Z of the NIM material.

3. Experimental and numerical results

3.1 Sample preparation and experimental measurements

The samples were fabricated by a commercial printed circuit board process with copper patterns on one or both sides of FR4 or Rogers 5880. The FR4 substrate was used for the wires only or rings only measurements, while the Rogers 5880 was used for the wire and ring combination. The boards were precision cut into strips, then stacked to create a slab structure. For the ring and wire combinations, Rohacell spacers were placed between the boards. Adhesive between the layers or tape at the outer edges may be used to hold the structure together. The samples were measured in a compact free-space microwave setup. The incident wave was polarized with the electric field aligned along the long dimension of the wires. The experimental setup permitted measurements of both transmission and reflection parameters using an HP 8510C network analyzer. Multipath nosie in the system was minimized by gating the analyzer. The error is estimated at ~ 0.1% of full scale for S₂₁ and ~0.5% full scale for S₁₁.

3.2 Scattering parameters

We initiated our experiments by determining the scattering parameters of a simple wire structure. The structure was similar to the one shown in Fig. 1 with the SRR removed. In such a structure there are two possible operational modes. In the first case the wires are directly connected to the electrical boundary. We will refer to this as the grounded case. In the second case the wires are not terminated at the electrical boundary; we will refer to this as the floating case. In the experiment we discovered that the grounded wire case corresponds to the illumination of the central region of the wire matrix only. This can be achieved by using a focus beam on the sample. The floating wire case corresponds to the full illumination of the wire matrix. In Fig. 2 (top) the computed and measured transmission Mod(S₂₁), of the wire matrix is shown as a function of the frequency for the grounded and floating wire cases. The agreement between the simulation and the experiment is quite good. Observe that in the grounded wire case the transmission is an increasing monotonic function of the frequency, as expected. For the floating wire case, the monotonic increase of the transmission is modulated. The discrete value of the frequency f_n/2 , for a minimum is given by f _n/2=(c/2 L_w )(n-1). Here, c is the speed of light in vacuum, L_w is the length of the wire and n (n≥2) is an even integer.

 

Fig. 2. Comparison of experimental and simulated transmission Mod(S21), of wires and rings. Electric / magnetic boundary conditions were used in the simulations. (Top) Wire only structure 3 cells deep in propagation direction having an effective length of the wires L_w =12.5cm. (Bottom) SRR only structure, similar to the 901 structure without wire, 3 cells deep in the direction of propagation. In the MWS simulations the wire and ring conductivity was 5.8×10⁷ (S/m) while the substrate dielectric constant ε_sub=3.7, and the loss tangent, tanδ=0.02.

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In Fig. 2 (bottom) the transmission for a SRR only structure is shown. As expected a stop-band is generated at approximately 11 GHz. The agreement between the simulation and the experiment is also quite good.

The single cell of the 901 HWD structure is shown in Fig. 3(a). The wire and ring dimensions are given in Fig. 1. The 901 HWD structure is derived from the 901 structure by adding an additional wire to the cell. The wire was added, to broaden the frequency region of negative dielectric constant. Figure 3(b) shows the comparison between the computed and measured Mod(S₂₁) parameters as a function of the frequency for the 901 HWD structure having both rings and wires. The band in which the structure displays negative index of refraction is between ~ 13.6 and ~ 14.8 GHz. The agreement between the measured and MWS computed modulus of S₂₁ appears to be satisfactory. In the numerical simulation the values for copper conductivity (5.8×10⁷ S/m) and a substrate dielectric constant (2.2) approximately equal to the nominal value for Rogers 5880 have been used. A loss tangent of 0.0009 was used for the Rogers 5880 cards, no adhesive was used to hold the structure together. The Rohacell spacers expanded the unit cell y-dimension from 0.33 cm to 0.51cm

 

Fig. 3. Comparison of computed and measured Mod(S₂₁) scattering parameters for a ring and wire structure. A Rogers 5880 substrate was used with Rohacell spacers and tape at the perimeter to hold the structure together. Periodic boundary conditions were used in the simulations. (a) 901 HWD structure showing additional wire in each unit cell. (b) Measured (blue) and computed (red) Mod(S₂₁) scattering parameter for 3 cells in propagation direction..

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3.3 Power losses

Since some previous NIM structures have exhibited low transmission we conducted an investigation [8] into the losses in NIMs. As for all materials, losses in NIMs will play a critical part in their possible applications. The structure insertion loss is defined as l=1-(S ₁₁ S*₁₁+S ₂₁ S*₂₁). The sources of losses are in the finite conductivity of the metallic (copper) layer and in the dielectric loss of the substrate, along with other materials that may be used to construct the NIM.

The loss tangent, assuming that μ′=1.0 and μ″=0, is defined as tan δ=ε″/ε′, where ε′ and ε″ are the real and imaginary part of the dielectric constant ε=ε′-iε″, respectively. The dielectric losses are concentrated in the high field regions, as the numerical simulations clearly show in Fig. 4. Here the dissipated power density is given in false colors, in various locations of the 901 HWD structure. It is obvious from this result that the small gaps concentrate the fields increasing the losses. It is also noticeable that the losses are limited to narrow regions surrounding the gaps. The removal of the dielectric in the region of high fields significantly reduces the losses. This is clearly shown in Fig. 5(a) where the loss tangent of the adhesive used to hold the structure together is varied. Reduction of adhesive loss tangent significantly increases the transmission for the case where the adhesive is placed directly over the azimuthal ring gap. If the adhesive is placed away from the gaps, as indicated in Fig. 5(a), then it has little effect on the transmission properties of the structure. Thus, if it is necessary to place an adhesive over the ring gaps, then it is critical that this adhesive have a low loss tangent. It is preferable to use construction methods that do not require adhesives over the gap regions. We also note, that due to the high dielectric constant of some adhesive, the NIM passband can be shifted to lower frequencies.

 

Fig. 4. Simulated losses at the pass-band peak for the 901 HWD structure. The thickness of the PC board was 0.025 cm. The nominal conductivity of Cu was given to the rings and wires. The dimensions of the 901 structure is given in Fig. 1.

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The copper conductive losses dominate at low values of the dielectric loss tangent [8]. Figure 5(b) shows that the metallic losses are reduced as the conductivity is increased. However, the loss reduction saturates above the conductivity of copper at 5.8×10⁷ S/m. The thickness of the metallic layer also affects the insertion loss. Approximately 5 skin depths, are needed to minimize the losses.

 

Fig. 5. Simulated transmission in the 901 HWD structure for (a) various adhesive loss tangents and (b) various metallic conductivities. The solid vertical bar indicates the conductivity of copper.

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3.4 Snell’s Law experiment

Several Snell’s Law experiments have been performed to show that the index of refraction of our structures becomes negative in a selected frequency region. These experiments reproduce and greatly extend the results reported previously [9]. The results for the 32⁰ wedge were reported previously [10]. Here we report the results for a 12⁰ wedge. This wedge was constructed using the low loss approaches discussed above. A Rogers substrate was used having copper wires and rings with Rohacell spacers without any adhesive at the Rogers / Rohacell interfaces. The structure was held together using shrink-wrap at the wedge external boundaries. An identical Teflon positive index material (PIM) wedge was used for calibration purposes. A focused beam illuminated the NIM and PIM wedges.

In Fig. 6 we present the measured normalized amplitude E_z (r, f), of the z-component of the electric field as function of the refraction angle r, and frequency f. The angle r is measured from the normal of the wedge exit face. The refraction angle r_Max is defined as [∂E_z (r, f)/∂r]_r=rMax=0. In Fig. 6 the Teflon wedge refracts the beam into positive angles. As expected, for the frequency range explored, the refraction angle r_Max is constant at ~17°. However, r_Max of the NIM wedge is strongly dependent on the frequency and appears at negative refractive angles.

 

Fig. 6. Surface plots of measured normalized peak amplitude of electric field component E_z (r, f) for the Teflon and 901 HWD NIM wedge. Note that the electric field refracted by the Teflon wedge peaks at a positive refractive angle of 17° (corresponding to an index of refraction of 1.4) and is independent of frequency. The electric field refracted by the NIM wedge however, peaks at negative refractive angles that are a function of the frequency

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We computed the index of refraction dispersion relation $n_{\mathit{\text{Th}}}^{\mathit{\text{NIM}}}$ (f), for the 901 HWD NIM structure, from the simulated values of the scattering parameters. The values of $n_{\mathit{\text{Th}}}^{\mathit{\text{NIM}}}$ obtained were compared with the experimental values $n_{\mathit{\text{Ex}}}^{\mathit{\text{NIM}}}$ , derived from Snell’s law $n_{\mathit{\text{Ex}}}^{\mathit{\text{NIM}}}$ =Sin (r_Max )/Sin (i), using the experimentally observed position of the refraction angle r_Max . The results shown in Fig. 7 agree within 20% over the frequency range tested (~13.6 to 14.8 GHz).

 

Fig. 7. Experimentally measured (data points) and simulated (solid line) dispersion relation of the index of refraction for the 901 HWD structure with substrate ε=2.2 and no adhesive used in the construction process. The measured and simulated values agree within 20%. The measured values were obtained from the Snell’s Law experiment and the simulated values were obtained from the simulated scattering parameters using retrieval procedures. Periodic boundary conditions were used in the simulations.

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4. Summary and conclusions

In summary we have designed, built and characterized several NIM structures operating in the 10 – 15 MHz region. We have also developed simulation and experimental procedures to test NIM systems. It has been found that the experimental performance of NIM structures closely follow the predictions from MWS numerical simulations. Performing a Snell’s Law experiment verified our procedures. Starting from rings and wires and leading to a Snell’s Law comparison of the index of refraction, the procedure produced results in agreement with the experimental results. Values of the simulated and measured index of refraction agree within 20% over the frequency range tested (~13.6 to 14.8 GHz). Our investigations have also included the study of the origin of the losses in NIM structures. These losses have been explored and approaches to mitigate them have been indicated and successfully implemented.