## Abstract

An improved planar structure of left-handed (LH) metamaterial is presented, and then designed and analyzed in microwave regime. In the anticipated LH frequency regime, the LH property is validated from the phenomena of backward wave propagation and negative refraction. To characterize the electromagnetic property of the planar metamaterial, we introduce the wedge method by constructing a wedge-shaped bulk LH metamaterial by stacking the planar LH metamaterials. The effective refractive index estimated by the wedge method is in excellent agreement with that retrieved by the inversion method from the transmission and reflection spectra.

© 2008 Optical Society of America

Since the hypothesis of left-handed (LH) materials [1] came true both theoretically [2] and experimentally [3] by artificially designed structures, the related interests in the LH metamaterials have grown explosively due to the peculiarity and potential applications, such as negative refraction [1], superlens [4], and phase compensation [5]. Great effort has been devoted to exploring an important task in the LH metamaterials, which is to realize the LH property at high frequency. Due to the difficulty in fabricating the LH metamaterials with the topological structure (i.e., combination of split-ring resonator and metal wire) [2, 3] in optical frequency regime, a new structure design by the coupled pairs of rods separated by a thin dielectric film [6] was pioneered for achieving the LH property, based on the idea proposed by Podolskiy *et al.* [7, 8]. For such planar LH metamaterials, negative magnetic permeability is provided by the excitation of asymmetric currents in pairs of either rods or strips and negative electric permittivity is usually achieved by the pairs of continuous metal wires[6]. Following such an idea, various planar structures have been designed successfully to show the LH property in the near infrared and even optical frequency regimes [9, 10, 11, 12, 13].

Here we propose and design a new planar LH metamaterial, which permits to operate for arbitrary polarized EM wave, because we deploy a symmetric unit cell design. We first calculate the reflection and transmission coefficients by the finite element method (FEM), and then retrieve the effective EM parameters (including permittivity, permeability, refractive index, and thickness) by the inversion method. The simulated field distributions at different moments validate that the phase velocity is in opposite direction to the energy propagation within the desired LH band. Importantly, the wedge configuration is introduced to characterize the EM property of the planar LH metamaterials. The simulation results show that the wedge geometry is a very reliable method, which can not only observe negative refraction at the interface between the LH metamaterial and vacuum but also estimate the effective refractive index. The results reveal that the effective refractive index given by the wedge method is in excellent agreement with that obtained the inversion method.

Figure 1 shows the structural details of the planar LH metamaterial we designed. The CGN500-NF-3006 commercial printed circuit board (PCB) is adopted. The PCB is a symmetric sandwiched structure composed of a dielectric interlayer (substrate) and two thin copper layers. The interlayer has a permittivity of 2.3, a low dissipation factor of 0.0008, and a thickness of *d _{S}*=0.5 mm. The two copper layers have the same thickness of

*d*=18

_{C}*µ*m. Our designed pattern can be manufactured through conventional commercial photolithography technique. The patterns of the two copper layers have the same shapes and no lateral displacement with each other, as shown in Fig. 1(b). In a single unit cell, the copper pattern is composed of one central and four corner squares, with the dimensions of

*b*×

*b*and

*a*×

*a*, where

*b*=7.8 mm and

*a*=1.625 mm, as shown in Fig. 1(c). In particular, there is an overlapping region with the dimension of

*g*×

*g*between the central and corner squares. The transmission spectra of the single-piece planar structure are calculated by FEM, at five different values of

*g*=-0.1,-0.05,0,0.05, and 0.1 mm.

It can be found, from the results furnished in Fig. 2, that the transmission spectra exhibit the distinct difference between the situations of *g*<0 (without overlapping between the central and corner copper squares) and *g*≥0 (with overlapping). The transmission spectrum is slightly changed as *g* varies, within the range of either *g*<0 or *g*≥0.

We now explore the distinct difference in transmission spectrum between *g*<0 and *g*≥0 will give rise to what inherent difference in physical property. Based on the inversion method [14, 15], from the reflection and transmission coefficients (*r*
_{11} and *t*
_{21}) calculated by FEM, we retrieve the effective parameters (refractive index *n*, permittivity *ε*, and permeability *μ*) of the single-piece planar metamaterial in terms of an effective homogeneous medium. The retrieved results reveal that when *g*≥0, the planar metamaterial exhibits the LH property within the frequency regime we anticipated; in contrast, the LH property disappears when *g*<0, because in this case the discontinuous copper layers cannot provide the negative electric response frequency regime to be the same as the negative magnetic response. Our attentions below focus on the case of *g*=0 for simplicity and without the lack of generality, so the dimension of the unit cell should be *D*×*D* with *D*=*b*+2*a*. Moreover, we restrict that the area of the cental copper square is larger than the total area of the four corner copper squares, i.e. *b*>2*a*.

Figure 3 shows the calculated parameters of the single-piece planar metamaterial as an effective homogeneous medium for the case of *g*=0 (*b*=7.8 mm, *a*=1.625 mm, and *D*=11.05 mm). Figure 3(a) plots the reflection and transmission (R-T) coefficients (*r*
_{11} and *t*
_{21}) calculated by using FEM as a function of frequency. The valley *V _{m}* located at

*ω*=13.88 GHz in

_{m}*t*

_{21}indicates the magnetic resonance. Figures 3(b)–(d) show the retrieved effective

*n*,

*ε*, and

*μ*, respectively, by the inversion method [14, 15]. The retrieved results indicate that our designed planar metamaterial acts as a LH material with low loss within the frequency regime from 14.11 to 14.38 GHz, while a nearly lossless RH material within the frequency regime from 16 to 17 GHz. The effective thickness of the single-piece planar metamaterial as an effective homogeneous medium can estimated to be

*d*=2.2 mm. Therefore, the single-piece planar metamaterial we designed can equivalently considered to be a homogeneous metamaterial slab with an effective

_{e}*n*,

*ε*,

*μ*, and thickness

*d*=2.2 mm, as the inset shown in Fig. 3(d). In addition, inasmuch as we adopt the very low-loss PCB in the microwave regime, our designed metamaterial has the high figure of merit (FOM) beyond 10 within the LH band from 14.1 to 14.3 GHz (in particular, at

_{e}*f*=14.194 GHz, the highest value of FOM is about 15.6 and the effective

*n*is -1.35).

We now would like to recognize intuitively the EM property by exploring the propagation behavior of the EM radiation in our designed metamaterials. For this propose, we construct a multi-piece planar metamaterial structure by stacking ten identical pieces with no lateral displacement each other. Each single-piece planar metamaterial occupies a dimension of *d _{e}* in the normal direction of the piece, and the period in the normal direction is also

*d*. Therefore, this stacked structure is equivalent to a thick homogeneous metamaterial slab with the effective

_{e}*n*,

*ε*,

*μ*, and thickness 10

*d*, from the effective medium model, as depicted in Fig. 4.

_{e}We simulate the propagation behaviors of the EM radiation in the normal incidence by FEM, from the initial physical parameters (instead of the retrieved effective EM parameters). We select two frequencies *f*=14.172 and *f*=16.77 GHz for simulations, the results of the electric field distributions at different moments are shown by the upper and lower panels in Fig. 5, respectively. One can find that the phase velocity (or wavefront **K**) propagates in the opposite direction to the energy flow (or Poynting vector **S**), implying that the phase velocity (or the effective *n*) is negative at *f*=14.172. In contrast, the phase velocity propagates in the same direction as the energy flow, suggesting that the phase velocity (or the effective *n*) is positive at *f*=16.77. The results furnished by FEM verify that the planar metamaterial we designed exhibits the LH and RH property at *f*=14.172 and *f*=16.77 GHz, respectively, which are in excellent agreement with the results retrieved by the inversion method. It should be pointed that the effective homogeneous metamaterial slab has *n*=-1.51 at *f*=14.172 GHz and *n*=1.04 at *f*=16.77 GHz, respectively.

Based on the Snell’s law, the refractive index of a material can be scaled through a refraction experiment. This method has also been extensively utilized in characterizing artificial materials including metamaterials [3, 17] and photonic crystals [18, 19]. We utilize a wedge-shaped configuration stacked by our designed single-piece planar metamaterials to characterize its effective refractive index and then validate its LH property again. The result simulated by FEM in Fig. 6(a) shows the phenomenon of negative refraction of the EM radiation at *f*=14.2 GHz at the interface between our designed metamaterial and vacuum, which shows indeed the LH property. Through our analysis on the transmitted field in vacuum in the far-field regime, we can easily determine the angle of refraction *β* and then to calculate the effective *n* by the Snell’s law (the angle of incidence *α* is 11.26°). Moreover, the effective *n* values at four different frequencies are also estimated by the wedge method, as shown by circles in Fig. 6(b). It can be found that the effective *n* estimated by the wedge configuration (by TEM) is in good agreement with that retrieved by the inversion method.

We now give some brief discussions. The negative magnetic response originates from the coupled pair of the copper patterns (as the split ring resonator in Ref. 3) separated by the dielectric interlayer. The non-resonance negative electric response is provided by the continuous copper strip (like the continuous metal wire in Ref. 3). The decrease of the period *D* will results in the relatively small blue shift of the resonance frequency of the negative magnetic response, when the size *b* of the central copper square in the unit cell is invariable. When the period *D* of the unit cell is fixed, the increase of the dimension of the central copper square in the unit cell will give rise to the large red shift of the resonance frequency of the negative magnetic response. In addition, inasmuch as the periodic boundary conditions are utilized, the EM property of the single unit cell can completely describe that of the infinite period planar metamaterial. The periodicity plays a role of translational symmetry (long-range order) as that in the atomic crystal. We find that the breakage in periodicity leads to the shift of the LH frequency band, although the shift is not so large.

In conclusion, we have presented and designed a planar metamaterial in microwave regime, with the structure simplicity, low lossy, and isotropy of polarization. By using the finite element method, the transmission and reflection spectra are simulated, based on which, we retrieve the effective electromagnetic parameters (including effective permittivity, permeability, refractive index, and thickness) of the our designed planar metamaterials. The results verify that such a kind of structure may exhibit the LH property within a frequency band. The wedge configuration is introduced in characterizing the effective refractive index of the planar metamaterials, and the results confirm the retrieved effective refractive index.

This work is supported by the Key Program of Ministry of Education of China under Grant 305007, and the National Basic Research Program of China under Grants 2006CB921805, 2004CB719801 and 2007CB310404.

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