## Abstract

We explored numerically a new negative index material in mid-infrared spectrum, based on a thin wire net pairs and a square ring pairs array, which exhibits simultaneously hyper-transmission, polarization independence and small period. The mechanism implementing the negative refractive index was analyzed using retrieved optical constants as well as a valid analytical expression of the effective permeability that was deduced by virtue of a simple equivalent circuit model and applied to account for the magnetic property of this metamaterial.

© 2012 OSA

## 1. Introduction

Negative index material (NIM) has been intensively researched especially in near infrared and visible light spectra because of its potential stunning optical applications. The representative work is the classic “fishnet” structure on account of its outstanding performance and simple preparation process, which was proposed firstly and demonstrated experimentally by S. Zhang et al in near infrared [1, 2]. The original design of the fishnet is physical connection of a two dimension array of wide infinite long stripe pairs with orthogonal thinner one, which is able to give a low absorptive loss in terms of the figure of merit (FOM) of 6.0 [1]. The experimental value of 3.0 was achieved by G. Dolling et al through replacing the constituent material and further optimizing geometrical data [3]. However, the designs, similarly to SRRs, suffer from polarization dependence that is unacceptable in some applications, for example, the proposed perfect lens based on isotropic NIM in which high symmetry is desirable [4].

On behalf of obtaining polarization independence, some effort have been afforded to optimize the fishnet using symmetric patterns such as square or circular holes [5, 6], square and rounding corner cross apertures [7, 8]. Unfortunately, in turn, the transmission property deteriorated substantially subsequently relative to that of the fishnet with rectangular holes at the same operating wavelength. This can be understood qualitatively. In order to obtain the symmetry, one had to substitute narrower strip for wider one. The substitution introduced the larger volume fraction of metal that lead to the blue-shift of electric plasma frequency (this can also be explained quantitatively in cutoff wavelength from the perspective of a hole waveguide [9]). Correspondingly, the impedance mismatch occurred and resulted in the more reflection loss, which was responsible for the degeneracy of transmission property.

For the sake of getting the impedance match back, a direct idea is to red-shift the electric plasma frequency through enlarging the aperture size [9]. At the same time, it is essential to expand the period for holding the magnetic resonance frequency, but the permeability also deteriorated subsequently due to the decreasing density of the “magnetic atoms” so that the mismatch still exist and the transmission properties could not be improved substantially. In a word, the dependence between electric and magnetic elements prevented a good transmission of fishnet with symmetric patterns [6]. Recently, an improved fishnet structure with polarization independence and high FOM (low absorptive loss) were reported by employing a second-order magnetic resonance, which becomes stronger than the first-order magnetic resonance (i.e., the above-mentioned common resonance) with the enlarging of apertures and period [10]. However, in another side, the large period make the fishnet more like photonic crystal, which is generally operated in their “Bragg regime” where the period is typically of the order of half a guided wavelength (namely *λ/a*~2) [6, 11], rather than metamterial in which the larger ratio of wavelength to period is safer for the effective homogeneous medium approximation. In fact, the most values of all the ratios in symmetric fishnet structures are less than 2.0 [6, 10]. This should be ascribed to the limitation about 2.0 of the ratio of cutoff wavelength (corresponding to the effective electric plasma frequency) to period [9]. Certainly, the ratio of wavelength to period could be designed to be lager by placing the magnetic resonance at those wavelengths that are much larger than the cutoff wavelength but we will face the impedance mismatch.

It is well known that the mid-infrared is an important regime in remote sensing technology, and yet in which there are few reports of NIMs. In this work, we explore firstly a NIM that is made up of a thin wire net pairs (WNP) and a square ring pairs (SRP) array and presents simultaneously hyper-transmission, polarization independence and small period in mid-infrared. We do not only analyze the mechanisms of achieving these performances, but also build a numerical model of the effective permeability based on a simple equivalent circuit model to interpret the magnetic property.

## 2. Design and simulation

Our NIM consists of three layers of films (metal-dielectric-metal) which were patterned by perforating holes to form a WNP and a SRP array configuration, as schematically shown in Fig. 1(a)
. The WNP is equivalent to a fishnet only composed of thin wires that was employed to provide the negative permittivity in present band, and the permittivity was tuned mainly by the period *a* and line width *w*_{n} as indicated in the unit cell in Fig. 1(b). The SRP array was expected to supply negative permeability originating from strong magnetic resonances. The permeability of the SRP can be tuned by scaling geometrical parameters independently or synergistically, which includes the period *a*, side length *l*, strip width *w*, metalization thickness *t* and dielectric spacer thickness *d* etc. The separate electric and magnetic elements allow us to overlap easily the negative permittivity and negative permeability in the present spectrum.

The optimization of our structure was performed by the aid of commercial program CST Microwave Studio, which using a finite-difference time domain method to determine complex scattering parameters, namely, reflection parameters S_{11} and transmission parameters S_{21}. With regard to the constituent material, a loss-free zinc sulfide (ZnS) that is transparent throughout the spectrum from ~1 μm to ~10 μm was chosen as dielectric spacer whose refractive index is nearly 2.24 in the region [12]. The metal element was defined as silver whose dielectric behavior is described by Drude dispersion model, i.e., $\epsilon (\omega )=1-{\omega}_{ep}^{2}/({\omega}^{2}+i{\nu}_{c}\omega ),$ where $i=\text{exception 25:},$the plasma frequency *ω _{ep}* = 2π × 21.75 × 10

^{14}rad/s and the collision frequency

*ν*= 2π × 4.35 × 10

_{c}^{12}rad/s [13]. For simplicity, in present work, we did not take the influence of the substrate on the electromagnetic behavior into account and assumed that transverse electromagnetic (TEM) wave impinges on the infinite surface of the structure as depicted in Fig. 1(a).

## 3. Results and discussion

We acquired a group of optimized geometrical data of a = 2.70 μm, *w _{n}* = 0.34 μm,

*l*= 1.60 μm

*w*= 0.47 μm,

*t*= 0.06 μm and

*d*= 0.22 μm through the electromagnetic calculation. The complex scattering parameters are displayed in Figs. 2(a) and 2(b). These scattering parameters both the magnitude and phase have the same features as that in the negative index region presented in the Ref [1]. and [14], where the dip in the phase of S

_{21}indicates the presence of a negative index band. The extracted complex refractive index using a classical retrieval procedure within Ref [14]. confirms that our metamaterial is a NIM ranging from 7.5 μm to 9.3 μm wavelengths, as is shown in Fig. 2(c), where

*n(λ)*is real part of the refractive index and

*κ(λ)*is extinction coefficient. The figure of merit as an absorptive loss indicter was also calculated by$\eta (\lambda )=-n(\lambda )/\kappa (\lambda ).$ Fig. 2(d) shows that the maximum value of η

*(λ)*is 6.9 around an operating wavelength of 8.1 μm where

*n(λ) =*−1. The corresponding transmission given by |S

_{21}|

^{2}is up to 90.2%. Furthermore, a ratio of 3.0 of operating wavelength to period is acquired. It should be mentioned that there is a bad transmission about 60% around 8.8 μm wavelength because of the large extinction coefficient although another

*n(λ) =*−1 is obtained here, which is also the case in literatures.

#### 3.1. Mechanism of implementing the NIM

It is desirable to analyze the permittivity, i.e., *ε _{eff}(λ) = ε_{1}(λ) + iε_{2}(λ)*, and the permeability, i.e.,

*μ*, in order to explain the fulfillment of the NIM. Figure 2(e) explicitly demonstrates that the NIM is double negative in the range of 7.5 μm - 8.7 μm and is single negative between 8.7 μm and 9.3 μm. And now, we analyze the contribution of the WNP and the SRP array to the permittivity and permeability. Figure 2(f) reveals the retrieved real part permittivity and permeability of the isolated WNP and the isolated SRP array (they are denoted as subscripts of “1n” and “1r”, respectively). Clearly, the WNP displays dispersive permittivity ε

_{eff}(λ) = μ_{1}(λ) + iμ_{2}(λ)_{1n}(λ) with a plasma frequency of 3.0 μm, which is mainly dominated by the cutoff wavelength of the hole waveguide in the WNP [9]; note that the SRP has also an electric response ε

_{1r}(λ) with a electric resonance of 3.25 μm rather than infinity, which stems from the additional depolarization effect due to its finite-long nature [15]. The two dispersive permittivity determine jointly the ε

_{1}(λ) of the combined system and lead to a new plasma frequency of 7.5 μm and a new resonance frequency around 3.1 μm in ε

_{1}(λ) (see Fig. 2(e)). Actually, the ε

_{1}(λ) is to first approximation equal to the sum of ε

_{1n}(λ) and ε

_{1r}(λ) [15]. That is to say, the “plasma wavelength” after introducing the SRP is 2.5 times that of the isolated WNP. This allows us to increase the ratio of operating wavelength to period. On the other hand, the invariant permeability μ

_{1n}(λ) of + 1.0 implies that the WNP is nonmagnetic in the present spectrum. Figures 2(e) and 2(f) illustrate that the permeability of the SRP array is almost same as the one of the combined system. This indicates that the magnetic property of the SRP array mostly determines that of the combined system and is not distinctly affected by the WNP. This is because the WNP is nonmagnetic and can also be explained by a numerical model of the permeability deduced from RLC equivalent circuit model shown in next section.

#### 3.2. Numerical model of permeability

To investigate the underlying magnetic property, we monitored computationally current density, electric and magnetic field distribution at magnetic resonance wavelength (8.7μm). For simplicity, we concentrate our attention on the SRP firstly. As shown in Fig. 3(a)
, two branches of parallel currents with mirror symmetry were independently excited on the surface of the square ring. Besides, another two branches of parallel currents flowing along reverse direction were also exited on the back-to-back square ring spaced by the dielectric layer. We divide the unit cell into two symmetric parts by the central line. Then, the SRP is divided into two groups of U shaped strip pairs (USSP). Each group of USSP sustains antiparallel currents resembling to cut-wire pairs [16, 17]. The antiparallel currents result from the coupling of electric dipole in the U shaped strip with its own image in the back one. The physical scene is revealed by the *z* component of E-field distribution in between the transversal metallic strips of the square rings and shown in Fig. 3(b). At the same time, the E-field closed the two branches of antiparallel currents sustained by USSP and then a virtual current loop between the metallic layers on a perpendicular plane to the incoming magnetic field formed. Furthermore, due to the accumulated opposite charges in the transversal strips, electric field is also expected to be confined within the separation gap between two nearest square rings. The E-field profiles in the spacer confirmed this picture, as is depicted in Fig. 3(c).

Considering all the description above, by analogy with cut-wire pairs [17, 18], the USSP can be mimicked by an equivalent circuit as shown in Fig. 4(a)
. The magnetic inductance *L _{m}* can be expressed as${L}_{m}={\mu}_{0}{l}_{eff}d/(2w),$where μ

_{0}is the permeability of vacuum,

*l*is the effective length of the vertical strip of the square rings under which the most of all magnetic energy was trapped, as depicted in Fig. 3(d). We regard roughly it as the average length of the square ring, i.e.,

_{eff}*l*. Since there are the larger charge distribution areas (the transversal strip) at the ends of USSP compared with cut-wire pairs, the plate capacitance

_{eff}= l-w*C*has a form of ${C}_{m}=\epsilon {}_{0}\epsilon {}_{r}f(0.5lw)/d,$where ε

_{m}_{0}is the permittivity of vacuum, ε

_{r}is the relative permittivity of the dielectric spacer, and

*f*is a fitting factor less than 1.0, which is used to adjust the effective charge distribution area of the transversal strip when the strip width vary, and 0.8 was chosen here. The gap capacitor approximated by two parallel wires of length 0.5

*l*and diameter

*t*is described as${C}_{g}=\pi {\epsilon}_{0}(0.5l)/\mathrm{ln}(g/t),$ where

*g = a-l*is separation distance between two nearest neighboring rings. Since one major source of the line broadening of optical constants originates from the damping of the constituent metal, dispersive resistances of the effective vertical strips are taken into account and have an expression of $R={l}_{eff}({\nu}_{c}-i\omega )/(wt{\epsilon}_{0}{\omega}_{p}^{2})$ [19].

If a virtual current loop mentioned above is viewed as a magnetic dipole, i.e. an artificial “magnetic atom”, the SRP array can be regarded as a large quantity of conformably oriented magnetic atoms. We simply provided that the loop current $I={I}_{0}{e}^{i\omega t}$ is induced by the external magnetic field $H={H}_{0}{e}^{i\omega t}$, and then the individual magnetic dipole moment is expressed as *IS*, where *S* = *l _{eff}d* is the area of the current loop. This leads the magnetization$M=NIS/V$, where

*N*is the number of magnetic atoms and

*V*is the volume of the array. One can easily derive$N/V=2/[{a}^{2}(2t+d)]$, so that the magnetic susceptibility ${\chi}_{eff}(\omega )=M/H$ is obtained. Finally, magnetic permeability can be calculated as${\mu}_{eff}(\omega )=1+{\chi}_{eff}(\omega )$. Now we need to find out the relationship between

*I*and

*H*. Applying the mentioned above equivalent circuit model illustrated in Fig. 4(a) and Kirchhoff’s voltage and current law (KVL and KCL), the circuit equations can be written as

*U*is the voltage drop over the parallel branch, and

_{p}*I*is the branch current in the parallel branch of

_{R}*R*and

*L*. Solve the Eqs. (2) and (3) above and yieldsWe denote$\beta =\frac{R+i\omega {L}_{m}}{1-{L}_{m}{C}_{g}{\omega}^{2}+iR{C}_{g}\omega}$, and then insert

_{m}*U*into Eq. (1), the expression of

_{p}*I*versus

*H*is immediately acquired as followingAs a result, the numerical model of analytical permeability is derived as

*w*of the WNP decrease to 0. Additionally, we find C

_{n}_{g}(≈7.6 aF) ~C

_{s}(≈6.1 aF),

*irC*~0 and

_{s}ω*R ~6r*, these explain why the WNP has few influence on the magnetic property of the SRP array.

Using Eq. (6), we are in a position predict intuitively the influences of the geometry on the permeability. Figure 5(a) sketches the analytical permeability with three different strip widths on the condition that the other geometries are held based on the previous optimized geometrical data of our NIM. We clearly observed the red-shift and attenuation of the analytical permeability with the narrowing of the metallic strip. The trend of the analytical results is comparatively consistent with that of the results extracted from the simulated scattering parameters shown in Fig. 5(b). This suggests that Eq. (6) can be used to describe the permeability of our NIM thus the foregoing explanation for the magnetic property is valid.

#### 3.3. Achievement of hyper-transmission

A better transmission performance could be expected by simultaneously optimizing two aspects: (i) smaller absorption loss; (ii) smaller reflection loss. Minimizing absorption is fulfilled by selecting low damping metal and transparent dielectric spacer. Minimizing reflection can be satisfied by matching impedance${Z}_{eff}(\lambda )={[{\mu}_{eff}(\lambda )/{\epsilon}_{eff}(\lambda )]}^{1/2}$ with free space. The generalized conditions of impedance match are *ε _{1}(λ) = μ_{1}(λ)* and

*ε*. As to the real part, we adjust the “magnetic plasma frequency” to equate the electric plasma frequency so that

_{2}(λ) = μ_{2}(λ)*μ*as close to

_{1}(λ)*ε*as possible (see Fig. 2(e), Fig. 5(b) also). Although the minus imaginary permittivity

_{1}(λ)*ε*originating from the disturbance of magnetic resonance is much weaker than imaginary permeability

_{2}(λ)*μ*around the magnetic resonance frequency, this does not prevent the accordance of the two imaginary part values at those frequencies far from the center frequency of the magnetic resonance. In Fig. 5(b), the devised NIM with a strip width of 0.47 μm exhibits better impedance match relative to the other two samples, thus present the best transmission performance in double negative regime marked by dashed box, as is shown in Fig. 5(c). It is worth mentioning that the magnetic plasma frequency is more sensitive than the electric plasma frequency to the changing of strip width, as is illustrated in Fig. 5(b). This implies that we can control relatively independently the permittivity and permeability, which allows us to match easily the impedance.

_{2}(λ)## 4. Summary

We proposed a new polarization-independent negative index material made up of a WNP and a SRP array. The hyper-transmission and small period were achieved simultaneously. The retrieved optical constants for the isolated WNP, SRP and combined system were employed to analyze the mechanism of implementing the NIM. We found that: (1) the permittivity of SRP and the one of WNP determine jointly the permittivity of the combined system. The addition of SRP broke through the limitation of electric plasma frequency of the WNP (corresponding to the cutoff wavelength of fishnet), and put it into lager wavelength; (2) the permeability can be tuned relatively independently by simply varying the strip width. The two aspects allow us to design the NIM with small period and matching impedance. In addition, we built a numerical model of the analytical permeability. The predicted results are good agreement with that obtained from retrieval method. Moreover, by virtue of the analytical expression, we explained why the WNP has few influence on the magnetic properties of the SRP array.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China (NSFC) (grants 51025208 and 61001026).

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