## Abstract

The wire medium consisting of an array of parallel thin metallic wires was previously studied by using an effective medium with spatial dispersion. In this paper, the validity of conventional effective model was examined analytically and numerically by studying a canonical structure of the wire medium. It is noted that the conventional model fails for high transversal spatial harmonics, which consequently results in discrepancy in the scattering between the effective model and the physical structure. In this study, we propose a new effective model to include higher order spatial dispersions: instead of the second-order expansion, the proposed dispersion equation is based on the fourth–order expansion of the dispersion equation of the photonic states. Compared with the 3D full-wave simulation results of the wire medium, the proposed model has demonstrated significant improvement in numerical accuracy in characterizing the EM behavior in this type of metamaterials.

© 2013 Optical Society of America

## 1. Introduction

The wire medium has been known for a long time [1–3] as a type of artificial materials, which consists of thin metallic wires (See Fig. 1) and possesses plasma-like frequency dependent permittivity.

In [4], the exact dispersion equation of ideally conducting thin wire array is obtained in closed form and is solved numerically. The only assumption is that the radii of wires are assumed to be much smaller than the lattice periods and the wavelength. The dispersion equation can be written as

*a*and

*b*denote the lattice periods, and

*r*denotes the radius of the wire. The symbol ${p}_{y}^{(0)}$ is the y component of the $n$

^{th}Floquet mode wave vector

Then in [4], Eq. (1) is expanded by using the second-order Taylor expansion of sin and cos functions of small arguments, yielding

Denote the wave vector in free space as $\overrightarrow{k}=\left({k}_{x},{k}_{y},{k}_{z}\right)$ . According to the boundary condition at the interface, ${q}_{y}^{}={k}_{y},\text{\hspace{1em}}{q}_{z}^{}={k}_{z}^{}$.From Eq. (3), a uniaxial effective model with strong spatial dispersion was proposed to character the wire medium under a further assumption of $a,b\ll \lambda $ [5]

The notation ${k}_{p}^{}$ plays the role of an equivalent “plasma frequency” which depends on the parameters *a*, *b* and *r*

For the commonly used case of the square grid (*a* = *b*), $F(1)=0.5275$.

By adopting an additional boundary condition [6], the transmission through an effective wire medium slab was theoretically studied [7]. Through comparison of the transmission- and reflection- coefficients, the spatial dispersion as shown in Eq. (4) in the effective wire medium is verified by numerical simulations [7].

It is possible to realize sub-wavelength imaging by using the wire medium [8]. Further study shows that the imaging system can be designed with literarily no limit of resolution if the thin wires are ideally conducting [9]. Besides, the wire medium slabs are able to transmit images to any distances, provided that the length *d* of the wires should be an integer number of half-wavelengths. This condition is necessary in order for the medium to fulfill the Fabry-Perot condition and thus eliminate unwanted reflections [7, 9]. However, if the thin wires are not made of PEC, theoretical and numerical studies show that the limit of resolution of wire medium slabs is determined by the skin depth of the metal [10].

Incorporating Eq. (4), a spatially dispersive FDTD is developed and its applications in sub-wavelength imaging are demonstrated both numerically [11, 12] and experimentally [9, 13, 14]. Studies in [15, 16] show that the transmission coefficient is a key factor in the image quality and the bandwidth of such an imaging system. The tapered array of wire medium with the separation between adjacent wires being radially enlarged is studied [17, 18]. Magnification or concentration of sub-wavelength field pattern in the microwave regime is demonstrated experimentally and numerically. Other interesting applications can be found in [19, 20].

The above literatures show that the transmission coefficient *T* is an important parameter of the wire medium, especially for a sub-wavelength imaging device where the transmission coefficient directly affects the quality of the image. However, it is noted that the conventional effective model becomes inaccuracy as the transverse spatial harmonics increase. Therefore, the discrepancy between the EM behaviors of the effective model and the physical structure occurs. In this study, we propose a new effective model to include higher order spatial dispersions: instead of the second-order expansion, the proposed dispersion equation is obtained based on the fourth–order expansion of the exact dispersion equation of the photonic states. Surface fitting is applied using the data from a 3-D full wave simulation. It is demonstrated that the proposed model has significant improvement in numerical accuracy when it is applied in characterizing metamaterials. This paper is organized as follows. In Sec. 2, the validity of the effective wire medium in terms of transmission coefficient is examined by comparing the theoretical result of the effective medium with the full wave simulation result of physical wire medium formed by parallel thin PEC wires. A criterion in the deviation of *T* of the effective model from that of the corresponding physical structure is given. Based on this criterion, the condition where the effective medium model becomes inaccurate is provided, and the origin of the discrepancy between the two results is discussed. In Sec. 3, a modified effective model based on a higher-order dispersive equation is proposed. Validation of the proposed model is presented in Sec. 4 before conclusions are drawn.

## 2. The validity of the effective wire medium

The PEC thin wire array (shown in Fig. 1) is simulated by using the FDTD method incorporating the sub-cell technique of thin wire [21]. The system is illuminated by a TM mode plane wave with an H_{z} excitation at operating frequency ${f}_{0}=3GHz$, with corresponding wave number denoted as ${k}_{0}$. The length of the thin wires is $d=\lambda /2$, where λ is the wavelength in free space. A fixed filling ratio $\pi {r}^{2}/{a}^{2}=0.001$ is adopted, indicating the physical thin wire array strictly meet the only assumption ${r}_{}\ll a,b,\lambda $ in [4]. A square lattice period $a=b$ is considered in this paper. According to Eq. (5), the lattice becomes denser proportionally with the increment of ${k}_{p}$. For example,${k}_{p}=9.683{k}_{0}$ corresponds to a dimension of $a=b=\lambda /40$. The transmission coefficient ${T}_{\mathrm{thin}wirearray}$ is calculated from the FDTD simulation result of the 3-D physical thin wire array by using Eq. (7):

The source and image planes are defined at different sides of the thin wire array in x-direction. The ${H}_{z\_imageplane}$ in Eq. (7) is the mean Hz on the image plane with the presence of thin wire array, while ${H}_{z\_sourceplane}$ is the mean Hz on the source plane in a counterpart domain without the wires, i.e., the free space model.

${T}_{\mathrm{effective}}$ is obtained analytically following Eq. (10) in [7] for the effective model, and is compared with the FDTD result of ${T}_{\mathrm{thin}wirearray}$. Two types of wire media with ${k}_{p}=4.8415{k}_{0}$ and ${k}_{p}=9.683{k}_{0}$ are studied with the result as a function of ${k}_{y}$ are presented in Fig. 2. For simplification, ${k}_{z}$ is set to zero both in the derivation of the analytical results and in the FDTD simulation.

It is revealed that *T* drops faster with the increment of ${k}_{y}$ in the physical thin wire array than that in the effective medium. Moreover, for the wire medium with a larger${k}_{p}$, ${T}_{\mathrm{effective}}$ in the effective model deviates slower from the FDTD result ${T}_{\mathrm{thin}wirearray}$ with the increment of${k}_{y}$.

In order to evaluate the validity of the effective medium model, we define the deviation ratio *DR*:

Figure 3 shows that the relation between ${k}_{y\mathrm{max}}$ and ${k}_{p}$ is nearly linear with a ratio of 0.71 during the ${k}_{p}$ range of $\left[2.42{k}_{0}-14.52{k}_{0}\right]$. That is, for a given ${k}_{p}$, the effective model can be considered as adequately accurate for harmonics ${k}_{y}\le 0.71{k}_{p}$ (equivalently ${k}_{y}a<0.11\pi $). This is a stricter criterion than the ${k}_{y}a<\pi $ in [7]. On the other hand, the transmission of the evanescent waves (${k}_{y}$>${k}_{0}$) can be calculated more accurately by using the effective model when the period lattice is smaller.

In the authors' opinion, there are two main reasons for such discrepancies in the result of *T* from the effective medium model and the physical structure. First, the analytical transmission coefficient of TM mode in [7] is deduced based on a homogeneous and uniaxial effective medium (see Eq. (4)). As a result, a TM excitation cannot excite a TE mode in the effective medium. Since the physical thin wire array is inhomogeneous in the transverse section, a TM excitation will generate TE mode propagation through the wire medium. In order to show the coupling of the TM and TE modes, the ratio of the average Ez and the average Hz on the image plane is plotted in Fig. 4. It can be seen that the coupling in wire medium increases with the increment of and with the decrement of . This coupling is a factor causing the deviation of the transmission from the effective model with the physical wire medium. This coupling cannot be considered in an effective homogeneous model unless the permittivity (or the permeability) as a non-diagonal tensor (electro- or magnetic- gyration) is introduced.

Second, the dispersion equation Eq. (3) is obtained by using the second-order Taylor expansion of sin and cos functions in Eq. (1) for small arguments. Since the permittivity of the effective model (see Eq. (4)) is derived from the 2nd order approximation (Eq. (3)), it is well understood that the effective model is sufficiently accurate only for small argument of ${k}_{y}$ (${k}_{y}a<0.11\pi $).

## 3. The proposed new effective uniaxial medium model

It is of significance for a sub-wavelength imaging system to be described accurately in calculating the transmission of the evanescent waves (${k}_{y}$>$k$) with an enlarged range of ${k}_{y}$. That is because the high-order spatial harmonics determines the details of an image in a sub‑wavelength imaging system. Therefore, a more accurate effective medium model is desirable.

Before proposing a new model of effective medium, it is worth briefly reviewing the derivation of ${T}_{\mathrm{effective}}$ in [7]. Denote ${k}_{x}=-j{\gamma}_{x}=-j\sqrt{{k}_{y}^{2}-{k}^{2}}$, and the propagation constant of TM wave in the wire medium as ${\gamma}_{TM}$.

By combining Eq. (3) with Eq. (9), ${\gamma}_{TM}$ can be written aswhere ${q}_{z}^{}={k}_{z}=0$ has been assumed. By solving the system of equations for the total magnetic field (ref [7].), the expression of ${T}_{\mathrm{effective}}$ can be obtained as follows:Obviously, by using higher-order Taylor expansion of sin and cos functions for small arguments of Eq. (1), a more accurate dispersion equation can be obtained, which corresponds to a propagation constant ${{\gamma}^{\prime}}_{TM}$ in a more complicated form. In the following, the fourth-order Taylor expansion is applied to expand the sin and cos functions in Eq. (1). Naturally, ${T}^{\prime}{}_{\mathrm{effective}}$ as a function of ${{\gamma}^{\prime}}_{TM}$ is expected to agree with ${T}_{\mathrm{thin}wirearray}$ better than the conventional one. However, a complete solving of a fourth-order dispersion equation in obtaining ${{\gamma}^{\prime}}_{TM}$ is cumbersome, and the expression of ${{\gamma}^{\prime}}_{TM}$ is implicit. By solving a quadratic equation with respect to ${q}_{x}$ adopting the 1st order Taylor expansion approximation, ${{\gamma}^{\prime}}_{TM}$ can be expressed explicitly using a polynomial with undetermined coefficients. Then after obtaining discrete data of ${{\gamma}^{\prime}}_{TM}$ through ${T}_{\mathrm{thin}wirearray}$ in Eq. (11) (for square lattice), a surface fitting tool of the commercial software, i.e. Origin, is applied to fit the surface of ${{\gamma}^{\prime}}_{TM}$ as a function of ${k}_{y}$ and ${k}_{p}$. Through fitting, the most important terms in the polynomial are found and a neat form for ${{\gamma}^{\prime}}_{TM}$ is obtained as

Next, the modified ${{T}^{\prime}}_{\mathrm{effective}}$ is obtained by submitting ${{\gamma}^{\prime}}_{TM}$ into Eq. (11), and it is compared with ${T}_{\mathrm{thin}wirearray}$ (see Fig. 6) in two types of wire media with ${k}_{p}=4.8415{k}_{0}$ and${k}_{p}=9.683{k}_{0}$. By using the accuracy criterion proposed above, ${{k}^{\prime}}_{y\mathrm{max}}$ is obtained and plotted in Fig. 7. It is obvious that valid range of the proposed effective model is significantly extended (${{k}^{\prime}}_{y}\le {k}_{p}$, equivalently ${{k}^{\prime}}_{y}a<0.16\pi $).

Because of the symmetry of the structure in the y- and z- directions, Eq. (12) can be easily generalized to a more general form

*A*and

*B*are provided above. Based on Eq. (13), the improved dispersion equation can be obtained as

Thus, by incorporating the assumption of Eq. (4a), and the dispersion equation for extraordinary plane waves (${E}_{x}\ne 0$) in this uniaxial dielectrics [5]

the improved effective model for the thin wire array can be expressed as## 4. Validation

First, examples of oblique incidence are considered. Plane waves polarized in the x-direction is illuminated on the wire medium with ${k}_{y}^{}={k}_{z}=\left(\sqrt{2}/2\right)\cdot {k}_{t}$, where ${k}_{t}$ denotes the transverse wave component. Two wire media with ${k}_{p}=4.8415{k}_{0}$ and ${k}_{p}=9.683{k}_{0}$ are examined. Figure 8 demonstrates that the proposed effective model is sufficiently accurate in a general way.

The above numerical experiments are based on wire medium of square lattice with the filling ratio fixed at 0.001. Next, the wire medium based on square lattice with different filling ratios (correspondingly, different ${k}_{p}$) and that based on rectangular lattice (*b = 1.5a*) are investigated with results plotted in Figs. 9 and 10 respectively. It clearly shows that the effective model describes the characteristics of wire medium accurately for various types of wire media.

## 5. Conclusion

In this paper, full wave simulation of the wave behavior in the parallel thin wire array is used to investigate the effectiveness of the effective model in a quantitative manner. Based on the Helmholtz equation and by assuming a homogeneous uniaxial electric property in the effective medium, a higher-order expansion is applied to the exact dispersion equation for the physical thin wire array. By surface fitting with the data from the FDTD results, a 4th order dispersion equation is finally obtained, which corresponds to an improved effective medium model. Comparing to the conventional model, the range of transverse spatial harmonics ${k}_{t}$ in which the transmission can be calculated with sufficient accuracy analytically is extended from $\left[0,\text{\hspace{0.17em}}0.11\pi /a\right]$ to $\left[0,\text{\hspace{0.17em}}0.16\pi /a\right]$. This improvement is of significance in determining the image quality in the sub-wavelength imaging applications. It has also been demonstrated that this proposed model is applicable for generalized cases, i.e., for wave incidence with arbitrary incident direction and with various type of wire media.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant no. 61102022) and by the Fundamental Research Foundation of Beijing Institute of Technology of China (Grant no. 20120542014).

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