## Abstract

In this paper we use spatially variant metamaterial substrate to manipulate the directivity of antennas. We show theoretically that by embedding a dipole at different locations inside this substrate, the emitted rays can be directed to different orientations as required. As a result, spatial multiplexing can be realized by carefully selecting proper parameters of this substrate. It can also be observed that the electric field received in this antenna system is enhanced when it is used for reception. Simulations based on finite element method are used to validate our theoretical analysis, showing a controllable high directive property. In order to simplify the physical realization process, we propose the reduced parameters for practical design and also study it with numerical simulations.

© 2008 Optical Society of America

## 1. Introduction

Since metamaterials were first investigated [1–3], there has been a particular interest in their potential applications [4–10], among which antenna is an important one. The first experimental demonstration of high directive emission with a metallic mesh of thin wires was reported in 2002 [7]. Afterwards, theoretical analysis and simulations of antennas based on metamaterials were also presented [8–10]. In 2006, J. B. Pendry introduced the idea of using metamaterial whose permittivity and permeability values are designed to vary spatially to direct the fields [11]. This idea inspires a new scheme in EM device design, such as cloak, superlens, etc., and can be applied to realize many extraordinary capabilities [12–20].

The conventional method of controlling the directivity of antenna with metamaterial substrate is based on adjusting the permittivity and permeability elements of the constitutive matrix [7, 8]. Through the control of the structure’s geometry, the frequencies at which *ε*
_{eff}=0 and *μ*
_{eff}=0 can be tuned to the desired specification to produce directional emission. With this metamaterial substrate, any obliquely incident wave will be totally reflected and the electric field will evanesce rapidly after passing the incidence surface. It is easily seen that no matter at which position the source is embedded, the radiated rays will always be confined to a small solid angle around the normal.

Different with the previous method, here we use a spatially variant metamaterial whose parameters are properly designed by a transformation approach. We show that some novel capabilities can be achieved with our proposed method. For instance, the directivity of the antenna can be manipulated by locating the source (or reception dipole) at different positions inside the slab, and ultimately, control the radiation pattern of the whole antenna system. We theoretically demonstrate that only the wave perpendicular to the interface of the substrate slab can reach the other boundary and the penetration depth of the incident wave decrease as the incident angle increases. In addition, in the metamaterial substrate with space dependent parameters we proposed, the received waves incident from a range of angles are distributed as standing waves. We also show that the electric field received in the substrate is enhanced and the enhancement can be controlled by choosing different factors in the constitutive parameter expressions. Reduced parameters are also proposed to simplify the experimental design and spatial multiplexing can be realized with the approach we suggested.

## 2. Field distribution in the metamaterial substrate

With a scale transformation between the Cartesian coordinate system and the cylindrical coordinate system *ρ*′=*α*
*x* , *φ*′=*y*/*l* , *z*′=*z*, which maps a cylinder to an infinite slab, the relative permittivity and permeability tensors of the transformation medium are designed to be

where *α* and *l* are definite constants. To begin with, we first examine the field distribution in this metamaterial substrate with the thickness of *d*. For TE polarization (the following idea is also applicable to TM polarization case), we write the electric field vector as *E*̄=*z*̂*E*
_{z}(*x*, *y*).

By substituting it into Maxwell’s equations, the wave equation for *E*
_{z}(*x*, *y*) is obtained

We find that the basis function of the solutions to the wave equation has been changed after the scale transformation between different coordinate system, and the general solution of Eq. (2) is no longer the combination of Fourier basis functions, but takes the form as *E*
_{z}=*B*
_{k,l}(*α*
*k*
_{0}
*x*)*e*
^{ik,y}, where *B*
_{n}(*ξ*) is the *n*–th order of Bessel function.

Suppose a TE polarized plane wave *E*
^{i}
_{z}=*E*
_{0}
*e*
^{-ikxx+ikyy}(Here *k*
^{2}
_{x}+*k*
^{2}
_{y}=*k*
^{2}
_{0}.) is incident upon the slab at the interface *x* = *d*, as shown in Fig. 1. By applying phase matching at the boundary, the reflected field (*x* > *d*), the transmitted field (*x* < 0), and the field inside the slab (0 < *x* < *d*) can be obtained respectively:

with *J*
_{n}(ζ) and *N*
_{n}(ζ) representing the Bessel functions of the first, and the second kind respectively. Here *R*, *T*, *A*, and *B* are all unknown coefficients. By applying the continuities of *E*
_{z} and *H*
_{z} at the two interfaces *x* = *d* and *x* = 0, we can get

where *a* = *J*
_{kyy} (*α*
*k*
_{0}
*d*) and *b* = *J*′_{kyl}(*α*
*k*
_{0}
*d*)*α*
*k*
_{0}
*d*/*k*
_{x}
*l* are both real numbers. It is interesting to see that the total transmitted field is zero and seen at the interface *x* = *d*, all the incident waves will travel back to the free space. In addition, only the wave normally (*k*
_{y} = 0) incident upon the substrate slab can result in non-zero electric field at the left interface (*E*
^{Slab}
_{z} = *AE*
_{0}
*J*
_{0}(0) = *AE*
_{0} at *x* = 0). Any wave obliquely incident at the right interface of the slab (*k*
_{y} ≠ 0) will be bent back completely before it get to the left interface and cannot reach the left boundary (*E*
^{Slab}
_{z} = *AE*
_{0}
*J*
_{kyl}(0)*e*
^{ikyy} = 0at *x* = 0). Based on this interesting phenomenon, we can see that a dipole embedded at the left boundary *x* = 0 of the substrate can only receive the normally incident wave. As a matter of fact, this can be easily understood by considering the dispersion relation of the metamaterial (*k*
^{2}
_{x} + *k*
^{2}
_{y}/*μ*
_{x} = *k*
^{2}
_{0} for TE wave). From the right boundary *x* = *d* to the left boundary *x* = 0, *μ*
_{x} varies spatially from *μ*
_{x} = *d*/*l* to *μ*
_{x} = 0. Consequently, for all the oblique incidences, *k*
_{x} should vary from real to imaginary since *k*
_{y} is kept a non-zero constant. And the switching point depends on the y component of the wavenumber vector, which is determined by the incident angle. As an example, we plot the *E*
_{z} distributions in a metamaterial slab with TE polarized incidences of three different angles (0°, 8°, and 30°) analytically in Fig. 2. In the case of normal incidence, the wave can reach the left boundary, as shown in Fig. 2(a). As the incident angle increases, the depth that the wave can penetrate the slab decreases, which can be observed in Fig. 2(b), and (c).

From Eq. 4 the enhancement of the antenna at the emitting direction can be calculated: ${|\frac{{E}_{z}^{\mathrm{Slab}}}{{E}_{0}}|}_{{k}_{y}=0}=\left|A\right|=\frac{2}{\sqrt{{a}^{2}+{b}^{2}}}$ with *a* = *J*
_{0} (*α*
*k*
_{0}
*d*) and *b* = *J*′_{0}(*α*
*k*
_{0}
*d*)α*d*/*l*. Obviously the enhancement can be controlled by the three parameters α, *l*, and *d*. Since the thickness *d* is always fixed in a practical realization, we can adjust α and *l* to modify the field distributions inside the slab and hence achieve the required properties. Fig. 2(d) depicts the *E*
_{z} distribution inside the 0.05 m thick metamaterial slab corresponding to three incident angles (0°, 8°, and 30°). Different with traditional homogeneous metamaterial with zero refractive index [7-8], in which the obliquely incident wave always decays exponentially at the incident interface, this metamaterial substrate can produce standing waves by the interference of the incident and reflected waves, and the electric field inside is enhanced instead of attenuated in small angle incidence cases. Only the normal incidence result in the largest enhancement at the boundary *x* = 0, as expected. As the incident angle increases to a large enough one (e.g., 30°), the electric field near the incident surface will tend to attenuate, which is more close to the situation in a traditional zero index metamaterial. In Fig. 2(d), we carefully selected the parameters (α and *l*) so that the *E*
_{z} peak of one incidence (0°) is approximately at the nodal point of the other (8°), which can be utilized in the wave multiplexing technique. Here if we place two dipoles in different positions (corresponding to the peak locations of two incidences) in the substrate, waves emitted from the two different directions can be well received at the same time.

## 3. Numerical simulation

In addition to EM wave reception, the proposed metamaterial can also be utilized for transmission. Eq. (1) shows both permittivity and permeability tensors of the slab are diagonal. From the reciprocal law, a beam emitted by a line source fed closed to the left interface of the slab can be reduced in the normal direction, but rays eradiated from sources at other positions will be steered to different directions. We use finite element method (FEM) based simulations to validate this design strategy. The simulation quantities are normalized to unity and all domain boundaries are assumed as perfectly matched layers in order to prevent reflections. The width and thickness of the slab is 1 m and 0.05m respectively and a TE polarized line source is used to excite the wave at 2GHz. Fig. 4(a) and (b) describe the near field *E*
_{z} distribution and when the metamaterial substrate is fed by z-polarized line source embedded at the different positions in the slab while Fig. 4(c) shows normalized power densities in the far field region with the corresponding two different locations of sources. As can be observed in the figures, when the source is at the left boundary, a beam with high directivity to the normal is produced. But as we move the line source along the normal, the radiation pattern changes. When the source is 0.025m from the left boundary, two narrow beams in ±8° directions are produced, which also confirms the results in Fig. 3. The black solid line corresponds to the case when the source embedded at the left boundary, showing a half-power beamwidth of about 4.8° with a 40 dB suppression of side-lobes. The red dashed line shows power peaks in two symmetric directions, and the suppression of side-lobes is about 30 dB.

## 4. Simplified approach for realization

Notice that in Eqs. (1), the three components of *ε* are all spatially varied (so as *μ*) while the *y* component is infinite at the origin. This increases the difficulty in fabrication and hence we need to figure out ways of simplifying the parameters to make it possible to realize. When the line source is TE polarized along z direction, only *ε*
_{z}, *μ*
_{x} and *μ*
_{y} in Eqs. (1) enter into Maxwell’s equations. Moreover, the dispersion properties and wave trajectory in the slab remain the same, as long as the value *ε*
_{z}
*μ*
_{x} and *ε*
_{z}
*μ*
_{y} are kept constant. This gives the ability to choose one of the three constitutive parameters arbitrarily to achieve some favorable condition [14]. One choice is to use the following reduced parameters

We can analyze this case is the similar way to the ideal case. It can be demonstrated that when the wave is normally incident upon the slab with electric field *E*
^{i}
_{z} = *E*
_{0}
*e*
^{-ik0x}, the fields are still distributed as standing wave inside the slab, but the transmitted coefficient is non-zero, indicating that the wave can transmit through the slab. When the wave is obliquely incident with electric field *E*
^{i}
_{z} = *E*
_{0}
*e*
^{-ikxx+ikyy}, the calculation shows that the transmitted fields are exactly equal to zero and penetration depth decreases as the incident angle increases, which is similar to the ideal case, but the fields distributed in the slab will no longer be enhanced.

The above idea is also evaluated with FEM based simulations. In the same simulation environment, we find that the slab antenna with reduced parameters still has a good performance, which is depicted by Fig. 4. Embedding the source at the left boundary produces a narrow beam in the normal direction, as shown in Fig. 4(a). Placing sources at other positions can change the directivity, and when the source is 0.04m from the left boundary, another optimum directivity can be achieved at ±28°, which can be observed in Fig. 4(b) and (c). Note that the half-power beamwidth is a little larger than the ideal case, but the side-lobe suppression is even better. The results demonstrate that the spatial variant metamaterial slab with reduced parameters is a good choice for physical realization and provides a simpler way to experimental demonstration, which is the future work of our research.

## 5. Conclusion

A metamaterial substrate is proposed in an antenna system to achieve the manipulation of directivity. The radiation beams can be steered to different directions by embedding the source at different positions inside the substrate, and we can select appropriate parameters to yield optimum radiation characteristics. As this antenna system is applied in reception, an enhancement of received electric field can be observed. The proposed reduced parameters simplify the design procedure and space-multiplexing can be realized in practical applications.

## Acknowledgments

This work is sponsored by the Chinese National Science Foundation under Grant Nos. 60531020 and 60671003, the NCET-07-0750, the ONR under Contract No. N00014-01-1-0713, the Department of the Air Force under Air Force Contract No.F19628-00-C-0002, and the excellent doctoral thesis foundation of Zhejiang University (No. 08009A).

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