## Abstract

The transformation optics technique for designing novel electromagnetic and optical devices offers great control over wave behavior, but is difficult to implement primarily due to limitations in current metamaterial design and fabrication techniques. This paper demonstrates that restricting the spatial transformation to a conformal mapping can lead to much simpler material parameters for more practical implementation. As an example, a flat cylindrical-to-plane-wave conversion lens is presented and its performance validated through numerical simulations. It is shown that the lens dimensions and embedded source location can be adjusted to produce one, two, or four highly directive planar beams. Two metamaterial designs for this lens that implement the required effective medium parameters are proposed and their behavior analyzed.

©2010 Optical Society of America

## 1. Introduction

The transformation optics design methodology allows for unprecedented flexibility in creating new electromagnetic and optical devices. Introduced in [1] and [2], the technique relies on the form-invariance of Maxwell's equations under a spatial coordinate transformation. Maxwell's equations, when evaluated for a transformed coordinate system, are valid and can be interpreted in the original coordinate system with permuted material parameters $\overline{\overline{\epsilon}}(x,y,z)$and$\overline{\overline{\mu}}(x,y,z)$. Given a desired field behavior, the permittivity and permeability can be determined for a material that will implement the transformation. The permittivity and permeability in the transformation medium force the electromagnetic fields to behave as if they were subject to the desired spatial mapping.

The electromagnetic cloak is one of the most well-studied applications enabled by the new transformation optics techniques [2–9], but other novel devices have been considered including beam collimators and shapers, field concentrators, and ideal far- and near-field focusing lenses [4,8,10,11]. A specialization of the focusing lens converts a diverging cylindrical wavefront to a highly directive collimated plane-wave beam [13–16].

Despite the recent advances in metamaterial design and synthesis, most of the existing transformation optics designs remain firmly in the realm of an academic exercise, with material parameters too complex to implement practically. Applications of the transformation optics design techniques have typically led to highly anisotropic and inhomogeneous solutions for the required permittivity and permeability of a device. Some papers have studied the effects of simplifying the material parameters for easier fabrication, such as the electromagnetic cloaks explored in [5,6,9,17], but these approximations often lead to significantly degraded performance. Another approach leverages transformation techniques to reduce the size of simple shaped near-field focusing lenses through basic transformations that are easier to implement [18]. Others have chosen quasi-conformal or conformal transformations to minimize either the anisotropic or inhomogeneous elements of the required material [19,20] in order to maximize the practicality of the design.

This paper presents conformal mappings as a class of transformations that result in much simpler material parameters than general transformation optics media. Using a conformal mapping, a flat cylindrical-to-plane-wave conversion lens with an embedded source that has practically realizable material parameters is designed and demonstrated via full-wave finite element simulations performed using Comsol. The same rectangular lens material can be configured to create one, two, or four highly collimated beams. Finally, two possible metamaterial specifications that implement the necessary parameters for the transformation media lens are presented.

## 2. Conformal mappings for transformation media

Many of the transformation optics examples in the literature require very complex material parameters that are difficult to implement in practice. By restricting the available transformations to those that satisfy the conditions of a conformal mapping, the resulting material parameters are much simpler than in the general case.

A function in the complex plane $\omega =f(z)=f(x+j\text{\hspace{0.05em}}y)=u(x,y)+\text{}j\text{\hspace{0.05em}}v(x,y)$is conformal, or angle-preserving, if it satisfies the requirements for an analytic function. A function in the complex plane is said to be analytic when it satisfies the Cauchy-Riemann equations given by

The constitutive parameters for a transformation medium expressed in free space are given by [14], [15]

where**A**is the Jacobian matrix for the transformation from the z-domain to the ω-domain. The two-dimensional conformal mapping can be extended to three dimensions by

Using the Cauchy-Riemann equations, the Jacobian matrix can be expressed in the form

## 3. The reciprocal transformation as a plane-wave collimator

As an example, consider a transformation that maps a cylindrical wave emanating from an embedded line source into a plane wave through a square lens. A version of this lens was introduced in [12], but it requires material parameters that would be extremely difficult to implement in a practical device. In order to simplify the material parameters while obtaining similar electromagnetic behavior, we replaced the transformation from [12] with the reciprocal transformation,

*ω*-domain. While this transformation may not lead to accurate beam collimation for arbitrarily large size square 2D lenses, it will be demonstrated that it does produce four highly collimated beams for small to moderate size lenses. Applying Eq. (5) to Eq. (6), the permittivity and permeability tensors for the lens are found to be:

*a*is valid in (8).The variables x and y are assumed to be measured in meters.

#### 3.1 Material simplification

The effective parameters of the metamaterial lens for the quad-beam cylindrical to plane wave converter are quite simple compared to a general transformation media, such as considered in [12]. For rectangular lenses centered at the origin with width w and height h, the material properties can be simplified even further by observing that the permittivity is very close to zero over the entire lens when w, h, and a are chosen such that $aw\le 0.4$ and $ah\le 0.4$. With this constraint, the maximum value of ϵ_{zz} is ${({(0.4/2)}^{2}+{(0.4/2)}^{2})}^{2}=0.0008$, occurring at the corners of the lens. By introducing the approximation${({x}^{2}+{y}^{2})}^{2}\approx 0$, the metamaterial lens may be implemented as a uniaxial homogeneous material with a reduced form of the permittivity and permeability tensors given by

_{zz}= μ

_{xx}= 0 anisotropic tensor elements can only be implemented through narrowband dispersive metamaterial structures, which will significantly limit the bandwidth of the lens in any real device. The material approximation produces a lens that can also be analyzed as a zero-index material with infinite effective phase velocity, resulting in equal phase fronts at the boundaries of the lens. The phase-collimating effect of sources embedded within zero-index materials was documented by Enoch et al. [21].

## 4. Simulation results

The conformal transformation described above can be used to create several distinct radiation scenarios by varying the size and form factor of the lens and the relative location of the embedded antenna. Simulations were performed using Comsol Multiphysics to explore the performance of the design. For Figs. 2 -3 and 5 -6 , the black squares indicate the boundaries of the metamaterial lens which surrounds a single embedded TE-mode line source at the origin. All simulations were performed at 3 GHz.

#### 4.1 Basic simulations

With a square lens centered on the antenna, the transformation medium produces the desired four orthogonal plane-wave beams, as demonstrated in Fig. 2. There is little scattering from the corners of the lens, and the beams are highly collimated.

Changing the lens from a square to a rectangle centered on the antenna increased the directivity, as shown in Fig. 2. The radiation from the short sides of the rectangular lens was reduced, while the radiation emitted from the longer sides of the lens was enhanced. Increasing the relative length of the long compared to the short axis of the lens enhances the focusing effect through increased attenuation of the side beams. By choosing an appropriate form factor for the rectangular lens, the relative strength of the horizontal and vertical beams can be controlled. Only the form factor, not the size of the lens, determines the relative power emitted from each lens face. The size of the lens does affect the directivity of the emitted beams and how well the cylindrical waves from the antenna are converted to directive plane waves at the lens boundary. The power radiated from a particular face is essentially related to the relative distance of the antenna from that face, compared to the distance from the remaining faces. More power is radiated from faces that are closer to the antenna. These observations suggest additional lens configurations.

If the antenna is placed in the corner of the square lens, instead of the center, then two perpendicular plane wave beams are created. When the antenna is placed much closer to one face than any other, a single highly directive plane wave is produced. The same constraints as to the relationship between size and form factor still apply - the form factor and relative distances from the antenna to the edge determine the radiating faces, and larger lenses create more directive, collimated beams. Figure 3 illustrates the performance of the lens in a perpendicular two-beam and a single-beam configuration.

## 5. Metamaterial design

#### 5.1. Split-ring resonator and electric LC resonator array

To realize the square lens, an anisotropic metamaterial composed of an array of modified split-ring resonators (SRR) and electric LC resonators (ELC) is designed and tuned for operation at 3GHz. The modified SRR is used to eliminate the bi-anisotropic property of the original SRR structure [22]. The simultaneous near-zero permittivity and permeability in the z-direction are produced by the ELC [23] and SRR, respectively. A single unit cell of a periodically repeating metamaterial is shown in Fig. 4(a)
with the geometric dimensions illustrated in Figs. 4(b) and 4(c). By periodically arranging the unit cell, it can approximate a homogeneous bulk metamaterial with permittivity and permeability tensor values approaching those specified in (9). The retrieved constitutive parameters under TE-mode illumination using the HFSS simulation package are ϵ_{zz} = 0.012 - j0.02, μ_{xx} = 1.1 - 0.021j and μ_{yy} = 0.9 - 0.023j.

To demonstrate the performance of this metamaterial as a square lens, the retrieved effective medium parameters were put into a COMSOL simulation without considering the imaginary parts of the z-directed permittivity and permeability. The results are shown in Fig. 5 and agree well with the theoretical predictions, despite the nonideal material parameters. By comparing Fig. 5 and Fig. 6 in which the imaginary component of ϵ_{zz} and μ_{zz} are included, it can be seen that these terms play a critical role in controlling the radiation efficiency of the metamaterial lens.

#### 5.2 Dipole array for TE-mode operation

The complete metamaterial implementation for the lens requires ϵ_{zz} = μ_{zz} = 0 for proper operation in both TE and TM modes. However, adequate performance can be obtained for single-mode operation by allowing either ϵ_{zz} = 1 for TM operation, or μ_{zz} = 1 for TE operation. The previously defined metamaterial achieves electric and magnetic zero-permittivity conditions through the separate operation of the z-oriented SRR and ELC structures. Eliminating the SRR and reducing the ELC to a simpler linear dipole produces the required anisotropic permittivity for TE-mode operation. Similarly, for TM operation the ELC structures could be removed to leave z-oriented SRR elements to create μ_{zz} = 0. Instead of the SRR and ELC resonator array, a grid of simple linear dipoles can produce the required z-oriented anisotropic permittivity. Figure 7(a)
shows a single layer of the TE-mode metamaterial lens comprised of an array of flat metallic strip dipole elements hosted by a grid of printed circuit boards. The linear dipole grid exhibits a z-oriented Lorenzian permittivity response that can be tuned to zero by optimizing the length of the wire elements for operation at a particular frequency. The 3D HFSS simulation uses symmetry planes to create an infinite stack of metamaterial layers with an infinitely long wire centered at the origin. The electric field cut through the center of the lens in Fig. 7(b) demonstrates the operation of the simplified four-beam lens excited by an ideal linear current source at the origin. The modified lens is simpler, easier to fabricate, and still produces good results.

## 6. Conclusions

The use of conformal mappings in the design of transformation media can reduce the complexity of the resulting material parameters in a final optical or electromagnetic device. Without sacrificing design flexibility, the use of conformal mappings can allow for better practical implementations of transformation optics designs. A transformation that produces a quad-beam cylindrical-to-plane-wave converter was presented as a simpler alternative to the approach taken in [12], and a possible metamaterial implementation satisfying the device requirements was also presented. Further simplification of the metamaterial structure by only considering TE-mode operation creates a lens that is easier to fabricate using printed circuit board technology and still provides good performance. Future efforts will focus on developing metamaterial lenses with lower losses for improved performance of cylindrical-to-plane-wave converters.

## Acknowledgments

This work was supported in part by the Penn State MRSEC under NSF grant no. DMR 0213623, and also in part by ARO-MURI award 50342-PH-MUR.

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