Design of Electromagnetic Cloaks ...
arXiv:0706.2452v1 [physics.optics] 17 Jun 2007 Design of Electromagnetic Cloaks and Concentrators Using Form-Invariant Coordinate Transformations of Maxwell���s Equations Marco Rahm, David Schurig, Daniel A. Roberts, Steven A. Cummer, David R. Smith Department of Electrical and Computer Engineering, Duke University, Box 90291, Durham, NC 27708, USA Sir John B. Pendry Department of Physics, The Blackett Laboratory, Imperial College, London SW7 2AZ, UK Abstract The technique of applying form-invariant, spatial coordinate transformations of Maxwell���s equations can facilitate the design of structures with unique electromag- netic or optical functionality. Here, we illustrate the transformation-optical approach in the designs of a square electromagnetic cloak and an omni-directional electro- magnetic field concentrator. The transformation equations are described and the functionality of the devices is numerically confirmed by two-dimensional finite ele- ment simulations. The two devices presented demonstrate that the transformation optic approach leads to the specification of complex, anisotropic and inhomogeneous materials with well directed and distinct electromagnetic behavior. Key words: Transformation Optical Design, Form-invariant Coordinate Transformations of Maxwell���s Equations, Electromagnetic Theory, Metamaterials, Cloaking, Anisotropic Media, Inhomogeneous Media, Numerical Full-Wave Simulations, Finite-Element Method PACS: 42.15.Eq, 42.25.-p, 42.25.Bs, 42.25.Fx, 02.40.-k, 02.70.Dh, 04.30.Nk 1 Introduction In a theoretical study, Pendry et al. reported a general method for the de- sign of electromagnetic materials based on form-invariant transformations of Preprint submitted to Elsevier 1 February 2008
Maxwell���s equations . In that paper, the methodology of transformation optics was applied to find the specification for an electromagnetic cloak-a complex material capable of rendering objects within its interior invisible to detection. Although just one example of the many intriguing structures pos- sible using the transformation optical approach, the proposed cloak design generated enormous interest in its own right. An approximation to the invis- ibility cloak based on metamaterials was subsequently realized by Schurig et al., who demonstrated the cloaking mechanism in microwave experiments . The transformation optical approach to invisibility is quite general, differing in scope from prior related work. Indeed, methods of reducing the electromag- netic scattering of objects at radar frequencies have long been a subject of intense research [3,4,5]. On the nanoscale, techniques have also been suggested to reduce the scattering of one or more multipole components of size-limited objects using tailored negative index or negative permittivity coatings [6,7]. More recently, a mathematically rigorous proof of an invisibility structure based on active devices was reported . Transformation optics provides for a conceptually simple approach to the de- sign of complex electromagnetic structures: one imagines warping space to achieve the desired electromagnetic functionality. The trajectories of electro- magnetic waves passing through a region of warped space must conform to the local metric, and this provides an alternative (though conceptual) means to control and manipulate electromagnetic fields. Once the desired design is determined, the coordinate transformation and its Jacobi matrix deter- mine the transformation of Maxwell���s equations and the constitutive rela- tions. The result provides the specification for an electromagnetic structure that is complex-being inhomogeneous and anisotropic-but realizable for exam- ple through artificially structured metamaterials. Indeed, because the fields in a volume bounding a transformation optical structure are identical to those that would exist where the structure is replaced by free space, anisotropy is necessary to circumvent uniqueness constraints . If the coordinate transformation can be realized exactly in the constitutive parameters, all aspects of wave propagation will be transformed by the struc- ture, including the near-fields. Adding constraints to the materials reduces the ultimate performance of the structure, but nevertheless can still allow for interesting and novel structures. Leonhardt, for example, has shown that if the materials are restricted to be isotropic, an approximate cloak can be con- structed that is valid in the geometrical optic limit . Likewise, constraints were employed for the metamaterial cloak utilized by Schurig et al. to ease the metamaterial design and fabrication, resulting in a structure that pro- duced significant reflection yet still demonstrated the cloaking mechanism for transmitted waves . Since the concept of transformation optics was introduced, there have been a 2
growing number of subsequent reports applying the method to a variety of elec- tromagnetic, acoustic and elasto-mechanical structures [10,11,12,13,14,15,16,17]. Full wave simulations have helped to confirm the expected behavior and have provided a platform to explore systematically the effects of absorption, imper- fections and other constraints that are inherent to fabricated realizations of the transformation optical structures [18,19]. In this paper we present two examples that demonstrate the general applica- bility of form-invariant coordinate transformations for the design of complex, inhomogeneous and anisotropic electromagnetic materials with well-defined functionality. For the first example, we derive the electromagnetic constitutive parameters corresponding to a two-dimensional electromagnetic cloak having square cross-section. The square shape has been chosen to illustrate the na- ture of the transformation-and the resulting design-for a structure that lacks rotational symmetry in the plane. In contrast to the cylindrical cloaks with circular cross-section previously presented, the square cloak design results in a non-orthogonal transformation producing a more complicated specification for the spatially dependent permittivity and permeability tensors. The method to design this structure, however, can be applied to the design of structures with arbitrary shape. For the second example, we derive the material properties of an electromag- netic field concentrator by the same approach. The purpose of the cylindrical concentrator is to focus incident electromagnetic waves with wave vectors per- pendicular to the cylinder axis, enhancing the electromagnetic energy density of incident waves in a given area. This example illustrates the strength of the transformation-optical approach for designing devices other than cloaks. 2 Transformation Equations In this section, the formulas describing the spatial coordinate transforma- tions and the calculation of the resulting material parameters, i. e. the electric permittivity tensor and the magnetic permeability tensor, are derived. The methodology used to compute the electromagnetic material properties is sim- ilar to the one reported in . For convenience, we denote Maxwell���s equations in the Minkowski form  ���[��F����] = 0 (1) �����G���� = j��, ��, ��, �� = 0, 1, 2, 3 (2) where the square brackets express an alternation among the indices  and 3