The technique of applying form-invariant, spatial coordinate transformations of Maxwell's equations can facilitate the design of structures with unique electromagnetic or optical functionality. Here, we illustrate the transformation-optical approach in the designs of a square electromagnetic cloak and an omni-directional electromagnetic field concentrator. The transformation equations are described and the functionality of the devices is numerically confirmed by two-dimensional finite element 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.

Through acoustic scattering theory we derive the mass density and bulk modulus of a spherical shell that can eliminate scattering from an arbitrary object in the interior of the shell--in other words, a 3D acoustic cloaking shell. Calculations confirm that the pressure and velocity fields are smoothly bent and excluded from the central region as for previously reported electromagnetic cloaking shells. The shell requires an anisotropic mass density with principal axes in the spherical coordinate directions and a radially dependent bulk modulus. The existence of this 3D cloaking shell indicates that such reflectionless solutions may also exist for other wave systems that are not isomorphic with electromagnetics.

Transformation optics offers an unconventional approach to the control of electromagnetic fields. A transformation optical structure is designed by first applying a form-invariant coordinate transform to Maxwell's equations, in which part of free space is distorted in some desired manner. The coordinate transformation is then applied to the permittivity and permeability tensors to yield the specification for a complex medium with desired functionality. The transformation optical structures proposed to date, such as electromagnetic "invisibility" cloaks and concentrators, are inherently reflectionless and leave the transmitted wave undisturbed. Here we expand the class of transformation optical structures by introducing finite, embedded coordinate transformations, which allow the electromagnetic waves to be steered or focused. We apply the method to the design of several devices, including a parallel beam shifter and a beam splitter, both of which exhibit unusual electromagnetic behavior as confirmed by 2D full-wave simulations. The devices are designed to be reflectionless, in accordance with a straightforward topological criterion.