ABSTRACT

The emerging technique of transformation optics (TO) has stim-

ulated widespread interest in electromagnetics, optics and other

fields related to wave physics [1-5]. Its mathematical foundations

can be traced back to the 1920s when the theory of general

relativity was intensively explored, or even to the earlier age

as Fermat’s principle was formulated [6, 7]. The investigation of

Maxwell’s equations in curved space-times lasted through the past

century, and the form invariance of these governing equations of

electromagnetic (EM) phenomena was then revealed by different

means [8-10]. However, the early investigation mainly stayed at the

level of mathematics and theoretical physics, and the applications

of this elegant property were rarely discussed. In 1996, aiming

at simplifying the computation of light propagation in complex

structures, Ward and Pendry studied Maxwell’s equations in a

general coordinate system [11]. It was found that the equations

retain their expressions in the Cartesian coordinates, but the

tensors of permittivity and permeability need to take a different

form. The new tensors can be derived from the transformation

that maps Cartesian frames into a new coordinate system, which

defines the “push forward” of the original medium following

mathematical terminology [3]. This equivalence between coordinate

transformations and changes of material properties establishes a

shortcut for computing fields in curved geometries. Soon afterward,

the transformation approach was applied to the study of the perfect

lens [12, 13], another theoretical milestone of metamaterials.