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.