  Negative refractive index materials (NRIM) make possible unique effects such
as a convergent at lens due to the reversed Snell-Descartes laws of refraction.
NRIM are also known to be able to support a host of surface plasmon states for
both polarizations of light which are responsible for the sub-wavelength image
resolution achieved by a slab of NRIM. A generalized lens theorem provides us
with a class of spatially varying slab lenses satisfying the prerequisite
symmetries to fold the optical space onto itself. This theorem can be derived
using powerful tools of transformational optics. A paradigm of such
complementary media are checkerboards consisting of alternating cells of
positive and negative refractive index that represent a very singular situation
in which the local density of modes at the corners are enormously enhanced. We
have considered several theoretical and numerical aspects of such structured
films including a finite slabs of multi-scale checkerboards of NRIM satisfying
the generalized lens theorem which are host of strongly enhanced
electromagnetic feld. Such checkerboards can be mapped using transformational
optics onto three-dimensional corner lenses consisting of semi-infinite
heterogeneous anisotropic regions of space satisfying the generalized lens
theorem. It is also possible to design three-dimensional checkerboards of
complementary media, the only restriction being that corresponding unfolded
structures in the plane are constrained by the four color theorem. Some of
these meta-surfaces in the plane display thin bridges of complementary media,
and this further enhances their plasmonic response. Since plasmonic metals
mimic the behaviour of NRIM at small length scales, opaque gold films
structured at sub-micron scales in a checkerboard fashion were fabricated using
focussed-ion-beam technologies and their scattering spectra measured.
