  Geometrical singularities in plasmonic metasurfaces have recently been
proposed for the enhancement of light-matter interactions, owing to their
broadband light-harvesting properties and extreme plasmon confinement. However,
the large plasmon momenta thus achieved lead to failure of local descriptions
of the optical response of metals. Here we study a class of metasurfaces
consisting of a periodic metal slab with a smooth modulation of its thickness.
When the thinnest part shrinks, the two surfaces almost touch, forming a
near-singular point. Using transformation optics, we show analytically how
nonlocal effects, such as a blueshift of the resonance peaks and a reduced
density of states, become important and cannot be ignored in this singular
regime. The method developed in this paper is very general and can be used to
model a variety of metasurfaces, providing valuable insight in the current
context of ultra-thin plasmonic structures.
