  We present a theoretical study of second-harmonic generation from a singular
metasurface. The singular metasurfaces strongly interact with the incident
light, where the large field enhancement forms an intense surface polarization
that generates the second-harmonic field. By using transformation optics, the
calculation of nonlinear optical response is converted from the metasurface
frame to that of a simple slab geometry, largely reducing the complexity of the
problem. In addition, the singular metasurface exhibits a weak dependence on
the incident angle of light, which can be potentially used as an all-angle
device for harmonic generations. Finally, we study the symmetry dependence of
second-harmonic generation in the far field for the singular metasurface and
show how to enhance the conversion efficiency under normal incidence by
breaking the surface inversion symmetry.
