  Maxwell electromagnetism, describing the wave properties of light, was
formulated 150 years ago. More than 60 years ago it was shown that interfaces
between optical media (including dielectrics, metals, negative-index materials)
can support surface electromagnetic waves, which now play crucial roles in
plasmonics, metamaterials, and nano-photonics. Here we show that surface
Maxwell waves at interfaces between homogeneous, isotropic media described by
real permittivities and permeabilities have a purely topological origin
explained by the bulk-boundary correspondence. Importantly, the topological
classification is determined by the helicity operator, which is generically
non-Hermitian even in lossless optical media. The corresponding topological
invariant, which determines the number of surface modes, is a Z4 number (or a
pair of Z2 numbers) describing the winding of the complex helicity spectrum
across the interface. Moreover, there is an additional pair of non-topological
Z2 indices, which describe zones of the TE and TM polarizations at the phase
diagram of surface modes. Our theory provides a new twist and insights for
several areas of wave physics: Maxwell electromagnetism, topological quantum
states, non-Hermitian wave physics, and metamaterials.
