  The discovery of two-dimensional topological photonic systems has transformed
our views on electromagnetic propagation and scattering of classical waves, and
a quest for similar states in three dimensions, known to exist in condensed
matter systems, has been put forward. Here we demonstrate that symmetry
protected three-dimensional topological states can be engineered in an
all-dielectric platform with the electromagnetic duality between electric and
magnetic fields ensured by the structure design. Magneto-electric coupling
playing the role of a synthetic gauge field leads to a topological transition
to an insulating regime with a complete three-dimensional photonic bandgap. An
emergence of surface states with conical Dirac dispersion and spin-locking is
unimpeded. Robust propagation of surface states along two-dimensional domain
walls defined by the reversal of magneto-electric coupling is confirmed
numerically by first principle studies. It is shown that the proposed system
represents a table-top platform for emulating relativistic physics of massive
Dirac fermions and the surface states revealed can be interpreted as
Jackiw-Rebbi states confined to the interface between two domains with opposite
particle masses.
