  Recently, high-order topological insulators (HOTIs), accompanied by
topologically nontrivial boundary states with codimension larger than one, have
been extensively explored because of unconventional bulk-boundary
correspondences. As a novel type of HOTIs, very recent works have explored the
square-root HOTIs, where the topological nontrivial nature of bulk bands stems
from the square of the Hamiltonian. In this paper, we experimentally
demonstrate 2D square-root HOTIs in photonic waveguide arrays written in glass
using femtosecond laser direct-write techniques. Edge and corner states are
clearly observed through visible light spectra. The dynamical evolutions of
topological boundary states are experimentally demonstrated, which further
verify the existence of in-gap edge and corner states. The robustness of these
edge and corner states is revealed by introducing defects and disorders into
the bulk structures. Our studies provide an extended platform for realizing
light manipulation and stable photonic devices.
