  The discovery of topological phases has recently led to a paradigm shift in
condensed matter physics, and facilitated breakthroughs in engineered photonics
and acoustic metamaterials. Topological insulators (TIs) enable the generation
of electronic, photonic, and acoustic modes exhibiting wave propagation that is
resilient to disorder, irrespective of manufacturing precision or unpredictable
defects induced by the operational environment, known as topological
protection. While originally limited to a dimensionality of the protected
states that is one dimension lower than the host TI material, the recent
discovery of higher-order topological insulators (HOTIs) provides the potential
to overcome this dimensionality limitations by offering topological protection
over an extended range of dimensionalities. Here we demonstrate 2D photonic
HOTI (PHOTI) with topological states two dimensions lower than the one of the
host system. We consider a photonic metacrystal of distorted Kagome lattice
geometry that exhibits topological bulk polarization, leading to the emergence
of 1D topological edge states and of higher order 0D states confined to the
corners of the structure. Interestingly, in addition to corner states due to
the nearest neighbour interactions and protected by generalized chiral symmetry
1, we discover and take advantage of a new class of topological corner states
sustained by long-range interactions, available in wave-based systems, such as
in photonics. Our findings demonstrate that photonic HOTIs possess richer
physics compared to their condensed matter counterparts, offering opportunities
for engineering novel designer electromagnetic states with unique topological
robustness.

Applied Physics;Materials Science;Optics