  Defect modes in two-dimensional periodic photonic structures have found use
in a highly diverse set of optical devices. For example, photonic crystal
cavities confine optical modes to subwavelength volumes and can be used for
Purcell enhancement of nonlinearity, lasing, and cavity quantum
electrodynamics. Photonic crystal fiber defect cores allow for supercontinuum
generation and endlessly-single-mode fibers with large cores. However, these
modes are notoriously fragile: small changes in the structure can lead to
significant detuning of resonance frequency and mode volume. Here, we show that
a photonic topological crystalline insulator structure can be used to
topologically protect the resonance frequency to be in the middle of the band
gap, and therefore minimize the mode volume of a two-dimensional photonic
defect mode. We experimentally demonstrate this in a femtosecond-laser-written
waveguide array, a geometry akin to a photonic crystal fiber. The topological
defect modes are determined by a topological invariant that protects
zero-dimensional states (defect modes) embedded in a two-dimensional
environment; a novel form of topological protection that has not been
previously demonstrated.
