  Recently, higher-order topological phases that do not obey the usual
bulk-edge correspondence principle have been introduced in electronic
insulators and brought into classical systems, featuring with in-gap
corner/hinge states. So far, second-order topological insulators have been
realized in mechanical metamaterials, microwave circuit, topolectrical circuit
and acoustic metamaterials. Here, using near-field scanning measurements, we
show the direct observation of corner states in second-order topological
photonic crystal (PC) slabs consisting of periodic dielectric rods on a perfect
electric conductor (PEC). Based on the generalized two-dimensional (2D)
Su-Schrieffer-Heeger (SSH) model, we show that the emergence of corner states
roots in the nonzero edge dipolar polarization instead of the nonzero bulk
quadrupole polarization. We demonstrate the topological transition of 2D Zak
phases of PC slabs by tuning intra-cell distances between two neighboring rods.
We also directly observe in-gap 1D edge states and 0D corner states in the
microwave regime. Our work presents that the PC slab is a powerful platform to
directly observe topological states, and paves the way to study higher-order
photonic topological insulators.
