  The discovery of photonic higher-order topological insulators (HOTIs) has
significantly expanded our understanding of band topology and provided
unprecedented lower-dimensional topological boundary states for robust photonic
devices. However, due to the vectorial and leaky nature of electromagnetic
waves, it is challenging to discover three-dimensional (3D) topological
photonic systems and photonic HOTIs have so far still been limited to two
dimensions (2D). Here, we report on the first experimental realization of a 3D
Wannier-type photonic HOTI in a tight-binding-like metal-cage photonic crystal,
whose band structure matches well with that of a 3D tight-binding model due to
the confined Mie resonances. By microwave near-field measurements, we directly
observe coexisting topological surface, hinge, and corner states in a single 3D
photonic HOTI, as predicted by the tight-binding model and simulation results.
Moreover, we demonstrate that all-order topological boundary states are
self-guided even in the light cone continuum and can be exposed to air without
ancillary cladding, making them well-suited for practical applications. Our
work thus opens routes to the multi-dimensional robust manipulation of
electromagnetic waves at the outer surfaces of 3D cladding-free photonic
bandgap materials and may find novel applications in 3D topological integrated
photonics devices.
