Fig. 1: Gapless environment for photonic crystal slabs.

a Schematic of a free-standing photonic crystal slab in a three-dimensional (3D) geometry. Photonic structures are inherently 3D and are usually surrounded by an homogeneous material that features a light cone. As such, out-of-plane radiative losses, depicted by the red arrows, are inherent to such structures. The photonic crystal is a triangular lattice with lattice constant a composed of dielectric rods, \({\bar{\epsilon }}_{jj}=14\) for j = x, y, z, of radius r = 0.37a and height t = 0.5a embedded in a gyro-electric material slab, \({\bar{\epsilon }}_{jj}=1\) and \({\bar{\epsilon }}_{xy}=-0.4i\), of thickness t = 0.5a. b Band structure of the photonic crystal slab in (a) over the first Brillouin zone, for a transverse magnetic-like polarization. The photonic band gap at around ω = 0.42[2πc/a] is the “topological band gap” known from extrapolation from the two-dimensional photonic crystal approximation. The shaded region depicts those frequencies and wavevectors that are at, or above, the light line of the surrounding air. The red line depicts the light line, ω = c∣k∣. Above the light line, photonic crystal slabs generally exhibit resonances, not bound states.