Fig. 1: Principle of construction of the compact flat-band multi-BIC laser.

a, Comparison of group indices and Q factors between cavity modes in a normal BIC device with infinite and finite sizes (top). The continuous band (grey curve) in an infinite-sized cavity transforms into a series of discrete modes (black dots) in a finite-sized cavity, with the corresponding group indices (red dots) and Q factors (blue dots) deviating from the values at the Γ point. Here the group index n_g = c/v_g, where v_g = dω_k/dk (v, group velocity; ω, angular frequency; k, wave vector), can reflect the in-plane feedback and localization capabilities in a photonic-crystal slab. A higher group index means a stronger in-plane feedback and localization. For a normal BIC mode in a finite-sized device, substantial radiation losses, scattering losses and side leakages exist in the cavity (bottom). b, Three-dimensional photonic band diagram of the hexagonal supercell in d with a flat-band BIC design (top). The side leakages can be well suppressed in this case enabled by the flat-band design (bottom). Inset: a schematic of the polarization singularity of the flat band with a topological charge of q = +2 at the Γ point. k_x and k_y represent the in-plane components of the wave vector along the x and y directions in the Brillouin zone, respectively. c, Schematic of the flat-band multi-BIC design in momentum space. The plus and minus signs represent a topological charge of +1 and −1, respectively. Both radiation and side-leakage losses are well suppressed when a multi-BIC is introduced into the flat-band cavity (bottom). d, A typical double-metal QCL configuration is used here. The air-hole photonic-crystal structure is constructed by drilling through the top Au/QCL layers. The top and bottom Au layers act as electrodes for current injection, as well as vertical confinement of the cavity mode. f, frequency.