Fig. 2

Three-dimensional simulations. a Schematic diagrams of the simulation geometry. The bottom substrate and the top metallic surface were simulated as perfect electric conductors (PEC) to mimic the metal layers. The active material was described by the refractive index n_eff = 3.50, the Cr pads with n₂ = 4.43 + 0.31i, while the open boundaries outside the mesa were modeled with an external region of air (n_Air = 1), surrounded by appropriate scattering boundary conditions. According to the defined reference system, the vertical and horizontal angles of emission are ϑ (blue shade), lying in the x–z plane, and φ (red shade), lying in the x–y plane. b Three-dimensional quality factors, Q_3D, computed at different eigenfrequencies for a sinusoidally corrugated resonator with a feedback periodicity Λ_fb = 13.6 µm, an average width of 40 µm, and a corrugation amplitude of 10 µm. The extraction hole pattern is neglected. The different photonic bandgaps associated with the fundamental mode (brown), the first-order lateral mode (magenta), and the second-order lateral mode (red), are shown. Inset: computed bandgap of the second-order lateral mode. c Total losses α_3D plotted as a function of the resonant frequencies for a 40-µm-wide wire cavity with η = 0.95, computed for a patterned surface array of circular air holes with periodicity Λ_e = 22.4 µm and radius r = 5 µm. Insets: real part of the electric field component E_z (x,y) for the fundamental band-edge modes above and below the bandgap. Red (blue) indicates negative (positive) values of E_z. d–f Spatial distributions of the normalized electric field |E(x,y)| of the lower-edge states at half height (z = 5 µm) of the sinusoidal wire laser cavity. d is the fundamental mode at ν = 3.12 THz, e is the first-order lateral mode at ν = 3.25 THz, and f is the second-order lateral mode at ν = 3.42 THz