Fig. 1

Design of a photonic metasurface exhibiting topological transition and band inversion above the light line. a, b Geometry of a metasurface based on a triangular lattice of hexamers of silicon pillars on a sapphire (Al₂O₃) substrate: a shrunken structure with a/R > 3; b expanded structure with a/R < 3. Here a is the lattice constant of distorted lattices. E_in, E_r, and E_t, are incident, reflected and transmitted fields. c Brillouin zone of an unperturbed honeycomb lattice a/R = 3 (black) and the triangular lattice under study a/R ≠ 3 (blue) obtained by the honeycomb lattice deformation. d Band structure of (infinitesimally) shrunk/expanded hexamers exhibits folding (of black dotted bands) near K and K′ points to Γ point in the new Brillouin zone (blue dotted bands). As a result of such symmetry reduction, valleys (pseudo-spins) mix, which leads to a topological transition. e–h Demonstration of band inversion: Complex photonic band structure for the four doublet bands of e shrunken a/R = 3.15 and g expanded a/R = 2.85, structures. Color encodes the radiative quality factor of the modes. Calculated extinction spectra 1 − T/T₀ of the shrunken (f) and expanded (h) metasurfaces. Spectra are normalized to the transmittance T₀ through Al₂O₃ substrate and are computed for fixed tangential wavenumber k_|| = 1.04 × 10⁶ m⁻¹. The peaks in far-field spectra correspond to the eigenmode frequencies for the given tangential wavenumber. i Simulated field profiles showing the E _z field component at the top surface of the unit cell for dipolar (p _x and p _y ) and quadrupolar (d _xy and \(d_{x^2 - y^2}\)) eigenmodes for expanded (topological) structure. j Side view of the field distribution in the unit cell under resonant excitation of the dipolar mode for the normal incidence of x-polarized light. Electric field magnitude is normalized to the maximum value