Fig. 4

Optical properties of two basic resonators and their NF and FF couplings. a A unit cell of the bi-layer metasurface containing one Au nano bar on top of a SiO₂ substrate and another Au nano bar buried inside the substrate at a depth h. Here the two resonators are of the same thickness (30 nm) and different lateral sizes, with their centers on different lateral positions on the xy-plane. b Spectra of reflectance/transmittance of the metasurface containing top resonators (with a₁ = 140 nm and b₁ = 340 nm) arranged in a hexagonal lattice with periodicity p = 600 nm, obtained by experiments (stars), simulations (circles) and CMT calculations (solid lines) with parameters computed by the LEM theory (f₁ = 203.32 THz, Γ_1r = 5.997 THz, Γ_1i = 6.140 THz). c Spectra of reflectance/ transmittance of the metasurface containing resonators (with a₂ = 320 nm and b₂ = 280 nm) buried inside the substrate at h = 236 nm arranged in a hexagonal lattice with periodicity p = 600 nm, obtained by experiments (stars), simulations (circles) and CMT calculations (solid lines) with parameters computed by the LEM theory (f₂ = 202.57 THz, Γ_2r = 4.985 THz, Γ_2i = 4.975 THz). White dashed lines and gray areas in b-c denote the frequencies and radiation damping of the resonant modes, calculated by LEM theory, and right panels in b-c are the SEM images of the fabricated samples with scale bars (white lines) of 500 nm. d LEM-calculated near-field coupling κ between two resonators versus their relative lateral configuration (d_x, d_y) with fixed vertical distance h = 236 nm. Points labeled with 1–4 represent the cases with α = 90^∘, 65^∘, 55^∘, 0^∘ on the circle \(l=\sqrt{d_x^2+{d}_y^2}=345\ \textrm{nm}\). e LEM-calculated inter-resonator couplings (X and κ) as functions of h, with fixed α = 0^∘ and \(l=\sqrt{d_x^2+{d}_y^2}=345\ \textrm{nm}\). Dashed lines labeled with #5–8 represent the cases with h = 140, 236, 450, 600 nm, respectively. The violet star represents the position of critical point with κ − Im(X) = 0 and θ_X = π/2 satisfied. Here, κ and X are scaled by \(\sqrt{\Gamma_{1r}{\Gamma}_{2r}}\), and the SiO₂ substrate has a thickness of 500 μm in experiments and is treated as a semi-infinite medium in simulations