Fig. 3: Analysis of time-varying LC resonant metasurface.

a, Generic illustration of a time-varying resonant LC metasurface. Each meta-atom is described by a time-modulated surface capacitance C(t) and a constant surface inductance L. The red arrow indicates a surface eigenmode whose amplitude is growing in time due to its wavenumber being inside the momentum bangdap of the metasurface. b, Upper: dispersion relations of a stationary metasurface for the two scaled values of the surface capacitance. Lower: band structure of the time-modulated resonant metasurface. Here, ℜ and ℑ represent real and imaginary parts of the complex frequency, respectively. The horizontal axis is normalized by k_r0 = ω_r0/c₀ where c₀ is the speed of light in vacuum. The grey region emphasizes the bandgap in the lower panel and the eigenwavenumber variation at ω = ω_m/2 in the upper panel. c,d, Magnetic field snapshots for momenta k_∣∣ = 5k_r0 and k_∣∣ = 10k_r0 above the metasurface at the time moment when modulation switches on (t = 0) and after some time passed (t = 30T_m, where T_m = 2π/ω_m). The complete field evolution animation is available in Supplementary Video 1. e, Time evolution of the magnetic field above the metasurface (z = 0) calculated with full-wave simulations and analytically from the band structure. The theoretical amplitude is calculated as \({H}_{y}(t)={H}_{y0}\exp [\Im (\omega )t]\) where H_y0 = 1 A m^–1 is the initial field at t = 0 and ℑ(ω) = 0.025ω_m. f, Imaginary part of the eigenfrequency for metasurfaces with different quality factors. The real part of eigenfrequency is fixed as ℜ(ω) = ω_m/2. The black dashed line separates propagating waves (p.w.) and surface waves (s.w.). The quality factor of the stationary metasurface can be qualitatively described as a quality factor of an RLC circuit (where \(R=\sqrt{{\mu }_{0}/{\varepsilon }_{0}}\) is the free-space characteristic impedance, which is connected in parallel to the metasurface equivalent circuit), that is, \(Q={\omega }_{{\rm{r0}}}{C}_{0}\sqrt{{\mu }_{0}/{\varepsilon }_{0}}\) (refer to section 6.1 of ref. ⁴¹). The quality factor is tuned by varying the values of C₀ and L while keeping \({\omega }_{{\rm{r0}}}=1/\sqrt{L{C}_{0}}\) constant. Importantly, such a regime of the open bandgap for propagating modes is inherent to metasurfaces with high quality factors and cannot occur in non-resonant metasurfaces such as those in ref. ¹⁴. g, The magnetic field evolution for a dipole source positioned 1.5 wavelengths above the time-varying LC metasurface with Q = 122. Due to the infinite bandgap, all momenta k_∣∣ are amplified by the metasurface. In Fig. 3c,d,g, the excitation source is turned off after modulation starts; m = 0.2 in all of the panels.