Fig. 4: Illustration of the momentum bandgap enhancement for surface waves.

a, A representative design of a PTC based on an optical time-varying resonant metasurface made of dielectric nanospheres. The nanospheres are arranged in an infinite square lattice with period a. The radius of each nanosphere is R. The purple arrow indicates a surface eigenmode whose amplitude grows in time due to its wavenumber being inside the momentum bandgap of the metasurface. b, Band structure of a time-invariant metasurface. The colour denotes the lowest singular value \({S}_{\min }\) of the matrix in equation (3). The white dashed lines represent the light lines. c, The two lowest-order Mie coefficients of an isolated time-invariant nanosphere. The vertical scale is the same as in b. d, Band structure of a time-varying metasurface with modulation frequency ω_m1 (non-resonant case). e, Magnified band structure in the green highlighted region of d. f, The corresponding imaginary part of the frequency for fixed ℜ(ω) = ω_m1/2. g, Band structure of a time-varying metasurface with modulation frequency ω_m2 (resonant case). h, Magnified band structure in the green highlighted region of g. i, The corresponding imaginary part of the frequency for fixed ℜ(ω) = ω_m2/2. j, Imaginary part of the eigenfrequency ℑ(ω) at the bandgap centre and the relative amplification momentum bandwidth \(\Delta {k}_{| | }^{{\rm{a}}}=2| ({k}_{| | }^{{\rm{a2}}}-{k}_{| | }^{{\rm{a1}}})/({k}_{| | }^{{\rm{a2}}}+{k}_{| | }^{{\rm{a1}}})|\) as a function of the damping factor γ. Here, \({k}_{| | }^{{\rm{a1}}}\) and \({k}_{| | }^{{\rm{a2}}}\) correspond to those values of k_∣∣ for a bandgap where ℑ(ω) = 0 (see Supplementary Fig. 7a). Note that the band structures shown in b, d and g accommodate the eigenmodes for both transverse magnetic and transverse electric polarizations; however, in e,f,h,i we optimize the metasurfaces for transverse-electric waves. Further note that the momentum bandgap in g is incomplete because bands exist at ℜ(ω) > ω_m2/2 within the bandgap. Nevertheless, in contrast to spatial photonic crystals with their energy bandgaps, the modes inside of an incomplete momentum bandgap are always dominant due to their amplifying nature⁴.