Fig. 4: Microscopic origin of negative refraction and emergence of macroscopic optical bulk response.
a Schematic illustrates the microscopic mechanism of negative refraction. An atomic polarisation wavepacket, excited by the incident beam, propagates along the y-axis over its lifetime t = 1/υ, where υ = υ(j)(qy, qz = 0) is the collective linewidth, accumulating a transverse displacement of D ≃vg,y × 1/υ. The transverse group velocity component vg,y = −∂δ(j)(qy, qz = 0)/∂qy is derived from the collective line shifts δ(j)(qy, qz = 0) for phase-matched quasimomenta qy in resonant band j. Green and grey wavepackets illustrate cases of negative and positive displacement, respectively. For b Nx = 25, c Nx = 50, and d Nx = 100 infinite layers, collective line shifts, in units of the single-atom linewidth γ, are presented for the lattice spacing a = 0.45λ, where λ is the resonance wavelength. The in-plane quasimomentum, indicative of the incident light’s tilting angle, is varied. The collective resonance linewidth is normalised to γ/(Nx − 1) on a colour-coded logarithmic scale. This choice highlights the linear dependence of the wavepacket lifetime, and displacement D, on sample thickness a(Nx − 1), as alluded to in (a). Anomalous bright dots correspond to resonances due to array edges in the x-direction.
