Fig. 5: Effects of finite lattice size and imperfections on negative refraction.

a Negative refraction of light through a 5 × 25 × 25 atomic array, denoted by green dots, with spacing a = 0.68λ, resonance wavelength λ, and an isolated σ⁺-polarised two-level transition. The light is incident in the xy plane at angle \(\theta=\arcsin (0.2)\) to the lattice normal, with laser detuning Δ = −0.1γ from the atomic resonance, scaled by the single-atom linewidth γ. The image shows the normalised light intensity profile I/I₀ (outside the medium) and atomic polarisation density \(| \langle {\hat{{{{{\bf{P}}}}}}{}^{+}}\rangle {| }^{2}/| \langle {\hat{{{{{\bf{P}}}}}}{}^{+}}\rangle {| }_{\max }^{2}\) (within the lattice delimited by the dashed green box) at plane z = 0, scaled by λ, where \({I}_{0}=2{\epsilon }_{0}c\,{\max }_{{{{\bf{r}}}}}| {{{{\boldsymbol{{\mathcal{E}}}}}}}^{+}({{{\bf{r}}}}){| }^{2}\) represents the maximum incident intensity. \(| \langle {\hat{{{{{\bf{P}}}}}}{}^{+}}\rangle {| }^{2}/| \langle {\hat{{{{{\bf{P}}}}}}{}^{+}}\rangle {| }_{\max }^{2}\) for point-like atoms is smoothed by convolution with a Gaussian of the root-mean-square widths σ_x = 0.25a and σ_y = 0.5a. The blue dashed lines trace the peak light intensity. b Collective line shifts δ^(j)(q_y, q_z = 0), in units of γ, for atomic Bloch wave resonances across bands indexed by j in the corresponding lattice with infinite in-plane layers. The in-plane quasimomentum q_y, indicative of the incident light’s tilting angle, is varied. The colour coding represents the collective resonance linewidth (see Methods), υ^(j)(q_y, q_z = 0), on a logarithmic scale, normalised to γ. In contrast with Fig. 3b, the larger lattice spacing gives rise to diffraction of phase-matched beams once q_y ≳0.48k, so we restrict the quasimomentum q_y ≲0.35k to lie well within this range. The green star denotes the dominantly excited resonance with linewidth ≃0.05γ in (a). c Power transmission T and lateral displacement D from the centre of the layer at the exit (not the displacement of the incident beam), in units of λ, as a function of the incident light’s wavevector y-component k_y. Perfect lattice with infinite layers (dashed lines) and finite-size layers (dotted lines); atomic position fluctuations with 1/e density width 0.074a about each site, obtained from stochastic simulations (diamonds); phenomenological model with corresponding imperfection parameter ζ_f = 0.975 with infinite layers (solid lines) and finite-size layers (stars).