Fig. 1: Negative refraction and transmission of light through an atomic medium.
a Schematic shows light transmission through a cubic atom array, stacked in infinite planar lattices along the x-axis, with the incident beam propagating towards x = ∞. The dominant wavevector k lies in the xy plane at angle \(\theta=\arcsin ({k}_{y}/k)\) to the lattice normal, tilting in the y direction. The transmitted beam undergoes lateral displacement D along the y-axis. b Negative refraction of a beam incident with θ = 0.2 × π and laser detuning Δ/γ = 0.73 from the atomic resonance, in units of single-atom linewidth γ, through a 25-layer atomic lattice for the \(J=0\to {J}^{{\prime} }=1\) transition. This is manifested in 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 resonance wavelength λ, where \({I}_{0}=2{\epsilon }_{0}c\,{\max }_{{{{\bf{r}}}}}| {{{{\boldsymbol{{\mathcal{E}}}}}}}^{+}({{{\bf{r}}}}){| }^{2}\) represents the maximum incident intensity. For visualisation, \(| \langle {\hat{{{{{\bf{P}}}}}}{}^{+}}\rangle {| }^{2}/| \langle {\hat{{{{{\bf{P}}}}}}{}^{+}}\rangle {| }_{\max }^{2}\) for point-like atoms is smoothed by convolution with a Gaussian of the root-mean-square widths σx = 0.25a and σy = 0.5a. The blue dashed lines trace the peak light intensity, while the connecting green line marks the effective trajectory in the medium. c D/λ and d power transmission T as a function of Δ/γ and ky/k across the transmission band. Green stars denote the parameters taken in b for the incident beam. e Variation of T and effective group refractive index \({n}_{{{{\rm{eff}}}}}^{{\prime} }\) with ky/k, for the same laser detuning as in (b).
