Negative refraction and photonic-crystal optics in a cold gas

We describe propagation of light in a gas with periodic density modulation, demonstrating photonic-crystal-like refraction with negative refraction angles. We address the role of poorly defined boundaries and damping and derive an optical analog of the quantum adiabatic theorem. For Cs atoms in an optical lattice, we show that relying on semiadiabatic propagation one can excite and spatially split positively and negatively refracting modes at experimentally available gas densities.

Figure 1

(a) Negative (red) and positive (gold) refraction at an interface. (b) Negative and positive Floquet-Bloch modes in a 1D photonic crystal. (c) Solid black line: EFS in a 1D photonic crystal. Green dashed circle: EFS for light in vacuum. Red and gold arrows show the propagation directions of the $N$ and $P$ modes. Points A, ${\mathrm{A}}^{\prime }$ denote the incident and outgoing light beams in free space, as discussed in the text. Points B, C, D, and E denote the $P,N,N_{\mathrm{ref}}$ and $P_{\mathrm{ref}}$ modes.

Figure 2

(a) Amplitudes $|c_P\left(L\right)|$ (gold, right scale) and $|c_N\left(L\right)|$ (red, left scale) in dependence on $L$ and $\delta$ with $L_{*}=20\mu \mathrm{m}.$ (b) $|c_N\left(L\right)|$ at $x=L$ as a function $k_z=\omega /csin\varphi$ for two sets of parameters.

Figure 3

Strong absorption $(\delta =2\gamma )$. Spatial dependence of the real (a) and imaginary (b) parts of $k_N$ and $k_P$ and of $c_N$ and $c_P$ (c). Dashed vertical lines mark the entrance and exit zones. (d) Beam propagation through the cloud. Gold shows the $P$ beam and red the $N$ beam. (e) Intensity profiles of the modes along $z$ at $L=L_{*}.$

Figure 4

Same as in Figs. 3 and 3 but for intermediate and weak absorption: $\delta =3\gamma$ [(a) and (b)] and $\delta =7\gamma$ [(c) and (d)].