Resonant-light diffusion in a disordered atomic layer

Light scattering in dense media is a fundamental problem of many-body physics, which is also relevant for the development of optical devices. In this work we investigate experimentally light propagation in a dense sample of randomly positioned resonant scatterers confined in a layer of subwavelength thickness. We locally illuminate the atomic cloud and monitor spatially resolved fluorescence away from the excitation region. We show that light spreading is well described by a diffusion process, involving many scattering events in the dense regime. For light detuned from resonance we find evidence that the atomic layer behaves as a graded-index planar waveguide. These features are reproduced by a simple geometrical model and numerical simulations of coupled dipoles.

Figure 1

Experimental setup. A layer of atoms is illuminated at its center with a resonant light beam of radius $R_0$. Photons initially emitted in the excitation region can be scattered several times in the atomic layer before exiting the system and being collected and imaged. (b) Typical fluorescence image at resonance and maximum density. The black circle, of diameter of $2R_0=5\mu \mathrm{m}$, delineates the excitation region. Intensity is not uniform inside the excitation region partially because of finite optical resolution but also, and more importantly, because of the light diffusion process itself, as described in the text. (c) Schematic of the imaging system. A pinhole is imaged on the atomic cloud using a microscope objective (MO1). The atomic fluorescence is imaged on a camera (CCD) using a second and identical objective (MO2) and a set of lenses. Fluorescence is spatially filtered using an iris and the detected polarization is chosen by adjusting a combination of a half-wave plate (HWP) and a polarization beam splitter (PBS).

Figure 2

Fluorescence decay. Binned and azimuthally averaged profile for the measured atomic fluorescence (squares). Circles show the small background signal observed without atoms. The shaded area represents the excitation region. Statistical errors bars due to photon counting are smaller than the size of the points. The solid lines are broken lines linking the points. The inset shows the same data outside the excitation region in a semilogarithmic plot to emphasize the exponential decay of the fluorescence signal. We attribute the deviation of the last two points from the exponential fit to the contribution from stray light. Data are averaged over 100 measurements.

Figure 3

(a) Measured escape radius $R_e$ for different densities at resonance ($\mathrm{\Delta }=0$). Vertical error bars correspond to the standard deviation of the results of exponential fits to the data obtained with a bootstrap approach [29]. Each data set is obtained from the average of 100 measurements. Horizontal error bars represent the statistical uncertainty on the atom number. The inset shows the relevant parameters of the simple model described in the main text. (b) Escape radius $R_e$ obtained from the analytical result of Eq. (4) (solid curve) and numerical simulations of a model of classical coupled dipoles (squares) as a function of density. Statistical error bars on $R_e$ obtained from the fitting procedure of the coupled dipole simulations are smaller than the size of the points. The inset shows the fluorescence signal computed with coupled dipole simulations for ${\rho }_{\text{3D}}{k}^{-3}=0.12$ and averaged over more than 1000 atomic distributions. The solid line is the exponential fit outside the illuminated region.

Figure 4

Escape radius versus detuning. (a) Experimental results obtained for a dense cloud with ${\rho }_{\text{3D}}{k}^{-3}\simeq 0.35$. The insets show the bending of the trajectories which depends on the sign of the detuning. The dashed lines in the insets correspond to the maximal emission angle a photon leaving the sample could have without considering the bending effect. Taking into account the bending effect, this maximal angle is smaller for negative detuning and larger for positive detunings. Each point is the average result of 100 measurements. (b) Escape radius computed from a numerical model of coupled dipoles (squares), with the escape radius $R_e=\ell /\sqrt{1-cos{\theta }_0}$ obtained by solving Eq. (5) for ${\theta }_0$ (solid line). All theory curves are computed for the largest density we are able to handle ${\rho }_{\text{3D}}{k}^{-3}=0.1$. Statistical error bars obtained from the fitting procedure for the coupled dipole simulations are smaller than the size of the points.