Dimensional crossover in Bragg scattering from an optical lattice

We study Bragg scattering at one-dimensional (1D) optical lattices. Cold atoms are confined by the optical dipole force at the antinodes of a standing wave generated inside a laser-driven high-finesse cavity. The atoms arrange themselves into a chain of pancake-shaped layers located at the antinodes of the standing wave. Laser light incident on this chain is partially Bragg reflected. We observe an angular dependence of this Bragg reflection which is different from what is known from crystalline solids. In solids, the scattering layers can be taken to be infinitely spread (three-dimensional limit). This is not generally true for an optical lattice consistent of a 1D linear chain of pointlike scattering sites. By an explicit structure factor calculation, we derive a generalized Bragg condition, which is valid in the intermediate regime. This enables us to determine the aspect ratio of the atomic lattice from the angular dependance of the Bragg scattered light.

Figure 1

The experimental setup consists of a cavity pumped at $811\mathrm{nm}$ and a diode laser at $780\mathrm{nm}$ for Bragg scattering. The Bragg-reflected light is observed on a CCD camera.

Figure 2

Bragg reflection in real space. The gray ellipsoids symbolize the atomic layers. Within the scattering plane the opening angle $2{\varphi }_2$ of the Bragg beam is found by projecting the widths $\delta k_x$ and $\delta k_z$ onto a plane $E$ orthogonal to the emission direction. The size of the angle is given by the larger of the two projections, here $\delta k_x$.

Figure 3

Bragg reflection in reciprocal space. The pictures show cuts through the Ewald sphere and the ellipsoid structure factor. In (a), the probe laser frequency and angle fulfill the Bragg condition. In (b), the lattice constant has changed. The outgoing wave vector $k_{s1}$ has been chosen such that ${\beta }_s=-{\beta }_i$, the wave vector ${\mathbf{k}}_{s2}$ such that $\cos {\beta }_s+\cos {\beta }_i=2{\lambda }_{\mathrm{brg}}∕{\lambda }_{\text{dip}}$. The light will be emitted in the direction where the structure factor reaches its maximum on the intersection with the Ewald sphere.

Figure 4

(a) and (b) CCD pictures of the Bragg-reflected light for two different lattice constants. By fitting a Gaussian curve to the picture (not shown here) the center position of the beam is determined. From the displacement of the beams the relative emission angle of the light beams is calculated. (c) Output angle as a function of lattice constant. The measured values (rings) are compared to various curves: The horizontal line is expected for ${\beta }_s=-{\beta }_i$, the dotted line for condition (12), and finally the solid line takes account of the finite lattice size and is a fit of Eq. (14) to the data points. The fitting parameter is the aspect ratio $\zeta$. The experimental error of the data points lies well within the plotted circles.

Figure 5

Intensity of the reflected light against emission angle. The data points (circles) are fitted with a Gaussian curve. The acceptance angle of the Bragg reflection equals the width of this fit. The error of the data points is due to shot-to-shot fluctuation and lies in the 20% range.