Optical Talbot carpet with atomic density gratings obtained by standing-wave manipulation

We report a method of generating atomic density gratings by standing-wave manipulation of ultracold Bose gases. By illuminating the gratings with a beam of homogeneous light, we observe one- and two-dimensional Talbot carpets. We further measure the autocorrelation function of the images, which varies with image plane position caused by Talbot effect. Periodically repeated self-images and subimages of the atomic density gratings are observed clearly. Our theoretical calculations and experimental results agree well with each other. The atomic density gratings hold promise for substituting for traditional gratings in many applications.

Figure 1

(a) Schematic diagram of one-dimensional atomic density gratings generation. The optical lattices represented by the red arrows in the horizontal direction are switched on adiabatically and maintain a period of time first. Then, a three-step sequence of standing-wave light pulses in the vertical direction is applied on the atom cloud. Periodic momentum distribution is generated in this process, and atomic density grating (red parts in the figure) with spatial periodicity is achieved after several milliseconds of ballistic expansion. A plane wave represented by brown regions is incident upon the gratings vertically to the grating plane, and the absorption images are detected by a CCD camera that can be moved along the plane wave propagation direction. (b)–(e) Four examples of 1D atomic density gratings generated in the experiments. Panel (b) shows an atomic density grating with the identical amplitudes, panel (c) shows a supergrating with stagger periods, panel (d) shows a grating where the specified peaks have been enhanced or weakened, and panel (e) shows a grating with different size and duty factors. The focal images of the atomic density gratings are shown in the figures. The shade of the color indicates the optical density of the atoms, with the maximum value normalized to 1. The integrated optical densities (OD) of the gratings are shown simultaneously, with the integrals normalized to 1. Different patterns in panels (b)–(d) arise from different sequences of standing-wave light pulses. The changing of the grating size in panel (e) arises from the different TOF. The parameters used for generating these gratings are listed in Table 1.

Figure 2

Talbot carpet with 1D atomic density gratings. (a) Talbot carpet directly measured in the experiment. The horizontal axis indicates the position of the CCD camera, and the vertical axis indicates the fringes of the images. The shade of the color indicates the optical density of the atoms, with the maximum value normalized to 1. Three vertical lines from left to right mark the position of $z=0$ (focal image), $z=3\mathrm{mm}(z_T/4)$, and $z=6\mathrm{mm}(z_T/2)$. The period number of the self-images reduces along with the distance of the CCD camera away from the image focus increase, and negative optical density areas appear at the edges of the carpet. (b) Talbot carpet obtained by theoretical calculations shows good agreement with the experimental results.

Figure 3

Correlations of the Talbot images. (a) Periodicity of the absorption images. Figures in rows (1), (2), and (3) show the integrated optical densities, ACFs, and the second-order derivative of the ACFs at $z=0, z=z_T/4=3\mathrm{mm}$, and $z=z_T/2=6\mathrm{mm}$, respectively. The integrals of the integrated optical densities in row (1) are normalized to 1. Talbot self-image at $z=z_T/2=6\mathrm{mm}$ is shifted by half the width of the grating period relative to $z=0$, although the ACF shows the same periodicity between them. The periodic signal at $z=z_T/4=3\mathrm{mm}$ is much weaker than the other two situations, and the halved period is shown clearly by calculating the second-order derivative of the ACF. The imaging resolution is $6.4\mu \mathrm{m}$ in our experiment. (b) Correlation parameter $\mathcal{C}$ versus position of CCD camera. The appearances of all self-images and subimages which are in the resolution range of the experiment are clearly displayed. Each of the local maxima corresponds to a self-image or a subimage of the atomic density grating. The red curve denotes the results of the theoretical calculations performed with the same imaging resolution as in the experiment (error bars denote the standard deviation of eight measurements). For comparison, the dashed gray line shows the results of the theoretical calculations in which the imaging resolution is reduced by half.

Figure 4

Talbot effect with 2D atomic density gratings. The images correspond to the CCD position at $z=0, z=z_T/8=1.5\mathrm{mm}, z=z_T/4=3\mathrm{mm}, z=3z_T/8=4.5\mathrm{mm}$, and $z=z_T/2=6\mathrm{mm}$, respectively. The curves at the side of the images are the integrated optical densities in two directions, with the integrals normalized to 1. The scales of the coordinates are identical among these curves. The Talbot self-image and the subimage are clearly shown at $z=z_T/2=6\mathrm{mm}$ and $z=z_T/4=3\mathrm{mm}$, respectively. The shade of the color indicates the optical density of the atoms with maximum values normalized to 1.