Theoretical analysis of quantum ghost imaging through turbulence

Atmospheric turbulence generally affects the resolution and visibility of an image in long-distance imaging. In a recent quantum ghost imaging experiment [P. B. Dixon et al., Phys. Rev. A 83, 051803 (2011)], it was found that the effect of the turbulence can nevertheless be mitigated under certain conditions. This paper gives a detailed theoretical analysis to the setup and results reported in the experiment. Entangled photons with a finite correlation area and a turbulence model beyond the phase screen approximation are considered.

Figure 1

(a) A generic quantum GI setup with a turbulent medium inserted in the object arm. (b) The Klyshko picture of a model setup for the theoretical description in this paper.

Figure 2

The experimental configurations considered in Dixon et al. [11]. In configurations 1 and 2, the turbulent layer (phase screen) is located in the object arm. In configurations 3 and 4, the layer is located in the reference arms. The distances $l_i$ and $d_j$ and the focal lengths of the lenses are chosen such that the four configurations give similar results (see the text).

Figure 3

Representative ghost images (cf. Ref. [11]) with theoretical fits using Eq. (22) and object transmission function (27). The left and the right columns represent the unshifted ($\Delta =0$) and the shifted ($\Delta =330$ mm) configurations, respectively. Note that the unshifted data (left) are fitted with an object of five slits whereas the shifted data (right) are fitted with an object of four slits. The top row shows images with no turbulence. The middle row shows images for turbulence between the lens and the object, 203 mm (right) and 229 mm (left) from the object. The bottom row shows images for turbulence between the crystal and the lens, 483 mm (right) and 432 mm (left) from the crystal. Shown also are the single counts of the reference detector [the envelope function of Eq. (22)] scaled by the factors adjacent to the plots.

Figure 4

Plots of the visibilities when the turbulence layer is placed (a) between the object and the lens and (b) between the nonlinear crystal and the lens. The solid curves are theoretical plots from Eq. (29) with $\lambda =2\pi /k=0.65$ $\mu$m, $a_0=0.362$ mm, ${\rho }_c=0.019$ mm, $d=0.124$ mm, and $\alpha =3.5$ mm${}^{-2}$.

Figure 5

The Klyshko picture for the extended version of the model with a turbulent layer of finite thickness $L$.