# Effects of Atmospheric Turbulence on Lensless Ghost Imaging with Partially Coherent Light
## 3 Numerical Examples and Analysis
For the case of monostatic turbulence, the beam wander and scintillation induced by the turbulence in two optical paths are exactly the same. In a sense, the intensity correlation from two paths is not destroyed by the turbulence. The quality of the image can keep unchanged in the monostatic turbulence. When the turbulence is bistatic, the beam wanders in two optical paths behave differently. At a certain moment, the position of the beam spot in one path drifts upwards, while in another path, it drifts downwards. The turbulence-induced scintillations in two paths are also not synchronous. Such turbulence acts as a disturber destroying the original intensity correlation between two paths. As a consequence, the quality of the image degrades. The visibility of the image is closely related to the coherence width of the beam. The smaller the coherence width is, the lower the visibility of the image is [13]. It is known that the turbulence deteriorates the transverse coherence width of the beam on propagation, while beam propagation will increase the coherence width due to diffraction. However, in the laboratory, the propagation distance is relatively short. The turbulence plays a dominant role in affecting the beam coherence width. Therefore, the visibility of the ghost image decreases with the increase of the strength of the turbulence. Figure 3: Visibility of lensless ghost image versus structure constant in monostatic turbulence (solid line) and in bistatic turbulence (dashed line). Figure 2: Lensless ghost image of \(g^{(k)}(u_{2},z)\) and \(\gamma^{(k)}(u_{2},z)\) for different values of the structure constant \(C_{n}^{2}\). Solid line: \(C_{n}^{2}\ =\ 0\); dashed line: \(C_{n}^{2}\ =\ 8\ \times\ 10^{-11}\ \mathrm{m}^{-2/3}\); dotted line: \(C_{n}^{2}\ =\ 2\ \times\ 10^{-10}\ \mathrm{m}^{-2/3}\). (**a,b**) bistatic turbulence; (**c,d**) monostatic turbulence. Let us now turn to study the characteristics of the GI when the distance from the light source to the object is an order of a hundred meters. Different from the short distance range, the propagation effect of the light may have significant effects on the visibility and quality of the image. In free space, the transverse coherence width \(\sigma_{g}(z)\) of the light will become large on propagation. Thus, the quality of image will be worse, and the visibility of image increases due to the increase of \(\sigma_{g}(z)\)[13]. While in the presence of turbulence, it will prevent the increase of \(\sigma_{g}(z)\) on propagation [48]. As a consequence, the ghost image can remain for a long propagation distance, compared to that in free space. Figure 4 shows the image of the double slit at several propagation distances, e.g., \(z=1\) m, 150 m, 300 m and 400 m through monostatic and bistatic turbulences. For comparison, the image in free space is plotted in Figure 4a. The parameters used are chosen to be \(\sigma_{I}\ =\ 15\) mm, \(\sigma_{g}\ =\ 1.5\) mm, \(C_{n}^{2}\ =\ 5\ \times\ 10^{-14}\) m\({}^{-2/3}\), \(a\ =\ 3\) mm and \(d\ =\ 6\) mm. In free space, the image of the double slit is blurred gradually as the propagation distance increases, as expected. This is caused by the increase of the transverse coherence width of the light on free-space propagation. In bistatic turbulence, the image degrades much faster than that in free space as the propagation distance increases due to the statistical independence of the turbulence in two paths. In monostatic turbulence, the quality of the image is better than that in free space in the propagation range from 0-400 m. Such turbulence displays the positive effects on the formation of the image. Figure 5 shows the dependence of the visibility of the ghost image on the propagation distance in free space, monostatic and bistatic turbulence. The parameters used in Figure 5 are the same as those in Figure 4. The visibility increases as the propagation distance increases since the coherence width of the beam increases due to beam diffraction. In the presence of turbulence, it will deteriorate the coherence width.