# Effects of Atmospheric Turbulence on Lensless Ghost Imaging with Partially Coherent Light
## 3 Numerical Examples and Analysis  
In this section, we investigate the visibility and quality of the ghost image both in bistatic and monostatic turbulences through numerical examples. The wavelength of light in the calculation is set as \(\lambda~{}=~{}632.8\) nm, and the constant structures \(C_{n}^{2}~{}=~{}C_{n1}^{2}~{}=~{}C_{n2}^{2}\) were fixed in the following numerical examples. The object is a double slit with slit width \(a~{}=~{}0.6\) mm, and the distance of two slits \(d~{}=~{}1.2\) mm; the beam width and the coherence width of the sources are \(\sigma_{I}~{}=~{}4\) mm and \(\sigma_{g}~{}=~{}0.3\) mm, respectively. Because the detector D\({}_{1}\) is a bucket detector, then after integrating over \(u_{1}\), the two normalized correlation functions are defined as:  
\[g^{(k)}(u_{2},z)=\frac{\int_{-\infty}^{\infty}G^{(2)(k)}(u_{1},u_{2},z)du_{1}} {\int_{-\infty}^{\infty}I_{4}^{(k)}(u_{1},u_{2},z)du_{1}},(k=m,b), \tag{30}\]  
\[\gamma^{(k)}(u_{2},z)=\frac{\int_{-\infty}^{\infty}\Gamma_{4}^{(k)}(u_{1},u_{ 2},z)du_{1}}{\left[\int_{-\infty}^{\infty}\Gamma_{4}^{(k)}(u_{1},u_{2},z)du_{ 1}\right]_{\max}},(k=m,b), \tag{31}\]  
the superscripts \(k=m\) and \(k=b\) denote the case of monostatic and bistatic turbulences, respectively. The subscript \"\(\max\)\" in Equation (31) denotes the maximum value of \(\Gamma_{4}\). \(g^{(k)}(u_{2},z)\) in Equation (30) stands for the FOCF normalized by the background noise, and \(\gamma^{(k)}(u_{2},z)\) denotes the normalized image information subtracting out the background noise. The visibility of ghost image is defined as [17]:  
\[V^{(k)}=\frac{\left[g^{(k)}(u_{2},z)\right]_{\max}-\left[g^{(k)}(u_{2},z) \right]_{\min}}{\left[g^{(k)}(u_{2},z)\right]_{\max}+\left[g^{(k)}(u_{2},z) \right]_{\min}}, \tag{32}\]  
where \"\(\min\)\" in Equation (32) denotes the minimum value of \(g^{(k)}(u_{2},z)\). The distance from the source to detector D\({}_{1}\) or D\({}_{2}\) in the calculation is 3 m, and the constant structure \(C_{n}^{2}\) is relatively large within the range \(10^{-10}\) m\({}^{-2/3}-10^{-12}\) m\({}^{-2/3}\), which is the typical parameter in the laboratory. Figure 2 illustrates the values of \(g^{(k)}(u_{2},z)\) and \(\gamma^{(k)}(u_{2},z)\) against \(u_{2}\) for different values of \(C_{n}^{2}\). For the convenience of comparison, the ghost image in free space (\(C_{n}^{2}~{}=~{}0\)) is also plotted in Figure 2 (solid lines). For the case of bistatic turbulence, one sees from Figure 2a,b that the quality of the image degrades as the turbulence strength increases. This result is consistent with that discussed in [32]. In monostatic turbulence, the quality of the image keeps unchanged with the increase of turbulence compared to that in free space (see in Figure 2d), if we subtract out the background noise. It seems that the image is immune to the turbulence, which agrees well with the experiment results in [39]. To display the visibility of the image of the double slit in two types of turbulence, we plot in Figure 3 the visibility of the image as a function of the constant structure \(C_{n}^{2}\). One can see that the visibility drops as the strength of the turbulence increases both in monostatic and bistatic cases. It decreases much faster in the monostatic case than that in bistatic case when \(C_{n}^{2}~{}<~{}1.5~{}\times~{}10^{-10}\) m\({}^{-2/3}\), while with the further increase of \(C_{n}^{2}\), the situation is inversed. Why are the visibility and quality of the image different in monostatic and bistatic turbulences? There are two main factors in atmospheric turbulence affecting the measured FOCF in the GI system due to the random fluctuation of the refraction index in the atmosphere. One is the beam wander, i.e., the random displacement of the beam position in the receiver plane. Another is the scintillation, which means random intensity fluctuation occurs. It is known that the ghost image arises from the intensity fluctuations of the light beams from test and reference paths in the GI system.