# Effects of Atmospheric Turbulence on Lensless Ghost Imaging with Partially Coherent Light
## 2 Theory of Ghost Imaging in Monostatic Turbulence and Bistatic Turbulence
By applying the Gaussian moment theorem [41, 42], the FOCF of the source can be expanded in terms of second-order correlation functions as follows:  
\[ \begin{array}{ll}\langle E(x_{1})E^{*}(x_{2})E(x_{3})E^{*}(x_{4})\rangle_{s}& =\langle E(x_{1})E^{*}(x_{2})\rangle_{s}\langle E(x_{3})E^{*}(x_{4})\rangle_{s }\\ &+\langle E(x_{1})E^{*}(x_{4})\rangle_{s}\langle E^{*}(x_{2})E(x_{3})\rangle_{s },\end{array}  \tag{6}\]  
On substituting Equation (6) into Equation (4), we obtain the expression:  
\[G^{(2)}(u_{1},u_{2})=I_{4}(u_{1},u_{2},z)+\Gamma_{4}(u_{1},u_{2},z), \tag{7}\]  
with:  
\[ \begin{array}{ll}I_{4}(u_{1},u_{2})&=\frac{1}{\lambda^{2}z_{1}z_{2}}\int \limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{ \infty}\int\limits_{-\infty}^{\infty}dx_{1}dx_{2}dx_{3}dx_{4}|H(u_{1})|^{2} \langle E(x_{1})E^{*}(x_{2})\rangle_{s}\langle E(x_{3})E^{*}(x_{4})\rangle_{s} \\ &\times F_{4}(x_{1},x_{2},x_{3},x_{4};u_{1},u_{2})\exp\left[-\frac{ik}{2z_{1} }(x_{1}-u_{1})^{2}\right]\exp\left[\frac{ik}{2z_{1}}(x_{2}-u_{1})^{2}\right] \\ &\times\exp\left[-\frac{ik}{2z_{2}}(x_{3}-u_{2})^{2}\right]\exp\left[\frac{ik }{2z_{2}}(x_{4}-u_{2})^{2}\right],\end{array}  \tag{8}\]\[ \begin{array}{ll}\Gamma_{4}(u_{1},u_{2})&=\frac{1}{\lambda^{2}z_{12}}\int\limits_{- \infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty}dx_{1}dx_{2}dx_{3}dx_{4}|H(u_{1})|^{2}\langle E(x_ {1})E^{*}(x_{4})\rangle_{s}\langle E^{*}(x_{2})E(x_{3})\rangle_{s}\\ &\times F_{4}(x_{1},x_{2},x_{3},x_{4};u_{1},u_{2})\exp\Bigl{[}-\frac{ik}{2z_{1 }}(x_{1}-u_{1})^{2}\Bigr{]}\exp\Bigl{[}\frac{ik}{2z_{1}}(x_{2}-u_{1})^{2} \Bigr{]}\\ &\times\exp\Bigl{[}-\frac{ik}{2z_{2}}(x_{3}-u_{2})^{2}\Bigr{]}\exp\Bigl{[} \frac{ik}{2z_{2}}(x_{4}-u_{2})^{2}\Bigr{]}.\end{array}  \tag{9}\]  
As we know from the GI scheme, the first term of Equation (7) represents the background noise, which does not contain any image information, while the second term contains the image information of the object. In atmospheric optics, it is usually assumed that the detector belongs to a \"slow\" detector. In this circumstance, three characteristic times satisfy the relation \(\tau_{s}~{}=~{}\tau_{u}~{}=~{}\tau_{a}\), with \(\tau_{s}\) being the characteristic time of the source fluctuations, \(\tau_{u}\) being the response time (integrated time) of the detectors and \(\tau_{a}\) being the characteristic time of the turbulence fluctuations. Thus, the detectors cannot perceive the instantaneous intensity fluctuations of the source due to \(\tau_{s}~{}=~{}\tau_{u}\), but they can detect the intensity fluctuations induced by the turbulence. Under this condition, the second term on the right side of Equation (7) can be neglected, and the first term is proportional to the turbulence-induced scintillation index [43]. Thus, no image information can be obtained since the information of the image is contained in the second term. In order to acquire the ghost image, the detectors must act as the \"fast detectors\", which satisfy the condition: \(\tau_{s}~{}=~{}\tau_{u}~{}=~{}\tau_{a}\). Thus, the FOCF includes the first and second terms in Equation (7) simultaneously.