# Effects of Atmospheric Turbulence on Lensless Ghost Imaging with Partially Coherent Light
## 2 Theory of Ghost Imaging in Monostatic Turbulence and Bistatic Turbulence  
Figure 1 shows the typical GI optical system [20; 21]. The light source is first split into two twin portions by the intensity beam splitter (BS), and then, two portions propagate through two distinct paths. One is a test path, which contains a bucket detector D\({}_{1}\) and an object. The object is close to D\({}_{1}\), and the distance from the source to D\({}_{1}\) is \(z_{1}\). Another is a reference path, where a high spatial resolution single-photon detector D\({}_{2}\) is located at a distance \(z_{2}\) from the source. Generally, both of two paths are in turbulence. The output signals from D\({}_{1}\) and D\({}_{2}\) are sent to a coincidence circuit to measure the forth-order correlation function (FOCF), i.e., intensity fluctuations. In order to observe the image information, the condition \(z_{2}=z_{1}=\mathrm{z}\) should be satisfied. As shown in Figure 1, the turbulence is introduced in two optical paths. Figure 1: Schematic setup for lensless ghost imaging through turbulence. BS, beam splitter; D\({}_{1}\), bucket detector; D\({}_{2}\), single-photon detector. Based on the optical coherence theory, the FOCF between the planes of \(u_{1}\) and \(u_{2}\) can be written as  
\[ \begin{array}{ll}G^{(2)}(u_{1},u_{2})&=\langle I(u_{1})I(u_{2})\rangle\\ &=\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{- \infty}^{\infty}\int\limits_{-\infty}^{\infty}\langle\langle E(x_{1})E^{*}(x_{ 2})E(x_{3})E^{*}(x_{4})\rangle_{s}\\ &\times\langle h_{1}(u_{1},x_{1})h_{1}^{*}(u_{1},x_{2})h_{2}(u_{2},x_{3})h_{2 }^{*}(u_{2},x_{4})\rangle_{m}dx_{1}dx_{2}dx_{3}dx_{4},\end{array}  \tag{1}\]  
where \(I(u)\) is the instantaneous intensity at point \(u\) in the transverse plane of \(u_{i}\) (\(i\) = 1, 2); \(\langle E(x_{1})E^{*}(x_{2})E(x_{3})E^{*}(x_{4})\rangle_{s}\) is the FOCF of the light source, with \(x\) being the coordinate in the source plane. The angle brackets with subscript \"\(s\)\" denote the ensemble average over the source fluctuations. \"\(s\)\" is the complex conjugate. \(\langle...m\rangle\) stands for the ensemble average over the statistics of turbulence. \(h_{1}\) and \(h_{2}\) are the response functions of Path 1 and Path 2, respectively, given by:  
\[h_{1}(u,x)=\left(\frac{i}{\lambda z_{1}}\right)^{1/2}H(u)\exp\left[-\frac{i \pi}{\lambda z_{1}}(u-x)^{2}\right]\exp\left[\psi_{1}(u,x)\right], \tag{2}\]  
\[h_{2}(u,x)=\left(\frac{i}{\lambda z_{2}}\right)^{1/2}\exp\left[-\frac{i\pi}{ \lambda z_{2}}(u-x)^{2}\right]\exp\left[\psi_{2}(u,x)\right], \tag{3}\]  
\(H(u)\) in Equation (2) represents the transmission function of the object. \(\lambda\) is the wavelength; \(\psi_{1}(u,x)\) and \(\psi_{2}(u,x)\) are the complex phase perturbations introduced by the turbulence in Path 1 and Path 2, respectively. On substituting Equations (2) and (3) into Equation (1), after some rearrangement, it turns out to be:  
\[ \begin{array}{ll}G^{(2)}(u_{1},u_{2})&=\frac{1}{\lambda^{2}z_{1}z_{2}}\int \limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty} ^{\infty}\int\limits_{-\infty}^{\infty}dx_{1}dx_{2}dx_{3}dx_{4}|H(u_{1})|^{2} \langle E(x_{1})E^{*}(x_{2})E(x_{3})E^{*}(x_{4})\rangle_{s}\\ &\times F_{4}(x_{1},x_{2},x_{3},x_{4};u_{1},u_{2})\exp\left[-\frac{ik}{2z_{1} }(x_{1}-u_{1})^{2}\right]\exp\left[\frac{ik}{2z_{1}}(x_{2}-u_{1})^{2}\right] \\ &\times\exp\left[-\frac{ik}{2z_{2}}(x_{3}-u_{2})^{2}\right]\exp\left[\frac{ik} {2z_{2}}(x_{4}-u_{2})^{2}\right],\end{array}  \tag{4}\]  
\(F_{4}(x_{1},x_{2},x_{3},x_{4};u_{2},u_{2})\) in Equation (4) is the fourth-order coherence function of the turbulent medium, given by:  
\[F_{4}(x_{1},x_{2},x_{3},x_{4};u_{1},u_{2})=\langle\exp[\psi_{1}(x_{1},u_{1})+ \psi_{1}^{*}(x_{2},u_{1})+\psi_{2}(x_{3},u_{2})+\psi_{2}^{*}(x_{4},u_{2})] \rangle_{m}, \tag{5}\]  
Assume that the light source obeys Gaussian statistics with a zero mean.