# Effects of Atmospheric Turbulence on Lensless Ghost Imaging with Partially Coherent Light
## 2 Theory of Ghost Imaging in Monostatic Turbulence and Bistatic Turbulence
### Bistatic Turbulence Case  
In this situation, the complex phase perturbations in Path 1 and Path 2 are statistically independent. \(F_{4}(x_{1},x_{2},x_{3},x_{4};u_{2},u_{2})\) can be simplified as:  
\[F_{4}(x_{1},x_{2},x_{3},x_{4};u_{1},u_{2})=\langle\exp[\psi_{1}(x_{1},u_{1})+ \psi_{1}^{*}(x_{2},u_{1})]\rangle_{m}\langle\exp[\psi_{2}(x_{3},u_{2})+\psi_{2 }^{*}(x_{4},u_{2})]\rangle_{m}. \tag{10}\]  
By applying the Kolmogorov turbulence and quadratic approximation, Equation (10) takes the form [32]:  
\[F_{4}(x_{1},x_{2},x_{3},x_{4};u_{1},u_{2})=\exp\Biggl{[}-\frac{(x_{1}-x_{2})^ {2}}{\rho_{01}^{2}}\Biggr{]}\exp\Biggl{[}-\frac{(x_{3}-x_{4})^{2}}{\rho_{02}^{ 2}}\Biggr{]}, \tag{11}\]  
where \(\rho_{01}~{}=~{}\bigl{(}0.545C_{n1}^{2}k^{2}z_{1}\bigr{)}^{-3/5}\) and \(\rho_{02}~{}=~{}\bigl{(}0.545C_{n2}^{2}k^{2}z_{2}\bigr{)}^{-3/5}\) are the coherence lengths of a spherical wave in Path 1 and Path 2, respectively. \(C_{n1}^{2}\) and \(C_{n2}^{2}\) are the structure constants indicating the turbulence strength. \(k\) is the wave number of the source. In order to evaluate Equation (7), we further assume that the light source is of a Gaussian-Schell model (GSM) beam, the mutual coherence function of which is expressed as [41; 42]:  
\[\left\langle E(x_{i})E^{*}\bigl{(}x_{j}\bigr{)}\right\rangle=\exp\Biggl{[}- \frac{x_{i}^{2}+x_{j}^{2}}{4\sigma_{i}^{2}}-\frac{\bigl{(}x_{i}-x_{j}\bigr{)}^{ 2}}{2\sigma_{g}^{2}}\Biggr{]}, \tag{12}\]  
where \(\sigma_{I}\) and \(\sigma_{g}\) are the transverse beam width and transverse coherence width, respectively. On substituting Equations (11) and (12) into Equations (8) and (9), with the help of a tensor method, we obtain the following expression:  
\[ \begin{array}{ll}I_{4}^{(b)}(u_{1},u_{2})&=\frac{1}{\lambda^{2}\bigl{[}\det \bigl{(}\widetilde{B}^{-1}\bigr{)}\bigr{]}^{1/2}}\int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty }^{\infty}\left|H(u_{1})\right|^{2}\exp\Bigl{(}-\frac{ik}{2}\widetilde{\mathbf{ r}}^{T}\widetilde{M}^{-1}\widetilde{\mathbf{r}}\Bigr{)}\exp\Bigl{(}-\frac{ik}{2} \widetilde{\mathbf{r}}^{T}\widetilde{\mathbf{S}}\widetilde{\mathbf{r}}\Bigr{)} \\ &\times\exp\Bigl{[}-\frac{ik}{2}\widetilde{\mathbf{r}}^{T}\widetilde{B} \widetilde{\mathbf{r}}-2\widetilde{\mathbf{r}}^{T}\widetilde{B}\widetilde{ \mathbf{u}}\Bigr{)}\Bigr{]}d\widetilde{\mathbf{r}},\end{array}  \tag{13}\]  
\[ \begin{array}{ll}\Gamma_{4}^{(b)}(u_{1},u_{2})&=\frac{1}{\lambda^{2}\bigl{[} \det\bigl{(}\widetilde{B}^{-1}\bigr{)}\bigr{]}^{1/2}}\int\limits_{-\infty}^{ \infty}\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\int\limits_{- \infty}^{\infty}\left|H(u_{1})\right|^{2}\exp\Bigl{(}-\frac{ik}{2}\widetilde{ \mathbf{r}}^{T}\widetilde{N}^{-1}\widetilde{\mathbf{r}}\Bigr{)}\exp\Bigl{(}- \frac{ik}{2}\widetilde{\mathbf{r}}^{T}\widetilde{\mathbf{S}}\widetilde{ \mathbf{r}}\Bigr{)}\\ &\times\exp\Bigl{(}-\frac{ik}{2}\widetilde{\mathbf{r}}^{T}\widetilde{B} \widetilde{\mathbf{r}}+ik\widetilde{\mathbf{r}}^{T}\widetilde{B}\widetilde{ \mathbf{u}}\Bigr{)}d\widetilde{\mathbf{r}},\end{array}  \tag{14}\]here, \(\widetilde{\mathbf{r}}^{T}=(x_{1},x_{2},x_{3},x_{4})\), \(\widetilde{\mathbf{u}}^{T}=(u_{1},u_{2},u_{3},u_{4})\) with the superscript being the transpose symbol and:  
\[\widetilde{M}^{-1}=\left( \begin{array}{cc}\overline{M}_{1}^{-1}&0I\\ 0I&\overline{M}_{1}^{-1}\end{array} \right),\widetilde{S}=\left( \begin{array}[] {cc}\overline{S}_{1}&0I\\ 0I&\overline{S}_{2}\end{array} \right), \tag{15}\]  
\[\widetilde{B}=\left( \begin{array}{cc}\overline{B}_{1}&0I\\ 0I&\overline{B}_{1}\end{array} \right),\widetilde{N}^{-1}=\left( \begin{array}[] {cc}\overline{N}_{1}^{-1}&\overline{N}_{2}^{-1}\\ \overline{N}_{2}^{-1}&\overline{N}_{1}^{-1}\end{array} \right), \tag{16}\]  
where \(\widetilde{M}^{-1}\), \(\widetilde{S}\), \(\widetilde{B}\) and \(\widetilde{N}^{-1}\) are \(4\times 4\) matrices; \(I\) is a \(2\times 2\) unit matrix.