the target’s reflection coefficient, the signal and reference fields at the source, and the turbulence on all three propagation paths.

Substituting for $ {\langle T^*({\bm\rho}_2)T({\bm\rho}_3) \rangle}$ from Eq.  and performing the ${\bm\rho}_2$ integral eliminates the turbulence term from the preceding $\langle C({\bm\rho}_p)\rangle$ expression, which shows that turbulence on the the target-to-bucket path has no impact on the ghost image’s spatial resolution or image contrast. Now, by back-propagating the field operators to the source—by means of Eq.  with $k_0a_0^2/2L \ll 1$—we have that $$\begin{aligned}
\lefteqn{{\langle {\hat{E}}'^\dagger_R({\bm\rho}_p,\tau_1) {\hat{E}}'^\dagger_T({\bm\rho},\tau_2) {\hat{E}}'_R({\bm\rho}_p,\tau_1) {\hat{E}}'_T({\bm\rho},\tau_2) \rangle} }\nonumber \\[.05in]
&=&\frac{1}{\lambda_0^4L^6}\int\! d{\bm\rho}_2 \, {\mathcal{T}}({\bm\rho}_2) \int\! d{\bm\rho}'_1 \int\!d{\bm\rho}'_2 \int\!d{\bm\rho}''_1 \int \! d{\bm\rho}''_2 \nonumber \\[.05in]
&\times& {\langle {\hat{E}}^\dagger_R({\bm\rho}'_1,\tau_1) {\hat{E}}^\dagger_S({\bm\rho}'_2,\tau_2) {\hat{E}}_R({\bm\rho}''_1,\tau_1) {\hat{E}}_S({\bm\rho}''_2,\tau_2) \rangle} \nonumber \\[.05in]
&\times& {\langle e^{\psi^*_R({\bm\rho}_p,{\bm\rho}'_1)}e^{\psi_R({\bm\rho}_p,{\bm\rho}''_1)} \rangle}{\langle e^{\psi^*_S({\bm\rho}_2,{\bm\rho}'_2)}e^{\psi_S({\bm\rho}_2,{\bm\rho}''_2)} \rangle} \nonumber \\[.05in]
&\times& e^{ik_0[{\bm\rho}_p\cdot({\bm\rho}'_1-{\bm\rho}''_1) + {\bm\rho}_2\cdot({\bm\rho}'_2 - {\bm\rho}''_2)]/L},\end{aligned}$$ where we have again invoked the statistical independence of the signal and reference fields at the source and the turbulence on the source-to-CCD, and source-to-target paths.

We can now directly evaluate all remaining moments. The turbulence moments are available from Eq. , and the fourth-order field moment can be expressed in terms of second-order moments with the Gaussian-state moment-factoring theorem [@Shapiro1994]: $$\begin{aligned}
\lefteqn{\hspace*{-.1in}{\langle {\hat{E}}^\dagger_R({\bm\rho}'_1,\tau_1) {\hat{E}}^\dagger_S({\bm\rho}'_2,\tau_2) {\hat{E}}_R({\bm\rho}''_1,\tau_1) {\hat{E}}_S({\bm\rho}''_2,\tau_2) \rangle} =  } \nonumber \\
&& 