that the correlation integration time is sufficiently long to capture a ghost image, so $T_I \gg T_0$ for the pseudothermal source, and $T_I \gg \Omega_B^{-1}$ for the SPDC source.

Resolution and Contrast {#sec:ResCon}
=======================

The spatial resolution and contrast of the ghost image are properties of the average cross correlation, $\langle C({\bm\rho}_p)\rangle$. To determine this ensemble-average cross correlation we: (1) write $\hat{i}_b(t)$ and $\hat{i}_p(t)$ in terms of the field operators that drive them, ${\hat{E}}_b({\bm\rho},t)$ and ${\hat{E}}_p({\bm\rho},t)$; (2) apply the canonical commutator relations to normally-order the resulting photocurrent-product expressions; and (3) use Eq.  to write the detected fields in terms of ${\hat{E}}'_R({\bm\rho},t)$ and ${\hat{E}}'_T({\bm\rho},t)$. From the linearity of expectation, we can bring the ensemble-average operation inside the time-domain integration to get $$\begin{aligned}
\label{eq:AvgCorr}
{\langle \hat{C}(&{\bm\rho}_p) \rangle} = q^2\eta^2 A_p \int\!d\tau_1\int\!d\tau_2 \int_{{\cal A}_b} \! d{\bm\rho}\, h(t-\tau_1)h(t-\tau_2) \nonumber \\[.05in]
&\times {\langle {\hat{E}}'^\dagger_R({\bm\rho}_p,\tau_1) {\hat{E}}'^\dagger_T({\bm\rho},\tau_2) {\hat{E}}'_R({\bm\rho}_p,\tau_1) {\hat{E}}'_T({\bm\rho},\tau_2) \rangle}.\end{aligned}$$ We then use Eqs.  and to back-propagate the target-return field operator to the target, and find $$\begin{aligned}
\lefteqn{
{\langle {\hat{E}}'^\dagger_R({\bm\rho}_p,\tau_1) {\hat{E}}'^\dagger_T({\bm\rho},\tau_2) {\hat{E}}'_R({\bm\rho}_p,\tau_1) {\hat{E}}'_T({\bm\rho},\tau_2) \rangle}}
\nonumber  \\[.05in]
&=&\int\! d{\bm\rho}_2 \!\int\! d{\bm\rho}_3\, {\langle T^*({\bm\rho}_2)T({\bm\rho}_3) \rangle}{\langle e^{\psi^*_T({\bm\rho},{\bm\rho}_2)}e^{\psi_T({\bm\rho},{\bm\rho}_3)} \rangle}\nonumber \\[.05in]
&\times& {\langle {\hat{E}}'^\dagger_R({\bm\rho}_p,\tau_1) {\hat{E}}'^\dagger_S({\bm\rho}_2,\tau_2) {\hat{E}}'_R({\bm\rho}_p,\tau_1) {\hat{E}}'_S({\bm\rho}_3,\tau_2) \rangle} \nonumber \\[.05in]
&\times& \frac{e^{-ik_0(|{\bm\rho}_2|^2-|{\bm\rho}_3|^2)/2L + ik_0{\bm\rho}\cdot({\bm\rho}_2-{\bm\rho}_3)/L}}{(\lambda_0L)^2},\end{aligned}$$ where we have used the statistical independence of 