deriving the spatial resolution and image contrast achieved in the presence of turbulence, although it *will be relevant to our signal-to-noise ratio analysis.**

The mutual coherence function of $e^{\psi_m({\bm\rho}',{\bm\rho})}$ is taken to be [@Ishimaru1978; @Osche2002] $$\begin{aligned}
\label{eq:TurbCorr}
{\langle &e^{\psi^*_m({\bm\rho}'_1, {\bm\rho}_1)}e^{\psi_m({\bm\rho}'_2, {\bm\rho}_2)} \rangle}\nonumber \\[.05in] &=e^{-(|{\bm\rho}'_1-{\bm\rho}'_2|^2+({\bm\rho}'_1-{\bm\rho}'_2)\cdot({\bm\rho}_1-{\bm\rho}_2) +|{\bm\rho}_1-{\bm\rho}_2|^2)/2\rho_m^2},\end{aligned}$$ which is the square-law approximation to the rigorous $5/3$-law behavior for uniformly-distributed turbulence [@footnote3]. In Eq. (\[eq:TurbCorr\]), $\rho_m$ is the turbulence coherence length on path $m$, given by $$\begin{aligned}
\label{eq:TurbCL}
\rho_m = (1.09 k_0^2 C_{n,m}^2 L)^{-3/5},\end{aligned}$$ in terms of the constant turbulence strength-parameter, $C^2_{n,m}$, along path $m$ [@Ishimaru1978].

We will assume that the path length $L$ satisfies the far-field coherence-propagation conditions for pseudothermal and SPDC light discussed in [@Erkmen2008Unified; @Erkmen2010AOP], viz., $k_0 a_0 \rho_0/2L \ll 1$, and $k_0 a_0^2/2L \ll 1$, respectively. For convenience, we also define $\rho_L = 2L/k_0 a_0$ and $a_L = 2L/k_0\rho_0$. These are the far-field, turbulence-free, on-target coherence length and intensity radius produced by the Gaussian-Schell autocorrelation function from Eq. (\[eq:AutoCorr\]).

Target Reflection
-----------------

Most real-world targets have surfaces that are sufficiently rough—on the scale of an optical wavelength—that reflections from them present fully-developed speckle patterns at appreciable distances from the surface [@Goodman2007]. We will model such a target as a 