Gaussian intensity in the first Rytov approximation

Let ${\psi }_R={\psi }_i{e}^{{\varphi }_1}$ with unit-modulus incidence and ${\varphi }_1={\int }_{D}K(\mathbf{r},{\mathbf{r}}^{\prime })V({\mathbf{r}}^{\prime })d{\mathbf{r}}^{\prime }$. For a centered real Gaussian random field $V$, the exponential moment of a Gaussian linear functional givesHere $C_V$ is the autocorrelation function of a random field. Both the covariance $\mathbb{E}|{\varphi }_1{|}^2$ and the generally nonzero term $\mathrm{Re}\mathbb{E}{\varphi }_1^2$ enter the exponent; dropping the latter is not justified for an arbitrary finite real random medium.