... series

$\left\vert \vec{r}- \vec{r}^\prime \right\vert \simeq r - \left( \vec{r} \cdot... ...eft[{r^\prime}^2/r- \left(\vec{r} \cdot \vec{r}^\prime \right)^2/r^3 \right]/2$ for $r \gg r^\prime$.

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...$-{\rm i}\omega \psi_0$

In other words, this corresponds to an additional phase shift in the incident wave: $\tilde{\phi}^0 \to \tilde{\phi}^0 {\rm e}^{-{\rm i}\omega \psi_0} \simeq \tilde{\phi}^0 \left( 1-{\rm i}\omega \psi_0 \right)$. But, $\tilde{\phi}^1$ is not changed under this shift since $\tilde{\phi}^1 \psi_0 \approx \mathcal{O}(U^2)$ is negligible. Hence, we are free to choose $\psi_0$.

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... fields

The correlation function due to electromagnetic scattering was exactly obtained, not only for weak fluctuation but also for strong fluctuation. See references, Ishimaru (1978), Tatarskii & Zavorotnyi (1980), for a detailed discussion.

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... function

We comment on the first term. In the previous work (M04), he includes the correlation between the scattered wave being second order of potential $\mathcal{O}(\psi^2)$ and the incident wave. Then, the correlation at s₁=0 is subtracted (see Eq. (23) in his paper), and ${\rm e}^{-{\rm i}\omega \vec{n} \cdot \vec{r}_1}$ in Eq. (14) is replaced by ${\rm e}^{-{\rm i}\omega \vec{n} \cdot \vec{r}_1} - {\rm e}^{-{\rm i}\omega z_1}$. Hence, the first term would vanish if we included the second order of $\psi$ in the scattered wave.

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