beams ${\bf x}_0^2 \gg
\Sigma_{z,y}^{-2}$. In this case the main contribution into the integral ${\bf I}({\bf x})$ (for $\eta^2=0$) in Eqs.(\[4.15\]),(\[4.18\]) at large $|{\bf x}| \simeq |{\bf
x}_0|$ (see Eq.(\[3.12\])) are given by large values $s \sim
{\bf x}_0^2 \gg \sigma_{z,y}^2$. Expanding the integrand over the powers $\sigma_{z,y}^2/s$ and keeping after integration the two main terms of the decomposition over $1/{\bf x}^2$ we get $${\bf I}^2({\bf x}) \simeq \frac{1}{{\bf
x}^2}\left[1+\frac{2}{({\bf x}^2)^2} (y^2-z^2)(\sigma_y^2 -
\sigma_z^2)\right].
 \label{5.20}$$ Expanding the function $1/({\bf x}_0+\mbox{\boldmath$\xi$})^2$ over the powers $\xi/x_0$ at the integration over $\mbox{\boldmath$\xi$}={\bf x}-{\bf x}_0$ in Eq.(\[3.12\])) we find $$\begin{aligned}
&&\int {\bf I}^2({\bf
x}_0+\mbox{\boldmath$\xi$})w_r(\mbox{\boldmath$\xi$})d^2\xi
\simeq \frac{1}{{\bf x}_0^2} \Bigg[1 + \frac{4}{({\bf
x}_0^2)^2}(z_0^2\Delta_z^2+y_0^2\Delta_y^2)
\nonumber \\
&& -\frac{\mbox{\boldmath$\Delta$}^2}{{\bf x}_0^2}+ \frac{2}{({\bf
x}_0^2)^2}(y_0^2-z_0^2)(\sigma_y^2 - \sigma_z^2)\Bigg], \quad
\mbox{\boldmath$\Delta$}^2=\Delta_z^2+\Delta_y^2.
\label{5.21}\end{aligned}$$ In this case the region $t \sim 1/{\bf x}_0^2 \ll \Sigma_{z,y}^2$ contributes into the integral $J$ Eq.(\[5.4\])). Expanding the integrand over the powers $t\Sigma_{z,y}^{-2}$ and keeping the two main terms of decomposition over $1/{\bf x}_0^2$ we have $$\begin{aligned}
&& J \simeq \frac{1}{\Sigma_z\Sigma_y {\bf x}_0^2}
\exp(z_0^2\Sigma_z^2+y_0^2\Sigma_y^2)
\Bigg\{1-\frac{\mbox{\boldmath$\sigma$}^2+\mbox{\boldmath$\Delta$}^2}{{\bf
x}_0^2}
\nonumber \\
&& +\frac{4}{({\bf x}_0^2)^2}\left[ z_0^2(\sigma_z^2+\Delta_z^2)
+y_0^2 (\sigma_y^2+\Delta_y^2)\right]\Bigg\}, \quad
\mbox{\boldmath$\sigma$}^2 =\sigma_z^2+\sigma_y^2.
\label{5.22}\end{aligned}$$ For the difference $J-J_{-}$ we obtain finally $$J-J_{-} =
\frac{1}{\Sigma_z\Sigma_y}\exp(z_0^2\Sigma_z^2+y_0^2\Sigma_y^2)
\frac{\mbox{\boldmath$\sigma$}^2}{({\bf x}_0^2)^2}.
 \label{5.23}$$

Narrow flat beams ($\sigma_z \ll \sigma_y, \Delta_z \ll \Delta_y)$
