shorter than the target beam length $$\frac{l_f}{l} \sim
\frac{1}{u}\left(1+\frac{\chi}{u}\right)^{-2/3}\frac{\gamma
\lambda_c }{l} < 1.
\label{2.6}$$ Besides it was supposed that one can neglect a variation of the impact parameter $\mbox{\boldmath$\varrho$}$ and therefore of the transverse electric field ${\bf
E}_{\perp}(\mbox{\boldmath$\varrho$})$ during the beam collision. This is true when disruption parameter is enough small $$D_i = \frac{2\alpha N_c \lambda_c l}{\gamma_r \sigma_i
(\sigma_z+\sigma_y)} \ll 1,\quad (i=z,y)
\label{2.7}$$

So, we consider the incoherent bremsstrahlung under following conditions: $$\chi \ll 1,\quad \frac{\chi}{u} \ll 1,\quad D_i \ll 1. \label{2.8}$$

[Spectral-angular distribution of the]{} 
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In this section we derive the basic expression for the incoherent bremsstrahlung probability at collision of two beams with bounded transverse dimensions.

We consider first the photon emission at collision of an electron with one particle with the transverse coordinate [**x**]{}. We select the impact parameter $\varrho_0=|\mbox{\boldmath$\varrho$}_0|$ which is small comparing with the typical transverse beam dimension $\sigma$ but which is large comparing with the electron Compton length $\lambda_c$ ($\lambda_c \ll \varrho_0 \ll \sigma$). In the interval of impact parameters $\varrho = |{\bf r}_\perp -{\bf x}| \geq \varrho_0$, where ${\bf r}_\perp$ is the transverse coordinate of emitting electron, the probability of radiation summed over the momenta of final particle can be calculated using 