  The dynamics and radiation of ultrarelativistic electrons in strong
counterpropagating laser beams are investigated. Assuming that the particle
energy is the dominant scale in the problem, an approximate solution of
classical equations of motion is derived and the characteristic features of the
motion are examined. A specific regime is found with comparable strong field
quantum parameters of the beams, when the electron trajectory exhibits
ultrashort spike-like features, which bears great significance to the
corresponding radiation properties. An analytical expression for the spectral
distribution of spontaneous radiation is derived in the framework of the
Baier-Katkov semiclassical approximation based on the classical trajectory. All
the analytical results are further validated by exact numerical calculations.
We consider a non-resonant regime of interaction, when the laser frequencies in
the electron rest frame are far from each other, avoiding stimulated emission.
Special attention is devoted to settings when the description of radiation via
the local constant field approximation fails and to corresponding spectral
features. Periodic and non-periodic regimes are considered, when lab
frequencies of the laser waves are always commensurate. The sensitivity of
spectra with respect to the electron beam spread, focusing and finite duration
of the laser beams is explored.
