Coherent bremsstrahlung in a bent crystal

Radiation spectrum from high energy $e^\pm$ in a bent crystal with arbitrary curvature distribution along the longitudinal coordinate is evaluated, based on the stationary phase approximation. For a uniformly bent crystal a closed-form expression for the spectrum is derived. Features such as sharp end of spectrum and volume reflection turnover at beginning of the spectrum are discussed. The coherence length in a bent crystal appears to depend only on the crystal geometry and not on the electron or photon energies, which is essential for interpretation of the results. Estimates of non-dipole radiation and multiple scattering effects are given. The value for the crystal bending angle at which the dipole coherent bremsstrahlung theory holds best appears to be $\sim 10^{-4}\mathrm{rad}$.