Radiative interaction of electrons in a bunch moving in an undulator

This paper presents investigations of the longitudinal radiative force in an electron bunch. The model of the electron bunch assumes line density distribution. General formulae are obtained for the calculation of the radiative force in the bunch moving along an arbitrary small-angle trajectory. The case of a motion in an undulator (wiggler) has been studied in detail. Analytical solutions are obtained for a rectangular and for a Gaussian bunch shape. It is shown that the rate of the bunch energy loss due to the radiative interaction is equal to the power of the coherent radiation in the far zone. Numerical estimations presented in the paper show that the effects of induced energy spread due to the radiative interaction can be important for free electron lasers operating in the infrared wavelength range.