On the coherent radiation of an electron bunch moving in an arc of a circle

Existing theories of coherent synchrotron radiation (CSR) are developed for the motion of an electron bunch on a circular orbit and do not describe the case of finite magnet length. In this paper we present the CSR theory for a bunch of any length moving in an are of a finite angle. The radiative interaction of the electrons in the bunch is analyzed for a line charge distribution using an ultrarelativistic approximation. It is shown in particular that this interaction is important not only inside the magnet but also on the straight part of the trajectory after the magnet. Detailed analytical study of the CSR effects in the electron bunch with a stepped distribution of the charge density has been performed. The simple analytical technique of the radiative force calculation has been developed. The analytical solutions in the form of elementary functions are obtained for the radiative interaction force, for the energy loss distribution along the bunch and for the total energy loss of the bunch. The latter result is confirmed with calculation of the energy of coherent radiation in the far zone. The criterium for the applicable region of the existing theories to the case of a finite magnet length is obtained.