Effects of the electron distribution on the radiation

Introduction

We calculated the synchrotron radiation emitted by a single particle on the nominal orbit. In other words, the position and angular direction of the moving electron were identical to the origin and axis we used to describe the synchrotron radiation. For an electron beam of finite emittance there is a spread in position and angle, with RMS values σx, σ′x and σy, σ′y around the nominal values. As a result, we observe radiation emitted at different angles and positions. As a first consequence, the emitted photon beam has an increased emittance, at least equal to the sum of the electron emittance and the natural photon emittance. Furthermore, since the frequency emitted by an electron depends on the angle of observation, the angular spread of the electrons tends to broaden the spectrum observed on the axis. This effect is negligible for standard synchrotron radiation but can be pronounced in the case of undulators with many periods.

Apart from a spread in angle and position, the electrons have also a spread in energy. The synchrotron-radiation frequency depends strongly on the electron energy. The spread of the latter leads again to a broadening of the synchrotron-radiation spectrum.