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.

Introduction

The radiation emitted by a charged particle moving with constant, relativistic speed on a circular arc is called synchrotron radiation. It is sometimes also called ordinary synchrotron radiation or bending-magnet radiation to distinguish it from the more general case of a non-circular trajectory like undulator radiation. On some occasions this radiation will be abbreviated as SR.

Some approximations are made in treating synchrotron radiation. First we assume that the radiation is emitted in a long magnet that has a constant field providing a constant curvature 1/ρ over a distance ℓr > 2ρ/γ. Secondly, the radiation is observed at a relatively large distance from the source rp ≫ ρ/γ. This approach is similar to the development of a general field into dipole, quadrupole, and higher-multipole components, for which the higher-order contributions become negligible at large distances. For an ultra-relativistic charge the opening angle of the emitted radiation is of order 1/γ and therefore very small. The radiation received originates from a small part of the trajectory and the observer is usually far away compared with this small source size. The third approximation assumes that the particle moves with ultra-relativistic velocity, γ ≫ 1, which leads to some simplifications. All three approximations are satisfied for most practical sources of synchrotron radiation.

Introduction

A wiggler is a set of dipole magnets located in a straight section that has a different field strength Bw from that of the bending magnets of the ring. They are arranged and powered in such a way that the overall bending and displacement of the electron orbit vanishes, a condition that is also realized in undulators. For this reason undulators and wigglers are often referred to as insertion devices since they are located in a straight section and can be powered at different field levels without disturbing the orbit in the rest of the machine. However, they provide some focusing of the electron beam, which might have to be corrected.

We distinguish basically between two types of wigglers: wavelength shifters, having only one period, and multipole wigglers with many periods. The wavelength shifter has a short and strong dipole magnet in the center, which is used as the main source of radiation, with longer and weaker magnets on each side to make the overall bending vanish, as illustrated in Fig. 10.1. Varying the field strength changes the critical energy and the radiated power. Usually the central field is much larger than that of the lattice bending magnets and has the purpose of providing very-short-wavelength radiation.

Introduction

We consider the radiation emitted by a charged particle moving with constant, relativistic velocity on a circular arc. It is called synchrotron radiation, or sometimes also ordinary synchrotron radiation, abbreviated as SR, to distinguish it from the related case of undulator radiation, abbreviated as UR. We start with a qualitative discussion of synchrotron radiation in order to obtain some insight into its physical properties such as the opening angle, spectrum, and polarization. This will also help us to judge the validity of some approximations used in later calculations.

The physical properties of synchrotron radiation have their basis in the fact that the charge moves with relativistic velocity towards the observer. The charge and the emitted radiation travel with comparable velocities in about the same direction. The fields created by the charge over a relatively long time are received by the observer within a much shorter time interval. This time compression determines the spectrum of synchrotron radiation.

The opening angle

We consider a charge moving in the laboratory frame F on a circular trajectory with radius of curvature ρ, Fig. 1.1. We go into a frame F′ that moves with a constant velocity that is the same as that of the charge at the instant it traverses the origin.