Undulator Radiation

For this discussion we use a reference frame with z along the undulator axis and x and y perpendicular to z. For a helical undulator, the field near the axis $z$ is given by

where e_x, e_y, and e_z are unit vectors along the x, y, and z axes. The electron velocity in this field, in units of c, is

The amplitude of the transverse velocity is a_w/. This quantity also represents the maximum angle between the electron trajectory and the undulator axis. For relativistic beams this quantity is much less than 1, and the longitudinal velocity is near 1. Notice that in the case of a helical undulator the axial component of the velocity,_z, remains constant. For a planar undulator only one of the two transverse components in equations for the B-field and the motion is nonzero. In this case the longitudinal velocity is not constant; in fact it can be written as the sum of a constant term plus one oscillating at twice the undulator period. The electron trajectory in a helical undulator is a helix of radius a_w0/2_z and period equal to the undulator period. For a planar undulator it is a sinusoid in the plane perpendicular to the magnetic field.

When traversing the undulator the electrons are subject to an acceleration and radiate electromagnetic waves. This spontaneous emission is fundamental to the operation of a FEL. One relativistic electron traversing an undulator magnet emits radiation in a narrow cone, with an angular aperture of order 1/ [13]. The radiation spectrum is a superposition of the synchrotron radiation emitted when the electron trajectory is bent in the undulator field, a narrow line introduced by the periodic nature of the motion in the same field, and its harmonics. The wavelength of the fundamental line is of the order of ₀/2², which can be interpreted as the undulator period ₀ doubly Doppler shifted by the electron motion.

The aperture 1/ of the synchrotron radiation cone can be compared with the angle between the electron trajectory and the undulator axis, a_w/. If a_w is smaller than 1, the emitted radiation is contained within the synchrotron radiation cone, and the emission is predominantly in a single line. If a_w is larger than 1, the synchrotron radiation cone sweeps an angle larger than its aperture. In this case the spectrum is rich in harmonics and approaches the synchrotron radiation spectrum when a_w is very large. When a_w > 1 the magnet is usually referred to as a wiggler, reserving the name undulator for the case a_w less than or of the order of one.

The fundamental wavelength seen by an observer looking at an angle to the undulator axis is =(₀/2²)(1+a_w²+²²), and is the wavelength at which there is positive interference between the radiation emitted in two undulator periods. This is also, when =0, the wavelength corresponding to the "synchronism condition''.

In a planar undulator the magnetic field has only one component, say along y. Then the velocity has components in the x and z directions. The wavelength of the spontaneous radiation is the same as that for a helical undulator if the vector potential a_w is replaced by its
rms value, a division by the square root of 2.

In a helical undulator the radiation on axis contains only the fundamental harmonic. In a planar undulator the longitudinal velocity is modulated twice at the undulator period. This leads to a richer spectrum: all the odd harmonics of the resonant wavelength appear on axis. For both type of undulators even and odd harmonics appear off axis.

The linewidth of the radiation is determined by the number N_w of undulator periods; the total length of the pulse emitted by one electron is that of a wavetrain with a number of periods equal to N_w, the number of undulator periods. The corresponding linewidth of the fundamental is

When more than one electron is producing the radiation, the pulse length and angular distribution can be affected by the electron beam characteristic. The pulse length is increased by the electron bunch length when this is larger than N_w.
The radiation linewidth is increased if the beam energy spread is larger than 1/2N_w, the single electron width. Similarly, the effective source radius and angular distribution can be increased by the electron beam.