# Difference between an electron wiggler and an undulator?

Both wigglers and undulators use periodic magnetic fields applied to stored relativistic electron beams to produce intense beams of UV or X-rays that can be used in a wide range of condensed matter physics and materials science experiments.

As for the difference between undulators and wigglers: synchrotron radiation has a characteristic opening angle (that goes as $$1/\gamma$$). In an undulator, the electron motion in the transverse direction is set to be on the order of the opening angle. In a wiggler, however, the motion is made to be larger than the opening angle and therefore a wider beam results.

In an undulator, radiation from the various periods interfere coherently. Sharp peaks are produced at harmonics of the resonant frequency, which depends on the electron energy, the undulation period and field strength, and the observation position. The optical wavelength is a Lorentz transformation of the undulation period into the beam frame followed by a relativistic Doppler shift back into the laboratory frame. The velocity used in the Lorentz transformation and the Doppler shift is the longitudinal electron velocity, which is less than the full electron velocity because of the electron’s curved path through the undulator.

Functionally speaking, it sounds like a wiggler could be characterized as an incoherent, underperforming undulator.

An undulator used with the wrong electron beam energy would be in effect a wiggler.

The narrow energy spectrum from a properly operated undulator, the result of the coherent addition in the forward direction would spread out to a broadband spectrum several orders of magnitude wider in energy and lower in brightness (energy per unit energy and solid angle) when operated with the wrong electron energy or the wrong alternating magnetic field strength.

Am I missing something fundamental, or does this pretty much describe the difference between the two devices?

note: I'm not asking for the difference between the two beams, I'm asking about the devices themselves, so if there are specific issues with their magnetic field designs that would be more interesting than differences in beam characteristics.

• this and this presentation (found here) may be helpful. – uhoh May 19 at 2:19

Short answer: Yes, you pretty much summed it up. However, I wouldn't go as far as describing a wiggler as an "underperforming" undulator.

Both these devices convert kinetic energy into radiation. A wiggler can produce magnitudes more radiation than an undulator, precisely because it radiates in a wider spectrum. Its overall output power is therefore much higher. (it's essentially a row of bending-magnets.)

An undulator provides a higher power per frequency, since its output is generally confined to a single frequency (and its harmonics). However, the output of a wiggler at that specific frequency can be comparable or higher (with the same electron beam / the same number of pair magnets). The undulator setup acts mainly as a filter, so unwanted frequencies are absorbed by the electrons and returned to kinetic energy of the electron beam. It does however produce a much narrower beam and thereby its brilliance can be much higher. The main advantage, if a specific frequency is desired, is in reducing heat load to monochromators after the insertion device.

Both undulator and wiggler are so similar, they are usually built as the same device. By increasing or decreasing magnetic field strength or moving permanent magnets closer or farther apart, the device can be set up as wiggler or undulator. The device can be classified as one or the other by its strength parameter:

$$K=\frac{e B \lambda_U}{2 \pi m_e c}$$

• K << 1 : classic undulator, waves can interfere and radiation will be a harmonic of the spatial magnet separation $$\lambda_U$$.
• K >> 1 : classic wiggler, electrons radiate independently and a wide high-power spectrum is the result.
• K ~ 1 : in between, generally results in a spectrum similar to that of a single bending magnet, though undulator/wiggler specific polarization.
• Excellent! This is exactly what I needed to know, thank you for the concise yet clear answer. – uhoh Jul 29 at 15:40