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# Undulator

An undulator is an insertion device from high-energy physics and usually part of a larger installation, a synchrotron storage ring. It consists of a periodic structure of dipole magnets (see dipole magnet). The static magnetic field is alternating along the length of the undulator with a wavelength λu. Electrons traversing the periodic magnet structure are forced to undergo oscillations and radiate. The radiation produced in an undulator is very intense and concentrated in narrow energy bands in the spectrum. It is also collimated on the orbit plane of the electrons. This radiation is guided through beamlines for experiments in various scientific areas.

The important dimensionless parameter

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

where e is the particle charge, B the magnetic field, me the electron rest mass and c the speed of light, characterizes the nature of the electron motion. For $K\ll1$ the oscillation amplitude of the motion is small and the radiation displays interference patterns which lead to narrow energy bands. If $K\gg1$ the oscillation amplitude is bigger and the radiation contributions from each field period sum up independently, leading to a broad energy spectrum. In this regime of fields the device is no longer called an undulator; it is called a wiggler.

The usual description of the undulator is relativistic but classic. This means that though the precision calculation is tedious the undulator can be seen as a black box. An electron enters this box and an electromagnetic pulse exits through a small exit slit. The slit should be small enough such that only the main cone passes, so that we do not have to deal with the side lobes.

Undulators can provide several orders of magnitude higher flux than a simple bending magnet and as such are in high demand at synchrotron radiation facilities. For an undulator with N periods, the brightness can be up to N2 more than a bending magnet. The intensity is enhanced up to a factor of N at harmonic wavelengths due to the constructive interference of the fields emitted during the N radiation periods. The usual pulse is a sine with some envelope. The second factor of N comes from the reduction of the emission angle associated with these harmonics, which is reduced as 1/N. When the electrons come with half the period, they interfere destructively, the undulator stays dark. The same is true, if they come as a bead chain.

The polarization of the emitted radiation can be controlled by using permanent magnets to induce different periodic electron trajectories through the undulator. If the oscillations are confined to a plane the radiation will be linearly polarized. If the oscillation trajectory is helical, the radiation will be circularly polarized, with the handedness determined by the helix.

If the electrons follow the Poisson distribution a partial interference leads to a linear increase in intensity. In the free electron laser the intensity increases exponentially with the number of electrons.

The figure of merit is spectral radiance.