Reproducing kernel Hilbert space

Working of the undulator. 1: magnets, 2: electron beam entering from the upper left, 3: synchrotron radiation exiting to the lower right

A multipole wiggler, as used in the storage ring at the Australian Synchrotron to generate synchrotron radiation

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. The static magnetic field is alternating along the length of the undulator with a wavelength $\lambda _{u}$. Electrons traversing the periodic magnet structure are forced to undergo oscillations and thus to radiate energy. 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

$K={\frac {eB\lambda _{u}}{2\pi \beta m_{e}c}}$Template:Cn

where e is the electron charge, B is the magnetic field, $\beta$ is the speed of the electron relative to the speed of light $(\beta =v/c)$, $m_{e}$ is the electron rest mass, and c is the speed of light, characterizes the nature of the electron motion. For $K\ll 1$ the oscillation amplitude of the motion is small and the radiation displays interference patterns which lead to narrow energy bands. If $K\gg 1$ 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 classical. 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 the side lobes may be ignored.

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 $N^{2}$ more than a bending magnet. The first factor of N occurs because 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.

An undulator's figure of merit is spectral radiance.

History

The first undulator was built by Hans Motz and his coworkers at Stanford in 1952. It produced the first manmade coherent infrared radiation, having a total frequency range was from visible light down to millimeter waves. The Russian physicist Vitaly Ginzburg showed theoretically that undulators could be built in a 1947 paper. Julian Schwinger published a useful paper in 1949. that reduced the necessary calculations to Bessel functions, for which there were tables. (The first UNIVAC I computer was not delivered until March 31, 1951.)

References

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External links

D. T. Attwood's page at Berkeley: Soft X-Rays and Extreme Ultraviolet Radiation. His lecture and viewgraphs are available online.