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| [[Image:Undulator.png|300px|thumb|Working of the undulator. 1: magnets, 2: electron beam entering from the upper left, 3: synchrotron radiation exiting to the lower right]]
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| [[Image:Aust.-Synchrotron,-Undulator,-14.06.2007.jpg|thumb|right|250px|A multipole wiggler, as used in the [[storage ring]] at the [[Australian Synchrotron]] to generate [[synchrotron radiation]]]]
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| An '''undulator''' is an [[insertion device]] from high-energy physics and usually part of a larger
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| installation, a [[synchrotron]] [[storage ring]]. It consists of a periodic structure of [[dipole magnet]]s. The static [[magnetic field]] is alternating along the length of the '''undulator''' with a wavelength <math>\lambda_u</math>. 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_light|collimated]] on the orbit plane of the electrons. This radiation is guided through [[beamlines]] for experiments in various scientific areas.
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| The important dimensionless parameter
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| <math>K=\frac{e B \lambda_u}{2 \pi \beta m_e c}</math>{{cn|date=October 2013}}
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| where ''e'' is the electron charge, ''B'' is the magnetic field, ''<math>\beta</math>'' is the speed of the electron relative to the speed of light ''<math>(\beta=v/c)</math>'', ''<math>m_{e}</math>'' is the electron rest
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| mass, and ''c'' is the speed of light, characterizes the nature of the electron motion. For
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| <math>K\ll1</math> the oscillation amplitude of the motion is small and the radiation displays interference patterns which lead to narrow energy bands. If <math>K\gg1</math> 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 (synchrotron)|wiggler]].
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| 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.
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| 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 [[Synchrotron_light_source#Brilliance|brightness]] can be up to <math>N^{2}</math> 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.
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| 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.
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| If the electrons follow the [[Poisson distribution]] a partial interference leads to a linear increase in intensity.
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| In the [[free electron laser]]<ref>Paolo Luchini, Hans Motz, ''Undulators and Free-electron Lasers'', Oxford University Press, 1990.</ref> the intensity increases exponentially with the number of electrons.
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| An undulator's [[figure of merit]] is [[spectral radiance]].
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| ==History==
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| The first undulator was built by [[Hans Motz]] and his coworkers at [[Stanford]]<ref>{{Cite doi|10.1063.2F1.1700002}}</ref><ref>{{Cite doi|10.1063/1.1721389}}</ref> in 1952. It produced the first manmade coherent infrared radiation, having a total frequency range was from visible light down to [[extremely high frequency|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.<ref name="js">{{Cite doi|10.1103/PhysRev.75.1912}}</ref> 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.)
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| ==References==
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| {{Reflist}}
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| ==External links==
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| D. T. Attwood's page at Berkeley: [http://www.coe.berkeley.edu/AST/sxreuv/ Soft X-Rays and Extreme Ultraviolet Radiation]. His lecture and viewgraphs are available online.
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| [[Category:Synchrotron instrumentation]]
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| [[fr:Synchrotron#Éléments d'insertion]]
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