Editing Undulator

{{See also||Wiggler (synchrotron)}}
[[File: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]]

An '''undulator''' is an [[insertion device]] from [[high-energy physics]] and usually part of a larger
installation, a [[synchrotron]] [[storage ring]], or it may be a component of a [[free electron laser]]. It consists of a periodic structure of [[dipole magnet]]s. These can be [[permanent magnet]]s or [[superconducting magnet]]s. The static [[magnetic field]] alternates 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 [[beamline]]s for experiments in various scientific areas.

The undulator strength parameter is:
:<math>K=\frac{e B \lambda_u}{2 \pi m_e c}</math>,

where ''e'' is the electron charge, ''B'' is the magnetic field, ''<math>\lambda_u</math>'' is the spatial period of the undulator magnets, ''<math>m_{e}</math>'' is the electron rest mass, and ''c'' is the speed of light.

This parameter characterizes the nature of the electron motion. For <math> K\ll 1</math> the oscillation amplitude of the motion is small and the transverse deflection nearly sinusoidal as a function of time, so that long undulators can have narrow on-axis bandwidth, and most of the radiated power at around the fundamental wavelength. For <math>1\ll K</math> the oscillation amplitude is large and the transverse deflection is no longer sinusoidal in time so that it contains Fourier components of high harmonics of the fundamental wavelength. This kind of device naturally has a much larger bandwidth and is typically called a wiggler.<ref>{{cite book |last1=Kim |first1=Kwang-Je |title=Synchrotron Radiation and Free-Electron Lasers |date=2017 |publisher=Cambridge University Press |isbn=978-1-107-16261-7 |pages=40–64 |ref=Kim_book}}</ref>

Away from the axis of the undulator, the radiation spectrum is broadened by the angle dependent Doppler effect, so to observe the naturally narrow bandwidth, one has to use a small aperture to select only the central radiation cone.

For a device with <math>N</math> periods and a small enough aperture, the [[Synchrotron light source#Spectral brightness|brightness]] of an undulator scales like <math>N^2</math> while the brightness of a wiggler only scales like <math>N</math>. The difference is due to the naturally narrower bandwidth of the undulator.<ref>{{cite journal |last1=Kim|first1=Kwang-Je |title=Brightness and coherence of synchrotron radiation and high-gain free electron lasers |journal=Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment |date=1987 |volume=261 |issue=1-2 |pages=44–53 |doi=10.1016/0168-9002(87)90560-2 |url=https://digital.library.unt.edu/ark:/67531/metadc1191669/m2/1/high_res_d/6806336.pdf |access-date=21 January 2025 |ref=Kim_article}}</ref> Since the radiation emitted from an undulator is incoherent, the power scales linearly with the number of electrons. In a [[Free-electron laser]], some coherence is achieved and the power can scale with a higher power of the number of electrons.

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.

An undulator's [[figure of merit]] is [[spectral radiance]].

==History==
The Russian physicist [[Vitaly Ginzburg]] showed theoretically that undulators could be built in a 1947 paper.<ref>{{cite journal |last1=Ginsburg |first1=Vitaly Lazarevich |title=ОБ ИЗЛУЧЕНИИ МИКРОРАДИОВОЛН И ИХ ПОГЛОЩЕНИИ В ВОЗДУХЕ |journal=Izv. AN USSR ser. phys. |date=1947 |volume=T. XI No.2 |pages=(pp. 165–182) |url=https://www.cia.gov/readingroom/document/cia-rdp80-00809a000600340656-6 |access-date=22 September 2024 |trans-title=On the emission of microradiowaves and their absorption in the air |publisher=USSR Academy of Sciences |language=Russian |issn=0367-6765}}</ref> [[Julian Schwinger]] published a useful paper in 1949<ref name="js">{{cite journal |last1= Schwinger |first1= Julian |title= On the Classical Radiation of Accelerated Electrons |doi= 10.1103/PhysRev.75.1912 |journal= Physical Review |volume= 75 |issue= 12 |pages= 1912 |year= 1949 |bibcode= 1949PhRv...75.1912S}}</ref> that reduced the necessary calculations to [[Bessel function]]s, for which there were tables. This was significant for solving the design equations as digital computers were not available to most academics at that time.

[[Hans Motz]] and his coworkers at [[Stanford University]] demonstrated the first undulator in 1952.<ref>{{cite journal |last1= Motz |first1= Hans |title= Applications of the Radiation from Fast Electron Beams |doi= 10.1063/1.1700002 |journal= Journal of Applied Physics |volume= 22 |issue= 5 |pages= 527 |year= 1951 |bibcode= 1951JAP....22..527M}}</ref><ref>{{cite journal |last1= Motz |first1= H. |last2= Thon |first2= W. |last3= Whitehurst |first3= R. N. |doi= 10.1063/1.1721389 |title= Experiments on Radiation by Fast Electron Beams |journal= Journal of Applied Physics |volume= 24 |issue= 7 |pages= 826 |year= 1953 |bibcode= 1953JAP....24..826M}}</ref> It produced the first manmade coherent infrared radiation. The design could produce a total frequency range from visible light down to [[extremely high frequency|millimeter waves]].

==References==
{{Reflist}}

==External links==
*D. T. Attwood's page at Berkeley: [https://people.eecs.berkeley.edu/~attwood/sxr2009// Soft X-Rays and Extreme Ultraviolet Radiation]. His lecture and viewgraphs are available online.

[[Category:Synchrotron instrumentation]]

[[fr:Synchrotron#Éléments d'insertion]]