Editing Free-electron laser (section)

== Beam creation ==

[[File:Undulator.FELIX.jpg|265px|thumb|The undulator of ''FELIX''.]]
To create an FEL, an [[electron gun]] is used. A beam of [[electron]]s is generated by a short [[laser]] pulse illuminating a [[photocathode]] located inside a [[microwave cavity]] and accelerated to almost the [[speed of light]] in a device called a [[photoinjector]]. The beam is further accelerated to a design energy by a [[particle accelerator]], usually a [[linear particle accelerator]]. Then the beam passes through a periodic arrangement of [[magnet]]s with alternating [[Dipole|poles]] across the beam path, which creates a side to side [[magnetic field]]. The direction of the beam is called the longitudinal direction, while the direction across the beam path is called transverse. This array of magnets is called an [[undulator]] or a [[Wiggler (synchrotron)|wiggler]], because the [[Lorentz force]] of the field forces the electrons in the beam to wiggle transversely, traveling along a [[Sine wave|sinusoidal]] path about the axis of the undulator.

The transverse acceleration of the electrons across this path results in the release of [[photons]], which are monochromatic but still incoherent, because the electromagnetic waves from randomly distributed electrons interfere constructively and destructively in time. The resulting radiation power scales linearly with the number of electrons. Mirrors at each end of the undulator create an [[optical cavity]], causing the radiation to form [[standing wave]]s, or alternately an external excitation laser is provided.
The radiation becomes sufficiently strong that the transverse [[electric field]] of the radiation beam interacts with the transverse electron current created by the sinusoidal wiggling motion, causing some electrons to gain and others to lose energy to the optical field via the [[ponderomotive force]].

This energy modulation evolves into electron density (current) modulations with a period of one optical wavelength. The electrons are thus longitudinally clumped into ''microbunches'', separated by one optical wavelength along the axis. Whereas an undulator alone would cause the electrons to radiate independently (incoherently), the radiation emitted by the bunched electrons is in phase, and the fields add together [[Coherence (physics)|coherently]].

The radiation intensity grows, causing additional microbunching of the electrons, which continue to radiate in phase with each other.<ref>{{Cite journal | doi = 10.1088/0953-4075/38/9/023| title = X-ray free-electron lasers| journal = Journal of Physics B | volume = 38| issue = 9| pages = S799| year = 2005| last1 = Feldhaus | first1 = J. | last2 = Arthur | first2 = J. | last3 = Hastings | first3 = J. B. |bibcode = 2005JPhB...38S.799F | s2cid = 14043530| url = https://digital.library.unt.edu/ark:/67531/metadc883674/}}</ref> This process continues until the electrons are completely microbunched and the radiation reaches a saturated power several orders of magnitude higher than that of the undulator radiation.

The wavelength of the radiation emitted can be readily tuned by adjusting the energy of the electron beam or the magnetic-field strength of the undulators.

FELs are relativistic machines. The wavelength of the emitted radiation, <math>\lambda_r</math>, is given by<ref>{{Cite journal | last1 = Huang | first1 = Z. | last2 = Kim | first2 = K.-J. | doi = 10.1103/PhysRevSTAB.10.034801 | title = Review of x-ray free-electron laser theory | journal = Physical Review Special Topics: Accelerators and Beams | volume = 10 | issue = 3 | pages = 034801 | year = 2007 | bibcode = 2007PhRvS..10c4801H | doi-access = free }}</ref>

: <math>\lambda_r = \frac{\lambda_u}{2 \gamma^2}\left(1+\frac{K^2}{2}\right)</math>

or when the wiggler strength parameter {{mvar|K}}, discussed below, is small

: <math>\lambda_r \propto \frac{\lambda_u}{2 \gamma^2}</math>

where <math>\lambda_u</math> is the undulator wavelength (the spatial period of the magnetic field), <math>\gamma</math> is the relativistic [[Lorentz factor]] and the proportionality constant depends on the undulator geometry and is of the order of 1.

This formula can be understood as a combination of two relativistic effects. Imagine you are sitting on an electron passing through the undulator. Due to [[Lorentz contraction]] the undulator is shortened by a <math>\gamma</math> factor and the electron experiences much shorter undulator wavelength <math>\lambda_u/\gamma</math>. However, the radiation emitted at this wavelength is observed in the laboratory frame of reference and the [[relativistic Doppler effect]] brings the second <math>\gamma</math> factor to the above formula. In an X-ray FEL the typical undulator wavelength of 1&nbsp;cm is transformed to X-ray wavelengths on the order of 1&nbsp;nm by <math>\gamma</math> ≈ 2000, i.e. the electrons have to travel with the speed of 0.9999998''c''.

=== Wiggler strength parameter K ===

{{mvar|K}}, a [[dimensionless]] parameter, defines the wiggler strength as the relationship between the length of a period and the radius of bend,{{cn|date=October 2019}}

: <math>K = \frac{\gamma \lambda_u}{ 2 \pi \rho } = \frac{e B_0 \lambda_u}{2 \pi m_e c}</math>

where <math>\rho</math> is the bending radius, <math>B_0 </math> is the applied magnetic field, <math>m_e </math> is the electron mass, and <math>e </math> is the [[elementary charge]].

Expressed in practical units, the dimensionless undulator parameter is
<math>K=0.934 \cdot B_0\,\text{[T]} \cdot \lambda_u\,\text{[cm]}</math>.

=== Quantum effects ===

In most cases, the theory of [[classical electromagnetism]] adequately accounts for the behavior of free electron lasers.<ref name="fain">{{Cite journal | last1 = Fain | first1 = B. | last2 = Milonni | first2 = P. W. | author-link2=Peter W. Milonni|doi = 10.1364/JOSAB.4.000078 | title = Classical stimulated emission | journal = Journal of the Optical Society of America B | volume = 4 | issue = 1 | pages = 78 | year = 1987 |bibcode = 1987JOSAB...4...78F }}</ref> For sufficiently short wavelengths, quantum effects of electron recoil and [[shot noise]] may have to be considered.<ref>{{Cite book | last1 = Benson | first1 = S. | last2 = Madey | first2 = J. M. J. | doi = 10.1063/1.34633 | chapter = Quantum fluctuations in XUV free electron lasers | title = AIP Conference Proceedings | volume = 118 | pages = 173–182 | year = 1984 }}</ref>