free-electron lasers
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This article was downloaded by: [Michigan State University]On: 27 September 2013, At: 05:08Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registeredoffice: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK
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Free-electron lasersS. Khan aa Centre for Synchrotronstrahlung (DELTA), Technische UniversitätDortmund, Dortmund, GermanyPublished online: 19 Dec 2008.
To cite this article: S. Khan (2008) Free-electron lasers, Journal of Modern Optics, 55:21,3469-3512, DOI: 10.1080/09500340802521175
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Journal of Modern OpticsVol. 55, No. 21, 10 December 2008, 3469–3512
TUTORIAL REVIEW
Free-electron lasers
S. Khan*
Centre for Synchrotronstrahlung (DELTA), Technische Universitat Dortmund,Dortmund, Germany
(Received 15 July 2008; final version received 1 October 2008)
This is a tutorial review of the basic principles and present status of free-electronlasers (FELs) with particular emphasis on high-gain FELs such as FLASH inHamburg. With their unprecedented intensity and a pulse duration of a few 10 fs,these novel accelerator-based radiation sources in the ultraviolet and X-rayregime will open new fields in physics, chemistry, biology and material science.Assuming that the reader is unfamiliar with FELs or particle accelerators ingeneral, this tutorial covers the basic concepts of particle accelerators and thegeneration of synchrotron radiation in some detail, and finally describes FELs inthe low-gain and high-gain regime.
Keywords: free-electron laser; accelerator; ultrafast; X-rays; synchrotronradiation
1. Introduction
1.1. What is a free-electron laser?
A free-electron laser (FEL) is a source of intense and coherent electromagnetic radiationwith tunable wavelength. The first operation of a FEL was reported in 1977 [1] in theinfrared regime at a wavelength of 3.7 mm, preceded as early as 1957 by a free-electron maserat 5mm, the ‘ubitron’, the history of which is reviewed in [2]. Meanwhile, several facilitieshave reached the ultraviolet regime. Ever since the year 2000, the world record in shortwavelength was held by the Tesla Test Facility at DESY in Hamburg, Germany [3], which isnow called FLASH and has reached a wavelength of 6.5 nm in 2007 [4]. Worldwide, a largenumber of FEL user facilities designed to produce vacuum-ultraviolet (VUV) and X-rayradiation have been proposed and some are presently under construction, such as the LinearCoherent Light Source (LCLS) [5], using a part of the normal-conducting SLAC linearaccelerator (or ‘linac’) at the Stanford University/USA and the European X-ray Laser FELat DESY [6], based on a superconducting linac. While a third generation of synchrotronlight sources emerged in the early 1990s with higher brilliance than before, the new FELs areconsidered to be the most prominent representatives of a fourth generation of radiationsources. The brilliance is defined as the number of photons per second, normalized to thesource size, usually in mm2, the radiation divergence in mrad2, and the bandwidth, i.e.a photon energy interval usually given as 0.1% of the photon energy under consideration.
*Email: [email protected]
ISSN 0950–0340 print/ISSN 1362–3044 online
� 2008 Taylor & Francis
DOI: 10.1080/09500340802521175
http://www.informaworld.com
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The peak brilliance is measured during a small fraction of the pulse duration, while the
average brilliance is measured over an extended period of time, e.g. one second. The term
‘brilliance’ tends to be more popular in Europe, whereas ‘brightness’ or ‘spectral brightness’
is often used in American publications. The properties of FELs include
. Short and tunable wavelength, where 6.5 nm has been achieved and 0.15 nm is
expected to be reached by LCLS in 2009,. A peak brilliance exceeding that of contemporary synchrotron light sources by
9–10 orders of magnitude (as shown in Figure 1) and an average brilliance higher
by 5–6 decades,. Full transverse and good longitudinal coherence,. A pulse duration well below 100 fs and the potential of producing sub-fs pulses.
Figure 1. Peak brilliance (as defined in the note 1) versus photon energy for synchrotron radiationfrom a dipole magnet, a wiggler (W125) and an undulator (U49) of a third-generation synchrotronlight source (BESSY in Berlin, Germany), and from an undulator (U29) at PETRA III, currentlyunder construction at DESY in Hamburg. The upper curves show the peak brilliance of FLASH,a free-electron laser (FEL) in operation at DESY, and two future FELs: the European X-ray FEL atDESY and LCLS at Stanford/USA. The lines represent calculations, the dots are measured data atthe fundamental wavelength of FLASH and at its third and fifth harmonic.
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Technically speaking, the FEL is a narrow-band amplifier for electromagnetic radiation
like the magnetron used e.g. in every microwave oven, or the klystron commonly used for
radio, television or radar transmitters. In either case, the energy of the electromagnetic
wave comes from electrons accelerated in some kind of vacuum tube [7]. What
distinguishes FELs from other electron tubes is that:
. The electron beam energy is in the regime of typical electron accelerators (between
several 10 MeV and the GeV regime),. The energy exchange between electrons and radiation requires a periodic magnetic
field provided by a so-called undulator,. The wavelengths range from infrared to the VUV, and will soon be extended to
the X-ray regime.
The amplification process only takes place at a particular wavelength, which is given by
the electron energy and by the strength and periodicity of the magnetic undulator field.
Undulators are arrangements of magnets with alternating field direction, in which
electrons follow a wiggly trajectory (see also Section 3.3). The amplified radiation may be
of different origin, as sketched in Figure 2. There are FEL oscillators, in which the
radiation field starts from noise and builds up between two mirrors forming an optical
cavity (just as in laser oscillators), while a train of electron bunches passes through the
undulator. These bunches may be supplied by a linear accelerator or may circulate in
a storage ring, passing the FEL repeatedly. In the case of VUV or X-ray radiation, no
mirrors with good reflectivity at normal incidence exist and the whole amplification
process must be completed within one pass of a single bunch through the undulator.
As will be discussed in Section 4 of this review, starting the radiation process from noise at
the beginning of a long undulator without external source is a well-proven and robust
technique. On the other hand, starting the FEL process with an external seed pulse at the
right wavelength may have advantages and several possibilities how to do so are currently
under investigation.
(a)
(b)
(c)
Figure 2. One type of free-electron laser (FEL) is the oscillator, (a) in which the radiation fieldbuilds up between two mirrors, while a train of electron bunches passes through an undulator, anarrangement of magnets with alternating magnetic field direction. In FEL amplifiers, a singleelectron bunch amplifies radiation which either starts from noise at the beginning of a longundulator (b) or is supplied by an external radiation source (c).
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Finally, the question may arise, whether free-electron lasers have anything to do with
lasers at all, given that they are more like a big electron tube and represent a new
generation of synchrotron light sources. FELs are indeed lasers (meaning ‘light
amplification by stimulated emission of radiation’), since they do amplify light and do it
by stimulated emission. Their energy reservoir is, however, not an ensemble of electrons
pumped to a higher atomic level by another laser or a flash lamp, but a relativistic beam of
free electrons with continuous energy transitions (hence their tunability) and pumped by
whatever is required to accelerate electrons. These transitions are stimulated by the electric
field of radiation, by which electrons are accelerated. The acceleration of electrons, in turn,
gives rise to the emission of electromagnetic waves. This mechanism is readily understood
in the framework of classical electrodynamics, whereas the transition between atomic
levels in conventional lasers calls for a quantum mechanical treatment.Since FELs are based on electron accelerators and are generally viewed in the
context of synchrotron radiation sources, with which they have much in common, the
following text approaches the topic of FEL physics by describing the basic concepts of
particle accelerators in Section 2 and reviewing synchrotron radiation sources in
Section 3. In Section 4, finally, low-gain and high-gain free-electron lasers will be
discussed.
2. Basics of particle accelerators
Since FELs use a beam of relativistic electrons, this section is meant to convey some basics
ideas of particle accelerator physics in a nutshell. For more details, the reader is referred to
the literature, e.g. [8–11]. Readers familiar with the subject may skip this section.
2.1. Acceleration of charged particles
As can be seen from the Lorentz force acting on an electron with charge �e
F ¼ �e � E� e v� Bð Þ , ð1Þ
only an electric field E can change the kinetic energy of the electron, while the magnetic
field B causes a centripetal force perpendicular to its velocity v. On the other hand,
focusing or deflecting an electron beam is usually done by magnets, since for v� c the
second term with an easily achievable magnetic field of 1T is as large as the first term with
an unrealistically high electric field of 300MVm�1. There are various ways to generate an
accelerating electric field:
. An electrostatic field is produced by separating and displacing charges, as e.g. in
Van-de-Graaf generators.. According to the law of induction, an electric field is created by a time-varying
magnetic field. Betatrons and induction linacs are based on this principle.. Radio-frequency (rf) fields with wavelengths of 0.1–1 m (the UHF band) confined
to a hollow metallic structure are the basis of most accelerators. This is true for
circular machines like microtrons or synchrotrons, as well as for linear
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accelerators. Advances in radar technology during World War II, particularly thedevelopment of the klystron as an efficient rf amplifier, have made UHF wavesuseful for particle acceleration with typical electric fields of several 10MVm�1.
. The electric field in a femtosecond laser pulse can reach 100GVm�1 and more,but being perpendicular to the direction of propagation, it cannot be used directlyto accelerate particles. The quest for higher gradients was recently highlighted byobtaining 1GeV electrons over a distance of only 3.3 cm with a laser-inducedplasma wave [12].
2.2. Acceleration by radio-frequency waves
While a static electric field accelerates a charged particle only once, standing orpropagating radiofrequency (rf) waves can increase the particle energy repeatedly. Incircular accelerators, particles pass through the same rf structure again and again, while ina linac the beam passes many identical structures one after the other. The achievable beamenergy is mainly given by economic limitations. Figure 3 shows the generic design ofaccelerators using rf structures. In a synchrotron, the magnetic field increasessynchronously with the beam energy (hence the name). Electron storage rings are quitesimilar in design, except that the rf structure only restores the energy lost by synchrotronradiation.
The principle of a cylindrical rf cavity is sketched in Figure 4 following [13]. In contrastto a capacitor with a DC voltage, an electric field oscillating with a frequency of several100MHz gives rise to an oscillating magnetic field. The time-varying magnetic field, inturn, modifies the electric field. As a result, the radial distribution of the electric fieldfollows Bessel’s function J0(kr) with wavenumber k¼ 2�f/c and radius r. For a frequencyof f¼ 500MHz (k¼ 10.5m�1), for example, the field is zero at r¼ 0.234m, sinceJ0(2.45)¼ 0. A metallic wall at this radius completes the cavity with a so-called transversemagnetic mode TM010 ringing inside (the subscripts indicate the number of longitudinal,radial and azimuthal nodes of the field).
A linac may be viewed as a succession of pillbox cavities. Alternatively, an electronlinac like the well-known two-mile accelerator at SLAC can be described as a cylindrical
(a) (b) (d)(c)
Figure 3. Schematic sketch of accelerators comprising rf structures. The shaded regions containa magnetic field perpendicular to the image plane, while the black lines represent the trajectories ofaccelerated particles. The cyclotron (a) and the microtron (b) use a time-independent magnetic field.In the synchrotron (c), the magnetic field increases with the particle energy to keep the trajectoryradius constant. This way, magnets are only required along the circumference, which makes largemachines possible. While the beam passes the same rf device repeatedly in circular machines, thelinear accelerator (d) consists of a succession of rf structures.
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waveguide with a propagating electromagnetic wave. Electrons ‘surfing’ on the wave arecontinuously accelerated, as long as their velocity matches the phase velocity of the wave.For a simple cylindrical tube, however, the phase velocity exceeds the velocity of light, seee.g. [14]. Slowing it down to the electron velocity (�c) requires a modification of theboundary conditions. In the case of the SLAC linac, equidistant copper irises witha properly chosen aperture and distance do the job. This structure is called a ‘disk-loadedwaveguide’.
2.3. Longitudinal particle dynamics
The deviation z in beam direction of a particle from its nominal position and the (relative)momentum deviation �p/p are known as longitudinal coordinates and form the so-calledlongitudinal phase space. Instead of z, the temporal deviation z/c or deviation in phase�’¼ 2�z/�rf with respect to an accelerating rf field of wavelength �rf may be used. Forhighly relativistic electrons, the momentum deviation is synonymous to the energy deviation�E/E or ��/�, where �¼ (1��2)�1/2 is the relativistic Lorentz factor with �¼ v/c. Fora sinusoidal rf field with voltage amplitude Vrf, the energy gain for an electron is
W ¼ �eVrf sin’s , ð2Þ
where ’s is the ‘design’ phase (called synchronous phase in storage rings), which is usuallynot chosen to yield the maximum energy eVrf for several reasons:
. Longitudinal focusing is provided when the arrival time at an rf structure dependson the momentum deviation such that a particle with higher (lower) momentumexperiences a lower (higher) accelerating voltage. This requires a slope of thevoltage, i.e. ’s 6¼�/2.
(a) (b) (d)(c)
(f)(e)
Figure 4. A parallel-plate capacitor with a DC voltage (a) has – neglecting the fringes –a homogeneous electric field, whereas an AC voltage (b) with wavenumber k causes the field E tofollow Bessel’s function J0(kr). A metallic wall at the radius of the first node of J0(kr) completesa simple cylindrical (‘pillbox’) rf cavity (c). The 2.9GHz disk-loaded waveguide structure of theSLAC linac is shown in (d) with a 2 �/3�mode wave, i.e. 120� phase between adjacent cavities,formed by copper disks (of which the SLAC linac contains more than 80,000). A bell-shaped cavity(e) is more elaborate to manufacture, but has several advantages. The superconducting 1.3GHzTESLA linac structure with a �-mode wave is sketched in ( f ).
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. The concept of bunch compression in linac-based FELs (see Section 4.6.2) also
requires a voltage slope in order to introduce an energy chirp along the bunch.
The energy-dependent path length in dipole magnets can then be used to reduce
the bunch length.
Deviations from ’s cause an oscillation in longitudinal position and energy, which is
called synchrotron oscillation and is much slower than the transverse particle motion
described in the next section. One oscillation period takes typically 102 revolutions in
a storage ring. Stable synchrotron motion is only possible within a limited range of energy
deviations, known as the energy acceptance of the machine, which depends as �V1=2rf on
the rf voltage.
2.4. Optics of charged particles
The guiding magnetic fields of an accelerator or storage ring define an ideal trajectory or
‘design orbit’ in the laboratory system, and particles deviating transversely from it will
perform so-called betatron oscillations about the design orbit. It is therefore convenient to
describe the motion of individual particles as time-dependent transverse deviations from
their ideal orbit.Magnets with fields up to 2 T used to deflect and focus a beam of relativistic particles
usually consist of water-cooled copper coils wound around an iron yoke. Alternatively,
permanent magnets are used for very compact arrangements like undulators.
Superconducting coils, usually made of NbTi and cooled by liquid helium, are employed
when higher fields are required (e.g. 8.5 T at the Large Hadron Collider at CERN,
deflecting a 7 TeV proton beam with a bending radius of 2.8 km).Decomposing e.g. the vertical magnetic field into multipole components yields
By ¼ B0 þdB
dxxþ
1
2
d2B
dx2x2 þ � � � ¼
p
e
1
RðsÞþp
ekðsÞxþ
p
2emðsÞx2 þ � � � , ð3Þ
i.e. a dipole field with bending radius R(s) for a particle of charge e and momentum p,
a quadrupole and a sextupole field with strength parameter k(s) and m(s), respectively,
where s is the longitudinal coordinate along the machine. The longitudinal coordinate s
should not be confused with the longitudinal deviation z, where z¼ 0 is a point moving
along the design orbit with the average particle speed. In most accelerators and storage
rings, the multipole components are realized in good approximation by individual magnets
with two, four and six poles, respectively, as sketched in Figure 5, although so-called
combined-function magnets also exist. Quadrupole magnets act as focusing lenses in one
plane and as defocusing lenses in the other, and an overall focusing effect is achieved by
using more than one quadrupole. Sextupole magnets are used to correct the focusing effect
of quadrupoles for particles deviating from their nominal energy. Only linear elements, i.e.
dipole and quadrupole magnets and drift spaces (sections with no magnetic field at all) are
covered by the matrix formalism described next.The horizontal and vertical deviation of a particle from its design orbit is described by
its phase space coordinates, (x, x0) and ( y, y0), respectively. The prime denotes the
derivative with respect to the longitudinal coordinate s, e.g. x0 ¼ dx/ds, which can be
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thought of as a transverse angle (in radian) or a transverse momentum normalized to the
total momentum x0 � px/p.To first order, the coupling of the horizontal and vertical motion due to sextupole
fields and misaligned magnets can be ignored, leading to separate equations of transverse
motion in the two planes, assuming here only vertical dipole fields:
x00ðsÞ þ1
R2ðsÞ� kðsÞ
� �xðsÞ ¼ 0 ,
y00ðsÞ þ kðsÞyðsÞ ¼ 0:
ð4Þ
The solution of these equations is an oscillation with varying frequency, which depends on
the ‘lattice’, i.e. the arrangement of magnets as a function of s. The particle vector (x, x0)
can be transformed from one position s0 to another position s1 by matrices in a similar
fashion as ABCD matrices, e.g. [15], transform light rays. A simple example is
a horizontally focusing quadrupole magnet of length L, where
x
x0
� �s1
¼cos� ðjkjÞ�1=2 sin�
�ðjkjÞ1=2 sin� cos�
!x
x0
� �s0
�1 0
�jkjL 1
� �x
x0
� �s0
ð5Þ
with �� (jkj)1/2L. In the last step, the ‘thin-lens’ approximation for small � emphasizes
the similarity to light ray optics, where jkjL is analogous to the inverse focal length of
a lens. The transfer matrix of successive elements is formed by matrix multiplication.
The general solution of Equation (4) is usually written as
xðsÞ ¼ ½"x�xðsÞ1=2 cos�xðsÞ , ð6Þ
where "x is a constant known as the Courant–Snyder invariant, �x(s) is the beta
function and �x(s) is the phase advance of the oscillation. Here and in the following,
the horizontal coordinate x is used and everything described applies to the vertical
plane as well.The beta function �x(s), its derivative expressed by �x(s)¼��
0x(s)/2, can be calculated
for a given lattice. Together with the the abbreviation �xðsÞ � ð1þ �2xðsÞÞ=�xðsÞ, they are
referred to as optical functions (or sometimes ‘Twiss parameters’), and the locus of
Figure 5. Schematic view of dipole, quadrupole and sextupole magnets, where the particle beam isperpendicular to the image plane. These devices are usually electromagnets with copper coils (black)wound around an iron yoke (gray). Technical drawings of magnets are shown in Figure 10.
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particles with the same "x is given by
"x ¼ �xðsÞ x2ðsÞ þ 2�xðsÞ xðsÞx
0ðsÞ þ �xðsÞx0 2ðsÞ , ð7Þ
i.e. a tilted ellipse in phase space, where ["x�x(s)]1/2 is the maximum spatial extension,
["x�x(s)]1/2 is the maximum angle, and �"x is the area enclosed by the ellipse. While "x is
a constant property of each particle, the shape of the ellipse, on which the particle is found,depends on the position s along the machine, as shown in Figure 6, and is given by theoptical functions. For a large beta function, the ellipse is elongated in space and small inangle, and vice versa for small �(s). For �(s)¼ 0, the ellipse is upright and the ensembleof trajectories with varying "x and random phase �x forms a beam waist or bulge.
The number of betatron oscillation periods for one revolution around a circularmachine is called ‘tune’. It is of the order of 10 and should be chosen away from integers,half-integers, etc. to avoid magnetic field imperfections to resonantly excite an oscillation(just as periodic bumps on a road can resonantly excite eigenmodes of a car). The betatronoscillation of an electron is damped by the emission of synchrotron radiation, whichcarries away longitudinal as well as transverse momentum, while the rf system restores
Figure 6. Top view (x versus s) of a beam waist in a drift space, i.e. no magnetic field, with fourparticles on the same phase space ellipse: two with maximum offset (white dots), two with maximumangle (black dots). The trajectories of all particles on or within that ellipse lie within the envelopeindicated by gray lines. The phase space distribution of a Gaussian beam is shown for threepositions. The gray ellipses mark one standard deviation of x and x 0. Their area has a constant valuedefined by �"x, where "x is the horizontal beam emittance.
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only the longitudinal component. This damping mechanism is an important property of
electron storage rings.In a storage ring, the horizontal orbit for particles with momentum deviation �p/p is
subject to an additional deviation �x¼D(s) ��p/p, where D(s) is the horizontal
dispersion, while the vertical dispersion is usually negligible.Sextupole magnets are required to correct the momentum-dependent focusing effect
of quadrupoles. With their magnetic field being nonlinear in x and y, they couple
horizontal and vertical motion and can give rise to chaotic trajectories for particles
venturing beyond the so-called dynamic aperture (as opposed to the physical aperture,
given by solid obstacles). The nonlinearity of the fields in x and y precludes the use of
transfer matrices to describe particle trajectories in sextupoles. Instead, they can be
approximated by kicks �x0 and �y0, which depend on the sextupole strength and on
the particle position in x and y.
2.5. From single particles to particle beams
Often, a particle beam can be assumed to be Gaussian in all phase space coordinates.
Consider the phase space ellipse of a hypothetical particle at exactly one standard
deviation in space and angle. The phase space area enclosed by the ellipse corresponding to
that particle is �"x, where "x is called the horizontal emittance of the beam. For a Gaussian
distributed beam, one may also define the horizontal emittance "x by a phase space ellipse
of area �"x enclosing 39% of the particles (occasionally, definitions based on a different
fraction can be found in the literature). Unfortunately, it is common practice to use the
same symbol " for the Courant–Snyder invariant (a property of a single particle) and for
the beam emittance (a property of the whole ensemble of particles).In electron storage rings, the horizontal beam emittance results from an equilibrium
between excitation due to the random nature of synchrotron radiation emission and the
damping effect of radiation. A typical number for to-date synchrotron light sources is
5� 10�9 rad m. The vertical emittance in electron storage rings is mainly given by field
errors and misaligned magnets. A ratio between vertical and horizontal emittance (called
‘the coupling’) of 0.01 to 0.001 can be achieved.For linear electron accelerators, the beam emittance in both planes depends on the
properties of the electron gun, and great care is taken in the design of the gun and
subsequent elements to avoid excessive blow-up of the emittance by Coulomb repulsion of
the electrons. During acceleration, the longitudinal momentum increases by the Lorentz
factor � while the transverse momentum does not. Thus, the particle distribution shrinks in
the angular coordinates x0 and y0, and so does the emittance (so-called adiabatic damping).
To compare different linacs, it is therefore customary to specify their ‘normalized’
emittance "x,y ¼ "x,y�.Particularly for synchrotron radiation sources and FELs, the emittance is an important
measure of the beam quality and is desired to be small, which implies a small size and a low
divergence of the electron beam. With a normalized emittance of the order of
1� 10�6 radm, a linac with 1GeV beam energy (�� 2000) yields a much better horizontal
emittance than a storage ring. One might argue, that the vertical emittance of a storage
ring is superior, but that may be less relevant because the ultimate size of the photon beam
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is dictated by the diffraction limit. For FELs, the decisive point in favor of a linac is the
high peak current.The horizontal root-mean-square (rms) size of a Gaussian beam is given by
�x¼ ("x�x)1/2. The rules by which optical functions are transformed from one position
in s to the other shall not be explained here, but to give an example, the beta function in
a drift space around a beam waist at s¼ 0 (as in Figure 6) is
�xðsÞ ¼ �xð0Þ þs2
�xð0Þ: ð8Þ
The resulting rms beam size is
�xðsÞ ¼ "1=2x �xð0Þ þ
s2
�xð0Þ
� �1=2
¼ �2xð0Þ þ"xs
�xð0Þ
� �2 !1=2
: ð9Þ
The similarity to the waist of a Gaussian laser beam with wavelength � [16]
wðzÞ ¼ w2ð0Þ þ�z
pwð0Þ
� �2 !1=2
ð10Þ
is obvious, but note that w(z) is defined as the radius at which the laser intensity drops to
1/e2 of its peak value, which corresponds to two standard deviations, whereas it is
customary in accelerator physics to quote one standard deviation �x(s) of the particle
density distribution.So far, only the interaction of beam particles with external fields (magnets and rf fields)
has been addressed. When speaking of beams with a high charge density, the interaction
between particles must be considered as well. The most important effects are
. Low-energy particles are subject to Coulomb repulsion, which is cancelled by an
attractive magnetic force, as the particle velocities approach c. Coulomb repulsion
at the electron gun limits the beam emittance in linac-based FELs.. Deviations of the vacuum beam pipe from a perfectly uniform tube with
infinite conductivity generates electromagnetic fields behind the beam particles.
These so-called wake fields act on trailing particles and may cause beam
instabilities.. Infrared coherent synchrotron radiation (CSR) is emitted in dipole
magnets and catches up with beam electrons in front of the emitting electron,
acting like a wake field on leading, rather than trailing particles. The CSR wake
from bunch compressors (see Section 4.6.3) in linac-based FELs is of particular
concern.. Electron beams ionize the residual gas atoms in the beam pipe and trap the
positive ions, which in turn may influence the beam. For this reason, some
synchrotron radiation sources use a positron beam.. Scattering between two electrons within a bunch limits the beam lifetime in
ring-based synchrotron radiation sources (Touschek effect).
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3. Synchrotron radiation
3.1. Radiation from accelerated charged particles
Accelerated charged particles emit electromagnetic radiation. The description of this well-
known fact using retarded potentials [17] is somewhat cumbersome, but there is a simple
argument, attributed to J.J. Thomson, that nicely illustrates the origin of this radiation
and some basic properties, see e.g. [18].Figure 7 shows the electric field of, say, a positron at time t after a small velocity
change �v (acceleration) within the time interval �t. A zone of transition between the field
before and after acceleration travels outward at the speed of light. This zone has
a thickness c�t and is r¼ ct away from the positron. For an angle � with respect to the
direction of acceleration, the ratio between angular and radial field component, as read
from Figure 7, is
E�
Er¼
�v t sin�
c�t: ð11Þ
With the radial field component given by Coulomb’s law
Er ¼1
4p"�
e
r2, ð12Þ
where "� is the free-space permittivity, the angular field component is
E� ¼1
4p"�
eð�v=�tÞ sin�
c2r¼
1
4p"�
e€r sin�
c2r: ð13Þ
The transition zone with kinks in the field lines is a pulse of electromagnetic radiation. Its
energy flow through a solid angle d� is given by
_W d� ¼ c "�E2�r
2d� ¼e2 €r2
16p2"�c3sin2 � d�: ð14Þ
Figure 7. Electric field of a charged particle at time t after an acceleration �v/�t, assuming that theparticle was at rest at t¼ 0. A transition zone (gray) travels outward at the speed of light. Anobserver at position A measures the field after acceleration, an observer at B is not yet aware of thevelocity change. A larger view of a field line at an angle � relative to the direction of accelerationillustrates the ratio between the angular and radial field components, see Equation (11).
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This is the well-known angular distribution of radiation from a dipole antenna.
Integration over solid angle yields Larmor’s formula for the radiated power
P ¼e2
6p"�c3€r2: ð15Þ
Prominent examples of electromagnetic radiation from accelerated charges are:
. Radio waves are emitted, when the drift velocity of electrons in an electrical
conductor is changed by applying a sinusoidal voltage.. X-rays are emitted, when an electron beam hits solid matter and is abruptly
decelerated. Apart from their penetration power, the usefulness of X-rays lies in
their short wavelength allowing one to retrieve spatial information on an atomic
scale, and a photon energy sufficient to excite electrons from inner atomic shells.. Radiation from relativistic electrons under the influence of a centripetal force was
first directly observation at the General Electric 70 MeV synchrotron in 1947 [19]
and is therefore called synchrotron radiation. It has largely replaced conventional
X-rays in scientific research due to its superior properties, which will be described
in the next section.
Synchrotron radiation is covered by many general textbooks on accelerator physics
and, owing to its importance, by several books devoted to its properties [20–22] and
applications [23,24].
3.2. Synchrotron radiation from dipole magnets
In the early days of synchrotron radiation research, synchrotron light was extracted
from dipole magnets in electron–positron colliders, which were operated for elementary
particle research. Even though undulators are now widely used (see next section), the
radiation from dipole magnets is still sufficient for many purposes.For the centripetal acceleration of a relativistic electron with rest mass me and
momentum p, Equation (15) can be written as
P ¼e2�2
6p"�m2ec
3_p2 with _p ¼ p! � p
c
R�
E
Rand � ¼
E
mec2, ð16Þ
where E is the electron energy, R is the bending radius and ! is the angular velocity,
see e.g. [20]. This, finally, leads to
P ¼e2c
6p" mec2ð Þ4
E4
R2: ð17Þ
The mass dependence shows why synchrotron radiation is usually not relevant for
accelerated particles other than electrons or positrons, and the factor E4/R2 explains why
circular eþe�-colliders have become as large as LEP at CERN/Geneva with R¼ 3000m
and have reached their economic limits. At 100GeV and a beam current of 6 mA, the
radiated power at LEP was about 18 MW [25]. As a practical rule, the energy in keV lost
by one electron per revolution is 88.5�E4/R with E in GeV and R in m.
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In a coordinate system co-moving with an electron, the angular distribution ofradiation is the same as for a dipole antenna given by Equation (14) with rotationalsymmetry about the centripetal acceleration vector. In the laboratory frame, however, thedistribution appears to be strongly ‘boosted’ in the forward direction, resulting in a narrowcone tangential to the electron trajectory. To estimate the typical divergence of this cone,consider a photon emitted perpendicularly to the direction of acceleration as well as to theelectron trajectory. If its momentum is p? in the co-moving frame, the transversemomentum in the laboratory system is still p?, while the Lorentz transformation results ina momentum pk¼ ��p? parallel to the electron direction. The angle between the totalphoton momentum and the electron direction is p?/p� 1/�. With �¼ 103–104, the typicaldivergence of synchrotron radiation is very much like that of a laser beam.
Since the passage through a dipole magnet is not an oscillatory motion, the radiationfrequency is not well defined and the spectrum is broad. The typical frequency can beestimated from !typ� 2�/�t, where �t is the time interval, during which a light coneof half-width 1/� tangential to a circular electron path illuminates a given observationpoint. The result of this estimate is not far from the so-called critical frequency!c¼ 3/2� c�3/R, which divides the spectrum into two parts of equal power. The spectraldensity is given by
dP
d!¼
P
!cS
!
!c
� �with SðxÞ ¼
9ffiffiffi3p
8px
ð1x
K5=3ð yÞ dy , ð18Þ
where P is given by Equation (17) and K5/3 ( y) is a modified Bessel function of the secondkind. It is remarkable, that the spectral shape is universally given by S(x), as depicted inFigure 8. The critical photon energy is typically of the order of a few keV. Harder X-rayscan be obtained by using superconducting magnets with smaller bending radius(sometimes called superbends or wavelength shifters).
3.3. Synchrotron radiation from undulators
Modern synchrotron radiation facilities usually comprise groups of bending and focusingmagnets, alternating with drift spaces of several meters in length. These straight sectionsprovide space for so-called insertion devices, which are usually wigglers or undulators.Both are arrangements of alternating dipole magnets, often made from permanent-magnet
Figure 8. Linear (left) and double-logarithmic plot (right) of the universal function S(!/!c) of thesynchrotron radiation spectrum of Equation (18), where !c is the critical frequency. The functionS(x) can be approximated by 1.333 x1/3 for x50.1 and by 0.777 x1/2/ex for x41, see e.g. [20].
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material like NeFeB or SmCo in combination with iron poles, see e.g. [26].The period length �u of typically a few cm is defined as the distance from one pole tothe next pole of equal polarity. The magnetic field on the beam axis is either fixed or tunedby mechanically changing the gap between the poles using powerful and precise motordrives. For period lengths of 20 cm or more, electromagnets are customary, which aretuned by varying the electric current. The highest fields are obtained using super-conducting coils. Superconducting wigglers with fields up to 7 T have been in operation fora while, whereas the experience with superconducting short-period undulators is stilllimited [27].
The (often vertical) magnetic field component perpendicular to the poles of wigglersand undulators is nearly sinusoidal, and so is the beam trajectory in the midplane betweenthe magnets:
x
x0
� �¼
K
�
� cosðkusÞ=ku
sinðkusÞ
� �with K �
�ue ~B
2pmec, ð19Þ
where s is the coordinate along the beam axis, ~B is the amplitude of the sinusoidalmagnetic field, K is called the field parameter or the undulator parameter, andku¼!u/c¼ 2�/�u. For an electron with velocity �c, the transverse excursions ina wiggler or undulator reduce its velocity _s made good along the beam axis to
_sðtÞ ¼ ð�cÞ2 � _x2� �1=2
¼ c 1�c2
�2�
_x2
c2
� �1=2
� c 1�1
2�2�
_x2
2c2
� �: ð20Þ
Inserting _x ¼ �cx0 with x0 from Equation (19), and using sin2(x)¼ (1� cos 2x)/2 finallyyields
_sðtÞ ¼ c 1�1
2�21þ
�2K2
2
� �� �þ c
�2K2
4�2cosð2!utÞ ¼ _�sþ�_sðtÞ: ð21Þ
The average velocity _�s is modulated by twice the frequency !u, because there are twovelocity minima at x¼ 0 and two maxima at x0 ¼ 0 per undulator period. In a co-movingframe with velocity _�s, the electron performs a figure-eight motion.
A wiggler with N periods can be viewed as a succession of 2N dipole magnets, and theradiation from these dipoles with the same properties as described in the previous paragraphjust adds up. In undulators, however, there is an interference between the radiation from thedipoles, leading to a line spectrum and a narrower angular distribution. Since interferencerequires sufficient overlap, the amplitude in x and x0 is smaller for an undulator than fora wiggler, which implies a smaller magnetic field and a smaller value of K. In the literature,an arbitrary distinction between wiggler and undulator is often made at K¼ 1, where theamplitude of x0 is 1/� and light cones with an assumed half-width of 1/� would overlapcontinuously forK51. In practice, however, the transition from an undulator to a wiggler isgradual. Figure 9 shows spectra of an undulator, starting with K¼ 0.5. Consider reducingthe gap between the magnetic structures, thus increasing the field on the beam axis:
. In a pure undulator, the transverse motion of the electrons is nearly harmonicwith a fixed frequency. Thus, the spectrum consists of a single line with its widthproportional to 1/N – the more periods, the better the frequency is defined.
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. As the field increases, the motion becomes more anharmonic and so does theelectromagnetic field experienced by a distant observer on the beam axis, givingrise to harmonics of the fundamental frequency. Due to the symmetry between themotion away from and back to the beam axis, only odd harmonics appear. If theobserver is not on the beam axis, even harmonics show up as well.
. As the field increases further, the fundamental frequency is shifted to a lowervalue and no longer dominates the spectrum. More and more harmonics show upand eventually merge into a broad spectrum with the same shape as from a dipolemagnet. This situation corresponds to a pure wiggler.
The fundamental wavelength of an undulator can be derived in several ways. Oneargument is that constructive interference between radiation from successive periodsrequires the electron to lag behind the electromagnetic wave by one wavelength � per
Figure 9. Photon flux from an undulator (solid lines) with N¼ 10 periods and an undulatorparameter K ranging from 0.5 to 4 (top to bottom) as a function of the photon energy (left column)and of the photon energy normalized to the critical energy Ec¼ �h!c of a dipole magnet with the samemagnetic field. The spectrum of such a dipole multiplied by 2N (dashed line) forms an envelope ofthe undulator harmonics.
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undulator period. In the time interval
�t ¼�uc¼�u � �
_�sð22Þ
radiation passes one undulator period, while the electron covers a distance of �u� �.Inserting _�s from Equation (21) with �� 1 and solving for � yields the resonance condition
� ¼�u2�2
1þK2
2
� �: ð23Þ
The electron is slower than the electromagnetic wave for two reasons, one being its mass(first term), the other its excursions in the undulator (second term). For radiation emittedat an angle � with respect to the beam axis, the electron lags behind even more, because itmoves in a different direction, and the resonant wavelength is red-shifted by �u�
2/2.The kth harmonic of undulator radiation from an electron comprises kN optical cycles
with a rectangular envelope. This temporal structure is linked to the spectrum via Fouriertransform. Centered at !k¼ k!1¼ k2�c/�, the spectral intensity is
Ið!Þ �sinx
x
� �2
with x � pN!� !k
!1, ð24Þ
where !1 is the fundamental frequency. The resulting line width is
�!k
!k�
1
kN: ð25Þ
An approximate rms value for the angular distribution of undulator radiation is
�� �1
�
1þ K2=2
2kN
� �1=2
: ð26Þ
Thus, the intensity of undulator radiation is N2 times larger than from a dipolemagnet–one factor N is gained from the spectral distribution, and a factor N1/2 from theangular distribution in horizontal and in vertical direction. For comparison, the intensityof wiggler radiation exceeds that of a dipole magnet only by a factor of 2N.
The emitted power from a wiggler or undulator is given by Equation (16). Using
px ¼E
cx0 ¼
EK
c�sin kusð Þ ¼
EK
c�sin !utð Þ and _px ¼
EK
c�!u cos !utð Þ ð27Þ
from Equation (19), inserting _px into Equation (16) and averaging over time withhcos2xi¼ 1/2 yields the instantaneous power emitted by a single electron
P ¼e2�2!2
uK2
12p"�c¼
pe2�2cK2
3"��2u: ð28Þ
Of practical interest is also the total power emitted for a given beam current I. In practicalunits:
Ptotal½W ¼ 7:26E2½GeV2
I ½AK2N
�u½cm2: ð29Þ
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Whether wiggler or undulator, the radiated power is the same. However, the power isdifferently distributed in spectrum and angle, which makes an undulator more useful formany applications.
This discussion was restricted to planar devices, i.e. wigglers or undulators that deflectthe electron beam in their midplane, producing linearly polarized light. There are alsoundulators in which the beam follows a helical trajectory, emitting circularly polarizedlight. In this case, on-axis radiation contains no higher harmonics. If circularly polarized
radiation from harmonics is desired (to reach shorter wavelengths), an elliptically helicaltrajectory is chosen and some linearly polarized background must be accepted.
3.4. Synchrotron radiation facilities
Since the first operating synchrotron at 8 MeV did not have a glass tube, synchrotronradiation was only observed at the second one, the 70 MeV synchrotron at GeneralElectric [19]. In the 1950s, this new type of radiation was used for first experiments inX-ray spectroscopy. The first generation of synchrotron radiation facilities emerged in the
1960s, extracting radiation from synchrotrons (like DESY in Hamburg) and storage rings(such as SPEAR at SLAC, DORIS in Hamburg, VEPP-3 in Novosibirsk and others)which primarily served other purposes in nuclear and particle physics.
The second generation of synchrotron light sources in the 1970s and 1980s wereelectron storage rings dedicated to and optimized for synchrotron radiation. Examples areAladdin at the University of Wisconsin, the Photon Factory in Tsukuba, Japan, BESSY inBerlin, Germany and SuperACO in Orsay, France.
Third-generation light sources have been constructed ever since the 1990s until today.Compared to the second generation, they are characterized by larger storage rings,allowing for higher beam energy (and thus shorter wavelength), smaller beam emittance(and hence higher brilliance) and a larger number of straight sections to accommodatewigglers and undulators. Figure 10 shows BESSY II in Berlin as a typical representative,and other examples are listed in Table 1. Worldwide, there are more than 50 synchrotronradiation sources in operation. Owing to the large user demand, new facilities are stillbeing built and older machines are remodeled to become modern radiation sources, such
as SPEAR3 at SLAC (recommissioned in 2004) or PETRA III, currently underconstruction at DESY in Hamburg.
The now emerging fourth generation of accelerator-based radiation sources is less
homogeneous than the previous ones. Apart from pushing storage ring parameters to thelimits, linear accelerators are now considered.
While storage rings offer little room for improvement in emittance and bunchlength, both given by an equilibrium between the disturbing and damping effects ofsynchrotron radiation, the beam properties in a linac depend mostly on the electronsource and can be further optimized. A disadvantage of linacs is their low bunchrepetition rate. The duty cycle, i.e. the ratio of time with and without beam is of theorder of one percent. This can be improved by the concept of energy recovery usinga superconducting linac in DC operation. Here, the electron beam is accelerated, andafter serving its purpose passes the linac again at the opposite rf phase and returnsmost of its energy to the rf field.
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Electrons from a linac may be used to generate undulator radiation with
much shorter pulse duration than conventional synchrotron light sources, e.g. SPPS
at SLAC [28]. However, full use of their superior beam properties can be made by free-
electron lasers, boosting the peak brilliance by up to 10 orders of magnitude. How this
Figure 10. Footprint of a typical third-generation synchrotron radiation source (here: BESSY II inBerlin, Germany). The beam is usually accelerated by a synchrotron (or, less commonly, by a linac),which is fed by a pre-accelerator, e.g. a small linac or a microtron. The accelerated beam istransferred to a storage ring, where it circulates for many hours. Most storage rings of synchrotronlight sources have an achromatic magnet structure, alternating with straight sections toaccommodate wigglers and undulators (an achromat, comprising two or three dipole magnets,starts and ends with zero dispersion). The radiation generated tangentially to the storage ring istransported to the experiments outside the radiation shield (typically 1 m of concrete) by complexphoton beamlines, in which the radiation is collimated, monochromatized and focused. (The colorversion of this figure is included in the online version of the journal.)
Table 1. Examples of third-generation synchrotron radiation sources in operation and one(PETRA III) presently under construction.
Facility name (location, first beam)Circumference
(m)Beam energy
(GeV)Current(mA)
Hor. emittance(nm rad)
ESRF (Grenoble, France, 1992) 844 6 200 4ALS (Berkeley, USA, 1993) 196.8 1.0–1.9 400 4.2–6.3ELETTRA (Trieste, Italy, 1993) 259.2 2.0–2.4 140–320 7.0–9.7APS (Argonne, USA, 1995) 1104 7 100 3SPring8 (Hyogo, Japan, 1997) 1436 8 100 3BESSY (Berlin, Germany, 1998) 240 0.9–1.9 300 5.2SLS (Villigen, Switzerland, 2000) 288 2.4 400 5SOLEIL (Gif-sur-Yvette, France, 2006) 354.1 2.75 500 3.7Diamond (Didcot, UK, 2006) 561.6 3 300 2.7PETRA III (Hamburg, Germany, 2009) 2304 6 100 1
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almost inconceivable factor of improvement is possible, will be described in the
following section.
4. Free-electron lasers
A free-electron laser (FEL) acts as a narrow-band amplifier for radiation, which
requires a mechanism that transfers energy from the electron beam to the radiation
field. This mechanism is explained in the next section, followed by a discussion of
low-gain and high-gain FELs. The theory of FELs is described in more detail in e.g.
[29–34].
4.1. Interaction between electrons and radiation
The energy transfer between electrons and radiation is the basic process in FELs.
Changing the electron energy requires a force in the direction of the electron velocity.
The force is given by the electric field of the radiation, which is perpendicular to its
direction of propagation. A non-zero energy transfer
dW ¼ F � ds ¼ �e E � ds ¼ �e E � v dt ð30Þ
is accomplished by overlapping electron bunches and radiation in an undulator, where the
electron velocity has a component parallel to the electric field, as sketched in Figure 11.
Consider a sinusoidal electric field in the horizontal direction
Exðz, tÞ ¼ E� cosðks� !tþ ’�Þ ð31Þ
(a)
(b)
(c)
Figure 11. Interaction between a radiation pulse and an electron on its sinusoidal trajectory in anundulator, denoted by the black line. In (a) the electric field (black arrows) as experienced by theelectron and its transverse velocity both point downwards, in (b) both are zero, and in (c) both pointupwards, giving rise to an energy transfer between electron and light field, which is oscillatory butdoes not change sign – see right figure and Equation (32) for the definition of sin �. The electricfield changes sign at the electron position, because the electron lags behind the radiation pulse by onewavelength per undulator period.
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with amplitude E�, wavenumber k, angular frequency ! and phase offset ’�. Given
a horizontally transverse electron velocity vx� cx0 with x0 from Equation (19), the energy
transfer per time is
dW
dt¼ �e E� cosðks� !tþ ’�Þ �
cK
�sinðkusÞ
¼ �ecE�K
2�sin ½kþ kus� !tþ ’�ð Þ � sin ½k� kus� !tþ ’�ð Þ
� �ecE�K
2�sin�þ � sin��
: ð32Þ
For the two sinusoidal terms to contribute over the whole length of the undulator, their
respective phase should be constant:
d�
dt¼ 0 ¼ ½k ku_s� ! ¼ kð_s� cÞ ku _s � �kc
1þ K2=2
2�2 kuc: ð33Þ
Here, _s was approximated by the average velocity _�s of Equation (21) with �� 1 in the first
term and by c in the second term. The resulting radiation wavelength is
� ¼2pk¼
2pku2�2
1þK2
2
� �¼�u2�2
1þK2
2
� �: ð34Þ
In the last step, the negative sign was dropped, since it would imply �50, which means
that only �þ can be made constant. The result is again the resonance condition as in
Equation (23), stating that the electron must lag behind the radiation by one wavelength
per undulator period. In other words, energy is exchanged with radiation of the same
wavelength as spontaneous undulator radiation. With �þ now constant, ��¼�þ� 2kus
oscillates twice per undulator period and cancels on average. Its occurrence in the term
{sin �þþ sin ��} of Equation (32) reflects the fact that the energy transfer is not constant.
For v?E, which happens twice per undulator period, the energy transfer is zero (see
Figure 11). For a homogeneous electron distribution, half of the electrons will gain and
half will lose energy, depending on their value of �þ, which is known as the
ponderomotive phase.If the Lorentz factor deviates by �� from the value that fulfills the resonance
condition, the ponderomotive phase �þ will not be constant any more. Subtracting
Equation (33) from the same equation for (�þ��) yields the rate, at which the
ponderomotive phase changes
d�þ
dt¼ �kc
1þ K2=2
2
1
ð� þ��Þ2�
1
�2
� �� �kc
1þ K2=2
2
�2 � ð� þ��Þ2
ð� þ��Þ2 �2
� �
� �kc1þ K2=2
2
�ð2� þ��Þ��
�4
� �� kuc�
2 2��
�3
� �¼ 2kuc
��
�: ð35Þ
Defining ���/�, the energy transfer rate from Equation (32) can be written as
dW
dt¼
d��
dtmec
2 ¼d
dt�mec
2: ð36Þ
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Using this expression, Equations (35) and (32) can be written as
d�
dt¼ 2kuc and
d
dt¼ �
eE�K
2mec�2sin� , ð37Þ
where ���þ and �� is ignored from now on. These two coupled first-order differentialequations describe the motion of an electron in a phase space (�, ) under the influence ofradiation. It should be emphasized that the phase � is not a fixed phase of the radiationfield (which slips forward by one wavelength per undulator period), but should rather beconsidered relative to an electron with ��¼ 0 at a longitudinal position for which energygain and loss cancel over the length of the undulator. It should also be noted that theamplitude E� of the electric field was assumed to be constant, which is not true once theenergy transfer between electrons and radiation becomes significant.
The two coupled Equations (37) can be combined to
€�þ�2 sin� ¼ 0 with �2 �eE�kuK
me�2, ð38Þ
which is known as the pendulum equation in classical mechanics. Consider, for example,a swing boat as shown in Figure 12. For small amplitudes, it acts like a harmonicoscillator. For larger amplitudes, the motion becomes anharmonic and the oscillationfrequency decreases. There is a contour in phase space known as separatrix, that mark thelimit between bounded and unbounded motion. Outside the separatrix, the swing boatperforms loopings.
It shall be noted without derivation, that the longitudinal oscillation of electrons in anundulator as described by Equation (21) causes a reduction of the coupling betweenelectrons and radiation, because the electron deviates from its optimum position in phase.This was neglected when approximating _s by _�s in Equation (33). A more elaboratetreatment results in a modification of the undulator parameter
K�!K J0K2
4þ 2K2
� �� J1
K2
4þ 2K2
� �� �, ð39Þ
Phase (rad) Phase (rad)
Figure 12. Electron motion in phase space (energy deviation �� versus ponderomotive phase �) assolutions of the pendulum equation. Within a given time interval, the electrons move from theirstarting positions (open circles) to positions (filled circles) which depend on their initialponderomotive phase. The separatrix is the borderline between bounded trajectories (cf. swingboat A) and unbounded motion (swing boat B). For starting points at �� ¼ 0 (open symbols in theleft plot), energy gain and energy loss are equal. When starting at ��40 (right plot), less energy isgained than lost by the electrons.
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where J0 and J1 are the zero- and first-order Bessel’s functions of the first kind. For K¼ 1,
as an example, the reduction factor in square brackets is 0.91. For large K, the factor
approaches 0.7.
4.2. Femtoslicing in electron storage rings
In synchrotron light sources with typical pulse durations of several 10 ps, the interaction
between electrons and radiation in an undulator as described above can be employed to
generate radiation pulses with a duration of 100 fs and shorter. This technique, known as
‘femtoslicing’ [35], makes use of ultrashort light pulses, e.g. from a Ti:sapphire laser
system, to modulate the electron energy within a ‘slice’ of an electron bunch. As shown in
Figure 13, the off-energy electrons are transversely displaced, e.g. by passing a dipole
magnet, such that their synchrotron radiation from a subsequent undulator can be selected
using an aperture. The time structure of this radiation is essentially that of the short laser
pulse, but – being incoherent radiation from a small fraction of the bunch – its intensity is
extremely low. Nevertheless, femtoslicing sources now exist at the ALS in Berkeley [36], at
BESSY in Berlin [37] and at the Swiss Light Source [38], and offer a good opportunity to
study laser-induced energy modulation in detail, which is part of the FEL process.
4.3. Low-gain free-electron lasers
Under the influence of electromagnetic radiation in an undulator, electrons move in
a phase space, spanned by the ponderomotive phase � and the energy deviation
¼��/�, as shown in Figure 12. For electrons with ¼ 0 and a homogeneous density
distribution in �, an equal amount of energy is gained and lost by the electrons, and
there is no net energy transfer from or to the radiation field. This is not yet a free-
electron laser.When starting with 40 (but with most electrons still within the separatrix) it is
obvious from Figure 12 that less energy is gained than lost by the electrons. It may be
assumed, that this energy is transferred to the radiation field, even though the increase of
the electric field is not part of the model yet. The energy transfer is only a second-order
Figure 13. Generation of ultrashort radiation pulses in a storage ring via ‘femtoslicing’: a short laserpulse, co-propagating with an electron bunch in an undulator (modulator) causes a periodic energymodulation within a ‘slice’ of the bunch. After passing a dipole magnet, the off-energy electrons aretransversely displaced and their radiation from a second undulator (radiator) can be extracted usingan aperture.
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effect, and the gain
g ��W
W, ð40Þ
i.e. the relative increase of energy in the radiation field, is small. The gain, the expression of
which will not be derived here, is given by
gðÞ ¼ �pe2K2N3�2une4"0mec2�3
�d
dx
sin2 x
x2
� �with x � 2pN ð41Þ
for an electron density ne and all other symbols defined as before. For the low-gain
FEL, the gain as a function of the initial value of is the negative derivative of the
undulator line shape given by Equation (24) – this is known as Madey’s theorem [39].
If 50, the gain is negative, i.e. energy is transferred to the electrons. This corresponds
to an inverse FEL, which is – in principle – a particle accelerator, albeit not a very
efficient one.Low-gain FELs are essentially FEL oscillators, where the radiation builds up slowly
within an optical cavity formed by two mirrors (see also Figure 2). These FELs usually
operate in the infrared regime with an electron beam from a linac. There are also a few
FEL oscillators embedded in storage rings. The shortest wavelength reached to-date
by a FEL oscillator is around 176 nm, obtained at the storage ring ELETTRA in Trieste,
Italy [40]. Further progress is inhibited by the fact that mirrors with good reflectivity at
normal incidence do not exist for shorter wavelengths. Table 2 lists a few examples of
low-gain FELs.
4.4. High-gain free-electron lasers
4.4.1. First-order differential equations
For low-gain FELs, the electric field amplitude E� in Equation (37) was assumed to be
constant. While such an approximation may be appropriate to show the basic features of
a low-gain FEL, where the field increases with every pass by only a few percent, it is
completely inadequate when the gain is larger. The need for larger gain arises when
considering small wavelengths, for which mirrors with sufficient reflectivity do not exist. In
this case, the whole amplification process must be completed within a single pass of an
electron bunch, because there is no way to make the radiation pulse interact with another
Table 2. Examples of low-gain free-electron lasers.
Facility name (location)Beam energy
(MeV)Wavelength
(mm) Electron source
EUFELE (Trieste, Italy) 1500 � 0.176 Storage ringFELBE (Dresden, Germany) 18 4–200 Superconducting linacJLab FEL (Newport News, USA) 80–200 1.5–14 Energy-recovery linacUVSOR FEL (Okazaki, Japan) 750 � 0.215 Storage ring
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bunch. In principle, larger gain can be accomplished by just making the undulator longer,
leading to an exponential growth of the radiation amplitude, until a certain limit
(saturation) is reached.In order to describe high-gain FELs, the constant electric field amplitude is replaced by
~E�ðsÞ, a function of the position s along the undulator. To allow for an arbitrary phase ’�,the horizontal field amplitude is written as a complex number ~E�ðsÞ ¼ E�ðsÞ expði’�Þ, andcomplex quantities shall be denoted by a tilde from now on.
Here, only the one-dimensional theory shall be outlined. In this approximation, the
electric field is subject to the wave equation
@2
@s2�
1
c2@2
@t2
� �~Exðs, tÞ ¼ ��
@~jx@t
with ~Exðs, tÞ ¼ ~E�ðsÞ expðiks� !tÞ , ð42Þ
where ~jx is the transverse current density, given by the transverse motion of electrons in the
undulator. The derivation of the solution can be found e.g. in [33]. Inserting ~Ex into the
wave equation and neglecting the second derivative of the field amplitude ~E00x (which is
called the slowly-varying amplitude approximation) finally yields
d ~E�ds¼ �
i��c
2!
@~jx@t
expð�iks� !tÞ ¼ ���Kc
4�~j1 ¼
��Kc2nee
2�hexpði�nÞi: ð43Þ
Here, ne is the electron density, i.e. the number of electrons per volume despite the
one-dimensional treatment. The transverse current density is replaced by ~jx � x0 ~js, where
~jsð�, sÞ ¼ j� þ ~j1ðsÞ expði�Þ ð44Þ
is the longitudinal current density, which is assumed to contain a constant term and
a periodic modulation with a complex amplitude ~j1ðzÞ, which changes as the bunch travels
along the undulator. In the last step of Equation (43), the constant term was dropped and
the modulation was expressed by hexp(i�n)i, the average of the phasors for all electrons. If
this so-called bunching factor is close to zero, the electrons are randomly distributed,
otherwise the electron density is modulated, giving rise to a change of the electric field
amplitude. Now, there are three coupled differential equations describing the rate, at
which the energy and the phase of the nth electron changes, as well as the amplitude of the
surrounding electromagnetic wave:
d�n
dsðsÞ ¼ f1 � nðsÞ
dndsðsÞ ¼ f2 � ~E�ðsÞ � sin�nðsÞ
e ~E�dsðsÞ ¼ f3 � hexp½i�nðsÞi , ð45Þ
where the factors fi, containing constant undulator and beam parameters, can be read
from Equations (37) and (43). From this point, there are two ways to proceed:
. By making further assumptions on the electron distribution, a third-order
differential equation for the electric field amplitude can be obtained. Its solution
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shows that the electric field grows indeed exponentially with longitudinal position
s and an analytical expression for the growth parameter can be obtained.. Representing the electrons by a smaller, but statistically significant number of
‘macro-particles’, their dynamics as well as the evolution of the electric field can
be simulated numerically. This not only demonstrates the exponential growth of
the field, but also allows one to study its startup, either from noise or from an
input field, as well as the saturation of the amplification process.
4.4.2. Analytical results
The coupled first-order differential equations presented in the previous section can under
further assumptions be combined to a third-order equation for the transverse electric field
as a function of the path length s in the undulator. There are three solutions corresponding
to exponential decay, exponential growth, and an oscillation. After startup, the
exponential growth soon dominates and the radiation power increases as
PðsÞ � es=Lg with Lg ¼1
31=24�3me
�0K2e2kune
� �1=3
ð46Þ
with the same symbols as before. In practice, the power gain length Lg will be somewhat
longer than predicted by the one-dimensional theory, in which the influence of space
charge, energy spread etc. is ignored. In the course of the FEL process, the electron
beam energy and its energy spread increases, eventually inhibiting further gain. Saturation
is typically reached at Lsat� 20 Lg [34]. The gain length is related to another useful
quantity by
¼�u
4p31=2Lg: ð47Þ
This dimensionless quantity is known as the FEL parameter or the Pierce parameter,
and is of the order of 10�3. It can be used to characterize several FEL properties, e.g. the
saturation length may be expressed as Lsat� �u/. Furthermore, near saturation the value
of corresponds to the relative FEL amplifier bandwidth, and the gain is significantly
reduced if the electron energy spread is larger than /2.For a given undulator, the gain length is proportional to the beam energy and depends
on the electron density �n�1=3e , which implies that the peak current should be large and the
beam emittance low. Another effect of a non-zero emittance is the beam divergence.
A non-zero transverse velocity reduces the longitudinal velocity. This way, the beam
divergence causes a spread of the longitudinal velocity, which in turn is equivalent to an
additional energy spread. Combining these and other considerations, it turns out that the
emittance should ideally be� �/4� for a given radiation wavelength �.
4.4.3. Simulation of the FEL process
Numerical simulations allow one to study the FEL process and the underlying electron
dynamics in more detail and to include effects which are inaccessible by analytical
methods. Figure 14 shows the result of a one-dimensional numerical calculation
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according to Equations (45) as an illustration, whereas realistic calculations are donewith three-dimensional FEL codes such as GENESIS [41]. If effects of the electron gunand acceleration process are studied, FEL codes are interfaced with other programs tosimulate space charge effects in the gun, e.g. ASTRA [42], and general beam dynamicscodes such as ELEGANT [43]. Examples of these ‘start-to-end’ FEL simulations aregiven in [44,45].
As shown in Figure 14, the motion of electrons under the influence of the radiationfield leads to a periodic modulation of the longitudinal charge density (‘micro-bunching’), which in turn gives rise to the coherent emission of radiation ata wavelength corresponding to the distance between adjacent micro-bunches. Whilethe modulation becomes more pronounced, the radiation power grows exponentially, asdescribed by the analytical result of Equation (46). From part (c) of Figure 14, it isimmediately clear that the electron motion eventually leads to a more complexsubstructure of the longitudinal charge density and saturation occurs, i.e. furthergrowth of the radiation field is inhibited.
A piece of FORTRAN code is given below to illustrate how to integrate the coupleddifferential equations of Equations (45). Here, this is done in small time steps istep usingthe fourth-order Runge-Kutta method [46] and with factor1, factor2, factor3
(a)
(a)
(b)
(b)
(c)
(c)
R
P
P
P P
L
Figure 14. Top: Electron motion in phase space (relative energy deviation versus ponderomotivephase �) as solutions of Equations (45) at the onset of the FEL amplification, (a) shortly beforesaturation, (b) and after saturation (c). The corresponding charge density is plotted below.The bottom figure shows the FEL power rising exponentially with increasing undulator length(in units of the gain length), until the process saturates.
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corresponding to f1, f2, f3, respectively. For each electron, labeled i, the position in phase
space is calculated, where phi(i) is the ponderomotive phase and eta(i) is the relative
energy deviation. The variables phi1, eta1, phi2, eta2 etc. contain intermediate results
inherent in the Runge-Kutta algorithm. From the phases of all electrons, the complex
bunching factor (bunching_real, bunching_imag) is calculated, which in turn is used to
update the complex electric field (e_real, e_imag). Apart from input and output
statements, this is all that is needed to produce Figure 14, showing the evolution of the
electron distribution in phase space as well as the exponential increase and saturation of
the radiation power (which is proportional to e_real**2þe_imag**2). The trajectories in
Figure 12 for the low-gain FEL were also calculated using this code, but setting f3 to zero,
i.e. ignoring changes of the electric field.
do istep ¼ 1, number of steps
bunching real ¼ 0:
bunchingimag ¼ 0:
do i ¼ 1, number of electrons
phi1 ¼ factor1 etaðiÞ
eta1 ¼ factor2 ðe real dcosð phiðiÞÞ
& þ e imag dsinð phiðiÞÞÞ
phi2 ¼ factor1 ðetaðiÞ þ eta1=2:Þ
eta2 ¼ factor2 ðe real dcosð phiðiÞ þ phi1=2:Þ
& þ e imag dsinð phiðiÞ þ phi1=2:ÞÞ
phi3 ¼ factor1 ðetaðiÞ þ eta2=2:Þ
eta3 ¼ factor2 ðe real dcosð phiðiÞ þ phi2=2:Þ
& þ e imag dsinð phiðiÞ þ phi2=2:ÞÞ
phi4 ¼ factor1 ðetaðiÞ þ eta3Þ
eta4 ¼ factor2 ðe real dcosð phiðiÞ þ phi3Þ
& þ e imag dsinð phiðiÞ þ phi3ÞÞ
phiðiÞ ¼ phiðiÞ þ ðphi1þ 2: ð phi2þ phi3Þ þ phi4Þ=6:
etaðiÞ ¼ etaðiÞ þ ðeta1þ 2: ðeta2þ eta3Þ þ eta4Þ=6:
bunching real ¼ bunching realþ dcosð phiðiÞÞ
bunching imag ¼ bunching imagþ dsinð phiðiÞÞ
end do
e real ¼ e realþ factor3 bunching real=electrons
e imag ¼ e imagþ factor3 bunching imag=electrons
end do
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4.5. Self-amplified spontaneous emission (SASE)
The low-gain and high-gain mechanisms, by which FELs amplify electromagneticradiation have been discussed, but little has been said so far about the amplifier input.In the case of the FEL oscillator, the radiation field starts from spontaneous undulatorradiation and builds up over time between two mirrors forming an optical cavity. Forhigh-gain FELs at wavelengths for which good mirrors do not exist, the amplificationprocess described above may also start from spontaneous radiation at the beginning ofa long undulator. Instead of describing the initial field as spontaneous radiation, one mayequivalently say that it starts from ‘noise’. The initial electron distribution is, of course, notcompletely homogeneous. Electrons are randomly distributed and have a non-zero densitymodulation at any wavelength, including the wavelength that is resonant to the FELundulator according to Equation (34). This startup from noise is called the SASE (self-amplified spontaneous emission) principle [47]. It has been proven to work reliably atseveral FEL facilities such as VISA [48] at Brookhaven and LEUTL [49] at the ArgonneLab in the visible regime as well as TTF/FLASH [3] in Hamburg and SCSS [50] at SPring-8 in Japan at shorter wavelengths. It is therefore also the basis of future X-ray FELs. Someexamples of SASE FELs are listed in Table 3.
4.6. FLASH – a SASE FEL for soft X-rays
FLASH is the first representative of a free-electron laser using the SASE principle at shortwavelengths. Initially, this machine was intended to be a testbed for superconducting rfcavities developed by the TESLA collaboration for a future linear collider (hence itsoriginal name TTF, TESLA Test Facility), where a SASE FEL was considered as onepotential application of the test linac [51]. In 2000, it was the first high-gain FEL operatingat wavelengths beyond the visible range (80 nm). In 2006, meanwhile reaching 13 nm,the facility was renamed to FLASH (Free-electron LASer in Hamburg), and in 2007 theto-date shortest wavelength of 6.5 nm was reached. Other FELs that are now coming intooperation, closely following the design of FLASH, which is shown in Figure 15 and will bedescribed in this section.
A SASE FEL comprises a source of low-emittance electron bunches, a linearaccelerator, some means to compress the bunches to sub-100 fs duration, a long undulator,
Table 3. Examples of SASE free-electron lasers. The electron source is either a normal-conducting(n.c.) or a superconducting (s.c.) linear accelerator.
Facility name (location, first beam)Beam energy
(GeV)Wavelength
(nm) Electron source
LEUTL (Argonne, USA, 2000) 0.217–255 385–530 n.c. linacFLASH (Hamburg, Germany, 2000) 0.986 6.5 s.c. linacSCSS (Hyogo, Japan, 2006) 0.250 49 n.c. linacSPARC (Frascati, Italy, �2008) 0.150–0.200 500 n.c. linacLCLS (Stanford, USA,�2009) 14.3 0.15 n.c. linacSPring-8 XFEL (Hyogo, Japan, �2010) 8 0.10 n.c. linacEuropean XFEL (Hamburg, Germany �2014) 17.5 0.10 s.c. linac
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and finally an electron beam dump and photon beamlines with experimental stations.Furthermore, beam diagnostics plays an important role and differs considerably from thatof conventional synchrotron light sources.
4.6.1. The electron source
The SASE process requires a small normalized emittance (a few 10�6 radm) and highpeak current (more than 1 kA) of the electron beam. The high current is achieved bya combination of high bunch charge (0.5–1 nC) and short duration (100 fs or less).Initially, the bunches are generated with a duration of several picoseconds usinga photocathode illuminated by a UV laser pulse. The laser system [52] developed at theMax-Born-Institute in Berlin and presently employed at FLASH comprises a Nd:YLFoscillator and four Nd:YLF amplifier stages (two pumped by diodes, two by
Figure 15. Layout of the high-gain free-electron laser FLASH at DESY in Hamburg as of summer2008. Top: footprint of the facility showing that part of the FLASH tunnel is inside a buildingcontaining the rf power stations, laboratories and workshops. The remaining tunnel is covered byearth. The numbers indicate the distance from the photocathode gun in meters. While the electronbeam is dumped 260m from the gun, the photon beam crosses the PETRA storage ring and entersan experimental hall accommodating several beamlines and a laser system (marked by *) employedfor pump–probe measurements. A laser laboratory 150 m from the gun is connected to the tunnel byseveral tubes. Another tube at 190m is foreseen for seeded-FEL radiation (see Section 4.7). Bulkyhardware (such as the 12 m long acceleration modules) can enter through the curved corridor at250m. Below: the enlarged view with exaggerated transverse dimensions shows the electron gun,followed by six superconducting rf modules (acc), accelerating the beam to 1GeV, and two magneticchicanes acting as bunch compressors (bc1, bc2). There are several undulators installed, twoelectromagnetic undulators for laser-based longitudinal diagnostics (ORS), six permanent-magnetdevices acting as one 27m long undulator for SASE, and an electromagnetic wiggler (IR) producinginfrared radiation for diagnostics purposes. Also shows is LOLA, a transversely deflecting rf cavity,and two stations for electro-optical sampling (EO). During accelerator tests, the beam may be sentthrough a bypass in order to protect the permanent-magnet undulators against radiation.
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flashlamps). A combination of an LBO and a BBO crystal converts the 1047 nm pulsesdown to a wavelength of 262 nm, which is required to liberate electrons from the CsTephotocathode. Pulses with an energy of up to 50 mJ are generated with a temporalpattern given by the linac duty cycle: a train of 1 to 800 pulses at a rate of 1MHz, andup to 10 pulse trains per second.
In order to meet the emittance requirements, Coulomb repulsion of the electrons mustbe minimized by immediately accelerating them, since the repelling force decreases withincreasing Lorentz factor like 1/�2. To this end, the photocathode is placed inside an rfcavity, as shown in Figure 16, which is directly followed by the first linear acceleratormodule. A solenoid field surrounding the photocathode focuses the electrons transversely.The laser pulses are shaped as to make the radial electron distribution rectangular ratherthan Gaussian. An ideal rectangular shape would result in a repelling force that increaseslinearly with radius, which in turn could be counteracted by quadrupole magnets.
Instead of using a photocathode gun, SASE was demonstrated in 2004 at a wavelengthof 49 nm using a thermionic gun [50] at SCSS in Japan. Yet another approach to producelow-emittance electron bunches, which is currently investigated [53], could be fieldemission from a microscopic conical object or an array of them, possibly assisted byphoto-emission.
4.6.2. The linear accelerator
Since the short duration and low emittance of the electron bunches would not be preservedin a synchrotron, SASE FELs are based on linear accelerators, either employing normal-conducting or superconducting rf structures as sketched in Figure 4. In contrast to the caseof a constant current, the resistance of a superconductor does not completely vanish in thepresence of an rf field, see e.g. [54]. It is nevertheless much smaller in superconductingniobium at e.g. 2K than for copper at room temperature. Therefore, the quality factor Qof a superconducting cavity (where Q/2� is the ratio of the energy stored in the system andthe energy dissipated per oscillation cycle) is of the order of 1010, compared to below 105
for copper cavities. On the other hand, for 1W of power lost in a superconducting cavityat 2K, the cryogenic system requires almost 1 kW to keep the temperature constant. Thistogether with the technological complexity and a fundamental limitation given by the
(a) (b) (c)
Figure 16. The photocathode of the electron gun is embedded in a 1-1/2-cell rf structure. Whena UV laser pulse hits the photocathode, (a) the accelerating electric field is at a maximum. It is zero,when the liberated electrons pass through the iris, (b) and assumes a maximum with opposite sign,when the electrons are at the center of the next cell (c). A longitudinal magnetic field from a solenoidcoil (not shown) surrounding the cavity helps to keep the beam emittance small.
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critical magnetic field makes the choice not so obvious. It required a committee ofinternational experts in 2004 to identify the superconducting TESLA technology as thebest solution for the International Linear Collider [55].
At FLASH, six acceleration modules each accommodating eight nine-cell TESLA-typecavities are presently used to accelerate the electron beam to 1GeV. With a cavity length of1m, the average electric field is about 20MVm�1. With a further optimized cavity designand new techniques to clean the cavity surfaces, fields exceeding 50MVm�1 have beendemonstrated [56].
The European X-ray FEL will employ TESLA cavities with a field of 21MVm�1.LCLS at Stanford, on the other hand, is based on the normal-conducting SLAC linac, andsome other projects, like the SCSS XFEL in Japan for example, have opted for normal-conducting rf structures as well.
4.6.3. Bunch compression
Electron bunches are created in the electron gun at moderate current in order to minimizespace charge effects that would otherwise blow up the beam emittance. A typical value is50 A, e.g. 0.5 nC with a duration of 10 ps. The FEL amplification process, on the otherhand, requires a peak current in the kA range. Therefore, the bunches are compressed oncethey have reached a Lorentz factor �� 1.
An electron bunch is compressed by causing electrons in its tail to catch up withelectrons in the head of the bunch. This is achieved by accelerating the bunches at an rfphase slightly away from the peak of the sinusoidal cavity voltage (‘off-crest’), such thatthe kinetic energy of tail electrons is higher than the energy of head electrons. This‘chirped’ electron bunch then passes a chicane formed by dipole magnets, where the tailelectrons catch up because their bending radius is larger and their path consequentlyshorter than that of the head particles. Being close to c, velocity differences between theelectrons are negligible. The path length differences are approximately given by
�l � L�2 �E
E, ð48Þ
where L and � are defined in Figure 17. However, the compression is not perfect, since theenergy acquired on the slope of the sinusoidal cavity voltage does not vary linearly alongthe bunch, as sketched in Figure 17. The resulting bunch comprises a short head (a few10 fs) with a high peak current, followed by a long tail (several ps), which carries most ofthe bunch charge (80–90%) but does not contribute to the lasing process. In future, thiscan be greatly improved, when a third-harmonic voltage is added to the accelerating rfvoltage, linearizing the sum voltage experienced by the bunches.
FLASH comprises two bunch compressors at different beam energies The reason isthat full compression in one stage at low energy would cause strong space charge effects,while one stage at high energy would require an undesirably large energy variation withinthe bunch.
The chicane magnets give rise to coherent synchrotron radiation (CSR). As the bunchfollows its curved path, CSR emitted by the bunch tail takes a straight shortcut and acts onthe bunch head, leading to detrimental effects which can be seen both in simulations aswell as in time-resolved images of the bunches [57].
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4.6.4. Undulators
The SASE lasing process takes place in a long undulator with a nearly sinusoidal magneticfield. The field may be produced by electromagnets or by permanent magnets. In the lattercase, the magnets may be outside (as at FLASH) or inside the vacuum vessel in order toachieve a higher field for a given free aperture (as at SCSS in Japan). The gap betweenopposite magnetic structures may be variable to change the field on the beam axis and tunethe radiation wavelength, or it may be fixed. At FLASH, a hybrid structure of iron poleswith NdFeB magnets between the poles is employed. The undulator period (from one poleto the next like-sign pole) is �u¼ 27mm, the fixed magnetic gap is 12mm, resulting ina field amplitude of 0.47 T at the beam. The corresponding undulator parameter is K¼ 1.2,leading with Equation (23) to a wavelength of about 6 nm at a beam energy of 1GeV(�¼ 1957). In order to achieve saturation of the lasing process, six 4.5 m long undulatorsare employed, interleaved with 0.6 m long sections for focusing and beam diagnostics.
Degradation of the permanent magnets by radiation is avoided by sending the electronbeam through a bypass (see Figure 15) during accelerator studies, when lasing is notrequired.
Apart from the SASE undulator, three electromagnetic undulators are presentlyinstalled at FLASH for diagnostics purposes, two undulators with five periodseach (�u¼ 200mm) tunable to 800 nm (the wavelength of Ti:sapphire lasers) for theoptical-replica experiment (see below), and one undulator with nine periods (�u¼ 400mm)for infrared radiation in the range of 1–200 mm.
In addition, 10m of permanent-magnet undulators with variable gap are foreseenfor seeding the FEL process with a high laser harmonic (HHG seeding is described inSection 4.7).
4.6.5. Beam diagnostics
In order to maintain and improve the quality of FEL radiation, the parameters of theelectron beam must be measured with high precision. The beam energy is known from the
Figure 17. Schematic view of a bunch compressor chicane formed by four dipole magnets, in whichelectrons with higher energy travel a shorter path than electrons with lower energy. Also shown arelongitudinal phase space distributions (�E/E versus z, ideal case in light gray) and correspondingcharge densities (z) before (left) and after compression (right). On the slope of the accelerating rfvoltage, the electrons acquire an energy ‘chirp’, i.e. an energy variation along the bunch. The densitydistribution (z) is not to scale: even in the non-ideal case (dark gray), the electron density increasestypically by a factor of 30.
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beampath through dipolemagnets and frommeasuring the radiation wavelength. The beamcurrent is measured using conventional current transformers. The beam position is eitherdeduced by subtracting the signals from opposite electrodes (beam position monitors),where a passing charge induces a current pulse, or by placing a screen in the beam path andimaging the light spot created by fluorescence or by optical transition radiation (OTR).The beam emittance, averaged over one bunch, can be inferred from measurements of thebeam size at positions with different beta functions, again using OTR screens.
So far, standard diagnostics tasks were described. Given that FELs require a high peakcurrent (i.e. very short bunches) and low emittance, there is an additional and verychallenging demand to resolve the temporal structure of a single bunch and to measure thevariation of the emittance along the bunch (the so-called slice emittance). Since longitudinaldiagnostics is essential for understanding and controlling the amplification process in FELs,some methods will be outlined in this paragraph.
When the electric field of an electron bunch sweeps over a crystal like GaP, it inducesa birefringent effect on a laser pulse traversing the crystal at the same time. This way, theshape of the bunch can be encoded in the intensity of that laser pulse after passinga polarizer. This technique is known as electro-optical sampling (EOS) and several single-shot methods to extract the encoded information have been devised at FLASH andelsewhere, reaching the intrinsic time resolution of about 50 fs [58].
A transversely deflecting rf cavity operating at 2.9 GHz was originally used at SLAC[59] and is now employed at FLASH to kick head and tail of a single electron bunchvertically in the opposite direction. This particular bunch is directed by a fast kickermagnet onto an OTR screen, where its charge distribution appears along the vertical axis(in a similar fashion as in an oscilloscope or streak camera). The horizontal distributionreflects the shape of the bunch, from which the slice emittance can be deduced. A temporalresolution of 20 fs was achieved [57,60] with the setup shown in Figure 18.
Bunch structures in the femtosecond range can be studied by measuring the spectrumof transition radiation [61] or synchrotron radiation [62]. Coherent emission occurs atwavelengths comparable or longer than the structure. There is no intrinsic limit of the time
Figure 18. Longitudinal bunch diagnostics using a transversely deflecting cavity. A fast kickermagnet directs the vertically tilted bunch onto a screen, where optical transition radiation (OTR) isemitted and imaged by a CCD camera. A sample picture and its projection onto the time axis isshown in the right part of the figure [57].
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resolution, but the relationship between the observed spectrum and the temporal
distribution is ambiguous, since only the radiation intensities and not the phases are
measured.Another novel method to obtain the longitudinal charge distribution with a resolution
of a few fs, the ‘optical-replica synthesizer’ (ORS) [63], was implemented at FLASH in
2007 and is presently being commissioned [64]. Here, a Ti:sapphire laser pulse at 800 nm
wavelength interacts with an electron bunch in an undulator, creating a periodic energymodulation. Passing a magnetic chicane, the energy modulation is converted into a density
modulation, which gives rise to coherent emission in a second undulator at the laser
wavelength or harmonics thereof. The resulting coherent light pulse is a ‘replica’ of theelectron bunch, and its shape can be determined uniquely and with excellent time
resolution using a commercially available FROG (frequency-resolved optical gating)
device [65].
4.6.6. Synchronization
At a FEL user facility, many components must be synchronized with respect to each otheron a very precise level, which ultimately limits the time resolution of experiments at the
photon beamline. These components include the rf photoinjector, the accelerating
structures, diagnostic devices, a seed laser (in the case of a seeded FEL, see below) and thelaser at the experiment for pump–probe studies.
At FLASH, an optical synchronization system was implemented and is being tested,
comprising a mode-locked Er-doped fiber laser as master oscillator and a distribution
system, based on optical fibers with length stabilization [66]. The fiber laser is superior tomicrowave oscillators at high frequencies (100 kHz and above), but is nevertheless phase-
locked to a microwave oscillator in order to reduce low-frequency phase noise due to
environmental effects. The length of an optical fiber can be controlled by measuring
the arrival time of reflected signals and by mechanically changing the fiber length usinga piezo stretcher. The reliable distribution of timing signals over kilometers with high
precision is particularly challenging for large machines like the European X-ray FEL, and
FLASH is an ideal testbed for this task.
4.6.7. Present status and results
In 2007, a radiation wavelength of 6.5 nm was achieved [4] with a beam energy slightly
below 1GeV. Exponential gain with increasing undulator length until saturation
was experimentally verified by kicking the beam with magnets installed between the
undulator modules and thus disrupting the amplification process at multiples of 4.5m.At 13.7 nm wavelength as a well-studied example [67], the average pulse energy is 40 mJ.
The pulse energy of the third and fifth harmonic is 250 and 10 nJ, respectively. Starting from
noise, SASE is subject to random fluctuations in energy and intensity. The single-shot
photon spectrum comprises several spikes, corresponding to ‘longitudinal modes’, i.e.individual wave trains with an estimated coherence time of 4.2 fs and little temporal overlap.
In the exponential gain regime, the intensity fluctuations follow a Gamma distribution
�(M), where M is the number of longitudinal modes (roughly, the bunch duration dividedby 4.2 fs). Experimentally, a value of M¼ 1.9 was found. Once saturation is achieved, the
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fluctuations are much smaller (18% rms). As for the transverse distribution, the TEM00
Hermite–Gaussian mode is expected to dominate, since it overlaps better with the electron
beam than other transverse modes and should grow faster. Very good transverse coherence
was demonstrated by performing double-slit experiments, e.g. [68].
4.7. Seeded free-electron lasers
It appears natural to use a FEL as an amplifier, i.e. supply it with a pulse of low-power
radiation and expect a high-power output pulse. This would be a ‘seeded’ FEL which has
a number of promising features compared to SASE FELs. An externally supplied input
allows better control of the output regarding its spectrum, time structure and arrival time.
The output is expected to be less prone to fluctuations and of better longitudinal coherence
compared to radiation starting from noise. However, high-gain FELs aim at small
wavelengths in the VUV or X-ray range, whereas the typical source of a coherent seed
pulse would be a laser with a wavelength in or near the visible regime. Thus, seeded FELs
rely on methods to convert a typical laser wavelength down to smaller wavelengths.The workhorse for short light pulses is the Ti:sapphire laser at a wavelength of about
800 nm. As illustrated by Figure 19, there are essentially two strategies to get from this
wavelength to the VUV regime, one is closer to accelerator technology, the other is well-
known in laser laboratories:
. High-gain harmonic generation (HGHG) [69]: the interaction of visible laser
pulses with an electron beam in an undulator (the ‘modulator’) results in
a modulation of the electron energy with the periodicity of the laser wavelength.
Passing through a dispersive section (typically, a chicane formed by four dipole
magnets), energy-dependent path length differences convert the energy modula-
tion into a density modulation. In a second undulator (the ‘radiator’), the
density-modulated beam emits coherent radiation, if the undulator is tuned to
Figure 19. Two laser-based FEL seeding schemes. Top: a laser pulse (assuming a Ti:sapphire laserwith 800 nm wavelength) is directed into a gas jet generating odd harmonics, one of which matchesthe wavelength of the FEL undulator and is amplified (here, the 27th harmonic at �30 nm). Bottom:in the high-gain harmonic generation (HGHG) scheme, a laser pulse (800 nm) causes a modulationof the electron energy in an undulator (modulator). A chicane (mb) causes ‘micro bunching’,converting the energy modulation into a density modulation, which gives rise to coherent radiationat even or odd harmonics (assuming here the third harmonic at 267 nm) in a second undulator(radiator). If the process is repeated, a ‘fresh-bunch’ chicane (fb) is required to overlap the radiationpulse with a yet unmodulated part of the electron bunch. In this example, the second HGHG stageleads to 89 nm wavelength.
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a harmonic (even or odd) of the laser wavelength. This was demonstrated for thefirst time at the Brookhaven Laboratory [70] and later at other facilities.
Typically, the output wavelength can be reduced by a factor of five, whereasfurther reduction would require ‘cascaded’ HGHG, i.e. using the radiation outputof one modulator–radiator combination to seed a second stage, and maybe evena third or a fourth. Cascading HGHG stages is the basis of several proposed FELfacilities but has not yet been demonstrated experimentally.
. High harmonic generation (HHG): rather than seeding a FEL with visible laserpulses, their harmonics may be employed. The generation of high harmonics byfocusing short and intense laser pulses into a gas jet or a gas-filled capillary hasbecome a well-established method to generate short VUV pulses [71].The resulting spectrum contains odd harmonics of the fundamental laserwavelength. Their intensity is several orders of magnitude lower than the initial
laser intensity but quite flat up to a characteristic cut-off energy. Recent progresshas led to pulse energies which are sufficient for FEL seeding down to 30 nm oreven smaller wavelengths [72]. A first experiment to this end has been conductedat SCSS in Hyogo, Japan [73] and others (at SPARC in Frascati [74] and FLASHin Hamburg [75]) are in preparation.
There is no doubt that seeded FELs can reach a wavelength of the order of 10 nm,either by two or three HGHG stages or by seeding with HHG pulses. For even lower
wavelength, the path is yet unclear. It is most likely that the two methods will be combinedto seed the electron beam at a small wavelength and go further down using the HGHGscheme. Finally, the spectrum of a FEL contains odd harmonics which can be usedfor experiments to reach even lower wavelengths, but with reduced intensity and stability.Some seeding experiments at existing facilities and proposed HGHG FELs are listedin Table 4.
4.8. Generation of sub-femtosecond pulses
Several schemes [76–82] have been devised to produce radiation pulses in a FEL witha duration below 1 fs, where the time structure in most of these ‘attosecond’ (as) pulse
Table 4. Examples of HHG/HGHG seeding experiments at existing facilities (with actual ortentative dates) and proposed free-electron lasers based on the HGHG scheme.
Facility name (location)Beam energy
(GeV)Wavelength
(nm) Type
ATF/BNL (Upton, USA, 1999) 0.04 5300 HGHGNSLS DUV FEL (Upton, USA, 2003) 0.177 266 HGHGSCSS (Hyogo, Japan, 2006) 0.250 160 HHGSPARC (Frascati, Italy, �2008) 0.150–0.200 113 HHGFLASH (Hamburg, Germany, �2009) 0.986 30 HHGBESSY FEL (Berlin, Germany) 1.0–2.3 51–1.24 HGHGFERMI@ELETTRA (Trieste, Italy) 1.2 100–10 HGHG
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schemes is defined by a very short laser pulse. An example of a scheme, where this is not
the case, is described in [76]. Here, an electron bunch with an energy chirp passes through
a magnetic chicane, introducing an energy-dependent transverse displacement, i.e. the
bunch is tilted with respect to its direction of propagation. The tilted bunch passes through
a foil, where a narrow slit defines a short longitudinal region of electrons, which will
radiate in a subsequent SASE-FEL, while the emittance of the other electrons is spoilt by
Coulomb scattering.In the laser-based schemes, the electron energy is modulated by an optical pulse just as
in the seeding processes described in the previous paragraph, but now the pulse contains
only a few optical cycles and their phase must be stabilized relative to the pulse envelope.
The production of carrier-envelope stabilized few-cycle pulses has been demonstrated
in several laboratories using Ti:sapphire laser systems, see e.g. [83]. In the resulting energy
modulation, a central maximum �Emax would exceed the neighboring maxima. If FEL
radiation from this maximum can be discriminated against radiation from the remaining
electron distribution, intense FEL radiation with a duration of a few 100 as can
be achieved.Since the radiation wavelength depends on the electron energy �1/�2 according to the
resonance condition in Equation (34), it appears feasible to select the attosecond radiation
component by setting a photon monochromator close to the largest wavelength [78]. An
additional undulator tuned to this wavelength can increase the power of the attosecond
pulse [79].Alternatively, the carrier-envelope stabilization can be tuned to a central zero crossing
of the electric field, resulting in a maximum of the energy gradient d(�E)/dz, which in turn
leads to a maximum of the electron density after passing a magnetic chicane, as shown in
Figure 20. The density in this ‘mirco-bunch’ will be larger than in the neighboring micro-
bunches and will dominate the radiation generated in a SASE FEL. It will also dominate
the radiation in a seeded FEL [82], using e.g. the HGHG scheme described above.In these schemes, pulses with a duration down to 100 as, and a power in the multi-MW
to GW regime are predicted. Using them in a pump–probe fashion and controlling the
delay between pump and probe pulse is yet another challenge for which only preliminary
ideas have been formulated. A bunch with a sub-fs density maximum may, for example,
pass a subsequent few-period undulator to produce a short pulse of coherent radiation
synchronized to the FEL pulse [82].
5. Outlook
There is a large gap between the properties of third-generation synchrotron light sources
and high-gain FELs – up to 10 orders of magnitude in peak brilliance, 3–4 orders of
magnitude in pulse duration. The superior properties of FEL radiation will certainly open
up new frontiers in physics, chemistry, biology and material science. On the other hand,these properties are not always required. Some experiments may e.g. not profit from the
high intensity of FEL pulses, some may not even be able to cope with it. It is therefore
presently believed that FELs will not replace conventional synchrotron light sources with
their large number of simultaneously usable beamlines, but will offer complementary
research opportunities.
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The success of FLASH at DESY in Hamburg as a user facility as well as promisingresults from other facilities have led to the proposal of other FELs. SASE FELs aiming atangstrom wavelengths are currently under construction. LCLS is based on the existingnormal-conducting SLAC linac and will be completed in 2009, the European X-ray FEL atDESY with a superconducting linac is scheduled for 2014.
Already at a stage at which only one FEL with sub-visible wavelength was worldwidein operation, a second generation of FELs with superior beam properties was considered.According to simulations and yet to be demonstrated, seeding by external radiation pulsespromises a cleaner spectral and temporal structure, better stability in intensity and arrivaltime and better longitudinal coherence. As for the conversion of near-visible seed laserpulses to the desired short FEL wavelengths, the emphasis has somewhat shifted fromHGHG schemes, where a factor of 3–5 in wavelength can be gained in one stage, toseeding with high laser harmonics (HHG), offering a factor of 30 or more. On the otherhand, it is believed – but not yet experimentally verified – that the output of one HGHGstage can be used to seed another one, and shorter wavelengths could be reached bycascading several HGHG stages.
A first HHG seeding experiment has been performed at SCSS in Japan [73] andothers are in preparation. While these experiments can be conceived as an add-on toexisting SASE FELs, the demonstration of cascaded HGHG stages requires muchmore dedicated hardware and several proposals to this end have been made. Whilereaching a wavelength of one angstrom appears to be straightforward using the SASE
T T
T
C
T
Figure 20. Top: electron energy modulation (in percent of the beam energy) simulated for a few-cycle phase-stabilized laser pulse at 800 nm wavelength. In this example, the pulse is sine-like, i.e.with a zero crossing of the electric field at its center. Bottom: longitudinal charge distribution of theelectron bunch (in arbitrary units). The left and right column shows the electron distribution beforeand after a magnetic chicane with energy-dependent path length differences of �l/(�E/E)¼ 40 mm.The result is a sub-femtosecond micro-bunch with higher charge density than the neighboring micro-bunches.
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principle, it is an open question, how this wavelength could be achieved with a seededFEL. A combination of HHG seeding and several cascaded HGHG stages may be thesolution.
Acknowledgements
Fruitful discussions with many colleagues at DESY (in Hamburg and Zeuthen) as well as theUniversity of Hamburg, particularly those involved in FLASH, and with my former colleagues atBESSY in Berlin are gratefully acknowledged. I benefited very much from a new textbookwritten by M. Dohlus (DESY), J. Roßbach and P. Schmuser (University of Hamburg), fromseveral recent dissertations written at DESY (e.g. from H. Delsim-Hashemi, M. Rohrs andA. Winter, to name a few) and the Ph.D. thesis of K. Goldammer (BESSY). I wish to thank theeditors of the Journal of Modern Optics, in particular J. Marangos, for their support andpatience.
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