fel-khan2

Journal of Modern OpticsVol. 00, No. 00, 00 Month 200x, 139

RESEARCH ARTICLE

Free-Electron Lasers: Tutorial Review

S. KHAN

Centre for Synchrotron Radiation, Technische Universitat Dortmund

Maria-Goeppert-Mayer-Str. 2, 44221 Dortmund, Germany

(v1.0 released April 2006)

This is a tutorial review of the basic principles and present status of free-electron lasers(FELs) with particular emphasis on high-gain FELs such as FLASH in Hamburg. With theirunprecedented intensity and a pulse duration of a few 10 fs, these novel accelerator-basedradiation sources in the ultraviolet and x-ray regime will open new fields in physics, chemistry,biology and material science. Assuming that the reader is unfamiliar with FELs or particleaccelerators in general, this tutorial covers the basic concepts of particle accelerators and thegeneration of synchrotron radiation in some detail, and finally describes FELs in the low-gainand high-gain regime.

1. Introduction1.1. What is a free-electron laser?

2. Basics of particle accelerators2.1. Acceleration of charged particles2.2. Acceleration by radio-frequency waves2.3. Longitudinal particle dynamics2.4. Optics of charged particles2.5. From single particles to particle beams

3. Synchrotron radiation3.1. Radiation from accelerated charged particles3.2. Synchrotron radiation from dipole magnets3.3. Synchrotron radiation from undulators3.4. Synchrotron radiation facilities

4. Free-electron lasers4.1. Interaction between electrons and radiation4.2. Femtoslicing in electron storage rings4.3. Low-gain free-electron lasers4.4. High-gain free-electron lasers4.5. Self-amplified spontaneous emission (SASE)4.6. FLASH a SASE FEL for soft x-rays4.7. Seeded free-electron lasers4.8. Generation of sub-femtosecond pulses

5. Outlook6. Acknowledgements

1. Introduction

1.1. What is a free-electron laser?

A free-electron laser (FEL) is a source of intense and coherent electromagneticradiation with tunable wavelength. The first operation of an FEL was reported in1977 (1) in the infrared regime at a wavelength of 3.7 m, preceded as early as 1957by a free-electron maser at 5 mm, the ubitron, the history of which is reviewedin (2). Meanwhile, several facilities have reached the ultraviolet regime. Ever sincethe year 2000, the world record in short wavelength was held by the Tesla Test

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ISSN: 0950-0340 print/ISSN 1362-3044 onlinec 200x Taylor & FrancisDOI: 10.1080/0950034YYxxxxxxxxhttp://www.informaworld.com

2 S. Khan

Figure 1. Peak brilliance (as defined in the footnote below) versus photon energy for synchrotron radi-ation from a dipole magnet, a wiggler (W125) and an undulator (U49) of a 3rd-generation synchrotronlight source (BESSY in Berlin/Germany), and from an undulator (U29) at PETRA III, currently underconstruction at DESY in Hamburg. The upper curves show the peak brilliance of FLASH, a free-electronlaser (FEL) in operation at DESY, and two future FELs: the European X-ray FEL at DESY and LCLS atStanford/USA. The lines represent calculations, the dots are measured data at the fundamental wavelengthof FLASH and at its 3rd and 5th harmonic.

Facility at DESY in Hamburg/Germany (3), which is now called FLASH and hasreached a wavelength of 6.5 nm in 2007 (4). Worldwide, a large number of FELuser facilities designed to produce vacuum-ultraviolet (VUV) and x-ray radiationhave been proposed and some are presently under construction, such as the LinearCoherent Light Source LCLS (5), using a part of the normal-conducting SLAClinear accelerator (or linac) at the Stanford University/USA and the EuropeanX-Ray Laser FEL at DESY (6), based on a superconducting linac. While a thirdgeneration of synchrotron light sources emerged in the early 1990s with higherbrilliance1 than before, the new FELs are considered to be the most prominentrepresentatives of a fourth generation of radiation sources. Their properties include

short and tunable wavelength, where 6.5 nm has been achieved and 0.15 nm isexpected to be reached by LCLS in 2009,

a peak brilliance exceeding that of contemporary synchrotron light sources by9-10 orders of magnitude (as shown in Fig. 1) and an average brilliance higherby 5-6 decades,

1The brilliance is defined as the number of photons per second, normalized to the source size, usually inmm2, the radiation divergence in mrad2, and the bandwidth, i.e. a photon energy interval usually givenas 0.1% of the photon energy under consideration. The peak brilliance is measured during a small fractionof the pulse duration, while the average brilliance is measured over an extended period of time, e.g. onesecond. The term brilliance tends to be more popular in Europe, whereas brightness or spectralbrightness is often used in American publications.

Free-electron lasers 3

Figure 2. One type of free-electron laser (FEL) is the oscillator (a), in which the radiation field buildsup between two mirrors, while a train of electron bunches passes through an undulator, an arrangementof magnets with alternating magnetic field direction. In FEL amplifiers, a single electron bunch amplifiesradiation which either starts from noise at the beginning of a long undulator (b) or is supplied by anexternal radiation source (c).

full transverse and good longitudinal coherence, a pulse duration well below 100 fs and the potential of producing sub-fs pulses.Technically speaking, the FEL is a narrow-band amplifier for electromagnetic

radiation like the magnetron used e.g. in every microwave oven, or the klystroncommonly used for radio, television or radar transmitters. In either case, the en-ergy of the electromagnetic wave comes from electrons accelerated in some kind ofvacuum tube (12). What distinguishes FELs from other electron tubes is that

the electron beam energy is in the regime of typical electron accelerators (betweenseveral 10 MeV and the GeV regime)

the energy exchange between electrons and radiation requires a periodic magneticfield provided by a so-called undulator,

the wavelengths range from infrared to the VUV, and will soon be extended tothe x-ray regime.

The amplification process only takes place at a particular wavelength, which isgiven by the electron energy and by the strength and periodicity of the magneticundulator field. Undulators are arrangements of magnets with alternating fielddirection, in which electrons follow a wiggly trajectory (see also Sect. 3.3). Theamplified radiation may be of different origin, as sketched in Fig. 2. There are FELoscillators, in which the radiation field starts from noise and builds up betweentwo mirrors forming an optical cavity (just as in laser oscillators), while a train ofelectron bunches passes through the undulator. These bunches may be supplied bya 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 normalincidence exist and the whole amplification process must be completed within onepass of a single bunch through the undulator. As will be discussed in Section 4of this review, starting the radiation process from noise at the beginning of along undulator without external source is a well-proven and robust technique. Onthe other hand, starting the FEL process with an external seed pulse at the rightwavelength may have advantages and several possibilities how to do so are currentlyunder investigation.

4 S. Khan

Finally, the question may arise, whether free-electron lasers have anything to dowith lasers at all, given that they are more like a big electron tube and representa new generation of synchrotron light sources. FELs are indeed lasers (meaninglight amplification by stimulated emission of radiation), since they do amplifylight and do it by stimulated emission. Their energy reservoir is, however, notan ensemble of electrons pumped to a higher atomic level by another laser or aflash lamp, but a relativistic beam of free electrons with continuous energy tran-sitions (hence their tunability) and pumped by whatever is required to accelerateelectrons. These transitions are stimulated by the electric field of radiation, bywhich electrons are accelerated. The acceleration of electrons, in turn, gives riseto the emission of electromagnetic waves. This mechanism is readily understood inthe framework of classical electrodynamics, whereas the transition between atomiclevels 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 basicconcepts of particle accelerators in Sect. 2 and reviewing synchrotron radiationsources in Sect. 3. In Sect. 4, finally, low-gain and high-gain free-electron laserswill be discussed.

2. Basics of particle accelerators

Since FELs use a beam of relativistic electrons, this Section is meant to conveysome basics ideas of particle accelerator physics in a nutshell. For more details, thereader is referred to the literature, e.g. (710). Readers familiar with the subjectmay 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 = e E e (v B) , (1)

only an electric field E can change the kinetic energy of the electron, while themagnetic field B causes a centripetal force perpendicular to its velocity v. On theother 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 1 T isas large as the first term with an unrealistically high electric field of 300 MV/m.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-varyingmagnetic field. Betatrons and induction linacs are based on this principle.

Radiofrequency (rf) fields with wavelengths of 0.1-1 m (the UHF band) confinedto a hollow metallic structure are the basis of most accelerators. This is truefor circular machines like microtrons or synchrotrons, as well as for linear ac-celerators. 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 10 MV/m.


Figure 3. Schematic sketch of accelerators comprising rf structures. The shaded regions contain a magneticfield perpendicular to the image plane, while the black lines represent the trajectories of acceleratedparticles. 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 trajectory radius constant. This way,magnets are only required along the circumference, which makes large machines possible. While the beampasses the same rf device repeatedly in circular machines, the linear accelerator (d) consists of a successionof rf structures.

Figure 4. A parallel-plate capacitor with a DC voltage (a) has neglecting the fringes a homogeneouselectric field, whereas an AC voltage (b) with wavenumber k causes the field E to follow Bessels functionJ0(kr). A metallic wall at the radius of the first node of J0(kr) completes a simple cylindrical (pillbox)rf cavity (c). The 2.9 GHz disk-loaded waveguide structure of the SLAC linac is shown in (d) with a 2pi/3-mode wave, i.e. 120 phase between adjacent cavities, formed by copper disks (of which the SLAC linaccontains more than 80,000). A bell-shaped cavity (e) is more elaborate to manufacture, but has severaladvantages. The superconducting 1.3 GHz TESLA linac structure with a pi-mode wave is sketched in (f).

The electric field in a femtosecond laser pulse can reach 100 GV/m and more, butbeing perpendicular to the direction of propagation, it cannot be used directlyto accelerate particles. The quest for higher gradients was recently highlightedby obtaining 1 GeV electrons over a distance of only 3.3 cm with a laser-inducedplasma wave (11).

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.In circular accelerators, particles pass through the same rf structure again andagain, while in a linac the beam passes many identical structures one after theother. The achievable beam energy is mainly given by economic limitations. Figure3 shows the generic design of accelerators using rf structures. In a synchrotron, themagnetic field increases synchronously with the beam energy (hence the name).Electron storage rings are quite similar in design, except that the rf structure onlyrestores the energy lost by synchrotron radiation.The principle of a cylindrical rf cavity is sketched in Fig. 4 following (13). In con-

trast to a capacitor with a DC voltage, an electric field oscillating with a frequencyof several 100 MHz gives rise to an oscillating magnetic field. The time-varying mag-netic field, in turn, modifies the electric field. As a result, the radial distribution of

6 S. Khan

the electric field follows Bessels function J0(kr) with wavenumber k = 2f/c andradius r. For a frequency of f = 500 MHz (k = 10.5 m1), for example, the field iszero at r = 0.234 m, since J0(2.45) = 0. A metallic wall at this radius completesthe cavity with a so-called transverse magnetic mode TM010 ringing inside (thesubscripts indicate the number of longitudinal, radial and azimuthal nodes of thefield).A linac may be viewed as a succession of pillbox cavities. Alternatively, an elec-

tron linac like the well-known two-mile accelerator at SLAC can be described asa cylindrical waveguide with a propagating electromagnetic wave. Electrons surf-ing on the wave are continuously accelerated, as long as their velocity matchesthe phase velocity of the wave. For a simple cylindrical tube, however, the phasevelocity exceeds the velocity of light, see e.g. (14). Slowing it down to the electronvelocity ( c) requires a modification of the boundary conditions. In the case of theSLAC linac, equidistant copper irises with a properly chosen aperture and distancedo the job. This structure is called a disk-loaded waveguide.

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 andform the so-called longitudinal phase space. Instead of z, the temporal deviation z/cor deviation in phase = 2z/rf with respect to an accelerating rf field of wave-length rf may be used. For highly relativistic electrons, the momentum deviation

is synonymous to the energy deviation E/E or /, where =(1 2)1/2 is

the relativistic Lorentz factor with = v/c. For a sinusoidal rf field with voltageamplitude Vrf , the energy gain for an electron is

W = eVrf sins , (2)

where s is the design phase (called synchronous phase in storage rings), whichis usually not chosen to yield the maximum energy eVrf for several reasons:

Longitudinal focusing is provided when the arrival time at an rf structure de-pends on the momentum deviation such that a particle with higher (lower) mo-mentum experiences a lower (higher) accelerating voltage. This requires a slopeof the voltage, i.e. s 6= /2.

The concept of bunch compression in linac-based FELs (see Sect. 4.6.2) alsorequires 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 reducethe bunch length.

Deviations from s cause an oscillation in longitudinal position and energy, whichis called synchrotron oscillation and is much slower than the transverse particlemotion described in the next Section. One oscillation period takes typically 102

revolutions in a storage ring. Stable synchrotron motion is only possible within alimited range of energy deviations, known as the energy acceptance of the machine,which depends as Vrf on the rf voltage.

2.4. Optics of charged particles

The guiding magnetic fields of an accelerator or storage ring define an ideal tra-jectory or design orbit in the laboratory system, and particles deviating trans-


Figure 5. Schematic view of dipole, quadrupole and sextupole magnets, where the particle beam is per-pendicular to the image plane. These devices are usually electromagnets with copper coils (black) woundaround an iron yoke (gray). Technical drawings of magnets are shown in Fig. 10.

versely 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 to deflect and focus a beam of relativistic particles

usually consist of water-cooled copper coils wound around an iron yoke. Alterna-tively, permanent magnets are used for very compact arrangements like undulators.Superconducting coils, usually made of NbTi and cooled by liquid Helium, are em-ployed when higher fields are required (e.g. 8.5 T at the Large Hadron Collider atCERN, 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 = B0 +dB

dxx+

1

2

d2B

dx2x2 + . . . =

p

e

1

R(s)+p

ek(s)x+

p

2em(s)x2 + . . . , (3)

i.e. a dipole field with bending radius R(s) for a particle of charge e and mo-mentum p, a quadrupole and a sextupole field with strength parameter k(s) andm(s), respectively, where s is the longitudinal coordinate along the machine1. Inmost accelerators and storage rings, these multipole components are realized ingood approximation by individual magnets with two, four and six poles, respec-tively, as sketched in Fig. 5, although so-called combined-function magnets alsoexist. Quadrupole magnets act as focusing lenses in one plane and as defocus-ing lenses in the other, and an overall focusing effect is achieved by using morethan one quadrupole. Sextupole magnets are used to correct the focusing effect ofquadrupoles for particles deviating from their nominal energy. Only linear elements,i.e. dipole and quadrupole magnets and drift spaces (sections with no magnetic fieldat all) are covered by the matrix formalism described next.The horizontal and vertical deviation of a particle from its design orbit is de-

scribed by its phase space coordinates, (x, x) and (y, y), respectively. The primedenotes the derivative with respect to the longitudinal coordinate s, e.g. x = dx/ds,which can be thought of as transverse angle (in radian) or transverse momentumnormalized to the total momentum x 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

1The longitudinal coordinate s should not be confused with the longitudinal deviation z. Note that s = 0is a particular position along the machine, while z = 0 is a point moving along the design orbit with theaverage particle speed

8 S. Khan

transverse motion in the two planes, assuming here only vertical dipole fields:

x(s) +(

1

R2(s) k(s)

)x(s) = 0

y(s) + k(s)y(s) = 0 . (4)

The solution of these equations is an oscillation with varying frequency, whichdepends on the lattice, i.e. the arrangement of magnets as function of s. Theparticle vector (x, x) can be transformed from one position s0 to another positions1 by matrices in a similar fashion as ABCD matrices, e.g. (15), transform lightrays. A simple example is a horizontally focusing quadrupole magnet of length L,where

(xx

)s1

=

(cos 1|k| sin

|k| sin cos) (

xx

)s0

(

1 0|k|L 1

) (xx

)s0

(5)

with |k|L. In the last step, the thin-lens approximation for small emphasizes the similarity to light ray optics, where |k|L is analogous to the inversefocal length of a lens. The transfer matrix of successive elements is formed bymatrix multiplication. The general solution of Eq. (4) is usually written as

x(s) =x x(s) cosx(s), (6)

where x is a constant known as Courant-Snyder invariant, x(s) is the betafunction and x(s) is the phase advance of the oscillation. Here and in the following,the horizontal coordinate x is used and everything said applies to the vertical planeas well.1

The beta function x(s), its derivative expressed by x(s) = x(s)/2, canbe calculated for a given lattice. Together with the the abbreviation x(s) (1 + 2x(s)

)/x(s), they are referred to as optical functions (or sometimes Twiss

parameters), and the locus of particles with the same x is given by

x = x(s)x2(s) + 2x(s)x(s)x

(s) + x(s)x2(s) , (7)

i.e. a tilted ellipse in phase space, wherexx(s) is the a maximum spatial

extension,xx(s) 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, asshown in Fig. 6, and is given by the optical functions. For a large beta function,the ellipse is elongated in space and small in angle, and vice versa for small (s).For (s) = 0, the ellipse is upright and the ensemble of trajectories with varyingx and random phase x forms a beam waist or bulge.The number of betatron oscillation periods for one revolution around a circu-

lar machine is called tune. It is of the order of 10 and should be chosen away

1In a storage ring, the horizontal orbit for particles with momentum deviation p/p is subject to anadditional deviation x = D(s)p/p, whereD(s) is the horizontal dispersion, while the vertical dispersionis usually negligible.


Figure 6. Top view (x versus s) of a beam waist in a drift space, i.e. no magnetic field, with four particleson the same phase space ellipse: two with maximum offset (white dots), two with maximum angle (blackdots). The trajectories of all particles on or within that ellipse lie within the envelope indicated by graylines. The phase space distribution of a Gaussian beam is shown for three positions. The gray ellipsesmark one standard deviation of x and x. Their area has a constant value defined by pix, where x is thehorizontal beam emittance.

from integers, half-integers, etc. to avoid magnetic field imperfections to resonantlyexcite an oscillation (just as periodic bumps on a road can resonantly excite eigen-modes of a car). The betatron oscillation of an electron is damped by the emissionof synchrotron radiation, which carries away longitudinal as well as transversemomentum, while the rf system restores only the longitudinal component. Thisdamping mechanism is an important property of electron storage rings.Sextupole magnets are required to correct the momentum-dependent focusing

effect of quadrupoles. With their magnetic field being nonlinear in x and y, theycouple horizontal and vertical motion and can give rise to chaotic trajectoriesfor particles venturing beyond the so-called dynamic aperture (as opposed to thephysical aperture, given by solid obstacles). The nonlinearity of the fields in x and yprecludes the use of transfer matrices to describe particle trajectories in sextupoles.Instead, they can be approximated by kicks x and y, which depend on thesextupole 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 coor-dinates. Consider the phase space ellipse of a hypothetical particle at exactly onestandard deviation in space and angle. The phase space area enclosed by the ellipsecorresponding to that particle is x, where x is called the horizontal emittance

1

1Unfortunately, it is common practice to use the same symbol for the Courant-Snyder invariant (aproperty of a single particle) and for the beam emittance (a property of the whole ensemble of particles).

10 S. Khan

of the beam. For a Gaussian distributed beam, one may also define the horizontalemittance 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 liter-ature). In electron storage rings, the horizontal beam emittance results from anequilibrium between excitation due to the random nature of synchrotron radia-tion emission and the damping effect of radiation. A typical number for to-datesynchrotron light sources is 5 109 radm. The vertical emittance in electron stor-age rings is mainly given by field errors and misaligned magnets. A ratio betweenvertical and horizontal emittance (called the coupling) of 0.01 to 0.001 can beachieved.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 ofthe gun and subsequent elements to avoid excessive blow-up of the emittance byCoulomb repulsion of the electrons. During acceleration, the longitudinal momen-tum increases by the Lorentz factor while the transverse momentum does not.Thus, the particle distribution shrinks in the angular coordinates x and y, and sodoes the emittance (so-called adiabatic damping). To compare different linacs, itis therefore customary to specify their normalized emittance x,y = 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 asmall size and a low divergence of the electron beam. With a normalized emittanceof the order of 1 106 radm, a linac with 1 GeV beam energy ( 2000) yieldsa much better horizontal emittance than a storage ring. One might argue, thatthe vertical emittance of a storage ring is superior, but that may be less relevantbecause the ultimate size of the photon beam 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 x =x x. 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 betafunction in a drift space around a beam waist at s = 0 (as in Fig. 6) is

x(s) = x(0) +s2

x(0). (8)

The resulting rms beam size is

x(s) =x

x(0) +

s2

x(0)=

2x(0) +

(xs

x(0)

)2. (9)

The similarity to the waist of a Gaussian laser beam with wavelength (16)

w(z) =

w2(0) +

(z

w(0)

)2(10)

is obvious.1 So far, only the interaction of beam particles with external fields

1Note 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 onestandard deviation x(s) of the particle density distribution.


Figure 7. Electric field of a charged particle at time t after an acceleration v/t, assuming that theparticle was at rest at t = 0. A transition zone (gray) travels outward at the speed of light. An observer atposition A measures the field after acceleration, an observer at B is not yet aware of the velocity change. Alarger view of a field line at an angle relative to the direction of acceleration illustrates the ratio betweenthe angular and radial field components, see Eq. (11).

(magnets and rf fields) has been addressed. When speaking of beams with a highcharge density, the interaction between particles must be considered as well. Themost important effects are

Low-energy particles are subject to Coulomb repulsion, which is cancelled by anattractive magnetic force, as the particle velocities approach c. Coulomb repul-sion at the electron gun limits the beam emittance in linac-based FELs.

Deviations of the vacuum beam pipe from a perfectly uniform tube with infiniteconductivity generates electromagnetic fields behind the beam particles. Theseso-called wake fields acts on trailing particles and may cause beam instabilities.

Infrared coherent synchrotron radiation (CSR) is emitted in dipole magnets andcatches up with beam electrons in front of the emitting electron, acting like awake field on leading, rather than trailing particles. The CSR wake from bunchcompressors (see Sect. 4.6.3) in linac-based FELs is of particular concern.

Electron beams ionize the residual gas atoms in the beam pipe and trap thepositive ions, which in turn may influence the beam. For this reason, some syn-chrotron radiation sources use a positron beam.

Scattering between two electrons within a bunch limits the beam lifetime inring-based synchrotron radiation sources (Touschek effect).

3. Synchrotron Radiation

3.1. Radiation from accelerated charged particles

Accelerated charged particles emit electromagnetic radiation. The description ofthis well-known fact using retarded potentials (17) is somewhat cumbersome, butthere is a simple argument, attributed to J. J. Thomson, that nicely illustrates theorigin 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 betweenthe field before and after acceleration travels outward at the speed of light. Thiszone has a thickness ct and is r = c t away from the positron. For an angle with respect to the direction of acceleration, the ratio between angular and radialfield component, as read from Fig. 7, is

12 S. Khan

EEr

=v t sin

ct. (11)

With the radial field component given by Coulombs law

Er =1

4e

r2, (12)

where is the free-space permittivity, the angular field component is

E =1

4e (v/t) sin

c2r=

1

4e r sin

c2r. (13)

The transition zone with kinks in the field lines is a pulse of electromagneticradiation. Its energy flow through a solid angle d is given by

Wd = c E2r2d =

e2r2

162c3sin2 d . (14)

This is the well-known angular distribution of radiation from a dipole antenna.Integration over solid angle yields Larmors formula for the radiated power

P =e2

6c3r2 . (15)

Prominent examples of electromagnetic radiation from accelerated charges are:

Radio waves are emitted, when the drift velocity of electrons in an electricalconductor is changed by applying a sinusoidal voltage.

X-rays are emitted, when an electron beam hits solid matter and is abruptlydecelerated. Apart from their penetration power, the usefulness of x-rays liesin their short wavelength allowing to retrieve spatial information on an atomicscale, and a photon energy sufficient to excite electrons from inner atomic shells.

Radiation from relativistic electrons under the influence of a centripetal forcewas first directly observation at the General Electric 70-MeV synchrotron in1947 (19) and is therefore called synchrotron radiation. It has largely replacedconventional x-rays in scientific research due to its superior properties, whichwill be described in the next paragraph.

Synchrotron radiation is covered by many general textbooks on acceleratorphysics and, owing to its importance, by several books devoted to its properties(2022) and applications (23, 24).

3.2. Synchrotron radiation from dipole magnets

In the early days of synchrotron radiation research, synchrotron light was extractedfrom dipole magnets in electron-positron colliders, which were operated for ele-mentary particle research. Even though undulators are now widely used (see nextSection), the radiation from dipole magnets is still sufficient for many purposes.


Figure 8. Linear (left) and double-logarithmic plot (right) of the universal function S(/c) of the syn-chrotron radiation spectrum of Eq. (18), where c is the critical frequency. The function S(x) can be

approximated by 1.333 x1/3 for x < 0.1 and by 0.777x/ex for x > 1, see e.g. (20).

For the centripetal acceleration of a relativistic electron with rest mass me andmomentum p, Eq. (15) can be written as

P =e22

6m2ec3p2 with p = p p c

R E

Rand =

E

mec2(16)

where E is the electron energy, R is the bending radius and is the angularvelocity, see e.g. (20). This, finally, leads to

P =e2c

6 (mec2)4

E4

R2. (17)

The mass dependence shows why synchrotron radiation is usually not relevantfor accelerated particles other than electrons or positrons, and the factor E4/R2

explains why circular e+e-colliders have become as large as LEP at CERN/Genevawith R = 3000 m and have reached their economic limits. At 100 GeV and a beamcurrent of 6 mA, the radiated power at LEP was about 18 MW (25). As a practicalrule, the energy in keV lost by one electron per revolution is 88.5 E4/R with Ein GeV and R in m.In a coordinate system co-moving with an electron, the angular distribution of

radiation is the same as for a dipole antenna given by Eq. (14) with rotational sym-metry about the centripetal acceleration vector. In the laboratory frame, however,the distribution appears to be strongly boosted in forward direction, resultingin a narrow cone tangential to the electron trajectory. To estimate the typical di-vergence of this cone, consider a photon emitted perpendicularly to the directionof acceleration as well as to the electron trajectory. If its momentum is p in theco-moving frame, the transverse momentum in the laboratory system is still p,while the Lorentz transformation results in a momentum p = p parallel to theelectron direction. The angle between the total photon momentum and the electrondirection is p/p 1/. With = 103-104, the typical divergence of synchrotronradiation is very much like that of a laser beam.Since the passage through a dipole magnet is not an oscillatory motion, the

radiation frequency is not well defined and the spectrum is broad. The typicalfrequency can be estimated from typ 2/t, where t is the time interval,during which a light cone of half-width 1/ tangential to a circular electron path

14 S. Khan

illuminates a given observation point. The result of this estimate is not far fromthe so-called critical frequency c = 3/2 c3/R, which divides the spectrum intotwo parts of equal power. The spectral density is given by

dP

d=

P

cS

(

c

)with S(x) =

93

8x

x

K5/3(y)dy , (18)

where P is given by Eq. (17) and K5/3(y) is a modified Bessels function of thesecond kind. It is remarkable, that the spectral shape is universally given by S(x),as depicted in Fig. 8. The critical photon energy is typically of the order of afew keV. Harder x-rays can be obtained by using superconducting magnets withsmaller bending radius (sometimes called superbends or wavelength shifters).

3.3. Synchrotron radiation from undulators

Modern synchrotron radiation facilities usually comprise groups of bending andfocusing magnets, alternating with drift spaces of several meters in length. Thesestraight sections provide space for so-called insertion devices, which are usuallywigglers or undulators. Both are arrangements of alternating dipole magnets, oftenmade from permanent-magnet material like NeFeB or SmCo in combination withiron poles, see e.g. (26). The period length u of typically a few cm is defined as thedistance from one pole to the next pole of equal polarity. The magnetic field on thebeam axis is either fixed or tuned by mechanically changing the gap between thepoles using powerful and precise motor drives. For period lengths of 20 cm or more,electromagnets are customary, which are tuned by varying the electric current. Thehighest fields are obtained using superconducting coils. Superconducting wigglerswith fields up to 7 T have been in operation for a while, whereas the experiencewith superconducting short-period undulators is still limited (27).The (often vertical) magnetic field component perpendicular to the poles of wig-

glers and undulators is nearly sinusoidal, and so is the beam trajectory in themidplane between the magnets:

(xx

)=

K

( cos(kus)/kusin(kus)

)with K ueB

2mec, (19)

where s is the coordinate along the beam axis, B is the amplitude of the sinusoidalmagnetic field, K is called field parameter or undulator parameter, and ku =u/c = 2/u. For an electron with velocity c, the transverse excursions in awiggler or undulator reduce its velocity s made good along the beam axis to

s(t) =(c)2 x2 = c

1 c

2

2 x

2

c2 c

{1 1

22 x

2

2c2

}(20)

Inserting x = cx with x from Eq. (19), and using sin2(x) = (1 cos 2x)/2finally yields

s(t) = c

{1 1

22

(1 +

2K2

2

)}+ c

2K2

42cos(2ut) = s+s(t) . (21)


The average velocity s is modulated by twice the frequency u, because thereare two velocity minima at x = 0 and two maxima at x = 0 per undulator period.In a co-moving frame 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 the radiation from these dipoles with the same properties as described in theprevious paragraph just adds up. In undulators, however, there is an interferencebetween the radiation from the dipoles, leading to a line spectrum and a narrowerangular distribution. Since interference requires sufficient overlap, the amplitudein x and x is smaller for an undulator than for a wiggler, which implies a smallermagnetic field and a smaller value of K. In the literature, an arbitrary distinctionbetween wiggler and undulator is often made atK = 1, where the amplitude of x is1/ and light cones with an assumed half-width of 1/ would overlap continuouslyfor K < 1. In practice, however, the transition from an undulator to a wiggler isgradual. Figure 9 shows spectra of an undulator, starting with K = 0.5. Considerreducing the gap between the magnetic structures, thus increasing the field on thebeam 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.

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 betweenthe motion away from and back to the beam axis, only odd harmonics appear.If the observer 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 showup and eventually merge into a broad spectrum with the same shape as from adipole magnet. 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 peri-ods requires the electron to lag behind the electromagnetic wave by one wavelength per undulator period. In the time interval

t =uc

=u

s(22)

radiation passes one undulator period, while the electron covers a distance ofu. Inserting s from Eq. (21) with 1 and solving for yields the resonancecondition

=u22

(1 +

K2

2

). (23)

The electron is slower than the electromagnetic wave for two reasons, one beingits mass (first term), the other its excursions in the undulator (second term). Forradiation emitted at an angle with respect to the beam axis, the electron lagsbehind even more, because it moves in a different direction, and the resonantwavelength 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 spec-

16 S. Khan

Figure 9. Photon flux from an undulator (solid lines) with N = 10 periods and an undulator parameterK ranging from 0.5 to 4 (top to bottom) as function of the photon energy (left column) and of the photonenergy normalized to the critical energy Ec = ~c of a dipole magnet with the same magnetic field. Thespectrum of such a dipole multiplied by 2N (dashed line) forms an envelope of the undulator harmonics

trum via Fourier transform. Centered at k = k1 = k2c/, the spectral intensityis

I() (sinx

x

)2with x N k

1, (24)

where 1 is the fundamental frequency. The resulting line width is

kk

1kN

. (25)

An approximate rms value for the angular distribution of undulator radiation is

1

1 +K2/2

2kN. (26)


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

N

from the angular distribution in horizontal and in vertical direction. For compar-ison, the intensity of wiggler radiation exceeds that of a dipole magnet only by afactor of 2N .The emitted power from a wiggler or undulator is given by Eq. (16). Using

px =E

cx =

EK

csin (kus) =

EK

csin (ut) and px =

EK

cu cos (ut) (27)

from Eq. (19), inserting px into Eq. (16) and averaging over time with cos2 x =1/2 yields the instantaneous power emitted by a single electron

P =e222uK

2

12c=

e22cK2

32u. (28)

Of practical interest is also the total power emitted for a given beam current I.In practical units:

Ptotal[W] = 7.26E2[GeV2]I[A]K2N

u[cm2]. (29)

Whether wiggler or undulator, the radiated power is the same. However, thepower is differently distributed in spectrum and angle, which makes an undulatormore useful for many applications.This discussion was restricted to planar devices, i.e. wigglers or undulators that

deflect the electron beam in their midplane, producing linearly polarized light.There are also undulators in which the beam follows a helical trajectory, emit-ting circularly polarized light. In this case, on-axis radiation contains no higherharmonics. If circularly polarized radiation from harmonics is desired (to reachshorter wavelengths), an elliptically helical trajectory is chosen and some linearlypolarized background must be accepted.

3.4. Synchrotron radiation facilities

Since the first operating synchrotron at 8 MeV did not have a glass tube, syn-chrotron radiation was nly observed at the second one, the 70-MeV synchrotronat General Electric (19). In the 1950s, this new type of radiation was used forfirst experiments in x-ray spectroscopy. The first generation of synchrotron radi-ation facilities emerged in the 1960s, extracting radiation from synchrotrons (likeDESY in Hamburg) and storage rings (such as SPEAR at SLAC, DORIS in Ham-burg, VEPP-3 in Novosibirsk and others) which primarily served other purposesin nuclear and particle physics.The second generation of synchrotron light sources in the 1970s and 1980s

were electron storage rings dedicated to and optimized for synchrotron radia-tion. Examples are Aladdin at the University of Wisconsin, the Photon Factoryin Tsukuba/Japan, BESSY in Berlin/Germany and SuperACO in Orsay/France.Third-generation light sources were constructed ever since the 1990s until to-

day. Compared to the second generation, they are characterized by larger storagerings, allowing for higher beam energy (and thus shorter wavelength), smaller beam

18 S. Khan

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 is transferredto a storage ring, where it circulates for many hours. Most storage rings of synchrotron light sources havean achromatic magnet structure, alternating with straight sections to accommodate wigglers and undu-lators (an achromat, comprising two or three dipole magnets, starts and ends with zero dispersion). Theradiation generated tangentially to the storage ring is transported to the experiments outside the radia-tion shield (typically 1 m of concrete) by complex photon beamlines, in which the radiation is collimated,monochromatized and focused.

emittance (and hence higher brilliance) and a larger number of straight sections toaccommodate wigglers and undulators. Figure 10 shows BESSY II in Berlin as atypical representative, and other examples are listed in Table 1. Worldwide, thereare more than 50 synchrotron radiation sources in operation. Owing to the largeuser demand, new facilities are still being built and older machines are remodeledto become modern radiation sources, such as SPEAR3 at SLAC (recommissionedin 2004) or PETRA III, currently under construction 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 parametersto the limits, linear accelerators are now considered.While storage rings offer little room for improvement in emittance and bunch

length, both given by an equilibrium between the disturbing and damping effectsof synchrotron radiation, the beam properties in a linac depend mostly on theelectron source and can be further optimized. A disadvantage of linacs is their lowbunch repetition rate. The duty cycle, i.e. the ratio of time with and without beamis of the order of one percent. This can be improved by the concept of energyrecovery using a superconducting linac in DC operation. Here, the electron beamis accelerated, and after serving its purpose passes the linac again at the oppositerf phase and returns most of its energy to the rf field.Electrons from a linac may be used to generate undulator radiation with much

shorter pulse duration than conventional synchrotron light sources, e.g. SPPS atSLAC (28). However, full use of their superior beam properties can be made byfree-electron lasers, boosting the peak brilliance by up to 10 orders of magnitude.How this almost inconceivable factor of improvement is possible, will be describedin the following Section.


Table 1. Examples of third-generation synchrotron radiation sources in operation and one (PETRA III)

presently under construction.

facility name (location, first beam) circumference beam energy current hor. emittance[m] [GeV] [mA] [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

4. Free-electron lasers

A free-electron laser (FEL) acts as a narrow-band amplifier for radiation, whichrequires a mechanism that transfers energy from the electron beam to the radiationfield. This mechanism is explained in the next Section, followed by a discussion oflow-gain and high-gain FELs. The theory of FELs is described in more detail ine.g. (2931, 33, 34, 61).

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 veloc-ity. The force is given by the electric field of the radiation, which is perpendicularto its direction of propagation. A non-zero energy transfer

dW = ~F d~s = e ~E d~s = e ~E ~v dt (30)

is accomplished by overlapping electrons bunches and radiation in an undulator,where the electron velocity has a component parallel to the electric field, as sketchedin Fig. 11. Consider a sinusoidal electric field in horizontal direction

Ex(z, t) = E cos(ks t+ ) (31)

with amplitude E, wavenumber k, angular frequency and phase offset .Given a horizontally transverse electron velocity vx cx with x from Eq. (19),the energy transfer per time is

dW

dt= eE cos(ks t+ ) cK

sin(kus)

= ecEK2

{sin ([k + ku]s t+ ) sin ([k ku]s t+ )}

ecEK2

{sin+ sin} (32)

For the two sinusoidal terms to contribute over the whole length of the undulator,their respective phase should be constant:

20 S. Khan

Figure 11. Interaction between a radiation pulse and an electron on its sinusoidal trajectory in an undu-lator, denoted by the black line. In (a), the electric field (black arrows) as experienced by the electron andits transverse velocity both point downwards, in (b) both are zero, and in (c) both point upwards, givingrise to an energy transfer between electron and light field, which is oscillatory but does not change sign see right figure and Eq. (32) for the definition of sin. The electric field changes sign at the electronposition, because the electron lags behind the radiation pulse by one wavelength per undulator period.

ddt

= 0 = [k ku]s = k(s c) kus kc1 +K2/2

22 kuc . (33)

Here, s was approximated by the average velocity s of Eq. (21) with 1 inthe first term and by c in the second term. The resulting radiation wavelength is

=2

k= 2

ku22

(1 +

K2

2

)=

u22

(1 +

K2

2

). (34)

In the last step, the negative sign was dropped, since it would imply < 0, whichmeans that only + can be made constant. The result is again the resonance con-dition as in Eq. (23), stating that the electron must lag behind the radiation byone wavelength per undulator period. In other words, energy is exchanged withradiation of the same wavelength as spontaneous undulator radiation. With +now constant, = + 2kus oscillates twice per undulator period and cancelson average. Its occurrence in the term {sin+ + sin} of Eq. (32) reflects thefact that the energy transfer is not constant. For ~v ~E, which happens twice perundulator period, the energy transfer is zero (see Fig. 11). For a homogeneous elec-tron distribution, half of the electrons will gain and half will lose energy, dependingon their value of +, which is known as the ponderomotive phase.If the Lorentz factor deviates by from the value that fulfills the resonance

condition, the ponderomotive phase + will not be constant any more. Subtract-ing Eq. (33) from the same equation for ( + ) yields the rate, at which theponderomotive phase changes

d+dt

= kc1 +K2/2

2

(1

( +)2 12

) kc1 +K

2/2

2

(2 ( +)2( +)2 2

)

kc1 +K2/2

2

((2 +)4

) kuc2

(2

3

)= 2kuc

. (35)


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 their startingpositions (open circles) to positions (filled circles) which depend on their initial ponderomotive phase. Theseparatrix is the borderline between bounded trajectories (cf. swing boat A) and unbounded motion (swingboat B). For starting points at = 0 (open symbols in the left plot), energy gain and energy loss areequal. When starting at > 0 (right plot), less energy is gained than lost by the electrons.

Defining /, the energy transfer rate from Eq. (32) can be written as

dW

dt=

d

dtmec

2 =d

dtmec

2 . (36)

Using this expression, Eqs. (35) and (32) can be written as

d

dt= 2kuc and

d

dt= eEK

2mec2sin , (37)

where + and is ignored from now on. These two coupled first-orderdifferential equations describe the motion of an electron in a phase space (, )under the influence of radiation. It should be emphasized that the phase is nota fixed phase of the radiation field (which slips forward by one wavelength perundulator period), but should rather be considered relative to an electron with = 0 at a longitudinal position for which energy gain and loss cancel over thelength of the undulator. It should also be noted that the amplitude E of theelectric field was assumed to be constant, which is not the true once the energytransfer between electrons and radiation becomes significant.The two coupled equations (37) can be combined to

+ 2 sin = 0 with 2 eEkuKme2

, (38)

which is known as the pendulum equation in classical mechanics. Consider, forexample, a swing boat as shown in Fig. 12. For small amplitudes, it acts like aharmonic oscillator. For larger amplitudes, the motion becomes anharmonic andthe oscillation frequency decreases. There is a contour in phase space known asseparatrix, that mark the limit between bounded and unbounded motion. Outsidethe separatrix, the swing boat performs loopings.It shall be noted without derivation, that the longitudinal oscillation of electrons

in an undulator as described by Eq. (21) causes a reduction of the coupling betweenelectrons and radiation, because the electron deviates from its optimum position inphase. This was neglected when approximating s by s in Eq. (33). A more elaboratetreatment results in a modification of the undulator parameter

22 S. Khan

Figure 13. Generation of ultrashort radiation pulses in a storage ring via femtoslicing: a short laser pulse,co-propagating with an electron bunch in an undulator (modulator) causes a periodic energy modulationwithin a slice of the bunch. After passing a dipole magnet, the off-energy electrons are transverselydisplaced and their radiation from a second undulator (radiator) can be extracted using an aperture.

K K[J0

(K2

4 + 2K2

) J1

(K2

4 + 2K2

)], (39)

where J0 and J1 are the zero- and first-order Bessels functions of the first kind.For K = 1, as an example, the reduction factor in square brackets is 0.91. For largeK, the factor approaches 0.7.

4.2. Femtoslicing in electron storage rings

In synchrotron light sources with typical pulse durations of several 10 ps, theinteraction between electrons and radiation in an undulator as described abovecan 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 aslice of an electron bunch. As shown in Fig. 13, the off-energy electrons aretransversely displaced, e.g. by passing a dipole magnet, such that their synchrotronradiation from a subsequent undulator can be selected using an aperture. The timestructure of this radiation is essentially that of the short laser pulse, but beingincoherent radiation from a small fraction of the bunch its intensity is extremelylow. 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 goodopportunity to study laser-induced energy modulation in detail, which is part ofthe FEL process.

4.3. Low-gain free-electron lasers

Under the influence of electromagnetic radiation in an undulator, electrons movein a phase space, spanned by the ponderomotive phase and the energy deviation = /, as shown in Fig. 12. For electrons with = 0 and a homogeneousdensity distribution in , an equal amount of energy is gained and lost by theelectrons, and there is no net energy transfer from or to the radiation field. This isnot yet a free-electron laser.When starting with > 0 (but with most electrons still within the separatrix)

it is obvious from Fig. 12 that less energy is gained than lost by the electrons. Itmay be assumed, that this energy is transferred to the radiation field, even thoughthe increase of the electric field is not part of the model yet. The energy transferis only a second-order effect, and the gain


Table 2. Examples of low-gain free-electron lasers.

facility name (location) beam energy wavelength electron source[MeV] [m]

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

g WW

, (40)

i.e. the relative increase of energy in the radiation field, is small. The gain, theexpression of which will not be derived here, is given by

g() = e2K2N32une40mec23

ddx

(sin2 x

x2

)with x 2N (41)

for an electron density ne and all other symbols defined as before. For the low-gain FEL, the gain as function of the initial value of is the negative derivativeof the undulator line shape given by Eq. (24) this is known as Madeys theorem(39). If < 0, the gain is negative, i.e. energy is transferred to the electrons. Thiscorresponds 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 Fig. 2). These FELsusually operate in the infrared regime with an electron beam from a linac. Thereare also a few FEL oscillators embedded in storage rings. The shortest wavelengthreached to-date by an FEL oscillator is around 176 nm, obtained at the storagering ELETTRA in Trieste/Italy (40). Further progress is inhibited by the factthat mirrors with good reflectivity at normal incidence do not exist for shorterwavelengths. 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 Eq. (37) was assumed tobe constant. While such an approximation may be appropriate to show the basicfeatures of a low-gain FEL, where the field increases with every pass by only a fewpercent, it is completely inadequate when the gain is larger. The need for largergain arises when considering small wavelengths, for which mirrors with sufficientreflectivity do not exist. In this case, the whole amplification process must becompleted within a single pass of an electron bunch, because there is no way tomake the radiation pulse interact with another bunch. In principle, larger gain canbe accomplished by just making the undulator longer, leading to an exponentialgrowth 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(s), a function of the position s along the undulator. To allow for anarbitrary phase , the horizontal field amplitude is written as a complex numberE(s) = E(s) exp(i), and complex quantities shall be denoted by a tilde fromnow on.

24 S. Khan

Here, only the one-dimensional theory shall be outlined. In this approximation,the electric field is subject to the wave equation

(2

s2 1c2

2

t2

)Ex(s, t) =

jxt

with Ex(s, t) = E(s)eikst , (42)

where jx is the transverse current density, given by the transverse motion ofelectrons 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 thefield amplitude Ex (which is called the slowly-varying amplitude approximation)finally yields

dEds

= ic2

jxt

eikst = Kc4

j1 =Kc2nee

2ein . (43)

Here, ne is the electron density, i.e. the number of electrons per volume despite theone-dimensional treatment. The transverse current density is replaced by jx xjs,where

js(, s) = j + j1(s)ei , (44)

is the longitudinal current density, which is assumed to contain a constant termand a periodic modulation with a complex amplitude j1(z), which changes as thebunch travels along the undulator. In the last step of Eq. (43), the constant termwas dropped and the modulation was expressed by ein, the average of the pha-sors for all electrons. If this so-called bunching factor is close to zero, the electronsare randomly distributed, otherwise the electron density is modulated, giving riseto a change of the electric field amplitude. Now, there are three coupled differentialequations describing the rate, at which the energy and the phase of the nth electronchanges, as well as the amplitude of the surrounding electromagnetic wave:

dnds

(s) = f1 n(s)dnds

(s) = f2 E(s) sinn(s)

eEds

(s) = f3 ein(s) , (45)

where the factors fi, containing constant undulator and beam parameters, canbe read from Eqs. (37) and (43). From this point, there are two ways to proceed:

By making further assumptions on the electron distribution, a third-order differ-ential equation for the electric field amplitude can be obtained. Its solution showsthat the electric field grows indeed exponentially with longitudinal position s andan analytical expression for the growth parameter can be obtained.

Representing the electrons by a smaller, but statistically significant number ofmacro-particles, their dynamics as well as the evolution of the electric field canbe simulated numerically. This not only demonstrates the exponential growth of


the field, but also allows to study its startup, either from noise of from an inputfield, as well as the saturation of the amplification process.

4.4.2. Analytical results

The coupled first-order differential equations presented in the previous paragraphcan under further assumptions be combined to a third-order equation for the trans-verse electric field as function of the path length s in the undulator. There are threesolutions corresponding to exponential decay, exponential growth, and an oscilla-tion. After startup, the exponential growth soon dominates and the radiation powerincreases as

P (s) es/Lg with Lg = 13

(43me

0K2e2kune

)1/3(46)

with the same symbols as before. In practice, the power gain length Lg will besomewhat longer than predicted by the 1-dimensional theory, in which the influenceof space charge, energy spread etc. is ignored. In the course of the FEL process, theelectron beam energy and its energy spread increases, eventually inhibiting furthergain. Saturation is typically reached at Lsat 20Lg (34). The gain length is relatedto another useful quantity by

=u

43Lg

. (47)

This dimensionless quantity is known as FEL parameter or Pierce parameter,and is of the order of 103. It can be used to characterize several FEL properties,e.g. the saturation length may be expressed as Lsat u/. Furthermore, nearsaturation the value of corresponds to the relative FEL amplifier bandwidth, andthe 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 n1/3e , which implies that the peak currentshould be large and the beam emittance low. Another effect of a non-zero emittanceis the beam divergence. A non-zero transverse velocity reduces the longitudinalvelocity. This way, the beam divergence causes a spread of the longitudinal velocity,which in turn is equivalent to an additional energy spread. Combining these andother considerations, it turns out that the emittance should ideally be /4 fora given radiation wavelength .

4.4.3. Simulation of the FEL process

Numerical simulations allow to study the FEL process and the underlying elec-tron dynamics in more detail and to include effects which are inaccessible by ana-lytical methods. Figure 14 shows the result of a 1-dimensional numerical calculationaccording to Eqs. 45 as an illustration, whereas realistic calculations are done with3-dimensional FEL codes such as GENESIS (41). If effects of the electron gun andacceleration process are studied, FEL codes are interfaced with other programsto simulate space charge effects in the gun, e.g. ASTRA (42), and general beamdynamics codes such as ELEGANT (43). Examples of these start-to-end FELsimulations are (44, 45).As shown in Fig. 14, the motion of electrons under the influence of the radiation

field leads to a periodic modulation of the longitudinal charge density (micro-bunching), which in turn gives rise to the coherent emission of radiation at a

26 S. Khan

Figure 14. Top: Electron motion in phase space (relative energy deviation versus ponderomotive phase) as solutions of Eqs. 45 at the onset of the FEL amplification (a), shortly before saturation (b), andafter saturation (c). The corresponding charge density is plotted below. The bottom figure shows the FELpower rising exponentially with increasing undulator length (in units of the gain length), until the processsaturates.

wavelength corresponding to the distance between adjacent micro-bunches. Whilethe modulation becomes more pronounced, the radiation power grows exponen-tially, as described by the analytical result of Eq. (46). From part c) of Fig. 14, itis immediately 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 coupled differential equations of Eqs. 45. Here, this is done in smalltime steps istep using the fourth-order Runge-Kutta method (46) and withfactor1,factor2,factor3 corresponding to f1, f2, f3, respectively. For each elec-tron, labeled i, the position in phase space is calculated, where phi(i) is theponderomotive phase and eta(i) is the relative energy deviation. The variablesphi1,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 updatethe complex electric field (e real,e imag). Apart from input and output state-ments, this is all what is needed to produce Fig. 14, showing the evolution of the


electron distribution in phase space as well as the exponential increase and satura-tion of the radiation power (which is proportional to e real**2+e imag**2). Thetrajectories in Fig. 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 = 1, number_of_steps

bunching_real = 0.

bunching_imag = 0.

do i = 1, number_of_electrons

phi1 = factor1 * eta(i)

eta1 = factor2 * ( e_real * dcos(phi(i))

& + e_imag * dsin(phi(i)) )

phi2 = factor1 * ( eta(i) + eta1/2. )

eta2 = factor2 * ( e_real * dcos(phi(i)+phi1/2.)

& + e_imag * dsin(phi(i)+phi1/2.) )

phi3 = factor1 * ( eta(i) + eta2/2.)

eta3 = factor2 * ( e_real * dcos(phi(i)+phi2/2.)

& + e_imag * dsin(phi(i)+phi2/2.) )

phi4 = factor1 * ( eta(i) + eta3 )

eta4 = factor2 * ( e_real * dcos(phi(i)+phi3)

& + e_imag * dsin(phi(i)+phi3) )

phi(i) = phi(i) + ( phi1 + 2. * (phi2 + phi3) + phi4 ) / 6.

eta(i) = eta(i) + ( eta1 + 2. * (eta2 + eta3) + eta4 ) / 6.

bunching_real = bunching_real + dcos(phi(i))

bunching_imag = bunching_imag + dsin(phi(i))

end do

e_real = e_real + factor3 * bunching_real / electrons

e_imag = e_imag + factor3 * bunching_imag / electrons

end do

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 amplifierinput. In the case of the FEL oscillator, the radiation field starts from spontaneousundulator radiation and builds up over time between two mirrors forming an opticalcavity. For high-gain FELs at wavelengths for which good mirrors do not exist, theamplification process described above may also start from spontaneous radiationat the beginning of a long undulator. Instead of describing the initial field as spon-taneous radiation, one may equivalently say that it starts from noise. The initialthe electron distribution is, of course, not completely homogeneous. Electrons arerandomly distributed and have a non-zero density modulation at any wavelength,including the wavelength that is resonant to the FEL undulator according to Eq.(34). This startup from noise is called the SASE (self-amplified spontaneous emis-sion) principle (47). It has been proven to work reliably at several FEL facilitiessuch as VISA (48) at Brookhaven and LEUTL (49) at the Argonne Lab in thevisible regime as well as TTF/FLASH (3) in Hamburg and SCSS (50) at SPring-8in Japan at shorter wavelengths. It is therefore also the basis of future x-ray FELs.Some examples of SASE FELs are listed in Tab. 3.

28 S. Khan

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 wavelength electron source[GeV] [nm]

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

Figure 15. Layout of the high-gain free-electron laser FLASH at DESY in Hamburg as of summer 2008.Top: Footprint of the facility showing that part of the FLASH tunnel is inside a building containing therf power stations, laboratories and workshops. The remaining tunnel is covered by earth. The numbersindicate the distance from the photocathode gun in meters. While the electron beam is dumped 260 m fromthe gun, the photon beam crosses the PETRA storage ring and enters an experimental hall accommodatingseveral beamlines and a laser system (marked by ) employed for pump-probe measurements. A laserlaboratory 150 m from the gun is connected to the tunnel by several tubes. Another tube at 190 m is foreseenfor seeded-FEL radiation (see Section 4.7). Bulky hardware (such as the 12 m long acceleration modules)can enter through the curved corridor at 250 m. Below: The enlarged view with exaggerated transversedimensions shows the electron gun, followed by six superconducting rf modules (acc), accelerating thebeam to 1 GeV, and two magnetic chicanes acting as bunch compressors (bc1, bc2). There are severalundulators installed, two electromagnetic undulators for laser-based longitudinal diagnostics (ORS), sixpermanent-magnet devices acting as one 27 m long undulator for SASE, and an electromagnetic wiggler(IR) producing infrared radiation for diagnostics purposes. Also shows is LOLA, a transversely deflectingrf cavity, and two stations for electro-optical sampling (EO). During accelerator tests, the beam may besent through a bypass in order to protect the permanent-magnet undulators against radiation.

4.6. FLASH a SASE FEL for soft x-rays

FLASH is the first representative of a free-electron laser using the SASE princi-ple at short wavelengths. Initially, this machine was intended to be a testbed forsuperconducting rf cavities developed by the TESLA collaboration for a futurelinear collider (hence its original name TTF, TESLA Test Facility), where a SASEFEL was considered as one potential application of the test linac (51). In 2000,


Figure 16. The photocathode of the electron gun is embedded in a 1-1/2-cell rf structure. When a UVlaser pulse hits the photocathode (a), the accelerating electric field is at a maximum. It is zero, whenthe liberated electrons pass through the iris (b), and assumes a maximum with opposite sign, when theelectrons are at the center of the next cell (c). A longitudinal magnetic field from a solenoid coil (notshown) surrounding the cavity helps to keep the beam emittance small.

it was the first high-gain FEL operating at 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 the to-date shortest wavelengthof 6.5 nm was reached. Other FELs that are now coming into operation, closelyfollowing the design of FLASH, which is shown in Fig. 15 and will be described inthis Section.A SASE FEL comprises a source of low-emittance electron bunches, a linear

accelerator, some means to compress the bunches to sub-100 fs duration, a longundulator, and finally an electron beam dump and photon beamlines with experi-mental stations. Furthermore, beam diagnostics plays an important role and differsconsiderably from that of conventional synchrotron light sources.

4.6.1. The electron source

The SASE process requires a small normalized emittance (a few 106 radm)and high peak current (more than 1 kA) of the electron beam. The high currentis achieved by a 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 severalpicoseconds using a photocathode illuminated by a UV laser pulse. The laser system(52) developed at the Max-Born-Institute in Berlin and presently employed atFLASH comprises a Nd:YLF oscillator and four Nd:YLF amplifier stages (twopumped by diodes, two by flashlamps). A combination of an LBO and a BBOcrystal converts the 1047 nm pulses down to a wavelength of 262 nm, which isrequired to liberate electrons from the CsTe photocathode. Pulses with an energyof up to 50 J are generated with a temporal pattern given by the linac duty cycle:a train of 1 to 800 pulses at a rate of 1 MHz, and up to 10 pulse trains per second.In order to meet the emittance requirements, Coulomb repulsion of the electrons

must be minimized by immediately accelerating them, since the repelling forcedecreases with increasing Lorentz factor like 1/2. To this end, the photocathodeis placed inside an rf cavity which is directly followed by the first linear acceler-ator module. A solenoid field surrounding the photocathode focuses the electronstransversely. The laser pulses are shaped as to make the radial electron distribu-tion rectangular rather than Gaussian. An ideal rectangular shape would resultin a repelling force that increases linearly with radius, which in turn could becounteracted by quadrupole magnets.Instead of using a photocathode gun, SASE was demonstrated in 2004 at a

wavelength of 49 nm using a thermionic gun (50) at SCSS in Japan. Yet anotherapproach to produce low-emittance electron bunches, which is currently investi-gated (53), could be field emission from a microscopic conical object or an arrayof them, possibly assisted by photo emission.

30 S. Khan

Figure 17. Schematic view of a bunch compressor chicane formed by four dipole magnets, in which elec-trons with higher energy travel a shorter path than electrons with lower energy. Also shown are longitudinalphase space distributions (E/E versus z, ideal case in light gray) and corresponding charge densities (z)before (left) and after compression (right). On the slope of the accelerating rf voltage, the electrons acquirean energy chirp, i.e. an energy variation along the bunch. The density distribution (z) is not to scale:even in the non-ideal case (dark gray), the electron density increases typically by a factor of 30.

4.6.2. The linear accelerator

Since the short duration and low emittance of the electron bunches would notbe preserved in a synchrotron, SASE FELs are based on linear accelerators, eitheremploying normal-conducting or superconducting rf structures as sketched in Fig.4. In contrast to the case of a constant current, the resistance of a superconductordoes not completely vanish in the presence of an rf field, see e.g. (54). It is neverthe-less much smaller in superconducting niobium at e.g. 2 K than for copper at roomtemperature. Therefore, the quality factor1 of a superconducting cavity is of theorder of 1010, compared to below 105 for copper cavities. On the other hand, for 1W of power lost in a superconducting cavity at 2 K, the cryogenic system requiresalmost 1 kW to keep the temperature constant. This together with the technolog-ical complexity and a fundamental limitation given by the critical magnetic fieldmakes the choice not so obvious. It required a committee of international expertsin 2004 to identify the superconducting TESLA technology as the best solution forthe International Linear Collider (55).At FLASH, six acceleration modules each accommodating eight 9-cell TESLA-

type cavities are presently used to accelerate the electron beam to 1 GeV. With acavity length of 1 m, the average electric field is about 20 MV/m. With a furtheroptimized cavity design and new techniques to clean the cavity surfaces, fieldsexceeding 50 MV/m have been demonstrated (56).The European X-ray FEL will employ TESLA cavities with a field of 21 MV/m.

LCLS at Stanford, on the other hand, is based on the normal-conducting SLAClinac, and some other projects, like the SCSS XFEL in Japan for example, haveopted 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 tominimize space charge effects that would otherwise blow up the beam emittance. Atypical value is 50 A, e.g. 0.5 nC with a duration of 10 ps. The FEL amplificationprocess, on the other hand, requires a peak current in the kA range. Therefore, the

1The quality factor Q = f/f is an important quantity to characterize an oscillating physical system. Itis the ratio of its resonant frequency and the width of the resonance curve. On the other hand, Q/2pi isthe ratio of the energy stored in the system and the energy dissipated per oscillation cycle.


bunches are compressed once they have reached a Lorentz factor 1.An electron bunch is compressed by causing electrons in its tail to catch up with

electrons in the head of the bunch. This is achieved by accelerating the bunchesat an rf phase slightly away from the peak of the sinusoidal cavity voltage (off-crest), such that the kinetic energy of tail electrons is higher than the energyof head electrons. This chirped electron bunch then passes a chicane formeddipole magnets, where the tail electrons catch up because their bending radius islarger and their path consequently shorter than that of the head particles. Beingclose to c, velocity differences between the electrons are negligible. The path lengthdifferences are approximately given by

l L2EE

, (48)

where L and are defined in Fig. 17. However, the compression is not perfect,since the energy acquired on the slope of the sinusoidal cavity voltage does notvary linearly along the bunch, as sketched Fig. 17. The resulting bunch comprisesa short head (a few 10 fs) with a high peak current, followed by a long tail (severalps), which carries most of the bunch charge (80-90%) but does not contribute tothe lasing process. In future, this can be greatly improved, when at third-harmonicvoltage is added to the accelerating rf voltage, linearizing the sum voltage experi-enced by the bunches.FLASH comprises two bunch compressors at different beam energies The reason

is that full compression in one stage at low energy would cause strong space chargeeffects, while one stage at high energy would require an undesirably large energyvariation within the bunch.The chicane magnets give rise to coherent synchrotron radiation (CSR). As the

bunch follows its curved path, CSR emitted by the bunch tail takes a straightshortcut and acts on the bunch head, leading to detrimental effects which can beseen both in simulations as well as in time-resolved images of the bunches (60).

4.6.4. Undulators

The SASE lasing process takes place in a long undulator with a nearly sinusoidalmagnetic field. The field may be produced by electromagnets or by permanentmagnets. In the latter case, the magnets may be outside (as at FLASH) or insidethe vacuum vessel in order to achieve a higher field for a given free aperture (as atSCSS in Japan). The gap between opposite magnetic structures may be variableto change the field on the beam axis and tune the radiation wavelength, or itmay be fixed. At FLASH, a hybrid structure of iron poles with NdFeB magnetsbetween the poles is employed. The undulator period (from one pole to the nextlike-sign pole) is u = 27 mm, the fixed magnetic gap is 12 mm, resulting in afield amplitude of 0.47 T at the beam. The corresponding undulator parameter isK = 1.2, leading with Eq. (23) to a wavelength of about 6 nm at a beam energyof 1 GeV ( = 1957). In order to achieve saturation of the lasing process, six 4.5m long undulators are employed, interleaved with 0.6 m long sections for focusingand beam diagnostics.Degrading of the permanent magnets by radiation is avoided by sending the

electron beam through a bypass (see Fig. 15) during accelerator studies, whenlasing is not required.Apart from the SASE undulator, three electromagnetic undulators are presently

installed at FLASH for diagnostics purposes, two undulators with 5 periods each(u = 200 mm) tunable to 800 nm (the wavelength of Ti:sapphire lasers) for the

32 S. Khan

Figure 18. Longitudinal bunch diagnostics using a transversely deflecting cavity. A fast kicker magnetdirects the vertically tilted bunch onto a screen, where optical transition radiation (OTR) is emitted andimaged by a CCD camera. A sample picture and its projection onto the time axis is shown in the rightpart of the figure (60).

optical-replica experiment (see below), and one undulator with 9 periods (u = 400mm) for infrared radiation in the range of 1-200 m.In addition, 10 m of permanent-magnet undulators with variable gap are foreseen

for seeding the FEL process with a high laser harmonic (HHG seeding is describedin Sect. 4.7).

4.6.5. Beam diagnostics

In order to maintain and improve the quality of FEL radiation, the parametersof the electron beam must be measured with high precision. The beam energyis known from the beam path through dipole magnets and from measuring theradiation wavelength. The beam current is measured using conventional currenttransformers. The beam position is either deduced by subtracting the signals fromopposite electrodes (beam position monitors), where a passing charge induces acurrent pulse, or by placing a screen in the beam path and imaging the lightspot created by fluorescence or by optical transition radiation (OTR). The beamemittance, 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 peak current (i.e. very short bunches) and low emittance, there is an addi-tional and very challenging demand to resolve the temporal structure of a singlebunch and to measure the variation of the emittance along the bunch (the so-calledslice emittance). Since longitudinal diagnostics is essential for understanding andcontrolling the amplification process in FELs, some methods will be outlined inthis paragraph.When the electric field of an electron bunch sweeps over a crystal like GaP,

it induces a birefringent effect on a laser pulse traversing the crystal at the sametime. This way, the shape of the bunch can be encoded in the intensity of that laserpulse after passing a polarizer. This technique is known as electro-optical sampling(EOS) and several single-shot methods to extract the encoded information havebeen devised at FLASH and elsewhere, reaching the intrinsic time resolution ofabout 50 fs (57)).A transversely deflecting rf cavity operating at 2.9 GHz was originally used at

SLAC (58) and is now employed at FLASH to kick head and tail of a single electronbunch vertically in opposite direction. This particular bunch is directed by a fastkicker magnet onto an OTR screen, where its charge distribution appears alongthe vertical axis (in a similar fashion an in an oscilloscope or streak camera).The horizontal distribution reflects the shape of the bunch, from which the slice


emittance can be deduced. A temporal resolution of 20 fs was achieved (59, 60).Bunch structures in the femtosecond range can be studied by measuring the spec-

trum of transition radiation (61) or synchrotron radiation (62). Coherent emissionoccurs at wavelengths comparable or longer than the structure. There is no intrin-sic limit of the time resolution, but the relationship between the observed spectrumand the temporal distribution is ambiguous, since only the radiation intensities andnot the phases are measured.Another novel method to obtain the longitudinal charge distribution with a res-

olution of a few fs, the optical-replica synthesizer (ORS) (63), was implementedat FLASH in 2007 and is presently being commissioned (64). Here, a Ti:sapphirelaser pulse at 800 nm wavelength interacts with an electron bunch in an undula-tor, creating a periodic energy modulation. Passing a magnetic chicane, the energymodulation is converted into a density modulation, which gives rise to coherentemission in a second undulator at the laser wavelength or harmonics thereof. Theresulting coherent light pulse is a replica of the electron bunch, and its shape canbe determined uniquely and with excellent time resolution using a commerciallyavailable FROG (frequency-resolved optical gating) device (65).

4.6.6. Synchronization

At an FEL user facility, many components must be synchronized with respectto each other on a very precise level, which ultimately limits the time resolutionof experiments at the photon beamline. These components include the rf photoin-jector, the accelerating structures, diagnostic devices, a seed laser (in the case of aseeded FEL, see below) and the laser 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 adistribution system, based on optical fibers with length stabilization (66). The fiberlaser is superior to microwave 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 fibercan be controlled by measuring the arrival time of reflected signals and by mechan-ically changing the fiber length using a piezo stretcher. The reliable distributionof timing signals over kilometers with high precision is particularly challenging forlarge machines like the European X-ray FEL, and FLASH is an ideal testbed forthis task.

4.6.7. Present status and results

In 2007, a radiation wavelength of 6.5 nm was achieved (4) with a beam energyslightly below 1 GeV. Exponential gain with increasing undulator length untilsaturation was experimentally verified by kicking the beam with magnets installedbetween the undulator modules and thus disrupting the amplification process atmultiples of 4.5 m.At 13.7 nm wavelength as a well-studied example (67), the average pulse energy is

40 J. The pulse energy of the third and fifth harmonic is 250 nJ and 10 nJ, respec-tively. Starting from noise, SASE is subject to random flu

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