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Lasers, Free-Electron

Claudio Pellegrini and Sven ReicheUniversity of California, •Los Angeles, CA 90095-1547, USA e-mail: [email protected];Q1

[email protected]

AbstractFree-electron lasers are radiation sources, based on the coherent emission of synchrotronradiation of relativistic electrons within an undulator or wiggler. The resonant radiationwavelength depends on the electron beam energy and can be tuned over the entirespectrum from micrometer to X-ray radiation. The emission level of free-electronlasers is several orders of magnitude larger than the emission level of spontaneoussynchrotron radiation, because the interaction between the electron beam and theradiation field modulates the beam current with the periodicity of the resonant radiationwavelength. The high brightness and the spectral range of this kind of radiation sourceallows studying physical and chemical processes on a femtosecond scale with angstromresolution.

Keywordsfree-electron laser; undulator; microbunching; SASE FEL; FEL oscillator; FELamplifier; FEL parameter; gain length.

1 Introduction 22 Physical and Technical Principles 52.1 Undulator Spontaneous Radiation 62.2 The FEL Amplification 72.3 The Small-signal Gain Regime 102.4 High-gain Regime and Electron Beam Requirement 112.5 Three-dimensional Effects 13

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2 Lasers, Free-Electron

2.6 Longitudinal Effects, Starting from Noise 142.7 Storage Ring–based FEL Oscillators 163 Present Status 173.1 Single-pass Free-electron Lasers 173.2 Free-electron Laser Oscillators 184 Future Development 20

Glossary 21References 23Further Reading 24

1Introduction

A free-electron laser (FEL) [1] transformsthe kinetic energy of a relativistic electronbeam produced by a particle acceleratorlike a microtron, storage ring, or radiofrequency (RF) linear •accelerator intoQ2

electromagnetic (EM) radiation. The trans-formation occurs when the beam goesthrough an alternating magnetic field, pro-duced by a magnet called an undulator [2],that forces the electrons to move in anoscillatory trajectory about the axis of thesystem, as shown in Fig. 1. An electromag-netic wave propagates together with theelectron beam along the undulator axis,and interacts with the electrons.

The undulator magnet is a periodicstructure in which the field alternatesbetween positive and negative values andhas zero average value. It produces anelectron trajectory having a transversevelocity component perpendicular to theaxis and parallel to the electric field of thewave, thus allowing an energy exchangebetween the two to take place. One caneither transfer energy from the beam to thewave, in which case the device is an FEL, orfrom the wave to the beam, in which case itis an inverse FEL (IFEL) [3]. In the secondcase, the system is acting like a particle

accelerator and can be used to acceleratethe electron beam to higher energies.

The energy transfer can take place onlyif a condition of synchronism between thewave and the beam oscillations is satisfied(Sect. 2.2). This condition gives a relation-ship between the radiation wavelength λ,the electron beam velocity βz along the un-dulator axis (measured in units of the lightvelocity c), and the undulator period λ0:

λ = λ0(1 − βz)

βz(1)

For relativistic electrons, this condition canalso be written in an approximate, butmore convenient, form using the beamenergy γ (measured in the rest energyunits E = mc2γ ):

λ =(

λ0

2γ 2

)(1 + a2

w). (2)

The dimensionless quantity

aw = eB0λ0

2πmc(3)

is the undulator vector potential normal-ized to the electron rest mass mc2; e is theelectron charge and B0 is the undulatorpeak magnetic field. (Here, and in the restof the paper, we use MKS units.) The quan-tity aw is called the undulator parameter,

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Lasers, Free-Electron 3

Undulator

Electronbeam

Optical cavity mirror

Partialmirror

Radiation field

Outcoupledfield

Fig. 1 Schematic representation of an FEL oscillator, showing its main component: the electronbeam, undulator magnet, and optical cavity. The undulator shown is a permanent-magnetplanar undulator. The arrows indicate the magnetic field direction. In an FEL amplifier of SASEFEL setup the optical cavity is omitted. The FEL amplifier is seeded by an external radiation field

which is the normalized vector potential ofthe undulator field. The undulator can beof two types: helical and planar (Sect. 2).Equation 2 is valid for a helical undulator.In the case of a planar undulator, Eq. 2is still valid if we replace the undulatorparameter with its rms value aw/

√2.

Because of the dependence of the radi-ation wavelength on the undulator period,magnetic field, and electron-beam en-ergy – quantities that can be easily andcontinuously changed – the FEL is a tun-able device that can be operated over avery large frequency range. At present, therange extends from the microwave to theUV [4]. A new FEL is now under construc-tion in the United States [5] to reach theX-ray region, about 0.1 nm. A similar pro-gram is being developed in Germany [6],and other FEL to cover the intermediateregion between 0.1 nm and the UV [7] arealso being considered by several countries.

The efficiency of the energy transferfrom the beam kinetic energy to the EMwave is between 0.1 to a few percent formost FELs, but it can be quite large, up toabout 40%, for specially designed systems.The beam energy not transferred to theEM wave remains in the beam and canbe easily taken out of the system, to bedisposed of, or recovered elsewhere. This

fact suggests that high- to average-powerFELs can be designed without the problem,common in atomic and molecular lasers,of heating the lasing medium.

The time structure of the laser beammirrors that of the electron beam. De-pending on the accelerator used, one candesign systems that are continuous-wave(cw) or with pulses as short as picosecondsor subpicoseconds.

Tunability, high efficiency, and timestructure make the FEL a very attractivesource of coherent EM power. In somewavelength regions, like the X-ray, theFEL is unique. Its applications range frompurely scientific research in physics, chem-istry, and biology to military, medical, andindustrial applications.

FELs originate in the work carried outin the 1950s and 1960s on the generationof coherent EM radiation from electronbeams in the microwave region [2, 8].As scientists tried to push power sourcesto shorter and shorter wavelengths, itbecame apparent that the efficiency ofthe microwave tubes, and the powerthey produced, dropped rapidly in themillimeter region. It was then realized thatthis problem could be overcome by usingan undulator magnet to modify the beamtrajectory [1], making it possible for the

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beam to interact with a wave, away fromany metallic boundary. Two pioneeringexperiments at the Stanford University [9,10] proved that the FEL is a useful sourceof coherent radiation.

The current disadvantage of FELs is thegreater complexity and cost associated withthe use of a particle accelerator. For thisreason, the use and development of FELsare mainly oriented to the following:

1. portions of the EM spectrum, like thefar infrared (FIR), or the soft andhard X-ray region, where atomic ormolecular lasers are not available orare limited in power and tunability;

2. large-average power, high-efficiencysystem.

An order-of-magnitude comparison ofthe peak brightness obtained or expected

TESLASASE FELs

DESYTTF-FEL(seeded)

DESYTTF-FEL

Spontaneous spectrumSASE FEL 4

TTF-FELspontan

BESSY-IIU-125

BESSY-IIU-49

PETRAundulator

APSundulator(Typ A)

ALS U5.0

TESLAspontaneous

undulator

SPring8undulator

(30m in vacuum)

ESRF-undulator(ID23)

Spontaneous spectrumSASE FEL 1

LCLS

15 GeV30 GeV

101

1035

1033

1031

1029

1027

1025

1023

1021

1019

102 103 104 105 106

Energy (eV)

Pea

k br

illia

nce

[Pho

t./(s

ec m

rad2

mm

2 0.

1% b

andw

.)]

Fig. 2 Peak brightness of free-electron lasers in the VUV and X-ray regime,compared to 3rd generation light sources [11]

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for the FEL, compared to other sources ofEM radiation, is given in Fig. 2, pointingout again the interest for FELs in the soft-to-hard X-ray regions.

In Sect. 2 of this article, we discussthe general physical principles of FELsand the properties and characteristics oftheir main components, that is, the un-dulator magnets, electron beams, andoptical systems. We also describe themain properties of the spontaneous ra-diation from an undulator, the processleading to the amplification of radia-tion, the high-gain and small-signal-gainregimes, diffraction effects, saturationeffects, the limits on the system effi-ciency, the dependence of gain on beamparameters.

In Sect. 3, we give a short review of theexperimental results obtained to date forvarious FELs. The future development ofFELs is described in Sect. 4.

2Physical and Technical Principles

An FEL has three fundamental compo-nents: an electron beam of given energyand intensity and the associated acceleratorused to produce it; the undulator magnet;and the EM wave and the associated opticalcomponents controlling its propagation.A schematic representation of an FEL isgiven in Fig. 1.

The accelerators used to provide the elec-tron beam are of many types: electrostatic,induction line, radio-frequency (rf) linearaccelerator, pulsed diode, or storage rings.Some of their basic characteristics, theirenergy ranges, and the FEL wavelengthsfor which they are more commonly usedare given in Table 1.

Undulator magnets are of two maintypes: helical or planar. In the first case,the magnetic field vector rotates aroundthe axis as a function of axial distance;in the second case, its direction is fixed,and its amplitude oscillates along the axis.These magnets can be, and have been,built using a wide variety of technologies:pulsed or DC electromagnets, permanentmagnets, and superconducting magnets.The field amplitude can vary from afraction of a tesla to over 1 T, and theperiod from 1 cm to many centimeters.A great effort is under way to developundulators with periods in the millimeterrange; these would allow a reduction ofthe beam energy for a given radiationwavelength, thus reducing the cost andcomplexity of the FEL. A helical undulatorwas used by Madey and coworkers [9]in the first FEL. This undulator was asuperconducting magnet, built with twohelical windings with current flowing inopposing directions.

The FEL can be operated as an oscillator,using an optical cavity to confine the radi-ation in the undulator region, as shown in

Tab. 1 Particle accelerators for FELs

Energy Peak current Pulse length FEL wavelength

Electrostatic 1–10 MeV 1–5 A 1–20 µs mm to 0.1 mmInduction linear

accelerator1–50 MeV 1–10 kA 10–100 ns cm to µm

Storage ring 0.1–10 GeV 1–1000 A 30 ps–1 ns 1 µm to nmRF linear accelerator 0.01–25 GeV 100–5000 A 0.1–30 ps 100 µm to 0.1 nm

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Fig. 1, or it can be used as an amplifier.Oscillator–amplifier combinations [mas-ter oscillator, power amplifier (MOPA)],as well as systems in which one amplifiesthe electron spontaneous radiation emittedin traversing the undulator [self-amplifiedspontaneous radiation (SASE) [12]], havebeen used.

The performance characteristics of someof the existing FELs are discussed in detailin Sect. 3.

2.1Undulator Spontaneous Radiation

For this discussion, we use a referenceframe with z along the undulator axisand x and y perpendicular to z. For ahelical undulator, the field near the axis zis given by

�B = B0

[�ex cos

(2π

z

λ0

)

+ �ey sin(

2πz

λ0

)], (4)

where �ex, �ey, and �ez are unit vectors alongthe x, y, and z axes. The electron velocityin this field, in units of c, is

�β = aw

γ

[�ex sin

(2π

z

λ0

)

+ �ey cos

(2π

z

λ0

)]+ �ezβz. (5)

The amplitude of the transverse velocityis aw/γ . This quantity also representsthe maximum angle between the electrontrajectory and the undulator axis. Forrelativistic beams, this quantity is muchless than 1, and the longitudinal velocity isnear 1. Notice that in the case of a helicalundulator, the axial component of thevelocity, βz, remains constant. For a planarundulator, only one of the two transversecomponents in Eqs. 4 and 5 is nonzero. In

this case, the longitudinal velocity is notconstant; in fact, it can be written as thesum of a constant term plus one oscillatingat twice the undulator period. The electrontrajectory in a helical undulator is a helix ofradius awλ0/2πγβz and period equal to theundulator period. For a planar undulator,it is a sinusoid in the plane perpendicularto the magnetic field.

When traversing the undulator, the elec-trons are subject to an acceleration andradiate EM waves. This spontaneous emis-sion is fundamental to the operation ofan FEL. One relativistic electron traversingan undulator magnet emits radiation in anarrow cone, with an angular aperture oforder 1/γ [13]. The radiation spectrum is asuperposition of the synchrotron radiationemitted when the electron trajectory is bentin the undulator field, a narrow line intro-duced by the periodic nature of the motionin the same field, and its harmonics. Thewavelength of the fundamental line is ofthe order of λ0/2γ 2, which can be inter-preted as the undulator period λ0 doublyDoppler shifted by the electron motion.

The aperture 1/γ of the synchrotronradiation cone can be compared withthe angle between the electron trajectoryand the undulator axis, aw/γ . If aw issmaller than 1, the emitted radiation iscontained within the synchrotron radiationcone, and the emission is predominantlyin a single line. If aw is larger than 1,the synchrotron radiation cone sweeps anangle larger than its aperture. In thiscase, the spectrum is rich in harmonicsand approaches the synchrotron radiationspectrum when aw is very large. Whenaw � 1, the magnet is usually referred toas a wiggler, reserving the name undulatorfor the case aw less than or on the orderof one. In this article, we use the genericterm ‘‘undulator’’ to describe a magnet

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with a periodic transverse field describedby Eq. 4.

The fundamental wavelength seen by anobserver looking at an angle θ to the undu-lator axis is λ = (λ0/2γ 2)(1 + a2

w + γ 2θ2),and is the wavelength at which there ispositive interference between the radiationemitted in two undulator periods. This isalso, when θ = 0, the wavelength corre-sponding to the ‘‘synchronism condition’’Eq. 1.

In a planar undulator, the magneticfield has only one component, say alongy. Then the velocity has components inthe x and z directions. The wavelength ofthe spontaneous radiation is the same asthat for a helical undulator if the vectorpotential aw is replaced by its rms value,aw/

√2.

In a helical undulator, the radiationon axis contains only the fundamentalharmonic given by Eq. 1. In a planarundulator, the longitudinal velocity ismodulated twice at the undulator period.This leads to a richer spectrum: all theodd harmonics of Eq. 1 appear on axis.For both type of undulators, even and oddharmonics appear off axis.

The linewidth of the radiation is deter-mined by the number Nw of undulatorperiods; the total length of the pulse emit-ted by one electron is that of a wave trainwith a number of periods equal to Nw, thenumber of undulator periods. The corre-sponding linewidth of the fundamental is

�ω

ω= 1

2Nw. (6)

When more than one electron is producingthe radiation, the pulse length and angulardistribution can be affected by the electron-beam characteristic. The pulse length isincreased by the electron bunch lengthwhen this is larger than Nwλ. The radiationlinewidth is increased if the beam energy

spread is larger than 1/2Nw, the singleelectron width. Similarly, the effectivesource radius and angular distribution canbe increased by the electron beam. Thiseffect can reduce the FEL gain, as we willdiscuss in the next section.

2.2The FEL Amplification

Undulators are widely used in storagerings as brilliant sources of synchrotronradiation. In this case, the superposition ofspontaneous radiation of many electronshas no phase correlation, so that the totalintensity is proportional to the electronnumber Ne. In an FEL, as in an atomiclaser, there is a phase correlation betweenemitting electrons. This correlation isobtained by modulating the longitudinalbeam density on the scale of the radiationwavelength, a process called bunching.When electrons are bunched together in•a distance that is short compared to the Q3

wavelength of the radiation, they emit inphase, thus increasing the intensity in theemitted line and simultaneously reducingits width.

The formation of bunching is caused bythe interaction between the electrons andthe radiation field. It can be summarizedby three actions, fundamental to any FELprocess:

1. the modulation of the electron energydue to the interaction with the radiationfield,

2. the change in the longitudinal positionof the electrons owing to the path lengthdifference of the trajectories within theundulator of electrons with differentenergies,

3. the emission of radiation by the elec-trons and, thus, growth in the radiationfield amplitude.

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Each point will be explained in detail be-low. The FEL amplification couples thesebasic processes and exploits the inherentinstability of this system. A stronger mod-ulation in the electron density increasesthe emission level of the radiation, which,in return, enhances the energy modulationwithin the electron bunch. With strongerenergy modulation, the formation of thebunching (modulation in the electron po-sitions) becomes even faster. The FELamplification is stopped, when the electrondensity modulation has reached a maxi-mum. At this point, all electrons have thesame phase of emission and the radiationis fully coherent. The free-electron laserhas reached saturation.

The FEL process is either initiated by aseeding radiation field or by the inherentfluctuation in the electron position at theundulator entrance. The first classifies thefree-electron laser as an FEL amplifier,while the latter is called a self-amplifyingspontaneous radiation (SASE) FEL, wherethe initial radiation level is produced bythe spontaneous radiation, as described inSect. 2.1•.Q4

In the following, we describe the basicFEL process. We start with a simplifiedmodel, where transverse effects suchas the diffraction of the radiation fielddue to its finite size is ignored. Thediscussion of these three-dimensionaleffects is postponed till Sect. 2.5. Let

�E = E0

[�ex cos

(2π

z

λ− ωt + �

)

+ �ey sin(

2πz

λ− ωt + �

)](7)

be the electric field of a plane wavepropagating along the undulator axis. Theinitial phase of the radiation field at t,z = 0 is �. This field is transverse to theundulator axis, and thus has componentsparallel to the transverse electron velocity,

as given by Eq. 5. An energy transferbetween the EM field and the electronbeam can take place. Using Eqs. 5 and 7,one finds

mc2 dγ

dt= eE0

aw

γsin ( + �) , (8)

where the phase is

= 2π

(1

λ+ 1

λ0

)z − ωt + 0. (9)

This quantity is called the ponderomotiveforce phase [14] and plays a key role in FELphysics. The ponderomotive force phasehas terms changing at the ‘‘fast’’ EM fieldfrequency and terms changing with the‘‘slow’’ undulator period. The time-averagevalue of the electron energy change (8),averaged over a few radiation periods,is 0 unless we satisfy a ‘‘synchronismcondition’’ d/ dt = 0. This conditioncan be written as λ = λ0(1 − βz)/βz withβz as the ‘‘resonant’’ velocity for agiven undulator period and radiation fieldwavelength. In the case of relativisticparticles, using the relationship βz =(1 − 1/γ 2 − β2

x − β2y )1/2, and assuming

γ � 1, βz, βy � 1, this condition canbe approximately written as λ = λ0(1 +a2

w)/2γ 2. These are Eqs. 1 and 2, givenin the introduction. If the synchronismcondition is satisfied, the ponderomotiveforce phase is constant, and the electronbeam, oscillating at the slow undulatorfrequency, can exchange energy with thefast oscillating EM wave.

If the longitudinal velocity βz differsfrom the resonant velocity βz, the electronslips in the ponderomotive force phase as

d

dt= 2πc

λ

(βz

βz− 1

). (10)

If the electron moves with the resonantvelocity, its ponderomotive force phase is

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constant according to the synchronismcondition. Particles with higher energymove forward with respect to the initialphase 0, while particles with lower energyfall back.

The distribution of the initial positionof the electrons within the bunch is de-termined by the electron source and theacceleration process. If the electron beamis produced by an accelerator using RFcavities to accelerate the beam, the dis-tribution is characterized by two scalelengths. One is the RF wavelength; theother is the radiation wavelength. In anFEL, the first is much larger than the sec-ond. The wavelengths used in RF cavitiesvary from a minimum of about 10 cmfor an RF linear accelerator to infinity foran electrostatic accelerator. At the scalelength defined by the RF system, the elec-trons are distributed longitudinally intobunches separated by one RF wavelength.The length of these bunches is a frac-tion of the wavelength of the acceleratingRF and varies from about 1 mm for thecase of a 3-GHz (λrf = 10 cm) RF linearaccelerator to a continuous beam for theelectrostatic case.

At the scale length of the radiation wave-length, the electron distribution is in goodapproximation uniform; there is no cor-relation between the longitudinal electronpositions at the scale of the radiation wave-length λ except for a residual, but smallbunching owing to the finite number ofelectrons per radiation wavelength. It rep-resents the spontaneous radiation level atthe given wavelength λ, which is amplifiedin an SASE FEL. Section 2.6 discusses indetail the properties of this device.

If the electrons have the resonantvelocity or are close to it, the interactionwith the radiation field results in asinusoidal modulation of the electronenergy (Eq. 9). Electrons, lying within the

phase interval [�, � + π ], gain energyand become faster ( d/ dt > 0), whileelectrons in the interval [� + π , � + 2π ]are slowed down ( d/ dt < 0). As a result,all electrons tend to drift toward thephase � + π and become bunched withthe periodicity of the radiation field. Theelectrons emit in phase and the radiationis coherent.

The radiation emitted by the electronsadds up to the interacting EM wave. Theamplitude E0 and the phase � of the EMfield change with the ongoing interactionbetween radiation field and electron beam.This is apparent because the electronstend to bunch at a phase different tothe radiation field. The radiation phase isslowly adapting to the new phase definedby the bunching phase of the electrons.For a self-consistent description of theFEL interaction, the Maxwell equationfor the EM wave has to be solved aswell. If the fast oscillation of the EMwave exp[i(2π/λ)z − iωt] is split fromthe amplitude and phase information,the complex amplitude E ≡ E0 exp[i�]changes slower than the ‘‘slow’’, transverseoscillation of the electrons (Eq. 5). Thechange is given by [15]:

(∂

c∂t+ ∂

∂z

)E = i

µ0

2aw

∑j

e−ij

γj(11)

with µ0 as the magnetic permeability andthe sum over all electrons. Equations 8,10, and 11 are the mathematical represen-tation of the three basic processes, listed atthe beginning of this chapter. Note that theright-hand side of Eq. 11 is determined bythe degree of bunching in the electron den-sity and is at maximum when all electronphases j are identical. It is also clear fromthe same equation that if more electronsare contributing to the FEL process, thegrowth in the radiation field is stronger.

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The strength of this coupling scales withthe FEL parameter [16]

ρ =(

aw

4γ

ωp

ω0

) 23

, (12)

where ωp = (4πrec2n0/γ )1/2 is the beamplasma frequency, n0 is the electron-beamdensity, and ω0 = 2πc/λ0. Typical valuesof ρ vary from 10−2 for a millimeter FELto 10−3 or less for a visible or UV FEL.

The solutions for Eqs. 9, 10, and 11are of type E ∝ exp[i�2ω0ρt], where thegrowth rate � depends on the electrondistribution in energy and the deviationof the mean energy from the resonance

energy γR =√

(λ0/2λ)·(1 + a2w). For the

simplest case of a mono-energetic beamwith <γ> = γR, � is the solution ofthe dispersion equation �3 = −1 [17].Although energy deviation or finite energyspread adds additional terms, the resultingdispersion equation is always cubic for thisone-dimensional model.

Of particular interest for the FEL processis the solution of the dispersion equation,which has a negative imaginary part. Itcorresponds to an exponentially growingmode. The maximum growth rate inthis one-dimensional model is −�e(i�) =√

3/2. Inserted into the •ansatz for theQ28

radiation field amplitude E, the e-foldinglength of the amplification is

Lg = λ0

2π√

3ρ, (13)

which is also referred to as the gainlength [18]. Besides the exponentiallygrowing mode, the other modes are eitheroscillating or decaying. In the beginning ofthe free-electron laser, all modes are of thesame magnitude and it requires a few gainlengths before the exponentially growingmode dominates. This retardation in the

amplification of the seeding field is calledlethargy regime [16].

The growth in the radiation field am-plitude stops at saturation, when themodulation in the electron distributionhas reached a maximum. At this point,the bunch is broken up into multiplemicrobunches. The spacing of the mi-crobunches is the radiation wavelength.Because the wavelength is tunable withthe electron beam energy, the free-electronlaser is also a device to modulate the cur-rent of a long pulse with free control onthe periodicity. This property is useful forsome experiments on plasma-beam inter-action [19].

2.3The Small-signal Gain Regime

The first FEL experiments had an enhance-ment of a few percent of the seedingradiation field. In this low-gain regime,the undulator length is shorter than thegain length. All modes – growing, decay-ing, and oscillating – have approximatelythe same amplitude and the radiationpower at the end of undulator as a resultof the interference of these modes. If thewavelength of the seeding radiation fieldfulfills the synchronism condition, the in-terference is completely destructive andthe gain – the relative change in the radi-ation field intensity – is zero. The growthrate of each mode depends differently onthe deviation from the synchronism con-dition (Eq. 1) and a slight deviation �ω

results in an enhancement of the EMfield.

The small-signal gain was first evalu-ated by Madey [1]. Although his originaltheoretical discussion of the FEL used aquantum-mechanical description, it wassoon realized that under most conditions,

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a classical description is sufficient to de-scribe the system, since the most relevantquantities, such as gain, Eq. 14, are inde-pendent of the Planck constant.

When the initial EM field amplitudeis small – and the relative change in theintensity is also small – the gain is calledthe small-signal gain. For an initially mono-energetic electron beam, the small-signalgain, calculated by Madey, is

Gs = 4√

2π

√λλ0

w20

a2w√

(1 + a2w)3

Ip

IAN3

w

× d

d(x/2)

sin2(x/2)

(x/2)2 , (14)

where x = 2πNw�ω/ω, Nw is the numberof undulator periods, Ip is the beamcurrent, IA = ec/re is the Alfven current(about 17 kA), re is the classical electronradius (re ≈ 2.8 × 10−15 m), and πw2

0 isthe transverse cross section of the EMwave (assumed to be larger than that of theelectron beam).

The gain curve in Eq. 14 is proportionalto the derivative of the spectrum of thespontaneous radiation. The gain is zero forx = 0, at synchronism, and is maximumfor x 2.6. The linewidth is on the orderof 1/Nw, the inverse of the number ofperiods in the undulator.

Since the EM radiation frequency andthe electron energy are related by thesynchronism condition, Eq. 2, we canalso redefine x, for a fixed frequency,as x = 4πNw�γ/γ . The gain linewidththen corresponds to an energy change�γ/γ on the order of 1/2Nw. Whenthe radiation field increases, the beamkinetic energy decreases; and when thechange is about 1/2Nw, the gain becomes0. Hence the maximum FEL efficiency,defined as ratio of the laser intensity to theinitial beam kinetic energy, is on the orderof 1/2Nw.

2.4High-gain Regime and Electron BeamRequirement

The FEL saturates when the electronbunch has achieved maximum modula-tion and all electrons radiate coherently.Early free-electron lasers could not reachsaturation with a single pass through anundulator. The gain can be accumulatedwhen the undulator is enclosed by an opti-cal cavity and the radiation field is reflectedback to seed the FEL for succeeding elec-tron bunches. On the other hand, opticalcavities restrict the FEL to radiation wave-length where the losses of the cavity aresmaller than the gain per single pass. Toavoid this restriction, the undulator lengthhas to be several gain lengths long to obtaina large gain and to reach saturation [18].

To reduce the overall undulator length,the gain length (Eq. 13) must be made asshort as possible. The undulator periodlength is typically limited to a fewcentimeters and found by optimizationof the gain length. Shorter values wouldreduce the undulator parameter and, thus,the coupling of the electron beam to theradiation field, while longer values wouldincrease the resonant wavelength (Eq. 2).The energy has to be increased to keepthe radiation wavelength constant, whichreduces the ρ-parameter.

The gain length depends on the electronbeam parameters as well. The gain lengthis reduced for a higher electron density,although there are limits on this, givenby the transverse velocity spread in theelectron beam. We discuss this andother three-dimensional effects in the nextsection.

The gain length of Eq. 13 is basedon a beam with no energy spread.Energy spread prevents bunching of allelectrons at the same ponderomotive force

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phase, because the electrons have differentlongitudinal velocities and the bunchingis smeared out. The bandwidth in energy,which contributes to the FEL performance,is given by the ρ-parameter. A largervalue means that the energy exchangebetween radiation field and electron beamis stronger. The undulator length, overwhich the synchronism condition must befulfilled, is shorter. This allows a largertolerance in the spread of the longitudinalvelocity and thus, in energy. To keep theincrease in the gain length reasonable, theinitial rms energy spread σγ must be

σγ

γR� ρ. (15)

Following the same argument, the devia-tion of the mean energy from the resonantenergy is limited to

∣∣∣∣<γ> − γR

γR

∣∣∣∣ � ρ. (16)

Note that in the high-gain regime, thegain is the largest if the beam energy ison resonance, as shown in Fig. 3. This isopposite to the small-signal gain regimewith no gain at <γ> = γR.

Only electrons that lie within the en-ergy acceptance bandwidth of the FELaround γR contribute to the FEL inter-action. The relative width is approximatelyρ. During the FEL amplification, energyis transferred from the electron beam tothe radiation field. The amplification stopswhen most of the electrons are droppedout of the bandwidth. After that the inter-action with the radiation field is negligiblebecause the synchronism condition is nolonger satisfied. Thus, the efficiency of theFEL, which is the relative amount of en-ergy transferred from the electron beamto the radiation field, is ρ. The satura-tion power of the radiation field [20] isgiven by

Psat = ρPb, (17)

−8 −6 −4 −2 0 2 4(<γ> − γR) /ρ<γ>

0.0

0.2

0.4

0.6

0.8

1.0

−Re(

iΛ)

Fig. 3 Growth rate of the FEL amplification as a function of the detuning–thedeviation of the mean energy of the electron beam from the resonant energy

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where Pb = <γ>Ipmc2/e is the electronbeam power and Ip is the peak current.Because of this scaling, it is of interestto maximize the ρ-parameter not onlyfor reducing the needed undulator lengthfor saturation but also for increasing theoutput power level of the FEL.

Because the FEL amplification is anexponential process, the major changein the electron energy occurs close tosaturation, when the electrons are pushedout of the FEL bandwidth. Although theFEL amplification stops at saturation,further energy can be extracted from theelectron beam. For this, the synchronismcondition has to be extended beyondsaturation. The radiation wavelength isfixed, so the undulator field or periodhas to be reduced to compensate for thebeam energy, which has been lost dueto the FEL interaction. This process togradually reduce the undulator field orperiod is called tapering. Although theenergy transfer during tapering is slowcompared to the exponential regime ofthe FEL, the process has no fundamentallimits beside the given length of theundulator. Experiments have shown that atapered system can enhance the efficiencyup to 40% [21].

2.5Three-dimensional Effects

The basic FEL process, as described inSect. 2.2, remains the same when three-dimensional effects are included in thediscussion, although the one-dimensionalFEL parameter ρ (Eq. 12) might notbe valid anymore. However, an effectivethree-dimensional FEL parameter can befound, which satisfies Eqs. 13, and 15to 17.

Two major effects contribute to theextended, three-dimensional model of the

FEL process, which are

1. the spread in the transverse velocity ofthe electron beam and

2. the diffraction of the radiation field.

A measure for the transverse spreadin the electron beam is the normalizedemittance εn [22], which is the area coveredby the electron beam distribution intransverse phase space. It is an invariantin linear beam optics. If the beam ismore focused, the spread in the transversemomenta is increased. Focusing alongthe undulator is necessary to preventthe growth of the transverse electronbeam size, reducing the electron densityand decreasing the gain. Electrons thatdrift away from the axis are deflectedback to the undulator axis. As a result,electrons perform an additional oscillationeven slower than the ‘‘slow’’ oscillationdue to the periodic magnetic field ofthe undulator (Eq. 5). Because part ofthe electron energy goes into this so-called betatron oscillation, the electron isslower in the longitudinal direction than anelectron without a betatron oscillation. Thespread in the betatron oscillation, whichscales with the normalized emittance, hasa similar impact as an energy spread.Thus, the requirement for the transverseemittance is

εn � 4λβ<γ>

λ0ρ, (18)

where the beta-function β [22] is a measurefor the beam size of the electron beam.Stronger focusing would increase theelectron density (β becomes smaller) andconsequently, the ρ parameter. But at thesame time, the impact of the emittanceeffect is enhanced, reducing the amountof electrons that can stay in phase with theradiation field. It requires the optimizationof the focusing strength to obtain the

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shortest possible gain length. A roughestimate for the optimum β-functionis β ≈ Lg ≈ λ0/4πρ. Inserting this intoEq. 18 shows that the beam emittanceε = εn/γ has to be smaller than the photonbeam ‘‘emittance’’ λ/4π . If this conditionis fulfilled, the electron beam does notdiverge faster than the radiation field andall electrons stay within the radiation field.

The diffraction of the radiation fieldtransports the phase information of theradiation field and the bunching of theelectron beam in the transverse direction.This is essential to achieve transverse co-herence of the radiation field, in particular,if the FEL starts from the spontaneous ra-diation (SASE FEL). SASE FELs do notguarantee the transverse coherence as aseeding laser signal of an FEL amplifierwould do. The betatron oscillation, whichyields a change in the transverse positionof the electron, contributes to the growthof transverse coherence as well.

As a degrading effect, the radiation fieldescapes from the electron beam in thetransverse direction. The field intensityat the location of the electron beam isreduced and the FEL amplification is in-hibited. The compensation for field lossesdue to diffraction is the FEL amplifica-tion itself. After the lethargy regime, theFEL achieves equilibrium between diffrac-tion and amplification. The transverseradiation profile becomes constant andthe amplitude grows exponentially. This‘‘quasi-focusing’’ of the radiation beam iscalled gain guiding [23] and the constantprofile of the radiation field is an eigen-mode of the FEL amplification.

Similar to the eigenmodes of the free-space propagation of a radiation field(e.g., Gauss–Hermite modes), there are aninfinite number of FEL eigenmodes [24].Each couples differently to the electronbeam and thus has a different growth

rate or gain length. That mode, which hasthe largest growth rate, dominates afterseveral gain lengths and the radiation fieldbecomes transversely coherent.

At saturation •gain, guiding vanishes Q5

and the electron beams radiate into mul-tiple modes. Typically, the fundamentalFEL-eigenmode is similar to that of free-space propagation and the resulting reduc-tion in transverse coherence at saturationis negligible.

The characteristic measures for diffrac-tion and FEL amplification are the Rayleighlength zR and gain length Lg. To calculatezR, we approximate the radiation size atits waist by the transverse electron beamsize � as the effective size of the radiationsource, resulting in zR = k�/2. For zR �Lg, the FEL amplification is diffraction-limited with a gain length significantlylarger than that in the one-dimensionalmodel (Eq. 13). In the opposite limit(zR � Lg), the one-dimensional model be-comes valid.

2.6Longitudinal Effects, Starting from Noise

The radiation field propagates faster thanthe electron beam and advances one ra-diation wavelength per undulator period.Thus, information cannot propagate fur-ther than the slippage length Nwλ, whereNw is the total number of undulator pe-riods. If the bunch is longer than theslippage length, parts of the bunch amplifythe radiation independently. While an FELamplifier is seeded by a longitudinal co-herent signal, this is not the case for anFEL, starting from the spontaneous emis-sion (SASE FEL). SASE FELs with electronbunches longer than the slippage lengthgenerate longitudinally incoherent signals.Further processing, succeeding the FEL, isrequired for fully longitudinal coherence

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(e.g., it can be achieved with a monochro-mator with a bandwidth smaller than theFourier limit of the electron bunch length).

An FEL amplifier has a coherent seedingsignal and the electron beam can be moreor less in resonance with it. The outputpower level depends on the deviation ofthe mean energy of the electron beam fromthe resonant energy γR. An SASE FEL, onthe other hand, uses the broadband signalof the spontaneous emission to start theFEL amplification. Independent of •theQ6

beam energy, the resonance conditionis always fulfilled, but the SASE FELamplifies all frequency components withinthe acceptance bandwidth of the FEL aswell. The relative width is �ω/ω = ρ andis typically much larger than the observedwidth of an FEL amplifier.

The initial emission level of the sponta-neous radiation depends on the fluctuationin the electron density. Because thereare only a finite number of electronsper radiation wavelength and the initialponderomotive force phase is random,electrons can cluster together at random.The radiation from parts of the bunch witha larger clustering will have larger inten-sity than that of other parts. The variationin these clusters and, thus, the fluctua-tion in the beam current, depend on thetotal number of electrons. Because of therandom nature of the electron positions,more electrons result in a smoother distri-bution. The chances for large clusters arereduced.

The effective power level, emitted be-cause of spontaneous radiation and thenfurther amplified by the FEL interaction,is called the shot noise power [25]. AnFEL amplifier, seeded with a power sig-nal smaller than the shot noise powerlevel does not operate as an FEL ampli-fier but as an SASE FEL instead. Typicalvalues are a few watts for free-electron

lasers in the IR regime to a few kilo-watts for FELs in the VUV and X-rayregime.

When the electron bunch length islonger than the slippage length, the ra-diation profile contains many spikes. Thespikes are also present in the frequencyspectrum. The origin is the randomfluctuation in the beam density. Theshot-to-shot fluctuation in the radiationpulse energy follows a Gamma distri-bution [26]. The only free parameter ofthis distribution, M, can be interpretedas the number of the spikes in theradiation pulse. The relative width ofthe distribution is 1/

√M. The length of

the spikes is approximately (λ/λ0)Lg [27].A shorter pulse would result in alarger fluctuation of the radiation en-ergy. The fluctuation of the instantaneouspower is given by a negative exponentialdistribution.

From the optical point of view, the elec-tron beam acts as a dielectric, dispersivemedium for the radiation field. The fieldamplitude E evolves as exp[i�2ω0ρt]. Theimaginary part of � results in the growth ofthe radiation field and thus, gain guiding,while the real part adds a phase advanceto the fast oscillating wave exp[ikz − iωt].The effective phase velocity differs from c,the speed of light. The electron beam isdielectric and the radiation field is focuseddue to the same principle of fiber opticcables. In addition, � depends on the de-viation of the radiation frequency fromthe frequency that fulfills the synchro-nism condition in Eq. 1. This dispersionreduces the group velocity below the speedof light [28] and spikes advance less thanone radiation wavelength per undulator pe-riod. Amplification stops at saturation andthe ‘‘dielectric’’ electron beam becomesnondispersive.

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2.7Storage Ring–based FEL Oscillators

Free-electron lasers are mainly based uponlinear accelerator and storage rings. Elec-tron beams, provided by linear acceler-ators, have the beam quality needed toreach saturation in a single pass. Becausethis setup exhibits a strong amplification,the single-pass FEL has properties that area disadvantage in comparison to quantumlasers or storage ring–based FEL oscilla-tors. Radiation of a single-pass FEL suffersfrom the inherent fluctuation of the beamparameters such as energy and positionjitter. In addition, an SASE FEL does notnecessarily provide longitudinal coherenceand has a spectral bandwidth, which islarge compared to standard lasers. Theaverage power is limited by the repetitionrate of a linear accelerator, small comparedto that of a storage ring. The single-passhigh-gain FEL is the only solution for ahigh-brightness radiation source for wave-length regions, where an FEL oscillatorsystem will not work owing to the lossesin the optical cavity.

The limitations for storage ring–basedFEL are given by the instabilities thatoccur in a storage ring. These instabilitiesyield an upper limit on the electronbeam current of typically a few hundredamperes [29]. In addition, the energyspread and normalized emittance arelarge compared to those in a linearaccelerator [30].

The FEL oscillator accumulates thesingle-pass gain of typically a few percent.As long as the amplitude of the EM fieldis small, the gain per pass is constant(small-signal gain) and the field growsexponentially with the number of passes.The oscillator reaches saturation whenthe FEL interaction is strong enough toinduce complete bunching within a single

pass. Because the electron emission is fullycoherent at saturation, the field cannot befurther amplified. The oscillator reachesan equilibrium between the losses of theoptical cavity and the reduced gain is closedto saturation.

When the FEL oscillator has reached sat-uration, the power level within the opticalcavity is small compared to that of a single-pass FEL because the effective ρ parameterand beam power are lower (Eq. 17). In ad-dition, only a fraction can be coupled outof the cavity for succeeding experiment be-cause the out-coupling contributes to thenet loss of the optical cavity. Higher, out-coupled radiation power requires a highergain to compensate for the losses. In theequilibrium state, the FEL oscillator’s tem-poral and frequency stability, longitudinalcoherence, and repetition rate are betterthan those of a linear accelerator–basedFEL. However, the official quality of linearaccelerator–based FELs can be improvedusing optical instruments, like monochro-mators.

The radiation of an FEL oscillator istypically longitudinal coherent, becausethe phase information of the radiationfield is distributed over the entire electronbunch mainly due to two reasons. Thefirst reason is the same as for a single-passFEL, because the radiation field propagatesfaster than the electron beam. The secondreason depends on the design of the opticalcavity. If the cavity is ‘‘detuned’’, the timeof the radiation pulse to pass the cavityis different from the arrival time of thenext electron bunch. The radiation fieldhas a longitudinal offset with respect tothe electron bunch. This offset allows afaster buildup of coherence.

The FEL oscillator can start from thespontaneous radiation. If the gain islarger than a few percent and the detun-ing is small, the random nature of the

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spontaneous emission phase can man-ifest in the occurrence of superradiantspikes [31]. The peak power can be con-siderably larger than the power level ofthe fully coherent signal, while the spikelength is shorter than the electron bunchlength. The nature of these superradiantspikes is the same as those occurring inthe temporal radiation profile of an SASEFEL.

3Present Status

The current status of the FEL researchprogram around the world can be clas-sified into two groups with differentobjectives. The single-pass free-electronlaser is aiming toward shorter wave-length, while storage ring–based FELsare developed to overcome the limita-tion of the losses of the optical cavi-ties. Free-electron lasers, which have fin-ished their research program, are oftenconverted to user facilities, where re-searchers can use the unique propertiesof the FEL radiation for research in theirfield.

3.1Single-pass Free-electron Lasers

Single-pass free-electron lasers are fre-quently based on linear accelerators. Be-cause the FEL does not rely on any opticalcavity to accumulate the gain, there is norestriction on the wavelength. A beam en-ergy of about 10 MeV corresponds to aresonant wavelength of about 10 microns,while a 10- to 25-GeV beam can drive anFEL in the VUV and X-ray region. Theenergy of the electron beam is determinedby the length of the linear accelerator.

The requirements on the beam qual-ity (Sect. 2.4 & 2.5) are more stringentfor FELs operating at shorter wavelength,and the first single-pass FELs started atwavelengths in the millimeter or infraredregime [32]. The major motivation of theongoing FEL research is to shorten thewavelength down to the angstrom level,where no alternative radiation sources withthe same radiation properties exist. The to-tal number of single-pass FEL is numerousand Table 2 lists a selection, namely, thosewith the shortest wavelength. The recordfor the shortest wavelength so far lies ataround 80 nm [33]. The majority of future

Tab. 2 Linear accelerator–based free-electron lasers

FEL UCLA VISA LEUTL TTF-FEL I

Energy [MeV] 18 71 217/255 250Energy

spread [%]0.25 0.1 0.4/0.1 0.06

Norm. emittance[mm·mrad]

5.3 2.3 8.5/7.1 6

Peak current [A] 170 250 630/180 1300

Wavelength [nm] 12 000 840 530/385 95Peak

power [MW]0.0001 15 500 1000

Pulse length[ps, RMS]

5.5 0.3 0.2/0.7 0.02

Gain length [cm] 25 18.7 97/76 67

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FELs will operate at shorter wavelengths(Sect. 4).

The FEL performance is limited bythe electron beam quality, the transverseemittance, the energy spread, and thebeam current. The linear accelerators ofall the latest single-pass free-electron lasersare based on RF photoelectron guns [34].A conventional laser in the visible or UVilluminates a cathode, embedded into anRF cavity. The photo effect generates theelectrons and the RF field accelerates themto minimize the beam ‘‘blowup’’ due tothe electric field of each electron (spacecharge field). The value of the transverseemittance arises mainly from this spacecharge field at the cathode region [35].At higher energy, the electron beam isrelativistic and is less sensitive to spacecharge effects.

Because a more intense pulse of thephoto gun laser produces a higher electrondensity, the space charge field is stronger.As a consequence, there is a limit on theachievable emittance and current from theRF photoelectron gun [36]. Increasing thecurrent would increase the emittance aswell. Typical values are an emittance ofaround 2 mm·mrad at a current of about50 A. The total charge of the electronbunch is on the order of 1 nC. Single-pass free-electron lasers typically requirebeam currents of 100 A and above to keepthe saturation length within reasonablelimits. This is achieved by including bunchcompressors in the beam line of the linearaccelerator.

A bunch compressor consists of bendingmagnets, which deflect the beam off theaxis of the linear accelerator and then bendit back to the axis. Because the bend angledepends on the electron energy, electronswith a lower energy have a strongerdistortion and, thus, a longer path length.The electron bunch is given a correlated

energy variation along its length by the RFstructure, so that electrons at the head ofthe bunch have lower energy than thosein the tail. The path-length difference inthe bunch compressor reduces the bunchlength and increases the beam current.The beam energy correlation is obtainedwhen the RF field is still building up(‘‘off-crest’’ injection). Electrons enteringfirst see a lower accelerating gradient thanthose entering later. At the cavity exit,the bunch has an energy correlation alongbunch, needed to compress the bunch inthe bunch compressor.

3.2Free-electron Laser Oscillators

Storage rings impose a limit on theachievable electron beam quality. Becausebending magnets are required to trans-port the electron bunches back to theundulator entrance, electrons emit sponta-neous radiation. When an electron loosesenergy owing to the spontaneous radi-ation, its orbit shifts with respect tothe design orbit. The energy losses arenot the same for every electron dueto quantum fluctuations of the emittedphotons. This results in a limitation ofthe achievable transverse emittance [37].To increase the performance of a stor-age ring–based FEL oscillator, this limithas to be reduced, which often requiresa special lattice of bending and fo-cusing magnets. Bends with more, butshorter, bending magnets alternating withquadrupole magnets, perform better thanthose with less magnets. This approach isnot only used for storage ring–based FELbut also for most third-generation lightsources [38].

Since electron bunches are kept in astorage ring for minutes or hours, evensmall effects can be accumulated and can

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degrade the beam quality. One effect isintrabeam scattering, in which, due toCoulomb collision electrons are gradu-ally kicked out of the orbit and lost inthe storage ring aperture [39]. Anothereffect is the ‘‘communication’’ betweenbunches by means of wakefields [40]. Eachstorage ring has RF cavities to com-pensate energy losses, which occur be-cause of spontaneous radiation and theFEL interaction. The modes of the cav-ities are excited whenever an electronbunch passes the cavities. These highermodes are not damped quickly enough,so that• the next bunch interacts withQ7

them. The resulting energy modulationis not constant over the entire bunchand dispersion – energy dependence ofthe orbit – tears the bunch apart, spoil-ing the beam quality or even dumpingpart of the beam in the vacuum cham-ber wall.

Besides an advanced lattice design ofthe storage ring to minimize the effectsdescribed above, storage ring limitationcan be diminished by improving theoptical cavity of the FEL oscillator. Thecurrent research on oscillators is focusedto improve the reflecting mirrors in twoareas such as

1. the extension of the wavelength rangein the VUV range by novel coating ormaterial,

2. increasing average power by improvedpower handling.

The shortest wavelength an FEL oscil-lator is operating is around 100 nm [41]with significant power emitted at higherharmonics as well. The FEL oscillator atJefferson Lab achieved an average powerin the IR region above 2 kW [42]. In slightcontrast to the single-pass FEL, whichare mostly FEL experiments to extendthe wavelength range, storage ring–basedFELs mostly serve as a user facility, wherethe radiation wavelength and power istuned to the needs of experiments fromall fields of science.

The limitations in beam current andbunch length can be overcome inan Energy Recovery Linear accelerator(ERL) [43], where the electron bunches areonly kept for a few turns in the storagering. The beam energy is extracted backto the accelerating field prior to the ejec-tion and dumping of the bunch. Since thepeak current in an ERL is higher thanthat in storage ring, the brightness and thesaturation power is increased.

Tab. 3 Free•-electron laser oscillatorsQ8

FEL Dukea Elletraa Cliob Felixc Jefferson Labd

Energy [MeV] 200–1200 1000 15–50 15–50 48Avg. current [mA] 115 25 2–40 2.5–100 4.8Bunch length [ps] <50 6.3 0.5–6 0.8 0.4Rep. rate [MHz] 13.95 4.63 62 25/1000 74.8Wavelength [µm] 0.19–0.4 0.19–0.20 3–90 4.5–250 3–6.2Avg. power [W] 0.025 0.01 3 25–5000 2000

aStorage ring.b10 µs electron bunch train from thermionic gun at 25 Hz.c10 µs electron bunch train from thermionic gun at 10 Hz.dEnergy recovery linear accelerator.

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4Future Development

All recent single-pass free-electron laserexperiments have not shown any fun-damental limits that would prevent aresonant wavelength at the angstrom level.The typical beam energy would be around10 to 25 GeV. The required undulatorlength would be around 100 m. Severalprojects have been proposed or are underconstruction to have single-pass FELs op-erating in the wavelength regime between1 and 1000 A [5–7]. Table 4 lists a selec-tion of them. All these FELs are plannedto be user facilities. Most of them haveseveral beam lines and undulators to servemultiple users simultaneously. To reducethe limitation on the average brightness,most of the linear accelerators are basedon superconducting technology. It allowsto lengthen the RF pulse and fill the beamline with multiple bunches per RF pulse,while operating on a lower power level ofthe RF system.

A combination of a storage ring witha linear accelerator is the Energy Recov-ery Linear accelerator (ERL) [43]. After theelectron bunch is accelerated and passesthe undulator, it is transported back to the

Tab. 4 VUV & X-ray free-electron lasers,planned or under construction (∗)

FEL Wavelength [A] Location

TESLA∗ 1–50 GermanySCSS 1–10 JapanLCLS∗ 1.5–15 USAMIT Bates FEL 5–1000 USAFERMI 12–100/400 ItalyBESSY 12–1500 GermanySPARX 15–140 ItalyTTF-FEL II∗ 60–1200 Germany4GLS 120–4000 UKNSLS DUV-FEL 200–2000 USA

beginning of the beam line. It propagatesto the accelerating cavities a second time,but with a 180◦ difference in the RF phase.The residual kinetic energy of the bunch istransferred back to the RF field, and thusis available for succeeding bunches. It re-quires superconducting cavities; otherwisethe stored energy will be lost by thermallosses before the next bunch enters thebeam line. Using this setup, the repetitionrate of the electron bunches are not limitedby the RF system and FELs can be operatedin cw mode, similar to storage ring but withthe advantage that there are no limitationson the beam current or emittance.

The research in FEL physics is shiftingfrom the proof of principle toward en-hancement of the radiation properties, inparticular, in the X-ray regime. SASE FELshave an intrinsic instability in the outputenergy of a few percent up to 100%. In ad-dition, the operation of a linear acceleratoris not as stable as that of a storage ring. Thesystem fluctuation adds up to the intrin-sic fluctuation of the SASE pulse. Anotherdisadvantage is the reduced longitudinalcoherence. Because of the short coopera-tion length of FELs in the VUV and X-rayregime, part of the bunch amplifies thespontaneous radiation independently. Thelongitudinal coherence can be achieved bya pulse stretcher, namely, monochroma-tor, with a bandwidth slightly smaller thanthe Fourier limit of the electron bunchlength. All spikes are stretched to thebunch length and add up coherently. Eachspike in the SASE pulse has 100% fluctu-ation, so has the resulting coherent pulseafter the monochromator. Stability of theoutput power is achieved if the monochro-mator is not placed after the SASE FEL butbetween, splitting the undulator in twoparts [44]. The second undulator is seededby the output signal of the monochro-mator and operates as an FEL amplifier

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instead of an SASE FEL. Because the sat-uration power of an FEL amplifier doesnot depend on the input signal, the outputpower level in the deep saturation regimeis stable despite the 100% fluctuation ofthe input signal.

Another focus is on short radiationpulses. Standard methods are pulse com-pression and slicing, similar to conven-tional lasers. It requires an energy correla-tion along the electron bunch (‘‘chirp’’),driving an SASE FEL. Because of thesquare dependence of the resonant wave-length on the energy (Eq. 1), the radia-tion frequency chirp is twice as large. Amonochromator selects only that part ofthe pulse in which frequency componentsfall within the bandwidth. The monochro-mator bandwidth has to be optimized inorder to avoid a pulse stretching for atoo-small bandwidth or large chirps fora too-wide bandwidth. For X-ray FELs, amonochromator bandwidth of 10−4, whichis roughly a tenth of the intrinsic FEL band-width, and a frequency chirp of around 1%,results in a 10-fs slice [45].

An alternative method for short bunchesis to condition the driving electron bunch.A stronger compression spoils the emit-tance and energy spread by wakefields andcoherent synchrotron radiation in the lin-ear accelerator and bunch compressor, sothat the electron pulse length is limitedto roughly 200 fs FWHM for the shortestgain length. On top of that, the beam canbe conditioned by various methods (RFcavities, wakefields, external laser pulses)to keep the beam quality good over only ashort subsection of the bunch. Althoughthe entire bunch emits spontaneous radia-tion, only the preserved part of the bunchamplifies it and reaches saturation [46].

A free-electron laser in the X-ray regimerequires about 100 m of undulator and alinear accelerator of about 1 km, using the

present technology. It is desirable to re-duce the size of the FEL facility or makeit even portable and robust. The FEL canbenefit from results of other branches inthe accelerator physics. Acceleration basedon plasmas are promising higher acceler-ation gradient and lower emittances [47],thus shortening both the undulator and thelinear accelerator. The FEL physics is un-changed, when the undulator is replacedby a device that also enforces a transverseoscillation of the electron bunch. A pos-sible solution would be a •high-intensity Q9

optical laser pulse [48]. Because the pe-riod length is several orders of magnitudesmaller than that for an undulator, the FELinteraction requires less energy to obtainthe same wavelength. Other proposals arebased on plasma columns [49], transversewakefields [50] or crystal lattices [51].

Glossary

3rd Generation Light Sources: Spontane-ous emission from dipoles and undulatorwith radiation pulse lengths in the picosec-ond regime and recirculating electronswith an energy between 3 and 10 GeV.

Bending Magnet: see dipole magnet.

β-function: Characteristic function of thefocusing property of a given beam line.Depending on the initial values of theβ-function and its derivative, it describesthe evolution of the beam size along thebeam line.

Bunch Compressor: A series of dipole mag-nets to deflect the electron beam from theorbit and then back onto the orbit. Theresulting orbit and thus the path length ofthis orbit bump depends on the electron

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energy. With a correlation between lon-gitudinal position and energy within anelectron bunch, the path-length differenceyields a reduction in the bunch length andan increase in the beam current.

Bunching: Density modulation of the elec-tron beam on a scale shorter than thebunch length. Also called microbunching.

Coherent Synchrotron Radiation: Strongenhancement of the synchrotron radiationfor wavelength longer than the bunchlength. The radiation can interact backwith the electron bunch and because of itslarge amplitude alter the electron energyand, thus, the orbit of the electron bunch.

Dipole Magnet: Magnet with a constant,transverse magnet field to deflect electronand thus, cause a bend in the orbit of theelectron bunch.

Emittance: Phase space area, covered bythe electron bunch distribution.

Energy Recovery Linear Accelerator: Cir-cular accelerator with continuous injectionof electron bunches. After acceleration,each electron bunch passes each RF cavityagain but with a 180◦ phase difference.The deceleration transfers back the kineticenergy of the electron bunch to the RFfield, which can be used to accelerate thesucceeding bunch.

Microbunching: See bunching.

Photoelectron Gun: RF cavity with anembedded photo cathode. A UV laser pulseilluminates the cathode and generates freeelectrons due to the photo effect. TheRF field accelerates these electrons torelativistic energies within the length ofthe RF gun.

Plasma Frequency: Measure for thestrength of the beam size growth due to therepulsive electrostatic field between eachelectron.

Ponderomotive Phase: Phase of the elec-tron–radiation interaction due to the beatwave of the transverse oscillation of theelectrons and the oscillation of the EMwave.

Quadrupole Magnet: Magnet with a lineargrowing transverse magnetic field withrespect to the orbit of the electron bunch.The field configuration yields a focusingin one transverse plane but a defocusingin the perpendicular transverse plane. Anet focusing in both planes is achieved bya series of quadrupoles with alternatingpolarity.

Linac: linear accelerator, where the elec-tron passes the beam line only once.In contrast to the storage ring, the lin-ear accelerator consists mainly of accel-erating structures, such as RF cavities.Quadrupole magnets keep the transversebeam size within the acceptable limits.

Radiofrequency Cavity: Cavity, which res-onantly holds an oscillating EM field atwell-defined frequencies. The fundamen-tal mode has a longitudinal electric fieldcomponent, which is used to accelerate anelectron beam, passing the cavity.

Shot noise Power: Power of the sponta-neous radiation, which falls within thebandwidth of the free-electron laser andwhich is amplified in the SASE FEL con-figuration.

Storage Ring: Circular accelerator to keepinjected electron bunches on a closed orbitfor many turns. RF cavities compensate

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the energy loss due to the spontaneousradiation within the dipole magnets andother transverse deflecting magnets (e.g.,undulators)

Superconducting Magnet: Electromagnet,based solely on superconducting windingsto overcome saturation effects of any yokematerial.

Undulator: Device with transverse, alter-nating magnetic field on the main axisin planar and helical configuration. In aplanar undulator, the transverse magneticfield is oscillating within only one trans-verse plane, while the magnetic field isconstant but rotates along the longitudinalaxis for a helical undulator.

Undulator Parameter: Normalized ampli-tude of the vector potential of the undulatorfield. It is proportional to the product ofpeak undulator field and undulator periodand can be regarded as the total deflectionstrength of the undulator field per halfperiod.

Undulator Taper: Variation of the undula-tor field and/or period along the longitudi-nal axis of the undulator.

Wakefield: Radiation, caused by the inter-action of the electric static field of anelectron with the vacuum chamber. Theradiation can interact with succeeding elec-trons, changing the electron energy ortransverse momentum.

Wiggler: See undulator.

References

[1] Madey, J. M. J. (1971), J. Appl. Phys. 42,1906.

[2] Motz, H. (1951), J. Appl. Phys. 22, 527.[3] Palmer, R. B. (1972), J. Appl. Phys. 43, 3014.[4] Freund, H. P., Granatstein, V. L. (1999),

Nucl. Inst. & Methods A429, 33; Col-son, W. B. (1999), Nucl. Inst. & MethodsA429, 37; online at http://sbfel3.ucsb.edu/www/vl fel.html

[5] Linac Coherent Light Source (LCLS), SLAC-R-521, UC-414 (1998).

[6] TESLA - Technical Design Report, DESY2001-011, ECFA 2001-209, TESLA Report2001-23, TESLA-FEL 2001-05, DeutschesElektronen Synchrotron, Hamburg, Ger-many (2001•). Q10

[7] Various proposals are listed in the Proc. ofthe 24th Free-Electron Laser Conference,Argonne, USA (2002•). Q11

[8] Phillips, R. M. (1960), IRE Trans. ElectronDevices 7, 231.

[9] Elias, R. L. •et al. (1976), Phys. Rev. Lett. 36, Q12717.

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Further Reading

A complete description of the FEL theory isgiven in the books Thy Physics of Free ElectronLasers by E. L. Saldin, E. A. Schneidmilller,M. V. Yurkov (Springer, Berlin, 2000) orFree-Electron Lasers by C.A. Brau (AcademicPress, Boston, 1990). A brief overview on thehigh gain FEL can be found in the paper‘‘Introduction to the Physics of the FEL’’by J.B. Murphy and C. Pellegrini (Proc. ofthe South Padre Island Conference, Springer(1986) 163).

The book An Introduction to the Physics ofHigh Energy Accelerators by D.A. Edwards andM.J. Syphers (John Wiley and Sons, NewYork, 1993) covers the physics, needed togenerate and accelerate an electron beam,driving the free-electron laser. A review ofundulator design is given by J. Spencer andH. Winick in Synchrotron Radiation Research(eds. H. Winick and S. Doniach, PlenumPress, New York, ch. 21, 1980).

The radiation process of relativistic particlesare covered in great detail by J.D. Jackson’sClassical Electrodynamics (John Wiley and Sons,New York, 1975) while the description of therandom nature of the SASE FEL follows thetreatment of Statistical Optics by J. Goodman(Wiley and Sons, New York, 1985).

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