theory and practice of free-electron lasers - us particle

11--11LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009

University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM

Theory and Practice ofFree-Electron Lasers

Particle Accelerator SchoolDay 1

Dinh Nguyen, Steven Russell& Nathan Moody

Los Alamos National Laboratory



Course Content

1. Introduction to Free-Electron Lasers2. Basics of Relativistic Dynamics3. One-dimensional Theory of FEL4. Optical Architectures5. Wigglers6. RF Linear Accelerators7. Electron Injectors

Chapter

ChapterChapter

ChapterChapter

Chapter

Chapter



Course Schedule

ElectronInjectorsRF LinacOptical

Architectures1-D FEL Theory

SimulationLab

SimulationLab

SimulationLab

SimulationLab

Final Exam

Lab Report DueRF LinacWigglersOptical

Architectures1-D FEL Theory

Final ExamRF LinacWigglersOptical Architectures

Intro. to FEL

Relativistic Dynamics

FridayThursdayWednesdayTuesdayMonday9:00

10:45

12:151:15

3:15

5:30

10:30

3:30



Chapter 1Introduction to Free-Electron Lasers



Introduction to Free-Electron Lasers

• The nature of light• Gaussian beam• Laser beam emittance• Longitudinal coherence• How a quantum laser works• How an FEL works• Basic features of FEL• RF-linac FEL• Fourth-generation Light Sources• Applications of FEL



We can treat the EM wave as a sinusoidal plane wave. In our convention, the electric field is in the x direction and magnetic field in the y direction. For a wave travelling in the positive z direction, the fields are given below

where k = wavenumber in m-1

= angular frequency in s-1

= phase in radians

Light can be described as bothparticles (photons) and waves

• Light consists of photons each having energy where h = Planck’s constant (h = 6.626 x 10-34 J-s) and = frequency of the light; Photon energy can be calculated from wavelength as follows

hvE

1.24( )

eV

E

v c

82.9979 10 mcs

B

E

• Light can also be described as a travelling electromagnetic (EM) wave.

c

12.4(Å)keV

E

0

0

ˆ( , ) cos( )ˆ( , ) cos( )

z t E kz tz t B kz t

E xB y



Gaussian Laser Beam

rms radius in x

-4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0

w

x

x

Intensity

.135 I0

I0

2 22 2

2

2( , )x yw wPI x y e e

w

2 xw

22

( , )

( , )x

I x y x dxdy

I x y dxdy

1/e2 radius

2 2ln 2 xFWHM

Full width at half max (FWHM)

FWHM

2.355 xFWHM



Gaussian Beam Propagation

0w

Diffraction limit

22 2

0 21R

zw wz

Parabolic expansion of 1/e2 radius with z

0w

At large z the divergence angle scales with /w0

The product of the waist radius and converging angle of a diffraction limited beam is the wavelength divided by . Focusing the beam to small spots requires large angles.

20

Rwz

Rayleigh length



Laser Beam Emittance

dxxdz

( , )x x

photonsz

' 4x x

Photon beam emittance

( , )x x

x

xconverging

x

xwaist

x

xdiverging

0' 4 4rms x x

wA

Light phase space area = times x

(rms radius) times x’ (rms angle)



Longitudinal Coherence

If and are the full-width at half max (FWHM), the transform limit becomes

4 2 0 2 40

5 108

1 109

1.5 109

2 109

g t( )

ttime (ps)

inte

nsity

(W/c

m2 )

Coherence length

Fourier transform

4 2 0 2 40

0.2

0.4

0.6

0.8

1

0

f ( )

55 frequency (THz)

An optical pulse with length is fully coherent if its coherence length ≥ 2 c

0.44 2 2

4ln 2

0 0t

t t

I I e I e

2

cL

Gaussian pulse



How a quantum laser works

An external source of energy excites electrons from the ground state to an excited state

Electrons from excited state decay to a metastable energy level with long lifetime (transition from this level to the ground state is quantum mechanically forbidden) → population inversion

A co-propagating light beam stimulates emission of radiation → amplification of co-propagating light beam (Light Amplification by Stimulated Emission of Radiation)

g

em

Absorption Population Inversion Stimulated Emission



How an FEL works

Electrons in an FEL are not bound to atoms or molecules. The “free” electrons traverse a series of alternating magnets, called a “wiggler,” and radiate light at wavelengths depending on electrons’ energy, wiggler period and magnetic field.

light (electromagnetic wave)

v┴

v║Bw

y z

x

wigglermagnets

electron trajectory

unbunched electron beam pulse

bunched electron beam pulse



How an FEL works (cont’d)

The wiggler induces transverse sinusoidal velocity in electron beam

Energy exchange occurs between the transverse electron current and transverse electric field of a co-propagating light beam

Depending on the phase of the light beam with the electrons’ wiggling motion, some electrons gain energy while others lose energy → energy modulation →bunching of electrons along the axial direction into microbunches with period equal to an optical wavelength

Microbunched electron beams radiate coherently at higher power →amplification of the co-propagating light beam.

Note: The subscript ┴ denotes transverse and s stands for signal.

sW e E



Basic features of FEL• Wavelength tunable• Diffraction limited optical beam• Longitudinally and transversely coherent• High power (GW peak, 100kW to MW average)• Efficient (with energy recovery)

0.1nm 1nm 10nm 100nm 1 10 100 1mm 10mm 100mm

Gamma X-rays VUV IR THz mm-wave waveVisible

Eb 10GeV 1GeV 100MeV 10MeV 1MeV 100keV



Wavelength Tunability

22 1

2w

wa

w wiggler period resonant wavelength

relativistic factoraw (also Krms) rms wiggler parameter

00

0

0.662w w

w

eBa B T cmk m c

20

1 2T T MeVm c

For electrons (m0c2 = 0.511 MeV)

Select coarse wavelength by choosing the electron beam energy, wiggler period and wiggler magnetic field. Fine-tune wavelength by adjusting electron beam energy or wiggler magnetic field.

Another convention uses peak parameter K2

2 12 2

w K

0

0

2 ww

eBK ak m c



Radio-frequency Linac FEL

Electron Injector

RF Linac

Bunch Compressor

Wiggler

FEL Beam

Beam Dump

Single-pass AmplifierSelf-Amplified Spontaneous Emission (SASE)

Electron Injector

Energy Recovery Linac

Beam Dump

Booster

Wiggler

Outcoupler High Reflector

FELOscillator



RF-Linac FEL Pulse Structure

FEL macropulseTmacro

1RF

RF

tf

n tRF

FEL micropulses

RF wave train



Temporal & Spectral StructuresSASE FEL have spiky temporal and spectral features.

Unsaturated oscillator/amplifier FEL have smooth temporal and spectral profiles.Oscillator/Amplifiertime domain

Oscillator/Amplifierspectraldomain

SASEtime domain

SASEspectraldomain



FEL optical beam properties

• Intensity

2p

x y

N hvB

t

20

2 pN hvI

w t

2p

x y

Nt

B

W/m2

W/cm2

photons/(m2 s 0.1% BW)

1

wN

• Brightness

• Spectral bandwidth

• Brilliance



InjectorInjectorat 2at 2--km pointkm point

1 km S1 km S--band band linaclinac

ee TransportTransport

UndulatorUndulatorExperiment HallExperiment Hall

4th Generation Light Source (4GLS)

Peak brilliance of linac-based 4th generation light sources (XFEL) is 8-10 orders of magnitude higher than that of 3rd generation light sources and >20 orders of magnitude above Bremsstrahlung sources.

Linac Coherent Light Source (LCLS)



Some examples of 4GLS

250 fs80 fs75 fsBunch length

1.5 cm1.3 (1.838)

50 m

3.56 cm2.33 (3.3)

200 m

3 cm2.62 (3.7)

55 m

Wiggler periodaw (K)Length

2 m1.4 m0.4 mrms emittance

1 nC1 nC0.25 nCBunch charge

Pulsed DC gunCeB6 thermionic

L-band RF gunCs2Te photocathode

NCRF, 2.856 GHzCu photocathode

Gun type, frequencyCathode

NCRF, 5.712 GHz0.75 km

SRF, 1.3 GHz3.4 km

NCRF, 2.856 GHz1 km

Linac type, frequencyLength

8 GeV20 GeV14.3 GeVBeam energy

0.1 nm12.4 keV

0.1 nm12.4 keV

0.15 nm8 keV

WavelengthX-ray energy

Spring-8HyogoJapan

DESYHamburgGermany

SLACPalo Alto, CA

USA

InstitutionLocationCountry

SCSSEuropean XFELLCLS



Peak brilliance of 4GLSPulse energy ~ 1 mJ

Photon energy ~ 1 keV

# of photons ~ 1013

rms emittance ~ 10-4 m

rms bunch length ~ 10-13 s

Energy spread ~ 0.01% BW

Brilliance ~ 1033 (s m2 0.1% BW)-1



High-average-power FEL

• Ground-based FEL Program (Boeing/LANL, LLNL/TRW)

• Energy-recovery FEL (e.g. Jefferson Lab FEL)

Jefferson Lab FEL holds the world record in cw average power (14 kW).



Applications of FEL and 4GLSFEL Features Wavelengths Examples of applications

• Ultrashort tunable pulses– Medicine 1-6 m Laser surgery– Physics XUV Ultrafast spectroscopy– Chemistry XUV, UV Chemical dynamics– Biology X-rays Protein structures

• High peak power– High-density physics X-rays Warm dense matter– Materials sciences near-IR Laser machining

• High average power– Directed energy IR Defense– Space near-IR Power beaming– Material processing UV Lithography



Chapter 2Basics of Relativistic Dynamics



Basics of Relativistic Dynamics• Special relativity• Lorentz transformation• Relativistic Doppler shifts• Wavelength dependence on angle• Relativistic velocity, momentum & energy• Lorentz force law• Curvilinear coordinate system• Linear beam dynamics• Emittance• Emittance & energy spread requirements



Special Relativity

1. All inertial frames are completely equivalent with regard to physical phenomena

2. The speed of light in vacuum is the same for all observers in inertial frames of reference.

Beam Framey’

x’

z’

v

Lab Framey

x

z

e- beam



Lorentz Transformation

( )( )

'''

'

x xy yz z ct

ct ct z

g b

g b

==

= -

= -2

11

c

gb

ub

=-

=Transverse dimensions are unchanged.

Lorentz factor

Velocity relative to c

Lengths of moving objects along direction of motion appear to becontracted in the Lab frame by a factor (Lorentz-FitzGerald contraction)

Clocks in the moving objects run slower by as observed in the Lab frame (time dilation).

x

y

z

y’

x’

z’

cBeam coordinates

Labcoordinates



Wiggler period contracts in beam frame

Lab frame

y

x

z

w

x’

z’

y’

Beam framew

' ww

ll

g=

Wiggler period in beam frame



Lorentz Transformation of Fields

( )( )

'

'

'

x x z y

y y z x

z z

E E B

E E B

E E

g u

g u

= +

= -

=

Electric field

Transverse electric and magnetic fields are different in the beam frame. Pure electric (and magnetic) fields in the Lab frame transform into mixed electric and magnetic fields in the beam frame. Longitudinal (along the direction of motion) electric and magnetic fields remain the same.

'2

'2

'

zx x y

zy y x

z z

B B Ec

B B Ec

B B

ug

ug

æ ö÷ç= - ÷ç ÷çè øæ ö÷ç= + ÷ç ÷çè ø

=

Magnetic field



Electromagnetic Field Transformation

x’

z’

y’

Beam frameF’ = -e (E’ + v’ x B)

B’

E’

F = -e (v x B)B

Lab framey

x

z

v

Wiggler magnetic field deflects electrons in x direction

Electromagnetic field deflects electrons in x’ direction

v’ ~ 0 in beam frameForce is almost entirely due to electric field

Force is due to magnetic field in Lab frame



Radiation in Beam Frame

x’

z’

y’

Beam frame Wiggler electromagnetic wave behaves like virtual photons impinging on the electrons

B’

E’

''ww

c cgn

l l» =

Real photons are scattered off the electrons. They can also be seen in the beam frame as circular waves radiated from the electrons at frequency ’

Lorentz contraction causes ’ to be increased by a factor of compared to Lab frame

View from the top



Useful Relativistic Relations

2

2

2

2

112

112

1 112

1 112

bg

bg

b g

b g

» -

- »

» +

- »

Approximations for ~ 1Exact relations

22

22

2 2 2

11

11

1

gb

bg

b g g

=-

= -

= -

2 2 2

1 1 1b g b

= -



Lorentz Transformation ofFrequency and Angle

( )

'

1 cosn

ng b q

=-

Relativistic Doppler shift depends on Lab frame observation angle

'

2n

ng

=

Forward ()

'2n gn=

Use approximation 2

112

bg

- »

Relativistic Doppler shift in the forward direction

Backward ( = )

( )' 1 cosn g b q n= -

For >>1 Lorentz transformation yields 1/ emission angle

( ) ( )22 '1 cos 1 cos 1 cosq g b q q- = - -1/

'1 cosqq

g-

=For small angles



Longitudinal Doppler Shift (Forward)

x’

z’

y’

Beam frame

2

2 ' 2

2w

w

c

c

gn gn g

l

gn

l

æ ö÷ç ÷= = ç ÷ç ÷çè ø

=

22wllg

=Combined effect of Lorentz contraction and Doppler shift gives a factor of 2 increase in frequency

2 'n gn=

Doppler effect causes up-shift in frequency and narrowing of emission angle

Consider radiation emitted in the forward direction (same direction as electrons)

Lab framey

x

z



Longitudinal Doppler Shift (Backward)

x’

z’

y’

Beam frame

' 12 2 2w w

c cn gn

g g l l

æ ö÷ç ÷= = =ç ÷ç ÷çè ø

2 wl l=

Lorentz contraction is negated by Doppler shift. Frequency is reduced by a factor of 2.

Doppler effect causes down-shift in frequency

Lab framey

x

z

Consider radiation emitted in the backward direction (opposite to beam direction)

'

2n

ng

=



Wavelength Dependence on Angle

( )2 22 1

2wll g qg

» +

Forward wave

( )1wl l b= +

Backward wave

The wavelength of wiggler (undulator) radiation depends on emission angle. Shortest wavelengths are radiated in the forward direction ( = 0). Radiation at larger angles have longer wavelengths. The opening half angle of wiggler radiation, is given by

w

2 waq

g=



Relativistic Energy & Momentum2 2 2

0 0E T m c mc m cg= + = =

Multiply by c and square

Energy is in unit of MeV or GeV.Momentum is in unit of MeV/c or GeV/c

Total energy

Kinetic energy

Momentum

( ) ( )2 20 0 1T m m c m c g= - = -

0p m m cu bg= =

( ) ( ) ( )( )2 22 2 2 2 2 20 01cp m c m cb g g= = -

Energy right triangle ( ) ( )222 20E cp m c= +

moc2

cpE



Parameter Variation Table

d

d

dpp

d

dpp

d

1

1

1

2

2 2

2

1 2 2

1

2

1

2



Relative velocity differencesbecome smaller at high energy

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9 10

2 2

1d db gb b g g

=

Most electron accelerators are speed-of-light (=1) machinesAt large , it becomes very hard to perform ballistic bunch compression because all electrons travel nearly at the speed of light.



Relative momentum change issame as energy spread at high energy

2

1dp dp

gb g

=

0

2

4

6

8

10

12

1 2 3 4 5 6 7 8 9 10

p (m

oc)

Bunch compression via momentum spread can be done at any energyGiven sufficient energy spread and dispersive elements such as magnetic chicanes, electron bunches can be compressed to ultrashort pulses.



Lorentz Force Law

e F E v B

In MKS units, e = 1.6 x 10-19 coulomb, electric field is in volts/m and magnetic field is in tesla.

Electric force acts on electrons along their direction of motionand thus changes the electrons’ kinetic energy.

Magnetic force is perpendicular to direction of motion and does not change the electrons’ kinetic energy. Magnetic field can be used to change momentum, i.e. bend electron beams.

T d e d F s E s

p dt e dt F B



Bending Relativistic Beams

0

tan

tan

1

b

b

b

pp

p F t e B t eB sEp m cc

s ecB sE

ecBE

q

u

bg

qr

r

^

^

=

= D =- D =- D

= =

D - D= =

=

bend radius

11 ( )299.8

b

B TmE MeV

1299.8 bB T m E MeV

Magnetic rigidity

incident beamdipole magnet

bent beam

Bend angle and radiuss

p┴

p║



Curvilinear Coordinate

y

x

y

Trajectory ofreference particle

s

x

Electrons travel in the s direction. Use (x, y, s) coordinate system to follow the reference electron, an ideal particle at the beam center with a curvilinear trajectory. The reference particle trajectory takes into account only pure dipole fields along the beam line. The x and y of the reference trajectory are thus affected only by the placement and strength of the dipole magnets.

For other electrons, define x’ and y’ as the slopes of x and y with respect to s

dxxds

dyyds



Paraxial Rays & Trace SpaceParaxial ray approximation deals with non-crossing trajectories near the axis.

z( , )x x

x

x’

x

x’

x

x’

In a drift space, converging beams come to a waist and then diverge



Lorentz Forces

20

y yecB eBx

m c p

0

x xeB eBym c p

Lorentz force in x2

0 2 y yd xm e B ecBdt

2

0 2 x xd ym e B ecBdt

Lorentz force in y

1dx dxxds c dt

2 2

2 2 2

1d x d xxds c dt

Slope of x with respect to s

Curvature of x with respect to s



Quadrupole LensA quadrupole is a focusing element in one plane (e.g., x) and defocusing in the other plane (e.g., y). Its magnetic field, and thus the focusing force, increases linearly with distance from the center. .

Quadrupole

x

y

Quadrupole field

Before quadrupole

x

x’

x

x’Quadrupole focusing After drift

x

x’



Linear Beam Dynamics

0xx K x 0yy K y

Mathieu-Hill Equations

Linear beam dynamics is valid if the restoring forces in x and y are linear.Quadrupoles are linear focusing (and defocusing) elements since the restoring forces are linear with distance from the center.

A system of alternating focusing and defocusing quadrupoles separated by drift space (abbreviated FODO) is used to transport electron beams.

Rx

Ry



Phase space concept

Beams are treated as a statistical distribution of particles in x’-x (also in y’-yand -ct) phase space (trace space, to be exact). We can draw an ellipsearound the particles such that 50% of the particles are found within the ellipse. The area of this ellipse is a measure of rms spread of electron distribution in phase space. The rms emittance is area of the ellipse divided by . Emittance has dimension of length (e.g. microns) since x’ is dimensionless. Traditionally, emittance has unit of mm-mrad.

rmsA

x

xwaist

x

xconverging

x

xdiverging



Beam Emittance

22 2,rms x x x xx

Emittance is defined using ensemble averages, denoted by < >, of x2 and x’ 2

and x’-x correlation. The correlation vanishes at the waist (upright ellipse) and rms beam emittance becomes xx’ where is the rms radius in x and is the rms spread in x’.

Ensemble average of x2 Ensemble average of x’2 x’-x Correlation

2x x

2x x

220

1

1 N

jj

x x xN

22

1

1 N

jj

x xN

01

1 N

j jj

xx x x xN

Root-mean-square x emittance (for y emittance, replace x with y)



Liouville’s Theorem

If the beam is accelerated, emittance (defined by x and x’) is not a conserved quantity because x’ decreases as the axial momentum increases by .

n u

Liouville’s theorem : In the absence of non-linear forces or acceleration, the beam ellipse area in x-px phase space is conserved. If the forces acting on the beam are linear, its emittance is also conserved.

.xx p const

px

pz

x’ = px/pzpx

pz

x’ = px/pz accelerated

By accelerating the beam (increasing pz), we reduce the “un-normalized”emittance (also known as Lab frame emittance). The conserved quantity is the normalized emittance, un-normalized emittance multiplied by . Normalized emittance is used to specify the quality of electron beams regardless of energy.



Electron Beam Emittance Requirement

4n

At a fixed wavelength and beam energy, the required normalized rms emittance for FEL is

Accelerating the electron beam reduces its un-normalized emittance (adiabatic damping). Beams with large (bad) normalized emittance need to be accelerated to high energy.

nu

Electrons’ phase-space area must be less than photons’ phase space area for efficient energy exchange between electrons and photons

x

x

photons

electrons

x

x

x

x

4e uA



Energy Spread Requirement

c t

Electron beam’s energy spread must be smaller than the electrons’velocity spread over the interaction length.

For oscillator FEL, interaction length ~ wiggler length

For SASE and amplifier FEL, interaction length ~ gain length

Uncompressed electron beams have small energy spread and low peak current. Compressed beams have high current and large energy spread.

c t

12 wN



Chapter 31-D Theory of FEL



One-dimensional Theory of FEL• Transverse motion in a wiggler• Figure 8 motion and harmonics• Pendulum equation• FEL bunching• Bunched beam radiation• Spontaneous emission spectrum• Madey’s theorem• Low-gain FEL• Synchrotron oscillation• Saturation• Extraction efficiency• High-gain FEL• Self-consistent FEL equations



Equations of Motion 0ˆ cosy wB yB k z

wwk

2

By

y

x

zvz

0 ˆˆ cosx z o wF m x e z y B k z

For most FEL, vx is much smaller than vz . We can ignore the second force equation and consider only motion in x (the wiggle plane).

Lorentz force laws

0 ˆ ˆ cosz x o wF m z e x y B k z



Equations of Motion (cont’d)

x

z

cc

x z x 2

z zdx x xdt

0

cosow

eBx k zm c

Small-angle approximation: transverse motion is small; axial velocity is almost c

Rewrite Lorentz force equation in term of second derivative with respect to z

Lorentz force equation

2

20

coszo w

ed xx B k zdt m

2

2 2 2z

d z x xxdz c

Transverse accelerationTransverse velocity

Second derivative of x with respect to z ( ) coso wB z B k z

Consider only on-axis magnetic field



Solution to Transverse EOMIntegrate Lorentz force equation once to obtain deflection angle

0

00

0

cos

sin

2 sin

ow

ow

w

ww

eBx k z dzm ceBx k z xk m c

ax k z x

Integrate again to obtain position

0

0 0

2 sin

2 cos

ww

ww

w

ax k z x dz

ax k z x z xk

Transverse motion is periodic with wiggler wavenumber kw. Wiggler magnetic force is harmonic oscillator’s restoring force. Transverse motion in the absence of field errors is given by

2 0wx k x 2w

w

k

x’0 = initial deflection anglex0 = initial position

Wiggler wavenumber



B field, deflection and position

Wiggler magnetic field

+ error - error

First integral of field (deflection)

Second integral of field (position)

2cos sinwo w w

aB k z dz k z

2cos coswo w w

w

aB k z dz dz k zk

coso wB k z



Transverse and longitudinal velocities

2 sinwx w

ca k z

2 2 2 2

22 2 2 2

2

22 2 2

2 2

2 sin

211 sin

z x

wz w

wz w

c

ac k z

ac k z

2 2

2 2

11 cos 2

2 2w w

z w

a ac k z

Transverse velocity is oscillatory with period equal to the wiggler period

Longitudinal velocity

Axial velocity oscillates with a period equal to one-half the wiggler period

vx

vz

v = c

2

2 cos 22

wz z w

ca k z

Find the square root and use small x approximation (1 + x)½ ≈ 1 + ½ x

( )( )2 22

11 1 2 sin2z w wc a k zug

é ùê ú= - +ê úë û

( ) ( )22sin 1 cos 2w wk z k z= -

Use sine squared identity



Figure 8 MotionIn the reference frame that travels at the electrons’ average axial velocity, vz as given by

2

2

11

2w

z

ac

Electrons’ transverse and axial motions are coupled. At zero crossing, transverse speed is at a maximum and axial speed a minimum. At the edges, transverse speed is zero and axial speed is at a maximum. Electrons’motion on the x-z plane follows the figure 8.

' ' '

2' ' '

' 2

2 sin

cos 22

x

z

ww

ww

w

ca k z

ca k zk

Motion in reference electron’s rest frame

Figure 8 motion gives rise to harmonicsin spontaneous (incoherent) radiation



Energy exchange betweenelectrons and FEL beam

sdW j Edt

sinww

ecaj ecx k z

Transverse electron current Plane-wave transverse electric field

,0( , ) cos( )s sE z t E kz t

2

0 0 sin cos( )ww

d m c eca E k z kz tdt

2

0 0 sin2

ww

d m c eca E k k z tdt

Rate of energy exchange depends on the phase of the “ponderomotive wave”



Resonance ConditionQuestion: How can an optical wave traveling at the speed of light interact with slower electrons in a fast wave device (e.g., FEL)?

Answer: If the optical wave slips ahead of the electrons exactly one wavelength every wiggler period, the sum of wiggler phase and optical phase is constant, and energy exchange can occur.

.

0

w

wz

k k z t constd k kdz

2

2

12

ww

ak k

2

2

12

ww

a

w

ww

z

c

Resonance wavelength satisfies this condition

2

22

2

1211

2

ww

w

akk k k ka



Ponderomotive phase = -/2

kwz = 0

kz - t = -/2

(kw + k)z – t = -/2

j

Eskwz =

kz - t = -3/2

(kw + k)z – t = -/2

Es

j

0dWdt

Electrons gain energy

j

Es

Electrons gain energy (light is absorbed)

Optical wave slips ahead by every w



Ponderomotive phase = 0

j

Es

j

Es

No energy gain or loss

Optical wave slips ahead one

j

Eskwz =

kz - t = -

(kw + k)z – t = 0

kwz = 0

kz - t = 0

(kw + k)z – t = 00dW

dt

No energy gain or loss



Ponderomotive phase = /2

j

Es

kwz = 0

kz - t = /2

kwz =

kz - t = -/2

j

Es

Electrons lose energy (FEL gains energy)

Optical wave slips ahead by every w

(kw + k)z – t = /2

(kw + k)z – t = /2

j

Es

0dWdt

Electrons lose energy



Ponderomotive WaveThe electrons interact with the so-called ponderomotive wave with frequency and wavenumber kw + k. The ponderomotive wave is synchronous with the resonant electrons, i.e. those at the zero phase of the ponderomotive wave. The ponderomotive phase velocity, divided by kw + k, is slightly less than the speed of light. The phase of the ponderomotive wave is defined by average arrival time of the electrons

wk k z t

wz

d k kdz

Taking derivative with respect to z

Average electron axial velocity 2

2

112

wz

ac

2k

where



Phase Equation

2 wR

d kdz

2 2

2 2

1 11

2 2w w

w w

a ad k k k kdz c

Evolution of phase along the wiggler

1R

R R

Define an energy difference relative to the resonant energy R

The phase of individual electrons evolves along the wiggler according to their energy difference relative to the resonance energy

2

2R

w wd k kdz

Using the definition for resonance condition in k space 22 1

2w wR

kk a

2

1 2R

R



Energy Exchange Equation

sins wcka addt

Rewrite the above equation in terms of derivative with respect to z of the energy difference relative to the resonant energy, R

2 sins w

R R

ka addz

Energy exchange rate depends on the phase of electrons in the ponderomotive potential. Electrons with phase between –and 0 gain energy. Electrons with phase between 0 and lose energy.

,02

0

ss

eEa

km cDefine a dimensionless signal field parameter, as

The energy of an electron relative to the resonance energy evolves according to the sine of its phase in the ponderomotive wave



Coupled First-OrderDifferential Equations

2 sin

2

s w

R R

wR

ka addz

d kdz

sinv a

v

Evolution of relative energy difference and phase along the wiggler

Define new variables, and a

22

2 w

R

s w

R

k

v

ka aa

Pendulum equations

Rate of energy gain/loss along z

Rate of phase change along z

angular phase

= angular velocity

|a| = height of potential well

= oscillation frequency



Hamiltonian SystemHamiltonian mechanics is useful in representing beam physics because it relies on something being conservative. In the case of a pendulum, the conserved quantity is the total energy of the system of two canonical conjugate variables , the angular momentum, and , the angular phase.

2

cos2

H a

Hamiltonian = Total energy

Potential energyKinetic energy

Hamiltonian equations

sinH a

H



Pendulum Equations

sinv a

sinv a

v

Coupled non-linear 1st order differential equations

Particles rotate clockwise in phase space as the rate of change of is proportional to -sin and the rate of change of is . Particles follow elliptical trajectories each of which corresponds to a constant energy. Higher energies occupy larger ellipses up to phase angle of ± .



Small-angle Solutions

2v

v

2 0

The small-angle oscillation frequency is known as the synchrotron frequency 0. The synchrotron frequency is proportional to the square root of dimensionless optical field (fourth root of intensity).

01

s wR

ka ag

W =

0sin

Small-angle approximation, i.e. sin ~ leads to harmonic solutions with oscillation frequency , square root of |a|

Second-order differential equation

and its solution



Large-angle Close-orbit Solutions

Solutions corresponding to large-angle oscillations can be solved numerically. The large-angle oscillation frequency is lower than the small-angle synchrotron frequency and approaches zero at = ± Oscillation frequency is given by

where K : elliptic function. 20 0

2

sin2

K

p

zW

=æ öæ öW ÷ç ÷ç ÷÷ç ç ÷÷ç ÷ç è øè ø

20

0

116zW

» -W

Oscillation frequency for initial angle up to



Separatrix

Motion at the two nodes, , vanishes. These are unstable equilibrium points, corresponding to the pendulum at the top. The separatrix is the boundary separating trapped and un-trapped trajectories. The region inside the separatrix is called the “bucket.” The bucket height is proportional to the square root of the optical field (fourth root of optical intensity).

Separatrix for a uniform wiggler

0 2 cos 1v

Bucket half-height

max 21s w

w

a ava



Laser Field and Bucket Height

,02

0

ss

eEa

km cDimensionless optical (signal) field parameter, as

The electric field of the FEL beam depends on the optical intensity and free space impedance ,0 02s LE Z I

Laser intensity depends on power and mode radius20

2 LL

PIw

1 x 10-3max

3 x 10-6as

6 x 1010Electric field (V/m)

5 x 1014Intensity (W/cm2)

1.5 x 1010Peak power (W)

X-ray FEL at 1.5 Å

0 377Z

max 21s w

w

a aa

Bucket half-height



Open Orbits

Motion has large angular velocity. The pendulum rolls over the top and librates about the pivot point. The corresponding phase space trajectories are not elliptical. These represent un-trapped electrons outside the “bucket.” The un-trapped electrons also provide FEL gain. The electrons at small phases near the top of the “bucket” flow down into the “troughs”and lose energy to the optical field. As the optical field grows, the bucket also grows in height and eventually capture these electrons.



2 0 220

10

0

10

20Phase Space

Theta

Ener

gy

20

20

PHSP.10

i

1 PHSP.11

i

2 0 220

10

0

10

20Phase Space

Theta

Ener

gy

20

20

PHSP.10

i

1 PHSP.11

i

4 6 820

10

0

10

20Phase Space

Theta

Ener

gy

20

20

PHSP.10

i

2.75 .75 PHSP.11

i

2 4 6 820

10

0

10

20Phase Space

Theta

Ener

gy

20

20

PHSP.10

i

2.6 .6 PHSP.11

i

2 4 620

10

0

10

20Phase Space

Theta

Ener

gy

20

20

PHSP.10

i

2.5 .5 PHSP.11

i

2 4 620

10

0

10

20Phase Space

Theta

Ener

gy

20

20

PHSP.10

i

2.422 .422 PHSP.11

i

2 4 620

10

0

10

20Phase Space

Theta

Ener

gy

20

20

PHSP.10

i

2.365 .365 PHSP.11

i

2 4 620

10

0

10

20Phase Space

Theta

Ener

gy

20

20

PHSP.10

i

2.3 .3 PHSP.11

i

Synchrotron Oscillation Animation

2w



Synchrotron Oscillation Animation

2 4 6

50

0

50

Phase Space

Theta

Ener

gy

60

60

PHSP.10

i

2.3 .3 PHSP.11

i

2 4 6

50

0

50

Phase Space

Theta

Ener

gy

60

60

PHSP.10

i

2.15 .15 PHSP.11

i

0 2 4 6

50

0

50

Phase Space

Theta

Ener

gy

60

60

PHSP.10

i

2. . PHSP.11

i

0 2 4

50

0

50

Phase Space

Theta

Ener

gy

60

60

PHSP.10

i

1.85 .15 PHSP.11

i

0 2 4

50

0

50

Phase Space

Theta

Ener

gy

60

60

PHSP.10

i

1.8 .2 PHSP.11

i

2w

Change scale



MicrobunchingThe FEL interaction causes the electrons to gain or lose energy, depending on their ponderomotive phase. Electrons with positive ponderomotivephase lose energy and migrate to the bottom of the bucket. Electrons with negative ponderomotive phase gain energy and move to the top of the bucket. The resulting energy modulation causes the electrons to develop density modulation with period of the radiation wavelength. The bunched electrons radiate higher power, i.e. it amplifies the electromagnetic wave. As the electric field of the electromagnetic wave increases, the height of the bucket also increases. When the electrons are completely bunched, FEL power is saturated. Microbunching is responsible for harmonic generation (the Fourier transform of short bunches has high frequency components).

Courtesy of S. Reiche



Radiation from bunched beam

e

incoherent e

E NI N

(a)

w

(b)2

e

coherent e

E N

I N

Electrons at the wiggler entrance are randomly distributed (a). Randomly distributed electrons radiate incoherently, i.e. the electric fields of Ne randomly distributed wave trains with Nw (Nw is the number of wiggler periods and is the wavelength) add incoherently. The total electric field is proportional with square root of Ne. The spontaneous radiation intensity scales with Ne.

Near saturation, the electrons are bunched into microbunches with bunch length z less than radiation wavelength (b). The electric fields of Ne wave trains scales with Ne, and the coherent radiation intensity scales with Ne

2.

Nw

Nw

lb lb lb

z



Spontaneous Emission

[ ]

222 2 22

20

sin 2( )2 1

w e ww

w

e N N adW JJ ad d c a

gw pe

æ öDæ ö ÷ç÷ ÷ç ç÷ ÷= ç ç÷ ÷ç ç÷ç ÷W + Dè ø ç ÷è ø

Spectral and angular energy fluence of spontaneous emission radiation from a planar wiggler as a function of frequency detuning from resonance condition

2 wN

Spontaneous emission is peaked at zero detuning (resonant wavelength)

Frequency detuning

20 15 10 5 0 5 10 15 200

0.125

0.25

f ( )

2sin 2

1

wNwwD

=

( ) ( ) ( )0 1JJ J Jx x x= -

( )2

12 4

JJ x xxé ù » - -ë û

( )2

22 1w

w

aa

x =+Approximation for small

Difference between J0 and J1 Bessel functions



Spontaneous Emission (cont’d)Consider only photons within coherent spectral bandwidth and solid angle

2

w wNpl

pql

=

2

0

14 137e

ca

pe= »

where = fine structure constant

[ ]2

22( )

1photon w

we w

N aJJ aN a

paæ ö÷ç ÷= ç ÷ç ÷ç +è ø

Number of coherent spontaneous photons per electron does not depend on Nw

For typical values of aw, on average we need 200 electrons to generate 1 spontaneous photon within coherent angle and bandwidth

1

wNwwD

=

Coherent spectral bandwidth Solid angle

wLl

q=

Coherent angle



Madey’s TheoremMadey’s Theorem: The small-signal gain spectrum (gain versus energy detuning) for a low-gain FEL is the derivative of the spontaneous emission spectrum. The small-signal gain is positive (amplification) at positive detuning, zero on resonance and negative (absorption) at negative detuning.

( )( )3

3

4 41 cos sin

2w

ss

Ng

pr æ öD ÷çD = - D- D÷ç ÷çè øD

2.6 14 5w w

EE N NpD

= »

Maximum gain is at = 2.6

10 7.5 5 2.5 0 2.5 5 7.5 100.5

0

0.5

g ( )

gss()

4 wEN

E

Maximum gain occurs at positive energy detuning (higher energy) than resonance, or at a fixed energy, longer wavelength.

( )3max 2 2 wssg Npr»



Visualization of Madey’s TheoremOn resonance R No gain or loss

12 wN

Positive detuning R Amplification1

5 wN

Negative detuning R Absorption

15 wN

R



Small-Signal Gain

[ ]23

2 wwss

w A

JJ aN Igk I

pg s

æ ö æ öæ ö ÷ ÷÷ ç çç ÷ ÷= ÷ ç çç ÷ ÷÷ ç çç ÷ ÷÷ ççè ø è øè ø

where

and IA (Alfven current) = 17 kA

The small-signal gain for a planar wiggler at the peak of the gain curve, assuming the electron beam radius is smaller than the optical beam, is

( )1out ss inP g P= +

Small-signal gain in a low-gain FEL is proportional to Nw

3

0 0.06 0.12 0.18 0.24 0.3 0.36 0.42 0.48 0.54 0.6

1h

Fundamental power

z (m)

Peak

Pow

er (W

) gssPin

Pin

Power versus z in a low-gain FEL

gss scales with z3

Pout

Wiggler length (m)

Pea

k P

ower

(W)

2w

w

k pl

=



Large-Signal Gain

( )1

ss

S

gg III

=æ ö÷ç ÷+ç ÷ç ÷çè ø

Large-signal gain

At high intensity, more electrons reside at the bottom of the bucket and FEL gain decreases. Saturation intensity is the intensity at which FEL gain reduces to one-half of gss.

z (m)

Peak

Pow

er (W

)Pe

ak In

tens

ity (W

/cm

2 )

Wiggler Length (m)

Large-signal gain

Saturation Intensity

Peak Intensity

Small-signal gain

[ ]

2 431 18S

w w w

mcIJJ a N

gp s l

æ ö æ ö÷ç ÷ç÷ ÷= ç ç÷ ÷ç ç÷ ÷ç÷ç è øè ø

FEL gain is reduced when optical intensity approaches the saturation intensity,



Synchrotron Oscillation

2

2

2

sin

2

s w

R R

wR

ka addz

d dkdz dz

Energy and phase equations

22

2 sin 0Sd Kdz

2nd-order differential equation of phase evolution with z

2 2

2 21

w s w s wS w

R w

k ka a a aK ka

Synchrotron oscillation wavenumber

212w w

Ss w

aa a

Synchrotron period

0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4

z (m)

Peak

Pow

er (W

)

Plot of power vs z showing synchrotron oscillations

S



Wiggler length ~ synchrotron period

Extraction Efficiency

2

max

max

12

2

2

w ww S

s w

ww

w

w

aLa a

L

L

At saturation, the wiggler length is about the same as a synchrotron oscillation period. The electrons rotate to the bottom of the “bucket.” The bucket half-height is inversely proportional to 2Nw.

max1

2 wN

1

2 wN



High-Gain FEL

[ ] ( ) ( )0 1JJ J Jx x= -

[ ]2 13 31

2w

w A

JJ a Ik I

rg s

æ ö æ ö÷ ÷ç ç÷ ÷= ç ç÷ ÷ç ç ÷÷ çç è øè ø

Dimensionless Pierce parameter as a function of kw (left) or w (right)

34w

GL

High gain FEL is applicable in a long wiggler driven by a high-brightness electron beam (one with high peak current and small emittance). The wiggler length must be significantly longer than the power gain length, given by

[ ]2 13 31

4 2w w

A

JJ a II

lr

g ps

æ ö æ ö÷ ÷ç ç÷ ÷= ç ç÷ ÷ç ç ÷÷ çç è øè ø

Recall JJ is the difference between J0 and J1 Bessel functions of argument

( )2

22 1w

w

aa

x =+

where[ ]

2

12 4

JJ x x» - -

Power gain length

( )2

0 14

J xx » - ( )1 2

J xx »



Power Growth in High-Gain FEL

GLzexpP)z(P 09

1

Power grows exponentially with distance by one e-folding (2.7) every power gain length. Starting from noise, the FEL saturates in 20 power gain lengths. FEL saturation power, Psat, is approximately times the electron beam power.

Power vs distance

bsat

IEPe

Saturation power0 0.4 0.7 1.1 1.4 1.8 2.2 2.5 2.9 3.2 3.6

1h

Fundamental power

z (m)

Log

Pow

er

Natural log of FEL power vs z (wiggler length)

Exponential growth

SynchrotronOscillation

Psat

P0

0

9ln satsat G

PL LP

Saturation length

Lsat



Slowly Varying Envelope ApproximationSo far, we’ve considered only the electron phase-space motion. To be complete, we must write self-consistent FEL equations for N electrons and the optical field. We’ll treat the optical field as a slowly varying phasor (ignoring the optical frequency oscillation). The phasor’s amplitude is the usual dimensionless optical field as. This is known as the Slowly Varying Envelope Approximation (SVEA).

( ) [ ]341 12

i

wb A

JJ I ea a iaz c t k I

qpg g

-é ùæ öé ù¶ ¶ ÷ç ê ú÷ê ú+ = -ç ÷ê úç ÷çê ú¶ ¶ S è øë û ë ûThe electron bunch is assumed to be many wavelengths long, so the beam current density is assumed to be independent of z over many wavelengths.

Wave equation without the fast time scale terms (e.g. 2nd order derivatives)

0( ) exp ( )E t E i kz t Optical electric field with fast oscillations

e sisa a f-=

SVEA phasor



Self-Consistent FEL Equations

The <cos> term corresponds to the real part of the e-beam’s susceptibility (refractive index) and <sin> term corresponds to the imaginary part (gain).

22 1 2 cos

2

sin

jw w w s j

j

j w sj

j

d kk a a a JJdz

d ka a JJdz

Evolution of the jth electron’s phase and energy

( ) [ ]

( ) [ ]

3

3

4 cos 12

4 sin2

ws

b A s

ws

b A

a JJd Idz k I a

a JJda Idz k I

pf qg g

p qg

é ùæ ö÷ç ê ú÷= -ç ÷ç ê ú÷çS è ø ë ûæ ö÷ç ÷= ç ÷ç ÷çS è ø

Evolution of optical phasor’s phase and amplitude



Scaled Variables

• Scaled axial position

• Dimensionless current density

• Scaled phasor equation

( )32 4 wj Np r=

( ) '0' ' '

02

i

i

da j edda j a e dd

q

tn t

t

t t t tt

-

-

=-

= -ò

w

zL

t =



Take the derivative of the last equation successively

Assuming solutions are of the form ei and at resonance condition, we obtain the characteristic cubic dispersion relation

Note: are roots of the cubic equation, not wavelength

Solutions of the cubic equation are of the form

Cubic Equation

( ) ( )3

3 2d a ja

dt t

t=-

3 02j

l + =

( ) 0ia a e ltt =



Solutions to Cubic Equation

11 13 13 3 32 2

2 22 2 2 2

3 3 30( )

3

jj j j

iiE eE e e e

Complex root

13

11 3

2 2 2j i

Complex root

13

3 2j

Real root

Three roots of the cubic equation

Solutions in electric field

13

21 3

2 2 2j i

growingmode

decayingmode

oscillatorymode

Im

Re-1½

32



Exponential GrowthIn the limit of large z, only the growing mode needs to be considered. The optical field vs scaled length is given by

Multiplying the electric field by its complex conjugate yields the FEL intensity versus the scaled length

1232 0( ) exp 3

9 2E jE

0 4 3( ) exp9 w

I zI z

1 13 3

23

2 2 2103( )

j ji

E E e e

Plug in the expressions for and j, we arrive at the expression for intensity vs. distance in the wiggler. This equation gives the exponential growth with wiggler length and the initial 1/9 reduction in signal intensity.



References

Free-Electron Lasers C.A. Brau

Particle Accelerator Physics I & II H. Wiedmann

Physics of Free Electron Lasers E.L. Saldin, E.A. Schneidmillerand M.V. Yurkov

“Free Electron Lasers” S. Khan (2008) J. of Modern Optics, 55:21,3469 – 3512

“Development of X-ray Free-Electron Lasers” C. Pellegrini and S. Reiche (2004)J. Quantum Electronics, 10(6) 1393-1404

Books and Articles

URLUC Santa Barbara WWW FEL http://sbfel3.ucsb.edu/www/vl_fel.html

Linac Coherent Light Source http://www-ssrl.slac.stanford.edu/lcls/

European XFEL http://xfel.desy.de/

theory and practice of free-electron lasers - us particle

Documents