Theory and Practice of Free-Electron Lasers 1.pdf · 2015. 3. 5. · Introduction to Free-Electron...
Transcript of Theory and Practice of Free-Electron Lasers 1.pdf · 2015. 3. 5. · Introduction to Free-Electron...
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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
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11--22LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
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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
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Chapter 1Introduction to Free-Electron Lasers
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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
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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
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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
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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
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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)
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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
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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
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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║Bwy z
x
wigglermagnets
electron trajectory
unbunched electron beam pulse
bunched electron beam pulse
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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
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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
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Wavelength Tunability
22 12w
wa
w wiggler period resonant wavelength
relativistic factoraw (also Krms) rms wiggler parameter
0 00
0.662w ww
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 2w K
0
0
2 ww
eBK ak m c
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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
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RF-Linac FEL Pulse Structure
FEL macropulseTmacro
1RF
RF
tf
n tRF
FEL micropulses
RF wave train
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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
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FEL optical beam properties
• Intensity
2p
x y
N hvB
t
20
2 pN hvIw t
2p
x y
Nt
B
W/m2
W/cm2
photons/(m2 s 0.1% BW)
1
wN
• Brightness
• Spectral bandwidth
• Brilliance
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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)
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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
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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
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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).
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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
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Chapter 2Basics of Relativistic Dynamics
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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
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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
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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
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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
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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
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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
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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
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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
= -
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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 211
2b
g- »
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 cosq
qg
-=
For small angles
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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
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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
=
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Wavelength Dependence on Angle
( )2 22 12wll 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 waqg
=
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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
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Parameter Variation Table
d
d
dpp
d
dpp
d
1
1
1
2
2 2
2
1 2 2
1
2
1
2
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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.
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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.
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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
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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 b
B T m E MeV
Magnetic rigidity
incident beamdipole magnet
bent beam
Bend angle and radiuss
p┴
p║
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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
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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
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Lorentz Forces
20
y yecB eBxm 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
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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’
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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
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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
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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
22 01
1 Nj
jx x x
N 22
1
1 Nj
j
x xN
01
1 Nj j
jxx x x x
N
Root-mean-square x emittance (for y emittance, replace x with y)
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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/pzpxpz
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.
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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
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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
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Chapter 31-D Theory of FEL
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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
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Equations of Motion 0ˆ cosy wB yB k z
wwk
2
Byy
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
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Equations of Motion (cont’d)
x
z
cc
x z x 2z zdx x xdt
0
coso weBx 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
cosz o wed xx B k z
dt 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 zConsider only on-axis magnetic field
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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
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B field, deflection and position
Wiggler magnetic field
+ error - error
First integral of field (deflection)
Second integral of field (position)
2cos sinwo w waB k z dz k z
2cos coswo w ww
aB k z dz dz k zk
coso wB k z
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Transverse and longitudinal velocities
2 sinwx wca 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 22w
z z wca k z
Find the square root and use small x approximation (1 + x)½ ≈ 1 + ½ x
( )( )2 2211 1 2 sin
2z w wc a k zu
g
é ùê ú= - +ê úë û
( ) ( )22sin 1 cos 2w wk z k z= -Use sine squared identity
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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
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Energy exchange betweenelectrons and FEL beam
sdW j Edt
sinw wecaj 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( )w wd m c eca E k z kz t
dt
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”
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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
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Ponderomotive phase = -/2kwz = 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
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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
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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
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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
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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
1RR 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 12w wRkk a
2
1 2RR
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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
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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
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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
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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 ± .
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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
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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 02
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
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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
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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
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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.
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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
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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
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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
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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 Ne2.
Nw
Nw
lb lb lb
z
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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
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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( ) 1
photon ww
e 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
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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»
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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
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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
=
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11--8787LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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,
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11--8888LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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 22 2
1w s w s w
S wR 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
-
11--8989LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
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11--9090LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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 1 4J xx » - ( )1 2
J xx »
-
11--9191LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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 GPL LP
Saturation length
Lsat
-
11--9292LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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 af-=
SVEA phasor
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11--9393LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
Self-Consistent FEL Equations
The term corresponds to the real part of the e-beam’s susceptibility (refractive index) and term corresponds to the imaginary part (gain).
22 1 2 cos2
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
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11--9494LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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 =
-
11--9595LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
( ) ( )33 2
d a jad
t tt
=-
3 02j
l + =
( ) 0 ia a e ltt =
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11--9696LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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
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11--9797LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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.
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11--9898LA-UR 09-01205US Particle Accelerator School 2009US Particle Accelerator School 2009
University of New Mexico University of New Mexico -- Albuquerque NMAlbuquerque NM
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/