Introduction to
Free-Electron Lasers
Neil Thompson
ASTeC
Introduction to FELs
Outline
 Introduction: What is a Free-Electron Laser?
 How does an FEL work?
 Choosing the required parameters
 Laser Resonators for FELs
 FEL Output Characteristics
 FEL vs Conventional Laser
 Current Trends
Introduction to FELs
What is a Free-Electron Laser?
Introduction to FELs
What is an FEL?
A beam of relativistic electrons
co-propagating with
an optical field
through
a spatially periodic magnetic field
 Undulator causes transverse electron oscillations
 Transverse e-velocity couples to E-component (transverse) of
optical field giving energy transfer.
 Interaction between electron beam and optical field causes
microbunching of electron beam on scale of radiation wavelength
leading to coherent emission
Introduction to FELs
What is an FEL?
Output is radiation that is
tunable
powerful
coherent
Introduction to FELs
Types of FEL
 AMPLIFIER (HIGH GAIN) FEL
 Long undulator
 Spontaneous emission from start of undulator interacts with electron
beam.
 Interaction between light and electrons grows giving microbunching
 Increasing intensity gives stronger bunching giving stronger emission
 >>> High optical intensity achieved in single pass
 OSCILLATOR (LOW GAIN) FEL
 Short undulator
 Spontaneous emission trapped in an optical cavity
 Trapped light interacts with successive electron bunches leading to
microbunching and coherent emission
 >>> High optical intensity achieved over many passes
Introduction to FELs
How does a Free-Electron Laser work?
Introduction to FELs
How does an FEL work?
y
x
B
E
vx
 Basic components
v
z
B field
S
N
S
N
N
S
N
S
S
N
Electron path
Introduction to FELs
E field
How does an FEL work?
 Basic physics: Work = Force x Distance
 Sufficient to understand basic FEL mechanism
 Electric field of light wave gives a force on electron and work is
done!
 No undulator = No energy transfer
i.e. If electron velocity is entirely longitudinal then v.E = 0
 Basic mechanism very simple!!
Introduction to FELs
How does an FEL work?
 Basic mechanism described explains energy transfer
between SINGLE electron and an optical field.
But in practice need to create right conditions for:
CONTINUOUS energy transfer
in the RIGHT DIRECTION
with a REAL ELECTRON BEAM
Introduction to FELs
How does an FEL work?
 Q. How do we ensure continuous energy transfer
over length of undulator?
 A. Inject at RESONANT ENERGY:
 This is the energy at which the electron slips back 1 radiation
wavelength per undulator period: relative phase between electron
transverse velocity and optical field REMAINS CONSTANT.
 Why does it slip back? Because electron longitudinal velocity < c:
 Not 100% relativistic
 Path length increased by transverse oscillations
Introduction to FELs
How does an FEL work?
Ex
 Q. Which way does
the energy flow?
 A. Depends on
electron phase
vx
vx
t1
 Depending on phase,
electron either:
 Loses energy to optical
field and decelerates:
GAIN
 Takes energy from
optical field and
accelerates:
ABSORPTION
t2
t3
Introduction to FELs
z
Does an FEL work?
 Q. What about a real e-beam?
 A. Electrons distributed evenly in phase:
 For every electron with phase corresponding to gain there is
another with phase corresponding to absorption
 So at resonant energy Net
gain is zero
So is this the end of the story??
Introduction to FELs
How does an FEL work
•No! There is a way to proceed.
•Energy modulation gives bunching.
•At resonant energy bunching is around
phase for zero net gain.
•By giving the electrons a bit of an energy
kick we can shift them along in phase a bit
and get bunching around a phase
corresponding to positive net gain.
Introduction to FELs
How does an FEL work?
 Bunching
Phase corresponding to
GAIN
Phase corresponding to
ABSORPTION
E = RESONANT ENERGY
t1
Bunching around phase
corresponding to
ZERO NET GAIN
t2
E > RESONANT ENERGY
t1
Bunching around phase
corresponding to
POSITIVE NET GAIN
t2
Introduction to FELs
A Simulation example
 Inject 20 electrons at resonance energy: zero detuning
absorption
gain
Detuning
gain
• Bunching around
phase
correponding to
zero gain
• 10 electrons lose
energy
0
• 10 electrons gain
energy
Phase
Introduction to FELs
A Simulation example
 Inject 20 electrons above resonance energy: +ve detuning
absorption
gain
Detuning
gain
• Bunching around
phase
correponding to
+ve gain
• 11 electrons lose
energy
+ve
• 9 electrons gain
energy
0
Phase
Introduction to FELs
How does an FEL work?
For
ForGAIN
GAIN>>ABSORPTION
ABSORPTIONinject
injectelectrons
electronsat
atenergy
energy
slightly
slightlyhigher
higherthan
thanresonant
resonantenergy
energy
‘POSITIVE
‘POSITIVEDETUNING’
DETUNING’
>>>>
>>>>NET
NETTRANSFER
TRANSFERof
ofenergy
energyto
tooptical
opticalfield
field
NB: this is only true for LOW GAIN FELs. For a HIGH GAIN FEL the
maximum gain occurs at a positive detuning much closer to zero. But that’s
another story….
Introduction to FELs
Gain
Gain
Small-Signal Gain Curve
2.6 2π
Spontaneous emission
Energy detuning δ
 Shows how gain varies as a
function of detuning
 Detuning parameter
 δ = 4πN(∆E/E)
 Maximum gain for δ = 2.6
 Madey Theorem
 “Gain curve is proportional to
negative derivative of spontaneous
emission spectrum”
Broadening of natural linewidth
causes gain degradation
Energy detuning δ
Introduction to FELs
The Oscillator FEL
electron bunch
output pulse
optical pulse
Electrons bunched at
radiation wavelength:
coherent emission
Random electron phase:
incoherent emission
Introduction to FELs
The Oscillator FEL
 For oscillator FEL the single pass gain is small.
 The emitted radiation is contained in a resonator to produce
a FEEDBACK system
 Each pass the radiation is further amplified
 Some radiation extracted, most radiation reflected
 Increasing cavity intensity strengthens interaction leading
to exponential growth:
Energy transfer depends on cavity intensity
Introduction to FELs
Saturation: Oscillator
 For ZERO cavity loss intensity would increase indefinitely!
 In practice we have passive loss (diffraction, absorption) and
active loss (outcoupling)





Power lost proportional to cavity intensity
As intensity rises so does power loss.
Finite extraction efficiency
Eventually power lost = power extracted from electrons
No more growth. SATURATION.
 (Equivalently gain falls until it equals cavity losses)
Introduction to FELs
Saturation: Oscillator
cavity intensity
Intensity
 Parameters: cavity loss = 4%
pass 150
gain
Gain = Losses
Intensity
Distance along undulator
pass 40
pass number
Distance along undulator
Introduction to FELs
Amplifier-like
saturation in
single pass:
electrons
have
continued to
rotate in
phase space
until they
start to
reabsorb
from the
optical field
Choosing the required parameters
Introduction to FELs
Required Parameters
 To achieve lasing with our Oscillator FEL the
following parameters must be optimised
 Electron beam parameters
 Energy, Peak Current, Emittance, Energy spread
 Undulator parameters
 K, Number of periods, Period
 Resonator parameters
 Length, mirror radius of curvature
For lasing must have
GAIN > LOSSES
Introduction to FELs
Required parameters
 To ‘first order’
 Select base parameters to optimise gain
 To ‘second order’
 Optimise other parameters to minimise gain degradation
Introduction to FELs
Gain Scaling
 We can get an idea of how the gain should scale with various
parameters by looking at our familiar equation
Total charge
depends on beam
current
Transverse
velocity depends
on K and beam
rigidity
Optical E-field
depends on area
of optical mode
Introduction to FELs
Interaction time
depends on
undulator length
Small Signal Gain
 In fact, small signal, single pass maximum gain is given by:
Undulator periods
Undulator K
Peak current
Filling factor:
averaged over undulator length
Beam Energy
Electron beam area
These parameters varied to tune FEL:
Introduction to FELs
Small Signal Gain
 To summarise, at a given wavelength we need:
 A long enough undulator
To allow sufficient interaction time
 A good peak current and
a tightly focussed electron beam
To provide high charge
density
 A small optical cross section
To provide high E field
Introduction to FELs
Electron beam quality
 Now we’ve selected parameters to optimise the gain,
we need to minimise the gain degradation
 Gain is degraded due to
 energy spread
 emittance
For optimum FEL performance a high quality
electron beam is required.
Introduction to FELs
Energy Spread
Gain
 SMALL SIGNAL
GAIN CURVE:
 Small energy range for
positive gain
 If energy spread is too
large electrons fall
outside detuning for
positive gain
2π
Energy detuning δ
Introduction to FELs
GAIN DEGRADATION
Energy Spread
 Derivation of Limit on Energy Spread:
 FEL wavelength given by:
Spontaneous emission
• Expand this in Taylor series
giving linewidth spread due
to energy perturbation
1/2N
• Require that
spread is less than
the natural line
halfwidth so that
no broadening
occurs and gain
is not reduced
Energy detuning δ
Introduction to FELs
ERLP IRFEL:
N=42
Gives rms energy spread
of < 0.1% for negligible
gain degradation
Emittance
 Emittance controls:
 1. BEAM DIVERGENCE: this effects overlap between
electron beam and optical mode
 2. BEAM SIZE: this affects quality of undulator field seen by
electrons
 We can do a separate analysis for each case to see how
small an emittance we need to avoid gain degradation
Introduction to FELs
Emittance 1: overlap
• We need the electron beam contained within optical beam
over whole interaction length
optical beam
small emittance
electron beam
good
Interaction length
large emittance
bad
Introduction to FELs
Emittance 1: overlap
 Electron beam envelope equation at a waist
gives electron beam Rayleigh length:
 Similarly, Gaussian beam equations in laser
resonator gives optical Rayleigh length:
 For electron beam confinement need:
ERLP IRFEL:
 So:
Introduction to FELs
Gives normalised
emittance of
< 10 mm-mrad
Emittance 2 : broadening
 As emittance increases beam size increases so electrons move
more off-axis
 Undulator field has a sinusoidal z-dependence on axis, so off
axis electrons experience a different field (because curl B = 0)
and thus a different K.
wavelength
shift
linewidth
broadening
gain
degradation
 By requiring that linewidth broadening is within the natural
linewidth the following restriction can be derived:
ERLP IRFEL:
Introduction to FELs
Gives normalised
emittance of
< 500 mm-mrad
Longitudinal effects
 So far we’ve only considered an ‘infinitely long’
electron beam: we haven’t worried about the ends.
 Known as the STEADY STATE solution
 In reality we have finite electron bunches
 Need to extend model accordingly to include effects
of PULSE PROPAGATION
2D MODEL
(transverse effects)
3D MODEL
(transverse + longitudinal
effects )
Introduction to FELs
Longitudinal effects
 Slippage
 Resonance condition: Electrons slip back by one radiation
wavelength per undulator period
 Slippage per undulator traverse = N x wavelength. This is
known as slippage length.
 For short electron bunch and/or long wavelength we have
slippage length ≈ bunch length
Effective interaction length reduced: GAIN DEGRADED.
Introduction to FELs
Pulse effects: Slippage
 Slippage
Long electron
bunch
Undulator length
optical pulse
electron bunch
Interaction length = undulator length
Short electron
bunch
Reduced Interaction length
Introduction to FELs
Slippage
length
Pulse effects: Lethargy
 The electrons slip back over the optical pulse:
 bunching increases, and maximum emission occurs at end of
undulator where bunching is strongest
 RESULT: optical pulse peaks at rear and centroid of pulse has
velocity < c.
 Synchonism between pulse and electron bunch on next pass is
not perfect
 Known as laser lethargy
ct
Pulse centroid
vt (v < c)
Introduction to FELs
Pulse effects: Lethargy
 Lethargy can be offset by reducing cavity length
slightly: CAVITY LENGTH DETUNING
 Centroid of optical pulse is then synchronised with e-bunch on
successive passes
 BUT: as intensity increases
 back of pulse saturates first
 then rest of pulse saturates, returning centroid velocity to
vacuum value c.
 So a single detuning can’t compensate for lethargy in both
growth and saturation phases.
 Different detunings for gain optimisation and power
optimisation
Introduction to FELs
Cavity length detuning
one cavity length
for maximum gain
another cavity length
for maximum power
Introduction to FELs
Summary
0D beam:
(single electron)
1D beam:
(Resonance condition)
(Injection above resonance for net gain)
2D beam:
transverse effects
(Emittance – beam overlap and broadening
3D beam:
longitudinal effects
(Pulse effects – slippage and lethargy)
Energy spread)
Introduction to FELs
Resonators for Free-Electron Lasers
Introduction to FELs
Optical Resonators
 Some general properties:
 Optical resonator consists of two spherical mirrors, radius of curvature R1 and R2,
separated by a distance D.
R1
R2
D
Introduction to FELs
Optical Resonators
 Eigenmode is not a plane wave, but a wave whose front is
curved.
 At mirrors radius of curvature matches mirror surface
 Fundamental TEM00 has a Gaussian transverse intensity
distribution
Z=0
planar wavefront
Gaussian profile
2w0
2w0 x √2
Introduction to FELs
mirror curvature = wavefront
curvature
Gaussian profile
Rayleigh length z = zR
maximum curvature
Optical Resonators
 More general properties:
 Length D has certain allowed values:
must have optical round trip time = bunch repetition frequency
 Mode size at waist (R1=R2=R) :
 Mode size z from waist:
 Rayleigh length:
 So for a particular wavelength and D we can choose R to give required waist:
defines both the Rayleigh length and the profile along the whole resonator.
 BUT only certain combinations of D, R are stable!
Introduction to FELs
Optical Resonators: Stability
 Wave propagation treated using element by element transfer
matrices (analogy with beam optics).
 For stability (i.e. existence of a stable periodic solution) must
have
where M is round trip transfer matrix (same as for storage rings)
 This gives the stability condition
Introduction to FELs
Resonators: Stability Diagram
 Plot of g1 vs g2 can be used
to show stable and unstable
regions [1]
 FEL cavities required to
focus beam for good
coupling, so typically
between confocal and
concentric
FELIX
LANL
CLIO
FELI
ERLP?
[1] Kogelnik and Li, Laser Beams and Resonators,
SPIE MS68, 1993
originally published:
Applied Optics, Volume 5, Issue 10, 1550Introduction to FELs
October 1966
Resonator Choice
CONFOCAL
PROPERTIES
R1
R2
Centres of curvature
far apart:
insensitive to mirror
alignment error
Large waist = small
filling factor
Introduction to FELs
Small mirror spot =
high power density +
low diffraction loss
Resonator Choice
 Confocal Angular Alignment Sensitivity
R1
R2
OPTICAL AXIS
OPTICAL AXIS
D
Introduction to FELs
Resonator Choice
CONCENTRIC
PROPERTIES
R1
Centres of curvature
close = sensitive to
mirror alignment
error
R2
small waist = large
filling factor
Introduction to FELs
large mirror spot =
low power density +
high diffraction loss
 Concentric Angular Alignment Sensitivity
OP
T IC
AL
A
R1
XIS
R2
OPTICAL AXIS
D
Introduction to FELs
Resonator Choice
 From gain scaling have that gain proportional to filling factor
 Filling factor is averaged along undulator length, giving an
optimum resonator configuration for maximum gain:
Undulator length L
Undulator length L
Undulator length L
optical
electron
ZR > L/3
ZR ~ L/3
OPTIMUM
Introduction to FELs
ZR < L/3
Resonator Choice
 For an FEL it is difficult to achieve a small cavity length:
 Restriction due to electron bunch frequency
 Space required for dipoles and quads
 So the small Rayleigh length for optimum coupling must be
achieved despite a large D:
 This is only possible if 2R~D, giving g=1-D/R ~ -1: i.e. on the
boundary of the stable region of the stability diagram
PERFORMANCE VS STABILITY COMPROMISE
Introduction to FELs
Resonators: outcoupling
 There are various methods for getting the light out of the cavity:
Outcoupling Method
Pros
Cons
Hole in Mirror
Broadband.
Simple.
Outcoupling fraction changes
with wavelength.
Mode distortion.
Partially Transmitting Mirror
No mode distortion.
Not broadband: only suitable
for certain wavelengths.
Brewster Plate
Variable outcoupling fraction.
No mode distortion.
Complicated.
Polarisation dependent.
Scraper Mirror
Variable outcoupling fraction.
Variable scraper position.
Complicated.
Introduction to FELs
Free-Electron Laser Output
Introduction to FELs
ERLP IRFEL:
FEL Output: power
I = 50A
E = 35MeV
N = 42
Gives P = 8MW
 δ = 4πN(∆E/E)
 Maximum gain for δ = 2.6
 Maximum energy that can be lost by an
electron injected at peak of gain curve is δ =
2.6: no more gain after that.
Gain
Gain
Gain curve can be used to estimate maximum output power:
2.6
2π
Energy detuning δ
Dividing by time, recognising that at
equilibrium
power extracted from electron beam =
output power
Introduction to FELs
FEL Output
 At saturation pulse length matches electron bunch length
 Linewidth depends on pulse length (Fourier)
 Brightness: output is coherent, so assuming diffraction
limited beam:
ERLP IRFEL:
P = 8 MW
Wavelength = 4 micron
 High power enables high brightness!
Gives B ~ 1026
Introduction to FELs
FEL Output
XUV-FEL
IR-FEL
2
Peak Brightness ph/(s.0.1%bp.mm .mrad )
1.E+30
VUV-FEL
2
1.E+26
1.E+22
Undulators
coherent enhancement
1.E+18
Bending Magnet
1.E+14
1.E+10
0.001
0.01
0.1
1
Photon energy, eV
Introduction to FELs
10
100
1000
FEL Output
TUNABLE OUTPUT!
A typical FEL facility operates at certain fixed beam
energies then tunes over wavelength sub-ranges by
varying undulator gap (and hence K)
Introduction to FELs
Free-Electron Lasers vs Conventional Lasers
Introduction to FELs
FEL vs Conventional Laser
 Conventional Laser
 Light Amplification by Stimulated Emission of Radiation
 Electrons in bound states - discrete energy levels
 Waste heat in medium ejected at speed of SOUND
Limited tunability
Limited power
Continuous tunability
High power
 Free-Electron Laser
 Light Amplification by Synchronised Electron Retardation ??
 Electrons free - continuum of energy levels
 Waste heat in e-beam ejected at speed of LIGHT
(and recovered in ERL)
Introduction to FELs
Current Trends and the way ahead
Introduction to FELs
Current trends
 Average power of FELs is increasing
 JLAB - record continuous average power
of 2.13kW (IR)
 JAERI - 2kW over a 1ms macropulse (IR)
 High average power goal is several tens of kW
 JLAB upgrade promises >10kW.
 Wavelengths decreasing
 190nm @ ELETTRA - storage ring oscillator FEL.
 Mirror technology limiting factor for oscillators
 80nm @ TTF1 - amplifier FEL
Introduction to FELs
Applications
 Medical
 FELs with high-peak and highaverage power are enabling
biophysical and biomedical
investigations of infrared tissue
ablation. A midinfrared FEL has
been upgraded to meet the
standards of a medical laser and is
serving as a surgical tool in
ophthalmology and human
neurosurgery.
Introduction to FELs
Applications
 Power Beaming and Remote Sensing
Introduction to FELs
Applications
 Micromachining
 A free-electron laser can be used for to fabricate three-dimensional
mechanical structures with dimensions as small as a micron
 The images above are 12-micron, micro-optical Fresnel Lens
components created by laser-micromachining.
Introduction to FELs
The End
Introduction to FELs