Introduction to FELs
Introduction to
Free-Electron LasersNeil Thompson
ASTeC
Introduction to FELs
Outline
Introduction: What is a Free-Electron Laser?How does an FEL work?Choosing the required parametersLaser Resonators for FELsFEL Output CharacteristicsFEL vs Conventional LaserCurrent Trends
Introduction to FELs
What is a Free-Electron Laser?
Introduction to FELs
What is an FEL?
Undulator causes transverse electron oscillationsTransverse 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
A beam of relativistic electronsco-propagating withan optical field
through a spatially periodic magnetic field
Introduction to FELs
What is an FEL?
Output is radiation that is tunable
powerfulcoherent
Introduction to FELs
Types of FEL
AMPLIFIER (HIGH GAIN) FELLong undulatorSpontaneous emission from start of undulator interacts with electron beam.Interaction between light and electrons grows giving microbunchingIncreasing intensity gives stronger bunching giving stronger emission>>> High optical intensity achieved in single pass
OSCILLATOR (LOW GAIN) FELShort undulatorSpontaneous emission trapped in an optical cavityTrapped 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?
Basic components
N S SNS
N S NSN
B field Electron path E field
BE
z
v
xy
vx
Introduction to FELs
How does an FEL work?
Basic physics: Work = Force x DistanceSufficient to understand basic FEL mechanismElectric field of light wave gives a force on electron and work is done!
No undulator = No energy transferi.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% relativisticPath length increased by transverse oscillations
Introduction to FELs
How does an FEL work?
Q. Which way does the energy flow?A. Depends on electron phase
Depending on phase, electron either:
Loses energy to optical field and decelerates: GAINTakes energy from optical field and accelerates: ABSORPTION
vx
z
Ex
t1
t2
t3
vx
Introduction to FELs
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 absorptionSo 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?
BunchingPhase corresponding to GAIN
Phase corresponding to ABSORPTION
E = RESONANT ENERGYBunching around phase corresponding toZERO NET GAINt2
t1
E > RESONANT ENERGYBunching around phase corresponding to POSITIVE NET GAINt2
t1
Introduction to FELs
A Simulation example
Inject 20 electrons at resonance energy: zero detuning
Det
unin
g
Phase
gain gainabsorption
0
• Bunching around phase correponding to zero gain
• 10 electrons lose energy
• 10 electrons gain energy
Introduction to FELs
A Simulation example
Inject 20 electrons above resonance energy: +ve detuning
Det
unin
g
Phase
gain gainabsorption
0
• Bunching around phase correponding to +ve gain
• 11 electrons lose energy
• 9 electrons gain energy
+ve
Introduction to FELs
How does an FEL work?
For GAIN > ABSORPTION inject electrons at energy slightly higher than resonant energy
‘POSITIVE DETUNING’
>>>> NET TRANSFER of energy to optical field
For GAINGAIN > > ABSORPTIONABSORPTION inject electrons at energy slightly higher than resonant energy
‘POSITIVE DETUNING’
>>>> NET TRANSFER of energy to optical 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
Small-Signal Gain Curve
Shows how gain varies as a function of detuningDetuning 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 linewidthcauses gain degradation
Energy detuning δ
Spon
tane
ous
emis
sion
Energy detuning δ
Gai
n
2.6 2πGai
n
Introduction to FELs
The Oscillator FEL
Electrons bunched at radiation wavelength: coherent emission
electron bunch
optical pulse
output pulse
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 systemEach pass the radiation is further amplifiedSome radiation extracted, most radiation reflectedIncreasing 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 intensityAs intensity rises so does power loss.Finite extraction efficiencyEventually power lost = power extracted from electronsNo more growth. SATURATION.(Equivalently gain falls until it equals cavity losses)
Introduction to FELs
pass number
gain
cavi
ty in
tens
ity
Gain = Losses
Saturation: Oscillator
Parameters: cavity loss = 4%
pass 40In
tens
ity
Distance along undulator
Distance along undulator
Inte
nsity pass 150
Amplifier-like saturation in single pass:
electrons have
continued to rotate in
phase space until they start to
reabsorb from the
optical field
Introduction to FELs
Choosing the required parameters
Introduction to FELs
Required Parameters
To achieve lasing with our Oscillator FEL the following parameters must be optimised
Electron beam parametersEnergy, Peak Current, Emittance, Energy spread
Undulator parametersK, Number of periods, Period
Resonator parametersLength, 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
Interaction time depends on
undulator length
Introduction to FELs
Small Signal Gain
In fact, small signal, single pass maximum gain is given by:
Beam Energy
Peak currentUndulator periods
Undulator K
Electron beam area
These parameters varied to tune FEL:
Filling factor: averaged over undulator length
Introduction to FELs
Small Signal Gain
To summarise, at a given wavelength we need:
A long enough undulator
A good peak current and a tightly focussed electron beam
A small optical cross section
To allow sufficient interaction time
To provide high charge density
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 spreademittance
For optimum FEL performance a high quality electron beam is required.
Introduction to FELs
Energy Spread
SMALL SIGNAL GAIN CURVE:Small energy range for positive gainIf energy spread is too large electrons fall outside detuning for positive gain
2π
Energy detuning δ
Gai
n
GAIN DEGRADATION
Introduction to FELs
Energy Spread
Derivation of Limit on Energy Spread:FEL wavelength given by:
• Expand this in Taylor series giving linewidth spread due to energy perturbation
• Require that spread is less than the natural line halfwidth so that no broadening occurs and gain is not reduced
Energy detuning δ
Spon
tane
ous
emis
sion
1/2N
ERLP IRFEL:N=42
Gives rms energy spread of < 0.1% for negligible
gain degradation
Introduction to FELs
Emittance
Emittance controls:
1. BEAM DIVERGENCE: this effects overlap between electron beam and optical mode2. 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
electron beam
good
bad
optical beam
Interaction length
small emittance
large emittance
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:
So: ERLP IRFEL:
Gives normalised emittance of
< 10 mm-mrad
Introduction to FELs
Emittance 2 : broadening
As emittance increases beam size increases so electrons move more off-axisUndulator 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.
By requiring that linewidth broadening is within the natural linewidth the following restriction can be derived:
wavelength shift
linewidthbroadening
gain degradation
ERLP IRFEL:
Gives normalised emittance of
< 500 mm-mrad
Introduction to FELs
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 solutionIn reality we have finite electron bunchesNeed to extend model accordingly to include effects of PULSE PROPAGATION
2D MODEL(transverse effects)
3D MODEL(transverse + longitudinal
effects )
Introduction to FELs
Longitudinal effects
SlippageResonance condition: Electrons slip back by one radiation wavelength per undulator periodSlippage 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
Reduced Interaction length
Short electron bunch
Undulator lengthSlippage
length
Interaction length = undulator length
Long electron bunch electron bunch
optical pulse
Introduction to FELs
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
Pulse centroid
ct
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 passesBUT: as intensity increases
back of pulse saturates firstthen 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 lengthfor maximum gain
another cavity lengthfor maximum power
Introduction to FELs
Summary
0D beam:(single electron)
1D beam:
2D beam: transverse effects
3D beam:longitudinal effects
(Resonance condition)
(Injection above resonance for net gain)
(Emittance – beam overlap and broadening
Energy spread)
(Pulse effects – slippage and lethargy)
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.
D
R1
R2
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 surfaceFundamental TEM00 has a Gaussian transverse intensity distribution
Gaussian profile
Gaussian profile
Z=0planar wavefront
2w0
Rayleigh length z = zRmaximum curvature
2w0 x √2
mirror curvature = wavefrontcurvature
Introduction to FELs
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
FELI
CLIOFELIX
ERLP?
LANL
[1] Kogelnik and Li, Laser Beams and Resonators,SPIE MS68, 1993 originally published:Applied Optics, Volume 5, Issue 10, 1550-October 1966
Introduction to FELs
Resonator Choice
R1
R2
CONFOCAL PROPERTIES
Large waist = smallfilling factor
Small mirror spot = high power density + low diffraction loss
Centres of curvature far apart:
insensitive to mirror alignment error
Introduction to FELs
Resonator Choice
Confocal Angular Alignment Sensitivity
D
R1
R2OPTICAL AXIS
OPTICAL AXIS
Introduction to FELs
Resonator Choice
R1 R2
CONCENTRIC PROPERTIES
small waist = largefilling factor large mirror spot =
low power density + high diffraction loss
Centres of curvature close = sensitive to
mirror alignment error
Introduction to FELs
Concentric Angular Alignment Sensitivity
D
R1
R2
OPTICAL AXIS
OPTICAL AXIS
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:
optical
electron
Undulator length L
ZR > L/3 ZR ~ L/3 ZR < L/3
OPTIMUM
Undulator length L Undulator length L
Introduction to FELs
Resonator Choice
For an FEL it is difficult to achieve a small cavity length:Restriction due to electron bunch frequencySpace 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:
Complicated.Variable outcoupling fraction. Variable scraper position.
Scraper Mirror
Complicated. Polarisation dependent.
Variable outcoupling fraction. No mode distortion.
Brewster Plate
Not broadband: only suitable for certain wavelengths.
No mode distortion. Partially Transmitting Mirror
Outcoupling fraction changes with wavelength.Mode distortion.
Broadband.Simple.
Hole in Mirror
ConsProsOutcoupling Method
Introduction to FELs
Free-Electron Laser Output
Introduction to FELs
FEL Output: power
Gain curve can be used to estimate maximum output power:
Energy detuning δ
Gai
n
2.6 2πGai
n
δ = 4πN(∆E/E)Maximum gain for δ = 2.6Maximum energy that can be lost by an electron injected at peak of gain curve is δ = 2.6: no more gain after that.
Dividing by time, recognising that at equilibrium
power extracted from electron beam = output power
ERLP IRFEL:
I = 50AE = 35MeV
N = 42Gives P = 8MW
Introduction to FELs
FEL Output
At saturation pulse length matches electron bunch lengthLinewidth depends on pulse length (Fourier)
Brightness: output is coherent, so assuming diffraction limited beam:
High power enables high brightness!
ERLP IRFEL:
P = 8 MWWavelength = 4 micron
Gives B ~ 1026
Introduction to FELs
FEL Output
1.E+10
1.E+14
1.E+18
1.E+22
1.E+26
1.E+30
0.001 0.01 0.1 1 10 100 1000Photon energy, eV
Peak
Brig
htne
ss p
h/(s
.0.1
%bp
.mm
2 .mra
d2 )
VUV-FEL
XUV-FEL
Bending Magnet
coherent enhancement
IR-FEL
Undulators
Introduction to FELs
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 LaserLight Amplification by Stimulated Emission of RadiationElectrons in bound states - discrete energy levelsWaste heat in medium ejected at speed of SOUND
Limited tunability Limited power
Continuous tunability High power
Free-Electron LaserLight 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 increasingJLAB - 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 decreasing190nm @ ELETTRA - storage ring oscillator FEL.
Mirror technology limiting factor for oscillators80nm @ TTF1 - amplifier FEL
Introduction to FELs
Applications
Medical
FELs with high-peak and high-average 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
MicromachiningA 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