Introduction to FELs

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 


What is a Free-Electron Laser? 


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 


What is an FEL? 

Output is radiation that is 

tunable 

powerful 

coherent 


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 


How does a Free-Electron Laser work? 



Basic components 

B 

y 

E 

v x 

x 

v 

S 

N S N S 

z 

N 

S 

N S N 

B field 

Electron path 

Introduction to FELs 

E field


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!! 



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 



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 



Q. Which way does 

the energy flow? 

E x 

v x 

v x 

A. Depends on 

electron phase 

t1 

z 

Depending on phase, 

electron either: 

Loses energy to optical 

field and decelerates: 

GAIN 

t2 

Takes energy from 

optical field and 

accelerates: 

ABSORPTION 

t3 


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?? 


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. 



Bunching 

Phase corresponding to 

GAIN 

Phase corresponding to 

ABSORPTION 

t 1 

t 2 

E = RESONANT ENERGY 

Bunching around phase 

corresponding to 

ZERO NET GAIN 

t 1 

t 2 

E > RESONANT ENERGY 

Bunching around phase 

corresponding to 

POSITIVE NET GAIN 


A Simulation example 

Inject 20 electrons at resonance energy: zero detuning 

Detuning 

gain 

absorption 

gain 

• Bunching around 

phase 

correponding to 

zero gain 

0 

• 10 electrons lose 

energy 

• 10 electrons gain 

energy 

Phase 


A Simulation example 

Inject 20 electrons above resonance energy: +ve detuning 

Detuning 

gain 

absorption 

gain 

• Bunching around 

phase 

correponding to 

+ve gain 

+ve 

• 11 electrons lose 

energy 

0 

• 9 electrons gain 

energy 

Phase 



For GAIN > ABSORPTION inject electrons at at energy 

slightly higher than resonant energy 

‘POSITIVE DETUNING’ 

>>>> NET TRANSFER of of energy to 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…. 


Small-Signal Gain Curve 

Gain 

Gain 

Spontaneous emission 

2.6 2π 

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 


The Oscillator FEL 

electron bunch 

output pulse 

optical pulse 

Random electron phase: 

incoherent emission 


Electrons bunched at 

radiation wavelength: 

coherent emission

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 


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) 


Saturation: Oscillator 

Parameters: cavity loss = 4% 

gain cavity intensity 

Gain = Losses 

Intensity 

Intensity 

pass 150 

Distance along undulator 

pass 40 

Amplifier-like 

saturation in 

single pass: 

electrons 

have 

continued to 

rotate in 

phase space 

until they 

start to 

reabsorb 

from the 

optical field 

pass number 


Distance along undulator

Choosing the required parameters 


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 


Required parameters 

To ‘first order’ 

Select base parameters to optimise gain 

To ‘second order’ 

Optimise other parameters to minimise gain degradation 


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 


Small Signal Gain 

In fact, small signal, single pass maximum gain is given by: 

Undulator periods 

Undulator K 

Peak current 

Beam Energy 

Electron beam area 

Filling factor: 

averaged over undulator length 

These parameters varied to tune FEL: 


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 


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. 


Energy Spread 

Gain 

2π 

SMALL SIGNAL 

GAIN CURVE: 

Small energy range for 

positive gain 

If energy spread is too 

large electrons fall 

outside detuning for 

positive gain 


GAIN DEGRADATION 


Energy Spread 

Derivation of Limit on Energy Spread: 

FEL wavelength given by: 

ERLP IRFEL: 

N=42 

Gives rms energy spread 

of < 0.1% for negligible 

gain degradation 

• Expand this in Taylor series 

giving linewidth spread due 

to energy perturbation 

Spontaneous emission 

1/2N 

• Require that 

spread is less than 

the natural line 

halfwidth so that 

no broadening 

occurs and gain 

is not reduced 



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 


Emittance 1: overlap 

• We need the electron beam contained within optical beam 

over whole interaction length 

optical beam 

electron beam 

good 

Interaction length 

small emittance 

large emittance 

bad 


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

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: 


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 ) 


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. 


Pulse effects: Slippage 

Slippage 

Undulator length 

Slippage 

length 

Long electron 

bunch 

optical pulse 

electron bunch 

Interaction length = undulator length 

Short electron 

bunch 

Reduced Interaction 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) 


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 


Cavity length detuning 

one cavity length 

for maximum gain 

another cavity length 

for maximum power 


Summary 

0D beam: 

(single electron) 

1D beam: 

2D beam: 

transverse effects 

(Resonance condition) 

(Injection above resonance for net gain) 

(Emittance – beam overlap and broadening 

Energy spread) 

3D beam: 

longitudinal effects 

(Pulse effects – slippage and lethargy) 


Resonators for Free-Electron Lasers 


Optical Resonators 

Some general properties: 

Optical resonator consists of two spherical mirrors, radius of curvature R1 and R2, 

separated by a distance D. 

R 1 

R 2 

D 



Eigenmode is not a plane wave, but a wave whose front is 

curved. 

At mirrors radius of curvature matches mirror surface 

Fundamental TEM 00 

has a Gaussian transverse intensity 

distribution 

Gaussian profile 

Z=0 

planar wavefront 

Rayleigh length z = z R 

maximum curvature 

mirror curvature = wavefront 

curvature 

Gaussian profile 

2w 0 

2w 0 

x √2 



More general properties: 

Length D has certain allowed values: 

must have optical round trip time = bunch repetition frequency 

Mode size at waist (R 1 

=R 2 

=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! 


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 


Resonators: Stability Diagram 

Plot of g 1 

vs g 2 

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 

FELI 

CLIO 

ERLP? 

[1] Kogelnik and Li, Laser Beams and Resonators, 

SPIE MS68, 1993 

originally published: 

Applied Optics, Volume 5, Issue 10, 1550- Introduction to FELs 

October 1966

Resonator Choice 

CONFOCAL 

PROPERTIES 

R2 

R1 

Centres of curvature 

far apart: 

insensitive to mirror 

alignment error 

Large waist = small 

filling factor 

Small mirror spot = 

high power density + 

low diffraction loss 



Confocal Angular Alignment Sensitivity 

R 1 

R 2 

OPTICAL AXIS 

OPTICAL AXIS 

D 



CONCENTRIC 

PROPERTIES 

R1 

R2 

Centres of curvature 

close = sensitive to 

mirror alignment 

error 

small waist = large 

filling factor 

large mirror spot = 

low power density + 

high diffraction loss 


Concentric Angular Alignment Sensitivity 

OPTICAL AXIS 

R 1 

R 2 

OPTICAL AXIS 

D 



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 

Undulator length L 



electron 

Z R 

> L/3 Z R 

~ L/3 Z R 

< L/3 

OPTIMUM 



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 


Resonators: outcoupling 

There are various methods for getting the light out of the cavity: 

Outcoupling Method 

Pros 

Cons 

Hole in Mirror 

Partially Transmitting Mirror 

Brewster Plate 

Scraper Mirror 

Broadband. 

Simple. 

No mode distortion. 

Variable outcoupling fraction. 

No mode distortion. 

Variable outcoupling fraction. 

Variable scraper position. 

Outcoupling fraction changes 

with wavelength. 

Mode distortion. 

Not broadband: only suitable 

for certain wavelengths. 

Complicated. 

Polarisation dependent. 

Complicated. 


Free-Electron Laser Output 


ERLP IRFEL: 

FEL Output: power 

I = 50A 

E = 35MeV 

N = 42 

Gives P = 8MW 

Gain curve can be used to estimate maximum output power: 

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 

2.6 2π 

Dividing by time, recognising that at 

equilibrium 

power extracted from electron beam = 

output power 



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: 

High power enables high brightness! 

P = 8 MW 

Wavelength = 4 micron 

Gives B ~ 10 26 


FEL Output 

1.E+30 

Peak Brightness ph/(s.0.1%bp.mm 2 .mrad 2 ) 

1.E+26 

1.E+22 

1.E+18 

1.E+14 

IR-FEL 

coherent enhancement 

XUV-FEL 

VUV-FEL 

Undulators 

Bending Magnet 

1.E+10 

0.001 0.01 0.1 1 10 100 1000 

Photon energy, eV 


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) 


Free-Electron Lasers vs Conventional Lasers 


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 

Continuous tunability 

Limited power 

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) 


Current Trends and the way ahead 


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 


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. 


Applications 

Power Beaming and Remote Sensing 


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. 


The End

Introduction to FELs

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