introduction to free-electron lasers - astec · introduction to free-electron lasers ... outline...

Introduction to FELs

Introduction to

Free-Electron LasersNeil Thompson

ASTeC


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


What is a Free-Electron Laser?


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


What is an FEL?

Output is radiation that is tunable

powerfulcoherent


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


How does a Free-Electron Laser work?


How does an FEL work?

Basic components

N S SNS

N S NSN

B field Electron path E field

BE

z

v

xy

vx



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!!



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% relativisticPath length increased by transverse oscillations



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


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??


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.



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


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


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



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….


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


The Oscillator FEL

Electrons bunched at radiation wavelength: coherent emission

electron bunch

optical pulse

output pulse

Random electron phase: incoherent 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 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


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)


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


Choosing the required parameters


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


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:

Beam Energy

Peak currentUndulator periods

Undulator K

Electron beam area

These parameters varied to tune FEL:

Filling factor: averaged over undulator length


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


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.


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


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


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


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


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-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


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 )


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.


Pulse effects: Slippage

Slippage

Reduced Interaction length

Short electron bunch

Undulator lengthSlippage

length

Interaction length = undulator length

Long electron bunch electron bunch

optical pulse


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)


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


Cavity length detuning

one cavity lengthfor maximum gain

another cavity lengthfor maximum power


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)


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.

D

R1

R2


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


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!


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 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


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


Resonator Choice

Confocal Angular Alignment Sensitivity

D

R1

R2OPTICAL AXIS

OPTICAL AXIS


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


Concentric Angular Alignment Sensitivity

D

R1

R2

OPTICAL AXIS

OPTICAL AXIS


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


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


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


Free-Electron Laser Output


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


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


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


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 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)


Current Trends and the way ahead


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


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.


Applications

Power Beaming and Remote Sensing


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.


The End

introduction to free-electron lasers - astec · introduction to free-electron lasers ... outline...

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