Classical and Quantum Free Electron Lasers
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Classical and Quantum Free Electron Lasers
Gordon Robb
Scottish Universities Physics Alliance (SUPA)
University of Strathclyde, Glasgow.
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Content1. Introduction – Light sources
2. The Classical FEL• Spontaneous emission• Stimulated emission & electron bunching• High-gain regime & collective behaviour• X-ray SASE FELs
3. The Quantum FEL (QFEL)• Model• Results & experimental requirements
4. Conclusions
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Useful References
• J.B. Murphy & C. Pelligrini, “Introduction to the Physics of the Free Electron Laser”, Laser Handbook, vol. 6 p. 9-69 (1990).
• R. Bonifacio et al, “Physics of the High-Gain Free Electron Laser & Superradiance”, Rivista del Nuovo Cimento, Vol. 13, no. 9 p. 1-69 (1990).
• Saldin E.L., Schneidmiller E.A., Yurkov M.V. The physics of free electron lasers. - Berlin et al.: Springer, 2000. (Advanced texts in physics, ISSN 1439-2674).
• Many, many other useful sources on web e.g. www.lightsources.org
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1. Introduction – Light Sources
En
En-1 1-nn EEhν
Pros : Capable of producing very bright, highly coherent light
Conventional (“Bound” electron) lasers
Cons : No good laser sources at short wavelengths e.g. X-ray
Synchrotrons Pros : Can produce short wavelengths e.g. X- rays
Cons : Radiation produced is incoherent
Free Electron Lasers offer tunability + coherence
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Attractive features of FELs
Tunable by varying electron energy or undulator parameters Bu and/or u
Spectral reach – THz, VUV to X-ray Cannot damage lasing medium (e--beam) High peak powers (>GW’s) Very bright (>~1030 ph/(s mm2 mrad2 0.1% B.W.)) High average powers – 10kW at Jefferson Short pulses (<100fs 100’s as (10-18s) )
1. Introduction – Light Sources
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Radiation from an accelerated charge
Most energy confined to the relativistic emission cone
Stationary electron
= -1
Relativistic electron
v <~ c
Energy emission confined to directions perpendicular to
axis of oscillation
2. Classical FELs – Spontaneous Emission
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N u periods
2. Classical FELs – Spontaneous Emission
Electrons can be made to oscillate
in an undulator or
“wiggler” magnet
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Undulator radiation simulation in 2-D (T. Shintake)
2. Classical FELs – Spontaneous Emission
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Consider a helical wiggler field :
wB
w
x
yz
The electron trajectory in a helical wiggler can be deduced from the Lorentz force
wBvF
e
where yksinxkcosBB wwww zz
and w
w
2k
2. Classical FELs – Spontaneous Emission
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Electron trajectory in a helical wiggler is therefore also helical
z
x
y
tvz,zkcosmk
Bey ,zksin
mk
Bex zw2
w
ww2
w
w
Electron trajectory in an undulator is therefore described by
2. Classical FELs – Spontaneous Emission
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The time taken for the electron to travel one undulator period:u
jzj
tv
A resonant radiation wavefront will have travelled u ru r rt c
Equating:
1 zju u rr u
zj zjv c
where: zjzj
v
c
e-
u
r
zv
z
2. Classical FELs – Resonance Condition
Resonant emission due to constructive interference
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Substituting in for the average longitudinal velocity of the electron, z :
22
ur K1
2γ
λλ
mc2
BλeK
RMSuu
is the “wiggler/undulator parameter” or
“deflection parameter”
2. Classical FELs – Resonance Condition
22
K12γ
11
z
z
where
then the resonance condition becomes
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The expression for the fundamental resonant wavelength shows us the origin of the FEL tunability:
u2
2
r λ2γ
K1λ
As the beam energy is increased, the spontaneous emission moves to shorter wavelengths.
For an undulator parameter K≈1 andu=1cm :
For mildly relativistic beams (≈3) : ≈ 1mm (microwaves) more relativistic beams (≈30) : ≈ 10m (infra-red) ultra-relativistic beams (≈30000) : ≈ 0.1nm (X-ray)
Further tunability is possible through Bu and u as K ∝ Buu
mc2
BλeK
RMSuu
2. Classical FELs – Resonance Condition
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Spontaneous emission is incoherent as electrons emit independently at random positions i.e. with random phases.
Now we consider stimulated processes
i.e. an electron beam moving in both a magnetostatic wiggler field and an electromagnetic wave.
Wiggler/undulatorelectron beam
EM wave (E,B)Bw
2. Classical FELs – Stimulated Emission
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BvEe-dt
vγmdF 0
vEe
dt
cγmd 20
Hendrick Antoon Lorentz
The Lorentz Force Equation:
The rate of change of electron energy
Can ca
lculat
e
How is the electron affected by resonant radiation ?
2. Classical FELs – Stimulated Emission
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u
e- e- e-
vE is +ve vE
is +ve vE
is +ve
Resonant emission – electron energy change
Energy of electron changes ‘slowly’ when interacting with a resonant radiation field.
2. Classical FELs – Stimulated Emission
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u
e- is -vevE
e-vE is +ve
For an electron with a different phase with respect to radiation field:Rate of electron energy change is ‘slow’ but changes periodically with respect to the radiation phase
2. Classical FELs – Stimulated Emission
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is +vexvLose
energy
Gain energy
Axial electron velocity
r
Electrons bunch at resonant radiation wavelength – coherent process*
vEe
dt
cγmd 20
E
2. Classical FELs – Stimulated Emission
Electrons bunch on radiation wavelength scale
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2
2 *
1 1 1 1
Radiation power j kj
N N N Nii
j j j kj j j k
E e E E E e
j k
Random
Power N Pi
Perfectly bunched
Power N2 Pi
For N~109 this is a huge enhancement !
2. Classical FELs – Stimulated Emission
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In the previous discussion of electron bunching, assumed that the EM field amplitude and phase were assumed to remain
constant.
Usually used at wavelengths where there are good mirrors: IR to UV
This is a good approximation in cases where FEL gain is low
e.g. in an FEL oscillator – small gain per pass, small bunching, highly reflective mirrors
2. Classical FELs – High Gain Regime
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Relaxing the constant field restriction allows us to study the fully coupled electron radiation interaction – the high gain FEL equations.
The EM field is determined by Maxwell’s wave equation
t
J
t
E
cE r
r
02
2
22 1
yαcosxαsin)(-EE rr z
where )(ztkz
The (transverse) current density is due to the electron motionIn the wiggler magnet.
j
jj trrveJ )(
Low gain is no use for short wavelengths e.g. X-rays as there are no good mirrors – need to look at high-gain.
2. Classical FELs – High Gain Regime
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Bunched electrons drive radiation
Radiation field bunches electrons
High-gain FEL mechanism
BvEe-F
t
Jμ
t
E
c
1E 02
2
22
2. Classical FELs – High Gain Regime
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The end result is the high gain FEL
equations :
N
j
ii
ij
jj
eeNzd
dA
ccAezd
pd
pzd
d
j
j
1
1
.).(
Newton-Lorentz(Pendulum) Equations
+Wave
equation
jwj tzkk
R
Rjjp
2
2
02A
mcn
E
R
wg
z
L
zz
4
32
4
1
w
p
ck
K
Ponderomotive phase
Scaled energy change
Scaled EM field intensity
Scaled position in wiggler
Interaction characterised by FEL parameter :
2. Classical FELs – High Gain Regime
34 1010~
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We will now use these equations to investigate the high-gainregime.
We solve the equations with initial conditions
2,0j(uniform distribution of phases)
0jp (cold, resonant beam)
1A (weak initial EM field)
and observe how the EM field and electrons evolve.For linear stability analysis see :
J.B. Murphy & C. Pelligrini“Introduction to the Physics of the Free Electron Laser”
Laser Handbook, vol. 6 p. 9-69 (1990).
R. Bonifacio et al“Physics of the High-Gain Free Electron Laser & Superradiance”
Rivista del Nuovo Cimento Vol. 13, no. 9 p. 1-69 (1990).
2. Classical FELs – High Gain Regime
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2. Classical FELs – High Gain Regime
High gain regime simulation (1 x )
Momentum spread at saturation : mcP )(
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The High Gain FEL
Usually used at wavelengths where there are no mirrors: VUV to X-ray
2. Classical FELs – High Gain Regime
No seed radiation field – interactions starts from electron beam shot noise
i.e. Self Amplified Spontaneous Emission (SASE)
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Strong amplification of field is closely linked to phase bunching of electrons.
Bunched electrons mean that the emitted radiation is coherent.For randomly spaced electrons : intensity NFor (perfectly) bunched electrons : intensity ~ N2
It can be shown that at saturation in this model, intensity N4/3
As radiated intensity scales > N, this indicates collective behaviour
Exponential amplification in high-gain FEL is an example of a collective instability.
High-gain FEL-like models have been used to describe collective synchronisation / ordering behaviours in a wide variety of systems in nature including flashing fireflies and rhythmic applause!
2. Classical FELs – High Gain Regime
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LCLS (Stanford) – first lasing at ~1Å reported in 2010
2. Classical FELs – X-ray SASE FELs
X-ray FELs under development at DESY (XFEL), SCSS (Japan) and elsewhere
Review : BWJ McNeil & N Thompson, Nature Photonics 4, 814–821 (2010)
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X-ray FELs : science case
X-ray FELs will have sufficiently high spatial resolution (<1A) and temporal resolution (<fs) to follow chemical & biological
processes in “real time” e.g. stroboscopic “movies” of molecular bond breaking.
High brightness = many photons, even for very short (<fs) pulses
2. Classical FELs – X-ray SASE FELs
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In 1960s, development of the (visible) laser opened up nonlinear optics and photonics
Intense coherent X-rays could similarly open up X-ray nonlinear optics (X-ray photonics?)
In the optical regime, many phenomena and applications are based on only a few fundamental nonlinear processes e.g.
Saturable absorption Optical Kerr effect
Q-switchingPulse shortening
Mode locking
holographyPhase conjugation
X-ray analogues of these processes may become possible
Optical information
2. Classical FELs – X-ray SASE FELs
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LCLSOperatingRanges
Courtesy of John N. Galayda – see LCLS website for more
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As SASE is essentially amplified shot noise, the temporal coherence of the FEL radiation in SASE-FELs is still poor in laser
terms i.e. it is far from transform limited
SASE Power output: SASE spectrum:
2. Classical FELs – High Gain Regime
Several schemes are under investigation to improve coherence properties of X-ray FELS e.g. seeding FEL interaction with a
coherent, weak X-ray signal produced via HHG.
In addition, scale of X-ray FELs is huge (~km) – need different approach for sub-A generation
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A useful parameter which can be used to distinguish between the different regimes is the “quantum FEL parameter”, .
k
p
k
mc
)(
21331
2 4L W
A Beam
aI
I
where
1 : Classical regime
1 : Quantum effects
Induced momentum spreadPhoton recoil momentum
3. The Quantum High-Gain FEL (QFEL)
Note that quantum regime is inevitable for sufficiently large photon momentum
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In classical FEL theory, electron-light momentum exchange is continuous and the photon recoil momentum is neglected
3. The Quantum High-Gain FEL (QFEL)
Classical induced momentum spread (mc)
1i.e. k
mc
where
one-photon recoil momentum(ħk)
>>
is the “quantum FEL parameter”
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Classical induced momentum spread (mc)
1i.e.k
mc R
where
one-photon recoil momentum(ħk)
<
We now consider the opposite case where
knP
Electron-radiation momentum exchange is now discrete i.e.
so a quantum model of the electron-radiation interaction is required.
3. The Quantum High-Gain FEL (QFEL)
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Procedure :
Describe N particle system as a Quantum Mechanical ensemble
Write a Schrödinger-like equation for macroscopic wavefunction:
Details in :R.Bonifacio, N.Piovella, G.Robb, A. Schiavi, PRST-AB 9, 090701 (2006)
3. The Quantum High-Gain FEL (QFEL) - Model
G. Preparata, Phys. Rev. A 38, 233(1988) (QFT treatment)
First quantum model of high-gain FEL :
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.).( ccAezd
pd
pzd
d
jij
jj
Electron dynamical equations
2
0
2
2
2
c.c.2
1
Aidez
A
Aeiz
i
i
iMaxwell-Schrodinger equations for electron
wavefunction and classical field A
c.c.2
2
iAeip
HSingle electron Hamiltonian ip ,
ip
3. The Quantum High-Gain FEL (QFEL) - Model
N
j
i jeN 1
1 ed i
22
0
Average in wave equation becomes
QM average
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Only discrete changes of momentum are now possible : pz= n (k) , n=0,±1,..
pzk
n=1n=0n=-1
Assuming electron wavefunction is periodic in
n
inn ezcz ),(),(
|cn|2 = pn = Probability of electron having momentum n(ħk)
Aicczd
dA
cAAccn
izd
dc
n
nn
nnnn
1*
1*
1
2
2M-S equations
in terms of momentum amplitudes
bunching
3. The Quantum High-Gain FEL (QFEL) - Model
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-15 -10 -5 0 5 100.00
0.05
0.10
0.15
(b)
n
p n
0 10 20 30 40 5010-9
10-7
10-5
10-3
10-1
101
=10, no propagation
(a)
z
|A|2
classical limit is recovered for
many momentum states occupied,
both with n>0 and n<0
1
Evolution of field, <p> etc.is identical to that of a classical
particle simulation
3. The Quantum High-Gain FEL (QFEL)
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Only 2 momentum states occupiedp=0
p=-ħk
Dynamical regime is determined by the quantum FEL parameter, _
Quantum regime (<1)_
3. The Quantum High-Gain FEL (QFEL)
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Classical regime: both n<0 and n>0 occupied
CLASSICAL REGIME: 5 QUANTUM REGIME: 1.0
Quantum regime: only n<0 occupied sequentially
3. The Quantum High-Gain FEL (QFEL)
Until now we have effectively ignored slippage i.e. that v<cWhen slippage / propagation effects included…
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/ 30cL L
quantum regimequantum regime classical regimeclassical regime 05.0 5
R.Bonifacio, N.Piovella, G.Robb, NIMA R.Bonifacio, N.Piovella, G.Robb, NIMA 543543, 645 , 645 (2005)(2005)
3. The Quantum High-Gain FEL (QFEL)
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ColdRb atoms
Pumplaser
Backscatteredfield
Behaviour similar to quantum regime of QFEL observed in experiments involving
backscattering from cold atomic gases(Collective Rayleigh backscatteringor Collective Recoil Lasing (CRL) )
~L
L
Main difference – negligible Doppler upshift of scattered field for atoms as v <<c.
QFEL and CRL described by similar theoretical models
pump light
See Fallani et al., PRA 71, 033612 (2005)
3. The Quantum High-Gain FEL (QFEL)
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kn k kn 1
Momentum-energy levels:Momentum-energy levels:(p(pzz=n=nħk, ħk, EEnnppzz
2 2 nn22))
1CLASSICAL REGIME:CLASSICAL REGIME:
→ → Many transitionsMany transitions→→ broad spectrumbroad spectrum
QUANTUM REGIME:QUANTUM REGIME:
→ → a single transitiona single transition
→→ narrow line spectrumnarrow line spectrum
1
Transition frequencies equally spaced byTransition frequencies equally spaced by
2
111 nEE nnn
1 4
Increasing the lines overlap for Increasing the lines overlap for 0.4 with widthwith width
3. The Quantum High-Gain FEL (QFEL)
Implications for the spectral properties of the radiation :
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0.1 1/ 10 0.2 1/ 5
0.3 1/ 3.3 0.4 1/ 2.5
,..]1,0n[2/)1n2(n
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Conceptual design of a QFEL Conceptual design of a QFEL
r
L
3. The Quantum High-Gain FEL (QFEL)
Easier to reach quantum regime if magnetostatic wiggler is replaced by electromagnetic wiggler (>TW laser pulse)
k
mc
As “wiggler” wavelength is now much smaller, allows much lower
energy beam to be used (smaller )
e.g. 10-100 MeV rather than > GeV
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)1(105
)(2
2/34
KE
E
rL
Energy spread < gain bandwidth:
Ar mL ,
Writing conditions for gain in terms of : ,, Lr
Bonifacio, Piovella, Cola, Volpe NIMA 577, 745 (2007)
In order to generate Å or sub- Å wavelengths with energy spread requirement becomes challenging (~10-4) for quantum regime .
mL 1
3. The Quantum High-Gain FEL (QFEL)
Experimental requirements for QFEL :
Condition may be also relaxed using harmonics :
Bonifacio, Robb, Piovella, Opt. Comm. 284, 1004 (2011)
May require e.g. ultracold electron sources such as those created byVan der Geer group (Eindhoven) by photoionising ultracold gases.
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FELs offer scientists a new tool that can light up previously dark corners of nature that have hitherto been unobservable.
The development of FEL’s has only really begun. We can expect advances in peak powers, average powers, shorter wavelengths and shorter pulses.
Currently an exciting time for X-ray FELs with LCLS (SLAC) already online and XFEL (DESY), SCSS (Japan) and others (e.g. MAX4) on the way – a whole new picture of nature awaits…
4. Conclusions
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CLASSICAL SASE-FELneeds:
GeV LinacLong undulator (100 m)
yields:High Power
Broad spectrum
QUANTUM SASE-FELneeds:
100 MeV Linac Laser undulator (~1m)Powerful laser (~10TW)
yields:Lower power but better
coherenceNarrow line spectrum
Quantum FEL - promising for extending coherent sources to sub-Ǻ wavelengths
4. Conclusions
![Page 51: Classical and Quantum Free Electron Lasers](https://reader034.fdocuments.us/reader034/viewer/2022051418/568157f2550346895dc56de3/html5/thumbnails/51.jpg)
Acknowledgements
Collaborators
Rodolfo Bonifacio (Milan/Maceio/Strathclyde)Nicola Piovella (Milan)Brian McNeil (Strathclyde)Mary Cola (Milan)Angelo Schiavi (Rome)