Classical and Quantum Free Electron Lasers. Gordon Robb Scottish Universities Physics Alliance (SUPA) University of Strathclyde, Glasgow. Content. Introduction – Light sources The Classical FEL Spontaneous emission Stimulated emission & electron bunching
Scottish Universities Physics Alliance (SUPA)
University of Strathclyde, Glasgow.
Conventional (“Bound” electron) lasers
Pros : Capable of producing very bright, highly coherent light
Cons : No good laser sources at short wavelengths e.g. X-ray
Pros : Can produce short
e.g. X- rays
Cons : Radiation produced
Free Electron Lasers offer tunability + coherence
Electrons can be made to oscillate in
an undulator or “wiggler” magnet
Consider a helical wiggler field :
The electron trajectory in a helical wiggler can be deduced
from the Lorentz force
Electron trajectory in an undulator is therefore described by
Electron trajectory in a
helical wiggler is therefore
Resonant emission due to constructive interference
The time taken for the electron to travel one undulator period:
A resonant radiation wavefront will have travelled
Substituting in for the average longitudinal velocity of the electron,bz:
is the “wiggler/undulator parameter” or
then the resonance condition becomes
The expression for the fundamental resonant wavelength shows us the origin of the FEL tunability:
As the beam energy is increased, the spontaneous emission
moves to shorter wavelengths.
For an undulator parameter K≈1 and lu=1cm :
For mildly relativistic beams (g≈3) : l ≈ 1mm (microwaves)
more relativistic beams (g≈30) : l ≈ 10mm (infra-red)
ultra-relativistic beams (g≈30000) : l ≈ 0.1nm (X-ray)
Further tunability is possible through Buand lu as K∝ Buu
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.
EM wave (E,B)
How is the electron affected by resonant radiation ?
The Lorentz Force Equation:
Hendrick Antoon Lorentz
The rate of change of electron energy
Resonant emission – electron energy change
Energy of electron changes ‘slowly’ when interacting with a resonant radiation field.
Rate of electron energy change is ‘slow’ but changes periodically with respect to the radiation phase
For an electron with a different phase with respect to radiation field:
Electrons bunch at resonant radiation wavelength – coherent process*
Axial electron velocity
Electrons bunch on radiation wavelength scale
Power N Pi
Power N2 Pi
For N~109 this is a huge enhancement !
In the previous discussion of electron bunching, assumed that the EM field amplitude and phase were assumed to remain constant.
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
Usually used at wavelengths where there are good mirrors: IR to UV
Low gain is no use for short wavelengths e.g. X-rays as
there are no good mirrors – need to look at high-gain.
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
The (transverse) current density is due to the electron motion
In the wiggler magnet.
The end result is the high gain FEL equations :
Scaled energy change
Scaled EM field intensity
Scaled position in wiggler
Interaction characterised by FEL parameter :
We will now use these equations to investigate the high-gain
We solve the equations with initial conditions
(uniform distribution of phases)
(cold, resonant beam)
(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).
High gain regime simulation (1 x l)
Momentum spread at saturation :
Strong amplification of field is closely linked to phase bunching
Bunched electrons mean that the emitted radiation is coherent.
For randomly spaced electrons : intensity N
For (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
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!
LCLS (Stanford) – first lasing at ~1Å reported in 2010
X-ray FELs under development at DESY (XFEL), SCSS (Japan) and elsewhere
Review : BWJ McNeil & N Thompson, Nature Photonics 4, 814–821 (2010)
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.
Optical Kerr effect
X-ray analogues of these processes may become possible
Courtesy of John N. Galayda – see LCLS website for more
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:
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
A useful parameter which can be used to distinguish between the different regimes is the “quantum FEL parameter”, .
Induced momentum spread
Photon recoil momentum
: Classical regime
Note that quantum regime is inevitable for
sufficiently large photon momentum
: Quantum effects
In classical FEL theory, electron-light momentum exchange
is continuous and the photon recoil momentum is neglected
momentum spread (gmcr)
is the “quantum FEL parameter”
We now consider the opposite case where
momentum spread (gmcr)
Electron-radiation momentum exchange is now discrete i.e.
so a quantum model of the electron-radiation interaction is required.
First quantum model of high-gain FEL :
G. Preparata, Phys. Rev. A 38, 233(1988) (QFT treatment)
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)
Electron dynamical equations
Average in wave equation becomes QM average
Single electron Hamiltonian
equations for electron
and classical field A
in terms of
Assuming electron wavefunction is periodic in q :
|cn|2 = pn = Probability of electron having momentum n(ħk)
Only discrete changes of momentum are now possible
: pz= n (k) , n=0,±1,..
is recovered for
many momentum states
both with n>0 and n<0
Evolution of field,
is identical to that of a classical
Dynamical regime is determined by the quantum FEL parameter, r
Quantum regime (r<1)
Only 2 momentum states occupied
Until now we have effectively ignored slippage i.e. that v When slippage / propagation effects included… QUANTUM REGIME: Quantum regime: only n<0 occupied sequentially Classical regime: both n<0 and n>0 occupied
When slippage / propagation effects included…
only n<0 occupied sequentially
both n<0 and n>0 occupied
R.Bonifacio, N.Piovella, G.Robb, NIMA 543, 645 (2005)
Behaviour similar to quantum regime of QFEL observed in experiments involving
backscattering from cold atomic gases
(Collective Rayleigh backscattering
or Collective Recoil Lasing (CRL) )
QFEL and CRL described by similar theoretical models
Main difference – negligible Doppler upshift of scattered field for atoms
as v < See Fallani et al., PRA 71, 033612 (2005)
See Fallani et al., PRA 71, 033612 (2005)
Implications for the spectral properties of the radiation :
(pz=nħk, Enpz2 n2)
Transition frequencies equally spaced by
Increasing the lines overlap for
→ a single transition
→narrow line spectrum
→ Many transitions
→ broad spectrum
Conceptual design of a QFEL
Easier to reach quantum regime if magnetostatic wiggler is
replaced by electromagnetic wiggler (>TW laser pulse)
As “wiggler” wavelength is now much smaller, allows much lower energy beam to be used (smaller g)
e.g. 10-100 MeV rather than > GeV
Experimental requirements for QFEL :
Writing conditions for gain in terms of :
Energy spread < gain bandwidth:
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 .
May require e.g. ultracold electron sources such as those created by
Van der Geer group (Eindhoven) by photoionising ultracold gases.
Condition may be also relaxed using harmonics :
Bonifacio, Robb, Piovella, Opt. Comm. 284, 1004 (2011)
Quantum FEL - promising for extending coherent sources to sub-Ǻ wavelengths
100 MeV Linac
Laser undulator (l~1mm)
Powerful laser (~10TW)
Lower power but better coherence
Narrow line spectrum
Long undulator (100 m)
Rodolfo Bonifacio (Milan/Maceio/Strathclyde)
Nicola Piovella (Milan)
Brian McNeil (Strathclyde)
Mary Cola (Milan)
Angelo Schiavi (Rome)