Classical and Quantum Free
Electron Lasers
Gordon Robb
Scottish Universities Physics Alliance (SUPA)
University of Strathclyde, Glasgow.
Content
1. 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
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
1. Introduction – Light Sources
Conventional (“Bound” electron) lasers
En
En-1
hν  En  En-1
Pros : Capable of producing very bright, highly coherent light
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
1. Introduction – Light Sources
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) )
2. Classical FELs – Spontaneous Emission
Radiation from an accelerated charge
Stationary electron
Relativistic electron
v <~ c
f
Energy emission confined to
directions perpendicular to
axis of oscillation
b
q
Most energy confined to the
relativistic emission cone
q = g -1
b
2. Classical FELs – Spontaneous Emission
Electrons can be
made to oscillate
in
an undulator or
“wiggler” magnet
2. Classical FELs – Spontaneous Emission
Undulator radiation simulation in 2-D (T. Shintake)
2. Classical FELs – Spontaneous Emission
Consider a helical wiggler field :
y

Bw
z
x
w
The electron trajectory in a helical wiggler can be deduced
from the Lorentz force

 
F   e v  Bw
where

Bw  Bw cosk w z xˆ  sink w z yˆ 
and
kw 
2
w
2. Classical FELs – Spontaneous Emission
Electron trajectory in an undulator is therefore described by
x
e Bw
gmkw
2
sink w z  , y 
y
e Bw
gmkw
2
cosk w z  , z  vz t
z
Electron trajectory in a
helical wiggler is therefore
also helical
x
2. Classical FELs – Resonance Condition
Resonant emission due to constructive interference
r
e-
vz
u
z
The time taken for the electron to travel one undulator period:
A resonant radiation wavefront will have travelled  u   r
tj 
vzj

 u  r
c
  r
1  b zj
b zj
u
where:
vzj
 tr 
Equating:
u
u
b zj 
vzj
c
 u  r
c
2. Classical FELs – Resonance Condition
Substituting in for the average longitudinal velocity of
the electron, bz :
1 bz
bz

1
 2 1 K2
2γ

where
e λu BuRMS
K
2m c
is the “wiggler/undulator parameter” or
“deflection parameter”
then the resonance condition becomes

λu
2
λr  2 1 K
2γ

2. Classical FELs – Resonance Condition
The expression for the fundamental resonant wavelength
shows us the origin of the FEL tunability:
1 K2 
 λu
λr  
2
 2γ 
e λu BuRMS
K
2m c
As the beam energy is increased, the spontaneous emission
moves to shorter wavelengths.
For an undulator parameter K≈1 and u=1cm :
For mildly relativistic beams (g≈3) :
 ≈ 1mm (microwaves)
more relativistic beams (g≈30) :
 ≈ 10mm (infra-red)
ultra-relativistic beams (g≈30000) :  ≈ 0.1nm (X-ray)
Further tunability is possible through Bu and u as K∝ Buu
2. Classical FELs – Stimulated Emission
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)
electron
beam
Bw
Wiggler/undulator
2. Classical FELs – Stimulated Emission
How is the electron affected by resonant radiation ?
The Lorentz Force Equation:
 d γm0 v 
  
F
 -e E  v B
dt

Hendrick Antoon
Lorentz
The rate of change
of electron energy
d γm0 c
dt
2

   e E  v
2. Classical FELs – Stimulated Emission
Resonant emission – electron energy change
   
 
 v changes
E  v is +veE  v is +ve
is +ve
Energy of E
electron
‘slowly’ when interacting with
a resonant radiation field.
e-
e-
u
e-
2. Classical FELs – Stimulated Emission
Rate
ofelectron
electronwith
energy
changephase
is ‘slow’
but changes
For an
a different
with
periodically
with respect
respect to radiation
field:to the radiation phase
 
E

v
is +ve
-
e
e-
 
E  v is -ve
u
2. Classical FELs – Stimulated Emission
Gain
energy


Electrons bunch at resonant radiation
 
d γm0 c 2
 e E v
wavelength – coherent process*
dt
E
Lose
energy
vx is +ve
Axial electron velocity
r
Electrons bunch on radiation wavelength scale
2. Classical FELs – Stimulated Emission
2
N
N N
 N
i f f 
if j 
2
Radiation power    E j e    E j   E j Ek* e j k
j 1
j 1 k 1
 j 1

jk
Random
Perfectly bunched
Power  N Pi
Power  N2 Pi
For N~109 this is a huge enhancement !
2. Classical FELs – High Gain Regime
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
2. Classical FELs – High Gain Regime
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


2

1  Er
J
2
 Er  2
 m0
2
c t
t

  kz  t  f (z )
Er  -Er ( z)sin α xˆ  cosα yˆ  where
The (transverse) current density is due to the electron motion
In the wiggler magnet.

  
J   e  v j  r  rj (t )
j
2. Classical FELs – High Gain Regime
High-gain FEL mechanism
Radiation field bunches electrons


  
F  -e E v B

Bunched electrons drive radiation


 1  E
J
2
 E  2 2  μ0
c t
t
2
2. Classical FELs – High Gain Regime
dq j
The end result
is the high
gain FEL
equations :
dz
d pj
 pj
iq j
 c.c.)
 iq j
 e  iq
 ( Ae
dz
1
dA

N
dz
N
e
j 1
q j  kw  k z  t j
g j g R
pj 
 gR
2

E
2
0
A 
ng R mc 2
z
4z
z

Lg
w

Ponderomotive phase
Scaled energy change
Scaled EM field intensity
Scaled position in wiggler
Interaction characterised by FEL parameter :
 ~ 104 103

Newton-Lorentz
(Pendulum)
Equations
+
Wave
equation
1  K p
  
g  4ckw



2
3
2. Classical FELs – High Gain Regime
We will now use these equations to investigate the high-gain
regime.
We solve the equations with initial conditions
q j  0,2 
pj  0
(uniform distribution of phases)
(cold, resonant beam)
(weak initial EM field)
A  1
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
High gain regime simulation (1 x )
Momentum spread at saturation :  ( P)  m cg
2. Classical FELs – High Gain Regime
The High Gain FEL
Usually used at wavelengths where there are no mirrors: VUV to X-ray
No seed radiation field – interactions starts from electron beam shot noise
i.e. Self Amplified Spontaneous Emission (SASE)
2. Classical FELs – High Gain Regime
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  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
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 – X-ray SASE FELs
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)
2. Classical FELs – X-ray SASE FELs
X-ray FELs : science case
High brightness = many photons, even for very short (<fs) pulses
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.
2. Classical FELs – X-ray SASE FELs
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
Q-switching
Pulse shortening
Mode locking
Optical Kerr effect
holography
Phase conjugation
Optical information
X-ray analogues of these processes may become possible
LCLS
Operating
Ranges
Courtesy of John N. Galayda – see LCLS website for more
2. Classical FELs – High Gain Regime
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
3. The Quantum High-Gain FEL (QFEL)
A useful parameter which can be used to distinguish
between the different regimes is the “quantum FEL
parameter”,  .
 m cg
  
 k
  ( p)


 k
1
where
1

2g
Induced momentum spread
Photon recoil momentum
 I   L aW 

  
I
4

 A 
Beam 
3
2
3
  1
: Classical regime
 1
: Quantum effects
Note that quantum regime is inevitable for
sufficiently large photon momentum
3. The Quantum High-Gain FEL (QFEL)
In classical FEL theory, electron-light momentum exchange
is continuous and the photon recoil momentum is neglected
Classical induced
one-photon
momentum spread (gmc) >> recoil momentum(ħk)
i.e.
  1
where
mc g

k
is the “quantum
FEL parameter”
3. The Quantum High-Gain FEL (QFEL)
We now consider the opposite case where
Classical induced
one-photon
momentum spread (gmc) < recoil momentum(ħk)
i.e.
 1
where
mc g R 

k
Electron-radiation momentum exchange is now discrete i.e.
P  n  k
so a quantum model of the electron-radiation interaction is required.
3. The Quantum High-Gain FEL (QFEL) - Model
First quantum model of high-gain FEL :
G. Preparata, Phys. Rev. A 38, 233(1988) (QFT treatment)
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
dq j
 pj
dz
Electron dynamical equations
dp j
dz
iq j
 ( Ae
 q , p  i

p2
Single electron Hamiltonian H   i  Aeiq  c.c.
2
Average in wave
equation becomes
QM average
1
N
Maxwell-Schrodinger
equations for electron
wavefunction 
and classical field A
N
e
 iq j
 c.c.)
p  i
2
2  iq
q

e
d
j 1
0

1  2
iq
i


i

Ae
 c.c. 
2
z
2  q

A

z
2
 iq

e
dq  iA

2
0


q
3. The Quantum High-Gain FEL (QFEL) - Model
Assuming electron
wavefunction is periodic in q :
 (q , z ) 

inq
c
(
z
,
)
e
 n
n  
|cn|2 = pn = Probability of electron having momentum n(ħk)
Only discrete changes of momentum are now possible
: pz= n (k) , n=0,±1,..
n=1
pz n=0
n=-1
M-S equations
in terms of
momentum
amplitudes
k

dcn
n2
 i
cn   Acn 1  A*cn 1
dz
2

bunching
dA
  cn c*n 1  iA
dz n  

3. The Quantum High-Gain FEL (QFEL)
=10,  no propagation
1
10
(a)
-1
10
2
|A|
classical limit
is recovered for
-3
10
  1
-5
10
-7
10
-9
10
0
10
20
30
40
50
z
many momentum states
occupied,
both with n>0 and n<0
0.15
(b)
pn
0.10
Evolution of field, <p> etc.
is identical to that of a classical
particle simulation
0.05
0.00
-15
-10
-5
0
n
5
10
3. The Quantum High-Gain FEL (QFEL)
_
Dynamical regime is determined by the quantum FEL parameter, 
_
Quantum regime (<1)
Only 2 momentum states occupied
p=0
p=-ħk
3. The Quantum High-Gain FEL (QFEL)
Until now we have effectively ignored slippage i.e. that v<c
When slippage / propagation effects included…
CLASSICAL REGIME:
 5
Classical regime:
both n<0 and n>0 occupied
QUANTUM REGIME:
  0.1
Quantum regime:
only n<0 occupied sequentially
3. The Quantum High-Gain FEL (QFEL)
quantum regime
L / Lc  30
  0.05
classical regime
  5
R.Bonifacio, N.Piovella, G.Robb, NIMA 543, 645
(2005)
3. The Quantum High-Gain FEL (QFEL)
Behaviour similar to quantum regime of
QFEL observed in experiments involving
backscattering from cold atomic gases
(Collective Rayleigh backscattering
or Collective Recoil Lasing (CRL) )
Pump
laser
L
Backscattered
Cold
field
~ Rb atoms
L
QFEL and CRL described by similar theoretical models
Main difference – negligible Doppler upshift of scattered field for atoms
as v <<c.
pump
light
See Fallani et al., PRA 71, 033612 (2005)
3. The Quantum High-Gain FEL (QFEL)
Implications for the spectral properties of the radiation :
Momentum-energy levels:
(pz=nħk, Enpz2 n2)
n k
k
n 1 k
1
1
 n  En  En1   n  
  2
Transition frequencies equally spaced by 1  with width 4 
Increasing  the lines overlap for   0.4
CLASSICAL REGIME:   1
→ Many transitions
→ broad spectrum
QUANTUM REGIME:   1
→
a single transition
→ narrow line spectrum
  0.1 1/  10
  0.3 1/  3.3
n  (2n 1) / 2 [n  0,1,..]
  0.2 1/  5
  0.4 1/  2.5
3. The Quantum High-Gain FEL (QFEL)
Conceptual design of a QFEL
r
L
Easier to reach quantum regime if magnetostatic wiggler is
replaced by electromagnetic wiggler (>TW laser pulse)
mc g

k
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
3. The Quantum High-Gain FEL (QFEL)
Experimental requirements for QFEL :
Writing conditions for gain in terms of
r , L , 
:
Energy spread < gain bandwidth:
 (E)
E
 5 10  4
 3/ 2
L r (1  K 2 )

r  A  , L m m
 
In order to generate Å or sub- Å wavelengths with L  1mm
energy spread requirement becomes challenging (~10-4) for quantum regime
Bonifacio, Piovella, Cola, Volpe NIMA 577, 745 (2007)
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)
4. Conclusions
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
Quantum FEL - promising for extending coherent
sources to sub-Ǻ wavelengths
CLASSICAL SASE-FEL
needs:
GeV Linac
Long undulator (100 m)
yields:
High Power
Broad spectrum
QUANTUM SASE-FEL
needs:
100 MeV Linac
Laser undulator (~1mm)
Powerful laser (~10TW)
yields:
Lower power but better
coherence
Narrow line spectrum
Acknowledgements
Collaborators
Rodolfo Bonifacio (Milan/Maceio/Strathclyde)
Nicola Piovella (Milan)
Brian McNeil (Strathclyde)
Mary Cola (Milan)
Angelo Schiavi (Rome)