Free Electron

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Introduction to Classical and Quantum High-Gain
FEL Theory
Rodolfo Bonifacio Gordon Robb University of
Strathclyde, Glasgow, Scotland.
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• Outline
• Introductory concepts
• Classical FEL Model
• Classical SASE
• Quantum FEL Model
• Quantum SASE regime Harmonics
• Coherent sub-Angstrom (g-ray) source
• Experimental evidence of QFEL in a BEC

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1. Introduction
The Free Electron Laser (FEL) consists of a
relativistic beam of electrons (vc) moving
through a spatially periodic magnetic field
(wiggler).
Relativistic electron beam
EM radiation
l ? lw /g2 ltlt lw
Magnetostatic wiggler field
(wavelength lw)

• Principal attraction of the FEL is tunability
• - FELs currently produce coherent light from
  microwaves
• through visible to UV
• X-ray production via Self- Amplified
  Spontaneous Emission (SASE) (LCLS 1.5Å)

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Exponential growth of the emitted radiation and
bunching
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• Ingredients of a SASE-FEL
• High-gain (single pass) (no
  mirrors)
• Propagation/slippage of radiation with respect
  to electrons
• Startup from electron shot noise (no seed
  field)

• Consequently, structure of talk is
• Recap of high-gain FEL theory (classical
  quantum)
• Propagation effects (slippage superradiance)
• SASE (classical quantum)

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Some references relevant to this talk
HIGH-GAIN AND SASE FEL with UNIVERSAL
SCALING Classical Theory (1) R.B, C. Pellegrini
and L. Narducci, Opt. Commun. 50, 373
(1984). (2) R.B, B.W. McNeil, and P. Pierini PRA
40, 4467 (1989) (3) R.B, L. De Salvo, P.Pierini,
N.Piovella, C. Pellegrini, PRL 73, 70 (1994). (4,
5) R.B. et al,Physics of High Gain FEL and
Superradiance, La Rivista del Nuovo Cimento vol.
13. n. 9 (1990) e vol. 15 n.11 (1992)
QUANTUM THEORY

• (6) R. B., N. Piovella, G.R.M.Robb, and M.M.Cola,
  Europhysics Letters, 69, (2005) 55 and
  quant-ph/0407112 .
• R.B., N. Piovella, G.R.M. Robb A. Schiavi,
  PRST-AB 9, 090701 (2006)
• R. B., N. Piovella, G.R.M.Robb, and M.M.Cola,
  Optics Commun. 252, 381 (2005)

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2. The High-Gain FEL
We consider a relativistic electron beam moving
in both a magnetostatic wiggler field and an
electromagnetic wave.
EM wave
electron beam
wiggler
Wiggler field (helical)
Radiation field (circularly polarised plane
wave)
where
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(details in refs. 4,5)
2.1 Classical Electron Dynamics
We want to know the beam-radiation energy
exchange
Energy of the electrons is
Rate of electron energy change is
This must be equal to work done by EM wave on
electrons i.e.
The canonical momentum is a conserved quantity.
i.e.
Consequently
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where
(wiggler EM field)
Now
EM field ltlt wiggler
no time dependence
so the only term of interest is
so
(1)
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Whether electron gains or loses energy depends on
the value of the phase variable
The EM wave (w,k) and the wiggler wave (0,kw)
interfere to produce a ponderomotive wave with
a phase velocity
From the definition of q, it can be shown that
(2)
where
is the resonant energy
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FEL resonance condition
(magnetostatic wiggler )
Let
Example for l1A, lw1cm, E5GeV
(electromagnetic wiggler )
Example for l1A, lpump1mm, E35MeV
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(details in refs. 4,5)
2.2 Field Dynamics
Radiation field (circularly polarised plane
wave)
The radiation field evolution is determined by
Maxwells wave equation
The (transverse) current density is due to the
motion of the (point-like) electrons in the
wiggler magnet.
where
Apply the SVEA
and average on scale of lr to give
where
(3)
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Classical universally scaled equations
A is the normalised S.V.E. A. of FEL rad. self
consistent
Ref 1.
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We will now use these equations to investigate
the high-gain regime. We solve the equations
with initial conditions
(uniform distribution of phases)
(cold, resonant beam)
(small input field)
and observe how the EM field and electrons evolve.
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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 ?
N For perfectly bunched electrons
intensity N2
z0
bltlt1
Ponderomotive potential
zgt0
b1
It can be shown that at saturation in classical
case, intensity ? N4/3
As radiated intensity scales gt N, this indicates
collective behaviour Exponential amplification
in high-gain FEL is an example of a collective
instability.
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In FEL and CARL particles self-organize to form
compact bunches l which radiate coherently.
Collective Recoil Lasing Optical gain
bunching
bunching factor b (0ltblt1)
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FEL instability animation
Steady State
Animation shows evolution of electron/atom
positions in the dynamic pendulum potential
together with the probe field intensity.
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Classical high-gain FEL
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Bonifacio, Casagrande Casati, Optics Comm. 40
(1982)
A fully Hamiltonian model of the classical FEL
Steady State
Defining
then
Defining
then the FEL equations can be rewritten as
where
Equilibrium occurs when
so
BUT
so
i.e. GAIN
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The scaled radiation power A2, electron
bunching b and the energy spread sp for the
classical high-gain FEL amplifier.
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Classical chaos in the FEL
If we calculate the distance, d (z), between
different trajectories in the 2-dimensional
phase-space
so
where
In the exponential regime
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Linear Theory (classical) Ref(1)
Linear theory
runaway solution
See figure (a)
Maximum gain at d0
Quantum theory different results (see later)
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For long beams (L gtgt Lc) Seeded Superradiant
Instability Ref(2)
Including propagation
CLASSICAL REGIME, LONG PULSE L 30LC , resonant
(d0)
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CLASSICAL SASE

• Ingredients of Self Amplified Spontaneous
  Emission (SASE)
• Start up from noise
• Propagation effects (slippage)
• SR instability
• ?
• The electron bunch behaves as if each
  cooperation
• length would radiate independently a SR spike
• which is amplified propagating on the other
  electrons
• without saturating. Spiky time structure and
  spectrum.

SASE is the basic method for producing coherent
X-ray radiation in a FEL
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DRAWBACKS OF CLASSICAL SASE
Time profile has many random spikes
Broad and noisy spectrum at short wavelengths
(x-ray FELs)
simulations from DESY for the SASE experiment (?
1 A)
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what is QFEL?QFEL is a novel macroscopic
quantum coherent effectcollective Compton
backscattering of a high-power laser wiggler by a
low-energy electron beam.The QFEL linewidth can
be four orders of magnitude smaller than that of
the classical SASE FEL
Phys. Rev. ST Accel. Beams 9 (2006) 090701
Nucl. Instr. And Meth. A 593 (2008) 69
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Why QUANTUM FEL theory?
In classical theory e-momentum recoil DP
continuous variable
QUANTUM THEORY
WRONG if one electron emits n photons
QUANTUM FEL parameter
If
CLASSICAL LIMIT
If
STRONG QUANTUM EFFECTS
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why QFEL requires a LASER WIGGLER?
and
for a laser wiggler
to lase at lr0.1 A
MAGNETIC WIGGLER lW 1cm, E 10 GeV r 10-6
, LW 1Km
LASER WIGGLER lL 1 mm, E 100 MeV r 10-4 ,
LW 1 mm
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Conceptual design of a QFEL
Compton back-scattering (COLLECTIVE)
lr
lL
If g ? 200 ( E ? 100 MeV) ? lr ? 0.3 Å !
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QUANTUM FEL MODEL
Procedure
Describe N particle system as a Quantum
Mechanical ensemble
Write a Schrödinger-like equation for macroscopic
wavefunction
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1D QUANTUM FEL MODEL
R.Bonifacio, N.Piovella, G.Robb, A. Schiavi,
PRST-AB (2006)
normalized FEL amplitude
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Madelung Quantum Fluid Description of QFEL
R. Bonifacio, N. Piovella, G. R. M. Robb, and A.
Serbeto,  Phys. Rev. A 79, 015801 (2009)
Let
and
See E. Madelung, Z. Phys 40, 322 (1927)
Classical limit
no free parameters
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Wigner approach for 1D QUANTUM MODEL
Introducing the Wigner function
Using the equation for we obtain
a finite-difference equation for
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for rgtgt1
The Wigner equation becomes a Vlasov equation
describing the evolution of a classical particle
ensemble
The classical model is valid when Quantum regime
for
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Quantum Dynamics
is momentum eigenstate corresponding to
eigenvalue
Only discrete changes of momentum are possible
pz n (?k) , n0,1,..
n1
pz
n0
n-1
probability to find a particle with pn(hk)
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steady-state evolution
classical limit is recovered for
many momentum states occupied, both with ngt0
and nlt0
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Quantum bunching
where
? relative phase
Momentum wave interference
Maximum interference
Maximum bunching when 2-momentum eigenstates are
equally populated with fixed relative phase
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Bunching and density grating
QUANTUM REGIME rlt1
CLASSICAL REGIME rgtgt1
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• The physics of the Quantum FEL

Momentum-energy levels (pznhk, En?pz2 ?n2)
(harmonics)
Frequencies equally spaced by
with width
Increasing the lines overlap for
CLASSICAL REGIME many momentum level
transitions ? many spikes
QUANTUM REGIME a single momentum level
transition ? single spike
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Quantum Linear Theory
Quantum regime for rlt1
Classical limit
max at
width
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discrete frequencies as in a cavity
max for
?
Continuous limit
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momentum distribution for SASE
Classical regime both nlt0 and ngt0 occupied
Quantum regime sequential SR decay, only nlt0
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SASE Quantum purification
R.Bonifacio, N.Piovella, G.Robb, NIMA(2005)
quantum regime
classical regime
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LINEWIDTH OF THE SPIKE IN THE QUANTUM REGIME
QUANTUM SINGLE SPIKE
CLASSICAL ENVELOPE
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QFEL requirements
Not necessary with plasma guiding (D. Jaroszynski
collaboration)
(thermal)
Emittance
Rosenzweig et al, NIM A 593, 39 (2008)
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Harmonics Production
Possible frequencies
One photon recoil
Larger momentum level separation
quantum effects easier
Extend Q.F. Model to harmonics
G Robb NIMA A 593, 87 (2008)
Results (a0 gt1)
Distance between gain lines
Gain bandwidth of each line
.
Separated quantum lines if
i.e.
Possible classical behaviour for fundamental BUT
quantum for harmonics
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3rd harmonic
5th harmonic
Fundamental
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0.1A
0.06A
0.3A
e.g.
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Main limitations in classical regime
1.
2.
3.
4.
Quantum FEL as above with
Quantum regime easier in the sub-A region and
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Parameters for QFEL
Electron beam
Laser beam
QFEL beam
Note 5th harmonic at 0.06 A
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Relaxed parameters with plasma channel (guiding)
Dino Jaroszynski
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FEL IN CLASSICAL\SASE CAN GO TO l1.5? (LCLS)
QUANTUM SASE WORKS BETTER FOR SUB-? REGION
QUANTUM SASE needs 100 MeV Linac Laser
undulator (l1mm) yields Lower power Very narrow
line spectrum
CLASSICAL SASE needs GeV Linac Long undulator
(100 m) yields High Power Broad and chaotic
spectrum
QFEL
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Quantum FEL and Bose-Einstein Condensates (BEC)
It has been shown 8 that Collective Recoil
Lasing (CARL) from a BEC driven by a pump laser
and a Quantum FEL are described by the same
theoretical model.
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Both FEL and CARL are examples of collective
recoil lasing
Pump field
CARL
llp
Cold atoms
Backscattered field (probe)
FEL
Electron beam
wiggler magnet (period lw)
At first sight, CARL and FEL look very different
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FEL
EM pump, lw (wiggler)
Connection between CARL and FEL can be seen
more easily by transforming to a frame (L)
moving with electrons
Backscattered EM field l lw
electrons
CARL
Pump laser
Connection between FEL and CARL is now clear
Backscattered field
Cold atoms
llp
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Experimental Evidence of Quantum Dynamics The
LENS Experiment

• Production of an elongated 87Rb BEC in a
  magnetic trap

• Laser pulse during first expansion of the
  condensate

• Absorption imaging of the momentum components
  of the cloud

Experimental values D 13 GHz w 750 mm P
13 mW
R. B., F.S. Cataliotti, M.M. Cola, L. Fallani, C.
Fort, N. Piovella, M. Inguscio, Optics Comm.
233, 155(2004) and Phys. Rev. A 71, 033612 (2005)
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LENS experiment
Temporal evolution of the population in the first
three atomic momentum states during the
application of the light pulse.
n-2
n0
n-1
p-4hk
p0
p-2hk
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MIT experiment
Superradiant Rayleigh Scattering from a BEC
S. Inouye et al., Science 285, 571 (1999)
Back scattered intensity for different laser
powers 3.8 2.4 1.4 mW/cm2 Duration 550 ms
Number of recoiled particles for different laser
intensity (25 45 mW/cm2). Total number of atoms
2 107
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Superradiant Rayleigh Scattering in a
BEC (Ketterle, MIT 1991)
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Summarising A BEC driven by a laser field shows
momentum quantisation and superradiant
backscattering as in a QFEL, being described by
the same system of equations.