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Introduction to Classical and Quantum High-Gain

FEL Theory

Rodolfo Bonifacio Gordon Robb University of

Strathclyde, Glasgow, Scotland.

- 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

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)

Exponential growth of the emitted radiation and

bunching

- 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)

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)

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

(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

where

(wiggler EM field)

Now

EM field ltlt wiggler

no time dependence

so the only term of interest is

so

(1)

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

FEL resonance condition

(magnetostatic wiggler )

Let

Example for l1A, lw1cm, E5GeV

(electromagnetic wiggler )

Example for l1A, lpump1mm, E35MeV

(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)

Classical universally scaled equations

A is the normalised S.V.E. A. of FEL rad. self

consistent

Ref 1.

13

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.

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.

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)

FEL instability animation

Steady State

Animation shows evolution of electron/atom

positions in the dynamic pendulum potential

together with the probe field intensity.

Classical high-gain FEL

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

The scaled radiation power A2, electron

bunching b and the energy spread sp for the

classical high-gain FEL amplifier.

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

Linear Theory (classical) Ref(1)

Linear theory

runaway solution

See figure (a)

Maximum gain at d0

Quantum theory different results (see later)

For long beams (L gtgt Lc) Seeded Superradiant

Instability Ref(2)

Including propagation

CLASSICAL REGIME, LONG PULSE L 30LC , resonant

(d0)

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

25

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)

26

26

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

27

27

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

28

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

29

29

Conceptual design of a QFEL

Compton back-scattering (COLLECTIVE)

lr

lL

If g ? 200 ( E ? 100 MeV) ? lr ? 0.3 Å !

30

QUANTUM FEL MODEL

Procedure

Describe N particle system as a Quantum

Mechanical ensemble

Write a Schrödinger-like equation for macroscopic

wavefunction

31

31

1D QUANTUM FEL MODEL

R.Bonifacio, N.Piovella, G.Robb, A. Schiavi,

PRST-AB (2006)

normalized FEL amplitude

32

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

Wigner approach for 1D QUANTUM MODEL

Introducing the Wigner function

Using the equation for we obtain

a finite-difference equation for

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

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)

36

steady-state evolution

classical limit is recovered for

many momentum states occupied, both with ngt0

and nlt0

37

Quantum bunching

where

? relative phase

Momentum wave interference

Maximum interference

Maximum bunching when 2-momentum eigenstates are

equally populated with fixed relative phase

38

Bunching and density grating

QUANTUM REGIME rlt1

CLASSICAL REGIME rgtgt1

39

- 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

40

Quantum Linear Theory

Quantum regime for rlt1

Classical limit

max at

width

discrete frequencies as in a cavity

max for

?

Continuous limit

42

42

momentum distribution for SASE

Classical regime both nlt0 and ngt0 occupied

Quantum regime sequential SR decay, only nlt0

43

43

SASE Quantum purification

R.Bonifacio, N.Piovella, G.Robb, NIMA(2005)

quantum regime

classical regime

44

44

45

45

LINEWIDTH OF THE SPIKE IN THE QUANTUM REGIME

QUANTUM SINGLE SPIKE

CLASSICAL ENVELOPE

46

46

QFEL requirements

Not necessary with plasma guiding (D. Jaroszynski

collaboration)

(thermal)

Emittance

Rosenzweig et al, NIM A 593, 39 (2008)

47

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

48

3rd harmonic

5th harmonic

Fundamental

49

0.1A

0.06A

0.3A

e.g.

Main limitations in classical regime

1.

2.

3.

4.

Quantum FEL as above with

Quantum regime easier in the sub-A region and

Parameters for QFEL

Electron beam

Laser beam

QFEL beam

Note 5th harmonic at 0.06 A

51

Relaxed parameters with plasma channel (guiding)

Dino Jaroszynski

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

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.

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

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

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)

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

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

Superradiant Rayleigh Scattering in a

BEC (Ketterle, MIT 1991)

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.

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