classical and quantum free electron lasers

Classical and Quantum Free Electron Lasers

Gordon Robb

Scottish Universities Physics Alliance (SUPA)

University of Strathclyde, Glasgow.

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

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

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

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

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

N u periods


Electrons can be made to oscillate

in an undulator or

“wiggler” magnet

Undulator radiation simulation in 2-D (T. Shintake)


C:\Documents and Settings\Gordon Robb\My Documents\ppoint\talks\John Adams Institute\Radiation2D.exe

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


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


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

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”


22

K12γ

11

z

z

where

then the resonance condition becomes

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


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

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 ?


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.


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


is +vexvLose

energy

Gain energy

Axial electron velocity

r

Electrons bunch at resonant radiation wavelength – coherent process*

vEe

dt

cγmd 20

E


Electrons bunch on radiation wavelength scale

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 !


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

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.


Bunched electrons drive radiation

Radiation field bunches electrons

High-gain FEL mechanism

BvEe-F

t

Jμ

t

E

c

1E 02

2

22


http://www.edinphoto.org.uk/0_P/0_photographers_0_early_pss_and_eps_members.htm

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 :


34 1010~

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).



High gain regime simulation (1 x )

Momentum spread at saturation : mcP )(

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)

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!


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)

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


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


LCLSOperatingRanges

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”, .

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

In classical FEL theory, electron-light momentum exchange is continuous and the photon recoil momentum is neglected


Classical induced momentum spread (mc)

1i.e. k

mc

where

one-photon recoil momentum(ħk)

>>

is the “quantum FEL parameter”

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.


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 :

.).( 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


N

j

i jeN 1

1 ed i

22

0

Average in wave equation becomes

QM average

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


-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


Only 2 momentum states occupiedp=0

p=-ħk

Dynamical regime is determined by the quantum FEL parameter, _

Quantum regime (<1)_


Classical regime: both n<0 and n>0 occupied

CLASSICAL REGIME: 5 QUANTUM REGIME: 1.0

Quantum regime: only n<0 occupied sequentially


Until now we have effectively ignored slippage i.e. that v<cWhen slippage / propagation effects included…

/ 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)


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)


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


Implications for the spectral properties of the radiation :

0.1 1/ 10 0.2 1/ 5

0.3 1/ 3.3 0.4 1/ 2.5

,..]1,0n[2/)1n2(n

Conceptual design of a QFEL Conceptual design of a QFEL

r

L


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

)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


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.

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

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

Acknowledgements

Collaborators

Rodolfo Bonifacio (Milan/Maceio/Strathclyde)Nicola Piovella (Milan)Brian McNeil (Strathclyde)Mary Cola (Milan)Angelo Schiavi (Rome)