Classical and Quantum Free Electron Lasers
This presentation is the property of its rightful owner.
Sponsored Links
1 / 51

Classical and Quantum Free Electron Lasers PowerPoint PPT Presentation


  • 147 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

Classical and Quantum Free Electron Lasers

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


Classical and quantum free electron lasers

Classical and Quantum Free Electron Lasers

Gordon Robb

Scottish Universities Physics Alliance (SUPA)

University of Strathclyde, Glasgow.


Content

Content

  • Introduction – Light sources

  • The Classical FEL

    • Spontaneous emission

    • Stimulated emission & electron bunching

    • High-gain regime & collective behaviour

    • X-ray SASE FELs

  • The Quantum FEL (QFEL)

    • Model

    • Results & experimental requirements

  • Conclusions


Useful references

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


Classical and quantum free electron lasers

1. Introduction – Light Sources

Conventional (“Bound” electron) lasers

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


Attractive features of fels

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 (>~1030ph/(s mm2 mrad2 0.1% B.W.))

  • High average powers – 10kW at Jefferson

  • Short pulses (<100fs 100’s as (10-18s) )


Radiation from an accelerated charge

2. Classical FELs – Spontaneous Emission

f

b

Radiation from an accelerated charge

Stationary electron

Relativistic electron

v <~ c

q

b

Most energy confined to the relativistic emission cone

Energy emission confined to directions perpendicular to axis of oscillation

q= g -1


Classical and quantum free electron lasers

2. Classical FELs – Spontaneous Emission

Nuperiods

Electrons can be made to oscillate in

an undulator or “wiggler” magnet


Undulator radiation simulation in 2 d t shintake

2. Classical FELs – Spontaneous Emission

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


Classical and quantum free electron lasers

2. Classical FELs – Spontaneous Emission

y

z

x

Consider a helical wiggler field :

lw

The electron trajectory in a helical wiggler can be deduced

from the Lorentz force

and

where


Classical and quantum free electron lasers

2. Classical FELs – Spontaneous Emission

y

z

x

Electron trajectory in an undulator is therefore described by

Electron trajectory in a

helical wiggler is therefore

also helical


Classical and quantum free electron lasers

2. Classical FELs – Resonance Condition

r

e-

u

where:

Resonant emission due to constructive interference

The time taken for the electron to travel one undulator period:

A resonant radiation wavefront will have travelled

Equating:


Classical and quantum free electron lasers

2. Classical FELs – Resonance Condition

Substituting in for the average longitudinal velocity of the electron,bz:

where

is the “wiggler/undulator parameter” or

“deflection parameter”

then the resonance condition becomes


Classical and quantum free electron lasers

2. Classical FELs – Resonance Condition

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∝ Buu


Classical and quantum free electron lasers

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)

Bw

electron

beam

Wiggler/undulator


Classical and quantum free electron lasers

2. Classical FELs – Stimulated Emission

Can calculate

How is the electron affected by resonant radiation ?

The Lorentz Force Equation:

Hendrick Antoon Lorentz

The rate of change of electron energy


Classical and quantum free electron lasers

2. Classical FELs – Stimulated Emission

is +ve

is +ve

is +ve

e-

e-

e-

Resonant emission – electron energy change

Energy of electron changes ‘slowly’ when interacting with a resonant radiation field.

u


Classical and quantum free electron lasers

2. Classical FELs – Stimulated Emission

is +ve

is -ve

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:

e-

e-

u


Classical and quantum free electron lasers

2. Classical FELs – Stimulated Emission

is +ve

Lose energy

Gain energy

Electrons bunch at resonant radiation wavelength – coherent process*

E

Axial electron velocity

r

Electrons bunch on radiation wavelength scale


Classical and quantum free electron lasers

2. Classical FELs – Stimulated Emission

Random

Perfectly bunched

Power  N Pi

Power  N2 Pi

For N~109 this is a huge enhancement !


Classical and quantum free electron lasers

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


Classical and quantum free electron lasers

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

where

The (transverse) current density is due to the electron motion

In the wiggler magnet.


High gain fel mechanism

2. Classical FELs – High Gain Regime

High-gain FEL mechanism

Radiation field bunches electrons

Bunched electrons drive radiation


Classical and quantum free electron lasers

2. Classical FELs – High Gain Regime

Newton-Lorentz

(Pendulum)

Equations

+

Wave

equation

The end result is the high gain FEL equations :

Ponderomotive phase

Scaled energy change

Scaled EM field intensity

Scaled position in wiggler

Interaction characterised by FEL parameter :


Classical and quantum free electron lasers

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

(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).


Classical and quantum free electron lasers

2. Classical FELs – High Gain Regime

High gain regime simulation (1 x l)

Momentum spread at saturation :


The high gain fel

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)


Classical and quantum free electron lasers

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!


Classical and quantum free electron lasers

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)


X ray fels science case

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 (l<1A)

and temporal resolution (<fs) to follow chemical & biological processes in “real time” e.g. stroboscopic “movies” of molecular bond breaking.


Classical and quantum free electron lasers

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

Optical Kerr effect

holography

Q-switching

Pulse shortening

Mode locking

Phase conjugation

Optical information

X-ray analogues of these processes may become possible


Classical and quantum free electron lasers

LCLS

Operating

Ranges

Courtesy of John N. Galayda – see LCLS website for more


Classical and quantum free electron lasers

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


Classical and quantum free electron lasers

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”, .

Induced momentum spread

Photon recoil momentum

where

: Classical regime

Note that quantum regime is inevitable for

sufficiently large photon momentum

: Quantum effects


Classical and quantum free electron lasers

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

momentum spread (gmcr)

one-photon

recoil momentum(ħk)

>>

is the “quantum FEL parameter”

where

i.e.


Classical and quantum free electron lasers

3. The Quantum High-Gain FEL (QFEL)

We now consider the opposite case where

Classical induced

momentum spread (gmcr)

one-photon

recoil momentum(ħk)

<

i.e.

where

Electron-radiation momentum exchange is now discrete i.e.

so a quantum model of the electron-radiation interaction is required.


Classical and quantum free electron lasers

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)


Classical and quantum free electron lasers

3. The Quantum High-Gain FEL (QFEL) - Model

Electron dynamical equations

Average in wave equation becomes QM average

p

2

ò

2

-

q

q

Y

i

d

e

0

Single electron Hamiltonian

Maxwell-Schrodinger

equations for electron

wavefunction Y

and classical field A


Classical and quantum free electron lasers

3. The Quantum High-Gain FEL (QFEL) - Model

M-S equations

in terms of

momentum

amplitudes

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,..

n=1

pz

n=0

n=-1

bunching


Classical and quantum free electron lasers

3. The Quantum High-Gain FEL (QFEL)

classical limit

is recovered for

many momentum states

occupied,

both with n>0 and n<0

Evolution of field, <p> etc.

is identical to that of a classical

particle simulation


Classical and quantum free electron lasers

3. The Quantum High-Gain FEL (QFEL)

_

Dynamical regime is determined by the quantum FEL parameter, r

_

Quantum regime (r<1)

Only 2 momentum states occupied

p=0

p=-ħk


Classical and quantum free electron lasers

3. The Quantum High-Gain FEL (QFEL)

CLASSICAL REGIME:

Until now we have effectively ignored slippage i.e. that v<c

When slippage / propagation effects included…

QUANTUM REGIME:

Quantum regime:

only n<0 occupied sequentially

Classical regime:

both n<0 and n>0 occupied


Classical and quantum free electron lasers

3. The Quantum High-Gain FEL (QFEL)

quantum regime

classical regime

R.Bonifacio, N.Piovella, G.Robb, NIMA 543, 645 (2005)


Classical and quantum free electron lasers

3. The Quantum High-Gain FEL (QFEL)

pump light

Pump

laser

Behaviour similar to quantum regime of QFEL observed in experiments involving

backscattering from cold atomic gases

(Collective Rayleigh backscattering

or Collective Recoil Lasing (CRL) )

lL

Backscattered

field

Cold

Rb atoms

l~lL

QFEL and CRL described by similar theoretical models

Main difference – negligible Doppler upshift of scattered field for atoms

as v <<c.

See Fallani et al., PRA 71, 033612 (2005)


Classical and quantum free electron lasers

3. The Quantum High-Gain FEL (QFEL)

Implications for the spectral properties of the radiation :

Momentum-energy levels:

(pz=nħk, Enpz2 n2)

Transition frequencies equally spaced by

with width

Increasing the lines overlap for

QUANTUM REGIME:

→ a single transition

→narrow line spectrum

CLASSICAL REGIME:

→ Many transitions

→broad spectrum


Classical and quantum free electron lasers

3. The Quantum High-Gain FEL (QFEL)

Conceptual design of a QFEL

lr

lL

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


Classical and quantum free electron lasers

3. The Quantum High-Gain FEL (QFEL)

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)


Classical and quantum free electron lasers

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…


Classical and quantum free electron lasers

4. Conclusions

Quantum FEL - promising for extending coherent sources to sub-Ǻ wavelengths

QUANTUM SASE-FEL

needs:

100 MeV Linac

Laser undulator (l~1mm)

Powerful laser (~10TW)

yields:

Lower power but better coherence

Narrow line spectrum

CLASSICAL SASE-FEL

needs:

GeV Linac

Long undulator (100 m)

yields:

High Power

Broad spectrum


Classical and quantum free electron lasers

Acknowledgements

Collaborators

Rodolfo Bonifacio (Milan/Maceio/Strathclyde)

Nicola Piovella (Milan)

Brian McNeil (Strathclyde)

Mary Cola (Milan)

Angelo Schiavi (Rome)


  • Login