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Towards Sub- Ångström Coherent Light Sources: The Quantum FEL . Gordon Robb & Rodolfo Bonifacio Scottish Universities Physics Alliance (SUPA), Department of Physics, University of Strathclyde, Glasgow , Scotland. Outline. Introduction Classical FEL and SASE Quantum FEL

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slide1

Towards Sub-Ångström Coherent Light Sources: The Quantum FEL

Gordon Robb & Rodolfo Bonifacio

Scottish Universities Physics Alliance (SUPA),

Department of Physics, University of Strathclyde,

Glasgow, Scotland.

slide2

Outline

  • Introduction
  • Classical FEL and SASE
  • Quantum FEL
  • Harmonic Generation in a Quantum FEL
  • Summary
slide3

1. Introduction

We consider classical and quantum regimes of SASE-FEL operation

The parameter we use to identify the different regimes is

the “quantum FEL parameter”

where

: Classical regime

: Quantum effects

slide4

2. Classical FEL and SASE

In usual classical FEL theory, photon recoil momentum is neglected

and electron-light momentum exchange is continuous.

Classical induced

momentum spread (gRmcr)

one-photon

recoil momentum(ħk)

>>

where

i.e.

Classical SASE-FELs produce VUV (DESY) and X-ray (LCLS) radiation :

High power

Broad spectrum / poor temporal coherence

slide5

3. Quantum FEL (QFEL)

We now consider the case where

Classical induced

momentum spread (gRmcr)

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.

SASE-QFEL may produce radiation with lower power than classical

SASE-FELs, but better temporal coherence, even at sub-A wavelengths.

slide6

3. Quantum FEL (QFEL)

Conceptual design of a QFEL :

Similar to (COLLECTIVE) Compton back-scattering

High power laser + (relatively) low energy e-beam

lL

lr

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

slide7

3. Quantum FEL (QFEL) - Model

Procedure :

Describe N particle system as a Quantum Mechanical ensemble

Write a Schrödinger-like equation for macroscopic wavefunction:

Details in :

G. Preparata, Phys. Rev. A 38, 233(1988)

R.Bonifacio, N.Piovella, G.Robb, A. Schiavi, PRST-AB 9, 090701 (2006)

slide8

3. Quantum FEL (QFEL) – 1D Model

Electron dynamical equations

Using scaled variables :

Single electron Hamiltonian

Maxwell-Schrodinger

equations for electron

wavefunctionY

and classical field A

bunching

slide9

3. Quantum FEL (QFEL) –1D 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 values of momentum are possible : pz= n (k) , n=0,±1,..

n=1

pz

n=0

n=-1

bunching

slide10

3. Quantum FEL (QFEL) –Linear Analysis

Linearising and looking for solutions :

Quantum term

Spacing =

Width=

i.e.

Continuous limit :

slide11

3. QFEL Physics

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

slide12

3. QFEL Physics – Momentum distribution evolution

CLASSICAL REGIME:

QUANTUM REGIME:

Quantum regime:

only n<0 occupied

Classical regime:

both n<0 and n>0 occupied

slide13

steady-state evolution:

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

slide14

3. QFEL Physics – Evidence of quantum dynamics

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 gas of

Rb atoms

l~lL

QFEL and CRL described by same theoretical model

Main difference from QFEL – negligible Doppler upshift of scattered field

See Bonifacio et al., Optics Comm. 233, 155(2004) and Fallani et al., Phys. Rev. A 71, 033612 (2005)

slide15

3. QFEL Physics - Quantum “Purification” of SASE spectrum

quantum regime

classical regime

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

slide17

3. QFEL Requirements

Writing conditions for gain in terms of :

Energy spread < gain bandwidth:

:e-beam radius

Beam current :

In order to generate Å or sub- Å wavelengths with

energy spread requirement becomes challenging for .

Is there a way of a reaching quantum regime

without having to use ?

Bonifacio, Piovella, Cola, Volpe NIMA 577, 745 (2007)

slide18

4. Quantum Harmonic Generation

n=0

Possible frequencies

Larger momentum level separation

for transitions involving harmonics

quantum regime easier

to attain?

.

Need to extend QFEL model to include harmonics

[G Robb NIMA A 593, 87 (2008)]

slide19

4. Quantum Harmonic Generation – Model

M-S equations

in terms of

momentum

amplitudes

Consider radiation field consisting of

fundamental + odd harmonics (h=1,3,5,…)

Following the same procedure as previously :

Maxwell-Schrodinger

equations for electron

wavefunctionY

and radiation field A

where

slide20

4. Quantum Harmonic Generation

Repeating linear analysis for harmonics :

Frequency separation between gain lines:

Gain bandwidth of each line :

Discrete emission lines if width (s) < separation (D) i.e.

Possible classical behaviour for fundamental

BUT quantum for harmonics

h=1 - classical

e.g. when :

h=3 – classical/quantum

h=5 – quantum

slide21

h=1

h=3

h=5

a0=2

parameters for qfel
Parameters for QFEL

QFEL beam (fundamental)

Electron beam

Laser beam

Fundamental at 0.3 Å will be in classical regime

5th harmonic at 0.06 Å will be in quantum regime – coherent g-rays

These parameters satisfy the

condition to neglect diffraction

This restriction can be relaxed using a plasma channel (guiding) : Dino Jaroszynski

slide23

5. Conclusion

Quantum FEL - promising for extending coherent sources to sub-Ǻwavelengths

QUANTUM SASE

needs:

100 MeV Linac

Laser undulator (l~1mm)

Powerful laser (~100TW)

yields:

Lower power but better coherence

Narrow line spectrum

CLASSICAL SASE

needs:

GeV Linac

Long undulator (100 m)

yields:

High Power

Broad spectrum