Modelling shot noise and coherent spontaneous emission in the FEL

Brian McNeil*, Mike Poole & Gordon Robb*

CCLRC Daresbury Laboratory, UK

*Department of Physics, University of StrathclydeGlasgow, UK

ICFA Future Light Sources Sub-PanelMini Workshop on

Start-to-End Simulations of X-RAY FELs 

August 18-22, 2003 at DESY-Zeuthen (Berlin, GERMANY)

Outline

• New model of electron shot-noise - derived from first principles

• Combined shot-noise/CSE numerical model – simulations

• New model for ultra high power/ultra short pulse radiation

• Simulation of such pulse propagation

New model of electron shot-noise

Previous Shot-Noise Model

1111

,)(),( zzbzzzAzz

Previously with the averaged equations…

ρ

zzi

Nzzb j

N

j 2exp

1, 1

11

bM

V(bM) = V(bR)

R

M

N

N3

MN

. .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … .

bR

RN. .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … .

-θj

z

Current

Spatial distribution

0 0|| ieasb

Density variation over a radiation wavelength acts as a source. This is not modelled in the averaged equations.

Coherent Spontaneous Emission - No SVEA approximation

Now with the un-averaged equations…

There is now no reference to any averaging over radiation wavelength scale – previous method is not valid.

Assume the arrival of electrons in a time interval is a Poisson process:

. .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … .

t

Some radiation period

Mean rate of e- arrival

Mean number of e-

Poisson prob. of Nn e-

Mean arrival time of an e-

Var. arrival time of an e-

. .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … .

realn

macron tVtV

Equating real and macro variances:

nN

tt

1212

22

macront

Macroparticle mean arrival time:

nn N

t

N

tt

. .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … .

t (<< some radiation period

t

. .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … .

So, in the interval t :

. .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … . . .. . … . . .. …. . . …. . .. .. …. .. ….. .. .. .. . . … . .. .. .. .. .. … .

Results for cold mono-energetic electron distribution:

Schematic of model for higher dimensional phase space - e.g. including energy spread

Macroparticle charge weighting:

Macroparticle position in phase space:

- Mean # electrons in phase space cell

- Charge weight assigned to jth macroparticle in cell is Poisson random variate

Uniform random variates:

Results for electron distribution with top-hat energy spread after a drift of 1z1 FWp

The following equations were solved using Finite Element Method:

- 1-D Pierce parameter

le - scaled electron pulse duration

N – Expectation of total # electrons in pulse

FEL Simulation

*

**Random variables

z1

1

z1

element

S2( ) z1S1( ) z1

A(z, ) = an(z) + an+1(z) S1( ) z1S2( ) z1

z1

Finite elements:

an an+1an-1 an+2

Simulation Results

In the following sequence of graphs:

• z is the scaled distance through the wiggler

• z1 is the scaled position within the pulse in radiation periods: = 0.1/4

• |A|2 is the scaled radiation power

• j/ 0 is the jth electron’s relativistic parameter scaled with respect to its

initial value

• P is the scaled spectral power

• f is the frequency scaled with respect to the resonant fundamental

• Ax,y are the scaled x-y components of the electric fields

Numerical solution including shot-noise and CSE:

• Top-hat electron pulse current

• Gaussian energy spread of

• FEL parameter

•Total charge

•Average macroparticle

6el

5.0p

08.04

1

nCQ 1

e51025.1

0.1 0.008

Averaged quantities

Bunching parameter |b| (x) and scaled intensity <|A|2> ( ) at a fixed position in the electron pulse

Shot-noise

SACSE

Numerical solution including shot-noise and CSE:

• Gaussian electron pulse current

• Gaussian energy spread of

• FEL parameter

•Total charge

•Average macroparticle

3;18 eel

5.0p

08.04

1

nCQ 1

e51025.1

0.1 0.008

New model for ultra high power/ultra short pulse radiation propagation in the FEL

Theory predicts that very short intense pulses of radiation can be generated in an FEL when the electrons emit superradiantly.

Electrons

N N NS S

S S SN N

Wiggler magnet Radiation pulses

4/1/1 pkI

pkI

We want to know :Is there a saturation mechanism in Ipk? – missing from current theoryIs there a limit to the duration of the high power radiation pulses? -Current theory breaks down when the ~ (frequency)-1

4/1/1 pkI

Why do we need another model for the FEL?

The Coupled 1-D Maxwell-Lorentz equations:

These equations are rewritten in a scaled form with the minimum number of assumptions necessary to model:

• large energy exchanges between electrons and radiation - >>1

• pulses of very short duration -

The 1-D (plane wave) approximation is made and space-charge effects are neglected.

4/1/1 pkI

- Wave equation

- Lorentz equation

- Current density

pkI

The fields in a helical wiggler FEL:

ONLY assumption outwith

1D and neglect of space charge

=> Neglect backward wave

- Wiggler magnetic

- Radiation electric

- Radiation magnetic

yixe ˆˆ2

1ˆ

Scaled parameters

With previous assumptions & fields we obtain:

Working equations

= 2ρpj

‘Old’ model

• Averaged equations

• SVEA valid

• 1|| 2 A

FEL

Numerical solution including shot-noise and CSE:

• Top-hat electron pulse current

• Gaussian energy spread of

• FEL parameter

•Total charge

•Average macroparticle

6el

5.0p

08.04

1

nCQ 1

e51025.1

0.1 0.008

New Model

Averaged model Non-averaged model

Scaled intensity |A|2

For Planar, short, cold electron pulse:

• Gaussian electron pulse

• FEL parameter

15.0; zel

076.0

Conclusions:• New model of electron shot-noise derived from first principles

• Model used to simulate simultaneous FEL start-up from CSE and shot-noise

• Demonstrated model for simulating ultra high power pulse propagation in Free Electron Laser

• Sub-wavelength radiation pulses are seen to propagate with quasi-unipolar fields

• No saturation effects yet observed for this preliminary study

• Exciting prospects for future analytical and numerical work in generation of exotic FEL radiation spikes and post-saturation modelling

Things we would like to develop:

• Changes in statistical nature of radiation ?

• Search for saturation mechanisms in radiation spiking

• Introduce 2-D – diffraction, space charge

• Can this method be adapted to model CSR in bending magnets ?