How an FEL works Brian McNeil,University of Strathclyde, Glasgow &

ASTeC, Daresbury Labs.

Some Sources of information• 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 p1-69 (1990) [see also Rivista del Nuovo Cimento, Vol. 15, no. 11 p1-52 (1992) ]

• 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).

• Z. Huang & K.-J. Kim, Review of x-ray free-electron laser theory, Phys. Rev. ST Accel. Beams 10, 034801 (2007).

• The World Wide Web Virtual Library: Free Electron Laser research and applications http://sbfel3.ucsb.edu/www/vl_fel.html

• W.B. Colson et al., Free Electron Lasers in 2009, Proceedings of the 31st International Free Electron Laser Conference (Liverpool, U.K.), WEPC43, 591-595 (2009).

Some figures

Some figures have been „borrowed‟ from other sources:

DESY (XFEL,FLASH) Group, Germany;

Riken/SPring-8 group, Japan;

LCLS group, USA.

What is an FEL and how does it work?

I – Spontaneous radiation

II – Coherent emission - FEL

What is a FEL?

NOT a quantum source!

En

En-1

e-

A classical source of tuneable, coherent electromagnetic

radiation due to accelerated charge (electrons)

vz

What is a FEL?

A classical source of tuneable, coherent electromagnetic

radiation due to accelerated charge (electrons)

NOT a quantum source!

En

En-1

1-nn EEhν

I – Spontaneous radiation:no interaction between electrons

and the light they emit

Generation of EM Radiation

Non-relativistic charge source

Relativistic Emission

Most energy confined to the

relativistic emission cone

Stationary electron

bq

b

qr = -1

Relativistic electron

v <~ c

Energy emission confined to

directions perpendicular to

axis of oscillation

qr

Planar Undulator or „Wiggler‟

Looking down the axis of an undulator

The electron trajectory in an

undulator

An electron trajectory

j j

j

d mve E v B

dt

The Lorentz force equation

for the j-th electron is written:

The independent variable of the electron trajectory here is t, and only t:

, ,j

j j j j j

drr t x t y t z t v

dt and so:

It is convenient to change the independent variable to z to get:

, , ; ??j j j j

dr z x z y z z t z

dt

0

0 0

lim

; ;

lim lim

1.

Now:

where: and

j zj j

t

j j zj

j j

zjz v z

j zj j

zj

zj zj zj zj

f t t f tdf

dt t

f t f z t f t t f z z z v t

f z z f z f z z f zdfv

dt z v z

vd d dv c

dt dz dz cb b b

t

jz t

t

jz

j

zj

dzv

dt

1

1

b b

An electron trajectory in an undulator

0 0j

j j

d eE E

dt mc

b , a constant, for

2

j j

j

z j

d eE c t B

d z mc

bb

b

z

yx

Consider a planar undulator field: 0,sin ,0 .u uB B k z

Rewriting the Lorentz equation:

From the Lorentz equation we can derive:

so that in an undulator, neglecting

radiation fields & space charge:0

j

j

z j

d eB

d z mc

bb

b

0 0 0

0 0

sin cos

cos sin

x j x j u uz j y u x j u

z j

j u uu j u

u

d d eB aeB k z k z

d z mc d z mc

d x a ak z x k z

dz k

b bb b

b

Can integrate again:

11 5; 2000; 2uu u GeV u

u

eBa a

mck where: - Typical values cm

1z jb

For 0 1,

A planar undulator

For x-component:

>> 1,

An electron trajectory in an undulator

Look at the z –component using: 0

0

1cos

1

ux j u

j j

ak zb

b b

and

2

2 2 2 2 2 2 2

0 0 2

0 0

222 2

2 2

0 0

222

2

0

2

11 1 cos 1 1 cos

1 cos 2 cos 21 11 1 1 1

2 2 2

cos 211 1

2 2

1 1 1

uz j x j u z j u u

u u uuz j u

u uuz j

z j z j z

ak z a k z

k z a k zaa

a k za

b b b

b

b

b b b

Now use

2 22 2

2 2

0 0

2 1 :

cos 2 cos 21 11 1 1 1

2 2 2 2 2 2

j z j

u u u uu uz j z j

a k z a k za a

b

b b

Can be averaged to give:2

2

0

11 1

2 2

uz j

ab

x

jq

z

An electron trajectory in an undulator

0

tan cosj u

j u

d x ak z

dzq

0

0

uj u

aaq

for

qr

x

z

The angle the electron makes

with respect to the undulator axis

can be approximated as:

The radiated power is confined

mainly to an angle01 .rq

Hence if: 1,j r uaq q i.e.

The emitted power behaves like

a „searchlight‟ when viewed at

end of the undulator.

- „Undulator‟

- „Wiggler‟1ua

1ua

>> au

>> 1,

>> 1

~ 1

u

e-

Non-resonant emission - destructive interference

e-

The radiation is not phase-matched to the electron trajectory.

Radiation Electric field

Resonant emission - constructive interference

r

e-e-

u

r

The radiation and electron trajectory are phase-matched.

+

+

=

+

+

=

Non-resonant emission Resonant phase matched emission

u

u

u

u

uλ

e-

r

Resonant phase matched emission by an electron

vz

Resonant phase matched emission for harmonics

r

e-

u

2nd Harmonic

3rd Harmonic

Harmonics of the fundamental are also phase-matched.

What are properties of radiation

from an undulator ?

Resonant emission - constructive interference

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

j

zj

tv

A resonant radiation wavefront will have travelledu r

u r rtc

Equating:

1 zju u rr u

zj zjv c

b

b

where: zj

zj

v

cb

e-

u

r

zv

z

Resonant emission - constructive interference

including harmonics and angle from undulator axis

Electron

d

u

q

Condition for constructive interference: cosuu

zj

d n

qb

Where: is an integer representing the harmonic number1, 2, 3,n

Undulator EquationSubstituting in for the average longitudinal velocity of

the electron, , for the earlier planar case:

For a 3 GeV electron passing through a 5 cm

period undulator with = 3, the wavelength of the

first harmonic (n = 1) on axis (q = 0) is ~ 4 nm

2

RMS

u uu

e Ba

mc

is the RMS “wiggler/undulator parameter”

- In this form also valid for helical undulators

zb

2

2

0

11 1 cos

2 2Substitute intou u

zj r u

zj

an

b q

b

2 2 2

02

0

12

ur ua

n

q

ua

Including angular

dependence

2

2 2 2

2( , )

4

n ri t

cd I en n n e dt

d d c

b

where ( )r t is the electron position at time t, and

We can calculate the spontaneous emission spectrum

in the frequency range d and solid angle d around the

observation direction by inserting the expression for the

electron trajectory into the standard formula for far-field

emission from an accelerated charged particle

(see e.g. “Classical Electrodynamics” by Jackson , ch. 14)

n

v

cb

Spontaneous spectrum

If we do this we find that the fundamental

spectrum on axis ( ) is :ˆn z

2 2

2

sinˆ( , )

d I xn z

d d x

Where: r

ru

ω

ωωπNx

uN is the number of undulator periods

2

0

2

2

1

ur

u

ck

a

is the central (resonance) frequency

Spontaneous spectrum

On axis fundamental spontaneous spectrum therefore looks like :

-10 -8 -6 -4 -2 0 2 4 6 8 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

d2I

dd

x

Main features :

• Spectrum strongly peaked at frequency r

i.e. at wavelength

• Width of spectrum

uN

1

ω

Δω

2

2

0

12

2

ur u

r

ac

uN

1

ω

Δω

Undulator Radiation

Undulator radiation (top) focused on a spot (bottom) by a refractive lens.

2 2 2

02

0

12

ur ua

n

q

Summarising

Undulator radiation

Setup>trajectory>undulator

Code available at: http://www-xfel.spring8.or.jp/

Electron bunching in a fixedradiation field

The electron-radiation interaction

j jF q E v B

22

02 2

1 E JE

c t t

The Lorentz force (electron dynamics)

Maxwell wave equation* (radiation evolution)

Both equations must be solved together simultaneously

(self-consistently) to fully describe the FEL interaction

*Neglect static fields (space charge effects) – Compton limit

0-

j j

j j

d m vF e E v B

dt

2

0j

j

d m ce E v

dt

Hendrick Antoon

Lorentz

The Lorentz Force Equation:

The rate of change

of electron energy

How the electron is effected by the resonant radiation

Slow energy exchange

The rate of change of electron energy: 2

0j

j

d m ce E v

dt

Consider plane-wave field: 0ˆ sin r rE x E k z t

0

cosux j u j

ak zb

Interacting with an electron on trajectory:

2

0

0

0

0

2

0 0

0

2

0 0

sin cos

sin cos

1sin sin

2

j uj r j r u j

j u

r j r u j

u

r u j r r u j r

d m c ae E v e E k z t k z

dt

d e a Ek z t k z

dt m c

e a Ek k z t k k z t

m c

Assuming:

0j

Slow energy exchange

0

2

0 0

1sin sin

2

j u

r u j r r u j r

d e a Ek k z t k k z t

dt m c

The first sin term on RHS is a wave with speed in z direction of:

r r rz z

r u r u r u

ck kv

k k k k k k

b

Recall previous result for resonance:

1 zj u rr u zj

zj r u r u

k

k k

b b

b

So, a resonant electron with average speed will have zjb constant

jd

dt

The second sin term on RHS is a wave with speed in z direction of:

1

fast, non-resonant, phase variation.

r r rz z

r u r u r u

ck kv

k k k k k k

b

u

e-

vE

is +ve

Resonant emission – electron energy change

u

e-

vE

is +ve

Resonant emission – electron energy change

u

e-

vE

is +ve

Resonant emission – electron energy change

Energy of electron changes

„slowly‟ when interacting with

a resonant radiation field.

u

e-

Resonant emission – electron energy change

is -vevE

e-vE

is +ve

For an electron with a different phase with

respect to radiation field:

u

e-

Resonant emission – electron energy change

is -vevE

e-vE

is +ve

Rate of electron energy change is „slow‟ but changes

periodically with respect to the radiation phase

is +vexv

Lose

energy

Gain

energy

Resonant interaction – electron bunching

Axial electron velocity

r

2

0j

j

d m ce E v

dt

Resonant interaction – electron bunching

Axial electron velocity

r

Electrons bunch at resonant radiation

wavelength – coherent process

Bunched electrons can exchange

energy coherently with radiation

2

2 *

1 1 1 1

Radiation power j kj

N N N Nii

j j j k

j j j k

E e E E E e

j k

If then the 2nd term >> 1st term as there are N 2 of them

and results in coherent emission.j k j

Resonant interaction – Even harmonics

e-

0.5u

3rd Harmonic

2nd Harmonic

Fundamental

Even harmonics do not allow a slow exchange of energy

Electron bunching in a self-consistent radiation field:

The FEL mechanism

1

The transverse current densityN

j

j

J e v r r t

Bunched electrons drive radiation

Radiation field bunches electrons

Basic FEL mechanism

-j jF e E v B

22

02 2

1 JEE

c t t

Bunched electrons drive radiation

Radiation field bunches electrons

Basic FEL mechanism

. .j

j

j

ij

dp

dz

dpAe c c

dz

q

q

,A A

b z tz t

1

1, j

Ni z

jpk t

I tb z t e

I N

q

j r u j r

j r

j

r

k k z t

p

q

2 rad

beam

PA

P

z

c

z c tt

l

b

These equations are assumed „slowly varying‟ i.e. any evolution is

assumed slow with respect to the radiation/undulator period. They can be

subsequently averaged over a radiation/undulator period. 4c rl

, , sin Radiation enveloper rA z t E z t k z t

Conventional laser Vs FEL pulses

Active medium

Undulator

Conventional laser pulse interacts with all of the active medium

FEL radiation pulse may interact with only a section of the active medium z

e-

t

0,b z t 0,A A

b z tz t

Partial form of wave equation

describes slippage of radiation

envelope through the electron pulse

z

Linear analysis

. .

,

j

j

j

ij

dp

dz

dpAe c c

dz

A Ab z t

z t

q

q

dbiP i b

d z

dPA i P

d z

dAb

d z

0

00; 0,2i

je Uq q

0jp

0 1A

0 1 etc.j j jq q q 1 1where: jq

2

1 ...2

x xe x

0

0

1

1

i

i

b i e

P p e

q

q

q

Steady-state

approx.: “No pulses”

Assume

that:

Where:

The steady-state approximation can

be thought of as the continuous e-

beam limit where the electron „pulse‟

has no beginning or end. In this case

one can see that the radiation field

can only be a function of the distance

through the undulator and no pulse

effects can be present.

0 1 0 1 0

11j j j j ji i i i

je e e e iq q q q q

q

Using:

1

1 N

j

j

x xN

<< 1,

<< 1

Linear analysis

2

2

3

3

d A dbiP

dz dz

d A dPi iA

dz dz

Differentiating linear equations:

0

3

3

Look for solutions:

1

1 3 1 31; ;

2 2 2 2

i zA z A e

i i

i i

Re

Im

First assume resonance: 0 Away from resonance: 0

3 2 1 0f

the dispersion relation is:

f

3 Real

1 Real2 Complex conjugate

1.89crit

Linear analysis

0

Solutions for 0:

1 3 1 3 for 1; ;

3 2 2 2 2

ji z

j j

j

AA z c e i i

Real parts give oscillatory solutions.

Imaginary parts give exponential growth:

and exponential decay:

3

2i

3

2i

A z

z

33

20 02

For 1

3 3

g

zz l

z

A AA z e e

0A

-20 0 20-1

0

1

2

3

Ga

in

Gain as a function of detuning from resonance

0

0

j r

j

r

p

2z

-20 0 20-4

-2

0

2

4x 10

-3

Gain

0.24z

-30 -20 -10 0 10 20 30

0

1000

2000

3000

4000

Gain

6z

2 2

0

2

0

GainA z A

A

Oscillatory

terms dominateOscillatory &

exponential

+ve exponential

term dominates

Constants of motion

. .

,

j

j

j

ij

dp

dz

dpAe c c

dz

dAb z t

d z

q

q

Two constants of motion can be obtained from

these equations in the steady-state limit:

2

2

2* *

2

constant

constant

A p

pi A b Ab A

Where the constant is the variables‟ initial values.

A

z

Linear

Numerical

The first constant above corresponds to conservation of

energy. The second, incorporating phase dependent

terms is related to the Hamiltonian of the system.

Opposite is plotted the linear and non-linear (numerical)

solutions of the equations for a resonant interaction

(δ = 0). From the definition of :

2 rad

beam

PA

P

and the saturated scaled field |Asat|~1, it is seen that ρ

is a measure of the efficiency if the interaction.

The pendulum equation

and phase-space

cos

1sin

da

dz

d

dz a

q

q

2 cos

j

j

j

j

dpa

dz

dp

dz

q

q

p

3

2

2

2

3

2

5

2

q

separatrix

The electrons can be thought of as a collection of pendula initially distributed

over a range of angles with respect to the vertical. The radiation field is

analogous to the gravitational field. The separartrix defines the boundary

between pendula that librate and rotate. Of course in the FEL equations

above, unlike a gravitational field, the radiation field can evolve in both

amplitude a, and phase .

Low Gain mechanism

1z

Low Gain – needs cavity feedback

p

3

2

2

2

3

2

5

2

q

-20 0 20-4

-2

0

2

4x 10

-3

Ga

in

0.24z

Oscillatory terms dominate

10opt

3

3

14 1 cos sin

2

2.6

2.6

Gain

where

The Gain is a maximum for

in the low-gain limit.opt

z

z

z

Here we consider the low gain

limit with 0.24z

The phase-space representation

of opposite will be used to look

at what happens with the

electrons and radiation during

the interaction.

FEL saturates when Gain=Cavity Losses

Low Gain – early stages

Beginning undulator

End undulator

0.4%Gain

Low Gain – intermediate

Beginning undulator

End undulator

0.3%Gain

Low Gain – saturated

10,000

10,005

10,010

Beginning undulator

End undulator

0.1%Gain

High Gain mechanism

1z

3

20

3

g

z

lAA z e

Single pass high-gain amplifier

q

1) e- begin to bunch about θ=3π/2

2) Radiation phase driven and shifts

3) Radiation amplitude is driven

F

cos

1sin

da

dz

d

dz a

q

q

2 cos

j

j

j

j

dpa

dz

dp

dz

q

q

Can assume periodic BC over one potential well:

i zA z a z e

Letting:

q3 2

0

q

1) e- begin to bunch about θ=3π/2

2) Radiation phase driven and shifts

3) Radiation amplitude is driven

F

cos

1sin

da

dz

d

dz a

q

q

2 cos

j

j

j

j

dpa

dz

dp

dz

q

q

Can assume periodic BC over one potential well:

i zA z a z e

Letting:

q3 2

0

Some resources

If you want to download some Fortran and Matlab codes that solve the

FEL equations and plot their solutions you can obtain them from:

http://phys.strath.ac.uk/eurofel/rebs/rebs.htm

2

sat

rad rad

beam beam

P PA

P P

Because the equations are universally

scaled (depend only on initial conditions)

and because , can see from

scaling of A that the rho parameter is a

measure of the efficiency of the high-

gain FEL amplifier:

21satA

Also see that the saturated radiated

power: sat

rad beamP P

As: ,beamP I and:1

3 ,I

see that: 4

3 ,sat

radP I demonstrating

that, in the steady-state, the high-gain

FEL interaction is a cooperative or

collective interaction.

Pulse effects in the high gain

Conventional laser Vs FEL pulses

Active medium

Undulator

Conventional laser pulse interacts with all of the active medium

FEL radiation pulse interacts with only a section of the active medium z

e-

t

0,b z t 0,A A

b z tz t

Partial form of wave equation

describes slippage of radiation

envelope through the electron pulse

z

An important parameter that defines how the FEL evolves within the pulses is the

cooperation length - the relative electron/radiation slippage in a gain length 4cl gl

FEL pulses starting from noise in

a High-Gain amplifier (SASE)

e- e-

vz<c

vz=c vz=c

vz<c

Regions of radiation pulse separated by evolve independently of

other regions. Hence there can be many regions that evolve independently

from different initial source terms due to noise* . This

leads to a noisy temporal and spectral radiation pulse.

0t 0t 0t z

(Note: in the scaled variables a radiation wavefront propagates a distance

with respect to the electron rest frame.)

z

0, constantb z t

~ 2 ct l

The spike widths are 2 cl

Puls

e e

nerg

ySaturation length – shot-

to-shot fluctuations.

Self Amplified Spontaneous Emission

(SASE) in the x-ray

SASE Power output: SASE spectrum:

Estimate of initial mean SASE powerThe radiation power evolves like:

And the saturation power:

From the analysis of *, the power in

the linear regime is:

N is # e- in radn. wavelength.

From last 2 eqns.:

(1)

(2)

(3)

(4)

Equating (1) & (3):

for:

and using (4):

*

Seeded FEL – improves SASE

e- e-

Longitudinal coherence of radiation pulse is inhereted

from that of seed if Pseed>>Pnoise

Region of seed with good longitudinal coherence :

undulator

Spoiling effects

Energy spread

0

0r j

j j

r

zp z

The effects of energy spread can be

investigated by introducing a spread in the

initial values of pj :

The steady state dispersion relation

becomes:

with solutions for the imaginary

part of lambda, determining the

high gain case, shown opposite.

Energy spread effects become

less important when:

1

f

2

Emittance*

http://www.fieldp.com/cpb.htmlCharged Particle Beams:*Stanley Humphries,

The beam emittance introduces two main effects:

1) The electron beam radius in a matched focussing

channel** is determined by the emittance via:

2) The emittance introduces an energy spread in the

resonant electron energy***. This can be added in

quadrature with the real energy spread to estimate

emittance effects in a 1D model:

β – betafunction

of focussing

lattice.

eff

***

Diffraction

The Rayleigh length lZR is that in which a beam diffracts to twice its

transverse mode area. In an FEL amplifier, if the gain length of the FEL

interaction is much greater than the Rayleigh length then diffraction can

cause reduced coupling and longer saturation lengths.

lZR

Good coupling

Reduced coupling

Gain length, lg

*

~

Conditions 2 & 3 yield the „Kim-Pellegrini‟ condition on the emittance: 1~ 4n

Thank You!

Real FEL designs

as taken from the 4GLS design*

http://www.4gls.ac.uk/documents.htm*4GLS Conceptual Design Report, Chapter 8:

4GLS CDR – April 2006

750-950MeV

600MeV