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