Synchrotron Radiation
Lecture 2
Undulators
Jim Clarke
ASTeC
Daresbury Laboratory
Undulators
For a sinusoidal magnetic field
bx is the relative transverse velocity
The energy is fixed so b is also fixed. Any variation in bx will
have a corresponding change in bs (by = 0)
2
The Condition for Interference
For constructive interference between wavefronts emitted by
the same electron the electron must slip back by a whole
number of wavelengths over one period
d
q
lu
Electron
Speed = distance/time = c
The time for the electron to travel one period is
In this time the first wavefront will travel the distance
3
Interference Condition
The separation between the wavefronts is
And this must equal a whole number of wavelengths for
constructive interference
Using
and
and
n is an integer – the
harmonic number
for small angles:
4
The Undulator Equation
We get
And the undulator equation
Example, 3GeV electron passing through a 50mm period
undulator with K = 3. First harmonic (n = 1), on-axis is ~4 nm.
cm periods translate to nm wavelengths because of the huge
term
5
Undulator equation implications
The wavelength primarily depends on the period and the
energy but also on K and the observation angle q.
If we change B we can change l. For this reason, undulators
are built with smoothly adjustable B field. The amount of the
adjustability sets the tuning range of the undulator.
Note. What happens to l as B increases?
6
Undulator equation implications
As B increases (and so K), the output wavelength increases
(photon energy decreases).
This appears different to bending magnets and wigglers where
we increase B so as to produce shorter wavelengths (higher
photon energies).
The wavelength changes with q2, so it always gets longer as
you move away from the axis
An important consequence of this is that the beamline
aperture choice is important because it alters the radiation
characteristics reaching the observer.
7
Harmonic bandwidth
Assume the undulator contains N periods
For constructive interference
For destructive interference to first occur
(ray from first source point exactly out of phase with centre
one, ray from 2nd source point out of phase with centre+1, etc)
Range over which there is some emission
Bandwidth (width of harmonic line):
8
Angular width
Destructive interference will first occur when
This gives
And using
We find, for the radiation emitted on-axis, the intensity falls to
zero at
Example, 50mm period undulator with 100 periods emitting
4nm will have
40 mrad, significantly less than
9
Diffraction Gratings
Very similar results for angular width and bandwidth apply to
diffraction gratings
This is because the
diffraction grating acts as a
large number of equally
spaced sources – very
similar concept as an
undulator (but no relativistic
effects!)
10
When does an undulator become a wiggler?
As K increases the number of harmonics increases:
K
1
5
10
20
Number of Harmonics
few
10s
100s
1000s
At high frequencies the spectrum smoothes out and takes on
the form of the bending magnet spectrum
At low frequencies distinct harmonics are still present and visible
There is in fact no clear distinction between an undulator and
a wiggler – it depends which bit of the spectrum you are
observing
11
Undulator or Wiggler?
The difference depends upon which bit of the spectrum you
use!
This example shows an undulator calculation for K = 15.
[Calculation truncated at high energies as too slow!]
Equivalent
MPW
spectrum
(bending
magnet like)
Looks like an
undulator here
Looks like a
wiggler here 12
Undulator On Axis (Spectral) Angular Flux Density
In units of photons/sec/mrad2/0.1% bandwidth
Where:
N is the number of periods
E is the electron energy in GeV
Ib is the beam current in A
Fn(K) is defined below (J are Bessel functions)
13
Undulator On Axis (Spectral) Angular Flux Density
In units of photons/sec/mrad2/0.1% bandwidth
As K increases we can see that the influence of the higher
harmonics grows
Higher harmonics
have higher flux
density
Only 1 harmonic
at low K
14
Example On Axis Angular Flux Density
An Undulator with 50mm period and 100 periods with a
3GeV, 300mA electron beam will generate:
Angular flux density of 8 x 1017 photons/sec/mrad2/0.1% bw
For a bending magnet with the same electron beam we get a
value of ~ 5 x 1013 photons/sec/mrad2/0.1% bw
The undulator has a flux density ~10,000 times greater
than a bending magnet due to the N2 term
15
Flux in the Central Cone
In photons/sec/0.1% bandwidth the flux in the central cone is
As K increases the higher
harmonics play a more significant
part but the 1st harmonic always
has the highest flux
16
Example Flux in the Central Cone
Undulator with 50mm period, 100 periods
3GeV, 300mA electron beam
Our example undulator has a flux of 4 x 1015 compared with
the bending magnet of ~ 1013
The factor of few 100 increase is dominated by the
number of periods, N (100 here)
17
Undulator Tuning Curves
Undulators are often described by “tuning curves”
These show the flux (or brightness) envelope for an
undulator.
The tuning of the undulator is achieved by varying the K
parameter
Obviously K can only be one value at a time !
These curves represent what the undulator is capable of as
the K parameter is varied – but not all of this radiation is
available at the same time!
18
Example Undulator Tuning Curve
Undulator with 50mm period, 100 periods
3GeV, 300mA electron beam
19
Example Undulator Tuning Curve
Undulator with 50mm period, 100 periods
3GeV, 300mA electron beam
20
Example Undulator Tuning Curve
Undulator with 50mm period, 100 periods
3GeV, 300mA electron beam
21
Undulator Brightness
All emitted photons have a position and an angle in phase space (x, x’)
Phase space evolves as photons travel but the area stays constant
(Liouville’s theorem)
The emittance of an electron beam is governed by the same theorem
Brightness is the phase space density of the flux – takes account of the
number of photons and their concentration
Brightness (like flux) is conserved by an ideal optical transport system,
unlike angular flux density for instance
Since it is conserved it is a good figure of merit for comparing sources
(like electron beam emittance)
x’
x’
1
Area stays constant 1
At s = 1
At s = 0
0
1
x
0
1
2
x
22
Undulator Brightness
To calculate the brightness we need the phase space areas
We need to include the photon and electron contributions
We add contributions in quadrature as both are assumed to
be Gaussian distributions
The photon beam size is found by assuming the source is the
fundamental mode of an optical resonator (called the
Gaussian laser mode)
For the peak undulator flux:
23
Undulator Brightness
The example undulator has a source size and divergence of
11mm and 28mrad respectively
The electron beam can also be described by Gaussian shape
(genuinely so in a storage ring!) and so the effective source
size and divergence is given by
The units of brightness are photons/s/solid area/solid angle/spectral
bandwidth
24
Example Brightness
Undulator brightness is the flux divided by the phase space
volume given by these effective values
For our example undulator, using electron beam parameters:
The brightness is
25
Brightness Tuning Curve
The same concept as the flux tuning curve
Note that although the
flux always increases
at lower photon
energies (higher K
values, remember
Qn(K)), the brightness
can reduce
26
Warning!
The definition of the photon source size and divergence is
somewhat arbitrary
Many alternative expressions are used – for instance we could
have used
The actual effect on the absolute brightness levels of these
alternatives is relatively small
But when comparing your undulator against other sources it is
important to know which expressions have been assumed !
27
Diffraction Limited Sources
Light source designers strive to reduce the electron beam size
and divergences to maximise the brightness (minimise
emittance & coupling)
But when
then there is
nothing more to be gained
In this case, the source is said to be diffraction limited
Example phase space volume
of a light source
Using electron parameters
In this example, for wavelengths
of >100nm, the electron beam
has virtually no impact on the
28
undulator brightness
Undulator Output Including Electron Beam Dimensions
Not including
electron beam size
and divergence
The effect of the electron beam is
to smear out the flux density
The total flux is unchanged
50mm period
100 periods
3GeV
300mA
K=3
l = 4 nm, 1st harmonic
Including electron beam
size and divergence
29
Effect of the Electron Beam on the Spectrum
Including the finite electron
beam emittance reduces the
wavelengths observed (lower
photon energies)
n=1
n=9
At the higher harmonics the
effects are more dramatic
since they are more sensitive
by factor n
30
Effect of the Electron Beam on the Spectrum
The same view of
the on-axis flux
density but over a
wider photon energy
range
2
The odd harmonics
stand out but even
harmonics are now
present because of
the electron beam
divergence
5
3
n=1
4
6
Even harmonics observed on
axis because of electron
divergence (q2 term in
undulator equation)
31
Flux Through an Aperture
The actual total flux observed
depends upon the aperture of
the beamline
As the aperture increases, the
flux increases and the
spectrum shifts to lower
energy (q2 again)
The narrower divergence of
the higher harmonic is clear
Zero electron emittance
assumed for plots
n=1
n=9
32
Flux Through an Aperture
As previous slide but
over a much wider
photon energy range
and all harmonics
that can contribute
are included – hence
greater flux values
Many harmonics will
contribute at any
particular energy so
long as large enough
angles are included
(q2 again!)
If a beamline can accept very wide
apertures the discrete harmonics are
replaced by a continuous spectrum
33
Polarised Light
A key feature of SR is its polarised nature
By manipulating the magnetic fields correctly any polarisation
can be generated: linear, elliptical, or circular
This is a big advantage over other potential light sources
Polarisation can be described by several different formalisms
Most of the SR literature uses the “Stokes parameters”
Sir George Stokes
1819 - 1903
34
Stokes Parameters
Polarisation is described by the relationship between two
orthogonal components of the Electric E-field
There are 3 independent parameters, the field amplitudes
and the phase difference
When the phase difference is zero the light is linearly
polarised, with angle given by the relative field amplitudes
If the phase difference is p/2 and
then the light
will be circularly polarised
However, these quantities cannot be measured directly so
Stokes created a formalism based upon observables
35
Stokes Parameters
The intensity can be measured for different polarisation
directions:
Linear erect
Linear skew
Circular
The Stokes Parameters are:
36
Polarisation Rates
The polarisation rates, dimensionless between -1 and 1, are
Total polarisation rate is
The natural or unpolarised rate is
37
Bending Magnet Polarisation
Circular polarisation is zero on
axis and increases to 1 at large
angles (ie pure circular)
But, remember that flux falls to
zero at large angles!
So have to make intensity and
polarisation compromise if
want to use circular polarisation
from a bending magnet
Linear
Circular
38
Qualitative Explanation
Consider the trajectory seen by an observer:
In the horizontal plane he sees the electron travel along a line (giving
linear polarisation)
Above the axis the electron has a curved trajectory (containing an element
of circular polarisation)
Below the axis the trajectory is again curved but of opposite handedness
(containing an element of circular polarisation but of opposite
handedness)
As the vertical angle of observation increases the curvature observed
increases and so the circular polarisation rate increases
39
Undulator Polarisation
Standard undulator
(Kx = 0) emits linear
polarisation but the
angle changes with
observation position
and harmonic order
Odd harmonics have
horizontal
polarisation close to
the central on-axis
region
40
Helical (or Elliptical) Undulators
Include a finite horizontal field of the same period so the
electron takes an elliptical path when viewed head on
Consider two orthogonal fields of equal period but of different
amplitude and phase
3 independent
variables
Low K regime so interference effects dominate
Q. What happens to the undulator wavelength?
41
Undulator Equation
Same derivation as before:
Find the electron velocities in both planes from the B fields
Total electron energy is fixed so the total velocity is fixed
Find the longitudinal velocity
Find the average longitudinal velocity
Insert this into the interference condition to get
Extra term now
Very similar to the standard result
Note. Wavelength is independent of phase
42
Polarisation Rates
For the helical undulator
Only 3 independent variables are needed to specify the
polarisation so a helical undulator can generate any
polarisation state
Pure circular polarisation (P3 = 1) when Bx0 = By0 and
Q. What will the trajectory look like?
43
Flux Levels for Helical Undulator
When
and
Angular flux density on axis (photons/s/mrad2/0.1% bandwidth)
And flux in the central cone (photons/s/0.1% bandwidth)
In this pure helical mode get circular polarisation only (P3 = 1) and only
the 1st harmonic is observed on axis
44
Power from Helical Undulators
Use the same approach as previously
For the case
rotating) and so
B is constant (though
Exactly double that produced by a planar undulator
The power density is found by integrating the flux density over
all photon energies
On axis
45
Helical Undulator Power Density
Example for 50 mm
period, length of 5 m
and energy of 3 GeV
Bigger K means lower
on-axis power density
but greater total
power
Get lower power
density on axis as only
the 1st harmonic is on
axis and as K
increases the photon
energy decreases
46
Summary
Undulators are periodic, low(er) field, devices which generate
radiation at specific harmonics
The distinction between undulators and multipole wigglers is
not black and white.
Undulator flux scales with N, flux density with N2
The effect of the finite electron beam emittance is to smear out
the radiation (in both wavelength and in angle)
The q2 term has an important impact on the observed radiation
 width of the harmonic
 the presence of even harmonics
 the effect of the beamline aperture
Undulators are excellent sources for experiments that require
variable, selectable polarisation
47