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Insertion Devices
Lecture 3
Undulator Radiation in Detail
Jim Clarke
ASTeC
Daresbury Laboratory
2
Recap from Lecture 2
Multipole wigglers are periodic, high field devices, used to
generate enhanced flux levels (proportional to the number of
poles)
Undulators are periodic, relatively low field, devices which
generate radiation at specific harmonics
Electron
d
lu
q
3
Flux on-axis
Previously we argued that light of the same wavelength was
contained in a narrow angular width
(interference effect)
Assuming that the SR is emitted in angle with a Gaussian
distribution with standard deviation then we can
approximate:
For a Gaussian
Integrating over all angles gives
4
Flux in the Central Cone
In photons/sec/0.1% bandwidth the flux in this central cone is
As K increases the higher
harmonics play a more significant
part but the 1st harmonic always
has the highest flux
5
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)
6
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!
7
Example Undulator Tuning Curve
Undulator with 50mm period, 100 periods
3GeV, 300mA electron beam
8
Example Undulator Tuning Curve
Undulator with 50mm period, 100 periods
3GeV, 300mA electron beam
9
Example Undulator Tuning Curve
Undulator with 50mm period, 100 periods
3GeV, 300mA electron beam
10
The Peak Flux
The peak flux does not actually occur at the exact harmonic
wavelength.
The peak flux wavelength occurs at a small detune
from the harmonic wavelength
For our example undulator we have a first harmonic
wavelength of 3.99nm and a detuned wavelength of 4.03nm.
The exact harmonic actually has half the total flux compared
with the peak value
11
The exact harmonic can
only receive contributions
from higher angles
The detuned, longer
wavelength, has a hollow
cone and can receive flux
from both higher and lower
angles
The hollow cone shape
gives greater flux but
lower brightness
The exact harmonic has a
narrower angular width
The Peak Flux
12
Undulator Brightness (or Brilliance)
For the peak flux condition:
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
13
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
14
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
15
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 !
16
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 the same electron
parameters as before
In this example, for wavelengths
of >100nm, the electron beam
has virtually no impact on the
undulator brightness
17
Undulator Output Including Electron Beam Dimensions
Including electron beam
size and divergence
Not including
electron beam size
and divergence
18
Undulator Output Including Electron Beam Dimensions
Including electron beam
size and divergence
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
19
Effect of the Electron Beam on the Spectrum
Including the finite electron
beam emittance reduces the
wavelengths observed (lower
photon energies)
At the higher harmonics the
effects are more dramatic
since they are more sensitive
by factor n
n = 1
n = 9
20
Effect of the Electron Beam on the Spectrum
The same view of
the on-axis flux
density but over a
wider photon energy
range
The odd harmonics
stand out but even
harmonics are now
present because of
the electron beam
divergence
n = 1
2
3
4 6
5
Even harmonics observed on
axis because of electron
divergence (q2 term in
undulator equation)
21
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
n = 1
n = 9
Zero electron emittance
assumed for plots
22
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
23
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
24
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
25
Stokes Parameters
The intensity can be measured for different polarisation
directions:
Linear erect
Linear skew
Circular
The Stokes Parameters are:
26
Polarisation Rates
The polarisation rates, dimensionless between -1 and 1, are
Total polarisation rate is
The natural or unpolarised rate is
27
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
28
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
29
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
30
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
Low K regime so interference effects dominate
Q. What happens to the undulator wavelength?
3 independent
variables
31
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
Very similar to the standard result
Note. Wavelength is independent of phase
Extra term now
32
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?
33
Flux Levels
For the helical case, when results are very similar
to planar case:
Angular flux density on axis (photons/s/mrad2/0.1% bandwidth)
where
34
Flux in the Central Cone
(photons/s/0.1% bandwidth)
where
In this pure helical mode get circular polarisation only (P3 = 1)
The Bessel function term in Qn(K) is always zero except when
n = 1, this means that only the 1st harmonic will be observed
on axis in helical mode
35
Flux Levels
These equations can be further simplified since
and so Y = 0.
Angular flux density on axis (photons/s/mrad2/0.1% bandwidth)
And flux in the central cone (photons/s/0.1% bandwidth)
36
Power from Helical Undulators
Use the same approach as previously
For the case B is constant (though
rotating) and so
Exactly double that produced by a planar undulator
The power density is found by integrating the flux density over
all photon energies
On axis
37
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
38
Summary
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
39
Just Out of Interest …
Although SR light sources always use electrons (or sometimes
positrons), all relativistic charged particles will generate SR
The Large Hadron Collider will accelerate protons to 7 TeV
before collision
The main dipoles have a field strength of 8.3T
What will be the critical wavelength of the SR emitted in the
dipoles by the protons?
The mass of the proton is 1836 times heavier than the electron
The rest mass energy is 1836 x 0.511 MeV = 938 MeV
The Lorentz g factor at 7 TeV is then 7000/0.938 = 7461
The bending radius is 2813 m
So the critical wavelength is 28.4 nm
40
What About the SR Power?
How much SR power will the LHC radiate if it stores 500mA of
proton beam?
Using
We find that the total radiated power is only 3320 W
If the LHC stored electrons instead of protons the radiated
power would be 18364 times larger or 3.8 x 1016 W (38 PW)
The world’s electricity consumption is currently estimated to be
2 TW. The total power from the sun striking the Earth is
~100PW.
This is one major reason why we talk of linear colliders!!!!
41
The LHC Undulators
Two undulators are installed in
the LHC, one for each circulating
beam
They have a peak field of 5 T
and just two periods of 280 mm
each
To an electron this would be a K
value of ~131 but to the proton it
is only a K value of ~ 0.07
42
The LHC Undulators
SR monitors are the only non-intercepting devices able to
produce 2-dimensional beam images, giving not only the
beam size along any axis but also any beam tilt
The difficulty is to provide enough photons over the whole
beam energy range
There is no problem to produce light with the normal
bending magnets above 1 TeV, but a dedicated undulator has
to be used for beam energies from 450 GeV to 1 TeV
To cover the full energy range, the emission spectrum has to
be kept wide, which implies a small number of periods
43
The LHC Undulators
The first harmonic wavelength changes from ~600 nm at
450GeV to ~55 nm at 1.5 TeV
The detector has a wide acceptance (-0.5 to +1.5 mrad) to
collect enough light
The small number of periods provides a very broad spectrum
so that the detector response range is always covered
Observation angle (mrad)1stH
arm
on
ic W
avele
ngth
(nm
)
