The Generation and Properties of Synchrotron Radiation Part 1

1
The Generation and Properties of Synchrotron
RadiationPart 1

• Jim Clarke
• ASTeC
• Daresbury Laboratory

2
Course Timetable

• 15th Jan 2 Lectures 10.30 11.45
• 22nd Jan 2 Lectures 10.30 11.45
• 1 Tutorial 15.15
• 29th Jan NO LECTURES FROM JAC
• 19th Feb 2 Lectures 10.30 11.45
• 26th Feb 2 Lectures 10.30 11.45
• 1 Tutorial 15.15

3
Course Syllabus

• Introduction History
• Fundamentals of SR Emission Part 1
• SR from a dipole
• Introduction to Insertion Devices
• SR from Wigglers
• SR from Undulators
• Generation of Polarised light
• Permanent Magnet Insertion Devices Part 2
• Electromagnet Insertion Devices
• Measurement Correction
• Impact of IDs on electron beam
• Special IDs for Free Electron Lasers
• Novel, exotic state of the art IDs

4
Course Book

• The Science and Technology of Undulators and
  Wigglers,
• J. A. Clarke, Oxford University Press

Available from all good bookshops (or
libraries!)
5
Introduction to SR

• SR is a relativistic effect
• Many features can be understood in terms of two
  basic processes
• Lorentz contraction
• Doppler shift
• Imagine that a relativistic charged particle is
  travelling through a periodic magnetic field (an
  undulator)
• In the particles rest frame it sees a magnetic
  field rushing towards it
• If in our rest frame the magnet period is then
  because of Lorentz contraction the electron sees
  it as
• g is the relativistic Lorentz factor

6
Lorentz Factor
v is the velocity of the particle E
Electron Energy (2000 MeV in SRS) Eo Electron
Rest Energy (0.511 MeV) So in the SRS, g
4000 This factor turns up again and again in SR !
7
Relativistic Doppler Shift

• In the relativistic case of the Doppler effect
  the frequency of light seen by an observer at
  rest is
• where f is the frequency emitted by the moving
  source, q is the angle at which the source emits
  the light.
• With the source travelling towards the observer
  so
• In terms of wavelength

8
Combining Lorentz and Doppler

• So the particle emits light of wavelength
• Since it is travelling towards us this wavelength
  is further reduced by a factor
• So the wavelength observed will be
• For GeV electron energies with g of 1000s, an
  undulator with a period of a few cm will provide
  radiation with wavelengths of nm (X-rays)

9
Angle of Emission

• In the moving frame of the electron, the electron
  is oscillating in the periodic magnetic field
  with simple harmonic motion
• It therefore emits in the familiar dipole pattern
  that has a distribution

Electric field lines due to a vertically
oscillating dipole
10
Angle of Emission

• A second consequence of Doppler is that the angle
  with which the observer views the source will
  also be affected
• So the point at which the electric dipole has
  zero amplitude is (q p/2) appears at the angle
• The peak of the emission is orthogonal to the
  direction of the particles acceleration so for a
  particle on a circular path the cone is emitted
  along the tangent

11
Effect of Relativity
Fernando Sannibale, USPAS, Jan 2006
12
Electromagnetic Radiation
Electromagnetic Radiation covers the spectrum
from Radio Waves through to Gamma rays
13
SR smoothly covers a wide part of the spectrum
14
SR from Bending Magnets
Observer
1/
g
R
R
1/
g
1/
g
Electron
The electrons in a synchrotron are accelerated as
they are forced to bend along a circular path in
a strong magnetic field.
15
Typical Wavelength
Pulse Length Time for electron along arc -
Time for photon along chord
So, Typical Wavelength
For SRS, R 5.5 m, g 4000 Wavelength 0.1 nm
16
Summary of the Three Basic Sources

• Bending magnet or dipole
• (Multipole) Wiggler
• Undulator

17
A Typical Spectrum
18
Definition of SR

• Synchrotron Radiation is electromagnetic
  radiation that is emitted by relativistic charged
  particles undergoing acceleration.
• Q. Apart from particle accelerators, where else
  might it be observed?

19
The First Observation
The Crab nebula is the expanding remains of a
star that was seen to explode by Chinese
astronomers in the year 1054AD.
At the heart of the nebula is a rapidly-spinning
neutron star, a pulsar, and it powers the
strongly polarised bluish 'synchrotron' nebula.
20
A Brief History of SR Early Theory

• 1897 Joseph Larmor derives expression for total
  instantaneous power radiated by an accelerated
  charged particle
• 1898 Alfred Lienard in Paris (before relativity!)
  derives the radiation due to charges moving close
  to c energy loss of an electron on a circular
  path

21
Lienards Paper

• Electric and Magnetic Field produced by an
  electric charge concentrated at a point and
  travelling on an arbitrary path

Prophetically published in the french journal
The Electric Light
22
A Brief History of SR - up to WW II

• 1908 G.A. Schott (Aberystwyth) confirmed results
  and also derived angular distribution,
  polarisation characteristics spectrum
• 1940 First Betatron operated with 2.3 MeV
  electrons
• Isaak Pomeranchuk in Russia looks again at
  radiation loss from high energy electrons start
  of significant Soviet studies

23
A Brief History of SR - The War Years

• 1941 Betatron with 20 MeV electrons operated
• 1945 Betatron with 100 MeV electrons operated

John Blewitt realised that EM should be emitted
but he looked at long wavelengths unaware of
the impact of relativity He was unlucky since
betatron vacuum chamber was opaque, otherwise he
would have seen the light He indirectly
observed the electrons lose energy (orbit
shrinking) This is why we dont have Betatron
Radiation
24
A Brief History of SR - Observation

• 1947 70MeV electron synchrotron at General
  Electric Research Lab in Schenectady, NY.
• Bluish-white light observed through glass vacuum
  chamber (24th April) thought to be sparking
  initially but measurement of linear polarisation
  confirmed SR

25
A Brief History of SR Schwingers contribution

• 1949 Julian Schwinger publishes seminal paper,
  repeating much of Schotts early work, but in a
  more useful formalism numerical calculations
  readily made
• Aside .
• 1965 Schwinger shares Nobel prize with Tomonaga
  Feynman but not for SR! (Quantum
  Electrodynamics)

26
A Brief History of SR The First Undulator

• 1951 Motz proposes undulator as source of
  quasi-monochromatic SR
• 1952 Motz makes experimental demonstration of
  undulator

27
A Brief History of SR First Use

• 1st Generation SR sources
• Electron synchrotrons start to be built for high
  energy physics use (rapidly cycling accelerators
  not Storage Rings!)
• Interest from other physicists in using the
  waste SR
• First users are parasitic

The first beamline on NINA at Daresbury
constructed in 1966/67 by Manchester
University NINA was a 6GeV electron synchrotron
devoted to the study of particle physics
28
A Brief History of SR Dedicated Facilities

• 2nd Generation SR sources
• Purpose built accelerators start to be built
  late 70s
• First users 1980 (at SRS, Daresbury)
• Based primarily upon bending magnet radiation

The VUV ring at Brookhaven in 1980 before the
beamlines are fitted Not much room for undulators!
29
A Brief History of SR Enhanced Facilities

• 3rd Generation SR sources
• Primary light source is the undulator
• First built in the late 80s/early 90s
• First users 1994

ESRF, Grenoble
Diamond, UK
30
A Brief History of SR The Next Generation

• 4th Generation SR sources
• Primary light source is the single pass Free
  Electron Laser
• First built 2000
• First users 2006

FLASH FEL facility at DESY
31
Synchrotron Radiation Good or Bad?

• For high energy accelerators it is bad
• Large amounts of energy are radiated
• So the energy given to the electrons to
  accelerate them is wasted
• Have to keep renewing this lost energy to
  maintain the electrons energy
• Where does the energy go?
• Must hit a cooled surface!
• Example
• SRS radiates about 50 kW (2GeV, 200mA)
• LEP radiated about 18 MW (100GeV, 6mA)
• LEP250 would have radiated 700 MW (250GeV, 6mA)
• This is why ILC is a linear accelerator!

32
Synchrotron Radiation Good or Bad?

• For SR users it is good!
• It is the most intense source of light available
  to scientists over a wide spectral range
• Especially important in the soft X-ray X-ray
  regions
• Smooth spectrum (BM) or Tuneable (Undulators)
• Selectable polarisation
• Experiments are carried out in all scientific
  areas
• Materials science
• Chemistry
• Biology
• Medicine
• ..

33
Materials Science
Fernando Sannibale, USPAS, Jan 2006
34
Semiconductor Development
Fernando Sannibale, USPAS, Jan 2006
35
Medicine
Fernando Sannibale, USPAS, Jan 2006
36
Protein Crystallography
Fernando Sannibale, USPAS, Jan 2006
37
Why X-rays are important
Fernando Sannibale, USPAS, Jan 2006
38
For more information

• 5th March 2 lectures
• Graham Bushnell-Wye
• Synchrotron Radiation Applications Academic and
  Commercial

39
SR Derivation

• Consider the geometry of the system for a
  relativistic electron on an arbitrary trajectory
• Relate the time of emission to the time of
  observation
• We know the accelerating charge emits EM
  radiation
• Use Maxwells equations to derive the electric
  and magnetic fields of the EM radiation as a
  function of time that the observer sees
• Convert this from time to frequency to predict
  the spectrum of EM radiation that is seen
• Apply these general results to specific cases
  bending magnets, wigglers and undulators

40
Emission Observation of SR

• Consider an electron on an arbitrary path moving
  with relativistic velocity (ie close to the speed
  of light, c)
• Electron emits photon in direction of observer
• When the photon arrives at the observer the
  electron has already moved position
• To consider the light arriving at the observer we
  must appreciate that it was emitted some time ago
• The electron emits at time t (retarded or
  emission time)
• The photon (travelling at speed c) arrives at the
  observer at time t (observation time)

41
Emission Observation of SR
42
Emission Observation of SR

• Use standard result
• To get
• And so
• Where n is the unit vector pointing along R(t),
  and so
• where

43
Electric Field at the Observer

• First introduce auxiliary variables into
  Maxwells eqns vector and scalar potentials
• Since is a standard vector analysis result
• is equivalent to
• We will call A(t) the vector potential
• Note that A could be any vector, not defined yet

44
Electric Field at the Observer

• Similarly
• can be written as
• Using another vector analysis result
• We can introduce the scalar potential such that
• And so again could be any scalar

45
Lienard-Wiechert potentials

• Apply a third condition to determine a consistent
  set of potentials the Lorentz condition
• The solution for the potentials are called the
  Lienard-Wiechert potentials

t implies that the variables are calculated at
the emission time
46
Electric Field at the Observer

• Now need to derive the electric field at the
  observer
• Long and tedious maths problem.
• When R is large we can ignore the first term

47
The Far Field Case

• Ignoring the first term
• Often see this in text books but remember this
  only holds in the far field. How far away is the
  far field?
• Most SR calculations can (and do!) ignore the
  near field.
• NOTEThe next steps assume this simplified
  far-field case

48
Fourier Transform of the Electric Field

• Now we know the E-field as a function of time,
  what does it mean? How do we interpret it?
• Convert from time to frequency with FT so we can
  understand the spectrum of EM radiation it
  contains

Transform integration variable to t using
earlier results
49
Fourier Transform of the Electric Field

• Additional assumption that R does not vary with
  time (dn/dt0)
• ok for far field
• Far field case of electron moving on arbitrary
  path

50
SR from a Bending Magnet

• Bending magnet is a uniform dipole
• Electron moves on purely circular path
• Angular velocity
• r is bending radius

51
SR from a Bending Magnet

• Position, velocity and acceleration can be
  written down as (x,y,s)
• The unit vector (assuming R constant over a short
  time) is
• Substitute these expressions into

52
SR from a Bending Magnet

• Make use of approximations since we assume
  is large and so is small.
• So and there is no component of the E field in
  the longitudinal direction

53
SR from a Bending Magnet

• In the x direction
• Change of variable for convenience
• and introduce critical frequency

odd function so integral 0
54
SR from a Bending Magnet
Similarly for the vertical plane
We can express these eqns using two well known
mathematical functions, Airy or Bessel functions.
Example
Usually Bessel functions are used because they
were easily found in standard mathematical tables
made quantitative results possible
55
Electric Field Expressed with Bessel Functions

• are known as modified Bessel functions and they
  are available directly in modern numerical codes

56
E Fields in x and y
is effectively the vertical angle of the radiation
On axis ( ) there is no E-field
At larger angles low frequencies (longer
wavelengths) become more dominant, short
wavelengths are no longer observed
Only Ex is observed the light is polarised
completely in the horizontal plane
57
Angular Power Distribution

• The Poynting vector gives the energy flow per
  unit area per unit time
• Substituting in the value for B, (derived in
  the same way as E) we have
• Define an area at the observer in terms of the
  solid angle
• then the energy radiated by the electron through
  this area
• in the time is

58
Angular Power Distribution

• The energy observed per unit solid angle will be
• Using Fourier Transform again to swap into
  frequency space we get
• The spectral angular distribution is
• (per revolution per electron)
• Where spectral means into a particular
  frequency band (ie not all frequencies)

59
Angular Power Distribution

• Spectral power angular density
• (taking into account number of
• revolutions per second)
• Using our results from earlier for Ex and Ey
• Integrate over all angles to get spectral power

the instantaneous power radiated by one electron
60
Critical Frequency

• If we integrate from 0 to then we find that it
  contains
• half the total power emitted
• In other words, splits the power spectrum for a
  bending magnet into two equal halves
• It is a useful parameter, and is frequently used
  to compare bending magnet sources
• Expressed as a wavelength or a photon energy

61
Examples
62
Angular Power Distribution

• Instead of integrating over angles to get
  spectral power, integrate over frequency to get
  angular power distribution

Horizontal Component
Vertical Component
63
Angular Power Distribution

• Maximum power is emitted on axis
• Power is symmetrical with vertical angle
• No vertically polarised power on axis (as seen
  earlier)

64
Number of Photons

• Convert from power to number of photons
• Number of photons emitted per second with energy
  
• is then the power emitted at that energy is
• So, number of photons/sec/solid angle by one
  electron into a relative bandwidth of is

65
How many electrons are there?

• Beam current is amount of charge passing a point
  in a unit time
• In a circular machine the same electron passes
  many times per second
• So the number of photons/sec/solid angle into a
  bandwidth by a beam current is

66
Spectral Angular Flux Density

• More correctly (following standard optics) -
  Spectral Intensity
• in photons/s/mrad2/0.1 bandwidth
• E is in GeV, Ib is in A
• On-axis only,

67
Example
3GeV, 300mA, 1.4T Dipole
68
Photon Flux
Similar approach gives number of photons per
electron per second Number of photons/sec into
all angles for a beam current Ib into a relative
bandwidth This is the spectral photon flux or
the vertically integrated spectral flux. in
photons/s/mrad horizontally/0.1 bandwidth
69
The Universal Curve

• Plot of gives universal curve
• All bending magnet radiation has the same
  characteristics
• Once the critical energy is calculated it is easy
  to find (scale off) the photon flux

log-log scale
70
Examples
log-linear scale, 200mA assumed for all sources
71
Vertical opening angle

• Different photon energies have quite different
  vertical angular distributions
• Useful to be able to estimate a typical angle
• This typical angle is called the vertical
  opening angle
• It is not exact, just a useful approximation
• Assume radiation has a Gaussian distribution
  not always a good approximation
• Symmetric about

72
Vertical opening angle

• Integrate over all vertical angles
• remember, integral of a Gaussian is equal to 1.
• Note

Note log-log scale
73
Examples

• Clearly, only a good approximation at high
  frequencies, short wavelengths

3GeV, 300mA, 1.4T Dipole
74
Bending Magnet Brightness

• All emitted photons have a position and angle in
  phase space
• Phase space evolves as photons travel but area
  stays constant (Liouvilles theorem)
• Emittance of electron beam governed by same
  theorem
• Brightness is the phase space density of the flux
  takes account of 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)

Area stays constant
75
Brightness

• To calculate the brightness we need the phase
  space areas
• Need to include the photon and electron
  contributions
• Horizontal angle considered separately since
  light emitted smoothly over full 2p
• Effective vertical angle
• Add in quadrature as both Gaussian distributions
• Horizontal and vertical effective sizes are
  similarly

76
Brightness

• Photon beam size is found by assuming source is
  fundamental mode of an optical resonator
  (Gaussian laser mode)
• Bending magnet brightness is then
• Each term contributes because rectangular
  function of equal area to Gaussian has width
• In general and then
• The units are photons/s/solid area/solid
  angle/spectral bandwidth

77
Power Emitted

• Virtually all SR sources have melted vacuum
  chambers or other components due to the SR
  hitting an uncooled surface
• Average power is high but power density is very
  high power concentrated in tight beam
• Even SRS has auto-detect and dump system to
  protect itself
• Instantaneous power emitted by a single electron
  is
• In a ring of circumference, C, with fixed bending
  radius, r0, the energy radiated per electron per
  turn is

78
Power

• Integral is and so
• For an electron beam,
• Or
• where power is in kW and E is in GeV.
• Another useful number is the power per horizontal
  angle (in W/mrad)

79
Power Density

• Use result found earlier
• to get on-axis power density
• Or in W/mrad2

80
Examples
81
Bending Magnet Spectrum

• In a storage ring of fixed energy, the spectrum
  can be shifted sideways along the photon energy
  axis if a different critical energy can be
  generated.
• Need to change r (B Field)
• Used especially to shift the rapidly falling edge
• Special insertion devices that do this are called
  wavelength shifters
• Alternative is to replace individual bending
  magnets (superbends) not popular but has been
  done

82
Wavelength Shifters

• Shift the critical energy by locally changing the
  bending magnet field
• The area under the curve is unchanged but the
  spectrum is shifted

SRS Example 1.2T BM 6T WS. 2GeV, 200mA.
83
Wavelength Shifters

• How can you put a high magnetic field into a
  ring?
• Dont want to disrupt the symmetry of the ring by
  replacing one of the dipoles
• Why not replace all the dipoles?
• Popular solution is to use 3 magnets to create a
  chicane type trajectory on the electron beam in a
  straight section
• Central magnet is the high field BM source

84
Electron trajectory in a Wavelength Shifter
Electron enters on axis and exits on axis Peak of
bump occurs at peak magnetic field when angle
is zero SR emitted here will travel parallel to
axis
85
Example magnetic field
Critical energy varies so exact characteristics
observed depend upon which part of the trajectory
the observer is looking at
86
Example of Wavelength Shifter
SRS 6T (central pole) wavelength shifter
What can you deduce about this magnet?
87
Example of Wavelength Shifter

• Spring-8 10T wiggler

88
Extension to Multipole Wigglers

• One wavelength shifter will give enhanced flux at
  high photon energies
• SR is emitted parallel to the axis at the peak of
  the main pole
• Imagine many WS installed next to each other in
  the same straight

89
Multiple Wavelength Shifters

• Each WS would be an independent source of SR
  all emitting in the forward direction.
• The observer on-axis would see SR from all 3
  Source points
• Observer will see 3 times more flux
• This is the basic concept for a multipole wiggler
  
• Three separate WS is not the most efficient use
  of the space, a better way of packing more high
  field emitters into a straight is

B field usually sinusoidal
90
Multipole Wigglers electron trajectory

• Electrons travelling in s direction
• For small angular deflections
• The equations of motion for the electron are
• Assume a MPW which only deflects in the
  horizontal plane
• Only has vertical fields on axis

91
Angular Deflection

• B field is sinusoidal with period
• Integrate to find which is the horizontal angular
  deflection from the s axis
• Therefore, peak angular deflection is
• Define the deflection parameter

92
Trajectory

• One more integration gives
• Peak angular deflection is
• Remember that SR is emitted with typical angle of
• So if the electron trajectory will overlap with
  the emitted cone of SR
• If there will be little overlap and the source
  points are effectively independent this is the
  case for a MPW
• The case of is an undulator
• The boundary between an undulator and a MPW is
  not clear!

93
Example MPW from SRS
94
MPW Flux

• Can be considered a series of dipoles, one after
  the other
• Two source points per period
• Flux is simply product of number of source points
  and dipole flux for that critical energy
• MPW has two clear advantages
• Critical energy can be set to suit the science
  need
• Flux is enhanced by twice number of periods

300mA, 3 GeV beam 1.4T dipole 6T WS 1.6T, MPW
with 45 periods
95
MPW Power

• Same starting point as before
• Instantaneous power emitted by single electron
• Integrate this over MPW of length L
• Convert radius to B instead, using

96
MPW Power

• Energy loss per pass (in eV) with beam energy in
  GeV
• Total power emitted by a beam of electrons
  passing through any magnet system is
• This is a general result can get bending magnet
  result from here.
• For a sinusoidal magnetic field with peak value
  integral is and so total power emitted is

97
Power Density

• Instantaneous power radiated per solid angle
  integrated over length of device
• More complicated analytically

98
Power Density
G(K)1 for K gt 1
Power contained in K/g horizontally for large
K Vertically, power contained in 1/g
99
On-Axis power density

• Peak is on-axis since
• In W/mrad2

Undulators MPWs
100
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)

101
The Undulator Equation

• Using
• And then using
• This is a constant with an oscillating cosine
  term. The relative average velocity in the
  forward direction is simply

102
Condition for Interference

• For interference between wavefronts emitted by
  the same electron the electron must slip back by
  a whole number of wavelengths over one period

Time for electron to travel one period is In this
time the first wavefront will travel the distance
Speed distance/time
103
Interference Condition

• The separation between the wavefronts is
• And this must equal a whole number of wavelengths
  for constructive interference
• Using
• We have
• Using and the small angle approximation .....
  .

104
Interference Condition

• 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.
• mm periods translate to nm wavelengths because of
  the huge term

105
Undulator equation implications

• Wavelength primarily depends on period and 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.
• Q. What happens to l as B increases?

106
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).
• Q. Why does a larger B give a longer wavelength?
  What is actually happening?
• Wavelength changes with q2, so always gets longer
  as move away from on-axis case.
• The beamline aperture choice is important because
  it alters the radiation characteristics reaching
  the observer.

107
Example

• 3GeV electron passing through a 50mm period
  undulator with K 3. First harmonic (n 1),
  on-axis is 4 nm.
• Note the wavelength can be varied quite
  significantly but this graph does not say now
  many photons you will observe!

108
Harmonic bandwidth

• Over the full device there are 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 centre1, etc)
• Range over which there is some emission
• Bandwidth

109
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

110
Diffraction Gratings

• Very similar results for angular width and
  bandwidth apply to diffraction gratings.
• Q. Why might this be so?
• eg Jenkins and White, Fundamentals of Optics

111
Odd and Even Harmonics

• There is an important difference in undulators
  between odd (n 1, 3, 5, ) and even (n 2, 4,
  6, ) harmonics
• On axis, only odd harmonics are observed
• Away from the axis, even harmonics are observed
  but their characteristics are different (poorer
  usually)
• Later we will derive the full undulator
  expressions which demonstrate that the even
  harmonics are absent on axis
• For now we will explore in a simple minded way
  why that might be

112
Case 1 K ltlt 1

• SR emitted in cone of
• Angular excursion of electron
• So observer sees all the emitted radiation
• Electric field experienced is a continuous
  sinusoidal one
• Fourier analysis of this shows it will be a
  single frequency, so a single fundamental
  harmonic (n 1)

113
Case2 K gtgt 1

• Observer only experiences electric field when
  electron emission cone flashes past
• On axis, electric field peaks are equally spaced
• Fourier Transform of evenly spaced peaks only
  contains odd harmonics
• Sharpness of spikes increases as K increases so
  radiation spectrum contains higher frequencies
  (higher harmonics)

Observation Range
114
Case 3 K gtgt 1 off axis

• Still only see flashes of electric field
• No longer evenly spaced
• Fourier Transform has to contain even harmonics

Observation Range
115
When does an undulator become a wiggler?

• We have just discussed K gtgt 1 but still looked at
  it in terms of harmonics (interference effects)
• Earlier we said that K gtgt 1 was a multipole
  wiggler with a smooth bending magnet spectrum
• In fact both are true !
• As K increases the number of harmonics increases
• At high frequencies the spectrum smoothes out and
  takes on the shape 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

116
Undulator vs Wiggler

• The difference depends upon which bit of the
  spectrum you use!
• Example shows undulator calculation for K 15.
  Calculation truncated at high energies as too
  slow!

Equivalent MPW spectrum
117
Undulator Angular Flux Distribution

• Return to earlier (general) result
• Include our expression for electric field
  generated by electron on arbitrary path
• We know it is periodic so we can split it up as

118
Undulator Angular Flux Distribution

• Separate out the phase terms
• This last term is again met in diffraction
  gratings (grating function)
• Example for N 5

119
Lineshape function

• This is the shape around one of the maxima
• Look at frequencies near to a harmonic
• Normalise by dividing by N2

N 20
For large N, it is independent of N FWHM
1 So as found earlier
120
Angular Flux Density continued

• Now we have

From the geometry we have Earlier we found
values for Can now evaluate cross products and
dot product
121
Angular Flux Density continued

• After quite a bit of algebra
• (J is a Bessel function)
• Integral is over a period so in general will
  equal zero
• (eikWt cos kWt i sin kWt)

122
Vertical Polarisation

• Integral will be zero unless k 0, ie
• In which case

123
Horizontal Polarisation

• Again have condition for qcosf term
• Have extra condition that has a finite integral
  for the bx term
• since get cos2 integral
• The infinite sums look bad but in fact only need
  around first 10 terms about zero for good
  accuracy

124
K 1 examples
n 1
n 3
There are n peaks in the horizontal plane The
even harmonics have zero intensity on
axis Remember that the wavelength changes with
angle so these plots are not at a fixed wavelength
n 2
n 4
125
On axis Flux Density

• q 0
• Terms become easier
• Vertical contribution 0
• All of the SR emitted on-axis is horizontally
  polarised
• Also
• The Bessel infinite sum also easy since is zero
  except when
• Consequently and since p is an integer, n
  must be odd
• Reinforcing the point that there are no even
  harmonics on axis

126
On axis Flux Density

• The spectral angular energy density is
• Using this can be rewritten as

127
Fn(K)

• As K increases we can see that the impact of the
  higher harmonics grows

128
Angular Flux Density

• Energy density emitted per electron per pass is
  converted to power density by multiplying by the
  number of electrons per second
• Power is converted to number of photons by
  dividing by the energy of each photon
• In practical units of photons/sec/mrad2/0.1
  bandwidth

129
Example

• Undulator with 50mm period, 100 periods
• 3GeV, 300mA electron beam
• Angular flux density 8 x 1017 photons/sec/mrad2/0.
  1 bw
• Earlier for a bending magnet with the same
  electron beam we had a value of 5 x 1013
  photons/sec/mrad2/0.1 bw
• Factor of 10,000 from N2 term

130
Flux on-axis

• Earlier we argued that light of the same
  wavelength was contained in a narrow angular
  width
• Assuming that the SR is emitted in a Gaussian
  distribution with standard deviation then we
  can approximate
• Remember that
• Integrating over all angles gives

131
Flux in the central cone

• So the flux contained in the central angular cone
  is
• where
• In photons/sec/0.1 bandwidth the flux in the
  central cone is
• Example undulator has flux of 4 x 1015 conpared
  with bending magnet of 1013
• Difference is dominated by number of periods, N
  (100 here)

132
Qn(K)

• As K increases the higher harmonics play a more
  significant part but the 1st harmonic always has
  the highest flux in the central cone

133
Undulator Tuning Curve

• Graph shows flux envelope for example undulator.
• K is varied to change photon energy.
• Not all of this flux is available at the same
  time!

134
Peak Flux

• In fact the peak flux does not 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.
• Exact harmonic actually has half the total flux
  compared with the peak value

135

• Exact harmonic can only receive contributions
  from higher angles
• Detuned, longer wavelength, has hollow cone and
  can receive flux from higher and lower angles
• Cone shape gives greater flux but lower
  brightness
• Exact harmonic has narrower angular width

136
Undulator Brightness

• We saw earlier for the peak flux condition
• Example undulator has source size and divergence
  of 11mm and 28mrad
• Electron beam can be described by gaussian shape
  also (much more accurately!) and so effective
  source size and divergence is given by

137
Example Brightness

• Undulator brightness is the flux divided by the
  phase space volume given by these effective
  values
• Brightness example with

138
Brightness tuning curve
139
Warning!

• The definition of the photon source size and
  divergence is somewhat arbitrary
• Many alternative expressions are used for
  instance we could have used
• Actual effect on absolute brightness levels is
  relatively small
• But when comparing different sources it is
  important to know which expressions have been
  assumed

140
Diffraction limited sources

• Light sources strive to reduce the electron beam
  size and divergences to maximise the brightness
• When then there is nothing more to be
  gained
• The source is said to be diffraction limited

Example phase space volume of light source Same
electron parameters as before Above about 100nm,
electron beam has little impact on brightness
141
Undulator Output Including Electron Beam
Dimensions
Electron beam smears out the flux density Total
flux unchanged
142
Effect of electron beam on spectrum

• Including electron beam reduces wavelength
  observed
• Higher harmonic effects more dramatic since more
  sensitive

n 1
n 9
143
Effect of electron beam on spectrum

• Same view of on-axis flux density but over a
  wider wavelength range
• Odd harmonics stand out but even harmonics also
  present because of electron beam divergence

5
3
n 1
6
4
2
144
Flux through an aperture
n 1

• How much total flux is observed depends upon
  beamline aperture
• As aperture increases, flux increases and shifts
  to lower energy detune effect
• Can see that higher harmonic has narrower
  divergence
• Plot has zero electron size and divergence, only
  single harmonic contribution included

n 9
145
Flux through an aperture

• As before but over a wider wavelength range and
  all harmonics that can contribute are included
  hence greater flux values
• Many harmonics will contribute at a particular
  energy provided large enough angles are observed

146
Polarised Light

• A key feature of SR is its polarised nature
• By manipulating the magnetic fields correctly any
  polarisation can be generated linear to circular
• This is a big advantage over other light sources
• Polarisation can be described by several
  different formalisms
• Most SR literature uses the Stokes parameters

Sir George Stokes 1819 - 1903
147
Stokes Parameters

• Polarisation is given by the relationship between
  two orthogonal components of the E field
• There are 3 independent parameters, the
  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 setup a formalism based upon
  observables

148
Stokes Parameters

• The intensity can be measured for different
  polarisation directions
• Linear erect
• Linear skew
• Circular
• The Stokes Parameters are

149
Stokes Parameters

• In terms of the electric field components
• Note the use of conjugates because electric
  fields are complex in general
• Reminder
• If A A0cos wt i A0sin wt
• then A A0cos wt - i A0sin wt
• AA A02
• If A is real then A A, if A is imaginary then
  A -A

150
Polarisation Rates

• The polarisation rates, dimensionless between -1
  and 1, are
• Total polarisation rate is
• The natural or unpolarised rate is

151
Bending Magnet Polarisation

• For bending magnets we already know what the
  electric fields are
• And so
• Since Ex is real and Ey is imaginary, implying a
  p/2 phase shift between them, we have and so
  S2 0

152
Bending Magnet Polarisation

• For circular polarisation
• Can change sign of P3 by changing the sign of
• So both left and right circular are available by
  changing observation angle from above axis to
  below axis

153
Bending Magnet Polarisation

• Circular polarisation is zero on axis and
  increases to 1 at large angles
• But, remember that flux falls to zero at large
  angles!
• So have intensity/polarisation trade-off if want
  to use bending magnet circular polarisation

154
Qualitative explanation

• Consider trajectory seen by observer
• In the horizontal plane he sees the electron
  travel along a line (ie linear polarisation)
• Above the axis the electron has a curved
  trajectory (ie contains some circular
  polarisation)
• Below the axis trajectory again curved but of
  opposite handedness (ie some circular of opposite
  handedness)
• As angle of observation increases the curvature
  observed increases and so the circular
  polarisation rate increases

155
Wiggler polarisation

• Horizontal linear polarisation on axis
• No circular polarisation away from the axis
• Q. Why is there no circular polarisation in a
  wiggler off-axis like there is in a bending
  magnet?
• Think about trajectory observer sees.
• Q. So if there is no circular polarisation off
  axis, but linear rate is decreasing, what is
  there instead?

156
Polarisation from a Planar Undulator (Bx 0)

• Horizontal and vertical electric fields are in
  phase so polarisation is always linear
• Orientation of polarisation is given by the
  ratio of the two fields

157
Odd harmonics have horizontal polarisation about
central on axis region
158
Special devices for generating different
polarisation rates

• Asymmetric Wiggler
• Alter the magnetic field so not symmetric
  (non-sinusoidal) then cancellation effect not
  equal and get net circular polarisation

Idealistic field compared with conventional
wiggler
159
Asymmetric Wiggler

• Field vs horizontal angle highlights effect of
  asymmetry
• At zero angle see 2T and -0.7T poles
  incomplete cancellation
• Polarisation rate changes with photon energy
  because at low energy low field pole dominates
  and at high energy the high field pole dominates
• Polarisation rate changes with both horizontal
  and vertical angle

0.2 mrad vertical 0.0 mrad horizontal angle
160
Elliptical Wiggler

• Introduce weak horizontal field with same period
  but p/2 out of phase with vertical field
• Generates elliptical trajectory when viewed head
  on
• This flips the apparent observation angle /-
  from one pole to the next so cancellation effect
  disappears
• All poles work together so more useful flux than
  asymmetric wiggler
• Designed to be viewed from on-axis
• Apparent vertical observation angle is

161
Helical Undulators

• Similar to elliptical wiggler
• Include horizontal field so electron takes
  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?

162
Undulator Equation

• Same derivation as before
• Find electron velocities in both planes from B
  fields
• Total energy fixed so total velocity fixed
• Find longitudinal velocity
• Find average longitudinal velocity
• Insert this into the interference condition to
  get
• Very similar to standard result
• Wavelength independent of phase

163
Polarisation Rates

• For the helical undulator
• Only 3 variables are needed to specify
  polarisation and so helical undulator can
  generate any polarisation state
• Pure circular polarisation (P3 1) when Bx0
  By0 and
• Q. What will the trajectory look like?

164
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

165
Flux in the central cone

• (photons/s/0.1 bandwidth)
• where
• In this pure helical mode get circular
  polarisation only (P3 1)
• Also Y 0, so only non-zero Bessel term in
• (when on-axis) is when n 1
• Q. What are the implications of this?

166
Crossed Undulators

• Two planar undulators are placed in the same
  straight orthogonal to each other
• Phase shifter is placed inbetween (electron
  delay)
• For the case with two identical undulators with N
  periods and phase delay the rates are

167
Crossed Undulators

• At zero detune rates only depend on phase delay
• Can change between linear at 45? to right
  circular to linear at 135? to left circular and
  back to 45? linear
• Away from the harmonic wavelength the
  polarisation state can change rapidly
• Crossed scheme relies upon interference effects
• How can light from first undulator interfere with
  second undulator when they are separated in time?
• Observer looks at single frequency after
  monochromator
• Monochromator stretches the pulses in time so
  that they overlap after the monochromator

168
Power from Helical Undulators

• Use same approach as before
• 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

169
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

170

• End of Part 1