Accelerator Physics:
Synchrotron radiation
Lecture 2
Henrik Kjeldsen – ISA
Synchrotron Radiation (SR)
• Acceleration of charged particles
– Emission of EM radiation
– In accelerators: Synchrotron radiation
• Our goals
– Effect on particle/accelerator
– Characterization and use
• Litterature
– Chap. 2 + 8 + notes
General Electric
synchrotron accelerator
built in 1946, the origin
of the discovery of
synchrotron radiation.
The arrow indicates the
evidence of arcing.
Emission of Synchrotron Radiation
• Goal
– Details (e.g.): Jackson – Classical Electrodynamics
– Here: Key physical elements
• Acceleration of charged particles: EM radiation
• Lamor: Non-relativistic, total power
• Angular distribution (Hertz dipole)
Relativistic particles
• Lorenz-invariant form
1
E
1
v
dt  d  dt ,  

, 
2
2

m0 c
c
1 
 dP

 d
• Result
2

1  dE 
 dp 
     2 

 d  c  d 

2
2
Linear acceleration
• Using dp/dt = dE/dx:
• Energy gain: dE/dx ≈ 15 MeV/m
– Ratio between energy lost and gain:
– h = 5 * 10-14 (for v ≈ c)
• Negligible
Circular accelerators
• Perpendicular acceleration:
– Energy constant...
– dp = pda → dp/dt = pw = pv/R
– E ≈ pc,  = E/m0c2
dv
v
dt
• In praxis: Only SR from electrons
Energy loss per turn
2R
E 4 [GeV]
E   Ps dt  Ps
 88.5
c
R[m]
• Max E in praxis: 100 GeV (for electrons)
Angular distribution I
• Similar to Hertz dipole in frame of electron
– Relativistic transformation
 pt   ES ' / c   ES ' / c 

  
 
 px   0   0 
P '     


p0 ' 
py
p0 '

  
 
 p   0   E ' / c 
 
S
 z 

py
p0 '
1
  tan  


p z p0 ' 
Spectrum of SR
• Spectrum: Harmonics of frev
• Characteristic/critical frequency
• Divide power in ½
ASTRID2
• Horizontal
emittance [nm]
Spectral Brightness
– ASTRID2:12.1
– ASTRID: 140
 R R '   4
Undulator, ASTRID2
1E+16
Ph/s*mm^2*mrad^2*0.1BW
• Diffraction limit:
1E+17
1E+15
Undulator
1E+14
2T 12 pol wiggler, ASTRID2
1E+13
Bend ASTRID2
1E+12
Bend, ASTRID1
1E+11
0.001
0.01
0.1
Photon Energy (keV)
1
10
Storage rings for SR
•
•
•
•
SR – unique broad spectrum!
0th generation: Paracitic use
1st generation: Dedicated rings for SR
2nd generation: Smaller beams
– ASTRID?
• 3rd generation: Insertion devices (straight sections), small beam
– ASTRID2
• 4th generation: FEL
Insertion devices
Wigglers and undulators
(Insertion devices)
•
•
•
•
•
•
The magnetic field configuration
Technical construction
Equation of motion
Wigglers vs. Undulators
Undulator radiation
The ASTRID undulator
Coordinate system
Magnetic field
• Potential:
• Solution:
• Peak field on axis:
Magnetic field on axis
Construction
a) Electromagnet; b) permanet magnets; c) hybrid magnets
Insertion devices
• Single period, strong field (2T / 6T)
– Wavelength shifters
• Several periods
– Multipole wigglers
– Undulators
• Requirement
– no steering of beam
Example (ASTRID2):
Proposed multi-pole wiggler (MPW)
•
•
•
•
•
B0 = 2.0 T
 = 11.6 cm
Number of periods = 6
K = 21.7
Critical energy = 447 eV
Summary – multi-pole wiggler
(MPW)
• Insertion device in straight section of
storage ring
• Shift SR spectrum towards higher
energies by larger magnetic fields
• Gain multiplied by number of periods
Equation of motion
 vx   0 
   
Set Bx = 0, vz = 0 F  ev  B  e 0    Bz 
v   B 
→ coupl. eq.
 s  s
Set s  0 (s  x ) and s  vs  c  constant
Undulator/wiggler parameter: K
• K – undulator/wiggler
parameter
– K < 1: Undulator
• w < 1/
– K > 1: Wiggler
• w > 1/
• Equation of motion: s(t)
Undulator radiation I
• Coherent superposition of radiation produced from each periode
• Electron motion in lab frame:
• Radiation in co-moving frame (c*):
• Radiation in lab:
Undulator radiation II
• If not K << 1: Harmonics of Ww
w,n
u  K 2
2 2
1 

  0 
2 
n  2 
2

Undulator radiation III
15
1.0x10
K = 2.3 (25 mm gap)
Integrated flux
2
2
2.0 mrad
2
2
1.0 mrad
2
2
0.5 mrad
2
2
0.25 mrad
14
Photon flux
8.0x10
14
6.0x10
14
4.0x10
14
2.0x10
0.0
0
50
100
Photon energy (eV)
150
200
Insertion devices: Summary
• Wiggler (K > 1,  > 1/)
– Broad broom of radiation
– Broad spectrum
– Stronger mag. field: Wavelength shifter (higher
energies!)
– Several periods: Intensity increase
• Undulator (K < 1,  < 1/)
– Narrow cone of radiation: Very high brightness
• Brightness ~ N2
– Peaked spectrum (adjustable)
• Harmonics if not K<<1
– Ideal source!
Use of SR
• Advantage: broad, intense spectrum!
• Examples of use:
– Photoionization/absorption
• e.g. hn + C+ → C++ + e-
– X-ray diffraction
– X-ray microscopy
– ...
Optical systems for SR I
• Purpose
– Select wavelength: E/DE ~ 1000 – 10000
– Focus: Spot size of 0.1∙0.1 mm2
Optical systems for SR II
• Photon energy: few eV’s to 10’s of keV
– Conventional optics cannot be used
• Always absorption
– UV, VUV, XUV (ASTRID/ASTRID2)
• Optical systems based on mirrors
– X-rays
• Crystal monochromators based on diffraction
Mirrors & Gratings
• Curved mirrors for
focusing
• Gratings for selection
of wavelength
• r and r’ – distances to
object and image
• Normally  ~ 80 – 90º
– Reflectivity!
Mirrors: Geometry of surface: Plane,
spherical, toriodal, ellipsoidal, hypobolic, ...
• Plane: No focusing (r’ = -r)
• Spherical: simplest, but not perfect...
1 1
2
 
– Tangential/meridian
r r ' Rt  cos( )
– Saggital
1 1 2  cos( )
• Toriodal: Rt ≠ Rs
 
• Parabola: Perfect focusing of parallel beam r r '
Rs
• Ellipse: Perfect focusing of point source
Focusing by mirrors: Example
Gratings
• kN = sin(a)+sin()
– NB:  < 0
– N < 2500 lines/mm
• Optimization
– Max eff. for k = (-)1
– Min eff. for k = 2, 3
• Typical max. eff. ≈ 0.2
Design of ‘beamlines’
• Analytically
– 1st order: Matrix formalism
– Higher orders: Taylor expansion
• Optical Path Function Theory (OPFT)
– Optical path is stationary
• Only one element
• Numerically
– Raytracing (Shadow)
Useful equations
•
Bending radius
•
Critical energy
•
Total power radiated by ring
•
Total power radiated by wiggler
•
Undulator/wiggler parameter
u  K 2
2 2


1



0 
2 
n  2 
2

•
Undulator radiation w,n
•
Grating equation
•
Focusing by curved mirror (targentical=meridian / saggital)
1 1
2
 
r r ' Rm  cos( )
  E  1240 nm  eV
1 1 2  cos( )
 
r r'
Rs