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

• Result

c

v

cm

Edtddt

,

1

1,

122

0

2

2

221

d

dE

cd

dp

d

dP

Linear acceleration

• Using dp/dt = dE/dx:

• Energy gain: dE/dx ≈ 15 MeV/m– Ratio between energy lost and gain:

– = 5 * 10-14 (for v ≈ c)• Negligible

Circular accelerators

• Perpendicular acceleration:– Energy constant...– dp = pd→ dp/dt = p = pv/R

– E ≈ pc, = E/m0c2

• In praxis: Only SR from electrons

vv

dt

d

Energy loss per turn

• Max E in praxis: 100 GeV (for electrons)

[m]

[GeV]5.88

2 4

R

E

c

RPdtPE ss

Angular distribution I

• Similar to Hertz dipole in frame of electron– Relativistic transformation

cE

p

cE

p

cE

p

p

p

p

P

S

SS

z

y

x

t

/'

'

0

/'

0

'

0

/'

'00

1

'

'tan

0

0 p

p

p

p

z

y

Spectrum of SR

• Spectrum: Harmonics of frev

• Characteristic/critical frequency

• Divide power in ½

Spectral Brightness

1E+11

1E+12

1E+13

1E+14

1E+15

1E+16

1E+17

0.001 0.01 0.1 1 10Photon Energy (keV)

Ph

/s*m

m^

2*m

rad

^2

*0.1

BW

Undulator, ASTRID2

Undulator

2T 12 pol wiggler, ASTRID2

Bend, ASTRID1

Bend ASTRID2

ASTRID2

• Horizontal emittance [nm]– ASTRID2:12.1– ASTRID: 140

• Diffraction limit:

4' RR

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

Constructiona) 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

Set Bx = 0, vz = 0

→ coupl. eq.

s

z

s

x

B

B

v

v

ee

0

0BvF

constant c and )( 0Set svsxss

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 w

2

02

2

2, 21

2

K

nu

nw

0 50 100 150 2000.0

2.0x1014

4.0x1014

6.0x1014

8.0x1014

1.0x1015

Ph

oto

n fl

ux

Photon energy (eV)

K = 2.3 (25 mm gap) Integrated flux

2.02 mrad2

1.02 mrad2

0.52 mrad2

0.252 mrad2

Undulator radiation III

-0.0010-0.0005

0.00000.0005

0.0010

0.00

0.05

0.10

0.15

0.20

0.25

-0.0006-0.0004

-0.00020.0000

0.00020.0004

0.0006

-0.0010-0.0005

0.00000.0005

0.0010

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

-0.0006-0.0004

-0.00020.0000

0.00020.0004

0.0006

-0.0010-0.0005

0.00000.0005

0.0010

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

-0.0006-0.0004

-0.00020.0000

0.00020.0004

0.0006

-0.0010-0.0005

0.00000.0005

0.0010

0.0

0.1

0.2

0.3

0.4

-0.0006-0.0004

-0.00020.0000

0.00020.0004

0.0006

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. h + 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!

sRrr

)cos(2

'

11

Mirrors: Geometry of surface: Plane, spherical, toriodal, ellipsoidal, hypobolic, ...

• Plane: No focusing (r’ = -r)• Spherical: simplest, but not perfect...

– Tangential/meridian– Saggital

• Toriodal: Rt ≠ Rs• Parabola: Perfect focusing of parallel beam• Ellipse: Perfect focusing of point source

)cos(

2

'

11

tRrr

Focusing by mirrors: Example

Gratings

• kN = sin()+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

• Undulator radiation

• Grating equation

• Focusing by curved mirror (targentical=meridian / saggital)

sRrr

)cos(2

'

11

)cos(

2

'

11

mRrr

2

02

2

2, 21

2

K

nu

nw eVnm 1240 E