Light Source II

Takashi TANAKA (RIKEN SPring-8 Center)

Cheiron2009: Light Source II

Characteristics of SR (2)• Electron Trajectory in the ID• Qualitative Description of Wiggler
Radiation• Qualitative Description of Undulator
Radiation

Cheiron2009: Light Source IITrajectory in the Undulator

Coordinate Systems

SR emitted by an electron moving at r = (x,y,z)
Observation of SR at R = (X,Y,Z)

If the far-field approximation (|r|<<Z) isapplicable, the radiation pattern depends
only on the observation angle θ = (θx,θy).
Cheiron2009: Light Source IITrajectory in the Undulator

Field Integrals

Equation of motion of an electron
moving in a magnetic field B: 1st and 2nd field integrals of the undulator
Cheiron2009: Light Source IITrajectory in the Undulator

Trajectory in an Ideal Undulator

magnetic field

velocity

position

Ｋ value, Deflection parametervvxvz

Ee=8GeV,K=1,λu=5cm：βxmax=64μrad,xmax=0.5μm
Cheiron2009: Light Source IITrajectory in the Undulator

Effects due to the Undulator Field
transverse
velocity
longitudinalvelocity

total velocity:average velocity

oscillating termUndulator field induces:• transverse(x) oscillation• longitudinal (z) oscillation• effective deceleration(Δβz=K2/4γ2)
Cheiron2009: Light Source IITrajectory in the Undulator

Electron Motion: Two Forms• Horizontal oscillation with a period of λu• Major contribution to radiation• Longitudinal oscillation with a period of λu/2
• Amplitude 1/γ times lower than βx.
• Minor contribution, but source of verticalpolarization observed vertically off-axis.
Cheiron2009: Light Source IITrajectory in the Undulator

General Form of Time Squeezing

Time squeezing takes place most significantly
when the direction of the electron motion
coincides with that of observation (β = θ ).
Cheiron2009: Light Source II

Characteristics of SR (2)• Electron Trajectory in the ID• Qualitative Description of Wiggler
Radiation• Qualitative Description of Undulator
Radiation

Cheiron2009: Light Source II

Radiation from Wigglers

Wiggler Radiation• Wiggler radiation (WR) is regarded asincoherent sum of SR at each position.
– Summation as photons in the framework ofgeometrical optics.

Flux:
Emittance:
Brilliance:
Cheiron2009: Light Source II

Radiation from Wigglers

Photon Distribution in Phase Spacex’,y’x’,y’x,yx’,y’x,yx’,y’x,yx’,y’x,yx,yphase ellipse ofdiffraction-limited light

Beam Waist Position• Larger N results in larger area ofphoton distribution in the phasespace, i.e., larger emittance.
• B does not linearly depend on N
Cheiron2009: Light Source IIRadiation from Wigglers

Polarization• No circular polarized radiation (CPR) is
observed unlike the BM radiation even off
axis, due to cancellation of CPR components.
right-hand CPR

left-hand CPR• EMPW is a special wiggler to utilize CPR byintroducing a vertical motion.
Cheiron2009: Light Source IIQualitative Descriptions of WR

Comparison with BM Radiation
BM (εc=29keV),

1710Wiggler (εc=38keV), N=37

Flux Density, Brilliance

Flux Density

Brilliance

16101510

obtained by Gaussian
approximation in thephase space

14101310310410510

Photon Energy (eV)610

310410510610

Photon Energy (eV)Cheiron2009: Light Source II

Characteristics of SR (2)• Electron Trajectory in the ID• Qualitative Description of WigglerRadiation• Qualitative Description of Undulator
Radiation

Cheiron2009: Light Source IIQualitative Descriptions of UR

Fundamental Wavelength
λuz
period of electron motion time squeezing= period of emitted lightθ

period of observed light

Fundamental Wavelength λ1

H. Kitamura et al.,J. Appl. Phys. 21 (1982) 1728

Cheiron2009: Light Source IIQualitative Descriptions of UR

UR with Infinite Periods• If the undulator length is infinite, the pulse
duration is infinitely long, and thus the radiationis completely monochromatic with line spectrum.

• In practice, the undulator length is finite, so theline spectrum is broadened.
infiniteperiods

finiteperiods
FFω1ω

ω1ω
Cheiron2009: Light Source IIQualitative Descriptions of UR

Effects due to Finite Periods
λ1/cElectric Field at Observert

T=Nλ1/c

Fourier Transform

Square of “sinc” function dominates the UR
Cheiron2009: Light Source IIQualitative Descriptions of UR

Brief Note on UR Formulae• In the previous derivations of UR spectral
function, no knowledge on electrodynamics is required.
• In practice, E0 is a complicated functionof θ and K, and needs to be calculated by
Fourier transforming the electric field
derived from the Lienard-Wiecherd
potential.
• However, the simple derivation gives us aclear understanding on UR properties.
Cheiron2009: Light Source IIQualitative Descriptions of UR

Energy and Angular Profile of UR

Energy Profile at θ = 0

Angular Profile at ω=αω1(0)Cheiron2009: Light Source IIQualitative Descriptions of UR

Flux Density (arb.unit)

Energy Profile: Example

1.0

N=5
N=10
N=50

0.5

0.0
-0.4

-0.2
0.0
0.2
Relative Energy ε0.4
Cheiron2009: Light Source IIQualitative Descriptions of UR

Angular Profile: Example

0.5

Θx0.0

0.5
1.0 -1.0-0.5

0.0

0.5

1.0

Θy

ω=0.9ω1(0)ω=ω1(0)ω=1.05ω1(0)lower energy

fundamentalenergy

higher energy
Cheiron2009: Light Source IIQualitative Descriptions of UR

Angular Divergence and Beam Size
Angular Profile at ω=ω1(0)approximation
Gaussian Profile with σr’

Angular Divergenceof UR (L=Nλu)Diffraction Limit (UR isSpatially Coherent)

Beam Size of UR
Cheiron2009: Light Source IIQualitative Descriptions of UR

Higher Harmonics• In addition to the fundamental radiation
at ω1, higher-energy radiation at nω1,
called higher harmonics, is observed.
The integer n is referred to as aharmonic number.
• This is a consequence of the fact thatthe time-squeezing factor depends on
the longitudinal electron position andthus the electric field in the time
domain is distorted.
Cheiron2009: Light Source IIQualitative Descriptions of UR

Interpretation of Higher Harmonics

1.0
0.8
0.6
0.4
K=0.1
K=0.5
K=1
K=30.2
0.0
0.0

0.5

1.0

1.5

2.0

2.5

z/λu

Large K value brings a modulationin the time squeezing factor

Electric Field

Time Squeezing (2γ2dτ/dt)-1

on-axis observation:n=(0,0,1)

K=0.1(x30)
K=0.5(x6)
K=1(x3)
K=30.00.5

1.01.5cτ / λ1

2.0

2.5

Distortions of the electric field
takes place due to the nonuniform
time squeezing. Due to symmetry,even harmonics do not appear.
Cheiron2009: Light Source IIQualitative Descriptions of UR

Examples of Higher Harmonics
18x10

Flux Density

10

K=0.1

K=0.5

K=1

K=355th
3rd

1st0050

100 050

100 050

100 050

100Photon Energy (eV)Cheiron2009: Light Source IIQualitative Descriptions of UR

Optical Properties of Higher Harmonics
For the n-th harmonic radiation,

band width
angular divergence

beam size
Cheiron2009: Light Source IIQualitative Descriptions of UR

1.0
0.8

Electric Field

2Time Squeezing (2γ dτ/dt)-1

Even Harmonics Horizontally Off Axis

0.6
0.4
0.2

K=1,γθy=0,γθx=0,0.1,0.3,0.5

0.0
0.00

0.25

0.50

0.75

1.00

0.00

0.25

0.50

0.75

1.00

z/λu

τ / λ1The position for the maximum
time squeezing is shifted due
to finite θx.

The symmetry of the electricfield is broken, resulting inappearance of even harmonics.
Cheiron2009: Light Source IIQualitative Descriptions of UR

Examples of Even Harmonics
18x108Flux Density6γθx=0.1γθx=0

K=1,γθy=0γθx=0.3

4γθx=0.5

20
10

15
20
Photon Energy (keV)

25
Cheiron2009: Light Source IIQualitative Descriptions of UR

Even Harmonics Vertically Off Axis
• Vertically off-axis observation does notbreak the symmetry of the E-field.
• Nevertheless, even harmonics areobserved due to the longitudinaloscillation in electron motion with aperiod of λu/2.
• Such even harmonics are verticallypolarized, reflecting the electron motion
projected onto the plane of observation.
Cheiron2009: Light Source IIQualitative Descriptions of UR

Mechanism of Vertical Polarization
verticalpolarizationyz

θy: period of oscillation=λu/2

no radiation
observed

Note: amplitude of oscillationis ~γ-1 smaller than that of βx
Cheiron2009: Light Source IIQualitative Descriptions of UR

Example of Vertical Polarization
K=1,γθx=018x10101.0

Flux Density8γθy=0.160.5γθy=0.3

40.0γθy=0.5-0.52

Stokes Parameter s1/s0γθy=0-1.00510152025
Photon Energy (keV)

30
Cheiron2009: Light Source IIQualitative Descriptions of URPolarization• As in the wigglers, no circular polarizedradiation (CPR) is observed due tocancellation of CPR components.
• The direction of the linear polarization
observed off axis is tilted due to thelongitudinal oscillation of electron
motion.

Cheiron2009: Light Source IIQualitative Descriptions of UR

Polarization:Examples
Examples of the direction of linear polarization forvarious observation angles.
H. Kitamura, JJAP 19 (1980) L185

Cheiron2009: Light Source IIPractical Knowledge on SR

Cheiron2009: Light Source IIPractical Knowledge on SR

Effects due to Finite Emittance (1)• Effects due to Finite Emittance of theElectron Beam– Injection to the undulator with angular andpositional offset
SR with Position and Slope Offset

θeff
Off-Axis Observation at θeff

Cheiron2009: Light Source IIPractical Knowledge on SR

Effects due to Finite Emittance (2)Off-axis observation at θeff
Peak shift tolower energy

Photon Fluxθeff=0larger θeff

EnergyCheiron2009: Light Source IIPractical Knowledge on SR

Effects due to the Energy Spread
Electron with an offset of Δγ
Energy shift of ω1

Photon Flux

Δγ=0Δγ < 0Δγ > 0Energy

Cheiron2009: Light Source IIPractical Knowledge on SR

Photon Flux Density (arb.unit)

ExamplesEmittance (m.rad)-1010-910-8
10-7
10

0.94

0.96

0.98

1.00

E-spread (%)
1
0.1
0.01
0.001

1.02

1.04 0.94

0.96

0.98

1.00

1.02

1.04

Relative Photon Energy(ω1)
Cheiron2009: Light Source IIPractical Knowledge on SR

Effects on the Higher Harmonics
Effects due to the ebeam are larger for
higher harmonics

Optical Emittance of UR: λ/4π
Bandwidth of UR: ~1/nN
191018Flux Density

101710161015100204060

Photon Energy (keV)

80100
Cheiron2009: Light Source IIPractical Knowledge on SR

Effective Beam Size and Divergence
θyσr'

σy'

σx'

θx
Under Gaussian approximation*effective beam size
*Note: Valid only near the resonance energy (nω1).

*effective divergence
Cheiron2009: Light Source IIPractical Knowledge on SR

Effective Flux Density and BrillianceSimple scheme to estimate the on-axis
flux density and brilliance.
Total Flux
on-axis flux density with zero-emittance beam

Effective
Flux Density

Effective
Brilliance
Cheiron2009: Light Source IIPractical Knowledge on SRHeat Load on Optical Elements• SR emitted from the light source isprocessed by several optical elements
before irradiation to the sample, such as thefocusing mirror, monochromator.
• These elements can be easily damaged by
the heat load brought by the SR.
• It is thus important to reduce the heat load
as much as possible without sacrificing theflux, which is actually done by the XY slit atthe front-end section.
Cheiron2009: Light Source IIPractical Knowledge on SR

Spatial Profile of Power and Flux
22Fluxy(mm)

Power
1

100-1-1-2-2-2-10x(mm)12-2-1012x(mm)The power profile is much broader than the flux. Extraction ofSR with an appropriate slit significantly reduces the heat load.
Cheiron2009: Light Source IIPractical Knowledge on SR

Optimum Slit Size?
700

1.0

150.8
500
0.6

4000.4300

Power2000.2Photon Flux (10 )600Power (W)

15Photon Flux (10 )

Flux

1.0
0.8
0.6
0.40.21000.0002468

Slit Size (Σx,Σy)

100.0
10000 11000 12000 13000 14000

Photon Energy (eV)Cheiron2009: Light Source IIPractical Knowledge on SRWiggler? Undulator? (1)• Wigglers are identical to undulator fromthe point of view of magnetic circuit.
• It is generally said that the K value
distinguishes between the two, however,this is not exactly correct.
• What we should take care is the regionof photon energy to be utilized forapplication.

Cheiron2009: Light Source IIPractical Knowledge on SR

Wiggler? Undulator? (2)15Photon Flux
(photons/sec/0.1%B.W.)

10wigglerundulator

14101310121011101010110210310410510

Photon Energy (eV)Cheiron2009: Light Source IIOther Topics Not Addressed• Quantitative descriptions of SR• Light sources for circular polarization andschemes for fast helicity switching– helical undulator & elliptic wiggler– chicanes&choppers, kicker magnets• Effects on the electron beam
– natural focusing– beam-axis fluctuation due to COD variation• R&Ds toward shorter magnetic period– superconducting undulators– cryogenic permanent magnet undulators• Coherent SR for intense THz light• Undulators for SASE-based X-ray FEL
Cheiron2009: Light Source II