presenter: 陳家祥 material edited by 周炳榮 2017.01 · Time dilation (stretching): For an...
Transcript of presenter: 陳家祥 material edited by 周炳榮 2017.01 · Time dilation (stretching): For an...
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How synchrotron light is generated
presenter: 陳家祥
material edited by 周炳榮
2017.01.19
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Important Notes to Students: The sole purpose of this lecture notes is meant for classroom use only. Some photographs and graphic illustrations are adapted from various reference literatures, which are NOT to be distributed beyond the classroom use.
[References]1. D. Attwood, Soft X‐rays and Extreme Ultraviolet Radiation: Principles and Applications,
(Cambridge Univ. , 1999) , Chap. 5.2. J. Freund, Special Relativity for Beginners (World Scientific, 2008)3. D. Attwood’s homepage, http://www.eecs.berkeley.edu/~attwood/
his class video can be found on Youtube (UCBerkeley: AST 210/EE 213)4. J.D. Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Chap. 14.5. Science and Technology of Future Light Sources-A White Paper, SLAC report SLAC‐R‐
917 (Dec. 2008).6. 吳大猷,狹義及廣義相對論(引論), (台灣中華書局, 1980) 7. H. Wiedemann, Synchrotron Radiation (Springer, 2003)
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•Primer of electromagnetics and special relativity
•Properties of synchrotron radiation
•Future prospect
Outline
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Primer of Electromagnetics and Special Relativity
Maxwell’s Equations in Homogeneous Medium
Maxwell’s equations in MKS units
0
BtBE
tDJH
D
)( BEqF
EJ
HB
ED
Ohm’s law
Lorentz force
surface
surface
JnHH
nDD
ˆ)(
ˆ)(
12
12 21 n̂
0ˆ)(
0ˆ)(
12
12
nEE
nBB
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-
x
y
z
EH
v
A polarized plane wave propagates along the z‐axis
polarization: direction of electric field
0
0)( 0
HE
eHE ztj
No attenuation
phase velocity:
1
0
wavelength: 0/2
Energy flow of electromagnetic wavesPoynting vector
Poynting vector: HES
Physical meaning: energy flowing out of the boundary surface per unit area per unit time.
, [ W/m2 ] where E and H are real quantities.
It has the dimensions of (energy/area/time).
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Two Postulates of Special Relativity:
1) The principle of relativity: The laws of physics are the same in all inertial reference frames.
2) The principle of constancy of the speed of light: The speed of light in free space has the same value c in all inertial reference frames.
x’
y’
S’
x
y
S
uObserver A
Considering two observers A and A’ in two inertial frames, each observer is stationary in the S frame and S’ frame respectively. The S’ frame is moving at a constant velocity relative to the inertial frame S.u
xxxuu ˆ//ˆ ,ˆ
Observer A’
Lab. frameCo‐moving frame
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2
2
1
1
ctxtc
ctxx
2
2
1
1
tcxct
tcxx
, where cu
Lorentz transformation of coordinates
Hendrik A. Lorentz, Nobel Prize 1902[Ref.] http://ca.wikipedia.org/wiki/Hendrik_Lorentz
211
Lorentz factor
zzyy
for relative motion in the x direction only.
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Time dilation (stretching): For an observer in the S’ frame, the time interval t2‐t1 between two events occurred at the same location x in the S frame is
212
121
tttt
The time interval t in the S frame measured by another observer moving in uniform velocity will become longeru
ttt
21
the readings of clocks at two locations in S’ frame21 , xx
Length contraction/shortening (Lorentz contraction): For a rod of rest length L’measured in the S’ frame, the length measured by another observer at rest in the S frame is
212
121
xxxx
the coordinates of the ends of rod measured in the S frame simultaneously
xx
The length xmeasured in the S frame will become shorter8
-
Lorentz force )( BEqF
tAVE
AB
The total energy of a particle
2
20
2
20
20
1
mc
cm
cm
cmTE
20
1
mm
Relativistic mass
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x’
y’S’
x
yS
u
observer
radiation source
Assuming a radiation source emitting E‐M fields in an inertial frame S’, which is moving at a constant velocity relative to the observer in an inertial frame S.
u
xxxuu ˆ//ˆ,ˆ
Relativistic Doppler Frequency Shift
’
10
-
)cos1(cos11
)cos1(2
x
uS
observer
)cos1(cos11 2
x
u observerS
relativistic Doppler effect
11
For a small angle between the observer and the wave source, i.e.
-
The observation angle is smaller than the emission angle ’ for > 0.
2tan
11
2tan
isotropic radiation in the rest frame of source
’
direction of wave propagation
anisotropic radiation in the lab frame as seen by an observerobserver
u
12
The direction of wave emission is different from the direction of propagation as detected by an observer in different reference frame.
Aberration of Light
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When the source is moving toward the observer
’u
observer
The emitted radiation pattern will be benttoward the direction of motion, i.e. benttoward the observer
S’S
x’
The observation angle (lab frame) is smaller than the emission angle (co‐moving frame).
2tan
11
2tan
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Properties of Synchrotron Radiation
Julian Schwinger at early years, American physicist, Nobellaureate in physics (quantum electrodynamics). He received hisPh.D. at age 21 from Columbia University, under the supervisionof I.I. Rabi. He shared the Nobel Prize with S. Tomonaga and R.Feynman in 1965 (quantum electrodynamics).
[Ref.] http://www.english.ucla.edu/ucla1960s/6465/mohajeri.htm[Ref.] http://nobelprize.org/nobel_prizes/physics/laureates/1965/schwinger‐bio.html
[Ref.]http://nobelprize.org/nobel_prizes/physics/laureates/1965/index.html
Julian Schwinger at later years. Fordetailed biography, please refer to thefollowing web sites. During WW II heworked at the Radiation Laboratory atMIT (radar project), where he carried outhis work on synchrotron radiation andmicrotron.
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radiation power/area ~ sin2total radiation power~
•An oscillating electric dipole:
a
tjeaa 0
20
4a
The electromagnetic radiation mainly concentrates in the direction perpendicular to the acceleration vector.
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■ Electromagnetic Radiation of an Oscillating Electric Dipole
-
acceleration
charge
electric field lines
longitudinal electric field component
magnetic field line
x
z
Ez
Ez
Hy
Hy
S
Sx
v
Assuming a positive charge moving in z direction
a
z (longitudinal direction)
Radiation power↔ Poynting vector
■ Physics of Synchrotron Radiation
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electric field lines
transverse electric field component
magnetic field line
acceleration
charge
x
z
ExEx
HyHySS
z (longitudinal direction)
x
v
Assuming a positive charge moving in z direction
a
Radiation power↔ Poynting vector
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22
3200
2
2
sin16
dtpd
cme
ddS
Direction of acceleration
The angular distribution of the radiation power:
2
3200
2
6
dtpd
cmeS
The total radiation power by an accelerated non‐relativistic electron:
Larmor formula
The angular distribution is the same as the Hertz dipole
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• The power radiated by non‐relativistic particles is very low and negligible.
■ Radiation from an accelerated charged particle at low velocities (v
-
2
2
22
2200
2
2
2
2
3200
2
1)(6
16
dtdE
cdtpd
cmce
ddE
cdpd
cmeS
The Lorentz invariant form for the total radiation power of a charged relativistic particle is:
220
2
1cm
p
ppp 2dtpd
dtdcm
dtdE
20
The radiation power depends strongly on the anglebetween the direction of particle velocity and the direction of acceleration!
試著自己推導一下
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■ Radiation from an accelerated charged particle at low velocities (v~ c)
-
Case 1) co‐linear acceleration Fp //
The radiation power emitted by longitudinally accelerated particles is given by
2
2200
2
)(6
dtpd
cmceS
from 22220
2 )( cpcmE differentiating both sides of the above equation w.r.t.
dpd
ddE
0mp
20
-
2
2200
2
)(6
dxdE
cmceS
For practical use it’s more convenient to the radiation power S as function of the accelerating gradient (energy gain per unit length), dE/dx.
dxdE
dtpd
F
The accelerating force:
velocity
acceleration
linac
Total radiation power:
[ref.] picture adapted from SLAC web site, www.slac.stanford.edu
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Case 2) transverse acceleration (circular orbit) Fp
This kind of radiation is mostly generated by particles moving through a magneticfield B. In a circular accelerator, the energy of the particle is not changed,
22
2200
2
)(6
dtpd
cmceS
2
)//()(
SSThe ratio of total radiation power:
In the lab frame,
22
-
2
44
0
2
2
420
4
2200
2
22222
00
2
22
2200
2
6
)(6
)(6
)(6
ce
mcm
ce
pcm
ce
dtpd
cmceS
p
epdtd
dedp
dtdpee
dtdpep
dtd
dtpd
ˆ
ˆ
ˆˆ)ˆ(
|| pdtpd
emepp
c
ˆˆ
0
Total radiation power (e‐ moving in a circular accelerator):
240
1m
S 23
Magnetic rigidity/
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In a circular accelerator, the total energy loss radiated by an electron per revolution,
)()]([108575.8 ][
3
3
2
42
4
200
32
0
342
meterGeVEMeVU
cmEe
e
SU
me 0.51 MeV/c2
mp 938.27 MeV/c2
m 105.66 MeV/c2
CESR(Cornell)
LEP(CERN)
Tevatron(Fermilab)
LHC(CERN)
e-e+ e-e+ pp
E [GeV] 6 100 1000 7000
L [km] 0.768 26.66 6.28 26.66
1for high energy electrons
• Rest mass of particles
• Parameters of circular colliders
pp
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For TLS, E= 1.5 GeV, I= 360 mA, = 3.495 m P= 46.19 kW
][][][654.2)(
)()]([108575.8
(Amp)(keV)(kW)
3
42
AIGeVEkGBmeter
mAIGeVE
JUP
The total power radiated by N electrons in a circular electron accelerator per turn:
The total radiation power can be rewritten as2
4
420
4
0
2
)(6
cmEceS
Summary of total radiation power in a circular accelerator:
2
40
4
1
1
S
mS
ES
134
1013.1)()(
e
p
mm
protonSelectronS
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transverse acceleration (circular orbit) Fp
),(ˆ//
ˆ//
ne
e
x
z
e
local center
y
z
xreference orbit
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200 400 600 800 1000
- 150- 100- 50
50100150
radiationconeHcircularLF= 0.9
max= 0
When >>1, 1/2 ~1/ searchlight beam.Synchrotron radiation is well collimated.
v
■ Angular distribution of radiation
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https://en.wikipedia.org/wiki/Synchrotron_radiation
in the co‐moving frame in the lab frame
After Lorentz transformation aberration of relativistic wave source
centripetal force
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centripetal force
-
2
2/12
21
11
)11(
cdcdcdT
32 c
T
From the property of Fourier transform, the maximum frequency of appreciable radiation power can be approximately estimated
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3 22
1
c
T
c
c
Pulse length of radiation emitted by an electron
Pulse length of synchrotron radiation is very short radiation spectrum is very broad
t
T~ 2.3×10‐19 s
= 3.495 mE= 1.5 GeV (TLS)
observer
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Trf
-
observer
x
v
|
y
z
n||e
||ene
2||||2
222
)()(4
AeAe
ce
ddId
The energy radiated per unit solid angle per unit frequency interval:
polarization parallel to the orbital plane
polarization approximately perpendicular to the orbit plane for small
■ Spectral distribution of radiation (instantaneously circular motion)
Assuming the observer is far away from the source,
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2/322
3
23/122
222
3/2222
2
22
2
)1(2
23
)(1
)()1(
c
c
c
c
KKddPd
In a circular accelerator, the radiation power spectrum from bending magnet is,
Horizontal polarization(‐mode)E‐field ⊥ deflecting field B
Vertical polarization(‐mode)E‐field // deflecting field B
c critical frequency: half of the total power is radiated above the critical frequency and the other half is radiated below
What does critical frequency mean?
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The resulting spectral photon density is
0
4
20
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36
)()/(
btot
cc
totph
IeNceP
SPddt
dN
Ptot is the total power radiated by N electrons during one revolution.S(c) is called the universal function and is expressed as following:
c
dxxKScc
)(8
39)( 3/5
The universal function fulfills the following normalization conditions,
21)(
1)(
1
0
0
dxxS
dxxS
This relation shows that the critical frequency divides the spectrum into two parts of equal power.
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0.0001 0.001 0.01 0.1 1 10w wc
0.01
0.1
1
10
Sww c
1.333 x1/3
0.777 x1/2e‐x
When c 1 S(x) ~0.777 x1/2e‐x
Universal function S(c)
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The critical photon energy is:
][[GeV]E2.218
23][
33
mckeVc
= 3.495 mE= 1.5 GeVTLS
-
Average power spectrum vs. wavelength during an acceleration cycle in 300‐MeV Cornell Synchrotron, Phys. Rev. v.102(1956)1423.
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How to increase the radiation intensity for an existing synchrotron light source?
btotph IP
ddtdN 4
)/(
from bending magnets:
• Increasing the beam current (collective instabilities)• Increasing the beam energy (need more rf power, magneticfield in dipole limited by saturation)
The radiation emitted from a bending magnetspread out into a flat horizontal fan. The angleextended by this radiation fan is as large as thebending angle . The sample to be studied is smalland this results in the use of a small portion of thesynchrotron radiation. Therefore, the effectiveradiation intensity is actually small.
Disadvantage of synchrotron radiation from bending magnets
(incoherent)
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x
[ref]: www.bessy.de/guided_tour/images/wiggler1_e.jpg
■ Radiation from Insertion Devices (wiggler & undulator)y
z
Using a large array of dipole magnets withalternating polarity one can increase thephoton flux of synchrotron radiation. Thelinear array of short dipole magnets arecalled wiggler or undulator.
e‐
B
B
B
B
B
B
B
B
z
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插件磁鐵
插件磁鐵放出同步輻射光的原理
正(北)極
負(南)極
電子由此進入磁場
電子團
同步輻射光束
U50聚頻磁鐵
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u
x
y
zx’
S
NSS
SS N
NN
N
g
w
electron beam
undulator periodAt the longitudinal positions
0,1,2,...) (n with 21
unz
the particle trajectory reaches the maximum deflecting angle w.r.t. the design orbit as given by:
K)
21( unx
when K≦1 it’s an undulatorwhen K>>1 it’s a wiggler
K is called the undulator or wiggler parameter
a periodic wiggling trajectory in the horizontal plane
x0= (K/)(u/2)
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Kcm
Benx uu02
ˆ)
2(
K: undulator or wiggler parameter
uuu
gcmeB
cmBeK
cosh22
ˆ
0
0
0
K decreases with increasing gap height g The undulator/wiggler parameter K is a non‐dimensional
parameter
in convenient units,
][][ˆ9337.0 cmTBK u
maximum deflecting angle
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-0.02-0.01 0.01 0.02
-0.15
-0.1
-0.05
0.05
0.1
0.15
z’
x’
K=1
K=0.5
uu
electron motion in lab framewiggling with a wavelength u
electron motion in co‐moving framefigure‐8 oscillation with a wavelength u/
e‐x
y
z
Electron motion in the undulator region
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For TLS: 1.5 GeV ( = 2935), U50 undulator, K= 2.99, u= 50 mm
mmx 8101.8ˆ 6
-
e‐
x
y
z
|
observer
x
z-0.02-0.01 0.01 0.02-0.15
-0.1
-0.05
0.05
0.1
0.15
e‐
x’
z’cz
co‐moving frame
lab frame
Like an oscillating dipole in the co‐moving frame
Dipole radiation
22 sin)( xeddP
-1 -0.5 0.5 1
-0.4
-0.2
0.2
0.4
x’
z’
is the angle between the dipole and the propagation vector (observer)
• the size of oscillating dipole
-
relativistic Doppler shift+ Lorentz contraction
is the angle between the propagation vector and the beam axis z
Property of undulator radiation
)2
1(2
222
21 Kuundulator equation:
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As the undulator parameter K increases, higher harmonics will be generated and the electron trajectory becomes non‐sinusoidal.
undulator radiation of higher harmonics:
,....3,2,1 )2
1(2
222
2 nK
nu
n
-
spectral lines broadening due to
1) off‐axis angle2) finite number N of e‐ oscillation3) beam emittance (size, angular divergence)4) beam energy spread
Doppler shift to lower frequencyThe net result is:
ddP
1
2
3
co‐moving frame
lab frame
non‐axial relativisticDoppler shift
near axis
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bandwidth of spectral line:N1
(N: the number of undulator period)
-
0.2 0.4 0.6 0.8 1
-0.06-0.04-0.02
0.020.040.06
x
z
NNcen 11
* N: undulator periods
angular distribution of undulator radiation:
bbph INKf
KIKN
ddtdN 22
22
222
)()21()/(
on‐axis spectral flux:
in practical units,
)(2
1)(306.1)( 2
222
GeVE
Kcmnm
u
22
2
2
21)(
)(9496.0)( Kcm
GeVEkeVE
u
ph
1st harmonic of undulator radiation(undulator equation)
photon energy of 1st harmonic undulator radiation
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Spectral brig
htne
ss
Photon energy (keV)0 4
K= 1
K= 5
K> 10
Undulator radiation (K≦ 1)
Wiggler radiation (K>> 1)
)2/1(4
3)2/1(
4
2
2
2
KKn
Kcn
c
u
cc
critical frequency:
)()/( c
bph SNKI
ddtdN
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e‐
undulator
e‐
e‐
wiggler
bending magnet
2K/
[Ref.] Opt. Eng. 34 (1995)342, K.J. Kim
•Spectral properties for bending magnet, wiggler, and undulator radiation
incoherent
partially coherent
incoherent
~ 1/
~ 1/
~ 1/
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[Ref.] “Wiggler and undulator magnets”, H. Winick et al., Phys. Today (May 1981), p.50.
•First observation of undulator radiation in an electron circular accelerator at Lebedev Physics Institute, Moscow
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The angular distribution is shown at a wavelength of 500 nm with polarization parallel () and perpendicular () to the orbital plane of electron motion. [ref.] Phys. Today, May 1981, pp.50‐63.
•Undulator radiation patterns for different polarization and higher harmonics
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同步加速器光源的特色
• 強度極高• 波長連續• 準直性佳• 光束截面積小• 具有時間脈波性與偏振性
以X光為例,同步加速器光源在這個波段的亮度比傳統X光機還要強百萬倍以上!過去需要幾個月才能完成的實驗,現在只需幾分鐘便能得到結果。以往因實驗光源亮度不夠而無法探測的結構,現在藉由同步加速器光源,都可分析得一清二楚,也因此得以開發新的研究領域。
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Present Status and Future Prospect
•Ring‐based Light SourceLarge circumference (e.g. Taiwan Photon Source)Ultimate storage rings
(PEP‐X in USA, Spring‐8 II in Japan)
•Free Electron LaserSASE X‐ray FEL
(USA‐‐operational, Europe, Japan, UK, Italy, Swiss)Hard X‐ray optic mirror using diamond crystal
(87% reflectivity, Argonne National Lab, USA)X‐ray FEL OscillatorHGHG X‐ray FEL
•Energy Recovery Linacbased on superconducting linear accelerator
A pilot project (5 GeV) to be built at Cornell Univ.
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Spectral Brightness
Science and Technology of Future Light Sources— A White Paper (SLAC‐R‐917)
[Ref.] http://www‐ssrl.slac.stanford.edu/aboutssrl/documents/future‐x‐rays‐09.pdfTPS
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感謝大家的聆聽,祝大家新年快樂Thanks for your listening and wish you
a happy Chinese new year
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