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USPAS Accelerator Physics June 2013
Accelerator Physics
Synchrotron Radiation
A. Bogacz, G. A. Krafft, and T. Zolkin
Jefferson Lab
Colorado State University
Lecture 8
USPAS Accelerator Physics June 2013
Synchrotron Radiation
'
'
'
'
ct ct z
x x
y y
z ct z
Accelerated particles emit electromagnetic radiation. Emission
from very high energy particles has unique properties for a
radiation source. As such radiation was first observed at one of
the earliest electron synchrotrons, radiation from high energy
particles (mainly electrons) is known generically as synchrotron
radiation by the accelerator and HENP communities.
The radiation is highly collimated in the beam direction
From relativity
USPAS Accelerator Physics June 2013
Lorentz invariance of wave phase implies kμ = (ω/c,kx,ky,kz) is a
Lorentz 4-vector
z
x x
y y
z z
k c
k k
k k
k c k
2 2 2 2
sin sin cos/ / /
/ / 1 cos /
/ / 1 cos /
x y x y z
z
z
k k k k k
c c c
c c k c
c c k c
USPAS Accelerator Physics June 2013
sin
sin1 cos
Therefore all radiation with θ' < π / 2, which is roughly ½ of the
photon emission for dipole emission from a transverse
acceleration in the beam frame, is Lorentz transformed into an
angle less than 1/γ. Because of the strong Doppler shift of the
photon energy, higher for θ → 0, most of the energy in the
photons is within a cone of angular extent 1/γ around the beam
direction.
USPAS Accelerator Physics June 2013
Larmor’s Formula
2 2
22
3 2 3
0 0
1 1
6 6
q eP t a p
c m c
For a particle executing non-relativistic motion, the total power
emitted in electromagnetic radiation is (Larmor, verified later)
Lienard’s relativistic generalization: Note both dE and dt are the
fourth component of relativistic 4-vectors when one is dealing
with photon emission. Therefore, their ratio must be an Lorentz
invariant. The invariant that reduces to Larmor’s formula in the
non-relativistic limit is
2
06
due duP
c d d
USPAS Accelerator Physics June 2013
2 2
6 2
06
eP t
c
For acceleration along a line, second term is zero and first term
for the radiation reaction is small compared to the acceleration
as long as gradient less than 1014 MV/m. Technically
impossible.
For transverse bend acceleration
2
4 4
2
06
e cP t
2
ˆc
r
USPAS Accelerator Physics June 2013
Fractional Energy Loss
23 4
06
eE
3 34
3
e
beam
rE
E
For one turn with isomagnetic bending fields
re is the classical electron radius: 2.82
10-13 cm
USPAS Accelerator Physics June 2013
Radiation Power Distribution
2
5/32
0 /
3
8cc
dP eK x dx
d
Consulting your favorite Classical E&M text (Jackson,
Schwinger, Landau and Lifshitz Classical Theory of Fields)
USPAS Accelerator Physics June 2013
Critical Frequency
Critical (angular) frequency is
33
2c
c
Energy scaling of critical frequency is understood from 1/γ
emission cone and fact that 1– β ~ 1/(2 γ2)
A B
1/γ
Atc
32Bt
c c c
3 32Bt t
c c c
USPAS Accelerator Physics June 2013
Photon Number
2 2
4
5/32 2
0 00 0
3
8 6c
dP e e cP d K x dxd
d
1dn dP
d d
0
0
8
15 3c
dnd
d
dnd
d
2
0
5 5 1
4 1372 3 2 3
c en n
c
USPAS Accelerator Physics June 2013
Insertion Devices (ID)
Often periodic magnetic field magnets are placed in beam path
of high energy storage rings. The radiation generated by
electrons passing through such insertion devices has unique
properties.
Field of the insertion device magnet
0ˆ, , cos 2 / IDB x y z B z y B z B z
Vector potential for magnet (1 dimensional approximation)
0ˆ, , sin 2 /2
IDID
BA x y z A z x A z z
USPAS Accelerator Physics June 2013
Electron Orbit
Field Strength Parameter
Uniformity in x-direction means that canonical momentum in
the x-direction is conserved
0
2
IDeBK
mc
v 1
cos 2 /v 2
x IDID
z z
Kx z dz z
v sin 2 /x ID
eA z Kz c z
m
USPAS Accelerator Physics June 2013
Average Velocity
zz xz
2
2
11
Energy conservation gives that γ is a constant of the motion
Average longitudinal velocity in the insertion device is
2
2
2
22*
2
11
Kz
Average rest frame has
2/11
12
2
2*
2*
K
USPAS Accelerator Physics June 2013
Relativistic Kinematics
In average rest frame the insertion device is Lorentz contracted,
and so its wavelength is
*** / ID
The sinusoidal wiggling motion emits with angular frequency
** /2 c
Lorentz transformation formulas for the wave vector of the
emitted radiation
***
*
*
***
cos
sinsin
cossin
cos1
kk
kkk
kkk
kk
z
yy
xx
USPAS Accelerator Physics June 2013
ID (or FEL) Resonance Condition
cos1
coscos
*
*
*
**
k
kz
Angle transforms as
Wave vector in lab frame has
cos1
2
cos1 *
*
**
*
ID
ckk
In the forward direction cos θ = 1
2
*2 21 / 2
2 2
ID IDe K
USPAS Accelerator Physics June 2013
Power Emitted Lab Frame
2224 *2
0
2 1
6 2ID
e KP c
Larmor/Lienard calculation in the lab frame yields
Total energy radiated after one passage of the insertion device
22 2 * 2
0
26 ID
eE NK
USPAS Accelerator Physics June 2013
Power Emitted Beam Frame
22* 2
*
0
2 2 1
6 2
eP cK
Larmor/Lienard calculation in the beam frame yields
Total energy of each photon is ħ2πc/λ*, therefore the total
number of photons radiated after one passage of the insertion
device
2
3
2NKN
USPAS Accelerator Physics June 2013
Spectral Distribution in Beam Frame
* 2*4 2 2 *
* 2
0
sin32
dP e ck a
d
Begin with power distribution in beam frame: dipole radiation
pattern (single harmonic only when K<<1; replace γ* by γ, β* by β)
Number distribution in terms of wave number
*2 *2
2
* *24
y zk kdNNK
d k
Solid angle transformation
*
22 1 cos
dd
USPAS Accelerator Physics June 2013
Number distribution in beam frame
22 2 2 2
2
44
sin sin cos cos
4 1 cos
dNNK
d
Number distribution as a function of normalized lab-frame energy
Energy is simply
2 1ˆ 1 cos 1 cosID
cE E
2
2 2
2 3 2
ˆ1
ˆ 4
dN ENK
dE
USPAS Accelerator Physics June 2013
Average Energy
Limits of integration
1 1ˆ ˆcos 1 cos 1 1 1
E E
Average energy is also analytically calculable
20 max
0
ˆˆ
2 /2ˆ
ˆ
ID
dNE dE
EdEE c
dNdE
dE
USPAS Accelerator Physics June 2013
dN/dEγ
Eγ 0 βγ2Eund (1+β)βγ2Eund
θ = π
sin θ = 1/ γ
θ = 0
Number Spectrum
2 /und IDE c
USPAS Accelerator Physics June 2013
Conventions on Fourier Transforms
1
2
i t
i t
f f t e dt
f t f e d
Results on Dirac delta functions
For the time dimensions
1
1
2
i t
i t
t e dt
t e d
USPAS Accelerator Physics June 2013
3
3
3
3 3
3
1
2
1
2
ik x
ik x
ik x
f k f x e d x
f x f k e d k
x x e d k
For the three spatial dimensions
USPAS Accelerator Physics June 2013
Green Function for Wave Equation
Solution to inhomogeneous wave equation
2 2 2 2
2 2 2 2 2
1, ; ,
4
G x t x tx y z c t
x x t t
Will pick out the solution with causal boundary conditions, i. e.
This choice leads automatically to the so-called Retarded Green
Function
, ; , 0 G x t x t t t
USPAS Accelerator Physics June 2013
In general
because there are two possible signs of the frequency for each
value of the wave vector. To solve the homogeneous wave
equation it is necessary that
i.e., there is no dispersion in free space.
3
, ; , 0
, ; ,
i k x t i k x t
G x t x t t t
G x t x t
A k e B k e d k t t
k k c
USPAS Accelerator Physics June 2013
Continuity of G implies
2
3
2
2
2
, ; ,1 4
4
2
t
i k x t i k x t
ik x i t
G x t x tx x
c t
i A k e i B k e d k
c x x
cA k e e
i
Integrate the inhomogeneous equation between and
i t i tA k e B k e
t t t t
USPAS Accelerator Physics June 2013
23
2
, ; ,
1
2
2 2
i k x x t t i k x x t t
ik x xi t t
G x t x t
ce e d k
i
t t
c e ce dk
x x
/0
ik x xi t te
e dk t tx x
x x c t t
x x
Called retarded because the influence at time t is due to the
source evaluated at the retarded time
/t t x x c
USPAS Accelerator Physics June 2013
Retarded Solutions for Fields
2 2 2 2
2 2 2 2 2
0
2 2 2 2
02 2 2 2 2
1
1
x y z c t
A Jx y z c t
3
0
30
,1, /
4
,, /
4
x tx t d x dt x x c t t
x x
J x tA x t d x dt x x c t t
x x
Tip: Leave the delta function in it’s integral form to do derivations.
Don’t have to remember complicated delta-function rules
USPAS Accelerator Physics June 2013
Delta Function Representation
/3
2
0
/30
2
,1,
8
, ,
8
i x x c t t
i x x c t t
x tx t d x dt d e
x x
J x tA x t d x dt d e
x x
f t t f t tt t
f x x f x xx x
Evaluation can be expedited by noting and using the symmetry of
the Green function and using relations such as
USPAS Accelerator Physics June 2013
Radiation From Relativistic Electrons
/3
2
0
/30
2
3 3
/
2
0
,1,
8
, ,
8
, , v
1,
8
i x x c t t
i x x c t t
i x r t c t t
x tx t d x dt d e
x x
J x tA x t d x dt d e
x x
x t q x r t J x t q t x r t
qx t dt d e
x r t
/0
2
v ,
8
i x r t c t ttqA x t dt d e
x r t
USPAS Accelerator Physics June 2013
Lienard-Weichert Potentials
0
0
0
0
/,
4
v /,
4
1,
4 ˆ1
v,
4 ˆ1
ret
ret
x r t c t tqx t dt
x r t
t x r t c t tqA x t dt
x r t
qx t
x r t n t
tqA x t
x r t n t
USPAS Accelerator Physics June 2013
EM Field Radiated
/
2
0
/0
2
32
02
0
1,
8
v ,
8
ˆ ˆˆ
4 4ˆ ˆ1 1
i x r t c t t
i x r t c t t
ret
qx t dt d e
x r t
tqA x t dt d e
x r t
AE B A
t
n nq n q
Ecn nR
3
ˆ /
ret
B n E c
R
Acceleration Field Velocity Field
USPAS Accelerator Physics June 2013
2
2
/
22
0
0
2
ˆ1ˆ
ˆ ˆ/ /ˆ ˆ1
ˆ 1 /,
8
v,
8
i x r t c t t
nx r t n
x r t x r t
dr dt n n dr dtd n c dn
dt dtx r t x r tx r t
n i x r t cqx t dt d e
x r t
t iqA x t dt d
t
/
/ /ˆ1
i x r t c t t
i x r t c t t i x r t c t t
ex r t
de i t n t e
dt
USPAS Accelerator Physics June 2013
/
22
0
/
22 20
2
ˆˆ,
8
integrate by parts to get final result
,8 ˆ1
ˆ ˆ ˆ ˆ1 2ˆ
i x r t c t t
i x r t c t t
vel
i nq nE x t dt d e
c x r tx r t
q eE x t dt d
n x r t
n n nn
2 2
22
2 22
ˆ ˆ
ˆ
ˆ ˆ ˆ ˆ1
n n n
n
n n n n
USPAS Accelerator Physics June 2013
/
22
0
/
22
0
,8 ˆ1
ˆ ˆ
ˆ ˆ1
ˆ ˆ8 ˆ1
i x r t c t t
acc
i x r t c t t
q eE x t dt d
c n x r t
n n
n n
q edt d n n
c n x r t
USPAS Accelerator Physics June 2013
Larmor’s Formula Verified
0
22
2 3
0 0
22
2
2 3
0
22
3
0
For small velocities can neglect retardation
ˆ ˆ, /4
ˆ ˆ16
v sin16
v6
acc
qE x t n n R
c
dP qn n
d c
q
c
qP
c
Classical Dipole
Radiation Pattern
Angle between acceleration
and propagation directions
USPAS Accelerator Physics June 2013
Relativistic Peaking
2
2
52
0
2 2 2
52
0
max
2 2 2 2
3 22 20
In far field after short acceleration
ˆ ˆ
16 ˆ1
sin
16 1 cos
1
2
For circular motions
sin cos1
16 1 cos 1 cos
n ndP t q
d c n
dP t q
d c
dP t q
d c
USPAS Accelerator Physics June 2013
Spectrum Radiated by Motion
2 2
2 2
0
2
2 22
0 0
ˆ ˆ ˆ1 2 / / / 1 2
1ˆ
ˆ ˆ ˆ ˆ1
8 ˆ ˆ1 1
i R n r t R n r t R c t t i R n r
dE dPdt E H nR dt E E R dt
d d c
n n n nq
t tc c n n
e e
2 2ˆ/ / /
2
22
0 0
clearing the unprimed time integral and omega prime
integral with delta represntation
ˆ ˆ ˆ ˆ2
8 ˆ ˆ1 1
t R n r t R c t t
dt d dt d dt
n n n nq
tc c n n
2
ˆ ˆ/ /
i n r t c t i n r t c t
t
e e dt dt d
USPAS Accelerator Physics June 2013
2
2 2ˆ /
23
0
22 2 2
ˆ /
3
0
ˆ ˆ
32 ˆ1
ˆ ˆ32
Factor of two difference from Jackson because he
combines positive frequency and
i n r t c t t
i t n r t c
n nd E q
e dtd d c n
d E qn n e dt
d d c
negative frequency
contributions in one positive frequency integral. I
don't like because Parseval's formula, etc. don't work!
We'll perform this calculation in new intensity regimes
of pulsed lasers.