USPAS Accelerator Physics January 2020
Accelerator Physics
Radiation Distributions
S. A. Bogacz, G. A. Krafft, S. DeSilva, B. Dhital
Jefferson Lab
Old Dominion University
Lecture 8
USPAS Accelerator Physics January 2020
Nonlinear Thomson Scattering
• Many of the the newer Thomson Sources are based on a PULSED laser (e.g. all of the
high-energy lasers are pulsed by their very nature)
• Have developed a general theory to cover radiation distribution calculations in the
general case of a pulsed, high field strength laser interacting with electrons in a
Thomson scattering arrangement.
• The new theory shows that in many situations the estimates people do to calculate flux
and brilliance, based on a constant amplitude models, need to be modified.
• The new theory is general enough to cover all “1-D” undulator calculations and all “1-
D” pulsed laser Thomson scattering calculations.
• The main “new physics” that the new calculations include properly is the fact that the
electron motion changes based on the local value of the field strength squared. Such
ponderomotive forces (i.e., forces proportional to the field strength squared), lead to a
red-shift detuning of the emission, angle dependent Doppler shifts of the emitted
scattered radiation, and additional transverse dipole emission that this theory can
calculate.
USPAS Accelerator Physics January 2020
Ancient History
• Early 1960s: Laser Invented
• Brown and Kibble (1964): Earliest definition of the field strength parameters K and/or a in the literature that I’m aware of
Interpreted frequency shifts that occur at high fields as a “relativistic mass shift”.
• Sarachik and Schappert (1970): Power into harmonics at high K and/or a . Full calculation for CW (monochromatic) laser. Later referenced, corrected, and extended by workers in fusion plasma diagnostics.
• Alferov, Bashmakov, and Bessonov (1974): Undulator/Insertion Device theories developed under the assumption of constant field strength. Numerical codes developed to calculate “real” fields in undulators.
• Coisson (1979): Simplified undulator theory, which works at low K and/or a, developed to understand the frequency distribution of “edge” emission, or emission from “short” magnets, i.e., including pulse effects
0 0 Undulators2
eBK
mc
0 0
2 Thomson Sources
2
eEa
mc
USPAS Accelerator Physics January 2020
Coisson’s Spectrum of a Short Magnet
2 2 222
2 2 222
1 / 21er cdEf B
d d
Coisson low-field strength undulator spectrum*
222
fff
22 2
22 2
2 2
2 2
1sin
1 1cos
1
1 1
f
f
*R. Coisson, Phys. Rev. A 20, 524 (1979)
USPAS Accelerator Physics January 2020
Dipole Radiation
Assume a single charge moves in the x direction
zytdxetzyx ),,,(
zytdxxtdetzyxJ ˆ),,,(
Introduce scalar and vector potential for fields.
Retarded solution to wave equation (Lorenz gauge),
0
0
0
1
1 ( ' / ), ', ' ' ' '
4
' ( ' / )1, ', ' ' ' '
4
, lim lim / 4
x x
r r
R e t t R cr t r t dx dy dz dt
R c R
d t t t R ceRA r t J r t dx dy dz dt
R c R
e AB A B d t R c E
R t
'' trrR
USPAS Accelerator Physics January 2020
Dipole Radiation
x
z
y
r
0
1
/ˆlim sin
4r
ed t r cB
cr
2
01
/ ˆlim sin4r
ed t r cE
c r
2 2
2
2 3 2
0 0
/1ˆsin
16
e d t r cE BI r
c r
Polarized in the plane containing and
2 2
2
2 3
0
/1sin
16
e d t r cdI
d c
nr
ˆ x
USPAS Accelerator Physics January 2020
Dipole Radiation
Define the Fourier Transform
dtetdd ti
)(
~
dedtd ti
~
2
1)(
22 4
2
3 3
0
( )1sin
32
e ddE
d d c
This equation does not follow the typical (see Jackson) convention that combines both positive and
negative frequencies together in a single positive frequency integral. The reason is that we would like to
apply Parseval’s Theorem easily. By symmetry, the difference is a factor of two.
With these conventions Parseval’s Theorem is
dddttd ~
2
1
22
2 2
22 4
2 3 3 3
0 0
/ 16 32
dE e ed t r c dt d d
d c c
Blue Sky!
USPAS Accelerator Physics January 2020
Dipole Radiation
2
2 4
3 3
0
1
32
e d ndE
d d c
For a motion in three dimensions
Vector inside absolute value along the magnetic field
2 2
2 4 2 4
3 3 3 3
0 0
1 1
32 32
e d n n e n d n ddE
d d c c
Vector inside absolute value along the electric field. To get
energy into specific polarization, take scalar product with the
polarization vector
USPAS Accelerator Physics January 2020
Co-moving Coordinates
• Assume radiating charge is moving with a velocity close to light in a direction taken to be the z axis, and the charge is on average at rest in this coordinate system
• For the remainder of the presentation, quantities referred to the moving coordinates will have primes; unprimed quantities refer to the lab system
• In the co-moving system the dipole radiation pattern applies
x
,z
y
'x
'z
'y
cz
USPAS Accelerator Physics January 2020
New Coordinates
Resolve the polarization of scattered energy into that
perpendicular (σ) and that parallel (π) to the scattering plane
'
'
'x
'z
'y
'n '
' 'x
'z
'y
'n
'
'
''
'ˆ'ˆ'sin'ˆ'sin'cos'ˆ'cos'cos'ˆ''ˆ
'ˆ'ˆ'cos'ˆ'sin'ˆ'/'ˆ''ˆ
'ˆ'cos'ˆ'sin'sin'ˆ'cos'sin'
zyxene
yxznzne
zyxn
USPAS Accelerator Physics January 2020
Polarization
It follows that
''~
'sin'sin'cos''~
'cos'cos''~
'ˆ''~
'cos''~
'sin''~
'ˆ''~
zyx
yx
ddded
dded
So the energy into the two polarizations in the beam frame is
' 2 42
3 3
0
2' 2 4
3 3
0
1 '' ' sin ' ' ' cos '
' ' 32
' ' cos 'cos ' ' ' cos 'sin '1 '
' ' 32 sin ' ' '
x y
x y
z
dE ed d
d d c
d ddE e
d d c d
USPAS Accelerator Physics January 2020
Comments/Sum Rule
• There is no radiation parallel or anti-parallel to the x-axis for x-dipole
motion
• In the forward direction θ'→ 0, the radiation polarization is parallel to the
x-axis for an x-dipole motion
• One may integrate over all angles to obtain a result for the total energy
radiated
' 2 42 2
3 3
0
' 2 42 2 2
3 3
0
1 '' ' ' ' 2
' 32
1 ' 2 8' ' ' ' ' '
' 32 3 3
x y
x y z
dE ed d
d c
dE ed d d
d c
2
2 4'
3 3
0
' ' '1 8
' 32 3
tot
e ddE
d c
Generalized Larmor
USPAS Accelerator Physics January 2020
Sum Rule
Parseval’s Theorem again gives “standard” Larmor formula
2 2 2 2'
3 3
0 0
' ' ' '1 1'
' 6 6
tote d t e a tdE
Pdt c c
Total energy sum rule
2
2 4
'
2 3
0
' ' '1
'12
tot
e d
E dc
USPAS Accelerator Physics January 2020
Energy Distribution in Lab Frame
By placing the expression for the Doppler shifted frequency and
angles inside the transformed beam frame distribution. Total energy
radiated from d'z is the same as d'x and d'y for same dipole strength.
222 4 2
3 3
2
22 4 2
3 3
' 1 cos sin1 cos
32 ' 1 cos cos
cos ' 1 cos cos
1 cos
1 cos cos' 1 cos sin
32 1 cos
sin' 1 cos
1 cos
x
y
x
y
z
dedE
d d c d
d
edEd
d d c
d
USPAS Accelerator Physics January 2020
Bend
e–
e–e–
Undulator Wiggler
white source partially coherent source powerful white source
Flu
x [
ph
/s/0
.1%
bw
]
Bri
gh
tnes
s
[ph
/s/m
m2/m
r2/0
.1%
bw
]
Flu
x [
ph
/s/0
.1%
bw
]
USPAS Accelerator Physics January 2020
Weak Field Undulator Spectrum
2
22
2
220
22
2
2
2
0
1 csin
2
coscos
2 1 co
1 co
os /
s
1 c
s
o
s
s /
1 co
e
e
z z
z
z z
z z
z
BdE r c
d d
BdE r c
c
c
d d
42
2 2 2 4
016e
er
m c
0
22
2
'/ˆ ˆ' ' ' '
'
zB cecd d x x
mc
dzezBkB ikz
~
2
222
2
111cos1
zz
Generalizes Coisson to arbitrary observation angles
USPAS Accelerator Physics January 2020
Strong Field Case
0dt
d
dm c ec B
dt
' '
z
x
ez B z dz
mc
Why is the FEL resonance condition?
0
2
2
21
2
K
USPAS Accelerator Physics January 2020
High K
zz xz
2
2
11
22 2
02 2 2
1 1 11 ' ' 1 1 cos 2
2 2 2 2 4
z
z
e K Kz B z dz k z
mc
Inside the insertion device the average (z) velocity is
21
2
11*
2
2
Kz
and the radiation emission frequency redshifts by the 1+K2/2 factor
USPAS Accelerator Physics January 2020
Thomson Scattering
• Purely “classical” scattering of photons by electrons
• Thomson regime defined by the photon energy in the electron rest frame being
small compared to the rest energy of the electron, allowing one to neglect the
quantum mechanical “Dirac” recoil on the electron
• In this case electron radiates at the same frequency as incident photon for low
enough field strengths
• Classical dipole radiation pattern is generated in beam frame
• Therefore radiation patterns, at low field strength, can be largely copied from
textbooks
• Note on terminology: Some authors call any scattering of photons by free
electrons Compton Scattering. Compton observed (the so-called Compton
effect) frequency shifts in X-ray scattering off (resting!) electrons that
depended on scattering angle. Such frequency shifts arise only when the
energy of the photon in the rest frame becomes comparable with 0.511 MeV.
Krafft, G. A. and G. Priebe, Rev. Accel. Sci. and Tech., 3, 147 (2010)
USPAS Accelerator Physics January 2020
Simple Kinematics
Beam Frame Lab Frame
e-z
cos1' 22 LLpe EmcEmcpp
2' ,0ep mc
' ' , 'p L Lp E E
2 ˆ, ep mc z
ˆ ˆ1,sin cosp Lp E y z
USPAS Accelerator Physics January 2020
cos1' LL EE
In beam frame scattered photon radiated with wave vector
'cos,'sin'sin,'cos'sin,1'
' c
Ek L
Back in the lab frame, the scattered photon energy Es is
cos1
'' cos1'
L
Ls
EEE
1 cos
1 coss LE E
USPAS Accelerator Physics January 2020
Electron in a Plane Wave
Assume linearly-polarized pulsed laser beam moving in the
direction (electron charge is –e)
xAxzyctAtxA xincˆˆcossin,
0,0,1,0
Polarization 4-vector (linearly polarized)
Light-like incident propagation 4-vector
zynincˆcosˆsin
0 incincinc nnn
cos,sin,0,1incn
Krafft, G. A., Physical Review Letters, 92, 204802 (2004), Krafft, Doyuran, and Rosenzweig , Physical Review E, 72, 056502 (2005)
USPAS Accelerator Physics January 2020
Electromagnetic Field
d
dAnn
x
A
x
AAAF
incinc
Our goal is to find xμ(τ)=(ct(τ),x(τ),y(τ),z(τ)) when the 4-velocity
uμ(τ)=(cdt/dτ,dx/dτ,dy/dτ,dz/dτ)(τ) satisfies duμ/dτ= –eFμνuν/mc
where τ is proper time. For any solution to the equations of motion.
ununuFn
d
undincincinc
inc 0
Proportional to amount frequencies up-shifted
in going to beam frame
USPAS Accelerator Physics January 2020
ξ is proportional to the proper time
fd
dcun
d
dfc
d
udinc
On the orbit
Integrate with respect to ξ instead of τ. Now
where the unitless vector potential is f(ξ) = ˗eA(ξ )/mc.
ucfu
unddxnct incinc /
USPAS Accelerator Physics January 2020
Electron Orbit
inc
inc
inc
inc
nun
fcn
un
ucfuu
2
22
'2
'
''
2
2
2
2
df
un
nc
dfun
cn
un
uc
un
ux
inc
inc
inc
inc
incinc
Direct Force from Electric Field Ponderomotive Force
USPAS Accelerator Physics January 2020
Energy Distribution
2
2 2
3 3
; , sin
sin; , cos32
1 cos
t
p
DdE e
Dd d c
2
2 2
3 3
cos ; , cos
1 cos
sin cos; , sin
32 1 cos 1 cos
cos sin ; ,
1 cos 1 cos
t
p
p
D
dE eD
d d c
D
USPAS Accelerator Physics January 2020
Effective Dipole Motions: Lab Frame
, ; ,1; ,
1 cos
i
t
eAD e d
mc
2 2
, ; ,
2 2
1; ,
1 cos 2
i
p
e AD e d
m c
And the (Lorentz invariant!) phase is
2 2
2 2 22
'sin cos '
1 cos, ; ,
'1 sin sin sin cos cos'
2
1 cos
1
1 co
c
s
os
eAd
mc
c e Ad
m c
USPAS Accelerator Physics January 2020
Summary
• Overall structure of the distributions is very like that from the general dipole motion,
only the effective dipole motion, including physical effects such as the relativistic
motion of the electrons and retardation, must be generalized beyond the straight
Fourier transform of the field
• At low field strengths (f <<1), the distributions reduce directly to the classical
Fourier transform dipole distributions
• The effective dipole motion from the ponderomotive force involves a simple
projection of the incident wave vector in the beam frame onto the axis of interest,
multiplied by the general ponderomotive dipole motion integral
• The radiation from the two transverse dipole motions are compressed by the same
angular factors going from beam to lab frame as appears in the simple dipole case.
The longitudinal dipole radiation is also transformed between beam and lab frame
by the same fraction as in the simple longitudinal dipole motion. Thus the usual
compression into a 1/γ cone applies
USPAS Accelerator Physics January 2020
Weak Field Thomson Backscatter
With Φ = π and f <<1 the result is identical to the weak field
undulator result with the replacement of the magnetic field
Fourier transform by the electric field Fourier transform
Undulator Thomson Backscatter
Driving Field
Forward
Frequency
zzx cE 1/cos1~
1 cos /y z zcB c
2
0
2
2
0
4
Lorentz contract + Doppler Double Doppler
USPAS Accelerator Physics January 2020
dN/dEγ
Eγ0 (1+β)γ2Elaser (1+β) 2γ2Elaser
θ = π
sin θ = 1/ γ
θ = 0
Compton
Edge
Photon Number Spectrum
USPAS Accelerator Physics January 2020
For a flat incident laser pulse the main results are very similar
to those from undulaters with the following correspondences
Undulator Thomson Backscatter
Field Strength
Forward
Frequency
a
'cos* z
21
2
2
2
0 K
21
4
2
2
0 a
Transverse Pattern 'cos1
K
NB, be careful with the radiation pattern, it is the same at small
angles, but quite a bit different at large angles
High Field Thomson Backscatter
USPAS Accelerator Physics January 2020
Forward Direction: Flat Laser Pulse
20-period
equivalent undulator: 000 20/2cos AAx
2 2 2
0 0 0 01 2 / 4 2 / , /z c c a eA mc
USPAS Accelerator Physics January 2020
2/1/1 2a
USPAS Accelerator Physics January 2020
Realistic Pulse Distribution at
High a
In general, it’s easiest to just numerically integrate the lab-
frame expression for the spectrum in terms of Dt and Dp. A
105 to 106 point Simpson integration is adequate for most
purposes. Flat pulses reproduce previously known results and
to evaluate numerical error, and Gaussian amplitude
modulated pulses.
One may utilize a two-timing approximation (i.e., the laser
pulse is a slowly varying sinusoid with amplitude a(ξ)), and
the fundamental expressions, to write the energy distribution
at any angle in terms of Bessel function expansions and a ξ
integral over the modulation amplitude. This approach
actually has a limited domain of applicability (K,a<0.1)
USPAS Accelerator Physics January 2020
Forward Direction: Gaussian Pulse
0
2
0
2 /2cos156.82/exp zAA peakx
Apeak and λ0 chosen for same intensity and same rms pulse length
as previously
/peak peaka eA mc
USPAS Accelerator Physics January 2020
Radiation Distributions: Backscatter
-0.05
-0.025
0
0.025
0.05
-2
0
2
0
5 10-42
1 10-41
1.5 10-41
dE
d d
-0.05
-0.025
0
0.025
0.05
0
5 10-42
1 10-41
1.5 10-41
dE
d d
-0.5
0
0.5
x
-1
0
1
y
0
5 10-42
1 10-41
1.5 10-41
dE
d d
-0.5
0
0.5
x
0
5 10-42
1 10-41
1.5 10-41
dE
d d
Gaussian Pulse σ at first harmonic peak
Courtesy: Adnan Doyuran (UCLA)
USPAS Accelerator Physics January 2020
Radiation Distributions: Backscatter
-0.05
-0.025
0
0.025
0.05
-2
0
2
0
5 10-42
1 10-41
1.5 10-41
dE
d d-0.05
-0.025
0
0.025
0.05
0
5 10-42
1 10-41
1.5 10-41
dE
d d
Gaussian π at first harmonic peak
-0.5
0
0.5
x
-0.5
0
0.5
y
0
5 10-42
1 10-41
1.5 10-41
dE
d d
-0.5
0
0.5
x
0
5 10-42
1 10-41
1.5 10-41
dE
d d
Courtesy: Adnan Doyuran (UCLA)
USPAS Accelerator Physics January 2020
Radiation Distributions: Backscatter
Gaussian σ at second harmonic peak
-2
0
2
x
-2
0
2
y
0
2 10-43
4 10-43
6 10-43
dE
d d
-2
0
2
x
Courtesy: Adnan Doyuran (UCLA)
USPAS Accelerator Physics January 2020
90 Degree Scattering
22 2
3 3
1; , sin ; , cos
32t p
dE eD D
d d c
2
2 2
3 3
cos ; , cos
1 cos
1 cos; , sin
32 1 cos
sin ; ,
1 cos
t
p
p
D
dE eD
d d c
D
USPAS Accelerator Physics January 2020
90 Degree Scattering
, ; ,1
; ,i
t
eAD e d
mc
2 2
, ; ,
2 2
1; ,
2
i
p
e AD e d
m c
And the phase is
2 2
2 2 2
'sin cos1 cos '
, ; ,'1 sin sin
'2
eAd
mc
c e Ad
m c
USPAS Accelerator Physics January 2020
Radiation Distribution: 90 Degree
Gaussian Pulse σ at first harmonic peak
-0.005
0
0.005
-2
0
2
0
2 10-42
4 10-42
6 10-42
8 10-42
dE
d d
-0.005
0
0.005
-0.5
0
0.5
x
-0.5
0
0.5
y
0
2 10-42
4 10-42
6 10-42
8 10-42
dE
d d
-0.5
0
0.5
x
0
2 10-42
4 10-42
6 10-42
8 10-42
dE
d d
Courtesy: Adnan Doyuran (UCLA)
USPAS Accelerator Physics January 2020
Radiation Distributions: 90 Degree
Gaussian Pulse π at first harmonic peak
-0.004
-0.002
0
0.002
0.004
-2
0
2
0
2 10-42
4 10-42
6 10-42
8 10-42
dE
d d
-0.004
-0.002
0
0.002
0.004
-0.4
-0.2
0
0.2
0.4
x
-0.4
-0.2
0
0.2
0.4
y
0
2 10-42
4 10-42
6 10-42
8 10-42
dE
d d
-0.4
-0.2
0
0.2
0.4
x
Courtesy: Adnan Doyuran (UCLA)
USPAS Accelerator Physics January 2020
Polarization Sum: Gaussian 90 Degree
-0.5
0
0.5
x
-0.5
0
0.5
y
0
2 10-42
4 10-42
6 10-42
8 10-42
dE
d d
-0.5
0
0.5
x
0
2 10-42
4 10-42
6 10-42
8 10-42
dE
d d
Courtesy: Adnan Doyuran (UCLA)
USPAS Accelerator Physics January 2020
Radiation Distributions: 90 Degree
Gaussian Pulse second harmonic peak
-1
-0.5
0
0.5
1
x
-1
-0.5
0
0.5
1
y
05 10
-461 10
-451.5 10
-452 10
-45
dE
d d
-1
-0.5
0
0.5
1
x
-1
-0.5
0
0.5
1
x
-1
-0.5
0
0.5
1
y
0
5 10-46
1 10-45
1.5 10-45
dE
d d
-1
-0.5
0
0.5
1
x
Second harmonic emission on axis from ponderomotive dipole!
Courtesy: Adnan Doyuran (UCLA)
USPAS Accelerator Physics January 2020
Compensating the Red Shift
• Add possibility of frequency modulation
(Ghebregziabher, et al.)
• Wave phase
• Wave (angular) frequency and wave number
cos 2 /xA a f
2 /f
kt z
USPAS Accelerator Physics January 2020
Double Doppler to Constant Frequency
• In beam frame
• Doppler shift to lab frame (θ=0)
• Compensation occurs when
* * * *
2 2
*2 *2 2
2
1 1
11 1 / 1
2 2
k k
a a
2 2* *2 21 1
d dC
d d
2
2
0 0
1/ 2
1 / 2f a d
a
USPAS Accelerator Physics January 2020
Compensation By “Chirping”
Terzic, Deitrick, Hofler, and Krafft,
Phys. Rev. Lett., 112, 074801 (2014)
USPAS Accelerator Physics January 2020
Hajima, et al.
Hajima, et al., NIM A, 608, S57 (2009)
TRIUMF Moly Source?
Uranium Detection
USPAS Accelerator Physics January 2020
THz Source
USPAS Accelerator Physics January 2020
Radiation Distributions for Short High-Field
Magnets
Krafft, G. A., Phys. Rev. ST-AB, 9, 010701 (2006)
0
2 2 2
0 0
0
/; , = exp , ; ,
/
1 1 ; , exp , ; ,
t
z
p
z z
a zD i z dz
a z
D i z dzz
And the phase is
0
0 0
0
2 2 2
0 0
1 cos 1 1, ; , '
'
' /sin cos '.
' /
z
z
z z z
z
z
zz dz
c c z
a zdz
c a z
USPAS Accelerator Physics January 2020
Wideband THz Undulater
Primary requirements: wide bandwidth and no motion and
deflection. Implies generate A and B by simple motion. “One
half” an oscillation is highest bandwidth!
2 2exp / 2x z z
2 2exp / 2eA z z
a z zmc
2
2 21 exp / 2peak
dA zB z B z
dz
peakeBK
mc
USPAS Accelerator Physics January 2020
THz Undulator Radiation Spectrum
USPAS Accelerator Physics January 2020
Total Energy Radiated
2 22 2 2 2
6 4 2
0 0
1 1
6 6
dE e e
dt c c
Lienard’s Generalization of Larmor Formula (1898!)
Barut’s Version22 2
3 2 2
0
1
6
d xdE e dt d x
d c d d d
2 22 2
2
0
1 cos6 2
e df f dfE d
d d
Usual Larmor term From ponderomotive
dipole
USPAS Accelerator Physics January 2020
Some Cases
2 22 2
06 2
e df f dfE d
d d
Total radiation from electron initially at rest
2 2
2 2
0
11 / 8
12
dE ea a
dt c
For a flat pulse exactly (Sarachik and Schappert)
USPAS Accelerator Physics January 2020
For Circular Polarization
Only other specific case I can find in literature completely
calculated has usual circular polarization and flat pulses. The orbits
are then pure circles
Sokolov and Ternov, in Radiation from Relativistic Electrons,
give
(which goes back to Schott and the turn of the 20th century!) and
the general formula checks out
2 2
2 2
0
' 1 '1
' 6
dE ea a
dt c
USPAS Accelerator Physics January 2020
Conclusions
• I’ve shown how dipole solutions to the Maxwell equations can be used to obtain
and understand very general expressions for the spectral angular energy
distributions for weak field undulators and general weak field Thomson
Scattering photon sources
• A “new” calculation scheme for high intensity pulsed laser Thomson Scattering
has been developed. This same scheme can be applied to calculate spectral
properties of “short”, high-K wigglers.
• Due to ponderomotive broadening, it is simply wrong to use single-frequency
estimates of flux and brilliance in situations where the square of the field
strength parameter becomes comparable to or exceeds the (1/N) spectral width of
the induced electron wiggle
• The new theory is especially useful when considering Thomson scattering of
Table Top TeraWatt lasers, which have exceedingly high field and short pulses.
Any calculation that does not include ponderomotive broadening is incorrect.
USPAS Accelerator Physics January 2020
Conclusions
• Because the laser beam in a Thomson scatter source
can interact with the electron beam non-colinearly with
the beam motion (a piece of physics that cannot happen
in an undulator), ponderomotively driven transverse
dipole motion is now possible
• This motion can generate radiation at the second
harmonic of the up-shifted incident frequency on axis.
The dipole direction is in the direction of laser incidence.
• Because of Doppler shifts generated by the
ponderomotive displacement velocity induced in the
electron by the intense laser, the frequency of the emitted
radiation has an angular asymmetry.
• Sum rules for the total energy radiated, which generalize
the usual Larmor/Lenard sum rule, have been obtained.