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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 calculations in the general
case of a pulsed high field strength laser interacting with electrons in acase 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, are just c cu e u d b ce, b sed o co s p ude ode s, e jusplain wrong.
. The new theory is general enough to cover all “1-D” undulater calculations and all 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 detuning of the emission angle dependent Dopplerstrength squared), lead to a detuning of the emission, angle dependent Doppler shifts of the emitted scattered radiation, and additional transverse dipole emission that this theory can calculate.
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Ancient HistoryEarly 1960s: Laser Invented. 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
eB λeE λ
Interpreted frequency shifts that occur at high fields as a “relativistic mass hif ”
rs Undulato2 2
00
mceBKπλ
=SourcesThomson 2 2
00
mceEaπλ
=
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.y p g
. Alferov, Bashmakov, and Bessonov (1974): Undulater/Insertion Device theories developed under the assumption of constant field strength. Numerical codes developed to calculate “real” fields in undulaters.Coisson (1979): Simplified undulater theory which works at low K and/or a. Coisson (1979): Simplified undulater 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
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Coisson’s Spectrum from a Short Magnet
( ) ( )( ) 222r cdE
Coisson low-field strength undulater spectrum*
( ) ( )( )2 2 2222 2 22 1 / 21er cdE f B
d dγ γ θ
ν πγ θ γν+= +
Ω%
222 222πσ fff +=
1
( )22 2
2 2
1 sin
1 1
1f
f
σγ θ
γ θ
φ
φ
=
⎛ ⎞−⎜ ⎟
+
( )22 2 2 2 cos1 1
fπγ
γ
θφ
γ θ⎜ ⎟+⎝ ⎠+=
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*R. Coisson, Phys. Rev. A 20, 524 (1979)
Dipole Radiation
Assume a single charge moves in the x direction
( )( ) ( ) ( )zytdxetzyx δδδρ −=)( ( )( ) ( ) ( )zytdxetzyx δδδρ −=),,,(
( ) ( )( ) ( ) ( )zytdxxtdetzyxJ δδδ −= ˆ),,,( &r
Introduce scalar and vector potential for fields. Retarded solution to wave equation (Lorenz gauge), ( )'' trrR rr
−=
( ) 1 ( ' / ), ', ' ' ' 'R t t R cr t r t dx dy dz e dtR c R
δφ ρ − +⎛ ⎞= − =⎜ ⎟⎝ ⎠∫ ∫
r r
( ) ( ) ')/'('''','1, dtRc
cRtttdedzdydxcRtrJ
RctrA xx ∫∫
+−=⎟
⎠⎞
⎜⎝⎛ −=
δ&rr
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Dipole Radiation&&
Θx
ˆ
Φ( )2
/ ˆsined t r c
Bc r−
= ΘΦ&&r
Φz
y
r Θ( )2
/ ˆsined t r c
Ec r−
= ΘΘ&&r
y( ) r
rccrtdeBEcI ˆsin/
41
42
23
22
Θ−
=×
=&&
rr
ππ
( )Θ
−=
Ω2
3
22
sin/41
ccrtde
ddI &&
π
Polarized in the plane containing and
Ω 4 cd π
nr r=ˆ x
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Dipole RadiationD fi h F i T fDefine the Fourier Transform
( ) dtetdd ti∫ −= ωω )(~ ( ) ωωπ
ω dedtd ti∫=~
21)(π2
With these conventions Parseval’s Theorem is
( ) ( ) ωω dddttd ~1 22 ∫∫ =( ) ( ) ωωπ
dddttd2
∫∫ =
( ) ( ) ωωω ddedtcrtdedE ~/24
22
2
∫∫ == &&
242 )(~1 dedE ωω
( ) ( ) ωωωππ
ddc
dtcrtdcd 8
/4 323 ∫∫ =−=
Ω
Θ=Ω
232 sin
)(
81
cdddE
πωThis equation does not follow the typical (see Jackson) convention that combines both positive and
i f i h i i l i i f i l h i h ld lik
Blue Sky!
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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.
Dipole Radiation
( )2
42~
1 dnedE ωωrr
×
For a motion in three dimensions
( )328
1cdd
dEπω
=Ω
Vector inside absolute value along the magnetic fieldVector inside absolute value along the magnetic field
( ) ( ) ( )2
422
42~~
1~
1 ndndendnedErrrrrrr
⎠⎞⎜
⎝⎛ ⋅−×
⎠⎞⎜
⎝⎛ × ωωωωω ( ) ( ) ( )
3232 81
81
ccdddE ⎠⎝=⎠⎝=Ω ππω
Vector inside absolute value along the electric field To getVector inside absolute value along the electric field. To get energy into specific polarization, take scaler product with the polarization vector
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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 systemy
. For the remainder of the presentation, quantities referred to the moving coordinates will have primes; unprimed quantities refer to the lab system
ˆ 'ˆx
ˆ
'x
'ˆcβ ,z
y
'z
'y
czβ
. In the co-moving system the dipole radiation pattern applies
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New Coordinates
'Θ'x 'Φ 'x
'r'Φ
Θ
'Φ'z
'ˆ
'nr 'Θ'z
'ˆ
'nr'Θ'θ
'φ
Resolve the polarization of scattered energy into that perpendicular (σ)
Φ'y 'y
Resolve the polarization of scattered energy into that perpendicular (σ)and that parallel (π) to the scattering plane
'ˆ'cos'ˆ'sin'sin'ˆ'cos'sin' θφθφθ ++= zyxnr
'ˆ'ˆ'sin'ˆ'sin'cos'ˆ'cos'cos'ˆ''ˆ
'ˆ'ˆ'cos'ˆ'sin'ˆ'/'ˆ''ˆ
θθφθφθ
φφφ
φφ
σπ
σ
=−+=×=
−=−=××=
zyxene
yxznzne
y
r
rr
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φφσπ y
Polarization
It follows that
( ) ( ) ( ) 'cos''~'sin''~'ˆ''~
φωφωω dded −=⋅r( ) ( ) ( )
( ) ( ) ( ) ( )''~'sin'sin'cos''~'cos'cos''~'ˆ''~
cossin
ωθφθωφθωω
φωφωω
π
σ
zyx
yx
ddded
dded
−+=⋅r
So the energy into the two polarizations in the beam frame is
( ) ( ) 242'
'cos''~'sin''~'1 φωφωωσ ddedE−= ( ) ( )
( ) ( ) 242'
32
'sin'cos''~'cos'cos''~'1
cossin8''
φθωφθωω
φωφωπω
π yx
yx
ddedE
ddcdd
+
−=Ω
( )32''~'sin 8'' ωθπω
π
z
y
dcdd −=
Ω
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Comments/Sum Rule
. There is no radiation parallel or anti-parallel to the x-axis for x-dipole motionIn the forward direction the radiation polarization is parallel to the x0'→θ. In the forward direction , 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
0→θ
( ) ( ) ⎟⎠⎞⎜
⎝⎛ += 2''~''~'
81
'22
3
42
2
'
πωωωπω
σyx dd
ce
ddE
( ) ( ) ( ) ⎥⎦⎤
⎢⎣⎡ +⎟
⎠⎞⎜
⎝⎛ +=
38''~
32''~''~'
81
'222
3
42
2
' πωπωωωπω
πzyx ddd
ce
ddE
( )3
8''~
'
81
' 3
242
2
' πωω
πω c
de
ddEtot
r
= Generalized Larmor
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38πω cd
Sum Rule
Total energy sum rule
( )~ 2
42 dr( )
∫∞
∞−
= ''''
31
3
42
' ωωω
πd
c
deEtot
Parseval’s Theorem again gives “standard” Larmor formula
( ) ( )3
22
3
22' ''32''
32
''
ctae
ctde
dtdEP tot
r&&r
===
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Energy Distribution in Lab Frame ( ) ( )( ) 2
2242 sincos1'~cos1 φθβωγθβγω −ddE ( ) ( )( )
( )( )2
32
242
coscos1'~sincos1
8cos1
φθβωγ
φθβωγπ
θβγωω
σ
−−
−−=
Ω y
x
d
dc
edd
dE
( )
( )( )2
2242
coscos1
coscos1'~ φθββθθβωγ
−−
−xd
d ( ) ( )( )32
2242
sin
sincos1
coscos1'~8
cos1
θ
φθββθθβωγ
πθβγω
ωπ
−−
−+−
=Ω yd
ce
dddE
B l i th i f th D l hift d f d l i id th
( ) ( )( )cos1'~cos1
sin θβωγθβγ
θ−
−− zd
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.
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Bend Undulater Wiggler
e–
e– e–ωh eωh ωh
%bw
]
ss 0.1%
bw]
%bw
]
Flux
[ph/
s/0.
1
Brig
htne
sh/
s/m
m2 /m
r2 /0
Flux
[ph/
s/0.
1
white source partially coherent source powerful white source
ωh
[ph
ωh ωh
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Weak Field Undulater Spectrum
( ) ( ) ( ) xcBmcecxdd z ˆ
'/'~
ˆ''~''~
22 ωγβωωω −==
r( ) ( ) dzezBkB ikz−∫=
~
( )( )( )
24
22 2 5 22
1 sin8
1 cos /z zBdE ed d
cσ
ωφ
β θ β=
Ω
−%
( )
( )( )
2 2 5
2 242
228
c
1 cos
os1 1 c /osz zz
zd d m c
BdE e cπ
φγ β θω π
ω θ β φβ θ β
Ω
⎛ ⎞−⎜ ⎟
−
−% ( )( )
22 222 5 cos
8 1 co1 cos sz
zzd d m c
π
γβ φ
βω π βθ θ= ⎜ ⎟Ω −⎝ ⎠−
42 e 0λ ( )( )
222 11 θγθβθβ +
422
cmere ≡ 2
0
2γλλ = ( )( ) 2
22
111cos1γθγθ
γβθβ +
≈++≈+− Kzz
Generalizes Coisson to arbitrary observation angles
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Generalizes Coisson to arbitrary observation angles
Strong Field Case
0=γdtddt
Becmd rrr×= ββγ Becm
dt×−= ββγ
( ) ( ) ''2 dzzBmcez
z
x ∫∞−
=γ
β
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High K
( ) ( )zz xz2
2
11 βγ
β −−=γ
2⎞⎛ z
( ) ( )22 ''11 ⎟⎟⎠
⎞⎜⎜⎝
⎛−−= ∫
∞−
dzzBmcez
z
z γγβ
( ) ( ) ( )zkKKdzzBezz
z 02
22
2
2
22 2cos111''111β −⎞
⎜⎜⎛+−=
⎞⎜⎜⎛
−−≈ ∫( ) ( ) ( )kdmcz 02222 cos
42222 γγγγβ
⎠⎜⎜⎝⎠
⎜⎝
∫∞−
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High K
Inside the insertion device the average (z) velocity is
⎞⎜⎛1 2K
⎟⎠
⎞⎜⎜⎝
⎛+−=
21
211* 2
Kz γ
β
with corresponding
1 γ2/1*1
1*22 Kz +
=−
=γ
βγ
To apply dipole distributions, must be in this frame to begin with
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"Figure Eight" Orbits
0
0.01
-0.00001 -0.000005 0 0.000005 0.00001
-0 03
-0.02
-0.01
x
K=0.5
K=1
K=2
-0.05
-0.04
-0.03
z
100 di t li d b λ / 2γ
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=100, distances are normalized by λ0 / 2πγ
Therefore
. Coisson’s Theory may be generalized to arbitrary observation angles by using the proper polarization decompositionthe proper polarization decomposition
. All kinematic parameters, including the angular distribution functions and frequency distributions, are just the same as in the weak field case, except unstarred quantities should be replaced by starred quantities
. In particular, the (FEL) resonance condition becomes
⎞⎜⎛ 2Knλ
⎟⎠
⎞⎜⎜⎝
⎛+=
21
2 20 Kn
n γλλ
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Thomson ScatteringP l “ l i l” tt i f h t b l t. 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 electronquantum 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 framep p g
. 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 S ch freq enc shifts arise onl hen thedepended on scattering angle. Such frequency shifts arise only when the energy of the photon in the rest frame becomes comparable with 0.511 MeV.
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Simple Kinematics
e-z ββ =
r
Φ θ
B F L b FBeam Frame Lab Frame
( )0,' 2mcp e =µ( )zmcpe ˆ ,2 γβγµ =( ),p eµ
( )LLp EEp ',''r
=µ( )zyEp Lp ˆcosˆsin,1 Φ+Φ=µ
( )Φ−==⋅ cos1' 22 βγLLpe EmcEmcpp
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( )Φ−= cos1' βγLL EE
b f d h di d i hIn beam frame scattered photon radiated with wave vector
( )'cos,'sin'sin,'cos'sin,1'' θφθφθEk L= ( )cos,sinsin,cossin,1 θφθφθµ ck
Back in the lab frame, the scattered photon energy Es is
( ) ( )θβγθβγ
cos1'' cos1'
−=+= L
LsEEE
( )( )1 cos1 coss LE E
ββ θ
− Φ−
=
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Electron in a Plane WaveAssume linearly polarized pulsed laser beam moving in theAssume linearly-polarized pulsed laser beam moving in thedirection (electron charge is –e)
zyninc ˆcosˆsin Φ+Φ=r
( ) ( ) ( )xAxzyctAtxA xinc ˆˆcossin, ξ≡Φ−Φ−=rr
P l i ti 4 t
yinc
( )0,0,1,0=µε
Polarization 4-vector
Light-like incident propagation 4-vector
( )ΦΦ= cossin01µn
0=⋅==⋅ incincinc nnn rrεεε µ
µ
( )ΦΦ= cos,sin,0,1incn
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Electromagnetic Field
εε µνµννµµν AAAAF∂∂
−∂∂
=∂−∂=
( ) ( )ξεε νµµν
νµ
dAnn
xx
incinc −=
∂∂
( ) ( )ξξdincinc
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= –eFµνu /mcuµ(τ) (cdt/dτ,dx/dτ,dy/dτ,dz/dτ)(τ) satisfies duµ/dτ eFµ uν/mc where τ is proper time. For any solution to the equations of motion.
( ) ( )µµµνµ
µund inc( ) ( )∞−=∴== µµ
µµν
µνµ
µ
τununuFn
d incincincinc 0
Proportional to amount frequencies up-shifted i t b f
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going to beam frame
ξ is exactly proportional to the proper time!
On the orbit
( ) ( ) ( ) µτξτττξ unddxnct ii =⋅−= /rr
( )Integrate with respect to ξ instead of τ. Now
( ) ( ) ( ) µτξτττξ unddxnct incinc /
( ) ( )( )τξτξτ
ε µµ
µµ f
ddcun
ddfc
dud
inc ==
where the unitless vector potential is f(ξ)=-eA(ξ )/mc2.
( )∞−=−∴ µµ
µµ εε ucfu
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Electron Orbit
( ) ( ) ( ) ( )( )
( )( )( )
µν
µµν
ννµµ ξεεξξ inc
iinc
i
nunfcn
unucfuu
∞−+
⎭⎬⎫
⎩⎨⎧
−∞−∞−
+∞−=2
22
( ) ( )( )νν incinc unun ∞⎭⎩ ∞ 2
Direct Force from Electric Field Ponderomotive Force
( ) ( ) ( ) ( ) '' ξξεεξξξµ
µν
νµ
µ dfcucu∫
⎪⎬⎫⎪
⎨⎧ ∞−∞−
Direct Force from Electric Field Ponderomotive Force
( ) ( )( )
( )( )( ) ( ) ( )
( )'
''
22
2
ξ
ξξξξ
ξµ
νν
µ
νν
νν
ν
µ
f
dfun
nunun
xinc
inc
incinc∫∞−⎪⎭
⎪⎬
⎪⎩
⎪⎨ ∞−
−∞−
+∞−
=
( )( )( ) '2
' 2
2
2
ξξξ
νν
µ
dfun
nc
inc
inc ∫∞−∞−
+
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Energy Distribution
( )
( )
2
32
22
cos;sinsin,;
8 φφθω
φφθωωσ
t
D
De
dddE
Φ=Ω ( ) ( )32 cos,;
cos18 φφθω
βγπω pDcdd
Φ−−Ω
2
( )2
coscos1
cos,; φθββθφθω
−−
tD
( ) ( )32
22
sincos1
cos,;cos1
sin8
φθββθφθω
βγπω
ωπ
−−
Φ−Φ
+=Ω pD
ce
dddE
( ) ( )cos1sin,;
cos1cos
θβγθφθω
ββ
−Φ−Φ−
+ pD
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Effective Dipole Motions: Lab Frame
( ) ( )( ) ( ) ξξ
βγφθω φθξωϕ de
mceAD i
t ∫Φ−= ,;,
2cos11,; ( )βγ
( ) ( )( ) ( ) ξξφθω φθξωϕ deAeD i∫= ,;,
221;( ) ( ) ξβγ
φθω decm
Dp ∫Φ− 422cos1,;
And the (Lorentz invariant!) phase is
( )
( )( ) ( )
( )2
'sin cos '1 cos
1 cos1 soc
eAd
mc
ξ ξθ φξ ξγ βωξ θ φ
β θβ −∞
⎛ ⎞− ∫⎜ ⎟− Φ⎜ ⎟
⎜ ⎟
−− Φ
( )
( )( )2 2
2 2 42
, ; ,'1 sin sin sin cos cos '
21 sco
c e Ad
m c
ξϕ ω ξ θ φ
ξθ φ θ ξγ β −∞
⎜ ⎟= ⎜ ⎟− Φ− Φ⎜ ⎟+ ∫⎜ − Φ⎝ ⎠⎟
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( )γ β⎝ ⎠
Summary
. Overall structure of the distributions is very like that from the general dipole motion only the effective dipole motion incuding physical effects such as themotion, only the effective dipole motion, incuding 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 integralinterest, 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 d po e case. e o g tud a d po e ad at o s a so t a s o ed betweebeam and lab frame by the same fraction as in the simple longitudinal dipole motion. Thus the usual compression into a 1/γ cone applies
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Weak Field Thomson BackscatterWith Φ = π and f <<1 the result is identical to the weak field undulater result with the replacement of the magnetic field Fourier transform by the electric field Fourier transform
Undulator Thomson Backscatter
( ) ( )( )( )~( )( )~Driving Field
ForwardF
( ) ( )( )( )zzx cE βθβω +− 1/cos1( )( )zzy cB βθβω /cos1−
20
2λλ ≈ 2
0
4λλ ≈
Frequency 22γ 24γ
Lorentz contract + Doppler Double Doppler
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High Field Strength Thomson Backscatter
For a flat incident laser pulse the main results are very similar to those from undulaters with the following correspondences
Undulater Thomson Backscatter
Field Strength
ForwardF
a
⎞⎜⎜⎛+≈ 1
2
20 Kλλ
⎞⎜⎜⎛+≈ 1
2
20 aλλ
K
Frequency
'cos* θβ +z
⎠⎜⎝ 22 2γ ⎠
⎜⎝ 24 2γ
Transverse Pattern 'cos1 θ+
NB, be careful with the radiation pattern, it is the same at small angles, but quite a bit different at large angles
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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 Dx , Dy, and Dz. A 105 to 106 point Simpson integration is adequate for most purposes We’ve done two typesSimpson integration is adequate for most purposes. We ve done two types of pulses, flat pulses to reproduce the previous results and to evaluate numerical error, and Gaussian Laser 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 y y g p ( ))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)
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Forward Direction: Flat Pulse20 i d20-periodequivalent undulater: ( ) ( ) ( ) ( )[ ]000 20/2cos λξξλπξξ −Θ−Θ= AAx
( ) 200
20
220 / ,/24/21 mceAaccz =≈+≡ λπγλπγβω
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( )2/1/1 2a+
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Forward Direction: Gaussian Pulse
( ) ( )( ) ( )02
02 /2cos156.82/exp λπξλξ zAA peakx −=
Apeakpeak andand λλ00 chosen for same intensity and samechosen for same intensity and same rmsrms pulse length as previous slidepulse length as previous slide
2/ mceAa peakpeak =
Apeakpeak and and λλ00 chosen for same intensity and same chosen for same intensity and same rmsrms pulse length as previous slidepulse length as previous slide
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Radiation Distributions: Backscatter
Gaussian Pulse σ at first harmonic peak
-0.052
0510-42110-411.510-41
dEdd0.05 0
511
1
0510-42110-41
1.510-41
dEdd0511
-0.025
0
0.025
-2
0
2
-0.025
0
0.025
-0.5
0
0.5
x
-1
0y
-0.5
0
0.5
x
0.050 05
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Radiation Distributions: Backscatter
Gaussian π at first harmonic peak
-0.052
0510-42110-411.510-41
dEdd05 0
51
0 5 0 5
0
510-42
110-41
1.510-41
dEdd
0 5
0
51
1
-0.025
0
0.025
-2
0
2
-0.025
0
0.025
-0.5
0
0.5
x
-0.5
0
0.5
y
-0.5
0
0.5
x
0.050 05
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Radiation Distributions: Backscatter
Gaussian σ at second harmonic peak
-2 20
210-43410-43
610-43
dEdd
-22
0
2
x
2
0
2
y
2
0
2
x
2 -22
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90 Degree Scattering
( ) ( )2
32
22
cos,;1sin,;8
φφθωγ
φφθωωω
σpt DDe
dddE
−=Ω 328 γπω pcdd Ω
2
( )2
coscos1
cos,; φθββθφθω
−−
tD
( )32
22
sincos1
cos,;18
φθββθφθω
γπω
ωπ
−−
+=Ω pD
ce
dddE
( ) ( )cos1sin,;
θβγθβφθω
−+ pD
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90 Degree Scattering
( ) ( ) ( ) ξξγ
φθω φθξωϕ demc
eAD it ∫= ,;,
2
1,;γ
( ) ( ) ( ) ξξφθω φθξωϕ deAeD i∫= ,;,221;( ) ξ
γφθω de
cmDp ∫ 422
,;
And the phase is
( )( ) ( )
⎟⎞
⎜⎜⎛
∫−−∞−
ξξξ
γφθθβξ
ωφθξ''cossincos1 2 d
mceA
( )( )
⎟⎟⎠
⎜⎜⎜⎜
⎝∫
−+
=
∞−
ξξξ
γφθ
φθξωϕ'
2'sinsin1
,;,
42
22
2 dcm
Aec
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⎠⎝
Radiation Distribution: 90 Degree
Gaussian Pulse σ at first harmonic peak
2
0210-42
410-42
610-42
810-42
dEdd
-0.5 0.50210-42
410-42
610-42
810-42
dEdd
-0.5021
4
6
-0.005
0
0.005
2
0
2
-0.005
0
0.005
0
x0
y0
x
-20.5 -0.50.5
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Radiation Distributions: 90 Degree
Gaussian Pulse π at first harmonic peak
42810-42
-0.0042
0210-42
410-42
610-42
810-42
dEdd
0.004-0.4
0 2 0 2
0.40
210-42
410-42
610-42
dEdd
-0.4
0 2-0.002
0
0.002
0
2
-0.002
0
0.002
-0.2
0
0.2
0 4
x
0 4
-0.2
0
0.2
y
-0.2
0
0.2
0 4
x
0.004-2
0.004
0.4 -0.40.4
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Polarization Sum: Gaussian 90 Degree
42
810-42
-0.5 0.50210-42
410-42
610-42
dEdd
-0.50214
6
0x
0y
0x
0.5 -0.50.5
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Radiation Distributions: 90 Degree
Gaussian Pulse second harmonic peak
1 510-45210-45
110-45
1.510-45
-1
-0.5
0x
0
0.5
1
y
0510-46110-45
1.510dE
dd
-1
-0.5
0x
-1
-0.5
0x
0
0.5
1
y
0510-46dEdd
-1
-0.5
0x
0.5
1
x
-1
-0.5y
0.5
1
x0.5
1
x
-1
-0.5y
0.5
1
x
σ πSecond harmonic emission on axis from ponderomotive dipole!
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Total Energy RadiatedLi d’ G li i f L F l (1898!)
⎥⎤
⎢⎡ ⎞⎜⎛+⎞⎜⎛⎥
⎤⎢⎡ ⎞⎜⎛+⎞⎜⎛
22
24
2226
2 22 ββββββ &rr&r&rr&r eedE
Lienard’s Generalization of Larmor Formula (1898!)
⎥⎦
⎢⎣ ⎠
⎞⎜⎝⎛ ⋅+
⎠⎞⎜
⎝⎛=⎥
⎦⎢⎣ ⎠
⎞⎜⎝⎛ ×+
⎠⎞⎜
⎝⎛= 246
33ββγβγβββγ
ccdt
B t’ V iBarut’s Version
2
2
2
2
3
2
32 µ
µ
dxd
dxd
ddte
ddE
= 2233 ττττ dddcd
( ) ξβγ ddffdfeE ∫∞
⎥⎤
⎢⎡ ⎞
⎜⎜⎛
+⎞
⎜⎜⎛
Φ=222
22
cos12 ( ) ξξξ
βγ ddd
E ∫∞− ⎥
⎥⎦⎢
⎢⎣ ⎠
⎜⎜⎝
+⎠
⎜⎜⎝
Φ−=2
cos13
Usual Larmor term From ponderomotivedi l
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dipole
Some Cases
dffdfe∫∞
⎥⎤
⎢⎡ ⎞
⎜⎛⎞
⎜⎛
22222
Total radiation from electron initially at rest
ξξξ
dddff
ddfeE ∫
∞− ⎥⎥⎦⎢
⎢⎣
⎟⎠
⎞⎜⎜⎝
⎛+⎟
⎠
⎞⎜⎜⎝
⎛=
232
For a flat pulse exactly (Sarachik and Schappert)
( )8/131 22
22
aaedtdE
+=ω ( )
3 cdt
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For Circular Polarization
( ) ( ) ( ) ( )[ ] zyxAAinc ˆsinˆcos/2sinˆ/2cos Φ+Φ−±= λπξλπξξξr
( )( ) βγγ µµ
AfuneE inc ⎥⎦
⎤⎢⎣
⎡+Φ−∞−= ∫
∞
∞−±
22
2
ˆsin
32
ξλπ
ξdA
dAd
⎥⎥⎤
⎢⎢⎡
⎠⎞
⎜⎝⎛+⎟
⎞⎜⎜⎛
×
⎦⎣∞
222
ˆ2ˆ 2/ˆ mceAA −= ξ
λξd ⎥⎦⎢⎣ ⎠⎜⎝⎠
⎜⎝
O l ifi I fi d i lit t l t l l l t d h
/ mceAA =
Only specific case I can find in literature completely calculated has sin Φ = 0 and flat pulses (dA/dξ = 0). The orbits are then pure circles
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F l i i iddl f lFor zero average velocity in middle of pulse
( ) ( ) ( )ˆ
1/ˆ 2
22 AcunAcn
incinc +=∞−→−=∞− ν
νγβγrr
Sokolov and Ternov, in Radiation from Relativistic Electrons,
( ) ( ) ( )22un inc
inc ∞− ννν
γβγ
, f ,give
( )2222
1'2' aaedE+=
ω
and the general formula (which goes back to Schott and the turn
( )13'
aacdt
+=
g ( gof the 20th century!) checks out
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Conclusions
. Recent development of superconducting cavities has enabled CW operation at energy gains in excess of 20 MV/m, and acceleration of average beam currents of 10s of mA.
. The ideas of Beam Recirculation and Energy Recovery have been introduced; how these concepts may be combined to yield a new class of accelerators that can be used in many interesting applications has been discussed. I’ve given you some indication about the historical development of recirculating SRFyou some indication about the historical development of recirculating SRF linacs.
. The present knowledge on beam recirculation and its limitations in a superconducting environment, leads us to think that recirculating accelerators p g , gof several GeV energy, and with beam currents approaching those in storage ring light sources, are possible.
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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 undulaters and general weak field Thomson Scatteringdistributions for weak field undulaters 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 theparameter 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 p , g y g p ycalculation that does not include ponderomotive broadening is incorrect.
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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 undulater), 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. 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 gy , gsum rule, have been obtained.
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