Nonlinear Thomson Scattering

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.

Operated by the Southeastern Universities Research Association for the U. S. Department of EnergyThomas Jefferson National Accelerator Facility

15 February 2005ODU Colloquium

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



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⎜ ⎟+⎝ ⎠+=



*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



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



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!



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



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



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



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



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



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

===



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.



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



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



Generalizes Coisson to arbitrary observation angles

Strong Field Case

0=γdtddt

Becmd rrr×= ββγ Becm

dt×−= ββγ

( ) ( ) ''2 dzzBmcez

z

x ∫∞−

=γ

β



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

⎠⎜⎜⎝⎠

⎜⎝

∫∞−



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



"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



=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



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.



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



( )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−

=



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



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



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



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 ∫∞−∞−

+



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



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⎝ ⎠⎟



( )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



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



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



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)



Forward Direction: Flat Pulse20 i d20-periodequivalent undulater: ( ) ( ) ( ) ( )[ ]000 20/2cos λξξλπξξ −Θ−Θ= AAx

( ) 200

20

220 / ,/24/21 mceAaccz =≈+≡ λπγλπγβω



( )2/1/1 2a+



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



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

Courtesy: Adnan Doyuran (UCLA)





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






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





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



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



⎠⎝

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





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





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





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!





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



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



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



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



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.



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.



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.



Nonlinear Thomson Scattering

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