General Theory of Intense Beam Nonlinear Thomson ScatteringG. A. Krafft Jefferson Lab A. Doyuran UCLA Thomas Jefferson National Accelerator Facility CASA Beam Physics Seminar 4 February

Thomas Jefferson National Accelerator Facility

CASA Beam Physics Seminar 4 February 2005

General Theory of Intense Beam Nonlinear Thomson Scattering

G. A. KrafftJefferson Lab

A. DoyuranUCLA



Outline

1. Ancient History2. Review of Thomson Scattering

1. Process2. Simple Kinematics3. Dipole Emission from a Free Electron

3. Solution for Electron Motion in a Plane Wave1. Equations of Motion2. Exact Solution for Classical Electron in a Plane Wave

4. Applications to Scattered Spectrum1. General Solution for Small a2. Finite a Effects3. Ponderomotive Broadening4. Sum Rules

5. Conclusions



Whats New in this Work. Many of the the newer Thomson Sources are based on a PULSED Laser (e.g.

all of the high-energy single-pulse lasers are pulsed by their very nature). Previously developed a general theory to cover the calculations in the general

case of a pulsed, high field strength laser interacting with electrons in a Thomson backscatter arrangement. Have extended this theory to cover more general scattering geometries

. 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 plain 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 Doppler shifts of the emitted scattered radiation, and additional transverse dipole emission that this theory can calculate.



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 Im 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): 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, developed to understand the frequency distribution of edge emission, or emission from short magnets, i.e., including pulse effects

rs Undulato2 2

00

mceBKπ

λ=SourcesThomson 2 2

00

mceEaπ

λ=



Coissons Spectrum from a Short Magnet

( ) ( )( )2222222222

2/1~1 γθγνθγγπν

++=Ω

Bfcrdd

dE e

Coisson low-field strength undulater spectrum*

222πσ fff +=

( )

( ) φθγθγ

θγ

φθγ

π

σ

cos11

11

sin1

1

22

22

222

222

+−

+=

+=

f

f

*R. Coisson, Phys. Rev. A 20, 524 (1979)



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.



Simple Kinematics

Beam Frame Lab Frame

e- z ββ =!

Φ

( )Φ−==⋅ cos1' 22 βγLLpe EmcEmcpp

( )0,' 2mcp e =µ

( )LLp EEp ','' !=µ

( )zmcpe ,2 γβγµ =

( )zyEp Lp cossin,1 Φ+Φ=µ

θ



( )Φ−= cos1' βγLL EE

In beam frame scattered photon radiated with wave vector

( )'cos,'sin'sin,'cos'sin,1'' θφθφθµ cEk L=

Back in the lab frame, the scattered photon energy Es is

( ) ( )θβγθβγ cos1'' cos1'

−=+= LLs

EEE

( )( )θβ

βcos1cos1

−Φ−= Ls EE



Cases explored

Backscattered

Provides highest energy photons for a given beam energy, or alternatively, the lowest beam energy to obtain a given photon wavelength. Pulse length roughly the ELECTRON bunch length

( )( ) 0at 4cos1

1 2 =≈−

+= θγθβ

βL

z

zLs EEE

π=Φ



Cases explored, contd.

Ninety degree scattering

2/π=Φ

( ) 0at 2cos11 2 =≈

−= θγ

θβ LzLs EEE

Provides factor of two lower energy photons for a given beam energy than the equivalent Backscattered situation. However, very useful for making short X-ray pulse lengths. Pulse length a complicated function of electron bunch length and transverse size.



Cases explored, contd.

Small angle scattered (SATS)

1



Dipole Radiation

Assume a single charge moves in the x direction

( )( ) ( ) ( )zytdxetzyx δδδρ −=),,,(

( ) ( )( ) ( ) ( )zytdxxtdetzyxJ δδδ −= ),,,( "!

Introduce scalar and vector potential for fields. Retarded solution to wave equation (Lorenz gauge),

( ) ( ) ')/'('''','1, dtRc

cRtttdedzdydxcRtrJ

RctrA xx ∫∫

+−=

−= δ

"!!

( ) ')/'(''','1, dtR

cRttedzdydxcRtr

Rtr ∫∫

+−=

−=Φ δρ !!

( )'' trrR !! −=



Dipole Radiation

Use far field approximation, r = | | >> d (velocity terms small)

Perform proper differentiations to obtain field and integrate byparts the delta function.

Long wave length approximation, λ >> d (source smaller than λ)

Low velocity approximation, (really a limit on excitation strength)

cd



Dipole Radiation

Θ

Φ

x

z

y

r Θ

Φ( ) ΦΘ−= sin/2rccrtdeB

""!

( ) ΘΘ−= sin/2rccrtdeE

""!

( ) rrc

crtdeBEcI sin/41

42

23

22

Θ−=×=""!!

ππ

Polarized in the plane containing and

( ) Θ−=Ω

23

22

sin/41

ccrtde

ddI ""

πnr != x



Dipole RadiationDefine the Fourier Transform

( ) dtetdd ti∫ −= ωω )(~ ( ) ωω

πω dedtd ti∫=

~21)(

Θ=Ω

23

242

2 sin)(~

81

c

de

dddE ωω

πωThis 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 Parsevals Theorem easily. By symmetry, the difference is a factor of two.

With these conventions Parsevals Theorem is

( ) ( ) ωωπ

dddttd ~21

22 ∫∫ =

( ) ( ) ωωωππ

ddc

edtcrtdc

eddE ~

8 /

424

32

22

3

2

∫∫ =−=Ω""

Blue Sky!



Dipole Radiation

( )3

242

2

~

81

c

dne

dddE ωω

πω

!!×=

Ω

For a motion in three dimensions

Vector inside absolute value along the magnetic field

( ) ( ) ( )3

242

23

242

2

~~

81

~

81

c

ndnde

c

ndne

dddE

!!!!!!!

⋅−

=×

×

=Ω

ωωω

π

ωω

πωVector 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 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

czβ



New Coordinates

Resolve the polarization of scattered energy into that perpendicular (σ) and that parallel (π) to the scattering plane

'Θ

'Φ

'x

'z'y

'n! 'Θ

'Φ 'x

'z'y

'n!'Θ

'Φ

'θ'φ

'''sin''sin'cos''cos'cos'''

'''cos''sin''/'''

''cos''sin'sin''cos'sin'

θθφθφθ

φφφθφθφθ

σπ

σ

=−+=×=

−=−=××=

++=

zyxene

yxznzne

zyxn

!

!!

!



Polarization

It follows that

( ) ( ) ( )( ) ( ) ( ) ( )''~'sin'sin'cos''~'cos'cos''~'''

~'cos''~'sin''~'''

~

ωθφθωφθωω

φωφωω

π

σ

zyx

yx

ddded

dded

−+=⋅

−=⋅!

!

So the energy into the two polarizations in the beam frame is

( ) ( )

( ) ( )( )

2

3

42

2

'

2

3

42

2

'

''~'sin

'sin'cos''~'cos'cos''~'8

1''

'cos''~'sin''~'8

1''

ωθ

φθωφθωωπω

φωφωωπω

π

σ

z

yx

yx

d

ddc

edd

dE

ddc

edd

dE

−

+=

Ω

−=Ω



Comments/Sum Rule

. There is no radiation parallel or anti-parallel to the x-axis for x-dipole motion

. 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

( ) ( )

( ) ( ) ( )

+

+=

+=

38''~

32''~''~'

81

'

2''~''~'8

1'

222

3

42

2

'

22

3

42

2

'

πωπωωωπω

πωωωπω

π

σ

zyx

yx

dddc

eddE

ddc

eddE

0'→θ

( )3

8''~

'

81

' 3

242

2

' πωω

πω c

de

ddEtot

!

= Generalized Larmor



Sum Rule

Parsevals Theorem again gives standard Larmor formula

( ) ( )3

22

3

22' ''32''

32

''

ctae

ctde

dtdEP tot

!""!===

Total energy sum rule

( )∫∞

∞−

= '''

~'

31

3

242

' ωωω

πd

c

deEtot

!



Relativistic Invariances

To determine the radiation pattern for a moving oscillating charge we use this solution plus transformation formulas fromrelativity theory. As an example note photon number invariance: The total number of photons emitted must be independent of the frame where the calculation is done. In particular,

∫∞

∞−

= ''

')'('~

31

3

42

2

ωω

ωω

πd

c

deNtot #

!

must be frame independent. Rewriting formulas in terms ofrelativistically invariant quantities can simplify formulas.



Wave Vector Transformation Law

Follows from relativistic invariance of wave phase, which implies is a 4-vector( )zyx kkkck ,,,/ωµ =

( ) ( )

θγβγωθφθφθφθφθ

θβγωθβγγωω

cos/'cos'sinsin'sin'sin'cossin'cos'sin'

cos1/cos//'

kckkkkk

ckcc

+−===

−=−=

and k = ω / c and k' = ω' / c are the magnitudes of the wave propagation vectors

'cos1'coscos

θββθθ

++= 'φφ =

Invert by reversing the sign of β



Solid Angle Transformation

φθββθφθ dddd ∧

−

−=∧cos1

cos''cos

( ) φθθββθβθβ dd ∧

−−+−= cos

cos1coscos1

2

2

( ) φθθβγ dd ∧

−= cos

cos11

22

( ) Ω

−=Ω dd 22 cos1

1'θβγ



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 for same dipole strength.

( ) ( )( )( )( )

( )

( )( )

( )( )

( ) ( )( )

2

32

2242

2

32

2242

cos1'~cos1

sin

sincos1

coscos1'~

coscos1

coscos1'~

8cos1

coscos1'~sincos1'~

8cos1

θβωγθβγ

θ

φθββθθβωγ

φθββθθβωγ

πθβγω

ω

φθβωγ

φθβωγπ

θβγωω

π

σ

−−

−

−−−+

−−−

−=Ω

−−

−−=Ω

z

y

x

y

x

d

d

d

ce

dddE

d

dc

edd

dE



Bend

e

e e

Undulater Wiggler

ω#ω# ω#

white source partially coherent source powerful white source

Flux

[ph/

s/0.

1%bw

]

ω#

Brig

htne

ss[p

h/s/

mm

2 /mr2

/0.1

%bw

]

ω#

Flux

[ph/

s/0.

1%bw

]

ω#



Weak Field Undulater Spectrum

( )( )( )

( )( )( ) φθβ

βθθβγ

βθβωπω

φθβγ

βθβωπω

π

σ

22

22

2

52

4

2

222

2

52

4

2

coscos1

coscos1

/cos1~

81

sincos1

/cos1~

81

−

−−

−=

Ω

−

−=

Ω

z

z

z

zz

z

zz

cB

cme

dddE

cB

cme

dddE

42

42

cmere ≡ 2

0

2γλλ =

( ) ( ) ( ) xcBmcecxdd z

'/'~''~''

~22 ω

γβωωω −==! ( ) ( ) dzezBkB ikz−∫=

~

( )( ) 222

22

111cos1γ

θγθγ

βθβ +≈++≈+− …zz

Generalizes Coisson to arbitrary observation angles



Strong Field Case

0=γdtd

Becmdtd !!! ×−= ββγ

( ) ( ) ''2 dzzBmcez

z

x ∫∞−

=γ

β



High K

( ) ( )zz xz 2211 βγ

β −−=

( ) ( )2

22 ''11

−−= ∫

∞−

dzzBmcez

z

z γγβ

( ) ( ) ( )zkKKdzzBmcez

z

z 02

22

2

2

22 2cos421

211''

21

211

γγγγβ −

+−=

−−≈ ∫

∞−



High K

Inside the insertion device the average (z) velocity is

+−=

21

211*

2

2

Kz γ

β

with corresponding

2/1*11*

22 Kz +=

−= γ

βγ

To apply dipole distributions, must be in this frame to begin with



Figure Eight

z

K*2

0

πβλ

γ

z

K*28

*2

02

πβγλγ

z'

x'



"Figure Eight" Orbits

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

-0.00001 -0.000005 0 0.000005 0.00001

z

x

K=0.5

K=1

K=2

=100, distances are normalized by λ0 / 2πγ



[ ] ( )( )θωωφθ

φω

σ nfnSSc

edd

dEnNnn

n ;cossin

sin/2

222

22

21

2, +=Ω

( )( ) ( )( )θωω

θθ

θθββθ

ωπ nf

nS

S

ce

dddE

nNn

z

zn

n ;

cossin

sincos*1*cos

22

2

2

12

,

+

−−

=Ω

fnN is highly peaked, with peak value nN, around angular frequency

( ) ( ) 0 as 2/12**2

cos*1*

02

2

020 →

+≈→

−= θωγωβγ

θβωβθω n

Knnn z

z

z



Energy Distribution in Lab Frame

[ ] ( )( )θωωφθ

φω

σ nfnSSc

edd

dEnNnn

n ;cossin

sin/2

222

22

21

2, +=Ω

( )( ) ( )( )θωω

θθ

θθββθ

ωπ nf

nS

S

ce

dddE

nNn

z

zn

n ;

cossin

sincos*1*cos

22

2

2

12

,

+

−−

=Ω

The arguments of the Bessel Functions are now

( )

( ) ( ) 222

0

0

8*

cos*1cos/''cos*

cos*1cossin/''cos'sin

βγβ

θβθωθβξ

γθβφθωφθξ

Kncdn

Kncdn

z

zzzz

zxx

−=+≡

−=≡



In the Forward Direction

In the forward direction even harmonics vanish (n+2k term vanishes when x Bessel function non-zero at zero argument, and all other terms in sum vanish with a power higher than 2 as the argument goes to zero), and for odd harmonics only n+2k=1,-1 contribute to the sum

( ) ( )( )0; sin2

222

22

, =

=

Ωθωωφγ

ωσ nf

nKF

ce

dddE

nNnn

( ) ( )( )0; cos2

222

22

, =

=

Ωθωωφγ

ωπ nf

nKF

ce

dddE

nNnn

( ) ( ) ( ) ( )2

2

2

212

2

212

2

2

2

2 2/142/14*141

+

−

+−

≈ +− KnKJ

KnKJKnKF nn

zn γβγ



Summary

. Coissons Theory may be generalized to arbitrary observation angles by using the proper polarization decomposition

. Emission (in forward direction) is at ODD harmonics of the fundamental frequency, in addition to the fundamental frequency emission. The strength of the emission at the harmonics is quantified by a Bessel function factor.

. All kinematic parameters, including the angular distribution functions and frequency distributions, are just the same as before except unstarred quantities should be replaced by starred quantities

. In particular, the (FEL) resonance condition becomes

+=

21

2

2

20 Kn

n γλλ



Finite Pulse Thomson Scattering

Generalize the work done so far to cover cases with

1. High field strength lasers

And

2. Finite energy spread from the pulsed photon beam itself

Roughly speaking, the conclusion is that the energy spectra of the scattered photons is increased by a width of order of 1/N, where N is the number of oscillations the electron makes for weak fields, but is considerably broader for strong fields.



Electron in a Plane WaveAssume linearly-polarized pulsed laser beam moving in thedirection (electron charge is e)

( ) ( ) ( )xAxzyctAtxA xinc cossin, ξ≡Φ−Φ−=!!

( )0,0,1,0=µεPolarization 4-vector

Light-like incident propagation 4-vector

zyninc cossin Φ+Φ=!

0=⋅==⋅ incincinc nnn!!εεε µµ

( )ΦΦ= cos,sin,0,1µincn



Electromagnetic Field

( ) ( )ξξ

εε

εε

νµµν

ν

µ

µ

νµννµµν

ddAnn

xA

xAAAF

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.

( ) ( )∞−=∴== µµµµνµνµµ

µ

τununuFn

dund

incincincinc 0

Proportional to amount frequencies up-shifted going to beam frame



ξ is exactly proportional to the proper time!

( ) ( )( )τξτξτ

ε µµ

µµ f

ddcun

ddfc

dud

inc ==

On the orbit

Integrate with respect to ξ instead of τ. Now

where the unitless vector potential is f(ξ)=-eA(ξ )/mc2.

( )∞−=−∴ µµµµ εε ucfu

( ) ( ) ( ) µµτξτττξ unddxnct incinc =⋅−= / !!



Electron Orbit

( ) ( ) ( ) ( )( )( )( )( )

µν

ν

µµν

ν

ννµµ ξεεξξ inc

incinc

inc

nunfcn

unucfuu

∞−+

−∞−∞−+∞−=

2

22

( ) ( )( )( )( )( ) ( ) ( )

( )( )( ) ξξ

ξξεεξξ

ξ

νν

µ

ξ

νν

µµ

νν

νν

νν

µµ

dfun

nc

dfuncn

unuc

unux

inc

inc

incinc

incinc

∫

∫

∞−

∞−

∞−+

∞−−

∞−∞−+

∞−∞−=

2'

''

2

2

2

2

Direct Force from Electric Field Ponderomotive Force



In Rest Frame of Electron

( ) ''2

''''''

42

22

ξξξξ

dcm

Aect ∫ ∞−+=

( ) '''''' ' 2 ξξξ d

mceAx ∫ ∞−=

( )( ) ''

2'''

cos1sin'

'

42

22

ξξβγ

ξd

cmAey ∫ ∞−Φ−

Φ=

( )( ) ''

2'''

cos1cos'

'

42

22

ξξβ

β ξ dcm

Aez ∫ ∞−Φ−−Φ=

( )( )( ) ( ) ( )( )( )

( ) ''' ''cos1/cos'cos1/sin''

',''

xAxzyctA

txA

x

inc

ξβββγ

≡Φ−−Φ−Φ−Φ−

=!!



Energy Distribution: Beam Frame

( ) ( ) 23222

'cos',';'''sin',';''8

'''

' φφθωφφθωπω

ωσ

yx DDce

dddE −=

Ω

( )( )

( )

2

32

22

'sin',';''

'sin'cos',';'''cos'cos',';''

8'

'''

θφθωφθφθωφθφθω

πω

ωπ

z

y

x

DD

D

ce

dddE

−

+=Ω



Effective Dipole Motions

( ) ( ) ( ) ( ) '''',';''',';'' ',';','2 ξξφθωφθω φθξωϕ de

mceADD itx ∫==

( ) ( ) ( ) '2

''',';'' ',';','4222

ξξφθω φθξωϕ decm

AeD ip ∫=

( ) ( ) ( )',';''cos1sin',';'' φθωβγ

φθω py DD Φ−Φ=

( ) ( )',';''cos1

cos',';'' φθωβ

βφθω pz DD Φ−−Φ=



Energy Distribution: Lab Frame

( )

( ) ( )

2

32

22

cos,;cos1

sinsin,;

8 φφθωβγ

φφθω

πω

ωσ

p

t

D

D

ce

dddE

Φ−Φ−

=Ω

( )

( ) ( )

( ) ( )

2

32

22

cos1sin,;

cos1cos

sincos1

cos,;cos1

sin

coscos1

cos,;

8

θβγθφθω

ββ

φθββθφθω

βγ

φθββθφθω

πω

ωπ

−Φ−Φ−+

−−

Φ−Φ+

−−

=Ω

p

p

t

D

D

D

ce

dddE



Effective Dipole Motions: Lab Frame

( ) ( )( ) ( ) ξξ

βγφθω φθξωϕ de

mceAD it ∫Φ−=

,;,2cos1

1,;

( ) ( )( ) ( ) ξξ

βγφθω φθξωϕ de

cmAeD ip ∫Φ−=

,;,42

22

2cos11,;

And the (Lorentz invariant!) phase is

( )

( )( ) ( )

( )

( )( )

∫Φ−

Φ−Φ−+

∫Φ−−

Φ−−

=

∞−

∞−

ξ

ξ

ξξβγ

θφθ

ξξβγ

φθβ

θβξωφθξωϕ

'2

'cos1

coscossinsinsin1

''cos1cossin

cos1cos1

,;,

42

22

22

2

dcm

Ae

dmc

eA

c



Weak Field Thomson BackscatterWith Φ = π and f



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 the relativistic motion of the electrons and retardation, must be generalized beyond the straight Fourier transform of the field

. At low field strengths (f



For a flat incident laser pulse the main results are very similar to those from undulators with the following correspondences

Undulator Thomson Backscatter

Field Strength

ForwardFrequency

a

'cos* θβ +z

+≈

21

2

2

20 Kγλλ

+≈

21

4

2

20 aγλλ

Transverse Pattern 'cos1 θ+

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 Strength Thomson Backscatter



Realistic Pulse Distribution at High aIn general, its 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. Weve 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 expressions, to write the energy distribution at any angle in terms of Bessel function expansions and a ξintegral over the modulation amplitude. This approach actually has a limited domain of applicability (K,a



Forward Direction: Flat, Undulator-like Pulse20-periodequivalent undulator: ( ) ( ) ( ) ( )[ ]000 20/2cos λξξλπξξ −Θ−Θ= AAx

( ) 20020220 / ,/24/21 mceAaccz =≈+≡ λπγλπγβω



( )2/1/1 2a+



Forward Direction: Gaussian Pulse

( ) ( )( ) ( )0202 /2cos156.82/exp λπξλξ zAA peakx −=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

2/ mceAa peakpeak =



Radiation Distributions: Backscatter

Flat Pulse σ at first harmonic peak

-0.002

0

0.002�

-2

0

2

�

0

1�10-41

2�10-41

3�10-41

4�10-41

dE��d�d�

-0.002

0

0.002�

-0.2

0

0.2

x-0.2

0

0.2

y

0

1�10-41

2�10-41

3�10-41

4�10-41

dE��d�d�

-0.2

0

0.2

x




Flat Pulse π at first harmonic peak

-0.002

0

0.002�

-2

0

2

�

0

1�10-41

2�10-41

3�10-41

dE��d�d�

-0.002

0

0.002�

-0.2

0

0.2

x-0.2

0

0.2

y

0

1�10-41

2�10-41

3�10-41

dE��d�d�

-0.2

0

0.2

x




-0.05

-0.025

0

0.025

0.05

�

-2

0

2

�

05�10-421�10-411.5�10-41

dE��d�d�

0.05

-0.025

0

0.025

0 05

�

05�11

-0.5

0

0.5

x

-1

0

1

y

05�10-421�10-411.5�10-41

dE��d�d�

-0.5

0

0.5

x

05�11

Gaussian Pulse σ at first harmonic peak




-0.05

-0.025

0

0.025

0.05

�

-2

0

2

�

05�10-421�10-411.5�10-41

dE��d�d�

05

-0.025

0

0.025

0 05

�

05�1

Gaussian π at first harmonic peak

-0.5

0

0.5

x

-0.5

0

0.5

y

0

5�10-421�10-411.5�10-41

dE��d�d�

-0.5

0

0.5

x

0

5�1

1




Gaussian σ at second harmonic peak

-2

0

2

x

-2

0

2

y

02�10-434�10-436�10-43

dE��d�d�

-2

0

2

x



90 Degree Scattering

( ) ( )2

32

22

cos,;1sin,;8

φφθωγ

φφθωπω

ωσ

pt DDce

dddE −=

Ω

( )

( )

( ) ( )

2

32

22

cos1sin,;

sincos1

cos,;1

coscos1

cos,;

8

θβγθβφθω

φθββθφθω

γ

φθββθφθω

πω

ωπ

−+

−−+

−−

=Ω

p

p

t

D

D

D

ce

dddE




( ) ( ) ( ) ξξγ

φθω φθξωϕ demc

eAD it ∫= ,;,21,;

( ) ( ) ( ) ξξγ

φθω φθξωϕ decm

AeD ip ∫= ,;,4222

21,;

And the phase is

( )( ) ( )

( )

∫−+

∫−−=

∞−

∞−

ξ

ξ

ξξγ

φθ

ξξγ

φθθβξωφθξωϕ

'2

'sinsin1

''cossincos1,;,

42

22

2

2

dcm

Ae

dmc

eA

c



For Flat Pulse

( ) ( )( )φθγθβλπφθω

sinsin14/cos1/2, 22

0

−+−=

ac



Radiation Distribution: 90 Degree

Flat Pulse σ at first harmonic peak

-0.2

0

0.2

x-0.2

0

0.2

y

0

1�10-41

2�10-41

3�10-41

4�10-41

dE��d�d�

-0.2

0

0.2

x

-0.002

0

0.002�

-2

0

2

�

0

1�10-41

2�10-41

3�10-41

4�10-41

dE��d�d�

-0.002

0

0.002�




Flat Pulse π at first harmonic peak

-0.002

0

0.002�

-2

0

2

�

0

1�10-41

2�10-41

3�10-41

dE��d�d�

-0.002

0

0.002�

-0.2

0

0.2

x-0.2

0

0.2

y

0

1�10-41

2�10-41

3�10-41

dE��d�d�

-0.2

0

0.2

x




Gaussian Pulse σ at first harmonic peak

-0.005

0

0.005

�

-2

0

2

�

0

2�10-424�10-426�10-428�10-42

dE��d�d�

-0.005

0

0.005

�

-0.5

0

0.5

x

-0.5

0

0.5

y

0

2�10-424�10-426�10-428�10-42

dE��d�d�

-0.5

0

0.5

x

0

2�1

4�

6



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�10-424�10-426�10-428�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�10-424�10-426�10-428�10-42

dE��d�d�

-0.4

-0.2

0

0.2

0.4

x



Polarization Sum: Gaussian 90 Degree

-0.5

0

0.5

x

-0.5

0

0.5

y

0

2�10-424�10-426�10-428�10-42

dE��d�d�

-0.5

0

0.5

x

0

2�14�

6



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�10-45

1.5�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�10-461�10-45

1.5�10-45

dE��d�d�

-1

-0.5

0

0.5

1

x

σ πSecond harmonic emission on axis from ponderomotive dipole!



THz Source



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!

( ) ( )22 2/exp σξσξ −−=x

( ) ( )22 2/exp σξσξξ −

=f

( ) ( )222

2/exp1 σξσξ

ξξ −

−=∝ peakBd

dfB



THz Undulater Motion Spectrum



Total Energy Radiated

⋅+

=

×+

=

22

24

2226

2

32

32 ββγβγβββγ "

!!"!"!!"!ce

ce

dtdE

Lienards Generalization of Larmor Formula (1898!)

Baruts Version

2

2

2

2

3

2

32

ττττµ

µ

dxd

dxd

ddt

ce

ddE =

( ) ξξξ

βγ dddff

ddfeE ∫

∞

∞−

+

Φ−=

2222

2

2cos1

32

Usual Larmor term From ponderomotivedipole



Some Cases

ξξξ

dddff

ddfeE ∫

∞

∞−

+

=

2222

232

Total radiation from electron initially at rest

( )8/131 2222 aa

ce

dtdE += ω

For a flat pulse exactly (Sarachik and Schappert)



''2'2

13

2'22222

ξξξ

dddff

ddffeE ∫

∞

∞−

+

+=

Total radiation from electron in the co-moving rest frame for flat laser pulse (Sarachik and Schappert)

( )8/31'31

'' 2222 aa

ce

dtdE += ω



Other Flat Pulse Cases

Backscatter

( )( )βγγωβ +++= 18/13

1 222222 aac

edtdE


( )8/131 222222 γγω aa

ce

dtdE +=



Undulater

( )8/13

2222222

γβγωβ KKc

edtdE +≈

Exact formula for the 1-D undulater, f=-eAx/mc2

For any practical undulater, with K



For Circular Polarization

( ) ( ) ( ) ( )[ ]{ }zyxAAinc sincos/2sin/2cos Φ+Φ−±= λπξλπξξξ!

( )( )

ξλπ

ξ

βγγ µµ

dAd

Ad

AfuneE inc

+

×

+Φ−∞−= ∫

∞

∞−±

222

22

2

2

sin

32

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

2/ mceAA −=



Sokolov and Ternov, in Radiation from Relativistic Electrons, give

and the general formula checks out

( )2222

1'32

'' aa

ce

dtdE += ω

For zero average velocity in middle of pulse

( ) ( ) ( ) 2

1/2

22

2 AcunAcunn

incinc

inc +=∞−→∞−

−=∞− νννν

γβγ!!



Conclusions. An introduction to Thomson Scatter source radiation calculations and a general

formula for obtaining the spectral angular energy distribution has been given. Ive 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 Insertion Devices 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. Anycalculation 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. 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.

General Theory of Intense Beam Nonlinear Thomson ScatteringG. A. Krafft Jefferson Lab A. Doyuran UCLA Thomas Jefferson National Accelerator Facility CASA Beam Physics Seminar 4 February

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