Accelerator Physics Radiation Distributions• The main “new physics” that the new calculations include properly is the fact that the electron motion changes based on the local

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


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.


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


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)


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


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


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!


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


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


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


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


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


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


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


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

]


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


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


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


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)


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


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


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)


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


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 /


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


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


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


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


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


dN/dEγ

Eγ0 (1+β)γ2Elaser (1+β) 2γ2Elaser

θ = π

sin θ = 1/ γ

θ = 0

Compton

Edge

Photon Number Spectrum


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


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


2/1/1 2a


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)


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


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)



-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




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



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


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


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



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



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



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!



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


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


Compensation By “Chirping”

Terzic, Deitrick, Hofler, and Krafft,

Phys. Rev. Lett., 112, 074801 (2014)


Hajima, et al.

Hajima, et al., NIM A, 608, S57 (2009)

TRIUMF Moly Source?

Uranium Detection


THz Source


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


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


THz Undulator Radiation Spectrum


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


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)


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


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.


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.

Documents

Accelerator Physics Radiation Distributions• The main “new physics” that the new calculations include properly is the fact that the electron motion changes based on the local