1. Energy Recovered Linacs and Ponderomotive Spectral Broadening
G. A. Krafft
Jefferson Lab

2. Outline Recirculating linacs defined and described
Review of recirculating SRF linacs
Energy recovered linacs
Energy recovered linacs as light sources
Ancient history: lasers and electrons
Dipole emission from a free electron
Thomson scattering
Motion of an electron in a plane wave
Equations of motion
Exact solution for classical electron in a plane wave
Applications to scattered spectrum
General solution for small a
Finite a effects
Ponderomotive broadening
Sum Rules
Conclusions

3. Schematic Representation of Accelerator Types
RF Installation
Beam injector and dump
Beamline
Ring
Recirculating
Linac
Linac

4. Why Recirculate? Performance upgrade of a previously installed linac
Stanford Superconducting Accelerator and MIT Bates doubled their energy this way
Cheaper design to get a given performance
Microtrons, by many passes, reuse expensive RF many times to get energy up. Penalty is that the average current has to be reduced proportional to 1/number passes, for the same installed RF.
Jefferson Lab CEBAF type machines: add passes until the “decremental” gain in RF system and operating costs no longer pays for an additional recirculating loop
Jefferson Lab FEL and other Energy Recovered Linacs (ERLs) save the cost of higher average power RF equipment (and much higher operating costs) at higher CW operating currents by “reusing” beam energy through beam recirculation.

5. Beam Energy Recovery
Recirculation path length in standard configuration recirculated linac. For energy recovery choose it to be (n + 1/2)λRF. Then

6. Beam Energy Recovery
Recirculation Path Length
Linac Cavity
Center
Recirculation path length in herring-bone configuration recirculated linac. For energy recovery choose it to be nλRF. Note additional complication: path length has to be an integer at each and every different accelerating cavity location in the linac.

7. Comparison between Linacs and Storage Rings
Advantage Linacs
Emittance dominated by source emittance and emittance growth down linac
Beam polarization “easily” produced at the source, switched, and preserved
Total transit time is quite short
Beam is easily extracted. Utilizing source laser control, flexible bunch patterns possible
Long undulaters are a natural addition
Bunch durations can be SMALL ( fsec)

8. Comparison Linacs and Storage Rings
Advantage Storage Rings
Up to now, the stored average current is much larger
Very efficient use of accelerating voltage
Technology well developed and mature
Disadvantage Storage Rings
Technology well developed and mature (maybe!)
The synchrotron radiation damping equilibrium, and the emittance and bunch length it generates, must be accepted

9. Power Multiplication Factor
Energy recovered beam recirculation is nicely quantified by the notion of a power multiplication factor:
where Prf is the RF power needed to accelerate the beam
By the first law of thermodynamics (energy conservation!) k < 1
in any linac not recirculated. Beam recirculation with beam deceleration somewhere is necessary to achieve k > 1
If energy IS very efficiently recycled from the accelerating to the decelerating beam

10. High Multiplication Factor Linacs
Recirculated Linacs
Normal Conducting Recirculators k<<1
LBNL Short Pulse X-ray Facility (proposed) k=0.1
CEBAF (matched beam load) k=0.99; (typical) k=0.8
High Multiplication Factor
Superconducting Linacs
JLAB IR DEMO k=16
JLAB 10 kW Upgrade k=33
Cornell/JLAB ERL k=200 (proposed)
BNL PERL k=500 (proposed)
Will use the words “High Multiplication Factor Linac” for those designs that feature high k.

11. Comparison Accelerator Types
Parameter
High Energy Electron Linac
High k Superconducting Linac
Ring
Accelerating Gradient[MV/m]
>50
10-20 NA
Duty Factor
<1% 1
Average Current[mA]
<1
10 going to 100
1000
Average Beam Power[MW]
0.5
1.0 going to 700
3000
Multiplication Factor
33 going to 200
Normalized Emittance[mm mrad] 4
Bunch Length
100 fsec
20 psec
Typical results by accelerator type

12. More Modern Reason to Recirculate!
A renewed general interest in beam recirculation has arisen due to the success of Jefferson Lab’s high average current energy recovered Free Electron Lasers (FELs), and the broader realization that it may be possible to achieve beam parameters “Unachievable” in storage rings or linacs without recirculation.
ERL synchrotron source: Beam power in a typical synchrotron source is (100 mA)(5 GeV)=500 MW. Realistically, even the federal govt. will be unable to provide a third of a nuclear plant to run a synchrotron source. Idea is to use the high multiplication factor possible in energy recovered designs to reduce the power load. Pulse lengths of order 100 fsec or smaller may in be possible in an ERL source; “impossible” at a storage ring. Better emittance may be possible too.
The limits, in particular the average current carrying capacity of possible recirculated linac designs, are not yet determined and may be far in excess of what the FELs can do!

13. Challenges for Beam Recirculation
Additional Linac Instability
Multipass Beam Breakup (BBU)
Observed first at Illinois Superconducting Racetrack Microtron
Limits the average current at a given installation
Made better by damping non-fundamental electromagnetic High Order Modes (HOMs) in the cavities
Best we can tell at CEBAF, threshold current is around 20 mA, measured to be several mA in the FEL
Changes based on beam recirculation optics
Turn around optics tends to be a bit different than in storage rings or more conventional linacs. Longitudinal beam dynamics gets coupled more strongly to the transverse dynamics and nonlinear corrections different
HOM cooling will perhaps limit the average current in such devices.

14. Challenges for Beam Recirculation
High average current sources needed to provide beam
Right now, looks like a good way to get there is with DC photocathode sources as we have in the Jefferson Lab FEL.
Need higher fields in the acceleration gap in the gun.
Need better vacuum performance in the beam creation region to increase the photocathode lifetimes.
Goal is to get the photocathode decay times above the present storage ring Toushek lifetimes
Beam dumping of the recirculated beam can be a challenge.

15. Recirculating SRF Linacs

16. The CEBAF at Jefferson Lab
Most radical innovations (had not been done before on the scale of CEBAF):
choice of Superconducting Radio Frequency (SRF) technology
use of multipass beam recirculation
Until LEP II came into operation, CEBAF was the world’s largest implementation of SRF technology.

17. CEBAF Accelerator Layout*
*C. W. Leemann, D. R. Douglas, G. A. Krafft, “The Continuous Electron Beam Accelerator Facility: CEBAF at the Jefferson Laboratory”, Annual Reviews of Nuclear and Particle Science, 51, (2001) has a long reference list on the CEBAF accelerator. Many references on Energy Recovered Linacs may be found in a recent ICFA Beam Dynamics Newsletter, #26, Dec. 2001:

18. CEBAF Beam Parameters Beam energy 6 GeV Beam current
A 100 μA, B nA, C 100 μA
Normalized rms emittance
1 mm mrad
Repetition rate
500 MHz/Hall
Charge per bunch
< 0.2 pC
Extracted energy spread
< 10-4
Beam sizes (transverse)
< 100 microns
Beam size (longitudinal)
<100 microns (330 fsec)
Beam angle spread
< 0.1/γ

19. Short Bunches in CEBAF
Wang, Krafft, and Sinclair, Phys. Rev. E, 2283 (1998)

20. Short Bunch Configuration
Kazimi, Sinclair, and Krafft, Proc LINAC Conf., 125 (2000)

21. dp/p data: 2-Week Sample Record
Energy Spread less than 50 ppm in Hall C, 100 ppm in Hall A
0.4
0.8
1.2
X Position => relative energy Drift
rms X width => Energy Spread
Primary Hall (Hall C)
Secondary Hall (Hall A)
Energy drift
0.4
0.8
1.2
X and sigma X in mm
Energy drift
Energy spread
1E-4
Energy spread
23-Mar
27-Mar
31-Mar
4-Apr
23-Mar
27-Mar
31-Mar
4-Apr
Date
Time
Courtesy: Jean-Claude Denard

22. Energy Recovered Linacs
The concept of energy recovery first appears in literature by Maury Tigner, as a suggestion for alternate HEP colliders*
There have been several energy recovery experiments to date, the first one in a superconducting linac at the Stanford SCA/FEL**
Same-cell energy recovery with cw beam current up to 10 mA and energy up to 150 MeV has been demonstrated at the Jefferson Lab 10 kW FEL. Energy recovery is used routinely for the operation of the FEL as a user facility
* Maury Tigner, Nuovo Cimento 37 (1965)
** T.I. Smith, et al., “Development of the SCA/FEL for use in Biomedical and Materials Science Experiments,” NIMA 259 (1987)

23. The SCA/FEL Energy Recovery Experiment
The former Recyclotron beam recirculation system could not be used to obtain the peak current required for FEL lasing and was replaced by a doubly achromatic single-turn recirculation line.
Same-cell energy recovery was first demonstrated in an SRF linac at the SCA/FEL in July 1986
Beam was injected at 5 MeV into a ~50 MeV linac (up to 95 MeV in 2 passes)
Nearly all the imparted energy was recovered. No FEL inside the recirculation loop.
T. I. Smith, et al., NIM A259, 1 (1987)

24. CEBAF Injector Energy Recovery Experiment
N. R. Sereno, “Experimental Studies of Multipass Beam Breakup and Energy Recovery using the CEBAF Injector Linac,” Ph.D. Thesis, University of Illinois (1994)

25. Instability Mechanism
Courtesy: N. Sereno, Ph.D. Thesis (1994)

26. Threshold Current Growth Rate where
If the average current exceeds the threshold current
have instability (exponentially growing cavity amplitude!)
Krafft, Bisognano, and Laubach, unpublished (1988)

27. Jefferson Lab IR DEMO FEL
Wiggler assembly
Neil, G. R., et. al, Physical Review Letters, 84, 622 (2000)

28. FEL Accelerator Parameters
Designed
Measured
Kinetic Energy
48 MeV
48.0 MeV
Average current
5 mA
4.8 mA
Bunch charge
60 pC
Up to 135 pC
Bunch length (rms)
<1 ps
0.40.1 ps
Peak current
22 A
Up to 60 A
Trans. Emittance (rms)
<8.7 mm-mr
7.51.5 mm-mr
Long. Emittance (rms)
33 keV-deg
267 keV-deg
Pulse repetition frequency (PRF)
18.7 MHz, x2
18.7 MHz, x0.25, x0.5, x2, and x4

29. ENERGY RECOVERY WORKS
Gradient modulator drive signal in a linac cavity measured without energy recovery (signal level around 2 V) and with energy recovery (signal level around 0).
Courtesy: Lia Merminga

30. Longitudinal Phase Space Manipulations
Simulation calculations of longitudinal dynamics of JLAB FEL
Piot, Douglas, and Krafft, Phys. Rev. ST-AB, 6, (2003)

31. Phase Transfer Function Measurements
Krafft, G. A., et. al, ERL2005 Workshop Proc. in NIMA

32. Longitudinal Nonlinearities Corrected by Sextupoles
Sextupoles Off
Nominal Settings
Basic Idea is to use sextupoles to get T566 in the bending arc to compensate any curvature in the phase space.

33. IR FEL Upgrade

34. IR FEL 10 kW Upgrade Parameters
Kinetic Energy
Average Current
Bunch Charge
Bunch Length
Transverse Emittance
Longitudinal Emittance
Repetition Rate
Design Value
160 MeV
10 mA
135 pC
<300 fsec
10 mm mrad
30 keV deg
75 MHz

35. ERL X-ray Source Conceptual Layout
CHESS / LEPP
ERL X-ray Source Conceptual Layout

36. Why ERLs for X-rays? ESRF 6 GeV @ 200 mA ERL 5 GeV @ 10-100 mA
CHESS / LEPP
Why ERLs for X-rays?
ESRF mA
ERL mA
εx = εy  0.01 nm mrad
B ~ 1023 ph/s/mm2/mrad2/0.1%BW
LID = 25 m
εx = 4 nm mrad
εy = 0.02 nm mrad
B ~ 1020 ph/s/mm2/mrad2/0.1%BW
LID = 5 m
ERL (no compression)
ESRF
ERL (w/ compression) t

37. Brilliance Scaling and Optimization
For 8 keV photons, 25 m undulator, and 1 micron normalized emittance, X-ray source brilliance
For any power law dependence on charge-per-bunch, Q, the optimum is
If the “space charge/wake” generated emittance exceeds the thermal emittance εth from whatever source, you’ve already lost the game!
BEST BRILLIANCE AT LOW CHARGES, once a given design and bunch length is chosen! Therefore, higher RF frequencies preferred
Unfortunately, best flux at high charge

38. ERL Phase II Sample Parameters
CHESS / LEPP
ERL Phase II Sample Parameters
Parameter
Value
Unit
Beam Energy
5-7
GeV
Average Current
100 / 10 mA
Fundamental frequency
1.3
GHz
Charge per bunch
77 / 8 pC
Injection Energy 10
MeV
Normalized emittance
2 / 0.2* μm
Energy spread %
Bunch length in IDs
0.1-2* ps
Total radiated power
400 kW *
* rms values

39. ERL X-ray Source Average Brilliance and Flux
CHESS / LEPP
ERL X-ray Source Average Brilliance and Flux
Courtesy: Qun Shen, CHESS Technical Memo , Cornell University

40. ERL Peak Brilliance and Ultra-Short Pulses
CHESS / LEPP
ERL Peak Brilliance and Ultra-Short Pulses
Courtesy: Q. Shen, I. Bazarov

41. Cornell ERL Phase I: Injector
CHESS / LEPP
Cornell ERL Phase I: Injector
Injector Parameters:
Beam Energy Range 5 – 15a MeV
Max Average Beam Current 100 mA
Max Bunch Rep. 77 pC 1.3 GHz
Transverse Emittance, rms (norm.) < 1b mm
Bunch Length, rms ps
Energy Spread, rms %
a at reduced average current
b corresponds to 77 pC/bunch

42. Beyond the space charge limit
CHESS / LEPP
Beyond the space charge limit
Cornell ERL Prototype
Injector Layout
Solenoids
2-cell SRF cavities
kV DC Photoemission Gun
Buncher
Merger dipoles
into ERL linac
Injector optimization
0.1 mm-mrad, 80 pC, 3ps
Courtesy of I. Bazarov

43. 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 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 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.

44. 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

45. Coisson’s Spectrum from a Short Magnet
Coisson low-field strength undulator spectrum*
*R. Coisson, Phys. Rev. A 20, 524 (1979)

46. Dipole Radiation Assume a single charge moves in the x direction
Introduce scalar and vector potential for fields. Retarded solution to wave equation (Lorenz gauge),

47. Dipole Radiation
Polarized in the plane containing and

48. Dipole Radiation Define the Fourier Transform
With these conventions Parseval’s Theorem is
Blue Sky!
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.

49. Dipole Radiation For a motion in three dimensions
Vector inside absolute value along the magnetic field
Vector inside absolute value along the electric field. To get energy into specific polarization, take scaler product with the polarization vector

50. 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

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

52. Polarization It follows that
So the energy into the two polarizations in the beam frame is

53. Comments/Sum Rule Generalized Larmor
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
Generalized Larmor

54. Sum Rule Total energy sum rule
Parseval’s Theorem again gives “standard” Larmor formula

55. 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.

56. Bend Undulator Wiggler e– e– e– white source partially coherent source
powerful white source
Flux [ph/s/0.1%bw]
[ph/s/mm2/mr2/0.1%bw]
Brightness

57. Weak Field Undulator Spectrum
Generalizes Coisson to arbitrary observation angles

58. Strong Field Case
Why is the FEL resonance condition?

59. High K Inside the insertion device the average (z) velocity is
and the radiation emission frequency redshifts by the 1+K2/2 factor

60. 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 MeV.

61. Simple Kinematics e-
Beam Frame
Lab Frame

62. In beam frame scattered photon radiated with wave vector
Back in the lab frame, the scattered photon energy Es is

63. Electron in a Plane Wave
Assume linearly-polarized pulsed laser beam moving in the direction (electron charge is –e)
Polarization 4-vector
Light-like incident propagation 4-vector
Krafft, G. A., Physical Review Letters, 92, (2004), Krafft, Doyuran, and Rosenzweig , Physical Review E, 72, (2005)

64. Electromagnetic Field
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.
Proportional to amount frequencies up-shifted going to beam frame

65. ξ is exactly proportional to the proper time
On the orbit
Integrate with respect to ξ instead of τ. Now
where the unitless vector potential is f(ξ)=-eA(ξ )/mc2.

66. Electron Orbit
Direct Force from Electric Field
Ponderomotive Force

67. Energy Distribution

68. Effective Dipole Motions: Lab Frame
And the (Lorentz invariant!) phase is

69. 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

70. 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
Lorentz contract + Doppler
Double Doppler

71. 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
Undulator
Thomson Backscatter
Field Strength
Forward
Frequency
Transverse Pattern
NB, be careful with the radiation pattern, it is the same at small angles, but quite a bit different at large angles

72. Forward Direction: Flat Laser Pulse
20-period
equivalent undulator:

74. 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)

75. Forward Direction: Gaussian Pulse
Apeak and λ0 chosen for same intensity and same rms pulse length as previously

76. Radiation Distributions: Backscatter
Gaussian Pulse σ at first harmonic peak
Courtesy: Adnan Doyuran (UCLA)

77. Radiation Distributions: Backscatter
Gaussian π at first harmonic peak
Courtesy: Adnan Doyuran (UCLA)

78. Radiation Distributions: Backscatter
Gaussian σ at second harmonic peak
Courtesy: Adnan Doyuran (UCLA)

79. 90 Degree Scattering

80. 90 Degree Scattering
And the phase is

81. Radiation Distribution: 90 Degree
Gaussian Pulse σ at first harmonic peak
Courtesy: Adnan Doyuran (UCLA)

82. Radiation Distributions: 90 Degree
Gaussian Pulse π at first harmonic peak
Courtesy: Adnan Doyuran (UCLA)

83. Polarization Sum: Gaussian 90 Degree
Courtesy: Adnan Doyuran (UCLA)

84. Radiation Distributions: 90 Degree
Gaussian Pulse second harmonic peak
Second harmonic emission on axis from ponderomotive dipole!
Courtesy: Adnan Doyuran (UCLA)

85. THz Source

86. Radiation Distributions for Short High-Field Magnets
And the phase is
Krafft, G. A., to be published Phys. Rev. ST-AB (2006)

87. 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!

88. THz Undulator Radiation Spectrum

89. Total Energy Radiated
Lienard’s Generalization of Larmor Formula (1898!)
Barut’s Version
From ponderomotive
dipole
Usual Larmor term

90. Some Cases Total radiation from electron initially at rest
For a flat pulse exactly (Sarachik and Schappert)

91. 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

92. 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 SRF linacs.
The present knowledge on beam recirculation and its limitations in a superconducting environment, leads us to think that recirculating accelerators of several GeV energy, and with beam currents approaching those in storage ring light sources, are possible.

93. 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.

94. 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.