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

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

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

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.

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

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.

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 (10-100 fsec)

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

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

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.

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

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!

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.

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.

Recirculating SRF Linacs

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.

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, 413-50 (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: http://icfa-usa/archive/newsletter/icfa_bd_nl_26.pdf

CEBAF Beam Parameters Beam energy 6 GeV Beam current A 100 μA, B 10-200 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

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

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

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

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)

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)

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)

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

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)

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

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

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

Longitudinal Phase Space Manipulations Simulation calculations of longitudinal dynamics of JLAB FEL Piot, Douglas, and Krafft, Phys. Rev. ST-AB, 6, 0030702 (2003)

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

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.

IR FEL Upgrade

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

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

Why ERLs for X-rays? ESRF 6 GeV @ 200 mA ERL 5 GeV @ 10-100 mA CHESS / LEPP Why ERLs for X-rays? ESRF 6 GeV @ 200 mA ERL 5 GeV @ 10-100 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

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

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 0.02-0.3 % Bunch length in IDs 0.1-2* ps Total radiated power 400 kW * * rms values

ERL X-ray Source Average Brilliance and Flux CHESS / LEPP ERL X-ray Source Average Brilliance and Flux Courtesy: Qun Shen, CHESS Technical Memo 01-002, Cornell University

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

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. Rate @ 77 pC 1.3 GHz Transverse Emittance, rms (norm.) < 1b mm Bunch Length, rms 2.1 ps Energy Spread, rms 0.2 % a at reduced average current b corresponds to 77 pC/bunch

Beyond the space charge limit CHESS / LEPP Beyond the space charge limit Cornell ERL Prototype Injector Layout Solenoids 2-cell SRF cavities 500-750 kV DC Photoemission Gun Buncher Merger dipoles into ERL linac Injector optimization 0.1 mm-mrad, 80 pC, 3ps Courtesy of I. Bazarov

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.

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

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

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),

Dipole Radiation Polarized in the plane containing and

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.

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

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

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

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

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

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

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.

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

Weak Field Undulator Spectrum Generalizes Coisson to arbitrary observation angles

Strong Field Case Why is the FEL resonance condition?

High K Inside the insertion device the average (z) velocity is 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.

Simple Kinematics e- Beam Frame Lab Frame

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

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, 204802 (2004), Krafft, Doyuran, and Rosenzweig , Physical Review E, 72, 056502 (2005)

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

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.

Electron Orbit Direct Force from Electric Field Ponderomotive Force

Energy Distribution

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

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 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 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

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

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 Apeak and λ0 chosen for same intensity and same rms pulse length as previously

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

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

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

90 Degree Scattering

90 Degree Scattering And the phase is

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

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

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

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

THz Source

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

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!

THz Undulator Radiation Spectrum

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

Some Cases Total radiation from electron initially at rest 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

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.

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.