Physical Review e 72, 056502 2005

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Pulsed-laser nonlinear Thomson scattering for general scattering geometries

G. A. KrafftCenter for Advanced Studies of Accelerators, Jefferson Laboratory, Newport News, Virginia 23606

A. Doyuran and J. B. RosenzweigUCLA Department of Physics and Astronomy, 405 Hilgard Ave., Los Angeles, California 90095

Received 26 May 2005; revised manuscript received 11 August 2005; published 23 November 2005

In a recent paper it has been shown that single electron Thomson backscatter calculations can be performed

including the effects of pulsed high intensity lasers. In this paper we present a more detailed treatment of the

problem and present results for more general scattering geometries. In particular, we present new results for

90 Thomson scattering. Such geometries have been increasingly studied as x-ray sources of short-pulse

radiation. Also, we present a clearer physical basis for these different cases.

DOI: 10.1103/PhysRevE.72.056502 PACS numbers: 41.75.Lx, 41.60.Ap, 41.60.Cr

INTRODUCTION

Thomson scattering of photons in an intense laser by rela-tivistic electrons has drawn considerable attention due to a

wide variety of compelling applications for the resultingshort-wavelength, narrow spectrum radiation. As the relativ-istic Thomson scattering also known as inverse Comptonscattering process yields a Doppler shifting of the photonwave-length proportional to 2, where = U/mec

2 is theelectrons normalized energy, possible scattered photonsrange from the soft x-ray 1 to MeV 2 or GeV gamma rays3.

In the x-ray region, the applications now driving the de-velopment of Thomson scattering sources include medicalimaging and therapy 4. These schemes rely heavily on thenearly monochromatic nature of the scattered radiation. Forinstance, dichromatic imaging employs digital subtraction oftwo x-ray images, one obtained with photon energy slightly

above a contrast medium absorption edge, and one slightlybelow. These dual energy images can be produced with littledifficulty by changing the electron beam energy in a Thom-son scattering source. Enhanced doses may similarly be de-livered to tissue containing contrast media irradiated by nar-row spectrum radiation.

Thomson scattering sources typically employ intenseultra-short picosecond and below photon and electronpulses, which are tightly focused, and produce ultra-shortscattered pulses with maximal photon number. In medicalapplications, only the number of photons issued over longtime scales many seconds are relevant. On the other hand,for uses in ultra-fast characterization in laser-excited materi-als, the pulse length is critical, as one needs to interrogatematerial dynamics at the 100s of femtosecond level. In ei-ther case, to achieve the scattered photon numbers demandedby such applications, one is driven to consider use of laserbeams of sufficient intensity at focus that the normalizedvector potential of the laser field, aL = eEL/mecL, ap-proaches or exceeds unity. In such scenarios, however, thespectrum of scattered photons is known to broaden throughthe addition of longer wavelength components, and also todevelop harmonics.

One of the more demanding applications of Thomsonscattering sources employs the scattering of circularly polar-

ized laser photons from many-GeV electron beams, to pro-duce circularly polarized 50100 MeV gamma rays, for useas an intermediate driver to produce polarized positrons used

in e+e linear colliders 5. In order to predict how to achievehigh levels of polarization, one must carefully calculate theangular spectrum, including harmonics, along with the cor-related final state polarization of the scattered photons. Asone demands high-scattered photon number, again scenarioswith aL not small compared to one have been proposed. Theefficacy of such scenarios is dependent on establishing boththe theoretical and experimental basis of the nonlinear Th-omson scattering interaction.

In order to test the physics of Thomson scattering in thenonlinear regime aL1 one should choose long wave-length aLL, and high intensity. Given the peak powerconstraints of existing lasers, one must focus the laser to very

tight spot sizes to achieve the needed intensities. With longwavelength lasers, this implies a highly diffractive laserbeam with an f-number focal length divided by lens diam-eter of only 2 or 3. In practice, to avoid the laser striking theelectrons final focusing elements quadrupoles this meansthat one is strongly pushed to use the 90 scattering geom-etry. In fact, such an experiment, which is just now begin-ning to explore nonlinear Thomson scattering at the UCLANeptune Advanced Accelerator Laboratory, is being per-formed using 90 laser-electron beam incidence.

It should also be pointed out that even in shorter wave-length, ultra-short pulse laser systems, where one may morereadily achieve high aL with collinear antiparallel inci-dence, one may often choose a 90 interaction geometry.This is because it is relatively straightforward to produce anelectron beam with, say, 30 m transverse width, while itis quite difficult to produce an electron pulse that is equiva-lently short. Thus the 90 geometry is critical for applicationsneeding the shortest scattered pulse. As the number of pho-tons produced naturally suffers in this geometry, one also isconstrained to look at higher aL scenarios to obtain sufficientflux.

Thus, in addition to the inherent interest that the nonlinearelectrodynamics of the Thomson interaction at high aL has,the combination of both nonlinearity and noncollinear inci-

PHYSICAL REVIEW E 72, 056502 2005

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dence angle in the Thomson interaction is of high currentinterest. As such, in this paper, we explore the expected spec-tra, in both wavelength and angle, of the Thomson scatteringflux, in more arbitrary geometries, with an emphasis on laserpolarization effects.

The results of a general theory of intense pulsed laserbeam Thomson scattering has been given in Ref. 6. Thispaper outlines a general calculation approach, but presents

detailed calculations only for a Thomson backscattering ar-rangement, i.e., when the scattering electron beam and thelaser beam are strictly antiparallel. The general conclusion ofthis calculation is that the usual spectral distribution may beconsiderably changed and broadened due to the ponderomo-tive effects of the pulsed laser on the electrons. In this paperwe apply the same theoretical framework to allow the moregeneral scattering geometries we are interested in to be stud-ied. Similar spectral broadening effects occur in the moregeneral cases. Also we find that additional transverse dipoleemission occurs because some of the ponderomotive motionis now transverse to the beam. These results generalize andextend those found in a publication by Ride, Esarey, and

Baine 7 to include cases where the field strength in thelaser beam varies throughout the laser pulse.The cases addressed in the present work examine only

linear incident laser polarization. The treatment of circularlypolarized scattering in arbitrary geometries is under study,and will be presented in a succeeding work.

ANALYSIS

We begin by noting some elementary facts about dipoleemission from a charge in non-relativistic motion. The farfield spectral energy distribution emitted from a single elec-tron undergoing such motion is 8

dE

dd=

2

82c3n d 2 =

e22

82c3n v2, 1

where E is the energy, is the angular frequency, d is thesolid angle, c is the velocity of light, n is the direction of

propagation, d is the Fourier time transform of the timederivative of the electric dipole moment of the charge distri-bution radiating, e is the charge of the electron, and v isthe Fourier time transform of the velocity of the electron. Inthis paper we choose not to follow the convention usuallyemployed in electrodynamics calculations whereby positiveand negative frequency components are combined into asingle positive frequency integral. This situation accounts forthe factor-of-two difference between Schwingers Eq.35.35 and Eq. 1. Sometimes, Eq. 1 is presented in asimpler manner by eliminating the cross product and pullinga sin2 factor out of the magnitude term. Such a procedure isstrictly valid only when the phases of the individual compo-nents of the vectors are the same, a situation that will notapply generally for our calculations. The vector inside theabsolute value sign is along the direction of the magneticfield. In order to obtain an expression in terms of the electricfield direction, it is merely necessary to add an additionalcross product with the propagation vector

dE

dd=

e22

82c3n v n2

=e22

82c3v n vn2 2

In order to obtain the energy into a specific polarization,the procedure is to take a scalar product between the vector

inside the complex amplitude and the polarization vector,before computing the complex amplitude. In the case ofmotion induced by a linearly polarized laser, a Lorentzinvariant and convenient resolution of the polarization isinto polarization perpendicular and polarization parallel to the plane of scattering. If the propagation vector isdescribed by the usual spherical coordinates co-latitudeand longitude with zero value on the x axis, thenn =sin cos ,sin sin ,cos , the perpendicular polar-ization vector is = sin ,cos , 0, and the parallel po-larization is =cos cos ,cos sin ,sin . Therefore

dE

dd

=e22

82

c3

vxsin + vycos 2

and

dE

dd=

e22

82c3vxcos cos

+ vycos sin vzsin 2. 3

Suppose now that in a frame at rest with a moving beamwith velocity in the positive z direction c, an electron isexecuting a nonrelativistic motion. There are some generalstatements that may be made regarding the spectral distribu-tion in the lab frame. Henceforward quantities in the labframe will be denoted without primes. If quantities within

the beam rest frame are denoted with primes, then the spec-tral distributions are

dE

d d=

e22

82c3 vxsin + vycos

2,

and

dE

d d=

e22

82c3vxcos cos

+ vycos sin vzsin 2. 4

The transformations between frames which allow recasting

of Eq. 4 in the lab frame are E=1 cos E,=1 cos , =, cos = cos /1 cos ,or equivalently sin =sin /1 cos , and finallyd=21 cos 2 d. Therefore, the energy distribu-tion in the lab frame is

dE

dd=

e22

82c3 vx1 cos sin

+ vy1 cos cos 2

and

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dE

dd=

e22

82c3vx1 cos cos

1 cos cos

+ vy1 cos cos

1 cos sin

vz1 cos sin

1 cos 2, 5

where the notation indicates replacement of by1 cos in v, the original function expressingthe Fourier time transform of velocity, as computed in thebeam frame. To get distributions into other polarizations, firstperform the scalar product between the polarization vector,in the beam frame, and v inside the complex amplitude,and transform in the same manner as above.

Next we show that the spectral distributions for intensepulsed linearly polarized laser beam scattering have formsvery similar to Eq. 5. The main additional complication isthat the Fourier transform of the velocity is replaced byangle-dependent generalized dipole moment vectors whose

frequency distribution is broadened by the actions of theponderomotive effects of the laser beam. These formulas in-clude correctly the relativistic motions of the electrons in-duced by very intense lasers and allow the field strengthparameter to vary within the laser pulse.

To begin, we review the classical motion of an electronin a plane wave. Early solutions employed solving therelativistic Hamilton-Jacobi equation 911. Recent deriva-tions evaluate the motion directly in terms of the electronconstants of the motion in the plane wave 12. Here, wegive a solution of the equations of motion and present aphysical discussion important for our subsequent work. For aplane wave solution to Maxwells equations the vector po-

tential of the wave can be written Ax ,y ,z , t=A, where= ct nkx, and nk is the unit vector in the direction ofpropagation. The fact that the waves are purely transverse isembodied in the requirement nk=0, and for an electronoutside of the plane wave to begin, will be proportional tothe proper time of the electron. Defining a polarization four-vector =0 , and a light-like propagation four-vectorn= 1 , nk, we see that the electromagnetic field is

F= A

x

A

x= An n. 6

We must first find the space-time four-vector components

x = (ct ,x ,y ,z) when u= cdt/d,dx/d, dy/d, dz/d satisfies du/d= eFu/mc

2. For

any solution to the equations of motion dnu/d

= nFu=0, because n

is light-like and the polarizationvector is orthogonal to the direction of propagation. We canevaluate the constant of the motion at a time before the wavearrives at the electron, calling this value nu

. Because = ct nkx on the electron orbit has d/d= nu

,

it must be a constant multiple of the proper time, and wecan integrate the equations of motion with respect to it,instead of .

Now, it also follows from use of the Lorentz forcelaw that du

/d= f(nu= fd/d, where

f= eA/mc2 is the normalized unitless vectorpotential, and there is an additional constant u

f=u

. Therefore, the force law may be integrated toobtain

u

= u

+ fu

nu n

+

f 2

2nu

n, 7

where u is the four-velocity before the arrival of thelaser pulse. Each of these terms has an easily understoodphysical interpretation. The first term is the initial conditionon the velocity before the laser pulse arrives. The secondterm, proportional to f, gives the velocity induced in theelectron by the direct action of the electric field of the laseron the electron. The first part of the second term accounts forthe fact that the effective electric field amplitude is decreasedwhen there is initial motion along the polarization vector.

The final term is due to the longitudinal ponderomotive ef-fect from the laser. Note that it is always positive, and di-rected in the direction of propagation of the plane wave, andit may act to change the average velocity of the electron.Including the change in the velocity of the electron due to theponderomotive effect of the laser beam as the field strengthchanges is the primary generalization in this treatment com-pared to previous ones.

A further integration with respect to gives

x =u

nu

+ u nu

2n

nu

fd +n

nu 2

f 2

2 d,

8

where by zeroing the initial conditions on x, it is implicitlyassumed, without loss of generality, that the coordinate sys-tem is chosen so that the electron arrives at the origin at thezero of ordinary time had there been no laser present. Nowthat we have the position and velocity as a function of, wecan perform the usual electrodynamics calculation to obtainthe spectral energy distribution of the radiation.

To continue with the calculation it is necessary to specifya coordinate system. In this particular calculation the mostexpeditious choice is to perform the electrodynamics calcu-lation within the frame in which the electron is initially atrest, even though it requires some extra work to set the cal-culation up there. Because f vanishes both before andafter the laser pulse and so the particle velocity vanishesbefore and after the pulse in this frame, the electrodynamicsintegrals performed over the velocity will be obviously con-vergent in this frame, avoiding the need for subtle argumentsrequiring cut-off procedures to obtain sensible results. Onetransforms back to the lab frame at the end of the calculation.This beam rest frame choice has an additional benefit in thatthe results there have a more obvious physical significance.

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Consider a right-handed lab frame coordinate system withz axis along the electron motion. Assume an intense, linearlyx-polarized plane electromagnetic wave is incident on anelectron with wave vector pointing along the direction nk=0,sin ,cos where is the angle between the incidentradiation and the z axis. The interaction between the waveand the electron is assumed to occur in the vicinity of theorigin of the coordinate system. As in Ref. 6 and above, thepolarization of the scattered radiation is resolved, in a Lor-entz invariant way, into polarization perpendicular and polar-ization parallel to the plane of scattering.

It is an exercise in the transformation formulas of light-like four-vectors to show the incident propagation four-vector in the primed beam frame before the arrival of thelaser is

nk = 1,nk

= 1,0,sin /1 cos,

cos /1 cos. 9

By Eq. 3, the position four-vector in the beam frame has

ct = +

e2Ax2

2m2c4d, x =

eAx

mc2d,

y =sin

1 cos

e2Ax2

2m2c4d, 10

z =cos

1 cos

e2Ax2

2m2c4d,

with

= ct siny

1 cos

cos z

1 cos.

To obtain the scattered energy distributions into the twopolarization states use the standard formula for the spectraldistribution 13, but change the integration variable from thetime to . Thus

dE

d d=

e22

82c3 Dx;,sin

+ D

y

;

,cos

2

,and

dE

dd=

e22

82c3Dx;,cos cos

+ Dy;,cos sin

Dz;,sin 2, 11

where

Dx;, = Dt;,

=

eAx

mc2ei,;,d,

Dy;, =sin

1 cosDp;, ,

Dz;, =cos

1 cosDp;,,

and

Dp;, = e2Ax

2

2m2c4ei,;,d. 12

The phase of the integrals in Eqs.12

is

,;, =

c + e2Ax

2

2m2c4d sin cos

eAx

mc2d

sin sin sin

1 cos

e2Ax2

2m2c4d

cos cos

1 cos

e2Ax2

2m2c4d . 13

So the full calculation of the scattering distribution for intense pulsed lasers involves generalizing from the Fourier transform

of the velocity vector to the vector D;, which now involves two integrals, the transverse moving dipole generated by

the direct action of the polarization, and the moving dipole generated by the ponderomotive effect of the laser which has bothtransverse and longitudinal components.

By transforming back into the laboratory frame, the final results for the lab-frame scattering distributions are

dE

dd=

e22

82c3 Dt;,sin + sin

1 cosDp;,cos 2 14

and


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dE

dd=

e22

82c3Dt;,cos

1 cos cos +

sin

1 cosDp;,

cos

1 cos sin

+ cos

1 cosDp;,

sin

1 cos

2

where

Dt;, =1

1 cos eAx

mc2 ei,;,d,

Dp;, =1

1 cos e

2Ax2

2m2c4ei,;,d, 15

and the Lorentz invariant phase is

,;, =

c 1 cos

1 cos

sin cos

1 cos

eAx

mc2d

+1 sin sin sin cos cos

21 cos2

e2Ax2

2m2c4d . 16

To effect the transformations leading to Eqs. 1416 therules =/1 cos and Ax=Ax =Ax1cos) are used so that only the lab-frame vector po-tential appears in the result. For convenience of expression,in Eq. 16 represents a dummy integration variable only.

Compared to the result in Ref. 6, this result has an al-most obvious physical interpretation: as before, the Dt termsgive the radiation from the direct x-dipole motion induced inthe electron. Also as before, the cos /1 cos fac-tors, which confine the radiation to small angles , come

from the relativistic transformation of the scattered wavevector between beam and lab frame, of the emission gener-ated by the x-motion. In addition, there is a transverse com-ponent from the ponderomotively driven dipole motion in-duced in the electron, which is quantified by the projection ofthe ponderomotive dipole motion Dl on the y axis. As thismotion is also directed transverse to the beam motion, one

expects the same factor tending to confine the y-directed di-

pole radiation. Finally, the motion in the z direction, which

transforms differently than the transverse dipole motions un-

der the Lorentz transformation, is quantified by the projec-

tion of the ponderomotive dipole motion on the z axis. There

is no perpendicular component generated by the z dipolebecause the polarization of the emission of z-directed dipole

is always in the plane of n and the z axis. Of course the

radiation from the ponderomotive dipole tends to occur at

frequencies twice as great as the direct radiation, and itdominates the pattern at large values of the field strength f.

The result of Ref. 6 follows exactly from these generalforms by taking =which means cos =1.

A new case, beyond that covered in Ref. 6 is the case ofintense 90 Thomson scattering. In this case =/2,

sin=1, cos= 0,

dE

dd=

e22

82c3 Dt;,sin + 1

Dp;,cos 2, 17

dE

dd=

e22

82c3Dt;,cos

1 cos cos +

1

Dp;,

cos

1 cos sin

+ Dp;,sin

1 cos

2

, 18

with

Dt;, =1

eAx

mc2ei,;,d, Dp;, =

1

e

2Ax2

2m2c4ei,;,d, 19


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and the phase is

,;,

=

c

1 cos sin cos

eAx

mc2d

+1 sin sin

2

e2Ax2

2m2c4d

.

20

In contrast to the Ref. 6 result, at high f the observedfrequency of the backscattered radiation has a dependenceon , to be expected due to the kinematical asymmetryintroduced by illumination from the side of theelectron beam. For flat laser pulses the fundamentalfrequency is

, =1 cos2c/0

1 cos + a2/421 sin sin , 21

where 0 is the laser wavelength. That this formula is correct

may be independently verified by calculating the relativisticDoppler shift from the emission of the dipole as it movesalong the incident wave vector due to the ponderomotiveforce.

II. Total energy sum rule

We conclude the discussion of our analyticalresults by noting that the relativistic Larmor Theoremmay be used to obtain an exact sum rule for the integral overall frequencies and solid angles for the energy of the scat-tered radiation. These sum rules generalize the single pe-riod integrals that appear in Ref. 7. The relativistic LarmorTheorem is

dE

d=

2

3

e2

c3dt

d

d2x

d2d2x

d222

where is the proper time. The force law and plane waveapproximation may be used to evaluate the energy emittedper electron exactly. As above, the proper time may be ex-pressed in terms of to obtain

E=2e2

3

21 cos dfd

2 + f22 df

d2d.

23

The first term in this equation is due to the direct action ofthe field on the electron, and the second term is due to theemission from the ponderomotive dipole. It is clear that un-less the field is so large that

21 cos f2

2, 24

the largest part of the emission will be from the directdipole. On the other hand, if the field strength exceeds thisvalue, the energy emitted by the ponderomotive dipole willbe dominant.

EXAMPLES OF DETAILED CALCULATIONS

The expressions in Eqs. 17 and 18 provide not onlyfrequency content of the scattering but also intensity distri-

bution in space coordinates. Because the longitudinal depen-dence of the field distribution of the laser can be arbitrarilyentered, it is possible to investigate the scattering with real-istic pulses such as Gaussian and flat top.

For the purposes of a relevant illustration, the interactionangle can be set to investigate the 90 scattering case, corre-sponding to an experiment in progress currently at theUCLA/Neptune Laboratory. In order to translate our generalresults to specific predictions concerning this and other ex-perimental scenarios, we have developed a MATHEMATICAprogram which numerically calculates the frequency and in-tensity distribution of the scattered photons.

Table I shows the design electron and CO 2 laser param-eters for the Thomson Scattering Experiment in Neptune

Laboratory 14. Table II shows the scattered photon param-eters for both head-on and transverse geometries. Since theexperiment is designed for transverse geometry, we numeri-cally calculated the on axis spectrum and intensity distribu-tion for each of the first three harmonics, using 90 scatteringgeometry, for both Gaussian and flat top shape laser pulses.

Figures 1a1d shows the scaled frequency /0content of the scattered photons into the and polarizations, for Gaussian and flat-top laser beams where0 == 0 ,= 0 in Eq. 21. The two cases have the same

TABLE I. Electron and laser beam parameters.

Parameter Value

Electron beam energy 14 MeV

Beam emittance 5 mm mrad

Electron beam spot size RMS 25 m

Beam charge 300 pC

Bunch length RMS 4 psLaser beam size at IP RMS 25 m

CO2 laser wavelength 10.6 m

CO2 laser Rayleigh range 0.75 mm

CO2 laser power 500 GW

CO2 laser pulse length 200 ps

TABLE II. Scattered photon parameters.

Parameter Head-on Transverse

Scattered photon wavelength 5.3 nm 10.7 nm

Scattered photon energy 235.3 eV 117.7 eV

Scattered photon pulse duration FWHM 10 ps 10 ps

Interaction time 5 ps 0.33 ps

Number of periods that electrons see N0 283 10

Number of photons emitted per electron N 3.34 0.11

Total number of photons 6.3109 2108

Half opening angle 2.7 mrad 15 mrad


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full width at half maximum FWHM pulse length and thesame peak laser intensity or field strength. Significant caremust be taken to accurately calculate the integrals here. Wechose about 800 trajectory points covering up to 8th har-monic to obtain good resolution. A working precision of 12is satisfactory in MATHEMATICA to complete the calculationof the on-axis spectra; less than this precison gives notableerrors in calculating the sudden dips in the spectrum. For the polarization, odd harmonics vanish on-axis, while for the polarization, even harmonics vanish on-axis, in bothGaussian and flat-top cases. We observe structure from theinterference of the radiation emitted at different longitudinal

FIG. 1. a On axis logarithmic spectrum of photons scatteredfrom a linearly polarized Gaussian laser beam into the polariza-

tion. b On axis spectrum of photons scattered from a linearlypolarized Gaussian laser beam into the polarization. c On axisspectrum of photons scattered from a linearly polarized flat-top la-

ser beam into the polarization. d On axis spectrum of photonsscattered from a linearly polarized flat-top laser beam into the

polarization.

FIG. 2. Color online a Intensity distribution of first harmonicscattered photons from a linearly polarized Gaussian laser pulse

into the polarization with a0 =1 on a screen located at z0 =

14 MeV. The screen is centered at =0 and screen size is in unitsof mm. b Intensity distribution of second harmonic scattered pho-tons from a linearly polarized Gaussian laser pulse into the po-

larization with a0 =1 on a screen located at z0 = 14 MeV. Thescreen is centered at =0 and screen size is in units of mm. cIntensity distribution of third harmonic scattered photons from a

linearly polarized Gaussian laser pulse into the polarization with

a0 =1 on a screen located at z0 = 14 MeV. The screen is centeredat =0 and screen size is in units of mm.


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locations in both cases and additional spectral broadening inGaussian case.

To calculate the intensity distributions, we choose a spe-cific harmonic and calculate the spectral energy by varying and coordinates. We vary half cone angle from 0 to

3 /by dividing the range into 45 steps and from to +by dividing the range into 20 steps. The calculation of theintegral does not require high accuracy in this case, so theworking precision may be set to 8 in MATHEMATICA for fastprocessing because the frequency is set to values correspond-ing to nominal spectral peaks. The intensity plots assume a

FIG. 3. Color online a Intensity distribution of first harmonicscattered photons from a polarized flat-top laser pulse into the

polarization with a0 =1 on a screen located at z0 = 14 MeV. Thescreen is centered at =0 and screen size is in units of mm. bIntensity distribution of second harmonic scattered photons from a

linearly polarized flat-top laser pulse into the polarization with

a0 =1 on a screen located at z0 = 14 MeV. The screen is centeredat =0 and screen size is in units of mm. c Intensity distributionof third harmonic scattered photons from a linearly polarized flat-

top laser pulse into the polarization with a0 = 1 on a screen located

at z0 = 14 MeV. The screen is centered at =0 and screen size isin units of mm.

FIG. 4. Color online a Intensity distribution of first harmonicscattered photons from a linearly polarized Gaussian laser pulse

into the polarization with a0 =1 on a screen located at z0 =

14 MeV. The screen is centered at =0 and screen size is in unitsof mm. b Intensity distribution of second harmonic scattered pho-tons from a linearly polarized Gaussian laser pulse into the po-

larization with a0 =1 on a screen located at z0 = 14 MeV. The

screen is centered at =0 and screen size is in units of mm. cIntensity distribution of third harmonic scattered photons from a

linearly polarized Gaussian laser pulse into the polarization with



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screen located at z0 = away, and we simply convert anglesinto Cartesian coordinates through =tan1x2 +y2/z0 and

=tan1y/x. Figures 2a2c show the scattered photondistribution at first three harmonics into the polarizationfor a Gaussian laser pulse. Figures 3a3c show the sameplots into the polarization for a flat-top laser pulse. Figures4a4c are the scattering intensity into the polarizationfor a Gaussian and Figs. 5a5c are the same into the polarization for flat-top laser pulses.

CONCLUSIONS

In this paper, we have presented an analysis of the spectraassociated with nonlinear Thomson scattering for linearlypolarized lasers incident on relativistic electrons at arbitraryangle. This formalism may be used to explore new, experi-mentally interesting scattering geometries, including, e.g.,small angle Thomson scattering, and 90 incidence scatter-ing. The overall structure of the distributions is similar tothat from nonrelativistic dipole emission, but must be gener-alized to include an angular dependence in the effective di-pole moments for the emission. Such a generalization is nec-essary to properly include ponderomotive effects on the

electron motion as the field strength changes throughout thelaser pulse, and the changes in retarded time that this motionengenders. The generalization involves two integrals, withproper retarded phase, over the vector potential of the inci-dent laser and its square. Given a measurement of the vectorpotential of a laser pulse, or a sufficiently accurate model, theformalism in this paper may be used to determine the spon-taneous emission spectrum in detail.

The formalism has been used in an illustrative analysis ofan on-going experiment at UCLA which explores 90 inci-dence scattering and large normalized vector potential, a0 1. The wavelength spectra obtained from this analysismethod display many experimentally interesting artifacts, in-

cluding harmonic generation and ponderomotive broadening.In addition, the profiles obtained in the angular spectra for agiven harmonic are rich in structure, and thus may serve asexperimental signatures.

ACKNOWLEDGMENTS

The analytical and computational methods employed inthis paper may straightforwardly be extended to the case ofcircularly polarized incident laser. An effort in this directionis now underway. This work was supported by the U.S. De-partment of Energy under Contract No. DE-AC05-84-ER40150 and Grant No. DE-FG03-92ER40693, and the UC

Office of the President under its Campus-Laboratory Col-laboration Program.

1 I. V. Pogorelsky et al., Phys. Rev. ST Accel. Beams 3, 0907022000.

2 I. Sakai et al., Phys. Rev. ST Accel. Beams 6, 091001 2003.3 Valery Telnov, Nucl. Instrum. Methods Phys. Res. A 335, 3

1995.

4 W. Thomlinson, Nucl. Instrum. Methods Phys. Res. A 319,295 1992.

5 T. Omori et al., Nucl. Instrum. Methods Phys. Res. A 500, 2322003.

6 G. A. Krafft, Phys. Rev. Lett. 92, 204802 2004.

FIG. 5. Color online a Intensity distribution of first harmonicscattered photons from a linearly polarized flat-top laser pulse into

the polarization with a0 =1 on a screen located at z0 =

14 MeV. The screen is centered at =0 and screen size is in unitsof mm. b Intensity distribution of second harmonic scattered pho-tons from a linearly polarized flat-top laser pulse into the polar-

ization with a0 =1 on a screen located at z0 = 14 MeV. Thescreen is centered at =0 and screen size is in units of mm. cIntensity distribution of third harmonic scattered photons from a

linearly polarized flat-top laser pulse into the polarization with



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