PHYSICAL REVIEW

VOLUME

1, NUMBER 10

15 MAY

19(0

Classical Theory of the Scattering of Intense Laser Radiation by Free Electrons

E. S.
SARAcHII& AND

G. T.

ScHAPPERT
OZ13P

National Aeronautics and Space Administration,

Electronics Research Center, Cambridge, Massachusetts

(Received 14 January 1970)

A complete discussion of the classical theory of high-intensity Thomson scattering by free electrons is presented. Neglecting the radiation reaction, the equations of motion for an electron in an arbitrarily intense, elliptically polarized, plane electromagnetic wave can be solved exactly. From the solutions for the electron motion, the radiated power, momentum, and harmonics are calculated in two special Lorentz frames: the laboratory frame and the frame in which the electron is on the average at rest. The diQerence between the radiated power measured by an observer and that emitted by the electron is discussed for each frame. A sum rule for the radiated harmonics is derived. The limitation due to the neglect of radiation reaction is considered. Finally, the high- and low-intensity behavior of the spectrum and angular distribution of the radiation is analyzed in both frames,

l.
+ ~URING

INTRODUCTION

the last ten years, the problem of the interaction of free electrons with intense electromagnetic fields has received considerable attention. The reason for this interest has been the development of high-power optical lasers capable of producing radiation fields whose power 'densities are many orders of rnagnitude greater than those possible with any other device. Thus the possibility naturally arises of the existence of effects that occur at large field intensities but at low photon energies. We refer the reader to a recent excellent review article by Eberly' for background and for a complete list of references. It should be borne in mind that not a single one of these effects can be said to have been unambiguously observed indeed, most of these experiments have not yet been attempted. In this paper, we present a complete discussion of the classical theory of high-intensity Thomson scattering from free electrons and the associated effects of harmonic production and intensity-dependent frequency shifts. While there have been several papers on this subject in the literature, none have presented detailed calculations as to what can be observed in the laboratory. The classical calculations' have been carried out (in perturbation theory) in the frame in which the electron is on the average at rest (the R frame). The quantummechanical calculations, expecially the one of Brown and Kibble, ' while performed exactly, refer to the radiation emitted by the electron in the laboratory frame (the L, frame). The radiation emitted by the electron in the frame is not the same as the radiation seen by an observer in the I. frame. The difIerence is due to the net motion of the average center of mass of the electron with respect to the observer. The distinction between the electron's point of view (retarded time) and the observer's is rather subtle and has not been suKciently discussed in the literature in connection with this ~ J. H. Eberly, Progress in Optics, edited by E. Wolf (NorthHolland, Amsterdam, 1969), Vol. 7. 2 Vachaspati, Phys. Rev. 128, 664 (1962); 130, 2598 (E) I'1963). ~'L. S. Brown and T. W. B. Kibble, Phys. Rev. 133, A705 (1964).

problem. We might mention that the classical and quanturn-mechanical treatments of this problem have been shown to be equivalent for high intensities and are reviewed in an article by Kibble. ' As a preliminary to the problem, let us recall what is meant by high-intensity electromagnetic radiation. In the usual classical treatment of Thomson scattering, 5 the electron is assumed to be set in motion by the electric force eE and the radiation produced by the subsequent motion is calculated by the nonrelativistic Larmor radiation formula. At high intensities, however, the full force eE+(e/c) (v xB) must be used to calculate the electron motion. The motion then becomes a nonlinear function of the driving field in addition to becoming relativistic. The problem therefore becomes vastly more complicated. The parameter characterizing this high-intensity region will be called q', and is defined by
q' = 2es(A'(t) )/ns'c'

= 2Irs)'/7rnzc'.

I.

The first expression is in terms of the square of the vector potential and the second in terms of the intensity (measured in W/cm'). The classical electron radius is ro, the wavelength of the radiation is 'A, c is the velocity of light, and e and ns are the electron charge and mass, respectively. To get a feeling for the magnitude of q' and the intensities involved, we list these quantities defor several typical electromagnetic-field-producing vices in Table I. The high-intensity eGects are de6ned as those for which q' is at least of order unity. We see region can be from Table I that the high-intensity focussing can be apentered if diffraction-limited proached. The held of high-intensity laser development is undergoing rapid expansion and there seems no reason why values of q' several orders of magnitude greater than those shown here could not become available. The plan of the paper is as follows. In Sec. II we discuss the classical motion of an electron in an elliptically polarized electromagnetic wave. The wave is as4T. W. B. Kibble, in Carghse Lectures in Physics, edited by
.

M. Levy (Gordon and Breach, New York, 1968), p. 299. ' J. D, Jackson, Classical Electrodynamics (Wiley, New York, 1962).

2738

CLASSICAL THEORY OF SCATTERING 8Y FREE ELECTRONS

sumed to be a pulse in the vector potential containing a large number of optical cycles. We then specialize the solution to two cases: motion in the R frame and motion in the frame. Neglecting the radiation reaction, the electron motion can be solved exactly for arbitrary in-

2739

I.

cident 6eld intensity. In Sec. III we discuss the radiation problem based on the solutions for the electron motion in the two frames of reference. In particular, we calculate the harmonic production as seen by observers in both frames and how they are related, and we show how the intensitydependent frequency shift comes about. The relation between the power radiated by the electron and the power observed in each frame is stressed, . The section closes with an estimate of the radiation reaction and the conditions under which it may be neglected in the equations of motion. Section IV is devoted to a detailed analysis of the results of Sec. III. We present a discussion of the angular distribution of the power in each harmonic, the distribution of harmonics, and the total power observed in the I. frame. These results are presented for both high and low intensities in the case of incident circular polarization, and for low intensities in the case of linear polarization. Some exact results, valid for arbitrary elliptically polarized incident radiation, are pointed opt. In the Sec. V we consider the limitations of our work due to quantum-mechanical and other effects.

beam incident upon it. The incident beam will be assumed to be transverse, plane, and arbitrarily elliptically polarized, and to be characterized by a wave vector k with frequency co = c k = ck. In order that the initial conditions be realistically included, the plane wave will be multiplied by a pulse-shape factor, so that the vector potential is written
~ ~

A(r,
where

1) = A(g)

P(rl),

(2 1)

g=cot k

(2 2)

is the I.orentz-invariant phase. We will assume that A(rl) is periodic in rl, and P(rl), the pulse-shape factor, is zero before and after the electron interaction = withe~the, laser"beam. Thus, P(&~) =0 and 1 P(0) . For mathematical convenience, we will further assume that P(rl) is square integrable. If we neglect the radiative reaction effects, the fully relativistic equation of motion for the electron in the plane-wave 6eld may be solved exactly. The Hamilton-Jacobi equation for this problem is

P,

LOS(r, 1) /W)'+m'c'= (1/c')

where S(r, t) is the Hamilton principal look for a solution in the standard form

0,

(2.3)

function. We

II. ELECTRON

S(r, t) =n r+pct+4

(rl),

(2.4)

EQUATIONS OF MOTION

We want to consider the motion of an electron, initially at rest at the origin, under the action of a laser

where n and. p are constants determined by the boundary conditions and C(rl) is a function determined by (2.3) . This function is easily found to be

~(n)

=l(~ k+p&) '

QQ

' p'+~'~' 2(%)rt. A(n)P(~)+(e'/~')~'(n)P'(v)]de

(2.5)

' 0.and equating this to the initial coordinate:

V

The equation for the motion is obtained by differentiating

the principal function with respect to the constants

S=ro=r(rl)+
r ro

(e/c) A( g) P(g)

'17Q

n k+pk

dq

2k

o
'QQ

n' P'+m'c'

(e/c) O.'A(rl) P (rl) + (e'/c') A'(rl) P'(rl) 2

(20. k+2pk)
s

d'gp

(2.6)
given by this equation,

and, with

cto=as/ap= ct

(k/&)

ro) & (r

'(rl no),

(2.7)

which just reflects the definition (2.2). The canonical momenta and energy are given by diQerentiating

the principal function with respect to the

coordinates:

P=p+(%)A=vS ~' p'+~'" 2(%)~

L. D. Landau
7
o

A(n)

P (n)+ (e'/~') ~'(n) P'(n)

Mass. , 1962).

2(n. k+Pk)
J. J. Sanderson, Phys. Letters 18, 114 (1965). J. H. Eberly and A. Sleeper, Phys. Rev. 176, 1570 (1968).
and

(2.8)

E. M. Lifshitz,

Classical Theory of Fields, 2nd ed. (Addison-Wesley,

Reading,

2740

E. S. SARACHIK
TABLE

AND

G.

T. 8C

HAP PERT
devices.

I. Characteristics

of some typical electromagnetic-6eld-producing

Wavelength

Power or
energy
104

Device
Microwave tube

(cm)

Pulse length

Diffractionfocused intensity (W/cm')

105 107

q8

10 2sec
CW

'7X10-5

He-Ne laser
High-brightness

6.3X10 5
Nd

50 mJ/sec 60 J

3X10

"

10

'

30 nsec

10"

0. 14

Q-switched glass laser

system
High-brightness

Nd

80 J

3 psec

4X10'i

3000

mode-locked glass laser system

and, in terms of this result,

The equation of motion (2.6) now becomes

~

E = BS/Bt=c[p+ (k/k) (n )j. (2.9) P,

are obtained by imposing boundary conditions on the electron. There are two conimon boundary conditions treated in the literature: the lab frame (quantities denoted by subscript L) in which the electron is initially at rest, and the "on the frame (subaverage at rest" or center-of-momentum script E), in which the electron undergoes periodic motion about a fixed center. We discuss these frames in turn.

k(it)
eA (q')

The complete solutions

P (g')

kz c~A2(g')P~(rt')
m2c4

mc'

2 kl,

labThe form of r(&) looks deceptively frame time t is in fact hidden in (2.13) through the deat fining equation rt=&azt kz, r, so th the orbit r(t) be quite complicated. In particular, although the may motion will contain no higher than second harmonic oscillations in terms of g, r(t) may contain al/ multiples
Of Mg.

the simple

(2.13)

A. Lab Frame

The electron is sitting at the origin before the laser beam is turned on. Thus, at t= P, =0, r=0, ~, 0c and E=mc'. Evaluating (2.8) at rt= therefore
yields

The momentum and energy are obtained by using the nz, and pz, in (2.8) and (2.9), yielding
p(n)
e = A(n) P(~)+

k2A2
2kl.

()P2() (2.14) mc'

(2.15)

'c')/(nz kz+Pzkz) nz= 2kz[(nz' -Pz'+m

j,

(2. 10)

0 which is consistent with re in (2.6). Noting that 0.1, xkL, =O, we see that o.Jhas no transverse part and the longitudinal part is given by

E (g) = mc'[1+ e'A'(g) P'(it) /2m'c4$.

(2. (2.13) 15) now

[nz (kz/kz)

+pz)'= m'c'.
1pz mc.

(2. 11)

The initial energy condition applied to (2.9) then 6xes the sign so that

nz (kz/kz)
Thus (2.12) and the only constraints ness can be traced principal function (2.4). Since nz, and

(2.12)

the fact that n is longitudinal are on these constants. This arbitrarito the arbitrary way in which the can be separated into the form pz, always enter in the combination given in (2.12), we can, without loss of generality, choose nz, =0 and pz, = to simplify the algebra. mc

provide a complete Equations solution to the electron motion problem in the lab frame. As the electromagnetic wave overlaps the electron, the electron acquires a harmonic motion transverse to the beam direction due to the first term in (2.14), and an acceleration along the beam direction due to the growth of P'(q) multiplying the zero-frequency part of A'(g). There will also be a longitudinal harmonic due to the 2q part of A'(g) which, as we shall see, vanishes in the particular case of circular polarization. After the P(it) factor reaches its constant value in the center of the pulse, the electron is undergoing various harmonic motions about a center that itself is drifting with respect to the lab. As P(rt) turns off, the harmonic motion dies down and the center decelerates until at g=+ ~, corresponding to t=+ ~, the pulse has passed and the

CLASSICAL THEORY OF SC'ATTERING

electron is again at rest in the laboratory. It is now displaced from its original position by a distance r(00) in the beam direction. This net displacement is pulse shape dependent but is easily calculated. For example, using a normalized Gaussian pulse of width T))col, ', the net displacement is
Applying (pic&=0 in

BY

FREE ELECTRONS
boundary
conditions
(A'(7J)

the R-frame

that

(2.8)
njp

yields
pg

'pm'c'+ (e'/c') crt ka+ pa4

(2.19)

r( ~ ) = (kz/4hz) e'(A'(q)
I

&/m'c'](-', ~) "'Tc~~q'Tc,

(2 16)
where terms of order exp( 1/T'o&z') have been neglected, and where g' is the previously defined intensity

so that 0,'& again is longitudinal, with only the combination e' k ) = m* (2.20) oz c &A'(g) ) ~+P~ I= m'"+

".

king

c'

parameter.

being determined. becomes

&grist

The electron motion in the R frame

To discuss the average motion along the beam direction, we defin a drift velocity (vz&) as the velocity of the frame in which the average momentum is zero. The time average is to be taken over a (laboratory) time that is long compared to the optical period col. ' but short compared to the pulse length T (since even picosecond pulses contain about 1000 optical cycles, this is easy to do at optical frequencies). Since p and E in (2. 14) and (2.15) form a four-vector and q is invariant under Lorentz transformation, va is the velocity of the Lorentz transformation that transforms the average three-momentum to zero:
(P(n)
&

(it)

A(g') dg'
e'

kg

L~'(I )

&&'i'~) )

d(22'&)

The integrals have been left indefinite since the initial

value is of no consequence for the present discussion. The motion is seen to be purely oscillatory with the transverse motion taking place at the frequency of A()1) and the longitudinal motion at twice this frequency. The zero-frequency part of A2()t) has been subtracted out in (2.21) and rid()t) is purely oscillatory. The position, time, and frequency in the R frame are, of course, related to the corresponding quantities in the lab frame by a Lorentz transformation. For example, the frequency of the laser beam in the E frame is given by the Doppler shift of the lab frame frequency (at the peak of the pulse):
(gz2

(/c')

&E(~) &=0,

(2.1'I)

which is solved to become

vD (kz/kz) I q'8'(q) c/I

4+ q'8'(g)

].
I

(2. 18)

We can follow the buildup and decline of VD with the onset and retreat of the pulse explicitly from (2. 18) in accordance with our previous discussion. It is easy to see that an electron at rest entering the leading edge of the laser pulse and picking up the velocity VD takes a time (1 z)D/c) 'T to have the trailing edge of the pulse catch up to it and therefore it travels a distance ~~q'Tc in the lab. This motion of the center of momentum was erst noted by Brown and Kibble' and most recently by Eberly and Sleeper. ' Finally we note from (2. 15) that the increased velocity of the center of momentum causes the electron to gain mass. ' When the electron is at the peak of the pulse, (E') (p')c'= nz*'c', so the new mass is (m*)'=m'+Am' where Dm'= 'q'm'

= ~~'L1/ e2(A2 ()t) &/m2c4]

C. Invariants

(2.22)

We will need some quantities that are the same in the L and E frames these are Lorentz invariants of the problem. We first consider the electron proper time and then the electron four-velocity and acceleration. The phase g is invariant. We will show that the proper time interval dr of the electron, also an invariant, is proportional to an interval d&. In any arbitrary frame, we have, from the definition (2.2),

B. Average

de= P~(dt/dr)

(dr/dr) k.

]dr

Rest Frame

= (1/mc)

(co(E/c)

ck p]dr.

(2.8) and

We can most easily discuss the oscillatory motion of the electron by working in the frame in which the average momentum is zero (the E frame) . This can be done by imposing diferent boundary conditions on (2.6) and (2.9) and solving for a new set of oz's and p's. To be specific, we consider a long pulse that has been turned on in the past and afterward remains constant at P(q) = 1. During the pulse buildup, the velocity of the E frame with respect to the lab increases until it reaches e~.

Now substituting the general expressions (2.9) for the energy and momentum gives
mcdg

= p+ (k/k)
I

oz]&udr,

(2.23)

and we see that the form of the proportionality factor depends on the frame we are in, although the numerical value is the same for all frames. For the specific case of the L and E frame, we find
d~

= ~zdr = ~+ (1+zq2) i&2dr

(2.22).

(2.24)

' T. W. B. Kibble,

Phys. Rev. 138, 8740 (1965).

which simply reflects the Doppler shift

2'142

E. S. SARACHIK

AND

G. T. SCHAPPERT

We can now easily calculate the various four-vectors by differentiating the coordinates with respect to r, or, equivalently, q. For example, the square of the fouracceleration, an invariant that proves useful in the discussion of the radiation, can be invariantly expressed as

a&

= (d'r /dr')

(d'r&/drI)
~

n+kPI' f
kIrIc

dA(g)~'
dg

1 &mc'

where (2.6), (2.7), and (2.24) have been used. This expression can be evaluated in terms of the quantities of any frame by using the appropriate n and P.

of the orbit and slowest on the round part. As a' gets small, the orbit shrinks into a one-dimensional harmonic oscillator common in low-intensity treatments of Thomson scattering. For circular polarization in the R frame, the orbit is a circle of radius V2ak& ' in the xy plane and the electron orbits with constant velocity V2 ac. Finally, we conclude this section by noting that we can no longer apply the superposition principle to the electron motion. We cannot describe the linear polarization case, say, as a linear combination of the right and left circularly polarized cases, since there would be no way of generating any oscillatory s motion by

D. Examples
Consider an elliptically polarized plane wave traveldirection: ing along the

+s

1.

4 2

A(g)
where ~, and

= AOLe, 8 cosg+ e(1

5') '~' sing),

(2.26)

1.

eare the transverse unit vectors and the constant parameter 8 characterizes the degree of elliptic polarization. Linear polarization corresponds to 6 = 0, ~1 and circular polarization to 8= ~2 '~'. In terms of the Ao defined by (2.26), the intensity parameter becomes q' = e'Ao'/III'c'. In the lab frame, we may insert (2.26) directly into (2.6) with P(g) = 1 and obtain

1.

,8-

.6-

kz, r

= ( sing) re, q8

5') + eLg (1 'I' cosg) $g+ (28' sin2g). + 'q'e, 1)

2

(2.27)

Then, using the de6nition of g, the s motion is written

kreis

vj)kgb+ (vz/2c) (28I

sin2g, 1)

(2.28)
kx
~

and we see explicitly that the electron drifts with velocity vz and has second harmonic oscillations (in g) in the s direction. Note that in the case of circular polarization, the oscillating s motion vanishes so that the resulting orbit is helical. We can perform the same operations in the E. frame and find'
k~r~

o2

2aL8, 8 sing "' cosg) e(1 8I)

-6(2.29)

+ e,
where the motion in

',

a'(2P sin2g, 1)

R frame is characterized by the

(2.30)

parameter

a'= q'(4+2'') '.

k~IgI) 16k Is&= k@&g&(4a&

-10-1.214I

For linearly polarized laser light, 8=1, the orbit in the R frame can be found by eliminating g from (2.29):

(2.31)
-.20
I

The orbit is a "figure eight" in the xs plane and is shown in Fig. 1 for various 6eld intensities 0&a &0.5. The velocity as a function of orbit position is
v~I

- IO

.10

.20

= c'P1 (2+2a' k~Ix') ') 4(1 2a')

(2.32)
Fio. 1. Orbital motion of the electron in the R frame for incident
linearly polarized light for various values of the intensity.

so that the electron moves fastest on the straight part

CLASSICAL THEORY OF SCATTERING 8 Y FREE ELECTRONS

circularly polarized light alone. The failure of superposition is not surprising because of the nonlinearity of the problem we have essentially solved for the electron motion under the complete nonlinear

2743

Since our equations of motion are expressed in terms of the parameter g, it is convenient to change the time variable to q, i.e.,

e[E+ (v/c) x Bj
force.

e)gt=g+k~ rn(g). We then define a vector J representing

k~
0

(3 2)
the integral

dpi(g)
exp fi[g+k& r&(g)

III.

RADIATION

We now turn to the radiation problem associated with the electron motion discussed in the Sec. II. As is well known, there are two different, but equivalent, ways of looking at the radiation problem. First, one can compute the electromagnetic fields radiated by the source and hence find 'the energy and momentum Sow at a space-time point (x, t) a large distance from the source. Second, one can calculate the energy-momentum loss rate of the source directly in terms of its velocity and acceleration without any reference to the radiated fields. If the source remains on the average at rest with respect to an observer, the rate of energy loss of the source equals the power measured by the observer. On the other hand, if there is a net average motion between the observer and the source, the relationship between the emitted power and the observed power is more subtle. Both situations will arise in the next few sections. In Secs. III A and III B we discuss the production and angular distribution of harmonics in the R and I. frame, respectively. Obviously, this requires the radiation-6eld point of view. In Sec. III C we calculate the energy-momentum loss rate of the charge directly and compare with the results of Secs. III A and III B. Section III D discusses the radiation reaction.

&(

k&n

r&(g)]Idg,

(3.3)
(3 4)

so that
dp~(~)/did,

= (e'cdirt'm'/87r'c) [I J I'

n' J I'j

If the general solution for kiter(g), (2.29), in terms of its x-y-s components is substituted into the above integral, the components of the integral become

J
where
X(1,2, 3)
0

Jz =

2u~vxlm)

2a(1

(2) 1/2~x

Z. = a'(2e ~x,-, 1)
(cosg, sing, cos2g)
&&

(3.5a) (3.5b) (3.5c)

exp[im(p(g) jdg,

(3.6)

with
q

(g) =g+a'(2l)'

sin'28 sin2g 1) +2a[8 sing cos(r 't' (1 (5')

cosg

cosg

(3.7) (3.8a) (3.8b) (3.8c)

and (see Fig. 2)

cosa=n e, =sin0 cosy,

cosP=n e=sino cos0=n e, .
sin(p,

A. Harmonic

Production

in

R Frame

The general formula for the radiation 6elds generated by an arbitrary electron motion are easily written down. However, since we are primarily interested in the scattered power, it is convenient to work directly with the radiation intensity. Furthermore, since the electron motion is harmonic in the R frame, we take as our starting point the expression for the average power per unit solid angle radiated into the mth harmonic (final polarization not measured)
dna(~)/dQ~
2m/ca~

In general, xp is a rather complicated function of a, 0, q, and 8, and is not expressible in simple form. Making repeated use of the generating function for a Bessel function"
exp(ip cosf)

P n=

(i)" exp(in/) J(p), (3.9)

': = [e'(ei,'m'/(2e. c) 'j

dr&

one can express the integrand in (3.6) as a triple sum over Bessel functions. The integral over g then collapses the triple sum to a double sum, with the result

&&

nx nx

dt

exp ime)I(

f
I

n rii(t)
c

&

dt(3 1)

Xi[ma'(2P

sin'(-'0) $ 1)

The vector n is a unit vector in the direction of observation and rir(t) is the electron coordinate in the average rest frame. The Fourier transform in (3.1) is with respect to the observer's time in the R frame; however, the integral is expressed in terms of the retarded time of the source. -. This"accountszfor the (n/c) rii in/the exponential. ~& is the frequency of the incident Geld.

G. N. Watson, 2 Treatise on the Theory of Besse/ Functions, 2nd ed. (Cambridge U. P., Cambridge, England, 1965).

E. S.

SARACHEK

AND

G. T. SCHAPPERT
(3.15)
reduces to
A (res)8 rLbs sinsn+ (1

qs~G,

dP~&"&/dQ~
n (

sin'Pg, bs)

(3.16)

SCATTERED BEAM DIRECTION)

ez ( NCIDENT BE

DIRECT ON)

FIG. 2. Angular geometry. 8 is the scattering angle of the scattered light.

where sr 2are(1 0s)'Is cosP and ss=2am8 cosa. One now applies the Graf addition theorem" to the sum over e,

which is the usual classical Thomson scattering result. Our general result (3.15) therefore constitutes the intensity corrections to ordinary Thomson scattering in the R frame and is exact to all orders of q', the intensity parameter of the incident laser beam. Because the expressions (3.12) contain infinite sums, the general result for eth harmonic scattering cannot usually be expressed in a tractable form. Only for the case of a circularly polarized incident laser beam does the sum in (3.12) collapse and in this case to a single

Bessel function. Thus, for incident circular polarization, the result becomes"
dP~ &"&/dojr

i)"J-(&)J-+n() Z (

2''A (c /(1+ o~s)

X f (cot'0/2a')

',

qs)

= P(st+iss)/(
and this reduces y( sult is
~

sr sss) )&I'J

((sr'+zs') 'I')

(3.11)

J '(2'~'am

sin0)
sin0)

to a single sum over l. The final re-

+J "(2'I'am

(3.17)

xi
&,.
X

which corresponds to synchrotron radiation from an electron in circular motion with radius V2 ah~ ' and

-)
~

'

J~Lmas(20s

sins(-'0) j 1)
'

j +'+"&~n+r(p) + i +
~

~~

&

(P)

if"+"+'J (p)+if"+" 'J+s( +, (+r

r (p

(a)+ f
where

" '~~

) ),

(I)

)~

(3.12)

velocity V2 ac. It is important to realize that even though the electron motion is periodic in the proper time with frequency csz (0s= s) or cs& and 2&os (Is=1, G), the radiation contains all the harmonics. This is due to both the retarded time factor n r in the exponent of (3.1) and the relation between the retarded time and the proper time. The closed form expression (3.17) for incident circular polarization can be summed over e and integrated over solid angles to give the total power radiated in the R frame:
dI'g&"~
dQ&

i(st iss) f= /(sr+ iss)

= (i/p')
p'

(2am sin0)'L(1

siny cosy7s 0s)'I' i8

(3.13)

de

=8

cop'

2q'

3.18

RIld

= s '+ ss' (2am

sin0)

'(0' cos'2y+ sin'y) . (3.14)

Substituting the expressions for into (3.4) gives the power per unit solid angle radiated into the nth harmonic in terms of the previously given p's:

where B(o&s) =esa&sqs/3c. This result provides us with a convenient sum rule for the harmonics. While for arbitrary incident polarization this summation and integration cannot be carried out explicitly, we will derive an expression for the total power radiated in the R frame. Equation (3.18) will then be a special case of this more general result.

dP &"&/de = $A ((ss') e'/(1+-'q') $

0') X Las x," )s+ (1 x,- [s gras(20s xs~ )s ])s
)
J

8 cosy sin0x, " 0')'~' + (1 siny sing xp a cos0 (20s xs" i'$, 1)
~

-',

(3.15)

where A (ce~s)

= es~~'qs/Ssc.

We note that in the limit

. W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical I'hysics (Springer-Verlag, New York, 1966).

"

the angular distribution of the harmonics radiated in the average rest frame E. In the L frame, the motion of the electron is sot periodic and hence one does not have pure harmonic generation. We could again calculate the radiation from the fields generated by the motion in the L frame, but since the motion in the L and R frames differ only by a drift velocity in the incident beam direction, we can more simply I.orentz-transform the E-frame results into the L frame.

B. Harmonic Production In the Sec. III A we calculated

in

L Frame

"J.Hanus

and

J. Ernest,

Phys. Letters 16, 262 (1965).

CLASSICAL THEORY OF SCATTERING BY FREE ELECTRONS

Consider the expression for the total average power observed in a solid-angle element dQ~ in the R frame:
dPz/=

2745

frame there is a contribution while in the from the uniform drift motion to give
6hz

to (rz, )

g (de(")/dna)dQ~. n=l

(3.19)

= LUz, '(1 cos8z) . PD

(3.29)

Associated with this energy per unit time is a momentum per unit time given by c 'dP&. But we know that energy and momentum per unit proper time are the components of a four-vector. Therefore, to transform (3.19) into the 1. frame, we first Lorentz-transform the energy-momentum per proper time interval and then correct for the time intervals. The L frame moves uniformly with a drift velocity e~ in the negative s direction with respect to the R frame. Hence the Lorentz transformation requires that
dPL

On the other hand, the time intervals associated with the electron motion (retarded times) in different frames are related by the Lorentz time dilation

(3.30)
so that the effective time interval ratio for two observers in the two frames becomes
dtz/dt/2

yD

(1 Pz)

co ) . s8z,

(3.31)

rDdP/2(1+pD

cos8B)de/dtz.

(3.20)

We now evaluate the right-hand side of terms of the L-frame quantities. With
pD

(3.20)

in

1 (4+ q2) 1 q2 0Dc

2)

(3.21)
(3 22)

for an This distinction between the transformation observer's time interval and the corresponding electron's retarded time interval was recently pointed out by Ginzburg' in connection with interpreting synchrotron radiation in astrophysical problems. The Anal expression for the average power per unit solid angle observed in the L frame becomes

one 6nds satisfies

yD

dPz
diaz,

(1 P

1/2

1/2 (1+2q2) (1+ tq2)

(1+-',q') ' [1+-',q'sin2(28z)$'=,

" dPz/'")
do/2

The frequency
o)z,

seen at the angle

0L,

in the

L frame

' &0/2'(1+-'2q') [1+-',q'

sin'(-'28z)

j ',

(3.23)

wh, ereas

the relation between the incident frequencies

(8=0)

is

~ 0 0(1+1qs)1/2 ~

where all quantities inside dPz2(~)/dQ~ are to be expressed in terms of their L-frame values. Similarly, if we calculate the average power per unit solid angle radiated by the electron in terms of its own retarded time in the L frame we do not correct for retardation but use dhz, ' instead of dtz in (3.20). This yields

(3.24)

dPI, '
dQI,

(1 y &q2) 2 ( 1+2q2)

1 ao

We must now distinguish the frequency of the incident radiation from the scattered frequency in each frame. We will only use the notation oP for the incident frequency when confusion is likely to arise. Angles in the two frames are related by
cos8/2= (cos8z

[1+-',q'

sin2(-'28z)

j' , dQ/2

(m)

(3.32b)

The frequency spectrum of the scattered radiation in the L frame does not consist of simple multiples of the frequency of the incident radiation. From (3.23), n&0~0 we obtain (3.24), and the E-frame result 0)/2
&d&(n)

(1 cos8z) PD) PD
(128&)

',

(3.25)

220)&0[1+1q2 sin2(18&) j 1

(3.33)

so that the solid-angle ratio becomes

dQ/2/dQz=

(1+12q ) [1+12q sin

. (3.26)

The evaluation of dt~/dhz is more subtle. The hh refers to the observer's time interval, which is related to the electron's retarded time interval dt' through
At

= h2t'

'), (1/c)n hr(t

with all times measured in the same known from the equations of the electron motion in a given frame and n is a unit vector in the direction of the observer from the origin of the r coordinate system. Since we are dealing with optical frequencies, we can only measure energies over a time interval containing many optical cycles we must therefore average. In the R frame (r/2(t') ) =0, so that

(3.27) frame. r(t') is

which for m=1 is commonly called the intensity-dependent frequency shift. The meaning of eth harmonic in connection with the I.-frame quantity dPz, &"'/diaz, then must be understood in the sense of (3.33). For q'~1 any frequency ~&~1,' is radiated we only have to look at the appropriate angle. At large q', however, the angular distribution in the L frame is sharply forward peaked so that only a small range of 01, contributes to the angular distribution. We will discuss this in more detail in Sec. IV. The relation (3.32a) between the average power observed in the L and E frames may be interpreted in a very physical way. Consider the number of photons observed with frequency cu& in a solid angle dQL, and a time interval At&. Clearly this same number of photons

htg

= Et'',

(3.28)

"U. L. Ginzburg, V. N. Sazonov, and S. I. Syrovatskii, Usp. Fiz. Nauk 94, 63 (1968) (Soviet Phys. Usp. 11, 34 (1968)g.

'

2746

E. S. SARACHIK
co+

AND

G. T. SCHAPP ERT
(2.26) substituted into (2.25), we
aa&= 'c-'(&or, ') q'[1
And

is observed in the R frame with frequency solid angle dOg and time interval At~. Hence

in the

X= (S~,/a~, ) (dP, /dn, ) dn, = (Atgg/5(o~) (de/de) de,

cos2q]. (28' 1)

(3.40)

(3.34)

which is equivalent to (3.32a). The relation between the power radiated and the power and number cross sections is given in the Appendix. For the case of incident circular, polarization, the solid angle integral over (3.32a) can be performed by transforming the integral back into E-frame variables. The result for the total power observed in the laboratory
1s

This then gives, for the instantaneous energy-momentum loss per proper timbre interval of the electron evaluated in an arbitrary frame of reference speci6ed by ~&,

dII&/dr=B(ul, ) [1 (28
The time component
stantaneous

cos2g](v&/c ). (3.41) 1)
in-

is related to the invariant energy emission rate

R=w(dIII'/dr)

= (c/y) (dII'/dr) =c(dII'/dt').

(3.42)

'q') Pg I (1+ ~q') '(1+ '+ 6'4 (1+~q2) '[(23/4) P~ 'q'+ (3v2/q) (1+-'q') 31' + Sq'+3
&&(1+2q')
l

q4

((&l) q+(1+lq')"'nI
(3.35 a)
limits to

and this reduces in the low- and high-q

P& Pp[1+ (7/80) q'],

This energy loss rate is with respect to the charge's retarded tim" we must still distinguish the observer's time t from the electron's retarded time t'. To obtain the average energy-momentum loss rate of the electron, we must average the instantaneous rates over times that are long compared to the periods of the motion. Thus the total energy-momentum lost by the charge along a world line containing many periods of the motion is

q'((1

(3.35b)
(3.35c)

II~(A) II~(B) =

&A

gQp

gg

d7

d7.

(3.43)

J.-I'~sq
',

' where Pa B(~1,) = (1+ q') B(~~') is given in (3.18) . The reasons for the difference between I'I, and I'~ will be discussed in more detail in Sec. IV.
C. Power Radiated by Electron
Up to now we have discussed the radiation problem from the electromagnetic 6eld point of view and have calculated the power received by observers in different frames of reference. Let us now consider the source loss point of view and calculate the energy-momentum of the electron directly from its acceleration and ve-

The average energy-momentum loss rate in a given frame is then the value of the above integral in that frame, divided by the electron's time interval corresponding to the proper interval along the world line: '~ dII~ = [f(rg) . dr. (3.44) t'(rs) ' dt'

DIP.
~

].

We can now easily evaluate the average using (2.24)

and

(2.29):

d(~'/dr

= y~

-'q') v~% = (1+ '~'[1+ a'(2P ' cos2g] 1) (3.45)

8') 1) + 2ae(1 'I' sing+ a'e, (28' ' cos2q],

and

locity. We begin the discussion with the energy-momentum four-vector rate at which radiation is leaving the
charge'4

dr~/cdr

= y~ p~ (1+-, '~'[ 8 cosy 'q') 2ae.

(3.46)

dIII'/dr

= ,(e'/c4) '

age" (~/c).

(3.36)
chlI g'/htp'
B((oz') [1+', q'

component refers to the energy and the @=1, 2, 3 components to the momentum vector. a& and v& are the acceleration and velocity four-vectors of the electron:

The

p,

=0

so that the average energy per unit time lost by the electron becomes

3.47) Sq'(2P 1) ']-(

S' (2 '. 1)

and the momentum

lost per unit time,

vl"

= dxl'/dr = (yc, yv (~) ),

(3.37) (3.38) (3.39)

aIIa/at g'
In the

e.B((op') (q'/Sc)

(3 48)

with
'Vp'V"=

(C 'y

)=

al'

= de~/dr.

We have already calculated the invariant u&a from the equations of motion. With the vector potential
'4 P. Rohrlich, Clussicul Reading, Mass. , 1965).

Churged

I'urticles

(Addison-Wesley,

there is no net average motion between observer and source. This results in equal-time intervals for observer and source, as shown in (3.27) and (3.28) . Therefore we can equate the average rate of energy loss (3.47) with the total average observed power in the E. frame given by the sum and solid-angle integral over (3.15). Hence (3.47) furnishes us with an exact sum rule for the harmonics even when the sum and integral over (3.15) cannot be performed. For

E frame

CLASSICAL THEOR

Y OF S CATTERIN G

8 Y FREE ELECTRONS
D. Radiation Reaction

2747

circularly polarized incident laser radiation in the E. frame, the identity between power loss of the electron (3.47) and the power observed (3.18) is explicitly demonstrated. We also note that the R-frame momentum loss rate (3.42) is zero for incident circular polarization (6'=-, ) which implies a symmetric angular distribution of the radiated energy with respect to the 0=-', x plane. In the L frame one again easily calculates (3.44). From (2.24) and (2.27) we And
dt's'/cdr

Our equations of motion for the electron in an electromagnetic field have up to now neglected radiation reaction effects. We will now estimate under what conditions this neglect is justified. Our discussion is based on the Lorentz-Dirac equation

ma" = F"+I'",
where

(3.55)
(3.56)

= yr, = wro/c = 1+~~q'L1+ (2P cos2g] 1)

(3.49)

F~= (c/c) (a~A" a"A~ ) v,

in terms of the vector potential (3.32), and
I'&

and the four-velocity

drr/cd' =yzPs,

't' sing qe(1 P) 1+ (28' cos2g]. (3.50) ' +e,,q'{ 1)

qe,
6 c osg

32

(e'/c') f(da&/d~)

aga"e&] (1/c')

(3 57)

The average rate of energy and momentum

laboratory frame then become, respectively,

c,
AIIz,

'

=B(cog') 1

',

',
-',
~

loss in the

q'(28' ' 1)

1+4q

(3.51)

and
Atr,

'

= e, '

'
c

1+-;q'

L1

(2P '] 1)

(3.52)

In the L frame, the average energy loss rate (3.51) is not equal to the total average power measured by an observer. In (3.29) we found that the retarded time interval is not equal to the observer's time interval, but is related to it by ddI, (1 cos01.). This implies LUr, Pz a relation between the emitted and observed power of the form

Pz

f d
kdQz

Ellz )
Atz
~

dQz
1
~

Pn cos0z

)c,
Ally

Atz'

.
~

3.53

The energy loss into a given solid angle, when integrated, satisfies
~IIO/At'=

Our previous discussion of the equations of motion corresponds to (3.55) without the I'&. The Abraham radiation reaction four-vector F& consists of two terms. We recognize the aiba"e& part as the four-vector negative of (3.36), the energy-momentum rate at which radiation is leaving the charge. Strictly speaking, this by itself is the radiation reaction. The da"/dr term, referred to as the Schott term, gives rise to a nonlocality in time in the formal solution of (3.55) . In the nonrelativistic limit of the theory, the daI'/dr term reduces to the familiar da/dt damping force which is derivable by considering the energy transfer from the incident field to the scattered radiation. We note that there have been many misconceptions about radiation and radiation reaction which have only recently been clarified '4 For our purposes, it suKces to calculate F& and require that its magnitude be small compared to that of the driving force FI'. From (3.37)-(3. 0) we easily 4 calculate the four-vector (expressed as a column matrix)
I'~ = -', (e'/c') (oz'

I { (d/dQ)

(Allo/dd') ]dQ,

(3.54)

and because there is a net forward momentum loss, (3.52), the angular distribution in (3.54) is forward peaked. This accounts for the inequality in (3.53) . The fact that the observed power radiated is larger than the energy loss rate of the source is explicitly demonstrated for circular polarization, where both rates have been calculated exactly { compare (3.35) with (3.51) after setting P= ', The above situation must not be thought to violate conservation of energy. Because of the net relative motion between electron and observer, the energy stored in the region between the observer and the electron continuously changes. When the electron comes toward us, the stored energy decreases and the observer measures the rate of energy emission from the source plus the decrease of the stored field energy. Because of the forward peaking of the radiation, we always observe more than the electron emits.

'q'L1+ (28' cos2g] cos~2g] 8q4L1 (28' 1)' 1)

qb

cosg {1+',q'Ll

cos2g] I (25' 1)

(2P 1) P) q(1 'I' sing {1+-'q'L1 cos2g])

].

' cos'2g] Sq'L1 (25' 1)

(3.58)
Similarly, the force due to the incident field becomes

-'q (1 25')
F~ = mccol, q
8 sing

sm2g

"

"' cosg (1 P) q(1 sin2g 2P)

-',

(3.59)

E. S.

SARA

Hl K AND

G.
(3.10) (3.17)

T.

S CHAP

PERT

Clearly one cannot require that each component of F& be less than the corresponding components of FI'. For incident circular polarization P' and F' are zero but not and I'. If we demand that the magnitude of the nonzero components of IiI' be greater than any of the components of F&, we obtain

I'

degenerates into a single Bessei function. Using and (3.32a), and expressing all angles in terms of I.-frame angles, yields the explicit result for incident circular polarization:
dI'z'"&/diaz
A(o& 2&o'/1+-', q' sin'(-', 8) $'J z ,')
&

ro/Xoz,

&(1,

q'( 1
qo)
1

(3.60a)

q' sin'8

roq'/Xoz((1,

(3.60b)
where

(4.3a)

where ro is the classical electron radius and Apl, is the laboratory wavelength of the incident field. The above restriction must then be satis6ed if the radiation reaction is to have a small e6ect on the electron motion. It should be realized, however, that even if this effect is small, it may, over a sufIiciently long time, induce a significant alteration to the electron's motion. Such radiation pressure corrections have been discussed by Sanderson~ and Kibble. '

e = q sin8/V2L1+

oqo sin'(zo8)

j.

(4.3b)

If we expand the Bessel functions for low q', we And the leading term
diaz(m&

dn,

=A(o&zo)

4 (q2) n

n 1(&o2

8-((~ 1)!)

(sin'8)"

'(1+ cos'8).
(4 4)

IV. LOW- AND HIGH-INTENSITY BEHAVIOR

We have seen in Sec. III that the power observed in the laboratory frame is not the same as the power lost by the electron in that frame, the difference being due to the retardation eGect introduced by the net motion of the electron's R frame with respect to the laboratory. Since the observed power is the experimentally important quantity, we will consider its behavior as a function of intensity and make specific lowand high-q2 predictions. Let us first note some exact results for arbitrary polarization and arbitrary intensity for forward and backward scattering. In the forward direction, scattering takes place only at the incident frequency, all harmonics higher than e= 1 not contributing:
(dI'z&"&/diaz)

We see that the power observed at low q' is symmetric with respect to the 0=90' plane and that, except for n = 1, it vanishes in the forward and backward direction in accordance with the discussion following (4.2). The integral over the solid angle in (4.4) can be performed using the beta functions" B(x, y) = 1'(&) 1'(X) /1'(&+X):
2
0

(sin'8)" '(1+cos'8)d(cos8) =B(o& r&)+B($, e).

(4.5)
The total observed low-q' eth harmonic power is therefore
Pz,

" = B(o&z') 3(~) '""(q') '(~+1) . (4.6) 2~

&

(2~+ 1) !

(8= 0) =8,&A (o&zo)

(4.1)

In the backward direction, scattering occurs only at the

odd harmonics
(where

n=2s+1):

(diaz"'/dDz)

(8= or) = A (o&z') L(2s+ 1)'/(1+-'q')

Xt (J,+J,+z)'

'j
(4.2)

4', J,

p&7.

The arguments

(4.2) are (2P (2s+1) go. For circular polarization, 1)

of all the Bessel functions appearing in

We note that in the low-q' limit, the leading term in I'I'"' is identical to the leading term in the corresponding result for the E. frame I'~&"), since the Lorentz transformation between frames is itself at least of order q'. We can expand the observed power beyond the leading term and in so doing test the prediction (3.35b) for the total observed I.-frame power to order q'. The results will also provide specific experimental predictions for the low-intensity region. Thus, to order q', we have radiation only at the first three harmonics:
dI'zo&/doz
A(o&z')

the expression in brackets becomes b, , p, so that none of the harmonics higher than e = 1 have either forward or backward scattering.

-'L1+ cos'8 'q'(11 cos8+ 6 cos'8 4

A. Circular Polarization

cos'8 + (1/768) q4 (997 cos8 864 cos48) 12 1248 + 27 cos'8 cos'8+ 891 cos'8+ 192 cos'8
+5 cos'8) $, (4.7a)
dP/diaz

Next we consider in detail the low- and high-q' behavior of the I.-frame observed power for the specific case of a circularly polarized incident beam. This case is, as we have already noted, the only incident polarization for which the infinite sum over Bessel functions in

= A (zoz') q'L1 q'(13 cos8 cos48 o 6

(4.7b) (4.7c)

"T. W. B. Kibble,

Phys. Letters 20, 627 (1966).

cos'8 cos'8+12 cos'8+2 cos'8) $, 6 9 cos'8 dI'z "&/dQz = A (o&z') q'$1 cos'8+ cos'8)

CLASSICAL THEORY OF SCATTERING BY FREE ELECTRONS

Integrating over the solid angle gives the total power observed at the first three harmonics to order q'.

2749

tions" for
(426)

e&1:
1/4

PL"' =B(~L') L1 (6/5) q'+ (g1/70) q') Pr t = B (4pr, 2) [(6/5) q' (83/35) q ),
Pr, /2'

(4)
/4

02 1
2

p )1/2

3~')

E1/2(22'4i

)+O(22

gbi,

(4. 11a)
1

82't'/4 f K j'(420) =
e
~

+0(/2 ( '24/' ')

' '),
(4.11b)

= B (4pr, 2) [(729/560) q').

(4.8c)
where
2(2/2

Summing these three expressions gives the total power observed in the L frame to order q':

ln

(1+ (1
~

()2) 1/2

'" (1 82)

(4. 11c)

Pr,

B(4dr, 2)

[1+(7/80) q'),

(4.9)

which agrees with our previous result (3.35b). Note that if we calculate the total power lost by the electron in the L frame according to (3.32b), we find that to order q it agrees with the sum rule (3.51). Turning now to the high-q' behavior, we find that if we consider nonforward angles (i.e., angles for which q&)1 implies q'sin'(-, 8)&)1), there is very little radiation at any harmonic. To make this explicit, let us consider the three regions in which we have approximate expressions for the Bessel functions, namely, for e((q, for e q, and for e))q. Using then the approximations to the Bessel functions for small argument and fixed order, for fixed argument and large order, and for large argument and large order, respectively, we find

and the E's are the modified Bessel functions of the second kind related to the Airy functions. The observed power at the 12th harmonic (for 22 large enough for the first term of the uniform asymptotic expansions of the Bessel functions to be valid) is
dQr,

=A(

2 )' 2r' (V") 'q' '8) 2[cosH sin'( )'

, [1+-'q' sin'(-'8)
sin28(1
1
q2

"),
+(

2e'

e2) 1/2
1

()2) 1/2
Q2

K2/2 (242f'

) .

(4.12)

dgl

( )

= A (4pr, 2)

4 (242) n (sjn28) n

dQr,

(q') "+'[2 sin'(-', 8) )"+'[(I !)' 1)

1
'

When the argument of the E functions becomes large compared to unity, the Efunctions a'ppearing in (4.12) become exponentially small according to" (4. 13) K1/2, 2/2(s) ~(2r/2s) 1/2e *.

1~q
dP~t~&

(4.10a)

dQr

= A (4pr, ')

16242(12e'

sin'8)

'

2r(q2)n+3[stn4(128))n+1

e~q))1
(4.10b)

We see therefore that the power radiated at the higher harmonics will be negligible if the condition 242''2/2&&1 obtains. We can verify from (4.10c) that, as a function 22ip/2 of qH, is of order unity except near qH=+8. At qH= +8, 8= 1 1/q' and 22''2/' becomes
f
2/2
~

[ (1q2)
2/2) 1
22

(4 14)

dQz

= A (4pr, 2)
2rq'

sin

(-', 8)

sin'8

)&

exp

2e

ln

v2 q sin-', 8
sin8

-i),

e&)q))1.
(4. 10c)

We conclude that there are no significant amounts of radiation at high q' for any nonforward angles. If we look, however, at forward angles (i.e., those for which 8 1/q), the situation changes. We will see that at high q', essentially a/l the radiation is emitted at the sa2/2e near forward angle 8p= (+8)/q. This situation may be traced to the inevitable forward peaking of radiation emitted from a rapidly moving particle due to I.orenz transformation rather than to any intrinsic structure in the angular distribution. In order to analyze the region q'))1, 8 1/q, we use the uniform asymptotic expansion of the Bessel func-

may be pi'2/2 violated. Away from forward directions, is itself of order unity so that for e)&1 the condition holds and there is negligible radiation [we already know this from our discussion of (4.10)). Near 8=8p (+8)/q, however, there will be significant amounts of radiation for all harmonics for which e&mp, since our condition is there violated, but insignificant radiation for e& sp, since there it is not. The radiation is thus confined to the near neighborhood of 8p and occurs only for those e's below the critical harmonic ep, dropping off exponentially above the critical harmonic. If we look precisely at the angle 8p= (+8)/q, the magnitude of the observed radiated power in the eth harmonic is
42-, f'2/2&)1

Let us consider how the condition

/PE(s)
dQI

(8=8p) =A

(4pr,

2),
242

,
1

(1

()2) 1/2

K2/2'
Sp

"Handbook
Abramowitz

(4 15)
edited by Mathematical Functions, and I. A. Stengun (Dover, New York, 1965).

of

M.

2750

E. S.

SARA

HI K AND G.
2

T.

CHAP P ERT
observed
in the laboratory
i~

where the first term of (4. 12) is of order q and has been neglected. Using the relation"
E'~~3(m/No)

smaller

total power becomes

dPr, /der,

frame thus

= I'( ) (2rso/e)
',

2i'

(1&&n(no)
Op.

(4. 16)

A ') {(1 0') (a&r.

sin'(-,'0) )'

q' /8[1+

sin'(-', 0) 7 )

gives the magnitude

of the observed power at

(dPI, &"&/dpi, ) (0 = Oo)

= A (~r,') [(P.41] ) '~@'/8$

(1e&~,). (4.17)

'q' [1+-,

(4.20)
This result is valid for arbitrary q and, in Fig. 3, we plot it for various values of the intensity. At large q', (4.20) can be written '~' t' A (( r,') / 7 ' 50' 00)'l dPr, (0
dQ

Notice that this expression is only trivially dependent on q' through the incident~beam term A(arL, '). The power cross section corresponding to (4. 17) (see the Appendix) is, in fact, independent of q'. We can estimate the total power at ep by summing over e up to the critical harmonic ep using the approxirnate formula (for large eo)

128

&12

4
0 0

(AO)'
2

j
(4.21)

+7(-' ')'I' 1+

5/2

(SO) 2

g rP~'= n=l
to obtain
(dPr/der, ) (0=00) =A

np

(no)"'

(a)1,

') (0.014) (q')'"

(4. 18)

The rapid increase in the total observed power with q2, indicated by this expression, is due to the rapid opening up of new harmonics rather than to any increase in power radiated in the individual harmonics. The power observed in each harmonic is constant [aside from the trivial q' dependence in A(idr, ')) but the number of radiating harmonics increases as q', thus leading to the q' dependence of the observed amplitude in (4.18). We will see below that the estimate (4.18) is quite a good approximation to the exact result. ' We can summarize the essential features of the angular distributions for the observed power at high q' as follows.

where 60= (Q8) jq'. We again see that the first term is smaller than the second by a factor of q '. At the peak 0 = Oo the estimate (4. 18) is seen to agree quite well with (4.21). If we integrate over the solid angle of the dominant term in (4.21), we get an estimate of the total observed power
PL,

') B((or,

7'

8X~'

q' '

(4.22)

(a) The radiation is confined to the region of the near forward angle Oo (+8) /q, all radiating harmonics peaking at this same angle. (b) Near this peak angle, all harmonics below the critical harmonic mo 3( ', q')3i' radiate strongly. Thus as the intensity of the incident laser beam increases, the spectrum of the scattered radiation shifts upward. (c) At the angle 00, the magnitude of dPr, /dQ increases as A (~i,') q5.
Let us now turn to the total power observed at a given angle. We can in fact sum (4.3) over all the harmonics using'
O') g ii'J.'(ii&) = &'(4+ e') /N(1 ", n=l

which agrees favorably with (3.35c) . Up to now we have assumed that the polarization of the scattered radiation is not measured. Each term in (4.3a) does, however, correspond to a unique linear polarization direction. If we take n to be the direction of the observer, the linear polarization vector corresponding to the second term of (4.3a) is perpendicular to both n and the scattering plane (i.e., it is in the plane transverse to the initial beam direction) . The polarization vector corresponding to the first term of (4.3a) is perpendicular to n and lies in the scattering plane. Since as we have seen, at high q' the second term of (4.3a) and (4.20) dominates near Oo, we conclude that essentially all of the observed scattered radiation is linearly polarized in the plane transverse to the initial

(4.19a)
(4.19b)

n=l

0')'i' Q e'J "(eB) = (4+39')/16(1

(Note that the 6rst of these expressions is incorrectly The angular distribution for the given in Watson.

")

beam direction. We have analyzed the observed scattered radiation, for circularly polarized incident radiation, by analyzing the radiation formula (4.3a). All the results we have obtained may be simply understood by recalling that an accelerating particle in relativistic motion synchrotron-radiates direction. along its instantaneous The motion at high q' for the case of circularly polarized incident radiation is a helix of pitch angle g~=mpvD where ~z is the transverse velocity and v& is the longitudinal drift velocity. Inserting the previously found values of these quantities yields 0& (+8)q '. It is clear therefore that only an observer directly along the pitch of the helix sees anything at all. This explains the characteristic peaking at the near forward angle (Q8) q '. Similar arguments for the width of this peak

CLA

SI CAL THEOR

Y OF

SCATTERING 8 Y FREE ELECTRONS

To order q',
dP&M dQr

2751

and for the polarization of the observed radiation can be given to reinforce the analysis of this section.

6rst three harmonics ~12 1+iq2 sjn2 i0 4 A

we find for the

B. Linear

Polarization

X I sin'n+q'(Ai
'

cosign ',

cos0)
&i!

We will briefly discuss the radiation produced by a linearly polarized incident laser beam. The mathematics for this situation is relatively intractable so that we can write the exact solution but can analyze it only in the case of low q'. We expect many of the features of the incident circular polarization case to still apply here at high q', since many of those results were characteristic of the drift velocity which is the same in this case. The exact result can of course be put on a computer if more detailed high-q' results are desired. The observed average power distribution in the laboratory frame for linearly polarized incident radiation is
de L~)
dQr,

Ai f +q [A2 siil (20) cos ~'~ + , sin'(20) cos'n cos'n sin'8 cos'u sin'(-', 8) cos0+ 4 cos'n cos0 sin'n sin'8] I, (4.25a) @'~
-',

where

Ai 4[ sin'a cos'n

sin'n sin'(-', 0) + cos'n cos0], A2= ~', sin'a. cos'a sin2(20) sin~n sin'(20) + (5/192) cos4n sin'n cos'n cos0 8 cos'n sin'(-', 0) cos0+ ~'~ sin'(-', 0) cos'u, 8'2
~'&

(~r,')

e'
[1+-',q'
sin'(-', 0) ]4

dPI &2i/dQr

4q2A

(~r 2)

(1+-', q') cos'n

3 X[Ci+q'(C2 sin'(-', 0) Ci cos4n cos0 4 +' cos'n sin'0]), (4.25b) ,

[1+-',q'

sin'(-', 0)

]'
P~ P2

where

cosn[cos0 sin'(-', 8) ] 2q'

[1+-'q2 sin2(-', 0) ]2

cos'n+ ', sin'8, Cg = 3 cos A cosg sin A cos cx sli1 8 cos A' 3 8
Ci sin'u cos'n
-',

dPI, /dQr,

= (81/64) A ((vr, ') q4

where

+",

cos0 cos'n) '+ sin'0 cos'a 6 cos0 cos'n) ]. (4.25c) 6 + cos'0 cos'u(1
X [i~ sin'n(1
sin'(-', 0) ',q' siii'(-', 1+ 8)

The corresponding

total powers radiated at these har-

j
p

monics are

gs cosa
gR cosa!

Pl,

&'&

= A ((ol,')

(8ir
~

1311m. 9ir q'+

3
5

&

1120

q4
~

+J2&~,

.. .,
]

. (4.23b)
Pr, & = A
&

4~ tt'+ + ( 3
35 k5 =A(, )[(621 /224)q], P,
~

227m

120

q',
~, j'

&4. 26a)

We can expand (4.23) to low order in q' an. d find the leading term for r&) 2. [For n=1 the leading term is simply the linear polarization version of Thomson scattering, i.e. , (3.16) with 02= 1, 0. The leading term then becomes
dPi&"&/dQr.

((ur. ')

(14ir

q'

138~ q' q
15

(4.26b)

(4.26c)

= r&'A (a)r,')
X [sin'n(Fi")
2

'q ,cosn cos0 Fi"Fp

i-q +
sin 0(F2")

~" (ecosn)" " " F "= (-'q)" " Q ( Pe sin'(-'0) ]' (B r & ll 21)

where

1)',

']

(4.24a)

(4.24b)
and is that value of 3 that makes either 0 or 1. For low q' we can expand beyond the leading term.

I.

e r 2l

where these quantities are written in such a way that the first parenthesis gives the result in the E. frame'~ and the second gives the corrections for transforming into the frame and for the difference between emitted and observed power. As we go to higher and higher q', the angular distribution changes from predominantly backward to more and more forward. This may be understood by looking at Fig. 1. At low q', the E- and I-frame results are the same since the drift velocity is of order q2. The electron radiates most on the curved portion of the figure eight, and on this portion, the electron is going backward. The radiation of the higher harmonics is

I.

'7 E. S. Sarachik and G. T. Schappert, Nuovo Cimento Letters 2, 7 (1969).

2752
225200
q*

E. S. SARACHIK

AND

G. T.

S CHAP

PERT

=50

175

I50
I25
b4

100-

50
25
I

20

00

60

80
ANGLE}

100'

l20

8 (SCATTERING

Frc. 3. Plot of the total observed power cross section in the

laboratory

q'=3

for various incident intensities. The scale for the and 10 curves is one-Qth that shown here.

therefore backward peaked at low q'. As q' increases, the drift velocity always increases faster than any internal velocity characteristic of the figure eight so that the radiation becomes more and more forward peaked. We expect the same qualitative behavior of the high-q radiation as in the circularly polarized case.

V. CONCLUSIONS
We have investigated the motion and subsequent radiation of an electron (or any charged particle) subject to an intense optical field. We have obtained an exact solution (3.12) and (3.15) for the expected observed power for arbitrary intensity and elliptic polarization of the incident optical field. Our two major assumptions have been that the problem could indeed be treated classically and that we could decouple the problem into a motion part and a radiation part within the classical approximation by neglecting radiation reaction. We will discuss these and other assumptions briefly in this section. That classical theory works for this problem has been shown by Brown and Kibble' and been reviewed by Kibble. 4 The incident frequency must be such that Acemc, a condition that is clearly satisfied for incident optical radiation. The meaning of the condition is that the incident photons must not individually have enough energy to cause the electron to recoil. So too in the emission of an eth harmonic photon, classical theory cannot account for the recoil of the electron so that we must impose the restriction eh~((mc' in order for our treatment to be valid. We see that for incident radiation of 5~~1 eV, our theory no longer holds for harmonics higher than about 10'. Note that this restriction is independent of the intensity. We have already seen that the neglect of radiation reaction requires that q'((X/ro which means that q'((10' at optical frequencies. This clearly is a very loose restriction and need not concern us further.

We have made some assumptions about the pulse length of the incident radiation that can be made more explicit. The pulse length must be taken long enough for there to be many optical cycles contained within it. Thus the pulse must be"longer than about 10 sec. For pulses shorter than "this, the averaging we performed can no"jonger be done. In addition to this averaging problem, we must realize that a pulse in the vector potential is no longer the same pulse in the electric field at these short times and we must begin to ask ourselves which one really comes out, of a pulsed laser. The pulse also cannot be too long the longitudinal drift distance 4'q'Tc of the electron must be small compared to the scale of the experimental apparatus so that the angle of the scattered radiation be well defined. Since at q'=1 a picosecond pulse corresponds to a drift distance of only 0.03 cm, this leaves us considerable leeway. However, an increase of intensity by several orders of magnitude would require still shorter pulses. Finally, let us remember that our calculations have dealt with scattering from a single electron. In an actual experiment, an electron beam or a plasma would be used to provide the electrons. When many electrons are present, we might expect the usual coherence eRects in the forward direction. Since we have seen that only the fundamental frequency is scattered forward, we conclude that there are no coherence effects for the higher harmonics. Other plasma eRects would be expected to enter only if the density of the electron beam approached that of a metal. In most experimental situations, the density would be such that the plasma frequency is far less than an optical frequency, so that plasma eRects would not enter to change the conclusions of this paper.

"

ACKNOWLEDGMENTS
We would like to thank Dr. Kenneth Billman for interesting us in this problem and for his constant encouragement and support. We would also like to thank Dr. David Burnham for many helpful discussions. One of us (GTS) wishes to thank Professor T. W. B. Kibble for valuable discussions.

APPENDIK

In the body of the paper, we have expressed our

in terms of the power angular distribution dP'"&/dQ. This quantity seemed the most natural one to use since it corresponds most closely to what an experiment would actually measure. Since other quan-

results

tities, namely, power cross sections and number cross sections, have appeared in the literature, we will here clarify the relations between all these quantities. In a given frame, the power cross section is defined as (1/I) dP/dD where I, the incident intensity, is given as e'q'&o'/Sn. cro'. We have written the equations in the text of the paper in such a way that replacing 2 (a&') by r02 and B(~') by (8/3) vrr0' changes the expressions for

CLASSICAL THEORY OF SCATTERING

I""& into those for do&"&/dQ and a&"& (the corresponding power cross sections), respectively. The third quantity, the number cross section, appears the least useful from the experimental point of view yet has appeared in some of the standard papers on the subject, since it is most natural from the point of view of quantum-mechanical calculations. The differential & number cross section in a given frame, dZ "&/ dQ, is defined as the number of eth harmonic photons detected in a solid angle dQ divided by the number of (firstharmonic) incident photons all in some unit time interval. In a given frame, the relation is easily seen to be
dI'~"&/dQ and
dZ

BY

FREE ELECTRONS

2753

In particular, in the
dpi.
n

I. frame

this relation becomes

= L1+ 'g'

'8) sin'( 7

JI.

dQz,

(A2)

" &/&dQ= (&o'/pp, p) (da &"&/dQ) = (&o'/&o. . ,) (1/I) (dI'&"&/dQ)

(A1)

If we compare our results to those of Brown and Kibble, ' we note that it is the number cross section derived from the power lost by the electron that agrees precisely with their results. They did not consider the extra time retardation that must be included when going from the power lost by the electron to the power observed in a given frame. It is the observed power rather than the power lost by the electron that is actually measured so that this extra retardation must be included when comparing theory to experiment.

HY8 XCAL RE VIEW

VOI UME 1, NUMBER 10

15 MA Y 1970

New Dynamical Grour& for the Relativistic Quantum Mechanics of Elementary Particles*

J. J. AGHASSI, P.

ROMAN,

AND

R. M.
1969)

SANTILLi
OZZ15

Depart@sent of Physics, Boston University,

Boston, Massachusetts

(Received 13 November

Nonrelativistic Galilean quantum mechanics and the standard transition to relativistic Poincare quantum mechanics is analyzed in terms of group theory. Special emphasis is given to the discussion of the relation between dynamics and geometry. Certain unsatisfactory features are pointed out and a new relativistic group g5 is suggested as the symmetry group of dynamics. g& contains both the nonrelativistic Galilei group and the Poincare group as subgroups, and it is a group extension of the restricted Lorentz group. For use in relativistic quantum mechanics, the central extension of 85 by a phase group must be employed. The Lie algebra of this relativistic quantum-mechanical Galilei group g5 contains an acceptable covariant space-time position operator and a nontrivial relativistic mass operator. The latter also serves to describe dynamical development. The irreducible unitary projective representations of g5 correspond to infinite towers of states with increasing spin.

I.

GROUP-THEORETICAL ANALYSIS OF NONRELATIVISTIC QUANTUM MECHANICS

ics. The underlying geometrical manifold is, to start with, the Euclidean space E3, where the identity cornponent of the group of isometrics is the connected Euclidean group, i.e. , the inhomogeneous rotation group ISO(3)=Tp&3SO(3). This space does not permit even the formulation of any dynamics. One therefore introduces the time as an additional kinematical variable and thereby changes the underlying manifold from E3 to E3&&E1. Note that no metric is introduced into this Cartesian product space. Next one demands that the laws of motion be invariant under the connected component of the Galilei group. This group we shall denote in what follows by the symbol b4. The carrier space of &&4 is Ep&&Ei and the group is obtained by adjoining to the transformations of ISO(3) the additional two sets of transformations

"NDOUBTEDI. Y, the most remarkable feature of

relativistic dynamics is that the invariance group of the dynamical law coincides with the group of rigid motions (essentially the group of isometrics) of the underlying geometrical manifold. In fact, the underlying geometrical manifold is the Minkowski space E3,1 where the identity component of the group of isometrics-is the connected Poincare group containing the identity, i.e. , the inhomogeneous Lorentz group' = T4&32+t. At the same time, the laws of ISOp(3, 1) motion are required to be invariant under ISOp(3, 1). The situation is very diGerent in nonrelativistic phys-

'For convenience, in this paper we shall use the symbol SOO(3, 1) for the restricted Lorentz group 2+~, even though this notation is not quite standard.

*Work supported AFOSR-67-0385B.

by the U. S. Air Force under

Grant No.

(1.1a)

t~t+r,

(1.1b)