New Effects in Relativistic Thomson Scattering
Prateek Sharma
Princeton Plasma Physics Laboratory1
PrincetonNJ 08543
Gennady Shvets
Illinois Institute of Technology, Chicago IL 60302
and
Fermi National Accelerator Laboratory, Batavia IL 60510
Abstract. We discuss two essential features of high intensity laser-plasma interactions: time
retardation and nonlinear figure-eight motion. We focus on the fundamental and 2-nd harmonic emission in the direction of laser polarization, and make two theoretical predictions.
First, the intensity of 2-nd harmonic (n = 2) radiation in the polarization direction varies nonmonotonically with laser field intensity, and even vanishes for laser intensities near a 0 = 2.2.
Second, there is a non-vanishing radiation at the fundamental (n — 1) harmonic in the polarization direction. These two effects are enabled by simultaneous occurrence of figure-eight
motion and time retardation. We argue that the most effective way of detecting relativistic
effects is by observing n = 1,2 harmonics in the direction of polarization. Suggestions for
the optimal acceptance angle of such a detector are also given.
INTRODUCTION
In a relativistically strong laser pulse (/ > 1018W/cm2) magnetic ev x B/c force
acting on a plasma electron can become comparable with the electric force eE. This
significance of the magnetic field defines the area of high-field science which has come
into prominence in the past decade [1]. The magnitude of the dimensionless parameter
a0 = eE/mcLjQ separates high-field (a^ > 1) from the low-field (al <C 1) science.
Relativistic Thompson scattering, topic the present paper revisits, is one such high field
effect.
Although the incoherent scattering of ultraintense linearly polarized laser pulse by
plasma has been studied extensively [2-4], some of the important issues, in particular,
the subtle interplay between figure-eight motion of an electron and frequency shift due
to time retardation, have not been emphasized. This interplay can give rise to fairly
1
) This work was supported by the US DOE under contract DE-AC02-76CH03073.
CP647, Advanced Accelerator Concepts: Tenth Workshop, edited by C. E. Clayton and P. Muggli
© 2002 American Institute of Physics 0-7354-0102-0/02/$19.00
760
counter-intuitive phenomena. For example, the intensity of 2-nd harmonic radiation
along the electric field of the laser (referred to as the polarization direction) depends nonmonotonically on the intensity of incident laser pulse and even vanishes near a 0 = 2.2.
Also, it turns out that a relativistic electron emits at the fundamental harmonic in the
polarization direction. Another interesting feature of strong field interaction is the asymmetry of radiation about the polarization direction. Nonrelativistically, radiation is symmetric about the polarization direction with zero intensity in the polarization direction.
This asymmetry helps to outline the parameter space of observation angles and angular
acceptances of the light-collecting detectors where experimentalists can look for the signatures of relativistic effects (figure-eight motion and time retardation) by collecting 1-st
and 2-nd harmonics of the Thompson-scattered light.
Sec. I is a review section based on Ref. [4]. In Sec. I analytic expressions for figureeight orbit and the radiation emitted due to electron motion in linearly polarized laser
field are reviewed. Sec. II cpntains^new results on emission of the fundamental and 2-nd
harmonic radiation along E and B directions. Angular distribution of radiation around
the polarization direction is computed and possible experimental tests for the theory are
suggested in Sec. III. Sec. IV concludes and summarizes important results of the paper.
I
ELECTRON ORBIT AND RADIATION
The normalized vector potential for a linearly polarized laser traveling in negative
z— direction is represented by
a — a0coskQTjex
(1)
where rj = z + ct and k0 = 27r/A 0 , the wavenumber of the laser field. Ions are assumed
to be cold, providing a neutralizing background. In the limit of a sufficiently low-density
plasma, such that up <C u0 = k0c, the high frequency motion of plasma electrons is
unaffected by the space charge fields of the plasma. The electron motion in the field a is
governed by the relativistic Lorentz equation with $ ^ 0, i. e.,
where f3 — v/cis the normalized electron velocity, u — p/mec — 7^ is the normalized
electron momentum, and 7 = (1 + u 2 ) 1 / 2 = (1 — /32)"1/2 is the relativistic factor.
Eq. (2) implies that transverse canonical momentum and energy in the wave frame are
conserved, i. e.,
MI = a_L
(3)
and
7
+ Uz - $ = 7o(l + A))
761
(4)
where prior to laser interaction (a_L = 0), wj_ = $ = 0, 7 = 70, and uz = j0/30 have
been assumed. Above equations allow the electron motion to be specified solely in terms
of fields. A single electron initially at rest receives a finite average drift velocity due
to ponderomotive force associated with the rise of incident laser pulse as pointed out in
Ref. [2]. But for the electrons in a moderately dense plasma such that LOP > r^1, where
TL is the laser pulse duration, there is no average axial motion of the electrons because
they are pulled back by the ions. From here on electrons are assumed, on average,
stationary in the z— direction. The appropriate initial conditions are uz0 = —a\l^h$ and
uxQ — a0. The electron orbit is given by
ux = CLQ cos koTj, x(jl) = xo H~ ^i sin ^o???
uy = 0, y(rj) = 2/0,
uz — [h,Q — (1 + a2, cos2 fc 0 ?y)]/2Ao, 2(77) = ZQ + /?i?y + z\ sin 2fc0/7,
(5)
1 2
where h0 = (1 + &0/2) / , TI = aQ/h0k0, and^i = —al/8hlk0.
Solving Eq. (5) numerically shows that the motion in x— direction consists of all the
harmonics whereas z— direction motion has only even harmonics. Hence only even
harmonics are predicted for radiation in E if time retardation is ignored. However, for
relativistic figure-eight motion retarded time effect causes power to scatter in all other
harmonics and results in n = 1 radiation in E. It is this interplay between the two
strong field effects (figure-eight motion and time retardation) that gives rise to non-zero
radiation for all harmonics in the direction of polarization.
Energy emission per unit solid angle per unit frequency by a single electron can be
calculated from the classic formula [5]:
sjl LT
\Ju
2 2
P
C UJ
i ,.+00
.
2
(6)
duod^l
where f = /3c and ? are electron velocity and coordinate. Due to time retardation,
an extra factor —n-r/cis present in the exponential in Eq. (6) as compared to the
nonrelativistic expression that has time t in the exponential. This extra term arises as
electrons become relativistic and the radiation observed at present time is actually the
radiation emitted by the electron at an earlier time. A thorough discussion on the origin
o f — n - r / c term in Eq. (6) because of time retardation is presented in Jackson [5].
It immediately follows from Eq. (6) that if the only nonvanishing component of electron velocity is /3X, then there is no emission along x— direction. The situation changes
dramatically when the motion in z— direction is taken into account. Assuming n = ex,
the integrand in front of the exponent is nonzero. Since the exponent in Eq. (6) contains
tf = t — n • r ( t ) / c , all harmonics are present in the radiation.
Following Esarey et al [4], on substituting the particle trajectory and velocity from
Eq. (5) into Eq. (6), one gets
2 2L2 r 2
Pn =
2°
[g*n(l - si*2 9 cos2 <f>) + C^ sin2 0 - CxnCzn sin 20 cos <£]
762
(8)
where
aZn)[Jn-2m-l(aXn}
+ Jn-2m+l(<**n)]
(9)
Czn = E^_^(-l) m 2fc 0 ^l Jm(azn)[Jn-2m-2(axn}
+ */n-2™+2 M]
(10)
and azn = nal(l + cos0)/$hl, axn = na0 sinflcos <^//i0. In Eq. (8) Pn is the power
radiated per unit area at nth harmonic and L is the length of the laser pulse. C%n term in
Eq. (8) represents the contribution to intensity due to motion in x— direction; C*n is the
radiation due to the motion in z— direction; CxnCzn is a cross term representing some
sort of interference between radiations due to motions in x— and z— directions. The
effect of time retardation is taken into account by the Bessel function terms in Cxn and
Czn. Instead of getting Cxn and Czn at u>0 and 2u>0 respectively, an infinite sum over all
harmonics is obtained. For aQ <C 1, C^ dominates and corresponds to classical dipole
radiation at cj0An insight into the role of time retardation in intensity distribution can be gained if one
neglects n-r/c in Eq. (6). In the absence of retarded time effect, the intensity distribution
is given by
\j ™ din x [n x /3]e*'
47T 2 C \J-oo
(11)
A calculation similar to one used to derive Eq. (8) from Eq. (6) shows that in the absence
of time retardation there is no odd-harmonic emission in the polarization direction. This
is consistent with the discussion earlier in this section, as motion in z— direction consists
only of even harmonics. Thus, time retardation is responsible for odd harmonics in the
polarization direction, in particular, the n = 1 harmonic. Fig. (Ib) shows the variation of
fundamental and second harmonic intensities in E and B directions with a 0 , neglecting
time retardation. It is evident from Fig. (Ib) that in the absense of time retardation there
is no difference between radiation intensities in E and B directions.
II
RADIATION IN E AND B
In this section incoherent scattering in E and B directions is examined, with special emphasis on scattering at 1-st and 2-nd harmonics. Experimental studies by Chen et
al [6] also agree with the theoretical prediction of non-vanishing intensity in the polarization direction. Fundamental power for a0 <C 1 scales as a^ and a^ in E and B directions
respectively. Second harmonic power scales as a^ in both E and B. For a0 <C 1, ra = 1
radiation in B dominates and corresponds to the classical dipole radiation. At sufficiently
high intensities a0 > 1, retarded time effect can cause most of the power to be present
in the higher harmonics. Design of tabletop x-ray sources is based on this effect [4].
Efficiency of such x-ray sources is very low as figure-eight trajectory and hence time
retardation cannot be engineered to produce most of the radiation at a desired frequency.
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a) radiation with retarded time
b) radiation without retarded time
-0.2
FIGURE 1. Variation of normalized intensities in E and B directions for n = 1 and n = i2. (a) includes
the effect of retarded time. The intensities in E are magnified 20 times. The dip in second harmonic
intensity in E can make it difficult to measure scattered radiation near a 0 = 2.2. (b) retarded time effect
is neglected. No dip is observed, 2-nd harmonic emission is the same in E and B directions.
Fig. (la) plots intensities PI and P2 in E and B directions using n = 1,2 in Eq. (8).
Naively one expects the intensities to increase with the incident laser intensity. But
the variation of second harmonic intensity in E shows a nontrivial behavior. Intensity
increases near a0 = 0, reaches a maxima at a0 = 1, then decreases and vanishes around
a0 = 2.2 and rises again for large values of a0. This dip in intensity is attributable to
time retardation which appears in the expression for intensity as a combination of Bessel
functions. Some particular combination of the Bessel functions can add to result in the
reduction of intensity, which corresponds to the destructive interference of radiation from
the electron at different points of its trajectory.
The integrand in Eq. (6) as a function of the variable 77 represents the contribution
to the radiation amplitude from different points along the electron trajectory. Fig. (2)
compares the amplitude of 2-nd harmonic radiation due to electron at each position of
its figure-eight orbit in E and B directions for a0 = 2.2,0.5. First consider the radiation
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comparison of radiation in E and B for a) aO=2.2, b) aO=0.5
0.01
0.3
0.2
-0.01
0.1
. -0.02
-0.1
-0.03
-0.2
-0.04
-0.3
-0.05
-0.4
0.2
0.4
0.6
-0.06
0.8
0.2
0.4
0.6
0.8
FIGURE 2. The second harmonic radiation in E and B directions for ao = 0.5 and GO = 2.2. The cancellation of individual amplitudes at each position of figure-eight orbit due to time retardation is responsible
for the vanishing of second harmonic intensity in E direction for a 0 = 2.2 in Fig. (la).
in E. For small intensities (e. g., a0 = 0.5) the contributions to the second harmonic
radiation from each point along the orbit add constructively. But as a 0 crosses a0 = 1 the
amplitudes from each position start to cancel as n • f/c becomes significant. At a0 = 2.2
the total second harmonic emission vanishes as the amplitudes from different points
along the electron figure-eight trajectory cancel. This can be observed from Fig. (2a):
the area under the solid curve adds up to zero. The contributions from all trajectory
points always add constructively in B because n • f = 0, and hence there is no effect of
time retardation. In Fig. (Ib) where the effect of retarded time is excluded, as expected,
there is no dip in n — 2 intensity in E.
Experimentalists have not realized the fact that second harmonic intensity in E varies
nontrivially and most of them did not operate in the regime where n — 2 intensity along
E is maximum. Chen et al [6] operated at a0 = 1.88, Malka et al [7] at a0 = 1.80,
and Bula et al [8] around a0 = 0.6. Because of an inappropriate choice of parameter a0
experimentalists were not able to measure a significant second harmonic intensity in E
direction.
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intensity variation with <|> for aQ=1.8,1
FIGURE 3. Variation of normalized intensity with (j> for 6 = Tr/2, ao = 1.88 and ao = 1 for n = 2.
Although intensity for a o = 1.88 is greater than GO = 1 for most angles, intensity for a o = 1 is more than
10 times greater at </> = 0, i. e., in the polarization direction.
As mentioned before, experiments done by Chen et al [6] used a0 — 1.88, Fig. (la)
shows that n = 2 intensity in E around a0 = 1.88 is very near zero. This is the reason
they were not able to conclusively measure any 2-nd harmonic radiation in E direction.
Fig. (3) shows the variation of second harmonic intensity in E — B plane for a0 = 1.88
and a0 = 1. Although a0 = 1.88 curve lies above the a0 = 1 curve for most of angles
(/>, that is not the case for intensity in the polarization direction. In fact, the intensity in
the E ((/> = 0), is more than ten times greater for a0 = 1 than for a0 = 1.88. Hence, in
order to measure the second harmonic intensity in E (which is an important signature of
the figure-eight motion), one needs to operate near a 0 = 1. This result seems surprising
as one expects scattered intensity to increase as the incident laser intensity is increased.
Ill ASYMMETRY IN RADIATION ABOUT POLARIZATION
DIRECTION
For nonrelativistic dipole motion of an electron, radiation is symmetric about the polarization direction. Situation changes dramatically when a0 > 1, as motion in z— direction becomes significant. For a figure-eight orbit of an electron there is an asymmetry
in particle motion along and opposite to the direction of propagation of the laser pulse as
seen in Fig. (4). This asymmetry causes the asymmetry in radiation about the polarization direction.
Fig. (4) shows a typical figure-eight orbit. n x , n2 are directions symmetric about the
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a typical figure eight orbit
FIGURE 4. A typical figure-eight orbit. HI and rc2 are two directions symmetric about polarization
direction, n. and r2 are the points opposite to each other in the figure-eight orbit, ft and ft are velocities
at these points.
polarization direction; r\, r2 are symmetric points on the figure-eight orbit; ft, ft are
the velocities at these points. We are interested in comparing the radiation in the two
directions fti and n2. For symmetric points n. and r2 on figure-eight orbit ft • r/c is the
same. The integrand in Eq. (6), [ft x (ft x (3)] for the points ri and r 2 in directions fti
and n 2 in terms of the angles 0, <j> and a as shown in Fig. (4) is /3cos(a-0 - 4>) [cos(9 +
^ _ sin(0 + <^] and -/? cos(a - 0 - <p)[cos(9 + <£)4 + sin(0 + ^)ez] respectively.
Thus the contribution by the symmetric points fi, r*2 in symmetric directions n^ n2 is
not the same for a figure eight orbit. But for pure dipole motion a = 0 and <f> = 7r/2 - 0,
thus [ft x (ft x /3)] vanishes in ex and is the same for HI and n2 directions.
One can use this asymmetry to calculate the angular acceptance of a light-collecting
detector used to measure intensity of a particular harmonic in the polarization direction.
Taylor expansion of formula in Eq. (8) around 0 = Tr/2 and </> = 0 gives
n
+ Bn(9 - 7T/2) + Cn(0 -
(12)
for intensity variation near 9 = n/1 and </> = 09 where
A
— r*2
/in — ^^n?
D
_ oynr
/~*
^^ — ^^aJTi^^n?
^
_
/-f2
^n — ^3771
767
_ ^2
^^n'
(13)
and Cxn and Czn are given by Eqs. (9) and (10) evaluated at 0 = it/I and <f> = 0.
Eq. (12) contains quadratic, linear, and constant terms in (0 — ?r/2) and c/>. The
quadratic terms are present even in the dipole approximation. The constant and linear
terms are due to strong field effects because, for a classical dipole there is no radiation
along the polarization direction, and radiation about the polarization direction is symmetric.
The solid angle about 0 = yr/2 over which flux due to relativistic terms An + Bn(0 —
7T/2) balances flux due to dipole terms Cn(0 — 7r/2)2 + Dn(/)2 can be taken as a benchmark
for the angular acceptances for detectors measuring a particular harmonic. Integration
of Eq. (12) over a cone with half-angle S9n about 9 = ;r/2 gives the radiation flux. With
the assumption that S0n is small, one gets for the half angle
=\ r k
The corresponding solid angle is
5ttn = 27r(l-cos£<9 n )
(15)
For n = 1,2 and a0 = 1 the angular acceptances for the light detectors measuring
fundamental and second harmonic intensities about the polarization direction is given by
<toi = 0.045Sr and <M2 = 0.115Sr.
An experiment which aims to measure the signatures of figure-eight motion (n =
1,2 radiation in E direction) should use the incident laser intensity near a 0 = 1. The
detectors used to measure 1-st and 2-nd harmonic intensity in E should have angular
acceptances of 0.045 Sr and 0.115 Sr respectively.
IV CONCLUSIONS
Relativistic Thompson scattering is reviewed, with the emphasis on radiation in polarization direction. This radiation is attributed to two relativistic effects: time retardation
and figure-eight motion. A new regime for the laser field amplitude a 0 = eE/maj is
suggested where the signatures of these relativistic effects are easier to measure. A thorough analysis of the effects of figure-eight motion and retarded time is made. Finally,
as an aid to experimentalists, we calculated the acceptance angle (Sfln) of a detector
which enables one to distinguish between the two contributions to radiation intensity:
regular dipole and strong-field. A detector with acceptance (£fi n ) positioned along the
polarization direction primarily intersects the radiation which is due to the relativistic
effects.
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