20 Further Applications of the ClassicalRadiation Theory

20
Further Applications of the Classical
Radiation Theory

20.1

RELATIVISTIC RADIATION AND THE STOKES VECTOR FOR A
LINEAR OSCILLATOR

In previous chapters we have considered the emission of radiation by nonrelativistic
moving particles. In particular, we determined the Stokes parameters for particles
moving in linear or curvilinear paths. Here and in Section 20.2 and 20.3 we reconsider these problems in the relativistic regime. It is customary to describe the velocity
of the charge relative to the speed of light by
¼ v=c.
For a linearly oscillating charge we saw that the emitted radiation was linearly
polarized and its intensity dependence varied as sin2 . This result was derived for the
nonrelativistic regime ð
( 1Þ. We now consider the same problem, using the relativistic form of the radiation field. Before we can do this, however, we must first
show that for the relativistic regime ð
$ 1Þ the radiation field continues to consist
only of transverse components, E and E , and the radial or longitudinal electric
component Er is zero. If this is true, then we can continue to use the same definition
of the Stokes parameters for a spherical radiation field.
The relativistic radiated field has been shown by Jackson to be
!
e
n
_
Eðx, tÞ ¼
ð20-1aÞ
 fðn À

Þ Â ð
Þg
4"0 c2 3 R
ret
where
¼1ÀnÁ


ð20-1bÞ

The brackets ½Á Á ÁŠret means that the field is to be evaluated at an earlier or retarded
time, t0 ¼ t À Rðt0 Þ=c where R/c is just the time of propagation of the disturbance
from one point to the other. Furthermore, c
is the instantaneous velocity of the
particle, c
_ is the instantaneous acceleration, and n ¼ R=R. The quantity  ! 1 for
nonrelativistic motion. For relativistic motion the fields depend on the velocity as

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.


Figure 20-1

Coordinate relations for an accelerating electron. P is the observation point
and O is the origin. (From Jackson.)

well as the acceleration. Consequently, as we shall soon clearly see the angular
distribution is more complicated.
In Fig. 20-1 we show the relations among the coordinates given in (20-1a)
We recall that the Poynting vector S is given by

1
S ¼ c"0 jEj2 n
2

ð20-2Þ

Thus, we can write, using (20-1a),
2
e2
1 
½ S Á nŠ ¼
n  ½ðn À
Þ Â
_Š
2
3
6
2
32 "0 c  R

!
ð20-3Þ
ret

There are two types of relativistic effects present. The first is the effect of the specific
spatial relationship between
and
_, which determines the detailed angular distribution. The other is a general relativistic effect arising from the transformation from
the rest frame of the particle to the observer’s frame and manifesting itself by the
presence of the factor  in the denominator of (20-3). For ultrarelativistic particles

the latter effect dominates the whole angular distribution.
In (20-3), S Á n is the energy per unit area per unit time detected at an observation point at time t due to radiation emitted by the charge at time t0 ¼ t À Rðt0 Þ=c.
To calculate the energy radiated during a finite period of acceleration, say from
t0 ¼ T1 to t0 ¼ T2 , we write
Z

Z

t¼T2 þRðT2 Þ=c

W¼

½S Á nŠret dt ¼
t¼T1 þRðT1 Þ=c

t0 ¼T2

ðS Á nÞ
t0 ¼T

1

dt 0
dt
dt0

ð20-4Þ

À
Á

The quantity ðS Á nÞ dt=dt0 is the power radiated per unit area in terms of the
charge’s own time. The terms t0 and t are related by
t0 ¼ t À

Rðt0 Þ
c

ð20-5Þ

Furthermore, as Jackson has also shown,
¼1þ

1 dRðt0 Þ
c dt0

Copyright © 2003 by Marcel Dekker, Inc. All Rights Reserved.

ð20-6Þ


Differentiating (20-5) yields
dt
¼
dt0

ð20-7Þ

The power radiated per unit solid angle is
dPðt0 Þ
dt

¼ R2 ðS Á nÞ 0 ¼ R2 S Á n
d
dt

ð20-8Þ

These results show that we will obtain a set of Stokes parameters consistent with
(20-8) by defining the Stokes parameters as
Â
Ã
1
S0 ¼ c"0 R2 E EÃ þ E EÃ
2
Â
Ã
1
S1 ¼ c"0 R2 E EÃ À E EÃ
2
Â
Ã
1
S2 ¼ c"0 R2 E EÃ þ E EÃ
2
Â
Ã
1
S3 ¼ c"0 R2 iðE EÃ À E EÃ Þ
2

ð20-9aÞ

ð20-9bÞ
ð20-9cÞ
ð20-9dÞ

where the electric field Eðx, tÞ is calculated from (20-1a).
Before we proceed to apply these results to various problems of interest, we
must demonstrate that the definition of the Stokes parameters (20-9) is valid for
relativistic motion. That is, the field is transverse and there is no longitudinal
component ðEr ¼ 0Þ. We thus write (20-1a) as
"Â
#
Ã
n  ðn Â
_Þ À ½n  ð
Â
_ފ
e
ð20-10Þ
Eðx, tÞ ¼
3
4"0 c2 R
ret

Because the unit vector n is practically in the same direction as ur , (20-10) is
rewritten as
È
É
e
Eðr, tÞ ¼
ð20-11Þ

½ur  ður Â
_ފ À ½ur  ð
Â
_ފ
2
3
4"0 c  R
The triple vector product relation can be expressed as
a  ðb  cÞ ¼ bða Á cÞ À cða Á bÞ
so (20-11) can be rewritten as
Â
e
Eðr, tÞ ¼
ur ður Á
_Þ À
_ður Á ur Þ À
_ður Á
_Þ þ