17Radiation Emitted by AcceleratingCharges

17
Radiation Emitted by Accelerating
Charges

17.1

STOKES VECTOR FOR A LINEARLY OSCILLATING CHARGE

We have shown how Maxwell’s equation gave rise to the equations of the radiation
field and the power emitted by an accelerating electron. We now discuss the polarization of the radiation emitted by specific electron configurations, e.g., bound
charges and charges moving in circular and elliptical paths.
At the beginning of the nineteenth century the nature of electric charges was not
fully understood. In 1895 the electron (charge) was discovered by J. J. Thompson.
Thus, the long-sought source of the optical field was finally found. A year after
Thompson’s discovery, P. Zeeman performed a remarkable experiment by placing
radiating atoms in a constant magnetic field. He thereupon discovered that the
original single spectral line was split into two, or even three, spectral lines.
Shortly thereafter, H. Lorentz heard of Zeeman’s results. Using Maxwell’s
theory and his electron theory, Lorentz then treated this problem. Lorentz’s calculations predicted that the spectral lines should not only be split but also completely
polarized. On Lorentz’s suggestions Zeeman then performed further measurements
and completely confirmed the predictions in all respects. It was only after the work of
Zeeman and Lorentz that Maxwell’s theory was accepted and Fresnel’s theory of
light replaced.
Not surprisingly, the importance of this work was immediately recognized, and
Zeeman and Lorentz received the Nobel Prize in physics in 1902. We should emphasize that the polarization predictions of the spectral lines played a key part in understanding these experiments. This prediction, more than any other factor, was one of
the major reasons for the acceptance of Maxwell’s theory into optics.
In this chapter we build up to the experiment of Zeeman and the theory of
Lorentz. We do this by first applying the Stokes parameters to a number of classical
radiation problems. These are the radiation emitted by (1) a charge oscillating along
an axis, (2) an ensemble of randomly oriented oscillating charges, (3) a charge
moving in a circle, (4) a charge moving in an ellipse, and (5) a charge moving in a

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


magnetic field. In the following chapter we then consider the problem of a randomly
oriented oscillating charge moving in a constant magnetic field—the Lorentz–
Zeeman effect.
We consider a bound charge oscillating along the z axis as shown in Fig. 17-1.
The motion of the charge is described by
d 2z
þ !02 z ¼ 0
dt 2

ð17-1Þ

The solution of (17-1) is
zðtÞ ¼ zð0Þ cosð!0 t þ Þ

ð17-2Þ

where z(0) is the maximum amplitude and  is an arbitrary phase constant. Because
we shall be using the complex form of the Stokes parameters, we write (17-2) as
zðtÞ ¼ zð0Þeið!0 tþÞ

ð17-3Þ

where it is understood that by taking the real part of (17-3), we recover (17-2); that is,
Re½zðtފ ¼ zð0Þ cosð!0 t þ Þ

ð17-4Þ

The radiation field equations are given by (16-8) and (16-9) in Section 16.1:
E ¼

e
½x€ cos  À z€ sin Š
4"0 c 2 R

ð16-8Þ

E ¼

e
½ y€ Š
4"0 c 2 R

ð16-9Þ

Recall that these equations refer to the observation being made in the xz plane, that
is, at  ¼ 0. The angle  is the polar angle in the observer’s reference frame.
Performing the differentiation of (17-3) to obtain z€, we have
z€ ¼ À!02 zð0Þeið!0 tþÞ

Figure 17-1

Motion of a linear oscillating charge.

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

ð17-5Þ

Substituting (17-5) into (16-8) yields
E ¼

e
½!02 zð0Þ sin eið!0 tþÞ Š
4"0 c 2 R

E ¼ 0

ð17-6aÞ
ð17-6bÞ

The Stokes parameters are defined in a spherical coordinate system to be
S0 ¼ E EÃ þ E EÃ

ð16-10aÞ

S1 ¼ E EÃ À E EÃ

ð16-10bÞ

S2 ¼ E EÃ þ E EÃ

ð16-10cÞ

S3 ¼ iðE EÃ À E EÃ Þ

ð16-10dÞ

Substituting (17-6a) and (17-6b) into (16-10) yields

2
ezð0Þ
!40 sin 2 
S0 ¼
4"0 c 2 R

2
ezð0Þ
!40 sin 2 
S1 ¼ À
4"0 c 2 R

ð17-7aÞ

ð17-7bÞ

S2 ¼ 0

ð17-7cÞ

S3 ¼ 0

ð17-7dÞ

We now arrange (17-7) in the form of the Stokes vector:
0
1

1

2
B À1 C
ezð0Þ
C
sin 2 !40 B
S¼
@ 0 A
2
4"0 c R
0

ð17-8Þ

Equation (17-8) shows that the observed radiation is always linearly vertically polarized light at a frequency !0, the fundamental frequency of oscillation of the bound
charge. Furthermore, when we observe the radiation parallel to the z axis ( ¼ 0 ),
the intensity is zero. Observing the radiation perpendicular to the z axis ( ¼ 90 ), we
note that the intensity is a maximum. This behavior is shown in Fig. 17-2. In order to
plot the intensity behavior as a function of , we set
IðÞ ¼ sin 2 

ð17-9aÞ

In terms of x() and z() we then have
xðÞ ¼ IðÞ sin  ¼ sin 2  sin 
2

zðÞ ¼ IðÞ cos  ¼ sin  cos 

ð17-9bÞ
ð17-9cÞ

The term ez(0) in (17-8) is recognized as a dipole moment. A characteristic of
dipole radiation is the presence of the sin2 term shown in (17-8). Hence, (17-8)

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


Figure 17-2

Plot of the intensity behavior of a dipole radiation field.

describes the Stokes vector of a dipole radiation field. This type of field is very
important because it appears in many types of radiation problems in physics and
engineering. Finally, we note that a linearly oscillating charge gives rise to linearly
polarized light. Thus, the state of polarization is a manifestation of the fundamental
motion of the electron. This observation will be confirmed for other types of radiating systems.
17.2

STOKES VECTOR FOR AN ENSEMBLE OF RANDOMLY
ORIENTED OSCILLATING CHARGES

In the previous section, we considered the radiation field emitted by a charge or
electron oscillating with an angular frequency !0 about an origin. Toward the end of
the nineteenth century a model was proposed for the atom in which an oscillating
electron was bound to a positively charged atom. The electron was believed to be
negative (from work with ‘‘free’’ electrons in gases and chemical experiments). The
assumption was made that the electron was attracted to the positively charged atom,
and the force on the electron was described by Hooke’s law, namely, -kr. This model

was used by H. Lorentz to solve a number of longstanding problems, e.g., the
relation between the refractive index and the wavelength, the so-called dispersion
relation.
The motion of the electron was described by the simple force equation:
m€r ¼ Àkr
ð17-10aÞ
or
r€ þ !02 r ¼ 0

ð17-10bÞ

where m is the mass of the electron, k is the restoring force constant, and the angular
frequency is !02 ¼ k=m. We saw in Part I that the nature of unpolarized light was not

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


well understood throughout most of the nineteenth century. We shall now show that
this simple model for the motion of the electron within the atom leads to the correct
Stokes vector for unpolarized light.
The treatment of this problem can be considered to be among the first successful applications of Maxwell’s equations in optics. This simple atomic model provides
a physical basis for the source term in Maxwell’s equations. The model leads to the
appearance of unpolarized light, a quantity that was a complete mystery up to the
time of the electron. Thus, an ensemble of oscillating charges bound to a positive
nucleus and randomly oriented gives rise to unpolarized light.
We now determine the Stokes vector of an ensemble of randomly oriented,
bound, charged oscillators moving through the origin. This problem is treated by
first considering the field emitted by a single charge oriented at the polar angle  and
the azimuthal angle
in the reference frame of the charge. An ensemble average is

then taken by integrating the radiated field over the solid angle sin  d d
. The
diagram describing the motion of a single charge is given in Fig. 17-3.
The equations of motion of the charged particle can be written immediately
from Fig. 17-3 and are
xðtÞ ¼ A sin  sin
ei!0 t

ð17-11aÞ

yðtÞ ¼ A sin  sin
ei!0 t

ð17-11bÞ

zðtÞ ¼ A cos ei!0 t

ð17-11cÞ

where !0 is the angular frequency of natural oscillation. Differentiating (17-11) twice
with respect to time gives
x€ ðtÞ ¼ À!02 A sin  cos
ei!0 t

ð17-12aÞ

y€ ðtÞ ¼ À!02 A sin  sin
ei!0 t

ð17-12bÞ

z€ðtÞ ¼ À!02 A cos ei!0 t

ð17-12cÞ

Figure 17-3 Instantaneous motion of an ensemble of oscillating charges.

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


Substituting (17-12) into the radiation field equations, we find that
E ¼ À

eA!02 ei!0 t
ðsin  cos
cos  À cos  sin Þ
4"0 c 2 R

ð17-13aÞ

E ¼ À

eA!02 ei!0 t
ðsin  sin
Þ
4"0 c 2 R

ð17-13bÞ

where  is the observer’s viewing angle measured from the z axis.

Recall that the Stokes parameters are defined by
S0 ¼ E EÃ þ E EÃ

ð16-10aÞ

S1 ¼ E EÃ À E EÃ

ð16-10bÞ

S2 ¼ E EÃ þ E EÃ

ð16-10cÞ

S3 ¼ iðE EÃ À E EÃ Þ

ð16-10dÞ

Substituting (17-13) in (16-10), we then find that the Stokes parameters are
S0 ¼ C ½sin 2  sin 2
þ sin 2  cos 2
cos 2 
À 2 sin  cos  cos
cos  sin  þ cos 2  sin 2 Š

ð17-14aÞ

S1 ¼ C ½sin 2  sin 2
À sin 2  cos 2
cos 2 
þ 2 sin  cos  cos

cos  sin  À cos 2  sin 2 Š

ð17-14bÞ

S2 ¼ C ½2ðsin 2  sin
cos
cos  À cos  sin  sin
sin ފ

ð17-14cÞ

S3 ¼ 0

ð17-14dÞ

where


2
eA
!40
C¼
4"0 c 2 R

ð17-14eÞ

The fact that S3 is zero in (17-14d) shows that the emitted radiation is always
linearly polarized, as we would expect from a model in which the electron only
undergoes linear motion.
In order to describe an ensemble of randomly oriented charges we integrate

(17-14) over the solid angle sin d d
:
Z 2 Z 
hÁ Á Ái ¼
ðÁ Á ÁÞ sin  d d

ð17-15Þ
0

0

where hÁ Á Ái is the ensemble average and ðÁ Á ÁÞ represents (17-14a), etc. Carrying out
the integration of (17-14) by using (17-15) and forming the Stokes vector, we find
that
0 1
1

2
B C
8
eA
4B 0 C
S¼
!0 @ A
ð17-16Þ
0
3 4"0 c 2 R
0

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