20 Further Applications of the ClassicalRadiation Theory

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 linearlypolarized and its intensity dependence varied as si

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 linearlypolarized and its intensity dependence varied as sin2
This result was derived for thetivistic form of the radiation field Before we can do this, however, we must firstshow that for the relativistic regime ð  1Þ the radiation field continues to consistonly of transverse components, E
and E, and the radial or longitudinal electriccomponent Eris 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

The brackets   ½ ret means that the field is to be evaluated at an earlier or retardedtime, t0¼t  Rðt0Þ=c where R/c is just the time of propagation of the disturbancefrom one point to the other Furthermore, c is the instantaneous velocity of theparticle, c _is the instantaneous acceleration, and n ¼ R=R The quantity ! 1 fornonrelativistic motion For relativistic motion the fields depend on the velocity as

well as the acceleration Consequently, as we shall soon clearly see the angulardistribution 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

In (20-3), S  n is the energy per unit area per unit time detected at an tion point at time t due to radiation emitted by the charge at time t0¼t  Rðt0Þ=c

observa-To calculate the energy radiated during a finite period of acceleration, say from

Before we proceed to apply these results to various problems of interest, wemust demonstrate that the definition of the Stokes parameters (20-9) is valid forrelativistic motion That is, the field is transverse and there is no longitudinalcomponent ðEr¼0Þ We thus write (20-1a) as

so (20-11) can be rewritten as

4"0c2 3R urður _Þ  _ðururÞ  _ður _Þ þ _ðurÞ ð20-13Þ

In spherical coordinates the field Eðr, tÞ is

Taking the dot product of both sides of (20-13) with urand using (20-14), we see that

Er¼ ður _Þ  ður _Þ  ðurÞður _Þ þ ður _ÞðurÞ ¼0 ð20-15Þ

so the longitudinal (radial) component is zero Thus, the radiated field is alwaystransverse in both the nonrelativistic and relativistic regimes Hence, the Stokesparameters definition for spherical coordinates continues to be valid.

The components E
and E are readily found for the relativistic regime Wehave

c ¼ ð _xsin
þ _zz cos
Þurþyyu_ þ ðx_cos
 _zz sin
Þu
ð20-18aÞ

c _ ¼ ð €xsin
þ €zz cos
Þurþyyu€ þ ðx€cos
 €zz sin
Þu
ð20-18bÞThe transverse components E
and Eare then

4"0c2 3R ðx€cos
 €zz sin
Þ 

€

x_zz  _x€zzc

For a linear charge that is accelerating along the z axis,  and _are parallel, so

Eðr, tÞ ¼ e

4"0c2R

ur ður_vvÞ3

ð20-21bÞ

According to (20-11) and (20-14), the field components of (20-21b) are

sin2
ð1   cos
Þ5

100

0B

@

1C

which is the well-known dipole radiation distribution In (20-25), the nonrelativisticresult, the minimum intensity I
ð Þis at
¼ 0
, where I 0ð
Þ ¼0, and the maximumintensity is at
¼ 90
, where I 90ð
Þ ¼I0

Equation (20-24), on the other hand, shows that the maximum intensity shiftstoward the z axis as  increases To determine the positions of the maximum andminimum of (20-24), we differentiate (20-24) with respect to
, set the result equal tozero, and find that

so
’ 0
We see that the maximum intensity has moved from
¼ 90
ð 1Þ to

¼0
ð ’1Þ, that is, the direction of the maximum intensity moves toward thecharge moving along the z axis

In Fig 20-2 the intensity contours for various values of  are plotted Thecontours clearly show the shift of the maximum intensity toward the z axis forincreasing  In the figure the charge is moving up the z axis from the origin, andthe horizontal axis corresponds to the y direction To make the plot we equated Ið
Þwith , so

Furthermore, using  notation,  ¼ _zz=c, we can express the velocity and acceleration

2c"0

ez04c

0B

@

1C

Thus, the radiation appears at the same frequency as the frequency of oscillation.With respect to the intensity distribution we now have radiation also appearingbelow the z ¼ 0 axis because the charge is oscillating above and below the xyplane Thus, the intensity pattern is identical to the unidirectional case but is nowsymmetrical with respect to the xy plane InFig 20-3we show a plot of the intensitycontour for  ¼ 0:4

Figure 20-3 Intensity contours for a relativistic oscillating charge  ¼ 0:4ð Þ

20.2 RELATIVISTIC MOTION OF A CHARGE MOVING IN

A CIRCLE: SYNCHROTRON RADIATION

In the previous section we dealt with the relativistic motion of charges moving in astraight line and with the intensity and polarization of the emitted radiation Thistype of radiation is emitted by electrons accelerated in devices known as linearaccelerators We have determined the radiation emitted by nonrelativistic chargesmoving in circular paths as well In particular, we saw that a charge moves in acircular path when a constant magnetic field is applied to a region in which the freecharge is moving

In this section we now consider the radiation emitted by relativistically movingcharges in a constant magnetic field The radiation emitted from highly relativisticcharges is known as synchrotron radiation, after its discovery in the operation of thesynchrotron A charge moving in a circle of radius a is shown in Fig 20-4

The coordinates of the electron are

Using the familiar complex notation, we can express (20-36) as

_x

€x

For the nonrelativistic case we saw that !, the cyclotron frequency, was given by

! ¼eB

where e is the magnitude of the charge, B is the strength of the applied magnetic field,

mis the mass of the charge, and c is the speed of light in free space We can obtain

Figure 20-4 Motion of a relativistic charge moving in a circle of radius a in the xy planewith an angular frequency !

the corresponding form for ! for relativistic motion by merely replacing m in (20-38),the rest mass, with the relativistic mass m by

The frequency ! in (20-40) is now called the synchrotron frequency

To find the Stokes vector of the emitted radiatin, we recall from Section 20-1that the relativistic field components are

4"0c2 3R ðx€cos
 €zz sin
Þ 

€

x_zz  _x€zzc

ð20-42bÞSubstituting (20-37b) and (20-37c) into (20-42), we find

2 cos

0BB

1C

where we emphasize that
is the observer’s angle measured from the z axis Equation(20-44) shows that for synchrotron radiation the radiation is, in general, ellipticallypolarized The Stokes vector (20-44) is easily shown to be correct because the matrixelements satisfy the equality:

We saw earlier when dealing with the motion of a charge moving in a circle forthe nonrelativistic case that the Stokes vector reduces to simpler (degenerate) forms

A similar situation arises with relativistically moving charges Thus, when we observethe radiation at
¼ 0
, the Stokes vector (20-44) reduces to

1

0BB

1C

0

0BB

1C

2 cos

0BB

1C

where ! ¼ eB=m ¼ !cis the cyclotron frequency This is the Stokes vector we found

in Section 17.3 for a charge rotating in the xy plane We now examine the intensity,orientation angle, and ellipticity of the polarization ellipse for synchrotron radiation(20-44)

The intensity of the radiation field, Ið
Þ, can be written from (20-44) asIð
Þ ¼e

The orientation angle and the ellipticity angle are ¼1

is further emphasized in Figs 20-6 and 20-7 Figure 20-6 shows (20-49) for

 ¼0:3, 0:4, and 0.5 Similarly,Figure 20-7shows (20-49) for  ¼ 0:6, 0:7, and 0.8.For , say, equal to 0.99 we have Ið0
Þ=Ið90
Þ ¼2  109, which is an extraordinarilynarrow beam.

InFig 20-8we have plotted the logarithm of the intensity Ið
Þ from
¼ 0
to

180
for  ¼ 0 to 0.9 in steps of 0.3

Figure 20-5 Relativistic intensity contours for  ¼ 0:0, 0:1, and 0.2

Figure 20-6 Relativistic intensity contours for  ¼ 0:3, 0:4, and 0.5

In order to plot the orientation angle , (20-50), as a function of
, wenote that, for  ¼ 0:0 and 1.0, ¼ 0 and =4, respectively In Fig 20-9we plot

as a function of
where the contours correspond to  ¼ 0:0, 0:1, , 1:0 fordecreasing

Figure 20-7 Relativistic intensity contours for  ¼ 0:6, 0:7, and 0.8

Figure 20-8 Logarithmic plot of the intensity for  ¼ 0:0 through 0.9

In Fig 20-10 the ellipticity angle , (20-51), is plotted for  ¼ 0 to 1.0 over arange of
¼ 0
to 180
For the extreme relativistic case (20-51) becomes

¼ 1

2sin

Figure 20-9 Orientation angle for synchrotron radiation

Figure 20-10 Ellipticity angle for synchrotron radiation

It is straightforward to show that (20-52) can be rewritten in the form of an equationfor a straight line, namely,

¼

245

ð20-53Þand this behavior is confirmed inFig 20-10 We see that Fig 20-10 shows that theellipticity varies from ¼ 45
(a circle) at
¼ 0
and ¼ 45
(a counterclockwisecircle) at
¼ 180

Finally, it is of interest to compare the Stokes vector for  ¼ 0 and for  ¼ 1.The Stokes vectors are, respectively,

S ¼ K

1 þ cos2
sin2
0

2 cos

0BB

1C

and

S0¼K0

1sin2
sin
cos
cos

0BB

1C

0B

@

1C

1001

0B

@

1C

S ¼ K

1100

0B

@

1C

1100

0B

@

1C

Thus, in the extreme cases of  ¼ 0 and  ¼ 1, the Stokes vectors—that is, thepolarization states—are identical! However, between these two extremes the polari-zation states are very different

Synchrotron radiation was first observed in the operation of synchrotrons.However, many astronomical objects emit synchrotron radiation, and it has beenassociated with sunspots, the crab nebula in the constellation of Taurus, and radia-tion from Jupiter Numerous papers and discussions of synchrotron radiations haveappeared in the literature, and further information can be found in the references

light in the medium Such radiation is called Cˇerenkov radiation, after its discoverer,

P A Cˇerenkov (1937) According to the great German physicist A Sommerfeld, theproblem of the emission of radiation by charged particles moving in an opticalmedium characterized by a refractive index n was studied as early as the beginning

of the last century

The emission of Cˇerenkov radiation is a cooperative phenomenon involving alarge number of atoms of the medium whose electrons are accelerated by the fields ofthe passing particle and so emit radiation Because of the collective aspects of theprocess, it is convenient to use the macroscopic concept of a dielectric constant "rather than the detailed properties of individual atoms

In this section our primary concern is to determine the polarization ofCˇerenkov radiation The mathematical background as well as additional informa-tion on the Cˇerenkov effect can be found in Jackson’s text on classical electro-dynamics Here, we shall determine the radiated field Eðx, tÞ for the Cˇerenkoveffect, whereupon we then find the Stokes parameters (vector)

A qualitative explanation of the Cˇerenkov effect can be obtained by ing the fields of the fast particle in the dielectric medium as a function of time Themedium is characterized by a refractive index n, so the phase velocity of the light isc/n, where c is the speed of light in a vacuum The particle velocity is denoted by v Inorder to understand the Cˇerenkov effect it is not necessary to include the refractiveindex in the analysis, however Therefore, we set n ¼ 1, initially At the end of theanalysis we shall see the significance of n

consider-If we have a charged particle that is stationary but capable of emitting sphericalwaves, then after the passage of time t the waves are described by

The maximum and minimum values of the spherical wave along the y axis arefound from the condition dy=dx ¼ 0 From Eq (20-59) we can then show that themaximum and minimum values of y occur at

It is of interest to plot (20-63) for  ¼ 0, 0.5, and 1.0 We see from (20-61b) that for

 ¼1 we have x¼0; that is, the trailing edges of the spherical wave fronts cide In Figs 20-11to20-13.we have made plots of (20-63) for  ¼ 0, 0:5, and 1.0.However, to describe the expansion of the spherical wave with the passage of time asthe particle moves, the coordinates of the x axis have been reversed That is, thelargest circle corresponds to 4 sec and appears first, followed by decreasing circles for

coin-3, 2, and 1 sec For completeness we have included a plot for  ¼ 0 The plot for

 ¼1, Fig 20-13, confirms that when the particle is moving with the speed of lightthe trailing edges, which are shown as the leading edges in the plot, coincide

Figure 20-11 Propagation of a spherical wave for a staionary particle ð ¼ 0Þ

Figure 20-13 is especially interesting because it shows that the wave fronts onlycoincide for  ¼ 1 The question now arises, what happens when  > 1? To answerthis question we return to (20-63) We observe that yðtÞ is imaginary if

Figure 20-12 Propagation of a spherical wave for a particle moving with a velocity  ¼ 0:5

Figure 20-13 Propagation of a spherical wave for a particle moving with a velocity  ¼ 1:0

Figure 20-14 Propagation of a spherical wave for a particle moving with a velocity  ¼ 1:5.

occur because there is a sudden change in the medium (the medium is unaffected),but because the waves, which were previously noninterfering ð 1Þ, now interfereð >1Þ InFig 20-16we have drawn the straight line from the origin through thetangents of the spheres.

The tangents line in Fig 20-16 is called a wake The normal to the wake makes

an angle
c, which is called the critical angle From the figure we see that it can beexpressed as

cos
c¼c

In free space a particle cannot propagate equal to or faster than the speed oflight However, in an optical medium the phase velocity of the light is less than c and

is given by c=n Thus, if a particle moves with a speed greater than c=n it will generate

an interference phenomenon exactly in the same manner as we have been describing.This behavior was first observed by Cˇerenkov, and, consequently, in optics thephenomenon is called the Cˇerenkov effect and the emitted radiation, Cˇerenkovradiation Furthermore, the critical angle
c is now called the Cˇerenkov angle; theshock wave is in the direction given by
c

The Cˇerenkov radiation is characterized by a cone Its most important tion is to measure the velocity of fast particles; that is,
c is measured by moving adetector such that the maximum intensity is observed At this condition
cis deter-mined, and v can then be immediately found

applica-With this background we now determine the intensity and polarization of theCˇerenkov radiation Our analysis draws heavily on Jackson’s treatment of theCˇerenkov effect and classical radiation in general

Figure 20-15 Propagation of a spherical wave for a particle moving with a velocity  ¼ 2:5

We restate the first two equations of the chapter The electric field emitted by

an accelerating charge is given by

accel-for the relations among the coordinates

The instantaneous energy flux is given by the Poynting vector:

The maximum and minimum values of the spherical wave along the y axis arefound from the condition dy=dx ¼ From Eq (20- 59) we can then show that themaximum and minimum values of y occur