Линейная поляризация излучения произвольно движущегося релятивистского заряда Текст научной статьи по специальности «Физика»

UDC 530.1; 539.1

LINEAR POLARIZATION OF RADIATION OF AN ARBITRARILY MOVING RELATIVISTIC CHARGE

V. G. Bagrov, V. A. Bordovitsyn, V. G. Bulenok, A. V. Kulikova

Department of Quantum Field Theory, Tomsk State University, Lenin pr., 30, 634050 Tomsk, Russia. Department of Theoretical Physics, Tomsk State Pedagogical University, Kievskaya Str., 60, 634061 Tomsk, Russia.

E-mail: vabord@sibmail.com

Exact methods of relativistic radiation theory have been used to construct indicatrixes of the angular distributions of instantaneous power radiated by an arbitrarily moving relativistic charge. It is assumed that at the moment of emission, the charge path is arbitrarily located relative to the coordinate system. The same technique has been used to study the linear polarization of radiation of an arbitrarily moving particle in the cases of curvature and fan-like radiation.

Keywords: radiation, relativistic particle, indicatrix of radiation, linear polarization, synchrotron radiation.

1 Angular distribution of instantaneous radiated power

In the general case of arbitrarily orientation of the velocity and acceleration vectors, the angular distribution of the total power of instantaneous radiation follows from [1,21

The indicatrixes of the total power of an arbitrarily-moving charges have the form:

p(0',y'; a,ß; n, A)

1

W =

a2 [1 - (nß)]2 + 2(na)(ßa) [1 - (nß)]

4nc3 _

(1 - ß2)(na)2

[1 - (nß)]5

[1 - (nß)]5

da

[1 - ß(sin 0' cos y' sin n + cos 0' cos n)]

I sin a

+ 2ß cos a -4

\ [1 - ß(sin 0' cos y' sin n + cos 0' cos n)]

x [(sin 0' cos y' cos n' - cos 0' sin n) cos A + sin 0' sin y' sin A]

(sin 0' cos y' sin n + cos 0' cos n) cos a \

+

Choosing

n = (sin 0 cos y, sin 0 sin y, cos 0),

fl = (0, 0, fl), a = (a sin a, 0, a cos a),

we obtain

p(0, y;a,ß) =

1

+ 2ß

(1 - ß cos 0)3

(sin 0 cos y sin a + cos 0 cos a) cos a

(1 - ßcos 0)4-

[1 — fl(sin 0' cos y' sin n + cos 0' cos n)] /

_(1 — fl 2)_

[1 — fl(sin 0' cos y' sin n + cos 0' cos n)]5 x ([(sin 0' cos y' cos n' — cos 0' sin n) cos A + sin 0' sin y' sin A] sin a + (sin 0' cos y sin n + cos 0' cos n) cos a)2.

In the particular case for n = 0, A = 0, we have an indicatrix of curvature radiation. For n = n/2, A = n/2 we have the indicatrix of fan-like radiation:

- (1 - ß2)

2 (sin 0 cos y sin a + cos 0 cos a)2 p(0', y'; a, ß; n =77, A = — ) =

1

(1 - ß cos 0)5

To find the angular distribution of radiation from charge arbitrarily located relative to the coordinate

nA

Fig. 1). The angular parameters of an arbitrarily moving charge are described by the equations

sin 0 cos y = (sin 0' cos y' cos n) cos A + sin 0' sin y' sin A, sin 0 sin y = (cos 0' sin n — sin 0' cos y' cos n) sin A + sin 0' sin y' cos A, cos 0 = cos 0' cos n + sin 0' cos y' sin n.

+2ß cos a

2'" 2' (1 - ßsin0'cosy')3 sin 0'(sin y' sin a + cos y' cos a)

+

(1 - ß sin 0' cos y')

^4

-(1 - ß2)

2 sin2 0'(sin y' sin a + cos y' cos a)2

(1 - ß sin 0' cos y')5

For a = n/2, this equation gives well known synchrotron radiation indicatrix [51

p(0',y'; a = —, ß; n = x,A = - )

(1 - ß sin 0' cos y')3

- (1 - ß2)

2 sin2 0' sin2 y'

(1 - ß sin 0' cos y')5

2

e

1

In the case of curvature radiation for a = n/2, we have wv = WSR3 sin2 a|a—n = 3 WSR

3,

,f'; a = 2 ,P; n = o, a = 0)

(1 - P cos 0')3

- (1 - ^2).sin2 COs2 f'

(1 - P cos 0')5'

, f ; a,p) =

p»(0, f; a,P)

(P — cos 0) cos f sin a + sin 0 cos a (1 — P cos 0)3 '

22 sin2 a sin2 f

(1 — P cos 0)3 '

b) Fan-like radiation (n = n/2, A = n/2)

,f; a,p) =

cos2 0(cos f cos a + sin f sin a)2 (1 — P sin 0 cos f)5

2

b [sin f cos a + (P sin 0 — cos f ) sin a]

M0, f ;a P) =-t,—-\5-'

T (1 — P sin 0 cos f)5

The corresponding indicatrixes are shown in Fig. 4. The integral characteristics of radiation have the form a) Curvature radiation (n = 0 A = 0)

We = Wsr(y2 cos2 a +

2

sin2 a

Wsr

W = Wsry2(1 — P2 sin2 a)|a-n/2 = Wsr; b) Fan-like radiation (n = n/2, A = n/2)

It can be seen that in int these two different cases (curvature and fan-like radiation) the indicatrixes of total angular distribution power have identical forms for the same a, £ and differ only in their orientation respect to the coordinate system (see Fig. 2).

2 Linearly polarized radiation from an arbitrarily moving charge

To calculate polarization we introduce unit vectors of linear polarization (see Fig. 3). Using these vectors and angles n A we find indicatrixes of polarization for a charge arbitrarily moving relative to the coordinate system. And for example the indicatrixes of linearly-polarized radiation in the cases of curvature and fanlike radiation have the form [3-5]

p(0, a, £) = ^(0, a, £) + ^(0, a, £).

a) Curvature radiation (n = 0 A = 0)

We

Wsr 8

{72(2 + 3P2 — P4)cos2 a

+(2 — P2)sin2 a} |a— n =

2 — P2

Wsr ,

W„

= {72(6 — 3P2 + P4)cos2 a 8

6 + P 2

+(6 + P2) sin2 a} |a— n = Wsr,

W = WSRy2(1 — P2 sin2 a

3 Conclusion

Wsr '

Thus, the indicatrixes of the angular distribution of instantaneous radiated power by an arbitrarily moving relativistic charge have been constructed. It is shown that all of them have the same form for different kinds of trajectory configurations on condition that the kinematic description of the motion of the emitting particle is identical. The total radiated power is described by-one and the same formula in all cases.

The same approach has been used to study the linear polarization radiation by an arbitrarily moving particle. It is shown that the integral components of linear polarization depend in a large degree on the spatial orientation of the particle trajectory with respect to the coordinate axes. This statement is supported by examples of curvature and fan-like radiation.

The results obtained here can be used to analyze the angular distribution and the polarization of radiation in the tasks for which the orientation of the radiating charge trajectory with respect to the used experimental equipment is essential. This investigation is also important in issues of cosmic radiation, for example, in constructing profiles of total and polarized pulsar radiation.

1

a

2

b

a

4

2

4

M

Z ? Q

X

Figure 1. Coordinate system of an arbitrarily moving charge and angular orientation of its radiation. Jet lies in XY plane

Figure 2. Angular distribution indicatrixes of the total power radiation by an arbitrarily moving charge for a = n/6, /3 = 0.9. a) n = 0 A = 0 (curvature radiation) b) n = n/2, À = n/2 (fan-like radiation) c)n = n/6, A = n/6 (arbitrarily radiation)

Figure 3. The kinematic configurations of radiation and unit vectors of the linear polarization: a) the case of curvature radiation (n = 0 A = 0), b) the case of fan-like radiation (n = n/2, A = n/2)

Figure 4. Polarization indicatrixes of an arbitrarily moving charge pe, p^, p = p^ + pe (top down) ft = 0.5 a = 0, 55, 75, 90 (left right): a) curvature radiation; b) fan-like radiation

References

[1] Bordovitsyn V. A. 1999 Synchrotron radiation theory and its development (World Scientific) ,447 p.

[2] Rohrlich F. 2007 Classical charged particles (World Scientific), 305 p.

[3] Bagrov V. G. and Markin U. A. 1967 Izv. Vuz. Fiz, 10 (5) 37.

[4] Bagrov V. G„ Bordovitsyn V. A. and Kopytov G. F. 1972 Izv. Vuz. Fiz. 15 (3) 30.

[5] Bagrov V. G. 1967 Izv. Vuz. Fiz. 10 (8) 135.

Received 23.10.2014

В. Г. Багров, В. А. Вордовицын, В. Г. Вуленок, А. В. Куликова

ЛИНЕЙНАЯ ПОЛЯРИЗАЦИЯ ИЗЛУЧЕНИЯ ПРОИЗВОЛЬНО ДВИЖУЩЕГОСЯ

РЕЛЯТИВИСТСКОГО ЗАРЯДА

Методы классической теории релятивистского излучения произвольно движущегося заряда используются для исследования общих свойств углового распределения полной мощности и соответствующих компонент линейной поляризации излучения.

Ключевые слова: релятивистское излучение заряда, угловое распределение мгновенной мощности излучения, индикатриса излучения, линейная поляризация излучения, кинематические характеристики движения заряда.

Багров В. Г., доктор физико-математических наук, профессор. Томский государственный университет. Пр. Ленина, 60, 634050 Томск, Россия. Институт сильноточной электроники СО РАН.

Пр. Академический, 2/3, 634055 Томск, Россия. E-mail: bagrov@phys.tsu.ru

Вордовицын В. А., доктор физико-математических наук, профессор. Томский государственный университет. Пр. Ленина, 60, 634050Томск, Россия. E-mail: vabord@sibmail.com

Вуленок В. Г., кандидат физико-математических наук. Томский государственный педагогический университет.

Ул. Киевская, 60, 634061 Томск, Россия. E-mail: vadim@tspu.edu.ru

Куликова А. В., аспирант.

Томский государственный университет.

Пр. Ленина, 60, 634050 Томск, Россия. E-mail: anasta-kulikova@yandex.ru