Scattering of electromagnetic wave by linear chain of charged particles
The Thomson scattering of a plane monochromatic electromagnetic wave by a linear chain of periodically spaced charged particles is investigated theoretically. It is obtained a functional dependence for the total power and angular distribution of this radiation as a function of the number of the char...
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2002
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| Zitieren: | Scattering of electromagnetic wave by linear chain of charged particles / V.V. Ognivenko // Вопросы атомной науки и техники. — 2002. — № 2. — С. 104-106. — Бібліогр.: 8 назв. — англ. |
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Ognivenko, V.V. 2015-04-12T06:56:27Z 2015-04-12T06:56:27Z 2002 Scattering of electromagnetic wave by linear chain of charged particles / V.V. Ognivenko // Вопросы атомной науки и техники. — 2002. — № 2. — С. 104-106. — Бібліогр.: 8 назв. — англ. 1562-6016 PACS: 41.60. Cr https://nasplib.isofts.kiev.ua/handle/123456789/80130 The Thomson scattering of a plane monochromatic electromagnetic wave by a linear chain of periodically spaced charged particles is investigated theoretically. It is obtained a functional dependence for the total power and angular distribution of this radiation as a function of the number of the charged particles and the distance between them. The coherence effects for a linear chain of pointlike bunches are discussed. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Theory and technics of particle acceleration Scattering of electromagnetic wave by linear chain of charged particles Рассеяние электромагнитной волны линейной цепочкой заряженных частиц Article published earlier |
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Scattering of electromagnetic wave by linear chain of charged particles |
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Scattering of electromagnetic wave by linear chain of charged particles Ognivenko, V.V. Theory and technics of particle acceleration |
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Scattering of electromagnetic wave by linear chain of charged particles |
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Scattering of electromagnetic wave by linear chain of charged particles |
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Scattering of electromagnetic wave by linear chain of charged particles |
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Scattering of electromagnetic wave by linear chain of charged particles |
| title_sort |
scattering of electromagnetic wave by linear chain of charged particles |
| author |
Ognivenko, V.V. |
| author_facet |
Ognivenko, V.V. |
| topic |
Theory and technics of particle acceleration |
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Theory and technics of particle acceleration |
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2002 |
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English |
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Вопросы атомной науки и техники |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Рассеяние электромагнитной волны линейной цепочкой заряженных частиц |
| description |
The Thomson scattering of a plane monochromatic electromagnetic wave by a linear chain of periodically spaced charged particles is investigated theoretically. It is obtained a functional dependence for the total power and angular distribution of this radiation as a function of the number of the charged particles and the distance between them. The coherence effects for a linear chain of pointlike bunches are discussed.
|
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1562-6016 |
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https://nasplib.isofts.kiev.ua/handle/123456789/80130 |
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Scattering of electromagnetic wave by linear chain of charged particles / V.V. Ognivenko // Вопросы атомной науки и техники. — 2002. — № 2. — С. 104-106. — Бібліогр.: 8 назв. — англ. |
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| first_indexed |
2025-11-26T15:09:26Z |
| last_indexed |
2025-11-26T15:09:26Z |
| _version_ |
1850625783671816192 |
| fulltext |
SCATTERING OF ELECTROMAGNETIC WAVE
BY LINEAR CHAIN OF CHARGED PARTICLES
V.V. Ognivenko
National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine
e-mail: ognivenko@kipt.kharkov.ua
The Thomson scattering of a plane monochromatic electromagnetic wave by a linear chain of periodically spaced
charged particles is investigated theoretically. It is obtained a functional dependence for the total power and angular
distribution of this radiation as a function of the number of the charged particles and the distance between them. The
coherence effects for a linear chain of pointlike bunches are discussed.
PACS: 41.60. Cr
1. INTRODUCTION
Investigations of the electromagnetic radiation
coherence from the charged particles bunches is of
considerable interest for a number of plasma and beam
physics applications (see, e.g. [1,2]). These
investigations are extremely impotent for the researches
directed on the short wavelength coherent radiation in
free – electron lasers (FEL), which are expected
generate coherent electromagnetic radiation at
wavelengths on the order of tens of angstroms [3,4]. For
a number of charged particles bunches, the dependence
of the total intensity of scattered electromagnetic wave
(EMW) on character distance between scatterers in
bunch has been investigated in [5-7]. In this paper the
results of investigations of the angular distribution and
the total intensity of radiation for a linear charged
particles chain under the Thomson scattering of EMW
are presented.
2. FORMULATION OF THE PROBLEM
As a model of the bunch configuration let us
consider a linear chain of a finite number of identical
charged particles scattering the plane monochromatic
linearly polarized EMW. The main physical reasons for
the choice of this model are follows. The Thomson
scattering of EMW by charged particles is the
fundamental physical process of electromagnetic field
emission. Secondly, for the ultra–relativistic beam
energy range a plane undulator field in the beam rest
frame is similar to the EMW. On another hand, by
varying the distance between point charge-radiators and
their total number can be control the level of radiation
coherence by the individual bunch, and by the chain of
pointlike bunches as well.
Let us consider a linear chain of N identical charged
particles, with mass m and charge q. In this chain the
distance between two neighboring particles is d. The
plane monochromatic EMW with the frequency ω, the
wave number k=ω/c and the amplitude E propagates
along the OZ axis (x=y=0), where the scatterers are
situated in: Eext(r,t)=ex E cos(ωt–kz). It is necessary to
determine the total radiation intensity of the scattered
radiation and angular distribution of the energy flux of
this radiation as a function of the number of scatterers N
and the period of their sequence d.
The angular distribution of the energy flux of the
scattered radiation and total intensity of this radiation
will be calculated from the formula for dipole radiation
of charges in its wave zone [8]. Indeed, at a distance
( )r x y z= + +2 2 2 1 2
, considerably larger than the
bunch linear dimension (r>>LN=(N–1)d), the energy
flux density of the radiation into the solid angle element
dΩ=sinϑdϑdϕ can be expressed in terms of the total
dipole momentum of all charges of the bunch D(t):
( ) ( )[ ]dI
c
t dtot ϑ Ω, 'ϕ
π
= 1
4 3
2
D n , (1)
( ) ( ) ( )[ ]D et qa t r kzx s
s
N
' cos ' cos= − −
=
∑ ω 1
1
ϑ .
Here ex is the unit vector along the OX axis,
( )a qE m= ω 2 is the amplitude of particle oscillations
in the EMW field, ϑ is the angle between the unit vector
n toward the observation point and positive direction of
the OZ axis, n=r/r, zs is the longitudinal coordinate of
the charge of the number s, the angular bracket mean a
time average over the field period T=2π/ω, t′(r)=t–r/c is
the retarded time.
Angular distribution of the radiation. In the
considered model of the bunch the right-hand side of
Eq. (1) is a rather simple function of the external bunch
parameters:
( ) ( ) ( ) ( )dI K dItot N inc
Nϑ ϑ ϑ, ; ; ,ϕ θ θ ϕ= , (2)
( ) ( ) ( )[ ]K
N
N s sN
s
N
ϑ ϑ; cos cosθ θ= + − −
=
∑1
2
1
1
, (3)
( ) ( ) ( )dI N I dinc
N ϑ ϑ Ω, sin cosϕ
π
ϕ= −3
8
11
2 2 . (4)
104 PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2002, № 2.
Series: Nuclear Physics Investigations (40), p. 104-106.
Here ( ) ( )dI inc
N ϑ ,ϕ is the angular distribution of
intensity of incoherent radiation of the bunch in solid
angle element dΩ, I
q a
c
q
mc
cE
1
2 2 4
3
2
2
2 2
3
8
3 8
= =
ω π
π
is
the intensity of individual charge radiation, θ=kd is the
dimensionless period of scatterer sequence in the chain,
KN(ϑ;θ) is the bunch coherence factor of the radiation
into the given solid angle element dΩ .
Performing the summation over the charge number s
in the right–hand side of the Eq. (3) we can obtain the
follow expression for the coherence factor:
( ) ( ) ( )[ ]
( )[ ] ( )
K ec N N
ec N
N
N
N
N ϑ ; cos sin cos
cos
cos cos
θ µ µ µ
µ
µ µ
= − − −
− − −
−
2 1 1
2
2 1 1
1
2
12 , (5)
where µ(ϑ;θ)=θ(1–cosϑ)/2, cosecz=1/sinz.
Total power. Integrating the right–hand side (2) over
the total solid angle (0≤ϑ≤π; 0≤ϕ≤2π) gives the
following explicit formula for the total radiation
intensity of the bunch considered [7]:
( ) ( ) ( ) ( )I K Itot N
tot
inc
Nθ θ= , ( )I N Iinc
N = 1 , (6)
( ) ( )K
s
N
N
tot
s s
s
s
s
N
s
s
θ
ρ ρ
ρ
ρ ρ
ρ
= +
+ −
+ −
=
∑
1
3 1 1
1
2cos sin cos . (7)
Here we expressed the total radiation intensity Itot(θ)
in terms of the total incoherent radiation intensity of the
charges ( )I inc
N , ρs=θs. In this way the coherence factor for
the total radiation intensity of the bunch defined by Eq.
(7), as in [7].
3. RESULTS AND DISCUSSION
Eqs. (2)–(7) present the complete solution of the
problem in the explicit analytical form. The last of them
generalizes the asymptotic results of the classical
Thomson scattering theory in case of the idealized
model of the bunch configuration considered (linear
periodic chain of scatterers). In the particular case of
two scatterers (N=2), these formulae are in good
agreement with those obtained earlier in [5]. In a theory
and applications of the microwave electronics based on
bremsstrahlung radiation the functional dependencies of
coherence factor on the external bunch parameters N
and θ are of fundamental interest. Below we will
describe these dependencies in more detail using their
analytical asymptotics and graphs (calculated for the
particular values of scatterers number N).
Analytical asymptotics. For small and large distances
between charges from the Eq. (7) follow the classical
results [8]. Indeed, for small bunch dimensions ( (N–1)θ
<<1 ) its right–hand side takes the form:
( ) ( ) ( )[ ]K N NN
tot θ θ= − −1 7 1 602 2 .
In the alternative limiting case (when the distance
between neighboring scatterers d is greater than
wavelength λ=2π/k, i.e. θ>>1) the overall contribution
of coherent interaction between charges-scatterers in the
bunch
( ) ( ) ( )δ θ θ
θ
θN N
tot
s
N
K
s N
s≡ − ≈ −
=
∑1 3
2
1 1 2
1
sin , (8)
is of the order θ–1<<1 .
For the values of the period d, multiplied to half
wave length (θ=πs; s=1,2,3,..), this contribution
decreases inversely to the square of this period:
( ) ( ) ( )( )δ π
π
ψ ψN s
s
N
N
C N= − − +
1
2
1
6 1
2 2 ' . (9)
Here C=0,577... is the Euler's constant; ψ(z) is Psi
(Digamma) function; ψ(z)≡Γ′(z)/Γ(z); Γ(z) is the
gamma function and ψ′(z)≡dψ/dz.
For the great numbers N (N>>1) the last formula
takes the form δN(πs)=1/(2s2)
Differentiating the right-hand side of the Eq. (7) over
θ we can see, that at points θ=θs=πs the derivative δN
′
(θ) is positive and increases linearly with the number of
particles N and decreases in inversely proportional to
the distance between them θs: δN
′ (θs)= 3(N–1)/(2θs).
The functional dependence of coherence factor in the
angular distribution of radiation ΚN(ϑ;θ) on the ϑ is
more complicated, including two external parameters –
θ and N (see (5)). Nevertheless, some general
characteristics of this dependence can be obtained in
analytical form directly from the Eq. (5). In fact, for
small bunch dimensions (LN << 1) the coherence factor
is at absolute maximum ΚN(ϑ;θ) = N for all values of the
angle ϑ. For a period of charge locations θ≈π the bunch
radiation anisotropy appears: in the directions ϑ and ϑ
* ≡ π – ϑ the values of coherence factor begin to differ
from one another: ΚN(ϑ*;θ) ≠ ΚN(ϑ;θ). Then the
distances between the charges are greater than
wavelength of scattered radiation (θ>2π), the coherence
factor ΚN(ϑ;θ) becomes substantially a non-monotonous
function of the angle ϑ. In particular, for the directions
ϑs , imposed by the equation
( )θ π1 2− =cos ϑ n n ; n=0,1,2,3,...,[θ/π], (10)
this factor is at its maximum
( )K NN nϑ ;θ = . (11)
Here the square bracket means the integer part of the
bracket number.
The half-widths of these extreme (in the angle ϑ) are
inversely proportional to the bunch length LN :
∆ ϑ ≈ −
−
θ N 2
1
1 . (12)
The exceptions are the angles ϑ=0 and ϑ=π for
distances between charges, multiplied to half-length of
the scattered wave: for them the half-widths of the
105
extrema decrease with increasing of the bunch length in
inverse proportion to the square root of this length:
∆ ϑ ≈ −
−
θ N 2
1 2
1 .
It should be noted that the coherence factor
( )K N
tot( ) θ drops quickly with increase of the bunch
dimension (the graphs for the coherence factor
( )K N
tot( ) θ as function of the number of scatterers N and
the distance between them d are presented in [7]).
Corresponding extreme in bremsstrahlung radiation
angular spectrum power of the bunch (see Eqs. (10),
(11) and figure) describe physically only the result of
the coherent summation of bremsstrahlung radiation
fields of individual scatterers phased by the regular field
of scattered wave. In the simplest case N=2 this effect
was investigated in details in [5].
The numerical calculations of the function ΚN(ϑ;θ)
have been carried out.
Figure shows the coherence factor ΚN(ϑ;θ) as a
function of the angle ϑ for two particular values of the
parameter θ for the number of scatterers N=6.
0,0 0,2 0,4 0,6 0,8 1,0
0,0
0,2
0,4
0,6
0,8
1,0
a)
K(ϑ ,θ )
ϑ /π
0,0 0,2 0,4 0,6 0,8 1,0
0,0
0,2
0,4
0,6
0,8
1,0
b)
K(ϑ ,θ )
ϑ /π
Coherence factor ΚN(ϑ;θ) as a function of the angle ϑ
for the parameters N=6, θ=4π/3 (a), 2π (b)
Using the formulae presented above one can
describe the coherence effects in periodic chain of
pointlike bunches separated by arbitrary distances. Let
N bunches, each of them contains M charged particles,
situated along the OZ axis so that the distance between
each two neighbors bunches in the chain is d. Then,
make the substitution q for Mq and m for Mm in
equation (4), and using Eqs. (2), (6) we obtain following
expressions for angular distribution of energy flux of the
scattered radiation:
( ) ( ) ( ) ( )dI M K dItot N inc
M Nϑ ϑ ϑ, ; ; ,ϕ θ θ ϕ= , (13)
and total intensity of this radiation
( ) ( ) ( ) ( )I M K Itot N
tot
inc
M Nθ θ= , (14)
( ) ( ) ( ) ( )dI M dIinc
M N
inc
Nϑ ϑ, ,ϕ ϕ= , ( ) ( )I M Iinc
M N
inc
N= ,
were ( ) ( )dI inc
M N ϑ ,ϕ , ( )I inc
M N are angular distribution and
total intensity of incoherent radiation by all charges in
the chain, respectively. The functions ΚN(ϑ;θ) and
( )K N
tot( ) θ in this equations have the form (3), (5) and (7).
It is show from Eqs. (13), (14) that coherence factors
in total intensity ( ) ( )KM N
tot θ and angular distribution
( )K M N ϑ ,θ for the considered periodic linear chain of
pointlike bunches of charged particles may be write as
( ) ( ) ( ) ( )K M KM N
tot
N
totθ θ= , ( ) ( )K M KM N Nϑ ϑ, ,θ θ= .
Thus, Eqs. (13) and (14) with expressions for
coherence factors (5)-(7) define the total intensity of the
scattered radiation and angular distribution of energy
flux of this radiation under the Thomson scattering of
EMW by periodic chain of pointlike bunches.
REFERENCES
1.A.V. Gaponov, M.I. Petelin. Relativistic High-
Frequency Electronics // Vestnik AN SSSR. 1979,
№ 4, p. 11–23 (in Russian).
2.T.C. Marshall. Free-Electron Lasers. M: Mir,
1987 (in Russian).
3.J.B. Murphy, C. Pellegrini. Free Electron Lasers
for the XUV Spectral Region // Nucl. Instr. and
Meth. 1985, v. A237, № 1,2, p. 159-167.
4.C. Pellegrini. Free Electron Lasers: Development
and Applications // Part. Accel. 1990, v. 33, part V,
p. 159-170.
5.V.I. Kurilko, V.V. Ognivenko. Coherence Effects
in Thomson Scattering // Zh. Eksp. Teor. Fiz. 1992,
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6.V.I. Kurilko, V.V. Ognivenko. Coherence of
electron bunch radiation in an ultrarelativistic
FEL. Proc. Kharkov International Seminar
Workshop on Plasma Laser and Linear Collective
Accelerators.– Kharkov (Ukraine), 1992, p. 178-
186.
7.V.I. Kurilko, V.V. Ognivenko. Scattering of
electromagnetic waves by charged particle clusters
// Fizika plasmy. 1994, v. 20, № 5, p. 475-482 (in
Russian).
106
8.L.D. Landau, E.M. Lifshits. Teorija Polya. M:
Fizmatgiz, 1960 (in Russian).
107
PACS: 41.60. Cr
REFERENCES
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