Excitation of harmonics by flows of the charged particles in the magnetic field
Results of research of the electromagnetic waves excitation by ensemble of relativistic oscillators in a magnetic field are presented. The basic attention is given to studying of generation efficiency of electromagnetic oscillations at harmonics of cyclotron frequency. It is shown, that presence o...
Gespeichert in:
| Datum: | 2009 |
|---|---|
| Hauptverfasser: | , , |
| Format: | Artikel |
| Sprache: | English |
| Veröffentlicht: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2009
|
| Schriftenreihe: | Вопросы атомной науки и техники |
| Schlagworte: | |
| Online Zugang: | http://dspace.nbuv.gov.ua/handle/123456789/88310 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Zitieren: | Excitation of harmonics by flows of the charged particles in the magnetic field / V.A. Buts, A.M. Yegorov, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2009. — № 1. — С. 107-109. — Бібліогр.: 9 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
irk-123456789-88310 |
|---|---|
| record_format |
dspace |
| spelling |
irk-123456789-883102015-11-12T03:02:14Z Excitation of harmonics by flows of the charged particles in the magnetic field Buts, V.A. Yegorov, A.M. Tolstoluzhsky, A.P. Плазменная электроника Results of research of the electromagnetic waves excitation by ensemble of relativistic oscillators in a magnetic field are presented. The basic attention is given to studying of generation efficiency of electromagnetic oscillations at harmonics of cyclotron frequency. It is shown, that presence of medium can essentially change efficiency of excitation of high numbers harmonics and all picture of development of oscillators instability. Increments of instability are received. Influence of thermal spread of oscillators is investigated. Викладено результати дослідження збудження електромагнітних хвиль ансамблем релятивістських осциляторів в магнітному полі. Основна увага приділена вивченню ефективності генерації електромагнітних коливань на гармоніках циклотронної частоти. Показано, що наявність середовища може істотно змінити ефективність збудження високих номерів гармонік і всю картину розвитку нестійкості осциляторів. Отриманo інкременти нестійкості. Дослідженo вплив теплового розкиду осциляторів. Изложены результаты исследования возбуждения электромагнитных волн ансамблем релятивистских осцилляторов в магнитном поле. Основное внимание уделено изучению эффективности генерации электромагнитных колебаний на гармониках циклотронной частоты. Показано, что наличие среды может существенно изменить эффективность возбуждения высоких номеров гармоник и всю картину развития неустойчивости осцилляторов. Получены инкременты неустойчивости. Исследовано влияние теплового разброса осцилляторов. 2009 Article Excitation of harmonics by flows of the charged particles in the magnetic field / V.A. Buts, A.M. Yegorov, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2009. — № 1. — С. 107-109. — Бібліогр.: 9 назв. — англ. 1562-6016 PACS: 05.45.-a; 05.45.-j http://dspace.nbuv.gov.ua/handle/123456789/88310 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| topic |
Плазменная электроника Плазменная электроника |
| spellingShingle |
Плазменная электроника Плазменная электроника Buts, V.A. Yegorov, A.M. Tolstoluzhsky, A.P. Excitation of harmonics by flows of the charged particles in the magnetic field Вопросы атомной науки и техники |
| description |
Results of research of the electromagnetic waves excitation by ensemble of relativistic oscillators in a magnetic field are
presented. The basic attention is given to studying of generation efficiency of electromagnetic oscillations at harmonics
of cyclotron frequency. It is shown, that presence of medium can essentially change efficiency of excitation of high
numbers harmonics and all picture of development of oscillators instability. Increments of instability are received.
Influence of thermal spread of oscillators is investigated. |
| format |
Article |
| author |
Buts, V.A. Yegorov, A.M. Tolstoluzhsky, A.P. |
| author_facet |
Buts, V.A. Yegorov, A.M. Tolstoluzhsky, A.P. |
| author_sort |
Buts, V.A. |
| title |
Excitation of harmonics by flows of the charged particles in the magnetic field |
| title_short |
Excitation of harmonics by flows of the charged particles in the magnetic field |
| title_full |
Excitation of harmonics by flows of the charged particles in the magnetic field |
| title_fullStr |
Excitation of harmonics by flows of the charged particles in the magnetic field |
| title_full_unstemmed |
Excitation of harmonics by flows of the charged particles in the magnetic field |
| title_sort |
excitation of harmonics by flows of the charged particles in the magnetic field |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2009 |
| topic_facet |
Плазменная электроника |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/88310 |
| citation_txt |
Excitation of harmonics by flows
of the charged particles in the magnetic field / V.A. Buts, A.M. Yegorov, A.P. Tolstoluzhsky // Вопросы атомной науки и техники. — 2009. — № 1. — С. 107-109. — Бібліогр.: 9 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT butsva excitationofharmonicsbyflowsofthechargedparticlesinthemagneticfield AT yegorovam excitationofharmonicsbyflowsofthechargedparticlesinthemagneticfield AT tolstoluzhskyap excitationofharmonicsbyflowsofthechargedparticlesinthemagneticfield |
| first_indexed |
2025-07-06T16:04:15Z |
| last_indexed |
2025-07-06T16:04:15Z |
| _version_ |
1836914160450928640 |
| fulltext |
EXCITATION OF HARMONICS BY FLOWS
OF THE CHARGED PARTICLES IN THE MAGNETIC FIELD
V.A. Buts, A.M. Yegorov, A.P. Tolstoluzhsky
NSC “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine,
E-mail: vbuts@kipt.kharkov.ua
Results of research of the electromagnetic waves excitation by ensemble of relativistic oscillators in a magnetic field are
presented. The basic attention is given to studying of generation efficiency of electromagnetic oscillations at harmonics
of cyclotron frequency. It is shown, that presence of medium can essentially change efficiency of excitation of high
numbers harmonics and all picture of development of oscillators instability. Increments of instability are received.
Influence of thermal spread of oscillators is investigated
PACS: 05.45.-a; 05.45.-j
1. INTRODUCTION
PROBLEMS OF ATOMIC SCIENCE AND TECHNOLOGY. 2009. № 1. 107
Series: Plasma Physics (15), p. 107-109.
In the previous researches (see, for example, [1-2]) it has
been shown, that charged oscillators which move in
periodically non-uniform media or in periodically non-
uniform potentials, can effectively radiate high numbers
of harmonics. Effectively radiate high numbers of
harmonics even nonrelativistic oscillators. The physical
reason of such radiation is existence in periodically non-
uniform media of not own virtual slow waves. When
phase velocity of these virtual waves appears close to the
velocity of the charged particles, efficiency of radiation
on high harmonics sharply increases. In the present paper
results of research of radiation, both the single charged
particles, and flows of the charged particles which move
in a constant external magnetic field are stated. It is
supposed that, medium in which particles move, can slow
down the electromagnetic waves excited by particles.
Without going into physical reason of such slowing, this
property is described by presence of constant dielectric
permeability. Let's note, that recently the big attention is
given to research of mechanisms of electromagnetic
waves generation in a submillimetric and infra-red range
of frequencies. One of such mechanisms is radiation
relativistic electrons, moving in a magnetic field, on
harmonics of cyclotron frequency. Below we for
simplicity shall be limited to consideration of ensemble of
relativistic particles in the medium with dielectric
permeability 0ε . Particles are in a constant external
magnetic field 0H
r
. Let's direct axis Z along the field 0H
r
.
Axis X we shall choose so, that Y component of a wave
vector is equal to zero, that is k
r
( ,0, )zk k k⊥=
r
, and in
space of impulses it is used cylindrical system of
coordinates ( , , )zp pϕ⊥ . We shall study excitation of
electromagnetic waves which extend across the magnetic
field. Distribution functions of ensemble oscillators we
shall choose in the form of:
2
0
2
( )
20
1( , ) ( )
2 2
T
p p
b pz z
T
nf p p p
p p eδ
π π
⊥ ⊥−
⊥
⊥
−= (1)
2. DISPERSION EQUATION.
INCREMENT
Instability of electromagnetic waves, propagating in the
medium with dielectric permeability 0ε , is described with
dispersion equation [3,4]):
2 2
2 2det 0i j
ij ij
k kk c
k
δ ε
ω
⎡ ⎤⎛ ⎞
− + =⎢ ⎜ ⎟
⎝ ⎠⎣ ⎦
⎥ , (2)
where 0
b
ij ij ijε ε δ ε= + ., b
ijε -- tensor of dielectric
permeability of oscillators.
We will investigate instability ensemble of oscillators
with respect to excitation of electromagnetic waves at the
harmonics of electron cyclotron frequency. For the
considered type of wave equation (2) can be reduce to the
type:
2 2 2
0 22 /b k cε ε ω⊥ 0+ − = . (3)
2.1. INCREMENT OF OSCILLATIONS, EXCITED
BY COLD OSCILLATORS
For cold oscillators in the case of transversal propagation
of waves ( )0zk = component 2,2
bε of tensor ,
b
i jε looks
like:
( )2 2 2 2
22 0 0
0
1b
ss
s
J P Jε λ λ
λ
∞
=−∞
′
Q
⎡ ⎤
′= + ′⎢ ⎥
⎣ ⎦
∑ , (4)
where
( ) ( )
2 2 2 2 2 2
0 0 0 2 0
02 22
0 0 0
1 1
, ,b b
H H
v
P Q
s cs
ω β ω β β
β
ω ω ω λ ω ω
⊥ ⊥ ⊥ ⊥
⊥
− −
= − = =
− −% %
,
2 ' 0 0
0 0 0 0
0 0
( )
1 , ( ), ,s
H H s s s
H
dJ k v
J J J
d
λ
ω ω β λ λ
λ ω
⊥ ⊥
⊥= − ≡ ≡ =%
%
.
We will consider frequency ω of the excited wave near
to frequency 0Hnω% . Then remaining in right part (4) only
main terms will get:
( ) ( )
( )2 22 2 22 2 2
00 0
0 2
0 00
1
0nb n
H H
Jk c J
n n
λω β β
ε
ω ω λ ωω ω ω
⊥ ⊥⊥
′⎛ ⎞′− ′⎜ ⎟− + −⎜ ⎟− −⎜ ⎟
⎝ ⎠
% %
= . (5)
Maximal increment of oscillations is arrived at equality
of eigen frequency 0 .k c 0ω ε⊥= to frequency of the
excited wave 0Hnω% , therefore, considering in (5)
0 0 0, ,H Hn nω ω δ ω ω δ ω= + ≈ <<% % , will get cube equaltion
for complex addition δ to frequency 0ω
( )2 22 2 2 2 2
00 0 02
0 2
0 0
1 1
2 0nb b
n
H H
J
J
n n
λω β β ω βδε
ω δ λ ω δ
⊥ ⊥ ⊥
′
′− −
0
+ − =′
% %
, (6)
where 0 0n 0λ β ε⊥= .
Remainig major members on the value 1δ − , find
increment of instability
( )
1
2 32
0 2 2
0 0 02
0
13Im ( )
2 2
b
H n
H
n J n
βωδ ω β β ε
ω ε
⊥
⊥ ⊥
⎡ ⎤−
′= ⎢
⎢
108
⎥
⎣ ⎥⎦
. (7)
From (7) evidently, that dependence of increment on the
number of the excited harmonic is concluded in
expression
n
1
2 3( , ) ( )nG n x nJ nx′⎡ ⎤= ⎣ ⎦ . (8)
Graphs of dependence on ( , )G n x x ( 0 0x β ε⊥≡ ) at
different values and dependence on at
different values
n ( , )G n x n
x are resulted on a Fig. 1.
a b
Fig.1. Dependences : ( , )G n x
a) on x at: 1- 10n = ,2- ; 20n =
b) on n at: 1- x =0.8, 2- x =1, 3- x =1.2
As be obvious from these pictures, increment, proportiona
1
2 3( , ) ( )nG n x nJ nx⎡ ′= ⎣ ⎤⎦ , quickly decreases at an increase n,
if 0 0 1β ε⊥ < . At 0 0 1β ε⊥ = increment very slowly falls
with growth . In addition, in this case, it is possible to
show that at . In case
n
10n > 1/ 9( ,1) 0.55G n n−≈
0 0 1β ε⊥ > dependence of increment on the number is
an almost periodic function. Thus, maximal increment is
arrived at at
n
0 0 1β ε⊥ ≥ .
2.2. INCREMENT OF OSCILLATIONS EXCITED
BY OSCILLATORS WITH THERMAL SPREAD
As in case of cold oscillators, including, that
0 0 0
0
, ,H H
k c
n nω ω δ ω ω δ ω
ε
⊥= + ≡ ≈ <<% % ( ) 1
Hnω ω −− % ,
receive the equation for finding of the complex additiveδ
to the frequency 0ω :
( )
2
0
22 2
2 2
00
12
1 2
( ) ( 1)1 exp
1 2
b
sJ x x x dx
ε
σ π
λ
γ γ σ σ
∞
Ω
Δ + ×
+ Δ
⎡ ⎤′ ⎛ ⎞− −
× + −⎢ ⎥⎜ ⎟− + Δ ⎝ ⎠ ⎣ ⎦
∫ 0,=
(9)
in which following dimensionless variables and
parameters are used:
0 0 0 0
2 2
20 0
02 2 2 2
, , , ,
1 , 1 ,
b T
b
H H
p px
n n p p
p p
0 0 .x n x
m c m c
ωδ σ
ω ω
γ γ λ β
⊥
⊥ ⊥
⊥ ⊥
⊥
Δ = Ω = = =
= + = + =
% %
ε
The equation (9) has been solved numerically for
0 3, 0.5b Hγ ω ω= = and various values 0.01σ = ,
0.001σ = , 0 0 1.2β ε⊥ = .
In Fig. 2a dependence of increment from number of
excited harmonic for cold oscillators and oscillators with
thermal spread is shown at 0.001σ = and 0 0 1β ε⊥ = .
One can see, that dependences practically coincide down to
40n ≈ . At increase of thermal spread ( 0.01σ = , Fig.2b)
increment quickly falls down with growth of number . n
a b
Fig.2. Dependences of increment on n at:
a) σ =0 and σ =0.001;
b) σ =0 and σ =0.01
On Fig. 3a increments for oscillators with small
thermal spread 0.001σ = and 0 0 1.2β ε⊥ = are resulted.
Increments for cold oscillators and oscillators with
thermal spread differ a little. In the case of
0 00.01; 1.2σ β ε⊥= = (Fig. 3b) increment for oscillators
with thermal spread falls to zero for harmonics with
numbers more than 50, but remains to enough greater in
the region of 20 40n< < .
a b
Fig.3. Dependences of increment on n at:
a) σ =0 and σ =0.001;
b) σ =0 and σ =0.01
3. THE ELEMENTARY MECHANISM OF
OSCILLATOR’S RADIATION
Radiating oscillator losses осциллятора in the
magnetic field are described by the formula [5]
2 2( / ) (n HP e v w nω ⊥= ) ,
0 02
2 2
0 2 0 0 0 0 2
0 0
( ) 2 (2 ) (1 ) ( )
n
n n
nw n J n J x dx
β ε
ε β β ε β ε
ε
⊥
⊥ ⊥ ⊥
⎡ ⎤
′⎢ ⎥= − −
⎢ ⎥⎣ ⎦
∫ ,
n - number of cyclotron harmonic.
Fig.4. Dependences on n at: ( )w n 0 0β ε⊥ =0.9428;
0 0β ε⊥ =1; 0 0β ε⊥ =1.03
In Fig. 4 graphics of dependence on number of the
harmonic for
( )w n
0γ =3 and various values of dielectric
permeability 0ε are resulted. It is visible, that efficiency
of radiation improves with increase of the harmonic’s
number in conditions, when parameter 0 0 1β ε⊥ ≥ .
4. BASIC RESULTS
109
Increments of waves amplitudes growth are
received. It is shown, that presence of medium with
refraction parameter essentially changes the picture of
oscillators instability development. So, increment quickly
falls at increasing of harmonic’s number , if 0 0 1β ε⊥ < .
In the case 0 0 1β ε⊥ = increment very slowly falls down
with growth of harmonic’s number as . In the case 1/ 9n−
0 0 1β ε⊥ > dependence of increment due to harmonic’s
number is almost periodic function, that is in this case
there are zones of effective excitation of electromagnetic
radiation. Thus, maximal increment is reached at
n
0 0 1β ε⊥ ≥ , it is possible to explain it as follows. With
increase of frequency of oscillations which are generated,
there is not only reduction of number of coherent particles
in a bunch, but also a simultaneous increase in radiating
losses of particles at radiation in this region of
frequencies. It is shown, that there are conditions in which
it is possible to excite effectively harmonics of cyclotron
frequency by oscillators with thermal disorder. However
the thermal disorder should not exceed 5 % from cross-
section speed of the beam. Above we were limited to
model of medium in which slowing-down of excited
electromagnetic waves has been caused by presence of
medium with dielectric permeability 0ε .
It is necessary to tell, that it is already enough for a
long time such media are used as nonlinear elements for
excitation of high numbers harmonic's (see, for example,
[6-9]). Thus it is possible to reach significant values of
susceptibilities even for high numbers of harmonics. For
obtaining of such results vapors of metal atoms are used.
On this way it is possible to excite due to nonlinear
effects ultra-violet, and even radiation in a soft X-ray
range (down to 141-harmonic). Efficiency of such
nonlinear transformation, however, is very insignificant.
Other opportunity consists in using periodically non-
uniform media (lattices of ideal crystals and
superlattices). At that the electromagnetic fields, waves
propagating in such media, contain spatial harmonics
which phase speed can be enough small.
Such possibility has been investigated in a series of
works (look [1-2] and the literature quoted there). For the
case considered in the present work it means, that
oscillators cooperate not with the basic mode, and with
the slow spatial harmonic.
REFERENCES
1. V.A. Buts, A.M. Egorov. Lasers on nonrelativistic
electrons // Uspekhi sovremennoj radioelectronici. 2006,
N 7, p. 3-17 (In Russian).
2. V.A. Buts, V.I. Marecha, A.P.Tolstoluzhsky. Excitation
of high numbers harmonics by charge particles in a
time-periodic electric field and a space-periodic
potential // Problems of Atomic Science and Technology.
Series “Nuclear Physics Investigations” (47). 2006, N 3,
p. 55-59.
3. E.Ya. Zlotnik // Izvestiya VUZov. Radiophysic. 1973,
v. 16, p. 1652 (In Russian).
4. A.I. Akhiezer et al. Plasma electrodynamics. Moscow:
“Nauka”, 1974 (In Russian).
5. J. Schwinger et al // Annals of Phys. 1976, v.96, p.303.
6. V.G. Arkhipkin, A.K. Popov. Nelinejnyye
preobrazovaniya sveta v gazakh. Novosibirsk:
“Nauka”, 1987(In Russian).
7. V.F. Lukinukh, S.A. Myslivets, A.K. Popov,
V.V .Slabko. Chetyrekhvolnovoye smesheniye chastot
v parakh krasitelej // Kvantovaya electronika. 1986,
v. 13, N 7, p. 1415-1422 (In Russian).
8. M.D. Perry, G. Mourou. Terawatt to Petawatt
Subpicosecond Lasers // Sciense. 1994, v. 264,
p. 917-924.
9. V.F. Lukinukh, S.A. Myslivets, A.K. Popov,
V. V .Slabko. Generatsiya VUV-izlucheniya v parakh
rtuti na nelinejnosti sed’mogo I devyatogo poryadkov //
Kvantovaya electronika. 1988, v.15, N 9, p. 1895-1904,
(In Russian).
Article received 22.09.08
ВОЗБУЖДЕНИЕ ГАРМОНИК ПОТОКАМИ ЗАРЯЖЕННЫХ ЧАСТИЦ В МАГНИТНОМ ПОЛЕ
В.А. Буц, А.М. Егоров, А.П. Толстолужский
Изложены результаты исследования возбуждения электромагнитных волн ансамблем релятивистских
осцилляторов в магнитном поле. Основное внимание уделено изучению эффективности генерации
электромагнитных колебаний на гармониках циклотронной частоты. Показано, что наличие среды может
существенно изменить эффективность возбуждения высоких номеров гармоник и всю картину развития
неустойчивости осцилляторов. Получены инкременты неустойчивости. Исследовано влияние теплового
разброса осцилляторов.
ЗБУДЖЕННЯ ГАРМОНІК ПОТОКАМИ ЗАРЯДЖЕНИХ ЧАСТИНОК У МАГНІТНОМУ ПОЛІ
В.О. Буц, О.М. Єгоров, О.П. Толстолужський
Викладено результати дослідження збудження електромагнітних хвиль ансамблем релятивістських
осциляторів в магнітному полі. Основна увага приділена вивченню ефективності генерації електромагнітних
коливань на гармоніках циклотронної частоти. Показано, що наявність середовища може істотно змінити
ефективність збудження високих номерів гармонік і всю картину розвитку нестійкості осциляторів. Отриманo
інкременти нестійкості. Дослідженo вплив теплового розкиду осциляторів.
1. INTRODUCTION
2. DISPERSION EQUATION.
|