Тhe superradiance of a bunch of rotating electrons
Equations describing the excitation of a TE wave by a beam of electrons rotating in an external magnetic field in a waveguide in two regimes were considered. In the first regime the interaction of the oscillators − rotating electrons in the magnetic field − was neglected. It was assumed that the b...
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| Cite this: | Тhe superradiance of a bunch of rotating electrons / V.M. Kuklin, D.N. Litvinov, A.E. Sporov // Вопросы атомной науки и техники. — 2018. — № 4. — С. 221-224. — Бібліогр.: 21 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859597968798121984 |
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| author | Kuklin, V.M. Litvinov, D.N. Sporov, A.E. |
| author_facet | Kuklin, V.M. Litvinov, D.N. Sporov, A.E. |
| citation_txt | Тhe superradiance of a bunch of rotating electrons / V.M. Kuklin, D.N. Litvinov, A.E. Sporov // Вопросы атомной науки и техники. — 2018. — № 4. — С. 221-224. — Бібліогр.: 21 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Equations describing the excitation of a TE wave by a beam of electrons rotating in an external magnetic field in
a waveguide in two regimes were considered. In the first regime the interaction of the oscillators − rotating electrons
in the magnetic field − was neglected. It was assumed that the beam electrons interact only with the waveguide
modes. In the second case, in the superradiance regime, the beam electrons interact with each other due to their
spontaneous radiation. In the article the basic features of the description of the gyrotron gain regime are discussed.
Розглянуто рівняння, що описують збудження пучком електронів, що обертаються в зовнішньому магнітному полі ТЕ-хвилі у хвилеводі в двох режимах. У першому режимі нехтуємо взаємодією випромінювачів
– електронів, що обертаються в магнітному полі. Вважаємо, що електрони пучка взаємодіють лише з хвилеводними модами. У другому випадку, у режимі надвипромінювання, електрони пучка взаємодіють один з
одним за рахунок їх спонтанного випромінювання. Обговорюються особливості опису режиму підсилення
гіротрона.
Рассмотрены уравнения, описывающие возбуждение пучком вращающихся во внешнем магнитном поле
электронов ТЕ-волны в волноводе в двух режимах. В первом режиме взаимодействием излучателей – вращающихся в магнитном поле электронов, пренебрегаем. Полагаем, что электроны пучка взаимодействуют
только с волноводными модами. Во втором случае, в режиме сверхизлучения, электроны пучка взаимодействуют друг с другом за счет их спонтанного излучения. Обсуждаются особенности описания режима усиления гиротрона.
|
| first_indexed | 2025-11-27T22:42:04Z |
| format | Article |
| fulltext |
ISSN 1562-6016. ВАНТ. 2018. №4(116) 221
THE SUPERRADIANCE OF A BUNCH OF ROTATING ELECTRONS
V.M. Kuklin, D.N. Litvinov, A.E. Sporov
V.N. Karazin Kharkiv National University, Kharkov, Ukraine
E-mail: v.m.kuklin@karazin.ua
Equations describing the excitation of a TE wave by a beam of electrons rotating in an external magnetic field in
a waveguide in two regimes were considered. In the first regime the interaction of the oscillators − rotating electrons
in the magnetic field − was neglected. It was assumed that the beam electrons interact only with the waveguide
modes. In the second case, in the superradiance regime, the beam electrons interact with each other due to their
spontaneous radiation. In the article the basic features of the description of the gyrotron gain regime are discussed.
PACS: 05.45.Xt, 52.40.Mj
INTRODUCTION
As a rule, in most papers the Larmor rotation radius
of beam electrons is small, less than the characteristic
size of the transverse field inhomogeneity, and is less
than or of the order of the beam thickness (the beam
electrons sufficiently uniformly fill a flat or a cylindrical
layer). Scientific school of A.V. Gaponov in the former
USSR has developed the methods for description of the
excitation of eigen oscillations of the waveguides in the
presence of an external magnetic field by an electron
beam [1, 2], which helped to develop a number of (vac-
uum) devices and equipment.
Later, it became necessary to study the processes of
excitation by charged-particle beams (usually electrons)
of cyclotron oscillations in plasma, for its further heat-
ing, mainly for controlled thermonuclear fusion purpos-
es. This led to a great cycle of the developments in
Kharkov (see, for example, [3 - 5]) in the nonlinear the-
ory of wave excitation in magnetoactive plasma media
(and waveguides) by the beams of charged particles, in
particular, and with finite values of the Larmor radius
[6, 7]. In these works it was also shown that the conser-
vation laws of the elementary effects of anomalous and
normal Doppler are also satisfied for more complicated
beam systems with stimulated radiation.
Similar original descriptions of nonlinear excitation
processes of oscillations by the beams of charged parti-
cles, where the main attention was focused on the rela-
tivistic effects and the spatial restrictions of the complex
resonant systems, were also considered by the authors
[8 - 14].
Great scientific interest to one of the most powerful
and popular generators − the gyrotron was associated
with the need to take into account the plasma medium in
its volume. Actually, if a certain value of the gyrotron
power is exceeded, the gas extraction from the elements
of the structure and its ionization takes place which
leads to the appearance of a plasma with comparatively
low density. The difficulty that deals with the impossi-
bility to separate waves of different polarizations in a
magnetoactive plasma was avoided in the work [15],
with the help of introducing the small parameter of their
coupling
( )2 2 2/ 1xyg c kω ε ⊥= << ,
here 2 2 2/ ( )xy pe B Biε ω ω ω ω ω= − is the transverse compo-
nent of the dielectric permittivity of a cold magnetoac-
tive plasma. Moreover, the ratio of the longitudinal and
transverse components of the wave vector was also
small ( )/ 1zk k⊥ < . Taking into account such low density
plasma made it possible to increase the efficiency of the
device (see, for example, [16]). But nevertheless, the
dynamics of particles at resonances requires further de-
tailed study [17 - 20]. In this work we dwell on the
problem of radiation of an group of interaction with
each other electrons under the conditions of the cyclo-
tron resonance with a TE wave.
1. EXCITATION OF THE TE-WAVE
BY THE NONINTERACTICE
WITH EACH OTHER ELECTRONS
The dispersion equation of such wave in metallic
waveguide can be written as:
2 2
2 2 2 1
2 2( , ) ( )( )z zD k k k k
c c
ω ωω −
⊥= + − −
, (1)
at that 2 2 1/22( )zk kD
cω
⊥+
= − , 2
zk zD k= ,
where k⊥ − is the transversal wavenumber that is de-
fined by the boundary conditions, and the group veloci-
ty along the waveguide is expressed as
2 2 1/2( ) / / ( )
( ) /
z
g z z
D kv k c k k
D
ω
ω ω ⊥
∂ ∂
= = +
∂ ∂
,
the longitudinal wave magnetic fields is written in the
form [7]
( ) exp{ }z m zB b J k r i t im ik zω θ⊥= ⋅ − + + , (2)
and the field equation has the following form [7]
'
1
1 ( ) exp{ 2 },
N
g j n j j
j
B BV i B B i a J a i
Nδ δ p ζ
τ x =
∂ ∂
+ − ∆ +Θ = −
∂ ∂ ∑
(3)
where tτ δ= , 0
2
02
Bnωµ
δβ⊥
= , 2
2
z B
ms
kR
k
ω
δ
= , 02 z bM k d Np= ⋅ ,
2 ,zk zx p=
2 2 2 2 2 2
0
2
1 2
2
4 [ ( )
(1 ) ] ( ),
b B e ms W m ms
m n ms C
ms
N e m c k r J x
m D J k r
x ω
δ ω
−
−
= ⋅ ⋅ ⋅ ⋅ ⋅
− ⋅ ⋅
0
0
2 2 '
( ) / 2,
z z B
C
k z t k z n t m
n m m
pζ pζ ω ω
p
± = ± = − ⋅ ± + + ⋅Φ +
+ − Φ +
( ) /m n ms C eB eb J k r m c δ−= ⋅ ⋅ ,
2 2
2 z
g
k cV p
ωδ
=
2
0 2
0
[( ) / ] (1 )z z B
ak v n
a
η ω ω δ µ= − + + ⋅ − ,
0bN −
is the number of the particles on the unit length of the
waveguide. The electrons motion equations that do not
interact with each other and interact only with the wave
have the following form:
ISSN 1562-6016. ВАНТ. 2018. №4(116) 222
2
22 ( ) [1 ] Cos(2 ),i
i n i i e
i
d nnB J a
d a
ζ
p = η + ⋅ ⋅ − ⋅ pζ + ϕ
τ
'/ ( ) Sin(2 )i n i i eda d n B J aτ = − ⋅ ⋅ ⋅ pζ + ϕ , (4)
'/ ( ) Sin(2 ).i i n i i ed d R B a J aη τ = − ⋅ ⋅ ⋅ ⋅ pζ + ϕ
2. SPONTANEOUS RADIATION
OT THE SINGLE ELECTRON
The case of the radiation of the single particle (from
its total number, that is equal to N) it is necessary to
consider in the following manner. The equation for the
field, that is radiated from a single particle, can be writ-
ten in the form
' ( ) exp{ 2 ' }exp{ } ( )j j
g n j j z j j
B a
v i J a i ik z z z
z N
p ζ δ
∂
= − − ⋅ −
∂
(5)
or ( )j
j
B
z z
z
λ δ
∂
= ⋅ −
∂
, where
' ( ) exp{ 2 ' }exp{ }j
n j j z j
g
a
i J a i ik z
Nv
λ p ζ= − −
its solution has the form ( )j jB C z zλ θ= + ⋅ − , where C −
is a constant which must be defined.
Since for the wave that is radiated by the oscillator
the equation ( , ) 0D kω = with the following roots
2
2 1/2
1,2 2Re (1 Im / Re ) ( ) (1 0)zk D i D D k i
c
ω
⊥= ± + ≈ ± − +
is valid, so for the wave that propagates in the jz z>
direction, the wavenumber 1 0z zk k= > and the con-
stant C must be chosen to be equal to zero for avoiding
field unlimited growth at the infinity. For the wave, that
propagates in the jz z< direction, the wavenumber
2 0z zk k= < and the value of the constant C for the same
reasons must be chosen to be equal to λ− . At the
same time the field amplitude can be expressed as
'( ) ( ) exp{ 2 ' }[exp{2 ( }
( ) exp{ 2 ( } ( )],
j
j n j j j
g
j j j
a
B i J a i i
NV
U i U
x p ζ p x x
x x p x x x x
= − − ⋅
− + − − ⋅ −
(6)
here ( ) 1U z = when 0z ≥ and ( ) 0U z = when 0z < . It
is necessary to mention that the direction of the longitu-
dinal component of the magnetic-field strength vector at
that case does not depend on the wave propagation di-
rection. Such behavior is caused by the suppression by
them the eigen magnetic field of the rotating electron
while the wave radiates in both directions.
3. THE EQUATIONS OF ELECTRONS
BEAM SUPERRADIANCE
It is obvious that for the system of N oscillators the
equation for the field can be written in the form
'
1
1( ) ( ) exp{ 2 ' }[exp{2 ( }
2
( ) exp{ 2 ( } ( )],
N
j n j j j
j
j j j
B i a J a i i
N
U i U
x p ζ p x x
J
x x p x x x x
=
= − − ⋅
⋅ − + − − ⋅ −
∑ (7)
where 02 / 2 /g b gV N M v dJ δ= = ⋅ − is the maximum in-
crement δ to the damping decrement due to radiation
from the ends of the system 2 /gv d ratio. It is necessary
to note, that in such notations wave energy on the sys-
tem length 2M zk dx p= to the particles energy ratio is
expressed as such expression 2 2
0
10
1| ( ) | /
M N
i
i
B d a
N
x
x x
=
∑∫ .
Since the ratio of the energy radiated from the system to
the total field energy at the time of order 1/ δ in the
system is equal to J, so the efficiency of the system (if
the quantity 1/ δ is chosen as the time unit) may be
evaluated as 2 2
0
10
1| ( ) | /
M N
i
iM
B d a
N
xJ x x
x =
∑∫ .
The beam electrons motion equation may be written
as:
( )
2
'
2
1
12 1 ( )
2
[Sin{2 ( )} ( )
Sin{2 ( )} ( )];
N
i
i n i j n j
j
i j i j
i j j i
d n n J a a J a
d Na
U
U
=
+ + + +
− − − −
ζ
p = η + − ⋅ τ J
⋅ p ζ − ζ ⋅⋅ ζ − ζ +
+ p ζ − ζ ⋅ ζ − ζ
∑
(8)
( )'
'
1
( )[Cos{2 ( )}
2
( ) Cos{2 ( )} ( )];
N
n ii
j n j i j
j
i j i j j i
J ada n a J a
d N
U U
p ζ ζ
τ J
ζ ζ p ζ ζ ζ ζ
+ +
=
+ + − − − −
= − − ⋅
⋅ − + − ⋅ −
∑
( )'
'
1
( )[Cos{2 ( )}
2
( ) Cos{2 ( )} ( )].
N
n ii
j n j i j
j
i j i j j i
RJ ad a J a
d N
U U
η p ζ ζ
τ J
ζ ζ p ζ ζ ζ ζ
+ +
=
+ + − − − −
= − − ⋅
⋅ − + − ⋅ −
∑
It is necessary to mention that the field values are
not needed for the calculations they are already included
in the right-hand parts of (8).
Nevertheless it is possible to calculate the value of the
longitudinal magnetic field (7), with the help of which it
can possible to restore all other wave field components at
any point as inside waveguide as outside it.
4. THE REGIME OF THE GYROTRON
SUPERRADIANCE
For the small values of the Larmor radius, for
exp{ / }i i iA a i nζ += ⋅ and for the exact resonance 0iη = ,
assuming that Bessel functions arguments are small
1 1( ) ( / 2) ( ); ' ( ) ( / 2)n n
n nJ x x J x x
n x
≈ = let obtain the
equation for the longitudinal magnetic wave field
1
1( ) ( ) exp{ 2 ' }
2 2
[exp{2 ( } ( )].
N
j n
j
j
j j
a
B i i
N
i U
x p ζ
J
p x x x x
=
= − ⋅
⋅ − ⋅ −
∑
(9)
For the case of the system of electrons that rotate in
the magnetic field and practically do not shift along the
system let show the motion equations
( )
2
2
2
0
1
( / ) (1 ) / 2
4
1 ( / 2) [Sin{2 ( )} ( )];
2
ni i i
i i i
i
N
n
j i j i j
j
d a naa n a a
nd a
a U
N
ζ µ
τ J
p ζ ζ ζ ζ
−+
+ + + +
=
= ⋅ − − ⋅
⋅ − ⋅ −∑
(10)
( ) 2
1
/ 2
( / 2) [Cos{2 ( )}
8
( )].
n N
i ni
i j i j
j
i j
ada na a
d N
U
p ζ ζ
τ J
ζ ζ
−
+ +
=
+ +
= − − ⋅
⋅ −
∑
The similar approach was used by the authors of
[1, 2] to create the theory of gyrotron. However, the
gyrotron theory was based on the neglecting of the elec-
trons interaction with each other due to their spontane-
ous radiation. These effects were neglected, considering
ISSN 1562-6016. ВАНТ. 2018. №4(116) 223
that the electrons interact only with waveguide modes.
In the dissipative regimes of generation 1δΘ >> , as was
noted earlier (see also the article of the authors in this
proceedings), the superradiance of interacting electrons
and the dissipative generation regime under the condi-
tions of neglecting of their interaction nevertheless lead
to the same characteristic times of the process develop-
ment and to the comparable generation intensities.
However, these are different processes and they have a
number of features, so we give the equations for the
superradiation of a gyrotron in the gyrotron traditional
gain regime. Moreover with the notation and terms
those are similar to the papers [1, 2]. Thus for example,
equations (10) can be written in a complex form
( ) 2
2
1
| | /2
(| | 1)}
| |
( ) exp{2 ( } ( )];
2
n
ii
i i
N
j n
i j i j
j
AdA i i A A GA
dZ N
A
i Up ζ ζ ζ ζ
−
=
= ∆ + − − ⋅
⋅ − ⋅ −∑
(11)
where the following quantities were used
2 2
04 /b b ee N Smω p= − averaged over the volume, the
Langmuir frequency of the beam particles, 2
WS rp= −
waveguide cross-section, 2 2
0 / 2B zZ z vω δ β⊥= ,
0
2
0 0
2( )B
B
n
n
ω ω
ω β⊥
−
∆ =
⋅
, exp{ / }i i iA a i nζ += ⋅
2 2( / )v cβ⊥ ⊥= , 2 2 2 1
0
2 2 2 2 2
0
( )
4 ( ) ( )
n
b m n ms C
m ms ms k
nV J k r aG
c J x x m D
ω
pβ
−
−
⊥
⋅
=
⋅ ⋅ ⋅ − ⋅
.
In this approach the only one complex equation (11)
where the values of the fields interacting with the parti-
cles have already been taken into account is completely
sufficient to calculate the gain regime. The calculation
details can be found in the book [21].
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Article received 04.06.2018
http://link.springer.com/journal/11232
http://link.springer.com/journal/11232
ISSN 1562-6016. ВАНТ. 2018. №4(116) 224
СВЕРХИЗЛУЧЕНИЕ СГУСТКА ВРАЩАЮЩИХСЯ ЭЛЕКТРОНОВ
В.М. Куклин, Д.Н. Литвинов, А.Е. Споров
Рассмотрены уравнения, описывающие возбуждение пучком вращающихся во внешнем магнитном поле
электронов ТЕ-волны в волноводе в двух режимах. В первом режиме взаимодействием излучателей – вра-
щающихся в магнитном поле электронов, пренебрегаем. Полагаем, что электроны пучка взаимодействуют
только с волноводными модами. Во втором случае, в режиме сверхизлучения, электроны пучка взаимодей-
ствуют друг с другом за счет их спонтанного излучения. Обсуждаются особенности описания режима уси-
ления гиротрона.
НАДВИПРОМІНЮВАННЯ ЗГУСТКА ЕЛЕКТРОНІВ, ЩО ОБЕРТАЮТЬСЯ
В.М. Куклін, Д.М. Литвинов, О.Є. Споров
Розглянуто рівняння, що описують збудження пучком електронів, що обертаються в зовнішньому магні-
тному полі ТЕ-хвилі у хвилеводі в двох режимах. У першому режимі нехтуємо взаємодією випромінювачів
– електронів, що обертаються в магнітному полі. Вважаємо, що електрони пучка взаємодіють лише з хвиле-
водними модами. У другому випадку, у режимі надвипромінювання, електрони пучка взаємодіють один з
одним за рахунок їх спонтанного випромінювання. Обговорюються особливості опису режиму підсилення
гіротрона.
|
| id | nasplib_isofts_kiev_ua-123456789-147441 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-11-27T22:42:04Z |
| publishDate | 2018 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Kuklin, V.M. Litvinov, D.N. Sporov, A.E. 2019-02-14T18:42:40Z 2019-02-14T18:42:40Z 2018 Тhe superradiance of a bunch of rotating electrons / V.M. Kuklin, D.N. Litvinov, A.E. Sporov // Вопросы атомной науки и техники. — 2018. — № 4. — С. 221-224. — Бібліогр.: 21 назв. — англ. 1562-6016 PACS: 05.45.Xt, 52.40.Mj https://nasplib.isofts.kiev.ua/handle/123456789/147441 Equations describing the excitation of a TE wave by a beam of electrons rotating in an external magnetic field in a waveguide in two regimes were considered. In the first regime the interaction of the oscillators − rotating electrons in the magnetic field − was neglected. It was assumed that the beam electrons interact only with the waveguide modes. In the second case, in the superradiance regime, the beam electrons interact with each other due to their spontaneous radiation. In the article the basic features of the description of the gyrotron gain regime are discussed. Розглянуто рівняння, що описують збудження пучком електронів, що обертаються в зовнішньому магнітному полі ТЕ-хвилі у хвилеводі в двох режимах. У першому режимі нехтуємо взаємодією випромінювачів – електронів, що обертаються в магнітному полі. Вважаємо, що електрони пучка взаємодіють лише з хвилеводними модами. У другому випадку, у режимі надвипромінювання, електрони пучка взаємодіють один з одним за рахунок їх спонтанного випромінювання. Обговорюються особливості опису режиму підсилення гіротрона. Рассмотрены уравнения, описывающие возбуждение пучком вращающихся во внешнем магнитном поле электронов ТЕ-волны в волноводе в двух режимах. В первом режиме взаимодействием излучателей – вращающихся в магнитном поле электронов, пренебрегаем. Полагаем, что электроны пучка взаимодействуют только с волноводными модами. Во втором случае, в режиме сверхизлучения, электроны пучка взаимодействуют друг с другом за счет их спонтанного излучения. Обсуждаются особенности описания режима усиления гиротрона. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нелинейные процессы Тhe superradiance of a bunch of rotating electrons Надвипромінювання згустка електронів, що обертаються Сверхизлучение сгустка вращающихся электронов Article published earlier |
| spellingShingle | Тhe superradiance of a bunch of rotating electrons Kuklin, V.M. Litvinov, D.N. Sporov, A.E. Нелинейные процессы |
| title | Тhe superradiance of a bunch of rotating electrons |
| title_alt | Надвипромінювання згустка електронів, що обертаються Сверхизлучение сгустка вращающихся электронов |
| title_full | Тhe superradiance of a bunch of rotating electrons |
| title_fullStr | Тhe superradiance of a bunch of rotating electrons |
| title_full_unstemmed | Тhe superradiance of a bunch of rotating electrons |
| title_short | Тhe superradiance of a bunch of rotating electrons |
| title_sort | тhe superradiance of a bunch of rotating electrons |
| topic | Нелинейные процессы |
| topic_facet | Нелинейные процессы |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147441 |
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