Nonlinear analysis of mm waves excitation by high–current REВ in dielectric resonator
A nonlinear self-consistent theory of excitation of millimeter wave lengths electromagnetic fields by high current relativistic azimuthally-modulated electron beam in cylindrical resonator with a dielectric rod is constructed. For generation of high frequency waves is used an electron beam. Nonlinea...
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
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| Цитувати: | Nonlinear analysis of mm waves excitation by high–current REВ in dielectric resonator / K.V. Galaydych, Yu.F. Lonin, A.G. Ponomarev, Yu.V. Prokopenko, G.V. Sotnikov // Вопросы атомной науки и техники. — 2012. — № 6. — С. 158-160. — Бібліогр.: 5 назв. — англ. |
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Galaydych, K.V. Lonin, Yu.F. Ponomarev, A.G. Prokopenko, Yu.V. Sotnikov, G.V. 2016-11-21T20:20:32Z 2016-11-21T20:20:32Z 2012 Nonlinear analysis of mm waves excitation by high–current REВ in dielectric resonator / K.V. Galaydych, Yu.F. Lonin, A.G. Ponomarev, Yu.V. Prokopenko, G.V. Sotnikov // Вопросы атомной науки и техники. — 2012. — № 6. — С. 158-160. — Бібліогр.: 5 назв. — англ. 1562-6016 PACS: 07.57.–c; 41.60 Bq; 94.05.Pt https://nasplib.isofts.kiev.ua/handle/123456789/109220 A nonlinear self-consistent theory of excitation of millimeter wave lengths electromagnetic fields by high current relativistic azimuthally-modulated electron beam in cylindrical resonator with a dielectric rod is constructed. For generation of high frequency waves is used an electron beam. Nonlinear numerical analysis is carried out. Построена нелинейная самосогласованная теория возбуждения электромагнитного излучения миллиметрового диапазона длин волн сильноточным релятивистским азимутально-модулированным электронным пучком в цилиндрическом резонаторе с диэлектрическим стержнем. Проведен нелинейный численный анализ. Побудовано нелінійну самоузгоджену теорію збудження електромагнітного випромінювання міліметрового діапазону довжин хвиль сильнострумовим релятивістським азимутально-модульованим електронним пучком у циліндричному резонаторі із діелектричним стрижнем. Проведено нелінійний чисельний аналіз. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Плазменная электроника Nonlinear analysis of mm waves excitation by high–current REВ in dielectric resonator Нелинейный анализ возбуждения миллиметровых волн сильноточным релятивистским электронным пучком в диэлектрическом резонаторе Нелінійний аналіз збудження міліметрових хвиль сильнострумовим релятивістським електронним пучком у діелектричному резонаторі Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Nonlinear analysis of mm waves excitation by high–current REВ in dielectric resonator |
| spellingShingle |
Nonlinear analysis of mm waves excitation by high–current REВ in dielectric resonator Galaydych, K.V. Lonin, Yu.F. Ponomarev, A.G. Prokopenko, Yu.V. Sotnikov, G.V. Плазменная электроника |
| title_short |
Nonlinear analysis of mm waves excitation by high–current REВ in dielectric resonator |
| title_full |
Nonlinear analysis of mm waves excitation by high–current REВ in dielectric resonator |
| title_fullStr |
Nonlinear analysis of mm waves excitation by high–current REВ in dielectric resonator |
| title_full_unstemmed |
Nonlinear analysis of mm waves excitation by high–current REВ in dielectric resonator |
| title_sort |
nonlinear analysis of mm waves excitation by high–current reв in dielectric resonator |
| author |
Galaydych, K.V. Lonin, Yu.F. Ponomarev, A.G. Prokopenko, Yu.V. Sotnikov, G.V. |
| author_facet |
Galaydych, K.V. Lonin, Yu.F. Ponomarev, A.G. Prokopenko, Yu.V. Sotnikov, G.V. |
| topic |
Плазменная электроника |
| topic_facet |
Плазменная электроника |
| publishDate |
2012 |
| language |
English |
| container_title |
Вопросы атомной науки и техники |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| format |
Article |
| title_alt |
Нелинейный анализ возбуждения миллиметровых волн сильноточным релятивистским электронным пучком в диэлектрическом резонаторе Нелінійний аналіз збудження міліметрових хвиль сильнострумовим релятивістським електронним пучком у діелектричному резонаторі |
| description |
A nonlinear self-consistent theory of excitation of millimeter wave lengths electromagnetic fields by high current relativistic azimuthally-modulated electron beam in cylindrical resonator with a dielectric rod is constructed. For generation of high frequency waves is used an electron beam. Nonlinear numerical analysis is carried out.
Построена нелинейная самосогласованная теория возбуждения электромагнитного излучения миллиметрового диапазона длин волн сильноточным релятивистским азимутально-модулированным электронным пучком в цилиндрическом резонаторе с диэлектрическим стержнем. Проведен нелинейный численный анализ.
Побудовано нелінійну самоузгоджену теорію збудження електромагнітного випромінювання міліметрового діапазону довжин хвиль сильнострумовим релятивістським азимутально-модульованим електронним пучком у циліндричному резонаторі із діелектричним стрижнем. Проведено нелінійний чисельний аналіз.
|
| issn |
1562-6016 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/109220 |
| citation_txt |
Nonlinear analysis of mm waves excitation by high–current REВ in dielectric resonator / K.V. Galaydych, Yu.F. Lonin, A.G. Ponomarev, Yu.V. Prokopenko, G.V. Sotnikov // Вопросы атомной науки и техники. — 2012. — № 6. — С. 158-160. — Бібліогр.: 5 назв. — англ. |
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2025-11-26T21:15:58Z |
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2025-11-26T21:15:58Z |
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158 ISSN 1562-6016. ВАНТ. 2012. №6(82)
NONLINEAR ANALYSIS OF MM WAVES EXCITATION BY
HIGH–CURRENT REВ IN DIELECTRIC RESONATOR
K.V. Galaydych1, Yu.F. Lonin1, A.G. Ponomarev1, Yu.V. Prokopenko2, G.V. Sotnikov1
1National Science Center “Kharkov Institute of Physics and Technology”, Kharkov, Ukraine;
2Usikov Institute of Radiophysics and Electronics, Kharkov, Ukraine
A nonlinear self-consistent theory of excitation of millimeter wave lengths electromagnetic fields by high current
relativistic azimuthally-modulated electron beam in cylindrical resonator with a dielectric rod is constructed. For
generation of high frequency waves is used an electron beam. Nonlinear numerical analysis is carried out.
PACS: 07.57.–c; 41.60 Bq; 94.05.Pt
INTRODUCTION
The problem of increasing the frequency of an
excited electromagnetic waves is a current and active
development task due to the fact that different physical
and technological applications require the availability of
resources millimeter and submillimeter diagnostics. For
example, using of these radiation sources allow
diagnostics of plasma with a density of the order of
11 1310 ...10 cm-3.
Effective use of such sources as the dielectric
waveguide structures, excited by relativistic electron
bunches, demonstrated experimentally in [1]. In similar
structures the main mechanism of generation is
transition radiation and Cherenkov radiation. Excitation
of high frequency in [1] was achieved by using
structures with the transverse dimensions of the order of
a few hundreds microns. In [2] it was suggested that the
azimuthally modulated electron beam to excite
oscillation modes with a high index number. It is
possible to use the structure with the transverse
dimensions of the order of a few centimeters.
In the present paper self-consistent theory of the
electromagnetic field excitation by azimuthally
modulated electron beam in a dielectric resonator is
constructed. We have demonstrated the possibility of the
mode selection by the special choice of the geometry of
the electron beam. In contrast to earlier papers on this
subject [3], in this theory an analytical expression for
the potential field, which is not taken into account
previously, is obtained.
STATEMENT OF THE PROBLEM AND
BASIC EQUATIONS
The structure under investigation is a cylindrical
metal resonator with dielectric rod placed inside. An
excitation source of the resonator is a multi-stream
electron beam, which is a set of cylindrical beams,
located equidistant in azimuth, and extending along the
axis of the resonator near the surface of a dielectric rod.
The side walls of the resonator are assumed to be a
closed with metal grids, transparent for the charged
particles and nontransparent for the excited
electromagnetic fields.
For the construction of the analytical theory of the
excitation of oscillation modes with high azimuthal
number, we start from the technique developed in [4],
where the general nonlinear theory of the excitation of
the dielectric resonators is built. We represent the
excited electric and magnetic field as a sum of
solenoidal and potential parts
, ,= + =t l tE E E H H (1)
where tE and tH are the solenoidal components of
electromagnetic field, and lE is the potential electric
field.
Solenoidal components of an excited
electromagnetic field will seek in the form of
decomposition by eigen solenoidal fields of empty
dielectric resonator:
s s s s
s s
A (t) ( ), i B (t) ( )= = −∑ ∑t tE E r H H r , (2)
where sE , sH are eigen solenoidal fields of resonator
without an electron beam, which satisfy the equations:
rot ( / ) , rot ( / ) ,s s s s s si c i cω ε ω μ= − =H E E H (3)
where s mnlω ω≡ are the eigenfrequencies of resonator;
, ,s n m≡ l numerate, respectively, radial, azimuthal and
axial indexes.
The current density of multi-stream electron beam,
with taking into account the geometry of the problem,
can be written as:
R
N
e p p p p
k 1 p V
p
1 q (r r (t)) (z z (t))
r
( (t) (k 1)2 / N),
= ∈
= δ − δ − ×
×δ ϕ−ϕ − − π
∑ ∑j v
(4)
where pq is the charge of the macroparticle; pr , pϕ ,
pz and pv are its time-dependent coordinates and
velocity. The summation in (4) is carried out over the
particles of N beams, being in the resonator volume
RV .
By using the orthonormality conditions of the
functions sE and sH [4]:
4 ,
R R
s s s s s ss
V V
dV dV Pε μ π δ∗ ∗
′ ′ ′= =∫ ∫E E H H (5)
for calculation the expansion coefficients ( )sA t and
( )sB t one can obtain the second–order differential
equations:
ISSN 1562-6016. ВАНТ. 2012. №6(82) 159
2 2
2 2
2 2, ,s s s
s s s s s s
d A dR d BA B R
dt dt dt
ω ω ω+ = − + = −
(6)
where *
s s p
s
1R ( )dV
P
= ⋅∫ ej E r .
Having solved Maxwell equations together with the
boundary conditions, for the axial components of the
electric and magnetic solenoidal fields we obtain:
im im
sz zm sz zmE e (r) cos k ze ,H h (r)sin k ze ,ϕ ϕ= =l l (7)
where zme (r) and zmh (r) are the functions, which
describes the radial structure of the electromagnetic
fields, in the dielectric – I ( r a≤ ) and in the vacuum – II
( a r b≤ < ). In present paper expressions for this
functions are omitted, and presented in full view in [4].
Substituting the expression for the current density of
the beam (4) to (6), and using found eigenfunctions of
electric field components (7), we obtain the expression
for s s sR Re R i Im R= + , which are in the right-hand
sides of the equations (6)
(
)
R
n, jN, p pz z, jN p p
n, jN, p V
pr r, jN p p p
p , jN p p p
NRe R q v e (r ) cos k z
P
v e (r )sin k z cos jN
v e (r )sin k z sin jN ,
∈
ϕ ϕ
⎡= +⎣
⎤ ϕ −⎦
ϕ
∑l l
l
l
l
(8)
(
)
R
n, jN, p pz z, jN p p
n, jN, p V
pr r, jN p p p
p , jN p p p
NIm R q v e (r )cos k z
P
v e (r )sin k z sin jN
v e (r )sin k z cos jN ,
∈
ϕ ϕ
− ⎡= +⎣
⎤ ϕ +⎦
ϕ
∑l l
l
l
l
(9)
here 0, 1,...j = ± And besides, n,m,R 0=l for m jN≠ .
Finally, for the solenoidal components of
electromagnetic field we obtain
t
z j n, jN, z, jN s
j,n,
t
z j n, jN, z, jN s
j,n,
E 2 A e (r)cos k z cos( jN ),
H 2 B h (r)sin k zsin( jN );
= ε ϕ+α
= ε ϕ+β
∑
∑
l l
l
l l
l
(10)
n, jN, n, jN,
n, jN, n, jN,
n, jN, n, jN,
Im A Im B
t g , t g .
Re A Re B
α = β =l l
l l
l l
(11)
The potential electric field, which can be presented
in the form = −∇ΦlE , satisfy the Poisson equation:
2 2
2 2 2
1 1 4+ +r
r r r r z
πε ρ
ε ϕ ε
∂ ∂Φ ∂ Φ ∂ Φ⎛ ⎞ = −⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠
(12)
with the boundary conditions consisting in that the
potential Φ on the resonator metal walls becomes zero
and continuity of the potential and radial component
electric induction vector. Eq. (12) is solved by the
method of eigenfunction expansion. The eigenvalues
and eigenfunctions are solutions of the corresponding
Sturm-Liouville problem. The final expression for the
potential Φ can be written as:
,22 2
0 , ,
,
8
( )sin
( )
( )sin cos ( ).
j p
n n jN l
j n l p n jN l n
n n jN p l p p
N q
R r k z
k L R
R r k z jN
ε
κ
κ
κ ϕ ϕ
=
Φ = ×
+
× −
∑∑∑
(13)
Eigenvalues κ are determined from the equation
' '( ) ( ) ( ) ( ).m m m mJ a Z a Z a J aκ κ ε κ κ=
(14)
The orthogonality of the eigenfunctions ( )mR r ,
which define dependence of the potential versus radius,
and their norms 2
mR defines as follows:
2
0
( ) ( ) ( ) ,
b
m m m mmrdr r R r R r Rε δ′ ′=∫ (15)
where ( )( ) ( ) , ( ) ( ) ,
( )
I IIm
m m
m
Z rR r J a R r J a
Z a
κκ κ
κ
= = and
( ) ( ) ( ) ( ) / ( ).m m m m mZ r J r J b Y r Y bκ κ κ κ κ≡ −
NUMERICAL INVESTIGATION
The main goal of numerical investigation of
excitation by multistream electron beam in a dielectric
resonator was to analyze the possibility of excitation
oscillations in the millimeter wavelength structure with
transverse dimensions on the order more of the excited
oscillations.The parameters of the resonator under study
were: radius of the dielectric rod 4.0 cma = , radius of
the resonator 7.5cmb = , resonator length 0.83 cmL = ,
permittivity of the rod 2.04ε = . The beam parameters
are: quantity of the beams 36N = , beam current and
energy was, respectively, 1.5kA and 300kV . Analysis
of an excited field in the resonator has shown that the
main mechanism of generation is monotron mechanism
[5]. The main contribution to the energy make
eigenmode with the frequency 34.117 GHz for which
the monotron mechanism efficiency is the best, under
the chosen parameters, and also the greatest coefficient
of coupling of mode with the beam (demonstrated on
Fig. 1).
Fig. 1. Radial distribution of "whispering gallery" mode
Fig. 2 shows dependence of the electromagnetic
field energy, stored in the resonator, on time.
160 ISSN 1562-6016. ВАНТ. 2012. №6(82)
Fig. 2. Energy of electromagnetic field stored in the
resonator
For a stationary regime of excitation is characterized
essentially nonlinear dynamics of the beam. Fig. 3 hows
the phase planes of the beam particles during the one
oscillation period.
Fig.3. Phase planes for different moments of time
corresponding to the energy saturation
Due to beam-excited field the modulation of the
beam by density and the bunching of the beam is take
place. Fig. 3 shows the formation of multystream flow
and overturning of the forward front of the perturbed
beam.
CONCLUSIONS
Analytical investigations and nonlinear numerical
analysis of an excitation of the dielectric resonator by
the high current relativistic azimuthally–modulated
electron beam are carried out. Demonstrated the
possibility of the mode selection and increasing the
frequency of an excited oscillations through the
modulation of the electron beam in azimuth.
REFERENCES
1. M.C. Tompson, H. Badakov, A.M. Cook, et al.
Breakdown Limits on Gigavolt-per-Meter Electron-
Beam-Driven Wakefields in Dielectric Structures //
Phys. Rev. Lett. 2008, v. 100, №21, p. 214801.
2. A.Ya. Kirichenko, Yu.F. Lonin, V.G. Papkovich,
et al. Microwave oscillator with "whispering gallery"
resonator // Problems of Atomic Science and
Technology. Series “Nuclear Physics Investigations”
(53). 2010, №2, p. 135-139.
3. L.A. Vainstein. Electromagnetic waves. 2nd ed. M.:
“Radio i svyaz”, 1988.
4. K.V. Galaydych, Yu.F. Lonin, A.G. Ponomarev, et al.
Mathematical model of an excitation
by electron beam of “whispering gallery” modes
in cylindrical dielectric resonator // Problems of Atomic
Science and Technology. Series: Plasma Physics (16).
2010, №6, p. 123-125.
5. V.A. Balakirev, V.O. Podobinsky. The theory of a
relativistic monotron // Problems of Atomic Science and
Technology. Series “Nuclear Physics Investigations”
(53). 2010. №2, p. 86-88.
Article received 13.09.12
НЕЛИНЕЙНЫЙ АНАЛИЗ ВОЗБУЖДЕНИЯ МИЛЛИМЕТРОВЫХ ВОЛН СИЛЬНОТОЧНЫМ
РЕЛЯТИВИСТСКИМ ЭЛЕКТРОННЫМ ПУЧКОМ В ДИЭЛЕКТРИЧЕСКОМ РЕЗОНАТОРЕ
К.В. Галайдыч, Ю.Ф. Лонин, А.Г. Пономарев, Ю.В. Прокопенко, Г.В. Сотников
Построена нелинейная самосогласованная теория возбуждения электромагнитного излучения
миллиметрового диапазона длин волн сильноточным релятивистским азимутально-модулированным
электронным пучком в цилиндрическом резонаторе с диэлектрическим стержнем. Проведен нелинейный
численный анализ.
НЕЛІНІЙНИЙ АНАЛІЗ ЗБУДЖЕННЯ МІЛІМЕТРОВИХ ХВИЛЬ СИЛЬНОСТРУМОВИМ
РЕЛЯТИВІСТСЬКИМ ЕЛЕКТРОННИМ ПУЧКОМ У ДІЕЛЕКТРИЧНОМУ РЕЗОНАТОРІ
К.В. Галайдич, Ю.Ф. Лонін, А.Г. Пономарьов, Ю.В. Прокопенко, Г.В. Сотніков
Побудовано нелінійну самоузгоджену теорію збудження електромагнітного випромінювання
міліметрового діапазону довжин хвиль сильнострумовим релятивістським азимутально-модульованим
електронним пучком у циліндричному резонаторі із діелектричним стрижнем. Проведено нелінійний
чисельний аналіз.
|