The relativistic E-beam interaction with plasma-filled coaxial corrugated waveguide
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| Опубліковано в: : | Вопросы атомной науки и техники |
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| Дата: | 1999 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
1999
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| Цитувати: | The relativistic E-beam interaction with plasma-filled coaxial corrugated waveguide / K.A. Lukin, I.N. Onishchenko // Вопросы атомной науки и техники. — 1999. — № 3. — С. 61-63. — Бібліогр.: 7 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859869069999603712 |
|---|---|
| author | Lukin, K.A. Onishchenko, I.N. |
| author_facet | Lukin, K.A. Onishchenko, I.N. |
| citation_txt | The relativistic E-beam interaction with plasma-filled coaxial corrugated waveguide / K.A. Lukin, I.N. Onishchenko // Вопросы атомной науки и техники. — 1999. — № 3. — С. 61-63. — Бібліогр.: 7 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| first_indexed | 2025-12-07T15:49:24Z |
| format | Article |
| fulltext |
THE RELATIVISTIC E-BEAM INTERACTION WITH PLASMA-FILLED
COAXIAL CORRUGATED WAVEGUIDE
K.A. Lukin, I.N. Onishchenko*
Institute of Radio-Electronics, NASU, *NSC KIPT, Kharkov, Ukraine
1. INTRODUCTION
The conventional FEL-devices are based on
relativistic electron beams and magnetic undulators. For
wavelength shortening and beam current arising the
plasma undulator has been proposed. Taking into
account the recent achievements in high power
generation (e.g. above 1012 W for the corrugated
systems) it is alluring to use these intense HF waves as
an undulator. Let the additional REB of energy mc2γ1
generates the eigenmode of the slow wave structure
(SWS) with frequency ω1 and wave number k1= 2π/D
which propagates oppositely to the main REB direction.
The main REB of energy mc2γ2 being undulated in the
fields of excited wave (by other words in the fields of
induced charges) irradiates the other eigenmode with
frequency ω2 and wave number k2. Using laws of energy
and momentum conservation
ω2 = ω1 + ωb , k2 = kb - k1
we can obtain the following relation for the frequency
of irradiated wave caused by the FEL mechanism:
ω2 = k1 ( v1 + v2 ) ( 1+ v2 /c ) γ2
2
where v1 and v2 are the velocities of the additional and
the main REB, respectively. It is evident that for the
ultrarelativistic case, i.e. v1 = v2 = c it gives a known
formula: ω2 = 2πc/D 4 γ2
2.
From the other side, the novel kind of SWS with
plasma assistance is being developed now. The vacuum
SWS, which are used in Cherenkov microwave
generators and amplifiers, have an essential
shortcoming due to the surface character of a slow
wave. The decreasing of longitudinal field component
from periphery to system axis causes the fall of the
coupling coefficient of the near-axis beam with a slow
wave. It leads to decreasing the instability in growth
rates. This shortcoming is especially strong in the high
frequency range. In the other kind of SWS -
homogeneous plasma waveguides [1] - the slow waves
are volumetric and have the maximal longitudinal
electrical field at the axis where charged particles move.
However, in the non-relativistic region of phase
velocities the plasma waves are quasi-longitudinal, with
small transversal components of fields, that complicates
the microwave energy input and output. Hybrid
systems, which are promizing in the non-relativistic
region of phase velocities, were offered for the first time
in KIPT. They combine the advantages of vacuum and
plasma systems and have no shortcomings marked
above. The hybrid structure uses a plasma waveguide as
the beam transition channel of vacuum SWS [2,3]. In
such a structure the beam-plasma interaction plays the
determining role in excitation of oscillations, and
periodic waveguide system is used for power output.
The coaxial systems have an additional advantage due
to the presence of a cable mode providing a wide
frequency band.
In this work we investigate the first stage of the
problem considered, namely the interaction of the
electron beam with the plasma-filled coaxial corrugated
waveguide. The dispersion equation, which describes
interaction of REB with SWS, is obtained. The non-
linear stage and excitation efficiency is considered.
2. THE DISPERSION EQUATION
The axially-symmetric waveguide, formed by two
coaxial ideally conducting cylinders is considered. The
inner cylinder is smooth, the external one is corrugated
sinusoidally with a period D. In the cylindrical system
of coordinates (r,ϕ,z) waveguide surfaces are given as:
Rs(z)=R0s=const, Rg(z)=R0g(1+δcos(k0z)), k0=2π/D,
where .10,20,,)1(00 < <<≤≤∞<<− ∞−< δπϕδ zRR gs
It is supposed, that waveguide is filled with
homogeneous plasma and is placed in a strong magnetic
field pc ω> >ω , where cω is the cyclotron frequency
and pω is the electron plasma frequency. Waves of E-
type ),,( ϕHEE rz are considered. We solve the
Maxwell equations system together with boundary
conditions for tangential component of electrical field
on a surface of waveguide and we find the dispersion of
electromagnetic waves )( 3kω proceeding from a
condition of existence of the non-trivial solution of this
system, as it was made in [4].
2.1. THE DISPERSION EQUATION OF PLASMA-
FILLED COAXIAL WAVEGUIDE
As the corrugated waveguide is periodical along Z-
axis, components of electromagnetic field, according to
the Flocke theorem, can be presented as a seriesof
spatial harmonics: ( ) ∑
∞
− ∞=
⊥⊥ =
n
zik
nn
nerfAzrF ||)(, , where
nkkk n 03|| += , 3k is the longitudinal wave number.
For axially-symmetric E-wave fields in the region
between internal and external cylinders are:
( )
( )
( ) .),,(
,),,(
,),,(
)(
1
)(
1
||
)(
0
||
||
||
∑∑
∑∑
∑∑
∞
− ∞=
ω−
⊥
⊥
∞
− ∞=
ϕϕ
∞
− ∞=
ω−
⊥
⊥
∞
− ∞=
∞
− ∞=
ω−
⊥
∞
− ∞=
ωε
−==
ε
−==
==
n
tzki
nn
nn
n
n
tzki
nn
n
n
n
rnr
n
tzki
nn
n
znz
n
n
n
erkFA
k
ci
HtzrH
erkFA
k
ki
EtzrE
erkFAEtzrE
(1)
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. №3.
Серия: Ядерно-физические исследования. (34), с. 61-64.
61
where ( )2
||
222
nn kck −ωε≡
⊥ , 221 ωω−=ε
p ,
epp men 22 4π=ω is the electron plasma frequency, pn
is the plasma density, e− and em are the charge and
mass of electron accordingly,
( ) ),()()()( 0000000 rkNRkJrkJRkNrkF nsnnsnn ⊥⊥⊥⊥⊥ −=
( ) ×= = ⊥⊥ )( 001 snn RkNrkF
);()()( 1001 rkNRkJrkJ nsnn ⊥⊥⊥ −× 1010 ,,, NNJJ are
cylindrical Bessel and Nejmann functions.
The boundary condition on corrugated surface of
waveguide can be written in components of electrical
field zE and rE in the following form:
( ) ( ) ( ) 0)()()( =θ⋅+ ztgzREzRE grgz , (2)
where )sin( 000 zkRkR
dz
dtg gg δ−==θ . Substituting
fields (1) into (2) after appropriate transformations we
received the following infinite system of algebraic
equations for nA :
∑
∞
− ∞=
∞<<− ∞=ω
n
mnn mkCA ,,0),( 3 (3)
[ ]
−ε
+=ω
⊥
−
−
∫ 2
0||)(
2/
2/
03
)(
1)(1),( 0
n
nzmnik
D
D
nmn k
mnkk
dzezfF
D
kC
(4)
where ( ))cos(1)( 00 zkRkzf gnn ⋅δ+≡ ⊥ . The system (3)
has the non-trivial solution, only if its determinant is
equal to zero. It defines the required dispersion
equation:
.0det =mnC (5)
2.2. THE DISPERSION EQUATION FOR
STRUCTURE WITH A BEAM
Let the axially-symmetric tubular infinitely thin
monoenergetic electronic beam propagates along the
system axis. Beam density [ ] ( )bbbb RrRevIrn −δπ= 2)( ,
where bv , I, bR are velocity of particles, current and
radius of beam, respectively. We consider fields in two
regions: I - between an internal cylinder surface and
beam, II - between beam and external corrugated
cylinder surface. The boundary conditions on beam for
fields zE and rE are:
[ ] ( )
[ ] ,0
,2
2
3
3
3
=
−ωγ
=
z
bbbe
z
r
E
vkvm
IEiekrE
(6)
where [ ] ( ) ( ) bbb RfRff γ−−+≡ ,00 is the relativistic
factor of beam. In the region 1 field can be presented in
the form (1). In the region 2 we have:
( ) ( )[ ]
( ) ( )[ ]∑
∑
∞
− ∞=
ω−
⊥⊥
⊥
∞
− ∞=
ω−
⊥⊥
+
ε
−=
+=
n
tzki
nnnn
n
n
nr
n
tzki
nnnnnz
n
n
erkNyrkJx
k
ki
AtzrE
erkNyrkJxAtzrE
.),,(
,),,(
)(
11
||2
)(
00
2
||
||
(7)
We matched fields from regions I and II on the
boundary bRr = taking into account boundary
conditions (6) and satisfying boundary conditions (2) on
corrugated waveguide surface. Similarly to the case
with no beam we obtained the dispersion equation in the
form (5) with the only difference, that [ ])(0 zfF n in (4)
should be replaced by:
( )
( )
( )
( )
( )bbn
gbn
bnbbbe
bsnn
gsn
RRkG
zRRkG
vkRvm
RRkIGek
zRRkGnkzT
,,
)(,,,,2
)(,,),,,(
1
0
2
||
3
00
0030
⊥
⊥
⊥⊥
⊥
−ωεγ
−
−≡ω
(8)
where
( ) ( ) ( ) ( ) ( )21021021,, RkNRkJRkJRkNRRkG ninninni ⊥⊥⊥⊥⊥ −≡
.
It is easy to notice, that when 0→I function
),,,( 30 nkzT ω turns into function [ ])(0 zfF n . The choice
of a thin beam allowed us to "extract" beam component
from arguments of cylindrical functions, where it would
enter in the case of REB with final thickness [4]. This
eliminates set of beam harmonics appearing in case of
beam with final thickness when bnvk ||→ω , and
simplifies the analysis of results.
3. EXCITATION EFFICIENCY
The linear stage of interaction of electronic beam
with a synchronous wave of corrugated coaxial line
continues until non-linear processes of multi-mode
interactions appear - disintegrations, modulation
instability etc. They redistribute microwave power of a
growing synchronous wave across a spectrum and that
stabilizes level of excited microwave oscillations. The
essential mechanism of instability stabilization of
microwave oscillations build-up, which is dominating,
is the beam trapping in a field of the main synchronous
wave (non-linearity of "wave - particle" type) [7].
Growth rate of this wave was determined in the linear
theory. In this case [7] the saturation takes place when
growth rate )Im(ω is comparable with frequency Ω of
trapped oscillations of beam particles in the wave field:
Ω≈ω )Im( , (9)
where ,3
3max
b
z
m
keE
γ
=Ω zmaxE is the saturation
amplitude of the longitudinal component of electrical
field intensity. From (9) it follows:
[ ]
3
32
max
)Im(
ek
mE b
z
γω
≅ . (10)
The received expressions allow to estimate efficiency of
excitation of microwave fields in the structure under
consideration. The efficiency of generation was defined
as the ratio of a microwave power flow to a flow of
electronic beam particles kinetic energy through the
waveguide cross section. To calculate the microwave
power flow the eigenwaves of coaxial corrugated
waveguide without beam were taken as:
,)()(
,)(
)(
1
11
10
00
0
010
)(
00
00
30
3
tzkizik
tzki
zzsz
ee
k
rkF
C
C
k
rkFA
c
iH
erkFAEEE
ω−−
−⊥
−⊥
−⊥
⊥
ϕ
ω−
⊥
⋅
−⋅ωε−=
⋅⋅===
(11)
i.e. it was supposed, that the basic contribution to a
longitudinal electrical field gives the harmonics with
number 0, and the power flow is determined by fields of
harmonics with numbers 0 and -1. As a result the
following expression for efficiency was obtained:
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. №3.
Серия: Ядерно-физические исследования. (34), с. 61-64.
61
[ ]
( )
,)()(
)Im(
1
0
2))cos(1(
1
11
10
00
0
01
0
2
0
42
3
4226
00
0
0∫ ∫
δ+
−
−⊥
−⊥
−⊥
⊥
⊥
−⋅×
×
ωεω
−γ
γ=η
D zkR
R
zik
bb
bg
g
s
e
k
rkF
C
C
k
rkFrdrdz
RkFcke
m
D
v
(12)
where gv is the group velocity of wave.
Results of investigations of influence of
various parameters of electron beam - SWS system on
the efficiency value are presented below. For calculation
of ω, 3k and Im(ω) the dispersion equation, taking into
account only harmonics with numbers 0 and -1, was
taken. The beam current was equal to 400 A, cmD 10= ,
cmR s 20 = , cmR g 40 = , cmRb 3= , 1,0=δ . The
frequency ω, appropriate to the wave number 3k of a
maximum of growth rate of the instability Im(ω) in case
with no beam, was substituted into expression (30). The
calculations were carried out for a case of vacuum
structure, and also for the following values of plasma
density: 1pn =1,3683·1010 cm-3, 2pn =2.2619·1010 cm-3,
3pn =2,8·1010 cm-3, 4pn =1,3·1010 cm-3.
Table
βb γb Sb
(MW)
η (%) for plasma densities:
np=0 np1 np2 np3 np4
0,6 1,25 51,2 1,08 1,17 1,23 1,27 2,84
0,7 1,4 82 1,69 1,74 1,79 1,81 2,44
0,8 1,67 136,5 2,69 2,71 2,72 2,73 2,98
0,9 2,29 265 2,55 2,62 2,60 2,63 2,75
In the Table the values of η for various densities
of plasma and for various values of cvbb =β , and the
appropriate values of beam energy flow Sb are
presented. Let us note the following: firstly, η increases
with the growth of plasma density, and with 15,1=γ b
for 311
4 103,1 −⋅= cmn p η becomes more than two and a
half times as much as the vacuum efficiency. Secondly,
with growth of bγ for vacuum case and for all plasma
cases except 4pn ,
η increases reaching the maximal
value with 67,1=γ b and then decreases with 29,2=γ b .
This work was partly supported by STCU Project
No. 365.
REFERENCES
1. Fainberg Ya.B., Gorbatenko M.F. ZhTF (in Russ.),
1959.-V.29, N 5.-P.549-562.
2. Berezin A.K., Buts V.A., Kovalchuk I.K. et al.
Preprint KhFTI 91-52 (in Russ.).
3. Fainberg Ya.B., Bezjazychnyj I.A., Berezin A.K. et
al. Plasma Physics and Controlled Nuclear Fusion
Research.-IAEA, Vienna, 1969.-V.2.-P.723-732.
4. Ostrovsky A.O., Ognivenko V.V. R. & E. (in Russ.),
1979.-V.24, N 12.-P.2470-2477.
5. Carmel Y., Minami K., Kehs R.A. et al. Phys. Rev.
Lett., 1989.-v.62.-p.2389-2392.
6. Fainberg Ya.B., Bliokh Yu.P., Kornilov Ye.A. et al.
Dokl. AN UcrSSR. Ser. À. Fiz.-mat. & techn. nauki
(in Russ.), 1990.-N11.-P.55-58.
7. Onishchenko I.N., Linetsky A.R., Matsiborko N.G. et
al. Pisma v ZhETF (in Russ.), 1970.- V.12, N 8.-P.407-
411.
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ. 1999. №3.
Серия: Ядерно-физические исследования. (34), с. 61-64.
61
1. INTRODUCTION
|
| id | nasplib_isofts_kiev_ua-123456789-81374 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T15:49:24Z |
| publishDate | 1999 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Lukin, K.A. Onishchenko, I.N. 2015-05-14T20:58:54Z 2015-05-14T20:58:54Z 1999 The relativistic E-beam interaction with plasma-filled coaxial corrugated waveguide / K.A. Lukin, I.N. Onishchenko // Вопросы атомной науки и техники. — 1999. — № 3. — С. 61-63. — Бібліогр.: 7 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/81374 This work was partly supported by STCU Project No. 365. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники The relativistic E-beam interaction with plasma-filled coaxial corrugated waveguide Взаимодействие релятивистского Е-пучка с плазмонаполненным коаксиальным гофрированным волноводом Article published earlier |
| spellingShingle | The relativistic E-beam interaction with plasma-filled coaxial corrugated waveguide Lukin, K.A. Onishchenko, I.N. |
| title | The relativistic E-beam interaction with plasma-filled coaxial corrugated waveguide |
| title_alt | Взаимодействие релятивистского Е-пучка с плазмонаполненным коаксиальным гофрированным волноводом |
| title_full | The relativistic E-beam interaction with plasma-filled coaxial corrugated waveguide |
| title_fullStr | The relativistic E-beam interaction with plasma-filled coaxial corrugated waveguide |
| title_full_unstemmed | The relativistic E-beam interaction with plasma-filled coaxial corrugated waveguide |
| title_short | The relativistic E-beam interaction with plasma-filled coaxial corrugated waveguide |
| title_sort | relativistic e-beam interaction with plasma-filled coaxial corrugated waveguide |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/81374 |
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