About generation of subharmonics and stochatic regimes arising in periodic plasma-filled waveguides excited by charged beam
Various mechanisms for generation of subharmonics in periodic plasma-filled waveguides excited by charged beam are considered. It is shown that beam particles seized by the excited electromagnetic wave can demonstrate under certain conditions the complex dynamics that results in generation of subhar...
Saved in:
| Published in: | Вопросы атомной науки и техники |
|---|---|
| Date: | 2000 |
| Main Authors: | , , , , |
| Format: | Article |
| Language: | English |
| Published: |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2000
|
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/81668 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | About generation of subharmonics and stochatic regimes arising in periodic plasma-filled waveguides excited by charged beam / А.А. Babaskin, А.V. Kirichok, V.М. Kuklin, S.Т. Mouharov, G.I. Zaginajlov // Вопросы атомной науки и техники. — 2000. — № 1. — С. 197-199. — Бібліогр.: 11 назв. — англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860022046249975808 |
|---|---|
| author | Babaskin, A.A. Kirichok, A.V. Kuklin, V.М. Mouharov, S.Т. Zaginajlov, G.I. |
| author_facet | Babaskin, A.A. Kirichok, A.V. Kuklin, V.М. Mouharov, S.Т. Zaginajlov, G.I. |
| citation_txt | About generation of subharmonics and stochatic regimes arising in periodic plasma-filled waveguides excited by charged beam / А.А. Babaskin, А.V. Kirichok, V.М. Kuklin, S.Т. Mouharov, G.I. Zaginajlov // Вопросы атомной науки и техники. — 2000. — № 1. — С. 197-199. — Бібліогр.: 11 назв. — англ. |
| collection | DSpace DC |
| container_title | Вопросы атомной науки и техники |
| description | Various mechanisms for generation of subharmonics in periodic plasma-filled waveguides excited by charged beam are considered. It is shown that beam particles seized by the excited electromagnetic wave can demonstrate under certain conditions the complex dynamics that results in generation of subharmonics and cause the transition to the stochastic regime of the instability.
|
| first_indexed | 2025-12-07T16:47:32Z |
| format | Article |
| fulltext |
ВОПРОСЫ АТОМНОЙ НАУКИ И ТЕХНИКИ 2000. №1.
Серия: Плазменная электроника и новые методы ускорения (2), с. 197-199.
197
UDK 533.9
ABOUT GENERATION OF SUBHARMONICS AND STOCHATIC
REGIMES ARISING IN PERIODIC PLASMA-FILLED WAVEGUIDES
EXCITED BY CHARGED BEAM
А.А. Babaskin, А.V. Kirichok, V.М. Kuklin, S.Т. Mouharov, G.I. Zaginajlov
Kharkov National University, Department of Physics and Technology, Kharkov, Ukraine
Various mechanisms for generation of subharmonics in periodic plasma-filled waveguides excited by
charged beam are considered. It is shown that beam particles seized by the excited electromagnetic wave can
demonstrate under certain conditions the complex dynamics that results in generation of subharmonics and cause
the transition to the stochastic regime of the instability.
1. Introduction
A variety of nonlinear phenomena in slowing electro-
dynamical plasma-filled structures and waveguides, ex-
citing by charged beams, was discussed in many papers
(see e.g. [1–3]).
However, a diversity of realizations and conditions for
exciting of oscillations in such systems can be explained
by a variety of physical mechanisms leading to complica-
tion of particle dynamics in a rather wide band of excited
oscillations. Below we shall discuss some aspects of this
problem and concentrate our attention on particular fea-
tures of nonlinear mechanisms of excitation of periodic
waveguides by charged beams. The interest to these sys-
tems is conditioned by great perspectives in their applica-
tion [4–6].
Below we shall consider the high-quality beams (in
terms of [1]), which are characterized by low velocity
spread.
2. Consequences of satellite instability
It is known that the slowly varying (in comparison
with the frequency of oscillations) factor in front of the
field amplitude can result in a well-known phenomenon
of period doubling and further complication of phase dy-
namics of particles, seized by the potential well. Really,
the particles, seized by the field of the wave, move along
the finite trajectories in a phase plane (e.g. in the space
''coordinate - velocity'' in the frame of references, where
the beam is in rest) with a mean trembling frequency trΩ .
Due to development of the satellite instability (see e.g.
[2]) the field amplitude oscillates in the beam rest refer-
ence frame with a frequency µΩ and can be represented
as a sum of three terms:
0
0 0 0
(1 cos )sin
1 1sin( ) sin( ).
2 2
A A t kz
A A kz t A kz t
µ
µ µ
= + µ Ω =
= + µ − Ω + µ + Ω
(1)
Following to [7], note that in addition to the main
band of separatrix cells with seized particles two extra
identical bands appear in this case, sliding relative to each
other (and to the central band). They are more narrow
than the central band and one of them is shifted upwards
and the second – downwards along the velocity axis rela-
tive to the central band on the value kµΩ . The particles
of the beam fall into potential wells of central, upper, or
lower bands depending on their velocities.
In a vicinity of the separatrices the so-called stochastic
layer appears. The behavior of a large part of particles in
this layer significantly depends on their initial state and,
generally speaking, can be considered as random from a
physical point of view. Just in this sense it is necessary to
understand the term ''weak chaos'' [7]. If the separatrix
bands are rather far apart in a phase space, they do not
exchange the particles. [7]. If the stochastic layers of dif-
ferent separatrix bands are superimposed, a probability of
particle exchange between them becomes finite and
groups of seized particles appears, which period of revo-
lution on the complicated trajectories can be doubled. The
doubling is promoted by a synchronizing role of relative
motion of the separatrix bands. The question is how large
a part of these particles and what a degree of their influ-
ence on the growth of field subharmonics. With further
converging of separatrix layers (that takes place in regime
of saturation of above discussed instabilities, when
trµΩ Ω! ) the motion of a noticeable part of particles
becomes complicated that leads to generation of next
subharmonics, and then to further chaotization of dy-
namics.
The authors of many works [8, 10, 11] noticed that
with increase of interaction length or beam duration a
more and more complicated dynamics was observed for
transition from the stationary regime to a regime with
gradual generation of subharmonics with lower and lower
frequencies (usually under the scenario of period dou-
bling), and, at last, to a regime with pronounced stochas-
tic origin. Really, the beam particles, seized to the poten-
tial well of oscillations, drifting with the beam, are lo-
cated in the interaction region during a finite time. There-
fore, the greater the interaction region, the greater time for
complex quasiperiodic motion in pulled together and
sliding relative to each other potential wells, which have
appeared because of modulation of oscillation amplitude.
Preliminary modulation of the beam (see e.g. [8]) not
only accelerates the instability, but also determines a re-
gime of instability development, having an influence on
the frequency and wave number shifts (velocity of enve-
lope drift). It is important that the beam modulation can
have an influence on the frequency of field modulation in
198
the beam rest reference frame and thereby on the bifurca-
tion thresholds and conditions of transition to the sto-
chastic regime [9].
However the regular mechanism of subharmonics
generation is also possible. The analysis of dynamics of
test particles in the field of spatial modes under consid-
eration has shown that when potential wells moves rela-
tively to each other with the velocity /tr kΩ , there is al-
ways exist a “resonance” beam particle located in a vicin-
ity of separatrix of the main mode. It moves with a spatial
period, which is two times greater than the wavelength of
the main mode and this motion remains stable during a
long time. This can result in excitation of subharmonics,
which frequency is a half of the main mode frequency.
The subharmonic field of finite amplitude can capture the
particles located in a phase vicinity of the “resonance”
particle. Thus the phase volume of beam particles inter-
acting with the subharmonic is finite.
3. About influence of beam spatial charge on
generation of subharmonics and transition of
instability to the stochastic regime
The inclusion of beam spatial charge leads to modifi-
cation of the expression for increment of the beam-plasma
instability. It is known that for low-density beams the
instability increment is proportional to 1/ 3
bn and for dense
beams the increment is proportional to 1/ 2
bn (in a case of
non-uniform beams the effective, e.g. mean density bn
should be assumed). With increase of beam density the
process of excitation of induced oscillations in the
waveguide becomes similar to the non-resonance insta-
bility. As this takes place, the motion of seized particles
in the potential well separates on two contradirectional
flows due to small phase shift of the excited wave during
the instability development (in the case of “pure” non-
resonance instability the phase shift is insignificantly
small and may be neglected). Then, two bunches of parti-
cles (two quasi-particles) are formed revolving in oppo-
site directions in the phase plane. Generally speaking,
their volume and speed of rotation (frequency of oscilla-
tion in the potential well or trembling frequency trΩ )
may take different values. Recall that in the case of low-
density beams there is only single quasi-particle, which
motion goes on at the initial moment of the instability in
the slowing phase of oscillations. The increase in beam
density leads to appearance of the second quasi-particle at
the initial moment of the instability, which motion goes
on in the accelerating phase of oscillations.
The presence of several bunches of seized particles
(quasi-particles) in the potential well of the main wave,
which frequencies in the reference frame, moving with a
phase velocity of the wave, are different, can result in the
case of multiple frequencies in generation of subharmon-
ics. This happens due to synchronism between spatial
period of oscillations of “resonance” beam particles
seized by the fields and subharmonic wavelengths. Note
that the oscillations with frequencies less than the fre-
quencies of proper waves, excited resonantly by the beam
in the waveguide system, are also unstable. Because of
this, the oscillations of seized particles result in appear-
ance of perturbations with frequencies trω± Ω and devel-
opment of low-frequency spectrum band of non-
resonance oscillations [2]. In more general case of ali-
quant frequencies trΩ , corresponding to different quasi-
particles, the spectra of non-resonance oscillations may
become practically continuous. Then, the particle motion
becomes very intricate and can’t be distinguished in prac-
tice from random.
4. About spatial spectrum of oscillations excited
in periodic waveguide systems
Consider some particular features of nonlinear re-
gimes of beam instabilities in periodic waveguide sys-
tems. The rigorous treatment of wave propagation in peri-
odic waveguides allows the conclusion to be made that
the number of independent oscillations with different fre-
quencies ω in the waveguide corresponds to the number
of full modulation periods of such waveguide parameters
as shape, size, etc. (needless to say that it concerns only
those oscillations, which are affected by this modulation).
However the oscillations having the same frequency pos-
sess due to a periodicity an infinite number of spatial
modes, which longitudinal wave-numbers are equal to
0k nk± , where k is the longitudinal wave-number of the
mode having a largest amplitude at this frequency,
0 2k L= π , L is the spatial period of waveguide pa-
rameter modulation, n N∈ . With growth of n the oscil-
lation amplitude decreases as nα , where α is the pa-
rameter proportional to the modulation depth (usually,
1α " ). As a rule, the beam can be synchronized only
with satellites 0k k+ or 02k k+ since the phase velocity
kω of proper modes (corresponding to proper modes of
the analogous smooth waveguide) often exceed the light
velocity in vacuum. The resonance interaction with the
satellite leads to the growth of as its amplitude as ampli-
tudes of coupled modes, including the main mode, which
contains the main power of the electromagnetic field.
However, the series expansion of the field in spatial
modes by using the small parameter α seems to be incor-
rect owing to the poor convergence of the series. In this
case other techniques for analysis of oscillations in peri-
odic waveguides should be used. In our opinion, the most
perspective method for this purpose was developed in
Kharkov National University (Ukraine).
5. About a possibility to increase the effectiveness
of generation in periodic waveguides
In the regime of instability saturation the beam parti-
cles seized by the field of the satellite ( 0k k+ ) are located
in the phase space “coordinate-velocity” in the velocity
range from 0( )tr k k−Ω + to 0( )tr k kΩ + (in the refer-
ence frame where the beam is in rest). It is obvious that
beside the main band of separatrix cells with particles
seized by the satellite field there is a lot of analogous
bands of different widths sliding relatively to each other
199
and displaced upward and downward with respect to the
band of the satellite ( 0k k+ ) along the velocity axis. We
are interesting in the band of the main mode (which am-
plitude exceeds the amplitude of the satellite by the factor
1 α ) located in the velocity range from
0 0( )trvk k k k− Ω α + to 0 0( )trvk k k k+ Ω α + . If these
bands will verge towards each other with increase in the
field amplitude at the frequency ω then the drifting parti-
cles and a part of particles seized by the satellite can oc-
cur in the field of the main mode. Under some conditions,
beside of the dynamics complication for this group of
particles and generation of subharmonics due to appear-
ance of closed periodic trajectories (see above) the regime
of so-called supercritical excitation of the main mode can
be realized. The growth of the main mode will result in
further overlapping of the separatrix bands and reseizing
of most of the beam particles by the main mode that will
provoke its further growth conditioned by this mecha-
nism. Obviously, that this mechanism manifests itself in
the case of periodic waveguide systems with low modu-
lation depth and excited by non-relativistic charged
beams.
6. About influence of non-sinusoidal parameter
modulation in periodic waveguides
Other mechanism for complication of dynamics of
seized particles and transition to stochastic regimes of the
instability is the resonance excitation of oscillations in a
vicinity of upper harmonics of the main mode [9]. Really,
the periodic system (a corrugated waveguide) contains
only one spatial harmonic as was assumed above. When
the depth of the corrugated waveguide is increased on
retention of a period it is necessary to take into account
the upper spatial harmonics. Here we restrict our consid-
eration by a role of the second spatial harmonic
2 0cos(2 )A k z , where 2A is its amplitude.
Obviously, the charged beam will interact effectively
with the first side mode, which longitudinal wave number
is shifted on 02k with respect to the longitudinal wave
number of the main mode with higher frequency. The
frequency of the latter almost twice exceeds the frequency
of oscillations considered above. Foregoing analysis, the
expressions obtained for instability increments, as well as
equations for the field turn out to be valid for oscillations
with higher frequency, which effective (even resonance)
excitation becomes possible due to presence of the second
spatial harmonic in the spectrum of the periodic structure.
To make sure in this, it is sufficiently to substitute α and
0k for 2α and 02k accordingly. Equation of motion for
beam particles in the case of resonance instability in this
case takes the different form (details see in [9]).
Now particles of the beam occur affected by the field
of two modes, synchronized with the beam and which
potential wells are differ on depth and longitudinal size.
The ratio of frequencies and longitudinal wave-numbers
are differ from 2 and some values are listed in the table:
d 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
1β 1.24 1.25 1.26 1.27 1.28 1.29 1.31 1.32 1.34 1.36 1.37
2β 1.05 1.05 1.06 1.06 1.06 1.06 1.06 1.06 1.06 1.06 1.06
2 1ω ω 1.86 1.85 1.85 1.84 1.83 1.83 1.82 1.81 1.80 1.79 1.79
1 2z zk k 2.18 2.19 2.20 2.21 2.22 2.24 2.25 2.26 2.28 2.29 2.31
where ph, ci ivβ = , ph, iv are the phase velocities of modes
under consideration.
The linear rates of the instability for these resonance
oscillations and also their amplitudes can be also esti-
mated [9].
The presence of two synchronized with the beam peri-
odic potential wells of different amplitudes and having,
generally speaking, aliquant spatial periods can be a rea-
son (together with the above-described mechanism) of
noticeable complication of dynamics of seized particles
(see, e.g. [7]).
References
1. Я.Б. Файнберг Некоторые вопросы плазменной
электроники //Физика плазмы, 1985, т.11, вып. 11,
с.1398-1410.
2. В.Д. Шапиро, В.И. Шевченко //Изв. Вузов, Ра-
диофизика, 1976, т.19, № 5-6, с. 787-791.
3. М.В. Кузелев, А.А. Рухадзе. Электродинамика
плотных электронных пучков в плазме.// М.:
"Наука", 1990.
4. J.A. Swegle et al., “Backward wave oscillators with
rippled wall resonators: Analytic theory and numeri-
cal simulation” //Phys. Fluids, 1985, vol. 28, p. 2882-
2894.
5. S.P. Bugaev et al.,” Relativistic multiwave Cheren-
kov generators” //IEEE Trans. Plasma Sci., 1990, vol.
18, p. 525-536.
6. W.Main et al., “Electromagnetic properties of open
and closed overmoded slow-wave resonators for in-
teraction with relativistic electron beams”// IEEE
Tran. Plasma Sci., 1994, vol. 22, № 5, p. 566-577.
7. G.M Zaslavsky, R.Z. Sagdeev, D.A. Usikov, A.A.
Chernikov, Weak Chaos and Quasi-Regular Patterns
M.: "Nauka", 1991.
8. V.A. Balakirev, N.I.Karbushev, A.O.Ostrovsky and
Yu.V. Tkach, Theory of Cherenkov's Amplifiers and
Generators on Relativistic Beams // Kiev: "Naukova
Dumka", 1993,
9. A.V. Kirichok, V.M. Kuklin, S.T. Mouharov, Physi-
cal mechanisms leading to appearance of subhar-
monics and upper harmonics in a periodic waveguide
system//Вестник Харьковского университета,
сер.физ., 1999, № 463, вып.5(9), с.39-46.
10. Ya. Bogomolov, V.L. Bratman, N.S. Ginzburg, M.I.
Petelin and A.D. Yanovsky // Opt. Commun., 1981,
vol.369, p.209.
11. S.J. Hahn, J.K. Lee // Phys. Lett. A, 1993, vol.176,
p.339-343.
UDK 533.9
ABOUT GENERATION OF SUBHARMONICS AND STOCHATIC
REGIMES ARISING IN PERIODIC PLASMA-FILLED WAVEGUIDES
EXCITED BY CHARGED BEAM
1. Introduction
2. Consequences of satellite instability
3. About influence of beam spatial charge on generation of subharmonics and transition of instability to the stochastic regime
4. About spatial spectrum of oscillations excited in periodic waveguide systems
5. About a possibility to increase the effectiveness of generation in periodic waveguides
6. About influence of non-sinusoidal parameter modulation in periodic waveguides
References
|
| id | nasplib_isofts_kiev_ua-123456789-81668 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1562-6016 |
| language | English |
| last_indexed | 2025-12-07T16:47:32Z |
| publishDate | 2000 |
| publisher | Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| record_format | dspace |
| spelling | Babaskin, A.A. Kirichok, A.V. Kuklin, V.М. Mouharov, S.Т. Zaginajlov, G.I. 2015-05-19T08:36:43Z 2015-05-19T08:36:43Z 2000 About generation of subharmonics and stochatic regimes arising in periodic plasma-filled waveguides excited by charged beam / А.А. Babaskin, А.V. Kirichok, V.М. Kuklin, S.Т. Mouharov, G.I. Zaginajlov // Вопросы атомной науки и техники. — 2000. — № 1. — С. 197-199. — Бібліогр.: 11 назв. — англ. 1562-6016 https://nasplib.isofts.kiev.ua/handle/123456789/81668 533.9 Various mechanisms for generation of subharmonics in periodic plasma-filled waveguides excited by charged beam are considered. It is shown that beam particles seized by the excited electromagnetic wave can demonstrate under certain conditions the complex dynamics that results in generation of subharmonics and cause the transition to the stochastic regime of the instability. en Національний науковий центр «Харківський фізико-технічний інститут» НАН України Вопросы атомной науки и техники Нерелятивистская плазменная элeктрoника About generation of subharmonics and stochatic regimes arising in periodic plasma-filled waveguides excited by charged beam Article published earlier |
| spellingShingle | About generation of subharmonics and stochatic regimes arising in periodic plasma-filled waveguides excited by charged beam Babaskin, A.A. Kirichok, A.V. Kuklin, V.М. Mouharov, S.Т. Zaginajlov, G.I. Нерелятивистская плазменная элeктрoника |
| title | About generation of subharmonics and stochatic regimes arising in periodic plasma-filled waveguides excited by charged beam |
| title_full | About generation of subharmonics and stochatic regimes arising in periodic plasma-filled waveguides excited by charged beam |
| title_fullStr | About generation of subharmonics and stochatic regimes arising in periodic plasma-filled waveguides excited by charged beam |
| title_full_unstemmed | About generation of subharmonics and stochatic regimes arising in periodic plasma-filled waveguides excited by charged beam |
| title_short | About generation of subharmonics and stochatic regimes arising in periodic plasma-filled waveguides excited by charged beam |
| title_sort | about generation of subharmonics and stochatic regimes arising in periodic plasma-filled waveguides excited by charged beam |
| topic | Нерелятивистская плазменная элeктрoника |
| topic_facet | Нерелятивистская плазменная элeктрoника |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/81668 |
| work_keys_str_mv | AT babaskinaa aboutgenerationofsubharmonicsandstochaticregimesarisinginperiodicplasmafilledwaveguidesexcitedbychargedbeam AT kirichokav aboutgenerationofsubharmonicsandstochaticregimesarisinginperiodicplasmafilledwaveguidesexcitedbychargedbeam AT kuklinvm aboutgenerationofsubharmonicsandstochaticregimesarisinginperiodicplasmafilledwaveguidesexcitedbychargedbeam AT mouharovst aboutgenerationofsubharmonicsandstochaticregimesarisinginperiodicplasmafilledwaveguidesexcitedbychargedbeam AT zaginajlovgi aboutgenerationofsubharmonicsandstochaticregimesarisinginperiodicplasmafilledwaveguidesexcitedbychargedbeam |