Nonlinear transition radiation of the modulated electron beam from the sharp vacuum-plasma border
The transition radiation of the radially unbounded modulated electron beam on the sharp vacuum-isotropic plasma boundary is studied up to quadratic nonlinearity approximation. Solution for the second harmonic radioemission and stationary forced fields is considered in details.
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| Дата: | 2003 |
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Національний науковий центр «Харківський фізико-технічний інститут» НАН України
2003
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| Назва видання: | Вопросы атомной науки и техники |
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| Цитувати: | Nonlinear transition radiation of the modulated electron beam from the sharp vacuum-plasma border / I.O. Anisimov, O.I. Kelnyk, V.K. Tyazhemov // Вопросы атомной науки и техники. — 2003. — № 4. — С. 88-90. — Бібліогр.: 4 назв. — англ. |
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irk-123456789-1109872017-01-08T03:03:41Z Nonlinear transition radiation of the modulated electron beam from the sharp vacuum-plasma border Anisimov, I.O. Kelnyk, O.I. Tyazhemov, V.K. Нелинейные процессы The transition radiation of the radially unbounded modulated electron beam on the sharp vacuum-isotropic plasma boundary is studied up to quadratic nonlinearity approximation. Solution for the second harmonic radioemission and stationary forced fields is considered in details. 2003 Article Nonlinear transition radiation of the modulated electron beam from the sharp vacuum-plasma border / I.O. Anisimov, O.I. Kelnyk, V.K. Tyazhemov // Вопросы атомной науки и техники. — 2003. — № 4. — С. 88-90. — Бібліогр.: 4 назв. — англ. 1562-6016 PACS: 52.35.Mw http://dspace.nbuv.gov.ua/handle/123456789/110987 en Вопросы атомной науки и техники Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
| topic |
Нелинейные процессы Нелинейные процессы |
| spellingShingle |
Нелинейные процессы Нелинейные процессы Anisimov, I.O. Kelnyk, O.I. Tyazhemov, V.K. Nonlinear transition radiation of the modulated electron beam from the sharp vacuum-plasma border Вопросы атомной науки и техники |
| description |
The transition radiation of the radially unbounded modulated electron beam on the sharp vacuum-isotropic plasma boundary is studied up to quadratic nonlinearity approximation. Solution for the second harmonic radioemission and stationary forced fields is considered in details. |
| format |
Article |
| author |
Anisimov, I.O. Kelnyk, O.I. Tyazhemov, V.K. |
| author_facet |
Anisimov, I.O. Kelnyk, O.I. Tyazhemov, V.K. |
| author_sort |
Anisimov, I.O. |
| title |
Nonlinear transition radiation of the modulated electron beam from the sharp vacuum-plasma border |
| title_short |
Nonlinear transition radiation of the modulated electron beam from the sharp vacuum-plasma border |
| title_full |
Nonlinear transition radiation of the modulated electron beam from the sharp vacuum-plasma border |
| title_fullStr |
Nonlinear transition radiation of the modulated electron beam from the sharp vacuum-plasma border |
| title_full_unstemmed |
Nonlinear transition radiation of the modulated electron beam from the sharp vacuum-plasma border |
| title_sort |
nonlinear transition radiation of the modulated electron beam from the sharp vacuum-plasma border |
| publisher |
Національний науковий центр «Харківський фізико-технічний інститут» НАН України |
| publishDate |
2003 |
| topic_facet |
Нелинейные процессы |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/110987 |
| citation_txt |
Nonlinear transition radiation of the modulated electron beam from the sharp vacuum-plasma border / I.O. Anisimov, O.I. Kelnyk, V.K. Tyazhemov // Вопросы атомной науки и техники. — 2003. — № 4. — С. 88-90. — Бібліогр.: 4 назв. — англ. |
| series |
Вопросы атомной науки и техники |
| work_keys_str_mv |
AT anisimovio nonlineartransitionradiationofthemodulatedelectronbeamfromthesharpvacuumplasmaborder AT kelnykoi nonlineartransitionradiationofthemodulatedelectronbeamfromthesharpvacuumplasmaborder AT tyazhemovvk nonlineartransitionradiationofthemodulatedelectronbeamfromthesharpvacuumplasmaborder |
| first_indexed |
2025-07-08T01:28:25Z |
| last_indexed |
2025-07-08T01:28:25Z |
| _version_ |
1837040251203223552 |
| fulltext |
NONLINEAR TRANSITION RADIATION OF THE MODULATED ELEC-
TRON BEAM FROM THE SHARP VACUUM-PLASMA BORDER
I.O.Anisimov, O.I.Kelnyk, V.K.Tyazhemov
Taras Shevchenko National University of Kyiv, Radio Physics Faculty, Kyiv, Ukraine, ioa@u-
niv.kiev.ua
The transition radiation of the radially unbounded modulated electron beam on the sharp vacuum-isotropic plas-
ma boundary is studied up to quadratic nonlinearity approximation. Solution for the second harmonic radioemission
and stationary forced fields is considered in details.
PACS: 52.35.Mw
1. INTRODUCTION
It has already known that the effectiveness of the
modulated electron beam transition radiation increases
with the beam current [1]. But large currents result to
appearance of the nonlinear effects. Their threshold
might be extremely low in plasmas. The “slow” nonlin-
earity was studied in [2-3]. It takes place in the weakly
inhomogeneous plasma due to the electrons’ pressing-
out from the local plasma resonance region by the high-
frequency electric field of the modulated electron beam.
In this article the “fast” nonlinear effects are studied for
the simplest model including modulated electron beam
of the infinite radius passing through the sharp vacuum-
isotropic plasma border.
2. MODEL DESCRIPTION, INITIAL
EQUATIONS AND METHOD OF SOLUTION
The charge compensated and obliquely modulated
electron beam of the infinite radius produces the current
wave
)cos(exp rtejj zm
χω −= , { }||;;0 χχχ ⊥=
,
0
|| v
ωχ = ,
where v0 is the beam velocity. This beam normally pass-
es through the plain border between vacuum and cold
homogeneous isotropic plasma.
Approximation of the given beam current is used.
Plasma perturbations caused by the beam are described
by the Maxwell’s equations and the motion equation for
the plasma electrons:
−−=
∂
∂−
∂
∂=∇+
∂
∂
∂
∂−−=
),(4)1(
;)(
;144
0
2
2
2exp
nne
t
A
c
div
t
A
mc
evv
t
v
t
A
c
vn
c
ej
c
Arotrot
π
ππ
(1)
where vector-potential is used to describe the electro-
magnetic field (calibration condition is taken in the form
ϕ=0). Non-linearity is caused by the quadratic terms in
the equations of motion and continuity for the plasma
electrons.
Magnitude of the beam current is considered to be a
small parameter. Than one can find the vector-potential
in plasma as a sum of series on the beam current magni-
tude:
...21 ++= AAA
(2)
(here index indicates the order of smallness).
The boundary conditions for finding out the transi-
tion radiation magnitude should be written taking into
account the surface charges on the vacuum-plasma
bound.
3. OBTAINING OF THE RECURRENT
EQUATIONS FOR ELECTROMAGNETIC
FIELD
By substituting (2) into (1) and picking out the terms
of the same order of smallness one can obtain the set of
recurrent equations. The first order perturbations are de-
scribed by the set:
=
∂
∂++
=
∂
∂=
.414
;
;
4
1
exp2
1
2
212
0
2
1
1
1
11
j
ct
A
c
A
mc
neArotrot
mc
Aev
Adiv
tec
n
ππ
π
(3)
The next order perturbations are described by the
equations:
∇+−=
=
∂
∂
++
∇−=
∂
∂=
∑ ∫∑
∑ ∫
−
=
−
−
=
−
−
=
−
1
1
0
1
1
2
2
2
2
1
1
)(44
1
)(
4
1
i
k
kik
i
k
kik
i
ipi
i
k
kik
i
i
ii
vvdt
c
envn
c
e
t
A
c
AArotrot
vvdt
mc
Aev
Adiv
tec
n
ππ
κ
π
(4)
Here the first equation includes a beam current in
the right side. Others have in the right sides the nonlin-
ear terms caused by the magnitudes of the lower orders
of smallness.
To obtain the transition radiation it is necessary to
take into account both the existing solutions of the equa-
tions (3)-(4) (that correspond to the eigen field of the
modulated electron beam) and the solutions of the ap-
propriate homogeneous equation (that describe the
waves moving from the border into plasma). In vacuum
the inhomogeneous wave equation remains linear.
Boundary conditions at the vacuum-plasma border
can be presented in the form:
H1=H2; Eτ1=Eτ2; En1-En2=4πσ,
where σ is the surface charge density.
4. THE FIRST APPROXIMATION SOLU-
TION
Magnitudes of the transition radiation into plasma
and vacuum calculated in the first approximation after
the beam current magnitude are in good agreement with
the well-known results [4]:
,
]))[()()((
])()[(4
]))[()()((
])())([(4
||
22
0||||
2
0
22
0
22
0
22
||
2
0
222
0||||||
2
||
22
0||||
2
0
22
0
22
0
22
||
2
0
22
0
2
||||||
2
vpvppp
ppvpm
v
vpvppp
pvppm
p
kkkkkkkkkkc
kkkkkkjA
kkkkkkkkkkc
kkkkkjA
−+−−−+
−++−−=
−+−−−+
−+−+−=
⊥
⊥
χχ
χχχπ χ
χχ
χχχκπ χ
(5)
(5)
where k0=ω/c, kp=ωp/c=(e/c)(4πn0/m)1/2; п0 – average
concentration of plasma electrons, χ2=χ//
2+χ⊥
2,
k//p=(k0
2-kp
2-χ⊥
2)1/2 and k//v=(k0
2-χ⊥
2)1/2 – longitudinal
wave numbers for the radiation into plasma and vacuum
correspondingly.
5. THE SECOND APPROXIMATION SOLU-
TION
Let us find out transition radioemission in the sec-
ond approximation. Using the relation
t
A
c
E
∂
∂−=
1 ,
the third equation of the set (4) can be transformed into:
.)(4)(41
112
0
1122
2
2
2
2
22 vv
c
envn
tc
eEk
t
E
c
Erotrot p
∇−
∂
∂=+
∂
∂+ ππ
(6)
There is a quadratic non-linearity in the right side of
equation (6) that results to appearance of the second
harmonic of the modulation frequency and the station-
ary field.
Components with the temporal dependence exp(i2ω
t) correspond to the second harmonic of the eigen field
of the beam and the sum mixed harmonic. Electromag-
netic radiation of the frequency 2ω is emitted both into
plasma and into vacuum.
The quadratic non-linearity produces also the spa-
tially inhomogeneous stationary field corresponding to
the difference mixed wave. It results to the appearance
of the constant component of the surface charge density.
Due to the different time dependence electromagnet-
ic fields of the second and zero harmonics can be con-
sidered separately.
6. WAVES AT THE SECOND HARMONIC
According to the boundary conditions one can write
down the set of equations to obtain the magnitude of the
second harmonic transition radiation into plasma and
vacuum in the second approximation after the beam cur-
rent magnitude:
−
−++
+−+
−
−
−+−+
+
×
×
−
−
−
−=
⊥
⊥
⊥
||
2
0
2
||||
2
||
||
2
0
22
||
22
0
2
0
0
2
0
222
0
22
2
0
2
0
2
0
24
||
22
'
'
||0
22
0'
||
0
)3)(2(
])()(2[
))(44(
)2(2
)(
4
4
4
χχχχ
χχχ
χχ
χ
χπ
kkk
kkkkkA
kkkk
kk
kkkmc
ejE
kk
kkE
k
k
pp
ppp
pp
p
p
m
p
p
p
v
v
−++
+−
−
⊥ ||
2
0
2
||||
2
||
2
||
2
0
22
0
2
||
0
)3)(2(
)]2)(([
χχχχ
χ
kkk
kkkkkA
pp
pppp
; (7)
(7)
−
−
++−
+−×
×
−++
−
−−+−−+
−++×
×
−
−=
⊥
⊥
⊥
2
0
2
||||
2
||||
2
0
2
||
2
0
2
0
2
||||
2
||
0
2
0
22
0
222
0
22
0
22
2
0
222
0
2
||
2
0
24
||
22
''
4
))()(2(22
)3)(2(
)4)()()(44(
)854)(2(
)(
8
kk
kkkkkk
kkk
A
kkkkkkkk
kkkk
kkmc
ejEE
p
ppp
p
pp
p
pppp
pp
p
m
pv
χχχ
χχχ
χχ
χχχ
χπ
−−+
−+−−
−
−
−+−−
⊥⊥
⊥
)4)((
)44()(2
)4(
)44)((
2
0
22
0
22
2
0
222
||||
2
||
2
0
2
2
0
2222
0
2
kkkk
kkkk
kk
kkkkk
pp
ppp
p
ppp
χ
χχχ
χ
χ
(8)
where
222
0
'
|| 4/ ⊥−−= χpp kkk ; mpp jcAA π4/0 = .
Solution of the set (7)-(8) is too bulky so it is not pre-
sented here.
Dependencies of the forced waves’ magnitudes ex-
cited in plasma due to the non-linearity on frequency
and transversal wave number are shown on Fig. 1.
Maximums at ω=ωp and ω=ωp/2 correspond to the
plasma resonance at the first and second harmonics of
the modulation frequency (the last maximum exists only
in plasma). Another maximums correspond to the van-
ishing of the denominators
( ) vpvp kkkkk //
2
////
2
04 −+′ and ( ) vpvp kkkkk //
2
////
2
0 −+
in the expression for the transition radiation magnitudes.
a
b
Fig. 1. Magnitude of the forced waves in plasma at
frequency 2ω (a – second harmonic of the beam
field; b – sum mixed harmonic of beam field and
radiation at frequency ω) in dependence on fre-
quency and transversal wave number
The dependencies of transition radiation magnitudes
in plasma and vacuum on frequency and transversal
wave number are shown on Fig.2.
a
b
Fig. 2. Magnitude of the z-component of transition
radiation into plasma (a) and into vacuum (b) at fre-
quency 2ω dependent on the modulation frequency
and transversal wave number
From comparison of fig.2 and fig.3 one can see that
maximums 1-3 are caused by the resonant growth of the
stimulated field of the beam. Maximum 4 corresponds
to the resonant excitation of some quasi-eigen mode
(see, e.g., [1]) of the vacuum-plasma boundary.
7. STATIONARY FIELD GENERATION
It was already noticed that the spatially inhomoge-
neous stationary field corresponding to the difference
mixed wave appears in the system due to the quadratic
non-linearity. The magnitude of this field can be pre-
sented in the form
)()(
)))(2((2
2
0
22
||
2
0
23
2
0
22
||||||
2
||
kkkkkmc
kkkAejE
ppp
pppm
z
−+−
−+−+= ⊥⊥
χ
χχχχχπ
Dependence of this magnitude upon the modulation
frequency and transversal wave number is presented on
fig. 3. Maximums of fig.3 are similar to the ones on
fig. 2.
Fig. 3. Magnitude of the difference mixed harmonic
(stationary in time field)
8. CONCLUSION
Nonlinear transition radiation from the obliquely
modulated electron beam of the infinite radius is calcu-
lated in the second approximation after the current wave
magnitude. It results to the appearance of the second
harmonic and stationary field of the difference wave
number (between the current wave and the electromag-
netic wave).
The next approximations result to the appearance the
corresponding harmonics in the radiation spectrum and
some modification of the main harmonic magnitude.
REFERENCES
1. V.A.Balakirev, G.L.Sidelnikov. Transition
radiation of the modulated electron beams in the
inhomogeneous plasma. Kharkiv: Preprint KIPT.
1994. (In Russian).
2. 1. I.O.Anisimov, O.A.Borisov, O.I.Kelnyk,
S.V.Soroka. Deformation of the plasma
concentration profile due to the modulated electron
beam // Probl. of Atomic Sci. and Technology.
Series "Plasma Physics". 2002, №5(8), p. 107-109.
3. I.O.Anisimov, S.M.Levitsky. Radioemission of the
modulated electron stream in the plasma with
nonlinear concentration profile // Ukr. Fiz. Zhurn.
1989, v.34, № 9, p. 1336. (In Russian).
4. V.L.Ginzburg, V.N.Tsytovich. Transition radiation
and transition scattering. Moscow: "Nauka", 1984.
(In Russian).
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