Download NONLINEAR INTERACTION OF WAVES IN PLASMA

NONLINEAR INTERACTION OF WAVES IN PLASMA-BEAM
SUPERHETERODYNE FEL OF THE DOPPLERTRON TYPE
WITH HELICAL ELECTRON BEAM
V.V. Kulish1, A.V. Lysenko2, G.A. Oleksiienko2
1
National Aviation University, Kiev, Ukraine
E-mail: [email protected];
2
Sumy State University, Sumy, Ukraine
E-mail: [email protected]
We have constructed a cubic-nonlinear theory of a plasma-beam superheterodyne free electron laser (SFEL) of
the dopplertron type with a helical electron beam. The saturation levels and mechanisms have been found for the
four operating modes. We have shown that the capture of beam electrons by plasma waves causes the signal saturation. We have found that among all operating modes the highest saturation level corresponds to the interaction mode
that uses an extraordinary wave as the signal. We have demonstrated the possibility of making of powerful coherent
electromagnetic radiation sources in the millimeter wavelength range on the basis of such SFEL.
PACS: 41.60.Cr; 52.35.Mw
INTRODUCTION
The devices that are capable to generate and amplify
coherent electromagnetic radiation in the millimeter
wavelength range constantly attract the attention of researchers. [1 - 4]. Plasma-beam SFELs belong to such
devices. Superheterodyne free-electron lasers stand out
among the other types of FELs due to the high amplification properties [1, 5 - 17]. An additional mechanism
of electromagnetic signal amplification is a cause of
these SFEL properties. The plasma-beam instability is
used as the additional amplification mechanism in the
plasma-beam SFEL (PBSFEL) [1, 2, 18]. Since the
plasma-beam instability has extremely high growth increments, then the plasma-beam SFEL is also characterized by exceptionally high amplification properties.
The idea of PBSFEL with pump in the form of slow
electromagnetic wave propagating in a magnetized
plasma-beam system was first proposed in [5]. Amplification properties of PBSFEL with the axial injection of
electron beam along the longitudinal magnetic field
were studied in the framework of the cubic nonlinear
theory in [6]. It has been shown that such devices can
work as powerful electromagnetic radiation sources in
the millimeter wavelength range.
In this paper we analyze the work PBSFEL of the
Dopplertron-type with helical electron beams. A great
number of articles are devoted to research of FEL properties with the helical electron beams. It is connected
with the fact that FEL with the helical electron beams
have a number of advantages [10 - 12, 19 - 22]. Plasmabeam SFELs of the dopplertron type with helical electron beams have been studied in the framework of the
quadratic approximation in [23, 24]. It was found that
PPSFEL can operate in four different modes. These
devices have higher growth increments of the signal
compared to PPSFEL with linear electron beams.
In this paper we continue to study the properties of
PBSFEL of the dopplertron type with helical electron
beams in the framework of cubic-nonlinear approximation. We find out levels and mechanisms of saturation
for different operation modes. The most effective operation modes have been found. We also have studied inISSN 1562-6016. ВАНТ. 2015. №6(100)
fluence of helical electron beam parameters on amplifying characteristics of PBSFEL.
MODEL
As a model of PBSFEL we consider the plasma with
the Langmuir frequency ωp and the electron beam with
the Langmuir frequency ωb, which pass through it. We
consider the case when ωb << ωp. The plasma-beam
system is located in the longitudinal magnetic field with
the induction vector B0. The magnetic field is directed
along the Z axis of the system. We consider that the
cyclotron frequency ωH of electron rotation in the magnetic field is much less than the plasma frequency
ωH << ωp. The electrons velocity vector of the helical
relativistic beam υb is directed at an angle β with respect to the Z axis. The refore tgβ = υb ⊥ / υbz , where
υb ⊥ and υbz are the transversal and the longitudinal
component of electron velocity. A circularly polarized
intense low-frequency electromagnetic wave with frequency ω2 and wave number k2 is used as a pump. The
pump wave propagates along Z axis of the system and in
the opposite direction to the electron beam. The frequency of the pump wave is less than the cyclotron frequency ω2 << ωH. In addition we feed a weak highfrequency circularly polarized electromagnetic signal
wave with frequency ω1 and wave number k1 to the system input. In the studied device the space-charge wave
(SCW) with frequency ω3 and wave number k3 is excited due to the parametric resonance of the interacting
waves. The condition of the parametric resonance can
be written as follows:
(1)
ω3 = ω1 − ω2 , k3 = k1 + k2 .
In SFEL a superheterodyne amplification effect is
realized. Its essence is to use an additional amplification
mechanism for the one of three waves that is involved in
a parametric resonance. The plasma-beam SFELs use
plasma-beam instability as an additional mechanism for
SCW amplification [1, 2, 18]. Parameters of the device
are chosen so that the growth increment of SCW is maximal. The growth increment of the plasma-beam instability is much greater than the growth increment of the
parametric instability. Therefore, the main objective of
83
the parametric resonance (1) is not the amplification of
interacting waves, but the transfer of amplification from
SCW to the high-frequency electromagnetic signal
wave. It is known that the growth increments of the
plasma-beam instability are extremely high. Therefore,
the gain of the electromagnetic wave in PBSFEL is also
large enough.
BASIC EQUATIONS
In order to describe the wave amplitudes dynamics
in the studied device, we use quasi-hydrodynamic equation [1, 2, 18], the continuity equation and Maxwell's
equations. The quasihydrodynamic equation we solve
using the method of averaged characteristics [1, 9]. The
continuity equation and the Maxwell's equations we
solve using the method of slowly varying amplitudes.
The velocity spread of electrons and the collision of
electrons are not taken into account. In accordance with
the method of averaged characteristics we carry out the
transition to characteristics of the quasihydrodynamic
equation. Then we express the components of the beam
electron velocity υbx and υby through the transverse
2
2
and the
component of the velocity υb ⊥ = υbx
+ υby
pb 0 of the electron beam:
υbx = υb ⊥ cos pb 0 , υby = υb ⊥ sin pb 0 .
phase
of rotation
This transition is due to the fact that unperturbed
electrons gyrate under the influence of the longitudinal
focusing magnetic field in the transverse plane. Considering the given transition the initial motion equations
have the following form:
dυb ⊥
υbz
e 


=
By  +
cos pb 0  E x −
dt
me γ b 
c


υ

 υ (Eυ) 
+ sin pb 0  E y + bz Bx  − b ⊥ 2  ,
c
c



dpb 0
e
ω
=− H +
dt
γ b me γ b υb ⊥

υbz 

Bx  −
cos pb 0  E y +
c



υ


− sin pb 0  E x − bz By  ,
c


υ
dυ bz
e

× E z + b⊥ ×
=
c
dt
me γ b 
(
(2)
)
× By cos pb 0 − Bx sin pb 0 −
υ bz (Eυ) 
,
c2 
(3)
d 2 E1x
dE1x
(6)
+ K1
= K3 E3 z E2 x + F1x ,
2
dt
dt
dE1 y
d 2 E1 y
K2
= K3η1η2 E3 z E2 y + F1 y ,(7)
+ K1
2
dt
dt
d 2 E2 x
dE2 x
(8)
M2
+ M1
= M 3 E3*z E1x + F2 x ,
2
dt
dt
d 2 E2 y
dE2 y
+ M1
= M 3η1η2 E3*z E1 y + F2 y ,(9)
M2
2
dt
dt
d 2 E3 z
dE
C2
+ C1 3 z + D3 E3 z =
2
dt
dt
*
(10)
= C3 ( E1x E2 x + E1 y E2* y ) + F3 z .
K2
It follows from the Eqs. (6) - (10) that circularly polarized electromagnetic waves are the eigenwaves for
the investigated system. In the Eqs. (6) - (10)
K1 = ∂D1 / ∂ (iω1 ) ;
K2 = 0.5 ⋅ ∂ 2 D1 / ∂ (iω1 )2 ;
M 2 = 0.5 ⋅ ∂ D2 / ∂ (iω2 ) ;
2
2
M1 = ∂D2 / ∂(iω2 ) , where
D1 = D(ω1 , k1 ) , D2 = D(ω 2 , k2 ) are the dispersion functions of transverse electromagnetic waves of the signal
(ω1, k1) and the pump (ω2, k2):
D(ω, k) =
b, p 
ωα2
i  2 2
2
×
k c − ω + ∑ 
2
cω 
α 
 γ α (Ω α + ηω H / γ α )
(

υ2
×  Ω α (Ω α + ηω H / γ α ) − α ⊥2 ω2 − k 2c 2

2c


) ,
 
(11)
where
η1 = E1 y /(iE1x ) = ±1 , η2 = E2 y /(iE2 x ) = ±1 ,
(4)
dγ b
e
[E xυb ⊥ cos pb0 + Ee υb ⊥ sin pb0 + E zυbz ] . (5)
=
dt me γ b
In Eqs. (2) - (5) E x , E y , E z are the components of
electric field strength vector; Bx and By are the components of the magnetic field induction vector;
ωH = eB0 /(mec) is the cyclotron frequency of electrons
rotation in a magnetic field; c is the speed of light in
2
vacuum; γ b = 1 / 1 − (υbz
+ υb2 ⊥ ) / c 2 is the relativistic
factor of the beam; e и me are the electron charge and
the electron mass respectively.
84
In accordance with the method of averaged characteristics [1, 9] we apply the procedure of asymptotic
integration to the system of equations (2) - (5). Thus we
find the velocities as the functions of the electric and
magnetic fields. Then we substitute the found solutions
in the continuity equation and Maxwell's equations,
which are solved by the method of slowly varying amplitudes.
As a result we obtain the set of differential equations
for the complex amplitude of x- and y- components of
the signal (E1x; E1y), x- and y- components of the pump
(E2x; E2y) and SCW (E3z) in the cubic approximation:
(12)
are the sign functions characterizing the rotation direction of the electric field strength vector of the circularly
polarized electromagnetic waves; Ω α = ω − kυαz . In
Eq. (10)
b, p 2
− iω3 
ωα (1 − υαz 2 / c 2 ) 
(13)
D3 =
1−
2

c 
α (ω3 − k3υαz ) γ α 
is the dispersion function of SCW, C1 = ∂D3 / ∂ (iω3 ) ;
∑
C2 = 0.5 ⋅ ∂ 2 D3 / ∂ (iω3 )2 . K3 , M 3 , C3 are the coefficients that depend on frequencies, wave numbers and
parameters of the investigated system. F1x , F1 y , F2 x ,
F2 y , F3 z are the cubic-nonlinear components of the
field amplitude of the signal, the pump and SCW reISSN 1562-6016. ВАНТ. 2015. №6(100)
spectively, taking into account the cubic-nonlinear interactions.
The system of equations (6) - (10) needs to be extended by the equations for constant components of the
velocity and the concentration
dυα / dt = Vα . dnα / dt = Nα .
(14)
Here Vα , Nα are the functions containing the cubic
nonlinear terms. They are dependent on the wave numbers, frequencies, amplitudes of the fields, the constant
components of velocities and densities (the subscript α
takes the values b and p ; the index b characterizes
beam parameters, the index p characterizes plasma
parameters).
As we mentioned above, the plasma-beam instability
is realized in the studied system. This means that the
dispersion equation for the space-charge wave
D3 (ω3 , k3 ) = 0 has complex solutions. Therefore, if we
substitute real frequencies and real wave numbers (the
real components of the complex solutions) in SCW dispersion function (13), then this function will not be
equal to zero D3 ≠ 0 . Left side of the equation (10) contains the term D3 which allows determining the growth
increment of the plasma-beam instability. So if we consider the equation (10) in the case of absence of the parametric resonance ( C3 ( E1x E2*x + E1 y E2* y ) = 0 ) and the
wave is less than the cyclotron frequency ωH. Note that
previously only one operation mode of PBSFEL of the
dopplertron type has been studied in the framework of a
cubic nonlinear approximation [6]. The right handed
circularly polarized electromagnetic wave (curve 1,
point A) participates in this interaction mode. The injection angle of the electron beam is equal to zero
( β = 0° ). The modes B, C and D were studied previously only in the quadratic approximation [24].
Parameters
Langmuir frequency of the
ma ( ω p ), [s–1]
Langmuir frequency of the beam ( ωb ),
[s–1]
Energy of the beam, [MeV]
The focusing magnetic field induction,
[G]
The amplitude of the first harmonic of
the pump electric field is
E2 = | E2 x |2 + | E2 y |2 , [V/m]
Values
1.0×1012
2.0×109
0.51
2.8×103
2.8×104
cubic-nonlinear components ( F3 z = 0 ), then the growth
rate will be determined by a plasma-beam instability
and will be approximately determined by the relation(here we suppose that
ship
(− D3 / C 2 )1 / 2
C2 d 2 E3 z / dt 2 , D3 E3 z >> C1dE3 z / dt ).
ANALYSIS
In this section, we analyze the wave dynamics of
PBSFEL with parameters given in Table. Using the dispersion relation for the transverse D(ω1 , k1 ) = 0 ,
D(ω2 , k2 ) = 0 and longitudinal D3 (ω3 , k3 ) = 0 waves,
we can determine the frequencies and wave numbers of
waves that are involved in the three-wave parametric
resonance (1). In [24] shown that the three-wave parametric interactions in the investigated plasma-beam
SFEL are possible for four different cases. Fig. 1 shows
the dispersion curves of the high-frequency electromagnetic signal waves (curves 1 and 2) and the SCW
(curves 3) for the case when the injection angle β of the
beam is equal to zero. Curve 1 corresponds to the righthanded circularly polarized electromagnetic wave
(η1 = -1), curve 2 corresponds to the left-handed circularly polarized electromagnetic wave (η1 = +1), when
viewed along the magnetic field direction. In Fig. 1
point O determines the frequency and the wave number
of the SCW with maximal growth increment (ω3, k3).
The points A, B, C and D determine the frequencies and
wave numbers of high-frequency electromagnetic signal
waves ((ω1, k1) that can participate in the resonant interactions (1). The parametric coupling of the signal waves
(ω1, k1) and the SCW ((ω3, k3) is caused by the pump
wave (ω2, k2). We assume that the frequency of this
ISSN 1562-6016. ВАНТ. 2015. №6(100)
Fig. 1 Dispersion curves of high-frequency
electromagnetic signal wave (curves 1 and 2)
and the SCW (curve 3)
Using standard numerical methods, we analyze dynamics of waves in the studied system in the framework
of the cubic-nonlinear approximation (6) - (10), (14) for
the different operation modes of PBSFEL (the modes
A , B , C and D ) and for the different injection angles
of the beam β .
Fig. 2 shows the dependences of the first harmonic
amplitudes of the electric field signal wave
E1 = | E1x | 2 + | E1 y, | 2 on the normalized time
τ = t ⋅ δω ,
(15)
for four possible operation modes. The injection angle
of the electron beam is β = 30° . In Eq. (15) δω is the
growth rate of the plasma-beam instability at an injection angle of the beam β = 0° [1, 2, 18]. It should be
noted that we consider the case when the beam energy
and the beam velocity modulus are constant. Therefore,
when the beam injection angle β is changing, then its
85
longitudinal velocity υ bz and the growth rate of the
plasma-beam instability are also changing. We take
constant multiplier as the normalization factor: the
growth rate of the plasma-beam instability at the beam
injection angle β = 0° . We do it in order to correctly
compare the wave dynamics at the different injection
angles β and the different operation modes in the studied system. In Fig. 2 the curve A corresponds to the
interaction mode A (Fig. 1), curve B corresponds to
the interaction mode B , curve C corresponds to the
interaction mode C , curve D corresponds to the interaction mode D .
harmonic amplitude of the plasma concentration
n p3 / n p0 .
Fig. 3. First harmonic amplitudes of the electric field
SCW E3 z as a function of normalized time τ = t ⋅ δω
Fig. 2. First harmonic amplitudes of the electric field
signal wave E1 as a function of normalized time
τ = t ⋅ δω . Curve A corresponds to the interaction
mode A , curve B corresponds to the mode B , curve
C corresponds to the mode C , curve D corresponds
to the mode D (Fig. 1)
We can see in Fig. 2 that the maximal saturation level of the signal wave electric field strength is achieved
in the mode D . Although the amplification rate of the
signal amplitude in the mode D at the initial interaction
stage is less than in the mode C . It should be noted that
the growth increment of the plasma-beam instability has
the same values for all modes shown in Fig. 2. The parametric growth increment is different in the modes A, B,
C and D. Therefore, the parametric interaction defines
the different signal wave dynamics for the different
modes. We also note that the signal electric field
strength reaches the value of ~ 2.5 MV/m in the saturation region of the mode D . The signal wavelength for
this mode is λ1 = 2πc / ω1 ≈ 1.8 mm. Thus, the PBSFEL
can work as a powerful source of electromagnetic millimeter wavelength range radiation in the interaction
mode D .
Let's find out the signal saturation mechanism in the
PBSFEL. Fig. 3 shows the dependences of the first
harmonic amplitudes of the electric field SCW E3 z on
normalized time τ = t ⋅ δω for the different operation
modes. Fig. 4 shows the dependence of the first harmonic amplitude of the plasma concentration n p 3 on
normalized time τ = t ⋅ δω for the modes A , B , C and
D . The plasma concentration is normalized to constant
component n p 0 . Comparing these figures, we see that
the dependences of the SCW electric field strength E3 z
are similar to the dependences of the normalized first
86
Fig. 4. First harmonic amplitude of the plasma concentration n p 3 is normalized to constant component n p 0
as a function of normalized time τ = t ⋅ δω
They are correlated for all modes of operation. This
means that the plasma electrons determine the dynamics
and the saturation level of the SCW electric field
strength. The beam electrons give the negligible contribution to the SCW electric field strength E3 z dynamics.
Also, Figs. 3 and 4 show that the SCW saturation levels
have the same values for all operation modes. The SCW
electric field E3 z reaches the sufficiently high values of
~ 5 MV/m at saturation. Therefore, the electron beam
with kinetic energy of 0.5 MeV is captured by plasma
wave at the saturation. The beam electrons translational
motion energy is the same for all four operation modes
of PBSFEL. So the electric field strength level has the
same value of saturation and capture of the electron
beam.
Figs. 3 and 4 also show that the SCW amplification
rate of mode D is the least compared to other modes
A , B and C . This happens due to the features of the
parametric interaction and the different coefficient values C3 in the equation (10) for the different operating
modes. Since the capture of the beam electrons in the
mode D occurs later than in other modes, then the
SCW electric field strength level is also achieved later.
The saturation time τ D in the mode D is longer than in
the modes A , B and C ( τ A , τ B and τC respectively,
see. Figs. 3 and 4). Therefore, the increase of the signal
wave is terminated in the modes A , B and C (Fig. 2)
ISSN 1562-6016. ВАНТ. 2015. №6(100)
earlier than in the mode D . The signal saturation level
is the highest in this mode (Fig. 2) because the electromagnetic signal rise time (the saturation time) of the
mode D is the largest. It should also be noted that this
saturation mechanism (when the beam electrons are
captured by the plasma waves) often occurs in different
plasma electronics devices [2].
Рис. 5. First harmonic amplitude of the electric field
signal wave as a function of normalized time τ = t ⋅ δω
at different beam injection angles β with respect to the
magnetic field in the interaction mode D . Curve 1 corresponds to injection angle β = 0° , curve 2 corresponds
to injection angle β = 10° , curve 3 corresponds to injection angle β = 20° , curve 4 corresponds to injection
angle β = 30°
Let us find out the changes of the electromagnetic
signal wave saturation levels depending on the electron
beam injection angle β . Fig. 5 shows the dependence of
the first harmonic amplitude of the electric field signal
wave on the normalized time τ = t ⋅ δω at the different
injection angles β of the beam with respect to the magnetic field in the interaction mode D. Curve 1 corresponds to the injection angle β =0°, curve 2 corresponds
to the injection angle β =10°, curve 3 corresponds to the
injection angle β =20°, curve 4 corresponds to the injection angle β =30°. From figure 5 follows that the gain of
the signal wave increases with the increase of injection
angle β at the initial interaction stage. The gain increase
is primarily associated with the change of the plasmabeam instability growth increment. The growth increment of the plasma-beam instability can be found from
the equation D3 = 0 (where D3 is given by Eq. (13)).
From the Eq. (13) follows that D3 depends on the longitudinal velocity of the beam υbz . The longitudinal
velocity of the beam decreases in the case of the off-axis
beam injection (the case of the constant beam energy). It
leads to the growth rate increase of the plasma-beam
instability. As a result the gain of the electromagnetic
signal wave increases. On the other hand, it leads to the
increase of the SCW electric field amplification rate
because the beam electrons are captured by the plasma
waves. As we see in Fig. 5, the signal wave saturation
level rises insignificantly with the increase of the beam
injection angle. In addition, the signal wave saturation at
the injection angle β = 30° happens earlier then at the
injection angle β = 0°. Thus, the use of the helical relativistic electron beams in PBSFEL is possible. It allows
ISSN 1562-6016. ВАНТ. 2015. №6(100)
strengthening of the powerful electromagnetic signals in
the millimeter wavelength range.
CONCLUSIONS
We have constructed the cubic nonlinear theory of
the wave interaction in a PBSFEL of the dopplertron
type with a helical electron beam. The saturation levels
are found for the four the operation modes. We found
that interaction mode D has the highest saturation levels of the high-frequency electromagnetic signal (the
parametric resonance interaction with extraordinary
circularly polarized electromagnetic signal wave)
among the other possible SFEL operation modes. Moreover, the saturation level of the signal in the mode D
exceeds the saturation levels in the operation modes A ,
B and C more than twice. We have found that the
saturation is associated with the capture of beam electrons by plasma waves in the investigated SFEL. We
have shown that the gain of the electromagnetic signal
rises with increase of the electron beam injection angle.
Such increase of the signal amplification is determined
by the increase of the growth rate of the plasma-beam
instability. It is associated with the decrease of the longitudinal electron energy. At the same time, the saturation level of the electromagnetic signal increases slightly with the increase of the injection angle of the electron
beam. Also, the saturation time decreases with the increase of electron beam injection angle. It means that
SFELs with the high-current helical electron beams may
use beams with shorter duration of the current pulse
compared to SFELs with coaxial beam. Thus, the plasma beam SFEL of the dopplertron type with the helical
electron beams can be used as a powerful source of coherent electromagnetic radiation in the millimeter wavelength range.
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ISSN 1562-6016. ВАНТ. 2015. №6(100)
НЕЛИНЕЙНЫЕ ВЗАИМОДЕЙСТВИЯ ВОЛН В СУПЕРГЕТЕРОДИННОМ ПЛАЗМЕННОПУЧКОВОМ ЛСЭ ДОПЛЕРТРОННОГО ТИПА С ВИНТОВЫМ ЭЛЕКТРОННЫМ ПУЧКОМ
В.В. Кулиш, А.В. Лысенко, Г.А. Алексеенко
Построена кубически-нелинейная теория плазменно-пучкового супергетеродинного лазера на свободных
электронах (СЛСЭ) доплертронного типа с винтовым электронным пучком. Определены уровни и механизм
насыщения для четырех режимов работы. Показано, что насыщение сигнала связано с захватом электронов
пучка плазменными волнами. Выяснено, что среди всех режимов работы СЛСЭ наиболее высоким уровнем
насыщения сигнала обладает режим взаимодействия, в котором в качестве сигнала используется необыкновенная электромагнитная волна. Продемонстрирована возможность создания на базе такого СЛСЭ источников мощного когерентного электромагнитного излучения в миллиметровом диапазоне длин волн.
НЕЛІНІЙНІ ВЗАЄМОДІЇ ХВИЛЬ У СУПЕРГЕТЕРОДИННОМУ ПЛАЗМОВО-ПУЧКОВОМУ ЛВЕ
ДОПЛЕРТРОННОГО ТИПУ З ГВИНТОВИМ ЕЛЕКТРОННИМ ПУЧКОМ
В.В. Куліш, О.В. Лисенко, Г.А. Олексієнко
Побудована кубічно-нелінійна теорія плазмово-пучкового супергетеродинного лазера на вільних електронах (СЛВЕ) доплертронного типу з гвинтовим електронним пучком. Визначені рівні та механізм насичення для чотирьох режимів роботи. З'ясовано, що насичення сигналу пов'язане із захопленням електронів пучка плазмовими хвилями. Показано, що серед усіх режимів роботи СЛВЕ найбільш високий рівень насичення
сигналу має режим взаємодії, у якому як сигнал використовується незвичайна електромагнітна хвиля. Продемонстрована можливість створення на базі такого СЛВЕ джерел потужного когерентного електромагнітного випромінювання в міліметровому діапазоні довжин хвиль.
ISSN 1562-6016. ВАНТ. 2015. №6(100)
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