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Nonlinear theory of plasma Cherenkov maser
Nuclear Instruments
and Methods in Physics
Research
A 375 (1996) 367-369
NUCLEAR INSTRUMENTS & METHODS IN PHVSICS RESEARCH SecttonA
Nonlinear theory of a plasma Cherenkov maser Jeong-Sik
Choi”‘“, Eun-Gi Heob, Byoung-Hee
Hong’, Duk-In Choib
*Department of Physics, Dongshin University, Naju, Chonnam 5.W 714, South Korea hDeparttnent of Ph>asics,Korea Advanced Institute of Science and Technology. Taejon 305.701. South Korea ‘Samsung Display Device Co.. Hwasung. Kyungi 415970. South Korea
A plasma Cherenkov maser (PCM), where a relativistic annular electron beam passes through a circular waveguide partially filled with a dense plasma, has been investigated experimentally in the Soviet Union [ I,21 and numerically developed into the particle simulation code [3]. There are several reasons for interest in the PCM as a practical, efficient source of microwave radiation. The plasma can neutralize the space charge and the current of the beam. This leads to the possibility of operation at very high beam currents and output powers. In addition, the magnitude of the radiation fields can be much larger in the PCM because the plasma does not suffer from dielectric breakdown. We have formulated the nonlinear analytic theory based on the cold-fluid Maxwell equation to analyze the nonlinear saturation characteristics for a PCM operation. The fluid model is applicable to the PCM using a circular waveguide (and circular symmetry for plasma density), together with the assumption that the electron beam motion is essentially one-dimensional along the cylindrical axis (strong external magnetic field, of IybyhwE,, @ 1) and electromagnetic waves are three-dimensional. The motion of plasma ions is neglected because of its large mass. Since there are no transverse perturbation currents, the fundamental TM modes with a nonzero IZZcan be excited. We adopt the Lagrangian variable 5 = t -Z/U and introduce the effective potential I/I = I$ - (u/c)A; = $,,(r) cos(w[), where #J is the scalar field and A: the vector field. The fundamental TM modes are written as E(r, z. t) = [E(r)e^. . . + E,(r)e^rl e
Ilk;,-w,,
At the final saturated state, we assume that all electrons of the beam are deeply trapped in the trough of most unstable TM,,,Z modes (/3 = ol!y). From the continuity equation, the final saturated density n, is obtained as
n,(r) =
n,(r) [,(p
-
p,,
(1)
’
where 1 r,(r) = G
w d5 (P-P,)
and the indices i = b. p are related to the beam and plasma electrons. After some manipulations, we can obtain the momentum conservation equations for the electrons of beam and plasma, respectively, the energy conservation equation and the wave equation as follows:
(4)
ia
[
y;rx-p*
a
1
~(~,5)=4ne]n,(l
-P&)
+ np( 1 - P&J1
where
1
(3
where the initial velocity of plasma electron &, = 0 and From the above nonlinear equations, Eqs. (2)-(5). we easily obtain the well-known linearized equation by introducing the linearized parameters S&. ?I&, fin,, and 6n, as a function of small perturbation I& The perturbation I& satisfies
p2 = k’ - o*/c’.
and B(r, Z, t) ZI B&)e^, e”‘:‘~“”
(6) *Corresponding 2909.
author. Tel. f82
0168-9002/96/$15.00 Copyright PII SOl68-9002(96)00098-8
613 30 3203, fax +82 613 33
where the dielectric
function
0 1996 Elsevier Science B.V. All rights reserved VI. HIGH POWER
FELs
J.-S. Choi et al. I NW/. Instr. ad
368
the boundary conditions, and D as
_?L
4
&.)=,_
Mcth. in Phys. Res. A 375 (1996) X7-369
wz’
&,(w - k&,c)' and the plasma frequency
of beam(plasma)
we can tind the constants A. B, C
electrons
Considering an annular electron beam (AR,IR, e I ) with a thin plasma (AR,IR, e I ) partially filling a waveguide, the density profiles are N,&r - R,) n,,,(r) =
and
T&l
n,,,,W =
NpS(r - R, 1 2vR ,
P
where N,,(N,,) is the number of beam (plasma) electrons per unit axial length and R,(R,) the radius of the beam (plasma) electrons. From OUT fluid model in which the entire beam electrons are treated as a single fluid component (p,, = /3 and .$ = 0). we finally rewrite the nonlinear governing equations of Eqs. (2)-(5) for the saturated state of the PCM as -cp,,(R,) + t = ~,(t
(7)
- PP,,,).
(8)
+ r&l ~ PP,) = 1 .
-v&,)
C=
D=
-f,(x,)K,Lq +r,(X,vqXb)
’
I,,(*, 1 4 I&,,,) I,,(Jr,W, (-qJ + 1,b-b)&,(.-qJ’
The nonlinear coupled equation, Eqs. (7)-(IO), will be numerically solved to find the excited frequency w, the wave number k;, and the final saturated energy of plasma electron 7,. The nonlinear efficiency of the conversion of the beam energy to the microwave field energy satisfies the relation as (II)
%E = Tlb - 77” 3 where the ratio of the energy transferred system
1 a
a
[ r ar Jr
p2
I
p,,(r) = 87re $+llo(1 (
-PP,)
>
3
(IO)
where the normalized effective potential 9,) = (e&,)/(mc’) and x,,+ = pR,,,. The general solution of the normalized effective potential from Eq. (10) is
+,\(r) =
i
A&,(I”) .
for 0 < r i
WP’)
for
+ CK,,( P’)
Dll,,frK,,(pR,)
- I,+ pR, K,,(pr)l
R, .
R, < i- < R,, .
> for R,
The electric field, that is ‘pa(r). is continuous at beamvacuum and vacuum-plasma interfaces. From the wave equation, we also have two more boundary conditions;
and the ratio of loss of the energy in plasma to the initial energy of a beam
VP =
’ -y, MU -1’
It is well-known that the saturation mechanism of the PCM is the electron trapping in the trough of the wave potential. Therefore, the fluid model is not sufficient for the description of the saturation characteristics since the beam and the plasma are treated as two single fluid components. In order to work around the limitation of the fluid model. we adopt the effective numbers including the kinetic effects such as electron trapping phenomena for the relativistic two-stream instability by Thode and Sudan [4]: the number of the beam electrons coherently trapped in the wave et*
nh = “bo where Ah = 4j,l( y2&,)), Ap = 4u,( 1 - pfl,), and the Budker’s parameter for beam and plasma electrons LJ, = P,/ respectively. From &, = N,,e’lmc” and up = N,e’lmc’,
by a beam to the
2‘
( > YbO
and the number of plasma electrons with the wave
coherently
responded
369
J.-S. Choi et al. I Nucl. Instr. and Meth. in Phys. Res. A 375 (1996) 367-369
Acknowledgements
Then, all the quantities related with n,,,(r) and n,,(r) in the above nonlinear governing equations Eqs. (7)~(IO) must be substituted with the “effective numbers”, i.e., n’,” and ,rf. Accordingly the efficiency from the beam energy to microwave field energy is written by
(12) Our nonlinear fluid model includes the energy between beam and plasma as well as that between beam and wave. Then we expect that the saturation mechanism of a PCM instabilities has a more close correspondence in that of the relativistics two stream instability than a DCM instability.
This paper was partially supported by Center for electroOptics at Korea Advanced Institute of Science and Technology.
References [I] L.S. Bogadankevich, M.V Kuzelev and A.A. Rukhadze, Usp. Fiz. Nauk 133 (1981) 3 [Sov. Phys. Usp. 24 (1981) 11. [2] M.V. Kuzelev, F.Kh. Mukhametzyanov, M.S. Rabinovich, A.A. Rukhadze, P.S. Strelkov and A.G. Shkavanmets, Zh. Eksp. Teor. Fiz. 83 (1982) 1358 [Sov. Phys. JETP 56 (1982) 7801; Dokl. Akad. Nauk SSSR 267 (1982) 829 [Sov. Phys. Dokl. 27 (1982) 1030]. [3] T.D. Pointon and J.S. De Grott, Phys. Fluids 31 ( 1988) 908. [4] L.E. Thode and R.N. Sudan. Phys. Fluids 18 (1975) 1552.
VI. HIGH POWER
FELs
and Methods in Physics
Research
A 375 (1996) 367-369
NUCLEAR INSTRUMENTS & METHODS IN PHVSICS RESEARCH SecttonA
Nonlinear theory of a plasma Cherenkov maser Jeong-Sik
Choi”‘“, Eun-Gi Heob, Byoung-Hee
Hong’, Duk-In Choib
*Department of Physics, Dongshin University, Naju, Chonnam 5.W 714, South Korea hDeparttnent of Ph>asics,Korea Advanced Institute of Science and Technology. Taejon 305.701. South Korea ‘Samsung Display Device Co.. Hwasung. Kyungi 415970. South Korea
A plasma Cherenkov maser (PCM), where a relativistic annular electron beam passes through a circular waveguide partially filled with a dense plasma, has been investigated experimentally in the Soviet Union [ I,21 and numerically developed into the particle simulation code [3]. There are several reasons for interest in the PCM as a practical, efficient source of microwave radiation. The plasma can neutralize the space charge and the current of the beam. This leads to the possibility of operation at very high beam currents and output powers. In addition, the magnitude of the radiation fields can be much larger in the PCM because the plasma does not suffer from dielectric breakdown. We have formulated the nonlinear analytic theory based on the cold-fluid Maxwell equation to analyze the nonlinear saturation characteristics for a PCM operation. The fluid model is applicable to the PCM using a circular waveguide (and circular symmetry for plasma density), together with the assumption that the electron beam motion is essentially one-dimensional along the cylindrical axis (strong external magnetic field, of IybyhwE,, @ 1) and electromagnetic waves are three-dimensional. The motion of plasma ions is neglected because of its large mass. Since there are no transverse perturbation currents, the fundamental TM modes with a nonzero IZZcan be excited. We adopt the Lagrangian variable 5 = t -Z/U and introduce the effective potential I/I = I$ - (u/c)A; = $,,(r) cos(w[), where #J is the scalar field and A: the vector field. The fundamental TM modes are written as E(r, z. t) = [E(r)e^. . . + E,(r)e^rl e
Ilk;,-w,,
At the final saturated state, we assume that all electrons of the beam are deeply trapped in the trough of most unstable TM,,,Z modes (/3 = ol!y). From the continuity equation, the final saturated density n, is obtained as
n,(r) =
n,(r) [,(p
-
p,,
(1)
’
where 1 r,(r) = G
w d5 (P-P,)
and the indices i = b. p are related to the beam and plasma electrons. After some manipulations, we can obtain the momentum conservation equations for the electrons of beam and plasma, respectively, the energy conservation equation and the wave equation as follows:
(4)
ia
[
y;rx-p*
a
1
~(~,5)=4ne]n,(l
-P&)
+ np( 1 - P&J1
where
1
(3
where the initial velocity of plasma electron &, = 0 and From the above nonlinear equations, Eqs. (2)-(5). we easily obtain the well-known linearized equation by introducing the linearized parameters S&. ?I&, fin,, and 6n, as a function of small perturbation I& The perturbation I& satisfies
p2 = k’ - o*/c’.
and B(r, Z, t) ZI B&)e^, e”‘:‘~“”
(6) *Corresponding 2909.
author. Tel. f82
0168-9002/96/$15.00 Copyright PII SOl68-9002(96)00098-8
613 30 3203, fax +82 613 33
where the dielectric
function
0 1996 Elsevier Science B.V. All rights reserved VI. HIGH POWER
FELs
J.-S. Choi et al. I NW/. Instr. ad
368
the boundary conditions, and D as
_?L
4
&.)=,_
Mcth. in Phys. Res. A 375 (1996) X7-369
wz’
&,(w - k&,c)' and the plasma frequency
of beam(plasma)
we can tind the constants A. B, C
electrons
Considering an annular electron beam (AR,IR, e I ) with a thin plasma (AR,IR, e I ) partially filling a waveguide, the density profiles are N,&r - R,) n,,,(r) =
and
T&l
n,,,,W =
NpS(r - R, 1 2vR ,
P
where N,,(N,,) is the number of beam (plasma) electrons per unit axial length and R,(R,) the radius of the beam (plasma) electrons. From OUT fluid model in which the entire beam electrons are treated as a single fluid component (p,, = /3 and .$ = 0). we finally rewrite the nonlinear governing equations of Eqs. (2)-(5) for the saturated state of the PCM as -cp,,(R,) + t = ~,(t
(7)
- PP,,,).
(8)
+ r&l ~ PP,) = 1 .
-v&,)
C=
D=
-f,(x,)K,Lq +r,(X,vqXb)
’
I,,(*, 1 4 I&,,,) I,,(Jr,W, (-qJ + 1,b-b)&,(.-qJ’
The nonlinear coupled equation, Eqs. (7)-(IO), will be numerically solved to find the excited frequency w, the wave number k;, and the final saturated energy of plasma electron 7,. The nonlinear efficiency of the conversion of the beam energy to the microwave field energy satisfies the relation as (II)
%E = Tlb - 77” 3 where the ratio of the energy transferred system
1 a
a
[ r ar Jr
p2
I
p,,(r) = 87re $+llo(1 (
-PP,)
>
3
(IO)
where the normalized effective potential 9,) = (e&,)/(mc’) and x,,+ = pR,,,. The general solution of the normalized effective potential from Eq. (10) is
+,\(r) =
i
A&,(I”) .
for 0 < r i
WP’)
for
+ CK,,( P’)
Dll,,frK,,(pR,)
- I,+ pR, K,,(pr)l
R, .
R, < i- < R,, .
> for R,
The electric field, that is ‘pa(r). is continuous at beamvacuum and vacuum-plasma interfaces. From the wave equation, we also have two more boundary conditions;
and the ratio of loss of the energy in plasma to the initial energy of a beam
VP =
’ -y, MU -1’
It is well-known that the saturation mechanism of the PCM is the electron trapping in the trough of the wave potential. Therefore, the fluid model is not sufficient for the description of the saturation characteristics since the beam and the plasma are treated as two single fluid components. In order to work around the limitation of the fluid model. we adopt the effective numbers including the kinetic effects such as electron trapping phenomena for the relativistic two-stream instability by Thode and Sudan [4]: the number of the beam electrons coherently trapped in the wave et*
nh = “bo where Ah = 4j,l( y2&,)), Ap = 4u,( 1 - pfl,), and the Budker’s parameter for beam and plasma electrons LJ, = P,/ respectively. From &, = N,,e’lmc” and up = N,e’lmc’,
by a beam to the
2‘
( > YbO
and the number of plasma electrons with the wave
coherently
responded
369
J.-S. Choi et al. I Nucl. Instr. and Meth. in Phys. Res. A 375 (1996) 367-369
Acknowledgements
Then, all the quantities related with n,,,(r) and n,,(r) in the above nonlinear governing equations Eqs. (7)~(IO) must be substituted with the “effective numbers”, i.e., n’,” and ,rf. Accordingly the efficiency from the beam energy to microwave field energy is written by
(12) Our nonlinear fluid model includes the energy between beam and plasma as well as that between beam and wave. Then we expect that the saturation mechanism of a PCM instabilities has a more close correspondence in that of the relativistics two stream instability than a DCM instability.
This paper was partially supported by Center for electroOptics at Korea Advanced Institute of Science and Technology.
References [I] L.S. Bogadankevich, M.V Kuzelev and A.A. Rukhadze, Usp. Fiz. Nauk 133 (1981) 3 [Sov. Phys. Usp. 24 (1981) 11. [2] M.V. Kuzelev, F.Kh. Mukhametzyanov, M.S. Rabinovich, A.A. Rukhadze, P.S. Strelkov and A.G. Shkavanmets, Zh. Eksp. Teor. Fiz. 83 (1982) 1358 [Sov. Phys. JETP 56 (1982) 7801; Dokl. Akad. Nauk SSSR 267 (1982) 829 [Sov. Phys. Dokl. 27 (1982) 1030]. [3] T.D. Pointon and J.S. De Grott, Phys. Fluids 31 ( 1988) 908. [4] L.E. Thode and R.N. Sudan. Phys. Fluids 18 (1975) 1552.
VI. HIGH POWER
FELs