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Spatial echoes in plasmas with pump
Volume 132, number 4
PHYSICS LETTERSA
3 October 1988
SPATIAL E C H O E S I N PLASMAS W I T H P U M P R. BLAHA, E.W. LAEDKE
In'stitutfiir TheoretischePhysik, Universit?itDiisseldorf D-4000Diisseldorf FRG and V.N. PAVLENKO
Institutefor NuclearResearch, Kiev, USSR Received 22 July 1987; revised manuscript received 15 July 1988; accepted for publication 15 July 1988 Communicatedby R.C. Davidson
The influence of an electrostaticpump field on the spatial echo in an electron plasma is investigated. Ballistic theory as well as the collectiveecho method including the self-consistentfield are used. We show, that the pump causes a splitting of a single echo into a systemof spatiallyseparated echo signals.
Subsequent years saw active research on echo phenomena in unbounded and bounded plasmas [ 1-9 ]. Concerning nonlinear processes in plasmas echo phenomena are of growing importance especially for plasma diagnostics and radio-physical applications [4,10-13 ]. In the present paper we investigate the influence of a high frequency electrostatic pump field on the spatial echo in a homogeneous collisionless unbounded electron plasma. There are two standard approaches to calculate echo phenomena in plasmas. The first one, which describes the so-called "collective echo", uses a nonlinear equation for the distribution function taking into account the self-consistent field. It is clear that the collective behavior is important when the characteristic frequencies correspond to the transparency region, i.e. to--tope, where tope denotes the electron plasma frequency. The second method, the so-called "ballistic theory" is based on the collisionless Boltzmann equation where only the field of external perturbations is incorporated. The ballistic approximation is obviously valid in the frequency range 09< tope or 09 >> tope [4-6 ], because in this case the modulated particle beams play the governing role in the excitation of echo oscillations. First the influence of an electrostatic p u m p field is studied in the ballistic approximation. Using the dipole approximation for the pump Eo (t)=¢,Eo sin tOot we split up the electric field in the form E = E o ( t ) d-Eext (2, t) .
( 1)
The external perturbations with frequencies tO~, o92>> tope are given by
EeX'( z, t) =Eo~ e-i°~tO( koz ) + Eo2 e i ~ t 0 [ k o ( z - z o ] + c . c . ,
(2)
where ko is a normalization constant. Assuming that the external perturbations are small compared to the pump, we solve the collisionless Boltzmann equation by the method of successive approximation (f=fo +
f~)+f~2)+ ...). After some algebra we obtain for the second-order electric field
E~2)(z,t)=
~
(E~)+ +Etm2),-),
(3)
0375-9601/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
179
n,m=
--oo
Volume 132, number 4
PHYSICS LETTERS A
3 October 1988
with
(z-zo+asincoot) E~2 +-=- 4~e3E°lE°2 2 2 m¢ko
a w - ~ w e x p [ - i ( ~ l +_~2)tlw-3j~(~la/w)J,,,(~2a/w) o
X e x p [ i ( z - l + n + a sin coo/) ((.Ol ___632)/W ] .
(4)
Here the abbreviations ¢bt =col +ncoo, ¢b2 =co2 +mcoo, w=v+ (eEo/mecoo)cos COot and a=eEo/meCO~ have been introduced. Eq. (4) describes the second-order echoes in a homogeneous infinite plasma under the influence of a pump field. The echo attains maximum intensity in the points
z..~l~,, =
ZO(~2/((~2
"]-O,~ 1 ) .
(5)
Note that for COo= 0 the electric field reduces to the usual ballistic second-order echo with maximum intensity in the point Z=ZoCO2/(CO2-091) (see e.g. refs. [4-6] ). Let us now take into account the self-consistent electric field, which has been neglected in the ballistic approximation. Solving the coupled set of the collisionless Boltzmann equation and the Poisson equation by the method of successive approximations we finally arrive at the following expression for the self-consistent field in the second approximation
£~2~(Z,CO)=
-ieCO~e 4~2no m e
~,, n,m=-oo
j7 dw j7 dk, 7 j dco, 7 j dke ik~ 0fo --oo
--~
--oo
--c~
Jm( ( k - k ' )a)J,(k' a)P.~k,(CO-CO'-mCOo)P.~xt(CO ' -nCOo) × (CO'-k'w+iO)(CO-kw+iO)Z~(k, CO)E(k',og') E(k-k',CO-CO') '
(6)
where E denotes the longitudinal dielectric permittivity of the plasma. After k, k' and CO' integration one obtains the expression for the second-order echo field. Note, that eq. (4) follows from (6) in the limit e-, 1. A detailed evaluation of formula (6) will be presented elsewhere. Summarizing the results, we have shown, that the influence of an electrostatic pump field on the usual second order echo leads to a new effect. From eqs. (4) and (5) it is clear, that the pump causes a splitting of a single echo into a system of spatially separated echo signals. Note, that this effect is not possible for the temporal echo, see ref. [9]. The distance between the position of the usual echo (o90=0) with frequency CO2-COl and the echo harmonics with frequencies 692 + ~1 is given by
l+, =lo0--lm, =
~2COl+-CO2~I
(CO2-o91 ) (~2 +-o51) Zo.
(7)
It is remarkable, that due to the influence of the pump the temporal harmonics corresponding to the frequencies coj and co2 are related to different points in space, as e.g. in holography. Our result shows, that already in the lowest nonlinear approximation several echoes of the same order of magnitude can appear, which may become important for rf-heating of plasmas. Note, that the location of the echo signal can be varied by changing the pump field frequency coo, without changing the positions and frequencies of the perturbation field sources. This feature may have useful applications in plasma diagnostics, e.g., for obtaining the velocity diffusion coefficient. Finally, the excitation of harmonics in space may be used for the aim of spectroscopy.
References [ 1 ] R.W. Gould, T.M. O'Neil and J.H. Malmberg, Phys. Rev. Lett. 19 ( 1967 ) 219. [2] B.B. Kadomtsev, Soy. Phys. Usp. 11 (1969) 328. [ 3 ] A.G. Sitenko, N.V. Chong and V.N. Pavlenko, Nucl. Fusion 10 (1970) 259.
180
Volume 132, number 4
PHYSICS LETTERS A
3 October 1988
[4] V.N. Pavlenko, Soy. Phys. Usp. 26 (1983) 931. [5] B.H. Ripin and R.E. Pechacek, Phys. Fluids 15 (1972) 1980. [6] R.W. Gould, Phys. Lett. A 25 (1967) 559. [7] A.G. Sitenko, V.N. Pavlenko and V.I. Zasenko, Sov. J. Plasma Phys. 2 (1976) 448. [ 8 ] V.N. Pavlenko and S.M. Revenchuk, Sov. J. Plasma Phys. 4 ( 1978 ) 384. [9] I.M. Aliev, O.M. Gradov and A.K. Nasarjan, Sov. J. Plasma Phys. 3 (1977) 523. [ 10] Ch. Moeller, Phys. Fluids 18 (1975) 89. [ I l ] V.F. Dryakhlushin and Yu.A. Romanov, Soy. J. Plasma Phys. 5 (1979) 657. [ 12] M.D. Leppert, K. Wiesemann and H. Schliiter, Plasma Phys. 24 1982) 1475. [ 13] I. Nycander, V.P. Pavlenko and S.M. Revenchuk, Soy. J. Plasma Phys. 12 (1986) 231.
181
PHYSICS LETTERSA
3 October 1988
SPATIAL E C H O E S I N PLASMAS W I T H P U M P R. BLAHA, E.W. LAEDKE
In'stitutfiir TheoretischePhysik, Universit?itDiisseldorf D-4000Diisseldorf FRG and V.N. PAVLENKO
Institutefor NuclearResearch, Kiev, USSR Received 22 July 1987; revised manuscript received 15 July 1988; accepted for publication 15 July 1988 Communicatedby R.C. Davidson
The influence of an electrostaticpump field on the spatial echo in an electron plasma is investigated. Ballistic theory as well as the collectiveecho method including the self-consistentfield are used. We show, that the pump causes a splitting of a single echo into a systemof spatiallyseparated echo signals.
Subsequent years saw active research on echo phenomena in unbounded and bounded plasmas [ 1-9 ]. Concerning nonlinear processes in plasmas echo phenomena are of growing importance especially for plasma diagnostics and radio-physical applications [4,10-13 ]. In the present paper we investigate the influence of a high frequency electrostatic pump field on the spatial echo in a homogeneous collisionless unbounded electron plasma. There are two standard approaches to calculate echo phenomena in plasmas. The first one, which describes the so-called "collective echo", uses a nonlinear equation for the distribution function taking into account the self-consistent field. It is clear that the collective behavior is important when the characteristic frequencies correspond to the transparency region, i.e. to--tope, where tope denotes the electron plasma frequency. The second method, the so-called "ballistic theory" is based on the collisionless Boltzmann equation where only the field of external perturbations is incorporated. The ballistic approximation is obviously valid in the frequency range 09< tope or 09 >> tope [4-6 ], because in this case the modulated particle beams play the governing role in the excitation of echo oscillations. First the influence of an electrostatic p u m p field is studied in the ballistic approximation. Using the dipole approximation for the pump Eo (t)=¢,Eo sin tOot we split up the electric field in the form E = E o ( t ) d-Eext (2, t) .
( 1)
The external perturbations with frequencies tO~, o92>> tope are given by
EeX'( z, t) =Eo~ e-i°~tO( koz ) + Eo2 e i ~ t 0 [ k o ( z - z o ] + c . c . ,
(2)
where ko is a normalization constant. Assuming that the external perturbations are small compared to the pump, we solve the collisionless Boltzmann equation by the method of successive approximation (f=fo +
f~)+f~2)+ ...). After some algebra we obtain for the second-order electric field
E~2)(z,t)=
~
(E~)+ +Etm2),-),
(3)
0375-9601/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
179
n,m=
--oo
Volume 132, number 4
PHYSICS LETTERS A
3 October 1988
with
(z-zo+asincoot) E~2 +-=- 4~e3E°lE°2 2 2 m¢ko
a w - ~ w e x p [ - i ( ~ l +_~2)tlw-3j~(~la/w)J,,,(~2a/w) o
X e x p [ i ( z - l + n + a sin coo/) ((.Ol ___632)/W ] .
(4)
Here the abbreviations ¢bt =col +ncoo, ¢b2 =co2 +mcoo, w=v+ (eEo/mecoo)cos COot and a=eEo/meCO~ have been introduced. Eq. (4) describes the second-order echoes in a homogeneous infinite plasma under the influence of a pump field. The echo attains maximum intensity in the points
z..~l~,, =
ZO(~2/((~2
"]-O,~ 1 ) .
(5)
Note that for COo= 0 the electric field reduces to the usual ballistic second-order echo with maximum intensity in the point Z=ZoCO2/(CO2-091) (see e.g. refs. [4-6] ). Let us now take into account the self-consistent electric field, which has been neglected in the ballistic approximation. Solving the coupled set of the collisionless Boltzmann equation and the Poisson equation by the method of successive approximations we finally arrive at the following expression for the self-consistent field in the second approximation
£~2~(Z,CO)=
-ieCO~e 4~2no m e
~,, n,m=-oo
j7 dw j7 dk, 7 j dco, 7 j dke ik~ 0fo --oo
--~
--oo
--c~
Jm( ( k - k ' )a)J,(k' a)P.~k,(CO-CO'-mCOo)P.~xt(CO ' -nCOo) × (CO'-k'w+iO)(CO-kw+iO)Z~(k, CO)E(k',og') E(k-k',CO-CO') '
(6)
where E denotes the longitudinal dielectric permittivity of the plasma. After k, k' and CO' integration one obtains the expression for the second-order echo field. Note, that eq. (4) follows from (6) in the limit e-, 1. A detailed evaluation of formula (6) will be presented elsewhere. Summarizing the results, we have shown, that the influence of an electrostatic pump field on the usual second order echo leads to a new effect. From eqs. (4) and (5) it is clear, that the pump causes a splitting of a single echo into a system of spatially separated echo signals. Note, that this effect is not possible for the temporal echo, see ref. [9]. The distance between the position of the usual echo (o90=0) with frequency CO2-COl and the echo harmonics with frequencies 692 + ~1 is given by
l+, =lo0--lm, =
~2COl+-CO2~I
(CO2-o91 ) (~2 +-o51) Zo.
(7)
It is remarkable, that due to the influence of the pump the temporal harmonics corresponding to the frequencies coj and co2 are related to different points in space, as e.g. in holography. Our result shows, that already in the lowest nonlinear approximation several echoes of the same order of magnitude can appear, which may become important for rf-heating of plasmas. Note, that the location of the echo signal can be varied by changing the pump field frequency coo, without changing the positions and frequencies of the perturbation field sources. This feature may have useful applications in plasma diagnostics, e.g., for obtaining the velocity diffusion coefficient. Finally, the excitation of harmonics in space may be used for the aim of spectroscopy.
References [ 1 ] R.W. Gould, T.M. O'Neil and J.H. Malmberg, Phys. Rev. Lett. 19 ( 1967 ) 219. [2] B.B. Kadomtsev, Soy. Phys. Usp. 11 (1969) 328. [ 3 ] A.G. Sitenko, N.V. Chong and V.N. Pavlenko, Nucl. Fusion 10 (1970) 259.
180
Volume 132, number 4
PHYSICS LETTERS A
3 October 1988
[4] V.N. Pavlenko, Soy. Phys. Usp. 26 (1983) 931. [5] B.H. Ripin and R.E. Pechacek, Phys. Fluids 15 (1972) 1980. [6] R.W. Gould, Phys. Lett. A 25 (1967) 559. [7] A.G. Sitenko, V.N. Pavlenko and V.I. Zasenko, Sov. J. Plasma Phys. 2 (1976) 448. [ 8 ] V.N. Pavlenko and S.M. Revenchuk, Sov. J. Plasma Phys. 4 ( 1978 ) 384. [9] I.M. Aliev, O.M. Gradov and A.K. Nasarjan, Sov. J. Plasma Phys. 3 (1977) 523. [ 10] Ch. Moeller, Phys. Fluids 18 (1975) 89. [ I l ] V.F. Dryakhlushin and Yu.A. Romanov, Soy. J. Plasma Phys. 5 (1979) 657. [ 12] M.D. Leppert, K. Wiesemann and H. Schliiter, Plasma Phys. 24 1982) 1475. [ 13] I. Nycander, V.P. Pavlenko and S.M. Revenchuk, Soy. J. Plasma Phys. 12 (1986) 231.
181