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Instability of incoherent photons
19 January
1998
PHYSICS
EISEWIER
LETTERS
A
Physics Letters A 237 ( 1998) 385-387
Instability of incoherent photons P.K. Shukla”, L. Stenflo’ B Fakultiitftir Physik und Astronomic, Ruhr-Universitiit Bochum. D-44780 Bochum, Germany b Department of Plasma Physics, Umerf University, S-90187 Urned, Sweden Received 22 July 1997; accepted for publication 4 August Communicated by V.M. Agranovich
1997
Abstract
A kinetic theory for the modulational instability of incoherent photons with a broad spectrum is presented. Explicit results for the growth rates are obtained in several limiting cases. @ 1998 Published by Elsevier Science B.V. PACS: 52.25.M~;
52.40.Db;
52.40.Nk
Thirty years ago, Vedenov, Gordeev and Rudakov [ I] considered the nonlinear propagation of an ensemble of randomly distributed Langmuir wave packets in a slowly varying unmagnetized plasma. Thus, the dynamics of the wave packets, which are treated as quasi-particles, is governed by a Liouville equation [ 21, whereas the slow plasma response (ion-acoustic disturbances) is affected by the ponderomotive force of the Langmuir turbulence. It has been found that the ion-acoustic waves are anomalously damped in the presence of a spectrum of random-phase Langmuir waves. Furthermore, when the latter have a Dirac-delta function spectrum, one encounters the envelope modulational instability [ 31 of coherent Langmuir waves. Recently, Bingham, Mendonca and Dawson [4] investigated Langmuir wave damping in the presence of a spectrum of photons, which obey a Liouville (or a wave kinetic) equation. In Ref. [ 41, the dynamics of Langmuir waves in the presence of photons has been obtained by means of the hydrodynamic and Poisson’s equations. In this Letter, we present a kinetic theory for the modulational instability of incoherent (broad band) 0375.9601/98/$19.00 @ 1998 Published PIf SO375-9601(97)00927-4
photons. The plasma motion is governed by the Vlasov-Poisson equations, whereas the photons are described by the Liouville equation. Let us consider the nonlinear propagation of random phase photons in an unmagnetized plasma. The photon dynamics is studied by means of the Liouville equation
11921 (1) where the number of photons is Nk = (I& 12)/4%Wk, Ek, k, and Wk = ( w2 + k2c2) ‘i2 are the electric field the wave vector, ai: the frequency of the photons: respectively, CL+ = (47rn,e2/m,)‘/2 is the electron plasma frequency, n, is the electron number density, e is the magnitude of the electron charge, m, is the electron mass, and c is the speed of light. The group velocity of the photons is us = kC2/ok. The angular bracket denotes the ensemble average. In ( 1) , we have adopted the “adiabatic approximation”, which is valid for lowfrequency (in comparison with the photon frequency), long wavelength modulational envelope electrostatic waves. The modulation occurs due to the effective po-
by Elsevier Science B.V. All rights reserved.
386
PK. Shukla, L. Stenfio/Physics
tential produced by the low-frequency density perturbations, i.e., Vwk = (w$/2nowk) Vnet , where 0~0 is the unperturbed plasma frequency and n,t (<< no) is the electron number density perturbation in the equilibrium plasma number density no. The electron density perturbation is given by n,i = J” fel due,where the perturbed electron distribution function is obtained from
Letters A 237 (1998) 385-387
2 nil = - kXirb9
where for a Maxwellian distribution susceptibility is given by
Xj
=
&G( 4
feo,io+ the plasma
(8)
Zj).
'Dj
Here where 4 is the self-consistent low-frequency ambipolar potential and I&is the effective photon ponderomotive potential given by *=-
27re meup
c k
Nk (1 + PA,2)‘/2’
Acj = Tj/4Tlloe2, 1
G(Z,)=Zj
=
(ZT)l/2
s
a/qUtj>
Here h, = c/wpc is the collisionless electron skin depth and fa is the unperturbed electron distribution function. Eqs. ( 1) and (2) are supplemented by Poisson’s equation
and utj = is the thermal velocity of the particle species j (j equals e for the electrons and i for the ions). Substituting for njt from (6) and (7) into q2+ = 4Ve(nil - GI ), we obtain Cp= x&/( l + xe + xi>. Using the latter in (6) we have
V24 = 47re(&i
n,1
(Tj/TTlj)li2
- nit),
(3)
where the ion number density perturbation is nil = s fii dvi, with the perturbed ion distribution determined from (4) where mi is the ion mass and fro is the unperturbed ion distribution function. In (4) we have neglected the contribution of the comparatively small ion ponderomotive potential. Letting Nk = @ + Ni, where Ni < @, and supposing that Ni and fet,tt are proportional to exp(iq * r - iat), where q and D are the wave vector and the frequency of the electrostatic modulations, we obtain from ( 1)
On the other hand, from (2) -( 4) we have
(6)
1 + Xi> 2meopoe
q2Xe( =
-
c k
NL
(9)
(1 + I%@‘/2
whereE=l+X,fXt. From (5) and (9) we readily obtain 1
;+_=---
1 +Xi X
J
(
L
q%:,
16WnaT, A2 dk 4. (i@/ak)
1 + k2A:)k.
q - (1 + k2A~)3h&opo/c2’
(10)
where ft = (ltik12) is the unperturbed intensity of random phased photons, and L is the length of the system. Eq. ( 10) is the general nonlinear dispersion relation which is appropriate for studying the modulational instability of arbitrary wavelength random-phased photons provided that the spectrum 1: is known. In the following, we present specific results for the kinetic modulational instability of random-phased photons by supposing a Gaussian intensity distribution, namely,
If =
10
(2n)‘i’k,L
expt-(k
-
kd2/2ktl,
(11)
PK. Shukla, L. StenJo/Physics
where lo = (]&,]*) is the maximum photon intensity corresponding to a mean photon wave vector ko, and k, represents the width of the photon spectrum. Substituting for I: from ( 11) into ( 10) and assuming that the photon wavelength is much larger than &, we obtain ;+_=
1 1+x
9*&P_ ,,,,WoC(50) we
3
(12)
where WO= (]E~012)/167r&r” < 1 and 50 = (k4 ko) lkw with k, = fl/qc cos tl I\e; 8 being the angle between q and ko. We note that (12) is valid when the photon frequency is close to 0~0. Supposing that the phase velocity of the lowfrequency modulation lies between the electron and ion thermal velocities, we take xe = l/q*&_ and and rewrite (12) in the form for Xi M -w$/fi* R < Wpi9
R2-
4%; = -$$woG(&o), w e
(13)
where
upi is the ion plasma frequency and c, = (T,/mi) ‘I* is the ion-acoustic velocity. For 1&u]<< 1, ( 13) predicts an oscillatory modulational instability for ko > k,,, where k,, = L&,,/qcosO A, and flfs = q*cf ( 1 - Wo/k$z). The growth rate of that instability is
Yk =
7r q*c$‘o -$ 2 2%
Ike - k,,l k3 weA2 .
387
which predicts an oscillatory instability for fi < qc,. The maximum growth rate of that hydrodynamic instability is qc cos 0 Wi’*. Finally, we see that for 0 N quG << qut, and /&I > 1, ( 12) admits a stimulated ion Compton scattering instability whose increment is yc = w;‘*q*&,u, cos 8 x Re[q*&
+ l/(1
+x1)1-‘/*,
(16)
where XI = ( 1/q*A&) G( kocq& cos f?/uh). In summary, we have presented a kinetic theory for the modulational instability of incoherent photons which are random phased. For this purpose, we have employed the Liouville equation for the photons and developed the kinetic plasma response associated with low-frequency electrostatic modulations in the presence of the ponderomotive force of the photons. When the latter has a broad band Gaussian distribution, we obtain new classes of photon modulational instabilities. The nonlinear saturation of a modulational instability can be studied with the help of the quasilinear description, as proposed in Ref. [4]. In conclusion, we stress that the results of the present investigation will contribute to the understanding of the nonlinear propagation of random phase electromagnetic waves through various space as well as astrophysical and laboratory (inertial confinement fusion) plasmas. They are thus of rather general interest.
(14)
Physically, the instability arises because of the resonant interaction between the quasi-particles (photons) and the modified ion-acoustic waves. The mechanism is thus analogous to the Cerenkov resonance interaction. On the other hand, for ]&I >> 1, ( 13) reduces to (fl-q*c;)(.fl-qckoA,cos8)* = q”c’cf WI)cos* 8,
Letters A 237 (1998) 385-387
(15)
This work was partially supported by the Deutsche Forschungsgemeinschaft (Bonn, Germany).
References [ I ] A.A. Vedenov. A.V. Gordeev and L.I. Rudakov, Plasma Phys. 9 (1967) 719. [2] B.B. Kadomtsev, Plasma Turbulence (Academic Press, New York, 1965 ) [ 31 W.L. Kruer, Physics of Laser Plasma Interactions (AddisonWesley, Redwood City, 1980). [4] R. Bingham. J.T. Mendonp and J. M. Dawson, Phys. Rev. Len. 78 (1997) 247.
1998
PHYSICS
EISEWIER
LETTERS
A
Physics Letters A 237 ( 1998) 385-387
Instability of incoherent photons P.K. Shukla”, L. Stenflo’ B Fakultiitftir Physik und Astronomic, Ruhr-Universitiit Bochum. D-44780 Bochum, Germany b Department of Plasma Physics, Umerf University, S-90187 Urned, Sweden Received 22 July 1997; accepted for publication 4 August Communicated by V.M. Agranovich
1997
Abstract
A kinetic theory for the modulational instability of incoherent photons with a broad spectrum is presented. Explicit results for the growth rates are obtained in several limiting cases. @ 1998 Published by Elsevier Science B.V. PACS: 52.25.M~;
52.40.Db;
52.40.Nk
Thirty years ago, Vedenov, Gordeev and Rudakov [ I] considered the nonlinear propagation of an ensemble of randomly distributed Langmuir wave packets in a slowly varying unmagnetized plasma. Thus, the dynamics of the wave packets, which are treated as quasi-particles, is governed by a Liouville equation [ 21, whereas the slow plasma response (ion-acoustic disturbances) is affected by the ponderomotive force of the Langmuir turbulence. It has been found that the ion-acoustic waves are anomalously damped in the presence of a spectrum of random-phase Langmuir waves. Furthermore, when the latter have a Dirac-delta function spectrum, one encounters the envelope modulational instability [ 31 of coherent Langmuir waves. Recently, Bingham, Mendonca and Dawson [4] investigated Langmuir wave damping in the presence of a spectrum of photons, which obey a Liouville (or a wave kinetic) equation. In Ref. [ 41, the dynamics of Langmuir waves in the presence of photons has been obtained by means of the hydrodynamic and Poisson’s equations. In this Letter, we present a kinetic theory for the modulational instability of incoherent (broad band) 0375.9601/98/$19.00 @ 1998 Published PIf SO375-9601(97)00927-4
photons. The plasma motion is governed by the Vlasov-Poisson equations, whereas the photons are described by the Liouville equation. Let us consider the nonlinear propagation of random phase photons in an unmagnetized plasma. The photon dynamics is studied by means of the Liouville equation
11921 (1) where the number of photons is Nk = (I& 12)/4%Wk, Ek, k, and Wk = ( w2 + k2c2) ‘i2 are the electric field the wave vector, ai: the frequency of the photons: respectively, CL+ = (47rn,e2/m,)‘/2 is the electron plasma frequency, n, is the electron number density, e is the magnitude of the electron charge, m, is the electron mass, and c is the speed of light. The group velocity of the photons is us = kC2/ok. The angular bracket denotes the ensemble average. In ( 1) , we have adopted the “adiabatic approximation”, which is valid for lowfrequency (in comparison with the photon frequency), long wavelength modulational envelope electrostatic waves. The modulation occurs due to the effective po-
by Elsevier Science B.V. All rights reserved.
386
PK. Shukla, L. Stenfio/Physics
tential produced by the low-frequency density perturbations, i.e., Vwk = (w$/2nowk) Vnet , where 0~0 is the unperturbed plasma frequency and n,t (<< no) is the electron number density perturbation in the equilibrium plasma number density no. The electron density perturbation is given by n,i = J” fel due,where the perturbed electron distribution function is obtained from
Letters A 237 (1998) 385-387
2 nil = - kXirb9
where for a Maxwellian distribution susceptibility is given by
Xj
=
&G( 4
feo,io+ the plasma
(8)
Zj).
'Dj
Here where 4 is the self-consistent low-frequency ambipolar potential and I&is the effective photon ponderomotive potential given by *=-
27re meup
c k
Nk (1 + PA,2)‘/2’
Acj = Tj/4Tlloe2, 1
G(Z,)=Zj
=
(ZT)l/2
s
a/qUtj>
Here h, = c/wpc is the collisionless electron skin depth and fa is the unperturbed electron distribution function. Eqs. ( 1) and (2) are supplemented by Poisson’s equation
and utj = is the thermal velocity of the particle species j (j equals e for the electrons and i for the ions). Substituting for njt from (6) and (7) into q2+ = 4Ve(nil - GI ), we obtain Cp= x&/( l + xe + xi>. Using the latter in (6) we have
V24 = 47re(&i
n,1
(Tj/TTlj)li2
- nit),
(3)
where the ion number density perturbation is nil = s fii dvi, with the perturbed ion distribution determined from (4) where mi is the ion mass and fro is the unperturbed ion distribution function. In (4) we have neglected the contribution of the comparatively small ion ponderomotive potential. Letting Nk = @ + Ni, where Ni < @, and supposing that Ni and fet,tt are proportional to exp(iq * r - iat), where q and D are the wave vector and the frequency of the electrostatic modulations, we obtain from ( 1)
On the other hand, from (2) -( 4) we have
(6)
1 + Xi> 2meopoe
q2Xe( =
-
c k
NL
(9)
(1 + I%@‘/2
whereE=l+X,fXt. From (5) and (9) we readily obtain 1
;+_=---
1 +Xi X
J
(
L
q%:,
16WnaT, A2 dk 4. (i@/ak)
1 + k2A:)k.
q - (1 + k2A~)3h&opo/c2’
(10)
where ft = (ltik12) is the unperturbed intensity of random phased photons, and L is the length of the system. Eq. ( 10) is the general nonlinear dispersion relation which is appropriate for studying the modulational instability of arbitrary wavelength random-phased photons provided that the spectrum 1: is known. In the following, we present specific results for the kinetic modulational instability of random-phased photons by supposing a Gaussian intensity distribution, namely,
If =
10
(2n)‘i’k,L
expt-(k
-
kd2/2ktl,
(11)
PK. Shukla, L. StenJo/Physics
where lo = (]&,]*) is the maximum photon intensity corresponding to a mean photon wave vector ko, and k, represents the width of the photon spectrum. Substituting for I: from ( 11) into ( 10) and assuming that the photon wavelength is much larger than &, we obtain ;+_=
1 1+x
9*&P_ ,,,,WoC(50) we
3
(12)
where WO= (]E~012)/167r&r” < 1 and 50 = (k4 ko) lkw with k, = fl/qc cos tl I\e; 8 being the angle between q and ko. We note that (12) is valid when the photon frequency is close to 0~0. Supposing that the phase velocity of the lowfrequency modulation lies between the electron and ion thermal velocities, we take xe = l/q*&_ and and rewrite (12) in the form for Xi M -w$/fi* R < Wpi9
R2-
4%; = -$$woG(&o), w e
(13)
where
upi is the ion plasma frequency and c, = (T,/mi) ‘I* is the ion-acoustic velocity. For 1&u]<< 1, ( 13) predicts an oscillatory modulational instability for ko > k,,, where k,, = L&,,/qcosO A, and flfs = q*cf ( 1 - Wo/k$z). The growth rate of that instability is
Yk =
7r q*c$‘o -$ 2 2%
Ike - k,,l k3 weA2 .
387
which predicts an oscillatory instability for fi < qc,. The maximum growth rate of that hydrodynamic instability is qc cos 0 Wi’*. Finally, we see that for 0 N quG << qut, and /&I > 1, ( 12) admits a stimulated ion Compton scattering instability whose increment is yc = w;‘*q*&,u, cos 8 x Re[q*&
+ l/(1
+x1)1-‘/*,
(16)
where XI = ( 1/q*A&) G( kocq& cos f?/uh). In summary, we have presented a kinetic theory for the modulational instability of incoherent photons which are random phased. For this purpose, we have employed the Liouville equation for the photons and developed the kinetic plasma response associated with low-frequency electrostatic modulations in the presence of the ponderomotive force of the photons. When the latter has a broad band Gaussian distribution, we obtain new classes of photon modulational instabilities. The nonlinear saturation of a modulational instability can be studied with the help of the quasilinear description, as proposed in Ref. [4]. In conclusion, we stress that the results of the present investigation will contribute to the understanding of the nonlinear propagation of random phase electromagnetic waves through various space as well as astrophysical and laboratory (inertial confinement fusion) plasmas. They are thus of rather general interest.
(14)
Physically, the instability arises because of the resonant interaction between the quasi-particles (photons) and the modified ion-acoustic waves. The mechanism is thus analogous to the Cerenkov resonance interaction. On the other hand, for ]&I >> 1, ( 13) reduces to (fl-q*c;)(.fl-qckoA,cos8)* = q”c’cf WI)cos* 8,
Letters A 237 (1998) 385-387
(15)
This work was partially supported by the Deutsche Forschungsgemeinschaft (Bonn, Germany).
References [ I ] A.A. Vedenov. A.V. Gordeev and L.I. Rudakov, Plasma Phys. 9 (1967) 719. [2] B.B. Kadomtsev, Plasma Turbulence (Academic Press, New York, 1965 ) [ 31 W.L. Kruer, Physics of Laser Plasma Interactions (AddisonWesley, Redwood City, 1980). [4] R. Bingham. J.T. Mendonp and J. M. Dawson, Phys. Rev. Len. 78 (1997) 247.