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On parametric absorption in laser plasma
Volume 78A, number 3
PHYSICS LETTERS
4 August 1980
ON PARAMETRIC ABSORPTION IN LASER PLASMA V.P. SILIN and V.T. TIKHONCHUK
Received 14 April 1980
A theory for the nonlinear saturation of convective parametric instabilities in inhomogeneous plasmas is proposed. The role of parametric processes in laser light absorption in a dense plasma corona is discussed on the basis of this theory.
The role of parametric processes in laser radiation absorption in a plasma has been vague until now. The results of the parametric turbulence theory of spatially uniform plasma have been used for estimates of these effects without any serious reasons. The question of a anomalous absorption of laser radiation in a plasma is investigated in the present report on the basis of a nonlinear theory of parametric instabilities in spatially inhomogeneous plasmas. The excitation of parametric instabilities in a real spatially inhomogeneous plasma takes place in a narrow region near the resonance line w = ad(x), kJ.= kd(x), where the decay conditions are fulfilled (w is the frequency of an excited wave, kL is the wave vector component in the plane perpendicular to the inhomogeneity axis x). The limitation of the amplitude E, 1 of the excited Langmuir wave in the instability regi’on is connected with the convection of the wave energy out of the unstable region (only for convective instabilities), or with its decay into the other Langmuir and ion-acoustic waves (for high pump field amplitudes). One can find the saturated amplitude E,,l by solving the kinetic equation for Langmuir waves. However, if the spectral width 6 w of the amplification region is small with respect to the change of the Langmuir wave frequency in every step of the secondary decay, we can use simple energy balance relations for the estimation of the nonlinear level of the saturation of the unstable wave amplitude E,,l as has been done for a spatially homogeneous plasma in ref. [ 1] . For a parametric instability in a homogeneous plasma the power Q, which is dissipated in a unit plasma 246
volume, is equal to the product of the growth rate of the instability 7 = Im w and the energy density of the excited waves in the instability region Ei I /47r: Q = -yEi I/47r (see ref. [l]). Similarly, for a convective instab’%ty the dissipated power per unit volume Q is equal to the product of the spatial growth rate $j= Im k, and the energy flux density of excited Langmuir waves ~,~Ez,~/47r: Q= &,,xE&/4n(upJ is the Langmuir wave group velocity along an inhomogeneity direction). The quasi-stationary wave distribution in the amplification region is achieved on the condition that the pumped power Q is equal to the power QI = ~1~a,xE~,21 4n, which is pumped out of the amplification region due to the secondary decay instability. (We neglect such small effects as a collisional damping of waves inside the amplification region and convection waves out of it). Here f’ is the spatial growth rate of a secondary decay instability Q + Q t s (it has a convective nature), Ea,2 is the amplitude of the Langmuir waves, arising as a result of secondary decay. If the amplitude of a pump field is large enough so that the amplification length of primary excited waves is less than the free path of the Langmuir waves in the x direction t-1 Q us,/? (7 is the damping coefficient of Langmuir waves), then the amplitudes of the primary (EQtl) and the secondary (E, 2) excited waves must be close to each other. Therefore from the energy balance condition Q = Ql in the amplification region one can obtain the equality of spatial growth rates of the primary and the secondary instabilities W,)
=
&(-%,l) .
As .$ is the known function
(1) of the amplitude
and wave
PHYSICSLETTERS
VolumeISA, number 3
vector of the excited waves, we can obtain the dependence of E,,L on g and so obtain the power Q(t), dissipated in a plasma, on a fixed pump amplitude EC,,by solving eq. (1). Such an approach permits us to analyze a wide range of phenomena, connected with nonlinear wave interaction in an inhomogeneous plasma. However, in the present report we confine ourselves only to the simple example of the normal incidence of an electromagnetic wave on a plasma slab with a characteristic scale length L in the critical density region. Since tI is a growing function of Es,L , one can see from eq. (1) that the instability with the maximum growth rate on the fixed amplitude of a pump field E. leads to a maximum absorption coefficient. In our problem such an instability is the convective ionacoustic decay t + Q+ s. Its growth rate was found in the weak pum field limit in refs. [2,3]. For stronger ! pump fields E0/4m,~ T, > 48$/o& (ys is the damping coefficient of ion-acoustic waves, oLl is the ion plasma frequency) one can find: (2) t 2: (keE& T,)l12 , where k = kd(x) is the decay wave number, K is the Boltzmann constant, T, is the electron temperature, and e is the electron charge. Because of the formal coincidence of growth rates of the primary (t + Q+ s) and the secondary (Q+ Qt s) instabilities the equality of the amplitudes E. and En,l (more exactly En,l = i E,,) follows from eq. (1). Therefore one can find the following expression for the effective collision frequency: uef = 4nQ/E; = 0.1 kv,
,
(3)
where VP is the amplitude of the oscillation velocity of an electron in the pump field. The region of validity of formula (3) is restricted from below by, the requirement of a sufficiently large amplification coefficient @x > ln Eo/Efluct (6x is the width of the amplification region, Efluct is the level of thermal fluctuations), and from above by the requirement of smallness of the amplification length as compared with the Langmuir wavelength eEofk k K T,. In the opposite case our picture of cascade energy transfer us unacceptable because a secondary aperiodic or oscillation two-stream unstability will occur directly in the amplification region. Let us note, that formula (3) for the effective colli-
sion frequency exceeds by a factor (kuPe/OLi)1’2 the well-known expression (vef z r), obtained earlier [l] in a spatially homogeneous plasma approximation. The explanation of this unexpected result is connected with the fact, that in a spatially inhomogeneous plasma the rate of pump energy absorption due to the decay t + Qt s is determined not by the product of the growth rate y1 and the energy density E$4n, but by the product of the spatial growth rate [L and the energy flux density vnrE$4n. Moreover, it is known that for a parametric instability, corresponding to excitation of a pair of different modes such an inequality is fulfilled: &uaJ > 71, where vQJ is the group velocity of the fastest wave. Formula (3) allows us to find the coefficient of the parametric absorption of laser light in a plasma (the optical plasma thickness): Xmax 7= s dx
V&t
;s
3
Xmin
where vtJ is the group velocity of the pump wave. The integration in this expression is over the whole excitation region of the t + Q+ s instability (the contribution to vef from other parametric instabilities is small). Using the well-known formulae for X,i, and xmax (see for example ref. [3]): Xmin 4xmax e $ L X In (Og/Vei) one can find: r = O-1(uP/vPe)LI&
3
(4)
where X0 is the vacuum wavelength of the heating radiation (the pump wave), bTe is the thermal electron velocity. The obtained result permits us to say that the supposition about the essential influence of the convective decay instability on laser light absorption in a plasma, which had been made earlier in refs. [3,4], is confirmed now. Even for picosecond laser pulses, according to formula (4) parametric processes can lead to absorption at a level,of about several percent under a pump wave energy flux density of ~10~~ W/cm2. Considerably large absorption can be achieved for long (nanosecond) laser pulses, providing that the scale length of the density variation L is much more than the heating radiation wavelength.
247
Volume 78A, number 3
PHYSICS LETTERS
References [l] V.Yu:Bychenkov,
V.P. Win and V.T. Tikhonchuk, Pis’ma Zh. Eksp. Teor. Fiz. 26 (1977) 309; V.P. SiIin and V.T. Tikhonchuk, Pis’ma Zh. Eksp. Teor. Fi$. 27 (1978) 504. [2] F.W. Perkins and J. Flick, Phys. Fluids 14 (197 1) 2012.
248
4 August 1980
[ 31 V. Yu. Bychenkov, A.N. Erokhin and V.P. Sihn, Kvantovajr Elektron. 6 (1979) 1399. (41 Yu.V. Afanas’ev et al., Vzaimodeistvie moshchnogo lazerno izluchenija s plasmoj (Interaction of powerful laser radiation with plasmas), VINITI Ser. Radiotechnique 17 (1978) p 5-2.
PHYSICS LETTERS
4 August 1980
ON PARAMETRIC ABSORPTION IN LASER PLASMA V.P. SILIN and V.T. TIKHONCHUK
Received 14 April 1980
A theory for the nonlinear saturation of convective parametric instabilities in inhomogeneous plasmas is proposed. The role of parametric processes in laser light absorption in a dense plasma corona is discussed on the basis of this theory.
The role of parametric processes in laser radiation absorption in a plasma has been vague until now. The results of the parametric turbulence theory of spatially uniform plasma have been used for estimates of these effects without any serious reasons. The question of a anomalous absorption of laser radiation in a plasma is investigated in the present report on the basis of a nonlinear theory of parametric instabilities in spatially inhomogeneous plasmas. The excitation of parametric instabilities in a real spatially inhomogeneous plasma takes place in a narrow region near the resonance line w = ad(x), kJ.= kd(x), where the decay conditions are fulfilled (w is the frequency of an excited wave, kL is the wave vector component in the plane perpendicular to the inhomogeneity axis x). The limitation of the amplitude E, 1 of the excited Langmuir wave in the instability regi’on is connected with the convection of the wave energy out of the unstable region (only for convective instabilities), or with its decay into the other Langmuir and ion-acoustic waves (for high pump field amplitudes). One can find the saturated amplitude E,,l by solving the kinetic equation for Langmuir waves. However, if the spectral width 6 w of the amplification region is small with respect to the change of the Langmuir wave frequency in every step of the secondary decay, we can use simple energy balance relations for the estimation of the nonlinear level of the saturation of the unstable wave amplitude E,,l as has been done for a spatially homogeneous plasma in ref. [ 1] . For a parametric instability in a homogeneous plasma the power Q, which is dissipated in a unit plasma 246
volume, is equal to the product of the growth rate of the instability 7 = Im w and the energy density of the excited waves in the instability region Ei I /47r: Q = -yEi I/47r (see ref. [l]). Similarly, for a convective instab’%ty the dissipated power per unit volume Q is equal to the product of the spatial growth rate $j= Im k, and the energy flux density of excited Langmuir waves ~,~Ez,~/47r: Q= &,,xE&/4n(upJ is the Langmuir wave group velocity along an inhomogeneity direction). The quasi-stationary wave distribution in the amplification region is achieved on the condition that the pumped power Q is equal to the power QI = ~1~a,xE~,21 4n, which is pumped out of the amplification region due to the secondary decay instability. (We neglect such small effects as a collisional damping of waves inside the amplification region and convection waves out of it). Here f’ is the spatial growth rate of a secondary decay instability Q + Q t s (it has a convective nature), Ea,2 is the amplitude of the Langmuir waves, arising as a result of secondary decay. If the amplitude of a pump field is large enough so that the amplification length of primary excited waves is less than the free path of the Langmuir waves in the x direction t-1 Q us,/? (7 is the damping coefficient of Langmuir waves), then the amplitudes of the primary (EQtl) and the secondary (E, 2) excited waves must be close to each other. Therefore from the energy balance condition Q = Ql in the amplification region one can obtain the equality of spatial growth rates of the primary and the secondary instabilities W,)
=
&(-%,l) .
As .$ is the known function
(1) of the amplitude
and wave
PHYSICSLETTERS
VolumeISA, number 3
vector of the excited waves, we can obtain the dependence of E,,L on g and so obtain the power Q(t), dissipated in a plasma, on a fixed pump amplitude EC,,by solving eq. (1). Such an approach permits us to analyze a wide range of phenomena, connected with nonlinear wave interaction in an inhomogeneous plasma. However, in the present report we confine ourselves only to the simple example of the normal incidence of an electromagnetic wave on a plasma slab with a characteristic scale length L in the critical density region. Since tI is a growing function of Es,L , one can see from eq. (1) that the instability with the maximum growth rate on the fixed amplitude of a pump field E. leads to a maximum absorption coefficient. In our problem such an instability is the convective ionacoustic decay t + Q+ s. Its growth rate was found in the weak pum field limit in refs. [2,3]. For stronger ! pump fields E0/4m,~ T, > 48$/o& (ys is the damping coefficient of ion-acoustic waves, oLl is the ion plasma frequency) one can find: (2) t 2: (keE& T,)l12 , where k = kd(x) is the decay wave number, K is the Boltzmann constant, T, is the electron temperature, and e is the electron charge. Because of the formal coincidence of growth rates of the primary (t + Q+ s) and the secondary (Q+ Qt s) instabilities the equality of the amplitudes E. and En,l (more exactly En,l = i E,,) follows from eq. (1). Therefore one can find the following expression for the effective collision frequency: uef = 4nQ/E; = 0.1 kv,
,
(3)
where VP is the amplitude of the oscillation velocity of an electron in the pump field. The region of validity of formula (3) is restricted from below by, the requirement of a sufficiently large amplification coefficient @x > ln Eo/Efluct (6x is the width of the amplification region, Efluct is the level of thermal fluctuations), and from above by the requirement of smallness of the amplification length as compared with the Langmuir wavelength eEofk k K T,. In the opposite case our picture of cascade energy transfer us unacceptable because a secondary aperiodic or oscillation two-stream unstability will occur directly in the amplification region. Let us note, that formula (3) for the effective colli-
sion frequency exceeds by a factor (kuPe/OLi)1’2 the well-known expression (vef z r), obtained earlier [l] in a spatially homogeneous plasma approximation. The explanation of this unexpected result is connected with the fact, that in a spatially inhomogeneous plasma the rate of pump energy absorption due to the decay t + Qt s is determined not by the product of the growth rate y1 and the energy density E$4n, but by the product of the spatial growth rate [L and the energy flux density vnrE$4n. Moreover, it is known that for a parametric instability, corresponding to excitation of a pair of different modes such an inequality is fulfilled: &uaJ > 71, where vQJ is the group velocity of the fastest wave. Formula (3) allows us to find the coefficient of the parametric absorption of laser light in a plasma (the optical plasma thickness): Xmax 7= s dx
V&t
;s
3
Xmin
where vtJ is the group velocity of the pump wave. The integration in this expression is over the whole excitation region of the t + Q+ s instability (the contribution to vef from other parametric instabilities is small). Using the well-known formulae for X,i, and xmax (see for example ref. [3]): Xmin 4xmax e $ L X In (Og/Vei) one can find: r = O-1(uP/vPe)LI&
3
(4)
where X0 is the vacuum wavelength of the heating radiation (the pump wave), bTe is the thermal electron velocity. The obtained result permits us to say that the supposition about the essential influence of the convective decay instability on laser light absorption in a plasma, which had been made earlier in refs. [3,4], is confirmed now. Even for picosecond laser pulses, according to formula (4) parametric processes can lead to absorption at a level,of about several percent under a pump wave energy flux density of ~10~~ W/cm2. Considerably large absorption can be achieved for long (nanosecond) laser pulses, providing that the scale length of the density variation L is much more than the heating radiation wavelength.
247
Volume 78A, number 3
PHYSICS LETTERS
References [l] V.Yu:Bychenkov,
V.P. Win and V.T. Tikhonchuk, Pis’ma Zh. Eksp. Teor. Fiz. 26 (1977) 309; V.P. SiIin and V.T. Tikhonchuk, Pis’ma Zh. Eksp. Teor. Fi$. 27 (1978) 504. [2] F.W. Perkins and J. Flick, Phys. Fluids 14 (197 1) 2012.
248
4 August 1980
[ 31 V. Yu. Bychenkov, A.N. Erokhin and V.P. Sihn, Kvantovajr Elektron. 6 (1979) 1399. (41 Yu.V. Afanas’ev et al., Vzaimodeistvie moshchnogo lazerno izluchenija s plasmoj (Interaction of powerful laser radiation with plasmas), VINITI Ser. Radiotechnique 17 (1978) p 5-2.