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Investigation of the effect of laser pulse length on the inverse bremsstrahlung absorption in laser–fusion plasma
High Energy Density Physics 16 (2015) 32e35
Contents lists available at ScienceDirect
High Energy Density Physics journal homepage: www.elsevier.com/locate/hedp
Investigation of the effect of laser pulse length on the inverse bremsstrahlung absorption in laserefusion plasma N. Firouzi Farrashbandi a, L. Gholamzadeh a, *, M. Eslami-Kalantari a, M. Sharifian b, A. Sid c a
Nuclear Physics Group, Faculty of Physics, Yazd University, Yazd, Iran Atomic Molecular Group, Faculty of Physics, Yazd University, Yazd, Iran c Laboratoire de physique des rayonnements et de leurs interactions avec la mati ere, Facult e des sciences, Universit e de Batna, Algeria b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 January 2015 Received in revised form 7 March 2015 Accepted 13 May 2015 Available online 23 June 2015
One of the most important aspects in laser fusion is optimizing of the laser energy in the fusion target. Using the kinetic theory, the inverse bremsstrahlung absorption has been investigated in the homogeneous hot plasmas. A pulsed laser is considered instead of a continuous laser and effect of the physical parameters such as pulse length, wavelength and temperature have been studied on the absorption value. In our calculations, the q-nonextensive distribution function (f0) is used for the electrons distribution. It has been shown that the absorption increases for both long pulse length, tL, and laser short wavelengths, and decreases with increasing of temperature. © 2015 Elsevier B.V. All rights reserved.
Keywords: Laserefusion plasma The inverse bremsstrahlung absorption The effect of laser pulse length
1. Introduction The inverse bremsstrahlung is the process in which an electron absorbs a photon while colliding with an ion or with another electron [1]. The inverse bremsstrahlung absorption (IBA) is an essential mechanism for coupling laser energy to the plasmas [2]. So the optimizing of laser absorption is very important. The linear and nonlinear approaches of IBA have largely been studied. Inverse bremsstrahlung absorption and evolution of the electron distribution function (EDF) studied in Refs. [3,4] and in the other work, collisional absorption was investigated with the quantum statistical methods [5]. Also the absorption was performed with simulation [6e8]. For high frequency laser fields, the quantum statistical expressions for absorption rate in terms of the Lindhard dielectric function are derived by Bornath [9]. IBA coefficient is determined using the Gaunt factor [10,11]. In earlier all studies, a non-pulsed laser is used to calculate of the IBA. A one-dimensional Lagrangian hydrodynamics code was used to show the absorption for long pulses, low intensities, and short wavelengths which favour inverse bremsstrahlung absorption [12]. Absorption using kinetic theory, in many reports has been taken into account [13e16].
* Corresponding author. E-mail address: [email protected] (L. Gholamzadeh). http://dx.doi.org/10.1016/j.hedp.2015.05.002 1574-1818/© 2015 Elsevier B.V. All rights reserved.
Here, IBA using kinetic theory is investigated in un-magnetized and homogeneous hot plasma that produced with circular polarized electric field. The q-nonextensive distribution function is used for the electron distribution. Our results show a further increase of absorption for long laser pulse length, low temperature and short wavelengths. Our calculations are simplified by high frequency approximation ðCee ≪Cei Þ and the Laurentz approximation ðnei ≪uL Þ [16]. In the next section in order to calculate the first anisotropy of electronic distribution function, f1, the basic equations have been established. In section three absorption calculations are discussed and related diagrams are shown. In fourth section, we present our results and discussion. Finally we end with the conclusions. 2. Theoretical model for calculating of f1 We used kinetic theory to calculate of IBA and electron FokkerePlanck (FP) equation for homogeneous un-magnetized plasma as [17]:
! ! vf e E vf v v2~I ! v! v vf $ $ ! þ Cee ðf Þ: ¼ A !$ vt me v! v3 v vv vv
(1)
Av=v! v $ðv2~I ! v Þ≡Cei ðf Þ and Cee(f), are electrov! v =v3 vf =v! neion (eei) and electroneelectron (eee) collisional operators. In the other hand A ¼ ð2pne Ze4 =m2e Þln L where, e, me and ne are the
N. Firouzi Farrashbandi et al. / High Energy Density Physics 16 (2015) 32e35
charge, mass and density of electron, respectively. Z is the ion charge number and ln L is the Coulomb logarithm [2]. f is the ! electron distribution function, E and ! v are the laser electric field and particle velocity, respectively. If f contains a slow-frequency part, fs, and a high-frequency part, h f , which oscillates at the laser frequency, uL, as [16]:
! ! fð r ;! v ; tÞ ¼ f ðsÞ ð r ; ! v ; tÞ þ f ðhÞ ð! v ÞeiuL t :
(2)
Expanding of fs, fh, on the Legendre polynomials, and considering of f at the first order, we get:
f ðv; m; 4; tÞ ¼ f0 ðvÞ þ mf1 ðv; 4; tÞ:
(3)
Using Laurentz approximation ðCee ≪Cei Þ and equations (1) and (3), the expansion in Legendre function simplifies the collision operator [17]:
vf1 vt
3. IBA absorption calculation ! The electric current in the plasma is J ðx; tÞ ¼ ene ! v ðx; tÞ and R 3! ne ¼ fd v . In the classical limit of a hot plasma ðħuL ≪Te Þ, the quantum and classical descriptions of lasereplasma interaction ! ! agree to each other, and absorption, Ab ¼ 〈 E $ J 〉 is expressed as
0 * 1 2q ! ! Z∞ 2 v4 2 q1 4pA e E ðx;tÞ$ E ðx;tÞ m v q e 1ðq1Þ Ab ¼ Re@ dvA; kB Te 2kB Te 2A 3 iu þ 3 L 0 v (9) !* ! where E ðx; tÞ is the complex conjugate of E ðx; tÞ. According to integration solution of Ab, we choose Aq with q > 1, therefore:
32p2 n2e Ze6 3u2L
Ab ¼
!! eE vf0 2A ¼ 3 f1 ; $ ! me v v v
(4)
f0 ðvÞ ¼ Aq
# 1 me v2 q1 ; 1 ðq 1Þ 2kB Te
(5)
1 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 G þ 1 þ q me ðq 1Þ 2 q 1 > > > q1 ne > > 1 2 2pkB Te > > G > < q1 ; Aq ¼ > 1 > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s > > G > > me ð1 qÞ 1q > > 1
! E ðx; tÞ ¼ E0 ðxÞexpðiuL tÞsin2
t ðb p y þ ib z Þ; tL
2A iuL f1 þ 3 f1 ¼ v eE0 eiðuL f1 ¼
eE0 eiuL
t
me
t tL
1 G 12 þ q1 1þq 2 1 G q1
(10) Using Maxwell equations and equation of motion for the electrons, we calculated E0:
(6)
! v! v ðx; tÞ e E ðx; tÞ ¼ nei ! v ðx; tÞ: vt me
(11)
(12)
The wave equation is derived with high frequency approximation, nei ≪uL as:
(7)
d2 E0 u2L þ 2 dx2 c
tþ4Þ
(8)
1þ
inei x in u2 E ¼ ei 2L E0 ; uL L 0 uL c
! 1 1 E 0 ðxÞ ¼ 2p2 r6 En expðsÞAiryðxÞ;
(13)
(14)
where 2
x ¼ r3
x þ is ; Ln
(15)
and
s¼
vy vz vf0 þi v v vv
where nei and L are electroneion collision frequency and distance of critical layer to vacuum, respectively. That this equation solved by the Airy-Hora's function [20,21] as follows:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2 p ttL 1 m2 vf0 : vv iuL þ 2A me v3
ln LE02 sin4 p
and
where tL, E0 and uL are laser pulse length, electric amplitude and laser frequency, respectively. Using spherical coordinates, (v, m ¼ vx/ v, 4 ¼ arctan vz/vy), and according to Eqs. (4), (5) and (7) relations, we will have f1 function as:
sin2 p ttL
! ! v E ðx; tÞ ! ! ; V B ðx; tÞ ¼ m0 J ðx; tÞ þ m0 30 vt
where G, kB and Te stand for the Gamma function, the Boltzmann constant and the electron temperature respectively. When pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi vmax ¼ 2kB Te =mðq 1Þ, in the limit of q / 1, goes to infinity and Eq. (6) reduces to the standard one-dimensional MaxwelleBoltzmann normalization constant. We start with pulsed laser circular-polarized electric field as:
!
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðq 1Þ : 2pkB Te m5e
where f0 is considered q-nonextensive distribution function [18,19]:
"
33
nei ; uL
r¼
Ln uL ; c
(16)
where En stand for the laser electric field magnitude in vacuum and c is the light speed. Ln z CstL is electron density gradient, whereas Cs and tL are the ion acoustic speed and the laser pulse length, respectively. If we calculate the mean absorption during a laser pulse length, we get:
34
Ab ¼
N. Firouzi Farrashbandi et al. / High Energy Density Physics 16 (2015) 32e35
32p2 Zn2e e6 8u2L
!
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 1 G 12 þ q1 1 1 L n uL 6 1þq ðq 1Þ 2 En expðsÞAiryðxÞ 2p ln L: 5 2 c 2pk B Te me 1 G q1
This equation expresses the absorption value in terms of laser pulse length, electron temperature and laser wavelength.
4. Results and discussion Using FokkerePlanck equation, we have derived the absorption relation for unmagnetized and homogeneous plasma for pulsed lasers and q-nonextensive distribution function. Here we have analytically calculated the effects of plasma and laser parameters such as laser pulse length, laser wavelength and temperature on the absorption. We plot the absorption as a function of x/Ln for different tL. The region of x/Ln > 0 shows the region of under dense plasma where the electron density is less than the critical density (ne < nc) and x/ Ln < 0 corresponds with the region of over dense plasma. Fig. 1 shows, for tL ¼ 0.5 ns, around critical layer (x ¼ 0) IBA is maximum and using Eq. (17), its value is about 1.2 1013 W/m3.
(17)
When we increase tL from 1 ns to 4 ns we will reach to Fig. 2 that is depicts larger absorption for larger pulse length in near of critical layer. For tL ¼ 4 ns, the absorption in critical layer is about 2.6 1013 W/m3. The absorption decreases away from the critical layer (x/Ln > 0). According to Fig. 2 the IBA value is maximum in near of critical layer, so comparison of absorption values for tL ¼ 1 ns to tL ¼ 4 ns in critical layer shows the transmission of laser energy by the electron of plasmaefusion laser will be maximum. So, if we draw absorption as function of tL (Fig. 3), we can see the effect of laser pulse length on the absorption directly. We consider critical layer and then calculate the absorption in this layer. As tL varies from 0.5 ns to 4 ns in critical layer, if the laser electric field magnitude and the laser wavelength value are 1015 V/m and 10.6 mm respectively, then we see that the absorption increases from 3.5 1028 W/m3 for tL ¼ 0.5 ns to 7 1028 W/m3 for tL ¼ 4 ns (Eq. (17)). In Fig. 4 we show the effect of laser wavelength on the absorption. This figure also depicts that the reduction of the
Fig. 1. Absorption as a function of x/Ln. In near of the critical layer (x ¼ 0) absorption is maximum. The laser and plasma parameters are: Z ¼ 4, q ¼ 2, Te ¼ 10 keV, tL ¼ 0.5 ns, lL ¼ 10.6 mm.
Fig. 3. Absorption versus tL in critical layer.
Fig. 2. Absorption versus function of x/Ln. The laser and plasma parameters are: Z ¼ 4, q ¼ 2, Te ¼ 10 keV, tL ¼ 1 ns to tL ¼ 4 ns, lL ¼ 10.6 mm.
Fig. 4. Absorption as a function of x/Ln for wavelengths of lL ¼ 0.353 mm, 0.53 mm and 10.6 mm, in tL ¼ 0.5 ns.
N. Firouzi Farrashbandi et al. / High Energy Density Physics 16 (2015) 32e35
35
5. Conclusions Using kinetic theory and FokkerePlanck equation in laserefusion plasma, an anisotropic distribution function, electrical current density and the absorption are obtained. We plotted the absorption as a function of laser pulse length, for tL ¼ 0.5e4 ns. The calculations showed that in near of critical layer, absorption has a maximum value. Also we investigated effect of laser wavelength and electron temperature on absorption. We used three wavelengths that they usually are used in inertial confinement fusion experiments. For shorter wavelengths, the absorption is higher and also if electron temperature increases then absorption decreases.
Fig. 5. Plot of absorption as function of electron temperature for tL ¼ 0.5 ns.
Fig. 6. Compare of absorption as a function of Te for tL ¼ 0.5 ns and tL ¼ 1 ns.
absorption is due to increasing of wavelength. As we can see, for the constant laser pulse length (tL ¼ 0.5 ns), using Eq. (17), the absorption value in the critical layer for lL ¼ 0.353 mm is about 7 1028 W/m3 while it is around 2.25 1025 W/m3 for lL ¼ 10.6 mm. The laser wavelength effect on the absorption is considered in Fig. 4. The picture shows that the absorption will increase with the shortness of wavelength. Fig. 5 describes absorption as a function of electron temperature for the constants laser pulse length, tL ¼ 0.5 ns. As we can see, increasing of temperature cause the decreasing of absorption. To compare absorption as function of Te for the various of tL ¼ 0.5 ns and tL ¼ 1 ns, we will have Fig. 6. Calculations show that the changing of absorption for the long pulse length is faster than short one while for the both of them the absorption decreases as electron temperature increases.
References [1] S. Pfalzner, An Introduction to Inertial Confinement Fusion, Taylor & Francis/ CRC Press, 2006. [2] S. Eliezer, The Interaction of High-Power Lasers with Plasmas, IOP Publishing Ltd, 2002. [3] S.-M. Weng, et al., Inverse bremsstrahlung absorption and the evolution of electron distributions accounting for electroneelectron collisions, Phys. Plasmas 13 (11) (2006) 113302. [4] S.-M. Weng, Z.-M. Sheng, J. Zhang, Inverse bremsstrahlung absorption with nonlinear effects of high laser intensity and non-Maxwellian distribution, Phys. Rev. E 80 (5) (2009) 056406. [5] P. Hilse, et al., Collisional absorption of dense plasmas in strong laser fields: quantum statistical results and simulation, Phys. Rev. E 71 (5) (2005) 056408. [6] S. Pfalzner, P. Gibbon, Direct calculation of inverse-bremsstrahlung absorption in strongly coupled, nonlinearly driven laser plasmas, Phys. Rev. E 57 (4) (1998) 4698e4705. [7] S.-M. Weng, Z.-M. Sheng, J. Zhang, Plasma currents and inverse bremsstrahlung absorption under strong dc/ac electric fields, J. Phys. Conf. Ser. 244 (2010) 022072. IOP Publishing. [8] H. Schmitz, R. Lloyd, R.G. Evans, Collisional particle-in-cell modelling of the generation and control of relativistic electron beams produced by ultraintense laser pulses, Plasma Phys. Control. Fusion 54 (8) (2012) 085016. [9] T. Bornath, et al., Nonlinear collisional absorption in dense laser plasmas, Phys. Rev. E 64 (2) (2001) 026414. [10] J. Stallcop, K. Billman, Analytical formulae for the inverse bremsstrahlung absorption coefficient, Plasma Phys. 16 (12) (1974) 1187. [11] A. Wierling, et al., Inverse bremsstrahlung of hot, weakly coupled plasmas, Phys. Plasmas 8 (2001) 3810. [12] C. Garban-Labaune, et al., Effect of laser wavelength and pulse duration on laser-light absorption and back reflection, Phys. Rev. Lett. 48 (15) (1982) 1018. [13] P. Porshnev, et al., Nonlinear inverse bremsstrahlung and highly anisotropic electron distributions, Phys. Rev. E 53 (1) (1996) 1100. [14] A. Bendib, J. Luciani, Collisional effects and dispersion relation of magnetic field structures, Phys. Fluids (1958e1988) 30 (5) (1987) 1353e1361. [15] A. Bendib, K. Bendib, A. Sid, Weibel instability due to inverse bremsstrahlung absorption, Phys. Rev. E 55 (6) (1997) 7522. [16] A. Sid, Nonlinear inverse bremsstrahlung absorption in laserefusion plasma corona, Phys. Plasmas 10 (2003) 214. [17] W.L. Kruer, The Physics of Laser Plasma Interactions, vol. 70, Addison-Wesley, New York, 1988. [18] J. Lima, R. Silva Jr., J. Santos, Plasma oscillations and nonextensive statistics, Phys. Rev. E 61 (3) (2000) 3260. [19] M. Sharifian, et al., Effect of q-non-extensive distribution of electrons on the plasma sheath floating potential, J. Plasma Phys. (2013) 1e12. [20] H. Hora, Laser Plasmas and Nuclear Energy, vol. 1, Plenum Press, New York, 1975. [21] M. Abramowitz, I.A. Stegun, in: Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, Courier Dover Publications, 1972.
Contents lists available at ScienceDirect
High Energy Density Physics journal homepage: www.elsevier.com/locate/hedp
Investigation of the effect of laser pulse length on the inverse bremsstrahlung absorption in laserefusion plasma N. Firouzi Farrashbandi a, L. Gholamzadeh a, *, M. Eslami-Kalantari a, M. Sharifian b, A. Sid c a
Nuclear Physics Group, Faculty of Physics, Yazd University, Yazd, Iran Atomic Molecular Group, Faculty of Physics, Yazd University, Yazd, Iran c Laboratoire de physique des rayonnements et de leurs interactions avec la mati ere, Facult e des sciences, Universit e de Batna, Algeria b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 14 January 2015 Received in revised form 7 March 2015 Accepted 13 May 2015 Available online 23 June 2015
One of the most important aspects in laser fusion is optimizing of the laser energy in the fusion target. Using the kinetic theory, the inverse bremsstrahlung absorption has been investigated in the homogeneous hot plasmas. A pulsed laser is considered instead of a continuous laser and effect of the physical parameters such as pulse length, wavelength and temperature have been studied on the absorption value. In our calculations, the q-nonextensive distribution function (f0) is used for the electrons distribution. It has been shown that the absorption increases for both long pulse length, tL, and laser short wavelengths, and decreases with increasing of temperature. © 2015 Elsevier B.V. All rights reserved.
Keywords: Laserefusion plasma The inverse bremsstrahlung absorption The effect of laser pulse length
1. Introduction The inverse bremsstrahlung is the process in which an electron absorbs a photon while colliding with an ion or with another electron [1]. The inverse bremsstrahlung absorption (IBA) is an essential mechanism for coupling laser energy to the plasmas [2]. So the optimizing of laser absorption is very important. The linear and nonlinear approaches of IBA have largely been studied. Inverse bremsstrahlung absorption and evolution of the electron distribution function (EDF) studied in Refs. [3,4] and in the other work, collisional absorption was investigated with the quantum statistical methods [5]. Also the absorption was performed with simulation [6e8]. For high frequency laser fields, the quantum statistical expressions for absorption rate in terms of the Lindhard dielectric function are derived by Bornath [9]. IBA coefficient is determined using the Gaunt factor [10,11]. In earlier all studies, a non-pulsed laser is used to calculate of the IBA. A one-dimensional Lagrangian hydrodynamics code was used to show the absorption for long pulses, low intensities, and short wavelengths which favour inverse bremsstrahlung absorption [12]. Absorption using kinetic theory, in many reports has been taken into account [13e16].
* Corresponding author. E-mail address: [email protected] (L. Gholamzadeh). http://dx.doi.org/10.1016/j.hedp.2015.05.002 1574-1818/© 2015 Elsevier B.V. All rights reserved.
Here, IBA using kinetic theory is investigated in un-magnetized and homogeneous hot plasma that produced with circular polarized electric field. The q-nonextensive distribution function is used for the electron distribution. Our results show a further increase of absorption for long laser pulse length, low temperature and short wavelengths. Our calculations are simplified by high frequency approximation ðCee ≪Cei Þ and the Laurentz approximation ðnei ≪uL Þ [16]. In the next section in order to calculate the first anisotropy of electronic distribution function, f1, the basic equations have been established. In section three absorption calculations are discussed and related diagrams are shown. In fourth section, we present our results and discussion. Finally we end with the conclusions. 2. Theoretical model for calculating of f1 We used kinetic theory to calculate of IBA and electron FokkerePlanck (FP) equation for homogeneous un-magnetized plasma as [17]:
! ! vf e E vf v v2~I ! v! v vf $ $ ! þ Cee ðf Þ: ¼ A !$ vt me v! v3 v vv vv
(1)
Av=v! v $ðv2~I ! v Þ≡Cei ðf Þ and Cee(f), are electrov! v =v3 vf =v! neion (eei) and electroneelectron (eee) collisional operators. In the other hand A ¼ ð2pne Ze4 =m2e Þln L where, e, me and ne are the
N. Firouzi Farrashbandi et al. / High Energy Density Physics 16 (2015) 32e35
charge, mass and density of electron, respectively. Z is the ion charge number and ln L is the Coulomb logarithm [2]. f is the ! electron distribution function, E and ! v are the laser electric field and particle velocity, respectively. If f contains a slow-frequency part, fs, and a high-frequency part, h f , which oscillates at the laser frequency, uL, as [16]:
! ! fð r ;! v ; tÞ ¼ f ðsÞ ð r ; ! v ; tÞ þ f ðhÞ ð! v ÞeiuL t :
(2)
Expanding of fs, fh, on the Legendre polynomials, and considering of f at the first order, we get:
f ðv; m; 4; tÞ ¼ f0 ðvÞ þ mf1 ðv; 4; tÞ:
(3)
Using Laurentz approximation ðCee ≪Cei Þ and equations (1) and (3), the expansion in Legendre function simplifies the collision operator [17]:
vf1 vt
3. IBA absorption calculation ! The electric current in the plasma is J ðx; tÞ ¼ ene ! v ðx; tÞ and R 3! ne ¼ fd v . In the classical limit of a hot plasma ðħuL ≪Te Þ, the quantum and classical descriptions of lasereplasma interaction ! ! agree to each other, and absorption, Ab ¼ 〈 E $ J 〉 is expressed as
0 * 1 2q ! ! Z∞ 2 v4 2 q1 4pA e E ðx;tÞ$ E ðx;tÞ m v q e 1ðq1Þ Ab ¼ Re@ dvA; kB Te 2kB Te 2A 3 iu þ 3 L 0 v (9) !* ! where E ðx; tÞ is the complex conjugate of E ðx; tÞ. According to integration solution of Ab, we choose Aq with q > 1, therefore:
32p2 n2e Ze6 3u2L
Ab ¼
!! eE vf0 2A ¼ 3 f1 ; $ ! me v v v
(4)
f0 ðvÞ ¼ Aq
# 1 me v2 q1 ; 1 ðq 1Þ 2kB Te
(5)
1 1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 G þ 1 þ q me ðq 1Þ 2 q 1 > > > q1 ne > > 1 2 2pkB Te > > G > < q1 ; Aq ¼ > 1 > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s > > G > > me ð1 qÞ 1q > > 1
! E ðx; tÞ ¼ E0 ðxÞexpðiuL tÞsin2
t ðb p y þ ib z Þ; tL
2A iuL f1 þ 3 f1 ¼ v eE0 eiðuL f1 ¼
eE0 eiuL
t
me
t tL
1 G 12 þ q1 1þq 2 1 G q1
(10) Using Maxwell equations and equation of motion for the electrons, we calculated E0:
(6)
! v! v ðx; tÞ e E ðx; tÞ ¼ nei ! v ðx; tÞ: vt me
(11)
(12)
The wave equation is derived with high frequency approximation, nei ≪uL as:
(7)
d2 E0 u2L þ 2 dx2 c
tþ4Þ
(8)
1þ
inei x in u2 E ¼ ei 2L E0 ; uL L 0 uL c
! 1 1 E 0 ðxÞ ¼ 2p2 r6 En expðsÞAiryðxÞ;
(13)
(14)
where 2
x ¼ r3
x þ is ; Ln
(15)
and
s¼
vy vz vf0 þi v v vv
where nei and L are electroneion collision frequency and distance of critical layer to vacuum, respectively. That this equation solved by the Airy-Hora's function [20,21] as follows:
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi sin2 p ttL 1 m2 vf0 : vv iuL þ 2A me v3
ln LE02 sin4 p
and
where tL, E0 and uL are laser pulse length, electric amplitude and laser frequency, respectively. Using spherical coordinates, (v, m ¼ vx/ v, 4 ¼ arctan vz/vy), and according to Eqs. (4), (5) and (7) relations, we will have f1 function as:
sin2 p ttL
! ! v E ðx; tÞ ! ! ; V B ðx; tÞ ¼ m0 J ðx; tÞ þ m0 30 vt
where G, kB and Te stand for the Gamma function, the Boltzmann constant and the electron temperature respectively. When pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi vmax ¼ 2kB Te =mðq 1Þ, in the limit of q / 1, goes to infinity and Eq. (6) reduces to the standard one-dimensional MaxwelleBoltzmann normalization constant. We start with pulsed laser circular-polarized electric field as:
!
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðq 1Þ : 2pkB Te m5e
where f0 is considered q-nonextensive distribution function [18,19]:
"
33
nei ; uL
r¼
Ln uL ; c
(16)
where En stand for the laser electric field magnitude in vacuum and c is the light speed. Ln z CstL is electron density gradient, whereas Cs and tL are the ion acoustic speed and the laser pulse length, respectively. If we calculate the mean absorption during a laser pulse length, we get:
34
Ab ¼
N. Firouzi Farrashbandi et al. / High Energy Density Physics 16 (2015) 32e35
32p2 Zn2e e6 8u2L
!
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !2 1 G 12 þ q1 1 1 L n uL 6 1þq ðq 1Þ 2 En expðsÞAiryðxÞ 2p ln L: 5 2 c 2pk B Te me 1 G q1
This equation expresses the absorption value in terms of laser pulse length, electron temperature and laser wavelength.
4. Results and discussion Using FokkerePlanck equation, we have derived the absorption relation for unmagnetized and homogeneous plasma for pulsed lasers and q-nonextensive distribution function. Here we have analytically calculated the effects of plasma and laser parameters such as laser pulse length, laser wavelength and temperature on the absorption. We plot the absorption as a function of x/Ln for different tL. The region of x/Ln > 0 shows the region of under dense plasma where the electron density is less than the critical density (ne < nc) and x/ Ln < 0 corresponds with the region of over dense plasma. Fig. 1 shows, for tL ¼ 0.5 ns, around critical layer (x ¼ 0) IBA is maximum and using Eq. (17), its value is about 1.2 1013 W/m3.
(17)
When we increase tL from 1 ns to 4 ns we will reach to Fig. 2 that is depicts larger absorption for larger pulse length in near of critical layer. For tL ¼ 4 ns, the absorption in critical layer is about 2.6 1013 W/m3. The absorption decreases away from the critical layer (x/Ln > 0). According to Fig. 2 the IBA value is maximum in near of critical layer, so comparison of absorption values for tL ¼ 1 ns to tL ¼ 4 ns in critical layer shows the transmission of laser energy by the electron of plasmaefusion laser will be maximum. So, if we draw absorption as function of tL (Fig. 3), we can see the effect of laser pulse length on the absorption directly. We consider critical layer and then calculate the absorption in this layer. As tL varies from 0.5 ns to 4 ns in critical layer, if the laser electric field magnitude and the laser wavelength value are 1015 V/m and 10.6 mm respectively, then we see that the absorption increases from 3.5 1028 W/m3 for tL ¼ 0.5 ns to 7 1028 W/m3 for tL ¼ 4 ns (Eq. (17)). In Fig. 4 we show the effect of laser wavelength on the absorption. This figure also depicts that the reduction of the
Fig. 1. Absorption as a function of x/Ln. In near of the critical layer (x ¼ 0) absorption is maximum. The laser and plasma parameters are: Z ¼ 4, q ¼ 2, Te ¼ 10 keV, tL ¼ 0.5 ns, lL ¼ 10.6 mm.
Fig. 3. Absorption versus tL in critical layer.
Fig. 2. Absorption versus function of x/Ln. The laser and plasma parameters are: Z ¼ 4, q ¼ 2, Te ¼ 10 keV, tL ¼ 1 ns to tL ¼ 4 ns, lL ¼ 10.6 mm.
Fig. 4. Absorption as a function of x/Ln for wavelengths of lL ¼ 0.353 mm, 0.53 mm and 10.6 mm, in tL ¼ 0.5 ns.
N. Firouzi Farrashbandi et al. / High Energy Density Physics 16 (2015) 32e35
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5. Conclusions Using kinetic theory and FokkerePlanck equation in laserefusion plasma, an anisotropic distribution function, electrical current density and the absorption are obtained. We plotted the absorption as a function of laser pulse length, for tL ¼ 0.5e4 ns. The calculations showed that in near of critical layer, absorption has a maximum value. Also we investigated effect of laser wavelength and electron temperature on absorption. We used three wavelengths that they usually are used in inertial confinement fusion experiments. For shorter wavelengths, the absorption is higher and also if electron temperature increases then absorption decreases.
Fig. 5. Plot of absorption as function of electron temperature for tL ¼ 0.5 ns.
Fig. 6. Compare of absorption as a function of Te for tL ¼ 0.5 ns and tL ¼ 1 ns.
absorption is due to increasing of wavelength. As we can see, for the constant laser pulse length (tL ¼ 0.5 ns), using Eq. (17), the absorption value in the critical layer for lL ¼ 0.353 mm is about 7 1028 W/m3 while it is around 2.25 1025 W/m3 for lL ¼ 10.6 mm. The laser wavelength effect on the absorption is considered in Fig. 4. The picture shows that the absorption will increase with the shortness of wavelength. Fig. 5 describes absorption as a function of electron temperature for the constants laser pulse length, tL ¼ 0.5 ns. As we can see, increasing of temperature cause the decreasing of absorption. To compare absorption as function of Te for the various of tL ¼ 0.5 ns and tL ¼ 1 ns, we will have Fig. 6. Calculations show that the changing of absorption for the long pulse length is faster than short one while for the both of them the absorption decreases as electron temperature increases.
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