Nonlinear absorption of ultrapower laser radiation by relativistic underdense plasma
NNonlinear absorption of ultrapower laser radiationby relativistic underdense plasma
H K Avetissian, A G Ghazaryan and G F Mkrtchian
Centre of Strong Fields Physics, Yerevan State University, 1 A. Manukian, Yerevan0025, ArmeniaE-mail: [email protected]
Abstract.
The nonlinear absorption of laser radiation of relativistic intensities in theunderdense plasma by a mechanism of stimulated bremsstrahlung of electrons on theions/nuclei is investigated in the low frequency approximation. Coefficient of nonlinearinverse-bremsstrahlung absorption is studied for relativistic Maxwellian plasma atasymptotically large values of laser fields and high temperatures of electrons.PACS numbers: 34.80.Qb, 52.38.Dx, 42.65.-k a r X i v : . [ phy s i c s . a t o m - ph ] M a y onlinear stimulated bremsstrahlung
1. Introduction
In the last two decades, laser technologies have made a giant leap forward such that lasersources of ultrarelativistic intensity level can be attained [1]. Near-infrared laser beamsare available up to intensities of 10 W / cm and much stronger lasers will be availablein near future [2]. For such lasers the dimensionless relativistic invariant parameterof intensity ξ ≡ eE /mcω >> e -elementary charge, m - electron mass, E , ω -electric field amplitude and frequency of a laser radiation, c -light speed in vacuum).The latter represents the work of the field on the one wavelength in the units of theparticle rest energy. Interaction of such lasers with the matter at extreme conditionsin ultrashort space-time scales have attracted broad interest over the last few yearsconditioned by a number of important applications, such as generation and probingof highenergy-density plasma [3], ions acceleration and inertial confinement fusion [4],vacuum nonlinear optics [5], compact laser-plasma accelerators [6], etc. Generally, theinteraction of such fields with the electrons in the presence of a third body makesavailable the revelation of many nonlinear relativistic electrodynamic phenomena. As athird body can serve ion and in the superintense laser fields one can observe relativisticabove threshold ionization [7] and high order harmonic generation [8], electron-positronpairs production on nuclei [9], and multiphoton stimulated bremsstrahlung (SB) ofelectrons on the ions/nuclei [10]. The latter is one of the fundamental processes at theinteraction of superstrong laser pulses with plasma and under the some circumstancesinverse-bremsstrahlung absorption may become dominant mechanism of absorption ofstrong electromagnetic (EM) radiation in underdense plasma.With the advent of lasers many pioneering papers have been devoted to thetheoretical investigation of the electron-ion scattering processes in gas or plasma inthe presence of a laser field using nonrelativistic [11–20] as well as relativistic [21–25]considerations. The appearance of superpower ultrashort laser pulses of relativisticintensities has initiated new interest in SB in relativistic domain [26–28], whereinvestigations were carried out mainly in the Born approximation over the scatteringpotential. Meanwhile for ions with the large charge and for the clusters [29], whenelectron interaction with the entire dense cluster ion core that composed of a largenumber of ions is dominant, the Born approximation is not applicable. The theoreticaldescription of SB in superstrong EM fields and scattering centers of large charges requiresone to go beyond the scope of Born approximation over the scattering potential andthe perturbation theory over laser field. In this context, when quantum effects areconsiderable, one can apply eikonal [23, 30] or generalized eikonal approximation [31].For the infrared and optical lasers, in the multiphoton interaction regime, one can applyclassical theory and the main approximation in the classical theory is low frequency(LF) or impact approximation [16, 17]. LF approximation have been generalized forrelativistic case in Refs. [24,25], where the effect of an intense EM wave on the dynamicsof SB and non-linear absorption of intense laser radiation by a monochromatic electronbeam due to SB have been carried out. Regarding the absorption of an EM of relativistic onlinear stimulated bremsstrahlung ξ .Hence, it is of interest to clear up how nonlinear SB effect will proceeds in the plasma,taking also into account initial relativism of plasma electrons.In the present paper inverse-bremsstrahlung absorption of an intense laser radiationin relativistic Maxwellian plasma is considered in the relativistic LF approximation[24, 25]. In this approximation one can consider as superstrong laser fields as well asscattering centers with large charges. The radiation power absorption in such plasmais investigated as for circularly polarized wave (CPW), as well as for linearly polarizedwave (LPW). We consider the dependence of the absorption coefficient on the intensityand polarization of the laser radiation, as well as on the temperature of the plasmaelectrons.The organization of the paper is as follows: In Sec. II the relativistic absorptioncoefficient of the EM wave of arbitrary polarization and intensity due to the mechanismof SB process is presented. In Sec. III we consider the problem numerically along withderivation of asymptotic formulas for absorption coefficient. Conclusions are given inSec. IV.
2. Nonlinear inverse-bremsstrahlung absorption coefficient
The absorption coefficient α for an EM radiation field of arbitrary intensity andpolarization, in general case of the homogeneous ensemble of electrons of concentration n e , with the arbitrary distribution function f ( p ) over momenta p , at the inversebremsstrahlung on the scattering centers with concentration n i , can be representedin the form: α = n e I (cid:90) d p f ( p ) W, (1)where W is the classical energy absorbed by a single electron per unit time from the EMwave of intensity I due to SB process on the scattering centers. For the homogeneousscattering centers W ∼ n i . For the generality, we assume Maxwellian plasma with therelativistic distribution function: f ( p ) = exp (cid:16) − E ( p ) k B T e (cid:17) πm ck B T e K ( mc /k B T e ) , (2)where k B is the Boltzmann’s constant, T e is the temperature of electrons in plasma, E ( p )is relativistic energy-momentum dispersion law of electrons, K ( x ) is the McDonald’sfunction; f ( p ) is normalized as (cid:90) f ( p ) d p = 1 . (3)To obtain W for the SB process, the electron interaction with the scatteringpotential and EM wave in the LF approximation can be considered as independentlyproceeding processes [17, 25], separated into the following three stages, schematically onlinear stimulated bremsstrahlung E and momentum p interacts withthe EM wave. The exact solution of the relativistic equation of motion of an electron ina plane EM wave is well known [10]. The energy and momentum in a plane EM wavefield can be written as: p ⊥ ( ψ ) = p ⊥ ( ψ ) − e A ( ψ ) − A ( ψ ) c , (4) ν p ( ψ ) = ν p ( ψ ) + 12 c ( E ( ψ ) − cν p ( ψ )) × (cid:2) e ( A ( ψ ) − A ( ψ )) − ec p ( ψ ) ( A ( ψ ) − A ( ψ )) (cid:3) , (5) E ( ψ ) = E ( ψ ) + cν ( p ( ψ ) − p ( ψ )) , (6)where A ( ψ ) = A ( ψ )( (cid:98) e cos ψ + (cid:98) e ζ sin ψ ) (7)is the vector potential of the EM wave of currier frequency ω and slowly varyingamplitude A ( ψ ). Here ψ = ωτ is the phase, τ = t − ν r /c , ν is an unit vector inthe EM wave propagation direction, (cid:98) e , are the unit polarization vectors, and arctan ζ is the polarization angle. At the second stage the elastic scattering of the electron inthe potential field takes place at the arbitrary, but certain phase ψ s of the EM wave.Thus, taking into account adiabatic turn on of the wave ( A ( ψ ) = 0) from Eqs. (4)-(6)before the scattering one can write p ⊥ ( ψ s ) = p ⊥ + e A ( ψ s ) c , (8) ν p ( ψ s ) = ν p + 12 c Λ (cid:2) e A ( ψ s ) + 2 ce p A ( ψ s ) (cid:3) , (9) E ( ψ s ) = E + cν ( p ( ψ s ) − p ) , (10)where Λ = E ( ψ s ) − cν p ( ψ s ) = E − cν p (11)is the integral of motion for a charged particle in the field of a plane EM wave. Themean energy of an electron in the wave-field before the scattering will be (cid:104)E ( ψ ) (cid:105) i = E + e (cid:104) A (cid:105) . (12)Then it takes place elastic scattering on the scattering center U ( r ). Due to theinstantaneous interaction of the electron with the scattering potential the wave phasedoes not change its value during the scattering. The electron with initial momentum p ( ψ s ) is acquired the momentum p (cid:48) ( ψ s ) ( E (cid:48) ( ψ s ) = E ( ψ s )) after the scattering directly atthe same phase ψ s of the wave, which can be defined from the generalized consideration onlinear stimulated bremsstrahlung Figure 1.
Schematic illustration of the SB process in the LF approximation. of the elastic scattering. Thus, measuring the scattering angle ϑ from a direction p ( ψ s ),with corresponding azimuthal angle ϕ for the scattered momentum one can write p (cid:48) x ( ψ s ) p (cid:48) y ( ψ s ) p (cid:48) z ( ψ s ) = p ( ψ s ) (cid:98) R sin ϑ cos ϕ sin ϑ sin ϕ cos ϑ , (13)where (cid:98) R = (cid:98) R z ( ϕ ) (cid:98) R y ( ϑ ), (cid:98) R y ( ϑ ) and (cid:98) R z ( ϕ ) are basic rotation matrices about the y and z axes: (cid:98) R = cos ϑ cos ϕ − sin ϕ sin ϑ cos ϕ cos ϑ sin ϕ cos ϕ sin ϑ sin ϕ − sin ϑ ϑ . (14)As a Oz axis we take wave propagation direction ν , θ is the polar angle and ϕ is theazimuthal angle in the wave-polarization plane.At the third stage the electron again interacts only with the wave, moving in thewave field with the momentum and energy defined from Eqs. (4)-(6): p ⊥ f ( ψ ) = p (cid:48) ( ψ s ) − e A ( ψ s ) − A ( ψ ) c , (15) ν p f ( ψ ) = ν p (cid:48) ( ψ s ) + 12 c Λ (cid:48) (cid:2) e ( A ( ψ s ) − A ( ψ )) − ce p (cid:48) ( ψ s ) ( A ( ψ s ) − A ( ψ ))] , (16) E f ( ψ ) = E ( ψ s ) + cν ( p ( ψ ) − p (cid:48) ( ψ s )) , (17)where Λ (cid:48) = E f ( ψ ) − cν p f ( ψ ) = E ( ψ s ) − cν p (cid:48) ( ψ s ) . (18)The mean energy of an electron in the wave field after the scattering will be (cid:104)E f ( ψ ) (cid:105) = E ( ψ s ) + 12Λ (cid:48) × (cid:2) e (cid:0) A ( ψ s ) + (cid:10) A (cid:11)(cid:1) − ce p (cid:48) ( ψ s ) A ( ψ s ) (cid:3) . (19)The energy change due to SB can be calculated as a difference of mean energy in thefield before and after the scattering:∆ E ( ϑ, ϕ, ψ s , p ) = (cid:104)E f ( ψ ) (cid:105) − (cid:104)E ( ψ ) (cid:105) i . onlinear stimulated bremsstrahlung E = e A ( ψ s ) + (cid:104) A (cid:105) (cid:18) (cid:48) − (cid:19) − ec p (cid:48) ( ψ s ) A ( ψ s )Λ (cid:48) + ec p ( ψ s ) A ( ψ s )Λ . (20)For the energy absorbed by a single electron per unit time from the EM wave due toSB process on the scattering centers one can write W = n i π (cid:90) π dψ s (cid:90) v ( ψ s ) ∆ E dσ ( ϑ, p ( ψ s )) , (21)where v ( ψ s ) = c p ( ψ s ) / E ( ψ s ) is the velocity of an electron in the wave-field, p ( ψ s ) = (cid:112) E ( ψ s ) − m c /c is the momentum, and dσ ( ϑ, p ( ψ s )) is the differential cross sectionof the elastic scattering in the potential field U ( r ). Taking into account that the maincontribution in the integral (21) comes from the small angle scatterings one can write W = n i π (cid:90) π dψ s (cid:90) v ( ψ s ) ∂ ∆ E ∂ ϑ dσ tr ( ϑ, p ( ψ s )) , (22)where dσ tr ( ϑ, p ( ψ s )) = (1 − cos ϑ ) dσ ( ϑ, p ( ψ s )) (23)is the transport differential cross section. For the Coulomb scattering centers withpotential energy U ( r ) = Ze r of electron interaction with ion of charge Ze , one can use relativistic cross section forelastic scattering at small angles [32] and make integration over ϑ and ϕ to obtain: W = n i Z e (cid:90) π dψ s m c Λ (cid:20) e A ( ψ s ) + (cid:104) A (cid:105) m c × (cid:0) E ( ψ s )Λ − m c (cid:1) + ce p ( ψ s ) A ( ψ s ) (cid:3) E ( ψ s ) p ( ψ s ) L cb , (24)where L cb = ln (cid:18) v ( ψ s ) p ( ψ s ) Ze ω (cid:19) (25)is the Coulomb logarithm. The latter has been obtained taking ρ min = Ze / v p as alower limit of the impact parameter, while for the upper limit we assume ρ max = v /ω .Taking into account Eqs. (1), (7), and (24) for the absorption coefficient we obtain: α = n i n e Z e I (cid:90) d p f ( p ) (cid:90) π dψ s (cid:20) e A ( ψ s ) + (cid:104) A (cid:105) m c × (cid:0) E ( ψ s )Λ − m c (cid:1) + ce p ( ψ s ) A ( ψ s ) (cid:3) m c E ( ψ s )Λ p ( ψ s ) L cb , (26) onlinear stimulated bremsstrahlung I = (1 + ζ ) ω A / πc. Thus, Eq. (26) represents nonlinear inverse-bremsstrahlung absorption coefficient α foran EM radiation field of arbitrary intensity and polarization, for homogeneous ensembleof electrons of concentration n e , with the arbitrary distribution function f ( p ) overmomenta p . Note that the LF approximation in the intense laser field is applicable when λ (cid:29) λ D , (27)where λ is the laser radiation wavelength and λ D = (cid:112) k b T e / πn e e Z is the Debyescreening length: λ D [cm] = 7 . × × (cid:115) T e [eV] Zn e [cm − ] . (28)Besides, for an underdense plasma one should take into account condition ω > ω p , where ω p = (cid:112) πn e e /m ∗ is the plasma frequency with ”effective mass” m ∗ of the relativisticelectron in the EM wave [33]: m ∗ = m (cid:112) (cid:104) ξ ( ψ ) (cid:105) . (29)Thus for CPW and for large ξ one can write n e < mξ ω / πe = 1 . × × ξ × λ − ( µ m) . (30)There is also limitation on the pulse duration τ of an EM wave. The SB should be themain mechanism, which is responsible for the absorption of the laser radiation in plasma.This condition is failed, if the influence of the strong EM wave lead to development of aninstability. Hence, pulse duration τ of an EM wave has to satisfy the condition µτ (cid:45) µ is the maximal increment of the instability of the plasma in the strong laserfield.In general, analytical integration over momentum p and scattering phase ψ s isimpossible, and one should make numerical integration. The latter along with derivationof asymptotic formulas for absorption coefficient α will be done in the next section.
3. Numerical Treatment: Asymptotic Formulas
As we are interested in superintense laser pulses of relativistic intensities, then it isconvenient to represent the absorption coefficient (26) in the form of dimensionlessquantities: αα = 12 π (1 + ζ ) ξ (cid:90) d p f ( p ) (cid:90) π dψ s γ ( ψ s ) p ( ψ s )Λ × (cid:18) ξ ( ψ s ) + (cid:104) ξ ( ψ s ) (cid:105) (cid:0) γ ( ψ s )Λ − (cid:1) + p ξ ( ψ s ) + ξ ( ψ s ) (cid:19) L cb . (31) onlinear stimulated bremsstrahlung α = 4 Z r e λ n i n e , (32)and r e is the classical electron radius. In Eq. (31) the dimensionless momentum, energy,and temperature were introduced as follows: p = p mc , γ ( ψ s ) = E ( ψ s ) mc , T n = k B T e mc , and the dimensionless relativistic intensity parameters of EM wave, ξ ( ψ s ) = ξ ( (cid:98) e cos ψ s + (cid:98) e ζ sin ψ s ) . The scaled relativistic distribution function is f ( p ) = 14 πT n K ( T − ) exp (cid:18) − γ T n (cid:19) , and Coulomb logarithm L cb = L + ln (cid:18) p ( ψ s ) γ ( ψ s ) (cid:19) , (33)where L = ln ( λ/ πr e Z ) . In the near-infrared and optical domain of frequencies, atthe Z = 1 − , L ≈
30. Taking into account that normalized absorption coefficient(31) depends on Z and ω through logarithm L , for the numerical simulations we willnot concretize Z and ω , assuming L ≈ γ ( ψ s ) and momentum p ( ψ s ) are constants,since ξ ( ψ s ) = const. Meanwhile in the LPW p ( ψ s ) oscillates and as a consequencesmall values of p ( ψ s ) give the main contribution in Eq. (31). The latter leads to morecomplicated behavior of the dynamics of SB at the linear polarization of a stimulatingstrong wave. Besides in the case of CPW, thanks to azimuthal symmetry one can makea step forward in analytical calculation and obtain explicit formula for the absorptioncoefficient at superstrong laser fields. Thus, taking into account azimuthal symmetry inthe case of CPW, one can make integration over phase ψ s , which results in αα = 12 (cid:90) d p f ( p ) (cid:18) γ ( p ) Λ + p ξ sin ϑ cos ϕ (cid:19) × γ ( p )Λ p ( p ) L cb , (34)where γ ( p ) = γ + 12Λ (cid:2) ξ + 2 p ξ sin ϑ cos ϕ (cid:3) , (35) p ( p ) = (cid:112) γ ( p ) − . (36) onlinear stimulated bremsstrahlung α / α ξ T n =0.1T n =0.5T n =1.0 Figure 2. (Color online) Total scaled rate of inverse-bremsstrahlung absorption (inarbitrary units) of circularly polarized laser radiation in Maxwellian plasma versus thedimensionless relativistic invariant parameter of the wave intensity for various plasmatemperatures. The wave is assumed to be circularly polarized.
At the large ξ >>
1, taking into account Eqs. (35) and (36), from Eq. (34) one canobtain α = α ξ (cid:90) d p f ( p ) 1Λ L ( c )cb , (37)where L ( c )cb = L + ln (cid:18) ξ (cid:19) . The formula (37) shows the suppression of the SB rate with increase of the waveintensity. Ignoring weak logarithmic dependence, we see that absorption coefficientinversely proportional to laser intensity: α ∼ /ξ . For the large ξ the dependence ofthe absorption coefficient on temperature comes from Λ in Eq. (37). In particular, forinitially nonrelativistic plasma T n << (cid:39)
1, which gives α ≡ α C = α ξ ln (cid:18) cZr e ω ξ (cid:19) . (38)The relation for the absorption coefficient in the case of LPW is complicated and evenfor large ξ one can not integrate it analytically. Therefore, for the analysis we haveperformed numerical investigations, making also analytic interpolation.The results of numerical investigations of Eq. (31) are illustrated in Figures 2-7both for the CPW and LPW.To show the dependence of the inverse-bremsstrahlung absorption rate on the laserradiation intensity in Fig. 2 it is shown scaled rate α/α versus relativistic invariantparameter of the wave intensity for various plasma temperatures. The wave is assumedto be circularly polarized. As is seen from this figure the SB rate is suppressed withincrease of the wave intensity and for the large values of ξ it exhibits a tenuousdependence on the plasma temperature. The behavior is also seen from Fig. 3, wheretotal scaled rate of inverse-bremsstrahlung absorption of CPW in plasma, as a functionof the plasma temperature T n is shown for various wave intensities. Here for large onlinear stimulated bremsstrahlung α / α T n ξ =1 ξ =2 ξ =3 ξ =4 Figure 3. (Color online) Total scaled rate of inverse-bremsstrahlung absorption (inarbitrary units) of circularly polarized laser radiation in plasma, as a function of theplasma temperature (in units of an electron rest energy mc ) is shown for various waveintensities. α / α C ’akt.dat’ u 2:1:3 2 4 6 8 10 12 14 ξ T n Figure 4. (Color online) Density plot of the total rate of inverse-bremsstrahlungabsorption scaled to asymptotic rate α C (in arbitrary units), as a function of theplasma temperature (in units of an electron rest energy mc ) and the dimensionlessrelativistic invariant parameter of the circularly polarized laser beam. α / α ξ T n =0.1T n =0.5T n =1.0 Figure 5. (Color online) Total scaled rate of inverse-bremsstrahlung absorption (inarbitrary units) of linearly polarized laser radiation in plasma versus the dimensionlessrelativistic invariant parameter of wave intensity for various plasma temperatures. onlinear stimulated bremsstrahlung α / α T n ξ =1 ξ =2 ξ =3 ξ =4 Figure 6. (Color online) Total scaled rate of inverse-bremsstrahlung absorption (inarbitrary units), as a function of the plasma temperature (in units of an electron restenergy mc ) is shown for various wave intensities. The wave is assumed to be linearlypolarized. α / α L ’aktlin.dat’ u 1:2:3 2 4 6 8 10 12 14 ξ T n Figure 7. (Color online) Density plot of the total rate of inverse-bremsstrahlungabsorption scaled to asymptotic rate α L (in arbitrary units), as a function of theplasma temperature (in units of an electron rest energy mc ) and the dimensionlessrelativistic invariant parameter of the linearly polarized laser beam. values of ξ we have a weak dependence on temperature, which is a result of lasermodified relativistic scattering of electrons irrespective of the initial state of electrons.The absorption coefficient α decreases as 1 /ξ in accordance with analytical result (37).To clarify the range of applicability of the asymptotic formula (38) in Fig. 4 densityplot of the total rate of inverse-bremsstrahlung absorption scaled to asymptotic rate α C ,as a function of the plasma temperature and the relativistic invariant parameter ξ isshown for CPW. As is seen in the wide range of T n and ξ one can apply asymptoticformula (38).In Fig. 5 and 6 it is shown the total scaled rate of inverse-bremsstrahlung absorptionof LPW in plasma versus the dimensionless relativistic invariant parameter of waveintensity and temperature, respectively. With the interpolation we have seen that α decreases as 1 /ξ / and exhibits a tenuous dependence on the plasma temperature. onlinear stimulated bremsstrahlung ξ we interpolate α by the followingformula: α (cid:39) α L = α ξ / ln (cid:18) cZr e ω ξ (cid:19) . (39)As is seen from Fig. 7, in the case of LPW and for the moderate temperatures, withthe well enough accuracy one can apply asymptotic rate (39).
4. Conclusion
We have presented a theory of inverse-bremsstrahlung absorption of an intenselaser radiation in relativistic Maxwellian plasma in the relativistic low-frequencyapproximation. The coefficient of nonlinear inverse-bremsstrahlung absorption has beencalculated for relativistic Maxwellian plasma. The simple analytical formulae have beenobtained for absorption coefficient at asymptotically large values of laser fields both forcircularly and linearly polarized radiations. The obtained results demonstrate that theSB rate is suppressed with the increase of the wave intensity and for large values of ξ absorption coefficient α decreases as 1 /ξ for circularly and as 1 /ξ / for the linearlypolarized one in contrast to nonrelativistic case where one has a dependence 1 /ξ [17].The SB rate is suppressed with increase of the plasma temperature but for the relativisticlaser intensities it exhibits a tenuous dependence on the plasma temperature.This work was supported by SCS of RA. References [1] Yanovsky V, Chvykov V, Kalinchenko G, Rousseau P, Planchon T, Matsuok T, Maksimchuk A,Nees J, Cheriaux G, Mourou G, Krushelnick K 2008
Opt. Express Science
Phys. Rev.Lett. Phys. Rev. Lett. Phys. Rev. Lett.
Rep. Prog. Phys. Rep. Prog. Phys. Rep. Prog. Phys .
47; Pukhov A 2006
Nature Phys. Rev. Mod. Phys. Phys. Rev. A Phys. Rev. A Phys. Rev. A Phys. Rev. A Phys. Rev. A Phys. Rev. A Phys. Lett. A
Adv. Atom. Mol. Opt. Phys. onlinear stimulated bremsstrahlung [9] Chen H, Wilks S C, Bonlie J D, Liang E P, Myatt J, Price D F, Meyerhofer D D, Beiersdorfer P2009 Phys. Rev. Lett.
Relativistic Nonlinear Electrodynamics (New York: Springer)[11] Dawson J, Oberman C 1962
Phys. Fluids Bell System Tech. J. Bell System Tech. J. Sov. Phys. JETP Sov. Phys. JETP Phys. Rev.
J. Phys. A Phys. Rev. E J. Phys. B Sov. Phys. Usp. Phys. Rev. A Phys. Rev. A Phys. Rev. A J. Phys. B Sov. Phys. JETP Sov. Phys. JETP J. Phys. A J. Phys. B J. Phys. B Phys. Rev. A Opt. Express Eur. Phys. J. D Phys. Rev. A Phys. Rev. A J. Phys.B Phys. Rev. Lett. Phys. Usp. Phys. Rev. A Phys. Rev. A Phys. Rev. A Teoria Polia (Moscow: Nauka)[33] Akhiezer A I, Polovin R V 1956
Sov. Phys. JETP Phys. Fluids13